Handbook of Mathematics, 4th ed

I.N. Bronshtein . K.A. Semendyayev. G. Musiol .H. Muehlig Handbook of Mathematics 4th Ed. With 745 Figures and Spring...

Author: I.N. Bronshtein | K.A. Semendyayev | G. Musiol | H. Muehlig | H. Mühlig

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I.N. Bronshtein . K.A. Semendyayev. G. Musiol .H. Muehlig

Handbook of Mathematics 4th Ed.

With 745 Figures and

Springer

142 Tables

Ilja N. Bronshtein t Konstantin A. Semendyayev t Gerhard Musiol Heiner Muehlig

Library of Congress Cataloging-in-Publication Data Spravochnik PO matematike. English. Handbook of mathematics I I.N. Bronshtein [et al.1.- 4th ed. p. cm. Based on the 5th German ed. published by Wissenschaftlicher Verlag Harri Deutsch GmbH, Frankfurt am Main under title: Taschenbuch der Mathematik. ISBN 3-540-43491-7 (acid-free Paper) 1. Mathematics--Handbooks, manuals, etc. I. Bronshtein, I. N. (Il'ia Nikolaevich) 11. Title.

...

QA40S6813 2003 510--dc22

2003062852

Based on the 5th German edition of BronsteinlSemendjajewlMusiollMiihlig "Taschenbuch der Mathematik", 2000. Published by Wissenschaftlicher Verlag Harri Deutsch GmbH, Frankfurt am Main.

ISBN 3-540-43491-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microhlm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com 0 Springer-Verlag Berlin Heidelberg 2004 Printed in Germany

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The authors and the publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibilty or liability for any errors or omissions that may be made. vpesetting by authors Final processing: PTP-Berlin Protago-TeX-Production GmbH, Berlin Cover-Design: design & production GmbH, Heidelberg Translation: Gabriela SzCp, Budapest Printed on acid-free paper 62I3020Yu - 5 4 3 z 1 o

Preface to the Fourth English Edition The .‘Handbookof Slathematics” by the mathematician, I. N. BRONSHTEIN and the engineer, K . A. SEMENDYAYEV was designed for engineers and students of technical universities. It appeared for the first time in Russian and was widely distributed both as a reference book and as a text book for colleges and universities. It was later translated into German and the many editions have made it a permanent fixture in German-speaking countries, where generations of engineers, natural scientists and others in technical training or already working with applications of mathematics have used it. On behalf of the publishing house Harri Deutsch, a revision and a substantially enlarged edition was prepared in 1992 by Gerhard Musiol and Heiner Muhlig, with the goal of giving ”Bronshtein” the modern practical coverage requested by numerous students, university teachers and practitioners. The original style successfully used by the authors has been maintained. It can be characterized as ‘(short, easily understandable, comfortable to use, but featuring mathematical accuracy (at a level of detail consistent with the needs of engineers)”’. Since 2000, the revised and extended fifth German edition of the revision has been on the market. Acknowledging the success that “BRONSTEIN” has experienced in the German-speaking countries, Springer-Verlag Heidelberg/Germany is publishing a fourth English edition! which corresponds to the improved and extended fifth German edition. In the work at hand, the classical areas of Engineering Mathematics required for current practice are presented, such as “Arithmetic” “Functions”, “Geometry”, “Linear Algebra”, “Algebra and Discrete Mathematics”. (including “Logic”. ”Set Theory”, “Classical .4lgebraic Structures”, “Elementary Number Theory” ’’ Cryptology” “Universal illgebra” , “Boolean Algebra and Swich .4lgebra” “Algorithms of graph Theory”. “Fuzzy Logic”), “Differentiation” “Integral Calculus”, “Differential Equations” “Calculus of Variations” “Linear Integral Equations”, “Functional Analysis”, “Vector Analysis and Yector Fields”. ”Function Theory”, “Integral Transformations“, “Probability Theory and Mathematical Statistics”. Fields of mathematics that have gained importance with regards to the increasing mathematical modeling and penetration of technical and scientific processes also receive special attention. Included amongst these chapters are “Stochastic Processes and Stochastic Chains” as well as “Calculus of Errors”, “Dynamical Systems and Chaos”. “Optimization”, “Numerical Analysis”. W s ing the Computer’’ and ‘‘ComputerAlgebra Systems”. The book is enhanced with over a thousand complementary illustrations and many tables. Special functions, series expansions, indefinite, definite and elliptic integrals as well as integral transformations and statistical distributions are supplied in an extensive appendix of tables. In order to make the reference book more effective, clarity and fast access through a clear structure were the goals, especially through visual clues as well as by a detailed technical index and colored tabs. An extended bibliography also directs users to further resources. We would like to cordially thank all readers and professional colleagues who helped us with their valuable statements. remarks and suggestions on the German edition of the book during the revision process. Special thanks go to Mrs. Professor Dr. Gabriela Szp (Budapest), who made this English debut version possible. Furthermore our thanks go to the co-authors for the critical treatment of their chapters. ~

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Dresden. June 2003 Prof. Dr. GERHARD MUSIOL

Prof. Dr. HEINERMUHLIG

Coauthors Some chapters and sections originated through a cooperation with the co-authors (see next page). ‘See Preface to the first russian edition

ChaDter or Section

Coauthor

Spherical Trigonometry (3.4.1-3.4.3) Spherical Curves (3.4.3.4) Logic, Set Theory, Classic Algebraic Structures, Applications of Groups, Rings and Fields, Vektor Spaces (5~1,5.2, 5.3, besides 5.3.4, 5.3.7.4-5.3.7.6) Groups Representations, Applications of Groups (5.3.4, 5.3.5.4-5.3.5.6) Elementary Number Theory, Cryptology, Graphs (5.4, 5.5, 5.8). Fuzzy-Logic (5.9) Non-Linear Partial Differential Equations, Solitonen (9.2.4) Linear Integral Equations (11) Functional Analyzis (12) Elliptic Functions (14.6) Dynamical Systems and Chaos (17) Optimization (18) Computer illgebra Systems (19.8.4, 20.)

Dr. H. NICKEL,Dresden Prof. L. MARSOLEK, Berlin Prof. Dr. J. BRUNNER, Dresden Prof. Dr. R. REIF,Dresden Doz. Dr. U. BAUMANN, Dresden Prof. Dr. A. GRAUEL,Soest Prof. Dr. P. ZIESCHE,Dresden Dr. I. STEINERT,Diisseldorf Prof. Dr. M. WEBER,Dresden Dr. N. M. FLEISCHER, Moscow Dr. V. REITMANN, Dresden Dr. I. STEINERT, Diisseldorf Prof. Dr. G. FLACH, Dresden

Contents List of Tables

XXXIX

1 Arithmetic 1.1 Elementary Rules for Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.1 Natural, Integer and Rational Numbers . . . . . . . . . . . . . . . 1.1.1.2 Irrational and Transcendental Numbers . . . . . . . . . . . . . . . 1.1.1.3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.4 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.5 Commensurability . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Methods for Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.1 Direct Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.2 Indirect Proof or Proof by Contradiction . . . . . . . . . . . . . . 1.1.2.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.4 Constructive Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Powers, Roots, and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.1 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.4 Special Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.2 iZlgebraic Expressions in Detail . . . . . . . . . . . . . . . . . . . 1.1.6 Integral Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.1 Representation in Polynomial Form . . . . . . . . . . . . . . . . . 1.1.6.2 Factorizing a Polynomial . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.3 Special Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.4 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6.5 Determination of the Greatest Common Divisor of Two Polynomials 1.1.7 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7.1 Reducing to the Simplest Form . . . . . . . . . . . . . . . . . . . . 1.1.7.2 Determination of the Integral Rational Part . . . . . . . . . . . . . 1.1.7.3 Decomposition into Partial Fractions . . . . . . . . . . . . . . . . 1.1.7.4 Transformations of Proportions . . . . . . . . . . . . . . . . . . . 1.1.8 Irrational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Definition of a Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Special Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5.1 Arithmetic Mean or Arithmetic Average . . . . . . . . . . . . . . 1.2.5.2 Geometric Mean or Geometric Average . . . . . . . . . . . . . . . 1.2.5.3 Harmonic Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5.4 Quadratic Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 1 2 2 3 4 4 5 5 5 6 6 6 7 7 7 8 9 9 10 10 11 11 11 11 12 12 14 14 14 15 15 17 17 18 18 18 19 19 19 19 20 20 20

1.3

1.4

1.5

1.2.5.5 Relations Between the Means of Two Positive Values . . . . . . . . Business Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Calculation of Interest or Percentage . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Calculation of Compound Interest . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.1 Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Amortization Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.1 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.2 Equal Principal Repayments . . . . . . . . . . . . . . . . . . . . . 1.3.3.3 Equal Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Annuity Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4.1 .Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4.2 Future Amount of an Ordinary Annuity . . . . . . . . . . . . . . . 1.3.4.3 Balance after n Annuity Payments . . . . . . . . . . . . . . . . . 1.3.5 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Pure Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.2 Properties of Inequalities of Type I and I1 . . . . . . . . . . . . . . 1.4.2 Special Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.1 Triangle Inequality for Real Numbers . . . . . . . . . . . . . . . . 1.4.2.2 Triangle Inequality for Complex Numbers . . . . . . . . . . . . . . 1.4.2.3 Inequalities for Absolute Values of Differences of Real and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.4 Inequality for Arithmetic and Geometric Means . . . . . . . . . . 1.4.2.5 Inequality for Arithmetic and Quadratic Means . . . . . . . . . . . 1.4.2.6 Inequalities for Different Means of Real Numbers . . . . . . . . . . 1.4.2.7 Bernoulli's Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.8 Binomial Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.9 Cauchy-Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . 1.4.2.10 Chebyshev Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.11 Generalized Chebyshev Inequality . . . . . . . . . . . . . . . . . . 1.4.2.12 Holder Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.13 Slinkowski Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Solution of Linear and Quadratic Inequalities . . . . . . . . . . . . . . . . . 1.4.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.2 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.3 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.4 General Case for Inequalities of Second Degree . . . . . . . . . . . Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Imaginary and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 1.5.1.1 Imaginary Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1.2 Complex Sumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Vector Representation . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2.2 Equality of Complex Numbers . . . . . . . . . . . . . . . . . . . . 1.5.2.3 Trigonometric Form of Complex Numbers . . . . . . . . . . . . . . 1.5.2.4 Exponential Form of a Complex Yumber . . . . . . . . . . . . . . 1.5.2.5 Conjugate Complex Numbers . . . . . . . . . . . . . . . . . . . . 1.5.3 Calculation with Complex Numbers . . . . . . . . . . . . . . . . . . . . . . 1.5.3.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . 1.5.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 21 21 22 22 22 23 23 23 24 25 25 25 25 26 28 28 28 29 30 30 30 30 30 30 30 30 31 31 31 32 32 32 33 33 33 33 33 34 34 34 34 34 34 34 35 35 36 36 36 36

Contents IX

1.6

1.5.3.3 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3.4 General Rules for the Basic Operations . . . . . . . . . . . . . . . 1.5.3.5 Taking Powers of Complex Numbers . . . . . . . . . . . . . . . . . 1.5.3.6 Taking of the n-th Root of a Complex Number . . . . . . . . . . . Algebraic and Transcendental Equations . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Transforming Algebraic Equations to Normal Form . . . . . . . . . . . . . . 1.6.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1.2 Systems of n Algebraic Equations . . . . . . . . . . . . . . . . . . 1.6.1.3 Superfluous Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Equations of Degree at !dost Four . . . . . . . . . . . . . . . . . . . . . . . 1.6.2.1 Equations of Degree One (Linear Equations) . . . . . . . . . . . . 1.6.2.2 Equations of Degree Two (Quadratic Equations) . . . . . . . . . . 1.6.2.3 Equations of Degree Three (Cubic Equations) . . . . . . . . . . . 1.6.2.4 Equations of Degree Four . . . . . . . . . . . . . . . . . . . . . . . 1.6.2.5 Equations of Higher Degree . . . . . . . . . . . . . . . . . . . . . 1.6.3 Equations of Degree n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3.1 General Properties of Algebraic Equations . . . . . . . . . . . . . 1.6.3.2 Equations with Real Coefficients . . . . . . . . . . . . . . . . . . . 1.6.4 Reducing Transcendental Equations to Algebraic Equations . . . . . . . . . 1.6.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4.2 Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4.3 Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4.4 Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . 1.6.4.5 Equations with Hyperbolic Functions . . . . . . . . . . . . . . . .

2 Functions

2.1

Notion of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.1 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.2 Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.3 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . 2.1.1.4 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.5 Further Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.6 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.7 Functions and Mappings . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Methods for Defining a Real Function . . . . . . . . . . . . . . . . . . . . . 2.1.2.1 Defining a Function . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.2 Analytic Representation of a Function . . . . . . . . . . . . . . . . 2.1.3 Certain Types of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.1 Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.2 Bounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.3 Even Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.4 Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.5 Representation with Even and Odd Functions . . . . . . . . . . . . 2.1.3.6 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.7 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 LimitsofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4.1 Definition of the Limit of a Function . . . . . . . . . . . . . . . . . 2.1.4.2 Definition by Limit of Sequences . . . . . . . . . . . . . . . . . . . 2.1.4.3 Cauchy Condition for Convergence . . . . . . . . . . . . . . . . . 2.1.4.4 Infinity as a Limit of a Function . . . . . . . . . . . . . . . . . . . 2.1.4.5 Left-Hand and Right-Hand Limit of a Function . . . . . . . . . . .

37

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38 38 38 38 38 39 39 39 39 40 42 43 43 43 44 45 45 45 46 46 46 47

47 47 47 47 47 47 47 47 48 48 48 48 49 49 50 50 50 50 50 51 51 51 52 52 52 52

2.2

2.3

2.4

2.5

2.6

2.7

2.1.4.6 Limit of a Function as z Tends to Infinity . . . . . . . . . . . . . . 2.1.4.7 Theorems About Limits of Functions . . . . . . . . . . . . . . . . 2.1.4.8 Calculation of Limits . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols . . . 2.1.5 Continuity of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5.1 Notion of Continuity and Discontinuity . . . . . . . . . . . . . . . 2.1.5.2 Definition of Continuity . . . . . . . . . . . . . . . . . . . . . . . 2.1.5.3 Most Frequent Types of Discontinuities . . . . . . . . . . . . . . . 2.1.5.4 Continuity and Discontinuity of Elementary Functions . . . . . . . 2.1.5.5 Properties of Continuous Functions . . . . . . . . . . . . . . . . . Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.2 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.3 Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.4 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . 2.2.2.5 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.6 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . 2.2.3 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Linear Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Quadratic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Polynomials of n-th Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Parabola of n-th Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Special Fractional Linear Function (Inverse Proportionality) . . . . . . . . . 2.4.2 Linear Fractional Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Curves of Third Degree, Type I . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Curves of Third Degree, Type I1 . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Curves of Third Degree, Type I11 . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Reciprocal Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Square Root of a Linear Binomial . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Square Root of a Quadratic Polynomial . . . . . . . . . . . . . . . . . . . . 2.5.3 Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Functions and Logarithmic Functions . . . . . . . . . . . . . . . . . . 2.6.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Error Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Exponential Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Generalized Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Product of Power and Exponential Functions . . . . . . . . . . . . . . . . . Trigonometric Functions (Functions of Angles) . . . . . . . . . . . . . . . . . . . . 2.7.1 Basic Notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1.1 Definition and Representation . . . . . . . . . . . . . . . . . . . . 2.7.1.2 Range and Behavior of the Functions . . . . . . . . . . . . . . . . 2.7.2 Important Formulas for Trigonometric Functions . . . . . . . . . . . . . . .

53 53 54 55 57 57 57 57 58 59 60 60 60 61 61 61 61 61 61 61 62 62 62 62 62 62 63 63 64 64 64 65 65 66 67 68 69 69 69 70 71 71 71 72 72 73 74 74 74 74 77 79

Contents

2.8

2.9

2.10

2.7.2.1 Relations Between the Trigonometric Functions of the Same Angle (Addition Theorems) . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2.2 Trigonometric Functions of the Sum and Difference of Two Angles 2.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle . . . . 2.7.2.4 Trigonometric Functions of Half-Angles . . . . . . . . . . . . . . . 2.7.2.5 Sum and Difference of Two Trigonometric Functions . . . . . . . . 2.7.2.6 Products of Trigonometric Functions . . . . . . . . . . . . . . . . 2.7.2.7 Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . 2.7.3 Description of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 2.7.3.2 Superposition of Oscillations . . . . . . . . . . . . . . . . . . . . . 2.7.3.3 Vector Diagram for Oscillations . . . . . . . . . . . . . . . . . . . 2.7.3.4 Damping of Oscillations . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Definition of the Inverse Trigonometric Functions . . . . . . . . . . . . . . . 2.8.2 Reduction to the Principal Value . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Relations Between the Principal Values . . . . . . . . . . . . . . . . . . . . 2.8.4 Formulas for Yegative Arguments . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Sum and Difference of arcsin z and arcsin y . . . . . . . . . . . . . . . . . . 2.8.6 Sum and Difference of arccos z and arccos y . . . . . . . . . . . . . . . . . . 2.8.7 Sum and Difference of arctan I and arctan y . . . . . . . . . . . . . . . . . . 2.8.8 Special Relations for arcsin z, arccos I,arctan z . . . . . . . . . . . . . . . . Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Definition of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Graphical Representation of the Hyperbolic Functions . . . . . . . . . . . . 2.9.2.1 Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.2 Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.3 Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.4 Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Important Formulas for the Hyperbolic Functions . . . . . . . . . . . . . . . 2.9.3.1 Hyperbolic Functions of One Variable . . . . . . . . . . . . . . . . 2.9.3.2 Expressing a Hyperbolic Function by Another One with the Same Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3.3 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . 2.9.3.4 Hyperbolic Functions of the Sum and Difference of Two Arguments (Addition Theorems) . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3.5 Hyperbolic Functions of Double Arguments . . . . . . . . . . . . . 2.9.3.6 De Moivre Formula for Hyperbolic Functions . . . . . . . . . . . . 2.9.3.7 Hyperbolic Functions of Half-Argument . . . . . . . . . . . . . . . 2.9.3.8 Sum and Difference of Hyperbolic Functions . . . . . . . . . . . . 2.9.3.9 Relation Between Hyperbolic and Ttigonometric Functions with Complex Arguments z . . . . . . . . . . . . . . . . . . . . . . . . . . . Area Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.1 Area Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.2 Area Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.3 Area Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1.4 Area Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Determination of Area Functions Using Natural Logarithm . . . . . . . . . 2.10.3 Relations Between Different Area Functions . . . . . . . . . . . . . . . . . . 2.10.4 Sum and Difference of Area Functions . . . . . . . . . . . . . . . . . . . . . 2.10.5 Formulas for Negative Arguments . . . . . . . . . . . . . . . . . . . . . . .

XI

79 79 79 80 81 81 82 82 82 82 83 83 84 84 84 85 86 86 86 86 87 87 87 88 88 88 88 89 89 89 89 89 89 90 90 90 90 91 91 91 91 91 92 92 92 93 93 93

.

Curves of Order Three (Cubic Curves) . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.11.1 Semicubic Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.11.2 Witch of Agnesi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 94 2.11.3 Cartesian Folium (Folium of Descartes) . . . . . . . . . . . . . . . . . . . . 2.11.4 Cissoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.11.5 Strophoide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.12 Curves of Order Four (Quartics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.12.1 Conchoid of Ncomedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.12.2 General Conchoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.12.3 Pascal's Limacon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.12.4 Cardioid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.12.5 Cassinian Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.12.6 Lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.13 Cycloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.13.1 Common (Standard) Cycloid . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.13.2 Prolate and Curtate Cycloids or Trochoids . . . . . . . . . . . . . . . . . . 100 2.13.3 Epicycloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.13.4 Hypocycloid and Astroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.13.5 Prolate and Curtate Epicycloid and Hypocycloid . . . . . . . . . . . . . . . 102 2.14 Spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.14.1 drchimedean Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 104 2.14.2 Hyperbolic Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14.3 Logarithmic Spiral . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . 104 104 2.14.4 Evolvent of the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14.5 Clothoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.16 Various Other Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.15.1 Catenary Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.15.2 Tractrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 106 2.16 Determination of Empirical Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 106 2.16.1.1 Curve-Shape Comparison . . . . . . . . . . . . . . . . . . . . . . 2.16.1.2 Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.16.1.3 Determination of Parameters . . . . . . . . . . . . . . . . . . . . . 107 2.16.2 Useful Empirical Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.16.2.1 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 108 2.16.2.2 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.16.2.3 Quadratic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . 109 109 2.16.2.4 Rational Linear Function . . . . . . . . . . . . . . . . . . . . . . . 2.16.2.5 Square Root of a Quadratic Polynomial . . . . . . . . . . . . . . . 110 2.16.2.6 General Error Curve . . . . . . . . . . . . . . . . . . . . . . . . . 110 110 2.16.2.7 Curve of Order Three, Type I1 . . . . . . . . . . . . . . . . . . . . 2.16.2.8 Curve of Order Three, Type I11 . . . . . . . . . . . . . . . . . . . 111 2.16.2.9 Curve of Order Three, Type I . . . . . . . . . . . . . . . . . . . . 111 2.16.2.10 Product of Power and Exponential Functions . . . . . . . . . . . . 112 112 2.16.2.11 Exponential Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.16.2.12 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.17 Scales and Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.17.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17.2 Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 115 2.17.2.1 Semilogarithmic Paper . . . . . . . . . . . . . . . . . . . . . . . . 2.17.2.2 Double Logarithmic Paper . . . . . . . . . . . . . . . . . . . . . . 115 2.17.2.3 Graph Paper with a Reciprocal Scale . . . . . . . . . . . . . . . . 116 2.11

Contents XI11 2.17.2.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.1 Definition and Representation . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.1.1 Representation of Functions of Several Variables . . . . . . . . . . 2.18.1.2 Geometric Representation of Functions of Several Variables . . . . 2.18.2 Different Domains in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.2.1 Domain of a Function . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.2.2 Two-Dimensional Domains . . . . . . . . . . . . . . . . . . . . . . 2.18.2.3 Three or Multidimensional Domains . . . . . . . . . . . . . . . . . 2.18.2.4 Methods to Determine a Function . . . . . . . . . . . . . . . . . . 2.18.2.5 Various Ways to Define a Function . . . . . . . . . . . . . . . . . . 2.18.2.6 Dependence of Functions . . . . . . . . . . . . . . . . . . . . . . . 2.18.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.3.2 Exact Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.3.3 Generalization for Several Variables . . . . . . . . . . . . . . . . . 2.18.3.4 Iterated Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.5 Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . 2.18.5.1 Theorem on Zeros of Bolzano . . . . . . . . . . . . . . . . . . . . . 2.18.5.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . 2.18.5.3 Theorem About the Boundedness of a Function . . . . . . . . . . . 2.18.5.4 Weierstrass Theorem (About the Existence of Maximum and Minimum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.1 Nomograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.2 Net Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.3 Alignment Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.3.1 .4lignment Charts with Three Straight-Line Scales Through a Point 2.19.3.2 Alignment Charts with Two Parallel and One Inclined Straight-Line Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.3.3 .4lignment Charts with Two Parallel Straight Lines and a Curved Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.4 Net Charts for More Than Three Variables . . . . . . . . . . . . . . . . . .

116 117 117 117 117 118 118 118 118 119 120 121 122 122 122 122 122 122 123 123 123 123

3 Geometry 3.1 Plane Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1 Point Line, Ray, Segment . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.3 Angle Between Two Intersecting Lines . . . . . . . . . . . . . . . . 3.1.1.4 Pairs of Angles with Intersecting Parallels . . . . . . . . . . . . . . 3.1.1.5 Angles Measured in Degrees and in Radians . . . . . . . . . . . . . 3.1.2 Geometrical Definition of Circular and Hyperbolic Functions . . . . . . . . . 3.1.2.1 Definition of Circular or Trigonometric Functions . . . . . . . . . . 3.1.2.2 Definitions of the Hyperbolic Functions . . . . . . . . . . . . . . . 3.1.3 Plane Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3.1 Statements about Plane Triangles . . . . . . . . . . . . . . . . . . 3.1.3.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Plane Quadrangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.1 Parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.2 Rectangle and Square . . . . . . . . . . . . . . . . . . . . . . . . .

128 128 128 128 128 128 129 130 130 130 131 131 131 132 134 134 135

2.18

2.19

.

123 123 123 123 124 125 125 126 127

3.2

3.3

3.4

3.5

3.1.4.3 Rhombus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.4 Trapezoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.5 General Quadrangle . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.6 Inscribed Quadrangle . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.7 Circumscribing Quadrangle . . . . . . . . . . . . . . . . . . . . . 3.1.5 Polygons in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5.1 General Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5.2 Regular Convex Polygons . . . . . . . . . . . . . . . . . . . . . . . 3.1.5.3 Some Regular Convex Polygons . . . . . . . . . . . . . . . . . . . 3.1.6 The Circle and Related Shapes . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6.1 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6.2 Circular Segment and Circular Sector . . . . . . . . . . . . . . . . 3.1.6.3 Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.1 Calculationsin Right-AngledTrianglesin the Plane . . . . . . . . 3.2.1.2 Calculations in General Triangles in the Plane . . . . . . . . . . . 3.2.2 Geodesic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.1 Geodetic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.2 Angles in Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.3 Applications in Surveying . . . . . . . . . . . . . . . . . . . . . . Stereometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Lines and Planes in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Edge, Corner, Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Polyeder or Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Solids Bounded by Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Basic Concepts of Geometry on the Sphere . . . . . . . . . . . . . . . . . . 3.4.1.1 Curve, Arc, and Angle on the Sphere . . . . . . . . . . . . . . . . 3.4.1.2 Special Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 3.4.1.3 Spherical Lune or Biangle . . . . . . . . . . . . . . . . . . . . . . 3.4.1.4 Spherical Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1.5 Polar Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1.6 Euler Triangles and Kon-Euler Triangles . . . . . . . . . . . . . . 3.4.1.7 Trihedral Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Basic Properties of Spherical Triangles . . . . . . . . . . . . . . . . . . . . . 3.4.2.1 General Statements . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2.2 Fundamental Formulas and Applications . . . . . . . . . . . . . . 3.4.2.3 Further Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Calculation of Spherical Triangles . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.1 Basic Problems, Accuracy Observations . . . . . . . . . . . . . . . 3.4.3.2 Right-Angled Spherical Triangles . . . . . . . . . . . . . . . . . . 3.4.3.3 Spherical Triangles with Oblique Angles . . . . . . . . . . . . . . . 3.4.3.4 Spherical Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector .4 lgebra and Analytical Geometry . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.1 Definition of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.2 Calculation Rules for Vectors . . . . . . . . . . . . . . . . . . . . . 3.5.1.3 Coordinates of a Vector . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.4 Directional Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.5 Scalar Product and Vector Product . . . . . . . . . . . . . . . . . 3.5.1.6 Combination of Vector Products . . . . . . . . . . . . . . . . . . .

135 135 136 136 136 137 137 137 138 138 138 140 140 141 141 141 141 143 143 145 147 150 150

150 151 154 158 158 158 160 161 161 162 162 163 163 163 164 166 167 167 168 169 172 180 180 180 181 182 183 183 184

Contents XV

3.6

3.5.1.7 Vector Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.8 Covariant and Contravariant Coordinates of a Vector . . . . . . . 3.5.1.9 Geometric Applications of Vector Algebra . . . . . . . . . . . . . . 3.5.2 Analytical Geometry of the Plane . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.1 Basic Concepts, Coordinate Systems in the Plane . . . . . . . . . . 3.5.2.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . 3.5.2.3 Special Notation in the Plane . . . . . . . . . . . . . . . . . . . . 3.5.2.4 Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.5 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.6 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.7 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.8 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.9 Quadratic Curves (Curves of Second Order or Conic Sections) . . . 3.5.3 Analytical Geometry of Space . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3.1 Basic Concepts, Spatial Coordinate Systems . . . . . . . . . . . . 3.5.3.2 Transformation of Orthogonal Coordinates . . . . . . . . . . . . . 3.5.3.3 Special Quantities in Space . . . . . . . . . . . . . . . . . . . . . . 3.5.3.4 Line and Plane in Space . . . . . . . . . . . . . . . . . . . . . . . 3.5.3.5 Surfaces of Second Order, Equations in Normal Form . . . . . . . . 3.5.3.6 Surfaces of Second Order or Quadratic Surfaces, General Theory . Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.1 Ways to Define a Plane Curve . . . . . . . . . . . . . . . . . . . . 3.6.1.2 Local Elements of a Curve . . . . . . . . . . . . . . . . . . . . . . 3.6.1.3 Special Points of a Curve . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.4 Asymptotes of Curves . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.5 General Discussion of a Curve Given by an Equation . . . . . . . . 3.6.1.6 Evolutes and Evolvents . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.7 Envelope of a Family of Curves . . . . . . . . . . . . . . . . . . . . 3.6.2 Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.1 Ways to Define a Space Curve . . . . . . . . . . . . . . . . . . . . 3.6.2.2 Moving Trihedral . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.3 Curvature and Torsion . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3.1 Ways to Define a Surface . . . . . . . . . . . . . . . . . . . . . . . 3.6.3.2 Tangent Plane and Surface Normal . . . . . . . . . . . . . . . . . 3.6.3.3 Line Elements of a Surface . . . . . . . . . . . . . . . . . . . . . . 3.6.3.4 Curvature of a Surface . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3.5 Ruled Surfaces and Developable Surfaces . . . . . . . . . . . . . . 3.6.3.6 Geodesic Lines on a Surface . . . . . . . . . . . . . . . . . . . . .

4 Linear Algebra 4.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Notion of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Square Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Arithmetical Operations with Matrices . . . . . . . . . . . . . . . . . . . . 4.1.5 Rules of Calculation for Matrices . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 187 189 189 189 190 191 194 197 198 200 203 205 207 207 210 212 214 220 223 225 225 225 225 231 234 235 236 237 238 238 238 240 243 243 244 245 247 250 250

251

251 251 252 253 254 257 258 258 259 259

4.3

4.4

4.5

4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Subdeterminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Rules of Calculation for Determinants . . . . . . . . . . . . . . . . . . . . . 4.2.4 Evaluation of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Transformation of Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 4.3.2 Tensors in Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Tensors with Special Properties . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3.1 Tensors of Rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3.2 Invariant Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Tensors in Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . 4.3.4.1 Covariant and Contravariant Basis Vectors . . . . . . . . . . . . . 4.3.4.2 Covariant and Contravariant Coordinates of Tensors of Rank 1 . . 4.3.4.3 Covariant, Contravariant and Mixed Coordinates of Tensors of Rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4.4 Rules of Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Pseudotensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5.1 Symmetry with Respect to the Origin . . . . . . . . . . . . . . . . 4.3.5.2 Introduction to the Notion of Pseudotensors . . . . . . . . . . . . Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Linear Systems, Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.2 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.3 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1.4 Calculation of the Inverse of a Matrix . . . . . . . . . . . . . . . . 4.4.2 Solution of Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 4.4.2.1 Definition and Solvability . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.2 Application of Pivoting . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.3 Cramer's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.4 Gauss's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Overdetermined Linear Equation Systems . . . . . . . . . . . . . . . . . . . 4.4.3.1 overdetermined Linear Systems of Equations and Linear Mean Square Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.2 Suggestions for Numerical Solutions of Mean Square Value Problems Eigenvalue Problems for Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 General Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Special Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.1 Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . . . 4.5.2.2 Real Symmetric Matrices, Similarity Transformations . . . . . . . 4.5.2.3 Transformation of Principal Axes of Quadratic Forms . . . . . . . 4.5.2.4 Suggestions for the Numerical Calculations of Eigenvalues . . . . . 4.5.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . .

5 Algebra and Discrete Mathematics

5.1 5.2

Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Propositional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Formulas in Predicate Calculus . . . . . . . . . . . . . . . . . . . . . . . . . SetTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Concept of Set, Special Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Operations with Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Relations and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Equivalence and Order Relations . . . . . . . . . . . . . . . . . . . . . . . .

259 259 260 261 262 262 262 264 264 265 266 266 266 267 268 268 269 270 271 271 271 271 272 272 272 272 274 275 276 277 277 278 278 278 278 278 280 281 283 285 286

286 286 289 290 290 291 294 296

5.3

5.4

5.5

5.23 Cardinality of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . 5.3.3.2 Subgroups and Direct Products . . . . . . . . . . . . . . . . . . . 5.3.3.3 Mappings Between Groups . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.2 Particular Representations . . . . . . . . . . . . . . . . . . . . . . 5.3.4.3 Direct Sum of Representations . . . . . . . . . . . . . . . . . . . . 5.3.4.4 Direct Product of Representations . . . . . . . . . . . . . . . . . . 5.3.4.5 Reducible and Irreducible Representations . . . . . . . . . . . . . 5.3.4.6 Schur’s Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.7 Clebsch-Gordan Series . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.8 Irreducible Representations of the Symmetric Group SA! . . . . . . 5.3.5 .4pplications of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5.1 Symmetry Operations, Symmetry Elements . . . . . . . . . . . . . 5.3.5.2 Symmetry Groups or Point Groups . . . . . . . . . . . . . . . . . 5.3.5.3 Symmetry Operations with Molecules . . . . . . . . . . . . . . . . 5.3.5.4 Symmetry Groups in Crystallography . . . . . . . . . . . . . . . . 5.3.5.5 Symmetry Groups in Quantum Mechanics . . . . . . . . . . . . . 5.3.5.6 Further .4pplications of Group Theory in Physics . . . . . . . . . . 5.3.6 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6.2 Subrings, Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6.3 Homomorphism, Isomorphism, Homomorphism Theorem . . . . . 5.3.7 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7.2 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7.3 Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7.4 Subspaces, Dimension Formula . . . . . . . . . . . . . . . . . . . . 5.3.7.5 Euclidean Vector Spaces, Euclidean Norm . . . . . . . . . . . . . . 5.3.7.6 Linear Operators in Vector Spaces . . . . . . . . . . . . . . . . . . Elementary Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.1 Divisibility and Elementary Divisibility Rules . . . . . . . . . . . . 5.4.1.2 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.3 Criteria for Divisibility . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.4 Greatest Common Divisor and Least Common Multiple . . . . . . 5.4.13 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Linear Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Congruences and Residue Classes . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Theorems of Fermat, Euler, and Wilson . . . . . . . . . . . . . . . . . . . . 54.5 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cryptology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Problem of Cryptology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Mathematical Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Security of Cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4.1 Methods of Conventional Cryptography . . . . . . . . . . . . . . .

298 ...

298 298 299 299 299 300 302 303 303 303 305 305 305 306 306 306 307 307 308 308 310 312 312 313 313 313 314 314 314 315 315 315 316 316 318 318 318 318 320 321 323 323 325 329 329 332 332 332 332 333 333

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5.6

5.7

5.8

5.9

5.5.4.2 Linear Substitution Ciphers . . . . . . . . . . . . . . . . . . . . . 5.5.4.3 Vigenbre Cipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4.4 Matrix Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Methods of Classical Cryptanalysis . . . . . . . . . . . . . . . . . . . . . . 5.5.5.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5.2 Kasiski-Friedman Test . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 One-Time Pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.7 Public Key Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.7.1 Diffie-Hellman Key Exchange . . . . . . . . . . . . . . . . . . . . 5.5.7.2 One-way Function . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.7.3 RSA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.8 DES .4lgorithm (Data Encryption Standard) . . . . . . . . . . . . . . . . . 5.5.9 IDEA Algorithm (International Data Encryption Algorithm) . . . . . . . . Universal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Congruence Relations, Factor Algebras . . . . . . . . . . . . . . . . . . . . 5.6.3 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Homomorphism Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.6 Term Algebras, Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . Boolean Algebras and Switch Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Finite Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.4 Boolean Algebras as Orderings . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5 Boolean Functions, Boolean Expressions . . . . . . . . . . . . . . . . . . . . 5.7 6 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.7 Switch Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms of Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Basic Notions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Traverse of Undirected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2.1 Edge Sequences or Paths . . . . . . . . . . . . . . . . . . . . . . . 5.8.2.2 Euler Trails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2.3 Hamiltonian Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Trees and Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3.2 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.4 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.5 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.6 Paths in Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.7 Transport Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Basic Notions of Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1.1 Interpretation of Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 5.9.1.2 Membership Functions on the Real Line . . . . . . . . . . . . . . . 5.9.1.3 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Aggregation of Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2.1 Concepts for Aggregation of Fuzzy Sets . . . . . . . . . . . . . . . 5.9.2.2 Practical Aggregator Operations of Fuzzy Sets . . . . . . . . . . . 5.9.2.3 Compensatory Operators . . . . . . . . . . . . . . . . . . . . . . . 5.9.2.4 Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2.5 Fuzzy Complement . . . . . . . . . . . . . . . . . . . . . . . . . .

334 334 334 335 335 335 336 336 336 337 337 337 338 338 338 338 339 339 339 339 340 340 341 341 341 342 343 344 346 346 349 349 350 351 352 352 353 354 355 355 356 358 358 358 359 361 363 363 364 366 366 366

5.9.3 Fuzzy-Valued Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.3.1 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.3.2 Fuzzy Product Relation R o S . . . . . . . . . . . . . . . . . . . . 5.9.4 Fuzzy Inference (Approximate Reasoning) . . . . . . . . . . . . . . . . . . . 5.9.5 Defuzzification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.6 Knowledge-Based Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . 5.9.6.1 Method of Mamdani . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.6.2 Method of Sugeno . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.6.3 Cognitive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.6.4 Knowledge-Based Interpolation Systems . . . . . . . . . . . . . .

367 367

367 369 370 371 372 372 373 373 375

6 Differentiation 377 6.1 Differentiation of Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . 377 6.1.1 Differential Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 6.1.2 Rules of Differentiation for Functions of One Variable . . . . . . . . . . . . . 378 6.1.2.1 Derivatives of the Elementary Functions . . . . . . . . . . . . . . . 378 6.1.2.2 Basic Rules of Differentiation . . . . . . . . . . . . . . . . . . . . 378 6.1.3 Derivatives of Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 6.1.3.1 Definition of Derivatives of Higher Order . . . . . . . . . . . . . . 383 6.1.3.2 Derivatives of Higher Order of some Elementary Functions . . . . . 383 6.1.3.3 Leibniz’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 383 6.1.3.4 Higher Derivatives of Functions Given in Parametric Form . . . . . 385 6.1.3.5 Derivatives of Higher Order of the Inverse Function . . . . . . . . . 385 6.1.4 Fundamental Theorems of Differential Calculus . . . . . . . . . . . . . . . . 386 6.1.4.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 6.1.4.2 Fermat’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 386 6.1.4.3 Rolle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 6.1.4.4 Mean Value Theorem of Differential Calculus . . . . . . . . . . . . 387 6.1.4.5 Taylor’s Theorem of Functions of One Variable . . . . . . . . . . . 387 6.1.4.6 Generalized Mean Value Theorem of Differential Calculus (Cauchy’s Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 6.1.5 Determination of the Extreme Values and Inflection Points . . . . . . . . . . 388 6.13.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . 388 6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 388 6.1.5.3 Relative Extreme Values of a Differentiable, Explicit Function . . . 389 6.1.5.4 Determination of Absolute Extrema . . . . . . . . . . . . . . . . . 390 6.1.5.5 Determination of the Extrema of Implicit Functions . . . . . . . . 390 6.2 Differentiation of Functions of Several Variables . . . . . . . . . . . . . . . . . . . . 390 6.2.1 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 6.2.1.1 Partial Derivative of a Function . . . . . . . . . . . . . . . . . . . 390 6.2.1.2 Geometrical Meaning for Functions of Two Variables . . . . . . . . 391 6.2.1.3 Differentials of zand f(z) . . . . . . . . . . . . . . . . . . . . . . 391 6.2.1.4 Basic Properties of the Differential . . . . . . . . . . . . . . . . . . 392 6.2.1.5 Partial Differential . . . . . . . . . . . . . . . . . . . . . . . . . . 392 6.2.2 Total Differential and Differentials of Higher Order . . . . . . . . . . . . . . 392 6.2.2.1 Notion of Total Differential of a Function of Several Variables (Complete Differential) . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 6.2.2.2 Derivatives and Differentials of Higher Order . . . . . . . . . . . . 393 6.2.2.3 Taylor’s Theorem for Functions of Several Variables . . . . . . . . 394 6.2.3 Rules of Differentiation for Functions of Several Variables . . . . . . . . . . 395 6.2.3.1 Differentiation of Composite Functions . . . . . . . . . . . . . . . 395 6.2.3.2 Differentiation of Implicit Functions . . . . . . . . . . . . . . . . . 396

.

6.2.4 Substitution of Variables in Differential Expressions and Coordinate nansformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4.1 Function of One Variable . . . . . . . . . . . . . . . . . . . . . . . 6.2.4.2 Function of Two Variables . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Extreme Values of Functions of Several Variables . . . . . . . . . . . . . . . 6.2.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5.2 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . 6.2.5.3 Determination of Extreme Values of Functions of Two Variables . . 6.2.5.4 Determination of the Extreme Values of a Function of n Variables . 6.2.5.5 Solution of Approximation Problems . . . . . . . . . . . . . . . . 6.2.5.6 Extreme Value Problem with Side Conditions . . . . . . . . . . . .

7 Infinite Series 7.1 Sequences of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Properties of Sequences of Numbers . . . . . . . . . . . . . . . . . . . . . . 7.1.1.1 Definition of Sequence of Kumbers . . . . . . . . . . . . . . . . . . . 7.1.1.2 Monotone Sequences of Kumbers . . . . . . . . . . . . . . . . . . 7.1.1.3 Bounded Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Limits of Sequences of Numbers . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Number Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 General Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.1 Convergence and Divergence of Infinite Series . . . . . . . . . . . . 7.2.1.2 General Theorems about the Convergence of Series . . . . . . . . . 7.2.2 Convergence Criteria for Series with Positive Terms . . . . . . . . . . . . . . 7.2.2.1 Comparison Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.2 D'Alembert's Ratio Test . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.3 Root Test ofCauchy . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.4 Integral Test of Cauchy . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 .4 bsolute and Conditional Convergence . . . . . . . . . . . . . . . . . . . . 7.2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3.2 Properties of Absolutely Convergent Series . . . . . . . . . . . . . 7.2.3.3 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Some Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4.1 The Values of Some Important Number Series . . . . . . . . . . . 7.2.4.2 Bernoulli and Euler Numbers . . . . . . . . . . . . . . . . . . . . 7.2.5 Estimation of the Remainder . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5.1 Estimation with Majorant . . . . . . . . . . . . . . . . . . . . . . 7.2.5.2 Alternating Convergent Series . . . . . . . . . . . . . . . . . . . . 7.2.5.3 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Function Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2.1 Definition, Weierstrass Theorem . . . . . . . . . . . . . . . . . . . 7.3.2.2 Properties of Uniformly Convergent Series . . . . . . . . . . . . . 7.3.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3.1 Definition, Convergence . . . . . . . . . . . . . . . . . . . . . . . 7.3.3.2 Calculations with Power Series . . . . . . . . . . . . . . . . . . . . 7.3.3.3 Taylor Series Expansion, Maclaurin Series . . . . . . . . . . . . . . 7.3.4 Approximation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Asymptotic Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5.1 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5.2 Asymptotic Power Series . . . . . . . . . . . . . . . . . . . . . . .

397 397 398 399 399 399 400 400 401 401 402

402 402 402 402 402 403 404 404 404 404 405 405 405 406 406 407 407 407 408 408 408 410 411 411 412 412 412 412 412 412 413 414 414 414 415 416 417 417 418

Contents XXI Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Trigonometric Sum and Fourier Series . . . . . . . . . . . . . . . . . . . . . 7.4.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1.2 Most Important Properties of the Fourier Series . . . . . . . . . . 7.4.2 Determination of Coefficients for Symmetric Functions . . . . . . . . . . . . 7.4.2.1 Different Kinds of Symmetries . . . . . . . . . . . . . . . . . . . . 7.4.2.2 Forms of the Expansion into a Fourier Series . . . . . . . . . . . . 7.4.3 Determination of the Fourier Coefficients with Numerical Methods . . . . . 7.4.4 Fourier Series and Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . 7.4.5 Remarks on the Table of Some Fourier Expansions . . . . . . . . . . . . . .

418 418 418 419 420 420 421 422 422 423

8 Integral Calculus 8.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Primitive Function or Antiderivative . . . . . . . . . . . . . . . . . . . . . . 8.1.1.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1.2 Integrals of Elementary Functions . . . . . . . . . . . . . . . . . . 8.1.2 Rules of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Integration of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 8.1.3.1 Integrals of Integer Rational Functions (Polynomials) . . . . . . . 8.1.3.2 Integrals of Fractional Rational Functions . . . . . . . . . . . . . . 8.1.3.3 Four Cases of Partial Fraction Decomposition . . . . . . . . . . . . 8.1.4 Integration of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . 8.1.4.1 Substitution to Reduce to Integration of Rational Functions . . . . 8.1.4.2 Integration of Binomial Integrands . . . . . . . . . . . . . . . . . . 8.1.4.3 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Integration of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 8.1.5.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5.2 Simplified Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.6 Integration of Further Transcendental Functions . . . . . . . . . . . . . . . 8.1.6.1 Integrals with Exponential Functions . . . . . . . . . . . . . . . . 8.1.6.2 Integrals with Hyperbolic Functions . . . . . . . . . . . . . . . . . 8.1.6.3 Application of Integration by Parts . . . . . . . . . . . . . . . . . 8.1.6.4 Integrals of Transcendental Functions . . . . . . . . . . . . . . . . 8.2 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Basic Notions, Rules and Theorems . . . . . . . . . . . . . . . . . . . . . . 8.2.1.1 Definition and Existence of the Definite Integral . . . . . . . . . . 8.2.1.2 Properties of Definite Integrals . . . . . . . . . . . . . . . . . . . . 8.2.1.3 Further Theorems about the Limits of Integration . . . . . . . . . 8.2.1.4 Evaluation of the Definite Integral . . . . . . . . . . . . . . . . . . 8.2.2 Application of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2.1 General Principles for .4 pplication of the Definite Integral . . . . . 8.2.2.2 Applications in Geometry . . . . . . . . . . . . . . . . . . . . . . 8.2.2.3 Applications in Mechanics and Physics . . . . . . . . . . . . . . . 8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals . . . . . . . . . . . . . 8.2.3.1 Generalization of the Notion of the Integral . . . . . . . . . . . . . 8.2.3.2 Integrals with Infinite Integration Limits . . . . . . . . . . . . . . 8.2.3.3 Integrals with Unbounded Integrand . . . . . . . . . . . . . . . . . 8.2.4 Parametric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4.1 Definition of Parametric Integrals . . . . . . . . . . . . . . . . . . 8.2.4.2 Differentiation Under the Symbol of Integration . . . . . . . . . . 8.2.4.3 Integration Under the Symbol of Integration . . . . . . . . . . . . 8.2.5 Integration by Series Expansion, Special Non-Elementary Functions . . . . .

425

7.4

425 425 426 426 427 430 430 430 430 433 433 434 435 436 436 436 437 437 438 438 438 438 438 438 439 441 443 445 445 446 449 451 451 452 454 457 457 457 457 458

XXII Contents 8.3

8.4

8.5

Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Line Integrals of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1.2 Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1.3 Evaluation of the Line Integral of the First Type . . . . . . . . . . 8.3.1.4 Application of the Line Integral of the First Type . . . . . . . . . . 8.3.2 Line Integrals of the Second Type . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.2 Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.3 Calculation of the Line Integral of the Second Type . . . . . . . . . 8.3.3 Line Integrals of General Type . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3.2 Properties of the Line Integral of General Type . . . . . . . . . . . 8.3.3.3 Integral Along a Closed Curve . . . . . . . . . . . . . . . . . . . . 8.3.4 Independence of the Line Integral of the Path of Integration . . . . . . . . . 8.3.4.1 Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4.2 Existence of a Primitive Function . . . . . . . . . . . . . . . . . . 8.3.4.3 Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . 8.3.4.4 Determination of the Primitive Function . . . . . . . . . . . . . . 8.3.4.5 Zero-Valued Integral Along a Closed Curve . . . . . . . . . . . . . Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1.1 Yotion of the Double Integral . . . . . . . . . . . . . . . . . . . . 8.4.1.2 Evaluation of the Double Integral . . . . . . . . . . . . . . . . . . 8.4.1.3 Applications of the Double Integral . . . . . . . . . . . . . . . . . 8.4.2 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2.1 Notion of the Triple Integral . . . . . . . . . . . . . . . . . . . . . 8.4.2.2 Evaluation of the Triple Integral . . . . . . . . . . . . . . . . . . . 8.4.2.3 Applications of the Triple Integral . . . . . . . . . . . . . . . . . . Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Surface Integral of the First Type . . . . . . . . . . . . . . . . . . . . . . . 8.5.1.1 Notion of the Surface Integral of the First Type . . . . . . . . . . . 8.5.1.2 Evaluation of the Surface Integral of the First Type . . . . . . . . 8.5.1.3 -4pplications of the Surface Integral of the First Type . . . . . . . 8.5.2 Surface Integral of the Second Type . . . . . . . . . . . . . . . . . . . . . . 8.5.2.1 Notion of the Surface Integral of the Second Type . . . . . . . . . 8.5.2.2 Evaluation of Surface Integrals of the Second Type . . . . . . . . 8.5.2.3 An Application of the Surface Integral . . . . . . . . . . . . . . . .

9 Differential Equations

9.1

Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 9.1.1.1 Existence Theorems, Direction Field . . . . . . . . . . . . . . . . . 9.1.1.2 Important Solution Methods . . . . . . . . . . . . . . . . . . . . . 9.1.1.3 Implicit Differential Equations . . . . . . . . . . . . . . . . . . . . 9.1.1.4 Singular Integrals and Singular Points . . . . . . . . . . . . . . . . 9.1.1.5 .4 pproximation Methods for Solution of First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Differential Equations of Higher Order and Systems of Differential Equations 9.1.2.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2.2 Lowering the Order . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2.3 Linear n-th Order Differential Equations . . . . . . . . . . . . . .

460 461 461 461 461 462 462 462 464 464 465 465 465 466 466 466 467 467 467 468 469 469 469 470 472 474 474 474 477 477 477 478 479 480 481 481 482 484 485 485 486 486 487 490 491

494 495 495 497 498

Contents XXIII

9.2

9.1.2.4 Solution of Linear Differential Equations with Constant Coefficients 500 9.1.2.5 Systems of Linear Differential Equations with Constant Coefficients 503 9.1.2.6 Linear Second-Order Differential Equations . . . . . . . . . . . . . 505 512 9.1.3 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 512 9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues . . . . 513 9.1.3.3 Expansion in Eigenfunctions . . . . . . . . . . . . . . . . . . . . . 514 514 9.1.3.4 Singular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 9.2.1 First-Order Partial Differential Equations . . . . . . . . . . . . . . . . . . . 515 9.2.1.1 Linear First-Order Partial Differential Equations . . . . . . . . . . 515 9.2.1.2 Non-Linear First-Order Partial Differential Equations . . . . . . . 517 9.2.2 Linear Second-Order Partial Differential Equations . . . . . . . . . . . . . . 520 9.2.2.1 Classification and Properties of Second-Order Differential Equations with Two Independent Variables . . . . . . . . . . . . . . . . . . . 520 9.2.2.2 Classification and Properties of Linear Second-Order Differential Equations with more than two Independent Variables . . . . . . . . . . 521 9.2.2.3 Integration Methods for Linear Second-Order Partial Differential Equa522 tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Some further Partial Differential Equations From Natural Sciences and Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 9.2.3.1 Formulation of the Problem and the Boundary Conditions . . . . . 532 9.2.3.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 9.2.3.3 Heat Conduction and Diffusion Equation for Homogeneous Media . 535 536 9.2.3.4 Potential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3.5 Schrodinger's Equation . . . . . . . . . . . . . . . . . . . . . . . . 536 9.2.4 Yon-Linear Partial Differential Equations: Solitons, Periodic Patterns and Chaos 544 9.2.4.1 Formulationofthe Physical-MathematicalProblem . . . . . . . . 544 9.2.4.2 Korteweg de Vries Equation (KdV) . . . . . . . . . . . . . . . . . 546 9.2.4.3 Yon-Linear Schrodinger Equation (NLS) . . . . . . . . . . . . . . 547 9.2.4.4 Sine-Gordon Equation (SG) . . . . . . . . . . . . . . . . . . . . . 547 9.2.4.5 Further Yon-linear Evolution Equations with Soliton Solutions . . 549

10 Calcuilus of Variations 550 10.1 Defining the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 551 10.2 Historical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 10.2.2 Brachistochrone Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 551 10.3 Variational Problems of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Simple Variational Problems and Extrema1 Curves . . . . . . . . . . . . . . 551 10.3.2 Euler Differential Equation of the Variational Calculus . . . . . . . . . . . . 552 10.3.3 Variational Problems with Side Conditions . . . . . . . . . . . . . . . . . . 553 10.3.4 Variational Problems with Higher-Order Derivatives . . . . . . . . . . . . . 554 10.3.5 Variational Problem with Several Unknown Functions . . . . . . . . . . . . 555 10.3.6 Variational Problems using Parametric Representation . . . . . . . . . . . . 555 10.4 Variational Problems with Functions of Several Variables . . . . . . . . . . . . . . . 556 10.4.1 Simple Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 10.4.2 More General Variational Problems . . . . . . . . . . . . . . . . . . . . . . 558 558 10.5 Numerical Solution of Variational Problems . . . . . . . . . . . . . . . . . . . . . . 559 10.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 First and Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 10.6.2 Application in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

I

XXIV Contents 11 Linear Integral Equations 561 11.1 Introduction and Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 11.2 Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . 562 11.2.1 Integral Equations with Degenerate Kernel . . . . . . . . . . . . . . . . . . 562 11.2.2 Successive Approximation Method, Neumann Series . . . . . . . . . . . . . 565 11.2.3 Fredholm Solution Method, Fredholm Theorems . . . . . . . . . . . . . . . 567 11.2.3.1 Fredholm Solution Method . . . . . . . . . . . . . . . . . . . . . . 567 569 11.2.3.2 Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Sumerical Methods for Fredholm Integral Equations of the Second Kind . . 570 11.2.4.1 Approximation of the Integral . . . . . . . . . . . . . . . . . . . . 570 572 11.2.4.2 Kernel .4pproximation . . . . . . . . . . . . . . . . . . . . . . . . 574 11.2.4.3 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 575 11.3 Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . 11.3.1 Integral Equations with Degenerate Kernels . . . . . . . . . . . . . . . . . . 575 11.3.2 Analytic Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 11.3.3 Reduction of an Integral Equation into a Linear System of Equations . . . . 578 11.3.4 Solution of the Homogeneous Integral Equation of the First Kind . . . . . . 579 11.3.5 Construction of Two Special Orthonormal Systems for a Given Kernel . . . 580 11.3.6 Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 583 11.4 Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 11.4.1 Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 11.4.2 Solution by Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Solution of the Volterra Integral Equation of the Second Kind by Yeumann 585 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.4 Convolution Type Volterra Integral Equations . . . . . . . . . . . . . . . . 585 11.4.5 Numerical Methods for Volterra Integral Equations of the Second Kind . . . 586 588 11.5 Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 11.5.1 Abel Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Singular Integral Equation with Cauchy Kernel . . . . . . . . . . . . . . . . .589 589 11.5.2.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 590 11.5.2.2 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2.3 Properties of Cauchy Type Integrals . . . . . . . . . . . . . . . . . 590 11.5.2.4 The Hilbert Boundary Value Problem . . . . . . . . . . . . . . . . 591 11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert 591 Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2.6 Solution of the Characteristic Integral Equation . . . . . . . . . . 592

594 12 Functional Analysis 594 12.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 12.1.1 Notion of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 12.1.2 Linear and .4ffine Linear Subsets . . . . . . . . . . . . . . . . . . . . . . . . 596 12.1.3 Linearly Independent Elements . . . . . . . . . . . . . . . . . . . . . . . . . 597 12.1.4 Convex Subsets and the Convex Hull . . . . . . . . . . . . . . . . . . . . . . 597 12.1.4.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 12.1.4.2 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 12.1.5 Linear Operators and Functionals . . . . . . . . . . . . . . . . . . . . . . . 598 12.1.5.1 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.5.2 Homomorphism and Endomorphism . . . . . . . . . . . . . . . . . 598 599 12.1.5.3 Isomorphic Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 599 12.1.6 Complexification of Real Vector Spaces . . . . . . . . . . . . . . . . . . . . 599 12.1.7 Ordered L'ector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.7.1 Cone and Partial Ordering . . . . . . . . . . . . . . . . . . . . . . 599

12.1.7.2 Order Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.7.3 Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.7.4 Vector Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Notion of a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1.1 Balls, Neighborhoods and Open Sets . . . . . . . . . . . . . . . . . 12.2.1.2 Convergence of Sequences in Metric Spaces . . . . . . . . . . . . . 12.2.1.3 Closed Sets and Closure . . . . . . . . . . . . . . . . . . . . . . . 12.2.1.4 Dense Subsets and Separable Metric Spaces . . . . . . . . . . . . . 12.2.2 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2.1 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2.2 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 12.2.2.3 Some Fundamental Theorems in Complete Metric Spaces . . . . . 12.2.2.4 Some Applications of the Contraction Mapping Principle . . . . . 12.2.2.5 Completion of a Metric Space . . . . . . . . . . . . . . . . . . . . 12.2.3 Continuous Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Notion of a Normed Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1.1 Axioms of a Normed Space . . . . . . . . . . . . . . . . . . . . . . 12.3.1.2 Some Properties of Normed Spaces . . . . . . . . . . . . . . . . . 12.3.2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2.1 Series in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . 12.3.2.2 Examples of Banach Spaces . . . . . . . . . . . . . . . . . . . . . 12.3.2.3 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Ordered Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Normed Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Notion of a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1.1 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1.2 Unitary Spaces and Some of their Properties . . . . . . . . . . . . 12.4.1.3 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2.1 Properties of Orthogonality . . . . . . . . . . . . . . . . . . . . . 12.4.2.2 Orthogonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Fourier Series in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3.1 Best Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3.2 Parseval Equation! Riesz-Fischer Theorem . . . . . . . . . . . . . 12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces . . . . . . . . . . . . . . . . 12.5 Continuous Linear Operators and Functionals . . . . . . . . . . . . . . . . . . . . . 12.5.1 Boundedness, Norm and Continuity of Linear Operators . . . . . . . . . . . 12.5.1.1 Boundedness and the Norm of Linear Operators . . . . . . . . . . 12.5.1.2 The Space of Linear Continuous Operators . . . . . . . . . . . . . 12.5.1.3 Convergence of Operator Sequences . . . . . . . . . . . . . . . . . 12.5.2 Linear Continuous Operators in Banach Spaces . . . . . . . . . . . . . . . . 12.5.3 Elements of the Spectral Theory of Linear Operators . . . . . . . . . . . . . 12.5.3.1 Resolvent Set and the Resolvent of an Operator . . . . . . . . . . . 12.5.3.2 Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . 12.5.4 Continuous Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.4.2 Continuous Linear Functionals in Hilbert Spaces, Riesz Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.4.3 Continuous Linear Functionals in LP . . . . . . . . . . . . . . . .

600

:602!! 602 603 604 604 605 605 605 606 606 606 608 608 609 609 609 610 610 610 610 611 611 612 613 613 613 613 613 614 614 614 615 615 616 616 617 617 617 617 618 618 620 620 620 621 621 622 622

XXVI Contents

12.6

12.7

12.8

12.9

12.5.5 Extension of a Linear Functional . . . . . . . . . . . . . . . . . . . . . . . . 12.5.6 Separation of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.7 Second Adjoint Space and Reflexive Spaces . . . . . . . . . . . . . . . . . . Adjoint Operators in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Adjoint of a Bounded Operator . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Adjoint Operator of an Unbounded Operator . . . . . . . . . . . . . . . . . 12.6.3 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.3.1 Positive Definite Operators . . . . . . . . . . . . . . . . . . . . . . 12.6.3.2 Projectors in a Hilbert Space . . . . . . . . . . . . . . . . . . . . . Compact Sets and Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 Compact Subsets of a Normed Space . . . . . . . . . . . . . . . . . . . . . . 12.7.2 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2.1 Definition of Compact Operator . . . . . . . . . . . . . . . . . . . 12.7.2.2 Properties of Linear Compact Operators . . . . . . . . . . . . . . 12.7.2.3 Weak Convergence of Elements . . . . . . . . . . . . . . . . . . . 12.7.3 Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.4 Compact Operators in Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 12.7.5 Compact Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . Non-Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.1 Examples of Non-Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 12.8.2 Differentiability of Non-Linear Operators . . . . . . . . . . . . . . . . . . . 12.8.3 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.4 Schauder's Fixed-point Theorem . . . . . . . . . . . . . . . . . . . . . . . . 12.8.5 Leray-Schauder Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.6 Positive Non-Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.7 Monotone Operators in Banach Spaces . . . . . . . . . . . . . . . . . . . . Measure and Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.1 Sigma Algebra and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.2 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.2.1 Measurable Function . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.2.2 Properties of the Class of Measurable Functions . . . . . . . . . . 12.9.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.3.1 Definition of the Integral . . . . . . . . . . . . . . . . . . . . . . . 12.9.3.2 Some Properties of the Integral . . . . . . . . . . . . . . . . . . . 12.9.3.3 Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . 12.9.4 LP Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5.1 Formula of Partial Integration . . . . . . . . . . . . . . . . . . . . 12.9.5.2 Generalized Derivative . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9.5.4 Derivative of a Distribution . . . . . . . . . . . . . . . . . . . . .

622 623 624 624 624 625 625 626 626 626 626 626 626 627 627 627 628 628 629 629 630 630 631 631 631 632 633 633 634 634 634 635 635 635 636 637 637 637 638 638 639

640 13 Vector Analysis and Vector Fields 640 13.1 Basic Notions of the Theory of Vector Fields . . . . . . . . . . . . . . . . . . . . . . 640 13.1.1 Vector Functions of a Scalar Variable . . . . . . . . . . . . . . . . . . . . . 640 13.1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1.2 Derivative of a Vector Function . . . . . . . . . . . . . . . . . . . 640 13.1.1.3 Rules of Differentiation for Vectors . . . . . . . . . . . . . . . . . 640 13.1.1.4 Taylor Expansion for Vector Functions . . . . . . . . . . . . . . . 641 641 13.1.2 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2.1 Scalar Field or Scalar Point Function . . . . . . . . . . . . . . . . 641 13.1.2.2 Important Special Cases of Scalar Fields . . . . . . . . . . . . . . 641

Contents XXVII 13.1.2.3 Coordinate Definition of a Field . . . . . . . . . . . . . . . . . . . 642 13.1.2.4 Level Surfaces and Level Lines of a Field . . . . . . . . . . . . . . 13.1.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3.1 Vector Field or Vector Point Function . . . . . . . . . . . . . . . . 643 13.1.3.2 Important Cases of Vector Fields . . . . . . . . . . . . . . . . . . 643 13.1.3.3 Coordinate Representation of Vector Fields . . . . . . . . . . . . . 644 13.1.3.4 Transformation of Coordinate Systems . . . . . . . . . . . . . . . 645 13.1.3.5 Vector Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 13.2 Differential Operators of Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 13.2.1 Directional and Space Derivatives . . . . . . . . . . . . . . . . . . . . . . . 647 13.2.1.1 Directional Derivative of a Scalar Field . . . . . . . . . . . . . . . 647 13.2.1.2 Directional Derivative of a Vector Field . . . . . . . . . . . . . . . 648 13.2.1.3 Volume Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 648 13.2.2 Gradient of a Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 648 13.2.2.1 Definition of the Gradient . . . . . . . . . . . . . . . . . . . . . . 13.2.2.2 Gradient and Volume Derivative . . . . . . . . . . . . . . . . . . . 649 13.2.2.3 Gradient and Directional Derivative . . . . . . . . . . . . . . . . . 649 13.2.2.4 Further Properties of the Gradient . . . . . . . . . . . . . . . . . . 649 13.2.2.5 Gradient of the Scalar Field in Different Coordinates . . . . . . . . 649 13.2.2.6 Rules of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 650 650 13.2.3 Vector Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 Divergence of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 13.2.4.1 Definition of Divergence . . . . . . . . . . . . . . . . . . . . . . . 651 13.2.4.2 Divergence in Different Coordinates . . . . . . . . . . . . . . . . . 651 13.2.4.3 Rules for Evaluation of the Divergence . . . . . . . . . . . . . . . 651 13.2.4.4 Divergence of a Central Field . . . . . . . . . . . . . . . . . . . . . 652 13.2.5 Rotation of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 13.2.5.1 Definitions of the Rotation . . . . . . . . . . . . . . . . . . . . . . 652 13.2.5.2 Rotation in Different Coordinates . . . . . . . . . . . . . . . . . . 653 13.2.5.3 Rules for Evaluating the Rotation . . . . . . . . . . . . . . . . . . 653 13.2.5.4 Rotation of a Potential Field . . . . . . . . . . . . . . . . . . . . . 654 654 13.2.6 Nabla Operator, Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . 13.2.6.1 Nabla Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 13.2.6.2 Rules for Calculations with the Nabla Operator . . . . . . . . . . . 654 13.2.6.3 Vector Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 13.2.6.4 Nabla Operator Applied Twice . . . . . . . . . . . . . . . . . . . . 655 13.2.6.5 Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 13.2.7 Review of Spatial Differential Operations . . . . . . . . . . . . . . . . . . . 656 13.2.7.1 Fundamental Relations and Results (see Table 13.2) . . . . . . . . 656 13.2.7.2 Rules of Calculation for Spatial Differential Operators . . . . . . . 656 13.2.7.3 Expressions of Vector Analysis in Cartesian, Cylindrical, and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 658 13.3 Integration in Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Line Integral and Potential in Vector Fields . . . . . . . . . . . . . . . . . . 658 13.3.1.1 Line Integral in Vector Fields . . . . . . . . . . . . . . . . . . . . . 658 13.3.1.2 Interpretation of the Line Integral in Mechanics . . . . . . . . . . . 659 13.3.1.3 Properties of the Line Integral . . . . . . . . . . . . . . . . . . . . 659 13.3.1.4 Line Integral in Cartesian Coordinates . . . . . . . . . . . . . . . 659 13.3.1.5 Integral Along a Closed Curve in a Vector Field . . . . . . . . . . . 660 13.3.1.6 Conservative or Potential Field . . . . . . . . . . . . . . . . . . . 660 13.3.2 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 13.3.2.1 Vector of a Plane Sheet . . . . . . . . . . . . . . . . . . . . . . . . 661

E;

I

XXVIII

Contents

13.3.2.2 Evaluation of the Surface Integral . . . . . . . . . . . . . . . . . . 13.3.2.3 Surface Integrals and Flow of Fields . . . . . . . . . . . . . . . . . 13.3.2.4 Surface Integrals in Cartesian Coordinates as Surface Integral of Second Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3.1 Integral Theorem and Integral Formula of Gauss . . . . . . . . . . 13.3.3.2 Integral Theorem of Stokes . . . . . . . . . . . . . . . . . . . . . . 13.3.3.3 Integral Theorems of Green . . . . . . . . . . . . . . . . . . . . . 13.4 Evaluation of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Pure Source Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Pure Rotation Field or Zero-Divergence Field . . . . . . . . . . . . . . . . . 13.4.3 Vector Fields with Point-Like Sources . . . . . . . . . . . . . . . . . . . . . 13.4.3.1 Coulomb Field of a Point-Like Charge . . . . . . . . . . . . . . . . 13.4.3.2 Gravitational Field of a Point Mass . . . . . . . . . . . . . . . . . 13.4.4 Superposition of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.4.1 Discrete Source Distribution . . . . . . . . . . . . . . . . . . . . . 13.4.4.2 Continuous Source Distribution . . . . . . . . . . . . . . . . . . . 13.4.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Differential Equations of Vector Field Theory . . . . . . . . . . . . . . . . . . . . . 13.5.1 Laplace Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Poisson Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . .

661 662 662 663 663 664 664 665 665 666 666 666 667 667 667 667 667 667 667 668

14 Function Theory 669 14.1 Functions of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 14.1.1 Continuity, Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 14.1.1.1 Definition of a Complex Function . . . . . . . . . . . . . . . . . . 669 669 14.1.1.2 Limit of a Complex Function . . . . . . . . . . . . . . . . . . . . . 14.1.1.3 Continuous Complex Functions . . . . . . . . . . . . . . . . . . . 669 14.1.1.4 Differentiability of a Complex Function . . . . . . . . . . . . . . . 669 670 14.1.2 iZnalytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2.1 Definition of Analytic Functions . . . . . . . . . . . . . . . . . . . 670 14.1.2.2 Examples of Analytic Functions . . . . . . . . . . . . . . . . . . . 670 14.1.2.3 Properties of Analytic Functions . . . . . . . . . . . . . . . . . . . 670 671 14.1.2.4 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 14.1.3 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3.1 Notion and Properties of Conformal Mappings . . . . . . . . . . . 672 673 14.1.3.2 Simplest Conformal Mappings . . . . . . . . . . . . . . . . . . . . 14.1.3.3 The Schwarz Reflection Principle . . . . . . . . . . . . . . . . . . 679 679 14.1.3.4 Complex Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 681 14.1.3.5 Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3.6 arbitrary Mappings of the Complex Plane . . . . . . . . . . . . . 682 683 14.2 Integration in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 14.2.1 Definite and Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1.1 Definition of the Integral in the Complex Plane . . . . . . . . . . . 683 14.2.1.2 Properties and Evaluation of Complex Integrals . . . . . . . . . . 684 686 14.2.2 Cauchy Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains . . . . . 686 14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains . . . . 686 687 14.2.3 Cauchy Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3.1 Analytic Function on the Interior of a Domain . . . . . . . . . . . 687 14.2.3.2 Analytic Function on the Exterior of a Domain . . . . . . . . . . . 687 687 14.3 Power Series Expansion of analytic Functions . . . . . . . . . . . . . . . . . . . . .

14.3.1 Convergence of Series with Complex Terms . . . . . . . . . . . . . . . . . . 14.3.1.1 Convergence of a Number Sequence with Complex Terms . . . . . 14.3.1.2 Convergence of an Infinite Series with Complex Terms . . . . . . . 14.3.1.3 Power Series with Complex Terms . . . . . . . . . . . . . . . . . . 14.3.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Principle of .4 nalytic Continuation . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 Laurent Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.5 Isolated Singular Points and the Residue Theorem . . . . . . . . . . . . . . 14.3.5.1 Isolated Singular Points . . . . . . . . . . . . . . . . . . . . . . . 14.3.5.2 Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . . . 14.3.5.3 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.5.4 Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.5.5 Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Evaluation of Real Integrals by Complex Integrals . . . . . . . . . . . . . . . . . . 14.4.1 Application of Cauchy Integral Formulas . . . . . . . . . . . . . . . . . . . 14.4.2 application of the Residue Theorem . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Application of the Jordan Lemma . . . . . . . . . . . . . . . . . . . . . . . 14.4.3.1 Jordan Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3.2 Examples of the Jordan Lemma . . . . . . . . . . . . . . . . . . . 14.5 Algebraic and Elementary Transcendental Functions . . . . . . . . . . . . . . . . . 14.5.1 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Elementary Transcendental Functions . . . . . . . . . . . . . . . . . . . . . 14.5.3 Description of Curves in Complex Form . . . . . . . . . . . . . . . . . . . . 14.6 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Relation to Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 Jacobian Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.3 Theta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.4 Weierstrass Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

687 687 688 688 689 689 690 690 690 691 691 691 692 692 692 693 693 693 694 696 696 696 699 700 700 701 703 703

15 Integral Transformations 15.1 Notion of Integral Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 General Definition of Integral Transformations . . . . . . . . . . . . . . . . 15.1.2 Special Integral Transformations . . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 Inverse Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.4 Linearity of Integral Transformations . . . . . . . . . . . . . . . . . . . . . 15.1.5 Integral Transformations for Functions of Several Variables . . . . . . . . . 15.1.6 Applications of Integral Transformations . . . . . . . . . . . . . . . . . . . 15.2 Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Properties of the Laplace Transformation . . . . . . . . . . . . . . . . . . . 16.2.1.1 Laplace Transformation, Original and Image Space . . . . . . . . . 15.2.1.2 Rules for the Evaluation of the Laplace Transformation . . . . . . 15.2.1.3 Transforms of Special Functions . . . . . . . . . . . . . . . . . . . 15.2.1.4 Dirac 6 Function and Distributions . . . . . . . . . . . . . . . . . 15.2.2 Inverse Transformation into the Original Space . . . . . . . . . . . . . . . . 15.2.2.1 Inverse Transformation with the Help of Tables . . . . . . . . . . . 15.2.2.2 Partial Fraction Decomposition . . . . . . . . . . . . . . . . . . . 15.2.2.3 Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2.4 Inverse Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Solution of Differential Equations using Laplace Transformation . . . . . . . 15.2.3.1 Ordinary Linear Differential Equations with Constant Coefficients 15.2.3.2 Ordinary Linear Differential Equations with Coefficients Depending on the Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . .

705 705 705 705 705 705 707 707 708

708 708 709 712 715 716 716 716 717 718 719 719 720

15.2.3.3 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 15.3 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Properties of the Fourier Transformation . . . . . . . . . . . . . . . . . . . 15.3.1.1 Fourier Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1.2 Fourier Transformationand Inverse Transformation . . . . . . . . 15.3.1.3 Rules of Calculation with the Fourier Transformation . . . . . . . 15.3.1.4 Transforms of Special Functions . . . . . . . . . . . . . . . . . . . 15.3.2 Solution of Differential Equations using the Fourier Transformation . . . . . 15.3.2.1 Ordinary Linear Differential Equations . . . . . . . . . . . . . . . 15.3.2.2 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 15.4 Z-Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Properties of the Z-Ttansformation . . . . . . . . . . . . . . . . . . . . . . . 15.4.1.1 Discrete Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1.2 Definition of the Z-Transformation . . . . . . . . . . . . . . . . . . 15.4.1.3 Rules of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1.4 Relation to the Laplace Transformation . . . . . . . . . . . . . . . 15.4.1.5 Inverse of the Z-Transformation . . . . . . . . . . . . . . . . . . . 15.4.2 Applications of the Z-Transformation . . . . . . . . . . . . . . . . . . . . . 15.4.2.1 General Solution of Linear Difference Equations . . . . . . . . . . 15.4.2.2 Second-Order Difference Equations (Initial Value Problem) . . . . 15.4.2.3 Second-Order Difference Equations (Boundary Value Problem) . . 15.5 Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.4 Discrete Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . . . . 15.5.4.1 Fast Wavelet Transformation . . . . . . . . . . . . . . . . . . . . . 15.5.4.2 Discrete Haar Wavelet Transformation . . . . . . . . . . . . . . . 15.5.5 Gabor Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Walsh Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.1 Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.2 Walsh Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

721 722 722 722 723 725 728 729 729 730 731 732 732 732 733 734 735 736 736 737 738 738 738 739 739 741 741 741 741 742 742 742

16 Probability Theory a n d Mathematical Statistics 16.1 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.3 Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.4 Collection of the Formulas of Combinatorics (see Table 16.1) . . . . . . . . . 16.2 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Event, Frequency and Probability . . . . . . . . . . . . . . . . . . . . . . . 16.2.1.1 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1.2 Frequencies and Probabilities . . . . . . . . . . . . . . . . . . . . 16.2.1.3 Conditional Probability, Bayes Theorem . . . . . . . . . . . . . . 16.2.2 Random Variables, Distribution Functions . . . . . . . . . . . . . . . . . . 16.2.2.1 Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2.2 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2.3 Expected Value and Variance, Chebyshev Inequality . . . . . . . . 16.2.2.4 Multidimensional Random Variable . . . . . . . . . . . . . . . . . 16.2.3 Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3.1 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3.2 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . .

743 743 743 743 744 745 745 745 745 746 748 749 749 749 751 752 752 753 754

16.2.3.3 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4 Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.2 Standard Normal Distribution, Gaussian Error Function . . . . . . 16.2.4.3 Logarithmic Normal Distribution . . . . . . . . . . . . . . . . . . 16.2.4.4 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.5 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.6 y, * (Chi-square) Distribution . . . . . . . . . . . . . . . . . . . . . 16.2.4.7 Fisher F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4.8 Student t Distribution . . . . . . . . . . . . . . . . . . . . . . . . 16.2.5 Law of Large Numbers, Limit Theorems . . . . . . . . . . . . . . . . . . . . 16.2.6 Stochastic Processes and Stochastic Chains . . . . . . . . . . . . . . . . . . 16.2.6.1 Basic Notions, Markov Chains . . . . . . . . . . . . . . . . . . . . 16.2.6.2 Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Statistic Function or Sample Function . . . . . . . . . . . . . . . . . . . . . 16.3.1.1 Population, Sample, Random Vector . . . . . . . . . . . . . . . . . 16.3.1.2 Statistic Function or Sample Function . . . . . . . . . . . . . . . . 16.3.2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2.1 Statistical Summarization and Analysis of Given Data . . . . . . . 16.3.2.2 Statistical Parameters . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3 Important Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3.1 Goodness of Fit Test for a Normal Distribution . . . . . . . . . . . 16.3.3.2 Distribution of the Sample Mean . . . . . . . . . . . . . . . . . . . 16.3.3.3 Confidence Limits for the Mean . . . . . . . . . . . . . . . . . . . 16.3.3.4 Confidence Interval for the Variance . . . . . . . . . . . . . . . . . 16.3.3.5 Structure of Hypothesis Testing . . . . . . . . . . . . . . . . . . . 16.3.4 Correlation and Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.4.1 Linear Correlation of two Measurable Characters . . . . . . . . . . 16.3.4.2 Linear Regression for two Measurable Characters . . . . . . . . . . 16.3.4.3 Multidimensional Regression . . . . . . . . . . . . . . . . . . . . . 16.3.5 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5.2 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5.3 Example of a Monte Carlo Simulation . . . . . . . . . . . . . . . . 16.3.5.4 Application of the Monte Carlo Method in Numerical Mathematics 16.3.5.5 Further Applications of the Monte Carlo Method . . . . . . . . . . 16.4 Calculus of Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Measurement Error and its Distribution . . . . . . . . . . . . . . . . . . . . 16.4.1.1 Qualitative Characterization of Measurement Errors . . . . . . . . 16.4.1.2 Density Function of the Measurement Error . . . . . . . . . . . . . 16.4.1.3 Quantitative Characterization of the Measurement Error . . . . . 16.4.1.4 Determining the Result of a Measurement with Bounds on the Error 16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy 16.4.1.6 Error Estimation for Direct Measurements with Different Accuracy 16.4.2 Error Propagation and Error Analysis . . . . . . . . . . . . . . . . . . . . . 16.4.2.1 Gauss Error Propagation Law . . . . . . . . . . . . . . . . . . . . 16.4.2.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

755 756 756 757 757 758 759 760 761 761 762 763 763 766 767 767 767 768 770 770 771 772 772 774 775 776 777 777 777 778 779 781 781 781 782 783 785 785 785 785 786 788 790 791 791 792 792 794

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17 Dynamical Systems and Chaos 17.1 Ordinary Differential Equations and Mappings . . . . . . . . . . . . . . . . . . . . 17.1.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1.2 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Qualitative Theory of Ordinary Differential Equations . . . . . . . . . . . . 17.1.2.1 Existence of Flows, Phase Space Structure . . . . . . . . . . . . . 17.1.2.2 Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . 17.1.2.3 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2.4 Invariant Slanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2.3 Poincarh Slapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2.6 Topological Equivalence of Differential Equations . . . . . . . . . . 17.1.3 Discrete Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.3.1 Steady States. Periodic Orbits and Limit Sets . . . . . . . . . . . . 17.1.3.2 Invariant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.3.3 Topological Conjugacy of Discrete Systems . . . . . . . . . . . . . 17.1.4 Structural Stability (Robustness) . . . . . . . . . . . . . . . . . . . . . . . 17.1.4.1 Structurally Stable Differential Equations . . . . . . . . . . . . . . 17.1.4.2 Structurally Stable Discrete Systems . . . . . . . . . . . . . . . . 17.1.4.3 Generic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Quantitative Description of Attractors . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Probability Measures on Attractors . . . . . . . . . . . . . . . . . . . . . . 17.2.1.1 Invariant Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1.2 Elements of Ergodic Theory . . . . . . . . . . . . . . . . . . . . . 17.2.2 Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2.1 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2.2 Metric Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.3 LyapunovExponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4.1 Metric Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4.2 Dimensions Defined by Invariant Measures . . . . . . . . . . . . . 17.2.4.3 Local Hausdorff Dimension According to Douady and OesterlC . . 17.2.4.4 Examples of Attractors . . . . . . . . . . . . . . . . . . . . . . . . 17.2.5 Strange Attractors and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.6 Chaos in One-Dimensional Mappings . . . . . . . . . . . . . . . . . . . . . 17.3 Bifurcation Theory and Routes to Chaos . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Bifurcations in Morse-Smale Systems . . . . . . . . . . . . . . . . . . . . . 17.3.1.1 Local Bifurcations in Neighborhoods of Steady States . . . . . . . 17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit . . . . . 17.3.1.3 Global Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Transitions to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2.1 Cascade of Period Doublings . . . . . . . . . . . . . . . . . . . . . 17.3.2.2 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2.3 Global Homoclinic Bifurcations . . . . . . . . . . . . . . . . . . . 17.3.2.4 Destruction of a Torus . . . . . . . . . . . . . . . . . . . . . . . .

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18 Optimization 18.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.1 Formulation of the Problem and Geometrical Representation . . . . . . . . 18.1.1.1 The Form of a Linear Programming Problem . . . . . . . . . . . . 18.1.1.2 Examples and Graphical Solutions . . . . . . . . . . . . . . . . . . 18.1.2 Basic Sotions of Linear Programming, Sormal Form . . . . . . . . . . . . .

844

795 795 795 797 798 798 799 801 804 806 808 809 809 810 811 811 811 812 812 814 814 814 815 817 817 817 818 820 820 822 824 825 826 827 827 827 828 833 836 a37 837 837 838 839 844 844 844 843 847

Contents XXXIII 18.1.2.1 Extreme Points and Basis . . . . . . . . . . . . . . . . . . . . . . 18.1.2.2 Normal Form of the Linear Programming Problem . . . . . . . . . 18.1.3 Simplex Slethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3.1 Simplex Tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3.2 Transition to the New Simplex Tableau . . . . . . . . . . . . . . . 18.1.3.3 Determination of an Initial Simplex Tableau . . . . . . . . . . . . 18.1.3.4 Revised Simplex Method . . . . . . . . . . . . . . . . . . . . . . . 18.1.3.5 Duality in Linear Programming . . . . . . . . . . . . . . . . . . . 18.1.4 Special Linear Programming Problems . . . . . . . . . . . . . . . . . . . . . 18.1.4.1 Transportation Problem . . . . . . . . . . . . . . . . . . . . . . . 18.1.4.2 Assignment Problem . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.4.3 Distribution Problem . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.4.4 Travelling Salesman . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.4.5 Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Yon-linear Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Formulation of the Problem Theoretical Basis . . . . . . . . . . . . . . . . 18.2.1.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 18.2.1.2 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1.3 Duality in Optimization . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Special Yon-linear Optimization Problems . . . . . . . . . . . . . . . . . . . 18.2.2.1 Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2.2 Quadratic Optimization . . . . . . . . . . . . . . . . . . . . . . . 18.2.3 Solution Methods for Quadratic Optimization Problems . . . . . . . . . . . 18.2.3.1 Wolfe's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.3.2 Hildreth-d'Esopo Method . . . . . . . . . . . . . . . . . . . . . . 18.2.4 Numerical Search Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.4.1 One-Dimensional Search . . . . . . . . . . . . . . . . . . . . . . . 18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space . . . . . 18.2.5 Methods for Unconstrained Problems . . . . . . . . . . . . . . . . . . . . . 18.2.5.1 Method of Steepest Descent . . . . . . . . . . . . . . . . . . . . . 18.2.5.2 Application of the Newton Method . . . . . . . . . . . . . . . . . 18.2.5.3 Conjugate Gradient Methods . . . . . . . . . . . . . . . . . . . . 18.2.5.4 Method of Davidon Fletcher and Powell (DFP) . . . . . . . . . . 18.2.6 Gradient Methods for Problems with Inequality Type Constraints) . . . . . 18.2.6.1 LIethod of Feasible Directions . . . . . . . . . . . . . . . . . . . . 18.2.6.2 Gradient Projection Method . . . . . . . . . . . . . . . . : . . . . 18.2.7 Penalty Function and Barrier Methods . . . . . . . . . . . . . . . . . . . . . 18.2.7.1 Penalty Function Method . . . . . . . . . . . . . . . . . . . . . . 18.2.7.2 Barrier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.8 Cutting Plane Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Discrete Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Discrete Dynamic Decision hlodels . . . . . . . . . . . . . . . . . . . . . . . 18.3.1.1 n-Stage Decision Processes . . . . . . . . . . . . . . . . . . . . . . 18.3.1.2 Dynamic Programming Problem . . . . . . . . . . . . . . . . . . . 18.3.2 Examples of Discrete Decision Models . . . . . . . . . . . . . . . . . . . . . 18.3.2.1 Purchasing Problem . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2.2 Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Bellman Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3.1 Properties of the Cost Function . . . . . . . . . . . . . . . . . . . 18.3.3.2 Formulation of the Functional Equations . . . . . . . . . . . . . . 18.3.4 Bellman Optimality Principle . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.5 Bellman Functional Equation Method . . . . . . . . . . . . . . . . . . . . .

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847 848 849 849 850 852 853 854 855 855 858 858 859 859 859 859 859 860 861 861 861 862 863 863 865 865 865 866 866 867 867 867 868 868 869 870 872 872 873 874 875 875 875 876 876 876 876 876 876 877 878 878

XXXIV Contents 18.3.5.1 Determination of Minimal Costs . . . . . . . . . . . . . . . . . . . 18.3.5.2 Determination of the Optimal Policy . . . . . . . . . . . . . . . . 18.3.6 Examples of Applications of the Functional Equation Method . . . . . . . . 18.3.6.1 Optimal Purchasing Policy . . . . . . . . . . . . . . . . . . . . . . 18.3.6.2 Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . .

878 878 879 879 880

19 Numerical Analysis 19.1 NumericalSolutionofNon-Linear Equations inaSingleUnknown . . . . . . . . . . 19.1.1 Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1.1 Ordinary Iteration Method . . . . . . . . . . . . . . . . . . . . . . 19.1.1.2 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1.3 Regula Falsi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2 Solution of Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . 19.1.2.1 Homer's Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2.2 Positions of the Roots . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Numerical Solution of Equation Systems . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1.1 Triangular Decomposition of a Matrix . . . . . . . . . . . . . . . . 19.2.1.2 Cholesky's Method for a Symmetric Coefficient Matrix . . . . . . . 19.2.1.3 Orthogonalization Method . . . . . . . . . . . . . . . . . . . . . . 19.2.1.4 Iteration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2 Non-Linear Equation Systems . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2.1 Ordinary Iteration Method . . . . . . . . . . . . . . . . . . . . . . 19.2.2.2 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2.3 Derivative-Free Gauss-Newton Method . . . . . . . . . . . . . . . 19.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.1 General Quadrature Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2 Interpolation Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2.1 Rectangular Formula . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2.2 Trapezoidal Formula . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2.3 Simpson's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2.4 Hermite's Trapezoidal Formula . . . . . . . . . . . . . . . . . . . 19.3.3 Quadrature Formulas of Gauss . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.3.1 Gauss Quadrature Formulas . . . . . . . . . . . . . . . . . . . . . 19.3.3.2 Lobatto's Quadrature Formulas . . . . . . . . . . . . . . . . . . . 19.3.4 Method of Romberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.4.1 Algorithmof the Romberg Method . . . . . . . . . . . . . . . . . 19.3.4.2 Extrapolation Principle . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Approximate Integration of Ordinary Differential Equations . . . . . . . . . . . . . 19.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.1.1 Euler Polygonal Method . . . . . . . . . . . . . . . . . . . . . . . 19.4.1.2 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . 19.4.1.3 Multi-Step Methods . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.1.4 Predictor-Corrector Method . . . . . . . . . . . . . . . . . . . . . 19.4.1.5 Convergence, Consistency, Stability . . . . . . . . . . . . . . . . . 19.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.2.1 Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.2.2 Approximation by Using Given Functions . . . . . . . . . . . . . . 19.4.2.3 Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Approximate Integration of Partial Differential Equations . . . . . . . . . . . . . . 19.5.1 Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

881 881 881 881 882 883 884 884 885 886 887 887 887 890 890 892 893 893 894 894 895 895 896 896 896 897 897 897 897 898 898 898 899 901 901 901 901 902 903 904 905 905 906 907 908 908

19.52 .4pproximation by Given Functions . . . . . . . . . . . . . . . . . . . . . . 19.5.3 Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Approximation, Computation of Adjustment, Harmonic Analysis . . . . . . . . . . 19.6.1 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.1.1 Newton's Interpolation Formula . . . . . . . . . . . . . . . . . . . 19.6.1.2 Lagrange's Interpolation Formula . . . . . . . . . . . . . . . . . . 19.6.1.3 .4itken-Neville Interpolation . . . . . . . . . . . . . . . . . . . . . 19.6.2 .4pproximation in Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.2.1 Continuous Problems, Normal Equations . . . . . . . . . . . . . . 19.6.2.2 Discrete Problems, Normal Equations, Householder's Method . . . 19.6.2.3 Multidimensional Problems . . . . . . . . . . . . . . . . . . . . . 19.6.2.4 Non-Linear Least Squares Problems . . . . . . . . . . . . . . . . . 19.6.3 Chebyshev Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.3.1 Problem Definition and the Alternating Point Theorem . . . . . . 19.6.3.2 Properties of the Chebyshev Polynomials . . . . . . . . . . . . . . 19.6.3.3 Remes Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.3.4 Discrete Chebyshev Approximation and Optimization . . . . . . . 19.6.4 Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.4.1 Formulas for Trigonometric Interpolation . . . . . . . . . . . . . . 19.6.4.2 Fast Fourier Transformation (FFT) . . . . . . . . . . . . . . . . . 19.7 Representation of Curves and Surfaces with Splines . . . . . . . . . . . . . . . . . . 19.7.1 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.1.1 Interpolation Splines . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.1.2 Smoothing Splines . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.2 Bicubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7.2.1 Use of Bicubic Splines . . . . . . . . . . . . . . . . . . . . . . . . 19.7.2.2 Bicubic Interpolation Splines . . . . . . . . . . . . . . . . . . . . . 19.7.2.3 Bicubic Smoothing Splines . . . . . . . . . . . . . . . . . . . . . . 19.7.3 Bernstein-BCzier Representation of Curves and Surfaces . . . . . . . . . . . 19.7.3.1 Principle of the B-B Curve Representation . . . . . . . . . . . . . 19.7.3.2 B-B Surface Representation . . . . . . . . . . . . . . . . . . . . . 19.8 Using the Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.1 Internal Symbol Representation . . . . . . . . . . . . . . . . . . . . . . . . 19.8.1.1 Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.1.2 Internal Number Representation . . . . . . . . . . . . . . . . . . . 19.8.2 Numerical Problems in Calculations with Computers . . . . . . . . . . . . . 19.8.2.1 Introduction, Error Types . . . . . . . . . . . . . . . . . . . . . . 19.8.2.2 Kormalized Decimal Numbers and Round-Off . . . . . . . . . . . . 19.8.2.3 .4ccuracy in Numerical Calculations . . . . . . . . . . . . . . . . . 19.8.3 Libraries of Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 19.8.3.1 NAG Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.3.2 IMSL Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.3.3 Aachen Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.4 Application of Computer Algebra Systems . . . . . . . . . . . . . . . . . . . 19.8.4.1 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8.4.2 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

909

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914 914 915 915 916 916 918 919 919 920 920 921 922 923 924 924 925 928 928 928 929 930 930 930 932 932 932 933 933 933 933 935 936 936 936 938 941 941 942 943 943 943 946

20 Computer Algebra Systems 950 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950 20.1.1 Brief Characterization of Computer Algebra Systems . . . . . . . . . . . . . 950 20.1.2 Examples of Basic Application Fields . . . . . . . . . . . . . . . . . . . . . 950 950 20.1.2.1 Manipulation of Formulas . . . . . . . . . . . . . . . . . . . . . .

XXXVI Contents 20.1.2.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . 20.1.2.3 Graphical Representations . . . . . . . . . . . . . . . . . . . . . . 20.1.2.4 Programming in Computer Algebra Systems . . . . . . . . . . . . 20.1.3 Structure of Computer Algebra Systems . . . . . . . . . . . . . . . . . . . . 20.1.3.1 Basic Structure Elements . . . . . . . . . . . . . . . . . . . . . . . 20.2 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Basic Structure Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2 Types of Numbers in Mathematica . . . . . . . . . . . . . . . . . . . . . . . 20.2.2.1 Basic Types of Numbers in Mathematica . . . . . . . . . . . . . . 20.2.2.2 Special Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2.3 Representation and Conversion of Numbers . . . . . . . . . . . . . 20.2.3 Important Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4.1 Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4.2 Nested Lists, Arrays or Tables . . . . . . . . . . . . . . . . . . . . 20.2.4.3 Operations with Lists . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4.4 Special Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.5 Vectors and Matrices as Lists . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.5.1 Creating Appropriate Lists . . . . . . . . . . . . . . . . . . . . . . 20.2.5.2 Operations with Matrices and Vectors . . . . . . . . . . . . . . . . 20.2.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.6.1 Standard Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.6.2 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.6.3 Pure Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.7 Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.8 Fuiictional Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.9 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.10 Supplement about Syntax, Information, Messages . . . . . . . . . . . . . . . 20.2.10.1 Contexts, Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.10.2 Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.10.3 Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Basic Structure Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1.1 Types and Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1.2 Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2 Types of Sumbers in Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2.1 Basic Types of Numbers in Maple . . . . . . . . . . . . . . . . . . 20.3.2.2 Special Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2.3 Representation and Conversion of Numbers . . . . . . . . . . . . . 20.3.3 Important Operators in Maple . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.4 Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.5 Sequences and Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.6 Tables, .4rrays, Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . 20.3.6.1 Tables and iZrrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.6.2 One-Dimensional Arrays . . . . . . . . . . . . . . . . . . . . . . . 20.3.6.3 Two-Dimensional Arrays . . . . . . . . . . . . . . . . . . . . . . . 20.3.6.4 Special Commands for Vectors and Matrices . . . . . . . . . . . . 20.3.7 Functions and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.7.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.7.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.7.3 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . 20.3.7.4 The Functional Operator map . . . . . . . . . . . . . . . . . . . .

951 952 952 952 952 953 953 954 954 955 955 956 956 956 957 957 958 958 958 959 960 960 960 961 961 962 963 964 964 964 965 965 965 965 966 967 967 967 967 968 969 969 970 970 971 971 972 972 972 973 973 974

Contents XXXVII

20.4

20.5

20.3.8 Programming in Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.9 Supplement about Syntax, Information and Help . . . . . . . . . . . . . . . 20.3.9.1 Using the Maple Library . . . . . . . . . . . . . . . . . . . . . . . 20.3.9.2 Environment Variable . . . . . . . . . . . . . . . . . . . . . . . . 20.3.9.3 Information and Help . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Computer .4lgebra Systems . . . . . . . . . . . . . . . . . . . . . . 20.4.1 Manipulation of Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . 20.4.1.1 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1.2 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.2 Solution of Equations and Systems of Equations . . . . . . . . . . . . . . . 20.4.2.1 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.2.2 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3 Elements of Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3.1 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3.2 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.4 Differential and Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . 20.4.4.1 Nathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.4.2 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphics in Computer Algebra Systems . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1 Graphics with Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1.1 Basic Elements of Graphics . . . . . . . . . . . . . . . . . . . . . . 20.5.1.2 Graphics Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1.3 Syntax of Graphical Representation . . . . . . . . . . . . . . . . . 20.5.1.4 Graphical Options . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1.5 Two-Dimensional Curves . . . . . . . . . . . . . . . . . . . . . . . 20.5.1.6 Parametric Representation of Curves . . . . . . . . . . . . . . . . 20.5.1.7 Representation of Surfaces and Space Curves . . . . . . . . . . . . 20.5.2 Graphics with Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.2.1 Two-Dimensional Graphics . . . . . . . . . . . . . . . . . . . . . . 20.5.2.2 Three-Dimensional Graphics . . . . . . . . . . . . . . . . . . . . .

974

E 975 975 975 976 976 978 981 981 983 984 984 986 989 989 992 994 995 995 995 995 996 998 999 1000 1002 1002 1004

21 Tables 1007 21.1 Frequently Used Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 21.2 Natural Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 21.3 Important Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009 1014 21.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 21.5.1 Integral Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 21.5.1.1 Integrals with X = ax b . . . . . . . . . . . . . . . . . . . . . . 1017 21.5.1.2 IntegralswithX = u x 2 t b x t c . . . . . . . . . . . . . . . . . . . 1019 1020 21.5.1.3 Integrals with X = u2 x 2 . . . . . . . . . . . . . . . . . . . . . . 1022 21.5.1.4 Integrals with X = u3 x3 . . . . . . . . . . . . . . . . . . . . . . 21.5.1.5 Integrals with X = u4 t x4 . . . . . . . . . . . . . . . . . . . . . . 1023 21.5.1.6 Integrals with X = a4 - x4 . . . . . . . . . . . . . . . . . . . . . . 1023 21.5.1.7 Some Cases of Partial Fraction Decomposition . . . . . . . . . . . 1023 21.5.2 Integrals of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . 1024 21.5.2.1 Integrals with f i and u2 b2x . . . . . . . . . . . . . . . . . . . . 1024 1024 21.5.2.2 Other Integrals with f i . . . . . . . . . . . . . . . . . . . . . . . 21.5.2.3 Integrals with ........................ 1025 21.5.2.4 Integrals with and . . . . . . . . . . . . . . . . . 1026 21.5.2.5 Integrals with d m. . . . . . . . . . . . . . . . . . . . . . . . 1027

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X X X V I I I Contents

21.6

21.7

21.8 21.9 21.10 21.11 21.12

21.13 21.14 21.15 21.16 21.17 21.18 21.19

21.5.2.6 Integrals with d m. . . . . . . . . . . . . . . . . . . . . . . . 1029 ........................ 1030 21.5.2.7 Integrals with 1032 21.5.2.8 Integrals with \/axz + bx + c . . . . . . . . . . . . . . . . . . . . . 21.5.2.9 Integrals with other Irrational Expressions . . . . . . . . . . . . . 1034 21.5.2.10 Recursion Formulas for an Integral with Binomial Differential . . . 1034 1035 21.5.3 Integrals of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 1035 21.5.3.1 Integrals with Sine Function . . . . . . . . . . . . . . . . . . . . . 21.5.3.2 Integrals with Cosine Function . . . . . . . . . . . . . . . . . . . . 1037 21.5.3.3 Integrals with Sine and Cosine Function . . . . . . . . . . . . . . . 1039 21.5.3.4 Integrals with Tangent Function . . . . . . . . . . . . . . . . . . .1043 21.5.3.5 Integrals with Cotangent Function . . . . . . . . . . . . . . . . . . 1043 21.5.4 Integrals of other Transcendental Functions . . . . . . . . . . . . . . . . . . 1044 21.5.4.1 Integrals with Hyperbolic Functions . . . . . . . . . . . . . . . . . 1044 21.5.4.2 Integrals with Exponential Functions . . . . . . . . . . . . . . . . 1045 21.5.4.3 Integrals with Logarithmic Functions . . . . . . . . . . . . . . . . 1047 21.5.4.4 Integrals with Inverse Trigonometric Functions . . . . . . . . . . . 1048 21.5.4.5 Integrals with Inverse Hyperbolic Functions . . . . . . . . . . . . . 1049 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 21.6.1 Definite Integrals of Trigonometric Functions . . . . . . . . . . . . . . . . . 1050 21.6.2 Definite Integrals of Exponential Functions . . . . . . . . . . . . . . . . . . 1051 21.6.3 Definite Integrals of Logarithmic Functions . . . . . . . . . . . . . . . . . . 1052 21.6.4 Definite Integrals of Algebraic Functions . . . . . . . . . . . . . . . . . . . . 1053 1055 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k ) , k = s i n a . . . . . . . . . . . . . 1055 21.7.1 Elliptic Integral of the First Kind F (9, 21.7.2 Elliptic Integral of the Second Kind E (9, k ) , k = s i n a . . . . . . . . . . . . 1055 1056 21.7.3 Complete Elliptic Integral, k = s i n a . . . . . . . . . . . . . . . . . . . . . . Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . . . . . . . . . . 1058 Legendre Polynomials of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . 1060 Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 1066 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 21.12.1 Fourier Cosine Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 21.12.2 Fourier Sine Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 1077 21.12.3 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 21.12.4 Exponential Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . 2 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 1083 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085 Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.15.1 Standard Normal Distribution for 0.00 5 z 5 1.99 . . . . . . . . . . . . . . 1085 21.15.2Standard Normal Distribution for 2.00 5 z 5 3.90 . . . . . . . . . . . . . . 1086 x2 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Fisher F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Student t Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090 1091 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 Bibliography

1092

Index

1103

M a t h e m a t i c Symbols

A

List of Tables XXXIX

List of Tables

8

1.1 1.2 1.3

Definition of powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal's triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Auxiliary values for the solution of equations of degree three . . . . . . . . . . . . .

2.1 2.2 2.3 2.4 2.5

77 Domain and range of trigonometric functions . . . . . . . . . . . . . . . . . . . . . Signs of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Values of trigonometric functions for 0". 30'. 45". 60" and 90". . . . . . . . . . . . . 78 Reduction formulas and quadrant relations of trigonometric functions . . . . . . . . 78 Relations between the trigonometric functions of the same argument in the interval 80 O 0 . . . 90 92 Domains and ranges of the area functions . . . . . . . . . . . . . . . . . . . . . . . For the approximate determination of an empirically given function relation . . . . 112

2.6 2.7 2.8 2.9

13 42

3.29

Names of angles in degree and radian measure . . . . . . . . . . . . . . . . . . . . . Properties of some regular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . Defining quantities of a right angled-triangle in the plane . . . . . . . . . . . . . . . Defining quantities of a general triangle, basic problems . . . . . . . . . . . . . . . Grade and Gon Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directional angle in a segment with correct sign for arctan . . . . . . . . . . . . . . Regular polpeders with edge length a . . . . . . . . . . . . . . . . . . . . . . . . . . Defining quantities of a spherical right-angled triangle . . . . . . . . . . . . . . . . First and second basic problems for spherical oblique triangles . . . . . . . . . . . . Third basic problem for spherical oblique triangles . . . . . . . . . . . . . . . . . . Fourth basic problem for spherical oblique triangles . . . . . . . . . . . . . . . . . . Fifth and sixth basic problemes for a spherical oblique triangle . . . . . . . . . . . . Scalar product of basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector product of basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scalar product of reciprocal basis vectors . . . . . . . . . . . . . . . . . . . . . . . Vector product of reciprocal basis vectors . . . . . . . . . . . . . . . . . . . . . . . \'ector equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric application of vector algebra . . . . . . . . . . . . . . . . . . . . . . . . Equation of curves of second order . Central curves ( b # 0) . . . . . . . . . . . . . . Equations of curves of second order . Parabolic curves (6 = 0) . . . . . . . . . . . . Coordinate signs in the octants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connections between Cartesian. cylindrical. and spherical polar coordinates . . . . Notation for the direction cosines under coordinate transformation . . . . . . . . . Type of surfaces of second order with 6 # 0 (central surfaces) . . . . . . . . . . . . Type of surfaces of second order with 6 = 0 (paraboloid, cylinder and two planes) . Tangent and normal equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \tctor and coordinate equations of accompanying configurations of a space curve . . Vector and coordinate equations of accompanying configurations as functions of the arclength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations of the tangent plane and the surface normal . . . . . . . . . . . . . . . .

241 246

5.1 5.2 5.3

Truth table of propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . NAND function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NOR function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

286 288 288

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28

129 139 141 144 145 145 154 168 170 171 172 173 186 186 186 186 188 189 205 206 208 210 211 224 224 226 241

XL Lzst of Tables 5.4 5.5 5.6 5.7 5.8 5.9

Primitive Bravais lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bravais lattice crystal systems, and crystallographic classes . . . . . . . . . . . . . Some Boolean functions with two variables . . . . . . . . . . . . . . . . . . . . . . Tabular representation of a fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . t- and s-norms, p E R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of operations in Boolean logic and in fuzzy logic . . . . . . . . . . . . .

6.1 6.2 6.3

Derivatives of elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Differentiation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Derivatives of higher order of some elementary functions . . . . . . . . . . . . . . . 385

7.1 7.2 7.3

The first Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 First Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 .4pproximation formulas for some frequently used functions . . . . . . . . . . . . . 417

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12

Basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important rules of calculation of indefinite integrals . . . . . . . . . . . . . . . . . . Substitutions for integration of irrational functions I . . . . . . . . . . . . . . . . . Substitutions for integration of irrational functions I1 . . . . . . . . . . . . . . . . . Important properties of definite integrals . . . . . . . . . . . . . . . . . . . . . . . Line integrals of the first type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curve elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane elements of area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of the double integral . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of the triple integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary regions of curved surfaces . . . . . . . . . . . . . . . . . . . . . . . . .

.

310 311 342 358 365 367

426 428 433 434 441 463 463 472 473 477 478 480

11.1 Roots of the Legendre polynomial of the first kind . . . . . . . . . . . . . . . . . . . 572 13.1 Relations between the components of a vector in Cartesian, cylindrical, and spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Fundamental relations for spatial differential operators . . . . . . . . . . . . . . . . 13.3 Expressions of vector analysis in Cartesian. cylindrical, and spherical coordinates . . 13.4 Line. surface, and volume elements in Cartesian, cylindrical, and spherical coordinates

646 656 657 658

14.1 Real and imaginary parts of the trigonometric and hyperbolic functions . . . . . . . 14.2 Absolute values and arguments of the trigonometric and hyperbolic functions . . . . 14.3 Periods, roots and poles of Jacobian functions . . . . . . . . . . . . . . . . . . . . .

698 698 702

15.1 Overview of integral transformations of functions of one variable . . . . . . . . . . . 706 15.2 Comparison of the properties of the Fourier and the Laplace transformation . . . . 728 16.1 16.2 16.3 16.4 16.5 16.6

Collection of the formulas of combinatorics . . . . . . . . . . . . . . . . . . . . . . Relations between events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X*test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confidence level for the sample mean . . . . . . . . . . . . . . . . . . . . . . . . . . Error description of a measurement sequence . . . . . . . . . . . . . . . . . . . . .

745 746 771 774 775 792

17.1 Steady state types in three-dimensional phase space

807

19.1 Helping table for FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . .

913 917

List of Tables XLI

19.3 19.4 19.5 19.6 19.7 19.8

Number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters for the basic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hlathematica, numerical operations . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematica, commands for interpolation . . . . . . . . . . . . . . . . . . . . . . . hlathematica. numerical solution of differential equations . . . . . . . . . . . . . Maple. options for the command fsolve . . . . . . . . . . . . . . . . . . . . . . . . .

20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 20.14 20.15 20.16 20.17 20.18 20.19 20.20 20.21 20.22 20.23 20.24 20.25 20.26 20.27 20.28 20.29

954 Mathematica, Types of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956 Mathematica. Important operators . . . . . . . . . . . . . . . . . . . . . . . . . . Slathematica Commands for the choice of list elements . . . . . . . . . . . . . . . 957 958 Mathernatica. Operations with lists . . . . . . . . . . . . . . . . . . . . . . . . . . 958 hlathematica, Operation Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959 hlathematica, Operations with matrices . . . . . . . . . . . . . . . . . . . . . . . . Slathematica. Standard functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 Mathematica, Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 966 hlaple. Basic types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966 Maple Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 Maple . Type of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968 Maple . Argument of function convert . . . . . . . . . . . . . . . . . . . . . . . . . Maple. Standard functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972 Maple. Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972 Mathematica: Commands for manipulation of algebraic expressions . . . . . . . . . 976 hlathematica . Algebraic polynomial operations . . . . . . . . . . . . . . . . . . . . 977 Maple, Operations to manipulate algebraic expressions . . . . . . . . . . . . . . . . 978 Mathematica, Operations to solve systems of equations . . . . . . . . . . . . . . . 982 Maple. Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 987 Maple . Operations of the Gaussian algorithm . . . . . . . . . . . . . . . . . . . . . 989 Mathematica. Operations of differentiation . . . . . . . . . . . . . . . . . . . . . . Slathematica Commands to solve differential equations . . . . . . . . . . . . . . . 991 994 Maple . Options of operation dsolve . . . . . . . . . . . . . . . . . . . . . . . . . . hlathematica. Two-dimensional graphic objects . . . . . . . . . . . . . . . . . . . 996 996 hlathematica . Graphics commands . . . . . . . . . . . . . . . . . . . . . . . . . . hlathematica. Some graphical options . . . . . . . . . . . . . . . . . . . . . . . . . 996 1001 hlathematica . Options for 3D graphics . . . . . . . . . . . . . . . . . . . . . . . . 1003 Maple. Options for P l o t command . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 hlaple. Options of command p l o t d d . . . . . . . . . . . . . . . . . . . . . . . . . .

21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10 21.11 21.12 21.13 21.14 21.15 21.16

Frequently Used Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natural Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . . . . . . . . . Legendre Polynomials of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z-Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xz Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

934 936 943

.

i:i

947

.

.

.

1007 1007 1009 1014 1017 1050 1055 1057 1058 1060 1061 1066 1080 1083 1085 1087

I

XLII List of Tables 21.17 Fisher F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.18 Student t Distribaon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.19 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1088 1090 1091

1 Arithmetic 1.1 Elementary Rules for Calculations 1.1.1 Numbers 1.1.1.1 Natural, Integer, a n d Rational Numbers 1. Definitions and Notation The positive and negative integers, fractions, and zero are together called the rational numbers. In relation to these we use the following notation (see 5.2.1, l., p. 290): Set of natural numbers: N={O. 1 . 2 , 3 , .. .}, Set of integers: z= { ...,-2,-1,0,1,2 Set of rational numbers: Q= {zlz = with p E Z, q E Z and q # 0 ) 4

The notion of natural numbers arose from enumeration and ordering. The natural numbers are also called the non-negative integers.

2. Properties of the Set of Rational Numbers The set of rational numbers is infinite. The set is ordered. i t . , for any two different given numbers a and b we can tell which is the smaller one.

The set is dense everywhere,Le., between any two different rational numbers a and b ( a < b) there is at least one rational number c (a < c < b). Consequently, there is an infinite number of other rational numbers between any two different rational numbers.

3. Arithmetical Operations The arithmeticaloperations (addition, subtraction, multiplication and division) can be performed with any two rational numbers, and the result is a rational number. The only exception is division b y zero, which is not possible: The operation written in the form a : 0 is meaningless because it does not have any result: If a # 0, then there is no rational number b such that b . 0 = a could be fulfilled, and if a = 0 then b can be any of the rational numbers. The frequently occurring formula a : 0 = m (infinity) does not mean that the division is possible; it is only the notation for the statement: If the denominator approaches zero and, e.g., the numerator does not, then the absolute value (magnitude) of the quotient exceeds any finite limit.

4. Decimal Fractions, Continued Fractions Every rational number a can be represented as a terminating or periodically infinite decimal fraction or as a finite continued fraction (see 1.1.1.4,p. 3). 5 . Geometric Representation If we fix an origin (the zeropoint)0 , a positive direction (orientation),and the unit of length l (measuring rule, see also 2.17.1, p. 114) and (Fig. l.l)3 thenevery rationalnumber a corresponds to acertain point on this line. This point has the coordinate a , and it is a so-called rational point. The line is called the numerical axis. Because the set of rational numbers is dense everywhere, between two rational points there are infinitely many further rational points.

-3-- 11 -2 --4

-

-1

0

1 23 2

w Figure 1.1

;3

* X

0

1

2

Figure 1.2

3

2

1. Arithmetic

1.1.1.2 Irrational and Transcendental Numbers The set of rational numbers is not satisfactory for calculus. Even though it is dense everywhere, it does not cover the whole numerical axis. For example if we rotate the diagonal AB of the unit square around A so that B goes into the point K : then K does not have any rational coordinate (Fig. 1.2). The introduction of irrational numbers allows us to assign a number to every point of the numerical axis. In textbooks we can find exact definitions for irrational numbers, e.g., by nests of intervals. For this survey it is enough to note that the irrational numbers take all the non-rational points of the numerical axis and every irrational number corresponds to a point of the axis, and that every irrational number can be represented as a non-periodic infinite decimal fraction. First of all, the non-integer real roots of the algebraic equation en an-lzn-lt . ' . aiz t a. = o ( n > 1, integer; integer coefficients), (1.la) belong to the irrational numbers. These roots are called algebraic irrationals. IA: The simplest examples of algebraic irrationals are the real roots of the equation zn - a = 0, as numbers of the form @, if they are not rational.

+

+

IB: f i = 1.414.. . , = 2.154.. . are algebraic irrationals The irrational numbers which are not algebraic irrationals are called transcendental. IA: T = 3.141592.. . e = 2.718281.. . are transcendental numbers. IB: The decimal logarithm of the integers, except the numbers of the form IOn, are transcendental. The non-integer roots of the quadratic equation x2 + alz + a0 = o ( a l ,a0 integers) (1.lb)

+

are called quadratic irrationals. They have the form (a b f l ) / c ( a ,b , c integers, c # 0; D > 0, square-free number). I The division of a line segment a in the ratio of the golden section x / a = (a - z)/z (see 3.5.2.3,3., p. 193) leads to the quadratic equation x2 z - 1 = 0, if a = 1. The solution z = (fi- 1)/2 is a quadratic irrational. It contains the irrational number fi

+

1.1.1.3 Real Numbers Rational and irrational numbers together form the set of real numbers, which is denoted by R.

1. Most Important Properties The set of real numbers has the following important properties (see also 1.1.1.1, 2., p. 1). It is: Finite. a Ordered. Dense everywhere. a Closed, Le., every point of the numerical axis corresponds to a real number. This statement does not hold for the rational numbers.

2. Arithmetical Operations Arithmetical operations can be performed with any two real numbers and the result is a real number, too. The only exception is division by zero (see 1.1.1.1. 3., p. 1). Raising to a power and also its inverse operation can be performed among real numbers; so it is possible to take an arbitrary root of any positive number; every positive real number has a logarithm for an arbitrary positive basis. except that 1 cannot be a basis. .-Z further generalization of the notion of numbers leads us to the concept of complex numbers (see 1.5, p. 34).

3. Interval of Numbers A connected set of real numbers with endpoints a and b is called an interval of numbers with endpoints a and b. where a < b and a is allowed to be -aand b is allowed to be +m. If the endpoint itself does

1.1 Elementary Rules for Calculations 3

not belong to the interval, then this end of the interval is open, in the opposite case it is closed. Lf-e define an interval by its endpoints a and b: putting them in braces. We use a bracket for a closed end of the interval and a parenthesis for an open one. We distinguish between open intervals (a, b ) , half-open (half-closed) intervals [a, b) or (a, b] and closed intervals [alb], according to whether none of the endpoints, one of the endpoints or both endpoints belong to it, respectively. h'e frequently meet the notation ]a. b[ instead of (a, b) for open intervals, and analogously [a, b[ instead of [al b). In the case of graphical representations, we denote the open end of the interval by a round arrow head, the closed one by a filled point.

1.1.1.4 Continued Fractions Continuedfractions are nested fractions, by which rational and irrational numbers can be represented and approximat,ed even better than by decimal representation (see 19.8.1.1, p. 934 and W A and IB on p. 4 ) .

1. Rational Numbers The continued fraction of a rational number is finite. For a positive rational number which is greater than 1 it has the form (1.2). We abbreviate it by the symbol P- = [ao:a l , a?, . . . ,an]with Q

ak

1

Lo+ a1 + a2 +

2 1 ( k = 1 , 2 , . . n).

.

1

(1.2)

1 '

,

1 +1 ~

an-l

+ a,

-

,

The numbers ah are calculated with the help of the Euclidean algorithm: (1.3a) (1.3b) (1.3~)

(1.3d) (1.3e) 61 7 1 1 I- = 2 + -= 2 + 7 =2+ 7 = [2;3.1,6]. 27 27 3+3+ 7 7 1t6

2. Irrational Numbers

Continued fractions of irrational numbers do not break off. We call them infinite continued fractions with [ao:a]. a 2 . . . If some numbers ah are repeated in an infinite continued fraction, then this fraction is called a periodic continuedfraction or recurring chain fraction. Every periodic continued fraction represents a quadratic irrationality, and conversely, every quadratic irrationality has a representation in the form of a periodic continued fraction. IThe number fi = 1.4142135 , , , is a quadratic irrationality and it has the periodic continued fraction representation fi = [I;2,2.2,. . .].

.I.

I

4

1. Arithmetic

3. Aproximation of Real Numbers If (Y = [ao;al,az,. . .] is an arbitrary real number, then every finite continued fraction P (Yk = [ao;a i , a 2 , . . . , a h ] = (1.4) 4 represents an approximation of a . The continued fraction is called the k-th approximant of a . It can be calculated by the recursive formula (Yk

=

pk = a k p k - 1 qk

akqk-1

+pk-2 +qk-2

(k 2 1; p-1 = 1,po = a0;q-I = o , q o = 1).

(1.5)

.4ccording to the Liouwille approximation theorem, the following estimat holds:

Furthermore, it can be shown that the approximants approach the real number a with increasing accuracy alternatively from above and from below. The approximants converge to cy especially fast if the numbers a, (i = 1 , 2 , . . . , k) in (1.4) have large values. Consequently, the convergence is worst for the numbers [l;1,1,.. .]. IA: From the decimalpresentationofa the continued fraction representation 7i = [3;7,15,1,292,.. .] follows with the help of (1.3a)-(1.3e). The corresponding approximants (1.5) with the estimat accord22 . 1 333 . 1 ing to (1.6) are: a1 = - with la -'cy11 < - z 2 . lo-', 02 = - with la - a21 < 7z 9 . 7 72 106 106 355 , 1 cy3 = - with I T - a31 < 8. The actual errors are much smaller. They are less than 113 1132 1.3. for a3. The approximants alra2 and cy3 represent better for a l , 8.4. for cy2 and 2.7. approximations for 7i than the decimal representation with the corresponding number of digits. IB: The formula of the golden section x / a = ( a - x ) / x (see 1.1.1.2, p. 2, 3.5.2.3, 3., p. 193 and 17.3.2.4,4.,p. 843) can be represented by the following two continued fractions: x = a[l; 1,1,.. .] and

x = a ( l + &) = a ( l + [2;4,4,4,. 2 2

. .I).

The approximant

cy4

delivers in the first case an accuracy of

0.018 a, in the second case of 0.000 001 a.

1.1.1.5 Commensurability We call two numbers a and b commensurable, Le., measurable by the same number, if both are an integer multiple of a third number c. From a = mc, b = nc ( m ,n E Z)it follows that

ab = z (zrational). Otherwise a and b are incommensurable. IA: In a pentagon the sides and diagonals are incommensurable segments (see IA in 3.1.5.3, p. 138). Today we consider that it was Hippasos from Metapontum (450 BC) who discovered irrational numbers by this example. IB: The length of a side and a diagonal of a square are incommensurable because their ratio is the irrational number 4. IC: The lengths of the golden section (see 1.1.1.2,p. 2 and 3.5.2.3, 3.,p. 193) are incommensurable, because their ratio contains the irrational number &.

1.1.2 Methods for Proof Mostly we use three types of proofs: e direct proof, e indirect proof,

1.1 Elementary Rules for Calculations 5

proof by (mathematical or arithmetical) induction. We also talk about constructive proof.

1.1.2.1 Direct Proof We start with a theorem which is already proved (premisep) and we derive the truth of the theorem we want to prove (conclusion q ) . The logical steps we mostly use for our conclusions are implication and equivalence.

1. Direct Proof by Implication

The implicationp + q means that the truth of the conclusion follows from the truth of the premise (see "Implication" in the truth table, 5.1.1, p. 286). a+b IProve the inequality 2 &for a > 0, b > 0. The premise is the well-known binomial formula 2 ( a + b)' = a2 + 2ab + b2. It follows by subtracting 4ab that (a t b)' - 4ab = ( a - b)' 2 0; we certainly obtain the statement from this inequality if we restrict our investigations only to the positive square roots because of a > 0 and b > 0. ~

2. Direct Proof by Equivalence The proof will be delivered by verifying an equivalent statement. In practice it means that all the arithmetical operations which we use for changing p into q must be uniquely invertible. 1 IProve the inequality 1 + a a' + . . . an < -for 0 < a < 1. 1-a Uultiplying by 1 - a we obtain: 1 - a t a - a2 + a2 - a3 iz ' . . + an - ant' = 1 - ant' < 1. This last inequality is true because of the assumption 0 < ant' < 1, and the inequality we started from also holds because all the arithmetical operations we used are uniquely invertible.

+

+

1.1.2.2 Indirect Proof or Proof by Contradiction To prove the statement q we start from its negation ij, and from q we arrive at a false statement r , Le.,

p + T (see also 5.1.1, 7., p. 288). In this case q must be false, because using the implication a false assumption can result only in a false conclusion (see truth table 5.1.1, p. 286). If q is false q must be true. IProve that the number 4 is irrational. Suppose, 4 is rational. Then the equality

4 = b holds for some integers a: b and b # 0. We can assume that the numbers a , b are coprime numbers, i.e., they a2 do not have any common divisor. We get (fi)' = 2 = - or a2 = 2b2, therefore, a' is an even number, b2 and this is possible only if a = 2n is an even number. We deduce that a' = 4n2 = 2b2 holds, and hence b must be an even number, too. It is obviously a contradiction to the assumption that a and b are coprime. 1.1.2.3 Mathematical Induction With this method, we prove theorems or formulas depending on natural numbers n. The principle of mathematical induction is the following: If the statement is valid for a natural number no, and if from the validity of the statement for a natural number n 2 no the validity of the statement follows for n+ 1, then the statement is valid for every natural number n 2 no. According to these, the steps of the proof are: 1. Basis of t h e Induction: We show that the statement is valid for n = no. Mostly we can choose no = 1. 2. Induction Hypothesis: We suppose n is an integer such that the statement is valid (premise p). 3. Induction Conclusion: We formulate the proposition for n 1 (conclusion q ) . 4. Proof of t h e Implication: p + q JVe call steps 3. and 4. together the induction step or logical deductionfrom n to n + 1.

+

6

1. Arithmetic

1 1 1 IProve the formula s, = - - 1.2 2'3 3.4 The steps of the proof by induction are:

+

1. n = 1 : s1 =

+.

+

'.

1 n n ( n t 1) n + l '

+---

is obviously true. 1+1 1 1 1 - ' t =!holds - for an n 2.3 3.4 n(n+l) n t l n+l 3. Supposing 2. we have to show: sn+l= =

1'2 1 2. Suppose s, = -t 1'2

+.

+

2 1.

n.+ 2-'

1 1 1 1 1 1 4. The proof s,+~ = -t -+ -+. . = s, t t 1'2 2'3 3'4 n(n t 1) ( n l ) ( n + 2) ( n l ) ( n 2) 1 n ( n t 1 ) Z -n+l - n2 2n t 1 n + l ( n t l ) ( n + 2 ) - (n+l)(n+2) - ( n t l ) ( n t 2 ) - n+2'

+-

+

+

+

+

+

-

1.1.2.4 Constructive Proof In approximation theory, for instance, the proof of an existence theorem usually follows a constructwe process, Le., the steps of the proof give a method of calculation for a result which satisfies the propositions of the existence theorem. IThe existence of a third-degree interpolation-spline function (see 19.7.1.1,1.) p. 928) can be proved in the following way: We show that the calculation of the coefficients of a spline satisfying the requirements of the existence theorem results in a tridiagonal linear equation system, which has a unique solution (see 19.7.1.1. 2., p. 929).

1.1.3 Sums and Products 1.1.3.1 Sums 1. Definition To briefly denote a sum we use the summation sign C : alta2t

n

. . . +a,

=

Cak.

(1.8)

k=l

m?th this notation we denote the sum of n summands ak (IC = 1 , 2 , ,. . n). We call k the running index or summation variable.

2. Rules of Calculation 1. S u m of Summands Equal t o Each Other , i.e.! a k = a for k = 1,2,. . . , n: n

1ak = nu.

(1.9a)

k=l

2.

Multiplication by a Constant Factor n

n

k=l

k=l

1cak = c C ah

(1.9b)

3. Separating a Sum n

m

n

C a k = x a k + k=1

4.

k=1

(1.9c)

(1<m
ak k=m+l

Addition of Sums with t h e Same Length t ( a k k=l

t bk t Ck t . ..) =

2 +2 Uk

k=l

k=l

bk

+

5 C k k=l

+ .. . .

(1.9d)

1.1 Elementar?! Rulesfor Calculations

7

5 . Renumbering n

m-n-1

n-m+l

n

Cak =

ak-m-1.

k=l

ak

k=m

=

(1.9e)

akim-1.

k=m

k=l

6. Exchange the Order of Summation in Double Sums n

m

m atk

n

=

~ = kl = l

a,k.

(1.9f)

k = l t=1

1.1.3.2 Products 1. Definition The abbreviated notation for a product is the product sign

n

n:

n

aIa2.,. a n =

(1.10)

ak.

k=l

\Vith this notation we denote a product of n factors a k ( k = 1;2 , . . . , n);where k is called the running index.

2. Rules of Calculation 1. Product of Coincident Factors , Le., a k

= a for k = 1:2 . . .

. , n: ( 1.1l a )

k=l

2. Factor out a Constant Factor

n

n

n

(cak)

= cn

k=l

(1.llb)

ak.

k=l

3. Separating into Partial Products n

m

n

fl

nak=na, k=l

k=l

ak

(1<m
(Lllc)

k=rntl

4. Product of Products

u n

n n n n

UpbkCk..

.=

k=l

n

Uk

t=l

n

bk

k=l

Ck.. .

.

(1.lld)

k=l

5 . Renumbering

fi

u

mtn-1 ak

k=l

=

ak-mtlr

k=m

fi

k=m

n

n-mtl

ak =

aktm-1.

(1.11e)

k=l

6. Exchange the Order of Multiplication in Double Products (1.llf) % = Ik = l

k = l z=l

1.1.4 Powers, Roots, and Logarithms 1.1.4.1 Powers The notation a” is used for the algebraic operation of razszng to a power. The number a is called the base. L is called the exponent or power, and a’ is called the power. Powers are defined as in Table 1.1.

I

For the allowed values of bases and exponents we have the following Rules of Calculation: (1.12) ax

ax b” =

(ab)”. ax : b“ = - =

(a“)Y=

(aY)X

b”

(;)’,

(1.13)

(1.14)

= axy,

(1.15) a” = ex”’’ ( a > 0). Here In a is the natural logarithm of a where e = 2.718281828459.. . is the base. Special powers are $1, if n even, 1, if n odd,

(1.16a)

’

( 1.16b)

a o = l forany a # o .

Table 1.1Definition of powers

base a

power a”

exponent x

-

0 arbitrary real, # 0 n = 1,2,3,.. .

1

an = a . a . a . . . . . a

( a to the power n)

n factors

1 n = -1, -2, -3,. . . an = a-”

positive real

rational: P 4 ( p , q integer, q irrational: lim Pk k+m Pk

I

0

I positive

a:=@

> 0) (q-th root of a to the power p ) lim

% aqk

k+m

Io

I

1.1.4.2 Roots According to Table 1.1the n-th root of a positive number a is the positive number denoted by (1.17a) fi ( a > 0, real; n > 0, integer). We call this operation taking of the root or extraction ofthe root, and a is called the radicand, n is called the radical or index. The solution of the equation (1.17b) zn= a ( a real or complex ; n > 0 , integer) is often denoted by z = fi.But we must not be confused: In this relation this notation denotes all the solutions of the equation, Le., it represents n different values x k ( k = 1 , 2 , .. . , n). In the cace of negative or complex values they are to be determined by (1.140b) (see 1.5.3.6, p. 38). IThe equation x2 = 4 has two real solutions, namely * 2 ~ Ithe equation^^ = -8hasthreerootsamongthecomplexnumbers: x1 = l + i d , x z = -2 and 2 3 = 1 - i d , but only one among the reals

1.1 Elementary Rules for Calculations 9

1.1.4.3 Logarithms 1. Definition The logarithm u of a positive number z > 0 to the base b > 0, b # 1, is the exponent of the power which has the value z with b in the base. We denote it by 1' 1 = log, 2. Consequently the equation bU = z

(1.18a)

log, z = u

yields

and conversely the second one yields the first one. In particular we have (1.18~) The logarithm of negative numbers can be defined only among the complex numbers. To take the logarithm of a given number means to find its logarithm. We take the logarithm of an expression when we transform it like (1.19a, 1.19b). The determination of a number or an expression from its logarithm is called raiszng to a power.

2. Some Properties of the Logarithm a) Every positive number has a logarithm to any positive base, except the base b = 1. b) For z > 0 and y > 0 the following Rules of Calculation are valid for any b (which is allowed to be a base): log(zy) = logz + logy, logsn = nlogz,

(i)

= logz - logy,

( 1.19a)

1 in particular log @ = - logz .

(1.19b)

log

-

With (1.19a. 1.19b) we can calculate the logarithm of products and fractions as sums or differences of logarithms. 3X2yjj 3x2 yjj ITake the logarithm of the expression -' log -= log (3z2+5) - log (zZu3) 2zu3 2Zu3 1 = log3 + 2 10gz + - logy - log2 - lOgZ - 310gu. 3 Often the reverse transformation is required, Le., we have to rewrite an expression containing logarithms of different amounts into one, which is the logarithm of one expression. I

c) Logarithms to different bases are proportional, Le., the logarithm to a base a can be change into a logarithm to the base b by multiplication: log, z = Af log, z where M = log, b = I log, a We call ,tf the modulus of the transformation.

.

(1.20)

1.1.4.4 Special Logarithms 1. The logarithm to the base 10 is called the decimal or Briggsian logarithm. We write log,,a: = lgz and log(zl0') = cy + logzis valid. 2. The logarithm to the base e is called the natural or Neperian logarithm. We write log, 2 = In z. The modulus of transformation to change from the natural logarithm into the decimal is

M

= loge =

In 10

= 0.4342944819,

(1.21) (1.22) (1.23)

10

1. Arithmetic

and to change from the decimal into the natural one it is 1 MI = - = In 10 = 2.3025850930. (1.24) .bf 3. The logarithm to base 2 is called the binary logarithm. We write (1.25) log,x = ldx or log,x = lbx. 4. We can find the values of the decimal and natural logarithm in logarithm tables. Some time ago the logarithm was used for numerical calculation of powers, and it often made numerical multiplication and division easier. Mostly the decimal logarithm was used. Today pocket calculators and personal computers make these calculations. Every number given in decimal form (so every real number), which is called in this relation the antilog, can be written in the form x = i l O k with 1 5 i < 10 (1.26a) by factoring out an appropriate power of ten: lok with integer k. This form is called the half-logarithmic representation. Here i is given by the sequence of figures of x,and lok is the order of magnitude of 2. Then for the logarithm we have

+

(1.26b) logx = k l o g i with 0 5 l o g i < 1, Le., log? = 0, Here k is the so-called characteristic and the sequence of figures behind the decimal point of log? is called the the mantissa. The mantissa can be found in logarithm tables. I1g324 = 2.5105, the characteristic is 2, the mantissa is 5105. If we multiply or divide this number by lon. for example 324000; 3240; 3.24; 0.0324, its logarithms have the same mantissa, here 5105, but different characteristics. That is why the mantissas are given in logarithm tables. In order to get the mantissa of a number z first we have to move the decimal point right or left to get a number between 1 and 10, and the characteristic of the antilog x is determined by how many digits k the decimal point was moved. 5 . Slide rule Beside the logarithm: the slide rule was of important practical help in numerical calculations. The slide rule works by the principle of the form (l.l9a), so we multiply and divide by adding and subtracting numbers. On the slide rule the scale-segments are denoted according to the logarithm values, so multiplication and division can be performed as addition or subtraction (see Scale and Graph Papers 2.17.1, p. 114).

1.1.5 Algebraic Expressions 1.1-5.1 Definitions 1. Algebraic Expression One or more algebraic quantities, such as numbers or symbols. are called an algebraic ezpression or term if they are connected by the symbols. , - , . , : , T , etc., as well as by different types of braces for fixing the order of operations. 2. Identity is an equality relation between two algebraic expressions if for arbitrary values of the symbols in them the equality holds.

+

3. Equation is an equality relation between two algebraic expressions if the equality holds only for a few values of the symbols. For instance an equality relation F ( x ) = f (XI (1.27) between two functions with the same independent variable is considered as an equation with one variable if it holds only for certain values of the variable. If the equality is valid for every value of I,we call it an identity, or we say that the equality holds identically, and we write F ( z ) f (x).

1.1 Elementary Rules for Calculations

11

4. Identical Transformations are performed in order to change an algebraic expression into another one if the two expression are identically equal. Our goal is to have another form, e.g., to get a shorter form or a more convenient form for further calculations. We often want to have the expression in a form which is especially good for solving an equation. or taking the logarithm, or for calculating the derivative or integral of it, etc.

1.1.5.2 Algebraic Expressions in Detail 1. Principal Quantities w'e call principal quantities the literal symbols occurring in algebraic expressions, according to which the expression is classified. They must be fixed in any single case. In the case of functions: the independent variables are the principal quantities. The other quantities not given by numbers are the parameters of the expression. In some expressions the parameters are called coeficients. I& e' talk about coefficients, e.g.. in the cases of polynomials, Fourier series, and linear differential equations, etc. An expression belongs to a certain class, depending on which kind of operations are performed on the principal quantities. Lsually, we use the last letters of the alphabet x, y, z , u , u principal quantities and the first letters a , b: c. . . . for parameters. The letters rn. IZ: p , . . . are usually used for positive integer parameter values, for instance for indices in summations or in iterations.

2. Integral Rational Expressions are expressions xhich contain only addition, subtraction, and multiplication of the principal quantities. The term also means powers of them with non-negative integer exponents.

3. Rational Expressions contain also division by principal quantities, Le., division by integral rational expressions, so principal quantities can have negative integers in the exponent.

4. Irrational Expressions contain roots. Le., non-integer rational powers of integral rational or rational expressions with respect to their principal quantities, of course. 5 . Transcendental Expressions contain exponential. logarithmic or trigonometric expressions of the principal quantities, Le., there can be irrational numbers in the exponent of an expression of principal quantities, or an expression of principal quantities can be in the exponent, or in the argument of a trigonometric or logarithmic expression.

1.1.6 Integral Rational Expressions 1.1.6.1 Representation in Polynomial Form Every integral rational expression can be changed into polynomial form by elementary transformations, as in addition, subtraction, and multiplication of monomials and polynomials. I(-a3 + 2a2x - x3)(4a2 8ax) + (a3x2 2a2x3 - 4az4) - (a5 4a3x2- 4ax4) = -4a5 8a4x - 4a2x3 - 8a4x + 16a3x2- 8ax4 a3x2 2a2x3 - 4ax4 - a5 - 4a3x2 4ax4 = -Sa5 13a3x2- 2a2x3 - 8ax4.

+

+

+ +

+

+

+

+

1.1.6.2 Factorizing a Polynomial Often we can decompose a polynomial into a product of monomials and polynomials. To do so. we can use factoring out, grouping, special formulas, and special properties of equations. IA: Factoring out: 8ar'y - 6bx3yZ+ 4cx5 = 2rc2(4ay - 3bzyZ + 2cx3). IB: Grouping: 62' r y - y2 - 10x2 - 5yz = 62' 3xy - 2xy - y2 - 10x2 - 5yz = 3x(2x y) y ( k y) - 5 Z ( 2 2 y) = ( 2 2 y)(32 - y - 5 2 ) . IC: Using of properties of equations (see also 1.6.3.1, p. 43): P ( x ) = x6 - 2x5 4x4 2x3 - 52'. a ) Factoring out 2'. b) Realizing that crl = 1 and a2 = -1 are the roots of the equation P ( x ) = 0

+

+

+

+

+

+

+

+

I

12

1. Arithmetic

and dividing P ( x ) by x Z ( x- 1)(z + 1) = z4- x2 we get the quotient x2 - 22 + 5. We can no longer decompose this expression into real factors because p = -2, q = 5 , p2/4 - q < 0, so finally we have the decomposition: x6 - 2x5 + 4x4 + 2x3 - 52;' = z'(2 - 1)(2 + 1)(z2- 22 + 5 ) .

1.1.6.3 Special Formulas

+

(1.28)

(xf y)2 = 22 f 2xy Y2, (x+ y + z)2 = 2 2 + y2 + z2 + 22y + 2xz + 2y2,

(x + y + 2 +. . . + t + u)2 = 2 2 + y2 + 2 + . ' . + t 2 + f f 1 2 (2

*

$ 2 2 ~f 2x2 t . .

y)3

= 23

(1.29)

+

+ XU + 2y2 + . + 25/21 + . . + 2 t ~ ,

* 3x2y + 32y2f y3.

'.

(1.30) (1.31)

We calculate the expression (x f y)" by the binomial formula (see (1.36a)-(1.37a)).

+

(1.32)

(x y)(x - y) = x2 - y2, 2"

- y" - xn-l

+ xn-'y + . . . + xyn-' + yn-',

(for integer n,and n > l ) ,

(1.33)

2 - Y

xn + Y" - xn-l - m-zy + . . . - X Y " - ~ --

+ yn-'

(for odd n,and n

- y"-'

(for even n,and n > 1).

> l),

(1.34)

X+Y

xn - y" xn-l - x"-2y X+Y

+ , . . + xy"-2

(1.35)

1.1.6.4 Binomial Theorem 1. Power of an Algebraic Sum of Two Summands (First Binomial Formula) The formula (a + b)" = an

n(n - 1lan-2b2 + nan-'b + 2!

+

n(n -

- 2Ian-3b3

3! (1.36a)

is called the binomial theorem, where a and b are real or complex values and n = 1,2, . . . . Using the binomial coeficients delivers a shorter and more convenient notation: (a+b)" = (:)an+

(;)a"-'b+

(:)o"-2b2+

(

n - 1 )abn-'

+ (:)bn(1.36b)

or (a

+ b)" = f:

(1.36~)

k=O

2. Power of an Algebraic Difference (Second Binomial Formula)

. (n- m + 1)$-"bm +. ' . + (-1)m n(n - 1 ) . .m! or (a - b)" =

f:(3(-l)ka"-kbk

k 0

+ . . . + (-1)"b"

(1.37a)

(1.37b)

1.1 Elernentam Rules for Calculations 13

3. Binomial Coefficients The definition is for non-negative and integer n and k:

n!

I 5 n), (1) where is the product of the positive integers from

(1.38a)

k

(0

=

n! 1 to n, and it is called n factorzal: n! = 1 . 2 . 3 . . . n, and by definition O! = 1. (1.38b) We can easily see the binomial coefficientsfrom the Pascal triangle in Table 1.2.The first and the last number is equal to one in every row; every other coefficient is the sum of the numbers standing on left and on right in the row above it. Simple calculations verify the following formulas:

(3

n!

= (n

k) =

m'

(3

(1.39a)

(1')

= 1,

(3

= 1.

= n,

(1.39b) (1.39~)

="(")

k

n-k+l

(1.39d)

'

n

n-k

9

(k

1) =

(1.39e)

k+l(k)'

(1.39f)

Coefficients

n

1

0 1

1 2

3 4 5

5

. .

.

.

.

.

.

1

6

15

20

.

5

10

15

.

1

4

10

6

1

1

6

4

1 1

1

3

3

1

6

cient

1 2

1

.

.

.

.

.

1

.

. . .

(3 :

(;)

- O(Q -

- l ) ( a - 2 ) . . , ( a- k k!

3!

+ 1) 16

for integer k and k 2 1,

(1.40)

I

14

1. Arithmetic

4.

Properties of the Binomial Coefficients

The binomial coefficients increase until the middle of the binomial formula (1.36b), then decrease. 0 The binomial coefficients are equal for the terms standing in symmetric positions with respect to the start and the end of the expression. 0

The sum of the coefficients at the odd positions is equal to the sum of the coefficients at the even positions. 5 . Binomial Series The formula (1.36a) of the binomial theorem can also be extended for negative and fraction exponents. If jbl < a, then ( a + b)" has a convergent infinite serzes (see also 21.3, p. 1009):

I

(1.41)

1.1.6.5 Determination of the Greatest Common Divisor of Two Polynomials

.

It is possible that two polynomials P ( x )of degree n and Q ( x )of degree m with n 2 m have a common polynomial factor, which contains z.The least common multiple of these factors is the greatest common divisor of the polynomials. IP(a:) = (x - 1)2(x- 2)(a: - 4 ) , &(a:) = (x - 1)(x - 2 ) ( 2 - 3); the greates common devisor is (x - 1)(x - 2). If P ( x )and Q ( x )do not have any common polynomial factor, we call them relativelyprime or coprime. In this case, their greatest common divisor is a constant. The greatest common divisor of two polynomials P ( x ) and Q ( x ) can be determined by the Euclidean algorithm without decomposing them into factors: 1. Division of P ( x ) by Q ( x ) = &(x) results in the quotient Tl(x) and the remainder Rl(a:): (1.42a) P ( 2 )= Q(x)Ti(r)+ Ri(a:). 2. Division of Q ( x ) by Rl(x) results in the quotient T2(x) and the remainder Rz(x): Q(2) = Ri(z)Tz(x) +Rz(x). (1.42b) 3. Division of R1(x) by &(x) results in T3(x) and &(a:), etc. The greatest common divisor of the two polynomials is the last non-zero remainder Rk(x).This method is known from the arithmetic of natural numbers (see 1.1.1.4. p. 3). We determine the greatest common divisor, for instance, when we solve equations, and we want to separate the roots with higher multiplicity, and when we apply the Sturm method (see 1.6.3.2. 2.. p. 44).

1.1.7 Rational Expressions 1.1.7.1 Reducing t o the Simplest Form Every rational expression can be written in the form of a quotient of two coprime polynomials. To do this. we need only elementary transformations such as addition, subtraction, multiplication and division of polynomials and fractions and simplification of fractions. 22 y 3a: + - * x + z IFind the most simple form of -y+-: a: (a:2

+

(3x2 + 22 + y)z2 (x39+ l ) z +

-Y*Z

+z +z z

+ $) 3 x 2 + 2 x 2 + yzZ + (x3z2+ a:)(-y2z + x + Z) 2323

+ xz

1.1 Elementary Rules for Calculations 15

+

3xz3 2 x 2

+ yz2 - x3y2z3- X Y ~ Z+ x4z2+ x2 t x3z3t xz x323 + X Z

1.1.7.2 Determination of the Integral Rational Part A quotient of two polynomials with the same variable x is a properfraction if the degree of the numerator is less than the degree of the denominator. In the opposite case, we call it an improperfraction. Every improper fraction can be decomposed into a sum of a proper fraction and a polynomial by dividing the numerator by the denominator, Le.. separating the integral rational part. 3x4 - 10ax3 + 22a2x2- 24a3x + loa4 , IDetermine the integral rational part of R ( z ) = x2 - 2ax + 3a2 -2a3x - 5a4 (3x4-10as3+22n'x'-24a3z t10u4) : (x' - 2ax + 3 2 ) = 32' - 4ax + 5a2 + x 2 - 2ax - 3a2 3z4- 6ax3+ 9a2x2 - ~ax3t13a'x2-24a3x - .lax3+ 8a2x2-12a3x 5a'z'-12a3x +loa4 5a2z2- 10a3x t 15a4 -2a3x - sa4 - 2a3x- sa4. R ( z ) = 32' - 4ax t 5a2 t x2 - 2ax + 3a2 ' The integral rational part of a rational function R(x) is considered to be as an asymptotzc approxzmatzon for R ( z )because for large values of 1x1, the value of the proper fraction part tends to zero. and R(x) behaves as its polynomial part.

1.1.7.3 Decomposition into Partial Fractions Every proper fraction (1.43) can be decomposed uniquely into a sum of partial fractions. In (1.43) the coefficients ao, a i , . . . .an, bo, b l , . , . b, are arbitrary real or complex numbers; we can suppose that bm = 1 , otherwise we can divide the numerator and the denominator by it. The partial fractions have the form

A

(x - N ) k

(1.444 ,

Dx+E

(x'

+ px + q)l

with

(i)'

-q < 0.

(1.44b)

If we restrict our investigation to real numbers, the following four cases, 1. 2, 3 and 4 can occur. If we consider complex numbers we have only two cases: cases 1 and 2. In the complex case, every fraction R(x) can be decomposed into a sum of fractions having the form (1.44a), where A and a are complex numbers. \Ye will use it when we solve linear differential equations.

1. Decomposition into Partial Fractions, Case 1 Suppose the equation Q(x) = 0 for the polynomial Q(x) in the denominator has m different simple roots a i , . .CY,. Then the decomposition has the form (1.45a) with coefficients (1.45b) dQ for x = c q . z = 0 2 , where in the denominator we have the substitution values of the derivative dx

I

16

1. Arithmetic

A:

6x2-z+1 23-z

P ( z ) = 62' - x

A

=z

B C +t -, 2-1 x+l

+ 1 , Q ( x ) = 32'

a1 = 0 ,a2 = $1 a n d a 3 = -1;

P(0) = -1, B = P(1) = 3 and C = P(-1) - 1, A = Q(0) Q'(1) Qq-1) - 43

P (-z ) ---+-+-. 1 3 4 Q(x) z 2-1 ~ + 1 Another possibility to determine the coefficients AI, Az, . . . , A , is called the method of comparzng coeficients or method of undetermined coeficzents. In the following cases we have to use this method. B A C A(z2- 1) B X ( Xt 1) CZ(Z - 1) B: 6 ~ -' z 1 --+-+-= 13-z x x-1 x + l x(x2 - 1) The coefficients of the corresponding powers of x must be equal on the two sides of the equality, so we obtain the equations 6 = A B t C , -1 = B - C , 1 = -A, which have the same solution for A, B and C as in example A .

+

+

+

+

2. Decomposition into Partial Fractions, Case 2 Suppose the polynomial Q ( x )in the denominator can be decomposed into a product of powers of linear factors; in the real case this means that the equation Q ( x ) = 0 has m real roots counted by multiplicity. Then the decomposition has the form

+ +

+

P(z) a,xn an-1xn-l . . . a0 -Q ( x ) (X - C Y ~ ) ~ ~ C Y( ZZ) ~ .~. .(X -

B1 x -CY2

t-

-

Ai

x - 01

+-+...+-AAz A (X - ai)' (x - a1)kl

+ , + . .BZ .+-+...+-

Bkz Lk, (1.46) (x - a z ) k z (x - a,)k, + 3. B3 The coefficients A i , B1, Bz, B3 can be determined

(z- 0 2 )

'

Ai + Bi + Bz I--" =z(x-1)3 x - i (x-1) (x-1) by the method of comparing coefficients. ~

3. Decomposition into Partial Fractions, Case 3 If the equation Q ( x ) = 0 for a polynomial with real coefficients in the denominator also has complex roots but only with multiplicity one, then the decomposition has the form

+

Q(x)

+ +

a,xn an-1xn-l . . . a0 - w ) ~ ~-(CxY Z ) ~ .~..(xZ+ p i ~ qi)(x2+ P Z X 9 2 ) . . . Dx+ E Fx+G Az - - Ai

P(x) - -

+

(Z

z- a1

+-----+...+ (z

22

-Cy1)2

+pix

+ 41

+

ZZt p z x

+ 92 +... .

(1.47)

The quadratic denominator z2t p x + q arises from the fact that if a polynomial with real coefficients has a complex root, the complex conjugate of this root is also a root, and they have the same multiplicity. 3x2 - 2 - A Dx+E -The coefficients A, D, E can be determined by the (22+z+l)(x+l) x+l zZ+x+l' method of comparing coefficients.

+-

4. Decomposition into Partial Fractions, Case 4 Analogously, if the real equation & ( E ) = 0 for the polynomial of the denominator has complex roots with a higher multiplicity, we decompose the fraction into the form P(x) - Q(x)

a,xn + an-lxn-l+ . . . + ao

z ~ + ~ ~ x + ~ ~ ). . . ~ ~ ( ~ ~ + ~ ~ ~ +

( x - a i ) k l ( ~ - a z ) .k. 2 .(

1.1 Elementary Rules for Calculations 17

(1.48) 52’ - 4x + 16 Dix+ El D2x =-+-.4 The coefficients A , D1, El, D2, E2 will I (x-3)(x2-x+1)2 s - 3 22-x+1 (22-2+1)2’ be determined by the method of comparing coefficients. +

+

1.1.7.4 Transformations of Proportions The equality ad = bc,

and furthermore c*d a*c c!cd afb--a*b =c 1 c b d ’ a From the equalities of the proportions an =-

(1.50a)

-

a

=d’

b5d a+b - c+d e-d‘ d ‘ a-b

it follows that

bn

jj =

a’

ba - dc

(1.49b)

(1.49~)

a1 .(1.50b) =bi

1.1.8 Irrational Expressions Every irrational expression can be written in a simpler form by 1. simplifying the exponent, 2. taking out terms from the radical sign and 3. moving the irrationality into the numerator. 1. Simplify the Exponent We can simplify the exponent if the radicand can be factorized and the index of the radical and the exponents in the radicand have a common factor. We divide the index of the radical and the exponents by their greatest common divisor.

If16(xl2

-

2xl1 + xlO) = $-/

=

(/-.

2. Moving the Irrationality There are different ways to move the irratzonalityinto the numerator.

1 x+fi-

I D : --

x2-xfi+%7 (s+fi)(s2-xfi+w)

= xz-xfi+w

z3+y

3. Simplest Forms of Powers and Radicals Also powers and radicals can be transformed into the simplest form.

18

1. Arithmetic

1.2 Finite Series 1.2.1 Definition of a Finite Series The sum s, = a0

+ + +".+a, a1

a2

n

=Ea,,

(1.51)

Z=O

is called a . W e series. The summands a, ( i = 0,1,2,. . . , n) are given by certain formulas, they are numbers, and they are the terms ojthe series.

1.2.2 Arithmetic Series 1. Arithmetic Series of First Order is a finite series where the terms form an arithmetic sequence: Le., the difference of two terms standing after each other is a constant: Aa, = a,+l - a, = d = const holds, so ai = a0 + id. (1.52a) With these we have: s, = a0 t (a0 + d ) t (a0 + 2d) + . ' ' t (a0 nd) (1.52b)

+

2. Arithmetic Series of k-th Order is a finite series, where the k-th differences Aka, of the sequence a,,, a l , a2, . . . ,a, are constants. The differences of higher order are calculated by the formula A"a, = Ay-lal+l - A"-laZ (V = 2,3,. . . , k). (1.53a) It is convenient to calculate them from the diference schema (also difference table or triangle schema):

(1.53b)

(1.53~) (1.53d)

1.2 FiniteSeries 19

1.2.3 Geometric Series The sum (1.51)is called a geometrzc sertes, if the terms form a geometrzc sequence, Le., the ratio of two successive terms is a constant: a,tl - = q = const holds, so a, = aoq'. (1.54a) aa

With these we have

s, = ( n t l)ao for q = 1.

(1.54~)

If n --t i y j (see 7.2.1.1, 2., p. 404). then we get an znfinite geometric series, which has a limit if 141 < 1, and it is s = - , ao

(1.54d)

1-q

1.2.4 Special Finite Series

+

n(n 1) 1 + 2 + 3 + , .. t ( n - 1) + TI = 2 ?

(1.55) (1.56)

1 + 3 + 5 + " ' + ( 2 n - 3) t (2n - 1) = n', 2 t 4 6 t . . . + (2n - 2) 2n = n (n + l ) , n (n l)(2n 1' 2' 3' t . ' . ( n - I)'+ n2 = 6

+ + +

+

+

+

13+23+33+...+(n-1)3+n3 =

+ + 52 + + + 33 + 53 + . . . +

l2 32 13

' '.

(1.57) (1.58)

+ 1)

(1.59)

>

n'(n +

(1.60)

~

4 ' n(4n2- 1) (2n - 1 ) 2 = 3 ' (2n - 113 = n 2 ( 2 n 2 - 11,

(1.61)

~

14+24t34t...+n4

=

(1.62)

n(n + 1)(2n t 1)(3nZ+ 3n - 1)

30 1 - ( n+ 1)Xn t n x n t l 1 t 2 2 + 3 2 + ' . ' + nxn-1 = (1 - X)Z

(2

f

1)

,

(1.63) (1.64)

1.2.5 Mean Values (See also 16.3.4.1, 1..p. 777 and 16.4, p. 785)

1.2.5.1 Arithmetic Mean or Arithmetic Average The arithmetic mean of the n quantities al, a2, alta2+...+an 1 XA =

n

=-Cak.

For two values a and b we have: a t b x* = 2 ' ~

I

.

.

,an is the expression

(1.65a)

k=l

(1.65b)

The values a , X A and b form an arithmetic sequence.

1.2.5.2 Geometric Mean or Geometric Average The geometric mean of n positive quantities al, a2, . . . , a n is the expression (1.66a) For two values a and b we have XG

=

6.

(1.66b)

The values a , ZG and b form a geometric sequence. If a and b are given line segments, then we can get /-------a segment with length XG = 6with the help of [F----b.\\\\\$J J L&& one of the constructions shown in Fig. 1.3a or in ' * * I Fig. 1.3b. a a A special case of the geometric mean is when we want to divide a line segment according to the a) b) golden section (see 3.5.2.3, 3., p. 193). Figure 1.3

1.2.5.3 Harmonic Mean The harmonic mean of n quantities a,, a2,. . . ,a, is the expression (1.67a)

For two values a and b we have (1.67b)

1.2.5.4 Quadratic Mean The quadratic mean of n quantities a l , a2,. . . , an is the expression (1.68a)

For two values a and b, we have " Q

=

JF,

(1.68b)

The quadratic mean is important in the theory of observational error (see 16.4, p. 785).

1.2.5.5 Relations Between the Means of Two Positive Values F o r z A = -a, x+Gb = & , 2 1. if a < b, then a < XH < XG < XA 2. if a = b, then a = X A = X G = XH

X H = ~ 2ab + , X Q =

/?we

have

< XQ < b ,

(1.69a)

= XQ = b.

(1.69b)

1.3 Business Mathematics 21

1.3 Business Mathematics Business calculations are based on the use of arithmetic and geometric series, on formulas (1.52a)(1.52~)and (1,54a)-(1,54d). However these applications in banking are so varied and special that a special discipline has developed using specific terminology. So business arithmetic is not confined only to the calculation of the principal by compound interest or the calculation of annuities. It also includes the calculation of interest, repayments, amortization, calculation of instalment payments, annuities, depreciation, effectiveinterest yield and the yield on investment. Basic concepts and formulas for calculations are discussed below. For studying business mathematics in detail, you will have to consult the specialist literature on the subject. Actuarial mathematics and risk theory use the methods of probability theory and mathematical statistics, and they represent a separate discipline, so we do not discuss them here.

1.3.1 Calculation of Interest or Percentage 1. Percentage or Interest P The expression p percent of K means --K, where K denotes the principal in business mathematics. 100 The symbol for percent is %, Le., we have the equalities (1.70)

2. Increment If K is raised by p%, we get the increased value (1.71) Relating the increment K- P to the new value K , then the proportion K- P : K = i, : 100, K contains 100 100 -p = - p.100 (1.72) 100 p percent of increment. IIf an article has a value of E 200 and a 15% extra charge is added, the final value is E 230. This price 15 100 contains i, = -= 13.04 percent increment for the user. 115

+

3. Discount or Reduction If we reduce the value K by p% rebate, we get the reduced value K=K(l-&)

(1.73)

If we compare the reduction K- P to the new value K , then we realize 100 p.100 p=(1.74) 100 - p percent of rebate. IIf an article has a value E 300, and they give a 10% discount, it will be sold for f 270. This price contains? = E = 11.11 percent rebate for the buyer. 90

22

1. Arithmetic

1.3.2 Calculation of Compound Interest 1.3.2.1 Interest Interest is either payment for the use of a loan or it is a revenue realized from a receivable. For a principal K , placed for a whole period of interest (usually one year),

K- P

(1.75) 100 interest is paid at the end of the period of interest. Here p is the rate of interest for the period of interest, and we say that p% interest is paid for the principal K .

1.3.2.2 C o m p o u n d Interest Compound interest is computed on the principal and on any interest earned that has not been paid or withdrawn. It is the return on the principal for two or more time periods. The interest of the principal increased by interest is called compound interest. We discuss different cases depending on how the principal is changing.

1. Single Deposit Compounded annually the principal K increases after n years up to the final value K,. At the end of the n-th year this value is:

K , = K (1 +

&),

(1.76)

P For briefer notation we substitute 1 t = q and we call q the accurnulatzonfactoror growth factor. 100 Interest may be compounded for any period of time: annually, half-annually, monthly, daily, and so on. If we divide the year into m equal interest periods the interest will be added to the principal K at the end of every period. Then the interest is K - P for one interest period, and the principal increases lOOm after n years with m interest period up to the value

(1.77) as the efectzve rate. The quantity 1 + - is known as the nominal rate. and (1 t &)m 1:o) I.4 principal of € 5000, with a nominal interest 7.2% annually, increases within 6 years a) compounded annually to K6 = 5000(1 t 0.072)6= E 7588.20, b) compounded monthly to K72 = 5000(1+ 0, 072/12)72= € 7691.74.

(

2. Regular Deposits Suppose we deposit the same amount E in equal intervals. Such an interval must be equal to an interest period. \$e' can deposit at the beginning of the interval, or at the end of the interval. At the end of the n-th interest period n e have the balance K,:

a) Depositing at the Beginning: qn

K , = Eq-.

-1

q-1

(1.78a)

b) Depositing at the End: qn - 1 Kn=E-. q-1

(1.78b)

3. Depositing in the Course of the Year A year or an interest period is divided into m equal parts. At the beginning or at the end of each of these time periods the same amount E is deposited and bears interest until the end of the year. In this way. after one year n e have the balance K1:

1.3 Business Mathematics 23

b) Depositing at the End:

a) Depositing a t the Beginning:

[ (m2id)P].

KI=E mt-

(1.79a)

(1.79b)

In the second year the total K1 bears interest, and further deposits and interests are added like in the first pear, so after n years the balance K , for midterm deposits and yearly interest payment is:

a) Depositing at the Beginning:

K,=E

[

m+-

( m + l)p qn - 1 -. (1.80a) 200 ] q - 1

b) Depositing at the End: ( m - 1)p 4, - 1 K, = E m+ -. 200 ] q - 1

[

(1.80b)

~

IAt a yearly rate of interest p = 5.2% a depositor deposits € 1000 at the end of every month. After how many years will it reach the balance E 500 OOO?

, follows the answer, n =

From (1 80b), for instance. from 500000 = 1000

22 42 years.

1.3.3 Amortization Calculus 1.3.3.1 Amortization Amortization is the repayment of credits. Our assumptions: 1. For a debt S the debtor is charged at p% interest at the end of an interest period. 2. After S interest period the debt is completely repaid. The charge of the debtor consists of interest and principal repayment for every interest period. If the interest period is one year. the amount to be paid during the whole year is called an annuity. There are different possibilities for a debtor. For instance, the repayments can be made at the interest date. or meanwhile: the amount of repayment can be different time by time. or it can be constant during the whole term.

1.3.3.2 Equal Principal Repayments The amortization instalments are paid during the year, but no midterm compound interest is calculated. \Ye use the following notation:

S m.'i

.V number of interest periods until the debt is fully repaid. Besides the principal repayments the debtor also has to pay the interest charges: a) Interest Z, for the n-thInterest Period:

;

zn -- _ [I - - ,;:

(n - - m2m+ l )]

'

(1.W

b) Total Interest Z to be Paid for a Debt S, m N Times, During N Interest Periods with an Interest Rate p% :

2=

5"

n=l

[-

p S "V-1 Z -- 100 2

A debt of E 60 000 has a yearly interest rate of 8%. The principal repayment of E 1000 for 60 months should be paid a t the end of the months. How much is the actual interest at the end of each year? The interest for every year is calculated by (1.81a) with S = 60000. p = 8, S = 5 and m = 12. They are enumerated in the annexed table.

1. Year: 2. year: 3. year: 4. year: 5 . year:

+m+l zi,= zz = z3 =

. (1.81b)

4360 E 3400 2440 z 4 = f 1480 z5 = E 520 E 12200

z=

f

24

1.

Arithmetic

8.60000 5 - 1 13 The total interest can be calculated also by (1.81b) as Z = --+ - = € 12200. 100 2 241

I

1.3.3.3 Equal Annuities For equal principal repayments T =

S the interest payable decreases over the course of time (see

mN

the previous example). In contrast to this, in the case of equal annuities the same amount is repaid for every interest period. A constant annuity A containing the principal repayment and the interest is repaid, Le., the charge of the debtor is constant during the whole period of repayment. With the notation

P the accumulation factor, q = 1+ -

100 after n interest periods the remaining outstanding debt S, is:

( m - 1)p qn - 1 (1.82) 200 q-1' Here the term Sqn denotes the value of the debt S after n interest periods with compound interest (see (1.76)). The second term in (1.82) gives the value of the midterm repayments a with compound interest (see (1.80b) with E = a ) . For the annuity:

[

S n = S q n - a m+-

]

[

A = a m+-

(1.83)

Here paying A once means the same as paying am times. From (1.83) it follows that A 2 ma. Because after N interest periods the debt must be completely repaid, from (1.82) for SN = 0 considering (1.83) we get: (1.84)

To solve a problem of business mathematics we can express, from (1.84), any of the quantities A, S,q or N if the others are known. IA: A loan of € 60 000 bears 8% interest per year, and is to be repaid over 5 years in equal instalments. How much is the yearly annuity A and the monthly instalment a? From (1.84) and (1.83) we get: 0 08 = € 1207.99. A = 60 O O O 1 L = € 15 027.39, a = l-12+1.085 200 IB: A loan of S = € 100 000 is to be repaid during N = 8 years in equal annuities with an interest rate of 7.5%. At the end of every year € 5000 extra repayment must be made. How much will the monthly 0.075 instalment be? For the annuity A per year according to (1.84) we get A = 1 0 0 0 0 0 =~ ~

1502:i3.98

l--

1.0758

€ 17 072.70. Because A consists of 12 monthly instalments a , and because of the € 5000 extra payment

[

at the end of the year, from (1.83) A = a 12 charge is a = € 972.62.

+

F] +

5000 = 17072.70 follows, so the monthly

1.3 Business Mathematics 25

1.3.4 Annuity Calculations 1.3.4.1 Annuities If a series of payments is made regularly at the same time intervals, in equal or varying amounts, at the beginning or at the end of the interval, we call it annuity payments. We distinguish: a ) Payments on a n Account The periodic payments, called rents, are paid on an account and bear compound interest. We use the formulas of 1.3.2. b) Receipt of Payments The payments of rent are made from capital bearing compound interest. We use the formulas of the annuity calculations in 1.3.3,where the annuities are called rents. If no more than the actual interest is paid as a rent, we call it a perpetual annuity. Rent payments (deposits and payoffs) can be made at the interest terms, or at shorter intervals during the period of interest, i.e. in the course of the year.

1.3.4.2 Future Amount of an Ordinary Annuity The date of the interest calculations and the payments should coincide. The interest is calculated at p% compound interest, and the payments (rents) on the account are always the same, R. The future oalue ofthe ordinary annuity R,, Le., the amount to which the regular deposits increase after n periods amounts to: qn

-

1

(1.85) with q = 1 + -.P 100 The present value of an ordinary annuity I& is the amount which should be paid at the beginning of the first interest period (one time) to reach the final value R,,with compound interest during n periods:

R, = R-

4- 1

R o = Rn -

with q = l + - . P (1.86) 100 W A man claims E 5000 at the end of every year for 10 years from a firm. Before the first payment the firm declares bankruptcy. Only the present value of the ordinary annuity & can be asked from the administration of the bankrupt's estate. With an interest of 4% per year the man gets: 1 qn- 1 1- q - n 1 - 1.04-'' = f 40 554.48. R' = -R= R= 5000 qn q - 1 q-1 0.04 qn

1.3.4.3 Balance after n Annuity Payments For ordinary annuity payments capital K is at our disposal bearing p% interest. After every interest period an amount r is paid. The balance K,, after n interest periods, Le., after n rent payments, is: qn - 1 (1.8ia) K , = Kqn - R, = Kq" - Twith q = 1 P 100 q-1 Conclusions from (1.87a):

+--.

r=K- P

100

(1.87b) Consequently K, = K holds, so the capital does not change. This is the case of perpetual annuity.

r > K- P 100

(1.87~) The capital will be completely used up after N rent payments. From (1.8ia) it follows for K N = O: (1.87d)

Ifmidterm interest is calculated and midterm rents are paid, and the original interest period is divided into m equal intervals, then in the formulas (1.85)-(1.87a) n is replaced by mn and accordingly q = P P 1 + - by = 1 + 100 lOOm

26

1. Arithmetic

IWhat amount must be deposited monthly at the end of the month for 20 years, from which a rent of E 2000 should be paid monthly for 20 years, and the interest period is one month with an interest rate of 0.5%. From (1.87d) we get for n = 20. 12 = 240 the sum K which is necessary for the required payments: 2ooo 1'005240 = E 279161.54. The necessary monthly deposits R are given by (1.85): 1.005240 0.005 1.005240 - 1 , Le., R = E 604.19. ~ 2 ~ = ~ 1= R . 5 0.005 4

1.3.5 Depreciation 1. Methods of Depreciation Depreciation is the term most often used to indicate that assets have declined in service potential in a given year either due to obsolescence or physical factors. Depreciation is a method whereby the original (cost) value at the beginning of the reporting year is reduced to the residual value at year-end. We use the following concepts: e A depreciation base, e,V useful life (given in years),

eR, residual value after n years ( n 5 N ) , ea, ( n = 1 , 2 . , , . N) depreciation rate in the n-th year. The methods of depreciation differ from each other depending on the amortization rate: e straight-line method, Le.. equal yearly rates, e decreasing-charge method, Le., decreasing yearly rates. 2. Straight-Line Method The yearly depreciations are constant, Le.. for amortization rates an and the remaining value R,, after n years we have: A-RN &=A-n( n = 1 , 2 , . . . ,A').(1.89) (1.88) N If we substitute RN = 0, then the value of the given thing is reduced to zero after .V years, Le., it is totally depreciated. IThe purchase price of a machine is A = € 50 000. In 5 years it should be depreciated to a value Rj = € l0000. With linear depreciation according to (1.88) and (1.89) we have the annexed amortization schedule: It shows that the percent42 000 age of accumulated depreci34 000 23.5 ation with respect to the ac30.8 tual initial value is increas44.4 ing. 3. Arithmetically Declining Balance Depreciation In this case the depreciation is not constant. It is decreasing yearly by the same amount d. by the so-called multzple. For depreciation in the n-th year we have: (1.90) a, = al - ( n - 1)d ( n = 2 , 3 , , , , ,V t 1: al and d a r e given). ~

Considering the equality A

N

-

RN = C a, from the previous equation it follows that: n=l

(1.91)

1.3 Business Mathematics 27

For d = 0 we get the special case of straight-line depreciation. If d > 0. it follows from (1.91) that (1.92) where a is the depreciation rate for straight-line depreciation. The first depreciation rate ul of the arithmetically-declining balance depreciation must satisfy the following inequality:

A - R,\r

A - RN ,v < a1 < 2- ,li

(1.93) '

IA machine of € 50 000 purchase price is to be depreciated to the value € 10 000 within 5 years by arithmetically declining depreciation. In the first year € 15 000 should be depreciated. The annexed depreciation schedule is calculated by the given formulas, and it shows that with the exception of the last rate the percentage of depreciation is fairly equal.

23 500

= 29.0 9.1

4. Digital Declining Balance Depreciation Digital depreciation is a special case of arithmetically declining depreciation. Here it is required that the last depreciation rate U N should be equal to the multiple d. From U N = d it follows that d=-

2(A - Rn.) 'V('V 1) '

(1.94a)

+

a1 = N d , a2 =

(N - l)d, . . . uN = d.

(1.94b)

~

IThe purchase price of a machine is € A = 50000. This machine is to be depreciated in 5 years to the value R5 = € 10 000 by digital depreciation. Year Depreciation Depreciation

Residual Depreciation in c value of the depr. base

= 4d = 10 668 25 997 = 3d 2d = 8001 5334 17996 12662

25997 17996

~4 a3

12662

Q,=

d = 2667

9995

29.6 30.8

The annexed depreciation schedule, calculated by the given formulas, shows that preciation the percentage is fairly of equal. the de-

21.1

5 . Geometrically Declining Balance Depreciation Consider geometrically declining depreciation where p% of the actual value is depreciated every year. For the residual value R, after n years we have:

-)

R, = A (1 - P n ( n = 1.2, . . .) .

(1.95) 100 Usually A (the acquisition cost) is given. The useful life of the asset is ili years long. If from the quantities R~\r3pand 'li. two is given, the third one can be calculated by the formula (1.95). IA: A machine with a purchase value € 50 000 is to be geometrically depreciated yearly by 10%. .4fter how many years will its value drop below € 10 000 for the first time? Based on (1.95).we get that

IB: For a purchase price of A = € 1000the residual value R,should be represented for n = 1 2. . . . , 10 years by a) straight-line. b) arithmetically declining, c) geometrically declining depreciation. The results are shown in Fig. 1.4. ~

I

28

1. Arithmetic

6. Depreciation with Different Types of Deprecation

lining

Since in the case of geometrically declining depreciation the residual value cannot become equal to zero for afinite n,it is reasonable after acertain time, e.g., after m years, to switch over to straight-line depreciation. We determine m so that from this time on the geometrically declining depreciation rate is smaller than the straight-line depreciation rate. From this requirement it follows that: 100 m>N---. (1.96)

P

Figure 1.4

Here m is the last year of geometrically declining depreciation and N is the last year of linear depreciation when the residual value becomes zero.

IA machine with a purchase value of € 50000 is to be depreciated to zero within 15 years, for m years by geometrically declining depreciation with 14% of the residual value, then with the straight100 line method. From (1.96) we get m > 15 - - = 7.76, Le., after m = 8 years it is reasonable to switch 14 over to straight-line depreciation.

1.4 Inequalities 1.4.1 Pure Inequalities 1.4.1.1 Definitions 1. Inequalities Inequalities are comparisons of two real algebraic expressions represented by one of the following signs: < (“smaller”) Type I1 Type I > (“greater”) Type IIIa <> (“greater or smaller”) Type 111 # (“not equal”) (“not smaller”) Type IVa pl Type IV 2 (“greater or equal”) Type Va 3 (“not greater”) Type V 5 (“smaller or equal”) The notation I11 and IIIa, IV and IVa, and V and Va have the same meaning, so they can be replaced by each other. The notation I11 can also be used for those types of quantities for which the notions of “greater” or “smaller” cannot be defined. for instance for complex numbers or vectors. but in this case it cannot be replaced by IIIa. 2. Identical Inequalities, Inequalities of the Same and of the Opposite Sense,

Equivalent Inequalities 1. Identical Inequalities are valid for arbitrary values of the letters contained in them. 2. Inequalities of t h e S a m e Sense belong to the same type from the first two, Le., both belong to type I or both belong to type 11. 3. Inequalities of t h e Opposite Sense belong to different types of the first two, Le.. one to type I, the other to type 11. 4. Equivalent Inequalities are inequalities if they are valid exactly for the same values of the unknowns contained in them.

3. Solution of Inequalities Similarly to equalities, inequalities can contain unknown quantities which are usually denoted by the last letters of the alphabet. The solutzon of an znequalzty or a system of inequalities means the determination of the limits for the unknowns between which they can change, keeping the inequality or system

1.4 Inequalities 29

of inequalities true. We can look for the solutions of any kind of inequality; mostly we have to solve pure inequalities of type I and 11.

1.4.1.2 Properties of Inequalities of Type I and I1 1. Change t h e Sense of t h e Inequality If a > b holds, then b < a is valid,

(1.97a)

< b holds, then b > a

(1.97b)

if a

2. Transitivity If a > b and b

isvalid.

> c hold, then a > c isvalid; then a < c isvalid.

(1.98a)

if a < b and b < c hold,

(1.98b)

3. Addition and Subtraction of a Quantity If a > b holds, then a zk c > b f c is valid;

(1.99a)

if a < b holds, then a & c < b k c isvalid. (1.99b) By adding or subtracting the same amount to the both sides of inequality, the sense of the inequality does not change. 4. Addition of Inequalities

then a + c > b + d is valid; < b and c < d hold, then a + c < b+ d isvalid.

If a > b and c > d hold,

(1.100a)

if a

(1.100b)

Two inequalities of the same sense can be added. 5. Subtraction of Inequalities

If a > b and c < d hold,

then a - c > b - d

isvalid;

( 1.10la)

(1.101b) if a < b and c > d hold, then a - c < b - d isvalid. Inequalities of the opposite sense can be subtracted; the result keeps the sense of the first inequality. Subtracting inequalities of the same sense is not allowed. 6. Multiplication and Division of an Inequality by a Quantity a b If a > b and c > 0 hold, then ac > be and - > - arevalid, c

a if a < b and c > 0 hold, then ac < be and c

c

b

< - arevalid, c

a

b

c

c

(1.102a) (1.102b)

if a > b and c < 0 hold, then ac < be and - < - arevalid,

(1.102c)

a b if a < b and c < 0 hold, then ac > be and - > -are valid.

(1.102d)

c

c

Multiplication or division of both sides of an inequality by a positive value does not change the sense of the inequality. Multiplication or division by a negative value changes the sense of the inequality. 7. Inequalities and Reciprocal Values 1 1 . If 0 < a < b or a < b < 0 hold, then - > - isvalid. ( 1.103) a b

I

30

1. Arithmetic

1.4.2 Special Inequalities 1.4.2.1 Triangle Inequality for Real Numbers For arbitrary real numbers a. b. a i , a 2 , . . . , a n rthere are the inequalities

+

1a1 t a2 + . . . t ani 5 /all la21 t ... t la,l. (1.104) la + bl I la1 lbl The absolute value of the sum of two or more real numbers is less than or equal to the sum of their absolute values The equality holds only if the summands have the same sign.

+

1.4.2.2 Triangle Inequality for Complex Numbers For n complex numbers ~

. . . .z, E C

1 . ~ 2 ,

(1.105)

1.4.2.3 Inequalities for Absolute Values of Differences ofReal and Complex Numbers For arbitrary real numbers a , b E R, there are the inequalities (1.106) la1 - lbl I la - bl I la1 + Ibl. The absolute value of the difference of two real numbers is less than or equal to the sum of their absolute values, but greater than or equal to the difference of their absolute values. For two arbitrary complex numbers 21. z2 E C (1.107) lIz1l - l b 2 l 1 I 121 - z2I I Ibll + 1221'

1.4.2.4 Inequality for Arithmetic and Geometric Means ai + a2 + "'ta, 2 i /m for a, > 0 n

(1.108)

The arithmetic mean of n positive numbers is greater than or equal to their geometric mean. Equality holds only if all the n numbers are equal.

1.4.2.5 Inequality for Arithmetic and Quadratic Means ( 1.109) The absolute value of the arithmetic mean of numbers is less than or equal to their quadratic mean

1.4.2.6 Inequalities for Different Means of Real Numbers For the harmonic, geometric, arithmetic, and quadratic means of two positive real numbers a and b with a < b the following inequalities hold (see also 1 . 2 5 5 , p. 20): (1.1loa) a < X H < X G < X A < X Q < b. Here (1.1lob)

1.4.2.7 Bernoulli's Inequality For every real number a 2 -1 and integer n 2 1 holds (1 + a)" 2 1 + n a .

(1.111)

1.4 Inequalities 31 The equality holds only for n = 1. or a = 0.

1.4.2.8 Binomial Inequality For arbitrary real numbers a , b E R, we have

( a bl

I $(a' + bz) .

(1.112)

1.4.2.9 Cauchy-Schwarz Inequality The Cauchy-Schwarz inequality holds for arbitrary real numbers Q,, bl E R :

or

5 ~ ~ ~ ~ + a ~ ~ + ~ ~ ~ + a , ~ ~ b ~(1.113a) ~ + b ~

lalbl+azbz+...+a,b,l -

+ +

+

(albl + azb2 . ' . anbn)' I (a1' az2 +. . + anz)(bl2t b2' + . ' . t b,'). (1.113b) For two finite sequences of n real numbers, the sum of the pairwise products is less than or equal to the product of the square roots of the sums of the squares of these numbers. Equality holds only if al : bl = a2 : b:, = = a, : bn, . as} and {bl, bzI b3) are considered as vectors in a Cartesian coordinate system, If n = 3 and { a ~az. then the Cauchy-Schwarz inequality means that the absolute value of the scalar product of two vectors is less than or equal to the product of absolute values of these vectors. If n > 3. then this statement can be extended for vectors in n-dimensional Euclidean space. Considering that for complex numbers 121' = z'z (z' is the complex conjugate of z ) , the inequality (1.113b) is valid also for arbitrary complex numbers z,, wl E C: (21w1 ZZW2 ' '. z,wn)'(zlwl zzwz ' . . + z,w,) I ( Z l * Z l + ZZ*ZZ ' ' . Zn*Z,)(W1*wl + wz*wz+ . ' . + w,*wn). An analogous statement is the Cauchy-Schwarz inequality for convergent infinite series and for certain integrals: ' 8 .

+

+ + + +

+

+

(1.114)

(1,115)

1.4.2.10 Chebyshev Inequality

.

If a l , a'>.. . , a n r bl. b z s . .. b, are real positive numbers. then we have the following inequalities:

for or

a1 a1

5 a2 I . . . I Q, and bl I bZ 5 . , . 5 b,, 2 a2 2 . . . 2 a, and bl 2 b2 2 , , 2 b,, ,

and ai+az+...+a,)

(bi+bz',--+bn)

albl+azbz t . . . + a , b , n

(1.116b)

for a1 5 a2 5 . . . 5 a, and bl 2 b:, 2 . . . 2 b,. For two finite sequences with n positive numbers, the product of the arithmetic means of these sequences is less than or equal to the aritmetic mean of the pairwise products if both sequences are increasing or

both are decreasing; but the inequality is valid in the opposite sense if one of the sequences is increasing and the other one is decreasing.

1.4.2.11 Generalized Chebyshev Inequality If a ] , a Z r . , .anr b l , bz, . . . , b, are real positive numbers, then we have the inequalities

and

1.4.2.12 Holder Inequality 1 1. Holder Inequality for Series Ifp and p are two real numbers such that P X I . ~ 2 . . ,E,, ~ . and y1, yz, . . . , y, are arbitrary 2n complex numbers, then we have:

+ -1 = 1 holds, and P

(1.118a) This inequality is also valid for countable infinite pairs of numbers: (1.118b) where from the convergence of the series on the right-hand side the convergence of the left-hand side follows. 2. Holder Inequality for Integrals If f(z)and g ( x ) are two measurable functions on the measure space (X, A, p) (see 12.9.2, p. 634), then we have:

( 1.118c)

1.4.2.13 Minkowski Inequality 1. Minkowski Inequality for Series Ifp 2 1 holds, and two sequences of numbers, then we have:

{X~):=Y and {yk}p=l with X k , yk E C are (1.119a)

2. Minkowski Inequality for Integrals If f ( x ) and g(z) are two measurable functions on the measure space (X, A:p ) (see 12.9.2, p. 634), then we have:

(1.119b)

1.4 Inequalities

33

1.4.3 Solution of Linear and Quadratic Inequalities 1.4.3.1 General Remarks During the solution of an inequality we transform it into an equivalent inequality. Similarly to the solution of an equation we can add the same expression to both sides; formally, it may seem that we bring a summand from one side to the other, changing its sign. Furthermore one can multiply or divide both sides of an inequality by a non-zero expression, where the inequality keeps its sense if this expression has a positive value, and changes its sense if this expression has a negative value. An inequality of first degree can always be transformed into the form Q X > b. (1.120) The simplest form of an inequality of second degree is

x2 > m

(1.12la)

or x2 < m

(1.121b)

(1.122a)

or ax2 + bx + c < 0.

(1.122b)

b and z < - for a

(1.123b)

and in the general case it has the form ax2 + bx t c > 0

1.4.3.2 Linear Inequalities Inequalities of first degree have the solution

b

z > - for a > O

(1.123a)

a

I 5 ~ + 3 < 8 ~ + 15 , ~ - 8 ~ < 1 - 3- 3~2 < - 2 ,

a

< 0.

2 3

z>-.

1.4.3.3 Quadratic Inequalities Inequalities of second degree in the form x2 > m

and z2 < m

(1.124a)

(1.124b)

have solutions a)

x2 > m : Form 2 0 the solution is z > fi and z < -fi (1x1> fi), for m < 0 the inequality holds identically.

b) x 2 < m : For m > 0 the solution is - fi< x

(1.125a) (1.125b)

< +fi (1x1 < fi),

for m 5 0 there is no solution.

(1.126a) (1.126b)

1.4.3.4 General Case for Inequalities of Second Degree ax2 + bx t c > 0

( 1.127a)

or ax2 + bx + c < 0.

(1.127b)

We divide the inequality by a. If Q < 0 then the sense of the inequality changes, but in any case it will have the form x2 t p z t q < 0

( 1.127~)

or z 2 + p z + q > 0 .

By completing the square it follows that

(x +

q

< (;)2

(1.127e)

-4

Denoting z + E by z and 2

(i)'

- q by m, we obtain the inequality

( 1.127d)

I

34

1. Arithmetic

z2 < m

or z’ > m.

(1.128a)

(1.128b)

Solving these inequalities. n e get the values for z.

IA: -21’

+ 142 - 20 > 0,x2 - 7 z + 1 0 < 0 ,

‘

9

3 2

7 2

3 2

(z-i) < z , -- - < z - - < - ,

3 7 3 7 -- t - < z < - t - . 2 2 2 2 The solution is 2 < x < 5. IB: x2 62 15 > 0, (x 3)’ > -6.

+ +

+

The inequality holds identically. 7 3 7 3 IC: -22’ t 142 - 20 < 0. (x-I)’>:. a:-->and x - - < - - . 2 2 2 2 The solution intervals are P > 5 and x < 2.

1.5 Complex Numbers 1.5.1 Imaginary and Complex Numbers 1.5.1.1 Imaginary Unit The imaginary unit is denoted by i, which represents a number different from any real number, and whose square is equal to -1. In electronics, instead of i the letter j is usually used to avoid accidently confusing it with the intensity ofcurrent. alsodenoted by i. The introduction of the imaginary unit leads to the generalization of the notion of numbers to the complex numbers, which play a very important role in algebra and analysis. The complex numbers have several interpretations in geometry and physics.

1.5.1.2 Complex Numbers The algebraic form of a complex number is z = a +ib. (1.129a) K’hen a and b take all possible real values, we get all possible complex numbers z. The number a is the real part. the number b is the imaginary part of the number z : a = Re(z): b = Im(z). (1.129b) For b = 0 we have z = a. so the real numbers form a subset of the complex numbers. For a = 0 we have z = i b. which is a “pure imaginary number”. The set of complex numbers is denoted by C . Remark: Functions u! = f ( z ) with complex variable z = z + i y will be discussed in function theory (see 14.1, p. 669 ff).

1.5.2 Geometric Representation 1.5.2.1 Vector Representation Similarly to the representation of the real numbers on the numerical axis. the complex numbers can be represented as points in the plane, the so-called Gaussian number plane: A number z = a + i b is represented by the point whose abscissa is a and ordinate is b (Fig. 1.5). The real numbers are on the axis of abscissae which is also called the real axis, the pure imaginary numbers are on the axis of ordinates which is also called the imaginary axis. On this plane every point is given uniquely by its position vector or radius vector (see 3.5.1.136., p. 180), so every complex number corresponds to a vector which starts at the origin and is directed to the point defined by the complex number. So, complex numbers can be represented as points or as vectors (Fig. 1.6).

1.5.2.2 Equality of Complex Numbers Two complex numbers are equal by definition if their real parts and imaginary parts are equal to each other. From a geometric viewpoint, two complex numbers are equal if the position vectors correspond-

1.5 Complex Numbers 35

ing to them are equal. In the opposite case the complex numbers are not equal. The notions "greater" and ..smaller" are meaningless for complex numbers.

Y

t

Figure 1.5

imag.axis

t

Y h a g . axis

Figure 1.6

Figure 1.7

1.5.2.3 Trigonometric Form of Complex Numbers The form z=a+ib (1.130a) is called the algebraic form of the complex number. When polar coordinates are used, we get the trigonometric form of the complex numbers (Fig. 1.7): z = COS p + i sin 9). (1.130b) The length of the position vector of a point p = /zI is called the absolute value or the magnitude of the complex number, the angle p: given in radian measure, is called the argument of the complex number and is denoted by arg 2: p = 121, p = argz = w + 2k7i with 0 5 p < m, -T < w 5 + T , k = 0 , f l . f 2 , . . . . (1.130~) Lye call p the principal value ofthe complex number. The relations between p . p and a , b for a point are the same as between the Cartesian and polar coordinates of a point (see 3.5.2.2, p. 191):

a = pcosp,

arccos

(1.131a)

aP

b = psinq.

for b 2 0. p

(1.131b)

- arccos - for b < 0, p > 0. P undefined for p = 0

(1.13Id)

for a > 0,

+-2

for a = 0, b > 0,

a

T

lp=

m,

b arctan T

> 0.

p=

__

for a = 0, b

2

(1.131~)

< 0,

b arctan - + 7r for a < 0. b 2 0 , a b arctan - - T for a < 0. b < 0.

a

(1.13le) The complex number

2

= 0 has absolute value equal to zero; its argument arg 0 is undefined.

1.5.2.4 Exponential Form of a Complex Number We call the representation

z = pe"+ (1.132a) the exponentzal form of the complex number, where p is the magnitude and 9 is the argument. The Euler relatzon is the formula el' = cos 4 + 1 sin 9 . ( 1.132b)

36

1. Arithmetic

IK e represent a complex number in three forms: a) t = 1

+ i & (algebraic form), b) z = 2

(trigonometric form),

c) z = 2 elf (exponential form), considering the principal value of it

If we do not restrict ourselves onlv to the principal value, we have

1.5.2.5 Conjugate Complex Numbers Two complex numbers z and z' are called conjugate complex numbers if their real parts are equal and their imaginary parts differ only in sign: (1.133a) Re@*)= Re(z), Im(z*) = -Im(z). The geometric interpretation of points corresponding to the conjugate complex numbers are points symmetric with respect to the real axis. Conjugate complex numbers have the same absolute value, their arguments differ only in sign: t = a f i b = p ( c o s q + i sinp) = pel+', (1.133b) z* = a - i b =

COS p - i sin p) = p e d v .

(1.1334

Instead of z* one often uses the notation Z for the conjugate of z.

1.5.3 Calculation with Complex Numbers 1.5.3.1 Addition and Subtraction Addition and subtraction of two or more complex numbers given in algebraic form is defined by the formula z1 t 22 - 2 3 ... = (a1 + i b l ) (a2 i b 2 ) - (a3 +ib3) *.. - (a1 +a:, - a3 ...) + i ( b l b2 - b3 ...). (1.134) We make the calculations in the same way as we do by the usual binomials. As a geometric interpretation of addition and subtraction we consider the addition and subtraction of the corresponding vectors (Fig.1.8). For these we use the usual rules for vector calculations (see 3.5.1.1, p. 180). For z and z', t z' is always real, and z - z' is pure imaginary.

+

+ + +

+ +

+

+

Figure 1.8

Figure 1.9

Figure 1.10

1.5.3.2 Multiplication The multiplication of two complex numbers z1 and z2 given in algebraic form is defined by the following formula zlzz = (a1 t i bl)(az + i bz) = (ala2 - blbz) i (albz blaz) . (1.135a) For numbers given in trigonometric form we have zizz = [pi(cosp1+ i sinp1)][pz(cosp2+ i sinp2)] = P I P Z [ C O ~ ( V I+ (02) + i sin(pl+ 4 1, (1.135b)

+

+

1.5 Complex Numbers 37

i.e., the absolute value of the product is equal to the product of the absolute values of the factors, and the argument of the product is equal to the sum of the arguments of the factors. The exponential form of the product is

zlzz = p1p2e1(”+v2). (1.135~) The geometric interpretation of the product of two complex numbers 21 and z2 is a vector such that we rotate the vector corresponding to z1 by the argument of z2 clockwise or counterclockwise according to the sign of this argument, and the length of the vector will be stretched by IzzI. The product t1z2 can also be represented with similar triangles (Fig. 1.9). The multiplication of a complex number 2 by i means a rotation by a/2 and the absolute value does not change (Fig. 1.10). For z and 2 ’ : zz* = p2 = lz12 = a’

+ bZ.

(1.136)

1.5.3.3 Division Division is defined as the inverse operation of multiplication. For complex numbers given in algebraic form, we have

+ +

+

- a1 i bl --ala2 blb2 + i a2b1 - albz a2 i b 2 aZ2 bz2 az2 bz2 ’ For complex numbers given in trigonometric form we have 21

22

+

( 1.137a)

+

(1.137b) Le., the absolute value of the quotient is equal to the ratio of the absolute values of the dividend and the divisor; the argument of the quotient is equal to the difference of the arguments. For the exponential form we get (1.137~) In the geometric representation we get the vector corresponding to 21/22 if we rotate the vector representing z1 by - arg22, then we make a contraction by 1221. Remark: Division by zero is impossible.

1.5.3.4 General Rules for the Basic Operations

+

’4s we can observe we can make our calculations with complex numbers z = a i b in the same way as we do with binomials, only we have to consider that i 2 = -1. We know how to divide binomials by a real number. So. on division of a complex number by a complex number, first we clear the denominator from the imaginary part of the divisor, and we multiply the numerator and the denominator of the fraction by the complex conjugate of the divisor. This is possible because (a + i b)(a - i b) = a2 bZ (1.138) is a real number. (3-4i)(-1+5i)’ 10+7i ( 3 - 4 i ) ( l - 1Oi -25) (10+7i)i - -2(3-4i)(12+5i) 1+3i 5i 1 + 3i 5i i 1 3i 7 - 1Oi - -2(56 - 33i)(l - 3i) 7 - 1Oi -2(-43 - 201i) 7 - 1Oi 1 -= -(50+19li) = 10+38.2i 5 (1 3 i ) ( 1 - 3i) 5 10 5 5

+

+--

+

+--

+--

+

+

+-

1.5.3.5 Taking Powers of Complex Numbers The n-th power of a complex number could be calculated using the binomial formula, but it would be very inconvenient. For practical reasons we use the trigonometric form and the so-called de Moivre formula: COS p i sin p)ln = COS ncp i sin ncp) (1.139a)

+

+

~

38

1. Arithmetic

Le.. the absolute value is raised to the a-th power, and the argument is multiplied by n. In particular, we have: i 2 =

-1,

i3=-i

,

i4=+1

(1.139b)

in general

( 1.139~)

i4n+k = i"

1.5.3.6 Taking of the n-th Root of a Complex Number

I

Taking of the n-th root is the inverse operation of taking powers. For 2 = p(cos p + i sin p) the notation

zlln=

6

with n > 0. integer,

(1.140a) z 4 Ytimag. axis

is the shorthand notation for the n different values

(12 = 0 , 1 , 2, . . . ,n - 1). (1.140b) ll'hile addition. subtraction, multiplication. division, and taking a power with integer exponent have unique results, taking of the n-th root has n different solutions wk. The geometric interpretations of the points uk are the vertices of a regular n-gon whose center is at the origin. In Fig. 1.11 the six values of @ are represented.

Figure 1.11

1.6 Algebraic and Transcendental Equations 1.6.1 Transforming Algebraic Equations to Normal Form 1.6.1.1 Definition The variable z in the equality F ( x ) = f (2) (1.141) is called the unknown if the equality is valid only for certain values X I ,22,.. . z, of the variable. and these values are called the solutions or the roots of the equation. Two equations are considered equivalent if they have exactly the same roots. An equation is called an algebraic equation if the functions F ( z ) and f(z)are algebraic. Le., they are rational or irrational expressions; of course one of them can be constant. Every algebraic equation can be transformed into normalform ( 1.142) P ( 2 ) = a,xn + an-12n-1 + ' ' ' + a12 + a0 = 0 by algebraic transformations. The roots of the original equation occur among the roots of the normal form. but under certain circumstances some are superfluous. The leading coeficient a, is frequently transformed to the value 1. The exponent n is called the degree ofthe equation. 2-3 = 1 -. The transformations IDetermine the normal form of the equation 3(. - 2) step by step are: = 32' - 62 32' - 152 t z(z- 1 + = 32(2 - 2) t 3(2 - 2)(s - 3), a:' - z +- X = 52' - 20a: + 18. '2 (2' - 6)= 25s4 - 2 0 0 2 + 5802 - 7202 + 324, 24x4 - 200z3+ 18, -2 5862' - 7202 + 324 = 0. The result is an equation of fourth degree in normal form. ~

'

m)

+

+

+

1.6.1.2 System of n Algebraic Equations Ever1 system of algebrazc equatzons can be transformed to normal form, Le.. into a system of polynomial equations:

1.6 A1,qebraic and Transcendental Equations 39 (1.143) Pl(r . . . ) = 0 % Pz(x,y,z....) = O s . . . P,(x,y,z , . . .) = O The P, (z = 1 , 2 , ,, , n ) are polynomials in x,y, 2,. . . . x 1 x-1 IDetermine the normal form of the equation system: 1. - = - 2. -= f i , 3. ry = z ~

.

~

The normal form is: 1. x2z2 - y = 0

~

2.

2'

8 Y-1 - 22 + 1 - y2z + 2y2 - z = 0, 3. xy - z = 0.

1.6.1.3 Superfluous Roots It can happen that after transformation of an algebraic equation into normal form P ( x ) = 0, it has solutions which are not solutions of the original equation. This happens in two cases: 1. Disappearing Denominator If an equation has the form of a fraction

( 1.144a) with polynomials P ( s )and Q ( x ) ,then we get the normal form by multiplying by the denominator: P ( 2 ) = 0. (1.144b) The roots of (1.144b) are the same as those of the original equation (1.144a), except if a root r = Q of the equation P ( x ) = 0 is also a root of the equation Q ( x )= 0. In this case we should simplify first by the term z - a , actually by (x - Q ) ~ Anyway . if we perform a so-called non-identical transformation, we still have to substitute the roots we get into the original equation (see also 1.6.3.1, p. 43).

x3 1 -- - 0 (1). Ifwe do not simplify by x - 1: then the root x1 = 1 satisfies 2-1 z-1 2-1 the equation x3 - 1 = 0. but it does not satisfy (l),because it makes the denominator zero. 23 - 322 + 32 - 1 = 0 (2). If we do not simplify by the term (x - l)', the equation does not B: 22-222+1 have any root, because (x- 1)3= 0 has the root xl = 1, but the denominator is also zero here. After simplification, (2) has a simple root r = 1. 2. Irrational Equations If in the equation we also have an unknown in the radicand. then it is possible that the normal form of the equation has roots which do not satisfy the original equation. Consequently every solution we get from the normal form must be substituted into the original equation in order to check whether it satisfies it or not. = 2x - 1, (l), z t 7 = (22 - 1)' or 42'- 52 - 6 = 0 (2). The I-+ 1 = 2 2 or solutions of (2) are x1 = 2, x2 = -3/4. The solution 2 1 satisfies (1).but the solution x2 does not. IA:

-e-

'

1.6.2 Equations of Degree at Most Four 1.6.2.1 Equations of Degree One (Linear Equations) 1. Normal Form az + b = 0 . 2. N u m b e r of Solutions There is a unique solution b x1 = -a

(1.145) (1.146)

1.6.2.2 Equations of Degree Two (Quadratic Equations) 1. N o r m a l Form ax2 t bx c = 0 or divided by a: 2 +pa: q = 0 .

+

+

(1.147a) (1.147b)

I

2. Number of Real Solutions of a Real Equation Depending on the sign of the discriminant

D=4ac-b2

or D = q - - -PZ ,

(1.148)

4

we have: for D > 0, there is no real solution (two complex roots). 3. Properties of the Roots of a Quadratic Equation If x1 and 2% are the roots of the quadratic equation (1.147a) or (1.147b), then the following equalities hold: C b a a 4. Solution of Quadratic Equations Method 1: Factorization of ax2 + bz + c = a(. - D)(Z - p) (1.150a)

z1 + 2 2 = -- = -p, Zl'Q = - = q .

(1.149)

or x2 S p x

if it is successful, immediately gives the roots x1=a, x2=p. x 2 z - 6 = 0,X' + z - 6 = (T + 3)(x - 2 ) , ~1 = -3, z2 = 2. Method 2: Using the solution formula a) for (1.147a) the solutions are

+ q = (z - a)(x - p) , (1.150b) (1.151)

+

21.2

-b rt d = 2a

m

-b2 *

or x1,2 =

(1.152a)

JF a

,

(1.152b)

where we use the second formula if b is an even integer; b) for (1.147b) the solutions are (1.153)

1.6.2.3 Equations of Degree Three (Cubic Equations) 1. Normal Form

+

ax3 + bz2 cz + d = 0

or after dividing by a and substituting y = z

(1.154a)

+

b - we have 3a

+ +

q* = 0 , y3 + 3py + 2q = 0 or in reduced form y3 p'y where bc d 3ac - bZ 2b3 q* = 2q = -- -+ - and p ' = 3 p = sa2 ' 27a3 3a2 a 2. Number of Real Solutions Depending on the sign of the discriminant D=q2+p3 we have :

(1.154b) (1.154~) (1.155)

for D = 0, one real solution (one real root with multiplicity three) in the case p = q = 0; or two real solutions (a single and a double real root) in the case p3 = -q2 # 0.

1.6 Algebraic and Transcendental Equations 41

3. Properties of t h e Roots of a Cubic Equation If equation (1.154a), then the following equalities hold:

2 1 , 22,

and x3 are the roots of the cubic

l +l tl = --c d XI 22 23 d I 21x2x3=-~' 4. Solution of a Cubic Equation M e t h o d 1: If it is possible to decompose the left-hand side into a product of linear terms ax3 bx2 cz d = a(x - a)(. - P ) ( x - y) n e immediately get the roots z1 +r2 + x 3

= --2

b

a

+ + +

x3

x1=a, xz=8, - 6~ = 0. x3

+ X*

x3=y. - 62 = X ( X

+ 2'

+ 3)(2- 2);

(1.156)

(1.157a) (1.157b)

21

= 0,

22

= -3,

23

= 2.

Method 2: Using the Formula of Cardano. By substituting y = u + 1~ the equation (1.154b) has the form u3 v3 (u V ) ( 3 U U + 3p) t 2q = 0. (1.158a) This equation is obviously satisfied if u 3 + u 3 = -29 and u u = -p (1.158b) hold. Ifwe write (1.158b) in the form

+ + +

u 3 + u3 = - 2 q , U3V3 = -p3 , (1.158~) then we have two unknowns u3 and d,and we know their sum and product. Therefore using Vieta's root theorem (see 1.6.3.1,3., p. 43) the solutions of the quadratic equation

+

wz- (u3 v 3 ) w t U3U3 = w2 t 2qw - p 3 = 0 can be calculated. We get w1=

u3

= -q

+ Jq2+p3,

w2

= u3 = -q -

(1.158d)

J-? q2 + p 3

(1.158e)

so for the solution y of (1.154b) the Cardanoformula results in (1.158f) Because the third root of a complex number means three different numbers (see (1.140b) on p. 38) we could have nine different cases, but because of 2/21 = -p, the solutions are reduced to the following three: y 1 = u1 + v1 (if possible, consider the real third roots ut and zll such that ulul = - p ) , (1.158g) 1

i 1 i + 5)d3+ul(-- -)A, 2 2

Y2

= u1(-2

Y3

= u d - 2 - ?)A+ VI(--

1

i

W y3 + 6y + 2 = 0 with p u =

=

1 2

(1.158h)

i + 2-)A.

(1.158i)

= 2, q = 1 and q2 + p 3 = 9 and u =

= -1.5874. The real root is y1 = u

= fi = 1.2599,

+ u = -0.3275,

the complex roots are 1 J;i Y2,3 = --(u v ) zk i-(u - v) = 0.1638 & i . 2.4659. 2 2 Method 3: If we have a real equation, we can use the auxiliary values given in Table 1.3. With p from (1.154b) we substitute

+

r=&fi.

(1.159)

I

where the sign of r is the same as the sign of q. With this, using Table 1.3, we can determine the value of the auxiliary variable 9 and with it we can tell the roots yl, yz and y3 depending on the signs of p and D = q2 t p3. Table 1.3 Auxiliary values for the solution of equations of degree three

I

COST = f

coship = 4 r3

sinhrp =

r3

2 q = 2 , qztp3<0,r=&.cosp=-=Oo.3849, p=67"22'. 3& YI = -2&~0~22'27' = -3.201, yz = 2&cos(6Oo - 22"27') = 2.747, y3 = 2&cos(60" t 22'227') = 0.455. Checking: y1 y2 y3 = 0.001 which can be considered 0 for the accuracy of our calculations Method 4: Numeric approximate solution, see 19.1.2, p. 884; numeric approximate solution by the help of a nomogram. see 2.19, p. 127.

Iy 3 - 9 y t 4 = 0 .

p=-3.

+ +

1.6.2.4 Equations of Degree Four 1. Normal Form ax4 t bx3 ex2 dx t e = 0. If all the coefficients are real. this equation has 0 or 2 or 4 real solutions. 2. Special Forms If b = d = 0 hold. then we can calculate the roots of ax4 c.2 e = 0 by the formulas -c* x1,2,3,4 = *fi> Y= 2a For a = e and b = d . the roots of the equation ax4 + bx3 ex2 t bx t a = 0 can be calculated by the formulas

+ +

+

+

@=iz

+

-b k dbz - 4ac t 8a2 2a 3. Solution of a General Equation of Degree Four Method 1: If we can somehow factorize the left-hand side of the equation ax4 + bx3 + exz t dx + e = 0 = a(z - a)(. - 4)(x - y)(z - 6) then the roots can be immediately determined: x1=cy. x2=0, 2 3 = 7 , 54=6. Ix4 - 2 2 - 2 + 2x = 0. ~ (- 2 I ) (-~2) = x(x - i ) (+~i ) (-~2): 21 = 0. 2 2 = 1, x3 = -1, x4 = 2. - @ * Y =

x1,2 3,4

.

Y=

(1.160)

(1.161a) (1.161b) (1.161~)

( 1.161d)

(1.162a) (1.162b)

1.6 Aloebraic and Transcendental Eauations 43

Method 2: The roots of the equation (1.162a) for a = 1 coincide with the roots of the equation

x2 + @ + A ) ?2+ (y

+

+

y)

(1.163a)

= 0,

where A = kJ8y b2 - 4c and y is one of the real roots of the equation of third degree 8y3 - 4cy2 (2bd - 8e)y e(4c - b2) - d2 = 0. Method 3: Approximate solution, see 19.1.2, p. 884.

+

+

(1.163b)

1.6.2.5 Equations of Higher Degree It is impossible to give a formula or a finite sequence of formulas which produce the roots of an equation of degree five or higher.

1.6.3 Equations of Degree n 1.6.3.1 General Properties of Algebraic Equations 1. Roots The left-hand side of the equation (1.164a) xn an_]xn-' . . . a0 = o is a polynomial Pn(z) of degree n, and a solution of (1.164a) is a root of the polynomial P,(x). If a is a root of the polynomial: then Pn(x) is divisible by ( x - a ) . Generally (1.164b) P,(x)= (z- .)Pn-l(.) Pn(.). Here P,_l(x) is a polynomial of degree n - 1. If Pn(z)is divisible by (x but it is not divisible by (xthen a is called a root of order k of the equation P,(x)= 0. In this case cy is a common root of the polynomial P,(z)and its derivatives to order ( k - 1).

+

+ +

+

2. Fundamental Theorem of Algebra Every equation of degree n whose coefficients are real or complex numbers has n real or complex roots, where the roots of higher order are counted by their multiplicity. If we denote the roots of P ( z ) by cy. 3.7,. , , and they have multiplicity k , 1, m, ., then the product representation ofthe polynomial is (1.165a) P ( z ) = (x - N ) y z - /3)+ - Y ) m . . . . The solution of the equation P ( x ) = 0 can be simplified if we can reduce the equation to another one, which has the same roots. but only with multiplicity one. In order to get this, we decompose the t .

polynomial into a product of two factors P ( x )= Q ( x ) T ( z ) , ( 1.165b) such that (1.165~) T ( x )= ( x - ajk-'(z - /3)'-'. . . , Q ( x ) = ( x - a)(. - B ) . . . . Because the roots of the polynomial P ( x )with higher multiplicity are the roots of its derivative P'(z). too, T ( x ) is the greatest common denominator of the polynomial P ( z ) and its derivative P'(x) (see 1.1.6.5, p.14). If we divide P ( x ) by T ( z )we get the polynomial Q ( x ) which has all the roots of P ( x ) and each root occurs with multiplicity one.

.

3. Theorem of Vieta About Roots 1 ~ x. .2. ,, x, and the coefficients of the equation (1.164a) are:

The relations between the n roots ~ q+x2+

. . . +2 ,

n

=CZi=-an-], i=l

z1x2+ 11x3

+ . . . + xn-lx, =

n

x,x3 = an-2, ,,,=1

i<,

~ ~ ~ +2 ~ x1 x3 2 + ~ 4. . . + xn-2xn-1x,

n

=

(1.166)

x,xJxl;= -an-3, t 9

k=l

1.6.3.2 Equations with Real Coefficients 1. Complex Roots Polynomial equations with real coefficients can also have complex roots but only pairwise conjugate complex numbers, Le., if a = a + i b is a root, then p = a - i b is also a root, and it has the same multiplicity. The expressions p = -(a

+ a) = -2a

and q = aP = a’

+ bZ satisfy

we have

(N2

-q

< 0, and

a)

(1.167) (z- a)(. - = z2+px + q . If we substitute the product corresponding to (1.167) for every pair of factors in (1.165a), then we get a decomposition of the polynomial with real coefficients with real factors. P ( z ) = (x - a1)kl(x - a 2 ) k Z . . . (z- .p ( 2+p1x+ql)m1(x2 +p2x+q2)m~’..(xz +p,x+q,)m‘. (1.168) Here a l ,a z ,. . . , a1 are the 1 real roots of the polynomial P(x). It also has r pairs of conjugate complex roots, which are the roots of the quadratic factors x2 +p,z + qi ( i = 1 , 2 , . . . , r ) . The numbers aJ ( j = 1.2,.. .,l),

pi

z);(

and qi (i = 1 , 2 , . . . , r ) are real and the inequality

- q2 < 0 holds.

2. Number of Roots of an Equation with Real Coefficients It follows from (1.167) that every equation of odd degree has at least one real root. We can determine the number of further real roots of (1.164a) between two arbitrary real numbers a < b, in the following way: a ) Separate t h e Multiple Roots: First we separate the multiple roots of P(x) = 0. so we get an equation which has all the roots of the original equation, but only with multiplicity one. Then we can continue to produce the form mentioned in the fundamental theorem. For practical reasons it is a good idea to start with the determination of the Sturm chain (the S t u n functions) by the Sturm method. This is almost the same as the Euclidean algorithm for determining the greatest common denominator, but we obtain some further information from it. If P, is not a constant then P ( z ) has multiple roots, which must be separated. In the following we can suppose P ( z ) = 0 is an equation without multiple roots. b) Creating t h e Sequence of S t u r m Functions: ( 1.169) P(z),PI(.), PI($),Pz(x),. . . ,P, = const. Here P ( z ) is the left-hand side of the equation, P’(x) is the first derivative of P ( z ) , Pl(x)is the remainder on division of P ( z )by p’(~), but with the opposite sign, Pz(x) is the remainder on division of P’(z)by Pl(z) similarly with the opposite sign, etc.; P, = const is the last non-zero remainder. but it must be a constant, otherwise P(x) and P’(x)have common denominators, and P ( z )has multiple roots. In order to simplify the calculations we can multiply the remainders by positive numbers; it does not affect our conclusions. c) Theorem of Sturm: If A is the number of changes in sign, i.e. the number of changes from to ‘I-” and vice versa, in the sequence (1.169) for z = a: and B is the number of changes in sign in the sequence (1.169) for x = b, then the difference A - B is equal to the number of real roots of P(x) = 0 in the interval [a, b]. If in the sequence some numbers are equal to zero, at counting we leave them out.

“+”

1.6 Akebraic and Transcendental Eouations 45

IDeterminethenumberofrootsoftheequationz4-5z2+8z-8 = Ointheinterval [O. 21. Thecalculation by theskrrnf~nctionsare:P ( z ) = x4-5x2+8x-8; P'(z) = 4x3--10z+8; Pl(z)= 52'-122+16; Pz(z)= -32 + 284;P3 = -1. Substituting z = 0 results in the sequence -8, $8,+16, $284, -1 with two changes in sign. substituting z = 2 results in +4,+20, $12, +278, -1 with one change in sign. so A - B = 2 - 1 = 1,Le., between 0 and 2 there is one root. d) Descartes Rule: The number of positive roots of the equation P ( z ) = 0 is not greater than the number of changes of sign in the sequence of coefficients of the polynomial P ( z ) .and these two numbers can differ from each other only by an even number. ILi'hat can we tell about the roots of the equation x4 + 2x3 - x2 + 5z - 1 = 0 7 The coefficients in the equation have signs + , + , - ! + , - , Le., there are three changes of sign. By the rule of Descartes the equation has either three or one roots. Because on replacing z by -z the roots of the equation change their signs, and on replacing zby z + h the roots are shifted by h, we can estimate the number of negative roots, or the roots greater than h with the rule of Descartes. In our example replacing zby - 2 we get z4 - 2x3 - 2' - 52 - 1 = 0, Le., the equation has one negative root. Replacing z by z + 1 we have x4 + 6x3 + 11z2+ 132 + 6 = 0, Le., every positive root of the equation (one or three) is smaller than 1. 3. Solution of Equations of Degree n Usually we can get only approximate solutions for equations of degree higher than four. In practice, we look only for approximate solutions also in the case of equations of degree three or four. It is possible to give approximate solutions for all the roots: including the complex roots of an algebraic equation of degree n by the Brodetsky-Smeal method. For the calculations to determine certain real roots of an algebraic equation we can use the general numerical procedures for non-linear equations (see 19.1: p. 881). In order to determine complex roots we can use the Bairstow method.

1.6.4 Reducing TranscendentalEquations to Algebraic Equations 1.6.4.1 Definition .4n equation F ( z ) = f(z)is transcendental if at least one ofthe functions F ( z ) or f(z)is not algebraic. IA: 3" = 4"-' 2". IB: 2 log, (32 - 1) - log, (122 + 1) = 0,IC: 3 cosh z = sinh z + 9, 1 ID: 25-1 = 8"-' - 4"-', W E: sin z = cos' z - -, IF: 5 cos z = sinz. 4 In some cases it is possible to reduce the solution of a transcendental equation to the solution of an algebraic equation, for instance by appropriate substitutions. In general, transcendental equations can be solved only approximately. Next we discuss some special equations which can be reduced to algebraic equations.

1.6.4.2 Exponential Equations Exponential equations can be reduced to algebraic equations in the following two cases, if the unknown x or a polynomial P ( z ) is only in the exponent of some quantities a, b, e , . . . :

a) If the powers a p l ( z ) : b p 2 ( " ) , . . . are connected by multiplication or division, then we can take the logarithm on an arbitrary base. 2 log 4 I3' = 4'-' 2'; z log 3 = (Z - 2) log 4 t z log 2; 5 = log4 - log3 + log2' b) If a, b, e , . . . are integer (or rational) powers of the same number k , Le., a = k". b = k", c = k ' , . . . . hold, then by substituting y = k" we get an algebraic equation for y, and after solving it we have the 1% Y solution z = -. log k 2X

I25-1 =

8'-2

232

22"

- 4 x - 2 , - = - - - Substitution of y = 2' results in y3 - 4y2 - 32y = 0 and ' 2 64 16 '

46

y1

1. Arithmetic

= 8. y2 =

-4,y3 = 0; 2" = 8, 2 x a = -4, 2"s = 0 , so 21 = 3 follows. There are no further real roots.

1.6.4.3 Logarithmic Equations Logarithmic equations can be reduced to algebraic equations in the following two cases. if the unknown z or a polynomial P ( z ) only is under the logarithm sign: a) If the equation contains only the logarithm of the same expression, then we can introduce this as a new unknown, and we can solve the equation with respect to it. The original unknown can be determined by using the logarithm.

Im[log, P(z)lZt n = ad-. The substitution y = log, P ( z ) results in the equation my2 + n = a m . After solving for y we get the solution for zfrom the equation P ( x ) = uv. b) If the equation is a linear combination of logarithms of polynomialsof 2,on the same base a, with integer coefficients m, n, . . ., Le.. it has the form m log, Pl(x)+nlog, Pz(z)t . . . = 0, then the left-hand side can be written as the logarithm of one rational expression. (The original equation may contain rational coefficients and rational expressions under the logarithm, or logarithms with different bases, if the bases are rational powers of each other.) (32 - 1 j 2 (32 - 1 j 2 I210g5(3x-1)-10g,(12~t1)=0:10g,= log, 1, - 1; ~1 = 0 , ~2 = 2. Sub122 + 1 122 t 1 stituting xl = 0 in the original equation we get negative values in the logarithm, Le.. this logarithm is a complex value. so there exists no real solution. ~

1.6.4.4 Trigonometric Equations Trigonometric equations can be reduced to algebraic equations if the unknown zor the expression n z t a with integer n is only in the argument of the trigonometric functions. After using the trigonometric formulas the equation will contain only one unique function containing 2 , and after replacing it by y we get an algebraic equation. The solution for z is obtained from the solutions for y. naturally taking the multi-valuedness of the solution into consideration. 1 1 3 Isinx = cos'x - - or sinz = 1 - s i n 2 z - - . Substitutingy = sinz we get y2 + y - - = 0 and 4 4 4 1 3 1 y1 = - , y2 = - - . The result y~ gives no real solution, because 1 sinxi 5 1 for all real 2 : from y1 = 2 2 2 57r we have x = t 2ki7 and x = - + 2 k i ~with k = 1 , 2 , 3 , . , . .

6

6

1.6.4.5 Equations with Hyperbolic Functions Equatzons vlzth hyperbolzc functzons can be reduced to algebraic equations if the unknown x is only in the argument of the hyperbolic functions. We can rewrite the hyperbolic functions as exponential 1 expressions. then substitute y = e" and - = e-", so we get an algebraic equation for y . After solving this we have the solution x = In y . 3(e" e-') e" - e-" 2 -t9;e"t2e-2-9=O:y+--9=0.y2-9y+Z=O: I3coshs = sinhs+9: 2 2 U 9-fl 9&V% z 2.1716, x2 = In -x -1.4784. : x l=lnYl,2 = 2 2

+

47

2 Functions 2.1 Notion of Functions 2.1.1 Definition ofa Function 2.1.1.1 Function If z and y are two variable quantities, and if there is a rule which assigns a unique value of y to a given value of x,then we call y a function of x,and we use the notation Y. I=. ( ! (2.1) The variable x is called the independent variable or the argument of the function y. The values of z, to which a value of y is assigned, form the domain D of the function f(x). The variable y is called the dependent variable: the values of y form the range Wof the function f(x) . Functions can be represented by the points (x,y) as curves, or graphs of the function. 2.1.1.2 Real Functions If both the domain and the range contain only real numbers we call the function y = f(x)a realfunction of a real variable. IA: y = xz with D : --x < x < m. M.' : 0 5 y < co. IB: y = f i i v i t h D : O < x < c o , W : O
2.1.1.3 Functions of Several Variables If the variable y depends on several independent variables ~ 1 ~ x 2. .,,x,, . then we use the notation y = !(XI,

(2.2)

2 2 % ,. , ,X")

as a function of several variables (see 2.18, p. 117).

2.1.1.4 Complex Functions If the dependent and independent variables are complex numbers w and 2 respectively, then w = f ( z ) means a complezfunction of a complex variable. (see 14.1, p. 669). Cornplez-valued functions w(x) are called complex functions even if they have real arguments x .

2.1.1.5 Further Functions In different fields of mathematics, for instance in vector analysis and in vector field theory (see 13.1, p. 640), we also consider other types of functions whose arguments and values are defined as follows: 1. The arguments are real - the function values are vectors. IA: Vector functions (see 13.1.1, p. 640). IB: Parameter representations of curves (see 3.6.2, p. 238). 2. The arguments are vectors the function values are real numbers IScalar fields (see 13.1.2, p. 641). 3. The arguments are vectors - the function values are vectors. IA: Vecto; fields (see 13.1.3, p. 643). IB: Parametric representations or vector forms of surfaces (see 3.6.3. p. 243). ~

2.1.1.6 Functionals If a real number is assigned to every function 2 = z ( t ) of a given class of functions: then it is called a functional.

IA: Ifx(t) is a giyen function which is integrableon [a,b],then f(x) =

/*x ( t )dt is a linear functional a

defined on the set of continuous functions x integrable on [a, b] (see 12.5, p. 617).

I

48

2. Functions

IB: Integral expressions in variational problems (see 10.1, p. 550).

2.1.1.7 Functions and Mappings Suppose there are given two non-empty sets X and Y . A mapping, which is denoted by

f:X-+Y, (2.3) is a rule. by which we assign a uniquely defined element y of Y to every element x of X . The element y is called the image of x, and we write y = f(x). The set Y is called the image space or range o f f , the set X is called the original space or domain o f f . IA: If both the original and the image spaces are subsets of real numbers, i.e., X = D c R and Y = W C R holds, then (2.3) defines a real function y = f(x) of the real variable x. IB: I f f is a matrix A = (aij) ( i = 1 , 2 , . . . , m ; j = 1 , 2 , .. . ,n)of type (m,n)and X = R" and Y = R", then (2.3) defines a mapping from R" into R". The rule (2.3) is given by the following system of m linear equations:

-

= Ax

or

y1 =

a1121

=

a2121

y2

......... .

+

+ ... +

a12x2

+

alnxn

+ ... + aznxn

a2222

2 .

.

ym = amlxl

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

+ arn2Xz + . . . + amnxn

i.e. Ax means the product of the matrix A and the vector x. Remarks: 1. The notion of mapping is a generalization of the notion of function. So, some mappings are sometimes called functions. 2. The important properties of mappings can be found in 5.2.3, 5. p. 296. 3. .A mapping, which assigns to every element from an abstract space X a unique element usually from a different abstract space Y, is called an operator. Here an abstract space usually means a function space. since the most important spaces in applications consist of functions. Abstract spaces are for instance linear spaces (see vector spaces 5.3.7, p. 314), metric spaces (see 12.2, p. 602) and normed spaces (see 12.3, p. 609).

2.1.2 Methods for Defining a Real Function 2.1.2.1 Defining a Function -4 function can be defined in several different ways, for instance by a table of values, by graphical representation, Le., by a curve, by a formula, which is called an analytic expression, or piece by piece with different formulas. Only such values ofthe independent variable can belong to the domainof an analytic expression for which the function makes sense, i.e., it takes a unique, finite real value. If the domain is not otherwise defined. we consider the domain as the maximal set for which the definition makes sense.

2.1.2.2 Analytic Representation of a Function Usually we use the following three forms: 1. Explicit Form: Y = f (XI.

m,

(2.4) -1 5 x 5 1, y 2 0. Here the graph is the upper half of the unit circle centered at

Iy = the origin. 2. Implicit Form:

F ( z ,y) = 0. (2.5) in the case when there is a unique y which satisfies this equation, or we have to tell which solution is considered to be the substitution value of the function.

2.1 Notion of Functions 49

+

Izz yz - 1 = 0, -1 5 z 5 $1,

y 2 0. Here the graph is again the upper half of the unit circle centered at the origin. We remark that x2 y2 1 = 0 itself does not define a real function. 3. Parametric Form:

+ +

5 = d t ) , Y = dJ(t). (2.6) The corresponding values of z and y are given as functions of an auxiliary variable t , which is called a parameter. The functions p(t) and $ ( t )must have the same domain. This representation defines a real function only if x = p(t) defines a one-to-one correspondence between z and t. Iz = p(t), y = $ ( t ) with p(t) = cost and $ ( t ) = sint, 0 5 t 5 K . Here the graph is again the upper half of the unit circle centered at the origin. Remark: Functions given in parametric form sometimes do not have any explicit or implicit parameterfree equation. Iz = t + 2 s i n t = ( n ( t ) , y = t - c o s t = y ( t ) . Examples for Functions Given Piece by Piece:

b)

a)

Figure 2.1

JL

e+

-1

a)

IA: y = E ( z ) = int(z) = [z] = n for n 5 z < n + 1 , n integer. The function E ( z )or int(z) (read “znteger part ofz”) means the greatest integer less than or equal to z. IB: The function y = frac(z) = z - [z] (read “fractzonalpart of?) gives the difference ofz and [z] (Fig. 2.lb). Fig. 2.la,b shows the corresponding graphical representations, where the arrow-heads mean that the endpoints do not belong to the curves. I C : y = ( z2 for x 5 0, (Fig. 2.2a). z for z 2 0. -1 for z < 0, ID: y = sign(z) = 0 for z = 0, +1 for z > 0, (Fig. 2.2b). By sign(z) (read “signum z”),we denote the signfunctzon.

{

Figure 2.2

2.1.3 Certain Types ofhnctions 2.1.3.1 Monotone Functions If a function satisfies the relations f(4 2 f(dor f(4 5 f(zl), (2.74 for arbitrary arguments z1 and z2 with 52 > z1 in its domain. then it is called monotonically zncreaszng or monotonzcally decreaszng (Fig. 2.3a,b).

1 Ax*be x1

a)

x2

x

b) Figure 2.3

If one of the above relations (2.7a) does not hold for every z in the domain of the function, but it is valid, e.g., in an interval or on a half-axis, then the function is said to be monotonzc zn thzs domain. Functions satisfying the relations f b z ) > fb1) or f ( 4 < f ( 4 , (2.2)

Le., when the equality never holds in (2.7a), are called strictly monotonically increasingor strictly monotonically decreasing. In Fig. 2.3a there is a representation of a strictly monotonically increasing function: in Fig. 2.3b there is the graph of a monotonically decreasing function being constant between z1 and z2. Iy = e-" is strictly monotonically decreasing, y = l n r is strictly monotonically increasing.

2.1.3.2 Bounded Functions A function is said to be bounded above if there is a number (called an upper bound) such that the values of the function never exceed it. A function is called bounded below if there is a number (called a lower bound) such that the values of the function are never less than this number. If a function is bounded above and below, we simply call it bounded. (If a function has one upper bound, obviously it has an infinite number of upper bounds, all numbers greater than this one. It can be proven that among the upper bounds there is always a smallest, the so-called least upper bound. Similar statements are valid for lower bounds.) IA: y = 1 - zz is bounded above (y 5 1). IB: y = e" is bounded below (y > 0).

IC: y = sinz is bounded (-1 5 y 5 +l).

ID: y = -is bounded (0 < y 5 4) 1 t x2

2.1.3.3 Even Functions Even functions (Fig. 2.4a) satisfy the relation If D isf(-the .) domain = f(.). of f l then

+

(2.8b) (X E D ) (-x E D ) (2.84 should hold. IA: y = COSZ: IB: y = x4 - 3z2t1.

$ ,,

~

b)

a)

2.1.3.4 Odd Functions

,Ay,

Figure 2.4

Oddfunctions (Fig. 2.4b) satisfy the relation

f(k.1

= -f(.).

(2.9a)

If D is the domain o f f . then (r E D ) =+ ( - X E D )

(2.9b)

should hold. IA: y = sinz, IB: y = z3- x.

2.1.3.5 Representation with Even and Odd Functions If for the domain D of a function f the condition "from z E D it follows that -z E D" holds. then f can be written as a sum of an even function g and an odd function u : 1 1 (2.10) with f(.) = g ( r ) t = -[f(.) t f ( b ) l > 41 . = ,[f(.) - f(k.11' 2 If ( ~ ) = e " = - (1e " + e - ~ ) + 2 ( 1e ' - - e - ' ) =coshz$sinhz(see2.9.ltp. 87) 2

4 . 1

2.1.3.6 Periodic Functions Periodic functions satisfy the relation f ( z t T ) = f ( r ) , Tconst, T # 0. (2.11) Obviously, if the above equality holds for some T , it holds for any integer multiple of T . The smallest positive number T satisfying the relation is called the period (Fig. 2.5).

yb

0

Figure 2.5

2.1 Notion of Functions 51

2.1.3.7 Inverse Functions If the function y = f(x)is a one-to-one function, Le., if zllz2 E D and z1 # ~2~ then f ( z l )# f(z2). thenthereisafunctiony = p(z) such that forevery pairofvalues (a, b) satisfyingtheequalityb = !(a), the equality a = p(b) will be valid, and for every pair of values for which a = p(b) holds, b = f ( a ) will be valid. The functions Y = Vk) and Y = f(.) (2.12) are inverse functions of each other. Obviously, every strictly monotonic function has an inverse function. The graph of an inverse function y = p(z) is obtained by reflection of the graph of y = f(z) with respect to the line y = z (Fig. 2.6).

a)

b) Figure 2.6

Examples of Inverse Functions: IA: y = f(x) = 2' withD: y = p(z) = fi with D : IB: y = f ( z ) = e " with D : y = p(z) = l n z with D : C: y = f ( 2 ) = sinz with D : y = p(z) = arcsinz with D:

K': y 2 0; w: y 2 0 . -cc < z < cc. W : y > 0; z > 0, w: -cc < y < cc. -7~12 5 z 5 7 ~ 1 2 , W : -1 5 y 5 1; w: - T I 2 5 y 5 a/2. -1 5 z 5 1, z 2 0, z 2 0,

In order to get the explicit form ofthe inverse function of y = f ( z )we exchange zand y in the expression, then from the equation z = f(y) we express y, so we have y = p(z). The representations y = f(z) and z = p(y) are equivalent. Therefore, we have two important formulas f(dY))= Y and d f ( z ) )= 2 . (2.13)

2.1.4 Limits of Functions 2.1.4.1 Definition of the Limit of a Function The function y = f ( z ) has the limit A at z = a limf(2) = A or f ( z ) + A for + a, x+a

(2.14)

if as z approaches the value a infinitely closely, the value of f ( z ) approaches the value A infinitely closely. The function f ( z ) does not have to be defined at a,and even if defined, it does not matter whether f(a)is equal to A. Precise Definition: The limit (2.14) exists, if for any given positive number E there is a positive number 7 such that for every z # a belonging to the domain and satisfying the inequality ( 2 - a(< 7% (2.15a)

I

the inequality

If(.) - A / < E (2.15b) holds eventually with the expection of the point a (Fig. 2.7). If a is an endpoint of a connected region, then the inequality - a1 < 7 is reduced either to a - 7 < z or to z < a q.

+

A+& A _______ A-E

2.1.4.2 Definition by Limit of Sequences (see 0 a-q a a+q 7.1.2, p. 403) Figure 2.7 A function f(z)has the limit A at z = a if for every sequence xl,52.. . . .z,.. . . of the values of z from the domain and converging to a (but being not equal to a ) , the sequence of the corresponding values of the function f ( z ~f(zz), ) ~ . . . , f(z,), . . . converges to A. 2.1.4.3 Cauchy Condition for Convergence 4 necessary and sufficient condition for a function f(z)to have a limit at z = a is that for any two values z1 # a and z2 # a belonging to the domain and being close enough to a , the values f(zl) and f ( x Z ) are also close enough to each other. Precise Definition: A necessary and sufficient condition for a function f(z)to have a limit at z = a is that for any given positive number E there is a positive number q such that for arbitrary values x1 and z2 belonging to the domain and satisfying the inequalities 0 < Izl - a1 < 7 and 0 < 1x2 - a1 < 7, (2.16a) the inequality (2.16b) If(z1) - f(.?)l < E holds.

2.1.4.4 Infinity as a Limit of a Function The symbol lim if(x)l = cc r+a

(2.17)

means that as 2 approaches a, the absolute value If(x)l does not have an upper bound, and the closer we are to a, the larger is its greatest lower bound. Precise Definition: The equality (2.17) holds if for any given positive number K there is a positive number 7 such that for any z # a from the interval a-q<x
’

if they are negative, we write lim f(z)= -m. r+a

(2.18e)

2.1.4.5 Left-Hand and Right-Hand Limit of a Function A function f(x) has a left-hand limit A- at z = a: if as z tends to a from the left, the value f(z) tends to A - : A- = lim f(z)= f ( a - 0 ) . (2.19a) xia-0

2.1 Notion of Functions 53

Similarly, a function has a right-hand limit A' if as 2 tends to a from the right, the value f(z)tends to A+ ; (2.19b) At = x)liof(~) = / ( a t 0). The equality lilif(z) = A is valid only if the left-hand and right-hand limits exist, and they are equal: A+ = A- = A .

IThe function f(z)=

(2.19c)

4

tends to different values from l + e Z theleftandfrorntherightforz + 1: f(1-0) = 1, f ( l + O ) = 0 (Fig. 2.8).

ai-*

-----1

I

- - -- - -- - - -.

X

Figure 2.8

2.1.4.6 Limit of a Function as z Tends t o Infinity Case a) A number

A

fk)

= xl&

(2.20a)

is called the limit of a function f(z) as z + t c o , if for any given positive number E there is a number A' > 0 such that for every z > N , the corresponding value f(z)is in the interval A - E < f(x) < A + E . Analogously A = xlj~x f(.) (2.20b) is the limit of a function f(z) as z + -m if for any given positive number E there is a positive number N > Osuch that foranyx < -iV thecorrespondingvalueoff(z)isintheintervalA-& < f(z)< A t & . x t 1 x+1 IA: r++m lim = 1, IB: lirn -= 1, IC: lirn ex = O . x x+-m z 51--m Case b) Assume that for any positive number K , there is a positive number N such that if z > ,V or < -,V then the absolute value of the function is larger then K . In this case we write lim If(r)l = co or lim If(z)l = co. (2.20c)

x

xitm

.*

Xi-m

-1 IA: x++m lim -= +co, IB: 23

IC:

1-23 lirn -- -co, I D :

x+tm

22

23 - 1lirn - -cot 22

z-t-m

lim

z-t-m

1- 2 3 ~

22

= tco.

2.1.4.7 Theorems About Limits of Functions 1. Limit of a Constant Function The limit of a constant function is the constant itself: limA = A. x+a

(2.21)

2. Limit of a S u m or a Difference If among a finite number of functions each has a limit, then the limit of their sum or difference is equal to the sum or difference of their limits (if this last expression does not contain co - m): (2.22) lim [f(z) ~ ( x-) 4 z ) ] = h f ( z ) t &p(x) - I i i $ ( z ) . x+a

+

3. Limit of P r o d u c t s If among a finite number of functions each has a limit, then the limit of their product is equal to the product of their limits (if this last expression does not contain a 0 m type):

(2.23)

54

2. Functions

4. Limit of a Quotient The limit of the quotient of two functions is equal to the quotient of their limits. in the case when both limits exist and the limit of the denominator is not equal to zero (and this last expression is not an ffi/x type):

(2.24)

Also if the denominator is equal to zero, we can usually tell if the limit exists or not, checking the sign of the denominator (the indeterminate form is O / O ) . Similarly, we can calculate the limit of a power by taking a suitable power of the limit (if it is not a Oo, l", or xotype). 5 . Pinching If the values of a function f(z)lie between the values of the functions +(x) and W ( X ) . i.e y ( s ) < f(2)< u ( x ) .and if p(x) = A and 4(z) = A hold, then f(x) has a limit, too, and

lili

lili

lim f(z)= A.

(2.25)

x-0.

2.1.4.8 Calculation of Limits The calculation of the value of a limit can be made by using the following transformations and the theorems of 2.1.4.7:

1. Suitable Transformations K e transform the expression into a form such that we can tell the limit. There are several types of recommended transformations in different cases; we show three of them. X3 - 1 IA: lim -= lim (2' + x + 1) = 3. X-1

z-1

z+i

tz f i -X 1

IB:

IC: lim-

sin 22

rio

lim

0-0

= lim

(fi-l)(G+l)

z+O

X ( r n + l )

2(sin 22) 22

1

= lim

x+Ofi+l

1 -2

sin 2 2 22

= lim -= 2 lim -- 2 . Here we refer to the well-known theorem

2

ria

2ri0

sin a =I a

~

2. Bernoulli-1'Hospital Rule o f f i - -

0 . x , x - m 0' xo, l", one often applies the 0' xBernoullz-l'Hospztalrule (usually called l'Hospztal rule for short): Suppose lim p(2) = 0 and lim y ( z ) = 0 or lim p(z) = x and lim V ( X )= ffi, and suppose that there x+a X+Q X+Q X+Q is an interval containing a such that the functions p(x) and $(x) are defined and differentiable in this In the case of indeterminate forms like

~

~

V ' ( X ) exists. , interval except perhaps at a. and ~ ' ( x#) 0 in this interval, and lim Then

y'(x)

(2.26)

Remark: If the limit of the ratio of the derivatives does not exist, it does not mean that the original limit does not exist. Maybe it does, but we cannot tell this using 1'Hospital's rule.

4 ( X ) is, still an indeterminate form, and the numerator and denominator satisfy the assumptions y'(x) of the above theorem. we can again use 1'Hospital.s rule.

If lim

~

2'+a

o w Case a ) I n d e t e r m i n a t e Forms -or -: We use the theorem after checking if the conditions are o

fulfilled.

w

2.1 Notion of Functions 55 2 cos 2x

2

Case b) Indeterminate Form 0

9

00:

If we have f(z)= q ( x )~ ( xand ) liliv(x) = 0 and

limy(.) = rn , than in order to use 1'Hospital's rule for z i u

i~f(x)we transform it into one of the forms

v(x)

o

m

lim Q(z) or lim -. so we reduce it to an indeterminate form - or - like in case a). 1 Z+Q 1 o m O(X) P(X) -2 T - 22 I lim (T - 22) t a n z = lim -- lirn - = 2 . S i T f 2 r i n / 2 cotx x + x / ~ -- 1 sin2x Casec)IndeterminateFormco-m: Iff(.) = cp(z)-$(x) and;licp(x) = m a n d lx-ra i m $ ( z ) = 03, o m then n e can transform this expression into the form - or - usually in several different ways; for instance r i a

as cp - v =

o x

($ i) /$. 1

-

Then we proceed as in case a) - - Applying 1'Hospital rule twice we get

Case d) Indeterminate Forms Oo , coo, lm: If f(x) = 'p(z)'(') and lirn p(x) = 0 and lim $(x) x+a ria

=

0, then we first find the limit A of l n f ( x ) = i ( x )l n p ( z ) , which has the form 0.rn (case b)). then we can find the value e.'. The procedures in the cases moand 1" are similar. In x Ilimx' = X.In? = z l n x . l i m x l n z = lim - = lim(-x) = 0. Le., A = I n X = 0, s i o ~ - ~r i o

x i 0

x-10

so X = 1,. and finally lim 2%= 1. xi0

3. Taylor Expansion Besides 1'Hospital's rule the expansion of functions of indeterminate form into Taylor series can be applied (see 6.1.4.5. p. 387). 2-

x - sin x Ilim -= lim 210

23

ai0

( 51 E---$

z3

I:

--...

)

; ) ;,

= lim - - - + . . . = X + O ( i !

2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols Comparing two functions. we often consider their mutual behavior with respect to a certain argument

z = a. It is also convenient to compare the order of magnitude of the functions. 1. A function f(z)tends to infinity with a higher order than a function g(x) at a if the quotient

and the absolute values of f(x)exceed any limit as z tends to a.

56

2. Functions

2. A function f(z)tends to zero with a higher order than a function g(z) at a if the absolute values of

f b) tends to zero as z tends to a. -

f(z),g(z) and the quotient

g(x)

3. Two functions f ( z ) and g(z) tend to zero or to infinity by the same order (or order of magnitude)

1 I

f < M holds for the absolute value of their quotient as z tends to a, where M is at a if 0 < m < g(x)

a finite number

4. Landau Order Symbols The mutual behavior of two functions at a point z = a can be described by the Landau order symbols 0 (“big 0”),or o (“small 0 ” ) as follows: If z -+ a then

f ( z ) = O(g(z)) means that

f =A lirn gb)

# 0, A = const,

(2.27a)

x+a

and

f(x) = 0, (2.27b) lim g(x) where a = 3xc is also possible. The Landau order symbols have meaning only if we assume the ztends to a given a. sin z IA: sinz = O ( x )for z -+ 0,because with f(z) = sinz and g(z) = z we have: lirn = 1 # 0,

f (z) = o(g(z)) means that

~

z+o

2

Le., sin z behaves like zin the neighborhood of z = 0. IB: For f ( z ) = 1 - cosz and g(z) = sinz the function f (z) vanishes with a higher order than g(z): f 1 - cosz lim - = lim -= 0, Le., 1- cosz = o(sinz) for z -+ 0. x+o!g(z)l x+oI sinz

I

IC: f(z)and g(z) vanish by the same order for f (z) = 1 - cosz, g(z) = x 2 :

, i.e., 1 - cosz = O ( 2 ) for z -+ 0. 5. Polynomial The order of magnitude of polynomials at kffi can be expressed by their degree. So the function f(z) = z has order 1, a polynomial of degree n + 1 has an order higher by one than a polynomial of degree n. 6. Exponential Function The exponential function tends to infinity more quickly than any high power zn ( n is a fixed positive number):

IeX1

lirn zn =co. The proof follows by applying 1’Hospital’srule for a natural number n: x+m

(2.28a)

(2.28b) 7. Logarithmic Function The logarithm tends to infinity more slowly than any small positive power

z’ ( a is a fixed positive number): (2.29) The proof is with the help of 1’Hospital’srule

2.1 Notion ojfinctions 57

2.1.5 Continuity of a Function 2.1.5.1 Notion of Continuity and Discontinuity Most functions occurring in practice are continuous, Le., for small changes of the argument x a continuous function y(z) changes also only a little. The graphical representation of such a function results in a continuous curve. If the curve is broken at some points, the corresponding function is discontinuous, and the values of the arguments where the breaks are, are the points of discontinuitg. Fig. 2.9 shows the curve of a function, which is piecevise continuous. The points of discontinuity are A, B , C, D,E , F and G . The arrow-heads show that the endpoints do not belong to the curve.

+\///, I

,

/

fi/

~

/ II ,!

? / 0 A / c DE/F G ’ B :

9

X

I ~

2.1.5.2 Definition of Continuity A function y = f(z)is called continuous at the point z = a if

Figure 2.9

1. f(x) is defined at a; 2. the limit j ( z )exists and is equal to f ( a ) .

lili

> 0 there is a 6 ( ~ >) 0 such that for every z with Iz - ul < S

This is exactly the case if for an arbitrary E

if(.) - f(a)l < E (2.30) holds. We also talk about one-sided (left- or right-hand sided) continuity, if instead of l i i f ( x ) = f ( u ) we consider only the one-sided limit lim f(x) or lim f(z)and this is equal to the substitution value x+a-0

x+a+O

f(a). If a function is continuous for every z in a given interval from a to b, then the function is called continuous in this interval, which can be open, half-open, or closed (see 1.1.1.3, 3., p. 2). If a function is defined and continuous at every point of the numerical axis, it is said to be continuous everywhere. A function has a point of discontinuity at z = a, which is an interior point or an endpoint of its domain, if the function is not defined here, or f ( a ) is not equal to the limit g i j ( z ) , or the limit does not exist. If the function is defined only on one side of z = a, e.g., +&for z = 0 and arccosx for z = 1 ,then it is not a point of discontinuity but it is a termination. A function f(x)is called piecewise continuous, if it is continuous at every point of an interval except at a finite number of points, and at these points it has finite jumps.

2.1.5.3 Most Frequent Types of Discontinuities 1. Values of the Function Tend t o Infinity The most frequent discontinuity is if the function tends to *w (points B , C, and E in Fig. 2.9).

1 A: f ( z ) = tanx, f

(4- )

(9 1

0 = +m, j - + 0 = -m. The type of discontinuity (see Fig. 2.34,

p. 76) is the same as at E in Fig. 2.9. For the meaning of the symbols f ( a - 0), j ( a + 0) see 2.1.4.5, p. 52.

1 B: f(z)= I f ( 1 - 0) = +w, f ( 1+ 0) = tco. The type of discontinuity is the same as at (x - 1 ) 2 ’ the point B in Fig. 2.9. 1 W C: f(x) = ex-1 f ( 1 - 0) = 0, f ( 1 + 0) = m. The type of discontinuity is the same as at C in Fig. 2.9, with the difference that this function f(z)is not defined at E = 1.

.

58

2. Functions

2. Finite J u m p Passing through z = a the function f(z)jumps from a finite value to another finite value (like at the points A , F , G in Fig. 2.9, p. 57): The value of the function f(z) for 2 = a may not be defined here, as at point G; or it can coincide with f(a - 0) or with f(a + 0) (point F);or it can be different from f(a - 0) and f ( a t 0) (point A ) . IA: f(z)= +s

f ( 1 - 0)= 1, f(1 + 0) = 0 (Fig. 2.8, p. 53).

I+e-1

IB: f(z)= E ( z ) (Fig. 2.lc, p. 49) f(n- 0) = n - 1, f(n+ 0) = n ( n integer). 1

1 C : f(z)= n+m lim f ( 1 - 0) = 1, f ( l +0) = 0, f(1) = - . 1 + z2n 2 3. Removable Discontinuitv

+

If it happens that l i i f ( z ) exists, Le., f(a - 0) = f ( a 0), but either the function is not defined for z = a or / ( a ) # h f ( z ) (point D in Fig. 2.9, p. 57)) this type of discontinuity is called removable, because defining f(a) = lilif(r) the function becomes continuous here. We add only one point to the curve, or we change the place only of one point a t D. The different indeterminate expressions for z = a, which have a finite limit examined by 1'Hospital's rule or with other methods, are examples of removable discontinuities. 0 1 is an undetermined - expression for z = 0, but lim f(z)= -: the function 1 f(.) = 0 x i 0 2

1 I -2 is continuous.

for z = 0

2.1.5.4 Continuity and Discontinuity of Elementary Functions The elementary functions are continuous on their domains; the points of discontinuity do not belong to their domain. We have the following theorems: 1. Polynomials are continuous everywhere.

2. Rational Functions '(') with polynomials P ( z ) and Q(z) are continuous everywhere except Q(X)

the points 2, where Q(z) = 0. If at z = a, &(a) = 0 and P(a) # 0, the function tends to fx on both sides of a : we call this point a pole. The function also has a pole if P(a) = 0, but a is a root of the denominator with higher multiplicity than for the numerator (see 1.6.3.1, 2., p. 43). Otherwise the discontinuity is removable. 3. Irrational Functions Roots of polynomials are continuous for every zin their domain. At the end of the domain they can terminate by a finite value if the radicand changes its sign. Roots of rational functions are discontinuous for such values of z where the radicand is discontinuous. 4.Trigonometric Functions The functions sinz and cos2 are continuous everywhere; t a n z and secz have infinite jumps at the points z =

~

+

2

' I T : the functions cot z and cosec z have infinite

jumps at the points I = n~ ( n integer). 5 . Inverse Trigonometric Functions The functions arctanz and arccot z are continuous everywhere, arcsin z and arccos z terminate at the end of their domain because of -1 z +1, and they are continuous here from one side. 6. Exponential Functions e" or ax with a > 0 They are continuous everywhere.

< <

2.1 Notion of Functions 59

7. Logarithmic Function log z with Arbitrary Positive Base The function is continuous for all positive z and terminates at z = 0 because of lim logz = -cc by a right-sided limit. z++o

8. Composite Elementary Functions The continuity is to be checked for every point z of every elementary function containing in the composition (see also continuity of composite functions in 2.1.5.5, 2., p. j 9 ) . 1 ex-2 1 IFind the points of discontinuity of the function y = The exponent has an zsin +'iZ' ~

infinite jump at x = 2: for z = 2 also e=

1

has an infinite jump:

( i, es-2

x=2-0

=o,

=m.

(e&) x=2+0

The function y has a finite denominator at z = 2. Consequently, at z = 2 there is an infinite jump of the same type as at point C in Fig. 2.9, p. 57. For x = 0 the denominator is also zero, just like for the values of z, for which sin is equal to zero. These last ones correspond to the roots of the equation = nT or z = 1 - n37r3,where n is an arbitrary integer. The numerator is not equal to zero for these numbers, so at the points z = 0, z = 1,x = 1 7r3, z = 1 5 87r3, z = 1 f 277r3, . . .the function has the same type of discontinuity as the point E in Fig. 2.9, p. 57.

*

2.1.5.5 P r o p e r t i e s of Continuous Functions 1. Continuity of Sum, Difference, Product and Quotient of Continuous Functions If f(z)and g(z) are continuous on the interval [a, b], then f ( z ) g(z) , f ( z )g(z) are also continuous, f ( x ) is . also continuous. and if g(z) # 0 on this interval, then g(x) 2. Continuity of Composite Functions y = f(u(z)) If u ( x ) is continuous at z = a and f ( u ) is continuous at u = .(a) then the composite function y = f ( u ( z ) is) continuous at z = a. and

*

(2.31) is valid. This means that a continuous function of a continuous function is also continuous.

Remark: The converse sentence is not valid. It is possible that the composite function ofdiscontinuous functions is continuous.

3. Bolzano Theorem If a function f ( x ) is continuous on a finite closed interval [a: b], and f ( a ) and f ( b ) have different signs, then f(z)has at least one root in this interval, Le., there exists at least one interior point of this interval c such t h a t : , f ( c ) = 0 with a < c < b. (2.32) The geometric interpretation of this statement is that the graph of a continuous function c a go from one side of the x-axis to the other side only if the curve has an intersection point with the z-axis.

4. Intermediate Value Theorem If a function f ( ~is continuous ) on a connected domain, and at two points a and b of this domain, where a < b, it has different values A and B , Le.! S(a) = A, f ( b ) = B , A # B; (2.33a) then for any value C between A and B there is at least one point c between a and b such that (2.33b) f ( c ) = C, ( a < c < b: A < C < B or A > C > B ) . In other words: The function f ( z ) takes every value between A and B on the interval ( a , b) at least once. Or: The continuous image of an interval is an interval.

I

60

2. Functions

5. Existence of an Inverse Function If a one-to-one function is continuous on an interval, it is strictly monotone on this interval. If a function f(z)is continuous on a connected domain I, and it is strictly monotone increasing or decreasing, then for this f(z)there also exists a continuous, strictly monotone increasing or decreasing inverse function p(z) (see also 2.1.3.7, p. 51), which is defined on domain I1 given by the substitution values of f(z)(Fig. 2.10).

Figure 2.10

Remark: In order to make sure that the inverse function of f(z)is continuous, f(z)must be continuous on an interval. If we suppose only that the function is strictly monotonic on an interval, and continuous at an interior point e, and j ( c ) = C, then the inverse function exists, but may be not continuous at C.

6. Theorem About the Boundedness of a Function If a function j ( z ) is continuous on a finite, closed interval [a,b] ,then it is bounded on this interval, i.e., there exist two numbers m and M such that m < f ( x ) < M for a < x < b .

(2.34)

7. Weierstrass Theorem If the function f(x) is continuous on the finite, closed interval [a,b] then f(z)has an absolute maximum ibf and an absolute minimum m, Le., there exists in this interval at least one point c and at least one point d such that for all z with a 5 z b: (2.35) VI= f(d) 5 f(x) 5 f ( c ) = M. The difference between the greatest and smallest value of a continuous function is called its variation in the given interval. The notion of variation can be extended to the case when the function does not have any greatest or smallest value.

<

2.2 Elementary Functions Elementary junctions are defined by formulas containing a finite number of operations on the independent variable and constants. The operations are the four basic arithmetical operations, taking powers and roots, the use of an exponential or a logarithm function, or the use of trigonometric functions or inverse trigonometric functions. We distinguish algebraic and transcendental elementary functions. As another type of function, we can define the non-elementaryfunctions (see for instance 8.2.5,p. 458).

2.2.1 Algebraic Functions In an algebraic function the argument x and the function y are connected by an algebraic equation. It has the form (2.36) PO(.) +P~(z)Y +pZ(z)yz + . ' . +pn(x)yn = o where PO:p l , .. . , p , are polynomials in z. I 3zy3 - 4x21 x3 - 1 = 0 , i.e., po(z) = 2 - 1 , pl(x) = -4x, pz(z)= O p3(z)= 3 x . If it is possible to solve an algebraic equation (2.36) for y1then we have one of the following types of the simplest algebraic functions.

+

~

2.2.1.1 Polynomials We perform only addition, subtraction and multiplication on the argument z: y = anxn + an-lxn-l+ . . . + ao. (2.37) In particular we distinguish y = a as a constant, y = az b as a linearfunction, and y = ax2 + bx + c as a quadratic function.

+

2.2 Elementaru Functions 61

2.2.1.2 Rational Functions A rational function can always be written in the form of the ratio of two polynomials: a,xn = b,xm

+ an-lxn-l + . . . + a0

(2.38a)

+ bm-lxm-l + . . . +bo

The special case ax + b cx d is called a homographic or linear fractional function.

y=-

(2.38b)

+

2.2.1.3 Irrational Functions Besides the operations enumerated for rational functions, the argument x also occurs under the radical sign. I A : y = m . I B : y = { m .

2.2.2 Transcendental Functions Transcendentalfunctzonscannot be given by an algebraic equation like (2.36). We introduce the simplest elementary transcendental functions in the following.

2.2.2.1 Exponential Functions The variable x or an algebraic function of x is in the exponent of a constant base (see 2.6.1, p. 71) IA: ? / = e x l IB: y = a x l IC: = 2 3 X 2 - 5 X

2.2.2.2 Logarithmic Functions The function is the logarithm with a constant base of the variable x or an algebraic function of x (see 2.6.2, p. 71). IC: y = 1 0 g , ( 5 ~-~ 32). IA: y = I n x . IB: y = I g z ,

2.2 -2.3 Trigonometric Functions The variable x or an algebraic function of x occurs under the symbols sin, cos, tan, cot, sec, cosec (see 2.7, p. 74). IA: y = sinx, IB: y = cos(2x + 3), IC: y = tan fi. In general, the argument of a trigonometric function is not only an angle or a circular arc as in the geometric definition, but an arbitrary quantity. The trigonometric functions can be defined in a purely analytic way without any geometry. For instance we can represent them by an expansion in a series, d2Y or, e g.. the sin function as the solution of the differential equation - y = 0 with the initial values dx2

+

dY = 1 at x = 0. The numerical value of the argument of the trigonometric function is equal y = 0 and -

dx

to the arc in units of radians. When we deal with trigonometric functions, the argument is considered to be given in radzan measure.

2.2.2.4 Inverse Trigonometric Functions The variable z or an algebraic function of 2 is in the argument of the inverse trigonometric functions (see 2.8. p. 84) arcsin, arccos, etc.

I

IA: y = arcsinx. IB: y = arccos G.

2.2.2.5 Hyperbolic Functions (see 2.9, p. 87).

2.2.2.6 Inverse Hyperbolic Functions (see 2.10, p. 91).

2.2.3 Composite Functions Composite functions are all possible compositions of the above algebraic and transcendental functions. Le.. if a function has another function as an argument.

IA: y = lnsinx,

I B: y =

Inx

+ &ZG

t sex Such composition of a finite number of elementary functions again yields an elementary function. The examples C in the previous types of functions are also composite functibns. x2

2.3 Polynomials 2.3.1 Linear Function The graph of the linearfunction y=ax+b (polynomial of degree 1) is a line (Fig. 2.11a). For a > 0 the function is monotone increasing, for a

(2.39)

< 0 it is monotone decreasing; for a

= 0 it is a

polynomial of degree zero, i.e., it is a constant function. The intercepts are at A (for details see 3.5.2.4, l., p. 194).'?Vith b = 0 we have direct proportionality y = ax: graphically it is a line running through the origin (Fig. 2.11b).

Figure 2.11

(2.40)

Figure 2.12

2.3.2 Quadratic Polynomial The polynomial of second degree y = a x 2 t bx t c

(2.41) b (quadratic polynomial) defines a parabola with a vertical axis of symmetry at z = -- (Fig. 2.12). 2a For a > 0 the function is first decreasing, it has a minimum, then it is increasing again. For a < 0 first

2.3 Polynomials 63

it is increasing, it has a maximum, then it is decreasing_ again. _ The intersection points A ., , A?-with the z-axis, if any. are at

~

2a

the intersection point B with the y-axis is at (0. e). The

extremum point of the curve is at C

(for more details about the parabola see 3.5.2.8,

p. 203)

b) Figure 2.13

2.3.3 Cubic Polynomials The polynomial of third degree y = ax3 62’ + cz + d (2.42) defines a cubic parabola (Fig. 2.13a,b,c). Both the shape of the curve and the behavior of the function depend on a and the discriminant A = 3ac - b2. If A 2 0 holds (Fig. 2.13a,b), then for a > 0 the function is monotonically increasing, and for a < 0 it is decreasing. If A < 0 the function has exactly one local minimum and one local maximum (Fig. 2 . 1 3 ~ )For . a > 0 the value of the function rises from --xuntil the maximum, then falls until the minimum, then it rises again to +co; for a < 0 the value of the function falls from +xuntil the minimum, then rises until the maximum, then it falls again to -m. The intersection points with the z-axis are at the values of the real roots of (2.42) for y = 0. The function can have one, two (then there is a point where the x-axis is the tangent line of the curve) or three real roots: AI, A2 and ‘43. The intersection point with the y-axis is at B(0,d ) , the extreme points b5 d 2b3 - Sabc iz (6ac - 2 b 2 ) G of the curve C and D , if any, are at -~ 3a ’ 27a2 b 2b3 - Sabc The inflection point which is also the center of symmetry of the curve is at E -- , 3a 27a2 + d ) ’

+

( a

+

(

At this point the tangent line has the slope tan cp =

1,

~

A

(2),=2

2.3.4 Polynomials of n-th Degree The integral rational function ofn-th degree y = a n i n an-lzn-l , . . a l z a0 (2.43) defines a curve o f n - t h degree or n-th order (see 3.5.2.3, 5., p. 194) of parabolic type (Fig. 2.14). Case 1, n odd: For a, > 0 the value of y changes continuously from --x to +x. and for a, < 0 from +xto -x. The curve can intersect or contact the z-axis up t o n times, and there is at least one intersection point (for the solution of an equation of n-th degree see 1.6.3.1. p. 4 3 and 19.1.2, p. 884). The function (2.43) has none or an even number up to n - 1 of extreme values, where minima and maxima occur alternately; the number of inflection points is odd and is between 1 and n - 2. There are no asymptotes or singularities. Case 2, n even: For a, > 0 the value of y changes continuously from +xthrough its minimum

+

+ +

+

64

2. Functions

- n odd

1'

... neven

Figure 2.14

Figure 2.15

until +w and for a, < 0 from -cc through its maximum until -w. The curve can intersect or contact the z-axis up to n times, but it is also possible that it never does that. The number of extrema is odd, and maxima and minima alternate; the number of inflection points is even, and it can also be zero. There are no asymptotes or singularities. If we want to sketch the graph of a function, it is recommended first to determine the extreme points, the inflection points, the values of the first derivative at these points, then to sketch the tangent lines at these points, and finally to connect these points continuously.

2.3.5 Parabola of n-th Degree The graph of the function y = axn (2.44) where R > 0, integer. is a parabola ofn-th degree, or ofn-th order (Fig. 2.15). 1. Special Case a = 1: The curve y = xn goes through the point (0,O) and ( 1 , l ) and contacts or intersects the z-axis at the origin. For even R we have a curve symmetric with respect to the y-axis, and with a minimum at the origin. For odd n the curve is symmetric with respect to the origin, and it has an inflection point there. There is no asymptote. 2. General Case a # 0: We get the curve of y = azn from the curve of y = zn by stretching the ordinates by the factor lal. For a < 0 we reflect y = Jalznwith respect to the z-axis.

2.4 Rational Functions 2.4.1 Special Fractional Linear Function (InverseProportionality) The graph of the function y=-

U

X

(2.45)

is an equilateral hyperbola, whose asymptotes are the coordinate axes (Fig. 2.16). The point of discontinuity is at x = 0 with y = *co.If a > 0 holds, then the function is strictly monotone decreasing in the interval (-co,0) with values from 0 to -w and also strictly monotone decreasing in the interval (0, +,m) with values from +m to 0 (curve in the first and third quadrants). If a < 0, then the function is increasing in the interval (-co , 0) with values from 0 to +cc and also increasing in the interval (0, +co) with values from -w to 0 (dotted curve in the second and fourth quadrants). The vertices A

2.L Rational Functions 65

(&a: +fi)(*&, -Ji;Ei>

and B are at and with the same sign for a > 0 and with different sign for a < 0. There are no extrema (for more details about hyperbolas see 3.5.2.7, p. 200). VI

Figure 2.16

Figure 2.17

2.4.2 Linear Fractional Function The graph of the function a1x bl y=azx b2 is an equilateral hyperbola, whose asymptotes are parallel to the coordinate axes (Fig. 2.17)

+ +

.

The parameter a in the equality (2.45) corresponds here to

(2.46)

A

--

a2

-~

2

~

where for A < 0 we take the same signs, for A > 0 different ones. The

point of discontinuity is at x =

b2

a2

a1 to For A < 0 the values of the function are decreasing from a2

a1 -co and from t c o to -.a1 For A > 0 the values of the function are increasing from to +m and from

a2

-co to

a1 -

a2

. There is no extremum.

a2

2.4.3 Curves of Third Degree, Type I The graph of the function y=a+b+c

(= ax2+ bx +

x

22

c

(2.47)

22

(Fig. 2.18) is a curve of third degree (type I). It has two asymptotes x = 0 and y = a and it has two branches. One of them corresponds to the monotone changing of y while it takes its values between a and t c o or -ea;the other branch goes through three characteristic points: the intersection point with

the asymptote y = a at A

, an extreme point at B

and an inflection point at

. The positions of the branches depend on the signs of b and c, and there are four cases (Fig. 2.18). The intersection points D, E with the x-axis, if any, are at

2. Functions

66

their number can he two, one (the x-axis is a tangent line) or none, depending on whether b2 - 4ac > 0, = 0 or < 0 holds.

Forb = 0 the function (2.47) becomes the function y = a +

22

(see (Fig. 2.21) the reciprocal power),

+

ax b and for c = 0 it becomes the homographic function y = -, as a special case of (2.46)

Yt

Yt

cO

c
c)

d) Figure 2.18

2.4.4 Curves of Third Degree, Type I1 The graph of the function 1

=

(2.48)

ax2 + hx + c

b

is a curue of thzrd degree (type 11) which is symmetric about the vertical line x = -- and the z-axis is 2a its asymptote (Fig. 2.19). because liymy= 0. Its shape depends on the signs of a and A = 4ac - b2. From the two cases a > 0 and a

'

= (-a)x2 - bx - c

Case a) A

< 0 we consider only the first one, because reflecting the curve of

with respect to the x-axis we get the second one.

> 0: The function is positive and continuous for arbitrary values of z and it is increasing h

on the interval (-m, - - ) ~ 2a

h

4a Here it takes its maximum, -, then it is decreasing again in the interval

(-%, co).The extreme point A of the curve is at

(- $);. c!

a

(-$ , );,

the inflection points B and C are at

: and for the corresponding slopes of the tangent lines (angular

coeficzents ) we

get tan 9 = F a 2 (:)3'2

(Fig, 2.19a).

Case b) A = 0: The function is positive for arbitrary values of x, its value rises from 0 to +m, at b x = --2a = 20 it has a point of discontinuity (a pole), where x+xo lim y = +-cy). Then its value falls from here back to 0 (Fig. 2.19b). Case c) A < 0: The value of y rises from 0 to +m, at the point of discontinuity it jumps to -cos and rises to the maximum, then falls back to --cy): at the other point of discontinuity it jumps to +cos then it falls to 0. The extreme point A of the curve is at at x =

( 2", . 2). --

-

The points of discontinuity are

- b & G (Fig. 2 . 1 9 ~ ) . 2a

Figure 2.19

2.4.5 Curves of Third Degree, Type I11 The graph of the function

'

= ax2

X

(2.49)

+ bx + c

is a czlrue of third degree (type 111) which goes through the origin, and has the x-axis (Fig. 2.20) as an asymptote. The behavior of the function depends on the signs of a and of A = 4ac - b2, and for A < 0 also on the signs of the roots N and p of the equation ax2 bx c = 0, and for A = 0 also on the sign of b. From the two cases: a > 0 and a < 0, we consider only the first one because reflecting the curve X with respect to the x-axis we get the second one. ofy= (-a).' - bx - c

+ +

Case a) A > 0: The function is continuous everywhere, its value falls from 0 to the minimum, then rises to the maximum, then falls again to 0. The extreme points of the curve, A and B , are at 4 , -b ; there are three inflection

(

E *ifi)

points (Fig. 2.20a).

Case b) A = 0: The behavior of the function depends on the sign of b, so we have two cases. In both b cases there is a point of discontinuity at 5 = -- ; both curves have one inflection point. 2a b > 0: The value of the function falls from 0 to -a, the function has a point of discontinuity, then the value of the function rises from -m to the maximum, then decreases to 0 (Fig. 2.20bl). The extreme point A of the curve is at il

(E'&) +

'1

\k-. A.

A=O,b > 0

A=O,b
bl)

b2)

A < 0, a and p negative

A 0, a and p positive c3)

C2)

Figure 2.20 0 b < 0: The value of the function falls from 0 to the minimum, then rises to +cc, running through the origin, then the function has a point of discontinuity, then the value of the function falls from +w to 0

( E$

(Fig. 2.20b2). The extreme point A of the curve is at A -

-~ 2 i - b ) .

Case c) A < 0: The function has two points of discontinuity, at z = cy and z = p; its behavior depends on the signs of cy and p. 0 The signs of cy and /3 are different: The value of the function falls from 0 to -m, jumps up to +m, then falls again from +m to -m, running through the origin, then jumps again up to +cc , then it falls tending to 0 (Fig. 2 . 2 0 ~ ~The ) . function has no extremum. 0 The signs of cy and /3 are both negative: The value of the function falls from 0 to -a, jumps up to +X , from here it goes through a minimum up to +cc again, jumps down to -cc , then rises to a maximum. then falls tending to 0 (Fig. 2.20~2). The extremum points A and B can be calculated with the same formula as in case a) of 2.4.5. 0 The signs of cy and /3 are both positive: The value of the function falls from 0 until the minimum, then rises to SX,jumps down to -co,then it rises to the maximum, then it falls again to -cc, then jumps up to fcc and then it tends to 0 (Fig. 2.20123). The extremum points A and B can be calculated by the same formula as in case a) of 2.4.5. In all three cases the curve has one inflection point.

2.4.6 Reciprocal Powers The graph of the function y=

= uz-n

(n > 0, integer)

(2.50)

2.5 Irrational Functions 69

Figure 2.21

Figure 2.22

is a curve ojhyperbolzc type with the coordinate axes as asymptotes. The point of discontinuity is at t = 0 (Fig. 2.21). Case a) For a > 0 and for even TI the value of the function rises from 0 to t w o then , it falls tending to 0, and it is always positive. For odd n it falls from 0 to -w, it jumps up to +w, then it falls tending to 0. Case b) For a < 0 and for even n the value of the function falls from 0 to -w, then it tends to 0, and it is always negative. For odd n it rises from 0 up to $30, jumps down to -a, then it tends to 0. The function does not have any extremum. The larger n is, the quicker the curve approaches the x-axis, and the slower it approaches the y-axis. For even n the curve is symmetric with respect to the y-axis, for odd n i t is centrosymmetric and its center of symmetry is the origin. The Fig. 2.21 shows the cases n = 2 a n d n = 3for a = 1.

2.5 Irrational Functions 2.5.1 Square Root of a Linear Binomial The union of the curve of the two functions

y=iVGTTl

(2.51)

is a parabola with the x-axis as the symmetry axis. The vertex A is at

a.

(9

--, 0 the semifocal chord ~

(see 3.5.2.8,p. 203) is p = The domain of the function and the shape of the curve depend on the 2 sign of a (Fig. 2.22) (for more details about the parabola see 3.5.2.8,p. 203).

2.5.2 Square Root of a Quadratic Polynomial The union of the graphs of the two functions y = d a x 2 t bx t c

is for a

(2.52)

< 0 an ellzpse, for a > 0 a hyperbola (Fig. 2.23). One of the two symmetry axes is the x-axis,

b the other one is the line z = -<. Lu.

The vertices A , C and B , D are at

,whereA=4ac-bZ.

The domain of the function and the shape of the curve depend on the signs of a and A (Fig. 2.23). For a < 0 and A > 0 the function has only imaginary values, so no curve exists (for more details about

a)

a
c)

a>O,A
Figure 2.23 the ellipse and hyperbola see 3.5.2.6, p. 198 and 3.5.2.7, p. 200)

2.5.3 Power Function We discuss the power function y = axk = azirnln

(m, n integer, positive. coprime)

(2.53)

for k > 0 and for k < 0 (Fig. 2.24). We restrict our investigation for the case a = 1. because for a # 1 the curve differs from the curve of y = xk only by a stretching in the direction of y-axis by a factor lal, and for a negative a also by a reflection to the z-axis.

'PI' 1

I2

Figure 2.24 Case a) IC > 0 , y = xrnln: The shape of the curve is represented in four characteristic cases depending on the numbers m and n in Fig. 2.24. The curve goes through the points (0,O)and (1,l). For k > 1 the z-axis is a tangent line of the curve at the origin (Fig. 2.24d), for k < 1 the y-axis is a tangent line also at the origin(Fig. 2.24a,b,c). For even n we may consider the union of the graph of functions y = hk: it has two branches symmetric to the x-axis (Fig. 2.24a,d), for even m the curve is symmetric to the y-axis (Fig. 2 . 2 4 ~ )If. m and n are both odd. the curve is symmetric with respect to the origin (Fig. 2.24b). So the curves can have a vertex, a cusp or an inflection point at the origin (Fig. 2.24). hone of them has any asymptote. Case b) IC < 0 , y = x - ~ / The ~ : shape of the curve is represented in three characteristic cases depending on m and n in Fig. 2.25. The curve is a hyperbolic type curve, where the asymptotes coincide with the coordinate axes (Fig. 2.25). The point of discontinuity is at z = 0. The greater (k( is the quicker the curve approaches the x-axis. and the slower it approaches the y-axis. The symmetry properties of the curves are the same as above for k > 0: they depend on whether m and n are even or

2.6 Exponential Functions and Lo.qarithmic Functions 71

yle . 1

01 1

x

Figure 2.25 odd. There is no extreme value.

2.6 Exponential Functions and Logarithmic Functions 2.6.1 Exponential Functions The graphical representation of the function y = a" = ebr ( a > 0. b = lna) (2.54) is the exponentzal curve (Fig.2.26). For a = e we have the natural exponentzal curwe y = ex (2.55) The function has only positive values. Its domain is the interval (-co.+co). For a > 1. Le., forb > 0, the function is strictly monotone increasing and takes its values from 0 until co. For a < 1, Le.. for b < 0. it is strictly monotone decreasing, its value falls from co until 0. The larger lbl is, the greater is the speed of growth and decay. The curve goes through the point ( 0 , l ) and approaches asymptotically the x-axis. for b > 0 on the right and for b < 0 on the left. and more quickly for greater values of lbl. The function y = a-' =

increases for a < 1 and decreases for a > 1

_____

Figure 2.26

y=log,x=lb x y=log,x=ln x y=log,,x=lg x

Figure 2.27

2.6.2 Logarithmic Functions The function y = logax (a > 0, a # 1) (2.56) defines the logarzthrnzc curve (Fig. 2.27); the curve is the reflection of the exponential curve with respect to the line y = x. For a = e we have the curve of the natural logarzthm y = lnx. (2.57)

The real logarithmic function is defined only for I > 0. For a > 1 it is strictly monotone increasing and takes its values from -m to +m, for a < 1 it is strictly monotone decreasing, and takes its values from +m to -m, and the greater 1 lnal is, the quicker the growth and decay. The curve goes through the point (1,O) and approaches asymptotically the y-axis, for a > 1 down, for a < 1 up, and again more quickly for larger values of I In ai.

2.6.3 Error Curve The function = (2.58) gives the error curve (Gauss error distribution curve) (Fig. 2.28). Since the function is even, the yaxis is the symmetry axis of the curve and the larger la1 is, the quicker it approaches asymptotically the z-axis. It takes its maximum at zero, and it is equal to one, so the extreme point A of the curve is

(

at (0, l ) ,the inflection points of the curve B , C are at f-

-

The angles of slopes of the tangent lines are here tan ~p = Fa&.

Y'L

A very important application of the error curve (2.58) is the description of the normal distributionproperties ofthe observational error (see 16.2.4.1,p. 756.): y = p(z) = -e

1

b

-1

0

(2.59)

202.

X

Figure 2.28

UJZ;;

2.6.4 Exponential Sum The function y = aebx + cedr

(2.60)

X

sign a=sign c sign b=sign d a)

X

sign a = sign c sign b sign d

+

b)

X

sign a 4 sign c sign b = sign d c)

sign a+ sign c sign b+ sign d d)

Figure 2.29 is represented in Fig. 2.29 for the characteristic sign relations. We get the result adding the ordinates of the curves, Le., the summands are y1 = aebz and y2 = cedz. The function is continuous. If none of the numbers a , b, c, d is equal to 0, the curve has one of the four forms represented in Fig. 2.29. Depending on the signs of the parameters it is possible that we have to reflect the graph in a coordinate axis. The intersection points A and B with the y-axis and with the x-axis are at y = a + c, and at z = 1

respectively, the extremum is at z = -In-

(-2)

(point C), and the inflection

2.6 Exponential Functions and Lo.qarithmic Functions 73

point is at z = -In d-b

( :::) --

(point D), in the case when they exist

Case a) The parameters a and c, and b and d have the same signs: The function does not change its sign, it is strictly monotone; its value is changing from 0 to +w or to -ffi or it is changing from +m or from -ffi to 0. There is no inflection point. The asymptote is the z-axis (Fig. 2.29a). Case b) The parameters a and c have the same sign, b and d have different signs: The function does not change its sign and either comes from t w and arrives at +w and has a minimum or comes from -ffi, goes to -w and has a maximum. There is no inflection point (Fig. 2.29b). Case c) The parameters a and c have different signs, b and d have the same signs: The function has one extremum and it is strictly monotone before and after. It changes its sign once. Its value changes whether from 0 until the extremum, then goes to $03 or -03 or it comes first from t f f i or -w, takes the extremum, then approaches 0. The z-axis is an asymptote, the extreme point of the curve is at C and the inflection point at D (Fig. 2 . 2 9 ~ ) . Cased) The parameters a and c and also b and d have different signs: The function is strictly monotone, its value rises from -m to +w or it falls from +w to -w. It has an inflection point D (Fig. 2.29d).

a) c > O

b) c < O Figure 2.30

2.6.5 Generalized Error Function The curve of the function (2.61)

= aebs+cs2 = ( a e - $ ) e 4 z + $ ) 2

can be considered as the generalization of the error function (2.58); it results in a symmetric curve with b respect to the vertical line x = --, it has no intersection point with the z-axis, and the intersection 2c point D with the y-axis is at (0, a) (Fig. 2.30a,b). The shape of the curve depends on the signs of a and e. Here we discuss only the case a > 0, because we get the curve for a < 0 by reflecting it in the z-axis. Case a) c > 0: The value of the function falls from +w until the minimum, and then rises again to +m. It is always positive. The extreme point

( 2”, (Fig.-”)

A of the curve is at -- , ae

to the minimum of the function; there is no inflection point or asymptote

Case b)

c

< 0:

4c

and it corresponds

2.30a).

The x-axis is the asymptote. The extreme point A of the curve is at

b (-%, ae-g)

and it corresponds to the maximum of the function. The inflection points B and C are at

I

2.6.6 Product of Power and Exponential Functions \We discuss the function y = axbeCx (2.62) only in the case a > 0, because in the case a < 0 we get the curve by reflecting it in the z-axis. For a non-integer b the function is defined only for x > 0, and for an integer b the shape of the curve for negative zcan be deduced also from the following cases (Fig. 2.31). Fig. 2.31 shows how the curve behaves for arbitrary parameters. For b > 0 the curve passes through the origin. The tangent line at this point for b > 1 is the z-axis, for b = 1 the line y = z.for 0 < b < 1 the y-axis. For b < 0 the y-axis is an asymptote. For c > 0 the function is increasing and exceeds any value, for c < 0 it tends asymptotically to 0. For different signs b of b and c the function has an extremum at x = -- (point A on the curve). The curve has either no or one or two inflection points at z = --

X

'A

(points C and D see Fig. 2.31c,e,f,g).

c>O,b=l

00, O
X

00, b
X

O ' c
2.7 Trigonometric Functions (Functions of Angles) 2.7.1 Basic Notion 2.7.1.1 Definition and Representation 1. Definition The trigonometric functions are introduced by geometric means. So in their definition and also in their arguments we use degree or radian measure (see 3.1.1.5, p. 130).

2. Sine The standard sane functzon y = sin x

(2.63)

2.7 Tkiqonometric Functions (Functions of Andes) 75

is a continuous curve with period T = 2a (see Fig. 2.32a)

Y4.

Figure 2.32

.

The intersection points Bo, B1, B-1: Bzl B-2,. . . with Bk = (kn,O)( k = 0, f l , 1 2 , . . .) of the standard sine curve and the x-axis are the inflection points of the curve. Here the angle of slope of the tangent line with the 1 axis is

1 4z . The extreme points of the curve are at CO,C1,

Cz, C-2:. . . with

Ck = ( ( k + f ) ~(-l)k) . ( k = 0, fl,f 2 , . . .). For every value of the function y there is -1 5 y 5 1. The general sine function (2.64) y = Asin(cjz po) with an amplitude IAl. frequency w ! and phase shift POis represented in Fig. 2.32b. Comparing the standard and the general sine curve (Fig. 2.32b) we see that in the general case the curve is stretched in the direction of y by a factor IAl, in the direction of z it is compressed by 1 PO 27T a factor -, and it is shifted to the left by a segment -. The period is T = - . The intersec-

+

Ld

d

tion points with the z-axis are Bk =

(" ~

", 0 ) (IC

bJ

= 0, *l, *2>. . .). Extreme values are Ck =

LL

3. Cosine The standard cosine function is represented in Fig. 2.33. The intersection points with the x-axis

B1,B2, , , , , Bk = ( ( k of the tangent line is

i) +

Figure 2.33 n,O) (IC = 11)1 2 . . . .) are also the inflection points. The angle of slope

4

The extreme points are CO,C1,.. Ck = ( k r . (-l)k) (k = O , H , 1 2 . . . .). The general eosane functaon y = il C O S ( d j Z po) can be transformed into the form .1

+

y = Asin

( ~t1 po

+ T) 2

~

i.e., the general sine function shifted left by p = 90".

(2.66a) (2.66b)

4. Tangent The tangent function y = tanx

(2.67) a

( k = 0, fl, 52,.. .) (Fig. 2.34). The

( k = 0, fl, 1 2 , . . .) and takes values from -cc to +ea. The curve has intersection points with the x-axis at Ao, AI, A-1, Az, A-2,. . ., Ab = (ka,0) (k = 0, f l , &2,. . .), these points are the inflection points and the angle of slope of the tangent line is T. 4

Figure 2.34

Figure 2.35

5 . Cotangent The cotangent function

(a : + - :)

y=cotx=-tan

(2.68)

has a graph which is the tangent curve reflected with respect to the z-axis and shifted to the left by T2 (Fig. 2.35). The asymptotes are x = k7r ( k = 0, kl,*2,. . .). Between 0 and a the function is monotone decreasing and takes its values from +ffi until -ea; the function has period T = T . The intersection points with the x-axis are at Ao, A I , A-1, Az, A - 2 , . . . with Ak =

((k + -:) a,O)

(k =

0 , *l> 1 2 , . . .), they are the inflection points of the curve and here the angle of the tangent line is

-E. 4

6. Secant The secant functzon 1 y = secx = cos x

:)

+ a ( k = 0, &l> 1 2 , . . .); and obviously IyI 2 1 ( The extreme points corresponding to the maxima of the function are Ao, Al, A-1,. . . with

has period T = 27r, the asymptotes are x = k holds.

(2.69) -

A k = ( ( 2 k + l)a,-1) ( k = 0, fl, 4 2 , . . .), the extreme points corresponding to the minima of the function are Bo, B1,B-l,. . . with Bk = (2ka, +1) ( k = O , H , & 2 > .. .) (Fig. 2.36).

7. Cosecant The cosecant function y = cosecx =

I

sin x

(2.70)

2.7 Tri.qonometric Functions (Functions of Angles) 77 has a graph which is the graph of the secant shifted to the right by z = I .The asymptotes are z = k7r ( k = 0, &I) f22.. .). The 2 extreme points corresponding to the maxima of the function are

(

Ao, A I , A I . . . with Ale = 4k; 37r,-1)

Y

( k = O , f l , k2,. . .)

and the the points corresponding to the minima of the function

Figure 2.36

Figure 2.37

['"2' / * I

,

1

Figure 2.38

\

's,+l) ( k = 0, f l , f 2 , . . .) (Fig. 2.37). are Bol B1: B-1,. . . with Bk = -

2.7.1.2 Range and Behavior of the Functions 1. Angle Domain 0 5 x 5 380' The six trigonometric functions are represented together in Fig. 2.38 in all the four quadrants for a complete domain of angles from 0" to 360" or for a complete domain of radians from 0 to 27r. In Table 2.1 there is a review of the domain and the range of these functions. The signs of the functions depend on the quadrant where the argument is taken from, and these are reviewed in Table 2.2. Table 2.1 Domain and range of trigonometric functions Domain -cc
Range

{

-1 -1

5 sinz 5 1 5 cosz 5 1

Domain

Range 7r

x # (2k + 1)1 x # k7r

( k = 0, f l , f 2 , . . .)

-cc < t a n z < cc -cc < c o t z < cc

Quadrant

I I1 I11 IV

Angle from On to 90n from 90n to 180" from 180" to 270" from270" to 360"

sin

cos

tan

cot

see

csc

+ +

+ -

+ +

+ + -

+ +

+ + -

-

t

-

-

Table 2.3 Values of trigonometric functions for On, 30°>45", 60" and 90".

Table 2.4 Reduction formulas and quadrant relations of trigonometric functions Function sin x cos x tan x cot x Argument 2 argument:

+ cos cy sin a

Ttana

sin a - cos a ftana *cot cy

- cos a & sin a 7 cot a 7 tan cy

1

- sin a

+ cos a -tan0 - cot cy

< 0: If the argument is negative, then we reduce the calculation of functions for positive

sin(-a) = - s i n a ,

(2.75)

cos( -CY) = cos a

(2.76)

tan(-a) = - tan a ,

(2.77)

cot(-a) = - c o t a .

(2.78)

A r g u m e n t 2 for 90' < 2 < 360': If 90" < z < 360" holds, then we reduce the arguments for an acute angle a by the reduction formulas given in Table 2.4. The relations between the values of the functions belonging to the arguments which differ from each other by 90°, 180" or 270" or which complete each other to go", 180° or 270" are called quadrant relatzons. The first and second columns of Table 2.4 give the complementary angle formulas, and the first and third ones give the supplementary angle fomulas. Because x = 90" - a is the complementary angle (see 3.1.1.2, p. 129) of a , we call the relations coscy = s i n s = sin(90" - c y )

complementary angle formulas.

, (2.79a)

sin a = cos x = cos(90" - a )

(2.79b)

2.7 Trigonometric Functions (Functions of Anqles) 79

+

For a z = 180" the relations between the trigonometric functions for supplementary angles (see 3.1.1.2. p. 129) s i n a = sinz = sin(l80O - a ) , (2.80a)

- coso

= cosz = ~os(l80"- a )

(2.80b)

are called supplementary angle formulas. Argument z for 0' < x < 90': We used to get the values of trigonometric functions for acute angles (0" < z < 90") from tables, but today we use calculators. Isin(-1000") = -sin 1000" = - sin(360" . 2 280") = - sin 280° = +cos loo = +0.9848.

+

4. Angles in Radian Measure The arguments given in radian measure, Le., in units of radians. can be easily converted by formula (3.2) (see 3.1.1.5. p. 130).

2.7.2 Important Formulas for Trigonometric Functions Remark: Trigonometric functions with complex argument z are discussed in 14.5.2, p. 697.

2.7.2.1 Relations Between the Trigonometric Functions of the Same Angle

(Addition Theorems) sin' a + cos' a = 1, see' a - tan'a = 1,

(2.82)

cosec' a - cot' a = 1,

(2.83)

sin a -= t a n a , cos a

(2.87)

(2.81)

sin a . cosec a = 1, cosa.seca = 1, tan a cot a = 1. cos cy

-= cot a. sin a

(2.84) (2.85) (2.86) (2.88)

Some important relations are summarized in Table 2.5 in order to create an easy survey. The square roots are always considered with the sign which corresponds to the quadrant where the argument is.

2.7.2.2 Trigonometric Functions of the Sum and Difference of Two Angles sin(a i3) = sin a cos ,3icos a s i n p,(2.89) tan(a

tana&tanO * 13) = 1rftancytanp'

(2.91)

cos(a i8)= cos a cos ,5 rf sin cy sin D,( 2.90)

cot(a f p) =

cotacotprf1 cotpicotcy '

sin(a + ,!? + y) = sin a c o s ~ c o ys + cosasin ,!?cosy t cos a cos p sin - sin a sin ,!? sin 7 , fi,

cos(a

(2.92)

(2.93)

+ ,!? + A/) = cos a cos p cos -/ - sin a sin B cos y -sin a cos psin y - cos a s i n 3 sin;!

(2.94)

2.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle sin 2a = 2 sin a cos a ,

(2.95)

cos 2a = cos' a - sin' a,

(2.97)

sin 3a = 3 sin a - 4sin3 a,

(2.96)

cos3a = 4 c o s 3 a - 3 c o s a 1

(2.98)

sin 4a = 8 cos3 a sin a

-

4 cos a sin a. (2.99)

cos4a = 8 cos4 a - 8 ~ 0 s a' + 1,

(2.100)

I

80

2. Functions

Table2.5 Relations between the trigonometricfunctionsof the same argument in the interval0

a

cos a

sin cy

tan a

L

cot a

tan a

sin a

Jm 1

cos a

JTTGZ

tan a

-

cot a

1 tan a

2tano 1 - tan2 a ! 3 t a n a - tan3 a tan3a = 1- 3tan2a ' 4 tan a - 4 tan3 a tanla = 1 - 6tanz a + tan4 a, tan2a =

~

c0Pa - 1 2cota ' cots, - 3cot a cot 3a = 3cot'a-1 ' cot4. - 6cot2a 1 cot4a = 4cot3a-4cotff .

(2.101)

cot2a =

(2.102)

~

+

(2.103)

(2.104) (2.105) (2.106)

For larger values of n in order to gain a formula for sin na and cos na we use the de Moivre formula.

(1) 2 (L)

Using the binomial coefficients cos na + i sin na =

(see 1.1.6.4, p. 12) we get:

ik c0sn-k a sink a = (cos a + i sin a)"

k 0

= COS" a

-

(i)

cos'-2 a sin* a - i

+ in

(E)

a sin a

COsn-3 a sin3 a +

(y)

COS"-4

a sin4 a

+ .. . ,

(2.107)

therefore, we get cos na = cos' a -

(y)

cos"-2 a sin2 a +

(2)

COSn-4

a sin4 a -

(:)

cos"-6 a sin6 a + . . . , (2.108)

sin na = n cos'-1 a sin a -

(:)

cos"-3 a sin3 a +

(:)

Cosn-5 a sin5 a - . . . .

(2.109)

2.7.2.4 Trigonometric Functions of Half-Angles In the following formulas the sign of the square root must be chosen positive or negative, according to the quadrant where the half-angle is.

2.7 TrigonometricFunctions (Functions of Angles) 81

sin 9 2 =

;

tan- = cot

cy = 2

d G , (2.110)

cos-2 =

/r -(l+coscr),

(2.111)

sina

1-cosa 1-cosa J 1 + cos a -

(2.112)

7 - 1 + cos a > sin a = -1 + cos a --

~

sina

~ - C O S ~

~ - C O S ~ '

2.7.2.5 Sum and Difference of Two Trigonometric Functions s i n a +sin@ = '2sin*cos*, 2 cos a

2

(2.114) s i n a - sinp = 2cos-sin-, a+p 2

+ cos 0= 2 cos 9cos CY-d,(2,116) 2

*

sin(a 0) cos (Y cos p

tana*tanp=-

tana+cotp=

2

2

sin(a ik p) sin a sin 4 '

+

cos(ff P) (2.120) cot a - t a n @= ___ sin a cos p '

cos a sin p '

2

a t p . a-P cos a - cos P = -2 sin -sin -,

(2.118) cot a f cot p = &-

cos(a - 8) ~

a-p

2

(2.115) (2.117) (2.119)

(2.121)

2.7.2.6 Products of Trigonometric Functions sinasin 3 = :[cos(a - p) - COS(^ 2

+ PI],

(2.122)

+ cos(a + P)], s i n a cos 13 = i[sin(a - p) + sin(a + PI], 2 cosacos 0 = : [ c o s ( a - P)

(2.123)

2

(2.124)

1 sinasinOsin-/ = - [ s i n ( a + P - y ) + s i n ( P + r - a ) 4 +sin(? a - p) - sin(a P r)],

(2.125)

1 . sinacospcosy = - [ s i n ( a + p - r ) - s i n ( p + y - a ) 4 +sin(y+a-O)+sin(a+p+y)],

(2.126)

1 sinasinflcosr = -[- cos(a p - y) cos(P 4 cos(y a - P) - cos(a

(2.127)

+ +

+

+

+

+

+ cosacospcos~/= A[cos(a+ p - y) + 4 + a - p) +COS(?

+y a) + a + y)],

COS(P+

+COS(@

-

y - a)

+ p + y)].

(2.128)

2.7.2.7 Powers of Trigonometric Functions sin2 cy = !(I2

cos2cy):

1 sin3cy= -(3sincy-sin3cy), 4

(2.129)

cos2 cy = !( 1 cos 2cy): 2

+

(2.130)

(2.131)

1 c0s3cy= -(cos3cy+3cosa),

(2.132)

4

1 1 sin4cy = -(cos4cy-4cos2cy+3), (2.133) cos4 cy = -(cos 4cy + 4 COS 2cy + 3). (2.134) 8 8 For large values of n we can express sin" cy and cosn cy from the formulas for cos no and sin ncy on page 80. ~

2.7.3 Description of Oscillations 2.7.3.1 Formulation of the Problem In engineering and physics we often meet quantities depending on time and given in the form (2.135) u = .4sin(ijt + p). We call also them sinusoidal quantities. Their dependence on time results in a hannonic oscillation. The graphical representation of (2.135) results in a general sine curve, as shown in Fig. 2.39.

Figure 2.40

Figure 2.39

The general sine curve differs from the simple sine curve y = sin 2: a) by the amplitude A, Le., the greatest distance between its points and the time axis t , b) by the period T =

-.277w

which corresponds to the wavelength (with w as the frequency of the oscilla-

tion, which is called the angular or radialfrequency in wave theory), c) by the initial phase or phase shift by the initial angle p # 0. The quantity u ( t ) can also be written in the form u = asinwt + bcoswt. (2.136) b Here for a and b we have .4 = and tan 'p = - . We can represent the amounts a, b, A and 'p as sides and angle of a right triangle (Fig. 2.40).

2.7.3.2 Superposition of Oscillations In the simplest case we call a superposition of oscillations the addition of two oscillations with the same frequency. It results again in a harmonic oscillation with the same frequency: (2.137a) AI sin(wt + p l ) + A2 sin(& 'p2) = A sin(& 'p) with A1 sin p1 A2 sin 'pz (2.137b) A= AzZ 2A1Azcos((pz - ql), t a n 9 =

+

AI' + +

+

+

AI COS ~1 + A2 COS ~2 '

2.7 Triqonometric Functions (Functions of Anqles) 83 where the quantities A and p can be determined by a vector diagram (Fig. 2.41a). .Us0 a lznear cornbznatzonof several sine functions with the same frequency again yields a general sine function (harmonic oscillation) with the same frequency:

~ ~sin(& ' 4 +~y t ) = Asin(wt + y ) .

Yf

a

Yt

(2.137~)

2.7.3.3 Vector Diagram for Oscillations Figure 2.41 The general sine function (2.135, 2.136) can be represented easily by the polar coordinates p = A, p and by the Cartesian coordinates z = a. y = b (see 3.52.1. p. 189) in a plane. The sum of two such quantities then behaves as the sum of two summand vectors (Fig. 2.41a). Similarly the sum of several vectors results in a linear combination of several general sine functions. iVe call this representation a vector diagram. The quantity u for a given time t can be determined from the vector diagram with the help of Fig. 2.41a: First we put the time axis O P ( t ) through the origin 0, which rotates clockwise around 0 by a constant angular velocity w. At start t = 0 the axes y and t coincide. Then at any time t the projection 0,'V of the vector Ci onto the time axis is equal to the absolute value of the general sine function u = A sin(wt + p). For time t = 0 the value ug = A sin p is the projection onto the y-axis (Fig. 2.41b).

2.7.3.4 Damping of Oscillations The function y = Ae-n' sin(wz + 90)

(a. z > 0)

E

(2.138) results in a curve of a damped oscillation (Fig. 2.42). The oscillation proceeds along the z-axis, while the curve asymptotically approaches the z-axis. The sine curve is enclosed by the exponential curves y = *Ae-a", and it contacts them in the points Al,Az, . . . ,Ak=

Figure 2.42 The intersection points with the coordinate axes are B = The extrema D1 Dz.. . . are at z = kn-po+2a

w

,

with t a n a =

k.ir

- po + a W

: and the inflection points El, E*, . . . are at

U

The logarzthmzc decrement of the damping is 6 = In of two consecutive extrema

z=

W -

where yz and yz+l are the ordinates

2.8 Inverse Trigonometric Functions We call the inverse of trigonometricfunctions also cyclometric functions. Because only a one-to-one function can have an inverse, we have to restrict the domain ofthe trigonometric functions to an interval such that the considered function should be a one-to-one function on it. There are infinitely many such intervals, and for each we can define an inverse. We will distinguish them by the index k according to the corresponding interval. Obviously the trigonometric functions are monotonic in these intervals.

I

Figure 2.43

Figure 2.44

Figure 2.45

Figure 2.46

2.8.1 Definition of the Inverse TrigonometricFunctions First we discuss the inverse of the sine function (Fig. 2.43). The usual notation for it is arc sine z,

+

< x 5 ka with 2 2 k = 0, H ,*2,. . . . Reflecting the curve of y = sinz in the line y = z we get the curves of the inverse functions (placed above each other) (2.139a) y = arck sin x with the domain and ranges

or arcsin I. The domain of y = sinz will split into monotonic intervals ka

-

and k a - T < y < k a + P , where k = 0 , % 1 , & 2> . . . . (2.139b) 2 2 The form y = arch sinz has the same meaning as z = siny. Similarly, we can get the other inverse trigonometric functions which are represented in Fig. 2.44-2.46. The domains and ranges of the inverse functions can be found in Table 2.6.

-1
2.8.2 Reduction to the Principal Value If we restrict the domain of the trigonometric functions to the intervals which belong to the index 0 in the definitions above, we call it the principal value of the function and in several textbooks a capital letter is used for the notation, e.g., Sinx. Similarly, the inverse trigonometric functions for = have the so-called

principal value, which is written without the index k, for instance arcsinx I arcosinx. In Fig. 2.47 the principal values of the inverse functions are presented. Remark: Calculators give the principal value of these inverses. We can get the values of different inverses from the principal value by the following formulas: arck sin x = ka + ( -l)k arcsin z. (2.140)

~

__________

AY

-....._.._._

arccot ...-...

----_------

' 9

______________.-_____ 2 Figure 2.47

2.8 Inverse Trigonometric Functions 85

{

(k odd), (k even).

arck cosz = ( k t 1). - arccosz k 7 + arccos z arch t a n 2 = ki7 arck cot z = k7r

(2.141)

+ arctanz. + arccot z.

(2.142) (2.143)

IA: arcsin 0 = 0 , arck sin 0 = k7r.

+ kn.

IB: arccot 1 = E , arck cot 1 = 4 1 7 7

4 1 7 r IC: arccos - = -, arck cos - = -- t ( k 2 3 2 -n 3

+ 1).

for odd k ,

= - i k ~ 3

Inverse function arc sine

y=arcksinz

arc cosine y = arch cos z

arc tangent y = arch tan x

arc cotangent y = arch cot x

] } } }

for even k .

Domain

Range

Trigonometric function with the same meaning

-15x51

k a - ?2<-y I k n + ? 2

x = siny

-1jxjl

knIy5(ktl)n

x = cos y

-m<x<m --x

< <

ks-?
2

x = tany

2

kn < y < ( k t 1)s

~

z = cot y

2.8.3 Relations Between the Principal Values - a r c c o s m (-1 5 x 5 o), - a r c c o s m (0 5 z 5 1).

arcsin z =

I - arccos z = arctan --

arccos z =

I - arcsin z = arccot

arctanz =

2

2

2

-

Ji-53- arccot z = arcsin 5

{

A - arcsin d i arcsin v ' F 2

-3

(0

< 0)

(2.145) (2.146)

m'

1 arccot - - 7 (z

1 5 z 5 oj, 5 z <_ 1).

(A -

(2.144)

-arCCOS-

1 (2

5 01,

arctanz =

(2.147) (z IT

> 0) = { arccos 1

d7-2

X

arccot z = - - arctan z = arccos -. 2

m

(z 2 0).

(2.148)

86

2. Functions

arccot z =

1 arctan - + x (z< 0)

K

1 - arcsin - ( x

m

(z> 0) = { arcsin 1 v G 2

5 01, (2.149)

(x 2 0).

2.8.4 Formulas for Negative Arguments arcsin(-z) = - arcsinz. arctan(-z) = -arctanz.

arccos(-x) = x arccot(-z) = x

(2.150) (2.151)

-

arccosz. arccotz.

(2.152) (2.153)

2.8.5 Sum and Difference of arcsin a and arcsin y

+

(zy 5 o or zz t yz 5 1) (2.154a) (-= x arcsin z 1 y2 + y m ) (z> 0, y > 0,z2 + y2 > 1). (2.154b) (-= -x - arcsin x 1 - y2 + y m ) (z< 0. y < 0,x 2 + y2 > 1). (2.154~) (-arcsin z - arcsin y = arcsin z 1 - y2 - y m ) (zy 2 0 oder zz + y2 5 1). (2.155a) (-= x arcsin z 1 - y2 )-y (z> 0,y < 0. x2 t y2 > 1). (2.155b) (-= -x - arcsin z 1 - y2 - y m ) (x < 0,y > 0,x2 + y2 > 1). (2.155~) (--

arcsin z arcsin y = arcsin z 1 - y2 t )-y -

~

-

-

-

2.8.6 Sum and Difference of arccos a and arccos y

( G)-

+

(zt y 2 0)

arccos z arccos y = arccos xy -

mm) (z + mm) (z2

t y < 0).

(2.156b)

y).

(2.157a)

= 277 - arccos (zy -

(

arccos z - arccos y = - arccos zy = arccos (zy t

~cT-)

(2.156a)

(z< y).

(2.157b)

2.8.7 Sum and Difference of arctan a and arctan y arctan z + arctan y = arctan ty 1 - z y (zy < l ) , =K

+ arctan z+y (z>O,zy>l), 1-zy

= -T

+ arctan zty 1 - xy

(zl).

2-Y arctanz - arctany = arctan (zy > -1). ltxy

(2.158a) (2.158b) (2.158~) (2.159a)

2.9 Hvperbolic Functions 87

= 7i t arctan (z > 0, xy

1+xy

=

-7

+ arctan x-y l+xy

< -l)>

(x
(2.159b) (2.159~)

2.8.8 Special Relations for arcsin z,arccos z,arctan z

(J-7 x ($? - < x 5 1) .

= K - arcsin 22 1

(2.160b) (2.160~)

2arccosx = arccos(22 - 1) (0 5 x 5 I), = 247 - arccos(2x2 - 1) (-1

22 2 arctanx = arctan 1- 2 2

1. (

5 x < 0).

< 111

(2.16l a ) (2.161b) (2.162a)

22 = 7i t arctan - (x > 11,

(2.162b)

22 = -T + arctan - (x < -1). 1- 2 2

(2.162~)

1-22

cos(narccosx) = T,(x)

(n 2 I ) ,

(2.163)

where n 2 1 can also be a fractional number and T,,(x) is given by the equation (2.164) For any integer n,T,(z) is a polynomial of x (a Chebyshev polynomial). To study the properties of the Chebyshev polynomials see 19.6.3, p. 920.

2.9 Hyperbolic Functions 2.9.1 Definition of Hyperbolic Functions Hyperbolzc szne. hyperbolzc coszne and hyperbolic tangent are defined by the following formulas: ex - e-z (2.165) sinhz = 2 ' ex + e-" coshx = 2 2 e" - e - x tanhx = -. ex + e-x

(2.166) (2.167)

The geometric definition in Chapter 3 Geometry (see 3.1.2.2, p. 131), is an analogy to the trigonometric functions.

88

2. Functions

Hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant are defined as reciprocal values of the above hyperbolic functions: 1 - ez+e-' (2.168) cothz = -- tanhz e" - e-" ' 1 2 (2.169) sechz = --coshz ez + e-= 1 2 (2,170) cosechz = sinhz = -. ez -e-" The shapes of curves of hyperbolic functions are shown in Fig. 2.48-2.52.

Ab.

-2-

Figure 2.48

lol 2

x

Figure 2.49

Figure 2.50

2.9.2 Graphical Representation of the Hyperbolic Functions 2.9.2.1 Hyperbolic Sine y = sinhz (2.165) is an odd strictly monotone increasing function between -03 and +m (Fig. 2.49). The origin is its symmetry center, the inflection point, and here the angle of slope of the tangent line is

p = I , There is no asymptote. 4

2.9.2.2 Hyperbolic Cosine y = coshz (2.166) is an even function, it is strictly monotone decreasing for x < 0 from +m to 1, and for z > 0 it is strictly monotone increasing from 1 until $03 (Fig. 2.50). The minimum is at z = 0 and it is equal to 1 (point A(0,l)); it has no asymptote. The curve is symmetric with respect to the X2

y-axis and it always stays above the curve of the quadratic parabola y = 1+ - (the broken-line curve). 2 Because the function demonstrate a catenary curve, we call the curve a catenoid (see 2.15.1, p. 105).

2.9.2.3 Hyperbolic Tangent y = tanhz (2.167) is an odd function, for --co < z < +m strictly monotone increasing from -1 to +1 (Fig. 2.51). The origin is the center of symmetry, and the inflection point, and here the angle of slope

2.9 Hyperbolic Functions 89

of the tangent line is 9 = !. The asymptotes are the lines y = f l . 4

4&b -3- 01 2 3 x __..__.____

Figure 2.51

Figure 2.52

2.9.2.4 Hyperbolic Cotangent y = cothx (2.168) is an odd function which is not continuous at z = 0 (Fig. 2.52). It is strictly monotone decreasing in the interval -cc < z < 0 and it takes its values from -1 until --x ; in the interval 0 < z < +m it is also strictly monotone decreasing with values from +co to +1 . It has no inflection point, no extreme value. The asymptotes are the lines z = 0 and y = f l .

2.9.3 Important Formulas for the Hyperbolic Functions We have similar relations between the hyperbolic functions as between trigonometric functions. We can show the validity of the following formulas directly from the definitions of hyperbolic functions, or if we consider the definitions and relations of these functions also for complex arguments, from (2.198)(2.205). we can calculate them from the formulas known for trigonometric functions.

2.9.3.1 Hyperbolic Functions of One Variable cosh' x - sinh2x = 1,

(2.171)

coth'z

+ tanh' z = 1,

(2.173)

tanhz .cothz = 1,

(2.174)

(2.175)

cosh z -- coth x. sinh 2

(2.176)

sech'z

sinh x -= tanhx, cosh x

- cosech'z

= 1,

(2.172)

2.9.3.2 Expressing a Hyperbolic Function by Another One with the Same Argument The corresponding formulas are collected in Table 2.7, so we can have a better survey.

2.9.3.3 Formulas for Negative Arguments sinh(-x) = -sinhz, tanh(-x) = - tanhx,

(2.177) (2.178)

cosh(-z) = coshz, coth(-z) = -cothz.

(2.179) (2.180)

2.9.3.4 Hyperbolic Functions of the Sum and Difference of Two Arguments (Addition Theorems) sinh(z f y) = sinh z cosh y f cosh 5 sinh y,

(2.181)

cosh(x * y ) = coshzcoshy*sinhzsinhy,

(2.182)

90

2. Functions

Table 2.7 Relations between two hyperbolic functions with the same arguments for x sinhx

1

coshx

sinh x

-

cosh x

&KZ

-

sinhx

dcosh' x - 1 coshx

tanhx

I

tanh x f tanh y 1 f tanhxtanhy

coth x

tanh x

1

J1-t..h2. & z G x

Jzzz \/sinh' x + 1 I cosh x coth x sinh x 1 Jcosh'x -1 tanh(x f y) =

tanh x

>0

'

1

coth x

JL-GE

dcoth' x - 1

-

1 coth x

1 tanh x

-

(2.183) coth(x 4Z y) =

1 f coth x coth y cothx f coth y '

(2.184)

2.9.3.5 Hyperbolic Functions of Double Arguments sinh 22 = 2 sinh x cosh x,

(2.185)

tanh2x = 1

cosh 2x = sinh' x + cosh' x,

(2'186)

~ 0 t 21 h =

2 tanh x tanh'x '

+

1 + coth' x 2cothx

(2.187) (2.188)

'

2.9.3.6 De Moivre Formula for Hyperbolic Functions (cosh x f sinh x ) =~ cosh nx f sinh nx.

(2.189)

2.9.3.7 Hyperbolic Functions of Half-Argument

{r

sinh 2 = i -(coshx

- 1),

(2.190)

coshz2 =

{G;

(2.191)

The sign of the square root in (2.190) is positive for x > 0 and negative for x < 0.

x

tanh- = 2

coshx - 1 - sinhx x sinhx - coshx + 1 (2.192) coth - = sinhx coshx+ 1 ' 2 coshx-1 sinhx ~

~

~

(2.193)

2.9.3.8 Sum and Difference of Hyperbolic Functions X f Y X F Y sinh x 4Z sinh y = 2 sinh cosh , 2 2

(2.194)

+ y cash 2-y , cash z t cash y = 2 cash x2 2

(2.195)

x t y sinh x-y , cosh x - cosh y = 2 sinh 2 2

(2.196)

2.10 Area Functions 91

*

tanh z tanh y =

sinh(z f y) cosh I cosh y

(2.197)

'

2.9.3.9 Relation Between Hyperbolic and Trigonometric Functions with Complex Arguments x sin z cos z tan z cot z

= -i sinh iz. = cosh iz, = -i tanh iz. = i coth iz,

(2.198) (2.199) (2.200) (2.201)

sinh z cosh z tanhz coth z

= -i sin iz, = cos iz, = -itaniz, = i cot iz.

(2.202) (2.204) (2.205)

Every relation between hyperbolic functions, which contains zor az but not az+b, can be derived from the corresponding trigonometric relation with the substitution i sinhz for sin cy and cosh z for cos cy. IA: cos2 a t sin' a = 1, cosh' z + iz sinh'z = 1 or cosh'z - sinh' I = 1. IB: sin 2cy = 2 sin cy cos cy, i sinh 2x = 2i sinhx cosh z or sinh 2x = 2 sinh zcoshz.

2.10 Area Functions 2.10.1 Definitions The area functzons are the inverse functions of the hyperbolic functions, i.e., the znwerse hyperbolzc functzons. The functions sinh z tanh z,and coth x are strictly monotone, so they have unique inverses without any restriction; the function cosh zhas two monotonic intervals so we can consider two inverse functions. The name area refers to the fact that the geometric definition of the functions is the area of certain hyperbolic sectors (see 3.1.2.2, p. 131). ~

2.10.1.1 Area Sine The function y = Arsinhz

(2.206)

(Fig. 2.53) is an odd, strictly monotone increasing function, with domain and range given in Table 2.8. It is equivalent to the expression z = sinh y . The origin is the center of symmetry and the inflection point of the curve, where the angle of slope of the tangent line is 'p = I . 4

2.10.1.2 Area Cosine The functions y = Arcosh z and y = - Arcosh z

(2.207)

(Fig. 2.54) or z = cosh y have the domain and range given in Table 2.8; they are defined only for z 2 1. The function curve starts at the point A(1,O) with a vertical tangent line and the function increases or decreases strictly monotonically respectively.

v4

Figure 2.53

Figure 2.54

92

2. Functions

Table 2.8 Domains and ranges of the area functions Area function

1

Domain

area sine y = Arsinh z -m 1

Figure 2.56

Range

Hyperbolic function with same meaning

-m
x = sinh y z = cosh y

-m < y <

co

--x < y < 0

x = tanhy z = cothy

o
Figure 2.56

2.10.1.3 Area Tangent The function y = Artanh z

(2.208) (Fig. 2.55) or z = tanh y is an odd function, defined only for 1x1 < 1, with domain and range given in Table 2.8. The origin is the center of symmetry and also the inflection point of the curve, and here the angle of slope of the tangent line is y = . The asymptotes are vertical, their equations are z = f l . 4

2.10.1.4 Area Cotangent The function y = Arcoth x

(2.209) (Fig. 2.56) or z = coth y is an odd function, defined only for 121 > 1, with domain and range given in Table 2.8. In the interval -m < z < -1 the function is strictly monotone decreasing from 0 until -m, in the interval 1 < x < +m it is strictly monotone decreasing from +m to 0. It has three asymptotes: their equations are y = 0 and x = rtl.

2.10.2 DeterminationofAreaF'unctions Using Natural Logarithm From the definition of hyperbolic functions ((2.165)-(2.170), see 2.9.1, p. 87) we can express the area functions with the logarithm function:

m) Arcoshz = In (z+ m) = In .4rsinhz = In (z+

(2.210)

~

1

-

(2.211)

2.11 Curves of Order Three (Cubic Curves) 93

1 1+z Artanhz = -In 2 1-2 ~

1.1(

1 z+1 Arcothz = -In - (1x1 > 1). 2 x-1

< l ) , (2.212)

(2.213)

2.10.3 Relations Between Different Area Functions Arsinhz = (sign z) Arcosh

= Artanh

rn, 2= Arcoth rn

(2.214)

2

I

= (sign x) ,4rtanh d7-3

Arcosh z = (sign z) Arsinh

X

m'

= (sign x ) Arcoth

(2.215) 1

.4rtanh z = (sign z) Arsinh

v C 2

= (signz) Arcoth X

1

= (signs) .4rcosh

1. (

~

(2.216)

< 1))

1 1 Arcoth z = Artanh - = (signz) .Arsinh -

dF-3

X

= (sign x ) Arcosh

~

Ix' d2-7

(1x1> 1).

(2.217)

2.10.4 Sum and Difference ofArea Functions .4rsinhx f Arsinh y = Arsinh

(2.218)

Arcoshz & .4rcosh y = Arcosh

(2.219)

XfY Artanh zf Artanh y = Artanh 1+xy'

(2.220)

2.10.5 Formulas for Negative Arguments .Arsinh(-z) = - Arsinhz,

(2.221)

.4rtanh(-z) = - Artanhz,

(2.222)

Arcoth(-z) = - Arcothz.

(2.223)

The functions Arsinh, Artanh and Arcoth are odd functions, and .4rcosh (2.211) is not defined for arguments z < 1.

2.11 Curves of Order Three (Cubic Curves) A curve is called an algebraic curve of order n if it can be written in the form of a polynomical equation F ( x ,y) = 0 of two variables where the left-hand side is a poynomial expression of degree n. The cardioid with equation (z2t y2)(z2t y2 - 2az) - a2y2= 0 (a > 0) (see 2.12.2,p. 96) is a curve of order four. The well-known conic sections (see 3.5.2.9,p. 205) result in curves of order two.

2.11.1 Semicubic Parabola The equation y = ax3''

( a > 0, z

2 0)

(2.224a)

94

2. Functions

or in parametric form r = t2, y = at3 ( a > 0,-cc < t < m) (2.224b) gives the semzcubzc parabola (Fig. 2.57). It has a cuspidal point at the origin, it has no asymptote. 6a The curvature K = 3,2 takes all the values between cc and 0. The arclength of the curve &(4 Sa r ) 1 between the origin and the point P(z.y) is L = -[(4 + 9 ~ ~ 2-) 81.~ ’ ~ 27a2

K;

+

Y

VI

Yt

0

Figure 2.57

Figure 2.58

Figure 2.59

2.11.2 WitchofAgnesi The equation

y=-

a3

a2

+

( a > 0, -cc

22

< z < co)

(2.225a)

determines the curve represented in Fig. 2.58, the witch of Agnesz. It has an asymptote with the equation y = 0, it has an extreme point at il(0, a ) , where the radiusof curvature is T = F. The inflection 2 3 4 points B and C are at i- - where the angles of slope of the tangent lines are tan 15 = F-. 8 The area of the region between the curve and its asymptote is equal to S = ra2. The witch of Agnesi (2.225,) is a special case of the Lorentz or Breit-TVigner curve

( k ): ~

~

a

( a > 0). (2.225b) b2 + (zIThe Fourier transform of the damped oscillation is the Lorentz or Breit-Wigner curve (see 15.3.1.4. p. 729). =

2.11.3 Cartesian Folium (Folium of Descartes) The equation z3 + y3 = 3ary ( a > 0) or in parametric form

3at

z= 1+t3’

3at2

y=i?

t =tan+POz

(2.226a) with

(a>O,-cc
and - l < t <M)

(2.226b)

gives the Carteszan folium curve represented in Fig. 2.59. The origin is a double point because the curve passes through it twice, and here both coordinate axes are tangent lines. .4t the origin the radius 3a

of curvature for both branches of the curve is r = - The equation of the asymptote is z+ y + a = 0. 2 ’ The vertex A has the coordinates A

3aZ

. The area of the loop is SI= -. The area S2between 2

2.11 Curves of Order Three (Cubic Curves) 95

the curve and the asymptote has the same value.

2.11.4 Cissoid The equation

13

y2 = - (a > 0), a-x at2

or in parametric form x = 1t t 2 y =

(2.227a) at3

iTF

with

a sin2 'p or with polar coordinates p = - ( a > 0) cos 9 (Fig. 2.60) describes the locus of the points P for which

(2.227~)

- _

OP = .VQ

(2.228)

is valid. Here M is the second intersection point of the line OP with the drawn circle of radius f,and 2 Q is the intersection point of the line OP with the asymptote x = a. The area between the curve and 3 the asymptote is equal to S = -xa2. 4

Figure 2.60

Figure 2.61

2.11.5 Strophoide Strophozde is the locus of the points Pi and Pz, which are on an arbitrary half-line starting a t A (-4 is on the negative x-axis) and for which the equalities

MP1=MP2=OM (2.229) is the intersection point with the y-axis (Fig.2.61). The equation of the strophoide are valid. Here ~\il in Cartesian. and in polar coordinates, and in parametric form is:

t2 - 1 y = atwith t = tan + P o x (a > 0, -m < t < 30). (2.230~) t2+1' t2 1 The origin is a double point with tangent lines y = f x . The asymptote has the equation I = a. The 1 vertex is A(-a. 0). The area of the loop is 5'1 = 2a2 - -xa2, and the area between the curve and the 2

x=a-

t2 - 1

+

96

2. Functions

1 asymptote is Sz = 2a2 t -Ta2. 2

2.12 Curvesof Order Four (Quartics) 2.12.1 Conchoid ofNicomedes The Conchozd ofNzcomedes (Fig. 2.62) is the locus of the points P , for which OP=OMf1 (2.231) holds, where M is the intersection point of the line between OPl and OP2 with the asymptote x = a. The sign belongs to the outer branch of the curve, the ‘ I - ” sign belongs to the inner one. The equations for the conchoid of Nicomedes are the following in Cartesian coordinates, in parametric form and in polar coordinates: (x - a)’(.’ yz) - L2x2 = o (a > 0,1 > 0 ) , (2.232a)

“+”

+

x = a + Icosp,

y = atancp + Isin’p (a > 0, right branch: -

p = 5c! E cos ‘p

(“+” sign: right branch,

“-”

3T T2 < ’p < I2 ,left branch: T2 < p < -), 2

sign: left branch,)

(2.232b) (2.232~)

1. Right Branch: The asymptote is z = a. The vertex A is at (a t 1,0),the inflection points B , C have as x-coordinate the greatest root of the equation z3- 3a2x t 2a(a2 - E’) = 0. The area between the right branch and the asymptote is S = m . 2. Left Branch: The asymptote is z = a . The vertex D is at (a - E , 0 ) . The origin is a singular point, whose type depends on a and 1: Case a) For 1 < a it is an isolated point (Fig. 2.62a). The curve has two further inflection points E and F , whose abscissa is the second greatest root of the equation x3 - 3a2x + 2a(az - L2) = 0. Case b) For 1 > a the origin is a double point (Fig. 2.62b). The curve has a maximum and a minimum

*-.

value at z = a - @.At the origin the slopes of the tangent lines are tan LY = ___ Here the radius of curvature is

=

1dF2 ~

2a Case c) For 1 = a the origin is a cuspidal point (Fig. 2 . 6 2 ~ ) .

2.12.2 General Conchoid The conchoad of Nzcomedes is a special case of the general conchoid. We get the conchoid of a given curve if we elongate the length of the position vector of every point by a given constant segment fl. If we consider a curve in a polar coordinate system with an equation p = f(’p), the equation of its conchoid is (2.233) P = f(9)+ 1. So, the conchoid of Nicomedes is the conchoid of the line.

2.12.3 Pascal’s Limaqon The conchozd of a czrcle is called the Pascal l i m a ~ o n(Fig. 2.63) if in (2.231) the origin is on the perimeter of the circle. which is a further special case of the general conchoid (see 2.12.2, p. 96). The equations in the Cartesian and in the polar coordinate systems and in parametric form are the following (see also (2.245c), p. 103): (2+ y2 - ax)’ = ?(x2 t y2) (a > 0, 1 > 0 ) , (2.234a)

2.12 Curves of Order Four (Quartics) 97

Figure 2.62

1?2a

aelc2a

a)

b) Figure 2.63

p=acosp+1

(2.234b)

(a>0, 1>0),

+

+

5 = a cos2 p 1 cos cp, y = acos psin p 1 sin p ( a > 0, 1 > 0, 0 5 cp < Z A ) (2.234~) with a as the diameter of the circle. The vertices A , E are at ( a & 1,O). The shape of the curve depends on the quantities a and 1, as we can see in Fig. 2.63 and 2.64. a) Extreme Points and Inflection Points: For a > 1 the curve has four extreme points C, D , E , F ;

for a 5 1 it has two; they are at

4a

. For a < 1 < 2a there exist two inflection

b) Double Tangent: For I < 2a, at the points I and K at

4a

tangent. c) Singular Points: The origin is a singular point: For a < 1 it is an isolated point, for a

> 1 it is a

double point and the slopes of the tangent lines are tancu = f is T O =

F

3here the radius of curvature

A mFor'. a = 1 the origin is a cuspidal point; then we call the curve a cardzoid (see also 2

2.13.3, p. 101). pa2 The area of the limacon is S = 7 d2,where in the case a > 1 (Fig. 2 . 6 3 ~the ) area of the inside L

+

loop is counted twice.

2.12.4 Cardioid The cardzozd (Fig. 2.64) can be defined in two different ways, as: 1. Special case of the Pascal l i m a p n with _ _ OP = O M a, (2.235) where a is the diameter of the circle. 2. Special case of the epzcyclozd with the same diameter a for the fixed and for the moving circle (see 2.13.3, p. 101). The equation is

*

+

a=l Figure 2.64

(2.236a) (z2 y2)2- 2az(z2 + yz) = a2y2 ( a > o), and the parametric form, and the equation in polar coordinates are: z= acosp(l+cosp), y = a s i n p ( l + c o s p ) (2.236b) (a > 0,o 5 $5 < 2 9 ) , (2.236~) p = a(1 cos p) ( a > 0).

+

1 The origin is a cuspidal point. The vertex A is at (2a, 0);extreme points C and D are at cos p = - with 2 3 4 3 . area is S = -paz, Le., six times the area of a circle with diameter a. coordinates (:a, & ~ a ) The 2 The length of the curve is L = 8a.

2.12.5 Cassinian Curve The locus of the points P , for which the product of the distances from two fixed points FI and F2 with coordinates ( c ,0) and (-c, 0) resp., is equal to a constant a' # 0, is called a Cassinzan curve (Fig. 2.65):

_ _

F I P . F2P = a'. The equations in Cartesian and polar coordinates are: ( 2+ yz)' - 2c2(z2- y2) = a4 - c4, ( a > 0, c > o), pz = cz cos 29 f 4 ~ cos2 4 2p

+ (a4 - c4)

(2.237) (2.238a)

(a > 0,c > 0).

(2.238b)

The shape of the curve depends on the quantities a and c : Case a > c f i : For a > c f i the curve is an oval whose shape resembles an ellipse (Fig. 2.65a). The intersection points A, C with the z-axis are ( f 0)) the intersection points B, D with the y-axis are (0.*-). ~

Case a = c f i : For a = c\/z the curve is of the same type with A , C ( ?C c d , 0) and B , D (0, *e)$ where the curvature at the points B and D is equal to 0. i.e., there is a narrow contact with the lines y =fc.

2.12 Curves of Order Four (Quartics)

99

Figure 2.65 Case c < a < c 4 : For c < a < c f i the curve is a pressed oval (Fig. 2.65b). The intersection points with the axes are the same as in the case a > c d , also the extreme points B. D, while there are c4 - 4 further extreme points E , G, K , I at ( k q , &") and there are four inflection points P,L , 2c

Case a = c: For a = c we have the lemniscate. Case a < c: For a < c there are two ovals (Fig. 2.65~).The intersection points A , C and P, Q with the x-axis are at ( 0) and ( k 0). The extreme points E , G, K , I are

* m,

(

m,

.

2a2p3

5 ; ) . The radius of curvature is r = where p satisfies the polar 2c c4 - a4 + 3p4 coordinate represent at ion.

at &- -,

2.12.6 Lemniscate The lemniscate (Fig. 2.66) is the special case a = c of the Cassinian curve satisfying the condition

-_

Fl Fz

(2.239)

where the fixed points F l , F' are at (*al 0). The equation in Cartesian coordinates is (x' + y2)* - 2a2(x2- y2) = 0 (a > 0) (2.240a)

____----

and in polar coordinates p = a J m

X

(a > 0).

(2.240b)

The origin is a double point and an inflection point at the same time, where the tangents are y = kz.

Figure 2.66

The intersection points A and C with the x-axis are at ( f a & 0) , the extreme points of the curve E ,

(

G, K , I are at i-

,

*-4) . The polar angle at these points is

'p =

& ? . The radius of curvature is 6

100

T

2. Functions

2a2 = - and the area of every loop is S = a' 3P

2.13 Cycloids 2.13.1 Common (Standard) Cycloid The cycloid is a curve which is described by a point of the perimeter of a circle while the circle rolls along a line without sliding (Fig.2.67). The equation of the usual cyclozd written in parametric form is the following: (2.241a) x = a(t - sint), y = a(1 -cost) with (a > 0, -m < t < m), where a is the radius of the circle and t is the angle 3: PClB in radian measure. In Cartesian coordinates we have: (a>o). (2.24113) x+\lyo=aarccosg The curve is periodic with period

\

/

wl = 27ra. At 0,01, 02,. . ., ok = ( a h a , 0) we have cusps, the ver-

\

/

' I b

' I

1

Figure 2.67

tices are at Ak = ((2k t l)aa, 2a). The arclength of OP is L = 8asin2(t/4), the length of one arch is L O ~ = l 8a. ~ lThe area of one arch is S = 37ra2. The radius of curvature is T = 4a sin i t , at the vertices rA = 4a. The evolute of a cycloid (see 3.6.1.6, p. 236) is a congruent cycloid, which is denoted in Fig. 2.67 by the broken line.

2.13.2 Prolate and Curtate Cycloids or Trochoids Prolate and curtate cycloids or trochoids are curves described by a point, which is inside or outside of a circle, fixed on a half-line starting from the center of the circle, while the circle rolls along a line without sliding. (Fig. 2.68). The equation of the trochoid in parametric form is (2.242a) 5 = a ( t - Xsint), (2.24213) y = a(l - XCOS~),

h
Figure 2.68

where a is the radius of the circle,t is the angle 3: PClM, and Xa = C1P. In the case X > 1 we have the prolate cycloid and for X < 1 the curtate one. The period of the curve is 001 = 27ra, themaximumpointsareat AI, Ai,.. ., Ak = ((2k t l)m,(1 +.\)a), the minimum points are at Bo, B1, B z , . . ., Bk = (2kaa, (1 - x)a).

2.13 Cvcloids 101

The prolate cycloid has double points at DO, D1,Dz, . . ., Dk = 2 k ~ aa 1 -

(

is the smallest positive root of the equation t = X sin t. The curtate cycloid has inflection points at

E l , Ez,

. . ., E,, =

~

X2 , where (7)

(

a (arccos X - A

1 dl

to2

m

to

) . a (1 - A’)

271

We calculate the length of one cycle by the integral L = a in Fig. 2.68 is S = 7ra2(2+ A”). For the radius of curvature we have rA =

-a-

(1

+

x

X)Z

T

=a

(1

t X2 - 2X cos t dt. The shaded area

+ X2 - 2x cos t ) 3 / 2 . which has at the maxima the value

X(c0s t - A) (1 - X)2 and at the minima the value T B = a-

A

’

2.13.3 Epicycloid A curve is called an epzcyclozd, if it is described by a point of the perimeter of a circle while this circle rolls along the outside of another circle without sliding (Fig. 2.69). The equation of the epicycloid in parametric form is A+a z = (A + a ) coscp - acos -p, y = ( A t a) s i n 9 - asin (--cx) < ‘p < co),(2.243)

Ata’p

where A is the radius of the fixed circle, a is the radius of the rolling one, and cp is the angle 3: COX.The

A

shape of the curve depends on the quotient m = a Form = 1 we get the cardzozd.

Figure 2.69

Case m integer: For an integer m the curve consists of m identically shaped branches surrounding the 2kT fixed curve (Fig. 2.69a). The cusps Ai, Az, . . . , A, are at p = A, cp = - (k = 0,1, . . . , m - 1) m the vertices B1,Bz,. . . , B, are at

m

Case m a rational fraction: If m is a non-integer rational number, the identically shaped branches follow each other around the fixed circle, overlapping the previous ones, until the moving point P returns back to the starting-point after a finite number of circuits (Fig. 2.69b).

I

Figure 2.70

Case m an irrational: For an irrational m the number of round trips is infinite; the point P never returns to the starting-point. The length of one branch is L A ~ B=~8A( A~ 'I. For an integer m the total length of the closed curve m = 8(A a ) . The area of the sector AIBIAzAl (without the sector of the fixed circle) is S = 4a( A+a) , Ay 4a(A a ) TU' The radius of curvature is r = sin -, at the vertices T B = 2a+A 2a 2a+A '

+

is L,,,,I

(y).

+

~

~

2.13.4 Hypocycloid and Astroid A curve is called a hypocycloid if it is described by a point of the perimeter of a circle, while this circle rolls along the inside of another circle without sliding (Fig. 2.70). The equation of the hypocycloid, the coordinates of the vertices, the cusps, the formulas for the arclength, the area, and the radius of curvature are similar to the corresponding formulas for the epicycloid, only we have to replace "+a'! by - a .. . The number of cusps for integer, rational or irrational m is the same as for the epicycloid (now m > 1 holds). Case m = 2: For m = 2 the curve is actually the diameter of the fixed circle. Case m = 3: Form = 3 the hypocycloid has three branches (Fig. 2.70a) with the equation: .t

+

z = a(2 cos y cos 2y), y = a(2 sin p - sin 29). (2.244a) We have: L,,,,I = 16a, S,,,,l = 27ra'. Case m = 4: For m = 4 (Fig. 2.70b) the hypocycloid has four branches, and it is called an astroid (or asteroid). Its equation in Cartesian coordinates and in parametric form is: 3?13

+ yZi3=

(2.244b)

IC

= Acos3p

y = A s h 3 'p (0 5 y

< T).

(2.244~)

3

We have: L,,,,, = 24a = 6A, S,,,,l = -7~4'. 8

2.13.5 Prolate and Curtate Epicycloid and Hypocycloid The prolate and curtate epzcycloid and the prolate and curtate hypocyclozd, which are also called the epztrochozdand hypotrochozd, are curves (Fig. 2.71 and Fig. 2.72) described by a point, which is inside or outside of a circle, fixed on a half-line starting at the center of the circle. while the circle rolls around the outside (epitrochoid) or the inside (hypotrochoid) of another circle. without sliding. The equation of the epitrochoid in parametric form is 1=

( A + a ) COS 9 - Xa COS ( Arap),

y=(A+a)sinp-Xasin

(' ~

0

'9) ,

(2.2458)

2.14 Spirals 103

Figure 2 71

Figure 2.72 where A is the radius of the fixed circle an& is the radius of the rolling one. For the hypocycloid we have to replace "+a" by "-a". For Xu = C P one of the inequalities X > 1 or X < 1 is valid, depending on whether we have the prolate or the curtate curve. For .4 = 2a. and for arbitrary X # 1 the hypocycloid with equation z = a ( l + X)coscp, y = a(1 - X)sincp (0 5 'p < 27~) (2.245b) describes an ellipse with semi-axes a(1 + A) and a(1 - A). For A = a we have the Pascal limacon (see also 2.12.3, p. 96) z = a ( 2 c o s p - X C O S ~ ~ ) , y=a(2sinq-Xssin2p) (2.245~)

Remark: For the Pascal limacon on 2.12 3, p. 96 the quantity denoted by a there is denoted by 2Xa here, and the 1 there is the diameter 2a here. Furthermore the coordinate system is different.

2.14 Spirals 2.14.1 Archimedean Spiral An Archtmedean spzralis a curve (Fig.2.73)described by a point which is moving with constant speed 2' on a ray. while this ray rotates around the origin at a constant angular velocity w . The equation of the Archimedean spiral in polar coordinates is p=ap,

ja>~,-x
a=? dJ

(2.246)

Figure 2.73

Figure 2.74

The curve has two branches in a symmetric position with respect to the y-axis. Every ray OK intersects the curve at the points 0, A I , A2,.. . ,A,,. . . ; their distance is A,A,+1 = 2na. The arclength OP is L = (y=+ Arsinhcp where for large 9 the expression 2L tends to 1. The area of the h

4

~

w2 sector P10P2 is a rg = 2'

S=

+

-(pZ3 - cp13). The radius of curvature is r = -a (92 1)3/2 and at the origin 6 $92 t 2 a2

2.14.2 Hyperbolic Spiral The equation of the hyperbolic spiral in polar coordinates is

,=aw

(a>O, -m
(2.247)

The curve of the hyperbolic spiral (Fig. 2.74) has two branches in a symmetric position with respect to the y-axis. The line y = a is the asymptote for both branches, and the origin is an asymptotic a2

point. The area of the sector Plopzis S = -

(-

i2),

a'

- - and ~ $ I S = - is valid. The radius of 2cpl

',

curvature is r =

2.14.3 Logarithmic Spiral The logarzthmzc spzralis a curve (Fig. 2.75) which intersects all the rays starting at the origin 0 at the same angle a. The equation of the logarithmic spiral in polar coordinates is p = aekp ( a > 0,--CO < 9 < m) (2.248) h

where k = c o t 0

The origin is the asymptotic point of the curve. The arclength PIPz is L =

d m ? ( p 2 - p l ) , the limit of the arclength OP calculated from the origin is Lo = mp. h

k radius of curvature is r = m

k

The

p = Lok.

Special case of a circle: For (Y = I we have k = 0, and the curve is a circle 2

2.14.4 Evolvent ofthe Circle

--

The euolvent of the c z d e is a curve (Fig. 2.76) which is described by the endpoint of a string while we roll it off a circle. and we always keep it tight, so that AB= BP. The equation of the evolvent ofthe

2.15 Various Other Curves 105 circle is in parametric form x = acos 9 + alpsinp! y = asin yo - aipcosip, (2.249) where a is the radius of the circle, and ip = 3: Box. The curve has two branches in symmetric position with respect to the x-axis. It has a cusp at A(a,0), and the intersection points with the x-axis are at 1 I=- a , where po are the roots of the equation tan ip = ip. The arclength of AP is L = -up2. The cos On 2 ," radius of curvature is r = ap = the centre of curvature B is on the circle, i.e., the circle is the evolute of the curve.

-

m;

Figure 2.75

Figure 2.76

Figure 2.77

2.14.5 Clothoid The clothoidis a curve (Fig. 2.77)such that at every point the radius of curvature is inversely proportional to the arclength between the origin and the considered point:

'a r=-

(2.250a)

(a>O).

The equation of the clothoid in parametric form is h

with t = L,s = OP.

(2.250b)

a&

The integrals cannot be expressed in terms of elementary functions; but for any given value of the parameter t = t o ,t l , .. . it is possible to calculate them by numerical integration (see 19.3, p. 895), so we can draw the clothoid pointwise. About calculations with a computer see the literature. The curve is centrosymmetric with respect to the origin, which is also the inflection point. At the inflection point the x-axis is the tangent line. At A and B the curve has asymptotic points with coordinates

(+$ .+e) and (-? , -*). The clothoid is applied, for instance in road construction, 2 2

where the transition between a line and a circular arc is made by a clothoid segment.

2.15 Various Other Curves 2.15.1 Catenary Curve The catenary curve is a curve wich has the shape of a homogeneous, flexible but inextensible heavy chain hung at both ends (Fig. 2.78) represented by a continuous line. The equation of the catenanJ

I

106

2. Functions

curve is

x

exla

+ e-x/Q

y = acosh - = a ( a > 0). (2.251) a 2 The parameter a determines the vertex A at (0, a). The curve is symmetric to the y-axis, and is always higher than the parabola y = a

22 +, which is represented by the broken line in Fig. 2.78. 2a

h

arclength of A P is L = a sinh - = a a a L = a2 sinh

- e-x/a

2 Y2 a

The

. The area of the region OAPM has the value S = L2 a

. The radius of curvature is T = - = acosh2 = a + - . a

Figure 2.78

Figure 2.79

The catenary curve is the evolute of the tractrix (see 3.6.1.6, p. 236), so the tractrix is the evolvent (see 3.6.1.6,p. 237) of the catenary curve with vertex A at ( 0 , a ) .

2.15.2 Tractrix The tractriz (the thick line in Fig. 2.79) is a curve such that the length of the segment PM of the tangent line between the point of contact P and the intersection point with a given straight line: here the x-axis, is a constant a. If we fasten one end of an inextensible string of length a to a material point P , and we drag the other end along a straight line, here the x-axis, then P draws a tractrix. The equation of the tractrix is

x = a Arcosh

aY

e

=a h

Y

(a>o).

(2.262)

The x-axis is the asymptote. The point A at (0,a) is a cusp. The curve is symmetric with respect to h

to the value a(1 - In2)

a.

For increasing arclength L the difference L - x tends Y 0.307a, where x is the abscissa of the point P . The radius of curvature is

the y-axis. The arclength of A P is L = a In

r = a cot ! . The radius of curvature

and the segment PE = b are inversely proportional: T b = u 2 . Y The evolute (see 3.6.1.6, p. 236) of the tractrix, i.e., the geometric locus of the centers of circles of curvature C. is the catenary curve (2.251), represented by the dotted line in Fig. 2.79.

2.16 Determination of Empirical Curves 2.16.1 Procedure 2.16.1.1 Curve-Shape Comparison If we have only empirical data for a function y = f(x),we can get an approximate formula in two steps. First we choose a formula for an approximation which contains free parameters. Then we calculate the

1.16 Determination of Empirical Curves 107

values of the parameters. If we do not have any theoretical description for the type of formula, then we first choose the approximate formula which is the simplest among the possible functions, comparing their curves with the curve of empirical data. Estimation of similiarity by eye can be deceptive. Therefore, after the choice of an approximate formula, and before the determination of the parameters, we have to check whether it is appropriate.

2.16.1.2

Rectification

Supposing there is a definite relation between 2 and y we determine, for the chosen approximate formula, two functions X = p(x,y) and Y = $(z, y) such that a linear relation of the form Y=AX+B (2.253) holds. where A and B are constant. If we calculate the corresponding X and Y values for the given I and y values, and we consider their graphical representation, it is easy to check if they are approximately on a straight line, or not. Then we can decide whether the chosen formula is appropriate. then we can substitute X = x , Y = !. and we get axtb’ Y 1 1 Y = aX t b. Another possible substitution is X = - Y = - . Then we get Y = a + b X . X Y IB: See semilogarithmic paper, 2.17.2.1, p. 115. IC: See double logarithmic paper, 2.17.2.2, p. 115 In order to decide whether empirical data satisfy a linear relation Y = AX + B or not, we can use linear regression or correlation (see 16.3.4, p. 777). The reduction of a functional relationship to a linear relation is called rectification. Examples of rectification of some formulas are given in 2.16.2. p. 107, and for an example discussed in detail, see in 2.16, p. 112.

I A: If the approximate formula is y

=

~

2.16.1.3 Determination of Parameters The most important and most accurate method of determining the parameters is the least squares method (see 16.3.4.2, p. 779). In several cases, however, even simpler methods can be used with success, for instance the mean value method.

1. Mean Value Method In the mean value method we use the linear dependence of “rectified variables X and Y , Le., Y = AX + B as follows: We divide the conditional equations I: = AXi B for the given values k;, X , into two groups, which have the same size, or approximately the same size. By adding the equations in the groups we get two equations, from which we can determine A and B . Then replacing X and Y by the original variables x and y again, we get the connection between 2 and y, which is what we were looking for. If we have not determined all the parcmeters, we have to apply the mean value method again with a rectification by other amounts X and Y (see for instance 2.16.2, 2.16.2.11,p. 112). Rectification and the mean value method are used above all when certain parameters occur in non-linear relations in an approximate formula, as for instance in (2.266b), (2.266~).

+

2. Least Squares Method When certain parameters occur in non-linear relations in the approximation formula, the least squares method usually leads to a non-linearfitting problem. Their solution needs a lot of numerical calculation and also a good initial approximation. These approximations can be determined by the rectification and mean value method.

2.16.2 Useful Empirical Formulas In this paragraph we discuss some of the simplest cases of empirical functional dependence, and we also represent the corresponding graphs. Each figure shows several curves corresponding to different parameter values involved in the formula. We discuss the influence of the parameters upon the forms of the curves.

I

For the choice of the appropriate function, we usually consider only part of the corresponding graph, which is used for the reproduction of the empirical data. Therefore, e.g., we should not think that the formula y = ax2 + bx + c is suitable only in the case when the empirical data have a maximum or minimum.

2.16.2.1 Power Functions 1. Type y = mb: Typical shapes of curve for different values of the exponent b y = axb (2.254a) are shown in Fig. 2.80. The curves for different values of the exponent are also represented in Figs. 2.15, 2.21, 2.24, 2.25 and Fig. 2.26. The functions are discussed on pages 64, 68 and 70 for the formula (2.44) as a parabola of order n, formula (2.45) as a reczprocal proportzonalzty and formula (2.50) as a reczprocal pozuerfunctzon. The rectification is made by taking the logarithm X = logx, Y = logy: Y = l o g a + bX. (2.254b)

+

2. Type y = axb c: The formula (2.255a) y = axb + c produces the same curve as in (2.254a), but it is shifted by c in the direction of y (Fig. 2.82). If b is given. we use the rectification: x=xb, Y = Y : Y = a X + c . (2.25513) If b is not known, first we determine c then we rectify X = logx, Y = log(y -c): Y = logs+ b X . (2.255~) In order to determine c. we choose two arbitrary abscissae x 1 , and ~ a third one, 2 3 = J .l.2,and the corresponding ordinates y1, y2, y3>and we assume c =

'lYz

Yl

- y32 . After we have determined a and

+ Y2 - 2Y3

b, we can correct the value of c: it can be chosen as the average of the amounts y - axb.

Figure 2.80

Figure 2.81

2.16.2.2 Exponential Functions 1. Type y = aebr: The characteristic shapes of the curves of the function y = aebx (2.256a) are shown in Fig. 2.81. The discussion of the exponentialfunction (2.54) and its graph (Fig. 2.26) is presented in 2.6.1 on p. 71. We rectify X = x. Y = logy: Y = loga + bloge .X. (2.256b)

2.16 Determination of Empirical Curves 109

x 0'

x U'

Figure 2.82

Figure 2.83

Figure 2.84

+

2. Type 9 = aebx c: The formula y = aeb" + c (2.257a) produce the same curve as (2.256a), but it is shifted by c in the direction of y (Fig. 2.83). First we determine c then we rectify by logarithm: (2.257b) Y = log(y - c), X = x : Y = log a + bloge . X . In order to determine c we choose two arbitrary abscissae x l , x2 and 23 = x 1 xz and the corresponding 2 ~

ordinates yI1yz. y3>and we assume c =

'lyZ

Y1

- 'Y32

+ Yz - 2Y3

+

. After the determination of a and b we can correct

c: it can be chosen as the average of the amounts y - aebs.

2.16.2.3 Quadratic Pofynomial Possible shapes of curves of the qzladratzc polynomzal y = ax2 + bx + c (2.258a) are shown in Fig. 2.84. For the discussion of quadratic polynomials (2.41) and their curves (Fig. 2.12) see 2.3.2, p. 62. Usually we determine the coefficients a, b and c by the least squares method; but also in this case we can rectify. Choosing an arbitrary point of data (x1,y1) we rectify

Y=Y=(b+axl)+aX. x - 21 If the given z values form an arithmetical sequence with a difference h, we rectify Y = Ay, X = x : Y = (bh + a h 2 ) i2ahX. In both cases we get c from the equation x=x.

cy = u c z '

+ b c x + ne

(2.258b) (2.258,) (2.258d)

after the determination of a and b. where n is the number of the given x values, for which the sum is calculated.

2.16.2.4 Rational Linear Function The ratzonal ltnearfunctzon ax + b ?/=cx

+d

(2.259a)

110

2. Functions

is discussed in (2.4) with (2.46) and graphical representation Fig. 2.17 (see p. 65) Choosing an arbitrary data point (xl,yl) we rectify x - x1

, X=x: Y=A+BX. (2.259b) Y - Y1 After determining the values A and B we write the relation in the form (2.259~).Sometimes the suggested forms are as (2.259d): y=-

y=y1+-

x - XI (2.259~) A+Bx'

y=-

X

cx + d

or u=-

1

cx

+d

(2.259d)

1 1 Then in the first case we rectify X = - and Y = - or X = x and Y = and in the second case X = x 5 Y Y 1

and Y =

Y

2.16.2.5 Square Root of a Quadratic Polynomial Several possible shapes of curves of the equation yz = ax2 + bx + c (2.260) are shown in Fig. 2.85.The discussion of the function (2.52) and its graph (Fig. 2.23) is on p. 69. If we introduce the new variable Y = y2, the problem is reduced to the case of the quadratic polynomial in 2.16.2.3, p. 109

Figure 2.85

yB

0

Figure 2.86

2.16.2.6 General Error Curve

The typical shapes of curves of the functions = aebx+cx2 or logy = loga + bz loge + ex2 loge (2.261) are shown in Fig. 2.86. The discussion of the function with equation (2.61) and its graph (Fig. 2.31) is on p. 74. If we introduce the new variable Y = logy, the problem is reduced to the case of the quadratic polynomial in 2.16.2.3. p. 109.

2.16.2.7 Curve of Order Three, Type I1 The possible shapes of graphs of the function 1

= ax2

+ bx + c'

(2.262)

are represented in Fig. 2.87. The discussion of the function with equation (2.48) and with graphs (Fig. 2.19)is on p. 66.

2.16 Determination ofEmpirica2 Curves 111

If we introduce the new variable Y = in 2.16.2.3. p. 109.

1 -)

Y

the problem is reduced to the case of the quadratic polynomial

Figure 2.87

Figure 2.88

2.16.2.8 Curve of Order Three, Type I11 Typical shapes of curves of functions of the type X

(2.263) ax2 + bx + c are represented in Fig. 2.88. The discussion of the function with equation (2.49) and with graphs (Fig. 2.20) is on p. 67. =

Introducing the new variable Y = 2.16.2.3. p. 109.

Y

we reduce the problem to the case of the quadratic polynomial in

2.16.2.9 Curve of Order Three, Type I Typical shapes of curves of functions of the type b c y=at-t2

(2.264)

X2

are represented in Fig. 2.89. The discussion of the function with equation (2.47) and with graphs (Fig. 2.18) is on p. 65. 1 Introducing the new variable X = - we reduce the problem to the case of the quadratic polynomial in 2.16.2.3, p, 109.

0'

X

Figure 2.89

I Figure 2.90

112 2. Functions

2.16.2.10 Product of Power and Exponential Functions Typical shapes of curves of functions of the type y = axbecx (2.265a) are represented in Fig. 2.90. The discussion of the function with equation (2.62) and with graphs (Fig. 2.31)is on p. 74. If the empirical values of z form an arithmetical sequence with difference h, we rectify Y = Alogy, X = Alogx: Y = hcloge+ bX. (2.265b) Here A logy and A log z denote the difference of two subsequent values of logy and log x respectively. If the z values form a geometric sequence with quotient q, then we rectify X = z, Y = A logy: Y = blogq c(q - 1)Xloge. (2.265~) After b and c are determined we take the logarithm of the given equation, and calculate the value of loga like in (2.258d). If the given z values do not form a geometric sequence, but we can choose pairs of two values of x such that their quotient q is the same constant, then the rectification is the same as in the case of a geometric sequence of z values with the substitution Y = A1 logy. Here A, logy denotes the difference of the two values of logy whose corresponding z values result in the constant quotient q (see 2.16.2.12, p. 112).

+

2.16.2.11 Exponential Sum Typical shapes of curves of the exponential sum y = aebx+ cedx (2.266a) are represented in Fig. 2.91. The discussion of the function with equation (2.60) and with graphs (Fig. 2.29)is on p. 72. If the values of z form an arithmetical sequence with difference h, and y, y1,yz are any three consecutive values of the given function, then we rectify

zY '

y = (ebb + edh)X -ebb. ,dh, Y .4fter we have determined b and d by this equation, we rectify again: X = e('-+ : y = ye-'', y = a 7 + c. y =

X = &:

(2.266b)

(2.266~)

2.16.2.12 Numerical Example Find an empirical formula to describe the relation between x and y iftheir values are given in Table 2.9. Table 2.9 For the approximate determination of an empirically given function relation -

Y -

0.7 0.8 0.9 1.0 1.1

1.78 3.18 3.19 2.54 1.77 1.14 0.69 0.40 0.23 0.13 0.07 0.04

X -

A Igx

Y 0.056 0.007 0.063 0.031 0.094 0,063 0.157 0.125 0.282 0.244 0.526 0.488 1.014 0.986 1.913 2.000 3.78 3.913 7.69 8.02 15.71 14.29 30.0

-1.000 -0.699 -0.523 -0.398 -0.301 -0.222 -0.155 -0.097 -0.046 0.000 0.041 0.079

0.250 0.502 0.504 0.405 0.248 0.057 -0.161 -0.398 -0.638 -0.886 -1.155 -1.398

- - -- -

0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 0.041 0.038

-

Yerr

0.252 f0.002 -0.099 -0.157 -0.191 -0.218 -0.237 -0.240 -0.248 -0.269 -0.243

-

0.252 -0.097 -0,447 -0.803 -1.134 -1.455

-

1.78 3.15 3.16 2.52 1.76 1.14 0.70 0.41 0.23 0.13 0.07 0.04

-

2.16 Determination of Empirical Curves 113

Y! 3.02.5 2.0

-

1.5.

1.00.5

-

Figure 2.91

Figure 2.93

Figure 2.92

Figure 2.94

Choice ofthe Approximation Function: Comparing the graph prepared from the given data (Fig. 2.92) with the curves discussed before, we see that formulas (2.263) or (2.265a) with curves in Fig. 2.88 and Fig. 2.90 can fit our case. Determination of Parameters: Using the formula (2.263) we have to rectify A T and z.The cal-

Y

culation shows, however, the relationship between z and A: is far from linear. To verify whether the Y formula (2.265a) is suitable we plot the graph of the relation between A logz and A logy for h = 0 , l in Fig. 2.93, and also between A, logy and z for p = 2 in Fig. 2.94. In both cases the points fit a straight line well enough, so the formula y = azbeczcan be used. In order to determine the constants a, b and c, we seek a linear relation between z and A1 logy by the method of mean values. Adding the conditional equations A, logy = blOg2 + cz loge in groups of three equations each, we obtain -0.292 = 0.903b t 0.2606~, -3.392 = 0.903b + 0.6514~, and we get b = 1.966 and c = -7.932. To determine a, we add the equations of the form logy = loga + blogz + cloge . z, which yields -2.670 = l2loga - 6.529 - 26.87, so from loga = 2.561, a = 364 follows. The values of y calculated from the formula y = 364z'1966e-703" are given in the last column of Table 2.9; they are denoted by yerr as an approximation of y. The error sum of squares is 0.0024. If we use the parameters determined by this rectification as initial values for the iterative solution of the non-linear least squares problem (see 19.6.2.4, p. 919) Ciz,[y, - az~eC'~]* = min!, we get a = 396.601 986, b = 1.998098, c = -8.000 0916 with the very small error sum of squares 0.000 0916.

2.17 Scales and Graph Paper 2.17.1 Scales The base of a scale is a function y = f(x). We construct a scale from this function so that on a curve, for instance on a line, we measure the substitution values of y as an arclength. but we mark them as the values of the argument x. We can consider a scale as a one-dimensional representation of a table of values. The scale equatzon for the function y = f(x) is: (2.267) Y = l[f(.) - f ( X 0 ) l . We fix the starting point xo of the scale. We choose the scale factor 2, because for a concrete scale we have only one given scale length. IA Logarithmic Scale: For l = 10 ern and xo = 1 the scale equation is y = lO[lgz - lg 11 = 1Olgx (in cm). For the table of values 111 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 110 x y = lgx 1 0 I 0.30 I 0.48 1 0.60 I 0.70 1 0.78 1 0.88 1 0.90 I 0.95 1 1

we get the scale shown in Fig. 2.95. 1 2

3

4

5

I

I

I

I

6 I

78910

I l l 1

Figure 2.95 IB Slide Rule: The most important application of the logarithmic scale, from a historical viewpoint, was the slzde rule. Here, for instance, multiplication and division were performed with the help of two identically calibrated logarithmic scales, which can be shifted along each other. From Fig. 2.96 we can read: y3 = y1 y2, Le., lgX3 = lgxl + Igx2 = Igxlx2, hence x3 = rl . x2; y1 = 23 23 y3 - y2. i.e., lgxl = Igr3 - 1gx2 = 1g-, so z1 = - .

+

52

x2

D

Figure 2.96 Figure 2.97 IC Volume Scale: We mark by a scale the lateral surface of a conical shaped jar, a filler. so that the volume could be read from it. The data of the filler are: Height H = 15 em. diameter D = 10 cm. 1 3

With the help of Fig. 2.97a we get the scale equation as follows: Volume I;= -r2.rrh, apothem s =

m,t a n 0 = hr

-

scale equation is s =

012 H

1 3

= - = - . From these h = 3T,

V Q -m fi

= r f i , I/ = -follows, so the

z 2 . 1 6 m . With the help of the following table of values we get the

calibration of the filler as in the figure:

2.17 Scales and Graph Paper 115

V 10 1 50 s

1

100

1

150

I

200

I

250

1

1

300

350

10 1 7.96 1 10.03 I 11.48 I 12.63 I 13.61 I 14.46 I 15.22

2.17.2 Graph Paper The most useful graph paper is prepared so that the axes of a right-angle coordinate system are calibrated by the scale equations 2 = h[g(u) - 9(uo)11 Y = lz[f(v) - f(vo)l. Here l1 and 12 are the scale factors; uo and uo are the initial points of the scale. (2.268)

2.17.2.1 Semilogarithmic Paper

I

If the x-axis has an equidistant subdivision, and the y-axis has a logarithmic one, then we talk about semzlogarzthmzc paper or about a semzlogarzthmic coordznate system. Scale Equations: 2 = I l [ u - uo] (linear scale), y = lZ[lgv - lgvo] (logarithmic scale). (2.269) The Fig. 2.98 shows an example of semilogarithmic paper. Representation of Exponential Functions: On semilogarithmic paper the graph of the exponential function y = aeBs ( 0 . 3constj (2.270a) IS a straight line (see rectification in 2.16.2,2.16.2.2, p. 108). We can use this property in the following way: If the measuring points, introduced on semilogarithmic paper, lie approximately on a line, we can suppose a relation between the variables as in (2.270a). With this line, estimated by eye, we can determine the approximate values of a and p: Considering two points Pl(z1,y1) and P~(x2. y ~ from ) this line we get 8 = In YZ - In YI and, e.g., a = yleDZ1 (2.270b) x2 - 51

'r

3w tc.103

190 40

30 20

45

10 4

3 2

1

0

5

10

15 20

25

X

Figure 2.98

'0

10

I

I

I

20

30

40

1

+

50 t

Figure 2.99

2.17.2.2 Double Logarithmic Paper If both axes of a right-angle x,y coordinate system are calibrated with respect to the logarithm function, then we talk about double logarithmic paper or log-log paper or a double logarithmic coordinate system. Scale Equations: The scale equations are (2.271) 2 = l1[lgu - lguo], y = l2[lgv - lgvo],

116

2. Functions

where 11, l2 are the scale factors and UO,2r0 are the initial points. Representation of Power Functions (see 2.5.3, p. 70): Log-log paper has a similar arrangement to semilogarithmic paper, but the x-axis also has a logarithmic subdivision. In this coordinate system the graph of the power function y = axo (ai, p const) (2.272) is a straight line (see rectification of a power function in 2.16.2, 2.16.2.1,p. 108). This property can be used in the same way as in the case of semilogarithmic paper.

2.17.2.3 Graph Paper with a Reciprocal Scale The subdivisions of the scales on the coordinate axes follow from (2.45) for the function of inverse proportionality (see 2.4.1, p. 64). Scale Equations: We have (2.273) where 11 and

12

are the scale factors, and U O ,vo are the starting points.

1 Concentration in a Chemical Reaction: For a chemical reaction we denote the concentration by c = c ( t ) >where t denotes time, and measuring c, we have the following results: t/min 1 5 1 10 I 20 I 40 C ' 103/mol/l 1 15.53 I 11.26 I 7.27 1 4.25

We suppose that we have a reaction of second order, i.e., the relation should be c(t) = co (co, k const). 1 + cokt

(2.274)

1 1 Taking the reciprocal value of both sides, we get - = - + k t , i.e., (2.274) can be represented as a line, if c

co

the corresponding graph paper has a reciprocal subdivision on the y-axis and a linear one on the x-axis. 1 The scale equation for the y-axis is, e.g., y = 10. - cm. V

It is obvious from the corresponding Fig. 2.99 that the measuring points lie approximately on a line, Le.. the supposed relation (2.274) is acceptable. From these points we can determine the approximate values of both parameters k (reaction rate) and co (initial concentration). We choose two points, e.g., P1(lO,lO) and Pz(30,5), and we get:

k=

l/Cl

t2

- l/cz - tl

0.005,

x 2 0 . ~ .

2.17.2.4 Remark There are several other possibilities for constructing and using graph paper. Although today in most cases we have high-capacity computers to analyse empirical data and measurement results, in everyday laboratory practice, when we have only a few data, graph paper is used quite often to show the functional relations and approximate parameter values needed as initial data for applied numerical methods (see the non-linear least squares method in 19.6.2.4, p. 919).

2.18 Functions of Several Variables 117

2.18 Functions of Several Variables 2.18.1 Definition and Representation 2.18.1.1 Representation of Functions of Several Variables .A variable value u is called a function of n independent variableszl, z2,. . . J,,iffor given values of the independent variables, u is a uniquely defined value. Depending on how many variable we have, two, three, or n,we write 21 = f ( z , y , z ) , 1 ' 1 = f(z1,zz,.. . ,%). 21 = f b , Y ) , If we substitute given numbers for the n independent variables, we get a value system of the variables, (2.275) which can be considered as a point of n-dimensional space. The single independent variables are called arguments; sometimes the entire n-tuple together is called the argument of the function.

Examples of Values of Functions: IA: u = f(z,y) = zy2 has for the value system z = 2, y = 3 the value f ( 2 , 3 ) = 2 3' = 18. IB: u = f ( z . y, z , t ) = z In(y - z t ) takes for the value system z = 3, y = 4, z = 3, t = 1 the value f ( 3 , 4 : 3: 1) = 3 ln(4 - 3 . 1) = 0.

2.18.1.2 Geometric Representation of Functions of Several Variables 1. Representation of the Value System of the Variables The value system of an argument of two variables z and y can be represented as a point of the plane given by Cartesian coordinates x and y. A value system of three variables x , y, z corresponds to a point given by the coordinates x , y, z in a three-dimensional Cartesian coordinate system. Systems of four or more coordinates cannot be represented obviously in our three-dimensional imagination. Similarly to the three-dimensional case we consider the system of n variables 21%2 2 , . . , .zn as a point of the n-dimensional space given by Cartesian coordinates 51, ~ 2 , ... ,zn.In the above example B, the four variables define a point in four-dimensional space, with coordinates z = 3, y = 4, z = 3 and t = 1.

2. Representation of the Function u = f ( z , g ) of Two Variables

f)yQy+ X

Figure 2.100

a) h function of two independent variables can be represented by a surface in three-dimensional space, similarly to the graph representation of functions of one variable (Fig. 2.100, see also 3.6.3, p. 243). If we consider the values of the independent variables of the domain as the first two coordinates, and the value of the function u = f ( z , y) as the third coordinate of a point in a Cartesian coordinate system, these points form a surface in three-dimensional space.

Examples of Surfaces of Functions: X Y W A: u = 1 - -represents a plane (Fig. 2.101a, see also 3.5.3.4,p. 214). 2 3

y2

W B: u = - + - represents an elliptic paraboloid (Fig. 2.101b, see also 3.5.3.5,5.; p. 222) 2 4 22

IC: u = 416 - z2 - yz represents a hemisphere with r = 4 (Fig. 2 . 1 0 1 ~ ) . b) The shape of the surface of the function u = f ( x , y) can be pictured with the help of intersection curves, which we get by intersecting the surface parallel to th coordinate planes. The intersection curves u = const are called level curves or nzveau lznes. In Fig. 2.101b,c the level curves are concentric circles (not denoted in the figure) Remark: A function with an argument of three or more variables cannot be represented in threedimensional space. Similarly to surfaces in three-dimensional space we introduce the notion of a hypersurface in n-dimensional space.

I

Figure 2.101

2.18.2 Different Domains in the Plane 2.18.2.1 Domain of a Function The domain of definition ofafunction (or domain of a function) is the set of the system ofvaluesor points which can be taken by the variables of the argument of the function. The domains defined this way can be very different. Mostly they are bounded or unbounded connected sets of points. Depending on whether the boundary belongs to the domain or not, the domain is closed or open. An open, connected set of points is called a domain. If the boundary belongs to the domain, we call it a closed domain, if it does not, sometimes we call it an open domain.

2.18.2.2 Two-Dimensional Domains Fig. 2.102 shows the simplest cases of connected sets of points of two variables and their notation. Domains are represented here as the shaded part; closed domains, Le., domains whose boundary belongs to them, are bounded by thick curves in the figures: open domains are bounded by dotted curves. Including the entire plane there are only simply connected domains or simply connected regions in Fig. 2.102.

Figure 2.102

2.18.2.3 Three or Multidimensional Domains These are handled similarly to the two-dimensional case. It concerns also the distinction between simply and multiply connected domains. Functions of more than three variables will be geometrically represented in the corresponding n-dimensional space.

2.18 Functions of Several Variables 119

2.18.2.4 Methods to Determine a h n c t i o n 1. Definition by Table of Values Functions of several variables can be defined by a table of values. .An example of functions of two independent variables are the tables of values of elliptic integrals (see 21.7. p. 1055). The values of the independent variables are denoted on the top and on the left-hand side of the table. The required substitution value of the function is in the intersection of the corresponding row and column. We call it a table with double entry. 2. Definition by Formulas Functions of several variables can be defined by one or more formulas. z t y forz>_O,y>O. Ic: 21= z - y forz>O.y
I

+

Figure 2.103

Examples for Domains: IA: u = x 2 t y2: The domain is the entire plane. 1 : The domain consists of all value systems z, y 3 satisfying the inequality I B : u = J16 - x2 - yz z2 + y2 < 16. Geometrically this domain is the interior of the circle in Fig. 2.103a. an open domain. IC: u = arcsin(x + y): The domain consists of all value systems z,y ) satisfying the inequality -1 5 z + y 5 +1. i.e.. the domain of the function is a closed domain, the stripe between the two parallel lines in Fig. 2.103b. ID: u = arcsin(2z - 1) + + f i + lnz: The domain consists of the value system 2. y, z , satisfying the inequalities 0 5 z 5 1, 0 5 y 5 1, z > 0. Le., it consists of the points lying above a square with side-length 1 shown in Fig. 2.103~.

d ed

Figure 2.104

2. Functions

120

ed

Figure 2.105 If from the interior of the considered part of the plane a point or a bounded, simply connected point set is missing, as shown in Fig. 2.104, we call it a doubly-connected domain or doubly-connected region. Multiply connected domains are represented in Fig. 2.105. A non-connected region is shown in Fig. 2.106.

2.18.2.5 Various Ways to Define a Function Functions of several variables can be defined in different ways, just as functions of one variable.

Figure 2.106

1. Explicit Representation A function is given or defined in an explicit way if its value (the dependent variable) can be expressed by the independent variables: u = f(z1,z2... . ,z,)

.

(2.276)

2. Implicit Representation A function is given or defined in an implicit way if the relation between its value and the independent variables is given in the form: (2.277)

F(zltz2,. , , ,zn,u) = 0, if there is a unique value of u satisfying this equality.

3. Parametric Representation A function is given in parametric form if the n arguments and the function are defined by n new variables, the parameters, in an explicit way, supposing there is a one-to-one correspondence between the parameters and the arguments. For a two-variable function, for instance

z = P(T,8) , Y = $J(Ts ) , u = x(r, s)

(2.278a)

and for a three-variable function 5

=dr,s,t),

Y = Nrls,t), z = x(r,s,t),

u = tc(r,s,t)

(2.278b)

etc.

4. Homogeneous Functions A function f(q, x2... . 2,) of several variables is called a homogeneous function if the relation ~

f(Azl,Axz,...,Azn) =Xrnf(21,z2,...,zn) holds for arbitrary A. The number m is the degree ofhornogeneity. IA: For u(x,y) = z2 - 32y

+ y2 + z

d +f, zy

- the degree of homogeneity is m = 2

(2.279)

2.18 hnctions of Several variables 121

g 9% 8.f

" " , short

o(f 'P)

D(u v)

(2,282a)

2or ?-.

D ( z ,!/I

D ( z ,Y) '

is not identically zero here. Analogously, in the case of n functions of n variables u1 = f l ( x l , . . . ,xn),. . . , un = f n ( X 1 ,

ax a y

-

o(f18fi,

I

3

fn)

+

D(xlrx21...,xn)

,(2,282b)

,

.

, $

xn):

If the number m of the functions u1, u2, . . . , u, is smaller than the number of variables 51, 2 2 , . . . , x,, these functions are independent if at least one subdeterminant of order m of the matrix (2.282~) is not identically zero:

The number of independent functions is equal to the rank T of the matrix (2.282~)(see 4.1.4, 7., p. 255). Here these functions are indepen(2.282~) dent, whose derivatives are the elements of the non-vanishing determinant of order r. If m > n holds, then among the given m functions at most n can be independent.

122

2. Functions

2.18.3 Limits 2.18.3.1 Definition .4 function of two variables u = f(z,y ) has a limit A at z = a, y = b if when z and y are arbitrarily close to a and b, respectively, then the value of the function f(z, y) approaches arbitrarily closely the value A . Then we write: i& f(z,Y) = A .

(2.283)

y-b

The function may not be defined at (a,b), or if it is defined here, may not have the value il.

2.18.3.2 Exact Definition A function of two variables u = f(x,y) has a limit A =

& f(z,y) w

if for

b

arbitrary positive E there is a positive 9 such that (Fig. 2.107) If(z,Y) - AI < E holds for every point (2, y) of the square

(2.284a) (2.284b)

Figure 2.107

2.18.3.3 Generalization for Several Variables a) The notion of limit of a function of several variables can be defined analogously to the case of two variables. b) We get criteria for the existence of a limit of a function of several variables by generalization of the criterion for functions of one variable, Le., by reducing to the limit of a sequence just as in the Cauchy condition for convergence (see 2.1.4.3, p. 52).

2.18.3.4 Iterated Limit If for a function of two variables f(z,y) we determine first the limit for z -+ a, Le., liliaf(z, y) for constant y, then for the function we obtain, which is now a function only of y, we determine the limit for y -+ b, the resulting number (2.285a) is called an iterated limit. Changing the order of calculations we get the limit

c = x+a lim (iimj(z, y i b y)) . In general B

(2.28513)

# C holds, even if both limits exist.

IFor the function f(zly) =

22

- y2

+ 1 3 + y3 for z + 0, y + 0 we get the iterated limits B = -1

t yz and C = +l. If the function f(z,y) has a limit A = & f(z:y), and both B and C exist, then B = C = A is valid. 52

u i b

The existence of B and C does not follow from the existence of A . From the equality of the limits B = C the existence of the limit A does not follow.

2.18.4 Continuity A function of two variables f(z,y) is continuous at 2 = a , y = b , Le., at the point (a, b), if 1. the point (a,b) belongs to the domain of the function and 2. the limit for z + a, y + b exists and is equal to the substitution value, Le., i&f(., Y) = f ( a , b). u-b

(2.286)

2.19 Nomography 123

Otherwise the function has a discontinuity at x = a, y = b. If a function is defined and continuous at every point of a connected domain, it is called continuous on this domain. We can define the continuity of functions of more than two variables similarly.

2.18.5 Properties of Continuous Functions 2.18.5.1 Theorem on Zeros ofBolzano If a function f(z,y) is defined and continuous in a connected domain, and at two points (zl,yl) and ( 5 2 . yz) of this domain the substitution values have different signs, then there exists at least one point ( 2 3 ; y3) in this domain such that f(z, y) is equal to zero there: f(231Y3) = 0, if f(Z1:Yl) > 0 and f(Z2,YZ) < 0 . (2.287)

2.18.5.2 Intermediate Value Theorem If a function f ( z ,y) is defined and continuous in a connected domain, and at two points ( z l 3 yl) and ( 2 2 ,yz) it has different substitution values A = f ( q , y1) and B = f ( ~yz), , then for an arbitrary value C between .4 and B there is at least one point ( 2 3 ) y3) such that: f(23.y3)=C, A < C < B or B < C < A . (2.288)

2.18.5.3 Theorem About the Boundedness of a Function If a function f ( z , y ) is continuous on a bounded and closed domain, it is bounded in this domain, Le., there are two numbers rn and M such that for every point (z, y) in this domain: (2.289) m I f(z,Y) 5 244.

2.18.5.4 Weierstrass Theorem (About the Existence of Maximum and

Minimum) If a function f ( z , y) is continuous on a bounded and closed domain, then it takes its maximum and minimum here, Le.. there is at least one point ( E ’ , y’) such that all the values f(z, y) in this domain are less than or equal to the value f(z’, y’)> and there is at least one point (z”,y”) such that all the values f(z,y) in this domain are greater than or equal to f(z”,y”): For any point (2,y ) of this domain (2.290) fk’, 34 2 f(.% Y) 2 fb”, Y”) is valid.

2.19 Nomography 2.19.1 Nomograms Nomograms are graphical representations of a functional correspondence between three or more variables. From the nomogram! the corresponding values of the variables of a given formula - the key formula - in a given domain of the variables can be immediately read directly. Important examples of nomograms are net charts and alignment charts. Nomograms are still used in laboratories, even in the computer age, for instance to get approximate values or starting guesses for iterations.

2.19.2 N e t Charts If we want t o represent a correspondence between the variables given by the equation

m,

(2.291) Y,2) = 0 (or in many cases explicitly by z = f(z,y)). then the variables can be considered as coordinates in space. The equation (2.291) defines a surface which can be visualized on two-dimensional paper by its level curves (see 2.18.1.2, p. 117). Here, a family of curves is assigned to each variable. These curves form a net: The variables 5 and y are represented by lines parallel to the axis, the variable z is represented by the family of level curves.

I

124 2. Functions

IOhm's law is U = R I . The voltage U can be represented by its level curves depending on two variables. If R and I are chosen as Cartesian coordinates, then the equation U = const for every constant corresponds to a hyperbola (Fig. 2.108). By looking at the figure one can tell the corresponding value of U for every pair of values R and I , and also I corresponding to every R, U ,and also R corresponding to every I and U . Of course, we always have to restrict our investigation to the domain which is interpreted: In Fig. 2.108 we have 0 < R < 10, 0 < I < 10 and 0 < U < 100. 3

f

TT

0 1 2 3 4 5 6 7 8 910 R-

2

1

~

Figure 2.108

3

R-

4 5 678910

Figure 2.109

Remarks: 1. By changing the calibration, the nomogram can be used for other domains. If we need the domain, for instance in (Fig. 2.108)the domain 0 < I < 1 but Rshould remain the same, then the hyperbolas of U are marked by U/lO. 2. By application of scales (see 2.17.1, p. 114) it is possible to transform nomograms with complicated curves into straight-line nomograms. Using uniform scales on the x and y axis, every equation of the form (2.292) .cp(Z) + Y N Z ) + x ( z ) = 0 can be represented by a nomogram consisting of straight lines. If function scales x = f ( z Z )and y = g(z2) are used, then the equation of the form (2.293) f(ZZ)LP(ZI) + g(zz)i(z1) + X(Z1) = 0 has a representation for the variables 21,252 and 23 as two families of curves parallel to the axis and an arbitrary family of straight lines. IBy applying a logarithmic scale (see 2.17.1, p. 114), Ohm's law can be represented by a straight-line nomogram. Taking the logarithm of R . I = U gives logR log I = log U . Substituting z = logR and y = log I results in x y = logU , Le., a special form of (2.293). The corresponding nomogram is shown in Fig. 2.109. ~

+

+

2.19.3 Alignment Charts A graphical representation of a relation between three variables z l r z 2 and z3 can be given by assigning a scale (see 2.17.1, p. 114) to each variable. The z, scale has the equation z,= ~ o z ( ~ z ) .=~ z&(zz) (i = 1,2,3).

(2.294)

2.19 Nomo.qraphu 125 The functions p, and $, are chosen in such a manner that the values of the three variables zl, z2 and 23 satisfying the nomogram equation should lie on a straight line. To satisfy this condition, the area of the triangle, given by the points (xI,yl), (x2,yz) and ( 2 3 , y3) ,must be zero (see (3.297) on p. 193), Le., (2.295)

PI(&) m191(4 1 ~ ~ ( 2 m292(22) 2 ) 1

~

3

=0,

(2.296)

m4 3 d 4 1

(

m-m3 Pl(4

+-m3-ml P2(4

ml-mz

+-=O

@(a)

(2.297a)

or (2.297b) Here C1. Cz and C3 are constants. 1 1 2. IThe equation - + - = - is a special case of (2.297b) and it is an important relation, for instance in a b f optics or for the parallel connection of resistances. The corresponding alignment nomogram consists of three uniformly scaled lines.

2.19.3.2 Alignment Charts with Two Parallel and One Inclined

Straight-Line Scales

One of the scales is put on the y-axis, the other one on another line parallel to it at a distance d. The third scale is put on a line y = mx.In this case (2.295) has the form (2.298)

(2.299a) -d - (O&) - md) = 0 oder f(zl) g(z3) - h ( z z ) = 0. P3(4 It is often useful to introduce measure scales El and E2 of the form ~ ( 2 1 ) ~

Ez

ElS(ZI)Fg(Z3) -

E2h(ZZ)= 0.

(2.299b)

(2.2994

126

2. Fzlnctions

Then, P 3 ( 4 =

E2

holds. The relation E2 : El can be chosen so that the third scale is pulled

1- $ ( 4

near a certain point or it is gathered. If we substitute m = 0, then E z h ( z z )= &(zZ) and in this case, the line of the third scale passes through both the starting points of the first and of the second scale. Consequently, these two scales must be placed with a scale division in opposite directions, while the third one will be between them. IThe relation between the Cartesian coordinats x and y of a point in the x,y plane and the corresponding angle 9 in polar coordinates is: yz = xz tan' p . (2.300) The corresponding nomogram is shown in Fig. 2.110. The scale division is the same for the scales of x and y but they are oriented in opposite directions. In order to get a better intersection with the third scale between them. their initial points are shifted by a suitable amount. The intersection points of the third scale with the first or with the second one are marked by ~p = 0 or p = 90" respectively. Iz = 3 y = 3.5, delivers 9 x 49.5'.

P* 71

4* r7

sI

4

Figure 2.110

Figure 2.111

2.19.3.3 Alignment Charts with Two Parallel Straight Lines and a

Curved Scale

If one of the straight-line scales is placed on the y-axis and the other one is placed at a distance d from it, then equation (2.295) has the form (2.301) Consequently: (2.302a) If we choose the scale El for the first scale and E2 for the second one, then (2.302a) is transformed into

2.19 Nomooraphu 127

(2.302~) holds. I The reduced third-degree equation z3 + p'z + q' = 0 (see 1.6.2.3, p. 40) is of the form (2.302b). After the substitutions El = E2 = 1 and f(z1) = q* , g(z2) = p* h(z3) = z , the formulas to calculate ~

d.2

23

the coordinates of the curved scale are z = 9 3 ( 2 ) = - and y = &(t) = -. 1+2 l + Z In Fig. 2.111 the curved scale is shown only for positive values of z . The negative values one gets by replacing z by - 2 and by the determination of the positive roots from the equation z3 + p * z - q* = 0. The complex roots u iv can also be determined by nomograms. Denoting the real root, which always exists. by zl.then the real part of the complex root is u = -z1/2, and the imaginary part v can be 3 determined from the equation 3u2 - u z p* = -2: - vz p* = 0. 4 Iy 3 + 2 y - 5 = 0. i.e.:p* = 2 , q * = -5. One readszl x 1.3.

+

+

+

2.19.4 Net Charts for More Than Three Variables To construct a chart for formulas containing more than three variables, the expression is to decompose by the help of auxiliary variables into several formulas: each containing only three variables. Here. every auxiliary variable must be contained in exactly two of the new equations. Each of these equations is to be represented by an alignment chart so that the common auxiliary variable has the same scale.

I

128

3. Geometry

3 Geometry 3.1 Plane Geometry 3.1.1 Basic Notation 3.1.1.1 Point, Line, Ray, Segment 1. Point and Line Points and straight lines are not defined in today's mathematics. We determine the relations between them only by axioms. We can imagine a line as a trace of a point moving in a plane along the shortest route between two different points without changing its direction. .4point is the intersection of two lines.

2. Closed Half-Line or Ray, and Segment A ray is the set of points of a line which are exactly on one side of a given point 0, including this point 0. bi'e can imagine a ray as the trace of a point which starts at 0 and moves along the line without changing its direction, like a beam of light after its emission until it is not led out of its way. A segment AB is the set of points of a line lying between two given points A and B of this line, including the points A and B. The segment is the shortest connection between the two points A and B in a -+ plane. The direction class of a segment is denoted by an arrowhead AB, or its direction starts at the first mentioned point A , and ends at the second B.

3. Parallel and Orthogonal Lines Parallel lines run in the same direction; they have no common points, i.e., they do not move off and do not approach each other, and they do not have any intersection point. The parallelism of two lines g and g' is denoted by gllg'. Orthogonal lines form a right angle at their intersection, Le., they are perpendicular to each other. Orthogonality and parallelism are mutual positions of two lines.

3.1.1.2 Angle 1. Notion of Angle

S

A. A

Figure 3.1

An angle is defined by two rays a and b starting at the same point S,so they can be transformed into each other by a rotation (Fig. 3.1). If A is a point on the ray a and B is on the ray b, then the angle in the direction given in Fig. 3.1 is denoted by the symbols ( a , b) or by 3: ASB, or by a Greek letter. The point S is called the vertex, the rays a and b are called sides or legs of the angle.

In mathematics, we call an angle positive or negative depending on the rotation being counterclockwise or clockwise respectively. It is important to distinguish the angle 3: ASB from the angle 3: BSA. Actually, 3: .4SB = - 3: B S A (05 3: ASB 5 180') and 3: ASB = 360"- 3: BSA (180" 5 3: ASB 5 360') holds. Remark:In geodesy the positive direction of rotation is defined by the clockwise direction (see 3.2.2.1, p. 143).

2. Names of Angles Angles have different names according to the different positions of their legs. The names given in Table 3.1 are used for angles (Y in the interval 0 5 (Y 5 360" (see also Fig. 3.2).

3.1.1.3 Angle Between Two Intersecting Lines At the intersection point of two lines g l , g2 there are four angles a , p, 7 , b (Fig. 3.3). We distinguish adjacent angles. vertex angles, complementary angles, and supplementary angles.

3.1 Plane Geometry 129

Table 3.1 Names of angles in degree and radian measure Kames of angles round (full) angle convex angle straight angle

acute angle

Degree a' = 360" a' > 180" x a = 180°

right angle

Radian a = 271 < a < 2x a=x

Degree Radian Names of angles right angle ao = 90" cy = x / 2 acute angle 0" < a' < 90" 0" < a < x / 2 obtuse angle 90" < a < 180" x / 2 < a < x

obtuse angle

straight angle

convex angle

round (full) angle

Figure 3.2 1. Adjacent Angles are neighboring angles at the intersection point of two lines with a common vertex S.and with a common leg; both non-common legs are on the same line, they are rays starting from S but in opposite directions, so adjacent angles sum to 180". IIn Fig. 3.3 the pairs (cyI p), (8, (rl6) and ( a ,6) are adjacent an&I gles. 2. Vertex Angles are the angles at the intersection point of two lines, opposite to each other, having the same vertex S but no common leg, and being equal. They sum to 180° by the same adjacent angle. I I IIn Fig. 3.3 ( a ,y) and (b,6) are vertex angles. Figure 3.3 A/),

3. Complementary Angles are two angles summing to 90". 4. Supplementary Angles are two angles summing to 180". IIn Fig. 3.3 the pairs of angles ( a I8) or (-y,6) are supplementary angles.

3.1.1.4 Pairs of Angles with Intersecting Parallels Intersecting two parallel lines p1,pZ by a third one g, we get eight angles (Fig. 3.4). Besides the adjacent and vertex angles with the same vertex S we distinguish alternate angles, opposite angles, and corresponding (exterior-interior) angles with different vertices. 1. Alternate Angles have the same size. They are on the opposite sides of the intersecting line g and of the parallel linesp, p2. The legs of alternate angles are in pairs oppositely oriented. IIn Fig. 3.4 the pairs of angles (cy1, yz), (b1,62), ( 7 1 . a and ~ ) ~(61,P~)are alternate angles. Figure 3.4 2 . Corresponding or Exterior-Interior Angles have the same size. They are on the same sides of the intersecting line g and of the parallel lines pl, pz. The legs of corresponding angles are oriented in pairs in the same direction. IIn Fig. 3.4 the pairs of angles (01, az), (PI,Pz). (71.72)~ and (61,6~)are corresponding angles. 3. Opposite Angles are on the same side of the intersecting line g but on different sides of the parallel lines pl, p2. They sum to 180". One pair of legs has the same orientation, the other one is oppositely oriented. IIn Fig. 3.4, e.g., the pairs of angles (al1&), (&, yz), (71,h),and (d1, a ~are ) opposite angles. ~

130

3. Geometry

3.1.1.5 Angles Measured in Degrees and in Radians In geometry, the measurement of angles is based on the division of the full angle into 360 equal parts or 360" (degrees). This is called measure an degrees. Further division of degrees is not a decimal one, it is sexagesimal: 1' = 60' (minute, or sexagesimal minute), 1' = 60" (second, or sexagesimal second). For grade measure see 3.2.2.2, p. 145 and the remark below. Besides measure in degrees we use radian measure to define the size of an angle. The size of the central angle cy of an arbitrary circle (Fig. 3.5a) is given as the ratio of the corresponding arclength 1 and the radius of the circle r: 1 Q

= -.

r

= 0.000291 rad, The unit ofradian measure is the radian (rad), i t . , the central = 0.000005 rad. angle belonging to an arc with length equal to the radius. In the table you will find approximate conversion values. If the measure of the angle is (YO in degrees and Q in radian measure, then for conversion we have:

, Formulas (3.2) refer to decimal In particular: 360" = 277, 180" = 77 90" = 77/2, 270" = 3 ~ / 2 etc. fractions and in the following examples we show how to make calculations with minutes and seconds. IA: Conversion of an angle given in degrees into radian measure: 52" 37'23" = 52.0.017453 37 0.000291 t 23 0.000005 = 0.918447 rad. IB: Conversion of an angle given in radians into degrees: 5.645 rad = 323 0.017453 + 26.0.000291 + 5 0.000005 = 323" 26'05". We have the result from: 5.645:0.017453 = 323 t 0.007611 0.007611:0.000291= 26 0.000025 0.000025:0.000005 = 5. $e\' usually omit the notation rad if it is obvious from the text that the number refers to the radian measure of an angle. Remark: In geodesy a full angle is divided into 400 equal parts, called grades. This is called measure in grades. A right angle is 100 gon. A gon is divided into 1000 mgon. On calculators the notation DEG is used for degree, GRAD for grade, and RAD for radian. For conversion between the different measures see Table 3.5 on p. 145. ~

+

+

3.1.2 Geometrical Definition of Circular and Hyperbolic F'unct ions 3.1.2.1 Definition of Circular or Trigonometric Functions 1. Definition by the Unit Circle The trigonometric functions of an angle (Y are defined for both the unit circle with radius R = 1 and the acute angles of a right-angled triangle (Fig. 3.5a,b) with the help of the adjacent side b, opposite side a , and hypotenuse c. In the unit circle we measure the angle between a fixed radius OA (with length l ) and a moving radius counterclockwise (positive direction):

Figure 3.5

3.1 Plane Geometry 131

sine

sin cy = BC =

:

C

- a t a n a = .4D = - , b sec a = OD = 4 , b

tangent. secant

a,

:

(3.3)

cosine :

(3.5)

cotangent :

(3.7)

cosecant :

b cos a = OB = c

(3.4)

~

b . a

(3.6)

csc cy = OF = C . a

(34

= EF =

Cot

-

-

2. Signs of Trigonometric Functions Depending in which quadrant of the unit circle (Fig. 3.5a) the moving radius have well-defined signs which can be taken from Table 2.2 (p. 78).

is, the functions

3. Definition of Trigonometric Functions by Area of Circular Sectors The functions sin cy. cos c y , tan a,cot cy are defined by the segments BC, OB, AD of the unit circle with R = 1 (Fig. 3.6), where the argument is the central angle cy = AOC. For this definition we could use the area z of the sector C O K , which is denoted in Fig. 3.6 by the shaded area. With the central angle 2a

+

1 measured in radians, we get z = -R22a = a 2 for the area with R = 1. Therefore, we have the same equations for sin z = BC , cos z = O B , t a n z = AD as in (3.3, 3.4, 3.5).

Figure 3.6

Figure 3.7

3.1.2.2 Definitions of the Hyperbolic Functions In analogy with the definition of trigonometric functions in (3.3,3.4,3.5)we can consider the area of a sector considering the equation z2 - y2 = 1 of a hyperbola (only the right branch in Fig. 3.7) instead of the equation z2+ y2 = 1. Denoting by z the area of COK, the shaded area in Fig. 3.7, the defining equations of the hyperbolic functions are: sinh z = (3.9) coshz = O B , (3.10) tanhz =AD. (3.11) -Calculating the area z by integration and expressing the results in terms of BC, OB, and n e get ~

~ = ~ ~ ( B c + J ~ ~ Z Z ) =&In* =2 ~1-AD' ~ ( O B + J S I ) (3.12) and so, from now on, we can express the hyperbolic functions in terms of exponential functions: -

B C = -ex

- e-x

2

= sinhz)

(3.13a)

- e"

OB=--

+ e-x - coshz, 2

e' - e-x - tanh z ex t e-"

AD=--

(3.13b)

(3.13~)

These equations represent the most popular definition of the hyperbolic functions

3.1.3 Plane Triangles 3.1.3.1 Statements about Plane Triangles 1. The Sum of Two Sides of a plane triangle is greater than the third one (Fig. 3.8): btc>a.

(3.14)

I

132

3. Geometry

2. T h e S u m of t h e Angles of a plane triangle is a+4+y=18O0. (3.15) 3. Unique Determination of Triangles A triangle is uniquely determined by the following data:

by one side and the two angles on it. If two sides and the angle opposite one of them are given, then they define two, one or no triangle (see the third basic problem in Table 3.4, 144).

a

Figure 3.8

Figure 3.9

Figure 3.10

4. Median of a Triangle

is a line connecting a vertex of the triangle with the midpoint of the opposite side. The medians of the triangle intersect each other at one point, at the center ofgravity of the triangle (Fig. 3.10), which divides them in the ratio 2 : 1 counting from the vertex. 5. Bisector of a Triangle is a line which divides one of the interior angles into two equal parts. The bisectors intersect each other at one point. 6. Incircle is the circle inscribed in a triangle, i.e., all the sides are tangents of the circle. Its center is the intersection point of the bisectors (Fig. 3.9). The radius of the inscribed circle is called the apothem or the short radius. 7. Circumcircle is the circle drawn around a triangle, Le., passing through the vertices of the triangle (Fig. 3.11). Its center is the intersection point of the three right bisectorsof the triangle. 8. Altitude of a Triangle is the perpendicular line that starts at a vertex and is perpendicular to the opposite side. The altitudes intersect each other at one point, the orthocenter. 9. Isosceles Triangle In an isosceles triangle two sides have equal length. The altitude, median, and bisector of the third side coincide. For a Figure 3.11 triangle the equality of any two of these sides is enough to make it isosceles.

@

10. Equilateral Triangle In an equilateral triangle with a = b = c the centers of the incircle and the circumcircle, the center of gravity, and the orthocenter coincide. 11. Median is a line connecting two midpoints of sides of a triangle; it is parallel to the third side and has the half of length as that side. 12. Right-Angled Triangle is a triangle that has aright angle (an angle of 90') (Fig. 3.31), p. 141.

3.1.3.2 Symmetry 1. Central Symmetry A plane figure is called centrosymmetric if by a rotation of the plane by 180" around the central point or the center of symmetry S it exactly covers itself (Fig. 3.12). Because the size and shape of the figure do not change during this transformation, we call it a congruent mapping. Also the sense class or orientation class of the plane figure remains the same (Fig. 3.12). Because of the same sense class we call such figures directly congruent.

3.1 Plane Geometru 133

The orientation of afigure means the traverse of the boundary of a figure in a direction: positive direction, hence'counterclockwise. negative direction, hence clockwise (Fig. 3.12, Fig. 3.13).

Figure 3.12

Figure 3.13

2. Axial Symmetry .A plane figure is called axially symmetric if the corresponding points cover each other after a rotation in space by 180" around a line g (Fig. 3.13). The corresponding points have the same distance from the axis g , the axis of symmetry. The orientation of the figure is reversed for axial symmetry with respect to the line g. Therefore, we call such figures indirectly congruent. We call this transformation a reflection in g . Because the size and the shape of the figures do not change, we also call it indirect congruent mapping. The orientation class of the plane figure is reversed under this transformation (Fig. 3.13). Remark: For space figures we have analogous statements.

3. Congruent Triangles, Congruence Theorems a ) Congruence: We call plane figures congruent if their size and shape coincide. Congruent figures can be transformed into a position of superimposition by the following three transformations, by translation. rotation, and reRection, and combinations of these. We distinguish between directly congruent figures and indirectly congruent figures. Directly congruent figures can be transformed into a covering position by translation and rotation. Because the indirectly congruent figures have a reversed sense class, an axially symmetric transformation with respect to a line is also needed to transform them into a covering position. Axially symmetric figures are indirectly congruent. To transform them into each other we need all three transformations. b) Laws of Congruence: We give the conditions for triangles to be congruent in the following theorems. Two triangles are congruent if they coincide for: three sides (SSS) or two sides and the angle between them (SAS), or one side and the interior angles on this side (ASA), or two sides and the interior angle being opposite to the longer one (SSA).

4. Similar Triangles, Similarity Theorems \We call plane figures similar if they have the same shape without having the same size. For similar figures there is a one-to-one mapping between their points such that every angle in one figure is the same as the corresponding angle in the other figure. An equivalent definition is the following: In similar figures the length of segments corresponding to each other are proportional. a ) Similarity of figures requires either the equality of all the corresponding angles or the equality of the ratio of all corresponding segments. b) Area The areas of similar plane figures are proportional to the square of the ratio of corresponding linear elements such as sides, altitudes, diagonals, etc. c) Laws of Similarity For triangles we have the following laws of similarity. Triangles are similar if they coincide for:

134

3. Geometru

two ratios of sides, two interior angles, the ratio of two sides and the interior angle between them, the ratio of two sides and the interior angle opposite of the longer one. Because in the laws of similarity only the equality of ratios of sides is required and not the equality of length of sides, therefore the laws of similarity require less than the corresponding laws of congruence. 5 . Intercept Theorems The intercept theorems are a consequence of the laws of similarity of a triangle. 1. First intercept theorem If two rays starting at the same point S are intersected by two parallels p 1 , p z . then the segments of one of the rays (Fig. 3.14a) have the same ratio as the corresponding segments on the other one:

(3.16)

fl

s

P1 Consequently, every segment on one of the rays is a) proportional to the corresponding segment on the other ray.

PI

P2

P2

b) Figure 3.14

2. Second Intercept Theorem If two rays starting at the same point S are intersected by two parallels pl,p2 then the segments of the parallels have the same ratio as the corresponding segments on the rays (Fig. 3.14a): ~

(3.17) The intercept theorems are also valid in the case of intersecting lines at the point S, if the point between the parallels (Fig. 3.14b).

S is

3.1.4 Plane Quadrangles 3.1.4.1 Parallelogram A quadrangle is called a parallelogram (Fig. 3.15) if it has the following properties: the sides opposite to each other have the same length. the sides opposite to each other are parallel, the diagonals intersect each other at their midpoints, the angles opposite to each other are equal. If we suppose only one of the previous properties for a quadrangle, or we suppose the equality and the parallelism of one pair of opposite sides, then all the other properties follow from it. The relations between diagonals, sides, and area are the following: d:

+ d i = 2(aZ+ b2).

Figure 3.15

(3.18)

h = bsina,

Figure 3.16

(3.19)

S = ah.

Figure 3.17

(3.20)

3.1 Plane Geometry 135

3.1.4.2 Rectangle and Square A parallelogram is a rectangle (Fig. 3.16), if it has: only right angles, or the diagonals have the same length. Only one of these properties is enough, because either of them follows from the other. It is sufficient to show that one angle of the parallelogram is a right angle, then all the angles are right angles. If a quadrangle has four right angles; it is a rectangle. The perimeter C and the area S of a rectangle are:

c' = 2(a + b).

S = ab.

(3.21a)

(3.21b)

If a = b holds (Fig. 3.17), the rectangle is called a square, and we have the formulas x 1.414a,

d = a&

(3.22)

4 x 0.707d.

a = d-

2

S = a'

(3.23)

=

dZ 2

-.

(3.24)

3.1.4.3 Rhombus A rhombus (Fig. 3.18) is a parallelogram in which all the sides have the same length, or the diagonals are perpendicular to each other, or the diagonals are bisectors of the angles of the parallelogram. .4ny of the previous properties is enough alone; all the others follow from it. We have: dl = 2acos

2

(3.25)

d2 = 2asin

e. 2

(3.26)

d:

+ d i = 4a2.

S = ah = a2sincu = -did2 ,

(3.27) (3.28)

2

3.1.4.4 Trapezoid A quadrangle is called trapezoid if it has two parallel sides (Fig. 3.19). The parallel sides are called bases. With the notation a and b for the bases, h for the altitude and m for the median ofthe trapezoid which connects the midpoints of the two non-parallel sides we have o+b m=-. 2

(3.29)

(a + b)h S= = mh. 2

(3.30)

hs =

+

h(a 2b) 3 ( a + b)

(3.31)

~

'

The centroid is on the connecting segment of the midpoints of the parallel basis a and b, at a distance h s (3.31) from the base a. For the calculation of the coordinates of the centroid by integration see 8.2.2.3, p. 451 For an zsosceles trapezozdnith d = c we get

S = (a-ccosr)csin?

= (b+ccosy)csin-/.

(3.32)

1 i a

a

Figure 3.18

Figure 3.19

Figure 3.20

I

136

3. Geometru

3.1.4.5 General Quadrangle The general quadrangle has no parallel sides, i.e., all four sides are of different length. If the diagonals lie fully inside the quadrangle, it is convex, otherwise concave. The general quadrangle is divisible by two diagonals d l , d2 in two triangles (Fig. 3.20).Therefore, in every quadrangle the sum of the interior angles is 2 . 180" = 360": 4

a, = 360'. I=

(3.33)

1

The length of the segment m connecting the midpoints of the diagonals (Fig. 3.20) is given by

+ +

a' b2 cz + d2 = d: + d; + 4m2. The area of the general quadrangle is

(3.34)

S = h l d 2 sin a. 2

(3.35)

3.1.4.6 Inscribed Quadrangle A quadrangle which can be circumscribed by a circumcircle is called an inscribed quadrangle (Fig. 3.21a) and its sides are chords of this circle. A quadrangle is an inscribed quadrangle if and only if the sums of its opposite angles are 180': a + y = /3 + b = 180". Ptolemy 's theorem is valid for the insribed quadrangle: ac + bd = dld2.

(3.36) (3.37)

-

The radius of the circumcircle of an inscribed ouadranale is

'qd~~;l~;bee~u~;,~ 1

R = -d(ab+cd)(ac+bd)(ad+cb).

The

b)

by the formulas

4s

' / d2 =

Figure 3.21

a d + be

'

ac + bd) (ad + be) ab cd

+

1 The area can be expressed in terms of the half-perimeter of the quadrangle s = - ( a 2

S = d ( s - a ) ( s - b ) ( s - c ) ( s- d ) .

(3.38)

(3.39a) (3.39b)

+ b + c +d): (3.40)

If the inscribed quadrangle is also a circumscribing quadrangle (see Fig. 3.21 and 3.1.4.7), then

S=Jabcd.

(3.41)

3.1.4.7 Circumscribing Quadrangle If a quadrangle has an inscribed circle (Fig. 3.21b),then it is called a circumscribing quadrangle, and the sides are tangents to the circle. A quadrangle has an inscribed circle if and only if the sum of the lengths of the opposite sides are equal, and this sum is also equal to the half-perimeter s: 1 s = - ( a + b + c + d ) = a + c = b + d. (3.42) 2 The area of the circumscribing quadrangle is S = ( a + C)T = ( b+ d ) r , (3.43)

3.1 Plane Geometru 137

where r is the radius of the inscribed circle,

3.1.5 Polygons in the Plane 3.1.5.1 General Polygon .4 closed plane figure bounded by straight-line segments as its sides, can be decomposed into n - 2 triangles (Fig. 3.22). The sums of the exterior angles /3,, and of the interior angles yt, and the number of diagonals D are n

n

J = 360".

c y z = 18Oo(n- 2),

(3.44)

(3.45)

1=1

z=1

b) Figure 3.22

Figure 3.23

3.1.5.2 Regular Convex Polygons Regular convex polygons (Fig. 3.23) have n equal sides and n equal angles. The intersection point of the mid-perpendiculars of the sides is the center M of the inscribed and of the circumscribed circle with radii T and R, respectively. The sides of these polygons are tangents to the inscribed circle and chords of the the circumcircle. They form a circumscribing polygon or tangent polygon for the inscribed circle and a inscribed polygon in the circumcircle. The decomposition of a regular convex n-gon (regular convex polygon) results in n isosceles congruent triangles around the center M. Central Angle

360" 9, = -. (3.47) n

Exterior Angle

pn = -, (3.49)

360° n

f) .go".

Base Angle

a, = (1 -

Interior Angle

^in = 180" - p,.

(3.48) (3.50)

1 4 2 sin n a, 180" 180° Inscribed Circle Radius r = - cot = R cos -. 2 n n a n = 2 d m = 2 R s i n e = 2 r t a n - Pn . Side Length 2 2 Perimeter U = na, . Circumcircle Radius

R=a ,RZ= r z + - a i . 1800 ,

(3.51)

(3.52)

~

(3.53) (3.54)

=R

SideLengthofthe2n-gon

a2,

Area of t h e n-gon

s, = p1a , r

= nr2 tan

Pn

-=

2

1 1 -nR2 sin 9, = p a : cot 2 2

e.(3.56)

138

3. Geometry

Areaofthean-gon

S2,=

(3.57)

3.1.5.3 Some Regular Convex Polygons The properties of some regular convex polygons are collected in Table 3.2. The pentagon and the pentagram deserve special attention since it is presumed that Hippasos of Metapontum (ca. 450 BC) recognized irrational numbers by the properties of these polygons (see 1.1.1.2, p. 2). We discuss them in an example.

I The diagonals of a regular pentagon Fig. 3.24 form an znscrzbed pentagram. Its sides enclose a regular pentagon again. In a regular pentagon, the proportion of a diagonal and a side is equal to the proportion of a side and the diagonal minus side: a. : al = al : (a0 - a l ) = a1 : a2, where a2 = a0 - al. B Ifwe consider smaller and smallernested pentagons with a3 = al-a2, a4 = a2 - as1 and a2 < ai. a3 < a2, a4 < a s . . . , we get ao : a1 = a1 : a2 = a2 : a3 = a3 : a4 = . . ' . The Euclidean algorithm for a0 and al never breaks off, since a0 = 1 . a1 + a2, al = 1 . a2 + a3, a2 = 1 a3 + ad,. . . , hence q, = 1. The side al and diagonal a0 of the regular pentagon are incommensurable. The continued fraction determined by a0 : a1 is identical to the golden section in I B on 1.1.1.4, 3., p. 4, Le., it results in an irrational number.

A

t . .

Figure 3.24

3.1.6 The Circle and Related Shapes 3.1.6.1 Circle The czrcle is the locus of the points in a plane which are at the same given distance from a given point. the center of the circle. The distance itself, and also the line segment connecting the center with any point of the circle, is called the radius. The circumference or periphery of the circle encloses the area ojthe czrcle. Every line passing through two points of the circle is called a chord. Lines having exactly one common point with the circle are called tangent lines or tangents of the circle. -- _ _ Chord Theorem (Fig. 3.26) AC . AD = AB AE = r2 - m2.

(3.58)

Secant Theorem (Fig. 3.27) A B . A E = AC . AD = m2 - r 2 .

(3.59)

-- _ _

Secant-Tangent Theorem (Fig. 3.27)

U = 27rr RZ 6,283r,

Perimeter

m2= AB. AE = U = ad RZ 3,142d,

n

U =2

= m2 - r 2 .

(3.60)

a FZ 3,545fi.

(3.61)

B P

L

Figure 3.25

Figure 3.26

Figure 3.27

3.1 Plane Geometry 139

Table 3.2 Properties of some regular polygons

R

a,

T

3-gon

5-gon

a5

=

R2 J i L Z

= 2rJ5

6-gon

a6 =

8-gon

a8

-

2 4

2 -r&

R

= -(&+ 1)

=r(&-l)

4

= !T&

3

= RJ2 -

“&GZ

= 10

3

Jz

= =

r d a

%(a+ 1) 2

= 2ai(Jz+ 1)

qG.3

=2

= 2

R2d

= 8~*(fi 1)

10-gon

a10

=

R 2

-(& - 1)

=2JKG Area

S=5

Radius

T

=

7 %~

3,~ 142r2,

c‘ x 0.159C. 257

ad2 4

Ud 4 ’

S = - % 0, 785d2, S = (3.63)

(3.62)

t

Diameter d = 2T = 2 - GZ 1.128fi.

(3.64)

For the following formulas with angles see the definition of the angle in 3.1.1.2. p. 128. Angle of Circumference (Fig. 3.25)

1 1 1 a = - BC= - BOC = -9. 2 2 2

A Special Case is the Theorem of Thales (see p. 141)

(o

= B O ” , i.e,

p

1 1 = - AC= - C O A , 2 2

Angle Between a Chord and a Tangent (Fig. 3.25) Interior Angle (Fig. 3.26) Exterior Angle (Fig. 3.27)

-

cy

= 90”.

+

1 1 + E D ) = ,(+BOC + +EOD). 2 1 1 cy = Z ( D E - BC) = ,(+EOC - +COB). T = -(CB

(3.65s) (3.65b) (3.66) (3.67) (3.68)

140

3. Geometru

-

1 1 - +BOT).(3.69) 2 2 Escribed Angle (Fig. 3.28, D and E are arbitrary points on the arcs to the left and to the right.) h 1 1 a = -(BDC - C E B ) = -(+BOG'- +COB) 2 2 1 = -(360° - 3: COB - 3: COB) = 180" - 0: COB. (3.70) 2

Angle Between Secant and Tangent (Fig. 3.27) p = - ( T E - T B ) = -(+TOE

3.1.6.2 Circular Segment and Circular Sector

I

Defining quantities: Radius T and central angle Q (Fig. 3.29). The amounts to determine are:

Chord

a=2JZF7?=

Central Angle

a = 2arcsin

a 2r

a

(3.71)

2rsin-. 2 (measured in degrees).

(3.72)

Height of the Circular Segment h = T -

(3.73)

8b-a 27rm Arc Length 1 = -x O.O1745ra, 1 x - or 360

(3.74)

3

Figure 3.28

Figure 3.29

Figure 3.30

m2a

Area of the Sector

S = -z 0.00873r2a.

Area of Circular Segment

S=

(3.75)

360

- a(r - h ) ] ,

S

h 15

-(sa

+ 8b).

(3.76)

3.1.6.3 Annulus Defining quantities of an annulus: Exterior radius R, interior radius r and central angle 1p (Fig. 3.30).

Exterior Diameter

D = 2 R.

Mean Radius

p=-

Area of the Annulus

+

2

,

(3.77)

Interior Diameter

d = 2r.

(3.79)

Breadth of the Annulus

6 = R - T . (3.80)

S = 7r(R2- r z ) = I ( D z - dz) = 27rp6. 4

Area of an Annulus Sector for a Central Angle cp (shaded area in Fig. 3.30) 'p" 2 - 'pr sv = 360 (Rz - T ) - 1440 (D' - dz) = y180 p 6 ,

(3.78)

(3.81)

(3.82)

3.2 Plane Trigonometry 141

3.2 Plane Trigonometry 3.2.1 Triangles 3.2.1.1 Calculations in Right-Angled Triangles in the Plane 1. Basic Formulas Notation (Fig. 3.31): c hypotenuse: a, b other sides, or legs of the right angle; a: and p the angles opposite to the sides a and b respectively; h altitude; p , q hypotenuse segments; S area. Sum of Angles cy B y = 180' with y = go", (3.83)

+ +

Calculation of Sides Pythagoras Theorem

a = c sin LY = ccos p = b t a n a = bcotp,

(3.84)

+ bZ = c2

(3.85)

a*

Figure 3.31

Thales Theorem The vertex angles of all triangles in a semicircle with hypotenuse as the base are right angles, i.e., all angles at circumference in this semicircle are right angles (see Fig. 3.32 and (3.6513))p. 139). Euclidean Theorems Area

h2 = p q , a' = p c ,

bZ = pc, (3.86)

ab a2 C2 S== - t a n p = -sin2P. 2 2 4

a, b, c, which are not all independent, of course), one angle, in Fig. 3.31 the angle y , is given as 90". A plane triangle can be determined by three defining quantities but these cannot be given arbitrarily (see 3.1.3.1, p. 132). So. in the case of a right-angled triangle only two more quantities can be given. The remaining three quantities can be determined from Table 3.3 and

(3.87)

Given

Figure 3.32

Calculation of t h e o t h e r quantities

e.g.a,a: p = 9 O 0 - o e'g' b" ci

' ''

b=acotcr

c=& sin 0

=

-

' a = b t a n c y '= cos a:

=

-

'

e.g,a,b !=tan@ b

= sin cy

= CCOS a

csin= L B = 90" - a cy

142

3. Geometry

A

A

6% v

B

C

A

Figure 3.33

4 '!

B

C

Figure 3.34

Figure 3.35

sides and angles we can get two further formulas by cyclic permutation the sides and angles according to Fig. - 3.34. a sin0 b sinp != nom - = (sine law) we get by cyclic permutation: - = -, b sin3 c sin? a sina a h c -- _Sine Law - 2R. (3.88) sin cy sin p - sin y ~

_.-

Projection Rule (see Fig. 3.35) c = acos p + bcos a .

(3.89)

Cosine Law or Pythagoras Theorem in General Triangles c2 = a' t bZ - 2ahcos y .

(3.90)

Mollweide Equations (a

+ b) sin -2 = ccos

(" 2 '), -

(3.91a)

at4

Tangent Law

a t h tan- 2 -a-b0-p' tan 2

(3.92)

~

Half-Angle Formulas tan - = 2 Tangent Formula tan cy =

(S

- b) (S - C) s(s-a)

asinp ~

C-UCOS~

(3.93) '

- asiny - ~. b-acosy

(3.94)

Additional Relations sin - = 2

(3.95,)

5

Height Corresponding t o t h e Side a

h, = bsiny = c s i n 9 .

ma = - dhz+ c2 + 2hc cos (Y . 2

Median of t h e Side a

Bisector of t h e Angle

2

Q

(3.96) (3.97)

2bccos 2 2 1, = h+c '

Radius of t h e Circumcircle R =

(3.95b)

(3.98)

a - h 2sincy 2sinp - 2siny C

(3.99) '

3.2 Plane Trigonometry 143

'

Radius of the Incircle r =

J

s - a)(s - b)(s - c)

= s tan

2

tan

2

tan

Y2

(3.100)

a P . 7 = 4Rsin -sin - sin - . 2

2

(3.101)

2

1 = 2R2sinasinSsiny = r s = J s ( s - a)(s - b)(s - c) 2 The formula S = ,,/s(s - a ) ( s - b ) ( s - c) is called Heron's formula.

Area

S = -absin*/

(3.102)

2. Calculation of Sides, Angles, and Area in General Triangles According to the congruence theorems (see 3.1.3.2, p. 133) a triangle is determined by three independent quantities. among which there must be at least one side. From here we get the four so-called basic problems. If from the six defining quantities (three angles a.3, yr and the sides opposite to them a , b, e) three independent quantities are given, we can calculate the remaining three with the equations in Table 3.4, p. 144. In contrast to spherical trigonometry (see the second basic problem, Table 3.9, p. 170) in a plane triangle there is no way to get any side only from the angles.

3.2.2 Geodesic Applications 3.2.2.1 Geodetic Coordinates In geometry we usually use a right-handed coordinate system to determine points (Fig. 3.170). In contrast with this, in geodesy left-handed coordinate systems are used.

1. Geodetic Rectangular Coordinates In a plane left-handed rectangular coordinate system (Fig. 3.37) the z-axis of abscissae is shown upward. the y-axis of ordinates is shown to the right. .4point P has coordinates yp, zp. The orientation of the z-axis follows from practical reasons. When measuring long distances, for which we mostly use the Soldner- or the Gauss-Krueger coordinate systems (see 3.4.1.2, p. E O ) , the positive z-axis points to grid North, the y-axis oriented to the right points to East. The enumeration of the quadrants follows a clockwise direction in contrast with the usual practice in geometry (Fig. 3.37, Fig. 3.38). If besides the position of a point in the plane we also consider its altitude, we can use a three-dimensional left-handed rectangular coordinate system ( y , q z ) >where the z-axis points to the zenith (Fig. 3.36).

Figure 3.36

Figure 3.37

Figure 3.38

2. Geodetic Polar Coordinates In the left-handed plane polar coordinate system of geodesy (Fig. 3.38) a point P is given by the directional (azimuthal) angle t between the axis of abscissae and the line segment s. and by the length of the line segment s between the point and the origin (called the pole). In geodesy the positive orientation of an angle is the clockwise direction. To determine the altitude we use the zenith dzstance C or the vertical angle respectively the angle oftzlt a. In Fig. 3.36 we can see that in a three-dimensional rectangular left-handed coordinate system (see

144

3. Geometrv

Table 3.4 Def ng quantities of a general triangle, basic problems

Given

Formulas for calculating the other quantities a sin P

y = 180" - - /3, b=sina ' a sin y c= S = - a bsiny. sina ' 2 2. 2 sides and the angle between them (a, b, 7 )

3. 2 sides and the angle opposite one of them (a, b, a )

a-P a-b y a+Ptan -- -cot-, --9O"--y; 2 a+b 2 2 cy and p come from a + /3 and a - P, a sin y 1 c= S = - a bsiny sin0 ' 2

1 2

b sin cy sinp = -. a [ f a 2 b holds, then p < 90" and is uniquely determined. If 2 < b holds, the following cases occur:

p has two values for bsincy < a (Pz = 180" - PI). 2. p has exactly one value (90") for bsin a = a. 1.

3. For b sin a > a there is no such triangle. a sin y 1 y = 180" - (0 P), c = - S = - a bsiny. sina ' 2

+

4. 3 sides ( a ,b, c)

S = r s = 4 s (s - a)(s - b)(s - c) .

also left and right-handed coordinate systems 3.5.3.1, p. 207), the zenith distance is measured between the zenith axis z and the line segment s, the angle of tilt between the line segment s and its perpendicular projection on the y, x plane.

3. Scale In cartography the scale factor M is the ratio of a segment S K I in a coordinate system K1 with respect to the corresponding segment S K Z in another coordinate system Kz . 1. Conversion of Segments With m as a modulus or scale and N as an index for the nature and K as the index of the map we have: M = 1 ; m= S K : SN. (3.103a) For two segments S K I , S K with ~ different moduli ml, mz we get: S K I ; s K 2 = m2 ; ml. (3.103b) 2. Conversion of Areas If the areas are calculated according to the formulas FK = aKbK, FN= aNbN, we have: FJ= ~ FKm2. (3.104a) For two areas Fl,F2 with different moduli m l , m2 : F K ~: F K = ~ mi ; m:. (3.104b)

3.2 Plane Trigonometrg 145

3.2.2.2 Angles in Geodesy 1. Grade or Gon Division In geodesy. in contrast to mathematics (see 3.1.1.5, p. 130) we use gradeor gon as a unit for angles. The perigon or full angle corresponds here to 400 grade or gon. The conversion between degrees and gons can be performed by the formulas in Table 3.5: Table 3.5 Grade and Gon Division 1 Full angle

= 360° = 27r rad

-

1 right angle = 90”

= !!rad

1 gon

=

Quadrant

2

I

200

= 400 gon = lOOgon

rad = 1000 mgon

I1

I11

IV

(3.105a)

AY IS. punched The quadrant of the angle t depends on the sign of AyaB and AZAB.If using a calculator Ax

in with the correct signs for Ay and Ax, then we get an angle t o by pressing the arctan button to which we have to add the gon-values given in Table 3.6 according to the corresponding quadrant. 2. Calculation of Rectangular Coordinates from Distances and Angles In a rectangular coordinate system we have to determine the coordinates of a point C by measuring in a local polar system (Fig. 3.40). Given: y4, X A : Y B , X B . Measured: a , S B C . Find: yc,zc. Solution:

146

3. Geometru

If we also measure SAB, then we consider the ratio of the locally measured distance and the distance computed from the coordinates by the scalefactor q , where q must be very close to 1:

-4

calculated distance -

I

(3.107a)

= measured distance

SAB

yc = YE + sBcqsintBc,

(3.107b)

IC = X B

+ sBcqcostBc.

Figure 3.39

(3.107~)

Figure 3.40

3. Coordinate Transformation Between Two Rectangular Coordinate Systems In order to locate a given point on a country-map we have to transform the local system y’, 2’ into the system y, zof coordinates ofthe map (Fig.3.41).The system y’, z’is rotated into y, zby an angle ‘p and is translated parallel by yo. xo.The directional angles in the system y’) z’are denoted by 19. The coordinates of A and B are given in both systems and the coordinates of a point C in the z’, y’-system. The transformation is given by the following relations: SAB

=

d

q = -S A B ”aB

w

’

tantAB = AYAB

AXAB’

,(3.108a)

,-d

(3.108b)

(3.108~)

p = tAB - 8ABi

(3.108d)

(3.108e)

tant9AB = AY>B ,

(3.108f)

s>B

AXLE

YO = YA - 4 2 ~sin ‘p - q y COS ~ p, (3.108g)

20 = z ~ + q y ~ s i n p - q z ~ c o s p(3.108h) ,

~ c = ~ ~ + q s i n , ~ ( & -+qcos’p(yk-?/A), z\)

zc = ZA

+ q cos ’p(&

=

- 2:) - qsin ‘p(yk - y;).

(3.108i) (3.108j)

Remark: The following two formulas can be used as a check. yc = yAtqs>,sin(p+19Ac),

(3.108k)

If the segment AB is on the 2’-axis, the formulas reduce to

zc = xa+qs>c C O S ( ~ + ~ A C(3.1081) ).

3.2 Plane Trigonometru 147

"t

"t

"'f

N

Figure 3.41

Figure 3.42

3.2.2.3 Applications in Surveying The determination of the coordinates of a point S to be fixed by triangulation is a frequent measuring problem in geodesy. The methods of solving it are intersection, three-point resection, arc intersection, free stationing and traversing. We do not discuss the last two methods here.

1. Intersection 1. Intersection of Two Oriented Lines or first fundamental problem of triangulation: To determine a point "V by two given points A and B with the help of a triangle ABN (Fig. 3.42). Given: y~~X A ; Y B , X B . Measured: a:p,if it is possible also y or y = 200 gon -a - p. Find: Yn.Xhr. Solution: (3.1loa)

SBN

Y,V

YW

sin a sin a =SAB. sin 7 sin (0

= SAB-

+

= YA SAX sin L A N = Y B = XA t SAX cos t A N = X B

+ 9)

+ +

SEN

(3.110~) '

sin t B N ,

SBN

cos t B N .

SA,^

sin p sin p (3.110d) = s,~B------sin? sin ( a p) '

= SAB-

+

(3.1log) (3.110h)

2. Intersection Problem for Non-Visible B If the point B cannot be seen from A , we determine the directional angles t A N and t B N with respect to reference directions to other visible points D and E whose coordinates are known (Fig. 3.43).

148

3. Geometrv

Figure 3.43

Figure 3.44

Given: Y A , X A ; YB, X B ; yo, ZD;YE,ZE. Measured: 6 in A, E in B , and if it is possible, also y . Find: y ~Z N; . Solution: We reduce it to the first fundamental problem, calculating tan t A B according to (3.110b) and: ~

tan t A D = AXAD

tan tA.V =

ZN =

’

AYNA AXNA ’

~

(3.111a)

tan t g E = -

(3.111b)

(3.11lg)

tantBN = -

(3.111h)

A X E B’

AXNB

AYBAt ZA sin tAN - ZB tan tBN tan t A N - tan t g N

s

(3.111i)

YN

= YB

+ (ZN -

Z B ) tantBN.

(3.111j)

2. Three-Point Resection 1. Snellius Problem of Three-Point Resection or to determine a point N by three given points .4, B , C; also called the secondf~ndamentalproblemof triangulation (Fig.3.44): Given: y ~X ,A ;YB, X B ; yc, xc . Measured: &,62 in N . Find: YN, X N . Solution:

AYAC AXAC’

tan tAC = -

AYBC, tan tBC = -

(3.112a)

(3.112b)

AXBC

(3.112c)

tan

’-’

~

2

a sin 62 tanX = b sin 61 ’

=t a n ~ ~ o t ( 4 5 ~ t X ) , ( 3 . 1 1 2 h )

2

SAN

= sin SI sin(&

+ ip),

(3.112g) (3.112i)

3.2 Plane Trzgonomety 149

SEX

=

b -sin(& sin h2

XN = XA

y , ~= YA

t $),

(3.112j)

SCN=

a -sincp= sin J1

b -sin@, sin S2

(3.112k)

+ S,AN cos tA,V = X B + S B N cos tBN, + SA,\, sin t,4N = yB + S E N sin t B N .

(3.1121) (3.112m)

Three-Point Resection by Cassini Given: Y A , . T , ~ ; Y B , Z B : ~ ~ , X CM. e a s u r e d S I , & in N . F i n d ~ N , X N . For this method we use two reference points P and Q, which are on the reference circles passing through A, C, P and B , C, Q so that both are on a line containing N (Fig. 3.45). The centers of the circles H1 and H2 are at the intersection points of the midperpendicular of AC and of with the segments PC and QC . Lye have the angles measured at N also at P and Q (angles of circumference). Solution: yp = YA + (XC - X A ) cot 61, (3.113a) XP = X A (yc - y ~ cot ) 61, (3.113b) 2.

YQ

= YE

+ (XB

- XC)cot

62, (3.113c) (3.113e)

YN

= YP + (Xn. - XP ) tantpQ

+ = X B + (YS- yc) cot&, X N = xp + YC - Y P + (ZC - Z P ) cot tPQ tan tpg + cot tpQ

(3.113d)

XQ

(tantpQ < CottpQ),

~

(3.1134

(3.113g)

(cot tpQ < tantpg), (3.113h) - XC)cot tpQ Dangerous Circle: When choosing the points we have to ensure they do not lie on one circle, because then there is no solution; we talk about a so-called dangerous circle. If the points are close to a circle, the method is nolonger accurate. y,v = yc

- (ZN

X

I;py

tANI .

Figure 3.45

Figure 3.46

3. Arc Intersection We get the required point N as the intersection point of the two arcs around the two points A and B with known coordinates and with measured radii S A N and S B N (Fig. 3.46). We calculate the unknown length S A B and the angle from the now already known three sides of the triangle ABN. A second solution - not discussed here - starts from the decomposition of the general triangle into two right-angled triangles. Given: ya. xq : YE,X B Measured: S A N ; S E N . Find: S A B , Y N , X N .

150

3. Geometry

Solution: SAB

=

d

tBA = AB

a

.

(3.114b)

(3.114a)

+ 200 gon,

(3.114~)

3.3 Stereometry 3.3.1 Lines and Planes in Space 1. Two Lines Two lines in the same plane have either one or no common point. In the second case they are parallel. If there is no plane such that it contains both lines, they are skew lznes. The angle between two skew lznes is defined by the angle between two lines parallel to them and passing through a common point (Fig. 3.47). The distance between two skew lines is defined by the segment which is perpendicular to both of them. (There is always a unique transversal line which is perpendicular to both skew lines and intersects them, too.) 2. Two Planes Two planes intersect each other in a line or they have no common point. In this second case they are parallel. If two planes are perpendicular to the same line or if both are parallel to every intersecting pair of lines in the other, the planes are parallel.

Figure 3.47

Figure 3.48

Figure 3.49

3. Line a n d Plane A line can be incident to a plane. it can have one or no common point with the plane. In this last case it is parallel to the plane. The angle between a line and a plane is measured by the angle between the line and its orthogonal projection on the plane (Fig. 3.48). If this projection is only a point, Le., if the line is perpendicular to two different intersecting lines in the plane, then the line is perpendzcular or orthogonal to the plane.

3.3.2 Edge, Corner, Solid Angle 1. Edge or dzhedral angle is a figure formed by two infinite half-planes starting at the same line (Fig.3.49). In everyday terminology the word edge is used for the intersection line of the two halfplanes. AS a measure of edges, we use the edge-angle ABC, the angle between two half-lines lying in the half-planes and being perpendicular to the intersection line D E at a point B .

3.3 Stereometry 151

2. Corner or polyhedral angle OABCDE (Fig. 3.50) is a figure formed by several planes, the lateral faces, which go through a common point, the vertex 0, and intersect each other at the lines OA, OB. . . . . Two lines which bound the same lateral face form a plane angle, while neighboring faces form a dzhedral angle. Polyhedra are equal to each other, i.e., they are congruent, if they are superposable. For this the corresponding elements, Le., the edges and plane angles at the vertex must be coincident. If the corresponding elements at the vertex are equal, but they have an opposite order of sequence, the corners are not superposable, and we call them s y m m e t n c corners, because we can bring them into a symmetric position as shown in Fig. 3.51. .A convex polyhedral angle lies completely on one side of each of its faces. The sum of the plane angles AOB t BOC t . . . t EOA (Fig. 3.50) is less than 360" for every convex polyhedron. 3. a i h e d r a l Angles are congruent if they coincide in the following elements: in three corresponding faces in the same order of sequence, in three corresponding dihedral angles in the same order of sequence. 4. Solid Angle .4 pencil of rays starting from the same point (and intersecting a closed curve) forms a solid angle in space (Fig. 3.52). It is denoted by R and calculated by the equality R = -S (3.115a) r2 ' Here S means the piece of the spherical surface cut out by the solid angle from a sphere whose radius is r , and whose center is at the vertex of the solid angle. The unit of solid angle is the steradian (sr). u'e have: 1m2 lsr=--. (3.115b) 1 m* Le., a solid angle of 1 sr cuts out a surface area of 1m2 of the unit sphere ( r = 1m). I A: The full solid angle is R = 47rrz/r2= 47r. I B: A cone with a vertex angle (also called an apex angle) (Y = 120" defines(determines) a solid angle R = 2;7r2(1- c0s(a/2))/r2 = 7r, where we have used the formula for a spherical cap (3.163).

Figure 3.50

Figure 3.51

Figure 3.52

Figure 3.53

3.3.3 Polyeder or Polyhedron In this paragraph we use the following notation: V volume, S total surface, M lateral area, h altitude, AG base area 1. Polyhedron is a solid bounded by plane polygons. 2. Prism (Fig. 3.53) is a polyhedron with two congruent bases, and having parallelograms as additional faces. .A rzght prasm has edges perpendicular to the base, a r e g u l a r p w m is a right prism with a regular polygon as the base. For the prism, we have:

152

3. Geometry

I/ = Ach, (3.116) M=pl, (3.117) s = .b! + 2Ac. (3.118) Here p is the perimeter of the section perpendicular to the edges, and l is the length of the edges. If the edges are still parallel to each other but the bases are not, the lateral faces are trapezoids. If the bases of a triangular prism are not parallel to each other, we can calculate its volume by the formula (Fig. 3.54): ( a b c)Q (3.119) V= 3 ’ where Q is a perpendicular cut, a, b, and care the lengths of the parallel edges. If the bases of the prism are not parallel, then its volume is (3.120) V=lQ,

+ +

where 1 is the length of the line segment E c o n n e c t i n g the centers of gravity of the bases, and Q is the cross-cut perpendicular to this line.

Figure 3.54

Figure 3.55

Figure 3.56

3. Parallelepiped is a prism with parallelograms as bases (Fig. 3-55), Le., it is bounded by six parallelograms. In a parallelepiped all the four body diagonals intersect each other at the same point, at their midpoint. 4. Rectangular Parallelepiped or block is a right parallelepiped with rectangles as bases. In a block (Fig. 3.56) the body diagonals have the same length. If a, b, and c are the edge lengths of the block and d is the length of the diagonal, then we have:

+ +

V = abc, (3.122) S=2(ab+bc+ca). d 2 = a2 b 2 c’, (3.121) 5. Cube (or regular hexahedron) is a block with equal edge lengths: a = b = c ,

(3.123)

V = a3,

(3.126)

d 2 = 3a2.

(3.124)

(3.125)

S= 6aZ.

6. Pyramid (Fig. 3.57) is a polyhedron whose base is a polygon and its lateral faces are triangles with a common point, the vertex. A pyramid is called right if the foot of the perpendicular from the vertex to the base AG is at the midpoint of the base. It is called regular if it is right and the base is a regular polygon (Fig. 3.58), and n facedif the base is an n-gon. Together with the base the pyramid has (n + 1) faces. We have:

(3.127) For the regular pyramid 1

M = ;ph,

(3.128)

with p as the perimeter of the base and h, as the altitude of a face. 7. Frustum of a Pyramid or truncated pyramid is a pyrazid whose vertex is cut away by a plane parallel to the base (Fig. 3.57, Fig. 3.59). Denoting by SO the altitude of the pyramid, Le., the

3.3 Stereometru 153 perpendicular from the vertex to the base, we have: (3.129) Area ABCDEF Area .41B1C1D1E1F1=

-

(E)'

(3.130)

If AD and Ac are the upper and lower bases, resp., h is the altitude of the truncated pyramid, i.e.. the distance between the bases, and G D and a~ are corresponding sides of the bases, then (3.131) The surface of a regular truncated pyramid is

JLJ =

PD ~

+ PC h,

(3.132)

>

where p~ and p~ are the perimeters of the bases, and h, is the altitude of the faces C

C

B

C Figure 3.57

Figure 3.58

Figure 3.59

0

A

_____------r B

Figure 3.61

Figure 3.60

0

T* q2 a' r2 0 p z b 2 1'2 = 42 p z 0 288 a2 b 2 e' 0 1 1 1 1

2-

1 1 1 1 0

(3.133)

9. Obelisk is a polyhedron whose lateral faces are all trapezoids. In the special case in Fig. 3.61

the bases are rectangles. the opposite edges have the same inclination to the base but they do not have

I

154

3. Geometry

a common point. If a, b and a l , bl are the sides of the bases of the obelisk and h is the altitude of it, then we have: h h V = - [(2a a l ) b ( 2 a l + a) b l ] = - [ab ( a al) ( b bl) albl] . (3.134) 6 6 10. Wedge is a polyhedron whose base is a rectangle, its lateral faces are two opposite isosceles triangles and two isosceles trapezoids (Fig. 3.62). For the volume we have 1 V = - (2a a l ) b h . (3.135) 6

+

+ +

+

+ +

+

11. Regular Polyeder have congruent regular polyeders as faces and congruent regular corners. The five possible regular polyheders are represented in Fig. 3.63;Table 3.7 shows the corresponding data. Table 3.7 Regular polyeders with edge length a

Name

Number and form of faces

Number of

Total area

GJGG

Sla2

Tetrahedron Cube

4triangles 6squares

6 12

4 8

fi = 1.7321

Octahedron

8t riangles

12

6

2& = 3.4641

Dodecahedron

12pentagons 30

20

Icosahedron

20triangles

12

30

6 = 6.0

= 20.6457

34-

Volume

Via3

Jz - = 0.1179 12 1 = 1.0

fi

- = 0.4714

3

l5+,7d 5(3 -k4

= 7.6631 = 2.1817

5 f i = 8.6603

12. Euler's Theorem on Polyeders If e is the number of vertices, f is the number of faces, and k is the number of edges of a convex polyhedron then e-lc+f=2. (3.136) Examples are given in Table 3.7.

3.3.4 Solids Bounded by Curved Surfaces In this paragraph we use the followingnotation: V volume, S total surface, M lateral surface, h altitude, AG base. 1. Cylindrical Surface is a curved surface which we get by parallel translation of a line, the generatzng lzne or generator along a curve, the so-called directzng curve (Fig. 3.64). 2. Cylinder is a solid bounded by a cylindrical surface with a closed directing curve, and by two parallel bases cut out from two parallel planes by the cylindrical surface. For every arbitrary cylinder

3.3 Stereometry 155

(Fig. 3.65) with the base perimeterp, with the perimeter s of the cut perpendicular to the apothem, whose area is Q , and with the length 1 of the apothem the following is valid: I’ = AG h = Q l , (3.137) M = p h = sl. (3.138)

3. Right Circular Cylinder has a circle as base, and its apothems are perpendicular to the plane of the circle (Fig. 3.66). With a base radius R we have: I/ = .rrRZh. (3.139) A4 = 2 ~ R h , (3.140) S = 2.ir R ( R + h ) . (3.141)

4. Obliquely Truncated Cylinder (Fig. 3.67) V = n R h +hz z L 2 ’

(3.142)

M

= 7T R (h,

+ hz) ,

(3.143)

(3.144)

Figure 3.65

Figure 3.64

Figure 3.66

Figure 3.67

5. Ungula of t h e Cylinder With the notation of Fig. 3.68 and with a = p/2 in radians we have:

h V = - [a(3R2 - a Z ) + 3 R Z ( b- R)a] = b 3b 2Rh M = -[(b - R). a] b where the formulas are valid even in the case b > R , ’p > r.

-acosa

+

(3.145) (3.146)

6. Hollow Cylinder With the notation R for the outside radius and T for the inside one, 6 = R - r for the differenceof the radii, and e =

R+r

for the mean radius (Fig. 3.69) we have: 2 V = x h ( R 2 - r 2 ) =7rh6(2R-6) = r h h ( Z r + h ) = 2 n h 6 ~ .

Figure 3.68

Figure 3.69

(3.147)

156

3. Geometry

7. Conical Surface arises by moving a line, the generating line, along a curve, the direction curve so that it always goes through a fixed point, the vertex (Fig. 3.70). 8. Cone (Fig. 3.71) is bounded by a conical surface with a closed direction curve and a base cut out from a plane by the surface. For an arbitrary cone

Figure 3.70

Figure 3.71

Figure 3.73

Figure 3.72

Figure 3.74

9. Right Circular Cone has a circle as base and its vertex is right above the centre of the circle (Fig. 3.72). With 1 as the length of the apothem and R as the radius of the base we have: 1 V = -TR'h, (3.149) M = T R ~ = T R (3,150) ~ ~ , S = a R ( R + l ) . (3.151) 3 10. Frustum of Right Cone or Truncated Cone (Fig. 3.73)

1=

Jm,

v = .rrh - (R' + r' 3

t Rr) ,

(3.152)

M=rl(R+r),

(3.154)

H=h+-

hr

R-r'

(3.153) (3.155)

11. Conic Sections see 3.5.2.9,p. 205.

12. Sphere (Fig. 3.74) with radius R and diameter D = 2R. Every plane section of it is a circle. A plane section through the center results in a great czrcle (see 3.4.1.1, p. 158) with radius R . Only one great circle can be fitted through two surface points of the sphere if they are not the endpoints of the same diameter. The shortest connecting surface curve between two surface points is the arc of the great circle between them (see 3.4.1.1,p. 158). Formulas for the surface and for the volume of the sphere:

3.3 Stereometry 157

S = 4nR'

x 12.57R2,

S= L'

=

%

A ~

S = A D'

(3.156a)

4.836 @, (3.156~)

D3 M 0.5236 D3, 6

'P

R = - - x 0.2821 2 7 7

RZ

(3.157b)

a,

3.142 D',

(3.156b)

4 V = - n R 3 x 4.189R3, 3

(3.157a)

x 0.094030,

(3.158a)

x 0.6204

a,

(3.157~) (3.158b)

I

13. Spherical Sector (Fig. 3.75)

S = A R(2 h + a ) ,

V = -2 ~ R ' h 3 '

(3.159)

(3.160)

14. Spherical Cap (Fig. 3.76) 1

a' = h(2 R

-

M = 27r R h

h),

= A

(u'

+ h') ,

I h(3u' t h') = ; h'(3 R - h ) ,(3.162)

(3.161)

V=

(3.163)

S = A (2 R h + u')

6

( + 2 u')

.

(3.164)

(3.165)

1 V = - ~ (3u' h + 3 b Z + h'),

(3.166)

(3.167)

S = 77(2Rh+u2 + b ' ) .

(3.168)

= A h'

15. Spherical Layer (Fig. 3.77)

R' =a' t

(

a2

-b' -hZ 2h

)',

= 2 71 R h,

6

If VI is the volume of a truncated cone written in a spherical layer (Fig. 3.78) and l is the length of its apothem then we have 1

V - v, = -7Thl'. 6

,

I-2a-I

(3.169)

I-2a-I

Figure 3.75

Figure 3.76

Figure 3.77

16. Torus (Fig. 3.79) is the solid we get by rotating a circle around an axis which is in the plane of the circle but does not intersect it.

S = 471' R r E: 39.48 Rr,

(3.170a)

S = 77' D d RZ 9.870 D d ,

(3.170b)

V = 2 ~ Rr' ' x 19.74 Rr',

(3.171a)

1 V = -n'Dd' 4

(3.171b)

2.467Dd'.

17. Barrel (Fig. 3.80) arises by rotation of a generating curve; a circular barrel by rotation of a circular segment, a parabolic barrel by rotation of a parabolic segment. For the circular barrel we have the approximation formulas

158

3. Geometry

Figure 3.78

V u 0.262h (2 D z + d z )

Figure 3.79 (3.172a) or T.'

Figure 3.80 FZ

0.0873 h ( 2 D + d)';

for the parabolic barrel .rrh 3 V = - ( 2 D 2 + D d + -4d z ) x 0.05236h ( 8 D z + 4 D d + 3 d 2 ) . 15

(3.172b)

(3.173)

3.4 Spherical Trigonometry For geodesic measures which extend over great distances we have to consider the spherical shape of the Earth. For this we need spherical geometry. In particular we need formulas for spherical triangles, Le., triangles lying on a sphere. This was also realized by the ancient Greeks, so besides the trigonometry of the plane the trigonometry of the sphere was developed, and we consider Hipparchus (around 150 BC) as the founder of spherical geometry.

3.4.1 Basic Concepts of Geometry on the Sphere 3.4.1.1 Curve, Arc, and Angle on t h e Sphere 1. Spherical Curves, Great Circle and Small Circle Curves on the surface of a sphere are called spherical curves. Important spherical curves are great circles and small circles. They are intersection circles of a plane passing through the sphere, the socalled intersecting plane (Fig. 3.81): If a sphere of radius R is intersected by a plane K at distance h from the center 0 of the sphere, then for radius T of the intersection circle we have r=d n (0 5 h 5 R). (3.174) For h = 0 the intersecting plane goes through the center of the sphere, and r takes the greatest possible value. In this case the intersection circle g in the plane r is called a great circle. Every other intersection circle, with 0 < h < R! is called a small circle, for instance the circle k in Fig. 3.81. For h = R the plane K has only one common point with the sphere. Then it is called a tangent plane. IOn the Earth the equatorand the meridianswith their countermeridians - which are their reflections with respect to the Earth's axis - represent great circles. The parallels of latitude are small circles (see also 3.4.1.2, p. 160).

2. Spherical Distance Through two points A and B of the surface of the sphere, which are not opposite points, i.e.. they are not the endpoints of the same diameter, infinitely many small circles can be drawn, but only one great circle (with the plane of the great circle 9 ) . Consider two small circles kl,kz through A and B and turn them into the plane of the great circle passing through A and B (Fig. 3.82). The great circle has the greatest radius and so the smallest curvature. So the shorter arc of the great circle is the shortest connection between A and B. It is the shortest connection between A and B on the surface of the sphere. and it is called the spherical distance.

3.4 Spherical Tv-igonornetrv 159

Figure 3.81

Figure 3.82

Figure 3.83

3. Geodesic Lines Geodeszc lines are the curves on a surface which are the shortest connections between two points of the surface (see 3.6.3.6, p. 250). In the plane the straight lines, on the sphere the great circles, are the geodesic lines (see also 3.4.1.2, p. 160).

4. Measurement of the Spherical Distance The spherical distance of two points can be expressed as a measure of length or as a measure of angle (Fig. 3.83). and 1. Spherical Distance as a Measure of Angle is the angle between the radii measured at the center 0. This angle determines the spherical distance uniquely, and in the following we denote it by a lowercase Latin letter. The notation can be given at the center or on the great circle arc. 2. Spherical Distance as a Measure of Length is the length of the great circle arc between A

-

and B. It is denoted by AB (arc AB). 3. Conversions from Measure of Angle into Measure of Length and conversely can be done by the formulas

,-.

AB=Rarce=Re, 0

(3.175a)

- e-. e =AB

(3.175b)

R

Here e denotes the angle given in degrees and arc e denotes the angle in radian (see radian measure 3.1.1.5, p. 130). The conversion factor Q is equal to 180' g = 1rad = -= 57.2958' = 3438' = 206265". (3.175,) T

The determinations of the distance as a measure of length or angle are equivalent but in spherical trigonometry the spherical distances are given mostly as a measure of angle. A: For spherical calculations on the Earth's surface we consider a sphere with the same volume as the biaxial reference ellipsoid of Krassowski. This radius of the Earth is R = 6371.110 km, and consequently we have 1" A 111.2km, 1' 1853.3m = 1oldseamile. Today 1seamile = 1852 m. h

B: The spherical distance between Dresden and St. Petersburg is AB = 1433 km or 1433 km e=57.3" = 12.89' = 12"53'. 6371 km 5 . Intersection Angle, Course Angle, Azimuth The zntersectzon angle between spherical curves is the angle between their tangent lines at the intersection point PI, If one of them is a meridian, the intersection angle with the curve segment to the north

160

3. Geometry

Figure 3.84 from PI is called the course angle (Y in navigation. To distinguish the inclination of the curve to the east or to the west, we assign a sign to the course angle according to Fig. 3.84a,b and we restrict it to the interval -90" < a 5 90". The course angle is an oriented angle, Le., it has a sign. It is independent of the orientation of the curve - of its sense. The orientation of the curve from PI to Pz,as in Fig. 3.84c, can be described by the azimuth 6: It is the intersection angle between the northern part of the meridian passing through PI and the curve arc from PI to Pz. We restrict the azimuth to the interval 0" 5 6 < 360". Remark: In navigation the position coordinates are usually given in sexagesimal degrees; the spherical distances and also the course angles and the azimuth are given in decimal degrees.

3.4.1.2 Special Coordinate Systems 1. Geographical Coordinates To determine points P on the Earth's surface we use geographical coordinates (Fig. 3.85), i.e., spherical coordinates with the radius of the Earth, the geographical longitude X and the geographzcal latitude (o. To determine the degree of longitude we subdivide the surface of the Earth by half great circles from the north pole to the south pole, by so-called meridians. The zero meridian goes through the observatory of Greenzuzch. From here we count the east longitude with the help of 180 meridians and the west longitude with 180 meridians. At the equator they are at a distance of 111km of each other. East longitudes are given as positive, west longitudes are given as negative values. So -180" < X 5 180".

Figure 3.85

Figure 3.86

To determine the degree of latitude the Earth's surface is divided by small circles parallel to the equator. Starting from the equator we count 90 degrees of latitude to the north, the northings, and 90 southern latitudes. Northings are positive, southern latitudes are negative. So -90" 5 (o 5 90'.

2. Soldner Coordinates The right-angled Soldner coordinates and Gauss-Kriiger coordinates are important in wide surface surveys. To map part of the curved Earth's surface onto a right-angled coordinate system in a plane, distance preserving in the ordinate direction, according to Soldner we place the x-axis on a meridian (we call it a central meridian), and the origin at a well-measured center point (Fig. 3.86a). The ordinate y of a point P is the segment between P and the foot of the spherical orthogonal (great circle) on the

3.4 Spherical Tkiqonometrg 161

central meridian. The abscissa 2 of the point Pis the segment of a circle between P and the main parallel passing through the center, where the circle is in a parallel plane to the central meridian (Fig. 3.86b). Transferring the spherical abscissae and ordinates into the plane coordinate system the segment AX is stretched and the directions are distorted. The coeficient of elongation a in the direction of the abscissa is AX a=-xI+yz R=6371km. (3.176) AX, 2R2 ' To moderate the stretching, the system may not be extended more than 64 km on both sides of the central meridian. A segment of 1km length has an elongation of 0.05 m at y = 64 km.

3. Gauss-Kriiger Coordinates To map part of the curved Earth's surface onto the plane with an angle-preserving (conformal) mapping, first we prepare the subdivision of the Gauss-Kriiger system into meridian zones. For Germany these mid-meridians are at 6", 9", 12") and 15" east longitude (Fig. 3.87a). The origin of every meridian zone is at the intersection point of the mid-meridian and the equator. In the north-south direction we consider the total range, in the east-west direction we consider a l"40' wide strip on both sides. In Germany it is ca. H O O km. The overlap is 20', which is here nearly 20 km.

y - elongation

Ax a) l e 5 0 km

-d

The coefficient of elongation a in the abscissa direction (Fig. 3.87b) is the same as in the Soldner system (3.176). To keep the mapping anglepreserving, we add to the ordinates the quantity b: 7.

Figure 3.87

b=

6RZ'

(3.177)

3.4.1.3 Spherical Lune or Biangle Suppose there are two planes rl and r2 passing through the endpoints A and B of a diameter of the sphere and enclosing an angle a (Fig. 3.88) and so defining two great circles gl and gz. The part of the surface of the sphere bounded by the halves of the great circles is called a spherical lune or bzangle or spherical digon. The sides of the spherical biangle are defined by the spherical distances between A and B on the great circles. Both are 180". We consider the angles between the tangents of the great circles g1 and g2 at the points A and B as the angles of the spherical biangle. They are the same as the so-called dihedral angle a between the planes rl and r2. If C and D are the bisecting points of both great circle arcs A and B , the angle (Y can be expressed as the spherical distance of C and D. The area Ab of the spherical biangle is proportional to the surface area of the sphere just as the angle a to 360". Therefore the area is Ab

=

47~R 2 a ~

360"

-

p

= 2 R2 a r c a with the conversion factor pas in (3.175~).

(3.178)

3.4.1.4 Spherical Triangle Consider three points A, B , and C on the surface of a sphere, not on the same great circle. If we connect every two of them by a great circle (Fig. 3.89), we get a spherical triangle ABC. The sides of the triangle are defined as the sphericaldistances of the points, Le., they represent the angles at the center between the radii OA, OB, and OC. They are denoted by a, b, and e, and in the following they are given in angle measure, independently of whether they are denoted at the center as angles or on the surface as great circle arcs. The angles of the spherical triangle are the angles between every two planes of the great circles. They are denoted by a, p, and y. The order of the notation of the points, sides, and angles of the spherical triangle follows the same

162

3. Geometry

Figure 3.88

Figure 3.89

Figure 3.90

scheme as for triangles of the plane. A spherical triangle is called a right-sided triangle if at least one side is equal to 90”. There is an analogy with the right-angled triangles of the plane.

3.4.1.5 Polar Triangle 1. Poles and Polar The endpoints of a diameter PI and Pz are called poles, and the great circle g being perpendicular to this diameter is called polar (Fig. 3.90). The spherical distance between a pole and any point of the great circle g is 90”. The orientation of the polar is defined arbitrarily: Traversing the polar along the chosen direction we have a left pole to the left and a right pole to the right. 2. Polar Triangle A’B’C’ of a given spherical triangle A, B , C is a spherical triangle such that the vertices of the original triangle are poles for its sides (Fig. 3.91). For every spherical triangle ABC there exists one polar triangle A’B’C’. If the triangle A’B’C’ is the polar triangle of the spherical triangle ABC , then the triangle ABC is the polar triangle of the triangle A’B’C’. The angles of a spherical triangle and the corresponding sides of its polar triangle are supplementary angles, and the sides of the spherical triangle and the corresponding angles of its polar triangle are supplementary angles: a’ = 180° - CY, b’ = 180° - p, C’ = 180° - yI (3.179a) CY’.

= 180” - a,

/3’ = 180” - b,

Figure 3.91

= 180” - c.

(3.179b)

Figure 3.92

3.4.1.6 Euler Triangles and Non-Euler Triangles The vertices A, B , C of a spherical triangle divide every great circle into two, usually different parts. Consequently there are several different triangles with the same vertices, e.g., also the triangle with sides a’, b, c and the shadowed surface in Fig. 3.92a. According to the definition of Euler we should always choose the arc which is smaller than 180” as a side of the spherical triangle. This corresponds

3.4 Spherical Trigonometry 163

to the definition of the sides as spherical distances between the vertices. Considering this, we call the spherical triangles all of whose sides and angles are less than 180" Euler trzangles, otherwise we call them non-Euler triangles. In Fig. 3.92b there is an Euler triangle and a non-Eder triangle.

3.4.1.7 Trihedral Angle This is a three-sided solid formed by three edge sa, sb, s, starting at a vertex 0 (Fig. 3.93a). As sides of the trihedral angle we define the angles a, b, and c, every of them enclosed by two edges. The regions between two edges are called the faces of the trihedral angle. The angles of the trihedral angle are a , p, and 7 , the angles between the faces. If the vertex of a trihedral angle is at the center 0 of a sphere, it cuts out a spherical triangle of the surface (Fig. 3.93b). The sides and the angles of the spherical triangle and the corresponding trihedral angle are coincident, so every theorem derived for a trihedral angle is valid for the corresponding spherical triangle, and conversely.

Figure 3.93

3.4.2 Basic Properties of Spherical Triangles 3.4.2.1 General Statements For an Euler triangle with sides a, b, and c, whose opposite angles are a, /3, and ?>the following statements are valid: 1. Sum of t h e Sides The sum of the sides is between 0" and 360": 0" < a b c < 360". (3.180) 2. Sum of two Sides The sum of two sides is greater than the third one, e.g., a+b>c. (3.181) 3. Difference of Two Sides The difference of two sides is smaller than the third one, e.g., la - bl < C. (3.182) 4. Sum of t h e Angles The sum of the angles is between 180" and 540': 180" < a + 9 + y < 540'. (3.183) 5. Spherical Excess The difference E = + /3 + 7 - 180° (3.184) is called the spherical excess. 6. Sum of Two Angles The sum of two angles is less than the third one increased by 180", e.g., a ,3 < Y 180'. (3.185) 7. Opposite Sides and Angles Opposite to a greater side there is a greater angle, and conversely. 8. Area The area AT of a spherical triangle can be expressed by the spherical excess E and by the radius of the sphere R with the formula

+ +

+

+

(3.186a)

I

164

3. Geometry

Here pis the conversion factor (3.175~).From the theorem of Girard, with As as the surface area of the sphere, we have AS AT = --E. (3.186b) 720” If we know the sides and not the excess, then E can be calculated by the formula of L’Huilier (3.201).

3.4.2.2 Fundamental Formulas and Applications The notation for the quantities of this paragraph corresponds to those of Fig. 3.89.

1. SineLaw sin a - sin a sinb sinfl’

(3.187a)

-sin =b-

sinc

sinp siny’

(3.187b)

sinc siny . (3.187~) sina sina

The equations from (3.187a) to (3.187~)can also be written as proportions, Le., in a spherical triangle the sines of the sides are related as the sines of the opposite angles: sina sinb sinc -- - - - -(3.187d) sincy sinp siny ’ The sine law of spherical trigonometry corresponds to the sine law of plane trigonometry.

2. Cosine Law, or Cosine Law for Sides cos a = cos bcos c + sin bsin ccosa, (3.188a)

cos b = cosccos a + sin csin a cos p, (3.18813)

(3.188~) cos c = cos acos b + sin asin bcos y. The cosine law for sides in spherical trigonometry corresponds to the cosine law of plane trigonometry. Ffom the notation we can see that the cosine law contains the three sides of the spherical triangle. 3. Sine-Cosine Law sin a cos,3 = cos bsin c - sin bcos ccoscy, (3.189a) sinacosy=coscsinb-sinccosbcosa. (3.18913) We can get four more equations by cyclic change of the quantities (Fig. 3.34). The sine-cosine law corresponds to the projection rule of plane trigonometry. Because it contains five quantities of the spherical triangle, we do not use it directly for solving problems of spherical triangles, but we usually use it for the derivation of further equations.

4. Cosine Law for Angles of a Spherical Triangle COS^=

-cos/3cosr+sinPsinycosa,

(3.190a)

cos3!, = -cos ycos a + sin y sina cos b, (3.190b) (3.190~) cos”/ = - cos cy cos p sin asin pcos c. This cosine rule contains the three angles of the spherical triangle and one of the sides. With this law we can easily express an angle by the opposite side with the angles on it, or a side by the angles; consequently every side can be expressed by the angles. Contrary to this, for plane triangles we get the third angle from the sum of 180”. Remark: It is not possible to determine any side of a plane triangle from the angles, because there are infinitely many similar triangles.

+

5. Polar Sine-Cosine Law (3.191a) sin cycos b = cos psin y +sin pcos y cos a, (3.191b) sin cy cos c = cosy sin p + sin y cos /3 cos a. Four more equations can be get by cyclic change of the quantities (Fig. 3.34). Just as for the cosine law for angles, also the polar sine-cosine law is not usually used for direct calculations for spherical triangles, but to derive further formulas.

3.4 Spherical Trigonometry 165 6 . Half-Angle Formulas To determine an angle of a spherical triangle from the sides we can use the cosine law for sides. The half-angle formulas allow us to calculate the angles by their tangents, similarly to the half-angle formulas of plane trigonometry: tan- = 2

sin(s - b) sin(s - c) (3.192a) sinssin(s-a) '

tan- = 2

tan- = 2

sin(s - a) sin(s - b) (3.192~) sin s sin(s - c) '

s = -,

sin(s - c) sin(s - a) , (3.19213) sins sin(s - b)

a+b+c 2

(3.192d) If from the three sides of a spherical triangle we want to determine all the three angles, the following calculations are useful: a k B k tan - = 7 (3.193a) tan - = (3.193b) 2 sin(s - a ) ' 2 sin(s - b)' ~

tan Y= with 2 sin(s - c)

(3.193~)

~

sin(s - a) sin(s - b ) sin(s - c) , (3.193d) sin s

k=

a+b+c 2 .

S=-

(3.193e)

7. Half-Side Formulas With the half-side formulas we can tell one side or all of the sides of a spherical triangle from its three angles: cos(0 - P ) COS(' - Y) ,(3,194a) - cos u cos(a - 0)

cot - = 2

cos(u - a ) cos(u - p )

cot - = 2

- cos u cos(u -

y)

, (3.194~)

cos(u - y) cos(0 - a ) , (3.194b) - cos u cos(u - P )

cot - = 2

g=-

@+P+Y 2

(3.194d)

'

or a k' cot - = 2 cos(u - a ) '

(3.195a)

~

c

cot - = kt 2 cos(u - y) ~

k' =

b k' cot - = 2 cos(u - P ) ' ~

with

(3.19513)

(3.195~)

cos(u - a ) cos(0 - p) cos(0 - y) (3.195d) - cos u ~

0

=

"+P+Y ~

2

,

(3.195e)

Because for the sum of the angles of a spherical triangle according to (3.183): 180" < 20. < 540" or 90" < u < 270" (3.196) holds; cos u < 0 must always he valid. Because of the requirements for Euler triangles all the roots are real.

I

166

3. Geometry

N

Figure 3.94

Figure 3.95

8. Applications of the Fundamental Formulas of Spherical Geometry With the help of the given fundamental formulas we can determine, for instance distances, the azimuth, and course angles on the Earth. IA: Determine the shortest distance between Dresden (A1 = 13"46', p1 = 51'16') and Alma .4ta (A, = 76'55'. ~2 = 43'18'). Solution: The geographical coordinates (Al. v1). (Az, 92) and the north pole N (Fig. 3.94) result in two sides of the triangle PlPplr a = 90" - 9 2 and b = 90" - 91 lying on meridians, and also the angle between them y = A2 - A I . For c = e it follows from the cosine law (3.188a) cos c = (cos a cos b sin a sin bcos c ) $ cose = cos(90° - p1) cos(90n - p2) t sin(90" - pl) sin(90" - 'p2) cos(Az - A,) = sin 91sin p2 cos p1cos 92 cos(A:, - A I ) , (3.197)

+

+

-

Le.. cose = 0.53498 t 0.20567 = 0.74065,e = 42.213'. The great circle segment PIPz has length 4694 km using (3.175a). IB: Calculate the course angles 61 and 62 at departure and at arrival, and also the distance in sea miles of a voyage from Bombay (XI = 72"48', p1 = 19'00') to Dar es Saalam (A, = 39"28', 9, = -6'49') along a great circle. Solution: The calculation of the two sides a = 90" - pl = 71"00', b = 90" - p2 = 96'49' and the enclosed angle y = A1 - A:, = 33"20' in the spherical triangle PlPziV with the help of the geographical coordinates (AI. p l ) ,(Az,pz) (Fig. 3.95) and the cosine law (3.188~)cosc = cose = cosacosb t h

h

sinasin bcos;, yields PIP.= e = 41.777'> and because 1' x I s m we have PlP2x 2507sm. With the cosine law for sides we get a = arccos

cos a - cos b cos c COS b - COS u COS c = 51.248" and 0 = arccos = 125.018" sin b sin c sin a sin c

Therefore, we get d1 = 360" - 3 = 234.982O and J2 = 180° + cy = 231,248".

Remark: It makes sense to use the sine law to determine sides and angles only if it is already obvious from the problem that the angles are acute or obtuse.

3.4.2.3 F u r t h e r Formulas 1. Delambre Equations Analogously to the hlollweide formulas of plane trigonometry the corresponding formulas of Delambre are valid for spherical triangles:

3.4 Spherical Trigonometry 167

a-3

cos +=-

sin 2

0-P sin -

sinpa + b

2,

(3.198a)

+=-

sin 2

cos 2

sine 2c r sin 2

(3.198b)

a+b a-6 sin a+p coscos - cos2 . (3.198~) 2 2 (3.198d) -/ sin cos cos cos 1 2 2 2 2 Because for every equation two more equations exist by cyclic changes, altogether there are 12 Delambre equations.

2. Neper Equations and Tangent Law

0.-3 0-a sin a + 6 costan =2 tan !, (3.199~) tan= cu tan (3.199d) 0.+P 2 2 sincos 2 2 These equations are also called Neper analogies. From these we can derive formulas analogous to the tangent law of plane trigonometry: a -b

~

~

a-b

a-a

tan 2 - (3.200a) 2 = a+b 0.tD' tan - tan 2 2 tan

~

tan e - a c+a tan 2

b-c tan -

3 5. B-y

tan 2= 2 - (3.200b) 6+c Btr' tan tan 2 2

7-0 tan ?+a' tan 2

2= 2-

(3.200~)

3. L'Huilier Equations We can calculate the area of a spherical triangle with the help of the excess 6 . It can be calculated from the known angles a , 8, y according to (3.184), or if the three sides a, b, c are known, with the formulas about the angles from (3.193a) to (3.193e). The L'Huilier equation makes possible the direct calculation of E from the sides: s s-a s-b s-c tan -tan -. 4 2 2 This equation corresponds to the Heron formula of plane trigonometry.

(3.201)

3.4.3 Calculation of Spherical Triangles 3.4.3.1 Basic Problems, Accuracy Observations The different cases occurring most often in calculations of spherical triangles are arranged into so-called basic problems. For every basic problem for acute-angled spherical triangles there are several ways to solve it. and it depends on whether we base our calculations only on the formulas from (3.187a) to (3.191b) or also on the formulas from (3.192a) to (3.201), and also whether we are looking for only one

I

168 3. Geometrv

or more quantities of the triangle. Formulas containing the tangent function yield numerically more accurate results, especially in comparison to the determination of defining quantities with the sine function if they are close to go", and with the cosine function if their value is close to 0" or 180". For Euler triangles the quantities calculated with the sine law are bivalued, since the sine function is positive in both of the first quadrants, while the results obtained from other functions are unique.

3.4.3.2 Right-Angled Spherical Triangles 1. Special Formulas In a right-angledspherical triangle one of the angles is 90". The sides and angles are denoted analogously to the plane right-angled triangle. If as in Fig. 3.96 y is a right angle, the side cis called the hypotenuse, a and 6 the legs and a and p are the leg angles. From the equations from (3.187a) to (3.201) it follows for y = 90": (3.202a)

sin b = sin p sin c,

(3.20213)

(3.202~)

cosc = cot a c o t p,

(3.202d)

tan a = cos 4 tan c,

(3.202e)

tan b = cos a tan c,

(3.2020

tan b = sina tan 4,

(3.202g)

tan a = sin b tan a ,

(3.202h)

cosa = sinpcosa,

(3.2021)

cos p = sin a cos 6.

(3.2023)

sin a = sin a sin c, COS c

= COS ucos 6,

n

90"-b

90'-a

90" C

Figure 3.96

Figure 3.97

If in certain problems other sides or angles are given, for instance instead of a ,p, y the quantities b, y, a , we get the necessary equations by cyclic change of these quantities. For calculations in a right-angled spherical triangle we usually start with three given quantities, the angle y = 90" and two other quantities. There are six basic problems, which are represented in Table 3.8.

Basic Given defining problem quantities 1.

2.

1 Hypotenuse and a leg c, a 1 Two legs a, b

N u m b e r of the formula t o determine other quantities

I a (3.202a), p (3.202e), b (3.202~) I a (3.202h),p (3.202g), c (3.202~)

3.

Hypotenuse and an angle c, a

a (3.202a), b (3.202f), @ (3.202d)

4.

Leg and the angles on it a, p

c (3.202e), b (3.202j), a (3.2021)

5.

Leg and the angle opposite to it a, a b (3.202h), c (3.202a), /3 (3.202i)

6.

Two angles a, p

a (3.202i), b (3.2023), c (3.202d)

3.4 Spherical Trigonometry 169

2. Neper Rule The Neper rule summarizes the equations (3.202a) to (3.202j). If the five determining quantities of a right-angled spherical triangle, not considering the right angle, are arranged along a circle in the same order as they are in the triangle, and if the legs are replaced by their complementary angles 90" - a, 90" - b. (Fig. 3.97), then the following is valid: 1. The cosine of every definingquantity is equal to the product of the cotangent values of its neighboring quantities. 2. The cosine of every defining quantity is equal to the product of the sine values of the non-neighboring quantities. tan b IA: cosu = cot(9O0- b) cot c = - (see (3.2024. tan c B: cos(90" - a ) = sincsincu = sina (see (3.202a)). IC: hlap the sphere onto a cylinder which contacts the sphere along a meridian, the so-called central merzdzan. This meridian and the equator form the axis of the Gauss-Kruger system (Fig. 3.98a,b).

Figure 3.98 Figure 3.99 Solution: .A point P of the surface of the sphere will have the corresponding point P'on the plane. The great circle g passing through the point P perpendicular to the central meridian is mapped into a line g' perpendicular to the x-axis, and the small circle k passing t,hrough P parallel to the given meridian becomes a line k' parallel to the x-axis (grid meridian). The image of the meridian m through P is not line but a curve rn' (true meridian). The upward direction of the tangent of m' at P' gives the geographical north, the upward direction of k' gives the grid north direction. The angle y between the two north directions is called the meridian convergence. In a right-angled spherical triangle QPN with c = 90n - p , and b = q we get y from u = 90" - y. The tan b tan 7 Neper rule yields cosu = -or cos(90" - y) = , sin y = tan 17 tan p. Because */ and tan c tan(90" - p) q are mostly small, we consider sin y x yI tan q x 7; consequently y = q tan p is valid. The length deviation y of this cylinder sketch is pretty small for small distances 7, and we can substitute 0 = y/R, where y is the ordinate of P . We get y = (y/R) tancp. The conversion of y from radian into degree measure yields a meridian convergence of y = 0.018706 or y = l"O4'19'' at = 50°, y = 100 km.

3.4.3.3 Spherical Triangles with Oblique Angles For three given quantities, we distinguish between six basic problems, just as we did for right-angled spherical triangles. The notation for the angles is cy, 8,y and a, b, c for the opposite sides (Fig. 3.99). Tables 3.9,3.10, 3.11, and 3.12 summarize, which formulas should be used for which defining quantities in the case of the six basic problems. Problems 3, 4, 5, and 6 can also be solved by decomposing the general triangle into two right-angled triangles. To do this for problems 3 and 4 (Fig. 3.100, Fig. 3.101) we use the spherical perpendicular from B to AC to the point D,and for problems 5 and 6 (Fig. 3.102) from C to A B to the point D.

170

3. Geometry

Figure 3.100

Figure 3.101

Figure 3.102

In the headings of Tables 3.9,3.10,3.11,and 3.12 the given sides and angles are denoted by S and W respectively. This means for instance SWS: Two sides and the enclosed angle are given. Table 3.9 First and second basic problems for spherical oblique triangles

First basic problem Given: 3 sides a, b, c Conditions: 0" < a + b + c < 360", a+b>c. a+c>b, b+c>a.

Isss/

Solution 1: Required a.

cos a + cos p cos y or sin p sin y cos(u - p) cos(u - y) cot - = ' 2 - cos 0 codu - a) cosa =

"d

sin(s - b) sin(s - c ) sin s sin(s - a ) ' a+b+c s =2 Solution 2: Required a , p, y . tan - = 2

"d

0

=-f f + P + Y

2 ' Solution 2: Required a, b, c . k' = cos(u - a)cos(0 - P)cos(u - 7) k = - cos u a ' k k a ' k' b k' cot - = tan- = tan - = cot - = ___ 2 sinis - a ) ' 2 sin(s 2 cos(u- a ) ' 2 cos(u' ~,- b) ' Y 'k ' r k' tan- = cot - = 2 sin(s- c ) ' 2 cos(0-y) ' Checking: ( s - a ) + ( s - b) + (s - c) = s , Checking: ( u - a ) (u - 0)+ (u - y) = u , a b c a tan P tan -r sins = k . tan cot -cot -cot -(- COS^) = k'. 2 2 2 2 2 2

a

~

~

~

a)

~

~

+

IA Tetrahedron: A tetrahedron has base ABC and vertex S (Fig. 3.103). The faces ABS and BCS intersect each other at an angle 74"18'. BCS and C.4S at an angle 63"40'. and CAS and ABS at 8O"OO'. How big are the angles between every two of the edges AS, B S , and CS? Solution: Froma spherical surface around the vertex Softhe pyramid, the trihedral angle (Fig. 3.103) cuts out a spherical triangle with sides a, b, c. The angles between the faces are the angles of the spherical triangle. the angles between the edges, we are looking for, are the sides. The determination of the angles a , b, c corresponds to the second problem. Solution 2 in Table 3.9 yields: u = 108"59', u - CY = 28"59', u - p = 34'41', 0 - y = 45"19', k' = 1.246983, ~0t(a/2) = 1.425514, cot(b/2) = 1.516440, cot(c/2) = 1.773328.

3.4 Spherical Triqonometru

Figure 3.103

171

Figure 3.104

Table 3.10 Third basic problem for spherical oblique triangles

Third basic problem jsws Given: 2 sides and the enclosed angle, e.g., a, b, y Condition: none , a-6 .-p-sin2 7 Solution 1: Required c, or c and a . tan- 2 , a+bCotT cos c = cos a cos b + sin a sin bcos 7 , 7 sin sin a sin y sina = ~. sin c (-goo'< .-P 2 < goo) a can be in quadrant I. or 11. a+p cy-p a+p a-p We apply the theorem: a=-+, p=--2 2 2 2 ' Larger angle is opposite Solution 4: Required a , p, c . to the larger side or y a-b checking- calculation: cos -cos a+P 2 2,tana in q. I. 2 a+b y N' cosa - cos bcosc 0 + a in q, 11, cos -sin 2 2 Solution 2: Required a , or cy and c . a-b y tanu = tanacosy cy - p sin-cos21 2 2 --tantan y sin u . a + b . y N' 2 tana = 7 sin sin ; sin(b - u ) ' a+'a tan@ - u ) (-90" < 7 < gon) tanc = cos a a Solution 3: Required a and (or) p . a = , p=---+ $ c y - B a-b 2 2 2 2 ' a + 3 cos 2 cot 2 . C z . c Z' tan -cos- = sin - = a+b 2 2 a+P' 2 ff-p cossin sin 2 2 2 Checking: Double calculation of c .

z

~

~

~

~

W B Radio Bearing: In the case of a radio bearing two fixed stations Pl(A1,pl) and P*(A*, pz)receive the azimuths 61 and 62 via the radio waves emitted by a ship (Fig. 3.104).We have to determine the geographical coordinates of the position Po of the ship. The problem, known by marines as the shore-

172

3. Geometrv

to-shzp bearing, is an intersection problem on the sphere, and it will be solved similar to the intersection problem in the plane (see 3.2.2.3, p. 147). 1. Calculation in triangle PlP2N:In triangle PIPz.N the sides P I N = 90" - p,, P2N = 90" - pz and the angle 0: PllYP2 = - XI = Ax are given. The calculations of the angle 3: ~1~ ~2 and the segment PIPz = e correspond to the third basic problem. 2. Calculation in the triangle PlP& Because t1= J1 - E ~
+

Table 3.11 Fourth basic problem for spherical oblique triangles

Fourth basic problem Given: O n e side a n d two adjacent angles, e.g., a,p, c Condition: none Solution 1: Required a/, or y and a . cos -/ = -cos acos P sin a sin Bcosc, a-b , sincsina (-900 < -< goo), sin a = v . 2 sin a = - +a -+, b a - b b = - -a-+- -b a can be in quadrant I or 11. We ap2 2 ply the theorem: The larger side is in Solution 4: Required a , b, y .2 opposite to the larger angle or checking calculation: a in q. I. cos cy + cos 13 cos -/ 0 --f cy in q, 11, Solution 2: Required a , or a and y . tan c cos p cotp = tanacosc. tana = COS( R - p ) ' cot($ - p) tan? = -. cos a Solution 3: Required a and (or) b a+b a-b a t b a-P b=--a=-+--, a + b cos2 2 2 tan=2 a+fl tan-, 2 2 cos -

/wsw

+

a-b 2 .

:

~

'

a-b 2 >

2

a -b

.-P

sin tan -= 2 tan 2 atP 2 sin

L

L

Checking: Double calculation of y.

3.4.3.4 Spherical Curves Spherical trigonometry has a very important application in navigation. One basic problem is to determine the course angle, which gives the optimal route. Other fields of application are geodesic surveys and also robot-movement planning.

3.4 Spherical Trigonometry 173 Table 3.12 Fifth and sixth basic problems for a spherical oblique triangle Fifth basic problem Given: 2 sides and a n angle opposite t o one of them, e.@;.,a, b, a Conditions: See distinction of cases. Solution: Required is any missing quantity. sin b sin a sinb = 2 values PI, PZare possible. sin a Let B1 be acute and 82 = 180" - obtuse. Distinction of cases: sin b sin a 1. > 1 Osolution. sin a sin b sin a 2. -- 1 1 solution p = 90" sin a sin b sin a 3. <1: sin a 3.1. sina > sin b : 3.1.1. b < 90" 1 solution PI. 3.1.2. b > 90n 1 solution 02. 3.2. sin a < sin b : ~

~

~

a

< 90" a > 90"

Further calculations with one angle or with two angles /3 : Method 1: Method 2:

Given: 2 angles and a side opposite t o one of them, e.g., a, a,/3 Conditions: See distinction of cases. Solution: Required is any missing quantity. sin a sin 0 sin b = 2 values b l , bz are possible. sin a Let bl be acute and b2 = 180" - Dl obtuse. Distinction of cases: sin a sin 4 1. > 1 0 solution. sin a sin a sin P 2. 7 = 1 1solution b = 90". sin a sin a sin p 3. 7 < 1 : sin a 3.1. sin a > sin P. 3.1.1. p < 90n 1 solution bl. 3.1.2. p > 90" 1 solution b2. 3.2. sin a < sin 0 : a < go", a < 90" 2 solutions 3'2'1' a > go", a > 90" bl, b2. a < 90" , a > 90" ~

~

Further calculations with one side or with two sides b :

a-b

a+P

tanu = tanbcosa, tanv = tanacoso, c =u+v, cot9 = c o s b t a n a , cot w = cos a tan B ,

Y

pivg

Sixth basic problem

c a + b c o S 2 tan - = tan -~ 2 2 a-P' cos 2 ff+P ,-bsin2 = tan -~ 2 . a-0' sin 7

=p+v'.

Checking: Double calculation of

y

a + B C O S T

tan - = cot -~ 2 . a+b' 2 sin 2 a-b a - p sin 2 = cot -2 , a + b ' sin -

Y and ;

1. Orthodrome 1. Notion The geodesic lines of the surface of the sphere -which are curves, connecting two points

A and B by the shortest path are called orthodromes or great czrcles (see 3.4.1.1, 3., p. 159). 2. Equation of t h e Orthodrome Moving on an orthodrome - except for meridians and the equator ~

-needs a continuous change of course angle. These orthodromes with position-dependent course angles a can be given uniquely by their point closest to the north pole J"(AN, p ~ )where , p~ > On holds. The orthodrome has the course angle CYN = 90" at the point closest to the north pole. The equation of the orthodrome through PN and the running point &(A, 9))whose relative position to Px is arbitrary, can be given by the Neper rule according to Fig. 3.105 as: =tanp (3.203) tanpNcos(X -A,)

174

3. Geometrv

Point Closest t o the North Pole: The coordinates of the point closest to the north pole Px(Xx. px) of an orthodrome with a course angle (YA (CYA # OD) at the point A(Xa, (FA) ( p #~ 90") can be calculated by the Yeper rule considering the relative position of PNand the sign of (YA according to Fig. 3.106 as: px = arccos(sin

IUA~

cos y ~ )(3.204a)

and Ax = XA + sign(cuA)

. (3.204b)

Remark: If a calculated geographical distance X is not in the domain -180" < X 5 180". then for X # ikk 180" (IC E W)the reduced geographical distance Xred is: (3.206) IVe call this the reduction of the angle in the domain.

Figure 3.105

Figure 3.106

Intersection Points with the Equator: The intersection points PE,( X E 0") ~ ~ and PE2(XE2, 0') of the orthodrome with the equator can be calculated from (3.203) because t a n p x COS(XE, - AN) = 0 ( u = 1.2) must hold: XE,, = 7 90" ( u = 1:2). (3.206) Remark: In certain cases an angle reduction is needed according to (3.205). 3. Arclength If the orthodrome goes through the points A(XA,p ~and ) B(XB.pB),the cosine law for sides yields for the spherical distance d or for the arclength between the two points: d = arccos[sin p~ sin ( F B t COS p~ COS ( F BCOS(XB - XA)]. (3.207a) We can convert this central angle into a length considering the radius of the Earth R: d = arccos[sin p~ sin p~

+

COS

p~ cos p~ COS(XB - A,)]

iTR -.180"

(3.207b)

4. Course Angle If we use the sine and cosine laws for sides to calculate sin CIA and cos CY*, we get

the final result for the course angle 0 . 4 after division: COS PACOS p~ sin(XB - A ), ' 04 = arctan (3.208) sin p~ - sin pAcos d Remark: With the formulas (3.207a), (3.208), (3.204a) and (3.204b), the coordinates of the point closest to the north pole PN can be calculated for the orthodrome given by the two points A and B. 5 . Intersection Point witha Parallel Circle For the intersection points Xl(Xx,,px) and X2(Xx2, px) of an orthodrome with the parallel circle p = px we get from (3.203): tan px Ax, = Ax arccos - ( u = 1 , 2 ) (3.209) tan p~

3.4 Spherical Riqonometru 175 From the Neper rule we have for the intersection angles ax, and axl, by which an orthodrome with a point closest to the north pole PN(XN,(PN)intersects the parallel circle cp = cpx: cos px lax, 1 = arcsin - (v = 1,2) (3.210) cos cpx The argument in the arc sine function must be extrema1 with respect to the variable cpx for the minimal course angle lamini.We get: sincpx = 0 + (PX = 0 , Le.. the absolute value of the course angle is minimal at the intersection points of the equator: (3.211) lax,,,l = 90" - 9 N . Remark 1: Solutions of (3.209) exist only for 1~x15 c p ~ . Remark 2: In certain cases a reduction of the angle is needed according to (3.205). 6. Intersection Point w i t h a Meridian We get for the intersection point Y(Xu,cpy) of an orthodrome with the meridian X = Xu according to (3.203): py = arctan[tanpvcos(Xy - A N ) ] . (3.212) 2. Small C i r c l e 1. Notion The definition of small a circle on the surface of the sphere must be discussed here in more detail than in 3.4.1.1. p. 158: The small circle is the locus of the points of the surface (of the sphere) ) 3.107). We at a spherical distance r ( r < 90") from a point fixed on the surface M ( X M ,c p ~ (Fig. denote the spherical center by M:r is called the spherical radius ofthe small circle. The plane of the small circle is the base of a spherical cap with an altitude h (see 3.3.4. p. 157). The spherical center M is above the center of the small circle in the plane of the small circle. In the plane the circle has the planar radius of the small circle ro (Fig. 3.108). Hence, parallel circles are special small circles with c p =~ *goo. IFor r + 90" the small circle tends to an orthodrome.

Figure 3.107

Figure 3.108

2. Equations of Small Circles As defining parameters we can use either M and r or the small circle ) the north pole and r. If the running point on the small circle is &(A, yo), we point Pv(Xv,p ~ nearest get the equation of the small circle with the cosine law for sides corresponding to Fig. 3.107: cos r = sin p sin phf t cos p cos pM cos(X - AM). (3.213a) From here. because of p~ = cpx - T and AM = AxI we get: COST = sinpsin(pN - r ) + coscpcos(cp~- r)cos(X - A N ) . (3.213b) IA: For paw = 90° we get the parallel circles from (3.213a), since COST = sin p sin(9Oo- r ) = sin p + p = const. IB: For r --t 90° we get orthodromes from (3.213b). 3. Arclength The arclength s between two points A(XA,c p ~ )and B(XB,c p ~ on ) a small circle k can s 2~ro cos d = cos' r + sin' r cosu be calculated corresponding to Fig. 3.108 from the equalities - = -, u 360"

176

3. Geometrv

and TO = R sin T:

.-

cos d - cosz T aR (3.214) sin'r 180° IFor r + 90n the small circle becomes an orthodrome, and from (3.214) and (3.20713) it follows that s = d. 4. Course Angle According to Fig. 3.109 theorthodromepassingthrough A(Xa,V A ) and M ( X u , v o ~ ) intersects perpendicularly the small circle with radius T. For the course angle (YOrth of the orthodrome after (3.208) s = sin r arccos

(3.215a) is valid. So. we get for the required course angle CXAof the small circle at the point A: a .= ~ ( / a O T t h / - 90') s i g n ( a o d .

Figure 3.109

(3.215b)

Figure 3.110

5. Intersection Points with a Parallel Circle For the geographical longitude of the intersection points X1(Ax,,px) and X2(Xxz5cpx) of the small circle with the parallel circle (o = (oxwe get from (3.213a): cos T - sin cpx sin c p ~ ( u = 1,2). (3.216) AX, = A.w 7 arccos cos (oxcos q?M

Remark: In some cases an angle reduction is needed according to (3.205). 6. Tangential Point The small circle is touched by two meridians, the tangential meridians, at the tangentialpointsTl(XT~,cpr) and T 2 ( X ~y, , ~(Fig. ) 3.110). Because for them the argument of the arc cosine in (3.216) must be extremal with respect to the variable (ox,we get: pr = arcsin ' - (3.217a) AT, = X,C~ arccos cos r

cos T - sin cpx sin c p ~ ( u = 1,2). (3.217b) cos (oxcos (ou

Remark: In certain cases an angle reduction is needed according to (3.205). 7. Intersection Points with a Meridian The calculation of the geographical latitudes of the intersection points Yl(Xy, cpy,) and Yz(Ay, cpy,) of the small circle with meridian X = Xu can be done according to (3.213a) with the equations -.4C 5 BJA2 + B2 - C2 (. = 1,2), (3.218a) A2 + B2 where we use the notation: . 4 = s i n c p ~ , B = c o s c p ~ c o ~ ( X ~ - A ~ )C, = - C O S T . (3.218b) For A' BZ > Cz, in general, there are two different solutions, from which one is missing if a pole is on the small circle. pyu = arcsin

+

3.4 Spherical Trigonometry 177

+

If A' B2 = Cz holds and there is no pole on the small circle, then the meridian touches the small circle at a tangential point with geographical latitude py, = 'py2 = (OT.

3. Loxodrome 1. Notion A spherical curve, intersecting all meridians with the same course angle, is called a lozodrome or spherical helix. So, parallels ( a = 90") and meridians (Q = On) are special loxodromes. 2. Equation of t h e Loxodrome Fig. 3.111 shows a loxodrome with course angle a through the running point &(A, 'p) and the infinitesimally close point P(X t d X , p+dp). The right-angled spherical triangle QCP can be considered as a plane triangle because of its small size. Then: Rcos 'p dX + dX=-. tancosQ$9dp R dp Considering that the loxodrome must go through the point A(XA,c p ~ ) therefore, , we get the equation of the loxodrome by integration: tancu =

~

tan (450

(3.219b)

+y )

In particular if A is the intersection point PE(XE,0") of the loxodrome with the equator, then:

(Q # go0)

(3.219c)

Remark: The calculation of XE can be done with (3.224).

Figure 3 11

Figure 3.112

3. Arclength From Fig. 3.1 1 we find the differential relation

(3.220a) Integration with respect to 'p results in the arclength s of the arc segment with the endpoints A ( X A , ' p A ) and B(XB,PB): (3.220b) If A is the starting point and B is the endpoint, then from the given values A, a and s we can determine step-by-step first ' p from ~ (3.220b), then X g from (3.219b). Approximation Formulas: According to Fig. 3.111, with Q = A and P = B we get an approximation for the arclength 1 with the arithmetical mean of the geographical latitudes with (3.221a) and (3.22113):

178

3. Geometry

(3.221a)

(3.221b) 4. Course Angle For the course anglecu of the loxodrome through the points A(Xa, PA)and B ( X B , ~ B ) , or through A(X.4, y ~and ) its equator intersection point PE(XE, 0") we have according to (3.219b) and (3.219c) :

(3.222a)

\

L /

(3.222b) 5. Intersection Point with a Parallel Circle Suppose a loxodrome passes through the point A ( X A . p A ) with a course angle cu. The intersection point X(Xx,px)of the loxodrome with a parallel circle p = px is calculated from (3.219b):

tan (45' t Xx = X , 4 + t a n a , I n

tan

(GO+

y ). _

1800

?)

( a # goo).

(3.223)

T

\Vith (3.223) we calculate the intersection point with the equator PE(XE, 0"):

3.-

(

t a n a . l n t a n 45 t -

(3.224)

o: l

Remark: In certain cases an angle reduction is needed according to (3.205) 6. Intersection Point with a Meridian Loxodromes - except parallel circles and meridians - wind in a spiral form around the pole (Fig. 3.112). The infinitely many intersection points Y,(Xi,, yyv) (v E Z) of the loxodrome passing through A(XA,PA) with course angle cy with the meridian X = Xu can be calculated from (3.219b): pyu

{ [

= 2arctan exp

Xy -

If A is the equator intersection point py, = 2 arctan exp

[

Xy

+

XA U . 360" . L] tan (45' t tancu 180"

1'

y)}

- 90"

(v E Z). (3.225a)

0") of the loxodrome, then we have simply:

- XE t 1/. 360" . - gon tancu 180"

(v E Z).

(3.225b)

4. Intersection Points of Spherical Curves 1. Intersection Points of Two Orthodromes Suppose the considered orthodromes have points

P~,(XN~,,~N~) and Pyz(X~2r ?x2) closest to the north pole, where PN,# Px, holds. Substituting the intersection point S(Xs. ps) in both orthodrome equations we get an equation system t a n p x i cos(Xs - X N ~ )= t a n y s , (3.226a)

tan yy2 cos(Xs - Ax,) = tan ps . (3.226b)

3.4 Spherical Trigonometry 179

By elimination of ps and using the addition law for the cosine function we get: tan pxl cos - tan p~~cos AN^ tan& = (3.227) tan q y l sin A N l - tan vNzsin ANz ' The equation (3.227) has two solutions Asl and Asz in the domain -180" < X 5 180"of the geographical longitude. The corresponding geographical latitudes can be got from (3.226a): ps, = arctan[tanpNl cos(Asu - AN1)] (v = 4 2 ) . (3.228) The intersection points SIand Sz are antipodal points, Le., they are the mirror images of each other with respect to the centre of the sphere. 2. Intersection Points of Two Loxodromes Suppose the considered loxodromes have equator intersection points P E(AE], ~ 0")and PE?(AE?, 0") and the course angles a1 und 02 (a1 # a2). Substituting the intersection point S(Xs, ps) in both loxodrome equations we get the equation system: (3.229a)

1'

By elimination of XS and expressing 'ps we get an equation with infinitely many solutions: X E ~- X E ~ V . 360' , psu = 2 arctanexp - goa (v E Z). (3.230) tan a2 - tan al 180" The corresponding geographical longitudes AS, can be found by substituting 'ps, in (3.229a):

+

(3.231)

Remark: In certain cases an angle reduction is needed according to (3.205).

I

180

3. Geometrv

3.5 Vector Algebra and Analytical Geometry 3.5.1 Vector Algebra 3.5.1.1 Definition ofvectors 1. Scalars and Vectors Quantities whose values are real numbers are called scalars. Examples are mass, temperature, energy, and work (for scalar invariant see 3.5.1.5,p. 184,3.5.3.1,p. 212 and 4.3.5.2, p. 270). Quantities which can be completely described by a magnitude and by a direction in space are called vectors. Examples are power, velocity, acceleration, angular velocity, angular acceleration, and electrical and magnetic force. We represent vectors by directed line segments in space. In this book we denote the vectors of three-dimensional Euclidean space by a', and in matrix theory by p (see also 4.1.3, p. 253).

2. Polar and Axial Vectors Polar vectors represent quantities with magnitude and direction in space, such as speed and acceleration; anal vectors represent quantities with magnitude, direction in space, and direction of rotation, such as angular velocity and angular acceleration. In notation we distinguish them by a polar or by an axial arrow (Fig. 3.113). In mathematical discussion we treat them in the same way.

3. Magnitude or Absolute Value and Direction in Space For the quantitative description of vectors a' or a, as line segments between the initial and endpoint A and B resp., we have the rnagnztude, Le., the absolute value ISI, the length of the line segment, and the dzrection zn space, which is given by a set of angles. 4. Equality of Vectors Two vectors Sand 6 are equal if their magnitudes are the same, and they have the same direction, Le., if they are parallel and oriented identically. Opposzte and equal vectors are of the same magnitude, but oppositely directed: + + + + AB = S 3 EA= -S but IABI = /BAI. Axial vectors have opposite and equal directions of rotation in this case.

Figure 3.113

Figure 3.114

(3.232)

Figure 3.115

5. Free Vectors, Bound or Fixed Vectors, Sliding Vectors A free vector is considered to be the same, i.e. its properties do not change, if it is translated parallel to itself, so its initial point can be an arbitrary point of space. If the properties of a vector belong to a certain initial point, we call it a bound orfixed vector. A slzding vector can be translated only along the line it is already in. In mathematics we deal with free vectors.

6. Special Vectors a) Unit Vector 8 = e' is a vector with length or absolute value equal to 1. With it we can express the vector a' as a product of the magnitude and of a unit vector having the same direction as a': S = e'(a'1. (3.233) 3

-

3

We often use the unit vectors i, j, k or 4 , e'J1Q (Fig. 3.114) to denote the three coordinate axes in the direction of increasing coordinate values.

3.5 VectorA1,qebra and Analytical Geometry 181

In Fig. 3.114 the directions given by the three unit vectors form an orthogonal triple. These unit vectors define an orthogonal coordinate system because for their scalar products &i ej '-- &iek ' = ejek +' =0 (3.234) is valid. Because also e',e', = = = 1, (3.235) holds, we call it an orthonormal coordinate system. (For more about scalar product see (3.247).) b) Null Vector or zero vector is the vector whose magnitude is equal to 0, Le., its initial and endpoint coincide! and it has no direction. i

c) Radius Vector i or position wectorofa point P i s the vector OP with the initial point at the origin and endpoint at P (Fig. 3.114). In this case we call the origin also a pole or polar point. The point P is defined uniquely by its radius vector. d) Collinear Vectors are parallel to the same line. e) Coplanar Vectors are parallel to the same plane. They satisfy the equality (3.259).

3.5.1.2 Calculation Rules for Vectors 1. Sum of Vectors i

--i

a) The Sum of Two Vectors AB = a' and AD = 6 can be represented also as the diagonal of the i

parallelogram ABCD, as the vector AC = c' in Fig. 3.115b. The most important properties of the sum of two vectors are the commutative law and the triangle inequality:

a'+6=6+a', /a'+615la'l+lEl. b) The Sum of Several Vectors a', 6, c', . . . , e' is the vector f

(3.236a) i

= AF, which closes the broken line composed of the vectors from a' to e'as in Fig. 3.115a. Important properties of the sum of several vectors are the commutative law and the associative law of addition. For three vectors we have: a' + 6 + c' = c' + 6 + a', (a' + 6) + c' = a' + (6 + c'). (3.236b)

c) The Difference of Two Vectors S - 6 can be considered as the sum of the vectors a' und +

-6, Le.,

+

a'- 6 = a'+ (-b) = d (3.236~) which is the other diagonal of the parallelogram (Fig. 3.115b). The most important properties of the difference of two vectors are: a - ~ = o ' (nullvector), ls-612 I ~ z I - ~ ~ ~ I . (3.236d)

Figure 3.116 2. Multiplication of a Vector by a Scalar, Linear Combination The products aa' and a'a are equal to each other and they are parallel (collinear) to a'. The length (absolute value) of the product vector is equal to lalla'l. For a > 0 the product vector has the same direction as a': for a < 0 it has the opposite one. The most important properties of the product of vectors by scalars are: a a' = s a , a as = 4a s ,

( a + p) a' = aa' + pa, a (a' + 6) = a a' + a 6 .

(3.237a)

I

182

3. Geometry

The linear combination of the vectors a', g, 6,

I

.

.

, d with the scalars c y , 0.. . . ,6is the vector

1; = aa'+$G t . . . + b d . 3. Decomposition of Vectors

(3.237b)

In three-dimensional space every vector a' can be decomposed uniquely into a sum of three vectors, which are parallel to the three given non-coplanar vectors fi, J , u7 (Fig. 3.116a,b): a'= c y f i + $ C + yu7. (3.238a) The summands aii, 3C and 7 4 are called the components of the decomposition, the scalar factors c y , @ and y are the coeficients. When all the vectors are parallel to a plane we can write a'=cyli-t$J (3.238b) with two non-collinear vectors fi and S being parallel to the same plane (Fig. 3.116c,d).

3.5.1.3 Coordinates ofa Vector i

Jji$ 1

?

Y

X

Figure 3.117

1. Cartesian Coordinates According to (3.238a) every vector A B = a' can be decomposed uniquely into a sum of vectors parallel to the basis 1;or 4 ,S j , Sk: vectors of the coordinate system

:,i,

-

+ + -

+

*

(3.239a) a' = a,i a j a,k = a,$ + a,$ +a,&, where the scalars a,> ay and a, are the Cartesian coordinates of the vector a' in the system with the unit vectors 4, Sj and 8. We also write a' = {a,. a,, a,} or a'(.,. ay, a,). (3.239b) The three directions defined by the unit vectors form an orthogonal direction triple. The components of a vector are the projections of this vector on the coordinate axes (Fig. 3.117).

The coordinates of a linear combination of several vectors are the same linear combination of the coordinates of these vectors, so the vector equation (3.237b) corresponds to the following coordinate equations: k , = aa, + B b, + ' ' + bd,, k, = a a y + Ob, + . . , t bd,, (3.240) a, + 3 b, + . . , + b d,. k, = For the coordinates of the sum and of the difference of two vectors c'=a'*G (3.241a) the equalities a, (3.241b) c, = a, b,, cy = ay b,, c, = a, are valid. The radius vector ?of the point P ( z ,y! 2 ) has the Cartesian coordinates of this point:

*

*

+

+

-

+

+

?=zityj+zk. (3.242) 2. Affine Coordinates are a generalization of Cartesian coordinates with respect to a system of linearly independent but not necessarily orthogonal vectors, Le., to three non-coplanar basis vectors SI.& & . The coefficients are a': a', a3, where the upper indices are not exponents. Similarly to (3.239a.b) we have for a' T',=X:

~,=y,

T,=z;

a' = a' 61+ a'& t a363

(3.243a)

or a' = { a ' , a', a 3 } or a' ( a ' , a', a3) .

(3.243b)

This notation is especially suitable as the scalars a ' , a', a3 are the contravariant coordinates of a vector (see 3.5.1.8. p. 187). For 61 = ,: 62 = 4 = k'the formulas (3.243a,b) become (3.239a,c). For the linear combination of vectors (3.237b) just as for the sum and difference of two vectors (3.241a,b) in

it

9.5 Vector Algebra and Analytical Geometry 183 analogy to (3.240) the same coordinate equations are valid: k'=aa'+Bb'+...+bd', IC2 = a a 2 + B b 2 t . . . + 6 d 2 ,

(3.244)

+

IC3 = cya3 t B b3 t . . . b d 3 ; C' = a ' 5 b l , c2 = a2 5 b2, c3 = a3 i b3.

(3.245)

3.5.1.4 Directional Coefficient The dzrectzonal coeficzent of a vector a' along a vector ab =

a'6'

'= where 6

6 is the scalar product

= /a'l cosp.

4 is the unit vector in the direction of lbl

(3.246) and p is the angle between a' and 6.

The directional coefficient represents the projection of a' on 6.

I In the Cartesian coordinate system the directionalcoefficients of the vector a'along the 2,y, z axes are the coordinates as>ay, a,. This statement is usually not true in a non-orthonormal coordinate system.

3.5.1.5 Scalar Product and Vector Product 1. Scalar product The scalar product or dot product of two vectors S a n d

6 is defined by the equation - , * a', b = a'b = (a'6)= la'l I6lcoscp, (3.247) where p is the angle between a' and 6 considering them with a common initial point (Fig. 3.118). The value of a scalar product is a scalar.

x w -+

a

Figure 3.118

a

Figure 3.119

2. Vector Product

or cross product of the two vectors a' and 6 is a vector c' such that it is perpendicular to the vectors a' and 6. and in the order a', 6, and c'the vectors form a right-hand system (Fig. 3.119). If the vectors have the same initial point, and we look at the plane of a' and 6 from the endpoint of c', then the shortest rotation of a' in the direction of 6 is counterclockwise. The vectors a', 6, and c'have the same arrangement as the thumb, the forefinger, and the middle finger of the right hand. Therefore this is called the rzght-hand rule. The vector product -

+

a'x b = [a'b] = c'

(3.248a)

has magnitude

~ ? l = 121161sinp,

(3.248b)

where p is the angle between a' and 6. Numerically the length of c' is equal to the area of the parallelogram defined b j the vectors a' and 6. 3. Properties of the Products of Vectors a) The Scalar Product is commutative: +

-

a'b = ba'. (3.249) b) The Vector Product is anticommutative (changes its sign if we interchange the factors): a'x

6 = -(6 x a').

(3.250)

I

184

3. Geometry

c) Multiplication by a Scalar Scalars can be factored out:

a(a'6)= ( a s )6,

(3.251a) a(a'x

6 ) = (.a')

x

6.

(3.251b)

d) Associativity The scalar and vector products are not associative:

a'(6c')# (a'6)c',

(3.252a) a' x

(6 x E) # (a' x 6 ) x c'.

(3.252b)

e) Distributivity The scalar and vector products are distributive over addition:

a ' ( 6 + E ) =a'6+a'E. a'x

(3.253a)

(6+c') = a' x 6 +a' x Z

and

(6+ Z ) x a'= 6 x a ' + E x a'.

(3.253b)

f ) Orthogonality of Two Vectors Two vectors are perpendicular to each other (a' i6) if the equality

a'6 = 0

holds, and neither a' nor 6 are null vectors.

(3.254)

g) Collinearity of Two Vectors Two vectors are collinear (a' 11 6) if the equality

a' x 6 = 6 holds, and neither a' nor 6 are null vectors. h) Multiplication of the same vectors: 5s = 5' = a', a' x a'= 6.

(3.255) (3.256)

i) Linear Combinations of Vectors can be multiplied in the same way as scalar polynomials (because of the distributive property), only we have to be careful with the vector product. If we interchange the factors, we have to change the sign. IA: ( 3a' +56 - 2s) (a' - 26 - 4E) = 38' =

3s' - 106'

IB: ( 3 s $56 - 23) x (a'-

+

48

+ 56s - 21% 6d -

-

106'

+ 41%

- 121c' - 2 0 k + 8c"

+ 8Z2 - & - 148c' - 166E.

~6 - 12s x c'-

26 - 4s) = 3 1 x a'+

56 x a'- 23 x a'- 6a' x

206 x c'+8E x c'= 0 - 5a' x

6 + 21 x E -

6 - 106 x 6 6 + 0 - 4 6 x c'

6a' x

- 12s xc'- 206 x c'+o = -1la'x 6 - 1oa'x c'- 246 x E = 116 x a'+ 1oc'x a'+ 24Ex 6. j) Scalar Invariant is a scalar quantity if it does not change its value under a translation or a rotation of the coordinate system. The scalar product of two vectors is a scalar invariant. IA: The coordinates of a vector a' = {Q, a', a3) are not scalar invariants, because in different coordinate systems they can have different values. IB: The length of a vector a'is a scalar invariant, because it has the same value in different coordinate systems. IC: Since the scalar product of two vectors is a scalar invariant, the scalar product of a vector by itself is also a scalar invariant, i.e., l a ' = ill' cosp = la')2, because p = 0.

3.5.1.6 Combination ofvector Products 1. Double Vector Product

The double vector product 2 x (b x E) results in a vector coplanar to 6 and c':

a' x

(6 x c') = 6 (2:)

2. Mixed Product

- c'(a'6).

(3.257)

The mzxed product (a' x 6 )E, which is also called the trzple product, results in a scalar whose absolute value is numerically equal to the volume of the parallelepipedon defined by the three vectors; the result is positive if a', 6, and c' form a right-hand system, negative otherwise. Parentheses and crosses can be

3.5 Vector Al.qebra and Analytical Geometrg 186

-

omitted: + ( a x b)c=a'bc'=bc'a'=c'a7;=-a'c'6=-6lc'=-c'6t. -

4

(3.258) The interchange of any two terms results in a change of sign: the cyclic permutation of all three terms does not affect the result. For coplanar vectors, Le., if a' is parallel to the plane defined by 6 and c', we have: +

S . (b x c') = 0.

(3.259)

3. Formulas for Multiple Products a ) Lagrange Identity: (a' x @(c'x d) = (a'?) (6d) - (63)( S i ) , (3'260) (3.261)

d

c'e'

c'g'

4. Formulas for Products in Cartesian Coordinates If the vectors a', b, c' are given by Cartesian coordinates as

a' = {az. a y ja,). 6 = { b x ,by, b,} , ? = {e,, cy, c,} , then we can calculate the products by the following formulas: 1. Scalar P r o d u c t :

ab' = arbx + a,b, + a,b,.

2. Vector Product:

a' x

6 = (a,b,

- a,b,)

:+

(3.262) (3.263)

(azbx - arbz)

I+(axby- ayb,) 1; (3.264)

3. Mixed P r o d u c t :

- II a r

a'bc'= b, ay by b, . c x cy c,

(3.265)

I

5 . Formulas for Products in Affine Coordinates 1. Metric Coefficients and Reciprocal System of Vectors If we have the affine coordinates of + two vectors S a n d b in the system of SI,8 ,& 3 i.e.,

a' = a1e'l + u2 Z2 + a3 4,b' = b' 61 + b2 62 + b3 153 are given, and we want to calculate the scalar product = u'b' e'' 61

+ a2 b' 62 e'' + a3b3 Q 4 + (a' b' + a'b') &e'' + (a' b3 + a3b')

(3.266)

6'2

Q + (a3 b'

+ a' b3) 4

6'1

(3.267)

or the vector product

a' x

6 = (a2b3 - a3 b')

Zz x

4 + (a3b'

- a' b3) S3 x e''

+ (a' b2 - a'b')

6' x

4,

(3.268a)

with the equalities

a,

e" x 61 = 6' x e" = 63 x 63 = (3.268b) then we have to know the products in pairs of coordinate vectors. For the scalar product these are the six metrzc coefficzents (numbers) 911 = e'' e'', $722 = s 2 h, Q33 = s3 4, 912 = = e'', 923 = 62 83 = 44,~ 3 = 1 4e'' = e', 4 (3.269)

I

186

3. Geornetru

and for the vector product the three vectors e" = R (e'2 x 4).e'* = R (4x S I ) , e'3 = R (SI x 4), which are the three reciprocal vectors with respect to Zl, &,4, where the coefficient

(3.270a)

1

1

a=-S162 4

(3.270b)

'

is the reciprocal value of the mixed product of the coordinate vectors. This notation serves only as a shorter way of writing in the following. With the help of the multiplication Tables 3.13 and 3.14 for the basis vectors calculations with the coefficients will be easy to perform. Table 3.13 Scalar product of basis vectors

Table 3.14 Vector product of basis vectors Multipliers

2. Application to Cartesian Coordinates The Cartesian coordinates are a special case of affine coordinates. From Tables 3.15 and 3.16 we have for the basis vectors - =. Y.. s2=j: 4= 1; (3.271a) el s i t h the metric coefficients 1 (3.271b) = -.-.-.-- , 911 = g22 = g33 = g12 = g23 = g31 = 0, ijk and the reciprocal basis vectors

e" =;, e'2 =j: 6°C. (3.271~) So the basis vectors coincide with the reciprocal basis vectors of the coordinate system, or, in other words, in the Cartesian coordinate system the basis vector system is its own reciprocal system. 3. Scalar Product of Vectors Given by Coordinates 3

3

=

gmnambn = gaoaa

fl

(3.272)

m = l n=l

Table 3.15 Scalar product of reciprocal basis vectors

Table 3.16 Vector product of reciprocal basis vectors Multidiers

3.5 Vector Algebra and Analytical Geometry 187

For Cartesian coordinates (3.272) coincides with (3.263). .4fter the second equality in (3.272) we applied a shorter notation for the sum which is often used in tensor calculations (see 4.3.1, 2.) p. 262): instead of the complete sum we write only a characteristic term so that the sum should be calculated for repeated indices, Le., for the indices appearing once down and once up. Sometimes the summation indices are denoted by Greek letters: here they have the values from 1 until 3. Consequently we have ga3aa b4 = glla' b' g12a' b2 g13a' b3 gZla2b' g22a2b2 t g23a2 b3 (3.273) g3'a3b' g32a3b2 g33a3 b3.

+ +

+ +

+ +

+

4. Vector Product of Vectors Given by Coordinates In accordance with (3.268a)

e" e'2 a' x

6 = e'' z24 ~

=

e'' e'?4

e'3

a1 a2 a3

b' b2 b3

~

[(a2b3 - a3 b2)e"

+ (a3b'

- a'

b3) e''

+ (a' b2 - a2 b' 1 1

(3.274)

is valid. For Cartesian coordinates (3.274) coincides with (3.264). 5. Mixed Product of Vectors Given by Coordinates In accordance with (3.268a) we have

(3.275)

For Cartesian coordinates (3.275) coincides with (3.265).

3.5.1.7 Vector Equations Table 3.17 contains a summary of the simplest vector equations. In this table a', 6, ?are given vectors. 2 is the unknown vector. a, D,y are given scalars, and 2 ,y, z are the unknown scalars we are looking for.

3.5.1.8 Covariant and Contravariant Coordinates of a Vector 1. Definitions The affine coordinates a', a', a3 of a vector a' in a system with basis vectors e'', &, S3,defined by the formula a' = a' e'' t a' Z2 a3 4 = an (3.276) are also called contravariant coordinates of this vector. The covariant coordinates are the coefficients in the decomposition with the basis vectors S1, S2, S3, i.e., with the reciprocal basis vectors of 81, $, $. With the covariant coordinates a ' , a2, a3 of the vector a'we have (3.277) a'= ale'' + a 2 S 2 a3S3 = a , P In the Cartesian coordinate system the covariant and contravariant coordinates of a vector coincide. 2. Representation of Coordinates with Scalar Product The covariant coordinates of a vector a' are equal to the scalar product of this vector with the corresponding basis vectors of the coordinate system: a1 = 561, a2 =a'?&! a3 = a ' 4 . (3.278) The contravariant coordinates of a vector a' are equal to the scalar product of this vector with the corresponding basis vectors: a' = a'"', a2 =a'@. a3 = a'@. (3.279) In Cartesian coordinates (3.278) and (3.279) are coincident:

+

sa

+

+ .+

a, = a i ,

3

ay = a'j,

-.

a, = a'k.

(3.280)

188

3. Geometry Table 3.17 Vector equations 2 unknown vector; S , 6 , 8 ,d given vectors; z , y , z unknown scalars; cy

Equation

Solution

1. 2 + S = 6

2=6-S x+= - S

2. @?=a'

p , y given scalars

cy

3.2Fi=cy

4. 2 x S = 6

(ha')

Indeterminate equation; if we consider all vectors 2 satisfying the equation, with the same initial point, then the endpoints form a plane perpendicular to the vector a'. Equation 3. is called the vector equation of this plane. Indeterminate equation; if we consider all vectors 2 satisfying the equation, with the same initial point, then the endpoints form a line parallel to a'. Equation 4. is called the vector equation_ofthis line. cyS+Sxb X= ( a = la'l) az

-

7. d = z S + y 6 + z c ' 8. d = z(6 x c') ty (c' x a') + z (a' x

+ -

where P,b, c'are the reciprocal vectors of a', 6, c' (see 3.5.1.6, l., p.,1_85). x=- dbc' y = - Sdc' z = - S 6 d $8' a'++' c' db dc' da' x=y=z=SGZ' a'bc" a'bc'

a'.$

6)

3. Representation of the Scalar Product in Coordinates The determination of the scalar product of two vectors by their contravariant coordinates yields the formula (3.272). The corresponding formula for covariant coordinates is:

a's

(3.281) = gap a, bp, where g"" = e'" e'" are the metric coefficients in the system with the reciprocal vectors. Their relation with the coefficients gmn is (3.282)

where A"" is the subdeterminant of the determinant in the denominator; we get it by deleting the row and column of the element gmn. If the vector S is given by covariant coordinates, and the vector 6 by contravariant coordinates, then their scalar product is (3.283) a'6 = a' bl + a 2 bz + a 3 b3 = a'b, and analogously we have

-

S b = a, ba.

(3.284)

3.5 Vector Algebra and Analytical Geometry 189

3.5.1.9 Geometric Applications of Vector Algebra In Table 3.18 we demonstrate some geometric applications of vector algebra. Other applications from analytic geometry, such as vector equations of the plane and of the line, are demonstrated in 3.5.1.7, p. 188 and 3.5.3.4,p. 214ff. and on the subsequent pages. Table 3.18 Geometric application of vector algebra Determination Length of the vector a' Area of the parallelogram determined by the vectors Sand b'

Vector formula

Formula with coordinates (in Cartesian coordinates)

a=@

a=

Jm

S=/Sxfl

L'olume of the parallelepiped determined by the vectors a', 6, C' Angle between the vectors a' and 6

d

:os9 = -

arbz

&s -4 cos 'p =

+ a,b, + a,b,

-4

3.5.2 Analytical Geometry of the Plane 3.5.2.1 Basic Concepts, Coordinate Systems in the Plane The position of every point P of a plane can be given by an arbitrary coordinate system. The numbers determining the position of the point are called coordinates. Mostly we use Cartesian coordinates and polar coordinates.

1. Cartesian or Descartes Coordinates The Cartesian coordinates of a point P are the signed distances ofthis point, given in a certain measure, from two coordinate axes perpendicular to each other (Fig. 3.120). The intersection point 0 of the coordinate axes is called the origin. The horizontal coordinate axis, usually the x-axis, is usually called the axis of abscissae, the vertical coordinate axis, usually the y-axis, is the axis ofordinates.

a x

Iv

b)

Figure 3.120 Figure 3.121 The positive direction is given on these axes: on the z-axis usually to the right, on the y-axis upwards. The coordinates of a point P are positive or negative according to which half-axis the projections of the point fall (Fig. 3.121). The coordinates z and y are called the abscissa and the ordinate of the point P , respectively. We define the point with abscissa a and ordinate b with the notation P ( a , b). The z, y plane is divided into four quadrants I, 11,111,and IV by the coordinate axes (Fig. 3.121,a).

190

3. Geometrw

2. Polar Coordinates The polar coordinates of a point P (Fig. 3.122)are the radius p, i.e.. the distance of the point from a given point, the pole 0, and the polar angle 'p, Le., the angle between the line OP and a given oriented half-line passing through the pole, the polar axis. The pole is also called the origin. The polar angle is positive if it is measured counterclockwise from the polar axis, otherwise it is negative.

Figure 3.122

Figure 3.123

3. Curvilinear Coordinate System This system consists of two one-parameter families of curves in the plane. the family of coordinate curves (Fig. 3.123).Exactly one curve ofbothfamiliespasses through every point of the plane. They intersect each other at this point. The parameters corresponding to this point are its curvilinear coordznates. In Fig. 3.123 the point P has curvilinear coordinates u = a1 and u = b S . In the Cartesian coordinate system the coordinate curves are straight lines parallel to the coordinate axes; in the polar coordinate system the coordinate curves are concentric circles with the center at the pole, and half-lines starting at the pole.

Figure 3.124

3.5.2.2

Figure 3.125

Coordinate Transformations

Under transformation of a Cartesian coordinate system into another one, the coordinates change according to certain rules.

1. Parallel Translation of Coordinate Axes Lye shift the axis of the abscissae by a, and the axis of the ordinates by b (Fig. 3.124). Suppose a point P has coordinates z, y before the translation, and it has the coordinates z', y' after it. The old coordinates of the new origin Of are a, b. The relations between the old and the new coordinates are the following: x=x'+a. y=y'+b, (3.285a) Z'=Z--, y'=~-b. (3.28513)

2. Rotation of Coordinate Axes Rotation by an angle p (Fig. 3.125)yields the following changes in the coordinates:

+

I = z'cosp - y'sin 9, y = z'sin y y f cosp, zf = z c o s p + y s i n y , y'=-zsinp++cosp.

(3.286a) (3.286b)

3.5 Vector Aloebra and Analwtical Geometrw 191

The coefficient matrix belonging to (3.286a)

D = (cosy sinp -cos9 s i n p ) with

(i)):()I

and

=

(i:)

= D-'(i),

(3.286~)

is called the rotatzon rnatriz. In general, transformation of a coordinate system into another can be performed in two steps, a translation and a rotation of the coordinate axes.

Figure 3.126

Figure 3.127

Figure 3.128

' a r c t a n -Y+ *

forlc<~,

X

p=

m.

Y arctan (3.287b)

X

p=<

forz > 0, for x = 0 and y > 0.

27r

_-

2

,indefined

(3.287c)

for z = 0 andy < 0, for x = y = 0.

3.5.2.3 Special Notation in t h e Plane 1. Distance Between Two Points If the two points given in Cartesian coordinates as Pi ( X I , y1) and Pz ( 5 2 , yz) (Fig. 3.127), then their distance is

J(Q

(3.288) d= -xi)* + ( ~ 2 ~1)'. If they are given in polar coordinates as PI ( p l , 91) and P2 ( p ~p2) , (Fig. 3.128), their distance is d = Jp:

+ pi - 2 ~ 1 ~ 2 (pz ~ 0 -s pi).

(3.289)

2. Coordinates of Center of Mass The coordinates (z. y) of the center of mass of a system of material points M, ( x ~yl), with masses m, (z = 1 , 2 . .. . n ) are calculated by the following formula:

.

(3.290)

192

3. Geometrw

3. Division of a Line Segment 1. Division in a Given Ratio The coordinates of the point P with division ratio

PIP m =-=X PP? n

=_

(Fig. 3.129a) of the line segment spZ are calculated by the formulas

+ mx2 --x 1 + Axz (3.291a) y=-- ~ Y +I my2 --y 1 + Xy2 (3.291b) nf m 1tX ’ n+m 1+X’ For the midpoint M of the segment because of X = 1, we have x1 + 52 y = - Y 1 + Yz x=(3.291~) (3.291d) 2 ’ 2 ’ and P P z can be defined. Their signs are positive or negative depending The sign of the segments on whether their directions are coincident with PIP2 or not. Then formulas (3.291a,b,c,d) result in a in the case X < 0. We call this an external division. If P is inside the point o u t e o f the segment segment PIP*,we call it an internal division. We define a) X = 0 if P = Pl, b) X = cc if P = Pz and c) X = -1 if P is an infinite or improper point of the line g, Le., if P is infinitely far from on g. The shape of X is shown in Fig. 3.129b. Pl p IFor a point P , for which Pz is the midpoint of the segment X = == -2 holds. PP2 2. Harmonic Division If the internal and external division of a line segment have the same absolute value 1x1, we call it harmonic division. Denote by PI and P, the points of the internal and external division respectively, and by A, and A, the internal and external devisions. Then we have - - - X 2---- Pl Pa - -A, (3.292a) or A, +A, = 0. (3.29213) PI P2 Pa P2 x=-- nxl

m,

m

m

w,

_ .

~

Yt

Figure 3.129 If M denotes the midpoint of the segment at a distance b from PI (Fig. 3.130), and the distances of P, and P, from ,Zf are denoted by x, and z, then we have

b + x ,-x,tb x, b , ~or - = - i.e., x,x, = b2. (3.293) b-x, x,-b b 2,’ The name harmonzc dzvzszonis in connection with the harmonicmean (see 1.2.5.3, p. 20). In Fig. 3.131 the harmonicfision is represented for X = 5 : 1, analogously to Fig. 3.14. The harmonic mean T of the segments PIP, = p and PIP, = q according to (3.292a) equals in accordance with (1.67b), p. 20, to 2Pq see Fig. 3.132. r =P+q

(3.294)

3.5 Vector Algebra and Analytical Geometry 193

Figure 3.131

Figure 3.132 3. Golden Section of a segment a is its division into two parts z and a - x such that the part z and the whole segment a have the same ratio as the parts a - z and z:

x - a-x a x

(3.295a)

In this case 5 is the geometric mean of a and a - x,and we have (see also golden section p. 2): 2

=

&&-q, (3.295b) c' ',...-

A

z = - a(&

-

2

1)

w 0.618. a.

(3.295~)

The part z of the segment can be geometrically constructed as shown in Fig. 3.133. The segment zis also the length of the side of a regular decagon whose circumcircle has radius a. The problem, to separate a square from a rectangle with the ratio of sides given as in (3.295a) so that for the remaining rectangle (3.295~)should be valid, also produces the equation of the golden section.

a

a

Figure 3.133

4. Areas 1. Area of a Triangle If the vertices are given by PI ( ~ ~ , y ~ ~ yz), 2 and ~ P3 ( 2 3 , y3) (Fig. 3.134), then we can calculate the

PZ (

[XI 2 3 Y3

P , ( X , Jl)

(Yz - Y3)

+ 2 2 (Y3 - Y1) f 2 3 (Y1 - Y2)I

1

P A X 2 TY2)

=

[(XI

- 2 2 ) (Yl

+ Yz) + ( 2 2 - 2 3 ) (Yz + Y3) (3.296)

Figure 3.134 Three points are on the same line if

(3.297) 2. Area of a Polygon If the vertices are given by PI ( X I , y ~ ) PZ , ( 2 2 , y ~ . .) .,~P, (x,,y,),

then the

area is

1

s = -2 [(XI - 52)(y1 t Yz) + ( 2 2 -

23)

(yz t y3)

+ ...+

(2, - 21) (Yn

+ Yl)]

'

(3.298)

The formulas (3.296) and (3.298) result in a positive area if the vertices are enumerated counterclockwise. otherwise the area is negative.

194

3. Geometry

5. Equation of a Curve Every equation F ( z ,y) = 0 for the coordinates x and y corresponds to a curve, which has the property that every point P satisfies the equation, and conversely. every point whose coordinates satisfy the equation is on the curve. The set of these points is also called the geometrzc locus or simply locus. If there is no real point in the plane satisfying the equation F ( x ,y) = 0,then there is no real curve, and we talk about an imagznary curve: IA: x2 yz 1 = 0, IB: y = In (1 - x 2 - coshz). The curve corresponding to the equality F ( x , y) = 0 is called an algebraic curve if F ( x ,y) is a polynomial, and the degree of the polynomial is the order or degree o f t h e curue (see 2.3.4, p. 63). If the equation of the curve cannot be transformed into the form F ( x ,y) = 0 with a polynomial expression P ( z ,y), then the curve is called a transcendental curue. The equation of a curve can be defined in the same way in any coordinate system. But from now on, we talk about the Cartesian coordinate system only, except when stated otherwise.

+ +

3.5.2.4 Line 1. Equation o f t he Line Every equation that is linear in the coordinates is the equation of a line, and conversely, the equation of every line is a linear equation of the coordinates. 1. General Equation of Line Az + B y + C = 0 ( A ,B , C const). (3.299) For A = 0 (Fig. 3.135) the line is parallel to the x-axis, for B = 0 it is parallel to the y-axis, for C = 0 it passes through the origin. 2. Equation of the Line with Slope (or Angular Coefficient) Every line that is not parallel to the y-axis can be represented by an equation written in the form y = kx + b (k,b const). (3.300) The quantity k is called the angular coeficient or slope of the line; it is equal to the tangent of the angle between the line and the positive direction of the z-axis (Fig. 3.136). The line cuts out the segment b from the y-axis. Both the tangent and the value of b can be negative, depending on the position of the line. 3. Equation of a Line Passing Through a Given Point The equation of a line which goes through a given point PI (zl,y l ) in a given direction (Fig. 3.137) is y - y1 = k ( z - X I ) , with k = t a n & (3.301)

Figure 3.135

Figure 3.136

Figure 3.137

4. Equation of a Line Passing Through Two Given Points If two points of the line PI (xl,yl), Pz (xzly2) are given (Fig. 3.138), then the equation of the line is v-y1

YZ - y1

-

x--1

x2 - 2 1

(3.302)

3.5 VectorAl.qebra and Analytical Geometry 195

5. Intercept Equation of a Line If a line cuts out the segments a and b from the coordinate axes, considering them with sign, the equation of the line is (Fig. 3.139) -X + -Y= l .

a

(3.303)

b

Figure 3.138

Figure 3.139

Figure 3.140

6. Normal Form ofthe Equation ofthe Line (Hessian Normal Form) W i t h p as the distance of the line from the origin, and with cy as the angle between the x-axis and the normal of the line passing through the origin (Fig. 3.140),with p > 0. and 0 5 a < 27r, the Hessian normal form is (3.304) z cosa + ysin a - p = 0. \Ye can get the Hesszan normalform from the general equation if we multiply (3.299) by the normalzzzng

factor /I =

*A d r n ’

(3.305)

The sign of /I must be the opposite to that of C in (3.299). 7. Equation of a Line in Polar Coordinates (Fig. 3.141)With p as the distance of the line from the pole (normal segment from the pole to the line), and with a as the angle between the polar axis and the normal of the line passing through the pole, the equation of the line is P (3.306) = cos ($5 - cy)

2. Distance of a Point from a Line \Ye get the distance d of a point Pi ( X I , y1) from a line (Fig. 3.140)by substituting the coordinates of the point into the left-hand side of the Hessian normal form (3.304): (3.307) d = x1 cos a + y1 sin a - p. If PIand the origin are on different sides of the line, we get d > 0, otherwise d < 0.

Figure 3.141

Figure 3.142

Figure 3.143

3. Intersection Point of Lines 1. Intersection Point of Two Lines In order to get the coordinates (xo, yo) of the intersection point of two lines we have to solve the system of equations given by the equation. If the lines are given by the equations (3.308a) Ais Biy Ci = 0, Azz B ~ Y Cz = 0

+

+

+

+

196

3. Geometry

then the solution is (3.308b)

! i:1

Ai Bi C = 0 holds, the lines are parallel. If - = - = 2 holds, the lines are coincident. If A2 Bz Cz 2. Pencil of Lines If a third line with equation

I

+

A35 + B3y C3 = 0 passes through the intersection point of the first two lines (Fig. 3.142), then the relation

(3.309a)

(3.309b) must be satisfied. The equation

+ +

+

+

+

(-w < X < +CO) (3.309~) (Ai2 Bly Ci) X (A22 B ~ Y C2) = 0 describes all the lines passing through the intersection point Po(z0,yo)of the two lines (3.308a). By (3.309~)we define a pencil oflines with center Po(z0,yo). If the equations of the first two lines are given in normal form, then for X = f l we get the equations of the bisectrices of the angles at the intersection point (Fig. 3.143).

Figure 3.144

Figure 3.145

4. Angle Between Two Lines In Fig. 3.144 there are two intersecting lines. If their equations are given in the general form

+

Alz + Bly C1 = 0 and then for the angle p we have

A22

+ Bly + C2 = 0,

(3.310a)

(3.310b)

+BIBz With the slopes k l and

k2

- ki tanp = 1 k1k2 ' k2

+

,

(3.310~)

-

(3.310d)

we have (3.310e)

3.5 Vector Algebra and Analytical Geometry 197

cosp =

+

1 kikz

J-d-’

(3.310f)

sin cp =

k2

- ki

(3.31Og)

Jl+lCl4-

Here we consider the angle cp in the counterclockwise direction from the first line to the second one. A B For parallel lines (Fig. 3.145a) the equalities A = -2 or kl = k2 are valid. Az B2 For perpendicular (orthogonal) lines (Fig. 3.145b) we have A1A2 BlB2 = 0 or k2 = - l / k l .

+

3.5.2.5

Circle

1. Definition of the Circle The locus of points at the same given distance from a given point is called a circle. The given distance is called the radius and the given point is called the center of the circle. 2. Equation of the Circle in Cartesian Coordinates The equation of the circle in Cartesian coordinates when its center is at the origin (Fig. 3.146a) is x2 + y 2 = R2. If the center is at the point C(x0,yo) (Fig. 3.146b), then the equation is

(3.311a)

+

( x - Io)2 ( y - y0)Z = R2. (3.31lb) The general equation of second degree ax2 + 2bxy t cy2 + 2dx + 2ey + f = 0 (3.312a) is the equation of a circle only if b = 0 and a = e. In this case the equation can always be transformed into the form x2 + y2 + 2mx + 2ny + q = 0. (3.312b) For the radius and the coordinates of the center of the circle we have the equalities

R=J-,

(3.313a)

xo = -m,

yo = -n.

(3.313b)

If q > m2+ n2 holds! the equation defines an imaginary curve, if q = m2 + n2 the curve has one single point W O , Yo).

YS

YS

Y4

I

01

Figure 3.146

xn

+ X

Figure 3.147

3. Parametric Representation of the Circle I = 10 Rcost: y = yo + Rsint, (3.314) where t is the angle between the moving radius and the positive direction of the x-axis (Fig. 3.147). 4. Equation ofthe Circle in Polar Coordinates in the general case corresponding to Fig. 3.148: (3.315a) p2 - 2ppo cos (p - yo) pi = RZ. If the center is on the polar axis and the circle goes through the origin (Fig. 3.149) the equation has the form p = 2Rcos9. (3.315b)

+

+

I

198

3. Geometru

Figure 3.148

Figure 3.149

Figure 3.150

5. Tangent ofaCircle The tangent ofacircle, given by (3.311a) at the point P(z0, yo) (Fig. 3.150) has the form xzo + yyo = RZ. (3.316)

I

3.5.2.6 Ellipse 1. Elements of the Ellipse In Fig. 3.151,AB = 2a is the major axis, = 2b is the minor axis, A , B , C, D are the vertices, F1, FZ are the foci at a distance c = on both sides from the midpoint, e = cia < 1 is the numerical eccentriczty, and p = b2/a is the semifocal chord. Le., the half-length of the chord which is parallel to the minor axis and goes through a focus.

L

2

a

-

i

Figure 3.151

Figure 3.152

2. Equation of the Ellipse If the coordinate axes and the axes of the ellipse are coincident, the equation of the ellipse has the normal form. This equation and the equation in parametric form are

x2

y2

z = acost, y = bsint. (3.317b) -+-=1, (3.317a) a2 b2 For the equation of the ellipse in polar coordinates see 6., p. 207. 3. Definition of the Ellipse, Focal Properties The ellipse is the locus of points for which the sum of the distances from two given points, the foci, is a constant, and equal to 20. These distances, which are also called the focal radii of the points of the ellipse, can be expressed as a function of the coordinate z from the equalities r l = FIP = a - ex, T Z = F2P= a + e x . T I + T Z = 2a. (3.318) Also here, and in the following formulas in Cartesian coordinates, we suppose that the ellipse is given in normal form. 4. Directrices of an Ellipse are lines parallel to the minor axis at distance d = a/e from it (Fig. 3.152).Every point P ( z ,y) of the ellipse satisfies the equalities (3.319) and this property can also be taken as a definition of the ellipse.

3.5 Vector Algebra and Analytical Geometry 199

5. Diameter of t h e Ellipse The chords passing through the midpoint of the ellipse are called dzameters o f t h e ellipse. The midpoint of the ellipse is also the midpoint of the diameter (Fig. 3.153). The locus of the midpoints of all chords parallel to the same diameter is also a diameter: it is called the conjugate dzameter of the first one. For k and k' as slopes of two conjugate diameters the equality -b2 kk' = (3.320) a2 holds. If 2al and 2bl are the lengths of two conjugate diameters and cy and 9 are the acute angles between the diameters and the major axis, where k = - tan cy and k' = tan p hold, then we have the Avollonius theorem in the form a l b l sin (0+ 3 ) = ab, a: + b: = a2 + bZ.

Figure 3.153

Figure 3.154

Figure 3.155

6. Tangent of the Ellipse at the point P(x0,yo) is given by the equation

ILo+E=1. a2 b2

(3.322)

The normal and tangent lines at a point P of the ellipse (Fig. 3.154) are bisectors of the interior and exterior angles of the radii connecting the point P with the foci. The line Ax + B y + C = 0 is a tangent line of the ellipse if the equation A2a2 + BZb2- C2 = 0 (3.323) is satisfied. 7. Radius of Curvature of t h e Ellipse (Fig. 3.154) If 1~ denotes the angle between the tangent line and the radius vector connecting the point of contact P(s0,yo) with a focus, then the radius of curvature is (3.324) At the vertices A and B (Fig. 3.151)and at C and D the radii are RA =

8. Areas of t h e Ellipse (Fig. 3.155) a ) Ellipse: S=rrab. b) Sector of t h e Ellipse BOP: ab (3.325b) bop = - arccos T , 2 a

Rg

b2

= - = p and Rc =

(3.325a) c) Segment of the Ellipse P B N :

S ~ B=Na barccos - x y .

(3.325~)

9. Arc and Perimeter of the Ellipse The arclength between two points A and B of the ellipse cannot be calculated in an elementary way as for the parabola, but only with an incomplete elliptic integral of the second kind E ( k , p) (see 8.2.2.2, 2., p. 447). The perimeter of the ellipse (see also 8.2.5, 7., 460) can be calculated by a complete elliptic integral of

( 1) one quadrant of the perimeter), and it is

the second kind E(e) = E e, - with the numericaleccentricity e = -/a

and with p =

2

(for

(3.326b) and an approximate value is

L x x [1,5(a+b) -

64 - 3X4 &%l ; L x x ( a + b)64 - 16Xz’

(3.326~)

IFor a = 1.5, b = 1 the formula (3.326~)results in the value 7.93. while the better approximation with the complete elliptic integral of the second kind (see 8.1.4.3, p. 436) results in the value 7.98.

3.5.2.7 Hyperbola 1. Elements of the Hyperbola In Fig. 3.156 = 2a is the real axis; A , B are the vertices; 0 the midpoint; F1 and Fz are the foci at a distance c > a from the midpoint on the real axis on both is the imaginary axis; sides; CD = 2b = -2 p = b2/a the semifocal chord of the hyperbola, Le., the half-length of the chord which is perpendicular to the real axis and goes through a focus; e = c / a > 1 is the numerical eccentricity. 2. Equation of the Hyperbola The equation of the hyperbola in normal form, i.e., for coincident x and real axes, and the equation in parametric form are

AB

X2 _ _ ”_ = 1, (3.327a)

a2

bZ

2b

Figure 3.156

x = acosht, y = bsinht (3.327b) or z = L, y = btant.(3.327c) cost

In polar coordinates see 3.5.2.9,6., p. 207 3. Definition ofthe Hyperbola, Focal Properties The hyperbola is the locus of points for which the difference of the distances from two given points, the foci, is a constant 2a. The points for which T I -rz = 2a belong to one branch of the hyperbola (in Fig. 3.156 on the left), the others with rZ -rl = 2a belong to the other brarich (in Fig. 3.156 on the right). These distances, also called the focal radzz, can be calculated from the formulas rl = *(ex - a ) ; r2 = &(ex + a); r2 - T I = 1 2 a , (3.328) where the upper sign is valid for the right branch, the lower one for the left branch. Here and in the following formulas for hyperbolas in Cartesian coordinates we suppose that the hyperbola is given in normal form.

3.5 Vector Alsebra and Analutical Geometru 201

Figure 3.157

Figure 3.158

4. Directrices of the Hyperbola are the lines perpendicular to the real axis at a distance d = a/c from the midpoint (Fig. 3.157).Every point of the hyperbola P ( z ,y ) satisfies the equalities

(3.329)

5. Tangent of the Hyperbola at the point P(x0, yo) is given by the equation

_ 2x0 - ?/o = 1, a2

(3.330)

b2

The normal and tangent lines of the hyperbola at the point P (Fig. 3.158) are bisectors of the interior and exterior angles between the radii connecting the point P with the foci. The line Ax By C = 0 is a tangent line if the equation AZa2- B2b2 - C2 = 0 (3.331)

+ +

is satisfied.

Figure 3.159

Figure 3.160

6. Asymptotes of the Hyperbola are the lines (Fig.3.159) approached infinitely closely by the

branches of the hyperbola for x + co. (For the definition of asymptotes see 3.6.1.4,p. 234.) The slopes of the asymptotes are k = k tan 6 = f b / a . The equations of the asymptotes are

y=i($)x.

(3.332)

A tangent is intersected by the asymptotes, and they form a segment of the tangent of the hyperbola, Le., the segment T T 1(Fig. 3.159). The midpoint of the segment of the tangent is the point of contact - P . so T P = TIP holds. The area of the triangle TOTl between the tangent and the asymptotes for any point of contact P is the same, and is So=ab.

(3.333)

202

3. Geometry

The area of the parallelogram OFPG,determined by the asymptotes and two lines parallel to the asymptotes and passing through the point P,is for any point of contact P (a2 h2) - c2 sp=--(3.334) 4 4’ 7. Conjugate Hyperbolas (Fig. 3.160)have the equations

+

(3.335) where the second is represented in Fig. 3.160 by the dotted line. They have the same asymptotes. hence the real axis of one of them is the imaginary axis of the other one and conversely.

Figure 3.161

Figure 3.162

8. Diameters of the Hyperbola (Fig. 3.161) are the chords between the two branches of the hyperbola passing through the midpoint, which is their midpoint too. Two diameters with slopes k and k‘ are called conjugate if one of them belongs to a hyperbola and the other one belongs to its conjugate. and kk‘ = h2/a2holds. The midpoints of the chords parallel to a diameter are on its conjugate diameter (Fig. 3.161). From two conjugate diameters the one with Ikl < h/a intersects the hyperbola. If the lengths of two conjugate diameters are 2ai and 2bl, and the acute angles between the diameters and the real axis are cy and 8 < c y , then the equalities a; - h 2 - a - h2, ah = alhl sin(cy - p) (3.336) are valid. 9. Radius of Curvature of t h e Hyperbola .kt the point P(z0,yo) the radius of curvature of the hyperbola is

(3.337a) where u is the angle between the tangent and the radius vector connecting the point of contact with a focus. At the vertices A and B (Fig. 3.156) the radius of curvature is h2 a 10. Areas in t h e Hyperbola (Fig. 3.162) a) Segment A P N :

RA = R g = p = - .

(:

S~p~=sy-ahln

+!b ) = r y - a h A r c o s h z

b) Area OAPG: ah ah 2d S O * p= ~ - + - In 4 2 c The line segment is parallel to the lower asymptote, cis the focal distance and d = E

(3.337b)

(3.338a)

(3.338b)

3.5 Vector Algebra and Analytical Geometry 203

11. Arc of the Hyperbola The arclength between two points A and B of the hyperbola cannot be calculated in an elementary way like the parabola, but we can calculate it by an incomplete elliptic integral of the second kind E ( k , y ) (see p. 447), analogously to the arclength of the ellipse (see p. 200). 12. Equilateral Hyperbola has axes with the same length a = b, so its equation is 22 - y2 = a2, (3.339a) The asymptotes of the equilateral hyperbola are perpendicular to each other. If the asymptotes coincide with the coordinate axes (Fig. 3.163), then the equation is zy=-

a’

(3.339b)

2 ’

Figure 3.163

Figure 3.164

Figure 3.165

3.5.2.8 Parabola 1. Elements of the Parabola In Fig. 3.164 the x-axis coincides with the axis ofthe parabola, 0 is the vertex of the parabola, F is the focus of the parabola which is on the x-axis at a distance p/2 from the origin, where p is called the semifocal chord of the parabola. We denote the directrix by N N ’ , which is the line perpendicular to the axis of the parabola and intersects the axis at a distance p/2 from the origin on the opposite side as the focus. So the semifocal chord is equal to half of the length of the chord which is perpendicular to the axis and passes through the focus. The numerical eccentricity of the parabola is equal to 1 (see 3.5.2.9,4., p. 207). 2. Equation of the Parabola If the origin is the vertex of the parabola and the x-axis is the axis of the parabola with the vertex on the left-hand side, then the normalfom ofthe equation ofthe parabola is y* = 2px. (3.340) For the equation of the parabola in polar coordinates see 3.5.2.9,6., p. 207. For a parabola with vertical axis (Fig. 3.165) the equation is y = ax2

+ bx + c.

(3.341a)

1 The parameter of a parabola given in this form is p = -. (3.341b) 2lal If a > 0 holds, the parabola is open up, for a < 0 it is open down. The coordinates of the vertex are b 4ac - b2 To = -(3.341~) 2 a ’ yo=- 4a 3. Properties of the Parabola (Definition of the Parabola) The parabola is the locus of points P ( x , y) whose distance from a given point, the focus, is equal to its distance from a given line: the directrix (Fig. 3.164). Here and in the following formulas in Cartesian coordinates we suppose the normal form of the equation of the parabola. Then we have the equation - P F = P K = x + -P, (3.342) 2

204

3. Geometry

where PF is the radius vector whose initial point is at the focus and endpoint is a point of the parabola. 4. Diameter of the Parabola is a line which is parallel to the axis of the parabola (Fig. 3.166). .4diameter of the parabola halves the chords which are parallel to the tangent line belonging to the endpoint of the diameter (Fig. 3.166). With slope k of the chords the equation of the diameter is

P

y=rc.

(3.343)

Yt. ,

Y).

Figure 3.166

Figure 3.167

Y4

Figure 3.168

5. Tangent of the Parabola (Fig.3.167)The equation of the tangent of the parabola at the point P(x0, YO) is (3.344) YYO = P (x + xO1 ' Tangent and normal lines are bisectors of the angles between the radius starting at the focus and the diameter starting at the point ofcontact. The tangent at the vertex, Le., the y-axis, halves the segment of the tangent line between the point of contact and its intersection point with the axis of the parabola, the x-axis: _ _ - TS=SP, m=OM=xo, TF=FP. (3.345) A line with equation y = kx b is a tangent line of the parabola if p=2bk. (3.346)

+

6. Radius of Curvature of the Parabola at the point P(xl, y1) with 1, as the length of the normal PN (Fig. 3.167) is (3.347a) and at the vertex 0 it is

R=p. 7. Areas in the Parabola (Fig. 3.168) a) Parabolic Segment PON: 2

S O p=~ -SUQNP(MQNP is a parallelogram). 3 ' b) Area OPR (Area under the Parabola Curve):

(3.347b)

(3.348a)

(3.348b)

8. Length of Parabolic Arc from the vertex 0 to the point P ( x ,y)

5.5 Vector Algebra and Analvtical Geometry 205

(3.34913) For small values of

Y

we have the approximation (3.349c)

Quantities 6 and A

Shape oft he curve Ellipse a) for A . S < 0: real,

AZO Central curves

6>0

6#0

6< 0

.

b) for A

. S > 0: imaginary’

A =0

A pair of imaginary’ lines with real common point

A#0 A =0

A pair of intersecting lines

Hyperbola

Normal form of the equation after the transformation

Required coordinate transformations 1. Translation of the origin to the center of the curve, whose coordinates are be - cd bd - ae

50

=

~

>

yo=-

6

,

2. Rotation of the coordinate axes by 2b the angle a with tan 2 a = -. a-c The sign of sin 2 0 must coincide with the sign of 2b. Here the slope of the new 2’-axis is c - a + \i(c - a)’ IC= 2b

+ 4b2

c’ =

a

+ c - d ( a - c)’ + 4bZ 2

(a’ and c‘ are the roots of the quadratic

equation

u2 - Su. + 6 = 0).

‘A, b and S are numbers given in (3.350b).

’ The equation of the curve corresponds to an imaginary curve. 3.5.2.9 Quadratic Curves (Curves of Second Order or Conic Sections) 1. General Equation of Quadratic Curves (Curves of Second Order or Degree) The ellipse, its special case, the circle, the hyperbola, the parabola or two lines as a singular conic section are defined by the general equation of a quadratic curve (curve of second order) (3.350a) az’ + 2 b x y + c y 2 + 2 d z + 2 e y + f = 0. We can reduce this equation to normal form with the help of the coordinate transformations given in Tables 3.19 and 3.20.

3. Geometry

206

Remark 1: The coefficients in (3.350a) are not the parameters of the special conic sections. Remark 2: If two coefficients ( a and 6 or 6 and c) are equal to zero, the required coordinate transformation is reduced to a translation of the coordinate axes. The equation cy2 + 2 d x t 2ey t f = 0 can be written in the form (y - yo)’ = 2 p ( x - xo): the equation a x 2 + 2 d x + 2ey + f = 0 can be written in the form ( x - xo)’ = 2p (y - yo). Table 3.20 Equations of curves of second order. Parabolic curves (6 = 0)

1 Shape of the curve

Quantities 6 and A Parabolic curves‘,

6= 0

A#0

Parabola

A=O Required coordinate transformation 1. Translation of the origin to the vertex of the parabola whose coordinates xo and yo are defined ad + be by the equations ax0 + 6yo + -= 0 and S

(

d + - d c ; 6),

xo+

(

e+- a e ; 6 d ) yo

+f =0 .

Double line for dZ - a f = 0, Imaginary* lines for d2 - a f < 0.

1

Normal form of the equation after the transformation

y ~ 2=

2px’ ae - 6d

2. Rotation of the coordinate axes by the angle a with t a n a = - 2 ; the sign of sina must differ 6 from the sign of a . Rotation of the coordinate axes by the angle a with

tan a =

-a6 ; the sign of sin a must differ

from the sign of a.

y’2 + -2

ad + be

&FP

y’ + f = 0 can be trans-

formed into the form (21’ - Yti) (Y’- Y : ) = 0.

(3.350b) They do not change during a rotation of the coordinate system, Le., if after a coordinate transformation the equation of the curve has the form (3.350~) a ‘ d 2 + 2b’z’y’ + c’y’’ + 2d’z’ + 2e‘y’ + f’ = 0, then the calculation of these three quantities A, 6, and S with the new constants will yield the same values

3.5 Vector A1,qebra and Analytical Geometry 207

3. Shape of the Quadratic Curves (Conic Sections) If a right circular cone is intersected by a plane, the result is a conic section. If the plane does not pass through the vertex of the cone, we get a hyperbola, a parabola, or an ellipse depending on whether the plane is parallel to two, one, or none of the generators of the cone. If the plane goes through the vertex. we get a singular conic section with A = 0. As a conic section of a cylinder, Le., a singular cone whose vertex is at infinity, we get parallel lines. We can determine the shape of a conic section with the help of Tables 3.19 and 3.20.

1& .8b

4. General Properties of Curves of Second Degree The locus of every point P (Fig. 3.169) with constant ratio e of the distance to a fixed point F , the focus, and the distance from a given line, the directrix, is a curve of second order with numerical eccentricity e. For e < 1 it is an ellipse, for e = 1 it is a parabola, for e > 1 it is a hyperbola. 5 . Determination of a Curve Through Five Points There is one and only one curve of second degree passing through five given points. If three of these points are on the same line, we have a singular or degenerate conic section.

_________._.

k:

s

Figure 3.169

6. Polar Equation of Curves of Second Degree A11 curves of second degree can be described by the polar equation p=-

P

1 t ecosp’

(3.351)

where p is the semifocal chord and e is the eccentricity. Here the pole is at the focus, while the polar axis is directed from the focus to the closer vertex. For the hyperbola this equation defines only one branch.

3.5.3 Analytical Geometry of Space 3.5.3.1 Basic Concepts, Spatial Coordinate Systems Every point P i n space can be determined by a coordinate system. The directions of the coordinate lines are given by the directions of the unit vectors. In Fig. 3.170a the relations of a Cartesian coordinate system are represented. We distinguish right-angled and oblique coordinate systems where the unit vectors are perpendicular or oblique to each other. Another important difference is whether it is a right-handed or a left-handed coordinate system. The most common spatial coordinate systems are the Cartesian coordinate system, the spherical polar coordinate system, and the cylindrical polar coordinate system.

1. Right- and Left-Handed Systems Depending on the successive order of the positive coordinate directions we distinguish right systems and left systems or right-handed and left-handed coordinate systems. A right system has for instance three non-coplanar unit vectors with indices in alphabetical order $, Zj, &. They form a right-handed system if the rotation of one of them around the origin into the next one in alphabetical order in the shortest direction is a counterclockwise rotation. This is represented symbolically in Fig. 3.34,p. 142; we substitute the notation a , b! c for the indices i, j, k . A left system consequently requires a clockwise rotation. Right- and left-handed systems can be transformed into each other by interchanging two unit vectors. The interchange of two unit vectors changes its orientation: A right system becomes a left system, and conversely, a left system becomes a right system. A very important way to interchange vectors is the cyclic permutation, where the orientation remains unchanged. As in Fig.3.34the interchange the vectors of a right system by cyclic permutation yields a rotation in a counterclockwise direction, i.e., according to the scheme (i --t j + k -+ i > j-+ k -+ i -+ j . k -+ i -+ j -+ k ) . In a left system the interchange of the vectors by cyclic permutation follows a clockwise rotation, Le., according to the scheme (i + k -+ j -+ i, k -+ j -+ i --t k , j -+ a -+ k -+ j ) .

208

3. Geometry

I

Figure 3.170

A right system is not superposable on a left system.

I

I1

I11

Y

-

+ -

+ +

z

t

+

+

+

-

X

+

ZI

I11

Iv--

II

i / ,-

/

--

/ fi,' I / ,/' I '1), --/---------. O,,' 1

/ I

/'

d ,/1

VI1 -----*

!

VI11

/

1

'

+-----VI

f I

b

IV

v

+ +

Octant

VI

VI1

VI11

-

-

-

+

-

+

of a point P are its distances from three mutually orthogonal planes in a certain measuring unit, with given signs. They represent the projections of the radius vector r' of the point P (see 3.5.1.1,6., p. 181) onto three mutually perpendicular coordinate axes (Fig. 3.170). The intersection point of the planes 0, which is the intersection point of the axes too, is called the origin. The coordinates x, y, and z are called the abscissa, ordinate, and applicate. The written form P(a,b, e) means that the point P has coordinates x = a , y = b, z = c. The signs of the coordinates are determined by the octant where the point Plies (Fig. 3.171, Table 3.21).

Next Page 3.5 Vector Alqebra and Analutical Geometru 209

Notice that in both cases the equations (3.352~) 6%x = 6, x SJ = e'k x 6k = 0' are valid. Usually we work with right-handed coordinate systems; the formulas do not depend on this choice. In geodesy we usually use left-handed coordinate systems (see 3.2.2.1, p. 143).

3. Coordinate Surfaces and Coordinate Curves Coordznate Surfaces have one constant coordinate. In a Cartesian coordinate system they are planes parallel to the other two coordinate axes. By the three coordinate surfaces z = 0, y = 0, and z = 0 three-dimensional space is divided into eight octants (Fig. 3.171). Coordinate lines or coordznate curues are curves with one changing coordinate while the others are constants. In Cartesian systems they are lines parallel to the coordinate axes. The coordinate surfaces intersect each other in a coordinate line. 4. Curvilinear Three-Dimensional Coordinate System arises if three families of surfaces are given such that for any point of space there is exactly one surface from every system passing through it. The position of a point will be given by the parameter values of the surfaces passing through it. The most often used curvilinear coordinate systems are the cylindrical polar and the spherical polar coordinate systems. 5 . Cylindrical Polar Coordinates (Fig. 3.172) are: 0 The polar coordinates p and p of the projection of the point P to the z, y plane and 0 the applicate z of the point P . The coordinate surfaces in a cylindrical polar coordinate system are: 0 The cylinder surfaces with radius e = const, the half-planes starting from the z-axis. p = const and 0 the planes being perpendicular to the z-axis, z = const. The intersection curves of these coordinate surfaces are the coordinate curves. The transformation formulas between the Cartesian coordinate system and the cylindrical polar coordinate system are (see also Table 3.22): x = ~ c o s p , y = psinp, z = z ; (3.353a)

e=

Jw, arctan;Y p=

Y

= arcsin -

e

for z

> 0.

(3.353b)

For the required distinction of cases with respect to p see ( 3 . 2 8 7 ~p.) ~191.

": Figure 3.172

Figure 3.173

6 . Spherical Coordinates or Spherical Polar Coordinates contain: 0 The length T of the radius vector ?of the point P , the angle 29 between the z-axis and the radius vector r' and 0 the angle p between the x-axis and the projection of i on the z, y plane. The positive directions (Fig. 3.173) here are for ?from the origin to the point P , for t9 from the z-axis

4 Linear Algebra 4.1 Matrices 4.1.1 Notion of Matrix 1. Matrices A of Size ( m n) , or Briefly A(m,,,) are systems of m times n elements, e.g., real or complex numbers, or functions, derivatives, vectors, arranged in m rows and n columns: all a12 . . . al, t 1st row a21 a22 t 2ndrow A = ( a t j )=

[

i

';'

. . amn T T 1'

am1 am2

'

1st 2nd

j

t

m-th row

n-th column.

IVith the notion s u e ofa matrzx matrices are classified according to their number of rows m and number of columns n: A of size (m,n). A matrix is called a spare matrzxif the number of rows and columns is equal, otherwise it is a rectangular rnatrzx.

2. Real and Complex Matrices Real matrzces have real elements, complex matrzces have complex elements. If a matrix has complex elements ail" t ib,, it can be decomposed into the form AtiB where A and B have real elements only. If a matrix A has complex elements, then its conjugate complex matrzx A* has elements aiv = Re (ailv)- i Im ( a p v ) .

(4.2a) (4.2b)

(4.2~)

3. Transposed Matrices AT Changing the rows and columns of a matrix A of size (m,n) we get the transposed matrzz AT. This has the size (n,m) and (awlT = is valid.

(%"I

(4.3)

4. Adjoint Matrices The adjoznt matrzz AHof a complex matrix A is the transpose of its conjugate complex matrix A' (see also 4.2.3, p. 260): A H = (A*)T. 5 . Zero Matrix A matrix 0 is called a zero matrix if it has only zero elements:

(4.4)

252

4. Linear Algebra

4.1.2 Square Matrices 1. Definition Square matrices have the same number of rows and columns, Le., m = n:

A = A(n,n)=

(

ail

;

'.'

ai,

)

, (4.6) ani . ' . ann The elements a,," of a matrix A in the diagonal from the left upper corner to the right lower one are the elements ofthe main diagonal. Their notation is all, a22,. . . ,ann,Le., they are all the elements aPy with p = I/. t..

2. Diagonal Matrices A square matrix D is called a diagonal matrix if all of the non-diagonal elements are equal to zero:

3. Scalar Matrix A diagonal matrix S is called a scalar matrix if all the diagonal elements are the same real or complex number e:

s=

(":;:!) . . . .

.

O O ' . ' C

4. Trace or Spur of a Matrix For a square matrix, the trace or spur of the matrix is defined as the sum of the main diagonal elements: Tr (A) = all + a22 + . . . + a n n =

2

a,,.

(4.9)

,=I

5 . Symmetric Matrices A square matrix A is symmetric if it is equal to its own transpose: A = AT. For the elements lying in symmetric positions with respect to the main diagonal a,,!, = au, is valid.

(4.10) (4.11)

6. Normal Matrices satisfy the equality A ~ = AA A ~ . (For the product of matrices see 4.1.4, p. 254.)

(4.12)

7. Antisymmetric or Skew-Symmetric Matrices are the square matrices A with the property: A = -AT. For the elements a,, of an antisymmetric matrix the equalities a,," = -av,,, a,, = 0 are valid, so the trace of an antisymmetric matrix vanishes: Tr (A) = 0.

(4.13a) (4.13b) (4.13~)

4.1 Matrices 253

The elements lying in symmetric positions with respect to the main diagonal differ from each other only in sign. Every square matrix A can be decomposed into the sum of a symmetric matrix A. and an antisymmetric matrix A,,: 1 1 with A = A, t A,, A - - ( A + A T ) ; A, = - ( A - AT). (4.13d) %-2 2 8. Hermitian Matrices or Self-Adjoint Matrices are square matrices A equal to their own adjoints : A = (A*)T= A H . (4.14) Over the real numbers the concepts of symmetric and Hermitian matrices are the same. The determinant of a Hermitian matrix is real.

9. Anti-Hermitian or Skew-Hermitian Matrices are the square matrices equal to their negative adjoints: A = -(A*jT = -AH. (4.15a) For the elements aiLVand the trace of an anti-Hermitian matrix the equalities a,, = -a;,,, a,p = 0: Tr (A) = 0 (4.1%) are valid. Every square matrix A can be decomposed into a sum of a Hermitian matrix Ah and an anti-Hermitian matrix Ash: 1 1 A = Ah + A,h with Ah = - ( A + A H ) ,A h = -(A - AH). (4.15~) 2 2 10. Identity Matrix I is a diagonal matrix such that every diagonal element is equal to 1 and all of the non-diagonal elements are equal to zero: /1 0 .'.o\ 0 for p # u, = (&,) with , S, = (4.16) 1 for p = u.

[

00

" '

The symbol 6," is called the Kronecker symbol.

11. Triangular Matrix 1. U p p e r Triangular Matrix, U, is a square matrix such that all the elements under the main diagonal are equal to zero: R = (rwVj with = 0 for p > u. (4.17) 2. Lower Triangular Matrix, L, is a square matrix such that all the elements above the main diagonal are equal to zero: L = ( l I v ) with l,, = 0 for p < v. (4.18)

4.1.3 Vectors Matrices of size (n,1)are one-column matrices or column vectors of dimension n. Matrices of size (1,n) are one-row matrices or row vectors of dimension n:

Column Vector: p =

[T )

, (4.19a)

Row Vector: a' = ( a l ,a2,. . . .an). (4.19b)

a,

By transposing, a column vector is changed into a row vector and conversely. A row or column vector

254

4. Linear Al.qebra

of dimension n can determine a point in n-dimensional Euclidean space R". The zero vector is denoted by 0 or QT respectively.

4.1.4 Arithmetical Operations with Matrices 1. Equality of Matrices Two matrices A = (abv)and B

= (bpY) are equal if they have the same size and the corresponding elements are equal: A = B. when apu = b,, for p = 1,.. . m; u = 1,. . . , n. (4.20) 2. Addition and Subtraction Matrices can be added or subtracted only if they have the same size. We get the sum/difference of two matrices by adding/subtracting the corresponding elements: (4.21a) A B = (a,,,) (bpY) = (apu5 bpu). ~

*

*

For the addition of matrices the commutative law and the associative law are valid: a) Commutative Law: A t B = B t A.

(4.21b)

b) Associative Law: (A + B) + C = A + (B + C).

(4.2IC)

3. Multiplication of a Matrix by a Number A matrix A of size (m:n) is multiplied by a real or complex number a by multiplying every element of A by CY: (4.22a) aA = a ( a p u )= (aa,,).

From (4.22a) it is obvious that we can factor out a constant multiplier contained by every element of a matrix. For the multiplication of a matrix by a scalar the commutative, associative and distributive laws for multiplication are valid: a) Commutative Law: aA = Aa; (4.22b)

b) Associative Law: a(BA) = (@)A; c) Distributive Law: ( a It @A = a A rt PA; CY(A rt B) = aA f aB.

(4.22~) (4.22d)

4. Division of a Matrix by a Number The division of a matrix b y a scalar y # 0 is the same as multiplication by a = l/y. 5. Multiplication of Two Matrices 1. The Product A B of two matrices A and B can be calculated only if the number of columns of the factor A on the left-hand side is equal to the number of rows of the factor B on the right-hand side. If A is a matrix of size (m.n ) ,then the matrix B must have size ( n , p )and , the product A B is a matrix C = (c,~) of size (m.p).The element c , ~is equal to the scalar product of the p-th row of the factor A on the left with the A-th column of the factor B on the right: n

A B = (xa,,b,A)

= (c,~) = C

( p = 1 , 2 , .. . , m ; X = 1 , 2 , .. . , p ) .

(4.23)

"=I

2. Inequality of Matrix Products Even if both products A B and B A exist, usually A B # B A, Le.: in general the commutative law for multiplication is not valid. If the equality A B = B A holds, then we say that the matrices A and B are commutable or commute with each other. 3. Falk Scheme Multiplication of matrices A B = C can be performed using the Falk scheme (Fig. 4.1). The element c , ~of the product matrix C appears exactly at the intersection point of

4.1 Matrices 255

the p-th row of A with the A-th column of B. IMultiplication of the matrices A(3,3) and B(3,2)is shown in Fig. 4.2 using the Falk scheme.

Figure 4.1

Figure 4.2

4. Multiplication of the Matrices K1 and Kz with Complex Elements For multiplication of two matrices with complex elements we can use their decompositions into real and imaginary parts according to (4.2b): K 1 = A I iB1, K2 = A2 t iB2. Here A I , Aa, B1, Bz are real matrices. After this decomposition. the multiplication results in a sum of matrices whose terms are products of real matrices. I( A i B ) ( A - i B ) = A' B2 i ( B A - A B ) . Of course when multiplying these matrices it must be considered that the commutative law for multiplication is not valid in general, Le., the matrices A and B do not usually commute with each other. 6. Scalar and Dyadic Product of Two Vectors If the vectors a and b a r e considered as one-row and one-column matrices, respectively, then there are two possibilities to multiply them according to the rules of matrix multiplication: If 5 has size (1.n) and b has size (n,1) then their product has size (1,1),i.e. it is a number. It is called the scalarproduct of two vectors. If conversely, a has size (n, 1) and b has size (1, m ) ,then the product has size ( n ,m ) ,Le.. it is a matrix. This matrix is called the dyadic product of the two vectors. 1. Scalar Product of Two Vectors The scalar product of a row vector aT= ( a l ,a2,. . . , a,) with a column vector fi = (b1>b2:. . . b,JT - both having n elements -is defined as the number

+

+

+ +

~

(4.24) The commutative law for multiplication is not valid for a product of vectors in general, so we must keep the exact order of aT and b. If the order of multiplication is reversed, then the product bT is a dyadic product 2. Dyadic Product or Tensor Product of Two Vectors The dyadzc product of a column vector a = ( a l ,a 2 . . . . . a,)T of dimension n with a row vector bT = ( b l ,b2,. . . , b,) of dimension m is defined as the following matrix: (4.25)

of size (n,m). Also here the commutative law for multiplication is not valid in general. 3. Hints on the Notion of Vector Products of Two Vectors In the domain of multi-vectors or alternating tensors there is a so-called outer product whose three-dimensional version is the wellknown vectorproduct or cross product (see 3.5.1.5,2., p. 183 ff). We do not discuss the outer product of multi-vectors of higher rank. 7 . Rank of a Matrix 1. Definition In a matrix A the number r of linearly independent column vectors is equal to the number of linearly independent row vectors. This number T is called the rank of the matrix and it is denoted by rank (A) = r .

256

4. Linear Aloebra

2. Statements about the Rank of a Matrix a) Because in a vector space of dimension m there exist no more than m linearly independent mdimensional row or column vectors (see 9.1.2.3, 2., p. 498), the rank r of a matrix A of size ( m ,n) cannot be greater, than the smaller of m and n: rank (A(m,n))= r 5 min (m,n ) . (4.26a) b) A square matrix A(n,n)is called a regular matrix if rank (A(n,nl)= r = n. (4.2613) A square matrix of size (n,n ) is regular if and only if its determinant differs from zero, Le., det A # 0 (see 4.2.3, 3., p. 260). Otherwise it is a singular matrix. c) Consequently for the rank of a singular square matrix A(n,n),Le., det A = 0

rank (A(n,n))= r < n is valid. d) The rank of the zero matrix 0 is equal to zero: Rg (0) = T = 0. e) The rank of the sum and product of matrices satisfies the relations /rank(A) - rank(B)/ 5 rank(A B) 5 rank(A) rank(B),

+

+

rank(AB) 5 min(rank(A),rank(B)).

(4.26~)

(4.26d) (4.26e) (4.26f)

3. Rules to Determine the Rank Elementary transformations do not change the rank of matrices.

Elementary transformations in this relation are: a) Interchanging two columns or two rows. b) Multiplication of a row or column by a number. c) Addition of a row to another row or a column to an other column. In order to determine their ranks every matrix can be transformed by appropriate linear combinations of rows into a form such that in the p-th row ( p = 2,3,. . . m ) ,at least the first p - 1elements are equal to zero (the principle of Gauss algorithm, see 4.4.2.4, p. 276). The number of row vectors different from the zero vector in the transformed matrix is equal to the rank r of the matrix. ~

8. Inverse Matrix For a regular matrix A = (a,,,) there is always an inverse matrix A-' (with respect to multiplication), i.e., the multiplication of a matrix by its inverse yields the identity matrix: A A - ~= A-'A = I. (4.27a) The elements of A-' = (p,,,) are (4.27b) where A,, is the cofactor belonging to the a,,, element of the matrix A (see 4.2.1, l., p. 259). For a practical calculation of A-' the method given in 4.2.3,2., p. 260 should be used. In the case of a matrix of size (2,2) we have: (4.28)

Remark: Why do we not define division among matrices but instead use the inverse for calculations? This is connected to the fact that division cannot be defined uniquely. The solutions of the equations BX1= A ( B regular), X2B = A are in general different.

Xi = B-'A

Xa = AB-'

(4.29)

4.1 Matrices 257

9. Orthogonal Matrices If the relation

A ~ = A - ~or AAT=ATA=I (4.30) holds for a square matrix A, then it is called an orthogonal matrzz, Le., the scalar product of a row and the transpose of another one. or the scalar product of the transpose of a column and another one are zero. while the scalar product of a row with its own transpose or of the transpose of a column with itself are equal to one. Orthogonal matrices have the following properties: a) The transpose and the inverse of an orthogonal matrix A are also orthogonal; furthermore. the determinant is det A = kl. b) Products of orthogonal matrices are also orthogonal. I The rotatzon matrza: D, which is used to describe the rotation of a coordinate system, and whose elements are the direction cosines of the new direction of axes (see 3.5.3.2, p. 211): is also an orthogonal matrix.

10. Unitary Matrix If for a matrix A with complex elements (A*)T= A-' or A(A*)T= (A*)TA= I (4.32) holds we call it a unitary matria. In the real case unitary and orthogonal matrices are the same.

4.1.5 Rules of Calculation for Matrices The following rules are valid of course only in the case when the operations can be performed, for instance the identity matrix I always has a size corresponding to the requirements of the given operation.

1. Multiplication of a Matrix by the Identity Matrix is also called the identical transformation: AI = IA = A. (4.33) (This does not mean that the commutative law is valid in general, because the sizes of the matrix I on the left- and on the right-hand side may be different.) 2. Multiplication of a Square Matrix A by a Scalar Matrix S or by the identity matrix I is commutative

AS = SA = cA with S given in (4.8),

(4.34a)

AI = IA = A.

(4.34h)

3. Multiplication of a Matrix A by the Zero Matrix 0 results in the zero matrix: AO=O. OA=O. (4.35) (The zero matrices above may have different sizes.) The converse statement is not true in general, Le., from AB = 0 it does not follow that A = 0 or B = 0.

4. Vanishing Product of Two Matrices The product of two matrices A and B can be the zero matrix even if neither of them is a zero matrix: A B = 0 or B A = 0 or both, although A

# 0, B # 0.

(4.36)

0 110 0

5. Multiplication of Three Matrices (AB)C = A (B C) Le., the associative law of multiplication is valid.

(4.37)

4. Linear A1,qebra

258

6. Transposing of a Sum or a Product of Two Matrices ( A + B)T = AT + BT, (AB)T= BTAT, (AT)T= A. For square invertible matrices A(","): (AT)-' = (A-l)T

(4.38a) (4.38b)

holds.

7. Inverse of a Product of Two Matrices (AB)-' = B-' A-l.

(4.39)

8. Powers of Matrices

AP = $A:. .4with p > 0, integer, p factors

A-P = (A-')p Apt9 = APAq

( p > 0, integeqdet A

(4.40a)

# 0),

( p ,q integer).

(4.40~) (4.40d)

9. Kronecker Product The Kronecker product of two matrices A = (a,,") and B = (a,,") is defined in the following way: (4.41) A 8 B = (a,,,,B). For the transpose and the trace we have the equalities: (ABB ) ~ = A BB ~ ~ , (4.42) Tr(A 8 B)=Tr(A).Tr(B).

(4.43)

4.1.6 Vector and Matrix Norms The norm of a vector or of a matrix can be considered as a generalization of the absolute value (magnitude) ofnumbers. Mieassignarealnumber Ilxll (Normx) tothevectorxor I/A//(NomA) toamatrix A. These numbers must satisfy the norm axioms (see 12.3.1.1,p. 609). For vectors x E R" they are: (4.44) 1. /IxlI 2 0 for every x; llxll = 0 if and only if x = 0. (4.45) 2. 1/Xx(I= 1x1 11x11 for every x and every real number A. 3. llx+ yll 5

11~1+ 1 ilrl1 for every x and y (triangle inequality) (see also 3.5.1.1, l., p. 181).

(4.46)

There are many different ways to define norms for vectors and matrices. But for practical reasons it is better to define a matrix norm IlAll and a vector norm 11x11 so that they might satisfy the inequality IlAxIl I IIAII llxll. (4.47) This inequality is very useful for error estimations. If the matrix and vector norms satisfy this inequality, then we say that they are conszstent vlzth each other. If there is a non-zero vector x for every A such that the equality holds, then we say the matria: norm //AI/is the subordinate to the vector n o m IlxlI.

4.1.6.1 Vector Norms If x = ( ~ 1 . ~ 2 ., .. ,x , ) ~is a real vector of n dimensions, i.e., x E R" , then the most often used vector norms are: 1. Euclidean Norm (4.48)

4.2 Determinants 259

2. Supremum or Uniform Norm Illill = l l l i l l m := maxis,/.

(4.49)

l
3. SumNorm

n

I l l i l l = I l l i l l 1 :=

(4.50)

IZil t=l

IIn R3, in elementary vector calculus 1/x112is considered as the magnitude of the vector magnitude 1x1 = 11x112 gives the length of the vector E.

x. The

4.1.6.2 Matrix Norms 1. Spectral Norm for Real Matrices IIAlI = IIAIIz :=

d m .

(4.51)

Here X,,,(ATA) denotes the greatest eigenvalue (see 4.5.1, p. 278) of the matrix ATA.

2. Row-Sum Norm

n

(4.52)

3. Column-Sum Norm (4.53) It can be proved that the matrix norm (4.51) is the sL-xdinate norm to the vector norm (4.48). The same is true for (4.52) and (4.49), and for (4.53) and (4.50).

4.2 Determinants 4.2.1 Definitions 1. Determinants are real or complex numbers uniquely associated with square matrices. The determinant of order n which is associated with the matrix A = (a,,,) of size (n,n ) ,

I a11 a12

D = detA = det (a,,)

’.

aln

I

=

(4.54)

is calculated in a recursive way using the Laplace expansion rule: n

det A =

a,,,A,,,

( p fixed, expansion along the p-th row),

(4.55a)

( u fixed, expansion along the u-th column).

(4.55b)

w=1

n

det A =

1a,,A,, p=1

Here A,, is the subdeterminant belonging to the element ab” multiplied by the sign factor (-l),+”. A,, is called the cofactor or algebraic complement.

4.2.2 Subdeterminants The subdetermznant of order ( n - 1) belonging to the element a,, of a determinant of order n is the determinant obtained by deleting the p-th row and the u-th column.

I

260

4. Linear Algebra

4.2.3 Rules of Calculation for Determinants Because of the Laplace expansion the following statements about rows are valid also for columns.

1. Independence of the Value of a Determinant The value of a determinant does not depend on which row was chosen.

2. Substitution of Cofactors If during the expansion of a determinant the cofactors of a row are replaced by the cofactors of another one, then we get zero:

2

(4.56) apuAhu = 0 ( p , X fixed; X # p ) . ”=l This relation and the Laplace expansion result in (4.57) Adj A = A A d j = (det A) I . The adjoint rnatrizof A , which is the transpose of the matrix made from the cofactors of A, is denoted by Adj. We must not confuse this adjoint with the transposed conjugate of a complex matrix A H (4.4). From the previous equality we get the inverse matrix A-1 = de:AAd”

(4.58)

3. Zero Value of a Determinant A determinant is equal to zero if a) a row contains zero elements only, or b) two rows are equal to each other, or c) a cow is a linear combination of the others. 4. Changes and Additions The value of the determinant does not change if a) its rows are exchanged for its columns, Le.. refiectzon in the main diagonal does not affect the value of it: (4.59) det A = det AT. b) any row is added to or subtracted from another one, or c) a multiple of any row is added to or subtracted from another one, or d) a linear combination of other rows is added to any row. 5 . Sign on Changing Rows If two rows are interchanged in a determinant, then the sign of the determinant changes.

6. Multiplication of a Determinant by a Number The value of a determinant will be multiplied by a if the elements of a row are multiplied by this number. The next formula shows the difference between this and the multiplication of a matrix A of size (n,n) by a number a (4.60) det (aA)= andet A .

7. Multiplication of Two Determinants The multiplication of two determinants can be reduced to the multiplication of their matrices: (4.61) (det A)(det B) = det (AB).

4.2 Determinants 261

Since det A = det AT (see (4.59)),we have the equalities (det A)(det B) = det (AB) = det (ABT) = det (ATB) = det (ATBT), (4.62) i.e.. it is permissible to take the scalar product of rows with columns, rows with rows, columns with rows or columns with columns.

8. Differentiation of a Determinant If the elements of a determinant of order n are differentiable functions of a parameter t , i.e., apv = apy(t), and we want to differentiate the determinant with respect t o t , then we differentiate one row at a time and finally we add the n determinants we have obtained.

I

!+1

I

I For a determinant of size ( 3 , 3 ) we have: all a12 a13 14, ai2 a h all a12 a13 021 a22 a23 = a21 a22 a23 ai1 a h a& a31 a32 a33 a31 a32 a33 a31 a32 a33 ~

11 +

a11 a12 a13 a21 a22 a:3 a$, a33

I

4.2.4 Evaluation of Determinants 1. Value of a Determinant of Second Order

l

all a12

a21 a22

1

= alla22 - a21a12.

(4.63)

2. Value of a Determinant of Third Order The Sarrus rule gives us a convenient scheme for the calculations, but it is valid only for determinants

1

of order three. It is the following: a 1I, a I 2" 1 3 r 1I,,,,.A 2 a21 ,,,..a22..p23,,.azl,a22 = allaz~a33+ a12a23a31 + w m a 3 2 a31 a32 a33 a31 a32 -(a31azza13 + (132u23a11+ a3ma12). The first two columns are copied after the determinant, then the sum of the products of the elements along the redrawn declining segments is calculated, then the sum of the products of the elements along the dotted inclining segments is subtracted. 3. Value of a Determinant of n-th Order By the expansion rule the calculation of the value of a determinant of order n is reduced to the evaluation of n determinants of order (n - 1). But for practical reasons (to reduce the number of required operations), first we transform the determinant with the help of the rules discussed above into a form such that it contains as many zeros as possible. 2 9 9 4 2 -312 8 I 4 8 3-5 1 2 6 4 (rule 4) (rule 6) = 0 (rule 3)

(rule 4)

Remark: An especially efficient method to determine the value of a determinant of order n can be obtained if we transform it in the same way we do in order to determine the rank of a matrix (see 4.1.4, 7., p. 255), Le.. all the elements under the diagonal all, a2z1.. . ,ann are equal to zero. Then the value of the determinant is the product of the diagonal elements of the transformed determinant.

262

4. Linear Algebra

4.3 Tensors 4.3.1 Transformation of Coordinate Systems 1. Linear Transformation By the linear transformation - = Ax

21= all21 t a1222 + a1323 x2 = a2121+ a2222 + aZ3x3 j.3 = a3121 + + a3323

or

(4.64)

a3222

a coordinate transformation is defined in three-dimensional space. Here I,,and i,,( p = 1.2,3) are the coordinates of the same point but in different coordinate systems K and k ,

2. Einstein’s Summation Convention Instead of (4.64) we can write 3

z,

ailVx,

=

(4.65a)

( p = 1,2,3)

u=1

or briefly by Einstein 2, = alrvxvr (4.65b) Le., we have to calculate the sum with respect to the repeated index v and put down the result for 1-1 = 1 , 2 ; 3. In general: the summation conventionmeans that if an index appears twice in an expression, then the expression is added for all values of this index. If an index appears only once in the expressions of an equation, for instance p i n (4.65b))then it means that the equality is valid for all possible values of this index.

3. Rotation of a Coordinate System If we get the Cartesian coordinate system K from K by rotation, then for the transformation matrix in (4.64) A = D is valid. Here D = (d,,”) is the orthogonal rotation matrix. The orthogonal rotation matrix D has the property D-’ = D T . (4.66a) The elements d,, of D are the direction cosines of the angles between the old and new coordinate axes. From the orthogonality of D , Le., from DDT=I and DTD=I, (4.66b) it follows that 3

3

xdiLZdYz=dllYl

xdk,dk~=&~

2=1

k=l

(pL,V=1,2.3).

(4.66~)

The equalities in (4.66~)show that the row and column vectors of the matrix D are orthonormalized, bcause d,, is the Kronecker symbol (see 4.1.2, lo., p. 253). The elements d,, of the rotation matrix can be determined by the Euler angles (see 3.5.3.2, p. 211). For rotation in the plane see 3.5.2.1, 3.5.2.2, p. 190; in space see 3.5.3.2,~.211.

4.3.2 Tensors in Cartesian Coordinates 1. Definition A mathematical or a physical quantity T can be described in a Cartesian coordinate system K by 3n elements til...ml the so-called translation invariants. Here the number of indices z, j , . . . , m is exactly equal to n ( n 1 0) . The indices are ordered. and every of them takes the values 1 , 2 and 3. If under a coordinate transformation from K to K for the elements ttj..,maccording to (4.64)

t,u,,.p=

3

3

t=l3=1

3

. . C a,iau3 . . apmt13...m , m=l

(4.67)

4.3 Tensors 263

is valid, then T is called a tensor of rank n, and the elements indices are the components ojthe tensor T .

(mostly numbers) with ordered

2. Tensor of Rank 0 .A tensor ojrank zero has only one component, Le., it is a scalar. Because its value is the same in every coordinate system, we talk about the invariance of scalars or about an invariant scalar. 3. Tensor of Rank 1 A tensor of rank 1 has three components t l , t2 and t 3 . The transformation law (4.67) is now 2

E,,

=

1QbLztz

(1= 1,2.3)

(4.68)

2=1

It is the transformation law for vectors, Le., a vector is a tensor of rank 1.

4. Tensor of Rank 2 If n = 2 . then the tensor T has nine components t,,, which can be arranged in a matrix

0 t l l t12 t13

T =T=

t2l

(4.69a)

t22 t23

t31 t 3 2 t 3 3

The transformation law (4.68) is now: (4.69b)

So, a tensor of rank 2 can be represented as a matrix. IA: The moment of inertia 0, of a solid with respect to the line g1which goes through the origin and has direction vector 5 = aT.can be represented in the form 0, = aT@. (4.70a) if we denote by (4.70b)

e,,

0, and 0, are the moments of inertia with respect to the coordithe so-called znertia tensor. Here nate axes, and O,,, Ox,and are the deviation moments with respect to the coordinate axes. IB: The load-up conditions of an elastically deformed body can be given by the tension tensor

e,,

(4.71) The elements 0 t k ( i ;k = 1,2,3)are determined in the following way: At a point P of the elastic body, we choose a small plane surface element whose normal vector points along the direction of the 21-axis of a right-angle Cartesian coordinate system. The power per surface unit on this element, depending on the material, is a vector with coordinates u l l , u12 and 013. The other components can be explained similarly. 5 . Rules of Calculation 1. Elementary Algebraic Operations The multiplication of a tensor by a number, and addition and subtraction of tensors of the same rank are defined componentwise, similarly to the corresponding operations for vectors and matrices. 2. Tensor Product Suppose there are given a tensor A of rank m and a tensor B of rank n with components ut,... and brs,,, respectively. Then the 3mtn scalars c,,,..~~,,,= at,,,,b, (4.72a) $...

264

4. Linear Algebra

give the components of a tensor C of rank m + n . We denote it by C = AB and we call it the tensor product of A and B.The associative and distributive laws are valid: ( A B ) C = A ( B C ) ,A ( B + C ) = A B +AC. (4.72b) 3. Dyadic Product The product of two tensors of rank 1 A = ( a l ,azra3) and B = (bl, bz, b3) gives a tensor of rank 2 with the elements c,,=aib, (i>j=l,2,3), (4.73a) i.e., the tensor product results in the matrix

0

albl albz alba azbl azbz ad3 . (4.73b) a h a d z ad3 This will be denoted as the dyadic product of the two vectors A and B. 4. Contraction If we make two indices equal to each other in a tensor of rank m ( m 2 2), and we sum with respect to them, then we get a tensor of rank m - 2, and we call this the contraction of the tensor. IThe tensor C of rank 2 of (4.73a) with c,j = sib,, which is the tensor product of the vectors A = (al,a2, a3) and B = (bl,bz, b3))can be contracted by the indices i and j , (4.74) oib, = albl a2bz a3b3 so we get a scalar, which is a tensor of rank 0. This gives the scalar product of vectors A and B.

+

+

4.3.3 Tensors with Special Properties 4.3.3.1 Tensors of Rank 2 1. Rules of Calculation For tensors of rank 2 the same rules are valid as for matrices. In particular, every tensor T can be decomposed into the sum of a symmetric and a skew-symmetric tensor: 1 1 T = - (T + T') + - (T - T') . (4.75a) 2 2 A tensor T = ( t i j ) is called symmetric if t,j = t,, for all i and j (4.75b) holds. In the case t,, = - t j z for all i and j (4.75c) we call it skew- or antisymmetric. Obviously the elements t l l , t 2 2 and t 3 3 of a skew-symmetric tensor are equal to zero. The notion of symmetry and antisymmetry can be extended for tensors of higher rank if we refer to certain pairs of elements.

2. Transformationof Principal Axes For a symmetric tensor T, Le., if t,, = t,, holds, there is always an orthogonal transformation D such that after the transformation the tensor has a diagonal form: (4.76a) The elements ill, iZ2 and &3 are called the eigenvalues ofthe tensorT. They are equal to the roots AI, X2 and X3 of the algebraic equation of third degree in A: (4.76b)

4.3 Tensors 265 The column vectors d, . d, and & of the transformation matrix D are called the eigenvectors corresponding to the eigenvalues, and they satisfy the equations Td, = Audu (V = 1.2,3). (4.76~) Their directions are called the directions of the principal axes, and the transformation T to diagonal form is called the transformation of the principal axes.

4.3.3.2 Invariant Tensors 1. Definition .A Cartesian tensor is called invariant if its components are the same in all Cartesian coordinate systems. Physical quantities such as scalars and vectors, which are special tensors, do not depend on the coordinate system in which they are determined; they must not change their value either under translation of the origin or rotation of a coordinate system K . We talk about translation invariance and about rotation invariance or in general about transfornation invariance. 2. Generalized Kronecker Delta If the elements i,, of a tensor of rank 2 are the Kronecker symbols, Le., = '1, =

{

1 for z = j 2 0 otherwise,

(4.77a)

then from the transformation law (4.69b) in the case of a rotation of the coordinate system considering (4.66~)we have (4.77b) tpu = d p t d u , = d p u , i.e.. the elements are rotation invariant. If we put them in a coordinate system so that they are independent of the choise of the origin, Le., they will be translation invariant, then the numbers 6,, form a tensor of rank 2, the so-called generalized Kronecker delta. 3. Alternating Tensor If &, C, and i'k are unit vectors in the directions of the axes of a right-angle coordinate system, then for the mixed product (see 33.1.6, 2., p. 184) we have

1, -1, 0,

if i , j , k cyclic (right-hand rule), if i.j. k anticyclic, otherwise.

(4.78a)

Altogether there are 33 = 27 elements, which are the elements of a tensor of rank 3. In the case of a rotation of the coordinate system from the transformation law (4.67) it follows that (4.7%) Le.. the elements are rotation invariant. If we put them in a coordinate system so that they are independent of the choise of the origin, i.e., they are translation invariant, then the numbers eijk form a tensor of rank 3. the so-called alternating tensor.

4. Tensor Invariants iVe must not be confused between tensor invariants and invariant tensors. Tensor invariants are functions of the components of tensors whose forms and values do not change during the rotation of the coordinate system.

A: If for instance the tensor T =

(tij)

is transformed in T =

(it,)by a rotation, then the trace

(spur) of it does not change:

+ +

+

Tr(T) = tll tz2 t33 = ill+ iZ2 i 3 3 . The trace of the tensor T is equal to the sum of the eigenvalues (see 4.1.2, 4., p. 252).

(4.79)

L. Linear Aloebra

266

B: For the determinant of the tensor T = ( t i j )

ill :I2

t l l t l 2 t13 ~

t2l t 2 2 t 2 3

~

=

~

t31 t 3 2 t 3 3

i13

111 t 2 2 i 2 3

(4.80)

~

t31 t 3 2 t33

,

is valid. The determinant of the tensor is equal to the product of the eigenvalues.

4.3.4 Tensors i n Curvilinear C o o r di n a t e S ys t e m s

4.3.4.1 Covariant and Contravariant Basis Vectors 1. Covariant Basis

We introduce the general curvilznear coordznates u, v,'w by the position vector i =?(?A, u, w) = 2(u,21. .)ZZ + y(u, 21, w)Zy + z ( u ,u. .)e;. (4.81a) We get the coordznate surfaces corresponding to this system by fixing the independent variables u, v,w in i ( u ,u , w),one at a time. There are three coordinate surfaces passing through every point of the considered region of space, and any two of them intersect each other in a coordznate line, and of course these curves pass through the considered point, too. The three vectors

ai -

ai -

au

au>

ai -

(4.81b)

aw

point along the directions of the coordinate lines in the considered point. They form the covarzant baszs of the curvilinear coordinate system.

2. Contravariant Basis The three vectors 1 ( -a xi - >a i )

1

D dv

D

aw

(ai

ai)

1

% x x 'D

(a i ai) dux%

(4.82a)

with the functional determinant (Jacobian determinant) (4.82b) are always perpendicular to the coordinate surfaces at the considered surface element and they form the so-called contravariant baszs of the curvilinear coordinate system. Remark: In the case of orthogonal curvilinear coordinates, Le., if

-a.i- a i = o , -a.i- =a i du

dv

au

aw

0,

ai ai

-.-=

av

aw

0,

(4.83)

then the directions of the covariant and contravariant basis are coincident.

4.3.4.2 Covariant and Contravariant Coordinates of Tensors of Rank 1 In order to be able to apply the summation convention of Einstein we introduce the following notation for the covariant and contravariant basis: (4.84)

(4.88)

4.3 Tensors 267

The components V kare the contravariant coordinates, the components V k are the covariant coordinates of the vector G. For these coordinates the equalities I.+ = gk1I.i and Vk =gklVl (4.86a) are valid, where g k , = glk = s;l. GI and gk' = dk = ik.i 1 (4.86b) respectively. Furthermore using the Kronecker symbol the equality d k 's" = bklr (4.87a) holds, and consequently 9kl g l m = b k m , (4.87b) The transition from LVk to i'k or from Vk to V k according to (4.86b) is described by raising or lowering the indices by oversliding. Remark: In Cartesian coordinate systems covariant and contravariant coordinates are equal to each other.

4.3.4.3 Covariant, Contravariant and Mixed Coordinates of Tensors of Rank 2 1. Coordinate Transformation In a Cartesian coordinate system with basis vectors Z 1 , Z z and Z3 a tensor T of rank 2 can be represented as a matrix

T=

0 ill t l 2 t13 t21 t22 t23

(4.88)

,

t31 t32 t33

BY ?= X l ( U l r U Z > u 3 ) ~t 1X Z ( U l r U Z > u 3 ) & t x 3 ( ~ l ? u 2 > u 3 ) ~ 3 we introduce curvilinear coordinates u l ru 2 , u 3 . The new basis is denoted by the vectors We have:

(4.89) il,

9; and c 3 . (4.90)

If we substitute 4 = $ , then we obtain 6and J Las covariant and contravariant basis vectors.

2. Linear Vector Function In a fixed coordinate system with the tensor T given as in (4.88) by the equality

d=Tv'

(4.91a)

with v'= = Vk(jkk,d = W k l j k = W k J k (4.91b) we define a linear relation between the vectors v' and 4. So we consider (4.91a) as a h e a r vector function.

3. Mixed Coordinates When we change the coordinate system, the equality (4.91a) will have the form

-.

- 3

Q = Ti..

(4.92a)

The relation between the components of T and T is the following: (4.92b) (4.92~)

268

4. Linear Alqebra

and we talk about mixed coordinates of the tensor; k contravariant index, 1 covariant index. For the components of vectors ;and tiwe have W k= TF’V’. (4.92d) If the covariant basis & is replaced by the contravariant basis d k ,then we get similarly to (4.92b) and (4.92~) (4.93a) and (4.92d) is transformed into Wk = Tk‘R. For the mixed coordinates Ti’ and T(”, we have the formula

TfI = gkmglnTAn.

(4.93b) (4.93c)

4. Pure Covariant and Pure Contravariant Coordinates If we substitute in (4.9313) for L( the relation r/; = glmVm,then we get

Wk = T,”gglmVm = TkmVm, (4.94a) if we also consider that Tk‘gglm = Tkm, (4.94b) The Tkm are called the covariant coordinates of the tensor T ,because both indices are covariant. Similarly we get the contravariant coordinates qkm = gm‘Tfl. (4.95) The explicit forms are: (4.96b)

4.3.4.4 Rules of Calculation In addition to the rules described on 4.3.2, 5., p. 264, the following rules of calculations are valid: 1. Addition, Subtraction Tensors of the same rank whose corresponding indices are both covariant or contravariant can be added or subtracted elementwise, and the result is a tensor of the same rank. 2. Multiplication The multiplication of the coordinates of a tensor of rank n by the coordinates of a tensor of rank m results in a tensor of rank m n . 3. Contraction If we make the indices of a covariant and a contravariant coordinates of a tensor of rank n (n 2 2) equal, then we use the Einstein summation convention for this index, we get a tensor of rank n - 2 . This operation is called contraction. 4. Oversliding Overslading of two tensors is the following operation: We multiply both, then we make a contraction so that the indices by which the contraction is made belong to different factors. 5. Symmetry A tensor is called symmetric with respect to two covariant or two contravariant standing indices if when exchanging them the tensor does not change. 6. Skew-Symmety A tensor is called skew-symmetric with respect to two covariant or two contravariant standing indices if when exchanging them the tensor is multiplied by -1. IThe alternating tensor (see 4.3.3.2, 3.,p. 265) is skew-symmetric with respect to two arbitrary covariant or contravariant indices.

+

4.3.5 Pseudotensors The reflection of a tensor plays a special role in physics. Because of their different behavior with respect to reflection we distinguish polarand m i d vectors (see 3.5.1.1,2., p. 180), altough mathematically they can be handled in the same way. Axial and polar vectors differ from each other in their determination,

4.3 Tensors 269 because axial vectors can be represented by an orientation in addition to length and direction. Axial vectors are also called pseudovectors. Since vectors can be considered as tensors, the general notion of pseudotensors is introduced.

4.3.5.1 Symmetry with Respect to the Origin 1. Behavior of Tensors under Space Inversion 1. Notion of Space Inversion The reflection of the position coordinates of points in space with respect to the origin is called space inversion or coordinate inversion. In a three-dimensional Cartesian coordinate system space inversion means the change of the sign of the coordinates: (4.97) .( Y,z ) --i ( - 2 , -Y, By this a right-hand coordinate system becomes a left-hand system. Similar rules are valid for other coordinate systems. In the spherical coordinate system we have: (4.98) (r>29,p) + ( - T , 7T - 19,p 7T). Under this type of reflection the length of the vectors and the angles between them do not change. The transition can be given by a linear transformation. 2. Transformation Matrix According to (4.64), the transformation matrix A = (afiv)of a linear transformation of three-dimensional space has the following properties in the case of space inversion: det A = -1. (4.99a) ab" = -dpv, For the components of a tensor of rank n (4.67)

.I-

+

tfiv.&? = (-l)"t,v..P (4.99b) holds. That is: In the case of point symmetry with respect to the origin a tensor of rank 0 remains a scalar, unchanged; a tensor of rank 1 remains a vector with a change of sign; a tensor of rank 2 remains unchanged. etc.

Figure 4.3

2. Geometric Representation The inversion of space in a three-dimensional Cartesian coordinate system can be realized in two steps (Fig.4.3): 1. By reflection with respect to the coordinate plane, for instance the z, z plane, the coordinate system z, y, z turns into the coordinate system z, -y, z . A right-hand system becomes a left-hand system (see 3.5 3.1. 1.. p. 207). 2. By a rotation of the system z, y, z around the y-axis by 180" we have the complete coordinate system z, Y,z reflected with respect to the origin. This coordinate system stays left-handed, as it was after the first step. Conclusion: Space inversion changes the orientation of a polar vector by NO", while an axial vector keeps its orientation.

270

4. Linear Alqebra

4.3.5.2 Introduction t o the Notion of Pseudotensors 1. Vector P r o d u c t u n d e r Space Inversion Under space inversion two polar vectors a and b

are transformed into the vectors - p and -b, Le., their components satisfy the transformation formula (4.99b) far tensors of rank 1. However, if we consider the vector product c = a x b as an example of an axial vector, then we get c = c under reflection with respect to the origin, Le., a violation of the transformation formula (4.99a) for tensors of rank 1. That is the reason why we call the axial vector c a pseudovector or generally a pseudotensor. The vector products f x v', ix F', V x v'= rot 17with the position vector r', the speed vector v', the power vector and the nabla operator V are examples of axial vectors, which have "false" behavior under reflection. 2. Scalar P r o d u c t under Space Inversion If we use space inversion for a scalar product of a polar and an axial vector, then we again have a violation of the transformation formula (4.99b) for tensors of rank 1. Because the result of a scalar product is a scalar, and a scalar should be the same in every coordinate system, here we have a very special scalar, which is called a pseudoscalar. It has the property that it changes its sign under space inversion. Pseudoscalars do not have the rotation invariance property of scalars. IThe scalar product of the polar vectors f (position vector) and v' (speed vector) by the axial vector 3 (angular velocity vector) results in the scalars 75 w' and v" w', which have the "false" behavior under reflection, so they are pseudoscalars. 3. Mixed P r o d u c t u n d e r Space Inversion The mixed product (a x b) (see 3.5.1.6.2., p. 184) of polar vectors a b, and c is a pseudoscalar according to (2.)) because the factor (a x b) is an axial vector. The sign of the mixed product changes under space inversion. 4. Pseudovector a n d Skew-Symmetric Tensor of Rank 2 The tensor product of axial vectors p = ( a l ,a2, a3)T and b = ( b l sbZ:b3)T results in a tensor of rank 2 with components ti, = aibj (i, j = 1,2.3) according to (4.72a). Since every tensor of rank 2 can be decomposed into a sum of a symmetric and a skew-symmetric tensor of rank 2 , according to (4.79) we have ~

t,, = $ ( a , b , + a j b , ) + i ( a2 , b , - a , b i )

(z,j=lt2,3).

(4.100)

The skewsymmetric part of (4.100) contains exactly the components of the vector product ( p x b) multiplied by so we can consider the axial vector c = (a x b) with components cl, c2, c3 as a skewsymmetric tensor of rank 2

4,

a2b3 - a362 = ~ 1 c31 = a3bl - alba = c2, c12 = a1bz - a2bl = c3, ~ 2 = 3

(4.101a) where -c13 -c23 0

,

(4.101b)

whose components satisfy the transformation formula (4.9913) for tensors of rank 2. Consequently we can consider every axial vector (pseudovector or pseudotensor of rank 1) as a skew-symmetric tensor C of rank 2 , where we have: c = (e1. c2, 0 c3 -c2 (4.102) 5. Pseudotensors of R a n k n The generalization of the notion of pseudoscalar and pseudovector is a pseudotensor of rank n. It has the same property under rotation as a tensor of rank n (rotation matrix D with det D = 1) but it has a (-1) factor under reflection through the origin. Examples of pseudotensors of higher rank can be found in the literature, e.g.. [4.1].

4.4 Systems of Linear Eauations 271

4.4 Systems of Linear Equations 4.4.1 Linear Systems, Pivoting 4.4.1.1 Linear Systems A h e a r system contains m linear forms y1 = allxl + a1222 + . . . t alnxn +

a1

+ a22x2 + t a2,x, + a2 ......................................... Y m = amlxl + amzxZ + . . . + amnxn t a,,, YZ

= azlxl

,..

or y=A&ta

(4.103a)

with

The elements aPyof the matrix A of size (m,n) and the components a@( p = 1 . 2 , . . . . m) of the column vectoraareconstants. Thecomponentsz, ( m = 1 , 2 , .. . . n ) ofthecolumnvectorxaretheindependent variables, the components y, ( p = 1 . 2 , . . . ,m) of the column vector y are the dependent variables.

4.4.1.2 Pivoting 1. Scheme of Pivoting If an element alk is not equal to zero in (4.103a) the variable yt can be exchanged for an independent one and the variable xk for a dependent one in a so-called pivoting step. The pivoting step is the basic element of pivoting, the method by which for instance systems of linear equations and linear optimization problems can be solved. The pivoting step is achieved by the schemes ~

(4.104)

where the left scheme corresponds to the system (4.103a).

2. Pivoting Rules The framed element in the scheme a,k (aik # 0 ) is called the pivot element; it is at the intersection of the pivot column and pivot row. The elements (Y", and cyP of the new scheme on the right-hand side will be calculated by the following pivoting rules: 1 azk

= -.

1.

fflk

3.

azv al" = -, azk

(4.105a)

(Y2 --

a,

--

atk

2-

apk

=apk (p=l, . . . .m:p#z),

aik

(v = 1,2.. . . . 71; v # k ) ,

(for every p # i and every v

(4.105b) (4.105~)

# k).

(4.105d)

272

L. Linear Aloebra

To make the calculations easier (rule 4) we write the elements aivin the ( m+ 1)-th row of the pivoting scheme (cellar row). With this pivoting rule we can change further variables.

4.4.1.3 Linear Dependence The linear forms (4.103a) are linearly independent (see 9.1.2.3,2., p. 498), if all yp can be changed for an independent variable 2, . The linear independence will be used, for instance to determine the rank of a matrix. Otherwise, the dependence relation can be found directly from the scheme.

I 21 2 2 2 3 2 4 y' 2 1 1 0 - 2 y2 1 - 1 0 0 2 y 3 1 5 2 0 0

After three pivoting steps (for instance y4 + x4, YZ + 21, YI + 2 3 ) we get:

No further change is possible because a32 = 0, and we can see the dependence relation y3 = 2y1 - 3y2

+

10. Also for another sequence of pivoting, there remains one pair of not exchangeable variables.

4.4.1.4 Calculation of the Inverse of a Matrix If A is a regular matrix of size (n,n ) ,then the inverse matrix A-' can be obtained after n steps using the pivoting procedure for the system y = Ax. Y3 Y1 Y2

. (We arrange the columns with respect

After rearranging the elements we get A-' =

to the indices of yz,the rows with respect to the indices of zk.)

4.4.2 Solution of Systems of Linear Equations 4.4.2.1 Definition and Solvability 1. System of Linear Equations

A system of m linear equations with n unknowns 21, 2 2 , . . . , E , all21 a2121

+ a1222 + ... + ainxn = a1 + a2222 + ... + aZnxn = a2 +

or briefly Ax = a,

(4.106a)

+ .

amlxl amzxz . . + amnxn = am is called a linear equation system. Here we denote:

(4.106b)

If the column vector a is the zero vector (a = Q), then the system of equations is called a homogeneous system, otherwise (a # Q) it is called an inhomogeneous system ofequations. The coefficients a,, of the system are the elements of the so-called matriz of coeficients A, and the components a, of the column vector 3 are the constant t e n s (absolute terns).

4.4 Systems of Linear Equations 273 2. Solvability of a Linear System of Equations A linear system of equations is called solvable or conszstent or compatzble if it has a solution. i.e., there exists at least one vector x = a such that (4.106a) is an identity. Otherwise, it is called inconsistent. The existence and uniqueness of the solution depend on the rank of the augmented matrzx (A, a). We get the augmented matrix attaching the vector p to the matrix A as its (n+ 1)-th column. 1. General Rules for Inhomogeneous Linear Equation Systems An inhomogeneous linear equation system Ax = p has at least one solution if rank (A) = rank (A,$ (4.107a) is valid. Furthermore, if T denotes the rank of A , i.e., T = rank ( A ) ,then a ) for T = n the system has a unique solution, (4.107b) ~

b) for r < n the system has infinitely many solutions.

(4.107~)

i.e.. the values of n - r unknowns as parameters can be chosen freely. IA: x1 - 2x2 t 3x3 - x4 2x5 = 2 The rank of A is 2. the rank of the augmented matrzz of co3x1 - x2 5x3 - 3x4 - 25 = 6 eficzents (A,p) is 3, i.e., the system is inconsistent. 2x1 + x2 + 2x3 - 2x4 - 3x5 = 8 IB: Both the matrices A and ( A , a ) have rank equal to 3. Bex1 - x 2 + 2x3 = 1 cause T = n = 3 the system has a unique solution: x1 = 21 - 2x2 - x3 = 2 10 1 3x1 - x2 t 5x3 = 3 - Q - x 3 = - -2 -221 t 2x2 t 3x3 = -4 7' 7' 7'

+

Ic: x1 - 22 13 - 14 = x1 - 22 - 53 + x4 = x1 - x2 - 2x3 + 2x4 =

+

+

1 0

-z1

Both the matrices A and ( A l a ) have rank equal to 2. The system is consistent but because r < n it does not have a unique solution. We can consider TI - r = 2 unknowns as free parameters: x2 = z1 23 = x4 1 - (xl, 24 arbitrary values). 2'

i,

ID:

+

+

We have the same number of equations as unknowns but 51 2x2 - 23 + x4 = 1 the system has no solution because rank(A) = 2, and 2x1 - x2 t 2x3 t 2x4 = 2 rank ( A ,a) = 3. 3x1 + 2 2 + 23 t 3x4 = 3 2 1 - 3x2 + 3x3 + xq = 0 2. Trivial Solution and Fundamental System of Homogeneous Systems a) The homogeneous system Ax = 0 of equations always has a solution, the so-called trivial solution 5 1 = 5 2 = . . . = x, = 0. (4.108a) (The equality rank (A) = rank ( A )Q)always holds.) b) If the homogeneous system has the non-trivial solutions a = (alr( ~ 2 , .. . , a,) and p = (bl,02,. . ., 6,). i.e , cy # 0 and p # 0, then x = s g tt p is also a solution with arbitrary constants s and t. Le., any linear combination of the solutions is a solution too. g2>.. . , m.Then these Suppose the system has exactly 1 non-trivial linearly independent solutions gl, solutions form a so-called fundamental system of solutions (see 9.1.2.3, 2., p. 498), and the general solution of the homogeneous system of equations has the form (4.108b) x = k 1 a l + kzp2+ . ' + !qa (ICl, k2$. . . ki arbitrary constants). If the rank of the coefficient matrix A of the homogeneous system of equations is less than the number of unknowns n, Le., T < n ,then the equation system has 1 = n - r linearly independent non-trivial solutions. If r = n, then the solution is unique, Le., the homogeneous system has only the trivial I

274

4. Linear Alqebra

solution. To determine a fundamental system in the case T < n we choose n-r unknowns as free parameters, and we express the remaining unknowns in terms of them. If we reorder the equations and the unknowns so that the subdeterminant of order T in the left upper corner is not equal to zero, then we get for instance: 21

=

~1(2r+lr~rt2r...rzn)

22

=

22(2rtlrZrt2r., ,>2n)

. . 2,

(4.109)

= Zr(Z,tlrZ,t2r..,rZ,).

Then we can have a fundamental system of solutions choosing the free parameters, for instance in the following way: Zrtl G t 2

1. fundamental solution: 1 2. fundamental solution: 0

0 1

, ..

...

( n - r)-th fundamental solution: 0

0

IE: 21 - 22 523 - 24 = 0 x1 + x2 - 2x3 t 3x4 = 0 321 - 22 + 823 + 2 4 = 0 21 322 - 923 724 = 0

+

+

+

2r+3

0 0 .. .

'' ' "'

5,

0

... 0, . .

(4.110)

.. .. 0 .'. 1

The rank of the matrix A is equal to 2. We can solve the 3 system for x1 and 22 and we get: x1 = - - 2 3 - x4, x2 = 2 7 - 2 3 - 2x4 ( ~ 3 ~ arbitrary). x4 Fundamental solutions are (yl= 2 3 7 (--,-,l,O)*and%= (-13-2,0,1)T. 2 2

4.4.2.2 Application of Pivoting 1. System of Linear Functions Corresponding to a Linear System of Equations = Ax - p to the equation system

In order to solve (4.106a), we assign a system of linear functions Ax = a so the use of pivoting (see 4.4.1.2, p. 271) is possible:

Ax=a

(4.111a) is equivalent to

y = A x - p = 0.

(4.111b)

The matrix A is of size (m,n).p i s a column vector with m components, Le., the number of equations m must not be equal to the number of unknowns n. After we finish pivoting we substitute y = 0. We can tell the existence and uniqueness of the solution of Ax = a directly from the last pivoting scheme.

2. Solvability of Linear Equation Systems The linear equation system (4.111a) has a solution if one of the following two cases holds for the corresponding linear functions (4.111b): Case 1: All ylt ( p = 1 , 2 , .. . , m) can be exchanged for some 2,. This means the corresponding system of linear functions is linearly independent. Case 2: At least one yc cannot be exchanged for any z, Le., (4.112) Yo = h Y 1 + h Y z + ' . ' + Any, + A0 holds and also Xo = 0. This means the corresponding system of linear functions is linearly dependent.

3. Inconsistency of Linear Equation Systems The linear equation system has no solution if in case 2 above A0 # 0 holds. In this case the system has contradictory equations.

4.4 I

+

Systems of Linear Equations 275

21 - 2x2 4x3 - 24 = 2 -321 f 3x2 - 3x3 4x4 = 3 2x1 - 3x2 + 5x3 - 354 = -1

+

x1 x 2 x3 x4 1 -4fter three pivoting steps (for instance y1 + 1 -2 4 -1 -2 21. y3 + 1 4 % y2 + 5 2 ) we get:

This calculation ends with case 1: y2 = y3 = 0, and x3 = t (-m 2 2 = 3t - 2.23 = t, 24 = 3.

y1, y2) y3

and

23

are independent variables. We substitute y1 =

< t < m) is a parameter. Consequently, the solution is: z1 = 2t + I,

4.4.2.3 Cramer's Rule We have a very important special case when the number of equations is equal to the number of unknowns allxl a1252 t . ' . t a l n z n = a1 a2121 a22x2 . , . t aznxn = a2 . . (4.113a) anlxl

+ + + . . + an2x2 t . ' . + annxn= a,

and the determinant of the coefficients does not vanish, Le., (4.113b) D = det A # 0. In this case the unique solution of the equation system (4.113a) can be given in an explicit and unique form: (4.1134

a11 a1 a13

Dz=

ai,

. . . a2n .,. ... ... ... ... .

a21 a2 a23

ani

an

an3

'.'

ann

(4.113d)

I

276

4. Linear Alqebra

:)

all a12 . . . ai, a1 [a?'.

"f'. 'I'.

[p

(4.114a)

... amn am

ami am2

all a12

.

... ai,

a1

!)

(4.114b)

. aps . . ;. ' a b., , .

0 aL2 . ' . ahn a&

... ...

...... 0

0

...

0

......

......... ... 0

(4.115)

o . . . o

........................ 0

0

......

0

o . . . o

3. Existence and Uniqueness of the Solution The Gauss steps are elementary row operations so they do not affect the rank of the matrix (A,a),consequently the existence and uniqueness of the solution and the solution itself do not change. From (4.115) we can tell the following:

Case 1: The system has no solution if any of the numbers ),a'!;

,)a';!

. . . . at-')

differs from zero.

Case 2: The system has a solution if)';a! = a!G)' = , . . = a;-1) = 0 is valid. Then we have two cases: a) r = n: The solution is unique. b) r < n: The solution is not unique; n - r unknowns can be chosen as free parameters.

4.4 Systems of Linear Equations

277

If the system has a solution we determine the unknowns in a successive way starting with the last row of the equation system with the matrix in row echelon form (4.115).

IB: -21 - 322 - 1 2 2 3 = -xi + 2x2 + 5x3 = 5x2 + 1723 = 3x1 - x2 + 2x3 = 7x1 - 4x2 - 2 3 =

-5 2 After two Gauss steps the 7 augmented matrix of coeffi1 cients has the form 0

[

-1 -3 -12 -5

0

1 'H i].

0

0 0

1. Overdetermined System of Equations Consider the linear system of equations (4.116) Ax=b with the rectangular matrix of coefficients A = ( a z j )(i = 1 , 2 , . . . m; j = 1,2,. . . n ; m 2 n). The matrix A and the vector b = ( b l , bz,. . . , b,)T on the right-hand side are given, and the vector x = (xl.x2.. . . x,JT is unknown. Because m 2 n holds we call this system an overdeteminedsystem. We can tell the existence and uniqueness of the solution and sometimes also the solution, for instance by pivoting. ~

2. Linear Mean Square Value Problem LVhen we represent a practical problem by the mathematical model (4.116) (i.e., A, b, x are reals), then because of measuring errors or because of other errors it is impossible to find an exact solution of (4.116) such that it satisfies all the equations. Substituting any vector we will have a residual vector = (rl,r 2 : . . . , rm)Tgiven as r = A ~ - b , r#Q. (4.117) We want to find a vector x to make the norm of the residual vector E as small as possible. Suppose now A, b, x are real. If we consider the Euclidean norm, then m

ET:= r T r =( A ~ - b ) ~ ( A x - =min b)

(4.118) z=l must be valid, i.e., the residual sum of squares must be minimal. Gauss already had this idea. We call of the residual vector is called (4.118) a linear mean squares value problem. The norm lip11 = the residue.

3. Gauss Transformation

The vector x is the solution of (4.118) if the residual vector r is orthogonal to every column of A. That is: A ~ I=: A ~ ( -Ab)~= or A~AZ = (4.119) Equation (4,119) is actually a linear equation system with a square matrix of coefficients. We refer to it as the system of normal equations. It has dimension n. The transition from (4.116) to (4.119) is called

278

4. Linear Alqebra

Gauss transformation. The matrix ATA is symmetric. If the matrix A has the rank n (because m 2 n all columns of A are independent), then the matrix ATA is positive definite and also regular, Le., the system of normal equations has a unique solution if the rank of A is equal to the number of unknowns.

4.4.3.2 Suggestions for Numerical Solutions of Mean Square Value Problems 1. Cholesky Method Because the matrix ATA is symmetric and positive definite in the case rank (A) = n, in order to solve the normal equation system we can use the Cholesky method (see 19.2.1.2, p. 890). Unfortunately this algorithm is numerically fairly unstable although it works fairly well in the cases of a “big” residue 1 1/1 and a “small” solution 1 l ~ l I . 2. Householder Method Numerically useful procedures in order to solve the mean square value problem are the orthogonalization methods which are based on the decomposition A = QR. Especially useful is the Householder method, where Q is an orthogonal matrix of size (m,m) and R is a triangular matrix of size (m,n ) (see 4.1.2, 11..p. 253).

3. Regularized Problem In the case of rank deficiency, Le., if rank (A) < n holds, then the normal equation system no longer has a unique solution, and the orthogonalization method gives useless results. Then instead of (4.118) we consider the so-called regularized problem rTr+ axTx= min! (4.120) Here cy > 0 is a regularization parameter. The normal equations for (4.120) are:

(ATA+ Q I )=~ATb. (4.121) The matrix of coefficients of this linear equation system is positive definite and regular for Q > 0, but the appropriate choice of the regularization parameter Q is a difficult problem (see [4.6]).

4.5 Eigenvalue Problems for Matrices 4.5.1 General Eigenvalue Problem Let A and B be two square matrices of size (n,n). Their elements can be real or complex numbers. The general ezgenvalue problem is to determine the numbers X and the corresponding vectors x # 0 satisfying the equation A x = XBx. (4.122) The number X is called an ezgenvalue, the vector x an ezgenvector corresponding to A. .4n eigenvector is determined up to a constant factor, because if is an eigenvector corresponding to X, so is a (c = constant) as well. In the special case when B = I holds, where I is the unit matrix of order n, Le., A x = Xx or (A-XI)x=Q, (4.123) the problem is called the speczal ezgenvalueproblem. We meet this form very often in practical problems, especially with a symmetric matrix A, and we will discuss it in detail in the following. More information about the general eigenvalue problem can be found in the literature (see [4.14]).

4.5.2 Special Eigenvalue Problem 4.5.2.1 Characteristic Polynomial The eigenvalue equation (4.123) yields a homogeneous system of equations which has non-trivial solutions 21 # Q only if det (A - XI) = 0. (4.124a)

4.5 Eigenvalue Problems f o r Matrices 279 By the expansion of det (A - XI) = 0 we get

det ( A - X I ) =

I:

all a21 ,

a12 a22 -

i

2 -3 1 3 1 3 , det(A-XI)= ( - 5 2-4)

HA:

. . . ain . . . azn ... ... ...

a13 a23

2-X

Choosing 5 3 = 1 the eigenvector is 3 = C2 A3

=

-3

(-J

0 , where Cz is an arbitrary constant.

-2: The corresponding homogeneous system yields:

7 3x2 = --x2. Choosing 2 2 = 3 the eigenvector is 3 = C3 3

MB:

(

31 04 -1I ) $ d e t ( A - X I ) = l 3-X 1 4 - :

-1 0 3 -1 The eigenvalues are X1 = 2, X2 = AB'= 4. X2 = 2: We get that 2 3 is arbitrary, xz =

x2

4 is arbitrary. z1 = -xz, x3 = -4x1 3

+

, where C3 is an arbitrary constant.

-~~=-X3+10X2-32X+32=0, 03-X

-23121

=

23

and choosing, for instance x3 = 1 the

280

4. Linear Alqebra

(

J

corresponding eigenvector is 3 = C1 -1 X2

=

A3

=

4: We get that

x2,q

, where C1 is an arbitrary constant.

are arbitrary,

x1 = - 2 3 .

x2 = C2 eigenvectors, e.g., for x 2 = 1,x3 = 0 and x2 = 0, x3 = 1: -

We have two linearly independent

(%),

3 = C3

( -!),

where C2,

C3 are arbitrary constants.

4.5.2.2 Real Symmetric Matrices, Similarity Transformations In the case of the special eigenvalue problem (4.123) for a real symmetric matrix A the following state-

I

ments hold:

1. Properties Concerning the Eigenvalue Problem 1. Number of Eigenvalues The matrix A has exactly n real eigenvalues X i (i = 1 , 2 , .. . , n ) ,count-

ing them by their multiplicity. 2. Orthogonality ofthe Eigenvectors The eigenvectorsxi and xjcorresponding to different eigenvalues X i # A, are orthogonal to each other, Le., for the scalar product of xiand xj (4.125) z:x, = (x$,xj)= 0 is valid. 3. Matrix with an Eigenvalueof Multiplicityp For an eigenvalue which has multiplicityp (A = X1 = Xz = = Xp),, there exist p linearly independent eigenvectors xl,xz,. . . x Because of (4.123) all the non-trivial linear combinations of them are also eigenvectors correspon&ng to A. Using the Gram-Schmidt orthogonalization process we can choose p of them such that they are orthogonal to each other. Summarizing: The matrix A has exactly n real orthogonal eigenvectors. t . .

~

, det (A - XI) = -A3

+ 3X + 2 = 0. The eigenvalues are

X1

= X2 = -1 and

X3

= 2.

XI = Xz = -1: From the corresponding homogenous equation system we get: 2, is arbitrary, x2 is arbitrary, x3 = -2, - 1 2 . Choosing first x1 = 1, x2 = 0 then x1 = 0, x2 = 1 we get the linearly inde-

pendent eigenvectors xl= C, X3 = 2: We get:

2, is

, where C1and C2 are arbitrary constants.

arbitrary, x2 = x l l z3 = xl,and choosing for instance x1 = 1 we get the

eigenvectors corresponding to different eigenvalues are orthogonal. 4. Gram-Schmidt Orthogonalization Process Let V, be an arbitrary n-dimensional Euclidean vector space. Let the vectors x l rx 2 , . . ,x,, E V, be linearly independent. Then there exists an orthogonal system of vectors y l , ~ 2. .s,y, . E V , which can be obtained as follows: (4.126)

Remarks: 1. Here (xkr y,) = x i y t is the scalar product of the vectors xk und yi. 2. Corresponding to theorthogonalsystemofthevectorsy,, y,, . . . ,y, weget theorthonormalsystem

4.5 Eigenvalue Problems for Matrices 281

2. Transformation of Principal Axes, Similarity Tkansformation For every real symmetric matrix A, there is an orthogonal matrix U and a diagonal matrix D such that A = UDUT. (4.127) The diagonal elements of D are the eigenvalues of A, and the columns of U are the corresponding normed eigenvectors. From (4.127) it is obvious that D = UTAU. (4.128) Transformation (4.128) is called the transformation of principal axes. In this way A is reduced to a diagonal matrix (see also 4.1.2, 2.,p. 252). If the square matrix A (not necessarily symmetric) is transformed by a square regular matrix G such a way that G - ~ AG = A

(4.129)

then it is called a similarity transformation. The matrices A and the following properties: 1. The matrices A and the eigenvalues.

A are called similar and they have

have the same eigenvalues, Le., the similarity transformation does not affect

2. If A is symmetric and G is orthogonal, then

A is symmetric, too:

= GTAG with GTG = I. (4.130) Now (4.128) can be put in the form: A real symmetric matrix A can be transformed orthogonally similar to a real diagonal form D.

4.5.2.3 Transformation of Principal Axes of Quadratic Forms 1. Real Quadratic Form, Definition A real quadratic form Q of the variables z1,z2,. . . z, has the form n

n

azjz,xj= xTAx,

Q=

(4.131)

i=l j=1

where 21 = (xl.x2.. . . z,)~is the vector of real variables and the matrix A = ( a t j )is a real symmetric matrix. The form Q is called positive definite or negative definite, if it takes only positive or only negative values respectively. and it takes the zero value only in the case z1 = 22 = . . . = z, = 0. The form Q is called positive or negative semidefinite, if it takes non-zero values only with the sign according to its name: but it can take the zero value for non-zero vectors, too. A real quadratic form is called indefinite if it takes both positive and negative values. According to

282

I . Linear Aloebra

the behavior of Q the associated real symmetric matrix A is called positive or negative definite, or semidefinite.

2. Real Positive Definite Quadratic Form, Properties 1. In a real positive definite quadratic form Q all elements of the main diagonal of the corresponding real symmetric matrix A are positive, i.e., a,, > 0 (i = 4 2 , ~. ,. n ) (4.132) holds. (4.132) represents a very important property of positive definite matrices. 2. .4 real quadratic form Q is positive definite if and only if all eigenvalues of the corresponding matrix A are positive. 3. Suppose the rank of the matrix A corresponding to the real quadratic form Q = r T A is~ equal to r . Then the quadratic form can be transformed by a linear transformation x = cg (4.133) into a sum of pure quadratic terms, into the so-called normal f o r m r

Q = ZTKZ = ~ p , Z , *

(4.134)

i=l

where p , = (sign X,)k, and k l , k?, . . . ,k, are arbitrary, previously given, positive constants. Remark: Regardless of the non-singular transformation (4.133) that transforms the real quadratic form of rank r into the normal form (4.134), the number p of positive coefficients and the number q = r - p of negative coefficients among the pi of the normal form are invariant (the inertia theorem of Sylvester). The value p is called the index of inertia of the quadratic form.

3. Generation of the Normal Form .A practical method to use the transformation (4.134) follows from the transformation of principal axes (4.128). First we perform a rotation on the coordinate system by the orthogonal matrix U , whose columns are the eigenvectors of A (i.e.. the directions of the axes of the new coordinate system are the directions of the eigenvectors). Then we have the form (4.135) Here L is a diagonal matrix with the eigenvalues of A in the diagonal. Then a dilatation is performed by the diagonal matrix D whose diagonal elements are d, =

E T.

given by the matrix C=UD, and we have: Q =%*A%=( U D f ) T A ( U D f ) = g T ( D T U T A U D ) g

The whole transformation now is (4.136)

=jiTDTL Dg = gTKg. (4.137) Remark: The transformation of principal axes of quadratic forms plays an essential role at the classification of curves and surfaces of second order (see 3.5.2.9, p. 205 and 3.5.3.6. p. 223).

4. Jordan Normal Form Let A be an arbitrary real or complex ( n ,n ) matrix. Then there exists a non-singular matrix T such that T’AT=J (4.138) holds, where J is called the Jordan matriz or Jordan normalform of A. The Jordan matrix has a block diagonal structure of the form (4.139)) where the elemnts J, of J are called Jordan blocks:

4.5 Eigenvalue Problems for Matrices 283

39)

J=

(4.140)

They have the following structure: 1. If A has only single eigenvalues A,, then J, = A, and k = n) i.e.. J is a diagonal matrix (4.140). 2. If A, is an eigenvalue of multiplicityp,. then there are 'J one or more blocks of the form (4.141) where the sum A, 1 0 of the sizes of all such blocks is equal to p, and we have . . . .. . J, = (4.141) E:=,p , = n. The exact structure of a Jordan block deA j 1 pends on the structure of the elementary divisors of the 0 A, characteristic matrix A - AI.

For further informationsee [4.13], [19.16] vol. 1.

4.5.2.4 Suggestions for the Numerical Calculations of Eigenvalues 1. Eigenvalues can be calculated as the roots of the characteristic equation (4.124b) (see examples on p. 279). In order to do this we have to determine the coefficients ai (i = 0.1.2,. . . , n - 1) of the characteristic polynomial of the matrix A. However, we should avoid this method of calculation, because this procedure is extremely unstable, Le., small changes in the coefficients ai of the polynomial result in big changes in the roots A,. 2. There are many algorithms for the solution of the eigenvalue problem of symmetric matrices. We distinguish between two types (see [4.6]): a) Transformation methods, for instance the Jacobi method, Householder tridiagona1ization.QR algorithm. b) Iterative methods, for instance vector iteration, the Rayleigh-Ritz algorithm. inverse iteration, the Lanczos method, the bisection method. .As an example the power method of Mises is discussed here. 3. The Power Method of Mises Assume that A is real and symmetric and has a unique dominant eigenvalue. This iteration method determines this eigenvalue and the associated eigenvector. Let the dominant eigenvalue be denoted by A I : that is, (4.142) 1x11 > l A 2 l 2 1x31 2 ." 2 IAnI. Let xl,x,.. . , ,x, be the associated linearly independent eigenvectors. Then: (4.143) 1. Ax, = A,xl (i = 1 ; 2 . . . . , n ) . 2. Each element 21 E R" can be expressed as a linear combination of these eigenvectors x,: (4.144) x = ~1x1 ~ 2 x 2 . , e n s , (e, const: z = 1 , 2 , . . . n). Multiplying both sides of (4.144) by A k times, then using (4.143) we have

+

+ +

~

From this relation and (4.142) we see that (4.146) This is the basis of the following iteration procedure: Step 1: Select an arbitrary starting vector x(O)E R"

284

4. Linear Alqebra

S t e p 2: Compute Akxiteratively: x(ktl)= Ax(k) ( k = 0 , 1 , 2 , . . . ;x(')is given). From (4.147) and keeping in mind (4.146) we have:

(4.147)

x ( ~=) A'x(') - % clXfx,. S t e p 3: From (4.147) and (4.148) it follows that X(k+l) = ~ ~ ( =k A(A"(O)), ) -

(4.148)

A(Akx('))E A(clX:xl) = clXf(Axl), cl(XtAXl) = Xl(clXf~l) % Xlx(')),therefore (4.149)

X ( k t l ) % /@),

that is, for large values of k the consecutive vectors ~ ( are~ X11 multiples of each other. S t e p 4: Relations (4.148) and (4.149) imply for X, and XI: (4.150)

x(0) x(I)

x(~) x ( ~ normalization )

1 3.23 14.89 88.27 0 -1.15 -18.12 -208.03

A1

I

1 7.58 67.75 -2.36 -24.93 -256.85

9.964

9.66 96.401 - 38.78 -394.09 11.67 117.78

~ ( ~ x 1 ( ~ normalization ) -

1 1

1 -3.79 10.177

l,22g)

1 10.09 102.331[ -4.09 -41.58 -422.49 -4.129 XI 1.22 12.38 125.73 10.16 10.161 x A1

Remarks: 1. Since eigenvectors are unique only up to a constant multiplier, it is preferable to normalize the 1 in the example. vectors ~ ( as~shown 2. The eigenvalue with the smallest absolute value and the associated eigenvector can be obtained by using the power method of Mises for A-'. If A-' does not exist, then 0 is this eigenvalue and any vector from the null-space of A can be selected as an associated eigenvector. 3. The other eigenvalues and the associated eigenvectors of A can be obtained by repeated application of the following idea. Select a starting vector which is orthogonal to the known vector zl, and in this subspace Xz becomes the dominant eigenvalue that can be obtained by using the power method. In order to obtain AB, the starting vector has to be orthogonal to both xl and xzrand so on. This procedure is known as matrza: dejatzon. 4. Based on (4.147) the power method is sometimes called vector zteration.

4.5 Eiqenvalue Problems for Matrices 285

4.5.3 Singular Value Decomposition 1. Singular Values and Singular Vectors Let A be a real matrix of size (m, n ) and its rank be equal to r . The matrices AAT and ATA have r non-zero eigenvalues A,, and they are the same for both of the matrices. The positive square roots d, = 6 ( u = 1 , 2 , . . . , T ) of the eigenvalues A, of the matrix ATA are called the singular values of the matrix A. The corresponding eigenvectors LI, of ATA are called right singular vectors of A, the corresponding eigenvectors y, of AAT left singular vectors: (4.15 1a) ,, ( u = 1,2.. . . , r ) . A ~ =A ~,y ~ ,AA%, = Ay The relations between the right and left singular vectors are: Au, = d,y,. ATy, = d,a,. (4.151b) A matrix A of size ( m ,n ) with rank r has r positive singular values d, (v = 1 . 2 , . . . r ) . There exist T orthonormalized right singular vectors u, and T orthonormalized left singular vectors L.Furthermore, there exist to the zero singular value n - T orthonormalized right singular vectors u,(u = r 1,. . . , n ) and m - r orthonormalized left singular vectors y, (v = r t 1... . , m). Consequently, a matrix of size (m, n) has n right singular vectors and m left singular vectors, and two orthogonal matrices can be made from them (see 4.1.4, 9., p. 257): (4.152) ~

+

2. Singular Value Decomposition The representation

\

A = VAUT (4.153a)

r rows

with A =

(4.153b) m - T rows

0 0

"'

r columns

0 ~0... 0 ) n - T columns

is called the singular value decomposition of the matrix A. The matrix A, as the matrix A, is of size (m, n) and has only zero elements except the first r diagonal elements a,, = d, (v = 1 , 2 . . . . , r ) . The values d, are the singular values of A. Remark: If we substitute ,4H instead of AT and consider unitary matrices 15'and V instead of orthogonals, then all the statements about singular value decomposition are valid also for matrices with complex elements. 3. Application Singular value decomposition can be used to determine the rank of the matrix A of size (m. n ) and to calculate an approximate solution of the overdetermined equation system Ax = b (see 4.4.3.1, p. 277) after the transformation according to the so-called regularization method, Le., to solve the problem (4.154) where a > 0 is a regularization parameter.

286

5. Aloebra and Dzscrete Mathematics

5 Algebra and Discrete Mat hematics 5.1 Logic 5.1.1 Propositional Calculus 1. Propositions .4 proposition is the mental reflection of a fact, expressed as asentence in a natural or artificial language. Every proposition is considered to be true or false. This is the principle oftluo-valuedness (in contrast to many-valued or fuzzy logic, see 5.9.1, p. 358). “True” and “false” are called the truth value of the proposition and they are denoted by T (or 1) and F (or 0), respectively. The truth values can be considered as propositional constants. 2. Propositional Connectives Propositional logic investigates the truth of compositions of propositions depending on the truth of the components. Only the extensions of the sentences corresponding to propositions are considered. Thus the truth of a composition depends only on that of the components and on the operations applied. So in particular. the truth of the result of the propositional operations “NOT A ’ (-A).

(5.1)

“A AND B” ( A A B ) ,

(5.2)

“A ORB” (A V B ) .

(5.3)

“IF A, THEN B” (A + B )

(5.4)

and

‘.A IF AND ONLY IF B“ (A e B ) (5.5) are determined by the truth of the components. Here “logical O R ’ always means “inclusive O R ’ , Le., “AND/OR’. In the case of implication, for A + B we also use the following verbal forms: A implies B.

B is necessary for A.

A is sufficient for B .

3. Truth Tables In propositional calculus. the propositions A and B are considered as variables (propositional variables) which can have only the values F and T. Then the truth tables in Table 5.1 contain the truth junctions defining the propositional operations. Table 5.1 Truth tables of propositional calculus

Negation

Conjunction

Disjunction

Implication

Equivalence

4. Formulas in Propositional Calculus Il’e can compose compound expressions (formulas) of propositional calculus from the propositional variables in terms of a unary operation (negation) and binary operations (conjunction, disjunction, implication and equivalence). These expressions, Le., the formulas, are defined in an inductive way: 1. Propositional variables and the constants T, F are formulas. (5.6) 2. If A and B are formulas: then (TA), ( AA B ) , (A V B ) , (A

+B)

~

(A u B )

(5.7)

5.1 Lopic 287

are also formulas. To simplify formulas we omit parentheses after introducing precedence rules. In the following sequence every propositional operation binds more strongly than the next one in the sequence: 7 ,

A,

v. =+.

@.

We often use the notation 2 instead of 'ITA" and we omit the symbol A. By these simplifications, for instance the formula ( ( AV ( 1 B ) )+ ( ( AA B ) V C ) )can be rewritten more briefly in the form:

AVB+ABVC 5. Truth Functions If we assign a truth value to every propositional variable of a formula. we call the assignment an znterpretatzon of the propositional variables. Using the definitions (truth tables) of propositional operations we can assign a truth value to a formula for every possible interpretation of the variables. C A V B ABVC A AB Thus for instance the formulagiven above de__ termines a truth function of three variables (a F F T F B o o L E a n function see 5.7.5. p. 358). T T T T W In this way, every formula with n proposiF F F T tional variables determines an n-place (or nT F T T ary) truth function, Le., a function which asF T F F signs a truth value to every n-tuple of truth T T T T values. There are 2'" n-ary truth functions, F T T T in particular these are 16 binary ones. T T T T

*

6. Elementary Laws in Propositional Calculus TKOpropositional formulas A and B are said to be logzcally equzvalent or sernantzcally equzvalent, denoted by A = B . if they determine the same truth function. Consequently, weJan check the logical equivalence of propositional formulas in terms of truth tables. So we get, e.g., A V B 3 ABVC = B v C , Le., the formula A V B + A B V C does not in fact depend on A. as follows from its truth table above. In particular, we have the following elementary laws of proposztzonal calculus: 1. Associative Laws ( A A B ) A C = A A ( B A C). (5.8a) ( A V B ) V C = A V ( B V C ) . (5.8b) 2. Commutative Laws A A B = 3 A A.

(5.9a)

AV B = B v A.

(5.9b)

3. Distributive Laws

( AV B ) C = ilC V BC.

(5.loa)

A B V C = ( AV C ) ( BV C ) .

(5.10b)

(5.11a)

AV A B = A.

(5.11b)

(5.12a)

AVA=A.

(5.12b)

(5.13a)

AvX=T.

(5.13b)

(5.14a)

AVB = AB.

(5.14 b)

4. Absorption Laws

A ( A V B ) = A. 5. Idempotence Laws .4A = A, 6. Excluded Middle

AX

= F.

7. De Morgan Rules -

AB = x V B ,

I

288

5. Algebra and Discrete Mathematics

8. Laws for T and F

AT = A,

(5.15a)

AVF=A,

(5.15b)

AF = F,

(5.15~)

AVT=T,

(5.15d)

-

T = F,

-

F=T.

(5.15e)

(5.15f)

9. Double Negation A = A. Using the truth tables for implication and equivalence, we get the identities

A+B=XVB

and A e B = AB V

(5.17a)

(5.16)

xz.

(5.17b)

Therefore implication and equivalence can be expressed in terms of other propositional operations. Laws (5.17a), (5.17b) are applied to reformulate propositional formulas.

+ AB V C = B V C can be verified in the following way: A V B + AB V C = AVBVAB v c = E V AB v c = ZBV AB v c = (ZVA)Bv c = TB v c = B v c.

I The identity A V i ?

10. Further Transformations

A ( 2 V B ) = AB,

A V Z B= A v B ,

(5.1sa)

( A V C ) ( BV C ) ( AV B ) = ( A V C ) ( BV

c),(5.18~)

(5.18b)

AC V BE V AB = AC V BE. (5.18d)

11. NAND Function and NOR Function As we know, every propositional formula determines a truth function. We can check the following converse of this statement: Every truth function can be represented as a truth table of a suitable formula in propositional logic. Because of (5.17a) and (5.17b) we can eliminate implication and equivalence from formulas (see also 5.7, p. 340). This fact and the De Morgan rules (5.14a) and (5.14b) imply that we can express every formula, therefore every truth function, in terms of negation and disjunction only, or in terms of negation and conjunction. There are two further binary truth functions of two variables which are suitable to express all the truth functions. They are called the NAND function or ShefTable 5.2 NAND function Table 5.3 NOR function fer function (notation L‘ I ”) and the NOR function or Peirce function (notation “J,”), with the truth tables given in Tables 5.2 T F T F and 5.3. Comparison of the truth tables for these operations with the truth tables of conjunction and disjunction makes the T T T T terminologies NAND function (NOT .4ND) and NOR function (NOT OR) clear.

v v

7. Tautologies, Inferences in Mathematics .4formula in propositional calculus is said to be a tautology if the value of its truth function is identically the value T. Consequently, two formulas A and B are called logically equivalent if the formula A @ B is a tautology. Laws of propositional calculus often reflect inference methods used in mathematics. As an example, consider the law of contraposition, i.e., the tautology A+BeB+x. (5.19a) This law, which also has the form A*B=B+;i, (5.19b) can be interpretegn this way: To show that B is a consequence of A is the same as showing that 2 is a consequence of B . Indirect proof(see also 1.1.2.2, p. 5) means the following principle: To show that

5.1 Loqic 289

B is a consequence of 4 , we suppose B to be false, and under the assumption that A is true, we derive a contradiction. This principle can be formalized in propositional calculus in several ways: A

+B = 4

A

+ B = AB=+F.

8 =+ 3

(5.20a)

or A

* B = AB+

B or

(5.20b)

(5.20~)

5.1.2 Formulas in Predicate Calculus For developing the logical foundations of mathematics we need a logic which has a stronger expressive power than propositional calculus. To describe the properties of most of the objects in mathematics and the relations between these objects the predicate calculus is needed.

1. Predicates We include the objects to be investigated into aset, i.e., into the domainX ofindividuums (or universe), e.g.. this domain could be the set N of the natural numbers. The properties of the individuums, as, e.g., ‘I n is a prime and the relations between individuums, e.g., m is smaller than n ”, are considered as predicates. An n-place predicate over the domain X of individuums is an assigment P : Xn --f {F,W}: which assigns a truth value to every n-tuple of the individuums. So the predicates introduced above on natural numbers are a one-place (or unary) predicate and a two-place (or binary) predicate. ‘*1

2. Quantifiers A characteristic feature of predicate logic is the use of quantifiers, i.e., that of a universal quantifier or ‘ifor every” quantifierv and existential quantifier or ‘Yorsome” quantifier 3. If P is a unary predicate, then the sentence “ P ( x )is true for every x in X ” is denoted by Vz P ( z )and the sentence‘’There exists an x in X for which P ( x )is true “is denoted by 3 x P ( z ) .Applying a quantifier to the unary predicate P , we get a sentence. If for instance N is the domain of individuums of the natural numbers and P denotes the (unary) predicate “nis a prime”, then Vn P(n) is a false sentence and 3 n P ( n ) is a true sentence.

3. Formulas in Predicate Calculus The formulas in predicate calculus are defined in an inductive way: 1. If ~1~ . . . , x, are individuum variables (variables running over the domain of individuum variables) and P is an n-place predicate symbol, then P ( q . . . . , x,) is a formula (elementaryfomula). (5.21) 2. If .4 and B are formulas: then (7.4)) ( A A B ) , ( A V B ) , (A + B ) ,( A @ B ) , (Vs A ) and ( 3 s A ) (5.22) are also formulas. Considering a propositional variable to be a null-place predicate, we can consider propositional calculus as a part of predicate calculus. An occurrence of an individuum variable z is bound in a formula if z is a variable in V x or in 3 x or the occurrence of z is in the scope of these types of quantifiers; otherwise an occurrence of x is free in this formula. A formula of predicate logic which does not contain any free occurrences of individuum variables is said to be a closedformula.

4. Interpretation of Predicate Calculus Formulas An interpretation of predicate calculus is a pair of a set (domain of individuums) and an assignment. which assigns an n-place predicate to every n-ary predicate symbol. For every prefixed value of free variables the concept of the truth evaluation of a formula is similar to the propositional case. The truth value of a closed formula is T or F. With a formula with free variables, we can associate the values of individuums for which the truth evaluation of the formula is true; these values constitute a relation (see 5.2.3. l . ,p. 294) on the universe (domain of individuums). I Let P denote the two-place relation 5 on the domain N of individuums. where K is the set of the

I

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5. Algebra and Discrete Mathematics

natural numbers then P ( z .y) characterizes the set of all the pairs (z, y) of natural numbers with z 5 y (two-place or binary relation on N); here z.y are free variables; V y P ( z ,y) characterizes the subset of N (unary relation) consisting of the element 0 only: here z is a free variable. y is a bound variable; 3 r V y P ( z .y) corresponds to the sentence ‘‘ There is a smallest natural number ” ; the truth value is true: here z and y are bound variables.

5. Logically Valid Formulas .4formula is said to be logically valzd (or a tautology) if it is true for every interpretation. The negation of formulas is characterized by the identities below: + z P ( z ) =3z+(z) or d z P ( r )=Vz+(z). Using ( 5 2 3 ) the quantifiers V and 3 can be expressed in terms of each other: Vz P ( r )= -32 +(z) or 3z P ( z ) = +z +(x). Further identities of the predicate calculus are: v z V y P ( z .y) = V y v z P ( z ,y),

323Y P b >Y) = 3Y 3a: P ( z ,Y),

(5.23) (5.24) (5.25) (5.26)

Vz P ( z )A Vz Q(z) = Vz ( P ( z )A Q(z)),

(5.27)

3 z P ( z ) v 3 z Q ( z )= 3 z ( P ( z ) v Q ( x ) ) .

(5.28)

The following implications are also valid: Vz P ( z )v Vz Q(z) + V r ( P ( z )V Q(z)),

(5.29)

3r(P(z)AQ(z))=+ 3zP(z)A3zQ(z). + Q(z)) (Vz P ( z ) + Vz Q(z)).

Vz ( P ( z )

*I).(&

Vz ( P ( z )

(5.30)

*

(5.31)

+ (Vz P ( z ) @ V ~ Q ( X ) ) ~

(5.32)

3 z V y P ( z ,y) =+ V y 32 P ( z ,y).

(5.33)

The converses of these implications are not valid, in particular, we have to be careful with the fact that the quantifiers V and 3 do not commute (the converse of the last implication is false).

6. Restricted Quantification Often it is useful to restrict quantification to a subset of a given set. So we consider Vz E X P ( z ) as a short notation of Vz (zE X + P ( z ) ) and

3 x E X P ( z ) as a short notationof 3z (zE X A P ( z ) ) ,

(5.34) (5.35)

5.2 Set Theory 5.2.1 Concept of Set, Special Sets The founder of set theory is Georg Cantor (1845-1918). The importance of the notion introduced by him became well known only later. Set theory has a decisive role in all branches of mathematics, and today it is an essential tool of mathematics and its applications.

1. Membership Relation 1. Sets and their Elements The fundamental notion of set theory is the membership relation. A set .4is a collection of certain different things a (objects, ideas, etc.) that we think belong together for certain reasons. These objects are called the elements of the set. We write “ a E A or “ a $ A to denote “ a is an element of A” or “ a is not an element of A ” , respectively. Sets can be given by enumerating their elements in braces. e.g.. M = {a, b. c} or U = {I, 3 , 5 , .. .}, or by a defining property possessed

5.6 Set Theorg 291

exactly by the elements of the set. For instance the set U of the odd natural numbers is defined and denoted by C‘ = {x 1 x is an odd natural number}. For number domains the following notation is generally used: IN = {0.1,2,.. .} set of the natural numbers, set of the integers. z = { O , l , -1.2. -2,. . .}

Q = ~ l p . y ~ Z ~ p i O } set of the rational numbers, R i Y

set of the real numbers, set of the complex numbers. 2. Principle of Extensionality for Sets Two sets A and B are identical if and only if they have exactly the same elements, Le., A = B @ V x (x E A * x E B ) . (5.36) IThe sets {3%1,3.7,2}and 11% 2,3,7} are the same. A set contains every element only “once”,even if it is enumerated several times.

C

2. Subsets 1. Subset If A and B are sets and V x (x E .4+ x E B) (5.37) holds. then A is called a subset of B , and this is denoted by A B . In other words: A is a subset of B , if all elements of A also belong to B. If for A B there are some further elements in B such that they are not in 4 , then we call A a proper subset of B , and we denote it by A C B (Fig. 5.1). Obviously. every set is a subset of itself A .4. ISuppose ‘4 = {2,4,6,8.10}is a set of even numbers and B = {1,2,3,. . . , l o } is a set of natural numbers. Since the set A does not contain odd numbers, A is a proper subset of B . 2. E m p t y Set or Void Set It is important and useful to introduce the notion of empty set or void set: 0, which has no element. Because of the principle of extensionality, there exists only one empty set. IA: The set {z/xE R A x2 + 2x 2 = 0) is empty. IB: 0 Jf for every set M ,Le., the empty set is a subset of every set M. For a set A the empty set and A itself are called the trivial subsets of A. 3. Equality of Sets Two sets are equal if and only if both are subsets of each other: .4 = B @ d C B A B A. (5.38) This fact is very often used to prove that two sets are identical. 4. Power Set The set of all subsets A of a set M is called the power set of M and it is denoted by P(,\I),Le., P(,lf) = { A 1 A M}. W For the set M = { a , b, c } the power set is P(.u)= {@,{ a } , { b ) , {e), { a . b } , { a , e}, {b. e}, { a , b, e ) } . It is true that: a ) If a set JI has m elements, its power set P(M)has 2m elements. b ) For every set M we have M ,0 E P(M),Le., M itself and the empty set are elements of the power set of .If. 5. Cardinal number The number of elements of a finite set M is called the cardinal number of M and it is denoted by cardM or sometimes by lM1. Lye also define the cardinal number of sets with infinitely many elements (see 5.2.5, p. 298).

+

5.2.2 Operations with Sets 1. Venn diagram The graphical representations of sets and set operations are the so-called V e n n diagrams, when we represent sets by plane figures. So, in Fig. 5.1, we represent the subset relation A B.

292

5. Algebra and Discrete Mathematics

Figure 5.1

Figure 5.2

Figure 5.3

2. Union, Intersection, Complement By set operations we form new sets from the given sets in different ways: 1. Union Let A and B be two sets. The union set or the union (denoted by A u B ) is defined by (5.39) A U B = {x jx E A V z E B } . We say ‘.A union B” or “Acup B”. If A and B are given by the properties El and E2 respectively, the union set A U B has the elements possessing at least one of these properties, Le., the elements belonging to at least one of the sets. In Fig. 5.2 the union set is represented by the shaded region. I{1,2,3}U {2,3,5: 6) = {1,2,3,5,6}. 2. Intersection Let A and B be two sets. The intersection set, intersection, cut or cut set (denoted by A n B ) is defined by A n B = { z l z E A A z E B}. (5.40) We say “ Aintersected by B” or “Acap B”. If A and B are given by the properties El and E2 respectively, the intersection A n B has the elements possessing both properties El and Ez, i.e., the elements belonging to both sets. In Fig. 5.3 the intersection is represented by the shaded region. IU‘ith the intersection of the sets of divisors T ( a ) and T(b)of two numbers a and b we can define the greatest common divisor (see 5.4.1.4, p. 321). For a = 12 and b = 18 we have T ( a )= {1,2,3,4,6,12} and T(b)= { 1,2,3:6: 9,18 }, so T(12) nT(18) contains the common divisors, and the greatest common divisor is g.c.d. (12,18) = 6. 3. Disjoint Sets Two sets A and B are called disjoint if they have no common element; for them AnB=0 (5.41) holds, Le., their intersection is the empty set. IThe set of odd numbers and the set of even numbers are disjoint; their intersection is the empty set, Le., {odd numbers} n {even numbers} = 0. 4. Complement If we consider only the subsets of a given set M, then the complementary set or the complement CM(A) of A with respect to M contains all the elements of M not belonging to A: CM(A= ) {Z 12 E M A Z A}. (5.42) We say “complement of A with respect to M”, and M is called the fundamental set or somet@es the universal set. If the fundamental set M is obvious from the coAsidered problem, the notation A is also used for the complementary set. In Fig. 5.4 the complement A is represented by the shaded region.

Figure 5.4

Figure 5.5

Figure 5.6

3. Fundamental Laws of Set Algebra These set operations have analoguous properties to the operations in logic. The fundamental laws of set algebra are:

5.2 Set Theory 293

1. Associative Laws (A n B ) n = An ( B n C),

c

(5.43)

(A U B ) U C = A U ( B J C).

(5.44)

(5.45)

A U B = B JA.

(5.46)

2. C o m m u t a t i v e Laws

A nB = B nA. 3. Distributive Laws

( AU B ) n C = ( A n C) U ( B n C), (5.47)

(An B ) U C = (A U C) n ( B U C). (5.48)

4. Absorption Laws

(5.49)

A U (A n B ) = A.

(5.51)

AUA=A

(5.53)

AuB=ZnB.

An2=0,

(5.55)

A U 2 = M (M fundamental set), (5.56)

A n M = A.

(5.57)

Au0=A,

(5.58)

An0=0,

(5.59)

A U M = M,

(5.60)

A n (4U B ) = A ,

(5.50)

5. Idempotence Laws

A n .4 = A, 6. D e M o r g a n Laws

AnB=XUB,

(5.54)

7. Some F u r t h e r Laws

-

121 = 0.

-

A

= A.

(5.61)

-

0 = M.

(5.62)

(5.63)

This table can also be obtained from the fundamental laws of propositional calculus (see 5.1.1, p. 286) if we make the following substitutions: A by n, V by U. T by M ,and F by 0. This coincidence is not accidental; it will be discussed in 5.7, p. 340. 4. Further Set Operations Besides the operations defined above there are defined some further operations between two sets A and B , the diflerence set or diflerence A \ B , the symmetric dzflerence A A B and the Cartesian product A x B: 1. Difference of T w o S e t s The set of the elements of A. not belonging to B is the diflerence set or difference of A and B : A \ B = {x I x € A A z $ B } . (5.64a) If A is defined by the property El and B by the property Ez, then A \ B contains the elements having the property El but not having property Ez. In Fig. 5.5 the difference is represented by the shaded region. I{I; 2 , 3 ! 4 } \ {3.4:5} = {1,2}. 2. S y m m e t r i c Difference of Two Sets The symmetric difference AAB is the set of all elements belonging to exactly one of the sets A and B: AAB = {x 1 (zE A A $ ~B ) v (zE B A X $ A)}. (5.64b) It follows from the definition that A N ? = (.I\ B ) u ( B \ A) = ( A u B ) \ (A n B ) , (5.64~)

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5. Aloebra and Dzscrete Mathematics

Le., the symmetric difference contains the elements which have exactly one of the defining properties El (for A ) and E2 (for B ) . In Fig. 5.6 the symmetric difference is represented by the shaded region. I11.2; 3; 4}A{3.4.5} = (1.2.5). 3. Cartesian Product of Two Sets The Cartesian product of two sets A x B is defined by A x B = {(a,b) 1 a E A A b E B } . (5.65a) The elements (a, b ) of A x B are called ordered pairs and they are characterized by a = c A b = d. (5.65b) ( a , b) = (e3d ) The number of the elements of a Cartesian product of two finite sets is equal to card ( A x B ) = (cardA)(cardB). (5~65~) IA: For.4= { 1 . 2 . 3 } a n d B = ( 2 . 3 ) w e g e t A x B = {(I:2).(l13).(2,2).(2.3),(3.2),(3,3)}and B x A = { ( 2 s 1 ) , ( 2 . 2 ) , ( 2 : 3 ) . ( 3 , 1 ) , ( 3 . 2 ) , ( 3 , 3 ) } u icardA th = 3 , cardB = 2, card(A x B ) = card(B x A ) = 6. IB: Every point of the 5 . y plane can be defined with the Cartesian product R x R (R is the set of real numbers). The set of the coordinates 2; y is represented by R x R, and we have:

R2 = R x R = {(x.y) 1

IE

R.y E R}

4. Cartesian Product of n Sets

From n elements. by fixing an order of sequence (first element, second element, . . , , n-th element) an ordered n-tuple is defined. If a, E A, ( i = 1 , 2 . . . . n) are the elements. the n-tuple is denoted by ~ a , ) )where a, is called the i-th component. For n = 3.4.5 we call these n-tuples triples, quadruples, and quintuples. The Cartesian product of n terms A1 x A2 x ' . x A, is the set of all ordered n-tuples ( a l ,a 2 , .. . ,a,) with a, E A, : (5.66a) ~

card(Al x '42 x ' ' x A,) = cardA1 cardA2 . . cardA,. Remark: The n times Cartesian product of a set A with itself is denoted by A"

(5.66b)

5.2.3 Relations and Mappings 1. n-ary Relations Relations define correspondences between the elements of one or different sets. An n-ary relation or n-place relation R between the sets AI A, is a subset of the Cartesian product of these sets. i.e.. R & A1 x . . . x A,. If the sets Ai. i = 1.. . . . n. are all the same set A. then R C A" holds and it is called an n-ary relation in the set A. 2. Binary Relations 1. Notion of Binary Relations of a Set The two-place (binary) relations in a set have special importance (see 5.2.3, 2.; p. 294). In the case of a binary relation the notation aRb is also very common instead of ( a , b) E R. I As an example, we consider the divisibility relation in the set A = {1,2,3.4}. Le.. the binary relation T = { ( a , b) 1 a, b E A A a is a divisor of b } (5.67a) (5.67b) = ((1;1).( 1 , 2 ) , (1.3): (134). (252). (2,4),(3,3),(4:4)1. 2. Arrow Diagram or Mapping Function Finite binary relations R in a set A can be represented by arrow functions or arrow diagrams or by relation matrices. The elements of A are represented as points of the plane and an arrow goes from a to b if aRb holds. Fig. 5.7 shows the arrow diagram of the relation T in A = {1,2,3:4). ~

5.2 Set Theory 295

I1 2 3 4

Figure 5.7

1 0 0 0

1 1 0 0

1 0 1 0

1 1 0 1

Scheme: Relation matrix

3. Relation Matrix The elements of A are used as row and column entries of a matrix (see 4.1.1, l., p. 251). At the intersection point of the row of a E A with the column of b E B there is an entry 1 if aRb holds, otherwise there is an entry 0. The above scheme shows the relation matrix for T in A = {1,2,3,4}.

3. Relation Product, Inverse Relation Relations are special sets. so the usual set operations (see 5.2.2, p. 291) can be performed between relations. Besides them, for binary relations, the relation product and the inverse relation also have special importance. Let R C A x E and S C B x C be two binary relations. The product R o S of the relations R. S is defined by

R o S = { ( a , C ) I 3 b ( b E B A aRb A bSc)}.

(5.68)

The relation product is associative, but not commutative. The inverse relation R-' of a relation R is defined by

R-'= {(b,aj I ( a , b ) E R}. For binary relations in a set .4 the following relations are valid:

(5.69)

c ( R0 T ) n (S0 T ) ,

( R U 5')o T = ( R o T)U ( S o T ) , (5.70)

( Rn S)0 T

( R u sj-' = R-1 u s-1,

(5.72)

(Rn s)-l= R-' n S-',

( R 0 s)-I= s-'0 R-1.

(5.74)

(5.71) (5.73)

4. Properties of Binary Relations A binary relation in a set A can have special important properties: R is called reflexive. if V Q E A aRa, (5.75) irreflexive. if V a E A iaRa, (5.76) (5.77) symmetric, if V a , b E A (aRb + bRa), antisymmetric, if Vu, b E A (aRb A bRa + a = b), (5.78) transitive, if V a t b, c E A (aRbA bRc =+ aRc), (5.79) linear: if V a , b E A (aRb V bRa). (5.80) These relations can also be described by the relation product. For instance: a binary relation is transitive if R o R 5 R holds. Especially interesting is the transitive closure tra(R) of a relation R. It is the smallest (with respect to the subset relation) transitive relation which contains R. In fact 'tra(R) =

U R" = R'

u R2u R3 u . ' . ,

n>l

where R" is the n times relation product of R with itself. ILet a binary relation R on the set {1.2,3,4,5} be given by its relation matrix M:

(5.81)

I

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5. Alaebra and Discrete Mathematics

2 0 0 0 1 0 3 0 0 1 0 1 4 0 1 0 0 1 5 0 1 0 0 0 TVe can calculate Mz by matrix multiplication where the values 0 and 1 are treated as truth values and instead of multiplication and addition we perform the logical operations conjunction and disjunction. So. M 2 is the relation matrix belonging to RZ. Similarly, we can calculate the relation matrices of R3. R4 etc. We get the relation matrix of R U RZ U R3 (the matrix on the Ai v M 2 v ,213 3 5 left) if we calculate the disjunction elementwise of the matrices M,M Z and M3. Since the higher powers of M contains no new 1-s, this matrix already coincides with the relation matrix of 0 1 1 1 1 tra(R). The relation matrix and relation product have important ap4 0 1 0 1 1 plications in search of path length in graph theory (see 5.8.2.1, 0 0 1 0 1 1 p. 349). In the case of finite binary relations, we can easily recognize the above properties from the arrow diagrams or from the relation matrices. We can recognize for instance the reflexivity from “self-loops” in the arrow diagram. and from the main diagonal elements 1 in the relation matrix. Symmetry is obvious in the arrow diagram if to every arrow there belongs another one in the opposite direction, or if the relation matrix is a symmetric matrix (see 5.2.3, 2., p. 294). We can see from the arrow diagram or from the relation matrix that the divisibility T is a reflexive but not symmetric relation. 5 . Mappings A mappzng (or function. see 2.1.1.1, p. 47) f from a set A to a set B with the notation f : A + B is a rule which assigns to every element a E A a unique element f ( a ) E B . We can consider a mapping f as a binary relation between A and B , ( f C A x B ) : f 5 A x B is called a mapping from A to B , if and V u E A 3 b E B ( ( a ,b) E f ) (5.82)

*

Va E

AVbl.bz E B ( ( a ,b i ) > (a,bz) E f

+ bi = bz)

(5.83)

hold. Here f is called a one-to-one (or injective) mapping if in addition (5.84) V a l . a2 E A V b E B ( ( a 1 b ) , (azl b) E f + a1 = a2) is valid. b’hile for a mapping we supposed only that an original element has one image, injectivity means also that every image has only one original element. Here f is called a mapping from A onto B (or surjective), if (5.85) V b E B 3 a E A ( ( a ,b ) E f ) holds. An injective mapping that is also surjective is called bijective. For a bijective mapping f : A + B there is an inverse relation which is also a mapping f-’: B + A , the so-called inverse mapping o f f . The relation product is used for composition of mappings: I f f : A + B and g : B i C are mappings, f o g is also a mapping from A to C, and is defined by (f o g ) ( a ) = g ( f ( a ) ) . (5.86) Remark: Be careful with the order off and g in this equation (it is treated differently in the literature!).

5.2.4 Equivalence and Order Relations The most important classes of binary relations with respect to a set A are the equivalence and order relations.

5.2 Set Theoru 297

1. Equivalence Relations A binary relation R with respect to a set A is called an equivalence relation if R is reflexive, symmetric, and transitive. For a R b we also use the notation a N R b or a b if the equivalence relation R is already known, and we say that a is equivalent to b (with respect to R ) . Examples of Equivalence Relations: IA: A = Z,m E W \ (0). a N R b holds exactly if a and b have the same remainder when divided by m (they are congruent modulo m). N

IB: Equality relation in different domains, e.g., in the set Q of rational numbers: Pl - = Pz - @ plqz = 41

pzql,

42

where the first equality sign defines an equality in Q, while the second one denotes an equality in

2. IC: Similarity or congruence of geometric figures. ID: Logical equivalence of expressions of propositional calculus (see 5.1.1, 6.. p. 287).

2. Equivalence Classes, Partitions 1. Equivalence Classes An equivalence relation in a set A defines a partition of A into non-empty pairwise disjoint subsets, into equzvalence classes. [ a ] := ~ { b 1 b E .4 A a N R b} (5.87) is called an equivalence class of a with respect to R. For equivalence classes the following is valid: (5.88) [ a ] #~ 0, a “ R b 9 [ a ]=~[ b ] ~ ,and a +R b H [ a ]n~[ b ]=~ 0. These equivalence classes form a new set, the quotzent set AIR: AIR = {[a],? 1 a E A}. (5.89) A subset Z C P(A) of the power set P(A) is called a partitzon of A if 0$Z, X.YEZAA’#Y+X~I’=~,

uX=A

(5.90)

XEZ

2. Decomposition Theorem Every equivalence relation R in a set A defines a partition Z of A , namely 2 = A/R. Conversely, every partition Z of a set A defines an equivalence relation R in A: a N R b o j l y E ( a E XA b E X). (5.91)

z

.4n equivalence relation in a set A can be considered as a generalization of the equality, where ‘‘ insignificant ” properties of the elements of A are neglected, and the elements, which do not differ with respect to a certain property, belong to the same equivalence class.

3. Ordering Relations .A binary relation R in a set A is called a partial ordering if R is reflexive, antisymmetric, and transitive. If in addition R is linear, then R is called a linear ordering or a chain. The set A is said to be ordered or linearly ordered by R. In a linearly ordered set any two elements are comparable. Instead of a R b we also use the notation a < E b or a < b, if the ordering relation R is known from the problem. Examples of Ordering Relations: IA: The sets of numbers N,2 , Q, R a r e completely ordered by the usual 5 relation. IB: The subset relation is also an ordering, but only a partial ordering. IC: The lezicographical order of the English words is a chain. Remark: If Z = {<4;B } is a partition of Q with the property a E A A b E B + a < b: then (A, B ) is called a Dedekind cut. If neither A has a greatest element nor B has a smallest element, so an irrational number is uniquely determined by this cut. Besides the nest of intervals (see 1.1.1.2, p. 2) the notion of Dedekind cuts is another way to introduce irrational numbers.

298

5. Al,qebra and Discrete Mathematics

4. Hasse Diagram Finite ordered sets can be represented by the Hasse diagram: Let an ordering relation be given on a finite set A. The elements of A are represented as points of the plane, where the point b E A is placed above the point a E A if a < b holds. If there is no c E A for which a < c < b, we say a and b are nezghbors or consecutive members. Then we connect a and b by a line segment. A Hasse diagram is a "simplified'' arrow diagram, where all the loops. arrow-heads, and the arrows following from the transitivity of the relation are eliminated. The arrow diagram of the divisibility relation T of the set A = I1.2.3,41 is airen in Fig. 5.7.-Talso denotes an ordering relation, which is represented dy t G Hasse diagram in Fig. 5.8.

<

O3

1.

Figure 5.8

5.2.5 Cardinality of Sets In 5.2.1, p. 290 the number of elements of a finite set was called the cardinality of the set. This notion of cardinality should can also be extended to infinite sets.

1. Cardinal Numbers Two sets A and B are called equznumerousif there is a bijective mapping between them. To every set .4 we assign a cardznal number IAJ or card A , so that equinumerous sets have the same cardinal number. A set and its power set are never equinumerous, so no '' greatest '' cardinal number exists. 2. Infinite Sets Infinite sets can be characterized by the property that they have proper subsets equinumerous to the set itself. The "smallest" infinite cardinal number is the cardinal number of the set DI of the natural numbers. This is denoted by No (aleph 0). .4 set is called enumerable or countable if it is equinumerous to N. This means that its elements can be enumerated or written as an infinite sequence a l , a * ) .. .. A set IS called non-countableif it is infinite but it is not equinumerous to N.Consequently every infinite set which is not enumerable is non-countable. IA: The set Z of integers and the set Q of the rational numbers are countable sets IB: The set R of the real numbers and the set C of the complex numbers are non-countable sets. These sets are equinumerous to P(N),the power set of the natural numbers, and their cardinality is called the contznuum.

5.3 Classical Algebraic Structures 5.3.1 Operations 1. wary Operations The notion of structure has a central role in mathematics and its applications. Now we investigate algebraic structures. Le., sets on which operations are defined. An n-ary operation 9 on a set A is a mapping p: A" i .4, which assigns an element of A to every n-tuple of elements of A.

2. Properties of Binary Operations Especially important is the case n = 2, which is called a binary operation, e.g., addition and multiplication of numbers or matrices, or union and intersection of sets. .4binary operation can be considered as a mapping * : A x A -+ A . where instead of the notation % ( a , b)" we use the infixform "a * b". .4 binary operation * in A is called associative if ( a * b) * c = a * ( b * c). (5.92) and commutative if a*b=b*a (5.93)

5.3 Classical Al.qebraic Structures 299

holds for every a. b. c E A . An element e E 4 is called a neutral element with respect to a binary operation * in A if a * e = e * a = a holds for every a E A .

(5.94)

3. Exterior Operations Sometimes we deal with exterior operations. That are the mappings from K x A to K , where K is an "exterior" and mostly already structured set (see 5.3.7, p. 314).

5.3.2 Semigroups The most frequently occurring algebraic structures have their own names. A set H having one associative binary operation * , is called a semigroup. The notation: is H = (H,*). Examples of Semigroups: IA: Number domains with respect to addition or multiplication. IB: Power sets with respect to union or intersection. IC: Matrices with respect to addition or multiplication. ID: The set A' of all words (strings) over an '' alphabet '' A with respect to concatenation (free semigroup). Remark: Except for multiplication of matrices and concatenation of words. all operations in these examples are also commutative; in this case we talk about a commutative semigroup.

5.3.3 Groups 5.3.3.1 Definition and Basic Properties 1. Definition .?I set G with a binary operation

*

is called a group if

* is associative.

* has a neutral element e. and for every element a E G there exists an inverse element a-' a * a-' = a-' * a = e.

such that (5.95)

A group is a special semigroup The neutral element of a group is unique, Le., there exists only one. Furthermore, every element of the group has exactly one inverse. If the operation * is commutative, then the group is called an Abelian group. If the group operation is written as addition, t,then the neutral element is denoted by 0 and the inverse of an element a by -a. Examples of Groups: IA: The number domains (except K)with respect to addition. IB: Q \ { 0 } , R \ {0}, and C \ {0} with respect to multiplication. IC: S,M := {f : .'d + W A f bijective} with respect to compositionofmappings (symmetric group). ID: Consider the set D, of all covering transformations of a regular n-gon in the plane. Here a covering transformation is the transition between two symmetric positions of the n-gon, i.e., the moving of the n-gon into a superposable position. If we denote by d a rotation by the angle 2 ~ / nand by u the reflection with respect to an axis, then D, has 2n elements: D, = { e ,d, d2,. . . , dn-': u, du, . . , dn-'u}. With respect to the composition of mappings D, is a group, the dihedral group. Here the equalities d" = uz = e and ud = d n - b hold. IE: All the regular matrices (see 4.1.4, p. 254) over the real or complex numbers with respect to multiplication. Remark: Matrices have a very important role in applications, especially in representation of linear transformations. Linear transformations can be classified by matrix groups. ~

300 5. Alqebra and Discrete Mathematics

2. Group Tables or Cayley's Tables For the representation of finite groups Cayley's tables or group tables are used: The elements of the group are denoted at the row and column headings. The element a * b is the intersection of the row of the element a and the column of the element b. IIf A4 = { 1 , 2 . 3 } , then the symmetric group S u is also denoted by S3. S3 consists of all the bijective mappings (permutati0ns)oftheset {1,2,3}andconsequentlyit has3! = 6elements (see 16.1.1,p. 743). Permutations are mostly represented in two rows, where in the first row there are the elements of Ad and under each of them there is its image. So, we get the six elements of S3 as the following: 1 2 3 1 2 3 1 2 3 3 2 1)' c = ( 1 2 3)' PI=(l 3 2). 4 (5.96) 1 2 3 1 2 3 1 2 3 ~ 3 = ( 2 1 3 ) ' p 4 = ( 2 3 1). ~ 5 = ( 3 1 2 ) . With the successive application of these mappings (binary operations) the following group table is obtained for S3: From the group table it can be seen that the identity perPlP2P3P4P5 mutation E is the neutral element of the group. E PI P2 P3 P4 P5 In the group table every element appears exactly once in PI Pi E P5 P4 P3 Pz (5,97) every row and in every column. P2 P2 P4 E P5 Pl P3 It is easy to recognize the inverse of any group element in P 3 P3 P5 P4 E P2 PI the table, Le., the inverse of p4 in S3 is the permutation p5, P4 P4 P2 P3 Pl P5 E because at the intersection of the row of p4 with the column P5 P5 P3 PI P2 E P4 of p5 is the neutral element E . If the group operation is commutative (Abelian group), then the table is symmetric with respect to the "main diagonal"; S3 is not commutative, since, e.g., pl op2 # pz o p l . The associative property cannot be recognized easily from the table.

5.3.3.2 Subgroups and Direct Products 1. Subgroups

Let G = (G. *) be a group and I; G. If b' is also a group with respect to *: then U = (U, *) is called a subgroup of G. .A non-empty subset U of a group (G, *) is a subgroup of G if and only if for every a, b E U , the elements a * b and are also in I; (subgroup criterion). 1. Cyclic Subgroups The group G itself and E = { e } are subgroups of G, the so-called trivial subgroups. Furthermore, a subgroup corresponds to every element a E G, the so-called cyclic subgroup generated by a: < a > = {. . . a-2: a - ' , e , a. a', . . .}. (5.98) If the group operation is addition, then we write the integer multiple k a as a shorthand notation of the k times addition of a with itself instead of the power a k las a shorthand notation of the k times operation of a by itself, Le., < a > = {. . , , (-2)a, -a,O,a, 2 a , . . .}. (5.99) Here < a > is the smallest subgroup of G containing a. If < a > = G holds for an element a of G, then G is called cyclic. There are infinite cyclic groups, e.g., Z with respect to addition, and finite cyclic groups, e.g., the set Z, the residue class modulo m with residue class addition (see 5.4.3,3., p. 325). IIf the number of elements of a finite G group is a prime, then G is always cyclic. 2. Generalization The notion of cyclic groups can be generalized as follows: If M is a non-empty subset of a group G: then the subgroup of G whose elements can be written in the form of a product of finitely many elements of M and their inverses, is denoted by < M >. The subset M is called the system of generators of < M >. If M contains only one element, then < M > is cyclic. ~

5.3 Classical Aloebraic Structures 301

3. Order of a Group, Left and Right Cosets In group theory the number of elements of a finite group is denoted by ord G. If the cyclic subgroup < a > generated by one element a is finite, then this order is also called the order of the element a, Le., ord < a > = ord a. If C is a subgroup of a group (G, *) and a E G, then the subsets (5.100) QC := { a * ulu E U } and Cra := { u * alu E U } of G are called left cosets and right cosets of U in G. The left or right cosets form a partition of G. respectively (see 5.2.4, 2., p. 297). .411 the left or right cosets of a subgroup U in a group G have the same number of elements, namely ord U . From this it follows that the number of left cosets is equal to the number of right cosets. This number is called the index of U in G. The Lagrange theorem follows from these facts. 4. Lagrange Theorem The order of a subgroup is a divisor of the order of the group. In general it is difficult to determine all the subgroups of a group. In the case of finite groups the Lagrange theorem as a necessary condition for the existence of a subgroup is useful. 2. Normal Subgroup or Invariant Subgroup For a subgroup 15'.in general, aU is different from Ua (however lalil = Ilia1 is valid). If aL' = Ua for all a E G holds, then U is called a normal subgroup or invariant subgroup of G. These special subgroups are the basis of forming factor groups (see 5.3.3.3, 3., p. 302). In Abelian groups. obviously, every subgroup is a normal subgroup. Examples of Subgroups and Normal Subgroups: IA: R \ { 0 } , Q \ {0} form subgroups of C \ (0) with respect to multiplication. IB: The even integers form a subgroup of 2 with respect to addition. IC: Subgroups of S3: according to the Lagrange theorem the group S3 having six elements can have subgroups only with two or three elements (besides the trivial subgroups). In fact, the group S3 has , = { E . P ~ ~ PS3. ~}, thefollowingsubgroups: E = {E}, UI= { E , P I } > UZ = { E . P Z } : U3 = { E , P ~ }Lr4 The non-trivial subgroups Lrls Uz, U3,and U4 are cyclic, since the numbers of their elements are primes. But the group S3 is not cyclic. The group S3 has only U4 as a normal subgroup, except the trivial normal subgroups. .4nyway, every subgroup U of a group G with lUI = lGl/2 is a normal subgroup of G. Every symmetric group SM and their subgroups are called permutation groups. ID: Special subgroups of the group GL(n)of all regular matrices of type (n,n) with respect to matrix multiplication: S L ( n ) group of all matrices A with determinant 1. O ( n ) group of all orthogonal matrices, SO(n) group of all orthogonal matrices with determinant 1. ThegroupSL(n) isanormalsubgroupofGL(n) (see5.3.3.3,3., p. 302) and SO(n)isanormalsubgroup of O ( n ) . IE: AS subgroups of all complex matrices of type (n,n) (see 4.1.4. p. 254): V ( n ) group of all unitary matrices, S G ( n ) group of all unitary matrices with determinant 1. 3. Direct Product 1. Definition Suppose A and B are groups, whose group operation (e.g., addition or multiplication) is denoted by .. In the Cartesian product (see 5.2.2. 4.. p. 294) A x B (5.65a) an operation * can be introduced in the following way: (5.10la) ( a ~b,), * ( Q Z ,b z ) = (a1 az,bl . b z ) . A x B becomes a group with this operation and it is called the direct product of A and B . ( e , e) denotes the unit element of A x B , (a-l, b-') is the inverse element of (a, b). For finite groups A : B (5.101b) ord (A x B) = ord A ord B

302

5. Alaebra and Discrete Mathematics

holds. Thegroups.4' := { ( a , e ) i a E A}andB' :={(el b)lb E B } arenormalsubsetsofAxBisomorphic to il and B. respectively. The direct product of Abelian groups is again an Abelian group. The direct product of two cyclic groups A, B is cyclic if and only if the greatest common divisor of the orders of the groups is equal to 1. I A: With Zz = {e, a } and Z3 = {e, b, b'}, the direct product Zz x Z3 = { ( e , e ) , (e. b), ( e , b z ) , ( a ,e ) % ( a , b). (a,b')}. is a group isomorphic to 2, (see 5.3.3.3: 2., p. 302) generated by ( a , b). IB: On the other hand ZZx Z z = { ( e , e ) , (e, b), ( a le), (a, b)} is not cyclic. This group has order 4 and it is also called Klein's four-group, and it describes the covering operations of a rectangle. 2. Fundamental Theorem of Abelian Groups Because the direct product is a construction which enables us to make "larger" groups from "smaller" groups, we can reverse the question: IVhen is it possible to consider a larger group G as a direct product of smaller groups A. B , Le., when will G be isomorphic to A x B 7 For Abelian groups, there exists the so-called fundamental theorem: Every finite Abelian group can be represented as direct product of cyclic groups with orders of prime powers.

5.3.3.3 Mappings Between Groups 1. Homomorphism and Isomorphism 1. Group Homomorphism Between algebraic structures we do not consider arbitrary mappings but only "structure keeping" mappings: Let GI = ( G I % *) and Gz = (Gz, 0 ) are two groups. .4mapping h: GI + G:!is called a group homomorphism. if for all a , b E GI h(a * b ) = h ( a ) o h(b) ("image of product = product of images") (5.102) is Yalid. I As an example, consider the multiplication law for determinants (see 4.2.3. 7., p. 260): det(AB) = (det A)(det B ) . (5.103) Here on the right-hand side there is the product of non-zero numbers: on the left-hand side there is the product of regular matrices. If h: GI + Gz is a group homomorphism, then the set of elements of GI, whose image is the neutral element of Gzris called the kernelof h, and it is denoted by ker h. The kernel of h is a normal subgroup OfGI. 2. Group Isomorphism If a group homomorphism h is also bijective, then h is called a group isomorphism. and the groups GI and Gz are said to be isomorphic to each other (notation: GI E Gz). Then ker h = E is valid. Isomorphic groups have the same structure. Le., they differ only by the notation of their elements. IThe symmetric group S3 and the dihedral group 0 3 are isomorphic groups of order 6 and describe the covering mappings of an equilateral triangle.

2. Cayley's Theorem The Cayley theorem says that every group can be interpreted as a permutation group (see 5.3.3.2, 2.: p. 301): Every group is isomorphic to a permutation group. The permutation group P , whose elements are the permutations xg (g E G) mapping a to (G, *)g, is a subgroup of SG isomorphic to G , *.

3. Homomorphism Theorem for Groups The set of cosets of a normal subgroup N in a group G is also a group with respect to the operation (5.104) aAVo biY = a b S . It is called the factor group of G with respect to ,V,and it is denoted by G I N . The folloiving theorem gives the correspondence between homomorphic images and factor groups of a group. because of what it is called the homomorphism theorem for groups:

5.3 Classical Alaebraic Structures 303

A group homomorphism h: GI + Gz defines a normal subgroup of G1, namely ker h = { a E Gljh(a) = e}. The factor group G1/ ker h is isomorphic to the homomorphic image h(G1)= {h(a)jaE G1}. Conversely. every normal subgroup N of GI defines a homomorphic mapping nath: GI + G1/N with nat.v(a) = u”V. This mapping natN is called a natural homomorphasm. ISince the determinant construction det: GL(n) + R \ (0) is a group homomorphism with kernel S L ( n ) ,S L ( n ) is a normal subgroup of GL(n) and (according to the homomorphism theorem): G L ( n ) / S L ( nis) isomorphic to the multiplicative group R\{O} ofreal numbers (for notation see 5.3.3.2. 2.) p. 301).

5.3.4 Group Representations* 5.3.4.1 Definitions 1. Representation X representation D(G) of the group G is a map (homomorphism) of G onto the group of non-singular linear transformations D on an n-dimensional (real or complex) vector space V,: D(G) : a -i D ( a ) 2 a E G. (5.105) The vector space L’, is called the representation space; n is the dimension of the representation (see also 12.1.3,2.: p. 597). Introducing the basis {e,} (i = 1 , 2 . . . . , n) in V, every vector x can be written as a linear combination of the basis vectors: n

x = C x , e , . x E Vn.

(5.106)

2=1

The action of the linear transformation D ( a ) , a E G: on x can be defined by the quadratic matrix (Dtic(u))(i. k = 1 , 2 . .. . n ) , which provides the coordinates of the transformed vector x’ within the basis ei: ~

(5.107) This transformation may also be considered as a transformation of the basis {e,}

+ {e:}:

n

e: = e,D(a) =

1Dkz(a)ek.

(5.108)

k=l

Thus, every element a of the group is assigned to the representation matrix (D,k(a)): D(G): a + ( & ( a ) ) (i, k = 1 , 2 , .. . , n ) , a E G. The representation matrix depends on the choice of basis.

(5.109)

2. Faithful Representation A representation is said to be faithful if G + D(G) is an isomorphism, Le., the assignment of the element of the group to the representation matrix is a one-to-one mapping. 3. Properties of the Representations .A representation has the following properties ( a , b E G, I : unit operator):

D ( u * b) = D ( a ) D(b), D(u-’) = D-l(a), D ( e ) = I .

(5.110)

5.3.4.2 Particular Representations 1. Identity Representation . b y group G has a trivial one-dimensional representation (identity representation). for which every element of the group is mapped to the unit operator I: a + I for all a E G. ‘In general, in this section vectors are not printed in bold symbols.

I

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5. Alqebra and Discrete Mathematics

2. Adjoint Representation The representation Dt(G) is said to be adjoint to D(G) if the corresponding representation matrices are related by complex conjugation and reflection in the main diagonal: (5.111)

D t ( G ) = D*(G).

3. Unitary Representation For a onztary representatzon all representation matrices are unitary matrices: D ( G ) . D f ( G ) = E. (5.112) where E is the unit matrix. 4. Equivalent Representations TKOrepresentations D(G) und D'(G) are said to be equivalent if for each element a of the group the corresponding representation matrices are related by the same similarity transformation with the nonsingular matrix T: (5.113) If such a relation does not hold two representations are called non-equzvalent. The transition from D(G) to D'(G) corresponds to the transformation T : {el, e*, . . . e,} + {e;) e;, . . . ,e;} of the basis in the representation space Y,: n

e' = e T ,

e: =

(z = 1 , 2 , .. . . n ) .

Tk,ek

(5.114)

k=l

Any representation of a finite group is equivalent to a unitary representation. 5 . Character of a Group Element In the representation D(G) the character x ( a ) of the group element a is defined as the trace of the representation matrix D(a) (sum of the matrix elements on the main diagonal): n

x ( a ) = Sp (D) =

(5.115)

&(a) ,=I

The character of the unit element e is given by the dimension n of the representation: x ( e ) = n. Since the trace of a matrix is invariant under similarity transformations, the group element a has the same character for equivalent representations. IWithin the shell model of atomic or nuclear physics two out of three particles with coordinates z1, 2 2 , 2 3 are supposed to be in the state 'pa while the third particle is in the state 'pp (configuration d B ) . The possible occupations ' p a ( z l ) ' p a ( d 9 & 3 ) = el: Wa(Zl)(F&2)%(23) = ez, $%(Zl)'pa(22)'Pa(Z3) = e3 form a basis {el: e 2 , e 3 } in the three-dimensional vector space V3 for a representation of the symmetric group S3. According to (5.108) the matrix elements of the representation matrices can be found by investigating the action of the group elements (5.92) on the coordiante subscripts in the basis elements e,. For example: plel = PlP0(Zl)9a(~2)9B(Z3) = pa(21)'pfl(22)'pa(z3)= D21(.%)e2j PI e2 = ~ 1 1 l ~ ()PO z l ( 4 p a( 2 3 ) = 'pa (21) ( ~ a ( ~ ) ' p p (=~ 03 1) 2 (PI )el, (5.116) n e 3 = P 1 ~ 3 ( z l ) % ( s Z ) ' p a ( r 3= ) ??8(21)%(22)9a(23)= D B ( P I ) ~ ~ . Altogether one finds: D(e)

=(:::). =(?:!),

D(j13) =

001

(loo), 001 010

D(pl)

(%),

D(p4) = 0 0 1 100

D(pz)

=(!::),

D(p5) =

100

(001) 1 00 . 010

(5.117)

5.3 Classical Aloebraic Structures 305

For the characters one has: x(e) = 3,

Y(P1)= X(Pz) = X ( P 3 ) = 1,

X(P4)

= X ( P 5 ) = 0.

5.3.4.3 Direct Sum of Representations The representations D(')(G), D(2)(G)of dimension nl and n2 can be composed to create a new representation D(G) of dimension n = n1 + nz by forming the direct sum of the representation matrices: (5.118)

The block-diagonal form of the representation matrix implies that the representation space V, is the direct sum of two invariant subspaces PI,, , Vn2:

Y,

= \',, @ V,,,

n = nl

+ nz.

(5.119)

.A subspace !m' ( m < n) of V, is called an invariant subspace if for any linear transformation D ( a ) . a E G. every vector x E ITm is mapped onto an element of V, again: I' = D ( ~ ) I with x,x' E I',. (5.120) The character ofthe representation (5.118) is the sum of the characters of the single representations: (5,121)

x ( a ) = + ' ) ( a ) t X(')(a).

5.3.4.4 Direct Product of Representations If e, ( i = 1 . 2 ; , . . , nl) and e; (IC = 1,2,. . . , nl) are the basis vectors of the representation spaces V,, and Yn2.respectively. then the tensor product (5.122) etk = {e,ek} ( i = 1 , 2 . . . . ni; k = 1 , 2 , . . . , n2)

.

forms a basis in the product space Yn,@Vn2of dimension nl '122. With the representations D(')(G) and D(2)(G)in Vnl and Ynz.respectively an ni . nz-dimensional representation D(G) in the product space can be constructed by forming the direct or (inner) Kronecker product of the representation matrices:

D(G) = D(')(G)8 D(')(G), ( D ( G ) ) %=~ ,@;)(a) ~[ .~$)(a) (5.123) with i . k = 1 . 2 , . . . , ni; j , 1 = 1.2,. . . ,712. The character of the Kronecker product of two representations is equal to the product of the characters of the factors p y a ) = /$)(a). #)(a). (5.124)

5.3.4.5 Reducible and Irreducible Representations If the representation space V, possesses a subspace V, ( m < n) invariant under the group operations the representation matrices can be decomposed according to

by a suitable transformation T of the basis in Y,. Dl(a) and Dz(a) themselfes are matrix representations of a E G of dimension m and n - m, respectively. A representation D(G) is called zrreduczble if there is no proper (non-trivial) invariant subspace in V,. The number of non-equivalent irreducible representations of a finite group is finite. If a transformation T of a basis can be found which makes V, to a direct sum of invariant subspaces, Le.. (5.126) Y, = VI @ ' ' ' €e Y,, ,

I

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5. Aloebra and Discrete Mathematics

then for every a E G the representation matrix D(a) can be transformed into the block-diagonal form (A = 0 in (5.125)):

T-'. D ( a ) . T = D(')(a)

. . @ D("j)(a)=

("('d'")

,'

D(n:)(a))

(5.127)

,

by a similarity transformation with T. Such a representation is called completely reduczble. Remark: For the application of group theory in natural sciences a fundamental task consists in the classification of all non-equivalent irreducible representations of a given group. IThe representation of the symmetric group S3 given in (5.117) is reducible. For example, in the transformation {el, ez. e3} + {e; = el + e2 + e3, e; = e2, e$ = e3) of the basis one obtains for the representation matrix of the permutation p3:

with A =

(i).

(

-1 0 Dl(p3) = 1 as the identity representation of S3 and Dz(p3) = -1 l ) .

5.3.4.6 Schur's Lemma 1 If C is an operator commuting with all transformations of an irreducible representation D of a group [C.D ( a ) ]= C . D(a)- D ( a ) C = 0, a E G, and the representation space V, is an invariant subspace of C, then C is a multiple of the unit operator, Le., a matrix (Czk)which commutates with all matrices of an irreducible representation is a multiple of the matrix E, C = X . E, X E C.

5.3.4.7 Clebsch-Gordan Series In general. the Kronecker product of two irreducible representations D(')(G),D(')(G)is reducible. By a suitable basis transformation in the product space D(')(G)@ D(')(G)can be decomposed into the direct sum of its irreducible parts D(") (a= 1.2.. . . n ) (Clebsch-Gordan theorem). This expansion is called the Clebsch-Gordan serzes: ~

(5.128) Here. m, is the multiplicity with which the irreducible representation D(")(G)occurs in the ClebschGordan series. The matrix elements of the basis transformation in the product space causing the reduction of the Kronecker product into its irreducible components are called Clebsch-Gordan coeflcients.

5.3.4.8 Irreducible Representations of the Symmetric Group SM 1. Symmetric Group SII.~ The non-equivalent irreducible representations of the symmetric group SW are characterized uniquely by the partitions of M. Le.. by the splitting of M into integers according to

[XI = [ X I , Xz, . . .

3

XM].

X1

+ A2 + + AM = M , ' '

'

XI

2 A* 2 ' . ' 2 AM 2 0.

The graphic representation of the partitions is done by arranging boxes in Young dzagrams.

(5.129)

5.3 Classical Alqebraic Structures 307 IFor the group Sd one obtains five Young dia111= grams as shown in the figure. The dimension of the representation [A] is given by (A, - A, j - 2 )

n

= Jf!-k

lTt=,(A,

141

+

+ k - i)!

'

(5.130)

"

HU

The Young diagram [i] conjugated to [A] is constructed by the interchange of rows and columns. In general, the irreducible representation of SM is reducible if one restricts to one of the subgroups S~t4-1,s.Zf-2, ' ' '.

IIn quantum mechanics for a system of identical particles the Pauli principle demands the construction of many-body ware functions that are antisymmetric with respect to the interchange of all coordinates of two arbitrary particles. Often, the wave function is given as the product of a function in space coordinates and a function in spin variables. If for such a case due to particle permutations the spatial part of the wave function transforms according to the irreducible representation [A] of the symmetric in order to get a total group it has to be combined with a spin function transforming according to [i] wave function which is antisymmetric if two particles are interchanged.

5.3.5 Applications of G r o u p s In chemistry and in physics. groups are applied to describe the "symmetry" of the corresponding objects. Such objects are, for instance, molecules, crystals, solid structures or quantum mechanical systems. The basic idea of these applications is the von Neumann principle: If a system has a certain group of symmetry operations, then every physical observational quantity of this system must have the same symmetry.

5.3.5.1 Symmetry Operations, Symmetry Elements A symmetry operation s of a space object is a mapping of the space into itself such that the length of line segments remains unchanged and the object goes into a covering position to itself. The set of fixed points of the symmetry operations is denoted by Fix s, Le., the set of all points of space which remain unchanged for s. The set Fix s is called the symmetry element of s. We use the Schoenfliess symbolism to denote the symmetry operation. Two types of symmetry operations are distinguished: Operations without a fixed point and operations with at least one fixed point. 1. Symmetry Operations without a Fixed Point, for which no point of the space stays unchanged. cannot occur for bounded space objects, but now we consider only such objects. A symmetry operation without a fixed point is for instance a parallel translation. 2. Symmetry Operations with at least One Fixed Point are for instance rotations and reflections. The following operations belong to them. a) Rotations Around an Axis by an Angle 9: The axis of rotation and also the rotation itself is denoted by C, for p = 27r/n. The axis of rotation is then said to be of n-th order. b) Reflection with Respect to a Plane: Both the plane of reflection and the reflection itself are denoted by u. If additionally there is a principal rotation axis, then we draw it perpendicularly and denote the planes of reflections which are perpendicular to this axis by uh (h from horizontal) and the planes of reflections passing through the rotational axis are denoted by uu (v from vertical) or g d (d means dihedral. if certain angles are halved). c) Improper Orthogonal Mappings: An operation such that after a rotation c, a reflection follows. is called an improper orthogonal mapping and it is denoted by S,. Rotation and reflection commute. The axis of rotation is then called an improper rotational axis of n-th order and it is also denoted by S,. This axis is called the corresponding symmetry element, although only the symme-

I

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5. Aloebra and Discrete Mathematics

try center stays fixed under the application of the operation S,. For n = 2, an improper orthogonal mapping is also called a point reflection (see 4.3.5.1, p. 269) and it is denoted by a.

5.3.5.2 Symmetry Groups or Point Groups For every symmetry operation S.there is an inverse operation S-’, which reverses S “back”,Le., (5.131) ss-1= s-1s = E, Here E denotes the identity operation, which leaves the whole space unchanged. The family of symmetry operations of a space object forms a group with respect to the successive application, which is in general a non-commutative symmetry group of the objects. The following relations hold: a) Every rotation is the product of two reflections. The intersection line of the two reflection planes is the rotation axis. b) For two reflections a and a’ aa,= ala (5.132) if and only if the corresponding reflection planes are identical or they are perpendicular to each other. In the first case the product is the identity E, in the second one the rotation Cz. c ) The product of two rotations with intersecting rotational axes is again a rotation whose axis goes through the intersection point of the given rotational axes. d) For two rotations C, and C: around the same axis or around axes perpendicular to each other: (5.133) czc; = c;cz . The product is again a rotation. In the first case the corresponding rotational axis is the given one, in the second one the rotational axis is perpendicular to the given ones.

5.3.5.3 Symmetry Operations with Molecules It requires a lot of work to recognize every symmetry element of an object. In the literature, for instance in [5.9]. [5.12].it is discussed in detail how to find the symmetry groups of molecules if all the symmetry elements are known. The following notation is used for the interpretation of a molecule in space: The symbols above C in Fig. 5.9 mean that the OH group lies above the plane of the drawing, the symbol to the right-hand side of C means that the group O C ~ H is S under C. The determination of the symmetry group can be made by the following method. 1. No Rotational Axis a) If no symmetry element exists, then G = { E } holds, Le., the molecule does not have any symmetry operation but the identity E . IThe molecule hemiacetal (Fig.5.9) is not planar and it has four different atom groups. b) If a is a reflection or i is an inversion, then G = { E , a} =: C, or G = { E , i} = C, hold, and with this it is isomorphic to 22. IThe molecule of tartaric acid (Fig.5.10) can be reflected in the center P (inversion).

OH

HO-

7

CH,

- C Ill

11.

I

H Figure 5.9

OC,H,

i

/-----

CII 1 1 CO,H 1.

tP -

CO,HbC-

-

H

Figure 5.10

2. There is Exactly One Rotational Axis C a) If the rotation can have any angle, Le., C = C,,

OH

- - - ~ - - -_-- - ~.- ~H= : - - - - I

H

_.I.

I

.*.;*-**--

I

Figure 5.11 then the molecule is linear, and the symmetry

group is infinite.

I A: For the molecule of sodium chloride (common salt) NaCl there is no horizontal reflection. The corresponding symmetry group of all the rotations around C is denoted by Cm,. IB: The molecule O2 has one horizontal reflection. The corresponding symmetry group is generated by the rotations and by this reflection, and it is denoted by D m h . b) The rotation axis is of n-th order, C = C, but it is not an improper rotational axis of order 2n. If there is no further symmetry element, then G is generated by a rotation d by an angle v/naround C,, Le.. G = < d > E 2,. In this case G is also denoted by C,. If there is a further vertical reflection ourthen G = < d, uu > D, holds (see 5.3.3.1, p. 299), and G is denoted by Gnu. If there exists an additional horizontal reflection ohl then G = < d, ou > 2, x 22 holds. G is denoted by C,h and it is cyclic for odd n (see 5.3.3.2, p. 300). IA: For hydrogen peroxide (Fig.5.11) these three cases occur in the order given above for 0 < 6 < ~ / 2 S. = 0 and S = 7712. IB: The molecule of water HzO has a rotational axis of second order and a vertical plane of reflection, as symmetry elements. Consequently. the symmetry group of water is isomorphic to the group Dz, which is isomorphic to the Klein four-group Vd (see 5.3.3.2. 3., p. 301). c) The rotational axis is of order n and at the same time it is also an improper rotational axis of order 2n. We have to distinguish two cases. a)There is no further vertical reflection, so G E 22, holds, and G is denoted also by 5’2,. IAn example is the molecule of tetrahydroxy allene with formula C3(OH)4 (Fig.5.12). p) If there is a vertical reflection, then G is a group of order 471,which is denoted by Dnh. I For n = 2 we get G E D4$Le.. the dihedral group of order eight. An example is the allene molecule (Fig.5.13). ~

Figure 5.12

Figure 5.13

Figure 5.14

3. Several Rotational Axes If there are several rotational axes, then we distinguish further cases. In particular, if several rotational axes have an order n 2 3, then the following groups are the corresponding symmetry groups. a ) Tetrahedral group Td:Isomorphic to 5’4,ord Td = 24 b) Octahedral group oh:Isomorphic to 5’4 x Z,, ord Oh = 48. c) Icosahedral group I h : Ord I h = 120. These groups are the symmetry groups of the regular polyeders discussed in 3.3.3. Table 3.7,p. 154, (Fig.3.63). IThe methane molecule (Fig.5.14) has the tetrahedral group Td as a symmetry group.

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5. Aloebra and Discrete Mathematics

5.3.5.4 Symmetry Groups in Crystallography 1. Lattice Structures In crystallography the parallelepiped represents, independently of the arrangment of specific atoms or ions, the elementary (unit) cell of the crystall lattice. It is determined by three non-coplanar basis vectors 4 startingfrom one lattice point (Fig. 5.15). The infinite geometric lattice structure is created by performing all primitive translations

LTkFJ

zn:

t', = nl& + n2& + n34, n = (121, nz:n3) n, E Z. Here. the coefficients ni (i = 1,2!. . .) are integers.

(5.134)

All the translations gn fixing the space points of the lattice L = {Cn} in terms of lattice vectors form the translation group T with the group element T(Cn),the inverse element T1(G)= and thecompositionlawT(t:)tT(t;) = T(t;+tl). The application of the group element T ( c )to the position vector i is described by:

-

~----------------____

'T-c)!

Figure 5.15

T ( & ) i= i+

c.

(5.135)

2. Bravais Lattices Taking into account the possible combinations of the relative lengths of the basis vectors 4 and the pairwise related angles between them (particularly angles 90" and 120") one obtains seven different types of elementary cellswith the corresponding lattices. the Bravais lattices (see Fig. 5.15, and Table 5.4). This classification can be extended by seven non-primitive elementary cells and their corresponding lattices by adding additional lattice points at the intersection points of the face or body diagonals, preserving the symmetry of the elementary cell. In this way one may distinguish one-side face-centered lattices. body-centered lattices, and all-face centered lattices. Tabelle 5.4 Primitive Bravais lattice Elementary cell Relative lengths Angles between of basis vectors basis vectors

'

rhombic

# a2 # a3 # a2 # a3 a1 # a2 # a3

trigonal

a1

hexagonal

a1 = a2

tetragonal

a1 = a2

cubic

a1

# 8 # 7 # 90" #s a = 8 = y = 90" CY = B = y < 120"# 90") CY = 13 = goo,/; = 120"

triclinic

a1

CY

monoclinic

a1

CY

= a2 = a3

# a3 # a3

= a2 = a3

= y = 900

a = 6 = y = 900 01

= 3 = y = 900

3. Symmetry Operations in Crystal Lattice Structures Among the symmetry operations transforming the space lattice to equivalent positions there are point group operations such as certain rotations, improper rotations, and reflections in planes or points. But not all point groups are also crystallographic point groups. The requirement that the application of a E L ( L is the set of all lattice points) group element to a lattice vector t i leads to a lattice vector again restricts the allowed point groups P with the group elements P ( R ) according to:

6

P = { R : Rt', E L }

t', E L.

(5.136)

5.3 Classical Aluebraic Structures 311

Here: R denotes a proper ( R E SO(3)) or improper rotation operator ( R = IR‘ E 0 ( 3 ) , R ’ E SO(3). I is the inversion operator with I? = -?, ? is a position vector). For example. only n-fold rotation axes with n = 1,2,3:4 or 6 are compatible with a lattice structure. Altogether, there are 32 cristallographic point groups P. The symmetry group of a space lattice may also contain operators representing simultaneous applications of rotations and primitive translations. In this way one gets gliding reflections, Le., reflections in a plane and translations parallel to the plane, and screws, Le., rotations through 27r/n and translations by m&/n ( m = 1.2,. . , n - 1, a‘ are basis translations). Such operations are called non-primitive translations q ( R ) .because they correspond to “fractional” translations. For a gliding reflection R is a reflection and for a screw R is a proper rotation. The elements of the space group G, for which the crystal lattice is invariant is composed of elements P of the crystallographic point group P , primitive translations T(t;) and non-primitive translations ~

iT(R): G = {{RI?(R) +t‘, : R E P,

t‘, E L } } .

(5.137)

The unit element of the space group is {elO} where e is the unit element of R. The element {elt;} means a primitive translation. {RIO} represents a rotation or reflection. ilpplying the group element {RIi,} to the position vector ?one obtains:

{Rlt;,}?= Rr’+ t;.

(5.138)

Table 5.5 Bravais lattice. crystal systems, and crystallographic classes Notation: C, - rotation about an n-fold rotation axis. D, dihedral group. T, - tetrahedral group, 0, - octahedral group. S, mirror rotations with an n-fold axis. ~

~

Lattlcetype rhombic tetragonal hexagonal trigonal cubic

4. Crystal Systems (Holohedry) From the 14 Bravais lattices, L = IC}, the 32 crystallographic point groups P = { R }and the allowed non-primitive translations q ( R )one can construct 230 space groups G = { R I q ( R )+ The point groups correspond to 32 crystallographic classes. Among the point groups there are seven groups that are not a subgroup of another point group but contain further point groups as a subgroup. Each of these seven point groups form a crystal system (holohedry). The symmetry of the seven crystal systems is reflected in the symmetry of the seven Bravais lattices. The relation of the 32 crystallographic classes to the seven crystal systems is given in Table 5.5 using the notation of Schoenfliess. R e m a r k : The space group G (5.137) is the symmetry group of the “empty” lattice. The real crystal is obtained by arranging certain atoms or ions at the lattice sites. The arrangement of these crystal

c}.

I

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5. Aloebra and Discrete Mathematics

constituents exhibits its own symmetry. Therefore, the symmetry group Go of the real crystal possesses a lower symmetry than G (G 3 Go),in general.

5.3.5.5 Symmetry Groups in Quantum Mechanics Linear coordinate transformations that leave the Hamiltonian H of a quantum mechanical system (see 9.2.3.5, l.,p. 536) invariant represent a symmetry group G, whose elements g commute with H:

a]

(5.139) [ g , = g H - H g = 0, g E G. The commutation property of g and H implies that in the application of the product of the operators g and H to a state p the sequence of the action of the operators is arbitrary: S ( H d = fi(gP).

(5.140)

Hence. one has: If p~~ ( a = 1,2,.. . , n) are the eigenstates of H with energy eigenvalue E of degeneracy n. Le., Ht+?En = Et+?Ea ( a = 1,2,. . . , n ) , then the transformed states gPEa are also eigenstates belonging to the same eigenvalue E :

(5.141)

(5.142) SH?Ea = H g 9 . h = EgPEa. The transformed states gpE0 can be written as a linear combination of the eigenstates p ~ ~ : (5.143) Hence. the eigenstates p~~form the basis of an n-dimensional representation space for the representation D ( G )of the symmetry group G of the Hamiltonian H with the representation matrices ( D a 8 ( g ) ). This representation is irreducible if there are no “hidden” symmetries. One can state that the energy eigenstates of a quantum mechaniccal system can be labeled by the signatures of the irreducible representations of the symmetry group of the Hamiltonian. Thus, the representation theory of groups allows for qualitative statements on such patterns of the energy spectrum of a quantum mechanical system which are established by the outer or inner symmetries of the system only. Also the splitting of degenerate energy levels under the influence of a pertubation which breaks the symmetry or the selection rules for the matrix elements of transitions between energy eigenstates follows from the investigation of representations according to which the participating states and operators transform under group operations. The application of group theory in quantum mechanics is presented extensively in the literature (see, e.g.. [5.6]. [ 5 . 7 ] .[5.8],[5.9],[5.10]).

5.3.5.6 Further Applications of Group Theory in Physics Further examples of the application of particular continuous groups in physics can only be mentioned here (see, e.g., [5.6],[5.9]). U (1): Gauge transformations in electrodynamics. SC’(2):Spin and isospin multiplets in particle physics. Sc’(3): Classification of the baryons and mesons in particle physics. Many-body problem in nuclear physics. SO(3): iZngular momentum algebra in quantum mechanics. Atomic and nuclear many-body problems. SO(4): Degeneracy of the hydrogen spectrum. Sc’(4). Wigner supermultiplets in the nuclear shell model due to the unification of spin and isospin degrees of freedom. Description of flavor multiplets in the quark model including the charm degree of freedom.

5.3' Classical Aloebraic Structures 313

SU(6): Multiplets in the quark model due to the combination of flavor and spin degrees of freedom. Nuclear structure models. L'(n):Shell models in atomic and nuclear physics. S C ( n ) ,SO(n):Many-body problems in nuclear physics. SC(2) @ G(1): Standard model of the electro weak interaction. SU(5) 2 SC(3)@ SU(2) @ U(1): Unification of fundamental interactions (GUT). Remark: The groups SU(n) and SO(n)are Lie groups, i.e. continuous groups that are not treated here (see, e.g.. [5.6]).

5.3.6 Rings and Fields In this section. we discuss algebraic structures with two binary operations.

5.3 -6.1 Definitions 1. Rings

A set R with two binary operations +, x is called a rzng (notation: ( R )+ , *)), if ( R .+) is an Abelian group, ( R ,1;) is a semigroup. and the dzstrzbutzve laws hold: a * ( b + c ) = ( a 1; b) + ( a x c). ( b + C) x a = ( b x a ) ( C * a ) . (5.144) If (R.*) is commutative or if (R:X ) has a neutral element, then (R,+, *) is called a commutative ring or a ring with identity (ring with unit element), respectively.

+

2. Fields

A ring is called afieldif ( R \ (0): *) is an Abelian group. So, every field is a special commutative ring with identity.

3. Field Extensions Let K and E be two fields. If K C E holds. E is called the extension fieldof K . Examples of rings and fields: W A: The number domains Z,Q, R, and C are commutative rings with identity with respect to addition and multiplication: Q , R, and C are also fields. The set of even integers is an example of a ring without identity. The set C is the extension field of R. IB: The set .\.I, of all square matrices of order n with real (or complex) elements is a non-commutative ring with the identity matrix as unit element. IC: The set of real polynomials p(z) = a,zn an-ld-l + . ' . + alz t a0 forms a ring with respect to the usual addition and multiplication of polynomials, the polynomial ring R[z]. More generally, instead of polynomials over R. polynomial rings over arbitrary commutative rings with identity element can be considered. ID: Examples of finite rings are the residue class rings 2, modulo m: Z, consists of all the classes [a],of integers having the same residue on division by m.([a],is the equivalence class defined by the natural number a with respect to the relation +JR introduced in 5.2.4, l . ,p. 297.) The ring operations e ,@ on 2, are defined by [a],@ [b], = [a b], and [a],0 [b], = [a b],. (5.145) If the natural number m is a prime, then (Z, , e ,0)is a field.

+

+

5.3.6.2 Subrings, Ideals 1. Subring SupposeR = (R.+,*)isaringandU is called a subrzng of R.

C R. IfUwithrespectto+and*isalsoaring,thenII

= (U)+.*)

314

5. Alaebra and Discrete Mathematics

A non-empty subset U of a ring (R,+, *) forms a subring of R if and only if for all a, b E U also a + (- b ) and a * b are in U . 2. Ideal A subring I is called an ideal if for all r E R and a E I also r * a and a * r are in I . These special subrings are the basis for the formation of factor rings (see 5.3.6.3, p. 314). The trivial subrings (0) and R are always ideals of R. Fields have only trivial ideals.

3. Principal Ideal If all the elements of an ideal can be generated by one element according to the subring cr’iterion, then it is called a principal ideal. All ideals of Z are principal ideals. They can be written in the form mZ = {mglg E Z) and we denote them by (m).

5.3.6.3 Homomorphism, Isomorphism, Homomorphism Theorem 1. Ring Homomorphism and Ring Isomorphism

+,

a) Ring Homomorphism: Let R1 = (R1, *) and Rz = (Rz,otr o*) be two rings. A mapping h: R1 + Rz is called a ring homomorphismif for all a: b E R1 (5.146) h(a + b) = h ( a ) 0, h(b) and h(a t b) = h(a)ot h(b) hold. b) Kernel: The kernel of h is the set of elements of R1 whose image by h is the neutral element 0 of (R2!+)>and we denote it by ker h: (5.147) ker h = { a E Rllh(a) = 0). Here ker h is an ideal of R1. c) Ring Isomorphism: If h is also bijective, then h is called a ring isomorphism, and the rings R1 and Rz are called isomorphic. d) Factor Ring: If I is an ideal of a ring ( R , then the sets of cosets { a Ila E R } of I in the additive group ( R ,+) of the ring R (see 5.3.3, l.,p. 300) form a ring with respect to the operations (5.148) ( a I ) o+ ( b I ) = ( a b) I and ( a I ) 0, ( b I ) = ( a * b) I . This ring is called the factor ring of R by I , and it is denoted by R / I . The factor ring of Z by a principal ideal (m) is the residue class ring Z, = Z/(,) (see examples of rings and fields on p. 313).

+, +

+

*))

+

+

+ +

+

+

2. Homomorphism Theorem for Rings If the notion of a normal subgroup is replaced by the notion of an ideal in the homomorphism theorem for groups. then the homomorphism theoremfor rings is obtained: A ring homomorphism h: R1 i Rz defines an ideal of R1. namely ker h = { a E R1 ih(a) = 0). The factor ring Ri/ ker h i s isomorphic to the homomorphic image h(R1) = {h(a)laE R1). Conversely, every ideal I of R1 defines a homomorphic mapping nat1: R1 i Rz/I with natI(a) = a+I. This mapping nail is called a natural homomorphism.

5.3.7 Vector Spaces * 5.3.7.1 Definition A vector space over a field F consists of an Abelian group V = (V, +) of ‘ I vectors ” written in additive form, of a field F = ( F ,+, t ) of ‘‘ scalars ” and an exterior multiplication F x V i V. which assigns to every ordered pair ( k . w) for k E F and u E V a vector ku E V. These operations have the following properties: ( V I ) ( u + v ) + w = u (v + w ) for all u, v )w E V. (V2) There is a vector 0 E V such that v t 0 = 2: for every u E V. (V3) To every vector u there is a vector - u such that w + (-v) = 0.

+

‘In this paragraph, generally, vectors are not printed in bold face.

(5.149) (5.150) (5.151)

5.3 Classical Aloebraic Structures 315

(V4) v + 21: = IC + v for every u , w E V. (V5) l v = c for every u E V,1 denotes the unit element of F.

(5.152)

(V6) r ( s u ) = ( r s ) u for every r, s E F and every u E V. (V7) ( T + s)v = T U + s u for every r, s E F and every w E V.

(5.154)

+

(5.153) (5.155)

+

(5.156) (V8) r(v w) = rv rw for every T E F and every v, w E V. If F = R holds, then it is called a real vector space. Examples of vector spaces: W A: Single-column or single-row real matrices of type ( n ,1) and (1,n ) ,respectively, with respect to matrix addition and exterior multiplication with real numbers form real vector spaces R" (the vector space of column or row vectors; see also 4.1.3, p. 253). IB: .A11 real matrices of type (m,n) form a real vector space. C: A l l real functions continuous on an interval [a,b] with the operations (f + g)(r) = f(.) + d.1 and (kf)(z)= k ' f(.) (5.157) form a real vector space. Function spaces have a fundamental role in functional analysis (1.)p. 594). For further examples see 12.1.2, p. 595.

5.3.7.2 Linear Dependence Let V be a vector space over F . The vectors v1, u 2 , . . ., u, E V are called linearly dependent if there are k l . I C 2 . . . .. k , E K not all of them equal to zero such that 0 = klui k2v2 . ' . k,v, holds. Otherwise they are linearly independent. Linear dependence of at least two vectors means that one of them can be expressed in terms of the other. If there is a maximal number n of linearly independent vectors in a vector space V , then the vector space V is called n-dimensional. This number n is uniquely defined and it is called the dimension. Every n linearly independent vectors of V form a basis. If such a maximal number does not exist, then the vector space is called infinite dimensional. The vector spaces in the above examples are n. m . n, and infinite dimensional. In the vector space R", n vectors are independent if and only if the determinant of the matrix, whose columns or rows are these vectors, is not equal to zero. If {vl.u 2 . . . . u,} form a basis of an n-dimensional vector space over F , then every vector 2: E V has a unique representation c = klvi kzu2 . . knun with k l , k ~ .. ,. k , E F . Every set of linearly independent vectors can be completed into a basis of the vector space.

+

+

~

+ +

+ +

5.3.7.3 Linear Mappings The mappings respecting the structure of vector spaces are called linear mappings. f: VI i V2 is called linear if for every u , v E VI and every k E F f ( u u ) = f(u) f ( u ) and f ( k u ) = k f(u) (5.158) are valid. IThe linear mappings f from R" into R" can be given by matrices A of type ( m ,n ) by f ( v ) = Au.

+

+

5.3.7.4 Subspaces, Dimension Formula 1. Subspace: Let V be a vector space and U a subset of V. If U is also a vector space with respect to the operations of I:. then c' is called a subspace of V. A non-empty subset c' of V is a subspace if and only if for every u1 u2 E U and every k E F also u1+ u2 and k . u1 are in C (subspace criterion). 2. Kernel, Image: Let V,,V2 be vector spaces over F . I f f : Vl + V2 is a linear mapping, then the linear subspaces kernel (notation: ker f ) and image (notation: im f ) are defined in the following way: (5.159) kerf = { u E Vlf(v) = 0). im f = {f(v)lv E V } .

316

5. Algebra and Discrete Mathematics

So, for example, the solution set of a homogeneous linear equation system Ax = 0 is the kernel of the linear mapping defined by the coefficient matrix A. 3. Dimension: The dimension dim ker f and dim im f are called the defect f and rank f l respectively. For these dimensions the equality (5.160) defect f + rank f = dim V, is valid and is called the dimension formula. In particular, if the defect f = 0, i.e., kerf = {0}, then the linear mapping f is injective, and conversely. Injective linear mappings are called regular.

5.3.7.5 Euclidean Vector Spaces, Euclidean Norm In order to be able to use notions such as length, angle, orthogonality in abstract vector spaces we introduce Euclidean vector spaces. 1. Euclidean Vector Space Let V be a real vector space. If p: V x V --t R is a mapping with the following properties (instead of p(w,w) we write v . w) for every u, v,w E V and for every r E R (Sl) 1I.w=w'w, (5.161) (5.162) (S2) ( u + w) ' w = u.w v ' W , (S3) r ( v . w ) = ( T U ) . w = 21. (rw), (5.163) (S4) w ' u > 0 ifand only i f v # 0, (5.164) then y is called a scalar product on V. If there is a scalar product defined on V , then we call V a Euclidean vector space. We use these properties to define a scalar product with similar properties on more general spaces, too (see 12.4.1.1. p. 613). 1. E u c l i d e a n N o r m The value 1 1 ~ 1 1= @denotes the Euclidean norm (length) of v. The angle Q between u , w from V is defined by the formula u . U' (5.165) cosa =

+

~

IIUII

. IIWII

If w u'= 0 holds, then v and w are said to be orthogonal to each other. IOrthogonality oflkigonornetrichnctions: In the theory of Fourier series (see 7.4.1.1,p. 418). we consider functions of the form sin kx and cos kx. We can consider these functions as elements of C[O. 2x1. In the function space C[a.b] the formula

defines a scalar product. Since

Jd Jd

2n

sinkz.sinlxdx=O

(kf1).

(5.167)

Jd

2ri

~oskx.coslxdz=O ( k f l ) ,

277

sin kz cos lx ciz = 0,

(5.168) (5.169)

the functions sin kx and cos lx for every k , 1 E IN are pairwise orthogonal to each other. This orthogonality of trigonometric junctions is used in the calculation of Fourier coefficients in harmonic analysis (see 7.4.1.1, p. 418).

5.3.7.6 Linear Operators in Vector Spaces 1. N o t i o n of L i n e a r O p e r a t o r s Let I*and W be two real vector spaces. .4mapping a from V into W is called a linear mapping or linear transformation or linear operator (see also 12.1.5.2, p. 598) from V into W if a(u + v ) = au + av for all u,w E r/,

(5.170)

5.3 ClassicalAlaebraic Structures 317

a(Xu) = Xau

for all u E V and all real A.

(5.171)

3

IA: The mapping av := J u ( t )dt. which transforms the space C[a,p]of continuous real functions Q

into the space of real numbers is linear. In the special case when W = R', as in the previous example, linear transformations are called h e a r functzonals. IB: Let I' = R" and let bV be the space of all real polynomials of degree at most n - 1 Then the mapping .(al, a 2 . . . ,a,) := al + a 2 x + a 3 x 2 . .+ a,xn-l is linear. In this case each n-element vector corresponds to a polynomial of degree 5 n - 1. IC: If I' = R" and bt' = R", then all linear operators a from V into W ( a : R" --+ R") can be characterized by a real matrix A = (atk) of type (m , n ). The relation Ax = y corresponds to the system of linear equations (4.103a)

+.

2. Sum and Product of two Linear Operators Let a: V -+ N', b: V + It' and e: W t U be linear operators. Then the sum a + b: I.' t Lt' is defined as ( a + b)u = au + bu for all u E V and the product ea: I'+ C: is defined as ( m ) u= C(QU) for all u E V.

(5.172)

(5.173) Remarks: 1. If a. b and c are linear! then a + b and ac are also linear operators. 2. The product (5.173) of two linear operators represents the consecutive application of these operators a and c. 3. The product of two linear operators is usually non-commutative even if the products exist: ea # ac . (5.174a) We have cornmutability if (5.174b) ca - ac = 0 holds. In quantum mechanics the left-hand side of this equation ea - ac is called the commutator. In the case (5.174a) the operators a and c do not commutate: therefore we have to be very careful about the order. IAs a particular example of sums and products of linear operators one may think of sums and products of the corresponding real matrices. ~

318

5. Alqebra and Discrete Mathematics

5.4 Elementary Number Theory Elementary number theory investigates divisibility properties of integers.

5.4.1 Divisibility 5.4.1.1 Divisibility and Elementary Divisibility Rules 1. Divisor An integer b E Z is divisible by an integer a without remainder iff * if there is an integer q such that qa = b (5.175) holds. Here a is a divisor of b in Z,and q is the complementary divisor with respect to a: b is a multiple of a. For “a divides b“ we write also alb. For “a does not divide b” Re can write alb. The divisibility relation (5.175) is a binary relation in Z (see 5.2.3. 2.. p. 294). Analogously, divisibility is defined in the set of natural numbers.

2. Elementary Divisibility Rules For every a E Z we have lla, ala and alO.

(5.176)

If alb, then (-a)lb and al(-b).

(5.177)

(DR3)

alb and bla implies a = b or a = -b.

(5.178)

(DR4)

all implies a = 1 or a = -1.

(5.179)

(DR5) (DR6)

alb and b # 0 imply la1 5 Ibl.

(5.180)

(DR7)

alb implies azlbz for every z

(DR8)

azlbz and z # 0 implies alb for every z

(DR1) (DR2)

alb implies alzb for every z E Z. E

(5.181)

Z.

(5.182) E

Z.

(5.185)

+

(DR11) alb and alc imply al(zlb z2c) for arbitrary zl, z2 E Z.

+

(5.183) (5.184)

(DR9) alb and bjc imply alc. (DR10) alb and cjd imply aclbd. (DR12) alb and al(b c) imply alc.

(5.186) (5.187)

5.4.1.2 Prime Numbers 1. Definition and Properties of Prime Numbers .A positive integer p ( p > 1) is called a prime number iff 1 and p are its only divisors in the set N of positive integers. Positive integers which are not prime numbers are called composite numbers. For every integer, the smallest positive divisor different from 1 is a prime number. There are infinitely many prime numbers. A positive integer p ( p > 1) is a prime number iff for arbitrary positive integers a. b. pl(ab) implies pia or plb.

2. Sieve of Eratosthenes By the method of the “Sieve of Eratosthenes“, every prime number smaller than a given positive integer n can be determined: a) LVrite down the list of all positive integers from 2 to n. b) Underline 2 and delete every subsequent multiple of 2. c) If p is the first non-deleted and non-underlined number, then underline p and delete every p t h number (beginning with 2p and counting the numbers of the original list). d) Repeat step c) for every p ( p 5 f i )and stop the algorithm. *if and only if

5.1 Elementarv Number Theoru 319 Every underlined and non-deleted number is a prime number. In this way, all prime numbers obtained. The prime numbers are called prime elements of the set of integers.

5 n are

3. Prime Pairs Prime numbers with a difference of 2 form prime pairs (twin primes). I(3.5). (5, 7). (11,13).(17,19). (29,31),(41.43), (59,61),(71,73), (101,103) are prime pairs.

4. Prime Triplets Prime triplets consist of three prime numbers occuring among four consecutive odd numbers I(5.7.11). ( 7 , l l . 13). (11,13:17), (13.17.19). (17,19,23).(37,41,43)are prime triplets. 5 . Prime Quadruplets If the first two and the last two of five consecutive odd numbers are prime pairs, then they are called a prime quadruplet. I(5.7.11. 13). (11. 13. 17,19),(101,103,107.109), (191,193,197,199) are prime quadruplets. The conjecture that there exist infinitely many prime pairs, prime triplets, and prime quadruplets, is not proved still. 6. Mersenne Primes If 2k - 1. k E N.is a prime number. then k is also a prime number. The numbers 2 P - 1 ( p prime) are called SIersenne numbers. .4 Llersenne prime is a Slersenne number 2 P - 1 which is itself a prime number. I2 P - 1 is a prime number for the first ten values ofp: 2;3, 5, 13,17, 19, 31: 61: 89. 107, etc.

7. Fermat Primes Ifanumber2k++, k E Ki,isanoddprimenumber,then kisapowerof2. Thenumbers2k+l, k E lK, are called Fermat numbers. If a Fermat number is a prime number. then it is called a Fermat prime. IFor k = 0 , l . 2 ; 3.4 the corresponding Fermat numbers 3 , 5 ?17,257,65537 are prime numbers. It is conjectured that there are no further Fermat primes.

8. Fundamental Theorem of Elementary Number Theory Every positive integer n > 1 can be represented as a product of primes. This representation is unique except for the order of the factors. Therefore n is said to have exactly one prime factorization. I360 = 2 . 2 . 2 , 3 . 3 . 5 = 2 3 , 3'. 5. Remark: .halogouslp, the integers (except -1! 0:1)can be represented as products of primeelements! unique apart from the order and the sign of the factors.

9. Canonical Prime Factorization It is usual to arrange the factors of the prime factorization of a positive integer according to their size, and to combine equal factors to powers. If every non-occurring prime is assigned exponent 0, then every positive integer is uniquely determined by the sequence of the exponents of its prime factorization. ITo 1533 312 = 27 . 32 . 113belongs the sequence of exponents ( 7 , 2 :0,0, 3,0,0, , . .). For a positive integer n. let p l , p 2 . . . . p , be the pairwise distinct primes divisors of n. and let ai, denote the exponent of a prime number P k in the prime factorization of n. Then rn

n=

flpik.

(5.188a)

k=l

and this representation is called the canonicalprimefactoriiation of n. It is often denoted by

n = l-JPUP(12)*

(5.188b)

P

ivhere the product applies to all prime numbers p i and where vp(n)is the multiplicity of p as a divisor of n. It always means a finite product because only finitely many of the exponents vp(n)differ from 0.

320

5. Alqebra and Discrete Mathematics

10. Positive Divisors If a positive integer n 2 1 is given by its canonical prime factorization (5.188a), then every positive divisor t of n can be written in the form m

t = flp?

with rk E {0,1,2,.. . , a k ) for IC = 4 2 , . . . ,m.

(5.189a)

k=l

The number r ( n )of all positive divisors of n is m

+

r ( n ) = n ( a k 1).

(5.189b)

k=l

IA: ~ ( 5 0 4 0 = ) ~ ( 2 3'~ 5. . 7) = (4 + 1)(2 + 1)(1+ 1)(1+1) = 60. IB: r(p1p2 ' .pr) = 2', if p l , p z , . . . ,p, are pairwise distinct prime numbers. The product P ( n )of all positive divisors of n is given by

I

P ( n )= n w , IA: P(20) = 203 = 8000. IB: P(p3)= p 6 , ifp is a prime number. C: P ( p q ) = p2q2.ifp and q are different prime numbers. The sum ~ ( nof)all positive divisors of n is

(5.189~)

(5.189d) IA: 4120) = ~ ( 23 .~5 ). = 15 4 . 6 = 360.

IB: u(p) = p

+ 1, if p is a prime number.

5.4.1.3 Criteria for Divisibility 1. Notation Consider a positive integer given in decimal form: n = (akak-1.' . a2a1a0)10 = U k l O k + ak-llok-' Then Ql(n)

=ao+al+az+...+ak

+. . + a2102 f al10 + ao. '

(5.190a) (5.190b)

and

Qi(n) = a o - ~ l + n z - + . . . t ( - l ) ~ a k (5.190~) are called the s u m of the digits (offirst order) and the alternating sum of the digits (offirst order) of n, respectively. Furthermore, and (5.190d) Q z ( ~ )= (a1a0)10 (a3aZ)IO ((15(14)10 + . ..

+

+

Q;(n) = (a1ao)lo- ( w 2 ) I O + ( w 4 ) l O - +...

(5.190e)

are called the sum of the digits and the alternating sum of the digits, respectively, of second order and (5.190f) Q3(n) = (aza1ao)lo t (%a4a3)10+ (a8a~%)10 + .' . and (5.190g) Q$(n) = (aza1ao)lo - (%a4a3)10 + (a8a7a6)IO- +. ' . are called the s u m of the digits and alternating s u m of the digits, respectively, of third order. IThe number 123 456 789 has the followingsum ofthe digits: Qi = 9+8+7+6+5+4+3+2+1 = 45, Q: = 9-8+7-6+5-4+3-2+1 = 5,Qz = 89+67+45+23+1= 225,Q: = 89-67$45-23+1= 45, Q3 = 789 456 t 123 = 1368 and Q$ = 789 - 456 + 123 = 456.

+

5.4 Elementary Number Theory 321

2. Criteria for Divisibility There are the following criteria for divisibility: DC-1:

31n H 3IQl(n),

(5191a)

DC-2:

7ln

DC-3:

9 / n e 91Qi(n)l

(5.191~)

DC-4:

l l l n H lllQ;(n),

(5.191d)

DC-5:

131n @ 131QL(n)

(5.191e)

DC-6:

37/n* 37/Q3(.),

(5.191f)

DC-7:

). 1011~1H l O l ~ Q ~ ( n (5.191g)

DC-8:

21n # 2Ia0,

(5.191h)

DC-9: 51n H DC-11:

(5.191i)

51~10,

71Q;(n),

(5.191b)

DC-10: 2kln @ 2k/(ak-lak-z...ala0)~0, (5.19lj)

jk1n e 5 k l ( ~ k - l ~ k _ z . . ' a l a o ) l (5.191k) o.

W A: a = 123456789 is divisible by 9 since Ql(a) = 45 and 9145, but it is not divisible by 7 since Q 3 ( a )= 456 and 7j456. W B: 91 619 is divisible by 11 since Q:(91619) = 22 and 11122. W C: 99 994 096 is divisible by 24 since 2414096.

5.4.1.4 Greatest Common Divisor and Least Common Multiple 1. Greatest Common Divisor

For integers al.a2.. . . ,a,. which are not all equal to zero, the largest number in the set of common divisors of ai.a2, . . . . a, is called the greatest common divisor of al.az, . . . ,anr and it is denoted by gcd(al, a 2 , ,. . .a,). If gcd(a1, az, . . . a,) = 1, then the numbers a l , a ' , . . . ,a, are called coprimes. To determine the greatest common divisor, it is sufficient to consider the positive common divisors. If the canonical prime factorizations a5 -- l - p * ( a t )

(5.192a)

P

of 01: a'.. . . .a, are given, then gcd(al>az. . . . , an) =

n P[

min

'

[.p'"i)']

(5.192b)

P

W For the numbers al = 15 400 = 23 .5' . 7 . 11,a2 = 7875 = 3' 53 7, a3 = 3850 = 2.5' . 7 . 1 1 , the greatest common divisor is gcd(a1, a2. a3) = 5' . 7 = 175. 2. Euclidean Algorithm The greatest common divisor of two integers a, b can be determined by the Euclidean algorithm without using their prime factorization. To do this, a sequence of divisions with remainder, according to the following scheme. is performed. For a > h let a0 = a , a1 = h. Then: a0 = y1a1

+ a2, +

a1 = q~a2 a3,

.. .

.. .

0 < a2 < a1, 0

< a3 < az,

..

(5.193a)

t

an-2 = qn-lan-l +a,,

0 < a,

< an-l>

an-i = The division algorithm stops after a finite number of steps, since the sequence azsu s , . . . is a strictly monotone decreasing sequence of positive integers. The last remainder a,, different from 0 is the greatest common divisor of a0 and a l .

322

5. Alqebra and Discrete Mathematics

Igcd(38, 105) = 1, since 105 = 2 . 3 8 + 29; 38 = 1 .29 + 9; 29 = 3 . 9 + 2; 9 = 4 . 2 + 1;2 = 2 1. By the recursion formula (5.193b) &(ai, a 2 , . . . , an) = gcd(gcd(a1, az,. . . , an-1)) an)) the greatest common divisor of n positive integers with n > 2 can be determined by repeated use of the Euclidean algorithm. Igcd(l50,105,56) = gcd(gcd(150,105),56) = gcd(l5,56) = 1. IThe Euclidean algorithm to determine the gcd (see also 1.1.1.4, l., 55 = 1 ' 34 + 21 p. 3) of two numbers has especially many steps, if the numbers are adja34 = 1 . 2 1 + 13 cent numbers in the sequence of Fibonacci numbers (see 5.4.1.5, p. 323). 21 = 1 . 1 3 + 8 The annexed calculation shows an example where all quotients are al13 = 1 . 8 + 5 8=1.5+3 ways equal to 1. 3. Theorem for the Euclidean Algorithm 5=1,3+2 For two natural numbers a, b with a > b > 0, let A(a, b) denote the 3=1.2+1 2=1.1+1 number of divisions with remainder in the Euclidean algorithm, and let 1=1.1. K ( b ) denote the number of digits of b in the decimal system. Then (5.194)

5 5 K(b). 4. Greatest Common Divisor as a Linear Combination x(a, b)

It follows from the Euclidean algorithm that a2 = a0 - q1al = cOaO t deal, a3 = a1 - qza2 = claO dial:

+

..

..

a, = an-2 - qn-lan-l = c,-zao

(5.195a)

+ d,-2al.

Here c,-2 and d,-z are integers. Thus the gcd(a0, al) can he represented as a linear combination of a0 and al with integer coefficients: (5.195b) gcd(a0. al) = c,-zao + d,-nal. Moreover gcd(al. a2,.. . , a,) can he represented as a linear combination of a l , az, . . . , a n rsince:

+

gcd(a1, az, . . . ,a,) = gcd(gcd(a1, a2,. . . . an-l)>a,) = e ' gcd(al, a z , . . . an-l) da,. (5.195~) Igcd(l50,105.56) = gcd(gcd(l50,105): 56) = gcd(l5,56) = 1 with 15 = (-2) . 150 + 3 . 105 and 1 = 15.15 + (-4) 56))thus gcd(l50,105;56) = (-30) . 150 + 45.105 (-4) 56.

+

5 . Least Common Multiple For integers a l , a2: . . . ,a,: among which there is no zero, the smallest number in the set of positive common multiples of al,a2, . , . a, is called the least common multiple of a l , a2,. . . , a,, and it is denoted by lcm(a1, azr. . . ,a,). If the canonical prime factorizations (5.192a) of al, a 2 > . . ,a, are given, then: ~

(5.196)

IFor the numbers a1 = 15400 = 2 3 , j 2 ,7 , l l , a z = 7876 = 32.S 3 . 7.a3 = 3850 = 2 . j z .7 . 1 1 the least common multiple is lcm(al, az, a3) = 23 . 32 . j3. 7 11 = 693 000.

6. Relation between g.c.d. and 1.c.m. For arbitrary integers a. b: lab1 = gcd(a, b) lcm(a, b).

(5.197)

5.1 Elementaru Number Theoru 323 Therefore, the lcm(a, b ) can be determined with the help of the Euclidean algorithm without using the prime factorizations of a and b.

5.4.1.5 Fibonacci Numbers 1. Fibonacci Sequence The sequence (Fn)nEN with F1 = F2 = 1 and Fn+2 = F, + Fntl (5.198) is called Fzbonaccz sequence. It starts with the elements 1,1,2.3,5l 8,13,21,34,55,89.144, 233.377.. . . IThe consideration of this sequence goes back to the question posed by Fibonacci in 1202: How many pairs of descendants has a pair of rabbits at the end of a year. if every pair in every month produces a new pair, which beginning with the second month itself produces new descended pairs? The answer is F11 = 377.

2. Fibonacci Recursion Formula Besides the recursive definition (5.198) there is an explicit formula for the Fibonacci numbers. (5.199) Some important properties of Fibonacci numbers are the following. For m, n E N:

+ FmFnil ( m > 1).

(5.200a)

(2) FmIFmn.

(5.200b)

(3) gcd(m. n) = dimplies gcd(Fm. F,) = F,+

(5.200~)

(4) gcd(Fn,Fntl) = 1.

(5.2OOd)

( 5 ) FmIFh holds iff mlk holds.

(5.200e)

(6)

(7) gcd(m. n) = 1 implies FmFnIFm,.

(5.200g)

(8) C F t = F n + 2 - 1. 1=1

(5.200h)

(5.2OOi)

(IO) F,'

+ F,+,

(5.2OOj)

(1) F m + n = F m - l F n

n

XF:

i=l

= FnFn+1.

(5.200f)

n

(9) F n F d -

(11)F;+2

-

Fit1 = ( - l ) n + I .

F; = FZn+2.

=

(5.200k)

5.4.2 Linear Diophantine Equations 1. Diophantine Equations

.An equation f(x1: x2., . . ~x,) = b is called a Diophantine equation in n unknowns iff f(xl! x 2 : . . . ,x,) is a polynomial in x l : x z I . . . , x , with coefficients in the set Z of integers, b is an integer constant and only integer solutions are of interest. The name "Diophantine" reminds us of the Greek mathematician Diophantus, who lived around 250 AD. In practice, Diophantine equations occur for instance: if relations between quantities are described. Until now. only general solutions of Diophantine equations of at most second degree with two variables are known. Solutions of Diophantine equations of higher degrees are only known in special cases.

2. Linear Diophantine Equations in n Unknowns A linear Diophantine equation in n unknowns is an equation of the form a l ~+ l ~2x2 + ' . anxn = b (ai E Z, b E Z), (5.201) where only integer solutions are searched for. .Z solution method is described in the following.

324

5. Aloebra and Discrete Mathematics

3. Conditions of Solvability If not all the coefficients ai are equal to zero, then the Diophantine equation (5.201) is solvable iff gcd(a1, az, . . . .a,) is a divisor of b.

I1142 + 315y = 3 is solvable, since gcd(ll4,315) = 3. If a linear Diophantine equation in n unknowns ( n > 1) has a solution and Z is the domain of variables, then the equation has infinitely many solutions. Then in the set ofsolutions there are n-1 free variables. For subsets of Z,this statement is not true. 4. Solution Method for n = 2 Let alxl + a222 = b ( a l , az) # (0,O) (5.202a) be a solvable Diophantine equation, Le., gcd(a1, az)lb. To find a special solution of the equation, the equation is divided by gcd(a1, az) and one obtains six: + a:.; = b’ with gcd(a:. a:) = 1. As described in 5.4.1,4., p. 322, gcd(a;, a;) is determined to obtain finally a linear combination of a; and a;: a;c: + a;c; = 1. Substitution in the given equation demonstrates that the ordered pair (c: b’, ckb’) of integers is a solution of the given Diophantine equation. I114~+315y= 6. Theequationisdivided by3,since3 = gcd(114,315). That implies38x+105y = 2 and 38 47 + 105. (-17) = 1 (see 5.4.1, 4.) p. 322). The ordered pair (47 2. (-17) . 2 ) = (94. -34) is a special solution of the equation 1142 + 315y = 6. The family of solutions of (5.202a) can be obtained as follows: If (zy, 2:) is an arbitrary special solution, which could also be obtained by trial and error, then (5.202b) {(x! t . a;,.; - t . a# E Z} is the set of all solutions. IThe set of solutions of the equation 1142 315y = 6 is {(94 t 315t, -34 - 114t)lt E Z}. 5 . Reduction Method for n > 2 Suppose a solvable Diophantine equation (5.203a) alxl a222 . ’ . a,x, = b with (al?a2,. . . a,) # (0.0,. . . , O ) and gcd(a1, az, . . . .a,) = 1 is given. Ifgcd(al, a z > .. . ,a,) # 1,then the equation should be divided by gcd(al, a z ,. . . .an). After the transformation (5.203b) alxl a222 . ’ . an-lxn-l= b - anin x, is considered as an integer constant and a linear Diophantine equation in n-1 unknowns is obtained, and it IS solvable iff gcd(al, azr. . . , an-l) is a divisor of b - a,$,. The condition (5.203~) gcd(a1,az. . . . .an_l)lb- a,x, is satisfied iff there are integers c, c,, such that: (5.203d) gcd(a1. a2,. . . , an-l) c ancn = b. This is a linear Diophantine equation in two unknowns, and it can be solved as shown in 5.4.2,4., .p. 324. If its solution is determined, then it remains to solve a Diophantine equation in only n - 1 unknowns. This procedure can be continued until a Diophantine equation in two unknowns is obtained, which can be solved with the method given in 5.4.2, 4., p. 324. Finally, the solution of the given equation is constructed from the set of solutions obtained in this way. ISolve the Diophantine equation (5.204a) 2 2 4y 32 = 3. This is solvable since gcd(2,4.3) is a divisor of 3. The Diophantine equation 2 s + 4y = 3 - 32 (5.204b)

+

+

+

+ +

I

+

+ +

+

+ +

5.1 Elementarv Number Theoru 325 in the unknowns z.y is solvable iff gcd(2,4) is a divisor of 3 - 32. The corresponding Diophantine equation 22‘ + 32 = 3 has the set of solutions ((-3 -+- 3t, 3 - 2t)/t E Z}. This implies, z = 3 - 2t, and now the set of solutions ot the solvable Diophantine equation 22 + 4y = 3 - 3(3 - 2t) or ~+2y=-3+3t (5.204~) is sought for every t E Z. The equation (5.204~)is solvable since gcd(l,2) = 11(-3 + 3t). Yow 1 . (-1) t 2 . 1 = 1 and 1 . (3 3t)+2.(-3$3t) = -3+3t. Thesetofsolutionis{((3-3t)+2~,(-3+3t)-s)ls E Z}. That implies x = (3 - 3t) 2s. y = (-3 + 3t) - s, and ((3 - 3t + 2s. -3 + 3t - s, 3 - 2t)Js,t E Z} so obtained is the set of solutions of (5.204a).

+

5.4.3 Congruences and Residue Classes 1. Congruences Let m be a positive integer m, m > 1. If two integers a and b have the same remainder, when divided by m, then a and bare called congruent modulo m, denoted by a I b mod m or a = b(m). 3 13 mod 5, 38 = 13 mod 5, 3 I -2 mod 5. Remark: Obviously, a b mod m holds iff m is a divisor of the difference a - b. Congruence modulo m is an equivalence relation (see 5.2.4, l . ,p. 297) in the set of integers. Note the following properties: a a mod m for every a E Z, (5.205a) a b mod m + b E a mod m, (5.205b) a b mod m A b I cmod m + a I c mod m. (5.205~)

2. Calculating Rules

d mod m + a + c = b + d mod m, a b mod m A c d mod m j a . c I b . d mod m, a . c z b . c m o d m A gcd(c,m)=l + a I b m o d m , a I b mod m A c

(5.206a) (5.206b) (5.206~) (5.206d)

3. Residue Classes, Residue Class Ring Since congruence modulo m is an equivalence relation in Z,this relation induces a partition of Z into residue classes modulo m: [a], = {zlz E Z A z E a mod m } . (5.207) The residue class ‘‘ a modulo m ” consists of all integers having equal remainder if divided by m. Now [a], = [b], iff a b mod m. There are exactly m residue classes modulo m, and normally they are represented by their smallest non-negative representatives: (5.208) [OI,, [I],, ’ ’ ,, [m- 11,. , of residue classes modulo m, residue class addition and residue class multiplication are In the set Z defined by (5.209) [a], @ [b], := [a b],. (5.210) [a], 0 [bIm := [ a . b],. These residue class operations are independent of the chosen representatives, Le., [a], = [a’], and [b], = [b’], imply (5.211) [a], @ [b], = [a’], @ [b’], and [a], 0 [b]m -- [a’], 0 [b’],. The residue classes modulo m form a ring with unit element, with respect to residue class addition and residue class multiplication (see 5.4.3, l.,p. 325), the residue class ring modulo m. If p is a prime number, then the residue class ring modulo p is a field (see 5.4.3, l.,p. 325).

+

326

5. Alqebra and Discrete Mathematics

4. Residue Classes Relatively Prime tom A residue class [a],,, with gcd(a, m) = 1 is called a residue class relatively prime to m . If p is a prime number. then all residue classes different from [O],are residue classes relatively prime t o p . The residue classes relatively prime to m form an Abelian group with respect to residue class multiplication. the so-called group of residue classes relatively prime to m. The order of this group is p(m), where (s is the Euler function (see 5.4.4, l.,p. 329). IA: [1]8, [3]8,[5]8.[7]8 are residue classes relatively prime to 8. IB: [115.[215. [315,[415 are residue classes relatively prime to 5. IC: p(8) = q(5) = 4 is valid. 5 . Primitive Residue Classes .A residue class [aim relatively prime to m is called a primitive residue class if it has order p(m) in the group of residue classes relatively prime to m. IA: [2]5 is a primitive residue class modulo 5, since ([2]5)’ = [4]5, ( [ 2 ] ~=) ~[3]5: = [l]5. IB: There is no primitive residue class modulo 8, since [ l ] S has order 1. and [3]8.[5]s,[7]8 have order 2 in the group of residue classes relatively prime to m. Remark: There is a primitive residue class modulo m. iff m = 2, m = 4. m = p k or m = 2pk, where p is an odd prime number and k is a positive integer. If there is a primitive residue class modulo m. then the group of residue classes relatively prime to m forms a cyclic group.

6. Linear Congruences 1. Definition If a. b and m > 0 are integers, then (5.212) ax c b(m) is called a linear congruence (in the unknown Z) 2. Solutions An integer x’ satisfying ax* 3 b(m) is a solution of this congruece. Every integer: which is congruent to x* modulo m. is also a solution. If all solutions of (5.212) we searched for, then it is sufficient to find the integers pairwise incongruent modulo m which satisfy the congruence. The congruence (5.212) is solvable iff gcd(a, m) is a divisor of b. In this case, the number of solutions modulo m is equal to gcd(a. m). In particular, if gcd(a, m)= 1 holds: the congruence modulo m has a unique solution. 3. Solution Method There are different solution methods for linear congruences. It is possible to transform the congruence ax c b(m) into the Diophantine equation az + my = b, and to determine a special solution (xO$yo) of the Diophantine equation a‘z + m’y = b’ with a’ = a/gcd(a: m).m’ = m/gcd(a. m ) ,b’ = b/gcd(a: m ) (see 5.4.2, l . ,p. 323). The congruence a’x E b’(m‘) has a unique solution since gcd(a’, m’) = 1 modulo m’, and 2 cxO(m’), The congruence ax c b(m) has exactly gcd(a. m) solutions modulo m:

(5.213a)

(5.213b) 2’. xo + m. xo t 2 m , . . . , I O + (gcd(a, m ) - 1)m. I1142 E 6 mod 315 is solvable, since gcd(ll4,315) is a divisor of 6; there are three solutions modulo 315. 382 E 2 mod 105 has a unique solution: I c 94 mod 105 (see 5.4.2. 4.. p. 324). 94, 199. and 304 are the solutions of 1142 c 3 mod 315.

7. Simultaneous Linear Congruences If finitely many congruences

(5.214) z E bl(ml),x E bZ(mz), . . . , Z E bt(mt) are given. then (5,214) is called a system of simultaneous linear congruences. A result on the set of solutions is the Chinese remainder theorem: Consider a given system x c bl(ml), x bz(mz),. . . , Z e

5.4 Elementaw Number Theorv 327 bt(mt).where ml. m2... . , mt are pairwise coprime numbers. If (5.215a) and x3 is choosen such that ajx3= b,(m,) for = 1 , 2 , . . . , t , then x' = alxl a2x2 ' . atxt (5.215b) is a solution of the system. The system has a unique solution modulo m, Le., if x' is a solution, then z" is a solution. too, iff a" I a'(m). ISolre the system a 5 1 (2). x 2 (3)) z = 4 (5). where 2,3,5 are pairwise coprime numbers. Then m = 30.al = 1 5 , = ~ 10,a3 = 6. The congruences 15x1 l ( 2 ) . 10x2 2(3), 6x3 4(5) have the special solutions a1 = 1, x2 = 2. $3 = 4. The given system has a unique solution modulo m: a 15 1 t 10 2 6 . 4 (30). Le., z = 29 (30). Remark: Svstems of simultaneous linear congruences can be used to reduce the problem of solving non-linear cdngruences modulo m to the problem of solving congruences modulo prime number powers (see 5.4.3. 9., p. 328).

+. +

+

+

8. Quadratic Congruences 1. Quadratic Residues Modulo m One can solve every congruence ax2 + bx + c I O(m)if one can solve every congruence x2 E a(m):

+

+

ax2 + bx c = O(m)H (2aa b)' 5 b2 - 4ac(m). (5.216) First quadratic residues modulo m are considered: Let m E IN,m > 1 and a E 2 , gcd(a, m ) = 1. The number a is called a quadratic reszdue modulo miff there is an x E 2 with x2 I a(m). If the canonical prime factorization of m is given, Le., X

m = flpp'.

(5.217)

G I

then T is a quadratic residue modulo m iff T is a quadratic residue modulo ppl for z = 1,2,3.. . . . If a is a quadratic residue modulo a prime number p . then this is denoted by non-residue modulo p . then it is denoted by

(%I

(%I

= 1, if a is a quadratic

= -1 (Legendre symbol).

IThe numbers 1.4.7 are quadratic residues modulo 9. 2. Properties of Quadratic Congruences (El)

(E2) (E3)

pfab and a

E

b(p) imply

(:)

(%)

=

(5.218a) '

(5.21 8b)

=l.

):(

=

E L

(E4)

(f) (i). (!)

(E5)

(:)

=

p2-1

=(-I)

8

.

(5.218~) in particular

($) (i) =

(5.218d) (5.218e)

I

5. Aloebra and Discrete Mathematics

328

(E6) Quadratic reciprocity law: Ifp and q are distinct odd prime numbers, then

(5) (i)

~@

=(-1) z

(5.2180

2

13

307

13

13'-I

- ( - l ) S = 1. In General: .4congruence x2 I 4 2 9 , gcd(a, 2) = 1. is solvable iff a = l(4) for a = 2 and a = l(8) for cy 2 3. If these conditions are satisfied, then modulo 2" there is one solution for a = 1, there are two solutions for cy = 2 and four solutions for a 2 3. A necessary condition for solvability of congruences of the general form (5.219a) x 2 I a(m),m = 2"p:'pF ' . ,pp', gcd(a,m) = 1, is the solvability of the congruences

a I l(4) for cy = 2,

a I l(8) for cy 2 3,

(i) (-$ = 1,

= 1,

. . . )

(2)

(5.219b)

= 1.

If all these conditions are satisfied, then the number of solutions is equal to 2t for cy = 0 and equal to Zt+' for a = 2 and equal to 2tS2 for cy 2 3.

cy

= 1,

9. Polynomial Congruences If ml, m2, . . . , mt are pairwise coprime numbers, then the congruence

+

a,zn an-lx'-l !(E) is equivalent to the system

+ . + a0 = ~ ( m l m z .m. t )

(5.220a)

'.

(5.220b) f(2) O(m1), f(x) = O(mz),. , f(.) I O(mt). If k, is the number of solutions of f(x) O(m,)for = 1 , 2 , . . . , t , then k l k z ' . kt is the number of solutions of f(z) O(n1rnz.. mt). This means that the solution of the congruence (5.220~) f(z) I 0 (p?'pF . ' . p a t ) , wherepl,p2,. . . , p t are primes, can be reduced to the solution of congruences f(z) I O(p"). Moreover, these congruences can be reduced to congruences f(x) I O(p) modulo prime numbers in the following way: a) .A solution of f(x) O(p") is a solution of f(x) O(p), too. b) A solution z I q ( p ) of f(z) E O(p) defines a unique solution modulo p" iff f'(z1) is not divisible by P: Suppose f ( x l ) O(p). Let z = z1+ p t l and determine the unique solution ti of the linear congruence

+

(5.221a) f'(2l)tl I O(p). P Substitute t: = ti t pt2 into x = z1 + p t l , then x = x 2 + p Z t z is obtained. Now, the solution tk of the linear congruence f(.l)

~

(5.221b)

+

+

has to be determined modulo p 2 . By substitution of t z = t; pt3 into x = zz p2t2 the result x = 23 + p3t3 is obtained. Continuing this process yields the solution of the congruence f(z) E 0 (p"). ISolve the congruence f(x) = x4 + 72 + 4 0 (27). f(z) = x4 + 7x + 4 3 0 (3) implies z E l ( 3 ) . Le., z = 1 + 3tl. Because of f'(z) = 4x3 + 7 and 3/f'(l) now the solution of the congruence f(1)/3 f'(1) t l I 4 l l t l = 0 (3) is searched for: tl 1 (3), Le., ti = 1 + 3t2 and z = 4 9t2.

+

+

+

5.1 Elementarv Number Theorv 329

Then consider f(4)/9

+ f‘(4) . t 2 I 0 (3) and the solution t 2

5

2 (3) is obtained, i.e., tz = 2 + 3t3 and

z = 22 + 27t3. Therefore, 22 is the solution of z4+ 7x + 4 I 0 (27), uniquely determined modulo 27.

5.4.4 Theorems ofFermat, Euler, and Wilson 1. Euler Function For every positive integer m with m > 0 one can determine the number of coprimes x with respect to m for 1 5 x 5 m. The corresponding function ’p is called the Euler function. The value of the function .p(m) is the number of residue classes relatively prime to m (s. 5.4.3, 4., p. 326). For instance, p(1) = 1, p(2) = 1, p(3) = 2, p(4) = 2, ~ ( 5 =) 4, ( ~ ( 6=) 2, ( ~ ( 7 = ) 6, p(8) = 4, etc. In general, y ( p ) = p - 1 holds for every prime number p and ( ~ ( p ” = ) p’ -pn-’ for every prime number power p”. If m is an arbitrary positiv integer, then ( ~ ( mcan ) be determined in the following way: (5.222a) where the product applies to all prime divisors p of m. Ip(360) = ~ ( 32 2 5) ~ = 360. (1 - . (1 (1 Furthermore P(4 = m

5)

i)

i ) = 96. (5.222b)

dlm

is valid. If gcd(m, n) = 1 holds, then we get p(mn) = (o(m)cp(n). Ip(360) = p p 3 .3’. 5) = ~ ( 2. (~(3’) ~ ) ( ~ ( 5=) 4 . 6 . 4 = 96. 8

2. Fermat-Euler Theorem The Fermat-Euler theorem is one of the most important theorems of elementary number theory. If a and m are coprime positive numbers, then d m )

E 1( m ) .

(5.223)

IDetermine the last three digits of 9’ in decimal notation. This means, determine z with z I 9’’ (1000) and 0 5 z 5 999. Now (~(1000)= 400, and according to Fermats theorem 9*0° I l(1000). Furthermore 9’ = (80 + 1)4 9 E ((:)80°. l4 (:)801 . 13). 9 = (1 4 80) . 9 I -79.9 I 89 (400). 8

+

Fromthat itfollowsthat999 E 989 = (10-1)89

E

+

(809)10°~(-l)89+(8p)101~(-1)88+(8~)102~(-l)87 =

-1 + 89. 10 - 3916 100 -1 - 110 + 400 = 289(1000). The decimal notation of 9” ends with the digits 289. Remark: The theorem above form = p , i.e., (o(p) = p - 1was proved by Fermat; the general form was proved by Euler. This theorem forms the basis for encoding schemes (see 5.4.5). It contains a necessary criterion for the prime number property of a positive integer: If p is a prime, then a p - l I l(p) holds for every integer o with p y a .

3. Wilson’s Theorem There is a further prime number criterion, called the Wilson theorem: Every prime number p satisfies ( p - l)! I -l(p). The inverse proposition is also true; and therefore: The number p is a prime number iff ( p - l)!I -l(p).

5.4.5 Codes 1. RSACodes R. Rivest, A. Shamir and L. Adleman (see [5.15]) developed an encryption scheme for secret messages on the basis of the Euler-Fermat theorem (see 5.4.4,2.). The scheme is called the RSA algorzthrn after the initials of their last names. Part of the key required for decryption can be made public without

330

5. Aloebra and Discrete Mathematics

endangering the confidentiality of the message; for this reason, the term public key code is used in this context as well. In order to apply the RSX algorithm the recipient B chooses two very large prime numbers p and q , calculates m = p q and selects a number T relatively prime to cp(m)= ( p - l ) ( q - 1)and 1 < r < cp(m). B publishes the numbers m and T because they are needed for decryption. For transmitting a secret message from sender A to recipient B the text of the message must be converted first to a string of digits that will be split into N blocks of the same length of less than 100 decimal positions. Now A calculates the remainder R of N P divided by m.

NP= R ( m ) . (5.224a) Sender A calculates the number R for each of the blocks N that were derived from the original text and sends the number to B. The recipient can decipher the message R if he has a solution of the linear congruence rs I 1 (cp(m)).The number :V is the remainder of R5 divided by m:

RS I ( . q s

Iy1+(4

E .IT. (,\w(m))k E

.v(~),

(5.224b)

Here. the Euler-Fermat theorem with N2T”(m) l ( m ) has been applied. Eventually, B converts the sequence of numbers into text. IA recipient B who expects a secret message from sender A chooses the prime numbers p = 29 and q = 37 (actually too small for practical purposes), calculates m = 29 . 37 = 1073 (and ~ ( 1 0 7 3 )= ~ ( 2 9 )~ ( 3 7 = ) 1008)).and chooses T = 5 (it satisfies the requirement of gcd(1008,5) = 1). B passes the values m = 1073 and r = 5 to A. .4 intends to send the secret message ,V = 8 to B. A encrypts N into R = 578 by calculating ,V‘ = 85 I 578 (1073). and just sends the value R = 578 to B. B solves the congruence 5 s I 1 (1008), arrives at the solution s = 605, and thus determines Rs = 57@05I 8 = N (1073). Remark: The security of the RSA code correlates with the time needed by an unauthorized listener to factorize m. Assuming the speed of today’s computers, a user of the RSA algorithm should choose the two prime numbers p and q with at least a length of 100 decimal positions in order to impose a decryption effort of approximately 74 years on the unauthorized listener. The effort for the authorized user. however, to determine an r relatively prime to p ( p q ) = ( p - l ) ( q - 1) is comparatively small.

2. International Standard Book Number (ISBN) A simple application of the congruence of numbers is the use of control digits with the International Standard Book Number ISBN. A combination of 10 digits of the form (5.225a) ISBN Q - bed - efghi - p . is assigned to a book. The digits have the following meaning: Q is the group number (for example, Q = 3 tells us that the book originates from Austria, Germany, or Switzerland), bcd is the publisher’s number, and efghi is the title number of the book by this publisher. .4 control digit p will be added to detect erroneous book orders and thus help reduce expenses. The control digit p is the smallest non-negative digit that fulfils the following congruence: (5.225b) 10a + 9b t 8c t 7d t 6e + 5f + 49 3h + 22 t p I O(11). If the control digit p is 10: a unary symbol such as X is used (see also 5.4.5, 4., p. 331). A presented ISBN can now be checked for a match of the control digit contained in the ISBN and the control digit determined from all the other digits. In case of no match an error is certain. The ISBS control digit method permits the detection of the following errors: 1. Single digit error and 2. interchange of two digits. Statisticalinvestigationsshowedthat by thismethod more than 90% of all actualerrorscan bedetected. All other observed error types have a relative frequency of less than 1%. In the majority of the cases the described method will detect the interchange of two digits or the interchange of two complete digit blocks.

+

5 4 Elementaru Number Theoru 331

3. Central Codes for Drugs and Medicines In pharmacy. a similar numerical system with control digits is employed for identifying medicaments. In Germany, each medicament is assigned a seven digit control code: (5.226a) abcde f p . The last digit is the control digit p . It is the smallest, non-negative number that fulfils the congruence 2a+3b+4c+5d+6e+7f

=p(ll).

(5.226b)

Here too, the single digit error or the interchange of two digits can always be detected.

4. Account Numbers Banks and saving banks use a uniform account number system with a maximum of 10 digits (depending on the business volume). The first (at most four) digits serve the classification of the account. The remaining six digits represent the actual account number including a control digit in the last position. The individual banks and saving banks tend to apply different control digit methods, for example: a) The digits are multiplied alternately by 2 and by 1, beginning with the rightmost digit. A control digit p will then be added to the sum of these products such that the new total is the next number divisible by 10. Given the account number abcd efghi p with control digit p , then the congruence (5.227) 2i + h + 2g+ f +2e + d + 2c t b + 2a t p 0 (mod 10). holds. b) .As in method a), however. any two-digit product is first replaced by the sum of its two digits and then the total sum will be calculated. In case a) all errors caused by the interchange of adjacent digits and almost all single-digit errors will be detected. In case b): however, all errors caused by the change of one digit and almost all errors caused by the interchange of two adjacent digits will be discovered. Errors due to the interchange of non-adjacent digits and the change of two digits will often not be detected. The reason for not using the more powerful control digit method modulo 11 is of a non-mathematical nature. The non-numerical sign X (instead of the control digit 10 (see 5.4.5,2.,p. 330)) would require an extension of the numerical keyboard. However, renouncing those account numbers whose control digit has the value of 10 would have barred the smooth extension of the original account number in a considerable number of cases.

5. European Article Number EAN EAS stands for European Article Number. It can be found on most articles as a bar code or as a string of 13 or 8 digits. The bar code can be read by means of a scanner at the counter. In the case of 13-digit strings the first two digits identify the country of origin, e.g., 40,41,42 and 43 stand for Germany. The next five digits identify the producer, the following five digits identify a particular product. The last digit is the control digit p . This control digit will be obtained by first multiplying all 12 digits of the string alternately by 1 and 3 starting with the left-most digit, by then totalling all values, and by finally adding a p such that the next number divisible by 10 is obtained. Given the article number abcdefghikmn p with control digit p , then the congruence

+

+

+ + +

(5.228) a + 3b + c + 3d+ e 3f + g 3h t i 3k rn 3n + p 0 (mod 10). holds. This control digit method always permits the detection of single digit errors in the EAN and often the detection of the interchange of two adjacent digits. The interchange of two non-adjacent digits and the change of two digits will often not be detected.

332

5. Aloebra and Discrete Mathematics

5.5 Cryptology 5.5.1 Problem of Cryptology Cryptology is the science of hiding information by the transformation of data The idea of protecting data from unauthorized access is rather old. During the 1970s together with the introduction of cryptosystems on the basis ofpublic keys, cryptology became an independent branch of science. Today! the subject of cryptological research is how to protect data from unauthorized access and against tampering. Beside the classical military applications, the needs of the information society gain more and more in importance. Examples are the guarantee of secure message transfer via email, electronic funds transfer (home-banking), the PIN of EC-cards. etc. Today. the fields of cryptography and cryptanalysis are subsumed under the notion of cryptology. Cryptography is concerned with the development of cryptosystems whose cryptographic strengths can be assessed by applying the methods of cryptanalysis for breaking cryptosystems.

5.5.2 Cryptosystems .4n abstract cryptosystem consists of the following sets: a set M of messages, a set C of ciphertexts. sets K and K' of keys, and sets E and D of functions. .4 message m E M will be encrypted into a ciphertext c E C by applying a function E E E together with a key k E K , and will be transmitted via a communication channel. The recipient can reproduce the original message m from c if he knows an appropriate function D E D and the corresponding key k' E K'. There are two types of cryptosystems: 1. Symmetric Cryptosystems: The conventional symmetric cryptosystem uses the same key k for encryption of the message and for decryption of the ciphertext. The user has complete freedom in setting up his conventional cryptosystem. Encryption and decryption should, however, not become too complex. In any case, a trustworthy transmission between the two communication partners is mandatory. 2. Asymmetric Cryptosystems: The asymmetric cryptosystem (see 5.5.7.1,p. 336) uses two keys, one private key (to be kept secret) and a public key. The public key can be transmitted along the same path as the ciphertext. The security of the communication is warranted by the use of so-called one-way functions (see 5.5.7.2. p. 337), which makes it practically impossible for the unauthorized listener to deduce the plaintext from the ciphertext.

5.5.3 Mathematical Foundation An alphabet A = {ao, al,.. . a n - l } is a finite non-empty totally ordered set, whose elements a, are called letters. IAl is the length of the alphabet. A sequence of letters w = ai.;. . .a;of length n E K and ai E il is called a word of length n over the alphabet A. A" denotes the set of all words of length n over A . Let n, m E N.let A , B be alphabets. and let S be a finite set. A4cryptofunction is a mapping t: A" x S + Bm such that the mappings t,: A" + Bm: w + t(w, s) are injective for all s E S.The functions t , and t;' are called the encryption and decryption function, respectively. w is called plaintext, t,(w) is the ciphertext. Given a cryptofunction t , then the one-parameter family { t s } s E S is a cryptosystem Ts. The term cryptosystem will be applied if in addition to the mapping t, the structure and the size of the set of keys is significant. The set S of all the keys belonging to a cryptosystem is called the key space. Then (5.229) T , = {t,: A" + A " ~ Es S} is called a cryptosystem on A". If Ts is a cryptosystem over An and n = 1, then t, is called a stream cipher; otherwise t, is called a block cipher. Cryptofunctions of a cryptosystem over An are suited for the encryption of plaintext of any length. To this end a plaintext will be split into blocks of length n prior to applying the function to each individual ~

5.5 Cryptology 333

block. The last block may need padding with filler characters to obtain a block of length n. The filler characters must not distort the plaintext. There is a distinction between context-free encryption, where the ciphertext block is only a function of the corresponding plaintext block and the key, and context sensitive encryption, where the ciphertext block depends on other blocks of the message. Ideally, each ciphertext digit of a block depends on all digits of the corresponding plaintext block and all digits of the key. Small changes to the plaintext or to the key cause extended changes to the ciphertext (avalanche effect).

5.5.4 Security of Cryptosystems Cryptanalysis is concerned with the development of methods for deducing from the ciphertext as much information about the plaintext as possible without knowing the key. According to A. Kerkhoff the security of a cryptosystem rests solely in the difficulty of detecting the key or, more precisely, the decryption function. The security must not be based on the assumption that the encryption algorithm is kept secret. There are different approaches to assess the security of a cryptosystem: 1. Absolutely Secure Cryptosystems: There is only one absolutely secure cryptosystem based on substitution ciphers, which is the one-time pad. This was proved by Shannon as part of his information theory. 2. Analytically Secure Cryptosystems: No method exists to break a cryptosystem systematically. The proof of the non-existence of such a method follows from the proof of the non-computability of a decryption function. 3. Secure Cryptosystems according to Criteria of Complexity Theory: There is no algorithm which can break a cryptosystem in polynomial time (with regard to the length of the text). 4. Practically Secure Cryptosystems: No method is known which can break the cryptosystem with available resources and with justified costs. Cryptanalysis often applies statistical methods such as determining the frequency of letters and words. Other methods are an exhaustive search, the trial-and-error method and a structural analysis of the cryptosystem (solving of equation systems). In order to attack a cryptosystem one can benefit from frequent flaws in encryption such as using stereotype phrases, repeated transmissions of slightly modified text, an improper and predictable selection of keys. and the use of filler characters.

5.5.4.1 Methods of Conventional Cryptography Besides the application of a cryptofunction it is possible to encrypt a plaintext by means of cryptologicul codes. .A code is a bijective mapping of some subset A' of the set of all words over an alphabet A onto the subset B' of the set of all words over the alphabet B . The set of all source-target pairs of such a mapping is called a code book. 0815 today evening tomorrow evening 1113 The advantage of replacing long plaintexts by short ciphertexts is contrasted with the disadvantage that the same plaintext will always be replaced by the same ciphertext. Another disadvantage of code books is the need for a complete and costly replacement of all books should the code be compromised even partially. In the following only encryption by means of cryptofunctions will be considered. Cryptofunctions have the additional advantage that they do not require any arrangement about the contents of the messages prior to their exchange. Transposition and substitution constitute conventional cryptoalgorithms. In cryptography, a transposition is a special permutation defined over geometric patterns. The substitutions will now be discussed in detail. There is a distinction between monoalphabetic and polyalphabetic substitutions according to how many alphabets are used for presenting the ciphertext. Generally, a substitution is termed polyal-

'

334

5. Alqebra and Discrete Mathematics

phabetic even if only one alphabet is used, but the encryption of the individual plaintext letter depends on its position within the plaintext. A further, useful classification is the distinction between monographic and polygraphic substitutions. In the first case. single letters will be substituted, in the latter case, strings of letters of a fixed length > 1.

5.5.4.2 Linear Substitution Ciphers Let il = { a o . a l , . . . , a n - l } be an alphabet and k , s E { O , l , . . . , n - l} with gcd(k. n) = 1. The permutation t f , which maps each letter a, to tf(ai) = a k i + s , is called a linear substitution cipher. There exist n p(n) linear substitution ciphers on A. Shift ciphers are linear substituting ciphers with k = 1. The shift cipher with s = 3 was already used by Julius Caesar (100 to 44 BC) and, therefore, it is called the Caesar cipher.

5.5.4.3 Vigenhre Cipher A n encryption called the Vigenere cipher is based on the periodic application of a key word whose letters are pairwise distinct. The encryption of a plaintext letter is determined by the key letter that has the same position in the key as the plaintext letter in the plaintext. This requires a key that is as long as the plaintext. Shorter keys are repeated to match the length of the plaintext.

A version of the \'igeni.re cipher attributed to L. Carroll utilizes the so-called Vigenkre tableau (see picture) for encryption and decryption. Each row represents the cipher for the key letter to its very left. The alphabet for the plaintext runs across the top. The encryption step is as follows: Given a key letter D and a plaintext letter C. then the ciphertext letter is found at the intersection of the row labeled D and the column labeled C: the ciphertext is F. Decryption is the inverse of this process.

.A B

C D E F

. , I

C D E F ...

A

B

.A B C D E F

B C D C D E D E F E F G F G H G H I

.

, I

. . .

.

. .

.

. .

E F G H I J .

. .

.

., .

F G H I

,,,

J

,,

K

,

. .

,

,,

,,,

.

. , .. I

Let the key be '' HCT ". Plaintext: 0 N C E U P 0 N A T I 51 E Key: H U T H U T H U T H U T H Ciphertext: V H V L 0 I V H T X C F L Formally. the Yigenkre cipher can be written in the following way: let a, be the plaintext letter and a3 be the corresponding key letter, then k = i + j determines the ciphertext letter a k . In the above example. the first plaintext letter is 0 = ~ 1 4 The . 15-th position of the key is taken by the letter H = u7. Hence. k = i + j = 14 + 7 = 21 yields the ciphertext letter a21 = V. I

5.5.4.4 Matrix Substitution Let A = {ao. a i , . . . . an-l} be an alphabet and S = (s,~),s , ~E (0:1.. . . m - 1). be a non-singular matrix of type (m.m ) with gcd(detS, n)= 1. The mapping which maps the block of plaintext at(l), at(m) to the ciphertext determined by the vector (all arithmetic modulo n. vectors transposed as required)

(s. [::/ti

1);

(5.230)

at(m)

is called the Hill cipher. This represents a monoalphabetic matrix substitution. Let the letters of the alphabet be enumerated a0 = A, a1 =B, . . . , a25 = 14 8 3 Z. For m = 3 and the plaintext AUTUMY. the strings AUT and UMN I S= '2) , correspond to the vectors (0,20,19) and (20,12,13).

(

5.5 Cruutoloau 335

Then S (0.20, 19)T = (217,138, 59)T E (9,8, 7)T(mod26) and S (20.12: 13)T = (415,246,97)' E (25.12, 19)T(mod26). Thus. the plaintext AUTUMN is mapped to the ciphertext JIHZMT.

5.5.5 Methods of Classical Cryptanalysis The purpose of cryptanalytical investigations is to deduce from the ciphertext an optimum of information about the corresponding plaintext without knowing the key. These analyses are of interest not only to an unauthorized "eavesdropper" but also help assess the security of cryptosystems from the user's point of view.

5.5.5.1 Statistical Analysis Each natural language shows a typical frequency distribution of the individual letters, two-letter combinations. words, etc. For example, in English the letter e is used most frequently: Letter E. T,.4, 0.1, N . S . H , R D. L C,C.M.lV.F,G.Y,P,B V. K. J. X, Q.Z

Relative frequency 12.7 % 56.9% 8.3 % 19.9% 2.2 %

Given sufficiently long ciphertexts it is possible to break a monoalphabetic, monographic substitution on the basis of the frequency distribution of letters.

5.5.5.2 Kasiski-Friedman Test Combining the methods of Kasiski and Friedman it is possible to break the Vignkre cipher. The attack benefits from the fact that the encryption algorithm applies the key periodically. If the same string of plaintext letters is encrypted with the same portion of the key then the same string of ciphertext letters will be produced. A length > 2 of the distance of such identical strings in the ciphertext must be a multiple of the key length. In the case of several reoccurring strings of ciphertext the key length is a divisor of the greatest common divisor of all distances. This reasoning is called the Kasiski test. One should. however. be aware of erroneous conclusions due to the possibility that matches may occur accidentally. The Kasiski test permits the determinationofthe key length at most as a multipleofthe true key length. The Friedman test yields the magnitude of the key length. Let n be the length of the ciphertext ofsome English plaintext encrypted by means of the Vignkre method. Then the key length 1 is determined by 0.027n (5.231a) I= ( n - 1)IC - 0.038n + 0.065' Here IC denotes the coincidence index of the ciphertext. This index can be deduced from the number n, of occurrences of the letter a, (i E {0,1,. . . ,25}) in the ciphertext: (5.231b) In order to determine the key, the ciphertext of length n is split into 1 columns. Since the Vignkre cipher produces the contents of each column by means of a shift cipher, it suffices to determine the equivalence of E on a column base. Should 1' be the most frequent letter within a column, then the Vignbre tableau points to the letter R

E (5.2314

R. .V

336

5. Aloebra and Discrete Mathematics

of the key. The methods described so far will not be successful if the Vignkre cipher employs very long keys (e.g.>as long as the plaintext). It is, however, possible to deduce whether the applied cipher is monoalphabetic. polyalphabetic with short period or polyalphabetic with long period.

5.5.6 One-Time Pad The one-time pad is the only substitution cipher that is considered theoretically secure. The encryption adheres to the principle of the Vignkre cipher, where the key is a random string of letters as long as the plaintext. Usually, one-time pads are applied as binary Vignkre ciphers: Plaintext and ciphertext are represented as binary numbers with addition modulo 2. In this particular case the cipher is involutory, which means that the twofold application of the cipher restores the original plaintext. A concrete implementation of the binary Vignkre cipher is based on shift register circuits. These circuits combine switches and storage elements, whose states are 0 or 1, according to special rules.

5.5.7 Public Key Methods Although the methods of conventional encryption can have efficient implementations with today’s computers, and although only a single key is needed for bidirectional communication, there are a number of drawbacks: 1. The security of encryption solely depends on keeping the next key secret. 2. Prior to any communication, the key must be exchanged via a sufficiently secured channel; spontaneous communication is ruled out. 3. Furthermore, no means exist to prove to a third party that a specific message was sent by an identified sender.

5.5.7.1 Diffie-Hellman Key Exchange The concept of encryption with public keys was developed by Diffie and Hellman in 1976. Each participant owns two keys: a public key that is published in a generally accessible register, and a private key that is solely known to the participant and kept absolutely secret. Methods with these properties are called asymmetric ciphers (see 5.5.2. p. 332). The public key KP, of the i-th participant controls the encryption step E,, his private key KS, the decryption step D,. The following conditions must be fulfilled: 1. D,o E, constitutes the identity 2. Efficient implementations for Ei and Di are known. 3. The private key KS, cannot be deduced from the public key KP, with the means available in the foreseeable future. If in addition 4. also E, o D, yields the identity, then the encryption algorithm qualifies as an electronic signature method with public keys. The electronic signature method permits the sender to attach a tamperproof signature to a message. If A wants to send an encrypted message m to B , then A retrieves B’s public key KPg from the register, applies the encryption algorithm E B ,and calculates E B ( ~=)c. A sends the ciphertext c via the public network to B who will regain the plaintext of the message by decrypting c using his private key K S B in the decryption function D g : DB(c)= D B ( E B ( ~=) )m.In order to prevent tampering of messages, A can electronically sign his message m to B by complying with an electronic signature method with the public key in the followingway: A encrypts the message m with his private key: D A ( ~ =) d. A attaches to d his signature “A” and encrypts the total using the public key of B: EB(DA(m), “A”) = E g ( d , “A”) = e. The text thus signed and encrypted is sent from A to B. The participant B decrypts the message with his private key and obtains D g ( e ) = D g ( E g ( d , “A”)) = ( d , “A”).Based on this text B can identify A as the sender and can now decrypt d using the public

key of A : Ea(d) = E~(Da(rn)) = m.

5.5.7.2 One-way Function The encryption algorithms of a method with public key must constitute a one-way function with a “trap door”. .A trap door in this context is some special, additional information that must be kept secret. An injective function f : X t Y is called a one-way function with a trap door, if the following conditions hold: 1. There is an efficient method to compute both f and f-’. 2. The calculation of f-’ cannot be deduced from f without the knowledge of the secret additional information. The efficient method to get f-’ from f cannot be made without the secret additional information.

5.5.7.3 RSA Method The RSA method described in the number theory section (see 5.4.5, l., p. 329) is the most popular asymmetric encryption method. 1. Prerequisites: Let p and q be two large prime numbers withpq > lozooand n = pq. The number of decimal positions ofp and q should differ by a small number; yet, the difference between p and q should not be too large. Furthermore, the numbers p - 1 and q - 1 should contain rather big prime factors, while the greatest common divisor of p - 1 and q - 1 should be rather small. Let e > 1 be relatively prime to (p- l ) ( q - 1) and let d satisfy d . e 1(mod(p- l)(q- 1)).Now n and e represent the public key and d the private key. 2. Encryption Algorithm: E: {0,1,....n - 1) -+ {0,1,.. . .n - 1) E ( z ) := zemodulo n. (5.232a) 3. Decyphering Operations: D {0,1,.. . , n - 1) -+ {0,1.. . . , n - 1) D ( z ) := zdmodulo n. (5.232b) Thus D ( E ( m ) )= E ( D ( m )= m for message m. The function in this encryption method with n > lozooconstitutes a candidate for a one-way function with trap door (see 5.5.7.2). The required additional information is the knowledge of how to factor n. Without this knowledge it is infeasible to solve the congruence d . e 1 (modulo(p - l)(q - 1)). The RSA method is considered practically secure as long as the above conditions are met. A disadvantage in comparison with other methods is the relatively large key size and the fact that RSA is 1000 times slower than DES.

5.5.8 DES Algorithm (Data Encryption Standard) The DES method was adopted in 1976 by the National Bureau of Standards (now NIST) as the official US encryption standard. The algorithm belongs to the class of symmetric encryption methods (see 5.5.2,p. 332) and still plays a predominant role among cryptographic methods. The method is, however, no longer suited for the encryption of top secret information because today’s technical means permit an attack by an exhaustive test trying all keys. The DES algorithm combines permutations and non-linear substitutions. The algorithm requires a 56-bit key. Actually, a 64-bit key is used, however, only 56 bits can freely be chosen; the remaining eight bits serve as parity bits, one for each of the seven-bit blocks to yield odd parity. The plaintext is split into blocks of 64 bits each. DES transforms each 64-bit plaintext block into a ciphertext block of 64 bits. First, the plaintext block will be subject to an initial permutation and is then encrypted in 16 rounds, each operating with a different subkey K1, Kz, . . . K16. The encryption completes with a final permutation that is the inverse of the initial permutation. Decryption uses the same algorithm with the difference that the subkeys are employed in reverse order K161 Klj,. , , 5 K1. The strength of the cipher rests on the nature of the mappings that are part of each round. It can be

I

338

5. Alqebra and Discrete Mathematics

shown that each bit of the ciphertext block depends on each bit of the corresponding plaintext and on each bit of the key Although the DES algorithm has been disclosed in full detail, no attack has been published so far that can break the algorithm without an exhaustive test of all 256 keys.

5.5.9 IDEA Algorithm (International Data Encryption Algorithm) The IDE.4 algorithm was developed by L.41 and M.ASSAY and patented 1991. It is a symmetric encryption method similar to the DES algorithm and constitutes a potential successor to DES. IDEA became known as part of the reputed software package PGP (Pretty Good Privacy) for the encryption of emails. In contrast to DES not only was the algorithm published but even its basic design criteria. The objective was the use of particularly simple operations (addition modulo 2, addition modulo 216, multiplication modulo 216+’). IDEA works with keys of 128 bits length. IDEA encrypts plaintext blocks of 64 bits each. The algorithm splits a block into four subblocks of 16 bits each. From the 128-bit key 52 subkeys are derived, each 16 bits long Each of the eight encryption rounds employs six subkeys. the remaining four subkeys are used in the final transformation which constructs the resulting 64-bit ciphertext. Decryption uses the same algorithm with the subkeys in reverse order. IDEA is twice as fast as DES, its implementation in hardware, however, is more difficult. No successful attack against IDEA is known. Exhaustive attacks trying all 256keys are infeasible considering the length of the keys.

5.6 Universal Algebra A universal algebra consists of a set, the underlying set: and operations on this set. Simple examples are semigroups, groups, rings, and fields discussed in sections 5.3.2, p. 299; 5.3.3; p. 299 and 5.3.6, p. 313. Universal algebras (mostly many-sorted, Le., with several underlying sets) are handled especially in theoretical informatics. There they form the basis of algebraic specifications of abstract data types and systems and of term-rewriting systems.

5.6.1 Definition Let R be a set of operation symbols divided into pairwise disjoint subsets R,, n E N.Ro contains the is called the type or constants, R,. n > 0, contain the n-ary operation symbols. The family (R,),,, signature. If ‘4 is a set, and if to every n-ary operation symbol w E R, an n-ary operation w A in A is assigned. then we call A = (A, {uAIwE R}) an R-algebra or algebra of type (or of signature) R. If R is finite, R = { w l l . . . , w k } , then we also write A = (A, w f , . . . w,”) for A. If a ring (see 5.3.6, p. 313) is considered as an R-algebra, then R is partitioned Ro = { w l } , , R1 = {UT}) R2 = { w g ,w4}l where to the operation symbols w1) w 2 ! wg, w4 the constant 0, taking the inverse with respect to addition, addition and multiplication are assigned. Let .4 and B be R-algebras. B is called an R-subalgebra of A , if B 2 A holds and the operations u B are the restrictions of the operations uA(w E R) to the subset B .

5.6.2 Congruence Relations, Factor Algebras If we want to construct factor structures for universal algebras, then we need the notion of congruence relation. .A congruence relation is an equivalence relation compatible with the structure: Let A = (A: {wAIw E R}) be an R-algebra and R be an equivalence relation in A. R is called a congruence relation in A. if for all w E R, ( n E N) and all at, b, E A with aiRbi (i = 1,. . . , n):

(5.233) d A ( a l . .. . a,) R w A ( b l , . . . , b,). The set of equivalence classes (factor set) with respect to a congruence relation also form an R-algebra with respect to representative-wise calculations: Let A = (A, {uAIwE R}) be an R-algebra and R be I

5.6 Universal Alqebra 339

a congruence relation in A. The factor set A I R (see 5.2.4, 2., p. 297) is an R-algebra A I R with the following operations wAIR (uE R,, n E N) with

(5.234) W " ' R ( [ a l ] R , . . . , [ a n ] R ) = [ w A ( a l , .. . ,a,)]R and it is called the factor algebra of A with respect to R. The congruence relations of groups and rings can be defined by special substructures - normal subgroups (see 5.3.3.2. 2. p. 301) and ideals (see 5.3.6.2,p. 313), respectively. In general, e.g., in semigroups, such a characterization of congruence relations is not possible.

5.6.3 Homomorphism Just as with classical algebraic structures, the homomorphism theorem gives a connection between the homomorphisms and congruence relations. Let A and B be R-algebras. A mapping h: A + B is called a homomorphism,if for every w E R, and all al.. . . , a, E .4: (5.235) h ( d A ( a l , .. . .a,)) = 2 ( h ( a , ) ,. . . h(a,)). If. in addition. h is bijective, then h is called an zsomorphzsm: the algebras A and B are said to be isomorphic. The homomorphic image h(A) of an R-algebra A is an R-subalgebra of B. Under a homomorphism h. the decomposition of A into subsets of elements with the same image corresponds to a congruence relation which is called the kernel of h: ker h = { ( a ,b) E A x Alh(a) = h(b)}. (5.236) ~

5.6.4 Homomorphism Theorem Let A and B be R-algebras and h: A -+ B a homomorphism. h defines a congruence relation ker h in A. The factor algebra A/ ker h is isomorphic to the homomorphic image h ( A ) . Conversely, every congruence relation R defines a homomorphic mapping d ~ A i:A I R with natR(a) = [ a ] R . Fig. 5.16 illustrats the homomorphism theorem. h(a) 0 0 0

5.6.5 Varieties

A

h

m

La],

h

o

/

\on0 on0

nut ker h

000 A/ker h

Figure 5.16

o

oh(A)

.A variety V is a class of 0-algebras, which is closed under forming direct products, subalgebras, and homomorphic images, Le., these formations do not lead out of V. Here the direct products are defined in the following way: If we consider the operations corresponding to R componentwise on the Cartesian product of the underlying sets of R-algebras, then we again get an Ralgebra, the direct product of these algebras. The theorem of Birkhoff (see 5.6.6, p. 339) characterizes the varieties as those classes of R-algebras, which can be equationally defined.

5.6.6 Term Algebras, Free Algebras Let (R,),,N be a type (signature) and X a countable set of variables. The set Tn(X) of R-terms over X is defined inductively in the following way: 1. xu R, Tn(X). 2. If t l . . , . . t, E Tn(A')and w E R, hold, then also ut1 . . . t, E Tn(X)holds. The set Tn(X)defined in this way is an underlying set of an R-algebra. the term algebraTn(X) of type

340

5. Alqebra and Discrete Mathematics

R over X . with the following operations: If t l , . . . ,t, E T n ( X )and w E 0, hold. then wTfl(x) is defined by L J ” ( X ) ( t l , . . . , t,) = Uti.. . t,. (5.237) Term algebras are the “most general” algebras in the class of all 0-algebras, Le., no “identities” are valid in term algebras. These algebras are called free algebras. An zdentzty is a pair ( ~ ( 2 1 , .. . ,x,), t(s1,.. . ,z,)) of 0-terms in the variables X I , .. . ,x,. An 0-algebra A satzsfies such a equation, if for every a l , . . . .a, E A we have: (5.238) sA(al... . a,) = t”(a1,. . . , a,). .A class of R-algebras defined by identities is a class of %algebras satisfying a given set of identities. Theorem of BirkhoE The classes defined by identities are exactly the varieties. I Varieties are for example the classes of all semigroups, groups, Abelian groups, and rings. But, e.g., the direct product of cyclic groups is not a cyclic group, and the direct product of fields is not a field. Therefore cyclic groups or fields do not form a variety, and cannot be defined by equations.

5.7 Boolean Algebras and Switch Algebra Calculating rules. similar to the rules established in 5.2.2, 3., p. 292 for set algebra and propositional calculus (5.1.1. 6., p. 287), can be found for other objects in mathematics too. The investigation of these rules yields the notion of Boolean algebra.

5.7.1 Definition A set B , together with two binary operations fl (“conjunction”) and U (“disjunction”),and a unary operation (“negation”),and two distinguished (neutral) elements 0 and 1 from B , is called a Boolean algebra B = ( B , n, U, -, 0, 1) if the following properties are valid: (1)Associative Laws: ( a n b)

n c = a n (b n c),

(5.239)

( a u b ) u c = a u ( b uc ) .

(5.240)

(5.241)

a U b = bu a.

(5.242)

(5.243)

a u ( a n b) = a.

(5.244)

(2) Commutative Laws: a n b = b f l a,

(3) Absorption Laws: a

n ( a U b)

= a.

(4) Distributive Laws: ( a U b)

n c = ( a fl c) U ( b n c), (5.245)

( a n b) U c = ( a U c) n ( b U e ) . (5.246)

( 5 ) Neutral Elements:

ani=i.

(5.247)

aUO=a,

(5.248)

anO=0,

(5.249)

a U 1 = 1,

(5.250)

(5.251)

a U E = 1.

(5.252)

(6) Complement:

ana=o,

A structure with the associative laws, commutative laws, and absorption laws is called a lattice. If the distributive laws also hold, then we call it a distributive lattice. So a Boolean algebra is a special distributive lattice.

5.7 Boolean Al.qebras and Switch Aloebra 341 Remark: The notation used for Boolean algebras is not necessarily identical to the notation for the operations in propositional calculus.

5.7.2 Duality Principle 1. Dualizing In the "axioms" of a Boolean algebra above we can discover the following duality: If we replace fl by U; u by n, 0 by 1, and 1 by 0 in an axiom, then we always get the other axiom in the same row. The axioms in a row are dual to each other. and the substitution process is called dualization. We get the dual statement from a statement of the Boolean algebra by dualization. 2. Duality Principle for Boolean Algebras The dual statement of a true statement for a Boolean algebra is also a true statement for the Boolean algebra. Le.: with every proved proposition, the dual proposition is also proved. 3. Properties LVe get, e.g., the following properties for Boolean algebras from the axioms (El) The Operations n and U are Idempotent: aflo=a,

(5.253)

aUa=a.

(5.254)

(5.255)

aub=zin6!

(5.256)

(E2)De Morgan Rules: anb=ziUb.

(E3)A further Property:

a = a. (5.257) It is enough to prove only one of the two properties in any line above, because the other one is the dual property. The last property is self-dual.

5.7.3 Finite Boolean Algebras All finite Boolean algebras can be described easily up to "isomorphism". Let algebras and f : B1 -+ B2 a bijective mapping. f is called an isomorphism if

B1,

Bz be two Boolean

fo

f ( a n b) = f ( a ) n f ( b ) . f ( a u b) = f(a)u f ( b ) and !(a) = (5.258) hold. Every finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. In particular every finite Boolean algebra has 2n elements, and every two finite Boolean algebras with the same number of elements are isomorphic. In the following, we denote by B the Boolean algebra with two elements { 0 21) with the operations

y y $ 1 0 1

1 1 1

If we define the operations f l : U. and -componentwise on the n-times Cartesian product B" = {0,1} x . ' . x (0.l},then Bn will be a Boolean algebra with 0 = (0,. . . 0) and 1 = (1,.. . ,1).We call Bn the n times direct product of B . Because Bncontains 2n elements, we get all the finite Boolean algebras in this way (up to isomorphism). ~

5.7.4 Boolean Algebras as Orderings We can assign an order relation to every Boolean algebra B: Here a 5 b holds if a n b = a is valid (or equivalently. if a U b = b holds). So every finite Boolean algebra can be represented by a Hasse diagram (see 5.2.4,4., p. 298).

5. Alaebra and Discrete Mathematics

342

(&:

ISuppose B is the set 11:2.3: 5,6,10,15,30} of the divisors of 30. We define the least common multiple and the greatest common divisor as binary opera0 and operation. 1. The corresponding The numbers Hasse 1 and di30 tions. correspond and taking to the the distinguished complement elements as unary agram is shown in Fig. 5.17.

*\Il/

5.7.5 Boolean Functions, Boolean Expressions 1. Boolean Functions We denote by B the Boolean algebra with two elements as in 5.7.3. An w a r y Boolean function f is a mapping from Bninto B . There are 2'' n-ary Boolean functions. The set of all n-ary Boolean functions with the operations

(f u g ) ( b ) = f ( b ) u g(bL

(5.259)

(f n g ) ( b ) = f ( b ) n g(bL -

Figure 5.17 (5.260)

fo.

f (b) = (5.261) is a Boolean algebra. Here b always means an n-tuple of the elements of B = {0, l}, and on the righthand side of the equations the operations are performed in B . The distinguished elements 0 and 1 correspond to the functions fo and fl with f o ( b ) = 0 , f l ( b ) = 1 for all b E B". (5.262) IA: In the case n = 1. Le., for only one Boolean variable b, there are four Boolean functions: f ( b ) = b, Negation Identity f ( b ) = 6, (5.263) Tautology f ( b ) = 1. Contradiction f ( b ) = 0. IB: In the case n = 2. Le., for two Boolean variables a and b, there are 16 different Boolean functions, among which the most important ones have their own names and notation. They are shown in Table 5.6

Table 5.6 Some Boolean functions with two variables a and b

~

Name of the Different function notation ~

~

Value table for

Different symbols ~

a.b

Sheffer or NAND

a l b

NASD ( a ,b)

a+b a i b

Peirce or Xntivalence or

~

XOR

Za iXOR b +b a $ b a@b

Equivalence

a6+ab a E b a+tb

Implication

~

+b a + b

ab

=

* -+-&

1,

0,

0,

0

0.

1,

1.

0

#

1,

0,

0.

1

1,

1.

0,

1

$+

NOR a. b

YOR

(i) (i), (0 (@, (i) 1,

1,

1,

0

~

2. Boolean Expressions Boolean expressions are defined in an inductive way: Let X = {x,y, z. . . .} be a (countable) set of

5.7 Boolean Alaebras and Switch Alaebra 343

Boolean varzables (which can take values only from ( 0 , l ) ) : 1. The constants 0 and 1just as the Boolean variables from A ' are Boolean expressions

(5.264)

2. If S and T are Boolean expressions, so are T , (Sn T ) .and (S U T ) ,as well. (5.265) If a Boolean expression contains the variables 21,. . . , z,,then it represents an wary Boolean function fT

Let b be a "valuation" of the Boolean variables zl.. . . ~z,, Le., b = ( b l , . . . , b,) E B". We assign a Boolean function to the expression T in the following way: 1. If T = 0, then f~ = fo: if T = 1 , then f~ = j l .

2. If T = z,,then fT(b)= b,; if T = 3,then fT(b)= fso. 3. If T = R n S,then f T ( b ) = fR(b) fl js(b). 4. If 7 = R U S.then fr(b) = fR(b) U f s ( b ) .

(5.266a) (5.266b) (5.266~) (5.266d)

On the other hand, every Boolean function f can be represented by a Boolean expression T (see 5.7.6). 3. Concurrent or Semantically Equivalent Boolean Expressions The Boolean expressions S and T are called concurrent or semantically equivalent if they represent the same Boolean function. Boolean expressions are equal if and only if they can be transformed into each other according to the axioms of a Boolean algebra. Under transformations of a Boolean expression we consider especially two aspects: Transformation in a possible "simple" form (see 5.7.7). Transformation in a "normal form'!.

5.7.6 Normal Forms 1. Elementary Conjunction, Elementary Disjunction Let B = ( B ,n. 0 , l ) be a Boolean algebra and {q, . . . ,z,} a set of Boolean variables. Every conjunction or disjunction in which every variable or its negation occurs exactly once is called an elementary conjunction or an elementary disjunction respectively (in the variables 51,. . . , z,). Let T ( r l ., , , 2,) be a Boolean expression. A disjunction D of elementary conjunctions nith D = T i s called a principal disjunctive normal form (PDNF) of T . .A conjunction C of elementary disjunctions with C = T i s called a principal conjunctive normalform (PCNF) of T . IPart 1: In order to show that every Boolean function f can be represented as a Boolean expression, we construct the PDNF form of the function f given in the annexed table: I y z f(x,y I z ) The PDNF of the Boolean function f contains the elementary conjunctions zn g n z , zfl g fl 2 . z fl y n z. These elementary conjunctions belong to the valuations b of the variables where the function f has the value 1. If a variable 'u has the value 1 in b, then we put 'u in the elementary conjunction, 01 10 0 0 otherwise we put is. IPart 2: The PDNF for the example of Part 1 is: (zn g n Z ) u (zn g n Z ) u (zn y n 2 ) . (5.267) 1 0 1 1 1 1 0 l The "dual" form for PDNF is the PCNF: The elementary disjunctions belong t o the valiations b of the variables for which f has the value 0. 1 1 11 0 If a variable 'u has the value 0 in b, then we put u in the elementary disjunction. otherwise D. So the P C h F is: (5.268) ( z u y u z ) n ( z u g u z ) n (zuguz) n ( T U Y U Z )n(zugu2). The PDNF and the PCNF off are uniquely determined, if the orderingof the variables and the ordering of the valuations is given, e.g., if we consider the valuations as binary numbers and we arrange them in increasing order. ~

.:_

344

5. Alqebra and Discrete Mathematics

2. Principal Normal Forms The principal normal form of a Boolean function f~ is considered as the principal normal form of the corresponding Boolean expression T . To check the equivalence of two Boolean expression by transformations often causes difficulties. The principal normal forms are useful: Two Boolean expressions are semantically equivalent exactly if their corresponding uniquely determined principal normal forms are identical letter by letter. W Part 3: In the considered example (see Part 1 and 2 ) the expressions ( g n z ) U (z n y fl z ) and (zU ((y U z ) n (y U z ) n ( g U F ) ) ) n ( E U ((y U z ) n (g U 2))) are semantically equivalent because the principal disjunctive (or conjunctive) normal forms of both are the same.

5.7.7 Switch Algebra A typical application of Boolean algebra is the simplification of series-parallel connections (SPC). We assign a Boolean expression to a SPC (transformation). This expression will be “simplified” with the transformation rules of Boolean algebra. Finally we assign a SPC to this expression (inverse transformation). As a result, we get a simplified SPC which produces the same behavior as the initial connection system (Fig. 5.18). A SPC has two types of contact points: the so-called “make contacts” and “break contacts”, and both types have two states: namely open or closed. Here we consider the usual symbolism: When the equip ment is put on, the make contacts close and the break contacts open. We assign Boolean variables to the contacts of the switch equipment.

transformation (modelling)

I

Boolean expression

I

sim lification by Booreanalgebra

.

inverse transformation

Figure 5.18 The position .‘off’or “on” of the equipment corresponds to the value 0 or 1of the Boolean variables. The contacts being switched by the same equipment are denoted by the same symbol, the Boolean variable belonging to this equipment. The contact value of a SPC is 0 or 1, according to whether the switch is electrically non-conducting or conducting. The contact value depends on the position of the contacts, so it is a Boolean function S (swztchfunctzon) of the variables assigned to the switch equipment. Contacts, connections, symbols, and the corresponding Boolean expressions are represented in Fig. 5.19. The Boolean expressions, which represent switch functions of SPC, have the special property that the negation sign can occur only above variables (never over subexpressions). ISimplify the SPC of Fig. 5.20. This connection corresponds to the Boolean expression S = (5n b) u ( a n b n E ) u (a n ( b u c)) (5.269) as switch function. According to the transformation formulas of Boolean algebra we get: S = (bn ( a U ( a n F ) ) ) u( a n ( b u c ) ) = ( b n (a u E ) ) u (a n ( b u c))

(~nb)u(bnc)u(an~) = (an b n C) u (an b n F) u ( b n E ) u ( a n b n F) u (a n c) u (a n T; n C) = (anc)u(bnz). =

(5.270)

5.7 Boolean Aloebras and Switch Alaebra 345

make contact (symbol:

series connection

Ja )

(symbol < )/-

break contact

parallel connection )

(symbol:

a

S = anb

S=a

(symbol:

S=%

€53) b S=aub

Figure 5.19

Figure 5.20

Figure 5.21

Here we get a n c from (an b n c) u (Tinc) u (anbnc ) and ~ bnzfrom (Tinbnz) u (b nz) u ( a n bnz). Finally we have the simplified SPC shown in Fig. 5.21. This example shows that usually it is not so easy to get the simplest Boolean expression by transformations. In the literature we can find different methods for this procedure.

346

5. Aloebra and Discrete Mathematics

5.8 Algorithms of Graph Theory Graph theory is a field in discrete mathematics having special importance for informatics, e.g.. for representing data structures. finite automata, communication networks, derivatives in formal languages, etc. Besides this there are applications in physics, chemistry, electrotechnics, biology and psychology. Moreover. flows can be applied in transport networks and in network analysis in operations research and in combinatorial optimization.

5.8.1 Basic Notions and Notation 1. Undirected and Directed Graphs 4 graph G is an ordered pair (V, E ) of a set V of vertices and a set E of edges. There is a mapping, defined on E : the incidence function, which uniquely assigns to every element of E an ordered or nonordered pair of (not necessarily distinct) elements of V. If a non-ordered pair is assigned then G is called an undirected graph (Fig. 5.22). If an ordered pair is assigned to every element of E , then the graph is called a directed graph (Fig. 5.23), and the elements of E are called arcs or directed edges. All other graphs are called mized graphs. In the graphical representation, the vertices of a graph are denoted by points, the directed edges by arrows. and undirected edges by non-directed lines.

Figure 5.22

Figure 5.23

Figure 5.24

IA: For the graph G in Fig. 5.24: V = {vll v 2 ,v3, v4. v5}> E = { e l , e2, e3. e4: e5. e6, e,}, f l ( e 1 ) = { ~ I , v z } , f l ( e 2 ) = { U I , V Z ) , h ( e 3 ) = ( 2 ' 2 , % ) , f l ( e 4 ) = (vg,v4), fl(e5) = (v3,u4): h ( e d = (04.v2). f~(e,)= ( ~ 5 ~ ~ 5 ) . W B: For the graph G in Fig. 5.23: V = {ultv2, v 3 ,v4, u5}: E' = { e ; , e;, e;: e;} fz(e'J = (vZ.'h), f d e 4 = ( 2 ' 4 . 4 , f i ( 4= ( ~ 4 : v 2 )fdek) , = (~5~2.5). IC: For the graph G in Fig. 5.22: V = { u l , u2, v3, vqI~ 5 ) ) E" = {e:, e,; e$,e:}, f 3 ( 4 = ( 1 ~ 2 ,~ 3 1 ,h ( e ; l ) = ( ~ 4031, ) f3(e%) = {u4,v2}: f 3 ( 4 = I 1 ~ 5 . d . 2. Adjacency If (v.w ) E E . then the vertex v is said to be adjacent to the vertex 20. Vertex u is called the initial point of (v. w).w is called the terminal point of (v, w), and v and w are called the endpoints of (v, w). Adjacency in undirected graphs and the endpoints of undirected edges are defined analogously.

3. Simple Graphs If several edges or arcs are assigned to the same ordered or non-ordered pairs of vertices, then they are called multiple edges. An edge with identical endpoints is called a loop. Graphs without loops and multiple edges and multiple arcs, respectively, are called simple graphs. 4. Degrees of Vertices The number of edges or arcs incident to a vertex v is called the degree &(v) of the vertex v. Loops are counted twice. Vertices of degree zero are called isolated vertices. For every vertex 2: of a directed graph G, the out-degree d:(v) and in-degree dG(v) of v are distinguished as follows: d & ( u ) = l{wl(v.w) E E}I, (5.271a) dG(v) = I{wl(wlv) E E}I. (5.271b)

5.8 Alaorithms of Graph Theorv 347 5 . Special Classes of Graphs Finite graphs have a finite set of vertices and a finite set of edges. Otherwise the graph is said to be infinite. In regular graphs of degree r every vertex has degree T . An undirected simple graph with vertex set V is called a complete graph if any two different vertices in L’ are connected by an edge. A complete graph with an n-element set of vertices is denoted by K,. If the set of vertices of an undirected simple graph G can be partitioned into two disjoint classes A‘ and Y such that every edge of G joins a vertex of X and a vertex of Y , then G is called a bipartite graph. A bipartite graph is called a complete bipartite graph, if every vertex of X is joined by an edge with every vertex of Y. If X has n elements and Y has m elements. then the graph is denoted by Kn,m. IFig. 5.25 shows a complete graph with five vertices. IFig. 5.26 shows a complete bipartite graph with a two-element set X and a three-element set Y.

Figure 5.25

Figure 5.26

Further special classes of graphs are plane graphs, trees and transport networks. Their properties will be discussed in later paragraphs.

6. Representation of Graphs Finite graphs can be visualized by assigning to every vertex a point in the plane and connecting two points by a directed or undirected curve, if the graph has the corresponding edge. There are examples in Fig. 5.27-5.30. Fig. 5.30 shows the Petersen graph, which is a well-known counterexample for several graph-theoretic conjectures, which could not be proved in general.

Figure 5.27

Figure 5.28

Figure 5.29

Figure 5.30

7. Isomorphism of Graphs .k graph GI = (\TI, El) is said to be isomorphic to a graph G1 = (Vz,E*) iff there are bijective mappings p from 1; onto V2 and v from El onto E2 being compatible with the incidence function, Le., if u, u are the endpoints of an edge or u is the initial point of an arc and u is its terminal point, then p(u) and y ( v ) are the endpoints of an edge and ~ ( uis) the initial point and ;p(u) the terminal point of an arc, respectively. Fig. 5.31 and Fig. 5.32 show two isomorphic graphs. The mapping p with ( ~ ( 1=) a , ( ~ ( 2 = ) b, ( ~ ( 3 = ) c, ~ ( 4 = ) d is an isomorphism. In this case, every bijective mapping of { 1! 2,3,4} onto { a . b, e, d } is an isomorphism, since both graphs are complete graphs with equal number of vertices.

8. Subgraphs, Factors If G = (V)E ) is a graph, then the graph G’ = (V’,E’) is called a subgraphof G, if V’ V and E’ C E . If E’ contains exactly those edges of E which connect vertices of V’,then G’ is called the subgraph ojG induced b y 1” (induced subgraph).

348

5. Aloebra and Discrete Mathematics

A subgraph G’ = ( V ,E’) of G = (V, E ) with V’ = V is called a partial graph of G A factor F of a graph G is a regular subgraph of G containing all vertices of G.

4m: 1

Figure 5.31

a

Ab Figure 5.32

9. Adjacency Matrix Finite graphs can be described by matrices: Let G = (V, E ) be a graph with V = { V I , vz, . . . ,v,} and E = {el, ez,. . . ,em}. Let m(v,,v,) denote the number of edges from ut to v,. For undirected graphs, loops are counted twice: for directed graphs loops are counted once. The matrix A of type (n,n) with A = (m(v,,v,)) is called an adjacency matrix. If in addition the graph is simple, then the adjacency matrix has the following form:

(5.272) i.e.. in the matrix A there is a 1 in the i-th row and j-th column iff there is an edge from v, to vJ. The adjacency matrix of undirected graphs is symmetric. IA: Beside Fig. 5.33 there is the adjacency matrix A(G1) of the directed graph GI. IB: Beside Fig. 5.34 there is the adjacency matrix A(G2) of the undirected simple graph Gz.

/ Az=

010101 101010 010101 101010 010101 101010

I I

Figure 5.33 Figure 5.34 10. Incidence Matrix For an undirected graph G = (V,E ) with V = { V I , v 2 , . . . , v,} and E = {el, ez, . . . ,em}, the matrix I of type ( n ,m) given by

0, u, is not incident withe,, (5.273) 1, v,is incident withe, ande, is not a loop, 2, v,is incident with e, ande, is a loop is called the incidence matrix. For a directed graph G = (V,E ) with V = {wI, 212,. . . , v,} and E = {el,ez,. . .,em}, the incidence matrix I is the matrix of type (n,m ) ,defined by 0, viis not incident withe,, v, is the initial point ofej and e j is not a loop, (5.274) I = (b!,) = -1.1, v, is the terminal point ofe, and e, is not a loop, -0, vi is incident to e? and ej is a loop.

[

11. Weighted Graphs If G = ( yE ) is a graph and f is a mapping assigning a real number to every edge, then (V,E , f) is called a weighted graph, and f(e) is the weight or length of the edge e.

5.8 Algorithms of Graph Theory 349

In applications, these weights of the edges represent costs resulting from the construction, maintenance or use of the connections.

5.8.2 Traverse of Undirected Graphs 5.8.2.1 Edge Sequences or Paths 1. Edge Sequences or Paths

of the eleIn an undirected graph G = (V,E ) every sequence F = ({ulI u z } , {vz, 213). . . . , { u s ) ments of E is called an edge sequence of length s. If u1 = c,+~,then the sequence is called a cycle, otherwise it is an open edge vz, . . . , v, are pairwise sequence. An edge sequence F is called a path iff distinct vertices. A closedpath is a circuit. A truilis a sequence of edges with- 5 2 out repeated edges. IIn the graphs in Fig. 5.35, F1 = ({1,2},{2,3}){3,5},{5)2},{214}) is an edge sequence of length 5, FZ = ({1,2},{2;3},{3,4},{4,2},{2)1}) is a cycle of length 5$ F3 = ({2)3}, {3,5}, {5,2}, {2,1}) is a path, F4 = 4 3 ({1,2}, {2.3}, {3,4}) is a path. An elementary cycle is given by Fs = Figure 5.35 (11% 21.I2.51, I5,11).

fa

2. Connected Graphs, Components If there is at least one path between every pair of distinct vertices w,w in a graph G, then G is said to be connected. If a graph G is not connected, it can be decomposed into components, Le., into induced connected subgraphs with maximal number of vertices.

3. Distance Between Vertices The distance b(v. w)between two vertices w,w of an undirected graph is the length of a path with minimum number of edges connecting v and w.If such a path does not exist, then let b(v, w)= m. 4. Problem of Shortest Paths Let G = (V,E , f) be a weighted simple graph with f ( e ) > 0 for every e E E . Determine the shortest path from u to w for two vertices v,w of G, Le., a path from u to w having minimum sum of weights of edges and arcs, respectively. There is an efficient algorithm of Dantzig to solve this problem, which is formulated for directed graphs and can be used for undirected graphs (see 5.8.6, p. 355) in a similar way. Every graph G = (V)E , f) with V = { v l , v ~., . . . v,} has a distance rnatrizD of type ( n , n ) : D = ( d t j ) with dij = b(u,,vj) ( i . j = 1 , 2 , .. . , n). (5.275) In the case that every edge has weight 1, Le., the distance between u and w is equal to the minimum number of edges which have to be traversed in the graph to get from v to w ,then the distance between two vertices can be determined using the adjacency matrix: Let vl, v2,. . . , u, be the vertices of G. The adjacency matrix of G is A = (aij),and the powers of the adjacency matrix with respect to the usual multiplication of matrices (see 4.1.4, 5., p. 254) are denoted by Am = ( a t ) ) m E W. There is a shortest path of length k from the vertex ui to the vertex wj (i # j ) iff a:j

#

0 and u b = 0 ( s = 1,2, . . . ,k

-

1).

(5.276)

W The weighted graph represented in Fig. 5.36 has the distance matrix D beside it. IThe graph represented in Fig. 5.37 has the adjacency matrix A beside it, and for m = 2 or m = 3 the matrices AZand A3 are obtained. Shortest paths of length 2 connect the vertices 1 and 3 , l and 4, 1 and 5 , 2 and 6 , 3 and 4 , 3 and 5 , 4 and 5. Furthermore the shortest paths between the vertices 1 and 6, 3 and 6,and finally 4 and 6 are of length 3.

350

5. Alqebra and Discrete Mathematics

/ O 2 3 5

D=

4/ \ 5

010000 A=l 010000

633\

2 0 1 3 4 x 3 1 0 2 3 f f i 0 3 2 0 1 f f i

111110

I b = Ill1110 I b =I151111

I

Figure 5.37

5.8.2.2 Euler Trails 1. Euler Trail, Euler Graph .4 trail containing every edge of a graph G is called an open or closed E d e r trail of G. A connected graph containing a closed Euler trail is an Euler graph. IThe graph GI(Fig. 5.38) has no Euler trail. The graph Gz (Fig. 5.39) has an Euler trail, but it is not an Euler graph. The graph G3 (Fig. 5.40) has a closed Euler trail, but it is not an Euler graph. The graph G4 (Fig. 5.41) is an Euler graph.

Figure 5.38

Figure 5.39

Figure 5.40

Figure 5.41

2. Theorem of Euler-Hierholzer .A finite connected graph is an Euler graph iff all vertices have positive even degrees. 3. Construction of a Closed Euler Trail If G is an Euler graph, then choose an arbitrary vertex u1 of G and construct a trail F1, starting at u l r which cannot be continued. If F1 does not contain all edges of G, then construct another path Fz induced by edges not in Fl starting at a vertex v2 E F l , until it cannot be continued. Compose a closed trail in G using Fl and Fz: Start traversing Fl at u1 until u2 is reached, then continue by traversing Fz, and finish by traversing the edges of F, not used before. Repeating this method a closed Euler trail is obtained in finitely many steps.

4. Open Euler Trails There is an open Euler trail in a graph G iff there are exactly two vertices in G with odd degrees. Fig. 5.42 shows a graph which has no closed Euler trail, but it has an open Euler trail. The edges are consecutively enumerated with respect to an Euler trail. In Fig. 5.43 there is a graph with a closed Euler trail 5.

Chinese Postman Problem

The problem. that a postman should pass through all streets in his service area at least once and return to the initial point and use a trail as short as possible, can be formulated in graph theoretical terms as

5.8 Algorithms of Graph Theorv 351

Figure 5.42

Figure 5.43

follows: Let G = (I,.E , f ) be a weighted graph with f(e) sequence F with minimum total length

L=

Figure 5.44

2 0 for every edge e E E . Determine an edge

1! ( e ) .

(5.277)

eEF

The name of the problem refers to the Chinese mathematician Kuan, who studied this problem first. To solve it two cases are distinguished: 1. G is an Euler graph - then every closed Euler trail is optimal and 2. G has no closed Euler trail. An effective algorithm solving this problem is given by Edmonds and Johnson (see [5.22]). ~

5.8.2.3 Hamiltonian Cycles 1. Hamiltonian Cycle .4 Hamiltonian cycle is an elementary cycle in a graph covering all of the vertices. In Fig. 5.44, lines in bold face show a Hamiltonian cycle. The idea of a game to constructHamiltonian cycles in the graph of a pentagondodecaeder. goes back to Sir \V.Hamilton. Remark: The problem of characterizing graphs with Hamiltonian cycles leads to one of the classical NP-complete problems. Therefore, an efficient algorithm to determine the Hamilton cycles cannot be given here.

2. Theorem of Dirac If a simple graph G = (\/’,E) has at least three vertices, and d G ( u ) 2 Ic‘1/2 holds for every vertex u of G, then G has a Hamiltonian cycle. This is a sufficient but not a necessary condition for the existence of Hamiltonian cycles. The following theorems with more general assumptions give only sufficient but not necessary conditions for the existence of Hamilton cycles, too.

e Figure 5.45

Fig. 5.45 shows a graph which has a Hamiltonian cycle. but does not satisfy the assumptions of the following theorem of Ore.

3. Theorem of Ore

If a simple graph G = (\I.E ) has at least three vertices. and d c ( u ) + d G ( W ) 2 IVl holds for every pair of non-adjacent vertices u , w:then G contains a Hamiltonian cycle.

4. Theorem of Posa Let G = (V,E ) be a simple graph with at least three vertices. There is a Hamiltonian cycle in G if the following conditions are satisfied:

1. For 1 5 k < (11’1 - 1)/2, the number of vertices of degree not exceeding k is less than k. 2. If 111 . is odd. then the number of vertices of degree not exceeding (Ib’i - l ) / 2 is less than or equal to (IVI - l ) / 2 .

352

5. Alaebra and Discrete Matherncitics

$ Figure 5.46

Q

grandchildren

Figure 5.47

great-grandchildren Figure 5.48

5.8.3 Trees and Spanning Trees

5.8.3.1 Trees 1. Trees An undirected connected graph without cycles is called a tree. Every tree with at least two vertices has at least two vertices of degree 1. Everv tree with n vertices has exactlv n - 1 edges. A directed graph is calledua tree if G is connected and does not contak any circiit (see 5.8.6,p. 355). IFig. 5.46 and Fig. 5.47 represent two non-isomorphic trees with 14 vertices. They demonstrate the chemical structure of butane and isobutane.

2. Rooted Trees .A tree with a distinguished vertex is called a rooted tree, and the distinguished vertex is called the root. In diagrams, the root is usually on the top, and the edges are directed downwards from the root (see Fig. 5.48). Rooted trees are used to represent hierarchic structures, as for instance hierarchies in factories, family trees, grammatical structures. IFig. 5.48 shows the genealogy of a family in the form of a rooted tree. The root is the vertex assigned to the father.

3. Regular Binary Trees If a tree has exactly one vertex of degree 2 and otherwise only vertices of degree 1 or 3, then it is called a regular binary tree. The number of vertices of a regular binary tree is odd. Regular trees with n vertices have (n + 1)/2 vertices of degree 1. The level of a vertex is its distance from the root. The maximal level occurring in a tree is the height of the tree. There are several applications of regular binary rooted trees, e.g., in informatics.

4. Ordered Binary Trees .Arithmetical expressions can be represented by binary trees. Here, the numbers and variables are assigned vertices of degree 1, the operations "+","-" ".'' correspond to vertices of degree > 1, and the left and right subtree, respectively, represents the first and second operand, respectively, which is, in general, also an expression. These trees are called ordered binary trees. The traverse of an ordered binary tree can be performed in three different ways, which are defined in a recursive way (see also Fig. 5.49): Traverse the left subtree of the root (in inorder traverse), Inorder traverse: visit the root, traverse the right subtree of the root (in inorder traverse). Preorder traverse: Visit the root, traverse the left subtree (in preorder traverse), traverse the right subtree of the root (in preorder traverse). POStoTdeT traverse: Traverse the left subtree of the root (in postorder traverse), traverse the right subtree of the root (in postorder traverse), visit the root.

5.8 Aluorzthms of Graph Theoru 353

Using inorder traverse the order of the terms does not change in comparision with the given expression. The term obtained by postorder traverse is called postfix notation PN or Polish notation. .4nalogously, the term obtained by preorder traverse is called prefiz notation or reversed Polish notation. Prefix and postfix expressions uniquely describe the tree. This fact can be used for the implementation of trees. IIn Fig. 5.49 the term a . ( b - e ) +d is represented by a graph. Inorder traverse yields a . b - c + d , preorder traverse yields . -bcad, and postorder traversal yields abc - .d+.

+

5.8.3.2 Spanning Trees 1. Spanning Trees A tree: being a subgraph of an undirected graph G, and containing all vertices of G,is called a spanning tree of G. Every finite connected graph G contains a spanning tree H : If G contains a cycle, then delete an edge of this cycle. The remaining graph GI is still connected and can be transformed into a connected graph Gz by deleting a further edge of a cycle of GI. if there exists such an edge. After finitely many steps a spanning tree of G is obtained. IFig. 5.51 shows a spanning tree H of the graph G shown in Fig. 5.50.

2

0

Figure 5.49

3 Figure 5.50

4

Figure 5.51

2. Theorem of Cayley

Every complete graph with n vertices ( n > 1) has exactly nn-' spanning trees. 3. Matrix Spanning Tree Theorem Let G = (V. E ) be a graph with V = {q,v 2 , . . . , v,} ( n > 1) and E = {el, e2,. . . ,em}. Define a matrix D = ( d Z j )of type (n,n ) : 0 for i # j , (5.278a) dlJ = d G ( u i ) for i = j : which is called the degree matrzz. The difference between the degree matrix and the adjacency matrix is the admittance matrix L of G: L=D-A. (5.278b) Deleting the z-th row and the i-th column of L the matrix L, is obtained. The determinant of L, is equal to the number of spanning trees of the graph G. W The adjacency matrix, the degree matrix and the admittance matrix of the graph in Fig. 5.50 are: 2-1-1 0 4000 2110 -1 3 - 2 0 0300 A= 0 0-1 1 0001 0010 Since detL, = 5, the graph has five spanning trees.

[

q,

[;

4. Minimal Spanning Trees Let G = (V> E , f ) be a connected weighted graph. A spanning tree H of G is called a minimum spanning tree if its total length f ( H ) is minimum: f(H)= f(e). (5.279)

1

eEH

Minimum spanning trees are searched for, e.g., if the edge weights represent costs, and one is interested in minimum costs. A method to find a minimum spanning tree is the Kruskal algorithm: a) Choose an edge with the least weight.

354

5. Algebra and Discrete Mathematics

b) Continue. as long as it is possible, choosing a further edge having least weight and not forming a cycle with the edges already chosen, and add such an edge to the tree. In step b) the choice of the admissible edges can be made easier by the following labeling algorithm: After adding an edge. the label of the endpoint with the larger label is changed to the value of the smaller endpoint label.

5.8.4 Matchings 1. Matchings

X set AI of edges of a graph G is called a matching in G, iff M contains no loop and two different edges of M do not have common endpoints. A matching M*of G is called a saturated matching, if there is no matching M in G such that M' C M. A matching M" of G is called a maximum matching, if there is no matching M in G such that lM1 > 1M**1. If AI is a matching of G such that every vertex of G is an endpoint of an edge of M, then M is called a 1 perfect matching. IIn the graph in Fig. 5.52 MI = {{2,3}, (5.6)) is a saturated matching and Mz = { { 1,2}, {3,4}, { 5 , 6 } } is a maximum matching 5 which is also perfect. Remark: In graphs with an odd number of edges there is no perfect Figure 5.52 matching.

b-4

2. Theorem of Tutte Let q(G - S) denote the number of the components of G - S with an odd number of vertices. A graph G = (\',E) hasaperfectmatchingiff /VI isevenandforeverysubset Softhevertexsetq(G-S) 5 /SI. Here G- Sdenotes the graph obtained from G by deleting the vertices of S and the edges incident with these vertices. Perfect machings exist for example in complete graphs with an even number of vertices, in complete bipartite graphs K,,, and in arbitrary regular bipartite graphs of degree r > 0.

3. Alternating Paths Let G be a graph with a matching M ..A path W in G is called an alternating path iff in W every edge e with E E ,If (or e # M)is followed by an edge e' with e' # M (or e E M). An open alternating path is called an increasing path iff none of the endpoints of the path is incident with an edge of M.

4. Theorem of Berge .4matching .Vin a graph G is maximum iff there is no increasing alternating path in G. If W is an increasing alternating path in G with corresponding set E ( W ) of traversed edges, then M' = ('2.1 \ E(\$')) U ( E ( W )\ M )forms a matching in G with IM'I = I MI+ 1. IIn the graph of Fig. 5.52 ({1.2}) {2.3}, ( 3 . 4 ) ) is an increasing alternating path with respect to matching M1. Matching ~11 .2with lMzl = lM11 + 1 is obtained as described above. 5. Determination of Maximum Matchings Let G be a graph with a matching M. a) First form a saturated matching M*with M M'. b) Chose a vertex u in G. which is not incident with an edge of M', and determine an increasing alternating path in G starting at u. c) If such a path exists, then the method described above results in a matching M' with \M'/ > \M*\. If there is no such path. then delete vertex 2: and all edges incident with u in G, and repeat step b).

5.8 Alaorithms of Graph Theoru 355

There is an algorithm of Edmonds, which is an effective method to search for maximum matchings, but it is rather complicated to describe (see [5.21]).

5.8.5 Planar Graphs Here, the considerations are restricted to undirected graphs, since a directed graph is planar iff the corresponding undirected graph is a planar one.

4lQ Figure ,553

Figure 5.54

1. Planar Graph .A graph is called a plane graph iff G can be d r a m in the plane with its edges intersecting only in vertices of G. A graph isomorphic with a plane graph is called a planar graph. Fig. 5.53 shows a plane graph GI.The graph G:, in Fig. 5.54 is isomorphic to GI,it is not a plane graph but a planar graph, since it is isomorphic with GI.

2. Non-Planar Graphs The complete graph K5 and the complete bipartite graph K3,3 are non-planar graphs (see 5.8.1! 5.. p. 347).

3. Subdivisions A subdivision of a graph G is obtained if vertices of degree 2 are inserted into edges of G. Every graph is a subdivision of itself. Certain subdivisions of K5 and K3,3 are represented in Fig. 5.55 and Fig. 5.56.

4. Kuratowski's Theorem .4 graph is non-planar iff it contains a subgraph which is asubdivision either of the complete bipartite graph K3.3or of the complete graph K5.

@ Figure 5.55

Figure 5.56

5.8.6 Paths in Directed Graphs 1. Arc Sequences A sequence F = (el:eZ1

e,) of arcs in a directed graph is called a chain of length s. iff F does not contain any arc twice and one of the endpoints of every arc ei for i = 2,3,. . . , s - 1 is an endpoint of the arc eL-l and the other one an endpoint of e,+l. A chain is called a directed chain iff for i = 1 , 2 , . . . , s - 1 the terminal point of the arc ei coincides with the initial point of ei+l. Chains or directed chains traversing every vertex at most once are called elementary chains and elementary directed chains, respectively. A closed chain is called a cycle. A closed directed path, with every vertex being the endpoint of exactly two arcs, is called a circuit. IFig. 5.57 contains examples for various kinds of arc sequences.

2. Connected and Strongly Connected Graphs A directed graph G is called connectedifffor any two vertices there is a chain connecting these vertices. The graph G is said to be strongly connectediffto every two vertices v , w there is is assigned a directed chain connecting these vertices.

3. Algorithm of Dantzig Let G = (I,,E , f ) be a weighted simple directed graph with f ( e ) > 0 for every arc e. The following algorithm yields all vertices of G, which are connected with a fixed vertex v l by a directed chain, together with their distances from V I : a) \'ertex v1 gets the label t ( q ) = 0. Let

S1 = {VI}.

356

5. Alqebra and Discrete Mathematics

chain

directed chain

elementary chain

elementary directed chain

cycle

circuit

Figure 5.57

b) Denote the set of the labeled vertices by S,. c) If C;, = {ele = (ut. vj) E E , v, E ,S , uj # S,} = 0, then finish the algorithm. d ) Otherwise choose an arc e' = (z*,y*) with minimum t(z') + f(e*). Label e* and y'. We set t ( y * ) = t(z*)+ f(e*) and also S ,+l = S, U {y'}, and repeat b) with m := m + 1. (If all arcs have weight 1, then the length of a shortest directed chain from a vertex v to a vertex w can be found using the adjacency matrix (see 5.8.2.1, 4.) p. 349)). V

VIO

'13

Figure 5.58

If a vertex u of G is not labeled, then there is no directed path from u1 to u. If v has label t(v), then t ( u ) is the length of such a directed chain. A shortest directed path from v1 to u can be found in the tree given by the labeled arcs and vertices, the distance tree with respect to VI. IIn Fig. 5.58, the labeled arcs and vertices represent the distance tree with respect to vl in the graph. The lengths of the shortest directed chains are: from u1 to v3 : 2 from v1 to u6 : 7 from u1 to v7 : 3 from v1 to vg : 7 from wl to 219 : 3 from v1 to 2114 : 8 from v1 to w2 : 4 from to v5 : 8 from v1 to 2110 : 5 from v1 to v12 : 9 from v1 to 214 : 6 from v1 to 2'13 : 10 fromvl tovll : 6. Remark: There is also a modified algorithm to find the shortest directed chains in the case that G = (V,E , f ) has arcs with negative weights.

5.8.7 Transport Networks 1. Transport Network .4connected directed graph is called a transport networkif it has two labeled vertices, called the source Q and sznk S which have the following properties: a) There is an arc u1 from S to Q,where u1 is the only arc with initial point S and the only arc with terminal point Q. b) Every arc u, different from u1 is assigned a real number c(uz)2 0. This number is called its capaczty. The arc u1 has capacity m. A function p,which assigns a real number to every arc, is called a pow on G, if the equality (5.280a)

5.8 Alaorzthrns

of

Graph Theorv 357

holds for every vertex u. The sum

E

P(Q;u)

(5.280b)

(Q.v)EG

is called the intensity of the flow. A flow 'p is called compatible to the capacities. if for every arc ui of G 0 5 p(l-1,)5 c ( u ~holds. ) For an example of a transport network see p. 357.

2. Maximum Flow Algorithm of Ford and hlkerson Using the maximum flow algorithm one can recognize whether a given flow y is maximal. Let G be a transport network and 'p aflow ofintensity u1 compatible with the capacities. The algorithm given below contains the following steps for labeling the vertices, and after finishing this procedure one can recognize how much the intensity of the flow could be improved depending on the chosen labeling steps. a) Label the source Q and set E ( Q ) = co. b) If there is an arc u, = (2, y) with labeled z and unlabeled y and p(zii) < c(u,), then label y and (z. y). and set ~ ( y = ) min{&(z)>c(u,) - 'p(u,)}, then repeat step b), otherwise follows step c). c) If there is an arc u, = (z. y) with unlabeled z and labeled y , ~ ( 2 1 %>) 0 and u, # u l r then we label z and (z, y), substitute ~ ( z=) min{e(y), 'p(u,)} and return to continue step b) if it is possible. Otherwise we finish the algorithm. If the sink S of G is labeled, then the flow in G can be improved by an amount of E(S).If the sink is not labeled, then the flow is maximal. Ihlaximum flow: For the graph in Fig. 5.59 the weights are written next to the edges. A flow with intensity 13, compatible to these capacities, is represented in the weighted graph in Fig. 5.60. It is a maximum flow

S

Figure 5.59

Figure 5.60

Transport network: A product is produced by p firms F l . F 2 , ,, ..Fp. There are q users Si,\!;%.. . V,. During a certain period there will be s, units produced by F, and t j units required by V,. c , ~units can be transported from F, to V, during the given period. Is it possible to satisfy all the requirements during this period? The corresponding graph is shown in Fig. 5.61.

.

Figure 5.61

I

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5. Aloebra and Discrete Mathematics

21

We can also write

PA(X1)

1x2

I PA(z2) 1

... ,'

'

1

2,

PA(%)

(5.285) In (5,285) the fraction bars and addition signs have only symbolic meaning. (E5)Ultra-fuzzy set: A fuzzy set. whose membership function itself is a fuzzy set, is called, after Zadeh, an ultra-fuzzy set.

5.9 Fuzzy Logic 359

3. Fuzzy Linguistics If we assign linguistic values, e.g., "small", "medium" or "big", to a quantity then we call it a linguistic quantity or linguistic variable. Every linguistic value can be described by a fuzzy set, for example, by the graph of a membership function (5.9.1.2) with a given support (5.283). The number of fuzzy sets (in the case of "small". "medium". "big" they are three) depends on the problem. In 5.9.1.2 the linguistic variable is denoted by 2. For example, I can have linguistic values for temperature, pressure. volume, frequency, velocity. brightness. age. wearing, etc., and also medical. electrical, chemical, ecological. etc. variables. IBy the membership function p ~ ( 5of) a linguistic variable. the membership degree of a fixed (crisp) value can be determined in the fuzzy set represented by p ~ ( 2 ) Namely, . the modeling of a "high" quantity, e.g.. the temperature, as a linguistic variable given by a trapezoidal membership function (Fig. 5.62) means that the given temperature a belongs to the fuzzy set "high temperature" with the degree of membership 3 (also degree of compatibility or degree of truth).

5.9.1.2 Membership Functions on the Real Line The membership functions can be modeled by functions with values between 0 and 1. They represent the different grade of membership for the points of the universe being in the given set.

1. Trapezoidal Membership Functions Trapezoidal membership functions are widespread. Pieceaise (continuously differentiable) membership functions and their special cases, e.g., the triangle shape membership functions described in the followingexamples! are very often used. Connecting fuzzy quantities we get smoother output functions if the fuzzy quantities were represented by continuous or piecewise continuous membership functions. IA: Trapezoidal function (Fig. 5.62) corresponding to (5.286). The graph of this function turns into a triangle function if az = a3 = a and al < a < a4. Choosing different symmetrical or asymmetrical trapezoidal functions, a symmetrical triangle function (a2 = a3 = a and la all = la4 - al) or asymmetrical triangle function (a2 = a3 = a a n d la - all # la4 Figure 5.62

all.

IB: hIembership function bounded to the left and to the right (Fig. 5.63) corresponding to (5,287):

W)t

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5. Aloebra and Discrete Mathematics

C: Generalized trapezoidal function (Fig. 5.64) corresponding to (5.288).

Figure 5.64

2. Bell-Shaped Membership Functions A: We get a class of bell-shaped, differentiable membership functions with the function f(x)from (5.289) ifwe choose an appropriate p(x): Ifp(z) = k(z - a)(b - x ) and, e.g.. k = 10 or k = 1 or k = 0.1, then we get a family of symmetrical curves of

f ( x )=

/

lo

I a,

0

x

e-l/p(s)

a < x < b, (5.289)

x 2 b.

/

different width with the membership function p , ~ . ( x= ) f(x) f - , where 1 f - is the (a;b) (a;b) normalizing factor (Fig. 5.65). We get the exterior curve with the value k = 10 and the interior one with k = 0.1. We get asymmetrical membership functions in [0,1]for example with p ( x ) = x ( 1 - x ) ( 2 - x) or with p(x) = z(1 - x)(x t 1) (Fig. 5.66), using appropriate normalizing factors. The factor (2 - x ) in the first polynomial results in the shifting of the maximum to the left and it yields an asymmetrical curve shape. Similarly, the factor (x t 1) in the second polynomial results in a shifting to the right and in an asymmetric form.

Figure 5.65

Figure 5.66

B: We can get examples for a more flexible class of membership functions by the formula (5.290)

5.9 FzlzzvLoaic 361

where f is defined by (5.289) with p(x) = (x - a)(b - z)and t is a transformation on [a,b]. If t is a smooth transformation on [a, b], Le., if t is differentiable infinitely many times in the interval [a,b] then Ft is also smooth. since f is smooth. If we require t to be either increasing or decreasing and to be smooth, then the transformation t allows us to change the shape of the curve of the membership function. In practice, polynomials are especially suitable for transformations. The simplest polynomial is the identity t ( z ) = I on the interval [a,b] = [0,1].

+

+

The next simplest polynomial with the given properties is t ( z ) = cx2 ( 1 - 5 ) z with a constant c E [-6.31. The choice c = -6 results in the polynomial of maximum curvature, its equation is q(z) = 4z3 - 6 2 + 31. If we choose for qo the identity function, i.e., qo(z) = z.then we can get recursively further polynomialsq by theformulaq, = qoqi-1 fori E N.Substituting the corresponding polynomial transformations qor q l , . , . into (5.290) for t , we get a sequence of smooth functions FqasF,, and Fqz (Fig. 5.67), which can be considered as membership functions p ~ ( z )where , F,, converges to a line. The trapezoidal membership function can be approximated by differentiable functions using the function Fq2.its reflection and a horizontal line (Fig. 5.68).

'0

/-1

,

0.5

0.25 0.5

0.75 1 x

'0

0.25

0.5

0.75

,

,

1x

Figure 5.67 Figure 5.68 Summary: Imprecise and non-crisp information can be described by fuzzy sets and represented by membership functions p ( z ) .

5.9.1.3 Fuzzy Sets 1. Empty and Universal Fuzzy Sets A over X is called empty if p ~ ( z=) 0 Va: E X holds. b) Universal fuzzy set: .4set is called universal if p ~ ( z=) 1 Vz E X holds. 2. Fuzzy Subset If pg(a:) 5 p ~ ( zV) I E X , then B is called afuzty subset of A (we write: B C A). 3. Tolerance Interval and Spread of a Fuzzy Set on t h e Real Line If A is a fuzzy set on the real line. then the interval (5.291) [ a b ] = {z E X l p ~ ( z=) 1) ( a ,b const, a < b) is called the tolerance interval of the fuzzy set A, and the interval [e,d] = cl(suppA) (c, d const,c < d ) is called the spread of A, where cl denotes the closure of the set. (The tolerance interval is sometimes also called the peak of set A , ) The tolerance interval and the kernel coincide only if the kernel contains a) E m p t y fuzzy set: A set

more then one point. IA: In Fig. 5.62 [a2, a31 is the tolerance interval, and [al.a41 is the spread. IB: If 02 = a3 = a (Fig. 5.62), then we get a triangle-shaped membership function p. In that case the triangular fuzzy set has no tolerance, but its kernel is the set { a } . If additionally a1 = a = a4 holds, too: then we have a crisp value; it is called a singleton. A singleton A has no tolerance, but ker(A) = supp(A) = { a } .

4. Conversion of Fuzzy Sets on a Continuous and Discrete Universe Let the universe be continuous, and let a fuzzy set be given on it by its membership function. Discretizing the universe, every discrete point together with its membership value determines a fuzzy singleton.

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5. Aloebra and Discrete Mathematics

Conversely, a fuzzy set given on a discrete universe can be converted into a fuzzy set on the continuous universe by interpolating the membership value between the discrete points of the universe. 5 . Normal and Subnormal Fuzzy Sets If A is a fuzzy subset of X,then its height is defined by

H ( A ) := max{pa(z)lz E X}. (5.292) A is called a normal fuzzy set if H(A) = 1, otherwise it is aubnormal. The notions and methods represented in this paragraph are limited to normal fuzzy sets, but it easy to extend them also to subnormal fuzzy sets. 6. Cut of a Fuzzy Set The a-cut A>" or the strong a-cut A'" of a fuzzy set A are the subsets of X defined by (5.293) A>" = {T E X l p ~ ( z>) a } , A'" = {X E XIpa(5) 2 a } , a E (0:11. and A Z O = el ('4'''). The a-cut and strong a-cut are also called a-level set and strong a-level set, respectively. 1. Properties a) The a-cuts of fuzzy sets are crisp sets. b) The support supp(A) is a special a-cut: supp(A) = A>'. c) The crisp 1-cut ,4>l = {z E X I p a ( z ) = 1) is called the kernelof A. 2. Representation Theorem To every fuzzy subset A of X we can assign uniquely the families of its a-cuts (A>Q)Qc[o,l) and its strong a-cuts (A>n)QEio,il. The a-cuts and strong a-cuts are monotone families of subsets from X,since:

a < i l j A>" 2 A>' and A'" 2 (5.294a) Conversely, if there exist the monotone families (UQ)aE[o,l) or (L<)ac(o,ll of subsets from X, then there are uniquely defined fuzzy sets b' and V such that li>" = C,; and V?" = V, and moreover p&)

=

sup{a E [O, 1))IzE U"},

p v ( z ) = sup{a E (0,lIla: E Va}.

(5.294b)

7. Similarity of the Fuzzy Sets A and B 1. The fuzzy sets A. B with membership functions P A , p g : X + [0,1] are called fuzzy similar if for every a E (0.11 there exist numbers a, with cy( E (0,1];(i = 1,2) such that: supp(alpa)n C SUPP(PB),;

S U P P ( ~ Z P BG ) ~S ~ P P ( P A ) ~ .

(pc)" represents a fuzzy set with the membership function (pc)" =

represents a fuzzy set with the membership function (Bpc) =

{

(5.295)

{ F(')~

t ~ ~ a@and ) (4pc) s ~

B iotherwise, f d z )>B

2. Theorem: Two fuzzy sets A, B with membership functions p a , p g : X they have the same kernel:

+ [0,1] are fuzzy-similar if (5.296a) (5.296b)

3. A , B with pal p~ : X the same kernel:

+ [O. 11 are called strongly fuzzy-similar if they have the same support and

5.9 Fuzzy Logic 363

5.9.2 Aggregation of Fuzzy Sets Fuzzy sets can he aggregated by operators. There are several different suggestions of how to generalize the usual set operations. such as union, intersection: and complement of fuzzy sets.

5.9.2.1 Concepts for Aggregation of Fuzzy Sets 1. Fuzzy Set Union, Fuzzy Set Intersection The grade of membership of an arbitrary element z E X in the sets A U B and A n B should depend only on the grades of membership p ~ ( zand ) p g ( z ) of the element in the two fuzzy sets A and B. The union and intersection of fuzzy sets is defined with the help of two functions (5.298) s . t : [O, 11 x [O, 11 --t [O, 11, and they are defined in the following way: / J . ~ J B ( := ~ ) s ( p A ( z ) i p B ( z ) )>

(5.299)

p A n B ( z ) := t

(pA(z),pB(z))

(5.300)

) p ~ ( zare ) mapped in a new grade of membership. The functions The grades of membership p ~ ( zand t and s are called the t-norm and t-conorm; this last one is also called the s-norm. Interpretation: The functions p.4uB and p ~ represent n ~ the truth values of membership, which is resulted by the aggregation of the truth values of memberships p ~ ( zand ) pg(z).

2. Definition of the &Norm: The t-norm is a binary operation t in [0,1]: (5.301) t : [0.1] x [O, 11 + 10%11. It is symmetric, associative, monotone increasing, it has 0 as the zero element and 1 as the neutral element. For z,y , z . u , w E [O, 11 the following properties are valid: ( E l ) Commutativity: t ( z ,y) = t(y, z). (5.302a)

(E2) Associativity: t ( a ,t ( y , z ) ) = t ( t ( z y, ) , t ) .

(5.302b)

(E3) Special Operations with Neutral and Zero Elements: t ( z :1) = zand because of (El): t(1,z) = z; t ( z ,0)= t ( 0 ,z)= 0. (E4) Monotony: If z 5 1~ and y 5 20, then t ( z ,y) 5 t ( u , 20) is valid.

(5.302~) (5.302d)

3. Definition ofthe s-Norm: The s-norm is a binary function in [0,1]: (5.303) s: [0.1] x [0,1] --f [O. 11. It has the following properties: (5.304a) ( E l ) Commutativity: s(z, y) = s(y,z). (5.304b) (E2) Associativity: s(z. s ( y , z)) = s(s(z, y). z ) . (E3) Special Operations with Zero and Neutral Elements: (5.304~) s(z.0) = S(0,S) = 2 : s(z.1) = s(1,z) = 1 (5.304d) (E4) Monotony: If z 5 u and y 5 w ,then s(z, y) 5 s ( u ,w) is valid LVith the help of these properties a class T of t-norms and a class S of s-norms can be introduced. Detailed investigations proved that the following relations hold: (5.304e) min{z. y} 2 t ( r ,y) V t E T. Vz3y E [O. 11 and (5.3049 rnax{z.y} s(z,y) V s E S, Vz,y E [0.1].

I

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5. Alaebra and Discrete Mathematics

5.9.2.2 Practical Aggregator Operations of Fuzzy Sets 1. Intersection of Two Fuzzy Sets The intersection A n B of two fuzzy sets A and B is defined by the minimum operation min(., .) on their membership functions p , ~ . ( zand ) p ~ ( z )Based . on the previous requirements, we get: C := A n E and p ~ ( z:= ) min ( p ~ ( z p) g, ( z ) ) Vz E X , where: min(a, b) :=

[ a,, iiff aa >5bb:.

(5.305a) (5.305b)

The intersection operation corresponds to the AND operation of two membership functions (Fig.5.69). The membership function pc(z) is defined as the minimum value of p ~ ( zand ) p~(z).

2. Union of Two Fuzzy Sets The unzon A U B of two fuzzy sets is defined by the maximum operation max(., .) on their membership functions p ~ ( zand ) p ~ g ( z )We . get: C := A U B and pc(z) := max(pA(z), p ~ ( x ) ) Vz E X , where: (5.306a) max(a, b) :=

[ a , iiffaa
(5.306b)

~

The union corresponds to the logical OR operation. Fig.5.70 illustrates p ~ ( zas) the maximum value of the membership functions p ~ ( zand ) p~(z). IThe t-norm t ( z ,y) = min{z, y} and the s-norm s(z, y) = max{z, y} define the intersection and the union of two fuzzy sets, respectively (see (Fig.5.71) and (Fig.5.72)).

Figure 5.69

Figure 5.70

Figure 5.71

Figure 5.72

3. Further Aggregations Further aggregations are the bounded, the algebraic, and the drastic sum and also the bounded dzflerence, the algebraic and the drastic product (see Table 5.8). The algebraic sum, e.g., is defined by (5.307a) C := A + B and p c ( z ) := p,~.(z) p ~ g ( z-) p ~ ( z. )p ~ ( z )for every z E X .

+

5.9 Fzlzzw Louzc 365 Table 5.8 t- and s-norms. D E R

1 1

I s-norm

Author

t-norm

Zadeh

intersection: t ( z , y) = min{z, y}

Lukasiewicz

bounded difference

tb(z.y) = max(0,a: + y algebraic product t,(s. Y) = ZY

drastic product

union: s(z, y) = max{x, y} bounded sum

-

1)

sb(z, y) = min{l,x

+ y}

algebraic sum s a ( z , Y) = z + Y - ZY drastic sum

Hamacher (P 2 0 ) Einstein

(P > 0. P # 1)

I-

(P 2 -1) Dubois

Remark to Table 5.8: For the values of the t- and s-norms listed in the table. the following ordering is valid: (5.30713) tdp 5 t b 5 t e 5 t a 5 t h 5 t 5 s 5 s h 5 s a 5 s e 5 s b 5 sds, Similarly to the union (5.306a,b), this sum also belongs to the class of s-norms. They are included in the right-hand column of Table 5.8. In Table 5.9 is given a comparision of operations in Boolean logic and fuzzy logic Analogously to the notion of the extended sum as a union operation, the intersection can also he extended for example by the bounded, the algebraic, and the drastic product. So. e.g., the algebraic

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5. Alqebra and Discrete Mathematics

product is defined in the following way: C := A B and p c ( z ) := ~ A ( I ). p ~ ( z )for every I E X . (5.3074 It also belongs to the class of t-norms, similarly to the intersection (5.305a,b), and it can be found in the middle column of Table 5.8.

5.9.2.3 Compensatory Operators Sometimes we need operators lying between the t- and the s-norms; they are called compensatory operators. Examples for compensatory operators are the lambda and the gamma operator. 1. Lambda Operator PAAB(1) = [pA(z)pB(z)]+ (1 - A) [!JA(I) -k P B ( z ) - /1A(z)pB(z)] with E [o. 11. (5.308) Case X = 0 : Equation (5.308) results in a form known as the algebraic sum (Table 5.8, s-norms): it belongs to the OR operators. Case X = 1: Equation (5.308) results in the form known as the algebraic product (Table 5.8, t-norms); it belongs to the AND operators. 2. Gamma Operator P.ATB(Z) = [pA(z)pB(z)I1-’[I - (1 - pA(Z )) (1 - p B ( z ) ) I 7 with 7 E [o, 11. Case ;t= 1: Equation (5.309) results in the representation of the algebraic sum. Case e/= 0: Equation (5.309) results in the representation of the algebraic product. The application of the gamma operator on fuzzy sets of any numbers is given by

(5.309)

(5.310) and xvith weights 6,:

5.9.2.4 Extension Principle In the previous paragraph; we discussed the possibilities of generalizing the basic set operations for fuzzy sets. Now, we want to extend the notion of mapping on fuzzy domains. The basis of the concept is the acceptance grade of vague statements. The classical mapping @ : Xn -+ Y assigns a crisp function value @ ( X I , . . . >zn) E Y to the point (11). . . ,x,) E Xn. This mapping can be extended for which assigns a fuzzy funcfuzzy variables as follows: The fuzzy mapping is 6 : F(X)” + F(Y): tion value 6(p13.. . pn) to the fuzzy vector variables ( q , ... !zn)given by the membership functions ~

( P l . , , , , P n ) E W)”.

5.9.2.5 Fuzzy Complement A function e : [O. 11 -+ [0,1]is called a complement function if the following properties are fulfilled for Vz. y E [O. 11:

(EK1) Boundary Conditions: c(0) = 1 and c(1) = 0. (EK2) Monotony: < Y =+ e(.) 2 C b ) . (EK3) Involutivity: c(c(z))= z.

(5.312a) (5.312b) (5.312~)

c(z) should be continuous for every z E [0,1]. ( 5.312d) (EK4) Continuity: IA: The most often used complement function is (continuous and involutive): (5.313) c(z) := 1 - 2 . IB: Other continuous and involutive complements are the Sugeno complement ~ ( z:=) (1- z)(1+ Ax)-’ with X E (-1, x)and the Yager complement cp(z) := (1 - Z P ) ” ~ with p E (0: x).

5.9 Fuzzu Loqic 367

Table 5.9 Comparison of operations in Boolean logic and in fuzzy logic

c = 7.4

p: = 1 - pA

(& as complement of P A )

5.9.3 Fussy-Valued Relations 5.9.3.1 Fuzzy Relations 1. Modeling fizzy-Valued Relations Uncertain or fuzzy-valued relations, as e.g. "approximately equal", "practically larger than", or "practically smaller than", etc.. have an important role in practical applications. A relation between numbers is interpreted as a subsets of RZ.So, the equality "=" is defined as the set (5.314)

A = {(z.y) E RZIz= y } ,

I

i.e., by a straight line y = x in RZ. Modeling the relation "approximately equal" denoted by R1,we can use a fuzzy subset on R2, the kernel of which is A, and we require that the membership function should decrease and tend to zero getting far from the line A. A linear decreasing membership function can be modeled by (5.315) 1 - l (x3 ~ ~ y) = max{O. 1 - a / z - yi} with a E R. a > 0. For modeling the relation Rz "practically larger than" it is useful to start with the crisp relation '$2''. The corresponding set of values is given by ~

(5,316) It describes the crisp domain above the line x = y. The modifier "practically" means that a thin zone under the half-space in (5.316) is still acceptable with some grade. So, the model of Rz is (5.317) If the value of one of the variables is fixed, e.g., y = yo3then Rz can be interpreted as a region with uncertain boundaries for the other variable. Handling the uncertain boundaries by fuzzy relations has practical importance in fuzzy optimization, qualitative data analysis and pattern classification. The foregoing discussion shows that the concept of fuzzy relations, Le., fuzzy relations between several objects, can be described by fuzzy sets. In the following. we discuss the basic properties of binary relations over universe which consists of ordered pairs.

2. Cartesian Product Let X and E' be two universes. Their 'kross product" X x Y , or Cartesian product, is a universe G: (5.318) G = X x E ' = { ( ~ , y ) l zE X A y E Y } . Then. a fuzzy set on G is a fuzzy relation, analogously to classical set theory, if it consists of the valued pair of universes ,Y and E'. A fuzzy relation R in G is a fuzzy subset R E F ( G ) ,where F ( G ) denotes the set of all the fuzzy sets over X x Y . R can be given by a membership function ~ R ( zy), which assigns a membership degree ~ R ( . c y) . from [O. 11 to every element of ( 2 ,y) E G.

3. Properties of fizzy-Valued Relations (El) Since the fuzzy relations are special fuzzy sets, all propositions stated for fuzzy sets will also be valid for fuzzy relations. (E2) .A11 aggregations defined for fuzzy sets can be defined also for fuzzy relations: they yield a fuzzy

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5. Aloebra and Discrete Mathematics

relation again. (E3) The notion of a-cut defined above can be transmitted without difficulties to fuzzy relations. (E4) The 0-cut (the closure of the support) of a fuzzy relation R E F ( G ) is a usual relation on G. (E5)We denote the membership value by /.LE($, y), Le., the degree by which the relation R between the pair (z, y) holds. The value p ~ ( xy), = 1 means that R holds perfectly for the pair (x,y), and the value p ~ ( zy). = 0 means that R does not at all hold for the pair (2, y). (E6) Let R E F ( G ) be a fuzzy relation. Then the fuzzy relation S := R-’, the inverse of R, is defined by (5.319) pS(X, 1/) = pR(yr z) for every (29 y) E G.

ITheinverse relation R;’ means “practically smaller than” (see 5.9.3.1, l., p. 367); the union RIUR;’ can be determined as “practically smaller or approximately equal”.

4. n-Fold Cartesian Product Let n be the number of universal sets. Their cross product is an n-fold Cartesian product. A fuzzy set on an n-fold Cartesian product represents an n-fold fuzzy relation. Consequences: The fuzzy sets, considered until now, are unary fuzzy relations, Le., in the sense of the analysis they are curves above a universal set. .A binary fuzzy relation can be considered as a surface over the universal set G. A binary fuzzy relation on a finite discrete support can be represented by a fuzzy relation matrix. IColour-ripe grade relation: The well-known correspondence between the colour zand the ripe grade y of a friut is modeled in the form of a binary relation matrix with elements {0,1}. The possible colours are X = {green, yellow, red} and the ripe grades are Y = {unripe, half-ripe, ripe}. The relation matrix (5.320) belongs to the table: unripe half-ripe ripe (5.320) 1 red Interpretation of this relation matrix: IF a fruit is green, THEN it is unripe. IF a fruit is yellow, THEN it is half-ripe. IF a fruit is red, THEN it is ripe. Green is uniquely assigned to unripe, yellow to half-ripe and red to ripe. If we want to formalize that a green fruit can be considered half-ripe in a certain percentage, then we can get the following table with discrete membership values: The relation matrix with p~ E [O, 11 p~ (green. unripe) = 1.0, ,UR (green, half-ripe) = 0.5, pR (green. ripe) = 0.0, p~ (yellow, unripe) = 0.25, is: 1.0 0.5 0.0 = 0.25, pR (yellow, half-ripe) = 1.0, p~ (yellow. ripe) pR (red. unripe) = 0.0, p R (red, half-ripe) = 0.5, 0.0 0.5 1.0 p~ (red. ripe) = 1.0.

5. Rules of Calculations The ASD-type aggregation of fuzzy sets, e.g. p1 : X -i [0,1] and /.LZ : Y + [0,1] given on different universes is formulated by the min operation as follows: p R ( z , y ) = min(pl(z),pZ(y))Or (11 bLZ)(z,g)= min(pl(z)!pZ(y)) with (5.322a)

[0,1], i where G = X x Y. p1 x pz: G -

(5.322b)

The result of this aggregation is a fuzzy relation R on the cross product set (Cartesian product universe offuzzysets) Gwith (2.y) E G. I f X a n d Y arediscretefinitesetsand~op~(x),/.LZ(y)canberepresented as vectors, then we get: (5.323) and P R - ~ ( Z Y) , := P R ( K z) v (z, Y) E G. PI x pz = PI 0 p; The aggregatzon operator o does not denote here the usual matrix product. The product is calculated here by the componentwise min operation and addition by the componentwise max operation.

5.9 Fuzzw Loaic 369 The validity grade of an inverse relation R-' for the pair (x,y) is always equal to the validity grade of R for the pair (y, 2). If the fuzzy relations are given on the same Cartesian product universe, then the rules of their aggregations can be given as follows: Let R1, Rs: X x Y + [0,1]be binary fuzzy relations. The evaluation rule of their AND-type aggregation uses the min operator, namely for V(z, y) E G: (5.324) P L R ~ ~ R ~= ( ~min(pRl(zs .Y) Y), P R ~ ( ~ , Y ) ) . .4corresponding evaluation rule for the OR-type aggregation is given by the max operation: P R , ~ R ~ ( ~= . Y ~) ~ P R , (YZ) , ,P R ~ ( ~ ~ Y ) ) . (5.325)

5.9.3.2 Fuzzy Product Relation R o S 1. Composition or Product Relation Suppose R E F ( X x Y) and S E F ( Y x Z)are two relations, and it is additionally assumed that R, S E F ( G ) with G C X x Z. Then the composition or the fuzzy product relation R o S is: (5.326) PRaS(z: z ) := supy,Y{min(PR(z, Y)) PS(y: z ) ) }v (212 ) E X z. If a matrix representation is used for a finite universal set analogously to (5.321), then the composition R c S is motivated as follows: Let X = { X I ) . . . ,zn}, Y = {y1 , y}, Z = {zl,. . . , z l } and R E F ( X x Y), S E F ( Y x 2 ) and let the matrix representations R , S be in the form R = ( T , ] ) and S = ( s j k ) f o r i = l , . . .% n : j = l .,. . ,m ; k = l ,. . . ,1,where rL3= P R ( G . Y ~ ) and s j k = P S ( Y ~z k, ) . (5.327) If the composition T = R o S has the matrix representation t z k , then (5.328) t,k = sup min{rtj, s J k } . 3

The final result is not a usual matrix product, since instead of the summation operation there is the least upper bound (supremum) operation and instead of the product we have the minimum operator. W With the representations for rZjand sjk and with (5.326), the inverse relation R - ' ( T % ,can ~ ) ~also ) be computed taking into consideration that R-l can be represented by the transpose matrix, Le.. R-' = (Tu

IT.

Interpretation: Let R be a relation from X to Y and S be a relation from Y to Z. Then the following compositions are possible: a ) If the composition R o S of R and S is defined as a max-min product, then the resulted fuzzy composition is called a max-min composition. The symbol sup stands for supremum and denotes the largest value, if no maximum exists. b) If the product composition is defined as with the usual matrix multiplication, then we get the maxprod composition. c) For max-average composition, "multiplication" is replaced by the average.

2. Rules of Composition The following rules are valid for the composition of fuzzy relations R, S,T E F ( G ) : (El)Associative Law: ( R0 S)0 T = R 0 (S0 T ) .

(5.329)

(E2)Distributive Law for Composition w i t h Respect to the Union: (5.330) R 0 (Su T ) = ( R 0 sj u ( R 0 T ) . (E3)Distributive Law i n a Weaker Form for Composition w i t h Respect to Intersection: R o ( S n T )E ( R o S ) n ( R o T ) . (5.331) (E4)Inverse Operations: ( R c S)-'= S-' c R-'!

( R US)-'= R-'

U S-'

and ( R n S)-' = R-' n S-'.

(5.332)

5. Alqebra and Discrete Mathematics

370

(E5) Complement a n d Inverse: (R-’)-l = R,

( R y = (R-’)C.

(5.333)

(E6) Monotonic Properties: R cS+RoTcSoT undToRs ToS

(5.334)

IA: Equation (5.326) for the product relation R o S is defined by the min operation as we have done for intersection formation. In general, any t-norm can be used instead of the min operation. IB: The @-cutswith respect to the union, intersection, and complement are: (AUB)>n = 4>aUB>a, ( A n B)>n = A>a n B’a, (Ac)>n= A<’-” = {x E X / p ~ ( x5) 1 - a } . Corresponding statements are valid for strong cr-cuts.

3. Fuzzy Logical Inferences It is possible to make a fuzzy inference, e.g., with the IF THEN rule by the composition rule p2 = ploR. The detailed formulation for the conclusion p2 is given by (5.335)

pZ(y) = maxzEX(min(pl(x), p R ( I , Y)))

with y

E

Y, p1: X

-+ [O. 11, pz: Y + [0,1], R: G + [O: 11 und G = X

x Y.

5.9.4 Fuzzy Inference (Approximate Reasoning) Fuzzy inference is an application of fuzzy relations with the goal of getting fuzzy logical conclusions with respect to vague information (see 5.9.6.3, p. 373). Vague information means here fuzzy information but not uncertain information. Fuzzy inference, also called implication, contains one or more rules, a fact and a consequence. Fuzzy inference, which is called by Zadeh, approximate reasoning, cannot be described by classical logic.

1. Fuzzy Implication, IF THEN Rule The fuzzy implication contains one IF THEN rule in the simplest case. The IF part is called the premise and it represents the condition. The THEN part is the conclusion. Evaluation happens by p2 = p1 o R and (5.335). Interpretation: p2 is the fuzzy inference image of p 1 under the fuzzy relation R: Le.: a calculation prescription for the IF THEN rule or for a group of rules.

2. Generalized Fuzzy Inference Scheme The rule IF .41 AND A2 AND A s . . . AKD A, THEN B with Ai: p t : Xi + [0,1] (i = 1 , 2 , . . . , n) and the membership function of the conclusion B: p : Y + [0,1]is described by an (n + 1)-valued relation (5.336a) R: X i x X2 x . . . X , x Y --t [0,1]. For the actual input with crisp values xi,x; ,xLthe rule (5.336a) defines the actual fuzzy output by P W ( Y ) =PR(z:,z:,...,z~,Y) = min(pLl(x;),p2(x;)1...rp,(xh),pB(~)) where Y E Y. (5.336b) Remark: The quantity min(pl(x;), pz(x;), . . . pn(xL)) is called the degree of fulfillment, and the quantities {pl(xi). p 2 ( 4 ) %. . . , pn(xh)}represent the fuzzy-valued input quantities. IForming the fuzzy relations for a connection between the quantities “medium” pressure and “high temperature (Fig. 5.73): pl(plT ) = p ~ ( pV) T E Xz with pl: XI + [0,1] is a cylindrical extension (Fig. 5 . 7 3 ~ )of the fuzzy set medium pressure (Fig. 5.73a). Analogously, pp(p,T).= pz(T) V p E XIwith p2 : X 2 + [0,1] is a cylindrical extension (Fig. 5.73d) of the fuzzy set hlgh temperature (Fig. 5.73b), where f i L , p 2 : G = X I x X Z + [0,1]. Fig. 5.74a shows the graphic result of the formation of fuzzy relations: In Fig. 5.74b the result of the composition medium pressure AND high temperature with the min operator p ~ ( p2’), = min(pl@), p z ( T ) )is represented, and (Fig. 5.74b) shows the result of the composition OR with the max operator p R ( p >T)= max(pl(p),p2(T)).

5.9 Fuzzu Loqic 371

Figure 5.74

5.9.5 Defuzzification Methods Often we have to get a crisp set from a fuzzy-valued set. This process is called defuzzification. There are different methods to do this. 1. Maximum-Criterion Method An arbitrary value 7 E Y is selected from the domain where the fuzzy set &":P,itn has the maximal membership degree. 2. Mean-of-Maximum Method (MOM) The output value is the mean value of the maximal membership values:

dY

Le., the set I.' is an interval. which should not be empty and it is characterized by (5.337).from which we get (5.338). 3. Center of Gravity Method (COG) In the center of gravity method, we take the abscissa value of the center of gravity of a surface with a fictitious homogeneous density of value 1.

WoG =

.

(5.339)

372

5. Alqebra and Discrete Mathematics

Center of Gravity Method (PCOG) The parametrized method works with the exponent y E R. From (5.340) it follows for y = 1, COG = COG and for 4. Parametrized

0, VPCOG = ~ U O h l ~ 5. Generalized Center of Gravity Method (GCOG) The exponent y is considered as a function of y in the PCOG method. Then (5.341) follows obviously. The GCOG method is a generalization of the PCOG method, where p(y) can be changed by the special weight y depending itself on y. 6 . Center of Area (COA) Method We calculate a line parallel to the ordinate axis so that the area under the membership function is the same on the left- and on the right-hand side of it. A/

7. Parametrized Center of Area (PCOA) Method

J

lp

Yinf

dY = i I % Y ) T

dY. (5.343)

8. Method of t h e Largest Area (LA) The significant subset is selected and one of the methods defined above, e.g., the method of center of gravity (COG) or center of area (COA) is used for this subset.

5.9.6 Knowledge-Based Fuzzy Systems There are several application possibilities of multi-valued fuzzy logic, based on the unit interval, both in technical and non-technical life. The general concept is that we fuzzify quantities and distinguish marks. we aggregate them in an appropriate knowledge base with operators, and if necessary, we defuzzify the possibly fuzzy result set.

5.9.6.1 Method of Mamdani The following steps are applied for a fuzzy control process: 1. Rule Base Suppose, for example, for the i-th rule (5.344) Rt : If e is E* AYD e is AEt THEY u is U’. Here e characterizes the error, e the change of the error and u the change of the (not fuzzy valued) output value. Every quantity is defined on its domain E , AE and U . Let the entire domain be E x AE x U . The error and the change of the error will be fuzzified on this domain, Le., they will be represented by fuzzy sets, where linguistic description is used. 2. Fuzzifying Algorithm In general, the error e and its change e are not fuzzy-valued, so they must be fuzzified by a linguistic description. The fuzzy values will be compared with the premisses of the IF THEY rule from the rule base. From this it follows, which rules are active and how large are their weights 3. Aggregation Module The active rules with their different weights will be combined with an algebraic operation and applied to the defuzzification. 4. Decision Module In the defuzzification process a crisp value should be given for the control quantity. With a defuzzification operation, a non-fuzzy-valued quantity is determined from the set of possible values. Le., a crisp quantity. This quantity expresses how the control parameters of the system should be set up to keep the deviation minimal. Fuzzy control means that the steps from 1. to 4. are repeated until the goal, the smallest deviation e and its change i. is reached.

5.9 Fuzzv Looic 373

5.9.6.2 Method of Sugeno The Sugeno method is also used for planning of a fuzzy control process. It differs from the Mamdani concept in the rule base and in the defuzzification method. It has the following steps: 1. R u l e Base: The rule base consists of rules of the following form: R' : IF .cl is At .4ND . . . AND zk is A;, THEN ui = pb + ptzl + p i z ~t . . p;zk. (5.345) The notations mean: A,: fuzzy sets, which can be determined by membership functions: z3:crisp input values as. e.g., the error e and the change of the error e , which tell us something about the dynamics of the system; p i : weights of zJ ( j = 1 , 2 , . . . IC); u,: the output value belonging to the i-th rule (i = 1 , 2 , . . . , n). 2. Fuzzifying Algorithm: A pi E [0,1]is calculated for every rule Rt. 3. Decision Module: A non-fuzzy-valued quantity is calculated from the weighted mean of ui, where the weights are pt from the fuzzyfication:

+

(5.346) Here u is a crisp value. The defuzzification of the hlamdani method does not work here. The problem is to get the weight parameters p(l available. These parameters can be determined by a mechanical learning method, e.g., by an artificial neuronetwork (ANN).

5.9.6.3 Cognitive Systems To clarify the method, the following known example will be investigated with the Mamdami method: The regulation of a pendulum that is perpendicular to its moving base (Fig. 5.75). The aim of the control process is to keep a pendulum in balance so that the pendulum rod should stand vertical, i.e., the angular displacement from the vertical direction and the angular velocity should be zero. It must be done by a force F acting a t the lower end of the pendulum. This force is the control quantity. The model is based on the activity of a human "control expert" (cognitive problem). The expert formulates its knowledge in linguistic rules. Linguistic rules consist, in general, of a premisse, Le., a specification of the measured values, and a conclusion which gives the appropriate control value. For every set of values XI, X2,. . . , X,, for the measured values and Y for the control quantity the appropriate linguistic terms are defined as "approximately zero", "small positive", etc. Here "approximately zero" with respect to the measured value (1 can have a different meaning as for the measured value <2. IInverse P e n d u l u m on a Moving Base (Fig. 5.75) 1. Modeling For the set XI (values of angle) and analogously for the input quantity X2 (values of the angular velocity) the seven linguistic terms, negative large (nl). negative medium (nm), negative small (ns), zero (z), positive small (ps), positive medium (pm) and positive large (pl) are chosen. For the mathematical modeling, a fuzzy set must be assigned by graphs to every one of these linguistic terms (Fig. 5.74), as was shown for fuzzy inference (see 5.9.4, p. 370). Determination of the Domain of Values Values of angles: @(-go0 < 0 < 90"): XI := [-goc, go"]. Values of angular velocity: d(-4sos-' 5 6 5 45" s-1) : ~2 := [-45's-'. ~ V a l u e s o f f o r c e F :( - l O K < F < l 0 N ) : Y := [-lON,lON].

L

2.

450 s-']

F Figure 5.75

374

5. Algebra and Discrete Mathematics

The partitioning of the input quantities X I and X Zand the output quantity Y is represented graphically in Fig. 5.76. Usually, the initial values are actual measured values. e.g., 0 = 36", d = -2.25O s-l.

Figure 5.76

I

3. Choice of Rules Considering the following table, there are 49 possible rules (7 x 7 ) but there are only 19 important in practice, and we discuss the following two, R1 and R2, from them.

R1: If 0 is positive small (ps) and d zero (2). then F is positive small (ps). For the degree offulfillment (also called the weight ofthe rules) of the premise with (Y = min { p ( ' ) ( O ) p(')(d)} : = min(0.4; 0.8) = 0.4 n e get the output set (5.347) by an a-cut, hence the output fuzzy set is positive small (ps) in the height a = 0.4 (Fig. 5 . 7 7 ~ ) . Table: Rule base with 19 practically meaningful rules O\O nl I nm 1 ns z I ps j pm 1 pl nl I 1 IPSIPl I I I nm (5.347) ns nm ns z nl nm ns PS otherwise.

I

I

lo

R2: If 0 is positive medium (pm) and d is zero (z), then F is positive medium (pm). For the performance score of the premise we get a = m i n { p ( 2 ) ( @ ) : p ( z ) @ } = min{0.6:0.8} = 0.6. the output set (5.348) analogously to rule R1 (Fig. 5.770.

(2 iy- 1

0.6 Output(RZ) p36i-225

=

'

2 3 - -y 5 0

2.5 5 y < 4, 4 5 ~ 5 6 , 6 < y 5 7.5.

(5.348)

otherwise.

i

4. Decision Logic The evaluation of rule R1with the min operation results in the fuzzy set in Figs. 5.77a-c. The corresponding evaluation for the rule R2 is shown in Figs. 5.77d-f. The control quantity is calculated finally by a defuzzification method from the fuzzy proposition set (Fig. 5.77g). \Ve obtain the fuzzy set (Fig. 5.77g) by using the max operation if we take into account the fuzzy sets (Fig. 5 . 7 7 ~ and ) (Fig. 5.77f). a) Evaluation of the fuzzy set obtained in this way, which is aggregated by operators (see max-min composition 5.9.3.2, 1.. p. 369). The decision logic yields: / 1OUtPUt 2 1 . . . . , ~ l7 n:

:Y

maxr€{l,...,k] {min{d:,!(zl),"

'r!$j(~n)~~t,(Y)}}.

(5.349)

5.9 Fuzzy Logic 375

b) For the function graph of the fuzzy set we get (5 350) after taking the maximum. c) For the other 17 rules we get a degree offulfillment equal to zero for the premise, Le.. it results in fuzzy sets, which are zeros themselves. 5 . Defuzzification The decision logic yields no crisp value for the control quantity. but a fuzzy set. That means. by this method, we get a mapping, which assigns a fuzzy set p:tP& ' of Y to every tuple (51 . . . 5,) E XI x X2 x ' . x X,of the measured values.

.

2

output !k,-2.25(~)=

=Y

for

0 5 y < 1,

0.4

for

15y

2 5 0.6

-y - 1 for

for

3 - f y for 5 0 for

< 3.5.

3.5 5 y < 4,

(5.350)

4 5 y < 6, 6

5 y 5 7.5,

otherwise.

Defuzzification means that we have to determine a control quantity using defuzzification methods. The center of gravity method and the maximum criterion method result in the value for control quantity F = 3.95 or F 5.0.

6. Remarks Figure 5.77 1. The -knowledge-based" trajectories should lie in the rule base so that the endpoint is in the center of the smallest rule deviation. 2. By defuzzification we start the iteration process, which leads finally to the middle of the partition space. i e.. which results in a zero control quantity. 3. Ever! non-linear domain of characteristics can be approximated with arbitrary accuracy by the choice of appropriate parameters on a compact domain.

5.9.6.4 Knowledge-Based Interpolation Systems 1. Interpolation Mechanism \Ye can build up interpolation mechanisms with the help of fuzzy logic. Fuzzy systems are systems

378

5. Alaebra and Discrete Mathematics

to process fuzzy information. With them we can approximate and interpolate functions. A simple fuzzy system, by which we investigate this property, is the Sugeno controller. It has n input variables & , . . . ,& and defines the value of the output variable y by rules R1,. . . , R, in the form

R,: IF

is A? and

.

'

and & is A!), THEN is y = f,(&,. . . , &)

(z = 1 , 2 , . . . , n).

(5.351)

The fuzzy sets A:'),. . . ,A(') here always partition the input sets X,. The conclusions ft(t1,. . . ,&) of the rules are singletons, which can depend on the input variables (1,. . . ,&. By a simple choice of the conclusions the expensive defuzzification can be omitted and the output value y will be calculated as a weighted sum. To do this, the controller calculates a degree of fulfillment a, for every rule R, with a t-norm from the membership grades of the single inputs and determines the output value (5.352) L,=lut

2. Restriction to the One-Dimensional Case For fuzzy systems with only one input x = (1, we often use fuzzy sets represented by triangle functions, intersected at the height 0.5. Such fuzzy sets satisfy the following three conditions: 1. For every rule R, there is an input x,,for which only one rule is fulfilled. For this input x,,the output is calculated by fi. By this, the output of the fuzzy system is fixed at N nodes z1.. . . ,z , ~ .Actually, the fuzzy system interpolates the nodes xl,. . . ,X N . The requirement that at the node X , only one rule R, holds. is sufficient for an exact interpolation, but it is not necessary. For two rules R1 and R2) as they will be considered in the following, this requirement means that cyI(x2) = cyz(z1) = 0 holds To fulfill the first condition, al(z2) = a 2 ( ~ 1= ) 0 must hold. This is a sufficient condition for an exact interpolation of the nodes. 2. There are at most two rules fulfilled between two consecutive nodes. If 21 and 5 2 are two such nodes with rules R1 and R2) then for inputs x E [x1,x2]the output y is

The actual shape of the interpolation curve between x1 and x2 is determined by the function g. The shape depends only on the satisfaction grades a1 and cy2, which are the values of the membership functions p (1) and p (1) at the point x, Le., cy1 = pAw(z)and cy2 = pap)(x) are valid, or in short form A,

4

ai = p I ( z ) and cy2 = p 2 ( ~ )The . shape of the curve depends only on the relation pI/pz of the membership functions. 3. The membership functions are positive, so the output y is a convex combination of the conclusions fi. For the given and for the general case hold (5.354) and (5.355), respectively: (5.355)

For constant conclusions, the terms f1 and f 2 cause only a translation and stretching of the shape of the curve g. If the conclusions are dependent on the input variables, then the shape of the curve is differently perturbed in different sections. Consequently, another output function can be found. If we apply linearly dependent conclusions and membership functions with constant sum for the input x, N

then the output is y = c E a,(z)f,(x) with a, depending on x and a constant c, so that the interpolation 2=1

functions are polynomials of second degree. These polynomials can be used for the construction of an interpolation method with polynomials of second degree. In general, choosing polynomials of n-th degree, we get an interpolation polynomial of the ( n + 1)th degree as a conclusion. In this sense fuzzy systems are rule-based interpolation systems besides conventional interpolation methods interpolating locally by polynomials, e.g., with splines.

377

6 Differentiation 6.1 Differentiation of Functions of One Variable 6.1.1 Differential Quotient 1. Differential Quotient or Derivative of a Function The dzflerentzal quotzent of a function y = f(x) at xo is equal to lim f(xo + AxiO

Ax

-

f(xo) if this limit

exists and is finite. The deriwatzwe functzon of a function y = f(x) with respect to the variable x is df(x) and its value for another function of x denoted by the symbols y’, i , Dy, ->dY f’(x),of(.),or -,

dx

dx

every z is equal to the limit of the quotient of the increment of the function Ay and the corresponding increment Ax for 4 x + 0, if this limit exists:

lr,,A*

’I

2. Geometric Representation of the Derivative If y = f(x) is represented as a curve in a Cartesian coordinate

a

X

Ax=dx

x

Figure 6.1

system as in Fig. 6.1, and if the x-axis and the y-axis have the same unit, then f’(x) = t a n a (6.2) is valid. The angle cy between the x-axis and the tangent line of the curve at the considered point defines the angular coeficientor slope ofthe tangent (see 3.6.1.2,2.,p. 227). The angle is measured from the positive x-axis to the tangent in a counterclockwise direction, and it is called the angle of slope or angle of inclination.

3. Differentiability &om the definition of the derivative it obviously follows that f(x) is differentiable with respect to z for the values of x where the differential quotient (6.1) has a finite value. The domain of the derivative function is a subset (proper or trivial) of the domain of the original function. If the function is continuous at x but the derivative does not exist, then perhaps there is no determined tangent line at that point, or the tangent line is perpendicular to the x-axis. In this last case the limit in (6.1) is infinity. For this case we use the notation f ’ ( x ) = +mor -m.

IA: f(z) = fi:f’(x) =

Figure 6.2 1 f’(0) = x. .4t the point 0 the limit (6.1) tends to infinity, so the 3x7

-,

derivative does not exist at the point 0 (Fig. 6.2a). W B: f(x) = xsin i f o r x X

the point z = 0 the limit of type (6.1) does not exist (Fig. 6.2b).

# 0, and f(0)

= 0: At

6. Differentiation

378

Y

4. Left-Hand and Right-Hand Differentiability If the limit (6.1) does not exist for a value x = a,but the left-hand limit or the right-hand limit exists, this limit is called the left-hand derivative or right-hand derivative respectively. If both exist, the curve has two tangents here: j’(a - 0) = tanct.1, !’(a 0) = tana2. (6.3) Geometrically this means that the curve has a knee (Fig. 6.2c, Fig. 6.3).

+

0

I‘

a ’ , Figure 6.3

x

Ij ( x ) =

4 for x # 0 and j ( 0 ) = 0. At the point x = 0 there 1+e;

is no limit of type (6.1). but there is a left-hand and a right-hand limit f’(-0) = 1 and f’(+O) = 0. i.e.. the curve has a knee here (Fig. 6 . 2 ~ ) .

6.1.2 Rules of Differentiation for Functions of One Variable 6.1.2.1 Derivatives of t h e Elementary Functions The elementary functions have a derivative on all their domains except perhaps some points, as represented in Fig. 6.2. A summary of the derivatives of elementary functions can be found in Table 6.1. Further derivatives of elementary functions can be found by reversing the results of the indefinite integrals in Table 8.1. Remark: In practice. it is often useful to transform the function into a more convenient form to perform differentiation, e.g., to transform it into a sum where parentheses are removed (see 1.1.6.1, p. 11) or to separate the integral rational part of the expression (see 1.1.7, p. 14) or to take the logarithm of the expression (see 1.1.4.3, p. 9). 1 2 5 3 - 3- 8 -2-3&+4@tx2 - -2 - 32 -2 t 4 x --3 + x ; -dY= - ~ X - ~ + - X 2--x 3 t 1 . IA: y = dx X 2 3

6.1.2.2 Basic Rules of Differentiation In the following u , u , w,and y are functions of the independent variable x, and u’. v’,w’,and y’ are the derivatives with respect to x. We denote by du,dv,dw,and dy the differential (see 6.2.1.3, p. 391). The basic rules of differentiation, which are explained separately, are summarized in Table 6.2, p. 384.

1. Derivative of a Constant Function The derivative of a constant function c is the zero function: d = 0. 2. Derivative of a Scalar Multiple .i\ constant factor c can be factored out from the differential sign: (cu)’ = cu’. d ( c u ) = cdu.

(6.4)

3. Derivative of a Sum If the functions u, v,w,etc. are differentiable one by one, their sum and difference is also differentiable, and equal to the sum or difference of the derivatives: (6.6a) ( u t v - w)’= u‘ t v’ - w’, (6.6b) d ( +~u - W ) = du + dv - dw. It is possible that the summands are not differentiable separately, but their sum or difference is. Then we calculate the derivative by the definition.

6.1 Differentiation of Functions of One Variable 379

Table 6.1 Derivatives of elementary functions in the intervals on which they are defined and the occurring numerators are not equal to zero

Derivative

Function

0

see x

1

cosec x

xn (nE R)

nxn-l

arcsin x (1x1< 1)

1

--1

arccos x (1x1 < 1)

Function

Derivative

~~

C (constant)

-

X

52

-- n

1

-

arctanx

p t l

5"

1

fi

arccot x

~

2&

( ~ E Rn .# O , x > O )

1

n

arcsec x

W

sin x cos2 x - cos x sin' x 1 V c ?

-~ 1

m

1 1 x' -- 1 1 +x2 1

+

X

V

e"

e"

arccosec x

-~

ebz ( b E R)

bebx

sinh x

cosh x

a" (a > 0)

a" In a

cosh x

ab' (b E R, a > 0)

babxIn a

tanhx

sinh x 1 -

In x

-

X

1

cothx (x # 0)

X

lgx (x > 0)

1 log, e = X xlna 1 0.4343 -1ge PZ -

sin x

cos x

cos x

- sin x

logax (a > 0. a

# 1, x > 0)

X

X

1 tanx (x # ( 2 k + l ) n , k E 2') -= sec2x 2 cos2 x -1 - - - - cosec' x cotx (x # kx, k E Z) sin2 x

.4rsinh x Arcoshx (x > 1)

m

cash' x

-~ 1 sinh' x 1

m 1

@=3

1 Artanhx (1x1 < 1) 1 - x2 1 Arcotha: (1x1 > 1)

[f(x)ln ( n E R) lnf(x) (f(x) > 0)

4. Jerivative of a Product If two, three, or n functions are differentiable one by one, then their product is differentiable. and can be calculated as follows: a) Derivative of the Product of Two Functions: (6.7a) (uv)'= U ' U + u v', d(uU) = 2: du udv. It is possible that the terms are not differentiable separately, but their product is. Then we calculate the derivative by definition.

+

380

6. Differentiation

b) Derivative of the Product of Three Functions: ( u u w)‘ = U’V w + u V’W + u V W ‘ , d ( u v w ) = v w d u + u w dv + u v dw. c) Derivative of the Product of n Functions:

(6.7b)

( q u 2 . . . u , ) ’ = ~ u l u z . . . u ~ . .21,..

(6.7~)

i=l

IA: y = x 3 c 0 s x , y’=3x2cosz-z3sinz. IB: y = 2%’ cos x, y’ = 3x2e5cos x + z3e2cos x - z3ersin x.

5. Derivative of a Quotient If both u and v are differentiable, and v(x) # 0, their ratio is also differentiable:

Iy = t a n x = - ,

I

sin x cos x

y’=

+

(cos x)(sin 2)’ - (sin x)(cos 2)’ - cos2 x sin’ z - 1 cos2 x cos2 x cos2x

,

6. ChainRule The composite function (see 2.1.5.5, 2., p. 59) y = u ( v ( x ) )has the derivative dY du dv - = u’(v) v’(x) = -dx dv dx ’ where the functions u = u ( v ) and v = v ( x ) are differentiable functions with respect to their own variables. u ( u ) is called the exterior function, and v ( x ) is called the interior function. -4ccording t o this,

du . dv - IS the exterior derivative and - is the interior derivative. It is possible that the functions u and dx dv v are not differentiable separately, but the composite function is. Then we get the derivative by the definition. We proceed similarly if we have a longer “chain”, Le., we have a composite function of several interrnediate variables. So we have, e.g., for y = u(v(w(z))): dy du dv dw (6.10) y = u ( v ( w ( x ) ) ) , y‘ = - = - - - , dx dv dw dx

7. Logarithmic Differentiation If y(x) > 0 holds, we can calculate the derivative y’starting with the function In y(z)>whose derivative (considering the chain rule) is: (6.11) From this rule (6.12) follows.

Remark 1: With the help of logarithmic differentiation it is possible to simplify some differentiation

6.1 Differentiation of Functions o f One Variable 381

problems. and there are functions such that this is the only way to calculate the derivative, for instance, when the function has the form y = u(x)'((") with u(x)> 0. The logarithmic differentiation of this equality follows from the formula (6.12)

d (lnuU)

I

Y =

Y

(6.13)

d(w1nu)

(6.14)

T

Iy = (2x t l)3s. In y = 321n(2x+ 1))

Y' = 31n(2z Y

-

+ 1) + -;2x3 2+. 21

q3=(ln(2x + 1) +

y' = 3 ( 2 2 t

Remark 2: Logarithmic differentiation is often used when we have to differentiate a product of several functions. 1 IA: y=d-. lny=;(3Inxt4z+lnsinx). L

cos x sin I y'

IB: y = u v: In y = lnu + lnv.

1

- = -ut 2 1 %

1 + -v'. u

From this identity it follows that y' = (u w)' =

w u'+ u v',so we get the formula for the derivative of a product (6.7a) (under the assumption u, w > 0). IC: y =

F.

111y

= lnu

-

lnv,

2'

Yl-1 1 - -u' - -w'. From this identity it follows that y' = y u 2:

(:)'

=

1121' - uv' , which is the formula for the derivative of a quotient (6.8) (under the assumption v* u2 u, v > 0). 8. Derivative of the Inverse Function If y = p(x) is the inverse function of the original function y = f(x),then both representations y = f(z) and x = p(y) are equivalent. For every corresponding value of z and y such that f is differentiable with respect to z.and 'pis differentiable with respect t o y , e.g., none of the derivatives is equal to zero,

21'

212"

~

u

between the derivatives off and its inverse function p we have the following relation: 1

f ' ( x ) = - or p'(y)

dY

1

dx

dz

(6.15)

-= -

dY IThe function y = f(x) = arcsina: for -1 < I < 1 is equivalent to the function - ~ / 2 < y < a/2. From (6.15) it follows that (arcsinx)' =

~

1

(siny)'

-

1

-cosy

1

=

~~

1-

1 1- x

IC

= p(y) = siny for

, because cosy # 0 for - ~ / 2 < y < 7112.

9. Derivative of an Implicit Function Suppose the function y = f(x) is given in implicit form by the equation F ( z ,y) = 0. Considering the rules of differentiation for functions of several variables (see 6.2, p. 390) calculating the derivative with respect to x we get

dF

i9F

-+-y=O ax dy

I

andso y ' = - - ,

FZ

F,

if the partial derivatiye F, differs from zero.

(6.16)

6. Differentiation

382

IThe equation 52 -

a2

22 -

a2

+ y2b2 = 1 of an ellipse with semi-axes a and b can be written in the form F ( z ,y) = -

y*

+- 1 = 0. For the slope of the tangent line at the point of the ellipse (x, y) we get according to b2

(6.16) yI

=

--/22 2y = a2

b2

b2x a2 y

10. Derivative of a Function Given in Parametric Form If a function y = f ( z ) is given in parametric form x = z ( t ) ,y = y(t), then the derivative y’ can be calculated by the formula

dy = f’(.) dx

=Y

(6.17)

X

dY dx . with the help of the derivatives Y(t) = - and i ( t ) = - with respect to the variable t , if of course dt dt i ( t ) # 0 holds. IPolar Coordinate Representation: If a function is given with polar coordinates (see 3.5.2.2,3., p. 191) p = p ( p ) )then the parametric representation is (6.18) 5 = p ( y ) cos p. y = p(p) sin p with the angle r; as a parameter. For the slope y’ of the tangent of the curve (see 3.6.1.2. 2., p. 227 or 6.1.1. 2.. p. 377) we get from (6.17)

I

p s i n p + p c o s + where ,b = dP . (6.19) pcos y - psin p dP Remarks: 1. The derivatives 5. y are the components of the tangent vector at the point ( x ( t ) .y(t)) of the curve. 2. It is often useful to consider the complex relation: z(t)+iy(t)=z(t). i(t)+iY(t)=i(t). (6.20) I

y = .

ICircular Movement: z ( t ) = reiwt ( T , W const), i ( t ) = &eiWt = rwei(wt+ The tangent vector runs ahead by a phase-shift ~ / with 2 respect to the position vector. given direction M N . To

11. Graphical Differentiation If a differentiable function y = f ( x ) is represented by its curve r in the Cartesian coordinate system in an interval a < z < b, then the curve r’ of its derivative can be constructed approximately. The construction of a tangent estimated by eye is pretty inaccurate. However, if the direction of the tangent .!-f.Y (Fig. 6.4) is given, then we can determine the point of contact 4 more precisely. 1. Construction ofthepoint ofontact of a Tangent We draw two secants Ml,V1 and M2N2parallel to the direction MN of

~~1

the tangent so that the curve is intersected in points being not far from each other. Then we determine the midpoints of the secants, and draw a MI straight line through them. This line PQ intersects the curve a t the point A. which is approximately the point, where the tangent has the Figure 6.4 check the accuracy. we draw a third line close to and parallel to the first two lines, and the line PQ should intersect it at the midpoint. 2. Construction of the Derivative Curve a) Choose some directions I I , . . .1, which could be the directions of some tangents of the curve

6.1 Dzfferentiatzonof Functzons of One Variable 383

y = f ( z ) in the considered interval as in Fig. 6.5, and determine the corresponding points of contact .41. & % .. . A,,. where the tangents themselves must not be constructed. b) Choose a point P . a "pole", on the negative side of the L-axis. where the longer the segment PO = a, the flatter the curve is. c ) Draw the lines through the pole P parallel to the directions 11, 1 2 . . , . 1,, and denote their intersection points with the y-axis by B1. B z ) . .. B,. d) Construct the horizontal lines B I C ~BzCz.. . . . , B,C, through the points B1:B z , .. . B, to the intersection points C 1 ,Cz.. . . . C, with the orthogonal lines from the points i l l , .42,. . . . '4,. e ) Connect the points C1. Cz, . . . , C, with the help of a curved ruler, The resulting curve satisfies the equation y = a!'(.). If the segment a is chosen so that it corresponds to the unit length on the y-axis, then the curve we get is the curve of the derivative. Otherwise, we have to multiply the ordinates of C1, ( 2 2 ) . . . , C, by the factor l j a . The points D1:Dz.. , . , D, given in Fig. 6.5 are on the correctly scaled curve r' of the derivative.

.

I

Figure 6.5

6.1.3 Derivatives ofHigher Order 6.1.3.1 Definition of Derivatives of Higher Order The derivative of y' = f'(x), ivhich means (y')' or

d dy , is called the second derivative of the dx (ds)

function y = f(z)and it is denoted by y", y, -,d2Y f"(x) or d 2 f ( x ) , Higher derivatives can be defined dx2 dx2 analogously. The notation for the n-th derivative of the function y = f ( x ) is: ~

=

dnM = f'"'(r) = d"f(x) dxn dxn

(n= 0 , l . . . . : y(O)(x)= f'"(x) = f ( x ) )

~

(6.21)

6.1.3.2 Derivatives of Higher Order of some Elementary Functions The n-th derivatives of the simplest functions are collected in Table 6.3.

6.1.3.3 Leibniz's Formula To calculate the n-th-order derivative of a product of two functions, the Leibniz formula can be used: n(n - 1) = ~n~ +:D% ~ n - 1 U D2u D n - 2 i ~l! 2! n(n - 1 ) .. . ( n - m 1) Dmu Dn-m~lt ' . t D"u u see also p. 383. (6.22) m! d" Here, we use the notation D(,) = - . If Dou is replaced by u and Dov by u. then we get the formula dx (6.23) whose structure corresponds to the binomial formula (see p. 12):

+

+

+

(6.23)

384

6. Diflerentiation

Table 6.2 Differentiation rules

Formula for the derivative

+ Constant function

c‘ = 0

Constant multiple

(cu)’ = cu’

(c const)

(c const)

(u f u)’ = u’ i v’

+U d

Product of two functions

(uu)’ = u’u

Chain rule for two functions

y = u ( v ( x ) :)

three functions

du dv y’ = - dv dx du dv dw y = u ( v ( w ( x ) ):) y’ = - - du dw dx

Power

Logarithmic differentiation p inverse function off, i.e. y = f(z)v 2 = p(y) :

Differentiation of the inverse function

Implicit differentiation Derivative in parameter form

z = z ( t ) , y = y(t) (t parameter):

Derivative in polar coordinates

+

IA: (x2cos az)@O):If v = x 2 , u = cos a2 are substituted, then we have u@)= &COS (ax k : ) , v’ = 2x. v” = 2. v”’ = d4)= ’ . = 0 . Except the first three cases, all the summands are equal to zero, so (uU)(’O) = Z2050COS (Qx

+ 50;) + 50 1

’

2xa4’ COS (02 + 49%)

+

. 2Q@

COS (aZ

+ 485)

6.1 Differentiatzon of Functzons of One Varzable 385

= a48[(2450- azxz)cosax -

(0

IB: ( x 3 e S ) ( ' 3 =

IOOaxsinaz].

~ x 3 e x + ( ~ ) ~ 3 x 2 (:).6zez+(:).6e' e"t

Table 6.3 Derivatives of higher order of some elementary functions Function

n-th-order derivative

Xrn

m(m - l ) ( m - 2). . . ( m - n t 1 ) P " (for integer m and n > m the n-th derivative is 0) 1 (-l)"-l(n - l)!-

In x

X"

( n - l)!1 l n a xn knekx (In a)"a" ( k In a)"akX

(-1)n-l-

log, x elcx a"

akx

sin x cos x sin kx cos kx

sinh x cosh x

n7r sin(x + -) 2 ng

cos(x + -) 2 nr t"sin(kx -) 2

+

k" cOs(kx + -)

ILII

2 sinh x for even n, cosh x for odd n cosh x for even n, sinh x for odd n

6.1.3.4 Higher Derivatives of Functions Given in Parametric Form If a function y = f ( x ) is given in the parametric form x = x ( t ) ,y = y(t), then its higher derivatives dy . dx (y", y"'. etc.) can he calculated by the following formulas, where G(t) = -, x(t) = -, % ( t )= dt dt d2y .. d2x - x = -, etc., denote the derivatives with respect to the parameter t: dt2' dt2 (6.24)

6.1.3.5 Derivatives of Higher Order of the Inverse Function If y = y ( z )is the inverse function of the original function y = f ( x ) , then both representations y = f ( x ) and x = p(y) are equivalent. Supposing pO'(y) # 0 holds, the relation (6.15) is valid for the derivatives of the function f and its inverse function p. For higher derivatives (y", y"', etc.) we get (6.25)

386

6. Dzferentiation

6.1.4 Fundamental Theorems of Differential Calculus 6.1.4.1 Monotonicity If a function f (z) is defined and continuous in a connected interval, and if it is differentiable at every interior point of this interval. then the relations f’(z) 2 0 for a monotone increasing function, (6.26a) j’(sj

5 0 for a monotone decreasing function

(6.26b)

are necessary and sufficient. If the function is strictly monotone increasing or decreasing, then the derivative function f ’ ( z ) must not be identically zero on any subinterval of the given interval. In Fig. 6.6b this condition is not fulfilled on the segment E. The geometrical meaning of monotonicity is that the curve of an increasing function never falls for increasing values of the argument, Le., it either rises or runs horizontally (Fig. 6.6a). Therefore the tangent line at any point of the curve forms an acute angle with the positive z-axis or it is parallel to it. For monotonically decreasing functions (Fig. 6.6b) analogous statements are valid. If the function is strictly monotone, then the tangent can be parallel to t h z - a x i s only at some single points, e.g., at the point A in Fig. 6.6a, Le.. not on a subinterval such as BC in Fig. 6.6b.

Figure 6.6

Figure 6.7

6.1.4.2 Fermat’s Theorem If a function y = j ( z j is defined on a connected interval, and it has a maximum or a minimum value at an interior point x = c of this interval (Fig. 6.7). Le., if for every x in this interval

f (4 > f (XI

(6.27a)

or

f (4 < f (XI,

(6.2713)

holds, and if the derivative exists at the point c, then the derivative must be equal to zero there: f’(c) = 0. (6.274 The geometrical meaning of the Fermat theorem is that if a function satisfies the assumptions of the theorem, then its curve has tangents parallel to the z-axis at A and B (Fig. 6.7). The Fermat theorem gives only a necessary condition for the existence of a maximum or minimum value at a point. From Fig. 6.6a it is obvious that having a zero derivative is not sufficient to give an extreme value: At the point A , f’(z) = 0 holds, but there is no maximum or minimum here. To have an extreme value differentiability is not a necessary condition. The function in Fig. 6.8d has a maximum at e, but the derivative does not exist here.

6.1.4.3 Rolle’s Theorem If a function y = f (z) is continuous on the closed interval [a, b], and differentiable on the open interval (a. b ) , and j ( a ) = 0. j ( b ) = 0 ( a < b) (6.28a) hold. then there exists at least one point c between a and b such that f’(c) = 0 (a < c < b) (6.28b) holds. The geometrical meaning of Rolle’s theorem is that if the graph of a function y = f(x) which is continuous on the interval ( a 2b) intersects the z-axis at two points A and B , and it has a non-vertical

6.1 Differentiationof Functions of One Variable 387

tangent at every point, then there is at least one point C between A and B such that the tangent is parallel to the x-axis here (Fig. 6.8a). It is possible, that there are several such points in this inter-

Yt I

”

Figure 6.8 val. e.g., the points C, D ,and E in Fig. 6.8b. The properties of continuity and differentiability are important in the theorem: in Fig. 6 . 8 ~ the function is not continuous at x = d, and in Fig. 6.8d the function is not differentiable at x = e. In both cases ! ’ ( E ) # 0 holds everywhere where the derivative exists.

/

6.1.4.4 Mean Value Theorem of Differential Calculus If a function y = f(x)is continuous on the closed interval [a,b],differentiable on the open interval ( a ,b) and it has a non-vertical tangent at every point, then there exists at least one point c between a and b such that

Y+

1

,I

I

,,/

5

~

,

j

I

!

0’ a

c

,,,‘A

I‘

I

,

I

!

b

Figure 6.9

= f’(c) (a < c < b) (6.29a) b-a holds. If we substitute b = a + h, and 0 means a number between 0 and 1, then the theorem can be written in the form f ( a + h) = f ( a ) + h f ’ ( a + O h ) (0< 0 < 1). (6.29b) 1. Geometrical Meaning The geometrical meaning of the theorem is that if a function y = f(x) satisfies the conditions of the theorem, then its graph has at least one point C between A and B such that the tangent line at this point is parallel to the line segment between A and B (Fig. 6.9). There can be several such points (Fig. 6.8b). f(b) -

The properties of continuity and differentiability are important in the theorem, as can be observed in Fig. 6.8c,d. 2. Applications The mean value theorem has several useful applications. I A: This theorem can be used to prove some inequalities in the form (6.30) I f ( b ) - f(a)l < K l b - al! where K is an upper bound of lf’(z)i for every x in the interval [a,b]. 1 . . I B: How accurate is the value of f(r)= if T is replaced by the approximate value 7i = 3.14? 1+rZ

We have:) . ( f i

-

between 0.092 084

f(i7)I =

/ r - iil 5 0.053 . 0.0016 = 0.000 085, which means

* 0.000 085

4 is l+T

6.1.4.5 Taylor’s Theorem of Functions of One Variable If a function y = f(x) is continuously differentiable (it has continuous derivatives) n - 1 times on the interval [a. a + h].and if also the n-th derivative exists in the interior of the interval. then the Taylor

388

6. Differentiation

formula or Taylor expansion is

hn-1

with 0 < 0 < 1. The quantity h can be positive or negative. The mean value theorem (6.29b) is a special case of the Taylor formula for n = 1.

6.1.4.6 Generalized Mean Value Theorem of Differential Calculus (Cauchy’s Theorem) If two functions y = f(z) and y = p(x) are continuous on the closed interval [a,b] and they are differentiable at least in the interior of the interval, and ’p’(z) is never equal to zero in this interval, then there exists at least one value c between a and b such that (6.32) The geometrical meaning of the generalized mean value theorem corresponds to that of the first mean value theorem. Supposing, e.g., that the curve in Fig.6.9 is given in parametric form x = p(t), y = f ( t ) , where the points A and B belong to the parameter values t = a and t = b respectively. Then for the point C (6.33) is valid. For p(z) = z the generalized mean value theorem is simplified into the first mean value theorem.

6.1.5 Determination of the Extreme Values and Inflection Points 6.1.5.1 Maxima and Minima The substitution value f(zo)of a function f(z)is called the relative maximum (M) or relative minimum (n) if one of the inequalities f(zo + h ) < f(z0) (for maximum), f(zo+ h ) > f(zo) (for minimum)

(6.34a) (6.34b)

holds for arbitrary positive or negative values of h small enough. At a relative maximum the value f(z0) is greater than the substitution values in the neighborhood, and similarly, at a minimum it is smaller. The relative maxima and minima are called relative or local extrema. The greatest or the smallest value of a function in an interval is called the global or absolute mazimurn or global or absolute minimum in this interval.

6.1.5.2 Necessar Conditions for the Existence of a Relative Extreme alue

t

A function can have a relative maximum or minimum only at the points where its derivative is equal to zero or does not exist. That is: At the points of the graph of the function corresponding to the relative extrema the tangent line is whether parallel to the z-axis (Fig. 6.10a) or parallel to the y-axis (Fig. 6.10b) or does not exist (Fig. 6 . 1 0 ~ ) Anyway, . these are not sufficient conditions, e.g., at the points A, B , C in Fig. 6.11 these conditions are obviously fulfilled, but there are no extreme values of the function. If a continuous function has relative extreme values, then maxima and minima follow alternately, that means, between two neighboring maxima there is a minimum, and conversely.

2. Method of Sign Change For values z-and z+, which are slightly smaller and greater than z,, and for which between 2, and xand x+ no more roots or points of discontinuity of f ’ ( x )exist, we check the sign of f‘(z).When during the transition from f’(z-) to f’(z+) the sign of f’(z)changes from ”+I’ to I ( - ” , then there is a relative

6.1 Differentiation of Functions of One Variable 389

Figure 6.10

\/iJj\ * x

0 Figure 6.11

6.1.5.3 Relative Extreme Values of a Differentiable Explicit Function y = f(z) 1. Determine the Points of Extreme Values Since f’(z) = 0 is a necessary condition where the derivative exists, after determining the derivative f’(x), first we calculate all the real roots xlr xZ1.. . , x,,. . . x, of the equation f‘(z) = 0. Then we check each of them, e.g., x, with the following method.

.

maximum of the function f(z)at z = x, (Fig. 6.12a): if it changes from “-’’ to “t”, then there is a relative minimum there (Fig. 6.12b). If the derivative does not change its sign (Fig. 6.12c,d), then there is no extremum at 5 = x,,but it has an inflection point with a tangent parallel to the x-axis.

3. Method of Higher Derivatives If a function has higher derivatives at x = 2 , . then we can substitute, e.g., the root z, into the second derivative f”(z).If f”(z,)< 0 holds, then there is a relative maximum at z,, and if f”(x,) > 0 holds, a relative minimum. If f”(zZ) = 0 holds. then x, must be substituted into the third derivative f”’(x). If f”’(z,)# 0 holds, then there is no extremum at x = x, but an inflection point. If still f”’(x,)= 0 holds, then we substitute it into the forth derivative, etc. If the first non-zero derivative at z = x, is an even one. then f(z)has an extremum here: If the derivative is positive. then there is minimum, if it is negative, then there is a maximum. If the first non-zero derivative is an odd one, then there is no extremum there (actually. there is an inflection point).

Yt

a)

b)

4 Figure 6.12

4. Further Conditions for Extreme Points and Determination of Inflection Points If a continuous function is increasing below xo and decreasing after, then it has a maximum there; if it is decreasing beloy and increasing after, then it has a minimum there. Checking the sign change of the derivative is a useful method even if the derivative does not exist at certain points as in Fig. 6.10b,c and Fig. 6.11. If the first derivative exists at a point where the function has an inflection point, then the first derivative has an extremum there. So, to find the inflection points with the help of derivatives, we have to do the same investigation for the derivative function as we have done for the original function to find its extrema. Remark: For non-continuous functions, and sometimes also for certain differentiable functions the

I

390

6. Differentiation

determination of extrema needs individual ideas. It is possible that a function has an extremum so that the first derivative exists and it is equal to zero, but the second derivative does not exist. and the first one has infinitely many roots in an arbitrary neighborhood of the considered point. so it is meaningless to say it changes its sign there. For instance f ( z ) = z2(2 sin ( l / x ) ) for x # 0 and f ( 0 ) = 0.

+

6.1.5.4 Determination of Absolute Extrema The considered interval of the independent variable is divided into subintervals such that in these intervals the function has a continuous derivative. The absolute extreme values are among the relative extreme values. or at the endpoints of the subintervals, if their endpoints belong to them. For noncontinuous functions or for non-closed intervals it is possible that no maximum or minimum exists on the considered interval. Examples of the Determination of Extrema: IA: y = e-"', interval [-1, $11. Greatest value at z = 0, smallest at the endpoints (Fig. 6.13a). IB: y = z3 - x 2 , interval [-1, $21. Greatest value at z = +2, smallest at z = -1, at the ends of the interval (Fig. 6.13b).

4% interval [-3, +3]. z # 0. There is no maximum or minimum. Relative minimum 1+e:

IC: y =

at z = -3, relative maximum at z = 3. If we define y = 1 for x = 0, then there will be an absolute maximum at z = 0 (Fig. 6.13~).

H D: y = 2 - x i . interval [-1, +1]. Greatest value at z = 0 (Fig. 6.13d, the derivative is not finite).

6.1.5.5 Determination of the Extrema of Implicit Functions If the function is given in the implicit form F ( z ,y) = 0, and the function F itself and also its partial derivatives F,, Fyare continuous, then its maxima and minima can be determined in the following way: 1. Solution of the Equation System F(m, y) = 0, F,(z,y) = 0 and substitution of the resulting values (zlryl), ( ~ ~ , y. . ~(xZ.yi):. ) ~ . . . in Fy and F,,. 2. Sign Comparison for Fy and FZz at the Point (mi,vi): When they have different signs. the function y = f ( x ) has a minimum at x i : when Fyand F,, have the same sign, then it has a maximum at 2,. If either Fvor F,, vanishes at (z2.y t ) , then we need further and rather complicated investigation.

y&y=;lY/; + = & -

-1

m2

m1

-

2 x -1

o a)

+I

x

m'

.

y m

M2

m2

m1

I

-3 b)

3 x

-1

c)

0

4

1

x

Figure 6.13

6.2 Differentiation of Functions of Several Variables 6.2.1 P a r t i a l Derivatives 6.2.1.1 Partial Derivative of a Function

The partial derivative of a function u = f(zl,2 2 , . . . , z,, . . . ,E,) with respect to one of its n variables, e.g., with respect to x1 is defined by (6.35)

6.2 Differentiation of Functions ofSeveral Variables 391 so only one of the n variables is changing, the other n - 1 are considered as constants. The symbols

a! &. .4function of n variables can have n first-order partial ax

du

for the partial derivatives are - , ui, -

ax

du au du derivatives - - -

ax, ax?. ax3.

dU

ax, . The calculation of the partial derivatives can be done following

,,’ *

the same rules we have for the functions of one variable.

6.2.1.2 Geometrical Meaning for Functions of Two Variables If a function u. = f(x.y) is represented as a surface in a Cartesian coordinate system. and this surface is intersected through its point P by a plane parallel to the x,u plane (Fig. 6-14), then we have dU

-= tana.

(6.36a)

ax

Tvhere a is the angle between the positive x-axis and the tangent line of the intersection curve at P . which is the same as the angle between the positive x-axis and the perpendicular projection of the tangent line into the x, u plane. Here, a is measured starting at the x-axis, and the positive direction is counterclockwise if we are looking toward the positive half of the y-axis. Analogously to a , 3 is defined with a plane parallel to the y, u plane: du - = tan 3 (6.36b) dY The derivative with respect to a given direction, the so-called directional derivative, and derivative with respect to volume, will be discussed in vector analysis (see 13.2.1. p. 647 and p. 648).

Figure 6.14

Figure 6.15

6.2.1.3 Differentials of iz and f(x) 1. The Differential dx of an Independent Variable x

is equal to the increment ax.Le., dx = Ax for an arbitrary value of Ax 2. The Differential dy of a Function y = f(z)of One Variable x is defined for a given value of x and for a given value of the differential d x as the product dy = f’(x) dx.

(6.37a)

(6.37b)

392

6. Differentiation

3. The Increment Ag from x to x

+ Ax of One Variable x

is the difference Ay = f(x

+ AX) - f (x). 4. Geometrical Meaning of the Differential

(6.37~)

If the function is represented by a curve in a Cartesian coordinate system, then dy is the increment of the ordinate of the tangent line for the change of x by a given increment dx (Fig. 6.1).

6.2.1.4 Basic Properties of the Differential 1. Invariance Independently of whether x is an independent variable or a function of a further variable t dy = f ' ( x )dx is valid.

2. Order of Magnitude

(6.38)

+

If dx is an arbitrarily small value, then dy and Ay = y(x Ax) - y(z) are also arbitrarily small, but AY = 1. Consequently, the difference between them is also arbitrarily equivalent amounts, Le., lim As+O dy small, but of higher order than dx, dy and Ax (except if dy = 0 holds). Therefore, we get the relation (6.39) which allows us to reduce the calculation of a small increment to the calculation of its differential. This formula is frequently used for approximate calculations (see 6.1.4.4, p. 387 and 16.4.2.1, 2., p. 793).

6.2.1.5 Partial Differential For a function of several variables u = f (2,y, . . .) we can form the partial differential with respect to one of its variables, e.g., with respect to x,which is defined by the equality (6.40)

6.2.2 Total Differential and Differentials of Higher Order 6.2.2.1 Notion of Total Differential of a Function of Several Variables (Complete Differential) 1. Differentiability

The function of several variable u = f(q, 2 2 , . . . , x,,. . . , 2), is said to be differentiable at the point Po(xl0, ZZO,. . . , xt0,. . . , x , ~ )if at a transition to an arbitrarily close point P(xlo+dxl, zZ0+dx2,. . . , z , ~ +dx,. . . . , x , ~ dz,) with the arbitrarily small quantities dxl, dx2,.. . dx,, . . . dx, the complete increment AU = f ( ~ 1 0 dxi, 2 2 0 dxz,. . . , Z,O dx,, . . . , ZO, + dz,) -f(x10.~20,.",~,0~".,~,0) (6.41a) of the function differs from the sum of the partial differentials of all variables

+

+

+

+

(6.41b) by an arbitrarily small amount in higher order than the distance = \/dx: + dx; + . . . + dxz . (6.41~) A continuous function of several variables is differentiable at a point if its partial derivatives, as functions of several variables, are continuous in a neighborhood of this point. This is a sufficient but not a

6.2 Dzfferentiatzonof Functzons of Several Varaables 393

necessary condition, while the simple existence of the partial derivatives at the considered point is not sufficient even for the continuity of the function.

2. Total Differential If u is a differentiable function, then the sum (6.41b) (6.42a) is called the total diferential of the function. With the n-dimensional vectors (6.42~) the total differential can be expressed as the scalar product du = (grad)T &.

(6.42d)

In (6.42b)! there is the gradient. defined in 13.2.2, p. 648, for n independent variables.

3. Geometrical Representation The geometrical meaning of the total differential of a function of two variables u = f(x,y ) , represented in a Cartesian coordinate system as a surface (Fig. 6.15),is that du is the same as the increment of the applicate (see 3.5.3.1,2., p. 208) ofthe tangent plane (at the same point) ifds and d y are the increments of I and y. From the Taylor formula (see 6.2.2.3, l., p. 394) it follows for functions of two variables that

+ 8.f

~ ( Z ~ =Yf () z o , ~ o ) G(GI.YO)(X - 2 0 )

+ -af (~o,Yo)(Y dY

-YO)

(6.43a)

+R1.

Ignoring the remainder R1: we have that 21

= f (eo; Yo)

+ af

+ I > Yo)(.

- 20)

af + -(% dY

(6.43b)

Yo)(Y - Yo)

gives the equation of the tangent plane of the surface u = f(e,y ) a t the point Po(e0,yo, 210).

4. The Fundamental Property of the Total Differential is the invariance with respect to the variables as formulated in (6.38) for the one-variable case. 5 . Application in Error Calculations In error calculations we use the total differential d u for an estimation of the error A u (see (6.41a)) (see, e.g.. 16.4.1.3, 5.,p. 790). From the Taylor formula (see 6.2.2.3, 1.) p. 394) we have (6.44) IAUI = ldu Ri1 I IduI t /Ri1 x IdUI, Le., the absolute error lhul can be replaced by ldul as a first approximation. It follows that du is a linear approximation for Au.

+

6.2.2.2 Derivatives a n d Differentials of Higher Order 1. Partial Derivatives of Second Order, Schwarz's Exchange Theorem . . . , x , , . . . , e,) can be calculated

The second-order partial derivative of a function u = f (q, 2 2 ,

azu aZu ax: ax;

, or with respect to another

with respect to the same variable as the first one was, Le., - -

8%

8%

'

I

' '

a2u

variable, i.e. - - -, . . . . In this second case we talk about mixed derivatives. If d ~ 1 a ~ ' 8x2823 2 ' 8238x1 at the considered point the mixed derivatives are continuous. then (6.45)

394

6. Differentiation

holds for given x1 and x2 independently of the order of sequence of the differentiation (Schwarz’s ex-

change theorem). d3u

8321

Partial derivatives of higher order such as, e.g., - -, . . are defined analogously. ax3 ’ dxdy2

2. Second-Order Differential of a Function of One Variable u = f ( z ) Thesecond-order differentialofafunctiony = f ( x ) ofonevariable, denoted by thesymbolsd’y, d2f(x), is the differential of the first differential: dzy = d(dy) = f”(x)dx2. These symbols are appropriate only if x is an independent variable, and they are not appropriate if x is given, e.g., in the form x = z(v). Differentials of higher order are defined analogously. If the variables ~ 1 ~ x. .2. ,I,, , . . . , x, are themselves functions of other variables, then we get more complicated formulas (see 6.2.4, p. 397). 3. Total Differential of Second Order of a Function of Two Variables u = f(z,y) a2 U d’U 8% (6.46a) d2u = d(dU) = -dx2 + 2dx dy + -dy’ 8x2 azay dY or symbolically

’

(6.46b)

4. Total Differential of n-th Order of a Function of Two Variables (6.47) 5 . Total Differential of n-th Order of a Function of Several Variables d dnU = -dxl -dx2 . . . ax2

(11+

+ +

(6.48)

6.2.2.3 Taylor’s Theorem for Functions of Several Variables 1. Taylor’s Formula for Functions of Two Variables a ) First Form of Representation:

(6.49a) Here ( a 3b) is the center of expansion and & is the remainder. Sometimes we write, e.g., instead of

The terms of higher order in (6.49a) can be represented in a clear way with the help of operators:

6.2 Differentiation of Functions of Several Variables 395

This symbolic form means that after using the binomial theorem the powers of the differential operators

d

d

- and - represent the higher-order derivatives of the function f(x,y). Then the derivatives must dx dy be taken at the point (a. b). b) Second Form of the Representation:

c) Remainder: The expression for the remainder is ntl

R

l)! ( b h t i k )

-- (n

f(x + Oh. y + O k )

(0 < 0

< 1).

(6.49d)

2. Taylor Formula for Functions of m Variables The analogous representation with differential operators is f(x + h, y + k , . . . t 1)

+

where the remainder can be calculated by the expression 1

a

R - -( - h - ( n + l ) ! ax

+ -dya k + . . . + g3tl )

ntl

f ( s + O h , y + 8 k , . . . ,t+Q1) (0

< Q < 1).

(6.5Ob)

6.2.3 Rules of Differentiation for Functions of Several Variables 6.2.3.1 Differentiation of Composite Functions 1. Composite Function of One Independent Variable 21 = f ( X l r X 2 . . . . ? X , ) , 21 = XI(<), 2 2 = xz(<).. . . , 2 , = x,(E) d~ - -d~ dxl du dxz t -a< 8 x 1 d< 8x2 d<

du dx +. . . + -2, ax, dS

(6.5la) (6.51b)

(6.52b)

396

6. Differentiation

6.2.3.2 Differentiation of Implicit Functions 1. A Function g = f(z) of One Variable is given by the equation F ( x , y) = 0. Differentiating (6.53a) with respect to x with the help of (6.51b) we get

F,

+ F,y’

=0

(6.53b)

Fz and y’= -F Y

(6.53a)

(F, # 0).

(6.53~)

Differentiation of (6.53b) yields in the same way

+

+

F,, 2FzYJ’ t FyY(y’)’F,y” = 0, so considering (6.53b) we have

(6.53d)

y” = 2FZF,FE, - (F,)’FZz - (Fz)2F,y

(6.53e) (FYI3 In an analogous way we can calculate the third derivative F,,, 3FZZyy’t 3FZ,,(y’)’ F,,s(y’)3 3F,,y” + 3F,,y’y” + F,y”’ = 0, (6.53f) from which y”’ can be expressed. 2. A Function u = f(z1,22,. ,zi, ,z,) of Several Variables is given by the equation F ( Z l , Z Z , . . . ,x,, . . . , z n , u )= 0. (6.54a) The partial derivatives

+

+

+

.. . ..

(6.54b) can be calculated similarly as we have shown above but we use the formulas (6.5213). The higher-order derivatives can be calculated in the same way. 3. Two Functions y = f(z)and z = cp(z) of One Variable are given by the system of equations F ( z ,y, z ) = 0 and @(x)yI z ) = 0. (6.55a) Then differentiation of (6.55a) according to (6.51b) results in F, + F,y’ + F,z’ = 0, @, t @,y) t @,z’ = 0, (6.55b) (6.55,) The second derivatives y” and z” are calculated in the same way by differentiation of (6.55b) considering y’ and z’. 4. n Functions of One Variable Let the functions yl = f(x),yz = p ( ~ .) . ~, yn. = $(x) be given by a system (6.56a) F ( z ,Y I Y~Z ’~ . , > yn) = 0% @(xs Y I Y~Z ~ ,, Yn) = 0, , . . , !P(z, y l j yz,. . . > yn) = O of n equations. Differentiation of (6.56a) using (6.51b) results in , j

F,

+ F,,yi + F,,y; +

@,+@,,y;

+@,*y;

’.

I

t F,,yk = 0,

+ ‘.+@,,Y:,

= 0,

i s i + i t i + : =o. P, + !Pv1y; + !P,*y; + . ’ . + !Pvk,y; = 0.

(6.56b)

Solving (636b) we get the derivatives yi, yi, . . . , yLI we are looking for. In the same way we can calculate the higher-order derivatives.

6.2 Dzfferentzatzon of Functzons of Several Varaables 397 5 . Two Functions u = f(z,y), v = q ( z ,y) of Two Variables are given by the system of equations F ( x , y. u.e) = 0 and @ ( x y.u,u) . = 0. (6.57a) Then differentiation of (6.57a) with respect to x and y with the help of (6.52b) results in

a u av au au , - and the system (6.57~)for -, - give the first-order partial ax ax aY aY derivatives. The higher-order derivatives should be calculated in the same way. 6. n Functions of m Variables Given by a System of n Equations The first-order and higher-order partial derivatives can be calculated in the same way as we did in the previous cases. Solving the system (6.57b) for

6.2.4 Substitution of Variables in Differential Expressions and Coordinate Transformat ions 6.2.4.1 Function of One Variable Suppose, given a function and a functional relation containing the independent variable, the function, and its derivatives: y = f(x),

(6.58b)

(6.58a)

If the variables are substituted, then the derivatives can be calculated in the following way: Case la: The variable x is replaced by the variable t . and they have the relation 2 = !p(t). Then we have

(6.59a) (6.59b)

{

d3Y - 1 d3Y - - - [p’(t)]’? - 3 ~ ’ ( V”(t)$ t ) dZY [3[p”(t)I2- ~ ’ ( t ) dx3 [p‘(t)15 dt Case l b : If the relation between the variables is not explicit but it is given in implicit form @(x. t ) = 0.

+

(6.60)

.

dy d2y . -are d3y then the derivatives calculated by the sameformulas, but the derivatives V ’ ( t ) ,cpf’(t), dx dx2 dx3 p”’(t) must be calculated according to the rules for implicit functions. In this case it can happen that the relation (6.58b) contains the variable x. To eliminate x, the relation (6.60) is used. Case 2: If the function y is replaced by a function u, and the relation between them is Y = Pb). (6.6la) then the calculation of the derivatives can be performed using the following formulas: du

d2y

(6.61 b) (6.61~)

398

6. Differentiation

Case 3: The variables x and y are replaced by the new variables t and u , and the relations between them are given by x = p ( t . u ) , y = V ( t .u ) . For the calculation of the derivatives the following formulas are used:

(6.62a)

(6.62b)

(6.62~)

(6.62d) azj

with A = - +

at

dydu --. du dt

B = - ap +-- dqdu at du dt

(6.62e)

(6.62f)

d3y can be done in an analogous nay. The determination of the third derivative dx3 I For the transformation from Cartesian coordinates into polar coordinates according to x=pcosp:. y=psinp the first and second derivatives should be calculated as follows: dy p' sin p t p cos q d2y - p2 + 2pt2- pp" - -(6.63b) -dx2 ( ~ ' C O S -~ ~ s i n '~ ) ~ dx p'cosp - psiny '

(6.63a)

(6.63~)

6.2.4.2 Function of Two Variables Suppose given a function and a functional relation containing the independent variables, the function and its partial derivatives: d = f(x. y),

(6.64a) (6.64b)

If x and y are replaced by the new variables u and v given by the relations x = p(u.v), y = y ( u , v ) , then the first-order partial derivatives can be expressed from the system of equations

(6.65a)

(6.6513) with the new functions A. B , C, and D of the new variables u and v

ad a,. ad -=.A-+B-, ax au at'

ad ay

aw

aw

au

av

- -- C - + D - .

(6.652)

6.2 Differentiation of Functions o f Several Variables 399

The second-order partial derivatives are calculated with the same formulas, only we do not use w in dd aw them but its partial derivatives - and -, e.g., dx ay d2w = d (%) dd = (A: + B z ) =,(AA%+B!?+!!&+E") au2 auav au au du du ~

~

The higher partial derivatives can be calculated in the same way. IExpress the Laplace operator (see 13.2.6.5, p. 655) in polar coordinates (see 3.5.2.132.. p. 190): (6.67a)

x = pcosp.

y = psinp.

(6.67b)

(6.67~)

Remark: If functions of more than two variables should be substituted, then similar substitution formulas can be derived.

6.2.5 Extreme Values of Functions of Several Variables 6.2.5.1 Definition A function u = f(zl9 2 2 , . . . . x,,. . . ,z,)has a relative extreme value at a point Po(z10,~ 2 0 , .., , z,~,. . . , xn0).if there is a number e such that for every point belonging to the domain 210 - E < z1 < x10 + E , 220 - E < 2 2 < 120 + E . . . . . zn0- E < z, < z o, + e and to the domain of the function but different

from Po. then for a maximum the inequality f(zl.zz,...lzn) < f(zlor~20>..~~~n0) holds, and for a minimum the inequality

(6.68a)

(6.68b) f ( x 1 , 221 ' ' ' > z,) > f(%, z20, ' ' ' > zno) holds. Using the terminology of several dimensional spaces (see 2.18.1, p. 117) a function has a relative maximum or a relative minimum at a point if it is greater or smaller there than at the neighboring points.

6.2.5.2 Geometric Representation In the case of a function of two variables, represented in a Cartesian coordinate system as a surface (see 2.18.1.2, p. 117). the relative extreme value geometrically means that the applicate (see 3.5.3.1,2., p. 208) of the surface in the point A is greater or smaller than the applicate of the surface in any other point in a sufficiently small neighborhood of A (Fig. 6.16).

400

6. Differentiation

Jm

"t

UI

0

Figure 6.16 If the surface has a relative extremum at the point Po which is an interior point of its domain, and if the surface has a tangent plane at this point, then the tangent plane is parallel to the x,y plane (Fig. 6.16a,b). This property is necessary but not sufficient for a maximum or minimum at a point Po. For example Fig. 6 . 1 6 ~shows a surface having a horizontal tangent plane at Po, but there is a saddle point here and not an extremum.

6.2.5.3 Determination of Extreme Values of Functions of Two Variables If u = f (x,y) is given, then we solve the system of equations fz = 0, fy = 0. The resulting pairs of yl), (x2,y2), . . . can be substituted into the second derivatives values (xI1 (6.69) Depending on the expression (6.70) it can be decided whether an extreme value exists: 1. In the case A > 0 the function f(z,y) has an extreme value at (zl,yt), and for fzz < 0 it is a maximum, for fzz > 0 it is a minimum (sufficient condition). 2. In the case A < 0 the function f (z, y) does not have an extremum. 3. In the case A = 0, we need further investigation.

6.2.5.4 Determination of the Extreme Values of a Function of n Variables If u = f(x1,x2,. .. equations

~

2,)

is given, then first we find a solution

..

(x10,z20,.,z,o)

of the system of the IZ

(6.71) fa = 0. fw = 0, ". I f i n = 0, because it is a necessary condition for an extreme value. We prepare a matrix of the second-order partial derivatives such that a,, = -. Then we substitute a solution of the system of equations into

az,azl

the terms, and we prepare the sequence of left upper subdeterminants (all, aI1a22 - a12a21r. . .). Then we have the following cases: 1. The signs of the subdeterminants follow the rule -, +, -, +, . . ., then there is a maximum there. 2. The signs of the subdeterminants follow the rule +, t,t,t,. . ., then there is a minimum there. 3. There are some zero values among the subdeterminants, but the signs of the non-zero subdeterminants coincide with the signs of the corresponding positions of one of the first two cases. Then further investigation is required: Usually we check the values of the function in a close neighborhood of x10.220.. . . . x,o.

6.2 Differentiationof Functions of Several Variables 401 4. The signs of the subdeterminants does not follow the rules given in cases 1. and 2.: There is no extremum at that point. The case of two variables is of course a special case of the case of n variables, (see [6.4])

6.2.5.5 Solution of Approximation Problems Several different approximation problems can be solved with the help of the determination of the extreme values of functions of several variables, e.g., fitting problems or mean squares problems. Problems to solve:

1,

ff

Determination of an approximate solution of an overdetermined linear system ofequations (see p. 890, 19.2.1.3). Methods: For these problems the following methods are used: Gaussian least squares method (see, e.g., 19.6.2,p. 916).

Calculus of observations (or fitting) (see 19.6.2, p. 916) and regression (see 16.3.4.2, l.,p. 778).

6.2.5.6 Extreme Value Problem with Side Conditions Suppose we have to determine the extreme values of a function 2~ = f(x1,2 2 , . . . , 2 , ) of n variables with the side conditions (6.72a) +(x1,22. . . . >x,) = o , $ ( 2 1 $ 2 2 > . . . , z,)=O , . . . , X ( Z 1 > 2 2 r. . . ,2,)=0. Because of the conditions, the variables are not independent, and if the number of conditions is k , obviously k < n must hold. One possibility to determine the extreme values of u is to express k variables with the others from the system of equations of the conditions, to substitute them into the original function, then the result is an extreme value problem without conditions for n - k variables. The other way is the Lagrange multiplier method. We introduce k undefined multipliers A, 1-1). . . , K ) and we compose the Lagrange junction (Lagrangian)of n + k variables X I , 5 2 , . . . , x,, A, p>. . . , n: CJ

(J1;ZZ:.

= f(21,

. . ,2,, 22:.

t , ,

A,P,. ..,K) 2,) A P("1, 2 2 ) . . . , 2 , )

+

+ pd421,

22,

. .. ,2,) + .' .

+nx(z1,x2,.. . ,%). (6.72b) A4nextremum of the function @ can be only at a point (210, ~ 2 0 , .. . , x , ~ ,XO, P O , .. . , K O ) which is the solution of the system of n k equations (6.71) p = o o , v = o ., . . , x = o , CJ,,=o, G2,=0 , . . . ,@,,=O (6.72~) with unknowns xl,2 2 , . . . , z,, A, p, . . . IG. Since the equations of the side conditions are fulfilled, the extreme value of CJ will be an extreme value also for f. So we have to look for the extremum points off amongthesolutionszlO,xzo,,, , ,z,oofthesystemofequations (6.72b). Todeterminewhether thereare really extreme values at these points fulfilling the necessary conditions requires further investigations, for which the general rules are fairly complicated. Usually we use some appropriate and individual calculations depending on the function f to prove if there is an extremum there, or not. IThe extreme value of the function u = f ( z ,y) with the side condition p(s,y) = 0 will be determined from the three equations

+

(6.73) There are three unknowns. x,y, A.

402

7. Infinite Series

7 Infinite Series 7.1 Sequencesof Numbers 7.1.1 Properties of Sequences of Numbers 7.1.1.1 Definition of Sequence of Numbers An znjinzte sequence of numbers is an infinite system of numbers a ] ,az, . . . , a n r. . . or briefly { a k } with k = 1,2, . . . , (7.1) arranged in a given order. The numbers of the sequence of numbers are called the terms ofthe sequence. Among the terms of a sequence of numbers the same numbers can occur several times. .4 sequence is considered to be defined if the law offormatzon, Le., a rule is given, by which any term of the sequence can be uniquely determined. Mostly there is a formula for the general term a,. Examples of Sequences of Numbers: I B :~ , = 4 + 3 ( n - 1 ) : 4 , 7 . 1 0 , 1 3 , 1 6 , . . . . IA: a n = n : 1,2,3.4,5. . . . .

(-i)

n-1

IC: a, = 3

:

3 3 3 3 3,---, --: -i,

Is,.. . .

ID: a, = (-l)n+l: l , - l ~ l , - l , l ~

1 1 3 7 3 11 (read 2- = -). IE: a, = 3 - 9: 1,2,2- 2- 2-, , . , 2‘ 4 ’ 8 4 4 1 1 n-1 IF: an = 3- - - 1 0 - T for odd n and 3 3 1 2 n an -- 3- + - .10-2 + 1 for even n: 3; 4; 3.3; 3.4; 3.33;3.34; 3.333;3.334;. . . 3 3 1 -1 -1 -1 IG *. an -- - : 1 13 -2’ IH: U, = (-l),+’n: 1, -2,3, -4,5, -6,. . . . 3’ 4 ’ 5 ” ” ’ n i l II: a, = -for odd n and a, = 0 for even n: -1,O, -2,O, -3,0, -4,0,. 2

1 IJ: a, = 3 - 7 for odd nand a, = 13 25-1

22 - 2

...

1 1 3 3 for even n: 1,11,2,12,2- 12- 2- 12-, . . . . 2’ 2 ’ 4 ’ 4

7.1.1.2 Monotone Sequences of Numbers A sequence ai,a2, . . . a,, . . . is monotone increasing if ~

al 5 a2 5 a3 5 . . . I an I . . . , is valid and it is monotone decreasing if

(7.2)

al 2 a2 2 a3 2 . , , 2 a, 2 , , , (7.3) is valid. We talk about a strictly monotone increasing sequenceor strictly monotone decreasing sequence, if equality never holds in (7.2) or (7.3). Examples of Monotone Sequences of Numbers: IA: Among the sequences from A to J the sequences A, B, E are strictly monotone increasing. IB: The sequence G is strictly monotone decreasing.

7.1.1.3 Bounded Sequences A sequence is called bounded if for all terms Ian1 < K , (7.4) is valid for a certain K > 0. If such a K (bound)does not exist, then the sequence is unbounded.

7.1 Seuuences of Numbers 403

IAmong the sequences from A to J the sequences C with K = 4. D with K = 2, E with K = 3, F with K = 5, G with K = 2 and J with K = 13 are bounded.

7.1.2 Limits of Sequences of Numbers 1. Limit of a Sequence of Numbers .An infinite sequence of numbers (7.1) has a limit A if for an unlimited increase of the index n the difference a, -A becomes arbitrarily small. Precisely defined this means: For an arbitrarily small E > 0 there exists an index no(&)such that for every n > no (7.58) la, - dl < E . The sequence has the limit +w (-co): if for arbitrary K > 0 there exists an index no(K)such that for every n > no a, > K (a, < -K). (7.5b)

2. Convergence of a Sequence of Numbers If a sequence of numbers {a,} satisfies (7.5a). then we say it converges to A. This is denoted by lim a, = A or a, + A . (7.6) n+cc

IAmong the sequences from A to J on the previous page, C with A = 0, E with A = 3, F with A = 3 f IG with A = 0 are convergent.

3. Divergence of a Sequence of Numbers Yon-convergent sequences of numbers are called divergent. We talk about proper diWeTgEnCE in the case of (7.5b), Le., if as n exceeds any value: a, exceeds any large positive number K ( K > 0) so that it never goes below, or if as n exceeds any value, a, goes below any negative number - K ( K > 0) with arbitrarily large magnitude and never increases above it; i.e., if it has the limit izm. We use the notation: a, = -w ( a , < -KVn > no). (7.7) nlirn i w a, = x (a, > KVn > no) or n lim i - x Otherwise the sequence is called imprOpETl2/ divergent.

Examples of Divergent Sequences of Numbers: IA: Among the sequences from A to J on the previous page, A and B tend +w, they are properly divergent. IB: Among the sequences D is improperly divergent. 4. Theorems for Limits of Sequences a) If the sequences {a,} and {b,} are convergent, then

+ b,) = hla, + dil b,, (7.8) hold, and if b, # 0 for every n,and ill b, # 0, then nlim i x (a,

nlim i m( a

b ) - (lim n+m a,)(dihb,)

(7.9)

(7.10)

Ail

b, = 0 and {a,} is bounded, then hl(a,b,) = 0 even if {a,} does not have any finite is valid. If limit. b) If )rla, = b, = A is valid and at least, from an index n1 and beyond, the inequality a, 5 c, 5 b, holds, then we also have il% c, = ‘4. (7.11)

All

c) .4 monotone and bounded sequence has a finite limit. If a monotone increasing sequence a1 5 a2 5 a3 5 . is bounded above. Le., a, 5 K1 for all n,then it is convergent. and its limit is equal to

404

7. Infinite Series

its least upper bound which is the smallest possible value for If1. If a monotone decreasing sequence a1 2 a2 2 a3 2 . . . is bounded below, i.e., a, 2 Kz, then it is convergent, and its limit is equal to its greatest lower bound which is the largest possible value for Kz.

7.2 Number Series 7.2.1 General Convergence Theorems 7.2.1.1 Convergence and Divergence of Infinite Series 1. Infinite Series and its Sum From the terms at of an infinite sequence {ak} (see 7.1.1.1, p. 402) the formal expression

+ a2 + . . . + a n + . .. = c a k m

al

(7.12)

k=l

can be composed and this is called an infinite series. The finite sums S1 = a l , S2 = a1 + a2, . . . Sn=

n

C ak

(7.13)

k=l

are called partial sums. 2. Convergent and Divergent Series A series (7.12) is called convergent if the sequence of partial sums {S,} is convergent. The limit

s = iil s, =

c m

(7.14)

ak

k=l

is called the sum, and ak is called the general term of the series. If the limit (7.14) does not exist or it is equal to h a ,then we call the series divergent. In this case the partial sums are not bounded or they oscillate. So the determination of the convergence of an infinite series is reduced to the determination of the limit of a sequence {Sn}. IA: The geometric series (see 1.2.3, p. 19) 1 1 1 1 (7.15) 1 + - + - + - +...+ 2 4 8 2n is convergent. IB: The harmonzc series (7.16) and the series (7.17) and (7.18)

+

1 ... 1 + -1+ - 1+ . . . + - + 2 3 n

(7.16)

+ + 1 + ... + 1 + ...

1 1

1 - 1 + 1 - . . . +(-l)n-l+... are divergent. For the series (7.16) and (7.17) &%S,, = co holds, (7.18) oscillates.

3. Remainder

c

(7.17) (7.18)

m

The remaznderof a convergent series S =

ak

is the difference between its sum S and the partial sum

k=l

Sn: ffi

R, = S - Sn=

C

ak

= a,+l

+an+2

+ ... .

(7.19)

k=n+l

7.2.1.2 General Theorems about the Convergence of Series 1. Leaving out the Initial Terms If we leave out finitely many initial terms of a series or introduce finitely many further terms into it at the begin or we change the order of finitely many terms, then its

7.2 Number Series 405

convergence behavior does not change. Exchange of the order of finitely many terms does not affect the sum if it exists. 2. Multiplication of Terms If all the terms of a convergent series are multiplied by the same factor e, then the convergence of the series does not change; its sum is multiplied by the factor c. 3. Termwise Addition or Subtraction If we add or subtract two convergent series 00

a l + a2 + . , , + a, + ' . . =

bl

ak = SI,(7.20a)

+ b:, + . . . t b, +

M

'.

=

k=l

bk =

sz

(7.20b)

k=l

term by term, then the result is a convergent series, and its sum is (7.20~) (alib,) + (a2 b:,) + . ' . (a, b,) t ' . = SI S2. 4. Necessary Criterion for the Convergence of a Series The sequence of terms of a convergent series is a null sequence: lim a, = 0. (7.21)

*

*

+

*

nim

This is only a necessary but not suficzent condztzon. W For the harmonic series (7.16) AI% a, = 0 holds, but

)rI S, = 03.

I

7.2.2 Convergence Criteria for Series with Positive Terms 7.2.2.1 Comparison Criterion Suppose we have two series al

+ ~2 + . . . +a, +

m ' '

=

ak.

(7.22a)

k=l

bl + bz t ... + bk + . . . =

m

bk

(7.22b)

k=l

with only positive terms (a, > 0, b, > 0).If a, 2 b, holds from a certain n, then the convergence of the series (7 22a) yields the convergence of the series (7.22b), and the divergence of the series (7.22b) yields the divergence of the series (7.22a). IA: Comparing the terms of the series 1 + -1- $ -1+ . . . + - +1. . . (7.23a) 22 33 nn with the geometric series (7.15), the convergence of the series (7.23a) follows. From n = 2 the terms of the series (7 23a) are smaller than the terms of the convergent series (7.15): 1 1 (7.23b) -< ( n 22). nn 2"-1 ~

IB: From the comparison of the terms of the series I + - 1+ - + .1. . + - + . . .1

J z d

fi

(7.24a)

with the terms of the harmonic series (7.16) follows the divergence of the series (7.24a). For n > 1 the terms of the series (7.24a) are greater than those of the divergent series (7.16): 1 1 (7.24b) - > - ( n > 1). & n

7.2.2.2 D'Alembert's Ratio Test If for the series

+ + ". +a, + . .

al a2

m

=

ak k=l

(7.25a)

406

7. Infinite Series

all the ratios an

gent:

are smaller than a number p < 1 from a certain n onwards, then the series is conver-

ant1

--
(7.25b)

an

If these ratios are greater than a number Q > 1 from a certain n onwards, then the series is divergent. From the two previous statements it follows that if the limit anAl

lim (7.25~) a =p exists. then for p < 1 the series is convergent. for p > 1 it is divergent. In the case p = 1 the ratio test gives no information whether the series is convergent or not. 1 2 3 IA : Theseries -+,+-+..,+:+... (7.26a) 2 2 23 2% is convergent, because 1 n+l n 1 t 1 p = nlim i x (3 : p) = ;;I -f = 5 < 1 holds. (7.26b) n i s

n+l 3 4 IB: Fortheseries 2 + - t - + . . . + - t . . . n2 4 9 the ratio test does not give any result about whether the series is convergent or not. because

(7.27a)

(7.27b)

7.2.2.3 Root Test of Cauchy If for a series

al 1a? + ' . ' + a,

+.'. =

5

(7.28a)

ak

k=l

from a certain n onwards for every value

6

i/;;; 1 holds, then the series is divergent. From the previous statements it follows that if lim fi = p n i x

(7.28b)

6is greater than a number Q (7.28~)

exists. in the case p < 1 the series is convergent, in the case p > 1 it is divergent. For p = 1 with this test we cannot tell anything about the convergence behavior of the series.

ITheseries 5+(:)4+(:)g+...+(-&)n2t 1

(7.29a)

is convergent because (7.29b)

7.2.2.4 Integral Test of Cauchy 1. Convergence If a series has the general term a, = f(n), and f(z)is a monotone decreasing

function such that the improper integral

7.2 Number Series 407

M

J

(see 8.2.3.2. l., p. 452)

f(2)ds

(7.30)

c

exists (it is convergent). then the series is convergent. 2. Divergence If the above integral (7.30) is divergent, then the series with the general term a, = f(n)is divergent. too. The loFver limit c of the integral is almost arbitrary but it must be chosen so that for c < z < m the function f(z)should be monotone decreasing. IThe series (7.27a) is divergent because (7.31)

7.2.3 Absolute and Conditional Convergence 7.2.3.1 Definition Along with the series

al

+ a2 + . . . + a , + . . . =

oc

(7.32a)

ak. kzl

whose terms can have different signs, we consider also the series 00

(7.32b)

~al~+~a~~+”’+~anl+”’=~lak~. k=l

whose terms are the absolute values of the terms of the original sequence (7.32a). If the series (7.3213) is convergent, then the original one (7.32a) is convergent, too. (This statement is valid also for series with complex terms.) In this case, the series (7.32a) is called absolutely convergent. If the series (7.32b) is divergent, then the series (7.32a) can be either divergent or convergent. In the second case, the series (7.32a) IS called condztzonally convergent. s i n a sin20 sin no (7.33a) IA : Theseries 2 22 2n where a is an arbitrarily constant number, is absolutely convergent, because the series of absolute values sin no is convergent. This is obvious by comparing it with the geometric series (7.15): with terms 2” - + - + + . . + - + + . . . .

I I ~

(7.3313) 1 1 1 (7.34) W B : The series 1 - - + - - . . . t (-l)n-l- + ” . 2 3 is conditionally convergent, because it is convergent according to (7.36b), and the series made of the 1

absolute values of the terms is the divergent harmonic series (7.16) whose general term is - = la,l.

7.2.3.2 Properties of Absolutely Convergent Series 1. Exchange of Terms a) The terms of an absolutely convergent series can be exchanged with each other arbitrarily (even infinitely many of them) and the sum does not change. b) Exchanging an infinite number of terms of a conditionally convergent series can change the sum and even the convergence behavior. Theorem of Riemann: The terms of a conditionally convergent series can be rearranged so that the sum will be equal to any given value, even to !cx

408

7. Infinite Series

2. Addition and Subtraction Absolutely convergent series can be added and subtracted term-by-term; the result is absolutely convergent.

3. Multiplication Multiplying a sum by a sum, the result is a sum of the products where every term of the first factor is multiplied by every term of the second one. These two-term products can be arranged in different ways. The most common way for this arrangement is made as if the series were power series, Le.:

+

+

(a1 + a2 + ...+a, + ...)(b1 bz t . . . b, + ...) = albl+azbl + a 1 b ~ + a s b l + a z b z + a l b 3 + . . . + a , b l w--

+a,..lbz+...+alb,+...

.

(7.35a)

If two series are absolutely convergent, then their product is absolutely convergent, so it has the same sum in any arrangement. If a, = Sa and b, = Sb hold, then the sum of the product is

s = sash. If two series a1

(7.3513) m

+ a2 + ...+a, + ... = n=l a, and bl + bz + ... + b, + ... = n=l b, are convergent, and W

at least one of them is absolutely convergent, then their product is also convergent, but not necessarily absolutely convergent.

7.2.3.3 Alternating Series 1. Leibniz Alternating Series Test (Theorem of Leibniz) For an alternating series al-a2+a3-...!ra,F..., (7.36a) where a, are positive numbers, a sufficient condition of convergence is if the following two relations hold: 1. lima,=O and 2. a l > a z > a 3 > ~ ~ ~ > a , > . . . (7.36b) nim

IThe series (7.34) is convergent because of this criterion.

2. Estimation of the Remainder of an Alternating Series If we consider the first n terms of an alternating series, then the remainder R, has the same sign as the first omitted term a,+], and the absolute value of & is smaller than la,+ll: sign& = sign(a,+l)

with & = S - S,, (7.37a)

IS - Snl < la,+ll.

(7.37b)

IConsidering the series I - - 1+ - - 1- + .1. . 2 3 4

n Ff...

- In 2 (7.38a) -

the remainder is

I In 2 - S,, < 1 . (7.38b) n+l

7.2.4 Some Special Series 7.2.4.1 The Values of Some Important Number Series (7.39) 1 - - 1+ - - -1+ . , 1. l! 2! 3!

1

1

IF”’= e n.

(7.40)

(7.41) 1 . . . + - +1 . . . I + -1+ -1+ - + 2 4 8 2,

= 2,

(7.42)

7.2 Number Series 409

(7.43) (7.44) 1 - - $ -1+ . . . 1 -+ 1'2 2 ' 3 3 ' 4

+-+...1 n(n t 1)

t

--$-+--$...

1 1.3

1 3'5

1 5'7

1 -+ 1'3

-t

1 2'4

7

1 3.0

= 1,

(7.45)

1 $ . . . = 12' (2n - 1)(2nt 1)

1 + . , . = 3+...t (n - l)(n 4' t 1)

1 1 1 1 + . . . = -1 - t - -t . . . t 3 . 5 7 . 9 11.13 (472- 1)(4n t 1) 2

+

1 1 1 + . . . = -1 t -+ . . . t 4' 1'2.3 2.3.1 n(n l ) ( n + 2) 1 1 1 1 . . .+...t t n . . . ( n+ 1 - 1) (1 - 1)(1 - l)!' 1 . 2 . .. 1 2 . 3 . . . (1 + 1)

+

~

(7.46) (7.47) (7.48) (7.49)

+

(7.51)

(7.52) (7.53) A4 I + - 1+ - + 1 - + . . 1. + - + . . . =124 34 44 n4 90 ' 1 1 1 7A4 I--+ --... =24 34 n4 720 1 1 1 1 A4 -+-+$,+...+-++. =14 34 04 (2n t 96 ' 1 1 1 1 .2k22k-l I + -+-+-+...+-+... =Bk , * 22k 32k 42k n2k (2k)!

*-*...

(7.54) (7.55) (7.56) (7.57)

(7.58) (7.59) 1 1 1--+---+.., 32ktl 52ktl

1 72ktl

'Bk are the Bernoulli numbers tEfi are the Euler numbers

(7.60)

410

7. In.finite Series

7.2.4.2 Bernoulli and Euler Numbers 1. First Definition of the Bernoulli Numbers The Bernoulli numbers Bk occur in the power series expansion of some special functions, e.g., in the trigonometric functions tan 2 , cot x and csc 2, also in the hyperbolic functions tanh z, coth x,and cosech x. The Bernoulli numbers Bk can be defined as follows

(7.61) and they can be calculated by the coefficient comparison method with respect to the powers of z.Their values are given in Table 7.1. Table 7.1 The first Bernoulli numbers

Bk

k

k

-1 6 1 30 1 42

1

4 a

6

k

Bk

1 30 5 66 691 2 730

B k

-7 6 3 617 510 43 867 798

7 8 9

k 10 11

Bk

174611 330 854 513 138

2. Second Definition of Bernoulli Numbers Some authors define the Bernoulli numbers in the following way:

-x

-x2

-5 = 1 + B1- + Bzex - 1 l! 2!

- X2n + . . . +Bzn+ ... (2n)!

(1x1< 2n).

So we get the recursive formula %= (B+l)kt’ ( k = 1 , 2 , 3 , . ..),

(7.63)

ahere after the application of the binomial theorem (see 1.1.6.4, l., p. 12) we have to replace B,, Le.. the exponent becomes the index. The first few numbers are: 1 - 1 1 - 1 B1 = --. B 2 = 6 , B4 = -30, B6 = 42’ -

Bs =

B16

=

1

--,30 3617

5 691 B1o = 66’ B12 = --. 2730

-

-

--, 510 . . . ,

_ _ -

B3 = B5 = B7 =

The following relation is valid: Bk = (-l)k’l& (k = 1 , 2 , 3 , .. .).

(7.62)

B14

.

7

= 6’

B”

by

(7.64)

= 0.

(7.65)

3. First Definition of E u l e r N u m b e r s The Euler numbers Ek occur in the power series expansion of some special functions. e.g., in the functions sec zand sech x. The EULER numbers Ek can be defined as follows (7.66) and they can be calculated by coefficient comparison with respect to the powers of x. Their values are given in Table 7.2.

7.2 Numberseries 411

4. Second Definition of Euler Numbers Analogously to (7.63) the Eder numbers can be defined with the recursive formula ( E + l ) k ( F - l)k = 0

+

(k = 1 , 2 , 3 , . . .),

(7.67)

where after the application of the binomial theorem we have to replace we get : _ _ E2 = -1. E4 = 5, E6 = -61, Ea = 1385. = -50521, El2 = 2 702 765, E14 = -199 360 981, _ _ _ E16 = 19391 512 145,. . . , El = E3 = Ej = ' . . = 0. The following relation is valid: Ek = (-l)kG ( k = 1 , 2 , 3 , ,. .).

by E. For the first values

(7.68)

(7.69)

Table 7.2 First Euler numbers

1 2 3 4

1 5 61 1385

5 6 7

50 521 2 702 765 199360981

5 . Relation Between the Euler and Bernoulli Numbers The relation between the Euler and

Bernoulli numbers is: (7.70)

7.2.5 Estimation of the Remainder 7.2.5.1 Estimation with Majorant In order to determine how well the n-th partial sum approximates the sum of the series. the absolute value of the remainder (7.71) X

X

ak must be estimated. For this estimation we use a majorant for

of the series

1lakl, usually a k=ntl

k=l

geometric series or another series which is easy to sum or estimate. m -l For the ratio amtl of two subsequent terms of this n! ' am m! 1 1 amtl series with m 2 n 1 we have: - = -= -5 - = q < 1. So the remainder am (m+1)! m t l n t 2 1 1 1 t -t . ' . can be majorized by the geometric series (7.15) with the R, = ( n t l)! (n 2)! ( nt 3)! 1 quotient q = -and with the initial term a = -. and it yields: n+2 ( n I)! 1 n + 2 1 n + 2 1 R, < = -- <--=(7.72) I-q ( n + l ) ! n + l n!n2+2n n.n!'

IEstimate the remainder of the series e =

n=n

+

~

+- +

+

a

412

7. Infinite Series

7.2.5.2 Alternating Convergent Series For a convergent alternating series, whose terms tend to zero with monotone decreasing absolute values, the easy estimation for the remainder is (see 7.2.3.3, l., p. 408): (7.73) Ihl= IS - Snl < Iantll'

7.2.5.3 Special Series For some special series. e.g., Taylor series, there are special formulas to estimate the remainder (see 7.3.3.3, p. 415).

7.3 Function Series 7.3.1 Definitions 1. Function Series is a series whose terms are functions of the same variable x: ~1 (z)t

(z)t

''

t fn (2) +

M

'

=

f n (E),

(7.74)

n=l

2. Partial S u m

S,(x)is the sum of the first n terms of the series (7.74):

(7.75) + fdx) -t. . . + f n ( x ) . 3. Domain of Convergence of a function series (7.74) is the set of values of x = a for which all the functions fn(z)are defined and the series of constant terms Sn(Z)

= fib)

m

(7.76) is convergent, Le., for which the limit ofthe partial sums Sn(a)exists: (7.77)

4. T h e S u m of the Series (7.74) is the function S(Z), and we say that the series converges to the function S(z). 5. Remainder & ( E ) is the difference between the sum S ( x )of a convergent function series and its partial sum Sn(x): (7.78) &(z) = S(z) - sn(x) = f n + l ( x ) t f n + 2 ( 2 ) + . ' + f n + m ( x ) + ' ' '

7.3.2 Uniform Convergence 7.3.2.1 Definition, Weierstrass Theorem According to the definition of the limit of asequence of numbers (see 7.1.2, p. 403 and 7.2.1.1,2., p. 404) the series (7.74) converges at a point x to S(z) if for an arbitrary E > 0 there is an integer N such that IS(z) - Sn(z)I < E holds for every n > N . For function series we distinguish between two cases: 1. Uniformly Convergent Series If there is a number N such that for every z in the domain of convergence of the series (7.74), IS(x) - Sn(x)I < E holds for every n > N , then the series is called uniformly convergent on this domain. 2. Non-Uniform Convergence of Series If there is no such number N which is good for every value of x in the domain of convergence, Le., there are such values of E for which there is at least one x in the domain of convergence such that IS(x) - Sn(x) I > E holds for arbitrarily large values of n, then the series is non-unzformly convergent. x x2 xn IA : Theseries l t - + - - + . . . + - - $ . . . (7.79a) l! 2! n!

7.3 Function Series 413

with the sum ex (see Table 21.3, p. 1011) is convergent for every value of z.The convergence is uniform for every bounded domain of z,and for every 1x1 < a using the remainder of the Maclaurin formula (see 7.3.3.3, 2., p. 416) the inequality (7.7913) is valid. Because ( n t l)!increases more quick than an+'. the expression on the right-hand side of the inequality, which is independent of z,for sufficiently large n will be less than E . The series is not uniformly convergent on the whole numerical axis: For any large n there will be a value of z such that ,p+l

l m e Q ' l

is greater than a previously given E .

+

+ +

+

(7.80a) IB : The series z + z(1 - z) z(1 - z)' ... z(1 - z ) n ... is convergent for every zin [0,1],because corresponding to the d'Alembert ratio test (see 7.2.2.2. p. 405) p = lim ,+XI

a,tl ~

a,

1

= 11 - z/< 1holds for 0

< z 5 1 (for z = 0 S = 0 holds).

(7.80b)

The convergence is non-uniform, because + (1 - z)n+' = (1 - z)n+l S(Z) - Sn(2)= z[(l is valid and for every n there is an zsuch that (1- z)ntlis close enough to 1, Le., it is not smaller than (7.80c) E . In the interval a 5 z 5 1 with 0 < a < 1 the series is uniformly convergent. 3. Weierstrass Criterion for Uniform Convergence The series (7.81a) fi(.) s f d z ) + " ' + f n ( z ) +'.. is uniformly convergent in a given domain if there is a convergent series of constant positive terms (7.81b) e1 + C' t .' . + c, t ' ' ' such that for every z in this domain the inequality lfn(~)I 5 cn (7.81~) is valid. M'e call (7.81~)a majorant of the series (7.81a).

+...I

7.3.2.2 Properties of Uniformly Convergent Series 1. Continuity If the functions fl(z),f2(z),. . ' , f n ( z )., ' . are continuous in a domain and the series f l ( z ) + f 2 ( z ) + ..+fn(z)+. ' '.isuniformlyconvergent in thisdomain, then thesum S(z) iscontinuous in the same domain. If the series is not uniformly convergent in a domain, then the sum S(z) may have discontinuities in this domain. IA: The sum of the series (7.80a) is discontinuous: S(z) = 0 for z = 0 and S(z) = 1 for z > 0. IB: The sum of the series (7.79a) is a continuous function: The series is uniformly convergent for any finite domain; it cannot have any discontinuity at any finite IC. 2. Integration and Differentiation of Uniformly Convergent Series In the domain [a, b] of uniform convergence it is allowed to integrate the series term-by-term. It is also allowed to differentiate term-by-term if the result is a uniformly convergent series. That is: (7.82a)

I

414

7. In.finite Series

7.3.3 Power series 7.3.3.1 Definition, Convergence 1. Definition The most important function series are the power series of the form a.

+ alx + a2x' + + Q,X" +. I

'

cc '.

anxn or

=

(7.83a)

n=O

a0 t a1(x - 2 0 ) + a2(x - 2 0 ) '

+ + a,(x '.

-~

0

+

m )' . ~=

an(x - x O ) ~ ,

(7.83b)

n=O

where the coefficients a, and the centre of expansion ZO are constant numbers 2. Absolute Convergence a n d Radius of Convergence .4 power series is convergent either only for x = xo or for all values of x or there is a number r > 0, the radius of convergence, such that the series is absolutely convergent for Ix - 201 < r and divergent for Ix - ZOI > r (Fig. 7.1). The radzus of convergence can be calculated by the formulas (7.84)

*

domain of convergence

x,,-r

if the limits exist. Ifthese limits do not exist, then we have to take the limit superior (lim) instead of the usual limit (see [7.6],Vol. I). At the endpoints z = t r and x = --T for the series (7.83a) and x = zo + r and x = 10 - r for the series (7.83b) the series can he either convergent or divergent.

xo+r

XO

Figure 7.1

3. Uniform Convergence .4 power series is uniformly convergent on every subinterval /z- 201 5 r0 < r of the domain of convergence (theorem ojiibel).

x x2 Xn 1 n+l (7.85) I For the series 1 + - + - + . ' . + - + ' . . we get - = lim -= 1 , Le., r = 1. n r nix n 1 2 Consequently the series is absolutely convergent in -1 < z < +1, for x = -1 it is conditionally convergent (see series (7.34) on p. 407) and for x = 1 it is divergent (see the harmonic series (7.16) on p. 404). According to the theorem of Abel the series is uniformly convergent in every interval [ - T I , + T I ] , where rl is an arbitrary number between 0 and 1.

7.3.3.2 Calculations with Power Series 1. S u m a n d P r o d u c t Convergent power series can be added, multiplied, and multiplied by a constant factor term-by-term inside of their common domain of convergence. The product of two power series is

+(&

+ aibz + azbi + a 3 b o ) X 3 +. . .

2. First Terms of Some Powers of Power Series: S = a + bx ex' dx3 ex4 j x 5 . .

+

S 2 = a'

+

+

+ 2abx + (b2 t 2ac)x'

+2(af

+ + . + 2(ad + bc)x3+ (c' + 2ae + 2bd)x4

+ be t ed)xj + . '.

(7.86) (7.87)

(7.88)

7.3 Function Series 415

(7.89)

(7.90)

(7.91)

3. Quotient of Two Power Series X

n=O anzn x:

b"xn

+ +

a0 1 a12 t azx2 -bo 1 $12 ~ Z X '

+

+ +

'.

=

' '

a.

-[I t (a1 - B1)x bo

+ (a2

+

n=O

- alBl

+ PIz p2)zz -

+ . .I.

(7.93) +(a3- azi31- a1D2- P3 - 6'13 0131~ t 2P1p2)z3 We get this formula by considering the quotient as a series with unknown coefficients, and after multiplying by the numerator we get the unknown coefficients by coefficient comparison. 4. Inverse of a Power Series If the series (7.94a) y = f(x) = ax bzZ cz3 t dx4 t ex5 t fz6t ' . ( a # 0) is given. then its inyerse is the series (7.94b) z = p(y) = i l y ByZt cy3t Dy4 t Ey5 Fy6 ' ' ' . Taking powers of y and comparing the coefficients we get 1 b 1 1 B = -A= C = -(2b2 - ac), D = -(5abc - azd - 5b3), a a3 ' a5 a7 1 (7.94c) E = -(6a2bd t 3aZc2t 14b4 - a3e - 21ab'c).

+

+

+

+

+

-.

o9

1 F = -(7a3be all

t 7a3cd + 84ab3c - a4f

-

28azb2d- 28azbcz - 42b5).

The convergence of the inverse series must be checked in every case individually.

7.3.3.3 Taylor Series Expansion, Maclaurin Series There is a collection of power series expansions of the most important elementary functions in Table 21.3 (see p. 1009). Usually, we get them by Taylor expansion.

1. Taylor Series of Functions of One Variable If a function f ( z ) has all derivatives at x = a, then it can often be represented with the Taylor formula as a power series (see 6.1.43. p. 388).

416

7. Infinite Series

a) First Form of the Representation:

(x - a)* f"(a) + ' . .

+T (X - a)" f (a) + . . . (n)

,

(7.968)

This representation (7.95a) is correct only for the x values for which the remainder I?, = f(x) - S, tends to zero if n + m. This notion of the remainder is not identical to the notion of the remainder given in 7.3.1, p. 412 in general, only in the case if the expressions (7.95b) can be used. There are the following formulas for the remainder: (x - a)n+' %=f'"")([)(a < < x) or (x < < a) (Lagrange formula), (7.95b) (n l)!

<

+

l X

R, =

/(x - t)"f("t"(t) d t

(Integral formula).

(7.95c)

a

b) Second Form of the Representation: k h2 h" f(a+ h ) = f ( a ) -!'(a) zf"(a) t ' . .t p ( a )

+

+ l!

+ .. '.

(7.96a)

The expressions for the remainder are: hntl

%=-

(0 < 0

f("+')(a t Oh) ( n I)!

+

l h

R, = -J/ ( h - t)nf(ntl)(a

< I),

(7.96b)

+ t )dt.

(7.96~)

0

2. Maclaurin Series The power series expansion of a function f(x) is called a Mucluzlrin series if it is a special case of the Taylor series with a = 0. It has the form

f(x) = f ( 0 ) + I f ' ( 0 ) l!

+ Zf"(0) 22 +.

' '

+ X" - p

( O )

+ .' '

(7.97a)

with the remainder xntl

%=f("")(Qx) ( nt I)!

(0

<8 <

l X

%= ; /(x - t)"f("t')(t) d t .

(7.97b) (7.97c)

0

The convergence of the Taylor series and Maclaurin series can be proven either by examining the remainder & or determining the radius of convergence (see 7.3.3.1, p. 414). In the second case it can happen that although the series is convergent, the sum S ( x ) is not equal to f(x). For instance for the function f(x) = e - 3 for x # 0, and f(0) = 0 the Maclaurin series is the identically zero series.

7.3.4 Approximation Formulas Considering only a neighborhood small enough of the centre of the expansion, we can introduce rational approximation formulas for several functions with the help of the Taylor expansion. The first few terms of some functions are shown in Table 7.3. The data about accuracy are given by estimating the remainder. Further possibilities for approximate representation of functions, e.g., by interpolation and fitting polynomials or spline functions, can be found in 19.6, p. 914 and 19.7, p. 928.

7.3 Function Series 417 Table 7.3 Approximation formulas for some frequently used functions

I

7.3.5 Asymptotic Power Series Even divergent series can be useful for calculation of substitution values of functions. In the following 1 we consider some asymptotic power series with respect to - to calculate the values of functions for large X

values of 1x1.

7.3.5.1 Asymptotic Behavior Two functions f (E) and g(z), defined for x0

< x < co,are called asymptotically equal for z -+ co if

+

f (XI = 1

(7.98b) (7.98a) or f ( ~=)g(z) o(g(z)) for 5 + m S(X) hold. Here, o(g(z)) is the Landau symbol 'little 0'' (see 2.1.4.9, p. 56). If (7.98a) or (7.98b) is fulfilled, then we write also f ( ~ )g(z). lim

~

x+33

N

418

7. Infinite Series

I B : ~ ! ~ I . IC:

I A : @ T ~ ~ X .

32 t 2 4x3 + x + 2

N -

3

4x2’

7.3.5.2 Asymptotic Power Series 1. Notion of Asymptotic Series a, . A series E,”=,is called an asymptotic power series of the function f ( z ) defined for z > zo if X”

(7.99) f ( z ) = C -a”+ O (?!I)v=o xu holds for every n = 0 , 1 , 2 . .. . . Here, 0 - is the Landau symbol “big 0”.For (7.99) we also write (z”:l)

2. Properties of Asymptotic Power Series a) Uniqueness: If for a function f ( z )the asymptotic power series exists, then it is unique. but a function is not uniquely determined by an asymptotic power series. b) Convergence: Convergence is not required from an asymptotic power series.

I A: e! x

“ ”=o

1 -is an asymptotic series, which is convergent for every x with 1x1 > zo

v!x”

I B: The integral f ( z ) =

/

00

0

e-”t

-dt

l f t

R,(z) = 0

“

~

0

(A),

(1

+

t)n+1

0).

(x > 0) is convergent for z > 0 and repeated partial inte-

1 gration results in the representation f(z) = x

n! with&(x) = (-l),-/ zn

(20>

l! - 22

2!

3!

+- - . + (-l),-’23 24

dt. Because of l&(z)l

?C

5

’.

dt

(n - I)! Xn

=

n!

:..l

+ &(XI we get

and with this estimation (7.100)

The asymptotic power series (7.100) is divergent for every z,because the absolute value of the quotient n t l of the (n+ 1)-th and of the n-th terms has the value -. However, this divergent series can be used for a reasonably good approximation of f(z).For instance, for z = 10 with the partial sums Sd(10) and Sj(l0) we get the estimation 0.09152 <

dt

< 0.09164.

7.4 Fourier Series 7.4.1 Trigonometric Sum and Fourier Series 7.4.1.1 Basic Notions 1. Fourier Representation of Periodic Functions Often it is necessary or useful to represent a given periodic function f(z) with period T exactly or approximatirely by a sum of trigonometric functions of the form s,(x) =

2 + a1 cos wx + a2 cos 2wz + . , . + a, cos nux 2 +bl s i n u s + bz sin 2wx + . . + b, sin nwz.

(7.101)

7.4 Fourier Series 419

2T

This is called the Fourzer expanszon. Here the frequency is w = - In the case T = 2n we have Y, = 1 T We can get the best approximation of f ( z ) in the sense given on 7.4.1.2, p. 419 by an approximation function s,(z). where the coefficients a k and b k ( k = 0,1,2,. . . , n) are the Fourier coefficients of the given function. They are determined with the Eulerformulas +f(-x)]coskdzdz,

(7.102a)

-f(-z)]sinlcwzdz,

(7.102b)

20

and

20

or approximatively with the help of methods of harmonic analysis (see 19.6.4, p. 924). 2. Fourier Series If there are z values such that the sequence of functions sn(z) tends to a limit s(z) for n + m, then the given function has a convergent Fourier series for these zvalues. This can be written in the form

+

+

+, + +

+

a1 coswz a2 cos 2wz . , + a , cosnwx 2 +b, s i n d z bz sin2wz . , . b, sin nwx ' . . and also in the form 00

s(z) = -

+

a0

s(z) = -

+

+

2 '41sin(wz t PI) A2 sin(2wz where in the second case:

A,, =

\/,2+bk2,

+

+ p2)+ . + A, sin(nuz + yn)+ . '

ak

tanpk = - ,

,,

~

(7.103b)

(7.103~)

bk

3. Complex Representation of the Fourier Series In many cases the complex form is very useful: CkeikW',

s(z) =

(7.104a)

k=-m

(7.104b)

I

;(a_,

+ ib-k)

fork < 0 .

7.4.1.2 Most Important Properties of t h e Fourier Series 1. Least Mean Squares Error of a Function If a function f ( z ) is approximated by a trigonometric sum sn(x) = 3 +

5

ak cos kdz

+

k=l

bk

sin kwx,

(7.105a)

k=l

also called the Fourzersurn, then the mean square error (see 19.6.2.1, p. 916, and 19.6.4.1, 2., p. 925)

' i[I(.)

F =T o

- sn(x)]2dx

(7.105b)

420

7. In.finite Series

is smallest if we define ak and bk as the Fourier coefficients (7.102a,b) of the given function.

2. Convergence of a Function in the Mean, Parseval Equation The Fourier series converges in mean to the given function, Le.,

/pljjij

-

sn(z)12di + 0 for n

+ rn

(7.106a)

0

holds. if the function is bounded and in the interval0 < x < T i t is piecewise continuous. A consequence of the convergence in the mean is the P a r s e d equation: 2 T a' - / [ f ( x ) ] ' d x = 0 x ( a k 2 t bk'). T O 2 k=l

+

(7.106b)

3. Dirichlet Conditions If the function f ( x ) satisfies the Dzrzchlet condztzons, Le., a) the interval of definition can be decomposed into a finite number of intervals where the function f(x) is continuous and monotone, and b) at every point of discontinuity of f(x)the values f ( x t 0) and f(x - 0) are defined. then the Fourier series of this function is convergent. At the points where f(x) is continuous the sum f(. - 0) + f(z + 0 ) is equal to f ( x ) .at the points of discontinuity the sum is equal to 2

4. Asymptotic Behavior of the Fourier Coefficients If a periodic function f(r)and its derivatives up to k-th order are continuous, then for n expressions annkt1and b,nktl tend to zero.

Figure 7.2

Figure 7.3

+ cc both

Figure 7.4

7.4.2 Determination of Coefficients for Symmetric Functions 7.4.2.1 Different Kinds of Symmetries 1. Symmetry of the First Kind If f ( x ) is an even function, i t . , if f(x) = f(-x) (Fig. 7.2), then for the coefficients we have

(7.107) 2. Symmetry of the Second Kind If f(x) is an odd function, i.e., if f(x) = - j ( - z ) (Fig. 7.3), then for the coefficients we have

(7.108)

7.L Fourier Series 421 3. Symmetry of the Third Kind If f(z+ T/2) = -f(x) holds (Fig. 7.4), then the coefficients are (7.109a)

(7.109b) 4. Symmetry of the Fourth Kind If the function f(x) is odd and also the symmetry of the third kind is satisfied (Fig. 7.5a), then the coefficients are r i d

ak = b2h = 0.

b2k+l =

8j f ( x ) s i n ( 2 k + 1)-2nx dx T

( k = 0,1,2:. . .).

(7.110)

T O

If the function f ( x ) is even and also the symmetry of the third kind is satisfied (Fig.7.5b), then the coefficients are

8

bk

= a Z k = 0, a2ktl = -

l4

T O

2x2 dx ( k = 0, 1 , 2 , .. .). T

f(x)cos(2k + 1)-

Figure 7.5

Figure 7.6

Figure 7.7

7.4.2.2 Forms of the Expansion into a Fourier Series Every function f ( x ) , satisfying the Dirichlet conditions in an interval 0 5 x 5 I (see 7.4.1.2,3., p. 420), can be expanded in this interval into a convergent series of the following forms: a0 27rx 2nx 2nx 1. fl(x) = - a l c o s - + a ~ c o s 2 - + ~ ~ ~ + a , c o s n - + ~ ~ ~ 2 1 1 1 2nx 27rx 2TX + bl sin + b:! sin 2- 1 + ' . + b, sin R- 1 + . . . (7.112a) 1

+

422

7. Infinite Series

The period of the function h ( x ) is T = 1; in the interval 0 < z < l the function fi(z)coincides with the function f(z)(Fig. 7.6). At the points of discontinuity the substitution values are f(x) =

I [ f ( z- 0) + f(z+ O ) ] . The coefficients of the expansion are determined with the Euler formulas

2 2T (7.102a.b) for w = -. 1 a0 -

KX

+

FX

+

+ +

FX

+

a,cosn' (7.112b) a1 cos - a2 cos 2'.. 2 1 1 1 The period of the function f 2 ( x ) is T = 21; in the interval 0 5 x 5 1 the function fZ(x) has a symmetry of the first kind and it coincides with the function f(x) (Fig. 7.7). The coefficients of the expansion of f 2 ( x )are determined by the formulas for the case of symmetry of the first kind with T = 21. 2. f 2 ( x ) =

TX

7iX

572

(7.112c) 3. f3(x)= b l s i n - + + b : ! s i n 2 - + + . . + b b , s i n n - + + . . . . 1 1 1 The period of the function f s ( x ) is T = 21; in the interval 0 < x < 1 the function f 3 ( x ) has a symmetry of the second kind and it coincides with the function f(x)(Fig. 7.8). The coefficients of the expansion are determined by the formulas for the case of symmetry of the second kind with T = 21.

1

TT 4 2

Figure 7.8

TX

Figure 7.9

7.4.3 Determination of the Fourier Coefficients with Numerical Methods If the periodic function f(x)is a complicated one or in the interval 0 5 x < T i t s values are known only kT . for a discrete system x k = - with k = 0: 1 . 2 , . . . , N - 1, then we have to approximate the Fourier N coefficients. Furthermore, e.g.: also the number of measurements N can be a very large number. In these cases we use the methods of numerical harmonic analysis (see 19.6.4, p. 924).

7.4.4 Fourier Series and Fourier Integrals 1. Fourier integral If the function f(x) satisfies the Dirichlet conditions (see 7.4.1.2, 3., p. 420) in an arbitrarily finite interval and. moreover: the integral

'SX

If(x)l dx is convergent (see 8.2.3.2,l., p. 452), then the following

-c€

integral): formula holds (FOURIER +m

f ( ~ 1 ) +~m= e ' ~~' d ; ~ f ( t ) e - ' w ' r l 1t = - ~ d+m w~f(t)cosw(t-x)dt. --s.

-c€

.4t the points of discontinuity we substitute

0

-x

(7.113a)

7.4 Fourier Series 423

2. Limiting Case of a Non-Periodic Function The formula (7.113a) can be regarded as the expansion of a non-periodic function f ( x )into a trigonometric series in the interval (-1, +1) for 1 + m. IVith Fourier series expansion a periodic function with period T is represented as the sum of harmonic 2i7

vibrations with frequency 3, = n-

T

.

with n = 1,2,. . . and with amplitude A,. This representation is

based on a discrete frequency spectrum. L\’*iththe Fourier integral the non-periodic function f ( 5 )is represented as the sum of infinitely many harmonic vibrations with continuously varying frequency w . The Fourier integral gives an expansion of the function f (x)into a continuous frequency spectrum. Here the frequency w corresponds to the density g(d) of the spectrum: -35

(7.113~) -m

The Fourier integral has a simpler form if f ( x ) is either a) even or b) odd: a)

---cosuxdw

f ( x )= 2 x

(7.114a)

0 m

b) f ( x ) = ~2 Jms i n ~ x d w S f ( t ) s i n w t d t .

(7.114b)

0

0

IThe density of the spectrum of the even function f ( x ) = e-l”l and the representation of this function are

7.4.5 Remarks on the Table of Some Fourier Expansions In Table 21.4 there are given the Fourier expansions of some simple functions, which are defined in a certain interval and then they are periodically extended. The shapes of the curves of the expanded functions are graphically represented. 1. Application of Coordinate Transformations Many of the simplest periodic functions can be reduced to a function represented in Table 21.4 when we either change the scale (unit of measure) of the coordinate axis or we translate the origin. W A function f(r) = f (-x) defined by the relations (7.116a)

(Fig. 7.9), can be transformed into the form 5 given in Table 21.4,if we substitute a = 1 and we 2i7x i7 - + -. By the substitution of the variables in T 2 2i7x Ti 27rx + = (-l), cos(2n + 1)we get for the function (7.116a) the series 5 ) because sin(2n 1) T 2 T expression 27ix 1 27rx 1 2KX (7.116b) y = 1 + - cos - - - cos3+ ;cos;- ”‘ . i ( T 3 T o T

introduce the new variables Y = y - 1 and X =

+

(-

-)

)

I

424

7. Infinite Series

2. Using the Series Expansion of Complex Functions Many of the formulas given in Table 21.4 for the expansion of functions into trigonometric series can be derived from power series expansion of functions of a complex variable.

I The expansion of the function I

+

-= 1 z t 22 t . . .

< 1)

(7.117)

2 = ae'p after separating the real and imaginary parts

(7.118)

1-Z

(121

yields for

1 + acosp + a2cos2p

1 -acos(o 1 - 2a cosp a2 ' (o for la1 < 1. . , = 1 - 2aa sin cos (o az

+. . . + ancosllp + , .

a s i n y + a2sin2(o+ . , . + ansinn(o

+.

*

=

+ +

(7.119)

8 Integral Calculus 1. Integral Calculus and Indefinite Integrals Integration represents the inverse operation of differentiation in the following sense: While differentiation calculates the derivative function f‘(.) of a given function f(x),integration determines a function whose derivative f’(z)is previously given. This process does not have a unique result, so we get the notion of an indefinite integral. 2. Definite Integral If we start with the graphical problem of the integral calculus, to determine the area between the curve of y = f(z) and the z-axis, and for this purpose we approximate it with thin rectangles (Fig. 8.1). then we get the notion of the definite integral. 3. Connection Between Definite a n d Indefinite Integrals The relation between these two types of integral is the fundamental theorem of calculus (see 8.2.1.2, l . , p. 440).

8.1 Indefinite Integrals 8.1.1 Primitive Function or Antiderivative 1. Definition Consider a function y = f(x) given on an interval [a, b]. F ( z ) is called a primitive function or antiderivative of f(x)if F ( s )is differentiable everywhere on [a, b] and its derivative is !(.): F ’ ( 2 )= f(.). (8.1) Because under differentiation an additive constant disappears, a function has infinitely many primitive functions, if it has any. The difference of two primitive function is a constant. So, the graphs of all primitive functions Fl(z),Fz(z),. . . , F,(z) can be got by parallel translation of a particular primitive function in the direction of the ordinate axis (Fig. 8.2).

Figure 8.2

Figure 8.3

2. E x i s t e n c e Every function continuous on a connected interval has a primitive function on this interval. If there are some discontinuities, then we decompose the interval into subintervals in which the original function is continuous (Fig. 8.3). The given function = f(z) is in the upper part of the figure; the function

426

8. Inteqral Calculus

y = F ( x ) in the lower part is a primitive function of it on the considered intervals.

8.1.1.1 Indefinite Integrals The indefinite integral of a given function f(z)is the set of primitive functions

F ( z )t C =

J f ( z )dz.

(8.2)

The function f ( x ) under the integral sign

I

is called the integrand, zis the integration variable, and C

is the integration constant. It is also a usual notation, especially in physics, to put the differential dx right after the integral sign and so before the function f ( z ) . Table 8.1 Basic integrals Exponential functions

Powers p t l

/zn dz = n+l

( n # -1)

/ez dx = ex

/$ = In 1x1

/ozdz = Hyperbolic functions

Trigonometric functions

/ sinh zdz / cosh zdz

S s i n x d z = -cos%

/ cos x dz

= sin z

= cosh z = sinh z

/tanzdz=-lnlcoszl

/tanhzdx=ln~coshzl

J

/cothzdz =ln/sinhzI

cot x d z = In 1 sinxi

/&

= tanz

A /&

= -cotx

/A

= tanhz

/A

=

Irrational functions

Fractional rational functions dx

dx

Ia2-z

-

ii1 arctan $

= i.4rtanh

$ = & In (for I zI < a )

/A x -iArcoth$= -a

-cothz

=

& l n l z l

dz

= Arcosh

5 = In Ix + d-1

(for I zI > a)

8.1.1.2 Integrals of Elementary Functions 1. Basic Integrals The integration of elementary functions in analytic form is reduced to a sequence of basic integrals. These basic integrals can be got from the derivatives of well-known elementary functions, since indefinite integration means the determination of a primitive function F ( z ) of the function f(z). The collection of integrals given in Table 8.1 comes from reversing the differentiation formulas in Table 6.1 (Derivatives of elementary functions). The integration constant C is omitted.

8.1 Indefinite Inteorals 427

2. General Case For the solution of integration problems, we try to reduce the given integral by algebraic and trigonometric transformations. or by using the integration rules to basic integrals. The integration methods given in section 8.1.2 make it possible in many cases to integrate those functions which have an elementary primitive function. The results of some integrations are collected in Table 21.5 (Indefinite integrals). The following remarks are very useful in integration: a) The integration constant is mostly omitted. Exceptions are some integrals, which in different forms can be represented with different arbitrary constants. b) If in the primitive function there is an expression containing In f(x),then we have to consider always In If(x)l instead of it. c) If the primitive function is given by a power series, then the function cannot be integrated in an elementary fashion. .A wide collection of integrals and their solutions are given in [8.1] and [8.2].

8.1.2 Rules of Integration The integral of an integrand of arbitrary elementary functions is not usually an elementary function. In some special cases we can use some tricks, and by practice we can gain some knowledge of how to integrate. Today we leave the calculation of integrals mostly to computers. The most important rules of integration, which are finally discussed here, are collected in Table 8.2.

1. Integrand with a Constant Factor .A constant factor a in the integrand can be factored out in front of the integral sign (constant multiple rule): /af(x)dx=

oi

s

f(z)dx.

2. Integration of a Sum or Difference The integral of a sum or difference can be reduced to the integration of the separate terms if we can tell their integrals separately (sum rule): /(u

+ IJ - w) dx =

/

u dz

+ / u dz -

1

wdz.

(8.4)

The variables u , IJ, w are functions of z.

s

I (x + 3)'(x2

+ 1)dx =

(x4 + 6x3+ 102'

3 10 + 6z + 9) dx = y3 t -z4 + -z3 + 32' + 92 + C . 2 3 25

3. Transformation of the Integrand The integration of a complicated integrand can sometimes be reduced to a simpler integral by algebraic or trigonometric transformations.

s s

I sin 22 cos zdz =

s:

-(sin 32 + sin r ) dx.

4. Linear Transformation in the Argument If f(x)dx = F ( x ) is known, e.g., from an integral table, then we get: 1

/!(ax) dx = - F ( a x ) + C,

/ f(x t b) dx

(8.58)

1 + b) dx = -F(az + b) + C. 1 sin ax dx = - - cos ax + C . IB:

= F(z

1

(8.5b) (8.5~)

//(ax IA:

+ b) + C,

1

ea'+*

1

dx = -eaxtb

+ c.

I

8. Intesral Calculus

428

dx

= arctan(x + a )

+ C.

Table 8.2 Important rules of calculation of indefinite integrals

Rule

Formula for integration

Integration constant

/f(x) dx = F ( z ) + C

(C const)

Integration and differentiation

I f(x)

( a const)

Constant multiple rule

/ a f ( x )dx = a

Sum rule

/[u(x)rt ~ ( x )dx] = /u(x)dx & / u ( x ) dx

Partial integration

1

dx

u(x)d(x)dx = u(x)w(x)-

1

u'(x)v(x) dx

x = u ( t ) or t = ~ ( x ) ; u und u are inverse functions of each other :

/ f(x)dx = / f ( u ( t ) ) u ' ( dtt )

Substitution rule

'(')

Special form of the integrand

or

dx = In If(x)l + C (logarithmic integration)

lm 1f'(x)f(x)

1 dx = -f2(x) C 2 u und w are inverse functions of each other : /u(x)dx = xu(.) - F ( u ( x ) )+ C1 with l. 2.

Integration of the inverse function

+

F ( x ) = /u(x)dx + C2 (C, , C2 const)

5 . Power and Logarithmic Integration If the integrand has the form of a fraction such that in the numerator we have the derivative of the denominator. then the integral is the logarithm of the absolute value of the denominator:

"

IA:

2'

+ 3x - 5 dx = ln(x2+ 32 - 5 ) + C. +

If the integrand is a product of a power of a function multiplied by the derivative of the function. and the power is not equal to -1, then

1 1

f'(x)f"(x)dx =

B:

2x+3

(xZ+ 32 - 5)3

I

fa+%) f a ( z ) df(x) = -

dx =

+

c,

a+l

1 ( - 2 ) ( x 2 + 32 - 5)'

+ c.

8.1 IndefinzteInteorals 429

6. Substitution Method If x = u ( f )where t = ~ ( xis) the inverse function of x = u ( t ) )then according to the chain rule of differentiation we get

I e"S1 I e"+ldxx I t + I -. e"

IA:

-

1

dx. Substituting z = lnt. (t

dx

1

dt

t

> 0), - = - , then taking the decomposition into

Dartial fractions we get: ex - 1 t-1dt -- = (L dt = 21n(e" t 1) - x t C . = 1t t t 1 t dx dt IB: Substituting 1$x2 = t , - = 22, then we get dx

1

A)

7. Partial Integration Reversing the rule for the differentiation of a product we get

I

u(x)z"(x)dx = ~ ( x~) ( x-) / u ' ( z ) u(x) dx,

(8.8)

) continuous derivatives. where u(x)and ~ ( xhave IThe integral

S

x e' dx can be calculated by partial integration where we choose u = x and u' = e',

so we get u' = 1 and 2' = e':

S ze'

dz = ze"

-

I

e' dx = (x - 1)e' t C .

8. Non-Elementary Integrals Integrals of elementary functions are not always elementary functions. These integrals are calculated mostly in the following three ways, where the primitive function will be approximated by a given accuracy. 1. Table of Values The integrals which have a particular theoretical or practical importance but cannot be expressed by elementary functions can be given by a table of values. (Of course, the table lists values of one particular primitive function.) Such special functions usually have special names. Examples are: IA: Logarithmic integral (see 8.2.5. 3., p. 458):

IB: Elliptic integral of the first kind (see 8.1.4.3, p. 435): dx = F ( k 9). J(1 - $)(1 - k Z 9 )

fnp

(8.10)

IC: Error function (see 8.2.5, 5., p. 459):

I'

e-tz d t = erf(z).

(8.11)

2. Integration by Series Expansion We take the series expansion of the integrand, and if it is uniformly convergent, then it can be integrated term-by-term.

IA:

1%?

IB:

I$

dx, (see also Sine integral p. 458).

dz. (see also Exponential integral p. 459)

430

8. Inte.qra1 Calculus

3. Graphical integration is the third approximation method, which is discussed in 8.2.1.4. 5.. p. 444.

8.1.3 Integration ofRational Functions Integrals of rational functions can always be expressed by elementary functions.

8.1.3.1 Integrals of Integer Rational Functions (Polynomials) Integrals of integer rational functions are calculated directly by term-by-term integration:

/(a,zn + an_lxn-l +.

’.

+ a12 + ao) dx (8.12)

8.1.3.2 Integrals of Fractional Rational Functions The integrand of an integral of a fractional rational function

I Bp;:i

-dx, where P ( x ) and Q ( x ) are

polynomials with degree rn and n, respectively, can be transformed algebraically into a form which is easy to integrate. \Ye perform the following steps’ 1. Il’e simplify the fraction by the greatest common divisor. so P(2)and Q ( x )have no common factor. 2. !Ye separate the integer rational part of the expression. If m 2 n holds, then we divide P(2) by Q ( x ) Then s e have to integrate a polynomial and a proper fraction. 3. !Ye decompose the denominator Q ( x )into linear and quadratic factors (see 1.6.3.2,p. 44): Q(2) = a n ( x - ~ ) k ( z - 3 ) “ ~ ~ ( ~ Z + p 2 + q ) P ( 2 2 + p ’ ~ + q ’ ) s ~ ~ ~ with

p’4 - q < 0, f4 - q’ < 0.. . . .

(8.13a) (8.13b)

4. IVe factor out the constant coefficient a, in front of the integral sign.

5 . \$‘e decompose the fraction into a sum of partial fractions: The proper fraction we get after the division. which can nolonger be simplified and whose denominator is decomposed into a product of irreducible factors, can be decomposed into a sum of partial fractions (see 1.1.7.3, p. l5), which are easy t o integrate.

8.1.3.3 Four Cases of Partial Fraction Decomposition 1.Case: All Roots of the Denominator are Real and Single Q(2) = (Z - a)(. - 9) ’ . . (2 - A)

(8.14a)

a) PVe form the decomposition:

B P(2) ‘4 +-+... Q(2) 2 - a 2-3

+-x -Lx

(8.14b)

b) The numbers A. B , C, . . . . L can also be calculated by the method of undetermined coefficients (see 1.1.7.3. 1..p. 16). c) Il’e integrate by the formula (8.14d)

8.1 Indefinite Inteqrals 431

21

II = /

+3 1

q x- 11513 2. Case: All Roots of the Denominator are Real, Some of them with a Higher Multiplicity Q ( x ) = ( E - .)'(E

- S),.

".

(8.15a)

a) R'e form the decomposition: P ( x ) - A1 +-+...+A2 Q(z) - (x - a) (x - a)'

A1

(x - a)' (8.15b)

b) Tle calculate the constants A I , Az.. . . , A ' , B1, Bz,. . . , B,. . . . by the method ofundetermined coefficients (see 1.1.7.3, l . ,p. 16) c ) Tle integrate by the rule = ill h ( z - a). 53

1

dx = Ak ( k > 1). x - .)k ( k - 1)(x - a)k-1

+1

(8.15~)

The method of undetermined

coefficients yields A + B1 = 1, -3A - 2B1+ B2 = 0 , 3 A + B1 - Bz + B3 = 0. - A = 1; A = -1. B1 = 2, Bz = 1. B3 = 2. The result of the integration is 1 2 1 1 1 2 dz = - I n x + 2 l n ( x - 1) - -- -+e I = [-3. + 2-1 (x-1)2 (x - 1 ) 2 z =In---

+

/

x

(x - 1 ) 2

+

1-

+ e.

3. Case: Some Roots of the Denominator are Single Complex Suppose all coefficients of the denominator Q ( x ) are real. Then. with a single complex root of Q ( x )its conjugate complex number is a root too and we can compose them into a quadratic polynomial.

Q(z) = (x- a)'(. - +),. with

t .

(x2+ p x

+ q)(x2 +p'x + 4'). . .

(8.16a)

2 < q . 2 < q' , . . . , 4 4

(8.16b)

because the quadratic polynomials have no real zeros. a) w'e form the decomposition:

P(x) Q(x)

A1

2-0

+-+...+A2 A1 Bi +-..+.+, ( z -BzP ) (.-a)' (x-a)' x-8 Cz+D Ex+ F t... .

+ x z t px + q

+-

+

22

+ p'x + q'

Bm (x - 0 ) m (8.16~)

432

8. InteoraE Calculus

b) We calculate the constants by the method of undetermined coefficients (see 1.1.7.3, l.,p. 16). c) We integrate the expression

cx by the formula +px +q +

x2

( C x + D ) dx = ln(x2 + px t q ) 2 x2 t p x t q

C

4dx

,

-

A

+PI2

D - Cpf2 +-

+-cx

(8.16d)

. The method of undetermined coefficients yields the

+

equations A + C = 0, D = 0, 4A = 4, A = 1, C = -1, D = 0. 1 1 x I= dx = lnx - -ln(x2+4) +lnCl = ln- ' l X 2 term arctan is missing

1(;

-)

m , where in this particular case the

4. Case: Some Roots of the Denominator are Complex with a Higher Multiplicity (8.17a) & ( x )= ( 2 - a)k(x- 8)". (x' + p x + q)"(xZ +p'x + qy.. .. I

a) We form the decomposition:

A +2 A Ak + Bi BZ BI p(,) - -1 + . . . +y+ ...+ Q(x) x - a (x-0)' (x-a)k 2 - p (x-p) (x - PI1 C ~ XD1 C Z X Dz Cmx D, (XZ px q ) m 2 2 +pr q (x2 +pa: qy EZX Fz E,x F, E I X Fi t x2 P'. q' (22 t p'x q')* (x' p'x q')n . b) We calculate the constants by the method of undetermined coefficients. Cmx D, for m > 1 in the following steps: c) We integrate the expression (x' px q)m a)We transform the numerator into the form

+

+

+

+

+

+ +

+ +

+

+ + +...+ + + + +"'+ + + + +

(8.17b)

+ + +

C,x

c m + p ) + (Dm - -) CmP + Dm = -(22 2 2

(8.17~)

.

p) We decompose the integrand into the sum of two summands, where the first one can be integrated directly:

cm

(22+P)dZ

S- 2 (x' + p z t

-

q)m

Cm

1

2(" - 1) (2'

+ px + q)m-1 .

(8.17d)

7) The second one will be integrated by the following recursion formula, not considering its coefficient:

+P / 2 dx (8.17e) +pa: t q)m-l ' CiX+D1 C Z X + D Z 2x2+22+13 - A 22' + 22 + 13 I=/ dx : 22 +1 (x - 2)(x2+ 1 ) 2 - x - 2 (2' + 1)Z .( - 2)(x2 t 1 ) 2 The method of undetermined coefficients results in the following system of equations: A+C1 = 0, -2C1+D1= 0 , 2A+C1-2Dl+CZ = 2 , -2C1tD1-2CZSDz = 2, A-2D1-202 = 13; the coefficients are A = 1, C1 = -1, D1 = -2, C2 = -3, DZ = -4,

+ 2(m -21)m(-q3- p2/4)

I

(2'

+-+-

I=/(

2+2

2 - 2

x'+1

3s+4)dx,

(xZt1)Z

8.1 Indefinite Integrals 433

dx

According to (8.17e) we get finally the result is I =

3 - 42 1 t -1n2 ( 9 + 1) 2

(2 - 2)2

~

x2+

1

- 4arctanx

+ C.

Table 8.3 Substitutions for integration of irrational functions I

I

Substitution

Integral *

where r is the lowest common multiple of the numbers m, n, . . . . ~

/ R (x.Jax2

+ bx + c)

One of the three Euler substitutions:

dx:

+c =t - fix dax2 t bx + c = xt + f i

1. For a > 0 t

Jazz+ bx

2 . For c > 0

+ +

3. If the the polynomial ax2 bx c has different real roots: ax2 + bx + c = a ( x - a ) ( . - p)

dax2 + bx + c = t ( x - a )

* The symbol R denotes a rational function of the expressions in parentheses. The numbers n, m, . . . are integers. t If a < 0, and the polynomial ax2 t bx + c has complex roots, then the integrand is not defined for any value of 2, since dax2 bx c is imaginary for every real value of x. In this case the integral is meaningless.

+ +

8.1.4 Integration of Irrational Functions 8.1.4.1 Substitution to Reduce to Integration of Rational Functions Irrational functions cannot always be integrated in an elementary way. Table 21.5 contains a wide collection of integrals of irrational functions. In the simplest cases we can introduce substitutions, as in Table 8.3. such that the integral can be reduced to an integral of a rational function.

R (x,d a x 2 t bx + c) dx can be reduced to one of the following three forms

The integral

1

m) dx.

R (x.

(8.1sa)

/ R (x,m) dx,

(8.18b) (8.18~)

because the quadratic polynomial ax2 + bx t c can always be written as the sum or as the difference of two complete squares. Then, we can use the substitutions given in Table 8.4.

W A : 4x2+16s+17=4

434

8. Integral Calculus

IB: zz + 3x t 1 = x2 + 32

+ -9 - -5 = (z +

IC: -x2

+

i)' ($)' ($)'

with xl = z + -3 2 2s = 1 - xz t 2x - 1 = 1' - (z - 1)' = l2 - xy with x1 = x - 1. 4

4

-

= z;-

Tabelle 8.4 Substitutions for integration of irrational functions I1

Integral

lR

(x.

Substitution

m) dx

I R (x,d-)

dx

/ R (x, d m ) dx

x=asinht

or x = c r t a n t

x=acosht

or z = a s e c t

x=asint

or z = a c o s t

8.1.4.2 Integration of Binomial Integrands An expression of the form zm(a bzn)P (8.19) is called a binomial integrand, where a and b are arbitrary real numbers, and m, n. p are arbitrary positive or negative rational numbers. The theorem of Chebyshev tells that the integral

+

11

+

xm(a bzn)Pdz

(8.20)

can be expressed by elementary functions only in the following three cases: Case 1: Ifp is an integer, then the expression ( a t b P ) P can be expanded by the binomial theorem, so the integrand after eliminating the parentheses will be a sum of terms in the form exk, which are easy to integrate. m+1 Case 2: If -is an integer, then the integral (8.20) can be reduced to the integral of a rational n function by substituting t = t -, where r is the denominator of the fraction p m+l Case 3: If + p is an integer, then the integral (8.20) can be reduced to the integral of a rational

-

function by substituting t =

Substitution t =

$/T!

- where r is the denominator of the fraction p .

(? x/ = (t3 = ,

= St4(4t3 - 7 ) -tc.

7

Because none of the three conditions is fulfilled, the integral is not an elementary function.

8.1 Indefinite Inteorals 435

8.1.4.3 Elliptic Integrals 1. Indefinite Elliptic Integrals Elliptic integrals are integrals of the form

1R (x,dax3 +

1

R ( x , 4 a x 4 + bx3 + ex2 + ex + f ) dx.

bx2 t cx + e ) dx,

(8.21)

Csually they cannot be expressed by elementary functions; if it is still possible, the integral is called pseudoellzptzc. The name of this type of integral originates from the fact that the first application of them was to calculate the perimeter of the ellipse (see 8.2.2.2, 2.) p. 447). The inverses of elliptic integralsare the elliptic functions (see 14.6.1, p. 700). Integralsofthe types (8.21), which are not integrable in elementary terms. can be reduced by a sequence of transformations into elementary functions and integrals of the following three types (see [21.1], [21.2], [21.6]): dt

(1 - k2t2)dt

1 JjiTqcFq

(0< k < 1)) (8.22a) dt

(0< k < 1). (8.2213)

(0< k < 1).

(8.22~)

Concerning the parameter n in (8.22~)one has to distinguish certain cases (see [14.1]). By the substitution t = s i n p

the integrals (8.22a,b,c) can be transformed into the

Legendre form: Elliptic Integral of t h e First Kind:

Elliptic Integral of the Second Kind: Elliptic Integral of the T h i r d Kind:

(8.23a)

-4

1

+

(1 nsin2 p

(8.2313)

dlp.

(8.23~)

dlp

~ .

)

m

2. Definite Elliptic Integrals Definite integrals with zero as the lower bound corresponding to the indefinite elliptic integrals are denoted by (8.24a) ]J=dt,L 1 - k2 sin’

1;

dz:

= E(k,p),

(8.24b)

0

= n ( n ,k , 9) (for all three integrals 0

< k < 1 holds). (8.24~)

We call these integrals incomplete elliptic zntegrals of the first, second, and third kind for p = L. The 2 first two integrals are called complete elliptic integrals, and we denote them by (8.25a)

8. Intesral Calculus

436

(8.25b) Tables 21.7.1, 2, 3 contain the values for incomplete and complete elliptic integrals of the first and second kind F, E and also K and E . IThe calculation of the perimeter of the ellipse leads to a complete elliptic integral of the second kind as a function of the numerical eccentricity e (see 8.2.2.2, 2., p. 447). For a = 1.5, b = 1 it follows that e = 0.74. Since e = k = 0.74 holds, we get from Table 21.7.3: sincy = 0.74, Le., cy = 47" and E ( k , T ) = E(0.74) = 1.33. It follows that U = 4aE(0.74) = 4aE(cy = 47") = 4 * 1 . 3 3 ~= 7.98. 2 Calculation with the approximation formula (3.326~)yields 7.93.

8.1.5 Integration of Trigonometric Functions 8.1.5.1 Substitution With the substitution 2dt Le., dz = 1+t2' an integral of the form

t

= tan

z) 2

, 2t 1- t 2 cosz = slnz=-1 + t * ' 1+t2'

1

(8.26)

(8.27)

R (sin z,cos z)dx

can be transformed into an integral of a rational function, where R denotes a rational function of its arguments.

tan*:

-

2 + tan - + - In tan - + C. In some special cases we can apply simpler substitutions. If the

4 2 2 2 integrand in (8.27) contains only odd powers of the functions sinz and cosz, then by the substitution t = tan z a rational function can be obtained in a simpler way.

8.1.5.2 Simplified Methods Case 1: /R(sinz)coszdz. Case 2: Case 3:

1 1

R (cos z) sin zdz.

Substitution t = sinz, coszdx = d t .

(8.28)

Substitution t = cos z, sinz dz = -dt.

(8.29) (8.30a)

sin" zdx :

a) n = 2m t 1, odd:

I

- c o s Z z ) m s i n z d z = - (1-t2)mdt

/sin"zdz=/(l

b) n = 2m, even: /sinnzdx =

1

[:(l -

COS^^)]

with t = c o s z .

(8.30b)

m

dz =

(1 -

with t = 22.

(8.30~)

8.1 Indefinite Inteorals 437 We halve the power in this way. After removing the parentheses in (1 - cost)" we integrate term-byterm.

Case 4:

1

(8.3la)

x dx.

COS"

a) n = 2m t 1, odd:

1

x dx =

COS"

/(1 - sin' x ) cos~ x dx

=

I

(1 - tz)mdt

b) n = 2m, even: m

/cosnxdx=/[~(1+cos2x)]

d x = - +2:1

with t = sin x.

(8.31b)

1

( l + ~ o s t ) ~ d t with t = 2 x .

(8.3IC)

We halve the power in this way. After removing the parentheses we integrate term-by-term.

1 1

(8.32a)

sin" x cosm x dx.

Case 5:

a) One of the numbers m or n is odd: We reduce it to the cases 1 or 2.

A:

I

sin' xcos5 xdx =

sin'x (1 - sin' x)'cosxdx = /?(1 - t2)' dt with t = sinx.

b) The numbers m and n are both even: we reduce it to the cases 3 or 4 by halving the powers using the trigonometric formulas sin 22 1 - cos 22 1 + cos 232 sin zcos x = , cos'x = (8.32b) sin' x = 2 ' 2 ' ~

~

~

J z l(1

1 - cos4x) dx = - sin3 22 48

Case 6:

1

tann x dx =

1

+

1 1 . -x - - sin4x 16 64

'J

sin' 2z(1+ cos 2 2 ) dx = 8

sin2 cos4 x dx = (sin x cos x)2 cos' x dx =

1 16

+C.

tan"-' x(sec*x - 1) dx =

1

sin222 cos 2z dz t

x (tanx)'dx - tan"-' x dx

1

tann-' x - -- tan"-' x dx. (8.33a) n-1 By repeating this process we decrease the power and depending on whether n is even or odd we finally get the integral /dx=z

or

I t a n x d x = -1ncosx

(8.33b)

respectively.

Case 7:

1

cot" 1 dx.

(8.34)

The solution is similar to case 6. Remark: Table 21.5. p. 1017 contains several integrals with trigonometric functions

8.1.6 Integration of Further Transcendental Functions 8.1.6.1 Integrals with Exponential Functions Integrals with exponential functions can be reduced to integrals of rational functions if it is given in the form

/ R (emz,

ens, .

. . . eps) dx,

(8.35a)

I

438

8. Inteqral Calculus

where rn. n . . . . . p are rational numbers. We need two substitutions to calculate the integral: 1. Substitution o f t = e' results in an integral

f R ( t m , t n :. . t P )dt.

(8.3513)

2. Substitution of z = 8. where r is the loweest common multiple of the denominators of the fractions m. n,. . . > p ,results in an integral of a rational function.

8.1.6.2 Integrals with Hyperbolic Functions Integrals with hyperbolic functions, Le., containing the functions sinh x,cosh x,tanhx and coth x in the integrand, can be calculated as integrals with exponential functions, if the hyperbolic functions are replaced by the corresponding exponential functions. The most often occurring cases cosh" x dx

.I

I

sinh" x dx

sinh" x coshmx dx can be integrated in a similar way to the trigonometric functions

(see 8.1.5, , p. 436)

8.1.6.3 Application of Integration by Parts If the integrand is a logarithm, inverse trigonometric function, inverse hyperbolic function or a product of xm with lnx, ea=) sinax or cosax or their inverses, then the solution can be got by a single or repeated integration by parts. In some cases the repeated partial integration results in an integral of the same type as the original integral. In this case we have to solve an algebraic equation with respect to this expression. We can calculate in this way, e.g., the integrals eaz cos bzdx, e sin ' bx dx: where we need integration by

1

1

parts twice. Il'e choose the same type of function for the factor 2~ in both steps, either the exponential or the trigonometric function. u'e also use integration by parts if we have integrals in the forms

1

P (x)ea' dx: P (x)sin bx dx and

P (x)cos bx dx;where P (z)is a polynomial. (Choosing 2~ = P (x)the degree of the polynomial will

be decreased at every step.)

8.1.6.4 Integrals of Transcendental Functions The Table 21.5,p. 1017: contains many integrals of transcendental functions

8.2 Definite Integrals 8.2.1 Basic Notions, Rules and Theorems 8.2.1.1 Definition and Existence of the Definite Integral 1. Definition ofthe Definite Integral The definite integral of a bounded function y = f (x)defined on a finite closed interval [a.b] is a number, which is defined as a limit of a sum, where either a < b can hold (case A) or a > b can hold (case B). In a generalization of the notion of the definite integral (see 8.2.3, p. 451) we will consider functions defined on an arbitrary connected domain of the real line, e.g., on an open or half-open interval, on a half-axis or on the whole numerical axis, or on a domain which is only piecewise connected, Le., everywhere, except finitely many points. These types of integrals belong to zmproper zntegrals (see 8.2 3. l., p. 451).

2. Definite Integral as the Limit of a Sum iYe get the limit. leading to the notion of the definite integral, by the following procedure (see Fig. 8.1, 425).

8.2 Definite Intearals 439

1. Step: The interval [ a ,b] is decomposed into n sobzntervals by the choice of n - 1 arbitrary points . . , xn-l so that one of the following cases occurs:

1 1 . x2,

a = zo < x1 < x 2

< . . . < z, < . . < z,-1 < E ,

a = xo > x1 > x 2 > . ' ' > 1,>

'.

> xn-l > E ,

=b

(case A) or

(8.36a)

=b

(case B).

(8.36b)

Et is chosen in the inside or on the boundary of each subinterval as in Fig. 8.4: T , - ~5 Et 5 x , (in case 4) or z,-~ 2 (, 2 z, (in case B). (8.36~)

2. Step: .4 point

41

5,

53

-004

m ... xl

b=xnT x,,.~ *

5"

'

5n

51

AXn-1

T

51

*..

Ax2 Axl Axo

* ... x3 T x,T x1'- "? T xo=a 53

52

ffi

(8)

51

Figure 8.4 3. Step: The value f (t,)of the function f ( x ) at the chosen point is multiplied by the corresponding difference Ax,-1 = E , - xt-13Le., by the length of the subinterval taken with a positive sign in case A and taken with negative sign in case B. This step is represented in Fig. 8.1, p. 425 for the case A. 4. Step: Then all the n products f (Ez) A E , - ~ are added 5. Step: The limit of the obtained zntegral approxzmatzon sum or Rzemann sum

(8.37) is calculated if the length of each subinterval 4zi-l tends to zero and consequently their number n as an infinitesimal quantity. tends to x. Based on this, we can also denote If this limit exists independently of the choice of the numbers z, and &, then it is called the definite Riemann integral of the considered function on the given interval. We write (8.38) The endpoints of the interval are called limits of integration and the interval [a>b] is the integration interval: a is the lower limit, b is the upper limit of integration: z is called the integration variable and f(x) is called the integrand.

3. Existence of the Definite Integral The definite integral of a continuous function on [a. b] is always defined, Le., the limit (8.38) always exists and is independent of the choice of the numbers xi and E,. Also for a bounded function having only a finite number of discontinuities on the interval [a. b] the definite integral exists. The function whose definite integral exists on a given interval is called an integrablefunction on this interval.

8.2.1.2 Properties of Definite Integrals The most important properties of definite integrals explained in the following are enumerated in Table 8.5, p. 441.

I

440 8. Inte9ral Calculus

1. Fundamental Theorem of Integral Calculus If the integrand f(x) is continuous on the interval [a,b], and F b

(2) is

a primitive function, then

b

/ f (x)dx = / F'(x)dx

=F

(x)18 = F (b) - F ( a )

(8.39)

a

a

holds, Le., the calculation of a definite integral is reduced to the calculation of the corresponding indefinite integral, to the determination of the antiderivative:

F (z) =

/ f (x)dx + C.

(8.40)

Remark: There are integrable functions which do not have any primitive function, but we will see that, if a function is continuous, it has a primitive function.

2. Geometric Interpretation and Rule of Signs 1. Area under a Curve For all x in [a,b] let f (x)2 0. Then the sum (8.37)can be considered as the total area of the rectangles (Fig. 8.1), p. 425,which approximate the area under the curve y = f (2). Therefore the limit of this sum and together with it the definite integral is equal to the area of the region A, which is bounded by the curve y = f (x),the z-axis, and the parallel lines x = a and x = b:

A=

f (z)dz

( a < b and

f (z) 2 0 for a 5 x 5 b).

(8.41)

a

2. Sign Rule If a function y = f(z) is piecewise positive or negative in the integration interval (Fig. 8.5),then the integrals over the corresponding subintervals, that is, the area parts, have positive or negative values, so the integration over the total interval yields the sum of signed areas. In Fig. 8.5a-d four cases are represented with the different possibilities of the sign of the area. VI

VA

Y

ys

f(x)>O, a>b

f(x)
f(x)b

c) Figure 8.5 IA:

/'==

sin zdx (read: Integral from x = 0 to z = T ) = (- cos 21: = (- cos F

+ cos 0) = 2.

2=0

W B: L'=*'sin zdz

(read: Integral from x = 0 to x = 2 ~ =) (- cos

= (- cos 2x

+ cos0) = 0.

3. Variable Upper Limit 1. Particular Integral If we consider the area depending on the upper limit as its variable (Fig. 8.6, region ABCD), then we have an area function in the form

/ f ( t )d t 2

S(z) =

(a < b and f(z) 2 0 for x 2 a ) .

a

We call this integral a particular integral.

(8.42)

8.2 Definite Inteorals 441

x

O a

x

The geometrical meaning of this theorem is that the derivative of the variable area S(z) is equal to the length of the segment YM (Fig. 8.7). Here, the area, just as the length of the segment. is considered according to the sign rule (Fig. 8 . 5 ) .

4. Decomposition of the Integration Interval The interval of integration [a, b] can be decomposed into subintervals. The value of the definite integral over the complete interval is

p a

f(x)dr

=

]f(x)dx + p f(.) a

dz.

(8.44)

C

This is called the interval rule. If the integrand has finitely many jumps, then the interval can be decomposed into subintervals such that on these subintervals the integrand will already be continuous. Then the integral can be calculated according to the above formula as the sum of the integrals on the subintervals. .4t the endpoints of the subintervals the function must be defined by its corresponding left or right-sided limit. if it exists. If it does not, then the integral is an improper integral (see 8.2.3.3, l.,p. 454) b] if we suppose Remark: The formula above is valid also in the case if c is outside of the interval [a, that the integrals on the right-hand side exist.

8.2.1.3 Further Theorems about the Limits of Integration 1. Independence of the Notation of the Integration Variable The value of a definite integral is independent of the notation of the integration variable: (8.45)

442

8. Intearal Culczllzls Table 8.5 Important properties of definite integrals

Property

Formula

1f(z)

dz = F ( z ) =:I

Fundamental theorem of the integral calculus ( f ( z )is continuous)

a

p

Interchange rule

~ ( a=)

1

f ( z ) dz = -

1

F(b) - F ( a ) with

dz + c or ~ ' ( z = ) f(x)

f ( z ) dx

b

a

Equal integration limits

Interval rule

p

Independence of the notation of the integration variable Differentiation with respect to the upper limit

~

Mean value theorem of the integral calculus

I/

f(x)dz =

a

p

p

f ( u )du = f(t ) d t

a

a

f ( t ) dt = f(z) with f(z)continuous

~

f(.)

dx = ( b - a ) f ( E )

(a < E

1

< b)

a

2. Equal Integration Limits If the lower and upper limits are equal, then the value of the integral is equal to zero:

p

f ( z ) dz = 0.

(8.46)

a

3. Interchange of the Integration Limits After interchanging the limits: the integral changes the sign (interchange rule): (8.47) a

4. Mean Value Theorem and Mean Value 1. Mean Value Theorem If a function f(z)is continuous on the interval [aIb]. then there is at least one value in this interval such that in the case A with a < E < band in the case B with a > E > b (see 8.2.1.1, 2..p.439)

<

j - i i x ) dx = ( b - a ) f ( E )

(8.48)

a

is valid. The geometric meaning of this theorem is that between the points a and b there exists at least one point

8.2 Definite Inteorals 443

E such that the area of the figure ABCD is equal to the area of the rectangle AB'C'D in Fig. 8.8. The value 1

b

/

m=f(x)dx b-ao

(8.49)

is called the mean value or the arithmetic average ofthe function f(x) in the interval [a, b] . 2. Generalized Mean Value Theorem If the functions f(x) and p(z) are continuous on the closed interval [a. b]. and ~ ( 2does ) not change its sign in this interval, then there exists at least one point such that b

/ f(x)&)

b

ds = f(E)

a

(a < E

dx

< b)

(8.50)

a

is valid. 5 . Estimation ofthe Definite Integral The value of a definite integral lies between the values of the products of the infimum m and the supremum M of the function on the interval [a,b] multiplied by the length of the interval:

/ f(x)dx 5 M ( b b

m(b - a ) 5

- a).

(8.51)

a

Iff is continuous. then m is the minimum and M is the maximum of the function. It is easy to recognize the geometrical interpretation of this theorem in Fig. 8.9.

Figure 8.9

8.2.1.4 Evaluation of the Definite Integral 1. Principal Method The principal method of calculating a definite integral is based on the fundamental theorem of integral calculus. Le., the calculation of the indefinite integral (see 8.2.1.2, 1..p. 440), e.g., using Table 21.5. Before substituting the limits we have to check if we have an improper integral. Nowadays we have computer algebra systems to determine analytically indefinite and definite integrals (see Chapter 20).

2. Transformation of Definite Integrals In many cases, definite integrals can be calculated by appropriate transformations, with the help of the substitution method or partial integration. W A: Use the substitution method for I =

I"m

d

Jo

z

First we substitute: x = p(t) = asint, t = $(x) = arcsin ,: li(0) = 0, $ ( a ) = I .We get: 2 I = L a J m d x = / a r c s ' n 1 a 2 ~ ~ c o s t =d at 2 ~ 4 c o s 2 t d t = a425(1+cos2t)dt. 1~ arcsin 0

R'ith the further substitution t = p(z) = f,z = @ ( t )= 2t, @(O) = 0. @(:) a2

I = -ti! 2

*

a2 + -/ 4 0

7r

2

iTa2

a'

get:

.a2

coszdz = - + - sinzl' - 4 4 0 4 '

W B: Method of partial integration:

= i~ we

1'

x ex dx = [xe']; -

1'

e"

dx = e - ( e - 1) = 1.

444

8. Inteoral Calculus

3. Method for Calculation of More Difflcult Integrals If the determination of an indefinite integral is too difficult and complicated, or it is not possible to express it in terms of elementary functions, then there are still some further ideas to determine the value of the integral in several cases. Here, we mention integration of functions with complex variables (see the examples on p. 692-695) or the theorem about the differentiation of an integral with respect to a parameter (see 8.2.4, p. 457):

d dt

1

(8.52)

T d x .

a

a

/#

1 b

f ( x ,t )dx =

'x-1 x d x . Introducing the parameter t: F ( t ) =

1

lxt-l -dx;

F ( 0 ) = 0; F(l) = I . lnx ' d xt-l 1xtlnx 1 dx = -dx = [ x t d x = -xttll' = dt o a t lnx o lnx t+l 0 tS1' t d t Integration: F ( t ) - F ( 0 ) = -= In(t t 1)Ifi= ln(t + 1). Result: I = F(l)= ln2. 0 t + l

II =

1 [-] 1

1

dF Using (8.52) for F ( t ) : - =

o

1

4. Integration by Series Expansion If the integrand f(x) can be expanded into a uniformly convergent series (8.53)

f ( x )= ' P 1 ( x ) + p z ( x ) + . . . + ' P n ( x ) + . . . in the integration interval [a, b],then the integral can be written in the form

In this way the definite integral can be represented as a convergent numerical series: (8.55) When the functions pk(x) are easy to integrate, if, e.g.. f ( x ) can be expanded in a power series, which is uniformly convergent in the interval [a,a], then the integral accuracy.

ICalculate the integral I =

1'''

l

f ( x )dx can be calculated to arbitrary

dx with an accuracy of 0.0001. The series e-"' = 1 -

e-"'

$+

$ $ + $ - . . . is uniformly convergent in any finite interval according to the Abel theorem (see -

7.3.3.1, p. 414), so /e-"' follows that I = /#"* e-"'

i

1

1

holds. With this result it

dx = x

dx = 1

i (1 1

-

-.

,

.). To achieve the accuracy 0.0001 for the calculation of the

= (1 + - zFBB + integral it is enough to consider the first four terms, according to the theorem of Leibniz about alternating series (see 7.2.3.3, l., p. 408):

I x i ( 1 - 0.08333 + 0.00625 - 0.00037) =

i '0.92255 = 0.46127,

rz e

x 2 dx

= 0.4613.

-

5 . Graphical Integration Graphical integration is a graphical method to integrate a function y = f ( x ) which is given by a curve Ai? (Fig. 8.10),i.e., to calculate graphically the integral

f ( x )dx, the area of the region MoABN:

8.2 Definite Inteurals 445

1.

The

interval

is

divided

by

the

points

~ I ~ z , x ~ , x ~ /. z. ,,~,-1,z,-1/2 xz,. into 2n equal parts, where the

result is more accurate if there are more points of division. we draw vertical lines intersecting the curve. The ordinate v a w o f the segments are denoted on the y-axis by OA1, OAz, . . . , OA,. 3. A segment oP of arbitrary length is placed on the negative x-axis, and P is connected with the points AI, Az, . . . ,A,. 4. Through the point ,440 a line segment is drawn parallel to PA1 to the intersection point with the line z = z&s is the segment M&'1. Through the point Mi the segment MlMz is drawn parallel to PA2 to the intersection with the line z = 2 2 , etc., until the last point M,,is reached with the abscissa zn. Theintegral .is numerically equal to the product of the length of OP and the length of 2. .At the points of division x l p , x 3 p , . . .

1

~

m:

-j ( z ) dx = OP . NM,.

Figure 8.10

(8.56)

a

By a suitable choice of the arbitrary segment oP the shape of our result can be influenced; the smaller the graph we want. the longer we should choose the segment

m.If oP = 1, then

l

f(z)dx =

m,

and the broken line ,Wo, Ml, M Z , . . . , M , represents approximately the graph of a primitive function of

f(x).Le., one of the functions given by the indefinite integral

I

j ( z ) dx.

6. Planimeter and Integraph A planimeter is a tool to find the area bounded by a closed plane curve, thus also to compute a definite integral of a function y = j(x) given by the curve. Special types of planimeters can evaluate not only

/

y dx, but also

S

y2 dx and

I

y3 dx.

An integraph is a device which can be used to draw the graph of a primitive function Y =

lz

f ( t ) dt if

the graph of a function y = j(x) is given (see [19.29]).

7. Numerical Integration If the integrand of a definite integral is too complicated, or the corresponding indefinite integral cannot be expressed by elementary functions, or we have the values of the function only at discrete points, e.g., from a table of values, then we use the so-called quadrature formulas or other methods of numerical mathematics (see 19.3.1. p. 895).

8.2.2 Application of Definite Integrals 8.2.2.1 General Principles for Application of the Definite Integral 1. We decompose the quantity we want to calculate into a large number of very small quantities, Le., into infinitesimal quantities: (8.57) A = a1 a2 ' ' . +a,. 2. We replace every one of these infinitesimal quantities a, by a quantity iL,, which differs only very slightly in value from a,, but which can be integrated by known formulas. Here the error CY, = a, - 5, should be an infinitesimal quantity of higher order than ai and 6%. 3. We express 6%by a variable z and a function f(z)so that Sdi has the form j(x,)Ax,. 4. \Ye evaluate the desired quantity as the limit of the sum

+ +

446

8. Intesral Calculus

(8.58) where Ax, 2 0 holds for every i. The lower and upper limit for x is denoted by a and b. IWe evaluate the volume V of a pyramid with base area S And height H (Fig. 8.11a-c): a) We decompose the required volume V by plane sections into frustums (Fig. 8.11a): \' = v1+ 11'2 + ' . ' + un. b) We replace every frustum by a prism, whose volume is iji. with the same height and with a base area of the top base of the frustum (Fig. 8.11b). The difference of their volumes is an infinitesimal quantity of higher order than v,. c ) We represent the volume ijt in the form iji = Si Ahi , where h, (Fig. 8 . 1 1 ~ )is the distance of the top surface from the vertex of the pyramid. Since S,: S = h: : H 2 Sh2 we can write: iji = 3Ah, .

H d) We calculate the limit of the sum

V= Figure 8.11

l i m e 6 - lim~-A Sh2 h , = / - - d hH=S-h- Z. ' - n + m 2=1 H2 H2

n + m ,t=1

SH 3

8.2.2.2 Applications in Geometry 1. Area of Planar Figures 1. Area of a Curvilinear Trapezoid Between B and C (Fig. 8.12a) if the curve is given by an equation in explicit form (y = f(z)and a 5 z 5 b) or in parametric form (x = z ( t ) , y = y(t). t l 5

t 5 tz):

pr!.)

(8.59a)

S,AB = ~ ~ dx = a

tl

2. Area of a Curvilinear Trapezoid Between G and H (Fig. 8.12b) if the curve is given by an equation in explicit form (x = g(y) and a 5 y 5 j3) or in parametric form (z= z ( t ) . y = y(t). tl 5

t 5 t2): (8.59b)

3. Area of a Curvilinear Sector (Fig. 8.12c), bounded by a curve between K and L , which is given by an equation in polar coordinates ( p = p(cp), cp1 5 p 5 cpz): (8.59~) Areas of more complicated figures can be calculated by partition of the area into simple parts, or by line integrals (see 8 3. p. 460) or by double integrals (see 8.4.1, p. 469).

8.2 Definite Integrals 447

a)

b) Figure 8.12

2. A r c l e n g t h o f P l a n e Curves 1. Arclength of a Curve Between T w o Points (I) A and B. given in explicit form (y = f ( z )or x = g(y)) or in parametric form (z= x ( t ) .y = y(t)) (Fig. 8.13a) can be calculated by the integrals:

L- = 4B

p

=

4-dz

a

1

J&jTldy

=

a

J4-k

(8.60a)

tl

\\lth the differential of the arclength dl we get

L=

/ dl

d12 = dx2 + dy’.

with

(8.60b)

The perimeter of the ellipse with the help of (8.60a): With the substitutions z = z ( t ) = a sin t , y =

y ( t ) = b c o s t wegetL- = AB

l:

da2

-

(a2 - b2) sin2t dt = a

6’

d t : where e = -/a

is the numerical eccentricity of the ellipse. Since z = 0. y = b and x = a , y = 0: the limits of the integral in the first quadrant are L4a

l”*d x

d

t

= a E(k,

ble 21.7 (see example on p, 436).

r ) with k

=

AB

=

2

E.

The value of the integral E ( k , -) T is . given in Ta2

2. Arclength of a Curve Between T w o Points (11) C and D. given in polar coordinates ( p = p(;r”)) (Fig. 8.13b):

(8.60~) LVith the differential of the arclength dl we get

L=

J

+

dl with d/* = p2dp2 dp2.

(8.60d)

3. S u r f a c e Area of a Body o f R e v o l u t i o n (see also First Guldin Rule, p. 451) 1. The area of the surface of a body given by rotating the graph of the function y = f(z)around the z-axis (Fig. 8.14a) is:

(8.61a)

I

448

8. Inteqrul Calculus

VI

Figure 8.13

Figure 8.14 2. The area of the surface of a body given by rotating x = f(y) around the y-axis (Fig. 8.14b) is:

S = 2 ; 4/ z d i = 2 r / x ( y ) ~ ~ d y (8.61b) a

n

To calculate the area of more complicated surfaces see the application of double integrals in 8.4.1.3, p. 472 and the application of surface integrals of the first kind, 8.5.1.3, p. 480. General formulas for the calculation of surface areas with double integrals are given in Table 8.9 (Application of double integrals), p. 473. 4. Volume (see also Second Guldin Rule, p. 461) 1. The volume of a rotationally symmetric body given by a rotation around the x-axis (Fig. 8.14a) is:

V

= K

j

y'dx.

(8.62a)

2. The volume of a rotationally symmetric body given by a rotation around the y-axis (Fig. 8.14b) is:

j

V = K x'dy.

(8.6213)

3. The volume of a body, whose section perpendicular to the x-axis (Fig. 8.15) has an area given by the function S = f ( z ) , is:

V=

j

f(z)dz.

(8.63)

8.2 Definite Integrals 449 General formulas t o calculate volumes by multiple integrals are given in Table 8.9 (Applications of double integrals, see p. 473) and Table 8.11 (Applications of triple integrals, see p. 478).

Figure 8.15

Figure 8.16

8.2.2.3 Applications in Mechanics and Physics 1. Distance Traveled by a Point The distance traveled by a moving point during the time from to until T with a time-dependent velocity v = f ( t ) is T

s=Judt

to

2. Work To determine the work in moving a body in a force field we suppose that the direction of the field and the direction of the moveme$ are constant and coincident. We define the z-axis in this direction. If the magnitude of the force F is changing, Le., IF1 = f(z),then we get for the work W necessary to move the body from the point I = a to the point z = b along the z-axis:

(8.65) In the general case, when the direction of the force field and the direction of the movement are not coincident, we calculate the work as a line integral (see (8.130), p. 467) of the scalar product of the force and the variation of the position vector at every point of i along the given path.

3. Pressure In a fluid at rest with a density e we distinguish between yravztational pressure and lateral pressure. This second one is exerted by the fluid on one side of a vertical plate immersed in the fluid. Both depend on the depth. 1. Gravitational Pressure The gravitational pressure P h at depth h is: Ph = @ Y h , (8.66) where y is the gravitational acceleration. 2. Lateral Pressure The lateral pressure p s , e.g., on the cover of a lateral opening of a container of some fluid Kith the difference of depth hl - hz (Fig. 8.16) is: ha

Ps = eY JzIy?(z) - Y1(2)1 dz. hi

The left and the right boundary of the Cover is given by the functions yl(z) and y*(z).

(8.67)

8. Intearal Calculus

450

4. Moments of Inertia 1. Moment of Inertia of an Arc The moment of inertia of a homogeneous curve segment y = f(z) with constant density pin the interval [a,b] with respect to the y-axis (Fig. 8.17a) is: (8.68)

If the density is a function ~ ( z )then , its analytic expression is in the integrand

Y

Figure 8.17 2. Moment of Inertia of a Planar Figure The moment of inertia of a planar figure with a homogeneous density e with respect to the y-axis, where y is the length of the cut parallel to the y-axis (Fig. 8.17b), is: b

Iy = e / x 2 y d z .

(8.69)

a

(See also Table 8.4.2.3: (Applications of line integral), p. 477.) If the density is position dependent, then its analytic expression must be in the integrand.

b)

a)

c)

Figure 8.18

5 . Center of Gravity, Guldin Rules 1. Center of Gravity of an Arc Segment The center of gravity C of an arc segment of a homogeneous plane curve y = f(x) in the interval [a, b] with a length L (Fig. 8.18a) considering (8.60a), p. 447. has the coordinates:

(8.70) Yc = a L L 2. Center of Gravity of a Closed Curve The center of gravity C of a closed curve y = f(z) (Fig. 8.18b) with the equations y1 = f~(z) for the upper part and y2 = f * ( x )for the lower part, and

xc=

a

8.2 Definite Intearals 451

with a length L has the coordinates: j&m+J-idx

xc =

a

3

L

Yc=a

L

. (8.71)

3. First Guldin Rule Suppose a plane curve segment is rotated around an axis which lies in the plane of the curve and does not intersect the curve. We choose it as the x-axis. The surface area S ,, of the body generated by the rotated curve segment is the product of the perimeter of the circle drawn by the centre of gravity at a distance yc from the axis of rotation. Le., 27ryc, and the length of the curve segment L : s,,,= L , 2iTyc. (8.72) 4. Center of Gravity ofa Trapezoid The center of gravity C of a homogeneous trapezoid bounded above by a curve segment between the points of the curve A and B (Fig. 8.18c), with an area Sof the trapezoid. and with the equation y = f ( x ) of the curve segment AB, has the coordinates: b

b

b/y2dx

/ W d X

xc = 0 .

yc=-

S

’

(8.73)

5 . Center of Gravity of an Arbitrary Planar Figure The center of gravity C of an arbitrary planar figure (Fig. 8.18d) with area S.bounded above and below by the curve segments with the equations y1 = f l ( x )and y2 = f 2 ( x ) ,has the coordinates

(8.74)

Formulas to calculate the center of gravity with multiple integrals are given in Table 8.9 (.4pplication of double integrals, p. 473) and in Table 8.11 (Application of triple integrals, p. 478). 6. Second Guldin Rule Suppose a plane figure is rotated around an axis which is in the plane of the figure and does not intersect it. We choose it as the x-axis. The volume V of the body generated by the rotated figure is equal to the product of the perimeter of the circle drawn by the center of gravity under the rotation, Le., 27ryc, and the area of the figure S: (8.75) vrot= s ’ 27ryc.

8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals 8.2.3.1 Generalization of t h e Notion of t h e Integral Thenotionofthedefiniteintegral (see 8.2.1.1, p. 438), as aRiemannintegra1 (see 8.2.1.1,2., p. 439), was introduced under the assumptions that the function f(z)is bounded, and the interval [a, b] is closed and finite. Both assumptions can be relaxed in the generalizations of the Riemann integral. In the following we mention a few of them.

1. Improper Integrals These are the generalization of the integral to unbounded functions and to unbounded intervals. We discuss integrals with infinite integration limits and integrals with unbounded integrands in the next paragraph.

2. Stieltjes Integral for Functions of One Variable bye start from two finite functions f ( x ) and g(z)defined on the finite interval [a, b]. We make a partition of the interval into subintervals, just as with the Riemann integral, but instead of the Riemann sum (8.37) we compose the sum

452

8. Inteqral Calculus

(8.76) If the limit of (8.76) exists, when the length of the subintervals tends to zero, and it is independent of the choice of the points x, and El, then this limit is called a dejnite Stieltjes integral (see [8.8]). IFor g ( x ) = x the Stieltjes integral becomes the Riemann integral.

3. Lebesgue Integral Another generalization of the integral notion is connected with measure theory (see 12.9,p. 633), where the measure of a set, measure spaces, and measurable functions are introduced. In functional analysis the Lebesgue integral is defined (see 12.9.3.2, p. 635) based on these notions (see [8.6]). The generalization with comparison to the Riemann integral is, e.g., the domain of integration can be a rather general subset of Rnand it is partitioned into measurable subsets. There are different notations for the generalizations of the integrals (see [8.8]).

8.2.3.2 Integrals with Infinite Integration Limits 1. Definitions a) If the integration domain is the closed half-axis [a, +co),and if the integrand is defined there, then

the integral is by definition

1

1 B

+X

f(x)dx = B-+W lim

a

(8.77)

f(x)dx.

a

If the limit exists, then the integral is called a convergent improper integral. If the limit does not exist, then the improper integral (8.77) is divergent. b) If the domain of a function is the closed half-axis (-m, b] or the whole real axis (-cc,+m), then we define analogously the improper integrals

c) At the limits of (8.78b) the numbers A and B tend to infinity independently of each other. If the limit (8.78b) does not exist, but the limit +A

lim A-tm

1f

(8.78~)

(x)dx,

-A

exists, then this limit (8.78~) is called the principal value ofthe improper integral. or Cauchy’s principal value. Remark:An obviously necessary but not sufficient condition for the convergence of the integral (8.77) f(x) = 0. is rlim i m

2. Geometrical Meaning of Integrals with Infinite Limits The integrals (8.77)) (8.78a) and (8.78b) give the area of the figures represented in Fig. 8.19. 33 dx B dx IA: - = lim - = lim 1nB = co (divergent). B+m 1 x B-tm 1 x B dx 1 1 m dx IB: - = lim - = lim = (convergent). 2 x2 B+m 2 x2 B+m 2 B 2

1

1

J

J

(- -) 1 =

lirn [arctan B - arctan A] =

8.2 Definite Integrals 453

Figure 8.19

3. Sufficient Criteria for Convergence Ifthedirect calculationofthelimits (8.77), (8.78a) and (8.78b) iscomplicated, or ifonlytheconvergence or divergence of an improper integral is the question. then one of the following sufficient criteria can be used. Here, only the integral (8.77) is considered. The integral (8.78a) can be transformed into (8.77) by substitution of z by - x : (8.79) -a

-X

The integral (8.78b) can be decomposed into the sum of two integrals of type (8.77) and (8.78a): tx

/

tx

f(z)dz t

f ( z ) dz =

-X

-X

/ f(z)

(8.80)

dz.

C

where c is an arbitrary number Criterion 1: If f(x) is integrable on any finite subinterval of [a, co)and if the integral (8.81) a

exists. then there exists also the integral (8.77). The integral (8.77) is in this case said to be absolutely convergent. and the function f(z) is absolute zntegrable on the half-axis [a, +m). Criterion 2: If for the functions f(z)and p(z) (8.82a) f(r)> 0, ~ ( 2>) 0 and f(z)5 p(z) for a 5 z < +m hold, then from the convergence of the integral

/ g(s)dx a

/ f(x)dx

tm

tco

(8.82b)

the convergence of the integral

a

(8.82~)

8. Inteqrul Calculus

454

follows, and conversely. from the divergence of the integral (8.82~)the divergence of the integral (8.82b) follo~vs. Criterion 3: If we substitute 1 (8.83a) P(X) = 2. and we consider that for a > 0,

(a- 1 ) u O - 1

a

cy

> 1 the integral (8.83b)

(a>O.cu>l)

is convergent and has the value of the right-hand side, and the integral of the left-hand side is divergent for cy 5 1. then we can deduce a further convergence criterion from the second one: If f ( z ) in a 5 x < co is a positive function. and there exists a number cy > 1 such that for x large enough (8.83~) f ( x ) x' < k < co ( k > 0, const) holds. then the integral (8.77) is convergent: if f ( z ) is positive and there exists a number cy 5 1 such that (8.83d) f ( x ) x" > c > 0 (c > 0, const) holds from a certain point, then the integral (8.77) is divergent. f m xsf2 dx

If we substitute

cy

=

1 x3~2 , then we get -2''' 2 1+x2

-

22

= - 3 1. The integral is

1 +x2

divergent.

4. Relations Between Improper Integrals and Infinite Series If x i , x2.. . . ,I,. . . . is an arbitrary, unlimited increasing infinite sequence, Le., if lim z, = co, a < z1 < 2 2 < ... < z, < ... with n+tx

(8.84a)

and if the function f(x) is positive for a 5 x < x. then the problem of conveigence of the integral (8.77) can be reduced to the problem of convergence of the series (8.84b) If the series (8.84b) is convergent, then the integral (8.77) is also convergent, and it is equal to the sum of the series (8.84b). If the series (8.84b) is divergent, then the integral (8.77) is also divergent. So the convergence criteria for series can be used for improper integrals, and conversely, in the integral criterion for series (see 7.2.2.4. p. 406) we use the improper integrals to investigate the convergence of infinite series.

8.2.3.3 Integrals with Unbounded Integrand 1. Definitions 1. Right O p e n or Closed Interval For a function f ( z ) ,which has a domain open on the right [a. b) or a domain closed on the right [a, b]. but at the point b it has the limit Kl0f(x) = x.we have the

definition of the improper integral in both cases.

1 b

a

1 b-E

f ( x ) dx = lim E

4

f ( x ) dx.

(8.85)

Q

If this limit exists and is finite. then the improper integral (8.86) exists, and we call it a convergent zmproper zntegrul. If the limit does not exist or it is not finite, then we call the integral a divergent zmproper zntegrul.

8.2 Definite Intearals 455

2. Left Open or Closed Interval For a function j ( z ) ,which has a domain open on the left ( a ,b] or a domain closed on the left [a.b], but at the point a it has the limit lim f ( s ) = CIC) we define the Z+l+O

improper integral analogously to (8.85). That is: (8.86) a

a+E

3. Two Half-Open Contiguous Intervals For a function j ( z ) ,which is defined on the interval [a,b] except at an interior point c with a < c < b, i.e., for a function j ( z ) defined on the half-open intervals [ a ,c) and (c. b], or is defined on the interval [a,b], but at the interior point c it has an infinite limit at least from one side lim f ( z ) = co or lim f ( z ) = M, the definition of the improper integral x+c-o

ZC ’ +O

is:

1 b

1 C-E

f ( z ) dz = lim

€40

f ( z ) dz

a

0

+F z

1f(.) b

dx.

(8.87a)

C+6

Here the numbers E and 6 tend to zero independently of each other. If the limit (8.87a) does not exist, but the limit

does, then we call the limit (8.87b) the principal value of the improper integral or Cauchy’s principal

value. 2. Geometrical Meaning The geometrical meaning of the integral of discontinuous functions (8.85), (8.86), and (8.87a) is to find the area of the figures bounded. e.g., from one side by a vertical asymptote as represented in Fig.8.20.

a) s. (8.85)

IA:

lb* 0

IB:

b) s. (8.86)

c) s. (8.87a)

Figure 8.20 :

Case (8.86), singular point at z = 0.

6

i”* 1‘’’

t a n z dz : Case (8.85), singular point at z = T . 2

tan zdz = lim

E 4 0

IT’*-‘ Q

t a n z ds = lim In cos 0 - In cos E+O

(f - E ) ]

=x

(divergent)

8. Inteqral Calculus

456

IC:

f 2: Case (8.87a), singular point at x = 0. 3 3 9 - 1) + lim -(4 - J213) = (convergent). s_" $ + limJd @ = lim -(&'I3 fi = lim\-' 2 @ 2 2 1 f i

€ 4-1

1

ID:

\

2

6-0

6

-

6+0

E-02

2xdx

-: Case (8.87a), singular point at x = ikl -1

-2 x2

= limln(x2 - 1) E+O

+

'.

= lim[ln(l €+O

+ 2~ + E'

- 1) - In31

+ ... = m

(divergent).

3. The Application of the Fundamental Theorem of Integral Calculus 1. Warning The calculation of improper integrals of type (8.87a) with the mechanical use of the formula

\ j ( x )dx b

= F ( x ) ;1

with F'(x)= j ( x )

(8.88)

a

(see 8.2.1.1,p. 438) usually results in mistakes if the singular points in the interval [a, b] are not taken into consideration. IE: Using formally the fundamental theorem we get for the example D

L2 2

2xdx

ln3 - ln3 = 0,

= ln(x2 - 1)

though this integral is divergent. 2. General Rule The fundamental theorem of integral calculus can be used for (8.87a) only if the primitive function of f ( x ) can be defined to be continuous at the singular point. IF: In the example D the function ln(x2 - 1) is discontinuous at x = 41, so the conditions are not 3 1 fulfilled. Consider the example C. The function y = - x213is such a primitive function of - on the 2 @ intervals [a. 0) and (0, b] which can be defined continuously at IC = 0, so the fundamental theorem can be used in the example C:

4. Sufficient Conditions for the Convergence of an Improper Integral with

Unbounded Integrand

1 j ( x ) l dx i," absolutely convergent integral 1. If the integral

exists, then the integral

I"

f(x)dx also exists. In this case we call it an

and the function f(x)is an absolutely integrable function on the consid-

ered interval. 2. If the function j ( x ) is positive in the interval [a, b ) , and there is a number a < 1 such that for the values of z close enough to b (8.89a) j ( 2 ) ( b - z ) <~ x holds, then the integral (8.87a) is convergent. But, if the function f(x)is positive in the interval [a,b), and there is a number a > 1 such that for the values of x close enough to b (8.89b) j ( x )( b - x)" > c > 0 (c const) holds. then the integral (8.87a) is divergent.

8.2 Definite Inteorals 457

8.2.4 Parametric Integrals 8.2.4.1 Definition of Parametric Integrals The definite integral

p

(8.90)

f(x. dx = F(Y)

a

is a function of the variable y considered here as a parameter. In several cases the function F ( y ) is no longer elementary, even if f(x,y) is an elementary function of x and y. The integral (8.90) can be an ordinary integral, or an improper integral with infinite limits or unbounded integrand f(x,y). For theoretical discussions about the convergence of improper integrals depending on a parameter see, e.g.,[8.3]. I Gamma Function or Euler Integral of the Second Kind (see 8.2.5,6., p. 459): m

(convergent for y > 0).

T(y) = / x v - l e P d x

(8.91)

0

8.2.4.2 Differentiation Under the Symbol of Integration 1. Theorem If the function (8.90) is defined in the interval c 5 y 5 e, and the function f(x,y) is continuous on the rectangle a 5 2 5 b, c 5 y 5 e and it has here a partial derivative with respect to y, then for arbitrary y in the interval [cl e]:

(8.92) This is called diferentiation under the symbol of integration. d

IFor arbitrary y > 0: dy

/ arctan 1

0

dx =

1 aY la

- (arctan

0

z)

dx = -

1

1

xdx

1 yz = - In -

2

1+yz

For y = 0 the condition of continuity is not fulfilled, and there exists no derivative. 2. Generalization for Limits of Integration Depending on Parameters The formula (8.92) can be generalized, if with the same assumptions we made for (8.92) the functions a(y)and p(y) are defined in the interval [c, e], they are continuous and differentiable there, and the curves x = a ( y ) ,x = B(y) do not leave the rectangle a 5 x 5 b, c 5 y 5 e: (8.93)

8.2.4.3 Integration Under the Symbol of Integration If the function f(x. y) is continuous on the rectangle a 5 x 5 b. c 5 y 5 e, then the function (8.90) is defined in the interval [c, e ] ,and (8.94) is valid. This is called integration under the symbol ufzntegratiun. IA: Integration of the function f ( x l y) = xu on the rectangle 0 5 x 5 1, a 5 y 5 b. The function xu is discontinuous at x = 0, y = 0, for a > 0 it is continuous. So we can change the order of integration:

I

8. Inteqral Calculus

458

[

[Jdl

xY dz] dy =

1' [[

l+b

xY dy] dx. On the left-hand side we get

-

I xb

Ins:dx. The indefinite integral cannot be expressed by elementary functions. Anyway,

handsidel

the definite integral is known, so we get: Jd'-dx=lnl t b (O
IB: Integration of the function f(x, y) =

~

8.2.5 Integration by Series Expansion, Special Non-Elementary

Functions It is not always possible to express an integral by elementary functions, even if the integrand is an elementary function. In many cases we can express these non-elementmy integrals by series expansions. If the integrand can be expanded into a uniformly convergent series in the interval [a,b], then we get

f ( t ) d t if we integrate it term by term.

also a uniformly convergent series for the integral

1. Integral Sine (1x1 < co,see also 14.4.3.2. 2.. p. 694) sint /7 dt 5

Si (z)=

0

K

1

2

x

m

.

=--

sint -dt t (8.95)

2. Integral Cosine (0 < z < 03)

=C

C=-

J

x2 24 (- l)nx2n + lnz - +- t .. + t ... 2 . 2 ! 4.4! 2n ( 2 n ) ! ,

eCt In t dt = 0.577 215 665, . .

3. Integral Logarithm (0 < 5 Li(z) = 0

~

with

(Euler constant).

(8.96a)

(8.96b)

< 1, for 1 < 2 < 00 as Cauchy Principal Value)

(In x)' (lnz)n dt Int = c t l n / l n x l + lnlzl+ -+ . . . t -t " 2.2! n . n!

(8.97)

8.2 Definite Inteoruls 459

4. Exponential Integral (-cc

j

Ei (2) =

d t = C t In

< z < 0,for 0 < z < 00 X2

Xn

2 ' 2!

n,n!

as Cauchy Principal Value)

1x1 t z t -t . ' . + -t . . .

-X

(8.98a)

Ei (lnz) = Li (z).

(8.9813)

5 . Gauss Error Integral and Error Function The Gauss error integral is defined for the domain 1x1 < m and it is denoted by O~The following definitions and relations are valid:

(8.99b) -1 2

(8.99~)

The function O(x)is the distribution function of the standard normal distribution (see 16.2.4.2, p. 757) and its values are tabulated in Table 21.15,p. 1085. The error function erf (z).often used in statistics (see also 16.2.4.2, p. 757), has a strong relation with the Gauss error integral: 2 x erf (z)= -/ e - " d t

fi0

= 200(zd).

(8.100a)

J&erf

(8.1OOb)

(x)= 1,

(-l)nz2n+l n! . (2n + 1)

(8.100~)

(8.100e)

6. Gamma Function and Factorial 1. Definition The gamma function, the Euler integral of the second kind (8.91), is an extension of the notion of factorial for arbitrary numbers z,even complex numbers, except zero and the negative integers. The curve of the function r ( z ) is represented in Fig. 8.21. Its values are given in Table 21.8, p. 1057. It can be defined in two ways:

1

4 x

Y

r(x)=

e-tt"-' dt

(x > 0 ) ,

.-2

(8.101a)

-3 -4 -5

0

nx . n! r(s)= n+m lim z ( x t 1)(zt 2) (x # 0. -1, -2,. . .).

t . .

(z t n)

(8.101b)

Figure 8.21

2. Properties of the Gamma Function

r(zt 1) = zT(sj,

(8.102a)

r ( n t 1) = n! (n = 0 , 1 , 2 , . . .),

(8.102b)

I

460

8. Inteoral Calculus

( n = O , l , 2 , . .),

(8.102e)

I

(8.1029 The same relations are valid for complex arguments z , but only if Re ( z ) > 0 holds. 3. Generalization of t h e Notion of Factorial The notion of factomal, defined until now only for positive integers n (see 1.1.6.4, 3., p. 13), leads to the function (8.103a) x! = i-(z 1) as its extension for arbitrary real numbers. The following equalities are valid:

+

Forpositiveintegersz: z ! = 1 . 2 . 3 . . . x ,

(8.103b) forz=O:

O!=r(l)=l,

for negative integers z: z! = rtm,

(8.103d) f o r z = f :

(i)!=r(i) =$,(8.103e)

f o r z = - ; :2

(8.103~)

( - f ) ! = r ( i ) = f (8.1034 i,

An approximate determination of a factorial can be performed for large numbers (> lo), also for fractions n with the Stirling formula: n ! x (a)"dG(l+--+-+...), 1 1 (8.103h) 12n 288n2 ln(n!) x ( n +

:)

Inn - n + In JZ;;.

(8.103i)

7. Elliptic Integrals For the complete elliptic integrals (see 8.1.4.3. 2., p. 435) the following series expansions are valid: (8.104)

k2 < 1. (8.105) The numerical values of the elliptic integrals are given in Table 21.7,p. 1055.

8.3 Line Integrals The notion of the integral can be generalized in different ways. While the domain of an ordinary definite integral is an interval on the numerical axis, for a line integral, the domain of integration is a segment of a planar or space curve. The curve, Le., the path of integration can also be closed; it is called also circuit integral and it gives the circulation of the function along the curve. We distinguish line integrals of the first type. of the second type, or of general type.

8.3 Line Inteorals 461

8.3.1 Line Integrals ofthe First Type

Y

8.3.1.1 Definitions The lzne zntegral ofthefirst type or zntegral over an arc is the definite integral

1f

(8.106)

( X >Y) ds,

(CI

-

where u = f(x.y) is a function of two variables defined on a connetted domain and the integration is performed over an arc C -AB of a plane curve given by its equation. The considered arc is in the same domain, and we call it the path ofzntegratzon. The numerical value of the line integral of the first type can be determined in the following way (Fig. 8.22):

c

b

X

Figure 8.22

1. We decompose the rectifiable arc segment AB into n elementary parts by points AI, Az, . . . , A,-1 chosen arbitrarily. starting at the initial point A A0 and finishing at the endpoint B I A,. h

2. We choose arbitrary points P, inside or at the end of the elementary arcs A,-lA,, with coordinates t,and rl,.

-

3. We multiply the values of the function f(JEI 7%)at the chosen points with the arclength Al-lA,= As,-, which should be taken positive. (Since the arc is rectificable, As,-, is finite.) 4. We add t h e n products f(&. qt)Ast-l. 5 . We evaluate the limit of the sum

(8.107a) as the arclength of every elementary curve segment As,-l tends to zero, while n obviously tends to m. If the limit of (8.107a) exists and is independent of the choice of the points A, and P,, then this limit is called the line zntegral of the first type, and we write n

(8.107b) We can define analogously the line integral of the first type for a function u = f(x,y , z ) of three variables, whose path of integration is a curve segment of a space curve: (8.107~)

8.3.1.2 Existence Theorem The line integral of the first type (8.107b) or (8.107~)exists if the function 12, Y) or [x,y , z ) and also the curve along the arc segment C are continuous, and the curve has a tanger which vi ies continuously. In other words: The above limits exist and are independent of the choice oi l i and P In this case, the functions f(x,y ) or f(x>y , z ) are said to be integrable along the curve C .

8.3.1.3 Evaluation of the Line Integral ofthe First Type We calculate the line integral of the first type by reducing it to a definite integral.

462

8. Inte.qral Calculus

1. The Equation of the Path of Integration is Given in Parametric Form If the defining equations of the path are z = z ( t )and y = y(t), then f(X>Y) ds = Splixlt!. (C)

(8.108a)

Y(t)l&%FTFmdt

t0

holds. and in the case of a space curve z = x ( t ) ,y = y ( t ) , and z = z ( t ) T

J f b ,Y>

2 ) ds

=

(CI

1

f[z(t),Y(t)l z(t)lJlz’(t)12

+ [Y’(t)I2+ [Z’(t)I2dt,

(8.108b)

t0

where t o is the value of the parameter t at the point A and T is the parameter value at B . The points A and B are chosen so that to < T holds.

2. The Equation of the Path of Integration is Given in Explicit Form We substitute t = z and get from (8.108a) for the planar case

1

b

f ( s > Y ) d s= / f P : Y ( z ) l ~ m d ~ :

(8.109a)

a

(CI

and from (8.108b) for the three dimensional case

1

b

f ( ~ . : ~ . ~ ) d ~ = ~ f I ~ , ~ ( ~ ) , ~ ( ~ ) l ~ 1 + i u ‘ ( ~ ) l ~ + [ ~ ’ (8.109b) ( ~ ) 1 ~ ~ ~

(C)

a

Here a and b are the abscissae of the points A and B , where the relation a < b must be fulfilled. We can consider x as a parameter if every point corresponds to exactly one point on the projection of the curve segment C onto the z-axis, i.e., every point of the curve is uniquely determined by the value of its abscissa. If this condition does not hold, we have to partition the curve segment into subsegments having this property. The line integral along the whole segment is equal to the sum of the line integrals along the subsegments.

8.3.1.4 Application of t h e Line Integral of t h e First Type Some applications of the line integral of the first type are given in Table 8.6. The curve elements needed for the calculations of the line integrals are given for different coordinate systems in Table 8.7.

8.3.2 Line Integrals ofthe Second Type 8.3.2.1 Definitions A line integral ofthe second type or an integral ouer a projection (onto the x-,y- or z-axis) is the definite integral (8.11Oh)

(8.11Oa) (C)

(C)

-

where f(x.y) or f ( z ,y, 2 ) are two or three variable functions defined on a connected domain. and we integrate over a projection of a plane or space curve C =AB (given by its equation) onto the z-. y-, or z-axis. The path of integration is in the same domain. We get the line integral of the second type similarly to the line integral of the first type, but in the third step the values of the function f(&,172) or f ( & ,qt, Cz) are not multiplied by the arclength of the n

elementary curve segments A,-lA,, hut by its projections onto a coordinate axis (Fig. 8.23).

8.3 Line Intearals 463

Table 8.6 Line integrals of the first type ~

Length of a curve segment C

1

Vass of an inhomogeneous curve segment C

L=

/ ds

I

M =

eds

(CI

Center of gravity coordinates

xc =

1

-

L

(e = f(z,y, z ) density function)

1

1

zeds

y~ =

I, = I, =

Moments of inertia of a space curve with respect to the coordinate axes

1

/ x'eds,

I

(y'

Iy=

+ z'),ods,

Iy=

I(.'

1

zeds

(C)

/ y'eds

(CI

I, =

1 yeds , zc = L

(CI

(C)

Moments of inertia of a plane curve in the z.u plane

/

+ z')pds.

(CI

+

(z' y')eds

In the case of homogeneous curves we substitute

e = 1.

Table 8.7 Curve elements

1

Cartesian coordinates z,y = y(z) Plane curve in Polar coordinates 9,p = p(p) the x. y plane Parametric form in Cartesian ds = J[z'(t)]'

-

1. Projection onto the x-Axis It'ith Pr, A,-lA,= z,

- x,-1 = h - 1

we get

(8.113a)

+ [y'(t)]' + [z'(t)]'dt

464

8. Inteoral Calculus

(8.113b)

(8.114)

8.3.2.2 Existence Theorem The line integral of the second type in the form (8.112a), (8.113a), (8.112b), (8.113b) or (8.114) exists if the function f(z,y) or f(z,y, z ) and also the curve are continuous along the arc segment C, and the curve has a continuously varying tangent there.

8.3.2.3 Calculation of the Line Integral of the Second Type We reduce the calculation of the line integrals of the second type to the calculation of definite integrals.

1. The Path of Integration is Given in Parametric Form With the parametric equations of the path of integration z = z ( t ) , y = y(t) and (for a space curve) z = z ( t ) we get the following formulas:

(8.115)

For (8.112a)

(8.116a)

For (8.113a)

(8.116b) m

For (8.112b)

(8.116c)

For (8.113b)

(8.116d)

For (8.114)

(8.116e)

Here, to and T are the values of the parameter t for the initial point A and the endpoint B of the arc segment. In contrast to the line integral of the first type, here we do not require the inequality to < T . Remark: If we reverse the path of the integral, Le., interchange the points A and B , the sign of the integral changes.

2. The Path of Integration is Given in Explicit Form In the case of a plane or space curve with the equations y=y(z) or y = y ( z ) , z =z(z)

(8.117)

8.3 Line Inteorals 465 as the path of integration, with the abscissae u and b of the points A and B , where the condition a < b is no longer necessary, the abscissa ztakes the place of the parameter t in the formulas (8.112a) - (8.114).

8.3.3 Line Integrals of General Type 8.3.3.1 Definition A line rntegral ofgeneral type is the sum of the integrals of the second type along all the projections of a curve. If two functions P ( z ,y) and Q(z,y) of two variables, or three functions P ( z ,y, z ) , Q(z,y , ~ ) , and R(z.y >z ) of three variables, are given along the given curve segment C, and the corresponding line integrals of the second type exist, then the following formulas are valid for a planar or for a space curve. 1. Planar Curve (8.1 1sa)

(8.118b)

The vector representation of the line integral of general type and an application of it in mechanics will be discussed in the chapter about vector analysis (see 13.3.1.1,p. 658).

8.3.3.2 Properties of t h e Line Integral of General Type 1. The Decomposition of the Path of the Integral

-

/4? a) A

by a point AI, which is on the curve, and it can even be outside of AB (Fig. 8.24). results in the decomposition of the integral into two parts:

4B

AM

MB

Figure 8.24

2. The Reverse of the Sense of the Path of Integration changes the sign of the integral:

-

(8.120)

/ ( ~ d z + ~ d y= )- / ( ~ d z + ~ d y ) . * h

BA

4B

3. Dependence on the Path In general, the value of the line integral is dependent not only on the initial and endpoints but also on the path of integration (Fig. 8.25):

-1

1

(Pdz+Qd~)#

Figure 8.25

ADB

I

(zydz + yzdy

A

(8.121)

h

AM B

1 A: I =

(Pdz+Qdy).*

+ z z d z ) , where C is one turn of the helix z = acost, y = asint, z = bt

(C)

(see Helix on p. 242) from to to T = 2 ~ : d

I = /OZn(-a3 sin't cos t + a'bt sin t cos t + ab% cost) dt = -'Similar formulas are valid for the three-variable case.

b

2

.

466

8. Inteoral Calczllzls

B: I

=

I

[y' dx

+ (zy - z') dy], where C is the arc of the parabola y2 = 9z between the points

(C)

A(O:O)andB(1!3): I = / 3 [ 2s y 3 + ( c3 - l ) ]4 9 81

dy=6- 3 20

8.3.3.3 Integral Along a Closed Curve 1. Notion of t h e Integral Along a Closed Curve .4 circuit integral or the circulation along a curwe is a line integral along a closed path of integration C , Le., the initial point A and the end point B are identical. We use the notation:

f ( P ~ +Z Q ~ Y )

f p d x + Q ~ +Y ~ d z ) .

or

(8.122)

(C)

(C)

In general. this integral differs from zero. But it is equal to zero if the conditions (8.127) are satisfied, or if the integration is performed in a conservative field (see 13.3.1.6, p. 660). (See also zero-valued circulation, 13.3.1.6,p. 660.) 2. The Calculation of t h e Area of a Plane Figure is a typical example of the application of the integral along a closed curve in the form

s = -2l f ( x d y - ydz),

(8.123)

(C)

where C is the boundary curve of the plane figure. The integral is positive if the path is oriented counterclockwise.

8.3.4 Independence of the Line Integral of the Path of Integration The condition for independence of a line integral of the path of integration is also called zntegrabzlzty of the total dzferentzal.

8.3.4.1 Two-Dimensional Case If the line integral

/W,Y) d x + Q(x, Y) d ~ l

(8.124)

with continuous functions P and Q defined on a simple connected domain depends only on the initial point A and the endpoint B of the path of integration, and does not depend on the curve connecting these points. Le., for arbitrary A and B and arbitrary paths of integration ACB and ADB (Fig. 8.25) the equality

1

(Pdz+Qdy)=

h

ACB

1

(Pdx+Qdy)

,-.

(8.125)

ADB

holds. then it is a necessary and sufficient condition for the existence of a function U ( z ,y) of two variables, whose total differential is the integrand of the line integral:

P dz + Q dy = d V ,

(8.126a)

(8.126b)

The function C(z,y) is a primitive function of the total differential (8.126a). In physics. the primitive

8.3 Line Intearals 467

function Cr(x,y) means the potential in a vector field (see 13.3.1.6,4., p. 661).

8.3.4.2 Existence of a Primitive Function .A necessary and sufficient criterion for the existence of the primitive junctzon. the integrability condition for the expression P dx + Q dy, is the equality of the partial derivatives d P dQ dy ax ’ where also the continuity of the partial derivatives is required.

(8.127)

8.3.4.3 Three-Dimensional Case The condition of independence of the line integral /[P(x. y. z ) dx t Q(x, y. z ) dy + R(x.y, 2) dz]

(8.128)

of the path of integration analogously to the two-dimensional case is the existence of a primitive function C(z. y , z ) for which (8.129a) P d x + Q d y + Rdz = dC‘, holds, i.e ,

p = -ac ,

ar;

ar; Q=R=--. 82 aY ’ The integrability condition is now that the three equalities for the partial derivatives dR - dP dP dQ -dQ = - dR -a2 dy ’ dx dz dy ax should be simultaneously satisfied. provided that the partial derivatives are continuous. dx

(8.129b)

(8.129~)

H The work 11’ (see also 8.2.2.3, 2., p. 449) is defined as the scalar product of force F(fl and displacement ?. In a conservative field the work depends only on the place ?, but not on the velocity .u’ With @ = P& t QZYt RZ2 = gradV and & = dx& t dyZYt dzCz the relations (8.129a). (8.129b) are satisfied for the potential V ( f l .and (8.129~)is valid. Independently of the path between the points PI and P2 we get:

IT =

[e(7)& 8

=

Ll

9

[ P d x t Qdy t Rdz] = V ( P 2 )- V ( P l ) .

Figure 8.26

(8.130)

Figure 8.27

8.3.4.4 Determination of the Primitive Function 1. Two-Dimensional Case If the integrability condition (8.127) is satisfied, then along an arbitrary path of integration connecting an arbitrary fixed point A(x0.yo) with the variable point P ( x ,y) (Fig.8.26) and passing through the

I

8. Inteoral Calculus

468

domain where (8.127) is valid, the primitive function U ( X y) , is equal to the line integral c'=

I

(8.131)

(Pdz+Qdy)

h

AP

In practice. it is convenient to choose a path of integration parallel to the coordinate axes, Le., one of the segments A K P or A L P , if they are inside the domain where (8.127) is valid. With these we have two formulas for the calculation of the primitive function V ( Xy) , and the total differential P d z + Q dy: (8.132a)

(8.132b) Here C is an arbitrary constant.

2. Three-Dimensional Case (Fig. 8.27) If the condition (8.129~)is satisfied, the primitive function can be calculated for the path of integration A K L P with the formulas

U = W X O , yo, 20) +

J +J +J

m m D X

/ P(<,

+

Y

Z

1

+ 1R(x, y, E ) dJ + C

(C arbitrary constant). (8.133) Yo 20 For the other five possibilities of a path of integration with the segments being parallel to the coordinate axes we get five further formulas. =

yo>zo) dE

Q(zlq,zo) dq

20

Application of the formula (8.13213) and the substitution of xo = 0, yo = 1 (20 = 0, yo = 0 may not be chosen since the functions P and Q are discontinuous at the point (0,O)) results in b' = -+ -+ U ( O ,1) = - arctan + c = arctan Y +~ 1 . Y

1'

)

-$ + (

&)

~ ~ 12 +dx ~t z dy iL zz dz. The relations (ZlY (8.129~)are satisfied. Application of the formula (8.133) and substitution of xo = 1, yo = 1. zo = 1 1 z z result in U = d t J;g 0 . dq d( C = arctan - - - t C .

B: P d X + Q d y + R d t

10.

+

=z

+

(-

--)

+

2

XY

8.3.4.5 Zero-Valued Integral Along a Closed Curve

The integral along a closed curve, Le., the line integral P d x + Q dy is equal to zero, if the relation (8.127) is satisfied, and if there is no point inside the curve where even one of the functions P, Q, 8P or

2

37

is discontinuous or not defined.

Remark: The value of the integral can be equal to zero also without this conditions, but then we get this value only after performing the corresponding calculations.

8.1 Multiple Inteorals 469

8.4 Multiple Integrals The notion of the integral can be extended to higher dimensions. If the domain of integration is a region in the plane or on a surface in space, then the integral is called a surface integral, if the domain is a part of space, then it is called a volume integral. For the different special applications we use different notations.

8.4.1 Double Integrals 8.4.1.1 Notion of the Double Integral 1. Definition The double integral of a function of two variables u = f ( z , y) over a planar domain S is denoted by

1f(.. ?/I 11f(T dS =

I/)

(8.134)

dI/ dz.

S

S

It is a number. if it exists. and it is defined in the following way (Fig. 8.28): 1. We consider a partition of the domain S into n elementary domain. 2. \$'e chose an arbitrary point Pi(z,:yi) in the interior or on the boundary of every elementary domain. 3. We multiply the value ofthe function u = f ( z z ,yz) at this point by the area AS,ofthe corresponding

elementary domain. 4. We add these products f(s,, y,)AS,. 5. We calculate the limit of the sum

kfb,.

YJlS,

(8.135a)

2=1

as the diameter ofthe elementary domains tends to zero, consequently AS',tends to zero, and so n tends to m. (The diameter of a set of points is the supremum of the distances between the points of the set.) The requirement AS tends to zero is not enough, because, e.g., in the case of a rectangle the area can be close to zero also if only one side is small and the other is not, so the considered points could be far from each other. u=f(x,y) Y

)d; ,@;-

,'

I ,I

,11 I ,111; ,111; 1111: '1,

X

Figure 8.28

X

AS^

1

Pi(Xi rYi)

Figure 8.29

If this limit exists independently of the partition of the domain S into elementary domains and also of the choice of the points P,(s,. y,)> then we call it the double integral of the function u = f ( z , y) over the domain S.the domain of integration. and we write: (8.135b)

2. Existence Theorem If the function f ( x , y) is continuous on the domain of integration including the boundary, then the double integral (8.135b) exists. (This condition is sufficient but not necessary.)

470

8. Integral Calculus

3. Geometrical Meaning The geometrical meaning of the double integral is the volume of a solid whose base is the domain in the x,y plane. whose side IS a cylindrical surface with generators parallel to the z-axis. and it is bounded above by the surface defined by u = f(s,y) (Fig. 8.29). Every term f(x,,y,)AS, of the sum (8.135b) corresponds to an elementary cell of a prism with base AS, and with altitude f ( z , . yz). The sign of the volume is positive or negative. according to whether the considered part of the surface u = f ( z , y) is above or under the z,y plane. If the surface intersects the z.y plane, then the volume is the algebraic sum of the positive and negative parts. If the value of the function is identically 1 ( f ( z , y) I 1).then the volume has the numerical value of the area of the domain S in the x.y plane.

8.4.1.2 Evaluation of the Double Integral The evaluation of the double integral is reduced to the evaluation of a repeated integral, Le., to the evaluation of two consecutive integrals.

1. Evaluation in Cartesian Coordinates If the double integral exists, then we can consider any type of partition of the domain of integration, such as a partition into rectangles. We divide the domain of integration into infinitesimal rectangles by coordinate lines (Fig. 8.30a). Then we calculate the sum of all differentials f ( x ,y)dS starting with all the rectangles along every vertical stripe, then along every horizontal stripe. (The interior sum is an integral approximation sum with respect to the variable y, the exterior one with respect to x.) If the integrand is continuous, then this repeated integral is equal to the double integral on this domain. The analytic notation is. (8.136a) h

Here y = qz(z)and y =

+1(x) are

the equations of the upper and lower boundary curves (AB),,,,,,

and of the surface patch 5’.Here a and b are the abscissae of the points of the curves to the very left and to the very right. The elementary area in Cartesian coordinates is

d S = dx dy. (8.136b) (The area of the rectangle is AxAy independently of the value of x.) For the first integration x is

0 Figure 8.30

Figure 8.31

Figure 8.32

handled as a constant. The square brackets in (8.136a) can be omitted, since according to the notation the interior integral is referred to the interior integration variable, the exterior integral is referred to the second variable In (8.136a) the differential signs dx and dy are at the end of the integrand. It is also usual to put these signs right after the corresponding integral signs, in front of the integrand. Il’e can perform the summation in reversed order, too, (Fig. 8.30b). If the integrand is continuous,

8.4 Multiple Inteorals 471

then it results also in the double integral: 9

/ f(.

Y) dS =

S

IA =

MZ(Y)

/ / f(x.

(8.136~)

Y) dx dY.

W(Y1

[

xy2 dS. where S is the surface patch between the parabola y = xz and the line y = 2 1 in 2s

Fig, 8.31.A = r r x y ' d y d x = l'xdr '9

A=

xz

/ f l x y 2 dzdy = Jd2y2dy '9

Y/2

[

[{]x2

2 4 :]v/2 =

=

32 [(8x4 - x 7 )dx = 5 or

):

l y 2 (y -

32 dy = 5 .

2. Evaluation in Polar Coordinates The integration domain is divided by coordinate lines into elementary parts bounded by the arcs of two concentric circles and two segments of rays issuing from the pole (Fig. 8.32). The area of the elementary domazn zn polar coordznatea has the form

pdpdp = dS. (8.137a) (The area of an elementary part determined by the same Ap and Ap is obviously smaller being close to the origin. and larger far from it.) With an integrand given in polar coordinates tu = f ( p . p) we perform a summation first along each sector. then with respect to all sectors: (8.137b)

-

where p = p l ( p ) and p = p Z ( p ) are the equations of the interior and the exterior boundary curves A

(AmB) and (AnB)of the surface S and and ;sz are the infimum and supremum of the polar angles of the points of the domain. The reverse order of integration is seldom used. psin' p dS. where S is a half-circle p = 3 cos p (Fig. 8.33):

3. Evaluation with Arbitrary Curvilinear Coordinates u and v The coordinates are defined by the relations x = z(u. u ) , y = y(u, w) (8.138) (see 3.6.3.1. p. 244). The domain of integration is partitioned by coordinate lines u = const and w = const into infinitesimal surface elements (Fig. 8.34) and the integrand is expressed by the coordinates u and v. Lye perform the summation along one strip, e.g., along v = const, then over all strips: (8.139) S

u1 U l ( U ) h

h

Here u = q ( u ) and w = wz(u) are the equations of the boundary curves AmB and AnB of the surface S.% e\' denote by u1 and u2 the infimum and supremum of the values of u of the points belonging to the

472

8. Integral Calculus

surface S. ID1 denotes the absolute value of the Jacobian determinant (functional determinant)

ax ax av D(u.2j) -19 au 31' dv _ _

D - D ( x Y) - au I

(8.140a)

The area of the elementary domain in curvilinear coordinates can be easily expressed: ID1dudu = dS. 1

0

I

I

Figure 8.33

Figure 8.34

Figure 8.35

The formula (8.13713) is a special case of (8.139) for the polar coordinates x = pcos 'p, y = psin 'p. The functional determinant here is D = p. We choose curvilinear coordinates so that the limits of integration in the formula (8.139) are as simple as possible. and also the integrand is not very complicated.

I Calculate A =

s s

f ( x , y ) dS for the case when S is the interior of an asteroid (see 2.13.4, p. 102),

with z = a cos3 1, y = asin3 t (Fig. 8.35). First we introduce the curvilinear coordinates 2 = u cos3 u, y = u sin3u whose coordinate lines u = c1 represents a family of similar asteroids with equations z = c1 cos3t and y = c1 sin3t. The coordinate lines v = c2 are rays with the equations y = kx, where k = tan3 e2 holds. We get D = 1 cos3 . u -321 cos2 v sin v = 3usin2ucos2u, A = j(z(u,v),y(u,~))3usin'vcos~ududu. s1n3 u 311 sin' u cos u 0 0

rS2'

~

8.4.1.3 Applications of the Double Integral Some applications of the double integral are collected in Table 8.9. The required areas of elementary domains in Cartesian and polar coordinates are given in Table 8.8 Tabelle 8.8 Plane elements of area

1

1

Coordinates Cartesian coordinates x , y

dS = dydx

Polar coordinates p, yo

dS = p d p d ' p

I .Arbitrary curvilinear coordinates u, v 1 dS

=

/DIdudv ( D Jacobian determinant)

[

8.4 Multiple Inteqrals 473

Table 8.9 Applications of the double integral General formula

1

1

Cartesian coordinates

Polar coordinates

1. Area of a plane figure:

2. Surface:

3. Volume of a cylinder:

4. Moment of inertia of a plane figure, with respect t o the z-axis:

Iz=/y2dS

=//y2dydr

~

=

11

p3 sin2 'p d p dp

C

5. Moment of inertia of a plane figure, with respect t o the pole 0:

=[Iedydx

=

Js

e p d p dcp

7. Coordinates of the center of gravity of a homogeneous plane figure:

""

/YdS YC =

S

474

8. Integral Calculus

8.4.2 Triple Integrals The trzple zntegralis an extension of the notion of the integral into three-dimensional domains. We also call it volume zntegral.

8.4.2.1 Notion of the Triple Integral 1. Definition We define the triple integral of a function f(z, y, z ) of three variables over a three-dimensional domain 1; analogously to the definition of the double integral. We write:

The volume 1’ (Fig. 8.36) is partitioned into elementary volumes A V . Then we form the products !(xi,yzrzJAr/, where the point Pi(zi,yi, zi) is inside the elementary volume or it is on the boundary. The triple integral is the limit of the sum of these products with all the elementary volumes in which the volume !,’ is partitioned, then the diameter of every elementary volume tends to zero, Le., their number tends to co.The triple integral exists only if the limit is independent of the partition into elementary volumes and the choice of the points Pi(xi, yzsz i ) . Then we have:

\ f(z, ~1

I’

2) dV

=

);yo

f(zii ~

$

(8.142)

AK.

2%) 3

n-tm 1=1

2. Existence Theorem The existence theorem for the triple integral is a perfect analogue of the existence theorem for the double integral.

dY Figure 8.36

Figure 8.37

8.4.2.2 Evaluation of the Triple Integral The evaluation of triple integrals is reduced to repeated evaluation of three ordinary integrals. If the triple integral exists. then we can consider any partition of the domain of integration.

1. Evaluation in Cartesian Coordinates The domain of integration can be considered as a volume V here. We prepare a decomposition of the domain by coordinate surfaces, in this case by planes, into infinitesimal parallelepipeds, Le., their diameter is an infinitesimal quantity (Fig. 8.37). Then we perform the summation of all the products f ( z ly , z ) dl’. starting the summation along the vertical columns, Le.. summation with respect to z , then in all columnsof one slice, Le., summation with respect to y, and finally in all such slices, i.e., summation with respect to z.Every single sum for any column is an approximation sum of an integral, and if the

8.1 Multiple Intearals 475

diameter of the parallelepipeds tends to zero, then the sums tend to the corresponding integrals, and if the integrand is continuous, then this repeated integral is equal to the triple integral. -4nalytically:

(8.143a) Here z = w1(x.y) and z = $z(z,y) are the equations of the lower and upper part of the surface bounding the domain of integration Ti (see limiting curve r in Fig. 8.37);dxdydz is the elementary volume in the Cartesian coordinate system. y = PI(.) and y = p2(z)are the functions describing the lower and upper part of the curve C which is the boundary line of the projection of the volume onto the x,y plane. and z = a and x = b are the extreme values of the z coordinates of the points of the volume under consideration (and also the projection under consideration). We have the following postulates for the domain of integration: The functions p](z) and p2(1c)are defined and continuous in the interval a 5 z 5 b. and they satisfy the inequality p1(x) I p~(x). The functions &(x,y) and &(x,y) are defined and continuous on the domain a 5 x I b: pl(z) 5 y 5 ~ ( z )and , also $l(x> y) 5 &(x,y) holds. In this way. every point (z, y, z ) in V satisfies the relations a

5 5 b,

PI(^)

I Y I Q(.).

~ ( z , YI ) 5 li)z(z,~).

(8.143b)

Just as with double integrals, we can change the order of integration, then the limiting functions will change in the same sense. (Formally: the limits of the outermost integral must be constants, and any limit may contain variables only of exterior integrals.)

I Calculate the integral I

=

(y2

+ z') dl/ for a pyramid bounded by the coordinate planes and the

plane z + y + z = 1: (y2

+ 2')

dz dy dx =

1'{ 11-'

[ll-i-y(yz

+ z') dz

11

1

dy dx = 30 .

zt

Figure 8.38

Figure 8.39

2. Evaluation in Cylindrical Coordinates The domain of integration is decomposed into infinitesimal elementary cells by coordinate surfaces p = const. p = const, z = const (Fig. 8.38). The volume of an elementary domain an cylindrical coordinates is dl' = p d z d p d p .

(8.144a)

476

8. Integral Calculus

After defining the integrand by cylindrical coordinates f ( p , (o, z ) the integral is: 192 P 2 ( V ) ZZ(P,19)

J i ( P : P>2) dl’ = V

JJ J

(8.144b)

f(P>(o, Z I P dzdpdip.

191 P l ( 1 9 ) Z I ( P , 1 9 )

ICalculate the integral I = the cylindrical surface x 2

dV for a solid (Fig. 8.39) bounded by the 2, y plane, the 2,t plane,

+ y 2 = a 2 and the sphere x 2 + y 2 +

[” { iaCo8’ [im

dt] p d p }

t 2=

a’:

z1

= 0,

z2

-4

=

=

a3

dp = 18 (377 - 4) . Since f ( p , (o, z ) = 1, the integral is equal to

the volume of the solid.

I X

Figure 8.40

Figure 8.41

3. Evaluation in Spherical Coordinates The domain of integration is decomposed into infinitesimal elementary cells by coordinate surfaces T = const, p = const, 29 = const (Fig. 8.40). The volume of an elementary domain in spherical coordinates is

dl.’ = sin 29 dr d29 dy. For the integrand f ( r >p>19) in spherical coordinates, the integral is:

(8.145a)

(8.145b)

ICalculate the integral I =

/

dV for a cone whose vertex is at the origin, and its symmetry axis

V

is the z-axis. The angle at the vertex is 2a, the altitude of the cone ish (Fig. 8.41). Consequently we h have: = 0,rZ = -’cos *’ 291 = 0,292 = a; (01 = 0, $92 = 2a. h l cos 8 cos 29 I = -T2sin8drd29dp= Jd27{j10cosliSinfi h/cosir dr] d 3 ) d(o r2

Jdy7[i

I[

8.5 Surface Intearals 477

= 2a h (1 - cos a ) .

4. Evaluation in Arbitrary Curvilinear Coordinates u, w , w The coordinates are defined by the equations (8.146) x = x(u,v , w ) . y = y(u, 21, u). z = z(u,v ) w ) (see 3.6.3.1, p. 244). The domain of integration is decomposed into infinitesimal elementary cells by the coordinate surfaces u = const, 2: = const, w = const. The volume of an elementary domazn zn arbztray coordznates is:

(8.147a)

Le.. D is the Jacobian determinant. For the integrand f(u,v,w) in curvilinear coordinates u, v , w . the integral is: (8.147b)

Remark: The formulas (8.144b) and (8.145b) are special cases of (8.147b). For cylindrical coordinates D = p holds, for spherical coordinates D = r2 sin v7 is valid. If the integrand is continuous, then we can change the order of integration in any coordinate system. M'e always try to choose a curvilinear coordinate system such that the determination of the limits of the integral (8.147b))and also the calculation of the integral, should be as easy as possible.

8.4.2.3 Applications of t h e Triple Integral Some applications of the triple integral are collected in Table 8.11. The elementary areas corresponding to different coordinates are given in Table 8.8. The elementary volumes corresponding to different coordinates are given in Table 8.10. Table 8.10 Elementary volumes

I s -

1

Coordinates 2, y,

z

~

Elementary volume

I

dV = dxdydz

1 dV = p dp d p dz I 1 dV = r'sint9drdddp Spherical coordinates r ,t9,p 1 Arbitrary curvilinear coordinates u, w,w 1 dV = /Dldudvdw ( D Jacobian determinant) I I Cylindrical coordinates p , p , z

8.5 Surface Integrals We distinguish surface integrals of the first type, of the second type, and of general type, analogously to the three different line integrals (see 8.3, p. 460).

8.5.1 Surface Integral ofthe First Type The surface zntegralor zntegral ouer a surface an space is the generalization of the double integral, similarly as the line integral of the first type (see 8.3.1, p. 461) is a generalization of the ordinary integral.

I

478

8. Inteqral Calculus

Table 8.11 Applications of the triple integral

General formula

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

1. Volume of a solid

2. Axial moment of inertia of a solid with respect t o the z-axis

I

I

3. Mass of a solid with the density function Q

4. Coordinates of the center of a homogeneous solid

111

/[Ir3sin’ 29 cos

x dz dy dx

111 sin 111

pz sin p d p d p dz

/I/

p d r d29 d v

29 d r d29 d p

r3 sin2 8sin p dr dt9 d p

111

r’ sin 8 dr dt9 d p

/// r3sin

111

T’

cos 29 dr dt9 d p

sin 19 d r d29 d p

8.5.1.1 Notion of the Surface Integral of the First Type 1. Definition The surface integral of thefirst type of a function u = f (z, y , z ) of three variables defined in a connected domain is the integral

over a region S of a surface. The numerical value of the surface integral of the first kind is defined in the following way (Fig. 8.42). 1. We decompose the region S in an arbitrary way into n elementary regions. 2. Lye choose an arbitrary point P,(x,, y z ,2,) inside or on the boundary of each elementary region.

8.5 Surface Integrals 479 3. \+'e multiply the value f (q, yi, ti) of the function at this point by the area ASl of the corresponding elementary region. 4. We add the products f ( ~ytI , zz)AStso obtained. 5. We determine the limit of the sum n

f

(Xi,Yi,

4 AS%

(8.149a)

i=l

as the diameter of each elementary region tends to zero, so AS, tends to zero, Figure 8.42 hence, their number n tends to co (see 8.4.1.1, l., p. 469). If this limit exists and is independent of the particular decomposition of the region S into elementary yi2z i ) , then it is called the surface integral of theJrst regions and also of the choice of the points Pi(xi, type of the function 2~ = f(x,y, z ) over the region S,and we write:

1

f(L!/>2s dS =

;;z0k

(8.149b)

Yz,21) AS,.

f(z1,

n i m

S

1=1

2. Existence Theorem If the function f ( ~y,, z ) is continuous on the domain, and the functions defining the surface have continuous derivatives here, the surface integral of the first type exists.

8.5.1.2 Evaluation of the Surface Integral of the First Type The evaluation of the surface integral of the first type is reduced to the evaluation of a double integral over a planar domain (see 8.4.1, p. 469). 1. Explicit Representation of t h e Surface If the surface 2 = z(x.y)

S is given by the equation (8.150) (8.15la)

S'is the projection of S onto the I,y plane and p and q are the partial derivatives az q = - Here we assume that to every point of the surface S there corresponds a unique point ay'

is valid, where a2

p = -,

ax

in S'in the L. y plane, i.e., the points of the surface are defined uniquely by their coordinates. If it does not hold. n e decompose S into several parts each of which satisfies the condition. Then the integral on the total surface can be calculated as the algebraic sum of the integrals over these parts of S. The equation (8.151a) can be written in the form (8.151b)

x - x Y - y z - z (see Tasince the equation of the surface normal of (8.150) has the form -= -- P 4 -1 ble 3.29, p. 246),since for the angle between the direction of the normal and the z-axis, cos7 = 1 holds. In evaluating a surface integral of the first type, this angle y is always considered dm-7

as an acute angle. so cos? > 0 always holds.

480

8. Inteoral Calculus

2. Parametric Representation of the Surface

If the surface S is given in parametric form by the equations x = x ( u ,ff), y = y(u,ff), z = z ( u ,ff), (Fig. 8.43), then

J

f(Xl

(8.152a)

Y>2) dS

S

=JJf[x(u,v),y(u,u),z(u,a)]~dudv,

(8.152b)

A

where the functions E , F , and G are the quantities given in 3.6.3.3, l., p. 245. The elementary region in parametric form is (8.152~)

Figure 8.43

and A is the domain of the parameters u and u corresponding to the given surface region. The evaluation is performed by a repeated integration with respect to t~and u:

S

w(u)

1/ @(u,u)-dvdu, 211

/@(u.v)dS=

(8.152d) @ = f[x(u,w),y(u,~),z(u,v)].

u1 Vl(U)

-

Here u1 and u2 are coordinates of the extreme coordinate lines u = const enclosing the region S h

(Fig. 8.43),and u = ul(u) and v = uz(u) are the equations of the curves AmB and AnB of the boundary of S . Remark: The formula (8.151a) is a special case of (8.152b) for

u=x, v = y ,

E=l+p',

F=pq,

G=1+q2.

(8.153)

3. Elementary Regions of Curved Surfaces The elementary regions of curved surfaces are given in Table 8.12. Table 8.12 Elementary regions of curved surfaces

Coordinates

Elementary region a2

at

Cartesian coordinates x , y, t = z ( x ,y ) dS = 1 + (-)' + (-)' dxdy ax ay dS= Rdpdz Cylindrical lateral surface, R (const. radius), coordinates cp, z dS = R' sin 19 d19 d p Spherical surface R (const. radius), coordinates ~ 9 ~p, Arbitrary curvilinear coordinates u , TJ dS = d ( E ,F, G see differential of arc, p. 245)

S =JdS S

m

d

u dv

(8.154)

8.5 Surface Inteqrals 481

2. Mass of an Inhomogeneous Curved Surface S

With the coordinate-dependent density e = f(z,u, z ) we have: (8.155)

8.5.2 Surface Integral of the Second Type The surface integral of the second type, also called an integral over a projection, is a generalization of the notion of double integral similarly to the surface integral of the first type.

8.5.2.1 Notion of the Surface Integral of the Second Type 1. Notion of an Oriented Surface A surface usually has two sides, and one of them can be chosen arbitrarily as the exterior one. If the exterior side is fixed, we call it an oriented surface. We do not discuss surfaces for which we cannot define two sides (see [8.7]). 2. Projection of an Oriented Surface onto a Coordinate Plane If we project a bounded part S of an oriented surface onto a coordinate plane, e.g., onto the z,y plane, we can consider this projection PrZyS as positive or negative in the following way (Fig. 8.44):

Figure 8.44

a) If the z.y plane is looked at from the positive direction of the z-axis, and we see the positive side of the surface S, where the exterior part is considered to be positive, then the projection PrZyS has a positive sign, otherwise it has a negative sign (Fig. 8.44 a,b). b) If one part of the surface shows its positive side and the other part its negative side, then the projection PrZyS is regarded as the algebraic sum of the positive and negative projections (Fig. 8 . 4 4 ~ ) . The Fig. 8.44d shows the projections PT& and PT,,S of asurface S;one of them is positive the other one is negative. The projection of a closed oriented surface is equal to zero.

482

8. Inteoral Calculus

3. Definition of the Surface Integral of the Second Type over a Projection

onto a Coordinate Plane The surface integral ofthe second type of a function f(z, y, z ) of three variables defined in a connected domain is the integral

over the projection of an oriented surface Sonto the x,y plane, where S is in the same domain where the function is defined, and if there is a one-to-one correspondence between the points of the surface and its projection. The numerical value of the integral is obtained in the same way as the surface integral of the first type except that in the third step the function value . f ( x I yE, , zt) is not multiplied by the elementary region AS,, but by its projection Pr,,AS,, oriented according to 8.5.2.1, 2., p. 481 on the x.y plane. Then we get: (8.157a) hVe define analogously the surface integrals of the second type over the projections of the oriented surface S onto the y, z plane and onto the z , z plane:

4. Existence Theorem for the Surface Integral of the Second Type The surface integral of the second type (8.157a,b,c) exists if the function f(z,y, z ) is continuous and the equations defining the surface are continuous and have continuous derivatives.

8.5.2.2 Evaluation of Surface Integrals of the Second Type The principal method is to reduce it to the evaluation of double integrals.

1. Surface Given in Explicit Form If the surface S is given by the equation 2

=P (.>

(8.158)

Y)

in explicit form, then the integral (8.15ia) is calculated by the formula

/f(z,Y>z)dzdY

f[.>Y>'p(x,Y)ldS,Y,

=

S

(8.159a)

Prz,S

where S,, = Pr,,S. The surface integral of the function f(z, y, z ) over the projections of the surface S onto the other coordinate planes is calculated similarly: /f(r.y,i)dydz = S

J

f[lCI(Y,~),yl~ldSYzr

(8.159b)

Pr,,S

where we substitute I = y(y, z ) , the equation of the surface S solved for z, and S,,= Pry,S. /f(x:y,z)dzdz = S

s

f[x,x(z,x),z]d&,

(8.159~)

Przd

where we substitute y = x ( z . z ) ,the equation of the surface S solved for y, and S,, = Pv,,S. If the orientation of the surface is changed, Le., if we interchange the exterior and interior sides, then the integral over the projection changes its sign.

8.5 Surface Integrals 483

2. Surface Given in Parametric Form If the surface is given by the equations L = I(U,t ) . y = y(u. v). 2 = z(u, v) (8.160) in parametric form.we calculate the integral (8.157a,b,c) with help of the following formulas:

(8.161a)

(8.16 1b)

00

are the Jacobian determinants of pairs of functions D(u. v) ’ D ( u ,v) ’ D ( u ,u ) x. y1z with respect to the variables u and u: A is the domain of u and v corresponding to the surface S.

Here the expressions

3. Surface Integral in General Form If P ( x . y. z ) ~Q(r.y. 2). R ( x .y >z ) are three functions of three variables defined in a connected domain and S is an oriented surface contained in this domain. the sum of the integrals of the second type taken over the projections on the three coordinate planes is called the surface integral in general form:

/ ( P dy dz + Qdzdz +Rdzdy) =

1

1

1R d z d y .

S

S

S

S

P d y dz +

Qdzdz +

(8.162)

The formula reducing the surface integral to a double integral is:

/ ( P d y d z t Qdzdx t Rdxdy) = S

where the quantities

1[.W+ Qv + D ( u ,v)

D(u. L )

Do DO w) and A have the same meaning, as above. D ( u ,v ) ’ D ( u ,v) ’ D ( u ,u )

Remark: The surface integral of vector-valued functions is discussed in the chapter about the theory of vector fields (see 13.3.2. p. 661).

4. Properties of the Surface Integrals 1. If the domain of integration, Le.. the surface

S.is decomposed into two parts SI and Sz,then

/ ( P d y d z t Qdzdz + R d x d y ) = [ ( P d y dz S

+ Q dzdx + R d x d y )

91

+

I

(Pdydz+Qdzdx+Rdxdy).

(8.164)

S2

If the orientation of the surface is reversed, Le., if we interchange the exterior and interior sides, the integral changes its sign: 2.

I

/(Pdydz+Qdzdr+Rdxdy) = - (Pdydz+Qdzdx+Rdxdy), s+ S-

(8.165)

484 8. Inteqral Calculus where St and S-denote the same surface with different orientation. 3. A surface integral depends, in general, on the line bounding the surface region S as well as on the surface itself. Thus the integrals taken over two different non-closed surface regions S l and Sz spanned by the same closed curve C are, in general, not equal (Fig. 8.45):

/ ( P dy dz t Q dz dz + R dz d y ) SI

# / ( P dy dz t Q dz dz t R dz dy).

(8.166) Figure 8.45

s2

8.5.2.3 An Application of the Surface Integral The volume V of a solid bounded by a closed surface S can be expressed and calculated by a surface integral

'J

V= -

3s

(zdydz+ydzdz+zdzdy),

where S is oriented so that its exterior side is positive.

(8.167)

485

9 Differential Equations 1. A Differential Equation is an equation, in which one or more variables, one or more functions of these variables, and also the derivatives of these functions with respect to these variables occur. The order of a differential equation is equal to the order of the highest occurring derivative. 2. Ordinary a n d Partial Differential Equations differ from each other in the number of their independent variables; in the first case there is only one, in the second case there are several.

9.1 Ordinary Differential Equations 1. General Ordinary Differential Equation of Order n in implicztform has the equation If this equation is solved for y(")(x), then it is the explicit form of an ordinary differential equation of order n.

2. Solution or Integral of a differential equation is every function satisfying the equation in an interval a 5 x 5 b which can be also infinite. .4 solution, which contains n arbitrary constants c l , c2, . . . ,e,, is called the general solution or general integral. If the values of these constants are fixed, a particular integralor a particular solution is obtained. The value of these constants can be determined by n further conditions. If the substitution values of y and its derivatives up to order n - 1 are prescribed at one of the endpoints of the interval. then the problem is called an initial value problem. If there are given substitution values at both endpoints of the interval, then the problem is called a boundary value problem. IThe differential equation -y'sin x + y cos x = 1 has the general solution y = cos x + c sin x. For the condition c = 0 we get the particular solution y = cosx.

3. Initial Value Problem If the n values y(xO), y ' ( ~ ~, ,) y("-')(xO) ~. are given at zo for the solution y = y(x) of an n-th order ordinary differential equation, then an initial value problem is given. The numbers are called the initial values or initial conditions. They form a system of n equations for the unknown constants c1, c2, . . . c, of the general solution of the n-th order ordinary differential equation. IThe harmonic motion of a special elastic spring-mass system can be modeled by the initial value problem y" + y = 0 with y(0) = yo, y'(0) = 0. The solution is y = yo cos x. 4. Boundery Value Problem If the solution of an ordinary differential equation and/or its derivatives are given at several points of its domain, then these values are called the boundary conditions. We call a differential equation with boundary conditions a boundary value problem. IThe bending line of a bar with fixed endpoints and uniform load is described by the differential equation y" = x - x2 with the boundary conditions y(0) = 0, y(1) = 0 (0 5 x 5 1). The solution is 23 24 x ~

y=6-z-E.

I

486

9. Differential Equations

9.1.1 First-Order Differential Equations 9.1.1.1 Existence Theorems, Direction Field 1. Existence of a Solution In accordance with the Cauchy existence theorem the differential equation

yl = f(.. Y) (94 has at least one solution in a neighborhood of 20 such that it takes the value yo at z = zo if the function f ( z , y) is continuous in a neighborhood G of the point (zo, yo). For example, we may select G as the region given by l z - 501< a and ly - yo1 < b with some a and b. 2. Lipschitz Condition The Lipschitz condition with respect to y is satisfied by f ( x ,y) if If(x.y1) - f(x,yz)l 5 d%l - Y2I (9.3) holds for all (x.yl) and (z.y2) from G, where N is independent of z,y1, and y2. If this condition is satisfied! then the differential equation (9.2) has a unique solution through (20:yo). The Lipschitz condition is obviously satisfied if f ( z . y) has a bounded partial derivative af/ay in this neighborhood. In 9.1.1.4, p. 491 we will see examples in which the assumptions of the Cauchy existence theorem are not satisfied.

3. Direction Field If the graph of a solution y = p(z) of the differential equation y’ = f(z,y) goes through the point P ( z ,y); then the slope dy/dz of the tangent line of the graph at this point can be determined from the differential equation. So, at every point (x.y) the differential equation defines the slope of the tangent line of the solution passing through the considered point. The collection of these directions (Fig. 9.1) forms the directionfield. An element of the direction field is a point together with the direction associated to it. Integration of a first-order differential equation geometrically means to connect the elements of a direction field into an integral curue, whose tangents have the same slopes a t all points as the corresponding elements of the direction field.

Figure 9.1

Figure 9.2

4. Vertical Directions If in a direction field we have vertical directions! i.e., if the function f(x,y ) has a pole, we exchange the role of the independent and dependent variables and we consider the differential equation

as an equivalent equation to (9.2). In the region where the conditions of the existence theorems are fulfilled for the differential equations (9.2) or (9.4), there exists a unique integral curve (Fig.9.2) through every point P(z0,yo).

9.1 Ordinaru DifferentialEauations 487

5 . General Solution The set of all integral curves can be characterized by one parameter and it can be given by the equation (9.5a) F ( s ,y, C) = 0 of the corresponding one-parameter family of curves. The parameter C, an arbitrary constant, can be chosen freely and it is a necessary part of the general solution of every first-order differential equation. .A particular solution y = p(z)>which satisfies the condition yo = ~(zo), can be obtained from the general solution (9.5a) if C is expressed from the equation (9.5b) F(z0,Yo; C) = 0.

9.1.1.2 Important Solution Methods 1. Separation of Variables If a differential equation can be transformed into the form (9.6a) .lf(z).V(y)dx P(x)Q(y)dy = 0, then it can be rewritten as (9.6b) R(xjdx t S(y)dy = 0, where the variables z and y are separated into two terms. To get this form, we divide equation (9.6a) by P(z)N(y).For the general solution we have

+

If for some values x = or y = g, the functions P ( x ) or X ( y ) or both are equal to zero: then the constant functions x = Z or/and y = are also solutions of the differential equation. They are called singular solutions. Izdy t ydz = 0;

J

t

t

/ d”c

= C;

ln 1y1t ln 1x1= c = In / c ~ ; yz = c. If we allow also c = o

I 0 and x I 0. 2. Homogeneous Equations If M ( x . y) and X ( x . y) are homogeneous functions of the same order (see 2.18.2.6, l., p. 121), then in

in this final equation, then we have the singular solutions y

the equation M(z.yjdz + :V(x: y)dy = 0 (9.8) we separate the variables by substitution of u = y/x. Iz(z- y)y’ + yz = 0 with y = u(x)x,we get (1 - u)u’ + u/x = 0. By separation of the variables

1

du = -

1:

-dx. After integration: In 1x1 + In / u - u1 = C = In IcI,

uz = ce’,

y = ceY/.”.

As we have seen in the preceding paragraph, Separation of Variables, the line x = 0 is also an integral curve.

3. Exact Differential Equations An exact diflerential equation is an equation of the form (9.9a) M(5.y)dz t ,V(z,y)dy = 0 or ,V(Z, y)y’ t M ( x ,y) = 0, if there exists a function @(xq y) of two variables such that (9.9b) M ( x . y)dx t N(x, y)dy I d@(z:y). i.e.. if the left side of (9.9a) is the total differential of a function @ ( E , y) (see 6.2.2.1, p. 392). If functions M(z,y) and “V(x,y j and their first-order partial derivatives are continuous on a connected domain G, then the equality (9.9c)

488

9. DifferentialEauations

is a necessary and sufficient condition for equation (9.9a) to be exact. In this case the general solution of (9.9a) is the function @(x. y) = C (C = const), (9.9d) which can be calculated according to (8.3.4), 8.3.4.4, p. 468 as the integral Y

X

@(LY) =

J WE>Y) dE + J N x o , a) drl,

(Me)

YO

XO

where xg and yo can be chosen arbitrarily from G. IExamples will be given later.

4. Integrating Factor A function p ( x , y) is called an integratingfactoror a multiplier if the equation (9.10a) Mdx + N d y = 0 multiplied by p(x,y) becomes an exact differential equation. The integrating factor satisfies the differential equation (9.10b) Every particular solution of this equation is an integrating factor. To give a general solution of this partial differential equation is much more complicated than to solve the original equation, so usually we are looking for the solution p(x, y) in a special form, e.g., p(x),p(y), p(xy) or p(x2 + y'). IWe now solve the differential equation (x'+ y) dx - x dy = 0. The equation for the integrating factor dlnp alnp dlnp - (x' + y) -= 2. An integrating factor which is independent of y must satisfy -x = is -2-

aY

dx

OX

1 -2, so p = - Multiplication of the given differential equation by p yields 1 1 '

1 - dx - -dy = 0.

( + J:

'

x

The general solution according to (9.9e) with the selection of xo = 1,yo = 0 is then:

5 . First-Order Linear Differential Equations A first-order linear diferential equation has the form

(9.lla) Y' + P(z)Y = Qb), where the unknown function and its derivative occur only in first degree, and P ( x ) and Q ( x )are given functions. If P ( z ) and Q(x) are continuous functions on a finite, closed interval, then the differential equation here satisfies the conditions of the Picard-Lindeliiftheorem (see p. 608). The integratingfactor is here (9.1lb) the general solution is

Y = exp

(-

P dx)

[QI (/P d x ) dx + C] exp

(9.1 IC)

If we replace the indefinite integrals by definite ones with lower bound xo and upper bound x in this formula. then for the solution y(x0) = C (see 8.2.1.2, l., p. 440). If y1 is any particular solution of the differential equation, then the general solution of the differential equation is given by the formula 2/ = Yl

+ Cexp (- J P d x ) .

(9.1Id)

9.1 Ordinaru Differential Equations 489

If yl(x) and y ~ ( x are ) two linearly independent particular solutions (see 9.1.2.3, 2., p. 498), then we can get the general solution without any integration as (9.1le) Y = Y1 + C(Yz - YI). ISolve the differential equation y' - y t a n z = cosz with the initial condition xo = 0, yo = 0. We calculate exp

(- 1% tanx dx) = cos x and get the solution according to (9.11~): [sinxc;x+x] cos x

-sinx 2

z +-2cosx'

6 . Bernoulli Differential Equations The Bernoulli diferential equation is an equation of the form (9.12) Y' + P(x)Y= Q(x)Y" ( n # O,n # 1)) which can be reduced to a linear differential equation if it is divided by yn and the new variable z = y-n+l is introduced. 4Y = x f i . Since n = 1/2, dividing by fi and introducing the ISolve the differential equation y' - X

dz 22 x new variable z = fi leads to the equation - - - = - , By using the formulas for the solution of a dx x 2 1 linear differential equation we have exp(J P d x ) = - and z = x2 22

so, finally. y = x4

1

(5 In 121 + c)'.

7. Riccati Differential Equations The Riccati diferential equation is the equation (9.13a) Y' = P(x)Y'+ Q(x)Y+ R(x), which cannot usually be solved by elementary integration. However, if we know a particular solution y1 of the Riccati differential equation, then we can reduce the equation to a linear differential equation for z by substituting 1

y=y1+-. 2

(9.13b)

If we also know a second particular solution yz, then 1

21

=YZ - Y l

(9.134

is a particular solution of the linear differential equation for the function z . so its solution can be simplified. If we know three particular solutions y1, yz, and y3, then the general solution of the Riccati differential equation is Y-Yz, Y - Y1 Y3 - Y l By the substitution of

(9.13d)

(9.13e) the Riccati differential equation can be transformed into the normal form du

- = 112 + R(x) dx

(9.13f)

490

9. DifferentialEquations

With the substitution u’ y = -P(x)v we get a second-order linear differential equation (see 9.1.2.6, l.,p. 505) from (9.13a)

Pv”-(P’+PQ)d+P2Rw=0.

(9.13h) 1

4 Solve the differential equation y’+ y2 -y - - = 0. We substitute y = z

+x

22

+ 4(s)into the equation,

+

and for the coefficient of the first power of z we get the expression 2p l/x,which disappears if we 15 substitute 9(s) = -1/2x In this way we get zf - z 2 - - = 0. We are looking for particular solutions 4x2 5 3 in the form z1 = 2 and we get the solutions a1 = - - > a2 = - by substitution, i.e., the two particular X 2 2 1 1 3 5 3 solutions are z1 = z2 = - . The substitution of z = - z1 = - - - results in the equation 22 22 u 21 22 321 1 z u’ + - = 1. Substituting the particular solution u1 = -= - we obtain the general solution X z2 - 21 4 z c 54 +c1 1 3 1 2x4 - 2c1 u=-+-=4x3 and therefore. y = - - - - - = 4 23 21 22 22 25+c1x ’

+

~

9.1.1.3 Implicit Differential Equations 1. Solution in Parametric Form Suppose we have a differential equation in implicit form (9.14) F ( z ,y ; y’) = 0. There are n integral curves passing through a point P(z0,yo) if the following conditions hold: a) The equation F(r0,y0,p) = 0 (p = dy/dz) has n real rootspl!. . . , p , at the point P(z0,yo). b) The function F ( z ,y l p ) and its first partial derivatives are continuous at x = 20.y = yo, p = p i ; furthermore d F / a p # 0. If the original equation can be solved with respect to y’, then it yields n equations of the explicit forms discussed above. Solving these equations we get n families of integral curves. If the equation can be written in the form z = p(y, y’) or y = $(x, y f ) , then putting y’ = p and considering p as an auxiliary variable, after differentiation with respect to y or zwe obtain an equation for dpldy or d p / d x which is solved with respect to the derivative. A solution of this equation together with the original equation (9.14) determines the desired solution in parametric form. Solve the equation z = yy‘ + y’*. We substitute y’ = p and get x = py p2. Differentiation with dx = -1 results in -1 = p (y 2p)-dP or dy PY - 2P2 Solving respect to y and substituting -dy P P dy dp 1 - p 2 - 1 - p 2 ’ c + arcsinp Substituting into the initial equation we get the this equation for y we get y = -p

+

+ +

+

J r ? ’ .

I

solution for x in parametric form.

2. Lagrange Differential Equation The Lagrange diflerential equation is the equation a(y’)z b(y’)y c(y’) = 0. The solution can be determined by the method given above. If for p = PO,

+

ab)

+

+ b ( p b = 0.

(9.14b) then a(p0)x

is a singular solution of (9.14a).

+ b(po)y + c(po) = 0

(9.14a)

(9.14c)

9.1 Ordinaru Differential Equations 491

3. Clairaut Differential Equation The Clazraut dzferentzal equatzon is the special case of the Lagrange differential equation if

4P)+ b(P)P = 0. and so it can be transformed into the form

(9.15a)

Y = Y’Z+ f (YO.

(9.15b)

The general solution is

+

(9.15c) y = cz f ( C ) Besides the general solution, the Clairaut differential equation also has a singular solution, which can be obtained by eliminating the constant C from the equations y = cz

+f(C)

(9.15d)

and 0 = 2

+ f’(C),

(9.15e)

\Ye can obtain the second equation by differentiating the first one with respect to C.Geometrically, the singular solution is the envelope (see 3.6.1.7, p. 237) of the solution family of lines (Fig. 9.3). W Solve the differential equation y = xy’+ Y’~. The general solution is y = C x C*, and we get the singular solution with the help of the equation x + 2C = 0 to eliminate C,and hence x2 4y = 0. Fig. 9.3shows this case.

+

+

\

Figure 9.3

Figure 9.4

9.1.1.4 Singular Integrals and Singular Points 1. Singular element .in element (Q. yo. yh) is called a singular element of the differential equation, if in addition to the differential equation (9.16a) F ( Z , y. y’) = 0 it also satisfies the equation (9.16b)

2. Singular Integral An integral curve from singular elements is called a singular integral curve; the equation P(X, Y) = 0 (9.16~) of a singular integral curve is called a singular integral. The envelopes of the integral curves are singular integral curves (Fig. 9.3);they consist of the singular elements. The uniqueness of the solution (see 9.1.1.1, l., p. 486) fails at the points of a singular integral curve.

3. Determination of Singular Integrals Usually we cannot obtain singular integrals for any values of the arbitrary constants of the general solution. To determine the singular solution of a differential equation (9.16a) with p = y’ we have to

492

9. Di.fferentialEauations

introduce the equation

-8F= o

(9.16d) 8P and to eliminate p . If the obtained relation is a solution of the given differential equation, then it is a singular solution. The equation of this solution should be transformed into a form which does not contain multiple-valued functions, in particular no radicals where the complex values should also be considered. Radicals are expressions we get by nesting algebraic equations (see 2.2.1, p. 60). If the equation of the family of integral curves is known, i.e., the general solution of the given differential equation is known, then we can determine the envelope of the family of curves, the singular integral, with the methods of differential geometry (see 3.6.1.7, p. 237). 4 8 IA: Solve thedifferentialequationz-y--y'+-~'~ = 0. After wesubstitute y' = p , thecalculation 9 27 8 8 of the additional equation with (9.16d) yields - - p + -p2 = 0. Elimination of p results in equation a) 9 9 4 x - y = 0 and b) x - y = - where a) is not a solution, b) is a solution, a special case of the general 27 solution (y - C)' = (x - C)3.The integral curves of a) and b) are shown in Fig. 9.4. IB: Solve the differential equation y'-In 1x1 = 0. We transform the equation into the form eP - 1x1 = dF 0. - ep = 0. We get the singular solution x = 0 eliminatingp. 8P 4. Singular Points of a Differential Equation Singular points of a differential equation are the points where the right side of the differential equation (9.17a) Y' = f ( X >Y) is not defined. This is the case, e.g., in the differential equations of the following forms: 1. Differential Equation with a Fraction of Linear Functions ax by dy (9.17b) (ae - bc # 0) dx c x + e y has an isolated singularpoint at (0,0 ) ,since the assumptions of the existence theorem are fulfilled almost at every point arbitrarily close to (0,O) but not at this point itself. The conditions are not fulfilled at the points where cx + ey = 0. We can force the fulfillment of the conditions at these points if we exchange the role of the variables and we consider the equation dx - cx + ey (9.17~) dy ax+by ' The behavior of the integral curve in the neighborhood of a singular point depends on the roots of the characteristic equation (9.17d) Xz - ( b + c)X t bc - ae = 0. We can distinguish between the following cases: Case 1: If the roots are real and they have the same sign, then the singular point is a branch point. The integral curves in a neighborhood of the singular point pass through it and if the roots of the characteristic equation do not coincide, except for one, they have a common tangent. If the roots coincide, then either all integral curves have the same tangent, or there is a unique integral curve passing through the singular point in each direction. dY 2Y IA: For the differential equation - = - the characteristic equation is Xz - 3X + 2 = 0, XI = 2, dx x XZ = 1. The integral curves have the equation y = C x 2 (Fig. 9.5). The general solution also contains ~

+

9.1 Ordinaru DifferentialEauatzons 493

the line x = 0 if we consider the form x2 = C1 y . dY X + Y IB: The characteristic equation for - = -is X2 - 2X + 1 = 0,XI = X2 = 1. The integral curves

dx

x

+ C x (Fig. 9.6). The singular point is a so-called node. dY = Y IC: The characteristic equation for is X2 - 2X + 1 = 0,XI = Xz = 1. The integral curves dx x are y = x In 1x1

are y = C x (Fig. 9.7). The singular point is a so-called ray point.

Figure 9.5

Figure 9.6

Figure 9.7

Case 2: If the roots are real and they have different signs, the singular point is a saddle poznt, and two of the integral curves pass through it.

dY = -Y is X2 - 1 = 0. XI = +1, X2 = -1. The integral curves D: The characteristic equation for dx x are z y = C (Fig. 9.8). For C = 0 we get the particular solutions x = 0, y = 0. Case 3: If the roots are conjugate complex numbers with a non-zero real part (Re(X) # 0), then the singular point is a spzralpoznt which is also called afocal poznt, and the integral curves wind about this singular point.

dy x + y . E: The characteristic equation for - = IS X2 - 2X + 2 dx X-y integral curves in polar coordinates are r = C eP (Fig. 9.9). ~

=

0,XI = 1 + i,

X2

= 1 - i. The

Yf

Figure 9.8

Figure 9.9

Figure 9.10

Case 4: If the roots are pure imaginary numbers, then the singular point is a central p o d , or center,

which is surrounded by the closed integral curves.

I

494

9. Dzferentzal Eauatzons

dY = -x is X2 + 1 = 0, A1 = i, Az = -i. The integral curves are IF: The characteristic equation for dx Y x2 + y2 = C (Fig. 9.10). 2. Differential Equation with the Ratio of Two Arbitrary Functions (9.18a) has the singular points for the values of the variables where (9.18b) P ( x . y) = Q ( x .y) = 0. If P and Q are continuous functions and they have continuous partial derivatives. (9.18a) can be written in the form

Here xo and yo are the coordinates of the singular point and Pl(x,y) and Q1(z,y) are infinitesimals of a higher order than the distance of the point (x.y) from the singular point (zo, yo). LVith these assumptions the type of a singular point of the given differential equation is the same as that of the approximate equation obtained by omitting the terms Pl and Q1. with the following exceptions: a) If the singular point of the approximate equation is a center, the singular point of the original equation is either a center or a focal point. a b b) If a e - b c = 0, i.e.. - = - or a = c = 0 or a = b = 0, then the type of singular point should be c

e

determined by examining the terms of higher order.

9.1.1.5 Approximation Methods for Solution of First-Order Differential Equations 1. Successive Approximation Method of Picard The integration of the differential equation (9.19a)

Y t = f (2. Y)

with the initial condition y = yo for z = zo results in the fixed-point problem (9.19b) If we substitute another function yl(z) instead of y into the right-hand side of (9.19b), then the result will be a new function yz(x), which is different from yl(z): if yl(z) is not already a solution of (9.19a). After substituting yz(z) instead of y into the right-hand side of (9.19b) we get a function y ~ ( x ) .If the conditions of the existence theorem are fulfilled (see 9.1.1.1. l., p. 486), the sequence of functions y l , y2! y3, . . . converges to the desired solution in a certain interval containing the point 50. This Picard method of successive approximation is an iteration method (see 19.1.1. p. 881). ISolve the differential equation y' = e" - y2 with initial values zo = 0, yo = 0. Rewriting the equation in integral form and using the successive approximation method with an initial approximation 1 yo(x) I 0. x e get: y1 = 1 ' e " d x = e" - 1. yz = [e5 - (e" - l)'] dx = 3e' - -e2" - z etc. 2 2

6"

E,

2. Solution by Series Expansion The Taylor series expansion of the solution of a differential equation (see 7.3.3.3. l., p. 415) can be given in the form y = yo

+ (x - x0)yol+

my{ + . . . (x 2

+ -yo(n)- .oIn n!

+ ...

(9.20)

9.1 Ordinaru Differential Eouations 495

if the values yo‘. yof’) , . . , yo(n),. . . of all derivatives of the solution function are known at the initial value xo of the independent variable. The values of the derivatives can be determined by successively differentiating the original equation and substituting the initial conditions. If the differential equation can be differentiated infinitely many times, the obtained series will be convergent in a certain neighborhood of the initial value of the independent variable. We can use this method also for n-th order differential equations. Remark: The above result is the Taylor series of the function, which may not represent the function itself (see 7.3.3.3, l . ,p. 415). It is often useful to substitute the solution by an infinite series with unknown coefficients, and to determine them by comparing coefficients. A: To solve the differential equation y‘ = e“ - y2>xo = 0, yo = 0 we consider the series y = alx+a2z2+a3z3+...+a,zn t... . Substitutingthisinto the equation consideringthe formula (7.88). p. 414 for the square of the series we get x2 2 3 al + 2a2x + 3 a 3 2 t ’ . [a12x2t 2ala2z3 (a2’ 2ala3)x4 ’ .] = 1 z - - . ’ .. 2 6 1 1 Comparing coefficients we get: a1 = 1, 2a2 = 1, 3a3 a12 = - , 4a4 t 2alaz = -, etc. Solving 2 6 these equations successively and substituting the coefficient values into the series representation we 1 2 23 5 get y = z + - - - - -z4 + . . .. 6 24 2 IB: The same differential equation with the same initial conditions can also be solved in the following way: If we substitute x = 0 into the equation, we get yo’ = 1. By successive differentiation we get y” = e” - 2yy’, yo” = 1; y”’ = e“ - 2y” - 2yy”, yo’’’ = -1, y(4) = e” - 6y’y’‘- 2yy”’, yo(4)= - 5 , From 22 23 5x4 the Taylor theorem (see 7.3.3.3, l . ,p. 415) we get the solution y = x + - - - - - + . . .. 4! 2! 3!

+

+

+

+

+ + + +

+

3. Graphical Solution of Differential Equations

Figure 9.11

The graphical integration of a differential equation is a method, which is based on the direction field (see 9.1.1.1, 3., p. 486). The integral curve in Fig. 9.11 is represented by a broken line which starts at the given initial point and is composed of short line segments. The directions of the line segments are always the same as the direction of the direction field at the starting point of the line segment. This is also the endpoint of the previous line segment.

4. Numerical Solution of Differential Equations The numerical solutions of differential equations will be discussed in detail in 19.4, p. 901. We use numerical methods to determine a solution of a differential equation, if the equation y‘ = f ( x , y) does not belong to the special cases dicussed above whose analytic solutions are known, or if the function f(z.y) is too complicated. This can happen if f ( x . y) is non-linear in y.

9.1.2 Differential Equations of Higher Order and Systems of Differential Equations 9.1.2.1 Basic R e s u l t s 1. Existence of a Solution 1. Reduction to a System of Differential Equations Every explicit n-th order differential equation (9.2la) y(n) = f (x, y. y’; . . . , y(n-1))

I

496

9. Differential Eauations

by introducing the new variables y1

=d,

. . , yn-l = y(n-1)

yz = 2//'>.

(9.21b)

can be reduced to a system of n first-order differential equations (9.21~) 2. Existence of a System of Solutions The system of n differential equations

dYt

, , , syn) dx = fz(z,yl,yz,

(i = 152,. ,,In)i

(9.22a)

which is more general than system (9.21c), has a unique system of solutions (9.22b) yt = 2/i(Z) (i = 1 , 2 , . . . , n)3 which is defined in an interval zo - h 5 z 5 zo h and for z = $0 takes the previously given initial values yz(zo)= yzo (i = 1,2,. . . , n ) ,if the functions fi(z,ylryz, . . . , yn) are continuous with respect to all variables and satisfy the following Lipschitz condition. 3. Lipschitz condition For the values z, and yz + Ayi, which are in a certain neighborhood of the given initial values, the functions f~satisfy the following inequalities: I ~ ~ ( ~ , Y I + A ~ / I ~ Y z + A Y z , -. ..fi(Zr~~>Y~r...r~n)/ .,~~+AY~) I K ( l A ~ ~ l + l A ~ z l + ~ ~ ~ + l A ~ n l (9.23a) ) with a common constant K (see also 9.1.1.1, 2., p. 486). This fact implies that if the function f(z, y, y', . . . , Y ( ~ - ' ) ) is continuous and satisfies the Lipschitz condition (9.23a): then the equation

+

y(n) = f ( 2 , y: y l l . .

.

~

y(n-1))

(9.23b)

= yo@-'), and it has a unique solution with the initial values ~ ( z o=) yo, ~ ' ( z o=) yo', . . . , ~(~-')(zo) is (n - 1) times continuously differentiable.

2. General Solution 1. The general solution of the differential equation (9.23b) contains n independent arbitrary constants: (9.24a) y = y(z.C1,Cz,. . ..Cn). 2. In the geometrical interpretation the equation (9.24a) defines a family of curves depending on n parameters. Every single one of these integral curves, i.e., the graph of the corresponding particular solution, can be obtained by a suitable choice of the constants C1, Cz, . . . , Cn. If the solution has to satisfy the above initial conditions, then the values C1,Cz, . . . , Cn are determined from the following equations:

Y(~o~CI,...,C~) =YO, (9.24b)

.........................

If these equations are inconsistent for any initial values in a certain domain, then the solution is not general in this domain, Le., the arbitrary constants cannot be chosen independently. 3. The general solution of system (9.22a) also contains n arbitrary constants. This general solution can be represented in two different ways: Either it is given in a form which is solved for the unknown functions (9.25a) ?/I = FI(S,CI,. , , ,Cn), YZ = F z ( ~ , C I , ,,,Cn), , , . , > 2 / n = Fn(Z,Cl,.. , > Cn)

9.1 Ordinaru DifferentialEquations 497

or in the form which is solved for the constants P I ( Z , YI,, . ~ n = ) C1, ’PZ(Z,Y I ) , . , In the case of (9.25b) each relation , j

~ n= )

CZ,. . . , c~n(x,Y I ~ ,,

Yn) = Cn.

(9.25b)

(9.25~) P ~ ( x * Y,I ,SYn) , = Ct is a first zntegral of the system (9.22a). The first integral can be defined independently of the general solution as a relation (9.25~).That is, (9.25~)will be an identity if we replace y1, yz,. . . , yn by any particular solution of the given system and we replace the constant by the arbitrary constant C, determined by this particular solution. If any first integral is known in the form (9.25c), then the function cpt(x,y1,. . . , yn) satisfies the partial different equation (9.25d) Conversely, each solution cp,(e,yl,. . . . yn) of the partial differential equation (9.25d) defines a first integral of the system (9.22a) in the form (9.25~).The general solution of the system (9.22a) can be represented as a system of n first integrals of system (9.22a), if the corresponding functions cp,(xl yl,. . . , gn) (z = 1.2. . . , n) are linearly independent (see 9.1.2.3, 2., p. 498).

9.1.2.2 Lowering the Order One of the most important solution methods for n-th order differential equations

f (G Y>Y’> , Y(nl) ’ ’ ’

(9.26)

=0

is the substitution of variables in order to obtain a simpler differential equation, especially one of lower order. We can distinguish between different cases. 1. f = f(y, y’, ,y‘”)), Le., z does not appear explicitly:

. ..

f (y, yl>.. . ,

p)= 0.

(9.27a)

By substitution dP (9.27b) d y = p . d2Y -p-&.... dx we can reduce the order of the differential equation from n to ( n - 1). I\Ve reduce the order of the differential equation yy” - yJ2 = 0 to one. With the substitution y’ = p.pdp/dy = y” it becomes a first-order differential equation y pdpldy - p 2 = 0, and y dpldy - p = 0 results in p = Cy = dg/dx, y = Clecz. Canceling p does not result in a loss of a solution, since p = 0 gives the solution y = C1, which is included in the general solution with C = 0. 2. f = f(z,y’, ,y@)), Le., y does not appear explicitly: -

. ..

f (z,yt, ...,y‘”’) = 0.

(9.28a)

The order of the differential equation can be reduced from n to (n- 1) by the substitution y’ = p . If the first k derivatives are missing in the initial equation, then a suitable substitution is

(9.28b)

y(k+‘l = P. (9.28~) IThe order of the differential equation y”-~y”’+(y”’)~= 0 will be reduced by the substitution y” = p,

E (2)3

so we get a Clairaut differential equationp-x-+ Therefore. y = @-*+CZX+C~. 6 2

= 0 whose general solution isp = Cl z+CI3.

From the singular solution of the Clairaut differential equation

I

9. Diferential Equations

498

= -2x 3 ~42

we get the singular solution of the original equation: y = -82'" 4 + Clz + C2. 315 3. f (z,y, y', ,y(.)) is a homogeneous function (see 2.18.2.4, 4., p. 120) 3

. .. in y, y', y", .. . ,y(.): f (x,y, y'. . . . . y'"')

= 0.

(9.29a)

We can reduce the order by the substitution Y' z = -. Y

i e., y = e . f z d z ~

(9.29b)

dz

W We transform the differential equation yy" - y" = 0 by the substitution z = y'/y. Then - = dx yy" - y'2 , so the order is reduced by one. We get 2 = C1, therefore, In IyI = C1.z C2,or y = CeC1'

+

Y2

with In IC1 = C,. f = f ( z ,y, y',

4.

... ,g(.))

is a function of only

5:

y(n) = f(x),

(9.30a)

We get the general solution by n repeated integrations. It has the form y = c1

+ c2x t e322 + + Cam-' + @(x)

(9.30b)

Y(X)=

11.

(9.30~)

'.'

with '

We mention here that xo is not an additional arbitrary constant, since the change in .zo results in the change of C,+because of the relation 1

C, = -y("-')(xo). (IC - l)!

(9.30d)

9.1.2.3 Linear n-th Order Differential Equations 1. Classification A differential equation of the form yen) + sly("-') + a z ~ ( ~ -+' ) ' .

+ an-ly' + any = F (9.31) is called an n-th order linear differential equation. Here F and the coefficients ai are functions of x, which are supposed to be continuous in a certain interval. If a l , a 2 , .. . ,a, are constants, we call it a diferential equation with constant coeficients. If F I 0, then the linear differential equation is homogeneous. and if F $ 0, then it is inhomogeneous. 2. Fundamental System of Solutions X system of n solutions y1, y2,. . . , y, of a homogeneous linear differential equation is called a fundamenta2 system if these functions are linearly independent on the considered interval, Le., their linear combination 9y1 C2 y2 . C, y, is not identically zero for any system of values C1:C2,. . . , C,, except for the values C1= C2 = . . . = C, = 0. The solutions y1, y2, . . . , y, of a linear homogeneous differential equation form a fundamental system on the considered interval if and only if their Wronskian determinant

+

+. +

(9.32)

9.1 Ordinary Differential Equations 499

is non-zero. For every solution system of a homogeneous linear differential equation the formula of Liouville is valid:

\t.(x)

= uT(z0)exp

(

-/al(z)dz)

.

(9.33)

It follows from (9.33) that if the Wronskian determinant is zero somewhere in the solution interval, then it can be only identically zero. This means: The n solutions y1, yz, . . . , y, of the homogeneous linear differential equation are linearly dependent if even for a single point zo of the considered interval W(zo)= 0. If the solutions yl, yz, . . . , y, form a fundamental system of the differential equation, then the general solution of the linear homogeneous differential equation (9.31) is given as (9.34) y = c1y1 t c, yz ' ' . C, yn. .4 linear n-th order homogeneous differential equation has exactly n linearly independent solutions on an interval, where the coefficient functions ai(z)are continuous. 3. Lowering the Order If a e know a particular solution y1 of a homogeneous differential equation, by assuming Y = 2/1(z)u(.) (9.35) we can determine further solutions from a homogeneous linear differential equation of order n - 1 for

+ +

u'(2).

4. Superposition Principle If y1 and yz are two solutions of the differential equation (9.31) for different right-hand sides F1 and F2. then their sum y = y1 + yz is a solution of the same differential equation with the right-hand side F = Fl + Fz. From this observation it follows that to get the general solution of an inhomogeneous differential equation it is sufficient to add any particular solution of the inhomogeneous differential equation to the general solution of the corresponding homogeneous differential equation. 5 . Decomposition Theorem If an inhomogeneous differential equation (9.31) has real coefficients and its right-hand side is complex in the form F = Fl + iFz with some real functions F1 and Fzl then the solution y = y1 + iyz is also complex, nhere y1 and yz are the two solutions of the two inhomogeneous differential equations (9.31) with the corresponding right-hand sides Fl and F2.

6. Solution of Inhomogeneous Differential Equations (9.31) by Means of Quadratures If the fundamental system of the corresponding homogeneous differential equation is already known, we have the following two solution methods to continue our calculations: 1. Method of Variation of Constants We are looking for the solution in the form (9.36a) L 4=C 1Y1 + C2Y2 + ' . ' + CnY, C,, . . . . C, are functions of z. There are infinitely many such functions: but ifwe require that where C1. they satisfy the equations Cl'Y1 + Cz'yz ' .. Cn'yn = 0, (9.36b) cl'yl' Cz'y2' . ' . + Cn'yn' = 0,

+

+ + +

...............

Cl'y,("-*)+ C;y2(n-2) + . . . + C,'yn(n-2) = 0 and we substitute y into (9.31) with these equalities we get cl'yl("-') + Cz'yz(n-') + . . . + Cn'y,(n-') = F. (9.36~) Because the ll'ronskian determinant of the coefficients in the linear equation system (9.36b) and (9.36~) is different from zero. we get a unique solution for the unknown functions C1', C2', . . . , C,', and their integrals give the functions C1, C, . . . , C,.

I

500

9. Differential Esuations

1

X

Iyll+-y’--y=x-l. 1-x 1-x In the interval 2 > 1 or x < 1 all assumptions on the coefficients are fulfilled. First we solve the 1 homogeneous equation g +L g - -g = 0. A particular solution is ’p1 = e‘. We are looking for a 1-2 1-2 secondonein theformpz = e%($), andwith thenotationu’(x) = ~ ( xweget ) thefirst-orderdifferential 1 equation u’ + 1 t - u = 0. A solution of this equation is w(x) = (1 - z)e-5, and therefore, 1-x) u(x)= u(x)dx = (1 - x)e-’ dx = ze-”. With this result we get ’pz = x for the second element

(

1

I

of the fundamental system. The general solution of the homogeneous equation is g(x) = CleZt Czx. The variation of constants is now: Y(Z)= u1(x)e5 t 4 x 1 2 , y’(x) = ul(x)ezt uZ(x)+ u1’(x)e5 + uz’(x)x, ul’(x)ezt u2’(x)x = 0, u1’(x)e” uZ‘(x)= x - 1, y”(z) = u1(x)e5+ ul’(x)ez + ~ ’ ( x ) ,

+

so

ul’(x) = xe-’, u*’(x)= -1, i.e., u1(z) = -(1 t x ) e P t C1, ~ ( z=)-x t CZ. \Vith this result the general solution of the inhomogeneous differential equation is: y(x) = -(I + 2 )+ CleQ+ (Cz - 1)x = -(I + 2 ) Cl*e5 Cz’z. 2. Method of Cauchy In the general solution (9.37a) Y = ClYl + CZYZ + . ’ . + CnYn of the homogeneous differential equation associated to (9.31) we determine constants such that for an arbitrary parameter cu the equations y = 0, y’ = 0,.. . , y(n-z) = 0,Y ( ~ - ’ ) = F ( a )are satisfied. In this way we get a particular solution of the homogeneous equation, denoted by ~ ( za,) ,and then

+

+

1 5

Y=

(9.3713)

Y(Xl 0) dcu

50

is a particular solution of the inhomogeneous differential equation (9.31). This solution and their derivatives up to order ( n - 1) are equal to zero at the point x = 20. IThe general solution of the homogeneous equation associated to the differential equation (9.36a) which we solved by the method variation of constants is y = Clez Czx. From this result we see that y(a) = Cleo + Cza = 0. CY) = CleQ+ Cz = cu - 1 and p ( x , a ) = ae-Qe” - z, so the particular solution y(x) of the inhomogeneous differential equation with y(x0) = y’(x0) = 0 is: y(x) =

+

ll(cue-oe5 - x) dcu =

+ l)eZ-’O

(20

t (xo- 1)x - x z - 1. Therefore, we can already get the general

solution y(z) = Cl*eZ+ Cz*x- (x2t 1) of the inhomogeneous differential equation.

9.1.2.4 Solution of Linear Differential Equations with Constant Coefficients 1. Operational Notation The differential equation (9.31) can be written symbolically in the form

Pn(D)y= ( D n + a ~ D n - ’ + a ~ D nt -~2~ ~ + a , - l D t a , ) y = F ,

(9.38a)

where D is a differential operator: dY dkY D Y =d x ’ DkY = dxk

(9.38b)

If the coefficients a, are constants, then Pn(D)is a usual polynomial in the operator D of degree n.

9.1 Ordinarv Differential Eauations 501

2. Solution of t h e Homogeneous Differential Equation with Constant Coefficients To determine the general solution of the homogeneous differential equation (9.38a) with F = 0, Le., Pn(D)y = 0 (9.39a) we have to find the roots T I , r2).. . , T, of the characteristic equation P,(r) = rn a1rn-'+ a Z P + ' . t a,-lr a, = 0. (9.39b) Every root r, determines a solution e'-' of the equation P,(D)y = 0. If a root r, has a higher multiplicity k. then ze',', z2eTi2. . . , &l er are also solutions. The linear combination of all these solutions is the general solution of the homogeneous differential equation:

+

+

ir

~

+. +

y = Clerlr + CZerzx

'.

erls

(c~ t ~ , + t~ .z. . t ~,+k-lx~-l) + ..'.

(9.39c)

If the coefficients a, are all real, then the complex roots of the characteristic equation are pairwise conjugate with the same multiplicity. In this case, for r1 = cy + iB and rz = cy - i o we can replace the corresponding complex solution functions eriz and erzs by the real functions ear cos px and ear sin ax. The resulting expression C1 cos Pz + Cz sin pz can be written in the form A cos(Pz + p) with some constants .4 and p. IIn the case of the differential equation y(4) - y" - y = 0, the characteristic equation is r6 t r4 - r2 - 1 = 0 with roots r1 = 1, r2 = -1, r3,4 = i, r5,6 = -i. The general solution can be given in two forms: y = CleZ+ Cze-" (C3 t C ~ Zcosx ) t (C, + c@) sinx, or y = Cle' Cze-" A1 cos(x cpl) t xA2 cos(z pz).

+

+

+ +

+

+

3. Hurwitz Theorem In different applications, e.g., in vibration theory, it is important to know whether a solution of a given homogeneous differential equation with constant coefficients tend to zero for x -+ +m or not. It tends to zero, obviously, if the real parts of the roots of the characteristic equation (9.39b) are negative. .kcording to the Hurwztz theorem an equation (9.40a) a,xn + an_lxn-l . ' . alx t a0 = O has only roots with negative real part if and only if all the determinants

+ +

, . . . , D,= (with a, = 0 for m > n)

(9.40b)

are positive. The determinants Dk haveon their diagonal thecoefficients al, az, . . . , ab (k = 1,2.. . . . n), and the coefficient-indices are decreasing from left to right. Coefficients with negative indices and also with indices larger than n are all put to 0. IFor a cubic polynomial the determinants have in accordance to (9.40b) the following form:

4. Solution of Inhomogeneous Differential Equations with Constant Coefficients can be determined by the method variation of constants. or by the method of Cauchy, or with the operator method (see 9.2.2.3,5., p. 532). If the right-hand side of the inhomogeneous differential equation (9.31) has a some special form, a particular solution can be determined very easily. 1. Form: F ( z ) = Aea', P,(a) # 0 (9.4la)

502

9. Differential Equations

.I\ particular solution is

y=-

Aeax

Pn(a) If (Y is a root of the characteristic equation of multiplicity m, i.e., if

(9.41b)

P,((Y) = P,'(a) = . . . = P,""(a) = 0, (9.41c) Axmeax then y = -is a particular solution. These formulas can also be used by applying the decompoP, (a) sition theorem, if the right side is F ( z ) = Aeax cos dz or Aeaz sin wz. (9.41d) The corresponding particular solutions are the real or the imaginary part of the solution of the same differential equation for

+

F ( z ) = Ae""(cosdz isinwz) = Ae(@'IW)" (9.41e) on the right-hand side. IA: For the differential equation y" - 6y' t 8y = eZ5,the characteristic polynomial is P(D) = D Z - 6 D + 8 w i t h P ( 2 ) = OandP'(D) = 2 0 - 6 w i t h P ' ( 2 ) = 2 . 2 - 6 = -2,sotheparticularsolution xeZ2 1sy=---. 2 IB: The differential equation y" y' y = e"sina results in the equation (0' t D t l ) y = e('+')". e(lti)z e"(cosz isinz) From its solution y = we get a particular solution y1 = 2t3i (1 t i ) 2 (1t i ) t 1 ex -(2 sin z - 3 cos x). Here y1 is the imaginary part of y 13 (9.42) 2. Form: F ( z ) = Qn(z)eLIz,Q n ( z ) is a polynomial of degree TI A particular solution can always be found in the same form, i.e., as an expression y = R(z)eaz. R ( z )is a polynomial of degree n multiplied by zmif a is a root of the characteristic equation with a multiplicity m.Il'e consider the coefficients of the polynomial R ( z )as unknowns and we substitute the expression into the inhomogeneous differential equation. It must satisfy the equation, so we get a linear equation system for the coefficients. and this equation system always has a unique solution. This method is very useful especially in the cases of F ( z ) = Qn(z) for a = 0 and F ( z ) = Qn(z)e'" coswz or F ( 5 ) = Q,(z)e'" sinwz for a = r i iw. There is a solution in the form y = zmerr[Mn(z)coswz ?J,(z) sinwz]. IThe roots of the characteristic equation associated to the differential equation y(4)+ 2y'" t y" = 6a: t 2zsinz are kl = k2 = 0: k3 = kd = -1. Because of the superposition principle (see 9.1.2.3, 4..p. 499), we can calculate the particular solutions of the inhomogeneous differential equation for the summands of the right-hand side separately. For the first summand the substitution of the given form yi = zZ(azt b ) results in a right-hand side 12a t 2b t 6az = 6z, and so: a = 1 und b = -6. For the second summand we substitute yz = ( c z t d ) s i n z t (fz+g) cosz. We get the coefficients by coefficient comparisonfrom(2gt2f-6ct2f~)sinz-(2ct2dt6ft2cz)cosz= 2 z s i n z . s o c = 0 . d = -3, f = 1. g = -1. Therefore, thegeneralsolutionisy = cl+c~z-6z2tz3t(c~ztc~)e-'-3sinz+(z-l)cosz. 3. Euler Differential Equation The Euler differential equation

+ +

+

+

+

n

C ak(cz t d ) k y ( k )= ~ ( z )

(9.43a)

k=O

with the substitution cz t d = et

(9.43b)

9.1 Ordinaru DifferentialEquations 503

can be transformed into a linear differential equation with constant coefficients. IThe differential equation x2y"- 5zy' + 8y = x 2 is a special case of the Euler differential equation for n = 2. IVith the substitution x = et it becomes the differential equation discussed earlier in

d2Y - 6-dy example A. on p. 502: dt2

dt

+ 8y = eZt. The general solution is y = CleZt+ C2e4t- !e2' 2

-

9.1.2.5 Systems of Linear Differential Equations with Constant

Coefficients 1. Normal Form The following simple case of a first-order linear differential equation system with constant coefficients is called a normal system or a normalform:

Y!' = a l l y l t alzy2 + ~ 2 = ' QIYI

+

t~ZZYZ

'.

+ alnyn:

' '.

t a2nyni

.................................

+

1

(9.44a)

yn' = aniZj! + a n ~ y z ' ' t a n n u n . To find the general solution of such a system, we have to find first the roots of the characteristic equation , , . a!, . . . a2, ~ ......................... anl an2 . , . ann - r

all - r

a12

aZ1

a22 - T

=o.

To every single root r, of this equation there is a system of particular solutions y! = Alertz. y2 = Azerxs,.. . , y, = Anerts, (9.44c) whose coefficients 4 k ( k = 1 , 2 , , . . , n) are determined from the homogeneous linear equation system

+ aiz.42 t . ' . t ain& = 0, ....................................... a , l A ~ + an2A2 + . ' . t (ann - rJA, = 0. (a11 - ?,)AI

(9.44d)

This system gives the relations between the values of Ah (see Trivial solution and fundamental system in 1.4.2.1, 2., p. 273). For every T , , the particular solutions we get in this way will contain an arbitrary constant. If all the roots of the characteristic equation are different, the sum ofthese particular solutions contains n independent arbitrary constants, so in this way we get the general solution. If a root r, has a multiplicity m in the characteristic equation. the system of particular solutions corresponding to this root has the form , , y,, = An(z)erLs, (9.44e) y1 = Al(x)erSs, y2 = A2(x)eP1Z:, where .41(x)~. . . , .4,(2) are polynomials of degree at most m - 1. We substitute these expressions with unknown coefficients of the polynomials Ak(x) into the differential equation system. We can first cancel the factor e''", then we can compare the coefficients of the different powers of x to have linear equations for the unknown coefficients of the polynomials, and among them m can be chosen freely. In this way, we get a part of the solution with m arbitrary constants. The degree of the polynomials can be less than m - 1. In the special case when the system (9.44a) is symmetric? Le., when aik = ah,>then it is sufficient to substitute A,(z) = const. For complex roots of the characteristic equation, the general solution can be transformed into a real form in the same way as shown earlier for the case of a differential equation with constant coefficients (see 9.1.2.4, p. 500). . . IFor the system yl' = 2yl t 2y2 - y3, y2' = -291 t 4yz y3. y3' = -3yl t 8y2 2y3 the characteristic ~

+

+

504

9. Differential Equations

equation has the form

1 ‘jT

4 rT! 2

~

= -(r

- 6)(r-

= 0.

For the simple root r1 = 6 we get -4A1+ 2A2 - A3 = 0, -2A1 - 2Az+ A3 = 0, -3A1 + 8A2 - 4A3 = 0. 1 From this system we have A1 = 0, A2 = -A3 = C1, y1 = 0, yz = Cle6”,y3 = 2C1e6’. For the multiple 2 root T Z = 1 we get y1 = (PIX &])ex,y2 = (PZZ Q2)e5, y3 = (P32 Q3)e2. Substitution into the equations yields

+

-I-(S+ + (P2 + Qz) p35 + (p3 + Q3)

Q1)

=

+

+ 2pz - P3)z + ( 2 Q 1 +

(2p1

+

2Q2 - Q 3 ) ,

+ P3)z + (-2Q1+ 4Qz + Q s ) , = (-3p1+ 8P2 + 2P3)2 + (-3Q1+ 8Q2 + 2Q3), , = 5C3 - 6C2, QZ= C3, = 7C3 - llCz. The which implies that PI = 5C2, P2 = C2, P3 = ~ C ZQ1 general solution is y1 = ( 5 G z t 5C3 - 6C2)e2, yz = C1e6” + (Czz + C3)e2, y3 = 2Cle6” + (7C2z + Pz.

= (-2P1+ 4Pz

Q3

7C3 - 11Cz)e”.

2. Homogeneous Systems of First-Order Linear Differential Equations with Constant Coefficients have the general form n

n

1ad&‘ + k=l 1blkyk = 0

(2

= 1, 2 , . . . , n).

(9.45a)

k=l

If the determinant det(a,k) does not disappear, i.e., d e t ( a d # 0, (9.45b) then the system (9.458) can be transformed into the normal form (9.44a). In the case of det(a,k) = 0 we need further investigations (see [9.15]). The solution can be determined from the general form in the same way as shown for the normal form. The characteristic equation has the form det(a,kr b,k) = 0. (9.45c) The coefficients A, in the solution (9.44~)corresponding to a single root T~ are determined from the equation system

+

n

l ( a , k r J + b z k ) A k = O ( 2 = 1 > 2 ,. . . , n ) . (9.45d) k=l Otherwise the solution method follows the same ideas as in the case of the normal form. IThe characteristic equation of the two differentialequations 5y1’+4y1-2yz’-y2 = 0, y1’+8~1-3yz = 0 is: 5T+4 - 2 - 1 =2r2+2~ 4 = 0, TI = 1, ~2 = -2. ~ + 8-3 We get the coefficients AI and Az for r1 = 1 from the equations-9Al - 3Az = 0, 9Ai - 3A2 = 0 so AZ = 3A1 = 3C1. For rz = -2 we get analogously A2 = 2A1 = 2C2. The general solution is y1 = Ciex+ C2e-2x,y2 = 3Clex + 2Cze-2x.

I

3. Inhomogeneous Systems of First-Order Linear Differerential Equations have the general form n

n

1azkyk’ + 1bzkyk = Ft(z) k=l k=l

(Z =

1, 2,. . . 2 n).

(9.46)

9.1 Ordinam Differential Eauations 505

1. Superposition Principle: If yj(') and yj(') ( j = 1,2,.. . , n) are solutions of inhomogeneous systems which differ from each other only in their right-hand sides F,(') and F,('), then the sum yJ = , n ) is a solution of this system with the right-hand side F,(z) = F,(')(z) + Ft(')(x). Because of this, to get the general solution of an inhomogeneous system it is enough to add a particular solution to the general solution of the corresponding homogeneous system. 2. T h e Variation of Constants can be used to get a particular solution of the inhomogeneous differential equation system. To do this we use the general solution of the homogeneous system, and we consider the constants C1, C2, . . . , C,, as unknown functions Cl(z),Cz(z),. . . , Cn(z).We substitute it into the inhomogeneous system. In the expressions of the derivatives of yk' we have the derivative of the new unknown functions Ck(z). Because y1. y 2 , . . . , yn are solutions of the homogeneous system, the terms containing the new unknown functions will be canceled; only their derivatives remain in the equations. We get for the functions Ck'(z) an inhomogeneous linear algebraic equation system which always has a unique solution. '4fter n integrations we get the functions C,(z), C Z ( Z ) .~.., Cn(z). Substitution them into the solution of the homogeneous system instead of the constants results in the particular solution of the inhomogeneous system. For the system of two inhomogeneous differential equations 5yl' 491 - 2y2' - y2 = e-", yl' i8y1 - 3y2 = X" the general solution of the homogeneous system is (see p. 504) y1 = Cle" + C2e-221 y2 = 3Cles -t 2Cze-". Considering the constants C1 and Cz as functions of z and substituting into the original equations we get 5Cl'e" t 5C2'e-'" - 6C1'e" - 4C2'e-2" = e-", Cl'e" -t C2'e-'= = 5e-" or C21e-2"- Cl'e-" = e-", Cl'e5 -t C2'e-2' = W". Therefore, 2Cl'e5 = 4e-', C1 = -e-'" + const, 2C2'e-'" = 6e-", Cz = 3e" + const. Since we are looking for a particular solution, we can replace every constant by zero and the result is y1 = 2e-", yz = 3 e P . The general solution is finally y1 = 2e-" t Cle" + C2e-2x3 y2 = 3e-" + 3Clex + 2Cze-2". 3. The Method of Unknown Coefficients is especially useful if on the right-hand side there are special functions in the form Qn(z)e0".The application is similar to the one we used for differential equations of n-th order (see 9.1.2.5,p. 503). 4. Second-Order Systems The methods introduced above can also be used for differential equations of higher order. For the system

+

n

1

a,kYk"

i;=l

n

n

k=l

k=l

+ 1b,kyk' 4-

C,kyk

=0

(2

= 1;2.. . . , n)

(9.47)

we can determine particular solutions in the form yi = Aie7*".To do this, we get T , from the characteristic equation det(a,kr2t b z k r + c,k) = 0, and we get A, from the corresponding linear homogeneous algebraic equations.

9.1.2.6 Linear Second-Order Differential Equations Many special differential equations belong to this class, which often occur in practical applications. We discuss several of them in this paragraph. For more details of representation, properties and solution methods see [9.15].

1. General Methods 1. The Inhomogeneous Differential Equation is Y" + P ( 2 ) Y ' + d S ) Y = F ( z ) .

(9.48a) a) The general solution of the corresponding homogeneous differential equation, Le., with F ( z ) = 0, is (9.48b) Y = ClYl -t C2Y2. Here y1 and y2 are two linearly independent particular solutions of this equation (see 9.1.2.3,2., p. 498). If a particular solution yl is already known, then the second one yz can be determined by the equation (9.48~)

506

9. Di.#erential Equations

which follows from the Liouville formula (9.33), where A can be chosen arbitrarily. b) .A particular solution of the inhomogeneous equation can be determined by the formula

where y1 and yz are two particular solutions of the corresponding homogeneous differential equation. c) 4 particular solution of the inhomogeneous differential equation can be determined also by variation ofconstants (see 9.1.2.3, 6., p. 499). 2. If in the Inhomogeneous Differential Equation (9.49aj 4x)Y/r+ P(X)Y/+ d X ) Y = F ( x ) the functions s(x))p(x), q(x) and F ( x ) are polynomials or functions which can be expanded into a convergent power series around xo in a certain domain, where s(x0) # 0, then the solutions of this differential equation can also be expanded into a similar series, and these series are convergent in the same domain. We determine them by the method of undetermined coefficients: The solution we are looking for as a series has the form (9.49b) y = a0 al(z - xoj az(x - xo)2 . . , and we substitute it into the differential equation (9.49a). Equating like coefficients (of the same powers of (x - xo)) results in equations to determine the coefficients aor a,, a2.. . . . ITo solve the differential equation y" xy = 0 we substitute y = a0 alx a2x2 a3x3 ' . . , y' = al 2a2x + 3a3xz ' . and y" = 2az 6a3x . ' . . We get 2az = 0, 6a3 a0 = 0 , . . . . a0 a1 The solution of these equations is a2 = 0, a3 = -- . a4 = -a5 = 0.. . . , so the solution is 2.3 3.4 X4 27 _ . . . + a 1 x---$-... 3.4 3.4.6.7 3. The Homogeneous Differential Equation XZY,' + xp(x)y/ q(2)y = 0 (9.50a) can be solved by the method of undetermined coefficients if the functionsp(x) and q(x) can be expanded as a convergent series of x. The solutions have the form y = x'(a0 + a,x t a252 + ' ' .). (9.50b) whose exponent T can be determined from the dejnzng eqzlatzon (9.50~) T ( T - 1) p(0jr q(0) = 0. If the roots of this equation are different and their difference is not an integer number, then we get two linearlj independent solutions of (9.50a). Otherwise the method of undetermined coefficients results only one solution. Then with the help of (9.48b) we can get a second solution or at least we can find a form from which we can get a second solution with the method of undetermined Coefficients. IFor the Bessel differential equation (9.51a) we get only one solution with the method of the undeter-

+

+

+

+

+.

+

~

+

+

+

+

+ +

+

~

) (

+

+

+

30

mined coefficients in the form y1 =

1akXntzk

(a0

# 0 ) , which coincides with Jn(x) up to a constant

k=O

we find a second solution by using formula (9.48~)

The determination of the unknown coefficients ck and dk is difficult from the Q ' S . But we can use this last expression to get the solution with the method of undetermined coefficients. Obviously this form is a series expansion of the function Yn(x) (9.52~).

9.1 Ordinaru DifferentialEouations 507

2. Bessel Differential Equation x2y” + zy’+ ( 2- n2)y = 0.

(9.51a)

1. The Defining Equation is in this case T ( T - 1) r - n2 T’ - n2 = 0, so, r = h. Substituting y = Zn(a0 alz ’ .) into this equation and equating the coefficients of xntk to zero we have k(2n k)ak ah-2 = 0. For k = 1 we get (2n 1)al = 0. For the values k = 2,3,. . . we obtain

+

+

+

+

a4 =

+.

+

a0

,

+

2 ’ 4 ’ (2n + 2)(2n 4 ) ” ”

a. is arbitrary.

(9.51b) (9.51~) (9.5ld)

(9.cile)

where r is 2niyn + 1)’ the gamma function (see 8.2.5, 6.,p. 459), is a particular solution of the Bessel differential equation (9.5la) for integer values of n. It defines the Besselor cylindricalfunction of the first kind of index n 2. Bessel or Cylindrical Functions The series obtained above for a0 =

Jn(z) = 2 , r c + 1)

X2

(l -

+ 2 . 4 , (2n+x42)(2n+4) (9.52a)

The graphs of functions Jo and 51 are shown in Fig. 9.12. The general solution of the Bessel differential equation for non-integer n has the form (9.52b) Y = CiJn(z) + GJ-n(z)> where J-n(z)is defined by the infinite series obtained from the series representation of J,(z) by replacing n with -n. For integer n. we have J-n(x) = (-l)nJn(z). In this case, the term J-n(x) in the general solution should be replaced with the Bessel function of the second kind Jrn(z)COSrnT - J-m(x) Yn(z)= lim (9.52~) I min sin mT which is also called the Weberfunction. For the series expansion of Y,(x) see, e.g., [9.15]. The graphs of the functions Yo and Yl are shown in Fig. 9.13. 3. Bessel Functions with Imaginary Variables In some applications we use Bessel functions with pure imaginary variables. In this case we have to consider the product i-nJn(ix) which will be denoted by I n ( z ) : (9.53a) The functions In(.)

+

z2yrr zy’

-

are solutions of the differential equation

( 2+ n2)y = 0.

(9.53b)

A second solution of this differential equation is the MacDonaldfunction (9.53c)

9. DifferentialEquations

508

Figure 9.12

Figure 9.13

If n converges to an integer number. this expression also converges. The functions In(x)and K n ( x )are called modified Besselfunctions. The graphs of functions IO and 11 are shown in Fig. 9.14; the graphs of functions KOand K1 are illustrated in Fig. 9.15. The values of functions Jo(x),J 1 ( x ) Y) o ( x ) Y, l ( x ) ,IO(.)) I l ( x ) ,K o ( x ) ,Kl(x) are given in Table 21.9, p. 1058.

Y

<:

/

i

:

-2. -3-4-5-

0.5 0

0.5 1.0 1.5 2.0

* -1.01 X

Figure 9.15

Figure 9.14

1 0.5

1

1.0

Figure 9.16

4. Important Formulas for the Bessel Functions &(z) Jn-l(x)

+ Jn+i(Z) = -2nJ n ( x )

I

dJn(x) = --Jn(x) dx

+ Jn-l(X).

(9.54a)

The formulas (9.54a) are also valid for the Weber functions Yn(x). - In+1(X)= 2nIn(x) d I h ) - In-l(Z)- ?I&), x ' dx X

(9.54b)

2nKn(z) dKn(x) = -Kn-l(x) - !Kn(.), Kn+l(x)- Kn-l(x)= x ' dx X

(9.54c)

In-l(x)

~

For integer numbers n the following formulas are valid: (9.54d)

9.1 Ordinaru Differential Eazlations 509

1 7712

~ ~ ~ += ~ ( zsin(z ) sin p) sin(2n

+ 1)yOdyO

(9.54e)

0

or, in complex form.

Jn(z)= -jeiscos+cosnpdp -(i)n

(9.54f)

0

The functions Jntljz(z) can be expressed by using elementary functions. In particular, J l j z ( X ) = p iTX si

nz)

(9.55a)

J-lp(x) = E c o s x .

(9.55b)

By applying the recursion formulas (9.54a)-(9.54f) the expression for Jn+l,l(z) for arbitrary integer n can be given. For large values of z we have the following asymptotic formulas: Jn(X)

;/

=

[cos (X -

y

I);(

,

[sin (z -

-

I n ( X ) = e" [l t 0

6

y,(z) =

Kn(z)=

E

-

I);(

$) t 0

,

(9.56a) (9.56b)

t0

(31 ,

Fe-. I ) ' ( 22

[l t 0

(9.56d)

1 The expression 0 - means an infinitesimal quantity of the same order as - (see the Landau symbol,

(5)

X

p. 56). For further properties of the Bessel functions see [21.1].

3. Legendre Differential Equation

Restricting our investigations to the case of real variables and integer parameters n = 0, 1 , 2 , . . . the Legendre differential equation has the form (9.57a) (1 - 2 ) y " - 2zy' + n(n t 1)y = O or ((I - x2)y')'+ n(n + 1)y = 0. 1. Legendre Polynomials or Spherical Harmonics of the First Kind are the particular solutions of the Legendre differential equation for integer n,which can be expanded into the power series m

y=

1a&.

By the method of undetermined coefficients we get the polynomials

"=O

(1x1< m ; n = 0 , 1 , 2 , .. .). (9.57b) (9.57c) where F denotes the hypergeometric series (see 4., p. 511). The first eight polynomials have the following simple form (see 21.10. p. 1060):

Po(s)= 1 .

(9.57d)

PI(X)

=x >

(9.57e)

510

9. DifferentialEquations

1 2

Pz(x)= - ( 3 2 - 1 ) .

(9.57f)

1

P ~ ( T=) -(35x4 - 302' 8

+ 3),

(9.57h)

P3(2)=

1 -(523 - 32), 2

(9.57g)

1 8

(9.57i)

P~(z) = -(63z5

-

70z3 + 1 5 ~ ) ,

1 1 P6 (x)= -( 2 3 1-315x4+105z2-5) ~~ ,(9.57j) 5 ( z ) = -(429z7-693z5+315r3 -352) .(9.57k) 16 16 The graphs of Pn(z) for the values from n = 1 to n = 7 are represented in Fig. 9.16. The numerical values can be calculated easily by pocket calculators or from function tables. 2. Properties of t h e Legendre Polynomials of t h e First Kind a ) Integral Representation:

(9.58a) The signs can be chosen arbitrarily in both equations. b) Recursion Formulas: ( n 1)Pn+1(x)= (2n t l)zPn(z)- nPn-l(.) (n2 1;Po(.) = 1, P l ( Z ) = z), dP (22 - 1)= n[xP,(.) - Pn-l(z)] (n2 1).

+

dx

(9.58b) (9.58~ )

c) Orthogonality Relation: form

# n>

form = n.

(9.58d)

-1

d) Root Theorem: All t h e n roots of P,(z) are real and single and are in the interval (-1.1). e) Generating Function: The Legendre polynomial of the first kind can be represented as the power series expansion of the function 1

J 1 - 2rx t r2

m

I

=

Pn(z)rn.

n=O

(9.58e)

For further properties of the Legendre polynomials of the first kind see [21.1]. 3. Legendre Functions or Spherical Harmonics of t h e Second Kind We get a second particular solution Qn(z), which is valid for 1x1 > 1 and linearly independent of P,(z)>see (9.58a), by the -(rat11

b,z":

power series expansion v=-m

(9.59a) The representation of Qn(z)valid for 1x1 < 1 is: (9.5913)

9.1 Ordinary Differential Equations 511

We call the spherical harmonics of the first and second kind also the associated Legendre functions (see p. 541). also (9.118~);

4. Hypergeometric Differential Equation The hypergeometric diferential equation is the equation z ( l - x)-d 2 y t [^, - ( a t 8t 1)4-dY - a3y = 0, (9.60a) dz2 dx where a.3. are parameters. It contains several important special cases. 1-2 a) For a = n t 1. 8 = -n, = 1: and z = -it is the Legendre differential equation. 2 b) If? # 0 or *, is not a negative integer, it has the hypergeometric series or hypergeometric function as a particular solution : t;

. ( a t n)P(B + 1 ) .. . (9 t n)Zntl + a (1a. t2 .1). . . ( n + 1 ) .$7 t 1) (Y t n) ,

I

.

+

,,,

(9.60b)

.

which is absolutely convergent for 1x1 < 1. The convergence for x = f l depends on the value of d = - a - 3. For z = 1 it is convergent if 6 > 0. it is divergent if 6 5 0. For x = -1 it is absolutely convergent if 6 < 0. it is conditionally convergent for -1 < 6 5 0. and it is divergent for 6 5 -1. c) For 2 - Y # 0 or not equal to a negative integer it has a particular solution y=z-'-'F(a+l-Y.3$1-y,2-~/,x). (9.60~) d) In some special cases the hypergeometric series can be reduced to elementary functions, e.g.,

F(1,3.3;xj = F ( a . 1. a;z) = I (9.61a) , F(-n, B.p; -z)= (1 t z)", 1-x ln(1 t x) F(1,l.2: -z) = -,

31%

)

(1 1 3 % arcsinz (9.61~) F - - - - z 2 = -, 2' 2' 2

(9.61b) (9.61d) (9.61e)

5 . Laguerre Differential Equation If we restrict our investigation to integer parameters ( n = 0,1,2,. . .) and real variables, the Laguerre dzflerentzal eqzlatzon has the form xy" t (a + 1 - x)y' t ny = 0. (9.62a) .As a particular 3olution we have the Laguerre polynornzal

(9.62b)

(9.62~) (9.62d)

(9.62e)

512

9. Di.ferential Equations

With

r we denote the gamma function (see 8.2.5, 6., p. 459).

6. Hermite Differential Equation Two defining equations are often used in the literature: a) Defining Equation of Type 1: y" -xy) + n y = 0 ( n = 0,1,2,.. .). (9.63a) b) Defining Equation of Type 2: yll- 2xy'+ny = 0 ( n = 0,1,2,. . .). (9.63b) Particular solutions are the Hermite polynomials,He,(x) for the defining equation of type 1,and H,(x) for the defining equation of type 2. a) Hermite Polynomials for Defining Equation of Type 1:

(9.63~)

For n 2 1 the following recursion formulas are valid: Hentl(.) = xHe,(x)

- nHe,-l(z),

(9.63d)

Heo(x) = 1, H e l ( x ) = x.

(9.63e)

The orthogonality relation is: f33

e-"2/2Hem(x)Hen(z) dx =

( 0n!&

form # n, form = n.

(9.630

b) Hermite Polynomials for Defining Equation of Type 2: H,(r) = ( - l ) , e Z Z -

d"

dxn

(e-q

( n E K).

The relation with the Hermite polynomials for defining equation of type 1 is the following: (9.63h)

9.1.3 Boundary Value Problems 9.1.3.1 Problem Formulation 1. Notion of the Boundary Value Problem In different applications, e.g., in mathematical physics, differentii. equations must be sc. ed as socalled boundary value problems (see 9.2.3, p. 532), where the solution we are looking for must satisfy previously given relations at the endpoints of an interval of the independent variable. A special case is the linear boundary value problem, where a solution of a linear differential equation should satisfy linear boundary value conditions. In the followings we restrict our discussion to second-order linear differential equations with linear boundary values.

2. Self-Adjoint Differential Equation Self-adpznt dzflerentzal equatzons are important special second-order differential equations of the form (9.64a) [PllT - QY + b y = f. The linear boundary values are the homogeneous conditions (9.64b) A o y ( a ) + Boy'(a) = 0, Aly(b) Bly'(b) = 0. The functions p(x),p'(x), ~ ( x )~, ( x )and , f(x) are supposed to be continuous in the finite interval a 5 x 5 b. In the case of an infinite interval the results change considerably (see [9.5]). Furthermore,

+

9.1 Ordinary Differential Equations 513

we suppose that p ( x ) > pa > 0, ~ ( z > ) > 0. The quantity A, a parameter of the differential equation. is a constant. For f = 0, it is called the homogeneous boundary value problem associated to the znhomogeneous boundary value problem. Every second-order differential equation of the form A?/"+ By'+ cy ARy = F (9.64~) can be reduced to the self-adjoint equation (9.64a) by multiplying it by p / A if in [a,b ] , A # 0, and performing the following substitutions

+

(9.64d) To find a solution satisfying the inhomogeneous conditions (9.64e) AOY(Q) + BOY'(^) = Co, A I Y ( ~+)Bi~'(b)= Ci we return to the problem with homogeneous boundary conditions, but we change the right-hand side f(x). We substitute y = z u where u is an arbitrary twice differentiable function satisfying the inhomogeneous boundary conditions and z is a new unknown function satisfying the corresponding homogeneous conditions.

+

3. Sturm-Liouville Problem For a given value of the parameter X there are two cases: 1. Either the inhomogeneous boundary value problem has a unique solution for arbitrary f(.), while the corresponding homogeneous problem has only the trivial, identically zero solution, or, 2. The corresponding homogeneous problem also has non-trivial, Le., not identically zero solutions, but in this case the inhomogeneous problem does not have a solution for arbitrary right-hand side: and if a solution exists, it is not unique. The values of the parameter A, for which the second case occurs, Le., the homogeneous problem has a non-trivial solution, are called the eigenvalues of the boundary value problem, the corresponding nontrivial solutions are called the eigenfunctions. The problem of determining the eigenvalues and eigenfunctions of a differential equation (9.64a) is called the Stwm-Liouville problem.

9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues 1. The eigenvalues of a boundary value problem form a monotone increasing sequence of real numbers (9.65,) A0 < XI < A 2 < . ' . < A, < . ' . , tending to infinity. 2. The eigenfunction associated to the eigenvalue A, has exactly n roots in the interval a < z < b. 3. If y(z) and z(x) are two eigenfunctions belonging to the same eigenvalue X, they differ only in a constant multiplier c, Le., z(.) = cy(.). (9.65b) 4. Two eigenfunctions yl(z) and yZ(z), associated to different eigenvalues A1 and 12.are orthogonal to each other with the weightfunction ~ ( z )

jgd4 yz(.)

).(oi

(9.65~)

d. = 0.

a

5 . If in (9.64a) the coefficients p ( z ) and q ( z ) are replaced by C(z) 2 p ( z ) and i ( z ) 2 q ( z ) ,then the eigenvalues will not decrease, Le., in 2 A, where and A, are the n-th eigenvalues of the modified and the original equations respectively. But if the coefficient e(z) is replaced by i(x) 2 ~ ( x )then , the eigenvalues will not increase, Le., in 5 A,. The n-th eigenvalue depends continuously on the coefficients of the equation. i.e., small changes in the coefficients will result in small variations of the n-th eigenvalue.

x,

9. DifferentialEquations

514

6. Reduction of the interval [a,b] into a smaller one does not result in smaller eigenvalues.

9.1.3.3 Expansion in Eigenfunctions 1. Normalization of the Eigenfunction For every A, an eigenfunction vn(x) is chosen such that

p,;.!x)I’P(x!

dx = 1.

(9.66a)

a

It is called a normalized eigenfunctzon.

2. Fourier Expansion To every function g(z) defined in the interval [a, b]. we can assign its Fourier series (9.66b) with the eigenfunctions of the corresponding boundary value problem, if the integrals in (9.66b) exist.

3. Expansion Theorem If the function g(z) has a continuous derivative and satisfies the boundary conditions of the given problem, then the Fourier series ofg(x) (in the eigenfunctions of this boundary value problem) is absolutely and uniformly convergent to g(x).

4. Parseval Eauation If the integral on the left-hand side exists, then m

/[g(x)120(x) dx = J

a

1c;

(9.66~)

n=O

is always valid. The Fourier series of the function g(z) converges in this case to g(x) in mean, that is (9.66d)

9.1.3.4 Singular Cases Boundary value problems of the above type very often occur in solving problems of theoretical physics by the Fourier method, however at the endpoints of the interval [a,b] some singularities of the differential equation may occur, e.g., p(x) vanishes. At such singular points we impose some restrictions on the solutions, e.g.. continuity or being finite or unlimited growth with a bounded order. These conditions play the role of homogeneous boundary conditions (see 9.2.3.3, p. 535). In addition, we often have the case where in certain boundary value problems homogeneous boundary conditions should be considered, such that they connect the values of the function or its derivative at different endpoints of the interval. \Ye often have the relations Y ( U ) = Y(b), p(a)y’(a)= P(b)Y’(b)> (9.67) which represent periodicity in the case ofp(a) = p ( b ) . For such boundary value problems everything we introduced above remains valid, except statement (9.65b). For further discussion of this topic see [9.5].

9.2 Partial DifferentialEauations 515

9.2 Partial Differential Equations 9.2.1 First-Order Partial Differential Equations 9.2.1.1 Linear First-Order Partial Differential Equations 1. Linear and Quasilinear Partial Differential Equations The equation

aZ

az az + x2t . . + x,=y ax, ax2 axn

x1-

(9.68a)

,

is called a linear first-order partial diflerential equation. Here z is an unknown function of the independent variables X I , , . . . x,, and X i , . . . , x,, Y are given functions of these variables. If functions X I , ,, , , X,, Y depend also on z , the equation is called a quasilinear partial diflerential equation. In the case of Y 0; (9.68b) the equation is called homogeneous.

2. Solution of a Homogeneous Partial Linear Differential Equation The solution of a homogeneous partial linear differential equation and the solution of the so-called characteristic system (9.69a) are equivalent. This system can be solved in two different ways: 1. .4ny xk, for which Xk # 0, can be chosen as an independent variable, so the system is transformed into the form (9.69b) 2. .A more convenient way is to keep symmetry and introduce a new variable t , and then we get

dt

= X,.

(9.69~)

Every first integral of the system (9.69a) is a solution of the homogeneous linear partial differential equation (9.68b), and conversely, every solution of (9.68b) is a first integral of (9.69a) (see 9.1.2.1, 2., p. 496). If the n - 1 first integrals pt(xl. . . . , x , ) = 0 ( z = 1 , 2 )n-l) (9.69d) are independent (see 9.1.2.3, 2., p. 498)) then the general solution is (9.69e) 2 = @(PI,. , qn-1). Here @ is an arbitrary function of the n - 1 arguments (pz and a general solution of the homogeneous linear differential equation. I

.

.

.

t

3. Solution of Inhomogeneous Linear and Quasilinear Partial Differential Equations To solve an inhomogeneous linear and quasilinear partial differential equation (9.68a) we try to find the solution z in the implicit form V ( x i > ,, . ,z,,z ) = C. The function V is a solution of the homogeneous linear differential equation with n + 1 independent variables

av +xz-av + . . . +x,-av xi-ax, ax2 ax,

av = 0,

+y-

at

(9.70a)

whose characteristic system (9.70b)

9. Di.#erential Equations

516

is called the characteristic system ofthe original equation (9.68a).

4. Geometrical Representation and Characteristics of the System In the case of the equation (9.71a) with two independent variables x1 = z and x~ = y, a solution z = f(x,y) is a surface in x,y, z space, and it is called the integral surface of the differential equation. Equation (9.71a) means that at every

(E; )

point of the integral surface z = f ( x , y) the normal vector - - , -1

is orthogonal to the vector

( P ,Q , R) given at that point. Here the system (9.70b) has the form dx - dY - d z (9.71b) P ( x ,Yr 2 ) &(.> Y , z ) R(x,Y, 2) . It follows (see 13.1.3.5, p. 646) that the integral curves ofthis system, the so-called characteristics, are tangent to the vector (P>&, R ) . Therefore, a characteristic having a common point with the integral surface z = f ( x l y) lies completely on this surface. Since the conditions for the existence theorem 13.1.3.5, 1.. p. 496 hold, there is an integral curve of the characteristic system passing through every point of space, so the integral surface consists of characteristics. 5 . Cauchy Problem There are given n functions of n - 1 independent variables t l , t z , . . . , t,-1: 5 1 = Zl(t1, t?!.. . , ta-l), xz = XZ(tl,tZ,. . . , t,-1), . . . , x, = X,(tl, tz,. . . , t,-1). (9.72a) The Cauchy problem for the differential equation (9.68a) is to find a solution z=cp(x1.xz, . . . :z,) (9.72b) such that if we substitute (9.72a), the result is a previously given function y(t1, t Z , . . . t,-l): p[xi(ti,tz,...,t,-i), xz(ti,tz,...,t,-i),...! xn(ti,tz,...,L-i)] = $(ti,tz,. ..:t,-i). (9.72~) In the case of two independent variables, the problem reduces to find an integral surface passing through the given curve. If this curve has a tangent depending continuously on a point and it is not tangent to the characteristics at any point, then the Cauchy problem has a unique solution in a certain neighborhood of this curve. Here the integral surface consists of the set of all characteristics intersecting the given curve. For more mathematical discussion on theorems about the existence of the solution of the Cauchy problem see [9.15]. at

IA: For the linear first-order inhomogeneous partial differential equation (mz - ny)-

dX

+ (nz -

dz dx dy l z ) - = ly - mx (1. m, n are constants), the equations of the characteristics are -= dY mz-ny nx-Iz dz , The integrals of this system are lx + my + nz = C1, xz + yz + zz = Cz. We get circles as ly - mx characteristics, whose centers are on a line passing through the origin, and this line has direction cosines proportional to 1, m, n. The integral surfaces are rotation surfaces with this line as an axis.

at +

IB: Determine the integral surface of the first-order linear inhomogeneous differential equation dX az

-

dY

=

2,

which passes through the curve x = 0, z = ~ ( y )The . equations of characteristics are

dz 9 = . The characteristics passing through the point (xo,yo, 1 2 -

20)

are y = x

-

dx

-

1

=

xo + yos z = zoez-"o.

A parametric representation of the required integral surface is y = x + yo, z = e"p(yo), if we substitute

9.2 Partial Differential Eouations 517

xo = 0, zo = p(y0). The elimination of yo results in z = e 5 v ( y - x ) .

9.2.1.2 Non-Linear First-Order Partial Differential Equations 1. General Form of First-Order Partial Differential Equation is the implicit equation (9.73a)

1. Complete Integral is the solution (9.7313) z = + $ X I ,... 2,: a l , . . . , a n ) , depending on n parameters al,. . . ,a, if at the considered values of 21,. . . , x,, z the functional determinant (or Jacobian determinant, see 2.18.2.6, 3., p. 121) is non-zero: (9.73c) 2. Characteristic Strip The solution of (9.73a) is reduced to the solution of the characteristic system

(9.73d) with

The solutions of the characteristic system satisfying the additional condition F(X1.. . . ,x,. t , p 1 , .. . ,pn)= 0 are called the characterzstzc strzps.

(9.730

2. Canonical Systems of Differential Equations Sometimes it is more convenient to consider an equation not involving explicitly the unknown function z. Such an equation can be obtained by introducing an additional independent variable xntl = z and an unknown function I’(x1, . . . , x n , x,+l), which defines the function z ( x l , x Z , .. . ) x n )with the equation (9.74a) V ( X l , ,. . . x,, z ) = c. aZ (i = 1 , . . . n) for - in (9.73a). Then At the same time. n e substitute the functions -axi we solve the differential equation (9.73a) for an arbitrary partial derivative of the function V . The corresponding independent variable will be denoted by x after a suitable renumbering of the other variables. Finally, we have the equation (9.73a) in the form ~

(9.74b) The system of characteristic differential equations is transformed into the system (9.74c) (9.74d) Equations (9.74~)represent a system of 2n ordinary differential equations, which corresponds to an arbitrary function H ( z 1 , .. . , x,, x,pl,.. . ,pn)with 2n + 1 variables. It is called a canonical system or a normal system of differential equations.

9. DifferentialEouations

518

Many problems of mechanics and theoretical physics lead to equations of this form. Knowing a complete integral L' = +(zl.. . . .x,. x,al,.. . ,a,) + a (9.74e) of the equation (9.74b) we can find the general solution of the canonical system (9.74~)) since the equa-

% = b,. dP = p, (z = 1 , 2 , . . . n) with 2n arbitrary parameters a, and b, determine a 2ntions ~

ax,

aa,

parameter solution of the canonical system (9.74~).

3. Clairaut Differential Equation If the given differential equation can be transformed into the form dz = xipi t zzpz t . . ' + w+lt j(p13 p z , . . . ,P,)) p , = - (i = 1, . . . , n ) ,

ax,

(9.75a)

it is called a Clairaut differential equation. The determination of the complete integral is particularly simple, because a complete integral with the arbitrary parameters al, az.. . . ,a, is (9.76b) z = a121 + a2x2 . + a,x, j(a1, a2,. . . ,a,). Two-Body Problem with Hamilton Function: Consider two particles moving in a plane under their mutual gravitational attraction according to the Newton field (see also 13.4.3.2, p. 667). We choose the origin as the initial position of one of the particles, so the equations of motion have the form d2x - 81' d2y - dV, V=- k2 (9.76a) dt2 8s' dt2- dy' If we introduce the Hamiltonian function 1 k2 (9.76b) H = -(p2 + 9') 2 the system (9.76a) is transformed into the normal system (into the system of canonical differential equations) dy - dH dp - aH dq dH -dx= - d H (9.76~) dt d p ' dt dq ' dt a x ' dt - dy with variables dx dy (9.76d) x,y,p=-, dt Now, the partial differential equation has the form

+.

~

+

-

m3

q=z.

-az+ - 1 [(a;)2+($)2] at

--=o.

k2

dx

2

(9.76e)

Introducing the polar coordinates p, 9 in (9.76e) we obtain a new differential equation having the solution

z

= -at

-

bv t c -

+ L:/T 2a

- - -dr

(9.76f)

with the parameters a, b. c. We get the general solution of the system (9.76~)from the equations

dz

- -- -to,

da

az ab - -Po.

4. First-Order Differential Equation in Two Independent Variables For z1 = z,z2 = y, pl = p , pz = q the characteristic strip (see 9.2.1.2, l., p. 517) can be geometrically interpreted as a curve at every point (x,y, z ) of which a plane p ( < - x) + q(v - y) = < - z being tangent

9.2 Partzal Differential Eauations 519

to the curve is prescribed. So, the problem of finding an integral surface of the equation (9.77) passing through a given curve. Le.. to solve the Cauchy problem (see 9.2.1 1, 5., p. 516), is transformed into another problem: To find the characteristic strips passing through the points of the initial curve such that the corresponding tangent plane to each strip is tangent to that curve. We get the values p and q at the points of the initial curve from the equations F ( z ,y. 2,p, q ) = 0 and pdx qdy = d z . u'e may have several solutions in the case of non-linear differential equations. Therefore, under the formulation of the Cauchy problem. in order to obtain a unique solution we assume two continuous functions p and q satisfying the above relations along the initial curve. For the existence of solutions of the Cauchy problem see [9.15]. IFor the partial differential equation p q = 1 and the initial curve y = z3,z = 2x2, we can choose p = z and q = l / x along the curve. The characteristic system has the form

+

dz dp -dt= 2 p q , -=oo, -dq dt dt= o The characteristic strip with initial values 20.yo. zo>poand qo for t = 0 satisfies the equations z = xo + qot. y = yo + p o t . z = 2poqot zo, p = po, q = 40.For the case ofpo = 2 0 . qo = l/zo the equation of the curve belonging to the characteristic strip that passes through the point ( 2 0 ,yo, zo) of the initial curve is

dx

-

&=4,

dy -d=t p ,

+

5

= zo

+ -t

~

20

y = xO3

+ iso, z = 2t + 2zO2.

Eliminating the parameters xo and t we get z2 = 4zy. For other chosen values of p and q along the initial curve we can get different solutions. Remark: The envelope of a one-parameter family of integral surfaces is also an integral surface. Considering this fact we can solve the Cauchy problem with a complete integral. We find a one-parameter family of solutions tangent to the planes given at the points of the initial curve. Then we determine the envelope of this family. Determine the integral surface for the Clairaut differential equation z - pa: - qy pq = 0 passing through the curve y = x. z = x'. The complete integral of the differential equation is z = az by - ab. Since along the initial curve p = q = z,we determine the one-parameter family of integral surfaces by 1 the condition a = b. Finding the envelope of this family we have z = - ( E y)'. 4

+

+

+

5 . Linear First-Order Partial Differential Equations in Total Differentials Equations of this kind have the form

+

+ +

(9.78a) dz = fidxi f i d x z . . fndXn, where f l ;f 2 ; . . . f, are given functions of the variables x l r 52,.. . , x,, z . The equation is called a completely integrableor exact diferential equationwhen there exists a unique relation between ~ 1 ~ x 2. .,,x,, . z with one arbitrary constant: which leads to equation (9.78a). Then there exists a unique solution z = z(xl,x2:. . , , x,) of (9.78a), which has a given value zo for the initial values z l O ,... , z n 0of the independent variables. Therefore, for n = 2 , z1 = I,z2 = y a unique integral surface passes through every point of space. n(n - 1) The differential equation (9.78a) is completely integrable if and only if the equalities 2 ~

~

aft

dXk

afk + f k - at = afk - + f*ax, a2 aft

(i, IC = 1. . , . , n)

(9.78b)

520

9. DifferentialEquations

in all variables x l rx2>. . . , x,, z are identically satisfied. If the differential equation is given in symmetric form fldxl t ' . . t fndx, = 0, then the condition for complete integrability is

(9.78~)

(9.78d) for all possible combinations of the indices i, j, k. If the equation is completely integrable, then the solution of the differential equation (9.78a) can be reduced to the solution of an ordinary differential equation with n - 1 parameters.

9.2.2 Linear Second-OrderPartial Differential Equations 9.2.2.1 Classification and Properties of Second-Order Differential Equations with Two Independent Variables 1. General Form of a linear second-order partial differential equation with two independent variables x , y and an unknown function u is an equation in the form

a2u a2U A+ 2B-axay + C-azu + a-au + b-au +cu = f . (9.79a) 8x2 dy2 ax ay where the coefficients A , B , C, a, b, c and f on the right-hand side are known functions of z and y . The form of the solution of this differential equation depends on the sign of the dzscrzmznant ~ = A c - B ~ (9.7913) in a considered domain. We distinguish between the following cases. 1.6 < 0: Hyperbolic type. 2 . 6 = 0: Parabolic type. 3 . 6 > 0: Elliptic type. 4. 6 changes its sign: Mixed type. 5. .4n important property of the discriminant 6 is that its sign is invariant with respect to arbitrary transformation of the independent variables, e.g., to introduction new coordinates in the x , y plane. Therefore, the type of the differential equation is invariant with respect to the choice of the independent variables. 2. Characteristics of linear second-order partial differential equations are the integral curves of the differential equation dy B t G 2Bdxdy + Cdx2 = 0 or - = (9.80) A ' dx For the characteristics of the above three types of differential equations the following statements are valid: 1. Hyperbolic type: There exist two families of real characteristics. 2. Parabolic type: There exists only one family of real characteristics. 3. Elliptic type: There exists no real characteristic. 4. A differential equation obtained by coordinate transformation from (9.79a) has the same characteristics as (9.79a). 5. If a family of characteristics coincides with a family of coordinate lines, then the term with the second derivative of the unknown function with respect to the corresponding independent variable is missing in (9.79a). In the case of a parabolic differential equation, the mixed derivative term is also missing.

Ad$

-

~

9.2 Partial DifferentialEauations 521

3. Normal Form or Canonical Form IVe have the following possibilities to transform (9.79a) into the normal form of linear second-order partial differential equations. 1. Transformation into Normal Form: The differential equation (9.79a) can be transformed into normal form by introducing the new independent variables E = 4 2 , Y) and 7 = v(x, Y) , (9.81a) which according to the sign of the discriminant (9.79b) belongs to one of the three considered types: d2u 8% - -+... dE2 872

=

0, 6 < 0, hyperbolic type;

d2 U + . . . = 0,

dV2

8% at2

-

d2U ++ . . . = 0,

8172

(9.81b)

6 = 0, parabolic type;

(9.81~)

6 > 0, elliptic type.

(9.81d)

The terms not containing second-order partial derivatives of the unknown function are denoted by dots. 2. Reduction of a Hyperbolic Type Equation to Canonical Form (9.81b): If, in the hyperbolic case, we chose two families of characteristics as the coordinate lines of the new coordinate system (9.81a), Le.: if n e substitute 51 = p(x, y)) 71 = $(x, y), where p(x, y ) = constant, $(x2 y) = constant are the equations of the characteristics, then (9.79a) becomes the form

8% t . . . = 0. d51d171 This form is also called the canonicalform ofa hyperbolic type diflerential equation. From here (9'81e) we get the canonical form (9.81b) by the substitution (9.81f) 5 = E1 + 171, 17 = 51 - V I . 3. Reduction of a Parabolic Type Equation to Canonical Form (9.81~): The only family of characteristics given in this case is chosen for the family 5 = const, where an arbitrary function of s and y can be chosen for 7 . which must not be dependent on 5. 4. Reduction of an Elliptic Type Equation to Canonical Form (9.81d): If the coefficients A(s. y), B ( x ,y). C ( x ,y) are analytic functions (see 14.1.2.1,p. 670) in the elliptic case, then the characteristics define two complex conjugate families of curves p(x>y) = constant, $(x,y ) = constant. If we substitute = p + y , and 7 = i(p - $J)> the equation becomes the form (9.81d). 4. Generalized Form Every statement for the classification and reduction to canonical form remains valid for equations given in a more general form

(

"1

(9.82) A ( X , Y ) G + 2B(x,y)t C ( s . y ) 7 + F z , y , u , - - = 0, dZu dx'dy du dxdy dY where F is a non-linear function of the unknown function u and its first-order partial derivatives du/dx d2U

8221

and duldy, in contrast to (9.79a).

9.2.2.2 Classification and Properties of Linear Second-Order Differential

Equations with More than Two Independent Variables 1. General Form A differential equation of this kind has the form (9.83)

I

522

9. DifferentialEquations

where uik are given functions of the independent variables and the dots in (9.83) mean terms not containing second-order derivatives of the unknown function. In general, the differential equation (9.83) cannot be reduced to a simple canonical form by transforming the independent variables. However, there is an important classification, similar to the one introduced above in 9.2.2.1, p. 520 (see [9.5]).

2. Linear Second-OrderPartial DifferentialEquations with Constant Coefficients If all coefficients a,k in (9.83) are constants, then the equation can be reduced by a linear homogeneous transformation of the independent variables into a simpler canonical form (9.84) where the coefficients K, are &1 or 0. We can distinguish between several characteristic cases. 1. Elliptic Differential Equation .411 the coefficients K, are different from zero, and they have the same sign. Then we have an elliptic differential equation. 2. Hyperbolic and Ultrahyperbolic Differential Equation All the coefficients K, are different from zero, but one has a sign different from the other’s. Then we have a hyperbolic differential equation. If both type of signs occur at least twice, then it is an ultrahyperbolic differential equation. 3. Parabolic Differential Equation One of the coefficients n, is equal to zero, the others are different from zero and they have the same sign. Then we have a parabolic differential equation. 4. Simple Case for Elliptic and Hyperbolic Differential Equations We have a relatively simple case if not only the coefficients of the highest derivatives of the unknown function are constants, but also those of the first derivatives. Then we can eliminate the terms of the first derivatives, for which K , # 0, by substitution. For this purpose, we substitute (9.85)

au where bk is the coefficient of - in (9.84) and the summation is performed for all K , # 0. In this way.

dxk every elliptic and hyperbolic differential equation with constant coefficients can be reduced to a simple form: d2v a ) Elliptic Case: av t kv = g. (9.86) b ) Hyperbolic Case: - - Av + kv = g. (9.87)

a2

Here 4 denotes the Laplace operator (see 13.2.6.5,p. 655).

9.2.2.3 Integration Methods for Linear Second-Order Partial Differential Equations 1. Method of Separation of Variables We can determine certain solutions of several differential equations of physics by special substitutions, and although these are not general solutions, we get a family of solutions depending on arbitrary parameters. Linear differential equations, especially those of second order, can often be solved if we are looking for a solution in the form of a product 4 2 1 , . . . x n ) = IP1(.1)(P2(.2)”.’Pn(5,). (9.88) Next, we try to separate the functions $&(Zk)l i.e., for each of them we want to determine an ordinary differential equation containing only one variable x k . This separation of uarzables is successful in many cases when the trial solution in the form of a product (9.88) is substituted into the given differential equation. In order to guarantee that the solution of the original equation satisfies the required homogeneous boundary conditions, it may appear to be sufficient that some of functions pl(xl), p2(x2), . . . , pn(xn)satisfy certain boundary conditions.

9.2 Partial DifferentialEavations 523

By means of summation, differentiation and integration, new solutions can be acquired from the obtained ones; the parameters should be chosen so that the remaining boundary and initial conditions are satisfied (see examples). Finally. we must not forget that the solutions obtained in this way, often infinite series and improper integrals. are only formal solutions. That is, we have to check whether the solution makes a physical sense, e.g., whether it is convergent, satisfies the original differential equation and the boundary conditions, whether it is differentiable termwise and whether the limit at the boundary exists. The infinite series and improper integrals in the examples of this paragraph are convergent if the functions defining the boundary conditions satisfy the required conditions, e.g., the continuity assumption for the second derivativs in the first and the second examples. IA: Equation of the Vibrating String is a linear second-order partial differential equation of hyperbolic type (9.89a) It describes the vibration of a spanned string. The boundary and the initial conditions are: (9.89b)

(9.89~) and after substituting it into the given equation (9.89a) we have (9.89d) The variables are separated, the right side depends on only x and the left side depends on only t , so each of them is a constant quantity. This constant must be negative, otherwise the bondary conditions cannot be satisfied. We get an ordinary linear second-order differential equation with constant coefficients for both variables. For the general solution see 9.1.2.4, p. 500. We denote this negative constant by -A2 and we get the linear differential equations

X " t xzx = 0,

(9.89e)

T f f+ aZX2T= 0.

(9.894

We have X ( 0 ) = X ( 1 ) = 0 from the boundary conditions. Hence X ( x ) is an eigenfunction of the Sturm-Liouville boundary value problemand is the correspondingeigenvalue (see 9.1.3.1,3., p. 513). Solving the differential equation (9.89e) for X with the corresponding boundary conditions we get nr X(z) = Csin Xz with sinX1 = 0, i.e., with X = (939g) 1 = An ( n = 1,2.. . .). Solving equation (9 89f) for T yields a particular solution of the original differential equation (9.89a) for every eigenvalue A,: naT t b, sin (9.89h) u, = (a, cos Requiring that for t = 0, m

u=

u,

is equal to f(x)

(9.89i) and

n=l

d "

-

u,

is equal to p(x),

(9.89j)

n=l

we get with a Fourier series expansion in sines (see 7.4.1.1. l., p. 418) a

" - 1

nirx 2 [f ( z )sin -dz, I 0

bn =

(9.89k)

I

524

9. Diferential Equations

IB: Equation of Longitudinal Vibration of a Bar is a linear second-order partial differential equation of hyperbolic type, which describes the longitudinal vibration of a bar with one end free and a constant force p affecting the fixed end. We have to solve the same differential equation as in example A (p. 523), Le., (9.90a) with the same initial but different boundary conditions: =0

(free end),

(9.9Oc)

(9.90d) These conditions can be replaced by the homogeneous conditions (9.90e) where instead of u we introduce a new unknown function kpxZ z=u------. 21 The differential equation becomes inhomogeneous: - = 02at2

ax2

+ -,

(9.9Of)

i

We are looking for the solution in the form 2 = u t w ,where w satisfies the homogeneous differential equation with the initial and boundary conditions for z , Le., (9.90h) and w satisfies the inhomogeneous differential equation with zero initial and boundary conditions. So, ka2ptz we get w = . Substituting the product form into the differential equation 21 21 = X ( x ) T ( t ) (9.9Oi) we get the separated ordinary differential equations as in the example A (p. 523)

-

_--

(9.9Oj)

Integrating the differential equation for X with the boundary conditions X f ( 0 ) = X f ( l ) = 0 we find the eigenfunctions nnx = cos (9.90k) 1 and the corresponding eigenvalues

x,

An2=T

n2T2

(n=0.1,2,~..).

(9.901)

Proceeding as in example A (p. 523) we finally obtain (9.90m)

9.2 Partial DifferentialEauations 525

where a, and b, ( n = 0 , 1 , 2 , ,, .) are the coefficients of the Fourier series expansion in cosines of the kpx2 1 functions f(x) and -p(x) in the interval ( 0 , l ) (see 7.4.1.1, l., p. 418). 2 aT IC: Equation of a Vibrating Round Membrane fixed along the boundary: The differential equation is linear, partial and it is of hyperbolic type. It has the form in Cartesian and in polar coordinates (see 3.5.3.1, 6., p. 209) ~

a z u +--++-=-iou 1 aZU -

(9.91a)

ap2

pap

p2a$

a2

a” a?

(9.91b) ’

The initial and boundary conditions are (9.91~)

= f(p.p),

fi~ = F(p,cp),

u l p == ~ 0.

(9.91d)

(9.91e)

t=O

The substitution of the product form (9.91f)

u = L’(P)@(P)T(t)

with three variables into the differential equation in polar coordinates yields

Three ordinary differential eauations are obtained for the seDarated variables analogously ” to examdes A (p. 523) a n i B (p. 524): ~

T”

+ a2X2T= 0.

p2U”

(9.91h)

+ pU’ U

2 2

@I1

+Xp=--=

~

@// + v% = 0. From the conditions @(O) = @(27~),@(O) = @ ( 2 ~it) follows that:

@(p) = a, cos np

+ b, sin np,

P

(9.91i) (9.91j)

u2 = n2 ( n = 0 , 1 , 2 , . . .).

n2 U and X will be determined from the equations [pU’]’ - -Lr

u2,

(9.91k)

= -X2pU

and U ( R ) = 0. Considering

the obvious condition of boundedness of U(p) at p = 0 and substituting Xp = z we get z2C”’

+ ZC“

t ( z 2- n2)U = 0,

( 3,

Le., U(p) = J,(z) = J, p-

(9.911)

P and Jn(p) = 0. The function where J,, are the Bessel functions (see 9.1.2.6, 2., p. 507) with X = R system unk(p) = Jn

(pnki)

(9.91m)

( k = 1,2, ’ ’

with pL,k as the Ic-th positive root of the function J,(z) is a complete system of eigenfunctions of the self-adjoint Sturm-Liouville problem which are orthogonal with the weight function p. The solution of the problem can have the form of a double series:

L’

=

E n=O k=l

[(a,, cos np + b,k sin np) cos R a h k t

+(c,k

3

cos np + d,k sin np) sin - J, pnh-

R

(

(9.91n)

9. Differential Equations

526

(9.910)

n=O k=l

R

where

(9.9 1r) In the case of n = 0, the numerator 2 should be changed to 1. To determine the coefficients c,k and dnk R we replace f ( p , p) by F(p. 9)in the formulas for ank and b,k and we multiply by ~

aPnk

ID: Dirichlet Problem (see 13.5.1, p. 668) for the rectangle 0 5 z 5 a, 0 5 y 5 b (Fig. 9.17): Find a function u(z,y) satisfying the elliptic type Laplace differential equation Au=O (9.92a) and the boundary conditions U ( 0 , Y ) = Pl(Y), 4 % Y ) = VZ(Y)> (9.92b) u(z,o)= q ~ ( z )u(z,b) , = qz(2).

Figure 9.17

First we determine a particular solution for the boundary conditions ipl(y) = p2(y) = 0. Substituting the product form u = X(z)Y(y) (9.92~) into (9.92a) we get the separated differential equations X// -- -Y _I' = - A 2 (9.92d)

X

Y

with the eigenvalue X analogously to examples A (p. 523) through C (p. 525). Since X ( 0 ) = X ( a ) = 0. we get n7i

X = Csin Ax.

X = - = X, a

( n = 1 , 2 , . . .).

(9.92e)

In the second step we write the general solution of the differential equation n2r2 n7i nr in the form Y = a, sinh - ( b - y) + b, sinh -y . (9.92g) - -y = 0 (9.920 02 a a From these equations we get a particular solution of (9.92a) satisfying the boundary conditions u(0. y) = u(u. y) = 0. which has the form u, = [a. sinh

a(b- y) + b, sinh n7i

(9.92h)

In the third step we consider the general solution as a series X

u=

u., n=l

(9.92i)

9.2 Partial Differential Equations 527

so from the boundary conditions for y = 0 and y = b we get u=

5

nT

- y)

(a,sinh -(b

+ b, sinh

(9.92j)

n=l

with the coefficients 2 a, = -/'v1(x)sin--2dx, nTb 0 a sinh -

nir

b, =

a

2

nTb a sinh -

~

la

n?r

&(z) sin -x

dx.

(9.92k)

Q

The problem with the boundary conditions yll(z) = yz ( z )= 0 can be solved in a similar manner, and taking the series (9.92j) we get the general solution of (9.92a) and (9.92b). IE: Heat Conduction Equation Heat conduction in a homogeneous bar with one end at infinity and the other end kept at a constant temperature is described by the linear second-order partial differential equation of parabolic type

which satisfies the initial and boundary conditions (9.93b) U l t d = f(.). u/,,o = 0 in the domain 0 5 x < +coo,t 2 0. We also suppose that the temperature tends to zero a t infinity. Substituting 11 = X ( z ) T ( t )

(9.93c)

into (9.93a) we obtain the ordinary differential equations (9.93d) whose parameter X is introduced analogously to the previous examples A (p. 523) through D (p. 526). We get

~ ( t=)CAe-xinit as a solution for T ( t ) .Csing the boundary condition X(0)= 0, we get

(9.93e)

X(x) = Csin Xx (9.930 and so u~ = CAe-A2a2t sin Xz, where X is an arbitrary real number. The solution can be obtained in the form

(9.934

u ( z .t ) =

/

r

C(X)e-A2a2t sin ~x d ~ .

(9.93h)

From the initial condition u l t = ~= f ( x ) $so

f(x) =

1-

C(X)sin ~x d ~ ,

(9.931)

which is satisfied if we substitute C(X) = z / m f ( s ) s i n X s d s

(9.93j)

T O

for the constant (see 7.4.1.1, 1..p. 418). Combining this equation and (9.93i) we get

~ ( 2 . t=) 'T/ =Of ( s )

(/me-"?"ztsinXssinXxdX) d s

(9.93k)

I

528

9. DifferentialEquations

or after replacing the product of the two sines with one half of the difference of two cosines ((2.122), p. 81) and using formula (21.27), p. 1052, we get m

(9.931)

2. Riemann Method for Solving Cauchy’s Problem for the Hyperbolic

Differential Equation

au + b-au +cu = F

a221

-t adxdy 8s

(9.94a)

ay

1. Riemann Function is a function u(x, y; (, q ) ,where ( and q are considered as parameters, satisfying the homogeneous equation 81; d(av) d(bv) (9.94b) +cv=o dxdy ax dy which is the adjoint of (9.94a) and the conditions

In general, linear second-order differential equations and their adjoint differential equations have the form 82U v) d(bu) atkb,+ cu = f (9.94d) and ,,k @(a - -2- + cv = 0. (9.94e) z,k dxtdxk , dxt dX&k , 8x1

+

au

2. Riemann Formula is the integral formula which is used to determine function .(E, 7) satisfying the given differential equation (9.94a) and taking the previously given values along the previously given curve (Fig. 9.18) together with its derivative in the direction of the curve normal (see 3.6.1.2, 2., p. 226):

‘6

r

w Figure 9.18 The smooth curve T (Fig. 9.18) must not have tangents parallel to the coordinate axes, i.e., the curve must not be tangent to the characteristics. The line integral in this formula can be calculated, since the values of both partial derivatives can be determined from the function values and from its derivatives in a non-tangential direction along the curve arc. dU

In the Cauchy problem, the values of the partial derivatives of the unknown function, e.g., - are often

3Y

given instead of the normal derivative along the curve. Then we use another Riemann formula:

9.2 Partial Differential Esuations 529

IElectric Circuit Equation (Telegraphic Equation) is a linear second-order partial differential equation of hyperbolic type

azU

au

at2

at

d2U

(9.95,) 8x2 where a > 0, b. and c are constants. The equation describes the current flow in wires. It is a generalization of the differential equation of a vibrating strin SVe replace the unknown function u(x,t ) by u = ze-(bTi)t, Then (9.95a) is reduced to the form a-++b-+cu=-

a22

d2Z

- = m2at2 ax2

+ n2z

( -,i m2 =

n 2--)

b2 - ac a2

(9.95b) '

Replacing the independent variables by (9.95c)

{=!(mt+x). q=!(mt-x) m m we finally get the canonical form

(9.95d) of a hyperbolic type linear partial differential equation (see 9.2.2.1, l., p. 521). The Riemann function v({. q; Eo, 70) should satisfy this equation with unit value a t = Eo and 7 = 70. If we choose the form (9.95e) = (E - Eo)(a - 70) for w in 2: = f ( w ) . then f(w) is a solution of the differential equation

d2f + df - -1f = 0 (9.959 dw2 dw 4 with initial condition f ( 0 ) = 1. The substitution w = a' reduces this differential equation to Bessel's differential equation of order zero (see 9.1.2.6, 2., p. 507) 2L'-

-d2f +

1 df - f = 0 -da2 cvda hence the solution is 2: =

lo

[J C Z G G ] .

(9.95h)

A solution of the original differential equation (9.958) satisfying the boundary conditions

can be obtained if we substitute the found value of u into the Riemann formula and then return to the original variables: 1 z(x,t ) = 2[f ( x - mt) f(. mt)]

+

+

ntIl x-mt

m

-'(')

('r -

-J

m2tZ

-

(s

- z)2)

]

ds.

(9.95j)

3. Green's Method of Solving the Boundary Value Problem for Elliptic Differential Equations with Two Independent Variables This method is very similar to the Riemann method of solving the Cauchy problem for hyperbolic differential equations.

I

530

9. Di.flerential Equations

If we want to find a function u(z, y) satisfying the elliptic type linear second-order partial differential equation au au d2u a2u (9.96a) f -t-ta-tb-tcu= ax2 dy2 ax dy in a given domain and taking the prescribed values on its boundary, first we determine the Green functzon G ( s . y , I ,q ) for this domain, where E and q are regarded as parameters. The Green function must satisfy the following conditions: 1. The function G(x, y: E. q ) satisfies the homogeneous adjoint differential equation

d(aG) d(bG) + c G = O dx ay everywhere in the given domain except at the point z = I , y = q. 2. The function G(x, y: E , q ) has the form a2G

d2G

azz

ay2

1 UIn- t 1.'

with r = d ( z - E)'

(9.96~)

r

(9.96b)

+ (Y - v ) ~ .

(9.96d)

where G has unit value at the point z = I . y = 7 and C and L' are continuous functions in the entire domain together with their second derivatives. 3. The function G(z. y; E , q ) is equal to zero on the boundary of the given domain. The second step is to give the solution of the boundary value problem with the Green function by the

where D is the considered domain, S is its boundary on which the function is assumed to be known and

a

- denotes the normal derivative directed toward the interior of D. dn

Condition 3 depends on the formulation of the problem. For instance, if instead of the function values the values of the derivative of the unknown function in the direction normal to the boundary of the domain are given, then in 3 we have the condition

_ aG _( a cos a + bcos ,!?)G= 0

(9.96f) dn on the boundary. (Y and /j' denote here the angles between the interior normal to the boundary of the domain and the coordinate axes. In this case, the solution is given by the formula 1 au 1 u ( < , q )= --/-Gds2a fGdzdy. (9.9%) 27 dn

-11

S

D

4. Green's Method for the Solution of Boundary Value Problems with Three

Independent Variables The solution of the differential equation au au du Auta-tb-tc-teu= f

ax ay az

(9.97a)

should take the given values on the boundary of the considered domain. As the first step, we construct again the Green function, but now it depends on three parameters E , q, and C. The adjoint differential equation satisfied by the Green function has the form (9.97b)

9.2 Partial DifferentialEouations 531

As in condition 2, the function G(x,y, z; E, q, C) has the form

c-T1 + L’

(9.97c)

with r = d ( z -

+ (y - q)’ + ( z - 02.

(9.97d)

The solution of the problem is:

Both methods, Riemann’s and Green’s, have the common idea first to determine a special solution of the differential equation, which can then be used to obtain a solution with arbitrary boundary conditions. An essential difference between the Riemann and the Green function is that the first one depends only on the form of the left-hand side of the differential equation, while the second one depends also on the considered domain. Finding the Green function is, in practice, an extremely difficult problem, even if it is known to exist; therefore, Green’s method is used mostly in theoretical research. IA: Construction of the Green function for the Dirichlet problem ‘Y of the Laplace differential equation (see 13.5.1, p. 668) Au=O (9.98a) for the case, when the considered domain is a circle (Fig. 9.19). The Green function is

1

G ( z ,y ; q ) = In r

!

Figure 9.19

m,

+ In -,R

rlp

(9.98b)

and R is the radius of the where T = p = OM,r1 = considered circle (Fig. 9.19). The points M and MIare symmetric with respect to the circle, Le., both points are on the same ray starting from the center and -O M .OM1 = R2. (9.98~) The formula (9.96e) for a solution of Dirichlet’s problem, after substituting the normal derivative of the Green function and after certain calculations, yields the so-called Poisson integral (9.98d)

The notation is the same as above. The known values of u are given on the boundary of the circle by ~(9) For . the coordinates of the point M ( < , q )we have: E = pcos$, q = psinv. H B: Construction of the Green function for the Dirichlet problem of the Laplace differential equation (see 133.1)p. 668) Au = 0. (9.99a) for the case when the considered domain is a sphere with radius R. The Green function now has the form 1 R (9.99b) G(z, Y, 2; €, v>0= ;- Tlp ?

with p = Jt2+ q2 + C2 as the distance of the point (t,q, C) from the center. T as the distance between the points (2.y. t)and (5, q1C). and r1 as the distance of the point (z,y , z ) from the symmetric point

(E, q>C) according to (9.98c), Le., from the point

-Rq RC . In this case, the Poisson integral ( P ) Pj P I has (with the same notation as in example A (p. 531))the form

of

(9.99c)

532

9. Differential Equations

5 . O p e r a t i o n a l Method Operational methods can be used not only to solve ordinary differential equations but also for partial differential equations (see 15.1.6, p. 707). They are based on transition from the unknown function to its transform (see 15.1, p. 705). In this process, we regard the unknown function as a function of only one variable and we perform the transformation with respect to this variable. The remaining variables are considered as parameters. The differential equation to determine the transform of the unknown function contains one less independent variable than the original equation. In particular, if the original equation is a partial differential equation of two independent variables, then we obtain an ordinary differential equation for the transform. If we can find the transform of the unknown function from the obtained equation, then we determine the original function either from the formula for the inverse function or from the table of transforms. 6. A p p r o x i m a t i o n Methods In order to solve practical problems with partial differential equations, we often use different approximation methods. We distinguish between analytical and numerical methods. 1. Analytical Methods make possible the determination of approximate analytical expressions for the unknown function. 2. Numerical Methods result in approximate values of the unknown function for certain values of the independent variables. We use the following methods (see 19.5. p. 908): a ) Finite Difference Method, or Lattzce-Pod Method The derivatives are replaced by divided differences, so the differential equation including the initial and boundary conditions becomes an algebraic equation system. A linear differential equation with linear initial and boundary conditions becomes a linear equation system. b) Finite Element Method, or briefly FEM, for boundary value problems: We assign a variational problem to the boundary value problem. We approximate the unknown function by a spline, whose coefficients should be chosen to get the best possible solution. We decompose the domain of the boundary value problem into regular subdomains. The coefficients are determined by solving an extreme value problem. c) Integral Equation Method (along a Closed Curve) for special boundary problems: We formulate the boundary value problem as an equivalent integral equation problem along the boundary of the domain of the boundary value problem. To do this, we apply the theorems of vector analysis, e.g., Green formulas. We determine the remaining integrals along the closed curve numerically by a suitable quadrature formula. 3. Physical Solutions of differential equations can be given by experimental methods. This is based on the fact that various physical phenomena can be described by the same differential equation. To solve a given equation, we first construct a model by which we can simulate the given problem, and we obtain the values of the unknown function directly from this model. Since such models are often known and can be constructed by varying the parameters in a wide range, the differential equation can also be investigated in a wide domain of the variables.

9.2.3 Some further Partial Differential Equations from Natural Sciences and Engineering 9.2.3.1 Formulation of the Problem and the Boundary Conditions 1. P r o b l e m F o r m u l a t i o n The modeling and the mathematical treatment of different physical phenomena in classical theoretical physics, especially in modeling media considered structureless or continuously changing, such as gases, fluids, solids, the fields of classical physics, leads to the introduction of partial differential equations. Examples are the wave (see 9.2.3.2,p. 534) and the heat equations (see 9.2.3.3,p. 535). Many problems in non-classical theoretical physics are also governed by partial differential equations. An important

9.2 Partial DifferentialEouations 533

area is quantum mechanics, which is based on the recognition that media and fields are discontinuous. The most famous relation is the Schrodinger equation. Linear second-order partial differential equations occur most frequently and they have special importance in today's natural sciences.

2. Initial and Boundary Conditions The solution of the problems of physics, engineering, and the natural sciences must usually fulfill two basic requirements: 1. The solution must satisfy not only the differential equation, but also certain initial and/or boundary conditions. There are problems with only initial condition or only with boundary conditions or with both. A11 the conditions together must determine the unique solution of the differential equation. 2. The solution must be stable with respect to small changes in the initial and boundary conditions, Le.. its change should be arbitrarily small if the perturbatzons of these conditions are small enough. Then a correct problemformulatzon is given. IVe can assume that the mathematical model of the given problem to describe the real situation is adequate only in cases when these conditions are fulfilled. For instance, the Cauchy problem (see 9.2.1.1,5.,p. 516) iscorrectlydefined withadifferentialequation of hyperbolic type for investigating vibration processes in continuous media. This means that the values of the required function, and the values of its derivatives in a non-tangential (mostly in a normal) direction are given on an initial manifold, Le., on a curve or on a surface. In the case of differential equations of elliptic type, which occur in investigations of steady state and equilibrium problems in continuous media, the formulation of the boundary value problem is correct. If the considered domain is unbounded, then the unknown function must satisfy certain given properties with unlimited increase of the independent variables.

3. Inhomogeneous Conditions and Inhomogeneous Differential Equations The solution of homogeneous or inhomogeneous linear partial differential equations with inhomogeneous initial or boundary conditions can be reduced to the solution of an equation which differs from the original one only by a free term not containing the unknown function, and which has homogeneous conditions. It is sufficient to replace the original function by its difference from an arbitrary twice differentiable function satisfying the given inhomogeneous conditions. In general. we use the fact that the solution of a linear inhomogeneous partial differential equation with given inhomogeneous initial or boundary conditions is the sum of the solutions of the same differential equation with zero conditions and the solution of the corresponding homogeneous differential equation with the given conditions To reduce the solution of the linear inhomogeneous partial differential equation d2 21 - - L[u]= g(s, t ) at2

(9.100a)

with homogeneous initial conditions (9.1OOb) t=O

to the solution of the Cauchy problem for the corresponding homogeneous differential equation, we substitute t

p(x2, t: 7 ) dr

u=

(9.1OOc)

0

Here p(x, t ;7) is the solution of the differential equation

_a2 U_ at2

L[u]= 0,

(9.100d)

534

9. Dzfferential Equations

which satisfies the boundary conditions

In this equation, z represents symbolically all the n variables

~

2

. . ,~2, .of the n-dimensional probdU

lem. L[u]denotes a linear differential expression, which may contain the derivative -. but not higherat order derivatives with respect to t .

9.2.3.2 Wave Equation The extension of oscillations in a homogeneous media is described by the wawe equation

8%

- - u2Au = Q ( x .t ) , (9.101a) at2 whose right-hand side Q ( x , t )vanishes when there is no perturbation. The symbol z represents the n variables zl,. , , ,zn of the n-dimensional problem. The Laplace operator A (see also 13.2.6.5, 655,) is defined in the following way:

(9.101b) The solution of the wave equation is the wavefunctzon u. The differential equation (9.lOla) is of hyperbolic type.

1. Homogeneous Problem The solution of the homogeneous problem with Q(z,t ) = 0 and with the initial conditions (9.102) is given for the cases n = 1.2,3 by the following integrals. Case n = 3 (Kirchhoff Formula):

where the integration is performed over the spherical surface Sat given by the equation (a1- x 1 ) 2+ (a2- ~ 2 +)(a3 ~ - x3)2= a2tz. Case n = 2 (Poisson Formula):

(9.103b) where the integration is performed along the circle Cat given by the equation a2tz. Case n = 1 (d'Alembert formula): u(q.t ) =

~ ( Z+ I at)

+ p(z1 2

- at)

+

'Yc:(a) da. 2azl-at

+ (a2- z2)25

(cy1 - z1)2

(9.103~)

9.2 Partial Differential Eouations 535

2. Inhomogeneous Problem In the case, when Q ( z ,t ) # 0, we have to add to the right-hand sides of (9.103a.b,c) the correcting terms: Case n = 3 (Retarded Potential): For a domain K given by r 5 at with

r = d((1- ~

1

+ (& - ~ 2 + )(E3 ~- z3)2sthe correction term is )

~

(9.104a)

(9.104b) where K is a domain of [ I 3 5 a2(t- 7 ) ' .

2 , space ~

defined by the inequalities 0 5

T

5 t . (E1 - x 1 ) 2+ (Ez - x2)2 5

(9.104~) where T is the triangle 0 5

7

5 t , It - x l / 5 alt - 71. a denotes the wave velocity of the perturbation.

9.2.3.3 Heat Conduction and Diffusion Equation for Homogeneous Media 1. Three-Dimensional Heat Conduction Equation The propagation of heat in a homogeneous medium is described by a linear second-order partial differential equation of parabolic type aU

- - aZAu= Q ( z , t ) , (9.105s) at where 4 is the three-dimensional Laplace operator defined in three directions of propagation 21, 2 2 . z3,determined by the position vector 3. If the heat flow has neither source nor sink, the right-hand side vanishes since Q ( x It ) = 0. The Cauchy problem can be posed in the following way: h'e want to determine a bounded solution u(2.t ) for t > 0, where ult=O= f ( 2 ) . The requirement of boundedness guarantees the uniqueness of the solution. For the homogeneous differential equation with Q(2,t ) = 0. we get the uavefunctzon tm+mtm

(9.105b) In the case of an inhomogeneous differential equation with Q(2,t) side of (9.105b) the following expression:

# 0, we have to add to the right-hand

(9.105~)

I

536

9. Dzferential Equations

The problem of determining u(z,t ) for t < 0, if the values u ( z ,0) are given, cannot be solved in this way, since the Cauchy problem is not correctly formulated in this case. Since the temperature difference is proportional to the heat, we often introduce u = T(?,t ) (temperature field) and a2 = Dw (heat diffusion constant or thermal conductivity) to get

m

- - D w A T = Qw(F, t ) .

dt 2. Three-Dimensional Diffusion Equation

(9.105d)

In analogy to the heat equation, the propagation of a concentration C in a homogeneous medium is described by the same linear partial differential equation (9.105a) and (9.105d), where Dw is replaced by the three-dzmensional daffusion coeficaent Dc.The daffusaon equation is: (9.106) We get the solutions by changing the symbols in the wave equations (9.105b) and (9.105~).

9.2.3.4 Potential Equation The linear second-order partial differential equation

Au = -4TQ (9.107a) is called the potential equation or Poisson differential equation (see 13.5.2, p. 668), which makes the determination of the potential u ( z )of a scalar field determined by a scalar point function p(z) possible, where z has the coordinates 21, Q, 2 3 and A is the Laplace operator. The solution, the potential u M ( z lz2, , z3)at the point M ,is discussed in 13.5.2, p. 668. We get the Laplace differential equation (see 13.5.1, p. 667) for the homogeneous differential equation with e 0 h u = 0. (9.107b) The differential equations (9.107a) and (9.107b) are of elliptic type. 9.2.3.5 Schrodinger’sEquation 1. Notion of the Schrodinger Equation 1. Determinationand Dependencies The solutions of the Schrodinger equation, the wauefunctions ql describe the properties of a quantum mechanical system, Le., the properties of the states of a particle. The Schrodinger equation is a second-order partial differential equation with the second-order derivatives of the wave function with respect to the space coordinates and first-order with respect to the time coordinate: d@ h2 (9.108a) + U(zl,zz,z3,t )$ = fi $J ih- = --A$ dt 2m -2 h d h (9.108b) H = P + U ( i , t ) , $E7- E-V, 2m I di i h Here, A is the Laplace operator, h = - is the reduced Planck’s constant, i is the imaginary unit and 277

V is the nabla operator. The relation between the impulse p of a free particle with mass m and wave length X is X = h / p . 2. Remarks: a) In quantum mechanics, we assign an operator to every measurable quantity. The operator occurring in (9.108a) and (9.108b) is called the Hamilton operator H (“Hamiltonian”). It has the same role as the Hamilton function of classical mechanical systems (see, e.g., the example on Two-Body Problem on p. 518). It represents the total energy of the system which is divided into kinetic and potential energy. The first term in H is the operator for the kinetic energy, the second one for the potential energy.

9.2 Partial Differential Eauations 537

b) The imaginary unit appears explicitly in the Schrodinger equation. Consequently, the wave functions are complex functions. Both real functions occurring in $ J ( ~ ) + are needed to calculate the observable quantities. The square lPIz of the wave function, describing the probability dw of the particle being in an arbitrary volume element dV of the observed domain, must satisfy special further conditions. c) Besides the potential of the interaction, every special solution depends also on the initial and boundary conditions of the given problem. In general, we have a linear second-order boundary value problem, whose solutions have physical meaning only for the eigenvalues. The squares of the absolute value of meaningful solutions are everywhere unique and regular, and tend to zero at infinity. d) The microparticles also have wave and particle properties based on the wave-particle duality, so the Schrodinger equation is a wave equation (see 9.2.3.2, p. 534) for the De Broglie matter waves. e ) The restriction to the non-relativistic case means that the velocity Y of the particle is very small with respect to the velocity of light c (v < c). The application of the Schrodinger equations is discussed in detail in the literature of theoretical physics (see, e.g., [9.15].[9.7],[9.10],[22.13]).In this chapter we demonstrate only some most important examples.

2. Time-Dependent Schrodinger Equation The time-dependent Schrodinger equation (9.108a) describes the general non-relativistic case of a spinless particle with mass m in a position-dependent and time-dependent potential field U ( x l , x 2 , 2 3 ,t ) . The special conditions, which must be satisfied by the wave function, are: a) The function $ must be bounded and continuous. b) The partial derivatives d $ / d q , d i / d x 2 , and d$~/ax3must be continuous. c) The function IyI2 must be integrable, Le.,

According to the normalization condition, the probability that the particle is in the considered domain must be equal to one. (9.109a) is sufficient to guarantee the condition, since multiplying $ by a constant the value of the integral becomes one. A solution of the time-dependent Schrodinger equation has the form -iEt (9.109b) v(Zl,%%t) = @(x1,x~,x3)e h . The state of the particle is described by a periodic function of time with angular frequency w = E/h. If the energy of the particle has the fixed value E = constant, then the probability dw of finding the particle in a space element dV is independent of time: (9.109c) dw = l,i12d V = $$* dV. Then we talk about a stationary state of the particle. 3. Time-Independent Schrodinger Equation If the potential U does not depend on time, Le., U = U(x1,x2,x3),then it is the time-independent Schrodinger equation and the wave function P ( Z ~ , Q ~is X sufficient ~) to describe the state. We can reduce it from the time-dependent Schrodinger equation (9.108a) with the solution (9.109b) and we get 2m (9.11Oa) AP + -(E - U ) P = 0 h2 In this non-relativistic case, the energy of the particle is 2

E=!.-

2m

(9.11Ob)

538

9. DifferentialEquations

h with impulse p = -. The wave functions P satisfying this differential equation are the eigenjunctions;

x

they exist only for certain energy values E , which are given for the considered problem of the special boundary conditions. The union of the eigenvalues forms the energy spectrum of the particle. If U is a potential of finite depth and it tends to zero at infinity, then the negative eigenvalues form a discrete spectrum. If the considered domain is the entire space, then it can be required as a boundary condition that P is quadratically integrable in the entire space in the Lebesgue sense (see 12.9.3.2,p. 635 and [&SI). If the domain is finite, e.g.. a sphere or a cylinder, then we can require, e.g., P = 0 for the boundary as the first boundary condition problem. We get the Helmholtz dzfeerential equation in the special case of U ( z )= 0:

2mE with the eigenvalue X = h2

(9.111b)

'

P

= 0 is often required here as a boundary condition. (9.111a) represents the initial mathematical equation for acoustic oscillation in a finite domain.

4. Force-Free Motion of a Particle in a Block 1. Formulation of t h e Problem A particle with a mass m is moving freely in a block with inpenetrable walls of edge lengths a, b, c, therefore, it is in a potential box which is infinitely high in all three directions because of the inpenetracy of the walls. That is, the probability of the presence of the particle. and also the wave function P,vanishes outside the box. The Schrodinger equation and the boundary conditions for this problem are -+ -+ -+ -EP 3x2 dy2 dz2 A2

= 0 , (9.112a)

P = 0 for

Solution Separating the variables @(IC, Y,z ) = @&) P,(Y)Pz(z) and substituting into (9.112a) we get

z = o , IC=a. y = 0, y = b, z=o, z=c.

(9.112b)

2.

(9.113a)

1 d2Pz 1 d2Py 1 d2Pz 2m -(9.113b) P= d22 + -~ + -- = --E = -B. Py dy2 P, dz2 h2 Every term on the left-hand side depends only on one independent variable. Their sum can be a constant -B for arbitrary IC, y, z only if every single term is a constant. In this case the partial differential equation is reduced to three ordinary differential equations: d2Pz @Pz d2@, - -k y2Pv,-= -kz2Pz (9.113~) - -kx 2 P,, dx2 dY dz2

The relation for the separation constants - k Z 2 , -k,',

+ +

kx2 kY2 k z 2 = B , consequently h2 E = -(kx2 + kY2+ k,').

-kZ2 is (9.113d)

(9.113e) 2m 3. Solutions of the three equations (9.113~)are the functions (9.114a) Pz= .4, sin k , IC, P, = A, sink, y. Pz= A, sin k, z with the constants A,. A,, A,. With these functions P satisfies the boundary conditions P = 0 for z = 0, y = Oand z = 0. sin k , a = sink, b = sin k , c = 0 (9.114b)

9.2 Partial DifferentialEauations 539

must be valid to satisfy also the relation !P = 0 for x = a, y = b and z = c, Le., the relations k - 3 k - T 2n k - - Tn, (9.114c) z - o ' Y - b ' z - c must be satisfied, where n,, ny. and n, are integers. \Ve get for the total energy

"." [ (?)' t (

E,,,,,,,, = 2m

+ (?)

2]

?)2

(% ny, n, = il, *2, . . .)

(9.114d)

It follows from this formula that the changes of energy of a particle by interchange with the neighborhood is not continuous, which is possible only in quantum systems. The numbers n,, ny, and n,, belonging to the eigenvalues of the energy, are called the quantum numbers. After calculating the product of constants A,AyA, from the normalization condition

111

(,4z~4yA,)2

000

m,x m,y . m 8 z sin2 -sin' sin -dx dy d z = 1 b ~

(9.114e)

we get the complete eigenjunctions of the states characterized by the three quantum numbers -

J-

8 , m Z x , m Y y , m,z (9.114f) sin -sin sin b - abc a C The eigenfunctions vanish at the walls since one of the three sine functions is equal to zero. This is always the case outside the walls if the following relations are valid a 20 (n, - 1). x=--. -...., n, n, n, 2b , . . . . (nu- l)b y=-b , (9.114g) nu nY nY c 2c (n, - l ) c z=--, n, nz n, So. there are n, - 1 and n, - 1 and n, - 1 planes perpendicular to the 2- or y- or z-axis, in which @ vanishes. These planes are called the nodal planes. 4. Special Case of a Cube, Degeneracy In the special case of a cube with a = b = c, a particle can be in different states which are described by different linearly independent eigenfunctions and they have the same energy. This is the case when the sum n,' t ny2t nZ2has the same value in different states. \Ye call them degenerate states, and if there are z states with the same energy, we call it 2-fold degeneracy. The quantum numbers n,, ny and n, can run through all real numbers, except zero. This last case would mean that the wave function is identically zero, i.e.. the particle does not exist a t any place in the box. The particle energy must remain finite. even if the temperature reaches absolute zero. This zero-poznt translatzonalenergy for a block is E+-('+-+-) h2 1 1 (9.114h) 2m a2 b2 c2 @nz,n,,,nz -

~

- . . . . )

5. Particle Movement in a Symmetric Central Field (see 13.1.2.2, p. 641) 1. Formulation of the Problem The considered particle moves in a central symmetric potential V ( r ) . This model reproduces the movement of an electron in the electrostatic field of a positively charged nucleus. Since this is a spherically symmetric problem, it is reasonable to use spherical coordinates (Fig. 9.20). \Ye have the relations

540

9. DifferentialEquations

r = J-,

x = rsindcosp,

t9 = arccos zT ,

y = rsinflsinp,

Y (0 = arctan - ,

z = r cos 19,

u.

X

(9.115a)

where r is the absolute value of the radius vector, 19 is the angle between the radius vector and the z-axis (polar angle) and p is the angle between the projection of the radius vector onto the z, y plane and the x-axis (azimuthal angle). For the Laplace operator we get

x

Figure 9.20

m

281 1 &p=-++-+--++Tar

cost9 1 a21 r2sin'at9+mv'

w

(9.1l5b)

so the time-independent Schrodinger equation is:

2. Solution We are looking for a solution in the form

(9.116a) @ ( T , 27,(0) = R~(r)k;~(', 9)) where Rl is the radial wave function depending only on r , and k;"(d, p) is the wave function depending on both angles. Substituting (9.116a) in (9.115~)we get

$;

( T ' Z )

qrn+ p2m[

E- V(T)]Rlk;rn (9.116b)

Dividing by RlKrnand multiplying by r' we get

R1 dr Equation (9.116~)can be satisfied if the expression on the left-hand side depending only on r and expression on the right side depending only on t9 and p are equal to a constant, Le., both sides are independent of each other and they are equal to the same constant. From the partial differential equation we get two ordinary differential equations. If the constant is chosen equal to 1(1 f l ) , then we get the so-called radzal equation depending only on T and the potential V ( r ) : (9.116d) We want to find a solution for the angle-dependent part also in the separated form k;"(',V) = Q ( W ( V ) . Substituting (9.116e) into (9.116~)we get 1 d'0 + 1 ( 1 + 1) = ---. sint9@dp2

( $)

}

(9.116e)

(9.116f)

If the separation constant is chosen as m2 in a reasonable way, then the so-called polar equation is 1 d , d Q m' (9.116g) -(sint9%) + l ( l + 1) - -- 0 sin' 29 Qsint9d6

9.2 Partial Differential Equations 541

and the azimuthal equation is d2@ -+ m2@ = 0. (9.116h) dq2 Both equations are potential-independent, so they are valid for every central symmetric potential. M’e have three requirements for (9.116a): It should tend to zero for r + co,it should be one-valued and quadratically integrable on the surface of the sphere. 3. Solution of the Radial Equation Beside the potential V ( r )the radial equation (9.116d) also contains the separation constant 1(1+ 1).We substitute U d T ) = ’ RdT), (9.117a) since the square of the function u ~ ( T )gives the last required probability Ivl(r)(2dr= /RI(r)l2r2dr of the presence of the particle in a spherical shell between r and r t d r . The substitution leads to the one-dimensional Schrodinger equation (9.117b) This one contains the effective potential (9.117~) l i = V ( r ) k$), which has two parts. The rotation energy 1(1+ l)h2 (9.117d) r/l(l) = \ ; o t ( l ) = 2mr2 is called the centrzf2lgalpotentzal. The physical meaning of / as the orbztal angular momentum follows from analogy with the classical rotation energy

+

~

(9.117e) a rotating particle with moment of inertia 0 = p2and orbital angular momentum

c2= / ( 1 t l)h2,

li21= hd-.

?= 03: (9.117f)

4. Solution of t h e polar equation The polar equation (9.116g), containing both separation constants 1(1 + 1) and m 2 ,is a Legendre differential equation (9.57a), p. 509. Its solution is denoted by 0r(B),and it can be determined by a power series expansion. Finite, single-valued and continuous solutions exist only for 1 ( / t 1) = 0,2,6,12,.. . . We get for 1 and m: 1 = 0,1.2, . . . * Iml 5 1. (9.118a) So, m can take the (21 t 1) values (9.118b) -1. (-It l ) , (-1 2), . . . , (1 - 2 ) , (1 - 1). 1. We get the corresponding Legendre polynomial for m # 0, which is defined in the following way.

+

PY(c0s 8)= -(

2‘1!

1 - cos2 8)m’Z

d”tm(cos28 - 1)1

(dcos Ig)ltrn

(9.118~)

\I’egettheLegendrefunctionofthefirstkind(9.57c),p.j09,asaspecialcase(1= 0. m = n, cos8 = z). Its normalization results in the equation P?(COS8) = ’vypln(cos8).

(9.118d)

5 . Solution of t h e Azimuthal Equation Since the motion of the particle in the potential field V ( r )

is independent of the azimuthal angle even in the case of the physical assignment of a space direction.

m

9. Differential Eouations

542

+

.

e.g by a magnetic field. we specify the general solution @ = a e i m p ,!?e-lrnv by fixing ~ ~ ( p = Ae*lmP, . ) because in this case I @ m12 is independent of p. The requirement for uniqueness is @ m ( P + 2n) = @m(lF), so m can take on only the values 0, kl,1 2 , . . .. It follows from the normalization

(9.11sa) (9.119b)

(9.119c) 0

that

The quantum number m is called the magnetzc quantum number. 6 . Complete Solution for the Dependency of the Angles In accordance with (9.116e). the solutions for the polar and the azimuthal equations should be multiplied by each other: (9.120a)

The functions qm(3.p) are the so-called surface spherical harmonics. \$'hen the radius vector ?is reflected with respect to the origin (i--t =r), the angle 29 becomes and p becomes 9 -t n. so the sign of qrnmay change:

qy71 - 6, $5 + n ) = ( - 1 ) y y I A p ) . IYe get the parity of the considered wave function P = (-1)l.

7r

- 29

(9.120b) (9.121a)

7. Parity The parity property serves the characterization of the behavior of the wave function under space inversion i + -i (see 4.3.5.1, l., p. 269). It is performed by the inversion or parity operator P: PP(?%t ) = P(-i,t ) . If we denote the eigenvalue of the operator by P , then applying P twice it must yield P P P ( ? ,t ) = P P P ( ? >t ) = @(?,t ) ,the original wave function. So: P2= 1, P = * l . (9.121b) We call it an euen wave function if its sign does not change under space inversion, and it is called an odd wave function if its sign changes. 6. Linear Harmonic Oscillator 1. Posing the Problem Harmonic oscillation occurs when the drag forces in the oscillator satisfy Hooke's law F = -kx. For the frequency of the oscillation, for the frequency of the oscillation circuit and for the potential energy we get:

-F 1

u = 2n

k m' (9.122a)

d

=

E,

1

dZ

E,,, = -2k x 2 = -x'. 2

(9.122b)

(9.122~)

Substituting into (9,11la),the Schrodinger equation becomes (9.123a) IVith the substitutions y

=.E.

(9.123b)

A=-

2E hw'

(9.123~)

9.2 Partial Diferential Equations 543 where X is a parameter and not the wavelength, (9.123a) can be transformed into the simpler form of the Weber differential equation d2P -+(Ay2)P = 0. dY2 2. Solution N e get a solution for the Weber differential equation in the form

P(y) = e-YZ”H(y). Differentiation shows that

(9.123d)

(9.124a)

(9.124b) Substitution into the Schrodinger equation (9.123d) yields dZH dH + ( A - l ) H = 0. dY2 dY We determine a solution in the form of a series

_ -2y-

(9.124~)

(9.125a) Substitution of (9.125a) into (9.124~)results in

Comparing the coefficients of y j we get the recursion formula

+

(9.125~) ( j 2 ) ( j + l)a,,z = [ 2 j - (A - 1)]a3 ( j= 0 , 1 , 2 , . . .). We get the coefficients a3 for even powers of y from ao, the coefficients for odd powers from a l . So, a. and al can be chosen arbitrarily. 3. Physical Solutions We want to determine the probability of the presence of a particle in the different states. This will be described by a quadratically integrable wave function P(z) and by an eigenfunction which has physical meaning, Le., normalizable and for large values of y it tends to zero. The exponential function exp(-y2/2) in (9.124a) guarantees that the solution @(y) tends to zero for y + x if the function H(y) is a polynomial. To get a polynomial, the coefficients aj in (9.125a), starting from a certain n! must vanish for every j > n: a, # 0, anti = an+2 = ant3 = . . . = 0. The recursion formula (9.125~)with j = n is

2n - (A - 1) (n+2)(n+l$, an+2 = 0 can be satisfied for a,

(9.126a)

an+2=

# 0 only if

2E 2n - (A - 1) = 0. X = - = 2 n + 1. (9.12613) hw The coefficients ant2, an+4,.. . vanish for this choice of A. Also a,-l = 0 rhust hold to make the coefficients an+lranta,. . . equal to zero. We get the Hermitepolynomialsfrom the second defining equation (see 9.1.2.6,6., p. 512) for the special choice of a, = 2”. = 0. The first six of them are: (9.126~)

9. Differential Equations

544

The solution @(y) for the vibratzon quantum numbern is

pn = .\Tne-v2/2Hn(y),

(9.127a)

where ,Vn is the normalizing factor. We get it from the normalization condition

LF

I

Pnzd y = 1 as

(see (9.123b), p. 542) with fi= -= 2nn! T From the terminating condition of the series (9.123~)we get ?Vn2=

En = Aw ( n +

;)

(9.127b)

( n = 0, 1 , 2 , . . .)

(9.127~)

for the eigenvalues of the vibration energy. The spectrum of the energy levels is equidistant. The summand +1/2 in the parentheses means that in contrast to the classical case the quantum mechanical oscillator has energy even in the deepest energetic level with n = 0, which is known as the zero-poznt vzbratzon energy. Fig. 9.21 shows a graphical representation of the equidistant spectra of the energy states, the corresponding wave functions from Po to P5 and also the function of the potential energy (9.122~).The points of the parabola of the potential energy represent the reversal points of the classical oscillator, which 1 are calculated from the energy E = -mw2a2as the amplitude 2 a=

ig,

The quantum mechanical probability of finding a

u'

particle in the interval (2, z+dz) is given by dw9,, = iP(z)I'dx. It is different from zero also outside of these points.

Figure 9.21

Ja

So we get for, e.g., n = 1, hence for E = ( 3 / 2 ) b , according to dwqu= 2 -e-xsz dx, the maximum of the probability of presence at

(9.127d)

For a corresponding classical oscillator, this is (9.127e) The quantum mechanical probability density function approaches the classical one for large quantum number n in its mean value.

9.2.4 Non-Linear Partial Differential Equations: Solitons, Periodic Patterns, and Chaos 9.2.4.1 Formulation of the Physical-Mathematical Problem 1. Notion of Solitons Solitons, also called solitary waves, from the viewpoint of physics, are pulses, or also localized disturbances of a non-linear medium or field; the energy related to such propagating pulses or disturbances is concentrated in a narrow spatial region. They occur:

9.2 Partial Differential Equations

545

conductors, in linear molecules. in classical and quantum field theory. Solitons have both particle and wave properties; they are localized during their evolution, and the domain of the localization, or the point around which the wave is localized, travels as a free particle; in particular it can also be at rest. A soliton has a permanent wave structure: based on a balance between nonlinearity and dispersion, the form of this structure does not change. SIathematically. solitons are special solutions of certain non-linear partial differential equations occurring in physics, engineering and applied mathematics. Their special features are the absence of any dissipation and also that the non-linear terms cannot be handled by perturbation theory. Important examples of equations with soliton solutions are: a) Korteweg de Vries ( K d V ) Equation (9.128) Ut ~ U U , ~,,, = 0,

+

+

+ u,,

b) Non-Linear Schrijdinger (NLS) Equation

iut

c) Sine-Gordon (SG) Equation

utt - u,,

& 21211% = 0,

+ sin u = 0.

(9.129) (9.130)

The subscripts x and t denote partial derivatives, e.g., u,, = d2u/dx2. We consider the one-dimensional case in these equations, Le., u has the form u = u(x,t ) ,where z is the spatial coordinate and t is the time. The equations are given in a scaled form, Le., the two independent variables x and t are here dimensionsless quantities. In practical applications, they must be multiplied by quantities having the corresponding dimensions and being characteristic of the given problem. The same holds for the velocity.

2. Interaction between Solitons If two solitons, moving with different velocities, collide, they appear again after the interaction as if they had not collided. Every soliton asymptotically keeps its form and velocity; there is only a phase shift. Two solitons can interact without disturbing each other asymptotically. This is called an elastic interaction which is equivalent to the existence of an N-soliton solution, where N ( N = 1 , 2 , 3 , .. .) is the number of solitons. Solving an initial value problem with a given initial pulse u ( z ,0) that disaggregates into solitons, the number of solitons does not depend on the shape of the pulse but on its total amount J-'," u(x.0) dx.

3. Non-Linear Phenomena in Dissipative Systems In dissipative systems (hence friction or damped systems), periodic patterns and non-linear waves can appear through the impact of external forces. One striking example is a fluid (gas or liquid), where the combined action of gravitation and a trnpereature gradient causes (with increasing temperature difference) a transition from a purely heat conducting state (without convection) to a very special, namely regular cell convection state finally up to turbulence. Depending on the magnitude of the temparature difference bifurcation and chaos can appear (see 17.3, p. 827). Important examples for equations of such phenomena are: ut - u - (1 ib)uzt (1 ic)(uI2u= 0, a) Ginsburg-Landau (GGL) Equation (9.131)

+

b) Kuramoto-Sivashinsky (KS) Equation ut

+ +

+ u,, + u,,,, + u: = 0.

(9.132)

In contrast to the dissipationless KdV, NLS, SG, equations, the equations (9.131) and (9.132) are non-linear dissipative equations, which have, besides spatio-temporal periodic solutions, also spatiotemporal disordered (chaotic) solutions.

4. Non-Linear Evolution Equation .4n evolution equation describes the evolution of a physical quantity in time. Examples for such evolution equations are the wave equation (see 9.2.3.2, p. 534)) the heat equation (see 9.2.3.3. p. 535) and the Schrodinger equation (see 9.2.3.5, 1.. p. 536). The solutions of the evolution equations are called evolution functions.

546

9. DifferentialEquations

In contrast to linearevolutionequations, thenon-linear evolutionequations (9.128), (9.129),and (9.130) ~ These ~ equations are (with the exception of (9.131)) contain non-linear terms udu/dx, 1 ~ and1 sinu. parameter-free. From the viewpoint of physics non-linear evolution equations describe structure formations like solitons (dispersive structures) as well as periodic patterns and non-linear waves (dissipative structures).

9.2.4.2 Korteweg de Vries Equation (KdV) 1. Occurrance The KdV equation is used in the discussion of problems of plasma physics and non-linear electric networks. 2. Equation and Solutions The KdV equation for the evolution function u is (9.133) t 62121, u,,,= 0. It has the soliton solution u u ( 5 :t ) = . (9.134) 2 cosh' [ f f i ( x - v t - p)]

+

b

-6

-4

-2

0' 2 x-vt-cp

4

6

Figure 9.22 This KdV soliton is uniquely defined by the two dimensionless parameters u (v > 0) and 'p. In Fig. 9.22 u = 1 is chosen. A typical non-linear effect is that the velocity of the soliton u determines the amplitude and the width of the soliton: KdV solitons with larger amplitude and smaller width move quicker than those with smaller amplitude and larger width (taller waves travel faster than shorter ones), The soliton phase 9 describes the position of the maximum of the soliton at timet = 0. Equation (9.133) also has N-soliton solutions. Such an N-soliton solution can be represented asymptotically for t --t km by the linear superposition of one-soliton solutions: N

~(z, t)

Un(x,

N

t),

(9.135)

n=l

Here every evolution function u,(x,t ) is characterized by a velocity u, and a phase (3.; The initial phases ' p i before the interaction or collision differ from the final phases after the collision cp,,' while the velocities v l l v 2 : .. . , uN have no changes, Le., it is an elastic interaction. For N = 2, (9.133) has a two-soliton solution. It cannot be represented for a finite time by a linear I

I

superposition, and with k - -fin and an = -&(z n-2 2

- vnt - 9,) (n = 1.2) it has the form:

Equation (9.136) describes for t --t --co asymptotically two non-interacting solitons with velocities v1 = 4k: and u2 = 4k;, which transform after their mutual interaction again into two non-interacting solitons with the same velocities fort --t +aasymptotically. The non-linear evolution equation (9.137a) ZL'~+ 6(wz)' w,,, = 0

+

FZ

where w = - has the following properties: F

9.2 Partial Differential Eouations

a) For F ( z ,t ) = 1 t ea,

1

a = - f i ( x - ut - 'p) 2

547

(9.13710)

it has a soliton solution and

(fly ;;)'

b) for F ( z .t ) = 1 t ea' t eaz t - e(o1tuz)

(9.137~)

it has a two-soliton solution. IVith 2w, = u the KdV equation (9.133) follows from (9.137a). Equation (9.136) and the expression w following from (9.137~)are examples of a non-linear superposition. If the term $ 6 2 1 ~is~replaced by -61111, in (9.133), then the right-hand side of (9.134) has to be multiplied by (-1). In this case the notation antisoliton is used.

9.2.4.3 Non-Linear Schrodinger Equation (NLS) 1. Occurrence The NLS equation occurs in non-linear optics, where the refractive index n depends on the electric field strength 3,as, e.g.. for the Kerr effect, where n(G) = no + nzId1' with no, n2 = constant holds, and in the hydrodynamics of self-gravitating discs which allow us to describe galactic spiral arms

2. Equation and Solution The SLS equation for the evolution function u and its solution are: iut tu,,

* 21u/'u = 0.

ei[2&+4(+~2)t-~:

(9.138)

Here u ( z ,t ) IS complex The NLS soliton is characterized by the four dimensionless parameters 17) 5 , p3 and x. The envelope of the wave packet moves with the velocity -4<: the phase velocity of the wave packet is 2(17' - ( ' ) / E . In contrast to the KdV soliton (9.134), the amplitude and the velocity can be chosen independently of each other In the case of .V interacting solitons, we can characterize them by 4V,' arbitrary chosen parameters: q n , E n . p n r x n ( n = 1 , 2, . . . ,N ) . If the solitons have different velocities, the N-soliton solution splits asymptotically for t + *co into a sum of N individual solitons of the form (9.139). Fig. 9.23 displays the real part of (9.139) with u = -4<, 17 = 112 and ( = 215.

u ( z ,t ) = 217

C O S ~ [ ~t V 4
(9.139)

-"lt "

Figure 9.23

9.2.4.4 Sine-Gordon Equation (SG) 1. Occurrence The SG equation is obtained from the Bloch equation for spatially inhomogeneous quantum mechanical two-level systems. It describes the propagation of

spin waves in superfluid helium 3 (3He). The soliton solution of the SG equation can be illustrated by a mechanical model of pendula and springs. The evolution function goes continuously from 0 to a constant value c. The SG solitons are often called

548

9. Differential Eouations

2. Equation and Solution The SG equation for the evolution function u is utt - u,, + sin u = 0. (9.140) It has the following soliton solutions: 1. Kink Soliton u(z.t ) = 4 arctan (9.141) 1 where 7 = -and -1 < v < +1. ~

The kink soliton (9.141) for v = 1/2 is given in Fig. 9.24. The kink soliton is determined by two dimensionless paramet,ersu and 20. The velocity is independent of the amplitude. The time and the position derivatives are ordinary localized solitons: ut = 21, = 27 -. (9.142) U cash Y(S - $0 - ut)

-10

-5

0 x-h-vt

5

10

b

Figure 9.24

kink solitons. If the evolution function changes from the constant value c to 0, it describes a so-called antikink soliton. Walls of domain structures can be described with this type of solutions. 2. Antikink Soliton u(z.t ) = 4 arctan e-7(z-zo-ut). (9.143) 3. Kink-Antikink Soliton We get a static kink-antikink soliton from (9.141,9.143) with u = 0: u(x,t ) = 4arctane*('-'o). Further solutions of (9.140) are: 4. Kink-Kink Collision

(9.144)

[

u ( z . t )= 4arctan vlt:;:::

(9.145)

5. Kink-Antikink Collision 1 sinh yvt u ( x , t ) = 4arctan -C O S 7z] ~

[

(9.146)

6. Double or Breather Soliton, also called Kink-Antikink Doublet

dl - w2 sinwt (9.147) u(x:t) = 4 arctan w c o s h w x Equation (9.147) represents a stationary wave, whose envelope is modulated by the frequency w. 7. Local Periodic Kink Lattice

I

[

u(x, t ) = 2arcsin zksn

(kg4] ~

+a.

(9.148a)

The relation between the wavelength X and the lattice constant k is

x =4K(k)kvT7 Fork = 1, Le., for --t m, we get

(9.148b)

u ( z ,t ) = 4 arctane*Y(S-ut), which is the kink soliton (9.141) and the antikink soliton (9.143) again, with 20 = 0.

(9.148~)

9.2 Partial DifferentialEauations 549

Remark: snx is a Jacobian elliptic function with parameter k and quarter-period K (see 14.6.2, p. 701): snx = sinp(z,k ) ,

(9.149a)

Equation (9.14913)comes from (14.102a), p. 701, by the substitution of sin Q = q . The series expansion of the complete elliptic integral is given as equation (8.104),p. 460.

9.2.4.5 Further Non-linear Evolution Equations with Soliton Solutions 1. Modified KdV Equation (9.150) ut 6u2u, + u,,, = 0.

*

The even more general equation Ut upu, u,,, = 0 has the soliton

+

+

u(2,t) =

l1

I

tu (P + 1)(P + 2)

(9.151) 1

P

(9.152)

cosh2 ( i p f i ( x - ut - p)]

as its solution. 2. Sinh-Gordon Equation utt - u,, + sinh u = 0.

3. Boussinesq Equation

(9.154) U x , - U t t t (u2),, + u,,, = 0. This equation occurs in the description of non-linear electric networks as a continuous approximation of the charge-voltage relation.

4. Hirota Equation ut

+ ~ ~ c Y ~ u+/ 8uz, ~ z L+, iou,,, + 6(uI2u= 0,

CUB= o6.

(9.155)

5 . Burgers Equation (9.156) ut - u,, 2121, = 0. This equation occurs when modeling turbulence. With the Hopf-Cole transformation it is transformed into the diffusion equation, Le., into a linear differential equation.

+

6. Kadomzev-Pedviashwili Equation The equation (ut + 621212 + uzzz), = u y y has the soliton

(9.157a) (9.157b)

as its solution. The equation (9.157a) is an example of a soliton equation with a higher number of independent variables, e.g.. with two spatial variables.

550

10. Calculus of Variations

10 CalculusofVariations 10.1 Defining the Problem 1. Extremum of an Integral Expression A very important problem of the differential calculus is to determine for which zvalues the given function y(x) has extreme values. The calculus of variations discusses the following problem: For which functions has a certain integral. whose integrand depends also on the unknown function and its derivatives, an extremum value? The calculus of variations concerns itself with determining all the functions y(z) for which the integral expression b

I[y]= J/ F ( z )y(z). y'(z), . . . . y(")(z))dz

(10.1)

a

has an extremum, if the functions y(z) are from a previously given class of functions. Here. we may define some boundary and side conditions for y(z) and for its derivatives.

2. Integral Expressions of Variational Calculus There can also be several variables instead of z in (10.1). In this case, the occurring derivatives are partial derivatives and the integral in (10.1) is a multiple integral. In the calculus of variations. the following types of integral expressions are discussed:

Here the unknown function is u = u ( z ,y). and R represents a plane domain of integration.

The unknown function is u = u ( z ,y , z ) , and R represents a space region of integration. Additionally, boundary values can be given for the solution of a variational problem, at the endpoints of the interval a and b in the one-dimensional case. and at the boundary of the domain of integration R in the twodimensional case. Besides, various further side conditions can be defined, e.g., in integral form or as a differential equation. A variational problem is called first-order or higher-order depending whether the integrand F contains only the first derivative y' or higher derivatives y(") ( n > 1) of the function y.

3. Parametric Representation of the Variational Problem A variational problem can also be posed in parametric form. If we consider a curve in parametric form 5 = x ( t ) , y = y ( t ) ( a 5 t 5 9).then, e.g., the integral expression (10.2) has the form 3

I [ z .Yl =

1

F ( z ( t ) ,y(t), 4 t ) ,Y(t)) dt.

a

(10.7)

10.2 Historical Problems 551

10.2 Historical Problems 10.2.1 Isoperimetric Problem The general zsoperzmetric problem is to determine the plane region with the largest area among the plane regions with a given perimeter. The solution of this problem, a circle with a given perimeter, originates from queen Dido, who was allowed, as legend has it, to take such an area for the foundation of Carthago which she could be surround by one bull’s leather. She cut the leather into fine stripes, and formed a circle with them. A special case of the isoperimetric problem is to find the equation of the curve y = y(x) in a Cartesian coordinate system connecting the points A ( a , 0) and B(b, 0) and having the given length 1, for which the area determined by the line segment and the curve is the largest possible (Fig. 10.1). The mathematical formalization is: \Ye have to determine a one-time continuously differentiable Figure 10.1 function y(x) such that b

I[y]= /y(x) dx = max

(10.8a)

a

holds, where the side condition (10.8b) and the boundary conditions ( 1 0 . 8 ~are ) satisfied: h

G[y] =

d

m

d

z =1

(10.8b)

y(a) = y(b) = 0

(10.8~)

a

10.2.2 Brachistochrone Problem The brachistochrone problem was formulated in 1696 by J. Bernoulli, and it is the following: A point mass descends from the point PO(z0, yo) to the origin in the vertical plane x,y only under the influence of gravity \Ve should determine the curve y = y(x) along which the point reaches the origin in the shortest possible time from Po (Fig. 10.2). Considering the formula for the time of fall, T , we get the mathematical description: We have to determine a one-time continuously differentiable function y = y(x), for which 00

dx = min,

(10.9)

(g is the acceleration due to gravity) and the boundary value conditions are Y(0) = 0. Y(Z0) = Yo. (10.10) R‘e see that there is a singularity for z = zo in (10.9).

xo Figure 10.2

x

10.3 Variational Problems of One Variable 10.3.1 Simple Variational Problems and Extrema1 Curves A simple uarzatzonal problem is to determine the extreme value of the integral expression given in the form b

/

I[Yl= F ( z ,Y(x)l

Y’(l))

dx,

(10.11)

a

where y(x) is a twice continuously differentiable function satisfying the boundary conditions y(a) = A and y(b) = B. The values a , b and A , B , and the function F are given.

552

10. Calculus of Variations

The integral expression (10.11) is an example of a so-called functional. A functional assigns a real number to every function y(x) from a certain class of functions. If the functional I[y] in (10.11) takes, e.g., its relative maximum for a function yo(z), then [[Yo1 2 4 Y l (10.12) for every twice continuously differentiable function y satisfying the boundary conditions. The curve y = yo(.) is called an extremal curve. Sometimes all the solutions of the Euler differential equation of the variational calculus are called extremal curves.

10.3.2 Euler Differential Equation of the Variational Calculus We get a necessary condition for the solution of the variational problem in the following way: We construct an auxiliary curve or comparable curve for the extremal yo(z) characterized by (10.12) (10.13) Y b ) = Yo(l) +t v b ) ) the special boundary conditions ~ ( a=) with a twice continuously differentiable function ~ ( xsatisfying q(b) = 0. t i s a real parameter. Substituting (10.13) in (10.11) we get a function dependingon E instead of the functional I [ y ] b

I ( € )=

/ F(x,

YO

t E 17, Y;t E 7’)dz,

(10.14)

a

and the functional I[y] has an extreme value for yo(z) if the function I ( € ) ,as a function of E, has an extreme value for E = 0. Yow, we deduce the variational problem to an extreme value problem with the necessary condition dl - = 0 for E = 0. (10.15) dt Supposing that the function F , as a function of three independent variables, is differentiable as many times as needed, by its Taylor expansion we get (see 7.3.3.3, p. 415) (10.16) The necessary condition (10.15) results in the equation (10.17)

By partial integration of this equation and considering the boundary conditions for q ( x ) ,we get (10.18) From the assumption of continuity and because the integral in (10.18) must disappear for any considerable ~ ( x ) , (10.19) must hold. The equation (10.19) gives a necessary condition for the simple variational problem and it is called the Euler diflerential equation of the calculus of variations. The differential equation (10.19) can be written in the form dF d2F d2F y’ - -y” d2F = 0. (10.20) dy dxdy’ dydy‘ dy’2 It is an ordinary second-order differential equation if Fy,y, # 0 holds.

10.3 Variational Problems o f One Variable 553

The Euler differential equation has a simpler form in the following special cases: Case 1: F ( x .y. y’) = F(y’),Le., z and y do not appear explicitly. Then instead of (10.19) we get (10.21a)

d

and

dF (%) = 0.

(10.21b)

Case 2: F ( x , y, y’) = F ( y , y’), Le., 5 does not appear explicitly. We consider dF d d (F = %y’ dF + @y” d F - y”- dF - y’- d dF dx ay/ dz = y’ ( d y -

y’g)

dF z (a,.))

(G)

and because of (10.19)?we get d

(F

-

y ’ g ) = 0,

(10.22b)

dx

Le.,

F

-

y’-

dF = c (c const) 8Y’

(10.22~)

as a necessary condition for the solution of the simple variational problem in the case F = F ( y , y’). W A: The functional to determine the shortest curve connecting the points Pl(a, A ) and &(b, B ) in the x.y plane is:

1[y1=

s”

~ l + y “ d z = min.

(10.23a)

a

It follows from (10.21b) for F = F(y’)=

that (10.23b)

so y” = 0. Le.. the shortest curve is the straight line. W B: We connect the points Pl(as A ) and P,(b,B ) by acurve y(x), and we rotate it around the x-axis. Then the surface area is

( 10.24a)

I[y]= 2 s a r y m d x

For which curve y(x) will the surface area be the smallest? It follows from (10.22~)with F = F ( y , y’) =

(t)’

or y” = - 1 with c1 = L . This differential equation is 2a 2a separable (see 9.2.2.3. l., p. 522), and its solution is 2 n y W that y =

y = ci cosh

(z+

c2)

(el,

e2

(10.24b)

const),

the equation of the so-called catenary curzle (see 2.15.1, p. 105). We determine the constants c1 and e2 from the boundary values y(a) = A and y(x) = B. We have to solve a non-linear equation system (see 19.2.2, p. 893), which cannot be solved for every boundary value.

10.3.3 Variational Problems with Side Conditions These problems are usually isoperimetric problems (see 10.2.1, p. 551): The simple variational problem (see 10.2 1.p. 551). given by the functional (lO.ll),is completed by a further side condition in the form

j C ( c , y ( z j . y / ( x l ) d x= 1

(1 const)

(10.25)

a

where the constant 1 and the function G are given. .4method to solve this problem is given by Lagrange (extreme values with side conditions in equation form, see 6.2.5.6, p. 401). We consider the expression (10.26) H ( z ,~ ( z )~. ’ bA)) = . F ( z ,Y(.c), ~ ’ ( x +) ) ( G ( ~ , Y ( x ) , Y ’ ( ~-) )

4,

I

554

10. Calculus of Variations

where X is a parameter. and we consider the problem

1

H ( x , y(x), y‘(x).

= extreme!,

(10.27)

a

Le.. an extreme value problem without side condition. The corresponding Euler differential equation is:

E -d dy

(”) = 0,

(10.28)

dx dy’

The solution y = y(x. A) depends on the parameter A, which can be determined by substituting y(x. A) into the side condition (10.25). IFor the isoperimetric problem 10.2.1, p. 551, we get (10.29a) H ( x ,Y(4,Y’(x)>4= Y + A m . Because the variable x does not appear in H , we get instead of the Euler differential equation (10.28), analogously to (10.22c), the differential equation

&

y+x---

-4 = c1 or

y’2 =

(cl c1 - y

(el const).

(10.29b)

whose solution is the family of circles (x - cz)’ (y - c1)’ = Xz (cl,cz,X const). (10.29~) The values cl, c2 and X are determined from the conditions y(a) = 0, y(b) = 0 and from the requirement that the arclength between A and B should be 1. We get a non-linear equation for A, which should be solved by an appropriate iterative method.

+

10.3.4 Variational Problems with Higher-Order Derivatives We consider two types of problems

1. F = F ( z ,y, g’, y”) The variational problem is:

1 b

I[y]= F ( q y, y’, 9”) dx = extreme!

(10.30a)

a

with the boundary values y ( a ) = A, y(b) = B , y‘(a) = A’, y’(b) = B’. (10.30b) where the numbers a, b, A , B , A’, and B‘, and the function F are given. Similarly as in 10.3.2, p. 552, we introduce comparable curves y(z) = yo(r) + E ~ ( with x ) q(a) = q(b) = q’(a) = a’(b) = 0. and we get the Eder dzflerentzal equation

dF dY -

d

dF (a,.) +g

(5) =0

(10.31)

as a necessary condition for the solution of the variational problem (10.30a). The differential equation (10.31) represents a fourth-order differential equation. Its general solution contains four arbitrary constants which can be determined by the boundary values (10.30b). IConsider the problem I [ y ] = il(~’’~ - cyy” - By’) dx = extreme!

(10.32a)

10.3 Variational Problems of One Variable 555

with the given constants a and I? for F = F(y, y’, y”) = y’” - ~ y- ’By2. ~ Then: d d2 Fy = -23y. Fyf= -2y’, FY”= 2y”, - (Fyt)= -2y”, - (Fv)t) = - 2 ~ ( ~ and ) , the Euler differential dx dx2 equation is

+

y (4) ay”- 3y = 0. (10.32b) This is a fourth-order linear differential equation with constant coefficients (see 9.1.2.3, p. 498).

...

2. F = F ( z ,y, y’, ,y‘“)) In this general case. when the functional I[y]of the variational problem depends on the derivatives of the unknown function y up to order n (n2 1))the corresponding Euler differential equation is

dF - d dF (10.33) dY + whose solution should satisfy the boundary conditions analogously to (10.30b) up to order n - 1.

2 (E)

(a,.)

10.3.5 Variational Problem with Several Unknown Functions Suppose the functional of the variational problem has the form b

.

I [ y i , YZ. ’ ’ . Y], =

1

F(L

YZ,.

. . . yn, ~

a

/

I

. . yk) dx,

1~ , 2 , .

(10.34)

.

where the unknown functions yl(x), yz(x)>.. . yn(x) should take given values at x = a and x = b. We consider n twice continuously differentiable comparable functions > n). (10.35) !/t(z) = yd.) + ~ t % ( Z ) (i = 132, where the functions vt(x) should disappear at the endpoints. (10.34) becomes I ( e l , c2. . . . . E , ) with (10.35). and from the necessary conditions I , ,

dI

-

=o

(2

=

4 2 . . . . ,n )

(10.36)

for the extreme values of a function of several variables, we get the n Euler differential equations

whose solutions yl(z). y2(z),. . . y,(x) must satisfy the given boundary conditions. ~

10.3.6 Variational Problems using Parametric Representation For some variational problems it is useful to determine the extremal, not in the explicit form y = y(x), but in the parametric form x = x ( t ) , y = y(t) ( t l 5 t 5 tz). (10.38) where t l and tz are the parameter values corresponding to the points ( a , A ) and (b, B ) . Then the simple variational problem (see 10.3.1, p. 551) is tz

I [ $ ,y] =

1F ( x ( t ) ,

y(t), k ( t ) ly(t))dt = extreme!

(10.39a)

tl

with the boundary conditions ~ ( t l= ) a. ~ ( t z = ) 6. y[tl) = A , ~ ( t z = ) B. (10.39b) Here x and j, denote the derivatives of x and y with respect to the parameter t, as usual in the parametric representation. The variational problem (10.39a) makes sense only if the value of the integral is independent of the

556

10. Calculus of Variations

parametric representation of the extrema1 curve. To ensure the integral in (10.39a) is independent of the parametric representation of the curve connecting the points (a, A ) and (b, B ) , F must be a positive homogeneous junctzon, Le., (10.40) F ( z ,Y . P% PY) = P F ( x . Y , 5 , Y) ( P > 0) must hold. Because the variational problem (10.39a) can be considered as a special case of (10.34),the corresponding Euler differential equations are (10.41) They are not independent of each other, but they are equivalent to the so-called Weierstrass form of the Euler differential equation: (10.42a) with (10.42b) Starting with the calculation of the radius of curvature R of a curve given in parametric representation (see 3.6.1.1, l.,p. 225), we calculate the radius of curvature ofthe eztremal curve considering (10.42a) with (10.42~) The isoperimetric problem (10.8a to 1 0 . 8 ~(see ) 10.2.1,p. 551) has the form in parametric representation: I [ x ,y ] = [’ y ( t ) i ( t ) d t= max! (10.43a) with G[z, y ] = tl

J”’ d

m

d

t = 1. (10.43b)

tl

This variational problem with the side condition becomes a variational problem without the side condition according to (10.26) with H = H ( x 3y , x,y ) = y x -tA m .

(10.434

w’e see that H satisfies the condition (10.40), so it is a positive homogeneous function of first degree. Furthermore, we have H i y = 1,

(10.43d)

so (10.42~) yields that the radius of curvature is R = 1x1. Since A is a constant, the extremals are circles.

10.4 Variational Problems with Functions of Several Variables 10.4.1 Simple Variational Problem One of the simplest problems with a function of several variables is the following variational problem for a double integral: I [ u ]=

// F ( z ,

y . u ( x ,y ) , u,, u y )dxdy = extreme!

(GI

(10.44)

10.1 Variational Problems with Functions o f Several Variables 557

Here. the unknown function u = u(z, y) should take given values on the boundary Analogously to 10.3.2. p 552, we introduce the comparablefunctzon in the form

r of the domain G

(10.45) 4 2 , Y) = uo(z,Y) + 4 z , ~ ) . where u o ( J , y) is a solution of the variational problem (10.44) and it takes the given boundary values. while ~ ( xy),satisfies the condition ~ ( 2y). = 0 on the boundary r (10.46) and together with uo(z.y), they are differentiable as many times as needed. The quantity 6 is a parameter. We determine a surface by u = u(x, y) which is close to the solution surface uo(z, y). I [ u ]becomes I ( € )with (10.45), Le., the variational problem (10.44) becomes an extreme value problem which must satisfy the necessary conditions

dl

-=

de

0

for

6

(10.47)

= 0.

\Ye get from this the Euler dzflerentzal eqzlatzon (10.48) as a necessary condition for the solution of the variational problem (10.44). W -4 free membrane, fixed a t the perimeter of a domain G of the z,y plane. covers a surface with area

r

II =

// dxdy

(10.49a)

@I

If the membrane is deformed by a load so that every point has an elongation u = u(x.y) in the direction. then its area is calculated by the formula I2 =

s/ d m d x d y .

2-

(10.49b)

(GI

If we linearize the integrand in (10.49b) using Taylor series (see 6.2.2.3, p. 394). then we get the relation (10.49~)

(10.49d) for the potential energy li of the deformed membrane, where the constant o denotes the tension of the membrane. We obtain the so-called Dzrzchlet uarzatzonal problem in this way: We have to determine the function u = u(x,y) so that the functional (10.49e)

r

should have an extremum, and u vanishes on the boundary of the plane domain G. The corresponding Euler differential equation is (10.49f)

558

10. Calculus of Variations

It is the Laplace differential equation for functions of two variables (see 13.5.1, p. 667).

10.4.2 More General Variational Problems We should consider two generalizations of the simple variational problem

1. F = F ( z , Y, 4 2 ,Y), u+, uy,%a, u + y , ‘Ilyy) The functional depends on higher-order partial derivatives of the unknown function v(z, y). If the partial derivatives occur up to second order, then the Euler differential equation is:

2. F = F ( Z l , Z Z , ..., 2,,u(21,...,2,),21,,,...,U,,) In the case of a variational problem with n independent variables equation is:

..

~ 1 ~ ~ 2,x,, , . the

Euler differential (10.51)

10.5 Numerical Solution of Variational Problems Most often two ways are used to solve variational problems in practice.

1. Solution of the Euler Differential Equation and Fitting the Found Solution to

the Boundary Conditions Usually, exact solution of the Euler differential equation is possible only in the simplest cases, so we have to use a numerical method to solve the boundary value problem for ordinary or for partial differential equations (see 19.5)p. 908 or 20.4.4: p. 989ff). 2. Direct Methods The direct methods start directly from the variational problem and do not use the Euler differential equation. The most popular and probably the oldest procedure is the Ritz method. It belongs to the socalled approximation methods which are also used for approximate solutions of differential equations (see 19.4.2.2, p. 906 and 19.5.2, p. 909), and we demonstrate it with the following example. ISolve numerically the isoperimetric problem

1’

y”(z)dx = extreme! (10.52a)

for

1’

y2(z)dz = 1 and y(0) = y(1) = 0. (10.52b)

The corresponding variational problem without side condition according to 10.3.3, p. 553, is: I[y] =

s1

[y”(z)dx - Xy2(x)] = extreme!

(10.52c)

\Ye want to find the best solution of the form (10.52d) y(z) = alz(x - 1) + a2x2(x - 1). Both approximation functions z(z-1) and x2(x-1) are linearly independent, and satisfy the boundary conditions. (10.52~)is reduced with (10.52d) to 1 2 1 (10.52e) I(a1, al) = -a; + -ai t -ala2 3 15 3 and the necessary conditions

#- # = 0 result in he homogeneous linear equation system a1

=

a2

105

(10.52f)

10.6 Supulementarv Problems 559

This system has a non-trivial solution only if the determinant of the coefficient matrix is equal to zero. So, we get: Xz - 52X + 420 = 0, Le., XI = 10, A2 = 42. (10.52g) For X = X I = 10 we get from (10.529 a2 = 0. a1 arbitrary, so the normed solution belonging to X I = 10 is: ( 10.52h) y = 5 . 4 8 ~-( ~1). To make a comparison: consider the Euler differential equation belonging to (10.524. We get the boundary value problem yll+ Xy = 0 with y(0) = y(1) = 0 (10.52i) with the eigenvalues Xk = k 2 d (k = 1 , 2 , . . .) and the solution yk = ck sin kax. The normed solution, e.g., for the case k = 1. Le.! X1 = 7r2 % 9.87 is

Jz

(10.52j) y= sin T Z : which is really very close to the approximate solution (10.52h). Remark: With today’s level of computers and science we have to apply, first of all, the finite element method (FEPVI) for numerical solutions of variational problems. The basic idea of this method is given in 19.5.3, p. 910, for numerical solutions of differential equations. The correspondence between differential and variational equations will be used there, e.g., by Euler differential equations or bilinear forms according to (19.146a,b). .Also the gradient method can be used for the numerical solution of variational problems as an efficient numerical method for non-linear optimization problems (see 18.2.6, p. 868).

10.6 Supplementary Problems 10.6.1 First and Second Variation In the derivation of the Euler differential equation with a comparable function (see 10.3.2, p. 552), we stopped after the linear term with respect to E of the Taylor expansion of the integrand of b

I ( E )=

\ F ( z , yo t

€7, yb

+ €4)dx.

(10.53)

a

If we consider also quadratic terms: then we get (10.54)

If we denote as 1. Variation 61 of the functional I[y]the expression (10.55) 2. Variation 6’1 of the functional 1[y] the expression

(10.56)

560

10. Calculus of Variations

then we can write: -

€2 r(o) N €61+ b2 I . 2

(10.57)

We can formalize the different optimality conditions with these variations for the functional I[y] (see [10.7]).

10.6.2 Application in Physics Variational calculus has a determining role in physics. We can derive the fundamental equations of Newtonian mechanics from a variational principle and arrive at the Jacobi-Hamilton theory. Variational calculus is also very important in both atomic theory and quantum physics. It is obvious that the extension and generalization of classical mathematical notions is undoubtedly necessary. So, the calculus of variations must be discussed today by modern mathematical disciplines, e.g., functional analysis and optimization. Unfortunately, we can only give a brief account of the classical part of the calculus of variations (see [10.4], [10.5], [10.7]).

11 Linear Integral Equations 11.1 Introduction and Classification 1. Definitions A n integral equation is an equation in which the unknown function appears under the integral sign. There is no universal method for solving integral equations. Solution methods and even the existence of a solution depend on the particular form of the integral equation. An integral equation is called linear if linear operations are performed on the unknown function. The general form of a linear integral equation is: b(C

g(z)d.(s)= f(z)+ A

J KhY)P(Y)

dY,

c

(11.1)

I z I d.

a(.)

The unknown function is ~ ( 2 the ) ~function K ( z ,y) is called the kernel of the integral equation, and f ( z ) is the so-called perturbation function. These functions can take complex values as well. The integral equation is homogeneous if the function f ( z ) is identically zero over the considered domain, i.e., f(z)= 0, otherwise it is inhomogeneous. X is usually a complex parameter. Two types of equation (11.1) are of special importance. If the limits of the integral are independent of z,Le., a(.) a and b(z) b, we call it a Fredholrn integral equation (11.2a,11.2b). If a(.) t a and b(z) = z,we call it a Volterra integral equation (11.2c, 11.2d). If the unknown function p(z) appears only under the integral sign, Le., g(z) I 0 holds, we have an integral equation ofthe first kindas (11,2a), (11.2~). The equation is called an integral equation ofthe secondkindifg(z) t 1 asin (11.2b). (11.2d).

2'

Pk)= f(.) a

+

J K ( z ,Y)P(Y) dY.

(11W

a

2. Relations with Differential Equations The problems of physics and mechanics relative rarely lead directly to an integral equation. These problems can be described mostly by differential equations. The importance of integral equations is that many of these differential equations, together with the initial and boundary values, can be transformed into integral equations. W From the initial value problem ~'(z) = f(z,y) with z 2 2 0 and y(z0) = yo by integration from zo to z we get

d.1

= Yo +

1:f(E>!/(
d<.

(11.3)

The unknown function y(z) appears on the left-hand side of (11.3) and also under the integral sign. The integral equation (11.3) is linear if the function f(<.y(<)) has the form f.((, ~ ( 0=)a ( < )y(E) b(<), i.e.. the original differential equation is also linear. Remark: In this chapter 11 we only deal with integral equations of the first and second kind of Fredholm and Volterra types. and with some singular integral equations.

+

562

11. Linear Intearal Eauations

11.2 Fredholm Integral Equations oft he Second Kind 11.2.1 Integral Equations with Degenerate Kernel If the kernelK(z. y ) of an integral equation is the finite sum of products of two functions of one variable, Le., one depends only on z and the other one only on y I it is called a degenerate kernelor a product kernel. 1. Solution in the Case of a Degenerate Kernel The solution of a Fredholm integral equation of the second kind with a degenerate kernel leads to the solution of a finite-dimensional equation system. Consider the integral equation (11.4a)

K ( z . y ) = c y ~ ( z ) & (+y )~ z ( ~ ) P z + ( v' )' '

+ an(x)Pn(y).

(11.4b)

The functions al ( z ) $... , cy,(x) and B1(x), . . . , p n ( x )are given on the interval [a, b] and are supposed to be continuous. Furthermore, the functions cy~(z),. . . , cy,(z) are supposed to be linearly independent of one another, Le., the equality (11.5) with constant coefficients c k holds for every x in [a,b] only if c1 = cz = K ( z .y) can be expressed as the sum of a smaller number of products. From (11.4a) and (11.4b) we get:

. . . = c,

= 0. Otherwise,

(11.6a) a

a

The integrals are nolonger functions of the variable z, they are constant values. Let's denote them by A k: b

Ak =

1

& ( y ) p ( y ) &, k = 1,.. . , n.

( 11.6b)

a

The solution function p(x), if any exists. is the sum of the perturbation function f(x) and a linear combination of the functions a l ( z ) ,. . . , a,(z): (11.6~) ~ ( 2= ) f ( ~t)X A ~ C Y ~ (XA~CQ(X) Z) . . . XA,an(x).

+

+ +

2. Calculation of the Coefficients of the Solution The coefficients A I . , , , A, are calculated as follows. Equation (11.6~)is multiplied by &(z) and its integral is calculated with respect to 1c with the limits a and b: ~

The left-hand side of this equation is equal to Ah according to (11.6b). Using the following notation (11.7b)

(11 . 7 ~ )

11.2 Fredholm Intearal Eauations of the Second Kind 563

It is possible that the exact values of the integrals in (11.7b) cannot be calculated. When this is the case. their approximate values must be calculated by one of the formulas given in 19.3, p. 895. The linear equation system (11.7~)contains n equations for the unknown values A I , . . . ,A,: A2 -

...

-kin

Ai +(1 - X ~ 2 2 ) A z-

...

-XCZ, .4, = bz

(1 - Xcl1)Al -XCZ~

-hi:!

A, = b l

~

~

(11.7d)

...................................................... -Xcni AI

-XC,Z

Az -

. . . +(1 - Xc,,)A,

= b, .

3. Analyzing the Solution, Eigenvalues and Eigenfunctions It is known from the theory of linear equation systems that (11.7d) has one and only one solution for Al, , , , A, if the determinant of the matrix of the coefficients is not equal to zero, Le., ~

~

D(X) =

-XCz1

(1 - Xczz) . . .

-XCzn

(11.8)

.................................... -Xc,z . . . (1 - Xc,,) -Xc,1

Obviously D(X) is not identically zero, as D(0) = 1 holds. So there is a number R > 0 such that D(X) # 0 if 1x1 < R. For further investigation we have to consider two different cases. Case D(X) # 0: The integral equation has exactly one solution in the form (11,6c), and the coefficients A I , . . . ,A, are given by the solution of the equation system (11.7d). If in (11.4a) we have a homogeneous integral equation, Le., f(x) 0, then bl = bz = . . . = b, = 0. Then the homogeneous equation system (11.7d) has only the trivial solution AI = .A:! = . . . = A, = 0. In this case only the function p(z) I 0 satisfies the integral equation. Case D(X) = 0: D(X) is a polynomial of no higher than n-th degree, so it can have at most n roots. For these values of X the homogeneous equation system (11.7d) with bl = bz = . . . = b, = 0 also has non-trivial solutions. so besides the trivial solution ( ~ ( zI ) 0 the homogeneous equation system has other solutions of the form p(x) = C . (Alal(x) A ~ c u ~ ( x ). . . A,a,(z)) (C is an arbitrary constant.) Because al(x)., , , . CY,(I)are linearly independent, p(x) is not identically zero. The roots of D(X) are called the eigenvalues of the integral equation. The corresponding non-vanishing solutions of the homogeneous integral equation are called the eigenfunctions belonging to the eigenvalue A. Several linearly independent eigenfunctions can belong to the same eigenvalue. If we have an integral equation with a general kernel, we consider all values of X eigenvalues, for which the homogeneous integral equation has 1. non-trivial solutions. Some authors call the X with D(X) = 0 the characteristic number, and p = - is X

+

+ +

1

b

called the eigenvalue corresponding to an equation form p p ( x ) =

a

K ( x ,y)p(y) d y .

4. Adjoint Integral Equation Now we need to investigate the conditions under which the inhomogeneous integral equation will have solutions if D(X) = 0. For this purpose we introduce the adjoint or transposed integral equation of (11.4a):

564

11. Linear Integral Equations

Let X be an eigenvalue and p(x) a solution of the inhomogeneous integral equation (11.4a). It is easy to show that X is also an eigenvalue of the adjoint equation. Now multiply both sides of (11.4a) by any solution ~ ( zof) the homogeneous adjoint integral equation and evaluate the integral with respect to x between the limits a and 6:

j

b

v ( z ) v ( x dx ) =

a

1f(x)ll(z) 1

dx +

p (i]

1

(11.9b)

K ( z ,Y)llb) dz V(Y) dY.

a

a

a

l

b

Assuming that ~ ( y =) X

K ( x ,y)@(zjdz, we get f(x)$(z) dx = 0 That is: The inhomogeneous integral equation (11.4a) has a solution for some eigenvalue X if and only if the perturbation function f(x) is orthogonalto every non-vanishing solution of the homogeneous adjoint integral equation belonging to the same A. This statement is valid not only for integral equations with degenerate kernels, but also for those with general kernels. a

IA: p(z) = z

+

1,

+ zy'

+I

(z'y

- xy)p(y) dy,

z z 1Q ~ ( z= )

( Y I ( ~ )=

2,

~ ( z =) --z, P ~ ( Y = )

y, &(y) = y') b3(y) = y. The functions c y k ( x ) are linearly dependent. This is why we transform the integral equation into the form p(z) = z+ have (

3 ( ~=)5

+

1:

[x'y

+ x(y' - y)]p(y)dy. For this integral equation we

a'(.) = x, pl(y) = y, Oz(y) = y2 - y. If any solution p(z) exists, it has the form A 1 2 t A:!2 .

~ ~ = ( 5x',)

2 2 2 With these values we have the equations for Al and A': Al - -A2 = - , --Al + 3 3 5 10 2 10 2 10 5 which in turn yield that Al = - , Az = - - and p(z) = x + -2' - -x = -z2 + -2. 21 7 21 7 21 7

I

IB: p(x) = x + X / n s i n ( x + y ) p ( y ) d y , i . e . : K ( z , y ) = s i n ( z + y ) =sinzcosy+cosxsiny, p(z) =z

+ Xsinx

1';

cosy p(y) dy

ell = i " s i n z c o s x d z = 0,

+ A cosx T

c12= [cos'xdz

bl = [zcosxdz

=2

= -2,

=7

cZ2= i * c o s x s i n z d z = 0, bz = [ z s i n z d z = T. 2' IVith these values the system (11.7d) is Al - ATAZ = -2, -ApAl+ A2 = K . It has a unique solution 2 2 cZ1 = L'sin'zdz

~

1

;;

1 -Afor any X with D(X) = 1 -A-

T '

= 1 - X'-

the solution of the integral equation is p(x) = x

K'

4

#

0. SO A1

A--2 = +z, 1- A'-

A2 = 4

T(l

- A)

~

1 - A'-

'x , and 4

+ ___ A Kz [ ( A ~ - 2 ) s i n x + ~ ( l - A ) c o s s 1 - A'-

eigenvalues of the integral equation are A1

2 =-

The homogeneous integral equation p(z) =

x

Ah

4 2 XZ = -- .

/'sin(z

+ y)p(y) dy has non-trivial solutions of the

11.2 Fredholm Inteoral Eouations of the Second Kind 565 2 form pk(z) = Xk(A1 sinz t Azcosz) ( k = 1,2). For XI = - we get AI = A2, and with an arbitrary ?r

constant A we have pl(z)= A(sinz t cosz). Similarlyfor

A2

2

= -- we get pz(z) = B(sin z K

- cos T )

with an arbitrary constant B. Remark: The previous solution method is fairly simple but it only works in the case of a degenerate kernel. By this method. however, we can get a good approximate solution in the case of a general kernel too if we can approximate the general kernel by a degenerate one closely enough.

11.2.2 Successive Approximation Method, Neumann Series 1. Iteration Method Similarly to the Pzcard zteratzon method (see 9.1.1.5, l., p. 494) for the solution ofordinary differential equations, an iterative method needs to be given to solve Redholm integral equations of the second kind. Starting with the equation b

P(2) = f(.)

i-

1

( 11.10)

K ( z ,Y ) d Y ) dY,

a

we define a sequence of functions .po(z), p1(z), p2(2),. . . . Let the first be po(z) = f(~).We get the subsequent pn(z)by the formula b

Pn(.) = f(z)+

1

K ( z ,Y)Pn-l(Y) dy ( n = 4 % .. . : d z ) = f(.)).

( 11.1l a )

a

Following the given method our first step is b

(11.11b)

P1(2)= f(z)i- x/K!T.Y)f(y)dY. a

.4ccording to the iteration formula this expression of p(y) is substituted into the right-hand side of (11.10). To avoid the accidental confusion of the integral variables, let’s denote y by q in (1l.llb). (1l.llc) b b

b

=

f(.) + A /

K ( z ,y ) f ( y ) dY + xz

a

SJK ( z ,

Y)K(Y.

a)f(o)dY d7).

( 11.1Id)

a a

1 b

Introducing the notation of K l ( z ,y) = K ( s ,y) and Kz(z,y) =

K ( z ,<)K(E.y) d<, and renaming q

as y , we can write yz(x) in the form

(1l.llf)

I

566

11. Linear Inte.qral Equations

we get the representation of the n-th iterated cp,(z): b

/

+

pn(x) =

b ~ l ( xY ), ~

( Y )dy + . ' ' +

J

~ " ( y 2 ) ,f ( y )

( 11.1lg)

dy.

a

a

We call Kn(x,y) the n-th zterated kernel of K ( z ,y).

2. Convergence of the Neumann Series To get the solution p(z),we have to discuss the convergence of the power series of X

f(x) +

5

n=l

1

Kn(x. Y ) f ( Y ) dy,

(11.12)

a

which is called the Yeumann series. If the functions K ( r ,y) and f ( z ) are bounded, Le., the inequalities (11.13a) IK(x,y)l < M ( a 5 x 5 b, a 5 y 5 b) and I f ( x ) i < N ( a 5 z 5 b ) , hold. then the series s.

I X M ( b - a)In

A'

(11.13b)

n=O

is a majorant series for the power series (11.12). This geometric series is convergent for all 1

IX' <

m,

(11.13~)

The Seumann series is absolute and uniformly convergent for all values of X satisfying (11.13~).By a sharper estimation of the terms of the Neumann series we can give the convergence interval more precisely. According to this, the Neumann series is convergent for (11.13d)

This limit for the parameter X does not mean that there are no solutions for any 1x1 outside the bounds set by (11,13d), but only that we cannot get it by the Neumann series. Let's denote by

c Xn-lKn(s: y) r

r ( z .y; A) =

(11.14a)

n=l

the resolvent or solving kernel of the integral equation. Using the resolvent we get the solution in the form b

=

+

J T ( x ,Y1 X)f(y) dy.

(11.14b)

a

IFor the inhomogeneous Fredholm integral equation of the second kind p(x) = x + X 1 1 we have Kl(x,y) = zy. Kz(z,y) = / l s i ) i ) y d y = ?xy, Ks(z,y) = -zy,. 9

and from this r ( z . y: X) = zy

c,

1x1 < 1, because i K ( x ,y)i 5 .d '

1'

xy p(y)dy

. . . Kn(z,y) =

XY

17 . With the limit (11.13~)the series is definitely convergent for = 1 holds. The resolvent T ( z ,y; A) = -is a geometric series

11.2 Fredholm Integral Equations of the Second Kind

which is convergent even for (XI < 3. Thus from (11.14b) we get cp(x)= x+X

%

dy=

(l-?)

567

X

1-3

Remark: If for a given X the relation (11.13d) does not hold, then we can decompose any continuous , where K ' ( x , y) is a kernel into the sum of two continuous kernels K ( z :y) = K ' ( z , y) + K Z ( zy): degenerat,e kernel, and K Z ( x y) , is so small that for this kernel (11.13d) holds. This way we have an exact solution method for any X which is not an eigenvalue.

11.2.3 Fredholm Solution Method, Fredholm Theorems 11.2.3.1 Fredholm Solution Method 1. Approximate Solution by Discretization A Fredholm integral equation of the second kind b

v(z) = f(z)+

1

(11.15)

K ( x ,Y)cp(Y) dY

a

can be approximately represented by a linear equation system. We need to assume that the functions K ( z .y) and f(x) are continuous for a 5 x 5 b, a 5 y 5 b. 1Ve willapproximate the integral in (11.15) with the so-calledleft-hand rectangular formula (see 19.3.2.1, p. 896). It is also possible to use any other quadrature formula (see 19.3.1, p. 895). With an equidistant partition yk=a+(k-1)h

( k = 1 . 2 . . . . , n; h = - b ; a )

(11.16a)

we get the approximation x f(x) + [ K ( xYI)P(YI) , + + K ( x ,~ n ) ~ ( ~ n ) l ' Let's replace p(z) in this expression by a function F ( x ) exactly satisfying (11.16b): I ,

(11.16b) ( 11 . 1 6 ~ ) [ K ( x ,Y I ) V ( Y I ) + , + K ( x ,y n ) ~ ( y n ) l . To determine this approximate solution, we need the substitution values of F(z) at the interpolation nodes xk = a + ( k . 1)h. If we substitute z = 51, z = x Z 1 .. ,. z = z, into (11.16c), we get a linear equation system for the required n substitution values of F(zk). Using the shorthand notation (11.17a) Kjk = K ( x j .Y k ) . Y'k = V(zk)r f k = f (zk) we get (1 . XhK11)$7, -AhK12 9 2 .... -XhK1n Pn = f l i ~ ( x=) f (x)+

-XhK2191 +(I . XhK22)~2....

I

-XhKzn pn = f z l

................................................. -XhKnl -XhKn2cp2 .... +(1 . XhKnn)qn= f, This system has the determinant of the coefficients (1 - XhKl1) -XhK12 . . -XhKIn -XhK21 (1 . XhK22) . . . -XhKzn

D,IX) = ........................................... -XhKnl -XhK,2 . . . (1 - XhK,,)

1

1.I

(11.17b)

(11 . 1 7 ~ )

This determinant has the same structure as the determinant of the coefficients in the solution of an integral equation with a degenerate kernel. The equation system (11.17b) has a unique solution for every X where Dn(X) # 0. The solution gives the approximate substitution values of the unknown function 'j (x)at the interpolation nodes. The values of X with Dn(X) = 0 are approximations of the

I

568

11. Linear Inteoral Eauations

eigenvalues of the integral equations. The solution of (11.17b) can be written in quotient form (see Cramer rule, 4.4.2.3, p. 275):

Here we get Dk(X) from Dn(X) by replacing the elements of the k-th column by f l , f2,. . . , fn.

2. Calculation of the Resolvent If n tends to infinity, so the number of rows and columns of the determinant Dk(X) and Dn(X), too. We use the determinant to get the solution kernel (resolvent) T ( z ,y; A) (see 11.2.2, p. 565) in the form (11.19b) It is true that every root of D(X) is a pole of T ( z ,y; A). Exactly these values of A. for which D(X) = 0, are the eigenvalues of the integral equation (11.15),and in this case the homogeneous integral equation has non-vanishing solutions, the eigenfunctions belonging to the eigenvalue A. In the case of D(X) # 0, knowing the resolvent T ( z ,y; A), we have an explicit form of the solution: b

d z ) = f(l)+xJr(z:Y;X)f(Y)dY = f(z)+mjD(z’”;”)f(Y)dy. A b

( 11.19c)

a

To get the resolvent, we need the power series of D(z, y ; A) and D(X) with respect to A:

’

where do = 1, Ko(z,y) = K ( z ,y), and we get further coefficients from the recursive formula:

1

_-

X7r

DdX) =

12

--&AT 12

0

0

I-- &Xa 24

_-X7r 24

_ 3x77 _

l--

24

system has thesolutionpz =

~

2--A

192

fixa 24

1 &a

6

I @ = -

v3 . IfX = 1,then(ol = 0, 2--A & 6

‘pz = 0.915, (p3

=

11.2 Fredholm Integral Equations of the Second Kind 569

1.585. The substitution values of the exact solution are: ~ ( 0 = ) 0,

(o

(T) = 1. p (T) = 1.732 6 3

In order to achieve better accuracy, the number of interpolation nodes needs to be increased ( 4 s~ x2)p(y)dy; do = 1, K ~ ( xy), = 42y - x 2 , d l = 4

4

1

(4xt-x2)(4ty-tZ)dt = ~ + 2 ~ ~ y - - z ~ - - x y , d=2 3 3

1 K2(z,y)= - ( 4 x y - z z ) - J 1 K ( x . t ) ~ ~ ( t , y ) d t= 0. Withthesethevaluesdj, K3(2,y)andallthefol18

lowingvaluesofdkandKk(x,y)areequaltozero. T(x,y:X) = l-X+-

*

XZ

XZ

18

From 1 - X + - = 0. we get the two eigenvalues Xl,z = 9 3fi. If X is not an eigenvalue, we have the l8

solution p(2) = z +

1

1

~ ( xY;,XU(Y) dy =

+

3~(2X- 3Xx 6) X2 - 18X 18 . +

11.2.3.2 Fredholm Theorems For the Fredholm integral equation of the second kind b

94x1 = f(2)+

I K(X>Y)P(Y)dY

(11.21a)

J

0

the correspondent adjoint integral equation is given by b

v ( z ) = g(2) +

J K(y, x)dJ(Y)dY.

(11.21b)

0.

For this pair of integral equations the following statements are valid (see also 11.2.1,p. 562). 1. A Fredholrn integral equation of the second kind can only have finite or countably infinite eigenvalues. The eigenvalues cannot accumulate in any finite interval, Le., for any positive R there are only a finite number of X for which /XI < R. 2. If X is not an eigenvalue of (11,21a), then both of the inhomogeneous integral equations have a unique solution for any perturbation function f(x) or g(x), and the corresponding homogeneous integral equations have only trivial solutions. 3. If X is a solution of (11,21a), then X is also an eigenvalue of the adjoint equation (11.21b). Both homogeneous integral equations have non-vanishing solutions, and the number of linearly independent solutions are the same for both equations. 4. For an eigenvalue X the homogeneous integral equation can be solved if and only if the perturbation function is orthogonal to every solution of the homogeneous adjoint integral equation, Le., for every solution of the integral equation

a

a

The Fredholm alternative theorem follows from these statements: Either the inhomogeneous integral equation can be solved for any perturbation function f ( x ) or the corresponding homogeneous equation

I

570

11. Linear Inteqral Equations

has non-trivial solutions

11.2.4 Numerical Methods for Fredholm Integral Equations ofthe

Second Kind Often it is either impossible or takes too much work to get the exact solution of a Fredholm integral equation of the second kind b

P(5) = f(z)+

1

K ( z :Y ) d Y ) dY

(11.23)

a

by the solution methods given in 11.2.1, p. 562, 11.2.2, p. 565 and 11.2.3, p. 567. In such cases certain numerical methods can be used for approximation. Three different methods are given below to get the numerical solution of an integral equation of the form (11.23).

11.2.4.1 Approximation of the Integral 1. Semi-Discrete Problem Working on the integral equation (11.23) we replace the integral by an approximation formula. These approximation formulas are called quadrature formulas. They take the form

1

f(z)dz SZ

Q[a,b](f)

a

=

5

(11.24)

ukf(zk).

k=l

Le.. instead of the integral we have a sum of the substitution values of the function at the interpolation nodes x k weighted by the values w k . The numbers w k should be suitably chosen (so as to be independent of f j . Equation (11.23) can be written in the approximate form: n

a f(z)+ xQ[a,b](K(z,‘)p(‘))= f(z)+

1

W k K ( z , Yk)’P(Yk).

(11.25a)

k=l

The quadrature formula &(a,b](K(z, .)p(.))also depends on the variable z.The dot in the argument of the function means that the quadrature formula will be used with respect to the variable y . Defining the relation n

a ( z )= f(x)t

WkK(z,Y k ) F ( Y k ) .

( 11.25b)

k=l

p(zj is an approximation of the exact solution p(x). &’e can consider (11.25b) as a semi-discreteproblem, because the variable y is turned into discrete values while the variable z can still be arbitrary. If the equation (11.25b) holds for a function V(z) for every z E [a>b], it must also be valid for the interpolation nodes z = z k : (11 . 2 5 ~ )

This is a linear equation system containing n equations for the n unknown values T ( z k ) . Substituting these solutions into (11.25b) we have the solution of the semi-discrete problem. The accuracy and the amount of calculations of this method depend on the quadrature formula used. For example if we use the left-hand rectangular formula (see 19.3.2.1, p. 896) with an equidistant partition y k = xk = a + h(k - l ) , h = ( b - a ) / n , ( k = 1,.. . .n): (11.26a)

(11.26b)

11.2 Fredholm Integral Eazlations of the Second Kind 571

the system (11.25~)has the form: - X h K 1 2 ~2 - . . . (1 - X h K 1 1 ) p l -XhK2: ~1 +(1 - XhK2292) - . . .

.................

- X h K 1 n 'Pn

= f1S

-XhK2n Pn = fi,

(11 . 2 6 ~ )

I...........................................

-XhKn2 ~2 - . . . +(1 - X h K n n ) i p n = fn. - X h K n l (31 We had the same system in the Fredholm solution method (see 11.2.3, p. 567). As the rectangular formula is not accurate enough, for a better approximation of the integral we have to increase the number of interpolation nodes, along with an increase in the dimension of the equation system. Hence we get the idea of looking for another quadrature formula.

2. Nystrom Method In the so-called Nystrom method we use the Gauss quadrature formula for the approximation of the integral (see 19.3.3, p. 897). In order to derive this, we consider the integral

I=

j

f(x)dx.

(11.27a)

We replace the integrand by a polynomial p(x), namely the interpolation polynomial of f(x) at the interpolation nodes x k : n

P(z) =

Lk(x)f(xk)

with

k=l

(z- 21) . . . (x - xk-I)(x - Q + l ) . . . (x - xn)

Lk(x)=

For this polynomial, p ( x k ) = f ( i k ) , results in the quadrature formula b

/ f(x)dx a

(11.27b) -z k + l ) . ..(xk -xn)' k = 1,.. . n. The replacement of the integrand f(x) by p(x)

(xk - 2:).. . (xk - X k - l ) ( x k

/

2

a

k=l

b

b

x

~

f(q)& ( x ) dx =

/p(x) dx = a

k=l

Idkf(2k)

b

with

wk

=

/ Lk(x)dx.

(11.27~)

a

For the Gauss quadrature formula the interpolation nodes cannot be chosen arbitrarily but we have to choose them by the formula: L

L

The n values t k are the n roots of the Legendre polynomial of the first kind (see 9.1.2.6, 3., p. 509) 1 d" [ ( t 2 - l)"] (11.28b) P,(t) = 2" n! dtn ' These roots are in the interval [-I, +1]. We calculate the coefficients w k by the substitution x - x k = b-a -(t - t k ) , so: 2

= (b - a ) A k . (11.29) In Table 11.1 we give the roots of the Legendre polynomial of the first kind and the weights A k for n = 1 , . . . , 6.

I

572

11. Linear Inteqral E~uations

Table 11.1Roots of the Legendre polynomial of the first kind

I

I n / t 1 tl= 0 2 1 tl = -0.5774 t 2 = 0.5774 ti = -0.7746 3 t2 = 0 t 3 = 0.7746 ti = -0.8612 4 t z = -0.3400 t 3 = 0.3400 t 4 = 0.8612

1

I

1

A Al=l A1 = 0.5 A2 = 0.5 Ai = 0.2778 A2 = 0.4444 A3 = 0.2778 Ai = 0.1739 A2 = 0.3261 A3 = 0.3261 A4 = 0.1739

I1

n 5

t

ti = -0.9062 t 2 = -0.5384 t3 = 0 t 4 = 0.5384 - t 5 = 0.9062 6 ti = -0.9324 tz = -0.6612 t 3 = -0.2386 t 4 = 0.2386 t 5 = 0.6612 tfi = 0.9324

A

'41 = 0.1185 A2

= 0.2393

A3

= 0.2844

A4

= 0.2393

A5

= 0.1185

Ai = 0.0857 A2 =

0.1804

A3 = 0.2340

A4 = 0.2340 A5 = 0.1804 A6 = 0.0857

ISolvetheintegralequationq(2) = c o s r t ~ (2 e2 '++ Rl Z) + - / l e z v p ( y ) d y b y t h e Y y s t r o m m e t h o d for n = 3. 12 = 3 : 2 1 = 0.1127. 2 2 = 0.5, 53 = 0.8873, A1 = 0.2778, A2 = 0.4444, A3 = 0.2778, f l = 0.96214, f 2 = 0.13087, j 3 = -0.65251. K11 = 1.01278, Kzz = 1.28403, K33 = 2.19746, Kl2 = K Z I= 1,05797, K13 = K31 = 1.10517. K23 = K32 = 1.55838. The equation system (11.25~)for P I , ' p z ! and 93 is 0.71864~1- 0.4701692 - 0.30702p3 = 0.96214. -0.2939091 t 0.4293892 - 0.43292~3= 0.13087, -0.30702~1- O.69254pz 0.38955~3= -0.65251.

+

The solution of the system is: 'p1 = 0.93651, p2 = -0.00144, 9 3 = -0.93950. The substitution values of the exact solution at the interpolation nodes are: p ( q ) = 0.93797, y ( z z )= 0; ( ~ ( 2=~-0.93797. )

11.2.4.2 Kernel Approximation Replace the kernel K ( z ,y) by a kernel R ( z ,y) so that K(z,y) x K ( z ,y) for a 5 z 5 b. a 5 y Try to choose a kernel making the solution of the integral equation

5 b.

b

4 . 1 = f b ) t x-/R(z:Y)F(Y)dY

-

(11.30)

a

the easiest possible.

1. Tensor Product Approximation A frequently-used approximation of the kernel is the tensor product approzzmatzon in the form (11.31a) with given linearly independent functions ~ ( z ). .,. , a,(z) and po(y), . . . % pn(y) whose coefficients dje must be chosen so that the double sum approximates the kernel closely enough in a certain sense.

11.2 FTedholm Intearal Eauations of the Second Kind 573

(11.31~) Functions cyo(z), . . . a,(.) and &(y), . . . , D,(y) should be chosen so that the coefficients d j k in (11.31a) can be calculated easily and also that the solution of (11.31~)isn't too difficult.

2. Special Spline Approach Let's choose

( 11.32)

lo

otherwise

for a special kernel approximation on the interval of integration [a,b] = [0,1]. The function

(

ah(.)

has

')

non-zero values only in the so called carrier interval -, - , (Fig. 11.1).

To calculate the coefficients d j k in (11,31a),consider x ( z ,y ) at the points z = l/n, y = i/n( I , i = 0 , 1 , . . . , n ) . We get 1 f o r j = 1, k = i, aj ak = 0 otherwise

(i) (i) { R (:, i) (4,i).

and consequently X ( l / n ,i / n ) = dli. Hence, we substitute dl, =

n

n

n

=K

-

Figure 11.1

n n

Now (11.31a) has the form

aj(z)Pk(Y). j=O

(11.34)

k=O

As we know, the solution of (11.31~)has the form

a(.)

+

+. +

(11.35) = f ( ~ ) Aoc~o(z) . . A,a,(z). The expression Aocuo(z) + . . . + A,a,(z) is a piecewise linear function with substitution values Ak at the points i k = k / n . Solving (11.31~)by the method given for the degenerate kernel, we get a linear equation system for the numbers Ao, . . . ,A,: -Xcol A1 - . . . -Xclo A0 + (1 - XcllA1) - . . .

(1 - Xco0)Ao

-XCO, A, = bo, A, = b l ,

....................................................... -XC,~ A1 - . . . + (1 - Xc,,)A, = b,, -&o AO where

(11.36a)

574

11. Linear Inteqral Equations

. . . +K ( ~n, ~n) ] a n ( z ) c Y k ( z ) d z .

= K(i,i)/ao(z)ak(z)dz+

(11.36b)

0

3n 1

I,, = /a,(z)ak(x) dx = 0

3n 1 6n ,0

for j = 0, k = 0 and j = n, k = n, for j = k , l < j < n , (11.36~) for j = k + l , j = k - 1 , otherwise.

(11.36d) Taking a matrix C with numbers c,k from (11.36a), a matrix B with the values K ( j / n ,k / n ) and a matrix A with the values I,, respectively, a vector b from the numbers bo,. . . , b,. and a vector a from the unknown values .'io2. . . A,, the equation system (11.36a) has the form (I - XC)a = (I - XBA)a = b. (11.36e) In the case when the matrix (I - XBA) is regular, this system has a unique solution a = (Ao,. . . , A n ) .

11.2.4.3 Collocation Method Suppose the n functions q l ( z ) ,. , , , pn(z)are linearly independent in the interval [a,b]. They can be used to form an approximation function $ ( E ) of the solution p(z): ( 11.37a) p(x) zz S ( x ) = a l p l ( z ) + azc~z(z)+ . . . + anpn(z). The problem is now to determine the coefficients al, . . . , a n . Usually, there are no values al... . , a n such that the function F(z)given in this form represents the exact solution p(z) = F(z)of the integral equation (11.23). Therefore, n interpolation points ~ 1 ,. .., z nare defined in the interval of integration, and it is required that the approximation function (11.37a) satisfies the integral equation at least at these points: (11.37b) ~ ( 5 ,= ) aiyi(zk) + . . . + anpn(zk) b

= f(xk)

+ ~ / ~ . ( z , , y[ a)l p l ( y )+ . + anpn(Y)l dy ,,

a

LVith some transformations this equation system takes the form:

( k = 1,.. . . n ) .

(11 . 3 7 ~ )

11.3 Fredholm Inteqral Equations ofthe First Kind

575

Then the equation system to determine the numbers al,. . . , a, can be written in matrix form: (A - AB) a = b. (11.37g)

Ip(x) =

$- + 1' &p(y) ."

+

d y . The approximation function is p(x) = alzZ uzx

+

u3,

y l ( z )=

x2,pz(x)= 2. p3(x)= 1. The interpolation nodes are x1 = 0, x2 = 0 . 5 , = ~ 1. 0 0 0 0 1

4 4 4 -

1 1 1

-

_

The system of equations is a3

=

0.

(4 - 7) al + (i- T) az + (1 - $)a3 = 2J;i 0 1 1 +-as= 7 a1 + z3 a2 3 2' 1

4

1

4

1

= - 0 . 8 1 9 7 ~+ ~ 1.80922, whose solutions are a l = -0.8197,a2 = 1.8092. a3 = 0 and with these and so q(0) = 0. g(O.5)= 0.6997, T(1) = 0.9895. The exact solution of the integral equation is p(z) = fiwith the values p(0) = 0, y(0.5) = 0.7071, p(1) = 1. In order to improve the accuracy in this example. it is not a good idea to increase the degree of the polynomial, as polynomials of higher degree are numerically unstable. It is much better to use different spline approximations, e.g., a piecewise linear approximationq(2) = u l y l ( s )+azyz(z) . '+anpn(z) with the functions introduced in 11.2.4.2

a(.)

+.

otherwise

a(.).

In this case. the solution p(2) is approximated by a polygon Remark: There is no theoretical restriction as to the choice of the interpolation nodes for the collocation method. In the case, however, when the solution function oscillates considerably in a subinterval, we have to increase the number of interpolation points in this interval.

11.3 Fredholm Integral Equations ofthe First Kind 11.3.1 Integral Equations with Degenerate Kernels 1. Formulation of the Problem Consider the Fredholm integral equation of the first kind with degenerate kernel b

f(.)

= /(ul(x!3liY)

+

"

'

+ % l ( ~ ) P n ( Y ) M YdY)

(c 5 z I 4,

(11.38a)

a

and introduce the notation similar to that used in 11.2. p. 562, b

.4, = / f l j ( ~ ) & ) d y a

(.i=1,2,..~,n).

(11.38b)

576

11. Linear Inteqral Equations

Then (11.38a) has the form (11 . 3 8 ~ ) f(z)= Alal(z) i- , + Anan(.), Le.. the integral equation has a solution only iff(.) is a linear combination of the functions a l ( z ) ,. . . a,(z).If this assumption is fulfilled, the constants A I , .. . , A , are known.

2. Initial Approach We are looking for the solution in the form ~ ( z=) C I & ( ~ )i- ' ,

+ cnPn(z)

(11.39a)

where the coefficients c l l . .. , c, are unknown. Substituting in (11.38b) (11.39b) a

a

and introducing the notation b

(11 . 3 9 ~ )

K*, = /3%(Y)PAY)dY a

we have the following equation system for the unknown coefficients cl,. . . ,:c, Kllcl t + Klnc, = A i . t..

+. . . + K,,c,

K,lcl

(11.39d) = A,.

3. Solutions The matrix of the coefficients is non-singular if the functions Pl(y),. . . , ,&(y) are linearly independent (see 12.1.3,p. 596). However, the solution obtained in (11.39a) is not the only one. Unlike the integral equations of the second kind with a degenerate kernel, the homogeneous integral equation always has a solution. Suppose @(z) is a solution of the homogeneous equation and p(z) is a solution of (11.38a). Then p(z) + @(z) is also a solution of (11.38a). To determine all the solutions of the homogeneous equation, let us consider the equation (11.38~)with f(z)= 0. If the functions a l ( z ) ,. . . , a,(z) are linearly independent, the equation holds if and only if b

Aj =

/ ~,(Y)v(Y)

=0

( j = 1,2,. . . I n ) l

(11.40)

a

Le.. every function y h ( y )orthogonal to every function Dj(y) is a solution of the homogeneous integral equation.

11.3.2 Analytic Basis 1. Initial Approach Several methods for the solution of Fredholm integral equations of the first kind b

f(.)

=

/ K(.?

Y)yo(Y) dy (c I z I 4

(11.41)

a

determine the solution p(y) as a function series of a given system offunctions (P,(y)) = {Pl(y), &(y), , . .}. i.e., we are looking for the solution in the form

c 00

Pb)=

C,P,(Y)

(11.42)

3=1

where we have to determine the unknown constants cj. When choosing the system of functions we have to consider that the functions (P,(y)) should generate the whole space of solutions, and also that the

11.3 Fredholm Intearal Eouations of the First Kind 577

calculation of the coefficients cj should be easy. For an easier survey, we will discuss only real functions in this section. All of the statements can be extended to complex-valued functions, too. Because of the solution method we are going to establish, certain properties of the kernel function K ( z ,y) are required. We assume that these requirements are always fulfilled. Next, we discuss some relevant information.

2. Quadratically Integrable Functions .4function q(y) is quadratzcally zntegrable over the interval [a,b] if

f

lV(Y)I2dY <

(11.43)

n

holds. For example, every continuous function on [a,b] is quadratically integrable. The space ofquadratically integrable functions over [a,b] will be denoted by L2[a,b].

3. Orthonormal System Two quadratically integrable functions a ( y ) , bj(y),y E [a,b] are considered orthogonal to each other if the equality (11.44a)

f3t(Y)bAY)dY = 0 a

holds LVe call a system of functions (pn(y)) in the space L2[a,b] an orthononnalsystem if the following equalities are true: ( 11.44b)

.4n orthonormal system of functions is complete if there is no function B ( g ) # 0 in LZ[a,b] orthogonal to every function of this system. A complete orthonormal system contains countably many functions. These functions form a baszs of the space L2[a,b]. To transform a system of functions (P,(y)) into an orthonormal system (3i(y)) we can use the Schmzdt orthogonalzzatzon procedure. This determines the coefficients bnl, b n z r , , , b, for n = 1 , 2 , . . . successively so that the function ~

(11 . 4 4 ~ ) j=1

is normalized and orthogonal to every function p;(y), . . . , PA-l(y). 4. Fourier Series If (bn(y)) is an orthonormal system and $(y) E L2[a,b], we call the series 00

Cd333(Y) = Y ( Y )

(11.4Sa)

j=1

the Fourier series of ~ ( y with ) respect to (Pn(y)), and the numbers dj are the corresponding Fourier coefficients. Based on (11.44b) we have: b

/ (1

ak(Y)@(Y)

dy =

dj

3=1

1

p~(?l)bkok(Y) d!4 = dk.

(11.45b)

a

If (&(y)) is complete, we have the Parseval equality (11.45~)

I

578

11. Linear Integral Equations

11.3.3 Reduction of an Integral Equation into a Linear System of Equations -4 linear equation system is needed in order to determine the Fourier coefficients of the solution function p(y) with respect to an orthonormal system. First, we have to choose a complete orthonormal system (,!-ln(y)).y E [a. b]. A corresponding complete orthonormalsystem (cyn(x))can be chosen for the interval x E [c, d ] . With respect to the system (cyn(x))the function f(z)has the Fourier series (11.46a) If the integral equation (11.41) is multiplied by cyl(z)and the integral is evaluated for 2 running from c to d alone, we get: d b

fi

=

11

K ( x ,YMY14z) dY dx

c a

(11.46b)

The expression in braces is a function of y with the Fourier representation

1

m

K ( x ,~ ) ~ t dz ( z =) K ~ ( Y=)

Kqb'j(y) with

(11.46~)

j=1

C

a c

b'ith the Fourier series approach 35

dY) = k=l C k 3 k ( Y )

(11.46d)

we get

(11.46e)

Because of the orthonormal property (11.44b), we have the equation system m

ft

=

1K I j c j

(i = 1,2,. . .).

(11.46f)

J=1

This is an infinite system of equations to determine the Fourier coefficients cl, c 2 . .. . . The matrix of coefficients of the equation system

(11.46g)

11.3 Fredholm Integral Equations of the First Kind 579

is called a kernel rnatrzz. The numbers f, and Kt3 (isj = 1 , 2 , . . .) are known, although they depend on the orthonormal system chosen. sin y If(x) = p(y) dy, 0 5 x 5 T.The integral is considered in the sense of the Cauchy T 0 cosy - cosx principal value. .4s a complete orthogonal system we use: . . 1 I. aa(x)= - a,(x) =

'ST ,/li

%

By (11.46d), the coefficients of the kernel matrix are

dxdy = 0 ( j = 1 , 2 , . . .),

-

??j~i~sinysinzycosrx dx dy = $ i' sin

y sin zy

'

cosix

d z } dy (z = 1 , 2 , . . .).

cos y - cos 5 For the inner integral the equation z3-7r7r

S 0

0

cosix sin i y dx = -Tcosy - cosz sin y

holds. Consequently Kzj =

-2T

(11.47)

0 sinjysiniy d y = O

1.1

-1 for i = j .

f(z)cu,(x)dx (i = 0 , 1 , 2 . . . .). The equation

The Fourier coefficients of f(x)from (11.46a) are fi =

[-:-: : :1 [!;) 0 0 0 ."

system is

=

solution only if the equality fa =

According to the first equation, the system can have any

[f(z)a~(x) dz = - f(x)dx = 0 holds. Then we have cj = S' A

-fi

for i # j :

0

(1 = 1.2.. . .), and p(y) = -

f (2)dx.

11.3.4 Solution ofthe Homogeneous Integral Equation ofthe First Kind If p(y) and $(y) are arbitrary solutions of the inhomogeneous and the homogeneous integral equation respectively, Le., h

h

then the sum p(y) + @(y) is a solution of the inhomogeneous integral equation. Therefore we have to determine all the solutions of the homogeneous integral equation. This problem is the same as determining all the non-trivial solutions of the linear equation system X

1K& j=1

= 0 ( 2 = 1 , 2 , ,. .)

(11.49)

580

11. Linear Inteqral Equations

.4s sometimes this system is not so easy to solve, we can use the following method for our calculations. take the functions If we have a complete orthonormal system (an(z)), d

K,(y) =

1

K ( z ,y)a,(x) dz (i = 1,2,. . .).

(11.50a)

C

If @(y) is an arbitrary solution of the homogeneous equation, Le.,

p

(11.50b)

K ( x ,y ) v h ( y )dy = 0

a

holds, then multiplying this equality by q ( z ) and performing an integration with respect to z,we get (11.50~) i.e., every solution ph(y)of the homogeneous equation must be orthogonal to every function K,(y). Ifwe replace the system (K,(y)) by an orthonormal system (K;l(y)) using an orthogonalization procedure, instead of (11.50~)we have (11.50d) a

If we extend the system (KG(y))into a complete orthonormal system, the conditions (11.50d) are obviously valid for every linear combination of the new functions. If the orthonormal system (KG(y))is already complete, then only the trivial solution ph(y)= 0 exists. We can calculate the solution system of the adjoint homogeneous integral equation in exactly the same way: (11.50e)

K ( z ,y)@(x)dx = 0. p(y) dy = 0, 0 5 z 5

1.2,. . .), K,(y) =

pi/

7r.

An orthonormal system is: a,(z) =

= sinzsinzz

T7r

0

cosy-cosx

(11.47) twice we get K,(y) = -

de. Applying

cos y - cos z

--s (sin(z - ~);~nysin(z+ 1)y) = $coszy

(z = 1 , 2 , . ..). The

1

system (K,(y)) is already an orthonormal system. The function Ko(y) = - completes this system.

fi

Consequently the homogeneous equation has only the solution: ph(y)= ' e

fi

= 2,

( e is arbitrary).

11.3.5 Construction of Two Special Orthonormal Systems for a Given Kernel 1. Preliminaries The solution of infinite systems of linear equations we saw in 11.3.3, p. 578, is not usually easier than the solution of the original problem. Choosing suitable orthonormal systems (a,(z))and (&,(g)) we can change the structure of the kernel matrix K in such a way that the equation system can be solved easily. By the following method we can construct two orthonormal systems such that the coefficients

11.3 Fredholm Inteqral Equations of the First Kind 581

Kt3of the kernel matrix are non-zero only for z = j and i = j + 1 Using the method given in the previous paragraph, we first determine two orthonormal systems (p,h(y)) and ( a h ( z ) ) ,the solution systems of the homogeneous and the corresponding adjoint homogeneous integral equations respectively. This means that we can give all the solutions of these two integral equations by a linear combination of the functions p,"(y) and ak(x). These orthonormal systems are not complete. By the following method we complete these systems step by step into complete orthonormal systems a J ( z ),!$(y) . (J = 1 , 2 , . . .).

2. Procedure First a normalized function a1(z) is determined, which is orthogonal to every function (ak(x)). Then the following steps are performed for j = 1 , 2 , . . .: 1. Determination of the function OJ(y) and the number vJ from the formulas (11.5la) d vJ$j(y)

= / K ( x $Y ) a j ( z ) d x - /13-1pJ-l(y)

(3 #

'11

(11.51b)

C

so that vj is never equal to zero and R j ( y ) is normalized. Then bJ(y) is orthogonal to the functions

((P,h(Y))>$*(Y)>

$j-l(Y)).

" ' 3

2. Determination of the function aJtl(z) and the number p, from the formula b PJaJ+1(")

=/K(5,y)OJ(~)dy-vJQ3(z)' a

There are two possibilities:

(11'51c)

a) pJ # 0: The function aJtl(z)is orthogonal to the functions ((ak(x)),al(z), . . . , a,(.)).

b) p J = 0: Then the function aJtl(z) is not uniquely defined. Here again we have two cases: bl) Thesystem ((ak(z)),a1(x),...,a~(2)) isalreadycomplete. Thenthesystem ( ( & ( y ) ) , b l ( y ) , . . . , $J(y)) is also complete. and the procedure is finished.

bz) The system ((ak(x)), al(z), . . . ,a,(.)) is not complete. Then again we choose an arbitrary function aJ+l(z)orthogonal to the previous functions. This procedure is repeated until the orthonormal systems are complete. It is possible that after a certain step the case b) does not occur during a countable number of steps, but the system of this countable number of functions ((ah(.)), al(z),. . .) is still not complete. Then again we can start the procedure by a function &I(.), which is orthogonal to every function of the previous system. If the functions aj(x), 3,(y) and the numbers v J spJ are determined by the procedure given above, we have the kernel matrix K in the form

(11.52)

I

582

11. Linear Inteqral Equations

The matrices K"' ( m = 1,2.. . .) are finite if during the procedure we have p p ) = 0 after a finite number of steps. They are infinite if for countably many values of 3 . p y ) # 0 holds. The number of zero rows and zero columns in K corresponds to the number of functions in the systems (ak(z))and ( O t ( y ) ) . We have a very simple case if the matrices Km contain one number vi"' = v, only, Le., all numbers p y ' are equal to zero. Using the notation of 11.3.3, p. 578, we have the solution of the infinite equation system under the assumptions f, = o for a 3 ( z )E (ak(x)) : (11.53)

11.3.6 Iteration Method To solve the integral equation b

/

f ( z ) = Kb.3 Y)P(Y) dy

(c

I z 5 4.

(11.54a)

a

starting with ag(z)= f ( z ) we determine the functions d

b

3,(y) = /K(z.y)a,:(z)dz

(11.54b)

and a,(z)= JK(z,y)&(y) dy,

(11.54~)

a

C

for n = 1.2. . . equalities hold:

.

If there is a quadratically integrable solution y(y) of (11.54a) then the following

b

b d

/ P(y)3n(21) J J dy =

a

~ ( y ) ~ (y)On-l(x) z, dzdy

a c

= jf(z)a,l(z)dz

( n = 1.2. . . .).

(11.54d)

e

By the orthogonalizationand normalizationofthe function systems that we have obtained from (11.54b), (11 54c) we get the orthonormal systems (a;(.))and (Oi(y)). Using the Schmidt orthogonalization method we have Di(y) in the form (11.54e) 3=1

\$'e need to show that the solution p(y) of (ll.54a) has the representation by the series

c JEi

P(Y) =

C,%(Y).

(1l.54f)

3=1

In this case we have for the coefficients c, regarding (11.54d): (1134g)

12.1 Volterra Intearal Eouations 583

To have a solution in the form (11.549 the following conditions are both necessary and sufficient: m

d

1.

/[f(3:)lZ d3:

= n=l

c

d

1 J ~ ( z ) Q ; ( zdxI2, )

m

(11.55a)

2.

1IcnI2 < m.

(11.55b)

n=l

11.4 Volterra Integral Equations 11.4.1 Theoretical Foundations A Volterra integral equation of the second kind has the form

d3:)= fb)+

1

K ( x .Y ) d Y ) dy.

(11.56)

a

The solution function p(z) with the independent variable z from the closed interval I = [a,b] or from the semi-open interval I = [a,w) is required. We have the following theorem about the solution of a L'olterra integral equation of the second kind: If the functions f(x) for z E I and K ( x ,y) on the triangular region 2 E I and y E [a,z]are continuous, then there exists a unzque solution p(x) of the integral equation such that it is continuous for z E I . For this solution yn(a) = ( a ) (11.57) holds. In many cases, the Volterra integral equation of the first kind can be transformed into an equation of the second kind. Hence, theorems about existence and uniqueness of the solution are valid with some modifications.

s

1. Transformation bv Differentiation Assuming that p(x), K ( z ,y). and K,(z, y) are continuous functions, we can transform the integral equation of the first kind (11.58a) a

into the form

by differentiation with respect to z.If K ( z .x) # 0 for all 3: E I , then dividing the equation by K ( z ,z) we get an integral equation of the second kind. 2. Transformation by Partial Integration -4ssuming that yn(x),K ( z ,y) and KY(z, y) are continuous, we can evaluate the integral in (ll.58a) by partial integration. Substituting J

a

gives

(11.59b)

I

584

11. Linear Inteqral Eauations

If K ( z ,z)# 0 for z E I , then dividing by K ( z ,z)we have an integral equation of the second kind: (11 . 5 9 ~ )

Differentiating the solution $(z)we get the solution p(z) of (11.58a).

11.4.2 Solution by Differentiation In some Volterra integral equations the integral vanishes after differentiation with respect to 2, or it can be suitably substituted. Assuming that the functions K ( z ,y), K,(z, y), and p(z) are continuous or. in the case of an integral equation of the second kind, p(z) is differentiable, and differentiating

f(.)

=

j K ( z .Y)P(Y)dY

(11.60a)

or P (.) = f(.)

1

+ K ( z ,Y)P(Y)dY

a

(11.60b)

a

with respect to z we get (11.60~)

d ( z ) = f ' b )+ K(z.z)p(z)+

/" a

zK(z,Y) d

Y ) dY.

(11.60d)

Q

1 x)

tiatingit twicewithrespect t o z we have y(z)cosz-

LX

1 . cos(z-2y)p(y) dy = -zsinz (I). Differen2 1 sin(z-Zy)p(y)dy = - ( s i n z + z c o s r ) (IIa), 2

I' I'

IFind thesolutionp(z) forz E 0, - oftheequation

1 . cos(z - 2y)(p(y)dy = cosz - -zsinz (IIb). The integral in the second equation 2 is the same as that in the original problem, so we can substitute it. We get $(z) COST = cosz and

and p'(z) cosz -

I ;)

because cos5 # 0 for z E 0, -

+

p'(z) = 1, so y ( z ) = z C.

To determine the constant C substitute z = 0 in (IIa) to obtain y ( 0 ) = 0. Consequently C = 0, and the solution of (I) is y ( z ) = z. Remark: If the kernel of a Volterra integral equation is a polynomial, then we can transform the integral equation by differentiation into a linear differential equation. Suppose the highest power of z in the kernel is n. After differentiating the equation (n t 1) times with respect to zwe have a differential equation of n-th order in the case of an integral equation of the first kind, and of the order n + 1 in the case of an integral equation of the second kind. Of course we have to assume that p(z) and f(z)are differentiable as many times as necessary. I X [ 2 ( z- y)* t l]p(y)dy = z3 (I*). After differentiating three times with respect to z we have p(z)t4[(z--y)p(y)

dy = 32' (II*a), y'(z)t4/'p(y) dy = 62 (II*b). 9"(5)+4p(z)= 6 (II*c).

3 The general solution of this differential equation is p(z) = A sin 2x + B cos 22 t -. Substituting z = 0 2 in (II*a) and (II*b) results in y ( 0 ) = 0, p'(0) = 0, so we have A = 0, B = -1.5. The solution of the 3 integral equation (I*) is ~ ( z=) -(1 - cos2z) 2

11.4 Volterru Integral Equations 585

11.4.3 Solution of the Volterra Integral Equation of the Second

Kind by Neumann Series Ll'e can solve Volterraintegral equations of the second kind by using Neumann series (see 11.2.3,p. 567). If we have the equation (11.61)

fory 5 x, for y > x.

(11.62a)

Il'ith this transformation (11.61) is equivalent to a Fredholm integral equation b

4 x 1I .=( !

J%

+

( 11.62b)

Y M Y ) dY,

a

allowing b = co as well. The solution has the representation m

=

b

I(.) + n=l X" Ja KL(x,y ) f ( y ) dY.

(11.62~)

The zteruted kernels K1,K z . . . . are defined by the following equalities: K1(z. Y) =

w. Y),

b

Kz(z,Y)

=

1 2

J"!.:

dwl> I/) dq =

K ( x ,M 1 ,Y) d% . .

(11.62d)

Y

a

and in general: (11.62e) The equalities K3(x,y) I 0 for y > x ( 3 = 1,2,. . .) are also valid for iterated kernels. Contrary to Fredholm integral equations if (11.61) has any solution, the Neumann series converges to it regardless of the value of X. I p(z) = 1 x 12e"-Y9(y) dy. ~ 1 ( xy), = ~ ( xy) ,= ex-21, ~ z ( zy), = eX-tleY-Y dV = ex-Y (x -

dx

+

ex-y

Y),,

' '

>

Kd.. Y) = -(x - y)"-'. ( n - I)!

r(x,y; A)

A" 1-(x O0

- y)" = e('-~)(Xtlj. It is well-known that n! this series is convergent for any value of the parameter A. edXt*)Ydy, in particular if X = -1: p(x) = We get p(z) = 1 + X e(2-gj(X+1) dy = 1 + Xe(At')"

Consequently the resolvent is:

= e2-Y

n=O

1'

1 - z,x

+ -1:

1 X f l

p(z) = -(1

+ Xe("'j2).

1'

11.4.4 Convolution Type Volterra Integral Equations If the kernel of a Volterra integral equation has the special form ~ ( z $ y=)

[ k(z 0

-

y)

for 0 5 y 5 x, for 0 5 x < y,

(11.63a)

586

11. Linear Inteqral Equations

we can use the Laplace transformation to solve the equations

j

k ( z - y ) p ( y ) dy = f ( z ) (11.63b)

or ~ ( z=) f(z)+ j k ( z - y)cp(y) dy.

0

(11.63~)

0

If the Laplace transforms L{p(z)} = @ ( p ) :L { f ( z ) }= F ( p ) , and L { k ( z ) }= K ( p ) exist, then the transformed equations have the form (see 15.2.1.2. ll., p. 711) K(P)@(P)= F (P )

+

(11.64a) or

@ ( p ) = F ( p ) K ( p ) @ ( p )resp.

(11.64b)

(11.64~)or

O(p)=

F ( p ) resp. 1 - K(P)

(11.64d)

From these we get ~

The inverse transformation gives the solution p(z) of the original problem. Rewriting the formula for the Laplace transform of the solution of the integral equation of the second kind we have (11.64e)

The formula (11.640 depends only on the kernel, and if we denote its inverse by h ( z ) ,the solution is

dz)= f(.) + j h ( .

-

Y).f(Y)

(11.64g)

dY.

0

The function h(z - y j is the resolvent kernel of the integral equation. 1 P-1 Ip(a) = f(z)+ i z e z - y p ( y )dy: @ ( p ) = F ( p ) + - @ ( p ) , Le., @ ( p ) = -F(p). P-1 P-2

The inverse

transformation gives cp(z). From H ( p ) = I it follows that h ( z )= e2‘. By (11.64g) the solution is P-2 p(z) = f ( z ) e2(”-y)f(y) dy.

+ 1’

11.4.5 Numerical Methods for Volterra Integral Equation of

the Second Kind \$’e are to find the solution for the integral equation 2

= f(.)

J

+ K ( 2 ,Y)cp(Y) dY

(11.65)

a

for 5 from the interval I = [alb]. The purpose of numerical methods is somehow to approximate the integral by a quadrature formula:

j

K ( z .y)u(u) du E ~ [ a , r l ( ~,jp(.)), (z,

(11.66a)

a

Both the interval of integration and the quadrature formula depend on z.This fact is emphasized by the index [a, z]of QI.,.,(.. .). We get the following equation as an approximation of (11.65): (11.66b) ~ ( x=)f ( z ) + Q[a,zl(K(z,.)F(.)).

11.1 Volterra Inteoral Eouations 587 The function v(z) is an approximation of the solution of (11.65). The number and the arrangement of the interpolation nodes of the quadrature formula depend on z,so as to allow little choice. If is an interpolation node of Qla,%;(K(z, . ) q ( . )then ) , (K(z,<)F(<)) and especially must be known. For this purpose, the right-hand side of (11.66b) should be evaluated first for z = <,which is equivalent to a quadrature over the interval [a. (1. As a consequence, the use of the popular Gauss quadrature formula is not possible. We solve the problem by choosing the interpolation nodes as a = zo < z1 < . . . < XI; < . . . and we use a quadrature formula &I.,., ! with the interpolation nodes zo,zl, . . . ,z,.The substitution values of the function at the interpolation nodes are denoted by the brief notation pk = F(Q) (k = 0,1! 2 , . . .). For po we have (see 11.3.1, p. 575)

a(<)

po = f ( z o )= f(a), (11.66~) and with this:

<

v1= f ( z l ) + Q[a,rll(K(zl, . ) q ( . ) ) .(11.66d)

has the interpolation points 20 and x1 and consequently it has the form (11.66e) Q ; R . z ~ ] (,)p(,) K ( ~ =~ woK(zi, , zo)Po t WK(zi,zi)pi with suitable coefficients wo and w1. Continuing this procedure, the values pk are successively determined from the general relation:

k = 1 , 2 , 3 , .. . . have the following form:

Pk = f ( Q ) -k Q [ o , z k ] ( K ( z k.)V(.)), r

The quadrature formulas

(11.66f)

( 11.66g) Hence, (11.66f) takes the form: k

Pk

= .f(zk)+ ~

W j k K ( ~ k , ~ j ) p j ~

(11.66h)

j=0

The simplest quadrature formula is the left-hand rectangular formula (see 19.3.2.1, p. 896). For this the coefficients are wjk = X j + l - Zj for j < k and W k k = 0. (11.66i) We have the system Po = f ( a ) , (11.67a) P1 = f(.l) + (21 - zo)K(zl,zo)po, Pz = f(zz) t (21 - zo)K(zz?xo)Po+ (z2- Z1)K(zzrZl)pl and generally k-1

Pk

= f(zk)+ E ( z j + l- z j ) K ( z k i x j ) p j .

(11.67b)

j=O

More accurate approximations of the integral can be obtained by using the trapezoidal formula (see 19.3.2.2, p. 896). To make it simple, we choose equidistant interpolation nodes zk = a t kh, k = 0,1.2. . . . :

(11 . 6 7 ~ ) Using this approximation for (11.66f) we get: Po = f(a)l

(11.67d) (11.67e)

588

11. Linear Inteqral Equations

Altough the unknown values also appear on the right-hand side of the equation, they are easy to express. Remark: With the previous method we can approximate the solution of non-linear integral equations as well. If we use the trapezoidal formula to determine the values 'Pk we have to solve a non-linear equation. We can avoid this by using the trapezoidal formula for the interval [a,~ k - ~ ]and . we use the rectangular formula for the interval [zk-l, zk]. If h is small enough, this quadrature error does not have a significant effect on the solution. ILet'sgivethe approximatevaluesofthesolutionoftheintegralequation ~ ( z=) 2+/'(z-y)q(y)

dy

by the formula (11.664 using the rectangular formula. The interpolation nodes are the equidistant points zk = k 0.1, and hence h = 0.1. rectangular trapezoidal formula formula rpn = 2. exact

I

etc.

0.8 1.0

2.6749 3.0862

2.0602 2.2030 2.4342 2.7629 3.2025

2.0401 2.1620 2.3706 2.6743 3.0852

In the table the values of the exact solution are given, as well as the approximate values calculated by the rectangular and the trapezoidal formulas, respectively, so the accuracies of these methods can be compared. The step size used is h = 0.1.

11.5 Singular Integral Equations .4n integral equation is called a szngular zntegral equatzon if the range of the integral in the equation is not finite, or if the kernel has singularities inside of the range of integration. We suppose that the integrals exist as improper integrals, or as Cauchy principal values (see 8.2.3, p. 451ff.). The properties and the conditions for the solutions of singular integral equations are very different from those in the case of "ordinary" integral equations. We will discuss only some special problems in the following paragraph. For further discussions see [11.8].

11.5.1 Abel Integral Equation

'LY

One of the first applications of integral equations for a physical problem was considered by Abel. A particle is moving in a vertical plane along a curve under the influence only of gravity from the point Po(z0,yo) to the point Pl(O,O)(Fig. 11.2). The velocity of the particle at a point of the curve is (11.68) Figure 11.2

By integration we calculate the time of fall as a function of yo:

(11.69a) If s is considered as a function of y, i.e., s = f(y), then (11.69b)

11.5 Sinaular Intearal Eauations 589

The next problem is to determine the shape of the curve as a function of yo if the time of the fall is given. By substitution

fi,T(Y0) = F ( Y o )

(11 . 6 9 ~ ) and f'(Y) = V(Y) and changing the notation of the variable yo into x, a Volterra integral equation of the first kind is obtained: (11.69d)

We will consider the slightly more general equation (11.70)

The kernel of this equation is not bounded for y = z.In (11.70), the variable y is formally replaced by E and the variable x by y. By these substitutions the solution is obtained in the form 'p = ~ ( z )If. both sides of (11.70) are multiplied by the term

~

(x - y)l-"

and integrated with respect to y between

the limits a and z,it yields the equation

jL(x

- y)'-"

(r""d()dy=/(.dy. (Y - Ela

xf ( Yy)'-" )

(11.71a)

Changing the order of integration on the left-hand side we have

The inner integral can be evaluated by the substitution y = E

F

dY ( x - y)I-"(y

-<)a

du = ./ uo(1- u11-a

-

+ (x- E)u:

sinras) '

(11 . 7 4

We substitute this result into (11.71b). After differentiation with respect to x we get the function p(x):

( 11.71d)

11.5.2 Singular Integral Equation with Cauchy Kernel 11.5.2.1 Formulation of the Problem Consider the following integral equation: (11.72) is a system consisting of a finite number of smooth, simple, closed curves in the complex plane such that they form a connected interior domain St with 0 E Stand an exterior domain S - . Traversing the

590

2 1 . Linear Integral Equations

curve we have S+always on the left-hand side of T . A function u(x)is Holder contznuous (or satisfies the Holder condztzon) over if for any pair 21, x2 E the relations

r

r

(11.73) lU(X1)- U ( X 2 ) J < Klx1 - z#, 0 < p 5 1, K > 0 are valid. We suppose that the functions a(z), f(z), and p(x) are Holder continuous with exponent pl, and K ( s .y) is Holder continuous with respect to both variables with the exponents pz > 81. The kernel K ( z .y)(y - s)-' has a strong singularity for x = y. The integral exists as a Cauchy principal value. mlth K ( x ,x) = b(s)and k ( z ,y) = K(x' - K ( z ' we have (11.72) in the form

Y-"

The expression (Cp)(z)denotes the left-hand side of the integral equation in abbreviated form. L is a singular operator. The function k ( z ,y) is a weakly singular kernel. It is assumed that the normality - b(x)* # 0,z E r holds. The equation condition (11.74b)

is the characterzstic equation pertaining to (11.74a). The operator LOis the characteristic part of the operator C. From the adjoint integral equation of (11.74a) we get the equality

11.5.2.2 Existence of a Solution The equation (Lp)(z)= f(z) has a solution p(z) if and only if for every solution V(y) of the homogeneous adjoint equation (CTw)(y) = 0 the condition of orthogonality

1

f(Y)V(Y)

(11.75a)

dY = 0

r

is satisfied. Similarly, the adjoint equation (CT$)(y) = g(y) has a solution if for every solution p(x) of the homogeneous equation (Cp)(s)= 0 the following is valid: (11.75b)

11.5.2.3 Properties of Cauchy Type Integrals We call the function (11.76a)

r

a Cauchy type integral over r. For z $ the integral exists in the usual sense and the result is a holomorphic function (see 14.1.2, p. 670). We also have @(m)= 0. For z = x E r in (11.76a) we consider the Cauchy principal value (11.76b)

11.5 Singular Integral Eouations 591

The Cauchy type integral @ ( z )can be extended continuously over F from St and from S-.Approaching the point z E r with z we denote the limit by Qt(z)and @-(z),respectively. The formulas of Plemelj and Sochozki are valid:

11.5.2.4 The Hilbert Boundary Value Problem 1. Relations The solution of the characteristic integral equation and the Hilbert boundary value problem strongly correlate. If 9(z) is a solution of (11.74b), then (11.76a) is a holomorphic function on St and S-with @(E) = 0. Because of the formulas of Plemelj and Sochozki (11.76~)we have: p(.) = @+(.) - @-(z). 2(%p)(z) = @+(z) @-(z)) z E (11.77a) With the notation

+

r.

(11.77b) the characteristic integral equation has the form: o+(z) = ~ ( z ) @ - (tz )g(z), z E r.

(11 . 7 7 ~ ) 2. Hilbert Boundary Value Problem We are looking for a function @ ( z )which is holomorphic on S+and S-, and vanishes at infinity, and satisfies the boundary conditions (11.77~)over r. 4 solution @ ( z )of the Hilbert problem can be given in the form (11.76a). So, as a consequence of the first equation of (11.77a), a solution p(z) of the characteristic integral equation is determined.

11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert Problem) 1. Homogeneous Boundary Conditions @'(z) = G ( z ) @ - ( z ) , z E r.

(11.78)

During a single circulation of the point z along the curve rl the value of logG(z) changes by 27riX1, where XI is an integer. The change of the value of the function log G ( z )during a single traverse of the complete curve system is

r

(11.79a) n

The number

K.

=

A/ is called the index of the Hilbert problem. We compose a function

1=0

GO(^) = (z- aO)-T(z)G(z)

( 11.79b)

with (11.794 n(z) = (z- al)'l(z - a#? ' . (z- a,)'", where a. E St and a1 (1 = 1,. . . n ) are arbitrarily fixed points inside f i . If r = ro is a simple closed curve ( n = 0), then we define n(z) = 1. With

.

(11.79d)

the following particular solution of the homogeneous Hilbert problems is obtained, which is called the fundamental solution: n-'(z)expI(z) for z E S+: (1 1.79e) X ( Z ) = ( z - ao)-K expI(2) for z E S-.

{

I

592

11. Linear Inteoral Eauations

This function doesn't vanish for any finite 2. The most general solution of the homogeneous Hilbert problem, which vanishes at infinity, for ti > 0 is @h(Z) = x ( z ) p ~ - l ( z ) , z E (11.80) with an arbitrary polynomial P,-l(z) of degree at most (ti - 1). For K 5 0 there exists only the trivial solution @ h ( Z ) = 0 which satisfies the condition @ h ( a ) = 0, so in this case PK-l(z) I 0. For K > 0 the homogeneous Hilbert problem has ti linearly independent solutions vanishing at infinity.

2. Inhomogeneous Boundary Conditions The solution of the inhomogeneous Hilbert problem is the following:

If IC < 0 holds, for the existence of a solution vanishing at infinity the following necessary and sufficient conditions must be fulfilled: (11.83)

11.5.2.6 Solution of the Characteristic Integral Equation 1. Homogeneous Characteristic Integral Equation If @,,(z) is the solution of the corresponding homogeneous Hilbert problem, from (11.77a) we have the solution of the homogeneous integral equation p h ( 5 ) = @ i ( z )- @ h ( z ) , x E (11.84a) For IC 5 0 only the trivial solution vh(z) = 0 exists. For ti > 0 the general solution is (11.84b) P h b ) = [X'(4 - X-(z)lPn-1(2) with a polynomial P,-l of degree at most ti - 1.

r.

2. Inhomogeneous Characteristic Integral Equation If @ ( z )is a general solution of the inhomogeneous Hilbert problem, the solution of the inhomogeneous integral equation can be given by (11.77a): (11.85a) p(z) = @+(2) - F ( z ) (11.85b) = X + ( T ) R + ( X )- X - ( z ) R - ( 2 )+ @ i ( z )- @jh(2), z E r Using the formulas of Plemelj and Sochozki (11.76~)for R ( z )we have

Substituting (11.85~)into (11.85a) and considering (11.76b) and g(z) = f(z)/(a(z) + b ( z ) )finally results in the the solution:

+

X+(x) X-(z)

dz)= 2(a(z)t b ( z ) ) X + ( zf)( 2 )

According to (11.83) in the case ti < 0 the following relations must hold simultaneously in order to ensure the existence of a solution:

11.5 Singular Integral Equations 593

W The characteristic integral equation is given with constant coefficients a and b: 6 d y ) dy = f(x). Here is a simple closed curve, Le., = (TI = 0). From (11.77b) a g ( x ) + ?ri

121-2

r a - b and g(x) = f (XI . G is a constant, consequently IG = 0. Therefore, n ( x ) = 1 and we get G = atb a+b

\

’

Since K. = 0 holds, the homogeneous Hilbert boundary value problem has only the function @ h ( z ) = 0 as the solution vanishing at infinity. From (11.86) we have

594

12. Functional Analusis

12 Functional Analysis 1. Functional Analysis Functional analysis arose after the recognition of a common structure in different disciplines such as the sciences, engineering and economics. General principles were discovered that resulted in a common and unified approach in calculus, linear algebra, geometry, and other mathematical fields, showing their interrelations.

2. Infinite Dimensional Spaces There are many problems, the mathematical modeling of which requires the introduction of infinite systems of equations or inequalities. Differential or integral equations, approximation, variational or optimization problems could not be treated by using only finite dimensional spaces. 3. Linear and Non-Linear Operators In the first phase of applying functional analysis - mainly in the first half of the twentieth century - linear or linearized problems were thoroughly examined, which resulted in the development of the theory of linear operators. More recently the application of functional analysis in practical problems required the development of the theory of non-linear operators, since more and more problems had to be solved that could be described only by non-linear methods. Functional analysis is increasingly used in solving differential equations, in numerical analysis and in optimization, and its principles and methods became a necessary tool in engineering and other applied sciences.

4. Basic Structures In this chapter only the basic structures will be introduced, and only the most important types of abstract spaces and some special classes of operators in these spaces will be discussed. The abstract notion will be demonstrated by some examples, which are discussed in detail in other chapters of this book, and the existence and uniqueness theorems of the solutions of such problems are stated and proved there. Because of its abstract and general nature it is clear that functional analysis offers a large range of general relations in the form of mathematical theorems that can be directly used in solving a wide variety of practical problems.

12.1 Vector Spaces 12.1.1 NotionofaVector Space A non-empty set V is called a vector space or h e a r space over the field F of scalars if there exist two operations on V - addition of the elements and multiplication by scalars from F - such that they have the following properties: 1. for any two elements z,y E pi, there exists an element z = z + y E V, which is called their sum. 2. For every z E V and every scalar (number) a E F there exists an element ax E V, the product of x and the scalar a so that the following properties, the anoms of vector spaces (see also 5.3.7.1, p. 314), are satisfied for arbitrary elements x,y, z E V and scalars a, D E F: (12.1) ( V l ) 2 (y 2 ) = (z y) 2 . (12.2) (V2) There exists an element 0 E V, the zero element, such that z + 0 = z.

+ +

+ +

(V3) To every vector x there is a vector - z such that x + (-z)= 0. (V4) x + y = y + z . (V5) l . z = 2 .

O.z=O.

(12.3) (12.4) (12.5)

(V6) a ( 0 z )= (aP)z.

(12.6)

+ +

(12.7) (12.8)

(V7) (a + O)z = az Pz. (V8) a ( z + y) = ax ay.

12.1 Vector Spaces 595

V is called a real or complex vector space, depending on whether F is the field R of real numbers or the field C of complex numbers. The elements of V are called either points or, according to linear algebra, vectors. In functional analysis, we do not use the vector notation Z or 2. We can also define in V the difference z - y of two arbitrary vectors x,y E V as x - y = z (-y). From the previous definition, it follows that the equation x+y = z can be solved uniquely for arbitrary elements y and z . The solution is x = z - y. Further properties follow from axioms (Vl)-(V8): the zero element is uniquely defined, ax = Ox and z # 0, imply a = az = ay and a # 0, imply x = y: 0 -(ax)= a ' (-x).

+

a,

12.1.2 Linear and Affine Linear Subsets 1. Linear Subsets A non-empty subset VOof a vector space V is called a linear subspaceor a linear manifoldof V if together with two arbitrary elements x,y E VOand two arbitrary scalars a , p E F, their linear combination ax t 3y is also in Vo. VOis a vector space in its own right, and therefore satisfies the axioms (V1)(V8). The subspace Vo can be V itself or only the zero point. In these cases the subspace is called trivial. 2. Affine Subspaces A subset of a vector space V is called an afine linear subspace or an afine manifoldif it has the form (12.9) {zo t 2/ : Y E Val, where xo E V is a given element and VO is a linear subspace. It can be considered (in the case 10 # 0) as the generalization of the lines or planes not passing through the origin in R3. 3. The Linear Hull The intersection of an arbitrary number of subspaces in V is also a subspace. Consequently, for every non-empty subset E c V, there exists a smallest linear subset lin(E)or [E]in V containing E , namely the intersection of all the linear subspaces, which contain E . The set lin(E)is called the linear hull of the set E . or the linear subspace generated by the set E . It coincides with the set of all (finite) linear combinations (12.10) C y 1 5 1 t azxz . . . a,x,, comprised of elements ~ 1 ~ x. 2. ., 5, E E and scalars a ] ,a2,. . . , a, E F. 4. Examples for Vector Spaces of Sequences I A Vector Space IFn: Let n be a given natural number and V the set of all n-tuples, Le., all finite sequences consisting of n scalar terms { (('1,. . . ,En) : ti E F, i = 1,.. . , n}. The operations will be defined componentwise or termwise, Le., if z = (Ell. . . E,) and y = ( ~ 1. ~. , 7.), are two arbitrary elements from V and a is an arbitrary scalar, a E F, then

+ +

~

a ' x = (aE1,.. . a<,).

(12.11b)

~

In this way, we get the vector space Fn.In the special case of n = 1 we get the linear spaces R or 6. This example can be generalized in two different ways (see examples B and C). I B Vector Space s of all Sequences: If we consider the infinite sequences as elements z = En E F and define the operations componentwise, similar to (12.11a) and (12.11b)!then we get the vector space s of all sequences. I C Vector Space cp(a1so COO) of all Finite Sequences: Let V be the subset of all elements of s containing only a finite number of non-zero components, where the number of non-zero components depends on the element. This vector space - the operations are again introduced termwise is denoted by cp or also by coo.and it is called the space of all finite sequences of numbers. ~

I

12. Functional Analysis

596

ID Vector Space m (also loo) of all Bounded Sequences: A sequence z = {&}F=lbelongs to m if and only if there exists C, > 0 with /,[I 5 C, V n = 1 , 2 , .. . . This vector space is also denoted by 1". IE Vector Space c of all Convergent Sequences: .4sequence z = {<,}F=lbelongs to c if and only if there exists a number (0 E F such that for V E > 0 there exists an index no = no(&)such that for all n > no one has (I, - to1< E (see 7.1.2, p. 403). IF Vector Space cg of all Null Sequences: The vector space co of all null sequences, Le., the subspace of c consisting of all sequences converging to zero ((0 = 0). IG Vector Space lp: The vector space of all sequences z = {&}p=l such that E,"==, I<,lP is convergent is denoted by lp (1 5 p < m). It can be shown by the Minkowski inequality that the sum of two sequences from 1P also belongs to lp, (see 1.4.2.13,p. 32). Remark: For the vector spaces introduced in examples A-G, the following inclusions hold: (12.12) p c co c c c m c s and p C lP c lq c CO, where 1 5 p < q < 03. 5. Examples of Vector Spaces of Functions IA Vector Space F ( T ) : Let V be the set of all real or complex valued functions defined on a given set T , where the operations are defined pointwise, Le., if z = z ( t ) and y = y(t) are two arbitrary elements of V and a E F is an arbitrary scalar, then we define the elements (functions) z+ y and c y . z by the rules ( 12.13a) (z y ) ( t ) = z ( t )+ y ( t ) v t E T , (cyz)(t)=cy.z(t)V t E T . (12.13b) We denote this vector space by F ( T ) . We introduce some of its subspaces in the following examples. IB Vector Space B ( T )or M ( T ) : The space B ( T )is the space of all functions bounded on T . This vector space is often denoted by M ( T ) . In the case of T = N,we get the space M(N) = m from example D of the previous paragraph. IC Vector Space C ( [ a ,b ] ) : The set C([a,b]) of all functions continuous on the interval [a,b] (see 2.1.5.1,p. 57). ~

+

ID Vector Space C(")([a,b ] ) : Let k E N, k 2 1. The set C ( k ) ( [ ab,] ) of all functions k-times continuously differentiable on [a,b] (see 6.1, p. 377-382) is a vector space. At the endpoints a and b of the interval [alb], the derivatives have to be considered as right-hand and left-hand derivatives, respectively. Remark: For the vector spaces of examples A-D of this paragraph, and T = [a, b] the following subspace relations hold: (12.14) C ( k ) ( [ abl). c c(b, 4)c B([a,bl) C F([albl). IE Vector Subspace of C ( [ a ,b]): For any given point to E [a, b ] , the set {z E C([a,b ] ] : z(to) = 0) forms a linear subspace of C ( [ a ,b]).

12.1.3 Linearly Independent Elements 1. Linear Independence A finite subset {XI, , , , ,E,} of a vector space V is called linearly independent if alxl+ . . . + cy,z, = 0 implies cy1 = . . = cy, = 0.

(12.15) Otherwise, it is called lznearly dependent. If a1 = . . . = cy, = 0. then for arbitrary vectors ~ 1 , ... ,x, from V, the vector alzl + . ' . + a,z, is trivially the zero element of V. Linear independence of the vectors zl,. . . ,z, means that the only way to produce the zero element 0 = cqzl+. . . + a , ~ ,is when all coefficients are zero cy1 = . . = cy, = 0. This important notion is well known from linear algebra

12.1 Vector Spaces 597

(see 5.3.7.2, p. 316) and was used for the definition of a fundamental system of homogeneous differential equations (see 9.1.2.3,2., p. 498). An infinite subset E c Vis called linearly independent ifevery finite subset of E is linearly independent. Otherwise, E is called linearly dependent. IIf we denote by ek the sequence whose k-th term is equal to 1 and all the others are 0, then ek is in the space cp and consequently in any space of sequences. The set { e l , ezr. . .} is linearly independent in every one of these spaces. In the space C([O,T I ) , e.g., the system of functions 1, sinnt, cosnt (n = 1,2,3,.. .) is linearly independent, but the functions 1,cos 2t, cos't are linearly dependent (see (2.97), p. 79). 2. Basis and Dimension of a Vector Space A linearly independent subset B from V, which generates the whole space V, i.e., lin(B)= V holds, is called an algebraic basis or a Hamel basis of the vector space V (see 6.3.7.2, p. 315). B = {z~: E E} is a basis of V if and only if every vector z E V can be written in the form z = q x ~where , the (€5

coefficients at are uniquely determined by x and only a finite number of them (depending on z)can be different from zero. Every non-trivial vector space V, i.e., V # {0}, has at least one algebraic basis, and for every linearly independent subset E of V,there exists at least one algebraic basis of V, which contains E . A vector space V is m-dimensional if it possesses a basis consisting of m vectors. That is, there exist m linearly independent vectors in V, and every system of m t 1vectors is linearly dependent. A vector space is infinite dimensional if it has no finite basis, Le., if for every natural number m there are m linearly independent vectors in V. The space F" is n-dimensional, and all the other spaces in examples B-E are infinite dimensional. The subspace /in({l: t , t 2 } ) c C ( [ a ,b])is three-dimensional. In the finite dimensional case, every two bases of the same vector space have the same number of elements. Also in an infinite dimensional vector space any two bases have the same cardinality, which is denoted by dim(\'). The dimension is an invariant quantity of the vector space, it does not depend on the particular choice of an algebraic basis.

12.1.4 Convex Subsets and the Convex Hull 12.1.4.1 Convex Sets A4subset C of a real vector space V is called convex if for every pair of vectors z,y

E C all vectors of the form Ax + (1 - X)y, 0 5 X 5 1, also belong to C. In other words, the set C is convex, if for any two elements z and y, the whole line segment (12.16) {Ax + (1 - X)y : 0 5 x 5 l}, (which is also called an interval), belongs to C. (For examples of convex sets in R2 see the sets denoted by A and B in Fig. 12.5, p. 624.) The intersection of an arbitrary number of convex sets is also a convex set, where the empty set is agreed to be convex. Consequently, for every subset E c V there exists a smallest convex set which contains E , namely, the intersection of all convex subsets of V containing E . It is called the convex hull of the set E and it is denoted by co ( E ) . eo ( E ) is identical to the set of all finite convex linear combinations of elements from E , Le.. eo ( E )consists of all elements of the form X l z l + . . . + Xnz,,where z i r . . , 2 , are arbitrary elements from E and A, E [0,1] satisfy the equality A1 + ' . . + A, = 1. Linear and affine subspaces are always convex.

12.1.4.2 Cones A non-empty subset C of a (real) vector space V is called a convex cone if it satisfies the following properties: 1. C is a convex set.

598

12. Functional Analusis

n o m z E C and X 2 0, it follows that Xz E C. 3. From z E C and -z E C , it follows that z = 0. .L\ cone can be characterized also by 3. together with z , y ~ Cand X , p > O imply X z t p y E C . (12.17) IA: The set R: of all vectors z = ([I,. . . ,En) with non-negative components is a cone in R". IB: The set C+ of all real continuous functions on [a, b] with only non-negative values is a cone in the space C ( [ a .b ] ) with only non-negative terms, Le., [,,2 0. V n , IC: The set of all sequences of real numbers is a cone in s. Analogously we obtain cones in the spaces of examples C-G on p. 593, if we consider the sets of non-negative sequences in these spaces. ID: The set C C 1P (1 5 p < m), consisting of all sequences such that for some a > 0 2.

(12.18) is a convex set in lp, but obviously. not a cone. IE: Examples from R2 see Fig. 12.1: a) convex set, not a cone, b) not convex, c) convex hull.

Yt

Figure 12.1

12.1.5 Linear O p e r a t o rs a n d h n c t i o n a l s 12.1.5.1 Mappings .A mapping T : D --t Y from the set D c X into the set Y is called injective, if (12.19) T ( z )= T(y) ==+ 5 = y: surjective, if for V y E Y there exists an element z E D such that T ( z )= y, (12.20) bijective, if T is both injective and surjective. D is called the domain of the mapping T and is denoted by DT or D ( T ) ,while the subset {y E Y : 3 z E DT with T ( z )= y} of Y is called the range of the mapping T and is denoted by R ( T )or Im(T).

12.1.5.2 Homomorphism and Endomorphism Let X and Y be two vector spaces over the same field F and D a linear subset of X. .4mapping T : D t Y is called linear (or a linear transformation, linear operator or homomorphism), if for arbitrary z,y E D and CY: 3 E F, (12.21) T ( a 5 t By) = cvTz PTy. For a linear operator T we prefer the notation 2 's. which is similarly used for linear functions, while the notation T ( z )is used for general operators. N ( T ) = {z E X : T z = 0 ) is the null space or kernel

+

12.1 Vector Spaces 599

of the operator T and is also denoted by ker(T). A mapping of the vector space X into itself is called an endomorphism. If T is an injective linear mapping, then the mapping defined on R ( T )by y c-t x. such that Tx = y, y E R ( T ) (12.22) is linear. It is denoted by T-': R ( T )-+ X and is called the inverse of T . If U is the vector space F: then a linear mapping f: X t F is called a linearfunctional or a linearfonn.

12.1.5.3 Isomorphic Vector Spaces A bijective linear mapping T X -i Y is called an isomorphism of the vector spaces X and Y. Two vector spaces are called isomorphic provided an isomorphism exists.

12.1.6 Complexification of Real Vector Spaces Every real vector space V can be extended to a complex vector space P. The set (z. y) with z,y

E

consists of all pairs

Pi. The operations (addition and multiplication by a complex number a + ib E C)

are defined as follows: 2/11

+ (zz; Yz) = (21 + 5 2 1 Y l + YZ),

(12.23a) (12.23b)

( a +ib)(z,y) = (az - b y , b x + ay).

Since the special relations (12.24) (z:y) = (x.0 ) (0, y) and i(y, 0) = (0 t il)(y, 0) = (0 y - 1 0, l y 0 . 0 ) = (0, y) hold, the pair (z, y) can also be written as z t iy. The set is a complex vector space, where the set Y is identified with the linear subspace 9, = {(z,0 ) : z E V}, Le.. z E V is considered as (z, 0) or as 5 t io. This procedure is called the complexificationof the vector space V. .4linearly independent subset in V is also linearly independent in 9. The same statement is valid for a basis in V, so dim(V)= dim(v).

+

+

12.1.7 Ordered Vector Spaces 12.1.7.1 Cone and Partial Ordering If a cone C is fixed in a vector space V,then an order can be introduced for certain pairs of vectors in W. Namely, if z - y E C for some z, y E V then we write z 2 y or y 5 z and say x is greater than or equal to y or y is smaller than or equal to z. The pair (V, C) is called an ordered vector space or a vector space partially ordered by the cone C. An element z is called positive, if z 2 0 or, which means the same, if x E C holds. Moreover = {z E r.: z 2 0 ) . (12.25)

c

If we consider the vector space R2ordered by its first quadrant as the cone C(= R t ) , then a typical phenomenon of ordered.vector spaces will be seen. This is referred to as "partially ordered or sometimes as "semi-ordered. Namely, only several pairs of two vectors are comparable. Considering the vectors z = (1,-1) and y = (0,2), neither the vector z - y = (1,-3) nor y - z = (-1.3) is in C, so neither x 2 y nor z 5 y holds. .4n ordering in a vector space, generated by a cone, is always only a partial ordering. It can be shown that the binary relation 2 has the following properties: (01) x 2 z V x E V (reflexivity). (12.26) ( 0 2 ) z 2 y and y 2 z imply z 2 z (transitivity).

(12.27)

(03) z 2 y and a 2 0,

(12.28)

(04) x1 2

y1

and zz 2

CY

YZ

E R,

imply

CY% 2

ay.

imply zl + z2 2 y1 + YZ.

(12.29)

600

12. Functional Analusis

Conversely, if in a vector space V there exists an ordering relation, Le., a binary relation 2 is defined for certain pairs of elemegts and satisfies axioms (01)-(04),and if one puts (12.30) Vt = {z E V: z O } , then it can be shown that V, is a cone. The order >v+ in V induced by Vt is identical to the original order >; consequently, the two possibilities of introducing an order in a vector space are equivalent. A cone C C X is called generating or reproducing if every element z E X can be represented as z = u - u with u , u E C. We also write X = C-C. A: .4n obvious order in the space s (see example B, p. 595) is induced by means of the cone (12.31) C = {z = {&}:=* : En 2 0 V n } (see example C, p. 598). We usually consider the natural coordinatewise order in the spaces of sequences (see (12.12), p 596). This is defined by the cone obtained as the intersection of the considered space with C (see (12.31), p. 600). The positive elements in these ordered vector spaces are then the sequences with non-negative terms. It is clear that we can define other orders by other cones, as well. Then we obtain orderings different from the natural ordering (see [12.17], [12.19]).

>

B: In the real spaces of functions F ( T ) , B ( T ) , C ( [ a ,b]) and C@)([a,b]) (see 12.1.2, 5., p. 596), we define the natural order z 2 y for two functions z and y by z ( t ) 2 y(t), V t E T , or V t E [a,b]. Then z 0 if and only if z is a non-negative function in 2'. The corresponding cones are denoted by Ft(T), B,(T),etc. WecanalsoobtainC+ = C + ( T )= F + ( T ) n C ( T ) i f T = [a,b].

>

12.1.7.2 Order Bounded Sets Let E be an arbitrary non-empty subset of an ordered vector space V. An element z E V is called an upper bound of the set E if for every z E E , z 5 z . An element u E V is a lower bound of E if u 5 z,V z E E . For any two elements z, y E V with z 5 y, the set (12.32) [z, y] = (2: E v: 2 5 v 5 y} is called an order interval or (0)-interval. Obviously, the elements z and y are a lower bound and an upper bound of the set [z, y], respectively, where they even belong to the set. A set E c V is called order bounded or simply (0)-bounded, if E is a subset of an order interval, Le., if there exist two elements u , z E V such that u 5 z 5 z , V z E E or, equivalently, E c [u,z]. A set is called bounded above or bounded below if it has an upper bound, or a lower bound, respectively.

12.1.7.3 Positive Operators A linear operator (see 112.21, [12.17]) T : X -+ Y from an ordered vector space X = (X,Xt) into an ordered vector space Y = (Y: Y,) is called positive, if T(X,) c Y,, Le., T z 2 0 for all z > 0. (12.33)

12.1.7.4 Vector Lattices 1. Vector Lattices In the vector space R' of the real numbers the notions of (0)-boundedness and boundedness (in the usual sense) are identical. It is known that every set of real numbers which is bounded from above has a supremum: the smallest of its upper bounds (or the least upper bound, sometimes denoted by lub). Analogously, if a set of reals is bounded from below, then it has an infimum, the greatest lower bound, sometimes denoted by glb. In a general ordered vector space, the existence of the supremum and infimum cannot be guaranteed even for finite sets. They must be given by axioms. An ordered vector space V is called a vector lattice or a linear lattice or a Riesz space, if for two arbitrary elements z,y E \.'there exists an element z E V with the following properties: 1. z 5 z and y 5 z , 2. if u E V with z 5 u and y 5 u, then z 5 u.

12.1 Vector Spaces 601

Such an element z is uniquely determined, is denoted by z V y , and is called the supremum of x and y (more precisely: supremum of the set consisting of the elements z and y). In a vector lattice, there also exists the infimum for any z and y, which is denoted by IC A y. For applications of positive operators in vector lattices see, e.g., [12.2],[12.3] [12.15]. A vector lattice in which every non-empty subset E that is order bounded from above has a supremum lub(E) (equivalently, if every non-empty subset that is order bounded from below has an infimum glb(E)) is called Dedekind complete or a K-space (Kuntorovzch space).

W A: In the vector lattice F([u,b]) (see 12.1.2, 5., p. 596), the supremum of two functions x,y is calculated pointwise by the formula ( Z V Y ) ( t ) =max{x(t),y(t)) V t E [a,bI. (12.34) In the case of [a,b] = [0,1],x(t) = 1 - it and y ( t ) = t2 (Fig. 12.2), we get l - i t , if o s t s ; , (12.35) (zvy)(t) = t 2 , if 5 t 5 1.

A

{

1

7

2

Figure 12.2

;

IB: The spaces C ( [ a ,b]) and B([a,b]) (see 12.1.2, 5., p. 596) are , is not also vector lattices, while the ordered vector space C ( ' ) ( [ ab]) a vector lattice, since the minimum or maximum of two differentiable functions may not be differentiable on [a,b], in general. A linear operator T : X t Y from a vector lattice X into a vector lattice Y is called a vector lattice homomorphzsm or homomorphzsrn of the vector lattice, if for all z, y E X

T ( z V y ) = Tx V T y and T ( x A y) = T z A Ty.

(12.36)

2. Positive and Negative Parts, Modulus of an Element For an arbitrary element z of a vector lattice V, the elements xt = x V 0, x- = (-x) V 0 and 1x1 = z+ t x(12.37) are called the positive part, negative part, and modulus of the element z, respectively. For every element x E V,the three elements x t r x-,1x1 are positive, where for z, y E V the following relations are valid: (12.38a) z 5 2+ 5 1x1, 2 = Z+ - 2-, Z+ A X - = 0, 1x1 = z V (-x),

(z+ Y ) t I z t + Y t ? (2 + Y)- 5 2- + Y-> x 5 y implies zt 5 yt and x- 2 yand for arbitrary a 2 0 (ax)+=ax+,

(0.)-

12

+ YI I I4 +

=ax-, lax/ =aIxI.

lYll

(12.38b) (12.38~) (12.38d)

Figure 12.3 In the vector spaces F ( [ ab]) , and C ( [ a ,b]), we get the positive part, the negative part, and the modulus of a function x ( t ) by means of the following formulas (Fig. 12.3):

I

602

12. Functional Analusis

(12.39a) (12.39b)

Izl(t)= Iz(t)l V t E

[a, b]

(12.394

12.2 Metric Spaces 12.2.1 Notion of a Metric Space Let X be a set. and suppose a real, non-negative function p(z, y) (z, y E X) is defined on X x X. If this function p : X x X -+ R\ satisfies the following properties (Ml)-(M3) for arbitrary elements x. y, z E X. then it is called a rnetrzc or dzstance in the set X, and the pair X = (X, p ) is called a rnetrzc space. The axioms of rnetrzc spaces are: (12.40) ( M l ) p(x. y j 2 0 and p ( i , y) = 0 if and only if z = y (non-negativity),

(M2) P(.. Y) = P(Y> (symmetry). (12.41) (M3) P k >Y) 5 P ( Z , 2) -t P(Zt Y) (triangle inequality). (12.42) X metric can be defined on every subset Y of a metric space X = (X, p ) in a natural way if we restrict the metric p of the space X to the set Y, Le., if we consider p only on the subset Y x Y. The space (Y, p ) of X x X is called a subspace of the metric space X. IA: The sets R" and C" are metric spaces with the Euclzdean rnetrzc defined for points 2 = ((1, " ' , E n ) and Y = (713 ". 37") as (12.43)

(12.44)

(t1,.. . .En) and y = (171,. . ., a )also defines a metric in R" and C", the so-called rnaxzrnurn metrzc. If 2 = ( i l , . . . .&) is an approximation of the vector z,then it is of interest to know

for vectors x =

how much is the maximal deviation between the coordinates: max

l
I&

- $1,

The function n

k k - 7lCI

p(zi y) =

(12.45)

k=l

for vectors x,y E R" (or C") defines a metric in R" and C, the so-called absolute value rnetrzc. The metrics (12.43), (12.44) and (12.45)are reduced in the case of TI = 1 to the absolute value 12 - yI in the spaces R = R' and C (the sets of real and complex numbers). IC: Finite 0-1 sequences, e.g., 1110 and 010110, are called words in coding theory. If we count the number of positions where two words of the same length n have different digits, Le., for z = ((1,. . . ,<,,)> y = ( ~ 1 . .. . ,vn), ( k , r)k E ( 0 , I}, we define p(z, y) as the number of the k E { I , . . . ,?I} values such that &, # V k , then the set of words with a given length n is a metric space, and the metric is the so-called Hammingdistance, e.g., p((lllO), (0100)) = 2. ID: In the set m and in its subsets c and co (see (12.12), p. 596) a metric is defined by p ( x . y) = SYP - 7781, (x = ([l) (2, , , y = 711 72) , . (12.46)

12.2 Metric Spaces 603

W E:

In the set 1P (1 5 p

< m) of sequences 2 = (.&, &, . . .) with absolutely convergent series

W

n=l

IEnlP

a metric is defined by (12.47)

(12.48)

W G : In the set C(')([a,b]) a metric is defined by k

p(z.y) =

1 max Iz(')(t) 1=o

-

(12.49)

y(')((t)/.

tEI4

where (see (12.14) C(')([a,b ] ) is understood as C ( [ a ,b])).

W H: Consider the set P ( Q (1 ) 5 p < m) of the equivalence classes of Lebesgue measurable functions which are defined almost everywhere on a bounded domain

R c R" and

I

n

lz(t)IPdp< co (see also

12.9, p. 633). A metric in this set is defined by P(Z?Y) =

\!

p

I 4 - Y(t)IPdP

(12.50)

12.2.1.1 Balls, Neighborhoods and Open Sets In a metric space X = (X, p ) , whose elements are also called points, the following sets

B ( z o :r ) = {x E X : p(xl 20) < T } . (12.51)

-

(12.52) B (z o ;r ) = {x E X : p(x, xo) 5 r } defined by means of a real number r > 0 and a fixed point EO,are called an open and closed ball with radius r and center at 2 0 , respectively. The balls (circles) defined by the metrics (12.43) and (12.44) and (12.45) in the vector space R2 are represented in Fig. 12.4a,b with 20 = 0 and r = 1.

Figure 12.4 .;2 subset li of a metric space X = (X, p ) is called a nezghborhoodof the point zoif U contains zo together with an open ball centered at 2 0 , in other words, if there exists an r > 0 such that B(z0;T ) c U . 4 neighborhood C' of the point z is also denoted by U ( z ) .Obviously, every ball is a neighborhood of its center; an open ball is a neighborhood of all of its points. A point 20 is called an znterzorpoint of a set A c X if zo belongs to A together with some of its neighborhood, i.e., there is a neighborhood N of zo such that xo E G c .4. A subset of a metric space is called open if all of its points are interior points. Obviously. X is an open set. The open balls in every metric space, especially the open intervals in R, are the prototypes of open sets.

604

12. Functional Analusis

The set of all open sets satisfies the following axzoms of open sets: If G, is open for Vcu E I , then the set U G, is also open. ,€I

n

If G1.Gz... . , G, are finitely many arbitrary open sets, then the set n Gk is also open. k=l The empty set 0 is open by definition. A subset A of a metric space is bounded if for a certain element zo(which does not necessarily belong to A) and a real number R > 0 the set A is in the ball B(z0;R), Le., p(z, zo) < R for all z E A.

12.2.1.2 Convergence of Sequences in Metric Spaces Let X = (X, p ) be a metric space, zo E X and {zn}r=l,z, E X a sequence of elements of X. The sequence {x,}:=~ is said to be convergent to the point zo if for every neighborhood U(z0)there is an index no = no(U(s0))such that for all n > no, x, E V(z0). We use the usual notation J, + T O (n co) i or hlx, = zo (12.53) and call the point zo the lzrnzt ofthe sequence {z~}:=~. The limit of a sequence is uniquely determined. Instead of an arbitrary neighborhood of the point 2 0 , it is sufficient to consider only open balls with arbitrary radii, so (12.53) is equivalent to the following: V E > 0 (we are at once thinking about the open ball B(z0;E ) ) , there is an index no = no(€),such that if n > no, then p(xn, 20) < E . Notice that (12.53) means p(x,, ZO)+0. LVith these notions introduced in special metric spaces we can calculate the distance between points and investigate the convergence of point sequences. This has a great importance in numerical methods and in approximating functions by certain classes of functions (see, e.g., 19.6, p. 914). In the space R", equipped with one of the metrics given above, convergence always means coordinatewise convergence. In the spaces B([a,b])and C ( [ a ,b ] ) , the convergence introduced by (12.48) means uniform convergence of the function sequence on the set [a,b] (see 7.3.2, p. 412). In the space L 2 ( Q )convergence with respect to the metric (12.50) means convergence in the (quadratic) mean, Le., x, --t xo if (12.54)

12.2.1.3 Closed Sets and Closure 1. Closed Sets

A subset P of a metric space X is called closed if X \ F is an open set. Every closed ball in a metric space. especially every interval of the form [alb], [a,co),(-coola] in R, is a closed set. Corresponding to the axioms of open sets, the collection of all closed sets of a metric space has the following properties: If F, are closed for Vcu E I , then the set F, is closed.

n

,€I

n

If P I , .. . , F, are finitely many closed sets, then the set U

Fk

is closed.

k=l

The empty set 0 is a closed set by definition. The sets 0 and X are oDen and closed at the same time. -4 point zo of a metricspace X is called a limit point of the subset A

c X if for every neighborhood

UXO), (12.55) c.(x0)n A # 0. If this intersection always contains at least one point different from 2 0 , then zo is called an accvrnvlatzon p o d of the set A. A limit point, which is not an accumulation point, is called an isolatedpoint. An accumulation point of A does not need to belong to the set A, e.g., the point a with respect to the set A = (a. b]. while an isolated point of A must belong to the set A.

22.2 Metric Spaces 605

-4 point 2 0 is a limit point of the set A if there exists a sequence {z,}~=~with elements z,from A, which converges to 20. If 2 0 is an isolated point, then z, = 20, V n 2 no for some index no.

2. The Closure of a Set Every subset A of a metric space X obviously lies in the closed set X. Therefore, there always exists a smallest closed set containing A, namely the intersection of all closedsets of X, which contain A. This set is called the closKe of the set A and it is usually denoted by .4. A is identical to the set of all limit points of A: weget .4 from the set A by adding all of its accumulation points to it. A is a closed set if and only if A = A. Consequently, closed sets can be characterized by sequences in the following way: .4 is closed if and only if for every sequence {z,}~=~of elements of A, which converges in X to an element zo(E X), the limit zo also belongs to A. Boundary points of A are defined as follows: 20 is a boundary point of A if for every neighborhood U(zo),U(Q) n A # 0 and also U ( z 0 ) n ( X \ A) # 0.zo itself does not need to belong to A. Another characterization of a closed set is the following: A is closed if it contains all of its boundary points. (The set of boundary points of the metric space X is the empty set.)

12.2.1.4 Dense Subsets and Separable Metric Spaces -4 subset A of a metric space X is called everywhere dense if 2 = X, Le., each point z E X is a limit point of the set A. That is, for each z E X, there is a sequence {z,} z, E A such that z, +z. IA: According to the Weierstrass approximation theorem, every continuous function on a bounded closed interval [a,b] can be approximated arbitrarily well by polynomials in the metric space of the space C([a.b ] ) :Le., uniformly. This theorem can now be formulated as follows: The set of polynomials on the interval [a,b] is everywhere dense in C ( [ a ,b]). IB: Further examples for everywhere dense subsets are the set of rational numbers Q and the set of irrational numbers in the space of the real numbers R . .4 metric space X is called separable if there exists a countable everywhere dense subset in X. A countable everywhere dense subset in R" is, e.g., the set of all vectors with rational components. The space 1 = 1' is also separable, since a countable everywhere dense subset is formed, for example, by the set of its elements of the form z = ( T I ) ~ 2 ). .. , T N . 0,O.. . .) , where r, are rational numbers and ,V = .V(z) is an arbitrary natural number. The space m is not separable.

12.2.2 Complete Metric Spaces 12.2.2.1 Cauchy Sequences Let X = (X, p ) be a metric space. A sequence {z,};=~ with z, E X is called a Cauchy sequence if for V i > 0 there is an index no = no(&)such that for V n: m > no there holds the inequality P(Zn,Zn) < E. (12.56) Every Cauchy sequence is a bounded set. Furthermore, every convergent sequence is a Cauchy sequence. In general, the converse statement is not true, as is shown in the following example. = IConsider the space 1' with the metric (12.46) of the space m. Obviously, the elements 1 1 1 (1. 2'?, . . . ,,; 0,O.. . belong to 1' for every n = 1 , 2 , .. . and the sequence {x(,)}:=~ is a Cauchy

.)

sequence in this space. If the sequence (of sequences) {z(")},"=l converges, then it has to be convergent also coordinate-wise to the element do)= to l', since

" 1 - = +m (see 7.2.1.1, 2., p. 404. harmonic series). n=l

n

) . However, do)does not belong

606

12. Functional Analusis

12.2.2.2 Complete Metric Spaces .4 metric space X is called complete if every Cauchy sequence converges in X. Hence, complete metric spaces are the spaces for which the Cauchy principle, known from real calculus, is valid: A sequence is convergent if and only if it is a Cauchy sequence. Every closed subspace of a complete metric space (considered as a metric space on its own) is complete. The converse statement is valid in a certain way: If a subspace Y of a (not necessary complete) metric space X is complete, then the set Y is closed in X.

ICompletemetricspacesare,e.g.,thespaces: m, LP(a. b) (1 5 p < co).

1P

(1 5 p < co),c,

B ( T ) ,C ( [ a , b ] ) ,

C @ ) ( [ ab,] ) ,

12.2.2.3 Some Fundamental Theorems in Complete Metric Spaces The importance of complete metric spaces can be illustrated by a series of theorems and principles. which are known and used in real calculus, and which we want to apply even in the case of infinite dimensional spaces.

1. Theorem on Nested Balls Let X be a complete metric space. If B(x1;r1) 3 B(z2;7-2) I). . ' 3 B(xn;rn) 3 . ' . (12.57) is a sequence of nested closed balls with r, t 0, then the intersection of all of those balls is nonempty and consists of only a single point. If this property is valid in some metric space for any sequence satisfying the assumptions, then the metric space is complete.

2. Baire Category Theorem m

Let X be a complete metric space and { F k } g l a sequence of closed sets in X with U

Fk

= X. Then

k=1

there exists at least one index ko such that the set F k o has an interior point.

3. Banach Fixed-point Theorem Let F be a non-empty closed subset of a complete metric space (X, p ) . Let Z? X operator on F , Le., there exists a constant q E [0,1) such that ~ ( T xTy) . 5 qp(z,y) for all z,y E F. Suppose, if z E F , then 2's E F . Then the following statements are valid: a) For an arbitrary initial point zo E F the iteration

+ X be a contracting

( n = 0.1.2,. . .) is well defined, Le., z, E F for every n. b) The iteration sequence {z~}:=~ converges to an element x' E F . Tx' = x*>Le., x* is a fixed point of the operator T . c) d) The only fixed point of T in F is x*. e) The following error estimation is valid: X,+l

:= T I ,

(12.58)

(12.59)

(12.60)

(12.61) The Banach fized-point theorem is sometimes called the contraction mapping principle

12.2.2.4 Some Applications of the Contraction Mapping Principle 1. Iteration Method for Solving a System of Linear Equations The given linear ( n ,R ) system of equations

+. . . + ainz1 = +. . . + aznxn = bz, ............... anlxl +an2x2 + . . . + annxn = b, +aizxz a21x1+a22x2 UIIZI

bl:

I.................

(12.62a)

12.2 Metn'c Spaces 607

can be transformed according to 19.2.1, p, 887, into the equivalent system z1 - ( I - al!)xl +a!zzz .. +ainzn = bi: x2 +azlxl -(1 - a22)z2 . +aznxn = bz,

+, +, ...................................................... 2, +anixl +an2z2 +. . -(1 - ann)$"= b,.

( 12.62b)

,

If the operator T : F" + Fn is defined by n

TX =

(

51

- XalkXk

+ b i , . . . ,Xn- xankXk + b,

kI1

(12.63)

k=l

then the last system is transformed into the fixed-point problem x = Tx (12.64) in the metric space F", where an appropriate met& is considered: The Euclidean (12.43), the maximum (12.44) or the absolute value metric p(x, y) =

? lzk - yhl (compare with (12.45)). If one of the

k=l

numbers I

n

(12.65) is smaller than one, then T turns out to be a contracting operator. It has exactly one fixed point according to the Banach fixed-point theorem, which is the componentwise limit of the iteration sequence started from an arbitrary point of F".

2. Fredholm Integral Equations The Fredholm integral equation of the second kind (see also 11.2, p. 562)

J

a

with a continuous kernel K(x,y) and continuous right-hand side f(z)can be solved by iteration. By means of the operator T: C([a,b]) +C([a,b]) defined as b

Td.)

=

1

K ( z ,Y)rP(Y) dY + f(.)

v 'p E C ( b >bl),

(12.67)

a

it is transformed into a fixed-point problem T y = p in the metric space C([a,b ] ) (see example A in

z;zb1lK(z,y)1dy < 1, then T is a contracting operator and the fixed-point b

12.1.2. 4.,p. 595). If

theorem can be applied. The unique solution is now obtained as the uniform limit of the iteration sequence {pn}T=l.where pn = Tpn-l, starting with an arbitrary function yo(.) E C([a,b]). It is clear that pn = Tnpo and the iteration sequence is {Tnpo}F=l.

3. Volterra Integral Equations The Volterra integral equation of the second kind (see 11.4, p. 583) 5

3(z)- /K!x. Y)rP(Y) dY = f(.),

E

1%

bl

(12.68)

a

with a continuous kernel and a continuous right-hand side can be solved by means of the Volterra integral operator (12.69)

I

608

12. Functional Analysis

and T y = f t Vp as the fixed-point problem T p = 'p in the space C ( [ a ,b]). 4. Picard-Lindelof Theorem Consider the differential equation x = f(t,x) (12.70) with a continuous mapping f : I x G -+R", where I is an open interval of R and G is an open domain of R". Suppose the function f satisfies a Lipschitz condition with respect to z (see 9.1.1.1, 2. p. 486)) Le., there is a positive constant L such that df(t.zi),f(t,zz))5 Le(zi,zz) v(t,zi),(t,zz)E I x G, (12.71) where g is the Euclidean metric in Rn. (Using the norm (see 12.3.1, p. 609) and the formula (12.81) e(z,y) = llx-y(l (12.71)canbewrittenasIlf(t,rl)-f(t,z2)II 5 I~zl-zzii.)Let (t0,zo) E I x G . Then there are numbers p > 0 and T > 0 such that the set R = { ( t ,z) E R x Rn: It - to1 5 /3, p(z,zo) 5 T } lies in I x G. Let M = maxn e(f(tlz), 0) and a = min{/3, L}.Then there is a number b

M

> 0 such

that for each i E B = {z E R": e(z, 2 0 ) 5 b}, the initial value problem x = f(t,.), x(t0) = i (12.72) hasexactlyonesolutionp(t,i),i.e., $ ( t , i )= f(t,y(t,i))forVtsatisfyingIt-toj 5 aandcp(t0,f) = 5. The solution of this initial value problem is equivalent to the solution of the integral equation t

p(t>i) = i t

1

f(s,p(s, i))ds,

t E [to - a , t o + a].

(12.73)

to

If X denotes the closed ball {p(t,z) : d(rp(t,x),20) 5 a] x B; R")with metric

T}

in the complete metric space C([to - a, t o t

4%Y ) = ( t , z ) E l , ~ ~ $ s ae('p(4~1, )xB N t , z)), then X is a complete metric space with the induced metric. If the operator T : X

(12.74)

+X is defined by

t

T d t ,). = 5 t

1

f ( S >C p ( S l 4 ) ds

(12.75)

to

then T is a contracting operator and the solution of the integral equation (12.73) is the unique fixed point of T which can be calculated by iteration.

12.2.2.5 Completion of a Metric Space Every (non-complete) metric space X can be completed; more precisely, there exists a metric space X with the following properties:

a) X contains a subspace Y isometric to X (see 12.2.3,2., p. 609).

b) Y is everywhere dense in X. c)

X is a complete metric space.

d) If Z is any metric space with the properties a)-c), then Z and X are isometric. The complete metric space, defined uniquely in this way up to isometry, is called the completzon of the space X.

12.2.3 Continuous Operators 1. Continuous Operators

Let T . X -iY be a mapping of the metric space X = ( X , p ) into the metric space Y = (U, e). T is said to be contznuous at the poznt zo E X if for every neighborhood V = V(y0) of the point yo = T(z0)

12.3 Normed Spaces 609

there is a neighborhood U = U(zo)such that: T ( z )E V for all z E U. (12.76) T is called contanuous on the set A c X if T is continuous at every point of A . Equivalent properties for T to be continuous on X are: a) For any point z E X and any arbitrary sequence { x ~ } ; = ~z, , E X with 2, +2 there always holds T(x,) i T ( z ) .Hence p(x,,zo) + 0 implies ~ ( T ( z ~ ) , T (+ z o0.) ) b) For any open subset G c Y the inverse image T-'(G) is an open subset in X. c ) For any closed subset F c Y the inverse image T - ' ( F ) is a closed subset in X. d) For any subset .4 C X one has T ( 2 )c T(A).

2. Isometric Spaces If there is a bijective mapping T : X

+Y for two metric spaces X = (X, p ) and Y = (Y, e) such that

P h Y) = e ( T ( x )T(Y)) , Vz, Y E X, then the spaces X and Y are called isometric, and T is called an isometry.

(12.77)

12.3 Normed Spaces 12.3.1 Notion of a Normed Space 12.3.1.1 Axioms of a Normed Space Let X be a vector space over the field F. A function I/ . 11 : X + R: is called a norm on the vector space X and the pair X = (X, /I 11) is called a normed space over the field F, if for arbitrary elements x,y E X and for any scalar a E F the following properties, the so-called azioms o f a normed space, are fulfilled: ( N l ) Ilz/I 2 0,

and llzli = 0 if and only if z = 0,

(N2) l l a 4 = la1 . ll4 (N3) Ilz

+ yI1 5 11x11 t llyll

(12.78)

(homogenity)

~

(triangle inequality)

4 metric can be introduced by means of 2, Y E X, P ( Z , Y) = llz - Yll,

(12.81) in any normed space. The metric (12.81) has the following additional properties which are compatible with the structure of the vector space: (12.82a) (12.82b) So, in a normed space there are available both the properties of a vector space and the properties of a metric space. These properties are compatible in the sense of (12.82a) and (12.82b). The advantage is that most of the local investigations can be restricted to the unit ball (12.83) B(O:I) = {z E X : llzll < 1) or B(O;1) = {z E X : 11x11 5 1) since (12.84) B ( z ;T ) = {y E X : Ily - zI1 < r } = z+ rB(0;l ) , V z E X and V r > 0. Moreover, the algebraic operations in a vector space are continuous, i.e., z, + z, yn --t y, a, --t a imply zn + Yn + 2 + Y, anzn + az, llznll + 1141. (12.85) In normed spaces instead of (12.53) we may write for convergent sequences Ilz, - 5011 -+0 ( n + m). (12.86)

610

12. Functional Analusis

12.3.1.2 Some Properties of Normed Spaces Among the linear metric spaces, those spaces are nomable (Le., a norm can be introduced by means of the metric, if one defines llzl1 = p(z, 0)) whose metric satisfies the conditions (12.82a) and (12.82b). Two normed spaces X and Y are called n o m isomorphic if there is a bijective linear mapping T: X + Y with IITzll = l/zIl for all z E X. Let /I . 111 and I! 112 be two norms on the vector space X, and denote the corresponding normed spaces by X1 and X2, i.e., X1 = (X, I/ . 111) and Xz = (X, 11 . 112). The norm 11 . 111 is stronger than the norm 11 . 112, if there is a number y > 0 such that /lzII:, 5 yllzlll, for all x E X. In this case, the convergence of a sequence to z with respect to the stronger norm 11 . Ill, Le., llzn - xIll t 0, implies the convergence to z with respect to the norm 11 112r Le., 3

llxn - 2112

t 0.

Two norms /I. 11 and 11 1. 1 1 I are called equivalent if there are two numbers y1 > 0, 72 > 0 such that Vz E X there holds n/1/1z/l5 lllzlll 5 y21lz/I. In a finite dimensional vector space all norms are equivalent to each other. A subspace of a normed space is a closed linear subspace of the space.

12.3.2 Banach Spaces A complete normed space is called a Banach space. Every normed space X can be completed into a Banach space X by the completion procedure given in 12.2.2.5,p. 608, and by the natural extension of its algebraic operations and the norm to X.

12.3.2.1 Series in Normed Spaces In a normed space X we can consider inJinite series. That means for a given sequence {zn}r=lof elements zn E X a new sequence {sk}p=1is constructed by ''' x k = sk-1 xk. (12.87) S1 = X I r 32 = x1 2 2 , . . . sk = If the sequence {Sk}& is convergent, i.e., /Isk - s/l --+ 0 ( k + co) for some 8 E then a convergent series is defined. The elements SI,s2.. . . , sk, . . . are called the partial sums of the series. The limit

+ +

+

+

x,

(12.88) oi

is the sum of the series, and we write s =

W

2.,

n=l

A series

n=l

zn is called absolutely convergent if the

M

/lznllis convergent. In a Banach space every absolutely convergent series is conver-

number series n=l

gent, and 1lsIl 5

cc

C llxnll holds for its sum s. n=l

12.3.2.2 Examples of Banach Spaces IA : Fn lvith ~~z~~ =

~

if 1 5 p < co; llzl/ =

ly2nI & j ,

if p = co.

(12.89a)

These normed spaces over the same vector space F" are often denoted by lP(n) (1 5 p 5 co). For 1 5 p < co,we call them Euclidean spaces in the case of F = R, and unztary spaces in the case of F = C. (12.89b) IB : m with 11x11 = s t p IQI.

IC : c and co with the norm from m.

(12.89~) (12.89d)

12.3 Normed Spaces 611

IE : C ( [ a ,b]) with IlzIl = max Ix(t)l.

(12.89e)

tE[a,bl

1

(12.89f) k

IG

C ( k ) ( [ ab,] ) with Ilz(I=

1max lx(')(t)12 where d0)(t)stands for z ( t ) .

(12.89g)

l=o t W 1

12.3.2.3 Sobolev Spaces R c R" be a bounded domain, Le., an open connected set, with a sufficiently smooth boundary dR. For n = 1 or n = 2,3 we can imagine R being something similar to an interval ( a ,b ) or a convex Let

set. d function f : + R is k-times continuously differentiable on the closed domain i f f is k-times continuously differentiable on R and each of its partial derivatives has a finite limit on the boundary, 1.e..if x approaches an arbitrary point of OR.In other words, all partial derivatives canJe continuously extended on the boundary of Q, Le., each partial derivative is a continuous function on 0. In this vector space (for p E [l.m))and with the Lebesgue measure X in Rn(see example C in 12.9.1, 2., p. 634) the following norm is defined:

a

/ / f l / k , p=

ilfil =

in1

If(x)IpdX+

(12.90)

Theresultingnormedspaceisdenoted byT.i/k,p(R)oralso byT.i/;(R) (incontrast to thespaceC(k)([a,b]) which has a quite different norm). Here cy means a rnultz-zndez, Le., an ordered n-tuple ( a l , .. . ,cy,) of non-negative integers, where the sum of the components of a is denoted by la1 = a1 a2 . . an. For a function f(x) = f ( G , . . .,En)with x = ( S I . . . . ,&) E we use the brief notation as in (12.90):

a

D"f=

+ + +

dI"1 f

a(:' . ,a(,".' ,

The normed space r n k , p ( R ) is not complete. Its completion is denoted by WkJ'(Q) or in the case of p = 2 by Hk(R)and it is called a Sobolev space.

12.3.3 Ordered Normed Spaces 1. Cones in a Normed Space Let X be a real normed space with the norm

11 11. A cone X+ C X (see 12.1.4.2,p. 597) is called a solzd, if X, contains a ball (with positive radius), or equivalently, X+ contains at least one interior point. IThe usual cones are solid in the spaces R, C( [a,b ] ) ,c, but in the spaces L p ( ( a ,b ) ) and 1P (1 5 p < m) they are not solid. .4cone X+ is called normalif the norm in X is sernzmonotonzc,Le., there exists a constant A l > 0 such that (12.92) 0 5 z I l/ 114 5 W Y l l . If X is a Banach space ordered by a cone X+, then every (0)-interval is bounded with respect to the norm if and only if the cone X+ is normal. IThe cones of the vectors with non-negative components and of the non-negative functions in the spaces R". rn. c. CO. C, IP and LP,respectively, are normal. A cone is called regular if every monotonically increasing sequence which is bounded above, 5 x* 5 . , , I x, 5 ' ' ' 5 2 (12.93)

-

612

12. Functional Analysis

is a Cauchy sequence in X. In a Banach space every closed regular cone is normal. IThe cones in R", lp and LP for 1 5 p < 03 are regular, but in C and m they are not.

2. Normed Vector Lattices and Banach Lattices Let X be a vect,or lattice: which is a normed space at the same time. X is called a normed lattice or normed vector lattice (see [12.15], [12.19], [12.22], [12.23]),if the norm satisfies the condition 1x1 5 IyI implies 11z11 5 lIy11 Vz,y E X (monotonicityofthe norm). (12.94) .A complete (with respect to the norm) normed lattice is called a Banach lattice. IThe spaces C ( [ a ,b ] ) , Lp, l p , B([a,b]) are Banach lattices.

12.3.4 Normed Algebras .4 vector space X over F is called an algebra, if in addition to the operations defined in the vector space X and satisfying the axioms (Vl)-(VB) (see 12.1.1, p. 594), a product z.y E Xis also defined for every two elements z,y E X,or with a simplified notation the product zy is defined so that for arbitrary z,y,z E X and cu E F the following conditions are satisfied: (AI) 4 ~= ()X Y ) ~ , (12.95) (A2) z(y + z ) = zy t Z Z , (12.96) (A3) (zt y)z = z z yz, (12.97) (A4) a ( z y ) = (az)y= z(ay). (12.98) .4n algebra is commutative if zy = yz holds for two arbitrary elements 2, y. A linear operator (see (12~21),p. 598) T: X t Y of the algebra X into the algebra Y is called an algebra homomorphism if for any 21. 1 2 E X: (12.99) T(z1 .z2)= Tzl . T x ~ . .An algebra X is called a normed algebra or a Banach algebra if it is a normed vector space or a Banach space and the norm has the additional property (12.100) llz ' YII 5 114 ' IlYIl. In a normed algebra all the operations are continuous, Le., additionally to (12.85), if z, -+z and yn -+ yI then also zny, -+ zy (see [12.20]). Every normed algebra can be completed to a Banach algebra, where the product is extended to the norm completion with respect to (12.100). IA: C ( [ a ,b]) with the norm (12.89e) and the usual (pointwise) product of continuous functions. IB: The vector space W([O,27~1)of allcomplex-valued functionsz(t) continuous on [ 0 , 2 ~and ] having an absolutely convergent Fourier series expansion, Le.,

+

cc

(12.101)

c,eint,

z(t) = n=-m

m

with the norm J / z J=J

IC,\

and the usual multiplication

n=--30

C: The space L(X) of all bounded linear operators on the normed space X with the operator norm and the usual algebraic operations (see 12.5.1.2, p. 617), where the product T S of two operators is defined as the sequential application, Le., T S ( z )= T ( S ( z ) ) z , E X. ID: The space L'(-ffi) m) of all measurable and absolutely integrable functions on the real axis (see 12.9, p. 633) with the norm (12.102) is a Banach algebra if the multiplication is defined as the convolution (z* y)(t) =

1

5.

-02

z(t - s)y(s) ds.

12.4 Hilbert Spaces 613

12.4 Hilbert Spaces 12.4.1 Notion of a Hilbert Space 12.4.1.1 Scalar Product .4vector space V over a field F (mostly F = C) is called a space wzth scalar product or an znnerproduct space or pre-Hzlbert space if to every pair of elements 2,y E V there is assigned a number (z> y) E F (the scalar product of z and y), such that the azzoms of the scalar product are satisfied, Le., for arbitrary z,y. z E V and cy E F: (Hl) (x,x) 2 0, (Le.. (x>x) is real), and ( 2 ,z)= 0 if and only if z = 0, (12.103) Y) = 4 5 , Y),

(H2)

(12.104)

(H3) (z+ Y. 2 ) = ( 2 , ~+)(Y, 2))

(12.105)

m.

(12.106) (H4) (2.y) = (Hereg denotes the conjugate of the complex number w , which is denoted by w* in (1.133~)Sometimes the notation of a scalar product is (2,y).) In the case of F = R, Le., in a real vector space, (H4) means the commutativity of the scalar product. Some further properties follow from the axioms: (12.107) (x,ay) = a(z.y) and ( 2 ,y z ) = (2,y) ( 2 , ~ ) .

+

+

12.4.1.2 Unitary Spaces and Some of their Properties In a pre-Hilbert space H a norm can be introduced by means of the scalar product as follows:

114 =

m

(zE HI.

.L\ normed space H = (H, 11

(12.108)

11)

is called unztary if there is a scalar product satisfying (12.108). Based on the previous properties of scalar products and (12.108) in unitary spaces the following facts are valid: a) Wangle Inequality: 112

+ YII' 5 (Ild+ llvll)' .

(12.109)

b) Cauchy-Schwarz Inequality or Schwarz-Buniakowski Inequality (see also 1.4.2.9, p. 31): (12.110)

c) Parallelogram Identity: This characterizes the unitary spaces among the normed spaces: (12.111)

(12.112)

12.4.1.3 Hilbert Space A complete unitary space is called a Hzlbert space. Since Hilbert spaces are also Banach spaces, they possess in particular. the properties of the last (see 12.3.1, p. 609; 12.3.1.2, p. 610: 12.3.2, p. 610). In addition they have the properties of unitary spaces 12.4.1.2, p. 613. A subspace of a Hilbert space is a closed linear subspace. IA: l2(n). 1' and L Z ( ( ab, ) ) with the scalar products (12.113)

I

614

12. Functional Analysis

IB: The space HZ(R)with the scalar product

/

(f,9) = f ( 4 g o d z + R

J D . f ( x ) D Q g ( x dx. )

(12.114)

1
IC: Let p(t) be a measurable positive function on [a,b]. The complex space L z ( ( a ,b ) , p) of all measurable functions. which are quadratically integrable with the weight function p on ( a . b), is a Hilbert space if the scalar product is defined as

12.4.2 Orthogonality Two elements x,y of a Hilbert space H are called orthogonal (we write x Iy) if (2, y) = 0 (the notions of this paragraph also make sense in pre-Hilbert spaces and in unitary spaces). For an arbitrary subset A c H , the set (12.116) A’ = {z E H: (z, y) = 0 V y E A } of all vectors which are orthogonal to each vector in A is a (closed linear) subspace of H and it is called the orthogonal space to A or the orthogonal complement of A. We write A I B if (z, y) = 0 for all z E A and y E B . If A consists of a single element z,then we write x 1B .

12.4.2.1 Properties of Orthogonality The zero vector is orthogonal to every vector of H. The folloning statements hold: a) z Iy and z 1z implyz I (cyy + bz) for any a,B E C. b) From z 1yn and yn + y it follows that z 1y. c) z 1A if and only if z 1lin(A),where lin(A)denotes the closed linear hull of the set A. d) If IA and A is afundamental set, Le., lin(A)is everywhere dense in H, then z = 0. e) Pythagoras Theorem: If the elements T I , . . . , x, are pairwise orthogonal, that is x k I zl for all k#l,tb; ~

I

n

(12.117)

f ) Projection Theorem: If Ho is a subspace of H, then each vector x E H can be written uniquely as z = 2’

+ d’, 2’ E Ho, 5’’ 1Ho.

(12.118)

g) ApproximationProblem: Furthermore, the equation /Iz’IJ= p(x,Ho) = inf,,a,{ Ilx-yll} holds,

and so the problem llz - yl1 -+ inf, Y E HO (12.119) has the unique solution z’in Ho. In this statement Ho can be replaced by a convex closed non-empty subset of H.The element z’is called the projection of the element z on Ho. It has the smallest distance from z (to Ho),and the space H can be decomposed: H = Ho E?H i .

12.4.2.2 Orthogonal Systems A set {zt:[ E E} of vectors from H is called an orthogonalsystem if it does not contain the zero vector and xF 1z7, ( # 7 , hence (q, z7)= dFq holds, where ( 12,120)

12.4 Hilbert Spaces 615

denotes the Kronecker symbol (see 4.1.2, lo., p. 253). An orthogonal system is called orthonomal if in addition llzcIl = 1 V<. In a separable Hilbert space an orthogonal system may contain at most countably many elements. In what follows we assume, therefore, E = N. IA: The system 1 -cos 1 -

$zJ;;

1 . t , -cos 1 t , -sin

J;;

J;;

1 . 2t, -sin 2t,

J;;

(12.121)

in the real space L'((-T, T ) ) and the system

. . .)

(12.122)

in the complex space L'((-T, T ) ) are orthonormal systems. Both of these systems are called trigonometric. IB: The Legendre polynomials of the first kind (see 9.1.2.6, 2., p. 510)

P,(t) =

d" --[(e - l),] d tn

( n = 0, 1,.. .)

(12.123)

form an orthogonal system of elements in the space L2((-1.1)). The corresponding orthonormal system is (12.124)

IC: The Hermite polynomials (see 9.1.2.6, 6.) p. 512 and 9.2.3.5, 3., 543) according to the second definition of the Hermite differential equation (9.63b) (12.125) form an orthogonal system in the space L2((-m, m)). I D : The Laguerre polynomials form an orthogonal system (see 9.1.2.6, 5., p. 511) in the space L'((0, a)). Every orthogonal system is linearly independent, since the zero vector was excluded. Conversely, if we have a system ~ 1 . ~ 2, ..z,, . , , . of linearly independent elements in a Hilbert space H, then there exist vectors e l , e2, . . . , e n r , . ., obtained by the Gram-Schmzdt orthogonalizatzon method (see 4.5.2.2, 1.. p. 280) which form an orthonormal system. They span the same subspace, and by the method they are determined up to a scalar factor with modulus 1.

12.4.3 Fourier Series in Hilbert Spaces 12.4.3.1 Best Approximation Let H be a separable Hilbert space and (12.126) {e,: n = 1 , 2 . . . .} a fixed orthonormal system in H. For an element z E H the numbers e,, = (z, e,) are called the Fourier coeflcients of z with respect to the system (12.126). The (formal) series m

1cnen I

( 12.127)

n=l

is called the Fourierserzes of the element zwith respect to the system (12.126) (see 7.4.1.1. l., p. 418). The n-th partial sum of the Fourier series of an element z has the property of the best approximation,

I

616

12. Functional Analgsis

Le., for fixed n, the n-th partial sum of the Fourier series n 0 ,

(12.128)

= k=l

gives the smallest value of llz -

5

among all vectors of H, = lin((e1,.. . ,en}). Furthermore,

akekll

k=l

z - un is orthogonal to H,,and there holds the Bessel inequality: m

1

ICnI'

5

cn =

IIzII's

(23en)

(n = 122,. .

(12.129)

n=l

12.4.3.2 Parseval Equation, Riesz-Fischer Theorem The Fourier series of an arbitrary element z E H is always convergent. Its sum is the projection of the element z onto the subspace Ho = lin({e,};.J. If an element x E H has the representation m

z = 1 a,e,, then a, are the Fourier coefficients of z (n = 1 , 2 , .. .). If {a,}:=, is an arbitrary n=l X

sequence of numbers with the property C Ianlz < m, then there is a unique element z in H,whose n=l

Fourier coefficients are equal to a, and for which the Parseval equation holds: 00

m

n=l

n=l

1/(x,e,)I' =

Ianl2= 11z/I2

(Riesz-Fischertheorem).

(12.130)

.An orthonormal system {e,} in H is called complete if there is no non-zero vector y orthogonal to every e,; it is called a basis if every vector z E H has the representation z =

m

anen,i.e., a, = (z, e,) and

n=l

z is equal to the sum of its Fourier series. In this case, we also say that x has a Fourier expansion. The following statements are equivalent: a ) {e,} is a fundamental set in H. b) {e,} is complete in H. c) {e,} is a basis in H. d) For V z,y E H with the corresponding Fourier coefficients c, and d, (n = 1 , 2 , .. .) there holds m

(12.131)

e) For every vector 5 E H,the Parseval equation (12.130) holds. IA: The trigonometric system (12.121) is a basis in the space L'((-a,a)). IB: The system of the normalized Legendre polynomials (12.124) p,(t) (n = 0 , 1 , . . .) is complete and consequently a basis in the space L2((-1, 1)).

12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces In every separable Hilbert space there exits a basis. From this fact it follows that every orthonormal system can be completed to a basis. Two Hilbert spaces H1and H:,are called zsometrzc or zsomorphzc as Hilbert spaces if there is a linear bijective mapping 7': HI t H2 with the property ( T z ,T ~ ) H=, (x,y ) ~ ,(that is, it preserves the scalar product and because of (12.108)also the norm). Any two arbitrary infinite dimensional separable Hilbert spaces are isometric, in particular every such space is isometric to the separable space 1'.

12.5 Continuous Linear Operators and Functionals 617

12.5 Continuous Linear Operators and Functionals 12.5.1 Boundedness, Norm and Continuity of Linear Operators 12.5.1.1 Boundedness and the Norm of Linear Operators Let X = (X, 11 .II) and Y = (U, 11 11) be normed spaces. In the following discussion, we omit the index X in the notation 11 . I/* emphasizing that we are in the space X, because from the text it will be always clear which norm and which space we are talking about. An arbitrary operator T : X +Y is called bounded if there is a real number X > 0 such that IlT(z)ll I X l M 'fzE X. (12.132) A bounded operator with a constant X "stretches" every vector at most X times and it transforms every bounded set of X into a bounded set of Y,in particular the image of the unit ball of X is bounded in Y. This last property is characteristic of bounded linear operators. A linear operator is continuous (see 12.2 3, p. 608) if and only if it is bounded. The smallest constant A, for which (12.132) still holds, is called the n o m of the operator T and it is denoted by llT/l. Le., (12.133) //T/l:= inf{X > 0 : IITxll 5 Xl1x11, x E X}. For a continuous linear operator the following equalities hold: (12.134) IlTlI = SUP IITxll = SUP IlT4 = SUP IITxll llxIlS1

/lxll<1

M=1

and, furthermore, the equality IITxll 5 IITII . It4 'fxE X. (12.135) Let T be the operator in the space C([a.b ] ) with the norm (12.89d),defined by the integral

( T z ) ( s )= ?/(SI=

I"

K ( s ,t ) x ( t )d t

(3 E

[a,all,

(12.136)

where K ( s ,t ) is a (complex-valued) continuous function on the rectangle { a 5 s, t 5 b}. Then T is a bounded linear operator, which maps C ( [ a ,b ] ) into C ( [ a ,b ] ) . Its norm is

'IT''

max

/b

= s€[a,b]

l ~ ( s , t )dt. l

(12.137)

a

12.5.1.2 The Space of Linear Continuous Operators

+

The sum U = S T and the multiple cuT of two linear (continuous) operators S,T : X -+Y are defined pointwise: G ( z ) = S ( z ) + T ( x ) , ( c u T ) ( x ) = a . T ( z ) ,V ~ E and X V~EF. (12.138) The set L(X, U), often denoted by B ( X ,U),of all linear continuous operators T from X into Y equipped with the operations (12.138) is a vector space, where IlTll (12.133) turns out to be a norm on it. So, L(X, Y)is a normed space and even a Banach space if Y is a Banach space. So the axioms (Vl)-(V8) and (Nl)-(N3) are satisfied. If Y = X. then a product can be defined for two arbitrary elements S,T 6 L ( X ,X) = L(X) = B ( X ) as ( S T ) ( x )= S(Tx) ( V l E X), (12.139) which satisfies the axioms (Al)-(A4) from 12.3.4,p. 612, and also the compatibility condition (12.100) with the norm. L(X) is in general a non-commutative normed algebra, and ifX is a Banach space, then it is a Banach algebra. Then for every operator T E L(X) its powers are defined by

TO = I , Tn = T n - 9 (n = 1 , 2 , .. .), where I is the identity operator Ix = x, V x E X. Then llTnll I IITIl" (n= 051:.

.)I

(12.140) (12.141)

I

618

12. Functional Analusis

and furthermore there always exists the (finite) limit

r ( ~=) nlim im

m,

(12.142)

which is called the spectral radzus of the operator T and satisfies the relations r ( T )5 I/TIl. T ( P ) = [ T ( T ) ] R , T(C2T)= lalr(T), r ( T )= T(T*), (12.143) where T* is the adjoint operator to T (see 12.6, p. 624, and (12.159)). If L ( X ) is complete, then for 1x1 > r ( T ) ,the operator (XI- T)-' has the representation in the form of a Neumann series

+

( X I - T)-1 = x-'I + X-2T + . . . X-nTn-'+ . . . , which is convergent for 1x1 > r(T)in the operator norm on L ( X ) .

(12.144)

12.5.1.3 Convergence of Operator Sequences 1. Pointwise Convergence of a sequence of linear continuous operators T,: X -+ Y to an operator T : X T,z t T z inY for each z E X.

---t

Y means that: (12.145)

2. Uniform Convergence The usual norm-convergence of a sequence of operators {T,}~xlin a space L ( X ,Y) to T , i.e.,

is the unzforrn convergence on the unit ball of X. It implies pointwise convergence, while the converse statement is not true in general.

3. Applications The convergence of quadrature formulas when the number n of interpolation nodes tends to co,the performence principle of summation, limiting methods, etc.

12.5.2 Linear Continuous Operators in Banach Spaces Suppose now X and Y to be Banach spaces

1. Banach-Steinhaus Theorem (Uniform Boundedness Principle) The theorem characterizes the pointwise convergence of a sequence {T,} of linear continuous operators T, to some linear continuous operator by the conditions: a) For every element from an everywhere dense subset D c X, the sequence {Tnz}has a limit in Y. b) there is a constant C such that I ~ T ,5~C, ~ V n.

2. Open Mappings Theorem The theorem tells us that a linear continuous operator mapping from X onto Y is open, Le., the image T ( G )of every open set G from X is an open set in Y.

3. Closed Graph Theorem An operator T : DT -+ Y with DT C X is called closed if z, E DT, z, -+ $0 in X and T z , -+ yo in Y imply zo E DT and yo = 2'20. A necessary and sufficient condition is that the graph of the operator T in the space X x Y, Le., the set (12.147) rj-= {(x,Tz): x E DT} is closed, where here (z, y) denotes an element of the set X x Y. If T is a closed operator with a closed domain DT. then T is continuous 4. Hellinger-Toeplitz Theorem Let T be a linear operator in a Hilbert space H. If (z, Ty) = ( T z ,y) for every 5,y E H, then T is continuous (here (z.Ty)denotes the scalar product in H).

12.5 Continuous Linear Operators and Functionals 619

5 . Krein-Losanovskij Theorem on the Continuity of Positive Linear

Operators

If X = (X,X,. 11 . 11) and Y = (Y,Yt, 11 11) are ordered normed spaces, where X, is a generating cone. then the set Lt(X, Y) of all positive linear and continuous operators T , i.e., T(X,) c Y,, is a cone in L(X,Y). The theorem ofKrein and Losanovskzjasserts (see [12.17]): If X and Y are ordered Banach spaces with closed cones Xt and Y+, and X, is a generating cone, then the positivity of a linear operator implies its continuity. 6. Inverse Operator Let X and Y be arbitrary normed spaces and let 2': X + Y be a linear, not necessarily continuous operator. T has a continuous inverse T-' : Y -i X, if T(X) = Y and there exists a constant m > 0 such that l/Tzl/2 rnllzll for each 2 E X . Then l~T-'l~ 5 1. m In the case of Banach spaces X, Y we have the

7. Banach Theorem on the Continuity of the Inverse Operator If T is a linear continuous bijective operator from X onto Y. then the inverse operator T-' is also continuous. An important application is, e.g., the continuity of (XI - T)-I given the injectivity and surjectivity of XI - T . This fact has importance in investigating the spectrum of an operator (see 12.5.3.2, p. 620). It also applies to the 8 . Continuous Dependence of the Solution on the right-hand side and also on the initial data of initial value problems for linear differential equations. We will demonstrate this fact by the following example. 1 The initial value problem

?(t)+ p l ( t ) i ( t ) + p 2 ( t ) x ( t )= q ( t ) , t E [ a b ] , z(t0)= E , i ( t 0 ) = E , t o E [a,b] (12.148a) with coefficients pl(t),pz(t) E C([a,b])has exactly one solution z from Cz([a,b])for every right-hand side q ( t ) E C([a,b]) and for every pair of numbers E , (. The solution zdepends continuously on q ( t ) , and ( in the following sense. If qn(t) E C([a, b]),En, E R' are given and z, E C([a, b]) denotes the solution of (12.148b) f n ( t ) + p l ( t ) & ( t ) + ~ 2 ( t ) z n ( t )= q n ( t ) , zn(a) = Ennr &(a) = En, for n = 1 . 2 . . . ., then:

in

qn(t) + q ( t ) in C([% bl),
5,

tn +E

1

implies that

2,

-+ z in the space C2([a,b]).

<

(12.148~)

9. Method of Successive Approximation to solve an equation of the form (12.149) a:-Tz=y with a continuous linear operator T in a Banach space X for a given y. This method starts with an arbitrary initial element xO,and constructs a sequence (2,) of approximating solutions by the formula (12.150) ',Z , = y + T z , ( n = 0 , 1 , . . .) . This sequence converges to the solution z' in X of (12.149). The convergence of the method, i.e., z, + z*, is based on the convergence of the series (12.144) with X = 1. Let 11TIl 5 q < 1. Then the following statements are valid: 1 a) The operator I - T has a continuous inverse with lI(1- T)-'Il 5 -, and (12.149) has exactly 1-4J one solution for each y

620

12. Functional Analysis

b) The series (12.144) converges and its sum is the operator ( I - T)-'. c) The method (12.150) converges to the unique solution z' of (12.149) for any initial element 2 0 , if the series (12.144) converges. Then the following estimation holds:

I",

- z*l/ 5 -1ITzo Q" 1-q Equations of the type

- zo/( (n= 1,2,. . .).

(12.151)

x - / L T z = ~ , Xz-Tz=y, p , X E F can be handled in an analogous way (see 11.2.2, p. 565, and [12.8]).

(12.152)

12.5.3 Elements of the Spectral Theory of Linear Operators 12.5.3.1 Resolvent Set and the Resolvent of an Operator For an investigation of the solvability of equations one tries to rewrite the problem in the form (12.153) ( I - T)" = y with some operator T having a possible small norm. This is especially convenient for using a functional analytic method because of (12.143) and (12.144). In order to handle large values of llTI1 as well, we investigate the whole family of equations (12.154) (XI - T ) z = y z E X, with X E C in a complex Banach space X. Let T be a linear, but in general not a bounded operator in a Banach space X. The set Q ( T )of all complex numbers X such that (XI - T)-' E B(X) = L(X) is called the resolvent set and the operator Rx = Rx(T) = (XI - T)-' is called the resolvent. Let T now be a bounded linear operator in a complex Banach space X. Then the following statements are valid: a) The set Q ( T is ) open. hlore precisely, if Xo E Q(T)and X E C satisfy the inequality

( 12.155) then Rx exists and

Rx = Rxa + (A - Xo)Ri, + (A

-

Xo)2Ri,+ . . . =

c(X- XO)~-'R?,. 30

(12.156)

k=l

b) { A E C : 1x1 > liTll} c Q ( T )More . precisely, V X E C with / X I > llTll, the operator Rx exists and I T TZ Rx = -- - - - - - , . , , (12.157) X

XZ

A3

X' (see 12.5.4.1, p. 621) and arbitrary x E X the function F(X) = f(Rx(z))is holomorphic on @). f ) For arbitrary A, p E p(T),and X # p one has: e) For an arbitrary functional f E

12.5.3.2 Spectrum of an Operator 1. Definition of the Spectrum

The set u(T)= C \ Q ( Tis) called the spectrum of the operator T . Since I - T has a continuous inverse (and consequently (12.153) has a solution, which continuously depends on the right-hand side) if and only if 1 E Q(T),we must know the spectrum u(T) as well as possible. From the properties of the

12.5 Continuous Linear Operators and Functionals 621

resolvent set it follows immediately that the spectrum u ( T )is a closed set of C which lies in the disk { A : ( X I 5 llT(i}lhowever. in many cases m(T) is much smaller than this disk. The spectrum of any linear continuous operator on a complex Banach space is never empty and r ( T )= sup 1x1. (12.159) XWT)

It is possible to say more about the spectrum in the cases of different special classes of operators. If T is an operator in a finite dimensional space X and if the equation (XI - T ) z = 0 has only the trivial solution (Le., XI - T is injective), then X E e(T) (Le., XI - T is surjective). If this equation has a non-trivial solution in some Banach space, then the operator XI - T is not injective and (XI - T)-' is in general not defined. The number X E C is called an eigenvalue of the linear operator T , if the equation Xz = T z has a nontrivial solution. All those solutions are called eigenvectors, or in the case when X is a function space (which occurs very often in applications), they are called eigenfunctions of the operator T associated to A. The subspace spanned by them is called the eigenspace (or characteristic space) associated to X. The set op(T)of all eigenvalues of T is called the point spectrum of the operator T .

2. Comparison to Linear Algebra, Residual Spectrum An essential difference between the finite dimensional case which is considered in linear algebra and the infinite dimensional case discussed in functional analysis is that in the first case o(T)= u p ( T )always holds, while in the second case the spectrum usually also contains points which are not eigenvalues of T . If XI - T is injective and surjective as well, then X E Q ( T )due to the theorem on the continuity of the inverse (see 12.5.2, 7.) p. 619). In contrast to the finite dimensional case where the surjectivity follows automatically from the injectivity, the infinite dimensional case has to be dealt with in a very different way. The set o,(T) of all X E @), for which XI - T is injective and Im(XI - T ) is dense in X, is called the continuous spectrum and the set a,(T) of all X with an injective XI - T and a non-dense image, is called the residual spectrum of operator T . For a bounded linear operator T in a complex Banach space X

4 T ) = u p m u o m u 40 where the terms of the right-hand side are mutually exclusive.

12.5.4 Continuous Linear F'unct ionals 12.5.4.1 Definition For Y = F we call a linear mapping a linear functional or a linear form. In the following discussions, for a Hilbert space we consider the complex case; in other situations almost every times the real case is considered. The Banach space L(X, F) of all continuous linear functionals is called the adjoint space or the dual space of X and it is denoted by X* (sometimes also by X'). The value (in F) of a linear continuous functional f E X* on an element 2 E Xis denoted by f(x),often also by (Lf ) emphasizing the bilinear relation of X and X' - (compare also with the Riesz theorem (see 12.5.4.2, p. 622). , tn be fixed points of the interval [a,b] and el, c2,. . . cn real numbers. By the formula n

C!d(tk)

f(z)=

(12.161)

k=l

a linear continuous functional is defined on the space C ( [ a ,b ] ) ;the norm off is l l f ( l = case of (12.161) for a fixed t E [a,b] is the 6 functional adz) = z ( t ) ( 2 E C([a,bl)).

n

C Ickl. h special k=l

(12.162)

12. Functional Analusis

622

IB: With an integrable function p(t) (see 12.9.3.1, p. 635) on [a,b] b

f(.)

(12.163)

P(t)Z(t)dt

=

is a linear continuous functional on C([a,b]) and also on B([a,b]) in each case with the norm

llfii

=

12.5.4.2 Continuous Linear Functionals in Hilbert Spaces.

Riesz Representation Theorem

In a Hilbert space H every element y E H defines a linear continuous functional by the formula f(z)= (z, y). where its norm is l l f i l = lly/l. Conversely, iff is a linear continuous functional on H, then there exists a unique element y E H such that S(.) = ( 2 ,Y1 'JzE H , (12.164) where l i f i l = /lyjl. .kcording to this theorem the spaces H and H' are isomorphic and might be identified. The Riesz representation theorem contains a hint on how to introduce the notion of orthogonality in an arbitrary normed space. Let A c X and A* c X*. We call the sets A' = {f E X: f(z)= 0 Vz E A} and A" = { E E X: f(x) = 0 V f E A*} (12.165) the orthogonal complement or the annulator of A and A*, respectively.

12.5.4.3 Continuous Linear F'unctionals in LP

1 1 Let p 2 1. The number q is called the conjugate exponent t o p if - + - = 1, where it is assumed that P Q q = cx in the case ofp = 1. IBased on the Holder integral inequality (see 1.4.2.12,p. 32) the functional (12.163) can be considered 1 1 also in the spaces Lp([a.b]) (1 5 p 5 co) (see 12.9.4, p. 637) if (o E Lq([a,b]) and - + - = 1. Its norm

is then

llfll = IIYII =

P

{ (s,"

lp(t)lqdt)'. if 1 < p 5 m, ess. sup Ip(t)I, if p = 1

q

(12.166)

tEIa,bl

(with respect to the definition ofess. sup Ipl see (12.2171,p. 637). To every linear continuous functional f in the space Lp([a,b])there is a uniquely (up to its equivalence class) defined element y E L*([a.b ] ) such that (12.167) f(.) = = 4t)Y(t)dt, E E LP and IIfII = I l Y I l q = lY(t)l4dt '

p

I/!,.(

(p

)'

For the case ofp = cx see [12.15].

12.5.5 Extension of a Linear Functional 1. Seminorm A mapping p: X +R of a vector space X is called a seminorm or pseudonorm, if it has the following properties: ("1)

P(Z)2

0,

(12.168)

("2)

P(oz1 = lQlP(5).

("3)

P(.

(12.169) (12.170)

+ Y) I A T ) +P(Y).

12.5 Continuous Linear Operators and Functionals 623

Comparison with 12.3.1.p. 609, shows that a seminorm is a norm if and only ifp(z) = 0 holds only for x = 0.

Both for theoretical mathematical questions and for practical reasons in applications of mathematics, the problem of the extension of a linear functional given on a linear subspace XO c X to the entire space (and, in order to avoid trivial and uninteresting cases) with preserving certain '' g o o d properties became a fundamental question. The solution of this problem is guaranteed by

2. Analytic Form of the Hahn-Banach Extension Theorem Let X be a vector space over F and p a pseudonorm on X. Let XObe a linear (complex in the case of F = C and real in the case of F = R) subspace of X, and let fo be a (complex-valued in the case of F = C and real-valued in the case of F = R) linear functional on Xo satisfying the relation lfo(.)l I P ( S ) V x E Xo. (12.171) Then there exists a linear functional f on X with the following properties: f(x) = fo(z) Va: E Xo, lf(.)l 5 P ( 5 ) v z E X. (12.172) So, f is an extension of the functional fo onto the whole space X preserving the relation (12.171). If Xo is a linear subspace of a normed space X and fo is a continuous linear functional on Xo2 then p ( z ) = lifoll . IlzIl is a pseudonorm on X satisfying (12.171), so we get the Hahn-Banach theorem on the extension of contznuous linear functionals. Two important consequences are: 1. For every element z # 0 there is a functional f E X* with f(x) = llzll and l l f i i = 1. 2. For every linear subspace XOC X and 10 !$ XOwith the positive distance d = infzEXo112 - xo((> 0 there is an f E X' such that 1 (12.173) f ( z ) = 0 Vx E XO, f(q) = 1 and llfil = -. d

12.5.6 Separation of Convex Sets 1. Hyperplanes .A linear subset L of the real vector space X,L # x, is called a hypersubspace or hyperplane through 0 if there exists an 50 E X such that X = Izn(x0,L ) . Sets of the form x + L ( L a linear subset) are affinelinear manifolds (see 12.1.2. p. 595). If L is a hypersubspace, these manifolds are called hyperplanes. There exist the following close relations between hyperplanes and linear functionals: a) The kernel f-'(O) = {z E X: f(x) = 0) of a linear functional f on Xis a hypersubspace in X, and = X and f-'(X) = xx f-'(O). for each number X E R there exists an element zx E X with f(q) b) For any given hypersubspace L C X and each xo $ L and X # 0 (A E R) there always exists a uniquely determined linear functional f on X with f-'(O) = L and f(z0) = A. The closedness of f-'(O) in the case of a normed space Xis equivalent to the continuity of the functional

+

f. 2. Geometric Form of the Hahn-Banach Extension Theorem Let X be a normed space, zo E X and L a linear subspace of X. Then for every non-empty convex open set K which does not intersect the affine-linear manifold 10 + L , there exists a closed hyperplane H such that xo + L c H and H n K = 0. 3. Separation of Convex Sets Two subsets A, B of a real normed space X are said to be separated by a hyperplane if there is a functional

f E X*such that: supf(z) I j $ f ( Y ) .

(12.174)

XEA

The separating hyperplane is then given by f - ' ( a ) with a = supzEAf ( ~which ) ~ means that the two sets are contained in the different half-spaces

I

624

12. Functional Analusis

A c {z E X: f(z) 5 a } and B c {x E X: f(x) 2 a}. (12.175) In Fig. 12.5b,c two cases of the separation by a hyperplane are shown. Their disjointness is less decisive for the separation of two sets. In fact, Fig. 12.5a shows two sets E and B. which are not separated although E and B are disjoint and B is convex. The convexity of both sets is the intrinsic property for separating them. In this case it is possible that the sets have common points which are contained in the hyperplane.

Figure 12.5

If A is a convex set of a normed space X with a non-empty interior Int(A)and B c X is a non-empty convex set with Int(A) n B = 0, then A and B can be separated. The hypothesis Int(A) # 0 in that statement cannot be dropped (see [12.3],example 4.47). A (real linear) functional f E X* is then called a supportingfunctionalto A of the set A at the point zo E A , if there is a real number X E R such that f(q) = A, and A c {z E X : f ( z ) 5 A}. f-'(X) is called the supporting hyperplane at the point 20. For a convex set K with a non-empty interior, there exists a supporting functional at each of its boundary points. Remark: The famous Kuhn-Tucker theorem (see 18.2, p. 860) which yields practical methods to determine the minimum of convex optimization problems (see [12.5]),is also based on the separation of convex sets.

12.5.7 Second Adjoint Space and Reflexive Spaces The adjoint space X* of a normed space X is also a normed space if it is equipped with the norm llfll = sup If(z)l. so (Xi)* = X**. The second adjoint space to X can also be considered. The canonical llxll51

embedding J : X tX** with J z = F,, where F,(f) = f(z) V f E X* (12.176) is a norm isomorphism (see 12.3.1,p. 609), hence X is identified with the subset J(X) C X". A Banach space X is called reflexive if J ( X ) = X". Hence the canonical embedding is then a surjective norm isomorphism. IEvery finite dimensional Banach space and every Hilbert space is reflexive, as well as the spaces Lp (1 5 p < m), however C ( [ a ,b ] ) , Li([O,l]), co are examples of non-reflexive spaces.

12.6 Adjoint Operators in Normed Spaces 12.6.1 Adjoint of a Bounded Operator For a given linear continuous operator T : X --+Y (X, Y are normed spaces) to every g

E Y* there is assigned a functional f E X' by f(z) = g ( T z ) , V z E X. In this way, we get a linear continuous operator T * : Y* tx', ( T * g ) ( z= ) g ( T z ) , V g E Y' and Vz E X, (12.177) which is called the adjoint operator of T and has the following properties: ( T + S ) *= T * + S * ,(ST)*= S*T', l/T*/l= IITII, wherefor thelinear continuousoperators'l': X + Y

12.6 Adjoint Operators in Normed Spaces 625 and S: Y + Z (X,Y, 2 normed spaces), the operator S T : X + 2 is defined in the natural way as S T ( z ) = S ( T ( z ) ) .With the notation introduced in 12.1.5, p. 598, and 12.5.4.2, p. 622, the following identities are valid for an operator T E B ( X ,Y):

Im(T)= ker(T*)', Im(T*)= ker(T)', (12.178) where the closedness of Im(T)implies the closedness of Im(T*). The operator T**: X** + Y**, obtained as (T*)*from T', is called the second adjoint of T . Due to (T**(F,))g = F,(T*g) = (T*g)(z) = g(Tz)= &(g) the operator T**has the following property: If Fx E X"', then T**Fz= FT, E Y'*. Hence, the operator T**:X" + Y** is an extension of T . In a Hilbert space H the adjoint operator can also be introduced by means of the scalar product (TI, y) = (zl T*y), 2, y E H. This is based on the Riesz representation theorem, where the identification of H and H**implies (AT)' = I T * ,I' = I and even T**= T . If T is bijective, then the same holds for T', and also (T*)-*= ( P I ) * . For the resolvents of T and T* there holds [RA(T)l*= W-*L ( 12.179) from which u(T*)= (1:A E u ( T ) }follows for the spectrum of the adjoint operator. IA: Let T be an integral operator in the space Lp([a, b]) (1 < p < m)

( T z ) ( s )= [ K ( s , t ) z ( t )d t

(12.180)

with a continuous kernel K ( s ,t ) . The adjoint operator of T is also an integral operator, namely (12.181) according with the kernel K * ( s ,t ) = K ( t ,s), where yg is the element from L* associated to g E (U)* to (12.167). W B: In a finite dimensional complex vector space the adjoint of an operator represented by the matrix A = ( a t j ) is defined by the matrix A* with a:j = q.

12.6.2 Adjoint Operator of an Unbounded Operator Let X and Y be real normed spaces and T a (not necessarily bounded) linear operator with a (linear) domain D ( T ) c X and values in Y. For a given g E Y*, the expression g(Tz), depending obviously linearly on z.is meaningful. Now the question is: Does there exist a well-defined functional f E X* such that f(I)= g(Tz) v z E D ( T ) . (12.182) Let D' c Y* be the set of all those g E Y' for which the representation (12.182) holds for a certain j E X*. If D(T) = X, then for the given g the functional f is uniquely defined. So a linear operator T' is defined by f = T*gwith D ( T ' ) = D'. Then for arbitrary I E D ( T ) and g E D(T*)one has dTI)= (T*g)(z). (12.183) The operator T' turns out to be closed and is called the adjoznt of T . The naturalness of this general procedure stems from the fact that D ( T ' ) = Y' holds if and only if T is bounded on D ( T ) .In this case T' E B(Y*.X') and llT*11 = liTll hold.

12.6.3 Self-Adjoint Operators An operator T E B(H) (H is a Hilbert space) is called self-adjoznt if T' = T . In this case the number ( T z ,z)is real for each z E H. One has the equality (12.184)

626

12. Functional Analysis

and with m = m(T) = inf ( T z ,z) and M = M ( T ) = sup ( T z ,z)also the relations llxJI=l

lIxll=1

IITII = $1 = max{lm/,M}. (12.185) The spectrum of a self-adjoint (bounded) operator lies in the interval [m,M] and m, Ad E u(T )holds. m(T)llzIl2 5 ( T z , ~5)M(T)llzllZ and

12.6.3.1 Positive Definite Operators A partial ordering can be introduced in the set of all self-adjoint operators of B ( H )if we define T 2 0 if and only if (Tz!x) 2 0 Vz E H. (12.186) An operator T with T 2 0 is called positive (or, more exactly positive dejnite). For any self-adjoint operator T (with (Hl)from 12.4.1.1, p. 613), ( T 2 z , z ,= ) ( T z , T z ) 2 0, so TZis positive definite. Every positive definite operator T possesses a square root, Le., there exists a unique positive definite operator W such that W 2 = 2’. Moreover, the vector space of all self-adjoint operators is a K-space (Kantorovich space, see 12.1.7.4,p. 600), where the operators 1 1 IT1 = V F . T+ = -(IT1 + T ) , T - = -(IT1 - T ) ( 12.187) 2 2 are the corresponding elements with respect to (12.37). They are of particular importance for the spectral decomposition and spectral and integral representations of self-adjoint operators by means of some Stieltjes integral (see 8.2.3.1, 2., p. 451, and [12.1],[12.11],[12.12],[12.15],[12.18]).

12.6.3.2 Projectors in a Hilbert Space Let Ho be a subspace of a Hilbert space H. Then every element z E H has its projection I’onto Ho according to the projection theorem (see 12.4.2, p. 614), and therefore, an operator P with Pz = z’is defined on H with values in HO. P i s called a projector onto Ho. Obviously, P i s linear, continuous, and lJPIJ = 1. .4continuous linear operator P in H is a projector (onto a certain subspace) if and only if: a) P = P*,Le.. P is self-adjoint, and b) P2 = P,Le., P is idempotent.

12.7 Compact Sets and Compact Operators 12.7.1 Compact Subsets of a Normed Space A subset A of a normed space+X is called compact, if every sequence of elements from A contains a convergent subsequence whose limit lies in

A, relatzvely compact or precompactif its closure (see 12.2.1.3,p. 604) is compact, Le., every sequence of elements from A contains a convergent subsequence (whose limit does not necessarily belong to A ) . This is the Bolzano-Weierstrass theorem in real calculus, and we say that such a set has the BolzanoWezerstrass property. Every compact set is closed and bounded. Conversely, if the space X is finite dimensional, then every such set is compact. The closed unit ball in a normed space X is compact if and only if X is finite dimensional. For some characterizations of relatively compact subsets in metric spaces (the Hausdorff theorem on the existence of a finite m e t ) and in the spaces s, C (Arzela-Ascoli theorem) and in the spaces LP(1 < p < m) see [12.15].

12.7.2 Compact Operators 12.7.2.1 Definition of Compact Operator An arbitrary operator T : X

tY

of a normed space X into a normed space Y is called compact if the

+It is enough that X is a metric (or an even more general) space. We do not use this generality in what follows.

12.7 Compact Sets and Compact Operators 627

image T ( A )of every bounded set A C X is a relatively compact set in Y. If, in addition the operator T is also continuous, then it is called completely continuous. Every compact linear operator is bounded and consequently completely continuous. For a linear operator to be compact it is sufficient to require that it transforms the unit ball of X into a relatively compact set in Y .

12.7.2.2 Properties of Linear Compact Operators characterization by sequences of the compactness of an operator from B(X,Y )is the following: For =~ a convergent subsequence. every bounded sequence {E,}:=~ from X the sequence { T X , } ~contains .A linear combination of compact operators is also compact. If one of the operators U E B(W, X), T E B ( X ,Y):S E B ( Y ,Z)in each of the following products is compact, then the operators TU and ST are also compact. If Y is a Banach space, then one has the following important statements. a) Convergence: If a sequence of compact operators {Tn}r=to=l is convergent in the space B ( X ,U), then its limit is a compact operator, too. b) Schauder Theorem: If T is a linear continuous operator, then either both T and T* are compact or both are not. c) Spectral Properties of a Compact Operator T i n a n (Infinite Dimensional) Banach Space X: The zero belongs to the spectrum. Every non-zero point of the spectrum g ( T ) is an eigenvalue with a finite dimensional eigenspace XX = {z E X : ( X I - T ) z = 0}, and V E > 0 there is always only a finite number of eigenvalues of T outside the circle { / X I 5 E } , where only the zero can be an accumulation point of the set of eigenvalues. If X = 0 is not an eigenvalue of T , then T-' is unbounded if it exists. .I\

12.7.2.3 Weak Convergence of Elements A sequence {zn}F=lof elements of a normed space X is called weakly convergent to an element 2 0 if for each f E X' the relation f(z,) fi ( q )holds (written as: z, -., 50). Obviously: z, i zo implies z, 50. If Y is another normed space and T : X + Y is a continuous

-

-

linear operator. then: a) 2, 2 .co implies T z , Tzo. b) if T is compact, then z, 2 zo implies T I , i Tzo. IA: Every finite dimensional operator is compact. From this it follows that the identity operator in an infinite dimensional space cannot be compact (see 12.7.1, p. 626). IB: Suppose X = 12, and let T be the operator in 1' given by the infinite matrix

(12.188)

If

30

lt,k12 =

,\f < co,then T is a compact operator from 1' into 1' with //TI/5 M .

k,n=l

IC: The integral operator (12.136)is a compact operator in the spaces C ( [ a ,b ] ) and U ( ( a , b ) ) (1 < P<X)

12.7.3 F'redholm Alternative Let T be a compact linear operator in a Banach space X. We consider the following equations (of the second kind) with a parameter X # 0: Xz-Tz = g > Xz-Tz = o , (12.189) Xf - T'f = 9 % Xf - T*f= 0.

I

628

12. Functional Analysis

The following statements are valid: a) dim(ker(X1- T ) )= dim(ker(X1- T'))< +m, i.e., both homogeneous equations always have the same number of linearly independent solutions. b) Im(XI - T ) = ker(XI - T*)' and t Im(X1- T ' ) = ker(X1- 2')' c) Im(XI - T ) = X if and only if ker(X1- T ) = 0. d) The Fredholrn alternatzve (also called the Riesz-Schauder theorem): a ) Either the homogeneous equation has only the trivial solution. In this case X E p(T),the operator (XI - T)-' is bounded, and the inhomogeneous equation has exactly one solution z = (XI - T)-'y for arbitrary y E X. 3 ) Or the homogeneous equation has at least one non-trivial solution. In this case Xis an eigenvalue of T , Le., X E o ( T) .and the inhomogeneous equation has a (non-unique) solution if and only if the righthand side y satisfies the condition f ( y ) = 0 for every solution f of the adjoint equation T*f = X f . In this last case every solution z of the inhomogeneous equation has the form z = 20+ h, where zo is a fixed solution of the inhomogeneous equation and h E ker(X1- 2'). Linear equations of the form T z = y with a compact operator T are called equations of the first kind. Their mathematical investigation is in general more difficult (see [12.11],[12.18]).

12.7.4 Compact Operators in Hilbert Space Let T : H ---+ H be a compact operator. Then T is the limit (in B(H)) of a sequence of finite dimensional operators. The similarity to the finite dimensional case can be seen from the following: If C is a finite dimensional operator and T = I - G, then the injectivity of T implies the existence of T-' and T-' E B(H). If C is a compact operator, then the following statements are equivalent: a) 3 T-' and it is continuous, b) z # 0 + T z # 0, Le., T i s injective, c) T(H) = H, Le., T is surjective.

12.7.5 Compact Self-Adjoint Operators 1. Eigenvalues A compact self-adjoint operator T # 0 in a Hilbert space H possesses at least one non-zero eigenvalue. More precisely, T always has an eigenvalue X with 1x1 = llTll. The set of eigenvalues of T is at most countable. Any compact self-adjoint operator T has the representation T =

&PAI.(in B(H)),where X k are the k

different eigenvalues of T and PA denotes the projector onto the eigenspace HA. We say in this case that the operator T can be diagonalized. From this fact it follows that T z = &(z,ek)ek for every k

z E H, where { e k } is the orthonormal system of the eigenvectors of 2'. If X

4 o(T)and y E H, then

the solution of the equation (XI - T ) z = y can be represented as z = Rx(T)y =

-(y,

1

k X-Xk

ek)ek.

2. Hilbert-Schmidt Theorem If T is a compact self-adjoint operator in a separable Hilbert space H, then there is a basis in H consisting of the eigenvectors of 2'. The so-called spectral (mapping) theorems (see [12.8], [12.10], [12.12],[12.13], [12.18]) can be considered as the generalization of the Hilbert-Schmidt theorem for the non-compact case of self-adjoint (bounded or unbounded) operators. $Here the orthogonality is considered in Banach spaces (see 12.5.4.2,p. 622).

I

12.8 Non-Linear Operators 629

12.8 Non-Linear Operators In the theory of non-linear operator equations the most important methods are based on the following principles: 1. Principle of the Contracting Mapping, Banach Fixed-point Theorem (see 12.2.2.3, p. 606, and 12.2.2.4. p. 606). For further modifications of this principle see [12.8],[12.11],[12.12], [12.18]. 2. Generalization ofthe Newton Method (see 18.2.5.2, p. 867 and 19.1.1.2, p. 882) for the infinite dimensional case. 3. Schauder Fixed-point Principle 4. Leray-Schauder Theory

Methods based on principles 1 and 2 yield information on the existence, uniqueness, constructivity etc. of the solution. while methods based on principles 3 and 4, in general, allow "only" the qualitative statement of the existence of a solution. If further properties of operators are known then see also 1 2 8.6, p. 631, and 12 8.7, p. 632.

12.8.1 Examples of Non-Linear Operators For non-linear operators the relation between continuity and boundedness discussed for linear operators in 125.1, p. 617 is no longer valid in general. In studying non-linear operator equations, e.g., nonlinear boundary value problems or integral equations, the following non-linear operators occur most often. Iteration methods described in 12.2.2.4, p. 606, can be succesfully applied for solving non-linear integral equations.

1. Nemytskij Operator Let R be an open measurable subset from R" (12.9.1, p. 633) and f : R x R --t R a function of two variables f ( z , s), which is continuous with respect to zfor almost every s and measurable with respect to s for every z (Caratheodory conditions). The non-linear operator N to F ( R ) defined as ( N u ) ( z= ) f[z, .()I .( E fl) (12.190) is called the Nemytskzj operator. It is continuous and bounded if it maps P ( R ) into L9(R), where I -

P

+ q-1 = 1. This is the case, e.g., if If(.$

or i f f :

s)I 5 a(.) + blslg with a(.) E L9(R) ( b > 0) R x R + R is continuous. The operator N is compact only in special cases.

(12.191)

2. Hammerstein Operator

R be a relatively compact s u k e t of R", f a function satisfying the Caratheodory condztzons and K ( z ,y ) a continuous function on R x 0. The non-linear operator 3t on F ( R ) Let

W)(z)=

1

K b , V ) f [ Y ?4 Y ) l dY .( E 0)

(12.192)

R

is called the Hammerstezn operator. 3t can be written in the form ?!I = IC IC determined by the kernel K

N with the integral operator (12.193)

R

If the kernel K ( z )y ) satisfies the additional condition (12.194) RXn

and the function f satisfies the condition (12.191), then 3t is a continuous and compact operator on

J!w).

630

12. Functional Analusis

3. Urysohn Operator Let R c R” be an open measurable subset and K ( z ,y,s) variables. Then the non-linear operator U on F(R) ( U u ) ( z )=

:

RxRxR

t

1

R a function of three (12.195)

K [ z ,Y, U(Y11 dY .( E 0 )

n

is called the Urysohn operator. If the kernel K satisfies the corresponding conditions, then U is a continuous and compact operator in C ( 0 ) or in Lp(R), respectively.

12.8.2 Differentiability of Non-Linear Operators Let X, Y be Banach spaces, D c X be an open set and T : D t Y. The operator T is called Fre‘chet dzflerentiable (or. briefly, differentiable) at the point x E D if there exists a linear operator L E B(X,Y) (in general depending on the point z)such that T ( z + h ) - T ( z )= L h + w ( h ) with ilw(h)ll= o(IIhl1) (12.196) or in an equivalent form

+

llT(z h ) - T ( z )- Lh/l = 0. ( 12.197) llhll Le.. V E > 0, 3 b > 0, such that Ilhll < 6 implies /lT(z h ) - T ( z )- Lhll 5 ~lihll.The operator L , which is usually denoted by T’(z),T‘(z:.) or T ’ ( z ) ( , )is, called the Fre‘chet derivative of the operator T at the point z.The value dT(z:h ) = T’(z)his called the Fre‘chet dzflerential of the operator T at the point z (for the increment h ) . The differentiability of an operator at a point implies its continuity at that point. If T E B(X, Y), i,e.) T itself is linear and continuous, then T is differentiable at every point, and its derivative is equal to T. lim

W+0

+

12.8.3 Newton’s Method Let X, D be as in the previous paragraph and T : D t X. Under the assumption of the differentiability of T at every point of the set D an operator T’ : D t B ( X ) is defined by assigning the element T’(z) E B(X) to every point z E D. Suppose the operator T’ is continuous on D (in the operator norm): in this case T is called continuously diflerentiable on D. Suppose Y = X and also that the set D contains a solution z*of the equation T ( z )= 0. (12.198) Furthermore, we suppose that the operator T’(z)is continuously invertible for each z E D. hence [T‘(z)]-’is in B(X). Because of (12.196) for an arbitrary zo E D one conjectures that the elements T ( z o )= T ( z o )- T ( z * )and T’(zo)(zo - z’)are “not far” from each other and therefore the element z1 defined as (12.199) X I = I o - [T’(zo)l-lT(zO), is an approximation of z’(under the assumption we made). Starting with an arbitrarpzo the so-called Newton approximation sequence (12.200) Z,+l = z, - [T’(zn)]-1T(zn)( n = 0 , 1 , .. .) can be constructed. There are many theorems known from the literature discussing the behavior and the convergence properties of this method. We mention here only the following most important result which demonstrates the main properties and advantages of Newton’s method: V E E (0:1) there exists a ball B = B(z0:6), 6 = b ( ~in) X, such that all points z, lie in B and the Yewton sequence converges to the solution x* of (12.198). Moreover, /Iz,- zo/1 5 c”11zg - z*ll which yields a practical error estimation. The modijed Newton’s method is obtained if the operator [T’(zo)]-’is used instead of [T’(z,,)]-lin

12.8 Non-Linear Operators 631

formula (12.200). For further estimations of the speed of convergence and for the (in general sensitive) dependence of the method on the choice of the starting point zo see [12.7], [12.12], [12.18].

12.8.4 Schauder’s Fixed-point Theorem Let T : D t X be a non-linear operator defined on a subset D of a Banach space X. The non-trivial question of whether the equation z = T ( z )has at least one solution, can be answered as follows: If X = R and D = [-l>11, then every continuous function mapping D into D has a fixed point in D. If X is an arbitrary finite dimensionalnormed space (dimX 2 2), then Brozlwer’s fixed-point theorem holds. 1. Brouwer’s Fixed-point Theorem Let D be a non-empty closed bounded and convex subset of a finite dimensional normed space. If T is a continuous operator, which maps D into itself, then T has at least one fixed point in D . The answer in the case of an arbitrary infinite dimensional Banach space X is given by Schazlder’s fixed-point theorem. 2. Schauder’s Fixed-point Theorem Let D be a non-empty closed bounded and convex subset of a Banach space. If the operator T : D ---+ Xis continuous and compact (hence completely continuous) and it maps D into itself, then T has at least one fixed point in D. By using this theorem. it is proved, e.g., that the initial value problem (12.70), p. 608, always has a local solution for t 2 0, if the right-hand side is assumed only to be continuous.

12.8.5 Leray-Schauder Theory For the existence ofsolutionsoftheequations z = T ( 2 )and ( I + T ) ( z )= y with acompletely continuous operator T . a further principle is found which is based on deep properties of the mapping degree. It can be successfully applied to prove the existence of a solution of non-linear boundary value problems. We mention here only those results of this theory which are the most useful ones in practical problems, and for simplicity we have chosen a formulation which avoids the notion of the mapping degree. Leray-Schauder Theorem: Let D be an open bounded set in a real Banach space X and let T : D : t X be a completely continuous operator. Let y E D be a point such that z+ XT(z) # y for each z E d D and X E [O. 11,where d D denotes the boundary of the set D. Then the equation ( I + T ) ( z )= y has at least one solution. The following version of this theorem is very useful in applications: Let T be a completely continuous operator in the Banach space X. If all solutions of the family of equations z = XT(2) ( A E [O, 11) (12.201) are uniformly bounded, Le., 3 c > 0 such that V X and V z satisfying (12.201) the a priori estimation llzil 5 c holds, then the equation z = T ( z )has a solution.

12.8.6 Positive Non-Linear Operators The successful application of Schauder’s fixed-point theorem requires the choice of a set with appropriate properties, which is mapped into itself by the considered operator. In applications, especially in the theory of non-linear boundary value problems, ordered normed function spaces and positive operators are often considered, Le., which leave the corresponding cone invariant, or isotone increasing operators, Le., if z 5 y + T ( z ) 5 T(y). If confusions (see, e.g., 12.8.7, p. 632) are excluded, we also call these operators monotone. Let X = (X, X,, 11 . 11) be an ordered Banach space, X+ a closed cone and [a,b] an order interval of X. If X, is normal and T is a completely continuous (not necessarily isotone) operator that satisfies T ( [ ab]) , c [a.b],then T has at least one fixed point in [a,b] (Fig. 12.6b). , C [a,b] automatically holds for any isotone increasing operator T , Notice that the condition T ( [ ab]) which is defined on an (0)-interval (order interval) [alb] of the space X if it maps only the endpoints a, b

632

12. Functional Analysis

=f(x)

Figure 12.6 into [a, b ] , Le.: when the two conditions T ( a ) 2 a and T(6)5 b are satisfied. Then both sequences 20 = a and x,+l = T(x,) ( n 2 0) and yo = b and yntl = T(y,) ( n 2 0) (12.202) are well defined, Le., x,, yn E [alb], n 2 0. They are monotone increasing and decreasing, respectively, Le., a = xo 5 x1 5 . . . 5 x, 5 . . . and b = yo 2 y1 2 . . . y, 2 . . .. A fixed point 2, of the operator T is called minimal, maximal, respectively, if for every fixed point z of T the inequalities x* 5 z , z 5 x, hold, respectively. Kow, we have the following statement (Fig. 12.6a)): Let X be an ordered Banach space with a closed cone X+ and T : D t X, D C X a continuous isotone increasing operator. Let [a,b] c D be such that T ( a )2 a and T(b)5 b. Then T ([a,b]) C [a,b], and the operator T has a fixed point in [a>b] if one of the following conditions is fulfilled: a) X+ is normal and T is compact; b) X+ is regular. Then the sequences and {yn}r=o, defined in (12.202), converge to the minimal and to the maximal fixed points of T in [a, b], respectively. The notion of the super- and subsolutions is based on these results (see [12.14]).

{~,}r=~

I12.8.7 Monotone Operators in Banach Spaces 1. Special Properties

An arbitrary operator T : D C X t Y (X, Y normed spaces) is called demi-continuousat the point 20 E D if for each sequence {x,};==, C D converging to xo (in the norm of X) the sequence {T(x,)}r=l converges weakly to T ( q )in Y. T is called demi-continuouson the set D if T is demi-continuous at every point of D. In this paragraph we introduce another generalization of the notion of monotonity known from real analysis. Let X now be a real Banach space, X* its dual, D C X and T : D t X' a non-linear operator. T is called monotone ifVx, y E D the inequality ( T ( x )- T(y), x - y) 2 0 holds. If X = H is a Hilbert space, then (., ,) means the scalar product, while in the case of an arbitrary Banach space we refer to the notation introduced in 12.5.4.1, p. 621. The operator T is called strongly monotone if there isaconstantc>Osuchthat(T(z)-T(y),x-y) > c l l x - y l l Z f o r V x , y ~D. A n o p e r a t o r T : X t X "

is called coercive if lim IIx++m

1/~11

= co.

2. Existence Theorems for solutions of operator equations with monotone operators are given here only exemplarily: If the operator T , mapping the real separable Banach space X into X', ( D T = X), is monotone demi-continuous and coercive, then the equation T ( x )= f has a solution for arbitrary f E X*. If in addition the operator T is strongly monotone, then the solution is unique. In this case the inverse operator T-' also exists. For a monotone, demi-continuous operator T : H t H in a Hilbert space H with DT = H, there holds

12.9 Measure and Lebesoue InteQral 633

Im(I t T ) = H, where ( I + T)-’ is continuous. If we suppose that T i s strongly monotone, then T-’ is bijective with a continuous T-’ . Constructive approximation methods for the solution of the equation T ( z )= 0 with a monotone operator T in a Hilbert space are based on the idea of Galerkin’s method (see 19.4.2.2, p. 906, or [12.10], [12.18]). By means of this theory set-valued operators T: X -+ 2” can also be handled. The notion of monotonity is then generalized by (f - g, 2 - y) 2 0. V 2 , y E Dr and f E T ( z ) g, E T(y).

12.9 Measure and Lebesgue Integral 12.9.1 Sigma Algebra and Measures The initial point for introducing measures is a generalization of the notion of the length of an interval in R. of the area, and of the volume of subsets of R2 and R3,respectively. This generalization is necessary in order to “measure” as many sets as possible and to “make integrable” as many functions as possible. For instance, the volume of an n-dimensional rectangular parallelepiped (12.203) Q = {X E R”: ah 5 ~k 5 bk ( k = 1 , 2 , . . . , n ) } has the value (bk - at). k=l

1. Sigma Algebra Let X be an arbitrary set. A non-empty system A of subsets from X is called a u-algebra if: a) A E A implies X \ A E A and (12.204a) b) A1,A2, ..,A,.. . . E A implies A, E A. (12.204b)

u

n=l

Every u-algebra contains the sets 0 and X, the intersection of countably many of its sets and also the difference sets of any two of its sets. In the following denotes the set R of real numbers extended by the elements {-cm} and {+E} (extended real line). where the algebraic operations and the order properties from R are extended to cm R in the natural way. The expressions (+cm) + ( ~ c m and ) - are meaningless, while 0 . (+m) and cm 0 (-co)are assigned the value 0.

2. Measure

A function p : A -t R+ = R+ U +cm, defined on a u-algebra A, is called a measure if a) p(A) 2 0 V z 4 E A, b) p(0) = 0.

c) ‘41, A?, . . .A,, . . . E A,

Ak

n Al = 0

( k # 1 ) implies p

(E An) n=l

=

p(An).

(12.205a) (12.205b) (12.205~)

n=l

The property c) is called 0-additivity of the measure. If p is a measure on A, and for the sets A: B E A, A c B holds, then p ( A ) 5 p ( B ) (monotonicity). If A, E A (n = 1,2:. . .) and A1 C A2 C . .., then p

U A,

(ny1

1

=

p(A,) (continuity from below)

Let A be a 0-algebra of subsets of X and p a measure on A. The triplet X = (X, A, p ) is called a measure space, and the sets belonging to A are called measurable or A-measurable. IA: Counting Measure: Let X be a finite set {q,x2,.. . , ZN}, A the u-algebra of all subsets of X, and let assign a non-negative number pk to each xk ( k = 1,.. . , N ) . Then the function I./ defined on A foreverysetA E A. A = {xn,,zn2, . . . ,xnk}byp(A) =pn, t p n , + . . . + p n kisameasurewhich takes on only finite values since p ( X ) = p 1 t ’ . t p~ < cm. This measure is called the counting measure. IB: Dirac Measure: Let A be a 0-algebra of subsets of a set X and a an arbitrary given point from X. Then a measure is defined on A by 1 if a E A, (12.206) = 0;if a A.

{

634

12. Functional Analwas

It is called the 6 function (concentrated on a ) . Obviously & ( A )= ~ & ( x A= ) X A ( U ) (see 12.5.4, p. 621), where X A denotes the characteristic function of the set A. IC: Lebesgue Measure: Let X be a metric space and B ( X ) the smallest 0-algebra of subsets of X which contains all the open sets from X. B ( X ) exists as the intersection of all the 0-algebras containing all the open sets, and is called the Bore! u-algebra of X. Every element from B ( X ) is called a Borel set (see [12.6]). Suppose now, X = R" ( n 2 1). Using an extension procedure we can construct a 0-algebra and a measure on it, which coincides with the volume on the set of all rectangular parallelepipeds in R". More precisely: There exists a uniquely defined 0-algebra A of subsets of R" and a uniquely defined measure X on A with the following properties: a) Each open set from R" belongs to A, in other words: B(Rn) c A. b) If A E A: X(A) = 0 and B c A then B E A and X(B) = 0.

c) If Q is a rectangular parallelepiped, then Q E A, and X(Q) =

n

fl (bk - ak) k=l

d) X is translation invariant, Le., for every vector z E R" and every set A E A one has z+ A = {z+ y : y E '4) E Aand X(z + A) = X(A). The elements of A are called Lebesgue measurable subsets of R". X is the (n-dimensional) Lebesgue measure in R". Remark: In measure theory and integration theory one says that a certain statement (property, or condition) with respect to the measure p is valid almost everywhere or p-almost everywhere on a set X, if the set, where the statement is not valid, has measure zero. We write a.e. or p-a.e.5 For instance, if X is the Lebesgue measure on R a n d A, B are two disjoint sets with R = A U B and f is a function on R with f(x) = 1, Vz E A and f(z)= 0, Vz E B , then f = 1, X-a.e. on R if and only if X(B)= 0.

12.9.2 Measurable F'unct ions 12.9.2.1 Measurable Function Let A be a u-algebra of subsets of a set X. A function f : X -+ is called measurable if for an arbitrary a E R the set f - ' ( ( a ,+m]) = {z : z E X, f(z)> a } is in A. A complex-valued function g t ih is called measurable if both functions g and h are measurable. If A is the 0-algebra of the Lebesgue measurable sets of R" and f : R" +R is a continuous function, then the set f-'((a. +m]) = f - ' ( ( a ,+m)),according to 12.2.3,p. 608, is open for every (Y E R, hence f is measurable.

12.9.2.2 Properties of the Class of Measurable Functions The notion of measurable functions requkes no measure but a 0-algebra. Let A be a u-algebra of subsets of the set X and let f , g, f n : X -+ R be measurable functions. Then the following functions (see 12.1.7.4,p. 600) are also measurable: a ) af for every a E R; f . g;

b) f+? f-. Ifl, f v 9 and f A 9; c) f + g1if there is no point from X where the expression (ha)+ ( ~ moccurs; ) d) supf,, inff,, limsupf, (= lim supfh), liminff,; n*m k>n

e) the pointwise limit lim fn, in case it exists; f ) iff 2 0 and p E R , p > 0, the f P is measurable. A function f : X + R is called elementary or simple if there is a finite number of pairwise disjoint sets §Here and in the following "a.e." is an abbrevation for "almost everywhere".

22.9 Measure and Lebesgue Inte.qra1 635

A I , . . . ..+ E AIand nreal numbers a l , .. . ,a, such that f =

n

akxkrwhere Xk denotes the characterk=l

istic function of the set 4 k . Obviously, each characteristic function of a measurable set is measurable, so every elementary function is measurable. It is interesting that each measurable function can be approximated arbitrarily well by elementary functions: For each measurable function f 2 0 there exists a monotone increasing sequence of non-negative elementary functions, which converges pointwise to f .

12.9.3 Integration 12.9.3.1 Definition of the Integral Let (X, A, p ) be a measurable space. The integral

I,f

dp (also denoted by

I

f dp; in this section,

except point 5. we prefer the latter notation) for a measurable function f is defined by means of the following steps: 1. Iff is an elementary function f =

n

k=l

akxk,then

(12.207) 2. I f f : X

1f 3. I f f : X

+a (f 2 0))then

[

dp = sup J g dp : g is an elementary function with 0

I

5 g(s) 5 f(s),V z E X . (12.208)

-+ R a n d f + ,f- are the positive and the negative parts o f f , then

J f dP = J f + d P J f- dP

(12.209)

-

under the condition that at least one of the integrals on the right side is finite (in order to avoid the meaningless expression w - w). 4. For a complex-valued function f = g + ih, if the integrals (12.209) of the functions g, h are finite, DUt

1f 1 dp =

g dp

+ i 1h dp.

( 12.210)

5. If for any measurable set A and a function f there exists the integral of the function f X A then put

Jfdir=JfxadiL.

(12.211)

4

The integral of a measurable function is in general a number from R. A function f : X

integrable or summable over X with respect to p if it is measurable and

I

If1

dp

-+ R is called

< M.

12.9.3.2 Some Properties of the Integral Let (X, A, p ) be a measure space, f , g : X --+ be measurable functions and a , p E R. 1. I f f is integrable, then f is finite a.e., i.e., p { z E X: If(s)l= +a}= 0. 2. Iff is integrable, then

f dpl 5

3. Iff is integrable and f 2 0, then 4. If 0

1I f 1 I

dp.

f dp 2 0

5 g(s)5 f ( E ) on X and f is integrable, then g is also integrable, and

J

g dp

5

J

f dp.

I

636

12. Functional Analvsis

5. Iff, g are integrable, then af

+ Pg is integrable, and

I

(of+ p g ) dp = a

6. If f,g are integrable on A E A, Le., there exist the integrals (12.211) and f = g p-a.e. on A, then

s,

1f + s,

f dp and

dp

p / . g dp.

g dp according to

f dp = i g d p .

If X = R” and X is Lebesgue measure, then we have the notion of the (n-dimensional) Lebesgue zntegral (see also 8.2.3.1, 3., p. 452). In the case n = 1 and A = [a,b], for every continuous function f on

[a,b] both the Riemann integral

l

f ( x ) dx (see 8.2.1.1,2.,p. 439) and the Lebesgue integral

-Ifa.ij

are defined. Both values are finite and equal to each other. Furthermore, iff is a bounded Riemann integrable function on [a,b], then it is also Lebesgue integrable and the values of the two integrals coincide. The set of Lebesgue integrable functions is larger than the set of the Riemann integrable functions and it has several advantages, e.g., when passing to the limit under the integral sign and f,If1 is Lebesgue integrable simultaneously.

12.9 -3.3 Convergence Theorems ?low Lebesgue measurable functions will be considered throughout.

1. B. Levi’s Theorem on Monotone Convergence be an a.e. monotone increasing sequence of non-negative integrable functions with values Let { i n R . Then lim

n+CC

1

f, dp =

1

lim f,dp.

( 12.212)

n+m

2. Fatou’s Theorem

Let { f,}LIbe a sequence of non-negative %valued measurable functions. Then

I f, Lbesgue’J Dominated Convergence Theorem lim inf f, dp 5 lim inf

dp.

( 12.213)

3. Let {f,}be a sequence of measurable functions convergent on X a.e. to some function f. If there exists an integrable function g such that If,l 5 g a.e., then f = limf, is integrable and lim/ f,dp = /(limf,) dp.

( 12.214)

4. Radon-Nikodym Theorem a) Assumptions: Let (X, A, p ) be a 0-finite measure space, i.e., there exists a sequence { A , } , A,

E

A

X

such that X = U A, and p ( A , ) < w for V n. In this case the measure is called o-finite. It is called ,=l

finite if p ( X ) < m, and it is called a probability measure if p ( X ) = 1. A real function p defined on A is called absolutely continuous with respect t o p if p ( A ) = 0 implies p(A) = 0. We denote this property by w + P. For an integrable function f,the function cp defined on A by p(A) = SA f dp is c7-additive and absolutely continuous with respect to the measure p. The converse of this property plays a fundamental role in many theoretical investigations and practical applications: b) Radon-Nikodym Theorem: Suppose a o-additive function cp and a measure p are given on a o-algebra A, and let ~p + p. Then there exists a p-integrable function f such that for each set A E A, (12.215) A

12.9 Measure and Lebesgue Inte.qra1 637 The function f is uniquely determined up to its equivalence class, and f 2 0 p-a.e.

’p is

non-negative if and only if

12.9.4 LP Spaces Let (X, A. p ) be a measure space and p a real number 1 5 p < cm. For a measurable function f, according to 12.9.2.2,p. 634, the function i f l p is measurable as well, so the expression

.%(f)=

(Jlf”d.)”

(12.216)

is defined (and may be equal to $00). A measurable function f:X +R i s calledp-thpowerintegrable, or an LP-function if Np(f)< +m holds or, equivalent to this, if IflP is integrable. For every p with 1 5 p < +m, we denote the set of all LP-functions, Le., all functions p t h power integrable with respect to p on X, by P ( p ) or by P ( X ) or in full detail L P ( X , A, p ) . For p = 1 we use the simple notation C(X). For p = 2 the functions are called quadratically integrable. We denote the set of all measurable p-a.e. bounded functions on X by P ( p ) and define the essential supremum of a function f as i’Vm(f) = ess. supf = inf{a E R : If(z)l5 a p-a.e.}. (12.217) P ( p ) (1 5 p 5 m) equipped with the usual operations for measurable functions and taking into consideration Minkowski inequality for integrals (see 1.4.2.13, p. 32), is a vector space and Np(.)is a semi-norm on P ( p ) . Iff 5 g means that f(z)5 g(z)holds p-a.e., then P ( p ) is also a vector lattice and even a K-space (see 12.1.7.4, p. 600). Two functions f , g E P ( p ) are called equivalent (or we declare them as equal) iff = g p-a.e. on X. In this way, functions are identified if they are equal pa.e. The factorization of the set P ( X ) modulo the linear subspace Ni’(0) leads to a set of equivalence classes on which the algebraic operations and the order can be transferred naturally. So we get a vector lattice (K-space) again, which is denoted now by LP(X, p ) or Lp(p). Its elements are called functions, as before, but actually they are classes of equivalent functions. It is very important that l i f ^l l p = Np(f)is now a norm on Lp(p) (f stands here for the equivalence class IlfilP) for every p with 1 5 p 5 +m is a o f f , which will later be denoted simply by f ) , and (LP(P)~ Banach lattice with several good compatibility conditions between norm and order. For p = 2 with g) =

(fl

fgdp as a scalar product, L*(p) is also a Hilbert space (see [12.12]).

Very often for a measurable subset R c R’ the space LP(R) is considered. Its definition is not a problem because ofstep 5 in (12.9.3.1,p. 639). The spaces LP(R,A), where X is the n-dimensional Lebesgue measure, can also be introduced as the completions (see 12.2.2.5, p. 608 and 12.3.2, p. 610) of the non-complete normed spaces C(R) of all continuous functions on the set R

c R” equipped with the integral norm lIzllp =

(1

lzlpdh)

’ (1 5 I

< m) (see [12.18]). Let X be a set with a finite measure, Le., p(X) < +m, and suppose for the real numberspl,pz, 1 5 p i < p z 5 +m. Then P ( X , p ) c LPl(X, p ) , and with a constant C = C ( p l , p z ,p ( X ) ) > 0 independent of z,there holds the estimation ~ ~5 zCllzllz ~ ~for z l E L P 2 (here //z11denotes k the norm of the space LPX(X,p) ( k = L2)). p

12.9.5 Distributions 12.9.5.1 Formula of Partial Integration For an arbitrary (open) domain R R”, Cr(R) denotes the set of all arbitrary many times in 0 differentiable functions p with compact support, i.e., the set supp(ip) = {z E R : cp(z) # 0) is compact in R” and lies in R. By Li,,(R) we denote the set of all locally summable functions with respect to

638

12. Functional Analysis

the Lebesgue measure in Rn,Le., all the measurable functions f (equivalent classes) on R such that If1 dX < +30 for every bounded domain w c R. Both sets are vector spaces (with the natural algebraic operations). There holds P ( R ) c L;,,(R) for 1 5 p 5 m, and L:oc(R) = L'(R) for a bounded R.If we consider the elements of C'((w) as the classes generated by them in U(R),then the inclusion Ck((w) c P ( R ) holds for bounded R,where C'((n) is at once iense. If R is unbounded, then the set C r ( R ) is dense (in this sense) in U(R).

J,

For a given function f E C'@) and an arbitrary function p E Cr(R) the formula of partial integration has the form

/f(x)D"p(z) dX = (-1)I"l /p(z)D"f(z) dX n

(12.218)

n

101 5 k (we have used the fact that D"&n = 0 ) , which can be taken as the starting point for the definition of the generalized derivative of a function f E L;,,(R).

Vcu with

12.9.5.2 Generalized Derivative Suppose f E L;,,(R). If there exists a function g E L;,,,(R) such that V some multi-index cy the equation

/ f ( z ) D " & ) dX

=

'p

E

CF(R)with respect to

(-1)'"' / g ( z ) ' p ( z )dX

n

(12.219)

n

holds. then g is called the generalized derivative (derivative in the Sobolev sense or distributional derivative) of order cu o f f . We write g = D" f for this as in the classical case. ~ the vector space C r ( R ) to 'p E CF(R) as follows: We define the convergence of a sequence { ~ p k } k m , in a) 3 a compact set K C R with supp(cpk) c K Vn pk (n if and Only if b) D"pk -+ D"'p uniformly on K for each multi-index cy. (12.220)

{

The set C r ( Q ) , equipped with this convergence of sequences, is called the fundamental space, and is denoted by D(R). Its elements are often called test functions.

12.9.5.3 Distributions A linear functional 1 on D(R) continuous in the following sense (see 12.2.3. p. 608): pk, 'p E D(R) and (Fk -+p implies 1(pk) -+t ( 9 ) is called a generalizedfunction or a distribution. N A: I f f E L;,,(R). then

(12.221)

is a distribution. .4 distribution, defined by a locally summablefunction as in (12.222). is called regular. Tworegulardistributionsareequal, i.e.,!f('p) = 1,(p) Vp E D(R).ifandonlyiff = 9a.e. withrespect to A. H B: Let a E R be an arbitrary fixed point. Then &,('p) = p(a)>'p E D(R) is a linear continuous functional on D(R), hence a distribution, which is called the Dirac distribution, 6 distribution or 6 function. Since e, cannot be generated by any locally summable function (see [12.11],[12.24]),it is an example for a non-regular distribution. The set of all distributions is denoted by D'(R). From a more general duality theory than that discussed in 12.5.4,p. 621, we get D'(Q) as the dual space ofD(R). Consequently, we should write D*(R) instead. In the space D'(Q), it is possible to define several operations with its elements and with functions from Cm(R), e.g.. the derivative of a distribution or the convolution of two distributions, which make

12.9 Measure and Lebes.que Inte.qra1 639

D'(Q) important not only in theoretical investigations but also in practical applications in electrical engineering, mechanics. etc. For a review and for simple examples in applications of generalized functions see, e.g., [12.11],[12.24]. Here. we discuss only the notion of the derivative of a generalized function.

12.9.5.4 Derivative of a Distribution If! is a given distribution, then the distribution D"!defined by

(D"4(Pl = ( - ~ ) ~ " ~ W O PY )E, D(f2), (12.223) is called the dzstrzbutional derzvative of order cr of e. Let f be a continuously differentiable function, say on R (so f is locally summable on R, and f can be considered as a distribution), let f' be its classical derivative and D'f its distributional derivative of order 1. Then: (12.224a) R

from which by partial integration there follows (12.224b) R

In the case of a regular distribution

(D"!,)(v) = (-l)l"le,(D'q)

!,

with f E L:o,(f2) by using (12.222) we obtain

= (-1)l"l

1

f(z)D"pdX,

(12.225)

n

and get the generalized derivative of the function f in the Sobolev sense (see (12.219)). IA: For the regular distribution generated by the obviously locally summable Heaviside function 1 for x 2 0, (12.226) O ( 2 )= 0 for z < 0 ne get the non-regular 6 distribution as the derivative. IB: In mathematical modeling of technical and physical problems we are faced with (in a certain sense idealized) influences concentrated at one point, such as a "point-like" force, needle-deflection, collision, etc.. which can be expressed mathematically by using the 6 or Heaviside function. For example. m6, is the mass density of a point-like mass m concentrated at one point a (0 < a < 1) of a beam of length 1. The motion of a spring-mass system on which at time to there acts a momentary external force F is described by the equation x + wzz = F&,. With the initial conditions x(0) = i(0)= 0 its solution is F . z ( t )= - sin(w(t - to))@(t- to)

[

iil

I

640

13. Vector Analvsis and Vector Fields

13 Vector Analysis andvector Fields 13.1 Basic Notions ofthe Theory of Vector Fields 13.1.1 Vector Functions ofa Scalar Variable 13.1.1.1 Definitions 1. Vector Function of a Scalar Variable t 4 vector function of a scalar variable is a vector a' whose components are real functions oft: a' = a'(t) = az(t)Zz+ ay(t)$ t a,(t)&

(13.1)

The notions of limit, continuity, differentiability are defined componentwise for the vector a'(t).

2. Hodograph of a Vector Function If we consider the vector function Z ( t ) as a position or radius vector i = i ( t )of a point P,then this function describes a space curve while t varies (Fig. 13.1). This space curve is called the hodograph of the vector function a'(t).

Figure 13.1

Figure 13.2

Figure 13.3

13.1.1.2 Derivative of a Vector Function The derivative of (13.1) with respect t o t is also a vector function oft: (13.2)

di dt

The geometric representation of the derivative - of the radius vector is a vector pointing in the direction of the tangent of the hodograph at the point P (Fig. 13.2). Its length depends on the choice of the parameter t. If t is the time, then the vector t ( t ) describes the motion of a point P in space (the

d? dt

space curve is its path), and - has the direction and magnitude of the velocity of this motion. If t = s is the arclength of this space curve, measured from a certain point, then obviously

13.1.1.3 Rules of Differentiation for Vectors da' d 6 d - (a'&b*c')=-k-&-

dt dt dt d dp da' - (pa')= -a't$odt dt dt d da'd6 -(a'b)=-b+Zdt ' dt dt

-

dc' dt '

(13.3a) (p is a scalar function oft),

(13.3b) (13.3~)

13.1 Basic Notions of the Theoru of Vector Fields 641 d da'd6 (a' x b) = - x b a' x dt dt dt d da' dp - a'[p(t)] = - ' dt dp dt +

-

+

(the factors must not be interchanged),

(13.3d)

(chain rule)

(13.3e)

da' da' If la'(t)l = const, Le., a"(t) = a'(t) a'(t) = const, then it follows from (13.3~) that I . - = 0, Le., dt dt and a' are perpendicular to each other. Examples of this fact: IA: Radius and tangent vectors of a circle in the plane and IB: position and tangent vectors of a curve on the sphere. Then the hodograph is a spherical curve.

13.1.1.4 Taylor Expansion for Vector Functions da' hZ d'a' h" Pa' q t + h ) = Z(t)S h 7 t -, (13.4) dt 2! d t n! dtn The expansion of a vector function in a Taylor series makes sense only if it is convergent. Because the limit is defined componentwise, the convergence can be checked componentwise, so the convergence of this series with vector terms can be determined exactly by the same methods as the convergence of a series with complex terms (see 14.3.2, p. 689). So the convergence of a series with vector terms is reduced to the convergence of a series with scalar terms. The differential of a vector function a'(t) is defined by: da' da' = -At. (13.5) dt

+

+...

+...

13.1.2 Scalar Fields 13.1.2.1 Scalar Field or Scalar Point Function If we assign a number (scalar value) U to every point P of a subset of space, then we write

u= C ( P ) and we call (13.6a) a scalarfield (or scalarfunctzon). IExamples of scalar fields are temperature, density, potential, etc., of solids. A scalar field Lr = U ( P )can also be considered as

c = by?),

(13.6b)

where ?is the position vector of the point P with a given pole 0 (see 3.5.1.1, 6., p. 181).

13.1.2.2 Important Special Cases of Scalar Fields 1. Plane Field \Ye have a plane field. if the function is defined only for the points of a plane in space.

2. Central Field If a function has the same value at all points P lying at the same distance from a fixed point C(31),called the center, then we call it a teed symrnetrzc field or also a central or sphericalfield. The function U depends only on the distance C P = Ifl:

u = f(1Gl). (13.7a) IThe field of the intensity of a point-like source, e.g., the field of brightness of a point-like source of light at the pole, can be described with = r as the distance from the light source: I: = 5 (c const) . r'

(13.7b)

642

13. Vector Analusis and Vector Fields

3. Axial Field If the function I; has the same value at all points lying at an equal distance from a certain straight line (axis of the field) then the field is called cylindrically symmetric or an axially symmetric field, or briefly an axial field.

13.1.2.3 Coordinate Definition of a Field If the points of a subset of space are given by their coordinates, e.g., by Cartesian, cylindrical, or spherical coordinates, then the corresponding scalar field (13.6a) is represented, in general, by a function of three variables: I; = @(x, Y, z ) , U = @(PlVI 2) or U = x(r,8% CF). (13.8a) In the case of a plane field, a function with two variables is sufficient. It has the form in Cartesian and polar coordinates: L' = @(z,y) or U = @(p, 9). (13.8b) The functions in (13.8a) and (13.8b), in general, are assumed to be continuous, except, maybe, at some points, curves or surfaces of discontinuity. The functions have the form

U ( 4 W )= U ( m )= U(T),

a) for a central field:

U

b) for a n axial field:

U = U ( 4 G ) = U ( p ) = U(rsin19).

=

(13.9a) (13.9b)

Dealing with central fields is easiest using spherical coordinates, with axial fields using cylindrical coordinates.

13.1.2.4 Level Surfaces and Level Lines of a Field 1. Level Surface A4level surface is the union of all points in space where the function (13.6a) has a constant value I; = const. (13.loa) Different constants Uo, UL, U2, . . .define different level surfaces. There is a level surface passing through every point except the points where the function is not defined. The level surface equations in the three coordinate systems used so far are: U = x(r,1 9 , ~ = ) const. I; = @(x> y, z ) = const, U = @ ( p ,q ,z ) = const, (13.10b) IExamples of level surfaces of different fields: A: U = c'i = c,z + cyy + c,z: Parallel planes. B: U = x2 + 2y2 + 42': Similar ellipsoids in similar positions. C : Central field: Concentric spheres. D: .4xial field: Coaxial cylinders.

2. Level Lines Level lines replace level surfaces in plane fields. They satisfy the equation U = const. (13.11) Level lines are usually drawn for equal intervals of U and each of them is marked by the corresponding value of I; (Fig. 13.3). IWell-known examples are the isobaric lines on a synoptic map or the contour lines on topographic maps. In particular cases, level surfaces degenerate into points or lines, and level lines degenerate into separate points. Y 1 IThe level lines of the fields a) I; = xy, b) U = 2, c) U = r 2 ,d) U = - are represented in Fig. 13.4.

19.1 Basic Notions of the Theory of Vector Fields 643

-1-2-3-4

-4-3-2 -1

d) Figure 13.4

13.1.3 Vector Fields 13.1.3.1 Vector Field or Vector Point Function If we assign a vector ? to every point P of a subset of space, then we denote it by

J = ?(P) (13.12a) and we call (13.12a) a vectorfield. IExamples of vector fields are the velocity field of a fluid in motion, a field of force, and a magnetic or electric intensity field. A vector field ? = < ( P ) can be regarded as a vector function

J = J(r')>

(13.12b)

where i is the position vector of the point P with a given pole 0. If all values of i as well as 3 lie in a plane, then the field is called a plane vector field (see 3.5.2, p. 189).

13.1.3.2 Important Cases of Vector Fields 1. Central Vector Field

In a central vector field all vectors ? lie on straight lines passing through a fixed point called the center

(Fig.13.5a). If we locate the pole at the center, then the field is defined by the formula ii = f(?)r'.

(13.13a)

644

13. Vector Analysis and Vector Fields

where all the vectors have the same direction as the radius vector 5. It often has some advantage to define the field by the formula 3 v = w(i))T. (13.13b)

-

where I#)

r’

is the length of the vector $ and - is a unit vector.

Figure 13.5

2. Spherical Vector Field A spherical vector field is aspecial case of a central vector field, where the length of the vector depends only on the distance (Fig. 13.5b). IExamples are the Newton and the Coulomb force field of a point-like mass or of a point-like electric charge: c c i V = -i = -- (e const). (13.14) r3 r2r The special case of a plane spherical vector field is called a czrcularfield.

-

3. Cylindrical Vector Field a) .411 vectors 3 lie on straight lines intersecting a certain line (called the axis) and perpendicular to it, and b) all vectors 3 at the points lying at the same distance from the axis have equal length, and they are directed either toward the axis or away from it (Fig. 13.5~). If we locate the pole on the axis parallel to the unit vector c‘, then the field has the form r” 3 =dP)- > (13.15a) P

where r”* is the projection of ?on a plane perpendicular to the axis: (13.15b) r” = c‘ x (ix 3). By intersecting this field with planes perpendicular to the axis, we always get equal circular fields.

13.1.3.3 Coordinate Representation of Vector Fields 1. Vector Field in Cartesian Coordinates The vector field (13.12a) can be defined by scalar fields VI(?), V2(3), and &(i) which are the coordinate functions of 3 , Le., the coefficients of its decomposition into any three non-coplanar base vectors 81, 62, and 4:

13.1 Basic Notions of the Theorv of Vector Fields 645

ii =

+~

5

+ ~~

(13.16a)

32 ~ 3 .

i,

If we take the coordinate unit vectors l, and k'as the base vectors and express the coefficients V,, V,, r/; in Cartesian coordinates, then we get:

+

(13.16b) = V'(Z,y,Z):+ V,(Z,Y,Z)j+ V,(z,y,t)l;. So, the vector field can be defined with the help of three scalar functions of three scalar variables. 2. Vector Field in Cylindrical and Spherical Coordinates In cylindrical and spherical coordinates, the coordinate unit vectors r' Z, 6+. Cz (= L), and & (= -), &, 6, ( 13.17a) are tangents to the coordinate lines at each point (Fig. 13.6, 13.7). In this order they always form a right-handed system. The coefficients are expressed as functions of the corresponding coordinates:

.3 = V,(P, P,4$ + V,(P, P,z)$ + V,(P, P,z ) & .3 = V,(T,29, p)C, + V+(T.19, 'p)e', + V,(T, 19, 'p)e'+j.

(13.17b)

I

(13.17~) At transition from one point to the other, the coordinate unit vectors change their directions, but remain mutually perpendicular.

p-p -t

X

Figure 13.6

Figure 13.7

Figure 13.8

13.1.3.4 Transformation of Coordinate Systems See also Table 13.1. Cartesian Coordinates in Terms of Cylindrical Coordinates V, = V, cos 'p - r/, sin 'p, r/, = V, sin cp + V, cos 'p, V , = V,. Cylindrical Coordinates in Terms of Cartesian Coordinates T/,=V,coscp+V,sincp, V,=-\Lsin'p+Vvcosrp, V,=V, Cartesian Coordinates in Terms of Spherical Coordinates V' = V, sin 19 cos cp - V, sin cp + V, cos cp cos 29, Vu = V, sin 19 sin 'p V, cos 'p + V, sin 'p cos 29, L< = VTcos 29 - V, sin 19. Spherical Coordinates in Terms of Cartesian Coordinates V , = r/, sin 17 cos p V, sin 29 sin 'p + V, cos 19, = '1 ; cos 19 cos 9 + V,cos $sin 'p - V ,sin 29,

(13.18) (13.19)

+

(13.20)

+

(13.21)

646

13. Vector Analvsis and Vector Aelds

T.5 =

-

T;. sin p + V, cos p.

5 . Expression of a Spherical Vector Field in Cartesian Coordinates

q = p((J.”+.2)(x?t

+y;

(13.22)

ZL).

6. Expression of a Cylindrical Vector Field in Cartesian Coordinates

iT = p((JiG&:+

(13.23)

y.);

In the case of a spherical vector field, spherical coordinates are most convenient for investigations, Le., the form 3 = V(r)Zr;and for investigations in cylindrical fields, cylindrical coordinates are most + convenient, Le., the form V = V(p)Z,. In the case of a plane field (Fig. 13.8),we have

3 = vz(z,Y):t

(13.24)

lry(z,y)l=Vp(P,P)%?+V,(P>P)%,

and for a circular field

3 = y ( r n ) ( z ? + y j ? = y(p)Z,.

(13.25)

Table 13.1 Relations between the components of a vector in Cartesian, cylindrical, and spherical coordinates

Cartesian coordinates

Cylindrical coord.

Spherical coordinates

vpsp+ V,$ + v*s,

V,Zr

= V, cos p - V, sin y

= V ,sin 6 cos (o

+ KjZa + V,Z, + V, cos 29 cos p - V, sin ‘p

= V, sin p

+ V, cos y

= r/;sin 8 sin p

+ \>cos 19 sin p

+ v,cos y v,

= Kcos29 - L5sin29

1;

=

C;cosp+L,sinp

= V,

= V ,sin 8

v, = v,

=

= Vpsin29+V,cos29

=

V, cos$cos (n + Li cost9sinp - V, sin ~p = \’,cos29 - V, sin29

=

-Li

sin p + V, cos y

T.: Vzsin $cos p

+ V, sin29sin y + V, cos8

-T.k sin IJ + L<
=

=

v,

+ Va cos 29

v,

= K COS 29

- L<9 sin

29

v, v, = v,

13.1.3.5 Vector Lines .4 curve C is called a lzne ofa vector or a vector hne of the vector field q(i)(Fig. 13.9) if the vector ?(<) is a tangent vector of the curve at every point P . There is a vector line passing through every point of the field. Vector lines do not intersect each other, except, maybe, at points where the function 3 is not defined, or where it is the zero vector. The differential equations of the vector lines of a vector field 3 given in Cartesian coordinates are

Figure 13.9

13.2 Differential Operators of Space 647

dx

a ) in general:

dy

dz

v, v, , (13.26a)

-= - = -

1:

dx

dy

vz

vy

b) for a plane field: - = - .

(13.26b)

To solve these differential equations see 9.1.1.2, p. 487 or 9.2.1.1. p. 515. W A: The vector lines of a central field are rays starting at the center of the vector field W B: The vector lines of the vector field 3 = c' x r' are circles lying in planes perpendicular to the vector c'. Their centers are on the axis parallel to c'.

13.2 Differential Operators of Space 13.2.1 Directional and Space Derivatives 13.2.1.1 Directional Derivative of a Scalar Field The directional derivative of a scalar field U = U(F) at a point P with position vector ?in the direction c' (Fig. 13.10) is defined as the limit of the quotient

ab-

-=

dc'

lim

C(i

+EZ) - U ( i )

(13.27)

E

E+O

If the derivative of the field U = U(i) at a point Fin the direction of the unit vector So of c'is denoted dC by -> then the relation between the derivative of the function with respect to the vector c'and with

azo

respect to its unit vector So at the same point is (13.28) The derivative

dC'

,

-with

azo

respect to the unit vector represents the speed of increase of the function U in

the direction of the vector

c" at the point r'. If 6 is the normal unit vector to the level surface passing dU

through the point r', and n' is pointing in the direction of increasing U , then -= has the greatest value dn among all the derivatives at the point with respect to the unit vectors in different directions. Between the directional derivatives with respect to n' and with respect to any direction So, we have the relation

dC

ar;

dU

(13.29) - = - COS(^'^. 5) = ;cos p = So . grad L' (see (13.35), p. 649) . dc'O dfi an In the following. directional derivatives always mean the directional derivative with respect to a unit vector. d?( 7 )

0

3 7+&?

J f V(7)

,/,'

p$

-+

a

r

--t

r

--t

Figure 13.10

Figure 13.11

I

648

13. Vector Analgsis and Vector Fields

13.2.1.2 Directional Derivative of a Vector Field The directional derivative of a vector field is defined analogously to the directional derivative of a scalar field. The directional derivative of the vector field 3 = ?(?) at a point P with position vector i (Fig. 13.11) with respect to the vector a' is defined as the limit of the quotient

a3 = lim S(i+Ea') aa'

E

-

3(?)

(13.30)

E

4

If the derivative of the vector field 3 = g(?)at a point ?in the direction of the unit vector a" of a' is denoted by

a3

a3

azo

-)

~

then

a3

- = lalaa' da'0' In Cartesian coordinates, i t . , for 3 = V2S2 V&,

+

+

(13.31)

+ V,SZ,,a' = a,& + alSv + a,&,

+

(a'. g r a d ) g = (a'. grad V,)e: (a'. grad V')e; (a'. grad Vz)ea. In general coordinates we have: + 1 (a'sgrad)V= -(rot(qxa')+grad(a'.?)+a'div?-?divZ-a'xrot?-$ 2

we have: (13.32a)

x rota'. ( 13.32b)

13.2.1.3 Volume Derivative Volume derivatives of a scalar field U = U(r') or a vector field 3 at a point r' are quantities of three forms, which are obtained as follows: 1. We surround the point ?of the scalar field or of the vector field by a closed surface E. This surface can be represented in parametric form r'= r'(u,v) = x(u,v)Z2 y(u, v)$ z(u, v)&, so the corresponding vectorial surface element is

+

-

ar'

ar'

au

av

dS = - x - d u d v .

+

(13.33a)

2. We evaluate the surface integral over the closed surface E. Here, the following three types of integrals can be considered:

(13.33b) 3. We determine the limits (if they exist) (13.33~) Here V denotes the volume of the region of space that contains the point with the position vector r' inside, and which is bounded by the considered closed surface C. The limits (13.33~)are called volume derivatives. The gradient of a scalarfield and the dzvergence and the rotatzon of a vector field can be derived from them in the given order. In the following paragraphs, we discuss these notions in details (we will even define them again.)

13.2.2 Gradient of a Scalar Field The gradient of a scalar field can be defined in different ways.

13.2.2.1 Definition of the Gradient The gradzent of a function U is a vector grad U , which can be assigned to every point of a scalar field U = U(r'), having the following properties:

13.2 Differential Operators of space 649

1. The direction of gradli is always perpendicular to the direction of the level surface U = const, passing through the considered point, 2. grad I; is always in the direction in which the function U is increasing,

dU

3. lgrad I;l = -, Le., the magnitude of grad U is equal to the directional derivative of U in the normal an direction. If the gradient is defined in another way, e.g., as a volume derivative or by the differential operator, then the previous defining properties became consequences of the definition.

13.2.2.2 Gradient and Volume Derivative The gradzent U of the scalar field U = U(5)at a point r'can be defined as its volume derivative. If the following limit exists, then we call it the gradient of U at r':

fl U d S

(13.34) grad U = lim (E) v+o Here V is the volume of the region of space containing the point belonging to Finside and bounded by the closed surface E. (If the independent variable is not a three-dimensional vector, then the gradient is defined by the differential operator.)

v

13.2.2.3 Gradient and Directional Derivative The directional derivative of the scalar field U with respect to the unit vector So is equal to the projection of grad U onto the direction of the unit vector So: (13.35) Le., the directional derivative can be calculated as the dot product of the gradient and the unit vector pointing into the required direction. Remark: The directional derivative at certain points in certain directions may also exist if the gradient does not exist there.

13.2.2.4 Further Properties of the Gradient 1. The absolute value of the gradient is greater if the level lines or level surfaces drawn as mentioned in 13.1.2.4,2., p. 642, are more dense. 2. The gradient is the zero vector (grad U = 6) if U has a maximum or minimum at the considered point. The level lines or surfaces degenerate to a point there.

13.2.2.5 Gradient of the Scalar Field in Different Coordinates 1. Gradient in Cartesian Coordinates dC'(z.Y, 2)

grad I! = i2.

ax

+

Y, "'5 I + dU(z, 8Y ~

dU(s,Y, 2) k' dz

Gradient in Cylindrical Coordinates (z = pcos cp, grad Cr = grad, US, + grad, US,

ac.

grad U = ->

aP

1d l i grad, U = --

+ grad, US,

Pap'

with

dU

grad,I; = dz

(13.36) y = p sin cp, z

=z) (13.37a) (13.37b)

3. Gradient in Spherical Coordinates (z = r sin 6 cos 9, y = r sin 6 sin cp, z = r cos 6) (13.38a) grad C = grad, UST t gradBUS$ + grad, US, with

650

13. Vector Analusis and Vector FaddS

au

1au

grad,U = -.

grad& = --,

ar

T

grad&'

aiy

1

au

(13.3813)

= --

r sin iJ a y

'

4. Gradient in General Orthogonal Coordinates ( E , q , C)

o;+

oi+

OC

If?(<, 1 7 1 0= X(6,17> Y(6,17s 4<> 171 then we get grad U = grad, USc t grad, US, + grad, US,, where 1

au

1

au

(13.39a) 1

au

(13.39b)

13.2.2.6 Rules of Calculations We assume in the followings that c'and c are constant. grad c = 0. grad (U1+ Uz) = grad UI grad Uz5 grad ( c V) = c grad U.

+

(13.40)

+

(13.41)

grad(UlU2) = U1gradUz UzgradUl, gradp(U) = -gradU. dP dU grad (51 . ?z)= (gl. grad)gz t (?z

. grad)q1 t ?I

+ qZx rot gl.

x rot q2

(13.42)

grad (?' c') = c'.

(13.43)

1. Differential of a Scalar Field as the Total Differential of the Function U

dU ax

dl; = grad U . d?= - d x

dU au +dy + -d z . ay a2

(13.44)

__-

_-(&.grad)?=

ax ay a2 at; av, av, --ax d y az

av, _ av, -av, _

(5)

,(13.47b)

gradV=

ax a y

az

av, av, at; --ax ay az avz _ -av, - ai/,

. (13.47~)

13.2 Differential Operators of Space 651

13.2.4 Divergence of Vector Fields 13.2.4.1 Definition of Divergence We can assign a scalar field to a vector field g(?)which is called its divergence. The divergence is defined as a space derivative of the vector field at a point i:

fl?.dg +

(E)

divV = lim (13.48) v+o ' If the vector field ? is considered as a stream field, then the divergence can be considered as the fluid output or source, because it gives the amount of fluid given in a unit of volume during a unit of time flowing by the considered point of the vector field 3. In the case d i v q > 0 the point is called a source, in the case div? < 0 it is called a sink.

v

~

13.2.4.2 Divergence in Different Coordinates 1. Divergence in Cartesian Coordinates -,

av, + av + av,

divV = az

ay

82

(13.49a)

with ?(z, y) z ) = V,i+

VY,+ V&.

(13.49b)

The scalar field div ? can be represented as the dot product of the nabla operator and the vector ? as div?=V,? and it is translation and rotation invariant, Le., scalar invariant (see 4.3.3.2, p. 265).

(13.49~)

2. Divergence in Cylindrical Coordinates

-

1 a(pv,) divV = -P

ap

1 av, av, + -+P

arp

(13.50a) with ?(p,cp, z ) = VpZp+ V,Zv

+ V&,.

(13.50b)

3. Divergence in Spherical Coordinates

with ?(r. 19, p) = V,&

+ V& + V,S,.

4. Divergence in General Orthogonal Coordinates (13.52a) with

D=

(13.52~)

and

13.2.4.3 Rules for Evaluation of the Divergence divc'=O,

d i v ( q 1 + q 2 ) = d i ~ ? ~ + d i v ? 2 , div(c?) =cdiv?.

div (C?) = Cr div 3 + 3

(13.53) (13.54)

652

13. Vector Analusis and Vector Fields

div (?I x

92)

= ?z. rot?1 - ?,. rot?z.

(13.55)

13.2.4.4 Divergence of a Central Field d i v i = 3, divip(r)i= 3yo(r)t r $ ( r ) ,

(13.56)

13.2.5 Rotation ofvector Fields 13.2.5.1 Definitions of the Rotation 1. Definition The rotation or curl of a vector field ? at the point ?is a vector denoted by rot ?, curl ? or with the nabla operator V x 3,and defined as the negative space derivative of the vector field:

(13.57)

2. Definition The vector field of the rotation of the vector field ?(?) can be defined in the following way: a) We put a small surface sheet S (Fig. 13.12)

'

through the point 3. We describe this surface sheet by a vector s' whose direction is the direction of the surface normal fi and its absolute value is equal to the area of this surface patch. The boundary of this surface is denoted by C. b) We evaluate the integral

f 3 . d t along the (CI

closed boundary curve C of the surface (the sense of the curve is positive looking to the surface from the direction of the surface normal (see Fig. 13.12).

c) We find the limit (ifit exists) lim s+o

f 5. d i , (C)

Figure 13.12

while the position of the surface sheet remains unchanged.

d) We change the position of the surface sheet in order to get a maximum value of the limit. The surface area in this position is S, and the corresponding boundary curve is Cmm. e) We determine the vector rot 3 at the point i,whose absolute value is equal to the maximum value found above and its direction coincides with the direction of the surface normal of the corresponding surface. We then get:

(13.58a) The projection of rot ? onto the surface normal ii of a surface with area S , Le., the component of the vector rot 3 in an arbitrary direction ii = i i s

15.2 Diferential Operators ofspace 653

(13.58b) The vector lines of the field rot? are called the curl lines ofthe vectorfield?.

13.2.5.2 Rotation in Different Coordinates 1. Rotation in Cartesian Coordinates r i 1 ; (13.59a)

vx

v, K

The vector field rot? can be represented as the cross product of the nabla operator and the vector ?: rot? = v x

?.

(13.59b)

2. Rotation in Cylindrical Coordinates

rot rot

+ rot v?S+, t rot ,?a, with av, - 1av, av, r o t , V = - - -av, V=---,

? = rot p?Sp

(13.60a)

.+

pap

a2

a2

3. Rotation in Spherical Coordinates t rot v?Sp rot ? = rot ,?ST + rot

.

ap

(13.60b)

(13.61a)

with

(13.61b)

4. Rotation in General Orthogonal Coordinates rot? = rot ?St t rot 'I ?Sq t rot ?Sc

(13.62a)

with

)

(13.62b)

(13.62~)

13.2.5.3 Rules for Evaluating t h e Rotation rot (V;+ Vz) = rot fl+ rot f z 3 rot (c?) = crot Q.

(13.63)

654

13. Vector Analusis and Vector Fields

rot

( U s ) = L' rot 9 + grad U x ?.

(13.64)

~zdiv\ r o t ( q 1 x q z ) = ( ~ ~ ~ g r a d ) ~ ~ - ( ~ ~ ~ g r a d ) ~ ~ t ~ ~ d i v ~ z - (13.65)

13.2.5.4 Rotation of a Potential Field This also follows from the Stokes theorem (see 13.3.3.2, p. 664) that the rotation of a potential field is identically zero: rot

? = rot (grad U) = 0'.

(13.66)

This also follows from (13.59a) for ? = grad U , if the assumptions of the Schwarz interchanging theorem are fulfilled (see 6.2.2.2, l . ,p. 393).

IFor r'= xi + y j + zk with r = I?/ = dx2t y2 t z2 we have: rot i = 0 and rot (q(r)F)= 6. where p(r) is a differentiable function of r . 3

.

4

.

.

13.2.6 Nabla Operator, Laplace Operator 13.2.6.1 Nabla Operator The symbolic vector V is called the nabla operator. Its use simplifies the representation of and calculations with space differential operators. In Cartesian coordinates we have:

aa;. -Ja;.t -k.

V = -1f

(13.67) ay az The components of the nabla operator are considered as partial differential operators, Le.. the symbol

ax

8 ax

- means partial differentiation with respect to 2 , where the other variables are considered as con-

stants. The formulas for spatial diferential operators in Cartesian coordinates can be obtained by formal multiplication of this vector operator by the scalar U or by the vector ?. For instance. in the case of the operators gradient, vector gradient, divergence, and rotation: grad U = V I; (gradient of U (see 13.2.2, p. 648)), (13.68a) grad

9 = V?

div? = V rot

5

?=V x ?

(vector gradient of?

(see 13.2.3, p. 650)),

(13.68b)

(divergence of?

(see 13.2.4, p. 651)),

(13.68~)

(rotation or curl of?

(see 13.2.5, p. 652)).

(13.68d)

13.2.6.2 Rules for Calculations with the Nabla Operator 1. If V stands in front of a linear combination a,X, with constants a, and with point functions X,. then, independently of whether they are scalar or vector functions, we have the formula:

V(Ca J z ) =

a,VX,.

(13.69)

2. If V is applied to a product of scalar or vector functions, then we apply it to each of these functions after each other and add the result. There is a -1 above the symbol of the function submitted to the operation -1 -1 -1 (13.70) V(XY2) = V ( X Y Z ) t V ( X Y Z ) + V ( X Y Z ) , Le.,

+

+

V ( X Y 2 )= (VX)YZ X(VY)2) XY(V2). We transform the products according to vector algebra so as the operator V is applied to only one factor with the sign -1 Having performed the computation we omit that sign.

13.2 Dafferential Operators of Space 655

4

-1

IA: dir(C'?) =V(C?)=V(U?) t V ( U $ ) = ? ~ V U + C ' V . $ = ~ . g r a d U + U d i v ? .

.1

+

-1

IB: grad (?1$2) = V(q~g2) = V( $1 ?2) V(?1q2). Because 6(a'C) = (d)c'+ a' x (6 x c') we 3, x (Vx 3,) (?1~)329, x (Vx 3,) get: grad (313,)= (320)3,

+

= (?zgrad)?l

+

$2

+

+

+ (q1grad)?2 + 3, x

x rot

rot ?2.

13.2.6.3 Vector Gradient The vector gradient grad? is represented by the nabla operator as grad? = 03.

( 13.71a)

!Ye get for the expression occurring in the vector gradient (a' 2(a'.V)?=rot(?xa')+grad(&)

V)?:

ta'div?-?diva'-a'x

r o t 3 - 5 ~r o t s .

(13.71b)

In particular we get for 3 = xi + y j t zk: .

.

-

3

(a' V)i = a'.

(13.71~)

'

13.2.6.4 Nabla Operator Applied Twice For every field 3:

V(V x 3 )= divrot?

= 0,

V(Vb') = divgrad C = AU.

(13.72a)

V x (VU)= rot grad U = 6,

(13.72b)

(13.72~)

13.2.6.5 Laplace Operator 1. Definition The dot product of the nabla operator with itself is called the Laplace operator:

v.v vz.

A= = (13.73) The Laplace operator is not a vector. It prescribes the summation of the second partial derivatives. It can be applied to scalar functions as well as to vector functions. The application to a vector function, componentwise, results in a vector. The Laplace operator is an znvarzant, Le.. it does not change during translation and/or rotation of the coordinate system. 2. Formulas for the Laplace Operator in Different Coordinates In the following, we apply the Laplace operator to the scalar point function U(?)?.Then the result is a scalar. The application of it for vector functions ?(?) results in a vector A? with components AV,, 4\;* A\; 1. Laplace Operator in Cartesian Coordinates

azu a2u a2U Ar;(s, y. z ) = -+ -t ax2 ay2 az2

(13.74)

2. Laplace Operator in Cylindrical Coordinates

(13.75)

656

13. Vector Analysis and Vector Fields

3. Laplace Operator in Spherical Coordinates (13.76) 4. Laplace Operator in General Orthogonal Coordinates

3. Special Relations between the Nabla Operator and Laplace Operator V(V

V(V

3 ) = grad div 3s

v x (V x 3)= rotrot?, 3)- V x (V x 3)= A$,

(13.78) (13.79) (13.80)

where

(13.81)

13.2.7 Review of Spatial Differential Operations 13.2.7.1 Fundamental Relations and Results (see Table 13.2) Table 13.2 Fundamental relations for spatial differential operators

Operator Gradient Vector gradient Divergence Rotation Laplace operator Laplace operator

I Symbol

I! vu

grad grad V div 3 rot 3 AU

A3

Meaning

Relation V3

vector 0 . 3 vector vector 0 x 3 ( V . V)U scalar ( V . V ) 3 vector

maximal increase tensor second order source, sink scalar curl vector potential field source scalar vector

13.2.7.2 Rules of Calculation for Spatial Differential Operators U, u,, ~

2 scalar , functions; c constant; 3, q1,q2vector functions: grad (U1 + U2) = grad Ul t grad UZ.

(13.82)

grad (cli) = c grad U. grad(U1Uz) = U1gradUz+UzgradUl.

(13.83)

gradF(U) = F’(U)gradU.

(13.85)

(13.84)

13.2 Differential Operators of Space 657

+

div (91+ ?2) = d i v q 1 div32.

(13.86)

div (c?) = cdiv?.

(13.87)

div (C?) = ? . grad U + Udiv ?.

(13.88)

rot ($1

+

= rot 31

?2)

+ rot 32.

(13.89)

rot (c?) = crot ?.

(13.90)

rot (U?) = U rot ? - ? x grad U.

(13.91)

.+

div rot V

E

rot grad U

(13.92)

0.

6

(13.93)

(zero vector).

div grad b' = AU.

(13.94)

rot rot? = grad div? div (q1x

32)= ?2

-

a?.

(13.95)

rot ?, - 9, . rot 52.

(13.96)

13.2.7.3 Expressions of Vector Analysis in Cartesian, Cylindrical, and

Spherical Coordinates (see Table 13.3) Table 13.3 Expressions of vector analysis in Cartesian, cylindrical, and spherical coordinates

F

Cartesian coordinates Cylindrical coordinates Spherical coordinates

+ $dU + Z,dz au au au e,-++-+SZ-

ds'= di &dx

gradU div?

rot?

4u

ax

yall

at

-+A+av, av av,

ax

ay

at

+ e'8rd19 + &,r sin t9d9 au 1 a u 1 au 4- + &-- + s,-T at9 rsint9 89 ar 1a i a --

+ &pdq + Zzdz au l a U av e' -.+ s,-- + s,p 89 at ap 1 8 lav, av, --(pV,) + -- + -

&dr

e',dp

P aP

az

I+-

T2ar(m +m G ( v 8 w 1

av,

658

29. Vector Analusis and Vector Fields

13.3 Integration in Vector Fields Integration in vector fields is usually performed in Cartesian. cylindrical or in spherical coordinate systems. Usually we integrate along curves, surfaces, or volumes. The line, surface, and volume elements needed for these calculations are collected in Table 13.4. Table 13.4 Line, surface, and volume elements in Cartesian, cylindrical, and spherical coordinates

1 Cartesian coordinates 1 Cylindrical coordinates I Spherical coordinates 1 +&r sin 8drdp

I

e'= = e', x e',

ZP = e',

gy = e'z x e'%

e, = e, x ep

6, = & x

e', = SP x e',

I

I* I

+

& = e'* x e',

x e',

+

e'* = spx

+

= 4 x e'*

I

The indices i and take the place of IC, y, z or p, q ,z or r, 19, cp. The volume is denoted here by u to avoid confusion with the absolute value of the vector function I?I= V.

13.3.1 Line Integral and Potential in Vector Fields 13.3.1.1 Line Integral in Vector Fields 1. Definition The scalar-valued curvilinear integral or line integral of a vector function ?(i) along h

a rectificable curve AB (Fig. 13.13) is the scalar value (13.97a)

,-.

AB

2. Evaluation of this Integral in Five Steps

/-.

a ) We divide the path AB (Fig. 13.13) by division points AI(&), Az(&), . . . , A,-l(?,-l) (A = Ao, B I A,) into n small arcs which are approximated by the vectors it- ic-l= Aic-,. b) We choose arbitrarily the points P,with position vectors itlying inside or at the boundary of each small arc. c) R'e calculate the dot product of the value of the function ?(it) at these chosen points with the corresponding A?,-,. d) We add all the n products. lL..

e) We calculate the limit of the sums got this way 2=1

obviously.

?(?J . A?%-] for Ai?,-,

+ 0, while n + co

13.3 Integration in Vector Fields 659

If this limit exists independently of the choice of the points Ai and P,, then it is called the line integral (13.97b) 4B

.A sufficient condition for the existence of the line integral (13.97a,b) is that the vector function q(F) h

and the curve AB are continuous and the curve has a tangent varying continuously. A vector function ?(i) is continuous if its components, the three scalar functions, are continuous.

Figure 13.13

Figure 13.14

13.3.1.2 Interpretation of the Line Integral in Mechanics If +(?) is a field of force, Le., g(i)= I?(?), then the line integral (13.97a) represents the work done by h

I? while a particle m moves along the path AB (Fig. 13.13J3.14).

13.3.1.3 Properties of the Line Integral

1 ?(<). / / g(l). 1q(l) / 1 +@(?)I . ,-. / q(l). / 1 1?(i), $(i) d i +

dl=

di.

h

h

h

ABC

AB

BC

. di=-

(Fig. 13.14). ?(l) di

h

h

AB

BA

[?(i)

dl=

di+

W(i) . d i

(13.99)

(13.100)

h

h

AB

AB

cg(r'),dF=

AB

di

c

h

h

AB

AB

(13.101)

13.3.1.4 Line Integral in Cartesian Coordinates In Cartesian coordinates, we have:

1g(i).

d l = /(Vzdz+Vydy+l',dz).

h

h

AB

AB

(13.102)

660

13. Vector Analusis and Vector Fields

13.3.1.5 Integral Along a Closed Curve in a Vector Field A line integral is called a contour integralif the path of integration is a closed curve. If the scalar value of the integral is denoted by P and the closed curve is denoted by C, then we use the notation: (13.103)

13.3.1.6 Conservative or Potential Field 1. Definition If the value P of the line integral (13.97a) in a vector field depends only on the initial point A and the endpoint B , and is independent of the path between them, then we call this field a conservative fieldor a potentialfield. The value of the contour integral in a conservative field is always equal to zero:

/?(r')

d i = 0.

(13.104)

A conservative field is always irrotational: (13.105) rot? = 6, and conversely, this equality is a sufficient condition for a vector field to be conservative. Of course, we have to suppose that the partial derivatives of the field function are continuous with respect to the corresponding coordinates, and the domain of ? is simply connected. This condition, also called the integrability condition (see 8.3.4, 8.3.4.2, p. 467), has the form in Cartesian coordinates

av, -- av, av, --3 3-3 _ ay ' ax az , ax ' az dy 2. Potential of a ConservativeField,

(13.106)

or its potential function or briefly its potential is the scalar function i

p ( i) =

p i ).

(13.107a)

di.

SO

We can calculate it in a conservative field with a fixed initial point A(?,-,) and a variable endpoint B ( q from the integral p(i)=

1

?(?). dr'.

(13.107b)

h

AB

Remark: In physics, the potential v*(i)of a function ?(i) at the point i i s considered with the opposite sign: i

p*(r')= -/$(i)'

d i = -p(?).

(13.108)

80

3. Relations between Gradient, Line Integral, and Potential If the relation ?(?) = grad U(i) holds, then U(i) is the potential of the field ?(?), and conversely, g(3) is a conservative or potential field. In physics we consider the negative sign corresponding to (13.108). 4. Calculation of the Potential in a Conser_vative_Fieid If the function ?(i) is given in Cartesian coordinates V = V,i+ Vuj+ Vz6,then for the total differential of its potential function:

13.3 Inte.gration in Vector Fields 661

dU = V, dx

+ V, dy + V, dz.

au

au

(13.109a) Here, the coefficients V,, V,, V, must fulfill the integrability condition (13.106). The determination of U follows from the equation system

au

(13.109b) - = v,, aZ = v,. 8X 8Y In practice, the calculation of the potential can be done by performing the integration along three straight line segments parallel to the coordinate axes and connected to each other (Fig. 13.15): - = v,,

J X

(13.110) Figure 13.15

13.3.2 Surface Integrals 13.3.2.1 Vector of a Plane Sheet The vector representation of the surface integral of general type (see 8.3.4.2, p. 483) requires to assign a vector s' to a plane surface region S, which is perpendicular to this region and its absolute value is equal to the area of S.Fig. 13.16a shows the case of a plane sheet. The positive direction in S is given by defining the positive sense along a closed curve C according to the nght-hand law (also called nght screw rule): If we look from the initial point of the vector into the direction of its final point, then the posztzve sense is the clockwise direction. By this choice of orientation of the boundary curve we fix the exterior side of this surface region, i.e., the side on which the vector lies. This definition works in the case of any surface region bounded by a closed curve (Fig. 13.16b,c).

interior side

exterior side

Figure 13.16

13.3.2.2 Evaluation of the Surface Integral The definition of a surface integral in scalar or vector fields is independent of whether the surface S is bounded by a closed curve or is itself a closed surface. The evaluation is performed in five steps: a) We divide the surface region S on the exterior side defined by the orientation of the boundary curve (Fig. 13.17) into n arbitrary elementary surfaces AS, so that each of these surface elements can be approximated by a plane surface element. We assign the vector A$ to every surface element AS, as given in (13.33a). In the case of a closed surface, the positive direction is defined so that the exterior side is where Agi should start.

662

13. Vector Analusis and Vector Fields

b) We choose an arbitrary point P,with the position vector r', inside or on the boundary of each surface element. c) w'e produce the products U ( 6 )ASi in the case of a scalar field and the product ?(4) . A$ or ?(r',) x A$i in the case of a vector field d) We add all these products. e) We evaluate the limit while the diameters of AS, tend to zero, Le., AGi --t 0 for n --t CG. So, the surface elements tend to zero in the sense given in 8.4.1, l., p. 469, for double integrals. If this limit exists independently of the partition and of the choice of the points r',, then we call it the surface integral of ? on the given surface. ZA

Figure 13.17

Figure 13.18

13.3.2.3 Surface Integrals and Flow of Fields 1. Vector Flow of a Scalar Field (13.111)

2. Scalar Flow of a Vector Field

Q=

lim ??(it) . As', = AS-0

/ ?(i) . ds'.

(13.112)

$=I

3. Vector Flow of a Vector Field (13.113)

13.3.2.4 Surface Integral in Cartesian Coordinates as Surface Integrals of Second Type /Lrds'= (SI

//UdydzT+ (%a)

//Udtdxjt

(&*I

//Udxdyg.

(%I

(13.114)

23.3 Inteqration in Vector Fields 663

(13.115)

/?

x dg =

(SI

//(L:j'-

Vvc)dydz

+

/ / ( V & - Vzi)dzdx + //(Vu'- V,;) dxdy.(13.116) (SWI

(SZ*)

(%XI

The existence theorems for these integrals can be given similarly to those in 8.5.2,4.. p. 482. In the formulas above, each of the integrals is taken over the projection S on the corresponding coordinate plane (Fig. 13.18),where one of the variables z. y or z should he expressed by the others from the equation of S. Remark: Integrals over a closed surface are denoted by

f c'dg = @ d i , (SI

(SI

f3 , d g = $9, dg,

f $ x dg =

(SI

(SI

(SI

I

W A: Calculate the integral $ =

(13.117 )

x dg. (SI

zyz dg, where the surface is the plane region x+y+z = 1bounded

(S)

hy the coordinate planes. The upward side is the positive side:

$ = //(l-y-z)yzdydz;+

l/(l-x-z)zzdzdzi+

(SYZ)

/ / ( 1 - y - z)yzdydz =

l1

//(l-z-y)zydzdyL;

(SZZI

1

-.120

L1-'(l - y - z)yzdydz =

We get the two further integrals

(Sua1

I - - analogously. The result is: P = -(i + j + k). 120 4

W B: Calculate the integral Q =

1

i. d g

same plane region as in A:

/ / z d y dz

11

z dy dz

=

+

(%I

(S)

= /l

0

I1-'(l 0

-z

11

y dz dx

+

-

y) dydz =

z dz dy over the (SWI

(SXD)

1

-.6

Both other integrals are

(SY'I

1 1 1 1 calculated similarly. The result is: Q = - t - + - = - . 6 6 6 2

W C: Calculate the integral

=

1

r ' x dg = / ( x i + y j + z L ) x (dydz;+ d z d z i + d x d y c ) , where

(SI

(SI

the surface region is the same as in A: Performing the computations we get

= 6.

13.3.3 Integral Theorems 13.3.3.1 Integral Theorem and Integral Formula of Gauss 1. Integral Theorem of Gauss or the Divergence Theorem The integral theorem of Gauss gives the relation between a volume integral of the divergence of ? over a volume u , and a surface integral over the surface S surrounding this volume. The orientation of the surfack (see 8.5.2.1, p. 481) is defined so that the exterior side is the positive one. The vector function ? should be continuous, their first partial derivatives should exist and be continuous. (13.1 18a)

664

13. Vector Analusis and Vector Fields

The scalar flow of the field d through a closed surface Sis equal to the integral of divergence of d over the volume v bounded by S. In Cartesian coordinates we get: (13.118b)

2. Integral Formula of Gauss In the planar case, the integral theorem of Gauss restricted to the x,y plane becomes the integral formula of Gauss. It represents the correspondence between a line integral and the corresponding surface integral:

11[

-

dP0 dY

(B)

[ P ( xy) , dx

+ &(x, y) dy]

(13.119)

B denotes a plane region which is bounded by C. P and Q are continuous functions with continuous first partial derivatives. 3. Sector Formula The sector formula is an important special case of the Gauss integral formula. We can calculate the area of plane regions with it. For Q = x, P = -y it follows that (13.120)

13.3.3.2 Integral Theorem of Stokes The integral theorem of Stokes gives the relation between a surface integral over an oriented surface region S defined in the vector field d ,and the integral along the closed boundary curve C of the surface S. The sense of the curve C is chosen so that the sense of traverse forms a right screw with the surface normal (see 13.3.2.1, p. 661). Thevector function3 should becontinuous and it should havecontinuous first partial derivatives. //rotg. (SI

ds=

f d .dr'.

(13.121a)

(C)

The vector flow of the rotation through a surface S bounded by the closed curve C is equal to the contour integral of the vector field 3 along the curve C. In Cartesian coordinates, we have:

f(Vzdx + V, dy + V, dz) (C)

In the planar case, the integral theorem of Stokes, just as that of Gauss, becomes into the integral formula (13.119) of Gauss.

13.3.3.3 Integral Theorems of Green The Green integral theorems give relations between volume and surface integrals. They are the applications of the Gauss theorem for the function 3 = Ul grad U.,where U, and U, are scalar field functions and 2: is the volume surrounded by the surface S. (13.122)

13.4 Evaluation of Fields 665

(13.123) In particular for C; = 1, we have: (13.124) In Cartesian coordinates the third Green theorem has the following form (compare 13.11%): (13.125)

f (x2y3dx + dy +

IA: Calculate the line integral I =

z dz) with C as the intersection curve of the

(C)

cylinder x2 + y2 = u2 and the plane z = 0. We get with the Stokes theorem: a6

(C)

(SI

+J

(S’)

r=O

r5coszpsin2pdrdp=--~ 8

with

rot? = -3x2y21;, dg = Gdxdy and the circle S*:x2 + y2 5 a’.

B: Determine the flux I =

f ? . d g in the drift space 3

= x3f+

y3j+ z31; through the surface S

(S)

of the sphere x2 + y2 + z2 = u2. The theorem of Gauss yields:

I = (SI f?~d~=///div?dv=3///(x2+yZ+z2)dxdydz=3 (VI

7]]

‘p=O

19=0 r=O

12 r 4 s i n d d r d d d p = -u577. 5

IC: Heat conduction equation: The change in time of the heat Q of a space region u containing no dv (specific heat-capacity e, density

heat source is given by

e, temperature T ) .while

dQ the corresponding time-dependent change of the heat flow through the surface S of v is given by - = dt X grad T . dg (thermal conductivity A). Applying the theorem of Gauss for the surface integral we

//

(SI

get from

///

[

C

Q

I

lv)

which has the form

dT

-~ div(XgradT) dv = 0 the heat conduction equation cX-

dT

-=

at

at

=

div(XgradT),

u2AT in the case of a homogeneous solid (e, e, X constants)

13.4 Evaluation of Fields 13.4.1 Pure Source Fields We call a field ?, a pure source field or an irrotational wurce field when its rotation is equal to zero everywhere. If the divergence is q(F), then we have: d i v g l = q(F),

- - .

rotV1 = 0.

(13.126)

666

13. Vector Analvsis and Vector Fields

In this case, the field has a potential U ,which is defined at every point P by the Poisson daflerential equation (see 13.5.2, p. 668)

?, = grad C;

div grad U = AU = q(r'),

(13.127a)

where r' is the position vector of P . (In physics, 31 = -grad U is used.) The evaluation of U comes from (13.127b) The integration is taken over the whole of space (Fig.13.19). The divergence of? must be differentiable and be decreasing quickly enough for large distances.

m1 or

m2or

41

Figure 13.19

e

Figure 13.20

13.4.2 Pure Rotation Field or Zero-Divergence Field A pure rotatzon (or curl) field or a solenozdalJeld is a vector field 52 whose divergence is equal to zero everywhere. If the rotation is G(r'),then we have: divqz = 0, rotqz = .i(r'). ( 13.128a) The rotation $(?) cannot be arbitrary: it must satisfy the equation divui = 0. With the assumptions ?,(r')=rotb;(?), divb;=O, we get according to (13.95)

Le., r o t r o t b ; = G

grad div b; - Ab; = G , Le., AX = -G.

(13.128b) (13.128~)

So, A(?) formally satisfies the Poisson differential equation just as the potential U of an irrotational and that is why it is called a vector potentzal. For every point P : field 1 G(P) (13.128d) 3, = rot b; with A = dv(r").

..

///

The meaning of r'is the same as in (13.127b); the integration is taken over the whole of space.

13.4.3 Vector Fields with Point-Like Sources 13.4.3.1 Coulomb Field of a Point-Like Charge The Coulomb field is an important example of an irrotational field, which is also solenoidal, except at the location of the point charge, the point source (Fig. 13.20). The field and potential equations are: +

E

E=-$, i-3

v=-! r

in physics also

2. r

(13.129a)

13.5 Diflerential Equations of Vector Field Theoru 667

The scalar flow is h e or 0, depending on whether the surface S encloses the point source or not: 4re, if S encloses the point source, (13.129b) E* , dg = 0; otherwise.

f

(SI

{

The quantity e is the source zntenszty or source strength.

13.4.3.2 Gravitational Field of a Point Mass The field of gravity of a point mass is the second example of an irrotational and at the same time solenoidal field, except at the center of mass. We also call it the Newton field. Every relation valid for the Coulomb field is valid analogously also for the Newton field.

13.4.4 Superposition of Fields 13.4.4.1 Discrete Source Distribution Analogously to the superposition of the fields of physics, the vector fields of mathematics superpose each other. The superposztzon law is: If the vector fields 5, have the potentials U,, then the vector field ? = E?" has the potential U = ELr,. For n discrete point sources with source strength e, (v = 1.2,. . . , n),whose fields are superposed, the resulting field can be determined by the algebraic sum of the potentials U,: n

?(3) = -grad

"=I

Vu

with

e, U, = -

l?-?"l'

(13.130a)

Here. the vector r' is again the position vector of the point under consideration, FV are the position vectors of the sources. If there is an irrotational field ?I and a zero-divergence field 5, together and they are everywhere continuous. then (13.130b) If the vector field is extended to infinity, then the decomposition of q(?) is unique if 1?(3)1 decreases sufficient rapidly for T = + co. The integration is taken over the whole of space.

13.4.4.2 Continuous Source Distribution If the sources are distributed continuously along lines, surfaces, or in domains of space, then, instead of the finite source strength e,,, we have infinitesimals corresponding to the density of the source distributions, and instead of the sums, we have integrals over the domain. In the case of a continuous space distribution of source strength, the divergence is q(3) = div?. Similar statements are valid for the potential of a field defined by rotation. In the case of a continuous space rotation distribution, the 'I rotation denszty " is defined by G(?)= rot 3.

13.4.4.3 Conclusion A vector field is determined uniquely by its sources and rotations in space if all these sources and rotations lie in a finite space.

13.5 Differential Equations of Vector Field Theory 13.5.1 Laplace Differential Equation The problem to determine the potential U of a vector field 3,= grad U containing no sources, leads to the equation according to (13.126) with q(F) = 0 div

= div grad C = 41:= 0,

(13.131a)

668

13. VectorAnalysis and Vector Fields

Le.. to the Laplace diflerential equation. In Cartesian coordinates we have: (13.131b)

Every function satisfying this differential equation and which is continuous and possesses continuous first and second order partial derivatives is called a Laplace or harmonzcfunctzon. We distinguish three basic types of boundary value problems: 1. Boundary value problem (for an interior domain) or Dirichlet problem: We determine a function U ( z ,y. z ) which is harmonic inside a given space or plane domain and takes the given values at the boundary of this domain. 2. Boundary value problem (for an interior domain) or Neumann problem: We determine a function

dU

U ( x ,y, z ) $which is harmonic inside a given domain and whose normal derivative - takes the given an values at the boundary of this domain. 3. Boundary value problem (for an interior domain): We determine a function U ( x ,y, z ) , which is harmonic inside a given domain and the expression dli aU + ,!I-- ( a ,B const, (Y’ p2 # 0 ) takes given values at the boundary of this domain. dn

+

13.5.2 Poisson Differential Equation The problem to determine the potential U of a vector field to the equation according to (13.126) with q(?) # 0

= grad U with given divergence, leads

(13.132a) d i v v l = div grad U = AU = q(?) # 0, Le., to the Poisson diflerential equation. Since in Cartesian coordinates: d2U d2U d2U A v = -+--+ (13.132b) 622 ’ 8x2 a y 2 the Laplacedifferential equation (13.131b)isa specialcase of the Poisson differential equation (13.13213). The solution is the Newton potential (for point masses) or the Coulomb potential (for point charges) (13.1324 The integration is taken over the whole of space. U(?) tends to zero sufficiently rapidly for increasing values. We can discuss the same three boundary value problems for the Poisson differential equation as for the solution of the Laplace differential equation. The first and the third boundary value problems can be solved uniquely: for the second one we have to prescribe further special conditions (see [9.5]).

669

14 Function Theory 14.1 Functions of Complex Variables 14.1.1 Continuity, Differentiability 14.1.1.1 Definition of a Complex Function .4nalogously to real functions, we can assign complex values to complex values, i.e., to the value z = z t i y we can assign a complex number w = u t i v, where u = u (z ,y) and v = v(z, y) are real functions of two real variables. We then write w = f(z). The function w = f ( z ) is a mapping from the complex z plane to the complex 20 plane. The notions of limit, continuity, and derivative of a complex function w = f ( z ) can be defined analogously to real functions of real variables.

14.1.1.2 Limit of a Complex Function The h u t of a function f ( z ) is equal to the complex number wg if for z approaching zg the value of the function f ( z ) approaches wg: 200 = f(2). (14.la) In other words: For any positive E there is a (real) 6 such that for every z satisfying (14.1b), except maybe zo itself, the inequality (14.1~)holds: 120 - zl < 6, (14.lb) Iwo - f(z)l < E . (14.1~) The geometrical meaning is as follows: .4ny point z in the circle with center zo and radius 6, except maybe the center zo itself, is mapped into a point w = f ( z ) inside a circle with center wg and radius E in the w plane where f has its range, as shown in Fig. 14.1.The circles with radii 6 and e are also called the neighborhoods UE(wg)and UJ(Z0).

Figure 14.1

14.1.1.3 Continuous Complex Functions .A function w = f ( z ) is continuous at zo if it has a limit there, a substitution value, and these two are equal, Le., if for an arbitrarily small given neighborhood U,(wg) of the point wg = f(zg) in the w plane there exists a neighborhood UJ(Z0) of zg in the z plane such that w = f ( z ) E UE(wg) for every z E Ud(z0). As represented in Fig. 14.1,U,(wo) is, e.g., a circle with radius E around the point wo.We then write lim f ( z ) = f ( z o ) or limf(z0 6) = f ( z 0 ) . (14.2) 610

t i 2 0

+

14.1.1.4 Differentiability of a Complex Function A function w = f ( z ) is differentiable at z if the difference quotient AW-f(~+Az)-f(~) A2

A2

(14.3)

670

14. Function Theoru

has a limit for A z + 0, independently of how A z approaches zero. This limit is denoted by f’(z) and it is called the derivative of f(z). IThe function f ( z ) = Re z = x is not differentiable at any point z = 20, since approaching zo parallel to the z-axis the limit of the difference quotient is one, and approaching parallel to the y-axis this value is zero.

14.1.2 Analytic Functions 14.1.2.1 Definition of Analytic Functions .4 function f ( z ) is called analytzc, regularor holornorphzcon a domain G, if it is differentiable at every point of G. The boundary points of G, where f’(z) does not exist, are singular points of f(z). The function f ( z ) = u(x. y)+iv(x, y) is differentiable in G if u and u have continuous partial derivatives in G with respect to z and y and they also satisfy the Cauchy-Riemann dzflerential equations:

all

-au_ - -av

av

_ -- _ (14.4) ax dy ’ dy dx’ The real and imaginary parts of an analytic function satisfy the Laplace differential equation: 8%

6221

A u(z, y) = + - = 0, 8x2 dy2

8% 8% A V ( X , y) = -+ - = 0.

(14.5a)

(14.5b) 8x2 dy2 The derivatives of the elementary functions of a complex variable can be calculated with the help of the same formulas as the derivative of the corresponding real functions. IA: f ( z ) = z3. f‘(z)= 32’; IB: f ( z ) = sinz, f ’ ( z ) = cosz.

14.1.2.2 Examples of Analytic Functions 1. Elementary Functions The elementary algebraic and transcendental functions are analytic in the whole z plane except at some isolated singular points. If a function is analytic on a domain, Le., it is differentiable, then it is differentiable arbitrarily many times. IA: The function w = z2 with u = x2 - y2, v = 21y is everywhere analytic IB: The function w = u + iv, defined by the equations u = 2s + y I v = I + 2y, is not analytic at any point. IC: The function f ( z ) = z3 with f ’ ( z ) = 32’ is analytic. ID: The function f ( z ) = sinz with f ’ ( z ) = cosz is analytic.

2. Determination of the Functions u and v If both the functions u and v satisfy the Laplace differential equation, then they are harmonic functions (see 13.5.1, p. 667). If one of these harmonic functions is known, e.g., u , then the second one. as the conjugate harmonic function u, can be determined up to an additive constant with the CauchyRiemann differential equations:

1

dU

- dy dX

+ q(x)

dq

(- + 1 du

with - = dz dy .4nalogously u can be determined if w is known.

v=

d dx

du

dy)

(14.6)

14.1.2.3 Properties of Analytic Functions 1. Absolute Value or Modulus of an Analytic Function The absolute value (modulus) of an analytic function is: (14.7) IW = l f ( z ) l = JMz, Y I P + >.(I Y)12 = v(x, Y). The surface IUJ = p(z.y) is called its relzef, Le., 1.u is the third coordinate above every point z = x+i y.

14.1 Functions of Complex Variables 671

IA: Theabsolutevalueofthefunctionsinz = sinzcoshyticoszsinhyisIsinzl = dsin2z +sinhZy. The relief is shown in Fig. 14.2a. IB: The relief of the function w = e'/' is shown in Fig. 14.2b. For the reliefs of several analytic functions see [14.10].

Figure 14.2

2. Roots Since the absolute value of a functions is never negative. the relief is always above the z plane, except the points where lf(z)l = 0 holds, so f ( z ) = 0. The z values, where f(z) = 0 holds. are called the roots of the function f ( z ) .

3. Boundedness A4function is bounded on a certain domain, if there exists a positive number N such that If(z)I < N is valid everywhere in this domain. In the opposite case, if no such number N exists, then the function is called unbounded. 4. Theorem about the Maximum Value If u, = f ( z ) is an analytic function on a closed domain. then the maximum of its absolute value is attained on the boundary of the domain. 5 . Theorem about the Constant (Theorem of Liouville) If w = f ( z ) is analytic in the whole plane and also bounded, then this function is a constant: f ( z ) = const.

14.1.2.4 Singular Points If a function w = f ( z ) is analytic in a neighborhood of z = a, Le., in a small circle with center a, except a itself, then f has a singularity at a. There exist three types of singularities: 1. f ( z ) is bounded in the neighborhood. Then there exists w = l i i f ( z ) , and setting f ( a ) = w the function becomes analytic also at a. In this case. f has a removable singularzty at a. 2. If i$lf(z)l = ea,then f has a pole at a. About poles of different orders see 14.3.5.1,p. 691 3. Iff has neither a removable singularity nor a pole, then f has an essentzal szngularity. In this case, for any complex w there exists a sequence zn+ a such that f(zn) + w.

IA: The function w =

z-a

has a pole at a.

672

14. Function Theory

IB: The function w = el/* has an essential singularity at 0 (Fig. 14.2b).

14.1.3 Conformal Mapping 14.1.3.1 Notion and Properties of Conformal Mappings 1. Definition .4mapping from the z plane to the w plane is called a conformal mapping if it is analytic and injective. In this case,

w = f ( z ) = u + iv, f'(z) # 0. The conformal mapping has the following properties:

(14.8)

The transformation dw = f ' ( z ) dz of the line element dz =

($1

is the composition of a dilatation by

u = If'(z)l and of a rotation by cu = argf'(z). This means that infinitesimal circles are transformed into almost circles, triangles into (almost) similar triangles (Fig. 14.3). The curves keep their angles of intersection, so an orthogonal family of curves is transformed into an orthogonal family (Fig. 14.4).

v

J

V

u Figure 14.4

Figure 14.3

Remark: Conformal mappings can be found in physics, electrotechnics, hydro- and aerodynamics and in other areas of mathematics.

2. The Cauchy-Riemann Equations The mapping between d z and dw is given by the affine differential transformation (14.9a) and in matrix form

dw=Adz

with

A=(uzuy).

(14.9b)

vx vy

According to the Cauchy-Riemann differential equations, A has the form of a rotation-stretching matrix (see 3.5.2.2, 2., p. 191) with CT as the stretching factor: (14.loa)

A = ( ~ ~ ~ ~ ) = u ( E Y ~ E - ~ ~ E ) , 21,

= v u = UCOSQ,

-uy = u, = usincu,

(14.10b) (14.10d)

4-

m,

CT

= If'(z)l =

(Y

= arg f'(z) = arg (us ius) .

=

+

(14.10~) (14.10e)

3. Orthogonal Systems The coordinate lines x = const and y = const of the z plane are transformed by a conformal mapping into two orthogonal families of curves. In general, we can generate a bunch of orthogonal curvilinear

14.1 Functions of Complex Variables 673

coordinate systems by analytic functions; and conversely, for every conformal mapping there exist an orthogonal net of curves which is transformed into an orthogonal coordinate system. IA: In the case of = 22 t y; u = I + 2y (Fig. 14.5),orthogonality does not hold. 0

Figure 14.5

IB: In the case w = tz the orthogonality is retained, except at the point t = 0 because here w’ = 0. The coordinate lines are transformed into two confocal families of parabolas (Fig. 14.6), the first quadrant of the z plane into the upper half of the w plane.

“t

’/ b)

\

\

Figure 14.6

14.1.3.2 Simplest Conformal Mappings In this paragraph, we discuss some transformations with their most important properties, and we give the graph of their zsometric net in the z plane, Le., the net which is transformed into an orthogonal Cartesian net in the w plane. The boundaries of the z regions mapped into the upper half of the w plane are denoted by shading. Black regions are mapped onto the square in the w plane with vertices ( O , O ) , (0,l ) , ( l , O ) , and (1,l)by the conformal mapping (Fig. 14.7).

1. Linear Function For the conformal mapping given in the form of a linear function

2c = az + b, the transformation can be done in three steps: a) Rotation of the plane by the angle a = arga according to : b) Stretching by the factor lal: c) Parallel translation by b:

( 14.1la)

w1 = einz. w2 = lalw1.

(14.11b)

w = w 2 + b.

b Altogether, every figure is transformed into a similar one. The points z1 = m and 22 = -for a # 1, 1-a b # 0 are transformed into themselves, and they are calledjxedpoints. Fig. 14.8 shows the orthogonal

674

14. Function Theory

net which is transformed into the orthogonal Cartesian net.

B-

O

Figure 14.8

Figure 14.7

Figure 14.9

2. Inversion The conformal mapping 1

w=-

(14.12)

2

represents an inversion with respect to the unit circle and a reflection in the real axis, namely, a point z of the z plane with the absolute value r and with the argument 'pis transformed into a point w of the w plane with the absolute value l / r and with the argument -9 (see Fig. 14.10).Circles are transformed into circles, where lines are considered as limitingcases of circles (radius + co).Points of the interior of the unit circle become exterior points and conversely (see Fig. 14.11).The point z = 0 is transformed into w = co.The points z = -1 and z = 1 are &xed points of this conformal mapping. The orthogonal net of the transformation (14.12) is shown in Fig. 14.9. Remark: In general a geometric transformation is called inversion with respect to a circle with radius r , when a point Pz with radius r2 inside the circle is transformed into a point PI on the elongation of

+

the same radius vector OPz outside of the circle with radius of the circle become exterior points and conversely.

Figure 14.10

= 71 = T

Points of the interior

~ / T ~ .

Figure 14.11

3. Linear Fractional Function For the conformal mapping given in the form of a linear fractional function XI

=

az+b cz d

~

+

(14.13a)

14.1 Functions of Complex Variables 675

the transformation can be performed in three steps: a) Linear function: w1 = cz + d.

b) Inversion: c)

Linear function:

wz=-.

1

(14.13b)

w1 a bc-ad w = - + -W2. c c

Circles are transformed again into circles (circular transformation), where straight lines are considered as limiting cases of circles with r + M. Fixed points of this conformal mapping are the both points satisfying the quadratic equation az+b z=(14.14) cztd If the points z1 and z2 are inverses of each other with respect to a circle K1 of the z plane, then their images w1and w2 in the w plane are also inversions of each other with respect to the image circle Kz of K1. The orthogonal net which has the orthogonal Cartesian net as its image is represented in Fig. 14.12.

Figure 14.12

Figure 14.13

Figure 14.14

4. Quadratic Function The conformal mapping described by a quadratic function w = 22

has the form in polar coordinates and as a function of 2 and y: = p2

@,

(14.15b)

w = u + iu = 2 - y2 + 2izy.

(14.15~)

It is obvious from the polar coordinate representation that the upper half of the z plane is mapped onto the whole w plane, Le., the whole image of the z plane will cover twice the whole w plane. The representation in Cartesian coordinates shows that the coordinate lines of the w plane, u = const and w = const. come from the orthogonal families of hyperbolas x2 - yz = u and 2x21 = w of the z plane (Fig. 14.13). Fixed points of this mapping are z = 0 and z = 1. This mapping is not conformal at z = 0. 5 . Square Root The conformal mapping given in the form as a square root of z , w=&, (14.16) transforms the whole z plane whether onto the upper half of the w plane or onto the lower half of it. Le., the function is double-valued. The coordinate lines of the w plane come from two orthogonal families of confocal parabolas with the focus at the origin of the z plane and with the positive or with the negative

676

14. Function Theoru

real half-axis as their axis (Fig. 14.14). Fixed points of the mapping are z = 0 and z = 1. The mapping is not conformal at z = 0. 6 . Sum of Linear and Fractional Linear Functions The conformal mapping given by the function

w=

(z

t

!-)

( k a real constant, k

> 0)

(14.17a)

can be transformed by the polar coordinate representation of z = p e ' q and by separating the real and imaginary parts according to (14.8): (14.17b) Circles with p = po = const of the z plane (Fig. 14.15a) are transformed into confocal ellipses u2 2 -+-=1 with (14.17~) a* b2 in the w plane (Fig. 14.15b). The foci are the points f k of the real axis. For the unit circle with p = po = 1 we get the degenerate ellipse of the w plane, the double segment (-k, + k ) of the real axis. Both the interior and the exterior of the unit circle are transformed onto the entire w plane with the cut (-kl t k ) , so its inverse function is double-valued: z=

uJ+@7?

(14.17d)

k

The lines cp = ipo of the z plane (Fig. 14.15~)become confocal hyperbolas u2 v2 (14.17e) _ -= 1 with (Y = kcoscpo, @ = ksinipo (Y2 32 with foci kk (Fig. 14.15d). The hyperbolas corresponding to the coordinate half-axis of the z plane (p = 0. 5.K , 2~ 3 , are degenerate in the axis u = 0 (v axis) and in the intervals (-m, -k) and (k, m) of the real axis running there and back.

Figure 14.15

7. Logarithm The conformal mapping given in the form of the logarithm function w=Lnz (14.18a) has the form for z given in polar coordinates: (14.18b) u = l n p , z=cp+2k.ir ( k = O , & l , f 2,...). We can see from this representation that the coordinate lines u = const and v = const come from concentric circles around the origin of the z plane and from rays starting at the origin of the z plane

14.1 Functions of Complex Variables 677

VI

Figure 14.16

Figure 14.16

(Fig. 14.16).The isometric net is a polar net. The logarithm function Lnz is infinitely many valued (see (14.74c), p. 697). If we restrict our investigation to the principal value In z of Lnz (-T < v 5 +T)>then the whole z plane is transformed into a stripe of the w plane bounded by the lines v = hi7,where w = i7 belongs to the stripe.

8. Exponential Function The conformal mapping given in the form of an exponential function (see also 14.5.2, l.,p. 696) w = e' (14.19a) has the form in polar coordinates: w = pel'. (14.19b) Wegetfromz=z+iy: p = e' and ~5 = y. (14.19~) If y changes from -T to + T ~and z changes from --x to + c q then p takes all values from 0 to m and $ from -T to s. .4 2i7 wide stripe of the z plane, parallel to the x-axis, will be transformed into the entire 'w plane (Fig. 14.17).

Y4

Figure 14.17

9. The Schwarz-Christoffel Formula By the Schwarz-Christoffel formula = c1

/

( t - W l ) W ( t - dt w p ( t - W,)""

+ c2

(14.20a)

the interior of a polygon of the z plane can be mapped onto the upper half of the w plane. The polygon has n exterior angles a l ~a, p , . . . , a , (Fig. ~ 14.18a,b).We denote by w,(i = 1 , .. . n) the points of the real axis in the 11: plane assigned to the vertices of the polygon, and by t the integration variable. The oriented boundary of the polygon is transformed into the oriented real axis of the w plane by this mapping.

678

1L. Function Theoru

Figure 14.18

For large values oft, the integrand behaves as l/t2 and is regular at infinity. Since the sum of all the exterior angles of an n-gon is equal to 2 ~we, get:

2

Cy”

= 2.

(14.20b)

V=l

The complex constants C1 and Cz yield a rotation, a stretching and a translation; they do not depend on the form of the polygon, only on the size and the position of the polygon in the z plane. Three arbitrary points 2c1,w2, w3 of thew plane can be assigned to three points zlr ~2~ 23 of the polygon in the z plane. If we assign a point at infinity in the w plane, Le., lul = *m to a vertex of the polygon in the z plane, e.g., to z = 21, then the factor (t - 2 ~ 1 ) ~is’ omitted. If the polygon is degenerate, e.g., a vertex is at infinity, then the corresponding exterior angle is T , so aCc= 1, Le., the polygon is a half-stripe.

Figure 14.19

IA: We want to map a certain region of the z plane. It is the shaded region in Fig. 14.19a. Considering a, = 2 we choose three points as the table shows (Fig. 14.19a,b). The formula of the mapping is: 2d dt = 2c1(fi- arctan f i )= i- (fi - arctan fi). For the determination of C1 we substitute t = pe’p - 1: id = C1 limpiO

s.0

(-1

A

c + pe’p) ’” ipe’pdp pel?

.d

C~T. Le.. C1 = I - . R

We get the constant C2 = 0 considering that the mapping assigns “z = 0 + w = 0”. IB: Mapping of a rectangle. Let the vertices of the rectangle be t1,4 = *K,~ 2 , 3= iK iK’. The points z1 and 22 should be transformed into the points w1 = 1 and w2 = l / k (0 < k < 1) of the real axis, z4 and 23 are reflections of z1 and z2 with respect to the imaginary axis. According to the Schwarz reflection principle (see 14.1.3.3, p. 679) they must correspond to the points w4 = -1 and w3 = -l/k (Fig. 14.20a,b). So, the mapping formula for a rectangle (a1= az = a3 = = 1/2) of the position

+

14.1 Functions of Complex Variables 679

“f

Figure 14.20 sketched above is: z = Ci

1%

dt

f i t - W l ) ( t - wz)(t - wWQ)(t - Wq)

dt

=cdw /&$q The

point z = 0 has the image w = 0 and the image of z = i K is w = m. With C1 = l / k we get 7 d8 dt = F ( 9 , k) (substituting t = sint9, w = sincp). ’= v’l- k2sin2t9

lw d w

~1 =

F ( p lk) is the elliptic integral of the first kind (see 8.1.4.3,p. 436). We get the constant Cz = 0 considering that the mapping assigns z = 0 -+ w = 0 ”.

14.1.3.3 The Schwarz Reflection Principle 1. Statement Suppose f ( z ) is an analytic complex function in a domain G, and the boundary of G contains a line segment gl. If the function is continuous on g1 and it maps the line g1 into a line g;, then the points lying symmetric with respect to g1 are transformed into points lying symmetric with respect to g; (Fig. 14.21).

Figure 14.21

Figure 14.23

2. ADDlication 1 1

The application of this principle makes it easier to perform calculations and the representations of plane regions with straight line boundaries: If the line boundary is a stream line (isolating boundary in Fig. 14.22).then the sources are reflected as sources, the sinks as sinks and curls as curls with the opposite sense of rotation. If the line boundary is a potential line (heavy conducting boundary in Fig. 14-23), then the sources are reflected as sinks, the sinks as sources and curls as curls with the same sense of rotation.

14.1.3.4 Complex Potential 1. Notion ofthe Complex Potential We will consider a field ? = $(z, y) in the z, y plane with continuous and differentiable components V,(s, y) and V,(z,y) of the vector ? for the zero-divergence and the irrotational case. a) Zero-divergence field with div ? = 0. i.e., % + % = 0: That is the integrability condition ax ay for the differential equation expressed with the field or stream function !P(x2y)

I

680

14. Function Theoru

dly = -Vy dx t V, dy = 0, (14.21a)

a*

and then V, = - , ay

V

- --a* . -

ax

(14.21b)

For two points PI Pz of the field ? the difference !P(Pz)- *(PI)is a measure of the flux through a curve connecting the points PI and Pz, in the case when the whole curve is in the field. ~

av av,

b) Irrotational field with rot? = 6,Le., 2 - - = 0: That is the integrability condition for the ax a y differential equation with the potential function @(z,y) d@ = r/,dx t V, dy = 0, (14.22a)

a@

and then V, = %,

a@

V - -. ydy

(14.22b)

The functions @ and P satisfy the Cauchy-Riemann differential equations (see 14.1.2.1, p. 670) and both satisfy the Laplace differential equation (A @ = 0, A P = 0). We combine the functions 0 and @ into the analytic function (14.23) IV = f(z) = @(x,y) t ily(z,y ) ) and this function is called the complex potential of the field ?. Then -@(x3y) is the potential of the vector field ? in the sense of the usual notation in physics and electrotechnics (see 13.3.1.6, 2., p. 660). The level lines of P and @forman orthogonal net. We have the following equalities for the derivative of the complex potential and the field vector ?: dW d@ ,d@ _ - - _ l-=vv,-iV (14.24) dW = f’(z)= V, +ivy. dz dx dy ” dz

2. Complex Potential of a Homogeneous Field The function W = az. (14.25) with real a is the complex potential of a field whose potential lines are parallel to the y-axis and whose direction lines are parallel to the x-axis (Fig. 14.24). A complex a results in a rotation of the field (Fig. 14.25).

@=const Figure 14.24

@=const Figure 14.25

3. Complex Potential of Source and Sink The complex potential of a field with a strength of source s > 0 at the point

w = S27rl n ( z - 20).

2

= 20 has the equation:

(14.26)

A sink with the same intensity has the equation:

w = -5 ln(z - 20). 277

(14.27)

14.1 Functions of Complex Variables 681

The direction lines run away radially from z = 20,while the potential lines are concentric circles around the point 20 (Fig. 14.26).

4. Complexes Potential of a Source-Sink System We get the complex potential for a source at the point zl and for a sink at the point 22, both having the same intensity, by superposition s

M.7

= -In 277

2-21

-

(14.28)

2-22

The potential lines @ = const form Apollonius circles with respect to I = const are circles through 21 and z2 (Fig. 14.27).

Figure 14.26 5 . Complex Potential of a Dipole The complex potential of a dipole with dipole moment M angle a with the real axis (Fig. 14.28),has the equation:

z1

and

22;

the direction lines

Figure 14.27

> 0 at the point 20,whose axis encloses an (14.29)

6 . Complex Potential of a Curl If the intensity of the curl is Irl with real r and its center is at the point 20,then its equation is:

r

W = -ln(z - zo). 27ri

(14.30)

In comparison with Fig. 14.26,the roles of the direction lines and the potential are interchanged. For complex r (14.30) gives the potential of a source of curl, whose direction and potential lines form two families of spirals orthogonal to each other (Fig. 14.29).

14.1.3.5 Superposition Principle 1. Superposition of Complex Potentials A system composed of several sources, sinks, and curls is an additive superposition of single fields, Le., we get its function by adding their complex potential and stream functions. hlathematically this is possible because of the linearity of the Laplace differential equations A @ = 0 and D I = 0.

2. Composition of Vector Fields 1. Integration A new field can be constructed from complex potentials not only by addition but also by integration of the weight functions. ILet a curl be given with density e(s) on a line segment 1. Then we get a Cauchy type integral (see

682

14. Function Theory

Yt

\

I

+

I

0)

X

Figure 14.28

I

Figure 14.29

14.2.3, p. 687) for the derivative of the complex potential:

(14.31) where C ( s ) is the complex parametric representation of the curve 1 with arclength s as its parameter. 2. Maxwell Diagonal Method If we want to compose the superposition of two fields with the potentials G1 and 12,then we draw the potential line figures [[11]] and [[12]] so that the value of the potential changes by the same amount h between two neighboring lines in both systems, and we direct the lines so that the higher 1 values are on the left-hand side. The lines lying in the diagonal direction to the net elements formed by [[11]] and [[12]] give the potential lines of the field whose potential is 1 = 11 t 12 or 1 = 11 - 12. We get the figure of [[@I + 1211 if the oriented sides of the net elements are added as vectors (Fig. 14.30a), and we get the figure of [[GI -1211 when we subtract them (Fig. 14.30b). The value of the composed potential changes by h at transition from one potential line to the next one. W Vector lines and potential lines of a source and a sink with an intensity quotient lelI/le21 = 3/2 (Fig. 14.31a,b).

[@I],

Figure 14.30

14.1.3.6 Arbitrary Mappings of the Complex Plane .4 function (14.32a) w = f ( z = z + i y) = u ( z ,y) + i v(z, y) is defined if the two functions u = u ( z ,y) and u = u(z,y) with real variables are defined and known. The function f ( z ) must not be analytic, as it was required in conformal mappings. The function to maps the z plane into the w plane, Le., it assigns to every point z, a corresponding point wv. a) Transformation of the Coordinate Lines z is a parameter; y = c t u = u ( z ,c), v = v(z, c),

11.2 Inteqration in the Complex Plane 683

Figure 14.31 y is a parameter. (14.32b) z = e1 i u = u(cl, y), v = v(cll y). b) Transformation of Geometric Figures Geometric figures as curves or regions of the z plane are usually transformed into curves or regions of the w plane: z = ~ ( t )y. = y(t) -iu = u ( z ( t ) , y ( t ) ) ,w = v(z(t),y(t)), t i s aparameter. (14.32~) IFor u = 2 2 + y, v = I + 2y, the lines y = c are transformed into u = 22 t c, v = z + 2c, hence into u 3 the lines v = - + -c. The lines I = c1 are transformed into the lines v = 221 - 3Cl (Fig. 14.5). The 2 2 shaded region in Fig. 14.5a is transformed into the shaded region in Fig. 14.5b. c) Riemann Surface If we get the same value w for several different z for the mapping w = f ( z ) then , the image of the function consists of several planes “lying on each other”. If we cut these planes and connect them along a curve, then we get a many-sheeted surface, the so-called many-sheeted Rzemann surface (see [14.13]).This correspondence can be considered also in an inverse relation. in the case of multiple-valued functions as, e.g., the functions fi,Ln z , Arcsin t.Arctan z .

Figure 14.32

Iw = 2 :While z = reip overruns the entire z plane, Le., 0 5 ~p < 27~,the values of w = eelv = rzei2p,cover twice the w plane. We can imagine that we place two w planes on each other, and we cut and connect them along the negative real axis according to Fig. 14.32. This surface is called the Riemann surface of the function w = 2’.

The zero point is called a branch point. The image of the function ez (see 14.69) is a Riemann surface of infinitely many sheets. (In many cases the planes are cut along the positive real axis. It depends on whether the principal value of the complex number is defined for the interval (-n>+T] or for the interval [O, 27r).)

14.2 Integration in the Complex Plane 14.2.1 Definite and Indefinite Integral 14.2.1.1 Definition of the Integral in the Complex Plane 1. Definite Complex Integral Suppose f ( z ) is continuous in a domain G, and the curve C is rectifiable, it connects the points A and B , and the whole curve is in this domain. We decompose the curve C between the points A and B by

684

14. Function Theory

choose a point CZ on every arc segment and form the sum

G

n

x f ( C , ) A z , with Az,=z,-z,-1.

(14.33a)

$=I

If the limit n+cc

F(z)=

I

f ( z ) dz t C with F'(z) = f ( z ) .

(14.35)

Here C is the integration constant which is complex, in general. The function F ( z ) is called an indefinite complex integral. The indefinite integrals of the elementary functions of a complex variable can be calculated with the same formulas as the integrals of the corresponding elementary function of a real variable.

w

A: / s i n z d z = - c o s z t C .

IB: J e z d z = e' t C.

3. Relation between Definite and Indefinite Complex Integrals If the function f ( z ) has an indefinite integral, then the relation between its definite and indefinite integral is f ( z ) dz = F ( z B )- F ( z a ) .

(14.36)

4B

14.2.1.2 Properties and Evaluation of Complex Integrals 1. Comparison with the curvilinear integral of the second type The definite complex integral has the same properties as the curvilinear integral of the second type (see 8.3.2. p. 462): a ) If we reverse the direction of the path of integration, then the integral changes its sign. b) If we decompose the path of integration into several parts, then the value of the total integral is the sum of the integrals on the parts.

2. Estimation of the Value of the Integral If the absolute value of the function f ( z ) does not exceed a positive number M at the points z of the 7.

path of integration AB which has the length s, then: (14.37)

14.2 Inte.qration in the Complex Plane 685

3. Evaluation of the Complex Integral in Parametric Representation h

If the path of integration AB (or the curve C) is given in the form

z= 4 t ) , Y = y(t) (14.38) and the t values for the initial and endpoint are t A and t g , then the definite complex integral can be calculated with two real integrals. We separate the real and the imaginary parts of the integrand and we get: B

B

B

/

/( dz - v d y ) t i /(v d z t

A

A

( C ) f ( z )dz =

u

u dy)

A

B

B

= / [ u ( t ) z ' ( t) v(t)y'(t)] d t t i/[v(t)z'(t) A

+ u(t)y'(t)]dt

(14.39a)

A

with f ( z ) = u ( z , y) t iv(z, y),

z = z+ i y .

(14.39b)

B

/

The notation (C) f ( z ) dz means that the definite integral is calculated along the curve C between the A

points A and B. The notation

II =

/

-/

f ( z ) dz and f ( z ) dz is also often used, and has the same meaning. (i.) AB i c i ( z - z ~ ) ~ d( zn E Z). Let the curve C be a circle around the point zo with radius T O :

z = 2 0 + T O cost, y

= yo

t T O sin t with 0 5 t 5 27r. For every point z of the curve C: z = z + i y =

+ COS t t isint), dz = ro(-sin t + icost) dt. After substituting these values and transforming (cosnt + isin nt)(- sin t t i cos t ) d t according to the de Moivre formula we get: I = ritl zo

27

=

,it'

1

.~

2n

[icos(n

+ 1)t - sin(n t l)t]dt =

4. Independence of the Path of Integration Suppose a function of a complex variable is defined in a simply connected domain. The integral (14.34) of the function can be independent of the path of integration. which connects the fixed points A(zA) and B ( z ~ A ) .sufficient and necessary condition is that the function is analytic in this domain, i.e., the Cauchy-Riemann differential equations (14.4) are satisfied. Then also the equality (14.36) is valid. If a domain is bounded by a simple closed curve, then the domain is szmply connected. 5 . Complex Integral along a Closed Curve Suppose f ( z ) is analytic in a simply connected domain. If we integrate the function f ( z ) along a closed curve C which is the boundary of this domain, the value of the integral according to the Cauchy integral theorem is equal to zero (see 14.2.2, p. 686):

ff ( z ) dz = 0.

(14.40)

If f ( z ) has singular points in this domain, then the integral is calculated by using the residue theorem (see 14.3.5.5.p. 692), or by the formula (14.39a). 1 IThe function f ( z ) = -, with a singular point at z = a has an integral value for the closed curve z-a directed counterclockwise around a (Fig.14.34)

dz f= 2ai. z-a (C)

686

11. Function Theorv

14.2.2 Cauchy Integral Theorem 14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains If a function f ( z ) is analytic in a simply connected domain, then we get two equivalent statements: a) The integral is equal to zero along any closed curve C:

f f ( z ) dz = 0.

(14.41)

b) The value of the integral

LB

f ( z ) dz is independent of the curve connecting the points A and B , Le.,

it depends only on A and B. This is the Cauchy integral theorem

14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains If C , C1, Cz,. . ., C, are simple closed curves such that the curve C encloses all the C,, (v = 1 , 2 , . . . , n!, none of the curves C, encloses or intersects another one, and furthermore the function f(z) is analytic in a domain G which contains the curves and the region between C and C,,, i.e., at least the region shaded in Fig.14.35, then

f f ( z ) dz = f f ( z ) dz + f f ( z ) dz + . . . + f f ( z ) dz,

(14.42)

(C“)

(CZ)

(Cl)

(C)

if the curves C, C1, . . ., C, are oriented in the same direction, e.g., counterclockwise. This theorem is useful for the calculation of an integral along a closed curve C, if it also encloses singular points of the function f ( z ) (see 14.3.5.5, p. 692).

[c

* X

Figure 14.35

Figure 14.34

ICalculate the integral

Figure 14.36

fz(tz+:) dz, where C is a curve enclosing the origin and the point z = -1 (C)

(Fig. 14.36). Applying the Cauchy integral theorem, the integral along C is equal to the sum of the integrals along C1 and Cz, where C1 is a circle around the origin with radius r1 = l / 2 and Cz is a circle around the point z = -1 with radius r2 = 1/2. The integrand can be decomposed into partial fractions.

f -+ 2 dz

f

:f

-2 dz f ZS1 zs1 in1 (C2) (Cl) (Compare the integrals with the example in 14.2.1.2,3.,p. 685.)

Then we get:

f

(C)

2-1 dz= -

$=0+4ri-2ri-O=2ri.

(C2)

14.3 Power Series Expansion of Analytic Functions 687

14.2.3 Cauchy Integral Formulas 14.2.3.1 Analytic Function on the Interior of a Domain If f ( z ) is analytic on a simple closed curve C and on the simply connected domain inside it, then the following representation is valid for every interior point z of this domain (Fig. 14.37):

(Cauchy integral formula),

(14.43)

where C traces the curve C counterclockwise. With this formula, the values of an analytic function in the interior of a domain are expressed by the values of the function on the boundary of this domain. The existence and the integral representation of the n-th derivative of the function analytic on the domain G follows from (14.43): (14.44) Consequently. if a complex function is differentiable, Le., it is analytic, then it is differentiable arbitrarily many times. In contrast to this, differentiability does not include repeated differentiability in the real case. The equations (14.43) and (14.44) are called the Cauchy zntegralformulas.

14.2.3.2 Analytic Function on the Exterior of a Domain If a function f ( z ) is analytic on the entire part of the plane outside of a closed curve of integration C, then the values and the derivatives of the function f ( z ) at a point t of this domain can be given with the same Cauchy formulas (14.43), (14.44), but the orientation of the curve C is now clockwise (Fig. 14.38). .&o certain real integrals can be calculated with the help of the Cauchy integral formulas (see 14.4, p. 692),

Figure 14.37

Figure 14.38

14.3 Power Series Expansion of Analytic Functions 14.3.1 Convergence of Series with Complex Terms 14.3.1.1 Convergence of a Number Sequence with Complex Terms .An infinite sequence of complex numbers zir2 2 , . . . , z,, . . . has a limit z

(2 =

lim z,) if for arbitrarily

nics

given positive E there exists an no such that the inequality ) z - t,l < E holds for every n > nor Le., from a certain no the points representing the numbers z,, zntll.. . are inside of a circle with radius E and center at z . H If the expression { G}means the root with the smallest non-negative argument, then the limit lim { @} = 1 is valid for arbitrary a (Fig. 14.39). n+x

688

14. Function Theory

21y/a=z1 .2 .3 .4

i+ I +I + I

2 4 8 I

01

Figure 14.39

+ X

Figure 14.40

14.3.1.2 Convergence of an Infinite Series with Complex Terms

A series a1 + a2 + ' . t a, t . ' . with complex terms a, converges to the number s, if s = lim (al + a2 + . . . + an) n+x

(14.45)

holds. The number s is the sum of the series. If we connect the points corresponding to the numbers s, = a1 t a2 . . . an in the z plane by a broken line, then convergence means that the end of the broken line approaches the point s. iz i3 i2 i3 i4 IB: i + - + - + ' . . (Fig. 14.40). IA: i + - + - + - + . . . . 2 22 2 3 4 A series is called absolutely convergent (see IB), if the series of absolute values of the terms (all t laz/ + la31 + . . . is also convergent. The series is called conditionally convergent (see IA) if the series is convergent but not absolutely convergent. If the terms of a series are functions j , ( z ) , like fi(z)+f~(z)+'.'+fn(z)+.'. 2 (14.46) then its sum is a function defined for the values z for which the series of the substitution values is convergent.

+ +

14.3.1.3 Power Series with Complex Terms 1. Convergence A power series with complex coefficients has the form P(z-zo) = a o + a l ( z - z o ) + a ~ ( z - z o ) 2 + ~ ~ ~ + a , ( z - z o ) " + ~ ~ ~ , (14.47a) where zo is a fixed point in the complex plane and the coefficients a, are complex constants (which can also have real values). For zo = 0 the power series has the form

+

+ +

+

(14.47b) ~ ( z=)a. + alz azzZ . . . a,zn . . . . If the power series P ( z - zo) is convergent for a value zlrthen it is absolutely and uniformly convergent for every z in the interior of the circle with radius r = 121 - 201 and center at 20.

2. Circle of Convergence The limit between the domain of convergence and the domain of divergence of a complex power series is a uniquely defined circle. We determine its radius just as in the real case, if the imits T

1 = lim -

1%~

or

r = n+m lim a

n+m

(14.48)

exist. If the series is divergent everywhere except at z = zo, then T = 0; if it is convergent everywhere, then T = cu. The behavior of the power series on the boundary circle of the domain of convergence should be investigated point by point. IThe power series P ( z ) =

30

n=l

zn

- with radius of convergence r = 1 is divergent for z = 1 (harmonic n

14.3 Pourer Series Expansion of Analutic Functions 689

series) and convergent for z = -1 (according to the Leibniz criteria for alternating series). This power series is convergent for all further points of the unit circle = 1 except the point z = 1.

3. Derivative of Power Series in the Circle of Convergence Every power series represents an analytic function f(z) inside of the circle of convergence. We get the derivative by a term-by-term differentiation. The derivative series has the same radius of convergence as the original one.

4. Integral of Power Series in the Circle of Convergence We get the power series expansion of the integral

J1: f(C)

d6 by a term-by-term integration of the power

series of f ( z ) . The radius of convergence remains the same.

14.3.2 Taylor Series Every function f ( z ) analytic in a domain G can be expanded uniquely into a power series of the form m

f(z) =

an(z

(Taylor series),

- 201,

(14.49a)

n=O

for any zo in G, where the circle of convergence is the greatest circle around zo which belongs entirely to the domain G (Fig. 14.41). The coefficients a, are complex numbers in general, and for them we get:

f

'"'

(20)

(14.49b)

a, = -

n! The Taylor series can be written in the form

f //(zo1

+

f'"'(zo) ( z - Z O ) ~ . . . . (14.49~) 2! n! Every power series is the Taylor expansion of its sum function in the interior of its circle of convergence. IExamples of Taylor expansions are the series representations of the functions e', sin z , cos z , sinh z , and coshz in 14.5.2, p. 696. f(Z)

f'(z0)

=f(zo) t - ( z - z o ) + - ( z - z o ) 2 + ~ ' ~ + l!

Figure 14.41

Figure 14.43

Figure 14.42

14.3.3 Principle of Analytic Continuation We consider the case when the circles of convergence KO around zo and K1 around z1 of the two power series m

fo(z) =

1an(z n=O

m ~

0

)

and ~

fi(z) =

bn(z

-

ziIn

(14.50a)

n=O

have a certain common domain (Fig. 14.42)and in this domain they are equal:

fo(z) = f l ( 2 ) .

(14.50b)

690

14. Function Theorv

Then both power series are the Taylor expansions of the same analytic function f ( z ) , belonging to the points zo and 21. The function f l ( z ) is called the analytic continuation into K1 of the function fo(z) defined only in KO.

c zn m

IThe geometric series fo(z) = fi(2)

=

1

2 "

with the circle of convergence KO (TO = 1) around zo = 0 and

n=n . .

z-i

l-i (G)with the circle of convergence K1

(TI

=

4)around z1 = i have the same

analytic function f ( z ) = l / ( l - z ) as their sum in their own circle of convergence, consequently also on the common part of them (doubly shaded region in Fig. 14.42) for z # 1. So, f l ( z ) is the analytic continuation of fo(z)from KO into K1 (and conversely).

14.3.4 Laurent Expansion Every function f ( z ) , which is analytic in the interior of a circular ring between two concentric circles with center zo and radii r1 and ~ 2 can , be expanded into a generalized power series, into the so-called Laurent series: m a-ktl f(z)= a n ( z - 2 0 ) n = ~ ~ ~ a-k t(2 - Z0)k ( 2 - 2o)k-l n=-m

c

+

+

+-2 a-120 + a0 +

al(2

-

- 20)

+

a2(2

- 20)2

+.

' . +a&

- z0)k

+ ..

'

.

(14.51a)

The coefficients a, are usually complex and they are uniquely defined by the formula (14.51b) where C denotes an arbitrary closed curve which is in the circular ring T I < /zl < r2, and the circle with radius T I is inside of it, and its orientation is counterclockwise (Fig. 14.43).If the domain G of the function f ( z ) is larger than the circular ring, then the domain of convergence of the Laurent series is the largest circular ring with center zo lying entirely in G. IDetermine the Laurent series expansion of the function f ( z ) =

1

( z - 1)(z - 2)

. around to= 0 in

the circular ring 1 <

121 < 2 where f ( z ) is analytic. First we decompose the function f ( z ) into partial 1 1 fractions: f ( z ) = -- -. Since 11/21 < 1 and 12/2l < 1 holds in the considered domain, the

2 - 2

2-1

two terms of this decomposition can be written as the sums of geometric series absolutely convergent in the entire circular ring 1 < 121 < 2. We get:

14.3.5 Isolated Singular Points and the Residue Theorem 14.3.5.1 Isolated Singular Points If a function f ( z ) is analytic in the neighborhood of a point zo but not at the point zo itself, then 20 is called an isolated singular point of the function f ( z ) . If f ( z ) can be expanded into a Laurent series in the neighborhood of zo

c m

f(z) =

n=-m

ZOln,

(14.52)

14.3 Power Series Expansion of Analvtic Functions 691

then the isolated singular point can be classified by the behavior of the Laurent series: 1. If the Laurent series does not contain any term with a negative power of ( z - ~ 0 Le.. ) ~a, = 0 for n < 0 holds, then the Laurent series is a Taylor series with coefficients given by the Cauchy integral formula (14.53) In this case, the function f ( z ) itself is either analytic at the point zo and f ( z 0 ) = a0 or zo is a removable singularity. 2. If the Laurent series contains a finite number of terms with negative powers of ( z - 20))Le., a, # 0, a, = 0 for n < m < 0; then z0 is called a pole, apole of orderm, or a pole ofmultiplicity rn. If we multiply by ( z - z0),, and not by any lower power, then f ( z ) is transformed into a function which is analytic at zo and in its neighborhood.

:( 2-)

If ( z ) = - z t - has a pole of order one at z = 0. 3. If the Laurent series contains an infinite number of terms with negative powers of ( z - zO),then zo is an essential singularity of the function f(z). Approaching a pole, lf(z)l tends to co.Approaching an essential singularity, f ( z ) gets arbitrarily close to any complex number c. O011 IThe function f(z)= e'/'>whose Laurent series is f ( z ) = has an essential singularity at --)

n! zn

z = 0.

14.3.5.2 Meromorphic Functions If an otherwise holomorphic function has only a finite number of poles as singular points, then it is called meromorphic. A meromorphic function can always be represented as the quotient of two analytic functions. IExamples of functions meromorphic on the whole plane are the rational functions which have a finite number of poles, and also transcendental functions such as tan z and cot z .

14.3.5.3 Elliptic Functions Elliptic functions are double periodic functions whose singularities are poles, Le., they are meromorphic functions with two independent periods (see 14.6, p. 700). If the two periods are w1 and w2, which are in a non-real relation, then f(z

+ mwl t nu2) = f ( z )

(3

* 2 , . . . ; Im -

( m , n=

# 0).

The range of f ( z ) is already attained in a primitive period parallelogram with the points

(14.54) O , w l , w1

t

Q, w2.

14.3.5.4 Residue of the power ( z - zo)-l in the Laurent expansion of f(z) is called the residue ofthe The coefficient function f ( z ) at the point 20: (14.55a) The residue belonging to a pole of order m can be calculated by the formula 7

Am-l

( 14.55b)

692

14. Function Theorv

If the function can be represented as a quotient f ( z ) = cp(z)/$(z),where the functions cp(z) and $ ( z ) are analytic at the point z = zo and zo is a simple root of the equation $ ( z ) = 0, Le., $(zo) = 0, ~ ' ( 2 0 ) # 0 holds, then the point z = zo is a pole of order one of the function f ( z ) . It follows from (14.55b) that (14.55~)

) . . . = $("'-')(z~) If zo is a root of multiplicity m of the equation $ ( z ) = 0, Le., $(zo) = ~ ' ( z o= = 0,d * ) ( z o ) # 0 holds, then the point z = 20 is a pole of order m of f ( z ) .

14.3.5.5 Residue Theorem With the help of residues we can calculate the integral of a function along a closed curve enclosing isolated singular points (Fig. 14.44). If the function f ( z ) is single valued and analytic in a simply connected domain G except at a finite number of points 20, zll 2 2 , . , , zn, and the domain is bounded by the closed curve C, then the value of the integral of the function along this closed curve in a counterclockwise direction is the product of 2 ~ i and the sum of the residues in all these singular points: ~

f f ( z ) d z = 2nif:Resf(z) lz=zk.

(14.56)

b=O

(K)

IThe function f ( z ) = e z / ( z 2+ 1) has poles of order one at z1,2 = f i . The corresponding residues have the sum sin 1. If K is a circle around the origin with radius r

> 1, then

f

z2: dz = P ~ i s i n1.

(K)

Y4

*

I

01

X

Figure 14.44

Figure 14.45

14.4 Evaluation of Real Integrals by Complex Integrals 14.4.1 Application of Cauchy Integral Formulas The value of certain real integrals can be calculated with the help of the Cauchy integral formula. IThe function f ( z ) = e*, which is analytic in the whole z plane, can be represented with the Cauchy integral formula (14.43), where the path of integration C is a circle with center z and radius r . The equation of the circle is = z + re'?. We get from (14.43)

<

14.4 Evaluation of Real Inte.qrals b y Complex Inte.qrals 693

Since the imaginary part is equal to zero, we get

r

2rrn ercoslp cos(rsin ~p - ncp) dcp = -. n!

14.4.2 Application ofthe Residue Theorem Several definite integrals of real functions with one variable can be calculated with the help of the residue theorem. If f ( z ) is a function which is analytic in the whole upper half of the complex plane including the real axis except the singular points z l , 2 2 , . . . , z, above the real axis (Fig. 14.45), and if one of the roots of the equation f ( l / z ) = 0 has multiplicity m 2 2 (see 1.6.3.1, l., p. 43), then +?o

J

n

(14.57)

f ( x ) d x =2~i~Resf(z)l,=,,. 1=1

--co

ICalculation of the integral

dx

Theequation f - =

1 ~

-

X6

= 0 has

1 a root of order six at x = 0. The function w = -has a single singular point z = i in the upper (1t 4 3 half-plane, which is a pole of order 3, since the equation (1 t z 2 ) 3= 0 has two triple roots at i and -i. The residue is according to f14.55b):

follows that Res 27ri

(

1 = 6(z + i)-51 '=I (1 t ~ 2 ) 3 1 ~ , ~

6 3 - - = --i, and with (14.57): - (21)s 16

1

tm

-m

f ( x )dx =

3;

= -K. For further applications of residue theory see, e.g., [14.12].

---I

14.4.3 Application of the Jordan Lemma 14.4.3.1 Jordan Lemma In many cases. real improper integrals with an infinite domain of integration can be calculated by complex integrals along a closed curve. To avoid the always recurrent estimations, we use the Jordan lemma about improper integrals of the form

1

f ( z ) e l a zdz,

(14.58a)

(CR

where C R is the half-circle arc with center at the origin and with the radius R in the upper half of the z plane (Fig. 14.46). The Jordan lemma distinguishes the following cases: a) a > 0: If f ( z ) tends to zero uniformly in the upper half-plane and also on the real axis for Iz/ + co and a is a positive number. then for R -+ 03

1

f(z)elaz

dz

-+0.

(14.58b)

(CR)

+ co,then the above statement is also valid in the case cy = 0. c) a < 0: If the half-circle is now below the real axis, then the corresponding statement is also valid for a < 0. d) The statement is also valid if only an arc segment is considered instead of the complete half-circle. b) a = 0: If the expression z f ( z ) tends to zero uniformly for IzI

I

14. Function Theory

694

e) The corresponding statement is valid for the integral in the form

1

f(z)eazdz,

(14.58~)

(C;)

where C; is a half-circle or an arc segment in the left half-plane with o > 0, or in the right one with a < 0.

Figure 14.46

Figure 14.47

Figure 14.48

14.4.3.2 Examples of the Jordan Lemma 1. Evaluation of the Integral

j

;.2"i;; dx. -

n

The following complex integral is assigned to the above real integral:

0-

R

R

2iJ-dx=iJ-d~+ +a2

-R

x sin ax x2 + a2

even function

x cos ax

-R

-R

= 0 (odd integrand)

The very last of these integrals is part of the complex integral

fz2 a2 dz. The curve C contains the zel"' +

(C)

CRhalf-circle defined above and the part of the real axis between the values -Rand R (R > lal). The complex integrand has the only singular point in the upper half-plane z = a i . We obtain from the ze'ot ( z - ai) = 2ni t-tai lim - = nie-", hence residue theorem: I = z + ai

1

(C)

I =

1

zeiar

dz +

1 R

xeiaz

d x = Tie-aa, It follows from lim I and from the Jordan lemma that R+W

-R

(CR)

( a > 0, a 2 0). 0

Several further integrals can be evaluated in a similar way Table 21.6,p. 1050. 2. Sine Integral (see also (8.95), p. 458) CC

d x is called the sine zntegralor the integral sine (see also 8.2.5, l., p. 458). Analo-

The integral 0

g

gously to the previous example, we investigate the complex integral - dz with the curve C according to Fig. 14.47. The integrand of the complex integral has a pole of first order at z = 0, so

14.4 Evaluation of Real Inte.qrals bv Complex Integrals 695

I = 2i JRsi:x -dx

= 2ai, hence

r

iJeir(cosi.+isinp) dq + J

/ 2 dz

= 2 ~ i This .

Ca

limit is evaluated as R + m>T -+ 0, where the second integral tends to 1 uniformly for r + 0 with respect to 9,Le.. the limiting process T + 0 can be done behind the integral sign. Then we get with the Jordan lemma: 2ij*

dx + ai = 2ai. hence

(14.59)

2

0

3. Step Function Discontinuous real functions can be represented as complex integrals. The so-called step function is an example: 1 fort>O, 1 $2 (14.60) F(t)= -dz= 27ri--+ z 0 for t < O .

/

The symbol --+ denotes a path of integration along the real axis (I R I+ m) going round the origin (Fig. 14.47). If t denotes time, then the function @(t)= cF(t - to) represents a quantity which jumps at timet = t o from 0 through the value 4 2 to the value c. We call it a step function or also a Heavisidefunctzon. It is used in the electrotechnics to describe suddenly occurring voltage or current impulses.

4. Rectangular Pulse A further example of the application of complex integrals and the Jordan lemma is the representation of the rectangular pulse: 0 for t < a and t > b, 1 ?i(b-t)z 1 &(a-t)z (14.61) P(t)=-dz-2dz= 112 for t = a and t = b. 2ai-_+ %-+

/

/

5 . Fresnel Fntegrals

To derive the Fresnel integral X

03

:@ 2

/sin(x2)dx=/cos(x2)dx= 0

0

we investigate the integral I = /e-’’ dz on the closed path of integration shown in Fig. 14.48.We J

K

get, according to the Cauchy integral theorem: I = I1 + I11 t I111 = 0 with I1 =

IR

e-” dx,

= i~ [I4 e-R2(cos2~tisinZp)tipd P3

;

1111 = e? l e i r z d r= -&(1

+ i)

[iR i

sinr’dr - i R c o s ~ ’ d r ] .

Estimation of 111: Since lil = leiT1= 1 ( T real) holds, we get: IIIII< R / r ’ 4 e-R2C0s2pd ‘p

<

1 - e-R2cosa

Ge-R2cosn

2

+

2Rsina

( 0 < (Y <

:).

Performing the limiting process lim I we get the R+W

696

11. Function Theoru

1 values of the integrals ZI and 111: limR+, ZI = -fi,lim 111 = 0. We get the given formulas (14.62) R+m 2 by separating the real and imaginary parts.

14.5 Algebraic and Elementary Transcendental Functions 14.5.1 Algebraic Functions 1. Definition A function which is the result of finitely many algebraic operations performed with z and maybe also with finitely many constants, is called an algebraic function. In general, a complex algebraic function W ( Z ) can be defined in an implicit way as a polynomial, just as its real analogue alzmlwnl$ a 2 z m z w n z + ~ ~ ~ + a k z m k= w O .n k (14.63) Such functions cannot always be solved for w. 2. Examples of Algebraic Functions Linear function:

20

= az

+ b.

Quadratic function: w = z2.

(14.64)

1 Inverse function: w = - .

(14.66)

Square root function: w =

(14.65)

Z

m.

zti Fractional linear function: w = 2-1

(14.67) (14.68)

14.5.2 Elementary Transcendental Functions The complex transcendental functions have definitions corresponding to the transcendental real functions, just as in the case of the algebraic functions. For a detailed discussion of them see, e.g., [21.1]or 121.101. 1. Natural Exponential Function z z3 e X = l +- + - + - + . . . , (14.69) l! 2! 3! The series is absolutely convergent in the whole z plane. a) Pure imaginary exponent iy: This is valid according to the Euler relation (see 1.5.2.4, p. 35): e'y = cosy f isiny with effl= -1. (14.70) b) General case z = z t iy: ( 14.71a) ez = el:+iu = ezeiu - e1:(cosy+isiny),

.

.

Im(ez) =eZsiny, le"I = e z , arg(e') = y . The function ez is periodic, its period is 2ri: e" = ez+2kffi (k = 0, hl. f 2 , . . .) .

(14.71b)

In particular: eo = eZkni = 1,

(14.71d)

Re(ez) =ercosy,

e('k+l)'i = -1.

(14.71~)

c) Exponential form of a complex number (see 1.5.2.4, p. 35): a t ib = peiP. d) Euler relation for complex numbers: e-" = cos z - i sin z . (14.73a) el2 = cos z + i sin z ,

(14.7313)

2. Natural Logarithm u l = L n z , if z = ew.

(14.74a)

(14.72)

14.5 Algebraic and Elementary Transcendental Functions 697

Since z = pe'q, we can write: Lnz = l n p + i ( p + 2kr) and Re(Lnz) = lnp, Im(Lnz) = q + 2k7r

( k = O,i.l,i.2,. . .).

(14.74 b) (14.74~)

Since Ln z is a multiple-valued function (see 2.8.2, p. 84), we usually give only the principal value ofthe logarithm In z : lnz=lnp+ip (-n
3. General Exponential Function (14.75a)

= ezLna,

a' ( a # 0 ) is a multiple-valued function (see 2.8.2, p. 84) with principal value =

( 14.75b)

elha,

4. Trigonometric Functions and Hyperbolic Functions ( 14.76a) ( 14.76b) ( 14.77a)

e' + e-' 22 24 (14.77b) c o s h z = -= 1+-+-+... , 2 2! 4! A11 four series are convergent on the entire plane and they are all periodic. The period of the functions (14.76a.b) is 27r, the period of the functions (14.77a.b) is 27ri. The relations between these functions for any real or complex z are: ( 14.78b) cos i z = cosh z , sin i z = i sinh z , ( 14.78a) sinh i z = i sin z ,

(14.79a)

cosh i z = cost.

(14.79b)

The transformation formulas of the real trigonometric and hyperbolic functions (see 2.7.2, p. 79, and 2.9.3, p. 89) are also valid for the complex functions. We calculate the values of the functions sinz, cosz, sinhz, andcoshzforthe argument z = s + i y w i t h t h e helpoftheformulassin(a+b), cos(a+b), sinh(a t b ) , and cosh(a + b) or we can do it by using the Euler relation (see 1.5.2.4, p. 35). (14.80) I c o s ( z + i y ) = coszcosiy -sina:siniy =coszcoshy-isinssinhy. Therefore. (14.81a) Re(cosz) = cosRe(z) coshIm(z), ( 14.81b) Im(cosz) = -sinRe(z) sinhIm(z). The functions tan z , cot z , tanh z , and coth z are defined by the following formulas: sinh z cosh z cos z sin z tan z = - cot z = (14.82a) tanh z = - cothz = - (14.82b) cosh z ' sinhz ' cos z sin z

.

~

5 . Inverse Trigonometric Functions and Inverse Hyperbolic Functions These functions are many-valued functions, and we can express them with the help of the logarithm function: Arcsinz= - i L n ( i z + m ) , (14.83a) Arsinhz=Ln(z+m), (14.83b)

Xrccosz= - i L n ( z + m ) ,

(14.84a)

Arcoshz=Ln(z+m),

(14.84b)

I

698

14. Function Theory

1 l+z 1 l+iz .4rtanh z = -Ln -, Arctan z = _Ln -, (14.85b) (14.85a) 2 1-2 21 1 - 1 2 1 z+l 1 iz+1 Arcoth z = -Ln -. (14.86b) Arccot z = -- Ln -, (14.86a) 2 2-1 2i 1 2 - 1 The prznczpal values of the inverse trigonometric and the inverse hyperbolic functions can be expressed by the same formulas using the principal value of the logarithm In z :

m),(14.87a) ln(z + m))(14.88a)

arcsinz = -i ln(iz + arccost = -i

1+iz arctan z = 7 In -, 21 1 - 12 1 iz+1 arccot z = -- In -, 21 I t - 1 1

(14.89a) (14.90a)

m), arcoshz = ln(z + m), arsinhz = ln(z +

1 l+z artanh z = - In -. 2 1-2 1 z+l arcoth z = - In -. 2 2-1

(14.87b) (14.88b) (14.89b) (14.90b)

6. Real and Imaginary Part of the Trigonometric and Hyperbolic Functions (See Table 14.1) 7. Absolute Values and Arguments of the Trigonometric and Hyperbolic Functions (See Table 14.2) Table 14.1 Real and imaginary parts of the trigonometric and hyperbolic functions

1

Function

w = f(z t it/) sin(x f iy) cos(z f iy) tan(2 & iy) sinh(x f iy) cosh(z f iy) tanh(z f iy)

Real part Re (w) sin x cosh y cos x cosh y sin 22 cos 22 cosh 2y sinh 5 cos y cosh x cos y sinh 22 cosh 2 2 cos 2y

+

+

Imaginary part Im (tu)

k cos zsinh y sin 2 sinh y sinh 2y *cos 22 cosh 2y fcosh 2 sin y sinh zsin y sin 2y *cosh 22 + cos 2w

+

*

Table 14.2 Absolute values and arguments of the trigonometric and hyperbolic functions

Function

sin(x & iy) cos(z f iy) sinh(x f iy) cosh(z f iy)

Absolute value Iw I

Argument arg w

JzGGE& =J

7 arctan(tan z tanh y)

&zGz&

-J

farctan(cot 2 tanh y)

* arctan(cothz tan y) * arctan(tanh x tan y)

14.5 A1,qebraic and Elementaru Transcendental Functions 699

14.5.3 Description of Curves in Complex Form A complex function of one real variable t can be represented in parameter form: (14.91) z = x ( t ) + iy(t) = f ( t ) . As t changes, the points z draw a curve z ( t ) . In the following, we give the equations and the corresponding graphical representations of the line, circle, hyperbola, ellipse, and logarithmic spiral. 1. Straight Line a) Line through a point (zlr'p) ( p i s the angle with the x-axis, Fig. 14.49a): (14.92a) z = z1 t te+' .' b) Line through two points zl: 22 (Fig. 14.49b): (14.92b) z = 21 + t ( z 2 - 21). 2. Circle a) Circle with radius T . center at the point zo = 0 (Fig. 14.50a): = reit (14 = T I . b) Circle with radius T , center at the point zo (Fig. 14.50b):

z = zo + relt

(12

(14.93a) (14.9313)

- 201 = r ) .

I

X

b)

a)

b) Figure 14.50

Figure 14.49

3. Ellipse

x2

y2

a) Ellipse, Normal Form - t - = 1 (Fig. 14.51a): a2 b2 z=acost+ibsint(14.94a) o r ~ = c e ' ~ + d e - ' ~ ( 1 4 , 9 4 bw) i t h e =

a+b 2

-,

a-b d = -, 2

(14.94~)

Le., c and d are arbitrary real numbers. b) Ellipse, General Form (Fig. 14.51b): The center is at 21, the axes are rotated by an angle. (14.95) z = z1 + ce'' + de-'t. Here c and d are arbitrary complex numbers, they determine the length of the axis of the ellipse and the angle of rotation. 12

y2

4. Hyperbola, Normal Form - - - = 1 (Fig. 14.52): , a2 b2 z = acosh t ibsinht

+

(14.96a)

or

z = ce' + Fe-', where c and E are conjugate complex numbers: a+ib a-ib c= 2 2 .

(14.9613) (14.96~)

14. Function Theoru

700

I' Figure 14.51

Figure 14.52

5 . Logarithmic Spiral (Fig. 14.53): 2 =

p,

(14.97)

where a and b are arbitrary complex numbers.

14.6 Elliptic Functions 14.6.1 Relation to Elliptic Integrals

m)

Figure 14.53 Integrals in the form (8.21) with integrands R (z, cannot be integrated in closed form if P ( x ) is a polynomial of degree three or four, except in some special cases, but they are calculated numerically as elliptic integrals (see 8.1.4.3,p. 435). The inverse functions of elliptic integrals are the ellipticfunctions. They are similar to the trigonometric functions and they can be considered as their generalization. As an illustration, let us consider the special case

](1

-

t 2 ) - i dt = x

(1.

5 1).

(14.98)

0

We can tell that a) there is a relation between the trigonometric function u = sin x and the principal value of its inverse function - 1 < z 5 1,-1 5 u 5 1, (14.99) 22 b) the integral (14.98) is equal to arcsinu. The sine function can be considered as the inverse function of the integral (14.98). Analogies are valid for the elliptic integrals. W The period of a mathematical pendulum, with mass m, hanging on a non-elastic weightless thread of length 1 (Fig. 14-54),can be calculated by a second-order non-linear differential equation. We get this equation from the balance of the forces acting on the mass of the pendulum:

u = sinz H z = arcsinu for

(14.100a) The relation between the length 1 and the amplitude s from the dwell is s = 119, so s = 18 and s = 18 hold. The force acting on the mass is F = mg, where g is the acceleration due to gravity, and it is decomposed into a normal component FN and a tangential component FT with respect to its path (Fig. 14.54). The normal component FN = mgcos19 is balanced by the thread stress. Since it is perpendicular to the direction of motion, it has no effect to the equation of motion. The tangential component FT yields the acceleration of the motion. FT = ms = m119 = -mg sin 19. The tangential component always points in the direction of dwell.

14.5 Algebraic and Elementaw Transcendental Functions 701 We get by separation of variables: d8

(14.100b)

Here, to denotes the time for which the pendulum is in the deepest position for the first time, Le., where t9(tO) = 0 holds. Q denotes the integration variable. We get the equation (14.100~)

8

8

after some transformations and with the substitutions sin - = k s i n q , k = sin -. Here F ( k ,9)is 2 2 an elliptic integral of the first kind (see (8.24a), p. 435). The angle of deflection 29 = O ( t ) is a periodic function of period 2T with (14.100d) where K represents a complete elliptic integral of the first kind (Table 21.7). T denotes the period dt9 of the pendulum, Le., the time between two consecutive extreme positions for which - = 0. If the dt

amplitude is small, Le., sin 29 FZ 29, then T = 2 ~ J 1 / 9 holds.

Figure 14.54

Figure 14.56

Figure 14.55

14.6.2 Jacobian h n c t ions 1. Definition It follows for 0 < k < 1 from the representation (8.23a) and (8.24a), 8.1.4.3, p. 435 for the elliptic integral of the first kind F ( k , y ) that = (1 - k2 sinZy ) - i > 0, dY Le.. F ( k . p) is strictly monotone with respect to 'p, so the inverse function 'p =

am(k.u) = p(u)

(14.102a)

Of

'

u = [ p &d* -

(14.101)

-u(p)

(14.102b)

exists. It is called the amplitudefunction. The so-called Jacobian functions are defined as: (14.103a) snu = sin9 = sinam(k, u ) (amplitude sine), cnu = cos p = cos am(k, u ) (amplitude cosine),

(14.103b)

702

14. Function Theory

dnu = dkzsn2.u

(14.103~)

(amplitude delta).

2. Meromorphic and Double Periodic Functions The Jacobian functions can be continued analytically in the z plane. The functions snz, cnz, and dnz are then meromorphicfunctions (see 14.3.5.2, p. 691), i.e., they have only poles as singularities. Besides, they are double periodic: Each of these functions f(z) has exactly two periods w1 and w2 with (14.104) f ( z + w1) = f ( Z L f ( z + wz) = f k ) . Here, w1 and wz are two arbitrary complex numbers, whose ratio is not real. The general formula (14.105) f ( z mu1 nwz) = f(z) follows from (14.104),and here m and n are arbitrary integers. Meromorphic double periodic functions are called elliptic functions. The set (14.106) (20 cYlbJl+ cvzwz: 0 5 cvl, a2 < l}, with an arbitrary fixed zo E 6,is called the period parallelogram of the elliptic function. If this function (Fig. 14.55) is bounded in the whole period parallelogram, then it is a constant. IThe Jacobian functions (14.103a) and (14.103b) are elliptic functions. The amplitude function (14.102a) is not an elliptic function.

+

+

+

3. Properties of the Jacobian functions The properties of the Jacobian functions given in Table 14.3 can be got by the substitutions

f) , K = F (k);

k 2 , K’ = F (k’>

Poles

Roots

Periods w1, wa

2.

(14.107)

2mK

+ 2niK’

snz

4K,2iK’

cnz

4K, 2(K + iK’)

(2m + l ) K + 2niK’

dnz

2K, 4iK’

(2m+ l ) K + (271 + 1)iK’

i

2mK

+ (2n + 1)iK’

+

sn(u+v) =

(snu)(cnv)(dnv) (snv)(cnu)(dnu) 1 - kz(snzu)(sn2v)

(14.109a)

cn(u + v) =

(cnu)(cnv) - (mu)(dnu)(snv)(dnv) 1 - k2(sn2u)(sn2v)

(14.109b)

dn(u

kZ (snu)(cnu)(snv)(cnv) + v) = (dnv)(dnv)1 --k2(sn2u)(sn2v) ,

3. (snz)’ = (cnz)(dnz), (dnt)’ = -k’(snz)(cnz).

(14.110a)

(14.109~) (cnz)’ = -(snz)(dnz),

(14.110b) (14.110~)

14.5 Algebraic and Elementary Tkanscendental Functions 703

For further properties of the Jacobian functions and further elliptic functions see [14.8],[14.12]

14.6.3 Theta Function We apply the theta function to evaluate the Jacobian functions (14.11 la) m

292(z,q) = 24:

qn(n+’)

(14.111b)

cos(2n t l)z,

n=O &(z,q)= 1

+2

m

qnz

( 14.11IC)

cos2nz,

n=l

m

(14.11Id)

&(z.q) = 1+2~(-l)nqnZcOs2nz. n=l

If 141 < 1 ( q complex) holds, then the series (14.111a)-(14.111d) are convergent for every complex argument z . We use the brief notation in the case of a constant q 2 9 k ( ~ ):= v?k(Tz,q) ( k = 1,2,3,4). ( 14.112) Then, the Jacobian functions have the representations:

( 14.113b)

and K , K’ are as in (14.107)

14.6.4 Weierstrass Functions The functions (14.114a)

p(z) = ~(Z,IJ~.W:,).

((2)

= s(z,wl>wz),

(14.114b)

4.) = 4 Z , ~ l r ~ 2 ) , (14.114~) were introduced by Weierstrass, and here w1 and w:, represent two arbitrary complex numbers whose quotient is not real. We substitute ( 14.115a) d m n = 2(mw1 t nul), where m and n are arbitrary real numbers, and we define p(z.Wlr(u‘2) = z - 2 t C ’ [ ( z - w m n , - ’ - w , ~ ] .

(14.115b)

m,n

The accent behind the sum sign denotes that the value pair m = n = 0 is omitted. The function p(zl wll 4) has the following properties: 1. It is an elliptic function with periods w1 and w2. 2. The series (14.115b) is convergent for every z # umn. 3. The function p(z, dl. w 2 ) satisfies the differential equation

704

14. Function Theory

p’* = 4p3 - gZp - g3

(14.116a)

with gz = 601’u;i, m,n

g3 = 1401’~;:.

(14.116b)

m,n

The quantities g2 and g3 are called the invariants of p(z, wl,wz). 4. The function u = p(z, u l , w 2 ) is the inverse function of the integral

( 14.117)

(14.118)

(14.11sa) (14.119b) are not double periodic, so they are not elliptic functions. The following relations are valid:

I.

(’(2)

2.

((-2)

3.

((2

= -p(z), = -((z),

+2

((2)=

(lnu(z)),

(14.120)

u(-2)

= -u(z),

(14.121) (14.122)

4 = ( ( 2 ) +2C(Wl),

C(z + 2wz) = ( ( 2 ) t 2C(uz),

(14.123) 5.

Every elliptic function is a rational function of the Weierstrass functions p(z) and C ( z ) .

705

15 Integral Transformat ions 15.1 Notion of Integral Transformation 15.1.1 General Definition of Integral Transformations Anlntegral transformation is a correspondence between two functions f ( t ) and F ( p ) in the form (15,la) -X

The function f ( t ) is called the originalfunction, its domain is the original space. The function F ( p ) is called the transform, its domain is the image space. The function K ( p ,t ) is called the kernel of the transformation. In general, t is a real variable, and p = u + iw is a complex variable. We may use a shorter notation by introducing the symbol Tfor the integral transformation with kernel K(P,t)' F(P ) = T{f(t)}. (15.1b) Then, we call it a 7 transformation.

15.1.2 Special Integral Transformations We get different integral transformations for different kernels K ( p , t ) and different original spaces. The most widely known transformations are the Laplace transformation, the Laplace-Carson transformation, and the Fourier transformation. We give an overview of the integral transformations of functions of one variable in Table 15.1. More recently, some additional transformations have been introduced for use in pattern recognition and in characterizing signals, such as the Wavelet transformation, the Gabor transformation and the Walsh transformation (see 15.6, p. 738ff.).

15.1.3 Inverse Transformations The inverse transformation of a transform into the original function has special importance in applications. IVith the symbol 7-' the inverse integral transformation of (15.la) is

f(t) = 7 - ' { W } .

(15.2a)

The operator 7-'is called the inverse operator of 7 ,so (15.2b) T-'V{f(t)l) = f(t). The determination of the inverse transformation means the solution of the integral equation (15.1a), where the function F ( p ) is given and function f ( t ) is to be determined. If there is a solution, then it can be written in the form f(t)= T-'{F(p)}. (15.2~) The explicit determination of inverse operators for different integral transformations, Le., for different kernels K ( p ,t ) . belongs to the fundamental problems of the theory of integral transformations. The user can solve practical problems by using the given correspondences between transforms and original functions in the corresponding tables (Table 21.11,p. 1061, Table 21.12,p. 1066, and Table 21.13, p. 1080)

15.1.4 Linearity of Integral Transformations If fi(t) and fz(t) are transformable functions, then (15.3) T{klfl(t)+ kzfz(t)) = klT{fl(t))+ kZT{fZ(t)), where k l and kz are arbitrary numbers. That is, an integral transformation represents a linear operation on the set T of the T-transformable functions.

I

706

15. Inteoral Transformations ~

Table 15.1 Overview of integral transformations of functions of one variable

Kernel K(p,t) Laplace transformation

for t for t

0 e-pt

Two-sided Laplace transformation Finite Laplace transformation Laplace-Carson transformation

<0 >0

e-Pt

0 for t < 0 e+ for 0 < t < a 0 for t > a l0 fort 0 mation.

Fourier

e-iwt

One-sided Fourier transformation Finite Fourier transformation

e-iwt

10

Fourier cosine transformation

(0

Fourier sine transformation

(0

Mellin

(0

\ Re

Stieltjes transformation

,

for t < O for t > 0

[elwt]

,

fort<^

\ Im [PIfor t > 0 (tp-'

Hankel transformation of order u

for t < 0 for 0 < t < a fort>a

0

(0

fort 0

fort 0

w

0

I~=u+iw W=O I &(ut) is the u-th order Bessel function of the first kind.

I

15.1 Notion of Inteqral Transformation 707

15.1.5 Integral Transformations for FunctionsofSeveralVariables Integral transformations for functions of several variables are also called multzple zntegral transfomnatzons (see [15.13]).The best-known ones are the double Laplace transformation, Le., the Laplace transformation for functions of two variables, the double LaplaceCarson transformation and the double Fourier transformation. The definition of the double Laplace transformation is c a m

P(p. q ) = Cz{f(z.y ) }

/ / e-ps-qyf(x,

y) dxdy.

(15.4)

3=0 y=o

The symbol L denotes the Laplace transformation for functions of one variable (see Table 15.1).

15.1.6 Applications of Integral Transformations 1. Fields of Applications Besides the great theoretical importance that integral transformations have in such basic fields of mathematics as the theory of integral equations and the theory of linear operators, they have a large field of application in the solution of practical problems in physics and engineering. Methods with applications of integral transformations are often called operator methods. They are suitable to solve ordinary and partial differential equations, integral equations and difference equations.

2. Scheme of the Operator Method The general scheme to the use of an operator method with an integral transformation is represented in Fig. 15.1. R’e get the solution of a problem not directly from the original defining equation; we first apply an integral transformation. The inverse transformation of the solution of the transformed equation gives the solution of the original problem.

-

Equation of the problem

i

- Solution of theequation

Transformation

Result

f

Inverse transformation

the transformed equation Figure 15.1 The application of the operator method to solve ordinary differential equations consists of the following three steps: 1. Transition from a differential equation of an unknown function to an equation of its transform. 2. Solution of the transformed equation in the image space. The transformed equation is usually no longer a differential equation, but an algebraic equation. 3. Inverse transformation of the transform with help of 7-’into the original space, Le., determination of the solution of the original problem. The major difficulty of the operator method is usually not the solution of the transformed equation, but the transform of the function and the inverse transformation.

708

15. Integral Transformations

15.2 Laplace Transformation 15.2.1 Properties of the Laplace Transformation 15.2.1.1 Laplace Transformation, Original and Image Space 1. Definition of the Laplace Transformation The Laplace transformation

1 P

L { f ( t ) }=

(15.5)

e - P t f ( t )dt = F ( p )

0

assigns a function F ( p ) of a complex variablep to a function f ( t ) of a real variable t , if the given improper integral exists. f ( t ) is called the original function, F ( p ) is called the t r a n s f o m of f(t). The improper integral exists if the original function f(t) is piecewise smooth in its domain t 2 0, in the original space, and for t --t co,suppose lf(t)l 5 Keet with certain constants K > 0, cy > 0. The domain of the transform F ( p ) is called the image space. In the literature one can find the Laplace transformation also introduced in the b'agner or LaplaceCarson form m

(15.6)

L w { f ( t ) }= p J e - P t f ( t ) d t = p ~ ( p ) . 0

2. Convergence The Laplace integral L { f ( t ) }converges in the half-plane Rep is an analytic function with the properties: 1. lim F ( p ) = 0.

> LY (Fig. 15.2). The transform F ( p ) (15.7a)

Repim

This property is a necessary condition for F ( p ) to be a transform. 2. pF(p)= A ,

bs

(15.7b)

(P-'CU)

if the original function f(t) has a finite limit

!iir f(t) = A . (t-10)

Figure 15.2

Figure 15.3

3. Inverse Laplace Transformation h'e can retrieve the original function from the transform with the formula

(15.8)

15.2 Laplace Transformation 709

The path of integration of this complex integral is a line Rep = c parallel to the imaginary axis, where Rep = c > cy. If the function f ( t )has a jump at t = 0, i.e., lim f(t) # 0, then the integral has the t++O

1 mean value -f(+O) 2

there.

15.2.1.2 Rules for t he Evaluation of the Laplace Transformation The rules for evaluation are the mappings of operations in the original domain into operations in the transform space. In the following, we denote the original functions by lowercase letters, the transforms are denoted by the corresponding capital letters.

1. Addition or Linearity Law The Laplace transform of a linear combination of functions is the same linear combination of the Laplace transforms. if they exist. With constants XI,. . . , A n we get: L{x,fl(t)+~zfz(t)+".+Xnfn(t)} = X,Fl(P) +XZFZ(P)+..'+X,F,(P). (15.9) 2. Similarity Laws The Laplace transform of f(at) (a > 0, a real) is the Laplace transform of the original function divided by a and with the argument p / a :

Analogously for the inverse transformation

( 15. lob) Fig. 15.3 shows the application of the similarity laws for a sine function. IDetermination of the Laplace transform of f(t) = sin(&). For the correspondence of the sine function we have L{sin(t)} = F ( p ) = l/(pz + 1). Application of the similarity law gives L{sin(wt)} = 1

1

W

w (p/w)2+1

-F(p/w) = -

1

W

~

-pZ+wZ'

3. Translation Law 1. Shifting t o t h e Right The Laplace transform of an original function shifted to the right by a ( a > 0) is equal to the Laplace transform of the non-shifted original function multiplied by the factor p p :

L{f(t- a)} = e-apF(p).

(15.1la)

2. Shifting t o t h e Left The Laplace transform of an original function shifted to the left by a is equal to eaP multiplied by the difference of the transform of the non-shifted function and the integral ]:f(t)e-ptdt:

L { f ( t + a ) } = eap F ( p ) - / e - P t f ( t )

[

oa

dt

1

.

(15.11b)

Figs. 15.4 and 15.5 show the cosine function shifted to the right and a line shifted to the left.

4. Frequency Shift Theorem The Laplace transform of an original function multiplied by e-bt is equal to the Laplace transform with the argument p + b ( b is an arbitrary complex number):

L{e-*'f(t)}= F ( p + b ) .

(15.12)

710

15. Inteqral Transformations

,

L{f’(t)l = P F b ) - f ( + % L{f”(t)l = PZF(P) - f(+O) P - f’(+OL

........................................................ L { p ( t ) }= p ” F ( p ) - f ( f 0 ) p n - l - f’(+o)p”-* - . . . - f(n-z)(+O)p- f(”-’)(+O)

f‘”’(+O)

=

’

(15.13)

with

t!yof‘”’(t).

/

In the original space. differentiation and integration act in converse ways if the initial values are zeros. 8. Integration in the Image Space m m 00 1 (15.18) T ( z - p)n-’F(z) dz. L = dPl JdP* ’ ’ ’ F(Pn) dPn =

{ T}J P

J

PI

P,-1

P

This formula is valid only if f ( t ) / t ” has a Laplace transform. For this purpose, f(x)must tend to zero fast enough as t -+ 0. The path of integration can be any ray starting at p, which forms an acute angle

15.2 Laplace Transformation 711

with the positive half of the real axis.

9. Division Law In the special case of n = 1 of (15.18) we have (15.19)

f ( t ) must also exist. For the existence of the integral (15.19);the limit lim -

t

t-0

10. Differentiation and Integration with Respect to a Parameter

C

{ y} aa

(15.20a)

=

,C

{ lf(t,

a ) da} = T F ( t ,a ) da.

01

(15.20b)

a1

Sometimes we can calculate Laplace integrals from known integrals with the help of these formulas.

11. Convolution 1. Convolution in t h e Original Space The convolution of two functions fl(z)and f2(z)is the integral

f * fz=jh(r)~fz(t-~)d7.

(15.21)

0

Equation (15.21) is also called the one-sided convolution in the interval (0, t ) . A two-sided convolution occurs for the Fourier transformation (convolution in the interval (-m, co)see 9., p. 727). The convolution (15.21) has the properties a) Commutative law: b) Associative law: c) Distributive law:

fl * fz = fz * f l ' ( f ~* f ~ *) f3 = fi * (fi * A). (fl

+ fz) * f3 = fl * f3 + fz * f3.

(15.22a) ( 15.22b) (15.22~)

In the image domain, the usual multiplication corresponds to the convolution: (15.23) L{fi * fz} = Fib).Fz(P). The convolution of two functions is shown in Fig. 15.6. We can apply the convolution theorem to determine the original function: a ) Factoring the transform F(P) = Fl(P) ' &(PI. b) Determining the original functions f l ( t ) and f Z ( t ) of the transforms 4 ( p ) and Fz(p) (from a table). c) Determining the original function associated to F ( p ) by convolution of f l ( t ) and f2(t) in the original space ( f ( t ) = h ( t )* fz(t)).

712

15. Intearal Transformations

2. Convolution in the Image Space (Complex Convolution)

xl-im

(15.24)

x2-icc

The integration is performed along a line parallel to the imaginary axis. In the first integral, 21 and p must be chosen so that z is in the half plane of convergence of C{fi} and p - z is in the half plane of convergence of L{f2}. The corresponding requirements must be valid for the second integral.

15.2.1.3 Transforms of Special Functions 1. Step Function The unit jump at t = t o is called a step function (Fig. 15.7) (see also 14.4.3.2,3., p. 695); it is also called the Heaviside unit step function: (15.25) IA: f ( t ) = u ( t - t o )sinwt,

F(p) = e+p

wcoswt~+psinwto p2 w2

(Fig. 15.8).

IB: f ( t ) = u ( t - to)sinw (t - to),

F(p) =

W p2 w2

(Fig. 15.9)

Figure 15.7

+

+

Figure 15.8

Figure 15.9

2. Rectangular Impulse .4rectangular impulse of height 1 and width T (Fig. 15.10)is composed by the superposition of two step functions in the form

0 uT(t - t o )= u(t - to) - u( t - to - T) = 1 0

for t < to, for to < t < to T , for t > t o + T ;

+

(15.26)

(15.27)

3. Impulse Function (Dirac 6 Function) (See also 12.9.5.4, p. 639.) The impulse function 6 ( t - t o )can obviously be interpreted as a limit of the rectangular impulse of width T and height 1/T at the point t = to (Fig. 15.11): 1 (15.28) b(t - t o )= lim -[ u ( t - to) - u(t - to - T ) ] . T+O T

15.2 Lavlace Transformation 713

Figure 15.10

Figure 15.11

For a continuous function h ( t ) , (15.29) Relations such as

6(t - to) = d’(t - t o ) ~

dt

C{ 6(t - to)} = e-top

’

(to 2 0)

(15.30)

are investigated generally in distribution theory (see 12.9.5.3, p. 638).

4. Piecewise Differentiable Functions The transform of a piecewise differentiable function can be determined easily with the help of the b function: If f ( t ) is piecewise differentiable and at the points t , (v = 1 , 2 , . . . , n) it has jumps a,. then its first derivative can be represented in the form

dfo = j i ( t )+ alb(t + a z ~ (-t t z ) + + a,b(t dt tl)

’.

- t,)

(13.31)

where f : ( t ) is the usual derivative of f ( t ) ,where it is differentiable. Ifjumps occur first in the derivative, then similar formulas are valid. In this way, we can easily determine the transform of functions which correspond to curves composed of parabolic arcs of arbitrarily high degree. e.g.. curves found empirically. In formal application of (15.13), we should replace the values f(+O). f‘(+O). . . . by zero in the case of a jump. W A: f ( t ) = [~~bff”bl~~~~,to~(Fig.15.12);f’(t)=au,(t)+b6(t)-(at~+b)6(t-t~)~C{f’ !(I - e 8 O p )

P

+ b - ( a t 0 + b) e?Op;

C { f ( t ) }=

W B:

1 for 0 < t < to, for 0 < t < to, f ( t ) = 2to - t for to < t < 2ta, (Fig.15.13);f ’ ( t ) = -1 for to < t < 2t0, (Fig. 15.14): 0 for t > 2t0, for t > 2t0, 0

i

it

f”(t ) = 6(t )- 6(t - t o )-6( t -t o )t 6(t - 2to): C{f ” ( t )} = 1- 2 e ~ ~ o ~ + e - ~ ~C{ o Pf (; t )} = for for otherwise,

0

< t < to,

$ Iio>{ 2$: (Fig.15.15);

( 1- e - t o P ) 2

PZ

714

15. Inteoral Transformations

;boob;pu; ~

~

2tO Figure 15.12

Figure 15.13

< t < to, < < - t o > (t T - to < t < T ,

Elto 0

for for

0

otherwise,

0

to

'

Figure 15.14

(Fig, 15.16);

a t

01 to

T-t, T

-t

Figure 15.15 ID: f(t) =

[ toWt2

f " ( t ) = -2211(t)

gEr:ii2

(Fig. 15.17); f'(t) =

T-t T

Figure 15.16

0 < t < 1, [ 1 - 2t forotherwise, (Fig. 15.18);

+ b(t) t q t - 1);

Figure 15.17

Figure 15.18

5 . Periodic Functions The transform of a periodic function f * ( t )with period T , which is a periodic continuation of a function f ( t ) , can be obtained from the Laplace transform of f(t) multiplied by the perioditation factor (1- e-Tp)-l. (15.32)

15.2 Laplace Transformation 715

IA: The periodic continuation of f(t) from example B (see above) with period T = 2to is f * ( t )with

IB: The periodic continuation of f ( t ) from example C (see above) with period T is f ' ( t ) with E (1 - ,-top) (1 - e-(T-toiP) '{f*(t)} = tOP2(1 - e-Tp)

15.2.1.4 Dirac S Function and Distributions In describing certain technical systems by linear differential equations, functions u ( t ) and 6 ( t ) often occur as perturbation or input functions, although the conditions required in 15.2.1.1, 1. p. 708, are not satisfied: u ( t ) is discontinuous, and 6 ( t ) cannot be defined in the sense of classical analysis. Distribution theory offers a solution by introducing so-called generalized functions (distrzbutions),so that with the known continuous real functions 6 ( t ) can also be examined, where the necessary differentiability is also guaranteed. Distributions can be represented in different ways. One of the best known representations is the continuous real linear form. introduced by L. Schwartz (see 12.9.5, p. 637). We can associate Fourier coefficients and Fourier series uniquely to periodic distributions, analogously to real functions (see 7.4, p. 418).

1. Approximations of the 6 Function hnalogously to (15.28), the impulse function b(t) can be approximated by a rectangular impulse of width E and height 1/E ( E > 0): (15.33a) Further examples of the approximation of b ( t )are the error curve (see 2.6.3, p. 72) and Lorentz function (see 2.11.2. p. 94): (15.3313) (15.33~)

These functions have the common properties:

1.

7

(15.34a)

f ( t . E ) dt = 1.

--3o

2.

f(-t,

E)

= f ( t , E ) , Le., they are even functions.

( 15.34b) (15.34~)

3.

2. Properties of the 6 Function Important properties of the 6 function are:

1.

T j ( t ) J ( x- t ) dt = f(x) (f is continuous, a > 0).

(15.35)

x-a

2.

1

6((Yz) = -6(z)

3. 6 ( g ( z ) )=

( a > 0).

n 1 1 -6(z ls'(xz)l ,=I

- 2,)

(15.36) with g ( z l )= 0 and g'(zt) # 0 (i = 1 , 2 , .. . , n).

(15.37)

I

716

15. Inteoral Transformations

Here we consider all roots of g ( x )and they must be simple. 4. n - t h Derivative of t h e 6 Function: After n repeated partial integrations of

f(")(z)= Taf(")(t)6(z- t ) dt,

(15.38a)

x-a

we obtain a rule for the n-th derivative of the b function:

(-l)"f'"'(z) = T f ( t )b(n)(z- t ) dt.

(15.38b)

x-a

15.2.2 Inverse Transformation into the Original Space To perform an inverse transformation, we have the following possibilities: 1. Csing a table of correspondences, Le., a table with the corresponding original functions and transforms (see Table 21.11, p. 1061). 2. Reducing to known correspondences by using some properties of the transformation (see 15.2.2.2, p. 716, and 15.2.2.3, p. 717). 3. Evaluating the inverse formula (see 15.2.2.4, p. 718).

15.2.2.1 Inverse Transformation with the Help of Tables The use of a table is shown here by an example with Table 21.11, p. 1061. Further tables can be found, e.g., in [15.3]. 1 = Fl(P). FZ(P),c-'{Fl(P)I = pz = t s i n w t = fl(t), F ( p ) = ( p + c)(p2 + d)

W l2 j

e-' {

'

{ L}

= e-ct = fi(t). We have to apply the convolution theorem (15.23):

LC-'{F2(p)} =

f(t)

P+C

= l-l{Fl(P)

=

it

'

FZ(P)}

f l (7) , f2(t

- 7 ) d7 =

1'

e-c(t-7)

csinwt -wcoswt sin w~ 1 -dr=-( W W cz t WZ

+ e-'') .

15.2.2.2 Partial Fraction Decomposition 1. Principle In many applications, we have transforms in the form F ( p ) = H(p)/G(p), where G(p) is a polynomial of p . If we already have the original functions for H ( p ) and l/G(p), then we get the required original function F ( p ) by applying the convolution theorem. 2. Simple Real Roots of G ( p ) If the transform 1/G(p) has only simple poles p , (v = 1,2,. . . , n),then we get the following partial fraction decomposition: (15.39) The corresponding original function is (15.40)

15.2 Laplace Transformation 717

3. The Heaviside Expansion Theorem If the numerator H(p)is also a polynomial ofp with a lower degree than G ( p ) ,then we can obtain the original function of F ( p ) with the help of the Heaviside formula (15.41)

4. Complex Roots Even in cases when the denominator has complex roots, we can use the Heaviside expansion theorem in the same way. We can also collect the terms belonging to complex conjugate roots into one quadratic expression. whose inverse transformation can be found in tables also in the case of roots of higher multiplicity.

F ( p ) = (p + c)(;2 + d)' The poles p l = -e, pz = iw,

f(t) = I e - c wz + cz

t

-I

e

i

Le., H ( p ) = 1, G ( p ) = (p

-iw are all simple. According to the Heaviside theorem we get

p3 =

w

2w(w - ic)

+ c)(p2 + w'), G'(p) = 3p2 + 2pc + 2.

-I e-iwt or by using partial fraction decomposition and 2w(w ic) 1

t

+

expressions are identical.

15.2.2.3 Series Expansion X

In order to obtain f ( t ) from F ( p ) , we can try to expand F ( p ) into a series F ( p ) =

n=O

Fn(p), whose

terms Fn(p)are transforms of known functions, Le., F,(p) = C{fn(t)}. 1. F ( p ) is an Absolutely Convergent Series If F ( p ) has an absolutely convergent series (16.42) for Ipl > R. where the values A, form an arbitrary increasing sequences of numbers 0 . . < A, < . ' . < ' . . + co,then a termwise inverse transformation is possible: d - 1

pm. P

f(t) =

< A0 < Ai <

L '

T denotes the gamma function (see 8.2.5, 6., p. 459). In particular, for A, = n + 1, Le., for F ( p ) = anti we get the series f(t) = %tn, which is convergent for every real and complex t. Fur. n=O pnt1 n=O n ! thermore, we can have an estimation in the form if(t)l < CeCltl(C, c real constants)

5

-)

I F ( p ) = -- - (1 +

m

P

$)

the original space we get f ( t ) =

-112

1

=

e (-nk) -& ,

After a termwise transformation into

n=O

=

Jo(t) (Bessel function of

n=O

0 order). 2. F ( p ) is a Meromorphic Function If P(p)is a meromorphic function, which can be represented as the quotient of two integer functions (of two functions having everywhere convergent power series expansions) which do not have common

718

15. Inteoral Transformations

roots. and so can be rewritten as the sum of an integer function and finitely many partial fractions. then we get the equality

(15.44) Here p , (v = 1 , 2 , . . . , n) are the first-order poles of the function F ( p ) ,b, are the corresponding residues (see 14.3.9.4,p. 691), y, are certain values and K, are certain curves, for example, half circles in the sense represented in Fig. 15.19.We get the solution f ( t ) in the form

1 etpF(p)

m

f(t) =

b,ePvt,

if

as y

dp

277i

"=l

--t

0

(15.49)

(K,)

+ co,what is often not easy to verify.

X

-Y1

Figure 15.19

Figure 15.20

In certain cases, e.g., when the rational part of the meromorphic function F ( p ) is identically zero, the above result is a formal application of the Heaviside expansion theorem to meromorphic functions.

15.2.2.4 Inverse Integral The inverse formula

f ( t ) = lim V"+==

1' T e t PF ( p )dp 2771

(15.46)

c-iy,

represents a complex integral of a function analytic in a certain domain. The usual methods of integration for complex functions can be used, e.g., the residue calculation or certain changes of the path of integration according to the Cauchy integral theorem.

F ( p ) = -e-@P P2

+u2

is double valued because of Jij. Therefore, we chose the following path of inte-

1 gration (Fig. 15.20): 277i

f e tp2p +wz p e - @

(K)

Res etPF(p)= e-'JW/' cos(wt - a*).

-

A

1 1 / ...+ / ...+ /

. . + . . + .. . +

dp = h

h

h

AB

CD

EF

-

DA

-

.. =

-

BE

FC

According to the Jordan lemma (see 14.4.3, p. 693), the

h

integral part over AB and C D vanishes as yn

+ 30.

The integrand remains bounded on the circular

arc E F (radius E ) , and the length of the path of integration tends to zero for E + 0; so this term of the integral also vanishes. We have to investigate the integrals on the two horizontal segments BE and E, where we have to consider the upper side ( p = re'") and the lower side ( p = re-'") of the negative real axis:

15.2 Laplace Transformation 719

Finally we get:

15.2.3 Solution of Differential Equations using Laplace Transformat ion We have already noticed from the rules of calculation of the Laplace transformation (see 15.2.1.2, p. 709), that using the Laplace transformation we can replace complicated operations, such as differentiation or integration in the original space, by simple algebraic operations in the image space. Here, we haye to consider some additional conditions, such as initial conditions in using the differentiation rule. These conditions are necessary for the solution of differential equations.

15.2.3.1 Ordinary Differential Equations with Constant Coefficients 1. Principle The n-th order differential equation of the form

+ c,-1

y'"'(t)

y y t )t

'

.

'

+

t c1 y'(t) co y ( t ) = f ( t )

(15.47a)

with the initialvalues y(+O) = yo, y'(+O) = yb, . . . , y(n-l)(+O) = yr-') can be transformed by Laplace transformation into the equation (15.47b) Here G ( p )=

5 ckpk = 0 is the characteristic equation of the differential equation (see 4.5.2.1, p. 279).

k=O

2. First-Order Differential Equations The original and the transformed equations are: y'(t)

+ c d t ) = f(t).

Y(+O) = YO,

(15.48a)

(Pt CO) Y ( P )- YO

= F(P),

(15.48b)

where co = const. For Y(p)we get (15.484 (15.49a)

Special case: For f ( t ) = X ept (A. p const) we get X Yo Y(P)= (P - P)(P + co) P t GI'

+-

(15.49b) (15.49~)

P + co

3. Second-Order Differential Equations The original and transformed equations are: Y(+O) = YO, y"(t) 2ay'(t) + b y ( t ) = f ( t ) ,

+

(pZ+ 2ap + b ) Y ( P )- 2 ~ -~(PYO 0 Jr Y;) = F(P). We then get for Y ( p )

Y'(+O) = Y;,

( 15.50a) ( 15.50b)

(15.504

720 15. Inteoral Transformations

(15.51a) (15.51b)

b) b = a 2 : G(p) = (p - a)’, c) b > a’:

q(t)= t eat.

(15.52a)

G(p)

has complex roots, 1 d t ) = -e-ot sin & 7 t .

(15.52b) (15.53a) (15.53b)

Jbl-;li

We obtain the solution y(t) as the convolution of the original function of the numerator of Y(p) and q(t). The application of the convolution can be avoided if we can find a direct transformation of the right-hand side.

+

+

+ 2y’(t) 10y(t) = 37cos 3t 9e-t 9 P+2 37P p2 2p 10 (p2 t 9)(p’ 2p 10) (p l)(pZ + 2p + 10) ’ 19 P 18 1 -p We get the representation Y(p) = p2 2p 10 (p2 t 2p 10) (PZ 9) (p’ 9) (p 1) by partial fraction decomposition of the second and third terms of the right-hand side but not separating the second-order terms into linear ones. We get the solution after termwise transformation (see Table 21.11, p. 1061) y(t) = (-cos 3t - 6sin 3t)e-t +cos 3t 6sin 3t e&.

I The transformed equation for the differential equation y”(t) with yo = 1 and yb = OisY‘(PI -

+ + + + +

+ + + + + +-+-++ +

+

+

+

4. n-th Order Differential Equations

The characteristic equation G(p) = 0 of this differential equation has only simple roots a l ,CY’, . . . ,an, and none of them is equal to zero. We can distinguish two cases for the perturbation function f ( t ) . 1. If the perturbation function f ( t )is the jump function ~ ( twhich ) often occurs in practical problems, then the solution is: 1 ” 1 for t > 0, (15.54a) y(t) = eaut . (15.54b) ‘(‘1 = 0 for t < 0, G(O) a,G’(a,)

+

[

~

2. For a general perturbation function f ( t ) ,we get the solution c(t)from (15.54b) in the form of the Duhamel formula which uses the convolution (see 15.2.1.2, ll., p. 711):

(15.55)

15.2.3.2 Ordinary Linear Differential Equations with Coefficients Depending on the Variable Differential equations whose coefficients are polynomials in t can also be solved by Laplace transformation. Applying (15.16), in the image space we get a differential equation, whose order can be lower than the original one. If the coefficients are first-order polynomials, then the differential equation in the image space is a firstorder differential equation and maybe it can be solved more easily.

I Bessel differential equation of 0 order: t-d 2 f

d f t tf = 0 +dt2 dt transformation into the image mace results in d dF(P) --[p’F(p) -pf(O) - f’(O)] +pF(p) - f(0) - dp = 0 or dP

(see (9.51a, p. 507) for n = 0). The

-~

dF P - = -dp p2 + lF(’)

15.2 Laplace Transformation 721

Separation of the variables and integration yields log F ( p ) = L

/ ?-!?!- + p2

(C is the integration constant), F ( t ) = CJo(t) m) = v5n-i

1

=

- log

+ log C,

(see example p. 717 with the

Bessel function of 0 order).

15.2.3.3 Partial Differential Equations 1. General Introduction The solution of a partial differential equation is a function of at least two variables: u = u ( z ,t ) . Since the Laplace transformation represents an integration with respect to only one variable, the other variable should be considered as a constant in the transformation:

/ e-Ptu(z,t ) m

C { u ( z .t ) } =

(15.56)

dt = U ( z > p ) .

0

z also remains fixed in the transformation of derivatives: (15.57)

For differentiation with respect to zwe suppose that they are interchangeable with the Laplace integral:

C

{ y]

= ;r{u(z:

t ) }= -U(z,p). d

(15.58)

dz

This way, we get an ordinary differential equation in the image space. Furthermore, we have to transform the boundary and initial conditions into the image space.

2. Solution ofthe One-Dimensional Heat Conduction Equation for a Homogeneous Medium 1. Formulation of the Problem Suppose the one-dimensional heat conduction equation with vanishing perturbation and for a homogeneous medium is given in the form (15.59a) u,, - a-2ut = u,, - uy = O in the original space 0 < t < co,0 < 2 < 1 and with the initial and boundary conditions (15.59b) u(+O, t ) = ao(t), u(1 - 0,t ) = a1(t). u ( z , SO) = u&), The time coordinate is replaced by y = at. (15.59a) is also a parabolic type equation, just as the three-dimensional heat conduction equation (see 9.2.3.3, p. 535). 2. Laplace Transformation The transformed equation is

d2U dx2 and the boundary conditions are G(+O,p) = Ao(p). U ( 1 - 0 , ~=)AI(P). The solution of the transformed equation is ~ ( z , p=)clezfi + c2e-xfi. It is a good idea to produce two particular solutions Ul and

(15.60a)

-= p U - u o ( z ) ,

C;(O.p) = 1, & ( l , p ) = 0,

(15.61a)

( 15.60b) (15.60~) U2

with the properties

Uz(0.p)= 0, U ~ ( 1 , p=) 1, Le..

(15.61b)

I

15. Inte.qra1 Trunsformations

722

(15.61d) The required solution of the transformed equation has the form (15.62) U ( ~ , P=) Ao(P)U I ( ~ ,+P2 4)1 ( ~ )UZ(X,P). 3. Inverse Transformation The inverse transformation is especially easy in the case of 1 + m:

1- (-g)

(15.63a) u ( z , t ) = 2fi

U ( z , p )= ao(p)e-'fi,

-

r3~2

exp

dr.

(15.63b)

15.3 Fourier Transformation 15.3.1 Properties of the Fourier Transformation 15.3.1.1 Fourier Integral 1. Fourier Integral in Complex Representation The basis of the Fourier transformation is the Fourier integral, also called the integral f o r m u l ~of Fourier If a non-periodic function f ( t ) satisfies the Dirichlet conditions (see 7.4.1.2, 3., p. 420) in an arbitrary finite interval, and furthermore the integral

T i f ( t ) idt

11

1 t'X+m e'"("-"f(r)d w d r is convergent, then f ( t ) = 2a

(15.64a)

(15.64b)

-m -m

--35

at every point where the function f ( t ) is continuous, and (15.64~) 0

--3o

at the points of discontinuity.

2. Equivalent Representations Other equivalent forms for the Fourier integral (15.64b) are: (15.65a) 'X

2. f ( t ) = /[a(.)

coswt

+ b(w) sinwt] dw

with the coefficients

(15.65b)

0

a(.)

=

1

1 +m ; f(t)coswtdt

(15.65~)

-'X

b(w) =

1 T f ( t ) sinwtdt.

(15.65d)

-W

'X

3. f ( t ) = 1 A ( w ) c o s [ . t + ~ ( w ) ] d w .

(15.66)

0

A(w)sin[wt+ip(w)]dw.

(15.67)

15.3 Fourier Transformation 723

The following relations are valid here:

(15.68~)

(15.68d) (15.68f)

15.3.1.2 Fourier Transformation and Inverse Transformation 1. Definition of the Fourier Transformation The Fourier transformation is an integral transformation of the form (15.1a)) which comes from the Fourier integral (15.64b) if we substitute

F ( d ) = T e - ' " ' f ( r ) dr.

(15.69)

-m

We get the following relation between the real original function f ( t )and the usually complex transform F(Ld): (15.70) -r;

In the brief notation we use E

/ e-'"tf(t)

tr;

F ( d ) = F{f ( t ) } =

dt

(15.71)

-ffi

The original function f ( t ) is Fourier transformable if the integral (15.69), Le., an improper integral with the parameter w , exists. If the Fourier integral does not exist as an ordinary improper integral, we consider it as the Cauchy principal value (see 8.2.3.3, l.,p. 455). The transform F ( w ) is also called the Fozlrzer transform: it is bounded, continuous, and it tends to zero for lw(+ co: lim F ( d ) = 0. (15.72) lw~+ffi

The existence and boundedness of F ( w ) follow directly from the obvious inequality (15.73) The existence of the Fourier transform is a sufficient condition for the continuity of F(w) and for the properties F ( w ) -+ 0 for 1wl + w. This statement is often used in the following form: If the function f ( t ) in (-co,co)is absolutely integrable, then its Fourier transform is a continuous function of w,and (15.72) holds. The following functions are not Fourier transformable: Constant functions, arbitrary periodic functions (e.g sin ij t , cos ~ t )power , functions, polynomials, exponential functions (e.g., e a t , hyperbolic functions). ~

2. Fourier Cosine and Fourier Sine Transformation In the Fourier transformation (15.71), the integrand can be decomposed into a sine and a cosine part. So. we get the sine and the cosine Fourier transformation.

724

15. Intearal Transformations

1. Fourier Sine Transformation 33

/

( 15.74a)

Fs(w)= FS{f ( t ) } = f ( t ) sin (ut)dt. 0

2. Fourier Cosine Transformation M

/

( 15.74b)

Fc(w)= FC{f ( t ) } = f ( t )cos (w t ) dt. 0

3. Conversion Formulas Between the Fourier sine (15.74a) and the Fourier cosine transformation (15.74b) on one hand, and the Fourier transformation (15.71) on the other hand, the following relations are valid: (15.75a) F ( w ) = Ft f(t) 1 = Fc{ f(t) + f(-4 I - iFS{ f(t)- f(-C 13 i

F'(w) = sF{ f(ltl)signt},

(15.75b)

1

FCb)= f{ f ( t ) I.

(15.75~)

For an even or for an odd function f ( t )we have the representation

f ( t ) even: F{f ( t ) 1 = 2 3 4 f ( t )I, f ( t ) odd: 3{f ( t )} = -2iFS{ f ( t ) }.

(15.75d)

3. Exponential Fourier Transformation Differently from the definition of F ( w ) in (15.71), the transform (15.76)

F,(w) = Fe{f(t)} = T e ' " ' f ( t )d t -M

is called the ezponentzal Fourzer transformatton, and we have F ( d ) = 2F,(-w).

(15.77)

4. Tables of the Fourier Transformation Based on formulas (15.75a,b,c) we either do not need special tables for the corresponding Fourier sine and Fourier cosine transformations, or we have tables for Fourier sine and Fourier cosine transformations and we may calculate F(w)with the help of (15.75a,b,c). In Table 21.12.1 (see 21.12.1, p. 1066) and Table 21.12.2 (see 21.12.2, p. 1072) the Fourier sine transforms Fs(w), the Fourier cosine transforms FC(w), and for some functions the Fourier transform 3 ( w ) in Table 21.12.3 (see 21.12.3, p. 1077) and the exponential transform F,(w) in Table 21.12.4 (see 21.12.4, p. 1079) are given. IThe function of the unipolar rectangular impulse f ( t ) = 1 for It1 < to, f ( t ) = 0 for It1 > to (.4.1) (Fig. 15.21) satisfies the assumptions of the definition of the Fourier integral (15.64a). Ac1 +to 2 . cos w t dt = -sin w to and b(w) = cording to (15.65c,d) we get for the coefficients a ( @ ) = T

1

2 sin w t dt = 0 (A.2) and so from (15.65b), f ( t ) = -

/ /

TW

-tn

m

sin wtocos w t

w 5. Spectral Interpretation of the Fourier Transformation 71

-to

T O

dw (.4.3).

Analogously to the Fourier series of a periodic function, the Fourier integral for a non-periodic function has a simple physical interpretation. A function f ( t ) ,for which the Fourier integral exists, can be represented according to (15.66) and (15.67) as a sum of sinusoidal vibrations with continuously changing frequency w' in the form

A ( w ) d w sin[wt + p(w)]>

(15.78a)

A(w)dwc o s [ w t + @ ( w ) ] .

(15.78b)

The expression A(w)dw gives the amplitudeof the wave components and p(w) and $ ( w ) are the phases.

15.9 Fourier Transformation 725

-to

0'

to

t

Figure 15.21

Figure 15.22

We have the same interpretation for the complex formulation: The function f ( t )is a sum (or integral) of summands depending on w of the form

L F (dw~eiut; ) 277

(15.79)

1 where the quantity -F(w) also determines the amplitude and the phase of all the parts. 27T

This spectral interpretation of the Fourier integral and the Fourier transformation has a big advantage in applications in physics and engineering. The transform

~ ( w =) IF(w)le'+) or l ~ ( w ) eiv@) l ( 15.80a) is called the spectrum or frequency spectrum of the function f ( t ) , the quantity IF(.)I = 7 7 4 w ) (15.80b) is the amplitude spectrum and io(.) and @(w)are the phase spectra of the function f(t). The relation between the spectrum F ( w ) and the coefficients (15.65c,d) is (15.81) F(w)= T [ a(.) - ib(w)], from which we get the following statements: 1. If f ( t ) is a real function, then the amplitude spectrum F ( w ) is an even function ofw, and the phase spectrum is an odd function of w. 2. If f ( t ) is a real and even function, then its spectrum F ( w ) is real, and if f ( t )is real and odd, then the spectrum F ( w ) is imaginary. I If we substitute the result (A.2) for the unipolar rectangular impulse function on p. 724 into (15.81), then we get for the transform F ( w ) and for the amplitude spectrum IF(u)l (Fig.15.22) sin w to sin u t o F ( w ) = F{ f ( t ) } = na(w) = 2(A.3), IF(u)I = 2 - (A.4). The points of contact of W I w ! 2 the amplitude spectrum lF(w)l with the hyperbola - are at wto = f ( 2 n t l ) ? ( n = 0 , 1 , 2 , .. .) . w 2

15.3.1.3 Rules of Calculation with the Fourier Transformation As we have already pointed out for the Laplace transformation, the rules of calculation with integral transformations mean the mappings of certain operations in the original space into operations in the image space. If we suppose that both functions f ( t ) and g(t) are absolutely integrable in the interval (--33, -33) and their Fourier transforms are FkJ)= F{f ( 4 1 and G(w)= F{ g(t)1 (15.82) then the following rules are valid.

1. Addition or Linearity Laws If cy and

B are two coefficients from (-m, ca),then:

F{af(t)+ Ps(t) } = ~ F ( w+ )PG(w).

(15.83)

726

15. Inteoral Transformations

2. Similarity Law For real cy # 0,

FI f ( t l a )1 = IQI F(aw). 3. Shifting Theorem

(15.84)

For real cy # 0 and real p,

F{ j(cyt + p) } = ( ~ / ae'PW'aF(w/cu) ) F{f ( t - t o )} = e-iwtoF(w).

(lL85a)

or

(15.85b)

If we replace to by -to in (15.85b), then we get

F{j ( t + t o )} = elWtoF(u).

(15.85~)

4. Frequency-Shift Theorem For real cy

> 0 and R E

(-m, ea),

F{e'"f(cyt) } = ( l / a ) F ( ( w- p ) / a ) or F{e'""f(t) } = F(u - wo).

(15.86a) (15.86b)

5. Differentiation in the Image Space If the function t n f ( t )is Fourier transformable, then

F{ t n f ( t ) }= i n W ( w ) ,

(15.87)

) the n-th derivative of F ( w ) . where F ( n ) ( w denotes 6. Differentiation in the Original Space 1. First Derivative If a function f ( t ) is continuous and absolutely integrable in (-m, 03) and it tends to zero fort -+ &coo,and the derivative f'(t) exists everywhere except, maybe, at certain points, and this derivative is absolutely integrable in (-ea,co), then (15.88a) F{ f'(4 1 = iwF{ f(t)}. 2. n-th Derivative If the requirements of the theorem for the first derivative are valid for all derivatives up to f ( n - l ) , then

(15.88b) F{f ( ' ) ( t )} = (iw)nF{f ( t )}. These rules of differentiation will be used in the solution of differential equations (see 15.3.2. p. 729).

7. Integration in the Image Space If the function t n f ( t )is absolutely integrable in (--03, ~ 0 then ) ~the Fourier transform of the function f ( t ) has n continuous derivatives, which can be determined for k = 1,2, . . . , n as +m

/

d k F ( w ) +m dk [ e-'"tf(t)] dt = (-l)k e @ t k f ( t ) dt: d u k = --3o -m

/

( 15.89a)

and we have dkF(d) lim -- 0. dwk With the above assumptions these relations imply that

(15.89b)

d i f c x

dnF(w) dwn '

F{tnf(t) } = in -

(15.89~)

15.3 Fourier Transformation 727

8. Integration in the Original Space and the Parseval Formula 1. Integration Theorem If the assumption

1

tso

f ( t )dt = 0

(l5.9Oa)

is fulfilled, then

F

(15.90b)

-X

2. Parseval Formula If the function f ( t ) and its square are integrable in the interval then

(-ffi.ffi),

(15.91) -w

9. Convolution The two-sided convolutzon

1

'X

fl(t)

* fdt) =

(15.92)

f1(r)fdt- 7 )d7

-X

is considered in the interval (-03, x)and exists under the assumptions that the functions h ( t ) and f2(t) are absolutely integrable in the interval (-03, 03). If fl(t) and fz(t) both vanish for t < 0, then we get the one-szded convolutzon from (15.92) (15.93)

So. it is a special case of the two-sided convolution. While the Fourier transformation uses the twosided convolution, the Laplace transformation uses the one-sided convolution. For the Fourier transform of a two-sided convolution we have (15.94)

(15.95) -X

-X

exist. Le.. the functions and their squares are integrable in the interval (-w, ffi) ICalculate the two-sided convolution$(t) = f ( t ) * f(t) =

:1

f ( r ) f ( t - r )dr (il.1)for the function

of the unipolar rectangular impulse function (A.1) on p. 724. Since y ( t ) =

1

to

f(t - r ) dr =

-60

for -2to 5 t 5 0. ~ ( t=)

1

ttta

:1

f(r)dr (A.2): we get fort < -2to and t > 2t0, w(u)= 0 and

dr = t t 2to. (A.3)

-60

Analogously. n e get for 0 < t 5 2to: y ( t ) =

1

to

t-to

dr = -t

+ 2to.

(.4.4)

.4ltogether, for this convolution (Fig. 15.23) we get t + 2t0 for -2to 5 t 5 0, ~ ( t=)f ( t ) * f ( t ) = -t 2to for 0 < t 5 2to. (A.5) (0 for It1 > 2to. n'ith the Fourier transform of the unipolar rectangular impulse function (.4.1) (p. 724 and Fig. 15.21)

+

728

15. Inteqral Transfomations

we get P(d)=

F{ v(t)}

=

sin' w to

F{ f ( t ) * f(t) = [ F ( w ) ] ' = 4-

spectrum of the function f ( t ) we have IF(w)I = 2

1

(A.6) and for the amplitude

w2

sin' w to and IF(w)I' = 4 . (A.7)

w

W2

-i. -2t0

~

2t,

-1

Figure 15.23

Figure 15.24

10. Comparing the Fourier and Laplace Transformations There is a strong relation between the Fourier and Laplace transformation, since the Fourier transformation is a special case of the Laplace transformation with p = iw. Consequently, every Fourier transformable function is also Laplace transformable, while the reverse statement is not valid for every f ( t ) . Table 15.2 contains comparisons of several properties of both integral transformations. Table 15.2 Comparison of the properties of the Fourier and the Laplace transformation

1

1 Laplace transformation

Fourier transformation ~ ( d=)

F{ f ( t ) } =

tm

J' e-'"'f(t) dt

~ ( p=) L{ f ( t ) , p } = r e - P t f ( t ) d t

--cc

is real, it has a physical meaning, e.g., frequency. One shifting theorem. interval: (-x.+m) Solution of differential equations, problems described by two-sided domain, e.g., the wave equation. Differentiation law contains no initial values.

p is complex, p = r

J

~

1

+ iz.

Two shifting theorems. interval: [ 0 , co) Solution of differential equations, problems described by one-sided domain. e.g., the heat conduction equation. Differentiation law contains initial values.

~~~

Convergence of the Fourier integral depends only Convergence of the Laplace integral can be improved by the factor e-@. on f ( t ) . It satisfies the one-sided convolution law. It satisfies the two-sided convolution law.

15.3.1.4 Transforms of Special Functions IA: Which image function belongs to the originalfunction f ( t ) = e-altl, Rea > 0 (A.l)? Considere-iWt-altldt= e-""-<dt+ ingthat It1 = -tfort < Oandltl = tfort > Owith(15.71)weget:

/_"

e-ARea

and Re a > 0. the limit exists for A

-+

2a co,so we get F ( w ) = F{ ecaltI } = -(A.3). a2

+ w2

B: Which image function belongs to the original function f ( t ) = e-at, Re a > O? The function is not Fourier transformable, since the limit A -+ co does not exist.

15.3 Fourier Transformation 729

C: Determinate the Fourier transform of the bipolar rectangular impulse function (Fig. 15.24) 1 for -2to < t < 0: -1 for 0 < t < 2t0, (C.1) 0 for It1 > 2to where p(t) can be expressed by using equation (A.l) given for the unipolar rectangular impulse on p. 724. since p(t) = f ( t + to) - f(t - to) (C.2). By the Fourier transformation according to (15.85b, 15.85~)we get @(w) = F{p(t) } = eiwtOF(w) - e-iwtoF(w), (C.3) from which. using (A.1).we have: sin2w t o 2 sin w to Q(wj = - e-id ) iJ = 4i((2.4). W

D: Transform of a damped oscillation: The damped oscillation represented in Fig. 15.25a is given 0 for t < 0, To simplify the calculations, the Fourier transformation is calculated with the complex function f * ( t ) = e(-a+ido)t,since f ( t ) = Re (f"( t ) ) .The Fourier transformation gives

F{f * ( t )} CY

CY2

=

is

+ i(w.0 - IJ) + (d- W o ) 2

n f(t)1=

e-iwte(-a+iwo)t

e-atei(w-wo)t

=

-a

I=

1

O0

+ i(w0 - w )

cy

-

- iwo - w)

The result is the Lorentz or Breit-Wigner curve (see also p. 94) '

cy (y2

dt

+( -.

(Fig. 15.25b). It has a unique peak in the frequency domain which cor-

responds to a damped oscillation in the time domain.

t b)

Figure 15.25

Figure 15.26

15.3.2 Solution of Differential Equations using the Fourier Transformat ion Analogously t o Laplace transformation. an important field of application of the Fourier transformation is the solution of differential equations. since these equations can be transformed by the integral transformation into a simple form. In the case of ordinary differential equations we get algebraic equations, in the case of partial differential equations we get ordinary differential equations.

15.3.2.1 Ordinary Linear Differential Equations The differential equation y'(t)

+ a y(t) = f(t)

with f ( t )=

1 for ltl < to, 0 for It1 2 to,

(15.96a)

730

15. Inteqral Transformations

i.e., with the function f ( t ) of Fig. 15.21,is transformed by the Fourier transformation FI Y(t) } = I~(d) into the algebraic equation sin wto 2 sin u to iuY+aY = , (15.96~) soweget Y(w)= 2- w(a iw) w The inverse transformation gives

+

y ( t ) = F-1{

Y ( w )} = F-l

and

dw

for -co

(15.96b) (15.96d)

(15.96e)

< t < -to,

Function (15.96f) is represented graphically in Fig. 15.26.

15.3.2.2 Partial Differential Equations 1. General Remarks The solution of a partial differential equation is a function of at least two variables: u = u ( z ,t ) . .4s the Fourier transformation is an integration with respect to only one variable, the other variable is considered a constant during the transformation. Here we keep as a constant the variable z and transform with respect to t:

/

tx

F{ u ( z ,t ) } =

e-lwtu(z,t) dt = G(z,w ) .

(15.97)

-X

During the transformation of the derivatives we also keep the variable z: (15.98) The differentiation with respect to z supposes that it is interchangeable with the Fourier integral: (15.99)

So, we get an ordinary differential equation in the image space. We also have to transform the boundary and initial conditions into the image space, of course.

2. Solution of the One-Dimensional Wave Equation for a Homogeneous Medium 1. Formulation of t h e P r o b l e m The one-dimensional wave equation with vanishing perturbation term and for a homogeneous medium is: (15.100a) u,, - utt = 0. Like the three-dimensional wave equation (see 9.2.3.2,~. 534), the equation (l5.100a) is a partial differential equation of hyperbolic type. The Cauchy problem is correctly defined by the initial conditions u(z.0) = f(z) (-cc < z < co), ut(z.0) = g(z) (0 5 t < co). (15.100b) 2. Fourier Tkansformation We perform the Fourier transformation with respect to z where the time coordinate is kept constant: F{U(X%t ) } = U(d.t ) . (15.101a)

15.4 2-Transformation 731

U'e get: dzu(d t )

(id)%(&. t ) - -= 0

with

dt2

(15,101h)

( 15.101e) (15.101d) 2 u + U'I = 0. (15.101e) The result is an ordinary differential equation with respect t o t with the parameter w of the transform. The general solution of this known differential equation with constant coefficients is u ( d . t ) = CleiVt+ C2e-iwt. (15.102a) \Ye determine the constants C1 and C2 from the initial values (15.102b) u ( d 30 ) = C1+ Cz = F ( w ) . d ( w , 0) = iwC1 - iwC2 = G(w) and we get 1 1 1 1 (15.102c) C1 = - [ F ( d ) t -G(u)]. C, = - [ F ( u ) - ,G(w)]. 2 1W 2 l(u' The solution is therefore 1 1 1 1 u(d,t) = -[ F ( J ) + -G(w)]e'"'+ -[F(w.)- TG(w)]e-'"t (15.102d) 2 1U 2 1w

F{ u(.. 0) } = U(W,0) = F{f(.) } = F ( d ) , F{ u~(z, 0) } = ~ ' ( 4 . 0 = ) F{ g(z) } = G(w)

3. Inverse Transformation We use the shifting theorem

F{f ( a z + b ) } = 1 / a . eibWIaF(u/a), for the inverse transformation of F ( w ) ,and we get

(15.103a)

e i W t F ( w }) = f ( z + t ) ) Applying the integration rule

(15.103b)

F 7-l

F-'[e - i W t ~ ( w )=] f(x - t ) .

1 } {-:

f(7)dr = -F(w) :i

{

7G(d)eid'} =

:1

after substituting s

(15.103~)

weget

] F-'{G(w)eiwf} ] d~ =

--si

-m

1

X t t

+ t ) dT =

g (T

g ( z )dz

(15.103d)

-m

+ t = z. analogously to the previous integral we get (15.103e)

Finally, the solution in the original space is (15.104)

15.4 Z-Transformation In natural sciences and also in engineering we can distinguish between continuous and discrete processes. jf'hile continuous processes can he described by differential equations, the discrete processes result mostly in diflereerence equations. The solution of differential equations mostly uses Fourier and Laplace transformations, however. to solve difference equations other operator methods have been

732

15. Inteqral Transformations

developed. The best known method is the z-transformation, which is closely related to the Laplace transformation.

15.4.1 Properties of the Z-Transformation 15.4.1.1 Discrete Functions

L e fo

f,

0 T 2T3T

t

Figure 15.27

If a function f ( t ) (0 5 t < m) is known only at discrete values t, = nT ( n = 0,1,2,. . . ; T > 0 is a constant) of the argument, then we write f ( n T ) = fn and we form the sequence {fn}. Such a sequence is produced, e.g., in electrotechnics by “scanning” a function f ( t )at discrete time periods t,. Its representation results in a step function (Fig. 15.27). The sequence {fn} and the function f ( n T )defined only at disCrete points of the argument, which is called a discrete function, are equivalent.

15.4.1.2 Definition of the Z-Transformation 1. Original Sequence and Transform We assign the infinite series (15.105) to the sequence {in}.If this series is convergent, then we call the sequence {f,} z-transformable, and we write F ( z ) = Z{fn}, (15.106) { f,} is called the original sequence, F ( z ) is the transform. z denotes a complex variable, and F ( z ) is a complex-valued function. Ifn = 1 (n = 0,1,2,, , .) . The corresponding infinite series is

F ( z )=

e (i)n.

n=O

(15.107) I ,

It1< 1 and its sum is

It represents a geometric series with common ratio 1/z, which is convergent if -

F(z)=

z, It is divergent for - 2 1. Therefore, the sequence (1) is z-transformable for - < 1, iil IkI 2-1

Le., for every exterior point of the unit circle 1x1 = 1 in the z plane. 2. Properties Since the transform F ( z ) according to (15.105) is a power series of the complex variable l / z , the properties of the complex power series (see 14.3.1.3,p. 688) imply the following results: a) For a z-transformable sequence {f,},there exists a real number R such that the series (15.105) is absolutely convergent for Iz/ > 1/R and divergent for IzI < 1/R. The series is uniformly convergent for IzI 2 l/Ro > 1/R. R is the radius of convergence of the power series (15.105) of l/z. If the series is convergent for every IzJ> 0, we write R = co. For non z-transformable sequences we write R = 0. b) If {fn} is z-transformable for IzI > 1/R, then the corresponding transform F ( z ) is an analytic func> 1/R and it is the unique transform of {f,}. Conversely, if F ( z ) is an analytic function tion for for > 1/R and is regular also at z = co, then there is a unique original sequence {f,} for F ( z ) . Here, F ( z ) is called regular at z = m, if F ( z ) has a power series expansion in the form (15.105) and F(m)= fo.

3. Limit Theorems Analogously to the limit properties of the Laplace transformation ((15.7b), p. 708), the following limit theorems are valid for the z-transformation:

15.4 Z- Transformation 733

a) If F ( z ) = Z { f n }exists, then fo = hlF(Z).

(15.108)

Here z can tend to infinity along the real axis or along any other path. Since the series 1 1

Z { F ( z )- fo} = fl + f i r + f32

zz

1 1 1 2' [ F ( z ) - f 0 - fi;) = f2 t f3; + f4'2

+. . +

' I '

'

,

(15.109)

>

(15.110)

are obviously z transforms. analogously to (15.108) we get f l = ~ ~ g F ( Z ) - . f o ~ ,f z = ~ ~ ~ Z ~ { F ( z ) -1f o - f l ~ } , . . .

(15.111)

IVe can determine the original function {fn} from its transform F ( z ) in this way. b) If fn exists, then lim

nim

fn =

lim ( z - 1)F(z).

(15.112)

ZilCO

\Ve can however determine the value

Ail fn from (15.112) only if its existence is guaranteed, since the

above statement is not reversible 1 lim (z-1)-=O,but z+l

If n = ( - l ) n ( n = 0 , 1 . 2 . . . . ) . T h e n Z { f , } = L a n d z t 1 not exist.

nlim(-l)ndoes im

2-+1+0

15.4.1.3 Rules of Calculations In applications of the z-transformation it is very important to know how certain operations defined on the original sequences affect the transforms, and conversely. For the sake of simplicity we will use the notation F ( z ) = Z { f n }for 121 > 1/R.

1. Translation We distinguish between forward and backward translations. 1. First Shifting Theorem: Z{fn-k} = z - ~ F ( z ) ( k = 0 , l . 2 , . . .) . here fn-k = 0 is defined for n - k < 0. k-1

2. Second Shifting Theorem: Z{fn+k}= zk [ p ( z ) -

fv "=O

(;)1 "]

(15.113)

( k = 1 , 2 , . . .) . (15.114)

2. Summation (15.115)

3. Differences For the dzferences Lfn = fn+l - fn, the following holds:

Z{Afn) Z{A'fn}

= =

Amfn = A(Am-'fn)

(VI=

1 , 2 , . , , : Aofn = f n )

1)F(2)- z f o , - ~ ) ' F ( z-) Z ( Z - I)fo - ~ A f o ,

(2(Z

-

(15.117 ) k-1

Z{Lkfn} =

(15.116)

(2 - l)kF(Z)

-2

(2 Y'O

1)k--.-'A.

f0.

734

15. Inteoral Transformations

4. Damping For an arbitrary complex number A 2{Xnfn} = F

# 0 and /zI > -:1x1

(i) .

R

(15.118)

5 . Convolution The convolution of two sequences {fn} and {gn} is the operation

fn *

n

gn =

(15.119)

fu gn-v

"=a

If the z-transformed functions 2{fn} = F ( z ) for then

Z{fn for

121

1x1 > 1/R1 and 2{gn}

= G ( z )for IzI

> l/Rz exist,

* gn} = P ( z ) G ( z )

> max

(+R

(15.120)

L). Relation (15.120) is called the convolution theorem of the z-transformation. Rz

It corresponds to the rules of multiplying two power series.

6. Differentiation of the Transform 2 { n f n }= -2- d F ( z ) dt

(15.121) '

We can determine higher-order derivatives of F ( z ) by the repeated application of (15.121).

7. Integration of the Transform Under the assumption fo = 0. (15.122)

15.4.1.4 Relation t o the Laplace Transformation If we describe a discrete function f ( t ) (see 15.4.1.1, p. 732) as a step function, then for nT 5 t < ( n + 1)T ( n = 0 , 1 , 2 , . . . : T > 0. T const). (15.123) f ( t ) = f ( n T ) = fn 118 can use the Laplace transformation (see 15.2.1.1, l . ,p. 708) for this piecewise constant function, and for T = 1 we get:

The infinite series in (15.124) is called the dzscrete Laplace transformatzon and it is denoted by

D:

X

D{f(t)} = D { f n } =

1fne-np.

(15.125)

n=O

If n e substitute e p = z in (15.125), then D{fn} represents a series with increasing powers of l/z. which is a so-called Laurent s e w s (see 14.3.4. p. 690). The substitution eP = z suggested the name of the z transformation. With this substitution from (15.124) we finally get the following relations between the Laplace and z-transformation in the case of step functions:

p F ( p ) = (1 -

1 --) F ( t ) (15.126a)

(

;)

or P L { f ( t ) }= 1 - - 2{fn}.

(15.12613)

In this way. we can transform the relations of z-transforms of step functions (see Table 21.13.p. 1080)

15.4 2-Transformation 735

into relations of Laplace transforms (see Table 21.11,p. 1061) of step functions, and conversely.

15.4.1.5 Inverse of t h e Z-Transformation The inverse of the z-transformation is to find the corresponding unique original sequence {fn} from its transform F ( z ) . We write

z - ' { F ( z ) } = {jn}.

(15.127)

There are different possibilities for the inverse transformation

1. Using Tables If the function F ( z ) is not given in tables. then we can try to transform it to a function which is given in Table 21.13.

2. Laurent Series of F ( z ) We get the inverse transform directly from the definition (15.105)) p. 732, if a series expansion of F ( z ) with respect to 1/z is known or if it can be determined.

I:(

3. Taylor Series of F is a series of increasing powers of 2 , from (15.105) and the Taylor formula we get (15.128)

4. Application of Limit Theorems Using the limits (15.108) and (Kill), p. 733, we can directly determine the original sequence {fn} from its transform F ( z ) . H F ( t )=

22

We use the previous four methods. - 2 ) ( 2 - 1)2 ' 1. By the partial fraction decomposition (see 1.1.7.3,p. 15) of F ( z ) / zwe obtain functions which are contained in Table 21.13. (2

F(2) -

-2

2 (2-2)(2-1)2 22

=

A 2 - 2

22

22

(2-1)2

2-1

+- B (2-1)2

+-.2 -C1

so

F ( 2 ) = -- -- - and therefore jn= 2 ( P - n - 1) for n 2 0. 2-2

2. F ( z ) will be a series with decreasing powers of 2 by division: 22 1 1 1 1 1 F ( z ) = z3 - 4 2 + 52 - 2 = 2-zZ t 8-23 t 22-z4 + 52- t 114-z6

From this expression we get fo = fi = 0, f z = 2, f 3 = 8, not obtain a closed expression for the general term f n .

3. For formulating F

f4

+ ... .

= 22, f 5 = 52.

(15.129) f6

=

114.

. . ., but we do

and its required derivatives, (see (15.128)) we consider the partial fraction

736

15. Intearal Transformations

decomposition of F ( z ) ,and get: 2 22 = 1-22 (1-2)2

4

- -

dz

(E)

d2F

dz2

i.e.,

42 4 -~ - - - (1 - 22)’ (1 - 2) 3 (1 - 2 ) ’ ’ -

d3F(:) dz3

2

1-2’

F ( ~ ) = o

for z = 0,

i.e., dz

16 122 12 (1 - 2 ~ ) (1~- z ) ~ (1 - z ) ~ ’

96

(1- 2 2 ) 4

482 (1 - z ) 5

48 (1 - ~

’

(’I

d3F i.e., 2-48 dz3

) 4

for z = O ,

from which we can easily obtain fo, f l , f 2 , f31 . . . considering also the factorials in (15.128). 4. .4pplication of the limit theorems (see 15.4.1.2, 3., p. 732) gives: 22 fo = hlF(z) = lim = 0, z+m 23 - 422 + 52 - 2 222 f l = ) m z ( F ( z ) - fo) = lim = 0, 2-m z3 - 4z2 + 52 - 2

- fo - fl - - f2 ?) = ;;I 23 (

--)

22 2 =8, ... - 42’ 52 - 2 2’ where the Bernoulli-l’Hospita1 rule is applied (see 2.1.4.8, 2., p. 54). We can determine the original sequence {fn} successively.

j3 =

p%

23 ( ~ ( 2 )

1

1

Z3

+

15.4.2 Applications of the Z-Transformation 15.4.2.1 General Solution of Linear Difference Equations A linear difference equation of order k with constant coefficients has the form (15.130) akyn+k + ak-lyn+k-l+ ’. + ~ Z Y ~ + Za l y n + l + aovn = gn (n = O,I,2 . . .). Here k is a natural number. The coefficients a, (z = 0 , l . . . . , k ) are given real or complex numbers and they do not depend on n. Here a0 and ak are non-zero numbers. The sequence {gn} is given, and the sequence {y,} is to be determined. To determine a particular solution of (15.130) the values yo, yl,. . . , yk-l have to be previously given. Then we can determine the next value y k for n = 0 from (15.130). We then get yk+l for n = 1 from y1. y ~ .. . . yk and from (15.130). In this way, we can calculate recursively all values yn. We can give however a general solution for the values yn with the z-transformation. We use the second shifting theorem (15.114) applied for (15.130) to get:

+

[

akzk Y(2) - yo - y l K 1 - .

. - yk-l~-(k-’)] + . . + a l z [ Y ( z ) - yo] + aoY ( 2 ) = G ( z ) (15.131) .

=p(z), HerewedenoteY(2) = Z(y,) andG(z) = Z(g,). Ifwesubstitutea~zkta~-lzk-’+~~~+alz+ao then the solution of the so-called transformed equation (15.131) is

(15.132)

As in the case of solving linear differential equations with the Laplace transformation, we have the similar advantage of the z-transformation that initial values are included in the transformed equation, so the solution contains them automatically. We get the required solution {y,} = 2-'{ Y ( z ) }from (15.132) by the inverse transformation discussed in 15.4.1.5, p. 735.

15.4.2.2 Second-Order Difference Equations (Initial Value Problem) The second-order difference equation is (15.133) Yn+2 + Q l Y n + l + QYn = Sn. where yo and y1 are given as initial values. Using the second shifting theorem for (15.133) we get the transformed equation 2'

[EI(z) -

If we substitute '2

YO

- YI;

ll

+ w [Y ( z )- yo] + aoY(z)= G ( z ) .

(15.134)

+ a l z + a0 = p ( z ) .then the transform is (15.135)

If the roots of the polynomialp(z) ar eal and az, then cy1 # 0 and cyz # 0, otherwise a. is to be zero, and then the difference equation could be reduced to a first-order one. By partial fraction decomposition and applying Table 21.13 for the z-transformation we get the following: for

a1

# QZ,

for

a 1

= CYZ,

(15.136a) Since pO = 0.by the second shifting theorem (15.136b) and by the first shifting theorem

Here we substitute p-l = 0. Based on the convolution theorem we get the original sequence with n

~n = C P n - l q n - u "=O

+ y o ( P n + l +alp") + ~

(15.136d)

1 ~ 1 .

Since p - 1 = PO = 0, this relation and (15.136a) imply that in the case of cy1

This form can be further simplified, since a 1 = -(a1 + az) and Vieta, 1.6.3.1. 3., p. 44).so

a0

# cyz

= alaz (see the root theorems of

(15.136f)

738

15. Inteoral Transformations

In the case of ai = a2 similarly n

gnJv

y, =

-

1)Q’;-2

- yoao(n

- 1)Cl-2

+ y1na;-’.

( 15.136g)

v=2

In the case of second-order difference equations the inverse transformation of the transform Y ( z )can be performed without partial fraction decomposition if we use correspondences such as, e.g.,

[

)

(15.137) sinh n and the second shifting theorem. By substituting al = -2acosh b, and an = a2 the original sequence of (15.135) becomes: 2-1

y

z 2 ’- 2az cosh b t a2

1

[

-7 - sinhb u=2

gn-&’

sinh(v - l ) b - yOan sinh(n - l)b

1

+ y1an-l sinhnb .

(15.138)

This formula is useful in numerical computations especially if a0 and a1 are complex numbers

Remark: Notice that the hyperbolic functions are also defined for complex variables.

15.4.2.3 Second-Order Difference Equations (Boundary Value Problem) It often happens in applications that the values yn of a difference equation are needed only for a finite number of indices 0 5 n 5 N . In the case of a second-order difference equation (15.133) both boundary values yo and y , ~are usually given. To solve this boundary value problem we start with the solution (15,136f) of the corresponding initial value problem. where instead of the unknown value y1 we have to introduce y ~ If. we substitute n = Y into (l5,136f), we can get y1 depending on yo and y , ~ :

If we substitute this value into (l5.136f) then we get

+-a? -1 a;

[ yo(a;a; - a;.,”,

+ yN(a;-

a;)].

(15.140)

The solution (15.140) makes sense only if Q? - a; # 0. Otherwise, the boundary value problem has no general solution, but analogously to the boundary value problems of differential equations, we have to solve the eigenvalue problem and to determine the eigenfunctions.

15.5 Wavelet Transformat ion 15.5.1 Signals If a physical object emits an effect which spreads out and can be described mathematieally, e.g.. by a function or a number sequence, then we call it a signal. Signal analysis means that we characterize a signal by a quantity that is typical for the signal. This means mathematically: The function or the number sequence, which describes the signal. will be mapped into another function or number sequence. from which the typical properties of the signal can be clearly seen. For such mappings, of course. some informations can also be lost. The reverse operation of signal analysis, Le.. the reconstruction of the original signal, is called signal synthesis. The connection between signal analysis and signal synthesis can be well represented by an example of Fourier transformation: A signal f ( t ) (t denotes time) is characterized by the frequency w. Then, formula (l5.141a) describes the signal analysis, and formula (15.141b) describes the signal synthesis:

15.5 Wavelet Transformation 739

/ e-'"'f(t) m

F ( d )=

dt

(15.141a)

and f ( t ) =

-X

&7

eiWtF(cj)dd.

(15.141b)

-m

15.5.2 Wavelets The Fourier transformation has no localization property, Le., if a signal changes at one position, then the transform changes everywhere without the possibility that the position of the change could be recognized "at a glance". The basis of this fact is that the Fourier transformation decomposes a signal into plane waves. These are described by trigonometric functions, which have arbitrary long oscillations with the same period. However, for wavelet transformations there is an almost freely chosen function w ,the wavelet (small localized wave), that is shifted and compressed for analysing a signal. Examples are the Haar wavelet (Fig. 15.28a) and the Mexican hat (Fig. 15.28b). IA Haar wavelet:

v=

{

if p5x5l. %herwise. (15.142) IB Mexican hat: d2 ~ ( x=) --e-"/* (15.143) a) dx2 - (1 - x 2 ) e - r 2 / 2 .(15.144)

-1

0

1

t

b) Figure 15.28

Every function Q comes into consideration as a wavelet if it is quadratically integrable and its Fourier transform Q(d) according to (15.141a) results in a positive finite integral (15.145) Concerning wavelets, the following properties and definitions can be mentioned: 1. For the mean value of the wavelet:

7

v(t) d t = 0

(15.146)

-X

2. The following integral is called the k-th moment of a wavelet y : t"(t) dt.

pic =

(15.147)

-m

The smallest positive integer n such that p,, # 0, is called the order of the wavelet v . IFor the Haar wavelet (15 142). n = 1, and for the Mexican hat (15.144), n = 2. 3. IYhen pk = 0 for every k , y has infinite order. Wavelets with bounded support always have finite order. 4. .4 wavelet of order n is orthogonal to every polynomial of degree 5 n - 1.

15.5.3 Wavelet Transformation For a wavelet y ( t )we can form a family of curves with parameter a: (15.148)

740

15. Intearal Transfornations

In the case of la1 > 0 the initial function $(t)is compressed. In the case of a < 0 there is an additional reflection. The factor 1 / f i is a scaling factor. The functions &(t) can also be shifted with a second parameter b. Then we get a two-parameter family of curves:

t-b (a, b real; a # 0). (e)

1 ca,b

= $0

(15.149)

The real shifting parameter b characterizes the first moment, while parameter a gives the deviation of the function & b ( t ) . The function & , b ( t ) is called a basis junction in connection to the wavelet transformatzon. The wavelet transformation of a function f ( t )is defined as:

L+j ( a . b) = c

7

m

f(t)qa,a(t)d t =

(15.150a)

-E

For the inverse transformation: (15.15Ob) Here cis a constant dependent on the special wavelet $. IUsing the Haar wavelets (15.144) we get 1 if b < t < b + a / 2 , -1 if b + a / 2 < t < b + a ,

and therefore & f ( a ; b) =

~

Ji;lT

(r’l ra

f(t) dt)

f ( t ) dt -

bia/2

(15.151)

, given in (15.151) represents the difference of the mean values of a function f ( t ) The value L V f ( a b) la1 connected at the point b. over two neighboring intervals of length -, 2 Remarks: 1. The dyadzc wavelet transjormatzon has an important role in applications. As a basis function we select the functions

(15.152) Le., we get different basis functions from one wavelet $(t) by doubling or halving the width and shifting by an integer multiple of the width. 2. A wavelet @ ( t ,) where the basis functions given in (15.152) form an orthogonal system, is called an orthogonal wavelet. 3. The Daubechies wavelets have especially good numerical properties. They are orthogonal wavelets with compact support, Le., they are different from zero only on a bounded subset of the time scale.

15.5 Wavelet Transformation 741

They do not have a closed form representation (see [15.9])

15.5.4 Discrete Wavelet Transformat ion 15.5.4.1 Fast Wavelet Transformation The integral representation (15.150b) is very redundant, and so the double integral can be replaced by a double sum without loss of information. We consider this idea at the concrete application of the wavelet transformation. We need 1. an efficient algorithm of the transformation: which leads to the concept of multi-scale analysis, and 2. an efficient algorithm of the inverse transformation, Le., an efficient nay to reconstruct signals from their wavelet transformations, which leads us to the concept of frames. For more details about these concepts see [l5.9],[15.1]. Remark: The great success of wavelets in many different applications, such as calculation of physical quantities from measured sequences pattern and voice recognition data compression in news transmission is based on "fast algorithms". Analogously to the FFT (Fast Fourier Transformation, see 19.6.4.2, p. 925) we talk here about FWT (Fast Wavelet Transformation).

15.5.4.2 Discrete Haar Wavelet Transformation As an example of a discrete wavelet transformation we describe the Haar wavelet transformation: The values f, ( i = 1 , 2 . . . . , N) are given from a signal. The detailed values di (i = 1 , 2 , .. . , N/2) are calculated as: S,

=

L(fzz-i

JT

+fit).

4

=

L(fzt-i

fi

- fit).

(15.153)

The values d, are to be stored while the rule (15.153) is applied to the values s,, i.e.. in (15.153) the values fi are replaced by the values s,. This procedure is continued, sequentially so that finally from (15.154) a sequence of detailed vectors is formed with components dj"). Every detailed vector contains information about the properties of the signals. Remark: For large values of N the discrete wavelet transformation converges to the integral wavelet transformation (15.15Oa).

15.5.5 Gabor Transformat ion Tzme-frequency analyszs is the characterization of a signal with respect to the contained frequencies and time periods when these frequencies occur. Therefore, the signal is divided into time segments (windows) and a Fourier transform is used. We call it a Windowed Fourier Transformation (WFT). The window function should be chosen so that a signal is considered only in the window. Gabor applied the window function

0.04

0

g ( t ) = -e

1

ti --

2g2

(15.155)

f i L l

-0.04

-30

0 Figure 15.29

30 t

(Fig. 15.29). This choice can be explained as g ( t ) , with the "total unit mass". is concentrated at the point t = 0 and the width of the window can be considered as a constant (about 2u).

I

742

15. Inteqral Transformations

The Gabor transformation of a function f(t) then has the form

Gf(.j. s) =

7

j ( t ) g ( t - s)e-'utdt.

(15.156)

-m

This determines, with which complex amplitude the dominant wave (fundamental harmonic) e l d t occurs during the time interval [s - q s t 01 in f , i.e., if the frequency J occurs in this interval. then it has the amplitude lGf(w,s)l

15.6 Walsh Functions 15.6.1 Step Functions In the approximation theory of functions orthogonal function systems have an important role. For instance. special polynomials or trigonometric functions are used since they are smooth, Le., they are differentiable sufficiently many times in the considered interval. However, there are problems. e.g., the transition of points of a rough picture, when smooth functions are not suitable for the mathematical description. but step functzons, piecewise constant functions are more appropriate. Walsh functions are very simple step functions. They take only two function values +1 and -1. These two function values correspond to two states, so the Walsh functions can be implemented by computers very easily.

15.6.2 Walsh Systems .4nalogously to trigonometric functions we can consider periodic step functions. We apply the interval I = [O. 1) as a period interval and we divide it into 2, equally long subintervals. Suppose S,is the set of periodic step functions with period 1 over such an interval. We can consider the different step functions belonging to S, as vectors of a finite dimensional vector space, since every function g E S, is defined by its values 90%gl, gZr., . , gZn-1 in the subintervals and it can be considered as a vector: gT = (gO,gl.g2,...,g2"-1). (15.157) The llalsh functions belonging to S,form an orthogonal basis with respect to a suitable scalar product in this space. The basis vectors can be enumerated in many different ways, so we can get many different Ilalsh systems. which actually contain the same functions. There are three of them which should be mentioned: LValsh-Kronecker functions, Lt'alsh-Kaczmarz functions and Walsh-Paley functions. The Wulsh transformation is constructed analogously to the Fourier transformation, where the role of the trigonometric functions is taken by the Walsh functions. We get, e.g., Walsh series, Walsh polynomials. LValsh sine and IValsh cosine transformations, Walsh integral, and analogously to the fast Fourier transformation there is a Fast Walsh Transformation. For an introduction in the theory and applications of \lalsh functions see [15.6].

743

16 Probability Theory and Mat hemat ical St at ist ics IYhen experiments or observations are made, various outcomes are possible even under the same conditions Probability theory and statistics deal with regularity of random outcomes of certain results with respect to given experiments or observations. (In probability theory and statistics, observations are also called experiments. since they have certain outcomes ) We suppose. at least theoretically, that these experiments can be repeated arbitrarily many times under the same circumstances; namely, these disciplines deal with the statistics of mass phenomena. The term stochastzcs is used for the mathematical handling of random phenomena.

16.1 Combinatorics \Ye often compose new sets, systems. or sequences from the elements of a given set, in a certain way. Depending on the way we do it. we get the notion of perrnutatzon,combznatzon. and arrangement. The basic problem of combinatorics is to determine how many different choices or arrangements are possible with the given elements

16.1.1 Permutat ions 1. Definition A perrnutatzon of n elements is an ordering of the n elements. 2. Number of Permutations without Repetition The number of different permutations of n different elements is P, = n!

(16.1) IIn a classroom 16 students are seated on 16 places. There are 16! different possible arrangements. 3. Number of Permutations with Repetitions The number Pn'k' of different permutations of n elements containing k identical elements ( k 5 n ) is

IIn a classroom 16 schoolbags of 16 students are placed on 16 chairs. Four of them are identical. There are 16!/4! different arrangements of the schoolbags.

4. Generalization The number Pn(kl.kz,,,,,km) of different permutations of n elements containing m different types of elements with multiplicities kl, kZ... . , k, respectively (kl + kz + . . . + k, = n ) is pnWz.....kml = n! kl!kz!. . .k,!

(16.3)

'

ISuppose we compose five-digit numbers from the digits 4,4,5; 5, 5. We can have P5'2,31 =

2 = 10 2!3!

different numbers.

16.1.2 Combinations 1. Definition A combination is a choice of k elements from n different elements not considering the order of them. \Ye call it a combination of k-th order and we distinguish between combinations with and without repetition.

16. Probability Theory and Mathematical Statistics

744

2. Number of Combinations without Repetition

The number Cn(k)of different possibilities to choose k elements from n different elements not considering the order is

Cn[')=

(L)

with 0 5 k 5 n (see binomial coefficient in 1.1.6.4, 3., p. 13),

(16.4)

if we choose any element at most once. We call this a combination without repetition. IThere are pants.

(3

= 27405 possibilities to choose an electoral board of four persons from 30 partici-

3. Number of Combinations with Repetition The number of possibilities to choose k elements from n different ones, repeating each element arbitrarily times and not considering the order is (16.5) In other words. we consider the number of different selections of k elements chosen from n different elements. where the selected ones must not be different. IRolling k dice, we can get C6@)=

G)

= 21 different

( + ') -

different results. Consequently, we can get Cg(*) =

results with two dice.

16.1.3 Arrangements 1. Definition An arrangement is an ordering of k elements selected from n different ones, i.e., arrangements are combinations considering the order.

2. Number of Arrangements without Repetition The number Vn(') of different orderings of Ic different elements selected from n different ones is (16.6)

IHow many different ways are there to choose a chairman, his deputy, and a first and a second assistant for them from 30 participants at an election meeting? The answer is

(3

4! = 657720.

3. Number of Arrangements with Repetition An ordering of k elements selected from n different ones, where any of the elements can be selected arbitrarily many times. is called an arrangement with repetition. Their number is \;(k) = n k , (16.7) IA: In a soccer-toto with 12 games there are 312 different outcomes. IB: We can represent 28 = 256 different symbols with the digital unit called a byte which contains 8 bits. see for example the well-known ASCII table.

16.2 Probability Theory 745

16.1.4 Collection ofthe Formulas of Combinatorics (see Table 16.1)

Type of choice or selection of kfrom

Number of possibilities without repetition with repetition

. 16.2.1 Event, Frequency and Probability

16.2 Probability Theory 16.2.1.1 Events 1. Different Types of Events

All the possible outcomes of an experiment are called events in probability theory, and they form the fundamental probabzlzty set A. if'e distinguish the certazn event. the zmposszble event and random events. The certain event occurs every time when the experiment is performed, the impossible event never occurs: a random event sometimes occurs. sometimes does not. All possible outcomes of the experiment excluding each other are called elementary events (see also Table 16.2). We denote the events of the fundamental probability set A by A. B. C.. . . , the certain event by I , the impossible event by 0. We define some operations and relations between the events; they are given in Table 16.2.

2. Properties of the Operations The fundamental probability set forms a Boolean algebra with complement, addition, and multiplication defined in Table 16.2, and it is called the field of events. The following rules are valid: 1. b)

AB=BA.

(16.10)

2. b)

AA=A.

(16.12)

3. b) A(BC) = (AB)C.

(16.14)

4. b)

AA=0.

(16.15)

(16.16)

5 . b)

A + BC = ( A + B ) ( A+ C).

(16 17)

6. a ) A + B = A B .

(16.18)

6. b)

m=X+B.

(16.19)

7. a) B - 4 = BX,

(16.20)

7. b) z = I - A .

8. a) A(B - C) = AB - AC.

(16.22)

8. b)

1. a ) i l + B = B + A ,

2.a)

A + A = A,

3. a) A

+ ( B + C)= ( A+ B ) + C ,

4.a) A+rl=I, 5. a) A ( B ~

+ C) = AB + AC. __

(16.8)

AB - C = ( A - C ) ( B- C).

(16.9)

(16.13)

(16.21) (16.23)

746

16. Probabilitu Theorv and Mathematical Statistics

9.a) O C A ,

(16.24)

9. b)

10. From A G B follows a) -4 = AB (16.26)

A

C I.

(16.25)

and b) B = A t BX

and conversely. (16.27)

11. Complete System of Events: .4 system of events A, (a E 0, 0 is a finite or infinite set of indices) is called a complete system of events if the following is valid: 11. a) A,Ay = 0 f o r a

# /3

(16.28)

and 11. b)

EAn= I .

(16.29)

aE8

Table 16.2 Relations between events -

Name

Notation

1. Complementary event of A: A 2. Sum of events A and B: AtB AB 3. Product of the events .4 and B: 4. Difference of the events A A - B and B: 5 . Event as a consequence of A G B the other: E 6. Elementary or simple event: 7. Compound event: 8. Disjoint or exclusive events AB = 0 - A and B:

Definition -

A occurs exactly if A does not. A t B is the event which occurs if A or B or both occur. AB is the event which occurs exactly if both A and B

occur. A - B occurs exactly if A occurs and B does not.

A G B means that from the occurrence of A follows the occurrence of B. From E = A t B it follows that E = A or E = B . Event, which is not elementary. The events 4 and B cannot occur at the same time.

mi

IA: Tossing two coins: Elementary events for the separate tossings: See the table on the right. Head Tail 1. Elementary event for tossing both coins. e.g.: First coin shows head, second shows tail: .411Azz. 2. Coin Azl A22 Compound event for tossing both coins: First coin shows head: -411= AiiAzi t AiiA2z 2. Compound event for tossing one coin, e.g., the first one: First coin shows head or tail: A l l t A12 = I . Head and tail on the same coin are disjoint events: AllAlz = 0. IB: Lifetime of light-bulbs. \\'e can define the elementary events A,: the lifetime t satisfies the inequalities ( n - 1)At < t 5 n 3 t (n = 1 , 2 . . . ., and At > 0, arbitrary unit of time). A,. Compound event A: The lifetime is at most nAt, i.e., A =

E

V=l

16.2.1.2 Frequencies and Probabilities 1. Frequencies Let A be an event belonging to the field of events A of an experiment. If event A occurred

n4 times while we repeated the experiment n times, then nA is called the frequency, and nA/n = ha is called the relatzve frequency of the event A. The relative frequency satisfies certain properties which can be used to built up an axiomatic definition of the notion of the probability P ( A ) of event A in the field of events A .

16.2 Probability Theory 747

2. Definition ofthe Probability A real function P defined on the field of events is called a probability if it satisfies the following properties: 1. For every event A E A we have 0 5 P ( A ) 5 1, and 0 5 h.4 5 1. (16.30) 2. For the impossible event 0 and the certain event I , we have P(0)= 0. P ( I ) = 1. and ho = 0, hr = 1. (16.31) 3. If the events .4E A (i = 1.2,. . .) are finite or countably many mutually exclusive events ( A , 4 k = 0 fori # IC). then P(AI+ '42 + . . .) = AI) t P(Azj t , and hA1+Az+.,. = h~~ hAz . . . . (16.32)

+

I . .

+

3. Rules for Probabilities 1. B C A yields P ( B ) 5 P ( A ) . 2.

(16.33)

P(A)+ P(2)= 1.

(16.34)

3. For two disjoint events A and B ( A B = 0),we have P ( A B) = P(A)t P ( B ) . 4. a) For arbitrary events A, (i = 1,., , , nj, we have P(Ai + . ' A,) = P(.4i)t ' . t P(A,) - P(A1A2) - . . - P(AIA,) -P(A2A3) - ' . - P(AzA,) - . . - P(A,-iA,) tP(A1.42A3) t . . P(AlAzA,) ". t P(An-2A,-1A,) -

+

(16.35)

+

+

+

+(-l)n-'P(A1A2.. . A , ) . (16.36a) 4.b) In particular for n = 2: P(A1 t A2) = P(Al) P(A2) - P(A1A2). (16.36b) 5 . Equally likely events: If every event Ai (i = 1 , 2 , . . . , n) of a finite complete system of events occurs with the same probability, then

+

P(A,) = A

(16.37)

If A is a sum of m (rn 5 n ) events with the same probability A, (z = 1 , 2 , . . . , n) of a complete system, then m P(.4)= - . (16.38) n

4. Examples of Probabilities 1 IA: The probability P ( A ) to get a 2 rolling a fair die is: P ( A ) = 6 IB: K h a t is the probability of guessing four numbers for the lotto "6 from 49", i.e.. 6 numbers are to be chosen from the numbers 1 , 2 , . . . ,49. If 6 numbers are drawn. then there are (:) possibilities to choose 4. On the other hand there are

(161;) =

(423)

possibilities for the false numbers. Altogether, there are draw 6 numbers. Therefore, the probability P(A4) is:

4():

different possibilities to

I

748

16. Probabilitv Theorv and Mathematical Statistics

Similarly. the probability P(AG)to get a direct hit is: 1

P ( & ) = - - 0 715. lo-' = 7.15.

('69)

%.

- ' C: What is the probability P ( A )that at least two persons have birthdays on the same day among k persons? (The years of birth must not be identical, and we suppose that every day has the same probability of being a birthday.) It is easier to consider the complementary event 2: All the k persons have different birthdays. We get:

365 365-1 365-2 P (A )= . -. -.

365 365 From this it follows that

365

. 365-k+1 " '

365

+

365.364 363 . . . . (365 - k 1) 36Sk k 1 10 20 23 30 60 P(A) 1 0.117 0.411 0.507 0.706 0.994 We can see that the probability that among 23 and more persons at least two have the same birthday is greater than 50 %.

P ( A )= 1 - P ( 2 )= 1 -

16.2.1.3 Conditional Probability, Bayes Theorem 1. Conditional Probability The probability of an event B , when it is known that some event A has already occurred, is called a condztzonal probabzlzty and it is denoted by P(BIA) or Pa(B) (read: The probability that B occurs given that A has occurred). It is defined by

P(.4B) P(BIA) = - P ( A ) # 0. P(A) The conditional probability satisfies the following properties: a) If P ( A ) # 0 and P ( B ) # 0 holds, then

(16.39)

P(BlA) - P(AIB) PPI P(A) ' b) If P ( A 1 A 2 A 3 ..A,) . # 0 holds, then P(A1A2... A,) = P(A1)P(A2/A1). . .P(AnIA1A2.. .A,-l).

(16.40b)

2. Independent Events The events A and B are independent events if P(A1B) = P ( A ) and P(BIA) = P ( B )

(16.41a)

(16.40a)

holds. In this case, we have

P(AB) = P ( A ) P ( B ) .

(16.41b)

3. Events in a Complete System of Events If A is a field of events and the events B, E A with P(B,) > 0 (i = 1 , 2 , .. .) form a complete system of events, then for an arbitrary event A E A the following formulas are valid: a) Total Probability Theorem P ( A )=

c P(.~IBz)P(B,). z

(16.42)

16.2 Probability Theory 749 b) Bayes Theorem with P ( A ) > 0

(16.43)

IThree machines produce the same type of product in a factory. The first one gives 20 % of the total production. the second one gives 30 % and the third one 50 %. It is known from past experience that 5 %. 4 %. and 2 % of the product made by each machine, respectively. are defective. Two types of questions often arise: a) IYhat is the probability that an article selected randomly from the total production is defective? b) If the randomly selected article is defective, what is the probability that it was made, e.g.. by the first machine? \Ye use the following notation: -4, denotes the event that the randomly selected article is made by the i-th machine (i = 1,2,3)with P(A1) = 0.2. P(A2) = 0.3; P(A3) = 0.5. The events A, form a complete system of eients: A,A, = 0, A i + A2 t A3 = I . A denotes the event that the chosen article is defected. P(illA1) = 0.05 gives the probability that an article produced by the first machine is defective: analogously P(AIA2)= 0.04 and P(AIA3) = 0.02 hold. Now. we can answer the questions: a) P ( A ) = P ( A 1 ) P ( A I A l+) P(AdP(AIA2)+ P ( . W ( A I A 3 ) = 0.2 0.05 + 0.3 .0.04 t 0.5 . 0.02 = 0.032.

16.2.2 Random Variables, Distribution Functions To apply the methods of analysis in probability theory. we introduce the notions of variable and function.

16.2.2.1 Random Variable If we assign numbers to the elementary events, then we define a random variable X . Then every random event can be described by this variable X . The random variable X can be considered as a quantity which takes its values z randomly from a subset R of the real numbers. If R contains finite or countably many different values, then we call X a discrete random variable: In the case of a continuous random variable, R can be the whole real axis or it may contain subintervals. For the precise definition see 16.2.2.2, 2., p. 750. There are also mized random variables. IA: If we assign the values 1,2.3:4 to the elementary events A l l . Alz. Azl, Azz, respectively, in example A, p. 746, then we define a discrete random variable X . IB: The lifetime T of a randomly selected light-bulb is a continuous random variable. The elementary event T = t occurs if the lifetime T is equal to t .

16.2.2.2 Distribution Function 1. Distribution Function and its Properties A random variable X can be defined by its distribution function F ( z ) = P ( X 5 z) for - co < z < co. (16.44) It determines the probability that the random variable X takes a value between -cc and z. Its domain is the whole real axis. The distribution function has the following properties: a) F(-x)= 0 , F(+co) = 1.

750

16. Probabilitu Theoru and Mathematical Statistics

b) F ( z )is a non-decreasing function of x. c) F ( z ) is continuous on the right. Remarks : 1. From the definition it follows that P ( X = a) = F ( a ) - lim F ( x ) . xia-0

2. In the literature, also the definition F ( x ) = P ( X P(X = a ) = lim F ( x ) - F ( a ) .

< x) is often used. In this case

s-a-0

2. Distribution Function of Discrete and Continuous Random Variables a) Discrete Random Variable: A discrete random variable X, which takes the values x, (i = 1;2 . . . .) with probabilities P ( X = xi) = pi (i = 1,2, , . .), has the distribution function F ( z )=

1P,.

(16.45)

X t 3

b) Continuous Random Variable: .4 random variable is called continuous if there exists a nonnegative function f ( x ) such that the probability P(X E S) can be expressed as P ( X E S) = Js f(x)dx for any domain S such that it is possible to consider an integral over it. This function is the so-called densityfunction. .Lz continuous random variable takes any given value zi with 0 probability so we rather consider the probability that X takes its value from a finite interval [a,b]: b

P(a 5 X 5 b) =

1

(16.46)

f ( t )dt.

a

.4 continuous random variable has an everywhere continuous distribution function:

1 X

F ( z ) = P(X 5 z)=

(16.47)

f ( t ) dt.

-m

F ' ( z ) = f ( z ) holds at the points where f(x) is continuous Remark: hThen there is no confusion about the upper integration limit, often the integration variable is denoted by z instead oft. 3. Area Interpretation of the Probability By introducing the distribution function and density function in (16.47), we can represent the probability P(X 5 z)= F ( z ) as an area between the density function f ( t ) and the x-axis on the interval --x < t 5 x (Fig. 16.la).

a)

b) Figure 16.1

Often there is given a probability value a. If (16.48) P ( X > x) = a holds, we call the corresponding value of the abscissa x = z, the p a n t i l e or the fractile of order (Y (Fig. 16.lb). This means the area under the density function f ( t ) to the right of z, is equal to cy.

16.2 Probability Theory 751

Remark: In the literature, the area to the left of 2, is also used for the definition of quantile. In mathematical statistics, for small values of CY, e.g., a = 5% or a = I%, we also use the notion of szgnzficance levelor type 1 error rate. The most often used quantiles for the most important distributions in practice are given in tables (Table 21.14,p. 1083, to Table 21.18, p. 1090).

16.2.2.3 Expected VaJue and Variance, Chebyshev Inequality. For a global characterization of a distribution, we mostly use the parameters expected value, denoted by p. and the varzance d of a random variable X. The expected value can be interpreted with the terminology of mechanics as the abscissa of the center of gravity of a surface bounded by the curve of the density function f(x) and the x-axis. The variance represents a measure of deviation of the random variable X from its expected value p.

1. Expected Value If g ( X ) is a single-valued function of the random variable X. then g ( X ) is also a random variable. Its expected value or ezpectatzon is defined as: m

E ( g ( X ) )= x g ( x k ) P k ,

a) Discrete Case:

if

the series

b

Ig(xk)lpk

exists.

(16.49a)

k=l

/ g(x)f(x)dx, ltm ig(x)lf(x)dx exists.

+-m

b ) Continuous Case: E ( g ( X ) )=

if

-m

m

( 16.49b)

The expected value of the random variable X is defined as

1 tm

px = E ( X ) = x

Z k p k k

Or

Zf(Z)

dx.

(16.5Oa)

--cc

if the corresponding sum or integral with the absolute values exists. We note that (16.49a,b) yields that (16.5Ob) E(aX + b) = a p +~ b (a, b const) is also valid. Of course, it is possible that a random variable does not have any expected value.

2. MomentsofOrder n iVe Introduce:

a) Moment of Order n: E ( X " ) .

(16.51a)

b) Central Moment of Order n: E ( ( X - p ~ ) " ) .

(16.5lb)

3. Variance and Standard Deviation In particular. for n = 2, the central moment is called the varzance or disperszon:

E ( ( X - p x ) 2 ) = D 2 ( X ) = 0; =

(16.52)

if the expected values occurring in the formula exist. The quantity ux is called the standard deviation. The following relations are valid: (16.53) D 2 ( X )= u; = E ( X 2 )- p i . D Z ( a X+ b ) = a Z D 2 ( X ) . 4. Weighted and Arithmetical Mean In the discrete case, the expected value is obviously the weighted m e a n E ( X ) = p , x , . . . p,z,

+ +

(16.54)

752

16. Probability Theory and Mathematical Statistics

of the values x i , . . . 2 , with the probabilities pk as weights (k = 1,.. . n). The probabilities for the uniform distribution are pl = pz = . . . = p, = l / n , and E ( X ) is the arithmetical mean of the values ~

xk:

E ( X )=

21 + 2 2

t . . . t 2,

(16.55)

n

In the continuous case, the density function of the continuous uniform distribution on the finite interval [a, 61 is

{-

6-a 0 and it follows that f(z)=

fora<x<6,

(16.56)

otherwise,

(16.57) 5 . Chebyshev Inequality If the random variable X has the expected value p and standard deviation o, then for arbitrary X > 0 the Chebyschev inequality is valid: 1 P(lX - pi 2 Ao) 5 (16.58) A* ' That is, it is very unlikely that the values of the random variable X are farther from the expected value p than a multiple of the standard deviation (A large).

16.2.2.4 Multidimensional Random Variable If the elementary events mean that n random variables X 1 , .. . , X, take n real values 2 1 , . . . , z, then a random vector X = ( X l ,X2... . ,X,) is defined (see also random vector, 16.3.1.1, 4., p. 768). The corresponding distribution function is defined by (16.59) F(z1.. . .,a,) = P(X15 Z l r . .. ,X, a,). The random vector is called continuous if there is a function f ( t 1 , . . . , t,) such that

<

F(z1... . , 2 , ) =

7. 7 '.

--s

f ( t l . . . . , t,) dtl . . , dt,

(16.60)

-m

holds. The function f ( t 1 , .. . , t,) is called the density junction. It is non-negative. If some of the variables a I 1 . .. a, tend to infinity, then we get the so-called marginal distributions. Further investigations and examples can be found in the literature. The random variables X I , . . . , X , are independent random variables if (16.61) F b l , . . . L n ) = F1(51)F2(22).. .F,(Z,), f(t1,. . . , t,) = f l ( t 1 ) . . . f n ( t n ) . ~

.

16.2.3 Discrete Distributions 1. Two-Stage Population and Urn Model Suppose we have a two-stage population with N elements, Le., the population we consider has two classes of elements. One class has M elements with a property A , the other one has N - &f elements which do not have the property A. If we investigate the probabilities P ( A ) = p and P ( A ) = 1 - p for randomly chosen elements. then we distinguish between two cases: When we select n elements one after the other, we either replace the previously selected element before selecting the next one, or we do not replace it. The selected n elements, which contain k elements with the property A , is called the sample. n being the site of the sample. This can be illustrated by the urn model.

16.2 Probability Theory 753

2. Urnmodel Suppose there are a lot of black balls and white balls in a container. The question is: What is the probability that among n randomly selected balls there are k black ones. If we put every chosen ball back into the container after we have determined its color, then the number k of black ones among the chosen n balls has a binomial distribution. If we do not put back the chosen balls and n 5 M and n 5 N - 11.1,then the number of black ones has a hypergeometric distribution.

16.2.3.1 Binomial Distribution Suppose we observe only th_e two events A and 2 in an experiment, and we do n independent experiments. If P ( A ) = p and P ( A ) = 1- p holds every time, then the probability that A takes place exactly k times is

W;(k) =

(3

p k ( l - P ) ” - ~ ( k = 0 , 1 , 2 , .. . , n ) .

(16.62)

For every choice of an independent element from the population, the probabilities are A1 - N - ‘If P(A)= -, P ( A )= =1-p=q.

s

(16.63)

~

121

The probability of getting an element with property A for the first k choices, then an element with the remaining property A for the n- k choices ispk(l-P)”-~, because the results of choices are independent of each other. \Ve get the same result assigning the k places any other way. We can assign these places n! (16.64)

(nk)

=

qFq

different ways, and these events are mutually exclusive, so we add

(3

equal numbers to get the re-

quired probability. A random variable X,, for which P ( X , = k ) = W,”(k)holds, is called binomially distributed with parameters n and p .

1. Expected Value and Variance E ( X ” )= p = n ’ p,

(16.65a)

DZ(Xn)= 2 = n . p ( 1 - p).

(16.65b)

2. Approximation of the Binomial Distribution by the Normal Disribution If X,has a binomial distribution, then (16.65~) ..

This means that, if n is large, the binomial distribution can be well approximated by a normal distribution (see 16.2.4.1.p. 756) with parameters px = E(X,) and o* = D2(X,), ifp or 1 - p are not too small. The approximation is the more accurate the closerp is to 0.5 and the larger n is, but acceptable if np > 4 and n(1 -p) > 4 hold. For very smallp or 1- p , the approximation by the Poisson distribution (see (16.68) in 16.2.3.3) is useful.

3. Recursion Formula The following recursion formula is recommended for practical calculations with the binomial distribution: n-k p Ul,”(k + 1) = -’ - ’ W,n(k). (16.65d) k+l q

4. Sum of Binomially Distributed Random Variables If X , and X, are both binomially distributed random variables with parameters n,p and m, p . then the random variable X = X , + X , is also binomially distributed with parameters n + m, p.

I

754 16. Probabilitv Theoru and Mathematical Statistics

Fig. 16.2alblc represents the distributions of three binomially distributed random variables with parameters n = 5; p = 0,s. 0.25, and 0.1. Since the binomial coefficients are symmetric, the distribution is symmetric for p = q = 0.5, and the farther p is from 0.5 the less symmetric the distribution is.

16.2.3.2 Hypergeometric Distribution Just as with the binomial distribution, we suppose that we have a two-stage population with "V elements, Le.. the population we consider has two classes of elements. One class has AI elements with a property A , the other one has Ai - M elements which do not have the property A. In contrast to the case of binomial distribution, we do not replace the chosen hall of the urn model. The probability that among the n chosen balls there are k black ones is

(16.66a)

0 5 k 5 n. k 5 .PIt n - k 5 A' - L\l.

( 16.66b)

If also n 5 -11 and n 5 - V , I hold, then the random variable X with the distribution (16.66a) is said to he hypergeometrically distributed.

0.7

f

0.7

f

0.7

f

0.4 0.5 0.6/,,[;;.

~ j l l l / O . l O

0.3 0.2 0.1 0.0

0.3 0.2 0.1

0 1 2 3 4 5

a)

k

0.1 0.0

0 1 2 3 4 5

b)

k

0.0

0 1 2 3 4 5

. k

c) Figure 16.2

1. Expected Value and Variance of the Hypergeometric Distribution (16.67a)

(16.67b)

2. Recursion Formula ( n- k ) ( M - k ) W;,~J(k t 1) = ( k + l)(.V - M - n t k t 1)K b , N ( k ) .

(16.67~)

16.2 Probability Theory 755

In Fig. 16.3a,b,c we represent three hypergeometric distributions for the cases N = 100, M = 50, 25 and 10, for n = 5. These cases correspond to the cases p = 0.5: 0.25, and 0.1 of Fig. 16.2a,b,c. There is no significant difference between the binomial and hypergeometric distributions in these examples. If also M and A' - M are much larger than n, then the hypergeometric distribution can be well approximated by a binomial one with parameters as in (16.63).

0.5 0.4 0.3 0.2 0.1 0.0

p=0.50

0 1 2 3 4 5

a)

p=o.10

k

0.2 0.1 0.0

0.3 0.2 0.1 0 1 2 3 4 5

k

b)

O'O 0 1 2 3 4 5 c)

k

Figure 16.3

16.2.3.3 Poisson Distribution If the possible values of a random variable X are the non-negative integers with probabilities Xk

(k = 0.1.2.. . . ; X > O), k! then it has a Pozsson dzstrzbutzon with parameter A.

P ( X = IC) = -e-A

(16.68)

1. Expected Value and Variance of the Poisson Distribution E ( X ) = A.

(16.69a)

D z ( X ) = A.

(16.69b)

2. Sum of Independent Poisson Distributed Random Variables If XIand Xz are independent Poisson distributed random variables with parameters XI and Xlr then the random variable X = X I + Xz also has a Poisson distribution with parameter X = XI t Xz.

3. Recursion Formula (16.69~) We can get the Poisson distribution as a limit of binomial distributions with parameters n and p if n + x.and p ( p + 0) changes with n so that np = X = const, Le., the Poisson distribution is a good approximation for a binomial distribution for large n and small p with X = np. In practice, we use it if p 5 0.08 and n 2 l5OOp hold, because the calculations are easier with a Poisson distribution. Table 21.14. p. 1083. contains numerical values for the Poisson distribution. Fig. 16.4a,b,c represents three Poisson distributions with X = np = 2.5, 1.25 and 0.5, Le, with parameters corresponding to Figs. 16.2 and 16.3. The number of independently occurring point-like discontinuities in a continuous medium can usually be described by a Poisson distribution. e.g., number of clients arriving in a store during a certain time interval; number of misprints in a book, etc.

750

:I,;, ;I,:.

16. Probabilitu Theow and Mathematical Statistics

0.5

0.1 0.0

*

0 1 2 3 4 5

a)

k

0.0

j!/,,

h=0.50

0.4 0.3 0.2

0.2

*

0 1 2 3 4 5

b)

k

0.0

0 1 2 3 4

k

c) Figure 16.4

16.2.4 Continuous Distributions 16.2.4.1 Normal Distribution 1. Distribution Function and Density Function h random variable X has a normal distrzbotion if its distribution function is (16.70) Then it is also called a normal variable, and the distribution is called a ( p , u ) noma1 distribution. The function

f ( t ) = -e

1

(16.71)

ad5

is the density function of the normal distribution. It takes its maximum at t = p and it has inflection points at p i.a (see (2.59), p. 72, and Fig. 16.5a).

Figure 16.5

2. Expected Value and Variance The parameters p and o2 of the normal distribution are its expected value and variance, respectively, i.e. (16.72a)

16.2 Probability Theory 757

If the normal random variables X 1 and X2 are independent with parameters p l , u1 and p2, uZ1resp., then the random variable X = k l X l kzX2 ( k l , kz real constants) also has a normal distribution with parameters p = k l p l

+ + k2p2. u = Jm.

t - p in (16.70),then the calculation of the substitution values of If we perform the substitution r = U

the distribution function of any normal distribution is reduced to the calculation of the substitution values of the distribution function of the ( 0 , l ) normal distribution, which is called the standard normal dzstrzbutzon Consequently, the probability P ( a 5 X 5 b) of a normal variable can be expressed by the distribution function @(z)of the standard normal distribution: (16.73)

16.2.4.2 Standard Normal Distribution, Gaussian Error Function 1. Distribution Function and Density Function \$'e get from (16.70) with p = 0 and uz = 1 the distribution function

P(X 5 z)= @(x)

1 m,

1 =-

t2

e-? dt =

j

p ( t ) dt

(16.74a)

-X

of the so-called standard normal distribution. Its density function is 1

u ( t )= -ee-?,

t2

( 16.74b)

Jz;;

it is called the Gaussian error curve (Fig. 16.5b). The values of the distribution function @(z)of the ( 0 , l ) normal distribution are given in Table 21.15. p. 1085. Only the values for the positive arguments x are given, while we get the values for the negative arguments from the relation (16.75) @(-x) = 1 - q z ) .

2. Probability Integral The integral @(x) is also called the probability integralor Gaussian error integral. We can also find in the literature the following definitions:

erf denotes the error function.

16.2.4.3 Logarithmic Normal Distribution 1. Density Function and Distribution Function The continuous random variable X has a logarithmic normal distribution, or lognormal distribution with parameters p L and u2 if it can take all positive values, and if the random variable Y , defined by Y = log (16.77) has a normal distribution with parameters p~ and ui. Consequently, the random variable X has the density function

x,

for t 5 0. (16.78)

758

16. Probability Theory and Mathematical Statistics

and the distribution function

for z 5 0,

(0

(16.79)

K e can use either the natural or the decimal logarithm in practical applications. 2. Expected Value and Variance Using the natural logarithm we get the expected value and variance of the lognormal distribution:

(16.80)

3. Remarks a) The density function of the lognormal distribution is continuous everywhere and it has positive values only for positive arguments. Fig. 16.6 shows the density functions of lognormal distributions for different

PL

and UL. Here we used the natural logarithm.

b) Here the values P L and 02 are not the expected value and variance of the lognormal random variable itself. but of the variable Y = logX. c) The values of the distribution function F ( z )of the lognormal distribution can be calculated by the distribution function @(z)of the standard normal distribution (see (16.74a)). in the following n a y (16.81)

d) The lognormal distribution is often applied in lifetime analysis of economical, technical. and biological processes. e) The normal distribution can be used in additive superposition of a large number of independent random variables, and the lognormal distribution is used for multiplicative superposition of a large number of independent random variables.

1-

pL=ln0.5 ;0,=0.5 ,p,=0 ;a,=l

0.5

Figure 16.6

Figure 16.7

16.2.4.4 Exponential Distribution 1. Density Function and Distribution Function A continuous random variable X has an exponentzal distrzbution with parameter X (A function is (Fig. 16.7) 0

for t < 0

Xe-At for

t 2 0,

> 0) if its density (16.82)

16.2 Probability Theory 759

consequently. the distribution function is

(16.83)

2. Expected Value and Variance p=-

1 A'

1 02=-

A2

(16.84) '

Usually. we can describe the following quantities by an exponential distribution: Length of phone calls, lifetime of radioactive particles. working time of a machine between two stops in certain processes, lifetime of light-bulbs or certain building elements.

16.2.4.5 Weibull Distribution 1. Density Function and Distribution Function The continuous random variable A ' has a Weibull distribution with parameters a and 3 (a> 0,/3 > 0), if its density function is for

t < 0, (16.85)

and so its distribution function is

for

z < 0. (16.86)

2. Expected Value and Variance (16.87) Here r ( z ) denotes the gamma function (see 8.2.5, 6.. p. 459):

/t 3c

r ( z )=

s-'

e -t dt for z > 0.

(16.88)

0

In (16.85), a is the shape parameter and 3 is the scale parameter (Fig. 16.8, Fig. 16.9). 1.o

0 1

""I .o

p=1

0.5

1.0

1.5

Figure 16.8

1

2.0 2.5

c(=2

t Figure 16.9

760

16. Probabilitv Theoru and Mathematical Statistics

Remarks: a) The Weibull distribution becomes an exponential distribution for CY = 1 with X = A.

3

b) The Weibull distribution also has a three-parameter form if we introduce a position parameter 7 . Then the distribution function is:

(16.89)

c) The Weibull distribution is especially useful in life expectancy theory, because, e.g., it describes the functional lifetime of building elements with great flexibility.

16.2.4.6 x2 (Chi-square) Distribution 1. Density Function and Distribution Function Let XI, Xz.. , , X , be n independent ( 0 , l ) normal random variables. Then the distribution of the ~

random variable

+ +

+

(16.90) x 2 = XI2 XZZ , , ' x,2 is called the xz dzstrzbzltzon with n degrees of freedom. Its distribution function is denoted by F,I(Z), and the corresponding density function by fXz ( t ) . (16.91a)

(16.91b)

2. Expected Value and Variance E ( x Z )= n3

D'(x2) = 2n.

(16.92a)

(16.92b)

3. Sum of Independent Random Variables If X1 and X2 are independent random variables both having a xzdistribution with n and m degrees of freedom, then the random variable X = XI+ Xz has a xz distribution with n + m degrees of freedom.

4. Sum of Independent Normal Random Variables If X I . Xz,. . . X,are independent, (0, a) normal random variables, then ~

X

=

5X,'

1 has the density function f ( t ) = -f

z

X

I n

=-

1X,*

n has the density function f ( t ) = --f

z

02

2=1

X= -EX'

has the density function f ( t ) =

(-)t (-)nt 02

UZ

z=1

a2

,

(16.93)

,

(16.94) (16.95)

z=1

5 . Quantile For the quantile (see 16.2.2.2, 3., p. 750) 16.10),

P(X >

= CY.

of the

x2 distribution with m degrees of freedom (Fig. (16.96)

16.2 Probability Theory 761

Figure 16.10 Quantiles of the

Figure 16.11

x2 distribution can be found in Table 21.16,p. 1087.

16.2.4.7 Fisher F Distribution 1. Density Function and Distribution Function If X1 and X z are independent random variables both having of freedom, then the distribution of the random variable

x2 distribution with ml and m2 degrees (16.97)

is a Fisher distribution or F distribution with ml, m2 degrees of freedom.

fort > 0, .fF(t)

=

(16.98a)

fort 5 0.

For I 5 0 we have F.P(z) = P(Fm,,m2 5 I) = 0, for I > 0:

F F ( ~=) P(Fm,,m25 I)

2. Expected Value and Variance

The quantiles (see 16.2.2.2, 3., p. 750) t,,,,,,, Table 21.17,p. 1088.

of the Fisher distribution (Fig. 16.11)can be found in

16.2.4.8 Student t Distribution 1. Density Function and Distribution Function If X is a ( 0 , l ) normal random variable and Y is a random variable independent from X and it has a x2 distribution with m = n - 1 degrees of freedom, then the distribution of the random variable T=-

X

(16.100)

762

16. Probability Theory and Mathematical Statistics

is called a Student t distribution or t distribution with m degrees of freedom. The distribution function is denoted by F ~ ( X and ) ~the corresponding density function by fs(t).

fdt) = 1 r(?)

(16.10 la)

1

m ( 16.101b)

Figure 16.12

2. Expected Value and Variance E ( T ) = 0: 3. Quantile The quantiles t,,m and P ( T > t,,,) = a

m P ( T )= - ( m > 2).

(16.102a)

m-2

(16.102b)

of the t distribution (Fig. 16.12a,b),for which (16.103a)

or

P ( /TI > t,p,m) = 0

(16.103b)

holds. are given in Table 21.18. p. 1090. The Student t distribution, introduced by Gosset under the name Student, is used in the case of samd e s with small samde size n, when only estimations can be given for the mean and for the standard heviation. The staniard deviation (16.102b) nolonger depends on the deviation of the population from where the sample is taken.

16.2.5 Law of Large Numbers, Limit Theorems The law of large numbers gives a relation between the probability P ( A ) of a random event .4 and its relative frequency na/n with a large number of repeated experiments.

1. Law of Large Numbers of Bernoulli

The following inequality holds for arbitrary given numbers E > 0 and 17 > 0 1 (16.104a) if n 2 4E217

(16.104b)

For other similar theorems see [16.5].

I How many times should we roll a not necessarily fair die if the relative frequency of the 6 should be closer to its probability than 0.1 with a probability of at least 95 % 7

16.2 Probabilitu Theoru 763

Now..E = 0.01 and q = 0.05, so 4811 = 2 . W5, and according to the law of large numbers of Bernoulli n 2 5 lo4 must hold. This is an extremely large number. We can reduce n, if we know the distribution function.

2. Central Limit Theorem of Lindeberg-Levy

If the independent random variables XI,. . . , X, all have the same distribution with an expected value p and a variance 0’.then the distribution of the random variable

(16.105) tends to the ( 0 , l ) normal distribution for n + co.Le., for its distribution function Fn(y) we get (16.106)

If n > 30 holds, then F,(y) can be replaced by the ( 0 , l ) normal distribution. Further limit theorems can be found in [16.5], [16.7]. We take a sample of 100 items from a production of resistors. We suppose that their actual resistance values are independent and they have the same distribution with deviation u’ = 150. The mean value for these 100 resistors is E = 1050 Q. In which domain is the true expected value p with a probability of99%? We are looking for an E such that P ( l x - pi 5 E ) = 0.99 holds. We can suppose (see (16.105)) that the random variable Y = -has a (0.1) normal distribution. From P(lY1 5 A) = P(-A 5

-’

ff/&

Y 5 A) = P(Y 5 A) - P ( Y < -A), and from P(Y 5 -A) P(lY1 5 A) = 2P(Y 5 A) - 1 = 0.99.

= 1 - P(Y

5 A) it follows that

S o . P ( Y < A) =@(A) =0.995andfromTable21.15,p.1085,wegetA=2.58. Sinceo/m=1.225, we get with a 99 % probability: 11050 - pi < 2.58.1.225, Le., 1046.80 < p < 1053.2 R.

16.2.6 Stochastic Processes and Stochastic Chains Many processes occurring in nature and those being studied in engineering and economics can be realistically described only by time-dependent random variables. IThe electric consumption of a city at a certain time t has a random fluctuation that is dependent on the actual demand of the households and industry. The electric consumption can be considered as a continuous random variable X. When the observation time t changes, electric consumption is a continuous random variable at every moment. so it is a function of time. The stochastic analysis of time-dependent random variables leads to the concept of stochastic processes, which has a huge literature of its own (see, e.g., [16.7],[16.8]). Some introductory notions will be given next.

16.2.6.1 Basic Notions, Markov Chains 1. Stochastic Processes A set of random variables depending on one parameter is called a stochastzc process. The parameter, in general, can be considered as timet, so the random variable can be denoted by Xt and the stochastic process is given by the set IXtIt E TI. (16.107) The set of parameter values is called the parameter space T , the set of values of the random variables is the state space Z.

I

16. Probabilitu Theoru and Mathematical Statistics

764

2. Stochastic Chains If both the parameter space and the state space are discrete, Le., the state variable X,and the parameter t can have only finite or countably infinite different values, then the stochastic process is called a stochastic chain. In this case the different states and different parameter values can be numbered: (16.108) Z = { I ) & .. . , i , i + 1,. . . , }

T = { t o , t ~. ., . ,t,,

. . .} with 0 5 to < tl < . . . < t, < tmtl < . . . .

t,+l,

(16.109)

The times t o ,t l , . . . are not necessary equally spaced.

3. Markov Chains, Transition Probabilities If the probability of the different values of X,,,, in a stochastic process depends only on the state at time t,, then the process is called a Markov chain. The Markov property is defined precisely by the requirement that P(X,,,,= i,+llXt, = io, = 2 1 , . . . ; xt, = 2,) = P(X,_+, = i,tl/X&" = i,) (16.110) for all m E {0,1,2,.. .} and for all i ~ , i l ,. .. , imtl E 2. The conditional probabilities Consider a Markov chain and times t , and (16.111) P(xt,,, = jI&, = = Ptj(t,,tfntl) are called the transitionprobabilitiesofthechain. The transition probability determines the probability by which the system changes from the state X,,= i at t , into the state Xtmtl= j at tmtl. If the state space of a Markov chain is finite, i.e., Z = {1,2,. . . . N } , then the transition probabilities pi,(tlrt z )between the states at times t l and t 2 can be represented by a quadratic matrix P(t1, t 2 ) , by the so-called transition matrix

x,,

4

(16.112)

The times tl and t 2 are not necessarily consecutive.

4. Time-Homogeneous (Stationary) Markov Chains If the transition probabilities of a Markov chain (16.111) do not depend on time, i.e., P,,(t,. t,+1) = P t J . (16.113) then the Markov chain is called tzme-homogeneous or statzonary. A stationary Markov chain with a finite state space Z = {1,2,. . . , N } has the transition matrix Pll P = [P:

PI2 P22

,.. ' ' '

PIN P2N)

PN1

pN2

,,.

PNN

,

( 16.114a)

where a) p,,

2 0 for all z , j and

cp2,1 for all i .

(16.114b)

N

b)

=

(16.114c)

3=1

Being independent of time p,, gives the transition probability from the state i into the state j during time unit. IThe number of busy lines in a telephone exchange can be modeled by a stationary Markov chain. For the sake of simplicity we suppose that we have only two lines. Hence, the states are i = 0,1,2. Let

16.2 Probabilitu Theow 765

the time unit be, e.g., 1 minute. Suppose the transition matrixpij is: (pv) =

10.7 0.3 O.O\ 0.2 0.5 0.3 0.1 0.4 0.5

[

In the matrix ( p z j )the first row corresponds to the state i = 0. The matrix element p12 = 0 , 3 (second row, third column) shows the probability that two lines are busy at timet, given that one was busy at tm-1.

Remark: Every quadratic matrix P = ( p i j ) of size N x N satisfying the properties (16.114b) and (16.114~)is called a stochastic matrix. Their row vectors are called stochastic vectors. Although the transition probabilities of a stationary Markov chain do not depend on time, the distribution of the random variable X t is given at a given time by the probabilities N

P ( X t = i) = p l ( t ) (i = 1.2.. . . . Y ) (16.115a)

with x p i ( t ) = 1

(16.115b)

2=1

since the process is in one of the states with probability one at any timet. Probabilities (16.115a) can be written in the form of a probability vector P=

(p,(t),P*(t),...,p~(t)T)'

(16.116)

The probability vector p i s a stochastic vector. It determines the distribution of the states of a stationary Slarkov chain at time period t.

5 . Probability Vector and Transition Matrix Let the transition matrixP ofastationary Markov chain be given (according to (16.114a,b,c)). Starting with the probability distribution at time period t determine the probability distribution at t + 1, that is, calculate p(t + 1) from P and p(t): (16.117 )

p(t

+ k ) = p(t) Pk. '

(16.118)

Remarks: 1. Fort = 0 it follows from (16.118) that P(k)

= p(O)P'",

( 16.119)

that is, a stationary Markov chain is uniquely determined by the initial distribution p(0)and the transition matrix P. 2. If matrices A and B are stochastic matrices, then C = AB is a stochastic matrix, as well. Consequently. if P is a stochastic matrix, then the powers Pkare also stochastic matrices. W X particle changes its position (state) X i (1 5 z I: 5) along a line in time periods t = 1 , 2 , 3 , .. . according to the following rules: a) If the particle is at .z = 2.3,4, then it moves to the right by a unit during the next time unit with probability p = 0.6 and to the left with probability 1 - p = 0.4. b) At points z = 1 and z = 5 the particle is absorbed, Le., it stays there with probability 1. c) At timet = 0 the position of the particle is x = 2. Determine the probability distribution p(3) at time period t = 3. By (16.119) the probability distribution p(3) = p(0) P3holds with p(0)= (0, 1,0, 0,O) and with the

766

16. Probubzlztv Theorv and Mathematical Statzstacs

transition matrix 1 0 0

P=

('i i :i 0

0.4

0

0 0.6

0

. Weget P3 =

0:6) p(3) = (0.496;0; 0.288; 0; 0.216) . and finally -

(

1 0 0 0 0.496 0 0.288 0 0.216 0.160 0.192 0 0.288 0.360 0 0.744 0.064 0 0.192 0 0 0 0 1O )

.

16.2.6.2 Poisson Process 1. The Poisson Process In the case of a stochastic chain both the state space 2 and the parameter space T are discrete, that is, the stochastic process is observed only at discrete time periods to, t l . tz.. . . . Now; we study a process nith continuous parameter space T . and it is called a Poisson process. 1. Mathematical Formulation Mathematical formulation of the Poisson process: a) Let the random variable X t be the number of signals in the time interval [O.t); b) let the probabilitypx(t) = P ( X t = z) be the probability of z signals during the time interval [O. t ) . .\dditionally. the following assumptions are required. which hold in the process of radioactive decay and many other random processes (at least approximately): c) The probablity P ( X t = x) of I signals in a time interval of length t depends only on x and t. and does not depend on the position of the time interval on the time axis. d) The numbers of signals in disjoint time intervals are independent random variables. e) The probability to get at least one signal in a very short interval of length At is approximately proportional to this length. The proportionality factor is denoted by X ( A > 0). 2. Distribution Function By properties a)-e) the distribution of the random variable X t is determined. \Ye get: (16.120) where = At is the expected value and uz = Xt the variance. 3. Remarks 1. From (16.120) we get the Poisson distribution as a special case for t = 1 (see 16.2.3.3, p.755). 2. To interpret the parameter X or to estimate its value from observed data the following properties are useful: X is the average number of signals during a time unit. 1. - is the average distance (in time) between two signals in a Poisson process.

I^

3. The Poisson process can be interpreted as the random motion of a particle in the state space 2 = {O. 1.2.. . .}. The particle starts in the state 0. and a t every sign it jumps from state i into the next state i + 1. Furthermore, for a small interval At the transition probability pt,%+l from state i into the state i 1 should be: Pz.z+l XAt. (16.121) X is called the transition rate. 4. Examples of Poisson Processes IRadioactive decay is a typical example of a Poisson process: The number of decays (signals) are registered with a counter and marked on the time axis. The observation interval should be relatively small nith respect to the half-period of the radiating matter. I Consider the number of calls registered in a telephone exchange until time t and calculate. e.g.. the probability that at most I calls are registered until timet with the assumption that the average number

+

16.9 Mathematical Statistics 767

of calls during a time unit is A. W In reliability testing. the number of failures of a reparable system is counted during a period of duty. In queuing theory we consider the number of customers arriving at the counter of a department store: to a booking office or to a gasoline station.

2. Birth and Death Processes One of the generalizations of the Poisson process is the following: We assume that the transition rate Ai in (16.121) depends on the state i . Another generalization is when the transition from state i into state i - 1 is allowed. The corresponding transition rate is denoted by pi. The state i can be considered, e.g., as the number of individuals in a population. It increases by one at transition from state i into state i + 1, and decreases by one a t transition from i into i - 1. These stochastic processes are called birth and death processes. Let p ( X t = i) = p i ( t ) be the probability that the process is in state i a t time t . Analogously to the Poisson process: a A,-,.&. from i - 1 into i : pZ-l,% (16.122) from z 1 into z : ~ ~x bz+1At, ~ 1 , ~ from z into z : pt,%x 1 - (A, p t ) 4 t Remark: The Poisson process is a pure birth process with a constant transition rate.

+

+

3. Queuing The simplest queuing system is considered as a counter where customers are served one by one in the order of their arrival time. The waiting room is sufficiently large, so no one needs to leave because it becomes full. The customers arrive according to a Poisson process, that is, the interarrival time between two clients is exponentially distributed with parameter A. and these interarrival times are independent. In many cases also the serving time has an exponential distribution with parameter p. The parameters X and p have the following meanings: A: average number of arrivals per time unit,

A. average interarrival time, A p: average number of served clients per time unit.

1:average serving time. P

Remarks: 1. Ifthenumber ofclientsstandingin thequeueisconsideredas thestateofthisstochastic process. then the above simple queuing model is a birth and death process with constant birth rate A and constant death rate p. 2. The above queuing model can be modified and generalized in many different ways, e.g., there can be several counters where the clients are served and/or the arrival times and serving times follow different distributions (see [16.8]: [16.17]).

16.3 Mat hematical St at istics hIathematica1 statistics provides an application of probability theory for given mass phenomena. Its theorems allow us to make statements with certain probability about properties of given sets: which statements are based on the results of some experiments whose number should be kept low for economical reasons.

16.3.1 Statistic Function or Sample Function 16.3.1.1 Population, Sample, Random Vector 1. Population The Population is the set of all elements of interest in a particular study. We can consider any set of things having the same property in a certain sense, e.g.. every article of a certain production process

I

768

16. Probability Theoru and Mathematical Statistics

or all the values of a measuring sequence occurring in a permanent repetition of an experiment. The number .Vof the elements of a population can be very large, even practically infinite. \\-e often use the word population to denote also the set of numerical values assigned to the elements. 2. Sample In order not to check the total population about the considered property, data are collected only from a subset, from a so-called sample of size n ( n 5 N ) . We talk about a random choice if every element of the population has the same chance of being chosen. A random sampleof size n from a finite population of size JV is a sample selected such that each possible sample of size n has the same probability of being selected. A random sample from an infinite population is a sample selected such that each element is selected independently. The random choice can be made by so-called random numbers. We often use the word sample for the set of values assigned to the selected elements.

3. Random Choice with Random Numbers It often happens that a random selection is physically impossible on the spot, e.g.. in the case of piled material, like concrete stabs. Then we apply random numbers for a random selection (see Table 21.19, p. 1091). hlost calculators can generate uniformly distributed random numbers from the interval [O. 11. Pushing the button R . 0 or RAND we get a number between 0.00,. . O and 0.99,. .9. The digits after the decimal point form a sequence of random numbers. We often take random numbers from tables. Two-digit random numbers are given in Table 21.19, p. 1091. If we need larger ones, then we can compose several-digit numbers from them by writing them after each other. I.4random sample is to be examined from a transport of 70 piled pipes. The sample size is supposed to be 10. We number the pipes from 00 to 69. A two-digit table of random numbers is applied to select the numbers. We fix the way we choose the numbers, e.g., horizontally, vertically or diagonally. If during this process random numbers occur repeatedly, or they are larger than 69: then they are simply omitted. The pipes corresponding to the chosen random numbers are the elements of the sample. If we have a several-digit table of random numbers, we can decompose them into two-digit numbers.

4. Random Vector .A random variable X can be characterized by its distribution function, by its parameters, where the distribution function itself is determined completely by the properties of the population. These are unknown at the beginning of a statistical investigation, so we want to collect as much information as possible with the help of samples. Usually we do not restrict our investigation to one sample but we apply more samples (with same size n if it is possible, for practical reasons). The elements of a sample are chosen randomly, so the realizations take their values randomly, i.e.: the first value of the first sample is usually different from the first value of the second sample. Consequently. the first value of a sample is a random variable itself denoted by X I . Analogously, we can introduce the random variables X z . X 3 , . . X , for the second, third,. . . . n-th sample values, and they are called sample variables. Together, they form the random vector

21 = ( X I , xz.. . . , Xn). Every sample of size n with elements xi can be considered as a vector x = ( 5 1 . z2:. . G I ) , as a realization of the random vector. . 1

16.3.1.2 Statistic Function or Sample Function Since the samples are different from each other, their arithmetic m e g s E are also different. We can consider them as realizations of a new random variable denoted by X which depends on the sample

16.3 Mathematical Statistics 769 varaables XI,X2.. . .

.X,.

. . . xl, with mean El. . . . 22, with mean ZZ. .. .. . .. .. . . . . . m-th sample: xmlrzm2, . . . x,, with mean Em. 1. sample: xll, 2. sample: xzl,

212,

x22, I

(16.123)

\$'e denote the realization of the j-th sample variable in the i-th sample by xij (i = 1,2,. . . , rn; j = 1 2, . . . n) . A function of the random vector = (XI)Xz,. . . , X,) is again a random variable, and it is called a statistic or sample function. The most important sample functions are the mean, variance, median and range. ~

~

1. Mean The mean

x of the random variables Xi is: .

-

n

1

'"

( 16.124a)

X=-CX,. 2=1

The mean Z of the sample (zlrxz,

t . .

2,)

is (16.124b)

It is often useful to introduce an estimate zo in the calculations of the mean. It can be chosen arbitrarily but possibly close to the mean E. If, e.g., x,,(i = 1 , 2 , . . .) are several-digit numbers in a long measuring sequence, and they differ only in the last few digits, it is simpler to do the calculations only with the smaller numbers (16.124~) 2, = 2, - zo. Then we get I = xo

+ -I 1n2 , = 50 + z.

(16.124d)

2=1

2. Variance The variance S2 of the random variables X, with mean 7 is defined by: l

n

SZ = -C(X,- 7 ) Z .

(16.125a)

n - 1 ,=I The realization of the variance with the help of the sample (21,$ 2 , . . . , 2), is (16.125b) It is proven that in the estimation of the variance of the original population we get a more accurate estimation by dividing n - 1 than by dividing n. With the estimated value 20 we get n

2

z

- 2=1

s -

n

n

2,2 -

2=l

n-1

zi2 - n(E - 2 0 ) 2

Ti

-

i=l

n-1

For z = xo the correction is z c z, = 0 because Z = 0 holds. 2=1

(16.125~)

770

2 6. Probabilitv Theorv and Mathematical Statistics

3. Median Let the n elements of the sample be arranged in ascending (or descending) order. If n is odd. then the n+l medzan d is the value of the --th item; if n is even, then the median is the average value of the 2 -th items, the two items on the middle.

. whose elements are arranged in ascending (or The median j: in a particular sample ( ~ 1 ~ x 2. .,, x,), descending) order, is if n = 2 m + 1, ,

2 4.

(16.126)

if n = 2 m .

Range R = max,Y, z

- minX, t

(i = 1 , 2 , .. . , n ) .

(16.127a)

The range R of a particular sample (a,. x2 (16.127b) R = a, - x,in. Every particular realization of a sample function is denoted by a lowercase letter. except the range R, and R. i.e., for a particular sample ( X I , xZ1 a,) we calculate the particular values E , s2:2%

ILVe take a sample of 15 loudspeakers from a running pro-

16.3.2 Descriptive Statistics 16.3.2.1 Statistical Summarization and Analysis of Given Data In order to describe statistically the properties of certain elements, we characterize them by a random variable X . Usually, the n measured or observed values x, of the property X form the starting point of a statistical investigation, which is made to find some parameters of the distribution or the distribution itself of X . Every measured sequence of size n can be considered as a random sample from an infinite population, if the experiment or the measurement could be repeated infinitely many times under the same conditions. Since the size n of a measuring sequence can be very large, we proceed as follows: 1.Protocol, P r i m e Notation The measured or observed values x, are recorded in a protocol list. 2. Intervals or Classes We consider an interval, which contains the n data 2, (i = 1 , 2 . . . . , n) of the sample, and divide it into k subintervals, so-called classes or class intervals. Usually 10-20 classes are selected with equal length h! and their boundaries are called class boundaries. The endpoints of the total interval are not uniquely defined; in general, we choose them approximately symmetricly with respect to the smallest and largest value of the sample, and class boundaries should be different from any sample value. 3. Frequencies and Frequency Distribution The absolutefrequencies hj ( j = 1 , 2 , .. . , k ) are the numbers h, of data (occupancy number) belonging to a given interval Axj.The ratios hj/n (in %) are called relative frequencies. If the values h,/n are represented over the classes as rectangles, then we get a graphical representation of the given frequency distribution. and this representation is called a histogram (Fig. 16.13a). The values h,/n can be considered as the empirical values of the probabilities or the density function f(x).

16.3 Mathematical Statistics 771 4. Cumulative Frequency Adding the absolute or relative frequencies, we get the cumulative abso-

lute or relative frequency Table 16.3 Frequency table

I

Class 50 - 70 71 - 90 91 -110 111 -130 131 - 150 151 -170 171 -190 191 -210 211 -230 231 -250

Fi (%)

12.0

7.2 4.8 2.4

22.4 40.0 64.0 85.6 92.8 97.6 100.0

Fj= h l + h + . " + h , % ( j = 1 , 2 > . . , k ) . (16.128) n If we represent the value F3 at the upper boundary and draw a horizontal line until the next boundary, then we get a graphical representation of the empirical distribution function, which can be considered as an approximation of the unknown underlying distribution function F ( z ) (Fig. 16.13b). I Suppose we perform n = 125 measurements during a study. The results spread in the interval from 50 to 270, so it is reasonable to divide this interval into k = 11 classes with a length h = 20. We get the frequency table Table 16.3.

Fi/% 100

70

110 150 190 230 270 x

70 110 150 190 230 270 x b)

a)

Figure 16.13

16.3.2.2 Statistical Parameters After summarizingand analyzing the data of the sample as given in 16.3.2.1,p. 770. we can approximate the parameters of the random variable by the following parameters:

1. Mean If we use directly the data of the sample, then the sample mean is (16.129a)

If we use the means E, and frequencies h, of the classes, then we get r i r

(16.129b)

2. Variance If we use directly the data of the sample, then the sample variance is (16.130a)

772

16. Probabilitv Theoru and Mathematical Statistics

If we use the means zj and frequencies hj of the classes, then we get (16.130b) The class midpoint u, (the midpoint of the corresponding interval) is also often used instead of ?EJ. 3. Median The median 5 of a distribution is defined by

P ( X < 5) = I.

2 The median may not be a uniquely determined point. The median of a sample is if n = 2m + 1, Xrntl, 5 = x,+1 +x, , ifn=2m. 2

{

4. Range R = x,

(16.131a)

( 16.131b)

(16.132)

- x,jn.

5 . Mode or Modal Value is the data value that occurs with greatest frequency. It is denoted by D.

16.3.3 Important Tests One of the fundamental problems of mathematical statistics is to draw conclusions about the population from the sample. There are two types of the most important questions: 1. The type of the distribution is known, and we want to get some estimate for its parameters. A distribution can be characterized mostly quite well by the parameters p and u2 (here p is the exact value of the expected value, and u2 is the exact variance), consequently one of the most important questions is how good an estimation can we give for them, based on the samples. 2. Some hypotheses are known about these parameters, and we want to check if they are true. The most often occurring questions are: a) Is the expected value equal to a given number or not? b) Are the expected values for two populations equal or not? c) Does the distribution of the random variable with p and u* fit a given distribution or not? etc. Because in observations and measurements, the normal distribution has a very important role, we discuss the goodness of fit test for anormal distribution. The basic idea can be used for other distributions, too.

16.3.3.1 Goodness of Fit Test for a Normal Distribution There are different tests in mathematicalstatistics to decide if the dataof a sample come from a normal distribution. We discuss a graphical one based on normal probability paper, and a numerical one based on the use of the chi-square distribution ( rix2test”).

1. Goodness of Fit Test with Probability Paper a) Principle of Probability Paper The x-axis in a right-angled coordinate system is scaled equidistantly, while the y-axis is scaled on the following way: It is divided equidistantly with respect to Z, but scaled by l

y = @(Z)= -

/ e-F Z

Jz;;-m

t2

dt.

(16.133)

16.3 Mathematical Statistics 773 If a random variable X has a normal distribution with exDected value LI and variance u2,then for its distribution function (see 16.2.4.2,p. 757)

F(x)= @

(7) =

@(Z)

(16.134a)

Z 0

holds, Le..

Z=x-P U

1 -1

(16.134b)

X ~

P

+

~

(16.134~)

P-ff

must be valid. and so there is a linear relation between x and Z and (16.134~).

+

b ) Application of Probability P a p e r 4 )Y We consider the data of the sample, we calz @ ( t ) , , , , , culate the cumulative relative frequencies according to (16 128), and sketch these onto the probability paper as the ordinates of the points with abscissae the upper class boundaries. If these points are approximately on a straight line, (with small deviations) then the random variable can be considered as a normal random variable (Fig. 16.14). As we see from Fig. 16.14, the distribution to which the data of Table 16.3 belong, can be considered as a normal distribution. We can also see that p x 176, c x 37.5 (from the x values be70 110 150 190 230 270 x longing to the 0 and i l values of 2). P* P P+O Remark: The values F, of the relative cumulative frequencies can be plotted more easily on the Figure 16.14 probability paper, if its scaling is equidistant with respect to y, which means a non-equidistant scaling for the ordinates. I

2. XZtest \%’e want to check if a random variable X can be considered normal. We divide the range of X into k classes and n e denote the upper limit of the j-th class ( j = 1 , 2 , . . . , k ) by[,. Let p j be the “theoretical” probability that X is in the j-th class, Le.. (16.135a) P j = F K j ) - F(G-1); where F ( X )is the distribution function of X ( j = 1 , 2 , . . . k ; (0 is the lower limit of the first class with F(
(16.135b) must hold, where @(z)is the distribution function of the standard normal distribution (see 16.2.4.2, p. 757). The parameters p and u2 of the population are usually not known. We use E and s2 as an approximation of them. We have to make the decomposition of the range so that the expected frequencies for every class should exceed 5, Le., if the size of the sample is n,then np, 2 5, Now, we consider the sample ( X I ,~ 2 . ... .z,)of size n and calculate the corresponding frequencies h, (forthe classes given above). Then the random variable

(16.135~) has approximately a x2 distribution with m = k - 1 degrees of freedom if we know p and u2,m = k - 2 if we estimated one of then1 from the sample, and m = k - 3 if we estimated both by ?E and 2.

774

16. Probability Theory and Mathematical Statistics

Now, n e determine a level a , which is called the signzficance level, and determine the quantile (i depends on the number of unknown parameters) of the corresponding x2 distribution. e.g., from Table 21.16;p. 1087. This means P ( x 2 2 xiikvi)= a holds. Then we compare the value n e got in (16.135~)and this quantile, and if

xi

< X a2 ; k - i

(16.135d) holds. we accept the assumption that the sample came from a normal distribution. This test is called the x2 test for goodness ojfit. IThe following x2 test is based on the example on p. 771. The sample size is n = 125, with the mean = 176.32 and variance s2 = 36.70. These values are used as approximations of the unknown according to (16.135~) parameters p and u2 of the population. We can determine the test statistic after performing the calculations according to (16.135a) and (16.135b))as shown in Table 16.4.p. 774. X S2

xi

Table 16.4 x2 test

-

p

ti 70 90 110 130 150 170 190 210 230 250 270

i}

13

9 15 22 30 27 9

;} 9

(V) 0.0019 0.0094 0.0351 0.1038 0.2358 0.4325 0.6443 0.8212 0.9279 0.9778 0.9946

-2.90 -2.35 -1.81 -1.26 -0.72 -0.17 0.37 0.92 1.46 2.01 2.55

Pi 0.0019 0.0075 0.0257 0.0687 0.1320 0.1967 0.2118 0.1769 0.1067 0.0499 0.0168

nPj

nPj

0.2375 12.9750

8.5857 16.5000 24.5875 26.4750 22.1125 13.3375 6.2375

0.00005 0.1635 0.2723 0.4693 1.0803 1.4106 0.0526

xi = 3.4486

-

-

(hj - npj)'

xi

It follows from the last column that = 3.4486. Because of the requirement np, 2 5. the number of classes is reduced from k = 11to k* = k -4 = 7. LVe calculated the theoretical frequencies np, with the estimated values z and s2 of the sample instead of p and uz of the population, so the degrees of freedom of the corresponding x2 distribution is reduced by 2. The critical value is the quantile x : , ~ . - ~ - For ~. a = 0.05 we get xi 05,4 = 9.5 from Table 21.16,p. 1087, so because of the inequality xi < there is no contradiction to the assumption that the sample is from a population with a normal distribution.

xi

16.3.3.2 Distribution of the Sample Mean Let X be a continuous random variable. Suppose we can take arbitrarily m a y samples of size n from the corresponding population. Then the sample mean is also a random variable X , and it is also continuous.

1. Confidence Probability of the Sample Mean If X has a normal distribution with parameters p aEd u2,then ff is also a normal random variable with parameters p and u 2 / n ,Le.. the density function j ( x ) of X is concentrated more around p than the density function f ( x ) of the population. For any value E > 0:

5 E)

= 20

(;)

-

jEt)

P ( / X - p I < E ) = 2 @ - -1.

(16.136)

16.3 Mathematical Statistics 775

It followsfrom this that with increasing samplesize n. the probability that the sample mean is a good approximation of p is also increasing. 1 1 We get for E = -u from (16.136) P 2

Table 16,5 Confidence level for the sample mean

- 1, and for different values of n we get the values listed

16

! :i 1

in Table 16.5. We see from Table 16.5,e.g., that with a sample size n = 49. the probability that the sample mean 2 differs from p

68.27 % 95.45 rc 98.76 99.96 %

1

~

by less than +-u is 99.95 %. 2

2. Sample Mean Distribution for Arbitrary Distribution of the Population The random variable 7 has an approximately normal distribution with parameters p and u z / nfor any distribution of the population with expected value 1.1 and variance u *. This fact is based on the central limit theorem.

16.3.3.3 Confidence Limits for the M e a n 1. Confidence Interval for the Mean with a Known Variance u2 If X is a random variable with parameters p and uz, then according to 16.3.3.2, p. 774. x i s approximately a normal random variable with parameters p and uz/n. Then substitution of -

- 'Y-p Z=-fi

(16.137)

U

yields a random variable

zwhich has approximately a standard normal distribution, therefore

1 E

P ( lzl 5 E ) =

;(x) dx = 2 @ ( ~ -) 1.

(16.138)

--E

If the given significance level is c y , namely, (16.139) P( I Zl 5 E ) = 1 - c y , is required, then E = ~ ( acan ) be determined from (16.138)) e.g.. from Table 21.15,p. 1085, for the standard normal distribution. From 1215 cy) and from (16.137) we get the relation p=T*-

U

( )

f i Ecy

The values Z f % ( c y )

J;;

(16.140) in (16.140) are called conjdence lzmztsfor the expected value and the interval

between them is called a confidence zntervalfor the ezpected value p with a known uz and given significance level cy. In other words: The expected value 1.1 is between the confidence limits (16.140) with a probability 1 - cy Remark: If the sample size is large enough, then we can use sz instead of u* in (16.140). The sample size is considered to be large. if n > 100, but in practice. depending on the actual problem. it is considered to be sufficiently large if n > 30. If n is not large enough, then we have to apply the t distribution to determine the confidence limits as in (16.143).

2. Confidence Interval for the Expected Value with an Unknown Variance u2 If the variance u* of the population is unknown, then we replace it by the sample ivariance sz and instead of (16.137) we get the random variable

x-/I

T=-fi.

(16.141)

I

776

16. Probabilitu Theoru and Mathematical Statistics

which has a t distribution (see 16.2.4.8, p. 761) with rn = n - 1 degrees of freedom. Here n is the size of the sample. If n is large, e.g., n > 100 holds, then T can be considered as a normal random variable as 2 in (16.137). We get for a given significance level a (16.142)

, = t,/z;n-l, where ta/2;n-1 is the quantile of the t distribution From (16.142) it follows that E = € ( a n) (with n - 1 degrees of freedom) for the significance level a (Table 21.18,p. 1090). It follows from IT1 = ta/2;n-l that p = zf S t , / z ; n - l .

(16.143)

fi

The values F & z t , p i n - l are called the confidence limitsfor the expected ualue 1.1 of the distribution of

J;;

the population with an unknown variance 0 ' and with a given significance level a.The interval between these limits is the confidence interval. I A sample contains the following 6 measured values: 0.842; 0.846; 0.835; 0.839; 0.843; 0.838. We get from this 2 = 0.8405 and s = 0.00394. LVhat is the maximum deviation of the sample mean from the expected value p of the population distribution. if the significance level cy is given as 5 % or 1 % ? 1. cy = 0.05: We read from Table 21.18,p. 1090, that ta/2;5 = 2.57, and we get - 5 2.57. 0.00394/& = 0.0042. Thus, the sample mean z = 0.8405 differs from the expected value p by less than f0.0042 with a probability 95 %. 2. a = 0.01: t a p 5 = 4.03; - 1-11 5 4.03.0.00394/& = 0.0065, Le., the sample mean differs from p by less than f0.0065with a probability 99 %.

lx

lx

16.3.3.4 Confidence Interval for the Variance If the random variable X has a normal distribution with parameters p and u2,then the random variable 52

XZ = ( n - 1)-

U2

has a xZ distribution with m = n - 1 degrees of freedom, where n is the sample size and sz is the sample deviation. fp (z) denotes the density function of the xZdistribution in Fig. 16.15,and we see that (16.145) P ( x 2< x:) = P ( x 2 > x:) = E 2. Thus, using the quantiles of the x2 distribution (see Table 21.16,p. 1087) we obtain that

XZ = X?-a/z;n-1> xt = ~ % / 2 ; n - 1 ,

a

Figure 16.15 (16.146)

Considering (16.144) we get the following estimation for the unknown variance o2 of the population distribution with a significance level a: (16.147) The confidence interval given in (16.147) for uz is fairly large for small sample sizes. IFor the numerical data of the example on p. 776 and for cy = 5% we get from Table 21.16,p. 1087,

16.9 Mathematical Statistics 777

x&~

= 12.8, so it follows from (16.147) that 0.625 s

= 0.831 and s = 0.00394.

5

u

5 2.453

s with

16.3.3.5 Structure of Hypothesis Testing A statistical hypothesis testing has the following structure: 1. First we develop a hypothesis H that the sample belongs to a population with some given properties, e.g., H: The population distribution has a normal distribution with parameters p and uz (or another given distribution), or H: The expected value is equal to a given value PO,or H: Two populations have the same expected value, p1 - pz = 0, etc. 2. We determine a confidence interval B , based on our hypothesis (in general with tables). The value of the sample function should be in this interval with the given probability, e.g., with probability 99% for cy = 0.01). 3. We calculate the value of the sample function and we accept the hypothesis if this value is in the given interval B , otherwise we reject it. ITest the hypotesis H: p-= po with a significance level a. The random variable T = - ” fihas a t distribution with m = n - 1 degrees of freedom according to 16.3.3.3,p. 775. It follows from this that we have to reject this hypothesis, if T is not in the confidence interval given by (16.143), Le., if IE - Pol

2 pL/z;n-1

(16.148)

J5i

holds. WB say that there is a signijcant dzference. For further problems about tests see [16.14].

16.3.4 Correlation and Regression Correlation analysis is used to determine some kind of dependence between two or more properties of the population from the experimental data. The form of this dependence between these properties is determined with the help of regression analysis.

16.3.4.1 Linear Correlation of two Measurable Characters 1. Two-Dimensional Random Variable In the following, we mostly use the formulas for continuous random variables, but it is easy to replace them by the corresponding formulas for discrete variables. Suppose that X and Y as a two-dimensional random variable ( X ,Y), have the joint distribution function ~

(16.149a) Fl(2) = P(X

5 2 . Y < m), Fz(y) = P ( X < co,Y 5 y).

(16.149b)

The random variables X and Y are said to be independent of each other if (16.150) F ( 2 ,Y) = Fib) . Fz(Y) holds. We can determine the fundamental quantities assigned to X and Y from their joint density function f(x, y): a) Expected Values m x

px = E ( X )

=//x f ( q y ) d z d y , (16.151a) -m-m

m m

py

= E(Y)

=//yf(.r,y)dzdy, -m-m

(16.151b)

778

16. Probabilitu Theoru and Mathematical Statistics

b) Variances u;. = E ( ( X - p x ) 2 ) ,

u; = E ( ( Y - p y ) 2 ) .

(16.152a)

c) Covariance

(16.162b)

d) Correlation Coefficient

oxy = E ((X - px)(Y- p y ) ) . (16.153)

OXY

exv = -.

(16.154)

Ox UY

We assume that every expected value above exists. The covariance can also be calculated by the formula

/ /xy f(z. x3c.

U ~ = Y

E ( X Y )- pxpy where E ( X Y ) =

y)

dx d y .

(16.155)

-X-X

The correlation coefficient is a measure of the linear dependence of X and Y ,because of the following facts: A11 points (X,Y)are on one line with probability 1 if = 1 holds. If A’ and Y are independent random variables, then their covariance is equal to zero, QXY = 0. From Q X . ~= 0, it does not follow that X and Y are independent, but it does if they have a two-dzmenszonalnormal dzstrzbutzon which is defined by the density function

(16.166)

2. Test for Independence of two Variable \Ve often have the question of whether the variables X and Y can be considered independent with QXY = 0, if the sample with size n and with the measured values (xi,9%)(i = 12,. . . , n) comes from a two-dimensional normal distributed population. The test is performed in the following way: a) We have the hypothesis H : =0. b) We determine a significance level CY and determine the quantile ta,mof the t distribution from Table 21.18, p. 1090, form = n - 2. c) K e calculate the empirical correlation coeficients T , and ~ calculate the test statistics (sample function) n - W Y % - 8) t=(16.157a) with T , ~= . (16.157b)

“’-

*\

d) LVe reject the hypothesis if It1 2

E(.%

%=I

holds.

16.3.4.2 Linear Regression for two Measurable Characters 1. Determination of the Regression Line If we have detected a certain dependence between the variables X and Y by the correlation coefficient, then the next problem is to find the functional dependence Y = f ( X ) . \Ye consider mostly linear dependence. The simplest case is linear regression; when we suppose that for any fixed value of zthe random variable Y in the population has a normal distribution with the expected value (16.158) E ( Y ) = a bz and a variance uz independent of x. The relation (16.158) means that the mean value of the random variable Y depends linearly on the fixed value of x. The values of the parameters a; b and 0’ of the

+

16.3 Mathematical Statistics 779 population are usually unknown, and we estimate them approximately by the least squares method from a sample with values (x2.yJ (i = 1 , 2 , . . . , n). The least squares method requires that

C[yt- (a+ bx,)12 = min!

(16.159)

2=1

and we get the estimates

(16.160a)

(16.160b) and the empirical correlation coefficient rZYis given in (16.157b). The coefficients 6 and 6 are called regression coefficients. The line y(x) = ii 6x is called the regression line.

"-

+

2. Confidence Intervals for the Regression Coefficients Our next question is, after the determination of the regression coefficients ii and estimates approximate the theoretical values a and 6. We form the test statistics tb

= ( b - b) ~

(16.161a)

S

Y

p

and t, = (ii- a )

s sy

X

,

m

J5i

Jq

b, how well do the .

(16.161b)

i=l

These are realizations of random variables having a t distribution with m = n - 2 degrees of freedom. We can determine the quantile t,/2;m taken from Table 21.18.p. 1090, for a given significance level cy and because P( It1 < tapim)= 1 - cy holds fort = t , and t = t b :

We can determine a conjdence region for the regression line y = a given in (16.162a.b) for a and b.

+ bx with the confidence interval

16.3.4.3 Multidimensional Regression 1. Functional Dependence

.

Suppose that there is a functional dependence between the characters XI, X 2 , . . . X,: and Y , which is described by the theoretical regression function S

.

y = f ( x I . x 2 . .. . xn) = Ca,g,(xl,

22,.

. . ,in).

(16.163)

j=0

.

The functions g3(x1.2 2 , . . . xn) are known functions of n independent variables. The coefficients a3 in (16.163) are constant multipliers in this linear combination. We also call expression (16.163) linear regression, although the functions g3 can be arbitrary. IThe function f(q, i z ) = a. alzl a222 a32l2 + aqxZ2+ a55152, which is a complete quadratic polynomial of two variables with go = 1, g1 = 21: g2 = x2,g3 = z l z ,g4 = x Z 2 ,and g5 = x1x2;is an example for a theoretical linear regression function.

+

+

+

780

16. Probabilitu Theoru and Mathematical Statistics

2. Writing in Vector Form It is useful to write formulas in vector form in the multidimensional case

x = ( 2 1 : 2 2 , , , , >Z")T1

(16.164)

so, (16.163) now has the form: S

(16.165)

3. Solution and Normal Equation System The theoretical dependence (16.163) cannot be determined by the measured values

(@, f J , ( 2 = 1 . 2 , . . . , 'V) because of random measuring errors. We are looking for the solution in the form Y = i(4=

(16.166a)

6393(21)

(16.166b)

j=O

and using the least squares method (see 16.3.4.2, l., p. 779) we determine the coefficients ii3 as the estimations of the theoretical coefficients a,, from the equation N

1[it- j (x('))I2= min! .

(16.166~)

1=1

Introducing the notation

(16.166d)

we get from (16.166~)the so-called normal equation system

G T G l = GTf (16.166e) to determine 5.The matrix GTG is symmetric, so the Cholesky method (see 19.2.1.2, p. 890) is especially good to solve (16.166e). I Consider the sample whose result is given in the next table. Determine the coefficients of the regression function (16.167): 5(21,22) J " ( Z ~ , Z ~ ) 1.5

3.5 6.2 3.2

= a0 + a l z 1

+~2x2.

(16.167)

From (16.166d) it follows that (16.168) and (16.166e) is 460 + 1661 + 1.6 62 = 14.4, l 6 & + 686l-t 6.462=58.6, 1.660 6.461 0.6862 = 5.32,

+

+

Le.,

E o = 7.0, 61=0.25, 62 = -11.

(16.169)

16.3 Mathematical Statistics 781

4. Remarks 1. To determine the regression coefficients we start with the interpolation f ( ~ ( ' 1 ) = fi (z = 1 , 2 . .. . , N ) , Le.. with Gl=f (16.170) In the case s < 5 . (16.170) is an overdetermined system of equations which can be solved by the Householder method (see 19.6.2.2,p. 918). The multiplication of (16.170) by GT to get (16.166e) is also called Gauss transformatton. If the columns of the matrix G are linearly independent, Le., rank G = s + 1 holds, then the normal equation system (16.166e) has a unique solution, which coincides with the result of (16.170) got by the Householder method. 2. Also in the multidimensional case. it is possible to determine confidence intervals for the regression coefficients with the t distribution, analogously to (16.162a,b). 3. lye can analyse the assumption (16.166b) by the help of the F distribution (see 16.2.4.7, p. 761), if there are some superfluous variables in it.

16.3.5 Monte Carlo Methods 16.3.5.1 Simulation Simulation methods are based on constructing equivalent mathematical models. These models are then easily analysed by computer. In such cases, we talk about dzgztal szmulatzon. A special case is given by Monte Carlo methods when certain quantities of the model are randomly selected. These random elements are selected by using random numbers.

16.3.5.2 Random Numbers Random numbers are realizations of certain random quantities (see 16.2.2, p. 749) satisfying given distributions.

1. Uniformly Distributed Random Numbers These numbers are uniformly distributed in the interval [0,1],they are realizations of the random variable X with the following density function fo(z)and distribution function Fo(z): 0 for 0 5 x, 1 for 0 5 2 5 1, (16.171) 0 otherwise: 1 for z 2 1. 1. Method of t h e Inner Digits of Squares A simple method to generate random numbers is suggested by J. v. Neumann. It is also called the method of the inner digits of squares, and it starts from a decimal number z E (0,l) which has 2n digits. Then we form z2, so we get a decimal number which has 4n digits. We erase its first and its last n digits, so we again have a number with 2n digits. To get further numbers, we repeat this procedure. In this way we get 2n digit decimal numbers from the interval [0,1]which can be considered random numbers with a uniform distribution. The value of 2n is selected according to the largest number representable in the computer. For example, we may choose 2n = 10. This procedure is seldom recommended, since it produces more smaller numbers than it should. Several other different methods have been developed. 2. Congruence Method The so-called congruence method is widely used: A sequence of integers zi (i = 0;1 . 2 . . . .) is formed by the recursion formula zttl I c z, modm. (16.172) Here zo is an arbitrary positive number and c and m denote positive integers, which must be suitably chosen. For z,+1 ne take the smallest non-negative integer satisfying the congruence (16.172). The numbers zi/m are between 0 and 1 and can be used for uniformly distributed random numbers. 3. Remarks a) We choose m = 2r: where T is the number of bits in a computer word, e.g., r = 40. Then the order of c is fi.

I

782

16. Probability Theory and Mathematical Statistics

b ) Random number generators using certain algorithms produce so-called pseudorandom numbers. c) On calculators and also in computers, “ran” or “rand” is used for generating random numbers.

2. Random Numbers with other Distributions To get random numbers with an arbitrary distribution function F ( z )we adopt the following procedure: Consider a sequence of uniformly distributed random numbers ( 2 , . . . from [0,1]. With these numbers we form the numbers 7%= F-’(<,) for i = 1,2,.. . . Here F-’(z) is the inverse function of the distribution function F ( z ) .Then we get:

P ( %I .) = P ( P ( & )5 ). = P ( & I F ( z ) )=

7

fo(t) d t = F ( z ) .

Le., the random numbers q1.

satisfy a distribution with the distribution function F(.r).

3. Tables and Applica

f Random Numbers

(16.173)

1. Construction Random number tables can be constructed in the following way. \Ve index ten identical chips by the numbers 0 , 1 , 2 , . . . .9. We place them into a box and shuffle them. One of them is then selected. and its index is written into the table. Then we replace the chip into the box, shuffle again, and choose the next one. In this way a sequence ofrandom numbers is produced, which is written in groups (for easier usage) into the table. In Table 21.19,p. 1091. four random digits form a group. In the procedure. we have to guarantee that the digits 0 , l . 2 9 always have equal probability. 2. Application of Random N u m b e r s The use of a table of random numbers is demonstrated with an example. ISuppose x e choose randomly n = 20 items from a population of :V = 250 items. Lye renumber the objects from 000 to 249. We choose a number in an arbitrary column or row in Table 21.19.p. 1091, and we determine a rule of how the remaining 19 random numbers should be chosen. e.g., vertically, horizontally or diagonally. We consider only the first three digits from these random numbers, and we use them only if they are smaller than 250.

16.3.5.3 Example of a Monte Carlo Simulation Lye consider the approximate evaluation of the integral

: ( 16.174) as an example of the use of uniformly distributed random numbers in a simulation. We discuss two solution methods.

1. Applying the Relative Frequency Il’e suppose 0 5 g(z) 5 1 holds. We can always guarantee this condition by a transformation (see (16.179). p. 783). Then the integral I is an area inside the unit square E (Fig. 16.16). If we consider the numbers of a sequence of uniformly distributed random numbers from the interval [0,1] in pairs as the coordinates of points of the unit square E . then we get n points P, ( t = 1.2.. . . , n ) . If we denote by m the number of points inside the area A , then considering the notion of the relative frequency (see 16.2.1.2. p. 746): 1

Figure 16.16

/g(o) dz x

-.m

( 16.175)

0

To achieve relatively good accuracy with the ratio in (16.175). we need a very large number of random numbers. This is the reason why we are looking for possibilities to improve the accuracy. One of these methods is the following Monte Carlo method. Some others can be found in the literature.

2. Approximation by the Mean Value

To determine (16 174), we start with n uniformly distributed random numbers (1, E 2 . . . . .En as the realizationof the uniformlydistributedrandomvariableX. Then thevaluesg, = g(&) (z = 1 , 2 , . . . n ) are realizationsof the random yariableg(X), whose expectation according to formula (16.49a,b),p. 751, is:

.

(16.176) This method. which uses a sample to obtain the mean value, is also called the usual Monte Carlo method.

16.3.5.4 Application of the Monte Carlo Method in Numerical Mat hemat ics 1. Evaluation of Multiple Integrals First, we have to show how to transform a definite integral of one variable

I' =

j

h(z)dz

(16.177)

into an expression which contains the integral

I=

/

g ( x ) d x with 0 5 g(z)

5 1.

(16.178)

Then we can apply the 1Ionte Carlo method given in 16.3.5.3. Introduce the following notation: T. = a + ( b - a)u. m = min h( z) . ,If = maxh(x). (16.179) xE[a,b;

r€[a.b]

Then (16.177) becomes (16.180)

h(a + ( b - a ) ~-) VL = g ( u ) satisfies the relation 0 5 g ( u ) 5 1. J4 - m IThe approximate evaluation of multiple integrals with Monte Carlo methods is demonstrated by an example of a double integral where the integrand

I' =

11

h ( z ,y) dxdy with h ( z .y) 2 0.

s

S denotes a plane surface domain given by the inequalities a 5 z 5

b and p1(z) 5 y 5 pz(z),where pl(x) and p2(z) denote given functions. Then V can be considered as the volume of a cylindrical solid K . which stands perpendicular to the z,y plane and its upper surface is given by h(x,y). If h ( z ,y) 5 e holds, then this solid is in a block Q given by the inequalities a 5 z 5 b, c 5 y 5 d, 0 5 z 5 e ( a >b. c, d, e const). After a transformation similar to (16.179), we get from (16.181) an expression

containing the integral

1'* = //g(u.u)dudu

with 0 5 g ( u ) u ) 5 1,

(16.182)

S'

where V*can be considered as the volume of a solid K' in the three-dimensional unit cube. The integral (16.182) is approximated by the Monte Carlo method in the following way: N'e consider the numbers of a sequence of uniformly distributed random numbers from the interval

784

16. Probabilitu Theow and Mathematical Statistics

[O: 11 in triplets as the coordinates of points P, (i = 1 , 2 , .. . , n) of the unit cube. and count how many points among P, belong to the solid K*. If m point belong to K*,then analogously to (16.175) v-* a -m, (16.183) n Remark: In definite integrals with one integration variable we should use the methods given in 19.3, p. 895. For the evaluation of multiple integrals, the Monte Carlo method is still often recommended.

2. Solution of Partial Differential Equations with the Random Walk Process The Monte Carlo method can be used for the approximate solution of partial differential equations with the random walk process.

a) Example of a Boundary Value Problem: Consider the following boundary value problem as an example: 6% a2u Au = -t - = 0 for (z,y) E G, (16.184a) ax2 ay2

4 z , Y) = f ( z ,Y) for (zl Y) E

r.

(16.184b)

Here G is a simply connected domain in the L , y plane; r denotes the boundary of G (Fig.16.17). Similarly to the difference method in paragraph 193.1. G is covered by a quadratic lattice, where we can assume, aithout loss of generality, that the step size can be chosen ash=l.

Figure 16.17

This way we get interior lattice points P ( z ,y) and boundary points 4. The boundary points 8, which are at the same time also lattice points, are considered in the following as points of the boundary r of G.Le.: u(R,) = f ( R J ( i = 1 , 2 , .. A T ) (16.185) t ,

b) SolutionPrinciple: \Ve imagine that aparticlestarts a random vralkfrom an interior point P ( z .y). That is: 1. The particle moves randomly from P ( z ,y ) to one of the four neighboring points. We assign to each of these four grid points 1/4: the probability to move into them. 2. If the particle reaches a boundary point 4 ,then the random walk terminates there with probability one. It can be proven that a particle starting at any interior point P reaches a boundary point R, after a finite number of steps with probability one. We denote by (16.186) P(P>8) = P ( ( T 2/15 4 ) the probability that a random walk starting at P ( z ,y) will terminate at the boundary point 4. Then we get p ( & , & ) = 1, p(Ri,R,) = 0 for i # j and (16.187)

The equation (16.188) is a difference equation for p ( ( z ,y). 4).If we start n random walks from the point P ( L ,y), from which m, terminates at 8 (m, 5 n ) ,then we get (16.189)

16.4 Calculus of Errors 785

The equation (16.189) gives an approximate solution of the differential equation (16.184a) with the boundary condition (16.185). The boundary condition (16.18413)will be fulfilled if we substitute

4 P ) = 41,Y) =

N

f(RtM(z3Y ) 1 8

1 3

(16.190)

2=1

because of (16.188). w(R,) =

N 2=1

f(R,,)p(R,,R,,) = f(R,).

To calculate v(z, y) we multiply (16.188) by f(R,,). After summation we get the following difference equation for v(z.y): 1

w(z,y) = -[v(z-1,y)+w(zt1,y)+w(z,y-1)+w(z,y+1)]. 4

(16.191)

If we start n random walks from an interior point P ( z .y), and among them m3terminate at the boundary point R,, (i = 1 . 2 , . , . , N ) , then we get an approximate value at the point P ( z ,y) of the boundary value problem (16.184a,b) by

16.3.5.5 Further Applications of the Monte Carlo Method Monte Carlo methods as stochastic simulation, sometimes called methods ofstatistical experiments,are used in many different areas. For example we mention: Operations research: Queueing systems, process design, inventory control, service For further details of these problem areas see for example [16.13].

16.4 Calculus of Errors Every scientific measurement, giving certain numerical quantities - regardless of the care which with the measurements are made - is always subject to errors and uncertainties. There are observational errors, errors of the measuring method, instrumental errors and often errors arising from the inherent random nature of the phenomena being measured. Together they compose the measunnent error. All measurement errors arising during the measuring process we call devzataons. As a consequence a measured quantity represented by a number of significant digits can be given only with a rounding error, Le.. with a certain statistical error, which we call the uncertazntyof the result. 1. The deviations of the measuring process should be kept as small as possible. On this basis we have to evaluate the possible best approximation, what can be done with the help of srnoothzng methods which have their origin in the Gaussian least squares method. 2. The uncertainties have to be estimated as well as possible, what can be done with the help of methods of mathematacal statastzcs. Because of the random character of the measuring results we can consider them as statistical samples (see 16.2.3. l.,p. 752) with its probability distribution, whose parameters contain the desired information. In this sence, measurement errors can be seen as sampling errors.

16.4.1 Measurement Error and its Distribution 16.4.1.1 Qualitative Characterization of Measurement Errors If we qualify the measurement errors by their causes, we can distinguish between the following three types of errors: 1. Rough errors are caused by inaccurate readings or confusion; they are excludable.

786

16. Probabilitv Theoru and Mathematical Statistics

2. Systematic measurement errors are caused by inaccurately scaled measuring devices and by the method of measuring, where the method of reading the data and also the measured error of the measurement system can play a role. They are not always avoidable. 3. Statistical or random measurement errors can arise from random changes of the measuring conditions that are difficult or impossible to control and also by certain random properties of the events observed. In the theory of measurement errors the rough errors and the systematic measurement errors are excluded and we deal only with the statistical properties and with the random measurement errors in the calculation of the rounding errors.

16.4.1.2 Density Function of t h e Measurement Error 1. Measurement Protocol UB suppose that in the characterization of the uncertainty we have the measured results listed in a measurement record as a prime notation and we have the relative frequencies or the density function f ( ~ ) ~ or the cumulative frequencies or the distribution function F ( z ) (see 16.3.2.1, p. 770) of the uncertain values. By z we denote the realization of the random variable X , which is under consideration.

2. Error Density Function Special assumptions about the properties of the measurement error result in certain special properties of the density function of the error distribution: 1. Continuous Density Function Since the random measurement errors can take any value in a certain interval, they are described by a continuous density function f(z). 2. Even Density Function If measurement errors with the same absolute value but with different signs are equally likely, then the density function is an even function: f(-z) = f(z). 3. Monotonically Decreasing Density Function If a measuring error with larger absolute value is less likely than an error with smaller absolute value, then the density function f(z)is monotonically decreasing for z > 0. 4. Finite Expected Value The expected value of the absolute value of the error must be finite: (16.193) -m

Different properties of the errors result in different types of density functions.

3. Normal Distribution of the Error 1. Density Function a n d Distribution Function In most practical cases we can suppose that the distribution of the measurement error is a normal distribution with expected value p = 0 and variance uz.Le.. the density function f(x) and the distribution function F ( z )of the measurement error are: 22

f(z)= 1 e - P U

(16.194a)

and F ( z ) = -

6

Here @(z) is the distribution function of the standard normal distribution (see (16.74a), p. 757, and Table 21.15, p. 1085). In the case of (16.194a,b) we speak about normal errors. 2. Geometrical Representation The density function (16.194a) is represented in Fig. 16.18a with inflection points and points at the center of gravity, and its behavior is shown in Fig. 16.18b when the variance changes. The inflection points are at the abscissa values * u ; the centers of gravity of the halfareas are at i.q. The maximum of the function is at 1 = 0 and it is l / ( u f i ) . The curve widens as u2 increases. the area under the curve always equals one. This distribution shows that small errors occur often. large errors only seldom.

16.4 Calculus of Errors 787

a)

-u -y

0 ty +u

-rl

+11

X

bF3

-2

0

1

2

3

x

Figure 16.18 4. Parameters to Characterize the Normally Distributed Error Beside the variance uz or the standard deviation u which is also called the mean square erroror standard error, there are other parameters to characterize the normally distributed error, such as the measure of accuracy h. the average error or mean error q1and the probable error y. 1. Measure of Accuracy Beside the variance u z ,the measure of accuracy 1 h=(16.195) U&

is used to characterize the width of the normal distribution. A narrower Gauss curve results in better accuracy (Fig. 16.18b). If we replace u by the experimental value of 6 or 6%obtained from the measured values, the measure of accuracy characteizes the accuracy of the measurement method. 2. Average or Mean Error The expected value 17 of the absolute value of the error is defined as oc

q = E(IX1) = 2 J z f ( z ) dz.

(16.196)

0

3. Probable Error The bound y of the absolute value of the error with the property P(lXl 5 7 ) =

;

(16.197a)

is called the probable error. It implies that

where @(z)is the distribution function of the standard normal distribution. 4. Given Error Bounds If an upper bound a > 0 of an error is given, then we can calculate the probability that the error is in the interval [-a, a] by the formula (16.198) 5. Relations between Standard Deviation, Average Error, Probable Error, and Accuracy If the error has a normal distribution, then the following relations hold with the constant factor e = 0.4769 :

(16.199a)

788

16. Probabilitu Theoru and Mathematical Statistics

(16.199b) and 1

@(a&) = -2 .

(16.200)

16.4.1.3 Quantitative Characterization of the Measurement Error 1. True Value and its Approximations The true value x, of a measurable quantity is usually unknown. We choose the expected value of the random variables, whose realizations are the measured values x, (i = 1 , 2 , . . . , n ) ,as an estimated value of 2., Consequently, the following means can be considered as an approximation of 2,: 1. Arithmetical Mean (16.201a)

or

k

-

x=

h,E,,

(16.201b)

3=1

if the measured values are distributed into k classes with absolute frequencies h, and class means

T, ( j= 1,2,.. . , k).

2. Weighted Mean

(16.202) Here the single measured values are weighted by the weighting factors p. 791).

gi

(gi

> 0 ) (see 16.4.1.6, l.,

2. Error of a Single Measurement in a Measurement Sequence 1. True Error of a Single Measurement in a Measurement Sequence is the difference between the true value x, and the measuring result. Because this is usually unknown, the true error E , of the 2-th measurement with the result x, is also unknown: E, = 2, - 2 , . (16.203a) 2. Mean Error of a Single Measurement in a Measurement Sequence is the difference of the arithmetical mean and the measurement result 2,: u,=z-xz. (16.203b) 3. Mean Square Error of a Single Measurement or Standard Error of a Single Measurement Since the expected value of the sum of the true errors E, and the expected value of the sum of the mean errors v, of n measurement is zero (independently of how large they are). we also consider the sum of the error squares: E2

=

Et2, i=l

(16.204a)

u2 =

v,2,

(16.20413)

i=l

From a practical point of view only the value of (16.204b) is interesting, since only the values of u, can be determined from the measuring process. Therefore, the mean square error of a single measurement of a measurement sequence is defined by (16.205) The value 6 is an approximation of the standard deviation u of the error distribution. We get for 5 = D in the case of normally distributed error: P(IEI5 5) = 2@(1) - 1 = 0.68.

(16.206)

That is: The probability that the absolute value of the true value does not exceed o.is about 68%. 4. Probable Error is the number 7 , for which 1 (16.207) P(lE1 5 7 ) = - . 2 That is: The probability that the absolute value of the error does not exceed y,is 50%. The abscissae iq divide the area of the left and the right parts under the density function into two equal parts (Fig. 16.18a). The relation beheen and 6 in the case of a normally distributed error is (16.208) 5 . Average Error is the number 11, which is the expected value of the absolute value of the error:

q = E(IE1) =

7

-m

lxlf(x)dx.

(16.209)

In the case of a normally distributed error we get 17 = 0.798. It follows from the relation (16.210) that the probability that the error does not exceed the value 11 is about 57.6 %. The centers of gravity of the left and right areas under the density function (Fig. 16.18a) are at abscissae ?q. We also get: (16.211) = 0.7978e x 0.85 = 0.84 e v t 2 / ( n- 1). ij = 1=1

3. Error of t h e Arithmetical Mean of a Measurement Sequence The error of the arithmetical mean E of a measurement sequence is given by the errors of the single measurement: 1. Mean Square Error or Standard Deviation (16.212) 2. Probable Error

: j 21 x3

cr2/n(n - 1) = --

(16.213)

3. Average Error

(16.214) 4. Accessible Level of Error Since the three types of errors defined above (16.212)-(16.214) are di-

rectly proportional to thecorresponding error ofthe single measurement (16.205), (16.208) and (16.211) and they are proportional to the reciprocal of the square root of n, it is not reasonable to increase the number of the measurements after a certain value. It is more efficient to improve the accuracy h of the measuring method (16.195). 4. Absolute a n d Relative Errors 1. Absolute Uncertainty, Absolute Error The uncertainty of the results of measurement is characterized by errors E,, u,, o,,-/$, 7t2or E , v , 0,y, 11, which measure the reliability of the method of

790

16. Probabilitu Theoru and Mathematical Statistics

measurement. The notion of the absolute uncertaznity, given as the absolute error, is meaningful for all types of errors and for the calculation of error propagation (see 16.4.2,p. 792). They have the same dimension as the measured quantity. The word ‘‘absolute” error is introduced to avoid confusion with the notion of relative error. We often use the notation Axi or Ax. The word “absolute“ has a different meaning from the notion of absolute value: It refers to the numerical value of the measured quantity (e.g., length, weight, energy), without restriction of its sign. 2. Relative Uncertainty, Relative Error The relative uncertainty, given by the relative error, is a measure of the quality of the method of measurement with respect to the numerical value of the measured quantity. In contrast to the absolute error. the relative errorhas no dimension, because it is the quotient of the absolute error and the numerical value of the measured quantity. If this value is not known. we replace it by the mean value of the quantity x:

( 16.215a) The relative error is given mostly as a percentage and it is also called the percentage error: 6xt/% = 62%.100%. (16.215b)

5 . Absolute and Relative Maximum Error 1. Absolute Maximum Error If the quantity z we want to determine is a function of the measured quantities 51, 2 2 , . . . ,z,,Le., z = f ( x l , x Z , . . . , z”),then the resulting error must be calculated taking also the function f into consideration. There are two different ways to examine errors. The first approach is that statistical error analysis is applied by smoothing the data values using the least squares method (min r ( z l - z ) ’ ) , and in the second approach, an upper bound Az,,, is determined for the absolute error of the quantities. If we have n independent variables I,,then: (16.216) where we should substitute the mean value E, for x,. 2. Relative Maximum Error We can get the relative maximum error if we divide the absolute maximum error by the numerical value of the measured value (mostly by the mean of z ) :

( 16.217)

16.4.1.4 Determining the Result of a Measurement with Bounds on the Error .A realistic interpretation of a measurement result is possible only if the expected error is also given; error estimations and bounds are components of measurement results. It must be clear from the data what is the type of the error: what is the confidence interval and what is the significance level. 1. Defining t h e Error The result of a single measurement is required to be given in the form (16.218a) x = I , Lk Ax % 2, 8, and the result of the mean has the form ( 16.218b) x = F i A X ~ , Wx F c! AM. Here Ax is the most often used distance, the standard deviation. p and ij could also be used. - 2, 2. Prescription of Arbitrary Confidence Limits The quantity T = -has a t distribution

*

x

DAM

(16.101b) with f = n - 1 degrees of freedom in the case of a population with a distribution N ( p , u 2 ) according to (16.100). For a required significance level cy or for an acceptance probability S = 1- CY we get the confidence limits for the unknown quantity x, = p with the t quantile t,jzif (16.219) p = p tap:f ’ 5A.M.

*

16.4 Calculus ofErrors 791

That is, the true value x, is in the interval given by these limits with a probability S = 1 - 0. We are mostly interested in keeping the size n of the measurement sequence at its lowest possible level. The length 2 t , p f 5 , 4 ~of the confidence interval decreases by a smaller value of 1 - (Y and also by a larger number n of measurements. Since decreases proportionally to 1 / f i and the quantile t,/z;f with f = n - 1 degrees of freedom also decreases proportionally to l / f i for values of n between 5 and 10 (see Table 21.18,p. 1090) the length of the confidence interval decreases proportionally to l / n for such values of n.

16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy If we can achieve the same variance ui for all n measurements, we talk about measurements with the same accuracy h = const. In this case, the least squares method results in the error quantities given in (16.205), (16.208))and (16.210). W Determine the final result for the measurement sequence given in the following table which contains n = 10 direct measurements with the same accuracy.

x, 1.592 1.581 1.574 1.566 1.603 1.580 1.591 1.583 1.571 1.559 0 -11 + 9 +21 v,.103 - 1 2 -3 -1 + 6 t 1 4 -23 vt2.1O6 144 1 36 196 529 0 121 9 81 441

Final result: z = f i6 a =~ 1.580 f 0.041

16.4.1.6 Error Estimation for Direct Measurements with Different Accuracy 1. Weighted Measurements If the direct measurement results 2, are obtained from different measuring methods or they represent means of single measurements, which belong to the same mean E with different variances 6,',we calculate a wezghted mean (16.220) where gz is defined as

2

gt=-q.

(16.221)

0%

Here 5 is an arbitrary positive value, mostly the smallest 6%. It serves as a weight unit of the deviations, i t . , for tL= 6 it is gZ = 1. It follows from (16.219) that a larger weight of a measurement results in a

smaller deviation cl.

2. Standard Deviations The standard deviation of the weight unit is estimated as (16.222) We have to be sure that 8 9 ) < 5.In the opposite case, if &) systematic deviations.

> 5,then there are z,values which have

I

792

16. Probabilitu Theoru and Mathematical Statistics

The standard deviation of the single measurement is (16.223) where 6t[g) < 6, can be expected. The standard deviation of the weighted mean is: (16.224)

3. Error Description The error can be described as it is represented in 16.4.1.4, p. 790, either by the definition of the error or by the t quantile with f degrees of freedom. H The final results of measurement sequences ( n = 5) with different means ?EL (i = 1,2,. . . ,5) and with different standard deviations AM, are given in Table 16.6. We calculate (?Et)m = 1.5830 and we choose 20 = 1.585 and 5 = 0.009. With z, = 2, - 20,g, = 5'/6,' we get T = -0.0036 and Z = zo 0.0088

< 5 and cfZ

+ Z = 1.582. The standard deviation is

&)'

=

\1

k g , a , 2 / ( n - 1) = 1=1

= C . A .= ~ ~0.0027. The final result is z = ?E f cfZ = 1.585 f 0.0027.

Table 16.6 Error description of a measurement sequence

AM; 1.573 1.580 1.582 1.589 1.591

0.010 0.004 0.005 0.009 0.011

-a A '-M;

1.0.10-4 1. 6 . m 5 2.5.1OW5 8.1.10-5 1.21.I 0-4

1.16.10-4 1.26.10-* 2.91.10-5 1. 6 . w 5 2.37.10-5

U ,=1

= 0.009

= 10.7

16.4.2 Error Propagation and Error Analysis Measured quantities appear in final results often after a functional transformation. If the error is small, we can use a linear Taylor expansion with respect to the error. Then we talk about error propagation.

16.4.2.1 Gauss Error Propagation Law 1. Problem Formulation Suppose we have to determine the numerical value and the error of a quantity z given by the function z = f ( qx, 2 , . . . ,z k )of the independent variables z j (3 = 1 , 2 , . , , , k ) . The mean value ?Ej obtained from n3 measured values is considered as realizations of the random variable x3,with variance uJ2.We wish to examine how the errors of the variables affect the function value f(zl,x2,.. . ,xt). It is assumed that the function f(x1,x 2 , . . . , zk)is differentiable, its variables are stochastically independent. However they may follow any type of distribution with different variances uJ'.

f 6.4 c a k u h s of Errors 793

2. Taylor Expansion Since the error represents relativelysmallchangesofthe independent variables, the function f(x1,52,. . . , q )can be approximated in the neighborhood of the mean TJ by the linear part of its Taylor expansion with the coefficients a,, so for its error Aj we have: ~f=f(.l~x21...rZk)-f(2lrTZr...,~k)r

(16.225a) (16.22513)

where the partial derivatives 8f/8x, are taken at (!&,Zz,. . . ,T k ). The variance of the function is af 2 = al2 u2,2

k + a22u222+. + ak2utk2= 1 a32u,,2. '.

(16.226)

3=1

3. Approximation of the Variance uj Since the variances of the independent variables x, are unknown, we approximate them by the variance of their mean, which is determined from the measured values x,~(1 = 1 , 2 , . , . , nl) of the single variables as follows: n

(16.227) With these values we form an approximation of uf2: k

E; =

1aJ26:,.

(16.228)

3=1

The formula (16.228) is called the Gauss error propagation law. 4. Special Cases 1. Linear Case .4n often occurring case is when we add the absolute values of the errors of sequentially occurring error quantities with a3 = 1:

i+f = JE:+ 6;+ . ' . + 6;.

(16.229)

IThe pulse length is to be measured a t the output of a pulse amplifier of a detector channel for spectrometry of radiation, whose error can be deduced for three components: 1. statistical energy distribution of the radiation of the part passing through the spectometer with an energy Eo, which is characterized by &tr, 2. statistical interference processes in the detector with 6Det, 3. electronic noice of the amplifier of the detector impulse eel. The total pulse length has the error Ef=

Jm

(16.230)

2. Power Rule The variables x3 often occur in the following form:

z = f ( x l , 2 2 , . , . xk) = azlbl. xZbz . . . x k b k . By logarithmic differentiation we get the relative error

(16.231)

(16.232)

794

16. Probabilitv Theorv and Mathematical Statistics

from which by the error propagation law we get for the mean relative error: (16.233)

ISuppose that the function f (xl.2 2 . ~ 3 )has the form f(q, ~ 2 deviations are ut,,us* and u Z 3 . The relative error is then 62

= L!!f =

\ (;g j2!3

~ x = 3 )&x22x331

and the standard

(3$)2,

5 . Difference to the Maximum Error Declaring the absolute or relative maximal error (16.216), (16.217) means that we do not use smoothing for the values of the measurement. For the determination of the relative and absolute error with the error propagation laws (16.228) or (16.231), smoothing between the measurement values xJ means that we determine for them a confidence interval for a previously given level. This procedure is given in 16.4.1.4, p. 790.

16.4.2.2 Error Analysis The general analysis of error propagation in the calculations of a function cp(xt),when quantities of higher order are neglected, is called error analysis. In the framework of the theory of error analysis we investigate using an algorithm, how an input error Ax, affects the value of ~ ( x , )In. this relation we also talk about differential error analysis. In numerical mathematics, error analysis means the investigation of the effect of errors of methods, of roundings. and of input errors to the final result (see [19.24]).

795

17 Dynamical Systems and Chaos 17.1 Ordinary Differential Equations and Mappings 17.1.1 Dynamical Systems 17.1.1.1 Basic Notions 1. The Notion of Dynamical Systems and Orbits A dynamical system is a mathematical object to describe the development of a physical, biological or another system from real life depending on time. It is defined by a phase space M ,and by a oneparameter family of mappings pt : M + 114, where t is the parameter (the time). In the following, the phase space is often R", a subset of it, or a metric space. The time parameter t is from R (time continuous system) or from 2 or from Z+ (timediscrete system). Furthermore, it is required for arbitrary z E .zI that a)).(OP = z and b) p t ( $ ( x ) ) = ~ ~ ' ~ for ( xall ) t:s. The mapping p1 is denoted briefly by p. In the following, the time set is denoted by hence, r = R, r = R,, r = Z or = Z+.If = R, then the dynamical system is also called a flour; if = 2 or r = Z,, then the dynamical system is discrete. In case r = R a n d = Z.the properties a) and b) are satisfied for every t 6 T ,so the inverse = p-t also exists, and these systems are called invertible dynamical systems. mapping If the dynamical system is not invertible, then @ ( A ) means the pre-image of A with respect to pt, for an arbitrary set A c M and arbitrary t > 0, Le., v - ~ ( A = ) {z E M :cp'(z) E A } . If the mapping pt:Jf i !If is continuous or k times continuously differentiable for every t E r (here M c R").then the dynamical system is called continuous or C'-smooth, respectively. For an arbitrary fixed z E !\f) the mapping t H pt (x),t E r, defines a motion of the dynamical system starting from x at time t = 0. The image $2) of a motion starting a t z is called the orbit (or the trajectory) through x,namely y(x) = {cpt(x)}tEr.Analogously, the positive semiorbit through x is defined by r+(z)= {pt(x)}tzOand, if # R+ or r # Z+2 then the negative semiorbit through z is defined by -,-(z) = {pt(x)}t 0, such that $+=(E) = ~'(z) for all t E r, and T E r is the smallest positive number with this property. The number T is called the period. 2. Flow of a Differential Equation Consider the ordinary linear planar differential equation x = f (x). (17.1) where f : ,21 -i R" (vectorfield) is an r-times continuously differentiable mapping and :M = R" or M is an open subset of R". In the following, the Euclidean norm 11 11 is used in R". Le.. for arbitrary

r,

r

r

r

r

r

r,

)zn), its norm is l/zl1 =

l--

z'. If the mapping f is written componentwise

i=l

(17.1) is a system of TI scalar differential equations xi = fi(zl,.. . , zn)>i = 1 , 2 : . . . .n. The Picard-Lindelof theorem on the local existence and uniqueness of solutions of differential equations locally and the theorem on the r-times diferentiability of solutions with respect to the initial values (see [17.11])guarantee that for every zo E M , there exist anumber E > 0, a sphere Bs(x0)= {x: Ilx-z011 < 6) in ,If and a mapping p: ( - E , E ) x Ba(xo) + M such that: 1. p(,),) is ( r t 1)-times continuously differentiable with respect to its first argument (time) and rtimes continuously differentiable with respect to its second argument (phase variable):

I

796

17. Dvnamical Svstems and Chaos

2. for every fixed I E B ~ ( z o'p) (., ~ 2) is the locally unique solution of (17.1) in the time interval

which starts from I at timet =

0, Le., -09 (t,z)

at

= $ ( t ! z )= f(p(t,z))holds for every t E

(-E, E ) (-&,E),

y(0. z) = I. and every other solution with initial point z at time t = 0 coincides with p(t,z) for all small It/. Suppose that every local solution of (17.1) can be extended uniquely to the whole of R. Then there exists a mapping p: R x M + M with the following properties: 1. p(0,1) = z for all I E M . 2. p(t + s, I) = 'p(t2p(s,z)) for all t , s E R a n d all z E M . 3. p(., .) is continuously differentiable ( r + 1)times with respect to its first argument and r times with respect to the second one. 4. For every fixed I E ,bf, p(.,z) is a solution of (17.1) on the whole of R. Then the C'-smooth flow generated by (17.1) can be defined by 'pt : = p(t, .). The motions p(.,z) : R i Af of a flow of (17.1) are called integral curves. The equation (17.2) X = ~ ( -yz). jr = TI - y - Z Z , i = ~y - bz is called a Lorenz system of convective turbulence (see also 17.2.4.3, p. 825). Here c > 0, r > 0 and b > 0 are parameters. The Lorenz system corresponds to a C" flow on M = R3. 3. Discrete Dynamical System Consider the difference equation Zttl = P(.t)> (17.3) which can also be written as an assignment z i--t p(z). Here 'p: M + M is a continuous or T times continuously differentiable mapping, where in the second case M C R". If p is invertible, then (17.3) defines an invertible discrete dynamical system through the iteration of p, namely,

-

pt = p o ~ ~ . o for p, t

t times,

> 0,

pt = p-'o...op-'

'-

-t times,

for t

< 0, po = id.

(17.4)

If p is not invertible, then the mappings pt are defined only for t 2 0. For the realization of 'pt see (5.86))p. 296. IA: The difference equation (17.5) It+l = azt (1 - I t ) , t = 0,1,. . . with parameter a E (0,4] is called a logistic equation. Here M = [0,1], and p: [0,1] + [0,1]is, for a fixed a , the function p(s) = a z ( 1 - z). Obviously, p is infinitely many times differentiable, but not invertible. Hence (17.5) defines a non-invertible dynamical system. B: The difference equation (17.6) ~ t + = l yt + 1 - ax:, yt+l = bzt, t = 0, k l , . . . , with parameters a > 0 and b # 0 is called a He'non mapping. The mapping p: RZ+ R* corresponding to (17.6) is defined by p(z,y ) = (y + 1 - az2,bz),is infinitely often differentiable and invertible.

4. Volume Contracting and Volume Preserving Systems The invertible dynamical system {'pt}tEr on M C R" is called dissipative (respectively volume-preserving

or conservative), if the relation vol('pt(A)) < vol(A) (respective vol(pt(A)) = vol(A)) holds for every set A c M with a positive n-dimensional volume vol(A) and every t > 0 (t E r). A: Let p in (17.3) be a C'-di$eomorphism (Le.: p : M + M is invertible, M C R" open, p and p-l are CT-smoothmappings) and let Dp(z) be the Jacobi matrix of p in z E M. The discrete system I I 1 in M . (17.3) is dissipative if I det Dp(z)/< 1 for all z E M , and conservative if I det

17.1 OTdinaT~Di,fferentialEquations and Mappings 797

B: For the system (17.6) Dp(z,y) =

( -? i) and so I det Dp(z, y)i

I

b. Hence, (17.6) is

dissipatike if (bl < 1.and conservative if Ibl = 1. The Henon mapping can be decomposed into three mappings (Fig. 17.1): First. the initial domain is stretched and bent by the mapping zf = z, y’ = y 1 - ax2 in a area-preserving way, then it is contracted in the direction of the d-axis by z” = bz’,y” = y’ (at Ibl < l ) ,and finally it is reflected with respect to the line y” = x’’by 2’’’= y”, y”‘ = d’.

+

Figure 17.1

17.1.1.2 Invariant Sets 1. CY- and w-Limit Set, Absorbing Sets Let { ~ ‘ } ~be€ardynamical system on M.The set A C M is invariant under {$}, if $((A) = A holds for all t E r. and positively invariant under {$}, if cpt(A) c A holds for all t 2 0 from r. For every z E M ,the w-limitset of the orbit passing through z is the set d(2) = {y E M :3 t, E r, t, + +x, ~ “ ( 2 )+ y as n + +x}. (17.7) The elements of w ( x ) are called w-limit points of the orbit. If the dynamical system is invertible, then for every I E M, the set CY(.) = {y E .kf: 3 t, E r, t, + -w, p“(z) + y as n -+ +E} (17.8) is called the a-limitset ojthe orbit passing through z; the elements of a(.) are called the d i m i t points of the orbit. For many systems which are volume decreasing under the flow there exists a bounded set in phase space such that every orbit reaching it stays there as time increases. .4bounded, open and connected set c’ c M is called absorbing with respect to if cp‘(c)c U holds for all positive t from r. is the closure of c’.) Consider the system of differential equations (17.9a) y = z t y ( 1 - z 2 - y2) .i = - y + 2 ( 1 - x 2 -y2), in the plane. Using the polar coordinates z = r cosl9, y = rsinl9. the solution of (17.9a) with initial state ( T O , 30)at timet = 0 has the form

(u

+

+

r ( t ,r O )= [I (TO’ - 1)e-2t]-1/2, 8(t,230) = t 190. This representation of the solution shows that the flow of (17.9a) has a periodic orbit with period 2 ~ , which can be given in the form y ( ( l . 0 ) ) = {(cost, sin t ) ,t E [0,27r]}.The limit sets of an orbit through p are:

+ y2 < r 2 } with r > 1 is an absorbing set for (17.9a). 2. Stability of Invariant Sets Let 4 be an invariant set of the dynamical system {pt}tEj-defined on (M. p). The set A is called stable. if every neighborhood c’ of A contains another neighborhood Ui c c‘ of A such that pt((crl)c U holds Every open sphere B, = {(z,y): x 2

798

17. Dynamical Systems and Chaos

for all t > 0. The set A, which is invariant under {iot}, is called asymptotically stable if it is stable and the following relations are satisfied: vx E .I1 (17.10) 3 4 > 0 dist(x, A ) < A } : dist(cpt(z), A) -+0 for t + f m . Here. dist(z. A ) = inf p ( x , y). YEA

3. Compact Sets Let (M. p ) be a metric space. A system { U Z } of ~ ~open I sets is called an open covering of dl if every point of .If belongs to at least one Ut. The metric space (M, p) is called compact if it is possible to choose finitely many Ut1 . . . , iYtTfrom every open covering {Uj}i,l of M such that M = C;,U . . U C;, holds. The set K c M is called compact if it is compact as a subspace.

4. Attractor, Domain of Attraction Let {pt}tGrbe a dynamical system on (M,p)and A an invariant set for {'it}. Then W(A) = {x E .\I: ~ ( 2c) A} is called the domain o j attraction of A. A compact set .I c M is called an attractor of {pt}ttron .tl if .I is invariant under {$} and there is an open neighborhood C; of 1 1 such that ~ ( z=) A for almost every (in the sense of Lebesgue measure) x E U .

I .2 = ~r((1.0))isanattractoroftheflowof(17.9a). HerelY(i2) = RZ\{(O~O)}.Forsomedynamical systems. a more general notion of an attractor makes sense. So, there are invariant sets '1 which have periodic orbits in every neighborhood of A which are not attracted by 11,e.g., the Feigenbaum attractor. The set .I may not be generated by a single limit set U . .4compact set A is called an attractor in the sense ojMilnorof the dynamical system {pt}tGron M if '2 is invariant under {pi} and the domain of attraction of .Icontains a set with positive Lebesgue measure.

17.1.2 Qualitative Theory of Ordinary Differential Equations 17.1.2.1 Existence of Flows, Phase Space Structure 1. Extensibility of Solutions Besides the differential equation (17.1). which is called autonomous, there are differential equations whose right-hand side depends explicitly on the time and they are called non-autonomous: j: = f ( t : x). (17.11) Let f : R x h ' + M be a C'-mapping with M C R". By the new variable xntl := t, (17.11) can be interpreted as the autonomous differential equation 3 = j ( z n t l l x ) 3 k n t l = 1. The solution of (17.11) starting from xo at time to is denoted by p(,>to,xo). In order to show the global existence of the solutions and with this the existence of the flow of (17.1))the following theorems are useful. 1. Criterion of Wintner and Conti If M = R" in (17.1) and there exists a continuous function

+ [I. +m).such that Ilf(x)ll 5

d:[0, +x)

(u'

(11x11)for all x E Rnand

+x iirnUtp) ~

dr =

holds, then

every solution of (17.1) can be extended onto the whole of Rt. For example, the following functions satisfy the criterion of Wintner and Conti: W ( T ) = Cr 1 or W ( T ) = Crl In 7.1 + 1. where C > 0 is a constant. 2. Extension Principle If a solution of (17.1) stays bounded as time increases, then it can be extended to the whole of R+. Assumption: In the following, the existence of the flow { p t } t Eof~(17.1) is always assumed.

+

2. Phase Portrait a) If;s(t) isasolutionof (17.1))then thefunctionp(t+c) withanarbitraryconstant cisalsoasolution.

b) Two arbitrary orbits of (17.1) have no common point or they coincide. Hence, the phase space of (17.1) is decomposed into disjoint orbits. The decomposition of the phase space into disjoint orbits is called a phase portrait.

27.1 Ordinarv Dzflerential Equations and Mappings 799

c) Every orbit, different from a steady state, is a regular smooth curve, which can be closed or not closed. 3. Liouville's Theorem Let {qt}tER be the flow of (17.1), D c M c Rnbe an arbitrary bounded and measurable set, Dt : = p t ( D ) and \; : = vol(Dt) be the n-dimensional volume of Dt (Fig. 17.2a). Then the relation d - V, = divf(z) dx holds for arbitrary t E R . For n = 3. Liouville's theorem states:

/

dt

Dt

(17.12)

Figure 17.2 Corollary: If divf(z) < 0 in M holds for (17.1)3then the flow of (17.1) is volume contracting. If divf(x) I 0 in holds, then the flow of (17.1) is volume preserving. IA: FortheLorenzsystem (17.2),divf(x.y.t) I - ( u + l + b ) . Sincea > Oand b > O,divf(z.y,z) <

0 holds. il'ith Liouville's theorem, - r/, =

dt

///

-(u

+ 1 + b) dxldxz dxg = - ( u + 1+ b) V, obviously

Dt

holds for any arbitrary bounded and measurable set D C R3. The solution of the linear differential so that V, + 0 follows for t + +m. equation 1; = -(a 1 t b) is \4 = Va e-(aS1+b)t,

+

dH IB: Let G c R" x R" be an open subset and H: b' -+ R a @-function. Then, x, = -(x, y), y, = dyi H -d_ ( 2 ,y) ( i = 1 , 2 ,. . . n)is called a Hamiltonian differential equation. The function H is called the ax, Hamiltonian of the system. I f f denotes the right-hand side ofthis differential equation, then obviously ~

divf(z,Y) =

1 m ( x " [ ":", i=l

; Y) - -(x,y)] dY,dX,

I 0.

Hence, the Hamiltonian differential equations are

volume preserving.

17.1.2.2 Linear Differential Equations 1. General Statements

Let A ( t ) = [aij(t)]Fj=l be a matrix function on R , where every component aij: R -+ R is a continuous function. and let b: R -+ R" be a continuous vector function on R . Then (17.13a) x = A(t)x b ( t ) is called an inhomogeneous linear first-order differential equation in R", and x = il(t)z (17.13b) is the corresponding homogeneous linear first-order differentialequation. 1. Fundamental T h e o r e m for Homogeneous Linear Differential Equations Every solution of (17.13a) exists on the whole of R. The set of all solutions of (17.13b) forms an n-dimensional vector subspace L H of the C'-smooth vector functions over R .

+

I

800 17. Dynamical Systems and Chaos

2. Fundamental Theorem for Inhomogeneous Linear Differential Equations The set of all solutions LI of (17.13a) is an n-dimensional affine vector subspace of the C'-smooth vector functions over R of the form L1 = po + L x , where cpo is an arbitrary solution of (17.13a). Let cpl!. . . , pn be arbitrary solutions of (17.13b) and CJ = [cpl, . . . ,cpn] the corresponding solution matriz. Then CJ satisfies the matriz diferential equation Z ( t ) = A ( t ) Z ( t ) on , R, where 2 E R"'". If the solutions P I , . . . pn form a basis of L H ,then @ = [cpl,. . . , cp,] is called the fundamental matrir of (17.13b). W(t) = detCJ(t) is the Wronskian determinant with respect to the solution matrix CJ of (17.13b). The formula of Liouville states that: ~

tP(t) = SpA(t) W ( t ) (t E R). (17.13~) For a solution matrix, either W(t) E 0 on R or W ( t )# 0 for all t E R . The system pl,. . . , 'pn is a basis of L H , if and only if det['pl(t),. . . , pn(t)]# 0 for a t (and so for all t ) . 3. Theorem (Variation of Constants Formula) Let CJ be an arbitrary fundamental matrix of (17.13b). Then the solution 'p of (17.13a) with initial point p at timet = 7 can be represented in the form

p(t) = C J ( t ) @ ( T ) - ' p +

/

@(t)CJ(s)-'b(s) ds ( t E R).

(17.13d)

T

2. Autonomous Linear Differential Equations Consider the differential equation

x =AX, (17.14) where A is a constant matrix of type (n,n). The operator norm (see also 12.5.1.1, p. 617) of a matrix A is given by lIAl1 = max{lIAz/l, z E R", I / X I ~ 5 l}, where for the vectors of R" the Euclidean norm is again considered. Let A and B be two arbitrary matrices of type (n, n). Then b) IlAAll = 1x1 IlAll (A E R), a) IlA + BII I IIAlI + IIBIL d) IlABIl I I l A l l l l ~ i l ~ e) IlAIl = G,where, ,A is the greatest eigenvalue of ATA . The fundamental matrix with initial value E, at timet = 0 of (17.14) is the matriz ezponentialfunction c) IlAzIl

5 IlAllII~llz E R"),

(17.15) with the following properties: a) The series of eAt is uniformily convergent with respect to t on an arbitrary compact time interval and absolutely convergent for every fixed t ; b) /leAtll 5 elAslt (t 2 0); d

c) z ( e A t ) = (eAt)' = AeAt = eAtA

(t E R);

d) e(t+s)A= etA esA ( s , t E R); e) eAt is regular for all t and (eAt)-' = e-At; f ) if A and B are commutative matrices of type (n, n ) ,Le., AB = B A holds, then B eA = eAB and €A-B - A E . -e e , g) if A and B are matrices of type (n, n ) and B is regular, then eBAB-' = B eA B-'.

17.1 Ordinary Diflerential Equations and Mappings 801

3. Linear Differential Equations with Periodic Coefficients We consider the homogeneous linear differential equation (17.13b), where A ( t ) = [aij(t)][j=lis a Tperiodic matrix function, Le., a z 3 ( t )= aij (t + T ) (Vt E R , i , j = 1 , .. . , n ) . In this case we call (17.13b) a linear T-periodic diflerential equation. Then every fundamental matrix @ of (17.13b) can be written in the form @ ( t )= G ( t ) e t R ,where G(t) is a smooth, regular T-periodic matrix function and R is a constant matrix of type (n,n) (Floquet's theorem). Let @(t)be the fundamental matrix of the T-periodic differential equation (17.13b), normed at t = 0, Le., @(O) = E,, and let @(t)= G ( t ) e t Rbe a representation of it according to Floquet's theorem. The matrix @ ( T )= em is called the monodromy matrix of (17.13b); the eigenvalues p3 of @ ( T )are the multipliers of (17.13b). A number p E C is a multiplier of (17.13b) if and only if there exists a solution p 0 of (17.13b) such that p(t + T ) = p p ( t ) (t E R) holds.

+

17.1.2.3 Stability Theory 1. Ljapunov Stability and Orbital Stability Consider the non-autonomous differential equation (17.11). The solution cp(t,to,xo) of (17.11) is said to be stable i n the sense of Lyapunov if

V t 2 tl.

The solution q(t,to, 20)is called asymptotically stable in the sense of Lyapunov, if it is stable and:

for t

+ +m.

For the autonomous differential equation (17.1) there are other important notions of stability besides the Lyapunov stability. The solution p(t,zo)of (17.1) is called orbztally stable (asymptotzcally orbztally stable). if the orbit ~ ( 5 0=) {cp(t,Q), t E R} is stable (asymptotically stable) as an invariant set. A solution of (17.1) which represents an equilibrium point is Lyapunov stable exactly if it is orbitally stable. The two types of stability can be different for periodic solutions of (17.1). ~

W Let a flow be given in R3,whose invariant set is the torus T 2 . Locally, let the flow be described in a rectangular coordinate system by 61 = 0, 62 = fz(@), where f2: R + R is a 2a periodic smooth function, for which: VO1 E R 3 Uel (neighborhood of 01) An arbitrary solution satisfying the initial conditions (01(0), 0 2 ( 0 ) )can be given on the torus by & ( t ) E 01(0), @z(t)= Qz(0) f*(Qi(O)t (t E R). From this representation it can be seen that every solution is orbitally stable but not Lyapunov stable (Fig. 17.2b).

+

2. Asymptotical Stability Theorem of Lyapunov A scalar-valued function V is called positive definite in a neighborhood U of a point p E M c R", if 1. V:II c M + R is continuous. 2. V(5) > 0 for all z E U \ { p } and V(p) = 0 . Let c' c M be an open subset and V : U -+ R a continuous function. The function V is called a Lyapunovfunctzon of (17.1) in U , if V ( v ( t )does ) not increase while for the solution p(t) E U holds. Let 1': C + R be a Lyapunov function of (17.1) and let V be positive definite in a neighborhood U of p . Then p is stable. If the condition V(cp(t,zo)) = constant (t 2 0) always yields cp(t,2 0 ) p for

I

802

17. Dynamical Systems and Chaos

a solution p of (17.1) with rp(t,z) E U (t 2 0), i.e., if the Lyapunov function is constant along a complete trajectory. then this trajectory can only be an equilibrium point, and the equilibrium point p is also asymptotically stable. IThe point (0,O)is a steady point of the planar diflerential equation x = y, y = -2 - z2y. The function V ( x ,y) = x2 y2 is positive definite in every neighborhood of (0,O) and for its derivative d - V ( x ( t ) , y(t)) = - 2 ~ ( t ) ' y ( t ) <~ 0 holds along an arbitrary solution for z ( t ) y ( t ) # 0. Hence, (0,O) is dt asymptotically stable.

+

3. Classification and Stability of Steady States Let xg be an equilibrium point of (17.1). In the neighborhood of 2 0 the local behavior of the orbits of (17.1) can be described under certain assumptions by the variational equation y = Df(xo)y, where D f(x0) is the Jacobian matrix o f f in 2 0 . If D f(z0)does not have an eigenvalue A, with Re A, = 0, then the equilibrium point zo is called hyperbolic. The hyperbolic equilibrium point zo is of type ( m ,k ) if D f(z0) has exactly m eigenvalues with negative real parts and k = n - m eigenvalues with positive real parts. The hyperbolic equilibrium point of type ( m ,k ) is called a sink, if m = n, a source, if k = n, and a saddle point, if m # 0 and k # 0 (Fig. 17.3). A sink is asymptotically stable; sources and saddles are unstable (theorem on stability in the first approximation). Within the three topological basic types of hyperbolic equilibrium points (sink, source, saddle point) further algebraic distinctions can be made. A sink (source) is called a stable node (unstable node) if every eigenvalue of the Jacobian matrix is real, and a stable focus (unstable focus) if there are eigenvalues with non-vanishing imaginary parts. For n = 3; we get a classification of saddle points as saddle nodes and saddle foci.

Type of equilibrium ooint

Sink

Source

Saddle point

Figure 17.3

4. Stability of Periodic Orbits Let p(t%xo) be a T-periodic solution of (17.1) and y(z0) = {p(t,zo),t E [O:T]} its orbit. Cnder certain assumptions. the phase portrait in a neighborhood of y(z0) can be described by the variational equation y = D f ( p ( t ,xo))y. Since A(t) = D f(p(t,20))is a T-periodic continuous matrix function of type (n, n ) , it follows from the Floquet theorem (see p. 801) that the fundamental matrix O,,(t) of the variational equation can be written in the form OZ0(t)= G(t)eRt,where G is a T-periodic regular smooth matrix function with G(0) = En,and R represents a constant matrix of type (n,n) which is not uniquely given. The matrix Ozn(7') = em is called the monodromy matriz of the periodic orbit y ( z ~ ) , and the eigenvalues p l , . . . , pn of em are called multipliers ofthe periodic orbit y(z0). If the orbit */(xo) is represented by another solution p(t,XI),i.e., if y(z0) = $ X I ) , then the multipliers of y(z0) and y(q) coincide. One of the multipliers of a periodic orbit is always equal to one (Andronov- Witt theorem). Let plI. . . , pn-1, pn = 1 be the multipliers of the periodic orbit y(z0) and let O,,(T) be the monodromy matrix of y(x0). Then

17.1 Ordinary Differential Equations and Mappings 803

-

e.r: divf(+(t, 2 0 ) ) d t ,

(17.17) T

.

Hence. if n = 2. then pz = 1 and p1 = e[, dlvf(dts )O' dt. ILet +(t5(1.0)) = (cost,sint) bea2a-periodicsolutionof (17.9a). ThematrixA(t) ofthevariational equation with respect to this solution is

The fundamental matrix @ ( 1 , 0 ) ( t ) normed at t = 0 is given by '(1,0)(t)

ecZtcost-sint cost -sint cost) = ( s i n t cost)

= (e-ztsint

e-2t

(0

o 1) 3

where the last product is a Floquet representation of 0(1,0)(t).Thus, pi = e-4a and pz = 1. The multipliers can be determined without the Floquet representation. For system (17.9a) divf(z: y) = 2 -42' -4y' holds, and hence divf(cos t , sin t )

-2. According to the formula above, pi = e-fo2r-2dt =

e-41;

5 . Classification of Periodic Orbits I f the periodic orbit y of (17.1) has no further multiplier on the complex unit circle besides pn = 1,then

7 is called hyperbolic. The hyperbolic periodic orbit is of tupe (m,k ) if there are m multipliers inside and k = n - 1 multipliers outside the unit circle. If m > 0 and k > 0, then the periodic orbit of type ( m ,k ) is called a saddle point. According to the Andronov-Witt theorem a hyperbolic periodic orbit y of (17.1) of type ( n - 1,O) is asymptotically stable. Hyperbolic periodic orbits of type (m,k ) with k > 0 are unstable. IA: .A periodic orbit 7 = { p ( t ) ,t E [0,TI} in the plane with multipliers pi and p~ = 1 is asymptoti-

cally stable if lpll < 1. i.e., if

dT

divf(cp(t)) d t < 0.

IB: If there is a further multiplier besides pn = 1 on the complex unit circle, then the AndronovWitt theorem cannot be applied. The information about the multipliers is not sufficient for the stability analysis of the periodic orbit. IC: .As an example, let the planar system x = -y + zf(z' + y2)> y = z+ y f(z' + y') be given by the smooth function f : (0, +XI) + R, which additionally satisfies the properties f(1) = f'(1) = 0 and ~ ( T ) ( T- 1) < 0 for all r # 1,r > 0. Obviously, p ( t ) = (cost, sin t ) is a 2a-periodic solution of the system and cost - sin t @(l,O)(t)= sint cost) is the Floquet representation of the fundamental matrix. It follows

(

(:y )

that p1 = pz = 1. The use of polar coordinates results in the system 1: = T f ( r 2 ) ,8 = 1. This representation yields that the periodic orbit y((1,O))is asymptotically stable.

6. Properties of Limit Sets, Limit Cycles The a-and w-limit sets defined in 17.1.1.2, p. 797, have with respect to the flow of the differential equation (17.1) with ,%f c R" the following properties. Let z E Ad be an arbitrary point. Then: a) The sets a(.) and d(z) are closed. b) If?+(.) (respectively y-(z)) is bounded, then w ( z ) # 0 (respectively a(.) # 0)holds. Furthermore, w ( z ) (respectively a(.)) are in this case invariant under the flow (17.1) and connected. IIf for instance, T+(Z) is not bounded, then ~ ( zis) not necessarily connected (Fig. 17.4a). For a planar autonomous differential equation (17.1),(Le., M c Rz)the PoincarC-Bendixson theorem is valid. PoincarB-Bendixson Theorem: Let cp(.,p) be a non-periodic solution of (17.1). for which yt(p) is bounded. If d ( p ) contains no equilibrium point of (17.1), then w ( p ) is a periodic orbit of (17.1).

17. Dynamical Systems and Chaos

804

Figure 17.4 Hence, for autonomous differential equations in the plane, attractors more complicated than an equilibrium point or a periodic orbit are not possible. A periodic orbit y of (17.1) is called a limit cycle, if there exists an z $ y such that either y c W(X)or c a(.) holds. A limit cycle is called a stable limit cycle if there exists a neighborhood U of y such that "/ = w ( z ) holds for all x E U , and an unstable limit cycle if there exists a neighborhood U of y such that Y = a ( z )holds for all x E U . IA: For the flow of (17.9a), the property y = w ( p ) for all p # (0,O) is valid for the periodic orbit */ = {(cost,sin t ) , t E [0,h ) }Hence, . C' = RZ\{(O; 0)) is a neighborhood of y such that with it, y is a stable limit cycle (Fig. 17.4b). IB: In contrast, for the linear differential equation x = -y, si = 2 , the orbit y = {(cost, sin t ) , t E [0, 27rl) is a periodic orbit, but not a limit cycle (Fig. 1 7 . 4 ~ ) .

7. m-Dimensional Embedded Tori as Invariant Sets -4 differential equation (17.1) can have an m-dimensional torus as an invariant set. An m-dimensional torus Tm embedded into the phase space M C R" is defined by a differentiable mapping g : R" + R", whichissupposed tobe2K-periodicineverycoordinateOiasafunction(Q1,. . ..Om) ++g ( O 1 , . . .,Om). IIn simple cases, the motion of the system (17.1) on the torus can be described in a rightangular coordinate system by the differential equations Qi = wi (i = 1 , 2 , .. . , rn). The solution of this system , Q,(O) at timet = 0 is @ ( t )= wit O,(O) (i = 1 , 2 , . . . , m ; t E R). Kith initial values (&(O) A continuous function f : R -+ R" is called quasiperiodic i f f has a representation of the form f ( t )= g ( w l t , w z t . .. . .unt))where g is also a differentiable function as above, which is 27r-periodic in every component, and the frequencies wi are incommensurable. Le., there are no such integers ni with

+

rn

n,' > 0 for which nlwl

+

'.

t nmw,

= 0 holds.

2=1

17.1.2.4 Invariant Manifolds 1. Definition, Separatrix Surfaces

-,

be a hyperbolic equilibrium point or a hyperbolic periodic orbit of (17.1). The stable manifold W ( ? )(respectively unstable manifold W'(y)) of y is the set of all points of the phase space such that pass through these points: the orbits tending t o y as t + +XI (resepctively t + -a) (17.18) W(-,)= {x E M :~ ( x=)y} and W"(y) = {x E M :a(.) = y}. Stable and unstable manifolds are also called separatriz surfaces. Let

I In the plane. the differential equation x=-x, y=y+x2 (17.19a) is considered. The solution of (17.19a) with initial state (XO,yo) at time t = 0 is explicitly given by XZ

p(t, zo,yo) = (e-'zo, etyo + 0(et - e-")). 3 For the stable and unstable manifolds of the equilibrium point (0,O) of (17.19a) we get:

(17.19b)

17.1 Ordinary Diferential Equations and Mappings 805

Figure 17.5 Let Af and .V be two smooth surfaces in R", and let L,M and L,,V be the corresponding tangent planes to I f and S through I. The surfaces M and N are transversal to each other if for all x E h 'n .V the following relation holds: dim L,M + dim L,N - n = dim (L,M n L,N). IFor the section represented in Fig. 17.5b we have dim L,M = 2: dimL,N = 1 and dim(L,M n L,.V) = 0. Hence, the section represented in Fig. 17.5b is transversal.

2. Theorem of Hadamard and Perron Important properties of separatrix surfaces are given by the Theorem of Hadamard and Perron: Let 7 be a hyperbolic equilibrium point or a hyperbolic periodic orbit of (17.1). a) The manifolds ! P ( y ) and W"(y) are generalized CT-surfaces,which locally look like CT-smoothelementary surfaces. Every orbit of (17.1),which does not tend t o y fort -+ +cc or t + -cc. respectively. leaves a sufficiently small neighborhood of 7 for t -+ +cx: or t -+ -m, respectively. b) If /; = 2 0 is an equilibrium point of type ( m ,k ) , then W s ( 2 0 )and W'(x0) are surfaces of dimension m and k. respectively. The surfaces W s ( e oand ) W"(z0)are tangent at 10 to the stable vector subspace

E' = {y E R": eDf(zo)ty--t 0 for t and the unstable vector subspace

+ +m}

of equation y = Df(zo)y

(17.20a)

E" = {y E R": eDf(to)ty-+ 0 for t -+ -m} of equation y = Df(zo)y> respectively. (17.20b) is a hyperbolic periodic orbit of type ( m ,k ) , then W ( 7 )and W"(y) are surfaces of dimension m + 1 and k + 1, respectively, and they intersect each other transversally along y (Fig. 17.6a). IA: To determine a local stable manifold through the steady state (0,O) of the differential equation 0)) has the following form: (17.19a) we suppose that W,S,,((O, Wly,sO,((O.O)) = {(xly):y = h(x),1x1 < A,h : (-A.A)+ R differentiable}. Let ( z ( t )y, ( t ) ) be a solution of (17.19a) lying in W,S,,((O,0)). Based on the invariance, for times s near c ) If

. differentiation and representation of k and j , from the system (17.19a) t o t we get y(s) = h ( z ( s ) ) By we get the initial value problem h'(x) (-x) = h ( z )+ x 2 , h(0) = 0 for the unknown function h ( s ) .If we are looking for the solution in the form of a series expansion h ( z ) =

sx2+ %x3 +. 2 3!

'

, where

h'(0) = 0

2 is taken under consideration, then we get by comparing the coefficients a2 = -- and ak = 0 for k 2 3. 3 W B: For the system (17.21) x = -y z ( l - 22 - y2), y = 2 y(1 - 22 - y2). i = az with a parameter cy > 0. the orbit Y = {(cost. sin t , 0). t E [0,27~]}is a periodic orbit with multipliers p 1 = e-4n. pz = eoz'i and p3 = 1. In cylindrical coordinates 5 = r cos 29, y = r sin 29, z = z , with initial values ( T O , 80.~0) a t time t = 0.

+

+

806

17. Dvnamical Svstems and Chaos

the solution of (17.21) has the representation ( r ( t ,r o ) ,d ( t , t90), solution of (17.9a) in polar coordinates. Consequently,

where r(t. T O ) and d ( t , do) is the

W ( ~ ) = { ( z . y . z ) : z = O )\ { ( O , O , O ) } and W U ( y ) = { ( z , y , z ) : z Z + y 2 = 1 } (cylinder). Both separatrix surfaces are shown in Fig. 17.6b.

Figure 17.6

3. Local Phase Portraits Near Steady States for n = 3 We consider now the differential equation (17.1) with the hyperbolic equilibrium point 0 for n = 3. Set A = D f(0) and let det[XE - A] = X3 + pXz + qX + T be the characteristic polynomial of A. LVith the notation 6 = p q - r and A = -p2q2 + 4p3r + 4q3 - 18pqr + 27r2 (discriminant of the characteristic

polynomial), the different equilibrium point types are characterized in Table 17.1.

4. Homoclinic and Heteroclinic Orbits Suppose 7 , and 7 2 are two hyperbolic equilibrium points or periodic orbits of (17.1). If the separatrix surfaces and intersect each other, then the intersection consists of complete orbits. For two equilibrium points or periodic orbits, the orbit y c Ws(yl) n WU(y2)is called heteroclznzc if # (Fig. 17.7a).and homoclznzcify = nyz. Homoclinic orbits ofequilibrium points are also called separatrzz loops (Fig. 17.7b). !&'"(my2)

Figure 17.7 W Consider the Lorenz system (17.2) with fixed parameters u = 10, b = 8/3 and with variable r. The equilibrium point (O,O,0) of (17.2) is a saddle for 1 < r < 13.926.. . , which is characterized by a twodimensional stable manifold W s and a one-dimensional unstable manifold W". If r = 13.926. . . , then there are two separatrix loops at ( O , O , 0); Le., as t -+ +xbranches of the unstable manifold return (over the stable manifold) to the origin (see [17.9]).

17.1.2.5 Poincard Mapping 1. Poincar6 Mapping for Autonomous Differential Equations Let -, = { y ( t . z o ) .t E [O,T]}be a T-periodic orbit of (17.1) and a (TI - 1)-dimensional smooth

hypersurface, which intersects the orbit y transversally in zo (Fig. 17.8a). Then, there is a neighborhood C of zo and a smooth function 7 : G + R such that T ( Q ) = T and 9(7(z)..) E for all z € U . The mapping P : c' n -+ with P ( z ) = p ( ~ ( zz) ) : is called the Poincare' mapping of y at 10.If the right-hand side f of (17.1) is r times continuously differentiable, then P is also r times continuously differentiable . The eigenvalues of the Jacobi matrix DP(z0) are the multipliers PI,.. . , pn-l of the periodic orbit. They do not depend on the choice of zo on /; and on the choice of the transversal surface.

17.1 Ordinary Differential Equations and Mappings 807

Parameter domain b > 0:T > 0. q>o

A

A< A>

Type of equilibrium point stable node stable focus

Roots ofthe charac- Dimension of teristic polynomial W *and Wu ImXj = 0 dim W s = 3. dim W” = 0 Xj
ReX1,2< 0 <0

A3

808

17. Dynamical Systems and Chaos

X system (17.3) in hl = U can be connected with the Poincar6 mapping, which makes sense until the iterates stay in G. The periodic orbits of (17.1) correspond to the equilibrium points of this discrete system. and the stability of these equilibrium points corresponds to the stability of the periodic orbits of (17.1).

Figure 17.8

IM'e consider for the system (17.9a) the transversal hyperplanes ~={(T;t9):r>o,t9=190}

in polar coordinate form. For these planes U = can be chosen. Obviously, r ( r ) = 27r (Vr > 0) and so ~ ( r=)[I (r-' - 1) where the solution representation of (17.9a) is used. It is also valid that P ( c ) = E, P(1) = 1 and P'(1) = e-4r < 1.

+

2. Poincar6 Mapping for Non-Autonomous Time-Periodic Differential Equations A non-autonomous differential equation (17.11), whose right-hand side f has period T with respect to t. Le.. for which f(t + T , z ) = f ( t , z) ( V t E R , Vz E M) holds, is interpreted as an autonomous differential equation x = f ( s ,z), s = 1 with cylindrical phase space M x {s mod 7'). Let SO E {s mod T } be arbitrary. Then, C = M x {so} is a transversal plane (Fig. 17.8b). The PoincarC mapping is given globally as P : --t over zo H q(s0 + T ,$0, zO)rwhere ip(t, so, 20)is the solution of (17.11) with the initial state zo at time SO.

17.1.2.6 Topological Equivalence of Differential Equations 1. Definition Suppose. besides (17.1) with the corresponding flow { ( o t } t e R , that a further autonomous differential equation x = Y(XL (17.22) is given, where y : N + R" is a C'-mapping on the open set N c R". Of course, the flow { y ' } t e of ~ (17.22) should exist. The differential equations (17.1) and (17.22) (or their flows) are called topologically equivalent if there exists a homeomorphism h: M + N (Le., h is bijective, h and h-' are continuous), which transforms each orbit of (17.1) to an orbit of (17.22) preserving the orientation, but not necessarily preserving the parametrization. The systems (17.1) and (17.22) are topologically equivalent if there also exists a continuous mapping r : R x M + R , besides the homeomorphism h : M + N , such that 7 is strictly monotonically increasing at every fixed z E M, maps R onto R, with ~ ( 0z), = 0 for all z E M and asatisfies the relation h ( p t ( z ) )= $ T ( t + ) ( h ( zfor ) ) all z E M and t E R . In the case of topological equivalence, the equilibrium points of (17.1) go over into steady states of (17.22) and periodic orbits of (17.1) go over into periodic orbits of (17.22),where the periods are not necessarily coincident. Hence! if two systems (17.1) and (17.22) are topologically equivalent, then the topological structure of the decomposition of the phase spaces into orbits is the same. If two systems (17.1) and (17.22) are topologicallyequivalent with the homeomorphism h: M --t N and if h preserves the parametrization, Le., h(cpt(z))= $ ( h ( z ) )holds for every t , z , then (17.1) and (17.22) are called

17.1 Ordinarv Differential Equations and Mappinqs 809

topologically conjugate. Topological equivalence or conjugacy can also refer to subsets of the phase spaces A4 and N . Suppose, e.g.. (17.1) is defined on C; c M and (17.22) on Uz c N . We say that (17.1) on U1 is topologically equivalent to (17.22) on U2 if there exists a homeomorphism h : U1 + U, which transforms the intersection of the orbits of (17.1) with Ul into the intersection of the orbits of (17.22) with U2 preserving the orientation. H A: Homeomorphisms for (17.1) and (17.22) are mappings where, e.g., stretching and shrinking of the orbits are allowed; cutting and closing are not. The flows corresponding to phase portraits of Fig. 17.9a and Fig. 17.9b are topologically equivalent; the flows shown in Fig. 17.9a and Fig. 1 7 . 9 ~are not.

Figure 17.9

B: Consider the two lznear planar diflerential equations (see [17.11])

S = Az and x

=

Bz with A =

(1;1;) and B = (: -:).

The phase portraits of these systems

close to (0,O) are shown in Fig. 17.10a and Fig. 17.10b. The homeomorphism h : R2 T :R x

+ R2 with h ( z ) = Rx.where R

=

-

JT

('1 -'),1

and the function

1

R2 + R with T ( t ,x) = - t transform the orbits of the first system into the orbits of the second

2 one. Hence, the two systems are topologically equivalent.

Figure 17.10

2. Theorem of Grobman and Hartman Let p be a hyperbolic equilibrium point of (17.1). Then, in a neighborhood ofp the differential equation (17.1) is topologically equivalent to its linearization y = Df(p)y.

17.1.3 Discrete Dynamical Systems 17.1.3.1 Steady States, Periodic Orbits and Limit Sets 1. Types of Steady State Points Let zo be an equilibrium point of (17.3) with M c R". The local behavior of the iteration (17.3) close to zo is given. under certain assumptions, by the variational equation yt+l = Dy(zo)y,, t E r. If Dp(zo)has no eigenvalue Xi with lX,I = 1,then the steady state point xo,analogously to the differential equation case, is called hyperbolic. The hyperbolic equilibrium point 20 is of type (m,k ) if D f ( x 0 ) has exactly m eigenvalues inside and k = n -m eigenvalues outside the complex unit circle. The hyperbolic equilibrium point of type (rn,k ) is called a sink for m = n, a source for k = n and a saddle point for

I

810 17. Dvnamical Systems and Chaos m > 0 and k > 0. .4 sink is asymptotically stable; sources and saddles are unstable (theorem on stability in the first approximation for discrete systems). 2. Periodic Orbits Let y(z0) = k = 0,. . ' , T - 1) be a T-periodic orbit (T 2 2) of (17.3). If zo is a hyperbolic equilibrium point of the mapping pT,then ~ ( z ois) called hyperbolic. The matrix Dp'(z0) = py(pT-'(zO)).. .Dp(zo) is called the monodromy matriq the eigenvalues pi of DqT(z0)are the multzplzers of y(zo). If all multipliers pz of y(z0) have an absolute value less than one, then the periodic orbit 5(zo) is asymptotically stable.

3. Properties of w -Limit Set Every w-limit set w ( z ) of (17.3) with M = R" is closed, and w(p(z))= ~ ( 2 )If .the semiorbit ? + ( E ) is bounded. then w ( x ) # 0 and .(E) is invariant under p. Analogous properties are valid for a-limit sets. ISuppose the difference equation zt+l = -zt: t = 0, f l . . . ', is given on R with p(z) = - 2 . Obviously. the relations w(1) = (1, -1}, w(p(1)) = a(-1) = ~ ( 1 )and ) ~ ( w ( 1 )= ) w(1) are satisfied for z = 1. We mention that u(1) is not connected. is different from the case of differential equations.

17.1.3.2 Invariant Manifolds 1. Separatrix Surfaces Let .CO be an equilibrium point of (17.3). Then W s ( r O=) {y E M : @(y) -+ zo for i -+ +m} is called a stable manifold and W'(x0) = {y E M : @(y) -+ zo for i -+ -m} an unstable manifoldof 2 0 . Stable and unstable manifolds are also called separatriz surfaces.

2. Theorem of Hadamard and Perron The theorem of Hadamard and Perron describes the properties of separatrix surfaces for discrete systems in N c R": If zo is a hyperbolic equilibrium point of (17.3) of type ( m ,k ) , then WS(zo)and W"(z0)are generalized Cr-smooth surfaces of dimension m and k , respectively, which locally look like C'-smooth elementary surfaces. The orbits of (17.3),which do not tend to EO for i -+ +w or i i -m, leave a sufficiently small neighborhood of zo for i i +m or i -+ -w, respectively. The surfaces W'(z0) and W"(z0) are tangent at EO to the stable vector subspace E' = {y E R" : [Dp(xo)]*y -+ 0 for i -+ - m } of yztl = Dp(z0)yzand the unstable vector subspace EU = {y E R" : [Dp(zo)]'y -+ 0 for i -+ -m}, respectively. ILye consider the following time discrete dynamical system from the family of HCnon mappings: (17.23) z,+1 = 5; + yz - 2, yz+l = 2 , ; i E z. Both hyperbolic equilibrium points of (17.23) are PI = (4,d) and P2 = (-J"i,-4). Determination of the local stable and unstable manifolds of PI: The variable transformation 2 , = 4. yz = 7, t 4 transforms system (17.23) into the system = [f 2 4 c, vi, 9i+1 = with the equilibrium point (0,O).The eigenvectors al = (d + fi,1) and a2 = (4- fi,1) of the Jacobian matrix D f ( ( 0 , O ) )correspond to the eigenvalues = J"i fi,so E s = {taz, t E R } and E" = { t a l , t E R}. Supposing that W&((O>O)) = { ( t , ~77 )=: p((), 151 < A,p:(-A,A) + Rdifferentiahle}, aearelookingforPintheformofapowerseriesB(<) = (fi-8) [ + k < 2 + , ... From (&,vi) E M'zc((O,O)), ( & + l , v z t l ) E follows. This leads to an equation for the coefficients of the decomposition of 9,where k < 0. The theoretical shape of the stable and unstable manifolds is shown in Fig. 17.11a (see [17.12]).

+

+

+

*

woc((O,O))

3. Transverse Homoclinic Points The separatrix surfaces W s ( x Oand ) WU(zo)of a hyperbolic equilibrium point zo of (17.3) can intersect each other. If the intersection W S ( q n) W'(z0) is transversal. then every point y E Ws(zo)n W"(z,) is called a transversal homoclinic point.

17.1 Ordinary Differential Equations and Mappings 811

Figure 17.11 Fact: I f y isa transversal homoclinicpoint, then theorbit {pi(y)}oftheinvertiblesystem (17.3) consists only of transversal homoclinic points (Fig. 17.11b).

17.1.3.3 Topological Conjugacy of Discrete Systems 1. Definition Suppose. besides (17.3), a further discrete system Zt-1 = 4.t) (17.24) with u : .I' + ,V is given; where :V c R" is an arbitrary set and $ is continuous (Mand .V can be general metric spaces). The discrete systems (17.3) and (17.24) (or the mappings p and q) are called topologically conjugate if there exists a homeomorphism h: M -+ N such that p = h-' o q o h. If (17.3) and (17.24) are topologically conjugated, then the homeomorphism h transforms the orbits of (17.3) into orbits of (17.24).

2. Theorem of Grobman and Hartman

If p in (17.3) is a diffeomorphism 9:R" + R": and 20 a hyperbolic equilibrium point of (17.3): then in a neighborhood of 20 (17.3) is topologically conjugate to the linearization yt+' = Dp(zO)yt.

17.1.4 Structural Stability (Robustness) 17.1.4.1 Structurally Stable Differential Equations 1. Definition

The differential equation (17.1),Le., the vector field f : M -+ R",is called structurally stableor robust, if small perturbations off result in topologically equivalent differential equations. The precise definition of robustness requires the notion of distance between two vector fields defined on M .We restrict our investigations to smooth vector fields on M , which have a common open connected absorbing set U c M.Let the boundary d C of C be a smooth ( n - 1)-dimensional hypersurface and suppose that it can be represented as dCT = {x E R": h ( z ) = 0}, where h : R" -+ R is a C'-function with gradh(z) # 0 in a neighborhood of ab'. Let X'(U) be the metric space of all smooth vector fields on M with the C' metrzc (17.25) p ( f , g ) =suPlIf(Z) -g(s)Il + s u p I l D f ( s ) - D s ( z ) / / . ZEC

XEC

(In the first term of the right-hand side 11 . 11 means the Euclidean vector norm, in the second one the operator norm.) The smooth vector fields f intersecting transversally the boundary dC in the direction U . Le., for which grad h ( ~ ) ~ f (#z 0.) (z E dU)and p'((z) E U (z E dU,t > 0) hold. form the set X:(C-) c X'(C-). The vector field f 6 X:(U) is called structurally stable if there is a 6 > 0 such that every other vector field g E Xi(C) with p ( f , g ) < 6 is topologically equivalent to f. H Consider the planar differential equation g (,, a )

x = -y + 2 ( a -xz

-

y",

y = 2 t y(a-22

- y2)

(17.26)

I

812

17. Dvnamical Svsterns and Chaos

with parameter a , where Jal < 1. The differential equation g belongs, e.g., to X:(U) with U = {(z, y): x* + y* < 2) (Fig. 17.12a). Obviously, p(g (., 0), g ( . , a ) )= la/ (4 1). The vector field g (., 0) is structurally unstable: there exist vector fields arbitrarily close to g (,, 0), which are not topologically equivalent to g (., 0) (Fig. 17.12b,c). This is clear if we consider the polar coordinate representation i = -r3 a r . 8 = 1 of (17.26). For a > 0 there always exists a stable limit cycle r = fi.

+

+

Figure 17.12

2. Structurally Stable Systems in the Plane Suppose the planar differential equation (17.1) with f E X:(U) is structurally stable. Then: a) (17.1) has only a finite number of equilibrium points and periodic orbits. b) A11 d-limit sets ~ ( xwith ) z E of (17.1) consist of equilibrium points and periodic orbits only. Theorem of Andronov and Pontryagin: The planar differential equation (17.1) with f E Xi(U)

u

is structurally stable if and only if a) All equilibrium points and periodic orbits in are hyperbolic. b) There are no separatrices, i.e., no heteroclinic or homoclinic orbits, coming from a saddle and tending to a saddle point.

17.1.4.2 Structurally Stable Discrete Systems In the case of discrete systems (17.3), i.e.. of mappings p: M --t M ,let U C M c R" be a bounded, open. and connected set with a smooth boundary. Let Diff ' ( U ) be the metric space of all diffeomorphisms on 121with the corresponding U defined C'-metric. Suppose the set Diff i ( U ) C Diff(U) consists of the diffeomorphisms p, for which p(V) C U is valid. The mapping ~p E Diff:(U) (and the corresponding dynamical system (17.3)) is called structurally stable if there exists a 6 > 0 such that every other mapping w E Diff \(L') with p(p, +) < 6 is topologically conjugate to 9.

17.1.4.3 Generic Properties 1. Definition .A property of elements of a metric space ( M .p) is called generic (or typical) if the set of the elements B of M with this property form a set of the second Baire category, i.e., it can be represented as B = n B,, where every set B, is open and dense in M . m=1,2 ... IA: The sets R and I C R (irrational numbers) are sets of second Baire category, but Q C R is not. IB: Density alone as a property of "typical" is not enough: Q C R and I C R are both dense, but they cannot be typical at the same time. IC: There is no connection between the Lebesgue measure X (see 12.9.1, 2., p. 634) of a set from R 1 1 and the Baire category of this set. The set B = n Bk with Bk = U a,, - -, a,, where k=1,2, ... n>o k 2" k 2" Q = {un}Fy0represents the rational numbers, is a set of second Baire category (see [17.5], [17.10]).

(

+ -),

17.1 Ordinary Differential Equations and Mappings 813

On the other hand. since Bi, I)Bktland A(&)

L

l

< scoalso X(B)= lim A(&) 5 lim - -= 0 kim

k+m

k 1 - 112

holds. 2. Generic Properties of Planar Systems, Hamiltonian Systems For planar differential equations the set of all structurally stable systems from X;(V) is open and dense in X:(c’). Hence. structurally stable systems are typical for the plane. It is also typical that every orbit (C) for increasing time tends to one of a finite number of equilibrium points of a planar system from and periodic orbits. Quasiperiodic orbits are not typical. Under certain assumptions, in the case of Hamiltonian systems, the quasiperiodic orbits of differential equation are preserved in the case of small perturbations. Hence, Hamiltonian systems are not typical systems. 8Ho 8Ho IGiven in R4 a Hamiltoniansystem in action-angle variables $1 = 0: 3 2 = 0, 6, = -, 02 = 8jl ?iZ where the Hamiltonian Ho(j1,jz) is analytical. Obviously. this system has the solutions jl = c l , j 2 = cg. 01 = d1t c3. 02 = uzt c4 with constants el, c4, where w1 and w2 can depend on c1 and c2. The relation ( j 1 - j ~=) (c1, cz) defines an invariant torus T Z .Consider now the perturbed Hamiltonian Ho(ji.jz)t i H i (ji,jz. @ , 6 ) instead of Ho, where HI is analytical and E > 0 is a small parameter. The Kolmogorov-Arno2d-Moser theorem (KAMtheorem) says in this case that if Ha is non-degenera-

Xi

-.

+

)::(

te. i.e.. det -

+

# 0, then in the perturbed Hamiltoniansystem most of the invariant non-resonant

tori will not vanish for sufficiently small E > 0 but will be only slightly deformed. “Xlost of the tori” means that the Lebesgue measure of the complement set with respect to the tori tends to zero i f s tends to 0. A torus, defined as above and characterized by w1 und ~2~ is called non-resonant if there exists a w1 P c holds for all positive integers p and q. constant c > 0 such that the inequality - - - 2 lw2

‘il

3. Non-Wandering points, Morse-Smale Systems Let {pt}tER be a dynamical system on the n-dimensional compact orientable manifold M. The point p E .I1 is called non-wandering with respect to {p‘} if V T > O 3t, ItlLT: pt(uP)nup#0 (17.27) holds for an arbitrary neighborhood Up C M of p . ISteady states and periodic orbits consist only of non-wandering points. The set Q(pt)of all non-wandering points of the dynamical systems generated by (17.1) is closed, invariant under {$} and contains all periodic orbits and all w -limit sets of points from M. The dynamical system { p t } t E on~M generated by a smooth vector field is called a Morse-Smale system if the following conditions are fulfilled: 1. The system has finitely many equilibrium points and periodic orbits and they are all hyperbolic. 2. A11 stable and unstable manifolds of equilibrium points and periodic orbits are transversal to each other. 3. The set of all non-wandering points consists only of equilibrium points and periodic orbits. Theorem of Palis and Smale: Morse-Smale systems are structurally stable. The converse statement of the theorem of Palis and Smale is not true: In the case of n 2 3, there exist structurally stable systems with infinitely many periodic orbits. For n 13, structurally stable systems are not typical.

814

17. Dvnamical Svstems and Chaos

17.2 Quantitative Description of Attractors 17.2.1 Probability Measures on Attractors 17.2.1.1 Invariant Measure 1. Definition, Measure Concentrated on the Attractor Let {pt}tcr be a dynamical system on (M, p ) . Let B be the u-algebra of Borel sets on ,Lf (12.9.1. 2.: p. 634) and let p : B i [0, +cw] be a measure on B. Every mapping pt is supposed to be p measurable. The measure p is called invariant under {pt}tErif ~ ( c p - ~ ( A =)p(A) ) holds for all A E B and t > 0. If the dynamical system {pt}tEris invertible. then the property of the measure being invariant under the dynamical system can be expressed as p(pt(A))= p(A) (A E B, t > 0). The measure p is said to be concentrated on the Borel set A C M if p(A4 \ -4) = 0. If A is also an attractor of { p t } t c rand p is an invariant measure under {pt},then it is concentrated at '2, if p ( B ) = 0 for every Borel set B with

.2 n B = 0.

+XI,

The support of a measure p : B --t [0, denoted by supp p1is the smallest closed subset of M on which the measure p is concentrated. IA: We consider the Bernoulli shift mapping on .21 = [0,1]: (17.28a) xttl = 2zt (modl). In this case the map p: [0,1] i [O. 11 is defined as =

{

22. 22 -

0 5 x 5 112,

(17.28b)

1, 112 < z 5 1.

The definition yields that the Lebesgue measure is invariant under the Bernoulli shift mapping. If a n "_

number x E [0,1)is written in dyadicform z =

a,.2-"

(a, = 0 or 1):then this representation can

n=l

be identified with z = . ~ l ~ z a 3. .. .The result of the operation 2 5 ( mod 1) can be written a s . a: a; . . . with ai = aztl , Le., all digits ak are shifted to the left by one position and the first digit is omitted. IB: The mapping P:[0,1] i [0,1]with (17.29) is called a tent mappzng and the Lebesgue measure is an invariant measure. The homeomorphism 2 h : [O. 1) i [0,1) with y = - arcsin transforms the mapping p from (17.5) into (17.29). Hence, ?r

in the case of a = 4, (17.5) has an invariant measure which is absolutely continuous. For the density p l ( y ) 5 1 of (17.29) and p(x) of (17.5) at a = 4 it is valid that pl(y) = p(h-'(y)) l(h-')'(y)l. It follows 1

directly that p(x) = 7r

Jm'

IC: If zo is a stable periodic point of period T of the invertible discrete dynamical system {p'}, 1 T-1 then p = 6,,(,,) is a probability measure for { q * } .Here, 6,, is the Dzrac measure concentrated 7- 2=0 at XO.

2. Natural Measure Let .I be an attractor of {pt}ter in A1 with domain of attraction W . For an arbitrary Borel set A and an arbitrary point xo E 14; define the number.

C

W

(17.30)

17.2 Quantitative Description o.fAttractors 815 Here. t ( T ,A, xo)is that part of the time T > 0 for which the orbit portion,{pt(zo)};=, lies in the set A. If p(A:ZO) = cy for A-a.e.* zo from W , then let p ( A ) := p(A; 20).Since almost all orbits with initial points zo E It’ tend to .Ifor t + +cq p is a probability measure concentrated at A.

17.2.1.2 Elements of Ergodic Theory 1. Ergodic Dynamical Systems A dynamical system {pt}tEron (M, p ) with invariant measure p is called ergodic (we also say that the measure is ergodic) if either p ( A ) = 0 or p(M\ A ) = 0 for every Borel set A with p;*(A) = A ( V t > 0). If {$} is a discrete dynamical system (17.3), yo : M + M is a homeomorphism and M is a compact metric space, then there always exists an invariant ergodic measure. IA: Suppose there is given the rotation mapping of the circle S’ xttl = xt Qj (mod 27r), t = 0,1,. . . , (17.31) with 9:[O. 227) + [0,27l),defined by p(x) = z + @(mod27r). The Lebesgue measure is invariant under 0 . . Qj p. If - is irrational, then (17.31) is ergodic; if - is rational, then (17.31) is not ergodic. 271 277 IB: Dgnamical systems with stable equilibrium points or stable periodic orbits as attractors are ergodic with respect to the natural measure. Birkhoff Ergodic Theorem: Suppose that the dynamical system {pt}ter is ergodic with respect to the invariant probability measure p. Then, for every integrable function h E L’(M, f3.p). the time

+

d t for flows and average along the positive semiorbits {pt(zo)}goLe. K(z0)= lim L I ’ h (p*(zo)) T-ttcoT 0 1 n-1 h (~‘(zo)) for discrete systems, coincide with the space average h d p for p-a.e. h(z0) = ~

-

I

Aiir

M ..

2=0

points xo E .\I.

2. Physical or SBR Measure The statement of the ergodic theorem is useful only if the support of the measure p is large. Let p : X + .21 be a continuous mapping, and p : B + R be an invariant measure. We call (see [17.9])p an SBR measure (according to Sinai, Bowen and Ruelle) if for any continuous function h: M + R the set of all points 20 E .VI3 for which 1 n-1

lim n 2=0 h($(xo)) = I h d p

n+x

(17.32a)

M

holds! has a positive Lebesgue measure for this. It is sufficient that the sequence of measures ( 17.32b) where 6, is the Dirac measure, weakly converges to p for almost all z E M ; Le.. for every continuous function

1

/ h d p as n +

M

.M

h dpn i

+m.

IFor some important attractors, such as the HBnon attractor, the existence of an SBR measure is proven. 3. Mixing Dynamical Systems A dpnamical system {pt}ter on (M, p ) with invariant probability measure p is called mixing if lim p ( An p - t ( B ) ) = p ( A ) p ( B )holds for arbitrary Borel sets A, B c M. For a mixing system, the t-itm

measure of the set of all points which are at t = 0 in A and under p*for large t in B , depends only on ’Here and in the following a.e. is an abbreviation for “almost everywhere”

I

816

17. Dynamical Systems and Chaos

the product p ( A ) p ( B ) . A mixing system is also ergodic: Let {pt} be a mixing system and A be a Bore1 set with P - ~ ( A = ) A ( t > 0). Then P ( A )=~jiim~(p-~(A) f l A ) = p ( A ) holds and p ( A ) is 0 or 1.

A flow {pt} of (17.1) is mixing if and only if the relation lim /[g(p'(z))

t-itm

- g ] [ h ( z)

711 dp = 0

(17.33)

M

holds for arbitrary quadratically integrable functions g, h E L Z ( M B, , p ) . Here, g and E denote the space average, which is replaced by the time average. IThe modulo mapping (17.28a) is mixing. The rotation mapping (17.31) is not mixing with respect

x

to the probability measure -.

2A

4. Autocorrelation Function Suppose the dynamical system {pt}tE,-on M with invariant measure p is ergodic. Let h : M -+ R be an arbitrary continuous function, { p ' ( ~ ) } be ~ >an~ arbitrary semiorbit and let the space average h be replaced by the time average, Le., by lim - h ( c p t ( ( z ) ) dt in the time-continuous case and by T+m T

'J

1 n-1 lim h ( ~ ' ( 2 ) )in the time-discrete case. With respect to h the autocowelation function along

nim

1

n t=O

the semiorbit { $ ( ~ ) } ~ > o to a time point 7 2 0 is defined for a flow by

".

C h ( 7 )=

lim

T+m

1 ]h (cp"'(z)) T

h (pt(z))dt -

z2

(17.34a)

and for a discrete system by (17.34b) The autocorrelation function is defined also for negative time, where Ch(.) is considered as an even function on R or Z. Periodic or quasiperiodic orbits lead to periodic or quasiperiodic behavior of Ch. A quicker descent of C h ( 7 ) for increasing 'T and arbitrary test function h refers to chaotic behavior. If Ch(7) decreases for increasing 7 with an exponential speed, then it means mixed behavior. 5 . Power Spectrum The Fourier transform of Ch(7)is called a power spectrum (see also 15.3.1.2, 5., p. 724) and is denoted by Ph(u). In the time-continuous case, under the assumption that

Ph(iu)= 7Ch(7)e-iw7d7= 2

ICh(7)Idr< m, we have (17.35a)

-x +W

In the time-discrete case, if C lCh(k)l < +co holds, then k=-m

Ph(W)

= Ch(0)

+2

m

C h ( k ) coswk.

(17.35b)

k=l

If the absolute integrability or summability of C h ( . ) does not hold, then, in the most important cases, can be considered as a distribution. The power spectrum corresponding to the periodic motions of

Ph

17.2 Quantitative Description of Attractors 817

a dynamical system is characterized by equidistant impulses. For quasiperiodic motions, there occur impulses in the power spectrum, which are linear combinations with integer coefficients of the basic impulses of the quasiperiodic motion. A “wide-band spectrum with singular peaks” can be considered as an indicator of chaotic behavior. IA: Let p be a T-periodic orbit of (17.1), h be a test function such that the time average of h(lp(t)) is zero, and suppose h ( v ( t ) )has the Fourier representation t m

akeikwot with

h(p(t))=

wg

=

2a

k=-m

Then we have, with 6 as the b distribution, tm

1 )cuk)*cos(kw,,~) and

C ~ ( T=)

t m

Iaklzb(w - kwo).

ph(d)= 2a

k=-m

k=-m

B: Suppose y is a quasiperiodic orbit of (17.1), h is a test function such that the time average is zero along 9,and let h(p(t))be the representation (double Fourier series) toc

t m

1

h ( q ( t ) )=

aklkze1(klwltlczw2)t.

ki=-m kz=-m

Then. t3c

t m

Ch(7) =

IQklkz12COS(klW1

+ k2uZ)T1

17.2.2 Entropies 17.2.2.1 Topological Entropy Let ( M , p ) be a compact metric space and {’pk}kEr be a continuous dynamical system with discrete time on M.A distance function pn on M for arbitrary n E W is defined by (17.36) Pn(Jc. 21) := P(c^%)> d ( Y ) ) .

oyz

Furthermore, let .V(E, pn) be the largest number of points from M which have in the metric pn a distance at least E from each other. The topological entropy of the discrete dynamical system (17.3) or of the 1

mapping 9 is h(p) = lim limsup - In N(E,pn). The topological entropy is a measure for the complexity E 4 n+m n of the mapping. Let ( M I .p1) be a further compact metric space and 91 : MI+ h31 be a continuous mapping. If both mappings 9 and p1 are topologically conjugate, then their topological entropies coincide. In particular. the topological entropy does not depend on the metric. For arbitrary n E W, h (pn)= n h (p) holds. If p is a homeomorphism, then h (pk)= Ik(h (p)for all k E Z. Based on the last property, we define the topological entropy h(p*) := h (p’) for a flow pt = p(t. ,) of (17.1) on

Ad c R“.

17.2.2.2 Metric Entropy Let {pt}tej-be a dynamical system on M with attractor h and with an invariant probability measure p concentrated on .I. For an arbitrary E > 0 consider the cubes Q ~ ( E ).,. . , Q n ( E ) ( ~of) the form {(XI,.. . ,zn) : k,c 5 z,< (ki t 1 ) ~ ( i = 1,2:. . . , n)} with ki E Z,for which p(Qi) > 0. For arbitrary z from a Qt the semiorbit {pt(z)}Eo is followed for increasing t. In time-distances of r > 0 (T = 1 in discrete systems), the N cubes, in which the semiorbit is found is denoted by i l . . . . , iN after each other. Let E,,,.,.,,,vbe the set of all starting points in the neighborhood of A whose semiorbits at the

I

818

17. Dunamical Susterns and Chaos

times t , = ZT ( i = 1.2,. . . , N ) are always in . . Qi, and let p ( i 1 , . ' , ZN) = p(Etl,...,E,v) be the probability that a (typical) starting point is in Eil,...,,N. The entropy gives the increment of the information on average by an experiment which shows that among a finite number of disjoint events which one has really happened. In the above situation this is

HAV =-

1

p ( i ] : . . . , i ~l )n p ( i l , . . . , i N ) ,

(17.37)

(t1,-,i,v)

) length N , which are realized by the where the summation is over all symbol sequences (il,. . ' , i ~ with orbits described above. The metric entropy or Kolmogorow-Sinai entropy h, of the attractor .? of {pt} with respect to the invariant measure p is the quantity h, = lim lim

N+m

E+O

-.HTNN

For discrete systems, the limit as E

+ 0 is

omitted. For the topological entropy h ( y ) of p : A -+ A the inequality h, 5 h(p) holds. In several cases h ( y ) = sup{h,: 1 invariant probability measure on A}. IA: Suppose A = { x o }is a stable equilibrium point of (17.1) as an attractor, with the natural measure p concentrated on 20. For these attractors h, = 0. IB: For the shift mapping (17.28a), h(p) = h, = In 2, where p is the invariant Lebesgue measure.

17.2.3 Lyapunov Exponents 1. Singular Values of a Matrix

Let L be an arbitrary matrix of type (n,n ) . The szngular values o1 2 u2 2 . . . 2 un of L are the non-negative roots of the eigenvalues a1 2 . . 2 a, 2 0 of the positive semidefinite matrix LTL. The eigenvalues a, are enumerated according to their multiplicity. The singular values can be interpreted geometrically. If K, is a sphere with center at 0 and with radius E > 0.then the image L(K,) is an ellipsoid with semi-axis lengths uE& (z = 1 , 2 , . . . , n ) (Fig. 17.13a).

Figure 17.13

2. Definition of Lyapunov Exponents Let {9'}tErbe a smooth dynamical system on M c R", which has an attractor A with an invariant ergodic probability measure p concentrated on A. Let u l ( t ,x ) 2 . . 2 gn(t,z) be the singular values of the Jacobian matrix D pt(x) of pt at the point x for arbitrary t 2 0 and x E 11.Then there exists 1 a sequence of numbers AI 2 ' . 2 A, the Lyapunov exponents, such that - In ul(t,x ) + A, for

t t -+ +mp-a.e. in the sense of L'. According to the theorem of Oseledec there exists p-a.e. a sequence of subspaces of R" (17.38) R" = Es", 2 Es", 3 ' . 2 E:,+, = { 0 } , such that for p-a.e. z the quantity

with respect to u E Es",\ E,",+1.

5t In llD yt(x)ull tends to an element A,,

E

{ A l , . . . ,A,}

uniformly

17.2 Quantitative Description 0.fAttractor.s 819

3. Calculation of Lyapunov Exponents Suppose u t ( t , x )are the semi-axis lengths of the ellipsoid got by deformation of the unit sphere with center at x by D pt(z). The formula xz(x) =

sup In oz(t, x) can be used to calculate the Lyat

punoi exponents, if additionally a reorthonormalization method, such as Housholder, is used. The function y(t. z.v) = D pt(z)vis the solution of the variational equation with v at t = 0 associated to the semiorbit ;/+(I)of the flow {$}. Actually, {pt}tER is the flow of (17.1),so the variational equation is y = Df(p'(x))y. The solution of this equation with initial v at time t = 0 can be represented as y(t.x: v) = O z ( t ) v .where O z ( t )is the normed fundamental matrix of the variational equation at t = 0, which is a solution of the matrix differential equation 2= Df(cpt(x))Z with initial Z(0) = En according to the theorem about differentiability with respect to the initial state (see 17.1.1.1,2., p. 795). 1 The number x ( z :v) = lim sup - In IlD pt(x)vJ1describes the behavior of the orbit Y ( I + E ~ )0, < E <( 1 t i m

t

with initial x +EL' with respect to the initial orbit y(x) in the direction v. If x ( r ,v) < 0, then the orbits move nearer to x for increasing t in the direction v. If, on the contrary, x(z,v ) > 0, then the orbits move away (Fig. 17.13b). Let .I be the attractor of the dynamical system {pt}tcr and p the invariant ergodic measure concentrated on it. Then, the sum of all Lyapunov exponents p-a.e. I E '1 is n

A, =

'i

!iirt

,=1

divf(ps(z))ds

(17.39a)

in the case of flows (17.1) and for a discrete system (17.3), it is (17.39b) Hence, in dissipative systems

n

A,

< 0 holds. Considering that one of the Lyapunov exponents is

%=I

equal to zero if the attractor is not an equilibrium point, the calculation of Lyapunov exponents can be simplified (see [17.9]). A: Let be x0 an equilibrium point of the flow of (17.1) and let cy, be the eigenvalues of the Jacobian matrix at xo. Il'ith the measure concentrated on 20. the following holds for the Lyapunov exponents: A, = R e a , (i = 1 , 2 . . . . , n ) . IB: Let $x0) = { p t ( x ~ )t l E [O:T]}be a T-periodic orbit of (17.1) and let pi be the multipliers of 1 r(r0).ll'ith the measure concentrated on y(x0) we have that X - -In IpiI for i = 1 , 2 . . . , n. " T

4. Metric Entropy and Lyapunov Exponents If {pt}teris a dynamical system on M C Rnwith attractor '1 and an ergodic probability measure 1.1 concentrated on .I,then the inequality h, 5 X i holds for the metric entropy h,, where in the sum X,>O

the Lyapunov exponents are repeated according to their multiplicity. The equality A,

h, =

(Pesin's formula)

(17.40)

X,>O

is not valid in general. If the measure p is absolutely continuous with respect to the Lebesgue measure and p: M + M is a C2-diffeomorphism, then Pesin's formula is valid.

I

820

17. Dynamical Systems and Chaos

17.2.4 Dimensions 17.2.4.1 Metric Dimensions 1. Fractals Attractors or other invariant sets of dynamical systems can be geometrically more complicated than point, line or torus. Fractals: independently of dynamics, are sets which distinguish themselves by one or several characteristics such as fraying, porousity, complexity, and self-similarity. Since the usual notion of dimension used for smooth surfaces and curves cannot be applied to fractals, we must give a generalized definition of the dimension. For more details see [17.2],[17.12]. I The interval Go = [0, I]is divided into three subintervals with the same length and the middle open third is removed. so we obtain the set G1 = [O,;] U 11. Then from both subintervals of G1 the open

[i,

i] [i,51 [i,i] [I,

middle third ones are removed, which yields the set Gz = [O) U U U 11. Continuing this procedure, Gk is obtained from Gk-1 by removing the open middle thirds from the subintervals. So, we get a sequence of sets Go 3 GI 3 . . 3 G, 3 * . . , where every G, consists of 2n intervals of 1

length k . 3" The Cantor set C is defined as the set of all points belonging to all G,, Le., C =

m

n G,.

The set C is

n=l

compact. uncountable. its Lebesgue measure is zero and it is perfect, i.e., C is closed and every point is an accumulation point. The Cantor set can be an example of a fractal.

2. Hausdorff Dimension The motivation for this dimension comes from volume calculation based on Lebesgue measure. If we suppose that a bounded set A c R3 is covered by a finite number of spheres B,, with radius T , 5 E , 4 so that U, B , 3 A holds, we get for A the "rough volume" - T T ~ . Now, we define the quantity 1

3

4

p E ( A )= i n f ( 1 - T T : } over all finite coverings of A by spheres with radius T , 5 E . If E tends to zero, * 3 then we get the Lebesgue outer measure X(A) of A. If A is measurable, the outer measure is equal to the volume vol(A). S U D D O S e :li is the Euclidean mace R" or. more generallv. a seuarable metric mace with metric LJ and let'i c hi be a subset of it. 6 r arbitrary parankers d 0 and E > 0, the quantity

>

:A

c

u

B,, diamB, 5 E

(17.41a)

is determined. where B, C M are arbitrary subsets with diameter diamB, = sup p(z,y). 2,BEB.

The Hausdorfl outer measure of dzmenszon d of A is defined by Pd(A) = l & p , E ( A ) = suPPd,EV)

(17.41b)

E>O

and it can be either finite or infinite. The Hausdor$dzmenszondH(A) of the set A is then the (unique) critical value of the Hausdorff measure: +x,if pd(A) # 0 for all d 2 0. (17.41~) dH(A) = inf . { d 2 0: pd(A) = 0 ) .

[

Remark: The quantities p d + ( A ) can be defined with coverings of spheres with radius T , 5 E or, in the case of Rn,of cubes with side length 5 E . Important Properties of t h e Hausdorff Dimension: (HD1) d ~ ( 0 =) 0. (HD2) If .4C R", then 0 5 dH(A)5 n.

17.2 Quantitative Description of Attractors 821

(HD3) From A (HD4) If A =

c B. it follows that ~ H ( A 5) & ( B ) . m

U At, then ~ H ( A =) s uz p d ~ ( A , ) .

2=1

(HD5) If A is finite or countable, then &(A) = 0. (HD6) If 9.AI' + AI' is Lipschitz continuous, Le., there exists aconstant L > 0 such that p(p(z).&)) 5 Lp(z. y) Vz. y E -21, then d ~ ( p ( A )5) &(A). If the inverse mapping 9-l exists as well. and it is also Lipschitz continuous. then &(A) = d ~ ( p ( A ) ) . I For the set Q of all rational numbers ~ H ( Q=) 0 because of (HD5). The dimension of the Cantor In 2 set C is ~ H ( C= )- x 0.6309.. . . In 3 3. Box-Counting Dimension or Capacity Let A be a compact set of the metric space ( X , p ) and let N t ( A ) be the minimal number of sets of diameter 5 E , necessary to cover A. The quantity In :YE(A) (17.42a) dB(A) = limsup E + ~ Inf is called the upper box-counting dimension or upper capacity. the quantity In.VE(A) (17.42b) d B ( A )= lim inf 7 E+O In; iscalled the lo~erbox-countingdimensionorlower capacity (then dc) ofA. IfaB(A) = &(A) := ~ B ( A ) holds. then dB(A) is called the box-counting dimensionof A. In Rn the box-counting dimension can be considered also for bounded sets which are not closed. For a bounded set A C R". the number ArE(A) in the above definition can also be defined in the following way: Let R" be covered by a grid from n-dimensional cubes with side length E . Then, XL(A) can be the number of cubes of the grid having a non-empty intersecting A. Important Properties of the Box-Counting Dimension: ( B D l ) d ~ ( ' 4 )5 ~ B ( A always ) holds. (BD2) For m-dimensional surfaces F C R" holds ~ H ( F=)~ B ( F=) m.

(BD3) dB(.4) = d B ( 3 ) holds for the closure 2 of A, while often d*(A) < d ~ ( 2is) valid. (BD4) If 4 = U A,, then, in general, the formula ~ B ( A = ) sup ~B(A,)does not hold for the boxn

n

counting dimension. 1 1 1 ISuppose '4 = {O. 1,-, -, . . . } . Then d ~ ( . 4 )= 0 and d ~ ( ' 4 )= - . 2 3 2 If A is the set of all rational points in [O. 11, then because of BD2 and BD3 ~ B ( A=) 1 holds. On the other hand d H ( A ) = 0. 4. Self-similarity Several geometric figures, which are called self-similar, can be derived by the following procedure: .4n initial figure is replaced by a new one which is composed o f p copies of the original, any of them scaled linearly by a factor y > 1. All figures that are k times scalings of the initial figure in the k-th step are handled as the in the first step.

-

IA: Cantor set: p = 2. y = 3. IB: Koch curve: p = 4, y = 3. The first three steps are shown in Fig. 17.14. Figure 17.14

I

822

17. Dynamical Systems and Chaos

IC: Sierpinski gasket: p = 3, q = 2. The first three steps are shown in Fig. 17.15. (The white triangles are always removed.) Figure 17.15

ID: Sierpinski carpet: p = 8, q = 3. The first three steps are shown in Fig. 17.16. (The white squares are always removed.) For the sets in the examples A-D: In D dg = d H = 2 In 4

Figure 17.16

17.2.4.2 Dimensions Defined by Invariant Measures 1. Dimension of a Measure Let p be a probability measure in (M,p), concentrated on '1 If z E .Iis an arbitrary point and & ( x ) is a sphere with radius 6 and center at 2 , then (17.43a) denotes the upper and p(B6 (17.43b) In6 > d,(z) is called the denotes the lower pointwise dimension of p in z.If &(x) = a,(.) := d P ( x )then dimension of the measure p in 2 . Young Theorem 1: If the relation d,(z) = cy holds for p-a.e.', 5 E A, then a = d H ( p ) := inf {dH(X)}. (17.44) d,(x) = liminf 6+0

xc'1.@[x)=I

The quantity d ~ ( pis) called the Hausdorf dimension ofthe measure p . ISuppose M = R" and let .1 c R" be a compact sphere with Lebesgue measure A(.!) restriction

x of p to .2 is p.1 = -.

> 0. The

Then

p ( & ( z ) ) 6" and d ~ ( p=) n . N

2. Information Dimension Suppose, theattractor.!of{pt}tcr iscovered by cubes Q1(&),. . . Qn(€)(c)OfsidelengthEasin 17.2.2.2, p. 817. Let p be an invariant probability measure on A. The entropy of the covering Q1 ( E ) , . . . , &,(,I( E ) is nk)

H ( E )= - EP,(E) Inpi(€). with p i ( & )= ~ ( Q , ( E ) ) (i = 1,.. . , I L ( E ) ) .

(17.45)

2=1

W E ) , If the limit d l ( p ) = - lim exists, then this quantity has the property of a dimension and is called E+O lne the information dimension.

'Here and in the following a.e is an abbreviation for "almost everywhere"

17.2 Quantitative Description of Attractors 823

Young T h e o r e m 2: If the relation d,(z) = cy holds for p-a.e. z E 11,then (17.46) a = dH(p) = di(p). A: Let the measure p be concentrated at the equilibrium point zo of {@}.Since H E ( p )= -1 In 1 =

0 always holds for E > 0, so dr(p) = 0. B: Suppose the measure p is concentrated on the limit cycle of holds and so d ~ ( p=) 1.

I$}. For E > 0, H E ( p )= - lne

3. Correlation Dimension Let { Y , } Z ~ = ~ be a typical sequence of points of the attractor '1 c R" of { $ } t E rp, an invariant probability measure on '1 and let m E K be arbitrary. For the vectors z,:= (ut!.. . , yitm) let the distance be defined as dist(z,.x ) .- max llyz+s- y3tsllrwhere 11 . 11 is the Euclidean vector norm. If 0 denotes 05sjrn

'-

the Heaviside function O ( x ) =

,O: then the expression

{O:

1 Crn(E)= limsup-card{(z,,z,): r++s.v2 I

= limsup N-toc

'

10

(E

dist(x,.z,)

<E}

- dist(x,.z,))

(17.47a)

J 2 $,]=I

is called the correlatzon integral. The quantity lnCm(E) 1nE (if it exists) is the correlation dimension. dK

= lim

(1i.47b)

~

E+O

4. Generalized Dimension Let the attractor .1of {.$}tGron ,ti' with invariant probability measure p be covered by cubes with side length E as in 17.2.2.2. p. 817. For an arbitrary parameter q E R, q # 1: the sum n(0

H,(E) = -In

1-q

1 p , ( ~ ) ,where p , ( ~ = ) p(Qi(~)) ,=I

(17.48a)

is called the generalized entropy ofq-th order with respect to the covering Q ~ ( E .) ., . . QncE)(~) The Re'nyi dimension o f q - t h orderis (17.48b) if this limit exists. Special Cases of the RBnyi Dimension: a ) q = 0: do = dc(suppp).

b) q = l :

(17.49a)

dl:=limd,=dI(p).

(17.49b)

4-1

C)

q=2:

d2=

(17.49~)

dK.

5 . Lyapunov Dimension Let {p'}terbe a smooth dynamical system on ,ti' C R" with attractor A (or invariant set) and with the invariant ergodic probability measure p concentrated on '4. If AI 2 A2 2 ' ' 2 A, are the Lyapunov exponents with respect to p and if k is the greatest index for which

k ,=I

A, 2 0 and

ktl t=1

A,

< 0 hold, then

I

17. Dynamical Systems and Chaos

824

(17.50) is called the Lyapunov damensaon of the measure p. If

2A, 2 0. then d L ( p ) = n;if XI

%=I

< 0, then d t ( p ) = 0

Ledrappier Theorem: Let {pt} be a discrete system (17.3) on M C R" with a C2-function 9 and p , as above, an invariant ergodic probability measure concentrated on the attractor . I of {pt}. Then d ~ ( p5) d ~ ( pholds. )

IA: Suppose the attractor A c RZof a smooth dynamical system {I$} is covered by .VEsquares with side length E . Let 0 1 > 1 > 0 2 be the singular values of D p. Then for the dB-dimensional volume of the attractor mdw N A'E . €dB holds. Every square of side length E is transformed by 'p approximately into a parallelogram with side length U ~ and E ~ I EIf. the covering is made by rhombi with side length then X,,,,

U ~ E ,

Y

N -01 holds. From the relation NE~dW N Nn2E(~02)dB we get directly 02

(17.51) This heuristic calculation gives a hint of the origin of the formula for the Lyapunov dimension. IB: Suppose the HBnon system (17.6) is given with a = 1.4 and b = 0.3. The system (17.6) has an attractor '1 (Henon attractor) with a complicated structure for these parameter values. The numerically determined box-counting dimension is d ~ ( i 1 )N 1.26. It can be shown that there exists an SBR measure for the HBnon attractor '2. For the Lyapunov exponents A1 and A 2 the formula A1 + A 2 = In 1 det D p(z)1 = In b = In 0.3 N -1.204 holds. With the numerically determined value XI N 0.42 we get X2 = -1.62. So, 0.42 d ~ ( pN) 1 + - N 1.26. (17.52) 1.62

17.2.4.3 Local Hausdorff Dimension According to Douady and Oesterl6 Let {ps'}tEr be a smooth dynamical system on M C R" and A a compact invariant set. Suppose that an arbitrary to 2 0 is fixed and let @ := ptO. Theorem of Douady a n d OesterlB: Let ul(z) 2 . . . 2 un(z) be the singular values of D @ ( s ) and let d E (0, n] be a number written as d = do s with do E (0, 1, . . . , n - 1) and s E [0,1]. If . . O d o ( z ) ~ & + , ( z ) ]< 1 holds, then &(A) < d. sup [o1(z)o2(z).

+

XE.2

Special Version for Differential Equations: Let {pt}tER be the flow of (17.1), .I be a compact invariant set and let ( Y ~ ( z2) ... 2 cx,(z) be the eigenvalues of the symmetrized Jacobian matria: 1 2 where do E (0,.

- [ D ~ ( Zt)D~f ( z ) ]at an arbitarary point z E A. If d E (0,n] is a number of the form d = do + s

..

~

n - l} and s E [0,1],and sup[cq(z) + ' . . + ado(z)+ sad,+l(Z)] < 0 holds, then XEA

dH('1) < d. The quantity 0. i f q ( z )< 0,

dDO(z) =

{ sup(& 0 5 d 5 n,

cyl(s)

+ . . + q d l ( 2 ) + (d - [d])a[d~+~(z) 2 0 ) otherwise, (17.53)

where z E .1 is arbitrary and [d] is the integer part of d , is called the Douady-OesterlC dimension at the point z. Under the assumptions of the Douady-OesterlB theorem for differential equations, d ~ ( ' 1 5) SUP dDo(z). ZE.1

17.2 Quantitative Description of Attractors 825

IThe Loren2 system (17.2), in the case of u = 10, b = 8/3, r = 28, has an attractor A (Lorenz attractor) with the numerically determined dimension &(A) = 2.06 (Fig. 17.17 is generated by Mathernatica). With the Douady-OesterlB theorem, for arbitrary > 1, u > 0 and r > 0 we get the estimation

b

~ H ( A 5) 3 - ___ u + b + l where

(17.54a)

17.2.4.4 Examples of Attractors IA: The horseshoe mapping p occurs in connection with Poincarh mappings containing the transversale intersections of stable and unstable manifolds. The unit square M = [0,1] x [0,1] is stretched linearly in one coordinate direction and contracted in the other direction. Finally, this rectangle is bent at the middle (Fig. 17.18). Repeating this procedure infinitely many times, we get a sequence of sets M 2 p(M) 2 ’ . ’ , for which

Figure 17.17

n m

A=

(17.55)

p k ( ~ )

k=O

of)

M

is a comnact set and an invariant set with respect to cp. A attracts all points of M. Apart from one point, il can be described locally as a product “line x Cantor set”.

q2(M)

Figure 17.18

IB: Let cy E (0,1/2) be a parameter and 12.I = [0,1]x [0,1]be the unit square. The mapping p: M M given by

+

(17.56a) is called thedissipative baker’s mapping. Two iterationsof the baker’smappingareshown in Fig. 17.19. The “flaky pastry structure” is recognizable. The set Y

1

n W

A=

(17.56b)

p k ( ~ )

k=O

l x

l x

lx

is invariant under p and all points from M are attracted by A. The value of the Hausdorff dimension is (17.56~)

Figure 17.19

For the dynamical system {pk} there exists an invariant measure p on M , which is different from the Lebesgue measure. .4t the points where the derivative exists, the Jacobian matrix is Dpk((z,y))=

(?d; :.),

Hence, the singular values are

u l ( k , (z, y))

= Z k , ~ ( k(z,, y)) =

ffk

and, consequently.

826

i7. Dvnamical Svstems and Chaos

the Lyapunov exponents are XI = In 2, A2 = In a (with respect to the invariant measure p). For the Lyapunov dimension we get In 2 (17.56d) d ~ ( p=) 1 + -= dH(A). -1na Pesin's formula for the metric entropy is valid here, i.e., h, = C A, = In 2. X,>O

IC: Let T be the whole torus with local coordinates (Q,x,y), as is shown in Fig. 17.20a.

Figure 17.20 Let a mapping p: T

+ T be defined by (17.57)

with parameter a E (0,1/2). The image p(T).with the intersections p(T)n D(8)and pZ(T)n D ( Q ) , is shown in Fig. 17.20b and Fig. 17.20~. The result of infinitely many intersections is the set A =

p*(T),which is called a solenoid. The attractor A consists of a continuum of curves in the length k=O

direction. and each of them is dense in A, and unstable. The cross-section of the A transversal to these curves is a Cantor set. In 2 The set A has a neighborhood which is a domain of atThe Hausdorff dimension is dH(A) = 1 - -. In a traction. Furthermore, the attractor A is structurally stable, i.e.; the qualitative properties formulated above do not change for @-small perturbations of p. ID: The solenoid is an example of a hyperbolic attractor.

17.2.5 Strange Attractors and Chaos 1. Chaotic Attractor Let {pt}terbe a dynamical system in the metric space ( M .p ) . The attractor A of this system is called chaotic if there is a sensitive dependence on the initial condition in 12. The property " sensitive dependence on the initial conditions " will be made more precise in different ways. It is given, e.g., if one of the two following conditions is fulfilled: a) All motions of {pt}on '2 are unstable in a certain sense. b) The greatest Lyapunov exponent of {p*}is positive with respect to an invariant ergodic probability measure concentrated on '2. ISensitive dependence in the sense of a) occurs for the solenoid. Property b) can be found, e.g.. for HCnon attractors.

2. Fractals and Strange Attractors An attractor A of {p"ttEris called fractal if it represents neither a finite number of points or a piecewise differentiable curve or surface nor a set which is bounded by some closed piecewise differentiable surface. An attractor is called strange if it is chaotic, fractal or both. The notions chaotic, fractal and strange are used for compact invariant sets analogously even if they are not attractors. h dynamical system is called chaotic if it has a compact invariant chaotic set.

17.3 Bifurcation Theory and Routes to Chaos 827 I Themapping

+

(17.58) z,+1 = 2 2 , + Y, (mod 11, Y"+I = 2, yn (mod 1) (Anosov difleomorphism) is considered on the unit square. The adequate phase space for this system is the torus 2'. It is conservative, has the Lebesgue measure as invariant measure, has a countable number of periodic orbits whose union is dense and is mixing. Otherwise, .A = T 2 is an invariant set with integer dimension 2.

3. Systems Chaotic in the sense of Devaney Let {pt}tErbe a dynamical system in the metric space (M,p)with a compact invariant set '4. The system {pt}ter (or the set A) is called chaotic in the sense of Devaney, if a) {pt}tEr is topologically transitive on A, Le., there is a positive semiorbit, which is dense in .A. b) The periodic orbits of {pt}tErare dense in A. c) {pt}tEr is sensitive with respect to the initial conditions in the sense of Guckenheimer on '2, i.e.! 3 F > o v X E 1 I t / 6 > 0 3 y E A f l U 6 ( 2 ) I t 2 0 :p(pt((.),pt(y))>& (17.59) where c6(z)= { z E M :p ( a , z ) < 6). W Consider the space of the 0-1-sequences =

{ s = SOSl5-2.. . , s, E {0,1} (i = 0 , l . . .)}

For two sequences s = sosls2.. . and s' = sbs{s;. . ., their distance is defined by 0. if s = st, p('% "1 = 2 - 3 , if s # st.

(

where j is the smallest index for which s3 # 51,. So, (E, p ) is a complete metric space which is also compact. The mapping p : s = soslsz.. . H~ ( s=) s' = s ~ s Z S ~ .. . is called a Bernoulli shift. The Bernoulli shift is chaotic in the sense of Devaney.

17.2.6 Chaos in One-Dimensional Mappings For continuous mappings of a compact interval into itself, there exist several sufficient conditions for the existence of chaotic invariant sets. We mention three examples. Shinai Theorem: Let p : I -i I be a continuous mapping of a compact interval I , e.g., I = [0,1] into itself. Then the system {pk} on I is chaotic in the sense of Devaney if and only if the topological entropy of cp on I . i.e., h(p),is positive. Sharkovsky Theorem: Consider the following ordering of positive integer numbers: 3 t 5 t 7 t . . . + 2 . 3 + 2 . 5 + . . . t 2 ' . 3 + 2 ' . 5 + . . . . . t 23 t 2* + 2 + 1. (17.60) Let p: I -i I be a continuous mapping of a compact interval into itself and suppose {pk} has an nperiodic orbit on I . Then {pk} also has an m-periodic orbit if n + m. Block, Guckenheimer and Misiuriewicz Theorem: Let p : I + I be a continuous mapping of the compact interval I into itself such that {pk} has a 2"m-periodic orbit ( m > 1, odd). Then In 2 h(p) 1holds. I

17.3 Bifurcation Theory and Routes to Chaos 17.3.1 Bifurcations in Morse-Smale Systems Let {p:}ter be a dynamical system generated by a differential equation or by a mapping on M C R", which additionally depends on a parameter E E V C R'. Every change of the topological structure of the phase portrait of the dynamical system for small changes of the parameter is called bifurcation. The parameter E = 0 E V is called the bifurcation value if there exist parameter values E E V in every

I

828

17. Dvnamical Svstems and Chaos

neighborhood of 0 such that the dynamical systems {cp:} and { cpk} are not topologically equivalent or conjugated on M. The smallest dimension of the parameter space for which a bifurcation can be observed is called the codzmenszon of the bifurcation. We distinguish between local bifurcations, which occur in the neighborhood of a single orbit of the dynamical system, and global bifurcations, which affect a large part of the phase space.

17.3.1.1 Local Bifurcations in Neighborhoods of Steady States 1. Center Manifold Theorem Consider a parameter-dependent differential equation

x = f(x,E) or & = fi(zl,.. . , x,, . . . , E L ) (z = 1,2,. . . , n) (17.61) with f : A1 x V + R". where h.3 C R" and V C RLare open sets and f is supposed to be r times continuously differentiable. Equation (17.61) can be interpreted as a parameter-free differential equation x = f(x.E ) , 6 = 0 in the phase space 12.3 x V . From the Picard-Lindelof theorem and the theorem on differentiability with respect to the initial values (see 17.1.1.1, 2., p. 795) it follows that (17.61) has a locally unique solution p ( , , p )E) with initial point p at timet = 0 for arbitraryp E M and E E I', which is r times continuously differentiable with respect t o p and E . Suppose all solutions exist on the whole of R. Furthermore. we suppose that system (17.61) has the equilibrium point x = 0 at E = 0. Le.. f(0,O) = 0 holds. Let XI,.

..

~

: :1

A, be the eigenvalues of Dzf(O,O) = -(O,

0)

I:,=

with ReX, = 0. Furthermore,

suppose, D,f(O, 0) has exactly m eigenvalues with negative real part and k = n - s - m eigenvalues with positive real part. According to the center manifold theorem for differential equations (theorem of Shoshitaishvili, see [17.12]),the differential equation (17.61),for E with a sufficiently small norm 1 1 ~ 1 1in the neighborhood of 0. is topologically equivalent to the system 5 = F(z,EE ) AX + g ( x , E ) , j , = -y, t= (17.62) where A is a matrix of type ( s , s) with eigenvalues XI,. . . , A,, and g with x E RS,y E R"' and z E Rk, represents a CT-functionwith g(0,O) = 0 and Dzg(O,0) = 0. It follows from representation (17.62) that the bifurcations of (17.61) in a neighborhood of 0 are uniquely described by the differential equation x =F(~,E). (17.63) Equation (17.63)represents the reduceddiferential equation to the local centermanifoldW& = {x,y: z : y = 0. z = 0) of (17.62). The reduced differential equation (17.63) can often be transformed into a relatively simple form, e.g., with polynomials on the right-hand side, by a non-linear parameterdependent coordinate transformation so that the topological structure of its phase portrait does not change close to the considered equilibrium point. This form is a so-called normalform. .4 normal form cannot be determined uniquely; in general, a bifurcation can be described equivalently by different normal forms.

2. Saddle-Node Bifurcation and Transcritical Bifurcation Suppose (17.61) is given with 1 = 1, where f is continuously differentiable at least twice and D,f(O, 0) has the eigenvalue XI = 0 and TI - 1 eigenvalues A, with ReXj # 0. According to the center manifold theorem. in this case, all bifurcations (17.61) near 0 are described by a one-dimensional reduced differ8F

entia1 equation (17.63). Obviously, here F(0,O) = -(O, 8X

a2

--F(O, ax2

0)

0 ) = 0. If, additionally, it is supposed that

8F

# 0, K(O, 0) # 0 and the right-hand side of (17.63) is expanded according to the Taylor

formula, then this representation can be transformed by coordinate transformation (see [17.6]) into

17.9 Bifurcation Theoru and Routes to Chaos 829

the normal form j: = a . + X 2 +

...

(17.64)

(for s ( O >. 0 0) ) or j: = a. - x2 + . . .

TX:

(

for -(O,O)

)

< 0 , where a. =

a(€)

is a differentiable

function with a(0) = 0 and the points indicate higher-order terms. For a < 0, (17.64) has two equilibrium points close to x = 0, among which one is stable, the other is unstable. For a. = 0, these equilibrium points fuse into x = 0, which is unstable. For a > 0, (17.64) has no equilibrium point near to 0 (Fig. 17.21b). The multidimensional case results in a saddle-node bijurcatzon in a neighborhood of 0 in (17.61). This bifurcation is represented in Fig. 17.22 for n = 2 and A1 = 0, A2 < 0. The representation of the saddle-node bifurcation in the extended phase space is shown in Fig. 17.21a. For sufficiently smooth vector fields (17.61), the saddle-node bifurcations are generic.

-

-

--:

a<0

a>O

Figure 17.21

a=O

Figure 17.22

dF If among the conditions which yield a saddle-node bifurcation for F , the condition -(O,O) de

dF

# 0 is

82 F

replaced by the conditions -(O. 0) = 0 and -(0,O) # 0, then we get from (17.63) the truncated dE dXdE normal form (without higher-order terms) x = ax - x2 of a transcrztical bzfurcatzon. For n = 2 and A 2 < 0. the transcritical bifurcation together with the bifurcation diagram is shown in Fig. 17.23. Saddle-node and transcritical bifurcations have codimension 1-bifurcations.

Figure 17.23

3. Hopf Bifurcation Consider (17.61) with n 2 2. 1 = 1 and T 2 4. Suppose that f(0.e) = 0 is valid for all E with / E / 5 EO (€0 > 0 sufficiently small). Suppose the Jacobian matrix DJ(0,O) has the eigenvalues X I = X2 = iw

830

17. Dynamical Systems and Chaos

with w # 0 and n - 2 eigenvalues X j with ReX, # 0. According to the center manifold theorm, the bifurcation is described by a two-dimensional reduced differential equation (17.63)of the form (17.65) i = a(E)z - w ( i ) y g * ( q 1/, E ) , y = W ( & ) Z a ( E ) y t gz(2. Y . E ) where cy. w ,g1 and g2 are differentiable functions and u(0)= w and also a(0)= 0 hold. By a non-linear complex coordinate transformation and by the introduction of polar coordinates ( r ,19). (17.65) can be written in the normal form i = O ( E ) T + a(c)r3 ’ . ’ , 8 = W ( E ) b(&)r2+ ’ . . (17.66) where dots denote the terms of higher order. The Taylor expansion of the coefficient functions of (17.66) yields the truncated normal form

+

+ +

+

+

+

1: = a ’ ( 0 ) ~ r a(0)r3, 8 = w ( 0 ) + ~ ’ ( O ) E b(0)r2. (17.67) The theorem of Andronov and Hopf guarantees that (17.67) describes the bifurcations of (17.66) in a neighborhood of the equilibrium point for E = 0. The following cases occur for (17.67) under the assumption cy’(0) > 0: 1. a(0) < 0 (Fig. 17.24a). : 2. a(0) > 0 (Fig. 17.2413) :

a)E > 0 : Stable limit cycle and a)E < 0 : Unstable limit cycle. unstable equilibrium point. b) E = 0 : Cycle and equilibrium point fuse b) E = 0 : Cycle and equilibrium point fuse into an unstable equilibrium point. into a stable equilibrium point. c) E < 0 : A l l orbits close to (0:0) tend c) E > 0 : Spiral type unstable as in b) fort + +x equilibrium point as in b). spirally to the equilibrium point (0,O).

6 a) E>O

& d

6 &SO

b)

E20

E
Figure 17.24 The interpretation of the above cases for the initial system (17.61) shows the bifurcation of a limit cycle of a compound equilibrium point (compound focus point of multiplicity 1))which is called a Hopf bijurcation (or Andronov-Hopf bijurcation). The case a(0) < 0 is also called supercritical. the case a(0) > 0 subcritical (supposing that a’(0) > 0). The case n = 3: X1 = XZ = i. X3 < 0. a’(0) > 0 and a(0) < 0 is shown in Fig. 17.25.

Figure 17.25 Hopf bifurcations are generic and have codimension 1. The cases above illustrate the fact that a supercritical Hopf bifurcation under the above assumptions can be recognized by the stability of a focus:

17.3 Bifurcation Theorv and Routes to Chaos 831

Suppose the eigenvalues XI(&) and & ( E ) of the Jacobian matrix on the right-hand side of (17.61) at 0 are pure imaginary for E = 0, and for the other eigenvalues A, ReX, # 0 holds. Suppose furthermore d that - ReX1(E),E=o> 0 and let 0 be an asymptotically stable focus for (17.61) at E = 0. Then there is dE a supercritical Hopf bifurcation in (17.61) at E = 0. The van der Pol differential equation x &(z2 - l ) i+ z = 0 with parameter E can be written as a planar differential equation (17.68) x = y. y = -&(2 - l ) y - 5. For E = 0, (17.68) becomes the harmonic oscillator equation and it has only periodic solutions and an equilibrium point, which is stable but not asymptotically stable. With the transformation u = f i x , u = f i y for E > 0 (17.68) is transformed into the planar differential equation

+

i = v , i.=-u-(U2-&)C. (17.69) For the eigenvalues of the Jacobian matrix at the equilibrium point (0,O) of (17.69): d 1 and so X1,2(0) = *i and - ReX1(E)i,,o = - > 0. X ~ . Z ( E )= 5 dE 2 As shown in the example of 17.1.2.3, 1.)p. 801. (0,O)is an asymptotically stable equilibrium point of (17.69) for i = 0. There is a supercritical Hopf bifurcation for E = 0, and for small E > 0, (0.0) is an unstable focus surrounded by a limit cycle whose amplitude is increasing as E increases.

4. Bifurcations in Two-Parameter Differential Equations 1. Cusp Bifurcation Suppose the differential equation (17.61) is given with r 2 4 and 1 = 2. Let the Jacobian matrix D J ( 0 . 0 ) have the eigenvalue XI = 0 and n - 1 eigenvalues A, with ReX, # 0 and

dF

suppose that for the reduced differential equation (17.63) F(O.0) = -(O,O) dX

l3

d3F

:= -(O.

ax3

a 2F

= -(O,O)

8x2

= 0 und

0) # 0. Then the Taylor espansion of F close to (0,O) leads to the truncated normal form

(without higher-order terms see. [17.1])

+

+

(17.70) 3 = cy1 cy22 sign /3z3 with the parameters aland a2. The set ( ( ~ az, 1 ~E ) : cy1 cyzz sign/& = 0 ) represents a surface in extended phase space and this surface is called a cusp (Fig. 17.26a).

+

+

Figure 17.26 In the following, we suppose l3 < 0. The non-hyperbolic equilibrium points of (17.70) are defined by the svstem a1 + cyzz - z3 = 0. a2 - 3z2 = 0 and thus they lie on the curves S1 and Sz,which are

832

17. Dvnamacal Svstems and Chaos

determined by the set {(aiI 0 2 ) : 27aT - 40; = 0) and form a cusp (Fig. 17.26b). If (a1. c y 2 ) = (0,O) then the equilibrium point 0 of (17.70) is stable. The phase portrait of (17.61) in a neighborhood of 0, e.g.. for n = 2: l 3 < 0 and XI = 0 is shown in Fig. 1 7 . 2 6 ~for Xz < 0 (triple node) and in Fig. 17.26d for Xz > 0 (triple saddle) (see [17.6]). At transition from ( a i Ia z ) = (0,O) into the interior of domain 1 (Fig. 17.26b) the compound nodetype non-hyperbolic equilibrium point 0 of (17.61) splits into three hyperbolic equilibrium points (two stable nodes and a saddle) (supercritical pitchfork bifurcation). In the case of the two-dimensional phase space of (17.61)the phase portraits are shown in Fig. 17.26c,e. When the parameter pair of Si \ { (0,O)) (i = 1,2) traverse from 1 into 2 then a double saddle nodetype equilibrium point is formed which finally vanishes. A stable hyperbolic equilibrium point remains.

Figure 17.27

2. Bogdanov-Takens Bifurcation Suppose, for (17,61)>n 2 2, l = 2, T 2 2 hold and the matrix D,f(O, 0) has two eigenvalues X I = Xz = 0 and n - 2 eigenvalues X j with ReXj # 0. Let the reduced two-dimensional differential equation (17.63) be topologically equivalent to the planar system (17.71) x = y , y = a1 t a2x t 22- xy. Then there is a saddle-node bifurcation on the curve SI= ((01,0 2 ) : ai - 4a1 = 0). r\t the transition on the curve Sz = {(ai,a?):a1 = 0, a2 < 0) from the domain cy1 < 0 into the domain a1 > 0 a stable limit cycle arises by a Hopf bifurcation and on the curve S3 = { ( a ~a ,~: a1 ) = -kai + . . .} ( k > 0, constant) there exists a separatrix loop for the original system (Fig. 17.27). which bifurcates into a stable limit cycle in domain 3 (see [17.6],[17.9]). This bifurcation is of a global nature and we say that a single periodic orbit arisesfrom the homoclinic orbit ofa saddle or a separatriz loop disappears. 3. Generalized Hopf Bifurcation Suppose that the assumptions of the Hopf bifurcation with r 2 6 are fulfilled for (17.61),and the two-dimensional reduced differential equation after a coordinate transformation into polar coordinates has the normal form i = e l r + ~~r~- r5 +. . , $ = 1 t. .. The bifurcation diagram (Fig. 17.28) of this system contains the lines1 = {(EL, E Z ) : E L = 0, ~2 # 0}, whose points represent a Hopf bifurcation (see [17.4], [17.6]). There exist two periodic orbits in domain 3, amongwhichoneisstable,theotheroneisunstable. On thecurveSz = { ( E ~ , E zE): +: ~ E Z > 0. EI < 0}, these non-hyperbolic cycles fuse into a compound cycle which disappears in domain 2. 5. Breaking Symmetry Some differential equations (17.61)have symmetries in the following sense: There exists a linear transformation T (or a group of transformations) such that f(Tz,E ) = T f ( 2 ,E) holds for all z € M and E E I/. An orbit y of (17.61) is called symmetric with respect t o T if T y = y. IVe talk about a symmetry breaking bifurcation at E = 0, e.g., in (17.61) (for 1 = l ) , if there is a stable equilibrium point or a stable limit cycle for E < 0, which is always symmetric with respect to T , and for E = 0 two further stable steady states or limit cycles arise, which are nolonger symmetric with respect to T .

17.3 Bifurcation Theory and Routes to Chaos 833

Figure 17.28 For system (17.61) with f ( z lE ) = c x - x 3 the transformation T defined as T : z H --z is a symmetry, since f ( - z , ~ = ) -f(z, E ) (zE R, E E R). For E < 0 the point x1 = 0 is a stable equilibrium point. For E > 0, besides z1 = 0, there exist the two other equilibrium points ~ 2 , 3= i&; both are nonsymmetric.

17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit 1. Center Manifold Theorem for Mappings Let y be periodic orbit of (17.61) for E = 0 with multipliers p1,. . . , pn-l, pn = 1. A bifurcation close to 7 is possible, if when changing E , at least one of the multipliers lies on the complex unit circle. The use of a surface transversal to y leads to the parameter-dependent PoincarC mapping z H P ( LE ) . (17.72) Then, with open sets E c Rn-' and V c RLlet P : E x V + Rn-' be a Cr-mapping where the mapping P : E x I' --t Rn-l x R' with P ( ~ , E=) ( P ( z ,E ) , & ) should be a Cr-diffeomorphism. Furthermore, let P(0,O) = 0 and suppose the Jacobian matrix D,P(O, 0) has s eigenvalues p1,. . . , ps with 1p,l = 1, m eigenvalues pstl.. . . , pstm with JpiI < 1 and k = n - s - m - 1 eigenvalues p s + m t l r . . , pn-1 with (pzl > 1. Then, according to the the center manijold theorem for mappings (see [17.4]),close to (0.0) E E x V . the mapping P is topologically conjugate to the mapping (17.73) (x,y. z , E ) +-+( F ( z ,E ) , A'y, A"z,E )

+

near (0.0) E Rn-' x RLwith F ( z ,E ) = ACz g ( z ,E ) . Here g is a Cr-differentiable mapping satisfying the relations g(0,O)= 0 and D,g(O. 0) = 0. The matrices A', As and A" are of type (s, s),( m ,rn) and ( k , k ) , respectively. It follows from (17.73) that bifurcations of (17.72) close to (0,O) are described only by the reduced mapping z HF ( z ,E ) (17.74) on the local center manijold &, = {(z,y. z ) : y = 0, t = 0).

;kg ;kg

;34$

a)

b)

U
a=O

a>O

Figure 17.29

2. Bifurcation of Double Semistable Periodic Orbits Let the system (17.61) be given with n 2 2, r 2 3 and 1 = 1. Suppose, at E = 0, the system (17.61) has periodic orbit y with multipliers pi = $1, lpzl # 1 (i = 2,3,. . . , n - 1) and pn = 1. According to the

834

1'7. Dvnamical Svstems and Chaos

center manifold theorem for mappings, the bifurcations of the Poinear6 mapping (17.72) are described 82 F dF by the one-dimensional reduced mapping (17.74) with A' = 1. If -(0,O)# 0 and - (0,O)# 0 is ax2

supposed, then it leads to the normal forms 2 HF ( z , a )= cy

+a:

or

$22

( E

z ~ c y + x - z ~ for-(O,O)<~

).

a€

(17.75a) (17.75b)

The iterations from (17.758) close to 0 and the corresponding phase portraits are represented in Fig. 17.29a and in Fig. 17.291, for different cy (see [17.6]). Close to z = 0 there are for cy < 0 a stable and an unstable equilibrium point, which fuse for cy = 0 into the unstable steady state z = 0 . For a > 0 there exists no equilibrium point close to z = 0. The bifurcation described by (17.75a) in (17.74) is called a subcritical saddle node bifurcation f o r mappings. In the case of the differential equation (17.61)1the properties of the mapping (17.75a) describe the bifurcation o f a double semistable periodic orbit: For (Y < 0 there exists a stable periodic orbit y1 and an unstable periodic orbit which fuse for cy = 0 into a semistable orbit 7 ,which disappears for cy > 0 (Fig. 17.30a,b).

Figure 17.30

3. Period Doubling or Flip Bifurcation Let system (17.61) be given with n 2 2, r 2 4 and 1 = 1. We consider a periodic orbit y of (17.61) at E = 0 with the multipliers p1 = -1, lpll # 1 (i = 2 , . . . , n - l ) ,and pn = 1. The bifurcation behavior of the Poinear6 mapping in the neighborhood of 0 is described by the one-dimensional mapping (17.74) with Ac = -1. if we suppose the normal form (17.76) z * B(z,c y ) = (-1 + cy)z + 2 3 . The steady state z = 0 of (17.76) is stable for small cy 2 0 and unstable for (Y < 0. The second iterated mapping B2 has for cy < 0 two further stable fixed points besides z = 0 for z1,2 = 5 6 + o(1aI). which are not fixed points of F, Consequently. they must be points of period 2 of (17.76). In general, for a C4-mapping(17.74) there is a two-periodic orbit at E = 0, if the following conditions are fulfilled (see [17.2]):

F(0,O) = 0,

82 F2 -(O.O) dXdE dF2

Since -(O,

dX

# 0.

dF -(o> ax d2F2

-(O,O)

8x2

dF2

0) = -1, = 0,

dF

0) = +1 holds (because of -(O, OX

-(O.O) dE

= 0.

d3F 2 -(O.O) 6x3

# 0.

(17.77)

0) = -1)$ the conditions for a pitchfork bzfurcation are

formulated for the mapping F 2 . For the differential equation (17.61) the properties of the mapping (17.76) imply that at cy = 0 a stable

17.3 Bifurcation Theory and Routes to Chaos 835

periodic orbit ye splits from y with approximately a double period (period doubling), where y loses its stability (Fig. 17.30~). I The logistic mapping pa:[0,1] -+ [0,1] is given for 0 < a 5 4 by ve(s)= a z ( 1 - s), Le., by the discrete dynamical system (17.78) Pt+l = azt(1 - st). The mapping has the following bifurcation behavior (see [17.10]): For 0 < a 5 1 system (17.78) has the equilibrium point 0 with domain of attraction [O,11. For 1 < a < 3, (17.78) has the unstable 1 equilibrium point 0 and the stable equilibrium point 1 - - , where this last one has the domain of a 1. attraction ( 0 , l ) . For a1= 3 the equilibrium point 1- - is unstable and leads to a stable two-periodic a orbit. the two-periodic orbit is also unstable and leads to a stable 22-periodic orAt the value a2 = 1 + 6, bit. The period doubling continues, and stable 29-periodic orbits arise at a = aq.Numerical evidence shows that cyp + a, x 3.570.. . as q -+ +m For cy = a x . there is an attractor F (the Feigenbaum attractor), which has a structure similar to the Cantor set. There are points arbitrarily close to the attractor which are not iterated towards the attractor, but towards an unstable periodic orbit. The attractor F has dense orbits and the Hausdorff dimension is dH(F) x 0.538 On the other hand, the dependence on initial conditions is not sensitive. In the domain a, < a < 4, there exists a parameter set A with positive Lebesgue measure such that system (17.78) has a chaotic attractor of positive Lebesgue measure for a E A . The set A is interspersed with windows in which period doublings occur. The bifurcation behavior of the logistic mapping can also be found in a class of unimodal mappings. Le.. of mappings of the interval I into itself, which has a single maximum in I . Although the parameter values cyzr for which period doubling occurs, are different from each other for different unimodal mappings. the rate of convergence by which these parameters tend to am,is equal: ak - a , x Cb-k, where b = 4.6692.. . is the Feigenbaum constant (C depends on the concrete mapping). The Hausdorff dimension is the same for all attractors F at a = a,: &(F)x 0.538.. . . 4. Creation of a Torus Consider system (17.61) with n 2 3, r 2 6 and 1 = 1. Suppose that for allE close to 0 system (17.61) has

aperiodicorbity,. Letthemultipliersof-/obep,,2 = e*%*with!P# ( O ,;- $ - .7T

}

3 , pJ

(- - 3 , . . . , n - l )

with lp31 # 1 and pn = 1. according to the center manifold theorem. in this case there exists a two-dimensional reduced C6mapping

z HF ( P . E ) (17.79) with F ( 0 ,E ) = 0 for E close to 0. If the Jacobian matrix D,F(O.E) has the conjugate complex eigenvalues P ( E ) and P(E) with Ip(0)l = 1 d for all E near 0. if d := - /p(e)l~,=o> 0 holds and p(0) is not a q-th root of 1 for q = 1,2,3.4. dE

then (17 79) can be transformed by a smooth E dependent coordinate transformation into the form z e P(z,E) = Fo(z, E ) + 0(11~1/~) (0Landau symbol), where Fois given in polar coordinates by (17.80) Here a , LU’ and b are differentiable functions. Suppose a(0) < 0 holds. Then, the equilibrium point r = 0 of (17.80) is asymptotically stable for all E < 0 and unstable for E > 0. Furthermore, for E > 0 there

I

836

17. Dunamical Sustems and Chaos

exists the circle r =

, which is invariant under the mapping (17.80) and asymptotically stable

(Fig. 17.31a).

Figure 17.31 The Neimark-Sacker Theorem (see [17.10],[17.1])states that the bifurcation behavior of (17.80) is similar to that of (supercritical Hopf bifurcation for mappings). IIn mapping (17.79), given by 1 (1+ € ) Iy +t2 2 - 2y2

(r) +-+ -(

fi

-5+(1+E)y+52-53

1

'

there is a supercritical Hopf bifurcation at E = 0. h'ith respect to the differential equation (1i.61)3the existence of a closed invariant curve of mapping (17.79) means that the periodic orbit yo is unstable for a(0) < 0 and for E > 0 a torus arises which is invariant with respect to (17.61) (Fig. 17.31b).

17.3.1.3 Global Bifurcation Besides the periodic creation orbit which arises if a separatrix loop disapears, (17.61) can have further global bifurcations. Two of them are shown in [17.12]by examples.

1. Emergence of a Periodic Orbit due to the Disappearance of a Saddle-Node IThe parameter-dependent system = z ( l - 2 2 - y2) + y ( 1 + 5 + c y ) > y = -5(1+ 5 + a ) + y ( l - 2 2 has in polar coordinates I = r cos 8,y = r sin 8the following form:

x

- y2)

r: = r ( l - .*), 9 = -(1 + a + r cos 8 ) . (17.81) Obviously, the circler = 1 is invariant under (17.81) for an arbitrary parameter a , and all orbits (except the equilibrium point (0,O)) tend to this circle for t + +m. For cy < 0 there are a saddle and a stable node on the circle, which fuse into a compound saddle-node type equilibrium point at cy = 0. For cy > 0, there is no equilibrium point on the circle, which is a periodic orbit (Fig. 17.32).

Figure 17.32

2. Disappearance of a Saddle-Saddle Separatrix in the Plane IConsider the parameter-dependent planar differential equation x = cy + 25y, y = 1 + 5 2 - yz.

(17.82)

17.3 Bifurcation Theorv and Routes to Chaos 837

For a = 0, equation (17.82) has the two saddles ( 0 , l ) and (0, -1) and the y-axis as invariant sets. The heteroclinic orbit is part of this invariant set. For small la1 # 0, the saddle-points are retained while the heteroclinic orbit disappears (Fig. 17.33).

Figure 17.33

17.3.2 Transitions to Chaos Often a strange attractor does not arise suddenly but as the result of a sequence of bifurcations, from which the typical ones are represented in Section 17.3.1. The most important ways to create strange attractors or strange invariant sets are described in the following.

17.3.2.1 Cascade of Period Doublings Analogously to the logistic equation (17.78), a cascade of period doublings can also occur in timecontinuous systems in the following way. Suppose system (17.61) has the stable periodic orbit 7:’)for E < E , . For E = E~ a period doubling occurs near $:), at which the periodic orbit 7:’) loses its stability for E > A periodic orbit 7::) with approximately double period splits from it. At E = ~ 2 there , is a new period doubling, where ntf) loses its stability and a stable orbit 7:;)with an approximately double period arises. For important classes of systems (17.61) this procedure of period doubling continues. so a sequence of parameter values { E ~ arises. } Numerical calculations for certain differential equations (17.61), e.g., for hydrodynamical differential equations such as the Lorenz system, show the existence of the limit lim J-++X

EJt1 - Ej Elf2 - E j t 1

= 6. where 6

(17.83)

is again the Feigenbaum constant. For E, = lim E j . the cycle with infinite period loses its stability, and a strange attractor appears. 31-

The geometric background for this strange attractor in (17.61)by acascade of period doubling is shown in Fig. 17.34a. The Poincari! section shows approximately a baker mapping, which suggests that a Cantor set-like structure is created.

Figure 17.34

17.3.2.2 Intermittency Consider a stable periodic orbit of (17.61)>which loses its stability for E = 0 when exactly one of the multipliers, for E < 0 inside the unit circle takes the value + l . According to the center manifold theorem. the corresponding saddle-node bifurcation of the Poincari! mapping can be described by a one-dimensional mapping in the normal form x e P ( x , a ) = a + z + x2 + . . . . Here a is a parameter

838

17. Dynamical Systems and Chaos

depending on E . Le., LY = a(&)with ( ~ ( 0 )= 0. The graph of p ( . )a)for positive a is represented in Fig. 17.34b. As can be seen in Fig. 17.3413, the iterates of ."(., a ) stay for a relatively long time in the tunnel zone for 0 2 0. For equation (17.61), this means that the corresponding orbits stay relatively long in the neighborhood of the original periodic orbit. During this time, the behavior of (17.62) is approximately periodic (laminar phase). After getting through the tunnel zone the considered orbit escapes, which results in irregular motion (turbulent phase). After a certain time the orbit is recovered and a new laminar phase begins. A strange attractor arises in this situation if the periodic orbit vanishes and its stability goes over to the chaotic set. The saddle-node bifurcation is only one of the typical local bifurcations playing a role in the intermittence scenario. Two further ones are period doubling and the creation of a torus.

17.3.2.3 Global Homoclinic Bifurcations 1. Smale's Theorem Let the invariant manifolds of the Poinear6 mapping of the differential equation (17.61) in R3 near the correspond periodic orbit y be as in Fig. 17.11b, p. 17.11. The transversal homoclinic points PJ(xo) to a homoclinic orbit of (17.61) to y. The existence of such a homoclinic orbit in (17.61) leads to e dependence on initial conditions. In connection with the considered PoincarC mapping. horseshoe mappings, introduced by Smale, can be constructed. This leads to the following statements: a) In every neighborhood of a transversal homoclinic point of the PoincarC mapping (17.74) there exists a periodic point of this mapping (Smale's theorm). Hence, in every neighborhood of a transversal homoclinic point there exists an invariant set of P"(m E N), .4,which is of Cantor type. The restriction of Pmto .2 is topologically conjugate to a Bernoulli shift. Le., to a mixing system. b) The invariant set of the differential equation (17.61) close to the homoclinic orbit is like a product of a Cantor set with the unit circle. If this invariant set is attracting, then it represents a strange attractor of (17.61).

2. Shilnikov's Theorem Consider the differential equation (17.61) in R3 with scalar parameter E . Suppose that the system (17.61) has a saddle-node type hyperbolic steady state 0 at E = 0, which exists so long as I E ~ remains small. Suppose. that the Jacobian matrix Dzf(O,0) has the eigenvalue A3 > 0 and a pair of conjugate complex eigenvalues XI,* = a & i w with a < 0. Suppose, additionally, that (17.61) has a separatrix loop 7 0 for E = 0, Le.: a homoclinic orbit which tends to 0 fort + -m and t -+ +m (Fig. 17.35a). Then, in a neighborhood of a separatrix loop (17.61) has the following phase portrait : a) Let A 3 + a < 0. If the separatrix loop breaks at E # 0 according to the variant denoted by A in (Fig. 17.35a). then there is exactly one periodic orbit of (17.61) for E = 0. If the separatrix loop breaks at E # 0 according to the variant denoted by B in (Fig. 17.35a), then there is no periodic orbit. b) Let X3 + a > 0. Then there exist countably many saddle-type periodic orbits at E = 0 (respectively. for small lcl) close to the separatrix loop yo (respectively, close to the destroid loop The Poincar6 mapping with respect to a transversal to the *io plane generates a countable set of horseshoe mappings at E = 0. from which there remain finitely many for small (€1 # 0. e,,,).

Figure 17.35

17.3 Bifurcation Theorv and Routes to Chaos 839

3. Melnikov's Method Consider the planar differential equation (17.84) i = f(x) c g ( t , x), where E is a small parameter. For E = 0, let (17.84) be a Hamiltonian system (see 17.1.4.3, 2., p. 813), dH dH i e , for f = ( f l . f 2 ) f l = - and f 2 = -- hold, where H : C' c R2 i R is supposed to be a C3ax2 axl function. Suppose the time-dependent vector field g: R x U + R2 is twice continuously differentiable, and T-periodic with respect to the first argument. Furthermore, let f and g be bounded on bounded sets Suppose that for E = 0 there exists a homoclinic orbit with respect to the saddle point 0. and the PoincarC section ztoof (17.84) in the phase space {(XI, 2 2 . t ) }f o r t = to looks as in Fig. 17.35b. The Poincark mapping P,,tO: Et, --t Et,, for small l ~ l has . a saddle point p , close to x = 0 with the invariant manifolds TVs(pE)and W ( p E ) If . the homoclinic orbit of the unperturbed system is given by p(t - t o ) , then the distance between the manifold W 9 ( p e )and W"(p,),measured along the line passing through 9(0)and perpendicular to f ( + ( O ) ) , can be calculated by the formula

+

(17.858) Here, M(.) is the Melnikovfunction which is defined by (17.85b)

M ( t o ) = T f ( p ( t - t o ) ) A g ( t , ~ (- to)) i dt. -m

(For a = ( a l , a2) and b = ( b l , b2): A means a A b = alb2 - a2bl.) If the Melnikov function M has a simple root at to3Le., M ( t 0 ) = 0 and M ' ( t 0 ) # 0 hold, then the manifolds W s ( p Eand ) W'.(pe) intersect each other transversally for sufficiently small&> 0. If M has no root, then W s ( p En) W"(p,) = 0, Le., there is no homoclinic point. Remark: Suppose the unperturbed system (17.84) has a heteroclinic orbit given by p(t - t o ) running , from a saddle point O1 in a saddle 02. Let p: and pz be the saddle points of the Poincark mapping P,,,, for small IEI. If )[I. calculated as above, has a simple root at to) then W s ( p l )and W " ( p : ) intersect each other transversally for small E > 0. I Consider the periodically perturbated pendulum equation x + sinx = E sinwt, Le.. the system x = y, k = - sinx t ~ s i n c j t in , which E is a small parameter and w is a further parameter. The 1 unperturbed system x = y) j, = - sinx is a Hamiltonian system with H ( z ,y) = -yz - cosx. It has 2 (among others) a pair of heteroclinic orbits through (-x,0) and ( T )0) (in the cylindrical phase space

S' x R these are homoclinic orbits) given by p*(t) = ( 1 2 arctan(sinh t ) ,1 2 -

)

(t E R ) . The cosh t 2 i sin ~ direct calculation of the hlelnikov function yields .M(to) = Since .Lf has a simple root cosh(.rrw/2) ' at to = 0, the Poincark mapping of the perturbed system has transversal homoclinic points for small E > 0.

.&

17.3.2.4 Destruction of a Torus 1. From Torus to Chaos 1. Hopf-Landau Model of Turbulence The problem of transition from regular (laminar) behavior to irregular (turbulent) behavior is especially interesting for systems with distributed parameters, which are described, e.g.. by partial differential equations. From this viewpoint, chaos can be interpreted as behavior irregular in time but ordered in space. On the other hand, turbulence is the behavior of the system, that is irregular in time and in space. The Hopf-Landau model explains the arising of turbulence by an infinite cascade of Hopf bifurcations: For

I

840

17. Dunarnica1 Sustems and Chaos

E = c1 a steady state befurcates in a limit cycle, which becomes unstable for E Z > E I and leads to a .At the k-th bifurcation of this type a k-dimensional torus arises, generated by non-closed torus TZ. orbits which wind on it. The Hopf-Landau model does not lead in general to an attractor which is characterized by sensitive dependence on the initial conditions and mixing. 2. Ruelle-Takens-Newhouse Scenario Suppose that in system (17.61) we have n 2 4 and 1 = 1. Suppose also that changing the parameter E , the bifurcation sequence “equilibrium point + periodic orbit + torus TZ+ torus T3” is achieved by three consecutive Hopf bifurcations. Let the quasiperiodic flow on T 3be structurally unstable. Then, certain small perturbation of (17.61) can lead to the destruction of T 3and to the creation of a structurally stable strange attractor. 3. Theorem of Afraimovich a n d Shilnikov on the Loss of Smoothness a n d the Destruction of t h e Torus T 2 Let the sufficiently smooth system (17.61) be given with n 2 3 and I = 2. Suppose that for the parameter value the system (17.61) has an attracting smooth torus T * ( Espanned ~) by a stable periodic orbit ys, a saddle-type periodic orbit 7%and its unstable manifold W’”(y,) (resonance torus). The invariant manifolds of the equilibrium points of the Poincark mapping computed with respect to a surface transversal to the torus in the longitudinal direction, are represented in Fig. 17.36a. The multiplier p of the orbit ys, which is the nearest to the unit circle, is assumed to be real and simple. Furthermore, let E ( . ) : [0,1] + V be an arbitrary continuous curve in parameter space. for which E(Oj = EO and for which system (17.61) has no invariant resonance torus for E = ~ ( 1 ) Then . the following statements are true: a) There exists a value s, E ( 0 , l ) for which TZ(&(s,)) loses its smoothness. Here, either the multiplier p(s,j is complex or the unstable manifold Wu(yujloses its smoothness close toys. b) There exists a further parameter values,, E (s., 1) such that system (17.61) has no resonance torus for s E (s*.. 11. The torus is destroyed in the following way: a ) The periodic orbit rS loses its stability for E = ~(s,,). A local bifurcation arises as period doubling or the creation of a torus. p ) The periodic orbits */u and ys coincide for E = E(s,,) (saddle-node bifurcation) and so they vanish. 7) The stable and unstable manifolds of -yu intersect each other non-transversally for E = E(s,,) (see the bijurcation diagram in Fig. 1 7 . 3 6 ~ ) .The points of the beak-shaped curve S1 correspond to the fused ;(s and yu (saddle-node bifurcation). The tip C1 of the beak-shaped curve is on a curve SO,which corresponds to a splitting of the torus.

Figure 17.36 The parameter points where the smoothness is lost, are on the curve S,,while the points on S3 characterize the dissolving of a T z torus. The parameter points for which the stable and unstable manifolds of yu intersect each other non-transversally, are on the curve S,. Let PObe an arbitrary point in the beaked shaped tip of the beak such that for this parameter value a resonance torus T Zarises. The transition from Po to PI corresponds to the case a)of the theorem. If the multiplier p becomes -1 on Sz.then there is a period doubling. .4 cascade of further period doublings can lead to a strange attractor. If a then it can result pair of complex conjugate multipliers pl,z arises on the unit circle passing through SZ. in the splitting of a further torus, for which the Afraimovich-Shilnikov theorem can be used again. The transition from Po to Pz represents the case p) of the theorem: The torus loses its smoothness, and on passing through on S1, there is a saddle-node bifurcation. The torus is destroyed, and a transition to chaos through intermittence can happen. The transition from Po to P3, finally, corresponds to the

17.3 Bifurcation Theorv and Routes to Chaos 841

case 7):After the loss of smoothness, a non-robust homoclinic curve forms on passing through on S,. The stable cycle ys remains and a hyperbolic set arises which is not attracting for the present. If ys vanishes, then a strange attractor arises from this set.

2. Mappings on the Unit Circle and Rotation Number 1. Equivalent and Lifted Mappings The properties of the invariant curves of the Poincark mapping play an important role in the loss ofsmoothness and destruction of a torus. If the Poincark mapping is represented in polar coordinates, then, under certain assumptions, we get decoupled mappings of the angular variables as informative auxiliary mappings on the unit circle. These are invertible in the case of smooth invariant curves (Fig. 17.36a) and in the case of non-smooth curves (Fig. 17.36b) they are not invertible. .4mapping F : R + R with F ( O + 1) = F ( O ) + 1for all 0 E R, which generates the dynamical system @ n + ~= F ( @ n ) , (17.86) is called equivariant. For every such mapping, an associated mapping of the unit circle f : SI i SI can be assigned where SI = R \ 2 = (0 mod 1,0E R}. Here f(z):= F ( 0 ) if the relation z = [e] holds for the equivalence class [O]. F is called a lified mapping off. Obviously, this construction is not unique. In contrast to (17.86) X t + l = f (4 (17.87) is a dynamical system on S'.

IFor two parameters w and K let the mapping F(*;w , K ) be defined by P(u;w , K ) = o+w - K sin u for all 7 E R. The corresponding dynamical system on+l= u n + w - K s i n o n (17.88) can be transformed by the transformation on = 2n0, into the system K On+, = 0, t R - -sin 27r0, (17.89) 2i7

K .

With F(0; R,K ) = 0 + R - - s i n 2 ~ an 0 equivariant mappingarises, which generates where R = 2. 27l 277 the cunonical form of the circle mapping. 2. Rotation Number The orbit y(0) = {Fn(0)} of (17.86) is a q-periodic orbit of (17.87) in S' if and only if it is a P- cycle of (17.86), Le., if there exists an integer p such that On+q= 0, 4

+ p , ( n E Z)

holds. The mapping f : S' + S' is called orientation preserving if there exists a corresponding lifted mapping F . which is monotone increasing. If F from (17.86) is a monotone inreasing homeomorphism,

F"(z) for every z E R, and this limit does not depend on z.Hence, then there exists the limit lim Inlim n Fn(z) can be defined. I f f : S' + S' is a homeomorphism and F and the expression p(F) := lim In/+-

TI

are two lifted mappings o f f , then p(F) = p(p)+ k , where k is an integer. Based on this last property, the rotation number p ( f ) of an orientation-preserving homeomorphism f : S' i S' can he defined as p ( f ) = p(F) mod 1. where F is an arbitrary lifted mapping o f f . I f f : SI i SI in (17.87) is an orientation-preserving homeomorphism, then the rotation number has the following properties (see [17.4]):

P

a) If (17.87) has a q-periodic orbit, then there exists an integer p such that p ( f ) = - holds. 4

b) If p ( f ) = 0, then (17.87) has an equilibrium point. c) Ifpjf) =

5.

wherep # 0 is an integer and q is a natural number ( p and q are coprimes), then (17.87)

has a q-periodic orbit.

I

842

17. Dunamical Systems and Chaos

d) p ( f ) is irrational if and only if (17.87) has neither a periodic orbit nor an equilibrium point Theorem of Denjoy: I f f : S' + S' is an orientation-preserving C2-diffeomorphismand the rotation number cy = p ( f ) is irrational, then f is topologically conjugate to a pure rotation whose lifted mapping is F ( x ) = x t a.

3. Differential Equations on the Torus Ta Let

61= f i ( Q i , @ ~ )Q~z = f z ( Q i , @ z ) (17.90) be a planar differential equation, where fi and f 2 are differentiable and one-periodic functions in both arguments. In this case (17.90) defines a flow, which can also be interpreted as a flow on the torus T 2 = S' x S' with respect to 01 and 02. If f l ( 0 1 , O Z ) > 0 for all (01,02),then (17.90) has no equilibrium points and it is equivalent to the scalar first-order differential equation (17.91) f2 LVith the relations 01 = t , 02 = 5 and f = -, (17.91) can be written as a non-autonomous differential

fl

equation

x = f ( t ,x) (17.92) whose right-hand side is one-periodic with respect to t and 5 . Let g ( , ,xo)be the solution of (17.92) with initial state 50 at timet = 0. So, a mapping PI(.) = p(1, ,) can be defined for (17.92),which can be considered as the lifted mapping of a mapping f : S'+ S'. ILet d1,d2E R be constants and 6,= wlr Q2 = . ~ 2a differential equation on the torus, which w2 is equivalent to the scalar differential equation x = - for w1 # 0. Thus, p(t, 20) = %:2t t 50 and w1

W1

4. Canonical Form of a Circle Mapping 1. Canonical Form The mapping F from (17.89) is an orientation-preserving diffeornorphism for

aF

0 5 K < 1, because - = 1 - Kcos2~19> 0 holds. For K = 1, F is nolonger a diffeomorphism,

a19

but it is still a homeomorphism, while for K > 1, the mapping is not invertible, and hence nolonger a homeomorphism. In the parameter domain 0 5 K 5 1. the rotation number p(R, K ) := p(F(8,R, K ) ) is defined for F ( . , R,K ) . Let K E ( 0 , l ) be fixed. Then p(.. K ) has the following properties on [0,1]: a) The function p(.. K ) is not decreasing, it is continuous, but it is not differentiable.

b) For every rational number P- E [0, 1) there exists an interval Iplqrwhose interior is not empty and 4

for which p(R, K ) = P- holds for all R E Ipiq 4 c) For every irrational number a E ( 0 , l ) there exists exactly one R with p(R, K ) = cy. 2. Devil's Staircase and Arnold Tongues For every K E (0, l ) ,p ( , 2K ) is a Cantor function. The graph of p ( . , K ) ,which is represented in Fig. 17.37b, is called the devil's staircase. The bifurcation diagram of (17.89) is represented in Fig. 17.37a. At every rational number on the R-axis: a beak-shaped region (Arnold tongue) with a non-empty interior starts, where the rotation number is constant and equal to the rational number. The reason for the formation of the tongue is a synchronization of the frequencies (frequency locking). a) For 0 5 K < 1, these regions are not overlapping. At every irrational number of the Q-axis, a continuous curve starts which always reaches the line K = 1. In the first Arnold tongue with p = 0, the dynamical system (17.89) has equilibrium points. If K is fixed and R increases, then two of these

17.3 Bifurcation Theoru and Routes to Chaos 843

equilibrium points fuse on the boundary of the first Arnold tongue and vanish at the same time. As a result ofsuch a saddle-node bifurcation, a dense orbit arises on 5’’. Similar phenomena can be observed when leaving other Arnold tongues. b ) For K > 1 the theory of the rotation numbers is nolonger applicable. The dynamics become more complicated, and the transition to chaos takes place. Here, similarly to the case of Feigenbaum constants, further constants arise, which are equal for certain classes of mappings to which also the standard circle mapping belongs. One of them is described in the following.

0.6 0.4

0.2

b)

0.4

0.6

0.8

Figure 17.37 &-1 3. Golden Mean, Fibonacci Numbers The irrational number II is called the golden mean L

and it has a simple continued fraction representation

&- 1 ~

2

1 = 1++ ~

= [l;1,1,.. .] (see 1.1.1.4,

1 + r

3.) p. 4). By successive evaluation of the continued fraction we get a sequence {r,} of rational numbers: 61 Fn which converges to . The numbers T, can be represented in the form T, = , where F, are 2 F,+l Fibonacci numbers, which are determined by the iteration F,+1 = F, +F,,-, (n = 1.2, . .) with initial &- 1 values F, = 0 and Fl = 1. Now, let R, be the parameter value of (17.89), for which p(R, 1) = 2 and let R, be the closest value to R,, for which p(R,, 1) = r, holds. Numerical calculation gives the R, - R*-*- -2.8336.. . . limit lim “+a: R,+l - n, ~

~

-

844

18. Optimization

18 Optimization 18.1 Linear Programming 18.1.1 Formulation of the Problem and Geometrical Represent at ion 18.1.1.1 The Form of a Linear Programming Problem 1. The Subject of lznearprogrammzngis the minimization or maximization of a h e a r objectwe junctzon (OF) of finitely many variables subject to a finite number of constraints (CT),which are given as linear equations or inequalities. Many practical problems can be directly formulated as a linear programming problem, or they can be modeled approximately by a linear programming problem.

2. General Form A linear programming problem has the following general form: OF:

f(x)= ~ 1 x +1 ' . . + c,x,

+ cr+lxr+l +. . + c,x,

= max!

(18.la)

(18.1b)

1

xi 2 0 , . . . ,x, 2 0; x r t l l . . . ,z, free. In a more compact vector notation this problem becomes: OF:

f ( r )= siTx1+cZTx2 = max!

(18.2a)

CT:

Allxi + A12x25 b', A z i ~ ' AzzxZ= bz, xi 2 0 , x2 free.

+

Here. we denote: (18.2~)

18.1 Linear Programming 845

3. Constraints with the inequality sign

.’

2 ’’ will have the above form if we multiply them by (-1).

4. Minimum Problem A minimum problem f(x) = min! becomes an equivalent maximum problem by multiplying the objective function by (-1) -f(x)= max!

(18.3)

5. Integer Programming Sometimes certain variables are restricted to be only integers. We do not discuss this discrete problem here.

6. Formulation with only Non-Negative Variables and Slack Variables In applying certain solution methods. we have to consider only non-negative variables, and constraints (18.1b). (18.2b) given in equality form. Every free variable xk must be decomposed into the difference of two non-negative variables xk = xi - OF:f(x)= c1z1+ . . . + c , ~ , = max! (18.4a) xi. The inequalities become equalities by adding non-negative variables; the): are called slack van- CT: a1,121 -t ’ + ai,n% = b l , 1 ables. That is. we can consider the problem in the form as given in (18.4a,b), where n is the increased number of variables. In vector form we have: “

OF: f ( x )= cTx= max! (18.5a) CT: Ax=b, x>Q. (18.5b) We can suppose that m 5 n. otherwise the system of equations contains linearly dependent or contradictory equations. 7. Feasible Set The set of all vectors satisfying constraints (18.2b) is called the feasible set of the original problem. If we rewrite the free variables as above, and every inequality of the form “5”into an equation as in (18.4a) and (18.4b), then the set of all non-negative vectors x 2 Q satisfying the constraints is called the feasible set M : M = {x E R” : x 0; Ax = b}. (18.6a) A point x* E :2f with the property f(x*)2 f(x) for every x E M (18.6b) is called the maximum point or the solution point of the linear programming problem. Obviously, the components of x not belonging to slack variables form the solution of the original problem.

>

18.1.1.2 Examples and Graphical Solutions 1. Example of the Production of Two Products Suppose we need primary materials Rl! R2, and RBto produce two products E1 and E*. Scheme 18.1 shows how many units of primary materials are needed to produce each unit of the products El and Ez, and there are given also the available amount of the primary materials. Scheme 18.1 Sellingone unit of the products El or E2 results in 20 or 60 units of profit: respectively ( P R ) . Determine a production program which yields maximum profit, if at least 10 units must be produced from product El. If we denote by x1and x2 the num-

I

846

18. Optimization

ber of units produced from El and Ez, then we get the following problem: OF: f(x) = 20x1 6Oz2 = max! CT:

1221 8x1

+ + 6x2 5 + 1222 5 10x2

21

5 2

630, 620, 350! 10.

Introducing the slack variables 2 3 > 2 4 , 2 5 , 2 6 , we get: OF: f(x) = 2021 60z2 0 . 2 3 0 . 2 4 0 x5 CT :

+ 1221 + 821

+ 622 +

+ 12x2

10x2

+

23

+

+

+ 0.z6 = max! = 630,

24

+

-21

25

+

= 620, = 350! 26 =

-10.

2. Properties of a Linear Programming Problem On the basis of this example, we can demonstrate some properties

"2,'

'

4 of the linear programming problem by graphical representation. To do this, we do not consider the slack variables; only the original two 35 ._ variables are used. a) X line alzl +a222 = b divides the 21)x2 plane into two half-planes. 25" The points ( q l2 2 ) satisfying the inequality alz1+a2x2 5 bare in one of these half-planes. The graphical representation of this set of points in a Cartesian coordinate system can be made by a line, and the halfXI plane containing the solutions of the inequalities is denoted by an -@+ arrow. The set of feasible solutions M.i.e., the set of points satisfying all inequalities is the intersection of these half-planes (Fig. 18.1). Figure 18.1 In this example the points of M form a polygonal domain. It may happen that ,PI is unbounded or empty. If more then two boundary lines go through a vertex of the polygon. we call this vertex a degenerate vertex (Fig. 18.2).

t,

r x

" I

Figure 18.2

b) Every point in the Z ~ , X Zplane satisfying the equality f ( 2 ) = 2021 + 6Oz2 = co is on one line. on the level line associated to the value eo. With different choices of eo, a family of parallel lines is defined, on each of which the value of the objective function is constant. Geometrically, those points are the solutions of the programming problem, which belong to the feasible set ,21and also to the level line 2021 + 60x2 = co with maximal value of eo. In this example, the solution point is (21, 2 2 ) = (25,35) on the line 2021 + 6Oz2 = 2600. The level lines are represented in Fig. 18.3,where the arrows point in the direction of increasing values of the objective function. Obviously. if the feasible set M is bounded, then there is at least one vertex such that the objective function takes its maximum. If the feasible set M is unbounded, it is possible that the objective function

18.1 Linear Programming 847

Figure 18.3

Figure 18.4

is unbounded. as well

18.1.2 Basic Notions of Linear Programming, Normal Form \Ye consider problem (18&,b) Kith the feasible set M

18.1.2.1 Extreme Points and Basis 1. Definition ofthe Extreme Point A point x E -21 is called an extreme point or vertex of M ,if for all E , , E A4 with x, # x,: 0 < x < 1, -x # AXl t (1 -A)&,. Le.. E is not on any line segment connecting two different points of M.

(18.7)

2. Theorem about Extreme Points The point x E .If is an extreme point of M if the columns of matrix A associated to the positive components of x are linearly independent. If the rank of A is m. then the maximal number of independent columns in A is m. So, an extreme point can have at most m positive components. The other components, a t least n - m: are equal to zero. In the usual case, there are exactly n positive components. If the number of positive components is less then m. we call it a degenerate extreme point.

3. Basis We can assign m linearly independent column vectors of the matrix A to every extreme point, the columns belonging to the positive components. This system of linearly independent column vectors is called the basis ofthe extreme point. Usually. exactly one basis belongs to every extreme point. However several bases can be assigned to a degenerate extreme point. There are at most

(3

possibilities to

choose m linearly independent vectors from n columns of A . Consequently, the number of different bases. and therefore the number of different extreme points is least one extreme point

. If M is not empty, then M has at

848

18. Optimization

OF:

f(x) =

CT:

221 21

t 352

+

22 22

-21

+ 4x3 = max!

+ I 2 1, 2, + 253 I2, 53

(18.8)

2x1 - 3x2 t 2x3 5 2.

The feasible set N determined by the constraints is represented in Fig. 18.4. Introduction of slack variables 2 4 ) 25) 2 6 , 2 7 leads to: CT: 21 .CZ ~3 - 5 4 = 1, 22 + 55 = 2, -21 t 223 x6 = 2, 221 - 322 223 27 = 2.

+

+

+

+

+

The extreme point PZ = (0,1,0) of the polyhedron corresponds to the point x = (zl1xz.z3,x4>z5,56, z7)= ( 0 , 1 , 0 , 0 , 1 , 2 , 5 )of the extended system. The columns 2 , 5 , 6 and 7of A form the corresponding basis. The degenerated extreme point Pl corresponds to (1,O , O , 0,2,3,0). .4basis of this extreme point contains the columns 1 , 5 , 6 and one of the columns 2,4 or 7. Remark: Here, the first inequality was a “2”inequality and we did not add but subtract x4. Frequently these types of additional variables both with a negative sign and a corresponding bi > 0 are called surplus variables, rather than slack variables. As we will see in 18.1.3.3,p. 852, the occurrence of surplus variables requires additional effort in the solution procedure.

4. Extreme Point with a Maximal Value of the Objective Function Theorem: If .Vi is not empty, and the objective function f(x) = cTxis bounded from above on rll. then there is at least one extreme point of M where it has its maximum. A linear programming problem can be solved by determining at least one of the extreme points with maximum value of the objective function. Usually, the number of extreme points of M is very large in practical problems, so we need a method by which we can find the solution in a reasonable time. Such a method is the simplez method which is also called the simplez algorithm or simplex procedure. ~

18.1.2.2 Normal Form of the Linear Programming Problem 1. Normal Form and Basic Solution The linear programming problem (18.4a,b) can always be transformed to the following form with a suitable renumbering of the variables:

OF:

CT:

f(x)= clzl +. ’ . + cn-,x,+,

+ cg = max!

Ui,iXi

+...+ ~ i , n - m ~ , - m t X,-,+i

am,lxl

+ ’ . + am,n-mxn-m

21:.

. . z,-, ~

2,-,+1,.

. . ,5, 2 0.

(18.9a)

!I

= bi,

+ X, = b,

(18.9b)

,

The last m columns of the coefficient matrix are obviously independent, and they form a basis. The basic solution ( X I 2, 2 ) . . . ,~,-,,x,-,~~,.. . ,x,) = (0,. . . , 0 , b i , . . . , b,) can be determined directly from the system of equations, but if b 2 0 does not hold, it is not a feasible solution. If b 2 0. then (18.9a.b) is called a normalform or canonicalform ofthe linearprogrammingproblem. In this case, the basic solution is a feasible solution, as well, Le., x 2 0, and it is an extreme point of .\I. The variables 21, . . . , zn-mare called non-basic variables and z,-,+l, . . . z,are called basic variables. The objective function has the value cg at this extreme point, since the non-basic variables are equal to zero.

I

~

18.1 Linear Programming 849

2. Determination of the Normal Form If an extreme point of ,\.Iis known, then we can get a normal form of the linear programming problem in the following nay. We choose a basis from the columns of A corresponding to the extreme point. Suppose the basic variables are collected into the vector xBand the non-basic variables are in x N .The columns associated to the basis form the basis matrix AB, the other columns form the matrix AN. Then. (18.10) Ax = ANSA, + ABXB= b. The matrix AB is non-singular and it has an inverse AB1, the so-called basas inverse. Multiplying (18.10) by AB1 and changing the objective function according to the non-basic variables results in the canonical form of the linear programming problem:

OF: f ( ~ =) CT,XN

+ CO,

( 18.1la)

CT: Ai1A1vxN+xB= Ai'b with

XN

2 0,

XB

2 0.

(18.11b)

Remark: In the original system (18.lb) has only constraints of type "5" and simultanously b 2 0. Then the extended system (18.4b) contains no surplus variables (see 18.1.2.1,p. 847). In this case a normal form is immediatlely known. Selecting all slack variables as basic variables xB we have AB = I and xB = b. xN = 0 is a feasible extreme point. IIn the above example x = (0, l , O , 0,1,2,5) is an extreme point. Consequently:

(18.12b)

+ + = 1, + + + + I;:} 521 + 523 - 354 t = 5. From f ( x ) = + 322 + 423. we get the transformed objective function f ( x )= + t 3x4 + 3, 22

-

-21

23

24

23

24

25

223

-21

26

(18.13)

27

221 -21

23

(18.14)

if we subtract the triple of the first constraint.

18.1.3 Simplex Method 18.1.3.1 Simplex Tableau The simple2 method is used to produce a sequence of extreme points of the feasible set with increasing values of the objective function. The transition to the new extreme point is performed starting from the normal form corresponding to the given extreme point, and arriving at the normal form corresponding to the new extreme point. In order to get a clear arrangement, and easier numerical performance, we put the normal form (18.9a,b) in the simplex tableau (Scheme 18.2a, 18.2b):

850

18. Optimization

Scheme 18.2a

Scheme 18.2b or briefly

1

e1

'.'

en-m

I-co

The k-th row of the tableau corresponds to the constraint xn-m+k a k . l X 1 + ' ' . ak,n-mxn-m= b k . ( 18.15a) We have for the objective function ClZl t .' ' + cn-mxn-m = f(x)- eo. (18.15b) , = (Qlb). We also get the value of From this simplex tableau, we can find the extreme point ( x NxB) the objective function at this point f(x)= eo. We can always find exactly one of the following three cases in every tableau: a) c3 5 0, j = 1, . . . , n- m: The tableau is optimal. The point (xN, xe) = (0,b) is the maximal point. b) There exists at least one j such that cj > 0 and aij 5 0, i = 1,.. . , m: The linear programming problem has no solution, since the objective function is not bounded on the feasible set; for increasing values of xj it increases without a bound. c) For every j with el > 0 there exists at least one i with at, > 0: We can move from the extreme point x to a neighboring extreme point with f(g)2 f(x). In the case of a non-degenerate extreme point x. the ">" sign always holds.

+

+

18.1.3.2 Transition to the New Simplex Tableau 1. Non-Degenerate Case If a tableau is not in final form (case c)), then we determine a new tableau (Scheme 18.3). We interchange a basic variable xp and a non-basic variable x, by the following calculations: a)

cPq= L.

(18.16a)

aPP

(18.16b) 6, = bp . Lip,. j # q> kPq, i # p. e, = -ep a,, (18.16~) d) 6ij = aij a,, . iitq3 i # p , j # q> 6, = bi b, hi,. i # p , j # q, Eo = co b, . E,. (18.16d) E3 = c3 a,j , E,, The element apqis called the pivot e l e m e n t , the p t h row is the pivot TOW, and the q-th column is the pivot column. We must consider the following requirements for the choice of a pivot element: a) Eo 2 co should hold; b) the new tableau must also correspond to a feasible solution, Le., h 2 0 must hold. Then. (&. &) = (0,b) is a new extreme point at which the value of the objective function f(g)= i.0 is not smaller than it was previously. These conditions are satisfied if we choose the pivot element in the following way: a) To increase the value of the objective function, a column with c, > 0 can be chosen for a pivot column; b) to get a feasible solution, the pivot row must be chosen as b)

6p3

C)

6,,= -ai,

= a,j , E,,

+

+

+

+

(18.17)

18.1 Linear Programming 851

If the extreme points of the feasible set are not degenerate, then the simplex method terminates in a finite number of steps (case a) or case b)). I The normal form in 18.1.2 can be written in a simplex tableau (Scheme 18.4a). Scheme 18.3

Scheme 18.4a

# %B

AN b - -co

27

-1 -1 5 -1

-1 2 0 5 -3 1 3 -3

Scheme 18.4b

24 26

27

-1 1 -1 2 0 -1

2 2 3 2 4 - 3 -6

2:2 8:2

This tableau is not optimal, since the objective function has a positive coefficient in the third column. The third column is assigned as the pivot column (the second column could also be taken under consideration). \\.'e calculate the quotients b,/a,* with every positive element of the pivot column (there is only one of them). The quotients are denoted behind the last column. The smallest quotient determines the pivot row. If it is not unique, then the extreme point corresponding to the new tableau is degenerate. After performing the steps of (18.16a)-(18.16d) we get the tableau in Scheme 18.4b. This tableau determines the extreme point (0,2,0,1,0,2,8),which corresponds to the point P7 in Fig. 18.4. Since . extreme point this new tableau is still not optimal, we interchange 26 and 2 3 (Scheme 1 8 . 4 ~ ) The of the third tableau corresponds to the point P6 in Fig. 18.4. After an additional change we get an optimal tableau (Scheme 18.4d) with the maximal point E* = (2,2,2,5,0,0,0),which corresponds to the point P5,and the objective function has a maximal value here: f(x*)= 18. Scheme 1 8 . 4 ~ Scheme 18.4d Scheme 18.5 Xn

2. Degenerate Case If the next pivot element cannot be chosen uniquely in a simplex tableau, then the new tableau represents a degenerate extreme point. .4degenerate extreme point can be interpreted geometrically as the coincident vertices of the convex polyhedron of the feasible solutions. There are several bases for such a vertex. In this case, it can therefore happen that we perform some steps without reaching a new extreme point. It is also possible that we get a tableau that we had before, so an infinite cycle may occur. In the case of a degenerate extreme point, one possibility is to perturb the constants bi by adding E' (with a suitable E , > 0) such that the resulting extreme points are nolonger degenerate. We get the solution from the solution of the perturbed problem, if we substitute E = 0. If the pivot column is chosen "randomly" in the non-uniquely determined case, then the occurrence of an infinite cycle is unlikely in practical cases.

852

18. Optimization

18.1.3.3 Determination of an Initial Simplex Tableau 1. Secondary Program, Artificial Variables If there are equalities among the original constraints (18.lb) or inequalities with negative h,, then it is not easy to find a feasible solution to start the simplex method. In this case, we start with a secondary program to produce a feasible solution, which can be a starting point for a simplex procedure for the original problem. We add an artificzal variable yh 2 0 (k = 1 , 2 , . . . , m) to every left-hand side of A s = b with b 2 0, and we consider the secondary program:

OF*: g(x,2)= -y1 - . . - ym = max!

CT':

a1,121

+...+ a1,nZn + YI

.,

~ 1 ,, . 2 ,

L 0;

~ 1 , ,.

(18.18a) = hi,

)

+ gm = h i ,

}

(18.18b)

J

. ,Ym 2 0.

For this problem, the variables yl,. . . , ym are basic variables, and we can start the first simplex tableau (Scheme 18.5). The last row of the tableau contains the sums of the coefficients of the non-basic variables, and these sums are the coefficients of the new secondary objective function OF*. Obviously, g(x,y) 5 0 always. If g(x*,y*)= 0 for a maximal point ( x ' , ~ 'of ) the secondary problem, then obviously y' = 0, and consequently x*is a solution of Ax = b. If g(x*,y') < 0, then Ax = b does not have any Glution.

2. Solution of the Secondary Program Our goal is to eliminate the artificial variables from the basis. We do not prepare a scheme only for the secondary program separately. We add the columns of the artificial variables and the row of the secondary objective function to the original tableau. The secondary objective function now contains the sums of the corresponding coefficients from the rows corresponding to the equalities, as shown below. If an artificial variable becomes a non-basic variable, we can omit its column, since we will never choose it again as a basis variable. If we determined a maximal point (x*,y*), then we distinguish between two cases: 1. g(x*,y') < 0: The system Ar = b has no solution, the linear programing problem does not have any feasible solution. 2. g(r*,y') = 0: If there are no artificial variables among the basic variables, this tableau is an initial tableau fir the original problem. Otherwise we remove all artificial variables among the basic variables by additional steps of the simplex method. By introducing the artificial variables, the size of the problem can be increased considerably. .4s we see, it is not necessary to introduce artificial variables for every equation. If the system of constraints before introducing the slack and surplus variables (see Remark on p. 848) has the form A1x 2 bl, Azr = bz.A3x 5 & with bI,b2,B2 0, then we have to introduce artificial variables only for the first two systems. For the third system the slack variables can be chosen as basic variables. IIn the example of 18.1.2, p. 848, only the first equation requires an artificial variable: OF': g(x,y)= - Y1 = max!

CT*:

x1

+

22 22

-21

2x1 -

322

+ 23 + 2x3

+ 2x3

- 24

+ y1

+ 25

= 1,

+ 26

= 2, =

2,

+ 2 7 = 2.

The tableau (Scheme 18.6b) is optimal with g(x*, y*)= 0. After omitting the second column we get the first tableau of the original problem.

18.1 Linear Programming 853

Scheme 18.6a

Scheme 18.6b

-1 -1 -1 1 - 1 0 2 0 5 3 5-3 -1-3 1 3 -3

OF

18.1.3.4 Revised Simplex Method 1. Revised Simplex Tableau Suppose the linear programming problem is given in normal form: co = max! OF: f(x)= c l z l + ' . . t C ~ - ~ X , - , , ,

+

CT:

+ .+ a 1 , n - m x n - m + Zn-m+l

0 1 1x1

2 1 2 0

'

=

(18.19a)

131,

1

. . . . ,x , > o .

Obviously. the coefficient vectors ( a = 1,.. . , n ) are the 2-th unit vectors. In order to change into another normal form and therefore to reach another extreme point, it is sufficient to multiply the system of equations (18.19b) by the corresponding basis inverse. (We refer to the fact that if AB denotes a new basis, the coordinates of a vector x can be expressed in this new basis as AB'X. If we know the inverse of the new basis. we can get any column as well as the objective function from the very first tableau by simple multiplication.) The simplex method can be modified so that we determine only the basis inverse in every step instead of a new tableau. From every tableau, we determine only those elements which are required to find the new pivot element. If the number of variables is considerably larger than the number of constraints ( n > 3m), then the revised simplex method requires considerably less computing cost and therefore has better accuracy. The general form of a revised simplex tableau is shown in Scheme 18.7. Scheme 18.7

1

I

I

1

c1 ' . ' en-m Cn-mtl ' . ' cn -cO cq The quantities of the scheme have the following meaning: z;, . . . x i : Actual basic variables (in the first step the same as xn-mtl ' . zn). el;. . , c, : Coefficients of the objective function (the coefficients associated to the basic variables are zeros). b l , . . . , b, : Right-hand side of the actual normal form. : \:slue of the objective function at the extreme point (x;,. . . ~ z i =) ( b l s . . . . bm). co

.

A*=

(

aipm+i

.' . 0 1 , ~ ;

j

2,-,+1,. am,n-m+l

' ''

A* are the columns of . . ,x, corresponding to the actual normal form;

) : actual basis inverse, where the columns of

am,n

1: = ( T I , . . . , T , ) ~ : ,Actual pivot column.

I

18. Optimization

854

2.

Revised Simplex Step

a) The tableau is not optimal when at least one of the coefficients e, ( j = 1 . 2 . . . . , n ) is positiv. Lye choose a pivot column q for a cq > 0. b) We calculate the pivot column 1: by multiplying the q-th column of the original coefficient matrix (18.19b) by A' and we introduce the new vector as the last vector of the tableau. \Ye determine the pivot row k in the same way as in the simplex algorithm (18.17). c) We calculate the new tableau by the pivoting step (18.16a-d), where utq is formally replaced by ri and the indices are restricted for n - m + 1 5 j 5 n. The column f is omitted. x q becomes a basic variable. For j = 1 , .. . . n - m, we get E, = e, t $E, where E = and gj is the j-th . ., column of the coefficient matrix of (18.19b). IConsider the normal form of the example in 18.1.2, p. 848. We want to bring x4 into the basis. The corresponding pivot column 1: = a is placed into the last column of the tableau (Scheme 18.8a) (initially A* is the unit matrix). Scheme 18.8a

Scheme 18.8b 21 2 3

x4

x2 2 5 2 6 2 7

0 0l0 0

x7

3 0 1

For j = 1.3.4 we have: E, = c, - 3~22,: (c1, c3, c4) = (2,4,0). The determined extreme point = (0,2,0,1,0,2,8)corresponds to the point P7 in Fig. 18.4. The next pivot column can be chosen for j = 3 = q . The vector 1:is determined by 1 1 0 0

[-;I[-!)

= ( r l , .. . r,) = A*% = ~

(00 03 1O0 01 ) . = and we place it into the very last column of the second tableau (Scheme 18.8b).We proceed as above analogously to the method shown in 18.1.3.2, p. 851. If n e want to return to the original method, then we have to multiply the matrix of the original columns of the non-basic variables by A* and we keep only these columns.

18.1.3.5 Duality in Linear Programming 1. Correspondence To any linear programming problem (primal problem) we can assign another unique linear programming problem (dual problem):

Przmal problem

OF: CT:

f(x)= $zl + cTx2= max! A, 1x1 + A1.2~25 bi,

+ -42.2112

A2,1~1

xl 2 0,

Dual problem (18.20a)

OF*: g(n)= bTnl + bTu2= min! (18.21a)

= b2,

x2free.

(18.20b)

The coefficients of the objective function of one of the problems form the right-hand side vector of the constraints of the other problem. Every free variable corresponds to an equation, and every variable with restricted sign corresponds to an inequality of the other problem.

18.1 Linear Programminq 855

2. Duality Theorems a) If both problems have feasible solutions, Le., M # 0,M*# 0 (where M and M' denote the feasible sets of the primal and dual problems respectively), then (18.22a) f ( x ) 5 g(u) for all 1~.E M, y E M*: and both problems have optimal solutions. b) The points x E M and u E M* are optimal solutions for the corresponding problem, if and only if f ( x )= d u ) . (18.22b) c ) If f ( x ) has no upper bound on M or g(u) has no lower bound on M*,then M*= 0 or M = 0;Le., the dual problem has no feasible solution. d) The points x E M and u E M* are optimal points of the corresponding problems if and only if (18.22~) uT(A1,lxl t A 1 . 2 ~ 11,) ~ = 0 and xT(AT1ult Allu2 - cl) = 0. Using these last equations, we can find a solution x of the primal problem from a non-degenerate optimal solution u of the dual problem by solving the following linear system of equations: (18.23a) (18.23b) (18.23~) We can also solve the dual problem by the simplex method.

3. Application of the Dual Problem Working with the dual problem may have some advantages in the following cases: a) If it is simple to find a normal form for the dual problem, we switch from the primal problem to the dual. b) If the primal problem has a large number of constraints compared to the number of variables. then we can use the revised simplex method for the dual problem. Consider the original problem of the example of 18.1.2, p. 848

Dual problem

Primal problem

OF:

CT:

f(x) = 2x1 + 3x2 + 4x3 = max! 5 x2 5 -21 t 2x3 5 2xl - 3x2 t 2x3 5 -21

-

22

-

23

-1. 2> 2) 2,

OF': CT':

g(u) = -ul

+

2u2 t 2u3 t 2u4 = min! - u3 t 2u4 2 2 , -211 t u2 - 3214 2 3. -u1 2u3 2uq 2 4, -ul

+

+

2111 u2.74ru4 2 0. 0. If the dual problem is solved by the simplex method after introducing the slack variables, then we get the optimal solution ut = (211, uzl 213,214) = (0,7,2/3,4/3) with g(G) = 18. We can obtain a solution x* of the primal problem by solving the system (Ax - b)i = 0 for ui > 0, Le., x2 = 2 , -xl + 2x3 = 2, 2x1 - 3x2 + 2x3 = 2, therefore: x* = ( 2 , 2 , 2 ) with f(x)= 18.

xl,X2,x3 2

18.1.4 Special Linear Programming Problems 18.1.4.1 Transportation Problem 1. Modeling A certain product, produced by m producers El, Ez,. . . , E, in quantities al, a 2 : . . . , a,, is to be transported t o n consumers \{. &, . . . V, with demands bl, b 2 : . . :, b,. Transportation cost of a unit product of producer 5, to consumer tJ is c,j. The amount of the produet transported from E, to 4is z , ~units. ~

856

18. Optimization

We are looking for an optimal transportation plan with minimum total transportation cost. We suppose the system is balanced, i.e., supply equals demand: m

n

E a , =Eb3.

(18.24)

3=1

t=1

Lye construct the matrix of costs C and the distribution matrix X:

(

=

v:

c1,1

'.

i

C1,n

j

cm,1 ' . ' cm,n

v, ... vn

)

E:

(

El

j

X=

)(18.25a)

Em

C:

Xi,i

. .. x1,n

;

xm,1

bl

i

. . . xm,n ... b,

)

c: ai j

.

(18.25b)

am

If condition (18.24) is not fulfilled, then we distinguish between two cases: a) If a, > b , , then we introduce a fictitious consumer V,+l with demand b,+l = a, - b3 and with transportation costs c,,,+1 = 0. b) If C a, < bl, then we introduce a fictitious producer E,,,,, with capacity am+l = b3 - a, and with transportation costs c,,,+,,~= 0. In order to determine an optimal program, we have to solve the following programming problem: m

n

OF: f ( X ) = E

ct3xt3= min!

(18.26a)

,=13=1

n

m

]=I

,=1

CT: E z t 3 = a t ( z = l ,. . . .m ) , c ~ , ~ =( ~b= ~1.,. . ,n ) , x t 3 > O ) .

(18.26b)

The minimum of the problem occurs at a vertex of the feasible set. There are m + n - 1 linearly independent constraints among the m + n original constraints, so, in the non-degenerate case, the solution contains m + n - 1 positive components zt3. To determine an optimal solution the following algorithm is used, which is called the transportation algorithm. 2. Determination of a Basic Feasible Solution Il'ith the Northwest corner rule we can determine an initial basic feasible solution: a) Choose zll = rnin(a1, b1). ( 18.27a)

b) If a, < bl, we omit the first row of X and proceed to the next source. If a, > bl, we omit the first column of X and proceed to the next destination. If al = bl, we omit either the first row or the first column of X.

(18.27b) (18.27~) (18.27d)

If there are only one row but several columns, we cancel one column. The same applies for the rows. a1 - xll and bl by bl - 211 and we repeat the procedure with the reduced scheme. The variables obtained in step a) are the basic variables, all the others are non-basic variables with zero values. c) We replace a] by

c=

( 58 32 21 71 ) : E: z , 9 2 6 3

1':

v, v, v3 v,

(

x=

E3

E:

x1,l

x 1 , ~ x1,3

xZ,1 x3,l

x2,2

x2,3

x2,4

x3,2

x3,3

x3,4

bl=

21,4

4 b2 = 6 b3 = 5 b4 = 7

Determination of an initial extreme point with the Northwest corner rule:

18.1 Linear Proqramminq 857

first step

x=(i' 4657

)

second step

further steps

x = ( j 5 - ) !ioo

9 5

3

2

0657

b

,

450 1 5 4 ) 1pgjo. I I3 3 0 $7

x=(

~

103 0 There are alternative methods to find an initial basic solution which also takes the transportation costs into consideration (see, e.g., the Vogel approximation method in [18.10]) and they usually result in a better initial solution. 3. Solution of the Transportation Problem with the Simplex Method If we prepare the usual simplex tableau for this problem, we get a huge tableau ( ( m+ n) x ( m . n ) ) with a large number of zeros: In each column, only two elements are equal to 1. So, we will work with a reduced tableau, and the following steps correspond to the simplex steps working only with the nonzero elements of the theoretical simplex tableau. The matrix of the cost data contains the coefficients of the objective function. The basic variables are exchanged for non-basic variables iteratively, while the corresponding elements of the cost matrix are modified in each step. The procedure is explained by an example.

0

a) Determination of the modified cost matrix C from C by (18.28a) Etj = c,,+p,+y, (i = 1,. . . ,m>j = 1, . . . ,n ) , with the conditions (18.28b) E,, = 0 for ( i , j ) if zt3is an actual basic variable. MC mark the elements of C belonging to basic variables and we substitute pl = 0. The other quantities p , and yj, also called potentials or simplex multiplicators, are determined so that the sum of p , ; qj and the marked costs c,? should be 0:

y 1 = -5 y2 = -3 b) We determine: Epq = n${ElJ}.

y3

= -2 q4 = -2

(18.28d)

If tpq2 0, then the given distribution X is optimal; otherwise zpqis chosen as a new basic variable. In our example: Epq = E32 = -2.

c) In C , we mark Epppand the costs associated to the basic variables. If C contains rows or columns with at most one marked element, then these rows or columns will be omitted. We repeat this procedure with the remaining matrix, until no further cancellation is possible. (18.28e) d) The elements ztj associated to the remaining marked elements E,, form a cycle. The new basic variable ipp is to be set to a positive value 6. The other variables Zzj associated to the marked elements E,, are determined by the constraints. In practice, we subtract and add 6 from or to every second element

858

18. Optimization

of the cycle. To keep the variables non-negative, the amount 6 must be chosen as 6 = z,,= min{x,, : 2 t j = ztj - 6}, where z,,will be the non-basic variable. In the example 6 = min{l,3} = 1. 4

(18.284

c9

5 t

5 4$6

1-6

X=

t

4.

6 +3-6

c

3

5 7 Then, we repeat this procedure with X = X but in the calculation of the new p , , qj values we always start with the original C. 4

6

(5)

(3)

q1 = -5 q2 = -3

2

q3

= -4

44

(0) 3

= -4

X=

\

l+b

-+

f

(

(0)

3 7 ) , f ( X ) = 49.

(18.281)

l

2-6)

The next matrix C does not contain any negative element. So, X is an optimal solution

18.1.4.2 Assignment Problem The representation is made by an example. In shipping contracts should be given to n shipping companies so that each company receives exactly one contract. We want to determine the assignment which minimizes the total costs, if the i-th company charges cZj for the j-th contract. An assignment problem is a special transportation problem with m = n and ai = bj = 1 for all i, j : n

OF:

n

f(x)=

(18.29a)

c,jzt3 = min! i = i 3=1

n

n

CT: c x i j = 1 ( i = l , . . . , n ) , J=1

= 1 ( j = 1 ,. . . > T I ) ,

zijE{O;l}.

(18.29b)

i=l

Every feasible distribution matrix contains exactly one 1 in every row and every column, all other elements are equal to zero. In a general transportation problem of this dimension, however, a nondegenerate basic solution would have 2n - 1 positive variables. Thus, basic feasible solutions to the assignment problem are highly degenerate, with n - 1 basic variables equal to zero. Starting with a feasible distribution matrix X,we can solve the assignment problem by the general transportation algorithm. It is time consuming to do so. However, because of the highly degenerate nature of the basic feasible solutions, the assignment problem can be solved with the highly efficient Hungarian method (see [18.6]).

18.1.4.3 Distribution Problem The problem is represented by an example.

18.2 Non-linear Optimization 859

Im products E l . E l . . . . em should be produced in quantities a l , az,. . . ,a,. Every product can be produced on any of n machines M I ,Mz,. . . , M,.The production of a unit of product E, on machine .$Ij needs processing time b,, and cost cij. The time capacity of machine M,is b,. Denote the quantity produced by machine .Lfj from product E, by x,j. We want to minimize the total production costs. K e get the following general model for this distribution problem:

OF:

m

n

1=1

,=I

f(x)=

C ~ , L ,= ~ min!

m

(18.30a) n

CT: x z L j = a l ( i = l > . . . % m ) ,x b , , x t 3 < b j ( j = l ,. . . ,n ) > x i j 2 0 forall i , j . 3=1

(18.30b)

1=1

The distribution problem is a generalization of the transportation problem and it can be solved by the simplex method. If all b,, = 1, then we can use the more effective transportation algorithm (see 18.1.4.1. p. 856) after introducing a fictitious product E,, (see 18.1.4.1, p. 856).

18.1.4.4 Travelling Salesman Suppose there are n places 01,02,.. . 0,. The travelling time from 0, to 0, is elj. Here, ctj # cji is possible. We want to determine the shortest route such that the traveller passes through every place exactly once, and returns to the starting point. Similarly to the assignment problem, we have to choose exactly one element in every row and column of the time matrix C so that the sum of the chosen elements is minimal. The difficulty of the numerical solution of this problem is the restriction that the marked elements cij should be arranged in order of the following form: (18.31) e,,,,,. czz,i3,.. . cln,i,,+] with i k # if for k # 1 and intl = il. The trayelling salesman problem can be solved by the branch and bound method. ~

~

18.1.4.5 Scheduling Problem n different products are processed on m different machines in a product-dependent order. At any time only one product can be processed on a machine. The processing time of each product on each machine is assumed to be known. Waiting times, when a given product is not in process, and machine idle times are also possible. An optimal scheduling of the processing jobs is determined where the objective function is selected as the time when all jobs are finished. or the total waiting time of jobs, or total machine idle time. Sometimes the sum of the finishing times for all jobs is chosen as the objective function when no waiting time or idle time is allowed.

18.2 Non-linear Optimization 18.2.1 Formulation ofthe Problem, Theoretical Basis 18.2.1.1 Formulation of the Problem 1. Non-linear Optimization Problem A non-linear optimization problem has the general form f(x)= min! subject to x E R" with (18.32a) gi(x) 50, i ~ I = { . l. . . ) m } ) h , ( x ) = o , j E J = { 1 .... ) r } (18.32b) where at least one of the functions f,gi, h, is non-linear. The set of feasible solutions is denoted by (18.33) .Lf = {x E Rn : gz(x)< 0, i E I , hj(x) = 0 , j E J } . The problem is to determine the minimum points.

I

860

18. Optimization

2. Minimum Points A point x* E M is called the global minimum point if

f(x*)5 f(x) holds for every x E M . If this relation holds for only the points x of a neighborhood U of x*>then x' is called a local minimum point. Since the equality constraints h,(x) = 0 can be expressed by two inequalities, -h, (x)I 0, h, (4 5 0, (18.34) the set can be supposed to be empty, J = 0. 18.2.1.2 Optimality Conditions 1. Special Directions a) The Cone of the Feasible Directions at x E M is defined by Z ( x ) = { d E R " : % > O : x + a d E M , 0 5 a < & } , EM,

(18.35) where the directions are denoted by d. If d E Z(x),then every point of the ray x + ad belongs to M for sufficient small values of a. b) A Descent Direction at a point x is a vector d E R" for which there exists an d > 0 such that f(x + ad) < f(3v1(2.E (0,Cu). (18.36) There exists no feasible descent direction at a minimum point. Iff is differentiable, then d is a descent direction V f ( ~ )<~0.dHere, V denotes the nabla operator, so Vf(x) represents the gradient of the scalar-valued function f at x.

2. Necessary Optimality Conditions I f f is differentiable and x' is a local minimum point, then ~ f ( x * )2~o d for every d E Z(x*). (18.37a) In particular. if x*is an interior point of M ,then Vf(x*= ) 0. (18.37b) 3. Lagrange Function and Saddle Point Optimality conditions (18.37a,b) should be transformed into a more practical form including the constraints. We construct the so-called Lagrange function or Lagrangian: m

L(x:u) = f(x)-t C u l g i ( x )= f(3 +UTg(3,

x E R, u E RT,

(18.38)

2=1

according to the Lagrange multiplier method (see 6.2.5.6, p. 401) for problems with equality constraints. A point (x',u') E R" x RT is called a saddle point of L , if L ( ~ * , u5) L(x':u*)5 L ( ~ , u * )for every E R", y E R;". (18.39)

4. Global Kuhn-Tucker Conditions .4 point x* E R" satisfies the global Kuhn-Tucker conditions if there is an U' E RT, i.e., u' 2 0 such that (x'.u') is a saddle point of L. For the proof of the Kuhn-Tucker conditions see 12.5.6: p. 623. 5 . Sufficient Optimality Condition If (x*, u*)E Rn x RT is a saddle point of L: then E' is a global minimum point of (18.32a,b). If the functions f and gz are differentiable, then we can also deduce local optimality conditions. 6 . Local Kuhn-Tucker Conditions .A point x* E M satisfies the local Kuhn-Tucker conditions if there are numbers ui 2 0, i E la(x*)such that -Vj(x') = u,Vgi(x*), where (18.40a) iE Io(?(' )

1,(x)= { i E { 1, . . . , m } : gt (x)= 0) is the index set of the active constraints at x. The point point.

x'

(18.40b) is also called a Kuhn-Tucker stationary

18.2 Non-linear Optimization 861 This means geometrically that a point x' E 'kf satisfies the local Kuhn-Tucker conditions. if the negative gradient -Vf(x') lies in the cone spanned by the gradients Vgi(x*) i E lo(x*)of the constraints active at x* (Fig. 18.5). The following equivalent formulation for (18.40a,b)is also often used: x* E R" satisfies the local Kuhn-Tucker conditions) if there is a u* E R;" such that dx*)5 0, (18.41a)

uZgt(x*) = 0, i = 1 , .. .,m,

g2(x)

(18.41b)

+ x u t V g t ( x * )= 0.

, /

/

,

level lines

f(x) = const

m

Vf(x*)

I , 0 /

(18.41~)

t=l

Figure 18.5

7. Necessary Optimality Conditions and Kuhn-Tucker Conditions If x' E ,21is a local minimum point of (18.32a.b) and the feasible set satisfies the regularity condition at x* : 3d E R" such that Vgz(x*)Td< 0 for every i E lo(x*), then x*satisfies the local Kuhn-Tucker conditions,

18.2.1.3 Duality in Optimization 1. Dual Problem With the associated Lagrangian (18.38) we form the maximum problem, the so-called dualof (18.32a,b):

L(s.u)= max! subject to ( x , ~E )M* with .21* = {(x,u)E R" x R;" : L(x,u)= min L(z,u)} ZER"

(18.42a) (18.42b)

2. Duality Theorems

(x2,uz)E M'. then a) L(x2
(18.42a,b).

18.2.2 Special Non-linear Optimization Problems 18.2.2.1 Convex Optimization 1. Convex Problem The optimization problem f(x)= min! subject to gz(x) 5 0 (i = 1:.. . , m) (18.43) is called a convex problem if the functions f and gi are convex. In particular. f and gt can be linear functions. The following statements are valid for convex problems: a ) Every local minimum o f f over M is also a global minimum. b) If JI is not empty and bounded, then there exists at least one solution of (18.43). c) I f f is strictly convex. then there is at most one solution of (18.43). 1. Optimality Conditions a) I f f has continuous partial derivatives, then x* E M is a solution of (18.43))if

(x- z * ) ~ v ~ ( z *2 )o for every x E M . (18.44) b) The Slater condition is a regularity condition for the feasible set ,tI. It is satisfied if there exists an x E A1 such that gi(x) < 0 for every non-affine linear functions gi.

I

862

18. Optimization

x' is a minimum point of (18.43) if and only if there exists a u* 2 0 such that (x*, u*)is a saddle point of the Lagrangian. Moreover, if functions f and gi are differentiable, then s*is a solution of (18.43) if and only ifx* satisfies the local Kuhn-Tucker conditions. d) The dual problem (18.42a,b) can be formulated easily for a convex optimization problem with differentiable functions f and gt: L(x,u)= max!, subject to (x,g)E M* with (18.45a) (18.45b) M' = {(x,p)E R" x RT : V,L(x,u) = Q}. The gradient of L is calculated here only with respect to 11. e) For convex optimization problems, the strong duality theorem also holds: If M satisfies the Slater condition and ifx' E M is a solution of (18.43), then there exists a 11' E RT; such that ( x * , g *is) a solution of the dual problem (18.45a,b), and f(x*)= minf(Z) = max L(x;u)= L(x*,u*). (18.46) c ) If the Slater condition is satisfied, then

ZE M

(X,U)€M'

18.2.2.2 Quadratic Optimization 1. Formulation of the Problem Quadratic optimization problems have the form f(x) = xTCx+ pTx= min! , subject to E hil c R" with M = .WT: M = {x E R" : Ax 5 b. x 2 Q}. Here, C is a symmetric ( n ,n) matrix, p E Rn, A is an (m,n) matrix, and b E R". The feasible set M can be written alternatively in the following way: '2il = .\ilI, : M = {x : Ax = b, x 2 Q}, M = A V l ~ ~ : = {x : Ax 5 b}. db'

2.

(18.47a) (18.47b)

(18.48a) ( 18.48b)

Lagrangian and Kuhn-Tucker Conditions

The Lagrangian to the problem (18.47a,b) is

L(x,u)= x T C x + p T x + g T ( A x - b ) . By introducing the notation dL

+

(18.49)

dL

v = - = p 2Cx+ ATy and y = -- = - A x + b ax du the Kuhn-Tucker conditions are as follows:

(18.50)

Case 11:

Case I:

Case 111:

a) A x + y = b ,

a) A x = b ,

a) A x + y = b .

b) 2Cx - 1 + ATu = -p,

b) 2Cx - y + ATu = -p, -

b) 2Cx + ATq = -p. (18.5lb)

c) x 2 0 . Y 2 0 , y 2 0 , u 2 0 , c) x20,Y 2 0 , d)

xTx+ y'u

= 0.

d) xTy= 0.

c)

112

0, y 2 0,

d) z T g = 0.

(18.5la)

(18.51~)

( 18.5Id)

3. Convexity The function f(x)is convex (strictly convex) if and only if the matrix C is positive semidefinite (positive definite). Every result on convex optimization problems can be used for quadratic problems with a positive semidefinite matrix C; in particular, the Slater condition always holds, so it is necessary and sufficient for the optimality of a point x*that there exists a point (x*, y, p,y),which satisfies the corresponding system of local Kuhn-Tucker conditions.

18.2 Non-linear Optimization 863

4. Dual Problem If C is positive definite, then the dual problem (18.45a) of (18.47a) can be expressed explicitly: L(2i.g) = max!. subject to ( x , ~E )M*, where (18.52a)

M" = {(x.u)E R" x RI; : x = - k - ' ( A T ~ + p ) } .

(18.52b)

2

1

If we substitute the expression x = --C-'(ATg 2 we get the equivalent problem 1

+ p) into the dual objective function L(x.u),then

+ blTp- a p ~ c --l p= max!,

q ( y )= - - ~ T A C - ~ A-T ~

4-

g2

0.

(18.53)

Hence: If x* E M is a solution of (18.47a,b),then (18.53) has a solution g' 2 0, and

f(x*)= du*).

(18.54)

Problem (18.53) can be replaced by an equivalent formulation: ~ ( g=)uTEu

+ hTu= min! ,

1

E = -AC-'AT 4

and

subject to g 2 0 where

(18.55a)

1

h = -AC-'p+b.

(18.55b)

2

18.2.3 Solution Methods for Quadratic Optimization Problems 18.2.3.1 Wolfe's Method 1. Formulation of the Problem and Solution Principle The method of Wolfe is to solve quadratic problems of the special form: f ( x )= xTCx+ eTx= min! subject to Ax = b! x 2 0. ~

'

(18.56)

We suppose that C is positive definite. The basic idea is the determination of a solution (x*, u*,y') of the corresponding system of Kuhn-Tucker conditions, associated to problem (18.56): Ax = b, (18.57a)

2Cx - 41 + ATg = -p, -

(18.57b)

x 2 0, v20;

(18.57~)

0. (18.58) Relations (18.57a,b,c)represent a linear equation system with m + n equations and 2n + m variables. i = 0 or ut = 0 (i = 1,2, . . . , n) must hold. Therefore, every solution Because of relation (18.581,either z of (18.57a.b:c), (18.58) contains at most m + n non-zero components. Hence, it must be a basic solution of (18.57a.b,c). xT41 =

2. Solution Process First, we determine a feasible basic solution (vertex) g of the system Ax = b. The indices belonging to the basis variables of form the set IB. In order to find a solution of system (18.57a,b,c),which also satisfies (18.58).we formulate the problem - p = min!, ( p E R); (18.59)

Ax = b,

(18.60a)

2 C x - y + A T g - p q-= -p- with q=2Cz+p,

x 2 0, 41 2 0, XTq: = 0.

p20;

(18.60b) (18.60~) (18.61)

I

864

18. Optimization

If (x,v>u> p ) is a solution of this problem also satisfying (18.57a,b,c) and (18.58), then p = 0. The vector ( ~ , v , u $ p=) (x,Q,Q, 1) is a known feasible solution of the system (18.60a,b,c), and it satisfies the relation (18.61), too. We form a basis associated to this basic solution from the columns of the coefficient matrix A 0 I denotes the unit matrix, 0 the zero matrix and 0 (18.62) 2 c -1 AT -g ' is the zero vector of the corresponding dimension,

0

in the following way: a) m columns belonging to x, with z E I B , b) n - m columns belonging to u, with z $ I B , c) all m columns belonging to u,, d) the last column, but then a suitable column determined in b) or c) will be dropped. If q = 0. then the interchange according to d) is not possible. Then g is already a solution. Now, we can construct a first simplex tableau. The minimization of the objective function is performed by the simplex method with an additional rule that guarantees that the relation sTy= 0 is satisfied: The variables x, and u, (i = 1 , 2 , .. . , n) must not be simultaneously basic variables. Thesimplexmethodprovidesasolutionofproblem (18.59),(18.60a,b,c),(18.61)withp = Ofor positive definite C considering this additional rule. For a positive semidefinite matrix C, based on the restricted pivot choice, it may happen that although p > 0, no more exchange-step can be made without violating the additional rules. We can show that in this case p cannot be reduced any further. If(x)= x: + 42; - loxl - 32x2 = min! with 21 + 2x2 2 3 = 7, 2x1 x2 2 4 = 8.

+

+ +

In this case C is positive semidefinite. A feasible basic solution of Ax = Ir is % = (0,0,7, 8)T, g = 2CZ + p = (-10, -32,0, O)T. We choose the basis vectors: a) columns 3 and 4 of

, b) columns 1 and 2 of

(

column Jg) instead of the first column of

( -4).

The basis matrix is formed from these columns, and the basis inverse is calculated (see 18.1, p. 844). Multiplying matrix (18.62) and the vectors

and d)

T :()

(-3 ~I

by the basis inverse, we get the first simplex tableau (Scheme 18.9). Only z1 can be interchanged with u2 in this tableau according to the complementary constraints. After a few steps, we get the solution x* = (2.5/2.0, 3/2)T. The last two equations of 2Cx v + ATu- pg = -pare: v3 = u1, u4 = u2. Therefore, by eleminating u1 and u2 the dimension of the problem can be reduced.

Scheme 18.9 21

x2

Ul

213

214

1 2

2 1

0 0 32

0 0 12

0 0 54

0 0 2 10

0 0

o--

0 0 - 1 1 2 10 10

0 - 1 0 1 10

7 8

0 0 0

1 -1

18.2 Non-linear Optimization 865

18.2.3.2 Hildreth-d'Esopo Method 1. Principle The strictly convex optimization problem f(x) = xTCxt pTx = min!, Ax 5 b has the dual problem (see l.,p. 862)

(18.63)

+

hTu = min! u 2 0 with 1 1 E = -AC-'AT, h = -AC-'p + b. 4 2 Matrix E is positive definite and it has positive diagonal elements e,, variables x and 11satisfy the following relation: 1 x = --C-'(ATu+p). 2 y(u) = uTEu

(18.64a) (18.64b)

> 0, (z

= 1 , 2 , .. m). The t ,

(18.65)

2. Solution by Iteration

The dual problem (18.64a), which contains only the condition u 2 0, can be solved by the following simple iteration method: a) Substitute u' 2 0, (e g., u' = Qjl k = 1. b) Calculate uf" for z = 1 , 2 , . . . , m according to

c) Repeat step b) with k + 1 instead of k until a stopping rule is satisfied, e.g., l y ( ~ ~ + - 'y)( a k ) l< E . E > O

Under the assumption that there is an x such that Ax < b, the sequence {$(uk)} converges to the minimum value pmlnand sequence { x k } given by (18.65) converges to the solution x* of the original problem. The sequence {uk}is not always convergent.

18.2.4 Numerical Search Procedures By using non-linear optimization procedures we can find acceptable approximate solutions with reasonable computing costs for several types of optimization problems. They are based on the principle of comparison of function values.

18.2.4.1 One-Dimensional Search Several optimization methods contain the subproblem of finding the minimum of a real function f(x) for I E [a. b]. It is often sufficient to find an approximation zof the minimum point x'.

1. Formulation of the Problem A function f(z),x E R, is called unimodal in [a,b] if it has exactly one local minimum point on every closed subinterval J [a,b]. Let f be a unimodal function on [a,b] and I * the global minimum point. Then we have to find an interval [c, d] C [a,b] with z* E [e,d] such that d - c < E , E > 0. 2. Uniform Search b-a E LVe choose a positive integer n such that 6 = -< -, and we calculate the values f ( x k )for xk = n t l 2 a + k6 ( k = 1 , .. . . n). If f(x) is the smallest value among these function values. then the minimum point z*is in the interval [x- 6, I t 61. The number of required function values for the given accuracy can be estimated by 2(b - a) (18.67) n>-1. E

866

18. Optimization

3. Golden Section Method, Fibonacci Method The interval [a.b] = [al.b l ] will be reduced step by step so that the new subinterval always contains the minimum point x*. m'e determine the points XI, p1 in the interval [al, b l ] as (18.68a) XI = al t (1 - r)(bl - al), p1 = ai t r(bl - al) with 1 (18.68 b) T = 2 (6 - 1) zz 0.618. This corresponds to the golden section. We distinguish between two cases: a) f ( A 1 ) < f(p1): We substitute a2 = al, bz = p1 and pz = XI. (18.69a) b) f ( A l ) 2 f ( p l ) : We substitute a2 = XI, bz = bl and XZ = p ~ . (18.69b) If b2 - a2 2 E , then we repeat the procedure with the interval [az,bz],where one value is already known, f(&) in case a) and f(pg) in case b), from the first step. To determine an interval [a,: b,], which contains the minimum point x*,we must calculate n function values altogether. From the requirement (18.70) E > bn - a, = T ' ( b 1 - ai) we can estimate the necessary number of steps n. By using the golden section method, at most one more function value should be determined compared to the Fibonacci method. Instead ofsubdividing the interval according to the golden section, we subdivide the interval according to the Fibonacci numbers (see 5.4.1.5, p. 323, and 17.3.2.4, 4., p. 843).

18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space The search for an approximation of the minimum point x* of the problem f(x) = min!, x E R", can be reduced to the solution of a sequence of one-dimensional optimization problems. We take (18.71a) a) x = xl, k = 1: where x' is an appropriate initial approximation of x*. b) h'e solve the one-dimensional problems (18.71b) ~(a,) = f(~!+'~ . . ., x;?;) zf +a,, x:+', . . . ,E:) = min! with a, E R for T = 1.2,. . . . n. If c& is an exact or approximating minimum point of the r-th problem, then we substitute E:+' = E," t &. c) If two consecutive approximations are close enough to each other, Le., with some vector norm,

llxk+' - xk/I < €1 or If(xk+')- f(xk)I< € 2 , (18.71~) then xkL' is an approximation of x*.Otherwise we repeat step b) with k + 1 instead of k . The onedimensional problem in b) can be solved, by using the methods given in 18.2.4.1, p. 865.

18.2.5 Methods for Unconstrained Problems The general optimization problem f(x)= min! for 11E R" (18.72) is considered with a continuously differentiable function f. Each method described in this section constructs, in general, an infinite sequence of points { x k } E R", whose accumulation point is a stationary point. The sequence of points will be determined starting with a point x' E R" and according to the formula X k ~l xk + a d k ( k = L 2 , . . .), (18.73) -

I

Le., we first determine a direction dk E R" at xk and by the step size ak E R we indicate how far xktl is from xk in the direction dk.Such a method is called a descent method, if f(Z'"+') < f(xk) ( k = 1 , 2 . . . .). (18.74)

18.2 Non-linear Optimization 867

The equality Vf(x) = 0. where V is the nablaoperator (see 13.2.6.1, p. 664). characterizes a stationary point and can be used as a stopping rule for the iteration method.

18.2.5.1 Method of Steepest Descent Starting from an actual point xk,the direction dk in which the function has its steepest descent is dk =

-Vf(xk)

(18.75a)

and consequently xkt' = xk - cykVf(xk).

(18.75b)

.A schematic representation of the steepest descent method with level lines f(x)= f(x')is shown in Fig. 18.6. The step size q is determined by a line search, Le., cXI; is the solution of the onedimensional problem ')

Figure 18.6 Xk-l = xk t ffk&

f ( x k+ adk)= min! cy 2 0. (18.76) This problem can be solved by the methods given in 18.2.4, p. 865. The steepest descent method (18.75b) converges relatively slowly. For every accumulation point x' of the sequence { x k } , Vf(x*) = 0. In the case of a quadratic objective function, Le., f(x)= xTCxt pTx, the method has the special form:

(18.77a) with dk = -(2@

I

+p) and

(Yk

= dkTdk

(18.77b)

2dkTCdk'

18.2.5.2 Application of the Newton Method Suppose we approximate the function f at the actual approximation point xk by a quadratic function: 1 (18.78) + (X- x k ) ' V f ( x k ) -(x - xk)'H(xk)(x- x~). q(x) = 2Here H(xk)is the Hessian matrix, Le., the matrix of second partial derivatives off at the point x k . If H(xk)is positive definite, then q ( x ) has an absolute minimum at xkt'with Vq(ykt') = 0, therefore we get the Newton method: xktl = xk - H-'(xk)Vf(xk) ( k = 4 2 , . , .), Le., (18.79a)

+

dk = - H - ' ( x ~ ) v ~ ( ~ and ' ) ak in (18.73). (18.79b) The Sewton method converges quickly but it has the following disadvantages: a) The matrix H(xk)must be positive definite. b) The method converges only for sufficiently good initial points. c ) \Ye cannot influence the step size. d) The method is not a descent method. e) The computational cost of computing the inverse of H-'(xk) is fairly high. Some of these disadvantages can be reduced by the following version of the damped Newton method: (18.80) xktl = zk- ~ y k H - * ( x ' ) V f ( ~ ~( k) = 1 , 2 , .. .) . The relaxation factor ah can be determined, for example, by the principle given earlier (see 18.2.6.1, p. 867).

18.2.5.3 Conjugate Gradient Met hods Two vectors d'.rJz E R" are called conjugate vectors with respect to a symmetric, positive definite matrix C. if dlTCdZ= 0.

(18.81)

I

868

18. Optimization

If d'. dz.,, . ,dn are pairwise conjugate vectors with respect to a matrix C, then the convex quadratic problem q ( ~ =) xTCx + pTx, x E Rn,can be solved in n steps if we construct a sequence xk+'= xk+akdkstarting from x', where (Yk is the optimal step size. Under the assumption that f(x)is approx1 imately quadratic in the neighborhood of x', Le., C M -H(x*),the method developed for quadratic 2 objective functions can also be applied for more general functions f(x), without the explicit use of the matrix H(x*). The conjugate gradient method has the following steps: a) x' E R". d' = -Of($), (18.82) where X' is an appropriate initial approximation for x*. b) xktl = xk + (Ykdk ( k = 1 , .. . n) with ak 2 0 so that f(xk+ adk)will be minimized. (18.83a) ~

dk+' = -Vf(xktl) + pkdk ( k = 1,.. . , n - 1) with

(18.83b) (18.83~)

c) Repeating steps b) with zn+l and #+'instead of E' and dl.

18.2.5.4 Method of Davidon, Fletcher and Powell (DFP) LVith the DFP method, we determine a sequence of points starting from x' E Rnaccording to the formula -x k - akMkVf(xk) ( k = 1 , 2 , . . .). (18.84) Here, Mk is a symmetric, positive definite matrix. The idea of the method is a stepwise approximation of the inverse Hessian matrix by matrices Mk in the case when f(x)is a quadratic function. Starting with a symmetric, positive definite matrix M1,e.g., M1 = I (I is the unit matrix), the matrix Mk is determined from Mk-1 by adding a correction matrix of rank two (18.85) - of(&') ( k = 2,3.. . .). w e get the step size ( Y k from with yk = xk - xk-' and y k = vf(xk) f(xk- cuMkVf(xk)))= min!, (Y 2 0. (18.86) If f(x)is a quadratic function, then the DFP method becomes the conjugate gradient method with Mi = I.

18.2.6 Gradient Method for Problems with Inequality

Type Constraints If the problem f($ = min! subject to the constraints gz(x)5 0 (z = 1,. . . , m ) (18.87) has to be solved by an iteration method of the type (18.88) Xk-l = xk f(Ykdk (k = 1 ~ 2 , . . . ) then we have to consider two additional rules because of the bounded feasible set: 1. The direction dk must be a feasible descent direction at xk. 2. The step size (Yk must be determined so that xk+' is in M. The different methods based on the formula (18.88) differ from each other only in the construction of the direction dk. To ensure the feasibility of the sequence {xk}C M, we determine a; and ai in the following nay:

18.2 N o n - h e a r Optimization 869

+

a; from f ( x k odk)= min! (Y 2 o ai = max{a E R : xk t adk E M}. ~

(18.89)

Then ok = min{a;.a;}.

(18.90)

If there is no feasible descent direction dkin a certain step k , then xk is a stationary point.

18.2.6.1 Method of Feasible Directions 1 . Direction Search Program A feasible descent direction dk at point xk can be determined by the solution of the following optimization problem: u = min!.

(18.91)

If u < 0 for the result d = dk of this direction search program, then (18.92a) ensures feasibility and (18.92b) ensures the descending property of dk. The feasible set for the direction search program is bounded by the normalizing condition (18.92~).If u = 0. then xk is a stationary point, since there is no feasible descent direction at xk. A direction search program. defined by (18.92a,b,c),can result in a zig-zag behavior of the sequence x k . which can be avoided if the index set 10(xk)is replaced by the index set which are the so-called E k active constraints in sk.Thus, we exclude local directions of descent which are going from xk and lead closer to the boundaries of M consisting of the &k active constraints (Fig. 18.7).

Figure 18.7 If

D

= 0 is a solution of (18.92a,b,c) after these modifications, then

xk is a stationary

point only if

(xk).Otherwise E k must be decreased and the direction search program must be repeated.

= IEk

f l ;@

-Vf(xk)

4 Figure 18.8

2. Special Case of Linear Constraints If the functions gz(x)are linear. Le., g z ( r ) = azTx- b,, then we can establish a simpler direction search method: = V f ( ~ ~=)min! ~ d with (18.94)

atTd lldll 51. 5 0,

z E 10(xk)or z E l E k ( x k )(18.95b) (18.95a) ,

= The effect of the choice of different norms J/d/l

max{ld,l} 5 1 or Fig. 18.8a,b.

lldll = @ 5 1 is shown in

870

18. Optimization

In a certain sense. the best choice is the norm lldll = lldiiz = @d, since by the direction search program we get the direction dk,which forms the smallest angle with - V f ( x k ) .In this case the direction search program is not linear and requires higher computational costs. With the choice 1 /dl1= 1 ldIlrn= max{ I d J } 5 1 we get a system of linear constraints -1 I d, I 1, (i = 1,.. . n ) ,so the direction search program can be solved. e.g.. by the simplex method. In order to ensure that the method of feasible directions for a quadratic optimization problem f(x)= x T C+~pTx = min! with Ax I b results in a solution in finitely many steps, the direction search program is completed by the following conjugate requirements: If ak-1 = holds in a step, Le.. xk is an "interior" point, then we add the condition ~

dk-IT Cd=O (18.96) to the direction search program. Furthermore we keep the corresponding conditions from the previous steps. If in a later step ah = ai we remove the condition (18.96). I f ( x ) = .r: + 42; - lox1 - 32xz = min! SI($ = -21 I 0, gz(x) = -22 I 0, g3(x) = 51 2x2 - 7 5 0, g4(x) = 2x1 22 - 8 5 0.

+

+

Step 1: Startingwith

x1 = (3.0)T, of@)= (-4. -4dl - 32d2 = min!

Direction search program:

-dz

IO. lldll, I 1

-32)T,

] =+ d'

lo(x') = (2). = (1,I)T.

: for z such that

hlaximal feasible step size. ai = min

2 ==+ 3

atTdk> 0 ,ai = $. a;' = 1

3

, 182 2 11 2 al=mln - - = - xz=(--) (5.31 3'3'3

Directionsearch program: 4 a2 = -

3

I

113

{

X2t

= (3.2)T

Step 3: V f ( x 3 )= (-4, -16)T, Direction search program:

(-I>

;IT,

152

- :dz = 2dl +dz 5 0. IIdIl, 5 1 -:dl

a$ = 1, a; = 3

IO = ($)

= {3,4}

23

=+a3 = 1, x4 =

The next direction search program results in the minimum point is x*= x4 (Fig. 18.9).

0 =

>:IT

.

(0. Here 2 -

1

0

b

* 2 1x

1 2 3 Figure 18.9

d 4x1

18.2.6.2 Gradient Projection Method 1. Formulation of the Problem and Solution Principle Suppose the convex optimization problem f($ = min! with a,T1lI b,,

(18.97)

18.2 Non-linear Optimization 871

for i = 1,, , , , m is given. A feasible descent direction dk at the point xk E M is determined in the following way: If -Of($) is a feasible direction, then dk = -Vf(xk) is selected. Otherwise xk is on the boundary of M and - V l ( x k ) points outward from M. The vector - V j ( x k ) is projected by a linear mapping Pk into a linear submanifold of the boundary of M defined by a subset of active constraints of xk. Fig. 18.10a shows a projection into an edge, Fig. 18.10b shows a projection into a face. Supposing non-degeneracy, Le., if the vectors a,, i E 1 0 ( ~ are ) linearly independent for every x E R", such a projection is given by

dk = -Pkvf(xk)= - (I - AkT(AkAkT)-'Ak)Of($).

(18.98)

Here. Ak consisrs of all vectors L ~whose , corresponding constraints form the submanifold, into which - V f ( x k )should be projected.

Figure 18.10

2. Algorithm The gradient projection method consists of the following steps. starting with x' E M and substituting k = 1 and proceeding in accordance to the following scheme: I: If - V f ( x k )is a feasible direction, then we substitute dk = - V f ( x k ) , and we continue with 111. Otherwise we construct Ak from the vectors atTwith z E l o ( x k )and we continue with 11.

11: R'e substitute dk = - (I - AkT(AkAkT)-*Ak)Vf(rk)).If dk # 0. then we continue with 111. If

dk = 0 and p = -(AkAhT)-'AkVf(xk)2 0. then zkis a minimum point. The local Kuhn-Tucker conditions -Of($) = C utat = AkTu are obviously satisfied

-

%EIo (x_k )

2: 0, then we choose an i with u, < 0, delete the i-th row from Ak and repeat 11. 111: Calculation of ak and xk+l = xk + Qkdkand returning to I with k = k + 1. If

3. Remarks on the Algorithm If -Of@) is not feasible. then this vector is mapped onto the submanifold of the smallest dimension which contains sk.If dk = 0, then - V f ( z k ) is perpendicular to this submanifold. If u 2 0 does not hold: then the dimension of the submanifold is increased by one by omitting one of the active constraints, somaybedk # Qcanoccur (Fig. 18.10b) (with projection onto a (latera1) face). Since Ak is often obtained from Ak-1 by adding or erasing a row. the calculation of (AkAkT)-'can be simplified by the use of (Ak-1Ak-lT)-l. Solution of the problem of the previous example on p. 870. Step 1: =(3.0)~. I: O f ( x ' ) = (-4, -32)T, -Vf(xl) is feasible, d' = (4, 32)T.

111: The step size is determined as in the previous example: a1 = Step 2:

1

16 8

I

872

18. OPtimization

I: Of(xz)=

18 96 [-T, -T) .

.

..

(not feasible), I&')

= {4}, A2 = (2 1).

8 16 111:

cy2

=

g5 , x3 = (3.2)T

Step 3: I: Of@) = (-4, -16)* (not feasible), I&)

,:(

= {3,4}, A3 = T ) : -

(.1

2 1), .

uz < 0 : A3 = (1 2).

16 8

111:

5 x4 = (2: 16' -

cy3 = -

%)'

Step 4:

I: Of($) = (-6, -12)T (not feasible), 10(x4) = {3}, A4 = A3. 11: P4 = P3. d4 = (0,O)T, u = 6 2 0. It follows that x4 is a minimum point.

18.2.7 Penalty Function and Barrier Methods The basic principle of these methods is that a constrained optimization problem is transformed into a sequence of optimization problems without constraints by modifying the objective function. The modified problem can be solved, e.g., by the methods given in Section 18.2.5. With an appropriate construction of the modified objective functions, every accumulation point of the sequence of the solution points of this modified problem is a solution of the original problem.

18.2.7.1 Penalty Function Method The problem

f(x)= min! subject to gz(x)5 0 (i = 1,2,. . . , m ) is replaced by the sequence of unconstrained problems H(&,pk) = !(E) + p k S ( x ) = min! with x E R", p k > 0 (k = 1 , 2 , . . .). Here, pk is a positive parameter, and for s(x)

(18.99) (18.100)

(18.101) . problem (18.100) holds, i.e., leaving the feasible set M is punished with a "penalty" term p k S ( ~ )The is solved with a sequence of penalty parameters pk tending to co.Then lim H ( x , p k ) = ! ( E ) , x E M. (18.102) kim

If xk is the solution of the Ic-th penalty problem, then: H ( d , p k ) 2 H(Xk-',pk-l), f(Xk)2 f(Xk-')9 and every accumulation point x' of the sequence {xk}is a solution of (18.99). If xk E solution of the original problem. For instance, the following functions are appropriate realizations of S(x): S(x)= maxr{0,g1(x)>.. . ,gm(x)} ( T = 1 , 2 , . . .) or

(18.103)

M ,then xk is a (18.104a)

18.2 Non-linear Optimization 873 m

maxP{O,gz(x)}( r = 1,2,.. .).

S(x)=

(18.104b)

1=1

If functions f(x)and gz(x)are differentiable, then in the case r > 1, the penalty function H(z,Pk) is also differentiable on the boundary of M, so analytic solutions can be used to solve the auxiliary problem (18.100). Fig. 18.11shows a representation of the penalty function method.

Figure 18.11

Figure 18.12

+

W f(x) = xf xi = min! for 51 + 2 2 2 1, ff(x,pk) = 2: The necessary optimality condition is: 221 - 2pkmax{0,1- 2 1 - Q}

+ + pk max2{0, 1 - zl - x2}. 2:

The gradient of H is evaluated here with respect to x. By subtracting the equations we have x1 = z2. The equation 221 - 2pk max(0.1

-

221} = 0 has a unique solution 2: = x; =

Pk 1 solution 2; = x; = lim = - by letting k kiW1$2pk 2 ~

A. We get the 1 f 2Pk

-+ co.

18.2.7.2 Barrier Method We consider a sequence of modified problems in the form

H(& qk) = f ( x ) + q&B(x)= 3 qk > 0 . (18.105) The term q k 8 ( x ) prevents the solution leaving the feasible set M ,since the objective function increases unboundedly on approaching the boundary of Ilf. The regularzty condition M ' = { ~ E M: g2(x)<0( z = 1 , 2 m ) } # 0 and @ = M must be satisfied, Le.. the interior of A4 must be non-empty and it is possible to get to any boundary (18'106) point by approaching it from the interior, Le., the closure of M o is M. The function B ( x ) is defined to be continuous on M'. It increases to co at the boundary of M.The modified problem (18.105) is solved by a sequence of barrier parameters q k tending to zero. For the solution xk of the k-th problem (18.105) holds ) . . . ,

f(xk)I f(xk-'), and every accumulation point x' of the sequence {xk}is a solution of (18.99). Fig. 18.12shows a representation of the barrier method.

(18.107)

I

874

18. Optimization

The functions. e.g.. m

(18.108a) (18.108b) are appropriate realizations of B ( x ) . Ij(x) = 2: 2; = min! subject to x1 x2 2 1, H(x,qk) = 1

+

21

t 2 2 > 1. VH(X. qk) =

with respect to x.

[

+

yz ')= (:)

- qkxl - 2x2 - q k 51 t 5 2 - 1 221

Subtracting the equations results in x1 = 2 2 ,

221

- qk-

,

21

2:

+ xi + qk(-

In(z1 t 2 2 - I ) ) ,

+ 2 2 > 1. The gradient of H is given

1 = 0, 2x1 - 1

21

>1 -)

2

*

2;

21 qk - -- - 2 4

The solutions of problems (18.100) and (18.105) at the k-th step do not depend on the solutions of the previous steps. The application of higher penalty or smaller barrier parameters often leads to convergence problems with numerical solution of (18.100) and (18.105), e.g., in the method of (18.2.4), in particular, if we do not have any good initial approximation. Using the result of the k-th problem as the initial solution for the (k + 1)-th problem we can improve the convergence behavior.

18.2.8 Cutting Plane Methods 1. Formulation of the Problem and Principle of Solution We consider the problem f(x)= cTx= min!, c E R" (18.109) over the bounded region M c R" given by convex functions gi(X) (i = 1 , 2 , . . . , m) in the form Si(&) 5 0. A problem with a non-linear but convex objective function j(x)is transformed into this form, if (18.110) f ( ~ -) 2nt1 5 0, 2 n + 1 E R is considered as a further constraint and (18.111) j(x)= xnt1 = min! for all = (x,zn+l) E Rnil is solved with gi(x) = g i b ) 5 0. The basic idea of the method is the iterative linear approximation of M by a convex polyhedron in the neighborhood of the minimum point x*,and therefore the original program is reduced to a sequence of linear programming problems. First, we determine a polyhedron (18.112) PI = {X E R" : atTz5 b,, i = 1 , .. . , s}. From the linear program j(x)= min! with X E P1 ( 18.113) an optimal extreme point x1of Pi is determined with respect to j b ) . If&' E M holds, then the optimal solution of the original problem is found. Otherwise, we determine a hyperplane H1 = {a : & + l T ~= b,tl, &tlTxl > bstl}, which separates the point 21' from M ,so the new polyhedron contains (18.114) pz = {X E 9 : as+iTX5 h+l}.

x

18.3 Discrete Dynamic Programming 875

Fig. 18.13 shows a schematic representation of the cutting plane method. V f(X)

Figure 18.13

2. Kelley Met hod The different methods differ from each other in the choice of the separating planes Hk. In the case of the Kelley method Hk is chosen in the following way: A jk is chosen so that gJk(xk)=mm{g2(xk) ( i = 1, . . . ,m ) } . (18.115) At the point x = xk,the function gjk(x) has the tangent plane

T(x)= gjk(Xk)+ (X- x ~ ) ) ' V Y ~ , ( X ~ )(18.116) . The hyperplane Hk = {X E R" : T ( x )= 0) separates the point zkfrom all points x with y J x(x)5 0. So, for the ( k + 1)-th linear program, T ( x )5 0 is added as a further constraint. Every accumulation point x*of the sequence {g}is a minimum point of the original problem. In practical applications this method shows a low speed of convergence. Furthermore, the number of constraints is always increasing.

18.3 Discrete Dynamic Programming 18.3.1 Discrete Dynamic Decision Models A wide class of optimization problems can be solved by the methods of dynamic programming. The problem is considered as a process proceeding naturally or formally in time, and it is controlled by time-dependent decisions. If the process can be decomposed into a finite or countably infinite number of steps, then n e talk about discrete dynamic programming,otherwise about continuous dynamic programming. In this section, we discuss only n-stage discrete processes.

18.3.1.1 n-Stage Decision Processes An n-stage process P starts at stage 0 with an initial s t a t e s = % and proceeds through the intermediate states xl. xz,. . . , z ~ into - ~ a final state xn = & E X , C Rm.The state vectors xj are in the state space X , R". To drive a state x ~ into - ~the state xj,a decision uj is required. All possible decision vectors IJ, in the state xj-lform the decision space U ~ ( X ~ -R'. ~ )From xj-lwe get the consecutive state xj by the transformation (Fig. 18.14)

x, = Y,(x,-bIJJ

(18.117 )

j = l(1)n.

Un(Xn.1)

Figure 18.14

18.3.1.2 Dynamic Programming Problem Our goal is to determine a polzcy (nl,.. . ,un)which drives the process from the initial state into the state considering all constraints so that it minimizes an objective function or cost functzon f ( f l ( % . g l.). . .fn(xn-lsgn)). The functions .fj(xj-],uj) are called stage costs. The standard form of the dynamic programming problem is

876

18. Optimization

( 18.118b) 1

.

, -

The first type of constraints ~r, are called dynamzc and the others s,uj are called s t a t ~ Similarly . to (18.118a).a maximum problem can also be considered. A policy (&,. . . ,un)satisfying all constraints is called jeaszble. The methods of dynamic programming can be applied if the objective function satisfies certain additional requirements (see 18.3.3,p. 876).

18.3.2 Examples of Discrete Decision Models 18.3.2.1 Purchasing Problem In the j-th period of a time interval which can be divided into n periods, a workshop needs v j units of a certain primary material. The available amount of this material at the beginning of period j is denoted by zj-], in particular, 20 = 2 , is given. We have to determine the amounts u j , for a unit price cJ,which should be purchased by the workshop at the end of each period. The given storage capacity K must not be exceeded, Le., t u, 5 K . We have to determine a purchase policy (ul,. . . , un), which minimizes the total cost. This problem leads to the following dynamic programming problem

OF:

j ( q , ... , u,) =

n

n

jj(u,) = C c , u 3

+min!

( 18.119a)

j=1

3x1

I

= ~ j - 1t u3 - v j j = l(l)n, .c~=x,, O
CT:

~j

+

n

f(zo,u1,.. .>2,-1,u,) = C(Cj.j j=1

+

(+I

t u , - vJ2) .l).

(18.120)

18.3.2.2 Knapsack Problem Il'e have to select some of the items A I , . . . ,A, with weights w l , . . . , w, and with values c l , . . . , c, so that the total weight does not exceed a given bound W,and the total value is maximal. This problem does not depend on time. It will be reformulated in the following way: At every stage we make a decision u, about the selection of item A,. Here, u, ,= l holds if A, is selected, otherwise u j = 0. The capacity still available at the beginning of a stage is denoted by ~ ~ - so 1 ,we get the following dynamic problem:

OF:

f ( u l . . . u,) = ~

n

c,u, --+ max!

(18.121a)

(18.121b)

18.3.3 Bellman Functional Equations 18.3.3.1 Properties of the Cost Function In order to state the Bellman functional equations, the cost function must satisfy two requirements:

18.3 Discrete Dynamic Programming 877

1. Separability The function f(fl(%,u,),. . . , f,,(~~-~,u~)) is called separable, if it can be given by binary functions HI, . . . . Hn-l and by functions F1,. . . , F,,in the following way:

f(f1(Eo.U l ) , . fn(xn-1r En)) = Fl (fl(Eo, Ul),. ' ' fn(x,,-114)). Fl (fl(Eor u1). . , fn(xn-l > u,)) = H l (fl(Eo,Ul),FZ(fZ(X1,nz),' ' ' * fn(Xn-1,u,,)))> . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . .... . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ... . . . . .. . . . . ' '

2

' '

1

Fn- I ( fn- 1 k n - 2 > ~ n - f n ( ~ n 1-I ~ n )= Hn-I ( f n - I ( ~ n - 2>

1)

Fn ( fn (xn11 u n1) = f n (&,,-I

I

un

un- 1) Fn(f n ( ~ n1- ~

1)1

l n

(18.122)

7

1.

2. Minimum Interchangeability A4function H(f(a),ki(f?)) is called minimum interchangeable,if min

(g,b)E..lxB

H

(io,~ ( b )=)min H (~ ( a min ) , ~ ( b ). ) BEE

(18.123)

&?EA

This property is satisfied, for example, if H is monotone increasing with respect to its second argument for every a E A . i.e., if for every a E A ,

H

(io,b)) 5 H (SM b,)) for F(b1)I &d.

(18.124)

Now, for the cost function of the dynamic programmingproblem, the separability off and the minimum interchangeability of all functions HI, j = l(1)n - 1 are required. The following often occurring type of cost function satisfies both requirements: (18.125)

(18.126)

(18.127)

18.3.3.2 Formulation of t h e Functional Equations We define the following functions:

k=j(l)n

@ntl(xn)= 0. If there is no policy (ul,., . .un)driving the state x ~ into - ~a final state & E X,. then n e substitute (18.129) 4J( x ~ -=~ )x.Using the separability and minimum interchangeability conditions and the dynamic constraints for = l(1)n w e get:

I

878

18. Optimization

(18.130) Equations (18.130) and (18.129) are called the Bellman junctzonal equatzons. value of the cost function f .

@I(%)

is the optimal

18.3.4 Bellman Optimality Principle The evaluation of the functional equation (18.131) corresponds to the determination of an optimal policy ($$,

,,

,$) minimizing the cost function, Le.,

(18.132) F j ( f j ( x j - 1 ) u j ) 3 . . f n ( l i n - 1 . U n ) ) -+min! based on the subprocess Pj starting at state x3-1and consisting of the last n - j t 1 stages of the total process P . The optimal policy of the process P1 with the initial state x3-l is independent of the decisions u,, . . . . u351 of the first j - 1 stages of P which have driven P to the state x3-l. To determine q ~ j ( ~ ~ we - ~ need ) . to know the value d3+1(x3).Now, if ($%. . . )u:) is an optimal policy for P3. then. obviously. ( u ; + ~. % . ,$). is an optimal policy for the subprocess Pjtl starting at xj = gj(xl-i,u;). This statement is generalized in the Bellman optimality principle. Bellman Principle: If (I$. . , . ,a;) is an optimal policy of the process P and (&.. . . , x;) is the corresponding sequence of states, then for every subprocess PI,j = l ( l ) n , with initial state the policy is also optimal (u;. . . . , I

18.3.5 Bellman Functional Equation Method 18.3.5.1 Determination of Minimal Costs Il'ith the functional equations (18.129). (18.130) and starting with &+1(xn) = 0 we determine every value q3(xj-1)with xI-l E X3-l in decreasing order of j . It requires the solution of an optimum problem over the decision space L'3(xj-l)for every xj-* E X3-1. For every xI-l there is a minimum point uj E C; as an optimal decision for the first stage of a subprocess PI starting at x3-1.If the sets Xj are not finite or they are too large: then the values can be calculated for a set of selected nodes E Xj-l. The intermediate values can be calculated by a certain interpolation method. &(a) is the optimal value of the cost function of process P . The optimal policy ($, . . . ,$) and the corresponding states (6%. . , f )can be determined by one of the following two methods.

.

18.3.5.2 Determination of the Optimal Policy 1. Variant 1: During the evaluation of the functional equations, the computed uj is also saved for After the calculation of $I(%), we get an optimal policy if we determine xy = every 53-1E X,-l. gl(&, u;) from xo = & and the saved ul = LI;~then from x; and the saved decision & we get $, etc. After every $3(x3-l) is known, we make 2. Variant 2: We save only q$(x3-!)for every x3-1E a forward calculation. Starting with J = 1 and x, = & we determine u; in increasing order of j by the evaluation of the functional equation

(18.133) During the forward calculation, we again have to solve an optimization Il'e obtain xi = gI(xi-l,g;), problem at every stage. 3. Comparison of the two Variants: The computation costs of variant 1 are less than variant 2 requires because of the forward calculations. However decision uj is saved for every state xj-i,which may require very large memory in the case of a higher dimensional decision space b'j(gl-l). while in

18.9 Discrete Dynamic Programming 879

Therefore, sometimes variant 2 is used

the case of variant 2, we have to save only the values q$(+J. on computers.

18.3.6 Examples for Applications of the Functional Equation Method 18.3.6.1 Optimal Purchasing Policy 1. Formulation of the Problem The problem from 18.3.2.1, p. 876, to determine an optimal purchasing policy

OF f ( u l , . . . . un) =

n

1c3u3 t min!

(18.134a)

3=1

(18.134b)

(18.135) (18.136)

j=l 2 3 4 5 6

~ , = 0 1 2 3 4 75 59 56 53 50 47 44 39 34 29 24 24 2 1 1 8 1 5 1 2 22 1 8 1 4 1 0 6 6 4 2 0 0

5

6

7

8

9 1 0

44 41 38 35 32 29 21 18 15 12 9 6 9 6 4 2 0 0 4 2 0 0 0 0 0 0 0 0 0 0

I

880

18. Optimization

18.3.6.2 Knapsack Problem 1. Formulation of the Problem Consider the problem given in 18.3.2.2,p. 876 n

OF : f ( u l . . . . , u,) =

1c,u, +max!

(18.137a)

,=1

CT:

X, = ~ zo =

~ -- w,u,, 1

3 = l(l)n, 3 = l(l)n,

w. 0 5 z,5 w,

(18.13713)

if < w,, Since we have a maximum problem, the Bellman functional equations are now O n t I ( X n ) = 0, u, = 0 ,

The decisions can be only 0 and 1, so it is practical to apply variant 1of the functional equation method. For j = n.n - 1 , . . . , 1 we get:

2. Numerical Example c3 = 3, cq = 1, c5 = 5, cg = 4, = 6, W4 = 3, W5 = 7, Wg = 6. Sincetheweightsw, areintegers, thepossiblevaluesforz, arez, E { O , l , . . . , l o } , j = l ( l ) n , and zo = ) every stage and and the actual decision U , ( X ~ - ~for 10 The table contains the function values for every state x , - ~ . For example. the values of ~ , & ( 2 5 ) ~4 3 ( 2 ) > 4+~3(6), and 43(8) are calculated:

W = 10, n = 6.

= 1,

c2

W1 = 2,

W2

c1

x,=O J=1 2 3 4 5 6

0: 0 0; 0 0: 0 0; 0 0: 0

= 2, = 4,

W3

1

2

3

3; 0 3; 0 3; 1 0; 0 0; 0

4; 1 3; 0 3; 1 0:0 0;0

7; 1 6; 1 3; 1 0; 0 0; 0

8

9

9; 0 10; 1 13; 1 13; 1 15; 0 9; 1 9; 0 10; 0 12; 1 15; 1 6; 0 9; 1 10: 1 10; 1 10; 1 6; 0 7; 1 7; 1 7 ; 1 7; 1 6; 1 6; 1 6; 1 6; 1 6; 1

16; 0 16; 1 13; 0 13; 1 6; 1

4

5

6

7

10 19; 0 19; 1 16; 0 16; 1 13; 1 6; 1

19 Numerical Analysis The most important principles of numerical analysis will be the subject of this chapter. The solution of practical problems usually requires the application of a professional nurnerzcal library of numerical methods, developed for computers. Some of them will be introduced at the end of Section 19.8.3. Special computer algebra systems such as Mathernatica and Maple will be discussed with their numerical programs in Chapter 20, p. 950 and in Section 19.8.4, p. 943. Error propagation and computation errors will be examined in Section 19.8.2, p. 936.

19.1 Numerical Solution of Non-Linear Equations in a Single Unknown Every equation with one unknown can be transformed into one of the normal forms: Zero form: f ( z ) = 0.

(19.1)

(19.2) Fixed point form: z = p(z). Suppose equations (19.1) and (19.2) can be solved. The solutions are denoted by z*.To get a first approximation of z*.WP can try to transfom the equation into the form f l ( z ) = f i ( z ) , where the curves of the functions y = f ~ ( zand ) y = f2(z) are more or less simple to sketch. f ( z ) = x2 - sin z = 0. We can see from the shapes of the curves y = x2 and y = sin z that z;= 0 and z;x 0.87 are roots (Fig. 19.1).

Yt

Figure 19.1

Figure 19.2

19.1.1 Iteration Method The general idea of iterative methods is that starting with known initial approximations ZR (k = 0,1,. . . , n ) we form a sequence of further and better approximations, step by step, hence we approach the solution of the given equation by iteratzon. by a convergent sequence. We try to create a sequence with convergence as fast as possible.

19.1.1.1 Ordinary Iteration Method To solve an equation given in or transformed into the fixed point form x = cp(z), we use the iteration rule (19.3) x,+~ = p(~,) (n = 0 , 1 , 2 , .. . ; zo given), which is called the ordznary zteratzon method. It converges to a solution z*if there is a neighborhood of z*(Fig. 19.2)such that (19.4)

I

882

19. Numerical Analusis

holds, and the initial approximation xo is in this neighborhood. If p(z) is differentiable, then the corresponding condition is (19.5) lp’(z)l 5 K < 1. The convergence of the ordinary iteration method becomes faster with smaller values of K .

I

1

sinx, 10.7643 10.7670 10.7681 10.7684 0.7686 0.7686 R e m a r k 1: In the case of complex solutions, we substitute z = u + iw. Separating the real and the imaginary part, we get an equation system of two equations for real unknowns u and w. R e m a r k 2: The iterative solution of non-linear equation systems can be found in 19.2.2, p. 893.

19.1.1.2 Newton’s Method 1. Formula of the Newton Method To solve an equation given in the form f(z) = 0, we mostly use the Newton method which has the formula

-

( n = 0,1,2,.. . ; xo is given), (19.6) f’(xn) Le.. to get a new approximation znil. we need the value of the function f(z) and its first derivative f’(z) at z,. 2. Convergence of the Newton Method The condition f‘(x) # 0 (19.7a) is necessary for convergence of the Newton method, and the condition z,+1

= 2,

- f(’n)

is sufficient. The conditions (19.7a.b) must be fulfilled in a neighborhood of z*such that it contains all the points I, and E* itself. If the Newton method is convergent, it converges very quickly. It has quadratic convergence, which means that the error of the (n+l)-st approximation is less than a constant multiple of the square of the error of the n-th approximation. In the decimal system, this means that after a while the number of exact digits will double step by step. IThe solution of the equation f ( z ) = x2 - a = 0. Le., the calculation of 2 = & (a > 0 is given), with the Newton method results in the iteration formula (19.8) We get for a = 2:

2,+1

= 2,

n

O

1

2

3

z, 1.5 1.4166666 1.414137 1.4142136

f‘(h)

- f(2n)

( m fixed, m < n).

(19.9)

19.1 Numerical Solution of Non-Linear Equations in a Single Unknown 883

Figure 19.3

Figure 19.4

The goodness of the convergence is hardly modified by this simplification.

5. Differentiable Functions with Complex Argument The Newton method also works for differentiable functions with complex arguments.

19.1.1.3 Regula Falsi 1. Formula for Regula Falsi To solve the equation f ( x ) = 0, the regula falsi method has the rule: (19.10) We need to compute only the function values. The method follows from the Newton method (19.6) by approximating the derivative f’(xn) by the finite difference of f ( x ) between x, and a previous approximation z, (rn < n ) .

2. Geometric Interpretation The geometric interpretation of the regula falsi method is represented in Fig. 19.4.The basic idea of the regula falsi method is the local approximation of the curve y = f ( x ) by a secant line.

3. Convergence The method (19.10) is convergent if we choose m so that f (5,) and f(x,) always have different signs. If the convergence already seems to be quick enough during the process, it will speed up if we ignore the change of sign, and we just substitute x, = zn-i. W f ( x ) = z2 - sinx = 0.

0 1

2

-0.3 0.0065

0.9 0.0267 0.87 -0.0074 0.8765 -0.000252 0.876729 0.000003 0.876726

-0.0341 0.007148 0.000255

0.8798 0.9093 0.8980

3 0.000229 -0.000003 4 If during the process the value of Ax,/Ayn only barely changes, we do not need to recalculate it again and again.

4. Steffensen Method Applying the regula falsi method with x, = x,-1 for the equation f ( x ) = x - p(x) = 0 we can often speed up the convergence. especially in the case ~ ’ ( x<) -1. This algorithm is known as the Stefensen

method. W To solye the equation z2 = sinx with the Steffensen method, we should use the form f ( x ) = x G = O .

I

884

19. Numerical Analusis

0 1 2 3

a,

xo

0.9

-0.03 0.006654

a,-l zoa:,-,

0.014942 -0.004259 0.876654 -0.000046 0.876727 0.000001 0.87

-0.019201 0.004213

1.562419 1.579397

an-? . . . a3 a2 a1 a0 ~ o a h -. .~. zoa; xoa; xoai zoab

(19.16)

(19.17)

j1 i ?;,

19.1 Numerical Solution of Non-Linear Equations in a Sinole Unknown 885

+ 2x3 - 3x2 - 7. The substitution value and derivatives of pd(z) are calculated at zo = 2 according to (19.16).

Ip4(x) = $4

1 2 -3 0 -7

We see: P4(2) = 13: P & P ) = 44, p&'(2) = 66, p1;'(2) = 60, p y ' ( 2 ) = 24.

1 4 5 10 13 1 6 17 44 1 8 33

Remarks: 1. We can rearrange the polynomialp,(z) with respect to the powers of z - 20,e.g., in the example above we get p4(x) = (x - 2)4 t 1O(z - 2)3 t 33(x - 2)' 44(x - 2) t 13. 2. The Horner scheme can also be used for complex coefficients a k . In this case for every coefficient we have to compute a real and an imaginary column according to (19.16).

+

2. Complex Arguments

If the coefficients ak in (19.11) are real, then the calculation ofp,(xo) for complex values zo = uo + ivo can be made real. In order to show this, we decompose p,(x) as follows: p , ( x ) = a,xn

+ an-lzn-l t . ' . + alx + a0 + . ' . + ub) t T ~ +Z ro

= (xz- px - q)(ak-zxn-2

with

(19.18a)

x2 - p x - q = ( x - xo)(z-TO), i.e., p = 2u0, q = -(uo2 + ~ 0 ' ) . ( 19.18b) Then. we get (19.18~) p,(xo) = T ~ t Z ro = ( ~ 1 t ~ ro) 0 t irlwo. To find (19.18a) we can construct the so-called two-row Horner scheme introduced by Collatz:

(19.18d)

W pd(x) = x4 t 2x3 - 32' - 7. Calculate the value of p4 at xo = 2 - i, Le., for p = 4 and q = -5.

19.1.2.2 Positions of the Roots 1. Real Roots, Sturm Sequence With the Cartesian rule of signs we can get a first idea of whether the polynomial equation (19.11) has a real root, or not. a) The number of positive roots is equal to the number of sign changes in the sequence of the cooefficients (19.19a) a,, a n - l , . . , . al, a0 or it is less by an even number.

886

19. Numerical Analysis

b) The number of negative roots is equal to the number of sign changes in the coefficient sequence uo. - u l . a2 , . . . . (-l)%, (19.19b) or it is less by an even number. Ips(x) = z5 - 6x4 t lor3 132’ - 152 - 16 has 1 or 3 positive roots and 0 or 2 negative roots. To determine the number of real roots in any given interval (a,b ) , Sturrn sequences are used (see 1.6.3.2. 2.) p. 44). After computing the function values yv = p,(z,) at a uniformly distributed set of nodes z,= zo t v . h (h constant) (which can be easily performed by using the Horner scheme) a good guess of the graph of the function and the locations of roots are obtained. If p,(c) and p,(d) have different signs. there is at least one real root between c and d.

+

2. Complex Roots In order to localize the real or complex roots into a bounded region of the complex plane consider the following polynomial equation which is a simple consequence of (19.11):

+ +

+

(19.20) f*(z)= lun_llrn-l t /an-21rn-2 ... / a l l r /ao/ = lu,lrn and we determine, e.g., by systematic repeated trial and error: an upper bound ro for the positive roots of(19.20). Then,forallrootszE ( k = 1 , 2 , . . . ,n)of(19.11), 1 4 5 To. (19.21) I f(x) = p4(x) = a4+4.4z3-20.01x2-50.12z+29.45 = 0, f*(z) = 4.4r3+20.01r2+50.12r+29.45 = r4. \Ye get for T = 6: f’(6) = 2000.93 > 1296 = r4. r = 7: f”(7)= 2869.98 > 2401 = T~~ r = 8: f*(8)= 3963.85 < 3096 = T ~ . From this it follows that Iz;~ < 8 (k = 1,2,3,4). Actually, for the root z;with maximal absolute value we hale: -7 < Z; < -6. Remark: A special method has been developed in electrotechnics in the so-called root locus theory for the determination of the number of complex roots with negative real parts. It is used to examine stability (see [19.11], [19.30]).

19.1.2.3 Numerical Methods 1. General Methods The methods discussed in Section 19.1.1, p. 881, can be used to find real roots of polynomial equations. The Newton method is well suited for polynomial equations because of its fast convergence, and the fact that the values of f(z,)and f’(z,) can be easily computed by using Horner’s rule. By assuming that an approximation z, of the root z*of a polynomial equation f(x) = 0 is sufficiently good, then the correction term 6 = z*- x, can be iteratively improved by using the fixed-point equation

2. Special Methods The Bazrstow method is well applicable to find root pairs, especially complex conjugate pairs of roots. The Horner scheme (19.18a-d) is used to find quadratic factors of the given polynomial having the root pair to determine the coefficients p and q which make the coefficients of the linear remainder ro and equal to zero (see I19.291, [19.11], [19.30]). If the computation of the root with largest or smallest absolute value is required, then the Bernoullz methodis the choice (see [19.19]). The Graefle method has some historical importance. It gives all roots simultaneously including complex conjugate roots; holyever the computation costs are tremendous (see [19.11]. [19 301).

19.2 Numerical Solution of Equation Svstems 887

19.2 Numerical Solution of Equation Systems In several practical problems, n e have m conditions for the n unknown quantities x, in the form of equations:

(2

= 1 , 2 , .. . , n)

(19.23) We have to determine the unknowns xi so that they form a solution of the equation system (19.23). Mostly m = n holds. Le.; the number of unknowns and the number of equations are equal to each other. In the case of m > n; we call (19.23) an overdetermined system: in the case of m < n it is an underdetermaned system. Overdetermined systems usually have no solutions. Then, we are looking for the "best" solution of (19.23). in the Euclidean metric with the least squares method m

C I ; " ( z 1 , z 2.....x,)=min!

(19.24)

i=l

or in other metrics as another extreme value problem. Usually. the values of n - m variables of an underdetermined problem can be chosen freely. so the solution of (19.23) depends on n - m parameters. We call it an ( n - m)-dimensional manifold ofsolutions. \Ye distinguish between linear and non-linear equation systems, depending on whether the equations are only linear or non-linear in the unknowns.

19.2.1 Systems ofLinear Equations Consider the linear equation system

az1xl

+ a12x2 + . . t alnx, = + ~ 2 2 x 2+ + aznx, = b2.

a,lxl

t an252

ullx1

bl.

,

,'.

+

'.

(19.25)

+ annxn= bn.

The sJstem (19.25) can be written in matrix form

Ax=b

(19.26a)

with (19.26b) Suppose the quadratic matrix A = (azk) ( 2 %k = 1 , 2 , . . . , n ) is regular, so system (19.25) has a unique solution (see 4.4.2.1, 2., p. 273). In the practical solution of (19.25), n e distinguish between two types of solution methods: 1. Direct Methods are based on elementary transformations, from which the solution can be obtained immediately. These are the pivoting techniques (see 4.4.1.2, p. 271) and the methods given in 19.2.1.119.2.1.3. 2. Iteration methods start with a known initial approximation of the solution, and forming a sequence of approximations that converges to the solution of (19.25) (see 19.2.1.4, p. 892).

19.2.1.1 Triangular Decomposition of a Matrix 1. Principle of t h e Gauss Elimination Method By elementary transformations

888

19. Numerical Analusis

1. interchanging rows, 2. multiplying a row by a non-zero number and 3. adding a multiple of a row to another row, the system A x = b is transformed into the so-called TOW echelon form

(19.27)

Since only equivalent transformations were made, the equation system Rx = c has the same solutions as A x = b. We get from (19.27): (19.28) The rule given in (19.28) is called backward substitution, since the equations of (19.27) are used in the opposite order as they follow each other. The transition from A to R is made by n - 1 so-called elimination steps, whose procedure is shown by the first step. This transforms matrix A into matrix AI: (1) 311

0 0

(19.29)

0

ilTe proceed as follows: 1. Lye choose an arl# 0 (according to (19.33)). If there is none, stop: A is singular. Otherwise a,l is called the pivot. 2. \le interchange the first and the r-th row of A. The result is A. 3. We subtract 1,l ( i = 2,3,. . . , n)times the first row from the i-th row of the matrix -4s a result we get the matrix A! and analogously the new right-hand side b, with the elements

x.

(19.30) bj” = $i - 1,161 ( i l k = 2 , 3 , . . . , n). The framed submatrix in AI (see (19.29)) is of type ( n - 1,n - 1) and it will be handled analogously to A.etc. This method is called the Gaussian elimination methodor the Gauss algorithm (see 4.4.2.4. p. 276).

2. Triangular Decomposition The result of the Gauss elimination method can be formulated as follows: To every regular matrix A there exists a so-called triangular decomposition or L Ufactorization of the form

PA=LR

(19.31)

19.8 Numerical Solution of Equataon Systems 889

with rii

R=(

r12

Ti3

TZZ

~ 2 3

733

... .

rln

I:],

::: rnn

1 (19.32)

I,=[:

41

LZ li

1

’.’

,o,

In,n-l

Here R is called an upper triangular matrix, L is a lower triangular matr% and P is a so-called pennutation matrix. A permutation matrix is a quadratic matrix which has exactly one 1 in every row and every column, and the other elements are zeros. The multiplication P A results in row interchanges in A, which comes from the choices of the pivot elements during the elimination procedure. The Gauss elimination method should be used for the system schematic form, where the coefficient matrix and the vector from the right-hand side are written next to each other (into the so-called extended coeficient matrix), we get:

1 0 0

P= 0 0 1 (0

1 0)

*PA=

3 1 6 1 1 1 , L = 1/3 1 (2 1 3) ( 2 i 3 1;2

:) 0

(i

,R=

0 2/3

:;2)’

In the extended coefficient matrices, the matrices A, Al and Az, and also the pivots are shown in boxes.

3. Application of Triangular Decomposition With the help of triangular decomposition, we can describe the solution of linear equation systems A x = in three steps: 1. P A = L R: Determination of the triangular decomposition and substitution Rx = c

L c = P b: Determination of the auxiliary vector c by forward substitution. 3. R E = c: Determination of the solution 11by backward substitution. 2.

If the solution of a system of linear equations is handled by the expanded coefficient matrix (A,b), as in the above example, by the Gauss elimination method, then the lower triangular matrix L is not needed explicitly. This can be especially useful if several systems of linear equations are to be solved after each other with the same coefficient matrix, with different right-hand sides.

4. Choice of the Pivot Elements Theoretically, every non-zero element a!!-’) of the first column of the matrix Ak-1 could be used as a pivot element at the k-th elimination step. In order to improve the accuracy of solution (to decrease the accumulated rounding errors of the operations), the following strategies are recommended. 1. Diagonal Strategy The successive diagonal elements are chosen as pivot elements Le., there is no row interchange. This kind of choice of the pivot element makes sense if the absolute value of the elements of the main diagonal are fairly large compared to the others in the same row. 2. Column Pivoting To perform the k-th elimination step, we choose the row index T for which:

890

19. Numerical Analysis

If r # k,then the r-th and the k-th rows will be interchanged. It can be proven that this strategy makes the accumulated rounding errors smaller.

19.2.1.2 Cholesky's Method for a Symmetric Coefficient Matrix In several cases, the coefficient matrix A in (19.26a) is not only symmetric, but also posztzwe definzte, i.e.. for the corresponding quadratic f o r m Q(x)holds n

Q(X) = LITAX =

E

n azkXIXk

>0

(19.34)

,=I k=1

for every x E R", # 0. Since for every symmetric positive definite matrix A there exists a unique triangular decomposition A=LL~ (19.35) with

(19.36a)

(19.36b)

,~= aij( k - 1 )

a(k)

- 1%k3)9 1 ( i , j = k t l , k + 2 , . . . ,n),

(19.36~)

the solution of the corresponding linear equation system A z = b can be determined by the Cholesky method by the following steps: 1. A = L LT: Determination of the so-called Cholesky decomposition and substitution LTz= c. 2. L c = b: Determination of the auxiliary vector c by forward substitution. 3. LTz= c: Determination of the solution x by backward substitution. For large values of n the computation cost of the Cholesky method is approximately half of that of the LC decomposition given in (19.31), p. 888.

19.2.1.3 Orthogonalization Method 1. Linear Fitting Problem Suppose an Overdetermined linear equation system

2

albxk

= b,

( i = 1 , 2 , . . . m; m > n ) ,

(19.37)

b=l

is given in matrix form (19.38) Ax = b. Suppose the coefficient matrix A = ( a , k ) with size ( m x n) has full rank n,Le.. its columns are linearly independent. Since an overdetermined linear equation system usually has no solution, instead of (19.37) we consider the so-called error equations

"

(19.39) with residues

ri,

and we require that the sum of their squares should be minimal: (19.40)

19.2 Numerical Solution of Equation Systems 891

The problem (19.40) is called a linearfitting problem or a linear least squares problem (see also 6.2.5.5, p. 401). The necessary condition for the relative minimum of the sum of residual squares F ( q , q.. . ., 2 , ) is

8F

-= 0

( k = 1 , 2 , .. . , n)

(19.41)

8Xk

722

A = QR

(19.44) with QTQ = E and R =

' ' '

7271

(19.43)

= rnn

(19.46)

without changing the sum of the squares of the residuals. From (19.46) it follows that the sum of the = i 2 = . = in= 0 and the minimum value is equal to the sum of the squares squares is minimal for i1 of ?,+I to im. \$'e get the required solution x by backward substitution

Rx = b,

(19.47)

892

19. Numerical Analusis

.!I,,

where is the vector with components &, , . . , &,, obtained from (19.46). There are two methods most often used for a stepwise transition of (19.39) into (19.46): 1. Givens transformation, 2. Householder transformation. The first one results in the QR decomposition of matrix A by rotatzons, the other one by reflectzons. The numerical implementations can be found in [19.23]. Practical problems in linear mean square approximations are solved mostly by the Householder transformation, where the frequently occurring special band structure of the coefficient matrix A can be used.

19.2.1.4 Iteration Methods 1. Jacobi Method Suppose in the coefficient matrix of the linear equation system (19.25) every diagonal element a,, (z = 1,2.. . . , n) is different from zero. Then the z-th row can be solved for the unknown x,, and we get immediately the following iteration rule, where p is the iteration index: (19.48) ( p = 0,1.2,. . . ; xy', xp',

. . . , xipl

are given initial values)

Formula (19.48) is called the Jacobi method. Every component of the new vector x(fitl) is calculated from the components of &')). If at least one of the conditions mtx

5: 1 1 < I

column sum crzterzon

(19.49)

5 1:1< &:)

row sum criterion

(19.50)

(S)

or max

'

1

holds, then the Jacobi method is convergent for any initial vector do).

2. Gauss-Seidel Method If the first component x p l ) is calculated by the Jacobi method, then this value can be used in the calculation of x p t l ) , While we proceed similarly in the calculation of the further components, we get the iteration formula (19.51)

(i = 1 ~ 2 , .. . , n; sy', xp13,., ,zip) given initial value; p = 0 , 1 , 2 , . . .) . Formula (19.51) is called the Gauss-Seidel method. The Gauss-Seidel method usually converges more quickly than the Jacobi method, but its convergence criterion is more complicated. I 10x1 - 3x2 - 4x3 + 2x4 = 14, -321 26x2 + 5x3 - 2 4 = 22, -4x1 + 5x2 + 16x3 + 5x4 = 17, 2x1 + 3x2 - 4x3 - 12x4 = -20. The corresponding iteration formula according to (19.51) is:

+

19.2 Numerical Solution of Equation Systems 893

Some approximations and the solution are given here:

10

zpl)= 1 (22 + 3spl)- 5xk)+ xp)), 26 5p1) =1 (17 + 4$+1) - 5 $ + 1 1 - 5xpl), 16 12

3. Relaxation Method The iteration formula of the Gauss-Seidel method (19.51) can be written in the so-called correction form

xjp+1) = 5jp) + dj’”) ( 2 = 1 , 2 , . . . ,n; p = 0 , 1 , 2 , . . .). By an appropriate choice of a relaxation parameter w and rewriting (19.52) in the form

(19.52)

z.l

$+lI = +4 4 ( i = 1 . 2 , . . .2 n ; p = o , l > 2 > . . . ) 2 (19.53) we can try to improve the speed of convergence. It can be shown that convergence is possible only for

O<w<2. (19.54) For w = 1 we retrieve the Gauss-Seidel method. In the case of w > 1, which is called overrelaxation, the corresponding iteration method is called the SOR method (successive overrelaxation). The determination of an optimal relaxation parameter is possible only for some special types of matrices. We apply iterative methods to solve linear equation systems in the first place when the main diagonal elements all of the coefficient matrix have an absolute value much larger than the other elements azk ( i # k ) (in the same row or column), or when the rows of the equation system can be rearranged in a certain way to get such a form.

19.2.2 Non-Linear Equation Systems Suppose the system of n non-linear equations F 2 ( z 1 , 2 *., . . 5,) = 0 ( 2 = 1 , 2 , . . . , n) (19.55) for the n unknowns xlr 52. . . . x, has a solution. Usually, a numerical solution can be given only by an iteration method. ~

~

19.2.2.1 Ordinary Iteration Method We can use the ordinary iteration method if the equations (19.55) can be transformed into a fixed-point form (19.56) 5, = ji(xl,xz~.. . . x n ) ( i = 1 , 2 , .. . n ) . ~

Then. starting from estimated approximations zp’, $),. 1. iteration with simultaneous steps

.., ~

~ we ~get 1the , improved values either by

z y = jl (5y Jp.. . . .xy) (z = 1 , 2 , . . . , n: p = 0, 1 , 2 , .. .) or by 2. iteration with sequential steps

(19.57)

894

19. Numerical Analysis

It is of crucial importance for the convergence of this method that in the neighborhood of the solution the functions f, should depend only weakly on the unknowns, Le., if fi are differentiable. the absolute values of the partial derivatives must be rather small. We get as a convergence condztzon

K < 1 with K = max

’

)l: 1

max (k:1

.

(19.59)

With this quantity K . the error estimation is the following:

Here. x, is the component of the required solution, zZ(r) and x?”) ( p + 1)-th approximations.

are the corresponding p-th and

19.2.2.2 Newton’s Method The Newton method is used for the problem given in the form (19.55). After finding the initial approximation values zp), x!), , , ,zho),the functions F, are expanded in Taylor form as functions of n independent variables 21. x2. . . z, (see p. 415). Terminating the expansion after the linear terms, from (19.55) we get a linear equation system, and with this we can get iterative improvements by the following formula:

(i = 1 , 2 : . . . 7%; p = 0.1,2;. . .). The coefficient matrix of the linear equation system (19.61),which should be solved in every iteration step. is ~

and it is called the Jacobian matrix. If the Jacobian matrix is invertible in the neighborhood of the solution, the Kewton method is locally quadratically convergent, Le., its convergence essentially depends on how good the initial approximations are. If we substitute z p l ) - x r ) = df’ in (19.61). then the Newton method can be written in the correction form (19.63) $+I) = x!) df’ ( 2 = 1:2,. . . n; p = 0 , l : 2 , . , .). To reduce the sensitivity to the initial values, analogously to the relaxation method, we can introduce a so-called damping or step length parameter ;t:

+

$+I)

= x!)

+y d y

(1

= 1 , 2 , .. . , n: p = 0 , l . 2..

. . ; y > 0).

(19.64)

Xethods to determine y can be found in [19.24].

19.2.2.3 Derivative-Free Gauss-Newton Method To solve the least squares problem (19.24), we proceed iteratively in the non-linear case as follows: 1. Starting from a suitable initial approximation xp),$I, . . . , z f j ) we approximate the non-linear functions F,(xl3x2... . 2,) (i = 1 , 2 , .. . m ) as in the Kewton method (see (19.61)) by linear approximations pt(xl,5 2 , . . . xn), which are calculated in every iteration step according to

.

(i = 1 . 2 , . . . : n ;p = 0,1,2.. . .).

(19.65)

19.3 Numerical Inteqration

895

in (19.65) and we determine the corrections dk' by using the Gaussian 2. K e substitute d!$ = rk least squares method. i.e., by the solution of the linear least squares problem m

C F:(rl.. . .

~

2,)

(19.66)

= min,

z=1

e.g.. with the help of the normal equations (see (19.42)): or the Householder method (see 19.6.2.2, p. 918). 3. Wt get approximations for the required solution by ,$tl)

+&)

(19.67a)

or

= x k( I )

( k = 1:2. . . . n ) , (19.67b) > 0) is a step length parameter similar to the Newton method. where ,; By repeating steps 2 and 3 with instead of I!) we get the Gauss-Newton method. It results in a sequence of approximation values, whose convergence strongly depends on the accuracy of the initial approximation. Il'e can reduce the sum of the error squares by introducing a length parameter y. p I;t i )

= x k(PI

+ */&)

~

(A,

xp')

aF,

(ZP)~

If the evaluation of the partial derivatives , , , , zk)) (i = 1,2, . . . . m; k = 1 , 2 , . . . , n) requires a xk too much work. we can approximate the partial derivatives by difference quotients :

-F,(zji") , . . . .

xtl . . . .$ r k ) ) ] ( i = 1 , 2, . . . , m; k = l , 2 , . . . , n;p=O:1,2 , . . .).

(19.68)

The so-called discretization step sizes h k ) may depend on the iteration steps and the values of the variables. If we use approximations (19.68): then we have to calculate only function values F, while performing the Gauss-Newton method, Le., the method is derivative free.

19.3 Numerical Integration 19.3.1 General Quadrature Formulas The numerical evaluation of the definite integral b

I ( f )=

JfiW

(19.69)

Q

must be done only approximately if the integrand f(x)cannot be integrated by elementary calculus, or it is too complicated. or when the function is known only at certain points I,, the so-called interpolation nodes from the integration interval [a,b]. We use the so-called quadrature formulas for the approximate calculation of (19.69). They have the general form n.

Q(f) =

+

n

C O ~ Y ~ "=O "=O

civd

+ .. +

n

cpvy!' "=O

(19.70) with yp) = f " ) ( r U()p = 1:2 , . , . , p ; u = 1 , 2 ; .. . n ) , yv = f(xv),and constant values of cIu. Obviously, (19.71) I(f)= Q ( f )+ R. where R is the error of the quadrature formula. We suppose in the application of quadrature formulas that the required values of the integrand f(r)and its derivatives at the interpolation nodes are known ~

I

896

19. Numerical Analysis

as numerical values. Formulas using only the values of the function are called mean value formulas: formulas using also the derivatives are called H e m i t e quadrature formulas.

19.3.2 Interpolat ion Quadratures The following formulas represent so-called interpolation quadratures. Here, the integrand f(z)is interpolated at certain interpolation nodes (possibly the least number of them) by a polynomial p(z)of corresponding degree, and the integral of f(z) is replaced by that of p ( z ) . The formula for the integral over the entire interval is given by summation. In the following, we give the formulas for the most practical cases. The interpolation nodes are equidistant: b-a r , = z o + ~ h ( ~ = 0 , 1 , 2. ., . ,n ) , z o = a , ~ , = b , h=-.

(19.72)

We give an upper bound for the magnitude of the error IR/ for every quadrature formula. Here, M,, means an upper bound of if(”)(z)l on the entire domain.

19.3.2.1 Rectangular Formula In the interval [zol 50+ h],f(z)is replaced by the constant function y = yo = f(zo), which interpolates f(z)at the interpolation node 20,which is the left endpoint of the integration interval. We get in this way the simple rectangular formula hZ

z i * f ( z )dz M h . yo; lR1 5 z M l .

(19.73a)

20

We get the left-sided rectangular formula by summation: (19.73b)

MI denotes an upper bound of lf’(z)Ion the entire domain of integration. We get analogously the rzght-szded rectangular sum, if we replace yo by y1 in (19.73a). The formula is: (19.74)

19.3.2.2 Trapezoidal Formula f(r)is replaced by a polynomial of first degree in the interval [.(I, zo + h],which interpolates f(z)at the interpolation nodes zo and z1 = 10 + h. We get: (19.75) LO

We get the so-called trapemidalformula by summation: (19.76)

Mz denotes an upper bound of If”(z)(, on the entire integration domain. The error of the trapezoidal formula is proportional to hZ,i.e., the trapezoidal sum has an error of order 2. It follows that it converges

19.3 Numerical Integration 897 to the definite integral for h + 0 (hence, n

--t

co),if we do not consider rounding errors.

19.3.2.3 Simpson’s Formula f(x) is replaced by a polynomial of second degree in the interval o.[s at the interpolation nodes 20,xi = xo + h and x2 = xo t 2h:

xo + 2h], which interpolates f(x) (19.77)

(19.78)

M4 is an upper bound for If(4)(2)/ on the entire integration domain. The Simpson formula has an error of order 4 and it is exact for polynomials up to third degree.

19.3.2.4 Hermite’s Trapezoidal Formula

f(x)is replaced by a polynomial of third degree in the interval [xo,20 + h], which interpolates f ( x ) and f’(x) at the interplation nodes xo and x1 = zo h:

+

(19.79) We get the Hermite trapezoidalformvlaby summation:

M4denotes an upper bound for 1f(4)(x)lon the entire integration domain. The Hermite trapezoidal formula has an error of order 4 and it is exact for polynomials up to third degree.

19.3.3 Quadrature Formulas of Gauss Quadrature formulas of Gauss have the general form (19.81) where not only the coefficients c, are considered as parameters but also the interpolation nodes x,. These parameters are determined in order to make the formula (19.81) exact for polynomials of the highest possible degree. The quadrature formulas of Gauss result in very accurate approximations, but the interpolation nodes must be chosen in a very special way.

19.3.3.1 Gauss Quadrature Formulas If the integration interval in (19.81) is chosen as [a,61 = [-I, 11, and we choose the interpolation nodes as the roots of the Legendre polynomials (see 9.1.2.6, 3., p. 509, 21.10, p. 1060), then the coefficients e,, can be determined so that the formula (19.81) gives the exact value for polynomials up to degree 2n + 1. The roots of the Legendre polynomials are symmetric with respect to the origin. For the cases n = 1 , 2 and 3 we get:

898

19. Numerical Analysis

n = 1:

n = 2:

20

= -XIr

cg = 1,

21

= A=0.577350269 . . . ,

c1 = 1.

d3

= 0.

21

x2 =

6

= 0.774 596 669. . .

1

n=3:

5 9' 8 c1= - , 9

cg = -

xg = -22,

21

22 23

c2

= cg.

cg

= 0.347 854 854. .

-

= -23. = -22. = 0.339 981 043.. . , = 0.861 136 311 . . . ,

20

(19.82)

~1 c2 cg

.,

= 0.652 145 154, . . , = c1. = cg.

Remark: A general integration interval [a.b] can be transformed into [-1.11 by the transformation b-a a+b t = -x +(t E [a,b1.2 E [-1, 11). Then 2 2 b

/

f ( t ) dt

b-a

a

with the values 2 ,

e)

c,f ($-%. + 2 2 u=o and e,, given above for the interval [-1.11

=Z -

(19.83)

19.3.3.2 Lobatto's Quadrature Formulas In some cases it is reasonable also to choose the endpoints of the integration interval as interpolation nodes. Then, we have 2n more free parameters in (19.81). These values can be determined so that polynomials up to degree 2n - 1 can be integrated exactly. We get for the cases n = 2 and n = 3: n = 2: n = 3: 1 1 2 0 = -1. 2 0 = -1, cg = -, = 6' 3 5 4 (19.84a) 21 = -22, X I = 0, c1 = j, = 6' (19.84b) x* = 1, c2 = c g . 22 = - 0.447213595.. ., c2 = cl,

4 $6-

23

= 1.

c3 = cg

The case n = 2 represents the Simpson formula.

19.3.4 Method of Romberg To increase the accuracy of numerical integration the method of Romberg can be recommended, where we start with a sequence of trapezoid sums, which is obtained by repeated halving of the integration step size.

19.3.4.1 Algorithm of the Romberg Method The method consists of the following steps:

1. Trapezoid sums determination We determine the trapezoid sum T(h,)according to (19.76) as approximationsofthe integral

lb

f ( x )dx

with the step sizes (19.85)

19.3 Numerical Integration 899

Here. we consider the recursive relation

[: + ( +

T ( h , )= T 1-' = h-l -!(a) (h2 2

)

+

+f(a 2 L l )

f

a

h;l)

3 + f ( a + hi-I) + f (u + 4 b) 2

+ . .. + f (a +

(19.86)

we need to compute the Recursion formula (19.86) tells that for the calculation of T(h,)from T(/L,-~) function values only at the new interpolation nodes.

2. Triangular Scheme We substitute Tot = T(h,) (z = 0,1,2,. . .) and we calculate recursively the values

Tkt = Tk-1.i

+ Tk-l,i4k--Tk-1,i-1 1

( k = 1 , 2, . . . ,m; i = k , k + 1 , . . .).

(19.87)

The arrangement of the values calculated according to (19.87) is most practical in a triangular scheme, whose elements are calculated in a column-wise manner: T(ho)= Too T(h1) = To1 Tll (19.88) T(h2)= To2 Tl2 T22 T(h3)=TO3 TI3 T23 T33

...

I..................

The scheme will be continued downwards (with a fixed number of columns) until the lower values at the right are almost the same. The values TI, (z = 1 , 2 , . . .) of the second column correspond to those calculated by the Simpson formula.

19.3.4.2 Extrapolation Principle The Romberg method represents an application of the so-called eztrapolatzon prznczple. This will be demonstrated by deriving the formula (19.87)for the case k = 1. We denote by I the required integral, by T ( h )the corresponding trapezoid sum (19.76). If the integrand of I is (2m + 2) times continuously differentiable in the integration interval, then it can be shown that an asymptotical ezpansion with respect to h is valid for the error R of the quadrature formula, and it has the form R(h)= I - T ( h ) = a l h 2 + ~ z h 4 + . . . + ~ , h 2 m + O ( h Z m + 2 ) (19.89a) or T ( h )= I - ~ l h ~ - a ~ h ~ - . . . - ~ , h ~ ~ + O ( h ~ ~ ' ~ ) . ( 19.89b) The coefficients a l , a2.. . . ,am are constants and independent of h. LVe form T ( h )and T - according to (19.89b) and consider the linear combination

+ (Y2T

Tl(h)= CYlT(h) If we substitute 01

+ CYZ = 1 and a1 + 042 = 0, then T l ( h )has an error of order 4, while T ( h )and T ( h / 2 ) -

both have errors of order only 2. We have (19.91)

I

19. Numerical Analusis

900

This is the formula (19.87) for k = 1. Repeated application of the above procedure results in the approximation Tkt according to (19.87) and

Tkz= I

+ O(hTkt2).

(19.92)

i:

-dx (integral sine, see 8.2.5, l., p. 458) cannot be obtained in an si:X elementary way. Calculate the approximate values of this integral (calculating for 8 digits). k=O I k=l I k=2 1 k=3

IThe definite integral I =

1

I

0.94451352 0.94608693 0.94608300 0.94569086 0.94608331 0.94608307 0.94608307. The Romberg method results in the approximation value 0.94608307. The value calculated for 10 digits of the error according to (19.92) is proved. is 0.9460830704. The order 0 ((1/8)8) % 6 . ~

2. Trapezoidal and Simpson Formulas: We can read directly from the scheme of the Romberg method that for hg = 1/8 the trapezoid formula has the approximation value 0.94569086 and the Simpson formula gives the value 0.94608331. The correction of the trapezoidal formula by Hermite according to (19.79) results in the value I %

+

0.94569086 0'30116868 64. 12 = 0.94608301. 3. Gauss Formula: By the formula (19.83) for , , we get -

n = 1:

I%

;

n = 2:

I

%

j[cof ( i z ~t

n = 3:

I

%

[Cof

;[

cof

(& t ;j

(

t

+Clf

(;XI

+ elf (;zl

+ij] t

+ c z f (;z2 t

;j+ . . + (1 + 131 '

Cgf

-z3

-

a)]

= 0.94604113; = 0.94608313; = 0.94608307.

We see that the Gauss formula results in an 8-digit exact approximation value for n = 3, Le., with only four function values. With the trapezoidal rule this accuracy would need a very large number (> 1000) of function values. Remarks: 1. Fourier analysis has an important role in integrating periodic functions (see 7.4.1.1, l., p. 418). The details of numerical realizations can be found under the title of harmonic analysis (see 19.6.4, p. 924). The actual computations are based on the so-called Fast Fourier Transformation FFT (see 19.6.4.2, p. 925). 2. In many applications it is useful to take the special properties of the integrands under consideration. Further integration routines can be developed for such special cases. A large variety of convergence properties, error analysis, and optimal integration formulas is discussed in the literature (see, e.g., [ 19,4]). 3. Numerical methods to find the values of multiple integrals are discussed in the literature (see, e.g., p9.261).

19.4 Approximate Inte.qration of Ordinaru Diflerential Equations 901

19.4 Approximate Integration of Ordinary Differential Equations In many cases. the solution of an ordinary differential equation cannot be given in closed form as an expression of known elementary functions. The solution, which still exists under rather general circumstances (see 9.1.1.1,p. 486), must bedetermined by numerical methods. These result only in particular solutions. but it is possible to reach high accuracy. Since differential equations of higher order than one can be either initial value problems or boundary value problems, numerical methods were developed for both types of problems.

yA

= y(z0

A) Y(Z1)

= YO

jY0

0

xo

jY1 XI

~

x2

Y2

jY3

x3

U W W

x

+ h)

+ f(zol yo)h + TY ” ( h0

z

(19.98)

with 20 < E < zo + h, then we see that the approximation y1 has an error of order h2. The accuracy can be improved by reducing the step size h. Practical calculations show that halving the step size h results in halving the error of the approximations yE.

902

19. Numerical Analysis

+

and the possible next point (xo h, yl) of the curve, and depending on the appropriate choice of these additional points we get more accurate values for y1. We have Runge-Kutta methods of different orders depending on the number and the arrangements of these "auxiliary" points. Here we show a fourthorder method (see 19.4.1.5, 1.. p. 904). (The Euler method is a first-order Runge-Kutta method.) The calculation scheme of fourth order for the step from 20 to 2 1 = xo h to get an approximate value for y1 of (19.93) is given in (19.99). The further 20 steps follow the same scheme. xo t hJ2 xo + h/2 The error of this Runge-Kutta method 2o + has order h5 (at every step) according to (19.99). so with an appropriate 1 2 1 = xo h YI = YO -6( k l + 2kz 2k3 + k4) choice of the step size we can have high accuracy. 1 1 W y' = - ( x z + y') with y(0) = 0. We determine y(O.5) k = i ( ~ 2+ y') 4 in one step. i.e. h = 0.5 (see the table on the right). The exact value for 8 digits is 0.01041860. 0.00781280

+

+

+

+

~

2. Remarks 1. For the special differential equation y'

= f(x)> this Runge-Kutta method becomes the Simpson formula (see 19.3.2.3, p. 897). 2. For a large number of integration steps, a change of step size is possible or sometimes necessary. The change of step size can be decided by checking the accuracy so that we repeat the step with a double step size 2h. If we have. e.g.. the approximate value yz(h) for y(x0 + 2h) (calculated by the single step size) and yp(2h) (calculated by the doubled step size), then we have the estimation for the error &(h) = ~ ( 2 + 0 2h) - yz(h):

(19.100) Information about the implementation of the step size changes can be found in the literature (see [19 241). 3. Runge-Kutta methods can easily be used also for higher-order differential equations, see [19.24]. Higher-order differential equations can be rewritten in a first-order differential equation system (see p. 495). Then, the approximation methods are performed as parallel calculations according to (19.99). as the differential equations are connected to each other.

19.4.1.3 Multi-Step Methods The Euler method (19.97) and the Runge-Kutta method (19.99) are so-called single-step methods. since we start only from yt in the calculation of y t t l . In general, linear multi-step methods have the form (19.101) ( j = 0 , 1 , , , . , k; ak = 1). The formula (19.101) is with appropriately chosen constants aJ and called a k-step method if 1 0 1 t I& # 0. It is called explicit, if 8 k = 0, since in this case the values ftt3 = f(xitl, ytTJ) on the right-hand side of (19.101) only contain the already known approximation values y,. yt+l.. , , yi+k-l. If p k # 0 holds. the method is called implicit, since then the required new value yz+k occurs on both sides of (19.101). IVe have to know the k initial values yo, y1,. . . , yk-] in the application of a k-step method. We can get these initial values. e.g., by one-step methods. b'e can derive a special multi-step method to solve the initial value problem (19.93) if n e replace the

19.4 Approximate Integration of Ordinary Differential Equations 903

derivative y'(x,) in (19.93) by a dzference formula (see 9.1.1.5, l., p. 494) or if we approximate the integral in (19.96) by a quadrature formula (see 19.3.1, p. 895). Examples of special multi-step methods are: 1. Midpoint Rule The derivative y'(xEtl) in (19.93) is replaced by the slope of the secant line between the interpolation nodes xi and xt+Z. We get: (19.102) yt+z - ~i = 2hfttl. 2. Rule of Milne The integral in (19.95) is approximated by the Simpson formula: h (19.103) YZ+Z - 212 = - ( f i -t4ft+l t f i + z ) . 3 3. Rule of Adams and Bashforth The integrand in (19.95) is replaced by the interpolation polynomial of Lagrange (see 19.6.1.2, p. 915) based on the k interpolation nodes x,,z,+~,.. . Z , + L - ~ . We integrate between xttk-1 and &+k and get: (19.104) The method (19.104) is explicit for yz+k. For the calculation of the coefficients Oj see [19.2]

19.4.1.4 Predictor-Corrector Method In practice, implicit multi-step methods have a great advantage compared to explicit ones in that they allow much larger step sizes with the same accuracy. But. an implicit multi-step method usually requires the solution of a non-linear equation to get the approximation value y,+k. This follows from (19.101) and has the form k

k-1

(19.105)

yaii

The solution of (19.105) is an iterative one. We proceed as follows: An initial value is determined by an explicit formula. the so-called predzctor. Then it will be corrected by an iteration rule y$;1) = F(y!!?) ( p = 0 . 1 , 2 , . . .). (19.106) which is called the corrector coming from the implicit method. Special predictor-corrector formulas are

The Simpson formula as the corrector in (19.108b) is numerically unstable and it can be replaced, e.g., by h (19.109) ~ 6.7ft-1 t 30.7fi t 8.lf$i). y y = 0 9yt - i + 0 . 1 t~ z(0.1fi-2

+

904

19. Numerical Analusis

19.4.1.5 Convergence, Consistency, Stability 1. Global Discretization Error and Convergence Single-step methods can be written generally in the form: (19.110) yZ+l = yi hF(z,, yt, h ) (i = 0, 1,2,. . ; yo given). Here F ( z :y, h) is called the incrementfunctionor progressive direction of the single-step method. The approximating solution obtained by (19.110) depends on the step size h and it should be denoted by y(z, h ) . Its difference from the exact solution y(z) of the initial value problem (19.93) is called the global discretization error g(z, h ) (see (19.111)), and we say: The single-step method (19,110) is convergent with orderp if p is the largest natural number such that (19.111) g ( z : h )= Y(2, h ) - Y ( 2 ) = O(hP)

+

.

z - 20

holds. Formula (19.111) says that the approximation y(z, h ) determined with the step size h = converges to the exact solution y(z) for every 2 from the domain of the initial value problem if h + 0. IThe Euler method (19.97) has order of convergence p = 1. For the Runge-Kutta method (19.99) p = 4 holds.

2. Local Discretization Error and Consistency The order of convergence according to (19.111) shows how well the approximating solution y(z, h ) approximates the exact solution y(z). Beside this, it is an interesting question of how well the increment function F ( r ,y. h ) approximates thederivative y’ = j ( z , y), For this purpose we introduce theso-called localdiscretization errorl(z. h) (see (19.112)) and we say: The single-step method (19,110)is consistent with orderp. ifp is the largest natural number with

’

h, - y(z) - F ( z ,y, h ) = O(hP), h It follows directly from (19.112) that for a consistent single-step method lim F ( z !y, h ) = f(z,y). h-0

l ( z ,h ) = y(z

(19.112) (19.113)

IThe Euler method has order of consistencyp = 1,the Runge -Kutta method has order of consistency p = 4.

3. Stability with Respect to Perturbation of the Initial Values In the practical performance of a single-step method, a rounding error O ( l / h ) adds to the global discretization error O(hP). Consequently, we have to select a not too small, finite step size h > 0. It is also an important question of how the numerical solution yi behaves under perturbations of the initial values or in the case z,+ co. In the theory of ordinary differential equations, an initial value problem (19.93) is called stable with respect to perturbations of its initial values i f ( 19.114) lG(x) - v(z)I I I G O - Yol. Here G(z)is the solution of (19.93) with the perturbed initial value $(q) = GO instead of yo. Estimation (19.114) tells that the absolute value of the difference of the solutions is not larger than the perturbation of the initial values. In general, it is hard to check (19.114). Therefore we consider the linear test problem (19.115) y’ = Xy with y(z0) =yo (A constant, X 5 0) which is stable, and a single-step method is applied to this special initial value problem. .A consistent method is called absolutely stable with step size h > 0 with respect to perturbed initial values if the approximating solution y1 of the above linear test problem (19.115) obtained by using the method satisfies the condition I Y L I I IYOI. (19.116)

19.4 Approximate Integration of Ordinary Differential Equations 905

IApplying the Euler polygon method for equation (19.115) results in the solution yttl = (1 + Xh)yt ( i = 0,1,. . .). Obviously, (19.116) holds if 11+Ah1 5 1, and so the step size must satisfy -2 5 Ah 5 0.

4. Stiff Differential Equations Many application problems, including those in chemical kinetics, can be modeled by differential equations whose solutions consist of terms converging to zero exponentially but in a high different kind of exponential decreasing. These equations are called stif differential equations. For example: y(z) = CleA1”+ CzexZs (Cl,Cz, XI, Xz const) ( 19.117) with X1 < 0, XZ < 0 and IX1,I < /Xzl, e.g., XI = -1, Xz = -1000. The term with Xz does not have a significant affect on the solution function, but it does in selecting the step size h for a numerical method. In such cases the choice of the most appropriate numerical method has special importance (see [19.23]).

19.4.2 Boundary Value Problems The most important methods for solving boundary value problems of ordinary differential equations will be demonstrated on the following simple linear boundary value problem for a differential equation of the second order: (19.118) y”(z)+ p(z)y‘(z) q(z)y(z)= f (x) (a 5 z 5 b) with y(a) = a , y ( b ) = p. The functions p(x), q(z) and f(x) and also the constants a and p are given. The given method can also be adapted for boundary value problems of higher-order differential equations.

+

19.4.2.1 Difference Method

+

We divide the interval [a, b] by equidistant interpolation points x, = zo vh (v = 0,1,2,. . . n;xo = a, z, = b) and we substitute the values of the derivatives in the differential equation at the interior interpolation points (19.119) Z I f f ( Z Y ) + P ( Z Y ) Y ’ ( Z Y ) + Q(ZY)Y(Z”) = f(zu) (v = 1 , 2 , ,, . . n - 1) by so-called finzte dzwzded dzfferences, e.g.: (19.120a)

= y:‘

Yutl -

2Y, -t Y Y - I

(19.120b) h2 This way, we get n - 1 linear equations for the n - 1 approximation values yu cz y(xu) in the interior of the integration interval [a,b],considering the conditions yo = a and y,, = p. If the boundary conditions also contain derivatives, they must also be replaced by finite expressions. Eigenvalue problems of differential equations (see 9.1.3.2, p. 513) are handled analogously. The application of the diflerence method, described by (19.119) and (19.120a,b), leads to a matrix eigenvalue problem (see 4.5, p. 278). IThe solution of the homogeneous differential equation y” + X 2 y = 0 with boundary conditions y(0) = y(1) = 0 leads to a matrix eigenvalue problem. The difference method transforms the differential equation into the difference equation yv+l - 2yu + y u - l + hZX2yu= 0. If we choose three interior points, hence h = 1/4, then considering yo = y(0) = 0, y4 = y(1) = 0 we get the discretized system ytf(z,)

(-2+;)Y1

+

=

Yz

= 0,

I

906

19. Numerical Analusis

This homogeneous equation system has a non-trivial solution only when the coefficient determinant is zero. This condition results in the eigenvalues XI’ = 9.37, X2’ = 32 and = 54.63. hmong them only the smallest one is close to its corresponding true value 9.87. Remark: The accuracy of the difference method can be improved by 1. decreasing the step size h, 2. application of a derivative approximation of higher order (approximations as (19.120a,b) have an error of order O ( h 2 ) ) , 3. application of multi-step methods (see 19.4.1.3, p. 902). If we have a non-linear boundary value problem, the difference method leads to a system of non-linear equations of the unknown approximation values yu (see 19.2.2, p. 893).

19.4.2.2 Approximation by Using Given Functions For the approximate solution of the boundary value problem (19.118) we apply a linear combination of suitably chosen functions gz(z),which are linearly independent and each one satisfies the boundary value conditions: n

Y(..)

= g(2) = C a , g t ( z ) .

(19.121)

Z=1

If we substitute g ( z ) into the differential equation (19.118), then we get an error, the so-called defect (19.122) 4 z ; a1, Q 2 . ’ ’ ’ > a,) = g ” ( z ) + p ( z ) g ’ ( z+) 9(z)g(z)To determine the coefficients a,, we can use the following principles (see also p. 910): 1. Collocation Method The defect is to be zero at n given points z,,the so-called collocatzon poznts. The conditions (19.123) ~(z,: a l , aZ1...,a,) = 0 (v = 1,2,. . . , T I ) , a < z1 < z2 < . . . < 2 , < b result in a linear equation system for the unknown Coefficients. 2. Least Squares Method We require that the integral b

~ ( a ~. . ., . ,aa,)~ = S E 2 ( 2 ; a 1 : a 2 , . . . . a , ) d z ,

(19.124)

a

depending on the coefficients, should be minimal. The necessary conditions aF - = 0 (i = 1,2,. . . , n)

da,

( 19.125)

give a linear equation system for the coefficients ai. 3. Galerkin Method We require that the so-called error orthogonality is satisfied, i.e.,

I.(.:

al, az,

.. .

~

a,)g&)

dz = 0 (i = 1 , 2 , . . . , n ) ,

(19.126)

a

and we get in this way a linear equation system for the unknown coefficients. 4. Ritz Method The solution y ( z ) often has the property that it minimizes the variational integral, h

(19.127) a

(see (10.4), p. 550). If we know the function H ( z , y , y ’ ) , then we replace y(z) by the approximation g(z) as in (19.121) and we minimize I[y] = I ( a l ,a’,. . .,a,). The necessary conditions

ai

- = 0 (i = 1,2, . . . , n) aa, result in n equation for the coefficients a,

(19.128)

19.4 Approximate Integration of Ordinary Differential Equations 907

ICnder certain conditions on the functions p , q, f and y, the boundary value problem -[P(4211(z)I‘+ q(z)y(z)= f(.) with y(a) = 0, Y(b) = and the variational problem

a

1[y] = [b(z)y”(z) a

+ q(z)y2(z)- 2f(z)y(z)]dz = min! with y(a) = a , y(b) = p

(19.129) (19.130)

are equivalent, so we can get H ( z ,y, y’) immediately from (19.130) for the boundary value problem of the form (19.129). Instead of the approximation (19.121), we often consider n

g ( z ) = sob) + CQtsz(4,

(19.131)

%=I

where go(z)satisfies the boundary values and the functions g%(z) satisfy the conditions (19.132) g,(a) = g,(b) = 0 (z = 1 , 2 , .. . , n). For the problem (19.118), we can choose, e.g., 3-(u (19.133) go(z) = a t -(z -a) b-a Remark: In a linear boundary value problem, the forms (19.121) and (19.131) result in a linear equation systems for the coefficients. In the case of non-linear boundary value problems we get non-linear equation systems, which can be solved by the methods given in Section 19.2.2, p. 893.

19.4.2.3 Shooting Method IVith the shooting method, we reduce the solution of a boundary value problem to the solution of an initial value problem. The basic idea of the method is described below as the szngle-target method.

1. Single-Target Method The initial value problem (19.134) y” p(z)?/lt q(z)y = f ( z ) with y ( a ) = a , y’(a) = s is associated to the boundary value problem (19.118). Here s is a parameter, which the solution y of the initial-value problem (19.134)depends on, Le., y = y(z, s) holds. The function y(z, s) satisfies the first boundary condition y(a, s) = (u according to (19.134). We have to determine the parameter s so that y(z. s) satisfies the second boundary condition y(b, s ) = p. Therefore, we have to solve the equation (19.135) F ( s j = y(b. s ) - B, and the regula falsi (or secant) method is an appropriate method to do this. It needs only the values of the function F ( s ) .but the computation of every function value requires the solution of an initial value problem (19.134) until z = b for the special parameter value s with one of the methods given in 19.4.1.

+

2. Multiple-Target Method In a so-called multzple-target method, the integration interval [a,b] is divided into subintervals, and we use the single-target method on every subinterval. Then, the required solution is composed from the solutions of the subintervals, where the continuous transition at the endpoints of the subintervals must be ensured. This requirement results in further conditions. For the numerical implementation of the multiple-target method. which is used mostly for non-linear boundary value problems. see. [19.24].

908

19. Numerical Analvsis

19.5 Approximate Integration of Partial Differential Equations In the following, we discuss only the principle of numerical solutions of partial differential equations using the example of linear second-order partial differential equations with two independent variables with the corresponding boundary or/and initial conditions.

19.5.1 Difference Method We consider a regular grid on the integration domain by the chosen points (xp,yu). Usually. this grid is chosen to be rectangular and equally spaced: (19.136) x P = 20 ph, yv = yo + ~l ( p , u = 1 , 2 , . . .). We get squares for 1 = h. If we denote the required solution by u(x,y), then we replace the partial derivatives occurring in the differential equation and in the boundary or initial conditions by finzte dzvzded dzflerences in the following way, where up,, denotes an approximate value for the function value

+

Finite Divided Difference

Order of Error

(19.137)

1

The error order in (19.137) is given by using the Landau symbol 0. In some cases; it is more practical to apply the approximation

with a fixed parameter u (0 5 u 5 1). Formula (19.138) represents a convex linear combination of two finite expressions obtained from the corresponding formula (19.137) for the values y = yv and y = yUtl. A partial differential equation can be rewritten as a diflerence equation at every interior point of the grid by the formulas (19.137), where the boundary and initial conditions are considered, as well. This equation system for the approximation values has a large dimension for small step sizes h and I , so usually, we solve it by an iteration method (see 19.2.1.4, p. 892).

W A: The function u(x,y) should be the solution of the differential equation Au = uzs+uyy= -1 for the points (x,y) with 1x1 < 1, IyI < 2 , i.e., in the interior of a rectangle, and it should satisfy the boundary conditions u = 0 for 1x1 = 1and lyl = 2. The difference equation corresponding to the differential equation for a square grid with step size h is:,+pu4 = uptl,u uIL,ytlt + up,u-lt h2. The step size h = 1 (Fig. 19.6) results in a first rough approximation for the function values at the three interior points: 4Uo,1 = 0 t 0 + 0 + uo,o 1, 4uo,o = O+Uo,i+O+Ua,-1 +I, 4Uo,~~=o+21o,o+o+o+1. 3 5 We get: uo,o= - FZ 0.429, ~ 0 =, U O~, - ~= - x 0.357

+

+

Figure 19.6

7

14

19.5 Approximate Inte.qration of Partial Diflerential Equations 909

IB: The equation system arising in the application of the difference method for partial differential equations has a very special structure. We demonstrate it by the following example which is a more general boundary value problem. The integration domain is the square G: 0 5 2 5 1, 0 5 y 5 1. We are looking for a function u(zly) with Au = u,, uya,= f (z, y) in the interior of G, u ( z ,y) = g(z, y) on the boundary of G. The functions f and g are given. The difference equation associated to this differential equation is. for h = 1 = l / n : u f i + ~ v + u f i , v + ~ t u f i - l , v t u ~ , v - l -= 4 u$f(z,,yv) ~,, ( p , v = 1 > 2 ,. . . ,n-1). Inthecaseofn = 5,the left-hand side of this difference equation system for the approximation values in the 4 x 4 interior points has the form (19.139):

+

Lrn Lmm - 4 1 0 1 - 4 1 0 1 - 4 0 0 1 -

1 0 0 0

0 0 1 4

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 - 4 1 0 1 0 0 1 - 4 1 0 1 0 0 1 - 4 0 0 1 0 0 1 -

0 0 0 1

0 0 1 4

1 0 0 0

0 1 0 0

0 0 1 0

0

0 0 0 1

I111 1 0 0 0

0 1 0 0

0 0 1 0

0 - 4 1 0 0 1 - 4 1 0 0 1 - 4 1 0 0 1 -

(19.139)

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

m1 1 0 0 0 1 0 0 0 1 0 0 0

0

0 0 1 4

0 0 0 1

-4 1 0 0

1 0 0 - 4 1 0 1 - 4 1 0 1 - 4

if we consider the grid row-wise from left to right, and considering that the values of the function are given on the boundary. We see that the coefficient matrix is symmetric and is a sparse rnatrzz. This form is called block-tridiagonal We can also see that the form of the matrix depends on how we select the grid-points. For different classes of partial differential equations of second order, such as elliptic, parabolic and hyperbolic differential equations, more effective methods have been developed, and also the convergence and stability conditions have been investigated. There is a huge number of books about this topic (see, Le., [19.22], [19.24]).

19.5.2 Approximation by Given Functions $e\'

approximate the solution u ( z ,y) by a function in the form n

+, Y) zz 4 2 , Y) = V O ( Z , Y ) + CwJ&Y).

(19.140)

1=1

Here, we distinguish between two cases: 1. uO(z.y) satisfies the given inhomogeneous differential equation, and the further functions v,(z>y) (i = 1 , 2 , , . , , n) satisfy the corresponding homogeneous differential equation (then we are looking for the linear combination approximating the given boundary conditions as well as possible). 2. uO(z,y) satisfies the inhomogeneous boundary conditions and the other functions v,(z,y) ( i = 1 , 2 . , . . , n) satisfy the homogeneous boundary conditions (then we are looking for the linear combination approximating the solution of the differential equation on the considered domain as well as

910

19. Numerical Analusis

possible). If we substitute the approximating function u(z3y) from (19.140) in the first case into the boundary conditions, in the second case into the differential equation. then in both cases we get an error term. the so-called defect E = E ( Z . y; a1, a ? , .. . , a n ) . (19.141) To determine the unknown coefficients a,, we can apply one of the following methods:

1. Collocation Method The defect E should be zero in n reasonably distributed points, a t the collocation points (xu,yy) (v = 1,2, . . . , n): (19.142) E ( Z ” , y,; a1: a * 2 . .. < a n )= 0 (I! = 1 , 2 , . . . , n). The collocation points in the first case are boundary points (we talk about boundary collocation): in the second case they are interior points of the integration domain (we talk about domain collocation). From (19.142) we get n equations for the coefficients. Boundary collocation is usually preferred to domain collocation. IWe apply this method to the example solved in 19.5.1 by the difference method, with the functions satisfying the differential equation: u(x. y; a l , a2, a3) = (z2+y2) +a1 +a2(x2 - y2)+a3(z4 - 6z2y2+y4). The coefficients are determined to satisfy the boundary conditions at the points (21%y1) = (1,0.5), (x2,y2) = (1,l.j)and ( ~ 3 ~ = ~ 3 (0.5.2) (boundary collocation). b’e get the linear equation system -0.3125 t a1 t 0 . 7 5 ~- 0 . 4 3 7 5 ~= 0; -0.8125 t a l - 1.25a2 - 7.4375a3 = 0, -1.0625 + a1 - 3 . 7 5 ~ 10.0625a3 = 0 with the solution al = 0.4562, a2 = -0.2000, a3 = -0.0143. We can calculate the approximate values of the solution a t arbitrary points with the approximating function. To compare the values with those obtained by the difference method: v(0,l) = 0.3919 and u(0,O) = 0.4562.

-a

+

2. Least Squares Method Depending on whether the approximation function (19.140) satisfies the differential equation or the boundary conditions, we require 1. either the line integral over the boundary C

I =

J Ez(z(t),y(t); a ] , . . .,a,) d t = min:

(19.143a)

(C)

where the boundary curve C i s given by a parametric representation z = z ( t ) !y = y(t). 2. or the double integral over the domain G (19.143b)

dI

From the necessary conditions, - = 0 (z = 1 , 2 , .. . , n ) , we get n equations for computing of the parameters al,a2,. . . , a,.

da,

19.5.3 Finite Element Method (FEM) After the appearance of modern computers the finite element methods became the most important technique for solving partial differential equations. These powerful methods give results which are easy to interpret. Depending on the types of various applications, the FEhl is implemented in very different ways, so here n e give only the basic idea. It is similar to those used in the Ritz method (see 19.4.2.2, p. 906)

)

19.5 Approximate integration of Partial DzfferentialEquations 911 for numerical solution of boundary value problems for ordinary differential equations and is related to spline approximations (see 19.7. p. 928). The finite element method has the following steps: 1. Defining a Variational Problem We formulate a variational problem to the given boundary value problem. The process is demonstrated on the following boundary value problem: l u = u,, uyy= f in the interior of G, u = 0 on the boundary of G. (19.144) It'e multiply the differential equation in (19.144) by an appropriate smooth function v ( z ,y) vanishing on the boundary of G, and we integrate over the entire G to get

+

(19.145) Applying the Gauss integral formula (see 13.3.3.1, l.,p. 663), where we substitute P ( z ,y) = -vuy and Q(z.y) = vu, in (13.119). we get the varzatzonal equatzonfrom (19.145) a(u,z.) = b(v) ( 19.146a) with (19.146b)

2. Triangularization The domain of integration G is decomposed into simple subdomains. Usually. we use a trzangularzratzon, where we cover G by triangles so that the neighboring triangles have a complete side or onlj a single vertex in common. Every domain bounded by curves can be approximated quite well by a union of triangles (Fig. 19.7). Remark: To avoid numerical difficulties, the triangularization should not contain obtuse-angled triangles. IA triangularization of the unit square could be performed as shown in Fig. 19.8. Here we start from the grid points with coordinates zp = ph, yv = vh ( p ,v = 0 , 1 , 2 , . . . , N ; h = l/N). We get (S- 1)' interior points. Considering the choice of the solution functions, it is always useful to consider the surface elements G,, composed of the six triangles having the common point ( x p ,y,,). (In other cases, the number of triangles may differ from six. These surface elements are obviously not mutually exclusive.) 3. Solution We define a supposed approximating solution for the required function u(x,y) in every triangle. A triangle with the corresponding supposed solution is called a jnzte element. Polynomials in 3 and y are the most suitable choices. In many cases, the linear approximation (19.147) G(z,y) = a1 t a22 + a3y

912

19. Numerical Analusis

is sufficient. The supposed approximating function must be continuous under the transition from one triangle to neighboring ones, so we get a continuous final solution. The coefficients a l , a2 and a3 in (19.147) are uniquely defined by the values of the functions u1, u2 and 113 at the three vertices of the triangle. The continuous transition to the neighboring triangles is ensured by this at the same time. The supposed solution (19.147) contains the approximating values u,of the required function as unknown parameters. For the supposed solution, which is applied as an approximation in the entire domain G for the required solution u ( z ,Y ) ~we choose (19.148) ,=l

,=l

n'e have to determine the appropriate coefficients cypy. The following must be valid for the functions uPv(z,y): They represent a linear function over every triangle of G,, according to (19.147) with the following conditions: 1 fork = p,1 = v, (19.149a) at any other grid point of GI".

2. u,&,

Y)

=0

for (z, Y)

# G,,.

(19.149b)

The representation of u,,(z,y) over G,, is shown in Fig. 19.9. The calculation of up,, over G,,, Le., over all triangles 1 to 6 in Fig. 19.8 is shown here only for triangle 1: uPv(z,y) = al + a22 a3 with (19.E O )

+

1 for z = z,,y = y, , u P L y ( z , y ) = 0 for z = ~ , - l , y = y , - ~ , 0 for z = z,,y = yY--l.

{

(19.151)

From (19.151) we have a1 = 1 - v, a2 = 0, a3 = l / h , and we get for triangle 1: u&,y)

Analogously. we have:

I

1-

(X

- p)

+ (f - v)

1+

(: v ) - p)

for triangle 2,

for triangle 4,

--

(i + (f

(19.152)

for triangle 3,

1-(;-p)

u,v(z,y) = ( 1 -

(: )

= 1t - - v

- v)

(19.153)

for triangle 5, for triangle 6.

4. Calculation of t h e Solution Coefficients We determine the solution coefficients cypv by the requirements that the solution (19.148) satisfies the variational problem (19.146a) for every solution function u , ~ .Le., we substitute G(z, y) for u(z. y) and u,,(z, y) for v(z,y) in (19.146a). This way, we get a linear equation system N-1 N - 1 ,=1

u=l

~ + v a ( u puki) ) = b ( ~ () k , 1 = 1 , 2 , . . . , N - 1)

(19.154)

19.5 Approximate Inteqration of Partial Differential Equations 913

for the unknown coefficients, where (19.155) In the calculation of u(upLy, ukl) we must pay attention to the fact that we have to integrate only in the cases of domains G ,, and Gkl with non-empty intersection. These domains are denoted by shadowing in Table 19.1. Table 19.1 Auxiliary table for FEM

Surface region

:;$ 1

~

1

1

2 3 4 5 6

2 3 4 5 6

1

pjj I

l

l

I% El6 4, p = k + l v=l+l

#

v i E--

Graphical Triangle of representation Gki G,, 1

;

1 5 2 4

2 6 3 5

-l/h -l/h

3 1

-l/h

4 2 5 1

n

0

llh

0

0 -l/h

5 3 6 2

6 4

7, p = k - l v=l-1

l 3

The integration is always performed over a triangle with an area h2/2, so for the partial derivatives with respect to z we get:

p1 ( 4 w - 2 a k t 1 , l - 2Qk-1,lj

hZ 2

-.

(19.156a)

914

19. Numerical Analysis

Analogously. for the partial derivatives with respect to y we have: (19.156b)

(19.157a) where V p is the volume of the pyramid over Since 1

Gkl

with height 1, determined by

ukl(z,y)

(Fig. 19.9).

1

(19.157b) VP= - 6 . -hZ we have b(ukl)FZ jklhz 3 2 So. the variational equations (19.154) result in the linear equation system . . , N - 1) ( k .1 = (19.158) 4~ - Q ~ + I , L ~ k - 1 ,~ O k , l t l - a k . l - 1 = hZ.fkl for the determination of the solution coefficients. Remarks: 1. If the solution coefficients are determined by (19.158), then C(z3y) from (19.148) represents an explicit approximating solution, whose values can be calculated for an arbitrary point (x,y) from G. 2. If the integration domain must be covered by an irregular triangular grid then it is useful to introduce triangular coordinates (also called barycentric coordinates). In this way, the position of a point can be easily determined with respect to the triangular grid, and the calculation of the multidimensional integral is made easier as in (19.155), because every triangle can be easily transformed into the unit triangle with vertices ( O , O ) , (0, l),(1,O). 3. If accuracy must be improved or also the differentiability of the solution is required, we have to apply piecewise quadratic or cubic functions to obtain the supposed approximation (see. e.g., [19.22]). 4. In practical applications, we usually obtain equation systems of huge dimensions. This is the reason why so many special methods have been developed, e.g., for automatic triangularization and for practical enumeration of the elements (the structure of the equation system depends on it). For detailed discussion of FEM see [19.13], [19.7], [19.22].

19.6 Approximation, Computation of Adjustment) Harmonic Analysis 19.6.1 Polynomial Interpolation The basic problem of interpolation is to fit a curve through a sequence of points (x,,, y,,) (v = 0,1, , . . n). This can happen graphically by any curve-fitting gadget, or numerically by a function g(x), which takes given values y,, at the points z, at the so-called interpolation points. That is g(z) satisfies the interpolation conditions (19.159) g(x,,)=y, ( v = 0 , 1 , 2 In the first place, we use polynomials as interpolation functions, or for periodic functions so-called trigonometric polynomials. In this last case we talk about trigonometric interpolation (see 19.6.4.1, 2., p. 924). If we have n + 1 interpolation points, the order of the interpolation is n, and the highest degree of the interpolation polynomial is at most n. Since with increasing degree of the polynomials, strong oscillation may occur, which is usually not required, we decompose the interpolation interval into subintervals and we perform a spline interpolation (see 19.7, p. 928). ~

I

19.6.1.1 Newton's Interpolation Formula To solve the interpolation problem (19.159)we consider a polynomial of degree n in the following form: g(z) = p,(z) = a0 al(z - zo) az(z - q ) ( z - 21) ..

+

+

+.

19.6 Approximation, Computation of Adjustment, Harmonic Analusis 915

+an(z - 2 0 ) ( 2 - 2 1 ) . . . (2 - & I ) . ( 19.160) This is called the Newton interpolation fornula, and it gives an easy calculation of the coefficients a, (i = 0 , l . . . . . n ) . since the interpolation conditions (19.159) result in a linear equation system with a triangular matrix. IFor n = 2 we get the annexed equation = Yo system from (19.159). The interpolation p z ( z O=) = Y1 polynomial pn(z)is uniquely determined pz(21)= + a1(21 p2('2) = 0' + 'l('2 - '0) + az(zz - '0)('2 - '1) = Y2 by the interpolation conditions (19.159). The calculation of the function values can be simplified by the Horner schema (see 19.1.2.1. p. 884).

19.6.1.2 Lagrange's Interpolation Formula Wecanfit apolynomialofn-thdegreethroughn+lpoints(z,, y,,) formula:

(u = 0,1,.

. . , n ) )withtheLagrange

n

g(2) = pn(.)

=

1~fiLfi(2).

(19.161)

=,O

Here L,(x) ( p = 0 , 1 , .. . , n ) are the Lagrange interpolation polynomials. Equation (19.161) satisfies the interpolation conditions (19.159), since 1 for p = u, (19.162) 0 for p # u. Here 6," is the Kronecker symbol. The Lagrange interpolation polynomials are defined by the formula

.

(19.163)

"#P

I\$e' fit a polynomial through the points given by the table b'e use the Lagrange interpolation formula (19.161) and we get:

5 17 L2(z) = - - 2 2 + -2 + 1. 6 6 The Lagrange interpolation formula depends explicitly and linearly on the given values y, of the function. This is its theoretical importance (see, e.g., the rule of Adams-Bashforth, 19.4.1.3, 3., p. 903). For practical calculation the Lagrange interpolation formula is rarely reasonable. p z ( 2 ) = 1 . Lo(2)

+3

'

Ll(2)

+2

'

19.6.1.3 Aitken-Neville Interpolation In several practical cases: we do not need to know the explicit form of the polynomial pn(z),but only its value at a given location z of the interpolation domain. We can get this function value in a recursive way due to Aitken and Neville. We apply the useful notation (19.164) pn(2) = PO,^ ...,,n(x)* in which the interpolation points 2 0 , xl, . . . , 2 , and the degree n of the polynomial are denoted. Notice that (2 - ZO)PI,Z,...,n(2) - ( 2 - Z ~ ) P O , ,..., ~ ,n-1(2) Z (19.165) P0,l ,...,n(x) = Xn - 20

I

916

19. Numerical Analysis

i.e.. the function value p o ~...,,(z) , can be obtained by linear interpolation of the function values of p l ~...,,(x) , and PO,~J,,.,,,-~(X), two interpolation polynomials of degree 5 n - 1. Application of (19.165) leads to a scheme which is given here for the case of n = 4: Yo =Po Y1 = PI Po1 Y Z = P2 PI2 Po12 !/3 = P3 P23 P123 PO123 y4 = P4 P34 P234 P1234 PO1234 = P4(x).

(19.166)

The elements of (19.166) are calculated column-wise. A new value in the scheme is obtained from its west and north-west neighbors (19.167a)

(19.167b) (19.167~)

For performing the Aitken-Neville algorithm on a computer we need to introduce only a vector p with n + 1 components (see [19.4]), which takes the values of the columns in (19.166) after each other according to the rule that the value ~ i - + k + l , . . . , ~ (i = k, k + 1, , n) of the k-th column will be the i-th component p, of p. The columns of (19.166)must be calculated from the top down. so we will have all necessary values. The algorithm has the following two steps: (19.168a) 1. For i = O . l , . . . . n set p,=y,. 2-x, 2. For k = 1,2,. . . , n and for i = n,n - 1,.. . , k compute pi = p i t L ( p , - pi-1).(19.168b) xi - Xi-k t . .

after finishing (19.168b) we have the required function value p,(x) at x in element p,.

19.6.2 Approximation in Mean The principle of approximation in mean is known as the Gauss least squares method. In calculations we distinguish between continuous and discrete cases.

19.6.2.1 Continuous Problems, Normal Equations The function f ( x ) is approximated by a function g(x) on the interval [a, b] so that the expression b

F = /.(.)If(.)

- 9(x)I2dz,

(19.169)

a

depending on the parameters contained by g(x), should be minimal. W(X)denotes a given weight function, such that ~ ( x >) 0 in the integration interval. If we are looking for the best approximation g(x) in the form n

g(x) = C a t g i ( x )

(19.170)

t=O

with suitable linearly independent functions go(x), gl(x), . . . , gn(x), then the necessary conditions

aF

-= 0 8Qt

(2

= O , l , . . .,n)

(19.171)

19.6 Approximation, Computation of Adjustment, Harmonic Analysis 917

for an extreme value of (19.169) result in the so-called normal equation system (19.172)

( 19.173a)

(19.173b) which are considered as the scalar products of the two indicated functions The system of normal equations can be solved uniquely, since the functions go(z),g l ( z ) ,. . . , gn(z) are linearly independent. The coefficient matrix of the system (19.172) is symmetric, so we could apply the Cholesky method (see 19.2.1.2. p. 890). The coefficients a, can be determined directly, without solving the equation system, if the system of functions gz(z)is orthogonal, that is, if = 0 for z

(gz.gk)

# k.

(19.174)

We call it an orthonormalsystem. if (19.175) R'ith (19.175). the normal equations (19.172) are reduced to a, = ( f . g Z ) (i = 0.1.. . . n).

(19.176)

~

Linearly independent function systems can be orthogonalized. From the power functions gt(z) = z2 (i = 0.1.. . . n ) ,depending on the weight function and on the interval, we may obtain the orthogonal polynomials in Table 19.2. Table 19.2 Orthogonal polynomials

.

i

' Name of the polynomials see p. Legendre polynomial P,(z) (19.177)

Chebysev polynomial Tn(z) Laguerre polynomial L,(z) Hermite polynomial H,(x)

511 512

With these polynomial systems we can work on arbitrary intervals: 1. Finite approximation interval. 2. Approximation interval infinite at one end, e.g., in time-dependent problems. 3. Approximation interval infinite at both ends, e.g., in stream problems. Every finite interval [a,b] can be transformed by the substitution

b+a b-a z = -t -t 2

2

(x E [a,b], t

E

[-1.11)

(19.178)

918

19. Numerical Analusis

into the interval [-1.11.

19.6.2.2 Discrete Problems, Normal Equations, Householder's Method Let S pairs of \slues (xu,y u ) be given. e.g., by measured values. We are looking for a function g(x). whose values g(lru)differ from the given values yv in such a way that the quadratic expression Y

F =

C[Yy- g(xv)12

(19.179)

"=l

is minimal. The value of F depends on the parameters contained in the function g(z). Formula (19.179) represents the classical sum ofresidualsquares. The minimizationof the sum of residual squares is called l3F the least squares method. From the assumption (19.170) and the necessary conditions - = 0 (i = aa, 0 . 1 , . . . n ) for a relative minimum of (19.179) we obtain a linear equation system for the coefficients: which is called the normal equations: n

a,[g,gkI = [ygkl ( k = 0%1.. . . , n).

(19.180)

0'2

Here we used the Gaussean sum symbols in the following notation: ,v

Y

[g&] =

gl(zv)gc(z,), (19.181a)

[ygk] =

1yvgk(z,)

(i, k = 0 , l . . . . , n).

(19.181b)

Lsually, n < Z. IFor the polynomial g(z) = a. + a l z + ' . + a,zn: the normal equations are a0[zk] al[zkt1] ' . u,[xktn] = [z"] ( k = 0:1.. . . n) with [zk]= Cf=, x V k l [xO]= N ! [xky] = zykyy: [y] = y,. The coefficient matrix of the normal equation system (19.180) is symmetric, so for the numerical solution we may apply the Cholesky method. The normal equations (19.180) and the residue sum square (19.179) have the following compact form:

+

+ +

~

Y1

a0

a= YN

I:]

(19.182a)

.

(19.18213)

an

If, instead of the minimalization of the sum of residual squares. we want to solve the interpolation problem for the S points ( z v 3yy), then we have to solve the following system of equations:

Gn = y.

(19.183)

This equation system is overdetermined in the case of n < M - 1, and usually it does not have any solution. \Ye get (19.180) or (19.182a) ifwe multiply (19.183) by GT.

19.6 Approximation, Computation of Adjustment, Harmonic Analysis 919

From a numerical viewpoint, the Householder method (see 4.4.3.2,2., p. 278) is recommended to solve equation (19.183), and this solution results in the minimal sum of residual squares (19.179).

19.6.2.3 Multidimensional Problems 1. Computation of Adjustments Suppose that there is a function f(zl,2 2 , . . . ,z,) of n inde, z,. We do not know its explicit pendent variables 21, 2 2 , form; only N substitution values fy are given, which are, in general, measured values. We can write these data in a table (see (19.184)). The formulation of the adjustment problem is clearer if we introduce the following vectors: I

x

.

z1

.

= ( X I )z2).. .~z,)~

:

&“, . . .

x(’) = (xi“’,

:

x2

i Xn

f

.. .. zp zp fl

fz

... z ”

’

i

(19.184) p

fw

Vector of n independent variables, Vector of the v-th interpolation node (v = 1,.. . , N ) , Vector of the N function values at the N interpolation nodes.

: f = (fl, j z , . . . f N ) T We approximate f(z1,xZr.. I,) = f(x)by a function of the form I

(19.185) Here, the m + 1 functions gz(zl,2 2 , . . . ,z,) = g,(x) are suitable, selected functions. IA: Linear approximation by n variables: g(zl, 2 2 , . , , , 2 , ) = a0 t alzl t a222 + * . t anz,. IB: Complete quadratic approximation with three variables: g(q.z2,z3)= a0 alzl a222 a313 a4z12 a5xZ2 a623’ a72122 t Q8z1z3t a92223.

+

+

+

+

+

+

+

The coefficients are chosen to minimize C,”=,[fy - g (zp),zr),. . . , z;))]’.

2. Normal Equation System Analogously to (19.182b) we form the matrix G, in which we replace the interpolation nodes z, by vectorial interpolation nodes x(”)(v = 1,2,. . . , N ) . To determine the coefficients, we can use the normal equation system GTGa = GTf (19.186) or the overdetermined equation system Ga = f . (19.187)

IFor an example of multidimensional regression see 16.3.4.3, 3., p. 780.

19.6.2.4 Non-Linear Least Squares Problems We show the main idea for a one-dimensional discrete case. The approximation function g(z) depends non-linearly on certain parameters. IA: g(z) = aOeal” t a2ea3’. This expression does not depend linearly on the parameters a1 and a3. IB: g(z) = aOeal‘ cos azz. This function does not depend linearly on the parameters a1 and az. We indicate the fact that the approximation function g(z) depends on a parameter vector a = (ao, a l , . . . , a,)T by the notation g = g ( r , a )= d z ; a o > a l , , , . , a n ) . (19.188) Suppose, N pairs of values ( z v ,yy) (v = 1 , 2 , . . . , N ) are given. To minimize the sum of residual squares (19.189)

I

920

19. Numerical Analusis

dF

the necessary conditions - = 0 (i = 0, 1,. . . , n) lead to a non-linear normal equation system which da, must be solved by an iterative method, e.g., by the Newton method (see 19.2.2.2, p. 894). .4nother way to solve the problem, which is usually used in practical problems, is the application of the Gauss-Newton method (see 19.2.2.3, p. 894) given for the solution of the non-linear least squares problem (19.24). We need the following steps to apply it for this non-linear approximation problem (19.189): 1. Linearization of the approximating function g(z, a) with the help of the Taylor formula with respect to a,. To do this, we need the approximation values ato) (i = 0 , 1,. . . , n):

2. Solution of the linear minimum problem N

C[yv- 5(zY,a)12= min!

(19.191)

"=I

with the help of the normal equation system

G T G h = GTAy -

(19.192)

or by the Householder method. In (19.192) the components of the vectors Aai = a, - a?)

and & are given as

(i = 0,1,2,. . . n) and ~

(19.193a)

(19.193b) Ayu = yu - g(z,,a(O)) (v = 1 , 2 , .. . , N). The matrix G can be determined analogously to G in (19.182b), where we replace g,(z,) by ( 2 = 0,1,. . . ,n; v = 1 , 2 , .. . , N). da, 3. Calculation of a new approximation

-(zu,ajO') ag

(19.194) a!') = ato)+ Aai or a!') = a!') + TAai (i = 0,1,2, . . . , n), where y > 0 is a step length parameter. By repeating steps 2 and 3 with a!') instead of a?), etc. we obtain a sequence of approximation values for the required parameters, whose convergence strongly depends on the accuracy of the initial approximations. We can reduce the value of the sum of residual squares with the introduction of the multiplyer y.

19.6.3 Chebyshev Approximation 19.6.3.1 Problem Definition and the Alternating Point Theorem 1. Principle of Chebyshev Approximation Chebyshev approximationor uniform approximationin the continuous case is the following: The function j ( z ) is to be approximated in an interval a 5 z 5 b by the approximation function g(x) = g(z; ao, a l , . . . ,an) so that the error defined by (19.195) max I j ( z ) - g(z; ao, a ] , . . . , an)l = O(ao,a ' , . . . ,an) asxsb should be as small as possible for the appropriate choice of the unknown parameters a, (i = 0, 1,. . . , n). If there exists such an approximating function for f(z),then the maximum of the absolute error value will be taken at least at n+ 2 points zv of the interval, at the so-called altematingpoints, with changing signs (Fig. 19.10).This is actually the meaning of the altematingpoint theorem for the characterization of the solution of a Chebyshev approximation problem.

19.6 Approximation, Computation of Adjustment, Harmonic Analysis 921

Figure 19.10

IIfwe approximate the function !(I) = x" on the interval [-1,1] by a polynomialofdegree 5 n- 1in the Chebyshev sense, then we get the ChebyshevpolynomialT,(z)as an error function whose maximum is normed to one. The alternating points, being at the endpoints and at exactly n - 1 points in the interior of the interval, correspond to the extreme points of T,(z) (Fig. 19.11a-f).

d)

e)

f)

Figure 19.11

19.6.3.2 Properties of the Chebyshev Polynomials 1. Representation T,(z) = cos(narccosz), 1

+ m)" +

(19.196a)

my]

- [(I (I , 2 cos nt, I = cost for l 2 1 < 1, (n = 1 , 2 , . . .), Tn(x) = coshnt, x = cosht for 1x1 > 1 T"(Z) =

{

(19.196b)

( 19.196~)

I

922

19. Numerical Analusis

2. Roots of Tn(z) ( p = 1 , 2 , . . , , n). 2n 3. Position of the e x t r e m e values of Tn(z)for z E [-1,1]

xp = cos (2p - l)?r

(19.197)

~

UT

( v = 0 . 1 , 2 ,...,n). n 4. Recursion Formula Tn+l= 2xTn(2) - Tn-l(Z)( n = 1 , 2 , . . . ; To(x)= 1, T,(z)= 2) From this recursion we have for example T2(2) = 2x2 - 1, T3(2)= 4x3 - 32, T~(Z) = 8x4- 82’+ 1, T5(x)= 16x5 - 20x3 52, 5u=cos-

T6(2) =

(19.198) (19.199)

(19.200a)

+

+

(19.200b)

32x6 - 48x4 182’ - 1,

(19.2OOc)

T,(x)= 64x’ - I l k 5 + 56x3 - 72, TB(z) = 1282’ - 2 5 6 + ~ 1~ 6 0 -~ 322’ ~ + 1, Tg(x)= 2562’ - 5762’ + 432x5 - 120x3 + 9x, Tlo(x)= 5 1 2 ~ ”- 12802’ + 1 1 2 0 -~ ~4 0 0 + ~ 50x’ ~ - 1.

(19.200d) (19.200e) (19.200f) (19.200g)

19.6.3.3 Remes Algorithm 1. Consequences of the Alternating Point Theorem The numerical solution of the continuous Chebyshev approximation problem originates from the alternating point theorem. We choose the approximating function (19.201) with n + l linearly independent known functions, and we denote by ai’ (i = 0,1,. . . , n)the coefficients of the solution of the Chebyshev problem and by Q = @(ao’,a ~ *. .~,an*) . the minimaldeviation according to (19.195). In the case when the functions f and gl (i = 0,1,.. . , n) are differentiable, from the alternating point theorem we have n

n

Cai’gi(2,)+(-1)”e=f(2,),

CaZ*gI(zv) = f’(x,) ( v = 1,2, . . . ,n + 2 ) .

i=O

i=O

(19.202)

The nodes x, are the alternating points with (19.203) a 5 x1 < x2 < . . . < x,+~5 b. The equations (19.202) give 2n + 4 conditions for the 2n + 4 unknown quantities of the Chebyshev approximation problem: n + 1 coefficients, n + 2 alternating points and the minimal deviation e. If the endpoints of the interval belong to the alternating points, then the conditions for the derivatives are not necessarily valid there.

2. Determination of the Minimal Solution according to Remes According to Remes, we proceed with the numerical determination of the minimal solution as follows: 1. We determine an approximation of the alternating points xu(0) (v = 1 , 2 , .. . , n 2) according to (19.203).e.g.! equidistant or as the positions of the extrema of Tn+l(x) (see 19.6.3.2, p. 920). 2. We solve the linear equation system

+

19.6 Approximation, Computation of Adjustment, Harmonic Analusis 923 and as a solution, we get the approximations aI(0) (i = 0 , 1 , . . . , n) and

eo.

3. \$’e determine a new approximation of the alternating points z Y ( l )(v = 1,2,. . . , n positions of the extrema of the error function f ( x ) -

e

+ 2), e.g., as

a,(’)g,(z). Now, it is sufficient to apply only

,=O

approximations of these points. By repeating steps 2 and 3 with z V ( l )and a,(’) instead of zY(”)and a,(’), etc. we obtain a sequence of approximations for the coefficients and the alternating points, whose convergence is guaranteed under certain conditions, which can be given (see [19.25]). We stop calculations if, e.g., from a certain iteration index p (19.204) holds with a sufficient accuracy.

19.6.3.4 Discrete Chebyshev Approximation and Optimization From the continuous Chebyshev approximation problem I

n

I

(19.205)

+

we get the corresponding discrete problem, if we choose N nodes X, (v = 1,2,. . . , N ; N 2 n 2) with the property a 5 $1 < 22 . < X N 5 b and require (19.206) We substitute (19.207) and obviously we have if(X”)

- kaigi(Xu)1I Y (v = 1 > 2 , .. . , N ) .

(19.208)

2=0

Eliminating the absolute values from (19.208) we obtain a linear inequality system for the coefficients a, and 7,so the problem (19.206) becomes a linear programming problem (see 18.1.1.1, p. 844):

I :I 7

y = min! subject to

+ ,E aig,(zu) 2

7-

f(XV),

c a,gi(z”) 2 - f ( z v )

( v = 1 , 2 ,..., N ) .

(19.209)

Equation (19.209) has a minimal solution with y > 0. For a sufficiently large number N of nodes and with some further conditions the solution of the discrete problem can be considered as the solution of the continuous problem. n

a,gi(z) a non-linear approximation

If we use instead of the linear approximation function g(z) = i=O

function g(z) = g(z: ao,a l , . . . , a n ) ,which does not depend linearly on the parameters ao, al, . . . a,, then we obtain analogously a non-linear optimization problem. It is usually non-convex even in the cases of simple function forms. This essentially reduces the number of numerical solution methods for

I

924

19. Numerical Analgais

non-linear optimization problems (see 18.2.2.1, p. 861).

19.6.4 Harmonic Analysis We want to approximate a periodic function f(z)with period 27r, which is given formally or empirically, by a trigonometric polynomial or a Fourier sum of the form a0 g(x) = - t C ( a k c o s k z t b k s i n k z ) ,

(19.210)

k=l

where the coefficients ao, a k and bk are unknown real numbers. The determination of the coefficients is the topic of harmonic analysis.

19.6.4.1 Formulas for Trigonometric Interpolation 1. Formulas for the Fourier Coefficients Since the function system 1, cos kx, sin kx ( k = 1 , 2 , .. . , n) is orthogonal in the interval [0,27~]with respect to the weight function w

I 1, we

get the formulas for the coefficients (19.211)

by applying the continuous least squares method according to (19.172). The coefficients ah and bk calculated by formulas (19.211) are called Fourier coeficienta of the periodic function f ( z ) (see 7.4, p. 418). If the integrals in (19.211) are complicated or the function f(z) is known only at discrete points, then the Fourier coefficients can be determined only approximately by numerical integration. Using the trapezoidal formula (see 19.3.2.2, p. 896) with N t 1 equidistant nodes 27T ( 19.212) 2, = vh (U = O , l , . . . N ) , h = -

N

we get the approximation formula 2 2 N 6k = - f ( Z v ) COS kz,, b k X gk = - f(X,) Sinkz, ( k = 0,1,2,. . . , n). (19.213) N u=l N w=l The trapezoidal formula becomes the very simple rectangular formula in the case of periodic functions. It has higher accuracy here as a consequence of the following fact: Iff z is periodic and (2m+ 2) times differentiable, then the trapezoidal formula has an error of order O(h'+'). 2. Trigonometric Interpolation Some special trigonometric polynomials formed with the approximation coefficients 6k and 6, have important properties. Two of them are mentioned here: 1. Interpolation Suppose N = 2n holds. The special trigonometric polynomial

ak

1

RZ

n-1 1 (ick cos kx t 6, sin kz) t -6, cos nz ijl(x)= $ t i o 2 2

(19.214)

k=l

with coefficients (19.213) satisfies the interpolation conditions (19.215) at the interpolation nodes z, (19.212). Because of the perodicity of f(z) we have f(zo) = f(z,v). 2. Approximation in M e a n Suppose N = 2n. The special trigonometric polynomial ij2(x)= sa0 1- t x ( 6 k c o s k x t i k s i n k z )

k=l

(19.216)

19.6 Approximation, Computation of Adjustment, Harmonic Analysis 925

form < nand with the coefficients (19.213) approximates the function f(z)in discrete quadratic mean with respect to the N nodes x, (19.212), that is, the residual sum of squares N

[fb") - Bz(r,)I2

F=

(19.217)

"=l

is minimal. The formulas (19.213) are the originatingpoint for the different ways of effective calculation of Fourier Coefficients.

19.6.4.2 Fast Fourier Transformation (FFT) 1. Computation costs of computing Fourier coefficients The sums in the formulas (19.213) also occur in connection with discrete Fourier transformation, e.g., in electrotechnics, in impulse and picture processing. Here N can be very large, so the occurring sums must be calculated in a rational way, since the calculation of the N approximating values (19.213) of the Fourier coefficients requires about N Zadditions and multiplications. For the special case of N = 2 P , the number of multiplications can be largely reduced from NZ(=22P) topN(= p2P) with the help of the so-called fast Fourier transformation FFT.The magnitude of this reduction is demonstrated on the example on the right-hand side. By this method, the computation costs and computation time are reduced so effectively that in some important application fields even a smaller computer is sufficient. The FFT uses the properties of the N-th unit roots, Le., the solutions of equation z N = 1to a successive sum up in (19.213).

2. Complex Representation of the Fourier Sum The principle of FFT can be described fairly easily if we rewrite the Fourier sum (19.210) with the formulas

into the complex form (19.220)

If we substitute

'/

U k - ibk ck = -, (19.221a) then because of (19.211) ck = 2 2.rr and (19.220) becomes the complex representation of the Fourier sum:

Ckeikx with

g(2) =

C-k

211

0

f(x)e-ikz2dx,

= Ek.

(19.221b)

(19.222)

k=-n

If the complex coefficients ck are known, then we get the required real Fourier coefficients in the following simple way: Uo = 2C0, Uk = 2Re(Ck), bk = -2Im(Ck) (k = 1,2,...,?I). (19.223)

3. Numerical Calculation of the Complex Fourier Coefficients For the numerical determination of ck we apply the trapezoidal formula for (19.221b) analogously to (19.212) and (19.213), and get the discrete complex Fourier coefficients Ek:

.

hT-1

N-1

(19.224a)

926

19. Numerical Analysis 1

fu

= lyf(”u)’

“u

2au N

Zni -W N = ~N .

= - ( u = 0 , 1 , 2 ,...,N - 1 ) ,

(19.224b)

Relation (19.224a) with the quantities (19.224b) is called the dzscrete complez Fourier transformation of length N of the values fv ( u = 0 , 1 , 2 , . . . , N - 1). The powers uk = z ( u = 0,1,2,. . . , N - 1) satisfy equation z N = 1. So, they are called the N-th unit roots. Since e-’‘’ = 1, W”+Z = w;, . . . . UN” = 1, w”+1 = (19.225) The effective calculation of the sum (19.224a) uses the fact that a discrete complex Fourier transforN mation of length N = 2n can be reduced to two transformations with length - = n in the following 2 way:

a) For every coefficient Ek with an even index, i.e., k = 21, we have In-1

E21

n-1

fyWp= 1 [ f u w 2t fn+uW;(n+”)]

= u=o

n-1

=

[fu

t f n + J w?.

(19.226)

”=O

u=o

wpwb

Here we use the equality w:(nt”) = = up. If we substitute Y, = f v t fn+v ( u = 0,1,2, , , , n - 1) and consider that w;, = wn, then j

n-1 E21

=

y”U:

( u = 0 , l : 2,. . . , n - 1)

(19.227)

(19.228)

”=O

is the discrete complex Fourier transformation of the values yu (u = 0 , 1 , 2 , . . . , n - 1) with length N

n=T. b) For every coefficient

with an odd index, i.e., with k = 22 + 1, we get analogously: (19.229)

(19.230) “-1

C2ltl

yn+&k

=

( u = 0,1,2:.

I . ,

n - 1)

(19.231)

”=O

which is the discrete complex Fourier transformation of the values yn+u ( u = 0, 1 , 2 , . . . , n - 1) with IY

length n = -. 2 The reduction according to a) and b), Le., the reduction of a discrete complex Fourier transformation to two discrete complex Fourier transformations of half the length, can be continued if N is a power of 2, Le., if N = 2 P (pis a natural number). The application of the reduction after p times is called the FFT. N Since every reduction step requires - complex multiplications because of (19.230), the computation 2 cost of the FFT method is

I

-.vp = 1 v log, N .

2

2

(19.232)

19.6 Avwoxzmation, Computation of Adiustment, Harmonic Analysis 927

4. Scheme for FFT For the special case N = 8 = 23, the three corresponding reduction steps of the FFT according to

Step 1

fo Yo = fo + f 4

+

fl

Y1

fi

YZ

= fl f 5 = f2 t f 6

f3

Y3

=f3

+f 7

Step 2

Step 3

yo := Yo + Yz Y1 := Y1 t Y3 Yz := (Yo - Y z ) 4

Yo := Yo f Y1 Y1 := (Yo - Y l ) 4

Y4

= (fo - f

4 ) 4

f5

Y5

= (fl - f

5 ) 4

f6

?j6

= ( f 2 - f6)lU’;

f7

Y7

y3

Y3

:= (Yz - Y 3 ) 4

:= Y4

Y4

:= Y4

Y5

:= (Y4 - Y 5 1 4

Y6

:= (Y4 - Y6)w40

y6

:= y 6 t 9 7

Y7

:= (Y5 - Y 7 ) 4

N

:= 4,

:= (Y1 - Y 3 ) 4

+ Y6 Y5 := Y5 + Y7

= (f3 - f,)wg3 277, IV = 8, n := 4, wg = e - T

:= YZ f

Y4

Y3

f4

92

+ Y5

Y7 := (Y6 - Y 7 ) 4 n := 2, w4 = w; N := 2, n := 1, w2 = wi

We can observe how terms with even and odd indices appear. In Scheme 2 (19.233) the structure of the method is illustrated. Scheme 2:

(19.233)

If the coefficients Ek are substituted into Scheme 1 and we consider the binary forms of the indices before step 1 and after step 3, then we recognize that the order of the required coefficients can be obtained by simply reverszng the order of the bzts of the binary form of their indices. This is shown in Scheme 3.

Scheme 3:

Index

Eo

000

6

Eo

E1

E2

E4

E5

OOL OLO OLL LOO LOL

E6

LLO

E7

LLL

E2

E4

Step 1 Step 2

Step 3 Index

6

000

E4

E4

LOO

E2

E2

OLO

26

E6

E6

LLO

E1

El

El

OOL

E3

E5

E5

E5

E3

E3

LOL OLL

E7

E7

E7

LLL

271 1 creteFouriertransformation. WechooseN = 8. Withz, = -, f - -f(x,) (v = 0,1,2,. . . , 7 ) , wg = 8 ’-8 e

_-Zni8

= 0.707107(1 - i). w,’= -is ui = -0.707107(1 t i ) we get Scheme 4:

I

928

19. Numerical Analusis

Scheme 4:

Step 1

Step 2

fo = 2.467401 yo = 3.701102

yo = 6.785353

Step 3 yo = 13.262281

fl

= 0.077106 y1 = 2.004763

y1 = 6.476928

y1 = 0.308425

= E4

fi

= 0.308425 y~ = 3.084251

yz = 0.616851

+ 2.467402 i = E2

f3

= 0.693957

f4

= 1.233701 y4 = 1.233700

U?

= 4.472165

y2

= 0.616851

un = 2.467402 i 214

= 1.233700

=&

y3 = 0.616851 - 2.467402 i = E6

y4 = 2.106058 + 5.9568331 = E 1

+2.467401 i f5

= 1.927657 y5 = -1.308537(1 - i)

y5

= 0,872358

ys = 0.361342 - 1.022031 i = E5

$3.489432 i fe = 2.775826 y6 = 2.467401 i

y6 =

1.233700

y6 = 0.361342 + 1.022031 i = E3

-2.467401 i f7

= 3.778208 y7 = 2.180895(1+ i)

y7 = -0.872358

y7 = 2.106058 - 5.9568331 = E7

+3.489432 i From the third (last) reduction step we get the required real Fourier coefficients according to (19.223). (See the right-hand side.) In this example, the general property CN-k

= ck

= 26.524 562 = 4.212 116 az = 1.233 702 a3 = 0.722684 a4 = 0.616850 a0

a1

= -11.913 666 bz = - 4.934 804 b3 = - 2.044 062 bd = 0 (19.234) bl

Of the discrete complex Fourier coefficients can be observed. For k = 1 , 2 , 3 , we see that E-, = & ,

4=

E2, E5 = E3.

19.7 Representation of Curves and Surfaces with Splines 19.7.1 Cubic Splines Since interpolation and approximation polynomials of higher degree usually have unwanted oscillations, it is useful to divide the approximation interval into subintervals by the so-called nodes and to consider a relatively simple approximation function on every subinterval. In practice, cubic polynomials are mostly used. We require a smooth transition at the nodes of this piecewise approximation.

19.7.1.1 Interpolation Splines 1. Definition of the Cubic Interpolation Splines, Properties Suppose there are given N interpolation points (xi,fi) (i = 1 , 2 , .. . , N ;z1 < 22 < . . . X N ) . The cubic interpolation spline S(z) is determined uniquely by the following properties: 1. S(z) satisfies the interpolation conditions S(zi)= fi (i = 1 , 2 , .. . , N). 2. S(z) is a polynomial of degree 5 3 in any subinterval [xi, (i = 1 , 2 , .. . , N - 1). 3. S(z) is twice continuously differentiable in the entire approximation interval [XI,ZN]. 4. S(z) satisfies the special boundary conditions: a) S”(z1) = S”(ZN) = 0 (we call them natural splines) or b) S’(z1) = fl’, s ’ ( z ~ = ) fN’ (fit and f ~ are ’ given values) or c) S(z1) = S ( X N in ) ~the cme of f1 = f N , S’(z1) = S ’ ( I N )and S”(z1) = S ” ( X N(we ) call them

19.7 Representation of Curves and Surfaces with Splines 929

periodic splines). It follows from these properties that for all twice continuously differentiable functions g ( z )satisfying the interpolation conditions g(zl)= ft (i = 1 , 2 , .. . , N ) (19.235) is valid (Holladay’s Theorem). Based on (19.235) we can say that S(z) has minimal total curvature, since for the curvature ti of a given curve, in a first approximation, ti M S”(see 3.6.1.2, 4.,p. 228). It can be shown that if we lead a thin elastic ruler (its name is spline) through the points (xi,j,) (i = 1:2 , . . . , N ) , its bending line follows the cubic spline S(z). 2. Determination of the Spline Coefficients We suppose that the cubic interpolation spline S(z) for z E [xi,z,+l]has the form:

+

+

(19.236) S(z) = Si(z)= a, bi(z - xi) t ci(z - xi)’ di(z - z ~ i ) (i ~ = 1 , 2 , .. . , N - 1). The length of the subinterval is denoted by hi = ziti - zi.We can determine the coefficients of the natural spline on the following way: 1. From the interpolation conditions we get a,=fi ( i = 1 , 2,...,N - 1 ) . (19.237) It is reasonable to introduce the additional coefficient a N = j N , which does not occur in the polynomials. 2. The continuity of Sff(z)at the interior nodes requires that

di-l - a‘ - c$-l (i = 2,3, . . . , N - 1). (19.238) 3h,-1 The natural conditions result in c1 = 0, and (19.238) still holds for i = N , if we introduce C N = 0. 3. The continuity of S(z) at the interior nodes results in the relation 2ci-l c, (19.239) b.1-1 - ai - ai-1 h,-l (i = 2 , 3 , . . .,A’). 3 hi-i 4. The continuity of S f ( z )at the interior nodes requires that

+

(i = 2,3,. . . ,N - 1). (19.240)

Because of (19.237), the right-hand side of the linear equation system (19.240) to determine the coefficients c, (i = 2,3, . . . , N - 1; c1 = c N = 0 ) is known. The left hand-side has the following form:

(19.241)

I

0

The coefficient matrix is tridzagonal, so the equation system (19.240) can be solved numerically very easily by an LR decomposition (see 19.2.1.1, 2., p. 888). We can then determine all other coefficients in (19.239) and (19.238) with these values c,.

19.7.1.2 Smoothing Splines The given function values j , are usually measured values in practical applications so they have some error. In this case, the interpolation requirement is not reasonable. This is the reason why cubic smoothang

I

930

19. Numerical Analusis

splines are introduced. We get this spline if in the cubic interpolation splines we replace the interpolation requirements by

2[1-’

+ X T[S”(r)]’dx = min!.

,=l

(19.242)

XI

We keep the requirements of continuity of S, S’and S“, so the determination of the coefficients is a constrained optimization problem with conditions given in equation form. The solution can be obtained by using a Lagrange function (see 6.2.5.6, p. 401). For details see [19.26]. In (19.242)X (A 2 0) represents a smoothingparameter, which must be given previously. For X = 0 we get the cubic interpolation spline, as a special case, for “large” X we get a smooth approximation curve, but it returns the measured values inaccurately, and for X = 03 we get the approximating regression line as another special case. A suitable choice of X can be made, e.g., on computer by screen dialog. The parameter q (q > 0) in (19.242) represents the standard deviation (see 16.4.1.3,2., p. 788) of the measurement errors, of the values fi (i = 1 , 2 , .. . , N ) . Until now, the abscissae of the interpolation points and the measurement points were the same as the nodes of the spline function. For large N this results in a spline containing a large number of cubic functions (19.236). A possible solution is to choose the number and the position of the nodes freely, because in many practical applications only a few spline segments are satisfactory. It is reasonable also from a numerical viewpoint to replace (19.236) by a spline of the form

c

rt2

S(x)=

aix,4(2).

(19.243)

i=l

Here r is the number of freely chosen nodes, and the functions Ne,4(x) are the so-called normalized Bsplines (basissplines) of order 4, i.e., polynomials of degree three, with respect to the i-th node. For details see [19.5].

19.7.2 Bicubic Splines 19.7.2.1 Use of Bicubic Splines Bicubic splines are used for the following problem: A rectangle R of the x,y plane, given by a 5 2 5 b, c 5 y 5 d, is decomposed by the grid points (xi,yj) ( i = 0, 1,. . . , n; j = 0,1,. . . , m) with (19.244) a = 2 0 < 21 < ’ . ’ < 2 , = b, c = YO < yi < ... < ym = d into subdomains h&, where the subdomain hl contains the points (x,y) with x, 5 x 5 xiflr yj 5 y 5 yjti (i = 0,1,. . . , n - 1; j = 0, 1,. . . m - 1). The values of the function j(x,y) are given at the grid points (19.245) f ( x i , yj) = jtJ (i = 0,1,. . . ,n; j = 0,1,. . . , m). A possible simple, smooth surface over R is required which approximates the points (19.245).

19.7.2.2 Bicubic Interpolation Splines 1. Properties The bicubic interpolation spline S ( x ,y) is defined uniquely by the following properties: 1. S(Z,y) satisfies the interpolation conditions S(Z,,Yj) = fil ( 2 = 0,1,. . . , n; j = 0,1,. . . , m). 2. S ( x ,y) is identical to a bicubic polynomial on every &j of the rectangle R, that is,

(19.246)

(19.247) on R,,3. So, S,,(z, y) is determined by 16 coefficients, and for the determination of S ( x ,y) we need 16 m n coefficients.

19.7 Representation of Curves and Surfaces with Splines 931

3. The derivatives

as as

82s

ax:

dsdy

-

ay’

(19.248)

are continuous on R. So, a certain smoothness is ensured for the entire surface. 4. S(z, y) satisfies the special boundary conditions:

dS

= p,,

z(z,,y,)

dS

%(xt,

d2S

&dy(z,l

for i = 0,n; j = O , l , .

y,) = qej for

y,) = T

, ~

. . ,m,

i = 0 , 1 , . . . n; j = O,m,

(19.249)

for i = 0, n;j = 0, m.

Here pi, qiJ and T , are ~ previously given values. We can use the results of one-dimensional cubic spline interpolation for the determination of the coefficients aykl. We see: 1. There is a very large number (271 + m + s) of linear equation systems but only with tridiagonal coefficient matrices. 2. The linear equation systems differ from each other only on their right-hand sides. In general. it can be said that bicubic interpolation splines are useful with respect to computation cost and accuracy, and so they are appropriate procedures for practical applications. For practical methods of computing the coefficients see the literature. ~

2. Tensor Product Approach The bicubic spline approach (19.247) is an example of the so-called tensor product approach having the form n

m

i=O

j=O

S(2,Y)= CCazjsl(z)MY)

(19.250)

and which is especially suitable for approximations over a rectangular grid. The functions si(.) (i = , , n ) and h,(y) ( j = 0 , 1 , . . . m) form two linearly independent function systems. The tensor product approach has the big advantage, from numerical viewpoint, that, e.g., the solution of a twodimensional interpolation problem (19.246) can be reduced to a one-dimensional one. Furthermore, the two-dimensional interpolation problem (19.246) is uniquely solvable with the approach (19.250) if 1. the one-dimensional interpolation problem with functions si(.) with respect to the interpolation nodes 2 0 , zl,. . . ,z,and 2. the one-dimensional interpolation problem with functions h,(y) with respect to the interpolation nodes YO,?/I,. . . , Ym are uniquely solvable. An important tensor product approach is that with the cubic B-splines: 0,1,,

.

~

(19.251) Here, the functions Nt,4(z) and Nj,4(y)are normalized B-splines of order four. Here r denotes the number of nodes with respect to 2 , p denotes the number of nodes with respect to y. The nodes can be chosen freely but their positions must satisfy certain conditions for the solvability of the interpolation problem. The B-spline approach results in an equation system with a band structured coefficient matrix, which is a numerically useful structure.

19. Numerical Analusis

932

For solutions of different interpolation problems using bicubic B-splines see the literature

19.7.2.3 Bicubic Smoothing Splines The one-dimensional cubic approximation spline is mainly characterized by the optimality condition (19.242). For the two-dimensional case we could determine a whole sequence of corresponding optimality conditions, however only a few special cases make the existence of a unique solution possible. For appropriate optimality conditions and algorithms for solution of the approximation problem with bicubic B-splines see the literature.

19.7.3 Bernstein-B6zier Representation of Curves and Surfaces 1. Bernstein Basis Polynomials The Bernstein-Bkzier representation (briefly B-B representation) of curves and surfaces applies the Bernstein polynomials

B,,,(t) =

(;)

tt(1 - t)n-,

(2

= 41,. . .,n)

(19.252)

and uses the following fundamental properties: 1. 0 5 Bi,,(t) 5 1 for 0 5 t 5 1,

(19.253)

n

2. CB,,,(t) = 1. a

(19.254)

d

Formula (19.254) follows directly from the binomial theorem (see 1.1.6.4, p. 12). IA: Bol(t) = 1 - t , B,,,(t)= t (Fig. 19.12). IB: B03(t)= (1 - t ) 3 ,B1,3(t)= 3t(l - t ) ' , B2,3(t)= 3t2(1- t ) B3,3(t)= t3 (Fig. 19.13).

-b

t

Figure 19.12

Figure 19.13

2. Vector Representation In the following, a space curve, whose parametric representation is I = z ( t ) ,y = y(t), z = z ( t ) ,will be denoted in vector form by (19.255) r'= i ( t )= I ( t )& + y(t) sv z ( t )&. Here t is the parameter of the curve. The corresponding representation of a surface is (19.256) r' = P(u, v) = z(u,w) & + y(u, v) Gy + z ( u ,v) &. Here, u and w are the surface parameters.

+

19.7.3.1 Principle of the B-B Curve Representation

Suppose there are given n + 1 vertices P, (i = O , l , . . . , n) of a three-dimensional polygon with the position vectors P,. Introducing the vector-valued function n

r'(t) =

CBi,,(t)Pi t=O

(19.257)

19.8 Using the Computer 933

bp:

we assign a space curve to these points, which is called the B-B curve. Because of (19.254) we can consider (19.257) as a “variable convex combination” of the given points. The three-dimensional curve (19.257) has the following important properties: 1. The pointsPo and Pnzre interpolated. andPn. 2. Vectors Popl and Pn-lP, are tangents to t ( t )at points PO The relation between a polygon and a B-B curve is shown in Fig. 19.14. The B-B representation is considered as a design of the curve, since we can easily influence the shape of the curve by changing p4 the polygon vertices. We often use normalized B-splines instead of Bernstein polyFigure 19.14 nomials. The corresponding space curves are called the B-spline curves. Their shape corresponds basically to the B-B curves with the following advantages: 1. The polygon is better approximated. 2. The B-spline curve changes only locally if the polygon vertices are changed. 3. In addition to the local changes of the shape of the curve the differentiability can also be influenced. So, it is possible to produce break points and line segments for example.

19.7.3.2 B-B Surface Representation Suppose there are given the points Pij (i = 0, 1,. . . , n; j = 0,1,. . . , rn) with the position vectors P i j , which can be considered as the nodes of a grid along the parameter curves of a surface. Analogously to the B-B curves (19.257), we assign a surface to the grid points by n

m

=

<(u,

Bi,n(u)Bj,m(u)*ij.

(19.258)

%=O j=O

Representation (19.258) is useful for surface design, since by changing the grid points we may change the surface. Anyway, the influence of every grid point is global, so we should change from the Bernstein polynomials to the B-splines in (19.258).

19.8 Using the Computer 19.8.1 Internal Symbol Representation Computer are machines that work with symbols. The interpretation and processing of these symbols is determined and controlled by the software. The external symbols, letters, cyphers and special symbols are internally represented in binary code by a form of bit sequence. A bit (binary digit) is the smallest representable information unit with values 0 and 1. Eight bits form the next unit, the byte. In a byte we can distinguish between 28 bit combinations, so 256 symbols can be assigned to them. Such an assignment is called a code. There are different codes; one of the most widespread is A S C I I (American Standard Code for Information Interchange).

19.8.1.1 Number Systems 1. Law of Representation Numbers are represented in computers in a sequence of consecutive bytes. The basis for the internal representation is the binary system, which belongs to the polyadic systems, similarly to the decimal system. The law of representation for a polyadic number system is n

a=

1 z,Bz t=-m

(rn > 0, n 2 0;m, n integer)

(19.259)

934 19. Numerical Analysis with B as basis and z, (0 5 zi < B ) a6 a cypher of the number system. The positions i 2 0 form the integers, those with a < 0 the fractional part of the number. W For the decimal number representation, i.e., B = 10, of the decimal number 139.8125 we get 139.8125 = 1 '10' + 3 10' + 9.10' + 8 . lo-' + 1 . lo-' + 2 . +5. The number systems occurring most often in computers are shown in Table 19.3. Table 19.3 Number systems -

Number system

Basis Corresponding cyphers

Binary system Octal system Hexadecimal system Decimal system

2. Conversion The transition from one number system to another is called convemon. If we use different number systems in the same time, in order to avoid confusion we put the basis as an index. W The decimal number 139.8125 is in different systems: 139.812510 = 10001011.1101~= 213.648 = 8B.D16. 1. Conversion of Binary Numbers into Octal or Hexadecimal Numbers The conversion of binary numbers into octal or hexadecimal numbers is simple. We form groups of three or four bits starting at the binary point to the left and to the right, and we determine their values. These values are the cyphers of the octal or hexadecimal systems. 2. Conversion of Decimal Numbers into Binary, Octal or Hexadecimal Numbers For the conversion of a decimal numbers into another system, we adept the following rules for the integer and for the fractional part separately: a) Integer Part: If G is an integer in the decimal system, then for the number system with basis B the law of formation (19.259) is:

cz,B~ n

G=

(19.260)

( n 2 0)

a=O

If we divide G by B , then we get an integer part (the sum) and a residue: (19.261) Here. zo can have the values 0,1,. . . , B - 1, and it is the lowest valued cypher of the required number. If we repeat this method for the quotients, we get further cyphers. b) Fractional Part: If g is a proper fraction, then the method to convert it into the number system with basis B is

+ c z.J-"+l, m

gB = z-1

(19.262)

a='

i t . , we get the next cypher as the integer part of the product gB. The values z-2, obtained in the same way.

z-3,

. . . can be

19.8 Usinq the Computer 935

IB: Conversion of a decimal fraction 0.8125 into a binary fraction. 0.8125.2 = 1.625 (1 = 2-1) 0.625 . 2 = 1.25 (1 = 2-2) 0.25 . 2 = 0.5 (0 = Z-3) 0.5 . 2 = 1.0 (1 = 2-4) 0.0 . 2 = 0.0

IA: Conversion of the decimal number 139 into a binary number. 139:2 = 69 residue 1 (1 = zo) 69: 2 = 34 residue 1 (1 = zl) 34:2 = 17 residue 0 (0 = 22) : 1 7 : 2 = 8 residue 1 8 : 2 = 4 residue 0 : 4 : 2 = 2 residue 0 : 2 : 2 = 1 residue 0 : 1 : 2 = 0 residue 1 (1 = z7)

0.812510 = 0.11012

13910 = 100010112 3. Conversion of Binary, Octal, a n d Hexadecimal N u m b e r s i n t o a Decimal N u m b e r The algorithm for the conversion of a value from the binary, octal, or hexadecimal system into the decimal system is the following, where the decimal point is after ZO: a=

2

( m > 0, n 2 0,integer).

Z,B%

(19.263)

t=-m

The calculation is convenient with the Horner rule. LLLOL = 1 .24 1 .23 + 1 . 2 ’ + 0 . 2 l + 1.2’ = 29. The corresponding Horner scheme is shown on the right.

+

111 0 1 21 2614; 1 3 7 14 29

19.8.1.2 Internal Number Representation Binary numbers are represented in computers in one or more bytes. We distinguish between two types of form of representation, the jmd-point numbers and the jloatzng-point numbers. In the first case, the decimal point is at a fixed place, in the second case it is “floating” with the change of the exponent.

-I

1. F i x e d - p o i n t Numbers binary number (t bits) The range for fixed-point numbers with the I given parameters is 0 5 1 a I 5 2t - 1. (19.264) Fixed-point numbers can be represented in the form of Fig. 19.15. sign IJ of the fixed-point number 2. F l o a t i n g - P o i n t Numbers Figure 19.15 Basically. two different forms are in use for the representation of floating-point numbers, where the internal implementation can vary in detail. 1. Normalized Semilogarithmic Form In the first form, the signs of the exponent E and the mantissa M of the number a are stored separately a = fMBiE. (19.26Sa) Here the exponent E is chosen so that for the sign u, sign IJ, mantissa 1/B I M < 1 (19.265b) of the exponent of the mantissa holds. We call it the normalized semilogarithFigure 19.16 micfonn (Fig. 19.16).

I-

The range of the absolute value of the floating-point numbers with the given parameters is: 2 4 5 I a 1 5 (1 - 2 - t )

. z(V-1).

(19.266)

I

936

19. Numerical Analysis

2. IEEE Standard The second (nowadays used) form of floating-point numbers corresponds to the IEEE (Institute of Electrical and Electronics Engineers) standard accepted in 1985. It deals with the requirements of computer arithmetic, roundoff behavior, arithmetical operators, conversion of numbers, comparison operators and handling of exceptional cases such as over- and underflow. The floating-point number representations are mantissaM pharacteristic C shown in Fig. 19.17. We get the characteristic C from the exponent E by addition of a suitable constant K . This is chosen so that we get only positive numbers in the characteristic. The representable number is sign IJ of the floating-point number u = (-1)”2’.l.blbz

. . .bt-1

with E = C - K.

(19.267)

Figure 19.17

Single precision Double precision

Parameter Word length in bits Maximal exponent E,, Minimal exponent E,,, Constant K Number of bits in exponent Number of bits in mantissa

32 $127 -126 $127 8 24

64 $1023 -1022 $1023 11 53

19.8.2 Numerical Problems in Calculations with Computers 19.8.2.1 Introduction, Error Types The general properties of calculations with a computer are basically the same as those of calculations done by hand, however some of them need special attention, because the accuracy comes from the r e p resentation of the numbers, and from the missing judgement with respect to the errors of the computer. Furthermore, computers perform many more calculation steps than human can do manually. So, we have the problem of how to influence and control the errors, e.g., by choosing the most appropriate numerical method among the mathematically equivalent methods. In further discussions, we will use the following notation, where x denotes the exact value of a quantity, which is mostly unknown, and j.is an approximation value of 2: Absolute error: 14x1 = 1x - 51.

(19.268)

Relative error:

lTl

ax =1 2 -T j. 1 .

(19.269)

The notation C(Z)= z

2-5 - j. and crel(x)= -

(19.270)

is also often used.

19.8.2.2 Normalized Decimal Numbers and Round-Off 1. Normalized Decimal Numbers Every real number z # 0 can be expressed as a decimal number in the form z = *O.blbZ.. . . loE (bi # 0).

(19.271)

19.8 Usina the Computer 937 Here 0, b l b z . . . is called the mantissa formed with the cyphers b, E (0, 1 , 2 , . . . ,9}.The number E is an integer, the so-called exponent with respect to the base 10. Since bl # 0, we call (19.271) a normalized decimal number. Since only finitely many cyphers can be handled by a real computer, we have to restrict ourselves to a fixed number t of mantissa cyphers and to a fixed range of the exponent E . So, from the number x given in (19.271) we obtain the number

i={

*O.blbz...bt. loE f ( 0 . b l b z ' . bt 10-t)lOE

+

for bt+l 5 5 (round-down), for bt+' > 5 (round-up),

(19.272)

by round-off (as it is usual in practical calculations). For the absolute error caused by round-off, =

IL

- j.1 5 0.5. 10-tlOE.

( 19.273)

2. Basic Operations and Numerical Calculations Every numerical process is a sequence of basic calculation operations. Problems arise especially with the finite number of positions in the floating-point representation. We give here a short overview. We suppose that x and y are normalized error-free floating-point numbers with the same sign and with a non-zero value: x = m l B E l , y = m2BE2 with (19.274a) t

m, =

1a!),B-k,

a!! # 0, and

(19.274b)

k=l

a!k=Oorlor

...

or B - 1 for k > l

(z=1,2).

(19.274~)

1. Addition If El > Ez>then the common exponent becomes El, since normalization allows us to make only a left-shift. The mantissas are then added.

I< 2 If B-' 51 ml +m2B-(E1-Ez)

(19.275a)

and

I ml +m2B-(E1-EZ) 12 1,

(19.27513)

then shifting the decimal point by one position to the left results in an increase of the exponent by one. 0.9604 lo3 + 0.5873.102 = 0.9604. lo3 + 0.05873. lo3 = 1.01913. lo3 = 0.1019. lo4. 2. Subtraction The exponents are equalized as in the case of addition, the mantissas are then subtracted. If and I ml - m2B-(E1-E2) /< B-', (19.276b) I ml - m2B-(E1-E2) I< 1 - B-t (19.276a) shifting the decimal point to the right by a maximum oft positions results in the corresponding decrease of the exponent. 0.1004. lo3 - 0.9988. lo2 = 0.1004 lo3 - 0.09988. lo3 = 0.00052. lo3 = 0.5200.10°. This example shows the critical case of subtractive cancellation. Because of the limited number of positions (here four), zeros are carried in from the left instead of the correct characters. 3. Multiplication The exponents are added and the mantissas are multiplied. If m1m2 < B-', (19.277) then the decimal point is shifted to the right by one position, and the exponent is decreased by one. (0.3176. lo3) . (02504. lo5) = 0.07952704 10' = 0.7953. lo7. 4. Division The exponents are subtracted and the mantissas are divided. If 9

(19.278) then the decimal point is shifted to the left by one position, and the exponent is increased by one. (0.3176 lO3)/(O.2504. lo5) = 1.2683706.. .lo-' = 0.1268 lo-'.

I

938

19. Numerical Analusis

5. Error of the Result The error of the result in the four basic operations with terms that are supposed to be error-free is a consequence of round-off. For the relative error with number of positions t and the base B , the limit is

B

(19.279)

--B-t.

2

6. Subtractive cancellation As we have seen above, the critical operation is the subtraction of nearly equal floating-point numbers. If it is possible, we should avoid this by changing the order of operations, or by using certain identities. = 0.4455 . lo2 - 0.4454. 10' = 0.1 . lo-' or I = Iz = - v%% = 1985 - 1984 = 0.1122~10-~

m

mtm

19.8.2.3 Accuracy in Numerical Calculations 1. Types of Errors Numerical methods have errors. There are several types of errors, from which the total error of the final result is accumulated (Fig. 19.18).

I Total error I Round-off error

Truncation error

Discretizationerror

Figure 19.18

2. Input Error 1. Notion of Input Error Input error is the error of the result caused by inaccurate input data. Slight inaccuracies of input data are also called perturbations. The determination of the error of the input data is called the dzrect problem of error calculus. The inverse problem is the following: How large an error the input data may have such that the final input error does not exceed an acceptable tolerance value. The estimation of the input error in rather complex problems is very difficult and is usually hardly possible. In general, for a real-valued function y = f(r) with x = ( X I , 52,.. . x , ) ~ for , the absolute value of the input error we have

IAYI = I f ( z l > z ~,j,,I ,n ) =

-f(zlrz2r., , $ z n ) l

IC -((El,Ez,...rEnn)(lt-~,)I af 2=1

ax,

(19.280)

if we use the Taylor formula (see 7.3.3.3, p. 415) for y = f (z)= f ( z l , z z ,... ,zn) with a linear residue.

El>( 2 , . . . ,En denote the intermediate values, j.1,&,. . . ,j.,denote the approximating values of ~ 1 ~ x 2 , . . . , 2 , . The approximating values are the perturbed input data. Here, we also consider the Gauss error propagation law (see 16.4.2.1,p. 792). 2. Input Error of Simple Arithmetic Operations The input error is known for simple arithmetical operations. With the notation of (19.268)-(19,270) for the four basic operations we get:

19.8 Using the Computer 939

4 x h Y) = +)

4), - ze(y) YZ

frel

(i)

(19.281) ~ ( x y = ) ye(x)

+

terms of higher order ine,

+

= erei(x) - crel(y)

terms of higher order in&.

+ xe(y) + ~(x)e(y),

(19.282) (19.283)

(19.286)

The formulas show: Small relative errors of the input data result in small relative errors of the result on multiplication and division. For addition and subtraction, the relative error can be very large if Ix YI 1x1 + 1211, 3. Error of t he Method 1. Notion of the Error of the Method The error of the method comes from the fact that we should be able to numerically approximate theoretically continuous phenomena in many different ways as limits. Hence, we have truncation errors in limiting processes (as, e.g., in iteration methods) and discretization errors in the approximation of continuous phenomena by a finite discrete system (as, e.g., in numerical integration). Errors of methods exist independently of the input and round-off errors; consequently, they can be investigated only in connection with the applied solution methodology. 2. Applying Iteration Methods If we use an iteration method, we should consider that both cases may occur: We can get a correct solution or also a false solution of the problem. It is also possible that we get no solution by an iteration method although there exists one. To make an iteration method clearer and safer, we should consider the following advice: a) To avoid “infinite” iterations, we should count the number of steps and stop the process if this number exceeds a previously given value (Le., we stop without reaching the required accuracy). b) We should keep track of the location of the intermediate result on the screen by a numerical or a graphical representation of the intermediate results. c) We should use all known properties of the solution such as gradient, monotonicity, etc. d ) We should investigate the possibilities of scaling the variables and functions. e) We should perform several tests by varying the step size, truncation conditions, initial values, etc 4. Round-off Errors R o u n d - o f errors occur because the intermediate results should be rounded. So, they have an essential importance in judging mathematical methods with respect to the required accuracy. They determine together with the errors of input and the error of the method, whether a given numerical method is strongly stable, weakly stable or unstable. Strong stability, weak stability, or instability occur if the total error, at an increasing number of steps, decreases, has the same order, or increases, respectively. At the instability we distinguish between the sensitivity with respect to round-off errors and discretization errors (numerical instability) and with respect to the error in the initial data at a theoretically exact calculation (natural instability). A calculation process is appropriate if the numerical instability is not greater than the natural instability. For the local error propagation of round-off errors, Le., errors at the transition from a calculation step to the next one! the same estimation process can be used as the one we have at the input error. 5 . Examples of Numerical Calculations We illustrate some of the problems mentioned above by numerical examples. IA: Roots of a Quadratic Equation: ax2 + bx + c = 0 with real coefficients a , b, c and D = bZ - 4ac 2 0 (real roots). Critical situations are the cases a) I 4ac I<< bZ and b) 4ac bZ. Recommended proceeding:

*

940

19. Numerical Analvsis

a) x1 = -

b t sign(b)@

, 22 = c (Vieta root theorem, see 1.6.3.1, 3., p. 44).

ax1 b) The vanishing of D cannot be avoided by a direct method. Subtractive cancellation occurs but the error in ( b t sign(bfl) is not too large since Ibl >> \/Is holds. IB: Volume of a Thin Conical Shell for h < r - r3 v = 471 ( r t h)3 because of ( r h) x r there is a case of subtractive cancellation. However in the 3 3r2h + 3rh2 t h3 equation V = 471 there is no such problem. 3 2a

+

IC: Determining t h e Sum S =

- (S = 1.07667.. .) with an accuracy of three signif-

k2 + 1 icant digits. Performing the calculations with 8 digits, we should add about 6000 terms. After the identical transformation

I - k2

s=c--cm l

k2

k=l

k=l

+1

k2

1

k2(k2+ 1)'

and

k 2 ( k 2+ 1)

we see that 1

7r*

S=--c6

k=l

+

k 2 ( k Z 1) '

By this transformation we have

to consider only eight terms.

ID: Avoiding t h e

00 Situation in the function t = (1 - d m ) x2m- y2 for x = y = 0.

Jm)

Multiplying the numerator and the denominator by (1 + we avoid this situation. E: Example for a n Unstable Recursive Process. Algorithms with the general form Y,,+~ = ay,

+ by,-l

;1 * d;'l -+b

( n = 1 , 2 , .. .) are stable if the condition -

case yntl = -3yn y2, y3) 1/4. y5, ye,.

+ 4y,-l

(TI

< 1 is satisfied. The special

= 1 , 2 , .. .) is unstable. If yo and y1 have errors E and - E , then for ~- 2, 5 ~ ,1 0 3 -4096, ~ ~ 16398, . . .. The process is instable for the

. . we get the errors 7

parameters a = -3 and b = 4. IF: Numerical Integration of a Differential Equation. For the first-order ordinary differential equation (19.287) Y' = f ( x 2u) with f(x,Y) = ay and the initial value y(x0) = yo we will represent the numerical solution. a ) N a t u ra l Instability. Together with the exact solution y(z) for the exact initial values y(xo) = yo let u(z) be the solution for a perturbed initial value. Without loss of generality, we may assume that the perturbed solution has the form 4 x 1 = Y ( 5 ) +Ea(.), (19.288a) where E is a parameter with 0 < E < 1 and ~ ( xis) the so-called perturbation function. Considering that d(x) = f(s,u ) we get from the Taylor expansion (see 7.3.3.3, p. 415) u'(z) = f(z, y(z) t E ~ ( x )= ) f(x,y) t E ~ ( xf,(x, ) y) terms of higher order (19.288b) which implies the so-called differential variation equation V ' k ) = fV(X,Y)V(.). (19.288~) The solution of the problem with f(x,y) = ay is

+

~(z) = 170 with 170 = ~ ( 2 0 ) . (19.288d) For a > 0 even a small initial perturbation v0 results in an unboundedly increasing perturbation ~(z). So, there is a natural instability. b) Investigation o f t h e Error of t h e Method in t h e Trapezoidal Rule. With a = -1, the stable

19.8 Usin.9 the Computer 941

differential equation y'(z) = -y(z) has the exact solution y(z) = yoe-(x-xO). where yo = y(z0). The trapezoidal rule is

T

y(z)dz RZ h*

21

(19.289a)

with h = zttl- z,.

2

(19.289b)

By using this formula for the given differential equation we get %,+I

k+l

= fii t

J

(-y)d.

-

di

+ di+l

= yt - h or

2

di+l

2-h2 t h

= -yi

or

2%

2-h

I, = (?Ti;) do. With z, = zo

+ ih, Le., with a =

(2,

(19.289~)

- zo/h we get for 0 5 h < 2

c(h) =

(19.289d) h If do = yo, then i, < y,, and so for h -+ 0, it also tends to the exact solution yoe-(x1-50). c) Input Error b'e supposed in b) that the exact and the approximate initial values coincide. Now, we investigate the behavior when yo # IO with I $0 - yo I< E O .

So, E , + ~ is at most of the same order as E O , and the method is stable with respect to the initial values. We have to mention that in solving the above differential equation with the Simpson method an artificial instability is introduced. In this case, for h + 0, we would get the general solution (19.290b) sz = Cle-'I + cz(-qte=J3. The problem is that the numerical solution method uses higher-order differences than those to which the order of the differential equation corresponds.

19.8.3 Libraries of Numerical Methods Over time, librarzes of functions and procedures have been developed independently of each other for numerical methods in different programming languages. An enormous amount of computer experimentation was considered in their development, so in solutions of practical numerical problems we should use the programs from one of these program libraries. Programs are available for current operating systems like WINDOWS, UNIX and LINUX and mostly for every computation problem type and they keep certain conventions, so it is more or less easy to use them. The application of methods from program libraries does not relieve the user of the necessity of thinking about the expected results. This is a warning that the user should be informed about the advantages and also about the disadvantages and weaknesses of the mathematical method he/she is going to use.

19.8.3.1 NAG Library The NAG library (Numerical Algorithms Group) is a rich collection of numerical methods in the form of functions and subroutines/procedures in the programming languages FORTRAN 77, FORTRAN 90,

I

942

19. Numerical Analysis

and C. Here is a contents overview: 14. Eigenvalues and eigenvectors 1. Complex arithmetic 15. Determinants 2. Roots of polynomials 3. Roots of transcendental equations 16. Simultaneous linear equations 17. Orthogonalization 4. Series 18. Linear algebra 5. Integration 6. Ordinary differential equations 19. Simple calculations with statistical data 20. Correlation and regression analysis 7. Partial differential equations 21. Random number generators 8. Sumeric differentiation 22. Non-parametric statistics 9. Integral equations 23. Time series analysis 10. Interpolation 11. .4pproximation of curves and surfaces from data 24. Operations research 12. hlinimum/maximum of a function 25. Special functions 26. Mathematical and computer constants 13. Matrix operations, inversion Furthermore the N.4G library contains extensive software concerning statistics and financial mathematics.

19.8.3.2 IMSL Library The IMSL library (International Mathematical and Statistical Library) consists of three synchronized parts: General mathematical methods, Statistical problems, Special functions. The sublibraries contain functions and subroutines in FORTRAN 77, FORTRAN 90 and C. Here is a contents overview: General Mathematical Methods 1. Linear systems 2. Eigenvalues 3. Interpolation and approximation 4. Integration and differentiation 5. Differential equations

6. 7. 8. 9. 10.

Statistical Problems 1. Elementary statistics 2. Regression 3. Correlation 4. Variance analysis 5. Categorization and discrete data analysis 6. Non-parametric statistics 7. Test of goodness of fit and test of randomness 8. Analysis of time series and forecasting 9. Covariance and factor analysis 10. Discriminance analysis 11. Cluster analysis Special Functions 1. Elementary functions 2. Trigonometric and hyperbolic functions 3. Exponential and related functions 4. Gamma function and relatives 5. Error functions and relatives

Transformations Non-linear equations Optimization Vector and matrix operations Auxiliary functions

12. 13. 14. 15. 16. 17. 18. 19. 20.

Random sampling Life time distributions and reliability Multidimensional scaling Estimation of reliability function, hazard rate and risk function Line-printer graphics Probability distributions Random number generators Auxiliary algorithms Auxiliary mathematical tools

Bessel functions Kelvin functions Bessel functions with fractional orders Weierstrass elliptic integrals and related functions 10. Different functions

6. 7. 8. 9.

19.8 Using the Computer 943

19.8.3.3 Aachen Library The Aachen library is based on the collection of formulas for numerical mathematics of G. EngelnMiillges (Fachhochschule Aachen) and F. Reutter (Rheinisch-Westfalische Technische Hochschule Aachen). It exists in the programming languages BASIC, QUICKBASIC, FORTRAN 77, FORTRAN 90, C, MODULA 2 and TURBO PASCAL. Here is an overview:

1. Numerical methods to solve non-linear and special algebraic eqda‘tiohs 2. Direct and iterative methods to solve systems of linear equations 3. Systems of non-linear equations 4. Eigenvalues and eigenvectors of matrices 5. Linear and non-linear approximation 6. Polynomial and rational interpolation, polynomial splines 7. Numerical differentiation 8. Numerical quadrature 9. Initial value problems of ordinary differential equations 10. Boundary value problems of ordinary differential equations The programs of the Aachen library are especially suitable for the investigation of individual algorithms of numerical mathematics.

19.8.4 Application of Computer Algebra Systems 19.8.4.1 Mathernatica 1. Tools for the Solution of Numerical Problems The computer algebra system Mathernatica offers a very effective tool that can be used to solve a large variety of numerical mathematical problems. The numerical procedures of Mathernatica are totally different from symbolic calculations. Mathernatica determines a table of values of the considered function according to certain previously given principles, similarly to the case of graphical representations, and it determines the solution using these values. Since the number of ppints must be finite, this could be a problem with badly ” behaving functions. Although Mathernatica tries to choose more nodes in problematic regions, we have to suppose a certain continuity on the considered domain. This can be the cause of errors in the final result. It is advised to use as much information as possible about the problem under consideration, and if it is possible, then to perform calculations in symbolic form, even if this is possible only for subproblems. In Table 19.5,we represent the operations for numerical computations: Table 19.5 Numerical operations NIntegrate calculates definite integrals n NSum calculates sums f ( i ) 1=1

NProduct NSolve NDSolve

1

calculates products numerically solves algebraic equations numerically solves differential equations

After starting Mathernatica the “Prompt ” In[ll := is shown; it indicates that the system is ready to except an input. Mathernatica denotes the output of the corresponding result by Out [l]. In general: The text in the rows denoted by Infnl := is the input. The rows with the sign Out [nl are given back by Mathernatica as answers. The arrow -> in the expressions means, e.g., replace z by the value a.

2. Curve Fitting and Interpolation 1. Curve Fitting Mathernatica can perform the fitting of chosen functions to a set of data using the least squares method (see 6.2.5, p. 399ff.) and the approximation in mean to discrete problems (see 19.6.2.2. p. 918). The general instruction is: (19.291) Fit[{yl, YZ, . . .I,funkt,zl.

944

19. Numerical Analvsis

Here the values yz form the list of data, funkt is the list of the chosen functions, by which the fitting should be performed, and x denotes the corresponding domain of the independent variables. If we choose funkt, e.g., as Table[x"z, {i, 0, n}] , then the fitting will be made by a polynomial of degree n. ILet the following list of data be given: InDI := l = {1.70045,1.2523,0.638803,0.423479,0.249091,0.160321,0.0883432,0.0570776, 0.0302744,0.0212794} With the input InC21 := fl = Fit[l,{l,e,x"2,xA3,x"4},x] we suppose that the elements of l are assignd to the values 1,2,.. . , 10 of 2. We get the following approximation polynomial of degree four: Out L21 = 2.48918 - 0.8534872 0.0998996~'- 0.00371393~~ - 0.0000219224~~ With the command Inf31 := Plot[ListPlot[l,{x3lo}], fl, {x,1, lo}, Axesorigin-> (0, O}] we get a representation of the data and the approximation curve given in Fig. 19.19a.

+

1.75 1.5

1.5 1.25

1.25

1

1.

0.75

a)

0.75

0.5

0.5

0.25

0.25

'

2

4

6

8 ' 1 0

b ) '

2

4

6

8

1

0

Figure 19.19 For the given data this is completely satisfactory. The terms are the first four terms of the series expansion of e1-0.5z. 2. Interpolation Mathernatica offers special algorithms for the determination of interpolation functions. They are represented as so-called interpolating function objects, which are formed similarly to pure functions. The directions for using them are in Table 19.6. Instead of the function values yI we can give a list of function values and values of specified derivatives at the given points. W With In[3] := Plot[Interpolation[l][x], { q l , lo}] we get Fig. 19.19b. We can see that Mathernatica gives a precise correspondence to the data list. Table 19.6Commands for interpolation Interpolation[{yl,yz,. ..)I

gives an approximation function with the values y, for the values x, = z as integers Interpolation[{{xl, yl}, {x2,yz}, . . .}] gives an approximation function for the point-sequence

3. Numerical Solution of Polynomial Equations As shown in 20.4.2.1,p. 981,Mathernatica can determine the roots of polynomials numerically. The command is NSolve[p[x] == O,x,n],where n prescribes the accuracy by which the calculations should be done. If we omit n, then the calculations are made to machine accuracy. We get the complete solution, Le., m roots, if we have an input polynomial of degree m. IInLfI := NSolve[z"6 + 3x"2 - 5 == 01 O ~ t [ l= ] {x-> -1.07432}, {z-> -0.867262 - 1.152921},{x-> -0.867262 + 1.152921},

19.8 Usin.9 the Computer 945

(2->

0.867262 - 1.152921}, (5-> 0.867262

+ 1.152921}, (z-> 1.07432)).

4. Numerical Integration For numerical integration Mathernatica offers the procedure NIntegrate. Differently from the symbolical method, here it works with a table of values of the integrand. We consider two improper integrals (see 8.2.3, p. 451) as examples. IA: In[lI := NIntegrate[Exp[-z"2], (5, - I n f i n i t y , Infinity}] Out [ll = 1.77245. B: Inf2l := NIntegrate[l/zA2,{z, -1, l}] Power::infy: Infinite expression 1 encountered. 0 N1ntegrate::inum: Integrand ComplexInfinity is not numerical at{z} = (0). Mathernatica recognizes the discontinuity of the integrand at z = 0 in example B and gives a warning. Mathernatica applies a table of values with a higher number of nodes in the problematic domain, and it recognizes the pole. However, the answer can be still wrong. Mathernatica applies certain previously specified options for numerical integration, and in some special cases they are not sufficient. We can determine the minimal and the maximal number of recursion steps, by which Mathernatica works in a problematic domain, with parameters MinRecursion and MaxRecursion. The default options are always 0 and 6. If we increase these values, although Mathernatica works slower, it gives a better result. IInC31 := NIntegrate[Exp[-~"21, (2, -1000, lOOO}] Mathernatica cannot find the peak at z = 0, since the integration domain is too large, and the answers is: N1ntegrate::ploss: Numerical integration stopping due to loss of precision. Achieved neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the following: highly oscillatory integrand or the true value of the integral is 0. Out [31 = 1.34946. If we require InC41 := NIntegrate[Exp[-~"21, (5, -1000, lOOO}, MinRecursion-> 3, MaxRecursion-> lo], then we get

Out f41 = 1.77245 Similarly. we get a result closer to the actual value of the integral with the command: NIntegrate[fun, {x,x,,x~,xz,. . .,z,}]. (19.292) We can give the points of singularities zi between the lower and upper limit of the integral to force Mathernatica to evaluate more accurately.

5. Numerical Solution of Differential Equations In the numerical solution of ordinary differential equations and also in the solution of systems of differential equations Mathematica represents the result by an InterpolatingFunction. It allows us to get the numerical values of the solution at any point of the given interval and also to sketch the graphical representation of the solution function. The most often used commands are represented in Table 19.7. Table 19.7 Commands for numerical solution of differential equations NDSolve[dgl,y >{x,z, ze}l InterpolatingFunction[lzste][z] Plot[Evaluate[y[z]/. los]],(2, z, z,}]

computes the numerical solution of the differential equation in the domain between z, and z, gives the solution at the point z scetches the gravical representation

I

946

19. Numerical Analysis

ISolution of a differential equation describing the motion of a heavy object in a medium with friction. The equations of motion in two dimension are

x=-y

J--

x2+y2.x,

y=-g-rJG.y.

The friction is supposed to be proportional to the velocity. If we substitute g = 10,y = 0.1,then the following command can be given to solve the equations of motion with initial values z(0) = y(0) = 0 and i ( 0 ) = 100, o(0) = 200: I n [ l ] := dg = NDSolve[{r"[t] == -O.1Sqrt[z'[tIA2 + y'[t]"2] z'[t], y"[t]== -10

-0.1~qrt[z'[t]'''2

+ y'[t]9] y'[t], 401 == y[O]== 0, z'[O]== 100,y'[O] == 2OO},

{z>Y>l{t,1511 Mathernatica gives the answer by the interpolating function: Out [ll = {{r-> InterpolatingFunction[{O.,15.},

<>I,

y-> InterpolatingFunction[{O.,15.},

<>I})

We can represent the solution rn[zl := ParametricPlot[{z[t],y[t]}/.dg,{ t ,0,2}, PlotRange-> All] as a parametric curve (Fig.19.20a). NDSolve accepts several options which affect the accuracy of the result. The accuracy of the calculations can be given by the command AccuracyGoal. The command PrecisionGoal works similarly. During calculations, Mathernatica works according to the so-called WorkingPre cision, which should be increased by five units in calculations requiring higher accuracy. The numbers of steps by which Mathernatica works in the considered domain is prescribed as 500. In general, Mathernatica increases the number of nodes in the neighborhood of the problematic domain. In the neighborhood of singularities it can exhaust the step limit. In such cases, it is possible to increase the number of steps by MaxSteps. It is also possible the prescribe Infinity for MaxSteps. I The equations for the Foucault pendulum are: i ( t )+ &(t) = 2ny(t), Y(t)t u2y(t)= -2Ri(t). With w = 1, R = 0.025 and the initial conditions z(0) = 0, y(0) = 10, i ( 0 ) = y(0) = 0 we get the solution: In[3] := dg3 = NDSolve[{z"[t] == -z[t] + 0.05y'[t],y " [ t ] == -y[t] - O.O5z'[t], z[O] == 0,y[O] == 10,z'[O] == y'[O] == 0 } , {z, y}, { t l 0,40}] Out[3] = {{z-> InterpolatingFunction[{O.,40.},<>I,

y-> InterpolatingFunction[{O.,40.},

<>I}}

With

In[@ := ParametricPlot[{z[t],y[t]}/.dg3,{ t ,0,40}, AspectRatio-> 11 we get Fig. 19.20b.

19.8.4.2 Maple The computer algebra system Maple can solve several problems of numerical mathematics with the use of built-in approximation methods. The number of nodes, which is required by the calculations, can be determined by specifying the value of the global variable Digits as an arbitrary n. But we should not forget that selecting a higher n than the prescribed value results in a lower speed of calculation.

1. Numerical Calculation of Expressions and Functions

After starting Maple, the symbol Prompt " > is shown, which denotes the readyness for input. Connected in- and outputs are often represented in one row, separated by the arrow operator +.

19.8 U5in.q the Computer 947

4

’

0.5

1

1.5

Figure 19.20 1. Operator evalf Numerical values of expressions containing built-in and user-defined functions which can be evaluated as a real number, can be calculated with the command (19.293) evalf (expr,n). expr is the expression whose value should be determined; the argument n is optional, it is for evaluation to n digits accuracy. Default accuracy is set by the global variable Digits. IPrepare a table of values of the function y = f(x) = f i lnx. First. we define the function by the arrow operator: > f := z -> sqrt(z) +I+); +f := z -> & + h i . Then we get the required values of the function with the command evalf (f(z));, where a numerical value should be substituted for z. A table of values of the function with steps size 0.2 between 1 and 4 can be obtained by > f o r x from 1 by 0.2 t o 4 do print(f[x] = evalf (f(z), 12)) od; Here, it is required to work with twelve digits. Maple gives the result in the form of aone-column table with elements in the form fp.21 = 2.95200519181. 2. Operator evalhf (ezpcpr): Beside evalf there is the operator evalhf. It can be used in a similar way to evalf, Its argument is also an expression which has a real value. It evaluates the symbolic expression numerically, using the hardware floating-point double-precision calculations available on the computer. A Maple floating-point value is returned. Using evalhf speeds up your calculations in most cases, but you lose the definiable accuracy of using evalf and Digits together. For instance in the problem in 19.8.2, p. 936, it can produce a considerable error.

+

2. Numerical Solution of Equations As discussed in Chapter 20, see 20.4.4.2, p. 992, by using Maple we can solve equations or systems of equations numerically in many cases. The command to do this is fsolve. It has the syntax fsolve(eqn, var,optzon). (19.294) This command determines real solutions. If eqn is in polynomial form, the result is all the real roots. If eqn is not in polynomial form, it is likely that f solve will return only one solution. The available options are given in Table 19.8. Table 19.8 Options for the command f solve complex maxsols = n fulldigits interval1

determines a complex root (or all roots of a polynomial) determines at least the n roots (only for polynomial equations) ensures that f solve does not lower the number of digits used during computations looks for roots in the given interval

I

948

19. Numerical Analysis

IA: Determination of all solutions of a polynomial equation z6+ 32' - 5 = 0. With > eq := zA6+ 3 1; x"2 - 5 = 0 : we get > f solve(ep, z); i -1.074323739, 1.074323739 Maple determined only the two real roots. With the option complex, we get also the complex roots: > f solve(eq, x, complex); 1,1529220121, -1.074323739, -0.8672620244 - 1,1529220121, -0.8672620244 0.8672620244 - 1.1529220121,0.8672620244 1.1529220121,1.074323739

+

+

IB: Determination of both solutions of the transcendental equation P3 - 4x2 = 0 After defining the equation > ep := exp(-z"3) - 4 1; x"2 = O we get > f solve(eq, x); +0.4740623572 as the positive solution. With > fsolve(eq, z, z = -2..0); + -0.5412548544 Maple also determines the second (negative) root.

3. Numerical Integration The determination of definite integrals is often possible only numerically. This is the case when the integrand is too complicated, or if the primitive function cannot be expressed by elementary functions. The command to determine a definite integral in Maple is e v a l f : e v a l f ( i n t ( f ( z ) , x = a..b),n). (19.295) Maple calculates the integral by using an approximation formula

L2 + l2

ICalculation of the definite integral

e-23 dx. Since the primitive function is not known, for the

integral command we get the following answeI

> int(exp(-xA3),z = -2..2);

e P 3 dz.

If we type > evalf(int(exp(-zA3),z = -2..2), 15); then we get 277.745841695583. Maple used the built-in approximation method for numerical integration with 15 digits. In certain cases this method fails, especially if the integration interval is too large. Then, we can try to call another approximation procedure with the call to a library readlib('evalf/int') : which applies an adaptive Newton method. IThe input > eva1f(int(exp(-zA2),z = -lOOO..lOOO)); results in an error message. With > r e a d l i b ( ' evalf / i n t ' ) :

> ' e v a l f / i n t ' (exp(-zA2),z = -1O00..1000,10,~NCrule); 1.772453851 we get the correct result. The third argument specifies the accuracy and the last one specifies the internal notation of the approximation method.

19.8 Usinq the Computer 949 4. Numerical Solution of Differential Equations We solve ordinary differential equations with the Maple operation dsolve given in 20.4.4, 5., p. 991. However. in most cases it is not possible to determine the solution in closed form. In these cases, we can try to solve it numerically, where we have to give the corresponding initial conditions. In order to do this, we use the command dsolve in the form dsolve(deqn, var, numeric) (19.296) with the option numeric as a third argument. Here, the argument deqn contains the actual differential equation and the initial conditions. The result of this operation is a procedure, and if we denote it, e.g., by j , for using the command f(t), we get the value of the solution function at the value t of the independent variable. Maple applies the Runge-Kutta method to get this result (see 19.4.1.2, p. 901). The default accuracy for the relative and for the absolute error is 10-D'g'tS+3.The user can modify these default error tolerances with the global symbols -RELERR and -ABSERR. If there are some problems during calculations, then Maple gives different error messages. IAt solving the problem given in the Rung-Kutta methods in 19.4.1.2, p. 902, Maple gives: > T := dsolve({diff(y(r),z) = (1/4) * ( 2 9 +y(z)"2),y(O) = O},y(a),numeric); r := proc 'dsolve/numeric/result2' (z, 1592392, [l])end With

> r(0.5);-+ ( 4 . 5 ) = 0.5000000000,y(2)(.5) = 0.01041860472) we can determine the value of the solution, e.g., at z = 0.5.

950

20. Computer Alqebra Systems

20 Computer Algebra Systems 20.1 Introduction 20.1.1 Brief Characterization of Computer Algebra Systems 1. General Purpose of Computer Algebra Systems The development of computers has made possible the introduction of computer algebra systems for "doing mathematics". They are software systems able to perform mathematical operations formally. These systems, such as Macsyrna, Reduce, Derive, Maple, Mathcad, Mathernatica, can also be used on relatively small computers (PC),and with their help, we can transform complicated expressions, calculate derivatives and integrals, solve systems of equations, represent functions of one and of several variables variables graphically, etc. They can manipulate mathematical expressions, i.e., they can transform and simplify mathematical expressions according to mathematical rules if this is possible in closed form. They also provide a wide range of numerical solutions to required accuracy, and they can represent functional dependence between data sets graphically. Most computer algebra systems can import and export data. Besides a basic offer of definitions and procedures which are activated at every start of the system, most systems provide a large variety of libraries and program packages from special fields of mathematics, which can be loaded and activated on request (see [20.4]). Computer algebra systems allow users to build up their own packages. However, the possibilities of computer algebra systems should not be overestimated. They spare us the trouble of boring, time-demanding, and mechanical computations and transformations, but they do not save us from thinking. For frequent errors see 19.8.2, p. 936.

2. Restriction to Mathematica and Maple The systems are under perpetual developing. Therefore, every concrete representation reflects only a momentary state. In the following, we introduce the basic idea and applications of these systems for the most important fields of mathematics. This introduction will help for the first steps in working with computer algebra systems. In particular, we discuss the two systems Mathernatica (version 2.2) and Maple V. These two systems seem to be very popular among users, and the basic structure of the other systems is similar. (Today Mathernatica version 4.1 [20.7, 20.111 and Maple V release 6 [20.6] are current).

3. Input and output in Mathematica and Maple In this book, we do not discuss how computer algebra systems are installed on computers. It is assumed that the computer algebra system has already been started by a command, and it is ready to communicate by command lines or in a Windows-like graphical environment. The input and output is always represented for both Mathernatica (see 19.8.4.1, l.,p. 943) and Maple (see 19.8.4.2, l., p. 946) in rows which are distinguished from other text-parts, e.g., in the form I n f f l := Solve[J z - 5 == 0, z] in Mathernatica, (20.1) > solve(3 * z- 5 = 0 , ~ ) in Maple. System specific symbols (commands, type notation, etc.) will be represented in typewriter style. In order to save space, we often write the input and the output in the same row in this book, and we separate them by the symbol 4.

20.1.2 Examples of Basic Application Fields 20.1.2.1 Manipulation of Formulas Formala manipulation means here the transformation of mathematical expressions in the widest sense, e.g., simplification or transformation into a useful form, representation of the solution of equations or equation systems by algebraic expressions, differentiation of functions or determination of indefinite

20.1 Introduction 951 integrals, solution of differential equations, formation of infinite series, etc. ISolution of the following quadratic equation: xz + ax t b = 0 with a, b E R. (20.2a) In Mathernatica, we type: (20.2b) Solve[z^2 t a z + b == 0 , 2 ] . After pressing the corresponding input key/keys (ENTER or SHIFT+RETURN, depending on the operation system), Mathematica replaces this row by (20.2c) In[1] := Solve[z"2 a x b == O,x] and starts the evaluation process. In a moment, the answer appears in a new row - a - Sqrt[a2 - 461 - a t Sqrt[a2 - 4b] 1). (20.2d) Uut[lI = {{x-> 1,{x-> 2 2 Mathernatica has solved the equation and both solutions are represented in the form of a list consisting of two sublists. Here Sqrt means the square root. In Maple the input has the following form: (20.3a) solve(z"2 + a * x t b = 0 , x ) ; The semicolon after the last symbol is very important. After the equation is entered with the ENTER key, Maple evaluates this input and the resulting output is displayed directly below the input

+ +

(20.3b) 1/2(-a + (a' - 4b)"'), 1/2(-a - (aZ- 4b)'") The result is given in the form of a sequence of two expressions representing the solutions. Except for some special symbols of the systems, the basic structures of commands are very similar. At the beginning there is a symbol, which is interpreted by the system as an operator, which is applied to the operand given in braces or in brackets. The result is displayed as a list or sequence of solutions or answers. Several operations and formula manipulations are represented similarly.

20.1.2.2 Numerical Calculations Computer algebra systems provide many procedures to handle numerical problems of mathematics. These are solutions of algebraic equations, linear systems of equations, the solutions of transcendental equations. calculation of definite integrals, numerical solutions of differential equations, interpolation problems, etc. IProblem: Solution of the equation (20.4a) x6 - 2x5 - 30x4 + 36x3 1 9 0 1 ~-~362 - 150 = 0. Although this equation of degree six cannot be solved in closed form, it has six real roots, which are to be determined numerically. In Mathernatica the input is: (20.4b) InC21 := NSolve[x"6 - 2xA5- 3 0 d 4 + 3 6 ~ +~ 139 0 ~ ~ 362 2 - 150 == O,Z] It results in the answer: Out [21= {{z-> -4.42228}, {z-> -2.14285}, {z-> -0.937397}, {z-> 0.972291},

+

(2->

3.35802}, {z-> 5.17217))

(20.4~)

This is a list of the six solutions with a certain accuracy which will be discussed later. The input in Maple is: (20.4d) fsolve(x"6 - 2 * x"5 - 30 * x"4 + 36 * x"3 + 190 * 2'2 - 36 * x - 150 = 0, E); Here, the input of 'I= 0" can be omitted, and the assignment of the variable "z"is also not necessary here, since it is the only one. Maple automatically considers the entered expression to be equal to zero.

952

20. Computer Alqebra Systems

The output is the sequence of the six solutions. The application of the command f solve tells to Maple that the result is wanted numerically in the form of floating-point decimal numbers.

20.1.2.3 Graphical Representations Most computer algebrasystems allow the graphical representation of built-in and self-defined functions. Usually, this is the representation of functions of one variable in Cartesian and in polar coordinates, parametric representation, and representation of implicit functions. Functions of two variables can be represented as surfaces in space; parametric representation is also possible. Further more curves can be demonstrated in three-dimensional space. In addition, there are further graphical possibilities to demonstrate functional dependence between data sets, e.g., in the form of diagrams. All systems provide a wide selection of options of representation such as line thickness of the applied elements, e.g., vectors, different colours, etc. Usually, the represented graphics can be exported in an appropriate format such as Postscript, Raster or Plotter and they can be built into other programs, or directly printed on a printer or plotter.

20.1.2.4 Programming in Computer Algebra Systems All systems allow useres to develop their own packages to solve special problems. This means both the use of well-known tools to build up procedures, e.g., DO, IF - THEN, WHILE, FOR, etc. and, on the other hand the application of the built-in methods of the system which allow elegant solutions for many problems. Self-constructed program blocks can be introduced into the libraries and they can be reactivated at any time.

20.1.3 Structure of Computer Algebra Systems 20.1.3.1 Basic Structure Elements 1. Type of Objects Computer algebra systems work with several different types of objects. Objects are the known family of numbers, variables, operators, functions, etc., which are loaded at the start of the system, or which are defined by the user according to a suitable syntax. Classes of objects, like type of numbers or lists, etc., are called types. Most of the objects are identified by their names, which one can think of as associated to an object class and which must satisfy the given grammatical rules. The user gives a sequence of objects in the input row, Le., their names, corresponding to the given syntax, closes the input with the corresponding special key and/or by a special system command, then the system starts evaluation and returns the result in a new row or rows. (Input can be spread over several lines.) The objects, object types and object classes described below are available in every computer algebra system, and their particular specialities are described in the manuals for the system. 2. Numbers Computer algebra systems usually use the number types integers, rational numbers, real numbers (Boating-point numbers), complex numbers; some systems also know algebraic numbers, radical numbers, etc. With different type-check operations, the type or certain properties of given numbers can be determined, like non-negative,prime, etc. Floating-point numbers can be determined with arbitrary accuracy. Usually, the systems work with a default precision, which can be changed on request. The systems know the special numbers, which have a fundamental importance in mathematics, such as e, T and co.They use these numbers symbolically, but they also know their numerical approximation to an arbitrary accuracy.

10.2 Mathematica 953

3. Variables and Assignments Variables have names represented by given symbols which are usually determined by the user. There are names that are predefined and reserved by the system; they cannot be chosen. While no value is assigned to the variable, the symbol itself stands for the variable. Values can be assigned to the variables by special assignment operators. These values can be numbers, other variables, special sequences of objects, or even expressions. In general, there exist some assignment operators which differ, first of all, in the time of their evaluation, Le., right after their input or for the first time at a later call of the variable. 4. Operators All systems have a basic set of operators. The usual operators of mathematics +, -, *, /, A (or * *), >, <, = belong to this set, for which the usual order of precedence is valid for evaluation. The znfiz form of an expression means that these operators are written between the operands. The set of operators written in prefix form - where the operator is written in front of the operand is dominant in all systems. This type of operator operates on special classes of objects, e.g., on numbers, polynomials, sets, lists, matrices, or on systems of equations, or they operate as functional operators. such as differentiation, integration, etc. In general, there are operators for organizing the form of the output, manipulating strings and further systems of objects. Some systems allow us to represent some operators in sufixform, Le., the operator is behind the operand. Operators often use optional arguments. 5 . Terms and Functions The notion of term means an arrangement of objects connected by mathematical operators, usually in infix form, hence, it means certain basic elements often occurring in mathematics. A basic task of computer algebra systems is transforming terms and solving equations. IThe following sequence (20.5) ~ " -4 5 * ~ " t3 2 * ~ " -2 8 is, e.g., a term, in which IC is a variable. Computer algebra systems know the usual elementary functions such as the exponential function, logarithmic function, trigonometric functions, and their inverses and several other special functions. These functions can be placed into terms instead of variables. In this way, complicated terms or functions can be generated. ~

6. Lists and Sets All computer algebra systems know the object class of lists, which is considered as a sequence of objects. The elements of a list can be reached by special operators. In general, the elements of a list can be lists themselves. So, we can get nested lists, which are used in the construction of special types of objects, such as matrices. tensors; all systems provides such special object classes. They make it possible to manipulate objects like vectors and tensors symbolically in vector spaces, and to apply linear algebra. The notion of set is also known in computer algebra systems. The operators of set theory are defined. In the following sections, the basic structure elements and their syntax will be discussed for the two chosen computer algebra systems, Mathematica 4.1 and Maple V.

20.2 Mathernatica Mathematica is a computer algebra system, developed by Wolfram Research Inc. A detailed description of Mathematica 4.1 can be found in j20.7, 20.111.

20.2.1 Basic Structure Elements In Mathematica the basic structure elements are called expressions. Their syntax is (we emphasize again, the current objects are given by their corresponding symbol, by their names): (20.6) obj,[obj,, obj,, . . . , obj,]

I

954

20. Computer Aloebra Svstems

obj, is called the head of the expression; the number 0 is assigned to it. The parts obji (i = 1,.. . , n) are the elements or arguments of the expression, and one can refer to them by their numbers 1,.. . , n. In many cases the head of the expression is an operator or a function, the elements are the operands or variables on which the head acts. Also the head, as an element of an expression, can be an expression, too. Square brackets are reserved in Mathernatica for the representation of an expression, and they can be applied only in this relation. IThe term xA2+2*x+1,which can also be entered in this infix form in Mathernatica, has the complete form (FullFonn) Plus[l,Times[2,x], Power[x,211 which is also an expression. Plus, Power and Times denote the corresponding arithmetical operations. The example shows that all simple mathematical operators exist in prefix form in the internal representation, and the term notation is only a facility in Mathernatica. Parts of expressions can be separated. This can be done with Part[expr,i], where i is the number of the corresponding element. In particular, i = 0 gives back the head of the expression. I If we enter in Mathernatica I~[I] := 2 9 + 2 * x 1 where the sign can be omitted, then after the ENTER key is pressed, Mathernatica answers Uut[ll= 1 + 2 x t x* Mathernatica analyzed the input and returned it in mathematical standard form. If the input had been terminated by a semicolon, then the output would have been suppressed. If we enter Inf2-l := FullForm[%] then the answer is Out [2l = Plus[l,Times[2,xI1Power[z,211 The sign % in the square brackets tells Mathernatica that the argument of this input is the last output. From this expression it is possible to get, e.g., the third element 1n[3] := Part[%, 31 for instance Out f31= Power[x,21 which is an expression in this case. Symbols in Mathernatica are the notation of the basic objects; they can be any sequence of letters and numbers but they must not begin with a number. The special sign $ is also allowed. Upper-case and lower-case letters are distinguished. Reserved symbols begin with a capital letter, and in compound words also the second word begins with a capital letter. Users should write their own symbols using only lower-case letters.

+

20.2.2 Types of Numbers in Mathematica

Type of number Integers Rational numbers Real numbers Complex numbers

Head Integer Rational Real Complex

Characteristic exact integer, arbitrarily long fraction of coprimes in form Integer/Integer floating-point number, arbitrary given precision complex number in the form number+number *I

Input nnnnn pppplqqqq

nnnn.mmmm

Real numbers, Le., floating-point numbers, can be arbitrarily long. If an integer nnn is written in the form nnn.,then Mathernatica considers it as a floating-point number, that is, of type Real.

20.2 Mathernatica 955

The type of a number z can be determined with the command Head[x]. Hence, I n f f l := Head[5l] results in Outff] = Integer, while In[2] := Head[51.] OutC21 = Real. The real and imaginary components of a complex number can belong to any type of numbers. A number such as 5.731 + 0 I is considered as a Real type by Mathernatica, while 5.731 0. I is of type Complex, since 0. is considered as a floating-point approximation of 0. There are some further operations, which give information about numbers. So, if z is a number Inf31 := NumberQ[x]Out C31 := True, (20.7a) Otherwise, if z is not a number, then the output is Uutf31 =False. Here, True and False are the symbols for Boolean constants. IntegerQ[z] tests if x is an integer, or not, so Inf41 := IntegerQ[2.] Uutf41 = False (20.7b) Similar tests can be performed for numbers with heads EvenQ, OddQand PrimeQ.Their meanings are obvious. So. we get (20.7~) Inf51 := PrimeQ[1075643] Out C51 = True, while Inf61 := PrimeQ[1075641] Uutf61 = False (20.7d) These last tests belong to a group of test operators, which all end by Q and always answer True or False in the sense of a logical test (in this case a type check).

+

20.2.2.2 Special Numbers In Mathernatica, there are some special numbers which are often needed, and they can be called with arbitrary accuracy. They include A with the symbol Pi, e with the symbol E, as the transformation 180" factor from degree measure into radian measure with the command Degree, I n f i n i t y as the symbol for w and the imaginary unit I.

20.2.2.3 Representation and Conversion of Numbers Numbers can be represented in different forms which can be converted into each other. So, every real number x can be represented by a floating-point number N[x, n]with an n-digit precision. INf71 := N[E, 201 yields Outf71 = 2.7182818284590452354 (20.8a) LVith Rationalize[x, dz], the number x with an accuracy dx can be converted into a rational number, i.e., into the fraction of two integers. 1457 (20.8b) I n f 8 1 := Rationalize[%, 10" - 51 OutC81 = 536 With 0 accuracy, Mathernatica gives the possible best approximation of the number x by a rational number. Numbers of different number systems can be converted into each other. With BaseForm[z, b], the number z given in the decimal system is converted into the corresponding number in the number system with base b. If b > 10, then the consecutive letters of the alphabet a, b, c, . . . are used for the further digits having a meaning greater than ten. A: Inf151 := BaseForm[255, 161 Uutf15l = //BaseForm = ff16 (20.9a) Inf16l := BaseForm[N[E, lo], 81 Outff6l = //BaseForm = 2.5576052138

(20.9b)

The reversed transformation can be performed by b""mmmm.

B: Inf171 := 8""735

Out[f7] = 477

(20.9~)

956

20. Computer Aloebra Susterns

Numbers can be represented with arbitrary precision (the default here is the hardware precision), and for large numbers so-called scientific form is used, Le., the form n.mmmml0” f qq.

20.2.3 Important Operators Several basic operators can be written in infix form (as in the classical form in mathematics) < symbl op symbz > . However, in every case, the complete form of this simplified notation is an expression. The most often occurring operators and their complete form are collected in Table 20.2. Table 20.2 Important Operators in Mathernatica ~

+

a b a b or a * b a“b alb u-> v r =s

Plus[a, b] Times[a, b] Power[a, b] Times[a, Power[b, -111 Rule[u, v] Set[?, s]

u == v Equal[u, v] w!= u Unequal[w, u]

r >t r >= t s s

<= t

Greater[r, t] GreaterEqual[r, t] Less[s, t] LessEqual[s, t]

Most symbols in Table 20.2 are obvious. For multiplication in the form ab, the space between the factors is very important. The expressions with the heads Rule and Set will be explained. Set assigns the value of the expression s on the right-hand side, e.g., a number, to the expression r on the left-hand side, e.g., a variable. From here on, r is represented by this value until this assignment is changed. The change can be done either by a new assignment or by x = . or Clear[x],Le., by releasing every assignment so far. The construction Rule should be considered as a transformation rule. It occurs together with the substitution operator

1.

,

Replace[t, u -> v] or t / . u-> v means that every element u which occurs in the expression t will be replaced by the expression v. IInf51 := x + y2 /. y-> a b Out 151 = x ( a b)’ It is typical in the case of both operators that the right-hand side is evaluated immediately after the assignment or transformation rule. So, the left-hand side will be replaced by this evaluated right-hand side at every later call. Here, we have to mention two further operators with delayed evaluation. (20.10a) u := v FullForm = SetDelayed[u,v] and (20.10b) u :> v FullForrn = RuleDelayed[u,v] The assignment or the transformation rule are also valid here until it is changed. although the lefthand side is always replaced by the right side, the right-hand side is evaluated for the first time only at the moment when the left one is called. The expression u == ti or Equal[u, v] means that u and v are identical. Equal is used, e.g., in manipulation of equations.

+

+ +

20.2.4 Lists 20.2.4.1 Notions Lists are important tools in Mathernatica for the manipulation of whole groups of quantities, which are important in higher-dimensional algebra and analysis. A list is a collection of several objects into a new object. In the list, each object is distinguished only by its place in the list. The definition of a list is made by the command (20.11) List[al, a2. a3, . . .] I {a& a2, a3, . . .}

20.2 Mathernatica 957

To explain the work with lists, a concrete list is used, denoted by 11: I n f f l := 11 = Lis t[al, a2, a3, a4, a5, a61 O u t f l l = {al, a2, 03, a4, a5, a6}

(20.12) Mathematica applies a short form to the output of the list: It is enclosed in curly braces. Table 20.3 represents commands which choose one or more elements from a list, and the output is a “sublist”. Table 20.3 Commands for the choice of list elements First [1] Last[1] Part[1, n] or 1[[n]] Part[1, {nl, n2, .}] 1[[{nl, n2, . . .}]I T&e[1, m] T&e[l, {m, .)I Drop[l. n] Drop[l, {m, .)I I

.

selects the first element selects the last element selects the n-th element gives a list of the elements with the given numbers equivalent to the previous operation gives the list of the first m elements of 1 gives the list of the elements from m through n gives the list without the first n elements gives the list without the elements from m through n

I For the list 11 in (20.11)we get, e.g., Inf21 := F i r s t [ l l ] Uutf21 = a1 Inf31 := l1[[3]] Uutf3]= a3 Inf41 := l1[[{2, 4, 6)]] Uutf41 = (a2, a4, as} Inf5l := Take[ll, 21 Uutf51 = {al, a2}

20.2.4.2 Nested Lists, Arrays or Tables The elements of lists can again be lists, so we get nested lists. If we enter, e.g., for the elements of the previous list 11 In[6] := a1 = { b l l , b12, b13, b14, b15) In[7] := a2 = (b21, b22, b23, b24, b25) Inf81 := a3 = (b31, b32, b33, b34, b35)

and analogously for a4, a5 and a6, then because of (20.12) we get a nested list (an array) which we do not represent here explicitly. We can refer to the j-th element of the i-th sublist with the command Part[1, i, j ] . The expression l[[i, j]]has the same result. In the above example, e.g., I n f f 2 l := 11[[3,411 yields Uutfl21 = b34 Furthermore, Part[1, { i l l 22 ...}, {jl,j 2 ...}I or l[[{il, i2, ...)${jl,j 2 , ...}]I results in a list consisting of the elements numbered with j l , j 2 . .. from the lists numbered with il, i2,. . .. IFor the above example Inff3-7:= 11[[{3,5), {2, 3, 4}]] Uut[131 = {{b32, b33, b34), (b52, b53, b54)) The idea of nesting lists is obvious from these examples. It is easy to create lists of three or higher dimensions, and it is easy to refer to the corresponding elements.

20.2.4.3 Operations with Lists Mathernatica provides several further operations by which lists can be monitored, enlarged or shortened (Table 20.4). IWith Delete, the list 11 can be shortened by the term a6: In[l4] := 12 = Delete[ll, 61 Uutf141 = {al, a2, a3, a4, as}, where in the output the ai are shown by their values - they are lists themselves.

I

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20. Computer Al.qebra &stems

Table 20.4 Operations with lists ~~

Position[l, a] MemberQ[l, a] FreeQ[l, a] Prepend[l, a] Append[&a1 Insert[l, a, i] Delete[l. {i, j , . . .}] Replacepart [ I , a, i]

gives a list of the positions where a occurs in the list checks whether a is an element of the list checks if a does not occur in the list changes the list by adding a to the front changes the list by appending a to the end inserts a at position z in the list delete the elements at positions z,j, . . . from the list replace the element at position z by a

20.2.4.4 Special Lists In Mathematica, several operations are available to create special lists. One of them, which often occurs in working with mathematical functions, is the command Table shown in Table 20.5. Table 20.5 Operation Table creates a list with imax values of f : f ( l ) , f(2), . . . , f(imax) Table[f, {imax}] Table[f, {i, imin,imaz}] creates a list with values off from imin to imax Table[f, {z, imin,imax, di}] the same as the last one, but by steps di ITable of binominal coefficients for n = 7: I n T l 5 l := Table[Binomial[7, 21, {i, 0, 7 } ] ] Uut[f51 = (1, 7, 21, 35, 35, 21, 7 , 1) With Table, also higher-dimensional arrays can be created. With the expression Table[f, (2, i l , i2}, ( j , jl, j2}, ...I we get a higher-dimensional, multiple nested table, Le.! entering InTl61 := Table[Binomial[i,j ] ) {z, 1, 7 } , {jl0, i} ] we get the binominal coefficients up to degree 7: OutT161 = ((1, 11, (1, 2,

11, (1, 3, 3, 11, (1, 4, 6, 4, 11,

(1, 5 , 10, 10, 5 , l}, (1, 6, 15, 20, 15, 6, l}, {II 7, 21, 35, 35, 21, 7, l}}

The operation Range produces a list of consecutive numbers or equally spaced numbers: Range[n] yields the list (1, 2, . . . n) Similarly, Range[nl, n2] and Range[nl, n2, dn]produce lists of numbers from n l to n2 with step-size 1 or dn respectively. ~

20.2.5 Vectors and Matrices as Lists 20.2.5.1 Creating Appropriate Lists Several special (list) commands are available for defining vectors and matrices. A one-dimensional list of the form (20.13) v = { v l , v2, . . . , vn} can always be considered as a vector in n-dimensional space with components v l , v2,. . . , vn. The special operation Array[v, n] produces the list (the vector) (v[1], 421,. . . , v[n]}. Symbolic vector operations can be performed with vectors defined in this way. The two-dimensional lists 11 of (20.2.4.2, p. 957) and 12 (20.2.4.3, p. 957) introduced above can be considered as matrices with rows i and columns j . In this case bij would be the element of the matrix

20.2 Mathernatica 959 in the i-th row and the j-th column. A rectangular matrix of type (6,5) is defined by 11, and a square matrix of type ( 5 , 5 ) by 12. With the operation Array[b, {n, m}] a matrix of type (n,m) is generated, whose elements are denoted by b[i, j]. The rows are numbered by i , i changes from 1 to n; by j the columns are numbered from 1 to m. In this symbolic form 11 can be represented as (20.14a)

11 = Array[b, {6, 5}], where

(20.14b) bji, j] = bij ( i = 1,.. . ,6; j = 1,.. . , 5 ) . The operation IdentityMatrix[n] produces the n-dimensional unit matrix. With the operation DiagonalMatrix[list] a diagonal matrix is produced with the elements of the list in its main diagonal. The operation Dimension[list] gives the size (number of rows, columns, . . . ) of a matrix, whose structure is given by a list. Finally, with the command MatrixForm[lzst],we get a matrix-type representation of the list. A further possibility to define matrices is the following: Let f(i, j ) be a function of integers i and j . Then, the operation Table[f, {i, n}, { j , m}]defines a matrix of type (n,m ) ,whose elements are the corresponding f ( i ,j ) .

20.2.5.2 Operations with Matrices and Vectors Mathernatica allows formal manipulation of matrices and vectors. The operations given in Table 20.6 can be applied. Table 20.6 Operations with matrices ca

a.b Det[a] Inverse[a] Transpose[a] MatrixPower[a, n] Eigenvalues[a] Eigenvectors[a]

matrix a is multiplied by the scalar c the product of matrices a and b the determinant of matrix a the inverse of matrix a the transpose of matrix a the n-th power of matrix a the eigenvalues of matrix a the eigenvectors of matrix a

Here, the transpose matrix rT of T is produced. If the general four-dimensional vector v is defined by Inf2Ol := u = Array[u, 41, then we get Outf2Ol = {u[l]>421, @I,

u[4]}

960

20. Computer Al.qebra Svsterns

' 'I

Now, the product of the matrix r and the vector v is again a vector (see Calculations with Matrices, 4.1.4, p. 254). Inf211 := r. v Out f 2 l l = { a 1, 1 u 11 a[l, 21 421 a[l, 31 u[3] a[l, 41 u[4] , a 2, 1 u 11 4 2 , 21 421 4 2 , 31 u[3] 4 2 , 41 u[4] , 4 3 , 11u 11 43, 21 421 43, 31 u[3] 43, 41 u[4] , 4 4 , 11411 4 4 , 21 421 4 4 , 31 u[3] 4 4 , 41 u[4] }. There is no difference between row and column vectors in Mathernatica. In general, matrix multiplication is not commutative (see Calculations with Matrices 4.1.4, p. 254). The expression T . v corresponds to the product in linear algebra when a matrix is multiplied by a column vector from the right, while u. T means a multiplication by a row vector from the left. I B: In the section on Cramer's rule (4.4.2.3, p. 275) the linear system of equations p t = b is solved with the matrix In1221 : = p = MatrixForm[{{2, 1, 3}, (1, -2, l}, (3, 2, 2}}] Out [22] = //MatrixForm = 2 1 3 1 -2 1 3 22

+

+ + +

+

+ + +

+

+ + +

and vectors Inf231 := t = Array[z, 31 Out f a 1 = (411, 421, z[3]} I n f q l := b = (9, -2, 7) Out [&I = (9, -2, 7). Since in this case det(p) # 0 holds, the system can be solved by t = p-'b. This can be done by In[25] := Inverseb]. b with the output of thesolutionvector Outf251 = {-1, 2, 3).

20.2.6 Functions 20.2.6.1 Standard Functions Mathernatica knows several mathematical standard functions, which are listed in Table 20.7 Table 20.7 Standard functions Exponential function Logarithmic functions Trigonometric functions Arc functions Hyperbolic functions Area functions

Exp[x] Log[x], Log[b,x] Sin[x],Cos[x], Tan[x], Cot[x], Sec[x], Csc[x] ArcSin[x], ArcCos[x], ArcTan[x], ArcCot[x], ArcSec[x], ArcCsc[x] Sinh[x], Cosh[x], Tanh[x], Coth[x], Sech[x], Csch[x] ArcSinh[x], ArcCosh[x], ArcTanh[x], ArcCoth[x], ArcSech[x], ArcCsch[x]

All these functions can also be applied with complex arguments In every case we have to consider the single-valuedness of the functions. For real functions we have to choose one branch of the function (if it is needed); for functions with complex arguments the principal value (see 14.5, p. 696) should be chosen.

20.2.6.2 Special Functions Mathernatica knows several special functions, which are not elementary functions. Table 20.8 lists some of these functions. Tabelle 20.8 Special functions Bessel functions & ( z ) and Y,(z) Modified Bessel functions I n ( z )and K , ( z ) Legendre polynomials P,(z) Spherical harmonic ~"'(t9,4)

BesselJ[n,z],BesselY[n,z] BesselI[n,z], BesselK[n,z] LegendrP[n,x] SphericalHarmonicY[l, m, theta, phi]

20.2 Mathernatica 961 Further functions can be loaded with the corresponding special packages (see also [17.1]).

20.2.6.3 Pure Functions Mathernatica supports the use of so-called pure functions. A pure function is an anonymous function, an operation with no name assigned to it. They are denoted by Function[z, body]. The first argument specifies the formal parameters and the second one is the body of the function, Le., body is an expression for the function of the variable 2 . InCf] := Function[x, 2"3 + 2"2] Out ClI = Function[z, z3+ 21' (20.15) and so In[2] := Function[z, zA3+ zA2][c] gives OutC21 = c3 + e'. (20.16) We can use a simplified version of this command. It has the form body &, where the variable is denoted by # . Instead of the previous two rows we can also write (20.17) 1nC31 := (#"3 + #^2) & [e] out C3I = c3 t c2 It is also possible to define pure functions of several variables: Function[ { q 5>2 , . . .}, body] or in short form body&, where the variables in body are denoted by the elements #1, #2:. . .. The sign & is very important for closing the expression, since it can be seen from this sign that the previous expression should be considered as a pure function.

20.2.7 Patterns Mathernatica allows users to define their own functions and to use them in calculations. (20.18) With the command InCfI := fk-1 := Polynom(z) with Polynom(s) as an arbitrary polynomial of variable x,a special function is defined by the user. In the definition of the function f , there is no simple 2 , but 2- (pronounced 2-blank) with a symbol for the blank. The symbol 5- means "something with the name 2 " . From here on, every time when the expression f [something] occurs, Mathernatica replaces it by its definition given above. This type of definition is called a pattern. The symbol blank- denotes the basic element of a pattern; y- stands for y as a pattern. It is also possible to apply in the corresponding definition only a ''-", that is y"-. This pattern stands for an arbitrary power of y with any exponent, thus, for an entire class of expressions with the same structure. The essence of a pattern is that it defines a structure. When Mathernatica checks an expression with respect to a pattern, it compares the structure of the elements of the expression to the elements of the pattern, Mathernatica does not check mathematical equality! This is important in the following example: Let 1 be the list (20.19) : 2"~)) 1n[21:= 1 = (1, y, y'a, y " ~ q r t [ z ] {f[y"(r/q)], If we write (20.20) 1nC3]:=1/.y~_-> j a then Mathernatica returns the list (20.21) 0utC31 = (1. y, j a , j a , {f[ja], Z'}} Mathernatica checked the elements of the list with respect to its structural identity to its pattern y"and in every case when it determined coincidence it replaced the corresponding element by j a . The elements 1 and y were not replaced, since they have not the given structure, even though yo = 1,y' = y holds. Remark: Pattern comparison always happens in FullForm. If we examine In&] := bly /. y"- -> j a then we get Out C4]= b j a This is a consequence of the fact that FullForm of b/y is Times[b, Power[y, -111, and for structure (20.22) comparison the second argument of Times is identified as the structure of the pattern. With the definition mc51 := f[zJ := 2"3 (20.23a)

I

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20. Computer Algebra Systems

Mathernatica replaces, corresponding to the given pattern, I n f 6 l = f [T] by Out f6l = r3 etc.

+

I n R 1 := f [a] t f [z] yields Out f71 = a3 x3

(20.2313) (20.23~)

If Inf81 := f[z] := z”3, so for the same input Inf71 := . . . then the output would be Out [71= f [a] + z3 In this case only the “identical” input corresponds to the definition.

(20.23d) (20.23e)

20.2.8 Functional Operations Functions operate on numbers and expressions. Mathernatica can also perform operations with functions, since the names of functions are handled as expressions so they can be manipulated as expressions. 1. Inverse Function The determination of the inverse function of a given function f(z) can be made by the functional operation InverseFunction. IA: I n f l l := InverseFunction[f] [z] O u t f l l = f-’ [z] Outf21 = Log IB: In[2] := InverseFunction[Exp] 2. Differentiation Mathernatica uses the possibility that the differentiation of functions can be considered as a mapping in the space of functions. In Mathernatica, the differentiation operator is Derivative[l][f]or in short form f’. If the function f is defined, then its derivative can be got by f’. IInf31 := f [z-] := Sin[z]Cos[z] With

1,1141

:= f ’

we get 0utf4]=

COS[#^]^

- ~ i n [ # l ]& *,

hence f ’ is represented as a pure function and it corresponds to 1nf5I := % [z] Out E] = C O S [ Z ] ~ - ~ i n [ z ] ’ 3. Nest The command Nest[!, E, n] means that the function f nested n times into itself should be applied on z. The result is f [f [. . . f [E]] . . .]. 4. NestList By NestList[f, z, n] a list {zl f [z],f [f [z]],. . .} will be shown, where finally f is nested n times. 5. Fixedpoint For FixedPoint[f, E], the function is applied repeatedly until the result does not change. 6. FixedPointList The functional operation FixedPointList[f, z] shows the continued list with the results after f is applied, until the value no longer changes. IAs an example for this type of functional operation the NestList operation will be used for the approximation of a root of an equation f(z) = 0 with Newton’s method (see 19.1.1.2,p. 882). We seek a root of the equation z cos z = sin z in the neighborhood of 3 ~ 1 2 : In[6] := f[z-] := z - Tan[z] Inf71 := f‘[z] Outf71 = 1 - Sec[zj2 I n D ] = g[z-] := z - f [z]/f’[z] Inf91 := NestList[g, 4.6, 41 // N Outf91 = (4.6, 4.54573, 4.50615, 4.49417, 4.49341) InflOl := FixedPoint[g, 4.61 OutflO] = 4.49341

A higher precision of the result can also be achieved.

20.2 Mathernatica 963 7. Apply Let f be a function which is considered in connection with a list { a , b, c, . . .}. Then, we get (20.24) APPlY[f, { a : 6,c, ' ' .}I f[Q,b, c, ' ' .I W In[l] := Apply[Plus, { u , u, w}]

Outfl] = u t w

+w

In[2] := Apply[List, Q + b + c] Outf2l = {a, b, c) Here, the general scheme of how Mathernatica handles expressions of expressions can be easily recognized. We write the last operation in FullForm: In[3] := Apply[List, Plus[a, b, c]] Out[3] = List[a, b, c] The functional operation Apply obviously replaces the head of the considered expression Plus by the required List. 8. Map With a defined function f the operation Map gives: (20.25) Map[!. {a, 6,c, . . -+ {fblIfP1, f[cI,. . .I Map generates a list whose elements are the values when f is applied for the original list. ILet f be the function f(z)= z2. It is defined by 1n[4] := f[z..] := 2%' With this f we get In[51 := Map[f, { u , u , tu}] Outf51 = { u * ) u2, w2} Map can be applied for more general expressions: In[6] := Map[f, Plus[a, 6,e]] OutC6l = a2 t b2 + c2

.}I

20.2.9 Programming Mathernatica can handle the loop constructions known from other languages for procedural programming. The two basic commands are Do[ezpr, {i, i l , 22, di}] (20.26a) and While[test, ezpr] (20.26b) The first command evaluates the expression ezpr, where i runs over the values from i l to 22 in steps di. If di is omitted, the step size is one. If i l is also missing, then it starts from 1. The second command evaluates the expression while test has the value True. IIn order to determine an approximate value of e', the series expansion of the exponential function is used: I n f l I := sum = 1.0; Do[sum = sum t (2%/2!), {i, 1,lo}]; (20.27) sum Out [ll = 7.38899 The Do loop evaluates its argument a previously given number of times, while the While loop evaluates until a previously given condition becomes false. Among other things, Mathematica provides the possibility of defining and using local variables. This can be done by the command (20.28) Module[{tl, t2,. . .}>procedure] The variables or constants enclosed in the list are locally usable in the module; their values assigned here are not valid outside of this module.

I

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20. Computer A1,qebra Systems

IA: We have to define a procedure which calculates the sum of the square roots of the integers from 1 to n. 1n[11 := sumq[n-] := Module[{sum = l.}, (20.29) Do[sum = sum N[Sqrt[i]], {i,2, n}]; sum 1;

+

The call sumq[30] results in 112.083. The real power of the programming capabilities of Mathematica is, first of all, the use of functional methods in programming, which are made possible by the operations Nest, Apply, Map and by some further ones. IB: Example A can be written in a functional manner for the case when an accuracy of ten digits is required: sump[n-] := N[Apply[Pl~~, Table[Sqrt[i],{i, 1,n}]], lo], sumq[30] results in 112.0828452. For the details, see [20.6].

20.2.10 Supplement about Syntax, Information, Messages 20.2.10.1 Contexts, Attributes Mathernatica must handle several symbols; among them there are those which are used in further program modules loaded on request. To avoid many-valuedness, the names of symbols in Mathernatica consist of two parts, the context and the short name. Short names mean here the names (see 20.2, p. 953) of heads and elements of the expressions. In addition, in order to name a symbol Mathernatica needs the determination of the program part to which the symbol belongs. This is given by the context, which holds the name of the corresponding program part. The complete name of a symbol consists of the context and the short name, which are connected by the ’ sign. When Mathernatica starts there are always two contexts present: System’ and Global’. We can get information about other available program modules by the command Contexts[ 1. All built-in functions of Mathematica belong to the context System’, while the functions defined by the user belong to the context Global’. If a context is actual, thus, the corresponding program part is loaded, then the symbols can be referred to by their short names. For the input of a further Mathernatica program module by << Namepackage, the corresponding context is opened and introduced into the previous list. It can happen that a symbol has already been introduced with a certain name before this module is loaded, and in this newly opened context the same name occurs with another definition. In this case Mathernatica gives a warning to the user. Then we can erase the previously defined name by the command Remove[Clobal‘name], or we can apply the complete name for the newly loaded symbol. Besides the properties that the symbols have per definition, it is possible to assign to them some other general properties, called attributes, like Orderless, Le., unordered, commutative, Protected, i.e., values cannot be changed, or Locked, i.e, attributes cannot be changed, etc. Informations about the already existing attributes of the considered object can be obtained by Attributes[f]. Some symbols can be protected by Protect[somesymbol]; then no other definition can be introduced for this symbol. This attribute can be erased with the command Unprotect.

20.2.10.2 Information Information can be obtained about the fundamental properties of objects by the commands

20.3 Maple 965

?symbol information about the object given by the name symbol, ??symbol detailed information about the object, ?B* information about all Mathernatica objects, whose name begins with B. It is also possible to get information about special operators, e.g., by ? := about the SetDelay operator.

20.2.10.3 Messages Mathernatica has a message system which can be activated and used for different reasons. The messages are generated and shown during the calculations. Their presentation has a uniform form: symbol : : tag, providing the possibility to refer to them later. (Such messages can also be created by the user.) Consider the following examples as illustrations. W A: In[lI := f[z_] := 1/x; Inf21 := f[O] 1 0

Power: :infy:Infinite expression - encountered.

Out[2] = ComplexInfinity

W B: In[3] := Log[$ 16, 251 Log: :argt:Log calledwith 3 arguments; 1 or 2 arguments are expected. Out[3] =Log[3,16,25] IC: In&] := Multply[x, zAn] General: :spelll: Possible spelling Error: new symbol name “Multply’ ’ i s s i m i l a r t o existing symbol “Multiply’ ’ . Out[4] = Multply[z, xAn] In example A. Mathematica warns us that during the evaluation of an expression it got the value m. The calculation itself can be performed. In example B the call of logarithm contains three arguments, which is not allowed according to the definition. Calculations cannot be performed. Mathernatica cannot do anything with the expression. In example C, Mathernatica finds a new symbol name, which is very similar to an existing one which is often used. A spelling error is supposed, and Mathernatica does not evaluate this expression. The user can switch off a message with Off[s : : tag]. With On the message will appear again. With Messages[symbol] all messages associated to the symbol with the name symbol can be recalled.

20.3 Maple The computer algebra system Maple was developed at the University of Waterloo (Ontario Canada). We discuss Maple V, release 4 by Waterloo Maple Software. A good intruduction can be found in the handbooks and in [20.6].

20.3.1 Basic Structure Elements 20.3.1.1 Types and Objects In Maple, all objects have a type which determines its proper affiliation in an object class. An object can be assigned to several types, as e.g., if a given object class contains a subclass defined by an additional relation. As an example it can be mentioned that the number 6 is of type integer and of type posint. By giving the type and arranging all objects in a hierarchy, a consistent formalization and evaluation of given classes of mathematical problems is guaranteed. The user can always ask about the type of an object with the question > whattype(obj); (20.30) The semicolon must be written at the end of the input. The output is the basic type of the object. Maple knows the following basic types of objects, collected in Table 20.9. The more detailed type structure can be determined by the help of commands like type(obj,typname), the values of which are the Boolean functions t r u e or f a l s e . Table 20.10 contains all type names known by Maple.

I

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20. Computer Al.qebra Systems

Table 20.9 Basic types in Maple

\+.

\

*

\A'

\

\ = \

'<>'

'<'

I'

.. \

' or'

not' exprseq float fraction function indexed integer procedure series set string table uneval

'and' list

Table 20.10 Types

*

**

-

A

+

PLOT algn algnumext and colourabl connected constant float expanded facint linear intersect laurent monomial name minus oddfunc numeric odd primeint positive posint radnum radfunext radical series relation scalar tree symmfunc taylor vector

< PLOT3D anything cubic fraction list negative operator procedure radnumext set trig

RootOf array digraph function listlist negint or quadratic range sqrt type

algebraic biconnect equation graph logical nonneg planar quartic rational square undigraph

I

<>

algext bipartite even indexed mathfunc nonnegint point radext ratpoly string uneval

algfun boolean evenfunc integer matrix not polynom radfun realcons subgraph union

It can be seen that the type-check functions themselves have a type, namely type.

20.3.1.2 Input and Output In Maple the input has the form obJ,(obJ,,obJ,,...,obJ~)~ (20.31) The first term. which is in front of the left parenthesis, is usually an operator, a command or a function, which acts on the parts in parentheses. In certain cases there are special options as arguments, which control the special application of operators or functions. The terminating semicolon is very important; it tells Maple that the input has ended. If the input is terminated by a :, this means that although the input is finished, the result should not be calculated. Symbols. Le., names in Maple. can consist of letters, numbers, and the Blank (-). Names cannot begin with numbers. In the first place numbers are not allowed. Upper- and lower-case letters are always distinguished. The blank is used by Maple for internal symbols; it should be avoided in user-defined symbols. Strzngs. Le., objects of type s t r i n g , must be input between single quotes: (20.32) > S := 'This is a string' S := This is a string The output is without the single quotes; type checking with whattype gives s t r i n g . While no value is assigned to a symbol, the symbol is of type s t r i n g or name, Le., the type checking (20.33) > type(symb,name); or type(symb, s t r i n g ) ; results in true. If the user does not knob, if a symbol in Maple is already reserved, then it can be asked for by Name. If Maple answers that it does not know this symbol, then it can be used freely. If a value is assigned to a symbol by the assignment operator :=, then the symbol automatically takes the type of the assigned value. ILet 21 be a symbol, which should be a variable. For the input > whattype(i1); Maple answers by s t r i n g

20.3 Maple 967

If an integer value is assigned to it: > s l : = 5 ; -+ X I : = 5 and then it is asked > whattype(z1); then the answer is integer. Maple knows a huge number of commands, functions, and operators. Not all of them can be called right after the start. Several special functions and operators are in different packages in the Maple library. There exist packages for linear algebra, for statistics, etc. These packages can be loaded by the command > with(packagename);if they are needed (see Supplement about Syntax, Information, Messages. 20.3.9.1, p. 975) then their operations and functions can be used as usual.

20.3.2 Types of Numbers in Maple 20.3.2.1 Basic Types of Numbers in Maple Maple knows the basic types of numbers introduced in Table 20.11.

Tabelle 20.11 Type of numbers in Maple

Number Integer Fraction Floating-point number

Type integer fraction float

Representation form

nnnnnn sequence of (almost) arbitrary number of digits ppplqqq fraction of two integer nn.mmm or in scientific notation n.mm * lO"(pp)

With the help of type control functions, some further properties of numbers can be asked: 1. Rational Numbers (Type rational): Rational numbers are in Maple the integers and fractions. A fraction, which becomes an integer with all common factors removed, will not be recognized by Maple as afraction (type f r a c t i o n ) . 2. Floating-point Numbers (Type f l o a t ) : If a decimal point is placed behind an integer (nnn.), it is automatically considered a floating-point number. 3. CommonProperties: All three typesofnumbers have the types realcons, numeric and constant. The last two types also belong to the complex numbers. 4. Complex Numbers: Complex numbers are formed with the imaginary unit I as usual. The number 1 is of type radnum, i.e. it is the root of a real number. Its internal definition is: alias(1 = (-1)~(1/2)) (20.34) The command a l i a s provides the possibility to rename functions or other mathematical symbols. (In electro-engineering the imaginary unit is denoted by j instead of I , or sometimes it is convenient to use a shorter name for a function than the one it has in the library.) It has the form > alias(gll!g/2,. . .); (20.35) Here, gli are equations, which define the new symbol by the corresponding Maple functions. If the function a l i a s is called, Maple shows all the other previously defined a l i a s with the new one. An alias can be removed by aliasing the name of itself a l i a s ( s y m = sym).

20.3.2.2 Special Numbers Maple knows several special numbers such as Pi, E, gamma.

20.3.2.3 Representation and Conversion of Numbers 1. Floating-Point Numbers The command evalf (number);results in a floating-point number with a previously given precision; the default value is 20 digits. The argument can be a rational number or a symbolic number given by an expression, and in this last case the result of the calculations is represented. I > evalf ( E ) ; 2.7182818284590452354

968

20. Computer Aloebra Systems

The precision is defined in Maple by the environment variable Digits.If the default value of the number of digits is not suitable for the concrete problem, then it can be changed by the command > Digits := m; m required number ofdigits (20.36) It remains valid until the next definition.

2. Numbers of Different Bases The conversion of decimal numbers into another base can be made by the command convert. The basic form of this command is

convert(ezpr,form,opt),

(20.37)

and it transforms certain types of expressions of one form into another form (if it has any meaning). The argument form can be a type enumerated in Table 20.12. Tabelle 20.12Argument of function convert

't' degrees factorial lessthan multiset rational vector

*' diff float lessequal name ratpoly

D double fraction list octal RootOf

array eqnlist

base binary equality exp GAMMA hex horner listlist I n matrix parfrac polar polynom series set sincos

confrac expln hostfile metric radians sqrfree

decimal expsincos hypergeom mod2 radical tan

The table shows that several forms are provided for conversion of numbers. IExamples of using the command convert: > convert(73, octal); > convert(73,binary); --+ 1001001

> convert(79,hex); --f 4F > convert(11001101,decimal,binary); ---f

+ 111 29 > convert(l.45,rational); + 20 > convert('FFAP',decimal,hex);

205

+ 65442

In the last one, the hexadecimal number is enclosed between backward quotes. With the command convert(list,base,basl,bas2) a number, which is given in the form of a list in basl, is converted into a number in bas2, and the result is given in the form of a list. Input in a form of a list means that the number has the form z = z1 * (bas)O t z2 * (bas)' t 23 * (bas)3t . . . and the list is[....~3~22,21].

IThe octal number 153 should be converted into a hexadecimal number: > convert([l,5,3],base,8,16);--f [9,14] The result is a list.

20.3.3 Important Operators in Maple Important operators are +, -, *, / 2 as the known arithmetical operations; = I <, <=, >, >=, <> as relational operators. The cat operator (concatenation operator) has special importance and it can be written in the short infix form as the point operator '.'. With this operator, two symbols can be connected. I > x := written : h := re :

> cat(h,z); + rewritten Instead of the input cat(h.z) it can be written ' '.h.z;, the result is rewritten. It is important to put an empty string in the first place, since Maple does not replace the very first operand by its value. With this construction, we can produce easily indexed variables. I > 2 := 1,2,3,4,5 : a sequence of integers

20.3 Maple 969

>

y.2;

connecting the sequence to a variable t y l , y2, y3, y4, y5

The result is a sequence of indexed variables.

20.3.4 Algebraic Expressions Expressions can be constructed from variables (symbols) with the help of arithmetical operators. All of them have the type algebraic, which includes the "subtypes" integer, f r a c t i o n , f l o a t , s t r i n g , indexed, s e r i e s , function, uneval, the arithmetical operator types and the point operator. We can see that a single variable (a string) also belongs to the type algebraic. The same is valid for basic number types, since algebraic expressions with the command subs are evaluated into them, in general. I > p:=x"3-4*2"2+3*x+5: Here, an expression, namely, a polynomial of degree 3 in z is defined. With the substitution operator subs a value can be assigned to the variable z in the polynomial (expression) and then the evaluation is performed: 347 > subs(z = 3/4,p); t > subs(z = 3,p); t 5 64 > subs(z = 1.543p);t 3.785864 The operator op displays the internal structure, the subexpressions from an expression. With oP(P); (20.38) we get the sequence (see next section) of subexpressions on the first level, 2 3 , -4x2,32, 5 (20.39) In the form op(i,p);, the i-th term is displayed, so, e.g., op(2,p) yields the term - 4 ~ ' . The number of terms of the expression is given by nops(p);.

20.3.5 Sequences and Lists In Maple, a sequence consists of consecutive expressions separated by commas. The order of the elements is important. Sequences with the same elements but in different orders are considered as different objects. The sequence is a basic type of Maple: exprseq. I > f l := z"3, -4 * ~ " 2 *~z,5 3 : (20.40a) defines a sequence, then (20.40b) > type(f1, exprseq); results in t r u e . With the command (20.41) > seq(f(i).i = 1.n): the sequence j ( l ) , f(2), . . . , j ( n ) is shown. IWith > seq(i2,i = 1..5);we get 1,4,9,16,25. The range operator range defines the range of integer variables represented in the form i = n..m, and it means that the index variable i takes the consecutive values n, n 1,n 2,. . . , m. The type of this structure is ' ..'. An equivalent form to generate a sequence is provided by the simplified form > f ( i ) $ i = n..m; n, n + which also generates f(n))j ( n + l), . . . , j ( m ) . Consequently, $n..m; results in the sequence (20.42) 1 , .. . , m and z$i;the sequence with i terms z. Sequences can be completed by further terms: sequence, a, b, . . . (20.43)

+

+

I

970

20. Computer Alqebra Swtems

If we put a sequence f into square brackets, then we get a list, which has a type l i s t . I > 1 := [i$i = 1..6)]; --il := [1,2,3,4,5,6] With the already known operator op, for the command op(1iste); we get the sequence back, which was the base of the list. A list can be completed if first it is changed back into a sequence, this sequence is completed, then it is changed into a new list by square brackets. Lists can have elements which are themselves lists; their type is l i s t l i s t . These types of constructions have an important role when matrices are constructed. The selection of one particular element of a list can be done by op(n, liste);.This command gives the n-th element of the list. If the list has a name, like L , then it is easier to type L[n];. For a double list we find the elements on a lower level with op(rn, op(n, L ) ) ;or with the equivalent call L[n][v];. There are no difficulties in building up lists with higher levels. IGenerating a simple list:

> L1 := [a,b,c,d,e,f]; -+ L1 := [a,b,c,d,e,f] Selecting the fourth element of this list: > op(4,L); or > L[4]; --+ d Generating a double nested list: > L2 := [ [ a ,b, c], [d, e , f]]: (output is suppressed!) Selecting the third element of the second sublist: > oP(3,OP(2,L)); or L[21[3]; + Generating a triple nested list:

f

> L3 := [[[a,b, cl, [d, e, fll, [[s, tl, [u,41)[[x,YI,

[wzlll :

20.3.6 Tables, Arrays, Vectors and Matrices 20.3.6.1 Tables and Arrays Maple knows the commands t a b l e and array to construct tables and arrays. With table(zfc, Izst)

(20.44) Maple generates a table-type structure. Here, zfc is an indexing function, list is a list of expressions, whose elements are equations. In this case Maple use the left-hand side of the equation as the numbering of the table elements and the right-hand side as the current table element. If the list contains only elements, then Maple uses the natural numbering, starting at one. I > T := table([a,b, c]); -+ T := table([ l=a 2=b 3=e])

> R := table([a = x,b = y,c = 21);

R

:= table([

a=x

I

b=y c=z])

The repeated call of T or R gives only the symbols T or R. With op(T);, the output is the table; for the call op(op(T)); we get the components of the table in the form fnc, a list of the equations for the table elements. Here, we can see that the evaluation principle for these structures is different from the general one. In general, Maple evaluates an expression until the end, Le., until no further

20.3 Maple 971

transformations are possible. However, while the definition is recognized in the example above, further evaluation is suppressed until it is explicitly required with the special command op. The indices of T form a sequence with the command indices(T);,a sequence of the elements can be obtained by entries ( T ) ; . IFor the previous examples > indices(T); yields [l],[2],[3] > indices(R); yields [a],[b],[c] and correspondingly > entries(R); yields [x]:[y],[t] With the command array(ifc, list); (20.45) special tables can be generated, which can be of several dimensions, but they can have only integer indices in every dimension.

20.3.6.2 One-Dimensional Arrays With array(l..5);,e.g., a one-dimensional field of length 5 is generated without explicit elements; with v := array(l.5,[a(l)>a(2), a(3).a(4), a(5)l);onegetsthesamebutwithgivencomponents.Theseonedimensional fields are considered by Maple as vectors. With the type check function type(v,vector); we get true. If we ask whattype(v);,then the answer is string. This is in connection with the special evaluation mentioned above. If we first give the command v l := eval(v);, then we can get for whattype(v1); the required answer table.

20.3.6.3 Two-Dimensional Arrays Two-dimensional arrays can be defined similarly with A := array(l..m,1 . q [[a(l,I), . . . ,a(l,n)ll.. . , [a(m,I), . . . ,a(m,n)]]); (20.46) The structure defined in this way is considered by Maple as a matrix of size m x R . The values of a ( i , j ) are the corresponding matrix elements. W > x := array(1..3,[xl,x2,x3]); x := [xlx2 231 results in a vector. A matrix is obtained, e.g., by > A := array(l..3,1..4, A := array(l..3,1..4, The input

[I); [I)

I

?[1,1] ?[1,2] ?[1,3]?[1,4]

> eval(A): yields the output ?[3,1] ?[3,2] ?[3,3] ?[3,4]

Maple characterizes the undefined values of the matrix by the question mark ?[,,31. to all or some of these elements, like > -4[1.1]:= 1 : A[2,2]:= 1 : A[3,3] := 0 : then the renewed call for A will be displayed with the given values:

> eval(A); +

1

1

?[2,11

?[I$

1

I

?[1,3] ?[1,4] ?[2,3] ?[2,4]

0 ?[3,4] With the command > B := array([[bll,b12,b131,[b21,b22.b23],[b31,b32,b33]]); ?[3,l] ?[3,2]

If a value is assigned

972

20. Computer Al.qebra Systems

c :=

c l l ~ 1 2e13 C [ i , d ] c21 c22 c[2,3c ][2,4] c[3,11

c[3,21

cY,31

c[3,41

c14,11

c[4,21 c[4,31

c[4,41

-

20.3 Maple 973

Table 20.14 Special functions Bessel functions Jn(z) and Y,(z) Modified Bessel functions In(z)and K , ( z ) Gamma function Integral exponential function

BesselJ(v, z ) , BesselY(v,z) BesselI(v, z ) , BesselK(v, z ) Gamma(x)

Ei(x)

The Fresnel functions can also be found among the special functions. The package for orthogonal polynomials contains among others the Hermite, Laguerre, Legendre, Jacobi and Chebyshev polynomials. For more details see [20.6].

20.3.7.2 Operators In Maple the functions behave as the so-called X functions in the programming language LISP. Slightly simplified this means that the name of a function, as defined in Maple, is considered as an operator. In other words, type(sin, operator); yields true. If the argument, or several arguments if this is needed, is attached to the operator in parentheses, then one gets the corresponding function of the given variables. I > type(cos, operator); yields t r u e and > type(cos, function); yields f a l s e . If we replace the argument cos by cos(x), then type checking will give the opposite results Maple provides the possibility to generate self-defined functions in operator form. A function can be defined by the arrow operator ->. With (20.47) > F := x-> mathexpr : and with mathexpr as an algebraic expression of the variable x, a new function with name F is defined in operator form. The algebraic expression can contain previously defined and/or built-in functions. If an independent variable is attached in parentheses to the operator symbol generated in this way, then it becomes a function of this independent variable. H > F := x-> sin($) * cos(z) t xA3* tan(z) t xA2 : > F(y);--+ F(y) := sin(y)cos(y)t y3 tan(y) yz If a numerical value (e.g., a floating-point number) is assigned to this argument, say, with the call > F(nn.mmm); Maple gives the corresponding function value. Conversely, it can be generated from a function (e.g.>from a polynomial of the variable x) the corresponding operator with the command unapply(function, uw).So, we get back from F ( y ) with > maPPlY(F(Y),Y); --+ F the operator with symbol F . It is possible to work with operators according to the usual rules. The sum and difference of two operators is again an operator. For multiplication we have to be careful that the product is again an operator. Maple uses the special multiplication symbol @ for operator multiplication. In general, this multiplication is not commutative. Let F := z-> cos(2 ii z)and G := E-> x"2. Then

+

> ( G Q F ) ( z ) ;t c0s2(2z), while

> ( F @ G ) ( z )t ; c0s(2z2) can be formed by the product oftwo functions given in operator representation (FrG)(z)= ( G * F ) ( x ) , which results in F ( z )ii G(z).

20.3.7.3 Differential Operators The differentiation operator in Maple is D. Its application on functions of operator form is D(F) or D[z](G). In the first case the derivative of a function of one variable is defined in operator form. The attachment of the variable in parentheses results in the derivative as a function. In another form, it

974

20. Computer Alqebra Swtems

can be written as D(F)(x) = d i f f ( F ( z ) , z ) . Higher derivatives can be got by repeated application of the operator D, which can be simplified by the notation ( D O Bn)(F) where B B n means the n-th “power” of the differential operator. If G is a function of several variables, then D[i](G)generates the partial derivatives of G with respect to the i-th variable. This result is also an operator. With D[i,j](G)we get D[i](D[j](G)),Le., the second partial derivative with respect to the j-th and i-th variables. Higher derivatives can be formed similarly. The rules of differential calculus (see 6.1.2.2, p. 378) are valid for the differential operator D, where F and H are differentiable functions: (20.48a) D(F + H ) = D(F)+ D(H),

D(F * H ) = (D(F)* H)+ (F * D(H)),

(20.48b)

D(FQH ) = D(F)Q H * D(H).

(20.48~)

20.3.7.4 The Functional Operator map The operator map can be used in Maple to apply an operator or a function to an expression or to its components. Let, e.g., F be an operator representing a function. Then map(F, z x”2 z * y) yields the expression F ( x ) t F ( x 2 )t F(xy). Similarly, with map(F; y * z ) the result is F ( y ) * F ( z ) .

+

+

map(f. [a>5, c, 4); + [f(a)sf(b), f(c), f(4l

20.3.8 Programming in Maple Maple provides the usual control and loopstructures in a special form to build procedures and programs. Case distinction is made by the i f command. Its basic structure is (20.49) i f cond. then stat1 e l s e stat2 f i The e l s e branch can be omitted. Before the e l s e branch, arbitrarily many further branches can be introduced with the structure (20.50) e l i f condi then stati f i Loops are generated with f o r and while, which require in the command part the form do. . . stat. . . od. In the f o r loop the running index must be written in the form i from n t o m by di where di is the step size. If the initial value and step size are missing, then they are automatically replaced by 1. In the while loop, the first part is f o r ind while cond. Also loops can be multiply nested into each other. In order to write a closed program, the procedure command is needed in Maple. It can have several rows, and if it is stored appropriately, then it can be recalled by name if it is needed. Its basic structure is: proc (args) local . . . (20.51) options . . . commands end; The number of arguments of the procedure is not necessarily equal to number of the variables used by the kernel of the procedure; in particular they can be completely missing. All variables defined by l o c a l are known only in the procedure.

20.4 Applications of Computer Algebra Systems 975

IWrite a procedure which calculates the sum of the square roots of the first n natural numbers: > sumqlli := proc(n) > l o c a l s, i; > s[O] := 0; > foriton do s[i] := s[z - 11 + s q r t ( i ) od; > > evalf ( ~ [ n ] ) ; > end; Maple displays the procedure defined in this way. Then the procedure can be called by name with the required argument n: > sumqw(30); Output: 112.0828452

20.3.9 Supplement about Syntax, Information and Help 20.3.9.1 Using the Maple Library Besides the innumerable commands available to the user after starting Maple, there exist so-called mixed library functions and commands which must be made available by the command r e a d l i b ( f ) . .4n enumeration and short description of these commands is in [20.11], Section 2.2. Maple has a huge library of packages. .4special package can be loaded with the command > with(name); (20.52) Here the name is the name of the current package, hence linalg for the package of linear algebra. After loading, Maple lists all the commands of the package and gives a warning if in the new definitions there are commands already available which were introduced earlier. If only one particular command is needed from a package, then it can be called by paket[command] (20.53)

20.3.9.2 Environment Variable The output of Maple is controled by several environment variables. We already introduced the variable Digits (see 20.3.2.3, l.,p. 967), by which the number of the displayed digits of floating-point numbers is defined. The general form of the output of the result is defined by p r e t t y p r i n t . Default is here > interface(prettyprint = true) (20.54) This provides centered output in mathematical style. If this option is defined f a l s e , then the output starts at the left-hand side with the form of input.

20.3.9.3 Information and Help Help about the meaning of commands and keywords is available by the input ?notion; (20.55) Instead of the question mark, help(notzon) can also be written. This results in a help screen, which contains the corresponding part of the library handbook for the required notion. If Maple runs under Windows, then the call for HELP opens a menu usually on the right-hand side, and the explanation about the required notion can be obtained by clicking on it with the mouse.

20.4 Applications of Computer Algebra Systems This section describes how to handle mathematical problems with computer algebra systems. The choice of the considered problems is organized according to their frequency in practice and also according to the possibilities of solving them with a computer algebra system. Examples will be given for

976

20. Computer Aloebra Svsterns

functions, commands, operations and supplementary syntax, and hints for current computer algebra systems. When it is important, the corresponding special package is also discussed briefly.

20.4.1 Manipulation of Algebraic Expressions In practice, further operations must usually be performed with the occurring algebraic expressions (see 1.1.5, p. 10) such as differentiation, integration, series representation, limitingor numerical evaluation, transformations, etc. In general, these expressions are considered over the ring of integers (see 5.3.6, p. 313) or over the field (see 5.3.6.1, 2., p. 313) of real numbers. Computer algebra systems can handle, e.g., polynomials also over finite fields or over extension fields (see 5.3.6.1, 3., p. 313) of the rational numbers. Interested people should study the special literature. The algebraic operations with polynomials over the field of rational numbers have special importance.

20.4.1.1 Mat hematica Mathernatica provides the functions and operations represented in Table 20.15 for transformation of algebraic expressions. Tabelle 20.15 Commands for manipulation of algebraic expressions EXPandbl EXPandb, TI PowerExpand[a] Factor[p] Collect[p,x] Collect[p, {x, y,. . .}] ExpandNumerator[r] ExpandDenominator[r] ExpandAll [r] Together [r] Apart[r] Cancel[r]

expands the powers and products in a polynomial p by multiplication multiplies only the parts in p , which contain r expands also the powers of products and powers of powers factorizes a polynomial completely orders the polynomial with respect the powers of x the same as the previous one, with several variables expands only the numerator of a rational expression expands only the denominator expands both numerator and denominator completely combines the terms in the expression over a common denominator represents the expression in partial fractions cancels the common factors in the fraction

1. Multiplication of Expressions The operation of multiplication of algebraic expressions can always be performed. The coefficients can also be undefined expressions. gives I I n [ l I : = Expand[(x + y - z)"4] Out[I] = x4 + 4 x 3 y + 6 x 2 y 2+ 4 2 y3 + y4 - 4x32 - 1 2 x 2 y z- 1 2 ~ - ~ 4 y 3~z 2 +6x2 z2 + 1 2 x y t ' + 6y2 z2 - 4 x z 3 - 4 y t 3 + t4 Similarly,

+

Inf21 : = Expand[(a x byA2)(cx"3 - d yA2)] Out [21 = a c x4 - a d x y2 b cx3yz - bd y4

+

2. Factorization of Polynomials Mathernatica performs factorization over the integer or rational numbers if it is possible. Otherwise the original expression is returned.

+

+

I I n [ 2 ] : = p = x"6 7xA5 12x"4 + 6x"3 - 252"2 - 30x - 25; In[3] := Factorb] , gives Out[3]= ( ( 5 + 2 ) ( 1 + x + x 2 ) (-5+x2+x3))

20.4 Applications of Computer Alqebra Sustems 977

Mathernatica decomposes the polynomial into three factors which are irreducible over the rational numbers. If a polynomial can be completely decomposed over the complex rational numbers, then this can be obtained by the option GaussianIntegers. I In[,$] := Factor[x2 - 22 t 51 --t Out[41= 5 - 22 t x2 , but In[5l := Factor[%, GaussianIntegers-> True] Out[51= ( - 1 - 2 1 + x ) ( - l t 2 1 t x )

3. Operations with Polynomials Table 20.16 contains a collection of operations by which polynomials can be algebraically manipulated over the field of rational numbers. Table 20.16 Algebraic polynomial operations

PolynomialLCM[pl,p2] PolynomialQuotient[pl,p2,x]

p l and p2 determines the least common multiple of the polynomials p l and p2 dividespl (as a function of x) by p2, the residue is omitted

ITwo polynomials are defined: I n [ l ] := p = x"6 t 7x"5 + 12x"4 + 6x"3 - 25x"2 - 30x - 25; q = ~ " +4 ~ " -3 6 ~ " 2- 72 - 7; LVith these polynomials the following operations are performed: In[2] := PolynomialGCD[p,q] t Out[2] = 1 x x2 I n [ ] := PolynomialLCM[p,q] a ~ t [ 3 1= 3 ( - 7 + ~ ) ( i t ~ t x ~ ) ( - 2 5 - 5 ~ t 5 ~ ~ + 6 ~ ~ + ~ ~ ) In&] := PolynomialQuotient [p, q: x] t Out [4] = 12 + 6x + x2 In[5] := PolynomialRemainder[pp,q . x] t Out[51 = 59 962 96x2 37x3 With the two last results we get 37x3 962' t 962 t 59 x6 + 7x5 t E x 4 t 6x3 - 252' - 302 - 25 = x2 t 62 12 t x4 + 2 3 - 6x2 - 7~ - 7 x4 x3 - 62' - 7x - 7

+ +

+

+

+

+

+

+

4. Partial Fraction Decomposition Mathernatica can decompose a fraction of two polynomials into partial fractions, of course, over the field of rational numbers. The degree of the numerator of any part is always less than the degree of the denominator. Using the polynomials p and q from the previous example we get -6 -55 11x 6s' InL61 := Apart[q/p] t Out C61 = 35 ( 5 + x ) $ 3 5 ( - 5 + x Z + x 3 )

+

+

~

5 . Manipulation of Non-Polynomial Expressions Complicated expressions, not necessarily polynomials, can often be simplified by the help of the command Simplify. Mathernatica will always try to manipulate algebraic expressions, independently of the nature of the symbolic quantities. Here, certain built-in knowledge is used. Mathernatica knows the rules of powers (see 1.1.4.1,p. 7 ) : I ~ [ I := I ~ i m p ~ i f y [ a " n / a " m )-+ ] o u t ~ =i a(-m+n) ~

(20.56)

I

978

20. Computer Alqebra Svstems

With the option T r i g -> True, the commands Expand and Factor can express powers of trigonometric functions by trigonometric functions with multiple arguments, and conversely. ISome trigonometric formulas (see 2.7.2.3, p. 79) can be verified with the following input: InL21 := Factor[Sin[4x],Trig-> True] t Uutf21 = 4 c 0 s ( x ) ~sin(x) - 4 cos(x) sin(^)^ In[31 := Factor[Cos[5x],Trig-> True] t Uutf31 = cos(x) (1 - 2 cos(22) + 2 cos(4x))

Beginning with version 2.2 of Mathernatica, the option Trig-> True can be directly called by the command TrigExpand. It is valid for several commands of the package Algebra' Trigonometry' which can be loaded. Finally, we mention that the command ComplexExpand[expr] assumes a real variable expr, while in ComplexExpand[expr, ( x l , x 2 , . . .}]the variables xi are supposed to be complex.

I

In[il := ComplexExpand[Sin[2 x], {x}] Uut [I] = Cosh[2 Im[x]]Sin[:! Re[z]] I Cos[2 Re[.]] Sinh[2 Im[x]]

+

20.4.1.2 Maple Maple provides the operations enumerated in Table 20.17 for transformation and simplification of algebraic expressions. Table 20.17 Operations to manipulate algebraic expressions expands the powers and the products in an algebraic expression p . The optional arguments qi prevent the further expansion of the subexpressions qz. factorizes the expression p. K is an optional RootOf argument. factor(p,K) simplify(p, q l , q2,. . .) applies built-in simplifying rules on p. In the presence of the optional arguments these rules are applied only for them. simplifies p containing radicals. radsimp(p) normalizes the rational expression p. normal(p) sorts the terms of polynomialp in order of decreasing degree. sort(p) determines the coefficient of xt. coeff (p, x,i) collects the terms of a polynomial of several variables containing the c o l l e c t ( p ,u ) variable u.

1. Multiplication of Expressions In the simplest case Maple decomposes the expression into the sum of powers of the variables: W > expand((x y - z)"4); 4 x3y - 4 x3z + 6 x'y' + 6 x z z 2+ 4 xy3 - 4xz3 - 4 y3z + 6 yzz2 - 4 yz3 + x4 + y4 + z4 -122'yz - 12zy*z + 12xyzZ IHere we can demonstrate the effect of the absence and presence of an optional argument on the Maple procedure. > expand((a * x"3 b * y"4) * sZn(3 * x) * cos(2 * x)); 8 ax3sin(x) COS(^)^ - 6 ax3 sin(x) cos(x)' + ax3 sin(x) + 8 by4 sin(x) -6 by4 sin(z) cos(x)' by4 sin(x) The expression is completely expanded. > expand((a * x"2 - b * y"3) 1; sin(3 * x) *cos(:! * x),a * x"2 - b * ~ " 3 ) ; 8 (axz- by3) sin(x) C O S ( Z-) ~6 (ax' - by3) sin(x) cos(z)' + (ax' - by3) sin(z) Maple kept the expression of the optional argument unchanged.

+

+

+

20.4 Applications of Computer Alqebra Svstems 979

The following example shows the effectiveness of Maple:

I > expand(ezp(2 * a * z)* sznh(2 * z)+ ln(x3)* sin(4 * z)); 2 eaZ2sinh(z) cosh(z) + 24 ln(x) sin(z) C O S ( X ) ~- 12 ln(z) sin(z) cos(z)

2. Factorization of Polynomials Maple can decompose polynomials over algebraic extension fields (if it is possible anyway). W

> p := ~

+ +

+

+

" 6 7 * ~ " 5 12 * ~ " 4 6 * ~ " -3 25 * ~ " -2 30 * x - 25: * *x -7:

q := ~ " 4 ~ " 3 6 ~ " -2 7 > p l := factor(p);

(z+ 5 ) (z2+ z + 1) (z3+ z2 - 5) and

> q l := factor(q); (2t z + 1) ( 2- 7) Here, Maple decomposed both polynomials into irreducible factors over the field of rational numbers. If we want to have a decomposition over an algebraic extension field, then we can continue:

I > p2 := factor(p, (-3)"(1/2)); (.3 + z 2 - 5 ) ( 2 z + 1 - ,m) ( 2 x + 1 + G)( z + 5 ) 4 Maple decomposed the second factor (in this case after a formal extension of the field by G). In general. we do not know if such an extension is possible. If the degrees of the factors are 5 4, then it is possible. With the operation RootOf, the roots can be determined as algebraic expressions.

I> r := RootOf (z3+ z2 - 5 ) : k := allvalues(r) : > k[l]:

1

-+-

- 113

> ki21: L

-

11

2

-

1

- 113

t

2 The call for k[3] results in the complex conjugate of k[2]. The procedure given in this example yields a sequence of floating-point numbers in case the polynomial can be decomposed only over the field of real or complex numbers.

3. Operations with Polynomials Besides the operations discussed above, the operations gcd and lcm have special importance. They find the greatest common divisor and the least common multiple of two polynomials. Correspondingly,

I

980

20. Computer A1,qebra Systems

quo(p, q , 2 ) yields the integer part of the ratio of polynomialsp and q, and rem(p, q, x) gives the residue.

I > p := x6 + 7 * x5 + 12 * x4 + 6 * x3 - 25* X' - 30* x - 25: q := x4 + x3 - 6 'X - 7 * x - 7 : > gcd(p, q ) ; X'+X+l

> Wp,q);

2102 + 5z6- 43x5 - 1 0 9 -~ 72x3 ~

+ 1502' + 175 + 72' + x8

With the command normal the ratio of two polynomials can be transformed into normal form over the field of rational numbers, Le., the quotient of two relatively prime polynomials with integer coefficients. IWith the polynomials from the previous example

> normal(p/q);

+

x4 + 6x3 5 X' - 5 2 - 25 2' - 7 With numer and denom the numerator and the denominator can be represented separately.

> factor(denom("), (7)"(1/2)); 4. Partial Fraction Decomposition The partial fraction decomposition in Maple is performed by the command convert, which is called with the option parfrac. IUsing the polynomials p and q from the previous examples we get

> convert(p/q, p a r f r a c , r ) ;

+

372 59 and + 6 ~12 > convert(q/p, p a r i r a c , i ) ; -~ 6 + -55 + 11x + 62' 352 + 175 35x3 + 352' - 175 ' 2

+ +

5. Manipulation of General Expressions The operations introduced in the following allow the transformation of algebraic and transcendental expressions with rational and algebraic functions containing functions which are self-defined or built-in in Maple. In general, optional arguments can be given, which modify the transformations under certain conditions. The command simplify is introduced here in an example. In the simple form simplify(exppr) Maple performs some built-in simplification rules on the expression. I

> t := sZnh(3 * 2 ) + Cosh(4 * X ) : > simplify(t); 4 sinh(z) cosh(x)' - sinh(x) t 8 c ~ s h ( x-) ~8 cosh(x)'

Similarly,

I > r := sin(2 * 2 ) * cos(3 * 2 ) : > simplify(r); 8 sin(z) C O S ( Z ) ~- 6 sin(x) cos(x)'

+1

20.4 Applications of Computer Alqebra Sustems 981

There exists a command combine, which is the reverse command of expand in a certain sense. I > t := t a n ( 2 * ~ ) ~ : > t l := expand(t); 4 sin(x)' cos(x)' t l := ( 2 cos(2)' - 1)' > combine(t1,trig); cos(22)4 - 1 Here, combine was called with the option t r i g , which provides the use ofthe basic rules of trigonometry. Using the command simplify we get cos(2 x)' - 1 > t2 := simplify(t); ; t cos(2x)2 Here, Maple reduced the tangent function to the cosine function. Transformations can be performed with the exponential, logarithmic and further functions.

20.4.2 Solution of Equations and Systems of Equations Computer algebra systems know procedures to solve equations and systems of equations. If the equation can be solved explicitly in the domain of algebraic numbers, then the solution will be represented with the help of radicals. If it is not possible to give the solution in closed form, then at least numerical solutions can be found with a given accuracy. In the following, we introduce some basic commands. The solution of systems of linear equations (see 20.4.3.2, l . ,p. 986) is discussed in a special section.

20.4.2.1 Mathernatica 1. Equations Mathernatica allows the manipulation and solution of equations within a wide range. In Mathernatica. an equation is considered as a logical expression. If one writes (20.57a) 1n[1I := g = x"2 t 22 - 9 == 0, then Mathernatica considers it as a definition of an identity. Giving the input In[21 := %/. x-> 2 , we get Out[2] = False, (20.57b) since with this value of x the left-hand side and right-hand side are not equal. The command Roots[g,x] transforms the above identity into a form which contains x explicitly. Mathematica represents the result with the help of the logical OR in the form of a logical statement: In[2] : = Roots[g,I] yields -2 - 2Sqrt[40] -2 Sqrt[40] Ilx== (20.57~) aut 121 = x == 2 2 In this sense, logical operations can be performed with equations. With the operation ToRules,the last logical type equations can be transformed as follows: In[31 : = {ToRules[%]} + -2 + Sqrt[40] -2 - 2Sqrt[40] (20.57d) Out [3i = {{x -> 1,{x -> 11 2 2

+

2. Solution of Polynomial Equations Mathernatica provides the command Solve to solve equations. In a certain sense, Solve perform the operations Roots and ToRules after each other. Mathernatica solves polynomialequations in symbolic form up to fourth degree, since for these equations solutions can be given in the form of algebraic expressions. However, if equations of higher degree can be transformed into a simpler form by algebraic transformations, such as factorization, then Mathematica

I

982 20. Computer Alqebra Sustems

provides symbolic solutions. In these cases, Solve tries to apply the built-in operations Expand and Decompose. In Mathernatica numerical solutions are also available. IThe general solution of an equation of third degree: In[41 := Solve[zA3+ a xA2t b z c == 0,2] Mathernatica gives -a aut[ql = {{x->

+

1

23 (-a2 3

3 -2 a3 t 9 a b - 27c + 35

(

(-2 a3 + 9 a b - 27c+ 39

4-(a2

+ 3 b) 1

b2)

+ 4 b3 + 4a3c - 1 8 a bc + 27c2):

4-(a2b2) + 4

b3

+ 4a3c - 1 8 a bc + 27c2)

32 Q

. . .}

The solution list shows only the first term explicitly because of the length of their terms. If we want to solve an equation with given coefficients a, b, c, then it is better to handle the equation itself with Solve than to substitute a, b, c into the solution formula. IA: For the cubic equation (see 1.6.2.3, p. 40) z3 + 62 + 2 = 0 we get: InL51 : = Solve[xA3+ 6 z 2 == 0, 21 Out[5] = {(z-> -0.32748}, (2-> 0.16374 + 2.46585 I}, (2-> 0.16374 - 2.46585 I}} IB: Solution of an equation of sixth degree:

+

+

In[6l : = Solve[zA6- 6 d 5 6 d 4 - 4zA3t 6 5 ~ -~ 382 2 - 120 == 0,2] Out[6] = ((2-> -l}, (2-> 4)) (2-> 3}, (2-> 2}, (2-> -1 - 2 I}, (E-> -1 2 I}} Mathernatica succeeded in factorizing the equation in B with internal tools; then it is solved without difficulty. If numerical solutions are required, then the command NSolve is recommended, since it is faster.

+

IThe following equation is solved by NSolve: In[7] : = NSolve[xA6- 42"5 + 6xA4- 52"3 + 3xA2- 42 2 == 0 , 2 ] Out [7] = ((x-> -0.379567 - 0.76948 I}, (2-> -0.379567 0.76948 I}, {x-> 0.641445}, (2-> 1. - 1. I}, (2-> 1. t 1. I}, {z-> 2.11769))

+ +

3. Solution of Transcendental Equations Mathernatica can solve transcendental equations, as well. In general, this is not possible symbolically, and these equations often have infinitely many solutions. In these cases, a domain should be given, where Mathernatica has to find the solutions. This is possible with the command FindRoot[g, (z,zS}], where 2, is the initial value for the search of the root. I In[8] : = FindRoot[z + ArcCoth[z] - 4 == 0, ( 2 )1.1}] Out[8] = 2-> 1.00502 t O.I} and In[9] : = FindRoot[z + ArcCoth[z] - 4 == 0, {x,5}] -i Out[9] = (2-> 3.72478 t 0.1)

4. Solution of Systems of Equations Mathernatica can solve simultaneous equations. The operations, built-in for this purpose, are represented in Table 20.18.and they present the symbolical solutions, not the numerical ones. Similarly to the case of one unknown, the command NSolve gives the numerical solution. The solution of systems of linear equations is discussed in 20.4.3, p. 984.

20.4 Applications of Computer Algebra Systems 983

Table 20.18 Operations to solve systems of equations Solve[{ll == rl, /2 == r 2 , .. .}, {z,yI . . .}] solves the given system of equations with respect to the unknowns eliminates the elements 2,. . . from the system of Eliminate[{ll = = T I , . . .},{z,. . .}] equations simplifies the system of equations and gives the possiReduce[{ll ==TI).. .},{z,. . .}] ble solutions

20.4.2.2 Maple 1. Important Operations The two basic operations in Maple to solve equations symbolically are solve and RootOf or r o o t s . With them. or with their variations with certain optional arguments, it is possible to solve several equations, even transcendental ones. If an equation cannot be solved in closed form, Maple provides numerical solutions. RootOf represents the roots of an equation of one variable: k := RootOf (2"s - 5 * z 7 , Z) + k := RootOf (-Z3 - 5 -Z 7 ) (20.58) In Maple k denotes the set of the roots of the equation x3 - 52 7 = 0. Here, the given expression is transformed into a simple form, if it is possible, and the global variable is represented by -Z. (Maple returns an unevaluated RootOf.) The command a l l v a l u e s ( k ) results in a sequence of the roots. The command solve yields the solution of an equation if any exists. W > k := solve(z"4 x"3 - 6 * x"2 - 7 * z - 7, x); 1 1 1 1 k : = - - + - l ~ , - - - - I ~ , ~ , - ~ ~ 2 2 2 2 If the equation entered is of degree greater then four, then answers are provided in terms of RootOf: > r := solve(x"6 4 * z"5 - 3 * z t 2, x);; --+ r := RootOf (-Z6 4-Z5 - 3-2 2) This equation has no solution for rational numbers. With a l l v a l u e s we get the approximate numerical solutions.

+

+

+

+

+

+

+

2. Solution of Equations with One Variable 1. Polynomial Equations Polynomial equations with one unknown, whose degree is 5 4, can be solved by Maple symbolically.

W

> solve(z"4 - 5 1; x"2 - 6); t I, -I, v%, -&

2. Equations of Degree Three Maple can solve the general third degree equation with general

coefficients. W > r := sohe(z"3 + a * x"2

> $1;

+ b * x + c, z):

J4b3-b2a2-18bact27c2+4ca3& 18 -b - -a2 i

-

0

.

n3

a

- -

3 b_a _ c_ -a3_ ~ 4 b 3 - b 2 a 2 - 1 8 b a c t 2 7 c 2 + 4 c a 3 &

d. 6_

2-

27' -

18

~.

We get the corresponding expression for the other roots r[2],r[3],too, but we do not give them here because of their length. If the input equation in solve has floating-point numbers for coefficierits,then Maple solves the equation numerically.

I

984

20. Computer Algebra Systems

I > solve(1. * z”3 t 6. * z + 2., 2); -3.27480002, .1637400010 - 2,465853271, .1637400010

+ 2.465853271

3. General E q u a t i o n of Degree Four The general solution is given by Maple also for polynomial equations of degree four. 4. O t h e r Algebraic Expressions Maple can solve equations containing radical expressions of the unknown. 5. Extra root We must be careful, when taking roots, because occasionally we may get equations whose roots are not roots of the original equation. These roots are called extra roots. Hence, every root offered by Maple should be substituted into the original expression. - 1 = 0 is to be determined. The input is IThe solution of the equation + > p := s q r t ( z t 7) t s q r t ( 2 3 z - 1) - 1 : 1 := solve(p = 0,z) : With > s := allvalues(l) : we obtain 1 1 1 1 s[l] := - + -(2 + fi)z and s[2]:= - t -(2 - fi)z 2 2 2 2 By > subs(z = s[i ]. p );i = 1 , 2 we can convince ourselves that only s[2] is a solution.

3. Solution of Transcendental Equations Equations containing transcendental parts can usually be solved only numerically. Maple provides the command f s o l v e for numerical solution of any kind of equation. With this command, Maple finds real roots of the equation. Usually, we get only one root. However, transcendental equations often have several roots. The command f s o l v e has an optional third argument, the domain where the root is to be found. I > f s o l v e ( z arccoth(z) - 4 = 0, E ) ; --t 3.724783628 but (20.59) > f s o l v e ( z + arccoth(z) - 4 = O,z, 1..1.5); + 1.005020099

+

4. Solution of Nonlinear System of Equations Systems of equations can be solved by the same commands solve and fsolve. The first argument contains all of the equations in curly braces, and the second one, also in curly braces, lists the unknowns for which the equations are to be solved: (20.60) > solve({gl1,9/2,. . .}, {z1,z2,.. .})I I

> solve((a”2 - yh2 = 2,2”2 + yh2 = 4}, {z, y}); {y=l,z=J?;},{y=l,z=-J?;},{y=-l,

(20.61) z=&},{y=-l,z=fl}

20.4.3 Elements of Linear Algebra 20.4.3.1 Mat hemat ica In 20.2.4, p. 956, the notion of matrix and several operations with matrices were defined on the basis of lists, Mathernatica applies these notions in the theory of systems of linear equations. In the followings (20.62) P = Arrayb, { m , defines a matrix of type (m,n) with elements pi, = p[[i,j]].Furthermore (20.63) z = Array[z, {n}] und b = Array[b, { m } ] are n- or m-dimensional vectors. With these definitions the general system of linear homogeneous or inhomogeneous equations can be written in the form (see 4.4.2, p. 272) p.z==b p.z==O (20.64)

I).

1. Special Case n = m, detp # 0 In the special case n = m, detp # 0, the system of inhomogeneous equations has a unique solution, which can be determined directly by s = Inverse[p].b (20.65)

20.4 Applications of Computer Algebra Systems 985

Mathematica can handle such systems of up to ca. 50 unknowns in a reasonable time, depending on the computer system. .An equivalent, but much faster solution is obtained by LinearSolveb, b].

2. General Case With the commands Linearsolve and NullSpace, all the possible cases can be handled as discussed in 4.4.2,p. 272, i.e., it can be determined first if any solution exists, and if it does, then it is calculated. In the following, we discuss some of the examples from Section 4.4.2, p. 272ff. A: The example in 4.4.2.1, 2., p. 274, is a system of homogeneous equations 21 - 2 2 + 5x3 - 2 4 = 0 x 1 + x2 - 2x3 3x4 = 0 3x1 - Z2 + 8x3 24 =0 21 3x2 - 9x3 7x4 = 0 which has non-trivial solutions. These solutions are the linear combinations of the basis vectors of the null space of matrixp. It is the subspace of the n-dimensional vector space which is mapped into the zero by the transformation p . A basis for this space can be generated by the command NullSpace[p]. lVith the input In[11:=p={{1,-1,5~-1},{1,1,-2,3},{3,-1~8,1},{1,3,-9,7}} we define the matrix whose determinant is actually zero, which can be checked by Det[p]. Now we take: 3 7 In121 := NullSpace[p] andget Uut121 = {{--,-,1,0},{-1~-2,0,1}} 2 2 is displayed, a list of two linearly independent vectors of four-dimensional space, which form a basis for the two-dimensional null-space of matrixp. An arbitrary linear combination of these vectors is also in the null-space, so it is a solution of the system of homogeneous equations. This solution coincides with the solution found in Section 4.4.2.1, 2., p. 274. B: Consider the example A in 4.4.2.1,2., p. 273, x1 - 2x2 + 3x3 - x4 t 2x5 = 2 3x1 - 2 2 523 - 3x4 - ~5 = 6 2xl + x2 t 2x3 - 2x4 - 3x5 = 8 with matrix m l of type ( 3 , 5 ) ,and vector bl Inr31 := m l = {{1, -2,3, -1,2}, (3, -1,5, -3, -1}, { 2 , 1 , 2 , -2, -3}}; In141 := b l = { 2 , 6 , 8 } ; For the command In141 := LinearSolve[ml, bl] the response is Linearsolve : : nosol: Linear equation encountered which has no solution. The input appears as output. C: According to example B from Section 4.4.2.1, l., p. 273, x 1 - x2 + 2x3 = 1 2 1 - 2x2 - 2 3 = 2 321 - x2 + 5x3 = 3 -221 + 2x2 3x3 = -4 the input is 1n[5] := m2 = {{1,-1,2},{1~-2,-1},{3,-1~5}~{-2,2,3}}; In161 := b2= { 1 , 2 , 3 , - 4 } ; If we want to know how many equations have independent left-hand sides, then we call InL71 := RowReduce[m2];+ U u t f 7 ] = {{1,0,0}, {0,1, O } , {0,0, I}, {0,0,0}}

+ + +

+

+

+

986

20. Computer Algebra systems

Then the input is In[8] := LinearSolve[m2, b2];

1 2 + Outf81 = {-107 ’ --7’ --} 7

The answer is the known solution.

3. Eigenvalues and Eigenvectors Eigenvalues and eigenvectors of matrices are defined in 4.5, p. 278. Mathernatica provides the possibility of determining eigenvalues and eigenvectors by special commands. So, the command Eigenvalues [m] produces a list of eigenvalues of a square matrix m, Eigenvectors[m] creates a list of the eigenvectors of m. If N[m]is substituted instead of m, then we get the numerical eigenvalues. In general, if the order of the matrix is greater than four ( n > 4), then no algebraic expression can be obtained, since the characteristic polynomial has degree higher than four. In this case, we should ask for numerical values. I In[9l := h = Table[l/(i + j - l),{i, 5}, {j,5}] This generates a five-dimensional so-called Hilbert matrix. 1 1 -1 -} 1 { -1 -1 -1 -1 -}.{1 1 -1 -1 -1 -} 1 {-1 -1 1 1 -} 1 { -1 -1 -1 -1 -}} 1 Out[g]= {{I ’ 2 3’4’5 ’ 2’3’4’5’6 3’4’5’6’7’ 4’5’6’7’8 ’ 5’6’7’8’9 With the command In[lOl := Eigenvalues[h] we get the answer Eigenva1ues::eival: Unable to find all roots of the characteristic polynomial. But with the command I n[ l l ] := Eigenvalues [ N [ h ] ] we get Out [ill = { 1.56705,0.208534,0.0114075,0.000305898,3.28793 ~

20.4.3.2 Maple The Maple library provides the special package l i n a l g . After the command > with (linalg) : (20.66) all the 100 commands and operations of this package are available for the user. For a complete list and description see [20.6]. It is important that matrices and vectors must be generated by the commands matrix and vector while using this package, and not by the general structure array. With matrix(m, n, s), an m x n matrix is generated. If s is missing, then the elements of this matrix are not specified, but they can be determined later by the assignments A[i,j]:= . . .. If s is a function f = f ( i , j ) of the indices, then Maple generates the matrix with these elements. Finally, s can be a list with elements, e.g., vectors. Analogously, the definition of vectors can happen by vector(n, e). The input of a vector is similar to a 1 x n matrix, but a vector is always considered to be a column vector. Table 20.19 gives the most important matrix and vector operations. Addition of vectors and matrices can be performed by the command add(%,u , k , l ) . This procedure adds the vectors or matrices u and v multiplied by the scalars k and 1. The optional arguments k and I can be omitted. The addition is performed only if the corresponding matrices have the same number of rows and columns. Multiplication of matrices can be performed by multiply(u, w) or with the short form (see 20.3.6.4, p. 972) &* as an infix operator.

1. Solution of Systems of Linear Equations To handle systems of linear equations Maple provides special operations contained in the linear algebra package. One of them is the command linsolve(A, c). It handles the system of linear equations in the form d . Z = C (20.67)

20.d Applications of Computer Aloebra Sustems 987

Table 20.19 Matrix operations determines the transpose of A determines the determinant of the square matrix A determines the inverse of the square matrix A determines the adjoint of the square matrix A, Le., A & * adjoint(A) = det(A) mulcol(A, s, expr) multiplies the s-th column of the matrix A by expr mulrow(A, T,expr) multiplies the r-th row by ezpr transpose(A) det(A) inverse(A) adjoint (A)

where A denotes the coefficient matrix and c is the vector on the right-hand side. If there is no solution, then the null-sequence Null is returned. If the system has several linearly independent solutions, they will be given in parametric form. The operation nullspace(A) finds a basis in the null space of matrix A , which is different from zero if the matrix is singular. It is also possible to solve a system of linear equations with the operators of multiplication and evaluation of the inverse. IA: Consider the example E from 4.4.2.1,2.,p. 274, of the homogeneous system 21 - 22 + 523 - x4 = 0 X2 - 2x3 324 = 0 IC1 321 - 22 + 8x3 24 = 0 z1 + 3x2 - 923 7x4 = 0 whose matrix is singular. This system has non-trivial solutions. In order to solve it, first we define matrix ‘4: > A := matrix([[l,-1,5, -11, [l,1, -2,3], [3, -1,8,1], [1,3, -9,711: (With det(A) we can be convinced that the matrix is singular.) With the command

+ + +

+

the list of two linearly independent vectors is determined. They form a basis in the two-dimensional null space of matrix A. For the general case, operations to apply the Gaussian algorithm are available in Maple. They are enumerated in Table 20.20. If the number of unknowns is equal to the number of equations and the coefficient matrix is non-singular, then the command linsolve is recommended. Table 20.20 Operations of the Gaussian algorithm row to the others, whose j-th column consists of zeros except A,, gausselim(A) generates the Gaussian triangle matrix by row-pivoting; the elements of A have to be rational numnbers gauss jord(A) generates a diagonal matrix according to the Gauss-Jordan method

B: The system from 19.2.1.4,2., p. 892, 10x1 - 322 - 4x3 + 2x4 = 14 -321 4- E x 2 + 5x3 - 2 4 = 22 -4x1 + 5 x 2 16x3 + 5x4 = 17 221 + 322 - 4x3 - 1224 = -20

+

988

20. Computer Algebra Systems

is to be solved. Now, the input is > A := rnatrix([[lO. -3, -4,2], [-3,26,5, -11, [-4,5,16,5], [ 2 , 3 ,-4, -1211):

> v := vector([l4, 22,17, -201): With l i n s o l v e we get

> l i n s o l v e ( A , v ) ; ;-+

[z 1 -f 2l

The Gaussian algorithm results in

>F

:= gaussjord(augment(A, u ) ) ;; --t

C: The inhomogeneous system of example

then by gaussjord the matrix F is transformed into an upper triangular form:

n _

1 0 0 0 ~ 2 0100 1 F := 1 00102 -00 0 1 2 -

10 100 7 1 > F1 := g a u s s j o r d ( F ) ; ; -i F1 := --7 2 O O L 7 The solution can be read from F1. 000 0

20.4 Applications of Computer A1,qebra Systems 989

With a l l v a l u e s , a sequence of the approximated values can be produced.

20.4.4 Differential and Integral Calculus 20.4.4.1 Mat hematica The notation of the derivative as a functional operator was introduced in 20.2.8, p. 962. Mathernatica provides several possibilities to apply the operations of analysis, e.g., determination of the differential quotient of arbitrarily high order, of partial derivatives, of the complete differential, determination of indefinite and definite integrals, series expansion of functions, and also solutions of differential equations. 1. Calculation of Differential Quotients 1. Differentiation Operator The differentiation operator (see Section 20.2.8, p. 962) has the name Derivative. Its complete form is (20.68) Derivative[nl,nz,.. .] The arguments say how many times the function is to be differentiated with respect to the current variables. In this sense. it is an operator of partial differentiation. Mathernatica tries to represent the result as a pure function. 2. Differentiation of Functions The differentiation of a given function can be performed in a simplified manner with the operator D. With D[f[x],x])the derivative of the function f(x) will be determined. D belongs to a group of differential operations, which are enumerated in Table 20.21. Tabelle 20.21 Operations of differentiation

I>.

D[f[XI, {x, yields the n-th derivative of function f(x) with respect to x D[f) {xl,nl}, {x2,nZ),...] multiplederivatives,n,-thderivativewithrespect tox, (z = 1 , 2 , . . . ) the complete differential of the function f D t [f1 df of the function f the complete derivative Wfl

XI

Dtlf, x1,zzr . .

dx

the complete derivative of a function of several variables

For both examples in 6.1.2.2, p. 380, we get IA : I n f f I : = D[Sqrt[z”3 Exp[4x] Sin[x]],x] E4’x2 (xCos[x] +3Sin[x] +4xSin[x]) outl-11 = 2 Sqrt[E4zx3 Sin[x]]

IB : I n f 2 l : = D [ ( 2 x t 1 ) ” ( 3 z ) , x ] + O u t f 2 1 = 6 x ( 1 t 2 x ) - ’ t 3 ’ + 3 ( 1 + 2 z ) 3 z L 0 g [ l t 2 z ] The command D t results in the complete derivative or complete differential. IC : Inf31 := Dt[x”3 + y”3] t Outf31 = 3xzDt[z] 3y2Dt[y]

+

ID : Inf41:=Dt[xA3+y”3,x] + Outf41=3x2+3y2Dt[y.x] In this last example, Mathernatica supposes y to be a function of x. which is not known, so it writes the second part of the derivative in a symbolic way. If Mathernatica finds a symbolic function while calculating a derivative, it leaves it in this general form, and expresses its derivative by f’. IE : Inf5I:=D[xf[x]”3,x] + Outf51=f[z]3+3xf[x]zf’[x] Mathernatica knows the rules for differentiation of products and quotients, it knows the chain rule, and it can apply these formally: IF : Inf61 := D[f[u[z]],z] Outf61 = f’[u[x]]u‘[z]

I

990

20. Computer Algebra Svstems

u'[z] - u[x] v'[z] IG : In[7] := D[u[x]/v[x],x] Out 171 = 4x1 v[xI'

2. Indefinite Integrals

I

LVith the command Integrate[!, $1, Mathernatica tries to determine the indefinite integral f(x) dx. If Mathernatica knows the integral, it gives it without the integration constant. Mathernatica supposes that every expression not containing the integration variable does not depend on it. In general, Mathernatica finds an indefinite integral, if there exists one which can be expressed in closed form by elementary functions, such as rational functions, exponential and logarithmic functions, trigonometric and their inverse functions, etc. If Mathernatica cannot find the integral, then it returns the original input. Mathernatica knows some special functions which are defined by non-elementary integrals, such as the elliptic functions. and some others. To demonstrate the possibilities of Mathernatica, some examples will be shown, which are discussed in 8.1, p. 425ff. 1. Integration of Rational Functions (see 8.1.3.3, p. 430ff.) IA : In[fl : = Integrate[(2x + 3)/(x"3 + x"2 - 2s),x] 5 Log[-l + 21 3 LOg[z] LOg[2 X] Out[fl = 2 6

+

IB : In[2] : = Integrate[(xA3+ l)/(x(x - 1 ) " 3 ) , ~ ] Uut[21 =

-

(-1

1 + 2 Log[-1 + x] - Log[x] + x)-2 - -1+x

(20.69)

2. Integration of Trigonometric Functions (see 8.1.5, p. 436ff.) IA: The example A in 8.1.5.2, p. 437, with the integral

1

sin' x cos5 x dx =

s

sin' x (1 - sin' x)' cos x dx =

1

t2(1 - t 2 ) 2dt with t = sin x

is calculated: 1n[31 : = 1ntegrate[~in[x]"2~os[x]~5,x] Sin[7z] 5 ~ i n [ x ] ~ i n [ J x ] 3 ~ i n [ 5 x -] Out[3] = -- -320 448 64 192 IB: The example B in 8 . 1 5 2 , p. 437, with the integral sin x dt dx = with t = cosx ~

1

1

is calculated: 1n[41 := ~ntegrate[Sin[z]/Sqrt[Cos[x]],x] -+Out 41 = -2Sqrt[Cos[x]] Remark: In the case of non-elementary integrals Mathernatica performs only a transformation. I1n[5] := Integrate[x"x, x] -i Out = 1ntegrate[x2,z]

[a

3. Definite Integrals With the command Integrate[!, {z,x,,xe}], Mathernatica can evaluate the definite integral of the function f ( z ) with a lower limit x, and upper limit 2 , .

IA : In[il := Integrate[Exp[-x'], (2, - I n f i n i t y , Infinity}] General: : i n t i n i t : Loading i n t e g r a t i o n packages.

+ Out [f1

= Sqrt[Pi]

After Mathernatica has loaded a special package for integration, it gives the value 7r (see Table 21.6, p. 1050, Nr. 9 for a = 1).

20.4 Applications of Computer Al.qebra Sgsterns 991

IB: It can happen, with earlier releases of Mathernatica, that if the input is 2 1n[2] := ~ n t e g r a t e [ l / z " Z {z, , -1, I}], we get 0ut[2] = -3 which is actually false since this integral is an improper integral. Mathernatica takes the primitive 1 1 function (antiderivative) of -, namely --, then it substitutes the limits and subtracts these values 2

22

The fact that the integrand is not finite is not considered. In version 2.2 of Mathernatica, this mistake is avoided. After a slightly longer working time, Mathernatica says that the integral cannot be calculated since it is improper. In the calculation of definite integrals, we should be careful. If the properties of the integrand are not known, it is recommended before integration to ask for a graphical representation of the function in the considered domain. 4. Multiple Integrals Definite double integrals can be called by the command (20.70) Integrate[f[z, YL { zsxor {Y. yar yell The evaluation is performed from right to left, so, first the integration is evaluated with respect to y. The limits ya and ye can be functions of z,which are substituted into the primitive function. Then the integral is evaluated with respect to 2. IFor the integral A, which calculates the area between a parabola and a line intersecting it twice, in 8.4.1.2, p. 470. we get 32 1n[3I:= Integrate[z y"2, {z,O, 2}, {y, x"2,22}] t Out DI =. :

GI,

.Also in this case, it is important to be careful with the discontinuities of the integrand. 5 . Solution of Differential Equations Mathernatica can handle ordinary differential equations symbolically if the solution can be given in closed form. In this case, Mathernatica gives the solution in general. The commands discussed here are listed in Table 20.22. The solutions (see 9.1, p. 485) are represented as general solutions with the arbitrary constants C[z]. Initial values and boundary conditions can be introduced in the part of the list which contains the equation or equations. In this case we get a special solution. Table 20.22 Commands to solve differential equations DSolve[dgl, y[z],z]

solves the differential equation for y[z] (if it is possible); y[z] may be given in implicit form gives the solution of the differential equation in the form of a DSolve[dgl, y >z] pure function DSolve[{dgll, dg12,.. .}, y, z] solves a system of ordinary differential equations As examples, two differential equations are solved here from 9.1.1.2, p. 487. IA: The,solution of the differential equation y'(z) - y(z) t a n z = cosz is to be determined. In[l] := DSolve[y'[z] - y[z] Tan[z] == Cos[z], y>z] Mathernatica solves this equation, and gives the solution as a pure function with the integrations constant C[1] sec[slot[l]] (4C(1) +Sin[2Slot[l]] + 2Slot[l]) Out[ll = {{y -+ Function[ Ill 4 The symbol Slot is for #, it is its FullForm. If it is required to get the solution for 2/[z],then Mathernatica gives Sec[z] ( 2 2 + 4 C ( 1 ) +Sin[Pz]) 1n[2] := y[z]/. 561 + Out [21 = { l 4

992

20. Computer Alqebra Swtems

We also could make the substitution for other quantities, e.g., for y’[x] or y[l]. The advantage of using pure functions is obvious here. IB: The solution of the differential equation y’(x)x(x - y(x)) + yz(z) = 0 (see 9.1.1.2,2., p. 487) is to be determined. In[3] := DSolve[y’[z]x(z - ~ [ x ]+) y[x]”2 == O,y[x],x] Mathernatica returns the input without any comment. The reason for doing so is that Mathernatica cannot solve this differential equation for y. The solution of this differential equation was given in implicit form (see 9.1.1.2, 2., p. 487). In such cases, the solutions can be found by numerical solutions (see 19.8.4.1, 5., p. 945). Also in the case of symbolic solutions of differential equations, like in the evaluation of indefinite integrals, the efficiency of Mathernatica should not be overestimated. If the result cannot be expressed as an algebraic expression of elementary functions, the only way is to find a numerical solution. Mathernatica 2.2 contains a special package which can solve partial differential equations with first partial derivatives of one single function.

20.4.4.2 Maple Maple provides many possib es to handle the problems of analysis. Besides differentiation of functions, it can also evaluate indefinite and definite integrals, multiple integrals, and expansion offunctions into power series. The basic elements of the theory of analytic functions are also provided. Several differential equations can be solved, as well. 1. Differentiation The operator of differentiation D was introduced in 20.3.7, p. 972ff. Its application with different optional arguments allows us to differentiate functions in operator representation. Its complete syntax is D[il(f) (20.71a) Here, the partial derivative of the (operator) function f is determined with respect to the i-th variable. The result is a function in operator representation. D[i, j](f)is equivalent to (20.71b) D[il(D[Jl(f))and D[l(f) = f. The argument f is here a function expression represented in operator form. The argument can contain self-defined functions besides the built-in functions, functions defined by the arrow operator, etc. ILet > f := (2, y) -> exp(x * y) sin(z y): . Then one sets

+

> > > >

D [I(f);-+

f

+

+

D IlI(f); + (z, Y) -> Y exp(x Y) cos(x + Y) D [2](f); -+ (z, Y)-> x exp(x Y) + cos(z Y) D [I,2](f); + (2, Y) -> exp(x 9 ) x Y exp(x y) - s i n ( x + Y)

+

+

Besides the differentiation operator there exists the operator d i f f with the syntax (20.72a) diff(ezpr, z l , x 2 , . . . ,zn) Here ezpr is an algebraic expression of the variables x l , x 2 , . . . . The result is the partial derivative of the expression with respect to the variables 21,. . . ,xn. If n > 1 holds, then the same result can be obtained by repeated application of the operation d i f f : (20.7213) d i f f (a, z1,22) = d i f f (diff(a, x1),22) Multiple differentiation with respect to the same argument can be got by the sequence operator $. I > d i f f ( s i n ( z ) , z $ j ) ; (E d i f f ( s i n ( x ) , x , z , x , x , x ) ) t COS(Z)

20.4 Applications of Computer Alyebra Sustems 993

a

If the function f ( z ) is not defined, then the operation d i f f gives the derivatives symbolically: -f(z). a d

ax

IA > d i f f ( f ( z ) / g ( z ) ) ; t IB

--f(.)

dz-

f(.)&4

dx)

9(x)2

a > d i f f ( z t f(z),z);t f(z) + z -f(x)

ax

2. Indefinite Integrals If the primitive function F ( z ) can be represented as an expression of elementary functions for a given function f (z), then Maple can usually find it with i n t ( f , z). The integration constant is not displayed. If the primitive function does not exist or is not known in closed form, then Maple returns the integrand. Instead of the operator i n t , the long form i n t e g r a t e can be used. 1. Integration of Rational Functions 3 1 5 IA : > i n t ( ( 2 * z + 3 ) / ( a A 3 + z " 2 - 2 * z ) , z ) t ; --ln(z) - - 1 n ( a : + 2 ) + - l n ( z - l ) 2 6 3 IB : > int((z"3 + l ) / ( z * (z - 1)'3), z); t - ln(z) -

4 -I + 2 ln(z - 1) (z-1) z-1

2. Integral of Radicands Maple can determine the indefinite integrals given in the table of indefinite integrals (see 21.5, p. 1017ff.). IIf the input is > X := sqrt(z"2 - a"2): then we have the output:

> i n t ( X , z ) ; --+ i > i n t ( X / a , z);t

z

v - 9~l n ( ~z +1- ~

vGFZ

2 - aarcsec

(20.73)

(a)

1 3 3. Integrals with Trigonometric Functions -a3x3 cos(az) 3a2x2sin(az) - 6 sin(az) 6az cos(az) IA : > int(z"3 * s i n ( a * 2))a ) ; a4 1 cos(az) 1 ln(csc(az) - cot(az) IB : > i n t ( l / ( s i n ( a * 2))"3,z); --+ +7 a 2 a s i n ( a ~ ) ~2

> i n t ( X * z,z);t -(zz -

+

+

Remark: In the case of non-elementary integrals, only a formal transformation is performed. I > i n t (z^z, z); the output is /z" dz, since this integral cannot be given in an elementary way.

3. Definite Integrals

To determine a definite integral the command i n t is used with the second argument z = a .. b. Here, z is the integration variable. and a .. bare the lower and upper limits of the integration interval. 1 1 26 (20.74) IA : > int(z"2,z = a h ) ; --+ -b3 - -a3 > int(z"2,z = 1..3); -+ 3 3 3 IB : > int(exp(-zA2),z = -infinity..infinity);+ fi W C : > int(1/zA4,z = -1..1); t co If Maple cannot solve the integral symbolically, it returns the input. In this case we can try a numerical integration (see 19.3, p. 895ff.) with the commands evalf ( i n t ( e x p r , v a r = a . . b ) ) or evalf ( I n t (expr,var=a. .b ) ) .

20. Computer Algebra S,ystems

994

4. Multiple Integrals Maple can even calculate multiple integrals ifit can be done explicitly. The operation i n t can be nested. IA : > int(int(z"2 t y"2 * exp(x t y),z),y);

IB :

> i n t ( i n t ( z * y"2,y

= 2^2.,2 * x ) , z = 0-2);

-+

32

T 3

5 . Solution of Differential Equations Maple provides the possibility of solving ordinary differential equations and differential equation systems with the different forms of the operator dsolve. The solution can be a general solution or a particular solution with given initial conditions. The solution is given either explicitly or implicitly as a function of a parameter. The operator dsolve accepts as a last argument the options shown in Table 20.23 Table 20.23 Options of operation dsolve explicit laplace series numeric

-

gives the solution in explicit form if it is possible applies the Laplace transformation for the solution the power series expansion is used for the solution the result is a procedure to calculate numerical values of the solution

1. General Solution > dsolve(diff(y(z).x)- y(z) * tan(s) = cos(z),y(s)); Y(X)

=

+

1 cos(x) sin(x) x t 2 -C1 cos(2)

5

(20.75a) (20.75b)

Maple gives the general solution in explicit form with a constant. In the following example the solution is given in implicit form, since y(z) cannot be expressed from the defining equation. The additional option e x p l i c i t has no effect here. (20.76a) > dsolve(diff (y(z), 2) * (x - y(z)) t y(r)^2,~ ( 2 ) ) ;

(20.76b) 2. Solution with Initial Values Consider the differential equation y' - e5 - y2 = 0 with y(0) = 0. Here we give the option s e r i e s . If we use this option, the initial values should belong to x = 0. The same is valid for the option laplace. (20.77a) > dsolve({diff(y(z),x)- exp(z) - y(x)"2, y(0) = 0}, y(x), s e r i e s ) ; y(2) = x

1 7 3 1 + -2 +1-z3 t -x4 + -x5 + O ( 2 ) 2 2 24 120

(20.7713)

The equation and the initialvalues have to be in curly braces. The same is true for systems of differential equations.

20.5 Graphics in Computer Algebra Systems By providing routines for graphical representation of mathematical relations such as the graphs of functions, space curves, and surfaces in three-dimensional space, modern computer algebra systems provide extensive possibilities for combining and manipulating formulas, especially in analysis, vector calculus,

20.5 Graphics in Computer Al.qebra Sgstems 995

and differential geometry, and they provide immeasurable help in engineering designing. Graphics is a special strength of Mathernatica.

20.5.1 Graphics with Mathernatica 20.5.1.1 Basic Elements of Graphics Mathernatica builds graphical objects from built-in (so-called) graphics primitives. These are objects such as points (Point), lines (Line) and polygons (Polygon) and properties of these objects such as thickness and colour. Mathernatica has several options to specify the environment for graphics and how the graphical objects should be represented. With the command Graphics[list], where list is a list of graphics primitives, Mathernatica is called to generate a graphic from the listed objects. The object list can follow a list of options about the appearance of the representation. With the following input I n [ l l := g = Graphics[{Line[{{O,O}>{ 5 , 5 } , {l0,3}}],Circle[{5,5),4], (20.78a) Text [FontForm[“Example”,“Helvetica-Bold” ,251 { 5, S}]}, AspectRat i o -> Automatic] (20.78b) a graphic is built from the following elements: a) Broken line of two line segments starting at the point (0,O) through the point ( 5 , 5 ) to the point (10,3). b) Circle with the center at (s35 ) and radius 4. c) Text with the content “Example”, written in Helvetica-Bold font (the text appears centered with respect to the reference point (5,6)). ~

Figure 20.1

With the call Show[g], Mathernatica displays the picture of the generated graphic (Fig.20.1). Certain options should be previously specified. Here the option AspectRatio is set to Automatic. By default Mathernatica makes the ratio of the height to the width of the graph 1 : GoldenRatio. It corresponds to a relation between the extension in the z direction to the one in the y direction of 1 : 1/1.618 = 1 : 0.618. With this option the circle would be deformed into an ellipse. The value of the option Automatic ensures that the representation is not deformed.

20.5.1.2 Graphics Primitives Mathernatica provides the two-dimensional graphic objects enumerated in Table 20.24.

20.5.1.3 Syntax of Graphical Representation 1. Building Graphic Objects If a graphic object is to be built from primitives, then first a list of the corresponding objects with their global definition should be given in the form {objectl, object?. . . .}, (20.79a) where the objects themselves can be lists of graphic objects

996

20. Computer Al.qebra Svstems

Table 20.24 Two-dimensional graphic objects Point[{z, YII Line[{{zl, yl}, { x 2 ,Y ~ } .~.}]. Rectangle[{zl,, ylu}, {z,, yvo}] Polygon[{{zl, yl}, { x z ,yz}, . . .}] Circle[{z, y}, TI Circle[{z, Y }T ,~{ ~ I , Q z ) ~ CircleKz, Y I I { a , bll Circle[{x, y}) { a , b}, {cq, az}] Disk[{z, y}, T ] , Disk[{x, y}, { a , b}] Textltezt, {x,y)]

point at position x,y broken line through the given points shaded rectangle with the given coordinates left-down, right-up shaded polygon with the given vertices circle with radius T around the center x,y circular arc with the given angles as limits ellipse with half-axes a and b elliptic arc shaded circle or ellipse writes text centered to the point 2 ,y

Besides these objects Mathernatica provides further primitives to control the appearance of the representation, the graphics commands. They specify how graphic objects should be represented. The commands are listed in Table 20.25. Table 20.25 Graphics commands point is drawn with radius a as a fraction of the total picture denotes the absolute radius b of the point (measured in pt (0.3515 mm)) Thickness[a] draws lines with relative thickness a Abs o l u t eThi cknes s [ b] draws lines with absolute thickness b (also in pt) Dashing[{al, az, a3,. . .}] draws a line as a sequence of stripes with the given length (in relative measure) AbsoluteDashing[{bl, b 2 , , ,.}] the same as the previous one but in absolute measure GrayLevellp] specifies the level of shade ( p = 0 is for black, p = 1 is for white) ~ointSize[a] AbsolutePointSize [b]

There is a wide scale of colors to choose from but their definitions are not discussed here.

20.5.1.4 Graphical Options Mathernatica provides several graphical options which have an influence on the appearance of the entire picture. Table 20.26 gives a selection of the most important commands. For a detailed explanation, see [20.5]. Table 20.26 Some graphical options AspectRatio -> w Axes -> True Axes -> False Axes -> {True, False} Frame -> True GridLines -> Automatic AxesLabel --> { Z S y n b o i , Ysymboi} Ticks -> Automatic

sets the ratio w of height and width. Automatic determines w from the absolute coordinates; the default setting is w = 1 : GoldenRatio draws coordinate axes does not draw coordinate axes shows only the z-axis shows frames shows grid lines denotes axes with the given symbols denotes scaling marks automatically; with None they can be suppressed scaling marks are placed at the given nodes

Let object 1 be, e.g., m[I] := 01 = {Circ~e[{5,5},{5,3}],Line[{{0,5},{10,5}}]} and corresponding to it 1nE1 := 02 = {Circle[{5,5}, 31)

80.5 Graphics an Computer Alqebra Sustems 997

Ifagraphic object, e.g., 02, is to be provided withcertain graphicalcommands. then it should be written into one list with the corresponding command In[31 := 03 = {Thickness[O.Ol],o2}. This command is valid for all objects in the corresponding braces, and also for nested ones, but not for the objects outside of the braces of the list. From the generated objects two different graphic lists are defined: In&] := g l = Graphics[{ol,o2}] ; 92 = Graphics[{ol,o3}];, which differs only in the second object by the thickness of the circle. With the call (20.79b) Show[gl] and Shou[g2,Axes -> True] we get the picture represented in Fig. 20.2. In the call of the picture in Fig. 20.2b,the option Axes -> True was activated. This results in the representation of the axes with marks on them chosen by Mathernatica and with the corresponding scaling.

Figure 20.2

1. Graphical Representation of Functions Mathernatica has special commands for the graphical representation of functions. With (20.80) Plot[f[zl. { z , ~ , , n , z m m l l the function f(z)is represented graphically in the domain between z = z,,, and z = z , , , . Mathematica produces a function table by internal algorithms and reproduces the graphic following from this table by graphics primitives. IIf the function sin 22 is to be graphically represented in the domain between -27r and 2 7 ~then , the input is In[5] := Plot[Sin[2z], {zI -2P1, ZPi}]

Figure 20.3

Mathernatica produces the curve shown in Fig. 20.3. It is obvious that Mathernatica uses certain default graphical options in the representation. So, the axes are automatically drawn. they are scaled and denoted by the corresponding z and y values. In this example, the influence of the default AspectRatio can be seen. The ratio of the total width to the total height is 1 : 0.618. With the command InputForm[%] the whole representation of the graphic objects can be shown. For the previous example we get:

Graphics[{{Line[{{-6.283185307179587,4.90059381963448* loA- 16). List of points from the function table calculated by Mathernatica {6.283185307179S37. -(4.90059381963448 * 10" - IS)}}]}},

I

998

20. Computer Alqebra Systems

{Plot Range-> Automatic AspectRatio-> GoldenRatio" (- l), ~

DisplayFunction:> $DisplayFunction,ColorOutput-> Automatic, Axes -> Automatic Axesorigin-> Automatic,PlotLabel-> None, ~

AxesLabel-> None,Ticks-> Automatic,GridLines-> None,Prolog-> { }

~

Epilog-> { } AxesStyle-> Automatic,Background-> Automatic, ~

DefaultColor-> Automatic,DefaultFont :> $Def aultFont, RotateLabel-> True,Frame-> False,Framestyle-> Automatic, FrameTicks-> Automatic,FrameLabel-> None,PlotRegion-> Automatic}] Consequently, the graphic object consists of two sublists. The first one contains the graphics primitive Line,with which the internal algorithm connects the calculated points ofthe curve by lines. The second sublist contains the options needed by the given graphic. These are the default options. If the picture is to be altered at certain positions: then the new settings in the Plot command must be set after the main input. With (20.81) In[6] := Plot[Sin[21],{x,-2Pi,2Pi}, AspectRatio-> 11 the representation would be done with equal I and y absolute scaling. It is possible to give several options at the same time after each other.With the input

Plot[{fl

"f

[I] >

' '

.I>{I>Iminr xrnaz}l

(20.82)

several functions are shown in the same graphic. With the command Show[plot,options] (20.83) an earlier picture can be renewed with other options.With (20.84) Show[GraphicsArray[list]l, (with lzst as lists of graphic objects) pictures can be placed next to each other, under each other, or they can be arranged in matrix form.

20.5.1.5 Two-Dimensional Curves .A series of curves from the chapter on functions and their representations (see 2.1,p. 47ff.) is shown as examples.

1. Exponential Functions A4family of curves with several exponential functions (see 2.6.1, p. 71) is generated by Mathernatica (Fig.20.4a) with the following input: I ~ [ I ] := f[z-] := 2"x; := 1O"x;

&:_I

In[2] := h[z-] := (1/2)"x;j[x-] := (l/E)"z;k[x-] := (1/10)"1;

These are the definitions of the considered functions. The function ez need not be defined. since it is built into Mathematica. In the second step the following graphics are generated: In[3] := pl = Plot[{f [I], h[x]}, {x,-4,4}, Plotstyle-> Dashing[{O.Ol,0.02}]] In&] :=p2 = Plot[{Exp[z],j[x]}, {I, -4,4}] In[5] := p3 = Plot[{g[x],k[x]}, {I,-4,4}, Plotstyle-> Dashing[{0.005,0.02,0.01,0.02}]]

The whole picture (Fig. 20.4a) can be obtained by: In[6] := Show[{pl,p2,p3},PlotRange-> {0,18}, AspectRatio-> 1.21 The question of how to write text on the curves is not discussed here. This is possible with the graphics primitive Text.

20.5 Graphics in Computer Alqebra Sustems 999

-11 1

-6

6

Figure 20.4

2. Function y = 2

.

+ Arcoth z

.

Considering the properties of the function .4rcothx discussed in 2.10, p. 91, the function y = z + Arcoth x can be graphically represented in the following way: In[ll := f l = Plot[x t ArcCoth[x], {x,1.000000000005,7)] In[2l := f 2 = Plot[x ArcCoth[x], {x,-7: -1.00000000000S}] In[] := 3Show[{fl, f2}, PlotRange-> {-lo! lo}, AspectRatio-> 1.2,Ticks->

+

{{{-6 -6}> (-1, -11, {1,1}. {6,6}}, {{2.5,2.5). {10,10)}) The high precision of the x values in the close neighborhood of 1 and -1 was chosen to get sufficiently large function values for the required domain of y. The result is shown in Fig. 20.4b. 3. Bessel Functions With the calls In[l] := bjO = Plot[{BesselJ[O, z],BesselJ[2,2],BesselJ[4,2]}) { z >0: lo}, PlotLabel-> “ J ( n z, ) n = 0,2,4”] In[2l := b j l = Plot[{BesselJ[l, z ] )BesselJ[3, z], BesselJ[5, z]}, { z , 0,lo}, (20.85) PlotLabel-> “ J ( n z, ) n = 1,3,5”] the graphics of the Bessel function Jn(z) for n = 0,2,4 and n = 1 , 3 , 5 are generated, which are then

represented by the call In[3] := Show[CraphicsArray[{bjO,bjl}]] next to each other in Fig. 20.5.

0.6

J(n,z)n=1,3,5

-0.2

b) Figure 20.5

20.5.1.6 Parametric Representation of Curves Mathernatica has a special graphics command, with which curves given in parametric form can be graphically represented. This command is: ParametricPlot[{f,(t), fy(t)}, {t,tl,tz}l. (20.86)

1000 20. Computer A1,qebra Systems

It provides the possibility of showing several curves in one graphic. A list of several curves must be given in the command. With the option AspectRatio-> Automatic, Mathernatica shows the curves in their natural forms. The parametric curves in Fig. 20.6 are the Archimedean spiral (see 2.14.1,p. 103) and the logarithmic spiral (see 2.14.3,p. 104). They are represented with the input I n f f I := ParametricPlot [{tCos[t],t Sin[t]},{ t ,0,3Pi}, AspectRatio-> Automatic] and I n f 2 1 := ParametricPlot[{Exp[O.lt]Cos[t],Exp[O.lt]Sin[t]},{ t ,0,3Pi}, AspectRatio-> Automatic] With I n f 3 1 := ParametricPlot[{t - 2 Sin[& 1 - 2 Cos[t]}, { t ,-Pi, llPi}, AspectRatio-> 0.31 a trochoid (see 2.13.2,p. 100)is generated (Fig. 20.7).

Figure 20.6

20.5.1.7 Representation of Surfaces and Space Curves Mathematica provides the possibility of representing three-dimensional graphics primitives. Similarly to the two-dimensional case, three-dimensional graphics can be generated by applying different options. The objects can be represented and observed from different viewpoints and from different perspectives. Also the representation of curved surfaces in three-dimensional space, Le., the graphical representation of functions of two variables, is possible. Furthermore it is possible to represent curves in three-dimensional space, e.g., if they are given in parametric form. For a detailed description of three-dimensional graphics primitives see [20.5].The introduction of these representations is similar to the two-dimensional case.

1. Graphical Representation of Surfaces The command Plot3D in its basic form requires the definition of a function of two variables and the domain of these two variables: I n f l := Plot3D[f[z,yl,{z,&,&},{y,ya,ye}1 (20.87) All options have the default setting. I For the function z = z2 + yz, with the input I n f f l := Plot3D[zA2t yA2,{z,- 5 , 5 } , {y, -5, 5},PlotRange-> {0,25}] we get Fig. 20.8a, while Fig. 20.8b is generated by the command In[2] := Plot3D[(1 - Sin[z]) (2 - Cos[2 y]), {z,-2,2}, {y, -2,2}] For the paraboloid. the option PlotRange is given with the required z values, because the solid is cut at z = 25. 2. Options for 3D Graphics The number of options for 3 D graphics is large. In Table 20.27, only a few are enumerated, where options known from 2 D graphics are not included. They can be applied in a similar sense. The option Viewpoint has special importance, by which very different observational perspectives can be chosen.

20.5 Graphics in Computer Al.qebra Svstems 1001

3. Three-Dimensional Objects in Parametric Representation Similarly to 2D graphics, three-dimensional objects given in parametric representation can also be represented. With (20.88) ParametricPlot3D[(f,[t, 4 f&, 4, fz[tl4}>it, ta,t e l , ua,4 1 a parametrically given surface is represented, with (20.89) ParametricPlot3D[{f,[tl, fy[tl,fz[tl),{t, ta,tell a three-dimensional curve is generated parametrically. Table 20.27 Options for 3D graphics default setting is True; it draws a three-dimensional frame around the surface Boxed Hiddensurf ace sets the non-transparency of the surface; default setting is True specifies the point (2, y, z ) in space, from where the surface is observed. DeViewpoint fault values are {1.3, -2.4,2} default setting is True; the surface is shaded; False yields white surfaces Shading {za,ze), {{za,ze}> {ya,ye}, {za, z , } } can be chosen for the values A l l . DePlotRange fault is Automatic

W The objects in Fig. 20.9a and Fig. 20.9b are represented with the commands I n f 3 1 := ParametricPlot3D[{Cos[t] Cos[u],Sin[t] Cos[u],Sin[%]},{ t ,0,2Pi},

1002

20. Computer Alqebra Systems

{ u , -Pi/2, Pi/2}]

In&]

:= ParametricPlot3D[{Cos[t],Sin[t],t/4},

(20.90)

{ t ,0,20}]

Mathernatica provides further commands by which density, and contour diagrams, bar charts and sector diagrams, and also a combination of different types of diagrams, can be generated. H The representation of the Lorenz attractor (see 17.2.4.3, p. 825) can be generated by Mathernatica.

20.5.2 Graphics with Maple 20.5.2.1 Two-Dimensional Graphics Maple can graphically represent functions with the command p l o t with several different options. The input functions can be explicit functions of one variable. functions given in parametric form and lists of two-dimensional points. Maple prepares a table of values from the input function by internal algorithms, and its points are connected by a spline method to get a smooth curve. There are several options by which the shape of the graphic can be influenced. The graphic itself is represented in a special environment, and it can be connected to the work document by the corresponding system commands, or it can be sent to the printer or plotter. The data can be saved in various formats, for example as a Postscript file.

1. Syntax of Two-Dimensional Graphics The two-dimensional plot command has the basic structure (20.91) plot(funct, hb, vb, options): The first argument f u n d can have the following meanings: a) a real function of one independent variable, e.g., f(x); b) an operator of a function, generated by, e.g., the arrow symbol; c) the parametric representation of a real function in the form of a list [ u ( t ) , v ( t ) , = t a..b], where t = a..b is the domain of the parameter; d) several functions enclosed in curly braces, which should be represented together; e ) a list of numbers, which are considered to be the coordinates (2,y) of the points to be represented. For completeness it should be mentioned that the first argument in this command can also be functions generated by procedures. The second argument hb is the domain of the independent variable; it has the form x = a..b. If no argument is given, then Maple automatically takes the domain -10..10. It is possible to assign to one or to both limits the values -co and/or co. In this case, Maple chooses a representation of the x-axis with arctan. The third argument vb directs the domain of the dependent variable (vertical). It should be given in the form y = a..b. If it is omitted, Maple takes the values determined from the equation of the function for the domain of the independent variable. It can cause problems if in this domain there is, e.g., a pole. Then, if it is necessary, this domain should be limited. One or several options can follow as further arguments. They are represented in Table 20.28. The representation of several functions by Maple in one graphic is made in general by different colors or by different line structure. Maple V/2 running under Windows provides the possibility of making changes directly on the graphic according to corresponding menus, e.g., the ratio of the horizontal and vertical measure, the frame of the picture, etc. 2. Examples for Two-Dimensional Graphics The followinggraphics are generated by Maple, then vectorized by Coreltrace and finished by Coreldraw!. This was necessary because the direct conversion of a Maple graphic in EPS data results in very thin lines and so unattractive pictures.

20.5 Graphics in Computer Al.qebra Systems 1003

Table 20.28 Options for Plot command yields the representation of a parametric input in polar coordinates (the first variable is the radius, the second one is the angle) sets the minimal number of the generated points (default 49) numpoints = n resolution = m sets the horizontal resolution of the representation in pixels (default m = 200) xtickmarks = p sets the number of scaling marks on the z-axis s t y l e = SPLINE generates the connection with cubic spline interpolation (default) generates linear interpolation s t y l e = LINE shows only the points s t y l e =POINT places a title for the graphic, T must be a string title =T coords = polar

Figure 20.10 1. Exponential a n d Hyperbolic Functions With the construction > plot({2"z, 1O^z,(1/2)"x, (1/1O)"x, exp(x), l/exp(z)}, x = -4..4, y = 0.20,

> xtickmarks = 2,ytickmarks = 2);

(20.92a) (20.92b)

we get the exponential functions represented in Fig. 20.10a. Similarly, the command (20.92c) > plot({sinh(x), cosh(z), tanh(z), coth(x)}, z = -2.1..2.1, y = -2.5-2.5): yields the common representation of the four hyperbolic functions (see 2.9.1, p. 87) in Fig. 20.10b. A4dditionalstructures, such as arrow heads on axes, captions, etc., are added subsequently with the help of graphic programs. 2. Bessel Functions With the calls (20.93a) > plot({BesselJ(O,z).BesselJ(Z, z),BesselJ(4, z ) } ,z = 0..10);

> plot({BesselJ(l,z),BesselJ(3,z),BesselJ(5,z ) } ,z = 0..10);

(20.93b)

we get the first three Bessel functions J(n,z ) with even n (Fig. 20.11a) and with odd n (Fig. 20.11b).

Figure 20.11 The other special functions built into Maple can be represented in a similar way.

1004

20. Computer Aloebra Susterns

3. Parametric Representation With the call

(20.94a) > plot([t * cos(t),t * sin(t),t = 0..3* Pi]); we get the curve represented in Fig. 20.12a. For the following two commands Maple gives a loop function similar to a trochoid Fig. 20.12b (compare: Curtate trochoid in 2.13.2, p. 100) and the hyperbolic spiral Fig. 2 0 . 1 2 ~(see 2.14.2, p. 104). (20.94b) > p l o t @ - sin(2 * t ) ,1 - cos(2 * t ) ,t = -2 * Pa.2 *Pi]);

> plot([l/t, t , t = 0..2 * PZ],rc= -.5..2,coords = polar);

(20.94~)

Because of the introduction the option coords, Maple interprets the parametric representation as polar coordinates at the execution of the command.

Figure 20.12

3. Special Package p l o t s The special package p l o t s with additional graphical operations can be found in the Maple library. In the two-dimensional case, the commands conformal and polarplot have special interest. With (20.95) polarplot ( L ,options) curves given in polar coordinates can be drawn. L denotes a set (enclosed in curly braces) of several functions r(p). Maple interprets the variable 'p as an angle and it denotes the curve in the domain between -7r 5 9 5 ?T if no other domain is prescribed. The command (20.96) conformal(F,rl, r2, optzons) displays the mapping of the standard grid lines of the plane to a set of grid curves with the help of a function F of complex variables. The new grid lines also intersect each other orthogonally. The domain r l determines the original grid lines. Default values are 0..1 + (-l)l/z.The domain r2 gives the size of the window in which the range lies. The default range here is calculated to completely enclose the resulting conformal lines.

20.5.2.2 Three-Dimensional Graphics Maple provides the command plot3d to represent functions of two independent variables as surfaces in space or to represent space curves. Maple represents the objects generated by this command analogously to two-dimensional ones in one window. The number of options for representation is usually larger, especially by the additional options about the viewpoint (of observation). 1. Syntax of plot3d Commands This command can be used in four different forms: a) plot3d(funct, z = a..b, y = c d ) . In this form, funct is a function of two independent variables, whose domain is given by rc = a..b and y = c..d. The result is a space surface. b) plot3d(f, a..b, c..d). Here f is an operator or a procedure with two arguments, e.g., generated by the arrow operator, the domains are associated to these variables. c) plot3d([u(s,t ) .u ( s , t ) )w(s, t ) ] ,s = a..b, t = c..d). The three functions u, v, 2u depending on the two parameters s and t define a parametric representation of a space surface, restricted to the domain of

20.5 Graphics in Computer Alqebra Sustems 1005

the parameters. d) plot3d( [f.g1h ] ,a..b, c..d). This is the equivalent form of the parametric representation, where j , g, h must be operators or procedures of two arguments. All further arguments of the operator plot3d are interpreted by Maple as options. The possible options are represented in Table 20.29. They should be used in the form option = value. Table 20.29 Options of command plot3d sets the minimal number of generated Doints (default number is n = 625) specifies that an m x n grid of equally spaced points is sampled (degrid[m, n] fault 25 x 25) indicates the labels used along the axes (string is required) l a b e l s = [x,y. z] s is a value from POINT, HIDDEN, PATCH, WIREFIRE. Here it defines style = s how the surface is represented f can have the values BOXED, NORMAL, FRAME or NONE. The represenaxes = f tation of the axes is specified specifies the required coordinate system. Values can be Cartesian, coords = c spherical, c y l i n d r i c a l . Default is Cartesian p takes values between 0 and 1 and it defines the observational perprojection = p spective. Default value is 1 (orthogonal projection) o r i e n t a t i o n = [theta,phi] specifies the angle of the point in space in a spherical coordinate system from which the surface is observed gives the domain of the t values for which the surface should be repview = 21.22 resented. Default is the total surface

numpoints = n

Y

In general. almost all options can be reached and appropriately set in the corresponding menu in the screen. In this way, the picture can subsequently be improved.

2. Additional Operations from Package p l o t s The library package p l o t s provides further possibilities for representation of space structures. Especially important is the representation of space curves with the command spacecurve. The first argument is a list of three functions depending on a parameter, the second argument must specify the

1006

20. Computer Aloebra Systems

domain of this parameter. So, the options of the command plot3d are kept until they have any meaning for this case. For further information about this package one should study the literature. \\.'ith the inputs (20.97a) > plot3d([cos(t)* cos(u), sin(t) * cos(u),sin(u)],t= 0..2 * Pi,u = 0..2 * Pz) (20.97b) > spacecurve([cos(t),sin(t),t/4], t = 0..7 * Pz) the graphics of a perspectively represented sphere (Fig. 20.13a) and a perspectively represented space spiral (Fig. 20.13b) are generated.

1007

21 Tables 21.1 Frequently Used Constants s141592654.. . K 1" = 0.017453293.. . 180 1' = B 0.000290888.. . 10800 1" = 1 7 0.000004848.. . 648000 1 rad=l 57.29577951,. , ' lo/o 0.01 B

1Oim e lge = M

0.001

2.718281828,. , 0.434294482. . .

1 In 10 = -

2.302585093.. .

C

0.577215665.. . Euler constant

M

21.2 Natural Constants The table contains the values of the physical constants, which are contained in "Determination of fundamental physical constants 1986" (E.R. Cohen and B.N. Taylor, Reviews of Modern Physics, 59, KO, 4, October 1987) and which are based on CODAT'4 Bulletin No. 63, November 1986. The numbers in the round parentheses represent the standard deviation of the last digits of the values. Avogadro constant iltomic mass unit Bohr magneton Bohr radius Boltzmann constant Compton wavelength of electron of neutron of proton Electronic charge Acceleration due to gravity Faraday constant Fine structure constant Gas constant, universal Gravitation constant Permeability of free space Permittivity of vacuum Nuclear magneton

= 6.022 136 7(36) IOz3mol-' = 1 U = (gmOl-')/ivA = 1.6605402(10)~

kg

= 931.49432(28) MeVc-*= 1822.89 me = eA/2me = 9.274015 4(31) JT-' = 5.788382 63(52). 10W5 eVT-' = h2/&(e)e2 = re/az= 0.529 177249(24). lo-'' m = & / N A = 1.380658(12) JK-' = 8.617385(73) . eVK-' = A/mc = 3.861 593 23(35) . m = 2.100 19445(19) m = 2.10308937(19).

m

= 1.602 17733(49) lo-'' C = 2.41798836(72) 1014 AJ-' = 9.806 65 ms-z = i"Jae = 96485.309(29) CmOl-' = poce2/2h = 7.297 353 08(33) = 137.035 9895(61) = NAk = 8.314510(70) Jmol-'K-' = 6.672 59(85) . lo-'' m 3 k g - k 2 = 4~ x NA-' = 12.566 370 614.. , . = l/p0cz = 8.854 187817. . . lo-'' Fm-' = eh/2m, = 5.050 786 6(17)

JT-'

NA-'

1008 2f. Tables

Yuclear radius Classical electron radius Velocity of light Loschmidt constant Magic numbers Magnetic moment electron proton neutron Molar volume Planck constant

= 3.152 451 66(28) . eVT-' = T o A ' / TO ~ ;= ( 1 , 2 . . .1,4) fm; 1 5 A 5 250: ro
R

Pe

PP Pn V, h h

Rest mass electron proton

me mp

neutron

m,

muon

mwi

pion

m,+ m,a

Rest energy electron proton neutron muon pion atomic mass unit Rydberg constant Rydberg energy Stefan-Boltzmann constant Thomson cross-section Interaction constants strong interaction electromagnetic interaction weak interaction gravitational interaction Characteristic impedance of vacuum

= 1.001 159652 1 9 3 ( 1 0 ) p = ~ 928.47701(31). = +2.792847386(63)pk = 1.41060761(47). = -1.913 042 75(45)Pk = 0.966 23707(40) . = R O T ~ / P= ~ 0.022 414 io(i9) m3moi-' = 6.626 075 5(40) . Js = 4.135 669 2(12) = h/2a = 1.054 572 66(63) . Js = 6.582 122 O(20) . eVs

JT-' JT-' JT-' 8

= 9.109389 7(54). kg = 5.485 79903(13). = 1.6726231(10).10-27kg= 1836.152701(37)me = 1.007276470(12)u = 1.674 928 6(10) . kg = 1838.683 662(40)me = 1.008664904(14)u = 1.883532 7(11). lO-"kg= 206.768262(30)me = 0.113428913(17) u = 2.488. kg = 273.19me = 2.406 kg = 264.20me

= 0.51099906(15) MeV = 938.27231(28) MeV &(n) = 939.56563(28) MeV Eo(bi)= 105.658389(34)MeV Eo(ai)= 139.57 MeV Eo(ao) = 134.972 MeV Eo(u) = 931.49432(28) MeV = po2me4c3/8h3= 10973731.534(13) rn-l R,

&(e) Eo(p)

~ C T ,

0

a0

= 13.605698 l(40) eV = (a2/60)k4/h3c2= 5.67051(19). = 8 ~ ~ , ' /= 3 0.665 246 16(18).

QC

= 0.08.. .14 = 11137

(YF aG

= 5.1 * 10-39

20

= 376.730 3 R

(Yk

=3.10-'2

Wm-2K-4 m2

eVs

u

21.3 Important Series Expansions 1009

21.3 Important Series Expansions Function

Convergence Region

Series Expansion Algebraic Functions Binomial Series

( a fx

) ~After transforming to the form am the following series:

(1fX)rn

Binomial Series with Positive Exponents m(m m(m - l ) ( m - 2) 2 3 + . . . l*ma:+2! 3! m(m - 1).. . ( m - n t 1) zn+ ... (fl)n n! 1 1'3 1.3.7 1.3.7.1lz4*... 1 f -z- -22 f4 4.8 4.8.12x3-4.8.12.16 1.2 1.2'5 -2 2 L L i X 4f.. . -23 1 f -12 - -22 3 3.6 3.6.9 3.6.9.12 1 . 1 f -23 1 . 1 . 3 - ----x4 1 . 1 . 3 . 5 f.. . 1 f -12 - -22 2 2.4 2.4.6 2.4.6.8 3 3.1 3.1.1 1 -2 + -22 -23 + -3224 .. 41 .. 61 .' 83 F..' 2 2.4 2.4.6 5 + -x2 5 . 3 f -23 5 ' 3 ' 1 - -x54 . 3 . 1 . 1 F.. 1 -5 . 2 2.4 2'4.6 2.4.6.8

' +

( m 0)

(1 42): (1 f x)i (1 f 2 ) ; (1f 2 ) :

(1i.Z):

*

* *

Binomial Series with Negative Exponents (1f X ) - m

'

( m 0)

lTrnx+---

+

m(m + 1) x2 F m(m + l3!) ( m + 2Ix3 2! m(m 1).. . ( m t n - 1)E " + ...

+

,, ,

+

n!

1 1.5 1.5.9 1 . 5 . 9 . 1 3 24F... ( l f z ) - i 1 7 -x+ - 2 F 23 4.8.12.16 4 4.8 4.8.12 1 1.4 1'4.7 ( l f z ) - f 1 7 -x + -2 F -23 + -3x14.. 64 .. 97 .. 11 20 7 . .. 3.6.9 3 3.6 1.3.5 1 1'3 (1f 2)-i 1 -zt - 2 2 -23 + -2214 .. 43 .. 65 .. 87 F..' 2 2.4 2.4.6 (1 f 2)-1 1 T 2 + 2 2 2 3 + 2 4 7 . . .

+

3 (13k z)-: 1 F -x 2 (1 f x ) - 2 1 F 22

3.5 3.5.7x3+3.5.7.9 + -22 7-24 2.4 2.4.6 2.4'6.8 + 322 7 4x3 + 5x4 T . ' .

7 . ..

(1k x)-4

*

(1 x)-3

(1 4x)-4

*

(1 x)-5

Convergence Region

Series Expansion

Function

5.7.9 5.7.9.11 5 5.7 1 F -x + -22 F -23 -24 2.4.6 2.4.6.8 2 2.4 1 1 F -(2 ’ 3x 7 3.422 4.5x3 ‘F 5 ’ 6x4 1.2

+

+

+

’

”

’.

.)

( 2 . 3 42 7 3 . 4 . 5 2 ’ 1.2.3 $4 5 . 6x3 7 5 . 6 7x4 + ...) 1 1 F ----(2,3,4,5~ F3.4.5.62’ 1.2.3.4

IF-

$4 5 . 6 . 7x3 7 5 . 6 . 7 . 8x4 t ...) 8

Trigonometric Functions sin x x2 sin a x3 cos a ;in(x + a) sina t xcosa - -- 2! 3! xn sin ( a + 7 ) x4 sin a ... +-+,..+ 4! n! XZn x2 1 4 2 6 1--+---+... + (-1)n-k(k)! cos x 2! 4! 6! x2 cos a x3 sin a :os(x + a) cosa - xsina - - 2! 3! X ~ C O S( a t 7 ) x4 cos a +--... 4! n! 1 2 x t -x3+ -25 + tan x 3 15

+

+

cot x 0 < 1x1 < T

sec x

1t

1 2

-22

277 61 + -24254 + -26 + -28 + .’ . 720 8064 -

21.3 Important Series Expansions 1011

cosec x

Convergence Region

Series Expansion

Function 1 x

7 + -x61 + -x3+ 360

31 x 19120

7

5t

127 x7 + ... 604800 0 < 1x1 < 77

Exponential Functions x x' 23 Xn 1 + -+--$,+...+-+... n! l! 2! 3.

e" = ex

az

in a

X ex - 1

1x1 < cx2

x In a ( x In a)' ( x In a)3 +-+-+...+2! 3! I! x B1 X' Bz x4 B3 1--+--+ - - .x6 .. 2 2! 4! 6!

( x In a)n n!

+

1x1 < 02

B xZn i., L .

1x1 < 277

1t-

+(-l)n+1

(2n)!

Logarithmic Functions

In x

+ (2n +(x1)(x - l)'n+l t

In x

In x

In (1 t x ) In (1 - x)

In =2

(. (. (.

- 1)' - 113 ( x - 1) - -+---+... 2 3

-

114

2-1+-

*

23

25

27

x>o

4

( x - l ) n & . .. +(-1)n+1 n (x-1)2 ( x - 1 ) 3 + , , . ; (x- l)n / . , . tX 2x2 3x3 nxn 22 23 24 + (_l)n+l-Xn ... x--+---+... n 2 3 4 2' x3 2 4 x5

(Z)

x2n+1

2 x + -+-+-+...+-+... [ 3 5 7 2n+1

Artanh x

+. .I

1

O<X52 1 x>2 -1<xI1

-l<x
Convergence

Series Expansion

Function

1 1 1 1 -+-+-+-+...+ x 3x3 5x5 7x7

(2n + 1) xZn+l

=2 Arcoth x In /sinxi

In l x / - - - - - -- . . . 6 180 2835 22

x4 12

x2 2

In cos x

26

24

17x8 2520

x6 45

XI

-...

n (271) !

.. - 22n-1

In ltan

2 2 n - 1 ~ 22n n

+

7 x4 -62 In 1x1 + -1 x2 t x 3 90 2835

6

(22n - 1)BnxZn_ . . . n(2n) !

+ ...

=I

0 < 1x1 < 2

Inverse Trigonometric Functions arcsin x

xt

23 -+-+ 2.3

1.325 2.4.5

1.3.52 t... 2.4.6.7 1 . 3 . 5 . ' (2n - 1)xZn+l+ . . . 2 . 4 . 6 . . . (2n)(2n 1)

~

+

+

arccos x

1.325

23

1.3'527

2 +

23

25

arctan x

a:--+---+...

arctan x

&7 - - - r+ - l- - +1 2 x 3x3

3

5

5 . . . (2n - 1) xZnt1 1. 3 2 . 4 . 6 . .. (2n)(2n 1)

+

27

x2n+l

+(-l)n-*".

7 1 5x5

2n+1

_1- . . . 7x7 t(-l)n+l

arccot x

+. .I

(2n + 1)x Z n t 1

+ (-1)n

x2ntl

*

2n+1

*.

f...

1

.

21.3 Important Series Expansions 1013

Function

Hyperbolic Functions x7 x2ntl -+-+-+...+-+... 3! 5! 7 ! (2n + 1)!

x3

sinh x cosh x tanh x

Convergence Region

Series Expansion

I+

x5

1 + -2 2+ - +2 4- + .26. . + - + . . .X2n 2 ! 4! 6! (2n) ! 1 2 17 z - - x 3 + -x5 - -x7 + -xQ62 3 15 315 2835

-.

,

.

coth x 0 < 1x1 < K sech x

1 x2 + 5 x4 - 61 x6 t 1385 x 8 - . . . 1- 2! 4! 6! 8!

+-(2n) ! Enxln 5 . . . cosech x

x 3. . . 31s' -1- - +x - - -7 + x

6

360

15120

(2+ 2(-1)" (2n) !

Arsinh x

- 1)

Area Functions 1 . 3 x5 - 1 . 3 . 5 x7 2.4'5 2'4.6.7

1 x--x3+2.3

~

n x2n-l

+

iZrcosh x 23

Artanh x Arcoth x

25

-3+ - +5

27

x2ntl

+

-t . . ' t 7 2n+ 1 1 1 1 1 1 -+-+-+-+...+ +... x 3x3 5x5 7x7 (2n 1)x*n+1

x+

+

+

,,

0 < 1x1 < x

1014

21. Tables

21.4 Fourier Series 1. y = x f o r O < x < 2 a

tY 2. y = x f o r O < x < . i r

y = 2a

-x

< x 5 271

for ?r

tY

y = - 2- -

cos32 cos52 a cosz+- 32 +- 5 2

'(

+...)

3. y = z f o r - T < x < n

tY 4. y = x f o r - T < x < z 22 y = 57 - z for

377 2

<x 5 -

2-

tY I

5. y = a f o r O < x < r y = -a for 7r

SY

< x < 2n sin32 sin52 y = - sin2 + - -t ..*) 3 5

3

+

21.4 Fourier Series 1015

6. y = 0 for 0 5 z < a and for a - cy < z 5 a + cyand 2a - LY < z 5 2a y = a for a < z < T - a y = -a for ~ + < az 5 2a - cy

3

1 3

y = - cosasinz+-cos3asin3z

+; 1 cos jcysin jz + . .) ,

ax 7. y=-for-a<x
y=-

a(.

-

a

x)

for T - a

y = -afor T + a

5 z5 a + a

1 +-sin

5 z 5 2a - cy

Especially, for cy = I holds: 3

1 . -sin3cysin3z 32

Tff

32

y=

5asin5z

+

'.

1 . sin 52

1 1 . +sin 7z - -isin llz+ . 72 11

=T2 -

(F +T+T

8. y = x2 for -a 5 z 5 T sy 2 t

-x O x 2 x 3 i l 4 x 5 ~ 6 x 7 i l

X

9. y = Z ( T - z ) for0 5 z 5 T

6

10.

g = T(T

- z)for

05z5T

y = (T- z)(h- z) for T

5 x 5 2a

cos42

cos6z

I

11. y =sinzforO 5 z 5 A

.tY 12. y = cosz for 0

< z < 71. 4 /2sin2z

13. y = sinz for 0 5 z

14. y = cos uz for --x

4sin4z

6sin6z

5 71.

5 5 71. +--u2-1 u2-4 (u arbitrary, but not integer number)

15. y = sin uz for

-A

\

212-9

+...) (u arbitrary, but not integer number)

16. y = z c o s z for -71. < z

y = cosz

1 1 + -COS22 + -COS32 + ... 2 3

1 2

y = cosz - - cos22

+ -31 cos3z - . .

1

19. y = -1n cot 2 for 0 < z < 71. 2

2

y = cos2

1 1 + -cos 32 + - cos 5 1 + . . 3 5

,

21.5 Indefinite Integrals 1017

21.5 Indefinite Integrals (For instructions on using these tables see 8.1.1.2,2., p. 427).

21.5.1 Integral Rational Functions 21.5.1.1 Integrals with X = ax + b

I Notation: X = ax + b ]

1. / X n d x =

1

xntl

( n # -1);

(for n = -1

see No. 2).

2. j ' sX= l lan x .

3. / x X n d x =

1 Xnt2 az(n + 2)

~

b xntl a2(n 1 )

+

(n # - l , # -2);

'I

4. /xmXn dx = am+' (X - b)mXndX

(n # -1,

(for n = -1, = -2

see No. 5 und 6).

# - 2 , . . . ,# -m).

The integral is used for m < n or for integer m and fractional n; in these cases ( X - b)m is expanded by the binomial theorem (see 1.1.6.4,p. 12).

a3

16.

3b

-

3b2 ( n - 2 ) ~ n - z ( n - 1)xn-l b3 +

1

( n # 1, # 2, # 3, # 4).

19.

dx /5

X ax ; + x).

1

= - p (In

dx

22. / E =dx- G + p ~l n - . a

x

23. /-=-a dx

dx dx

26.

dx /x3X

=

-1 [a2 b3

dx

- - -+ x 2aX x 2x2

3a21nX x

28.

dx / 23x3

29.

I-=-dx

(n 2 2).

nalnz]

=

.x".+ x" - K] 2x2 x ,

-; - + x [6u2 In

X x

,

4a3x

[ : ( )

a4x2 X 2 - - -- 2x2 2x2 x

+

1 n+l n + 1 ( - u ) W 2 a2X2 +--bnt2 i (i -2)X'-2 2x2

(n+ 1)aX X

+

t-lnz] n(n l ) a 2 2 30.

(n23).

/2mXn A= bm+n-1 -i=O

If the denominators of the terms behind the sum vanish, then such terms should be replaced by (-a)m-' In. ;X

(,,my T ')

21.5 Indejnite Inte.qrals 1019

I Notation: A = bf

x dx 33’ . / ( a x t b ) ( j x + g )

35.

36’

J ( a t x )x(dbx+ x ) ’

-

b (a - b ) ( b t x )

a

--

( a - b)’

In-

-1

1

--(-t1 -

38. J ( u + x ) ’ ( b t x ) 2

(a-b)’

a:x

b:x)

-1 x2 dx -( a t x ) 2 ( b t x ) 2 ( a - b)2 ( a ? x

1

b2 - 2ab + I)t ln(b + x) (b - a)2

21.5.1.2 Integrals with X = az2

I Notation: X 2 -

2ax t b

A)t m

a

(for A < 0).

=

2a

2ax+b ( n - 1)AXn-1

(afb)

(a#b)

+ bx t c : A = 4ac - b2 1

(for A < 0),

2ax+ b

a+x btx

(a # b).

+ bz + c

= ax2

-7z dx

In-

(a # b).

(for A > 0),

a

62arctan 2ax + b Artanh VQ 1 In2ax-tb-2 a x + b + a

JFdx

t ya tl b n - a t x (a-b) b+x 2ab

+

( a # b)

2 a+x In( ~ - b ) ~b t x

1 b-tx

t-)t-m(z

43.

a+x btx

x 2 dx b2 a’ tln(a (a+x)(btx)’ (b - a ) ( b + z ) ( b - a)’

40.

(A # 0)

f

dx 37’ / ( a - t x ) 2 ( b + x ) 2

39.

1

b ln(ax + b) - 9 In(fx -t g)]

1

x dx

- ag

(see No. 40).

dx +- (2n-3)2a ( n 1)A Xn-1 -

1

(see No. 40).

(see No. 40).

56'

dx

1 - 2(cf2 - gbf

+ g2a) [

7 1 (fx + SI2

In

2ga - bf +2(cfZ - gbf g2a)

+

dx

Jx

(see No. 40).

21.5.1.3 Integrals with X = u2 k x 2

I

Notation: X = a2 & x2, arctan

for the

"+" sign,

for the "-"sign and 1x1 < a, x 1 x t a Arcoth - = - In -for the "-"sign and 1x1 > a. a 2 x-a If there is a double sign in a formula, then the upper one belongs to X = a2 and the lower one to X = a2 - xz, a > 0.

+ 2'.

21.5 Inde.finiteInteorals 1021

dx

1

x

dx

1

ST=dx

59.

x 32 +-+-Y. 4a2X2 8a4X

3 8a5

dx

61. /$=f21nX. 1

/ g =2x~ - . / g 4= ~ ~ , x 1

62. 63.

1

x dx

1

(n # 0).

65. l F = * x ~ a Y . 1 66. /%=?-*-Y. x 2X 2a 67. / % = T $ * ~ *x ~ Y 1.

68.

l--=~--&-/ x2 dx

dx

X.

( n # 0).

x2 a2 - -1nX. 2 2 a2 1 --+-InX. 2X 2 = &-

69.

70. 71.

L 1 2 n ~ n 2n

~ n + l

1%-

/ x3dx x = -1 za2+ s .

x3 dx 72. I s = -

1 a2 t2(n - l)Xn-l 2nXn

73. I g = 3 1 1n z .x2 74.

1

dx

1 2a2X 1

1s = dx

1 2a4

= -+-In-.

x2

X

1 1 75' t - -In 2a4X 2a6 76. /&=-a2ria3y 1 1

+

22

-.X

( n > 1).

21.5.1.4 Integrals with X = a3 f z3 Notation: a3 i x3 = X ; if there is a double sign in a formula, the upper sign belongs to X = a3 + x3,the lower one to X = a3 - x3. 83. 84.

/ dx

1

(six)'

6a2

a2Fax+x2

= +---In

a&

'

Is=1

85. /$=-ln

86.

1 2x a +arctan a2&

6a

(see No. 83). a2Tax+x2 1 22 F a f -arctan (a+x2) a d a& ' 1

/$=x+~/$ 22

(see No. 85).

87. J Tx=2 id-x1 n X 1. 3 1 3x 89.

I x3 - -dx= + ~ ~ ~dx~ J x X

(see No. 83). (see No. 83).

(see No. 85). (see No. 85).

21.5 Indefinite Inteorals 1023

dx

(see No. 83). (see No. 83).

21.5.1.5 Integrals with X = a4 t x4 dx

97.

I=-=

-

I

In

ax Jz +-2 a13 4 arctan aZ-x2

x2+axfi+a2 x2-axa++2

21.5.1.6 Integrals with X = a4 - x4 101.

dx 1 -= -1n I a 4 - x 4 4a3

103.

I

x2dx

104.

1

x3dx

1

= =

a+x a-x

~

1 +arctan 2a3 a

a+x 1 In a-2- arctan I.

-41 ln(a4 - x4).

21.5.1.7 Some Cases of Partial Fraction Decomposition 105. 106.

1

1

1 --A (x$a)(x+b)(x+c)-x+a

B +-+x+b

where it holds A = 107.

1 -- A (x+a)(x+b)(z+c)(s+d) - x + a

108.

1 (a + b x 2 ) ( f gx2) E

+

C x+c' 1 (b-a)(c-a)'

B +-+-+x+b

C

B=

1 (a-b)(c-b)'

C=

1 (a-c)(b-c)'

D

x+c s+d' 1 1 B= usw. where it holds A = (a - b)(c- b)(d - b) ( b - a)(c - a)(d - a ) ' 1

21.5.2 Integrals of Irrational Functions 21.5.2.1 Integrals with &and u2 f b2x I Notation: arctan !@ for the sign "+", a -1nfor the sign "-". 2 a-b& If there is a double sign in a formula, then the upper one belongs to X = a2 + b2x,the lower one to X = a2 - b2x.

113.

2 dx JW = ;17Y.

116.

dx /m

2 3 b 2 f i 3b T -Y. a 2 X f i T a 4 X a5

= --

21.5.2.2 Other Integrals with 6 117.

/=fidx a4-22

120.

= --

2a&

ln

1 a& +arctan -. a& a2-x

1 a+& 1 fi -In -- - arctan 2a a - & a

J (a4 dxx2 I f i --In-2a3' -

x+a&+a2 x-a&+a'

-

'+fi+Larctan--. a - fi a3

fi

21.5 Indefinite Inteorals 1025

21.5.2.3 Integrals with d

a

I Notation: X = ax + b I 121. / a d z =

L a. 3a 2(3ax - 2 b ) m 15a2 '

122. / x f i d x =

2(15a2z2- 12abx + 8 b 2 ) m 105a3

123. / r 2 & d x = 124.

125. 126.

/E dx = a 2 6 x dx - 2(ax - 2b)

1%

=

+

2(3a2x2- 4abx 8 b ' ) a 15a3 forb > 0, f o r b < 0.

128.

/g d x

129.

/m=----

= 2&

+b /

dx

dx -

(see No. 127).

XVT

a dx 2 b / z

bx

(see No. 127).

(see No. 127).

130.

2 m 132. / m d x = 5a ' 133. / x a d x = -

%a2

(5 dX 7- -7bd-X)5

134. 135.

136.

/Tds=

-2 +f2 lb f i + b 2 / 3

dx

XVT

/s=L(fi+-$=), a2

(see No. 127).

(see No. 127). (see No. 1 2 7 ) .

21.5.2.4 Integrals with

Jn and d

[Notation: X = ax

m + b, Y = f x + g, A = bf - ag 1

(see No. 146).

149.

I%=

dx

'

2

f d

J-af arctan m a-m L r n h ffO+rn

forAf < 0 ,

(

A +2aY 150. / m d x = ___ 4af

forAf>O. (see No. 146).

21.5 Indefinite Inteqrals 1027

(see No. 146). (see No. 149).

153,

I!?!?-fi dx

154.

155. J a Y ' d x = 156.

Yn-'dx

2 - (2n+ 1).

1 ~

P n + 3)f

(

2fiY"+' t A

IF=( J17 --

( n - 1)f

2"JL). fiyn-1

Yn-1

21.5.2.5 Integrals with d

158. l x f i d x =

m

1 Notation: X = a2 - x2 1

-3\/57j. 1

159. / x 2 f l d x =

m m

160. / r 3 a d x = --a2-. 5 3 161. / g d x = 162. 163. 164. 165. 166.

$dx

1 1J17 1~

a-ah--.a t f i

-e

- arcsin

fi

g d z = -dx

x dx

2x2

= arcsin =

z.

1 a +In -. 2a

t 0 x

z.

-e.

--a t 2

x'dx x -= 0

167.

=

I$$--fl 3

a2

x

arcsin -.

a2fi.

Y" dx JT )

(see No. 153).

168. 169.

1zJI? -. /a=-dx

1

a + o

= --In

dx

a2x ' dx

172. / x @ d x

=

1 -;m.

x f l 173. / x 2 m d x = --

+-a 224x f l +

6

JsTi

a2fl

7

5

m 175. J $dz = -+ 177.

/z

178. 179.

.

a + O

- a3 In -.

3

1

+

= -- -

174. / x 3 @ d x

176.

a 4 x 0 a6 x - - arcsin -. 16 16 a

m -3- x c J f l -

dx = --

x

2

X

3

-a2arcsin 2

@ 3 J f ? 3a d x = -- - -In 2x2 2 2 dx

+

I.

a+dX

-.

/*m - E ' 1

-

arcsin 2.

a

181. /*=d?+g x3 dx

Jrr' 182.

184.

I

dx

Jm=--

/- dx

1 a 2 0

1 -In a3

1 2a2x2Jrr

= -~

-a. + d X x

3 3 +- -In-.

2a4dX

2a5

a+dX

x

21.5 Indefinite Inteqrals 1029

21.5.2.6 Integrals with

4 [Notation: X = x2 + a2

185.

/ v%dx

=

1

(xv% + a2Arsinhz) + C c1. + a)+]

1 = - [ x a a2 In (x 2

+

=

186. /W??dx

I

id%

a2fl Jx3 - -

188. /x3v%dx =

3

'

/$ dx a-ah-, a + J s ? Js? 190. / 5dx -z+ Arsinh I + C =

189.

=

191.

0 123

192.

/ Edx

193.

/$=a .

Js?

dx = -2x2

-

I a+Js? - In ---x 2a

+ C = In (x + a)+ C1.

= Arsinh

/ $ iv%- 22 Arsinh E + C 195. /$= -3 194.

Js? +In (x + a)+ C1.

= --

=

=2

az

-

2 (x -In

+ 0)+ 4.

aza.

196. / a =dx- - l n - ,1 a 197.

a+Js? x

/nJs? / m Js? dx

= --

aZx '

198.

dx

= --

2a2x2

1 +-ln-. 2a3

a+Js? x +C

1 200. / x m d x = ;v%.

x m

aZx@ a4x0 201. / x 2 m d x = -- -- -16 16 6 24 a6 a Z x m- a 4 x 0- xJl?s - -16 16 6 24

gArsinhz +c a

202 203 204

/$

F 3 3 - x a -a2 Arsinh C x 2 2 - --f l + ! x f i + ! a 2 l n ( x + f i ) + C 1 . x 2 2

d dx = --

+

+

+

205

206 207 208 209

210 211 212

21.5.2.7 Integrals with

4 I Notation: X = x2 - a21

213. / a d z = 2.1 (Xfl-a'Arcosh?)

+C

q +cl,

21.5 Indefinite Inteqrals 1031

215. / x 2 d f d x =

216. /x3&dx

=

rn + a 2 m 3

,

220.

/ Js? dx f i aarccos fl. 0dx Jls + .4rcosh + C 0+In (x + a)+ /7 / $dx 2x2 + 2a arccos fl. / Edx Arcosh + In (x + 6) + 9.

222.

/ x x5d?+

217. 218. 219.

-

=

= --

= --

0

1 -

= --

C=

=

x2dx

a2 ?a+ -2l n ( x + a) +C1

a’

-ArcoshC 2 +C = 2

=

223. 224. 225. 226.

0 dx

/a /=dx

C1.

3 =

1

- arccos

.!

=-

I==dx

229. /x2@dx=

a2x

0+ 1 a arccos -.

2a2x2

2a3

X

x@ a2xfl -+

-230. / x 3 m d x

’

6

24

6

24

fl+ a Z m = ~

7

5

a 4 x 0 fArcosh 2 16

+

16

a

+

231. J ~ d x = ~ - a 2 f i + a 3 a r c c o sa - . 3 232.

1-m -g

233.

1

m3 3 dx = -- x G - -a2 Arcosh 2 2 2 - -~ @+!xfi-!a21n(x+fi)+C1. 2 2 2

23

dx

234.

+

+C

fl+ 3 0 3 - -aarccos E, 2 2

dx = -2x2 X

= --

a2JIS' x dx 1 = --

235.

1-

236.

I

237.

I$$=&---.

0'

x'dx

= --

x

+ Arcosh -X + C = -z + In (x + 0)+ C1. 0

0

a2

Jr? 238. 239. 240.

J

1

dx dx x2vF dx

1 a -a3 arccos -. X

-m

=

I-=-----

')

1 0 a4 1 3 2 ~ 2 x 2 0 2a4Jr?

+ Jr? -.

3 2a5

a arccos -. X

+ +c

21.5.2.8 Integrals with dux2 bx Notation: X = ax2

J;;

241.

dx

+

+ bx + e ,

A = 4ac- b 2 , k

1 2ax b -Arsinh -+ c1

fora>O, A > O ,

+ b)

fora>O, A = O ,

/ E = J;; .1

-ln(2ax

J;;

-- 1

a 2ax

+b

4a

=-

fora
A

21.5 Indefinite Inte.qrals 1033

(see No. 241). (see No. 241). (see No. 241).

(see No. 241).

(see No. 244). (see No. 241). (see No. 241). (see No. 241). (see No. 246). (see No. 248). (see No. 245).

dx

1 --Arsinh

+

bx 2c -

bx+ 2c _f 1_ ilnT1

bx

forc>O, A > O , forc>O, A = O ,

+ 2c (see No. 258). (see No. 241 and 258).

(see No. 241 and 258).

264.

-1

dx , x - a = arcsin -. ax - x x dx

267'

1

-

x-a

-J'ZZG?+ a arcsin -.

dx (ax2 b ) J m =

1

&Jm

+

( a s - bf > 0 ) .

21.5.2.9 Integrals with other Irrational Expressions

269.

1-W dx

n ( a x + b)

1

= ~( n - 1)a

W'

dx

2

= - arccos -

~

21.5.2.10 Recursion Formulas for an Integral with Binomial Differential 273. / x m ( a x n -

+ b)Pdx

m+np+l 1 ~

bn(p + 1) 1

+

( m 1)b

[xmil(axn

+ b ) P + npb

[-xm-'(axn+b)P~l+(m+n+np+l)/xm(axn+b)P+'dx

+

xm+'(axn b)pt' - a ( m

1,

+ n + np + 1)/xmtn(axn + b)pdx],

21.5 Indefinite Inteqrals 1035

-

1

+ + 1) [zm-n+'(azn+ b)pt'

- ( m- n

a(m np

+l)b

I

zm-"(uzn

+ b)Pdz

21.5.3 Integrals of Trigonometric Functions Integrals of functions also containing sin z and cos z together with hyperbolic and exponential functions are in the table of integrals of other transcendental functions (see 21.5.4, p. 1044).

21.5.3.1 Integrals with Sine Function

278.

1 1 1 1 1

280.

1

274. 275. 276. 277.

1 sin ax dx = - - cos ax. 1 1 . sin' ax dz = -z - - sin 2az. 2 4a 1 1 sin3 az dz = -- cos az t - cos3 az 3a 3 1 1 sin4ax dx = -z- - sin 2az -sin 4az. 8 4a 32a

+

sinn-' ax cos ax na sinaz zcosaz 279. I x s i n a z d z = -- -.

sinn-' az dz

sinn az dx = -

(ninteger numbers, > 0).

a2

:( $) (5 ): :(

22 . z' sin ax dz = -sin az a'

281. / z 3 s i n a z d z =

-

-

-

sinaz -

Xn

282. /x"sin az dz = -- cos az t

The definite integral

cos az -

2)

cosaz.

zn-' cos az dz

(n> 0).

-dt is called the sine integral (see 8.2.5, l.,p. 458) and it is denoted si(z).

si:t For the calculation of the integral see 14.4.3.2. 2., p. 694. The power series expansion is 23 27 si(z) = z + x5 - + ' . . ; see 8.2.5, l.,p. 458. 3.3! 5 . 5 ! 7 . 7 ! sin az a cos dz 284. y d z : = -(see No. 322). ~

1

~

+

285.

I%&=---

286.

1

287.

I

1

2

1 sinaz n-1zn-'

~

a n-1

,[ zl!;c

dr

dx 1 a5 1 -= cosec az dz = - In tan - = - In(cosec az cot ax) sin ax 2 a dx 1 -= --cot ax. sin'ax a

(see No. 324).

dx

289.

1-

290.

I*=-

cosax

1 ax +--Intan--. 2

1 cosax dx = sinnax a(n - 1)sin'-' a z

sinax

(

1 (ax)3 ax+-+-+3.3! a'

+-nn --21 I sin'-'dx a z

( n > 1).

~

7 ( a ~ ) ~3 1 ( a ~ ) ~ 3.5.5! 3'7'7! 2(22n-' -

+-+...+ 127(

')Bn(ax)''+l (2n + I)!

3.5.9! B, denote the Bernoulli numbers (see 7.2.4.2, p. 410). 1 xdx x 291. = - a c o t a x + -1nsinaz. a' 1 x dx x cos ax

+ ...

/&

293.

1

dx l+sinax

( n > 2).

a

dx

296.

298.

1

dx sin az( 1 sin ax)

*

a

(a

dx - _ _1t a n 299' l ( 1+sinax)' - 2a

300. 301.

1 1

(1 sinaxdx

=

1 cot

- _ 1_

7)

--- -1 tan3 6a

(f - y ) +

(1 + sin ax)2 - 2a tan

1

Cot3

I-=-

dx 1+ & a x

2JZa

-

.

(f - 7) + sa1 tan3

=--cot(%-'i.)+sacot3(~--). 1 ax 1

303.

(f

7) 7)

(f -

arcsin

(3

)

sin' ax - 1 sin' ax + 1

ax

21.5 Inde.finateInte.qrals 1037

305.

/ sin ax sin bx dx

=

+ +

sin(a - b)x - sin(a b)x 2(a - b) 2(a b) 2

dx -306' / b + c s i n a x a -

m

b tan ax12 + c

arctan

m

1 b tan ax12 a m L nb tanax12 x b dx

for la1 = /bl see No.275).

(la1 f Ibl ;

+c +c +

for b2 > c2),

for b2

/- sinaxdx dx 1 ax In tan 4 / dx 308. / sinax(b+csinax) ab 2 b b+csinax c cos ax b dx dx 309. / (b+csinax)2 a(b2-c2)(b+csinax) + m / b + c s i n a x sinaxdx cos ax dx 310. / ( b + c ~ i n a x ) -a(c2-b2)(b+csinax) ~ + Ab+csinax / -tan ax dx arctan (b > 0). b 311' / b2 + sin2ax abm t a n a x dx (bZ> arctan b 312' / c2sin2ax ab-tanax+ b 307.

=-

c2).

(see No. 306). (see No. 306).

--

-

b2

-

b

--

1

--

1

-

1

cz

c2,

In -tan

2ab-

ax - b

21.5.3.2 Integrals with Cosine Function 1 .

313. Jcosaxdx = -sinax.

/ cos2ax dx = -x2 + 4a sin 2ax. 1 315. / cos3 ax dx sin ax sin3ax. 3a 3 1 316. / cos4 ax dx -x + sin 2ax + sin4ax. 8 4a 32a ax sin ax n 1 + / cos"-2 ax dx. 317. / cos" ax dx na 1

314.

1

==

(see No. 306).

1 -

--

1 -

-

-

=

cosax xsinax 318. /xcosaxdx = - -. a2 22 x2 cos a z dx = - cos ax + a2

+

:(

5)

320. / x 3 c o s a x d x = ( s - $ ) c o s a x +

x" sin ax n 321. /zncosaxdx = -- -

/

xn-l

sin ax.

(:-$)sinax. sin ax dx.

(see No. 306).

b > 0),

(2> b2, b > 0).

1038

322.

21. Tables

1

(ax)2 (ax)4e)..( E d x = ln(ax) - - -- -t . . 2.2! 4.4! 6.6!

+

dt is called the cosine integral (see 14.4.3.2, p. 694) and it is denoted by

The definite integral -

22

+

24

26

+. . see 8.2.5,2., p. 458;

Ci(x). The power series expansion is Ci(x)= C t In x - - -- 252! 4 . 4 ! 6 . 6 ! C denotes the Euler constant (see 8.2.5, 2., p. 458). --cos ax -

(see No. 283).

X

(see No. 285). 325.

I-=- '

326.

a

dx cosax dx

1 Artanh Artanh(sin ax) = - In tan a 1 = tan ax.

dx dx

329.

( n > 1). En(ax)2n+2 t . , (2n 2) (2n!)

+

cos ax

.)

E, denote the Euler numbers (see 7.2, p. 411).

332.

/* cos2ax 1% 1

333.

1

334.

I---

330. 331.

cosnax

1

=

f tan ax + - In cos ax.

=

1 x sin ax (n- l)acosn-'ax (n- l ) ( n - 2)a2cos"-2ax

a

a2

dx 1 ax = - t a n -. l+cosax a 2 dx 1 ax = --cot -. 1 - cosax Q 2 x ax 2 ax xdx - -tan - - lncos -. 1t c o s a x - a 2 a2 2 x dx x ax 2 ax ---cot-+-Insin--. 1 -cosax a 2 a2 2 cos ax dx 1 ax = x - - t a n -. 1t COSUZ a 2 1 ax cos ax dx = -x - -cot -, 1 - cos ax a 2 dx 1 1 ax = - lntan - - tan cos ax(1 cos ax) a a 2

~

~

335.

I--

336.

/-

+

~

+

(: + y )

x dx

(n > 2).

21.5 Inde.finite Inte.qrals 1039

339.

1

dx cos ax(1 - cos ax)

a

dx 1 ax 1 ax = -tan - + - tan3 340' s ( 1 t cosax)2 2a 2 6a 2 ax 1 ax - - - Cot32 6a 2

1 dx = --cot 341' s ( 1 -cosax)' 2a

1 6a

ax 2

cosaxdx = -cot 1 ax - 1 343' / ( I - cosax)' 2a 2 6a

ax 2

cosazdx 342'/(l+cosax)'

344.

I---

(

1 - 3 cos' ax dx arcsin 1 +cosZax 2 4 % 1 +cos'ax

346. 1cos ax cos b s dx =

347'

ax 2

1 2a

= - tan - - - tan3 -

1+ b

sin(a - b)x sin(a t 2(a - b) 2(a

dx -ccos ax a

)

'

+ b)x + b)

( b - c) tan ax/2

2

m

+m

( c - b) tanax/2 m In (c - b) tan ax/2 cosaxdx = b dx c -c btccosax b+ccosaz 1 dx = -1ntan cos ax(b + c cos ax) ab

a

1

349' 350'

351'

352' 353'

1 1+ 1 1+ 1

(for b' > e')

d F 7

1

-

(y + t)

(for bZ < e').

m

(see No. 347).

J

dx

+

dx

cosaxdx b sin ax ( b + ccosax)' - a(b2 - c2)(b+ ccosax)

dx

b'

dx -cZcosZax abdx

b2 - c2cos2ax

1

b tan ax arctan -

b tan ax -~ 1 arctan ab1 btanax 2 a b m In b tan ax

1 . sin ax cos ax dx = - sin2 ax. 2a

(see No. 347). (see No. 347).

(b > 0).

m

21.5.3.3 Integrals with Sine and Cosine Function

1

(see No. 347).

CCOSax

c sin ax dx ccosax)2 a(cz - bz)(b + ccosax)

(b

+

354.

(for la1 = J b / see No. 314).

( l a # lbl);

(b2 > c 2 ; b

> 0) ,

(e' > b2 b > 0 ) . ~

355.

1

356.

/ sinn ax cos ax dx

357. 358.

x sin4ax sin' ax cos' ax dx = - - 8 32a '

1 1

=sinn+laa:

a(n

( n + -1).

+ 1)

= - -coSn+lax

sin ax cosn ax

a(n

sinn ax cos"' ax dx = -

+ 1)

( n + -1).

sinn-' ax COS"'+' ax a(n m)

+

n- 1

+n + m /"

ax COS"' ax dx

(lowering the exponent n ; m and n > 0), -

sinn+' ax cosm-' ax a ( n m)

+

/

m-1 +n+m

sinn ax c o P 2 ax dx

(lowering the exponent m ;m and n > 0).

359. 360. 361. 362. 363. 364. 365. 366. 367.

368. 369.

dx 1 = -In tan ax. J sinaxcosaz a

1 1 1 1

dx sin' ax cos ax dx sin ax cos' ax dx sin3ax cos ax dx sin ax cos3 ax

I I I

=

= - (1i n t a n - + ax a 2 =

cosax

).

1a (Intanax - 7). 1 2sin ax

= - (1I n t a n a x + 2cosZax a l )' 2 dx = --cot2ax. sin' ax cos2 ax a dx sin'axcos3ax dx sin3axcos2ax 'dx dx 1 sin ax cosn-' ax sin ax COS^ ax a(n - 1) cosn-1 ax

1

I I

-1.

1 [lntan (i+ y ) - 1 a sin ax

+I I

( n # 1) (see No.361 and 363).

1 dx dx -_ a(n - 1) sin'-1 ax sin'-2 ax cos ax sin' ax cos ax 1 1 dx - --, a ( n - 1) sin'-' ax cos"'-1 ax sinn ax c o p ax

( n # l)(see No. 360 and 362).

+

+

%I

dx sin"-' ax cosm ax

(lowering the exponent n; m > 0, n > l ) , --.

1 1 a(m - 1) sinn-1 ax cos"'-1 ax

n+m-2 +

I

dx

7sin" a x cos"'-2

(lowering the exponent m; n > 0,m > 1).

ax

21.5 1nde.finite1nte.qrals 1041

370.

I---=

sinaxdx -

1 acosax

sin ax dx

372.

373. 374.

375.

1-

sin ax dx COS"

ax

a

1 2a

1

-

-1 sec ax. + C = -tanZax+Cl.

1 a(n - 1)cosn-1 a x '

I-

sin'axdx 1 1 =--sinax+-lntan cos ax a

-1 -1

sin2axdx = -1 C O S ~ ~ Z

sin' ax dx COS"

ax

-

[- sinax

- - ~1n t a n ( : + y ) ] . 2

a 2cos'ax

dx

sin ax a(n - 1)COS"-1 ax

( n # 1)

(see No. 325,326, 328).

cos ax sin3ax dx cos ax 1

379. 1-dx 380.

sinn-' ax a(n - 1)

= -~

ax dx = = cosrnax

382. 383.

I---

dx

sinn-' ax

cos ax dx 1 = -~ sin3ax 2a sin2ax

( m # 1).

cot' ax +c=-+GI . 2a

cosaxdx 1 a(n - 1)sinn-' a x ' sin ax a

(-

-)

ax cos'axdx 1 cosax - -- In tan . 2 sin3ax - 2a sin'ax

-1

(n # 1). dx

cos ax dx - -- 1 - --1 cosec ax. asinax a

sinax 385.

sin:;:

sin"-' ax

-

1 1-

1 1:

sinni1 ax

1

-_ -

381.

+

1

386.

I----

387.

-1

388.

I--

cos2ax dx sinn ax

+JL) ( n # 1) sinn-' ax

(see No. 289).

(

1 , - -- (sinax a

/--A[cos3axdx -

390.

=

sinax

I GFT

cosn ax dx

+ -)sin1ax 1

ax a(n - 1)

c0P-l ~

--

. -

a ( n - 3)

sin ax

391.

___ cosaz

( n - 1) a sinn-' a x

cos3axdx 1 cos2ax -- sinax - a 2

cos3axdx

389.

(

-

ax

+

1

( n - 1)sinn-' ax

J ;;:c o:s

dx

( n # 1).

cosn+1ax a(m - 1)sinm-1 ax cosn-1 ax

(m#

111

( m# n ) , ( m # 1).

dx

1

dx

1

394.

1 1f c o s a x J cos ax(sin1axidxcos ax) = -In a cos ax

395.

1

,-

1 lisinax cosaxdx = --ln7, sin ax(1 f sin ax) a sin ax

sinaxdx

397. 398,

-

1

I

cosaxdx -1 1 ax f-lntan-. 2 2 4 1 & cos ax) 2a sin a x ( l & cos ax)

J

sinaxdx x 1 = - 7 - ln(sinax i cosax). sinax f cosax 2 2a cos ax dx dx dx = illn (1 l+cosax+sinax a dx bsinax

--

+ ccosax - a

1 m

* tan y ). ax t e In tan 2

and t a n e =

with sin0 = ~

!.

b

21.5 Indefinite Integrals 1043

/ 404. / 403.

sinaxdx 1 = - - In(b b+ccosax ac b

+ c cos ax).

1 cosaxdz = -ln(b+csinax). csinaz ac

+

d (x

dx

405'

/

/b+

+ fsinaz =

btccosax

+

:)

m s i n ( a z +e) with sin0 =

dx

406'

/

407'

1

408.

/

b2

b2

1 abc

= -arctan

cos2 ax + cz sin2ax

tan ax)

(see No. 306).

(a2 # b2); for a = b

(see No. 354).

.

ctanax+b c tan ax - b'

1 dx =-In cos2 ax - c2 sin' ax 2abc

sin ax cos bx dx = -

(i

C

r n and t a n 0 = 2f

~

COS(^ + b)x - COS(^ - b ) ~ 2(a

+ b)

2(a - b)

21.5.3.4 Integrals with Tangent Function 1

409. / t a n az dz = -- In cos ax. tan ax 410. / t a n Z ax dx = -- 2 .

/ 412. / 411.

1 tan3 az dz = - tan' ax 2a

+ -1 In cos ax.

tann ax dz = -tann-' ax a(n - 1)

1

tann-' ax dx.

22n(22n - 1 ) a2n-l ~ 2n+l ax3 a3x5 2a5x7 17a7x9 n x 413. /z tanaxdx = - + - -t -t ... (2n + I)! 3 15 105 2835

+

+

+..

B, denote the Bernoulli numbers (see 7.2.4.2, p. 410). tan ax dx

414.

/-

=

ax+-+-

22"(22n - l)B,(ax)'n-l +t ...+ 2205 (2n 1)(2n!)

2 ( a ~ ) 1~ 7 ( a ~ ) ~

9

75

-

/ t a n adxx & 1 *-x2 + -2a1 ln(sinax cosaz). tanazdz x 1 417. /- - - ln(sin ax cos ax). t a n a s k 1 2 2a 416.

~

=

&

zt

21.5.3.5 Integrals with Cotangent Function 418.

/ cot a z dz

1

= - In sin ax.

+...

cot ax --x.

419. /cot'axdx=

1 1 420. / c o t 3 a x d x = --cot'ax--1nsinax. 2a

421.

1

cotn ax dx = -cotn-' ax a(n - 1)

/ cotn-' ax dx

(n # 1).

22ngna2n-lX2n+l x ax3 a3x5 -*.. 422. I x c o t axdx = - - - - -- ... 9 225 (2n + I)! a

B, denote the Bernoulli numbers (see 7.2.4.2, p. 410). (ax)3 --2 ( a ~ _) ~. . . - 2'"Bn (a2)'n-l 423, cotaxdx = _ 1__ ax _ _4725 (2n - 1)(2n)! X ax 3 135

1-

cotn ax 424. / x d x = -cotn+lax a(n t 1)

425.

I-=/*

dx 1 cot ax

-...

( n # -1).

tan ax dx tanax % 1

(see No. 417).

21.5.4 Integrals of other Transcendental Functions 21.5.4.1 Integrals with Hyperbolic Functions

429.

1 1 1 1

430.

1

426. 427. 428.

1 sinh ax dx = - cosh ax. 1 cosh ax dx = - sinh ax. 1 1 sinh' ax dx = - sinh ax cosh ax - -x. 2a 2 1 . 1 cosh' ax dx = - sinh ax cosh ax + -x. 2a 2

sinhn ax dx 1 . an

= - sinhn-' ax cosh ax -

1

n+2 sinhn+l ax cosh ax - - sinhn+' ax dx n+1 a(n + 1)

431.

coshn ax dx

1

n-1 +coshn-' ax dx n + 2 j - -sinh ax coshntl ax + - coshnt2 ax dx n+l a(n t 1) an

433.

(for n < 0) (n # -1).

1

1 sinh . ax coshn-' ax =-

432.

(for n > 0) ,

1 1

dx

=

a1 In tanh -.ax2

dx 2 -= - arctan eaz. coshax a

(for n

>O),

(for n < 0) (n # -1)

22.5 Indefinite Inte.qrals 1045 1 1 434. [ z sinh a z dz = -zcosh a z - - sinh az. a2

1 . 1 435. [z cosh ax dx = -x sinh a z - - cosh az. a2

436.

1

1 tanh ax dx = - lncosh ax.

[coth ax dx = -1 In sinh ax. tanh ax 438. [tanh2ax dx = x coth ax 439. [coth2 a z ds = a: - -. 440. [sinh a z sinh bx dz = -(asinhbzcoshax-bcoshbzsinhaz) a2 - b2 437.

-,

441.

1

cosh a z cosh bz dz = -( a sinh ax cosh bx - bsinh bx cosh ax) a2 - b2 1

442. I c o s h a z s i n h b z d z = -(asinhbxsinhax a2 - b2

- bcoshbzcoshaz)

1 2a

443. [sinhaxsinazdx = -(coshazsinax - sinhaxcosax). 444.

1

1 . cosh ax cos a z da: = -(sinh a z cos ax + cosh ax sin ax). 2a

1 445. [sinhaxcosaxdx = -(coshazcosax +sinhazsinax). 2a

446.

1 [cosh ax sin ax da: = -(sinh 2a

ax sin a z - cosh a z cos ax).

21.5.4.2 Integrals with Exponential Functions 447.

1

1 ear dz = -eax.

eax a2

- 1).

448. [zeaxdz = -(ax 449.

[z2eardx = ea' :(

22 7 +

-

$-)

1

1

450. /zneax dz = -xneaz - - xn-'ear dx 451. / $ d z

ax l.l!

(ax)' (ax)3 +... ++2.2! 3 . 3 !

=lnz+ -

(a2 # b 2 ) . (a2 #

b2)

(a2 # b2).

21. Tables

1046

dt is called the exponential function integral (see 8.2.5, 4., p. 459) and it is --m

denoted by Ei(x). For x > 0 the integrand is divergent at t = 0; in this case we consider the principal value of the improper integral Ei(x) (see 8.2.5,4.,p. 459). ;dt = C + l n i x l + --oc

x +x= 23 xn t-+ ... + +. . . 1.1! 2.2!

3.3!

C denotes the Euler constant (see 8.2.5, 2., p. 458)

dx dx 454' 455.

/ enzdx

456.

1

1 ens - -1na 1 t ens' x 1 In(b t tea").

dx

beax

=

1 ; ln(b t ce"')

+ ce-ar

1

=

arctan ( e n s 4

I c t e a z G 2 a a In c - ens-

--

xens dx

--

ear x

1 a

easInx a

458. /en"Inxdx = -- - 1 - d x ;

(see No. 451).

en' a2 t b2 ear ear cos bx dx = -(acos bx + bsin bx). a2 + bZ

459. /ear sin bx dx = -(asinbx - bcosbx). 460. 461.

1 1

ear sinn x dx =

Sinn-l

+

a2 n2

'(a sin x - n cos x) - 1) +-n(n /enx sinn-' x dz ; az t nz

462.

1

ens cosn x dx =

a 2 t n2

+ 463. /senzsinbxdx =

x

ear

+

a2 n2

(a cos x + n sin x) ens

x dx ;

xeax ( a sin bx - b cos bx) a2 t b2

(see No. 447 and 459).

(see No. 447 and 460). ear ~

(a'

+ b 2 ) 2 [(a2 - b2) sin bx - 2ab cos bx]

,

21.5 Indefinite Inteqrals 1047

xeax

ea'

464. / r e n ' cos bx dx = -(acosbztbsinbz)---a2 + b2

[(a'

b2)2

-

b2) cos bz t 2absin bx].

21.5.4.3 Integrals with Logarithmic Functions 465. / l n r d x = x l n z - x . 466. /(lnx)2dz = z(1nx)' - 221nx + 22.

I

467. ( l n ~ ) ~ =d zx ( l n ~ -)3~z ( l n x ) 2 + 6 z I n z - 6 z . 468. 469.

S(lnz)"

1

S

dz = z(1nz)" - n (lnz)"-'dz

= In lnx

( n # -1).

(lnz)' (1112)~ + In x t t-t..,. 2.2! 3.3!

1

is called the logarithm integral (see 8.2.5,p. 458) and it is denoted by Li(z). In t For x > 1 the integrand is divergent at t = 1. In this case we consider the principal value of the improper integral Li(x) (see 8.2.5, p. 458). The relation between the logarithm integral and the exponential function integral (see 8.2.5, p. 459) is: Li(x) = Ei(ln2).

The definite integral

( n # 1) ;

[-

1-

1 In x 471. 1 x " l n x d z = xmtl m + 1 - ( m t l ) '

472. /xm(lnx)"dz =

474.

l$dx=-

475.

1

xm

xmdx 476. II,,=/,dy

dx = e-31

( m # -1).

zm+'(lnz)" n - -/xm(Inz)"-' dx ( m # -1 m t l m+l

1 In' ( m - 1)xm-1 ( m - 1 ) 2 2 m - 1

(see No. 469)

~

n # -1 ; (see No. 470).

( m # 1).

(Inz)"

( m # 1);

(see No. 474).

with y = -(m t 1 ) l n x ;

(see No. 451).

478.

1%dx

479.

( n - 1)2(lnz)2 ( n dx Ja = In lnx - ( n - 1)l n z t 2.2!

= lnlnx. -

~)~(inz)~ +..'. 3.3!

25

22n-IB n x Z n t l

900

n(2n t l)!

482. / l n s i n x d x = x l n x - x - - - - -x3 s..-

18

-...

B, denote the Bernoulli numbers (see 7.2.4.2, p. 410). 2%-1

483. / l n c o s x d x = - - - -2-3- - . .2. -5

6

484. I l n t a n x d x = x l n x - x

315

+23 t 7x5 t ... + 2 2 y p - 1 - 1)Bnx2n+l , . . 9

485.

/ sin In x dx

2n

(2 - 1)Bnx2n+l - . . . n(2n + l)!

27

60

450

n(2n

+ l)!

+

= :(sin In x - cos In x)

2

486. S c o s l n x d x = z(sinlnx+coslnx). 2 487.

1 1 ear In x dx = -ear In x - a

1

eez

(see No. 451).

;

-dx x

21.5.4.4 Integrals with Inverse Trigonometric Functions 488. 489. 490.

/ arcsin z dx x arcsin a + m. / x arcsin F dx (g <) arcsin + % d m . 1 / x2arcsin 2a dx 3 arcsin xa + 9-(x2 +2 a 2 ) m . =

23

=-

,

491, /arCSi:fdx

492. 493. 494. 495.

-

=

arcs:z:

1 1

arccos

-

1 . 3 d' 1 . 3 . 5 -x7+ . . . , 2 . 4 . 5 . 5 a5 2 . 4 . 6 . 7 . 7 a 7

1 x3 2 . 3 3a3

x a

--+--+--

dx

I

1 a

x a

1

+

= -- arcsin - - -In X

:dx = x arccos

x arccos F dx =

-d

(5 ): -

a t d F 7 X

m .

arccos ! -

2m.

x 23 x 1 x2 arccos - dx = - arccos - - -(x2 t 2 a 2 ) d n . a 3 a 9 arccos!dx

x

1

13

1.3

25

1.3'5

27

2f.5 Indefinite Integrals 1049

498.

/ arctan

x a dx = x arctan - - - ln(az + xz). a

x

2

1

x

ax

499. / x arctan - dx = -(x2 t a') arctan - - -. a 2 a 2

x

ax2 a3

x

x3

500. Jx' arctan - dx = - arctan - - a 3 a 6

+ln(a2 + x'). 6 ( n # -1).

,

a r c t ~dx 2~

503.

504.

arctzi=

507.

dx

510,

-

z

a

1 J x arccot x- dx = -(x' a

2

/ / xn arccot -a dx

+2

+

x + -.ax

+ a z ) arccot a

2

x x3 x ax' a3 x2 arccot - dx = - arccot - + - - - ln(a' a 3 a 6 6 X

508.

aZ+x2

1

1 dx arctan - + ( n - 1)xn-1 a n - 1 xn-l(a2+z2) x a arccot dx = x arccot - - ln(a' x').

505. 506.

x

1

= -- arctan - - - In x2 ' X a 2a

arcc;2'

a

dx

+ x').

xntl

= -arccot

( n # -1).

n+1

1 x 1 =--arccot-+-In2 a 2a

aZ+x2 x2 '

21.5.4.5 Integrals with Inverse Hyperbolic Functions 512.

Arsinh dx = x Arsinh

513.

Arcosh

514.

1

515.

/ Arcoth

Artanh

dx = x Arcosh

-

d m .

z d m . -

x a dx = x Artanh - + - In(a2 - x'), a

2

x

a

a

2

dx = x Arcoth - + - ln(x2 - a').

(n # 1).

21.6 Definite Integrals 21.6.1 Definite Integrals of Trigonometric Functions For natural numbers m. n: 1. Jsinnxdx = 0.

(21.1)

2. Jcosnxds = 0. (21.2)

0

0 form # n, x form = n.

4. Jsin nx sin mx dx = 0

3. Jsinnxcosmxdx = 0.

(21.3)

0

0

25

(21.4) 5 . [cosnzcosrnxdx = 0

0 for m # n, (21.5) form = n.

T

(21.6)

(21.7a)

B(x, Y) = r(x)r(y) denotes the beta function or the Euler integral of the first kind, r ( x ) denotes the r ( x + Y) gamma function or the Euler integral of the second kind (see 8.2.5, 6., p. 459). The formula (21.7a) is valid for arbitrary (Y and 9;we use it, e.g., to determine the integrals ~

*I2

1

TI2

G d x ,

1

G d x ,

0

0

1

' I z dx

etc.

0

For positive integer cy, 3:

1 TI2

7b.

sinZat1 x coszDf1 x dx =

0

for a

X

9.

I---

cos ax dx

- cc

.!P!

2(.

+ p + l)!

< 0.

( a arbitrary).

(21.7b) '

(21.8)

(21.9)

0

'x

tan ax dx

0

11, ~ c o s n x - cXo s h x n

for a b dx = ln-a

< 0.

(21.10)

(21.11)

21.6 Definite Integrals 1051

I for la1 < 1,

sin x cos ax

(21.12)

0

0 for ( a (> 1.

(21.13)

14.

7-

dx = iIe-labl (the sign is the same as the sign of b) , 2

0

(21.14)

(21.15)

(21.16)

/ sin(x2)dx / cos(x2)dx E.

tcc

17.

tm

=

=

(21.17)

-m

-CX

(21.18)

(21.19) 20,

7'

sinzzdz

1 k2

= -(K

odGmiG

-

E) for Ik/ < 1 .

(21.20)

Here, and in the following, E and K mean complete elliptic integrals (see 8.1.4.3, 2., p. 435):

E = E (k, 21.

j.;.

22,

1

);

, K =F

cosZxdx

J1 - k2sinZx

(k) 5) (see also the table of elliptic integrals 21.7, p. 1055). 1 kZ

= -[E - (1 - k z ) K].

cosaxdx 1 - 2 b c 0 s ~+ bz

-

nb" for integer a 2 0, Ibl < 1. 1 - b2

21.6.2 Definite Integrals of Exponential Functions (partially combined with algebraic, trigonometric, and logarithmic functions)

n! - -p+I

for a > 0, n = 0,1,2,. . . .

(21.21)

(21.22)

r ( n )denotes the gamma function (see 8.2.5, 6., p. 459); see also the table of the gamma function 21.8, p. 1057).

r (+)

P

for a > 0, n > -1,

dx = 2a(+)

24. /xne-"' 0

(2 1.24a)

(21.2413)

k! --

for n = 2 k + 1 ( k = O , 1 , 2 , . . .) , a > 0 .

2ak+1

/

(21.24~)

P

25.

J;; for a > 0 e-a2xzdx = 2a

0

(21.25)

m

J;; for a 26. /x2e-nazadx = 4a3

0

1

> 0.

(21.26)

X

27.

cos bx dx = J". e-b2/4a2 for a > 0 2a

e-""

0

(2 1.27)

(21.28)

(21.29)

30,

7%

sin x

1

dx = arccot a = arctan - for a > 0. U

0

In x dx = -C

31.

M

-0,5772

(21.30)

(21.31)

0

C denotes the Euler constant (see 8.2.5, 2.)p. 458).

21.6.3 Definite Integrals of Logarithmic Functions (combined with algebraic and trigonometric functions) 32. ]In

1 lnx I

dx = -C = -0,5772

(reduced to Nr. 21.31).

(21.32)

0

C is the Euler constant (see 8.2.5, 2., p. 458). 33.

12-i lnx

0

34.

1 0

R2

dx = 6

(reduced to Nr. 21.28)

(21.33)

dx = -- (reduced to Nr. 21.29) 12

(21.34)

R2

21.6 Definite Inteorals 1053

(21.35)

(21.36)

(2 1.37)

r ( z ) denotes the gamma function (see 8.2.5, 6., p. 459; see also the table of the gamma function 21.8, p. 1057).

38.

1

1

0

0

lnsinzdz =

lncosxdz = - p l n 2 . 2

(21.38)

n2 In 2 39. j z l n s i n z d z = -2 '

(21.39)

40. ~ s i n z l n s i n x d = r ln2 - 1.

(21.40)

0

0

41. j l n ( a

* bcosz) dz = Tln a +

0

42. ]ln(a2 - 2abcosz + bZ) dx = 0

2

fora 2 b.

(21.41)

P a l n a for (a 2 b > 0), 2alnb for (b 2 a > 0 ) .

43. ?'In tan z dz = 0.

(21.42)

(2 1.43)

0

(21.44)

21.6.4 Definite Integrals of Algebraic Functions 45. ]P(1 - z)P dx = 2 0

=

i

zZa+l(1 - " 2 ) B

0

dz =

r(cP+ l ) r ( p+ 1)

+ +

r ( a p 2)

B ( a t 1, /3 + l ) , (reduced to Nr. 21.7a).

(21.45)

B(z,y) = r(z)r(y) denotes the beta function (see 21.6.1, p. 1050) or the Euler integral of the first ~

r ( x Jr Y)

kind, r ( z ) denotes the gamma function (see 8.2.5, 6., p. 459) or the Euler integral of the second kind. W

46.

for a < 1.

(21.46)

I

=-?rcotan

fora
(21.47)

forO
(21.48)

xa-l

48. J = d x = A

bsin b

0

(21.49) r ( x ) denotes the gamma function (see 8.2.5, 6., p. 459; see also the table of the gamma function 21.8, p. 1057).

(21.50)

51.

7

dx - -a 1 + 2 x c o s a + x 2 sinx

@
(21.51)

21.7 Elliptic Inteqrals 1055

2 1.7 Elliptic Integrals 21.7.1 Elliptic Integral ofthe First Kind F (9,I C ) , k = sin a 'p

I" -

---

10

20

30

0.0000 0.1745 0.3491 0.5236 0.6981 0.8727 1.0472 1.2217 1.3963 1.5708

0.0000 0.1746 0.3493 0.5243 0.6997 0.8756 1.0519 1.2286 1.4056 1.5828

0.0000 0.1746 0.3499 0.5263 0.7043 0.8842 1.0660 1.2495 1.4344 1.6200

0.0000 0.1748 0.3508 0.5294 0.7116 0.8982 1.0896 1.2853 1.4846 1.6858

0

~

0 10 20 30 40 50 60 70 80 90

--

60

70

0.0000 0.1752 0.3545 0.5422 0.7436 0.9647 1.2126 1.4944 1.8125 2.1565

0.0000 0.1753 0.3555 0.5459 0.7535 0.9876 1.2619 1.5959 2.0119 2.5046

-

80

90

0.0000 0.1754 0.3561 0.5484 0.7604 1.0044 1.3014 1.6918 2.2653 3.1534

0.0000 0.1754 0.3564 0.5493 0.7629 1.0107 1.3170 1.7354 2.4362

--

30

21.7.2 Elliptic Integral of the Second Kind E (cp, k) , k = sin a -

'I"

0 -0 10 20 30 40 50 60 70 80 90

-

0.0000 0.1745 0.3491 0.5236 0.6981 0.8727 1.0472 1.2217 1.3963 1.5708

10

I

-- -20

30

40

50

60

70

0.0000 0.1743 0.3473 0.5179 0.6851 0.8483 1.0076 1.1632 1.3161 1.4675

0.0000 0.1742 0.3462 0.5141 0.6763 0.8317 0.9801 1.1221 1.2590 1.3931

0.0000 0.1740 0.3450 0.5100 0.6667 0.8134 0.9493 1.0750 1.1926 1.3055

0.0000 0.1739 0.3438 0.5061 0.6575 0.7954 0.9184 1.0266 1.1225 1.2111

0.0000 0.1738 0.3429 0.5029 0.6497 0.7801 0.8914 0.9830 1.0565 1.1184

1

80

90

0.0000 0.1737 0.3422 0.5007 0.6446 0.7697 0.8728 0.9514 1.0054 1.0401

0.0000 0.1736 0.3420 0.5000 0.6428 0.7660 0.8660 0.9397 0.9848 1.0000

1056

21. Tables

21.7.3 Complete Elliptic Integral, k = sin a E K E I" a /" xI" K ff

K

E

-2.1565 1.2111 2.1842 1.2015 2.2132 1.1920 2.2435 1.1826 2.2754 1.1732

0 1 2 3 4

1.5708 1.5709 1.5713 1.5719 1.5727

1.5708 1.5707 1.5703 1.5697 1.5689

30 31 32 33 34

1.6858 1.6941 1.7028 1.7119 1.7214

1.4675 1.4608 1.4539 1.4469 1.4397

60 61 62 63 64

5

1.5738 1.5751 1.5767 1.5785 1.5805

1.5678 1.5665 1.5649 1.5632 1.5611

35

1.7312 1.7415 1.7522 1.7633 1.7748

1.4323 1.4248 1.4171 1.4092 1.4013

65

66 67 68 69

2.3088 2.3439 2.3809 2.4198 2.4610

1.1638 1.1545 1.1453 1.1362 1.1272

1.5828 1.5854 1.5882 1.5913 1.5946

1.5589 1.5564 1.5537 1.5507 1.5476

40

1.7868 1.7992 1.8122 1.8256 1.8396

1.3931 1.3849 1.3765 1.3680 1.3594

70 71 72 73 74

2.5046 2.5507 2.5998 2.6521 2.7081

1.1184 1.1096 1.1011 1.0927 1.0844

1.5981 1.6020 1.6061 1.6105 1.6151

1.5442 1.5405 1.5367 1.5326 1.5283

45

1.8541 1.8691 1.8848 1.9011 1.9180

1.3506 1.3418 1.3329 1.3238 1.3147

75

76 77 78 79

2.7681 2.8327 2.9026 2.9786 3.0617

1.0764 1.0686 1.0611 1.0538 1.0468

1.6200 1.6252 1.6307 1.6365 1.6426

1.5238 1.5191 1.5141 1.5090 1.5037

50

1.9356 1.9539 1.9729 1.9927 2.0133

1.3055 1.2963 1.2870 1.2776 1.2681

80 81 82 83 84

3.1534 3.2553 3.3699 3.5004 3.6519

1.0401 1.0338 1.0278 1.0223 1.0172

1.6490 1.6657 1.6627 1.6701 1.6777

1.4981 1.4924 1.4864 1.4803 1.4740

55

2.0347 2.0571 2.0804 2.1047 2.1300

1.2587 1.2492 1.2397 1.2301 1.2206

85

3.8317 4.0528 4.3387 4.7427 5.4349 m

1.0127 1.0080 1.0053 1.0026 1.0008 1.0000

6 7 8 9 10 11

12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29

- --

36 37 38 39 41 42 43 44 46 47 48 49 51 52 53 54 56 57 58 59

86 87 88 89 90

21.8 Gamma Function 1057

21.8 Gamma Function X

X

X

X

1.00 01 02 03 04

1.00000 1.25 0.90640 1.50 0.88623 26 0.90440 51 0.88659 0.99433 27 0.90250 52 0.88704 0.98884 28 0.90072 53 0.88757 0.98355 29 0.89904 54 0.88818 0.97844

1.75 76 77 78 79

0.91906 0.92137 0.92376 0.92623 0.92877

1.05 06 07 08 09

0.97350 0.96874 0.96415 0.95973 0.95546

0.89747 1.55 0.88887 56 0.88964 0.89600 0.89464 57 0.89049 0.89338 58 0.89142 0.89222 59 0.89243

1.80 81 82 83 84

0.93138 0.93408 0.93685 0.93969 0.94261

1.30 31 32 33 34

1.10 0.95135 1.35 0.89115 1.60 0.89352 1.85 0.94561 11 0.94740 61 0.89468 86 0.94869 36 0.89018 62 0.89592 87 0.95184 12 0.94359 37 0.88931 63 0.89724 88 0.95507 38 0.88854 13 0.93993 64 0.89864 89 0.95838 39 0.88785 14 0.93642 0.90012 1.90 0.96177 0.90167 91 0.96523 0.90330 92 0.96877 0.90500 93 0.97240 94 0.97610 0.90678

1.15 16 17 18 19

0.93304 1.40 0.88726 41 0.88676 0.92980 0.92670 42 0.88636 0.92373 43 0.88604 44 0.88581 0.92089

1.20 21 22 23 24

0.91817 1.45 0.88566 1.70 0.90864 1.95 0.97988 46 0.88560 71 0.91057 96 0.98374 0.91558 72 0.91258 97 0.98768 47 0.88563 0.91311 98 0.99171 48 0.88575 73 0.91467 0.91075 49 0.88592 74 0.91683 99 0.99581 0.90852

1.25 0.90640

1.65 66 67 68 69

1.50 0.88623 1.75 0.91906 2.00 1.00000

The values of the gamma function for z < 1 (z # 0, -1, -2,. . .) and z > 2 can be calculated by the following formula: r(x+ 1) r ( z )= (z - 1) r ( z - 1). r ( z )= , 2

r(1'7) 0'90864 A: r(0.7) = =- 1.2981, 0.7 0.7 IB: r(3.5) = 2.5. r(2.5) = 2.5.1.5' ~ ( 1 . 5 = ) 2.5.1.5.0.88623 = 3.32336.

21.9 Bessel Functions (Cylindrical Functions) X

0.0 +1.0000 0.1 0.9975 0.2 0.9900 0.9776 0.3 0.4 0.9604 0.5 $0.9385 0.9120 0.6 0.7 0.8812 0.8 0.8463 0.8075 0.9 1.0 +0.7652 1.1 0.7196 1.2 0.6711 1.3 0.6201 1.4 0.5669 1.5 t0.5118 0.4554 1.6 1.7 0.3980 0.3400 1.8 0.2818 1.9 2.0 t0.2239 2.1 0.1666 0.1104 2.2 0.0555 2.3 2.4 0.0025 2.5 -0.0484 0.0968 2.6 0.1424 2.7 2.8 0.1850 0.2243 2.9 3.0 -0.2601 0.2921 3.1 0.3202 3.2 0.3443 3.3 3.4 0.3643 3.5 -0.3801 0.3918 3.6 3.7 0.3992 0.4026 3.8 3.9 0.4018 4.0 -0.3971 4.1 0.3887 4.2 0.3766 4.3 0.3610 4.4 0.3423 4.5 -0.3205 0.2961 4.6 4.7 0.2693 0.2404 4.8 4.9 0.2097

+o.oooo

0.0499 0.0995 0.1483 0.1960 +0.2423 0.2867 0.3290 0.3688 0.4059 +0.4401 0.4709 0.4983 0.5220 0.5419 +0.5579 0.5699 0.5778 0.5815 0.5812 +0.5767 0.5683 0.5560 0.5399 0.5202 f0.4971 0.4708 0.4416 0.4097 0.3754 t0.3391 0.3009 0.2613 0.2207 0.1792 t0.1374 0.0955 0.0538 t0.0128 -0.0272 -0.0660 0.1033 0.1386 0.1719 0.2028 -0.2311 0.2566 0.2791 0.2985 0.3147

-m

-1.5342 1.0181 0.8073 0.6060 -0.4445 0.3085 0.1907 -0.0868 +0.0056 +0.0883 0.1622 0.2281 0.2865 0.3379 $0.3824 0.4204 0.4520 0.4774 0.4968 10.5104 0.5183 0.5208 0.5181 0.5104 +0.4981 0.4813 0.2605 0.4359 0.4079 +0.3769 0.3431 0.3070 0.2691 0.2296 +0.1890 0.1477 0.1061 0.0645 t0.0234 -0.016s 0.0561 0.0938 0.129€ 0.1632 -0.1947 0.2235 0.2494 0.2723 0.2921

--03

-6.4590 3.3238 2.2931 1.7809 -1.4715 1.2604 1.1032 0.9781 0.8731 -0.7812 0.6981 0.6211 0.5485 0.4791 -0.4123 0.3476 0.2847 0.2237 0.1644 -0.1070 -0.0517 +0.0015 0.0523 0.1005 +0.1459 0.1884 0.2276 0.2635 0.2959 t0.3247 0.3496 0.3707 0.3879 0.4010 +0.4102 0.4154 0.4167 0.4141 0.4078 $0.3979 0.3846 0.3680 0.3484 0.3260 +0.3010 0.2737 0.2445 0.2136 0.1812

+1.000 1.003 1.010 1.023 1.040 1.063 1.092 1.126 1.167 1.213 1.266 1.326 1.394 1.469 1.553 1.647 1.750 1.864 1.990 2.128 2.280 2.446 2.629 2.830 3.049 3.290 3.553 3.842 4.157 4.503 4.881 5.294 5.747 6.243 6.785 7.378 8.028 8.739 9.517 10.37 11.30 12.32 13.44 14.67 16.01 17.48 19.09 20.86 22.79 24.91

0.0000 +0.0501 0.1005 0.1517 0.2040 0.2579 0.3137 0.3719 0.4329 0.4971 0.5652 0.6375 0.7147 0.7973 0.8861 0.9817 1.085 1.196 1.317 1.448 1.591 1.745 1.914 2.098 2.298 2.517 2.755 3.016 3.301 3.613 3.953 4.326 4.734 5.181 5.670 6.206 6.793 7.436 8.140 8.913 9.759 10.69 11.71 12.82 14.05 15.39 16.86 18.48 20.25 22.20

m

2.4271 1.7527 1.3725 1.1145 0.9244 0.7775 0.6605 0.5653 0.4867 0.4210 0.3656 0.3185 0.2782 0.2437 0.2138 0.1880 0.1655 0.1459 0.1288 0.1139 0.1008 0.08927 0.07914 0.07022 0.06235 0.05540 0.04926 0.04382 0.03901 3.03474 3.03095 3.02759 3.02461 3.02196 3.01960 3.01750 3.01563 3.01397 3.01248 0.01116 0.00998C 0.008927 0.007988 0.007146 0.00640C 0.00573C 0.005132 0.004595 0.004119

oc;

9.8538 4.7760 3.0560 2.1844 1.6564 1.3028 1.0503 0.8618 0.7165 0.6019 0.5098 0.4346 0.3725 0.3208 0.2774 0.2406 0.2094 0.1826 0.1597 0.1399 0.1227 0.1079 0.09498 0.08372 0.07389 0.06528 0.05774 0.05111 0.04529 0.04016 0.03563 0.03164 0.02812 0.02500 0.02224 0.01979 0.01763 0.01571 0.01400 0.01248 0.01114 0.00993E 0.008872 0.00792: 0.007078 0.006325 0.005654 0.00505~ 0.004521

21.9 Bessel Functions (Cylindrical Functions) 1059

X

5.0

5.1 5.2 5.3 5.4

5.5

5.6 5.7 5.8 5.9

6.0

6.1 6.2 6.3 6.4

6.5

6.6 6.7 6.8 6.9

7.0

7.1 7.2 7.3 7.4

7.5

7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5

8.6 8.7 8.8 8.9

9.0

9.1 9.2 9.3 9.4

9.5

9.6 9.7 9.8 9.9

10.0

-0.1776 0.1443 0.1103 0.0758 0.0412 -0.0068 $0.0270 0.0599 0.0917 0.1220 +0.1506 0.1773 0.2017 0.2238 0.2433 +0.2601 0.2740 0.2851 0.2931 0.2981 $0.3001 0.2991 0.2951 0.2882 0.2786 t0.2663 0.2516 0.2346 0.2154 0.1944 t0.1717 0.1475 0.1222 0.0960 0.0692 t0.0419 t0.0146 -0.0125 0.0392 0.0653 -0.0903 0.1142 0.1367 0.1577 0.1768 -0.1939 0.2090 0.2218 0.2323 0.2403 -0.2459

-0.3276 0.3371 0.3432 0.3460 0.3453 -0.3414 0.3343 0.3241 0.3110 0.2951 -0.2767 0.2559 0.2329 0.2081 0.1816 -0.1538 0.1250 0.0953 0.0652 0.0349 -0.0047 +0.0252 0.0543 0.0826 0.1096 t0.1352 0.1592 0.1813 0.2014 0.2192 t0.2346 0.2476 0.2580 0.2657 0.2708 +0.2731 0.2728 0.2697 0.2641 0.2559 +0.2453 0.2324 0.2174 0.2004 0.1816 t0.1613 0.1395 0.1166 0.0928 0.0684 f0.0435

-0.3085 0.3216 0.3313 0.3374 0.3402 -0.3395 0.3354 0.3282 0.3177 0.3044 -0.2882 0.2694 0.2483 0.2251 0.1999 -0.1732 0.1452 0.1162 0.0864 0.0563 -0.0259 t0.0042 0.0339 0.0628 0.0907 t0.1173 0.1424 0.1658 0.1872 0.2065 t0.2235 0.2381 0.2501 0.2595 0.2662 t0.2702 0.2715 0.2700 0.2659 0.2592 t0.2499 0.2383 0.2245 0.2086 0.1907 t0.1712 0.1502 0.1279 0.1045 0.0804 t0.0557

t0.1479 0.1137 0.0792 0.0445 tO.O1O1 -0.0238 0.0568 0.0887 0.1192 0.1481 -0.1750 0.1998 0.2223 0.2422 0.2596 -0.2741 0.2857 0.2945 0.3002 0.3029 -0.3027 0.2995 0.2934 0.2846 0.2731 -0.2591 0.2428 0.2243 0.2039 0.1817 -0.1581 0.1331 0.1072 0.0806 0.0535 -0.0262 tO.OO1l 0.0280 0.0544 0.0799 t0.1043 0.1275 0.1491 0.1691 0.1871 t0.2032 0.2171 0.2287 0.2379 0.2447 t0.2490

27.24 29.76 32.58 35.6; 39.01 42.66 46.74 51.li 56.04 61.38 67.23 73.6E 80.72 88.46 96.96 106.3 116.5 127.8 140.1 153.7 168.6 185.0 202.9 222.7 244.3 268.2 294.3 323.1 354.7 389.4 427.6 469.5 515.6 566.3 621.9 683.2 750.5 824.4 905.8 995.2 1094 1202 1321 1451 1595 1753 1927 2119 2329 2561 2816 -

24.34 26.68 29.2: 32.08 35.18 38.56 42.33 46.44 50.95 55.9c 61.34 67.32 73.85 81.1C 89.03 97.74 107.3 117.8 129.4 142.1 156.0 171.4 188.3 206.8 227.2 249.6 274.2 301.3 331.1 363.9 399.9 439.5 483.0 531.0 583.7 641.6 705.4 775.5 852.7 937.5 1031 1134 1247 1371 1508 1658 1824 2006 2207 2428 2671 -

0.00 3691 3308 2966 2659 2385 2139 1918 1721 1544 1386 1244 1117 1003 09001 08083 07259 06520 05857 05262 04728 04248 03817 03431 03084 02772 02492 02240 02014 01811 01629 01465 01317 01185 01066 009588 008626 007761 006983 006283 005654 005088 004579 004121 003710 003339 003036 002706 002436 002193 001975 001778

0.00 4045 3619 3239 2900 2597 2326 2083 1866 1673 1499 1344 1205 1081 09691 08693 07799 06998 06280 05636 05059 04542 04078 03662 03288 02953 02653 02383 02141 01924 01729 01554 01396 01255 01128 01014 009120 308200 307374 306631 305964 305364 304825 304340 303904 303512 303160 302843 302559 302302 302072 301865

1060

21. Tables

21.10 Legendre Polynomials ofthe First Kind Pl(2)= 2 ;

Po(.) = 1; 1 Pz(.) = - ( 3 2 - 1); 2 I

P~(x) = - ( 3 h 4 - 302'

8 1 P ~ ( Z=) -(231d 16

P3(2)=

+ 3);

1 -(523 2

- 32);

' ( 6 3 ~-~ 70x3 + 152); 8 1 P,(z)= -(429x7 - 6 9 3 +~ 3~ 1 5 -~ 352). ~ 16 P5(2)=

+

- 3 1 5 ~ 1052' ~ - 5);

x =S ( x ) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

-0.3000 -0.4962 -0.4850 -0.4662 -0.4400 -0.4062 -0.3650 -0.3162 -0.2600 -0.1962 -0.1250 -0.0462 +0.0400 0.1338 0.2350 0.3438 0.4600 0.5838 0.7150 0.8538 1.0000

0.0000 -0.0747 -0.1475 -0.2166 -0.2800 -0.3359 -0.3825 -0.4178 -0.4400 -0.4472 -0.4375 -0.4091 -0.3600 -0.2884 -0.1925 -0.0703 +0.0800 0.2603 0.4725 0.7184 1.0000

0.3750 0.3657 0.3379 0.2928 0.2320 0.1577 $0.0729 -0.0187 -0.1130 -0.2050 -0.2891 -0.3590 -0.4080 -0.4284 -0.4121 -0.3501 -0.2330 -0.0506 +0.2079 0.5541 1.0000

0.0000 0.0927 0.1788 0.2523 0.3075 0.3397 0.3454 0.3225 0.2706 0.1917 +0.0898 -0.0282 -0.1526 -0.2705 -0.3652 -0.4164 -0.3995 -0.2857 -0.0411 +0.3727 1.0000

-0.3125 -0.2962 -0.2488 -0.1746 -0.0806 +0.0243 0.1292 0.2225 0.2926 0.3290 0.3232 0.2708 0.1721 $0.0347 -0.1253 -0.2808 -0.3918 -0.4030 -0.2412 $0.1875 1.oooo

0.0000 -0.1069 -0.1995 -0.2649 -0.2935 -0.2799 -0.2241 -0.1318 -0.0146 +0.1106 0.2231 0.3007 0.3226 0.2737 +0.1502 -0.0342 -0.2397 -0.3913 -0.3678 $0.0112 1.0000

21.11 Laplace Transformation 1061

2 1.11 Laplace Transformation (see 15.2.1.1, p. 708) m

F(p) =

/

e P t " f t ) dt, f ( t ) = 0 for t

< 0.

0

C is the Euler constant:

No.

C = 0.577216 (see 8.2.5, 2., p. 458).

-

1

2 3

0

0

-1 P 1 -

1 in-1

(n- I)! tn-1

Pn 1

4

( n - I)! ePt

5

eQt

- eat

P-. BePt - cue"

6

P-. e-"t

7

d~

8

i n at

9

jin(.t

10

+ P)

P

p2

t 2ap + P 2

p2

+

P

11

12

sin

:os a t .2

p c o s p - cusinp p2 t QZ

:os(& t p)

13

jinh cut

14

:osh cut

15 -

t

No.

-

16 17 18

( a - a)2 a sin Bt - p sin at

19

aP(ff2- P') cos Pt - cos at

20

(a2- PZ)

21

cos2 at

22

sin2 at

23

cosh' at

24

2a2 P(P2 - 4 4

sinh' at

25

2dp p4 + 4a4

sin at . sinh at

26 27 28 29 30 31 -

+

a(p2 2a2)

p4 t 4a4 a(p2 - 2 2 ) p4 + 4a4

sin at . cosh at cos a t . sinh at

P3

p4

+ 4a4 ffP

(p2

+ a')'

t .

- sin at 2

t

- sinh

2

at

B sinh at - a sinh Pt a2 - p2

21.11 Laplace Transformation 1063

No. P

32 33 34 35 36

n! 4n in-+ -(2n)!J;;

1

Pnfi 1

(n> 0 , integer)

m

37 sin at -

38

t

g gcos sin at

39

at

40

E s i n , rrt

41

gcosh

42

m

44

.)m m

46

at

1

43

45

m

1

(P+ P

1

dF=

&(at) (Bessel function of order 0, p. 507)

1

47

Io(rrt) (modified Bessel function of order 0, p. 507)

1064

21. Tables

No. 48 49 50

51

a

arctan P

t 2 - sin at . cos Pt t

52 53

arctan

54

1nP P

55 56

P2-ffP+P

a8

eat 2 sin Pt t -C - l n t

Inp

$(n) = - t

pn+l

(lnt+C)’--

P

1(,Pt

57

-1 + ... + -1- C 2

T2

6

- eat)

t

58

a P+ff In -= 2artanhP-ff P

59

p2 a2 In P2 + P2

60

pz - cy2 In P2 - P2

+

2 . - sinh at t

2.

cos pt - cos f f t

t

e-a2/4t

61 1

62

--e-Qfi Rea20

63

(m - P)” m , R e v > -1

~

v+

a”Jy(at) (see Bessel function, p. 507)

21.11 Laplace Transformation 1065

-

No. 64

(P -

m)” , R e v > -1

Cy”l,,(at) (see Bessel function, p. 507) 0 fort < p 1 fort > ,8

65

66

67

68

69

70

71

for t < p

- e-P&3

e-PP

for t < p

Pa -

72 1 - e-QP

73 74

P

e-QP - ,-PP

P

(-1

a t2 -

0 for t > Q 1 for 0 < t < Q 0 for 0 < t < cy 1 for Q < t < /3 0 for t > p

2

fort > p

21.12 Fourier Transformation The symbols in the table are defined in the following way: C: Euler constant (C = 0.577215.. . ) r(z) =

e-?'-' m

'J~)

dt,

Re z > 0

(Gamma function, see 8.2.5, 6., p. 459),

(-l)n(;&tzn

(Bessel functions, see 9.1.2.6, 2., p. 507),

= zn!r(v+n+i) L

(modified Bessel functions, see 9.1.2.6, 3., p. 507), C(x)

(Fresnel integrals, see 14.4.3.2, 5.: p. 695);

S(x) Si(x) - i z y d t si(x)

=

Ci(x)

-

(Integral sine, see 14.4.3.2, 2., p. 694),

- 1 y d t = si(x)--

2 (Integral cosine, see 14.4.3.2, 2., p. 694).

The abbreviations for functions occurring in the table correspond to those introduced in the corresponding chapters.

21.12.1 Fourier Cosine Transformation F,(w) = J f ( t ) c o s ( t w ) dt Oa

O2 3.

~

p' t '

1

0.

sin(a i d )

-) w-2 id

4 cos w sin* ( 2

Oa

t>a

-Ci(aw)

21.12 Fourier Transformation 1067

4

00

t>a

+ t)-'

( a - t)-'

e 11.

:a2

~

O
(a

10.

= 1 f ( t ) c o s ( t w ) dt

FJW)

a>O

,

[- si (aw) sin (aw) - Ci ( a u )cos ( m ) ]

a>O A e-aw _

+ t2)-'

2

a

--sin(au) A

2

b b2 + ( a - t ) 2

b

+

b2

I2+ ( a t ) 2 + b2

(a2 - P - i ,

0,

A

e-& cos (au)

A

e-bw sin(au)

a-t

a+t

+

+ ( a t t)2 + (a

-

t)'

w

0< t < a t>a

t-",

0 < Rev < 1

~

a

,-at

a' ?-bt

- e-at

t

.-at

-

fi

+ w2

I

m

F,(w) = J f ( t ) c o s ( t w ) d t

No. 20.

tn e-at

21.

tu-1

22.

-

23.

e-at2

e-at

1 (-1 - -1 + -) 1 t 2 t et-1

1 --ln(12

e-2aw)

2

24.

25.

s e- ot

t-2

In t

26.

27.

,

0,

Ol

Si (w)

--

w

-F

In t -

2w ( C + f + l n ? w )

&

~

7rl - - (sin (aw) Ci (w) - cos (w) si (w))

28.

2 a

29.

(t2- aZ)-' In ( b t )

30.

-

31.

In lb-tl

32. -

1

t

a1 - - {sin (aw) [Ci (au) - In (ab)]- cos (w) si(w)} 2 a

In (1 + t )

a+t

:-at

In t

1 [f [cos (bw)- cos (aw)] W +cos (bw)Si (bw)+ cos (w) Si (aw) -sin (awl -

ci (awl - sin ( b ~ c)i (bw))

a 2 [a. + ln (a2 + + w arctan 0 a2 i- w2 1 w2j

21.12

FOUTaeT

Ransformation

Q3

F,(w) = J f ( t ) c o s ( t w ) d t

No.

0

T

33.

-

W

34. - 2 si(aw) ~

35.

36.

In ( a Zt t')

d7TF

37.

38.

1 - cos (aw)

71

W

R

39.

40.

41.

sin (at) ~

t

45.

2 -

sin (at) t ( P + bZ)

t

w
e-ab

t sin (ut) t2 + b2

42. e-bt sin (at) e - t sin t 43.

44.

2 ' R 4'

2

cosh (bw),

w
e-& sinh (ab) ,

w>a

T b-'

(1 - e-ab cosh (bw)),

w
T b-'

e-& sinh (ab) ,

w>a

2 2

1 2 - arctan ( 2 ) 2

sin' (at)

t

4

sin (at) sin@)

t

1

(a+b)'- wz

5 In~(a-b)Z-wzl

1069

-

46.

3

sin2 ( a t ) t2

p ),

(a- 1

0,

-

47.

m

= J f ( t ) cos(tw)dt

F&)

No.

1 - { (w 8

sin3 ( a t ) t2

w

< 2a

w

> 2a

+ 3a) In (w t 3a)

+(w - 3a) In Iw - 3a/ - (w -(w - a ) In ~w - a i }

-

?! ( 3 2 - w') ,

O<w
8

48.

iwz,

sin3 ( a t )

16 0:

49.

(3a-w)2,

a<w

< 3a

w

> 3a

1 - cos ( a t )

t

-

50.

w=a

t3

2

1 - cos (at) t2

-

51.

T e-ab c o s h ( h ) , 2 b T e-& cosh(ab) , 2 b

cos ( a t ) b2 + t Z

52.

:-bt

1 b2 + ( a - d)'

cos(at)

-

a'

1 & Te - -

53.

54. -

t +

cot (at) 55. b2 t2 -

w>a

+ b2 + (a + w)*

+ w2 4b

w
cosh

(E)

+ eZab)-l

R

cosh (bw)(1

R

cosh (h) (ezab- l)-'

+ a ) In (w + a )

21.12 Fourier Transformation 1071 -

I F&)

No.

:E

56.

sin (at')

57.

sin [a(l - t')]

58. 59.

60.

= b ( t ) cos(to)dt

~

(g)

(cos

-sin

(a+"+jl) 4 4a

-iEc0s

sin (at')

t' sin (at')

t

e-

sin (bt') '

61.

(z))

sin

[

arctan

(a) 1-"

- 4 (a'

+ b2)

LF (g)I ) : (

cos (at')

2

2a [cos

+sin

62. cos [a(l - t')] -

63.

e-

cos ( bt') cos

64.

-

- - arctan 2

(a)]

1t sin j:( 1 . - sin

(:)

1 67. - cos -

(;)

65.

~

[?(ap: b2)

4

66.

4

68.

17T [cos ( 2 2 2w

- sin ( 2

+e - 2 6 1

m

No.

F,(w) = ./ f ( t ) cos(t w ) dt

69.

2~

70.

[. (g)

sin

ecbt sin ( a d )

.xcos

t

72.

1 - cos ( a 4)

73.

e- at -c o s ( b 4 )

-S

2

sin

4

fi

74.

75.

e- a Ji -[cos ( a 4)- sin ( a d ) ]

d

21.12.2 Fourier Sine Transformation 00

F,(w) =

No. 1. 2.

3. ~

-1

t' 0.

Oa

Si (aw)

cos

I);(

-- - arctan

[4(bfJw2)

[s (%)+ c (31 E (: + g)

sin ( a d )

71.

(g) (g) ($1

,S f ( t ) sin(tw) dt

11.11 Fourier Runsformation

00

F,(w) = J f(t) sin(t w ) d t O
0: -1

5.

-si (aw)

t>a

- + ' 6.

O
7. - 0.

t>a O
a.

t>a

-

&

9. -

10.

Ia+t)-'

(a>0)

-t)-'

(a > 0)

[sin (aw)Ci (aw) - cos (w) si (aw)]

11.

a

I

12.

t 2'

-

!!e-w

+t2

2

1 - [sin (w)Ci(uu) - cos (w)Si(aw)

13. 14.

b

!I'

+ ( a - t)'

5'

+ ( a + t)'

15. -

a t t

b

-

-

b'

+ ( a t t)'

b'

+ ( a - t)'

a-t

K

e+ sin (au)

w e-bw cos (aw)

-? cos(aw)

16.

2

-

17.

1 L (a* - t') 1

L (a'

+ t')

x 1 -cos ( a u ) 2 a2 T l-e-w _-

2

a'

1073

-

No.

19.

”.

t-

O
-

20.

e-

W

at

a2

e-

+ w2

Rt

21.

arctan

e-ot

22.

(t) (-)

-

[iw In b2 + w 2 + b arctan

t2

-

23. e-

24.

at

Jt

-

25.

tn e-at

-

26.

tY-1

e-at

1 -- tanh ( T U ) 2

(1 - e-t)-l

27. -

28. -

29. -

30.

31. -

In t , 0.

O
t>l

Ci(w) - C - In w W

():

- a arctan

(t)] 1

21.12 Fourier Trans.formation 1075

-

No. In t

32.

-

33.

In t -

-

t

&

34.

t (t' -

In (bt)

35.

t (t'

-2 - l

In (-)

36.

e-at

In t

!! [cos (aw) (In (ab) - Ci(aw)) - sin(aw) si(aw)] 2

t

j:(

1 - [a arctan a2

+ w2

1

- cw - - w In (a2 2

1

+ wz)

1 {In (5j + cos ( b ~ ci(bW) ) - cos (awl Ci(aw) W

+sin (bw)Si@) - sin (aw) Si(aw)

37.

+? [sin (bw) +sin (w)]] 2

38. 2T

- [I - cos (au) - aw si(aw)]

39.

40.

In

41.

42.

-2ne

1 I:;

2 - [C In ( a ~-) cos ( a ~ Ci(aw) ) - sin (aw) Si(aw)]

In 1 - -

In

a2

W

+

++ ( b - tt )) 22)

sin

+

W

2n - e-aw sin (bw) W

43.

-n

44.

i~

-

-.a j:(

(W)

cij:(

[C + In (aw) - Ci(aw)]

-

No

sin ( a t ) 45. t 46.

fw,

sin ( a t ) t'

~ < w < a

7r

2"'

w>a

47.

sin (7rt) 1- t 2

48.

A ecab - sinh (bw), 2 b _T2 -e-bw sinh (ab) , b

sin ( a t ) bZ t2

+

[

-

49. e-bt sin ( a t ) e- bt sin ( a t ) 50. t -

Ab 2 b2

i

In

2

sin' ( a t )

t

-

4' 8' 0, 0,

sin ( a t ) sin (bt)

t

-

-

(U

(E)

b K

54.

-

+ (a - w)' b2 + + u)' (b2b2 ++ (w(w +a)' - a)2)

-pe -

53.

w>a

1 T - 4 I bW sinh

51. -

52.

1

0 <w
sin2 ( a t ) t2

-

4' 0,

O<w<2a w = 2a

w > 2a O<w
< a+b

w>a+b

21.12 Fourier Transformation

!

No. sin' ( a t ) t3

cos ( a t ) t

:

1

w > 2a

fa',

0,

O<w
i7

-

w=a

4' K

-

w>a

2 '

57.

t cos ( a t ) bZ + t2

- i7 - e-ab sinh (bw), 2 2

e-bw cosh(ab) ,

0<w

w>a

-

58. sin (at') sin (at') 59. t 60.

cos (at')

61.

-

cos (at')

t

62.

No. f ( t ) 1.

b ( t ) (Dirac 6 function)

F(w) =

7 e-Iwtf(t)dt

-m

1

b(n) (t)

3.

6(")(t- a )

(iw).e-'-

( n = 0 , 1 , 2 , . . .)

1077

1078

21. Tables

1 1

tn

H ( t ) = 1 for t > 0 H ( t ) = 0 for t < 0 (Heaviside unit step function)

7.

~

8.

1 - t ab(w) iw

tn H ( t ) e-at H ( t ) = e-a* e-at H ( t ) = 0

for for

t >0 t <0

1

atiw

10. ~

t -

c;

1

15.

k ~

+ a2 H ( t + a) - H ( t - a) = 1

t2

H(t+a)-H(t-a)=O

for jtl < a for I t l > a

l-

2sinaw w

,i ai

2a6(w - a)

cos at

7r[b(W

+ a) + 6(w - a)]

sin at 1 cash t

1 sinh t

19.

;in at2

20.

:os at2

aw

-i7r tanh 2

E($;)

(a>O)

21.18 Fourier Transformation 1079

21.12.4 Exponential Fourier Transformation although the exponential Fourier transformation F,(w)can be represented by the Fourier transforma1 tion F ( w ) according to (15.77), Le., F,(w)= ;F(-w), here we give some direct transforms.

I

I

f ( t ) = A for a < t < b f ( t )= 0 otherwise

f(t)=t" 2.

f(t) = 0

for Ostsb otherwise ( n = 1 , 2 , ...)

Rev > 0

Rev

>0

T

for w > 0

0

for w

0

for w > 0

w v - l e-w

<0

21.13 2 Transformation For definition see 15.4.1.2,p. 732, for rules of calculations see 15.4.1.3, p. 733, for inverses see p. 735 Convergence Region

No. Original Sequence f,, 1

1

2 -

2-1

z zil

2

3

n

4

n2

5

n3

6

ean

7

an

8

an n!

9

n an

10

n2an

11

(3

12

13

z (2

- 1)2

+

z ( z 4 2 t 1) - 114

(2 2

z - ea 2 -

z-a

za

( z -a)? az(z + a )

( z - a)3 Z

(2

- l)k+'

G)

(1

+

sin bn

z sin b z2 - 22 COS b + 1

14 cos bn -

y

Z(Z

- COS b)

z2 - 22

COS

bt 1

21.13 Z Transformation 1081

Transform

Original Sequence f,,

=

F )(. 15

~

Zen

ean sin b n

16

ean cos b n

17

sinh bn

z ( z - e” cos b) z2 - 2ze4 cos b

z sinh b

Z(Z

18 cosh b n

2’ -

an sinh b n

20

an cosh b n

21

22

23

= 0,

f2n

= 0,

f2n

25

26

27

28

z ( z - a cosh b) z2 - 2za cosh b a2

+

f2n+l

= 2(2n

=2

+ 1)

I-

(ZZ

+

1) - 1)2

2422

Li 2-1

2 =-

2n + 1

nn cos

-5-

( n + 1) ean

+1

za sinh b

= 0,

f2nt1

+1

z2 - 2za cosh b t a2

fk = 1

f2n

+ eZa

- cash b)

22 cash b

fn = 0 f”ur n # k ,

f2n+l

24

sin b b + eZa

z 2 - 22120 cos

z2 - 22 cash b

19

Convergence Region

(fn)

1082

21. Tables

No. Original Sequence fn

Convergence Region

In t 2-1

30

~

(2n

+ 1) ! 1

1

cos

7

21.14 Poisson Distribution

21.14 Poisson Distribution For the formula of the Poisson distribution see 16.2.3.3, p. 755.

x k

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7

0.904837 0.090484 0.004524 0.000151 0.000004

0.818731 0.163746 0.016375 0.001091 0.000055 0.000002

0.740818 0.222245 0.033337 0.003334 0.000250 0.000015 0.000001

0.670320 0.268128 0.053626 0.007150 0.000715 0.000057 0.000004

0.606531 0.303265 0.075816 0.012636 0.001580 0.000158 0.000013 0.000001

0.548812 0.329287 0.098786 0.019757 0.002964 0.000356 0.000035 0.000003

2.0

3.0

0.135335 0.270671 0.270671 0.180447 0.090224 0.036089 0.012030 0.003437 0.000859 0.000191 0.000038 0.000007 0.000001

0.049787 0.149361 0.224042 0.224042 0.168031 0.100819 0.050409 0.021604 0.008102 0.002701 0.000810 0.000221 0.000055 0.000013 0.000003 0.000001

__

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

t x

0.496585 0.347610 0.121663 0.028388 0.004968 0.000696 0.000081 0.000008 0.000001

0.449329 0.359463 0.143785 0.038343 0.007669 0.001227 0.000164 0.000019 0.000002

0.9

1.0

0.406570 0.365913 0.164661 0.049398 0.011115 0.002001 0.000300 0.000039 0.000004

0.367879 0.367879 0.183940 0.061313 0.015328 0.003066 0.00051 1 0.000073 0.000009 0.000001

I

1083

1084

21. Tables

(Continuation)

x k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 -

4.0

5.0

6.0

7.0

8.0

9.0

0.018316 0.073263 0.146525 0.195367 0.195367 0.156293 0.104194 0.059540 0.029770 0.013231 0.005292 0.001925 0.000642 0.000197 0.000056 0.000015 0.000004 0.000001

0.006738 0.033690 0.084224 0.140374 0.175467 0.175467 0.146223 0.104445 0.065278 0.036266 0.018133 0.008242 0.003434 0.001321 0.000472 0.000157 0.000049 0.000014 0.000004 0.000001

0.002479 0.014873 0.044618 0.089235 0.133853 0.160623 0.160623 0.137677 0.103258 0.068838 0.041303 0.022529 0.011264 0.005199 0.002228 0.000891 0.000334 0.000118 0.000039 0.000012 0.000004 0.000001

0.000912 0.006383 0.022341 0.052129 0.091126 0.127717 0.149003 0.149003 0.130377 0.101405 0.070983 0.045171 0.026350 0.014188 0.007094 0.003311 0.001448 0.000596 0.000232 0.000085 0.000030 0.000010 0.000003 0.000001

0.000335 0.002684 0.010735 0.028626 0.057252 0.091604 0.122138 0.139587 0.139587 0.124077 0.099262 0.072190 0.048127 0.029616 0.016924 0.009026 0.004513 0.002124 0.000944 0.000397 0.000159 0.000061 0.000022 0.000008 0.000003 0.000001

0.000123 0.001 111 0.004998 0.014994 0.033737 0.060727 0.091090 0.117116 0.13 1756 0.131 756 0.118580 0.097020 0.072765 0.050376 0.032384 0.019431 0.010930 0.005786 0.002893 0.001370 0.000617 0.000264 0.000108 0.000042 0.000016 0.000006 0.000002 0.000001

21.15 Standard Normal Distribution

21.15 Standard Normal Distribution For the formula of the standard normal distribution see 16.2.4.2, p. 757.

21.15.1 Standard Normal Distribution for 0.00 5 z X

@(XI

2

X

0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79

@(x) 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79

0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

@(XI

~

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

D.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19

0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

2 - @(XI

1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10

0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19

0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

0.8643

0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 X

1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39

0.5793 0.40 0.5832 0.41 0.5871 0.42 0.5910 0.43 0.5948 0.44 0.5987 0.45 0.6026 0.46 0.6064 0.47 0.6103 0.48 0.6141 0.49 D.6179 0.50 0.6217 0.51 0.6255 0.52 0.6293 0.53 0.6331 0.54 0.6368 0.55 0.6406 0.56 0.6443 0.57 0.6480 0.58 0.59 0.6517 @(XI

0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69

D.6915 0.70

0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99

@(E)

0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8079 0.8106 0.8133 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 -

5

X

1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 0.9032 1.50 0.9049 1.51 0.9066 1.52 0.9082 1.53 0.9099 1.54 0.9115 1.55 0.9131 1.56 0.9147 1.57 0.9162 1.58 0.9177 1.59 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

5 1.99 2

3.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

D.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

1085

1086

21. Tables

21.15.2 Standard Normal Distribution for 2.00 5 2

5 3.90

X 0 x X @(E) @(XI @(XI @(XI 0.9773 2.20 0.9881 2.40 0.9918 2.60 3.9953 2.80 D.9974

X - @(x)

2.00

2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 -

0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.10

0.9821

2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 -

0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9864 0.9857

x

@(x)

2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 -

0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.30

0.9893

2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39

0.9896 0.9894 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

x

@(x)

2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 -

0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69

0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89

2.50 0.9938

2.70

1.9965

2.90 0.9981

2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59

0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79

0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99

0.9982 0.9983 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

x

@(x)

x

@(x)

x

@(x)

3.00 0.9987

3.20 0.9993

3.40 0.9997

3.10

3.30

3.50

0.9990

0.9995

0.9998

3.60 0.9998

3.70

0.9999

0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

3.80 0.9999

3.90

0.9999

21.16 x 2 Distribution

21.16

x2Distribution

For the formula of the x2 distribution see 16.2.4.6, p. 760.

x2 Distribution: Quantile Degree

Probability a

of Freedom m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100

0.99 0.00016 0.020 0.115 0.297 0.554 0.872 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54 10.2 10.9 11.5 12.2 12.9 13.6 14.3 15.0 22.2 29.7 37.5 45.4 53.5 61.8 70.1

0.975 0.00098 0.051 0.216 0.484 0.831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.26 6.91 7.56 8.23 8.91 9.59 10.3 11.0 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8 24.4 32.4 40.5 48.8 57.2 65.6 74.2

0.95

005

0025

001

0.0039 3.8 5.0 6.6 6.0 7.4 0.103 9.2 7.8 9.4 11.3 0.352 9.5 11.1 13.3 0.711 11.1 12.8 15.1 1.15 12.6 14.4 16.8 1.64 14.1 16.0 18.5 2.17 15.5 17.5 20.1 2.73 16.9 19.0 21.7 3.33 18.3 20.5 23.2 3.94 19.7 21.9 24.7 4.57 21.0 23.3 26.2 5.23 22.4 24.7 27.7 5.89 23.7 26.1 29.1 6.57 25.0 27.5 30.6 7.26 26.3 28.8 32.0 7.96 27.6 30.2 33.4 8.67 28.9 31.5 34.8 9.39 30.1 32.9 36.2 10.1 31.4 34.2 37.6 10.9 32.7 35.5 38.9 11.6 12.3 33.9 36.8 40.3 13.1 35.2 38.1 41.6 36.4 39.4 43.0 13.8 14.6 37.7 40.6 44.3 38.9 41.9 45.6 15.4 40.1 43.2 47.0 16.2 41.3 44.5 48.3 16.9 42.6 45.7 49.6 17.7 43.8 47.0 50.9 18.5 55.8 59.3 63.7 26.5 67.5 71.4 76.2 34.8 43.2 79.1 83.3 88.4 90.5 95.0 100.4 51.7 101.9 106.6 112.3 60.4 113.1 118.1 124.1 69.1 124.3 129.6 135.8 77.9

1087

21.17 Fisher F Distribution For the formula of the Fisher F distribution see 16.2.4.7, p. 761. Fisher F Distribution: Quantile - 1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 125 m

3

4

for cr = 0.05

8 12 24 5 6 - - -

161.4 199.5 115.7 224.6 230.2 !34.0 238.9 143.9 18.51 19.00 19.16 19.25 19.30 .9.33 19.37 L9.41 9.12 9.01 8.94 8.85 8.74 10.13 9.55 9.28 7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.91 5.19 5.05 4.95 4.82 4.68 6.61 5.79 5.41 5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.00 4.12 3.97 3.87 3.73 3.57 5.59 4.74 4.35 3.84 3.69 3.58 3.44 3.28 5.32 4.46 4.07 5.12 4.26 3.86 3.63 3.48 3.37 3.23 3.07 4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.91 4.84 3.98 3.59 3.36 3.20 3.09 2.95 2.79 3.26 3.11 3.00 2.85 2.69 4.75 3.89 3.49 4.67 3.81 3.41 3.18 3.03 2.92 2.77 2.60 4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.53 4.54 3.68 3.29 3.06 2.90 2.79 2.64 2.48 4.49 3.63 3.24 3.01 2.85 2.74 2.59 2.42 4.45 3.59 3.20 2.96 2.81 2.70 2.55 2.38 4.41 3.55 3.16 2.93 2.77 2.66 2.51 2.34 2.90 2.74 2.63 2.48 2.31 4.38 3.52 3.13 2.87 2.71 2.60 2.45 2.28 4.35 3.49 3.10 4.32 3.47 3.07 2.84 2.68 2.57 2.42 2.25 2.82 2.66 2.55 2.40 2.23 4.30 3.44 3.05 4.28 3.42 3.03 2.80 2.64 2.53 2.37 2.20 4.26 3.40 3.01 2.78 2.62 2.51 2.36 2.18 2.76 2.60 2.49 2.34 2.16 4.24 3.39 2.99 2.74 2.59 2.47 2.32 2.15 4.23 3.37 2.98 2.73 2.57 2.46 2.31 2.13 4.21 3.35 2.96 2.71 2.56 2.45 2.29 2.12 4.20 3.34 2.95 2.70 2.55 2.43 2.28 2.10 4.18 3.33 2.93 2.69 2.53 2.42 2.27 2.09 4.17 3.32 2.92 2.61 2.45 2.34 2.18 2.00 2.84 4.08 3.23 2.53 2.37 2.25 2.10 1.92 4.00 3.15 2.76 2.44 2.29 2.17 2.01 1.83 3.92 3.07 2.68 2.37 2.21 2.10 1.94 1.75 3.84 3.00 2.60 -

30

249.0 250.0 19.45 19.46 8.64 8.62 5.77 5.75 4.53 4.50 3.84 3.81 3.41 3.38 3.12 3.08 2.90 2.86 2.74 2.70 2.61 2.57 2.51 2.47 2.42 2.38 2.35 2.31 2.29 2.25 2.24 2.19 2.19 2.15 2.15 2.11 2.11 2.07 2.08 2.04 2.05 2.01 2.03 1.98 2.00 1.96 1.98 1.94 1.96 1.92 1.95 1.90 1.93 1.88 1.91 1.87 1.90 1.85 1.89 1.84 1.79 1.74 1.70 1.65 1.60 1.55 1.52 1.46

40 !51.0 9.47 8.59 5.72 4.46 3.77 3.34 3.05 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.85 1.84 1.82 1.80 1.79 1.69 1.59 1.49 1.39 -

x, 154.3 -9.50 8.53 5.63 4.36 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 1.39 1.25 1.00 ~

21.17 Fisher F Distribution 1089 for cy = 0,Ol Fisher F Distribution: Quantile ja,ml,mz

m~ ma

1 2 3 4 a

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 125 !x

-

1

2

3

4

5

6

8

12

24

30

40

4052 18.50 14.12 !1.20 .6.26 .3.74 .2.25 .1.26 .0.56

4999 j9.00 i0.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 4.98 4.78 4.60

5403 j9.17 l9.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.13 3.94 3.78

5625 39.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.65 3.48 3.32

5764 39.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.70 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.86 3.82 3.78 3.76 3.73 3.70 3.51 3.34 3.17 3.02

5859 39.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.12 2.95 2.80

5981 39.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.82 2.66 2.51

6106 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.50 2.33 2.18

6235 99.46 26.60 13.93 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.80 2.75 2.70 2.66 2.62 2.58 2.55 2.52 2.49 2.47 2.29 2.12 1.94 1.79

6261 19.47 i6.50 .3.84 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.54 2.50 2.47 2.44 2.41 2.38 2.20 2.03 1.85 1.70

6287 39.47 16.41 13.74 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.30 2.11 1.94 1.75 1.59

.0.04

9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7.88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.08 6.84 6.63

co 6366 39.50 26.12 13.46 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.16 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.06 2.03 2.01 1.80 1.60 1.37 1.00

1090 21. Tables

21.18 Student t Distribution For the formula of the Student t distribution see 16.2.4.8, p. 761

Student t Distribution: Quantile

Degree of Freedom

&,, or t a p r n

Probability a for Two-sided Problem

m

0.10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 co

6.31 12.7 2.92 4.30 2.35 3.18 2.13 2.78 2.01 2.57 1.94 2.45 1.89 2.36 1.86 2.31 1.83 2.26 1.81 2.23 1.80 2.20 1.78 2.18 1.77 2.16 1.76 2.14 1.75 2.13 1.75 2.12 1.74 2.11 1.73 2.10 1.73 2.09 1.73 2.09 1.72 2.08 1.72 2.07 1.71 2.07 1.71 2.06 1.71 2.06 1.71 2.06 1.71 2.05 1.70 2.05 1.70 2.05 1.70 2.04 1.68 2.02 1.67 2.00 1.66 1.98 1.64 1.96 0.05

0.05

0.025

0.02

0.01

0.002

0.001

31.82 63.7 6.97 9.92 4.54 5.84 3.75 4.60 3.37 4.03 3.14 3.71 3.00 3.50 2.90 3.36 2.82 3.25 2.76 3.17 2.72 3.11 2.68 3.05 2.65 3.01 2.62 2.98 2.60 2.95 2.58 2.92 2.57 2.90 2.55 2.88 2.54 2.86 2.53 2.85 2.52 2.83 2.51 2.82 2.50 2.81 2.49 2.80 2.49 2.79 2.48 2.78 2.47 2.77 2.46 2.76 2.46 2.76 2.46 2.75 2.42 2.70 2.39 2.66 2.36 2.62 2.33 2.58

318.3 22.33 10.22 7.17 5.89 5.21 4.79 4.50 4.30 4.14 4.03 3.93 3.85 3.79 3.73 3.69 3.65 3.61 3.58 3.55 3.53 3.51 3.49 3.47 3.45 3.44 3.42 3.40 3.40 3.39 3.31 3.23 3.17 3.09

637.0 31.6 12.9 8.61 6.86 5.96 5.40 5.04 4.78 4.59 4.44 4.32 4.22 4.14 4.07 4.01 3.96 3.92 3.88 3.85 3.82 3.79 3.77 3.74 3.72 3.71 3.69 3.66 3.66 3.65 3.55 3.46 3.37 3.29

0.01

0.005

0.001

0.0005

Probability a for One-sided Problem

21.19 Random Numbers 1091

21.19 Random Numbers For the meaning of random numbers see 16.3.5.2, p. 781. 4730 0612 0285 7768 4450 17332 4044 0067 5358 0038 8344 7164 7454 3454 0401 6202 8284 9056 9747 2992 6170 3265 0179 1839 2276 4146 3526 3390 4806 7959 8245 7551 5903 9001 0265

1530 8004 2278 8634 1888 9284 9078 3428 8085 8931 6563 4013 1643 9005 7697 9278 5256 7574 4772 0449 2271 4689 7492 5157 7616 8021 6292 0067 7414 3186 0195 1077 0338 4286 0151 7260 3840 7921 8836 3342 4595 2539 8619 0814 3949 6995 6042 9650 8078 9973 9952 7945 3809 5523 7825 7012 9286 5051 5983 0204 9611 0641 4915 2913 2744 7318 4521 5070 3305 3814

7993 2549 3672 2217 3162 7406 5969 4765 3219 6906 3835 8731 2995 5579 3081 7406 5969 4765 3219 6906 7592 5133 3170 3024 4398 5207 0648 9934 4651 4325 7024 9031 7614 4150 0973

3141 3737 7033 0293 9968 4439 9442 9647 2532 8859 2938 4980 7868 9028 5876 4439 9442 9647 2532 8859 1339 7995 9915 0680 3121 1967 3326 7022 1580 5039 3899 9735 5999 5059 4958

0103 7686 4844 3978 6369 5683 7696 4364 7577 5044 2671 8674 0683 5660 8150 5683 7696 4364 7577 5044 4802 8030 6960 1127 7749 7325 1933 2260 5004 7342 8981 7820 1246 5178 4830

4528 0723 0149 5933 1256 6877 7510 1037 2815 8826 4691 4506 3768 5006 1360 6877 7510 1037 2815 8826 5751 7408 2621 8088 8191 7584 6265 0190 8981 7252 1280 2478 9759 7130 6297

7988 4505 7412 1032 0416 2920 1620 4975 8696 6218 0559 7262 0625 8325 1868 2920 1620 4975 8696 6218 3785 2186 6718 0200 2087 3485 0649 1816 1950 2800 5678 9200 6565 2641 0575

4635 6841 6370 5192 4326 9588 4973 1998 9248 3206 8382 8127 9887 9677 9265 9588 6973 1998 9248 3206 7125 0725 4059 5868 8270 5832 6177 7933 2201 4706 8096 7269 1012 7812 4843

8478 1379 1884 1732 7840 3002 1911 1359 9410 9034 2825 2022 7060 2169 3277 3002 1911 1359 9410 9034 4922 5554 9919 0084 5233 8118 2139 2906 3852 6881 7010 6284 0059 1381 3437

9094 6460 0717 2137 6525 2869 1288 1346 9282 0843 4928 2178 0514 3196 8465 2869 1288 1346 9282 0843 8877 5664 1007 6362 3980 8433 7236 3030 6855 8828 1435 9861 2419 6158 5629

9077 1869 5740 9357 2608 3746 6160 6125 6572 9832 5379 7463 0034 0357 7502 3746 6160 6125 6572 9832 9530 6791 6469 6808 6774 0606 0441 6032 5489 2785 7631 2849 0036 9539 3496

5306 5700 8477 5941 5255 3690 9797 5078 3940 2703 8635 4842 8600 7811 6458 3690 9797 5078 3940 2703 6499 9677 5410 3727 8522 2719 1352 1685 6386 8375 7361 2208 2027 3356 5406

4357 5339 6583 6564 4811 6931 8755 6742 6655 8514 8135 4414 3727 5434 7195 2705 1547 3424 8969 5225 6432 3085 0246 8710 5736 2889 1499 3100 3736 7232 8903 8616 5467 5861 4790

8353 6862 0717 2171 3763 1230 6120 3443 9014 4124 7299 0127 5056 0314 9869 6251 4972 1354 3659 8898 1516 8319 3687 6065 3132 2765 3068 1929 0498 2483 8684 5865 5577 9371 9734

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22 Bibliography 1. Arithmetic

BECKENBACH, E.; BELLMANN, R.: Inequalities. - Springer-Verlag 1983. BOSCH,K.: Finanzmathernatik. - Oldenbourg-Verlag 1991. HARDY,G.: A Course in Pure Mathematics. - Cambridge University Press 1952. HEILMANN, W.-R.: Grundbegriffe der Risikotheorie.- Verlag Versicherungswirtschaft 1986. F.; MUNZER,H.: Lebensversicherungsmathematik fur Praxis und Studium. ISENBART, Verlag Gabler, 2nd ed. 1986. GELLERT,W . ; KASTNER,H.; NEUBER,S.: Fachlexikon ABC Mathematik. - Verlag H. Deutsch 1978. HEITZINGER, W.: TROCH, I.; VALENTIN, G.: Praxis nichtlinearer Gleichungen. - -C. Hanser Verlag 1984. PFEIFER, A,: Praktische Finanzmathematik. - Verlag H. Deutsch 1995. 2. Functions

[2.1]

[2.2] [2.3] [2.4] [2.5] [2.6]

FETZER, A.; FRANKEL, H.: Mathematik Lehrbuchfur Fachhochschulen, Bd. 1.-VDI-Verlag 1995. FICHTENHOLZ, G.M.: Differential- und Integralrechnung, Bd. 1.- Verlag H. Deutsch 1994. HARDY,G.: A Course in Pure Mathematics. -Cambridge University Press 1952. Handbook of Mathematical, Scientific and Engineering Formulas, Tables, Functions, Graphs, Transforms. - Research and Education Association 1961. PAPULA, L.: Mathematik fur Ingenieure, Bd. 1, 2, 3. - Verlag Vieweg 199441996, SMIRNOW, W.I.: Lehrbuch der hoheren Mathematik, Bd. 1. - Verlag H. Deutsch 1994.

3. Geometry

[3.1] [3.2] [3.3] [3.4] [3.5] [3.6] [3.7] [3.8] [3.9] [3.10] [3.11] [3.12] [3.13] [3.14]

[3.15] [3.16] [3.17] [3.18]

BAR, G.: Geometrie. - B. G. Teubner 1996. BERGER,M.: Geometry, Vol. 1, 2. - Springer-Verlag 1987. BOHM,J.: Geornetrie, Bd. 1, 2. - Verlag H. Deutsch 1988. DRESZER,J.: Mathematik Handbuch fur Technik und Naturwissenschaft. - Verlag H. Deutsch 1975. A.; NOVIKOV,S.: Modern Geometry, Vol. 1-3. -Springer-Verlag DUBROVIN, B.; FOMENKO, 1995. EFIMOW,N.V.: Hohere Geometrie, Bd. 1, 2. - Verlag Vieweg 1970. FISCHER, G.: Analytische Geornetrie. - Verlag Vieweg 1988. JENNINGS, G.A.: Modern Geometry with Applications. - Springer-Verlag 1994. LANG,S.; MURROW,G.: A High School Course. - Springer-Verlag 1991. GELLERT.W.; KUSTNER,H.; KASTNER,H . (Eds.): Kleine Enzyklopadie Mathematik. Verlag H. Deutsch 1988. KLINGENBERG, W.: Lineare Algebra und Geometrie. - Springer-Verlag 1992. KLOTZEK,B.: Einfiihrung in die Differentialgeornetrie, Bd. 1 , 2 . - Verlag H. Deutsch 1995. KOECHER,M.: Lineare Algebra und analytische Geornetrie. - Springer-Verlag 1997. H. v . ; KNOPP.K.: Einfuhrung in die hohere Mathematik, Bd. 11. - S. Hirzel MANGOLDT, Verlag 1978. MATTHEWS.V.: Vermessungskunde Teil 1,2. - B. G. Teubner 1993. RASCHEWSKI, P.K.: Riemannsche Geornetrie und Tensoranalysis. -Verlag H. Deutsch 1995. SIGL,R.: Ebene und spharische Trigonometrie. - Verlag H. Wichrnann 1977. SINGER.D. A.: Plane and Fancy. - Springer-Verlag 1998.

[3.19] STEINERT,K.-G.: Spharische Trigonometrie. - B. G. Teubner 1977. 4. Linear Algebra

[4.1] [4.2] [4.3] [4.4] [4.5] [4.6] [4.7] [4.8] [4.9]

BERENDT,G.: WEIMAR,E.: Mathematik fur Physiker, Bd. 1, 2. - VCH 1990. BLYTH,T. S.; ROBERTSON, E. F.: Basic Linear Algebra. - Springer-Verlag 1998. CURTIS,C. W.: Linear Algebra. An Introductory Approach. - Springer-Verlag 1984. FADDEJEW, D.K.; FADDEJEWA, W.N.: Numerische Methoden der linearen Algebra. Deutscher Verlag der Wissenschaften 1970. JANICH,K.: Lineare illgebra. - Springer-Verlag 1996. KIELBASI~K A,; I , SCHWETLICK, H.: Numerische lineare Algebra. Eine computerorientierte Einfuhrung. - Verlag H. Deutsch 1988. KLINGENBERG, W .: Lineare Algebra und Geometrie. - Springer-Verlag 1992. KOECHER,M.: Lineare algebra und analytische Geometrie. - Springer-Verlag 1997. LIPPMANN, H.: .4ngewandte Tensorrechnung. Fur Ingenieure, Physiker und Mathematiker. Springer-Verlag 1993. RASCHEWSKI: P.K.: Riemannsche Geometrie und Tensoranalysis. -Verlag H. Deutsch 1995. SMIRNOW, W.I.: Lehrbuch der hoheren Mathematik, Teil II1,l. - Verlag H. Deutsch 1994. SMITH,L.: Lineare Algebra. - Springer-Verlag 1998. S.: Matrizen und ihre Anwendung - 1. Grundlagen. - Springer-Verlag ZURMUHL, R.; FALK, 1997. R.: Praktische Mathematik fur Ingenieure und Physiker. - Springer-Verlag 1984. ZURMUHL,

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[4.10] [4.11] [4.12] [4.13] [4.14]

5. Algebra a n d Discrete Mathematics

[5.5]

A ) Algebra and Discrete Mathematics, General AIGNER,M.: Diskrete Mathematik. - Verlag Vieweg 1993. BURRIS,S . ; SANKAPPANAVAR, H. P . : A Course in Universal Algebra. - Springer-Verlag 1981. EHRIG,H.; MAHR,B.: Fundamentals of Algebraic Specification 1. - Springer-Verlag 1985. F.: A Logical Approach to Discrete Mathematics. - Springer-Verlag GRIES,D.; SCHNEIDER, 1993. WECHLER,W.: Universal Algebra for Computer Scientists. - Springer-Verlag 1992.

[5,6]

B) Algebra a n d Discrete Mathematics, Group Theory FASSLER, A.; STIEFEL,E.: Group Theoretical Methods and their Applications. -Birkhauser

[Ll] [5.2]

I5.31 . . [5.4]

1992. HEIN,W.: Struktur und Darstellungstheorie der klassischen Gruppen. - Springer-Verlag 1990. [5.8] HEINE,V.: Group Theory in Quantum Mechanics. - Dover 1993. C.: Symmetries in Physics. Group Theory Applied to Physical Prob[5.9] LUDWIG, W . , FALTER, lems. - Springer-Verlag 1996. [5.10] VARADARAJAN, V.: Lie Groups, Lie Algebras and their Representation. - Springer-Verlag 1990. [5.11] WALLACE, D.: Groups, Rings and Fields. - Springer-Verlag 1998. [5.12] ZACHMANN. H.G.: Mathematik fur Chemiker. - VCH 1990. [5.7]

C) Algebra and Discrete Mathematics, N u m b e r Theory G.A.; JONES, J.M.: Elementary Number Theory. - Springer-Verlag. [5.13] JONES, [5.14] NATHANSON. M.: Elementary Methods in Number Theory. - Springer-Verlag.

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[5.15] RIVEST,R.L.; SHAMIR,A.; ADLEMAN, L.: A Method for Obtaining Digital Signatures and Public Key Cryptosystems. - Comm. ACM 21 (1978) 12-126. [5.16] [5.17] [5.18] [5.19]

D ) Algebra a n d Discrete Mathematics, Cryptology BAUER,F. L.: Decrypted Secrets. - Springer-Verlag. BEUTELSPACHER, A.: Cryptology. The Mathematical Association of iZmerica 1996. SCHNEIDER, B.: bpplied Cryptology. - John Wiley 1995. STALLINGS, W.: Cryptology and Network Security. Prentice Hall 1998. - Addison Wesley Longman 1997.

E) Algebra and Discrete Mathematics, G r a p h Theory [5.20] DIESTEL.R.: Graph Theory. - Springer-Verlag. [5.21] EDMONDS, J.: Paths, Trees and Flowers. - Canad. J. Math. 17 (1965) 449-467. J., JOHNSON,E.L.: Matching, Euler Tours and the Chinese Postman. - Math. [5.22] EDMONDS, Programming 5 (1973) 88-129. [5.23] HARARY,F.: Graph Theory. -Addison Wesley.

F) Algebra a n d Discrete Mathematics, Fuzzy-Logik [5.24] BANDEMER, H.; GOTTWALD, S.: Einfiihrung in Fuzzy-Methoden - Theorie und Anwendungen unscharfer Mengen. - Akademie-Verlag 1993. M.: An Introduction to Fuzzy Control. [5.25] DRIANKOV, D.; HELLENDORN, H.; REINFRANK, Springer-Verlag 1993. H.: Fuzzy Sets and System Theory and Applications. -Academic Press [5.26] DUBOIS,D.; PRADE, 1980. [5,27] GRAUEL;A,: Fuzzy-Logik. Einfiihrung in die Grundlagen mit Anwendungen. - B.I. Wissenschaftsverlag 1995. j5.281 HORDESON, J.N.; NAIR,N.S.: Fuzzy Mathematics. -An Introduction for Engineers and Scientists. - Physica Verlag 1998. J.; KLAWONN: F.: Fuzzy-Systeme. - B. G. Teubner 1993. [5.29] KRUSE,R.; GEBHARDT, [5.30] WANG,Z.; KLIR,G.T.: Fuzzy Measure Theory. - Plenum Press 1992. [5,31] ZIMMERMANN, H-J.: Fuzzy Sets. Decision Making and Expert Systems. - Verlag KluwerNijhoff 1987. 6. Differential Calculus

[6.1] [6.2]

[E:] [6.5] [6.6] [6.7] [6.8] [6.9]

R.: Introduction to Calculus and .4nalysis, Vols. 1 and 2. - Springer-Verlag 1989. COURANT, FETZER,A , ; FRANKEL, H.: Mathematik. - Lehrbuch fur Fachhochschulen, Bd. 1 , 2 . -VDIVerlag 1995. FICHTENHOLZ, G.M.: Differential- und Integralrechnung, Bd. 1-3. -Verlag H. Deutsch 1994. LANG,S.: Calculus of Several Variables. - Springer-Verlag 1987. KNOPP,K.: Theorie und Anwendung der unendlichen Reihen. - Springer-Verlag 1964. MANGOLDT, H. v.; KNOPP,K.: Einfiihrung in die hohere Mathematik, Bd. 2, 3. - S. Hirzel Verlag 1978-81. PAPULA, L.: Mathematik fur Ingenieure, Bd. 1-3. Verlag Vieweg 1994-1996. SMIRNOW, W.I.: Lehrbuch der hoheren Mathematik, Bd. 11,111. - Verlag H. Deutsch 1994. ZACHMANN, H.G.: Mathematik fur Chemiker. - VCH 1990. -

7. Infinite Series A.: Tables of Integrals and Series. - Verlag H. Deutsch 1996. [7.1] APELBLAT, COURANT, R.: Introduction to Calculus and Analysis, Vols. 1 and 2. - Springer-Verlag 1989. [7.2] A.: FRANKEL, H.: Mathematik. - Lehrbuch fur Fachhochschulen, Bd. 1 , 2 . - VDI[7.3] FETZER, Verlag 1995.

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FICHTENHOLZ, G.M.: Differential- und Integralrechnung, Bd. 1-3. - Verlag H. Deutsch 1994. KNOPP:K.: Theorie und Anwendung der unendlichen Reihen. - Springer-Verlag 1964. MANGOLDT, H. v.; KNOPP,K.; HRG. F. LOSCH: Einfiihrung in die hohere Mathematik, Bd. 1-4. - S. Hirzel Verlag 1989. PAPULA. L.: h'lathernatik fur Ingenieure, Bd. 1-3. - Verlag Vieweg 1994-1996. PLASCHKO, P.; BROD,K.: Hohere mathematische Methoden fur Ingenieure und Physiker. -Springer-Verlag 1989. SMIRNOW, W.I.: Lehrbuch der hoheren Mathematik, Bd. 11,111.- Verlag H. Deutsch 1994.

8. Integral Calculus

[8.1] [8.2]

APELBLAT, A.: Tables of Integrals and Series. - Verlag H. Deutsch 1996. BRYTSCHKOW, J.A.; MARITSCHEW, 0.1.; PRUDNIKOV, A.P.: Tabellen unbestimmter Integrale. - Verlag H. Deutsch 1992. [8.3] COURANT, R.: Introduction to Calculus and Analysis, L'ols. 1and 2. - Springer-Verlag 1989. [8.4] FICHTENHOLZ, G.M.: Differential- und Integralrechnung, Bd. 1-3. -Verlag H. Deutsch 1994. [83] KAMKE,E.: Das Lebesgue-Stieltjes-Integral. - B. G. Teubner 1960. [8.6] KNOPP,K.: Theorie und Anwendung der unendlichen Reihen. - Springer-Verlag 1964. [8.7] MANGOLDT, H. v.; KNOPP,K.; HRSG. F. LOSCH: Einfiihrung in die hohere Mathematik, Bd. 1-4. - S. Hirzel Verlag 1989. L.: Mathematik fur Ingenieure, Bd. 1-3. - Verlag Vieweg 1994-1996. [8.8] PAPULA. [8.9] SMIRNOW, W.I.: Lehrbuch der hoheren Mathematik. Bd. 11,111.- Verlag H. Deutsch 1994. [8.10] ZACHMANN,H.G.: Mathematik fur Chemiker. VCH 1990. ~

9. Differential Equations

i9.1I] [9.12] [9.13] [9.14]

A) Ordinary and Partial Differential Equations BRAUN.M.: Differentialgleichungen und ihre Anwendungen. - Springer-Verlag 1991. CODDINGTON, E.; LEVINSON,N.: Theory of Ordinary Differential Equations. - McGraw Hill 1955. COLLATZ, L.: Differentialgleichungen. - B. G. Teubner 1990. COLLATZ, L.: Eigenwertaufgaben mit technischen Anwendungen. - .4kademische Verlagsgesellschaft 1963. COURANT, R.; HILBERT,D.: Methoden der mathematischen Physik, Bd. 1. 2. - SpringerVerlag 1968. EGOROV,Yu.: SHUBIN,M.: Partial Differential Equations, Vols. 1-4. - Encyclopaedia of Mathematical Sciences. Springer-Verlag 1991. FRANK, PH.; MISES, R. v.: Die Differential- und Integralgleichungen der Mechanik und Physik. Bd. 1. 2. - Verlag Vieweg 1961. GREINER.W.: Quanten Mechanics. An Introduction. - Springer-Verlag 1994. KAMKE,E.: Differentialgleichungen, Losungsmethoden und Losungen, Teil 1, 2. - BSB B. G. Teubner 1977. LANDAU. L.D.; LIFSCHITZ, E.M.: Quantenmechanik. - Verlag H. Deutsch 1992 P OLJ ANIN A.D .; SAIZEW, V. F. : Sammlung gewohnlicher Differentialgleichungen.- Verlag H. Deutsch 1996. REISSIG,R . ; SANSONE, G.; CONTI,R.: Yichtlineare Differentialgleichungen hoherer Ordnung. - Edizioni Cremonese 1969. SMIRNOW, W.I.: Lehrbuch der hoheren Mathematik, Teil2. -Verlag H. Deutsch 1994. SOMMERFELD, A,: Partielle Differentialgleichungen der Physik. - Verlag H. Deutsch 1992. ~

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[9.15] STEPANOW:W.W.: Lehrbuch der Differentialgleichungen. senschaften 1982.

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B) Non-Linear Partial Differential Equations [9.16] DODD,R.K.; EILBECK, J.C.; GIBBON,J.D.; MORRIS,H.C.: Solitons and Non-Linear Wave Equations. - Academic Press 1982. [9.17] DRAZIN, P.G., JOHNSON, R.: Solitons. An Introduction. -Cambridge University Press 1989. [9.18] G U CHAOHAO (Ed.): Soliton Theory and its Applications. - Springer-Verlag 1995. [9.19] LAMB,G.L.: Elements of Soliton Theory. - John Wiley 1980. [9.20] MAKHANKOV, V.G.: Soliton Phenomenology. - Verlag Kluwer 1991. [9.21] REMOISSENET, S.: Waves Called Solitons. Concepts and Experiments. - Springer-Verlag 1994. [9.22] TODA, M.: Nonlinear Waves and Solitons. - Verlag Kluwer 1989. [9.23] VVEDENSKY, D.: Partical Differential Equations with Mathematica. - Addison Wesley 1993. 10. Calculus of Variations [10.1] BLANCHARD, P.; BRUNING,E.: Variational Methods in Mathematical Physics. - SpringerVerlag 1992. [10.2] GIAQUINTA, M.; HILDEBRANDT, S.: Calculus of Variations. - Springer-Verlag 1995. E.: Variationsrechnung. - BI-Verlag 1988. [10.3] KLINGBEIL, [10.4] KLOTZLER,R.: Mehrdimensionale Variationsrechnung. - Birkhauser 1970. [10.5] KOSMOL,P.: Optimierung und Approximation. - Verlag W. de Gruyter 1991. [10.6] MICHLIN, S.G.: Numerische Realisierung von Variationsmethoden. -Akademie-Verlag 1969. [10.7] ROTHE. R.: Hohere Mathematik fur Mathematiker, Physiker, Ingenieure, Teil VII. B. G. Teubner 1960. 11. Linear Integral Equations [11.1] CORDUNEANU, I.C.: Integral Equations and Applications. - Cambridge University Press

1991. [11.2] ESTRADA, R . ; KANWAL, R.P.: Singular Integral Equations. -John Wiley 1999. [11.3] HACKBUSCH, W.: Integral Equations: Theory and Numerical Treatment. - Springer-Verlag 1995. [11.4] KANWAL, R.P.: Linear Integral Equations. - Springer-Verlag 1996. [ll.5] KRESS,R.: Linear Integral Equations. - Springer-Verlag 1999. S.: Singular Integral Operators. - Springer-Verlag 1986. [11.6] MICHLIN, S.G.; PROSSDORF, [11.7] MICHLIN,S.G.: Integral Equations and their Applications to Certain Problems in Mechanics. - MacMillan 1964. [11.8] MUSKELISHVILI, N.I.: Singular Integral Equations: Boundary Problems of Functions Theory and their Applications to Mathematical Physics. - Dover 1992. [11.9] PIPKIN, A.C.: A Course on Integral Equations. - Springer-Verlag 1991. [ll.lO] POLYANIN, A.D.; MANZHIROV, A.V.: Handbook of Integral Equations. - CRC Press 1998. 12. Functional Analysis [12.1] ACHIESER,N.I.; GLASMANN, I.M.: Theory of Linear Operators in Hilbert Space. - M. Yestell. Ungar. 1961. [12.2] ALIPRANTIS, C.D.; BURKINSHAW, 0.: Positive Operators. - Academic Press 1985. [12.3] ALIPRANTIS,C.D.; BORDER,K.C.; LUXEMBURG, W.A.J.: Positive Operators, Riesz Spaces and Economics. - Springer-Verlag 1991. [12.4] ALT. H.W .: Lineare Funktionalanalysis. - Eine anwendungsorientierte Einfuhrung. - Springer-Verlag 1976.

1097 [12.5] BALAKRISHNAN, A.V.: Applied Functional Analysis. - Springer-Verlag 1976. [12.6] BAUER,H.: MaB- und Integrationstheorie. - Verlag W. de Gruyter 1990. [12.7] BRONSTEIN, I.N.; SEMENDAJEW, K.A.: Erganzende Kapitel zum Taschenbuch der Mathematik. - BSB B. G. Teubner 1970; Verlag H. Deutsch 1990. [12.8] DUNFORD, N.; SCHWARTZ, J.T.: Linear Operators, Vols. I, 11,111.-1ntersciences 1958,1963, 1971. [12.9] EDWARDS, R.E.: Functional Analysis. - Holt, Rinehart and Winston 1965. [12.10] GAJEWSKI, H.; GROGER,K.; ZACHARIAS, K.: Nichtlineare Operatorengleichungen und Operatordifferentialgleichungen.- Akademie-Verlag 1974. [12.11] HALMOS,P . R . : A Hilbert Space Problem Book. -Van Nostrand 1967. [12.12] HUTSON,V.C.L.; PYM,J.S.: Applications of Functional Analysis and Operator Theory. Academic Press 1980. [12.13] HEWITT,E.; STROMBERG, K.: Real and Abstract Analysis. - Springer-Verlag 1965. [12.14] JOSHI. M.C.; BOSE,R.K.: Some Topics in Nonlinear Functional Analysis. - Wiley Eastern 1985. [12.15] KANTOROVICH, L.V.; AKILOW,G.P.: Functional .4nalysis - Pergamon Press 1982. S.W.: Introduction toFunctional Analysis. -Graylock Press [12.16] KOLMOGOROW, A.N.; FOMIN, 1961. [12.17] KRASNOSEL'SKIJ, M.A.; LIFSHITZ,J.A., SOBOLEV,A.V.: Positive Linear Systems. Heldermann-Verlag 1989. [12.18] LUSTERNIK, L.A.; SOBOLEW,V.I.: Elements of Functional Analysis. - Gordon and Breach 1961, Hindustan Publishing Corporation Delhi 1974, in German: Verlag H. Deutsch 1975. [12.19] MEYER-NIEBERG, P.: Banach Lattices. - Springer-Verlag 1991. [12.20] NAIMARK, M.A.: Normed Rings. - Wolters-Noordhoff 1972. [12.21] RUDIN,W . : Functional Analysis. - McGraw-Hill 1973. [12.22] SCHAEFER, H.H.: Topological Vector Spaces. - Macmillan 1966. [12.23] SCHAEFER, H.H.: Banach Lattices and Positive Operators. - Springer-Verlag 1974. [12.24] YOSIDA,K.: Functional Analysis. - Springer-Verlag 1965.

13. Vector Analysis a n d Vector Fields [13.1] DOMKE,E.: Vektoranalysis: Einfiihrung fur Ingenieure und Naturwissenschaftler. Verlag 1990. [13.2] JANICH.K.: Vector Analysis. - Springer-Verlag 1999. R.: Vektoranalysis fur Ingenieurstudenten. - Verlag H. Deutsch 1992. [13.3] SCHARK,

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14.Function Theory [14.1] ABRAMOWITZ, M.; STEGUN,I. A.: Pocketbook of Mathematical Functions. - Verlag H. Deutsch 1984. [14.2] BEHNKE.H.; SOMMER.F.: Theorie der analytischen Funktionen einer kornplexen Veranderlichen. - Springer-Verlag 1976. [14.3] FICHTENHOLZ, G.M.: Differential- und Integralrechnung, Bd. 2. - Verlag H. Deutsch 1994. E.; BUSAM,R.: Funktionentheorie. - Springer-Verlag 1994. [14.4] FREITAG, [I451 GREUEL,0.;KADNER,H.: Komplexe Funktionen und konforme Abbildungen. -B. G. Teubner 1990. [14.6] JAHNKE, E.; EMDE,F.: Tafeln hoherer Funktionen. - B. G. Teubner 1960. [14.7] JANICH,K.: Funktionentheorie. Eine Einfiihrung. - Springer-Verlag 1993. [14.8] KNOPP,SONI,R.: Funktionentheorie. - Verlag W. de Gruyter 1976. [14.9] MAGNUS.W.; OBERHETTINGER, F.: Formulas and Theorems for the Special Functions of mathematical Physics, 3rd ed. - Springer-Verlag 1966. F.; MAGNUS,W.: Anwendung der elliptischen Funktionen in Physik und [14.10] OBERHETTINGER, Technik. -- Springer-Verlag 1949.

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[14.11] SCHARK, R.: Funktionentheorie fur Ingenieurstudenten. - Verlag H. Deutsch 1993. [14.12] SMIRNOW: Lehrbuch der hoheren Mathematik, Bd. 111. - Verlag H. Deutsch 1994. [14.13] SPRINGER, S.: Introduction to Riemann Surfaces. - Chelsea Publishing Company 1981. 15. Integral Transformations [l5.l] BLATTER.C.: Wavelets. - Eine Einfuhrung. - Vieweg 1998. [15.2] DOETSCH.G.: Introduction to the Theory and Application of the Laplace Transformation. Springer-Verlag 1974. [ K 3 ] DYKE,P.P.G.: .4n Introduction to Laplace Transforms and Fourier Series. - Springer-Verlag 2000. V.: Integral-Transformationen. - Hiithig 1977. [ K 4 ] FETZER. 0.: Laplace- und Fourier-Transformation. - Hiithig 1993. [l5.5] FOLLINGER, [l5.6] GAUSS,E.: WALSH-Funktionen fur Ingenieure und Naturwissenschaftler. - B. G. Teubner 1994. B.B.: Wavelets. Die Mathematik der kleinen Wellen. - Birkhauser 1997. [l5.7] HUBBARD, R.C.: Fourier Transforms and Convolutions for the Experimentalist. -Pergamon [l5.8] JENNISON, Press 1961. [15.9] LOUIS,A. K.; MAASS.P.; RIEDER,A.: Wavelets. Theorie und Anwendungen. - B. G. Teubner 1994. [15.10] OBERHETTINGER, F.: Tables of Fourier Transforms of Distributions. - Springer-Verlag 1990. F.; BADIL,L.: Tables of Laplace Tkansforms. - Springer-Verlag 1973. [15.11] OBERHETTINGER: [15.12] PAPOCLIS, A.: The Fourier Integral and its Applications. - McGraw Hill 1962. j15.131 SCHIFF,J.L.: The Laplace Transform. - Theory and Applications. Springer-Verlag 1999. [15.14] SIROVICH, L.: Introduction to Applied Mathematics. Springer-Verlag 1988. R.; AN, M.; Lu, C.: Mathematics of Multidimensional Fourier Transforms Al[15.15] TOLIMIERI. gorithms. Springer-Verlag 1997. R . ; AN: M.; Lu, C.: Algorithms for Discrete Transform and Convolution. [15.16] TOLIMIERI, Springer-Verlag 1997. [15.17] VOELKER,D.; DOETSCH,G.: Die zweidimensionale Laplace-Transformation. - Birkhauser 1950. [15.18] WALKER,J.S.: Fast Fourier Transforms. - Springer-Verlag 1996. ~

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16. Probability Theory and Mathematical Statistics

[16.1] BERGER,M.,A.: An Introduction to Probability and Stochastic Processes. - Springer-Verlag 1993. [16.2] BEHNEN, K.; NEUHAUS, G.: Grundkurs Stochastik. - B. G. Teubner 1995. [16.3] BRANDT,S.: Data Analysis. Statistical and Computational Methods for Scientists and Engineers. Springer-Verlag 1999. [16.4] CLAUSS,G.; FINZE, F.-R.; PARTZSCH, L.: Statistik fur Soziologen, Padagogen, Psychologen und Mediziner, Bd. 1. - Verlag H. Deutsch 1995. [16.5] FISZ, M.: Wahrscheinlichkeitsrechnung und mathematische Statistik. - Deutscher Verlag der Wissenschaften 1988. [16.6] GARDINER, C.W.: Handbook of Stochastic Methods. - Springer-Verlag 1997. [16.7] GNEDENKO, B.W.: Lehrbuch der Wahrscheinlichkeitstheorie. - Verlag H. Deutsch 1997. [16.8] HUBNER,G.: Stochastik. - Eine anwendungsorientierte Einfuhrung fur Informatiker, Ingenieure und Slathematiker. - Vieweg, 3rd ed. 2000. [l6.9] KOCH,K.-R.: Parameter Estimation and Hypothesis Testing in Linear Models. SpringerVerlag 1988. [16.10] RINNE:H.: Taschenbuch der Statistik. - Verlag H. Deutsch 1997. [16.11] SHAO,JUN:Mathematical Statistics. - Springer-Verlag 1999. ~

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1099 [16.12] SINIAI, J.G.: Probability Theory. - Springer-Verlag 1992. [16.13] SOBOL.I.M.: Die Monte-Carlo-Methode. - Verlag H. Deutsch 1991. [16.14] STORM. R.: Wahrscheinlichkeitsrechnung, mathematische Statistik und statistische Qualitatskontrolle. - Fachbuchverlag 1995. J.R.: An Introduction to Error Analysis. - University Science Books 1982, VCH [16 151 TAYLOR, 1988. G.R.: Mathematical Statistics. A Unified Introduction. - Springer-Verlag 1999. [16.16] TERRELL. [16.17] WEBER,H.: Einfuhrung in die Wahrscheinlichkeitsrechnung und Statistik fur Ingenieure. B. G. Teubner 1992. 17. Dynamical Systems a n d Chaos (17.11 ARROWSMITH, D.K.; PLACE, C.M.: An introduction to Dynamical Systems. - Cambridge University Press 1990. K.: Fractal Geometry. - John Wiley 1990. [17.2] FALCONER, [17.3] GUCKENHEIMER, J . ; HOLMES.P.: Non-Linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. - Springer-Verlag 1990. [17.4] HALE.J.; KOFAK:H.: Dynamics and Bifurcations. - Springer-Verlag 1991. B.: Introduction to the Modern Theory of Dynamical Systems. [17.5] KATOK,A , ; HASSELBLATT, - Cambridge University Press 1995. [17.6] KUZNETSOV, Yu.A.: Elements of Applied Bifurcation Theory. No. 112 in: Applied Mathematical Sciences. - Springer-Verlag 1995. [17.7] LEONOV,G.A.; REITMANN, V.; SMIRNOVA, V.B.: Non-Local Methods for Pendulum-Like Feedback Systems - B. G. Teubner 1987. 17.8 MANE, R.: Ergodic Theory and Differentiable Dynamics. - Springer-Verlag 1994. I.: Chaotic Behaviour of Deterministic Dissipative Systems. 17.9 MAREK.M.; SCHREIBER. Cambridge University Press 1991. [17.10] MEDVED',M.: Fundamentals of Dynamical Systems and Bifurcations Theory. -Adam Hilger 1992. [17.11] PERKO, L.: Differential Equations and Dynamical Systems. - Springer-Verlag 1991.

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[17.12] PESIN, YA., B.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics. - The University of Chicago Press 1997. 18. Programming, Optimization [18.1] BELLMAN, R.: Dynamic Programming. Princeton University Press 1957. [18.2] BERTSEKAS, D.P.: Nonlinear Programming. - Athena Scientific 1999. [18.3] CHVATAL, V: Linear Programming. - W.H. Freeman 1983. [18.4] DANTZIG, G.B.: Linear Programming and Extensions. - Princeton University Press 1998. R.B.: Numerical Methods for Unconstrained Optimization and [18.5] DENNIS,J . E . ; SCHNABEL, Nonlinear Equations. - SIAM 1996. [18.6] KUHN,H.W.: The Hungarian Method for the Assignment Problem. Naval. Res. Logist. Quart., 2 (1995). [18.7] MURTY,K.G.: Operations Research: Deterministic Optimization Models. - Prentice Hall 1995. [18.8] ROCKAFELLAR, R.T.: Convex Analysis. - Princeton University Press 1996. [18.9] SHERALI.H.D.; BAZARAA, M.S.; SHETTY,C.M.: Nonlinear Programming: Theory and Algorithms. -John Wiley 1993. M.N.: DANTZIG,G.B.: Linear Programming 1: Introduction. - Springer-Verlag [18.10] THAPA, 1997. [18.11] WOLSEY,L.A.: Integer Programming. - John Wiley 1998. -

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22. Bzblioqraphv

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19. Numerical Analysis 119.11 BRENNER.S.C.; SCOTT, L.R.: The Mathematical Theory of Finite Element Methods. Springer-Verlag 1994. 119.21 CHAPRA.S.C.: CANALE.R.P.: Numerical Methods for Engineers. - McGraw Hill 1989. 119.31 COLLATZ,L.: Numerical'Treatment of Differential Equations. - Springer-Verlag 1966. [19.4] DAVIS,P.J.; RABINOWITZ, P : Methods of Numerical Integration. -Academic Press 1984. [19.5] DE BOOR,C.: A Practical Guide to Splines. - Springer-Verlag 1978. [19.6] GOLUB,G.; ORTEGA,J.M.: Scientific Computing. - B. G. Teubner 1996. [19.7] GROSSMANN, CH.; ROOS,H.-G.: Numerik partieller Differentialgleichungen. - B. G. Teubner 1992. 119.81 HACKBUSCH, W.: Elliptic Differential Equations. - Springer-Verlag 1992. j19.91 HAMMERLIN, G.; HOFFMANN, K.-H.: Numerische Mathematik. - Springer-Verlag 1994. [19.10] HAIRER,E.; NORSETT,S.P.; WANNER,G.: Solving Ordinary Differential Equations. Vol. 1: Nonstiff Problems. Vol. 2: Stiff and Differential Problems. Vol. 3: Algebraic Problems. Springer-Verlag 1994. 119.111 HEITZINGER. I.: VALENTIN. G.: Praxis nichtlinearer Gleichunnen. - C. Hanser , W.:, TROCH. Verlag 1984. [19.12] KIELBASINSKI: A.; SCHWETLICK, H.: Numerische lineare Algebra. Eine computerorientierte Einfiihrung. - Verlag H. Deutsch 1988. [19.13] KNOTHE,K.; WESSELS,H.: Finite Elemente. Eine Einfiihrung fur Ingenieure. - SpringerVerlag 1992. [19.14] KRESS,R.: Numerical Analysis. - Springer-Verlag 1998. [19.15] LANCASTER, P.; SALKAUSKA, S.K.: Curve and Surface Fitting. - Academic Press 1986. [19.16] MAESS,G.: Vorlesungen iiber numerische Mathematik, Bd. 1, 2. - Akademie-Verlag 19841988. G.: Approximation von Funktionen und ihre numerische Behandlung. - Sprin[19.17] MEINARDUS, ger-Verlag 1964. [19.18] NURNBERGER, G.: Approximation by Spline Functions. - Springer-Verlag 1989. [19.19] PAO,Y.-C.: Engineering Analysis. - Springer-Verlag 1998. [19.20] QUARTERONI, A , ; VALLI,A.: Numerical Approximation of Partial Differential Equations. Springer-Verlag 1994. [19.21] REINSCH,CHR.: Smoothing by Spline Functions. - Numer. Math. 1967. [19.22] SCHWARZ, H.R.: Methode der finiten Elemente. - B. G. Teubner 1984. [19.23] SCHWARZ, H.R.: Numerische Mathematik. - B. G. Teubner 1986. [19.24] SCHWETLICK, H.; KRETZSCHMAR, H.: Numerische Verfahren fur Naturwissenschaftler und Ingenieure. Fachbuchverlag 1991. R.: Introduction to Numerical Analysis. - Springer-Verlag 1993. [19.25] STOER,J.; BULIRSCH, [19.26] STROUD,A.H.: Approximate Calculation of Multiple Integrals. - Prentice Hall 1971. W.: Numerische Mathematik fur Ingenieure und Physiker, Bd. 1, 2. - Springer[19.27] TORNIG, Verlag 1990. 19.28 UBERHUBER: C.: Numerical Computation 1, 2. - Springer-Verlag 1997. 19.29 WILLERS,F.A.: Methoden der praktischen Analysis. - Akademie-Verlag 1951. 19.30 ZURMUHL, R.: Praktische Mathematik fur Ingenieure und Physiker. - Springer-Verlag 1984. 1

L

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20. Computer Algebra Systems I20.11 BENKER.M.: Mathematik mit Mathcad. - Springer-Verlag 1996. [20.2] BURKHARDT, W . : Erste Schritte mit Mathematica. - Springer-Verlag, 2nd ed. 1996. [20.3] BURKHARDT, W.: Erste Schritte mit Maple. - Springer-Verlag, 2nd ed. 1996. [20.4] CHAR;GEDDES;GONNET;LEONG;MONAGAN: WATT: Maple V Library, Reference Manual. - Springer-Verlag 1991.

[20.5] DAVENPORT, J.H.; SIRET,Y.; TOURNIER, E.: Computer Algebra. -Academic Press 1993. H.: Maple V. - Verlag Markt & Technik 1993. [20.6] GLOGGENGIESSER, [20.7] GRABE,H.-G.; KOFLER.M.: Mathematica. Einfuhrung, Anwendung,Referenz. - AddisonWesley 1999. [20.8] JENKS. R.D.; SUTOR,R.S.: Axiom. Springer-Verlag 1992. [20.9] KOFLER.M.: Maple V, Release 4. -Addison Wesley (Deutschland) GmbH 1996. [20.10] MAEDER,R.: Programmierung in Mathematica. Addison Wesley, 2nd ed. 1991. [20.11] WOLFRAM,S.: The Mathematica Book. - Cambridge University Press 1999. ~

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21. Tables [21.1] ABRAMOWITZ, M.; STEGUN,I.A.: Pocketbook of Mathematical Functions. - \’erlag H. Deutsch 1984. [21 21 APELBLAT. A.: Tables of Integrals and Series. - Verlag H. Deutsch 1996. [21.3] BRYTSCHKOW. Ju.A.; MARITSCHEW, 0.1.;PRUDNIKOW. A.P.: Tabellen unbestimmter Integrale. Verlag H. Deutsch 1992. [21.4] EMDE,F.: Tafeln elementarer Funktionen. - B. G. Teubner 1969. I.M.: Summen-, Produkt- und Integraltafeln, Bd. 1, 2. Verlag [21.6] GRAD STEIN.^.^.; RYSHIK, H. Deutsch 1981. [21.6] GROBNER,W . ; HOFREITER,N.: Integraltafel, Teil 1: Unbestimmte Integrale, Teil 2: Bestimmte Integrale. Springer-Verlag, Teil l, 1975: Teil2, 1973. E.; EMDE,F.; LOSCH,F.: Tafeln hoherer Funktionen. - B. G. Teubner 1960. 21.7 JAHNKE. 21.8 MADELUNG, E.: Die mathematischen Hilfsmittel des Physikers. - Springer-Verlag 1964. F.: Formulas and Theorems for the Special Functions of [21.9] MAGNUS, W . ; OBERHETTINGER, mathematical Physics, 3rd ed. - Springer-Verlag 1966. [21.10] MULLER, H.P.; NEUMANN,P.; STORM, R.: Tafeln der mathematischen Statistik. C. Hanser Verlag 1979. [21.11] POLJANIN. A.D.; SAIZEW,V.F.: Sammlung gewohnlicher Differentialgleichungen. L’erlag H. Deutsch 1996. [21.12] SCHULER,M.: Acht- und neunstellige Tabellen zu den elliptischen Funktionen, dargestellt mittels des Jacobischen Parameters q. - Springer-Verlag 1955. [21.13] SCHULER,M.; GEBELEIN, H.: Acht- und neunstellige Tabellen zu den elliptischen Funktionen. - Springer-Verlag 1965. [21.14] SCHUTTE,K.: Index mathematischer Tafelwerke und Tabellen. Munchen 1966. ~

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22. Handbooks, Guide Books and Reference Books M.; STEGUN:I. A.: Pocketbook of Mathematical Functions. - Verlag [22.1] ABRAMOWITZ, H. Deutsch 1984. [22.2] BAULE,B.: Die Mathematik des Naturforschers und Ingenieurs, Bd. 1,2.- Verlag H. Deutsch 1979. [22.3] BERENDT,G.; WEIMAR, E.: Mathematik fur Physiker, Bd. 1, 2. - VCH 1990. 22.4 BOURBAKI, N.: The Elements of Mathematics, Vols. Iff. - Springer-Verlag 199Off. 22.5 BRONSTEIN, J.N.; SEMENDJAJEW, K.A.: Taschenbuch der Mathematik. - B. G. Teubner 1989,24.,Auflage; Verlag H. Deutsch 1989, [22.6] BRONSTEIN, J.N.; SEMENDJAJEW, K.A.: Taschenbuch der Mathematik, Erganzende Kapitel. - Verlag H. Deutsch 1991. [22.7] DRESZER, J.: Mathematik. - Handbuch fur Technik und Naturaissenschaft. - Verlag H. Deutsch 1975. [22.8] GELLERT,W . ; KASTNER,H.; NEUBER,S . (Eds.): Fachlexikon ABC Mathematik. - Verlag H. Deutsch 1978. [22.9] FICHTENHOLZ, G.M.: Differential- und Integralrechnung, Bd. 1,3. Verlag H. Deutsch 1994.

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22. Biblioqraphu

[22.11] GELLERT.W.: KUSTNER,H.; HELLWICH,M.; KASTNER,H. (Eds.): Kleine Enzyklopadie Mathematik. - Verlag Enzyklopadie, Leipzig 1965. [ 22. lo] JOOS.G.: RICHTER,E.W.: Hohere Mathematik fur den Praktiker. - Verlag H. Deutsch 1994. [ 22,121 MANGOLDT, H. v.; KNOPP,K.; HRG. F. LOSCH: Einfuhrung- in die hohere Mathematik, Bd. 1-4. - S. Hirzel-Verlag 1989. H.: MURPHY.G.M.: The Mathematics of Phvsics and Chemistrv. Vols. 1.2. [22.13] MARGENAU. Van Nostrand 1956; Verlag H. Deutsch 1965, 1967. L.: hlathematik fur Ingenieure, Bd. 1-3. - Verlag Vieweg 1994-1996. [22.14] PAPULA, P.; BROD,K.: Hohere mathematische Methoden fur Ingenieure und Physiker. [22.15] PLASCHKO, Springer-Verlag 1989. M.; VOIT, K.; KRAFT, R.: Mathematik fur Nichtmathematiker, Bd. 1, 2. [ 22,161 PRECHT, Oldenbourg-Verlag 1991. [ 22.171 ROTHE: R.: Hohere Mathematik fur Mathematiker, Physiker, Ingenieure, Teil I-IV. B. G. Teubner 1958-1964. E.: Grundlagen der theoretischen Physik, Bd. 1,4.- Deutscher Verlag der Wis[ 22.181 SCHMUTZER, senschaften 1991. W.I.: Lehrbuch der hoheren hlathematik, Bd. 1-5. - Verlag H. Deutsch 1994. [ 22,191 SMIRNOW, [ 22.201 ZEIDLER,E. (ED.): Teubner Taschenbuch der Mathematik. - B. G. Teubner, Teil 1, 1996; Teil 2. 1995. ~

23. Encyclopedias [23.1] EISENREICH, G.. SUBE,R.: Worterbuch Mathematik, Vols. 1, 2 (English, German, French, Russian) - Verlag Technik 1982; Verlag H. Deutsch 1982. [23.2] Encyclopaedia Britannica. [23.3] Encyclopaedia of Mathematical Sciences (Transl. from the Russian). Springer-Verlag 1990. [23.4] Encyclopaedia of Mathematics (Revised Transl. from the Russian). - Kluwer 1987-1993. ~

INDEX

Index Terms

Links

A Abel group

299

integral equation

588

theorem

414

abscissa plane coordinates

189

space coordinates

208

absolute term

272

absolutely continuous

636

convergent

453

integrable

453

absorbing set

797

absorption law Boolean algebra

340

propositional logic

287

sets

293

account number system uniform accumulation factor

331 331 22

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

accumulation point

604

accuracy, measure of

787

addition complex numbers computer calculation polynomials rational numbers

36 937 11 1

addition theorem area functions

93

hyperbolic functions

89

inverse trigonometric functions

86

trigonometric functions

79

additivity, σ-additivity

633

adjacency

346

matrix

81

348

adjacent side

130

adjoint

259

admittance matrix

353

a.e. (almost everywhere)

634

aggregation operator

368

algebra

251

286

340

745

Boolean finite

341

classical structures

298

commutative

612

factor algebra

338

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

algebra (Con.) free

339

linear

251

nórmed

612

Ω algebra

338

Ω-subalgebra

338

σ-algebra

633

Borelian

634

switch algebra

340

term algebra

339

universal algebra

338

344

algorithm Aitken–Neville

916

Dantzig

355

Euclidean

3

Ford-Fulkerson

357

Gauss

276

Gauss algorithm

888

graph theory

346

Kruskal

353

maximum flow

357

QR

283

Rayleigh-Ritz

283

Remes

922

Romberg method

898

theorem for the Euclidean algorithm

322

14

321

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

alignment chart

123

nomogram

125

α -cut

124

362

strong

362

α-level set

362

strong

362

α-limit set

797

alternating point

920

theorem

920

alternating tensor

265

803

810

altitude cone

156

cylinder

154

polyhedron

151

triangle

132

amortization calculus

23

amplitude function sine function spectrum

701 75 725

analysis functional

594

harmonic

914

multi-scale analysis

741

numerical

881

924

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

analysis (Cont.) vector

640

analytical geometry plane

189

space

207

angle

128

acute

129

between vectors

189

central

130

chord and tangent

139

circumference

139

convex

129

directional

145

escribed circle

140

Eulerian

211

exterior

139

full

129

geodesy

145

inclination

377

interior

139

line and plane

219

lines, plane

128

lines, space

219

measure in degrees

130

names

128

notion

128

140

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

angle (Cont.) nutation

211

obtuse

129

perigon

130

plane

151

notion

128

plane curves

228

planes, space

216

precession

211

radian measure

130

reduction

174

right

129

round

129

secant and tangent

140

slope

377

tangent

227

solid

151

space curves

246

straight

129

tilt

143

angles adjacent

129

alternate

129

at parallels

129

complementary

129

corresponding

129

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

angles (Cont.) exterior–interior

129

opposite

129

sum plane triangle

132

spherical triangle

163

supplementary

129

vertex

129

angular coefficient, curve third degree coefficient, plane frequency annuity

66 194 82 23

calculation

25

payment

25

perpetual

25

annulator

622

annulus

140

area

140

Anosov diffeomorphism

827

anticommutativity, vector product

183

antiderivative

425

antikink soliton

548

antilog

24

10

antisoliton

547

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Apollonius circle

681

theorem

199

apothem

132

applicate

208

approximate equation

494

approximate formula, empirical curve

106

approximation

914

asymptotic, polynomial part

15

best, Hilbert space

615

bicubic

932

Chebyshev

920

δ function

715

formulas series expansion in mean

416 401

Liouville theorem

4

numbers

4

partial differential equations

909

problem

614

solution by extreme value

916

401

successive Banach space

619

differential equations, ordinary

494

integral equation

565

uniform

920

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

approximation (Cont.) using given functions

906

Weierstrass theorem

605

ellipse

200

graph

346

arc

chain

355

hyperbola

203

intersection

149

length circular segment

140

line integral, first type

462

parabola

204

plane curve

140

447

space curve

246

462

sequences

355

Archimedean spiral

103

area annulus sector

140 140

circle

139

circular sector

140

circular segment

140

cosine

91

cotangent

92

curved surface

480

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

area (Cont.) curvilinear bounded

446

curvilinear sector

446

double integral

472

ellipse

199

formula, Heron’s

143

function

91

hyperbola

202

parabola

204

parallelogram

134

parallelogram with vectors

189

planar figures

446

polyeder with vectors

189

polygon

193

rectangle, square

135

rhombus

135

similar plane figures

133

sine

91

subset

633

surface patch

247

tangent

92

triangle

193

plane

141

143

spherical

163

167

argument function of one variable

47

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

argument (Cont.) function of several variables arithmetic sequence

117 1 18

Arnold tongue

842

arrangement

743

with repetition

744

without repetition

744

744

arrow diagram

294

function

294

article number, European

331

ASCII (American Standard Code for Information Interchange) assignment problem

933 858

associative law Boolean algebra

340

matrices

254

propositional logic

287

sets

293

tensors

264

associativity, product of vectors

184

astroid

102

472

curve

231

234

definition

234

asymptote

This page has been reformatted by Knovel to provide easier navigation.

Index Terms attractor

Links 798

chaotic

826

fractal

826

Hénon

824

hyperbolic

826

Lorenz

825

Strange

826

autocorrelation function

816

axial field

642

826

axioms algebra

612

closed set

604

metric spaces

602

normed space

609

open set

604

ordered vector space

599

pseudonorm

622

scalar product

613

vector space

594

axis abscissae plane coordinates

189

space coordinates

208

ordinates plane coordinates

189

space coordinates

208

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

axis (Cont.) parabola

203

azimuth

159

azimuthal equation

541

B backward substitution

888

Baire, second Baire category

812

Bairstow method

886

ball, metric space

603

Banach Fixed-point theorem

629

space

610

example

610

series

610

theorem, continuity, inverse operator band structure, coefficient matrix

619 892

barrel

157

circular

157

parabolic

157

base

7 power

7

vector

186

reciprocal vector space

185

186

315

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

basic formulas plane trigonometry

141

problems plane trigonometry

143

spherical trigonometry

167

theorems propositional logic basis

287 597

contravariant

266

covariant

266

inverse

849

vector contravariant

266

covariant

266

vector space Bayes theorem

597 748

B-B representation curve

932

surface

932

Berge theorem

354

Bernoulli inequality

30

numbers

410

shift

827

Bernoulli–1’Hospital rule

54

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Bernstein polynomial

Links 932

Bessel differential equation

507

differential equation, linear, zero order

720

function

507

imaginary variables

507

modified

507

inequality beta function biangle, spherical

616 1050 161

bifurcation Bogdanov-Takens

832

codimension

827

cusp

831

flip bifurcation

834

global

827

homoclinic

838

Hopf bifurcation

829

generalized

832

local

827

mappings, subcritical saddle node

834

pitchfork

834

supercritical

836

832

saddle node

829

transcritical

829

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

binary number

814

binary system

934

binomial

934

69

coefficient

13

distribution

753

formula

12

linear

69

quadratic

69

theorem

12

binormal, space curve

239

birth process

767

bisection method

283

bisector

142

triangle bit

241

132 933

reversing the order

927

block

152

body of revolution, lateral surface

447

Bolzano theorem one variable several variables Bolzano–Weierstrass property

59 123 626

Boolean algebra finite expression

340

745

341 342

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Boolean (Cont.) function n-ary variable

287

342

342 342

Borel set

634

σ-algebra

634

bound function

50

sequence

402

boundary collocation

910

condition

550

uncertain for some variables (fuzzy)

367

boundary value conditions

485

512

problem

485

512

Hilbert

591

Hilbert, homogeneous

591

homogeneous

512

inhomogeneous

512

linear

512

bounded set (order-bounded)

905

600

boundedness of a function one variable several variables

60 123

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Bravais lattice

310

break of symmetry, bifurcation

832

Breit–Wigner curve

729

Brouwer fixed-point theorem

631

business mathematics byte

21 933

C calculation complex numbers determinants

36 260

numerical accuracy

938

basic operations

937

tensors

263

calculus differentiation

377

errors

785

integral

425

observations, measurement error

785

propositional

286

variations

550

canonical form, circle-mapping

841

canonical system

517

Cantor function

842

Cantor set

820

842

821

825

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

cap, spherical

157

capacity, edge

356

Caratheodory condition

629

Cardano formula

41

cardinal number

291

cardinality, set

298

cardioid

298

98

carrier function

573

Carson transformation

705

Cartesian coordinates plane

189

space

208

folium

94

cartography scale

144

cascade, period doubling

837

Cassinian curve category, second Baire category

840

98 812

catenary curve

88

catenoid

88

105

553

Cauchy integral

590

integral formula

687

integral formulas, application

692

method, differential equations of higher order

500

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Cauchy (Cont.) principal value improper integral

455

principal value, improper integral

452

principle

606

problem

516

sequence

605

theorem

388

Cauchy–Riemann differential equations, partial

670

Cayley table

300

theorem

302

353

center circle

197

curvature

230

spherical

175

center manifold theorem differential equations

828

mappings

833

center of area method

372

center of gravity

213

arbitrary planar figure

451

arc segment

450

closed curve

450

double integral

472

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

center of gravity (Cont.) line integral, first type

462

method

371

generalized

372

parametrized

372

trapezoid

451

triangle

132

triple integral

478

center of mass

191

213

central angle

130

curves

205

field

641

limit theorem, Lindeberg–Levy

763

surface

220

centroid, points having masses

191

chain

297

directed, elementary

355

graph

355

elementary Markov

355 764

stationary

764

time-homogeneous

764

rule

349

641 composite function

stochastic

380 763

764

This page has been reformatted by Knovel to provide easier navigation.

Index Terms chaos

Links 795

attractor, strange

826

from torus to chaos

839

one-dimensional mapping

827

routes to chaos

827

through intermittence

840

transition to chaos

837

839

character group element

304

representation of groups

305

characteristic characteristic strip

10 517

Chebyshev approximation

920

continuous

920

discrete

923

formula

87

inequality

31

polynomials

921

theorem

434

Chinese remainder theorem

326

2

760

2

773

x distribution x test

774

Cholesky decomposition

890

method

278

890

This page has been reformatted by Knovel to provide easier navigation.

Index Terms chord

Links 140

theorem

138

circle Apollonius

680

area

138

center

197

chord

138

circumference

138

convergence

688

curvature

230

plane curve

228

dangerous

149

definition

138

197

equation Cartesian coordinates

197

polar coordinates

197

great circle

158

173

intersection

158

mapping

841

parametric representation

197

periphery

138

plane

138

197

radius

138

197

small circle

158

175

tangent

138

198

842

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

circuit directed, graph

355

Euler

350

Hamilton

351

circuit integral

466

being zero

468

circular field

644

point

249

sector

140

segment

140

circumcircle quadrangle

136

triangle

132

142

circumscribing quadrangle

136

triangle

132

cissoid

95

Clairaut differential equation, ordinary

491

differential equation, partial

518

class defined by identities

339

equivalence class

297

midpoint

771

statistics

770

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Clebsch–Gordan coefficient

306

series

306

theorem

306

closure closed linear

614

linear

595

set

605

transitive

295

clothoid

105

code

329

ASCII

933

ASCII (American Standard Code for Information Interchange)

933

public key

329

RSA (Rivest, Shamir, Adleman)

329

codimension

827

coding

329

coefficient Clebsch–Gordan comparison

333

11 306 16

Fourier

418

leading

38

matrix, extended

889

metric

185

vector decomposition

182

186

This page has been reformatted by Knovel to provide easier navigation.

Index Terms collinearity, vectors

Links 184

collocation boundary

910

domain

910

method

574

906

points

906

910

column pivoting

889

column sum criterion

892

combination

743

with repetition

743

without repetition

743

conibinatorics commensurability

910

743 4

commutative law Boolean algebra

340

matrices

254

propositional logic

287

sets

293

vectors

255

commutativity, scalar product

183

commutator

317

comparable function

556

complement

292

algebraic

259

fuzzy

366

fuzzy set

363

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

complement (Cont.) orthogonal

614

sets

292

Sugeno complement

366

Yager complement

366

complementary angles formulas

78

completeness relation

616

completion, metric space

608

complex analysis

669

complex function

47

pole

622

671

complex number plane complex mapping complex-valued function

34 34 682 47

complexification

599

composition

369

max-average

369

maw-(t-norm)

369

compound interest

22

calculation

22

computation of adjustment

914

computer basic operations

937

error of the method

939

error types

936

938

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

computer (Cont.) internal number representation

935

internal symbol representation

933

numerical accuracy

938

numerical problems in calculation

936

use of computers

933

computer algebra systems applications

975

basic structure elements

952

differential and integral calculus

989

elements of linear algebra

984

equations and systems of equations

981

functions

953

graphics

994

infix form

953

list

953

manipulation of algebraic expressions

976

numbers

952

object

952

operator

953

prefix form

953

programming

952

purpose

950

set

953

suffix form

953

terms

953

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

computer algebra systems (Cont.) type

952

variable

953

concave, curve

228

conchoid Nicomedes

96

general

96

of the circle

96

of the line

96

conclusion

370

concurrent expressions

343

condition boundary value, ordinary differential equation

485

Caratheodory

629

Cauchy (convergence)

122

Dirichlet (convergence)

420

initial value, ordinary differential equation

485

Kuhn-Tucker

860

Lipschitz, higher-order differential equation

496

Lipschitz, ordinary differential equation

486

number

891

regularity condition

873

cone

156

central surface

224

circular

156

221

611

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

cone (Cont.) generating

600

imaginary

224

normal

611

regular

611

solid

611

truncated

156

vector space

597

confidence interval

779

regression coefficient

779

variance

776

limit for the mean

775

prescription

790

probability

774

region

779

congruence algebraic

325

corners

151

linear

326

method

781

plane figures

133

polynomial congruence

328

quadratic

327

relation

338

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

congruence (Cont.) kernel

339

simultaneous linear

326

system simultaneous linear

326

theorems

133

congruent directly

132

indirectly

133

mapping

132

133

conic section

156

205

singular

207

conjugate complex number conjunction elementary

207

36 286 343

consistency integration of differential equation

904

order p

904

constant Euler in polynomial

458 60

propositional calculus

286

term

272

constants (often used), table continuation, analytic continued fractions

1007 689 3

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

continuity composite functions

59

elementary function

58

from belon

633

function one variable several variables Hölder

57 122 589

continuous, absolutely

636

continuum

298

contour integral vector field

660

contracting principle

606

contraction

264

tensor

268

contradiction, Boolean function

342

control digit

330

convergence absolute complex terms

407 688

alternating series test of Leibniz

408

Banach space

610

circle of convergence

688

condition of Cauchy

52

conditional complex terms

414

407 688

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

convergence (Cont.) in mean

420

infinite series, complex terms

688

integration of differential equation

904

non-uniform

412

order p

904

sequence of numbers

403

complex terms series complex terms

687 404

406

687

uniform

412

uniformly, function sequences

604

weak

627

Weierstrass criterion

413

convergence criterion comparison criterion

405

D’Alembert’s ratio test

405

integral test of Cauchy

406

root test of Cauchy

406

convergence theorem

404

measurable function

636

conversion, number systems

934

convex, curve

228

convolution Fourier transformation

727

Laplace transformation

711

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

convolution (Cont.) one-sided

727

two-sided

727

z-transformation

734

coordinate inversion

269

coordinate line

209

266

coordinate surface

209

266

coordinate system double logarithmic

115

Gauss–Krueger

143

left-hand

207

orientation

207

orthogonal

180

orthonormal

180

plane

189

right-hand

207

semilogarithmic

115

Soldner

143

spatial

207

transformation

262

coordinate transformation

210

267

645

equation central curves second order

205

quadratic curve (parabolic)

206

coordinates affine

185

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

coordinates (Cont.) axis

189

barycentric

914

Cartesian

186

plane

189

space

208

Cartesian, transformation

191

contravariant

187

268

covariant

187

268

curvilinear

190

244

three dimensional cylindrical vector field

266

209 209 645

Descartes

189

equation, space curve

241

Gauss

244

Gauss–Krüger

160

geodetic

143

geographical

160

mixed

267

point

189

polar, plane

190

polar, spherical

209

representation with scalar product

187

Soldner

160

spherical

209

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

coordinates (Cont.) vector field

645

transformation

190

triangle coordinates

914

vector

182

coprime

5

corner convex

151

figure

150

northwest corner rule

856

symmetric

151

trihedral

151

correction form

893

corrector

903

correlation

777

analysis

777

coefficient

778

empirical

778

cosecant geometric definition

130

hyperbolic

87

trigonometric

76

coset left

301

right

301

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

cosine geometric definition hyperbolic geometric definition

130 87 131

law

142

law for sides

164

rule, spherical triangle

164

trigonometric

75

cotangent geometric definition

130

hyperbolic

87

trigonometric

76

Coulomb field, point-like charge

644

counterpoint

158

course angle

159

covariance, two-dimensional distribution

778

covering transformation

299

covering, open

798

Cramer rule

275

credit

666

22

criterion convergence sequence of numbers

403

series

404

series with positive terms

405

uniform, Weierstrass

413

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

criterion (Cont.) divisibility

321

subspace

315

cross product

183

cryptanalysis, classical, methods

335

Kasiski-Friedman test

335

statistical analysis

335

cryptography, conventional methods linear substitution ciphers substitution

333 334 333

monoalphabetic

333

monographic

333

polyalphabetic

333

polygraphic

333

transposition cryptology

333 332

classical Hill cypher method

334

matrix substitution method

334

Vigenere cypher method

334

cryptosystem

332

DES algorithm

337

Diffie–Hellman key exchange

336

encryption context free

332

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

cryptology (Cont.) context sensitive

332

IDEA algorithm

338

mathematical foundation

332

one-time pad

336

one-way function

337

public key methods

336

RSA method

337

security of cryptosystems

333

subject

332

crystal class

311

crystal system

311

crystallography lattice

310

symmetry group

310

cube

152

curl

666 density

667

field pure

666

zero-divergence field

666

line

653

curvature center

230

circle

230

curves on a surface

247

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

curvature (Cont.) Gauss surface

249

mean, surface

249

minimal total curvature

929

plane curve

228

radius

230

curve on a surface

247

principal

247

space curve

240

surface

247

constant curvature total

249

249 243

curve algebraic n-th order

93 234

arc cosine

84

arc cotangent

84

arc sine

84

arc tangent

84

Archimedean spiral

194

103

area cosine

91

area cotangent

92

area sine

91

area tangent

92

astroid

102

asymptote

231

234

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

curve (Cont.) asymptotic point

232

B–B representation

932

cardioid

98

Cartesian folium

94

Cassinian

98

catenary

105

cissoid

95

clothoid

105

concave

228

conchoid of Nicomedes

96

convex

228

corner point

232

cosecant

76

cosine

75

cotangent

76

curvature

228

cuspidal point

232

cycloid

100

damped oscillation

83

directing

154

double point

232

empirical

106

envelope

236

epicycloid

101

epitrochoid

102

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

curve (Cont.) equation complex form

699

plane

194

second degree

205

second order

205

space

214

error curve

72

evolute

236

evolvent

236

of the circle

225

237

104

exponential

71

fourth order

96

Gauss error curve

757

general discussion

235

hyperbolic cosine

88

hyperbolic cotangent

89

hyperbolic sine

88

hyperbolic tangent

88

hyperbolic type

68

hyperbolice spiral

104

hypocycloid

102

hypotrochoid

102

imaginary

194

inflection point

231

involute

237

70

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

curve (Cont.) isolated point

232

Koch curve

821

lemniscate

99

length, line integral, first type logarithmic logarithmic spiral loops

462 71 104 1004

Lorentz curve

94

multiple point

233

normal, plane

226

n-th degree

63

194

n-th order

63

194

parabolic type

63

Pascal limaçon

96

plane

225

angle

228

direction

225

vertex

232

quadratic

205

radius of curvature

228

representation with splines

928

secant

76

second degree

205

second order

205

semicubic parabola

229

93

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

curve (Cont.) sine

75

space

238

spherical

158

spiral

103

strophoide

641

95

tacnode

232

tangent

76

plane

226

terminal point

232

third degree

65

third order

93

tractrix

106

transcendent

194

trochoid

100

witch of Agnesi

172

94

curves family of, envelope

237

second order central curves

205

parabolic curves

206

polar equation

207

spherical, intersection point cut

178 292

Dedekind’s

297

fuzzy sets

362

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

cut (Cont.) set

292

cutting plane method

874

cycle

349

chain

355

limit

831

cycloid

100

basis

100

common

100

congruent

100

curtate

100

prolate

100

cylinder

154

circular

155

elliptic

223

hollow

155

hyperbolic

223

223

invariant signs elliptic

224

hyperbolic

224

parabolic

224

parabolic

223

cylindrical coordinates

209

cylindrical function

507

cylindrical surface

154

213

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

D d’Alembert formula

534

damping parameter

894

damping, oscillations

83

Darboux vector

243

data type

338

De Morgan law

293

rule

287 Boolean algebra

death process debt

341 767 23

decay, radioactive

766

decimal number

934

normalized

936

representation

934

system decoding

934 329

decomposition orthogonal partial fractions vectors

614 15 182

decomposition theorem differential equation, higher order

499

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Dedekind cut

297

defect

906

vector space

316

definite, positive

890

defuzzification

371

degeneracy of states

539

910

degree curve, second degree

205

curve, n-th degree

194

homogeneity

120

in-degree

346

matrix

353

measure in degrees

130

out-degree

346

Delambre equations

166

δ distribution

638

δ function

634

application

715

approximation

715

Dirac

712

δ functional

621

density function

750

multidimensional dependence, linear

638

752 272

315

deposit in the course of the year

22

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

deposit (Cont.) regular

22

single

22

depreciation

26

arithmetically declining

26

digital

27

geometrically declining

27

straight-line

26

derivative complex function

669

constant

378

directional

647

scalar field

647

vector field

648

distribution

639

exterior

380

fraction

380

Fréchet

630

function composite

380

elementary

378

implicit

381

inverse

381

parametric representation

382

several variable

390

generalized

638

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

derivative (Cont.) higher order

383

inverse function

385

parametric representation

385

interior

380

left-hand

378

logarithmic

380

mixed

393

one variable

377

partial

390

product

379

quotient

380

right-hand

378

scalar multiple

378

Sobolev sense

638

space

647

sum

378

table

379

vector function

640

volume differentiation

648

Derive (computer algebra system) Descartes rule

950 45

descent

895

descent method

866

determinant

259

differentiation

393

261

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

determinant (Cont.) evaluation

261

functional

121

Jacobian

121

multiplication

260

reflection

260

rules of calculation

260

Wronskian

498

zero value

260

800

determination of extrema absolute extremum

390

implicit function

390

deviation, standard deviation

751

devil’s staircase

842

diagonal matrix

252

diagonal method, Maxwell

682

diagonal strategy

889

diameter parabola

204

circle

139

conjugate ellipse

199

hyperbola

202

ellipse

199

hyperbola

202

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

diffeomorphism

796

Anosov

827

orientation-preserving

842

difference bounded

364

finite expression

905

sets

293 symmetric

293

significant

777

symmetric sets

293

z-transformation

733

difference equation

731

boundary value

738

linear

736

partial differential equations

908

second-order

737

difference method

905

partial differential equations

738

908

difference quotient

895

difference schema

18

differentiability complex function

669

function of one variable

377

function of several variables

392

with respect to the initial conditions

795

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differntiable continuously

630

Fréchet

630

differential arc plane

226

surface

245

complete

392

first-order

378

higher order

392

integrability

466

notion

391

partial

392

principal properties

392

quotient (see also derivative)

377

second-order

394

total

392 n-th order

394

second-order

394

differential calculus fundamental theorem differential equation

393

377 386 485

boundary value problem

512

eigenfunction

513

eigenvalue

513

eigenvalues

539

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differential equation (Cont.) flow

795

Fourier transformation

729

Laplace transformation

719

numerical solution

495

operational notation

500

order

485

orthogonality

513

self-adjoint

512

singular solution

487

stiff

905

topological equivalent

808

Weber

543

differential equation, higher order

495

constant coefficients

498

decomposition theorem

499

Euler, n-th order

502

fundamental system

498

n-th order

498

lowering the order

497

normal form

503

quadrature

499

superposition principle

499

system

495

system of linear

503

system of solutions

496

499

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differential equation, higher order (Cont.) variation of constants

499

differential equation, linear autonomous, on the torus

801

first-order

799

fundamental theorem

799

homogeneous

799

homogeneous systems

504

inhomogeneous

799

inhomogeneous systems

504

matrix-differential equation

799

non-autonomous, on the torus

842

periodic coefficients

801

second order

505

Bessel

507

defining equation

506

Hermite

512

hypergeometric

511

Laguerre

511

Legendre

509

method of unknown coefficients

505

differential equation, ordinary

485

approximate integration

901

autonomous

798

Bernoulli

489

boundary value conditions

485

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differential equation, ordinary (Cont.) boundary value problem

485

center

493

central point

493

Clairaut

491

direction field

486

element, singular

491

exact

487

existence theorem

486

explicit

485

first-order

486

approximation methods

494

important solution methods

487

flow

795

fraction of linear functions

492

general solution

485

general, n-th order

485

graphical solution

495

homogeneous

487

implicit

485

initial value conditions

485

initial value problem

485

integral

485

integral curve

486

integral, singular

491

integrating factor

488

490

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differential equation, ordinary (Cont.) Lagrange

490

linear

729

constant coefficients

719

first-order

488

variable coefficient

720

linear, planar

795

multiplier

488

non-autonomous

798

notion

485

particular solution

485

point, singular

491

radicals

491

ratio of arbitrary functions

494

Riccati

489

separation of variables

487

series expansion

494

singular point

492

solution

485

successive approximation

494

van der Pol

831

differential equation, partial

515

approximate integration

908

Cauchy–Riemann

528

characteristic system

516

Clairaut

518

809

670

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differential equation, partial (Cont.) completely integrable

519

eigenfunction

539

electric circuit

529

elliptic type

520

constant coefficients

552

Euler, calculus of variations

552

field theory

667

first-order

515

canonical system

517

characteristic strip

517

characteristic system

515

linear

515

non-linear

517

non-linear, complete integral

517

total differentials

519

two variables

518

Fourier transformation

730

Hamilton

799

839

heat conduction equation one dimensional

529

one-dimensional

527

three-dimensional

535

Hélmholtz

538

hyperbolic type

520

integral surface

516

522

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differential equation, partial (Cont.) Laplace

667

Laplace transformation

721

linear

515

non-linear

544

Schrödinger

547

normal system

517

notion

485

parabolic type

520

constant coefficients

522

Poisson

536

quasilinear

515

reduced

828

reduced, normal form

828

second-order

520

constant coefficients

665

668

522

separation of variables

538

ultrahyperbolic type, constant coefficients

522

differential operation review

656

vector components

657

differential operator divergence

651

656

gradient

648

649

Laplace

655

656

nabla

654

656

non-linear

630

656

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differential operator (Cont.) relations

656

rotation

652

rules of calculations

656

space

647

vector gradient

650

differential transformation, affine

672

differentiation

377

complex function

669

composite function

395

656

654

function elementary

378

implicit

381

inverse

381

one variable

377

parametric representation

382

several variables

390

graphical

382

higher order inverse function

385

parametric representation

385

implicit function

396

logarithmic

380

several variables

395

under the integration sign

457

volume

648

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

differentiation rules basic rules

378

derivative of higher order

383

function one variable

378

several variables

378

table

384

vector function

640

vectors

640

diffusion coefficient

536

diffusion equation

549

three-dimesnsional

393

535

digon, spherical

161

dihedral angle

150

dihedral group

299

dimension

820

capacity

821

correlation

823

defined by invariant measures

822

Douady–Oesterlé

824

formula

315

generalized

823

Hausdorff

820

information

822

lower pointwise

822

Lyapunov

823

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

dimension (Cont.) measure

822

metric

820

Renyi

823

upper pointwise

822

vector space

315

597

Dirac distribution

638

measure

634

theorem

351

directing curve

814

154

direction cosine, space

210

plane curve

225

space

210

space curve

238

directional angle

145

derivative

649

directrix ellipse

198

hyperbola

201

parabola

203

Dirichlet condition

420

problem

526

668

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

discontinuity

57

function

57

removable

58

discount

21

discretization error global

904

local

904

discretization step interval

895

discriminant

520

disjoint

292

disjunction

286

elementary

343

dispersion

751

dissolving torus

839

distance

449

Hamming distance

602

line–point

195

metric space

602

planes

217

parallel

217

point–line, space

218

point–plane, space

215

spherical

158

two lines, space

218

two points

191

space

212

This page has been reformatted by Knovel to provide easier navigation.

Index Terms distribution

Links 637

binomial

753

χ

760

2

continuous

756

derivative

638

Dirac

638

discrete

752

exponential

758

Fisher

761

frequency

770

function

749

function, continuous

750

hypergeometric

753

logarithmic normal

757

yognormal distribution

757

measurement error density

786

normal

756

Poisson

755

regular

638

standard normal

757

Student

761

t distribution

761

theory

712

Weibull

759

distribution problem

638

715

754

858

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

distributive law Boolean algebra

340

matrices

254

propositional logic

287

ring, field

313

sets

293

tensors

264

distributivity, product of vectors

184

divergence

665

central field

652

definition

651

different coordinates

651

improper

403

proper

403

remark

648

sequence of numbers

403

series

407

theorem

663

vector components

657

vector field

651

divisibility

318

criteria

320

321

division complex numbers

37

computer calculation

937

external

192

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

division (Cont.) golden section

192

harmonic

192

in extreme and mean ratio

192

internal

192

line segment, plane

192

polynomial rational numbers segment, space divisor

14 1 212 318

greatest common (g.c.d.) integer numbers

321

linear combination

322

polynomials positive

14 320

dodecahedron

154

domain

118

closed

118

convergence, function series

412

doubly-connected

118

function

47

image, set

48

multiply-connected

118

non-connected

118

of attraction

798

of individuals, predicate calculus

289

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

domain (Cont.) open

118

operator

598

set

48

simply-connected

118

three or multidimensional

118

two-dimensional

118

values, set

685

48

dot product

183

double integral

469

application

472

notion

469

double line

205

dual

341

duality linear programming

854

non-linear optimization

861

theorem, strong

862

duality principle, Boolean algebra

341

dualizing

341

Duhamel formula

720

E eccentricity, numerical curve second order

207

ellipse

198

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

eccentricity, numerical (Cont.) hyperbola

200

parabola

203

edge angle

150

figure

150

graph

346

length

348

valuation

348

multiple

346

sequence

349

cycle

349

directed circuit

349

isolated edge

349

open

349

path

349

effective rate

22

eigenfunction

539

differential equation

513

integral equation

563

normalized

514

eigenvalue

278

differential equation

513

integral equation

563

normalized

514

568

539

568

This page has been reformatted by Knovel to provide easier navigation.

Index Terms eigenvalue

Links 278

differential equation

513

integral equation

563

operator

620

539

568

eigenvalue problem general

278

matrices

278

special

278

eigenvector

264

Einstein’s summation convention

262

element

290

finite

911

generic

812

linearly independent

596

neutral

298

positive

599

element of area, plane, table

472

element of surface, curved, table

480

element of volume, table

477

278

620

elementary cell crystal lattice

310

non-primitive

310

primitive

310

elementary formula, predicate logic

289

elementary surface, parametric form

480

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

elementary volume arbitrary coordinates

477

Cartesian coordinates

474

cylindrical coordinates

475

spherical coordinates

476

elimination method, Gauss

276

elimination step

888

ellipse

198

arc

200

area

199

diameter

198

equation

198

focal properties

198

focus

198

perimeter

200

radius of curvature

199

semifocal chord

198

tangent

199

transformation

205

vertex

198

ellipsoid

220

central surface

224

cigar form

220

imaginary

224

lens form

220

of revolution

220

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

ellipsoid (Cont.) surface second order

224

embedding, canonical

624

encoding method, RSA

329

encyphering

332

endomorphism, linear operators

599

endpoint

346

energy particle

537

spectrum

538

system

536

zero-point translational energy

539

zero-point vibration energy

544

entropy generalized

823

metric

817

topological

817

827

envelope

236

237

epicycloid

101

curtate

102

prolate

102

epitrochoid

102

equality asymptotic complex numbers matrices

417 34 254

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

equality relation identity equation

10 10

algebraic

38

43

278

492

40

63

plane

194

225

second degree

205

second order

205

characteristic cubic curve

defining degree Diophantine linear ellipse

506 38 323 323 198

exponential, solution

45

first degree

39

fourth degree

42

homogeneous

627

hyperbola

200

hyperbolic functions, solution inhomogeneous irrational Korteweg de Vries

46 627 39 546

line plane

194

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

equation (Cont.) space

217

linear

39

logarithmic, solution

46

logistic

796

835

non-linear fixed point

881

numerical solution

881

normal form

38

n-th degree

43

operator equation

627

parabola

203

Parseval

420

pendulum

839

plane

214

general

214

Hessian normal form

214

intercept form

215

plane curve

514

616

194

polynomial numerical solution

884

quadratic

39

root, definition

38

62

Schrödinger linear

536

non-linear

545

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

equation (Cont.) second degree

39

sine–Gordon

547

solution, general space curve vector form sphere

38 214

238

238 243

surface normal form

220

second order

223

space

213

system of

38

term algebra

339

third degree

40

transcendent

38

trigonometric, solution

46

vector

243

187

equation system linear overdetermined

271 890

non-linear

887

numerical method, iteration

887

direct method

887

893

887

overdetermined

277

row echelon form

887

underdetermined

887

887

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

equations Delambre

166

L’Huilier

167

Mollweide

142

Neper

167

equilibrium point hyperbolic equivalence

795 802 286

class

297

logic

287

proof

5

relation Eratosthenes sieve

296 318

error absolute

789

absolute maximum error

790

apparent

788

average

787

computer calculation

938

defined

790

density function

786

discretization

939

equation

890

estimation, iteration method

894

input error

938

least mean squares

419

936

789

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

error (Cont.) mean square

787 788

normally distributed

786

percentage

790

probable

787

relations between error types

787

relative

790

relative maximum error

790

round-off

939

single measurement

788

standard

788

true

788

truncation error

939

type 1 error rate

751

type, measurement errors

785

error analysis

789

789

936

794

error calculus direct problem

938

inverse problem

938

error curve

72

error estimation, mean value theorem

387

error function

459

Gauss

757

error integral, Gauss

459

error orthogonality

906

This page has been reformatted by Knovel to provide easier navigation.

Index Terms error propagation law

Links 792 793

error types, computer calculation

938

ess, sup (essential supremum)

637

estimate

769

Euclidean algorithm

3

norm

316

vector norm

258

vector space

316

14

Euler angle

211

broken line method

901

circuit

350

constant

458

differential equation n-th order

502

variational calculus

552

formula

247

function

329

graph

350

integral of the second kind

457

integral, first kind

1050

integral, second kind

1050

numbers

410

polygonal method

901

418

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Euler (Cont.) relation complex numbers

35 696

theorem

154

trail

350 open

350

Euler–Hierholzer theorem

350

event

745

certain

745

complete system of

746

elementary

745

impossible

745

independent

748

random

745

set of events

745

simple

745

evolute

748

236

evolution equation

545

function

545

evolvent of circle

236

237

104

excess, spherical triangle

163

exchange theorem

393

exchange, cyclic, sides and angles

141

excluded middle

287

This page has been reformatted by Knovel to provide easier navigation.

Index Terms existential quantifier

Links 289

expansion Fourier expansion, forms

421

Laplace expansion

259

Laurent expansion

690

Maclaurin

416

Taylor

415

Taylor expansion

387

expectation

751

expected value

751

bivariate distribution

777

two-dimensional distribution

777

exponent exponential distribution exponential function

7 758 71

complex

677

general

697

natural

696

exponential integral exponential sum

459 72

expression algebraic manipulation analytic

10 976 48

domain

48

explicit form

48

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

expression (Cont.) implicit form

48

parametric form

49

Boolean

342

concurrent

343

explicit form finite partial differential equations

48 905 980

implicit form

48

integral rational

11

irrational

11

non-polynomial, manipulation parametric form propositional logic rational

977 49 286 11

semantically equivalent

343

tautology

288

transcendent

17

14

11

vector analysis

657

extension principle

366

extension theorem of Hahn

623

extension, linear functional

622

extraction of the root complex numbers real numbers extrapolation principle

38 8 899

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

extremal, radius of curvature

556

extreme value

388

absolute

388

determination

389

higher derivatives

389

side conditions

401

sign change

388

function

399

relative

388

face, corner

151

factor algebra

338

factor group

302

factor ring

314

F

factor, polynomial, product representation

43

factorial

13

generalization of the notion

460

factoring out

11

Falk scheme

254

feasible set

845

Feigenbaum constant

835

FEM (finite element method)

910

FFT (fast Fourier transformation)

925

837

Fibonacci numbers

323

843

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Fibonacci (Cont.) sequence field

323 313

axial field

642

central symmetric

641

circular

644

conservative

660

Coulomb field, point-like charge

644

cylinder symmetric

642

extension

313

flow

662

function

679

gravitational, point mass

667

Newton field, point-like mass

644

potential

660

scalar field

641

source field

665

spherical field

641

666

field theory basic notions

640

differential equations, partial

667

fields, superposition

667

finite difference method

532

finite element method

532

910

fitting problem different versions

401

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

fitting problem (Cont.) linear

890

non-linear

107

fixed point conformal mapping

673

flip bifurcation

834

fixed-point number

935

floating-point number

935

IEEE standard

936

Maple

967

Mathematica

954

semilogarithmic form

935

674

Floquet representation

803

theorem

801

flow differential equation

795

edge

356

scalar field

662

vector field scalar flow

662

rector flow

662

focal line, tractrix

106

focus

807

compound

830

ellipse

198

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

focus (Cont.) hyperbola

200

parabola

203

saddle

802

saddle focus

807

stable

802

form quadratic

890

saddle

249

formula binomial

12

Cardano

41

closed, predicate logic

289

d’Alembert

534

Duhamel

720

Euler

247

Heron’s

143

interpretation, predicate logic

289

Kirchhoff

534

Leibniz

383

Liouville

499

manipulation

950

418

800

de Moivre complex number

37

hyperbolic functions

90

trigonometric functions

79

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Pesin

819

Plemelj and Sochozki

591

Poisson

534

predicate logic

289

rectangular

896

Riemann

528

Simpson

897

Stirling

460

tangent

142

tautology

290

826

Taylor one variable

387

several variables

395

trapezoidal

896

formulas Frenet

243

predicate logic

289

four-group, Klein’s

302

Fourier analysis

418

Fourier coefficient

418

asymptotic behavior

420

determination

401

numerical methods

422

Fourier expansion

924

418

complex functions

424

forms

421

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Fourier expansion (Cont.) symmetries

420

Fourier integral

422

complex representation

722

equivalent representations

722

Fourier series

419

best approximation

615

complex representation

419

Hilbert space

615

Bessel inequality Fourier sum complex representation

616 419

722

addition law

725

comparing to Laplace transformation

728

convolution

727

definition

924

925

Fourier transformation

two-sided

722

711 723

differentiation image space

726

original space

726

discrete complex

925

fast

925

Fourier cosine transformation

723

Fourier exonential transformation

724

Fourier sine transformation

723

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Fourier transformation (Cont.) frequency-shift theorem

726

integration

726

image space

726

original space

727

inverse

723

linearity law

725

shifting theorem

726

similarity law

726

spectral interpretation

724

survey

705

tables

724

transform

728

fractal

820

fractile

750

fraction continued

3

decimal

1

improper

15

proper

15

fractional part of x frames

49 741

Fréchet derivative

630

differential

630

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Fredholm alternative

627

alternative theorem

569

integral equation

561

first kind

575

solution method

567

theorems

567

Frenet formulas frequency angular/radial

243 75 82

distribution

770

locking

842

relative

746

sine

607

82

spectrum

725

continuous

423

discrete

423

statistics cumulative Fresnel integral

746

770

771 695

frustum cone

156

pyramid

152

function absolutely integrable algebraic

47 453

456

60

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

function (Cont.) amplitude function, elliptic

701

analytic

670

area

91

arrow

294

autocorrelation

816

Bessel

507

modified

507

beta function

1050

Boolean

287

bounded

50

circular, geometric definition

130

comparable

556

complement

366

complex

47

algebraic

696

bounded

671

linear

673

linear fractional

674

quadratic

675

square root

675

complex variable

669

complex-valued

47

composite

62

derivative

380

intermediate variable

380

342

669

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

continuity in interval

57

one-sided

57

piecewise

57

continuous, complex

669

cosecant

76

cosine

75

cotangent

76

cyclometric

84

cylindrical

507

density

750

dependent

121

discontinuity removable

57 58

discrete

732

distribution

749

double periodic

702

elementary

60

elementary, transcendental

696

elliptic

435

entire rational

60

error function

459

Euler

329

even

50

exponential

61

exponential, complex

691

702

71

677

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

discontinuity (Cont.) exponential, natural

696

function series

412

gamma

459

generalized

637

Green

530

Hamilton

518

799

harmonic

667

670

Heaviside

639

Hermite

614

halomorphic

670

homogeneous

120

homographic

61

65

hyperbolic

87

697

geometric definition

130

131

impulse

712

638

715

function (continued II) increment

904

independent

121

integrable

439

integral rational

60

first degree

62

n-th degree

63

second degree

62

third degree

63

inverse

51

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

function (continued II) (Cont.) complex, hyperbolic

697

complex, trigonometric

697

derivative

381

derivative of higher order

385

existence

60

hyperbolic

91

trigonometric

61

84

irrational

61

69

Jacobian

701

Lagrangian

860

Laguerre

614

Laplace

667

limit

51

at infinity

53

infinity

52

iterated

122

left-hand

52

right-hand

52

Taylor expansion

55

limit theorems

53

linear

60

62

linear fractional

61

65

local summable

637

logarithm, complex

676

logarithmic

61

71

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

function (continued II) (Cont.) loop function

1004

MacDonald

507

matrix exponential

507

mean value

442

measurable

634

meromorphic

691

702

717

monotone decreasing

49

increasing

49

non-elementary

60

notion

47

odd

50

of angle

74

one variable

47

order of magnitude

55

parametric representation derivative of higher order

385

periodic

50

piecewise continuous

57

point of discontinuity

57

finite jump

58

tending to infinity

57

positive homogeneous

556

power primitive

714

70 425

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

parametric representation (Cont.) quadratic random variable

60 749

rational

61

real

47

regular

670

Riemann

528

sample function

768

secant

76

several variables

47

sign of

49

sine

74

special fractional linear fractional

64

step

732

stream

679

strictly monotone

676

summable

635

switch

344

tangent

76

theory

669

theta

703

transcendental

61

trigonometric

61

truth

117

49

sum of linear and fractional linear functions

geometric definition

64

74

697

130 286

287

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

parametric representation (Cont.) Weber

507

Weierstrass

703

function system orthogonal

917

orthonormal

917

function theory

669

functional

551

definition linear

47 317

linear continuous p

L space functional determinant

617

598

599

266

472

621 622 121

fundamental form first quadratic, of a surface

245

second quadratic, of a surface

248

fundamental formulas spherical trigonometry

164

fundamental laws, set algebra

292

fundamental matrix

800

fundamental problem first, triangulation

147

second, triangulation

148

fundamental system differential equation, higher order

498

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

fundamental theorem Abelian groups algebra

302 43

elementary number theory

319

integral calculus

440

future value

25

fuzzy control

372

inference

370

linguistics

359

logic

358

logical inferences

370

product relation

369

relation

367

relation matrix

368

system

375

systems, applications

372

valuation

367

fuzzy set aggregation

363

aggregation operator

368

complement

363

composition

369

cut (representation theorem)

362

degree

362

empty

361

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

fuzzy set (Cont.) intersection

363

intersection set

364

level set

362

normal

362

peak

361

similarity

362

subnormal

362

subset

361

support

358

tolerance interval

361

union

363

universal

361

fuzzy sets cut

362

intersection

364

union

364

G g.c.d. (greatest common divisor)

322

g.c.d. and l.c.m., relation between

322

Gabor transformation

741

Galerkin method

906

gamma function

457

459

276

888

Gauss algorithm

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Gauss (Cont.) coordinates

244

curvature, surface

249

elimination method

276

error curve

757

error function

757

error integral

459

error propagation law

793

integral theorem

663

least squares method

401

plane

34

step

276

transformation

277

Gauss–Krüger coordinates

161

Gauss–Newton method

894

derivative free

895

Gauss–Seidel method

892

generating line

154

generator

154

ruled surface geodesic line

887

891

916

781

891

222 250

geodesy angle

145

coordinates

143

polar coordinates

143

geometric sequence

19

This page has been reformatted by Knovel to provide easier navigation.

Index Terms geometry

Links 128

analytical

180

plane

189

space

207

differential

226

plane

128

Girard theorem

164

golden section gon

2

4

192

145

gradient definition

649

different coordinates

649

remark

648

scalar field

648

vector components

657

vector gradient

650

Graeffe method

649

655

886

graph alternating way

354

arc

346

bipartite

347

complete

347

complete bipartite

347

components

349

connected

349

cycle

355

355

This page has been reformatted by Knovel to provide easier navigation.

843

Index Terms

Links

graph (Cont.) directed

346

directed circuit

355

directed edge

346

edge

346

Euler

350

flow

356

increasing way

354

infinite

347

isomorphism

347

loop

346

mixed

346

non-planar

355

partial

347

planar

355

plane

347

regular

347

simple

346

special classes

347

strongly connected

355

subdivision

355

subgraph

347

transport network

347

tree

347

undirected

346

vertex

346

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

graph (Cont.) weighted

348

graph paper double logarithmic

115

log–log paper

115

notion

115

reciprocal scale

116

semilogarithmic

115

graph theory, algorithm

346

gravitational field, point mass

667

great circle

158

173

greatest common divisor (g.c.d.) integer numbers

321

linear combination

322

polynomials

14

Green function

530

integral theorem

664

method

529

group

530

299

Abelian

299

dihedral

299

element

299

character

304

inverse

299

factor group

302

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

group (Cont.) permutation

301

point group

308

representation

303

irreducible

305

reducible

305

subgroup

300

symmetry group

308

group table Cayley table grouping

300 300 11

groups applications

307

direct product

301

homomorphism

302

homomorphism theorem

302

isomorphism

302

growth factor

22

Guldin’s first rule

451

Guldin’s second rule

451

H half-angle formulas plane trigonometry

142

spherical trigonometry

165

half-line, notion

128

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

half-side formulas

165

Hamel basis

597

Hamilton circuit

351

differential equation

813

differential equation, partial

799

839

function

518

799

system

813

Hamiltonian

536

Hamming distance

602

Hankel transformation

705

harmonic analysis

914

harmonic, spherical first kind

509

second kind

510

Hasse diagram

298

heat conduction equation one-dimensional

527

three-dimensional

535

721

Heaviside expansion theorem

717

function

639

695

unit step function

712

1078

helix

242

Helmholtz, differential equation

538

Hénon mapping

796

810

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Hermite differential equation

512

polynomial

512

trapezoidal formula

897

543

Hessian matrix

867

normal form line equation, plane

195

plane equation, space

214

hexadecimal number

934

system

934

Hilbert boundary value problem

591

homogeneous

591

inhomogeneous

592

matrix

986

space

613

isomorphic

616

pre-Hilbert space

613

scalar product

613

histogram, statistics

770

hodograph, vector function

640

Hölder condition

589

continuity

589

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Hölder (Cont.) inequality

32

integrals

32

series

32

Holladay, theorem

929

holoedry

311

homeomorphism and topological equivalence

808

conjugate

811

orientation preserving

841

homomorphism

302

algebra

612

groups

302

linear operators

598

natural

302

ring

314

theorem

339

groups

302

ring

314

vector lattice

601

Hopf bifurcation

829

Hopf–Landau model, turbulence

839

339

314

Horner rule

935

scheme

884

two rows

935

885

This page has been reformatted by Knovel to provide easier navigation.

Index Terms horseshoe mapping

Links 825

Householder method

278

tridiagonalization

283

1’Huilier equations

918

167

hull convex

597

linear

595

Hungarian method

858

hyperbola

200

arc

203

area

202

asymptote

201

binomial

69

conjugate

202

diameter

202

equation

200

equilateral

64

focal properties

200

focus

200

radius of curvature

202

segment

202

semifocal chord

200

tangent

201

transformation

205

vertex

200

203

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

hyperbolic cosecant

87

cosine

87

cotangent

87

secant

87

sine

87

tangent

87

hyperbolic function

697

geometric definition

131

inverse, complex

697

hyperboloid one sheet central surface two sheets central surface hyperplane

220 220

222

224 220 224 623

of support

624

hypersubspace

623

hypersurface

117

hypocycloid

102

curtate

102

prolate

102

hypotenuse

130

hypothesis testing

777

hypotrochoid

102

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

I icosahedron

154

ideal

313

principal ideal

313

idempotence law Boolean algebra

341

propositional logic

287

sets

293

identically valid

10

identity

10

Boolean function

342

representation of groups, identity matrix

253

IEEE standard

936

IF–THEN rule

370

iff (if and only if)

318

image function

48

set

48

space

48

subspace

705

315

imaginary number

34

part

34

unit

34

This page has been reformatted by Knovel to provide easier navigation.

Index Terms implication proof

Links 286 5

impulse function

712

impulse, rectangular

712

incidence function

346

incidence matrix

348

incircle quadrangle

136

triangle

132

143

4

804

21

392

linear

272

596

path of integration

466

685

potential field

660

incommensurability increment independence

induction step inequality

5 28

arithmetic and geometric mean

30

arithmetic and quadratic mean

30

Bernoulli

30

Bessel binomial Cauchy–Schwarz Chebyshev Chebyshev, generalized

616 31 31 31

752

32

This page has been reformatted by Knovel to provide easier navigation.

Index Terms different means

Links 30

first degree

33

Hölder

32

linear

33

first degree

33

solution

33

Minkowski

32

product of scalars

31

pure

28

quadratic

33

solution

33

Schwarz–Buniakowski

613

second degree

33

solution

28

special

30

triangle

181

norm

609

triangle inequality complex numbers

30

real numbers

30

infimum

600

infinite denumerable

298

non-denumerable

298

infinitesimal quantity

439

infinity

1 This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

infix form

298

inflection point

231

initial conditions

485

initial value problem

485

inner product

613

388

389

901

inscribed circle quadrangle

136

triangle

132

inscribed pentagram

138

instability, round-off error numerical calculation insurance mathematics

939 21

integer non-negative part of x programming

1 49 845

integrability complete

519

condition

467

conditions

466

differential

466

quadratic

577

660

integral absolutely convergent

456

antiderivative

425

basic This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

integral (Cont.) notion

426

calculus

425

circuit

460

466

comp1ex definite

683

indefinite

684

complex function measurable function

635

elementary functions

426

error integral

459

Euler

457

exponential integral

459

Fourier integral

422

Fresnel

695

interval of integration

439

Lebesgue integral

635

comparison with Riemann integral

722

451

limits of integration

439

line integral

460

first type, applications

459

462

lorarithmic integral

429

lower limit

439

non-elementary

429

parametric

457

primitive function

425

458

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

integral (Cont.) probability integral

757

Riemann

439

comparison with Stieltjes integral Stieltjes

451 626

comparison with Riemann integral

451

notion

451

surface integral

477

triple integral

474

upper limit

439

volume integral

474

integral calculus

662

425

fundamental theorem

440

mean value theorem

442

integral cosine

458

integral curves

796

456

integral equation Abel

588

adjoint

563

590

approximation successive

565

characteristic

590

collocation method

574

eigenfunction

563

eigenvalue

563

first kind

561

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

integral equation (Cont.) fredholm

561

degenerate kernel

575

first kind

575

second kind

562

general form

561

homogeneous

561

inhomogeneous

561

iteration method

565

kernel

561

degenerate

562

iterated

585

product

562

kernel approximation tensor product

582

572 572

linear

561

Nystrom method

571

orthogonal system

577

perturbation function

561

quadratically integrable function

577

quadrature formula

570

second kind

561

singular

588

Cauchy kernel

607

580

589

transposed

563

Volterra

561

607

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

integral equation (Cont.) convolution type

585

first kind

583

second kind

583

integral equation method, closed curve integral exponential function, table

532 1045

integral formula Cauchy application Gauss integral logarithm, table

687 692 664 1047

integral norm

637

integral surface

516

integral test of Cauchy

406

integral theorem

663

Cauchy

686

Gauss

663

Green

664

Stokes

664

integral transformation

705

application

707

Carson

705

definition

705

Fourier

705

Gabor

741

Hankel

705

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

integral transformation (Cont.) image space

705

inverse

705

kernel

705

Laplace

705

linearity

705

Mellin

705

multiple

707

one variable

705

original space

705

several variables

707

special

705

Stieltjes

705

Walsh

742

wavelet

738

fast

741

integral, definite

438

differentiation

441

notion

425

particular integral

440

table

739

1050

algebraic functions

1053

exponential function

1051

logarithmic function

1052

trigonometric functions

1050

This page has been reformatted by Knovel to provide easier navigation.

Index Terms integral, elliptic

Links 435

first kind

429

second kind

435

series expansion

460

table

435

1055

third kind

435

integral, improper

438

Cauchy’s principal value

452

455

convergent

452

454

divergent

452

454

infinite integration limits

451

notion

451

principal value

452

unbounded integrand

451

integral, indefinite

425

basic integral

426

cosine function, table

1037

cotangent function, table

1043

elementary functions

701

426

elementary functions, table

1017

exponential function, table

1045

hyperbolic functions, table

1044

inverse hyperbolic functions, table

1049

inverse trigonometric function, table

1048

irrational functions, table

1024

logarithmic functions, table

1047

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

integral, indefinite (Cont.) notion

426

other transcendental functions, table

1044

sine and cosine function, table

1039

sine function, table

1035

table

1017

tangent function, table

1043

trigonometric functions, table

1035

integral, logarithmic

429

integrand

426

integrating factor

488

439

integration approximate ordinary differential equation

901

partial differential equation

908

complex plane

683

constant

426

function, non-elementary

458

graphic

430

graphical

444

in complex

692

interval

439

limit depending on parameter

457

lower

439

upper

439

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

integration (Cont.) logarithmic

428

numerical

895

multiple integrals

900

partial

429

power

428

rational functions

430

rules by series expansion

429

by substitution

429

constant multiple rule

427

definite integrals

441

general rule

427

indefinite integrals

428

interchange rule

442

interval rule

441

series expansion

444

sum rule

427

under the integration sign

457

variable

439

variable, notion

426

vector field

658

volume

448

integrator

445

intensity, source

666

interaction, soliton

545

458

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

intercept theorem interest

134 22

calculation

21

compound

22

intermediate value theorem one variable

59

several variables

123

intermediate variable

380

intermittence

837

Internationale Standard Book Yumber ISBN

330

840

interpolation Aitken–Neville

915

condition

914

formula Lagrange

915

Newton

914

fuzzy system

375

knowledge-based

378

node

895

equidistant

376

901

points

914

quadrature

896

spline

914

bicubic

930

cubic

928

trigonometric

914

928

924

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

interpretation formula, predicate logic

289

variable

287

intersection

292

angle

159

by two oriented lines

147

fuzzy set

363

on the sphere

171

point four planes

216

lines

195

plane and line

218

three planes

216

two lines, space

219

set, fuzzy set

364

sets

292

without visibility

147

interval convergence numbers

414 2

order (0)–interval

600

rule

441

statistics

770

invariance rotation invariance

265

transformation invariance

265

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

invariance (Cont.) translation invariance invariant

265 704

quadratic curve

206

scalar

212

scalar invariant

184

surface second order

224

inverse

599

inverse function

51

hyperbolic

91

trigonometric

84

inverse transformation

705

inversion Cartesian coordinate system

269

conformal mapping

674

space

269

involute

237

irrational algebraic

2

quadratic

2

isometry, space

609

isomorphic vector spaces isomorphism

599 302

Boolean algebra

341

graph

347

339

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

isomorphism (Cont.) groups

302

surjective norm

624

iteration

881

inverse

283

method

283

881

881

893

ordinary sequential steps

893

simultaneous steps

893

vector

284

892

J Jacobi function

701

method

283

892

determinant

121

266

matrix

894

Jacobian

Jordan matrix

282

normal form

282

jump, finite

58

K KAM (Kolmogorov–Arnold–Moser) theorem

813

ker

339

699

This page has been reformatted by Knovel to provide easier navigation.

Index Terms kernel

Links 302

approximation integral equation

572

spline approach

573

tensor product

572

congruence relation

339

homomorphism

339

integral equation

561

degenerate

572

iterated

566

resolvent

568

solving

566

integral transformation

705

operator

599

ring

314

subspace

315

key equation

123

kink soliton

548

Kirchhoff formula

534

Klein’s four-group

302

Koch curve

821

Korteweg de Vries equation

545

585

Kronecker generalized delta

265

product

258

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Kronecker (Cont.) symbol

253

Kuan

360

Kuhn-Tucker conditions

860

reference Kuratowski theorem

265

624 355

L 1.c.m. (least common multiple)

322

Lagrange function

401

identity, vectors

185

interpolation formula

915

method of multiplier

401

theorem

301

Lagrangian

401

860

Laguerre differential equation, linear, second order

511

polynomial

511

Lanczos method

283

Landau, order symbol

56

Laplace differential equation, partial

536

expansion

259

wave equation

534

667

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Laplace operator different coordinates

655

polar coordinates

399

vector components

657

Laplace transformation

708

addition law

709

comparing to Fourier transformation

728

convergence

708

convolution

711

one-sided

711

convolution, complex

712

definition

708

differential equation, ordinary linear, constant coefficients

719

linear, variable coefficients

720

differential equation, partial

721

differential equations

719

differentiation, image space

710

differentiation, original space

710

discrete

734

division law

711

frequency-shift theorem

709

image space

708

integration, image space

710

integration, original space

710

inverse

708

716

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Laplace transformation (Cont.) inverse integral

718

linearity law

709

original function

708

original space

708

partial fraction decomposition

716

piecewise differentiable function

713

series expansion

717

similarity law

709

step function

712

survey

705

table

1061

transform

708

translation law

709

largest area method

372

lateral area cone

156

cylinder

154

polyhedron

151

lateral face

151

latitude, geographical

160

lattice

340

Banach

612

Bravais

310

crystallography

310

distributive

340

244

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

lattice (Cont.) vector

612

Laurent expansion

690

series

690

cosine

164

734

law

spherical triangle

164

large numbers

762

Bernoulli

762

limit theorem, Lindeberg–Levy

763

sine–cosine polar layer, spherical

164 164 157

least common multiple (l.c.m.), integer numbers least squares method

322 107

277

910

916

calculus of observations

785

Gauss

401

regression analysis

778

least squares problem, linear

891

887

Lebesgue integral comparison with Riemann integral measure

452

635

451 634

814

This page has been reformatted by Knovel to provide easier navigation.

906

Index Terms

Links

left singular vector

285

left-hand coordinate system

207

leg or side of an angle

128

Legendre differential equation

509

polynomial associated

511

first kind

509

second kind

510

symbol

327

Leibniz alternating series test

408

formula

383

lemma Jordan

693

Schur

306

lemniscate

99

234

length interval

633

line integral, first type

462

reduced

174

vector

189

length, arc space curve

246

level curve

117

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

level (Cont.) line

642

surface

642

level set (fuzzy set)

362

library (numerical methods)

941

Aachen library

943

IMSL library

942

NAG library

941

limit cycle

804

stable

804

unstable

804

definite integral

438

831

function complex variable one variable several variables theorems

669 51 122 53

function series

412

integration, depending on parameter

457

partial sum

412

sequence of numbers

403

sequence, in metric space

604

series

404

superior

414

This page has been reformatted by Knovel to provide easier navigation.

Index Terms line

Links 62

curvature, surface

248

geodesic

159

imaginary

205

notion

128

space

150

vector equation

188

214

line element surface

245

vector components

658

line equation plane

194

Hessian normal form

195

intercept form

195

polar coordinates

195

through a point

194

through two points

194

slope, plane

194

space

217

line integral

460

Cartesian coordinates

659

first type

461

applications

462

general type

465

second type

462

vector field

658

This page has been reformatted by Knovel to provide easier navigation.

Index Terms linear combination, vectors linear form continuous

Links 83

181

598

599

184

621

linear programming assignment problem

858

basic notions

847

basic solution

848

basis of the extreme point

847

constraints

844

degenerate extreme point

847

distribution problem

858

dual problem

854

duality

854

extreme point

847

normal form

847

objective function

844

primal problem

854

round-tour problem

859

surplus variable

848

848

linearly dependent

315

independent

315

lines angle between, plane

196

intersection point, plane

195

orthogonal

128

197

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

lines (Cont.) parallel

128

pencil

196

perpendicular

128

skew

150

150

197

197

Liouville approximation theorem

4

formula

499

theorem

799

800

Lipschitz condition higher-order differential equation

496

ordinary differential equation

486

locus, geometric

194

logarithm binary Briggsian complex function

10 9 676

decimal

9

definition

9

natural

9

Neperian

9

principal value table taking of an expression logarithmic decrement logarithmic integral

696

696 10 9 83 429

458

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

logarithmic normal distribution

757

logic

286

fuzzy

358

predicate

289

propositional

286

longitude, geographical

160

loop function

1004

loop, graph

346

Lorentz curve

729

Lorenz system

796

loxodrome

177

arclength

177

course angle

178

intersection point

178

intersection point of two loxodromes

179

p

L space

244

799

837

637

LU factorization (lower and upper triangular matrix)

888

Lyapunov exponent

818

function

801

M MacDonald function

507

Maclaurin series expansion

416

Macsyma (computer algebra system)

950

This page has been reformatted by Knovel to provide easier navigation.

Index Terms majorant series

Links 411

manifold center manifold theorem local

833

mappings

833

of solutions

887

stable

804

810

unstable

804

810

manipulation algebraic expressions

976

non-polynomial expressions

977

mantissa

10

manyness set

298

Maae (computer algebra system)

950

936

Maple addendum to syntax

975

algebraic expressions

969

manipulation

978

multiplication

978

array

970

attribute

975

basic structure elements

965

context

975

conversion of numbers, different bases

968

differential equations

994

differential operators

973

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Maple (Cont.) differentiation

992

eigenvalues, eigenvectors

988

elements of linear algebra

986

environment variable

975

equation one unknown

983

transcendental

984

factorization of polynomials

979

floating-point number, conversion

967

formula manipulation, introduction

950

functions

972

graphics

1002

introduction

952

three-dimensional

1004

two-dimensional

1002

help and information

975

input, output

946

input–output

950

966

integral definite

993

indefinite

993

multiple

994

lists

969

manipulation, general expressions

980

matrices

970

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Maple (Cont.) numerical calculations, introduction

951

numerical mathematics

946

differential equations

949

equations

947

expressions and functions

946

integration

948

object classes

965

objects

965

operations important

983

polynomials

979

operators functions

973

important

968

partial fraction decomposition

980

programming

974

sequences

969

short characteristc

950

special package plots

1004

system description

965

system of equations

984

systems of equations, linear

986

table structures

970

types

965

types of numbers

967

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Maple (Cont.) vectors

970

mapping

294

between groups

302

bijective

296

complex number plane

682

conformal

672

circular transformation

674

exponential function

677

fixed-point

673

inversion

674

linear fractional

674

linear function

673

logarithm

676

quadratic function

675

square root

675

296

598

674

sum of linear and fractional linear functions

676

contracting

606

equivalent

841

function

48

Hénon mapping

810

horseshoe

825

injective

296

inverse mapping

296

kernel

302

294

598

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

mapping (Cont.) lifted

841

linear

315

modulo

814

one-to-one

296

Poincaré

825

Poincaré mapping

806

reduced

833

regular

316

rotation mapping

815

shift

818

surjective

296

tent

814

topological conjugate

811

unit circle

841

marginal distribution

316

598

598

752

mass double integral

472

line integral, first type

462

triple integral

478

matching

354

maximal

354

perfect

354

saturated

354

Mathcad (computer algebra system)

950

Mathematica (computer algebra system)

950

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Mathematica 3D graphics

1000

algebraic expressions, manipulation

976

algebraic expressions, multiplication

976

apply

963

attribute

964

basic structure element

953

context

964

curves parametric representation

999

two-dimensional

998

differential and integral calculus

989

differential equation

991

differential quotient

989

differentiation

962

eigenvalues, eigenvectors

986

elements

953

elements of linear algebra

984

equation

981

manipulation

981

transcendent

982

expression

953

factorization, polynomials

976

FixedPoint

962

FixedPointList

962

floating–point number, conversion

955

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Mathematica (Cont.) formula manipulation, introduction

950

function, inverse

962

functional operations

962

functions

960

graphics

995

functions

997

introduction

952

options

996

primitives

995

head

953

input and output

943

input–output

950

954

integral definite

990

indefinite

990

multiple

991

lists

956

manipulation with matrices

959

manipulation with vectors

959

Map

963

matrices as lists

958

messages

965

Nest

962

NestList

962

numerical calculations, introduction

951

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Mathematica (Cont.) numerical mathematics

943

curve fitting

943

differential equations

945

integration

945

interpolation

944

polynomial equations

944

objects, three-dimensional

1001

operations, polynomials

977

operators, important

956

partial fraction decomposition

977

pattern

961

programming

963

short characteristc

950

surfaces

1000

surfaces and space curves

1000

syntax, additional information

964

system description

953

system of equations

982

general case

985

special case

984

types of numbers

954

vectors as lists

958

mathematics, discrete

286

matrices arithmetical operations

254

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

matrices (Cont.) associative law

254

commutative law

254

distributive law

254

division

254

eigenvalue

278

eigenvalue problem

278

eigenvectors

278

equality

254

multiplication

254

powers

258

rules of calculation

257

matrix

255

251

adjacency

348

adjoint

251

admittance

353

anti-Hermitian

253

antisymmetric

252

block tridiagonal

909

complex

251

conjugate

251

deflation

284

degree

353

diagonal

252

distance

349

extended coefficient

889

260

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

matrix (Cont.) full rank

277

fundamental

800

Hermitian

253

Hessian

867

identity

253

incidence

348

inverse

256

inverse calculation

272

Jacobian

894

Jordan

282

lower triangular

889

main diagonal element

252

monodromy

801

normal

252

of coefficients

272

augmented

260

810

273

orthogonal

257

rank

255

real

251

rectangular

251

regular

256

rotation

257

scalar

252

self-adjoint

253

singular

256

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

matrix (Cont.) size

251

skew-symmetric

252

spanning tree theorem

353

sparse

909

spur

252

square

251

stochastic

764

symmetric

252

trace

252

transposed

251

triangle decomposition

887

triangular

253

unitary

257

upper triangular

889

zero

251

matrix exponential function

800

matrix product, disappearing

257

max-min composition

369

252

maximum absolute

60

global

388

point

845

relative

388

maximum-criterion method

371

Maxwell, diagonal method

682

123

388

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

mean arithmetic

19

geometric

20

golden

20

quadratic

20

sample function

769

statistics

771

weighted

788

788

791

843

harmonic

value

751

19 751

mean squares problem different versions

401

non-linear

919

mean squares value problem linear

277

rank deficiency case

278

mean value formula

896

mean value function, integral calculus

442

mean value method

107

mean value theorem differential calculus

387

generalized

388

integral calculus

442

generalized

443

concentrated on a set

814

measure

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

measure (Cont.) counting

633

degrees

130

dimension

822

Dirac

634

ergodic

815

function, convergence theorems

636

Hausdorff

820

invariant

814

Lebesgue

634

natural

814

physical

815

probability

636

invariant

814

814

814

SBR (Sinai–Bowen–Ruelle)

815

σ-algebra

633

σ-finite

636

support

814

measured value

785

measurement error

785

error density

786

error, characterization

785

protocol

786

measuring error, distribution

785

This page has been reformatted by Knovel to provide easier navigation.

Index Terms median

Links 142

sample function

770

statistics

772

triangle

132

Mellin transformation

705

Melnikov method

839

membership degree

358

function

358

bell-shaped

360

trapezoidal

359

relation meridian

359

290 160

convergence

169

tangential

176

244

method Bairstow

886

barrier

873

Bernoulli

886

bisection

283

broken line, Euler

901

center of area

372

center of gravity

371

generalized

372

parametrized

372

Cholesky

278

890

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

method (Cont.) collocation boundary value problem

906

integral equation

574

partial integral equation

910

cutting plane

874

descent

866

difference ordinary differential equations

905

partial differential equations

908

extrapolation

902

Fibonacci

866

finite difference

532

finite element

532

Galerkin

906

Gauss elimination

887

Gauss–Newton

894

derivative free

559

910

895

Gauss–Seidel

892

gradient

559

Graeffe

886

Green

529

Hildreth–d’Esopo

865

Householder

278

Hungarian

858

inner digits of squares

781

530

918

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

method (Cont.) integrating factor

488

iteration

283

881

881

893

Jacobi

283

892

Kelley’s, optimization

875

Lagrange multiplier

860

Lanczos

283

largest area

372

least squares

107

887

916

918

ordinary

Mamdani

372

maximum-criterion

371

mean-of-maximum

371

Monte Carlo

781

multi-step

902

multiple-target

907

Newton

882

modified

882

non-linear operators

630

operator

707

orthogonalization

278

parametrized center of area

372

pivoting

271

polygonal, Euler

901

predictor–corrector

903

887

892

906

910

894

890

891

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

method (Cont.) regula falsi

883

relaxation

893

Riemann

528

Ritz

558

Romberg

898

Runge–Kutta

901

separation of variables

487

shooting

907

single–target

907

SOR (successive overrelaxation)

893

statistical experiment

785

steepest descent

867

Steffensen

883

906

successive approximation Banach space

619

Picard

494

Sugeno

373

transformation

283

undetermined coefficients

16

variation of constants

499

Wolfe

863

metric

602

Euclidean

602

surface

247

Meusnier, theorem

247

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

midpoint line segment plane

192

space

213

midpoint rule

903

minimal surface

249

minimum absolute

60

global

388

point

860

global

860

local

860

problem

845

relative

388

123

388

Minkowski inequality

32

integrals

32

series

32

mixed product

187

modal value, statistics

772

mode, statistics

772

model Hopf–Landau, turbulence

839

urn model

753

module of an element

601

modulo congruence

297

325

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

modulo (Cont.) mapping modulus, analytic function

814 670

de Moivre formula complex number

37

hyperbolic functions

90

trigonometric functions

79

Mollweide equations

142

moment of inertia

263

double integral

472

line integral, first type

462

triple integral

478

moment, order n

751

monodromy matrix

801

monotonicity

633

monotony

386

function sequence, numbers Monte Carlo method

810

402 781 783

usual

783

example

802

49

application in numerical mathematics

Monte Carlo simulation

450

781 782

Morse–Smale system

813

multi-index

611

multi-scale analysis

741

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

multi-step method

902

multiple integral

469

transformation

707

multiple-target method

907

multiplication complex numbers computer calculation polynomials rational numbers

36 937 11 1

multiplicity, of divisors

318

multiplier

488

Lagrange method

801

802

401

N nabla operator definition

654

divergence

654

gradient

654

rotation

654

twice applied

655

vector gradient

654

NAND function

288

nautical, radio hearing

171

navigation

172

negation

286

Boolean function

655

342

This page has been reformatted by Knovel to provide easier navigation.

810

Index Terms

Links

negation (Cont.) double

288

neighborhood

603

point

603

Neper equations

167

logarithm

9

rules

169

chart

123

net

more than three variables

127

three variahles

123

isometric

673

points

930

Neumann problem

668

series

566

585

Newton field, point-like mass

644

interpolation formula

914

Newton method

882

adaptive

948

modified

882

non-linear operators

630

approximation sequence non-linear optimization

894

830 867

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Newton method (Cont.) damped

867

nonlinear operators modified

630

niveau line

117

nodal plane

539

node

807

approximation interval

928

differential equation, ordinary

492

saddle node

802

stable

802

triple

832

nominal rate

22

nomogram

123

nomography

123

non-negative integer NOR function

807

1 288

norm axioms, linear algebra

258

Euclidean

316

integral

637

isomorphism

624

matrix

258

column sum norm

259

row sum norm

259

spectral norm

259

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

norm (Cont.) subordinate

259

uniform norm

259

operator norm

617

pseudonorm

622

residual vector

277

seminorm

622

s-norm

363

space

609

t-norm

363

vector

258

Euclidean norm

258

matrix norm

258

sum norm

258

normal distribution

756

logarithmic normal distribution observational error standard normal distribution standard, table two-dimensional

800

757 72 757 1085 778

normal equation

891

916

system

780

917

normal form

918

343

equation of a surface

220

Jordan

282

principal conjunctive

343

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

normal form (Cont.) principal disjunctive normal plane, space curve

343 238

241

normal vector plane

211

plane sheet

661

surface

244

normal, plane curve

226

normalization condition

539

normalizing factor

195

northern direction geodesical

169

geographical

169

northwest corner rule

856

notation Polish notation

353

postfix notation

353

prefix notation

353

reversed polish

353

n-tuple

117

number

1

approximation

4

cardinal

291

complex

34

absolute value

35

addition

36

298

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

number (Cont.) algebraic form

34

argument

35

division

37

exponential form

35

main value

35

multiplication

36

power

37

subtraction

36

taking the root

38

trigonometric form

35

composite

318

conjugate complex

36

imaginary

34

integer

1

interval

2

irrational

2

natural

1

non-negative

1

random

768

rational

1

real

2 taking of root

sequence

297

3

8 402

convergent

604

metric space

604

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

number (Cont.) transcendental number line, extended number plane, complex, Gauss

2 633 34

number representation, computer internally

933

number system

933

binary

934

decimal

934

hexadecimal

934

octal

934

number theory, elementary

935

318

numbers Bernoulli

410

Euler

410

Fermat, prime

319

Fibonacci

323

Mersenne, prime

319

prime

318

843

numerical analysis axis

881 1

calculations, computer

936

library (numerical methods)

881

nutation angle

211

Nyström method

571

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

O obelisk

153

objective function, linear

844

observational value

785

occupancy number, statistics

770

octahedron

154

octal number

934

octal system

934

octant

209

ω-limit set

797

operation

298

algebraic arithmetical

803

810

263 1

associative

298

binary

298

commutative

298

exterior

299

n-ary

298

operational method

532

operational notation, differential equation

500

operator adjoint

624

AND

366

bounded

617

closed

618

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

operator (Cont.) coercivity

632

compact

626

compensatory

366

completely continuous

626

continuous

608

inverse

619

contracting

606

demi-continuous

632

differentiable

630

finite dimensional

626

gamma

366

Hammerstein

629

idempotent

626

inverse

599

isotone

631

lambda

366

linear

316

bounded

617

continuous

617

linear, notion

316

linear, permutability

317

linear, product

317

monotone

632

Nemytskij

629

non-linear

629

598

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

operator (Cont.) norm

800

notion

48

OR

366

positive

600

positive definite

626

positive non-linear

631

self-adjoint

625

singular

590

strongly monotone

632

Urysohn

630

operator method

707

opposite side

130

optimality condition

860

sufficient

861

860

optimality principle, Bellman

878

optimization

844

optimization method Bellman functional equation

878

conjugate gradient

867

cutting plane method

874

damped

867

DFP (Davidon–Fletcher–Powell)

868

feasible direction

869

Fibonacci method

866

golden section

866

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

optimization method (Cont.) Hildreth–d’Esopo

865

Kelley

875

penalty method

872

projected gradient

870

unconstrained problem

866

Wolfe

863

optimization problem convex

623

dynamic

875

non-linear

923

optimization, non-linear

859

barrier method

873

convex

861

convexity

862

descent method

866

direction search program

869

duality

861

861

gradient method inequality constraints

868

Newton method

867

numerical search procedure

865

quadratic

862

saddle point

860

steepest descent method

867

This page has been reformatted by Knovel to provide easier navigation.

Index Terms orbit

Links 795

double, periodic

834

heteroclinic

806

homoclinic

806

periodic

795

hyperbolic

803

saddle type

803

839

order (0)-interval

600

curve, second order

205

curve, n-th order

194

differential equation

485

interval

600

of magnitude, function

55

second order surface

220

wavelet

739

order relation

296

linear

297

order symbol, Landau ordering

56 297

lexicographical

297

linear

297

partial

297

599

ordinate plane coordinates

189

space coordinates

208

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Ore theorem

351

orientation

269

coordinate system numerical axis

207 1

origin numerical axis

1

plane coordinates

189

space coordinates

208

original function

705

original space

705

set

48

orthocenter, triangle

132

orthodrome

173

arclength

174

course angle

173

intersection point

174

intersection point of two orthodromes

178

point, closest to the north pole

173

175

orthonornal function system

917

polynomials

917

space

614

spherical

169

orthogonality eigenvalues, differential equation

513

Hilbert space

614

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

orthogonality (Cont.) lines

128

real vector space

316

trigonometric functions

316

vectors

184

weight

513

orthogonality conditions line–plane

219

lines in space

219

planes

216

orthogonalization method

890

Givens

892

Gram–Schmidt

615

Householder

278

Schmidt

577

orthogonalization process

280

Gram–Schmidt

280

891

892

orthogomal function system orthonormalization, vectors

917 262

oscillation duration

82

harmonic

82

oscillator, linear harmonic

542

osculating plane, space curve

238

osculation point, curve

232

241

This page has been reformatted by Knovel to provide easier navigation.

Index Terms oversliding

Links 266

P pair, ordered

294

parabola

203

arclength

204

area

204

axis

203

binomial

69

cubic

63

diameter

204

directrix

203

equation

203

focus

203

intersection figure

221

n-th degree

64

parameter

203

quadratic polynomial radius of curvature semicubic

62 204 93

semifocal chord

203

tangent

204

transformation

205

vertex

203

paraboloid

221

elliptic

221

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

paraboloid (Cont.) hyperbolic central surface

222 224

invariant signs elliptic

224

hyperbolic

224

parabolic

224

of revolution

222

parallel circle

244

parallelepiped

152

rectangular

152

parallelism lines

128

parallelism conditions line–plane

219

lines in space

219

planes

216

parallelogram

134

parallelogram identity, unitary space

613

parameter

11

parabola

203

statistical

771

parameter space, stochastic

763

parametric integral

457

parametric representation, circle

197

parametrized center of area method

372

49

This page has been reformatted by Knovel to provide easier navigation.

Index Terms parity

Links 542

Parseval equation

420

formula

727

partial fraction decomposition special cases

616

15 1023

partial ordering

297

partial sum

404

partition

297

Pascal limaçon

96

Pascal triangle

13

path

514

349 closed

349

integration

461

PCNF (principal conjunctive normal form)

343

PDNF (principal disjunctive normal form)

343

pencil of lines

196

pendulum equation

839

Foucault

946

mathematical

700

period

701

pentagon, regular pentagram pentagram, regular

138 4 138

This page has been reformatted by Knovel to provide easier navigation.

Index Terms percentage

Links 21

calculation

21

performance score

370

perimeter circle

138

ellipse

200

polygon

137

period

50

secant

76

sine

75

tangent

76

period doubling

834

cascade

835

period parallelogram

702

permutability, linear operators

317

permutation

743

cyclic, vectors

207

group

301

matrix

889

with repetition

743

without repetition

743

82

840

perturbation

533

938

Pesin formula

819

826

phase initial portrait, dynamical systems

82 798

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

phase (Cont.) shift

75

82

sine

75

82

space, dynamical systems

795

spectrum

725

Picard iteration method integral equation successive approximation method

565 494

Picard–Lindelöf theorem

608

pivot

888

column

271

850

element

271

850

row

271

850

scheme

271

pivoting

271

column pivoting

889

step

271

274

plane equation general

214

in space

214

geometry

128

intersecting

158

rectifying

239

241

space

150

214

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

plane (Cont.) vector equation

188

planes orthogonality, conditions

216

parallel

150

distance parallelism, conditions

217 216

planimeter

445

Poincaré mapping

806

Poincaré section

839

point

128

accumulation

604

asymptotic, curve

232

boundary

604

circular

249

coordinates

189

corner

232

cuspidal, curve

232

discontinuity

57

double, curve

232

fixed, conformal mapping

674

fixed, stable

834

focal, ordinary differential equation

493

improper

192

infinite

192

interior

603

808

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

point (Cont.) isolated

604

isolated, curve

232

limit

237

limiting

604

multiple, curve

233

n-dimensional space

117

nearest approach

237

neighborhood

603

non-wandering

813

notion

128

plane curve

231

rational

1

regular, surface

244

saddle, ordinary differential equation

493

singular

225

isolated

492

ordinary differential equation

492

surface

244

spectrum

620

spiral, ordinary differential equation

493

232

245

surface point circular

249

elliptic

248

hyperbolic

249

parabolic

249

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

point (Cont.) spherical

248

umbilical

249

terminal, curve

232

transversal homoclinic

810

umbilical

249

Poisson differential equation, partial

536

distribution

755

formula

534

integral

531

polar

665

668

162

angle

190

axis

190

coordinates, plane

190

coordinates, spherical

209

coordinates, transformation

191

distance

244

equation

540

curve second order

207

subnormal

227

subtangent

227

complex function

671

pole

function left

58 162

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

pole (Cont.) multiplicity m, complex function

691

on the sphere

162

order m, complex function

691

origin

190

right

162

Polish notation

353

polyeder

151

polygon area 2n-gon

138

n-gon

137

base angle

137

central angle

137

circumcircle radius

137

circumscribing

137

exterior angle

137

inscribed circle radius

137

interior angle

137

perimeter

137

plane

137

regular, convex

137

side length

137

similar

133

polyhedral angle

151

138

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

polyhedron

151

convex

154

regular

154

polynomial

62

Bernstein

932

characteristic

278

Chebyshev formula equation, numerical solution first degree Hermite integral rational function

87 884 62 543

615

60

interpolation

914

Laguerre

511

615

Legendre

509

614

n-th degree

63

product representation

43

quadratic

62

second degree

62

third degree

63

trigonometric polynomials

924 11

Chebyshev

921

orthogonal

917

population

767

two-stage

752

Posa theorem

351

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

position coordinate, reflection

269

positive definite

890

postfix notation

353

postman problem, Chinese

350

potential complex

679

equation

536

field

660

conservative

660

rotation

654

retarded

535

power complex number

37

notion

7

real number

7

reciprocal power series

68 414

asymptotic

417

complex terms

688

expansion, analytic function

687

inverse

415

power set

298

power spectrum

816

precession angle

211

predicate

289

logic

289

418

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

predicate (Cont.) n-ary

289

predictor

903

predictor–corrector method

903

pre-Hilbert space

613

present value pressure

25 449

prime coprime

5

decomposition, canonical

319

element

318

factorization

319

canonical

319

Fermat

319

Mersenne

319

notation (measurement protocol)

786

number

318

pair

319

quadruplet

319

relatively

14

triplet prime formula, predicate logic principal (a sum of money)

14

319 289 21

principal axis direction

265

transformation

264

This page has been reformatted by Knovel to provide easier navigation.

Index Terms principal ideal

Links 313

principal normal section, surface

248

space curve

239

principal quantities

241

11

principal value integral, improper

452

inverse hyperbolic function

698

inverse trigonometric function logarithm

84

698

696

698

principle Cauchy

606

contracting mapping

606

extensionality

291

extrapolation principle

899

Neumann

307

of bivalence

286

prism

151

lines

151

regular

151

probability area interpretation

750

conditional

748

definition

747

total

748

probability integral

757

This page has been reformatted by Knovel to provide easier navigation.

Index Terms probability measure

Links 814

ergodic

818

invariant

814

probability paper

772

probability theory

743

probability vector

765

745

problem Cauchy

516

Dirichlet

526

discrete

918

inhomogeneous

535

668

multidimensional computation of adjustments

919

Neumann

668

regularized

278

scheduling

859

shortest way

349

Sturm–Liouville

513

two-body

518

process birth process

767

death process

767

orthogonalization process

280

Poisson process

766

stochastic

763

This page has been reformatted by Knovel to provide easier navigation.

Index Terms product

Links 7

algebraic

364

Cartesian fuzzy sets

367

n-ary

368

sets

294

cross

183

derivative

379

direct group

301

Ω-algebra

339

dot

183

drastic

364

dyadic

255

dyadic, tensors

264

inner product

613

Kronecker

258

n times direct

341

product sign

7

rules of calculation

7

scalar

183

vector

183

programming

255

844

computer algebra system

952

continuous dynamic

875

discrete dynamic

876

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

programming (Cont.) Bellman functional equation

876

Bellman functional equation method

878

Bellman optimality principle

878

constraint dynamic

876

constraint static

876

knapsack problem

876

problem

875

purchasing problem

876

state vector

875

linear

844

linear, transportation problem

855

Maple

974

Mathematica

963

problem dual problem

854

primal problem

854

programming, discrete continuous dynamical

875

cost function

876

decision

875

decision space

875

dynamic

875

functional equation

877

Bellman

876

method

878

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

programming, discrete (Cont.) knapsack problem

876

minimum interchangeability

877

minimum separability

877

n-stage decision process

875

optimal policy

878

optimal purchasing policy

879

purchasing problem

876

state costs

875

state vector

875

880

programming, linear constraints

844

forms

844

general form

844

properties

846

scheduling problem

859

projection sides

142

projection theorem, orthogonal space

614

projector

626

proof by contradiction

5

constructive

6

direct

5

indirect implication mathematical induction

288 5 5

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

proof (Cont.) step from n to n + 1

5

proportionality direct

62

inverse

64

proportions

17

proposition

286

dual

341

propositional logic

286

basic theorems

287

expression

286

propositional operation

286

extensional

286

propositional variable

286

protocoll

770

pseudonorm

622

pseudorandom numbers

781

pseudoscalar

270

pseudotensor

268

270

pseudovector

268

270

Ptolemy’s theorem

136

pulse, rectangular

695

pyramid

152

frustum

152

n-faced

152

regular

152

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

pyramid (Cont.) right

152

truncated

152

Pythagoras general triangle

142

right-angled triangle

141

Q QR algorithm

283

QR decomposition

891

quadrangl

134

circumscribing

136

concave

136

convex

136

inscribed

136

quadrant relations, trigonometric functions quadrant, Cartesian coordinates

891

136

77 189

quadratic curve

205

surface

223

quadratic form first fundamental, of a surface

245

index of inertia

282

real positive definite

282

second fundamental, surface

248

transformation

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

quadratic form (Cont.) principal axis

281

quadrature formula

895

Gauss type

897

Hermite

896

integral equation

570

interpolation quadrature

896

Lobatto type

898

quadruple (ordered four-tuple)

294

quantification, restricted

290

quantifier

289

quantile

750

quantity, infinitesimal

439

quantum number

539

energy

539

magnetic

542

orbital angular momentum

541

vibration quantum number

544

quartic

445

96

quasiperiodic

804

queuing

767

theory quintuple (ordered 5-tuple) quotient

767 294 1

derivative

380

differential

377

12

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

quotient (Cont.) set

297

R radial equation

540

radian definition

130

measure

130

radicals, ordinary differential equation radicand

491 8

radius circle

139

circumcircle

142

convergence

414

curvature

230

curve

229

space curve

240

curvature, extremal

556

polar coordinates

190

197

principal curvature surface

247

short

132

torsion, space curve

242

vector

181

raising to a power complex numbers

37

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

raising to a power (Cont.) real numbers

7

9

random number

768

781

application

782

congruence method

781

construction

782

different distributions

782

pseudorandom

781

uniformly distributed

781

random numbers, table random variable

1091 749

continuous

749

discrete

749

independent

752

mixed

749

multidimensional

752

two-dimensional

777

750

777

random vector mathematical statistics

768

multidimensional random variable

752

random walk process range

784 47

operator

598

sample function

770

statistics

772

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

rank matrix

255

tensor

262

vector space

316

rate effective

22

nominal

22

of interest

22

ray point

493

ray, notion

128

Rayleigh–Ritz algorithm

283

reaction, chemical, concentration

116

real part (complex number)

34

rebate

21

rectangle

135

rectangular formula, simple

896

rectangular impulse

712

rectangular pulse

695

rectangular sum

896

rectification

107

Reduce (computer algebra system)

950

reduction

21

angle

174

reduction formula, trigonometric functions

77

reflection orincide, Schwarz

679

reflection, position coordinate

269

This page has been reformatted by Knovel to provide easier navigation.

Index Terms region

Links 118

multiply-connected

118

non-connected

118

simply-connected

118

two-dimensional

118

analysis

777

coefficient

779

line

778

linear

778

multidimensional

779

regula falsi

883

regularity condition

861

regularization method

285

regularization parameter

278

relation

294

binary

294

congruence relation

338

equivalence relation

296

fuzzy-valued

367

inverse

295

less or equal than (≤ relation)

289

matrix

294

n-ary

294

n-place

294

order relation

296

product

295

779

873

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

relaxation method

893

relaxation parameter

893

reliability testing

767

relief, analytic function

670

remainder estimation

411

series

404

term

404

Remes algorithm rent

922 25

ordinary, constant representation of groups

25 303

adjoint

304

direct product

305

direct sum

305

equivalent

304

faithful

303

identity

303

irreducible

305

non-equivalent

304

particular

303

properties

303

reducible

305

reducible, complete

306

representation matrix

303

representation space

303

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

representation of groups (Cont.) subspace

305

true

303

unitary

304

representation theorem (fuzzy logic)

362

resection Cassini

149

Snellius

148

residual spectrum

621

residual sum of squares

277

residual vector

277

891

residue quadratic modulo m

327

complex function

691

theorem

692

residue class

325

addition

325

multiplication

325

primitive

326

relatively prime

326

ring

313

ring, modulo m

325

residue theorem

690

application

693

residuum

277

325

890

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

resolvent set

566

568

585

620

resonance torus

840

reversing the order of the bits

927

rhombus

135

Riemann formula

528

function

528

integral

439

comparison with Lehesgue integral

451

comparison with Stieltjes integral

451

method

528

sum

439

surface, many-sheeted

683

theorem

407

right screw rule

661

right singular vector

285

right-hand coordinate system

207

right-hand rule

183

ring

313 factor ring

314

homomorphism

314

theorem

314

isomorphism

314

subring

313

risk theory

661

21

This page has been reformatted by Knovel to provide easier navigation.

620

Index Terms

Links

Ritz method

558

Rn, n-dimensional Euclidean vector space

254

Romberg method

898

906

root complex function

671

complex number

38

equation

43

non-linear notion N-th of unity real number square root, complex theorem of Vieta

881 8 926 8 675 43

root locus theory

886

root test of Cauchy

406

rotation definition

652

different coordinates

653

mapping

815

potential field

654

remark

648

vector components

657

vector field

652

rotation angle

212

rotation field pure

666

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

rotation field (Cont.) zero-divergence field

666

rotation invariance

265

rotation matrix

190

orthogonal

262

rotation number

841

rotator, rigid

540

round-off

937

error

939

measurement error method

257

785 939

round-tour problem

859

row echelon form, system of linear equations

276

row sum criterion

892

Ruelle–Takens–Newhouse scenario

840

rule

903 Adams and Bashforth

903

Bernoulli-I’Hospital

54

Cartesian rule of signs

885

Cramer

275

De Morgan

287

Descartes

211

45

Guldin, first rule

451

Guldin, second rule

451

linguistic

373

Milne

903

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

rule (Cont.) Sarrus ruled surface

261 250

rules composition

369

divisibility, elementary

318

Neper

169

Runge–Kutta method

901

S saddle

802

saddle form

249

809

saddle point differential equation

493

Lagrange function

860

sample

752

function

768

random

768

size

752

summarizing the sample

770

variable

768

Sarrus rule

768

261

scalar invariant

263

notion

180

This page has been reformatted by Knovel to provide easier navigation.

Index Terms scalar field

Links 641

axial field

642

central field

641

coordinate definition

642

directional derivative

647

gradient

648

plane

641

scalar matrix

252

scalar product

183

Hilbert space

613

representation in coordinates

186

rotation invariance property

270

two functions

917

vectors

255

649

316

188

scale cartography

144

equation

114

factor

114

logarithmic

114

notion

114

semilogarithmic

115

scenario, Ruelle–Takens–Newhouse

840

Schauder fixed-point theorem

631

scheduling problem

859

146

scheme Falk

254

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

scheme (Cont.) Young

306

Schmidt, orthogonalization method

577

Schoenflies symbolism

307

Schrödinger equation non-linear, partial

545

time-dependent

537

time-independent

537

547

Schrödinger’s equation linear

536

Schur’s lemma

306

Schwarz exchange theorem

393

reflection principle

679

Schwarz–Buniakowski inequality

613

Schwarz–Christoffel formula

677

screw left

242

right

242

search procedure, numerical

865

secant geometric definition hyperbolic theorem

130 87 138

trigonometric

76

secant–tangent theorem

138

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

section, golden

2

sector formula

664

sector, spherical

157

4

192

segment normal

227

notion

128

polar normal

227

polar tangent

227

tangent

227

self-similar

821

semantically equivalent

287

expressions

343

semifocal chord ellipse

198

hyperbola

200

parabola

69

semigroup free

299 299

semimonotone

611

seminorm

622

semiorbit

795

sense class

133

of a figure

203

132

sensitive, with respect to the initial values

827

sentence

286

sentence-forming functor

286

This page has been reformatted by Knovel to provide easier navigation.

843

Index Terms

Links

sentential operation

286

separable sets

623

separation constant

538

theorems (convex sets)

623

variables

487

522

loop

806

838

surface

804

810

538

separatrix

sequence

402

bounded

596

Cauchy sequence

605

convergence

604

finite

595

in metric space

604

infinite

402

numbers

402

bounded

402

bounded above

403

bounded below

403

convergence

403

converging to zero

596

divergence

403

finite

595

law of formation

402

limit

403

596

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

sequence (Cont.) monotone

402

term

402

series

18

alternating

408

arithmetic

18

Banach space

610

Clebsch–Gordan

306

comparison criterion

405

constant term

404

convergence

406

absolute

414

non-uniform

412

uniform

412

convergence theorems

404

convergent

404

D’Alembert’s ratio test

405

definite

402

414

18

divergence

407

divergent

404

expansion

415

Fourier

418

Laplace transformation

717

Laurent

690

Maclaurin

416

power series

414

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

series (Cont.) Taylor finite Fourier complex representation function domain of convergence general term geometric infinite

387

415

18 419 419 412 412 404 19 19

harmonic

404

hypergeometric

511

infinite

402

integral test of Cauchy

406

Laurent

690

Maclaurin

416

Neumann

566

Neumann series

618

partial sum

404

power

414

expansion

415

inverse

415

remainder

404

root test of Cauchy

406

sum

404

Taylor

387

404

404

412

415

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

series (Cont.) uniformly convergent continuity

413

differentiation

413

integration

413

properties

413

Weierstrass criterion, uniform convergence set

413 290

absorbing

797

algebra, fundamental laws

292

axioms of closed sets

604

axioms of open sets

604

Borel

634

bounded in metric space

604

cardinality

298

closed

604

closure

605

compact

626

complex numbers

798

34

convex

597

dense

605

denumerable infinite

298

disjoint

292

element

290

empty

291

equality

291

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

set (Cont.) equinumerous

298

fundamental

292

fuzzy

358

image

48

infinite

298

integers

1

invariant

797

chaotic

826

fractal

826

stable

797

irrational numbers

2

linear

595

manyness

298

measurable

633

natural numbers

614

1

non-denumerable infinite

298

notion of set

290

open

603

operation Cartesian product

294

complement

292

difference

293

intersection

292

symmetric difference

293

union

292

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

set (Cont.) operations

291

order-bounded

600

original space

48

power

298

power set

291

quotient set

297

rational numbers

1

real numbers

2

relative compact

626

subset

291

theory

290

universal

292

void

291

sets

290 coordinates x, y

294

difference, symmetric

293

sexagesimal degree

130

shift mapping

818

shooting method

907

simple

907

shore-to-ship bearing

171

side or leg of an angle

128

side-condition

553

Sierpinski carpet

822

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Sierpinski (Cont.) gasket

822

sieve of Eratosthenes

318

σ-additivity

633

σ-algebra

633

Borelian

634

sign of a function

49

signal

738

analysis

738

synthesis

738

signature, universal algebra

338

significance

777

level similarity transformation

751

774

280

281

848

849

simplex method revised

853

step, revised

854

tableau

849

revised Simpson’s formula

853 897

simulation digital

781

Monte Carlo

781

geometric definition

130

sine

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

sine (Cont.) hyperbolic geometric definition trigonometric

87 131 74

sine integral

458

694

sine law

142

164

sine–cosine law

164

polar

164

sine–Gordon equation

545

single-target method

907

singleton

361

singular value

285

decomposition singular value decomposition

547

818

285 285

singularity analytic function

671

essential

671

isolated

690

pole

691

removable

671

sink

802 vector field

651

vertex

356

sinusoidal amount

691

809

82

slack variable

845

Slater condition

861

This page has been reformatted by Knovel to provide easier navigation.

Index Terms slide rule logarithmic scale

Links 10 114

slope plane

194

tangent

227

small circle

158

arclength

175

course angle

176

intersection point

176

radius, plane

175

radius, spherical

175

175

smoothing continuous problem

916

parameter

930

spline

929

cubic

929

Sobolev space

611

solenoid

826

solid angle

151

soliton antikink

548

antisoliton

547

Boussinesq

549

Burgers

549

differential equation, partial, non-linear

544

Hirota

549

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

soliton (Cont.) interaction

545

Kadomzev-Pedviashwili

549

kink

548 lattice

548

kink-antikink

548

collision

548

doublet

548

kink-kink collision

548

Korteweg de Vries

546

non-linear, Schrödinger

547

solution point

845

product form, method separation of variables 522 SOR method (successive overrelaxation)

893

source

802

809

distribution continuous

667

discrete

667

field irrotational

665

pure

665

vector field

651

vertex

356

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

space abstract

48

Banach

610

complete, metric

605

directions

210

finite-dimensional

626

higher-dimensional

117

Hilbert

613

infinite-dimensional

597

isometric

609

Kantorovich

601

linear

594

P

L space

637

metric

602

completion

608

convergence of sequence

604

normable

610

separable

605

non-reflexive

624

606

626

normed axiom

609

properties

610

ordered normed

611

orthogonal

614

reflexive

624

Riesz

600

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

space (Cont.) second adjoint

624

separable

605

Sobolev

611

unitary

613

vector

594

space curve

238

binormal

239

coordinate equation

241

curvature

240

direction

238

equation

214

moving trihedral

239

normal plane

238

241

osculating plane

238

241

principal normal

239

241

radius of curvature

240

radius of torsion

242

tangent

238

torsion

242

vector equation

238

space inversion

241

238

241

241

269

mixed product

270

scalar product

270

spectral radius

618

spectral theory

620

This page has been reformatted by Knovel to provide easier navigation.

Index Terms spectrum

Links 725

amplitude

725

continuous

621

frequency

725

linear operator

620

phase

725

sphere

156

ellipsoid

220

equation, three forms

243

spherical biangle

161

coordinates

209

diagon

161

distance

158

field

641

helix

177

lune

161

Archimedean

103

hyperbolic

104

logarithmic

104

233

spline basis spline

930

bicubic

930

approximation

932

interpolation

930

cubic

928

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

spline (Cont.) interpolation

928

smoothing

929

interpolation

914

natural

928

normalized B-spline

930

periodic

928

smoothing

929

matrix

252

tensor

265

928

spur

square

135

stability absolutely stable

904

first approximation

802

integration of differential equation

904

Lyapunov stability

801

orbital

801

periodic orbit

802

perturbation of initial values

904

round-off error, numerical calculation

939

structural

811

812

deviation

751

789

error

788

normal distribution

757

standard 791

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

state degenerate

539

particle

536

space, stochastic

763

stationary

537

statistics

743

descriptive

770

estimate

769

mathematical

743

sample function

768

steady state

795

Steffensen method

883

767

step from n to n + 1

5

function

695

interval parameter

894

size

901 change

712

732

902

steradian

151

stereometry

150

Stieltjes integral

626

comparison with Riemann integral

451

notion

451

transformation

705

Stirling formula

460

This page has been reformatted by Knovel to provide easier navigation.

742

Index Terms

Links

stochastic basic notions

763

chains

764

process

763

processes

763

stochastics

743

Stokes, integral theorem

664

strangling

264

stream function

679

strip, characteristic

517

strophoide

95

structure algebraic

286

classical algebraic

298

Sturm chain

44

function

44

sequence

886

theorem

44

Sturm–Liouville problem

513

subdeterminant

259

subdomain

930

subgraph

347

induced subgroup criterion

347 300 300

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

subgroup (Cont.) cyclic

300

invariant

301

normal

301

trivial

300

subinterval

439

subnormal

227

subring

313

trivial subset

313 291

affine

595

linear

595

open

603

subspace

315

affine

595

criterion

315

invariant, representation of groups

305

subtangent

595

227

subtraction complex numbers computer calculation polynomials rational numbers subtractive cancellation sum

36 937 11 1 937 6

algebraic

364

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

sum (Cont.) drastic

364

of the digits alternating, first order

320

alternating, second order

320

alternating, third order

320

first order

320

second order

320

third order

320

residual squares

918

Riemann

439

rules of calculation

6

summation sign

6

transverse

320

vectors

181

summation convention, Einstein’s

262

superposition fields

667

law, fields

667

non-linear

547

oscillations

82

principle

681

differential equation, linear

505

differential equations, higher order

499

supplementary angle formulas

79

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

support compact

740

measure

814

membership function

358

supporting functional

623

supremum

600

surface

243

area, double integral

472

B–B representation

932

barrel

157

block

152

cone

156

conical

156

constant curvature

249

cube

152

curvature of a curve

247

cylinder

154

developable

250

element

247

element, vector components

658

equation

223

in normal form

220

space

213

equation, space

243

first quadratic fundamental form

246

Gauss curvature

249

214

249

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

surface (Cont.) geodesic line

250

harmonic

542

integral

477

first type

477

general form

483

second type

481

line element

245

line of curvature

248

metric

247

minimal

249

normal

244

normal vector

244

oriented

481

patch, area

247

polyhedron

151

principal normal section

248

pyramid

152

quadratic

223

radius of principal curvature

247

rectangular parallelepiped

152

rectilinear generator

222

representation with splines

928

rotation symmetric

213

ruled

250

second order

220

661

662

482

246

223

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

surface (Cont.) central surfaces

224

invariant signs

224

types

223

second quadratic fundamental form

248

sphere

156

tangent plane

244

torus

157

transversal

804

surplus variable, linear programming

246

848

switch algebra

340

function

344

value

344

Sylvester, theorem of inertia

344

282

symbol internal representation (computer)

933

Kronecker

265

Legendre

327

symmetry axial

133

central

132

element

307

Fourier expansion

420

group

308

applications in physics

312

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

symmetry (Cont.) crystallography

310

molecules

308

quantum mechanics

312

operation

307

crystal lattice structure

310

improper orthogonal mapping

307

reflection

307

rotation

307

without fixed point

307

with respect to a line

133

system chaotic according to Devaney

827

cognitive

373

complete

616

differential equation, higher order

495

linear

503

differential equations, partial canonical system

517

normal system

517

dynamical

795

chaotic

827

conservative

796

continuous

795

r

C -smooth

795

discrete

796

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

system (Cont.) dynamical (dynamical) dissipative

796

ergodic

815

invertible

795

mixing

815

motion

795

time continuous

795

time discrete

795

time dynamical

796

volume decreasing

796

volume preserving

796

volume shrinking

796

equation numerical solution

887

four points

213

generators

300

knowledge based interpolation

375

linear equations

271

compatible

273

consistent

273

fundamental system

273

homogeneous

272

inconsistent

273

inhomogeneous

272

overdetermined

277

272

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

system (Cont.) pivoting

274

row echelon form

276

solvable

273

trivial solution

273

mixing

827

Morse–Smale

813

normal equations

277

orthogonal

614

orthonormal

615

term-substitutions

338

trigonometric

614

table with double entry

119

tacnode, curve

232

T

tangent circle

198

formula

142

geometric definition

130

hyperbolic geometric definition

87 131

law

142

plane

158

393

surface

244

246

plane curve

226

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

tangent (Cont.) polygon

137

space curve

238

trigonometric

241

76

tautology Boolean function

342

predicate logic

290

propositional logic

288

Taylor expansion

55

one variable

415

several variables

395

vector function

641

formula

395

two variables

394 387

analytic function

689

one variable

415

several variables

395

theorem several variables

415

387 394

telegraphic equation

529

telephone-call distribution

767

tensor

262

addition, subtraction

415

387

several variables

series

387

268

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

tensor (Cont.) alternating

265

antisymmetric

264

components

262

contraction

264

definition

262

dyadic product

264

eigenvalue

264

generalized Kronecker delta

265

inertia

263

invariant

265

multiplication

268

oversliding

268

product

263

vectors

268

255

rank n

262

rank 0

263

rank 1

263

rank 2

263

rules of calculation

263

264

skew-symmetric

264

268

spur

265

symmetric

264

tension

263

trace

265

tensor product approach

268

931

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

tent mapping

814

term algebra

339

test problem, linear

904

test, statistical

772

tetraeder

213

tetragon

136

tetrahedron

153

Thales theorem

139

141

theorem Abel

414

Afraimovich–Shilnikov

840

alternating point

920

Andronov–Pontryagin

812

Andronov–Witt

802

Apollonius

199

Arzeli–Ascoli

626

Baire category

606

803

Banach continuity, inverse operator

619

fixed-point

629

Banach–Steinhaus

618

Bayes

748

Berge

354

binomial

12

Birkhoff

340

Block–Guckenheimer–Misiuriewicz

827

749

815

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

theorem (Cont.) Bolzano one variable several variables

59 123

boundedness of a function one variable

60

several variables

123

Cauchy integral theorem

686

Cayley

302

353

center manifold differential equations

828

mappings

833

Chebyshev

434

Chinese remainder

326

Clebsch–Gordan

306

closed graph

618

constant analytic function

671

convergence, measurable function

636

decomposition

297

Denjoy

842

differentiability, respect to initial conditions

795

Dirac

351

Douady–Oesterlé

824

Euclidean theorems Euclidean algorithm

141 322

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

theorem (Cont.) Euler

154

Euler–Hierholzer

350

Fatou

636

Fermat

329

Fermat–Euler

329

329

386

fixed point theorem Banach

606

fixed-point Banach

629

Brouwer

631

Schauder

631

Floquet

801

fundamental integral calculus

456

Girard

164

Grobman–Hartman

809

811

Hadamard–Perron

805

810

Hahn (extension theorem)

623

Hellinger–Toeplitz

618

Hilbert–Schmidt

628

Holladay

929

Hurwitz

501

intermediate value one variable several variables KAM (Kolmogorov–Arnold–Moser)

59 123 813

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

theorem (Cont.) Krein–Losanovskij

619

Kuratowski

355

Lagrange

301

Lebesgue, majorized convergence

636

Ledrappier

824

Leibniz

408

Lerey–Schauder

631

Levi, B.

636

limits functions

403

sequences of numbers

403

Liouville

671

Lyapunov

801

maximum value, analytic function

671

Meusnier

247

nested balls

606

Ore

351

Oseledec

818

Palis–Smale

813

Picard–Lindelöf

608

Poincaré–Bendixson

803

Posa

351

Ptolemy’s

136

799

795

Pythagoras general triangle

142

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

theorem (Cont.) orthogonal space

614

right-angled triangle

141

Radon–Nikodym

636

Riemann

407

Riesz

622

Riesz–Fischer

616

Rolle

386

Schauder

627

Schwarz, exchange

393

Sharkovsky

827

Shilnikov

838

Shinai

827

Shoshitaishvili

828

Smale

838

stability in the first approximation

802

Sturm

44

superposition law

667

Sylvester, of inertia

282

Taylor

387

one variable

415

several variables

394

Thales

139

total probability

748

Tutte

354

variation of constants

800

141

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

theorem (Cont.) Weierstrass one variable several variables

412

605

60 123

Wilson

329

Wintner–Conti

798

Young

822

823

theory distribution

712

elementary number

318

field

667

function

669

graph, algorithms

346

probability

743

risk

21

set

290

spectral

620

vector fields

640

theta function

703

time frequency analysis

741

tolerance

362

topological conjugate

808

equivalent

808

torsion, space curve

242

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

torus

157

differential equation, linear, autonomous

801

dissolving

839

formation

835

invariant set

804

losing smoothness

840

resonance torus

840

total curvature

826

827

840

243

trace matrix

252

tensor

265

tractrix

106

trail

349 Euler open

350 350

trajectory

795

transform

705

transformation Cartesian into polar coordinates

398

covering

299

determinant

211

element of area

472

element of curved surface

480

geometrical

141

Hopf–Cole

549

identical

11

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

transformation (Cont.) invariance

265

linear

262

method

283

orthogonal coordinates

210

principal axes

264

quadratic form

281

similarity

280

wavelet transformation

738

316

598

281

281

transition matrix

764

probability, stochastic

764

transitivity

765

29

translation invariance

265

primitive

310

transport, network

356

transportation problem

855

transposition law

288

trapezoid

135

Hermite’s trapezoidal formula

897

trapezoidal formula

896

sum

896

traversing

147

This page has been reformatted by Knovel to provide easier navigation.

Index Terms tree

Links 352

hight

352

ordered binary

352

regular binary

352

rooted

352

spanning

352

minimum

353

353

triangle altitude

132

area

193

bisector

132

center of gravity

132

circumcircle

132

congruent

133

coordinates

914

equilateral

132

Euler

162

incircle

132

inequality

181

axioms of norm complex numbers

258 30

metric space

602

norm

609

real numbers

30

unitary space

613

isosceles

132

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

triangle (Cont.) median

132

orthocenter

132

Pascal

13

plane

131

area

141

basic problems

143

Euclidean theorems

141

general

141

incircle

143

radius of circumcircle

142

right-angled

132

tangent formula

142

polar

162

similar

133

spherical

161

basic problems

167

calculation

167

Euler

162

oblique

169

right-angled

168

143

141

triangular decomposition

888

matrix

253

lower

253

upper

253

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

triangularization, FEM (finite element method)

911

triangulation, geodesy first fundamental problem

147

second fundamental problem

148

trigonometry plane

141

spherical

158

trihedral angle

151

moving

238

triple (ordered 3-tuple)

294

triple integral

474

application

163

478

trochoid

100

truncation, measurement error

785

truth function

286

conjunction

286

disjunction

286

equivalence

286

implication

286

NAND function

288

negation

286

NOR function

288

table

286

value

286

287

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

turbulence

796

Tutte theorem

354

two lines, transformation

205

two-body problem

518

type, universal algebra

338

839

U umbilical point

249

uncertainty absolute

789

fuzzy

358

relative

790

ungula, cylinder

155

union fuzzy sets

363

sets

292

universal quantifier

289

universal substitution

436

urn model

753

364

V vagueness

358

valence in-valence

346

out-valence

346

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

value expected, measurent

751

system (function of several variables)

117

true, measurement

788

van der Pol differential equation

831

variable artificial

852

basic

848

Boolean

342

bound variable, predicate logic

289

dependent free, predicate logic independent

47 289 47

linguistic

359

non-basic

848

propositional

286

random

749

variance

271

117

271

751

distribution

751

sample function

769

statistics

771

two-dimensional distribution

778

variation function of constants, method

60 499

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

variation of constants differential equation, linear

505

theorem

800

variational calculus

550

auxiliary curve

552

comparable curve

552

Euler differential equation

552

side-condition

550

variational equation equilibrium point points variational problem

802

819

911

809 550

brachistochrone problem

551

Dirichlet

557

extremal curves

551

first order

550

first variation

559

functional

551

higher order

550

higher order derivatives

554

isoperimetric, general

551

more general

558

numerical solution

558

direct method

558

finite element method

559

gradient method

559

Ritz method

558

906

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

variational problem (Cont.) parameter representation

550

parametric representation

555

second variation

559

several unknown functions

555

side-conditions

553

simple one variable

551

several variable

556

variety

339

vector absolute value

180

affine coordinates

182

axial

180

reflection behavior base

268 186

base vector reciprocal

185

bound

180

Cartesian coordinates

182

column

253

components

646

conjugate

867

coordinates

182

Darboux vector

243

decomposition

182

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

vector (Cont.) diagram, oscillations

83

differentiation rules

640

direction in space

180

direction, vector triple

180

directional coefficient

183

expansion coefficient

183

fields

640

free

180

left singular

285

length

189

line

646

magnitude

180

matrix

253

metric coefficients

185

notion

180

null vector

181

polar

180

reflection behavior

268

pole, origine

181

position vector

181

complex-number plane radius vector complex-number plane

34 181 34

reciprocal

186

reciprocal basis vectors

186

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

vector (Cont.) residual

277

right singular

285

row

253

scalar invariant

655

sliding

180

space

595

stochastic

764

unit

180

zero vector

181

vector algebra

180

geometric application

189

notions and principles

180

vector analysis

640

vector equation line

188

plane

188

space curve

238

vector equations

187

vector field

643

Cartesian coordinates

644

central

643

circular field

644

components

646

contour integral

660

coordinate definition

644

241

795

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

vector field (Cont.) cylindrical

644

cylindrical coordinates

645

directional derivative

648

divergence

651

point-like source

666

rotation

652

sink

651

source

651

spherical

644

spherical coordinates

645

vector function

640

derivative

640

differentiation

640

hodograph

640

linear

267

scalar variable

640

Taylor expansion

641

vector gradient

650

vector iteration

283

vector lattice

600

homomorphism

643

284

601

vector potential

666

vector product

183

hints

255

representation in coordinates

187

This page has been reformatted by Knovel to provide easier navigation.

Index Terms vector space

Links 314

all null sequences

596

bounded sequences

596

B(T)

596

C([a, b])

596

(κ)

C ([a, b])

596

complex

595

convergent sequences

596

Euclidean

316

finite sequence of numbers

595

n

F

595

F(T)

596

functions

596

infinite-dimensional

315

L I

p

594

597

637

p

596

n-dimensional

315

n-dimensional Euclidean

254

ordered by a cone

599

partial ordering

599

real

314

s of all sequences

595

sequences

595

595

vector subspace stable

805

810

unstable

805

810

This page has been reformatted by Knovel to provide easier navigation.

Index Terms vectors

Links 180

253

angle between

189

collinear

181

collinearity

184

commutative law

255

coplanar

181

cyclic permutation

207

double vector product

184

dyadic product

255

equality

180

Lagrange identity

185

linear combination

181

184

mixed product

184

187

Cartesian coordinates orthogonality

185 184

products affine coordinates

185

Cartesian coordinates

185

products, properties

183

scalar product

255

Cartesian coordinates

185

representation in coordinates

186

sum

181

tensor product

255

triple product

184

vector product This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

vectors (Cont.) representation in coordinates Venn diagram verification, proof

187 291 5

vertex angle

128

degree

346

ellipse

198

graph

346

hyperbola

200

initial

346

isolated

346

level

352

parabola

203

plane curve

232

sink

356

source

356

terminal

346

vertical angle

143

vertices, distance

349

vibration, differential equation bar

524

round membrane

525

string

523

Vieta, root theorem

43

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Volterra integral equation

Links 561

first kind

583

second kind

585

607

volume barrel

157

block

152

cone

156

cube

152

cylinder

154

double integral

472

element, vector components

658

hollow cylinder

155

obelisk

153

parallelepipedon with vectors

189

polyhedron

151

prism

152

pyramid

152

rectangular parallelepiped

152

sphere

156

subset

633

tetraeder

213

torus

157

triple integral

478

wedge

154

volume derivative

649

volume integral

474

This page has been reformatted by Knovel to provide easier navigation.

Index Terms volume scale

Links 114

W Walsh functions

742

systems

742

wave amplitude

82

frequency

82

length

82

period

82

phase

82

plane

739

wave equation n-dimensional

534

one-dimensional

730

Schrödinger equation

537

wave function classical problem

534

heat-conduction equation

535

Schrödinger’s equation

536

wavelet

739

Daubechies

740

orthogonal

740

transformation

738

discrete

739

741

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

wavelet (Cont.) discrete, Haar

741

dyadic

740

fast

741

Weber differential equation

543

function

507

wedge

154

Weibull distribution

759

Weierstrass approximation theorem

605

criterion uniform convergence

413

function

703

theorem

412

one variable several variables

60 123

weight measurement

791

of orthogonality

513

statistical

751

weighting factor, statistical witch of Agnesi word

788 94 602

work (mechanics) general

467

special

449

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Wronskian determinant

Links 498

800

Y Young scheme

306

Z zenith

143

distance

143

zero matrix

251

zero-point translational energy

539

zero-point vibration energy

544

zeropoint

1

z-transformation

731

convolution

734

damping

734

definition

732

difference

733

differentiation

734

integration

734

inverse

735

original sequence

732

rules of calculation

733

summation

733

transform

732

translation

733

z-transformable

732

This page has been reformatted by Knovel to provide easier navigation.

Mathematical Sumbols A

MATHEMATICAL SYMBOLS Relational Symbols = equal to G identically equal to := equal to by definition << much less than + partial order relation

5 less than or equal to 2 greater than or equal to

approximately equal to less than > greater than >> much greater than + partial order relation

M

<

# unequal to, different from corresponding to

Greek Alphabet A a Alpha H 7 Eta N u Nu T r Tau

B 4 Beta B 829 Theta E Xi T u Upsilon

l'y I1

0o @ .p

Gamma Iota Omicron Phi

4b K

K

IT R Xx

Delta Kappa Pi Chi

E

E

AX P p Ii

Epsilon Lambda Rho Psi

Z

6

Zeta

M p

Mu

Cu

Sigma Omega

Rw

Constants const R = 3.14159,

constant amount (constant) ratio of the perimeter of the circle to the diameter

C = 0.57722.. . e = 2.71828.. .

Euler constant base of the natural logarithms

Algebra A, B 7.4, ;.i AAB,n AVB,U

A*B A*B A, B , C , . . . -

propositions negation of the proposition A conjunction, logical AND disjunction, logical OR implication, IF A T H E 3 B equivalence, A IF AND ONLY IF B

set of natural numbers sets set of the integers closure of the set A or complement of set of the rational numbers A with respect to a universal set set of the real numbers A is a proper subset of B AcB set of the positive real numbers A is a subset of B AEB n-dimensional Euclidean vector space difference of two sets A\B set of the complex numbers symmetric difference AAB relation product Cartesian product AxB z is not an element of A z is an element of A xEA empty set, zero set cardinal number of the set A card A intersection of n sets A, intersection of two sets AnB union of n sets A, union of two sets AUB there exists an element 2 for all elements z V I {z : p ( z ) } , set of all z with the {z E X : p(z)} subset of all z from X of the property p(z) tzlp(z)} property ~ ( 4 :'2 X f Y mapping T from the space X I isomorphy of groups -R equivalence relation into the space Y residue class addition 0 residue class multiplication H = HIe H2 orthogonal decomposition of space H A @ B Kronecker product SUPP support supremum: least upper bound of the non-empty set M ( M C R) bounded above sup M infimum: greatest lower bound of the non-empty set M ( M C R) bounded helow inf M

A

B Mathematical Sumbob [a,bl (a. b).]a,b[ (a.b].]a,b] [a,b ) , [a,b[

closed interval, Le., open interval, Le., interval open from le!?, i.e, interval open from right, Le.,

{z E R: a 5 z {z E R: a < 2 {z E R: a < z

5 b} < b] 5 b} {z E R: a 5 z < b}

sign of the number a, e.g., sign (13)= *l, sign0 = 0 absolute value of the number a a to the power m, a to the m-th square root of a n-th root of a logarithm of the number a to the base b, e.g., log, 32 = 5 decimal logarithm (base 10) of the number a, e.g., lg 100 = 2 natural logarithm (base e ) of the number a, e.g., h e = 1

a I bmodm. a I b(m) g.c.d.(al,al,,. . , a,) l.c.m.(al. a,,. . . , a n )

a is a divisor of b, a devides b, the ratio of a to b a is not a divisor of b a is congruent to b modulo m, Le., b - a is divisible by rn greatest common divisor of a l , a,,. . . ,a, least common multiple of a l , a,, . . . , a,, binomial coefficient, n over k

(3

Legendre symbol

n! = 1 2 . 3 ' . . . n factorial, e.g., 6! = 1 2 ' 3 4 5 , 6 = 720; specially: O! = l! = 1 (Zn)!!= 2 , 4 , 6 . . . . ( 2 n ) = 2" n!; in particular: O!! = l!!= 1 ( 2 n l)!!= 1' 3 , 5 , . , . , ( 2 n t 1)

+

A = (atj) AT

A-1 detA, D E = (6q 0 6t,

matrix 4 with elements a,j transposed matrix inverse matrix determinant of the square matrix A unit matrix zero matrix Kronecker symbol: 6,, = 0 for i # j and

6i,

= 1 for i = j

column vector in R" unit vector in the direction of (parallel to) a norm of a vectors in R3 basis vectors (orthonormed) of the Cartesian coordinate system coordinates (components) of the vector a' absolute value, length of the vector a' multiplication of a vector by a scalar scalar product, dot product vector product, cross product parallelepipedal product, mixed product (triple scalar product) zero vector

T G = (1: E )

tensor graph with the set of vertices V and the set of edges E

Mathematical Svmbob C

Geometry I/

orthogonal (perpendicular) equal and parallel

I

# A

N

parallel similar, e.g., AABC portional angle, e.g., SABC radian

N

ADEF; pre

triangle 3: r. arc segment, e.g., AB the arc between A and B rad degree minute as measure of angle and circular arc, e.g., 32" 14' 11.5" second the line segment between A and B the directed line segment from A to B , the ray from A to B

h

I

I I,

AB

A3B

Complex Numbers i (sometimes j) Re ( z ) Izl Z or z'

imaginary unit (i2 = -1) real part of the number z absolute value of z complex conjugate of z , e.g., z = 2 3i, Z=2-3i

I Im (2) argz Ln z

+

imaginary unit in computer algebra imaginary part of the number z argument of the number z logarithm (natural)of a complex number z

Trigonometric Functions, Hyperbolic Functions sin tan sec arcsin arctan

sine tangent secant principal value of arc sine (sine inverse) principal value of arc tangent (tangent inverse)

arcsec

principal value of arc secant (secant inverse)

sinh tanh sech Arsinh Artanh Arsech

hyperbolic sine hyperbolic tangent hyperbolic secant area-hyperbolic sine area-hyperbolic tangent area-hyperbolic secant

cos cot cosec arccos arccot

cosine cotangent cosecant principal value of arc cosine (cosine inverse) principal value of arc cotangent (cotangent inverse) arccosec principal value of arc cosecant (cosecant inverse) cosh hyperbolic cosine coth hyperbolic cotangent cosech hyperbolic cosecant Arcosh area-hyperbolic cosine Arcoth area-hyperbolic cotangent Arcosech area-hyperbolic cosecant

Analysis lim f(z)= B

A is the limit of the sequence (xn). We also write zn-+ A as n -+ m; e.g., Iim (I + =e n-bm B is the limit of the function f(x) as x tends to Q

f = o(g) for x -+ a f = O(g) for x --t Q

Landau symbol "small 0" means: f(x)/g(x) --t 0 as z --t a Landau symbol "big 0" means: f(x)/g(x) --t C (C= const, C # 0) as x

E >E:=, 1=1

sum of n terms for i equals 1 to n

lim x, = A

n+m

A)"

=-+a

n

rI>n:=,

product of n factors for i equals 1 to n

f ( 1) v( 1

notation for a function, e.g., 21 = f(x), u = q(x,y,z ) difference or increment, e.g., A x (delta x) differential, e.g., dx (differential of x)

,=l

A d d _ d2 dn _ dx' dx2' ""dxn

f'(z),f"(X)> f"'(z), f'4'(x), . . . , f ( " )2) ( or

j r , 1,.. . , y(*)

determination of the first, second, . . ., n-th derivative with respect to x

I

first, second,. . . ,n-th derivative of the function f(x) or of the function 21

-ia

3 Mathematical Svmbols

determination of the first, second, . . ., n-th partial derivative determination of the second partial derivative first with respect to I , then with respect to y first, second,

. . . partial derivative of function f ( x , y)

differential operator, e.g., Dy = u', D2y = y" gradient of a scalar field (gradip = Vip) divergence of a vector field (div J = V J )

.

rotation or curl of a vector field (rot J = V x J )

a - ja -+ -ka aZ ax -ay

V = -i+

A = - +a'- + ax2

a2

a2

ay2

aZ2

nabla operator, here in Cartesian coordinates (also called the Hamiltonian differential operator, not to be confused with the Hamilton operator in quantum mechanics) Laplace operator directional derivative, Le., derivative of a = a' g r a d 9 scalar field 'p into the direction a':

9 as

definite integral of the function f between the limits a and b line integral of the first kind with respect to the space curve C with arclength s integral along a closed curve (circulatory integral)

double integral over a planar region S

surface integral of the first kind over a spatial surface S (see (8.152b), p. 480) triple integral or volume integral over the volume V

surface integrals over a closed surface in vector analysis

A = may! A = max

expression A is to be maximized, similarly min!, extreme! expression A is maximal, similarly min, extreme.

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