Tables of integrals, series, and products

Table of Integrals, Series, and Products Seventh Edition

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Table of Integrals, Series, and Products Seventh Edition

I.S. Gradshteyn and I.M. Ryzhik Alan Jeffrey, Editor University of Newcastle upon Tyne, England

Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA

Translated from Russian by Scripta Technica, Inc.

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK ∞ This book is printed on acid-free paper. 

c 2007, Elsevier Inc. All rights reserved. Copyright  No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: [email protected]. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” For information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com ISBN-13: 978-0-12-373637-6 ISBN-10: 0-12-373637-4 PRINTED IN THE UNITED STATES OF AMERICA 07 08

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Contents Preface to the Seventh Edition Acknowledgments The Order of Presentation of the Formulas Use of the Tables Index of Special Functions Notation Note on the Bibliographic References 0

Introduction 0.1 0.11 0.12 0.13 0.14 0.15 0.2 0.21 0.22 0.23–0.24 0.25 0.26 0.3 0.30 0.31 0.32 0.33 0.4 0.41 0.42 0.43 0.44

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xxi xxiii xxvii xxxi xxxix xliii xlvii

Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sums of powers of natural numbers . . . . . . . . . . . . . . . . Sums of reciprocals of natural numbers . . . . . . . . . . . . . . Sums of products of reciprocals of natural numbers . . . . . . . Sums of the binomial coefficients . . . . . . . . . . . . . . . . . Numerical Series and Infinite Products . . . . . . . . . . . . . . The convergence of numerical series . . . . . . . . . . . . . . . Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . Examples of numerical series . . . . . . . . . . . . . . . . . . . Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of infinite products . . . . . . . . . . . . . . . . . . . Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions and theorems . . . . . . . . . . . . . . . . . . . . . . Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . Certain Formulas from Differential Calculus . . . . . . . . . . . . Differentiation of a definite integral with respect to a parameter . The nth derivative of a product (Leibniz’s rule) . . . . . . . . . . The nth derivative of a composite function . . . . . . . . . . . . Integration by substitution . . . . . . . . . . . . . . . . . . . . .

Elementary Functions Power of Binomials . . . . 1.11 Power series . . . . . . . 1.12 Series of rational fractions 1.2 The Exponential Function

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1.21 1.22 1.23 1.3–1.4 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.41 1.42 1.43 1.44–1.45 1.46 1.47 1.48 1.49 1.5 1.51 1.52 1.6 1.61 1.62–1.63 1.64 2

Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The basic functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) . . . . . . . . . . . . . . . . . The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions . . . . . . . . . . . Certain sums of trigonometric and hyperbolic functions . . . . . . . . . . . . . Sums of powers of trigonometric functions of multiple angles . . . . . . . . . . Sums of products of trigonometric functions of multiple angles . . . . . . . . . Sums of tangents of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . Sums leading to hyperbolic tangents and cotangents . . . . . . . . . . . . . . . The representation of cosines and sines of multiples of the angle as finite products The expansion of trigonometric and hyperbolic functions in power series . . . . Expansion in series of simple fractions . . . . . . . . . . . . . . . . . . . . . . Representation in the form of an infinite product . . . . . . . . . . . . . . . . . Trigonometric (Fourier) series . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of products of exponential and trigonometric functions . . . . . . . . . . Series of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Lobachevskiy’s “Angle of Parallelism” Π(x) . . . . . . . . . . . . . . . . . . . The hyperbolic amplitude (the Gudermannian) gd x . . . . . . . . . . . . . . . The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of logarithms (cf. 1.431) . . . . . . . . . . . . . . . . . . . . . . . . . . The Inverse Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . The domain of definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Indefinite Integrals of Elementary Functions Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.00 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.01 The basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.02 General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 General integration rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11–2.13 Forms containing the binomial a + bxk . . . . . . . . . . . . . . . . . . . . . . 2.14 Forms containing the binomial 1 ± xn . . . . . . . . . . . . . . . . . . . . . . 2.15 Forms containing pairs of binomials: a + bx and α + βx . . . . . . . . . . . . . 2.16 Forms containing the trinomial a + bxk + cx2k . . . . . . . . . . . . . . . . . . 2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x . . . . 2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx 2.2 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20 Introduction . . . . . . . . . . . . . . . . .√. . . . . . . . . . . . . . . . . . . 2.21 Forms containing the binomial a + bxk and x . . . . . . . . . . . . . . . . .

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26 27 27 28 28 28 31 33 36 37 38 39 39 41 42 44 45 46 51 51 51 52 53 53 55 56 56 56 60 63 63 63 64 65 66 66 68 74 78 78 79 81 82 82 83

CONTENTS

2.22–2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.3 2.31 2.32 2.4 2.41–2.43 2.44–2.45 2.46 2.47 2.48 2.5–2.6 2.50 2.51–2.52 2.53–2.54 2.55–2.56 2.57 2.58–2.62 2.63–2.65 2.66 2.67 2.7 2.71 2.72–2.73 2.74 2.8 2.81 2.82 2.83 2.84 2.85

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 n Forms containing √ (a + bx)k . . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing √a + bx and the binomial α + βx . . . . . . . . . . . . . . Forms containing √a + bx + cx2 . . . . . . . . . . . . . . . . . . . . . . . . Forms containing √a + bx + cx2 and integral powers of x . . . . . . . . . . . Forms containing √a + cx2 and integral powers of x . . . . . . . . . . . . . . Forms containing a + bx + cx2 and first- and second-degree polynomials . . Integrals that can be reduced to elliptic or pseudo-elliptic integrals . . . . . . The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing eax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The exponential combined with rational functions of x . . . . . . . . . . . . . Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of sinh x, cosh x, tanh x, and coth x . . . . . . . . . . . . . . . . . . Rational functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . Algebraic functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . Combinations of hyperbolic functions and powers . . . . . . . . . . . . . . . Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . Sines and cosines of multiple angles and of linear and more complicated functions of the argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational functions of√the sine and cosine . . . . . . . . . . . . . . . . . . . . √ Integrals containing a ± b sin x or a ± b cos x . . . . . . . . . . . . . . . . Integrals reducible to elliptic and pseudo-elliptic integrals . . . . . . . . . . . Products of trigonometric functions and powers . . . . . . . . . . . . . . . . Combinations of trigonometric functions and exponentials . . . . . . . . . . . Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . Logarithms and Inverse-Hyperbolic Functions . . . . . . . . . . . . . . . . . . The logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . Arcsines and arccosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The arcsecant, the arccosecant, the arctangent, and the arccotangent . . . . Combinations of arcsine or arccosine and algebraic functions . . . . . . . . . . Combinations of the arcsecant and arccosecant with powers of x . . . . . . . Combinations of the arctangent and arccotangent with algebraic functions . .

3–4 Definite Integrals of Elementary Functions 3.0 Introduction . . . . . . . . . . . . . . . 3.01 Theorems of a general nature . . . . . . 3.02 Change of variable in a definite integral . 3.03 General formulas . . . . . . . . . . . . . 3.04 Improper integrals . . . . . . . . . . . . 3.05 The principal values of improper integrals 3.1–3.2 Power and Algebraic Functions . . . . . 3.11 Rational functions . . . . . . . . . . . .

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CONTENTS

3.12 3.13–3.17 3.18 3.19–3.23 3.24–3.27 3.3–3.4 3.31 3.32–3.34 3.35 3.36–3.37 3.38–3.39 3.41–3.44 3.45 3.46–3.48 3.5 3.51 3.52–3.53 3.54 3.55–3.56 3.6–4.1 3.61 3.62 3.63 3.64–3.65 3.66 3.67 3.68 3.69–3.71 3.72–3.74 3.75 3.76–3.77 3.78–3.81 3.82–3.83 3.84 3.85–3.88 3.89–3.91 3.92 3.93 3.94–3.97 3.98–3.99 4.11–4.12 4.13

Products of rational functions and expressions that can be reduced to square roots of first- and second-degree polynomials . . . . . . . . . . . . . . . . . . . Expressions that can be reduced to square roots of third- and fourth-degree polynomials and their products with rational functions . . . . . . . . . . . . . . Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions . . . . . . . . . . . . . . . . . . . . . Combinations of powers of x and powers of binomials of the form (α + βx) . . Powers of x, of binomials of the form α + βxp and of polynomials in x . . . . . Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponentials of more complicated arguments . . . . . . . . . . . . . . . . . . . Combinations of exponentials and rational functions . . . . . . . . . . . . . . . Combinations of exponentials and algebraic functions . . . . . . . . . . . . . . Combinations of exponentials and arbitrary powers . . . . . . . . . . . . . . . . Combinations of rational functions of powers and exponentials . . . . . . . . . Combinations of powers and algebraic functions of exponentials . . . . . . . . . Combinations of exponentials of more complicated arguments and powers . . . Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of hyperbolic functions and algebraic functions . . . . . . . . . . Combinations of hyperbolic functions and exponentials . . . . . . . . . . . . . Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . . Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational functions of sines and cosines and trigonometric functions of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . Powers of trigonometric functions and trigonometric functions of linear functions Powers and rational functions of trigonometric functions . . . . . . . . . . . . . Forms containing powers of linear functions of trigonometric functions . . . . . Square roots of expressions containing trigonometric functions . . . . . . . . . Various forms of powers of trigonometric functions . . . . . . . . . . . . . . . . Trigonometric functions of more complicated arguments . . . . . . . . . . . . . Combinations of trigonometric and rational functions . . . . . . . . . . . . . . Combinations of trigonometric and algebraic functions . . . . . . . . . . . . . . Combinations of trigonometric functions and powers . . . . . . . . . . . . . . . Rational functions of x and of trigonometric functions . . . . . . . . . . . . . . Powers of trigonometric with other powers . . . . . . . . .  functions combined √ 2 2 2 Integrals containing 1 − k sin x, 1 − k cos2 x, and similar expressions . . Trigonometric functions of more complicated arguments combined with powers Trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . . Trigonometric functions of more complicated arguments combined with exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric and exponential functions of trigonometric functions . . . . . . . Combinations involving trigonometric functions, exponentials, and powers . . . Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . . Combinations involving trigonometric and hyperbolic functions and powers . . . Combinations of trigonometric and hyperbolic functions and exponentials . . . .

254 254 313 315 322 334 334 336 340 344 346 353 363 364 371 371 375 382 386 390 390 395 397 401 405 408 411 415 423 434 436 447 459 472 475 485 493 495 497 509 516 522

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4.14 4.2–4.4 4.21 4.22 4.23 4.24 4.25 4.26–4.27 4.28 4.29–4.32 4.33–4.34 4.35–4.36 4.37 4.38–4.41 4.42–4.43 4.44 4.5 4.51 4.52 4.53–4.54 4.55 4.56 4.57 4.58 4.59 4.6 4.60 4.61 4.62 4.63–4.64 5

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Combinations of trigonometric and hyperbolic functions, exponentials, and powers Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithms of more complicated arguments . . . . . . . . . . . . . . . . . . . . Combinations of logarithms and rational functions . . . . . . . . . . . . . . . . Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . . Combinations of logarithms and powers . . . . . . . . . . . . . . . . . . . . . . Combinations involving powers of the logarithm and other powers . . . . . . . . Combinations of rational functions of ln x and powers . . . . . . . . . . . . . . Combinations of logarithmic functions of more complicated arguments and powers Combinations of logarithms and exponentials . . . . . . . . . . . . . . . . . . . Combinations of logarithms, exponentials, and powers . . . . . . . . . . . . . . Combinations of logarithms and hyperbolic functions . . . . . . . . . . . . . . . Logarithms and trigonometric functions . . . . . . . . . . . . . . . . . . . . . . Combinations of logarithms, trigonometric functions, and powers . . . . . . . . Combinations of logarithms, trigonometric functions, and exponentials . . . . . Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of arcsines, arccosines, and powers . . . . . . . . . . . . . . . . . Combinations of arctangents, arccotangents, and powers . . . . . . . . . . . . . Combinations of inverse trigonometric functions and exponentials . . . . . . . . A combination of the arctangent and a hyperbolic function . . . . . . . . . . . Combinations of inverse and direct trigonometric functions . . . . . . . . . . . A combination involving an inverse and a direct trigonometric function and a power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of inverse trigonometric functions and logarithms . . . . . . . . . Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of variables in multiple integrals . . . . . . . . . . . . . . . . . . . . . Change of the order of integration and change of variables . . . . . . . . . . . Double and triple integrals with constant limits . . . . . . . . . . . . . . . . . . Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Indefinite Integrals of Special Functions Elliptic Integrals and Functions . . . . . . . . . . . . . . . . Complete elliptic integrals . . . . . . . . . . . . . . . . . . . 5.11 5.12 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . 5.14 Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . 5.2 The Exponential Integral Function . . . . . . . . . . . . . . 5.21 The exponential integral function . . . . . . . . . . . . . . . 5.22 Combinations of the exponential integral function and powers 5.23 Combinations of the exponential integral and the exponential 5.3 The Sine Integral and the Cosine Integral . . . . . . . . . . . 5.4 The Probability Integral and Fresnel Integrals . . . . . . . . . 5.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . 5.1

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x

CONTENTS

6–7 Definite Integrals of Special Functions 6.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Forms containing F (x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Forms containing E (x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Integration of elliptic integrals with respect to the modulus . . . . . . . . . . 6.14–6.15 Complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 The theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2–6.3 The Exponential Integral Function and Functions Generated by It . . . . . . . 6.21 The logarithm integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.22–6.23 The exponential integral function . . . . . . . . . . . . . . . . . . . . . . . . 6.24–6.26 The sine integral and cosine integral functions . . . . . . . . . . . . . . . . . 6.27 The hyperbolic sine integral and hyperbolic cosine integral functions . . . . . 6.28–6.31 The probability integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.32 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Gamma Function and Functions Generated by It . . . . . . . . . . . . . 6.41 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.42 Combinations of the gamma function, the exponential, and powers . . . . . . 6.43 Combinations of the gamma function and trigonometric functions . . . . . . . 6.44 The logarithm of the gamma function∗ . . . . . . . . . . . . . . . . . . . . . 6.45 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . 6.46–6.47 The function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5–6.7 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.51 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.52 Bessel functions combined with x and x2 . . . . . . . . . . . . . . . . . . . . 6.53–6.54 Combinations of Bessel functions and rational functions . . . . . . . . . . . . 6.55 Combinations of Bessel functions and algebraic functions . . . . . . . . . . . 6.56–6.58 Combinations of Bessel functions and powers . . . . . . . . . . . . . . . . . . 6.59 Combinations of powers and Bessel functions of more complicated arguments 6.61 Combinations of Bessel functions and exponentials . . . . . . . . . . . . . . . 6.62–6.63 Combinations of Bessel functions, exponentials, and powers . . . . . . . . . . 6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.66 Combinations of Bessel, hyperbolic, and exponential functions . . . . . . . . . 6.67–6.68 Combinations of Bessel and trigonometric functions . . . . . . . . . . . . . . 6.69–6.74 Combinations of Bessel and trigonometric functions and powers . . . . . . . . 6.75 Combinations of Bessel, trigonometric, and exponential functions and powers 6.76 Combinations of Bessel, trigonometric, and hyperbolic functions . . . . . . . 6.77 Combinations of Bessel functions and the logarithm, or arctangent . . . . . . 6.78 Combinations of Bessel and other special functions . . . . . . . . . . . . . . . 6.79 Integration of Bessel functions with respect to the order . . . . . . . . . . . . 6.8 Functions Generated by Bessel Functions . . . . . . . . . . . . . . . . . . . . 6.81 Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.82 Combinations of Struve functions, exponentials, and powers . . . . . . . . . . 6.83 Combinations of Struve and trigonometric functions . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

631 631 631 632 632 632 633 635 636 636 638 639 644 645 649 650 650 652 655 656 657 658 659 659 664 670 674 675 689 694 699

.

708

. . . . . . . . . . . . .

711 713 717 727 742 747 747 748 749 753 753 754 755

CONTENTS

6.84–6.85 6.86 6.87 6.9 6.91 6.92 6.93 6.94 7.1–7.2 7.11 7.12–7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.21 7.22 7.23 7.24 7.25 7.3–7.4 7.31 7.32 7.325∗ 7.33 7.34 7.35 7.36 7.37–7.38 7.39 7.41–7.42 7.5 7.51 7.52 7.53 7.54 7.6 7.61 7.62–7.63 7.64 7.65

xi

Combinations of Struve and Bessel functions . . . . . . . . . . . . . . . . . . . Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of Mathieu, hyperbolic, and trigonometric functions . . . . . . . Combinations of Mathieu and Bessel functions . . . . . . . . . . . . . . . . . . Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . Associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of associated Legendre functions and powers . . . . . . . . . . . Combinations of associated Legendre functions, exponentials, and powers . . . Combinations of associated Legendre and hyperbolic functions . . . . . . . . . Combinations of associated Legendre functions, powers, and trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A combination of an associated Legendre function and the probability integral . Combinations of associated Legendre and Bessel functions . . . . . . . . . . . . Combinations of associated Legendre functions and functions generated by Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of associated Legendre functions with respect to the order . . . . . Combinations of Legendre polynomials, rational functions, and algebraic functions Combinations of Legendre polynomials and powers . . . . . . . . . . . . . . . . Combinations of Legendre polynomials and other elementary functions . . . . . Combinations of Legendre polynomials and Bessel functions . . . . . . . . . . . Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of Gegenbauer polynomials Cnν (x) and powers . . . . . . . . . . Combinations of Gegenbauer polynomials Cnν (x) and elementary functions . . . Complete System of Orthogonal Step Functions . . . . . . . . . . . . . . . . . Combinations of the polynomials C νn (x) and Bessel functions; Integration of Gegenbauer functions with respect to the index . . . . . . . . . . . . . . . . . Combinations of Chebyshev polynomials and powers . . . . . . . . . . . . . . . Combinations of Chebyshev polynomials and elementary functions . . . . . . . Combinations of Chebyshev polynomials and Bessel functions . . . . . . . . . . Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of hypergeometric functions and powers . . . . . . . . . . . . . . Combinations of hypergeometric functions and exponentials . . . . . . . . . . . Hypergeometric and trigonometric functions . . . . . . . . . . . . . . . . . . . Combinations of hypergeometric and Bessel functions . . . . . . . . . . . . . . Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . Combinations of confluent hypergeometric functions and powers . . . . . . . . Combinations of confluent hypergeometric functions and exponentials . . . . . Combinations of confluent hypergeometric and trigonometric functions . . . . . Combinations of confluent hypergeometric functions and Bessel functions . . .

756 760 761 763 763 763 767 767 769 769 770 776 778 779 781 782 787 788 789 791 792 794 795 795 797 798 798 800 802 803 803 806 808 812 812 814 817 817 820 820 822 829 830

xii

CONTENTS

7.66 7.67 7.68 7.69 7.7 7.71 7.72 7.73 7.74 7.75 7.76 7.77 7.8 7.81 7.82 7.83

Combinations of confluent hypergeometric functions, Bessel functions, and powers Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of confluent hypergeometric functions and other special functions Integration of confluent hypergeometric functions with respect to the index . . Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of parabolic cylinder functions, powers, and exponentials . . . . . Combinations of parabolic cylinder and hyperbolic functions . . . . . . . . . . . Combinations of parabolic cylinder and trigonometric functions . . . . . . . . . Combinations of parabolic cylinder and Bessel functions . . . . . . . . . . . . . Combinations of parabolic cylinder functions and confluent hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of a parabolic cylinder function with respect to the index . . . . . . Meijer’s and MacRobert’s Functions (G and E) . . . . . . . . . . . . . . . . . Combinations of the functions G and E and the elementary functions . . . . . Combinations of the functions G and E and Bessel functions . . . . . . . . . . Combinations of the functions G and E and other special functions . . . . . . .

8–9 Special Functions 8.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12 Functional relations between elliptic integrals . . . . . . . . . . . . . . . . . . . 8.13 Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.14 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.15 Properties of Jacobian elliptic functions and functional relationships between them 8.16 The Weierstrass function ℘(u) . . . . . . . . . . . . . . . . . . . . . . . . . . 8.17 The functions ζ(u) and σ(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.18–8.19 8.2 The Exponential Integral Function and Functions Generated by It . . . . . . . . 8.21 The exponential integral function Ei(x) . . . . . . . . . . . . . . . . . . . . . . 8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x . . . 8.23 The sine integral and the cosine integral: si x and ci x . . . . . . . . . . . . . . 8.24 The logarithm integral li(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.25 The probability integral Φ(x), the Fresnel integrals S (x) and C (x), the error function erf(x), and the complementary error function erfc(x) . . . . . . . . . 8.26 Lobachevskiy’s function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Euler’s Integrals of the First and Second Kinds . . . . . . . . . . . . . . . . . . 8.31 The gamma function (Euler’s integral of the second kind): Γ(z) . . . . . . . . 8.32 Representation of the gamma function as series and products . . . . . . . . . . 8.33 Functional relations involving the gamma function . . . . . . . . . . . . . . . . 8.34 The logarithm of the gamma function . . . . . . . . . . . . . . . . . . . . . . . 8.35 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . . 8.36 The psi function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.37 The function β(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.38 The beta function (Euler’s integral of the first kind): B(x, y) . . . . . . . . . . 8.39 The incomplete beta function Bx (p, q) . . . . . . . . . . . . . . . . . . . . . . 8.4–8.5 Bessel Functions and Functions Associated with Them . . . . . . . . . . . . . .

831 834 839 841 841 841 842 843 844 845 849 849 850 850 854 856 859 859 859 863 865 866 870 873 876 877 883 883 886 886 887 887 891 892 892 894 895 898 899 902 906 908 910 910

CONTENTS

8.40 8.41 8.42 8.43 8.44 8.45 8.46 8.47–8.48 8.49 8.51–8.52 8.53 8.54 8.55 8.56 8.57 8.58 8.59 8.6 8.60 8.61 8.62 8.63 8.64 8.65 8.66 8.67 8.7–8.8 8.70 8.71 8.72 8.73–8.74 8.75 8.76 8.77 8.78 8.79 8.81 8.82–8.83 8.84 8.85 8.9 8.90 8.91 8.919 8.92 8.93 8.94

xiii

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral representations of the functions Jν (z) and Nν (z) . . . . . . . . (1) (2) Integral representations of the functions Hν (z) and Hν (z) . . . . . . Integral representations of the functions Iν (z) and Kν (z) . . . . . . . . Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic expansions of Bessel functions . . . . . . . . . . . . . . . . Bessel functions of order equal to an integer plus one-half . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential equations leading to Bessel functions . . . . . . . . . . . . . Series of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . Expansion in products of Bessel functions . . . . . . . . . . . . . . . . . The zeros of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomson functions and their generalizations . . . . . . . . . . . . . . . Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anger and Weber functions Jν (z) and Eν (z) . . . . . . . . . . . . . . . Neumann’s and Schl¨afli’s polynomials: O n (z) and S n (z) . . . . . . . . Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathieu’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periodic Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . (2n) (2n+1) (2n+1) (2n+2) Recursion relations for the coefficients A2r , A2r+1 , B2r+1 , B2r+2 Mathieu functions with a purely imaginary argument . . . . . . . . . . . Non-periodic solutions of Mathieu’s equation . . . . . . . . . . . . . . . Mathieu functions for negative q . . . . . . . . . . . . . . . . . . . . . Representation of Mathieu functions as series of Bessel functions . . . . The general theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic series for large values of |ν| . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special cases and particular values . . . . . . . . . . . . . . . . . . . . Derivatives with respect to the order . . . . . . . . . . . . . . . . . . . Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . The zeros of associated Legendre functions . . . . . . . . . . . . . . . . Series of associated Legendre functions . . . . . . . . . . . . . . . . . . Associated Legendre functions with integer indices . . . . . . . . . . . . Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of products of Legendre and Chebyshev polynomials . . . . . . . Series of Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . Gegenbauer polynomials Cnλ (t) . . . . . . . . . . . . . . . . . . . . . . The Chebyshev polynomials Tn (x) and Un (x) . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

910 912 914 916 918 920 924 926 931 933 940 941 942 944 945 948 949 950 950 951 951 952 953 953 954 957 958 958 960 962 964 968 969 970 972 972 974 975 980 981 982 982 983 988 988 990 993

xiv

CONTENTS

8.95 8.96 8.97 9.1 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.2 9.20 9.21 9.22–9.23 9.24–9.25 9.26 9.3 9.30 9.31 9.32 9.33 9.34 9.4 9.41 9.42 9.5 9.51 9.52 9.53 9.54 9.55 9.56 9.6 9.61 9.62 9.63 9.64 9.65 9.7 9.71 9.72

The Hermite polynomials H n (x) . . . . . . . . . . . . . . . . . . . . . . . . . Jacobi’s polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of elementary functions in terms of a hypergeometric functions . Transformation formulas and the analytic continuation of functions defined by hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A generalized hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . The hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . Riemann’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . Representing the solutions to certain second-order differential equations using a Riemann scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypergeometric functions of two variables . . . . . . . . . . . . . . . . . . . . A hypergeometric function of several variables . . . . . . . . . . . . . . . . . . Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The functions Φ(α, γ; z) and Ψ(α, γ; z) . . . . . . . . . . . . . . . . . . . . . . The Whittaker functions Mλ,μ (z) and Wλ,μ (z) . . . . . . . . . . . . . . . . . . Parabolic cylinder functions Dp (z) . . . . . . . . . . . . . . . . . . . . . . . . Confluent hypergeometric series of two variables . . . . . . . . . . . . . . . . . Meijer’s G-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A differential equation for the G-function . . . . . . . . . . . . . . . . . . . . . Series of G-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connections with other special functions . . . . . . . . . . . . . . . . . . . . . MacRobert’s E-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation by means of multiple integrals . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s) Definition and integral representations . . . . . . . . . . . . . . . . . . . . . . Representation as a series or as an infinite product . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singular points and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Lerch function Φ(z, s, v) . . . . . . . . . . . . . . . . . . . . . . . . . . . The function ξ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernoulli Numbers and Polynomials, Euler Numbers . . . . . . . . . . . . . . . Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y) . . . . . . . . . . Euler polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

996 998 1000 1005 1005 1005 1006 1008 1010 1010 1014 1017 1018 1022 1022 1022 1023 1024 1028 1031 1032 1032 1033 1034 1034 1034 1035 1035 1035 1036 1036 1037 1037 1038 1039 1040 1040 1040 1041 1043 1043 1044 1045 1045 1045

CONTENTS

9.73 9.74

xv

Euler’s and Catalan’s constants . . . . . . . . . . . . . . . . . . . . . . . . . . Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Vector Field Theory 10.1–10.8 Vectors, Vector Operators, and Integral 10.11 Products of vectors . . . . . . . . . . 10.12 Properties of scalar product . . . . . . 10.13 Properties of vector product . . . . . . 10.14 Differentiation of vectors . . . . . . . . 10.21 Operators grad, div, and curl . . . . . 10.31 Properties of the operator ∇ . . . . . 10.41 Solenoidal fields . . . . . . . . . . . . 10.51–10.61 Orthogonal curvilinear coordinates . . 10.71–10.72 Vector integral theorems . . . . . . . . 10.81 Integral rate of change theorems . . .

1046 1046

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1049 1049 1049 1049 1049 1050 1050 1051 1052 1052 1055 1057

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1059 1059 1059 1060 1061

12 Integral Inequalities 12.11 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . 12.111 First mean value theorem . . . . . . . . . . . . . . . . . . 12.112 Second mean value theorem . . . . . . . . . . . . . . . . . 12.113 First mean value theorem for infinite integrals . . . . . . . 12.114 Second mean value theorem for infinite integrals . . . . . . 12.21 Differentiation of Definite Integral Containing a Parameter 12.211 Differentiation when limits are finite . . . . . . . . . . . . . 12.212 Differentiation when a limit is infinite . . . . . . . . . . . . 12.31 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . 12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals . . . 12.312 H¨older’s inequality for integrals . . . . . . . . . . . . . . . 12.313 Minkowski’s inequality for integrals . . . . . . . . . . . . . 12.314 Chebyshev’s inequality for integrals . . . . . . . . . . . . . 12.315 Young’s inequality for integrals . . . . . . . . . . . . . . . 12.316 Steffensen’s inequality for integrals . . . . . . . . . . . . . 12.317 Gram’s inequality for integrals . . . . . . . . . . . . . . . . 12.318 Ostrowski’s inequality for integrals . . . . . . . . . . . . . 12.41 Convexity and Jensen’s Inequality . . . . . . . . . . . . . . 12.411 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . 12.412 Carleman’s inequality for integrals . . . . . . . . . . . . . . 12.51 Fourier Series and Related Inequalities . . . . . . . . . . . 12.511 Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . . 12.512 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . 12.513 Parseval’s theorem for trigonometric Fourier series . . . . . 12.514 Integral representation of the nth partial sum . . . . . . . .

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1063 1063 1063 1063 1063 1064 1064 1064 1064 1064 1064 1064 1065 1065 1065 1065 1065 1066 1066 1066 1066 1066 1067 1067 1067 1067

Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Algebraic Inequalities 11.1–11.3 General Algebraic Inequalities . . . . . . . . . . 11.11 Algebraic inequalities involving real numbers . . 11.21 Algebraic inequalities involving complex numbers 11.31 Inequalities for sets of complex numbers . . . .

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xvi

CONTENTS

12.515 12.516 12.517

Generalized Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bessel’s inequality for generalized Fourier series . . . . . . . . . . . . . . . . . Parseval’s theorem for generalized Fourier series . . . . . . . . . . . . . . . . .

1067 1068 1068

13 Matrices and Related Results 13.11–13.12 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.111 Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.112 Identity matrix and null matrix . . . . . . . . . . . . . . . . . . . . . . . 13.113 Reducible and irreducible matrices . . . . . . . . . . . . . . . . . . . . . . 13.114 Equivalent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.115 Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.116 Adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.117 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.118 Trace of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.119 Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.120 Skew-symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.121 Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.122 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.123 Hermitian transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . 13.124 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.125 Unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.126 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . 13.127 Nilpotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.128 Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.129 Positive definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.130 Non-negative definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.131 Diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.21 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.211 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.212 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.213 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.214 Positive definite and semidefinite quadratic form . . . . . . . . . . . . . . 13.215 Basic theorems on quadratic forms . . . . . . . . . . . . . . . . . . . . . 13.31 Differentiation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 13.41 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.411 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1069 1069 1069 1069 1069 1069 1069 1070 1070 1070 1070 1070 1070 1070 1070 1070 1071 1071 1071 1071 1071 1071 1071 1071 1072 1072 1072 1072 1072 1073 1074 1074

14 Determinants 14.11 14.12 14.13 14.14 14.15* 14.16 14.17 14.18 14.21 14.31

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1075 1075 1075 1075 1076 1076 1076 1077 1077 1077 1078

Expansion of Second- and Third-Order Determinants Basic Properties . . . . . . . . . . . . . . . . . . . . Minors and Cofactors of a Determinant . . . . . . . . Principal Minors . . . . . . . . . . . . . . . . . . . . Laplace Expansion of a Determinant . . . . . . . . . Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . Hadamard’s Theorem . . . . . . . . . . . . . . . . . Hadamard’s Inequality . . . . . . . . . . . . . . . . . Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . Some Special Determinants . . . . . . . . . . . . . .

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CONTENTS

14.311 14.312 14.313 14.314 14.315 14.316 14.317 15 Norms 15.1–15.9 15.11 15.21 15.211 15.212 15.213 15.31 15.311 15.312 15.313 15.41 15.411 15.412 15.413 15.51 15.511 15.512 15.61 15.611 15.612 15.71 15.711 15.712 15.713 15.714 15.715 15.81–15.82 15.811 15.812 15.813 15.814 15.815 15.816 15.817 15.818 15.819 15.820 15.821 15.822

xvii

Vandermonde’s determinant (alternant) . . . . . Circulants . . . . . . . . . . . . . . . . . . . . . Jacobian determinant . . . . . . . . . . . . . . Hessian determinants . . . . . . . . . . . . . . Wronskian determinants . . . . . . . . . . . . . Properties . . . . . . . . . . . . . . . . . . . . Gram-Kowalewski theorem on linear dependence

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1078 1078 1078 1079 1079 1079 1080

Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . General Properties . . . . . . . . . . . . . . . . . . . . . . . Principal Vector Norms . . . . . . . . . . . . . . . . . . . . The norm ||x||1 . . . . . . . . . . . . . . . . . . . . . . . . The norm ||x||2 (Euclidean or L2 norm) . . . . . . . . . . . The norm ||x||∞ . . . . . . . . . . . . . . . . . . . . . . . . Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . General properties . . . . . . . . . . . . . . . . . . . . . . . Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . Natural norm of unit matrix . . . . . . . . . . . . . . . . . . Principal Natural Norms . . . . . . . . . . . . . . . . . . . . Maximum absolute column sum norm . . . . . . . . . . . . . Spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . Maximum absolute row sum norm . . . . . . . . . . . . . . . Spectral Radius of a Square Matrix . . . . . . . . . . . . . . Inequalities concerning matrix norms and the spectral radius Deductions from Gerschgorin’s theorem (see 15.814) . . . . Inequalities Involving Eigenvalues of Matrices . . . . . . . . . Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . Inequalities for the Characteristic Polynomial . . . . . . . . . Named and unnamed inequalities . . . . . . . . . . . . . . . Parodi’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Corollary of Brauer’s theorem . . . . . . . . . . . . . . . . . Ballieu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Routh-Hurwitz theorem . . . . . . . . . . . . . . . . . . . . Named Theorems on Eigenvalues . . . . . . . . . . . . . . . Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . Sturmian separation theorem . . . . . . . . . . . . . . . . . Poincare’s separation theorem . . . . . . . . . . . . . . . . . Gerschgorin’s theorem . . . . . . . . . . . . . . . . . . . . . Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . Wielandt’s theorem . . . . . . . . . . . . . . . . . . . . . . Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . . . . First theorem due to Lyapunov . . . . . . . . . . . . . . . . Second theorem due to Lyapunov . . . . . . . . . . . . . . .

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1081 1081 1081 1081 1081 1081 1081 1082 1082 1082 1082 1082 1082 1082 1083 1083 1083 1083 1084 1084 1084 1084 1085 1086 1086 1086 1086 1087 1087 1087 1087 1088 1088 1088 1088 1088 1088 1089 1089 1089

xviii

CONTENTS

15.823 15.91 15.911 15.912

Hermitian matrices and diophantine relations rational angles due to Calogero and Perelomov Variational Principles . . . . . . . . . . . . . . Rayleigh quotient . . . . . . . . . . . . . . . . Basic theorems . . . . . . . . . . . . . . . . .

involving . . . . . . . . . . . . . . . . . . . .

circular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1089 1091 1091 1091

16 Ordinary Differential Equations 1093 16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations . . . . . . . 1093 16.11 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 16.111 Solution of a first-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093 16.112 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 16.113 Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093 16.114 Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.21 Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094 16.211 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.212 Comparison of approximate solutions of a differential equation . . . . . . . . . 1094 16.31 First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.311 Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.312 Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 16.313 Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095 16.314 Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 16.315 Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096 16.316 First-order linear differential equation . . . . . . . . . . . . . . . . . . . . . . . 1096 16.317 Linear systems of differential equations . . . . . . . . . . . . . . . . . . . . . . 1096 16.41 Some Special Types of Elementary Differential Equations . . . . . . . . . . . . 1097 16.411 Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.412 Exact differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.413 Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.414 Homogeneous differential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.51 Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 16.511 Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098 16.512 Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 16.513 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 16.514 The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 16.515 Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Solution of a second-order linear differential equation . . . . . . . . . . . . . . 1100 16.516 16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100 16.611 First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100 16.622 Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.623 Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.624 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.625 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.626 Szeg¨ o’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.627 Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 16.628 Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 16.629 Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 16.71 Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103 16.711 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103

CONTENTS

16.712 16.81–16.82 16.811 16.822 16.823 16.91 16.911 16.912 16.913 16.914 16.92 16.921 16.922 16.923 16.924 16.93

xix

Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . . Kneser’s non-oscillation theorem . . . . . . . . . . . . . . . . . Comparison theorem for non-oscillation . . . . . . . . . . . . . . Necessary and sufficient conditions for non-oscillation . . . . . . Some Growth Estimates for Solutions of Second-Order Equations Strictly increasing and decreasing solutions . . . . . . . . . . . . General result on dominant and subdominant solutions . . . . . Estimate of dominant solution . . . . . . . . . . . . . . . . . . . A theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . Boundedness Theorems . . . . . . . . . . . . . . . . . . . . . . All solutions of the equation . . . . . . . . . . . . . . . . . . . . If all solutions of the equation . . . . . . . . . . . . . . . . . . . If a(x) → ∞ monotonically as x → ∞, then all solutions of . . . Consider the equation . . . . . . . . . . . . . . . . . . . . . . . Growth of maxima of |y| . . . . . . . . . . . . . . . . . . . . . .

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1103 1103 1103 1104 1104 1104 1104 1104 1105 1105 1106 1106 1106 1106 1106 1106

17 Fourier, Laplace, and Mellin Transforms 17.1–17.4 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . 17.11 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . 17.12 Basic properties of the Laplace transform . . . . . . . . . . . . . . 17.13 Table of Laplace transform pairs . . . . . . . . . . . . . . . . . . 17.21 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.22 Basic properties of the Fourier transform . . . . . . . . . . . . . . 17.23 Table of Fourier transform pairs . . . . . . . . . . . . . . . . . . . 17.24 Table of Fourier transform pairs for spherically symmetric functions 17.31 Fourier sine and cosine transforms . . . . . . . . . . . . . . . . . . 17.32 Basic properties of the Fourier sine and cosine transforms . . . . . 17.33 Table of Fourier sine transforms . . . . . . . . . . . . . . . . . . . 17.34 Table of Fourier cosine transforms . . . . . . . . . . . . . . . . . . 17.35 Relationships between transforms . . . . . . . . . . . . . . . . . . 17.41 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.42 Basic properties of the Mellin transform . . . . . . . . . . . . . . 17.43 Table of Mellin transforms . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

1107 1107 1107 1107 1108 1117 1118 1118 1120 1121 1121 1122 1126 1129 1129 1130 1131

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

18 The z-Transform 18.1–18.3 Definition, Bilateral, and 18.1 Definitions . . . . . . . 18.2 Bilateral z-transform . . 18.3 Unilateral z-transform . References Supplemental references Index of Functions and Constants General Index of Concepts

Unilateral z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

1135 1135 1135 1136 1138 1141 1145 1151 1161

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Preface to the Seventh Edition Since the publication in 2000 of the completely reset sixth edition of Gradshteyn and Ryzhik, users of the reference work have continued to submit corrections, new results that extend the work, and suggestions for changes that improve the presentation of existing entries. It is a matter of regret to us that the structure of the book makes it impossible to acknowledge these individual contributions, so, as usual, the names of the many new contributors have been added to the acknowledgment list at the front of the book. This seventh edition contains the corrections received since the publication of the sixth edition in 2000, together with a considerable amount of new material acquired from isolated sources. Following our previous conventions, an amended entry has a superscript “11” added to its entry reference number, where the equivalent superscript number for the sixth edition was “10.” Similarly, an asterisk on an entry’s reference number indicates a new result. When, for technical reasons, an entry in a previous edition has been removed, to preserve the continuity of numbering between the new and older editions the subsequent entries have not been renumbered, so the numbering will jump. We wish to express our gratitude to all who have been in contact with us with the object of improving and extending the book, and we want to give special thanks to Dr. Victor H. Moll for his interest in the book and for the many contributions he has made over an extended period of time. We also wish to acknowledge the contributions made by Dr. Francis J. O’Brien Jr. of the Naval Station in Newport, in particular for results involving integrands where exponentials are combined with algebraic functions. Experience over many years has shown that each new edition of Gradshteyn and Ryzhik generates a fresh supply of suggestions for new entries, and for the improvement of the presentation of existing entries and errata. In view of this, we do not expect this new edition to be free from errors, so all users of this reference work who identify errors, or who wish to propose new entries, are invited to contact the authors, whose email addresses are listed below. Corrections will be posted on the web site www.az-tec.com/gr/errata. Alan Jeffrey [email protected] Daniel Zwillinger [email protected]

xxi

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Acknowledgments The publisher and editors would like to take this opportunity to express their gratitude to the following users of the Table of Integrals, Series, and Products who, either directly or through errata published in Mathematics of Computation, have generously contributed corrections and addenda to the original printing. Dr. A. Abbas Dr. P. B. Abraham Dr. Ari Abramson Dr. Jose Adachi Dr. R. J. Adler Dr. N. Agmon Dr. M. Ahmad Dr. S. A. Ahmad Dr. Luis Alvarez-Ruso Dr. Maarten H P Ambaum Dr. R. K. Amiet Dr. L. U. Ancarani Dr. M. Antoine Dr. C. R. Appledorn Dr. D. R. Appleton Dr. Mitsuhiro Arikawa Dr. P. Ashoshauvati Dr. C. L. Axness Dr. E. Badralexe Dr. S. B. Bagchi Dr. L. J. Baker Dr. R. Ball Dr. M. P. Barnett Dr. Florian Baumann Dr. Norman C. Beaulieu Dr. Jerome Benoit Mr. V. Bentley Dr. Laurent Berger Dr. M. van den Berg Dr. N. F. Berk Dr. C. A. Bertulani

Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.

J. Betancort-Rijo P. Bickerstaff Iwo Bialynicki-Birula Chris Bidinosti G. R. Bigg Ian Bindloss L. Blanchet Mike Blaskiewicz R. D. Blevins Anders Blom L. M. Blumberg R. Blumel S. E. Bodner M. Bonsager George Boros S. Bosanac B. Van den Bossche A. Bostr¨ om J. E. Bowcock T. H. Boyer K. M. Briggs D. J. Broadhurst Chris Van Den Broeck W. B. Brower H. N. Browne Christoph Bruegger William J. Bruno Vladimir Bubanja D. J. Buch D. J. Bukman F. M. Burrows

xxiii

Dr. R. Caboz Dr. T. Calloway Dr. F. Calogero Dr. D. Dal Cappello Dr. David Cardon Dr. J. A. Carlson Gallos Dr. B. Carrascal Dr. A. R. Carr Dr. S. Carter Dr. G. Cavalleri Mr. W. H. L. Cawthorne Dr. A. Cecchini Dr. B. Chan Dr. M. A. Chaudhry Dr. Sabino Chavez-Cerda Dr. Julian Cheng Dr. H. W. Chew Dr. D. Chin Dr. Young-seek Chung Dr. S. Ciccariello Dr. N. S. Clarke Dr. R. W. Cleary Dr. A. Clement Dr. P. Cochrane Dr. D. K. Cohoon Dr. L. Cole Dr. Filippo Colomo Dr. J. R. D. Copley Dr. D. Cox Dr. J. Cox Dr. J. W. Criss

xxiv

Dr. A. E. Curzon Dr. D. Dadyburjor Dr. D. Dajaputra Dr. C. Dal Cappello Dr. P. Daly Dr. S. Dasgupta Dr. John Davies Dr. C. L. Davis Dr. A. Degasperis Dr. B. C. Denardo Dr. R. W. Dent Dr. E. Deutsch Dr. D. deVries Dr. P. Dita Dr. P. J. de Doelder Dr. Mischa Dohler Dr. G. Dˆ ome Dr. Shi-Hai Dong Dr. Balazs Dora Dr. M. R. D’Orsogna Dr. Adrian A. Dragulescu Dr. Eduardo Duenez Mr. Tommi J. Dufva Dr. E. B. Dussan, V Dr. C. A. Ebner Dr. M. van der Ende Dr. Jonathan Engle Dr. G. Eng Dr. E. S. Erck Dr. Jan Erkelens Dr. Olivier Espinosa Dr. G. A. Est´evez Dr. K. Evans Dr. G. Evendon Dr. V. I. Fabrikant Dr. L. A. Falkovsky Dr. K. Farahmand Dr. Richard J. Fateman Dr. G. Fedele Dr. A. R. Ferchmin Dr. P. Ferrant Dr. H. E. Fettis Dr. W. B. Fichter Dr. George Fikioris Mr. J. C. S. S. Filho Dr. L. Ford Dr. Nicolao Fornengo

Acknowledgments

Dr. J. France Dr. B. Frank Dr. S. Frasier Dr. Stefan Fredenhagen Dr. A. J. Freeman Dr. A. Frink Dr. Jason M. Gallaspy Dr. J. A. C. Gallas Dr. J. A. Carlson Gallas Dr. G. R. Gamertsfelder Dr. T. Garavaglia Dr. Jaime Zaratiegui Garcia Dr. C. G. Gardner Dr. D. Garfinkle Dr. P. N. Garner Dr. F. Gasser Dr. E. Gath Dr. P. Gatt Dr. D. Gay Dr. M. P. Gelfand Dr. M. R. Geller Dr. Ali I. Genc Dr. Vincent Genot Dr. M. F. George Dr. P. Germain Dr. Ing. Christoph Gierull Dr. S. P. Gill Dr. Federico Girosi Dr. E. A. Gislason Dr. M. I. Glasser Dr. P. A. Glendinning Dr. L. I. Goldfischer Dr. Denis Golosov Dr. I. J. Good Dr. J. Good Mr. L. Gorin Dr. Martin G¨ otz Dr. R. Govindaraj Dr. M. De Grauf Dr. L. Green Mr. Leslie O. Green Dr. R. Greenwell Dr. K. D. Grimsley Dr. Albert Groenenboom Dr. V. Gudmundsson Dr. J. Guillera Dr. K. Gunn

Dr. D. L. Gunter Dr. Julio C. Guti´errez-Vega Dr. Roger Haagmans Dr. H. van Haeringen Dr. B. Hafizi Dr. Bahman Hafizi Dr. T. Hagfors Dr. M. J. Haggerty Dr. Timo Hakulinen Dr. Einar Halvorsen Dr. S. E. Hammel Dr. E. Hansen Dr. Wes Harker Dr. T. Harrett Dr. D. O. Harris Dr. Frank Harris Mr. Mazen D. Hasna Dr. Joel G. Heinrich Dr. Sten Herlitz Dr. Chris Herzog Dr. A. Higuchi Dr. R. E. Hise Dr. Henrik Holm Dr. Helmut H¨ olzler Dr. N. Holte Dr. R. W. Hopper Dr. P. N. Houle Dr. C. J. Howard Dr. J. H. Hubbell Dr. J. R. Hull Dr. W. Humphries Dr. Jean-Marc Hur´e Dr. Ben Yu-Kuang Hu Dr. Y. Iksbe Dr. Philip Ingenhoven Mr. L. Iossif Dr. Sean A. Irvine ´ ´Isberg Dr. Ottar Dr. Cyril-Daniel Iskander Dr. S. A. Jackson Dr. John David Jackson Dr. Francois Jaclot Dr. B. Jacobs Dr. E. C. James Dr. B. Jancovici Dr. D. J. Jeffrey Dr. H. J. Jensen

Acknowledgments

Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.

Edwin F. Johnson I. R. Johnson Steven Johnson I. Johnstone Y. P. Joshi Jae-Hun Jung Damir Juric Florian Kaempfer S. Kanmani Z. Kapal Dave Kasper M. Kaufman B. Kay Avinash Khare Ilki Kim Youngsun Kim S. Klama L. Klingen C. Knessl M. J. Knight Mel Knight Yannis Kohninos D. Koks L. P. Kok K. S. K¨ olbig Y. Komninos D. D. Konowalow Z. Kopal I. Kostyukov R. A. Krajcik Vincent Krakoviack Stefan Kramer Tobias Kramer Hermann Krebs J. W. Krozel E. D. Krupnikov Kun-Lin Kuo E. A. Kuraev Konstantinos Kyritsis Velimir Labinac A. D. J. Lambert A. Lambert A. Larraza K. D. Lee M. Howard Lee M. K. Lee P. A. Lee

xxv

Dr. Todd Lee Dr. J. Legg Dr. Armando Lemus Dr. S. L. Levie Dr. D. Levi Dr. Michael Lexa Dr. Kuo Kan Liang Dr. B. Linet Dr. M. A. Lisa Dr. Donald Livesay Dr. H. Li Dr. Georg Lohoefer Dr. I. M. Longman Dr. D. Long Dr. Sylvie Lorthois Dr. Y. L. Luke Dr. W. Lukosz Dr. T. Lundgren Dr. E. A. Luraev Dr. R. Lynch Dr. R. Mahurin Dr. R. Mallier Dr. G. A. Mamon Dr. A. Mangiarotti Dr. I. Manning Dr. J. Marmur Dr. A. Martin Sr. Yuzo Maruyama Dr. David J. Masiello Dr. Richard Marthar Dr. H. A. Mavromatis Dr. M. Mazzoni Dr. K. B. Ma Dr. P. McCullagh Dr. J. H. McDonnell Dr. J. R. McGregor Dr. Kim McInturff Dr. N. McKinney Dr. David McA McKirdy Dr. Rami Mehrem Dr. W. N. Mei Dr. Angelo Melino Mr. Jos´e Ricardo Mendes Dr. Andy Mennim Dr. J. P. Meunier Dr. Gerard P. Michon Dr. D. F. R. Mildner

Dr. D. L. Miller Dr. Steve Miller Dr. P. C. D. Milly Dr. S. P. Mitra Dr. K. Miura Dr. N. Mohankumar Dr. M. Moll Dr. Victor H. Moll Dr. D. Monowalow Mr. Tony Montagnese Dr. Jim Morehead Dr. J. Morice Dr. W. Mueck Dr. C. Muhlhausen Dr. S. Mukherjee Dr. R. R. M¨ uller Dr. Pablo Parmezani Munhoz Dr. Paul Nanninga Dr. A. Natarajan Dr. Stefan Neumeier Dr. C. T. Nguyen Dr. A. C. Nicol Dr. M. M. Nieto Dr. P. Noerdlinger Dr. A. N. Norris Dr. K. H. Norwich Dr. A. H. Nuttall Dr. Frank O’Brien Dr. R. P. O’Keeffe Dr. A. Ojo Dr. P. Olsson Dr. M. Ortner Dr. S. Ostlund Dr. J. Overduin Dr. J. Pachner Dr. John D. Paden Mr. Robert A. Padgug Dr. D. Papadopoulos Dr. F. J. Papp Mr. Man Sik Park Dr. Jong-Do Park Dr. B. Patterson Dr. R. F. Pawula Dr. D. W. Peaceman Dr. D. Pelat Dr. L. Peliti Dr. Y. P. Pellegrini

xxvi

Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.

Acknowledgments

G. J. Pert Nicola Pessina J. B. Peterson Rickard Petersson Andrew Plumb Dror Porat E. A. Power E. Predazzi William S. Price Paul Radmore F. Raynal X. R. Resende J. M. Riedler Thomas Richard E. Ringel T. M. Roberts N. I. Robinson P. A. Robinson D. M. Rosenblum R. A. Rosthal J. R. Roth Klaus Rottbrand D. Roy E. Royer D. Rudermann Sanjib Sabhapandit C. T. Sachradja J. Sadiku A. Sadiq Motohiko Saitoh Naoki Saito A. Salim J. H. Samson Miguel A. Sanchis-Lozano J. A. Sanders M. A. F. Sanjun P. Sarquiz Avadh Saxena Vito Scarola O. Sch¨ arpf A. Scherzinger B. Schizer Martin Schmid J. Scholes Mel Schopper H. J. Schulz G. J. Sears

Dr. Kazuhiko Seki Dr. B. Seshadri Dr. A. Shapiro Dr. Masaki Shigemori Dr. J. S. Sheng Dr. Kenneth Ing Shing Dr. Tomohiro Shirai Dr. S. Shlomo Dr. D. Siegel Dr. Matthew Stapleton Dr. Steven H. Simon Dr. Ashok Kumar Singal Dr. C. Smith Dr. G. C. C. Smith Dr. Stefan Llewellyn Smith Dr. S. Smith Dr. G. Solt Dr. J. Sondow Dr. A. Sørenssen Dr. Marcus Spradlin Dr. Andrzej Staruszkiewicz Dr. Philip C. L. Stephenson Dr. Edgardo Stockmeyer Dr. J. C. Straton Mr. H. Suraweera Dr. N. F. Svaiter Dr. V. Svaiter Dr. R. Szmytkowski Dr. S. Tabachnik Dr. Erik Talvila Dr. G. Tanaka Dr. C. Tanguy Dr. G. K. Tannahill Dr. B. T. Tan Dr. C. Tavard Dr. Gon¸calo Tavares Dr. Aba Teleki Dr. Arash Dahi Taleghani Dr. D. Temperley Dr. A. J. Tervoort Dr. Theodoros Theodoulidis Dr. D. J. Thomas Dr. Michael Thorwart Dr. S. T. Thynell Dr. D. C. Torney Dr. R. Tough Dr. B. F. Treadway

Dr. Ming Tsai Dr. N. Turkkan Dr. Sandeep Tyagi Dr. J. J. Tyson Dr. S. Uehara Dr. M. Vadacchino Dr. O. T. Valls Dr. D. Vandeth Mr. Andras Vanyolos Dr. D. Veitch Mr. Jose Lopez Vicario Dr. K. Vogel Dr. J. M. M. J. Vogels Dr. Alexis De Vos Dr. Stuart Walsh Dr. Reinhold Wannemacher Dr. S. Wanzura Dr. J. Ward Dr. S. I. Warshaw Dr. R. Weber Dr. Wei Qian Dr. D. H. Werner Dr. E. Wetzel Dr. Robert Whittaker Dr. D. T. Wilton Dr. C. Wiuf Dr. K. T. Wong Mr. J. N. Wright Dr. J. D. Wright Dr. D. Wright Dr. D. Wu Dr. Michel Daoud Yacoub Dr. Yu S. Yakovlev Dr. H.-C. Yang Dr. J. J. Yang Dr. Z. J. Yang Dr. J. J. Wang Dr. Peter Widerin Mr. Chun Kin Au Yeung Dr. Kazuya Yuasa Dr. S. P. Yukon Dr. B. Zhang Dr. Y. C. Zhang Dr. Y. Zhao Dr. Ralf Zimmer

The Order of Presentation of the Formulas The question of the most expedient order in which to give the formulas, in particular, in what division to include particular formulas such as the definite integrals, turned out to be quite complicated. The thought naturally occurs to set up an order analogous to that of a dictionary. However, it is almost impossible to create such a system for the formulas of integral calculus. Indeed, in an arbitrary formula of the form  b f (x) dx = A a

one may make a large number of substitutions of the form x = ϕ(t) and thus obtain a number of “synonyms” of the given formula. We must point out that the table of definite integrals by Bierens de Haan and the earlier editions of the present reference both sin in the plethora of such “synonyms” and formulas of complicated form. In the present edition, we have tried to keep only the simplest of the “synonym” formulas. Basically, we judged the simplicity of a formula from the standpoint of the simplicity of the arguments of the “outer” functions that appear in the integrand. Where possible, we have replaced a complicated formula with a simpler one. Sometimes, several complicated formulas were thereby reduced to a single, simpler one. We then kept only the simplest formula. As a result of such substitutions, we sometimes obtained an integral that could be evaluated by use of the formulas of Chapter Two and the Newton–Leibniz formula, or to an integral of the form  a f (x) dx, −a

where f (x) is an odd function. In such cases, the complicated integrals have been omitted. Let us give an example using the expression 

π/4 0

p−1

(cot x − 1) sin2 x

ln tan x dx = −

π cosec pπ. p

By making the natural substitution u = cot x − 1, we obtain  ∞ π up−1 ln(1 + u) du = cosec pπ. p 0 Integrals similar to formula (0.1) are omitted in this new edition. Instead, we have formula (0.2).

xxvii

(0.1)

(0.2)

xxviii

The Order of Presentation of the Formulas

As a second example, let us take 

π/2

ln (tanp x + cotp x) ln tan x dx = 0.

I= 0

The substitution u = tan x yields 

∞

I= 0

ln (up + u−p ) ln u du. 1 + u2

If we now set υ = ln u, we obtain  ∞  ∞   pυ υeυ ln (2 cosh pυ) −pυ dυ. dυ = ln e + e υ I= 2υ 2 cosh υ −∞ 1 + e −∞ The integrand is odd, and, consequently, the integral is equal to 0. Thus, before looking for an integral in the tables, the user should simplify as much as possible the arguments (the “inner” functions) of the functions in the integrand. The functions are ordered as follows: First we have the elementary functions: 1. 2. 3. 4. 5. 6. 7.

The function f (x) = x. The exponential function. The hyperbolic functions. The trigonometric functions. The logarithmic function. The inverse hyperbolic functions. (These are replaced with the corresponding logarithms in the formulas containing definite integrals.) The inverse trigonometric functions.

Then follow the special functions: 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Elliptic integrals. Elliptic functions. The logarithm integral, the exponential integral, the sine integral, and the cosine integral functions. Probability integrals and Fresnel’s integrals. The gamma function and related functions. Bessel functions. Mathieu functions. Legendre functions. Orthogonal polynomials. Hypergeometric functions. Degenerate hypergeometric functions. Parabolic cylinder functions. Meijer’s and MacRobert’s functions. Riemann’s zeta function.

The integrals are arranged in order of outer function according to the above scheme: the farther down in the list a function occurs, (i.e., the more complex it is) the later will the corresponding formula appear

The Order of Presentation of the Formulas

xxix

in the tables. Suppose that several expressions have the same outer function. For example, consider sin ex , sin x, sin ln x. Here, the outer function is the sine function in all three cases. Such expressions are then arranged in order of the inner function. In the present work, these functions are therefore arranged in the following order: sin x, sin ex , sin ln x. Our list does not include polynomials, rational functions, powers, or other algebraic functions. An algebraic function that is included in tables of definite integrals can usually be reduced to a finite combination of roots of rational power. Therefore, for classifying our formulas, we can conditionally treat a power function as a generalization of an algebraic and, consequently, of a rational function.∗ We shall distinguish between all these functions and those listed above, and we shall treat them as operators. Thus, in the expression sin2 ex , we shall think of the squaring operator as applied to the outer function, sin x+cos x namely, the sine. In the expression sin x−cos x , we shall think of the rational operator as applied to the trigonometric functions sine and cosine. We shall arrange the operators according to the following order: 1. 2. 3. 4.

Polynomials (listed in order of their degree). Rational operators. Algebraic operators (expressions of the form Ap/q , where q and p are rational, and q > 0; these are listed according to the size of q). Power operators.

Expressions with the same outer and inner functions are arranged in the order of complexity of the operators. For example, the following functions [whose outer functions are all trigonometric, and whose inner functions are all f (x) = x] are arranged in the order shown: sin x,

sin x cos x,

1 = cosec x, sin x

sin x = tan x, cos x

sin x + cos x , sin x − cos x

sinm x,

sinm x cos x.

Furthermore, if two outer functions ϕ1 (x) and ϕ2 (x), where ϕ1 (x) is more complex than ϕ2 (x), appear in an integrand and if any of the operations mentioned are performed on them, the corresponding integral will appear [in the order determined by the position of ϕ2 (x) in the list] after all integrals containing only the function ϕ1 (x). Thus, following the trigonometric functions are the trigonometric and power functions [that is, ϕ2 (x) = x]. Then come • combinations of trigonometric and exponential functions, • combinations of trigonometric functions, exponential functions, and powers, etc., • combinations of trigonometric and hyperbolic functions, etc. Integrals containing two functions ϕ1 (x) and ϕ2 (x) are located in the division and order corresponding to the more complicated function of the two. However, if the positions of several integrals coincide because they contain the same complicated function, these integrals are put in the position defined by the complexity of the second function. To these rules of a general nature, we need to add certain particular considerations that will be easily 1 understood from the tables. For example, according to the above remarks, the function e x comes after 1 1 ex as regards complexity, but ln x and ln are equally complex since ln = − ln x. In the section on x x “powers and algebraic functions,” polynomials, rational functions, and powers of powers are formed from power functions of the form (a + bx)n and (α + βx)ν . ∗ For any natural number n, the involution (a + bx)n of the binomial a + bx is a polynomial. If n is a negative integer, (a + bx)n is a rational function. If n is irrational, the function (a + bx)n is not even an algebraic function.

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Use of the Tables∗ For the effective use of the tables contained in this book, it is necessary that the user should first become familiar with the classification system for integrals devised by the authors Ryzhik and Gradshteyn. This classification is described in detail in the section entitled The Order of Presentation of the Formulas (see page xxvii) and essentially involves the separation of the integrand into inner and outer functions. The principal function involved in the integrand is called the outer function, and its argument, which is itself usually another function, is called the inner function. Thus, if the integrand comprised the expression ln sin x, the outer function would be the logarithmic function while its argument, the inner function, would be the trigonometric function sin x. The desired integral would then be found in the section dealing with logarithmic functions, its position within that section being determined by the position of the inner function (here a trigonometric function) in Gradshteyn and Ryzhik’s list of functional forms. It is inevitable that some duplication of symbols will occur within such a large collection of integrals, and this happens most frequently in the first part of the book dealing with algebraic and trigonometric integrands. The symbols most frequently involved are α, β, γ, δ, t, u, z, zk , and Δ. The expressions associated with these symbols are used consistently within each section and are defined at the start of each new section in which they occur. Consequently, reference should be made to the beginning of the section being used in order to verify the meaning of the substitutions involved. Integrals of algebraic functions are expressed as combinations of roots with rational power indices, and definite integrals of such functions are frequently expressed in terms of the Legendre elliptic integrals F (φ, k), E (φ, k) and Π(φ, n, k), respectively, of the first, second, and third kinds. The four inverse hyperbolic functions arcsinh z, arccosh z, arctanh z, and arccoth z are introduced through the definitions 1 arcsinh(iz) i 1 arccos z = arccosh(z) i 1 arctan z = arctanh(iz) i arccot z = i arccoth(iz) arcsin z =

∗ Prepared

by Alan Jeffrey for the English language edition.

xxxi

xxxii

Use of the Tables

or 1 arcsin(iz) i arccosh z = i arccos z 1 arctanh z = arctan(iz) i 1 arccoth z = arccot(−iz) i arcsinh z =

The numerical constants C and G which often appear in the definite integrals denote Euler’s constant and Catalan’s constant, respectively. Euler’s constant C is defined by the limit   s  1 − ln s = 0.577215 . . . . C = lim s→∞ m m=1 On occasion, other writers denote Euler’s constant by the symbol γ, but this is also often used instead to denote the constant γ = eC = 1.781072 . . . . Catalan’s constant G is related to the complete elliptic integral  π/2 dx  K ≡ K(k) ≡ 0 1 − k 2 sin2 x by the expression  ∞  (−1)m 1 1 K dk = = 0.915965 . . . . G= 2 0 (2m + 1)2 m=0 Since the notations and definitions for higher transcendental functions that are used by different authors are by no means uniform, it is advisable to check the definitions of the functions that occur in these tables. This can be done by identifying the required function by symbol and name in the Index of Special Functions and Notation on page xxxix, and by then referring to the defining formula or section number listed there. We now present a brief discussion of some of the most commonly used alternative notations and definitions for higher transcendental functions. Bernoulli and Euler Polynomials and Numbers Extensive use is made throughout the book of the Bernoulli and Euler numbers Bn and En that are defined in terms of the Bernoulli and Euler polynomials of order n, B n (x) and E n (x), respectively. These polynomials are defined by the generating functions ∞  tn text = for |t| < 2π B n (x) t e − 1 n=0 n! and ∞  2ext tn = for |t| < π. E n (x) t e + 1 n=0 n! The Bernoulli numbers are always denoted by Bn and are defined by the relation Bn = Bn (0)

for n = 0, 1, . . . ,

when B0 = 1,

1 B1 = − , 2

B2 =

1 , 6

B4 = −

1 ,.... 30

Use of the Tables

xxxiii

The Euler numbers En are defined by setting



1 En = 2 E n for n = 0, 1, . . . 2 The En are all integral, and E0 = 1, E2 = −1, E4 = 5, E6 = −61, . . . . An alternative definition of Bernoulli numbers, which we shall denote by the symbol Bn∗ , uses the same generating function but identifies the Bn∗ differently in the following manner: 1 t2 t4 t = 1 − t + B1∗ − B2∗ + . . . . t e −1 2 2! 4! This definition then gives rise to the alternative set of Bernoulli numbers n

B1∗ = 1/6, B2∗ = 1/30, B3∗ = 1/42, B4∗ = 1/30, B5∗ = 5/66, B6∗ = 691/2730, B7∗ = 7/6, B8∗ = 3617/510, . . . . These differences in notation must also be taken into account when using the following relationships that exist between the Bernoulli and Euler polynomials: n 1  n n = 0, 1, . . . Bn−k E k (2x) k 2n k=0

 x  x+1 2n E n−1 (x) = Bn − Bn n 2 2

B n (x) =

or E n−1 (x) = and

 x  2 B n (x) − 2n B n n 2

n = 1, 2, . . .

 n  n−k 2 − 1 Bn−k B n (x) n = 2, 3, . . . k 2 k=0 There are also alternative definitions of the Euler polynomial of order n, and it should be noted that some authors, using a modification of the third expression above, call

  x  2 B n (x) − 2n B n n+1 2 the Euler polynomial of order n. E n−2 (x) = 2

n

−1

n−2 

Elliptic Functions and Elliptic Integrals The following notations are often used in connection with the inverse elliptic functions sn u, cn u, and dn u: 1 sn u sn u sc u = cn u sn u sd u = dn u

ns u =

1 cn u cn u cs u = sn u cn u cd u = dn u nc u =

1 dn u dn u ds u = sn u dn u dc u = cn u

nd u =

xxxiv

Use of the Tables

The elliptic integral of the third kind is defined by Gradshteyn and Ryzhik to be  ϕ   da 2 Π ϕ, n , k =   2 2 0   1 − n sin a 1 − k 2 sin2 a −∞ < n2 < ∞  sin ϕ dx  = 2 2 (1 − n x ) (1 − x2 ) (1 − k 2 x2 ) 0 The Jacobi Zeta Function and Theta Functions The Jacobi zeta function zn(u, k), frequently written Z(u), is defined by the relation   u E E 2 zn(u, k) = Z(u) = dn υ − dυ = E(u) − u. K K 0 This is related to the theta functions by the relationship ∂ zn(u, k) = ln Θ(u) ∂u giving  πu  π ϑ1 2K cn u dn u  πu − (i). zn(u, k) = 2K ϑ sn u 1 2K  πu  ϑ 2 π dn u sn u  2K − (ii). zn(u, k) = 2K ϑ πu cn u 2 2K  πu  ϑ π 3 2K sn u cn u  − k2 (iii). zn(u, k) = 2K ϑ πu dn u 3 2K  πu  π ϑ4 2K  (iv). zn(u, k) = 2K ϑ πu 4 2K Many different notations for the theta function are in current use. The most common variants are the replacement of the argument u by the argument u/π and, occasionally, a permutation of the identification of the functions ϑ1 to ϑ4 with the function ϑ4 replaced by ϑ. The Factorial (Gamma) Function In older reference texts, the gamma function Γ(z), defined by the Euler integral  ∞ Γ(z) = tz−1 e−t dt, 0

is sometimes expressed in the alternative notation Γ(1 + z) = z! = Π(z). On occasions, the related derivative of the logarithmic factorial function Ψ(z) is used where 

(z!) d (ln z!) = = Ψ(z). dz z!

Use of the Tables

xxxv

This function satisfies the recurrence relation Ψ(z) = Ψ(z − 1) + and is defined by the series Ψ(z) = −C +

∞  n=0

1 z−1

1 1 − n+1 z+n

.

The derivative Ψ (z) satisfies the recurrence relation Ψ (z + 1) = Ψ (z) − and is defined by the series Ψ (z) =

1 z2

∞ 

1 . (z + n)2 n=0

Exponential and Related Integrals The exponential integrals E n (z) have been defined by Schloemilch using the integral  ∞ E n (z) = e−zt t−n dt (n = 0, 1, . . . , Re z > 0) . 1

They should not be confused with the Euler polynomials already mentioned. The function E 1 (z) is related to the exponential integral Ei(z) through the expressions  ∞ E 1 (z) = − Ei(−z) = e−t t−1 dt z

and



z

dt = Ei (ln z) [z > 1] . ln t 0 The functions E n (z) satisfy the recurrence relations  1  −z e − z E n−1 (z) E n (z) = [n > 1] n−1 and E n (z) = − E n−1 (z) with li(z) =

E 0 (z) = e−z /z. The function E n (z) has the asymptotic expansion  

 e−z 3π n n(n + 1) n(n + 1)(n + 2) − + · · · |arg z| < E n (z) ∼ 1− + z z z2 z3 2 while for large n,     n e−x n(n − 2x) n 6x2 − 8nx + n2 1+ + + + R(n, x) , E n (x) = x+n (x + n)2 (x + n)4 (x + n)6 where

1 −4 −0.36n ≤ R(n, x) ≤ 1 + [x > 0] . n−4 x+n−1 The sine and cosine integrals si(x) and ci(x) are related to the functions Si(x) and Ci(x) by the integrals  x π sin t Si(x) = dt = si(x) + t 2 0 and

xxxvi

Use of the Tables



(cos t − 1) dt. t 0 The hyperbolic sine and cosine integrals shi(x) and chi(x) are defined by the relations  x sinh t shi(x) = dt t 0 and  x (cosh t − 1) chi(x) = C + ln x + dt. t 0 Some authors write  x (1 − cos t) Cin(x) = dt t 0 so that Cin(x) = − Ci(x) + ln x + C. The error function erf(x) is defined by the relation  x 2 2 erf(x) = Φ(x) = √ e−t dt, π 0 and the complementary error function erfc(x) is related to the error function erfc(x) and to Φ(x) by the expression erfc(x) = 1 − erf(x). The Fresnel integrals S (x) and C (x) are defined by Gradshteyn and Ryzhik as  x 2 S (x) = √ sin t2 dt 2π 0 and  x 2 C (x) = √ cos t2 dt. 2π 0 Other definitions that are in use are  x  x πt2 πt2 dt, C 1 (x) = dt, S 1 (x) = sin cos 2 2 0 0 and  x  x sin t cos t 1 1 √ dt, √ dt. S 2 (x) = √ C 2 (x) = √ t t 2π 0 2π 0 These are related by the expressions      2 = S 2 x2 S (x) = S 1 x π and      2 = C 2 x2 C (x) = C 1 x π x

Ci(x) = C + ln x +

Hermite and Chebyshev Orthogonal Polynomials The Hermite polynomials H n (x) are related to the Hermite polynomials He n (x) by the relations

x −n/2 He n (x) = 2 Hn √ 2 and  √ H n (x) = 2n/2 He n x 2 .

Use of the Tables

xxxvii

These functions satisfy the differential equations dHn d2 H n + 2n H n = 0 − 2x 2 dx dx and d2 He n d He n + n He n = 0. −x dx2 dx They obey the recurrence relations H n+1 = 2x H n −2n H n−1 and He n+1 = x He n −n He n−1 . The first six orthogonal polynomials He n are He 0 = 1, He 1 = x, He 2 = x2 − 1, He 3 = x3 − 3x, He 4 = x4 − 6x2 + 3, He 5 = x5 − 10x3 + 15x. Sometimes the Chebyshev polynomial U n (x) of the second kind is defined as a solution of the equation  d2 y  dy + n(n + 2)y = 0. 1 − x2 − 3x dx2 dx Bessel Functions A variety of different notations for Bessel functions are in use. Some common ones involve the replacement of Y n (z) by N n (z) and the introduction of the symbol −n 1 Λn (z) = z Γ(n + 1) J n (z). 2 1 In the book by Gray, Mathews, and MacRobert, the symbol Y n (z) is used to denote π Y n (z) + 2 (ln 2 − C) J n (z) while Neumann uses the symbol Y (n) (z) for the identical quantity. (1) (2) The Hankel functions H ν (z) and H ν (z) are sometimes denoted by Hsν (z) and Hiν (z), and some 1 authors write Gν (z) = πi H (1) ν (z). 2 The Neumann polynomial O n (t) is a polynomial of degree n + 1 in 1/t, with O 0 (t) = 1/t. The polynomials O n (t) are defined by the generating function ∞  1 = J 0 (z) O 0 (t) + 2 J k (z) O k (t), t−z k=1 giving n−2k+1 [n/2] 1  n(n − k − 1)! 2 O n (t) = for n = 1, 2, . . . , 4 k! t k=0   where 12 n signifies the integral part of 12 n. The following relationship holds between three successive polynomials:   2 n2 − 1 nπ 2n O n (t) = sin2 . (n − 1) O n+1 (t) + (n + 1) O n−1 (t) − t t 2

xxxviii

Use of the Tables

The Airy functions Ai(z) and Bi(z) are independent solutions of the equation d2 u − zu = 0. dz 2 The solutions can be represented in terms of Bessel functions by the expressions 







2 3/2 2 3/2 2 3/2 1√ 1 z Ai(z) = z z K 1/3 z z I −1/3 − I 1/3 = 3 3 3 π 3 3





 2 3/2 2 3/2 1√ Ai(−z) = z z z J 1/3 + J −1/3 3 3 3 and by





 2 3/2 2 3/2 z z z + I 1/3 , I −1/3 3 3 3 



 2 3/2 2 3/2 z z z Bi(−z) = − J 1/3 . J −1/3 3 3 3 Bi(z) =

Parabolic Cylinder Functions and Whittaker Functions The differential equation  d2 y  2 + az + bz + c y = 0 dz 2 has associated with it the two equations



1 2 1 2 d2 y d2 y z z + + a y = 0 and − + a y = 0, dz 2 4 dz 2 4 the solutions of which are parabolic cylinder functions. The first equation can be derived from the second by replacing z by zeiπ/4 and a by −ia. The solutions of the equation

1 2 d2 y z +a y =0 − dz 2 4 are sometimes written U (a, z) and V (a, z). These solutions are related to Whittaker’s function D p (z) by the expressions U (a, z) = D −a− 12 (z) and

  1 1 D −a− 12 (−z) + (sin πa) D −a− 12 (z) . V (a, z) = Γ +a π 2 Mathieu Functions There are several accepted notations for Mathieu functions and for their associated parameters. The defining equation used by Gradshteyn and Ryzhik is  d2 y  + a − 2k 2 cos 2z y = 0 with k 2 = q. dz 2 Different notations involve the replacement of a and q in this equation by h and θ, λ and h2 , and √ b and c = 2 q, respectively. The periodic solutions sen (z, q) and cen (z, q) and the modified periodic solutions Sen (z, q) and Cen (z, q) are suitably altered and, sometimes, re-normalized. A description of these relationships together with the normalizing factors is contained in: Tables Relating to Mathieu Functions. National Bureau of Standards, Columbia University Press, New York, 1951.

Index of Special Functions Notation β(x) Γ(z) γ(a, x), Γ(a, x) Δ(n − k) ξ(s) λ(x, y) μ(x, β), μ(x, β, α) ν(x), ν(x, α) Π(x) Π(ϕ, n, k) ζ(u) ζ(z, q), ζ(z)  πu   πu  , Θ1 (u) = ϑ⎫ Θ(u) ⎧ = ϑ4 2K 3 2K ⎪ ⎨ ϑ0 (υ | τ ) = ϑ4 (υ | τ ), ⎪ ⎬ ϑ1 (υ | τ ), ϑ2 (υ | τ ), ⎪ ⎪ ⎩ ⎭ ϑ3 (υ | τ ) σ(u) Φ(x) Φ(z, s, υ) Φ(a, c; x) = 1 F 1 (α; γ; ⎧ ⎫ x) Φ (α, β, γ, x, y) ⎪ ⎪ ⎨ 1 ⎬ Φ2 (β, β  , γ, x, y) ⎪ ⎪ ⎩ ⎭ Φ3 (β, γ, x, y) ψ(x) ℘(u) am(u, k) Bn B n (x) B(x, y) Bx (p, q) bei(z), ber(z)

Name of the function and the number of the formula containing its definition

Lobachevskiy’s angle of parallelism Elliptic integral of the third kind Weierstrass zeta function Riemann’s zeta functions Jacobian theta function

8.37 8.31–8.33 8.35 18.1 9.56 9.640 9.640 9.640 1.48 8.11 8.17 9.51–9.54 8.191–8.196

Elliptic theta functions

8.18, 8.19

Weierstrass sigma function See probability integral Lerch function Confluent hypergeometric function

8.17 8.25 9.55 9.21

Degenerate hypergeometric series in two variables

9.26

Euler psi function Weierstrass elliptic function Amplitude (of an elliptic function) Bernoulli numbers Bernoulli polynomials Beta functions Incomplete beta functions Thomson functions

8.36 8.16 8.141 9.61, 9.71 9.620 8.38 8.39 8.56

Gamma function Incomplete gamma functions Unit integer pulse function

continued on next page

xxxix

xl

Index of Special Functions

continued from previous page

Notation C C (x) C ν (a) C λn (t) C λn (x) ce2n (z, q),

ce2n+1 (z, q)

Ce2n (z, q),

Ce2n+1 (z, q)

chi(x) ci(x) cn(u) D(k) ≡ D D(ϕ, k) D n (z), D p (z) dn u e1 , e2 , e3 En E (ϕ, k)   E(k) = E E(k ) = E E(p; ar : q; s : x) Eν (z) Ei(z) erf(x) erfc(x) = 1 − erf(x) F (ϕ, k) (α , . . . , αp ; β1 , . . . , βq ; z) F p q 1 (α, β; γ; z) = F (α, β; γ; z) F 2 1 (α; γ; z) = Φ(α, γ; z) F 1 1 FΛ (α :β1 , . . . , βn ; γ1 , . . . . . . , γn : z1 , . . . , zn ) F1 , F2 , F3 , F4   fen (z, q), Fen (z, q) . . . Feyn (z, q), Fekn (z, q) . . . G g2 , g3 gd x   gen (z, q), Gen (z, q) Geyn (z, q), Gekn (z, q)  ! ! ,... ,ap x ! ab11,... G m,n p,q ,bq

Name of the function and the number of the formula containing its definition Euler constant 9.73, 8.367 Fresnel cosine integral 8.25 Young functions 3.76 Gegenbauer polynomials 8.93 Gegenbauer functions 8.932 1 Periodic Mathieu functions (Mathieu 8.61 functions of the first kind) Associated (modified) Mathieu functions of 8.63 the first kind Hyperbolic cosine integral function 8.22 Cosine integral 8.23 Cosine amplitude 8.14 Elliptic integral 8.112 Elliptic integral 8.111 Parabolic cylinder functions 9.24–9.25 Delta amplitude 8.14 (used with the Weierstrass function) 8.162 Euler numbers 9.63, 9.72 Elliptic integral of the second kind 8.11–8.12 Complete elliptic integral of the second 8.11-8.12 kind MacRobert’s function 9.4 Weber function 8.58 Exponential integral function 8.21 Error function 8.25 Complementary error function 8.25 Elliptic integral of the first kind 8.11–8.12 Generalized hypergeometric series 9.14 Gauss hypergeometric function 9.10–9.13 Degenerate hypergeometric function 9.21 Hypergeometric function of several 9.19 variables Hypergeometric functions of two variables 9.18 Other nonperiodic solutions of Mathieu’s 8.64, 8.663 equation Catalan constant 9.73 Invariants of the ℘(u)-function 8.161 Gudermannian 1.49 Other nonperiodic solutions of Mathieu’s equation

8.64, 8.663

Meijer’s functions

9.3 continued on next page

Index of Special Functions

xli

continued from previous page

Notation h(n) heiν (z), herν (z) H (1) ν (z),

H (2) ν (z)  πu  H (u) = ϑ1 2K  πu H 1 (u) = ϑ2 2K H n (z) Hν (z) I ν (z) I x (p, q) J ν (z) Jν (z) kν (x) K(k) = K, K(k ) = K K ν (z) kei(z), ker(z) L(x) Lν (z) Lα n (z) li(x) M λ,μ (z) O n (x) P μν (z),

P μν (x)

P ν (z), P ν (x) ⎫ ⎧ ⎨a b c ⎬ P α β γ δ ⎩  ⎭ α β γ P (α,β) (x) n Q μν (z),

Q μν (x)

Q ν (z), Q ν (x) S (x) S n (x) s μ,ν (z), S μ,ν (z) se2n+1 (z, q), se2n+2 (z, q) Se2n+1 (z, q),

Se2n+2 (z, q)

shi(x) si(x) sn u T n (x) U n (x)

Name of the function and the number of the formula containing its definition Unit integer function 18.1 Thomson functions 8.56 Hankel functions of the first and second 8.405, 8.42 kinds Theta function 8.192 Theta function 8.192 Hermite polynomials 8.95 Struve functions 8.55 Bessel functions of an imaginary argument 8.406, 8.43 Normalized incomplete beta function 8.39 Bessel function 8.402, 8.41 Anger function 8.58 Bateman’s function 9.210 3 Complete elliptic integral of the first kind 8.11–8.12 Bessel functions of imaginary argument 8.407, 8.43 Thomson functions 8.56 Lobachevskiy’s function 8.26 Modified Struve function 8.55 Laguerre polynomials 8.97 Logarithm integral 8.24 Whittaker functions 9.22, 9.23 Neumann’s polynomials 8.59 Associated Legendre functions of the first 8.7, 8.8 kind Legendre functions and polynomials 8.82, 8.83, 8.91 Riemann’s differential equation

9.160

Jacobi’s polynomials Associated Legendre functions of the second kind Legendre functions of the second kind Fresnel sine integral Schl¨ afli’s polynomials Lommel functions Periodic Mathieu functions Mathieu functions of an imaginary argument Hyperbolic sine integral Sine integral Sine amplitude Chebyshev polynomial of the 1st kind Chebyshev polynomials of the 2nd kind

8.96 8.7, 8.8 8.82, 8.83 8.25 8.59 8.57 8.61 8.63 8.22 8.23 8.14 8.94 8.94 continued on next page

xlii

Index of Special Functions

continued from previous page

Notation U ν (w, z), V ν (w, z) W λ,μ (z) Y ν (z) Z ν (z) Zν (z)

Name of the function and the number of the formula containing its definition Lommel functions of two variables 8.578 Whittaker functions 9.22, 9.23 Neumann functions 8.403, 8.41 Bessel functions 8.401 Bessel functions

Notation Symbol

Meaning

x 

The integral part of the real number x (also denoted by [x]) 

(b+) a

(b−) a

" C

PV

Contour integrals; the path of integration starting at the point a extends to the point b (along a straight line unless there is an indication to the contrary), encircles the point b along a small circle in the positive (negative) direction, and returns to the point a, proceeding along the original path in the opposite direction. Line integral along the curve C

"

Principal value integral

z = x − iy

The complex conjugate of z = x + iy

n!

= 1 · 2 · 3 . . . n,

(2n + 1)!!

= 1 · 3 . . . (2n + 1). (double factorial notation)

(2n)!!

= 2 · 4 . . . (2n). (double factorial notation)

0!! = 1 and (−1)!! = 1

(cf. 3.372 for n = 0)

00 = 1

(cf. 0.112 and 0.113 for q = 0)

p

p! p(p − 1) . . . (p − n + 1) = , 1 · 2...n n!(p − n)! [n = 1, 2, . . . , p ≥ n] =

n x

k=m  n

# ,

= a(a + 1) . . . (a + n − 1) =

0

= 1,

p n

=

p! n!(p − n)!

Γ(a+n) Γ(a)

(Pochhammer symbol) n  = um + um+1 + . . . + un . If n < m, we define uk = 0

uk ,

p

= x(x − 1) . . . (x − n + 1)/n! [n = 0, 1, . . . ]

n (a)n n 

0! = 1

 m,n

$

k=m

Summation over all integral values of n excluding n = 0, and summation over all integral values of n and m excluding m = n = 0, respectively. # $ An empty has value 0, and an empty has value 1 continued on next page

xliii

xliv

Notation

continued from previous page

Symbol  1 i=j δij = 0 i=

j

Meaning Kronecker delta

τ

Theta function parameter (cf. 8.18)

× and ∧

Vector product (cf. 10.11)

·

Scalar product (cf. 10.11)

∇ or “del”

Vector operator (cf. 10.21)

∇2

Laplacian (cf. 10.31)

∼

Asymptotically equal to

arg z

The argument of the complex number z = x + iy

curl or rot

Vector operator (cf. 10.21)

div

Vector operator (divergence) (cf. 10.21)

F

Fourier transform (cf. 17.21)

Fc

Fourier cosine transform (cf. 17.31)

Fs

Fourier sine transform (cf. 17.31)

grad

Vector operator (gradient) (cf. 10.21)

hi and gij

Metric coefficients (cf. 10.51)

H

Hermitian transpose of a vector or matrix (cf. 13.123)

 0 x<0 H(x) = 1 x≥0

Heaviside step function

Im z ≡ y

The imaginary part of the complex number z = x + iy

k

The letter k (when not used as an index of summation) denotes a number in the interval [0, 1]. This notation is used in√integrals that lead to elliptic integrals. In such a connection, the number 1 − k 2 is denoted by k  .

L

Laplace transform (cf. 17.11)

M

Mellin transform (cf. 17.41)

N

The natural numbers (0, 1, 2, . . . )

O(f (z))

The order of the function f (z). Suppose that the point z approaches z0 . If there exists an M > 0 such that |g(z)| ≤ M |f (z)| in some sufficiently small neighborhood of the point z0 , we write g(z) = O(f (z)). continued on next page

Notation

xlv

continued from previous page

Symbol

Meaning

q

The nome, a theta function parameter (cf. 8.18)

R

The real numbers

R(x)

A rational function

Re z ≡ x

The real part of the complex number z = x + iy

Snm

Stirling number of the first kind (cd. 9.74)

Sm n

Stirling number of the second kind (cd. 9.74)

⎧ ⎪ ⎨+1 sign x = 0 ⎪ ⎩ −1

x>0 x=0 x<0

The sign (signum) of the real number x

T

Transpose of a vector or matrix (cf. 13.115)

Z

The integers (0, ±1, ±2, . . . )

Zb

Bilateral z transform (cf. 18.1)

Zu

Unilateral z transform (cf. 18.1)

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Note on the Bibliographic References The letters and numbers following equations refer to the sources used by Russian editors. The key to the letters will be found preceding each entry in the Bibliography beginning on page 1141. Roman numerals indicate the volume number of a multivolume work. Numbers without parentheses indicate page numbers, numbers in single parentheses refer to equation numbers in the original sources. Some formulas were changed from their form in the source material. In such cases, the letter a appears at the end of the bibliographic references. As an example, we may use the reference to equation 3.354–5: ET I 118 (1) a The key on page 1141 indicates that the book referred to is: Erd´elyi, A. et al., Tables of Integral Transforms. The Roman numeral denotes volume one of the work; 118 is the page on which the formula will be found; (1) refers to the number of the formula in this source; and the a indicates that the expression appearing in the source differs in some respect from the formula in this book. In several cases, the editors have used Russian editions of works published in other languages. Under such circumstances, because the pagination and numbering of equations may be altered, we have referred the reader only to the original sources and dispensed with page and equation numbers.

xlvii

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0 Introduction 0.1 Finite Sums 0.11 Progressions 0.111 Arithmetic progression. n−1  n n (a + kr) = [2a + (n − 1)r] = (a + l) 2 2

[l = a + (n − 1)r is the last term]

k=0

0.112 Geometric progression. n  a (q n − 1) aq k−1 = q−1

[q = 1]

k=1

0.113 Arithmetic-geometric progression.   n−1  rq 1 − q n−1 a − [a + (n − 1)r]q n + (a + kr)q k = 1−q (1 − q)2 k=0

0.1148



n−1 

k 2 xk =



−n + 2n − 1 x 2

k=1

n+2

[q = 1, n > 1]  2  n+1 + 2n − 2n − 1 x − n2 xn + x2 + x (1 − x)3

JO (5)

0.12 Sums of powers of natural numbers 0.121

n  k=1

1.

n  k=1

2.

3.

n 

nq 1 q nq+1 1 q 1 q + + B2 nq−1 + B4 nq−3 + B6 nq−5 + · · · q+1 2 2 1 4 3 6 5 nq+1 nq qnq−1 q(q − 1)(q − 2) q−3 q(q − 1)(q − 2)(q − 3)(q − 4) q−5 = + + − n n + − ··· q+1 2 12 720 30, 240 [last term contains either n or n2 ] CE 332

kq =

k=

n(n + 1) 2

CE 333

n(n + 1)(2n + 1) 6 k=1  2 n  n(n + 1) 3 k = 2 k2 =

CE 333

CE 333

k=1

1

2

Finite Sums

4.

5.

6.

7.

n  k=1 n  k=1 n  k=1 n 

k4 =

1 n(n + 1)(2n + 1)(3n2 + 3n − 1) 30

CE 333

k5 =

1 2 n (n + 1)2 (2n2 + 2n − 1) 12

CE 333

k6 =

1 n(n + 1)(2n + 1)(3n4 + 6n3 − 3n + 1) 42

CE 333

k7 =

1 2 n (n + 1)2 (3n4 + 6n3 − n2 − 4n + 2) 24

CE 333

k=1

0.122

0.122

n 

 2q q+1 1  q q−1 1  q q−3  3 n 2 B2 nq−1 − 2 2 − 1 B4 nq−3 − · · · − q+1 2 1 4 3

(2k − 1)q =

k=1

[last term contains either n or n2 .] 1.

2.

3. 4.11 5.10 6.10

n  k=1 n  k=1 n  k=1 n  k=1 n  k=1 n 

(2k − 1) = n2 1 n(4n2 − 1) 3

(2k − 1)2 =

(2k − 1)3 = n2 (2n2 − 1) (mk − 1) =

n 

(mk − 1)2 =

1 n[m2 (n + 1)(2n + 1) − 6m(n + 1) + 6] 6

(mk − 1)3 =

1 n[m3 n(n + 1)2 − 2m2 (n + 1)(2n + 1) + 6m(n + 1) − 4] 4

k(k + 1)2 =

k=1

0.124 1. 2.10

1 n(n + 1)(n + 2)(3n + 5) 12

q    1   k n2 − k 2 = q(q + 1) 2n2 − q 2 − q 4

k=1 n 

k(k + 1)3 =

k=1

0.125 0.126

JO (32b)

n [m(n + 1) − 2] 2

k=1

0.123

n  k=1 n  k=0

JO (32a)

  1 n(n + 1) 12n3 + 63n2 + 107n + 58 60

k! · k = (n + 1)! − 1 (n + k)! = k!(n − k)!

[q = 1, 2, . . .]



e K 1 π n+ 2

AD (188.1)



1 2

WA 94

0.142

Sums of the binomial coefficients

3

0.13 Sums of reciprocals of natural numbers ∞ n   1 1 Ak = C + ln n + − , k 2n n(n + 1) . . . (n + k − 1)

0.13111

k=1

JO (59), AD (1876)

k=2

where  1 1 x(1 − x)(2 − x)(3 − x) · · · (k − 1 − x) dx k 0 1 1 A2 = , A3 = 12 12 19 9 A4 = , A5 = , 120 20  3  n  2 − 1 B4 1 B2 1 7 0.132 = (C + ln n) + ln 2 + 2 + + ... 2k − 1 2 8n 64n4 k=1 n  3 2n + 1 1 = − 0.133 k2 − 1 4 2n(n + 1) Ak =

JO (71a)a JO (184f)

k=2

0.14 Sums of products of reciprocals of natural numbers 1.

n  k=1

2.

n  k=1

3.

4.

n 1 = [p + (k − 1)q](p + kq) p(p + nq)

GI III (64)a

n(2p + nq + q) 1 = [p + (k − 1)q](p + kq)[p + (k + 1)q] 2p(p + q)(p + nq)[p + (n + 1)q]

GI III (65)a

n 

1 [p + (k − 1)q](p + kq) . . . [p + (k + l)q] k=1

 1 1 1 − = (l + 1)q p(p + q) . . . (p + lq) (p + nq)[p + (n + 1)q] . . . [p + (n + l)q] n  k=1

1 1 = [1 + (k − 1)q][1 + (k − l)q + p] p

%

n  k=1

1 − 1 + (k − 1)q

n  k=1

1 1 + (k − 1)q + p

&

AD (1856)a

GI III (66)a

0.142

n  k2 + k − 1 k=1

(k + 2)!

=

1 n+1 − 2 (n + 2)!

JO (157)

0.15 Sums of the binomial coefficients Notation: n is a natural number.



m  n+k n+m+1 1. = n n+1 k=0 n n 2. 1+ + + . . . = 2n−1 2 4

KR 64 (70.1) KR 62 (58.1)

4

Finite Sums

3. 4.

n 1 m 

+

n

(−1)k

3 n k

k=0

0.152 1. 2. 3. 0.153 1. 2. 3. 4.

+

n 5

0.152

+ . . . = 2n−1

= (−1)m

n−1 m

KR 62 (58.1)

[n ≥ 1]

KR 64 (70.2)

nπ 1 n 2 + 2 cos 3 3 0 3 6

n n n 1 (n − 2)π + + + ... = 2n + 2 cos 3 3 1 4 7

n n n 1 (n − 4)π n + + + ... = 2 + 2 cos 3 3 2 5 8 n

+

n

+

n

+ ... =

KR 62 (59.1) KR 62 (59.2) KR 62 (59.3)

n 1  n−1 nπ 2 + 2 2 cos 2 4 0 4 8 n n n n 1  n−1 nπ 2 + + + ... = + 2 2 sin 2 4 1 5 9  n n  n n 1 n−1 nπ 2 + + + ... = − 2 2 cos 2 4 2 6 10 n n  n n 1  n−1 nπ 2 + + + ... = − 2 2 sin 2 4 3 7 11

n

+

n

+

n

+ ... =

KR 63 (60.1) KR 63 (60.2) KR 63 (60.3) KR 63 (60.4)

0.154 1.

n 

(k + 1)

n

k=0

2.

n 

k

(−1)k+1 k

n k

k=1

3.

N 

(−1)k

k=0

4.

n 

(−1)k

5.

n 

(−1)k

N  k=0

0.155 1.

(−1)k

k n−1 = 0

n k N k

=0

KR 63 (66.1)

[n ≥ 2]

KR 63 (66.2)

[N ≥ n ≥ 1;

00 ≡ 1]

kn = (−1)n n!

[n ≥ 0;

00 ≡ 1]

(α + k)n = (−1)n n!

[n ≥ 0;

00 ≡ 1]

(α + k)n−1 = 0

n  (−1)k+1  n n = k+1 k n+1

k=1

[n ≥ 0]



k

k=0

6.

N k

n

k=0

= 2n−1 (n + 2)

[N ≥ n ≥ 1,

00 ≡ 1 N, n ∈ N + ]

KR 63 (67)

0.159

2.

Sums of the binomial coefficients

1  n 2n+1 − 1 = k+1 k n+1

n  k=0

3.

5

KR 63 (68.1)

n  αk+1  n (α + 1)n+1 − 1 = k+1 k n+1

KR 63 (68.2)

k=0

4.

n n  (−1)k+1  n  1 = k k m m=1

KR 64 (69)

k=1

0.156 1.



p   n m n+m = k p−k p

[m is a natural number]

KR 64 (71.1)

k=0

2.

n−p  k=0

n k



n p+k

0.157 1.

n   n

k

k=0

2.

2n 

2

(−1)k

k=0

3.

2n+1 

(−1)k

k=0

4. 0.15810 1.

2.

3.

4.

= 2n k



=

2n n

(2n)! (n − p)!(n + p)!

KR 64 (72.1)



2

= (−1)n

2n + 1 k

2n n



1.

2

=0

n n  (2n − 1)! 2 k = 2 k [(n − 1)!] k=1







 n   2n 2n − k 2n − k − 1 k k+1 n 2 −2 k=4 − n n−k n−k−1 k=1





 n   2n 2n − k 2n − k − 1 3 · 4n 2k − 2k+1 k 2 = 4n − n n−k n−k−1 k=1





 n   2n 2n − k 2n − k − 1 2k − 2k+1 k 3 = (6n + 13)4n − 18n n n−k n−k−1 k=1





 n   2n 2n − k 2n − k − 1 − (60n + 75)4n 2k − 2k+1 k 4 = (32n2 + 104n) n n−k n−k−1 



  n   2n 2n 2n 1 n − k= 4 − n n−k n−k−1 2

k=0

KR 64 (72.2)



k=1

0.15910

KR 64 (71.2)

KR 64 (72.3)

KR 64 (72.4)

6

Numerical Series and Infinite Products

2.

3.

0.160



  

n   2n 2n 2n 1 − k2 = (2n + 1) − 4n n−k n−k−1 2 n k=0





n   2n 2n (3n + 2) n 1 2n ·4 − − k3 = (3n + 1) n−k n−k−1 4 2 n k=0

0.16010 1.



 k

2n n−1  2n α 1 2n (1 + α)2n−1 (1 − α)  2k 1 = (1 + α)2n αk + αn + 2 k k 2 n 2 (1 + α) 2

k=n+1

2.

k=0

 n Γ (r + b) B (n + a − b, b) = (−1)r r Γ (r + a) Γ (a − b) r=0

n 

0.2 Numerical Series and Infinite Products 0.21 The convergence of numerical series The series ∞  0.211 uk = u1 + u2 + u3 + . . . k=1

is said to converge absolutely if the series ∞  0.212 |uk | = |u1 | + |u2 | + |u3 | + · · · , k=1

composed of the absolute values of its terms converges. If the series 0.211 converges and the series 0.212 diverges, the series 0.211 is said to converge conditionally. Every absolutely convergent series converges.

0.22 Convergence tests Suppose that lim |uk |

k→∞

1/k

=q

If q < 1, the series 0.211 converges absolutely. On the other hand, if q > 1, the series 0.211 diverges. (Cauchy) 0.222 Suppose that ! ! ! uk+1 ! !=q lim !! k→∞ uk ! ! ! ! uk+1 ! ! Here, if q < 1, the series 0.211 converges absolutely. If q > 1, the series 0.211 diverges. If !! uk ! approaches 1 but remains greater than unity, then the series 0.211 diverges. (d’Alembert) 0.223

Suppose that

! 

! ! uk ! ! ! lim k ! −1 =q k→∞ uk+1 ! Here, if q > 1, the series 0.211 converges absolutely. If q < 1, the series 0.211 diverges. (Raabe)

0.229

Convergence tests

7

0.224

Suppose that f (x) is a positive decreasing function and that   ek f ek lim =q k→∞ f (k) #∞ for natural k. If q < 1, the series k=1 f (k) converges. If q > 1, this series diverges. (Ermakov) 0.225 Suppose that ! ! ! uk ! q |vk | ! ! ! uk+1 ! = 1 + k + k p , where p > 1 and the |vk | are bounded, that is, the |vk | are all less than some M , which is independent of k. Here, if q > 1, the series 0.211 converges absolutely. If q ≤ 1, this series diverges. (Gauss) 0.226 Suppose that a function f (x) defined for x ≥ q ≥ 1 is continuous, positive, and decreasing. Under these conditions, the series ∞  f (k) k=1

converges or diverges accordingly as the integral  ∞ f (x) dx q

converges or diverges (the Cauchy integral test). 0.227 Suppose that all terms of a sequence u1 , u2 , . . . , un are positive. In such a case, the series 1.

∞ 

(−1)k+1 uk = u1 − u2 + u3 − . . .

k=1

is called an alternating series. If the terms of an alternating series decrease monotonically in absolute value and approach zero, that is, if 2.

3.

uk+1 < uk and lim uk = 0, k→∞

the series 0.227 1 converges. Here, the remainder of the series is ! !∞ ∞ n ! !   ! ! (−1)k−n+1 uk = ! (−1)k+1 uk − (−1)k+1 uk < un+1 ! ! ! k=n+1

0.228 1.

k=1

(Leibniz)

k=1

If the series ∞ 

vk = v1 + v2 + . . . + vk + . . .

k=1

2.

converges and the numbers uk form a monotonic bounded sequence, that is, if |uk | < M for some number M and for all k, the series ∞  uk vk = u1 v1 + u2 v2 + . . . + uk vk + . . . FI II 354 k=1

converges. (Abel) 0.229 If the partial sums of the series 0.228 1 are bounded and if the numbers uk constitute a monotonic sequence that approaches zero, that is, if

8

Numerical Series and Infinite Products

! ! n ! ! ! ! vk ! < M ! ! !

[n = 1, 2, . . .]

0.231

and lim uk = 0,

k=1

FI II 355

k→∞

then the series 0.228 2 converges (Dirichlet).

0.23–0.24 Examples of numerical series 0.231

Progressions ∞ 

1.

aq k =

k=0 ∞ 

2.

a 1−q

(a + kr)q k =

k=0

0.232 1.

∞ 

∞ 

1 = ln 2 k

(−1)k+1

 π 1 1 =1−2 = 2k − 1 (4k − 1)(4k + 1) 4

3.

(cf. 0.113)

(cf. 1.511)

k=1

∞  ka k=1

[|q| < 1]

∞

k=1

∗

rq a + 1 − q (1 − q)2

(−1)k+1

k=1

2.

[|q| < 1]

(cf. 1.643) ⎤

⎡

a i  1  (−1)j (a + 1)!(i − j)a ⎦ 1 ⎣ = bk (b − 1)a+1 i=1 ba−i j=0 j!(a + 1 − j)!

[a = 1, 2, 3, . . . , 0.233 1.

∞  1 1 1 = 1 + p + p + . . . = ζ(p) kp 2 3

b = 1]

[Re p > 1]

WH

[Re p > 0]

WH

k=1

2.

∞ 

(−1)k+1

k=1

3.10

∞  1 22n−1 π 2n |B2n |, = k 2n (2n)!

k=1

4.

∞ 

(−1)k+1

k=1

5.

∞  k=1

6.

1 = (1 − 21−p ) ζ(p) kp

∞  k=1

∞  1 π2 = k2 6

1 (22n−1 − 1)π 2n |B2n | = k 2n (2n)!

1 (22n − 1)π 2n |B2n | = 2n (2k − 1) 2 · (2n)! (−1)k+1

FI II 721

k=1

1 π 2n+1 |E2n | = 2n+2 2n+1 (2k − 1) 2 (2n)!

JO (165)

JO (184b)

JO (184d)

0.236

0.234 1.

2.

3.

4.

5.

6.

7. 8.6 9.∗

Examples of numerical series

∞  k=1 ∞  k=1 ∞  k=0 ∞  k=1 ∞  k=1 ∞  k=1 ∞ 

(−1)k+1

1 π2 = 2 k 12

9

EU

1 π2 = 2 (2k − 1) 8

EU

(−1)k =G (2k + 1)2

FI II 482

(−1)k+1 π3 = (2k − 1)3 32

EU

1 π4 = 4 (2k − 1) 96

EU

(−1)k+1 5π 5 = (2k − 1)5 1536

EU

(−1)k+1

k=1 ∞ 

k π2 − ln 2 = (k + 1)2 12

1 = 2 − 2 ln 2 k(2k + 1) k=1   ∞  √ Γ n + 12 = π ln 4 2 n Γ (n) n=1

0.235 Sn =

∞  k=1

S1 =

(4k 2 1 , 2

1 n − 1)

S2 =

π2 − 8 , 16

S3 =

32 − 3π 2 , 64

S4 =

π 4 + 30π 2 − 384 768 JO (186)

0.236 1.

2.

3.

4.

5.

∞  k=1 ∞  k=1 ∞  k=1 ∞ 

1 = 2 ln 2 − 1 k (4k 2 − 1)

BR 51a

3 1 = (ln 3 − 1) 2 k (9k − 1) 2

BR 51a

3 1 = −3 + ln 3 + 2 ln 2 k (36k 2 − 1) 2 k

=

2

(4k 2 − 1) k=1 ∞  k=1

1 k (4k 2

2

− 1)

1 8

=

BR 52, AD (6913.3)

BR 52

3 − 2 ln 2 2

BR 52

10

6.

Numerical Series and Infinite Products ∞ 

2

k=1

7.6

12k 2 − 1

∞  k=1

k (4k 2 − 1)

0.237

= 2 ln 2

AD (6917.3), BR 52

1 π2 − 2 ln 2 =4− 2 k(2k + 1) 4

0.237 1.

∞  k=1

2.

∞  k=1

3.

∞  k=2

4.

1 1 = (2k − 1)(2k + 1) 2 1 π 1 = − (4k − 1)(4k + 1) 2 8 3 1 = (k − 1)(k + 1) 4

∞   k=1,k=m

5.

AD (6917.2), BR 52

∞   k=1,k=m

[cf. 0.133],

3 1 =− 2 (m + k)(m − k) 4m

[m is an integer]

AD (6916.1)

3 (−1)k−1 = (m − k)(m + k) 4m2

[m is an even number]

AD (6916.2)

0.238 1.

∞  k=1

2.

∞  k=1

3. 0.239 1.11 2.7

3.

1 (−1)k+1 = (1 − ln 2) (2k − 1)2k(2k + 1) 2

∞ 

1 1 π 1 = − ln 3 + √ (3k + 1)(3k + 2)(3k + 3)(3k + 4) 6 4 12 3 k=0

π √ + ln 2 3 k=1

∞  1 π 1 √ − ln 2 = (−1)k+1 3k − 1 3 3 k=1 ∞ 

∞  k=1

4.

1 1 = ln 2 − (2k − 1)2k(2k + 1) 2

∞  k=1

(−1)k+1

(−1)k+1

1 1 = 3k − 2 3

GI III (93)

GI III (94)a

GI III (95)



√ , 1 + 1 = √ π + 2 ln 2+1 4k − 3 4 2

k+3 1 π 1 (−1)[ 2 ] = + ln 2 k 4 2

GI III (85), BR∗ 161 (1)

BR∗ 161 (1)

BR∗ 161 (1)

GI III (87)

0.241

Examples of numerical series ∞ 

5.

k+3 (−1)[ 2 ]

k=1 ∞ 

6.

k+5 (−1)[ 3 ]

k=1 ∞ 

7.

k=1

11

1 π = √ 2k − 1 2 2 1 5π = 2k − 1 12

GI III (88)

1 π √ 1 = − 2+1 (8k − 1)(8k + 1) 2 16

0.241 ∞  1 = ln 2 2k k

1.

JO (172g)

k=1 ∞ 

2.

k=1

3.11

1 2k k 2

=

∞  2n n

n=0

1 π2 2 − (ln 2) 12 2

pn = √

JO (174)



1 1 − 4p

0≤p<

4.10

 p ∞  pn ln(1 − x) π2 − dx = 2 n 6 x 1 n=1

5.10







 i   2i 2i − j 2i − (j + 1) 2j − 2j+1 j = 4i − i i − j i − (j + 1) j=1 + n

6.10

7.10



2j

j=1

8.10

9.10

10.

10

2i − j i−j



− 2j+1

2i − (j + 1) i − (j + 1)



[0 ≤ p ≤ 1]

m





 i   2i 2i − j 2i − (j + 1) − 3 · 4i 2j − 2j+1 j 2 = 4i i i − j i − (j + 1) j=1 + n i  

1 4



m

, = 0,

m<0

= 0,

m<0

,

j 3 = (6i + 13)4i − 18i + n

2i i

,

= 0,

m<0

m





i    2  2i 2i − j 2i − (j + 1) j j+1 4 2 −2 j = 32i + 104i − (60i + 75)4i i − j i − (j + 1) i j=1

 i

n−1 2n  2n k 1 2n (1 + k)2n−1 (1 − k)  2i 1 = (1 + k)2n kj + kn + 2 j i 2 n 2 (1 + k) 2 j=n+1 i=0

i  i+k k=0

k

2i−k = 4i

12

11.

Numerical Series and Infinite Products

10

i  i+k h

k=0

12.

10

i  2i k

k=0

13.10

i  2i k

k=0

14.10

i  2i k

k=0

0.242

∞ 

1.

∞  k=1

k = (i + 1)4 − (2i + 1) i

2i i



  2i 1 i = 4 + i 2 k=

i i 4 2

k2 = (2i + 1)i4i−1 −

(−1)k

k=0

0.243

i−k

0.242

i2 2



2i i



1 n2 = n2k n2 + 1

[n > 1]

1 1 1 = [p + (k − 1)q](p + kq) . . . [p + (k + l)q] (l + 1)q p(p + q) . . . (p + lq) (see also 0.141 3)

2.7

3.

 1 i tp−1 (1 − t)t x = dt [p + (k − 1)q][p + (k − 1)q + 1][p + (k − 1)q + 2] . . . [p + (k − 1)q + l] l! 0 1 − xtq k=1   p > 0, x2 < 1 BR∗ 161 (2), AD (6.704) ∞ 

∞  k=0

k−1

1 (2k + 1)3



   

(2k + 1)πx 1 (2k + 1)π π3 tanh + x tanh = x 2 2x 16

0.244 1.

∞  k=1

2.

∞ 

1 1 = (k + p)(k + q) q−p (−1)k+1

k=1

3.10

∞  k=1

1 = p + (k − 1)q



1 0



1 0

xp − xq dx 1−x

[p > −1,

tp−1 dt 1 + tq

[p > 0,

q  1 1 1 = (k + p)(k + q) q − p m=p+1 m

Summations of reciprocals of factorials 0.245 1.

∞  1 = e = 2.71828 . . . k!

k=0

2.11

∞  (−1)k k=0

k!

=

1 ≈ 0.1839397 . . . 2e

q > −1, q > 0]

[q > p > −1,

p = q]

GI III (90)

BR∗ 161 (1)

p and q integers]

0.249

3.

4.

5.

6.

7.

8.

Examples of numerical series ∞  k=1 ∞ 

1.

2.

3.

4.

5.

6.

k=0 ∞  k=0 ∞ 

∞  k=0 ∞  k=0 ∞  k=0 ∞ 

0.248

(−1)k = cos 1 = 0.54030 . . . (2k)! (−1)k−1 = sin 1 = 0.84147 . . . (2k − 1)! 1 (k!)

2

k=0 ∞ 

∞  k=1 ∞  k=1

= I 0 (2) = 2.27958530 . . .

1 = I 1 (2) = 1.590636855 . . . k!(k + 1)! 1 = I n (2) k!(k + n)! (−1)k 2

k=0 ∞ 

k=0

0.247

1 k = = 0.36787 . . . (2k + 1)! e

k =1 (k + 1)! k=1

∞  1 1 1 = e+ = 1.54308 . . . (2k)! 2 e k=0

∞  1 1 1 = e− = 1.17520 . . . (2k + 1)! 2 e

k=0

0.246

(k!)

= J 0 (2) = 0.22389078 . . .

(−1)k = J 1 (2) = 0.57672481 . . . k!(k + 1)! (−1)k = J n (2) k!(k + n)! 1 k! = (n + k − 1)! (n − 2) · (n − 1)! kn = Sn , k! S1 = e, S5 = 52e,

0.2497

13

∞  (k + 1)3 k=0

k!

= 15e

S2 = 2e, S6 = 203e,

S3 = 5e, S7 = 877e,

S4 = 15e S8 = 4140e

14

Numerical Series and Infinite Products

0.250

0.25 Infinite products 0.250

Suppose that a sequence of numbers a1 , a2 , . . . , ak , . . . is given. If the limit lim

n→∞

n -

(1 + ak )

k=1

exists, whether finite or infinite (but of definite sign), this limit is called the value of the infinite product ∞ (1 + ak ), and we write k=1

1.

lim

n→∞

n -

(1 + ak ) =

k=1

∞ -

(1 + ak )

k=1

If an infinite product has a finite nonzero value, it is said to converge. Otherwise, the infinite product is said to diverge. We assume that no ak is equal to −1. FI II 400 0.251 For the infinite product 0.250 1. to converge, it is necessary that lim ak = 0. FI II 403 k→∞

0.252 If ak > 0 or ak < 0 for all values of the index k starting with some# particular value, then, for the ∞ product 0.250 1 to converge, it is necessary and sufficient that the series k=1 ak converge. ∞ ∞ 0.253 The product (1 + ak ) is said to converge absolutely if the product (1 + |ak |) converges. k=1

k=1

FI II 403

0.254 0.255

Absolute convergence of an infinite product implies its convergence. ∞ ∞  (1 + ak ) converges absolutely if, and only if, the series ak converges abThe product k=1

k=1

solutely.

FI II 406

0.26 Examples of infinite products 0.261

∞ k=1

0.262 1.

∞ -

(−1)k+1 1+ 2k − 1

1−

k=2

2.

3.

∞ -

1 k2

=

=

√ 2

1 2 =

k=1

0.263 1. 2.∗

EU

FI II 401



2 π k=1

∞ 1 π 1− = 2 (2k + 1) 4 1−

1 (2k)2



1/2

1/4

1/8 4 6·8 10 · 12 · 14 · 16 2 e= · ... 1 3 5·7 9 · 11 · 13 · 15 1/2 2 1/3 3 1/4 4 4 1/5 2 2 2 ·4 2 ·4 e= ··· 3 1 1·3 1·3 1 · 36 · 5

FI II 401

FI II 401

0.303

3.

∗

Definitions and theorems

π = 2

1/2 2 1/4 3 1/8 4 4 1/16 1 2 2 ·4 2 ·4 ··· 2 1·3 1 · 33 1 · 36 · 5

where the nth factor is the (n + 1)th root of the product 0.264 1.

2.∗

15

$n

k=0 (k

+ 1)(−1)

k+1

( nk ) .

√ k e 1 k=1 1 + k 1/2 2 1/3 3 1/4 4 4 1/5 2 2 2 ·4 2 ·4 eC = ··· 1 1·3 1 · 33 1 · 36 · 5 eC =

∞ -

FI II 402

$n k+1 n where the nth factor is the (n + 1)th root of the product k=0 (k + 1)(−1) ( k ) . Here C is the Euler constant, denoted in other works by γ. / .  .  0  0 2 1 1 1 1 11 1 1 1 1 0.265 = · + · + + ... FI II 402 π 2 2 2 2 2 2 2 2 2 ∞  k 1 1 + x2 = [0 < x < 1] 0.2668 FI II 401 1−x k=0

0.3 Functional Series 0.30 Definitions and theorems 0.301 1.

The series ∞ 

fk (x),

k=1

the terms of which are functions, is called a functional series. The set of values of the independent variable x for which the series 0.301 1 converges constitutes what is called the region of convergence of that series. 0.302 A series that converges for all values of x in a region M is said to converge uniformly in that region if, for every ε ≥ 0, there exists a number N such that, for n > N , the inequality ! ! ∞ !  ! ! ! fk (x)! < ε ! ! ! k=n+1

holds for all x in M . 0.303

If the terms of the functional series 0.301 1 satisfy the inequalities: |fk (x)| < uk (k = 1, 2, 3, . . .) , throughout the region M , where the uk are the terms of some convergent numerical series ∞  uk = u1 + u2 + . . . + uk + . . . , k=1

the series 0.301 1 converges uniformly in M . (Weierstrass)

16

Functional Series

0.304

0.304 Suppose that the series 0.301 1 converges uniformly in a region M and that a set of functions gk (x) constitutes (for each x) a monotonic sequence, and that these functions are uniformly bounded; that is, suppose that a number L exists such that the inequalities 1.

|gn (x)| ≤ L

2.

hold for all n and x. Then, the series ∞  fk (x)gk (x) k=1

converges uniformly in the region M . (Abel)

FI II 451

0.305 Suppose that the partial sums of the series 0.301 1 are uniformly bounded; that is, suppose that, for some L and for all n and x in M , the inequalities ! n ! ! ! ! ! fk (x)! < L ! ! ! k=1

hold. Suppose also that for each x the functions gn (x) constitute a monotonic sequence that approaches zero uniformly in the region M . Then, the series 0.304 2 converges uniformly in the region M . (Dirichlet) FI II 451 6

0.306 If the functions fk (x) (for k = 1, 2, 3, . . . ) are integrable on the interval [a, b] and if the series 0.301 1 made up of these functions converges uniformly on that interval, this series may be integrated termwise; that is,   b  ∞ ∞  b  fk (x) dx = fk (x) dx [a ≤ x ≤ b] FI II 459 a

k=1

k=1

a

 0.307 Suppose that the functions fk (x) (for k = 1, 2, 3, . . . ) have continuous derivatives #∞ fk (x) on the interval [a, b]. If the series 0.301 1 converges on this interval and if the series k=1 fk (x) of these derivatives converges uniformly, the series 0.301 1 may be differentiated termwise; that is,  ∞ ∞   fk (x) = fk (x) FI II 460 k=1

k=1

0.31 Power series 0.311 1.

A functional series of the form ∞ 

ak (x − ξ)k = a0 + a1 (x − ξ) + a2 (x − ξ)2 + . . .

k=0

is called a power series. The following is true of any power series: if it is not everywhere convergent, the region of convergence is a circle with its center at the point ξ and a radius equal to R; at every interior point of this circle, the power series 0.311 1 converges absolutely, and outside this circle, it diverges. This circle is called the circle of convergence, and its radius is called the radius of convergence. If the series converges at all points of the complex plane, we say that the radius of convergence is infinite (R = +∞).

0.315

0.312 is,

Power series

17

Power series may be integrated and differentiated termwise inside the circle of convergence; that 

x



ξ

∞ 

 ak (x − ξ)

k

k=0

d dx



∞ 

∞  ak (x − ξ)k+1 , k+1

dx =

k=0 ∞ 

 ak (x − ξ)k

=

k=0

kak (x − ξ)k−1 .

k=1

The radius of convergence of a series that is obtained from termwise integration or differentiation of another power series coincides with the radius of convergence of the original series. Operations on power series 0.313

Division of power series. ∞  k=0 ∞ 

bk xk = ak xk

∞ 1  ck xk , a0 k=0

k=0

where

1 cn−k ak − bn = 0, a0 n

cn +

k=1

or

⎡

k=0

where c0 =

a0 a1 a2 .. .

0 a0 a1 .. .

···

⎢ ⎢ (−1) ⎢ ⎢ cn = ⎢ .. an0 ⎢ . ⎢ ⎣an−1 b0 − a0 bn−1 an−2 an−3 · · · an b 0 − a0 b n an−1 an−2 · · · Power series raised to powers. n ∞ ∞   k ak x = ck xk , n

0.314

a1 b 0 − a0 b 1 a2 b 0 − a0 b 2 a3 b 0 − a0 b 3 .. .

an0 ,

cm

m 1  = (kn − m + k)ak cm−k ma0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ ⎥ ⎥ ⎥ a0 ⎦ a1

AD (6360)

k=0

for m ≥ 1

[n is a natural number]

AD (6361)

k=1

0.315

The substitution of one series into another. ∞ ∞ ∞    bk y k = ck xk y= ak xk ; k=1

c 1 = a1 b 1 ,

k=1

c2 = a2 b1 + a21 b2 ,

k=1

c3 = a3 b1 + 2a1 a2 b2 + a31 b3 ,

c4 = a4 b1 + a22 b2 + 2a1 a3 b2 + 3a21 a2 b3 + a41 b4 ,

...

AD (6362)

18

0.316

Functional Series

Multiplication of power series ∞ ∞ ∞    ak xk bk xk = ck xk k=0

k=0

cn =

k=0

0.316

n 

ak bn−k

FI II 372

k=0

Taylor series 0.317 If a function f (x) has derivatives of all orders throughout a neighborhood of a point ξ, then we may write the series 1.

(x − ξ)  (x − ξ)2  (x − ξ)3  f (ξ) + f (ξ) + f (ξ) + . . . , 1! 2! 3! which is known as the Taylor series of the function f (x).

f (ξ) +

The Taylor series converges to the function f (x) if the remainder 2.

Rn (x) = f (x) − f (ξ) −

n  (x − ξ)k k=1

k!

f (k) (ξ)

approaches zero as n → ∞. The following are different forms for the remainder of a Taylor series: 3.

Rn (x) =

4.

Rn (x) =

5.

(x − ξ)n+1 (n+1) f (ξ + θ(x − ξ)) (n + 1)!

[0 < θ < 1]

(Lagrange)

(x − ξ)n+1 (1 − θ)n f (n+1) (ξ + θ(x − ξ)) [0 < θ < 1] n! ψ(x − ξ) − ψ(0) (x − ξ)n (1 − θ)n (n+1) f Rn (x) =  (ξ + θ(x − ξ)) ψ [(x − ξ)(1 − θ)] n! [0 < θ < 1] ,

(Cauchy)

(Schl¨ omilch)

where ψ(x) is an arbitrary function satisfying the following two conditions: (1) It and its derivative ψ  (x) are continuous in the interval (0, x − ξ); and (2) the derivative ψ  (x) does not change sign in that interval. If we set ψ(x) = xp+1 , we obtain the following form for the remainder: (x − ξ)n+1 (1 − θ)n−p−1 (n+1) f (ξ + θ(x − ξ)) (p + 1)n!  1 x (n+1) Rn (x) = f (t)(x − t)n dt n! ξ Rn (x) =

6. 0.318 1.11

0 < θ < 1]

(Rouch´e)

Other forms in which a Taylor series may be written: f (a + x) =

∞  xk k=0

2.

[0 < p ≤ n;

∞  xk k=0

k!

k!

f (k) (a) = f (a) +

f (k) (0) = f (0) +

x  x2 f (a) + f  (a) + . . . 1! 2!

x  x2 f (0) + f  (0) + . . . 1! 2!

(Maclaurin series)

0.323

0.319

Fourier series

19

The Taylor series of functions of several variables:

∂f (ξ, η) ∂f (ξ, η) + (y − η) f (x, y) = f (ξ, η) + (x − ξ) ∂x ∂y 

2 2 ∂ f (ξ, η) ∂ 2 f (ξ, η) 1 2 2 ∂ f (ξ, η) + (y − η) + 2(x − ξ)(y − η) + + ... (x − ξ) 2! ∂x2 ∂x ∂y ∂y 2

0.32 Fourier series 0.320 Suppose that f (x) is a periodic function of period 2l and that it is absolutely integrable (possibly improperly) over the interval (−l, l). The following trigonometric series is called the Fourier series of f (x):

∞ a0  kπx kπx + + bk sin , 1. ak cos 2 l l k=1

2. 3.11

the coefficients of which (the Fourier coefficients) are given by the formulas   1 α+2l 1 l kπt kπt ak = dt = dt (k = 0, 1, 2, . . .) f (t) cos f (t) cos l −l l l α l   1 α+2l 1 l kπt kπt dt = dt (k = 1, 2, . . .) bk = f (t) sin f (t) sin l −l l l α l

Convergence tests 0.321

The Fourier series of a function f (x) at a point x0 converges to the number f (x0 + 0) + f (x0 − 0) , 2 if, for some h > 0, the integral  h |f (x0 + t) + f (x0 − t) − f (x0 + 0) − f (x0 − 0)| dt t 0 exists. Here, it is assumed that the function f (x) either is continuous at the point x0 or has a discontinuity of the first kind (a saltus) at that point and that both one-sided limits f (x0 + 0) and f (x0 − 0) exist. FI III 524 (Dini) 0.322 The Fourier series of a periodic function f (x) that satisfies the Dirichlet conditions on the interval [a, b] converges at every point x0 to the value 12 [f (x0 + 0) + f (x0 − 0)]. (Dirichlet) We say that a function f (x) satisfies the Dirichlet conditions on the interval [a, b] if it is bounded on that interval and if the interval [a, b] can be partitioned into a finite number of subintervals inside each of which the function f (x) is continuous and monotonic. 0.323 The Fourier series of a function f (x) at a point x0 converges to 12 [f (x0 + 0) + f (x0 − 0)] if f (x) is FI III 528 of bounded variation in some interval (x0 − h, x0 + h) with center at x0 . (Jordan–Dirichlet) The definition of a function of bounded variation. Suppose that a function f (x) is defined on some interval [a, b], where z < b. Let us partition this interval in an arbitrary manner into subintervals with the dividing points a = x0 < x1 < x2 < . . . < xn−1 < xn = b and let us form the sum

20

Functional Series n 

0.324

|f (xk ) − f (xk−1 )|

k=1

Different partitions of the interval [a, b] (that is, different choices of points of division xi ) yield, generally speaking, different sums. If the set of these sums is bounded above, we say that the function f (x) is of bounded variation on the interval [a, b]. The least upper bound of these sums is called the total variation of the function f (x) on the interval [a, b]. 0.324 Suppose that a function f (x) is piecewise-continuous on the interval [a, b] and that in each interval of continuity it has a piecewise-continuous derivative. Then, at every point x0 of the interval [a, b], the Fourier series of the function f (x) converges to 12 [f (x0 + 0) + f (x0 − 0)]. 0.325 A function f (x) defined in the interval (0, l) can be expanded in a cosine series of the form ∞

1.

a0  kπx + , ak cos 2 l k=1

where 2. 0.326 1.

ak =

2 l



l

f (t) cos 0

A function f (x) defined in the interval (0, l) can be expanded in a sine series of the form ∞ 

bk sin

k=1

where 2.

kπt dt l

bk =

2 l



kπx , l l

f (t) sin 0

kπt dt l

The convergence tests for the series 0.325 1 and 0.326 1 are analogous to the convergence tests for the series 0.320 1 (see 0.321–0.324). 0.327 The Fourier coefficients ak and bk (given by formulas 0.320 2 and 0.320 3) of an absolutely integrable function approach zero as k → ∞. If a function f (x) is square-integrable on the interval (−l, l), the equation of closure is satisfied:  ∞  1 l 2 a20   2 + ak + b2k = f (x) dx (A. M. Lyapunov) FI III 705 2 l −l k=1

0.328 Suppose that f (x) and ϕ(x) are two functions that are square-integrable on the interval (−l, l) and that ak , bk and αk , βk are their Fourier coefficients. For such functions, the generalized equation of closure (Parseval’s equation) holds:  ∞ 1 l a0 α0  + (ak αk + bk βk ) = f (x)ϕ(x) dx FI III 709 2 l −l k=1 For examples of Fourier series, see 1.44 and 1.45.

0.411

Differentiation of a definite integral with respect to a parameter

21

0.33 Asymptotic series 0.330 Included in the collection of all divergent series is the broad class of series known as asymptotic or semiconvergent series. Despite the fact that these series diverge, the values of the functions that they represent can be calculated with a high degree of accuracy if we take the sum of a suitable number of terms of such series. In the case of alternating asymptotic series, we obtain greatest accuracy if we break off the series in question at whatever term is of lowest absolute value. In this case, the error (in absolute value) does not exceed the absolute value of the first of the discarded terms (cf. 0.227 3). Asymptotic series have many properties that are analogous to the properties of convergent series, and, for that reason, they play a significant role in analysis. The asymptotic expansion of a function is denoted as follows: ∞  An z −n f (z) ∼ n=0

∞  An This is the definition of an asymptotic expansion. The divergent series is called the asymptotic zn n=0 expansion of a function f (z) in a given region of values of arg z if the expression Rn (z) = z n [f (z) − Sn (z)], n  Ak , satisfies the condition lim Rn (z) = 0 for fixed n. FI II 820 where Sn (z) = zk |z|→∞ k=0 A divergent series that represents the asymptotic expansion of some function is called an asymptotic series. 0.331 Properties of asymptotic series

1.

The operations of addition, subtraction, multiplication, and raising to a power can be performed on asymptotic series just as on absolutely convergent series. The series obtained as a result of these operations will also be asymptotic.

2.

One asymptotic series can be divided by another, provided that the first term A0 of the divisor is not equal to zero. The series obtained as a result of division will also be asymptotic. FI II 823-825

3.

An asymptotic series can be integrated termwise, and the resultant series will also be asymptotic. FI II 824 In contrast, differentiation of an asymptotic series is, in general, not permissible.

4.

A single asymptotic expansion can represent different functions. On the other hand, a given function can be expanded in an asymptotic series in only one manner.

0.4 Certain Formulas from Differential Calculus 0.41 Differentiation of a definite integral with respect to a parameter 0.410 0.411 1. 2.

 ϕ(a)  dψ(a) d d ϕ(a) dϕ(a) − f (ψ(a), a) + f (x, a) dx f (x, a) dx = f (ϕ(a), a) da ψ(a) da da da ψ(a) In particular,  d a f (x) dx = f (a) da b  d a f (x) dx = −f (b) db b

FI II 680

22

Certain Formulas from Differential Calculus

0.430

0.42 The nth derivative of a product (Leibniz’s rule) Suppose that u and v are n-times-differentiable functions of x. Then, dn v  n du dn−1 v  n d2 u dn−2 v  n d3 u dn−3 v dn u dn (uv) =u n + + + + ··· + v n n n−1 2 n−2 3 n−3 dx dx dx 1 dx dx 2 dx dx 3 dx dx or, symbolically, dn (uv) = (u + v)(n) dxn

FI I 272

0.43 The nth derivative of a composite function 0.430 1.

If f (x) = F (y)and y = ϕ(x), then dn U1  U2  U3  Un (n) F (y) + F (y) + F (y) + . . . + F (y), f (x) = dxn 1! 2! 3! n! where n dn k k dn k−1 k(k − 1) 2 dn k−2 k−1 k−1 d y y y y − y + y − . . . + (−1) ky AD (7361) GO dxn 1! dxn 2! dxn dxn (l) k i  j  h  y y y dm F y  dn n! f (x) = ··· , dxn i!j!h! . . . k! dy m 1! 2! 3! l! # Here, the symbol indicates summation over all solutions in non-negative integers of the equation i + 2j + 3h + . . . + lk = n and m = i + j + h + . . . + k.

Uk =

2.

0.431 1.

dn (−1) F dxn n







1 1 (n) 1 n − 1 n (n−1) 1 = 2n F + 2n−1 F x x x x 1! x (n − 1)(n − 2) n(n − 1) (n−2) 1 F + + ... x2n−2 2! x ⎡

2.

(−1)n

AD (7362.1)

 n  a n−1  n  a n−2 dn a 1 a ⎣  a n x = x e e + (n − 1) + (n − 1)(n − 2) dxn xn x x x 1 2 ⎤  n  a n−3 + (n − 1)(n − 2)(n − 3) + . . .⎦ x 3 AD (7362.2)

0.432 1.

  n(n − 1)   dn  2  (2x)n−2 F (n−1) x2 F x = (2x)n F (n) x2 + n dx 1!   n(n − 1)(n − 2)(n − 3) + (2x)n−4 F (n−2) x2 + 2!   n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) + (2x)n−6 F (n−3) x2 + . . . 3! AD (7363.1)

0.440

Integration by substitution

23

⎡ n(n − 1) n(n − 1)(n − 2)(n − 3) d ax + e = (2ax)n eax ⎣1 + 2 dxn 1! (4ax2 ) 2! (4ax2 ) ⎤ n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) + + · · ·⎦ 3 3! (4ax2 ) n

2.

2

2

AD (7363.2)

p p(p − 1)(p − 2) . . . (p − n + 1)(2ax) d  1 + ax2 = n−p dxn (1 + ax2 )  

2 n(n − 1) 1 + ax2 n(n − 1)(n − 2)(n − 3) 1 + ax2 × 1+ + + ... , 1!(p − n + 1) 4ax2 2!(p − n + 1)(p − n + 2) 4ax2 n

3.

n

AD (7363.3)

4. 5.

m− 12 (2m − 1)!! dm−1  sin (m arccos x) 1 − x2 = (−1)m−1 m−1 dx m



n+1 

2k n a a n+1 b n ∂ k (−1) (−1) = n! n 2 2 2 2 2k ∂a a +b a +b a

AD (7363.4) (3.944.12)

0≤2k≤n+1

6. 0.433 1.

(−1)n

∂n ∂an



b 2 a + b2



= n!

a 2 a + b2

n+1  0≤2k≤n

(−1)k

n+1 2k + 1

2k+1 b a

(3.944.11)

√ √ √ dn √  F (n) ( x) n(n − 1) F (n−1) ( x) (n + 1)n(n − 1)(n − 2) F (n−2) ( x) F x = − + √ √ n+1 √ n+2 − . . . n dxn 1! 2! (2 x) (2 x) (2 x) AD (7364.1)

2.

0.434 0.435

n−1 √ 2n−1 dn  (2n − 1)!! a 1 2 √ 1 + a x = − a dxn 2n x x    2

n 1 n 1 n n y n−p dn p p−1 d y p−2 d y y y =p − + − ... n dxn dxn dxn 1 p−1 2 p−2    n 1 dn y  n 1 dn  y 2  dn  y 3  dn n ln y = − + x − ... 1 1 · y dxn dxn dxn 2 2 · y 2 dxn

AD (7364.2)

AD (737.1)

AD (737.2)

0.44 Integration by substitution 0.44011 Let f (g(x)) and g(x) be continuous in [a, b]. Further, let g  (x) exist and be continuous there.  b  g(b) Then f [g(x)]g  (x) dx = f (u) du a

g(a)

This page intentionally left blank

1 Elementary Functions 1.1 Power of Binomials 1.11 Power series ∞   q q(q − 1) 2 q(q − 1) . . . (q − k + 1) k xk x + ··· + x + ··· = k 2! k! k=0 If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for |x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1. For x = 1, the series converges absolutely for q > 0. For x = −1, it converges absolutely for q > 0 and diverges for q < 0. If FI II 425 q = n is a natural number, the series 1.110 is reduced to the finite sum 1.111. n   n xk an−k 1.111 (a + x)n = k

1.110 (1 + x)q = 1 + qx +

k=0

1.112 1.

(1 + x)−1 = 1 − x + x2 − x3 + · · · =

∞ 

(−1)k−1 xk−1

k=1

(see also 1.121 2) 2.

(1 + x)−2 = 1 − 2x + 3x2 − 4x3 + · · · =

∞ 

(−1)k−1 kxk−1

k=1

3.11 4.

1·1 2 1·1·3 3 1·1·3·5 4 1 x + x − x + ... (1 + x)1/2 = 1 + x − 2 2·4 2·4·6 2·4·6·8 1·3 2 1·3·5 3 1 (1 + x)−1/2 = 1 − x + x − x + ... 2 2·4 2·4·6 ∞

 x = kxk (1 − x)2

1.113



x2 < 1



k=1

1.114 1.



  √ q q  x q(q − 3)  x 2 q(q − 4)(q − 5)  x 3 q 1+ 1+x =2 1+ + + ... + 1! 4 2! 4 4  2 3!  x < 1, q is a real number AD (6351.1)

25

26

The Exponential Function

 2.

x+



1 + x2

q

1.121

     ∞  q 2 q 2 − 22 q 2 − 42 . . . q 2 − (2k)2 x2k+2 =1+ (2k + 2)! k=0  2    ∞  2 2  q −1 q − 32 . . . q 2 − (2k − 1)2 2k+1 +qx + q x (2k + 1)! k=1  2  x < 1, q is a real number

AD(6351.2)

1.12 Series of rational fractions 1.121 ∞

1.

∞

 2k−1 x2  x2 x = k−1 = 2 1−x 1+x 1 − x2k k−1

k=1

k−1

x2 < 1



AD (6350.3)

k=1

∞

2.



 2k−1 1 = x−1 x2k−1 + 1



x2 > 1



AD (6350.3)

k=1

1.2 The Exponential Function 1.21 Series representation 1.211 1.11

ex =

∞  xk k=0

2.

ax =

k!

∞ k  (x ln a)

k!

k=0

3.

e−x = 2

∞  k=0

4.∗

ex = lim

n→∞

x2k k! x n

(−1)k

 1+

n

∞ 

xk (k + 1) k! k=0 ∞  x x B2k x2k 1.213 = 1 − + ex − 1 2 (2k)! k=1

2 5x3 15x4 2x ex + + + ... 1.214 e = e 1 + x + 2! 3! 4! 1.215 3x4 8x5 3x6 56x7 x2 1. esin x = 1 + x + − − − + + ... 2! 4! 5! 6! 7!

4x4 31x6 x2 + − + ... 2. ecos x = e 1 − 2! 4! 6! 1.212 ex (1 + x) =

[x < 2π]

FI II 520 AD (6460.3)

AD (6460.4) AD (6460.5)

1.232

Series of exponentials

etan x = 1 + x +

3.

27

3x3 9x4 37x5 x2 + + + + ... 2! 3! 4! 5!

AD (6460.6)

1.216 2x3 5x4 x2 + + + ... 2! 3! 4! x3 7x4 x2 − − + ... =1+x+ 2! 3! 4!

1.

earcsin x = 1 + x +

AD (6460.7)

2.

earctan x

AD (6460.8)

1.217 1.

π

∞  1 eπx + e−πx = x πx −πx 2 e −e x + k2

(cf. 1.421 3)

AD (6707.1)

(cf. 1.422 3)

AD (6707.2)

k=−∞

∞  (−1)k 2π = x πx −πx e −e x2 + k 2

2.

k=−∞

1.22 Functional relations 1.221 1. ax = ex ln a 1

aloga x = a logx a = x

2. 1.222 1.

ex = cosh x + sinh x

2.

eix = cos x + i sin x 

1.223 eax − ebx = (a − b)x exp

 ∞  1 (a − b)2 x2 (a + b)x 1+ 2 2k 2 π 2 k=1

1.23 Series of exponentials 1.231

∞ 

akx =

k=0

1 1 − ax

[a > 1 and x < 0 or 0 < a < 1 and x > 0]

1.232 1.

∞ 

(−1)k e−2kx

[x > 0]

(−1)k e−(2k+1)x

[x > 0]

tanh x = 1 + 2

k=1

2.

sech x = 2

∞  k=0

3.

cosech x = 2

∞ 

e−(2k+1)x

k=0

%

4.

∗

∞  cos2n x sin x = exp − 2n n=1

[x > 0] & [0 ≤ x ≤ π]

MO 216

28

Trigonometric and Hyperbolic Functions

1.311

1.3–1.4 Trigonometric and Hyperbolic Functions 1.30 Introduction The trigonometric and hyperbolic sines are related by the identities sinh x =

1 sin(ix), i

sin x =

1 sinh(ix). i

The trigonometric and hyperbolic cosines are related by the identities cosh x = cos(ix),

cos x = cosh(ix).

Because of this duality, every relation involving trigonometric functions has its formal counterpart involving the corresponding hyperbolic functions, and vice versa. In many (though not all) cases, both pairs of relationships are meaningful. The idea of matching the relationships is carried out in the list of formulas given below. However, not all the meaningful “pairs” are included in the list.

1.31 The basic functional relations 1.311

 1  ix e − e−ix 2i = −i sinh(ix)

1.

sin x =

2.

sinh x =

3.

cos x =

4.

cosh x =

5. 6. 7. 8.

 1 x e − e−x 2 = −i sin(ix)

 1  ix e + e−ix 2 = cosh(ix)  1 x e + e−x 2 = cos(ix)

1 sin x = tanh(ix) cos x i 1 sinh x = tan(ix) tanh x = cosh x i cos x 1 cot x = = = i coth(ix) sin x tan x 1 cosh x = = i cot (ix) coth x = sinh x tanh x tan x =

1.312 1.

cos2 x + sin2 x = 1

1.314

2.

The basic functional relations

cosh2 x − sinh2 x = 1

1.313 1. sin (x ± y) = sin x cos y ± sin y cos x 2.

sinh (x ± y) = sinh x cosh y ± sinh y cosh x

3.

sin (x ± iy) = sin x cosh y ± i sinh y cos x

4.

sinh (x ± iy) = sinh x cos y ± i sin y cosh x

5.

cos (x ± y) = cos x cos y ∓ sin x sin y

6.

cosh (x ± y) = cosh x cosh y ± sinh x sinh y

7.

cos (x ± iy) = cos x cosh y ∓ i sin x sinh y

8.

cosh (x ± iy) = cosh x cos y ± i sinh x sin y tan x ± tan y tan (x ± y) = 1 ∓ tan x tan y tanh x ± tanh y tanh (x ± y) = 1 ± tanh x tanh y tan x ± i tanh y tan (x ± iy) = 1 ∓ i tan x tanh y tanh x ± i tan y tanh (x ± iy) = 1 ± i tanh x tan y

9. 10. 11. 12. 1.314 1. 2. 3. 4. 5. 6. 7. 8. 9.∗

1 1 (x ± y) cos (x ∓ y) 2 2 1 1 sinh x ± sinh y = 2 sinh (x ± y) cosh (x ∓ y) 2 2 1 1 cos x + cos y = 2 cos (x + y) cos (x − y) 2 2 1 1 cosh x + cosh y = 2 cosh (x + y) cosh (x − y) 2 2 1 1 cos x − cos y = 2 sin (x + y) sin (y − x) 2 2 1 1 cosh x − cosh y = 2 sinh (x + y) sinh (x − y) 2 2 sin (x ± y) tan x ± tan y = cos x cos y sinh (x ± y) tanh x ± tanh y = cosh x cosh y     1 π 1 π sin x ± cos y = ±2 sin (x + y) ± sin (x − y) ± 4 4 2 2 1 π 1 π = ±2 cos (x + y) ∓ cos (x − y) ∓  2 4  2 4 1 π 1 π = 2 sin (x ± y) ± cos (x ∓ y) ∓ 2 4 2 4 sin x ± sin y = 2 sin

29

30

Trigonometric and Hyperbolic Functions

. 10.∗

a sin x ± b cos x = a 1 +

2   b b sin x ± arctan a a

 11.∗

±a sin x + b cos x = b

1+

[a = 0]  a 2 b

+  a , cos x ∓ arctan b

[b = 0]  2  r r 1 a sin x ± b cos y = q 1 + sin (x ± y) + arctan q 2 q     1 1 [q = 0] q = (a + b) cos (x ∓ y) , r = (a − b) sin (x ∓ y) 2 2    s   s 2 1 a cos x + b cos y = t 1 + cos (x ∓ y) + arctan [t = 0] t 2 t .  2  t t 1 cos (x ∓ y) − arctan = −s 1 + [s = 0] s 2 s     1 1 s = (a − b) sin (x ± y) , t = (a + b) cos (x ± y) 2 2 .

12.∗

13.∗

1.315 1.

sin2 x − sin2 y = sin(x + y) sin(x − y) = cos2 y − cos2 x

2.

sinh2 x − sinh2 y = sinh(x + y) sinh(x − y) = cosh2 x − cosh2 y

3.

cos2 x − sin2 y = cos(x + y) cos(x − y) = cos2 y − sin2 x

4.

sinh2 x + cosh2 y = cosh(x + y) cosh(x − y) = cosh2 x + sinh2 y

1.316 1. 2. 1.317 1. 2. 3. 4. 5.

1.315

n

(cos x + i sin x) = cos nx + i sin nx n

(cosh x + sinh x) = sinh nx + cosh nx  x 1 sin = ± (1 − cos x) 2 2  x 1 sinh = ± (cosh x − 1) 2 2  x 1 cos = ± (1 + cos x) 2 2  x 1 cosh = (cosh x + 1) 2 2 1 − cos x sin x x = tan = 2 sin x 1 + cos x

[n is an integer] [n is an integer]

1.321

6.

Trigonometric and hyperbolic functions: expansion in multiple angles

tanh

31

cosh x − 1 sinh x x = = 2 sinh x cosh x + 1

The signs in front of the radical in formulas 1.317 1, 1.317 2, and 1.317 3 are taken so as to agree with the signs of the left-hand members. The sign of the left hand members depends in turn on the value of x.

1.32 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) 1.320 1.

2.

n−1

  2n 2n n−k sin (−1) 2 cos 2(n − k)x + k n k=0 n−1

 2n 2n (−1)n  2n n−k sinh x = 2n (−1) 2 cosh 2(n − k)x + k n 2 2n

1 x = 2n 2

k=0

3.

4.

5.

6.

sin2n−1 x =



n−1 

1 22n−2

(−1)n+k−1

k=0

8.

cos2n−1 x =

cosh

2n−1

n−1 

1 22n−2

x=

k=0

1 22n−2

2n − 1 k

3. 4.

KR 56 (10, 4)

n−1  k=0

2n − 1 k

cos(2n − 2k − 1)x

cosh(2n − 2k − 1)x

1 (− cos 2x + 1) 2 1 sin3 x = (− sin 3x + 3 sin x) 4 1 sin4 x = (cos 4x − 4 cos 2x + 3) 8 1 sin5 x = (sin 5x − 5 sin 3x + 10 sin x) 16 sin2 x =

KR 56 (10, 1)



1.321

2.

sin(2n − 2k − 1)x



n−1 (−1)n−1  n+k−1 2n − 1 sinh x = 2n−2 (−1) sinh(2n − 2k − 1)x k 2 k=0 n−1

  2n 2n 1 2n cos x = 2n 2 cos 2(n − k)x + k n 2 k=0 n−1

  2n 2n 1 2n cosh x = 2n 2 cosh 2(n − k)x + k n 2

Special cases

1.



2n−1

k=0

7.

2n − 1 k

KR 56 (10, 2)

KR 56 (10, 3)

32

5. 6.

Trigonometric and Hyperbolic Functions

1 (− cos 6x + 6 cos 4x − 15 cos 2x + 10) 32 1 sin7 x = (− sin 7x + 7 sin 5x − 21 sin 3x + 35 sin x) 64

sin6 x =

1.322 1.

sinh2 x =

2.

sinh3 x =

3.

sinh4 x =

4.

sinh5 x =

5.

sinh6 x =

6.

sinh7 x =

1 (cosh 2x − 1) 2 1 (sinh 3x − 3 sinh x) 4 1 (cosh 4x − 4 cosh 2x + 3) 8 1 (sinh 5x − 5 sinh 3x + 10 sinh x) 16 1 (cosh 6x − 6 cosh 4x + 15 cosh 2x + 10) 32 1 (sinh 7x − 7 sinh 5x + 21 sinh 3x + 35 sinh x) 64

1.323 1.

cos2 x =

2.

cos3 x =

3.

cos4 x =

4.

cos5 x =

5.

cos6 x =

6.

cos7 x =

1 (cos 2x + 1) 2 1 (cos 3x + 3 cos x) 4 1 (cos 4x + 4 cos 2x + 3) 8 1 (cos 5x + 5 cos 3x + 10 cos x) 16 1 (cos 6x + 6 cos 4x + 15 cos 2x + 10) 32 1 (cos 7x + 7 cos 5x + 21 cos 3x + 35 cos x) 64

1.324 1.

cosh2 x =

2.

cosh3 x =

3.

cosh4 x =

4.

cosh5 x =

5.

cosh6 x =

6.

cosh7 x =

1 (cosh 2x + 1) 2 1 (cosh 3x + 3 cosh x) 4 1 (cosh 4x + 4 cosh 2x + 3) 8 1 (cosh 5x + 5 cosh 3x + 10 cosh x) 16 1 (cosh 6x + 6 cosh 4x + 15 cosh 2x + 10) 32 1 (cosh 7x + 7 cosh 5x + 21 cosh 3x + 35 cosh x) 64

1.322

1.332

Trigonometric and hyperbolic functions: expansion in powers

33

1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions 1.331 1.7

n n sin nx = n cosn−1 x sin x − cosn−3 x sin3 x + cosn−5 x sin5 x − . . . ; 3 5 ⎧

⎨ n − 2 n−3 2 cosn−3 x = sin x 2n−1 cosn−1 x − ⎩ 1 ⎫



⎬ n − 3 n−5 n − 4 n−7 + 2 cosn−5 x − 2 cosn−7 x + . . . ⎭ 2 3 AD (3.175) [(n+1)/2]

2.



sinh nx = x

n 2k − 1



k=1 [(n−1)/2]

= sinh x



sinh2k−2 x coshn−2k+1 x

(−1)k

k=0

3.

cos nx = cosn x −

n

n − k − 1 n−2k−1 coshn−2k−1 x 2 k n

cosn−4 x sin4 x − . . . ; 4

n n−3 n n − 3 n−5 n−1 n n−2 =2 cos x − 2 cos x+ cosn−4 x 2 1 2 1

n n − 4 n−7 − cosn−6 x + . . . 2 3 2 2

cosn−2 x sin2 x +

AD (3.175)

4.3

[n/2] 

n sinh2k x coshn−2k x 2k k=0

[n/2]  n n−1 k1 n−k−1 cosh x + n (−1) =2 2n−2k−1 coshn−2k x k k−1

cosh nx =



k=1

1.332 1.



  2   4n − 22 4n2 − 42 4n2 − 22 3 5 sin 2nx = 2n cos x sin x − sin x + sin x − . . . 3! 5!

2n − 2 2n−3 2n−3 2 = (−1)n−1 cos x 22n−1 sin2n−1 x − sin x 1! (2n − 3)(2n − 4) 2n−5 2n−5 2 sin x + 2!  (2n − 4)(2n − 5)(2n − 6) 2n−7 2n−7 2 − sin x + ... 3!

AD (3.171)

AD (3.173)

34

2.

3.

Trigonometric and Hyperbolic Functions

1.333

(2n − 1)2 − 12 sin3 x sin(2n − 1)x = (2n − 1) sin x − 3!     (2n − 1)2 − 12 (2n − 1)2 − 32 5 sin x − . . . AD (3.172) + 5!

2n − 1 2n−4 2n−3 2 = (−1)n−1 22n−2 sin2n−1 x − sin x 1! (2n − 1)(2n − 4) 2n−6 2n−5 2 sin x + 2!  (2n − 1)(2n − 5)(2n − 6) 2n−8 2n−7 2 − sin x + ... AD (3.174) 3!      4n2 4n2 − 22 4n2 4n2 − 2 4n2 − 42 4n2 sin2 x + sin4 x − sin6 x + . . . cos 2nx = 1 − 2! 4! 6!

2n 2n−3 2n−2 2 = (−1)n 22n−1 sin2n x − sin x 1!

AD (3.171)

 2n(2n − 3) 2n−5 2n−4 2n(2n − 4)(2n − 5) 2n−7 2n−6 2 2 + sin x− sin x + ... 2! 3! AD (3.173)a

4.

(2n − 1)2 − 12 sin2 x cos(2n − 1)x = cos x 1 − 2!     (2n − 1)2 − 12 (2n − 1)2 − 32 4 sin x − . . . + 4!

2n − 3 2n−4 2n−4 2 = (−1)n−1 cos x 22n−2 sin2n−2 x − sin x 1! (2n − 4)(2n − 5) 2n−6 2n−6 2 sin x + 2!  (2n − 5)(2n − 6)(2n − 7) 2n−8 2n−8 2 − sin x + ... 3!

AD (3.172)

AD (3.174)

By using the formulas and values of 1.30, we can write formulas for sinh 2nx, sinh(2n − 1)x, cosh 2nx, and cosh(2n − 1)x that are analogous to those of 1.332, just as was done in the formulas in 1.331. Special cases 1.333 1.

sin 2x = 2 sin x cos x

2.

sin 3x = 3 sin x − 4 sin3 x   sin 4x = cos x 4 sin x − 8 sin3 x

3. 4. 5.

sin 5x = 5 sin x − 20 sin3 x + 16 sin5 x   sin 6x = cos x 6 sin x − 32 sin3 x + 32 sin5 x

1.337

6.

Trigonometric and hyperbolic functions: expansion in powers

sin 7x = 7 sin x − 56 sin3 x + 112 sin5 x − 64 sin7 x

1.334 1.

sinh 2x = 2 sinh x cosh x

2.

sinh 3x = 3 sinh x + 4 sinh3 x   sinh 4x = cosh x 4 sinh x + 8 sinh3 x

3.11 5.11

sinh 5x = 5 sinh x + 20 sinh3 x + 16 sinh5 x   sinh 6x = cosh x 6 sinh x + 32 sinh3 x + 32 sinh5 x

6.

sinh 7x = 7 sinh x + 56 sinh3 x + 112 sinh5 x + 64 sinh7 x

4.

1.335 1.

cos 2x = 2 cos2 x − 1

2.

cos 3x = 4 cos3 x − 3 cos x

3.

cos 4x = 8 cos4 x − 8 cos2 x + 1

4.

cos 5x = 16 cos5 x − 20 cos3 x + 5 cos x

5.

cos 6x = 32 cos6 x − 48 cos4 x + 18 cos2 x − 1

6.

cos 7x = 64 cos7 x − 112 cos5 x + 56 cos3 x − 7 cos x

1.336 1.

cosh 2x = 2 cosh2 x − 1

2.

cosh 3x = 4 cosh3 x − 3 cosh x

3.

cosh 4x = 8 cosh4 x − 8 cosh2 x + 1

4.

cosh 5x = 16 cosh5 x − 20 cosh3 x + 5 cosh x

5.

cosh 6x = 32 cosh6 x − 48 cosh4 x + 18 cosh2 x − 1

6.

cosh 7x = 64 cosh7 x − 112 cosh5 x + 56 cosh3 x − 7 cosh x

1.337 1.∗ 2.∗ 3.∗ 4.∗ 5.∗ 6.∗

cos 3x cos3 x cos 4x cos4 x cos 5x cos5 x cos 6x cos6 x sin 3x cos3 x sin 4x cos4 x

= 1 − 3 tan2 x = 1 − 6 tan2 x + tan4 x = 1 − 10 tan2 x + 5 tan4 x = 1 − 15 tan2 x + 15 tan4 x − tan6 x = 3 tan x − tan3 x = 4 tan x − 4 tan3 x

35

36

7.∗ 8.∗ 9.∗ 10.∗ 11.∗ 12.∗ 13.∗ 14.∗ 15.∗ 16.∗

Trigonometric and Hyperbolic Functions

1.341

sin 5x = 5 tan x − 10 tan3 x + tan5 x cos5 x sin 6x = 6 tan x − 20 tan3 x + 6 tan5 x cos6 x cos 3x = cot3 x − 3 cot x sin3 x cos 4x = cot4 x − 6 cot2 x + 1 sin4 x cos 5x = cot5 x − 10 cot3 x + 5 cot x sin5 x cos 6x = cot6 x − 15 cot4 x + 15 cot2 x − 1 sin6 x sin 3x = 3 cot2 x − 1 sin3 x sin 4x = 4 cot3 x − 4 cot x sin4 x sin 5x = 5 cot4 x − 10 cot2 x + 1 sin5 x sin 6x = 6 cot5 x − 20 cot3 x + 6 cot x sin6 x

1.34 Certain sums of trigonometric and hyperbolic functions 1.341 1.

n−1  k=0

2.

3.

ny y n−1 y sin cosec sin(x + ky) = sin x + 2 2 2

ny 1 n−1 y sinh sinh(x + ky) = sinh x + 2 2 sinh y k=0 2

n−1  ny y n−1 y sin cosec cos(x + ky) = cos x + 2 2 2

AD (361.8)

n−1 

AD (361.9)

k=0

4.

5.

ny 1 n−1 y sinh cosh(x + ky) = cosh x + 2 2 sinh y k=0 2

2n−1  y 2n − 1 y sin ny sec (−1)k cos(x + ky) = sin x + 2 2 n−1 

JO (202)

k=0

6.

n(y + π) y n−1 (y + π) sin sec (−1)k sin(x + ky) = sin x + 2 2 2

n−1  k=0

AD (202a)

1.351

Sums of powers of trigonometric functions of multiple angles

37

Special cases 1.342 1. 2.10

n  k=1 n 

sin kx = sin

nx x n+1 x sin cosec 2 2 2

AD (361.1)

nx x n+1 x sin cosec + 1 2 2 2     sin n + 12 x nx n+1 x 1 = cos sin x cosec = 1+ 2 2 2 2 sin x2

cos kx = cos

k=0

AD (361.2)

3.

4.

n  k=1 n 

sin(2k − 1)x = sin2 nx cosec x 1 sin 2nx cosec x 2

cos(2k − 1)x =

k=1

1.343 1.

2.

3.

  1 (−1)n cos 2n+1 2 x (−1) cos kx = − + x 2 2 cos k=1 2 n  sin 2nx (−1)k+1 sin(2k − 1)x = (−1)n+1 2 cos x n 

k=1 n 

k

cos(4k − 3)x +

k=1

AD (361.7)

n 

JO (207)

AD (361.11)

AD (361.10)

sin(4k − 1)x = sin 2nx (cos 2nx + sin 2nx) (cos x + sin x) cosec 2x

k=1

JO (208)

1.344 1.

n−1  k=1

2.

n−1  k=1

3.

n−1  k=0

sin

π πk = cot n 2n

√  nπ nπ n 2πk 2 = 1 + cos − sin sin n 2 2 2 √  nπ nπ n 2πk 2 = 1 + cos + sin cos n 2 2 2

AD (361.19)

AD (361.18)

AD (361.17)

1.35 Sums of powers of trigonometric functions of multiple angles 1.351 1.

n  k=1

1 [(2n + 1) sin x − sin(2n + 1)x] cosec x 4 n cos(n + 1)x sin nx = − 2 2 sin x

sin2 kx =

AD (361.3)

38

2.

Trigonometric and Hyperbolic Functions n  k=1

1.352

n−1 1 + cos nx sin(n + 1)x cosec x 2 2 n cos(n + 1)x sin nx = + 2 2 sin x

cos2 kx =

AD (361.4)a

3.

n 

sin3 kx =

n+1 nx x 1 3(n + 1)x 3nx 3x 3 sin x sin cosec − sin sin cosec 4 2 2 2 4 2 2 2

JO (210)

cos3 kx =

n+1 nx x 1 3(n + 1) 3nx 3x 3 cos x sin cosec + cos x sin cosec 4 2 2 2 4 2 2 2

JO (211)a

sin4 kx =

1 [3n − 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] 8

JO (212)

cos4 kx =

1 [3n + 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] 8

JO (213)

k=1

4.

n  k=1

5.

n  k=1

6.

n  k=1

1.352 1.11

2.11

  n cos 2n−1 sin nx 2 x − 2 sin x2 4 sin2 x2 k=1   n−1  n sin 2n−1 1 − cos nx 2 x k cos kx = − 2 sin x2 4 sin2 x2 n−1 

k sin kx =

AD (361.5)

AD (361.6)

k=1

1.353 1.

n−1 

pk sin kx =

k=1

2.

n−1 

p sin x − pn sin nx + pn+1 sin(n − 1)x 1 − 2p cos x + p2

pk sinh kx =

k=1

3.

n−1  k=0

4.

n−1  k=0

pk cos kx =

p sinh x − pn sinh nx + pn+1 sinh(n − 1)x 1 − 2p cosh x + p2

1 − p cos x − pn cos nx + pn+1 cos(n − 1)x 1 − 2p cos x + p2

pk cosh kx =

AD (361.12)a

1 − p cosh x − pn cosh nx + pn+1 cosh(n − 1)x 1 − 2p cosh x + p2

AD (361.13)a¡

JO (396)

1.36 Sums of products of trigonometric functions of multiple angles 1.361 1.

n 

sin kx sin(k + 1)x =

1 [(n + 1) sin 2x − sin 2(n + 1)x] cosec x 4

JO (214)

sin kx sin(k + 2)x =

1 n cos 2x − cos(n + 3)x sin nx cosec x 2 2

JO (216)

k=1

2.

n  k=1

1.381

3.

Sums leading to hyperbolic tangents and cotangents

2

39

n(x + 2y) x + 2y n+1 x sin cosec sin kx cos(2k − 1)y = sin ny + 2 2 2

k=1 n(2y − x) 2y − x n+1 x sin cosec − sin ny − 2 2 2 n 

JO (217)

1.362 1.

2.

n  

x 2  n x 2 = 2 sin − sin2 x 2k 2n k=1

2

2 n  1 1 x x 2 sec k = cosec x − cosec n 2k 2 2n 2 2k sin2

AD (361.15)

AD (361.14)

k=1

1.37 Sums of tangents of multiple angles 1.371 1.

n  1 x 1 x tan k = n cot n − 2 cot 2x k 2 2 2 2

AD (361.16)

n  1 x 22n+2 − 1 1 2 x tan = + 4 cot2 2x − 2n cot2 n 22k 2k 3 · 22n−1 2 2

AD (361.20)

k=0

2.

k=0

1.38 Sums leading to hyperbolic tangents and cotangents 1.381

⎞

⎛

⎟ 1

⎟ ⎠ 2k + 1 π n sin2 4n = tanh (2nx) 2 tanh x

1+ 2k + 1 π tan2 4n ⎞ ⎛

⎜ tanh ⎜ ⎝x 1.

n−1  k=0

⎜ tanh ⎜ ⎝x 2.

n−1  k=1



1

kπ 2n 2 tanh x

1+ kπ 2 tan 2n n sin2

JO (402)a

⎟

⎟ ⎠ = coth (2nx) −

1 (tanh x + coth x) 2n

JO (403)

40

Trigonometric and Hyperbolic Functions

⎞

⎛

⎟

⎟ ⎠ 2k + 1 π (2n + 1) sin2 tanh x 2(2n + 1) = tanh (2n + 1) x − 2n + 1 tanh2 x

1+ 2k + 1 π tan2 2(2n + 1) ⎞ ⎛

⎜ tanh ⎜ ⎝x 3.

n−1  k=0

⎜ tanh ⎜ ⎝x 4.

1.382

2

2

kπ 2(2n + 1) tanh2 x

1+ kπ 2 tan (2n + 1)

n  k=1

(2n + 1) sin2

JO (404)

⎟

⎟ ⎠ = coth (2n + 1) x −

coth x 2n + 1

JO (405)

1.382 1.

n−1  k=0

2.

n−1  k=1

3.

n−1  k=0

4.

n  k=1



⎛

1

⎞ = 2n tanh (nx)  x ⎟ 1 ⎟ + tanh 2 2 ⎠

JO (406)

⎞ = 2n coth (nx) − 2 coth x  x ⎟ 1 ⎟ + tanh 2 2 ⎠

JO (407)

2k + 1 π ⎜ sin 4n ⎜ ⎝ sinh x 2

⎛



⎛



kπ 2 ⎜ sin 2n ⎜ ⎝ sinh x

2

⎜ sin ⎜ ⎝

⎛





1

1

⎞ = (2n + 1) tanh 2k + 1 π  x ⎟ 1 2(2n + 1) ⎟ + tanh sinh x 2 2 ⎠

kπ 2 ⎜ sin 2n + 1 ⎜ ⎝ sinh x



1

⎞ = (2n + 1) coth +

 x ⎟ 1 ⎟ tanh 2 2 ⎠



(2n + 1)x 2

(2n + 1)x 2

− tanh

− coth

x 2

x 2

JO (408)

JO (409)

1.395

Representing sines and cosines as finite products

41

1.39 The representation of cosines and sines of multiples of the angle as finite products 1.391 1.

sin nx = n sin x cos x

n−2 2 k=1

2.

3.

4.

1.392 1.

2.

n 2 -

⎛

⎛

⎞ 2

sin x ⎟ ⎜ ⎝1 − kπ ⎠ sin2 n ⎞

sin2 x ⎟ ⎠ (2k − 1)π 2 k=1 sin 2n ⎛ ⎞ n−1 2 2 sin x ⎟ ⎜ sin nx = n sin x ⎝1 − kπ ⎠ k=1 sin2 n ⎛ ⎞ n−1 2 2 - ⎜ sin x ⎟ cos nx = cos x ⎝1 − ⎠ (2k − 1)π 2 k=1 sin 2n cos nx =

⎜ ⎝1 −

[n is even]

JO (568)

[n is even]

JO (569)

[n is odd]

JO (570)

[n is odd]

JO (571)a



kπ sin x + n k=0

n 2k − 1 π cos nx = 2n−1 sin x + 2n

sin nx = 2n−1

n−1 -

JO (548)

JO (549)

k=1

1.393 1.

1 2k π = n−1 cos nx cos x + n 2 k=0  n 1  = n−1 (−1) 2 − cos nx 2

n−1 -

[n odd] [n even] JO (543)

2.11

n−1 -

sin x +

k=0



n−1

(−1) 2 2k π = sin nx n 2n−1 n (−1) 2 = n−1 (1 − cos nx) 2

[n odd] [n even] JO (544)

1.394

n−1 - k=0



2kπ x − 2xy cos α + n

1.395 1.

cos nx − cos ny = 2n−1

+y

n−1 - k=0





2

2

= x2n − 2xn y n cos nα + y 2n



 2kπ cos x − cos y + n

JO (573)

JO (573)

42

2.

Trigonometric and Hyperbolic Functions

cosh nx − cos ny = 2

n−1

n−1 - k=0

1.396 1.

n−1 -

x2 − 2x cos

kπ +1 n

x2 − 2x cos

2kπ 2n + 1

x2 + 2x cos

2kπ 2n + 1

k=1

2.

n -

k=1

3.

n -

k=1

4.

n−1 -

x2 − 2x cos

k=0

1.396



 2kπ cosh x − cos y + n

JO (538)



x2n − 1 x2 − 1

x2n+1 − 1 +1 = x−1

x2n+1 − 1 +1 = x+1 =

(2k + 1)π +1 2n

KR 58 (28.1)

KR 58 (28.2)

KR 58 (28.3)

= x2n + 1

KR 58 (28.4)

1.41 The expansion of trigonometric and hyperbolic functions in power series 1.411 1.

sin x =

∞ 

(−1)k

k=0

2.

sinh x =

x2k+1 (2k + 1)!

∞  x2k+1 (2k + 1)!

k=0

3.

cos x =

∞ 

(−1)k

k=0

4.

x2k (2k)!

∞  x2k cosh x = (2k)! k=0   ∞  22k 22k − 1 |B2k |x2k−1 tan x = (2k)!

6.11

  π2 x2 < 4 k=1   ∞  22k 22k − 1 2x5 17 7 x3 + − x + ··· = B2k x2k−1 tanh x = x − 3 15 315 (2k)! k=1   π2 2 x < 4

7.

1  22k |B2k | 2k−1 x cot x = − x (2k)!

5.

∞



x2 < π 2



FI II 523

FI II 523a

k=1

∞

8.

coth x =

2x5 1  22k B2k 2k−1 1 x x3 + − + − ··· = + x x 3 45 945 x (2k)! k=1  2  x < π2

FI II 522a

1.414

9.

Trigonometric and hyperbolic functions: power series expansion

sec x =

∞  |E2k | k=0

(2k)!

  π2 x2 < 4

x2k

43

CE 330a

∞

10.

sech x = 1 −

 E2k 5x4 61x6 x2 + − + ··· = 1 + x2k 2 24 720 (2k)! k=1

11.

  ∞ 1  2 22k−1 − 1 |B2k |x2k−1 cosec x = + x (2k)!

  π2 x2 < 4 

x2 < π 2



CE 330

CE 329a

k=1

12.

  ∞ 7x3 31x5 1  2 22k−1 − 1 B2k 2k−1 1 1 − + ··· = − x cosech x = − x + x 6 360 15120 x (2k)! k=1  2  x < π2

JO (418)

1.412 1.

sin2 x =

∞ 

(−1)k+1

k=1

2.

cos x = 1 − 2

∞ 

22k−1 x2k (2k)!

(−1)k+1

k=1

JO (452)a

22k−1 x2k (2k)!

JO (443)

∞

3.

4.

32k+1 − 3 2k+1 1 x (−1)k+1 4 (2k + 1)! k=1  2k  ∞ + 3 x2k 1 3 k 3 cos x = (−1) 4 (2k)!

sin3 x =

JO (452a)a

JO (443a)

k=0

1.413 1.

sinh x = cosec x

∞ 

(−1)k+1

k=1

2.

cosh x = sec x + sec x

∞ 

(−1)k

k=1

3.

sinh x = sec x

∞ 

(−1)[k/2]

k=1

4.

cosh x = cosec x

∞  k=1

1.414 1.

22k−1 x4k−2 (4k − 1)! 22k x4k (4k)!

2k−1 x2k−1 (2k − 1)!

(−1)[(k−1)/2]

2k−1 x2k−1 (2k − 1)!

 2     ∞ , +    n + 02 n2 + 22 . . . n2 + (2k)2 2k+2 x cos n ln x + 1 + x2 = 1 − (−1)k (2k + 2)! k=0  2  x <1

JO (508)

JO (507)

JO (510)

JO (509)

AD (6456.1)

44

Trigonometric and Hyperbolic Functions

1.421

 2  2   2  2k+1 ∞ , +  2 2 2   n n . . . n x + 1 + 3 + (2k − 1) k+1 sin n ln x + 1 + x2 = nx − n (−1) (2k + 1)! k=1  2  x <1 AD (6456.2)

2.

Power series for ln sin x, ln cos x, and ln tan x see 1.518.

1.42 Expansion in series of simple fractions 1.421 ∞

1.

tan

πx 4x  1 = 2 π (2k − 1)2 − x2

BR* (191), AD (6495.1)

k=1

∞

2.10

tanh

4x  1 πx = 2 π (2k − 1)2 + x2 k=1

3.

cot πx =

∞ ∞ 2x  x  1 1 1 1 + + = πx π x2 − k 2 πx π k(x − k) k=1

AD (6495.2), JO (450a)

k=−∞ k=0

∞

4.

2x  1 1 + coth πx = πx π x2 + k 2

(cf. 1.217 1)

k=1

∞

5.

tan2

 πx 2(2k − 1)2 − x2 = x2 2 2 2 2 (12 − x2 ) (32 − x2 ) . . . [(2k − 1)2 − x2 ]

JO (450)

k=1

1.422 1.

2.

∞ πx 4 2k − 1 sec = (−1)k+1 2 π (2k − 1)2 − x2 k=1  ∞ 1 4  πx 1 = 2 sec2 + 2 π (2k − 1 − x)2 (2k − 1 + x)2

AD (6495.3)a

JO (451)a

k=1

∞

3.

cosec πx =

2x  (−1)k 1 + πx π x2 − k 2

(see also 1.217 2)

AD (6495.4)a

k=1

∞ ∞ 1 1  1 2  x2 + k 2 = + 2 π2 (x − k)2 π 2 x2 π2 (x2 − k 2 ) k=−∞ k=1

4.

cosec2 πx =

5.

 (−1)k+1 1 + x cosec x 1 = 2− 2 2x x (x2 − k 2 π 2 )

JO (446)

∞

JO (449)

k=1

6.

cosec πx =

∞ 2  (−1)k π x2 − k 2

JO (450b)

k=−∞

∞

1.423

π π 1  π 1 π2 + cot − = cosec2 2 2 4m m 4m m 2 (1 − k 2 m2 ) k=1

JO (477)

1.439

Representation in the form of an infinite product

45

1.43 Representation in the form of an infinite product 1.431 1.

∞ x2 1− 2 2 k π k=1

∞ x2 sinh x = x 1+ 2 2 k π k=1

∞ 4x2 cos x = 1− (2k + 1)2 π 2 k=0

∞ 4x2 cosh x = 1+ (2k + 1)2 π 2 sin x = x

2.

3.

4.

EU

EU

EU

EU

k=0

1.432 1.

11

2.

 



∞ x2 x2 x2 2 y cos x − cos y = 2 1 − 2 sin 1− 1 − 2 y 2 (2kπ − y)2 (2kπ + y) k=1



∞ y x2 x2 x2 cosh x − cos y = 2 1 + 2 sin2 1+ 1 + y 2 (2kπ + y)2 (2kπ − y)2

AD (653.2)

AD (653.1)

k=1

 ∞  πx πx (−1)k x − sin = 1.433 cos 1+ 4 4 2k − 1 k=1 %

2 &2 ∞ π + 2x 1 2 2 1.434 cos x = (π + 2x) 1− 4 2kπ k=1



∞ x+a sin π(x + a) x x = 1.435 1− 1+ sin πa a k−a k+a k=1 % &

∞ 2 x sin2 πx = 1− 1.436 1 − 2 k−a sin πa k=−∞ %

2 & ∞ 2x sin 3x =− 1.437 1− sin x x + kπ k=−∞ %

2 & ∞ x cosh x − cos a = 1.438 1+ 1 − cos a 2kπ + a

BR* 189

MO 216 MO 216

MO 216

MO 216

MO 216

k=−∞

1.439 1.

sin x = x

∞ k=1

2.

cos

x 2k

∞   x  sin x 4 = 1 − sin2 k x 3 3 k=1

[|x| < 1]

AD (615), MO 216

MO 216

46

Trigonometric and Hyperbolic Functions

1.441

1.44–1.45 Trigonometric (Fourier) series 1.441 ∞  sin kx

1.

k=1

k

∞  cos kx

2.

k=1

k

=

π−x 2

1 = − ln [2 (1 − cos x)] 2

∞  (−1)k−1 sin kx

3.

∞ 

4.

(−1)k−1

k=1

FI III 539

[0 < x < 2π]

FI III 530a, AD (6814)

x 2

[−π < x < π]

FI III 542

 x cos kx = ln 2 cos k 2

[−π < x < π]

FI III 550

FI III 541

=

k

k=1

[0 < x < 2π]

1.442 1.11

∞  sin(2k − 1)x k=1

2k − 1

∞  cos(2k − 1)x

2.

k=1

2k − 1

=

π sign x 4

[−π < x < π]

=

1 x ln cot 2 2

[0 < x < π] BR* 168, JO (266), GI III(195)

∞ 

3.

(−1)k−1

1 sin(2k − 1)x = ln tan + 2k − 1 2 4 2

+ π π, − <x< 2 2

(−1)k−1

cos(2k − 1)x = 2k − 1

+ π π, − <x< 2  2 3π π <x< 2 2

k=1

4.10

∞  k=1

π

π 4 π =− 4

x

BR* 168, JO (268)a

BR* 168, JO (269)

1.443 1.

8

∞  cos kπx k=1

2.

k 2n

∞  sin kπx k=1

k 2n+1

2n π 2n  2n = (−1) 2 B2n−k ρk k (2n)! k=0 x 2n n−1 1 (2π) B 2n = (−1) 2 (2n)! 2 + 0 ≤ x ≤ 2, n−1 2n−1

ρ=

2n+1 π 2n+1  2n + 1 B2n−k+1 ρk k (2n + 1)! k=0 x 2n+1 n−1 1 (2π) B 2n+1 = (−1) 2 (2n + 1)! 2 + 0 < x < 1;

x : x ;, − 2 2

CE 340, GE 71

= (−1)n−1 22n

ρ=

x : x ;, − 2 2

CE 340

1.445

3.

Trigonometric (Fourier) series

∞  cos kx k=1

4.

∞ 

k2

k=1

5.

∞  sin kx k=1

6.

∞  cos kx k=1

7.

1.

k4

∞  sin kx k=1

1.444

k3

k5

πx x2 π2 − + 6 2 4

[0 ≤ x ≤ 2π]

FI III 547

x2 cos kx π2 − = k2 12 4

[−π ≤ x ≤ π]

FI III 544

=

(−1)k−1

=

x3 π 2 x πx2 − + 6 4 12

[0 ≤ x ≤ 2π]

=

π 2 x2 πx3 x4 π4 − + − 90 12 12 48

[0 ≤ x ≤ 2π]

AD (6617)

=

πx4 x5 π 4 x π 2 x3 − + − 90 36 48 240

[0 ≤ x ≤ 2π]

AD (6818)

∞  sin 2(k + 1)x k=1

k(k + 1)

  = sin 2x − (π − 2x) sin2 x − sin x cos x ln 4 sin2 x [0 ≤ x ≤ π]

2.

∞  k=1

3.

∞  k=1

4.

∞ 

∞  k=0

BR* 168, GI III (190)

π   cos 2(k + 1)x = cos 2x − − x sin 2x + sin2 x ln 4 sin2 x k(k + 1) 2 [0 ≤ x ≤ π] ! x !! x sin(k + 1)x ! = sin x − (1 + cos x) − sin x ln !2 cos ! (−1)k k(k + 1) 2 2 (−1)k

k=1

5.

47

(−1)k

! x !! x cos(k + 1)x ! = cos x − sin x − (1 + cos x) ln !2 cos ! k(k + 1) 2 2 sin(2k + 1)x π = x (2k + 1)2 4 π = (π − x) 4

BR* 168 MO 213

MO 213

+ π π, − ≤x≤ 2  2 π 3 ≤x≤ π 2 2 MO 213

6.6

cos(2k − 1)x π π − |x| = 2 (2k − 1) 4 2

[−π ≤ x ≤ π]

FI III 546

1 π cos 2kx = − sin x (2k − 1)(2k + 1) 2 4

+ π, 0≤x≤ 2

JO (591)

∞  k sin kx π sinh α(π − x) = k 2 + α2 2 sinh απ

[0 < x < 2π]

BR* 157, JO (411)

∞  1 cos kx π cosh α(π − x) − 2 = 2 2 k +α 2α sinh απ 2α

[0 ≤ x ≤ 2π]

BR* 257, JO (410)

∞  k=1

7.

∞  k=1

1.445 1.

k=1

2.

k=1

48

3.

Trigonometric and Hyperbolic Functions ∞  (−1)k cos kx

k 2 + α2

k=1

4.

∞ 

(−1)k−1

k=1

5.

=

1 π cosh αx − 2 2α sinh απ 2α

k sin kx π sinh αx = 2 2 k +α 2 sinh απ

1.446

[−π ≤ x ≤ π]

FI III 546

[−π < x < π]

FI III, 546

+ ,  if x = 2mπ, then ··· = 0

∞  k sin kx sin {α[(2m + 1)π − x]} =π 2 2 k −α 2 sin απ

k=1

[2mπ < x < (2m + 2)π,

6.

∞ 

[2mπ ≤ x ≤ (2m + 2)π, (−1)k

k=1

8.

α not an integer]

MO 213

∞ 

+ ,  if x = (2m + 1)π, then ··· = 0 ,

k sin kx sin[α(2mπ − x)] =π k 2 − α2 2 sin απ

[(2m − 1)π < x < (2m + 1)π, α not an integer] (−1)k

k=1

9.∗

MO 213

∞  cos kx 1 π cos [α {(2m + 1)π − x}] = − 2 2 2 k −α 2α 2 α sin απ

k=1

7.

α not an integer]

FI III 545a

cos kx 1 π cos[α(2mπ − x)] = − k 2 − α2 2α2 2 α sin απ [(2m − 1)π ≤ x ≤ (2m + 1)π, α not an integer]

FI III 545a

∞ 

einα π eiβ(α−2π) sinh(γα) + eiβα sinh [γ(2π − α)] = (n − β)2 + γ 2 γ cosh(2πγ) − cos(2πβ) n=−∞ [0 ≤ α ≤ 2π]

1.446

∞  k=1

π (−1)k+1 cos(2k + 1)x 1 = cos2 x − cos x (2k − 1)(2k + 1)(2k + 3) 8 3

+ π π, − ≤x≤ 2 2

BR* 256, GI III (189)

1.447 1.

∞ 

pk sin kx =

k=1

2.

∞ 

pk cos kx =

k=0

3.

1+2

∞  k=1

p sin x 1 − 2p cos x + p2 [|p| < 1]

FI II 559

[|p| < 1]

FI II 559

[|p| < 1]

FI II 559a, MO 213

1 − p cos x 1 − 2p cos x + p2

pk cos kx =

1 − p2 1 − 2p cos x + p2

1.449

Trigonometric (Fourier) series

49

1.448 1.

∞  pk sin kx

k

k=1

2.

∞  k=1

3.

∞  k=1

4.

∞  k=1

5.

∞  k=1

6.

= arctan

p sin x 1 − p cos x



 1  pk cos kx = − ln 1 − 2p cos x + p2 k 2 2p sin x p2k−1 sin(2k − 1)x 1 = arctan 2k − 1 2 1 − p2 p2k−1 cos(2k − 1)x 1 1 + 2p cos x + p2 = ln 2k − 1 4 1 − 2p cos x + p2 (−1)k−1 p2k−1 sin(2k − 1)x 1 1 + 2p sin x + p2 = ln 2k − 1 4 1 − 2p sin x + p2

∞  (−1)k−1 p2k−1 cos(2k − 1)x

2k − 1

k=1

=

1 2p cos x arctan 2 1 − p2











0 < x < 2π,

p2 ≤ 1

0 < x < 2π,

p2 ≤ 1

0 < x < 2π,

p2 ≤ 1

0 < x < 2π,

p2 ≤ 1

0 < x < π,

p2 ≤ 1

0 < x < π,

p2 ≤ 1







FI II 559







FI II 559

JO (594)

JO (259)

JO (261)

JO (597)

1.449 1.

∞  pk sin kx

k!

k=1

2.

∞  k=0

= ep cos x sin (p sin x)

pk cos kx = ep cos x cos (p sin x) k!

Let S(x) = − x1 cos x + 3.∗

∞  n=1

4.∗

5.∗



n2

1 x

and C(x) =

 1 x

p2 ≤ 1

p2 ≤ 1





JO (486)

JO (485)

sin x.

n π S(nx) = [C(ax) − cot(πa)S(ax)] 2 −a 2

[0 < x < 2π,

a = 0, ±1, ±2, . . .]

[0 ≤ x ≤ 2π,

a = 0, ±1, ±2, . . .]

∞ 

1 1 π [S(ax) − cot(πa)C(ax)] C(nx) = 2 − 2 − a2 n 2a 2a n=1 ∞ 

(−1)n−1 n π S(nx) = cosec(πa)S(ax) 2 − a2 n 2 n=1

[−π < x < π,

a = 0, ±1, ±2, . . .]

50

Trigonometric and Hyperbolic Functions

6.∗

∞  (−1)n−1 1 π cosec(πa)C(ax) C(nx) = − 2 + 2 − a2 n 2a 2a n=1

7.∗

 πa , 2n − 1 π+ S(nx) = C(ax) + tan S(ax) (2n − 1)2 − a2 4 2 n=1

8.∗

1.451

a = 0, ±1, ±2, . . .]

[−π < x < π,

∞ 

[0 < x < π, ∞  πa ,  1 π + S(ax) − tan C(ax) C(nx) = − (2n − 1)2 − a2 4a 2 n=1 ∞ 

 πa

9.∗

(−1)n−1 π sec S(nx) = 2 − a2 (2n − 1) 4a n=1

10.∗

∞  πa  (−1)n−1 (2n − 1) π sec C(nx) = C(ax) (2n − 1)2 − a2 4 2 n=1

2

S(ax)

a = 0, ±1, ±2, . . .]

[0 ≤ x ≤ π, a = 0, ±1, ±2, . . .] + π , π − ≤ x ≤ , a = 0, ±1, ±2, . . . 2 2 + π π − ≤x≤ , 2 2

, a = 0, ±1, ±2, . . .

Fourier expansions of hyperbolic functions 1.451 1.

2.

    ∞  2  1 + 02 12 + 22 . . . 12 + (2k)2 sinh x = cos x sin2k+1 x (2k + 1)! k=0     ∞  2  1 + 12 12 + 32 . . . 12 + (2k − 1)2 sin2k x cosh x = cos x + cos x (2k)!

JO (504)

JO (503)

k=1

1.452 1.

sinh (x cos θ) = sec (x sin θ)

∞  x2k+1 cos(2k + 1)θ

(2k + 1)!

k=0

2.

cosh (x cos θ) = sec (x sin θ)

∞  k=0

3.

sinh (x cos θ) = cosec (x sin θ)

x2k cos 2kθ (2k)!

cosh (x cos θ) = cosec (x sin θ)



x2 < 1

x2 < 1



JO (391)



JO (390)

∞  x2k sin 2kθ k=1

4.



(2k)!



x2 < 1,

x sin θ = 0

x2 < 1,

x sin θ = 0



JO (393)

∞  x2k+1 sin(2k + 1)θ k=0

(2k + 1)!





JO (392)

1.480

Lobachevskiy’s “Angle of Parallelism”

51

1.46 Series of products of exponential and trigonometric functions 1.461 1.

∞ 

e−kt sin kx =

k=0

2.

1+2

∞ 

sin x 1 2 cosh t − cos x

e−kt cos kx =

k=1

1.4629

⎡

∞  sin kx sin ky

k

k=1

sinh t cosh t − cos x

e−2k|t|

[t > 0]

MO 213

[t > 0]

MO 213

⎤ x+y 2 + sinh t 1 ⎢ ⎥ 2 = ln ⎣ ⎦ 4 2 2 x−y + sinh t sin 2 sin2

MO 214

1.463 x cos ϕ

∞  xn cos nϕ cos (x sin ϕ) = n! n=0

1.

e

2.

ex cos ϕ sin (x sin ϕ) =

∞  xn sin nϕ n! n=1

 

x2 < 1 x2 < 1

 

AD (6476.1)

AD (6476.2)

1.47 Series of hyperbolic functions 1.471 1.

∞  sinh kx k=1

2.

∞  cosh kx k=0

3.

k!

∞  k=0

k!

= ecosh x sinh (sinh x) .

JO (395)

= ecosh x cosh (sinh x) .

JO (394)

  1 1 (2m + 1)πx (2m + 1)π π3 tanh + x tanh = (2k + 1)3 x 2 2x 16

1.472 1.

∞ 

pk sinh kx =

p sinh x 1 − 2p cosh x + p2



pk cosh kx =

1 − p cosh x 1 − 2p cosh x + p2



k=1

2.

∞  k=0

p2 < 1 p2 < 1

 

JO (396)

JO (397)a

1.48 Lobachevskiy’s “Angle of Parallelism” Π(x) 1.480 1.

Definition. Π(x) = 2 arccot ex = 2 arctan e−x

[x ≥ 0]

LO III 297, LOI 120

52

2. 1.481

Trigonometric and Hyperbolic Functions

Π(x) = π − Π(−x)

2.

Functional relations 1 sin Π(x) = cosh x cos Π(x) = tanh x

3.

tan Π(x) =

1.

[x < 0]

1.481

LO III 183, LOI 193

LO III 297 LO III 297

4.

1 sinh x cot Π(x) = sinh x

5.

sin Π(x + y) =

sin Π(x) sin Π(y) 1 + cos Π(x) cos Π(y)

LO III 297

6.

cos Π(x + y) =

cos Π(x) + cos Π(y) 1 + cos Π(x) cos Π(y)

LO III 183

LO III 297 LO III 297

1.482

Connection with the Gudermannian. π gd(−x) = Π(x) − 2 (Definite) integral of the angle of parallelism: cf. 4.581 and 4.561.

1.49 The hyperbolic amplitude (the Gudermannian) gd x 1.490 1. 2. 1.491

Definition.  x gd x =

dt = 2 arctan ex − 0 cosh t  gd x gd x dt = ln tan + x= cos t 2 0

cosh x = sec(gd x)

2.

sinh x = tan(gd x)

3.

ex = sec(gd x) + tan(gd x) = tan

4.

tanh x = sin(gd x)

1 x gd x tanh = tan 2 2 1 arctan (tanh x) = gd 2x 2

6.

π 4

JA

JA

Functional relations.

1.

5.

π 2

1.492 If γ = gd x, then ix = gd iγ 1.493 Series expansion.

AD (343.1), JA



π gd x + 4 2

AD (343.2), JA

=

1 + sin(gd x) cos(gd x)

AD (343.5), JA AD (343.3), JA AD (343.4), JA AD (343.6a) JA

∞

1.

x gd x  (−1)k = tanh2k+1 2 2k + 1 2 k=0

JA

1.513

Series representation ∞

2.

x  1 = tan2k+1 2 2k + 1



k=0

3. 4.

53

1 gd x 2

JA

x5 61x7 x3 + − + ··· 6 24 5040 (gd x)5 61(gd x)7 (gd x)3 + + + ... x = gd x + 6 24 5040 gd x = x −

JA

+ π, gd x < 2

JA

1.5 The Logarithm 1.51 Series representation ∞

1.511

 xk 1 1 1 ln(1 + x) = x − x2 + x3 − x4 + · · · = (−1)k+1 2 3 4 k k=1

[−1 < x ≤ 1] 1.512 1.

∞

 (x − 1)k 1 1 (−1)k+1 ln x = (x − 1) − (x − 1)2 + (x − 1)3 − · · · = 2 3 k k=1

%

2.

3.

x−1 1 + ln x = 2 x+1 3

ln x =

x−1 1 + x 2

4.∗

ln x = lim

→0

1.513 1.



x −1  



x−1 x+1

x−1 x

3

2 +

1 + 5

1 3





∞

ln

 1 1+x =2 x2k−1 1−x 2k − 1



x−1 x+1

x−1 x

3

5

&

[0 < x ≤ 2]

+ ... = 2

∞  k=1

1 2k − 1



x−1 x+1

[0 < x]

k 1 x−1 + ··· = k x k=1   x ≥ 12

2k−1

∞ 



x2 < 1



AD (644.6)

FI II 421

k=1 ∞

2.

ln

 1 x+1 =2 x−1 (2k − 1)x2k−1



x2 > 1



AD (644.9)

k=1

∞

3.

 1 x = ln x−1 kxk

[x ≤ −1 or x > 1]

JO (88a)

[−1 ≤ x < 1]

JO (88b)

[−1 ≤ x < 1]

JO (102)

k=1 ∞

4.

ln

 xk 1 = 1−x k k=1

∞

5.

 1 xk 1−x ln =1− x 1−x k(k + 1) k=1

54

6.

The Logarithm

∞ k   1 1 1 ln = xk 1−x 1−x n n=1

1.514



x2 < 1



JO (88e)

k=1

∞

7.

 xk−1 1 (1 − x)2 1 3 = + ln − 2x3 1−x 2x2 4x k(k + 1)(k + 2)

[−1 ≤ x < 1]

AD (6445.1)

k=1

1.514

1.515 1.11

2.

3.

4.

  ln x + 1 + x2 = arcsinh x k=1   2 (see 1.631, 1.641, 1.642, 1.646) x ≤ 1, x cos ϕ = 1

∞    cos kϕ k x ; ln 1 − 2x cos ϕ + x2 = −2 k

  1·1 2 1·1·3 4 1·1·3·5 6 x − x + x − ... ln 1 + 1 + x2 = ln 2 + 2·2 2·4·4 2·4·6·6 ∞  (2k − 1)! 2k (−1)k = ln 2 − 2x 22k (k!) k=1  2  x ≤1   1 1 1·3 ln 1 + 1 + x2 = ln x + − + − ... x 2 · 3x3 2 · 4 · 5x5 ∞ (2k − 1)! 1  (−1)k 2k−1 = ln x + + x 2 · k!(k − 1)!(2k + 1)x2k+1 k=1  2  x ≥1 ∞     22k−1 (k − 1)!k! 2k+1 x 1 + x2 ln x + 1 + x2 = x − (−1)k (2k + 1)! k=1  2  x ≤1 √   ∞ 2   2  ln x + 1 + x2 22k (k!) 2k+1 √ x x ≤1 = (−1)k 2 (2k + 1)! 1+x

MO 98, FI II 485

JO (91)

AD (644.4)

JO (93) JO (94)

k=0

1.516 1.

∞ k k+1 k+1   (∓1) x 1 1 2 {ln (1 ± x)} = 2 k+1 n n=1



∞ k n  (−1)k+1 xk+2  1  1 1 3 {ln(1 + x)} = 6 k+2 n + 1 m=1 m n=1



x2 < 1



JO (86), JO (85)

k=1

2.

x2 < 1



AD (644.14)

k=1

∞ 2k−1   2  x2k  (−1)n+1 x <1 k n=1 n k=1  ∞  xk−1 1 + + 2 ln(1 − x) = 2x (2k − 1)2k(2k + 1)

3.

− ln(1 + x) · ln(1 − x) =

4.

1 4x

√ 1+x 1+ x √ √ ln 1− x x

JO (87)

k=1

[0 < x < 1]

AD (6445.2)

1.521

Series of logarithms (cf. 1.431)

1.517 1.

6

1 2x

55

 

∞ √ (−1)k+1 xk−1 1−x 1 − ln(1 + x) − √ arctan x = (2k − 1)2k(2k + 1) x k=1

[0 < x ≤ 1] 2.

3.

1+x 1 arctan x ln = 2 1−x

∞  k=1

4k−2 2k−1 

x 2k − 1

n=1

(−1)n−1 2n − 1

∞ 2k    (−1)k+1 x2k+1  1 1 arctan x ln 1 + x2 = 2 2k + 1 n n=1

 

x2 < 1

x2 ≥ 1

 

AD (6445.3) BR* 163

AD (6455.3)

k=1

1.518 1.

ln sin x = ln x − = ln x +

x4 x6 x2 − − − ... 6 180 2835 ∞  (−1)k 22k−1 B2k x2k k(2k)!

k=1

[0 < x < π] 2.3

3.

2

4

6

x x 17x x − − − − ... 2 12  45 2520  ∞ ∞ 2k−1 2k 2 2 − 1 |B2k | 2k 1  sin2k x x =− =− k(2k)! 2 k k=1 k=1   π2 x2 < 4

ln cos x = −

7 x2 62 6 127 8 + x4 + x + x + ... 3 90 18, 900  2835  ∞  22k−1 − 1 22k B2k x2k = ln x + (−1)k+1 k(2k)!

+ π, 0<x< 2

1.52 Series of logarithms (cf. 1.431)

1.

2.

FI II 524

ln tan x = ln x +

k=1

1.521

AD (643.1)a

8

ln 1 −

4x2 = ln cos x (2k − 1)2 π 2 k=1

∞  x2 ln 1 − 2 2 = ln sin x − ln x k π ∞ 

k=1

+ π π, − <x< 2 2 [0 < x < π]

AD (643.3)a

56

The Inverse Trigonometric and Hyperbolic Functions

1.621

1.6 The Inverse Trigonometric and Hyperbolic Functions 1.61 The domain of definition The principal values of the inverse trigonometric functions are defined by the inequalities: 1. 2.

π π ≤ arcsin x ≤ ; 0 ≤ arccos x ≤ π 2 2 π π − < arctan x < ; 0 < arccot x < π 2 2 −

[−1 ≤ x ≤ 1]

FI II 553

[−∞ < x < +∞]

FI II 552

1.62–1.63 Functional relations 1.621 1.

The relationship between the inverse and the direct trigonometric functions. + π, π arcsin (sin x) = x − 2nπ 2nπ − ≤ x ≤ 2nπ + 2 2 + π, π = −x + (2n + 1)π (2n + 1)π − ≤ x ≤ (2n + 1)π + 2 2

2.

arccos (cos x) = x − 2nπ = −x + 2(n + 1)π

3.

arctan (tan x) = x − nπ

4.

arccot (cot x) = x − nπ

[2nπ ≤ x ≤ (2n + 1)π] [(2n + 1)π ≤ x ≤ 2(n + 1)π] + π, π nπ − < x < nπ + 2 2 [nπ < x < (n + 1)π]

1.622 The relationship between the inverse trigonometric functions, the inverse hyperbolic functions, and the logarithm. 1. 2. 3. 4. 5. 6. 7. 8.

1  1  ln iz + 1 − z 2 = arcsinh(iz) i i 1  1  arccos z = ln z + z 2 − 1 = arccosh z i i 1 1 + iz 1 arctan z = ln = arctanh(iz) 2i 1 − iz i 1 iz − 1 arccot z = ln = i arccoth(iz) 2i iz + 1 1   arcsinh z = ln z + z 2 + 1 = arcsin(iz) i   arccosh z = ln z + z 2 − 1 = i arccos z arcsin z =

1 1 1+z ln = arctan(iz) 2 1−z i 1 z+1 1 arccoth z = ln = arccot(−iz) 2 z−1 i arctanh z =

1.624

Functional relations

57

Relations between different inverse trigonometric functions 1.623 1. 2. 1.624 1.

2. 3.

4.

5.

6. 7. 8.

9.

π 2 π arctan x + arccot x = 2 arcsin x + arccos x =

NV 43 NV 43

 1 − x2  = − arccos 1 − x2

arcsin x = arccos

arcsin x = arctan √

[0 ≤ x ≤ 1]

NV 47 (5)

[−1 ≤ x ≤ 0]

NV 46 (2)



x 1 − x2

√ 1 − x2 √ x 1 − x2 −π = arccot x  arccos x = arcsin 1 − x2  = π − arcsin 1 − x2 arcsin x = arccot

√ 1 − x2 x√ 1 − x2 = π + arctan x x arccos x = arccot √ 1 − x2 x arctan x = arcsin √ 1 + x2 1 arctan x = arccos √ 1 + x2 1 = − arccos √ 1 + x2 arccos x = arctan

1 x 1 = − arccot − π x

arctan x = arccot

1 10.11 arccot x = arcsin √ 1 + x2 1 = π − arcsin √ 1 + x2 x 11. arccot x = arccos √ 1 + x2

x2 < 1



[0 < x ≤ 1] [−1 ≤ x < 0]

NV 49 (10)

[0 ≤ x ≤ 1] [−1 ≤ x ≤ 0]

NV 48 (6)

[0 < x ≤ 1] [−1 ≤ x < 0]

NV 48 (8)

[−1 ≤ x < 1]

NV 46 (4) NV 6 (3)

[x ≥ 0] [x ≤ 0]

NV 48 (7)

[x > 0] [x < 0]

NV 49 (9)

[x > 0] [x < 0]

NV 49 (11) NV 46 (4)

58

12.

1.625 1.

The Inverse Trigonometric and Hyperbolic Functions

1 x 1 = π + arctan x

arccot x = arctan

1.625

[x > 0] [x < 0]

NV 49 (12)

   arcsin x + arcsin y = arcsin x 1 − y 2 + y 1 − x2    = π − arcsin x 1 − y 2 + y 1 − x2    = −π − arcsin x 1 − y 2 + y 1 − x2

  

xy ≤ 0 or x2 + y 2 ≤ 1



x > 0,

y > 0 and x2 + y 2 > 1

x < 0,

y < 0 and x2 + y 2 > 1

 

NV 54(1), GI I (880)

2.

3.





 1 − x2 1 − y 2 − xy   = − arccos 1 − x2 1 − y 2 − xy

arcsin x + arcsin y = arccos

 √ x 1 − y 2 + y 1 − x2  arcsin x + arcsin y = arctan √ 1 − x2 1 − y 2 − xy  √ x 1 − y 2 + y 1 − x2  +π = arctan √ 1 − x2 1 − y 2 − xy  √ x 1 − y 2 + y 1 − x2  −π = arctan √ 1 − x2 1 − y 2 − xy

  

[x ≥ 0,

y ≥ 0]

[x < 0,

y < 0]

xy ≤ 0 or x2 + y 2 < 1

NV 55



x > 0,

y > 0 and x2 + y 2 > 1

x < 0,

y < 0 and x2 + y 2 > 1

  NV 56

4.

   arcsin x − arcsin y = arcsin x 1 − y 2 − y 1 − x2    = π − arcsin x 1 − y 2 − y 1 − x2    = −π − arcsin x 1 − y 2 − y 1 − x2

  

xy ≥ 0 or x2 + y 2 ≤ 1



x > 0,

y < 0 and x2 + y 2 > 1

x < 0,

y > 0 and x2 + y 2 > 1

 

NV 55(2)

5.

6.

7.11

   arcsin x − arcsin y = arccos x 1 − x2 1 − y 2 + xy   = − arccos 1 − x2 1 − y 2 + xy    arccos x + arccos y = arccos xy − 1 − x2 1 − y 2    = 2π − arccos xy − 1 − x2 1 − y 2    arccos x − arccos y = − arccos xy + 1 − x2 1 − y 2    = arccos xy + 1 − x2 1 − y 2

[xy > y] [x < y]

NV 56

[x + y ≥ 0] [x + y < 0]

NV 57 (3)

[x ≥ y] [x < y]

NV 57 (4)

1.627

8.

Functional relations

x+y 1 − xy x+y = π + arctan 1 − xy x+y = −π + arctan 1 − xy

arctan x + arctan y = arctan

59

[xy < 1] [x > 0,

xy > 1]

[x < 0,

xy > 1] NV 59(5), GI I (879)

9.

x−y 1 + xy x−y = π + arctan 1 + xy x−y = −π + arctan 1 + xy

arctan x − arctan y = arctan

[xy > −1] [x > 0,

xy < −1]

[x < 0,

xy < −1] NV 59(6)

1.626 1.

  2 arcsin x = arcsin 2x 1 − x2   = π − arcsin 2x 1 − x2   = −π − arcsin 2x 1 − x2

  1 |x| ≤ √ 2   1 √ <x≤1  2  1 −1 ≤ x < − √ 2 NV 61 (7)

2.

3.





[0 ≤ x ≤ 1]

2 arccos x = arccos 2x2 − 1   = 2π − arccos 2x2 − 1

[−1 ≤ x < 0]

2x 1 − x2 2x +π = arctan 1 − x2 2x −π = arctan 1 − x2

2 arctan x = arctan

NV 61 (8)

[|x| < 1] [x > 1] [x < −1] NV 61 (9)

1.627 1.

2.

arctan x + arctan

arctan x + arctan

1 π = 2 x π =− 2

[x > 0] [x < 0]

1−x π = 4 1+x 3 =− π 4

GI I (878)

[x > −1] [x < −1]

NV 62, GI I (881)

60

The Inverse Trigonometric and Hyperbolic Functions

1.628

1.628 1.

arcsin

2x = −π − 2 arctan x 1 + x2 = 2 arctan x

[x ≤ −1] [−1 ≤ x ≤ 1]

= π − 2 arctan x

[x ≥ 1] NV 65

1−x = 2 arctan x 1 + x2 = −2 arctan x 2

2.

arccos

[x ≥ 0] [x ≤ 0]

NV 66



1 2x − 1 2x − 1 − arctan tan π = E (x) 1.629 2 π 2 1.631 Relations between the inverse hyperbolic functions.  x 1. arcsinh x = arccosh x2 + 1 = arctanh √ x2 + 1 √  x2 − 1 2 2. arccosh x = arcsinh x − 1 = arctanh x x 1 1 3. arctanh x = arcsinh √ = arccosh √ = arccoth 2 2 x 1−x 1−x    2 4. arcsinh x ± arcsinh y = arcsinh x 1 + y ± y 1 + x2   5. arccosh x ± arccosh y = arccosh xy ± (x2 − 1) (y 2 − 1) 6.

arctanh x ± arctanh y = arctanh

GI (886)

JA

JA JA JA JA

x±y 1 ± xy

JA

1.64 Series representations 1.641 1.

2.

1 3 1·3 5 1·3·5 7 π − arccos x = x + x + x + x + ... 2 2·3 2 · 4 ·5 2 · 4 · 6 · 7 ∞  1 1 3 2 (2k)! x2k+1 = x F , ; ;x = 2k (k!)2 (2k + 1) 2 2 2 2 k=0  2  x ≤1

arcsin x =

1 3 1·3 5 x + x − ...; arcsinh x = x − 2·3 2·4·5 ∞  (2k)! x2k+1 (−1)k = 2 2k 2 (k!) (2k + 1) k=0   = x F 12 , 12 ; 32 ; −x2



x2 ≤ 1



FI II 479

FI II 480

1.645

Series representations

61

1.642 1.

1 1 1·3 1 − + ... 2 2x2 2 · 4 4x4 ∞  (2k)!x−2k (−1)k+1 = ln 2x + 2 22k (k!) 2k k=1

arcsinh x = ln 2x +

[x ≥ 1] AD (6480.2)a

2.

arccosh x = ln 2x −

∞  (2k)!x−2k

[x ≥ 1]

2

k=1

22k (k!) 2k

AD (6480.3)a

1.643 1.

x3 x5 x7 + − + ... 3 5 7 ∞  (−1)k x2k+1 = 2k + 1

arctan x = x −



k=0

2. 1.644 1.

arctanh x = x +

x5 x3 + + ··· = 3 5

∞  k=0



x2k+1 2k + 1

k ∞  x2 x (2k)! arctan x = √ 1 + x2 k=0 22k (k!)2 (2k + 1) 1 + x2

1 1 3 x2 x , ; ; =√ F 2 2 2 1 + x2 1 + x2

x2 ≤ 1 x2 < 1



FI II 479



AD (6480.4)

  2 x <∞ AD (641.3)

2.

arctan x =

1 1 π 1 1 π − + 3 − 5 + 7 − ··· = − 2 x 3x 5x 7x 2

∞ 

(−1)k

k=0

1 (2k + 1)x2k+1

AD (641.4)

1.645 ∞

1.

π 1 1 1·3 π  (2k)!x−(2k+1) − − − − ··· = − 2 3 5 2 x 2 · 3x 2 · 4 · 5x 2 (k!) 22k (2k + 1) k=0

1 1 3 1 1 π , ; ; = − F 2 x 2 2 2 x2

arcsec x =



x2 > 1

 AD (641.5)

2. 3.

∞ 2   2  22k (k!) x2k+2 x ≤1 AD (642.2), GI III (152)a (2k + 1)!(k + 1) k=0



3! 3! 1 1 1 3 (arcsin x) = x3 + 32 1 + 2 x5 + 32 · 52 1 + 2 + 2 x7 + . . . 5! 3 7! 3 5  x2 ≤ 1 2

(arcsin x) =

BR* 188, AD (642.2), GI III (153)a

62

The Inverse Trigonometric and Hyperbolic Functions

1.646

1.646 ∞

1.

arcsinh

 1 (−1)k (2k)! x−2k−1 = arcosech x = 2 x 22k (k!) (2k + 1) k=0

∞ 

(2k)!

2.

arccosh

2 1 = arcsech x = ln − x x

3.

arcsinh

2  (−1)k+1 (2k)! 2k 1 x = arcosech x = ln + 2 x x 22k (k!) 2k

2

22k (k!) 2k k=1

x2k



x2 ≥ 1



AD (6480.5)

[0 < x ≤ 1]

AD (6480.6)

[0 < x ≤ 1]

AD (6480.7)a

∞

k=1

4. 1.647 1.

2.

arctanh

1 = arccoth x = x

∞  k=0

−(2k+1)

x 2k + 1



x2 > 1



AD (6480.8)

⎛

   ∗ n ∗  (−1)j−1 22j − 1 24n−2j+4 − 1 B2j−1 B4n−2j+3 ⎝2 = (2k − 1)4n+3 2 (2j)!(4n − 2j + 4)! j=1 k=1  2n+2 2 ∗ 2 ⎞ n (−1) 2 − 1 B2n+1 ⎠ + 2 [(2n + 2)!] n = 0, 1, 2, . . . , ⎛ ⎞ ∞ n−1 j ∗ ∗ k−1 4n+1 ∗ n ∗ 2   (−1) B B 2B4n (−1) B2n ⎠ (−1) sech(2k − 1) (π/2) π 2j 4n−2j + + , = 4n+3 ⎝2 (2k − 1)4n+1 2 (2j)!(4n − 2j)! (4n)! [(2n)]!2 j=1 ∞  tanh(2k − 1) (π/2)

π

4n+3

k=1

n = 1, 2, . . . (The summation term on the right is to be omitted for n = 1.) (See page xxxiii for the definition of Br∗ .)

2 Indefinite Integrals of Elementary Functions 2.0 Introduction 2.00 General remarks We omit the constant of integration in all the formulas of this chapter. Therefore, the equality sign (=) means that the functions on the left and right of this symbol differ by a constant. For example (see 201 15), we write  dx = arctan x = − arctan x 1 + x2 although π arctan x = − arctan x + . 2 we !integrate certain functions, we obtain the logarithm of the absolute value (for example, ! √ " When √ dx = ln !x + 1 + x2 !). In such formulas, the absolute-value bars in the argument of the logarithm 1+x2 are omitted for simplicity in writing. In certain cases, it is important to give the complete form of the primitive function. Such primitive functions, written in the form of definite integrals, are given in Chapter 2 and in other chapters. Closely related to these formulas are formulas in which the limits of integration and the integrand depend on the same parameter. A number of formulas lose their meaning for certain values of the constants (parameters) or for certain relationships between these constants (for example, formula 2.02 8 for n = −1 or formula 2.02 15 for a = b). These values of the constants and the relationships between them are for the most part completely clear from the very structure of the right-hand member of the formula (the one not containing an integral sign). Therefore, throughout the chapter, we omit remarks to this effect. However, if the value of the integral is given by means of some other formula for those values of the parameters for which the formula in question loses meaning, we accompany this second formula with the appropriate explanation. The letters x, y, t, . . . denote independent variables; f , g, ϕ, . . . denote functions of x, y, t, . . . ; f  ,  g , ϕ , . . . , f  , g  , ϕ , . . . denote their first, second, etc., derivatives; a, b, m, p, . . . denote constants, by which we generally mean arbitrary real numbers. If a particular formula is valid only for certain values of the constants (for example, only for positive numbers or only for integers), an appropriate remark is made, provided the restriction that we make does not follow from the form of the formula itself. Thus, in formulas 2.148 4 and 2.424 6, we make no remark since it is clear from the form of these formulas themselves that n must be a natural number (that is, a positive integer).

63

64

Introduction

2.01 The basic integrals 



1. 2. 3. 4. 5. 6.

xn+1 (n = −1) x dx = n+1  dx = ln x x  ex dx = ex  ax ax dx = ln a  sin x dx = − cos x  cos x dx = sin x  dx = − cot x sin2 x  dx = tan x cos2 x n

11

7. 8.11

 16. 17. 18. 19. 20. 21. 22.11 23. 24. 25. 26.

dx 1 1+x = arctanh x = ln 2 1−x 2 1−x  dx √ = arcsin x = − arccos x 1 − x2    dx √ = arcsinh x = ln x + x2 + 1 x2 + 1    dx √ = arccosh x = ln x + x2 − 1 x2 − 1  sinh x dx = cosh x  cosh x dx = sinh x  dx = − coth x sinh2 x  dx = tanh x cosh2 x  tanh x dx = ln cosh x  coth x dx = ln sinh x  x dx = ln tanh sinh x 2

9.  10. 

sin x dx = sec x cos2 x cos x dx = − cosec x sin2 x tan x dx = − ln cos x

11.  12.

cot x dx = ln sin x 

13. 14. 15.

x dx = ln tan sin x 2  π x dx = ln tan + = ln (sec x + tan x) cos x 4 2  dx π = arctan x = − arccot x 2 1+x 2

General formulas

65

2.02 General formulas 

1.



af dx = a f dx 







[af ± bϕ ± cψ ± . . .] dx = a f dx ± b ϕ dx ± c ψ dx ± . . .  d f dx = f dx  f  dx = f    f ϕ dx = f ϕ − f ϕ dx [integration by parts]   (n+1) (n)  (n−1)  (n−2) n (n) n+1 f ϕ dx = ϕf − ϕ f +ϕ f − . . . + (−1) ϕ f + (−1) ϕ(n+1) f dx   f (x) dx = f [ϕ(y)]ϕ (y) dy [x = ϕ(y)] [change of variable]

2. 3. 4. 5. 6. 7.

 11

8.

9. 10. 11. 12. 13. 14. 15.

16. 17.

(f )n f  dx =

(f )n+1 n+1

For n = −1   f dx = ln f f  (af + b)n+1 (af + b)n f  dx = a(n + 1) √   f dx 2 af + b √ = a af + b    f f ϕ−ϕf dx = ϕ2 ϕ   f f ϕ − ϕ f dx = ln fϕ ϕ    dx dx dx =± ∓ f (f ± ϕ) fϕ ϕ (f ± ϕ)     f dx  = ln f + f 2 + a f2 + a    a b f dx dx dx = − (f + a)(f + b) a − b (f + a) a − b (f + b) For a = b    dx dx f dx −a = (f + a)2 f +a (f + a)2    f dx dx ϕ dx = − (f + ϕ)n (f + ϕ)n−1 (f + ϕ)n  qf f  dx 1 arctan = 2 2 2 p +q f pq p

[n = −1]

66

Rational Functions

2.101



qf − p f  dx 1 ln = q 2 f 2 − p2 2pq qf + p   f dx dx = −x + 1−f 1−f    1 f dx 1 f dx f 2 dx + = f 2 − a2 2 f −a 2 f +a  f f  dx  = arcsin a a2 − f 2   1 f f dx = ln 2 af + bf b af + b  f 1 f  dx  = arcsec 2 2 a a f f −a    (f ϕ − f ϕ ) dx f = arctan 2 2 f +ϕ ϕ  1 f −ϕ (f  ϕ − f ϕ ) dx = ln f 2 − ϕ2 2 f +ϕ

18. 19. 20. 21. 22. 23. 24. 25.

2.1 Rational Functions 2.10 General integration rules F (x) , where F (x) and f (x) are polynomials with f (x) no common factors, we first need to separate out the integral part E(x) [where E(x) is a polynomial], if  there is an integral part, and then to integrate separately the integral part and the remainder; thus: F (x) dx ϕ(x) = E(x) dx + dx. f (x) f (x) Integration of the remainder, which is then a proper rational function (that is, one in which the degree of the numerator is less than the degree of the denominator) is based on the decomposition of the fraction into elementary fractions, the so-called partial fractions. 2.102 If a, b, c, . . . , m are roots of the equation f (x) = 0 and if α, β, γ, . . . , μ are their corresponding ϕ(x) can be decomposed into the following multiplicities, so that f (x) = (x−a)α (x−b)β . . . (x−m)μ , then f (x) partial fractions: ϕ(x) Aα Bβ Aα−1 A1 Bβ−1 B1 = + + ... + + ... + + + ... + α α−1 β β−1 f (x) (x − a) (x − a) x − a (x − b) (x − b) x−b Mμ−1 M1 Mμ , + + ... + + μ μ−1 (x − m) (x − m) x−m 2.101

To integrate an arbitrary rational function

where the numerators of the individual fractions are determined by the following formulas: (k−1)

(a) ψ1 , (k − 1)! ϕ(x)(x − a)α ψ1 (x) = , f (x)

Aα−k+1 =

(k−1)

(b) ψ2 , (k − 1)! ϕ(x)(x − b)β ψ2 (x) = , f (x)

Bβ−k+1 =

(k−1)

..., ...,

ψm (m) , (k − 1)! ϕ(x)(x − m)μ ψm (x) = f (x)

Mμ−k+1 =

2.104

General integration rules

67

TI 51a

If a, b, . . . , m are simple roots, that is, if α = β = . . . = μ=1, then ϕ(x) A B M = + + ··· + , f (x) x−a x−b x−m where ϕ(a) ϕ(b) ϕ(m) A=  , B=  , ... , M=  . f (a) f (b) f (m) If some of the roots of the equation f (x) = 0 are imaginary, we group together the fractions that represent conjugate roots of the equation. Then, after certain manipulations, we represent the corresponding pairs of fractions in the form of real fractions of the form M1 x + N 1 M2 x + N 2 Mp x + N p + p. 2 + ... + x2 + 2Bx + C (x2 + 2Bx + C) (x2 + 2Bx + C)  g dx ϕ(x) reduces to integrals of the form 2.103 Thus, the integration of a proper rational fraction f (x) (x − a)α  Mx + N or p dx. Fractions of the first form yield rational functions for α > 1 and logarithms (A + 2Bx + Cx2 ) for α = 1. Fractions of the second form yield rational functions and logarithms or arctangents:   d(x − a) g g dx =g =− 1. (x − a)α (x − a)α (α − 1)(x − a)α−1   d(x − a) g dx =g = g ln |x − a| 2. x−a x−a  Mx + N N B − M A + (N C − M B)x 3. p dx = p−1 (A + 2Bx + Cx2 ) 2(p − 1) (AC − B 2 ) (A + 2Bx + Cx2 ) (2p − 3)(N C − M B) dx + p−1 2(p − 1) (AC − B 2 ) (A + 2Bx + Cx2 )    dx 1 Cx + B 4. =√ arctan √ for AC > B 2 2 2 A + 2Bx + Cx2 AC − B Ac − B ! ! ! Cx + B − √B 2 − AC !   1 ! ! √ = √ ln ! ! for AC < B 2 2 2 ! ! 2 B − AC Cx + B + B − AC  (M x + N ) dx 5. A + 2Bx + Cx2 !   NC − MB M !! Cx + B ln A + 2Bx + Cx2 ! + √ = arctan √ for AC > B 2 2 2 2C C AC − B AC − B ! ! ! Cx + B − √B 2 − AC ! !   N C − M B M !! ! ! √ √ ln A + 2Bx + Cx2 ! + = ln ! ! for AC < B 2 2 2 ! ! 2C 2C B − AC Cx + B + B − AC

The Ostrogradskiy–Hermite method 

ϕ(x) dx f (x) without finding the roots of the equation f (x) = 0 and without decomposing the integrand into partial fractions: 2.104

By means of the Ostrogradskiy-Hermite method, we can find the rational part of

68

Rational Functions

2.110

 M N dx ϕ(x) dx = + FI II 49 f (x) D Q Here, M , N , D, and Q are rational functions of x. Specifically, D is the greatest common divisor of f (x) the function f (x) and its derivative f  (x); Q = ; M is a polynomial of degree no higher than m − 1, D where m is the degree of the polynomial D; N is a polynomial of degree no higher than n − 1, where n is the degree of the polynomial Q. The coefficients of the polynomials M and N are determined by equating the coefficients of like powers of x in the following identity: 

ϕ(x) = M  Q − M (T − Q ) + N D f  (x) where T = and M  and Q are the derivatives of the polynomials M and Q. D

2.11–2.13 Forms containing the binomial a + bxk 2.110 1.

Reduction formulas for zk = a + bxk and an explicit expression for the general case.   amk xn+1 zkm + xn zkm dx = xn zkm−1 dx km + n + 1 km + n + 1 p (ak)s (m + 1)m(m − 1) . . . (m − s + 1)zkm−s xn+1  = m + 1 s=0 [mk + n + 1][(m − 1)k + n + 1] . . . [(m − s)k + n + 1]  (ak)p+1 m(m − 1) . . . (m − p + 1)(m − p) xn zkm−p−1 dx + [mk + n + 1][(m − 1)k + n + 1] . . . [(m − p)k + n + 1] LA 126(4)

  3.  4.  5.  6. 7.∗

8.∗





km + k + n + 1 xn zkm+1 dx ak(m + 1) ak(m + 1)  bkm xn+1 zkm n m − x zk dx = xn+k zkm−1 dx n+1 n+1  xn+1−k zkm+1 n+1−k − xn zkm dx = xn−k zkm+1 dx bk(m + 1) bk(m + 1)  xn+1−k zkm+1 a(n + 1 − k) − xn−k zkm dx xn zkm dx = b(km + n + 1) b(km + n + 1)  xn+1 zkm+1 b(km + k + n + 1) n m − x zk dx = xn+k zkm dx a(n + 1) a(n + 1)   b c k−i  k i  b k n +n nk  (−1) k! Γ a+1 n b  a+1  xa+1+ib x nx + c dx = b i=0 (k − i)! Γ b + i + 1 xn zkm dx =

2.

−xn+1 zkm+1

+



 xn zkm dx =

 a m−i

LA 126 (6)

LA 125 (2)

LA 126 (3)

LA 126 (5)

[a, b, k ≥ 0 are all integers]

m xk(J+i+1) bm  (−1)i m!J! xk + b k i=0 (m − i)!(J + i + 1)!

J=

n+1 −1 k

[a, b, k, m, n real,

k = 0,

m ≥ 0 an integer]

Forms containing the binomial a + bxk

2.114

Forms containing the binomial z1 = a + bx 2.111 1.

2.

3.8

4. 5. 6. 2.113



z1m+1 b(m + 1) For m = −1  1 dx = ln z1 z1 b  n  n−1 dx na xn x dx x − = m−1 z1m z1m z1 (n + 1 − m)b (n + 1 − m)b For n = m − 1, we may use the formula  m−1  dx 1 xm−2 dx xm−1 x + = − m−1 z1m z1 (m − 1)b b z1m−1 For m = 1  n x dx axn−1 an−1 x (−1)n an xn a2 xn−2 − = + − . . . + (−1)n−1 + ln z1 2 3 z1 nb (n − 1)b (n − 2)b 1 · bn bn+1  n n−1 n−1 kak−1 xn−k an x dx  n−1 n+1 na = (−1)k−1 + (−1) + (−1) ln z1 2 z1 (n − k)bk+1 bn+1 z1 bn+1 k=1  a x dx x = − 2 ln z1 z1 b b  2 ax a2 x dx x2 − 2 + 3 ln z1 = z1 2b b b z1m dx =



1.  2.  3. 2.114 1. 2. 3. 4.6



dx 1 =− 2 z1 bz1 x dx x 1 a 1 =− + 2 ln z1 = 2 + 2 ln z1 z12 bz1 b b z1 b x2 dx x a2 2a = − − 3 ln z1 2 2 3 z1 b b z1 b

dx 1 =− z13 2bz12  +x a , 1 x dx + = − z13 b 2b2 z12    2 2ax 3a2 1 x dx 1 = + + 3 ln z1 z13 b2 2b3 z12 b  3   3 x a 2 a2 5 a3 1 a x dx + 2 2x − 2 3x − = − 3 4 ln z1 3 2 4 z1 b b b 2 b z1 b

69

70

2.115 1. 2. 3. 4. 2.116 1. 2. 3. 4. 2.117

Rational Functions



dx 1 =− z14 3bz13  +x a , 1 x dx + 2 3 =− 4 z1 2b 6b z1  2   2 x ax a2 1 x dx + = − + z14 b b2 3b3 z13    3 3ax2 x dx 9a2 x 11a3 1 1 = + + + 4 ln z1 z14 b2 2b2 6b4 z13 b 

dx 1 =− 5 z1 4bz14  +x a , 1 x dx + = − z15 3b 12b2 z14  2   2 x 1 ax x dx a2 + 2+ =− z15 2b 3b 12b3 z14  3   3 x 3ax2 x dx a2 x a3 1 + = − + + z15 b 2b2 b3 4b4 z14 

1.  2.  3.  4.

dx −1 b(2 − n − m) = + xn z1m a(n − 1) (n − 1)axn−1 z1m−1 dx 1 =− m z1 (m − 1)bz1m−1 1 dx 1 + = m−1 xz1m z1 a(m − 1) a



dx xz1m−1

n−1  (−1)k bk−1 dx (−1)n bn−1 z1 = + ln xn z1 (n − k)ak xn−k an x k=1

2.118 1. 2. 3. 2.119 1.



dx 1 z1 = − ln , xz1 a x  b dx 1 z1 + 2 ln =− 2 x z1 ax a x  b2 1 b z1 dx =− + 2 − 3 ln 3 2 x z1 2ax a x a x 



dx 1 1 z1 = − 2 ln 2 xz1 az1 a x

dx xn−1 z1m

2.115

Forms containing the binomial a + bxk

2.124

  1 2b 1 dx 2b z1 + =− + 3 ln x2 z12 ax a2 z1 a x    3b2 1 dx 1 3b 3b2 z1 = − + 2 + 3 − 4 ln 2 3 2 x z1 2ax 2a x a z1 a x

 2. 3. 2.121 1. 2. 3. 2.122 1. 2. 3. 2.123 1.11 2. 3. 2.124 1.

2.

  3 dx bx 1 1 z1 + = − 3 ln 3 2 2 xz1 2a a z1 a x    2 1 9b dx 3b x 1 3b z1 + =− + 3 + 4 ln x2 z13 ax 2a2 a z12 a x    9b2 dx 1 2b 6b3 x 1 6b2 z1 = − + 2 + 3 + 4 − 5 ln 3 2 3 2 x z1 2ax a x a a z1 a x 

  11 5bx b2 x2 1 dx 1 z1 + = + 3 − 4 ln xz14 6a 2a2 a z13 a x    1 22b 10b2 x 4b3 x2 1 dx 4b z1 + = − + + + 5 ln x2 z14 ax 3a2 a3 a4 z13 a x    2 3 4 2 1 55b dx 1 5b 25b x 10b x 10b2 z1 + = − + + + − ln x3 z14 2ax2 2a2 x 3a3 a4 a5 z13 a6 x 

  25 13bx 7b2 x2 b3 x3 1 1 z1 dx + = + + − 5 ln xz15 12a 3a2 2a3 a4 z14 a x    2 3 2 4 3 1 125b 65b x 35b x dx 1 5b x 5b z1 − = − − − − + 6 ln 5 4 2 2 3 4 5 x z1 ax 12a 3a 2a a z1 a x    2 3 4 2 5 3 1 125b dx 1 3b 65b x 105b x 15b x 15b2 z1 = − + 2 + + + + − 7 ln 5 4 3 2 3 4 5 6 x z1 2ax a x 4a a 2a a z1 a x



Forms containing the binomial z2 = a + bx2 .   b dx 1 if [ab > 0] (see also 2.141 2) = √ arctan x z2 a ab √ a + xi ab 1 √ if [ab < 0] (see also 2.143 2 and 2.1433) = √ ln 2i ab a − xi ab  x dx 1 =− (see also 2.145 2, 2.145 6, and 2.18) m z2 2b(m − 1)z2m−1

71

72

Rational Functions

2.125

Forms containing the binomial z3 = a + bx3  a Notation: α = 3 b 2.125  n  n−3 dx (n − 2)a xn−2 x dx x − = 1. m−1 m m z3 z3 z3 (n + 1 − 3m)b b(n + 1 − 3m)  n  n n+1 x dx x n + 4 − 3m x dx 2. = − m m−1 z3 3a(m − 1) 3a(m − 1)z3 z3m−1 2.126



1.

 2.



3. 4. 5. 2.127

 

2.  3.  4. 2.128 1. 2.

 √  √ (x + α)2 dx α 1 x 3 ln = + 3 arctan z3 3a 2 x2 − αx + α2 2α − x

 2 √ (x + α) 2x − α α 1 √ ln 2 + 3 arctan = 3a 2 x − αx + α2 α 3 x dx 1 =− z3 3bα

2

(x + α) 1 ln − 2 x2 − αx + α2

√

(see also 2.141 3 and 2.143)  2x − α √ 3 arctan α 3 (see also 2.145 3. and 2.145 7)

2

 −3



x dx 1 ln 1 + x3 α = z3 3b  3  x dx x a dx = − z3 b b z3  4  2 x dx a x dx x − = z3 2b b z3

1.

LA 133 (1)

dx x 2 = + z32 3az3 3a



x2 1 x dx = + 2 z3 3az3 3a



x3 dx x 1 =− + 2 z3 3bz3 3b

(see 2.126 2)

(see 2.126 1)

x dx z3



1 ln z3 3b (see 2.126 1)

dx z3

x2 dx 1 =− z32 3bz3

=

dx z3

(see 2.126 2)

(see 2.126 1)



 dx 1 b(3m + n − 4) dx = − − m m−1 n n−3 n−1 x z3 a(n − 1) x z3m (n − 1)ax z3   dx 1 n + 3m − 4 dx = + xn z3m 3a(m − 1) xn z3m−1 3a(m − 1)xn−1 z3m−1

LA 133 (2)

Forms containing the binomial a + bxk

2.133

2.129 1. 2. 3. 2.131 1. 2. 3.



x3 1 dx ln = xz3 3a z3   b x dx 1 dx − = − x2 z3 ax a z3   dx 1 b dx =− − x3 z3 2ax2 a z3

(see 2.126 2) (see 2.126 1)



1 1 x3 dx = + ln xz32 3az3 3a2 z3     2 1 1 4bx dx 4b x dx + = − − x2 z32 ax 3a2 z3 3a2 z3     1 5bx 1 5b dx dx =− + 2 − 2 2 3 2 x z3 2ax 6a z3 3a z3

(see 2.126 2) (see 2.126 1)

Forms containing the binomial z4 = a + bx4   a a 4  α = 4 − Notation: α = b b 2.132  √ √   2 2 + αx 2 + α 2 x α αx dx √ 1.8 ln = √ + 2 arctan 2 for ab > 0 z4 α − x2 4a 2 x2 − αx 2 + α2 

x α x + α + 2 arctan for ab < 0 = ln 4a x − α α   b 1 x dx 2 for ab > 0 (see = √ arctan x 2. z4 a 2 ab √ a + x2 i ab 1 √ for ab < 0 (see = √ ln 4i ab a − x2 i ab  √ √   2 x dx x2 − αx 2 + α2 1 αx 2 √ √ 3. ln = + 2 arctan 2 for ab > 0 z4 α − x2 4bα 2 x2 + αx 2 + α2

 x x + α 1 − 2 arctan  ln for ab < 0 =− 4bα x − α α  3 1 x dx ln z4 = 4. z4 4b 2.133 1. 2.

73



xn+1 4m − n − 5 xn dx = + m m−1 z4 4a(m − 1) 4a(m − 1)z4

(see also 2.141 4) (see also 2.143 5) also 2.145 4) also 2.145 8)



xn dx z4m−1  n  n−4 dx (n − 3)a x dx xn−3 x − = m−1 z4m z4m z4 (n + 1 − 4m)b b(n + 1 − 4m)

LA 134 (1)

74

2.134

Rational Functions



1.  2.  3.  4. 

x 3 dx = + z42 4az4 4a



x dx x2 1 = + z42 4az4 2a



x2 dx x3 1 = + 2 z4 4az4 4a

dx z4



2.134

(see 2.132 1)

x dx z4

(see 2.132 2)

x2 dx z4

(see 2.132 3)

x3 dx x4 1 = =− z42 4az4 4bz4

dx 1 b(4m + n − 5) 2.135 =− − m m−1 n n−1 x z4 (n − 1)a (n − 1)ax z4 For n = 1   dx dx dx 1 b = − m m−1 −3 xz4 a xz4 a x z4m 2.136  dx 1 x4 ln x ln z4 1. − = ln = xz4 a 4a 4a z4   2 b x dx dx 1 − 2. =− 2 x z4 ax a z4



dx xn−4 z4m

(see 2.132 3)

2.14 Forms containing the binomial 1 ± xn 2.141  1.  2.11

4.

dx = arctan x = − arctan 1 + x2



1 x

(see also 2.124 1)

√ 1+x x 3 1 1 dx (see also 2.126 1) = ln √ + √ arctan 1 + x3 3 2−x 3 1 − x + x2 √ √  dx 1 + x 2 + x2 x 2 1 1 √ √ √ ln arctan = + 1 + x4 1 − x2 4 2 1 − x 2 + x2 2 2 

3.

dx = ln(1 + x) 1+x

(see also 2.132 1)  2.142

n

−1

2 dx 2 = − Pk cos 1 + xn n

k=0



n



2 −1 2k + 1 2k + 1 2 π + π Qk sin n n n

k=0

for n a positive even number n−3 2

=

2 1 ln(1 + x) − Pk cos n n k=0





n−3 2

2k + 1 2 π + Qk sin n n k=0





TI (43)a

2k + 1 π n for n a positive odd number TI (45)

Forms containing the binomial 1 ± xn

2.145

where

2.143 1. 2. 3. 4. 5.

75





2k + 1 1 2 Pk = ln x − 2x cos π +1 2 n     x − cos 2k+1 x sin 2k+1 π π n2k+1  = arctan  2k+1n  Qk = arctan 1 − x cos n π sin n π



dx = − ln(1 − x) 1−x  dx 1 1+x = arctanh x [−1 < x < 1] (see also 2.141 1) = ln 2 1−x 2 1−x  1 x−1 dx = ln = − arccoth x [x > 1, x < −1] 2 x −1 2 x+1 √ √  dx 1 1 + x + x2 x 3 1 + √ arctan (see also 2.126 1) = ln 1 − x3 3 1−x 2+x 3  dx 1 1 1+x 1 + arctan x = (arctanh x + arctan x) = ln 4 1−x 4 1−x 2 2 (see also 2.132 1)

2.144



1.

n

−1

n

−1

2 2 2 2 dx 1 1+x 2k 2k ln − π + = P cos Qk sin π k 1 − xn n 1−x n n n n

k=1

k=1

for n a positive even number



1 2k + 1 π+1 , where Pk = ln x2 + 2x cos 2 n  2.

n−3

n−3

k=0

k=0

1. 2. 3.

x + cos 2k+1 n π sin 2k+1 n π

2 2 2 2 dx 1 2k + 1 2k + 1 ln(1 − x) + π + π = − P cos Qk sin k n 1−x n n n n n



1 2k 2 where Pk = ln x − 2x cos π + 1 , 2 n 2.145

Qk = arctan

TI (47)

for n a positive odd number Qk = arctan

x − cos

2k n π

sin 2k n π



x dx = x − ln(1 + x) 1+x   x dx 1  = ln 1 + x2 2 1+x 2  (1 + x)2 2x − 1 1 1 x dx ln = − + √ arctan √ 1 + x3 6 1 − x + x2 3 3

(see also 2.126 2)

TI (49)

76

Rational Functions

2.146



4. 5. 6. 7. 8. 2.146 1.

x dx 1 = arctan x2 1 + x4 2  x dx = − ln(1 − x) − x 1−x   x dx 1  = − ln 1 − x2 2 1−x 2  (1 − x)2 x dx 2x + 1 1 1 ln = − − √ arctan √ 1 − x3 6 1 + x + x2 3 3  x dx 1 1 + x2 = ln 1 − x4 4 1 − x2

(see also 2.126 2) (see also 2.132 2)

For m and n natural numbers.

  m−1 n x 2k − 1 dx 1  mπ(2k − 1) 2 ln π + x 1 − 2x cos = − cos 1 + x2n 2n 2n 2n k=1 n 2k−1 x − cos 2n π mπ(2k − 1) 1 arctan sin + n 2n sin 2k−1 2n π k=1

[m < 2n]  2.

n 

m−1

1 dx ln(1 + x) x − = (−1)m+1 2n+1 1+x 2n + 1 2n + 1 k=1 n 2k−1  x − cos 2n+1 π mπ(2k − 1) 2 arctan sin + 2k−1 2n + 1 2n + 1 sin 2n+1 π k=1

 3.11

TI (44)a

 2k − 1 mπ(2k − 1) 2 ln 1 − 2x cos π+x cos 2n + 1 2n + 1

[m ≤ 2n]

TI (46)a



n−1  kπ 1  1  kmπ xm−1 dx m+1 2 ln + x (−1) 1 − 2x cos = ln(1 + x) − ln(1 − x) − cos 1 − x2n 2n 2n n n k=1   n−1 kπ x − cos n 1 kmπ + arctan sin n n sin kπ n k=1

 4.

[m < 2n]

TI (48)

m−1

dx x 1 ln(1 − x) =− 1 − x2n+1 2n + 1

n 2k − 1 1  mπ(2k − 1) m+1 2 +(−1) ln 1 + 2x cos π+x cos 2n + 1 2n + 1 2n + 1 k=1 n 2k−1 x + cos 2n+1 π 2  mπ(2k − 1) arctan sin +(−1)m+1 2k−1 2n + 1 2n + 1 sin 2n+1 π k=1 [m ≤ 2n]

2.147    1 xm dx 1 xm dx xm dx = + 1. 1 − x2n 2 1 − xn 2 1 + xn  m−2  xm−1 x dx xm dx 1 m−1 · 2. + n =− 2n − m − 1 (1 + x2 )n−1 2n − m − 1 (1 + x2 )n (1 + x2 )

TI (50)

LA 139 (28)

Forms containing the binomial 1 ± xn

2.149

 3.  4.

 5. 2.148  1.

2. 3.

4.

xm−1 xm − dx = 1 + x2 m−1



77

xm−2 dx 1 + x2

 m−2 xm−1 x dx 1 m−1 xm dx = − n n−1 2 2 2 2n − m − 1 (1 − x ) 2n − m − 1 (1 − x )n (1 − x )  m−1 x 1 m−1 xm−2 dx = − 2n − 2 (1 − x2 )n−1 2n − 2 (1 − x2 )n−1 m

m−1

x dx x + =− 2 1−x m−1



LA 139 (33) m−2

x dx 1 − x2

dx 1 1 2n + m − 3 − n =− m − 1 xm−1 (1 + x2 )n−1 m−1 xm (1 + x2 )



dx n xm−2 (1 + x2 )

LA 139 (29)

For m = 1   1 dx 1 dx LA 139 (31) n = n−1 + n−1 2 2 2n − 2 x (1 + x ) (1 + x ) x (1 + x2 ) For  m = 1 and n = 1 x dx = ln √ 2 x (1 + x ) 1 + x2   1 dx dx = − − m 2 m−1 m−2 x (1 + x ) (m − 1)x x (1 + x2 )   dx x dx 1 2n − 3 = + FI II 40 n 2n − 2 (1 + x2 )n−1 2n − 2 (1 + x2 )n−1 (1 + x2 )  n−1 dx x  (2n − 1)(2n − 3)(2n − 5) · · · (2n − 2k + 1) (2n − 3)!! arctan x = + n−1 n n−k 2 k 2 2n − 1 2 (n − 1)! (1 + x ) k=1 2 (n − 1)(n − 2) . . . (n − k) (1 + x ) TI (91)

2.149  1.

2.

3.

dx 1 2n + m − 3 n =− n−1 + m−1 2 m−1 xm (1 − x2 ) (m − 1)x (1 − x )



dx n xm−2 (1 − x2 )

LA 139 (34)

For m = 1   dx 1 dx LA 139 (36) n = n−1 + n−1 2 x (1 − x2 ) 2(n − 1) (1 − x ) x (1 − x2 ) For  m = 1 and n = 1 x dx = ln √ 2 x (1 − x ) 1 − x2   x dx 1 2n − 3 dx = + LA 139 (35) n n−1 2 2 2n − 2 (1 − x ) 2n − 2 (1 − x2 )n−1 (1 − x )  n−1 1+x dx x  (2n − 1)(2n − 3)(2n − 5) . . . (2n − 2k + 1) (2n − 3)!! ln + n n = n−k k 2 2n − 1 2 · (n − 1)! 1 −x (1 − x2 ) k=1 2 (n − 1)(n − 2) . . . (n − k) (1 − x ) TI (91)

78

Rational Functions

2.151

2.15 Forms containing pairs of binomials: a + bx and α + βx Notation: t = α + βx; Δ = aβ − αb  z = a + bx; n+1  mΔ tm z n m − 2.151 z t dx = z n tm−1 dx (m + n + 1)b (m + n + 1)b 2.152  z bx Δ 1. dx = + 2 ln t t β β  βx Δ t dx = − 2 ln z 2. z b b  m  m−1 tm dx t dx 1 mΔ t 2.153 = − zn (m − n + 1)b z n−1 (m − n +  1)b zn tm+1 1 (m − n + 2)β tm dx = − n−1 (n − 1)Δ z n−1 (n − 1)Δ  m−1z tm 1 mβ t =− + dx (n − 1)b z n−1 (n − 1)b z n−1  1 t dx = ln 2.154 zt Δ z   1 1 (m + n − 2)b dx dx = − − 2.155 z n tm (m − 1)Δ tm−1 z n−1 (m − 1)Δ tm−1 z n 1 1 (m + n − 2)β dx = + m z n−1 (n − 1)Δ tm−1 z n−1 (n − 1)Δ t

 1 a α x dx = ln z − ln t 2.156 zt Δ b β

2.16 Forms containing the trinomial a + bxk + cx2k 2.160 1. 2. 3.

2.161

Reduction formulas for Rk = a + bxk + cx2k .    xm Rkn+1 (m + k + nk)b (m + 2k + 2kn)c m−1 n m+k−1 n x − Rk dx = Rk dx − x xm+2k−1 Rkn dx ma ma ma    bkn xm Rkn 2ckn m−1 n m+k−1 n−1 − Rk dx = Rk dx − x x xm+2k−1 Rkn−1 dx m m m    xm−2k Rkn+1 (m − 2k)a (m − k + kn)b xm−1 Rkn dx = − xm−2k−1 Rkn dx − xm−k−1 Rkn dx (m + 2kn)c (m + 2kn)c (m + 2kn)c   2kna xm Rkn bkn + = xm−1 Rkn−1 dx + xm+k−1 Rkn−1 dx m + 2kn m + 2kn m + 2kn Forms containing the trinomial R2 = a + bx2 + cx4 . b 1 2 b 1 2 − b − 4ac, g = + b − 4ac, 2 2 2 2     a h = b2 − 4ac, q = 4 , l = 2a(n − 1) b2 − 4ac , c

Notation: f =

b cos α = − √ 2 ac

Forms containing a + bx + cx2 and powers of x

2.172

 1.

dx R2

   2  dx dx − h >0 LA 146 (5) 2 2 cx + f cx + g 

 2  x2 − q 2 1 α α x2 + 2qx cos α2 + q 2 h < 0 LA 146 (8)a ln arctan = + 2 cos sin α 4cq 3 sin α 2 x2 − 2qx cos 2 + q 2 2 2qx sin α2   2  x dx cx2 + f 1 h >0 LA 146 (6) ln 2 = R2 2h cx + g 2 2  2  x − q cos α 1 h <0 LA 146 (9)a arctan = 2 2 2cq sin α q sin α  2    2  f x dx g dx dx − h >0 = LA 146 (7) 2 2 R2 h cx + g h cx + f      bcx3 + b2 − 2ac x b2 − 6ac dx bc x2 dx dx = + + R22 lR2 l R2 l R2  2   2   3 bcx + b − 2ac x (4n − 7)bc x dx 2(n − 1)h2 + 2ac − b2 dx dx = + + n 2 n−1 R2 lRn−1 l l R2 R2n−1 =

2.

3. 4. 5.

79

 6.9

c h



1 (m + 2n − 3)b dx =− − xm R2n (m − 1)a (m − 1)axm−1 R2n−1



[n > 1] (m + 4n − 5)bc dx − xm−2 R2n (m − 1)a



LA 146

dx xm−4 R2n LA 147 (12)a

2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x Notation: R = a + bx + cx2 ; Δ = 4ac − b2 2.171    am xm Rn+1 b(m + n + 1) m+1 n m−1 n − R dx = R dx − 1. x x xm Rn dx c(m + 2n + 2) c(m + 2n + 2) c(m + 2n + 2)  2. 3. 4.





TI (97)

R dx R b(n − m + 1) R dx c(2n − m + 2) R dx =− + + LA 142(3), TI (96)a m+1 m x amx am xm am xm−1   dx b + 2cx (4n − 2)c dx = + TI (94)a Rn+1 nΔRn nΔ Rn  n−1 n dx (2cx + b)  2k(2n + 1)(2n − 1)(2n − 3) . . . (2n − 2k + 1)ck dx n (2n − 1)!!c = + 2 n+1 k+1 n−k n R 2n + 1 n(n − 1) · · · (n − k)Δ R n!Δ R n

n+1

n

n

k=0

 2.17211

TI (96)a

√

1 −Δ − (b + 2cx) −2 b + 2cx dx √ =√ ln =√ arctanh √ R −Δ (b + 2cx) + −Δ −Δ −Δ −2 = b + 2cx b + 2cx 2 = √ arctan √ Δ Δ

for [Δ < 0] for [Δ = 0, b and c non-zero] for [Δ > 0]

80

Rational Functions

2.173

2.173   dx b + 2cx 2c dx + 1. = R2 ΔR Δ R

   dx 1 b + 2cx 3c 6c2 dx 2. = + + R3 Δ 2R2 ΔR Δ2 R 2.174 1.

2. 2.175 1. 2. 3. 4. 5. 6.

 m−1  m−2 xm dx dx dx xm−1 (n − m)b (m − 1)a x x = − − + n n−1 n n R (2n − m − 1)cR (2n − m − 1)c R (2n − m − 1)c R For m = 2n − 1, this formula is inapplicable. Instead, we may use  2n−1    x dx 1 x2n−3 dx a x2n−3 dx b x2n−2 dx = − − Rn c Rn−1 c Rn c Rn 

 1 b x dx dx = ln R − R 2c 2c R   x dx b 2a + bx dx − =− R2 ΔR Δ R   x dx 2a + bx 3b(b + 2cx) 3bc dx − 2 =− − R3 2ΔR2 2Δ2 R Δ R  2  x dx x b b2 − 2ac dx = − 2 ln R + R c 2c 2c2 R  2   2  ab + b − 2ac x 2a dx x dx + = R2 cΔR Δ R      2 2 ab + b − 2ac x 2ac + b2 (b + 2cx) x dx + = + R3 2cΔR2 2cΔ2 R

7.

 8.

 9.  2.176

1.

(see 2.172)





2.177

(see 2.172)

(see 2.172) (see 2.172) (see 2.172) (see 2.172) 2ac + b2 Δ2

  b b2 − 3ac x3 dx x2 bx b2 − ac dx = − 2 + ln R − 3 3 R 2c c 2c 2c R



dx R

(see 2.172)

(see 2.172)     2 a 2ac − b + b 3ac − b x b 6ac − b2 x dx 1 dx − = 2 ln R + 2 2 2 R 2c c ΔR 2c Δ R 

3

3

x dx =− R3 dx xm Rn



(see 2.172)

=



2

2

2

x abx 2a + + c cΔ cΔ



−1 (m − 1)axm−1 Rn−1

1 x2 b dx = ln − xR 2a R 2a



dx R



1 3ab − 2R2 2cΔ



(see 2.172)

dx (see 2.173 1) R2   dx dx b(m + n − 2) c(m + 2n − 3) − − m−1 n m−2 a(m − 1) x R a(m − 1) x Rn (see 2.172)

2.177

Quadratic trinomials and binomials

 2.

 3.

1 1 x2 dx + = ln 2 2 xR 2a R 2aR

5.

dx 1 b 1 1 x − = + 2 + 3 ln xR3 4aR2 2a R 2a R 2a





b − 2 2a



2ac 1+ Δ



dx R

(see 2.172)   b b dx dx dx − 2 − 3 R3 2a R2 2a R (see 2.172, 2.173)



dx b 1 b − 2ac dx x = − 2 ln − + (see 2.172) x2 R 2a R ax 2a2 R  2 

  b − 3ac (b + 2cx) dx a + bx 1 b4 b x2 6b2 c 6c2 dx − + − = − ln − + 2 2 3 2 2 3 2 x R a R a xR a ΔR Δ a a a R 

6. 7.

b(b + 2cx) 1− Δ

2



4.

81

2

2

dx 1 3b =− − x2 R3 axR2 a



5c dx − xR3 a



(see 2.172) dx R3

(see 2.173 and 2.177 3)

   b 3ac − b2 ac − b2 x2 b 1 dx dx =− + 2 − ln + 3 3 2 3 x R 2a R a x 2ax 2a R 

8.

 9.

(see 2.172) 2

  3b dx 1 1 3b 2c 9bc dx dx + = − + − + x3 R2 2ax2 2a2 x R a2 a xR2 2a2 R2

dx = x3 R3



−1 2b + 2 2ax2 a x



1 + R2



2

6b 3c − a2 a



(see 2.173 1 and 2.177 2)  10bc dx dx + 2 xR3 a R3 (see 2.173 2 and 2.177 3)

2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx Notation: R = a + bx + cx2 ; B = bβ − 2cα;  1. 2.

z = α + βx;

A = aβ 2 − αbβ + cα2 ;

Δ = 4ac − b2

 (m − 1)A R dx − z z m−2 Rn dx (m + 2n + 1)c  n  n−1 Rn dx R dx R 1 2nA = − − zm (m − 2n − 1)β z m−1 (m − 2n − 1)β 2 zm nB Rn−1 dx − ; LA 184 (4)a (m − 2n − 1)β 2 z m−1  n  n n+1 R −β (m − n − 2)B R dx (m − 2n − 3)c R dx = − − LA 148 (5) m−1 (m − 1)A z (m − 1)A  z m−1 (m − 1)A z m−2 Rn Rn−1 dx Rn−1 dx 1 nB 2nc =− + + LA 418 (6) m−1 2 m−1 2 (m − 1)β z (m − 1)β z (m − 1)β z m−2 (m + n)B βz m−1 Rn+1 − z R dx = (m + 2n + 1)c (m + 2n + 1)c m

n



m−1

n

82

Algebraic Functions

 3.

z m−1 β (m − n)B z m dx = − n n−1 R (m − 2n + 1)c R (m − 2n + 1)c b + 2cx z 2(m − 2n + 3)c − (n − 1)Δ Rn−1 (n − 1)Δ m

=





2.201

z m−1 dx (m − 1)A − n R (m − 2n + 1)c

m

Bm z dx − Rn−1 (n − 1)Δ





z m−2 dx Rn LA 147 (1)

z

m−1

dx Rn−1 LA 148 (3)

 4.3

1 β (m + n − 2)B dx =− − m n m−1 n−1 z R (m − 1)A z R (m − 1)A

=

1 β B − 2(n − 1)A z m−1 Rn−1 2A





dx (m + 2n − 3)c − m−1 n z R (m − 1)A

(m + 2n − 3)β dx + z m−1 Rn 2(n − 1)A

2



dx z m−2 Rn LA 148 (7)

dx z m Rn−1 LA 148 (8)

For m = 1 and n = 1   β z2 B dx dx = ln − zR 2A R 2A R For A = 0   1 β (m + 2n − 2)c dx dx =− − z m Rn (m + n − 1)B z m Rn−1 (m + n − 1)B z m−1 Rn

LA 148 (9)

2.2 Algebraic Functions 2.20 Introduction



r

s αx + β αx + β 2.201 The integrals R x, , , . . . dx, where r, s, . . . are rational numbers, γx + δ γx + δ can be reduced to integrals of rational functions by means of the substitution αx + β = tm , FI II 57 γx + δ where m is the common denominator of the fractions r, s, . . . .  p 2.202 Integrals of the form xm (a + bxn ) dx,∗ where m, n, and p are rational numbers, can be 

expressed in terms of elementary functions only in the following cases: (a)

When p is an integer; then, this integral takes the form of a sum of the integrals shown in 2.201;

(b)

is an integer: by means of the substitution xn = z, this integral can be transformed m+1 1 to the form (a + bz)p z n −1 dz, which we considered in 2.201; n

(c)

When

When

m+1 n

m+1 n

+ p is an integer; by means of the same substitution xn = z, this integral can be

p  m+1 a + bz 1 z n +p−1 dz, considered in 2.201; reduced to an integral of the form n z

For reduction formulas for integrals of binomial differentials, see 2.110. ∗ Translator:

The authors term such integrals “integrals of binomial differentials.”

2.214

Forms containing the binomial a + bxk and

2.21 Forms containing the binomial a + bxk and

√

√

x

x

Notation: z1 = a + bx.   dx bx 2 √ = √ arctan [ab > 0] 2.211 a √ z1 x ab a − bx + 2i xab 1 [ab < 0] = √ ln z1 i ab  m√ m m+1  √  (−1)k ak xm−k x x dx m+1 a √ dx = 2 x + (−1) 2.212 z1 (2m − 2k + 1)bk+1 bm+1 z1 x k=0

(see 2.211) 2.213

√  √  dx x dx 2 x a √ 1. − (see 2.211) = z1 b b z1 x  √  a √ x x dx  x a2 dx √ − 2 2 x+ 2 (see 2.211) 2. = z1 3b b b z1 x 2

 2√  √ x xa x x dx a2 a3 dx √ − 2 + 3 2 x− 3 (see 2.211) 3. = z1 5b 3b b b z1 x √   dx x 1 dx √ √ 4. = (see 2.211) + az1 2a z1 x z12 x √  √  x dx x 1 dx √ 5. (see 2.211) =− + 2 z1 bz1 2b z1 x √  √  √ x x dx x dx 2x x 3a 6. = − (see 2.213 5) z12 bz1 b z12 2

√  √  2√ x 5ax 2 x 5a2 x dx x x dx − 2 = + 2 (see 2.213 5) 7. 2 z1 3b 3b z1 b z12

  √ 1 dx 3 3 dx √ √ 8. = (see 2.211) + x + 2az12 4a2 z1 8a2 z1 x z13 x

 √  √ x dx 1 1 1 dx √ 9. (see 2.211) = − + x + 3 2 z1 2bz1 4abz1 8ab z1 x √  √  √ x x dx x dx 2x x 3a 10. =− + (see 2.213 9) z13 bz12 b z13 2

√  √  2√ x 5ax 2 x 15a2 x dx x x dx + = − (see 2.213 9) 11. 3 2 2 2 z1 b b z1 b z13   a a  4 2 , α = 4 − . Notation: z2 = a + bx , α = b b % √ √ & +  , 2 a x + α 2x + α 1 α 2x dx √ = √ ln >0 + arctan 2 2.214 √ z2 α −x b z2 x bα3 2 √ √

+ ,  a x 1 α − x <0 = ln  √ − 2 arctan  3 2bα α b α + x

83

84

Algebraic Functions

2.215

% √ √ &  √ x dx x + α 2x + α2 1 α 2x 2.215 = √ − ln + arctan 2 √ z2 z2 α −x bα 2  √ √   1 x α − x = ln  √ + 2 arctan  2bα α α + x 2.216



1.  2.  3.  4.  5.  6.  7.  8.  9.  10.

√ √  2 x a x x dx dx √ − = z2 b b z2 x √ √  √ x dx x2 x dx 2x x a − = z2 3b b z2 √  x dx 3 dx √ = √ + 2az2 4a z2 x z22 x √ √  √ x dx x dx x x 1 = + z22 2az2 4a z2 √ √  dx x x x dx 1 √ =− + z22 2bz2 4b z2 x √ √  √ x dx x2 x dx x x 3 = − + z22 2bz2 4b z2

 √ 1 dx 7 21 dx √ = √ + x+ 2 3 2 2 4az2 16a z2 32a z2 x z2 x

√  √ √ 1 x dx x dx 5 5 = + x + x z23 4az22 16a2 z2 32a2 z2  2 √ √  bx − 3a x x x dx dx 3 √ = + 3 z2 16abz22 32ab z2 x √ √  √ x dx 2x x 3a x2 x dx =− + z23 5bz22 5b z23

2.22–2.23 Forms containing Notation: z = a + bx.  n   √ n l lm+f 2.220 x z dx = k=0

k=0

2.222 1.



dx 2√ √ = z b z

b +a b

, >0 , <0

(see 2.214) (see 2.215) (see 2.214) (see 2.215) (see 2.214) (see 2.215) (see 2.214) (see 2.215) (see 2.214) (see 2.216 8)

 n (a + bx)k

 √   l (−1)k nk z n−k ak l z l(m+1)+f ln − lk + l(m + 1) + f bn+1

The square root  n  √     (−1)k n z n−k ak 2 z 2m+1 √ n k 2.221 x z 2m−1 dx = 2n − 2k + 2m + 1 bn+1

+a

Forms containing the binomial a + bxk and

2.225

3.

x

√ 2 z 1 z−a 3 b2

√  2 1 2 2 2 z x dx √ = z − az + a2 5 3 b3 z 

2.

√

x dx √ = z



2.223  dx 2 √ =− √ 1. b z z3  x dx 2 √ = (z + a) 2 √ 2. 3 b z z 2

 2 x dx z 2 √ √ − 2az − a2 3. = 3 3 b z z3 2.224 1.

2.

√  z m dx z m dx zm z 2m − 2n + 3 b √ √ = − + (n − 1)axn−1 2(n − 1) a xn−1 z xn z ⎧  m ⎨ √ 1 z dx m √ = −z z n ⎩ (n − 1)axn−1 x z



⎫ (2m − 2n + 3)(2m − 2n + 5) . . . (2m − 2n + 2k + 1) bk ⎬ + 2k (n − 1)(n − 2) . . . (n − k − 1)xn−k−1 ak+1 ⎭ k=1  (2m − 2n + 3)(2m − 2n + 5) . . . (2m − 3)(2m − 1) bn−1 z m dx √ + 2n−1 (n − 1)!x an−1 x z n−2 

3. 4. 5.6

2.225 1. 2. 3.

For n = 1  m−1  m 2z m z z √ dx = √ +a √ dx x z (2m − 1) z x z  m  m  z 2am−k z k dx √ dx = √ + am √ x z (2k − 1) z x z k=1 !√ √ !  ! z − a! 1 dx ! √ = √ ln ! √ √ ! x z a z + √a ! 2 z =√ arctan √ −a −a  √  √ z dx dx √ =2 z+a x x z √  √  b dx z dx z √ + =− 2 x x 2 x z √ √   √ b2 dx z dx z3 b z √ − =− + x3 2ax2 4ax 8a x z

[a > 0] [a < 0]

(see 2.224 4) (see 2.224 4) (see 2.224 4)

85

86

2.226 1. 2. 3.

Algebraic Functions

  √ 3 √ z dx  z dx √ = + a 2 z + a2 x 3 x z √  √ 3  √ 3 z dx z5 z dx 3b + =− 2 x ax 2a x

√  √ 3  √ 3 1 z dx z dx b 3b2 5 =− + 2 z + 2 x3 2ax2 4a x 8a x

2.226

(see 2.224 4) (see 2.226 1)

(see 2.226 1)  2.227

 m−1  dx 2 1 dx √ √ = √ + m m m−k k a xz z (2k + 1)a z z x z

(see 2.224 4)

k=0

2.228 1. 2. 2.229 1. 2. 3.

√  b dx z dx √ √ − = − 2 ax 2a x z x z

  √ 1 dx 3b 3b2 dx √ √ = − + 2 z+ 2 2ax2 4a x 8a x3 z x z 

(see 2.224 4) (see 2.224 4)



 dx 1 2 dx √ = √ + √ (see 2.224 4) a x z a z x z3

  1 dx 3b 3b 1 dx √ √ = − − 2 √ − 2 (see 2.224 4) 2 3 ax a 2a z x z x z

  1 15b2 15b2 1 5b dx dx √ √ = − √ + + + 2ax2 4a2 x 4a3 8a3 x z z x3 z 3 (see 2.224 4)

Cube root 2.231 1.

2.

3.

4.

 √   3 (−1)k nk z n−k ak 3 z 3(m+1)+1 3n − 3k + 3(m + 1) + 1 bn+1 k=0   n     (−1)k n z n−k ak 3 xn dx k √ √ = 3 3n − 3k − 3(m − 1) − 2 bn+1 3 z 3(m−1)+2 z 3m+2 k=0  √  n    √ 3  (−1)k n z n−k ak 3 z 3(m+1)+2 3 n k z 3m+2 x dx = 3n − 3k + 3(m + 1) + 2 bn+1 k=0  n      (−1)k n z n−k ak xn dx 3 k √ √ = 3 3 3m+1 n+1 3n − 3k − 3(m − 1) − 1 z b z 3(m−1)+1  n  √  3 n 3m+1 z x dx =

k=0

Forms containing the binomial a + bxk and

2.236

√

x



 1 z n+ 3 3n − 3m + 4 b z n dx z n dx √ √ = − + 3 m−1 (m − 1)ax 3(m − 1) a xm−1 3 z 2 xm x2 For m = 1  n−1  n dx 3z n z z dx √ √ √ = + a 3 3 3 2 2 x z (3n − 2) z x z2 √ √   3 z dx dx 33z 1 √ 6. = + 3 n (3n − 1)az a xz n xz n z 2   √ √ √ √  3 z− 3a √ 33z 3 1 dx √ √ √ ln − 3 arctan √ 2.232 = √ 3 3 3 3 x z+23a x z2 a2 2 5.

2.233 1. 2. 3.

 √ 3

 √ z dx dx √ =33z+a (see 2.232) 3 x x z2 √  √  3 b√ dx z dx z3z b 3 √ + = − z + (see 2.232) x2 ax a 3 x 3 z2

 √  3 √ z dx 1 b b2 √ b2 dx 3 3 √ = − + z − z − z 3 2 2 2 x 2ax 3a x 3a 9a x 3 z 2 √ 3

 4. 5. 2.234



z 2b dx dx √ √ − =− 3 3 2 ax 3a z x z2     1 5b √ 5b2 dx dx 3 √ √ = − + z + 3 2ax2 6a2 x 9a2 x 3 z 2 x3 z 2 x2

(see 2.232) (see 2.232) (see 2.232)

√  3 z n dx zn z2 3n − 3m + 5 b z n dx √ √ 1. = − + 3 (m − 1)axm−1 3(m − 1) a xm−1 3 z xm z 2 For m = 1:  n  n−1 z dx dx 3z n z √ √ √ 2. = + a x3z (3n − 1) 3 z x3z √   √ 3 3 3 z2 z 2 dx 1 dx √ = + 3. 3 n n (3n − 2)az a xz n xz z   √ √ √ √  3 z− 3a √ 33z 1 dx 3 √ = √ √ √ ln + 3 arctan √ 2.235 3 3 3 x3z x z+23a a2 2 2.236 1. 2.



 √ 3

 3√ z 2 dx dx 3 √ = z2 + a x 2 x3z √  √  3 3 b√ z 2 dx z5 2b dx 3 2+ √ + = − z 2 x ax a 3 x3z

(see 2.235) (see 2.235)

87

88

Algebraic Functions

 √ 3 3.

 4.  5.

   z 2 dx 1 b b2 √ b2 dx 3 5/3 2− √ = − + − z z x3 2ax2 6a2 x 6a2 9a x 3 z

dx √ x2 3 z dx √ 3 z

x3

√  3 b z2 dx √ − =− ax 3a x 3 z    1 2b √ 2b2 dx 3 √ = − + 2 z+ 2 2 2ax 3a x 9a x3z

2.24 Forms containing Notation: z = a + bx, 2.241 1. 2.

2.241

(see 2.235) (see 2.235) (see 2.235)

√ a + bx and the binomial α + βx

t = α + βx,

Δ = aβ − bα.



 m−1 n √ z m tn dx t dx 2 (2m − 1)Δ z √ √ tn+1 z m−1 z + = (2n + 2m + 1)β (2n + 2m + 1)β z z

 n m k n  n−k k   √ k z k−p ap n α β t z dx p √ = 2 z 2m+1 (−1) p 2k − 2p + 2m + 1 k bk+1 p=0 z k=0

2.242 1.



11

 2.  3.

 4.  5.  6.

 7.  8.

√ z 2√z 2α z t dx √ = +β −a b 3 b2 z 2

√ √ z 2√z z 2 z 2 2α2 z t2 dx 2 2 √ = + 2αβ −a − za + a + β b 3 b2 5 3 b3 z

√ √ √ 2 z z z2 2 z 2 t3 dx 2α3 z 2 2 2 √ = + 3α β −α − za + a + 3αβ 2 b 3 b √ 5 3 b3 z 3 2 z 2 z 3z a − + za2 − a3 +β α 7 5 b4 √ √ z a 2 z3 2α z 3 tz dx √ = +β − 3b 5 3 b2 z √ √ 2

√ z z a 2 z3 2za a2 2 z 3 2α2 z 3 t2 z dx 2 √ + 2αβ − − + = +β 3b 5 3 b2 7 5 3 b3 z √ √ 2

√ z z a 2 z3 2za a2 2 z 3 t3 z dx 2α3 z 3 2 2 √ = + 3α β − − + + 3αβ 3b 5 3 b2 7 5 3 b3 z √ 3

2 2 3 z 3z a 3za a 2 z3 − + − +β 3 9 7 5 3 b4 √ √ z a 2 z5 2α z 5 tz 2 dx √ +β − = 5b 7 5 b2 z √ √ 2

√ z z a 2 z5 2za a2 2 z 5 2α2 z 5 t2 z 2 dx 2 √ + 2αβ − − + = +β 5b 7 5 b2 9 7 5 b3 z

LA 176 (1)

2.244

Forms with

√ a + bx and α + βx

89

√ √ 2

√ z z a 2 z5 2za a2 2 z 5 t3 z 2 dx 2α3 z 5 2 2 √ + 3α β − − + = + 3αβ 5b 7 5 b2 9 7 5 b3 z √ 3

2 2 3 5 z 3z a 3za a 2 z − + − +β 3 11 9 7 5 b4 √ √  3 z a 2 z7 2α z 7 tz dx √ +β − = 7b 9 7 b2 z √ √ 2

√  2 3 z z a 2 z7 2za a2 2 z 7 2α2 z 7 t z dx 2 √ + 2αβ − − + = +β 7b 9 7 b2 11 9 7 b3 z √ √ √

 3 3 z z2 a 2 z7 2za a2 2 z 7 t z dx 2α3 z 7 2 2 √ + 3α β − − + = + 3αβ 7b 9 7 b2 11 9 7 b3 z √ 3

2 2 3 7 z 3z a 3za a 2 z − + − +β 3 13 11 9 7 b4 

9.

10. 11. 12.

2.243



1.

 tn dx tn+1 √ 2 (2n − 2m + 3)β tn dx √ √ = z − (2m − 1)Δ z m (2m − 1)Δ zm z z m−1 z  tn √ tn−1 dx 2nβ 2 √ z+ =− m (2m − 1)b z (2m − 1)b z m−1 z LA 176 (2)

 2. 2.244



1.  2.  3.  4.  5.  6.  7.

2 tn dx √ =√ 2m−1 zm z z



k n  n−k k   k z k−p ap n a β p (−1) p 2k − 2p − 2m + 1 k bk+1 p=0

k=0

t dx 2a 2β(z + a) √ =− √ + √ z z b z b2 z

2 − 2za − a t dx 2α 4αβ(z + a) √ =− √ + √ √ + z z b z b2 z b3 z   2 z2 2 3 z3 2 2 3 6αβ 2β 3 3 2 − 2za − a − z a + 3za + a 3 5 t dx 2α 6α β(z + a) √ =− √ + √ √ √ + + z z b z b2 z b3 z b4 z   2β z − a3 2a t dx √ √ √ = − − z2 z 3b z 3 b2 z 3    a2 2 2 a z 2β + 2az − 2 2 4αβ z − 3 3 2a t dx √ =− √ − √ √ + 3 2 3 3 3 z2 z 3b z b z b z   3 2   2 6αβ z 2 + 2za − a3 2β 3 z3 − 3z 2 a − 3za2 + 6α2 β z − a3 2α3 t3 dx √ =− √ − √ √ √ + + z2 z 3b z 3 b2 z 3 b3 z 3 b4 z 3   2β z3 − a5 2α t dx √ =− √ − √ z3 z 5b z 5 b2 z 5 2

2

2β 2



z2 3

a3 3



90

Algebraic Functions

 8.  9.

2.245 1.

2.

2

t dx √ z3 z t3 dx √ z3 z

 2za a2 2 2 z 2β − + 4αβ 3 − 3 5 2α √ √ =− √ − − 5 2 5 3 5 5b z b z b z    2za a2 2 2 z a 2 z 6αβ − + 3 6α β 3 − 5 3 5 2α √ √ =− √ − − 5 2 5 3 5 5b  z b z b z 3 2β 3 z 3 + 3z 2 a − za2 + a5 √ + b4 z 5 a 5





 m−1 z m dx z m−1 √ dx 2 (2m − 1)Δ z √ =− √ z− (2n − 2m − 1)β tn−1 (2n− 2m − 1)β tn z tn z z m−1 z m−1 √ 1 (2m − 1)b √ dx =− z + (n − 1)β tn−1 2(n − 1)β tn−1  zm m √ z 1 (2n − 2m − 3)b z dx √ =− z− n−1 n−1 (n − 1)Δ t 2(n − 1)Δ t z ⎡  m √ 1 1 z dz √ = −z m z ⎣ n−1 n (n − 1)Δ t t z n−1  k=2

−

4.

z

2

+

3.

2.245

(2n − 2m − 3)(2n − 2m − 5) . . . (2n − 2m − 2k + 1)b 2k−1 (n − 1)(n − 2) . . . (n − k)Δk

k=0

2.246

2.247

√ √ 1 dx β z − βΔ √ √ √ ln √ [βΔ > 0] t z βΔ β z + βΔ √ 2 β z =√ arctan √ [βΔ < 0] −βΔ −βΔ √ 2 z [Δ = 0] =− bt   m  dx βm 2 β k−1 z k dx √ √ = m−1 √ + + m k m Δ (2m − 2k + 1) Δ z tz z z t z k=1

(see 2.246) 2.248 1.



k−1

(2n − 2m − 3)(2n − 2m − 5) . . . (−2m + 3)(−2m + 1)bn−1 2n−1 · (n − 1)!Δn

For n = 1   m zm 2 Δ z m−1 dx z dx √ = √ √ + (2m − 1)β z β t z t z  m  m−1  Δk z m−k Δm z dx dx √ =2 √ √ + k+1 m (2m − 2k − 1)β β t z z t z 

LA 176 (3)

2 β dx √ = √ + Δ tz z Δ z



dx √ t z

(see 2.246)



⎤ 1 ⎦

tn−k

z m dx √ t z

2.248

Forms with

 2.  3.

 4.  5.  6.

 7.

 8.  9.

 10.  11.  12.  13.  14.  15.  16.

2 2β β2 dx √ √ √ = + + Δ2 tz 2 z 3Δz z Δ2 z



√ a + bx and α + βx

dx √ t z

2 2β 2β 2 β3 dx √ + 3√ + 3 √ = √ + 3 2 2 Δ tz z 5Δz z 3Δ z z Δ z √

b z dx √ =− − Δt 2Δ z

t2



91

(see 2.246) 

dx √ t z (see 2.246)

dx √ t z

dx 1 3b 3bβ √ = − √ − 2√ − 2Δ2 t2 z z Δt z Δ z

(see 2.246) 

dx √ t z

1 5b 5bβ 5bβ 2 dx √ √ √ √ = − − − − 2Δ3 t2 z 2 z Δtz 2 z 3Δ2 z z Δ3 z

(see 2.246) 

dx √ t z

(see 2.246)  dx 1 7b 7bβ 7bβ 7bβ 3 dx √ − 4√ − √ √ =− √ − √ − 2Δ4 t z t2 z 3 z Δtz 2 z 5Δ2 z 2 z 3Δ3 z z Δ z 2

√ √  dx 3b2 z 3b z dx √ √ + = − + 2Δt2 4Δ2 t 8Δ2 t z t3 z dx 1 5b 15b2 15b2 β √ √ √ √ = − + + + 8Δ3 t3 z z 2Δt2 z 4Δ2 t z 4Δ3 z

(see 2.246) (see 2.246) 

dx √ t z

(see 2.246) √  2 dx 1 7b z 35b 35b2 β 35b2 β 2 dx √ + √ + √ + √ √ =− √ + 4 3 2 2 2 2 4 8Δ t z z 2Δt z z 4Δ tz z 12Δ z z 4Δ z t z (see 2.246) dx 1 9b 63b2 21b2 β 63b2 β 2 63b2 β 3 √ √ √ √ √ √ = − + + + + + 8Δ5 t3 z 3 z 2Δt2 z 2 z 4Δ2 tz 2 z 20Δ3 z 2 z 4Δ4 z z 4Δ5 z (see 2.246) √  Δ 2 z z dx dx √ = √ + (see 2.246) β β t z t z √ √  2Δ z z 2 dx 2z z Δ2 dx √ = √ + (see 2.246) + 2 2 3β β β t z t z √ √ √  2Δz z 2z 2 z z 3 dx 2Δ2 z Δ3 dx √ = √ + (see 2.246) + + 5β 3β 2 β3 β3 t z t z √ √  z dx b z b z z dx √ √ =− + + (see 2.246) 2 Δt βΔ 2β t z t z √ √ √  bz z 3b z z2 z z 2 dx 3bΔ dx √ √ + + = − (see 2.246) + Δt βΔ β2 2β 2 t z t2 z



dx √ t z

92

Algebraic Functions

 17.

 18.3  19.

 20.

2.249

√ √ √ √  bz 2 z 5bz z z3 z 5bΔ z 5Δ2 b z 3 dx dx √ √ =− + + + + 2 3 3 2 Δt βΔ 3β β 2β t z t z

z dx √ t3 z z 2 dx √ t3 z

(see 2.246) √ √ 2 2  b z z z bz z b dx √ − =− (see 2.246) + + 2Δt2 4Δ2 t 4βΔ2 8βΔ t z √ √ √ √  b2 z z 3b2 z2 z bz 2 z 3b2 z dx √ + + =− + + 2 2 2 2 2 2Δt 4Δ t 4βΔ 4β Δ 8β t z √

(see 2.246) √ √ √ √ √  3 2 2 2 z dx 3b z z 15b2 z z z 3bz z 5b z z 15b2 Δ dx √ √ + + = − + + + 2 2 2 2 3 3 3 2Δt Δ t 4βΔ 4β Δ 4β 8β t z t z 3

3

(see 2.246) 2.249 1.

2.

√  z 2 (2n + 2m − 3)β dx dx √ = √ + m n z m−1 z (2m − 1)Δ tn−1 z (2m − 1)Δ z m tn z t √  z 1 (2n + 2m − 3)b dx √ =− − m n−1 n−1 (n − 1)Δ z t 2(n − 1)Δ t zm z ⎡ √  1 z ⎣ −1 dx √ = m n−1 m n z (n − 1)Δ t z t z 

LA 177 (4)

⎤ k−1 (2n + 2m − 3)(2n + 2m − 5) . . . (2n + 2m − 2k + 1)b 1 + (−1)k · n−k ⎦ 2k−1 (n − 1)(n − 2) . . . (n − k)Δk t k=2 n−1  (2n + 2m − 3)(2n + 2m − 5) . . . (−2m + 3)(−2m + 1)b dx √ +(−1)n−1 n−1 n−1 m 2 (n − 1)!Δ tz z n−1 

For n = 1   1 2 β dx dx √ √ √ = + m m−1 m−1 (2m − 1)Δ z Δ tz z t z z z

2.25 Forms containing

√

a + bx + cx2

Integration techniques  2.251

It is possible to rationalize the integrand in integrals of the form

  R x, a + bx + cx2 dx by

using one or more of the following three substitutions, known as the “Euler substitutions”: √ √ a + bx + cx2 = xt ± a for a > 0; 1. √ √ a + bx + cx2 = t ± x c for c > 0; 2.  c (x − x1 ) (x − x2 ) = t (x − x1 ) when x1 and x2 are real roots of the equation a + bx + cx2 = 0. 3.

2.252

Forms containing

√

a + bx + cx2

93

2.252

Besides Euler substitutions, there is also the following method of calculating integrals of   the  the form R x, a + bx + cx2 dx. By removing the irrational expressions in the denominator and performing simple algebraic operations, we can reduce the integrand to the sum of some rational function P (x) √ 1 , where P 1 (x) and P 2 (x) are both polynomials. of x and an expression of the form P 2 (x) a + bx + cx2 P 1 (x) By separating the integral portion of the rational function from the remainder and decomposing P 2 (x) the latter into partial fractions, we can reduce the integral of these partial fractions to the sum of integrals, each of which is in one of the following three forms:  P (x) dx √ , where P (x) is a polynomial of some degree r; 1. a + bx + cx2  dx √ 2. ; k (x + p) a + bx + cx2

 (M x + N ) dx a b  , a1 = , b 1 = 3. . m c c (a + βx + x2 ) c (a1 + b1 x + x2 ) In more detail:    P (x) dx dx √ 1. = Q(x) a + bx + cx2 + λ √ , where Q(x) is a polynomial of 2 a + bx + cx a + bx + cx2 degree (r − 1). Its coefficients, and also the number λ, can be calculated by the method of undetermined coefficients from the identity

2.

3.

 1  P (x) = Q (x) a + bx + cx2 + Q(x)(b + 2cx) + λ LI II 77 2  P (x) dx Integrals of the form √ (where r ≤ 3) can also be calculated by use of formulas a + bx + cx2 2.26.  P (x) dx √ , where the degree n of the polynomial P (x) is Integrals of the form k (x + p) a + bx + cx2 1 lower than k can, by means of the substitution t = , be reduced to an integral of the form x + p  P (t) dt  . (See also 2.281). a + βt + γt2  (M x + N ) dx  can be calculated by the following Integrals of the form m (α + βx + x2 ) c (a1 + b1 x + x2 ) procedure: • If b1 = β, by using the substitution x=

< 2 t − 1 (a1 − α) − (αb1 − a1 β) (β − b1 )

a1 − α + βb1 t+1

β − b1

94

Algebraic Functions

2.260



P (t) dt  , where P (t) m + p) c (t2 + q)  P (t) dt  is a polynomial of degree no higher than 2m − 1. The integral can be 2 + p)m (t t2 + q   t dt dt reduced to the sum of integrals of the forms and .   k k 2 2 2 (t + p) t +q (t + p) t2 + q  P (t) dt  • If b1 = β, we can reduce it to integrals of the form by means of the m 2 c (t2 + q) (t + p) b1 substitution t = x + . 2  t dt can be evaluated by means of the substitution t2 + q = The integral  k 2 (t + p) c (t2 + q) u2 .  t dt = can be evaluated by means of the substitution  The integral  k 2 2 2 t +q (t + p) c (t + q) υ (see also 2.283). FI II 78-82 we can reduce this integral to an integral of the form

2.26 Forms containing

(t2

√ a + bx + cx2 and integral powers of x

Notation: R = a + bx + cx2 , Δ = 4ac − b2 For simplified formulas for the case b = 0, see 2.27. 2.260  √  √ √ (2m + 2n + 1)b xm−1 R2n+3 m 2n+1 1. x − R dx = xm−1 R2n+1 dx (m + 2n + 2)c  2(m + 2n + 2)c √ (m − 1)a − xm−2 R2n+1 dx (m + 2n + 2)c

2. 3.

TI (192)a

 2cx + b √ 2n+1 2n + 1 Δ √ 2n−1 R R R2n+1 dx = + dx TI (188) 4(n + 1)c 8(n + 1) c  √   √ n−1  (2n + 1)(2n − 1) . . . (2n − 2k + 1) Δ k+1 R (2cx + b) n n−k−1 R + R2n+1 dx = R 4(n + 1)c 8k+1 n(n − 1) . . . (n − k) c k=0 n+1  Δ (2n + 1)!! dx √ + n+1 8 (n + 1)! c R  √

TI (190)

2.26111



For n = −1  √  1 dx 2 cR + 2cx + b √ = √ ln √ c R Δ

2cx + b 1 √ = √ arcsinh c Δ 1 = √ ln(2cx + b) c

2cx + b −1 = √ arcsin √ −c −Δ

[c > 0]

TI (127)

[c > 0,

Δ > 0]

DW

[c > 0,

Δ = 0]

DW

[c < 0,

Δ < 0]

TI (128)

2.263

2.262 1. 2. 3.

Forms containing

√  Δ dx (2cx + b) R √ + R dx = 4c 8c R √  √  3 (2cx + b)b √ R bΔ x R dx = − R− 3c 8c2 16c2 2

√  √ x 5b 5b x2 R dx = − R3 + − 4c 24c2 16c2

 √

 4.

5.

6.

2.263 1.



2

x 7bx 7b 2a − + − 2 3 2 5c 40c 48c 15c 3

 7b 3ab Δ dx √ − − 2 32c3 8c 8c R

R3 dx =

2

R 3Δ + 8c 64c2



95

(see 2.261) dx √ (see 2.261) R √ 2

 5b a (2cx + b) R a Δ dx √ + − 4c 4c 16c2 4c 8c R

√

R3 −

√ 3Δ2 (2cx + b) R + 128c2



(see 2.261) √

7b3 3ab (2cx + b) R − 2 32c3 8c 4c

(see 2.261) dx √ R

(see 2.261) √

 √ b √ 3 R5 3Δb √ 3Δ2 b dx √ − (2cx + b) R + R − x R3 dx = 5c 16c2 128c3 256c3 R

(see 2.261) 





√ 3 √ x 7b2 R 7b √ 5 3Δ √ a b 2 3 x R dx = − + R + − R 2x + 6c 60c2 24c2 6c c 8 64c

2  Δ2 7b dx √ −a + 3 4c 256c R (see 2.261) 2

 √ x 3bx 3b2 2a √ 5 − x3 R3 dx = + − R 2 3 7c 28c 40c 35c2   √ 3



3b 3Δ √ R3 ab b + − − 2 R 2x + 16c3 4c c 8 64c

2  3Δ2 b 3b dx √ −a − 4c 512c4 R (see 2.261) 

8.

√ x3 R dx =

 √



7.

√ a + bx + cx2 and xn





xm dx xm−1 (2m − 2n − 1)b √ √ = − 2n+1 2n−1 2(m − 2n)c R (m − 2n)c R



(m − 1)a xm−1 dx √ − 2n+1 (m − 2n)c R



xm−2 dx √ R2n+1 TI (193)a

2.

For m = 2n    x2n dx x2n−1 b 1 x2n−1 x2n−2 √ √ √ √ =− − dx + dx 2c c R2n+1 (2n − 1)c R2n−1 R2n+1 R2n−1

TI (194)a

96

Algebraic Functions



dx

√ R2n+1

3.  4.

dx √ R2n+1

2.264

 dx 2(2cx + b) 8(n − 1)c √ √ = + 2n−1 (2n − 1)Δ (2n − 1)Δ R R2n−1   n−1  ck k 8k (n − 1)(n − 2) . . . (n − k) 2(2cx + b) √ 1+ = R (2n − 3)(2n − 5) . . . (2n − 2k − 1) Δk (2n − 1)Δ R2n−1 k=1

[n ≥ 1] . 2.264



1.

3. 4.



5. 6. 7. 8.

TI (191)

(see 2.261)

√  x dx b R dx √ = √ (see 2.261) − c 2c R R 2

  2 x 3b x dx 3b √ a dx √ √ = (see 2.261) − 2 R+ − 2 2c 4c 8c 2c R R 2 3

  3 x 5b 5bx x dx 5b2 2a √ 3ab dx √ √ = − + 3− 2 R− − 2 2 3 3c 12c 8c 3c 16c 4c R R

 2.

dx √ R

TI (189)

(see 2.261)

dx 2(2cx + b) √ √ = 3 Δ R R  x dx 2(2a + bx) √ √ =− 3 Δ R R    2  Δ − b2 x − 2ab 1 x dx dx √ √ √ + (see 2.261) =− 3 c cΔ R R R      3  cΔx2 + b 10ac − 3b2 x + a 8ac − 3b2 3b x dx dx √ √ − 2 √ = 2c c2 Δ R R R3

 √ 2n+1 R 2.265 xm

(see 2.261) √  √ 2n+1 R2n+3 R (2n − 2m + 5)b dx = − + dx (m − 1)axm−1 √ 2(m − 1)a xm−1  R2n+1 (2n − m + 4)c dx + (m − 1)a xm−2 TI (195)

For√m = 1 √  √ 2n−1   R2n+1 R2n+1 b √ 2n−1 R dx = + dx R dx + a x 2n + 1 2 x For
TI (198)

LA 169 (3)

2.267

Forms containing

 8

2.266

√ dx 2a + bx + 2 aR 1 √ = − √ ln x a x R 1 2a + bx =√ arcsin √ −a x b2 − 4ac 1 2a + bx =√ arctan √ √ −a 2 −a R 2a + bx 1 = − √ arcsinh √ a x Δ 2a + bx 1 = − √ arctanh √ √ a 2 a R x 1 = √ ln a√ 2a + bx 2 bx + cx2 =− bx

2a + bx 1 √ √ arccosh = a x −Δ

√ a + bx + cx2 and xn

[a > 0] [a < 0,

TI (137)

Δ < 0]

TI (138)

[a < 0] [a > 0,

97

LA 178 (6)a

Δ > 0]

DW

[a > 0] [a > 0,

Δ = 0]

[a = 0,

b = 0]

LA 170 (16)

[a > 0, Δ < 0]

2.267  √   b dx R dx √ dx √ + √ (see 2.261 and 2.266) = R+a 1. x x R 2 R √  √   dx b R dx R dx √ √ 2. + c (see 2.261 and 2.266) + = − 2 x x 2 x R R For a = 0 √  √  dx bx + cx2 2 bx + cx2 √ + c dx = − (see 2.261) 2 x x bx + cx2 2

  √ √ 1 b c R dx b dx √ − 3. =− + R− x3 2x2 4ax 8a 2 x R (see 2.266)

4.

5.

For a = 0 < 3  √ 2 2 (bx + cx2 ) bx + cx dx = − x3 3bx3 √ √     b 12ac − b2 2bcx + b2 + 8ac √ R3 R3 dx dx 2 √ + √ dx = + R+a x 3 8c 16c x R R (see 2.261 and 2.266) √     √ 3 √ 3 4ac + b2 R5 R cx + b √ 3 3 dx dx 3 √ + √ + dx = − R + (2cx + 3b) R + ab 2 x ax a 4 2 8 x R R (see 2.261 and 2.266) For a = 0 < 3  (bx + cx2 ) x2

< =

3

(bx + cx2 ) 2x

+

3b  3b2 bx + cx2 + 4 8



dx √ bx + cx2

(see 2.261)

98

6.

Algebraic Functions

2.268

 

√  √ 3 1 R b bcx + 2ac + b2 √ 3 3 bcx + 2ac + b2 √ 5 dx = − + 2 R + R + R 2 x3 2ax2 4ax 4a  4a  3 dx dx 3 √ + bc √ + 4ac + b2 8 x R 2 R (see 2.261 and 2.266) For a = 0 < 3  (bx + cx2 ) x3

dx =



 dx 2b  3bc √ c− bx + cx2 + x 2 bx + cx2 (see 2.261)

 2.268

xm

dx 1 √ √ =− 2n+1 m−1 2n−1 R (m − 1)ax R  (2n + 2m − 3)b (2n + m − 2)c dx dx √ √ − − 2(m − 1)a (m − 1)a xm−1 R2n+1 xm−2 R2n+1 TI (196)



For m = 1 For a = 0 

2.269



1.

3.

√

dx R2n+1

+

1 a



dx √ x R2n−1

dx 2 < < =− 2n+1 2n−1 xm (bx + cx2 ) (2n + 2m − 1)bxm (bx + cx2 )  dx (4n + 2m − 2)c < − (2n + 2m − 1)b 2n+1 xm−1 (bx + cx2 ) (cf. 2.265)

dx √ x R

(see 2.266)

√  dx b dx R √ √ =− − (see 2.266) 2 ax 2a x R x R For a = 0

 2 2c  dx 1 √ = bx + cx2 − 2+ 2 3 bx b x x2 bx + cx2 2

  √ 3b 1 dx 3b c dx √ √ = − + R + − 2ax2 4a2 x 8a2 2a x3 R x R

 2.

dx 1 b √ √ = − 2n+1 2n−1 2a x R (2n − 1)a R



(see 2.266)

4.

For a = 0

 2 4c 8c2  dx 1 √ = bx + cx2 − 3+ 2 2− 3 5 bx 3b x 3b x x3 bx + cx2     2 bcx − 2ac + b2 1 dx dx √ √ √ + =− (see 2.266) 3 a x R aΔ R x R For a = 0

TI (199)

2.271

Forms containing

  5.11

dx

< 3 x (bx + cx2 )

2 = 3



4c 8c2 x 1 − + 2 + 3 bx b b

3b A dx √ = −√ − 2 2 3 R 2a x R



dx √ x R

where A =

6.



√

a + cx2 and xn

99

1 √ bx + cx2



     b 10ac − 3b2 c 8ac − 3b2 x 1 − − − ax a2 Δ a2 Δ

(see 2.266)

For a = 0

 1 dx 8c2 2 2c 16c3 x 1 < √ = − 2+ 2 − 3 − 4 5 bx b x b b 3 bx + cx2 x2 (bx + cx2 )  dx √ 3 x R3      bc 15b2 − 52ac x 15b2 − 12ac 15b4 − 62acb2 + 24a2 c2 1 1 5b dx √ + √ − = − 2+ 2 − 3 3 3 ax 2a x 2a Δ 2a Δ 8a 2 R x R (see 2.266) For a = 0

 1 64c3 2 8c 16c2 128c4 x 1 dx < √ = + − 3+ 2 2− 3 + 4 5 7 bx 5b x 5b x 5b 5b 3 bx + cx2 x3 (bx + cx2 )

2.27 Forms containing

√ a + cx2 and integral powers of x

√ Notation: u = a + cx2 .  √  1 I1 = √ ln x c + u c  1 c = √ arcsin x − a −c √ u− a 1 √ I2 = √ ln 2 a √ u+ a a−u 1 = √ ln √ 2 a a + u  1 1 c a 1 =√ − arcsec x − = √ arccos a x c −a −a 2.271 1. 2. 3. 4.

[c > 0] [c < 0 and a > 0] [a > 0 and c > 0] [a > 0 and c > 0] [a < 0 and c > 0]



1 5 5 5 5 xu + axu3 + a2 xu + a3 I1 6 24 16 16  3 1 3 u3 dx = xu3 + axu + a2 I1 4 8 8  1 1 u dx = xu + aI1 2 2  dx = I1 u u5 dx =

DW DW DW DW

100

Algebraic Functions

 5.  6.

dx u2n+1

 7. 2.272

dx 1x = u3 au

n−1 1  (−1)k n − 1 ck x2k+1 = n a 2k + 1 k u2k+1

x2 u3 dx = x2 u dx =

2.  3.  4.  5.  6.  7.

 4.  5.  6. 

DW

1 x2 x dx = − + I1 u3 cu c

DW

x2 1 x3 dx = u5 3 au3

DW



n−2 x2 dx 1  (−1)k n − 2 ck x2k+3 = n−1 u2n+1 a 2k + 3 k u2k+3 k=0

3

x dx 1 a =− + u2n+1 (2n − 3)c2 u2n−3 (2n − 1)c2 u2n−1

x4 u dx =

3.

DW

DW

 

DW

1 xu 1 a x2 dx = − I1 u 2 c 2c

x4 u3 dx =

2.

1 axu3 1 a2 xu 1 a3 1 xu5 − − − I1 6 c 24 c 16 c 16 c

1 axu 1 a2 1 xu3 − − I1 4 c 8 c 8 c



1.

7.

k=0

1 x dx =− 2n+1 u (2n − 1)cu2n−1



2.273

DW



1.

2.272

axu5 1 x3 u5 a2 xu3 3a3 xu 3a4 − + + + I1 8 c 16c2 64c2 128c2 128c2

axu3 1 x3 u3 a2 xu a3 − + + I1 2 2 6 c 8c 16c 16c2

DW

DW DW

1 x3 u 3 axu 3 a2 x4 dx = − + I1 u 4 c 8 c2 8 c2

DW

3 a x4 1 xu ax dx = + 2 − I1 3 2 u 2c c u 2 c2

DW

1 x3 x4 x 1 dx = − 2 − + 2 I1 5 u c u 3 cu3 c

DW

x4 1 x5 dx = u7 5 au5

DW



n−3 x4 dx 1  (−1)k n − 3 ck x2k+5 = n−2 u2n+1 a 2k + 5 k u2k+5 k=0

2.275

Forms containing

 8. 2.274

x6 u3 dx =  x6 u dx =

2.  3.  4.  5.  6.  7.  8.  9. 2.275



1.  2. 3. 4. 5. 6. 7. 8.

a + cx2 and xn

1 2a a2 x5 dx = − + − u2n+1 (2n − 5)c3 u2n−5 (2n − 3)cu2n−3 (2n − 1)c3 u2n−1



1.

√

101

DW

ax3 u5 1 x5 u5 a2 xu5 a3 xu3 3a4 xu 3 a5 − + − − − I1 2 3 3 3 10 c 16c 32c 128c 256c 256 c3

5 ax3 u3 1 x5 u3 5a2 xu3 5a3 xu 5 a4 − + − − I1 8 c 48 c2 64c3 128c3 128 c3

1 x5 u 5 ax3 u x6 5 a2 xu 5 a3 dx = − + − I1 u 6 c 24 c2 16 c3 16 c3

DW

5 ax3 15 a2 x 15 a2 x6 1 x5 − − + dx = I1 3 2 u 4 cu 8 c u 8 c3 u 8 c3

DW

x6 1 x5 10 ax3 5 a2 x 5 a dx = + + − I1 u5 2 cu3 3 c2 u3 2 c3 u3 2 c3

DW

x6 23 x5 7 ax3 a2 x 1 dx = − − − + 3 I1 7 5 2 5 3 5 u 15 cu 3c u c u c

DW

x6 1 x7 dx = 9 u 7 au7

DW



n−4 x6 dx 1  (−1)k n − 4 ck x2k+7 = u2n+1 an−3 2k + 7 k u2k+7 k=0

7

x dx 1 3a 3a2 a3 = − + − + u2n+1 (2n − 7)c4 u2n−7 (2n − 5)c4 u2n−5 (2n − 3)c4 u2n−3 (2n − 1)c4 u2n−1

DW

u5 1 u5 dx = + au3 + a2 u + a3 I2 x 5 3

DW

u3 u3 dx = + au + a2 I2 x 3  u dx = u + aI2 x  dx = I2 xu  n−1  1 dx 1 = I + 2 2n+1 n xu a (2k + 1)an−k u2k+1 k=0  5 5 15 u u5 15 dx = − + cxu3 + acxu + a2 I1 2 x x 4 8 8  3 3 u 3 3 u dx = − + cxu + aI1 x2 x 2 2  u u dx = − + cI1 x2 x

DW DW DW

DW DW DW

102

Algebraic Functions

 9. 2.276



1.  2.  3.  4.  5.  6.  7.  8.  9.  10. 2.277



1.  2.  3.  4.

dx 1 = − n+1 x2 u2n+1 a



2.276

n u  (−1)k+1  n k  x 2k−1 + c x 2k − 1 k u



k=1

5 u5 u5 5 5 dx = − 2 + cu3 + acu + a2 cI2 3 x 2x 6 2 2

DW

u3 3 u3 3 dx = − 2 + cu + acI2 3 x 2x 2 2 u c u dx = − 2 + I2 x3 2x 2 dx u c =− − I2 x3 u 2ax2 2a 3c 3c dx 1 − − =− I2 x3 u3 2ax2 u 2a2 u 2a2 5 c dx 1 5 c 5 c − =− − − I2 x3 u5 2ax2 u3 6 a2 u 3 2 a3 u 2 a3

DW DW DW DW DW

u5 au3 2acu c2 xu 5 + + acI1 dx = − − x4 3x3 x 2 2

DW

u3 u3 cu + cI1 dx = − 3 − 4 x 3x x

DW

u u3 dx = − x4 3ax3 dx 1 = n+2 x4 u2n+1 a

DW



cu  (−1)k u3 − 3 + (n + 1) + 3x x 2k − 3 n+1

k=2



n+1 k

ck

 x 2k−3



u

u3 u3 3 cu3 3 c2 u 3 2 + c I2 dx = − 4 − + 5 2 x 4x 8 ax 8 a 8

DW

u u 1 cu 1 c2 I2 dx = − − − x5 4x4 8 ax2 8 a

DW

u dx 3 cu 3 c2 = − + + I2 x5 u 4ax4 8 a2 x2 8 a2

DW

5 c 15 c2 15 c2 dx 1 + + + =− I2 5 3 4 2 2 3 x u 4ax u 8 a x u 8 a u 8 a3 DW

2.278



1.  2.

u3 u5 dx = − x6 5ax5

DW

u u3 2 cu3 dx = − + 6 5 x 5ax 15 a2 x3

DW

2.284

Forms containing

 3.  4.

√

a + bx + cx2 and polynomials

103



2 cu3 c2 u u5 − DW − 5+ 5x 3 x3 x  





n+2 n + 2 c2 u  (−1)k n + 2 k  x 2k−5 1 1 n + 2 cu3 u5 + = n+3 − 5 + − c 2 a 5x 3 1 x3 x 2k − 5 k u

1 dx = 3 6 x u a dx x6 u2n+1

2.28 Forms containing

k=3

√ a + bx + cx2 and first- and second-degree polynomials

Notation: R = a + bx + cx2 See also 2.252   tn−1 dt dx 3 √ =−  2.281 (x + p)n R c + (b − 2pc)t + (a − bp + cp2 ) t2  t= 2.282 1.

3

    √   dx x dx dx R dx √ = c √ + (b − cp) √ + a − bp + cp2 x+p R R (x + p) R 

2. 3. 4. 5.





[x + p > 0]

1 1 dx dx dx √ = √ + √ q − p p − q (x + p)(x + q) R (x + p) R (x + q) R √   √  √ 1 1 R dx R dx R dx = + (x + p)(x + q) q−p x+p p−q x+q √  √  √  R dx (x + p) R dx = R dx + (p − q) x+q x+q    (rx + s) dx s − pr s − qr dx dx √ = √ + √ q − p p − q (x + p)(x + q) R (x + p) R (x + q) R 

2.283

1 >0 x+p

(Ax + B) dx A √ = n c (p + R) R



2Bc − Ab du n + 2c (p + u2 )



n−1 dυ 1 − cυ 2  n , 2 b − cpυ 2 p+a− 4c 

√ b + 2cx √ . where u = R and υ = 2c R  2Bc − Ab A Ax + B √ dx = I1 +  I2 , 2.284 c (p + R) R c2 p [b2 − 4(a + p)c] where . R 1 [p > 0] I1 = √ arctan p p √ √ 1 −p − R √ = √ [p < 0] ln √ 2 −p −p + R



104

Algebraic Functions



b + 2cx p √ b2 − 4(a + p)c R  b + 2cx p √ = − arctan b2 − 4(a + p)c R  √ √ 4(a + p)c − b2 R + p(b + 2cx) 1 √ ln  = √ 2i 4(a + p)c − b2 R − p(b + 2cx)  √ √ b2 − 4(a + p)c R − −p(b + 2cx) 1 √ ln  = √ 2i b2 − 4(a + p)c R + −p(b + 2cx)

I2 = arctan

2.290

  2  p b − 4(a + p)c > 0,

p<0

  2  p b − 4(a + p)c > 0,

p>0

  2  p b − 4(a + p)c < 0,

p>0

  2  p b − 4(a + p)c < 0,

p<0

   

2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals  2.290

Integrals of the form

  R x, P (x) dx, where P (x) is a third- or fourth-degree polynomial,

can, by means of algebraic transformations, be reduced to a sum of integrals expressed in terms of elementary functions and elliptic integrals (see 8.11). Since the substitutions that transform the given integral into an elliptic integral in the normal Legendre form are different for different intervals of integration, the corresponding formulas are given in the chapter on definite integrals (see 3.13, 3.17). "   2.291 Certain integrals of the form R x, P (x) dx, where Pn (x) is a polynomial of not more than "   fourth degree, can be reduced to integrals of the form R x, k Pn (x) dx with k ≥2. Below are examples of this procedure.     dz dx 1 √ =− √ x2 = 1. 1 + z2 1 − x6 3 + 3z 2 + z 4   dz dx 1 √ √ 2. = 2 4 6 2 2 a + bx + cx + dx az + bz + cz 3 + dz 4  2  x =z   1 ±1/3  3 z 2 A± 3 dz dx = 3. a + 2bx + cx2 + gx3 2 B ⎤ ⎡ 3   2 + (z 3 − a) c  b −b + ⎣a + 2bx + cx2 = z 3 , A = g + z 3 , B = b2 + (z 3 − a) c⎦ c  dx √ 4. 6 a + bx + cx2 + dx3 + cx4 + bx5 + ax   + ,  dx dz 1 1   −√ x = z + z2 − 1 = −√ 2 2 (z + 1)p (z − 1)p + ,  d 1 dz 1   +√ x = z − z2 − 1 = −√ 2 2 (z + 1)p (z − 1)p    where p = 2a 4z 3 − 3z + 2b 2z 2 − 1 + 2cz + d.

2.292

Integrals reducible to elliptic integrals





105

dy √ [x = y] √  2 3 4 y a + by + cy + by + ay , +  1 dz dz 1   =− √ + √ y = z + z2 − 1 2 2 (z + 1)p 2 2 (z − 1)p + ,  dz dz 1 1   = √ − √ y = z − z2 − 1 2 2 (z + 1)p 2 2 (z − 1)p   where p = 2a 2z 2 − 1 + 2bz + c.       a√ dx 18 a dt 8 √ √√ 6. = t ; x = 2 c c t ab t2 + at4 a + bx4 + cx8    1  + ,  dz a dz 1  −  t = z + z2 − 1 =− √ 8 2 2 c (z + 1)p (z − 1)p     + ,  1 8 a dz dz  =− √ +  t = z − z2 − 1 2 2 c (z + 1)p (z − 1)p  2   where p = 2a 2z − 1 + b1 ; b1 = b ac .     x dx z 2 dz √ √ 7. a + bx2 + cx4 = z 4 , A = b2 − 4ac, B = 4c = 2 4 a + bx2 + cx4 A + Bz 4         4 2 R2 z 4 z 2 dz b2 − a (c − z 4 ) + b dx 2  √ 8. z dz = R1 z z dz +  , = 4 a + 2bx2 + cx4 (c − z 4 ) b2 − a (c − z 4 ) b2 − a (c − z 4 )     where R1 z 4 and R2 z 4 are rational functions of z 4 and a + 2bx2 + cx4 = x4 z 4 . "   2.292 In certain cases, integrals of the form R x, P (x) dx, where P (x) is a third- or fourth-degree polynomial, can be expressed in terms of elementary functions. Such integrals are called pseudo-elliptic integrals. Thus, if the relations



1 1 − k2 x 1−x (x) = f (x) = f f1 (x) = f1 , f , f , 2 2 3 3 k2 x k 2 (1 − x) 1 − k2 x 5.

dx 1 √ = 2 4 6 8 2 a + bx + cx + bx + ax

hold, then   f1 (x) dx  = R1 (z) dz 1. x(1 − x) (1 − k 2 x)   f2 (x) dx  2. = R2 (z) dz x(1 − x) (1 − k 2 x)   f3 (x) dx  3. = R3 (z) dz x(1 − x) (1 − k 2 x) where R1 (z), R2 (z), and R3 (z) are rational functions of z.

+ ,  z = x(1 − x) (1 − k 2 x) %

&  x (1 − k 2 x) √ z= 1−x % &  x(1 − x) z= √ 1 − k2 x

106

The Exponential Function

2.311

2.3 The Exponential Function 2.31 Forms containing eax 

eax a 2.312 ax in the integrands should be replaced with ex ln a = ax 2.313  dx 1 [mx − ln (a + bemx )] 1. = a + bemx am  dx ex 2. = ln = x − ln (1 + ex ) x 1+e 1 + ex 

 dx a 1 mx 2.314 [ab > 0] √ arctan e = aemx + be−mx b m ab √ ! ! ! b + emx −ab ! 1 ! [ab < 0] √ √ = ln !! 2m −ab b − emx −ab ! √ √  dx a + bemx − a 1 √ 2.315 = √ ln √ [a > 0] √ mx mx m a a + be a + be √ + a 2 a + bemx √ = √ arctan [a < 0] m −a −a eax dx =

2.311

PE (410) PE (409)

PE (411)

2.32 The exponential combined with rational functions of x 2.321   n xn eax − xn−1 eax dx 1. xn eax dx = a a  n      (−1)k k! n n ax ax n−k k 11 x x e dx = e 2. ak+1 k=0

2.322

x 1 − 2 a a 2

 x 2x 2 2 ax ax − 2 + 3 2. x e dx = e a a a 3

 2 x 3x 6x 6 − 2 + 3 − 4 3. x3 eax dx = eax a a a a 4

 3 2 x 4x 12x 24x 24 4 ax ax 10 − 2 + 3 − 4 + 5 4. x e dx = e a a a a a  m P (k) (x) eax  (−1)k , 2.323 P m (x)eax dx = a ak 1.





xeax dx = eax

k=0

where P m (x) is a polynomial in x of degree m and P (k) (x) is the k th derivative of P m (x) with respect to x.

2.325

2.324

Exponentials and rational functions of x

107



  ax  1 e dx eax dx eax = + a − xm m−1 xm−1 xm−1



 ak−1 eax an−1 ax Ei(ax) dx = −e + xn (n − 1)(n − 2) . . . (n − k)xn−k (n − 1)!

1. 2.

n−1

k=1

2.325



eax dx = Ei(ax) x  ax e eax + a Ei(ax) dx = − 2 x x  ax e a2 eax aeax + Ei(ax) dx = − − x3 2x2 2x 2  ax a3 eax aeax a2 eax e + Ei(ax) dx = − − − x4 3x3 6x2 6x 6  ±axn  ±axn   ±axn 1 e e e dx = dx − m−1 ± na xm m−1 x xm−n  axn (−1)z+1 az Γ (−z, −axn ) e dx = m x n (−1)z+1 az ∞ e−t = dt z+1 n −axn t

1. 2. 3. 4.∗ 5.∗ 6.∗

z= 

[m = 1]

m−1 , n

8.∗



9.∗ 10.

∗

11.

∗

12.∗

 



[n = 0]

n

Ei (axn ) eax dx = x n #z−1 az−k−1  axn az Ei (axn ) e k=0 k! xn(k+1) axn + dx = −e xm nz! nz!  a = 0,

7.∗

for Γ(α, x) see 8.350.2

n

n

n

n

n

eax aeax a2 Ei (axn ) eax dx = − − + xm 2nx2n 2nxn 2n n

2

n = 0]

m−1 = 1, 2, . . . , n

 a = 0,

z=

 m = 2, 3, . . .

m−1 =1 n

m−1 =2 z= n

 

n

eax eax a2 eax a3 Ei (axn ) eax dx = − − − + xm 3nx3n 6nx2n 6nxn 6n

√  √ eax eax + aπ erfi ax dx = − 2 x x

z=

 a = 0,

n

eax a Ei (axn ) eax dx = − n + m x nx n n

[a = 0,

 a = 0,

z=

m−1 =3 n

2

where erfi(z) =

erf(iz) i



108

13.

The Exponential Function

∗



2 1 e(ax +2bx+c) dx = 2

 2.326 2.33 1.



8

2.∗ 3.∗

e





π exp a



ac − b2 a



2.326

√ b ax + √ erfi a [a = 0]

ax

ax

xe dx e = 2 (1 + ax)2 a (1 + ax) −(ax2 +2bx+c)

1 dx = 2



[a = 0]

π exp a



b2 − ac a





√ b ax + √ erf a



[a = 0]



√  π erf(iz) erfi ax where erfi(z) = a i 



 2 √ 2 ac − b 1 π b exp ax + √ eax +bx+c dx = erfi 2 a a a 2

eax dx =

1 2

where erfi(z) = 4.∗

5.∗

∗



7.

8.∗ 9.

∗

10.∗

11.∗

n

n

xm eax dx =

6.

∗

n





m axn

eax dx = n

n



n

n

xm eax dx =



eax n

n

m axn

x e 

% (γ − 1)!

γ−1 

(−1)k+1−γ

k=0

eax dx = n



xn 1 − 2 a a

n

xnk k!aγ−k



x2n 2xn 2 − 2 + 3 a a a



x3n 3x2n 6xn 6 − 2 + 3 − 4 a a a a

Γ (γ, βxn ) γ nβ  ∞ 1 tγ−1 e−t dx =− γ nβ βxn

xm e−βx dx = − n



[a = 0]

xm−n e±ax dx

n

eax dx = na

x e



eax n

m axn

x e 

m+1−n xm+1−n ∓ na na

xm e±ax dx = ±



erf(iz) i

[a = 0]



&

[a = 0,

n = 0]

 a = 0,

γ=

 a = 0,

γ=

 a = 0,

γ=

 a = 0,

γ=

 a = 0,

γ=

 m+1 = 1, 2, . . . n  m+1 =1 n  m+1 =2 n  m+1 =3 n  m+1 =4 n

for Γ(α, x) see 8.350.2  m+1 , β = 0, γ= n &

%γ−1  xnk (γ − 1)! exp (−βxn ) xm exp (−βxn ) dx = − n k!β γ−k k=0   m+1 = 1, 2, . . . γ= n

 n = 0

2.33

12.

Exponentials and rational functions of x

∗

13.∗ 14.

∗

15.∗

16.∗ 17.∗

18.∗ 19.∗

20.∗ 21.∗

22.∗

23.∗



exp (−βxn ) x exp (−βx ) dx = − nβ m



n

xm exp (−βxn ) dx = − 

exp (−βxn ) n

exp (−βxn ) x exp (−βx ) dx = − n



xn 1 + 2 β β



  m+1 =1 γ= n   m+1 =2 γ= n

109

x2n 2xn 2 + 2 + 3 β β β   m+1 =3 γ= n

 3x2n exp (−βxn ) x3n 6xn 6 + 2 + 3 + 4 xm exp (−βxn ) dx = − n β β β β   m+1 =4 γ= n     n 1 π erf βx [β = 0] e−βx dx = 2 β  β z Γ (−z, βxn ) exp (−βxn ) dx = − xm  n β z ∞ e−t =− dt n βxn tz+a m−1 z= n  Ei (−βxn ) exp (−βxn ) dx = x n  z−1 n  z−k−1 z exp (−βxn ) z exp (−βx ) k! β z β Ei (−βxn ) dx = (−1) (−1) + (−1) n(k+1) xm nz! nz! x k=0   m−1 = 1, 2, . . . , m = 2, 3, . . . z= n    n n n exp (−βx ) β Ei (−βx ) exp (−βx ) m−1 =1 dx = − − z= xm nxn n n  exp (−βxn ) β exp (−βxn ) β 2 Ei (−βxn ) exp (−βxn ) dx = − + + m x 2nx2n 2nxn 2n   m−1 =2 z= n  exp (−βxn ) β exp (−βxn ) β 2 exp (−βxn ) β 3 Ei (−βxn ) exp (−βxn ) dx = − + − − xm 3nx3n 6nx2n 6nxn 6n   m−1 =3 z= n        exp −βx2 exp −βx2 − dx = − βπ erf βx x2 x m

n

110

Hyperbolic Functions

2.411

2.4 Hyperbolic Functions 2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x 

2.411

2.412  1.

 sinhp+1 x coshq−1 x q − 1 + sinh x cosh x dx = sinhp x coshq−2 x dx p+q p+q  sinhp−1 x coshq+1 x p − 1 = − sinhp−2 x coshq x dx p+q p+q  sinhp−1 x coshq+1 x p − 1 = − sinhp−2 x coshq+2 x dx q+1 q+1  sinhp+1 x coshq−1 x q − 1 = − sinhp+2 x coshq−2 x dx p+1 p+1  sinhp+1 x coshq+1 x p + q + 2 = − sinhp+2 x coshq x dx p+1 p+1  sinhp+1 x coshq+1 x p + q + 2 =− + sinhp x coshq+2 x dx q+1 q+1 p

q

⎡ p+1 x sinh ⎣ cosh2n−1 x sinhp x cosh2n x dx = 2n + p

3.

4.

⎤

(2n − 1)(2n − 3) . . . (2n − 2k + 1) cosh2n−2k−1 x⎦ (2n + p − 2)(2n + p − 4) . . . (2n + p − 2k) k=1  (2n − 1)!! + sinhp x dx (2n + p)(2n + p − 2) . . . (p + 2) This formula is applicable for arbitrary real p, except for the following negative even integers: −2, −4, . . . , −2n. If p is a natural number and n = 0, we have



 m−1  2m x 2m sinh(2m − 2k)x 1 2m m k + 2m−1 (−1) TI (543) sinh x dx = (−1) m 22m k 2 2m − 2k k=0

 m 2m + 1 cosh(2m − 2k + 1)x 1  2m+1 k ; sinh x dx = 2m (−1) TI (544) k 2 2m − 2k + 1 k=0 m  m cosh2k+1 x  = (−1)n (−1)k GU (351) (5) k 2k + 1 k=0  sinhp x cosh2n+1 x dx   n  sinhp+1 x 2k n(n − 1) . . . (n − k + 1) cosh2n−2k x 2n = cosh x + 2n + p + 1 (2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1) k=1 This formula is applicable for arbitrary real p, except for the following negative odd integers: −1, −3, . . . , −(2n + 1). +

2.

n−1 

2.414

Powers of hyperbolic functions

2.413  1.

111

⎡ p+1 x cosh ⎣ sinh2n−1 x coshp x sinh2n x dx = 2n + p

⎤ 2n−2k−1 (2n − 1)(2n − 3) . . . (2n − 2k + 1) sinh x ⎦ + (−1)k (2n + p − 2)(2n + p − 4) . . . (2n + p − 2k) k=1  (2n − 1)!! n +(−1) coshp x dx (2n + p)(2n + p − 2) . . . (p + 2) This formula is applicable for arbitrary real p, except for the following negative even integers: −2, −4, . . . , −2n. If p is a natural number and n = 0, we have

 m−1  2m sinh(2m − 2k)x 2m x 1 2m + 2m−1 TI (541) cosh x dx = m 22m k 2 2m − 2k k=0

 m 1  2m + 1 sinh(2m − 2k + 1)x cosh2m+1 x dx = 2m TI (542) k 2 2m − 2k + 1 k=0 m   m sinh2k+1 x = GU (351) (8) k 2k + 1 k=0 ⎡  p+1 x cosh ⎣ sinh2n x coshp x sinh2n+1 x dx = 2n + p + 1 ⎤ n 2n−2k k  2 n(n − 1) . . . (n − k + 1) sinh x ⎦ + (−1)k (2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1) n−1 

2.

3.

4.

k=1

This formula is applicable for arbitrary real p, except for the following negative odd integers: −1, −3, . . . , −(2n + 1). 2.414



1.  2.  3.  4.  5.  6.

1 cosh ax a x 1 sinh2 ax dx = sinh 2ax − 4a 2 1 1 3 cosh 3x = cosh3 x − cosh x sinh3 x dx = − cosh x + 4 12 3 1 1 3 3 1 3 sinh4 x dx = x − sinh 2x + sinh 4x = x − sinh x cosh x + sinh3 x cosh x 8 4 32 8 8 4 5 1 5 sinh5 x dx = cosh x − cosh 3x + cosh 5x 8 48 80 4 1 4 = cosh x + sinh4 x cosh x − cosh3 x 5 5 15 15 3 1 5 sinh6 x dx = − x + sinh 2x − sinh 4x + sinh 6x 16 64 64 192 5 1 5 5 = − x + sinh5 x cosh x − sinh3 x cosh x + sinh x cosh x 16 6 24 16

sinh ax dx =

112

Hyperbolic Functions

2.415

 7. 

8. 9. 10. 11. 12.

sinh7 x dx = − 35 64 cosh x +

7 64

cosh 3x −

7 320

= − 24 35 cosh x +

8 35

cosh3 x −

6 35

1 448

cosh 7x

cosh x sinh4 x +

1 7

cosh x sinh6 x

1 sinh ax a  1 x cosh2 ax dx = + sinh 2ax 2 4a  1 cosh3 x dx = 34 sinh x + 12 sinh 3x = sinh x + 13 sinh3 x  1 cosh4 x dx = 38 x + 14 sinh 2x + 32 sinh 4x = 38 x + 38 sinh x cosh x +  5 1 cosh5 x dx = 58 sinh x + 48 sinh 3x + 80 sinh 5x cosh ax dx =

=  13.

4 5

sinh x + +

15 64

sinh 2x +

=

5 16 x

+

5 16

sinh x cosh x +

4 15

sinh 4x +

sinh x +

7 64

sinh 3x +

7 320

=

24 35

sinh x +

8 35

sinh3 x +

6 35

1 192

sinh 5x +

1 448

cosh(a + b)x cosh(a − b)x + 2(a + b) 2(a − b)

sinh ax cosh ax dx =

1 cosh 2ax 4a

sinh2 x cosh x dx =

1 3

sinh3 x

sinh3 x cosh x dx =

1 4

sinh4 x

sinh4 x cosh x dx =

1 5

sinh5 x

sinh x cosh2 x dx =

1 3

cosh3 x

sinh x cosh5 x

sinh 7x

sinh x cosh4 x +

sinh ax cosh bx dx =

1 6

1 7

sinh x cosh6 x

 5.  6. 

7. 8. 9.

1 x + sinh 4x 8 32    sinh3 x cosh2 x dx = 15 sinh2 x − 23 cosh3 x  1 1 1 x sinh4 x cosh2 x dx = − sinh 2x − sinh 4x + sinh 6x 16 64 64 192 sinh2 x cosh2 x dx = −

sinh x cosh3 x

sinh 6x

 4.

1 4

sinh3 x

sinh x cosh3 x +

5 24

35 64

 3.

3 64

cosh7 x dx =

 2.

cosh4 x sinh x +

5 16 x



1.

1 5

cosh6 x dx = 

14.

2.415

cosh 5x +

2.416

Powers of hyperbolic functions

113

 sinh x cosh3 x dx =

10. 

cosh4 x

1 4

sinh2 x cosh3 x dx =

11.

1 5



  cosh2 x + 23 sinh3 x

3 sinh3 x cosh3 x dx = − 64 cosh 2x +

12.

1 192 4

cosh 6x =

1 48

cosh3 2x −

1 16

cosh 2x

cosh6 x cosh4 x sinh6 x sinh x + = − 6 4 6 4      sinh4 x cosh3 x dx = 17 sinh3 x cosh4 x − 35 cosh2 x − 25 = 17 cosh2 x + 25 sinh5 x  sinh x cosh4 x dx = 15 cosh5 x  1 1 1 x sinh2 x cosh4 x dx = − − sinh 2x + sinh 4x + sinh 6x 16 64 64 192      sinh3 x cosh4 x dx = 17 cosh3 x sinh4 x + 35 sinh2 x − 25 = 17 sinh2 x − 25 cosh5 x  1 1 3x sinh4 x cosh4 x dx = − sinh 4x + sinh 8x 128 128 1024 =

13. 14. 15. 16. 17. 2.416



1.10

⎡ p

p+1

x⎣ sinh x sinh sech2n−1 x dx = 2n − 1 cosh2n x

⎤ (2n − p − 2)(2n − p − 4) . . . (2n − p − 2k) sec h2n−2k−1 x⎦ + (2n − 3)(2n − 5) . . . (2n − 2k − 1) k=1  (2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p) + sinhp x dx (2n − 1)!! " This formula is applicable for arbitrary real p. For sinhp x dx, where p is a natural number, see 2.412 2 and 2.412 3. For n = 0 and p a negative integer, we have for this integral: ⎡  cosh x ⎣ dx = − cosech2m−1 x sinh2m x 2m − 1 ⎤ m−1 k  (m − 1)(m − 2) . . . (m − k) 2 + cosec h2m−2k−1 x⎦ (−1)k−1 · (2m − 3)(2m − 5) . . . (2m − 2k − 1) n−1 

2.

k=1

 3.

⎡ cosh x ⎣ − cosech2m x = 2m sinh2m+1 x dx

+

⎤ (2m − 1)(2m − 3) . . . (2m − 2k + 1) cosec h2m−2k x⎦ (−1)k−1 · 2k (m − 1)(m − 2) . . . (m − k)

m−1  k=1

+(−1)m

(2m − 1)!! x ln tanh (2m)!! 2

114

Hyperbolic Functions

2.417  1.

2.417

⎡ sinhp x sinhp+1 x ⎣ sech2n x dx = 2n cosh2n+1 x

⎤ (2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1) sec h2n−2k x⎦ + 2k (n − 1)(n − 2) . . . (n − k) k=1  (2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) sinhp x dx + 2n n! cosh x This formula is applicable for arbitrary real p. For n = 0 and p integral, we have  m  sinh2m+1 x (−1)m+k dx = sinh2k x + (−1)m ln cosh x cosh x 2k k=1 m  (−1)m+k  m cosh2k x + (−1)m ln cosh x [m ≥ 1] = 2k k n−1 

2.

 3.

k=1

2m

sinh x dx = cosh x



2.418  1.

dx (−1)k cosech2m−2k+2 x + (−1)m ln tanh x = 2m − 2k + 2 x cosh x k=1

2m+1

 (−1)k cosech2m−2k+2 x dx + (−1)m arctan sinh x = 2m 2m − 2k + 1 sinh x cosh x k=1 m

⎡ coshp x coshp+1 x ⎣ cosech2n−1 x dx = − 2n − 1 sinh2n x

⎤ (−1)k (2n − p − 2)(2n − p − 4) . . . (2n − p − 2k) + cosec h2n−2k−1 x⎦ (2n − 3)(2n − 5) . . . (2n − 2k − 1) k=1  (−1)n (2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p) + coshp x dx (2n − 1)!! " This formula is applicable for arbitrary real p. For the integral coshp x dx, where p is a natural number, see 2.413 2 and 2.413 3. If p is a negative integer, we have for this integral:    m−1  2k (m − 1)(m − 2) . . . (m − k) dx sinh x 2m−1 2m−2k−1 sech sech x+ x = 2m − 1 (2m − 3)(2m − 5) . . . (2m − 2k − 1) cosh2m x k=1    m−1  (2m − 1)(2m − 3) . . . (2m − 2k + 1) dx sinh x sech2m x + sech2m−2k x = 2m 2k (m − 1)(m − 2) . . . (m − k) cosh2m+1 x k=1 (2m − 1)!! arctan sinh x + (2m)!! n−1 

2.

3.

sinh2k−1 x + (−1)m arctan (sinh x)

m 

sinh

5.

k=1

2k − 1

[m ≥ 1]

 4.

m  (−1)m+k

2.423

2.419

Powers of hyperbolic functions



1.

115

⎡ coshp x coshp+1 x ⎣ cosech2n x dx = − 2n sinh2n+1 x

⎤ (−1)k (2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1) cosec h2n−2k x⎦ + 2k (n − 1)(n − 2) . . . (n − k) k=1  (−1)n (2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) coshp x dx + 2n n! sinh x This formula is applicable for arbitrary real p. For n = 0 and p an integer  m  x cosh2k−1 x cosh2m x dx = + ln tanh sinh x 2k − 1 2 k=1  m  cosh2m+1 x cosh2k x dx = + ln sinh x sinh x 2k k=1 m   m sinh2k x = + ln sinh x k 2k n−1 

2.

3.

 4.  5. 2.421

k=1

 sech2m−2k+1 x x dx + ln tanh = 2m 2 sinh x cosh x k=1 2m − 2k + 1 m

 sech2m−2k+2 x dx = + ln tanh x sinh x cosh2m+1 x k=1 2m − 2k + 2 m

In formulas 2.421 1 and 2.421 2, s = 1 for m odd and m < 2n + 1; in all other cases, s = 0. GI (351)(11, 13)

 1.10  2.

n 2k−m+1   m−1 x sinh2n+1 x n+k n cosh dx = + s(−1)n+ 2 (−1) m k cosh x 2k − m + 1 k=0 k= m−1 2



n m−1 2

ln cosh x



n  n sinh2k−m+1 x  n cosh2n+1 x dx = + s ln sinh x m−1 sinhm x 2k − m + 1 k 2 k=0 k= m−1 2

2.422 1.

2.



m+n−1  (−1)k+1 m+n−1 dx = tanh2k−2m+1 x 2m − 2k − 1 k sinh2m x cosh2n x k=0

 m+n  (−1)k+1 m + n

m+n dx 2k−2m m x + (−1) ln tanh x = tanh m 2m − 2k k sinh2m+1 x cosh2n+1 x k=0 

k=m

GI (351)(15)

2.423  1.

x 1 cosh x − 1 dx = ln tanh = ln sinh x 2 2 cosh x + 1

116

Hyperbolic Functions

 2.  3.  4.  5.  6.

 7. 

8. 9.

dx = − coth x sinh2 x cosh x 1 x dx =− − ln tanh 3 2 2 2 sinh x 2 sinh x 1 dx cosh x 2 =− + coth x = − coth3 x + coth x 4 3 3 3 sinh x 3 sinh x x dx cosh x 3 cosh x 3 =− + + ln tanh 5 4 2 8 8 2 sinh x 4 sinh x sinh x dx 4 cosh x 4 coth3 x − coth x =− + 6 5 15 5 sinh x 5 sinh x 1 2 = − coth5 x + coth3 x − coth x 5 3

dx 1 x cosh x 5 15 5 ln tanh = − − + − 8 16 2 sinh7 x 6 sinh2 x sinh4 x 4 sinh2 x

3 1 dx = coth x − coth3 x + coth5 x − coth7 x 8 5 7 sinh x  dx = arctan (sinh x) cosh x = arcsin (tanh x) = 2 arctan (ex ) = gd x 

10.  11.  12.  13.  14.

 15.  16. 17.

dx = tanh x cosh2 x sinh x 1 dx = + arctan (sinh x) cosh3 x 2 cosh2 x 2 dx sinh x 2 = + tanh x cosh4 x 3 cosh3 x 3 1 = − tanh3 x + tanh x 3 dx sinh x 3 sinh x 3 = + + arctan (sinh x) 5 4 2 cosh x 4 cosh x 8 cosh x 8 dx 4 sinh x 4 tanh3 x + tanh x = − 6 5 5 cosh x 5 cosh x 15 1 2 = tanh5 x − tanh3 x + tanh x 5 3

1 dx sinh x 5 15 5 arctan (sinh x) = + + + 8 16 cosh7 x 6 cosh2 x cosh4 x 4 cosh2 x

dx 3 1 = − tanh7 x + tanh5 x − tanh3 x + tanh x 8 7 5 cosh x  sinh x dx = ln cosh x cosh x

2.423

2.423

Powers of hyperbolic functions

 18.  19.



20. 21. 22. 23. 24. 25.

26. 27.

28. 29. 30. 31. 32. 33. 34.

sinh2 x dx = sinh x − arctan (sinh x) cosh x sinh3 x 1 dx = sinh2 x − ln cosh x cosh x 2 1 = cosh2 x − ln cosh x 2

1 sinh4 x dx = sinh3 x − sinh x + arctan (sinh x) cosh x 3  sinh x 1 dx = − cosh x cosh2 x  sinh2 x dx = x − tanh x cosh2 x  sinh3 x 1 dx = cosh x + cosh x cosh2 x  1 sinh4 x 3 dx = − x + sinh 2x + tanh x 2 2 4 cosh x  sinh x 1 dx = − cosh3 x 2 cosh2 x 1 = tanh2 x 2  2 sinh x sinh x 1 dx = − + arctan (sinh x) 3 2 cosh x 2 cosh x 2  3 sinh x 1 dx = − tanh2 x + ln cosh x 3 2 cosh x 1 = + ln cosh x 2 cosh2 x  sinh x 3 sinh4 x dx = + sinh x − arctan (sinh x) 2 cosh x 2 cosh3 x  sinh x 1 dx = − cosh4 x 3 cosh3 x  1 sinh2 x dx = tanh3 x 4 3 cosh x  3 sinh x 1 1 + dx = − 4 cosh x cosh x 3 cosh3 x  4 sinh x 1 dx = − tanh3 x − tanh x + x 3 cosh4 x  cosh x dx = ln sinh x sinh x  cosh2 x x dx = cosh x + ln tanh sinh x 2

117

118

Hyperbolic Functions

 35. 

36. 37. 38. 39. 40. 41.

42. 43.

44. 45. 46. 47. 48. 49. 50. 51.

1 cosh3 x dx = cosh2 x + ln sinh x sinh x 2

cosh4 x 1 x dx = cosh3 x + cosh x + ln tanh sinh x 3 2  cosh x 1 dx = − 2 sinh x sinh x  2 cosh x dx = x − coth x sinh2 x  1 cosh3 x dx = sinh x − 2 sinh x sinh x  1 cosh4 x 3 dx = x + sinh 2x − coth x 2 4 sinh2 x  cosh x 1 dx = − sinh3 x 2 sinh2 x 1 = − coth2 x 2  2 cosh x x cosh x dx = − + ln tanh 2 sinh3 x 2 sinh2 x  3 cosh x 1 dx = − + ln sinh x 3 sinh x 2 sinh2 x 1 = − coth2 x + ln sinh x 2  4 x cosh x cosh x 3 dx = − + cosh x + ln tanh 3 2 2 2 sinh x 2 sinh x  1 cosh x dx = − sinh4 x 3 sinh3 x  cosh2 x 1 dx = − coth3 x 3 sinh4 x  3 cosh x 1 1 dx = − − 4 sinh x 3 sinh3 x sinh x  cosh4 x 1 dx = − coth3 x − coth x + x 3 sinh4 x  dx = ln tanh x sinh x cosh x  x dx 1 + ln tanh = 2 cosh x 2 sinh x cosh x  dx 1 = + ln tanh x 3 sinh x cosh x 2 cosh2 x 1 = − tanh2 x + ln tanh x 2

2.423

2.424

Powers of hyperbolic functions

 52.  53.  54.  55.  56.  57.  58.  59.  60.  61.  62.  63.  64. 2.424 1. 2.

1 dx 1 x + = + ln tanh 4 3 cosh x 2 sinh x cosh x 3 cosh x dx 1 − arctan sinh x =− sinh x sinh2 x cosh x dx = −2 coth 2x sinh2 x cosh2 x sinh x 1 3 dx =− − − arctan sinh x sinh2 x cosh3 x 2 cosh2 x sinh x 2 dx 1 8 = − coth 2x sinh2 x cosh4 x 3 sinh x cosh3 x 3 dx 1 =− − ln tanh x sinh3 x cosh x 2 sinh2 x 1 = − coth2 x + ln coth x 2 dx cosh x x 1 3 − =− − ln tanh 3 2 2 cosh x 2 2 sinh x cosh x 2 sinh x dx 2 cosh 2x =− − 2 ln tanh x 3 3 sinh x cosh x sinh2 2x 1 1 = tanh2 x − coth2 x − 2 ln tanh x 2 2 dx 1 x 2 cosh x 5 − =− − − ln tanh 3 4 2 2 cosh x 3 cosh x 2 sinh x 2 2 sinh x cosh x 1 dx 1 − = + arctan sinh x sinh x 3 sinh3 x sinh4 x cosh x 1 8 dx =− + coth 2x 4 2 3 sinh x cosh x 3 cosh x sinh x 3 1 dx 2 sinh x 5 − = + + arctan sinh x 4 3 3 2 sinh x 3 sinh x 2 cosh x 2 sinh x cosh x dx 8 = 8 coth 2x − coth3 2x 4 4 3 sinh x cosh x

 tanhp−1 x + tanhp−2 x dx [p = 1] p−1  n  1 (−1)k−1  n + ln cosh x tanh2n+1 x dx = 2k k cosh2k x k=1 n  tanh2n−2k+2 x =− + ln cosh x 2n − 2k + 2 

tanhp x dx = −

k=1



3.

4.

119

n 

tanh2n−2k+1 x +x 2n − 2k + 1 k=1   cothp−1 x + cothp−2 x dx cothp x dx = − p−1 tanh2n x dx = −

GU (351)(12)

[p = 1]

120

Hyperbolic Functions



2.425

n  1 1 n + ln sinh x 2n k sinh2k x k=1 n  coth2n−2k+2 x =− + ln sinh x 2n − 2k + 2

coth2n+1 x dx = −

5.

k=1

 coth2n x dx = −

6.

n  k=1

coth2n−2k+1 x +x 2n − 2k + 1

GU (351)(14)

For formulas containing powers of tanh x and coth x equal to n = 1, 2, 3, 4, see 2.423 17, 2.423 22, 2.423 27, 2.423 32, 2.423 33, 2.423 38, 2.423 43, 2.423 48. Powers of hyperbolic functions and hyperbolic functions of linear functions of the argument 2.425



1.

sinh(ax + b) sinh(cx + d) dx =

 2.

sinh(ax + b) cosh(cx + d) dx =

 3.

4. 5. 6. 2.426 1.

cosh(ax + b) cosh(cx + d) dx =

1 sinh[(a + c)x + b + d] 2(a + c) 1 − sinh[(a − c)x + b − d] 2(a − c)  2  a = c2

GU (352)(2a)

1 cosh[(a + c)x + b + d] 2(a + c) 1 + cosh[(a − c)x + b − d] 2(a − c)  2  a = c2

GU (352)(2c)

1 sinh[(a + c)x + b + d] 2(a + c) 1 + sinh[(a − c)x + b − d] 2(a − c)  2  a = c2

GU (352)(2b)

When a = c:  1 x sinh(2ax + b + d) sinh(ax + b) sinh(ax + d) dx = − cosh(b − d) + 2 4a  1 x cosh(2ax + b + d) sinh(ax + b) cosh(ax + d) dx = sinh(b − d) + 2 4a  1 x sinh(2ax + b + d) cosh(ax + b) cosh(ax + d) dx = cosh(b − d) + 2 4a  sinh ax sinh bx sinh cx dx =

GU (352)(3a) GU (352)(3c) GU (352)(3b)

cosh(a + b + c)x cosh(−a + b + c)x − 4(a + b + c) 4(−a + b + c) cosh(a − b + c)x cosh(a + b − c)x − − 4(a − b + c) 4(a + b − c) GU (352)(4a)

2.428

Powers of hyperbolic functions

 2.

sinh ax sinh bx cosh cx dx =

sinh ax cosh bx cosh cx dx =

cosh(a + b + c)x cosh(−a + b + c)x − 4(a + b + c) 4(−a + b + c) cosh(a − b + c)x cosh(a + b − c)x + + 4(a − b + c) 4(a + b − c) GU (352)(4c)

 4.

sinh(a + b + c)x sinh(−a + b + c)x − 4(a + b + c) 4(−a + b + c) sinh(a − b + c)x sinh(a + b − c)x − + 4(a − b + c) 4(a + b − c) GU (352)(4b)

 3.

121

cosh ax cosh bx cosh cx dx =

sinh(a + b + c)x sinh(−a + b + c)x + 4(a + b + c) 4(−a + b + c) sinh(a − b + c)x sinh(a + b − c)x + + 4(a − b + c) 4(a + b − c) GU (352)(4d)

2.427



1 sinh x sinh ax dx = p+a p

1. 

sinhp x sinh(2n + 1)x dx =

2.

 sinhp x sinh 2nx dx =

3.



 p−1 x cosh(a − 1)x dx sinh px cosh ax − p sinh Γ(p + 1)   Γ ⎡p+3 2 +n   n−1  Γ p+1 + n − 2k 2 sinhp−2k x cosh(2n − 2k + 1)x ×⎣ 22k+1 Γ (p − 2k + 1) k=0 ⎤  p−1  Γ 2 + n − 2k − 2k+2 sinhp−2k−1 x sinh(2n − 2k)x⎦ 2 Γ(p − 2k)    Γ p+3 2 −n + 2n sinhp−2n x sinh x dx 2 Γ(p + 1 − 2n) [p is not a negative integer]

Γ(p + 1) p  Γ 2 +⎡n + 1   n−1  Γ p2 + n − 2k ⎣ sinhp−2k x cosh(2n − 2k)x × 22k+1 Γ(p − 2k + 1) k=0 ⎤   Γ p2 + n − 2k − 1 − sinhp−2k−1 x sinh(2n − 2k − 1)x⎦ 22k+2 Γ(p − 2k) [p is not a negative integer]

2.428 1.



1 sinh x cosh ax dx = p+a p



 p p−1 x sinh(a − 1)x dx sinh x sinh ax − p sinh

GU (352)(5)a

122

Hyperbolic Functions

 sinhp x cosh(2n + 1)x dx =

2.

Γ(p + 1)   Γ ⎧p+3 2 +n ⎡   ⎨ n−1  Γ p+1 + n − 2k 2 ⎣ sinhp−2k x sinh(2n − 2k + 1)x × ⎩ 22k+1 Γ(p − 2k + 1) k=0 ⎤  p−1  Γ 2 + n − 2k − 2k+2 sinhp−2k−1 x cosh(2n − 2k)x⎦ 2 Γ(p − 2k) ⎫    ⎬ Γ p+3 − n 2 sinhp−2n x cosh x dx + 2n ⎭ 2 Γ(p + 1 − 2n) [p is not a negative integer]

 sinhp x cosh 2nx dx =

3.

2.429

Γ(p + 1) p  Γ ⎧2 + n⎡+ 1   ⎨n−1  Γ p2 + n − 2k ⎣ sinhp−2k x sinh(2n − 2k)x × ⎩ 22k+1 Γ(p − 2k + 1) k=0 ⎤   Γ p2 + n − 2k − 1 − sinhp−2k−1 x cosh(2n − 2k − 1)x⎦ 22k+2 Γ(p − 2k) ⎫ p   ⎬ Γ −n+1 + 2n 2 sinhp−2n x dx ⎭ 2 Γ(p + 1 − 2n) [p is not a negative integer]

2.429

 coshp x sinh ax dx =

1.  2.

1 p+a

GU (352)(6)a



 coshp x cosh ax + p coshp−1 x sinh(a − 1)x dx ⎡

 p+1  n−1 Γ(p + 1) ⎣  Γ 2 + n − k  cosh x sinh(2n + 1)x dx =  p+3 coshp−k x cosh(2n − k + 1)x 2k+1 Γ(p − k + 1) Γ 2 +n k=0 ⎤  p+3   Γ 2 + n coshp−n x sinh(n + 1)x dx⎦ 2 Γ(p − n + 1) p

[p is not a negative integer] p   n−1 Γ(p + 1) ⎣  Γ 2 + n − k p   coshp−k x cosh(2n − k)x cosh x sinh 2nx dx = 2k+1 Γ(p − k + 1) Γ p2 + n + 1 k=0 ⎤ p   Γ 2 +1 + n coshp−n x sinh nx dx⎦ 2 Γ(p − n + 1) ⎡

3.

[p is not a negative integer]

GU (352)(7)a

2.433

2.431

Powers of hyperbolic functions

 coshp x cosh ax dx =

1.

1 p+a

 coshp x cosh(2n + 1)x dx =

2.

 3.



 coshp x sinh ax + p coshp−1 x cosh(a − 1)x dx ⎡

 p+1  n−1 Γ(p + 1) ⎣  Γ 2 + n − k  p+3  coshp−k x sinh(2n − k + 1)x 2k+1 Γ(p − k + 1) Γ 2 +n k=0 ⎤  p+3   Γ 2 + n coshp−n x cosh(n + 1)x dx⎦ 2 Γ(p − n + 1)

[p is not a negative integer]  n−1 Γ(p + 1) ⎣  Γ 2 + n − k p  coshp−k x sinh(2n − k)x cosh x cosh 2nx dx =  p 2k+1 Γ(p − k + 1) Γ 2 +n+1 k=0 ⎤ p  Γ 2 +1 coshp−n x cosh nx dx⎦ + n 2 Γ(p − n + 1) ⎡

p

[p is not a negative integer]

2.432 1. 2. 3. 4. 2.433

123

GU (352)(8)a



1 sinhn x sinh nx n  1 sinh(n + 1)x coshn−1 x dx = coshn x cosh nx n  1 cosh(n + 1)x sinhn−1 x dx = sinhn x cosh nx n  1 cosh(n + 1)x coshn−1 x dx = coshn x sinh nx n sinh(n + 1)x sinhn−1 x dx =



1.

 sinh(2n − 2k)x sinh(2n + 1)x dx = 2 +x sinh x 2n − 2k n−1

k=0

 2.  3.

sinh 2nx dx = 2 sinh x

n−1  k=0

sinh(2n − 2k − 1)x 2n − 2k − 1

GU (352)(5d)

 cosh(2n − 2k)x cosh(2n + 1)x dx = 2 + ln sinh x sinh x 2n − 2k n−1

k=0

 4.  5.

cosh 2nx dx = 2 sinh x

n−1  k=0

x cosh(2n − 2k − 1)x + ln tanh 2n − 2k − 1 2

 cosh(2n − 2k)x sinh(2n + 1)x dx = 2 + (−1)n ln cosh x (−1)k cosh x 2n − 2k n−1

k=0

GU (352)(6d)

124

Hyperbolic Functions

 6.

 sinh 2nx cosh(2n − 2k − 1)x dx = 2 (−1)k cosh x 2n − 2k − 1

2.433

n−1

GU (352)(7d)

k=0

 7.

 sinh(2n − 2k)x cosh(2n + 1)x dx = 2 + (−1)n x (−1)k cosh x 2n − 2k n−1

k=0

 8.  9.

10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

cosh 2nx dx = 2 cosh x

n−1 

(−1)k

k=0

sinh(2n − 2k − 1)x + (−1)n arcsin (tanh x) 2n − 2k − 1

2 sinh 2x dx = − sinhn x (n − 2) sinhn−2 x

For n = 2:  sinh 2x dx = 2 ln sinh x sinh2 x  sinh 2x dx 2 = coshn x (2 − n) coshn−2 x For n = 2:  sinh 2x dx = 2 ln cosh x cosh2 x  cosh 2x x dx = 2 cosh x + ln tanh sinh x 2  cosh 2x dx = − coth x + 2x sinh2 x  x cosh x 3 cosh 2x dx = − + ln tanh 2 sinh3 x 2 sinh2 x 2  cosh 2x dx = 2 sinh x − arcsin (tanh x) cosh x  cosh 2x dx = − tanh x + 2x cosh2 x  sinh x 3 cosh 2x dx = − + arcsin (tanh x) cosh3 x 2 cosh2 x 2  sinh 3x dx = x + sinh 2x sinh x  sinh 3x x dx = 3 ln tanh + 4 cosh x 2 2 sinh x  sinh 3x dx = −3 coth x + 4x sinh3 x  4 1 sinh 3x dx = − n−3 coshn x (3 − n) cosh x (1 − n) coshn−1 x For n = 1 and n = 3:  sinh 3x dx = 2 sinh2 x − ln cosh x cosh x

GU (352)(8d)

2.442

Rational functions of hyperbolic functions



24. 25.

26. 27. 28. 29. 30.

sinh 3x 1 dx = + 4 ln cosh x 3 cosh x 2 cosh2 x  1 4 cosh 3x + dx = n−3 sinhn x (3 − n) sinh x (1 − n) sinhn−1 x For n = 1 and n = 3:  cosh 3x dx = 2 sinh2 x + ln sinh x sinh x  cosh 3x 1 dx = − + 4 ln sinh x 3 sinh x 2 sinh2 x  cosh 3x dx = sinh 2x − x cosh x  cosh 3x dx = 4 sinh x − 3 arcsin (tanh x) cosh2 x  cosh 3x dx = 4x − 3 tanh x cosh3 x

2.44–2.45 Rational functions of hyperbolic functions 2.441



1.

2. 3.

2.442 1.

2.

cosh x A + B sinh x aB − bA · n dx = (n − 1) (a2 + b2 ) (a + b sinh x)n−1 (a + b sinh x)  1 (n − 1)(aA + bB) + (n − 2)(aB − bA) sinh x + dx n−1 (n − 1) (a2 + b2 ) (a + b sinh x)

For n = 1:   B aB − bA A + B sinh x dx dx = x − a + b sinh x b b a + b sinh x √  a tanh x2 − b + a2 + b2 dx 1 √ =√ ln a + b sinh x a2 + b2 a tanh x2 − b − a2 + b2 a tanh x2 − b 2 =√ arctanh √ a2 + b 2 a2 + b 2 

(see 2.441 3)

 A + B cosh x B dx dx = − + A n n n−1 (a + b sinh x) (a + b sinh x) (n − 1)b (a + b sinh x)

For n = 1:   B dx A + B cosh x dx = ln (a + b sinh x) + A a + b sinh x b a + b sinh x (see 2.441 3)

125

126

2.443

Hyperbolic Functions



1.

2. 3.

2.444 1.

2.11 2.445 1.

2.

2.443

A + B cosh x sinh x aB − bA · n dx = n−1 2 2 (n − 1) (a − b ) (a + b cosh x) (a + b cosh x) 1 (n − 1)(aA − bB) + (n − 2)(aB − bA) cosh x + dx n−1 (n − 1) (a2 − b2 ) (a + b cosh x)

For n = 1:   B aB − bA A + B cosh x dx dx = x − a + b cosh x b b a + b cosh x  1 dx b + a cosh x =√ arcsin a + b cosh x a + b cosh x b 2 − a2 b + a cosh x 1 arcsin = −√ a + b cosh x b 2 − a2 √ x a2 − b2 tanh a + b + 1 2 =√ ln √ a2 − b2 a + b − a2 − b2 tanh x 2

(see 2.443 3)  2 b > a2 ,

x<0

 2 b > a2 ,

x>0

 

  2 a > b2

  dx x+a x−a = cosech a ln cosh − ln cosh cosh a + cosh x 2 2  a x = 2 cosech a arctanh tanh tanh 2 2   x a dx = 2 cosec a arctan tanh tan cos a + cosh x 2 2 



B B sinh x dx = − n (a + b cosh x) (n − 1)b(a + b cosh x)n−1 For n = 1:  B B sinh x dx = ln (a + b cosh x) a + b cosh x b

[n = 1]

(see 2.443 3)

In evaluating definite integrals by use of formulas 2.441–2.443 and 2.445, one may not take the integral over points at which the integrand becomes infinite, that is, over the points  a x = arcsinh − b in formulas 2.441 or 2.442 or over the points  a x = arccosh − b in formulas 2.443 or 2.445. Formulas 2.443 are not applicable for a2 = b2 . Instead, we may use the following formulas in these cases:

2.449

2.446

Rational functions of hyperbolic functions



1.

127

A + B cosh x n dx (ε + cosh x) =

n−1  (2n − 2k − 3)!! (n − 1)! B sinh x n B sinh x + εA + n n−1 (2n − 1)!! (n − k − 1)! (1 − n) (ε + cosh x) k=0

×

2. 2.447

εh (ε + cosh x)

n−k

[ε = ±1,

For n = 1:  cosh x − ε A + B cosh x dx = Bx + (εA − B) ε + cosh x sinh x

b sinh x dx a ln cosh x + arctanh bx a a cosh x + b sinh x = a2  − b2 a bx − a ln sinh x + arctanh b = b 2 − a2 For a = b = 1:  x 1 sinh x dx = + e−2x cosh x + sinh x 2 4 For a = −b = 1:  x 1 sinh x dx = − + e2x cosh x − sinh x 2 4

n > 1]

[ε = ±1]



1.

2.

3. 2.448



1.

2.

3. 2.449 1.6

  ax − b ln cosh x + arctanh ab cosh x dx = a cosh x + b sinh x a2 −b2  −ax + b ln sinh x + arctanh ab = b 2 − a2

[a > |b|] [b > |a|]

MZ 215

MZ 215

[a > |b|] [b > |a|]

For a = b = 1:  x 1 cosh x dx = − e−2x cosh x + sinh x 2 4 For a = −b = 1:  x 1 cosh x dx = + e2x cosh x − sinh x 2 4 

1 dx n =  n 2 (a cosh x + b sinh x) (a − b2 )

MZ 214, 215



dx

b sinhn x + arctanh a  1 dx  = n a (b2 − a2 ) coshn x + arctanh b

[a > |b|] [b > |a|]

128

2.

3.

4. 2.451

Hyperbolic Functions

For n = 1: !

!  ! b !! 1 dx ! =√ arctan !sinh x + arctanh a cosh x + b sinh x a ! a2 − b 2 ! a !! ! x + arctanh 1 b! =√ ln !tanh ! 2 b 2 − a2 !





3.

[a > |b|] [b > |a|]

For a = b = 1:  ax = −e−x = sinh x − cosh x cosh x + sinh x For a = −b = 1:  dx = ex = sinh x + cosh x cosh x − sinh x

1.

2.

2.451

MZ 214

A + B cosh x + C sinh x n dx (a + b cosh x + c sinh x) Bc − Cb + (Ac − Ca) cosh x + (Ab − Ba) sinh x 1 = n−1 + (n − 1) (a2 − b2 + c2 ) (1 − n) (a2 − b2 + c2 ) (a + b cosh x + c sinh x)  (n − 1)(Aa − Bb + Cc) − (n − 2)(Ab − Ba) cosh x − (n − 2)(Ac − Ca) sinh x × dx n−1 (a + b cosh x + c sinh x)   2 a + c2 = b2   A n(Bb − Cc) Bc − Cb − Ca cosh x − Ba sinh x (n − 1)! = + (c cosh x + b sinh x) n + 2 a (n − 1)a (2n − 1)!! (n − 1)a (a + b cosh x + c sinh x) n−1  (2n − 2k − 3)!! 1 × (n − k − 1)!ak (a + b cosh x + c sinh x)n−k k=0  2  a + c2 = b 2

Cb − Bc A + B cosh x + C sinh x dx = 2 ln (a + b cosh x + c sinh x) a + b cosh x + c sinh x b − c2

 Bb − Cc dx Bb − Cc x + A − a + 2 2 2 2 b −c b − c  a + b cosh x + c sinh x (see 2.451 4) b2 = c2    A (B ∓ C) b C ∓B A + B cosh x + C sinh x dx = (cosh x ∓ sinh x) + − x a + b cosh x ± b sinh x a 2a2 2a  C ±B A (C ∓ B) b ± − + ln (a + b cosh x ± b sinh x) 2b a 2a2 [ab = 0]

2.452

Rational functions of hyperbolic functions

 4.

dx a + b cosh x + c sinh x (b − a) tanh x2 + c 2 arctan √ =√ b 2 − a2 − c 2 b 2 − a2 − c 2 √ x (a − b) tanh a2 − b 2 + c 2 1 2 −c+ √ =√ ln x a2 − b2 + c2 (a − b) tanh 2 − c − a2 − b2 + c2 x 1  = ln a + c tanh c 2 2 = (a − b) tanh x2 + c

129

 

b2 > a2 + c2 and a = b b2 < a2 + c2 and a = b

 

[a = b and c = 0]  2  b = a2 + c 2 GU (351)(18)

2.452 1.



A + B cosh x + C sinh x dx (a1 + b1 cosh x + c1 sinh x) (a2 + b2 cosh  x + c2 sinh x)  a1 + b1 cosh x + c1 sinh x dx dx = A0 ln + A1 + A2 a2 + b2 cosh x + c2 sinh x a1 + b1 cosh x + c1 sinh x a2 + b2 cosh x + c2 sinh x where

GU (351)(19)

! ! !a1 b1 c1 ! ! ! !A B C! ! ! !a2 b2 c2 ! !2 ! ! ! !b1 c1 !2 !c1 b1 !! ! ! +! − !! b2 ! b 2 c2 ! c2

A0 = ! !2 !a1 a1 !! ! !a2 a2 ! ! ! ! a1 ! !! ! ! b 1 ! ! c1 ! ! !! C B ! ! C A ! ! B A !! !! ! ! ! ! !! !!c2 b2 ! !c2 a2 ! !b2 a2 !! ! ! ! a2 b2 c2 ! A2 = ! ! ! ! , ! ! !a1 b1 !2 !b1 c1 !2 !c1 a1 !2 ! ! +! ! −! ! !a2 b2 ! !b2 c2 ! !c2 a2 !  2.

! ! ! ! a1 !! ! ! b1 ! ! c1 !! !!b1 c1 ! !c1 a1 ! !a1 b1 !! !! ! ! ! ! !! !! B C ! ! C A ! ! A B !! ! ! ! a2 b2 c2 ! A1 = ! ! ! ! , ! ! !a1 b1 !2 !b1 c1 !2 !c1 a1 !2 ! ! +! ! −! ! ! a2 b 2 ! ! b 2 c2 ! ! c 2 a2 !

%! ! a1 ! ! a2

!2 ! !b b1 !! + !! 1 ! b2 b2

!2 ! !c c1 !!

= !! 1 ! c2 c2

!2 & a1 !! . a2 !

A cosh2 x + 2B sinh x cosh x + C sinh2 x dx a cosh2 x + 2b sinh x cosh x + c sinh2 x ⎧ ⎨ 1 [4Bb − (A + C)(a + c)]x = 2 4b − (a + c)2 ⎩   + [(A + C)b − B(a + c)] ln a cosh2 x + 2b sinh x cosh x + c sinh2 x ⎫ ⎬   + 2(A − C)b2 − 2Bb(a − c) + (Ca − Ac)(a + c) f (x) ⎭

130

Hyperbolic Functions

2.453

where

√ 1 c tanh x + b − b2 − ac √ f (x) = √ ln 2 b2 − ac c tanh x + b + b2 − ac c tanh x + b 1 arctan √ =√ ac − b2 ac − b2 1 =− c tanh x + b 2.453



1.

 2.

GU (351)(24)



 b2 > ac



 b2 < ac



b2 = ac



   ! dx 1 x !! (A + B sinh x) dx ! = A ln !tanh ! + (aB − bA) sinh x (a + b sinh x) a 2 a + b sinh x (see 2.441 3) ! !

 ! ! ! a + b cosh x ! dx x! A (A + B sinh x) dx ! ! ! +B = 2 a ln !tanh ! + b ln ! sinh x (a + b cosh x) a − b2 2 sinh x ! a + b cosh x (see 2.443 3) 2

3. 4. 2.454 1. 2.

2

For a = b = 1: !

 A (A + B sinh x) dx x !! 1 x ! 2 x = ln !tanh ! − tanh + B tanh sinh x (1 + cosh x) 2 2 2 2 2

 ! ! (A + B sinh x) dx A x! 1 x x ! = − ln !coth ! + coth2 + B coth sinh x (1 − cosh x) 2 2 2 2 2 ! !  ! a + b sinh x ! 1 (A + B sinh x) dx ! ! = 2 (Aa + Bb) arctan (sinh x) + (Ab − Ba) ln ! cosh x (a + b sinh x) a + b2 cosh x ! ! !

  ! ! sinh x ! dx (A + B cosh x) dx 1 x !! ! ! ! − Ab = A ln !tanh ! + B ln ! sinh x (a + b sinh x) a 2 a + b sinh x ! a + b sinh x 

(see 2.441 3) 2.455 1.



! !  ! ! a + b cosh x ! x !! (A + B cosh x) dx 1 ! ! ! = 2 (Aa + Bb) ln !tanh ! + (Ab − Ba) ln ! sinh x (a + b cosh x) a − b2 2 sinh x !

For a2 = b2 = 1:  x !! A − B x A + B !! (A + B cosh x) dx = ln !tanh ! − tanh2 2. sinh x (1 + cosh x) 2 2 4 2  A+B x x A−B (A + B cosh x) dx = coth2 − ln coth 3. sinh x (1 − cosh x) 4 2 2 2 ! !    ! a + b sinh x ! dx (A + B cosh x) dx A ! +B ! 2.456 = 2 a arctan (sinh x) + b ln ! cosh x ! cosh x (a + b sinh x) a + b2 a + b sinh x (see 2.441 3)

2.459

2.457

Rational functions of hyperbolic functions



1.

131

   1 (A + B cosh x) dx dx = A arctan sinh x − (Ab − Ba) cosh x (a + b cosh x) a a + b cosh x (see 2.443 3)

2.458



1.

dx a + b sinh2 x

 b = − 1 tanh x arctan a a(b − a)   1 b = arctanh 1 − tanh x a a(a − b)   1 b = arccoth 1 − tanh x a a(a − b) 1





b >1 a



  b b a 0 < < 1 or < 0 and sinh2 x < − a a b   b a < 0 and sinh2 x > − a b MZ 195

 2.

dx a + b cosh2 x . 

1 b = arctan − 1+ coth x a −a(a + b)   1 b = arctanh 1 + coth x a a(a + b)   b 1 arccoth 1 + coth x = a a(a + b)



b < −1 a



  b a −1 < < 0 and cosh2 x > − a b   b b a 2 > 0 or − 1 < < 0 and cosh x < − a a b MZ 202

2

3. 4.

5. 6. 2.459 1.

2

For a = b = 1:  dx = tanh x 1 + sinh2 x  √ dx 1 2 tanh x = √ arctanh 2 1 − sinh x 2 √ 1 2 tanh x = √ arccoth 2  √ dx 1 2 coth x = √ arccoth 2 1 + cosh x 2  dx = coth x 1 − cosh2 x 

 

sinh2 x < 1 sinh2 x > 1

 

   b sinh x cosh x 1 dx = + (b − 2a)  2 2a(b − a) a + b sinh2 x a + b sinh2 x a + b sinh2 x dx

(see 2.458 1)

MZ 196

132

Hyperbolic Functions

 2.

 3.

2.461

   1 dx b sinh x cosh x + (2a + b) −  2 = 2a(a + b) a + b cosh2 x a + b cosh2 x a + b cosh2 x dx

⎡

(see 2.458 2)

MZ 203





p tanh x 2 1 ⎣ 3 3 2 3 − 2 + 4 arctan (p tanh x) + 3 − 2 − 4  3 = 3 2 8pa p p p p 1 + p2 tanh2 x a + b sinh x ⎤

2p tanh x 2 1 + 1 + 2 − 2 tanh2 x  2 ⎦ p p 1 + p2 tanh2 x   b 2 p = −1>0 a ⎡



1 ⎣ q tanh x 2 3 3 2 = 3 + 2 + 4 arctanh (q tanh x) + 3 + 2 − 4 8qa3 q q q q 1 − q 2 tanh2 x ⎤

2q tanh x 1 2 + 1 − 2 + 2 tanh2 x  2 ⎦ q q 1 − q 2 tanh2 x   b 2 q =1− >0 a dx

MZ 196

⎡

4.





 p coth x 2 dx 1 ⎣ 3 3 2 3 − 2 + 4 arctan (p coth x) + 3 − 2 − 4  3 = 3 2 8pa p p p p 1 + p2 coth2 x a + b cosh x ⎤

2p coth x 2 1 + 1 + 2 − 2 coth2 x  2 ⎦ p p 1 + p2 coth2 x   b p2 = −1 − > 0 a ⎡



1 ⎣ q coth x 2 3 2 3 = 3 + 2 + 4 ϕ(x)∗ + 3 + 2 − 4 3 8qa q q q q 1 − q 2 coth2 x ⎤

2q coth x 1 2 + 1 − 2 + 2 coth2 x  2 ⎦ q q 1 − q 2 coth2 x   b q2 = 1 + > 0 a

2.46 Algebraic functions of hyperbolic functions 2.461  √ √ √ tanh x dx = arctanh tanh x − arctan tanh x 1. ∗ In 2.459.4, if b < 0 and cosh2 x > − a , then ϕ(x) = arctanh (q coth x). If a b ϕ(x) = arccoth (q coth x).

MZ 221 b a

< 0, but cosh2 x < − ab , or if

b a

> 0, then

2.462

2. 2.462

Algebraic functions of hyperbolic functions

 √



1.

coth x dx = arccoth

133

√ √ coth x − arctan coth x

  cosh x sinh x dx  = arcsinh √ = ln cosh x + a2 + sinh2 x a2 − 1 a2 + sinh2 x   cosh x = ln cosh x + a2 + sinh2 x = arccosh √ 1 − a2 = ln cosh x

MZ 222

 2  a >1  2  a <1  2  a =1



2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

  cosh x sinh x dx  = arcsin √ sinh2 x < a2 2 2 2 a +1 a − sinh x    sinh x dx cosh x  = arccosh √ = ln cosh x + sinh2 x − a2 a2 + 1 sinh2 x − a2   sinh2 x > a2    sinh x cosh x dx  = ln sinh x + a2 + sinh2 x = arcsinh a a2 + sinh2 x    sinh x cosh x dx  sinh2 x < a2 = arcsin 2 a a2 − sinh x    cosh x dx sinh x  = ln sinh x + sinh2 x − a2 = arccosh a sinh2 x − a2   sinh2 x > a2    cosh x sinh x dx  = arcsinh = ln cosh x + a2 + cosh2 x a a2 + cosh2 x    cosh x sinh x dx  cosh2 x < a2 = arcsin a a2 − cosh2 x    sinh x dx cosh x  = ln cosh x + cosh2 x − a2 = arccosh a cosh2 x − a2   cosh2 x > a2    sinh x cosh x dx  = arcsinh √ = ln sinh x + a2 + cosh2 x a2 + 1 a2 + cosh2 x    sinh x cosh x dx  = arcsin √ cosh2 x < a2 2 2 a −1 a2 − cosh x    2 cosh x dx sinh x  = arccosh √ a >1 2 2 2 a −1 cosh x − a  2  = ln sinh x a =1

MZ 199

MZ 215, 216

MZ 206

134

Hyperbolic Functions

 √ coth x dx b √ = 2 a arccoth 1 + sinh x a a + b sinh x  √ b = 2 a arctanh 1 + sinh x a .

√ b = 2 −a arctanh − 1 + sinh x a   √ tanh x dx b √ = 2 a arccoth 1 + cosh x a a + b cosh x  √ b = 2 a arctanh 1 + cosh x a .

√ b = 2 −a arctanh − 1 + cosh x a

2.463

 13.

14.

[b sinh x > 0,

a > 0]

[b sinh x < 0,

a > 0]

a<0

[b cosh x > 0,

a > 0]

[b cosh x < 0,

a > 0]

[a < 0] MZ 220, 221

2.463  √ sinh x a + b cosh x dx 1. p + q cosh x . =2 . =2 . =2  2.

aq − bp arccoth q aq − bp arctanh q bp − aq arctanh q

√ cosh x a + b sinh x dx p + q sinh x . =2 . =2 . =2

. .

q (a + b cosh x) aq − bp

.

q (a + b cosh x) aq − bp q (a + b cosh x) bp − aq

 b cosh x > 0,

aq − bp >0 q

 b cosh x < 0, 

aq − bp <0 q



aq − bp >0 q

 

MZ 220

aq − bp arccoth q aq − bp arctanh q bp − aq arctanh q

. . .

q (a + b sinh x) aq − bp q (a + b sinh x) aq − bp q (a + b sinh x) bp − aq

 b sinh x > 0,

aq − bp >0 q

 b sinh x < 0, 

aq − bp <0 q



aq − bp >0 q

 

MZ 221

2.464   dx dx  =  = F (arcsin (tanh x) , k) 1. 2 2 2  k + k cosh x 1 + k  2 sinh2 x [x > 0]



  1 dx dx  2. =  = F arcsin ,k cosh x cosh2 x − k 2 sinh2 x + k  2 [x > 0]

BY (295.00)(295.10)

BY (295.40)(295.30)

2.464

Algebraic functions of hyperbolic functions

 3.





tanh x  = F arcsin ,k k 1 − k  2 cosh2 x

  1 0 < x < arccosh  k

dx

Notation: In 2.464 4–2.464 8, we set α = arccos  4.

5.

 √ 

6.  7. 8.

1 dx √ F (α, r) = 2a sinh 2ax 1 1 [F (α, r) − 2 E (α, r)] + sinh 2ax dx = 2a a

1 1 − sinh 2ax ,r= √ 1 + sinh 2ax 2

11. 12. 13. 14. 15.

<

  sinh 2ax 1 + sinh2 2ax BY (296.53)

1 + sinh 2ax

BY (296.51)

2

1 (1 − sinh 2ax) dx √ = [2 E (α, r) − F (α, r)] 2 2a (1 + sinh 2ax) sinh 2ax  √ sinh 2ax dx 1 2 = 4a [F (α, r) − E (α, r)] (1 + sinh 2ax) 

BY (296.55)

BY (296.54)

1 cosh 2ax − 1 ,r= √ cosh 2ax 2

1 dx √ = √ F (α, r) a 2 cosh 2ax  √ sinh 2ax 1 cosh 2ax dx = √ [F (α, r) − 2 E (α, r)] + √ a 2 a cosh 2ax  1 dx √ = √ [2 E (α, r) − F (α, r)] 3 a 2 cosh 2ax  tanh 2ax dx 1 √ = √ F (α, r) + √ 5 3 2a 3a cosh 2ax cosh 2ax √  √ 2 1 sinh2 2ax dx √ F (α, r) + sinh 2ax cosh 2ax =− 3a 3a cosh 2ax √  2 2 tanh 2ax tanh 2ax dx √ F (α, r) − √ = 3a cosh 2ax 3a cosh 2ax √    1 cosh 2ax dx = √ Π α, p2 , r p2 + (1 − p2 ) cosh 2ax a 2 Notation: In 2.464 16–2.464 20, we set: √ a2 + b2 − a − b sinh x , α = arccos √ a2 + b2 + a + b sinh x . √ + a + a2 + b 2 √ r= a > 0, b > 0, 2 a2 + b 2 

16.

[ax > 0]

1 cosh2 2ax dx = E (α, r) 2√ 2a (1 + sinh 2ax) sinh 2ax



10.

BY (295.20)

BY (296.50)

Notation: In 2.464 9–2.464 15, we set α = arcsin 9.

135

dx 1 √ F (α, r) = √ 4 2 a + b sinh x a + b2

[x = 0]: BY (296.00) BY (296.03) BY (296.04) BY (296.04)

BY (296.07)

BY (296.05)

BY (296.02)

x > − arcsinh

a, b BY (298.00)

136

Hyperbolic Functions

√ 2b cosh x a + b sinh x a + b sinh x dx = + [F (α, r) − 2 E (α, r)] + √ a2 + b2 + a + b sinh x √  √  a + b sinh x a2 + b 2 − a 4 2 + b2 E (α, r) − √ a F (α, r) dx = 4 2 2 cosh2 x √ 2 a +b √ √ a2 + b2 − a − b sinh x a + b sinh x a + a2 + b 2 ·√ − · b cosh x a2 + b2 + a + b sinh x 

17. 18.

2.464

√

 4

a2

b2

BY (298.02)

BY (298.03)

 19.

20.

2

1 cosh x dx = √ E (α, r) √ 2 √ 2 4 a2 + b 2 2 2 b a + b + a + b sinh x a + b sinh x √  a + b sinh x dx 1 √  E (α, r) √ 2 = − √ 4 2 2 a +b a2 + b 2 − a a2 + b2 − a − b sinh x √ b cosh x a + b sinh x +√ · 2 2 2 a + b − a a + b2 − (a + b sinh x)2 

21. 22. 23. 24. 25.11

27.

28. 29.

BY (298.04)

 a−b x ,r= [0 < b < a, x > 0]: Notation: In 2.464 21–2.464 31, we set α = arcsin tanh 2 a+b  dx 2 √ BY (297.25) =√ F (α, r) a + b cosh x a+b  √ √ x√ a + b cosh x dx = 2 a + b [F (α, r) − E (α, r)] + 2 tanh a + b cosh x BY (297.29) 2 √  2 x√ cosh x dx 2 2 a+b √ E (α, r) + tanh a + b cosh x BY (297.33) =√ F (α, r) − b b 2 a + b cosh x a+b √  tanh2 x2 2 a+b √ [F (α, r) − E (α, r)] BY (297.28) dx = a−b a + b cosh x √ √  tanh4 x2 sinh x2 a + b cosh x 2 2 a+b √ [(3a + b) F (α, r) − 4a E (α, r)] + dx = x 3(a − b)2 3(a − b) a + b cosh x cosh3 2 

26.

BY (298.01)

BY (297.28)

2 +

, √ x √ tanh a + b cosh x − a + b E (α, r) 2

cosh x − 1 √ dx = b a + b cosh x √  2 (cosh x − 1) 4 a+b √ [(a + 3b) E (α, r) − b F (α, r)] dx = 3b2+ a + b cosh x , x√ x 4 + 2 b cosh2 − (a + 3b) tanh a + b cosh x 3b 2 2  √ √ a + b cosh x dx = a + b E (α, r) cosh x + 1 √  2b a+b dx √ √ E (α, r) − = F (α, r) a−b (cosh x + 1) a + b cosh x (a − b) a + b

BY (297.31)

BY (297.31) BY (297.26) BY (297.30)

2.464

Algebraic functions of hyperbolic functions

 30.

137

 1 dx √ = b(5b − a) F (α, r) 2√ (cosh x + 1) a + b cosh x 3(a − b)2 a + b  sinh x2 √ 1 · + (a − 3b)(a + b) E (α, r) + a + b cosh x 6(a − b) cosh3 x2 297.30)

 31.

  (1 + cosh x) dx 2 √ =√ Π α, p2 , r 2 [1 + + (1 − p ) cosh x] a + b cosh x a+b Notation: In 2.464 32–2.464 40, we set:  a − b cosh x α = arcsin a−b  + a, a−b r= 0 < b < a, 0 < x < arccosh a+b b p2

BY (297.27)



32. 33.

dx 2 √ =√ F (α, r) a − b cosh x a+b  √ √ a − b cosh x dx = 2 a + b [F (α, r) − E (α, r)] 

34.  35.  36.  37.  38.  39.

BY (297.54)

√ 2 cosh x dx 2 a+b √ E (α, r) − √ = F (α, r) b a − b cosh x a+b √ √ cosh2 x dx 2 2(b − 2a) 4a a + b √ E (α, r) + sinh x a − b cosh x = √ F (α, r) + 2 3b 3b a − b cosh x 3b a + b √ (1 + cosh x) dx 2 a+b √ E (α, r) = b a − b cosh x

dx 2b a−b √ ,r = √ Π α, a cosh x a − b cosh x a a+b x√ dx 1 1 √ tanh a − b cosh x =√ E (α, r) − a+b 2 (1 + cosh x) a − b cosh x a+b

BY (297.56)

BY (297.56)

BY (297.51) BY (297.57) BY (297.58)

1 dx =  [(a + 3b) E (α, r) − b F (α, r)] √ (1 + cosh x) a − b cosh x 3 (a + b)3 √ tanh x2 a − b cosh x 1 [2a + 4b + (a + 3b) cosh x] − 3(a + b)2 cosh x + 1 2

 40.

BY (297.50)

BY (297.58)



dx 2 √ √ = Π α, p2 , r (a − b − ap2 + bp2 cosh x) a − b cosh x (a − b) a + b Notation: In 2.464 41 –2.464 47, we set:  b (cosh x − 1) , α = arcsin b cosh x − a  a+b [0 < a < b, x > 0] r= 2b



BY (297.52)

138

Hyperbolic Functions

 2 dx √ F (α, r) = b b cosh −a   √ √ 2 2b sinh x F (α, r) − 2 2b E (α, r) + √ b cosh x − a dx = (b − a) b b cosh x − a   2 1 dx < [2b E (α, r) − (b − a) F (α, r)] = 2 · 2 b − a b 3 (b cosh x − a)   2 1 dx < [(b − 3a)(b − a) F (α, r) + 8ab E (α, r)] = 2 2 2 b 5 3 (b − a ) (b cosh x − a) sinh x 2b ·< + 2 2 3 (b − a ) 3 (b cosh x − a)

2.464

 41. 42. 43.

44.

 45. 46.

47.

48. 49. 50. 51. 52. 53.

BY (297.05)

BY (297.06)

BY (297.06)

2 sinh x 2 [F (α, r) − 2 E (α, r)] + √ BY (297.03) b b cosh x − a   2 (cosh x + 1) dx 2 < E (α, r) = BY (297.01) b−a b 3 (b cosh x − a)  √   b cosh x − a dx 2  = Π α, p2 , r BY (297.02) 2 2 p b − a + b (1 − p ) cosh x b .  b cosh x − a 2b and r = for Notation: In 2.464 48–2.464 55, we set α = arcsin b (cosh x − 1) a+b + a, 0 < b < a, x > arccosh : b  dx 2 √ BY (297.75) =√ F (α, r) b cosh x − a a+b  √ √ x√ b cosh x − a dx = −2 a + b E (α, r) + 2 coth b cosh x − a BY (297.79) 2 √  coth2 x2 dx 2 a+b √ E (α, r) BY (297.76) = a−b b cosh x − a √  √ b cosh x − a dx = a + b [F (α, r) − E (α, r)] BY (297.77) cosh x − 1 √  1 a+b dx √ E (α, r) − √ BY (297.78) = F (α, r) a−b (cosh x − 1) b cosh x − a a+b   1 dx √ = (a − 2b)(a − b) F (α, r) 2√ (cosh x − 1) b cosh x − a 3(a − b)2 a + b  cosh x2 √ a+b · + (3a − b)(a + b) E (α, r) + b cosh x − a 6b(a − b) sinh3 x2 

54.



BY (297.00)

cosh x dx √ = b cosh x − a

√ dx 1 2 b cosh x − a √ =√ [F (α, r) − E (α, r)] + (a + b) sinh x a+b (cosh x + 1) b cosh x − a

BY (297.78) BY (297.80)

2.471

Hyperbolic functions and powers

 55.

139

 1 dx =  (a + b) F (α, r) 2√ (cosh x + 1) b cosh x − a 3 (a + b)3 √ 

b cosh x − a a + 3b 2 x − tanh − (a + 3b) E (α, r) + 2 3(a + b) sinh x a+b 2 BY (297.80)

56. 57.

Notation: In 2.464 56–2.464 60, we set √ 4 2 b − a2 α = arccos √ , a sinh x + b cosh x   a 1 0 < a < b, − arcsinh √ <x r= √ 2 b 2 − a2   dx 4 √ F (α, r) BY (299.00) = 4 2 b − a2 a sinh x + b cosh x   √ 2 (a cosh x + b sinh x) a sinh x + b cosh x dx = 4 4 (b2 − a2 ) [F (α, r) − 2 E (α, r)] + √ a sinh x + b cosh x 

58.

59.

60.

.

BY (299.02)

4 BY (299.03) 3 [2 E (α, r) − F (α, r)] (b2 − a2 ) .  a cosh x + b sinh x 4 dx 14 2 < ·< = F (α, r) + 3 (b2 − a2 )5 3 (b2 − a2 ) 5 3 (a sinh x + b cosh x) (a sinh x + b cosh x) dx < = 3 (a sinh x + b cosh x)

4

   √ 2 b − a2 + a sinh x + b cosh x dx 4 < =24 2 E (α, r) b − a2 3 (a sinh x + b cosh x)

BY (299.03) BY (299.01)

2.47 Combinations of hyperbolic functions and powers 2.471  1. xr sinhp x coshq x dx =

 1 (p + q)xr sinhp−1 x coshq−1 x (p + q)2 

−rxr−1 sinhp x coshq x + r(r + 1) xr−2 sinhp x coshq x dx    + rp xr−1 sinhp−1 x coshq−1 x dx + (q − 1)(p + q) xr sinhp x coshq−2 x dx  1 = (p + q)xr sinhp−1 x coshq+1 x (p + q)2  −rxr−1 sinhp x coshq x + r(r − 1) xr−2 sinhp x coshq x dx    − rq xr−1 sinhp−1 x coshq−1 x dx − (p − 1)(p + q) xr sinhp−2 x coshq x dx GU (353)(1)

140

Hyperbolic Functions





 m−1  xn+1 2m 1 k + x sinh x dx = (−1) (−1) xn cosh(2m − 2k)x dx k 22m (n + 1) 22m−1 k=0

  m  2m + 1 1 (−1)k xn sinh2m+1 x dx = 2m xn sinh(2m − 2k + 1)x dx k 2

 2.

3.

n

2m

m



5.

k=0



m−1  2m  xn+1 1 x cosh x dx = + xn cosh(2m − 2k)x dx k 22m (n + 1) 22m−1 k=0

  m  2m + 1 1 xn cosh2m+1 x dx = 2m xn cosh(2m − 2k + 1)x dx k 2

 4.

2m m



n

2m

2m m

k=0

2.472



1.

 xn sinh x dx = xn cosh x − n xn−1 cosh x dx = x cosh x − nx n

 2.

n−1

= x sinh x − nx

5.11

6. 7. 8. 9. 10. 2.473 1.

n−1

 cosh x + n(n − 1) xn−2 cosh x dx

 n n   x2k x2k−1 cosh x − sinh x x sinh x dx = (2n)! (2k)! (2k − 1)! k=0 k=1   n  x2k+1 x2k 2n+1 cosh x − sinh x x sinh x dx = (2n + 1)! (2k + 1)! (2k)! k=0   n  n  x2k  x2k−1 2n sinh x − cosh x x cosh x dx = (2n)! (2k)! (2k − 1)! k=0 k=1   n  x2k+1 x2k 2n+1 x sinh x − cosh x cosh x dx = (2n + 1)! (2k + 1)! (2k)! k=0  x sinh x dx = x cosh x − sinh x    x2 sinh x dx = x2 + 2 cosh x − 2x sinh x  x cosh x dx = x sinh x − cosh x    x2 cosh x dx = x2 + 2 sinh x − 2x cosh x 



4.

 sinh x + n(n − 1) xn−2 sinh x dx

 xn cosh x dx = xn sinh x − n xn−1 sinh x dx n

3.

2.472

2n

Notation: z1 = a + bx  1 b z1 sinh kx dx = z1 cosh kx − 2 sinh kx k k

2.474

Hyperbolic functions and powers

 2.

1 b z1 sinh kx − 2 cosh kx k k

1 2b2 2bz1 2 2 z1 sinh kx dx = z1 + 2 cosh kx − 2 sinh kx k k k

2 1 2b 2bz1 z12 cosh kx dx = z12 + 2 sinh kx − 2 cosh kx k k k



2 z1 6b 2b2 3b z13 sinh kx dx = z12 + 2 cosh kx − 2 z12 + 2 sinh kx k k k k



2 z1 6b 2b2 3b 3 2 3 z1 cosh kx dx = z1 + 2 sinh kx − 2 z1 + 2 cosh kx k k k k



1 12b2 24b4 6b2 4bz1 cosh kx − 2 z12 + 2 sinh kx z14 sinh kx dx = z14 + 2 z12 + 4 k k k k k



2 4 1 12b 2 24b 6b2 4bz1 4 4 2 sinh kx − 2 z1 + 2 cosh kx z1 cosh kx dx = z1 + 2 z1 + 4 k k k k k



z1 20b2 b4 b2 b4 5b z15 sinh kx dx = z14 + 2 z12 + 120 4 cosh kx − 2 z14 + 12 2 z12 + 24 4 sinh kx k k k k k k



2 4 2 z1 b 2 b b 2 b4 5b 5 4 4 z1 cosh kx dx = z1 + 20 2 z1 + 120 4 sinh kx − 2 z1 + 12 2 z1 + 24 4 cosh kx k k k k k k

2 4 6 1 b b b z16 sinh kx dx = z16 + 30 2 z14 + 360 4 z12 + 720 6 cosh kx k k k k

b2 2 b4 6bz1 4 − 2 z1 + 20 2 z1 + 120 4 sinh kx k k k

2 4 1 b 4 b 2 b6 6 6 z1 cosh kx dx = z1 + 30 2 z1 + 360 4 z1 + 720 6 sinh kx k k k

k 2 4 b b 6bz1 − 2 z14 + 20 2 z1 + 120 4 cosh kx k k k

z1 cosh kx dx = 

3.  4.  5.  6.  7.  8.  9.  10.  11.

 12.

2.474

141



 n/2 xn−2k n!  xn−2k−1 xn+1 + sinh 2x − 2k+1 cosh 2x x sinh x dx = − 2(n + 1) 4 22k (n − 2k)! 2 (n − 2k − 1)! n

1.

2

k=0

GU (353)(2b)



n/2

n!  xn+1 + x cosh x dx = 2(n + 1) 4 n

2.

2

k=0

GU (353)(3e)

 3. 4.

 xn−2k xn−2k−1 sinh 2x − 2k+1 cosh 2x 22k (n − 2k)! 2 (n − 2k − 1)!

1 x2 1 x sinh 2x − cosh 2x − 4 8 4

 x3 1 1 x x2 sinh2 x dx = x2 + sinh 2x − cosh 2x − 4 2 4 6 x sinh2 x dx =

MZ 257

142

Hyperbolic Functions

2.475

 5. 6. 7.

1 x2 x sinh 2x − cosh 2x + 4 8 4

 x3 1 1 x x2 cosh2 x dx = x2 + sinh 2x − cosh 2x + 4 2 4 6  xn sinh3 x dx x cosh2 x dx =

MZ 261



 n/2 xn−2k cosh 3x sinh 3x n!  xn−2k−1 = − 3 cosh x − − 3 sinh x 4 (n − 2k)! 32k+1 (n − 2k − 1)! 32k+2 k=0

GU (353)(2f)

 xn cosh3 x dx

8.

=



 n/2 xn−2k sinh 3x cosh 3x n!  xn−2k−1 + 3 sinh x − + 3 cosh x 4 (n − 2k)! 32k+1 (n − 2k − 1)! 32k+2 k=0

GU (353)(3f)

 9. 10. 11. 12. 2.475

1 3 x 3 sinh x − sinh 3x − x cosh x − cosh 3x 4 36 4 12 2

2

 3x 3 1 x x 3x x2 sinh3 x dx = − + + sinh x − sinh 3x. cosh x + cosh 3x + 4 2 12 54 2 18  1 3 x 3 cosh 3x + x sinh x + sinh 3x x cosh3 x dx = − cosh x − 4 36 4 12

2

 3 2 3 1 x x 3 3 2 x cosh x dx = x + + cosh 3x sinh x + sinh 3x − x cosh x − 4 2 12 54 2 18 x sinh3 x dx =



1.

(p − 2) sinhq x + qx sinhq−1 x cosh x sinhq x dx = − xp (p − 1)(p − 2)xp−1   sinhq−2 x sinhq x q(q − 1) q2 + dx + dx p−2 (p − 1)(p − 2) x (p − 1)(p − 2) xp−2

MZ 257

MZ 262

[p > 2] GU (353)(6a)

 2.

 3.  4.

coshq x (p − 2) coshq x + qx coshq−1 x sinh x dx = − p x (p − 1)(p − 2)xp−1   q(q − 1) q2 coshq−2 x coshq x − dx + dx (p − 1)(p − 2) xp−2 (p − 1)(p − 2) xp−2 sinh x 1 dx = − x2n x(2n − 1)!

sinh x 1 dx = − 2n+1 x x(2n)!

n−2  (2k + 1)! k=0

n−1  (2k)! k=0

x2k

x2k+1

cosh x +

cosh x +

n−1  k=0

n−1  k=0

GU (353)(7a)



(2k)! sinh x x2k 

[p > 2]

+

1 chi(x) (2n − 1)!

(2k + 1)! 1 shi(x) sinh x + 2k+1 x (2n)!

GU (353)(6b) GU (353)(6b)

2.476

Hyperbolic functions and powers

 5.  6.

n−2  (2k + 1)!

cosh x 1 dx = − x2n x(2n − 1)!

1 cosh x dx = − 2n+1 x (2n)!x

k=0

n−1  (2k)! k=0

x2k

x2k+1

sinh x +

n−1  k=0

sinh x +

n−1  k=0

143

 (2k)! 1 shi(x) cosh x + x2k (2n − 1)! 

(2k + 1)! 1 chi(x) cosh x + 2k+1 x (2n)!





m−1  2m 1 sinh2m x (−1)m 2m dx = 2m−1 (−1)k chi(2m − 2k)x + 2m ln x k m x 2 2 k=0

 m 2m + 1 1  sinh2m+1 x dx = 2m (−1)k shi(2m − 2k + 1)x k x 2

GU (353)(7b) GU (353)(7b)

 7.

8.

GU (353)(6c)

GU (353)(6d)

k=0



m−1  2m

2m 1 cosh2m x 1 dx = 2m−1 GU (353)(7c) chi(2m − 2k)x + 2m ln x k m x 2 2 k=0

 m 1  2m + 1 cosh2m+1 x dx = 2m GU (353)(7c) chi(2m − 2k + 1)x k x 2 k=0

 sinh2m x (−1)m−1 2m dx = x2 22m x m 

m−1  2m cosh(2m − 2k)x 1 k+1 − (2m − 2k) shi(2m − 2k)x + 2m−1 (−1) k 2 x 

9.

10.

11.

 12.

k=0

sinh

2m+1

x

x2

 13.

cosh2m x dx x2 =−

 14.

dx =

2m+1

cosh x2

1 22m x

x

2m + 1 k 22m

k=0  sinh(2m − 2k + 1)x × − (2m − 2k + 1) chi(2m − 2k + 1)x x



1

2m m

m 



(−1)k+1

−

1 22m−1

m−1  k=0

2m k



 cosh(2m − 2k)x − (2m − 2k) shi(2m − 2k)x x

dx



m cosh(2m − 2k + 1)x 1  2m + 1 − (2m − 2k + 1) shi(2m − 2k + 1)x = − 2m k 2 x k=0

2.476 1.



  sinh kx 1 ka ka dx = shi(u) − sinh chi(u) cosh a + bx b  b

b

 1 ka ka = exp − Ei(u) − exp Ei(−u) 2b b b

  k u = (a + bx) b

144

Hyperbolic Functions

2.477

  1 ka cosh kx ka dx = chi(u) − sinh shi(u) cosh a + bx b  b

 b

  1 ka ka k = exp − Ei(u) + exp Ei(−u) u = (a + bx) 2b b b b   sinh kx 1 sinh kx k cosh kx + dx (see 2.476 2) dx = − · (a + bx)2 b a + bx b a + bx   cosh kx 1 cosh kx k sinh kx · + dx (see 2.476 1) dx = − (a + bx)2 b a + bx b a + bx   k 2 sinh kx sinh kx sinh kx k cosh kx + dx dx = − − 2 (a + bx)3 2b(a + bx)2 2b (a + bx) 2b2 a + bx

 2.

3. 4. 5.

 6.  7.  8.  9.

 10.

 11.

 12.

2

cosh kx k cosh kx k sinh kx + 2 dx = − − 2 3 2 (a + bx) 2b(a + bx) 2b (a + bx) 2b

(see 2.476 1)



cosh kx dx a + bx

(see 2.476 2)  sinh kx k 3 cosh kx sinh kx k cosh kx k sinh kx + dx dx = − − − (a + bx)4 3b(a + bx)3 6b2 (a + bx)2 6b3 (a + bx) 6b3 a + bx 2

(see 2.476 2)  cosh kx k 3 sinh kx cosh kx k sinh kx k cosh kx + dx dx = − − 2 − 3 (a + bx)4 3b(a + bx)3 6b (a + bx)2 6b (a + bx) 6b3 a + bx 2

(see 2.476 1) 2

sinh kx sinh kx k cosh kx k sinh kx dx = − − − 5 4 2 3 (a + bx) 4b(a + bx) 12b (a+ bx) 24b3 (a + bx)2 3 4 k k cosh kx sinh kx + dx − 24b4 (a + bx) 24b4 a + bx (see 2.476 1) cosh kx cosh kx k sinh kx k 2 cosh kx dx = − − − 5 4 2 3 (a + bx) 4b(a + bx) 12b (a+ bx) 24b3 (a + bx)2 3 4 k k sinh kx cosh kx + dx − 24b4 (a + bx) 24b4 a + bx (see 2.476 2) sinh kx sinh kx k cosh kx k 2 sinh kx k 3 cosh kx dx = − − − − (a + bx)6 5b(a + bx)5 20b2 (a +bx)4 60b3 (a + bx)3 120b4 (a + bx)2 k5 k 4 sinh kx cosh kx + dx − 120b5 (a + bx) 120b5 a + bx (see 2.476 2) cosh kx cosh kx k sinh kx k 2 cosh kx k 3 sinh kx dx = − − − − (a + bx)6 5b(a + bx)5 20b2 (a +bx)4 60b3 (a + bx)3 120b4 (a + bx)2 4 5 k k cosh kx sinh kx + dx − 120b5 (a + bx) 120b5 a + bx (see 2.476 1)

2.477

2.477 1.

2.

3.

Hyperbolic functions and powers

145



 p(p − 1) xp dx −pxp−1 sinh x − (q − 2)xp cosh x xp−2 + dx = q q−1 sinh x (q − 1)(q − 2) sinhq−2 x (q  − 1)(q − 2) sinh x p q−2 x dx − q − 1 sinhq−2 x [q > 2]   pxp−1 cosh x + (q − 2)xp sinh x xp−2 dx xp dx p(p − 1) = − q q−1 cosh x (q − 1)(q − 2) coshq−2 x (q −1)(q − 2) cosh x p q−2 x dx + q − 1 coshq−2 x [q > 2]    ∞  2 − 22k B2k n+2k xn dx = x [|x| < π, n > 0] sinh x (n + 2k)(2k)! 

4.  5.

6.11

 7.

 8.

 9.

GU (353)(10a) GU(353)(8b)

k=0 ∞

 E2k xn+2k+1 xn dx = cosh x (n + 2k + 1)(2k)! k=0

+ π |x| < , 2

n≥0

[|x| < π,

n ≥ 1]

, GU (353)(10b)

n−1 dx −1 n 2 = − [1 + (−1) Bn ln x ] n x sinh x n! ∞  2 − 22k B2k x2k−n + (2k − n)(2k)! k=0 k= n 2



GU (353)(8a)

dx = xn cosh x

∞  k=0 k= n−1 2

GU (353)(9b)

E2k 1 En−1 x2k−n+1 + [1 + (−1)n ] + ln x (2k − n + 1)(2k)! 2 (n − 1)! + π, |x| < 2

GU (353)(11b)

∞  22k B2k xn n xn+2k−1 coth x + n dx = −x 2 (n + 2k − 1)(2k)! sinh x k=0

∞ 

 2k



[n > 1,

22k − 1 B2k n+2k−1 2 xn x dx = xn tanh x − n 2 (n + 2k − 1)(2k)! cosh x k=1 + n > 1,

|x| < π]

|x| <

π, 2

GU (353)(8c)

GU (353)(10c)

dx 2n n coth x n Bn+1 ln x − [1 − (−1) ] = − 2 n x (n + 1)! xn sinh x ∞ k n  B2k − n+1 (2x)2 x (2k − n − 1)(2k)! k=0 k= n+1 2

[|x| < π]

GU (353)(9c)

146

Hyperbolic Functions

 10.

  2n 2n+1 − 1 n dx tanh x n Bn+1 ln x + [1 − (−1) ] − = xn (n + 1)! xn cosh2 x   ∞ 22k − 1 B2k k n  + n+1 (2x)2 x (2k − n − 1)(2k)! k=1 k= n+1 2

 11.

 12.

x 2n−1

x

15.  16.  17. 18. 19.

x cosh



n−1 

(2n − 3)(2n − 5) . . . (2n − 2k + 1) (−1)k (2n − 2)(2n − 4) . . . (2n − 2k) k=1

  x cosh x 1 x dx n−1 (2n − 3)!! + × + (−1) 2n−2k 2n−2k−1 (2n − 2)!! sinh x sinh x (2n − 2k − 1) sinh x GU (353)(8e) (see 2.477 15)

n−1  (2n − 2)(2n − 4) . . . (2n − 2k + 2) x dx = 2n (2n − 1)(2n − 3) . . . (2n − 2k + 1) cosh x k=1 

 x dx x sinh x 1 (2n − 2)!! + + × 2n−2k+1 2n−2k (2n − 1)!! cosh2 x cosh x (2n − 2k) cosh x GU (353)(10e) (see 2.477 18)

 14.

GU (353)(11c)

dx =



+ π, |x| < 2

n−1  (2n − 2)(2n − 4) . . . (2n − 2k + 2) x (−1)k dx = 2n (2n − 1)(2n − 3) . . . (2n − 2k + 1) sinh x k=1

  x cosh x 1 x dx n−1 (2n − 2)!! + × + (−1) (2n − 1)!! sinh2 x sinh2n−2k+1 x (2n − 2k) sinh2n−2k x GU (353)(8e) (see 2.477 17)

sinh

13.

2.477

2n−1

x

dx =

n−1 

(2n − 3)(2n − 5) . . . (2n − 2k + 1) (2n − 2)(2n − 4) . . . (2n − 2k) k=1

  x sinh x 1 x dx (2n − 3)!! + × + 2n−2k 2n−2k−1 (2n − 2)!! cosh x cosh x (2n − 2k − 1) cosh x GU (353)(10e) (see 2.477 16)

∞

 2 − 22k x dx = B2k x2k+1 sinh x (2k + 1)(2k)!

|x| < π

GU (353)(8b)a

π 2

GU (353)(10b)a

k=0 ∞

 E2k x2k+2 x dx = cosh x (2k + 2)(2k)! k=0

x dx = −x coth x + ln sinh x sinh2 x  x dx = x tanh x − ln cosh x cosh2 x   1 x dx x cosh x 1 x dx − =− − 3 2 2 sinh x 2 sinh x sinh x 2 sinh x

|x| <

MZ 257 MZ 262

(see 2.477 15)

MZ 257

2.478

Hyperbolic functions and powers

 20.  21.  22.  23.  24.

147

 1 x dx x sinh x 1 x dx + (see 2.477 16) = + 3 2 2 cosh x 2 cosh x cosh x 2 cosh x x cosh x 1 2 2 x dx =− − + x coth x − ln sinh x 4 3 2 3 3 sinh x 3 sinh x 6 sinh x 2 x dx x sinh x 1 2 = + + x tanh x − ln cosh x 4 3 2 3 3 cosh x 3 cosh x 6 cosh x  3 x dx x cosh x 1 3x cosh x 3 x dx + =− − + + 5 4 3 2 8 sinh x 8 sinh x sinh x 4 sinh x 12 sinh x 8 sinh x (see 2.477 15)  3 x dx x sinh x 1 3x sinh x 3 x dx + = + + + 5 4 3 2 cosh x 4 cosh x 12 cosh x 8 cosh x 8 cosh x 8 cosh x

MZ 262 MZ 258 MZ 262

MZ 258

(see 2.477 16) 2.478



1.  2.  3. 4. 5. 6. 7.

xn sinh x dx xn m =− m−1 (a + b cosh x) (m − 1)b (a + b cosh x)



xn−1 dx (a + b sinh x)

m−1

[m = 1]  n xn−1 dx + (m − 1)b (a + b cosh x)m−1

MZ 263

[m = 1]

MZ 263

x x x dx = x tanh − 2 ln cosh 1 + cosh x 2 2  x x x dx = x coth − 2 ln sinh 1 − cosh x 2 2  x x sinh x dx x 2 = − 1 + cosh x + tanh 2 (1 + cosh x)  x x x sinh x dx 2 = 1 − cosh x − coth 2 (1 − cosh x)  1 x dx = [L(u + t) − L(u − t) − 2 L(t)] cosh 2x − cos 2t 2 sin 2t [u = arctan (tanh x cot t) , 

8.

xn cosh x dx xn n m =− m−1 + (m − 1)b (a + b sinh x) (m − 1)b (a + b sinh x)

MZ 262







1 u+t u−t x cosh x dx υ+t = L −L +L π− cosh 2x − cos 2t 2 sin t 2

2 2

 υ−t t π−t + L − 2L − 2L 2 2 

2

t t x x u = 2 arctan tanh · cot , υ = 2 arctan coth · cot ; 2 2 2 2

MZ 262-264

t = ±nπ] LO III 402

 t = ±nπ

LO III 403

148

2.479 1.

2.

Hyperbolic Functions



  sinh2m x m+k m dx = (−1) k coshn x m

2.479



xp dx (see 4.477 2) coshn−2k x k=0  m 2m+1    x sinh x p sinh m+k m dx = dx x (−1) xp n k cosh x coshn−2k x k=0 xp

 xp

3.

xp sinh x p dx = − + n n−1 cosh x (n − 1) cosh x n−1



[n > 1] xp−1 dx coshn−1 x [n > 1]

 4.

5. 6.

cosh2m x dx = sinhn x

xp tanh x dx =

7.

xp coth x dx = 

∞  k=1

 8.

10.

(see 2.477 2)

∞  k=0

m k

GU (353)(13c)

 2k  2 − 1 B2k p+2k 2 x (2k + p)(2k)!

+ p > −1,

|x| <

22k B2k xp+2k (p + 2k)(2k)!

[p ≥ +1,

|x| < π]

2k

π, 2

x x cosh x x dx = ln tanh − 2 2 sinh x sinh x  x sinh x x + arctan (sinh x) dx = − cosh x cosh2 x

2.48 Combinations of hyperbolic functions, exponentials, and powers 2.481

 eax sinh(bx + c) dx =

1.

 2.

GU (353)(12)



xp cosh x (see 2.477 1) sinhn−2k x k=0   p m   m cosh2m+1 x x cosh x dx = dx (see 2.479 6) xp n−2k k sinhn x sinh x k=0   xp p xp−1 dx p cosh x dx = − x + n sinh x (n − 1) sinhn−1 x n − 1 sinhn−1 x [n > 1] (see 2.477 1) xp



9.

m  

(see 2.479 3)

eax cosh(bx + c) dx =

eax [a sinh(bx + c) − b cosh(bx + c)] a2 − b 2  2  a = b2 eax [a cosh(bx + c) − b sinh(bx + c)] a2 − b 2  2  a = b2

GU (353)(12d)

GU (353)(13d)

MZ 263

2.483

3. 4. 5. 6. 2.482

Hyperbolic functions, exponentials, and powers

For a2 = b2 :  1 1 eax sinh(ax + c) dx = − xe−c + e2ax+c 2 4a  1 1 e−ax sinh(ax + c) dx = xec + e−(2ax+c) 2 4a  1 1 eax cosh(ax + c) dx = xe−c + e2ax+c 2 4a  1 1 e−ax cosh(ax + c) dx = xec − e−(2ax+c) 2 4a  xp eax sinh bx dx =

1.

1 2



 xp e(a+b)x dx −

149

MZ 275-277

 xp e(a−b)x dx 

2.

3. 4. 5. 2.483 1.

2.

3.

 a2 = b2 

   1 xp eax cosh bx dx = xp e(a+b)x dx + xp e(a−b)x dx 2  2  a = b2 For a2 = b2 :   1 xp+1 xp eax sinh ax dx = xp e2ax dx − 2 2(p + 1)   p+1 1 x xp e−ax sinh ax dx = − xp e−2ax dx 2(p + 1) 2   1 xp+1 + xp eax cosh ax dx = xp e2ax dx 2(p + 1) 2

(see 2.321) (see 2.321) (see 2.321)

MZ 276, 278





 a2 + b 2 2ab ax − 2 sinh bx − bx − cosh bx a − b2 a2 − b 2  2  a = b2 



  eax a2 + b 2 2ab xeax cosh bx dx = 2 ax − cosh bx − bx − sinh bx a − b2 a2 − b 2 a2 − b 2  2  a = b2 %  &    2 a2 + b 2 2a a2 + 3b2 eax 2 ax 2 x e sinh bx dx = 2 x+ ax − sinh bx 2 a − b2 a2 − b 2 (a2 − b2 ) &  %     2 2b 3a2 + b2 4ab 2 x+

b2 cosh x a = − bx − 2 2 2 2 a − b2 (a − b )



xeax sinh bx dx =

eax 2 a − b2

150

Hyperbolic Functions

 2 ax

4.

5. 6. 7. 8. 9. 10. 11. 2.484 1. 2. 3.

4.

5. 6. 7.

x e

   & 2 a2 + b 2 2a a2 + 3b2 x+ ax − cosh bx 2 a2 − b 2 (a2 − b2 )   & 2b 3a2 + b2 4ab 2 x+ sinh x − bx − 2 2 a − b2 (a2 − b2 )

eax cosh bx dx = 2 a − b2 %

2.484

%

2



For a2 = b2 :

 e2ax 1 x2 ax xe sinh ax dx = x− − 4a 2a 4

 e−2ax 1 x2 xe−ax sinh ax dx = x+ + 4a 2a 4

 2 2ax e x 1 + xeax cosh ax dx = x− 4 4a 2a

 2 −2ax e x 1 − xe−ax cosh ax dx = x+ 4 4a 2a

 2ax 1 e x x3 x2 eax sinh ax dx = x2 − + 2 − 4a a 2a 6

 −2ax 1 e x x3 x2 e−ax sinh ax dx = x2 + + 2 + 4a a 2a 6

 e2ax 1 x3 x + x2 eax cosh ax dx = x2 − + 2 6 4a a 2a 

 2  1 dx = {Ei[(a + b)x] − Ei[(a − b)x]} a = b2 x 2   2  1 dx = {Ei[(a + b)x] + Ei[(a − b)x]} a = b2 eax cosh bx x 2  dx eax sinh bx 1 + {(a + b) Ei[(a + b)x] − (a − b) Ei[(a − b)x]} eax sinh bx 2 = − x 2x 2  2  a = b2  dx eax cosh bx 1 + {(a + b) Ei[(a + b)x] + (a − b) Ei[(a − b)x]} eax cosh bx 2 = − x 2x 2  2  a = b2 eax sinh bx

For a2 = b2 :  1 dx = [Ei(2ax) − ln x] eax sinh ax x 2  1 dx = [ln x − Ei(−2ax)] e−ax sinh ax x 2  1 dx = [ln x + Ei(2ax)] eax cosh ax x 2

a2 = b2



MZ 276, 278

2.510

Powers of trigonometric functions

151



8. 9. 10.

 dx 1  2ax e =− − 1 + a Ei(2ax) x2 2x   dx 1  1 − e−2ax + a Ei(−2ax) e−ax sinh ax 2 = − x 2x   dx 1  2ax e eax cosh ax 2 = − + 1 + a Ei(2ax) x 2x eax sinh ax

MZ 276, 278

2.5–2.6 Trigonometric Functions 2.50 Introduction 2.501



Integrals of the form

R (sin x, cos x) dx can always be reduced to integrals of rational functions x by means of the substitution t = tan . 2 2.502 If R (sin x, cos x) satisfies the relation R (sin x, cos x) = −R (− sin x, cos x) , it is convenient to make the substitution t = cos x. 2.503 If this function satisfies the relation R (sin x, cos x) = −R (sin x, − cos x) , it is convenient to make the substitution t = sin x. 2.504 If this function satisfies the relation R (sin x, cos x) = R (− sin x, − cos x) , it is convenient to make the substitution t = tan x.

2.51–2.52 Powers of trigonometric functions 

2.510

 sinp−1 x cosq+1 x p − 1 + sinp−2 x cosq+2 x dx q+1 q + 1 sinp−1 x cosq+1 x p − 1 =− + sinp−2 x cosq x dx p+q p+q  sinp+1 x cosq+1 x p + q + 2 + = sinp+2 x cosq x dx p+1 p +1 sinp+1 x cosq−1 x q − 1 = + sinp+2 x cosq−2 x dx p+1 p + 1 sinp+1 x cosq−1 x q − 1 + = sinp x cosq−2 x dx p+q p+q  sinp+1 x cosq+1 x p + q + 2 =− + sinp x cosq+2 x dx q+1 q + 1

 q−1 sinp−1 x cosq−1 x = sin2 x − p+q p+q−2  (p − 1)(q − 1) p−2 sin x cosq−2 x dx + (p + q)(p + q − 2)

sinp x cosq x dx = −

FI II 89, TI 214

152

2.511

Trigonometric Functions

2.511

 sinp x cos2n x dx

1.



n−1 

(2n − 1)(2n − 3) . . . (2n − 2k + 1) cos2n−2k−1 x cos x+ (2n + p − 2)(2n + p − 4) . . . (2n + p − 2k) k=1  (2n − 1)!! + sinp x dx (2n + p)(2n + p − 2) . . . (p + 2)

sinp+1 x = 2n + p

2.

2n−1



This formula is applicable for arbitrary real p, except for the following negative even integers: −2, −4, . . . , −2n. If p is a natural number and n = 0, we have:  sin2l x dx   l−1  (2l − 1)(2l − 3) . . . (2l − 2k + 1) cos x 2l−1 2l−2k−1 sin sin x+ x =− 2l 2k (l − 1)(l − 2) . . . (l − k) k=1 (2l − 1)!! x (see also 2.513 1) + 2l l!  sin2l+1 x dx = −

3.

 p

4.

sin x cos

2n+1

cos x 2l + 1

 sin2l x +

l−1  k=0

sinp+1 x x dx = 2n + p + 1



2k+1 l(l − 1) . . . (l − k) sin2l−2k−2 x (2l − 1)(2l − 3) . . . (2l − 2k − 1) (see also 2.513 2)

 cos

TI (232)

2n

x+

n  k=1

TI (233)

2k n(n − 1) . . . (n − k + 1) cos2n−2k x (2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1)



This formula is applicable for arbitrary real p, except for the negative odd integers: −1, −3, . . . , −(2n + 1). 2.512 1.

 cosp x sin2n x dx



n−1 

(2n − 1)(2n − 3) . . . (2n − 2k + 1) sin2n−2k−1 x x+ sin (2n + p − 2)(2n + p − 4) . . . (2n + p − 2k) k=1  (2n − 1)!! cosp x dx + (2n + p)(2n + p − 2) . . . (p + 2)

cosp+1 x =− 2n + p

2.

2n−1



This formula is applicable for arbitrary real p, except for the following negative even integers: −2, −4, . . . , −2n. If p is a natural number and n = 0, we have    l−1  (2l − 1)(2l − 3) . . . (2l − 2k + 1) sin x 2l 2l−1 2l−2k−1 cos x dx = cos cos x+ x 2l 2k (l − 1)(l − 2) . . . (l − k) k=1 (2l − 1)!! x + 2l l! (see also 2.513 3) TI (230)

2.513

Powers of trigonometric functions

 3.

sin x cos2l+1 x dx = 2l + 1

 cos2l x +

l−1  k=0

 2k+1 l(l − 1) . . . (l − k) cos2l−2k−2 x (2l − 1)(2l − 3) . . . (2l − 2k − 1) (see also 2.513 4)

 cosp x sin2n+1 x dx

4.

153

cosp+1 x =− 2n + p + 1

 2n

sin

x+

n  k=1

TI (231)

2k n(n − 1) . . . (n − k + 1) sin2n−2k x (2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1)



This formula is applicable for arbitrary real p, except for the following negative odd integers: −1, −3, . . . , −(2n + 1). 2.513

 2n

1.

sin

1 x dx = 2n 2



2n n





n−1 2n sin(2n − 2k)x (−1)n  k (−1) x + 2n−1 k 2 2n − 2k

(see also 2.511 2)

k=0

TI (226)



 1 n+1 (−1) (−1)k 22n n

sin2n+1 x dx =

2.



k=0

2n + 1 k



cos(2n + 1 − 2k)x 2n + 1 − 2k

(see also 2.511 3) TI (227)

 cos2n x dx =

3.

1 22n

 cos2n+1 x dx =

4.  5.  6.  7.  8.  9.



1 22n

2n n

x+

n  k=0

n−1 

1 22n−1

2n + 1 k



k=0

2n k



sin(2n − 2k)x 2n − 2k (see also 2.512 2)

TI (224)

(see also 2.512 3)

TI (225)

sin(2n − 2k + 1)x 2n − 2k + 1

1 1 1 1 sin2 x dx = − sin 2x + x = − sin x cos x + x 4 2 2 2 3 1 1 sin3 x dx = cos 3x − cos x = cos3 x − cos x 12 4 3 3x sin 2x sin 4x sin4 x dx = − + 8 4 32 1 3 3 = − sin x cos x − sin3 x cos x + x 8 4 8 5 1 5 sin5 x dx = − cos x + cos 3x − cos 5x 8 48 80 1 4 4 = − sin4 x cos x + cos3 x − cos x 5 15 5 15 3 1 5 sin6 x dx = x− sin 2x + sin 4x − sin 6x 16 64 64 192 1 5 5 5 = − sin5 x cos x − sin3 x cos x − sin x cos x + x 6 24 16 16

154

Trigonometric Functions



7 7 1 35 cos x + cos 3x − cos 5x + cos 7x 64 64 320 448 1 6 8 24 = − sin6 x cos x − sin4 x cos x + cos3 x − cos x 7 35 35 35 x 1 1 1 cos2 x dx = sin 2x + = sin x cos x + x 4 2 2 2 3 1 1 sin 3x + sin x = sin x − sin3 x cos3 x dx = 12 4 3 1 1 3 3 1 3 cos4 x dx = x + sin 2x + sin 4x = x + sin x cos x + sin x cos3 x 8 4 32 8 8 4 5 1 4 4 5 1 cos5 x dx = sin x + sin 3x + sin 5x = sin x − sin3 x + cos4 x sin x 8 48 80 5 15 5 15 3 1 5 cos6 x dx = x+ sin 2x + sin 4x + sin 6x 16 64 64 192 5 5 5 1 = x+ sin x cos x + sin x cos3 x + sin x cos5 x 16 16 24 6 7 7 1 35 sin x + sin 3x + sin 5x + sin 7x cos7 x dx = 64 64 320 448 24 8 6 1 = sin x − sin3 x + sin x cos4 x + sin x cos6 x 35 35 35 7

cos3 x 1 1 cos 3x + cos x = − sin x cos2 x dx = − 4 3 3

sin7 x dx = −

10.  11.  12.  13.  14.  15.  16.  17. 

sin x cos3 x dx = −

18. 

19. 20. 21. 22.

23. 24. 25.

cos4 x 4

cos5 x 5

 sin3 x 1 1 2 sin 3x − sin x = sin x cos x dx = − 4 3 3

 1 1 sin2 x cos2 x dx = − sin 4x − x 8 4

 1 1 1 2 3 sin x cos x dx = − sin 5x + sin 3x − 2 sin x 16 5 3



sin3 x 2 sin3 x 5 2 = − sin2 x cos x + = 5 3 5 3  1 1 1 x sin2 x cos4 x dx = + sin 2x − sin 4x − sin 6x 16 64 64 192

 sin4 x 1 1 sin3 x cos x dx = cos 4x − cos 2x = 8 4 4

 1 1 1 3 2 sin x cos x dx = cos 5x − cos 3x − 2 cos x 16 5 3 1 1 = cos5 x − cos3 x 5 3 sin x cos4 x dx = −

2.513

2.516

Powers of trigonometric functions

155

1 3 cos 6x − cos 2x 26. 6 2

 1 2 3 27. sin3 x cos4 x dx = cos3 x − − sin2 x + sin4 x 7 5 5  5 sin x 28. sin4 x cos x dx = 5  1 1 1 1 x− sin 2x − sin 4x + sin 6x 29. sin4 x cos2 x dx = 16 64 64 192

 2 3 1 + cos2 x − cos4 x 30. sin4 x cos3 x dx = sin3 x 7 5 5  1 1 3 31. sin4 x cos4 x dx = x− sin 4x + sin 8x 128 128 1024  sinp x 2.514 dx cos2n x   n−1  (2n − p − 2)(2n − p − 4) . . . (2n − p − 2k) sinp+1 x 2n−1 2n−2k−1 = sec sec x+ x 2n − 1 (2n − 3)(2n − 5) . . . (2n − 2k − 1) k=1  (2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p) sinp x dx + (2n − 1)!! " This formula is applicable for arbitrary real p. For sinp x dx, where p is a natural number, see 2.511 2, 3 and 2.513 1, 2. If n = 0 and p is a negative integer, we have for this integral: 2.515    l−1  2k (l − 1)(l − 2) . . . (l − k) dx cos x 2l−1 2l−2k−1 cosec cosec 1. =− x+ x 2l − 1 (2l − 3)(2l − 5) . . . (2l − 2k − 1) sin2l x 

1 sin x cos x dx = 32 3



3

k=1

 2.



dx sin2l+1 x



TI (242)

l−1  (2l − 1)(2l − 3) . . . (2l − 2k + 1) cos x cosec2l−2k x cosec2l x + 2l 28k (l − 1)(l − 2) . . . (l − k) k=1 x (2l − 1)!! ln tan + 2l l! 2

=−

TI (243)

2.516 1.

2.



sinp x dx cos2n+1 x

  n−1  (2n − p − 1)(2n − p − 3) · · · (2n − p − 2k + 1) sinp+1 x 2n 2n−2k sec x + sec x = 2n 2k (n − 1)(n − 2) · · · (n − k) k=1  (2n − p − 1)(2n − p − 3) · · · (3 − p)(1 − p) sinp x dx + 2n n! cos x

This formula is applicable for arbitrary real p. For n = 0 and p a natural number, we have  l  sin2k x sin2l+1 x dx =− − ln cos x cos x 2k k=1

156

Trigonometric Functions

 3.

2.517

π x  sin2k−1 x sin2l x dx =− + ln tan + cos x 2k − 1 4 2 l

k=1

2.517



1.  2. 2.518 1.

2.

m  1 dx + ln tan x = − sin2m+1 x cos x (2m − 2k + 2) sin2m−2k+2 x k=1 m π x  1 dx − + ln tan = − 4 2 sin2m x cos x (2m − 2k + 1) sin2m−2k+1 x k=1

 sinp−1 x sinp x dx = − (p − 1) sinp−2 x dx cos2 x cos x ⎧  cosp x dx cosp+1 x ⎨ = cosec2n−1 x 2n − 1 ⎩ sin2n x 

⎫ ⎬ (2n − p − 2)(2n − p − 4) . . . (2n − p − 2k) + cosec2n−2k−1 x ⎭ (2n − 3)(2n − 5) . . . (2n − 2k − 1) k=1  (2n − p − 2)(2n − p − 4) . . . (2 − p)(−p) + cosp x dx (2n − 1)!! " This formula is applicable for arbitrary real p. For cosp x dx where p is a natural number, see 2.512 2, 3 and 2.513 3, 4. If n = 0 and p is a negative integer, we have for this integral: n−1 

2.519 1.

2.



 2k (l − 1)(l − 2) . . . (l − k) 2l−2k−1 sec sec x+ x (2l − 3)(2l − 5) . . . (2l − 2k − 1) k=1    l−1  sin x (2l − 1)(2l − 3) . . . (2l − 2k + 1) dx 2l 2l−2k = sec sec x + x cos2l+1 x 2l 2k (l − 1)(l − 2) . . . (l − k) k=1 π x (2l − 1)!! ln tan + + 2l l! 4 2 

sin x dx = 2l cos x 2l − 1

2l−1

l−1 

TI (240)

TI (241)

2.521  1.

⎧ cosp+1 x ⎨ cosp x dx =− cosec2n x ⎩ 2n sin2n+1 x

⎫ ⎬ (2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1) 2n−2k + cosec x ⎭ 2k (n − 1)(n − 2) . . . (n − k) k=1  p (2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) cos x dx + 2n · n! sin x n−1 

This formula is applicable for arbitrary real p. For n = 0 and p a natural number, we have

2.526

Powers of trigonometric functions

 2.

157

cos2l+1 x dx  cos2k x = + ln sin x sin x 2k l

k=1

 3. 2.522



1.  2.

2l

cos x dx = sin x

l  k=1

x cos2k−1 x + ln tan 2k − 1 2

 dx 1 = + ln tan x 2m+1 sin x cos x (2m − 2k + 2) cos2m−2k+2 x m

k=1

dx = sin x cos2m x

m  k=1

x 1 + ln tan (2m − 2k + 1) cos2m−2k+1 x 2

GW (331)(15)

 cosm x cosm−1 x − (m − 1) cosm−2 x dx dx = − sin x sin2 x 2.524 In formulas 2.524 1 and 2.524 2, s = 1 for m odd and m < 2n + 1; in other cases, s = 0.

 n  2k−m+1  m+1 n x sin2n+1 x k+1 n cos 2 dx = + s(−1) 1. (−1) ln cos x m−1 k 2k − m + 1 cosm x 2 

2.523

k=0 k= m−1 2

GU (331)(11d)

 2.

n  2k−m+1  m−1 x cos2n+1 x k n sin dx = + s(−1) 2 (−1) m k 2k − m + 1 sin x k=0 k= m−1 2

2.525



1.  2.

m+n−1  m + n − 1 tan2k−2m+1 x dx = k 2k − 2m + 1 sin2m x cos2n x k=0



n m−1 2

ln sin x

TI (267)

m+n  m + n tan2k−2m x m + n

dx + ln tan x = m k 2k − 2m sin2m+1 x cos2n+1 x k=0

TI (268), GU (331)(15f)

2.526  1.  2.  3.  4.  5.

dx x = ln tan sin x 2 dx = − cot x sin2 x x 1 cos x 1 dx =− + ln tan 3 2 2 sin x 2 2 sin x 1 dx cos x 2 =− − cot x = − cot3 x − cot x 4 3 3 sin x 3 sin x 3 dx cos x 3 cos x 3 x =− − + ln tan 5 4 2 2 sin x 4 sin x 8 sin x 8

158

Trigonometric Functions

 6.

 7.  8.  9.  10.  11.  12.  13.  14.  15.  16.  17.  18. 

19. 20. 21. 22. 23. 24.

dx 4 cos x 4 cot3 x − cot x =− − 5 sin6 x 5 sin5 x 15 1 2 = − cot5 x − cot3 x − cot x 5 3

dx cos x 1 5 15 x 5 = − + + ln tan + 7 2 4 2 8 16 2 sin x 6 sin x sin x 4 sin x

dx 1 3 cot7 x + cot5 x + cot3 x + cot x =− 7 5 sin8 x  π x π x dx 1 + sin x = ln tan + = ln cot − = ln cos x 4 2 4 2 1 − sin x dx = tan x cos2 x π x 1 sin x 1 dx = + ln tan + cos3 x 2 cos2 x 2 4 2 sin x 2 1 dx = + tan x = tan3 x + tan x cos4 x 3 cos3 x 3 3 x π sin x 3 sin x 3 dx = + + ln tan + cos5 x 4 cos4 x 8 cos2 x 8 2 4 sin x 4 1 dx 4 2 = + tan3 x + tan x = tan5 x + tan3 x + tan x cos6 x 5 cos5 x 15 5 5 3 x π sin x 5 sin x 5 sin x 5 dx = + + + ln tan + cos7 x 6 cos6 x 24 cos4 x 16 cos2 x 16 2 4 1 dx 3 = tan7 x + tan5 x + tan3 x + tan x 8 cos x 7 5 sin x dx = − ln cos x cos x π sin2 x x dx = − sin x + ln tan = cos x 4 2

sin2 x 1 sin3 x dx = − − ln cos x = cos2 x − ln cos x cos x 2 2  x π 4 sin x 1 dx = − sin3 x − sin x + ln tan + cos x 3 2 4  2 1 sin x dx = 2 cos x cos x  2 sin x dx = tan x − x cos2 x  sin3 x dx 1 = cos x + 2 cos x cos x  4 1 3 sin x dx = tan x + sin x cos x − x cos2 x 2 2

2.526

2.526

Powers of trigonometric functions

 25.  26.  27. 

28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

1 1 sin x dx = = tan2 x cos3 x 2 cos2 x 2 π x sin x 1 sin2 x dx = − ln tan + cos3 x 2 cos2 x 2 4 2 1 sin x sin3 x dx = + ln cos x cos3 x 2 cos2 x

x π 1 sin x 3 sin4 x dx = + sin x − ln tan + cos3 x 2 cos2 x 2 2 4  1 sin x dx = cos4 x 3 cos3 x  1 sin2 x dx = tan3 x cos4 x 3  1 1 sin3 x dx =− + 4 cos x cos x 3 cos3 x  1 sin4 x dx = tan3 x − tan x + x cos4 x 3  cos x dx = ln sin x sin x  x cos2 x dx = cos x + ln tan sin x 2  3 2 cos x cos x dx = + ln sin x sin x 2  x 1 cos4 x dx = cos3 x + cos x + ln tan sin x 3 2  cos x 1 dx = − sin x sin2 x  cos2 x dx = − cot x − x sin2 x  1 cos3 x dx = − sin x − 2 sin x sin x  3 cos4 x 1 dx = − cot x − sin x cos x − x 2 2 2 sin x  1 cos x dx = − sin3 x 2 sin2 x  cos x 1 x cos2 x dx = − − ln tan 2 sin3 x 2 sin2 x 2  3 1 cos x dx = − − ln sin x 3 sin x 2 sin2 x  1 cos x 3 x cos4 x dx = − − cos x − ln tan 3 2 2 sin x 2 2 sin x

159

160

Trigonometric Functions

 45.  46.  47.  48.  49.  50.  51.  52.  53.  54.  55.  56.  57.  58.  59.  60.  61.  62.  63.  64.

cos x 1 dx = − sin4 x 3 sin3 x cos2 x 1 dx = − cot3 x 4 3 sin x 1 1 cos3 x dx = − 4 sin x 3 sin3 x sin x 1 cos4 x dx = − cot3 x + cot x + x 3 sin4 x dx = ln tan x sin x cos x dx 1 x = + ln tan sin x cos2 x cos x 2 dx 1 = + ln tan x sin x cos3 x 2 cos2 x dx 1 1 x = + + ln tan sin x cos4 x cos x 3 cos3 x 2  dx π x + = ln tan − cosec x 4 2 sin2 x cos x dx = −2 cot 2x sin2 x cos2 x

π x 1 3 3 1 dx − + ln tan + = 2 cos2 x 2 sin x 2 4 2 sin2 x cos3 x 8 dx 1 − cot 2x = 2 3 4 3 sin x cos x 3 sin x cos x dx 1 =− + ln tan x 3 sin x cos x 2 sin2 x

1 x dx 1 3 3 = − − + ln tan 3 2 2 cos x 2 sin x 2 2 2 sin x cos x dx 2 cos 2x =− + 2 ln tan x sin3 x cos3 x sin2 2x 1 cos x x dx 2 5 + − = + ln tan cos x 3 cos3 x 2 sin2 x 2 2 sin3 x cos4 x  dx 1 1 x π − + =− + ln tan sin x 3 sin3 x 2 4 sin4 x cos x 1 8 dx =− − cot 2x sin4 x cos2 x 3 cos x sin3 x 3 x π dx 1 5 2 sin x − + ln tan + = − + 4 3 2 sin x 3 sin x 2 cos x 2 2 4 sin x cos3 x 8 dx = −8 cot 2x − cot3 2x 3 sin4 x cos4 x

2.527

2.532

2.527 1. 2.

Sines and cosines of multiple angles and functions of the argument

 tanp−1 x − tanp−2 x dx [p = 1] tan x dx = p−1  n n  1 − (−1)n ln cos x tan2n+1 x dx = (−1)n+k k 2k cos2k x k=1 n  (−1)k−1 tan2n−2k+2 x − (−1)n ln cos x = 2n − 2k + 2



p

k=1



3. 4. 5.

161

n 

tan2n−2k+1 x + (−1)n x 2n − 2k + 1 k=1   cotp−1 x p − cotp−2 x dx cot x dx = − [p = 1] p−1  n n  1 + (−1)n ln sin x cot2n+1 x dx = (−1)n+k+1 k 2k sin2k x k=1 n  cot2n−2k+2 x + (−1)n ln sin x = (−1)k 2n − 2k + 2 tan2n x dx =

GU (331)(12)

k=1

 cot2n x dx =

6.

(−1)k−1

n  k=1

(−1)k

cot2n−2k+1 x + (−1)n x 2n − 2k + 1

GU (331)(14)

For special formulas for p = 1, 2, 3, 4, see 2.526 17, 2.526 33, 2.526 22, 2.526 38, 2.526 27, 2.526 43, 2.526 32, and 2.526 48.

2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated functions of the argument 2.531 1. 2. 2.532



1 sin(ax + b) dx = − cos(ax + b) a  1 cos(ax + b) dx = − sin(ax + b) a 

1.

sin(ax + b) sin(cx + d) dx =

 2.8

sin[(a − c)x + b − d] sin[(a + c)x + b + d] − 2(a − c) 2(a + c)   2 a = c2

sin(ax + b) cos(cx + d) dx = −

cos[(a − c)x + b − d] cos[(a + c)x + b + d] − 2(a − c) 2(a + c)   2 a = c2

162

Trigonometric Functions

 3.

cos(ax + b) cos(cx + d) dx =

5. 6. 2.533 1.

8

2.8

GU (332)(3)



 2  cos(a + b)x cos(a − b)x − a = b2 2(a + b) 2(a − b) ⎧  cos(b + c − a)x 1 ⎨ cos(a − b + c)x + sin ax sin bx sin cx dx = − 4 ⎩ a−b+c b+c−a ⎫ cos(a + b − c)x cos(a + b + c)x ⎬ − + a+b−c a+b+c ⎭ sin ax cos bx dx = −



 4.

⎧ 1 ⎨ cos(a + b + c)x

PE (376)

cos(b + c − a)x b+c−a ⎫ cos(a + b − c)x cos(a + c − b)x ⎬ + + a+b−c a+c−b ⎭

sin ax cos bx cos cx dx = −

3.

cos ax sin bx sin cx dx =

 5.

sin[(a − c)x + b − d] sin[(a + c)x + b + d] + 2(a − c) 2(a + c)   2 a = c2

For c = a:  sin(2ax + b + d) x sin(ax + b) sin(ax + d) dx = cos(b − d) − 2 4a  cos(2ax + b + d) x sin(ax + b) cos(ax + d) dx = sin(b − d) − 2 4a  x sin(2ax + b + d) cos(ax + b) cos(ax + d) dx = cos(b − d) + 2 4a

4.

2.533

cos ax cos bx cos cx dx =

4⎩

a+b+c

⎧ 1 ⎨ sin(a + b − c)x

−

PE (378)

sin(a + c − b)x a+c−b ⎫ sin(a + b + c)x sin(b + c − a)x ⎬ − − a+b+c b+c−a ⎭ 4⎩

a+b−c

⎧ 1 ⎨ sin(a + b + c)x

+

PE (379)

sin(b + c − a)x b+c−a ⎫ sin(a + c − b)x sin(a + b − c)x ⎬ + + a+c−b a+b−c ⎭ 4⎩

a+b+c

+

PE (377)

2.535

2.534

Sines and cosines of multiple angles and functions of the argument



1.  2. 2.535

 



cos px + i sin px dx = −2i cos nx 1 sin x sin ax dx = p+a p

1. 2.

cos px + i sin px dx = −2 sin nx

⎧ ⎪ ⎪ ⎪ ⎨

= (2n + 1)

×

z p+n−1 dz 1 − z 2n

[z = cos x + i sin x]

Pe (374)

[z = cos x + i sin x]

Pe (373)



 p p−1 x cos(a − 1)x dx − sin x cos ax + p sin

sinp x sin(2n + 1)x dx





z p+n−1 dz 1 − z 2n

⎪ ⎪ ⎪ ⎩

163

GU (332)(5a)

   (2n + 1)2 − 12 (2n + 1)2 − 32 . . .   n  . . . (2n + 1)2 − (2k − 1)2 p+1 k sin x dx + (−1) (2k + 1)! k=1

⎫ ⎬ sin2k+p+1 x dx ⎭ TI (299)

⎧ ⎡   ⎨ n−1  (−1)k−1 Γ p+1 + n − 2k Γ(p + 1) 2 ⎣

sinp−2k x cos(2n − 2k + 1)x = ⎩ p+3 22k+1 Γ(p − 2k + 1) k=0 +n Γ 2 ⎤  p−1  Γ + n − 2k 2 + (−1)k 2k+2 sinp−2k−1 x sin(2n − 2k)x⎦ 2 Γ(p − 2k) ⎫  ⎬ (−1) Γ 2 − n + 2n sinp−2n+1 x dx ⎭ 2 Γ(p − 2n + 1) n

 p+3

GU (332)(5c)

164

Trigonometric Functions

⎧ ⎨ sinp+2 x

 sinp x sin 2nx dx = 2n

3.

+

2.536

⎩ p+2 

n−1 

(−1)k

k=1

⎫     ⎬ 4n2 − 22 4n2 − 42 . . . 4n2 − (2k)2 sin2k+p+2 x ⎭ (2k + 1)!(2k + p + 2)

TI (303) ⎧ p  n−1 ⎨ k−1  (−1) Γ 2 + n − 2k Γ(p + 1) sinp−2k x cos(2n − 2k)x = p 2k+1 ⎩ 2 Γ(p − 2k + 1) +n+1 Γ k=0 2 ⎫ p  ⎬ k (−1) Γ 2 + n − 2k − 1 p−2k−1 − sin x sin(2n − 2k − 1)x ⎭ 22k+2 Γ(p − 2k)

[p is not equal to − 2, −4, . . . , −2n] GU (332)(5c)

2.536



1.  2.

1 sin x cos ax dx = p+1 p

  p−1 x sin(a − 1)x dx sin x sin ax − p sin p

GU (332)(6a)

sinp x cos(2n + 1)x dx

     n 2 2 (2n + 1)2 − 32 . . . (2n + 1)2 − (2k − 1)2 sinp+1 x  k (2n + 1) − 1 + (−1) = p+1 (2k)!(2k + p + 1) k=1

× sin2k+p+1 x TI (301) ⎧ ⎡  p+1  n−1 ⎨ k  (−1) Γ 2 + n − 2k Γ(p + 1) ⎣  sinp−2k x sin(2n − 2k + 1)x =  p+3 ⎩ 22k+1 Γ(p − 2k + 1) Γ 2 +n k=0

⎤   (−1)k Γ p−1 + n − 2k 2 sinp−2k−1 x cos(2n − 2k)x⎦ + 22k+2 Γ(p − 2k) ⎫  ⎬ (−1) Γ 2 − n sinp−2n x cos x dx + 2n ⎭ 2 Γ(p − 2n + 1) n

 p+3

[p is not equal to −3, −5, . . . , −(2n + 1)] GU (332)(6c)

2.537

Sines and cosines of multiple angles and functions of the argument

165

 sinp x cos 2nx dx

3.

 =

p

sin x dx +

n 

2 k 4n

(−1)

k=1

    · 4n2 − 22 . . . 4n2 − (2k − 2)2 sin2k+p x dx (2k)!

⎧ ⎡ p  n−1 Γ(p + 1) ⎨  ⎣ (−1)k Γ 2 + n − 2k  sinp−2k x sin(2n − 2k)x = p 22k+1 Γ(p − 2k + 1) Γ 2 +n+1 ⎩

TI (300)

k=0

⎤   (−1)k Γ p2 + n − 2k − 1 sinp−2k−1 x cos(2n − 2k − 1)x⎦ + 22k+2 Γ(p − 2k) ⎫   ⎬ (−1)n Γ p2 − n + 1 p−2n sin x dx + ⎭ 22n Γ(p − 2n + 1) GU (332)(6c)

2.537

 cosp x sin ax dx =

1.  2.

1 p+a



 − cosp x cos ax + p cosp−1 x sin(a − 1)x dx

GU (332)(7a)

cosp x sin(2n + 1)x dx ⎧ ⎨ cosp+1 x = (−1)n+1 ⎩ p+1

+

n  k=1

⎫      ⎬ 2 2 2 2 2 2 (2n + 1) − 1 (2n + 1) − 3 . . . (2n + 1) − (2k − 1) cos2k+p+1 x (−1)k ⎭ (2k)!(2k + p + 1) ⎧ ⎨

=

n−1 

 p+1



TI (295)

Γ 2 +n−k Γ(p + 1)

− cosp−k x cos(2n − k + 1)x ⎩ p+3 22k+1 Γ(p − 2k + 1) k=0 +n Γ 2 ⎫    ⎬ Γ p+3 2 + n cosp−n x sin(n + 1)x dx ⎭ 2 Γ(p − n + 1) [p is not equal to − 3, −5, . . . , −(2n + 1)] GU (332)(7b)a

166

Trigonometric Functions

⎧ ⎨ cosp+2 x

 cosp x sin 2nx dx = (−1)n

3.

+

⎩ p+2 

n−1 

(−1)k

k=1

⎫     ⎬ 4n2 − 22 4n2 − 42 . . . 4n2 − (2k)2 cos2k+p+2 x ⎭ (2k + 1)!(2k + p + 2) ⎧ ⎨

=

2.538

n−1 

p

TI (297)



Γ 2 +n−k Γ(p + 1) − p cosp−k x cos(2n − k)x k+1 Γ(p − k + 1) ⎩ 2 +n+1 Γ k=0 2 ⎫ p   ⎬ Γ 2 +1 + n cosp−n x sin nx dx ⎭ 2 Γ(p − n + 1) [p is not equal to − 2, −4, . . . , −2n] GU (332)(7b)a

2.538  1. cosp x cos ax dx =  2.

1 p+a



 cosp x sin ax + p cosp−1 x cos(a − 1)x dx

cosp x cos(2n + 1)x dx = (−1)n (2n + 1)

+

n  k=1

 ×

(−1)k

⎧ ⎨ ⎩

GU (332)(8a)

cosp+1 x dx

     (2n + 1)2 − 12 (2n + 1)2 − 32 . . . (2n + 1)2 − (2k − 1)2 (2k + 1)!

⎫ ⎬ cos2k+p+1 x dx ⎭

⎧  p+1  n−1 Γ(p + 1) ⎨  Γ 2 + n − k  cosp−k x sin(2n − k + 1)x =  p+3 2k+1 Γ(p − k + 1) Γ 2 +n ⎩

TI (293)

k=0

⎫    ⎬ Γ p+3 2 cosp−n x cos(n + 1)x dx + n ⎭ 2 Γ(p − n + 1) GU (332)(8b)a

2.541

Sines and cosines of multiple angles and functions of the argument

167

 cosp x cos 2nx dx    2   2  n 2 2 2   n p k 4n 4n − 2 . . . 4n − (2k − 2) 2k+p = (−1) (−1) x dx cos x dx + cos (2k)!

3.

k=1

=

⎧ ⎨ n−1 

TI (294)

p



Γ 2 +n−k Γ(p + 1) p  cosp−k x sin(2n − k)x k+1 Γ(p − k + 1) Γ 2 + n + 1 ⎩ k=0 2 p

+



Γ 2 +1 2n Γ(p − n + 1)



⎫ ⎬ cosp−n x cos nx dx ⎭ GU (332)(8b)a

2.539



1.  2.  3.  4.  5.  6.  7.  8. 2.541 1. 2.

 sin 2kx sin(2n + 1)x dx = 2 +x sin x 2k n

k=1

sin 2nx dx = 2 sin x

n  sin(2k − 1)x

2k − 1

k=1

GU (332)(5e)

 cos 2kx cos(2n + 1)x dx = 2 + ln sin x sin x 2k n

k=1

cos 2nx dx = 2 sin x

n  k=1

x cos(2k − 1)x + ln tan 2k − 1 2

GI (332)(6e)

 cos 2kx sin(2n + 1)x dx = 2 + (−1)n+1 ln cos x (−1)n−k+1 cos x 2k n

k=1

sin 2nx dx = 2 cos x

n 

(−1)n−k+1

k=1

cos(2k − 1)x 2k − 1

GU (332)(7d)

 sin 2kx cos(2n + 1)x dx = 2 + (−1)n x (−1)n−k cos x 2k n

k=1

cos 2nx dx = 2 cos x

n 

(−1)n−k

k=1

π x sin(2k − 1)x + (−1)n ln tan + . 2k − 1 4 2

GU (332)(8d)



1 sinn x sin nx n  1 sin(n + 1)x cosn−1 x dx = − cosn x cos nx n sin(n + 1)x sinn−1 x dx =

BI (71)(1)a BI (71)(2)a

168

Trigonometric Functions

2.542

 3. 4. 5. 6. 2.542

1 sinn x cos nx n  1 cos(n + 1)x cosn−1 x dx = cosn x sin nx n  , + π π 1 − x sinn−1 x dx = sinn x cos n −x sin (n + 1) 2 n 2  , + π π 1 − x sinn−1 x dx = − sinn x sin n −x cos (n + 1) 2 n 2 cos(n + 1)x sinn−1 x dx =



1.

2. 2.543 1.

2. 2.544

For n = 2:  sin 2x dx = 2 ln sin x sin2 x 

sin 2x dx 2 = cosn x (n − 2) cosn−2 x For n = 2:  sin 2x dx = −2 ln cos x cos2 x



1.  2.  3.  4.  5.  6.  7.  8.  9.

2 sin 2x dx = − n sin x (n − 2) sinn−2 x

cos 2x dx x = 2 cos x + ln tan sin x 2 cos 2x dx = − cot x − 2x sin2 x x cos x 3 cos 2x dx =− − ln tan 2 sin3 x 2 sin2 x 2  π x cos 2x dx = 2 sin x − ln tan + cos x 4 2 cos 2x dx = 2x − tan x cos2 x π x sin x 3 cos 2x dx = − + ln tan + cos3 x 2 cos2 x 2 4 2 sin 3x dx = x + sin 2x sin x sin 3x x dx = 3 ln tan + 4 cos x 2 2 sin x sin 3x dx = −3 cot x − 4x sin3 x

BI (71)(3)a BI (71)(4)a BI (71)(5)a BI (71)(6)a

2.548

Sines and cosines of multiple angles and functions of the argument

169

2.545  sin 3x 4 1 1. dx = − n n−3 cos x (n − 3) cos x (n − 1) cosn−1 x For n = 1 and n = 3:  sin 3x 2. dx = 2 sin2 x + ln cos x cos x  sin 3x 1 3. dx = − − 4 ln cos x 3 cos x 2 cos2 x 2.546  cos 3x 1 4 1. − dx = n−3 sinn x (n − 3) sin x (n − 1) sinn−1 x For n = 1 and n = 3:  cos 3x dx = −2 sin2 x + ln sin x 2. sin x  cos 3x 1 3. dx = − − 4 ln sin x 3 sin x 2 sin2 x 2.547  1.  2.  3.  4. 2.548 1.



 (k − n)π x    + sin 2n 2k + 1 sinm x dx 1  2(2n + 1) 2   = π ln (−1)n+k cosm k+n+1 x sin(2n + 1)x 2n + 1 2(2n + 1) k=0 π− sin (2n + 1) 2 

2.

3.

  sin nx sin(n − 1)x dx sin(n − 2)x dx dx = 2 − p p−1 cos x cos x cosp x cos 3x dx = sin 2x − x cos x π x cos 3x dx = 4 sin x − 3 ln tan + cos2 x 4 2 cos 3x dx = 4x − 3 tan x cos3 x

(−1)n sin2m x dx = sin 2nx 2n

 ln cos x +

n−1 

(−1)k cos2m

k=1

⎧  π x (−1)n ⎨ sin2m+1 x dx = − ln tan sin 2nx 2n ⎩ 4 2 +

n−1 

(−1)k cos2m+1

k=1

[m a natural number ≤ 2n]

 kπ 2 kπ 2 ln cos x − sin 2n 2n [m a natural number ≤ n]

⎫ 



 n+k x n−k x ⎬ kπ ln tan π− π− tan 2n 4n 2 4n 2 ⎭ [m a natural number < n]

TI (378)

TI (379)

170

Trigonometric Functions

 4.

π x  (−1) sin x dx = − + (−1)k ln tan cos(2n + 1)x 2n + 1 ⎩ 4 2 2m

n+1

× cos2m  5.

⎧ ⎨

2.549

n

k=1 ⎫ 



⎬ 2n + 2k + 1 x 2n − 2k + 1 x kπ ln tan π− π− tan 2n + 1 4(2n + 1) 2 2(2n + 1) 2 ⎭

(−1)n+1 sin2m+1 x dx = cos(2n + 1)x 2n + 1

 ln cos x +

n  k=1

TI (381) [m a natural number ≤ n]

 kπ kπ ln cos2 x − sin2 (−1)k cos2m+1 2n + 1 2n + 1

[m a natural number ≤ n]  2k − 2n + 1 x    π+ sin 2n−1 2k + 1 1  sinm x dx 8n 2   = π ln (−1)n+k cosm 2k + 2n + 1 x cos 2nx 2n 4n k=0 π− sin 8n 2 

6.

 7.

 8.

1 cos2m+1 x dx = sin(2n + 1)x 2n + 1

 ln sin x +

k=1

1 cos2m x dx x = ln tan sin(2n + 1)x 2n + 1 ⎩ 2 n 

(−1)k cos2m

 10.



1 cos2m+1 x dx = sin 2nx 2n

1 cos2m x dx = sin 2nx 2n

ln tan

[m a natural number < 2n] TI (377)

 kπ kπ ln sin2 x − sin2 2n + 1 2n + 1 [m a natural number ≤ n]

k=1

9.

(−1)k cos2m+1

⎧ ⎨

+



n 

x + 2

 ln sin x +



n−1 

(−1)k cos2m+1

k=1

TI (375) [m a natural number ≤ n] 



 x kπ x kπ kπ ln tan + − tan 2n 2 4 2 4n

[m a natural number < n]

 kπ kπ ln sin2 x − sin2 2n 2n

TI (374)

[m a natural number ≤ n]  2k + 1 x π+ sin 2k + 1 4n 2   π ln 2k + 1 x 2n π− sin 4n 2

TI (373)

n−1 

(−1)k cos2m

k=1

n−1 1 cosm x dx = (−1)k cosm cos nx n k=0

[m is a natural number ≤ n] 2.549   π 2 S (x) 1. sin x dx = 2

TI (376)

⎫ 



⎬ x kπ kπ x kπ ln tan + − tan 2n + 1 2 4n + 2 2 4n + 2 ⎭



11.

TI (382)a

TI (372)

2.552

Rational functions of sine and cosine



 2

2.

cos x dx = 

3.

11



π C (x) 2

  sin ax2 + 2bx + c dx =







cos ax2 + 2bx + c dx =

4.11



π 2a





 ax + b ax + b ac − b2 ac − b2 √ √ S C cos + sin a a a a

[a > 0]



 2 ax + b ax + b π ac − b ac − b2 √ √ C S cos − sin 2a a a a a [a > 0]



5. 6.

171

x (sin ln x − cos ln x) 2  x cos ln x dx = (sin ln x + cos ln x) 2 sin ln x dx =

PE (444) PE (445)

2.55–2.56 Rational functions of the sine and cosine 2.551  1.

⎡ A + B sin x 1 ⎣ (Ab − aB) cos x n dx = (n − 1) (a2 − b2 ) (a + b sin x)n−1 (a + b sin x)  +

⎤

(Aa − Bb)(n − 1) + (aB − bA)(n − 2) sin x n−1

(a + b sin x)

dx⎦ TI (358)a

2. 3.

2.552 1.

2.

For n = 1:   B Ab − aB A + B sin x dx dx = x + a + b sin x b b a + b sin x  a tan x2 + b 2 dx =√ arctan √ a + b sin x a2 − b 2 a2 − b2√ x a tan 2 + b − b2 − a2 1 =√ ln  b2 − a2 a tan x + b + b2 − a2 2 

(see 2.551 3)  

a2 > b 2 a2 < b 2

TI (342)

 

 B dx A + B cos x n dx = − n n−1 + A (a + b sin x) (a + b sin x) (n − 1)b (a + b sin x) (see 2.552 3)

TI (361)

(see 2.551 3)

TI (344)

For n = 1:   B dx A + B cos x dx = ln (a + b sin x) + A a + b sin x b a + b sin x

172

Trigonometric Functions

 3.

2.553

⎡ b cos x dx 1 ⎣ n = (n − 1) (a2 − b2 ) (a + b sin x)n−1 (a + b sin x) ⎤  (n − 1)a − (n − 2)b sin x ⎦ + dx n−1 (a + b sin x)

(see 2.551 1) TI (359)

2.553



1.

 B dx A + B sin x dx = + A n n n−1 (a + b cos x) (a + b cos x) (n − 1)b (a + b cos x) (see 2.554 3)

2.

3.∗

For n = 1:   B dx A + B sin x dx = − ln (a + b cos x) + A a + b cos x b a + b cos x 

2.554  1.

TI (355)

(see 2.553 3∗ ) dx a + b cos x    (a − b) tan x2 2 √ arctan =√ a2 − b 2 a2 − b 2 ! ! ! (b − a) tan  x  + √b2 − ba ! 2 ! !  x2  √ =√ ln ! ! 2 2 2 a ! a −b (b − a) tan 2 − b − b !     (a − b) tan x2 2 √ =√ arctanh b 2 − a2 b 2 − a2    (a − b) tan x2 2 √ =√ arccoth b 2 − a2 b 2 − a2

 

a2 > b 2 b 2 > a2

 

!  x !  , ! ! ! < b 2 − a2 !(b − a) tan 2 !  x !  + , ! ! b2 > a2 , !(b − a) tan ! > b 2 − a2 2 (compare with 2.551 3) +

b 2 > a2 ,

⎡ A + B cos x 1 ⎣ (aB − Ab) sin x n dx = (n − 1) (a2 − b2 ) (a + b cos x)n−1 (a + b cos x)  +

TI (343)

(Aa − bB)(n − 1) + (n − 2)(aB − bA) cos x n−1

(a + b cos x)

⎤ dx⎦ TI (353)

2.557

2.

3.

Rational functions of sine and cosine

173

For n = 1:   B Ab − aB A + B cos x dx dx = x + (see 2.553 3) a + b cos x b b a + b cos x ⎧  ⎨ b sin x 1 dx n =− 2 2 (n − 1) (a − b ) ⎩ (a + b cos x)n−1 (a + b cos x) ⎫  (n − 1)a − (n − 2)b cos x ⎬ − dx (see 2.554 1) n−1 ⎭ (a + b cos x)

TI (341)

TI (354)

In integrating the functions in formulas 2.551 3 and 2.553 3, we may not take integration over points  the a in formula 2.551 3 or at which the integrand becomes infinite, that is, over the points x = arcsin − b  a over the points x = arccos − in formula 2.553 3. b 2.555 Formulas 2.551 3 and 2.553 3 are not applicable for a2 = b2 . Instead, we may use the following formulas in these cases:  n−2     n − 2 tan2k+1 π ∓ x A + B sin x 1 4 2 1. 2B n dx = − n−1 k 2 2k + 1 (1 ± sin x) k=0 π x

n−1 2k+1  n − 1 tan 4 ∓ 2 ± (A ∓ B) k 2k + 1 k=0

 2.



TI (361)a n−2 



 2k+1 π

π

 x

∓ 4−2 2k + 1 k=0    

n−1  n − 1 tan2k+1 π ∓ π − x 4 4 2 ± (A ∓ B) k 2k + 1

A + B cos x 1 n dx = n−1 2 (1 ± cos x)

2B

n−2 k

tan

4

k=0

TI (356)

3.11 4.

For n = 1 :  π x A + B sin x dx = ±Bx + (B ∓ A) tan ∓ 1 ± sin x 4 2  +  π π x , A + B cos x dx = ±Bx ± (A ∓ B) tan ∓ − 1 ± cos x 4 4 2

2.556  1. 2.

 

1 − a2 dx 1+a x tan = 2 arctan 1 − 2a cos x + a2 1−a 2

 1+a x (1 − a cos x) dx x + arctan tan = 1 − 2a cos x + a2 2 1−a 2

2.557  1.

1 dx n =  n 2 (a cos x + b sin x) (a + b2 )

 sin

n



dx

TI (250) TI (248)

[0 < a < 1,

|x| < π]

FI II 93

[0 < a < 1,

|x| < π]

FI II 93

a b (see 2.515)

x + arctan

MZ 173a

174

Trigonometric Functions

2.558

  ax − b ln sin x + arctan ab sin x dx = a sin x + b cos x a2 + b 2    ax + b ln sin x + arctan ab cos x dx = a cos x + b sin x a2 + b 2 1    ln tan 2 x + arctan ab dx √ = a cos x + b sin x a2 + b 2    cot x + arctan ab 1 a sin x − b cos x dx =+ 2 · 2 =− 2 + b2 2 a cos x + b sin x a a + b (a cos x + b sin x) 

2.

6

3. 4. 5.

MZ 174a

MZ 174a

2.558  A + B cos x + C sin x 1. n dx (a + b cos x + c sin x) (Bc − Cb) + (Ac − Ca) cos x − (Ab − Ba) sin x 1 = n−1 + (n − 1) (a2 − b2 − c2 ) 2 2 2 (n − 1) (a − b − c ) (a + b cos x + c sin x)  (n − 1)(Aa − Bb − Cc) − (n − 2) [(Ab − Ba) cos x − (Ac − Ca) sin x] × dx n−1 (a + b cos x + c sin x)   2 2 2

n = 1, a = b + c A n(Bb + Cc) Cb − Bc + Ca cos x − Ba sin x + = (−c cos x + b sin x) n + a (n − 1)a2 (n − 1)a (a + b cos x + c sin x) n−1 1 (n − 1)!  (2n − 2k − 3)!! · × n−k k (2n − 1)!! (n − k − 1)!a (a + b cos x + c sin x) k=0   n = 1, a2 = b2 + c2

11

2.

For n = 1 :  A + B cos x + C sin x Bc − Cb Bb + Cc dx = 2 ln (a + b cos x + c sin x) + 2 x a + b cos x + c sin x b + c2 b + c2

 dx Bb + Cc a + A− 2 2 b +c a + b cos x + c sin x 

3.

4.

(see 2.558 4) GU (331)(18)



dx d(x − α) n = n, (a + b cos x + c sin x) [a + r cos(x − α)] where b = r cos α, c = r sin α (see 2.554 3)  dx a + b cos x + c sin x (a − b) tan x2 + c 2 arctan √ a2 − b 2 − c 2 a2 − b 2 − c 2 √ (a − b) tan x2 + c − b2 + c2 − a2 1 =√ ln  b2 + c2 − a2 (a − b) tan x + c + b2 + c2 − a2 2 x 1  = ln a + c · tan c 2 −2 = x c + (a − b) tan 2

=√

 

a2 > b 2 + c 2 a2 < b 2 + c 2

 

TI (253), FI II 94 TI (253)a

[a = b] 

a2 = b 2 + c 2



TI (253)a

2.561

Rational functions of sine and cosine

   c (a sin x − c cos x) x 1 2 = c3 a (1 + cos x) + c sin x − a ln a + c tan 2 [a (1 + cos x) + c sin x]  A + B cos x + C sin x dx (a1 + b1 cos x + c1 sin x) (a2 + b2 cos x+ c2 sin x)  a1 + b1 cos x + c1 sin x dx dx = A0 ln + A1 + A2 a2 + b2 cos x + c + 2 sin x a1 + b1 cos x + c1 sin x a2 + b2 cos x + c2 sin x GU (331)(19) (see 2.558 4)

2.559  1. 2.

175

dx

where ! ! !A B C! ! ! !a1 b1 c1 ! ! ! !a2 b2 c2 ! A0 = ! ! ! ! , ! ! !a1 b1 !2 !b1 c1 !2 !c1 a1 !2 ! ! −! ! +! ! !a2 b2 ! !b2 c2 ! !c2 a2 ! ! ! ! ! !! !! !! C B ! ! C A ! ! A B !! ! ! ! ! !! !! !!c2 b2 ! !c2 a2 ! !a2 b2 !! ! ! ! a1 b1 c1 !! ! ! a2 b2 c2 ! A2 = ! !2 ! !2 ! !2 , !a1 b1 ! ! ! ! ! ! ! − !b1 c1 ! + !c1 a1 ! !a2 b2 ! !b2 c2 ! !c2 a2 !  3.

! ! ! ! !! !! !! B C ! ! A C ! ! B A !! ! ! ! ! !! !! !!b1 c1 ! !a1 c1 ! !b1 a1 !! ! ! ! a1 b1 c1 !! ! ! a2 b2 c2 ! A1 = ! !2 ! !2 ! !2 , ! a1 b 1 ! ! ! ! ! ! ! − ! b 1 c 1 ! + ! c 1 a1 ! ! a2 b 2 ! ! b 2 c2 ! ! c 2 a2 !

%! ! a1 ! ! a2

!2 ! !c b1 !! + !! 1 b2 ! c2

!2 ! !b a1 !!

= !! 1 a2 ! b2

!2 & c1 !! c2 !

A cos2 x + 2B sin x cos x + C sin2 x dx a cos2 x + 2b sin x cos x + c sin2 x

⎧ ⎨ 1 = 2 [4Bb + (A − C)(a − c)]x + [(A − C)b − B(a − c)] 4b + (a − c)2 ⎩   × ln a cos2 x + 2b sin x cos x + c sin2 x ⎫ ⎬   + 2(A + C)b2 − 2Bb(a + c) + (aC − Ac)(a − c) f (x) ⎭

where

√ 1 c tan x + b − b2 − ac √ f (x) = √ ln 2 b2 − ac c tan x + b + b2 − ac c tan x + b 1 arctan √ =√ 2 ac − b ac − b2 1 =− c tan x + b

GU (331)(24)



 b2 > ac



 b2 < ac



b2 = ac



2.561   A x Ba − Ab dx (A + B sin x) dx = ln tan + 1. sin x (a + b sin x) a 2 a a + b sin x (see 2.551 3)

TI (348)

176

Trigonometric Functions

 2.

2.561

 A a + b cos x (A + B sin x) dx x = a ln tan + b ln 2 sin x (a + b cos x) a2 − 2 sin x b dx +B a + b cos x

(see 2.553 3) TI (349)

3. 4. 5.

2

2

2

2

For a = b (= 1) :

  A 1 (A + B sin x) dx x x = ln tan + + B tan sin x (a + b cos x) 2 2 1 + cos x 2

  (A + B sin x) dx A 1 x x = ln tan − − B cot sin x (1 − cos x) 2 2 1 − cos x 2

   (A + B sin x) dx 1 a + b sin x π x = 2 + − (Ab − aB) ln (Aa − Bb) ln tan cos x (a + b sin x) a − b2 4 2 cos x TI (346)

6. 7.

For a = b (= 1):  π x A±B (A + B sin x) dx A∓B = ln tan + ∓ cos x (1 ± sin x) 2 4 2 2 (1 ± sin x)   π x B a + b cos x A (A + B sin x) dx = ln tan + + ln cos x (a + b cos x) a  4 2 a cos x Ab dx − a a + b cos x 

8.  9.

10. 11.

A x B a + b sin x Ab (A + B cos x) dx = ln tan − ln − sin x (a + b sin x) a 2 a sin x a



(see 2.553 3) TI (351)a

dx a + b sin x

(see 2.551 3)

 1 a + b cos x (A + B cos x) dx x = 2 + (Ab − Ba) ln (Aa − Bb) ln tan sin x (a + b cos x) a − b2 2 sin x

(see 2.551 3)

13.

TI (345)

For a2 = b2 (= 1) :  A∓B A±B x (A + B cos x) dx =± + ln tan sin x (1 ± cos x) 2 (1 ± cos x) 2 2

    A a + b sin x (A + B cos x) dx π x dx = 2 + − b ln a ln tan + B 2 cos x (a + b sin x) a −b 4 2 cos x a + b sin x 2

12.

TI (352)

TI (350)

2

For a = b (= 1):  π x A±B A∓B (A + B sin x) dx = ln tan + ∓ cos x (1 ± sin x) 2 4 2 2 (1 ± sin x)   π x Ba − Ab  A (A + B cos x) dx dx = ln tan + + cos x (a + b cos x) a 4 2 a a + b cos x (see 2.553 3)

TI (347)

2.563

2.562

Rational functions of sine and cosine



1.

 2.

 a+b tan x a   sign a a+b tan x = arctanh − a −a(a + b)   a+b sign a tan x arccoth − = a −a(a + b)

dx sign a arctan = a + b sin2 x a(a + b)

− sign a dx = arctan 2 a + b cos x a(a + b)

4. 5. 6. 2.563



  

b > −1 a



b < −1, a

a sin x < − b

b < −1, a

a sin x > − b



2



2

MZ 155

  

b > −1 a



b < −1, a

a cos x < − b

b < −1, a

cos2 x > −



2

a b



MZ 162

√

2 tan x

dx 1 = √ arctan 1 + sin2 x 2  dx = tan x 1 − sin2 x  √ dx 1 √ = − arctan 2 cot x 1 + cos2 x 2  dx = − cot x 1 − cos2 x



   1 b sin x cos x dx + (2a + b)  2 = 2a(a + b) a + b sin2 x a + b sin2 x a + b sin2 x



  (2a + b)

1.

2.



a+b cot x a   − sign a a+b = cot x arctanh − a −a(a + b)   − sign a a+b cot x = arccoth − a −a(a + b)

 3.



177

dx

dx 2

(a + b cos2 x)

=

1 2a(a + b)

(see 2.562 1)  b sin x cos x dx − a + b cos2 x a + b cos2 x

MZ 155

(see 2.562 2)

MZ 163

178

Trigonometric Functions

 3.

⎡



2 1 ⎣ 3 3 + = + arctan (p tan x)  3 8pa3 p2 p4 a + b sin2 x ⎤



2p tan x p tan x 3 1 2 2 + 3+ 2 − 4 + 1 − 2 − 2 tan2 x  2 ⎦ p p p p 1 + p2 tan2 x 1 + p2 tan2 x   b p2 = 1 + > 0 a ⎡

1 ⎣ 2 3 = 3 − 2 + 4 arctanh (q tan x) 3 8qa q q ⎤



2q tan x q tan x 3 1 2 2 + 3− 2 − 4 + 1 + 2 + 2 tan2 x  2 ⎦ q q q q 1 − q 2 tan2 x 1 − q 2 tan2 x dx

 b q 2 = −1 − > 0, a  4.

2.564

a sin x < − ; b 2

a for sin x > − , change arctanh (q tan x) to arccoth (q tan x) b MZ 156

⎡

dx 3

(a + b cos2 x)

=−



2 1 ⎣ 3 3 + + arctan (p cot x) 8pa3 p2 p4

⎤



2p cot x p cot x 3 1 2 2 + 3+ 2 − 4 + 1 − 2 − 2 cot2 x  2 ⎦ p p p p 1 + p2 cot2 x 1 + p2 cot2 x   b 2 p =1+ >0 a ⎡

1 ⎣ 2 3 =− 3 − 2 + 4 arctanh (q cot x) 8qa3 q q ⎤



2p cot x q cot x 2 3 1 2 + 1 + 2 + 2 cot2 x  + 3− 2 − 4 2 ⎦ q q q q 1 − q 2 cot2 x 1 − q 2 cot2 x

 b q 2 = −1 − > 0, a



2

a cos x < − ; b 2

a for cos x > − , change arctanh (q cot x) to arccoth (q cot x) b 2

MZ 163a

2.564    ln cos2 x + m2 sin2 x tan x dx = 1. 2 (m2 − 1) 1 + m2 tan2 x  tan α − tan x dx = sin 2α ln sin(x + α) − x cos 2α 2. tan α + tan x  1 tan x dx = 2 3. {bx − a ln (a cos x + b sin x)} a + b tan x a + b2 %  &   b b dx 1 x− arctan tan x 4. = a−b a a a + b tan2 x

LA 210 (10) LA 210 (11)a PE (335)

PE (334)



2.571

Integrals with

2.57 Integrals containing

√

a ± b sin x or

√ a ± b cos x

179

√ √ a ± b sin x or a ± b cos x

Notation:  α = arcsin  γ = arcsin 2.571



1.

 2.

3.

b (1 − sin x) , a+b . (a + b) (1 − cos x) δ = arcsin , 2 (a − b cos x)

β = arcsin

b (1 − cos x) , a+b

dx −2 √ =√ F (α, r) a + b sin x a + b

1 2 F β, =− b r

+ a > b > 0, + 0 < |a| < b,

 r=

2b a+b

π, π ≤x< 2 2 π, a − arcsin < x < b 2

−

BY (288.00, 288.50)

sin x dx √ a + b sin x

√ + π π, 2a 2 a+b a > b > 0, − ≤ x < BY (288.03) E (α, r) = √ F (α, r) − 2 2 b b a + b 



 + a π, 2 1 1 0 < |a| < b, − arcsin < x < BY (288.54) = F β, − 2 E β, b 2 b r r   √  √ 2 2a2 + b2 sin2 x dx 4a a + b 2 √ √ cos x a + b sin x E (α, r) − = F (α, r) − 2 2 3b 3b + a + b sin x 3b a + b π π, a > b > 0, − ≤ x < 2 2 

 √ 1 1 2 4a 2a + b 2 E β, F β, cos x a + b sin x = − − b 3b r 3b r + 3b π, a 0 < |a| < b, − arcsin < x < b 2 

4.

 5.



1 − sin x , 2

BY (288.03, 288.54)

x

dx 2 √ ,r =√ F 2 a + b cos x a + b

1 2 = F γ, b r

[a > b > 0, + b ≥ |a| > 0,

0 ≤ x ≤ π]  a , 0 ≤ x < arccos − b BY (289.00)

dx 2 √ =√ F (δ, r) a − b cos x a+b

[a > b > 0,

0 ≤ x ≤ π]

BY (291.00)

180

Trigonometric Functions

 6.

2.572

 x   x cos x dx 2 √ ,r − aF ,r = √ (a + b) E 2 2 a + b cos x b a + b

0 ≤ x ≤ π]

[a > b > 0, 



 2 1 1 = 2 E γ, − F γ, b r r

BY (289.03)

+ b > |a| > 0,

 a , 0 ≤ x < arccos − b

BY (290.04)



7.6

8.

    cos x dx 2 √ = √ (b − a) Π δ, r2 , r + a F (δ, r) a − b cos x b a+b BY (291.03) [a > b > 0, 0 ≤ x ≤ π]      2 √  2  x 2 x cos x dx 2 √ 2a + b2 F , r − 2a(a + b) E ,r sin x a + b cos x + = 2√ 2 2 3b a + b cos x 3b a + b [a > b > 0, 0 ≤ x ≤ π] 1 = 3b

BY (289.03) 



 √ 2 1 1 2 sin x a + b cos x (2a + b) F γ, − 4a E γ, + b r 3b +r  a , b ≥ |a| > 0, 0 ≤ x < arccos − b BY (290.04)

 9.

   2 cos2 x dx 2 √ = 2√ 2a + b2 F (δ, r) − 2a(a + b) E (δ, r) a − b cos x 3b a + b a + b cos x 2 + sin x √ [a > b > 0, ] 3b a − b cos x BY (291.04)a

 2.572

tan2 x dx √ a + b sin x a 1 √ F (α, r) + E (α, r) =√ a+b (a − b) a + b b − a sin x √ − 2 a + b sin x 2 (a − b ) cos x

 2a + b 1 2 1 ab F β, E β, = + 2 b 2(a + b) r a − b2 r √ b − a sin x a + b sin x − 2 (a − b2 ) cos x

+ π, π 0 < b < a, − < x < 2 2 + 0 < |a| < b,

− arcsin

π, a <x< b 2

BY(288.08, 288.58)

2.573  1.

1 − sin x dx 2 ·√ = 1 + sin x a + b sin x a − b

√

⎫ ⎬  π x √ − a + b E (α, r) − tan a + b sin x ⎭ 4 2 + π π, 0 < b < a, − ≤ x < 2 2

BY (288.07)

2.575

Integrals with

 2.

2.574

(2 −

 2.

a ± b sin x or

√ a ± b cos x

181

√ 1 − cos x dx x√ 2 a + b x 2 √ tan E ,r a + b cos x − = 1 + cos x a + b cos x a − b 2 a−b 2 [a > b > 0, 0 ≤ x < π]



1.

√

p2

+

p2

  dx 1 √ Π α, p2 , r =− a+b sin x) a + b sin x + 0 < b < a,

dx 1 √ =− a+b (a + b − p2 b + p2 b sin x) a + b sin x



2 1 Π β, p2 , b + r



x

(2 −

4.

− arcsin

π, a <x< b 2 BY (288.52)





π, π ≤x< 2 2 BY (288.02)



0 < |a| < b,

3.

−

BY (289.07)

p2

+

p2

dx 1 √ , p2 , r =√ Π 2 cos x) a + b cos x a+b



BY (289.02) [a > b > 0, 0 ≤ x < π]

2 dx 1 √ Π γ, p2 , √ = 2 2 r (a + b − p b + p b cos x) a + b cos x (a + b) b +  a , b ≥ |a| > 0, 0 ≤ x < arccos − b

√

BY (290.02)

2.575 1.



2b cos x dx 2 < √ √ = 2 E (α, r) − 2 ) a + b sin x 3 a+b (a − b (a − b) + (a + b sin x) 0 < b < a,

−

π, π ≤x< 2 2

BY (288.05) 

 2b 1 2 1 cos x 1 2b F β, = E β, ·√ − + 2 b b 2 − a2 r a+b b − a2 a + b sin x +r π, a 0 < |a| < b, − arcsin < x < b 2 BY (288.56)

182

Trigonometric Functions

 2.

2.575

 2   dx 2 < a − b2 F (α, r) − 4a(a + b) E (α, r) = 2√ 2 2 5 3 (a − b ) a + b (a + b sin x)   2b 5a2 − b2 + 4ab sin x < + cos x 2 3 3 (a2 − b2 ) (a + b sin x) + π π, 0 < b < a, − ≤ x < 2 2 BY (288.05) 



 2 1 1 1 =− (3a − b)(a − b) F β, + 8ab E β, 2 2 2 b r r 3 (a − b )   2b a2 − b2 + 4a (a + b sin x) < + cos x 2 3 3 (a2 − b2 ) (a + b sin x)

+ 0 < |a| < b,

− arcsin

a π, <x< b 2

BY (288.56)

 3.

x 2b dx 2 sin x < √ ,r − 2 = ·√ E 2 2 a −b 3 (a − b) a + b a + b cos x (a + b cos x) 1 = 2 a − b2

[a > b > 0,

0 ≤ x ≤ π]

BY (289.05) 



 2 sin x 1 1 2b ·√ (a − b) F γ, + 2b E γ, + 2 b r r b − a2 a + bcos x , + a b ≥ |a| > 0, 0 ≤ x < arccos − b BY (290.06)

 4.

dx 2 < √ = E (δ, r) 3 (a − b) a + b (a − b cos x)

[a > b > 0,

0 ≤ x ≤ π]

(291.01)

2.578

Integrals with



√ 2 a+b

dx

−

a ± b sin x or

√ a ± b cos x

183

 x  x  4a E , r − (a − b) F ,r 2 2

< = 2 5 3 (a2 − b2 ) (a + b cos x)

5.

√

2b 2

3 (a2 − b2 )

·

5a2 − b2 + 4ab cos x < sin x 3 (a + b cos x) [a > b > 0,





 2 1 1 1 = (a − b)(3a − b) F γ, + 8ab E γ, 2 b r r 3 (a2 − b2 ) +

  2b 5a2 − b2 + 4ab cos x sin x < 2 3 3 (ab − b2 ) (a + b cos x)

+ b ≥ |a| > 0,

0 ≤ x ≤ π] BY (289.05)

 a , 0 ≤ x < arccos − b

BY (290.06)

2.576



1.

√

x √ ,r a + b cos x dx = 2 a + b E 2

[a > b > 0,

0 ≤ x ≤ π] BY (289.01)





 2 1 1 = (a − b) F γ, + 2b E γ, b r r+

 a , 0 ≤ x < arccos − b

b ≥ |a| > 0,

BY (290.03)

 2. 2.577 1.

3

√

√ 2b sin x a − b cos x dx = 2 a + b E (δ, r) − √ a − b cos x

[a > b > 0,

0 ≤ x ≤ π]

 √ 2(a − b) a − b cos x 2ap √ dx = ,r Π δ, 1 + p cos x (a + b)(1 + p) (1 + p) a + b [a > b > 0,

0 ≤ x ≤ π,

 . 2.3



2(a − b) a − b cos x dx =  Π δ, −r2 , 1 + p cos x (1 + p)(a + b)

. 2(ap + b) (1 + p)(a + b)

2.578

tan x dx 1 √ arccos =√ 2 b −a a + b tan x

√

b−a √ cos x b

[b > a,

p = −1] BY (291.02)



[a > b > 0, 

BY (291.05)

0 ≤ x ≤ π,

b > 0]

p = −1] PE (333)

184

Trigonometric Functions

2.580

2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals 2.580   , +  dϕ dψ c √ 1. =2  ϕ = 2ψ + α, tan α = , p = b2 + c2 b a + b cos ϕ + c sin ϕ a − p + 2p cos2 ψ   dx dϕ  =2 √ 2. 2 2 A + Bx + Cx2 − Dx3 + Ex4 a + b cos ϕ + c sin ϕ + d cos ϕ + e sin ϕ cos ϕ + f sin ϕ + , ϕ tan = x, A = a + b + d, B = 2c + 2e, C = 2a − 2d + 4f, D = 2c − 2e, E = a − b + d 2  Forms containing 1 − k2 sin2 x  √ Notation: Δ = 1 − k 2 sin2 x, k  = 1 − k 2 2.581  1.

sinm x cosn xΔr dx

⎧ ⎨

  1 sinm−3 x cosn+1 xΔr+2 + (m + n − 2) + (m + r − 1)k 2 2 (m + n + r)k ⎩ ⎫   ⎬ × sinm−2 x cosn xΔr dx − (m − 3) sinm−4 x cosn xΔr dx ⎭ ⎧ ⎨ , + 1 m+1 n−3 r+2 2 2 = x cos xΔ + (n + r − 1)k − (m + n − 2)k sin (m + n + r)k2 ⎩ ⎫   ⎬ 2 sinm x cosn−4 xΔr dx × sinm x cosn−2 xΔr dx + (n − 3)k  ⎭

=

[m + n + r = 0]

2.

3.

4. 5.

For r = −3 and r = −5:  sinm x cosn x sinm−1 x cosn−1 x dx = 3 Δ k 2Δ  n − 1 sinm x cosn−2 x m − 1 sinm−2 x cosn x dx + dx − 2 k Δ k2 Δ  sinm x cosn x sinm−1 x cosn−1 x dx = Δ5 3k 2Δ3  m − 1 sinm−2 x cosn x n − 1 sinm x cosn−2 x − dx + dx 3k 2 Δ3 3k 2 Δ3 For m = 1 or n = 1:  2  cosn−1 xΔr+2 (n − 1)k  cosn−2 x sin xΔr dx − sin x cosn xΔr dx = − (n + r + 1)k 2 (n + r + 1)k 2   sinm−1 xΔr+2 m−1 m r sin x cos xΔ dx = − + sinm−2 x cos xΔr dx (m + r + 1)k 2 (m + r + 1)k 2 For m = 3 or n = 3:

2.583

Elliptic and pseudo-elliptic integrals



  (n + r + 1)k 2 cos2 x − (r + 2)k2 + n + 1 sin x cos xΔ dx = cosn−1 xΔr+2 4 (n + r + 1)(n + r + 3)k   2 (r + 2)k 2 + n + 1 (n − 1)k  − cosn−2 x sin xΔr dx (n + r + 1)(n + r + 3)k 4 3

6.



185

n

r

sinm x cos3 xΔr dx

7.

=

, + 2 (m + r + 1)k 2 sin2 x − (r + 2)k2 − (m + 1)k 

(m + r + 1)(m++ r + 3)k 4 , 2 (r + 2)k2 − (m − 1)k  (m − 1)  × sinm−1 xnm−1 xΔr+2 + sinm−2 x cos xΔr dx (m + r + 1)(m + r + 3)k 4

2.582      n−1  n−2 2 − k2 1 − k2 Δn−2 dx − Δn−4 dx 1. Δn dx = n n k2 + sin x cos x · Δn−2 n  2. 3.

n−3 1 dx − Δn−1 n − 1 k 2   sinn−3 x sinn x n − 2 1 + k 2 sinn−2 x dx = dx cos x · Δ + Δ (n − 1)k 2 n − 1 k2 Δ  n−3 sinn−4 x dx − (n − 1)k2 Δ 

4.

 5.

 6.

dx k2 sin x cos x n−22−k =− + n+1 Δ n − 1 k 2 (n − 1)k  2 Δn−1

2

cosn−3 x cosn x n − 2 2k 2 − 1 dx = sin x · Δ + 2 Δ (n − 1)k n − 1 k2 2  n−4 cos x n−3k dx + 2 n−1 k Δ 

(n − 2) 2 − k tann−3 x Δ tann x dx = 2 cos2 x −  Δ (n − 1)k (n − 1)k  2  n−3 tann−4 x − dx 2  Δ (n − 1)k



 cotn−1 x Δ n−2  cotn x dx = − − 2 − k2 2 Δ n − 1 cos x n − 1 n − 3  2 cotn−4 x k dx − n−1 Δ

LA (316)(1)a

dx Δn−3

LA 317(8)a

LA 316(1)a

cosn−2 x dx Δ

 2 





LA 316(2)a

tann−2 x dx Δ

LA 317(3)

cotn−2 x dx Δ

LA 317(6)

2.583  1. Δ dx = E (x, k)

186

Trigonometric Functions



2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13.

14. 15.

16.

17.

Δ cos x k  − ln (k cos x + Δ) 2 2k  Δ sin x 1 Δ cos x dx = + arcsin (k sin x) 2 2k  2 k Δ 2k 2 − 1 Δ sin2 x dx = − sin x cos x + 2 F (x, k) + E (x, k) 3 3k 3k 2  Δ3 Δ sin x cos x dx = − 2 3k  2 k Δ k2 + 1 sin x cos x − 2 F (x, k) + Δ cos2 x dx = E (x, k) 3 3k 3k 2  2k 2 sin2 x + 3k 2 − 1 3k 4 − 2k 2 − 1 Δ sin3 x dx = − Δ cos x + ln (k cos x + Δ) 2 8k 8k 3  2k 2 sin2 x − 1 1 Δ sin2 x cos x dx = Δ sin x + 3 arcsin (k sin x) 2 8k 8k  4 2 2 2 2k cos x + k k Δ sin x cos2 x dx = − Δ cos x + 3 ln (k cos x + Δ) 2 8k 8k  2 2 2 2 2k cos x + 2k + 1 4k − 1 Δ cos3 x dx = Δ sin x + arcsin (k sin x) 2 8k 8k 3  3k 2 sin2 x + 4k 2 − 1 Δ sin4 x dx = − Δ sin x cos x  4 15k2 2  2 2k − k − 1 8k 4 − 3k 2 − 2 − F (x, k) + E (x, k) 4 15k 15k 4  3k 4 sin4 x − k 2 sin2 x − 2 Δ Δ sin3 x cos x dx = 15k 4  3k 2 cos2 x − 2k 2 + 1 Δ sin2 x cos2 x dx = − Δ sin x cos x  15k 2   2 2 k 1 + k 2 k4 − k2 + 1 F (x, k) + E (x, k) − 15k 4 15k 4    3k 4 sin4 x − k 2 5k 2 + 1 sin2 x + 5k 2 − 2 3 Δ sin x cos x dx = − Δ 15k 4  3k2 cos2 x + 3k 2 + 1 Δ cos4 x dx = Δ sin x cos x  15k 2 2 2 2k  k  − 2k 2 3k 4 + 7k 2 − 2 F (x, k) + E (x, k) + 15k 4 15k 4    −8k 4 sin4 x − 2k 2 5k 2 − 1 sin2 x − 15k 4 + 4k 2 + 3 Δ sin5 x dx = Δ cos x 48k 4 6 4 2 5k − 3k − k − 1 ln (k cos x + Δ) + 16k 5  8k 4 sin4 x − 2k 2 sin2 x − 3 1 Δ sin x + arcsin (k sin x) Δ sin4 x cos x dx = 4 48k 16k 5 2

Δ sin x dx = −

2.583

2.583

Elliptic and pseudo-elliptic integrals



  8k4 sin4 x − 2k 2 k 2 + 1 sin2 x − 3k 4 + 2k 2 − 3 Δ sin x cos x dx = Δ cos x 48k 4   4 k k2 + 1 + ln (k cos x + Δ) 16k 5   −8k 4 sin4 x + 2k 2 6k 2 + 1 sin2 x − 6k 2 + 3 2 3 Δ sin x Δ sin x cos x dx = 48k 4 2 2k − 1 arcsin (k sin x) + 16k 5   −8k 4 sin4 x + 2k 2 7k 2 + 1 sin2 x − 3k 4 − 8k 2 + 3 4 Δ cos x Δ sin x cos x dx = 48k 4 6 k ln (k cos x + Δ) − 16k 5   8k4 sin4 x − 2k 2 12k 2 + 1 sin2 x + 24k 4 + 12k 2 − 3 5 Δ sin x Δ cos x dx = 48k 4 4 2 8k − 4k + 1 arcsin (k sin x) + 16k 5 2 k k2 2 2 Δ3 dx = F (x, F ) + Δ sin x cos x 1 + k  E (x, k) − 3 3 3 3

18.

 19.

 20.

 21.

 22. 

2



24. 25.

26. 27.

28.

29.

3k 2k 2 sin2 x + 3k 2 − 5 Δ cos x − ln (k cos x + Δ) 8 8k 4

Δ3 sin x dx =

23.

3 −2k 2 sin2 x + 5 Δ sin x + arcsin (k sin x) 8 8k   2 k  3 − 4k2 3k2 sin2 x + 4k 2 − 6 2 3 Δ sin x dx = Δ sin x cos x + F (x, k) 15 15k 2 4 2 8k − 13k + 3 E (x, k) − 15k 2  Δ5 Δ3 sin x cos x dx = − 2 5k   2 2 k k2 + 3 −3k sin2 x + k 2 + 5 3 2 Δ sin x cos x − Δ cos x dx = F (x, k) 15 15k 2 4 2 2k − 7k − 3 E (x, k) − 15k 2    8k4 sin4 x + 2k 2 5k 2 − 7 sin2 x + 15k 4 − 22k 2 + 3 3 3 Δ cos x Δ sin x dx = 48k 2 6 4 2 5k − 9k + 3k + 1 ln (k cos x + Δ) − 16k 3  −8k 4 sin4 x + 14k 2 sin2 x − 3 Δ3 sin2 x cos x dx = Δ sin x 48k 2 1 + arcsin (k sin x) 16k 3 Δ3 cos x dx =

187

188

Trigonometric Functions

  −8k 4 sin4 x + 2k 2 k 2 + 7 sin2 x + 3k 4 − 8k2 − 3 Δ sin x cos x dx = 48k 2 6 k ×Δ cos x + ln (k cos x + Δ) 16k 3    8k4 sin4 x − 2k 2 6k 2 + 7 sin2 x + 30k 2 + 3 3 3 Δ sin x Δ cos x dx = 48k 2 2 6k − 1 arcsin (k sin x) + 16k 3  1 Δ + cos x Δ dx = − ln + k ln k (k cos x + Δ) sin x 2 Δ − cos x  Δ dx k  Δ + k  sin x = ln + k arcsin (k sin x) cos x 2 Δ − k  sin x  Δ dx 2 = k  F (x, k) − E (x, k) − Δ cot x sin2 x  1 1 − Δ k Δ + k Δ dx = ln + ln sin x cos x 2 1+Δ 2 Δ − k  Δ dx = F (x, k) − E (x, k) + Δ tan x cos2 x   sin x k Δ + k Δ dx = Δ tan x dx = −Δ + ln cos x 2 Δ − k   1 1−Δ cos x Δ dx = Δ cot x dx = Δ + ln sin x 2 1+Δ  2 Δ + cos x Δ dx Δ cos x k ln =− + 3 2 4 Δ − cos x sin x 2 sin x  1 + k 2 Δ − k  sin x Δ dx −Δ − ln = 2 sin x 2k  Δ + k  sin x sin x cos x  Δ 1 Δ + cos x Δ dx = + ln 2 sin x cos x cos x 2 Δ − cos x  Δ dx Δ sin x 1 Δ + k  sin x = +  ln 3 2 cos x 2 cos x 4k Δ − k  sin x  Δ Δ sin x dx = − k ln (k cos x + Δ) cos2 x cos x  Δ cos x dx Δ − k arcsin (k sin x) =− sin x sin2 x  Δ sin2 x dx Δ sin x 2k2 − 1 k  Δ + k  sin x =− + arcsin (k sin x) + ln cos x 2 2k 2 Δ − k  sin x  Δ cos x k 2 + 1 1 Δ + cos x Δ cos2 x dx = + ln (k cos x + Δ) + ln sin x 2 2k 2 Δ − cos x        Δ dx 1 2 = −Δ cot3 x + k 2 − 3 Δ cot x + 2k  F (x, k) + k 2 − 2 E (x, k) 4 3 sin x 

30.

31.

32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

3

2

2.583

2.583

Elliptic and pseudo-elliptic integrals



48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66.

k2 − 2 1 + Δ Δ dx Δ k Δ + k ln ln + = − + 2 Δ − k 4 1−Δ sin3 x cos x 2 sin2 x

 2 1 Δ dx 1 + k = tan x − cot x Δ + 2 F (x, k) − E (x, k) 2 2 2 sin x cos2 x k k  Δ 1 1 + Δ 2 − k2 Δ + k Δ dx = − ln + ln 3 2 sin x cos x 2 cos x 2 1 − Δ 4k  Δ − k  + ,   2   2  1 Δ dx 2 2 2 k = tan x − 2k − 3 tan x Δ + 2k F (x, k) + k − 2 E (x, k) cos4 x 3k  2  Δ k2 Δ + k sin x Δ dx = + ln cos3 x 2 cos2 x 4k  Δ − k   cos x Δ k2 1 + Δ ln Δ dx = − + 4 1−Δ sin3 x 2 sin2 x   2 sin x Δ dx = tan2 xΔ dx = Δ tan x + F (x, k) − 2 E (x, k) cos2 x   cos2 x 2 cot2 xΔ dx = −Δ cot x + k  F (x, k) − 2 E (x, k) 2 Δ dx = sin x  sin3 x k 2 sin2 x + 3k 2 − 1 k Δ + k Δ dx = − Δ + ln 2 cos x 3k 2 Δ − k  k 2 sin2 x − 3k 2 − 1 cos3 x 1 1−Δ Δ dx = − Δ + ln sin x 3k 2 2 1+Δ  2   2  2 k − 3 sin x + 2 k k2 + 3 Δ dx Δ + cos x ln = cos xΔ + 16 Δ − cos x sin5 x 8 sin4 x  2   2 3 − k sin x + 1 Δ dx k Δ − k  sin x =− Δ − ln 4 3 2 Δ + k  sin x sin x cos x 3 sin x  Δ dx 3 sin2 x − 1 k 2 − 3 Δ − cos x ln = Δ + 4 Δ + cos x sin3 x cos2 x 2 sin2 x cos x  2 2k2 − 3 Δ + k  sin x Δ dx 3 sin x − 2 Δ− ln = 2 2 3 2 sin x cos x 4k  Δ − k  sin x sin x cos x  2  2  2k − 3 sin x − 3k 2 + 4 Δ dx 1 Δ + cos x = Δ + ln 2 4  3 sin x cos x 2 Δ − cos x 3k cos x  2  2  2 2k − 3 sin x − 4k + 5 Δ dx 4k 2 − 3 Δ + k  sin x = ln sin xΔ − 2 cos5 x Δ − k  sin x 8k  cos4 x 16k  3    2 2 2 4 2 − 2k + 1 k sin x + 3k − k + 1 sin x Δ dx = Δ 4 cos x 3k  2 cos3 x  cos x Δ3 4 Δ dx = − sin x 3 sin3 x  sin x 2k 2 − 1 Δ + k  sin x sin2 x Δ dx = Δ + − k arcsin (k sin x) ln cos3 x 2 cos2 x 4k  Δ − k  sin x

189

190

Trigonometric Functions

 67.  68.  69.  70.

 71.

2.584 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.



cos2 x cos x k2 + 1 Δ + cos x ln − k ln (k cos x + Δ) Δ dx = − Δ − 4 Δ − cos x sin3 x 2 sin2 x sin3 x sin2 x − 3 3k2 − 1 Δ dx = − Δ− ln (k cos x + Δ) 2 cos x 2 cos x 2k cos3 x 2k2 + 1 sin2 x + 2 Δ − arcsin (k sin x) Δ dx = − 2 sin x 2k sin2 x sin4 x 2k 2 sin2 x + 4k 2 − 1 Δ dx = − Δ sin x cos x 8k 2 4 2 k  Δ + k  sin x 8k − 4k − 1 ln arcsin (k sin x) + + 8k 3 2 Δ − k  sin x −2k 2 sin2 x + 5k 2 + 1 cos4 x Δ dx = Δ cos x sin x 8k 2 4 1 Δ + cos x 3k + 6k 2 − 1 + ln (k cos x + Δ) + ln 2 Δ − cos x 8k 3

dx = F (x, k) Δ  1 Δ − k cos x 1 sin x dx = ln = − ln (k cos x + Δ) Δ 2k Δ + k cos x k  1 1 k sin x cos x dx = arcsin (k sin x) = arctan Δ k k Δ  2 1 1 sin x dx = 2 F (x, k) − 2 E (x, k) Δ k k  Δ sin x cos x dx =− 2 Δ k  2 1 cos2 x dx k = 2 E (x, k) − 2 F (x, k) Δ k k  3 sin x dx cos xΔ 1 + k 2 = − ln (k cos x + Δ) Δ 2k 2 2k 3  sin2 x cos x dx sin xΔ arcsin (k sin x) =− + Δ 2k 2 2k 3  2 cos xΔ sin x cos2 x dx k =− + ln (k cos x + Δ) Δ 2k 2 2k 3  cos3 x dx sin xΔ 2k 2 − 1 = + arcsin (k sin x) Δ 2k 2 2k 3    2 1 + k2 sin x cos xΔ 2 + k2 sin4 x dx = + F (x, k) − E (x, k) Δ 3k 2 3k 4 3k 4   sin3 x cos x dx 1  = − 4 2 + k 2 sin2 x Δ Δ 3k

2.584

2.584

Elliptic and pseudo-elliptic integrals



13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23. 24.

25. 26.

27. 28.

sin x cos xΔ 2 − k 2 sin2 x cos2 x dx 2k 2 − 2 =− + E (x, k) + F (x, k) Δ 3k 2 3k 4 3k 4  1  sin x cos3 x dx 2 = − 4 k 2 cos2 x − 2k  Δ Δ 3k  4 cos x dx sin x cos xΔ 4k 2 − 2 3k 4 − 5k 2 + 2 = + E (x, k) + F (x, k) Δ 3k 2 3k 4 3k 4  sin5 x dx 2k 2 sin2 x + 3k 2 + 3 3 + 2k 2 + 3k 4 = cos xΔ − ln (k cos x + Δ) 4 Δ 8k 8k 5  sin4 x cos x dx 2k 2 sin2 x + 3 3 =− sin xΔ + 5 arcsin (k sin x) Δ 8k 4 8k  sin3 x cos x dx 2k 2 cos2 x − k 2 − 3 k 4 + 2k 2 − 3 = cos xΔ − ln (k cos x + Δ) 4 Δ 8k 8k 5  sin2 x cos3 x dx 2k 2 cos2 x + 2k 2 − 3 4k 2 − 3 =− sin xΔ + arcsin (k sin x) Δ 8k 4 8k 5  sin x cos4 x dx 3 − 5k 2 + 2k 2 sin2 x 3k 4 − 6k 2 + 3 = cos xΔ − ln (k cos x + Δ) 4 Δ 8k 8k 5  cos5 x dx 2k 2 cos2 x + 6k 2 − 3 8k 4 − 8k 2 + 3 = sin xΔ + arcsin (k sin x) Δ 8k 4 8k 5  sin6 x dx 3k 2 sin2 x + 4k 2 + 4 = sin x cos xΔ Δ 15k 4 4 2 8k 4 + 7k 2 + 8 4k + 3k + 8 F (x, k) − E (x, k) + 15k 6 15k 6  3k 4 sin4 x + 4k 2 sin2 x + 8 sin5 x cos x dx =− Δ Δ 15k 6  sin4 x cos x dx 3k 2 cos2 x − 2k 2 − 4 = sin x cos xΔ Δ 15k 4 4 2 2k 4 + 3k 2 − 8 k + 7k − 8 F (x, k) − E (x, k) + 6 15k 15k 6    3k 4 sin4 x − 5k 4 − 4k 2 sin2 x − 10k 2 + 8 sin3 x cos3 x dx = Δ Δ 15k 6  sin2 x cos4 x dx 3k 2 cos2 x + 3k 2 − 4 =− sin x cos xΔ Δ 15k 4 4 2 3k 4 − 13k 2 + 8 9k − 17k + 8 F (x, k) − E (x, k) + 15k 6 15k 6  2 sin x cos5 x dx −3k 4 cos4 x + 4k 2 k  cos2 x − 8k 4 + 16k 2 − 8 = Δ Δ 15k 6  cos6 x dx 3k 2 cos2 x + 8k 2 − 4 = sin x cos xΔ Δ 15k 4 6 4 2 23k 4 − 23k 2 + 8 15k − 34k + 27k − 8 F (x, k) + E (x, k) + 15k 6 15k 6

191

192

Trigonometric Functions

 29.

 30.  31.

 32.

 33.

 34.

 35.

 36.



37. 38. 39. 40.11 41. 42.

  sin7 x dx 8k 4 sin4 x + 10k 2 k 2 + 1 sin2 x + 15k 4 + 14k 2 + 15 = cos xΔ Δ  4   2 48k 6 2 5k − 2k + 5 k + 1 − ln (k cos x + Δ) 16k 7 8k 4 sin4 x + 10k 2 sin2 x + 15 sin6 x cos x dx 5 =− sin xΔ + arcsin (k sin x) Δ 48k 6 16k 7   sin5 x cos2 x dx −8k 4 sin4 x + 2k 2 k 2 − 5 sin2 x + 3k 4 + 4k 2 − 15 = cos xΔ Δ 48k 6 6 4 2 k + k + 3k − 5 ln (k cos x + Δ) − 16k 7   sin4 x cos3 x dx 8k 4 sin4 x − 2k 2 6k 2 − 5 sin2 x − 18k 2 + 15 = sin xΔ Δ 48k 6 2 6k − 5 arcsin (k sin x) + 16k 7   sin3 x cos4 x dx 8k 4 sin4 x − 2k 2 6k 2 − 5 sin2 x + 3k 4 − 22k 2 + 15 = cos xΔ Δ 48k 6 6 4 2 k + 3k − 9k + 5 ln (k cos x + Δ) − 16k 7   sin2 x cos5 x dx −8k 4 sin4 x + 2k 2 12k 2 − 5 sin2 x − 24k 4 + 36k 2 − 15 = sin xΔ Δ 48k 6 4 2 8k − 12k + 5 arcsin (k sin x) + 16k 7   sin x cos6 x dx −8k 4 sin4 x + 2k 2 13k 2 − 5 sin2 x − 33k 4 + 40k 2 − 15 = cos xΔ Δ 48k 6 6 5k ln (k cos x + Δ) + 16k 7   cos7 x dx 8k 4 sin4 x − 2k 2 18k 2 − 5 sin2 x + 72k 4 − 54k 2 + 15 = sin xΔ Δ 48k 6 16k 6 − 24k 4 + 18k 2 − 5 arcsin (k sin x) + 16k 7

dx 1 k2 sin x cos x =  2 E (x, k) −  2 3 Δ Δ k k  sin x dx cos x =− 2 Δ3 k Δ  cos x dx sin x = Δ3 Δ  2 sin x dx 1 1 1 sin x cos x =  2 2 E (x, k) − 2 F (x, k) −  2 3 Δ k Δ k k k  sin x cos x dx 1 = 2 Δ3 k Δ  2 1 1 sin x cos x cos x dx = 2 F (x, k) − 2 E (x, k) + Δ3 k k Δ

2.584

2.584

Elliptic and pseudo-elliptic integrals

 43.  44.  45.  46.  47.  48.  49.  50.  51.  52.9  53.  54.  55.  56. 

57.

58.

59. 60.

sin3 x dx cos x 1 = − 2  2 + 3 ln (k cos x + Δ) 3 Δ k k Δ k sin2 x cos x dx 1 sin x = 2 − 3 arcsin (k sin x) Δ3 k Δ k sin x cos2 x dx 1 cos x = 2 − 3 ln (k cos x + Δ) 3 Δ k Δ k 1 k  sin x cos3 x dx + 3 arcsin (k sin x) = − Δ3 k2 Δ k 2

k + 1 2 sin x cos x sin4 x dx = E (x, k) − 4 F (x, k) − 3 Δ k k 2 k4 k2 k 2 Δ 2

sin3 x cos x dx 2 − k 2 sin2 x = Δ3 k4 δ 2 − k2 2 sin x cos x sin2 x cos2 x dx = F (x, k) − 4 E (x, k) + 3 Δ k4 k k2 Δ k 2 sin2 x + k2 − 2 sin x cos3 x dx = Δ3 k4 Δ cos4 x dx k + 1 2k  k sin x cos x = E (x, k) − 4 F (x, k) − 3 4 Δ k k k2 Δ 2

2

2

k2 + 3 k 2 k  sin2 x + k2 − 3 sin5 x dx cos x + = ln (k cos x + Δ) 2 3 4  Δ 2k 5 2k k Δ 2

sin4 x cos x dx 3 −k 2 sin2 x + 3 sin x − 5 arcsin (k sin x) = Δ3 2k 4 Δ 2k sin3 x cos2 x dx −k 2 sin2 x + 3 k2 − 3 = cos x + ln (k cos x + Δ) 4 Δ 2k Δ 2k 5 sin2 x cos3 x dx 2k 2 − 3 k 2 sin2 x + 2k 2 − 3 sin x − = arcsin (k sin x) Δ3 2k 4 Δ 2k 5 3k  k 2 sin2 x + 2k 2 − 3 sin x cos4 x dx cos x + = ln (k cos x + Δ) 3 4 Δ 2k Δ 2k 5 2

cos5 x dx 4k2 − 3 −k 2 sin2 x + 2k 4 − 4k 2 + 3 sin x + = arcsin (k sin x) Δ3 2k 4 Δ 2k 5  2 2  dx −k 2 sin x cos x 2k k + 1 sin x cos x 1 = − −  2 F (x, k) 2 4 5  3  Δ 3k Δ 3k 3k Δ 2 2 k + 1 + E (x, k) 3k  4  sin x dx 2k 2 sin2 x + k 2 − 3 = cos x Δ5 3k  4 Δ3  cos x dx −2k 2 sin2 x + 3 = sin x Δ5 3Δ3

193

194

Trigonometric Functions



61.

62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79.

sin2 x dx k 2 + 1 1 =  4 2 E (x, k) −  2 2 F (x, k) 5 Δ 3k k 3k k  k 2 k 2 + 1 sin2 x − 2 + sin x cos x 3k  4 Δ3  1 sin x cos x dx = 2 3 Δ5 3k Δ    k 2 2k 2 − 1 sin2 x − 3k 2 + 2 1 2k 2 − 1 cos2 x dx = 2 F (x, k) + E (x, k) + sin x cos x Δ5 3k 3k 2 k  2 2k  2 Δ  2   3k − 1 sin2 x − 2 sin3 x dx = cos x Δ5 3k  4 Δ3  sin2 x cos x sin3 x dx = Δ5 3Δ3  sin x cos2 x cos3 x dx = − 2 5 Δ 3k  Δ3  2   − 2k + 1 sin2 x + 3 cos3 x dx = sin x Δ5 3Δ3  1 Δ + cos x dx = − ln Δ sin x 2 Δ − cos x  dx 1 Δ − k  sin x = −  ln Δ cos x 2k Δ + k  sin x   dx 1 + cot2 x dx = F (x, k) − E (x, k) − Δ cot x = Δ Δ sin2 x   dx dx 1 1−Δ 1 Δ + k = (tan x + cot x) = ln +  ln Δ sin x cos x Δ 2 1 + Δ 2k Δ − k     dx 1 dx 1 = 1 + tan2 x = F (x, k) −  2 E (x, k) +  2 Δ tan x 2 Δ cos x Δ k k    dx 1 sin x dx Δ+k = tan x =  ln cos x Δ Δ 2k Δ − k   dx 1 1−Δ cos x dx = cot x = ln sin x Δ Δ 2 1+Δ  dx Δ cos x 1 + k 2 Δ + cos x ln =− − 3 4 Δ − cos x Δ sin x 2 sin2 x  Δ dx 1 Δ − k  sin x = − − ln sin x 2k  Δ + k  sin x Δ sin2 x cos x  Δ dx 1 Δ − cos x = 2 + ln 2 Δ sin x cos x k cos x 2 Δ + cos x  Δ sin x dx 2k 2 − 1 Δ − k  sin x = 2 ln + 3 2 Δ cos x Δ + k  sin x 2k cos x 4k  3  sin x dx Δ = 2 cos2 x Δ k cos x

2.584

2.584

Elliptic and pseudo-elliptic integrals

 80.  81.  82.  83.  84.

 85.

 86.

 87.



88. 89. 90. 91. 92. 93. 94.

Δ cos x dx =− sin x sin2 x Δ 1 Δ + k  sin x 1 sin2 x dx =  ln − arcsin (k sin x) cos x Δ 2k Δ − k  sin x k cos2 x dx 1 Δ + cos x 1 = ln + ln (k cos x + Δ) sin x Δ 2 Δ − cos x k        dx 1 −Δ cot3 x − Δ 2k 2 + 3 cot x + k 2 + 2 F (x, k) − 2 k 2 + 1 E (x, k) = 4 3 Δ sin x   dx  dx = tan x + 2 cot x + cot3 x 3 Δ Δ sin x cos x Δ Δ + k k2 + 2 1 + Δ 1 ln =− − +  ln 2 Δ − k 4 1−Δ 2 sin x 2k   dx  2 dx = tan x + 2 + cot2 x 2 2 Δ Δ sin x cos x

tan x k2 − 2 = − cot x Δ + E (x, k) + 2 F (x, k) k 2 k 2    dx dx = cot x + 2 tan x + tan3 x 3 Δ sin x cos x Δ Δ 1 1 + Δ 2 − 3k 2 Δ + k  + = − 2 ln − ln Δ − k 2k cos2 x 2 1 − Δ 4k  3 ⎧   1 ⎨ dx 5k2 − 3 = Δ tan x − 3k 2 − 2 F (x, k) Δ tan3 x − 2 2 4   Δ cos x 3k ⎩ k ⎫  2  ⎬ 2 2k − 1 + E (x, k) 2 ⎭ k 

  dx Δ Δ + k k2 tan x 1 + tan2 x = 2 ln − Δ 2k cos2 x 4k  3 Δ − k   Δ cos x dx k2 1 + Δ =− ln − 3 2 4 1−Δ sin x Δ 2 sin x   2 2 tan x Δ 1 sin x dx = dx =  2 tan x −  2 E (x, k) cos2 x Δ Δ k k   cos2 x dx cot2 x = dx = −Δ cot x − E (x, k) Δ sin2 x Δ  sin3 x dx Δ 1 Δ + k = 2 +  ln cos x Δ k 2k Δ − k  Δ cos3 x dx 1 1+Δ = 2 − ln sin x Δ k 2 1−Δ      3 1 + k 2 sin2 x + 2 dx 3k 4 + 2k 2 + 3 Δ + cos x ln = − Δ cos x + 5 2 16 Δ − cos x Δ sin x 8 sin x sin x dx = cos3 x Δ

195

196

Trigonometric Functions

 95.  96.  97.  98.  99.  100.  101.  102.  103. 

104. 105. 106. 107. 2.585 1.

2.585

  3 + 2k 2 sin2 x + 1 1 Δ − k  sin x dx =− Δ −  ln 4 3 2k Δ + k  sin x Δ sin x cos x 3 sin x   2 3 − k 2 sin2 x − k  dx k 2 + 3 Δ − cos x ln = Δ + 4 Δ + cos x Δ sin3 x cos2 x 2k  2 sin2 x cos x  2  2 2 2 3 − 2k sin x − 2k dx 4k − 3 Δ + k  sin x ln Δ − = Δ − k  sin x Δ sin2 x cos3 x 2k  2 sin x cos2 x 4k  3  2  2 2 5k − 3 sin x − 6k + 4 1 Δ + cos x dx Δ − ln = 4 Δ sin x cos x 2 Δ − cos x 3k  4 cos3 x   3 2k 2 − 1 sin2 x − 8k 2 + 5 dx 8k 4 − 8k 2 + 3 Δ + k  sin x = ln Δ sin x + Δ cos5 x Δ − k  sin x 8k  4 cos4 x 16k  5 sin x dx 2k 2 cos2 x − k =− Δ 4 cos x Δ 2k  4 cos3 x 2

2k 2 sin2 x + 1 cos x dx = − Δ sin4 x Δ 3 sin3 x Δ sin x Δ + k  sin x sin2 x dx 1 = ln − 2 3 3  2  cos x Δ Δ − k  sin x 2k cos x 4k Δ cos x k  Δ + cos x cos3 x dx = − ln + 3 2 4 Δ − cos x sin x Δ 2 sin x 2

sin3 x dx Δ 1 = 2 + ln (k cos x + Δ) cos2 x Δ k cos x k  −Δ 1 cos3 x dx = − arcsin (k sin x) sin x k sin2 x Δ  sin4 x dx Δ sin x 1 Δ + k  sin x 2k 2 + 1 = − + ln arcsin (k sin x) cos x Δ 2k 2 2k  Δ − k  sin x 2k 3  Δ cos x 1 Δ + cos x 3k 2 − 1 cos4 x dx = + + ln ln (k cos x + Δ) sin x Δ 2k 2 2 Δ − cos x 2k 3 

(a + sin x) Δ =

p+3

dx

1 (p + 2)k2

 p

(a + sin x) cos xΔ   p+2   (a + sin x)p+1 dx dx (a + sin x) + (p + 1) 1 + k 2 − 6a2 k 2 +2(2p + 3)ak2 Δ  Δ  (a + b sin x)p dx  2 2 2 −a(2p + 1) 1 + k − 2a k Δ &  p−1     (a + sin x) dx − p 1 − a2 1 − a 2 k 2 Δ   1 p = −2, a = ±1, a =

± k For p = n a natural number, this integral can be reduced to the following three integrals:

2.586

Elliptic and pseudo-elliptic integrals

 2.  3. 4.6

5. 2.586 1.

2.

3.

197

1 Δ − k cos x a + sin x dx = a F (x, k) + ln Δ 2k Δ + k cos x 2

1 + k 2 a2 1 a Δ − k cos x (a + sin x) dx = F (x, k) − 2 E (x, k) + ln Δ k2 k k Δ + k cos x

  1 1 sin x dx dx  , = Π x, 2 , k −  2 (a + sin x) Δ a a a − sin2 x Δ where √ √  −1 1 − a2 Δ − 1 − k 2 a2 cos x sin x dx   =  √ ln √ a2 − sin2 x Δ 1 − a2 Δ + 1 − k 2 a2 cos x 2 (1 − a2 ) (1 − a2 k 2 ) % dx cos xΔ 1 − = n n−1 (a + sin x) Δ (n − 1) (1 − a2 ) (1 − a2 k 2 ) (a + sin x)    dx −(2n − 3) 1 + k 2 − 2a2 k 2 a n−1 (a + sin x) Δ   2 2  dx −(n − 2) 6a k − k 2 − 1 n−2 (a + sin x) Δ &   dx dx 2 2 − (10 − 4n)ak − (n − 3)k n−3 n−4 (a + sin x) Δ (a + sin x) Δ   1 n = 1, a = ±1, a = ± k This integral can be reduced to the integrals:      dx 1 dx cos xΔ 2 2 2 = − a 1 + k − 2a k − 2 2 2 2 a + sin x (a + sin x) Δ (a + sin x) Δ (1 − a ) (1 − a k ) &   2 dx (a + sin x) dx (a + sin x) + k2 − 2ak2 Δ Δ (see 2.585 2, 3, 4) ⎡     1 dx dx ⎣ − cos xΔ − 3a 1 + k 2 − 2a2 k 2 = 3 2 2 (a + sin x) Δ 2 (1 − a2 ) (1 − a2 k 2 ) (a + sin x) (a + sin x) Δ ⎤    2 2 dx + 2ak2 F (x, k)⎦ − 6a k − k 2 − 1 (a + sin x) Δ



(see 2.585 4 and 2.586 2)

4.

For a = ±1, we have: %     dx dx 1 cos xΔ 2 = ∓ n n + (n − 1) 1 − 5k n−1 2  (1 ± sin x) Δ (2n − 1)k (1 ± sin x) (1 ± sin x) Δ &   dx dx − (n − 2)k2 + 2(2n − 3)k2 n−2 n−3 (1 ± sin x) Δ (1 ± sin x) Δ GU (241)(6a)

This integral can be reduced to the following integrals:

198

Trigonometric Functions

2.587

 5. 6.

∓ cos xΔ dx 1 = 2 + F (x, k) −  2 E (x, k) (1 ± sin x) Δ k (1 ± sin x) k     2 1 − 5k 2 cos xΔ 1 dx k cos xΔ = ∓ 2 2 ∓ 4 1 ± sin x (1 ± sin x) Δ 3k  (1 ± sin x)      2 + 1 − 3k 2 k  F (x, k) − 1 − 5k 2 E (x, k)

GU (241)(6c)

GU (241)(6b)

7.

1 For a = ± , we have k      dx dx 1 k cos xΔ 2 = + (n − 1) 5 − k ± n n n−1 (1 ± k sin x) Δ (2n − 1)k  2 (1 ± k sin x) (1 ± k sin x)& Δ   dx dx − 2(2n − 3) + (n − 2) n−2 n−3 (1 ± k sin x) Δ (1 ± k sin x) Δ GU (241)(7a)

8. 9.

This integral can be reduced to the following integrals:  k cos xΔ dx 1 = ± 2 +  2 E (x, k) (1 ± k sin x) Δ k (1 ± k sin x) k %    2 k 5 − k2 cos xΔ 1 dx kk cos xΔ = ± 2 2 ± 4 1 ± k sin x (1 ± k sin x) Δ 3k  (1 ± k sin x)    2 − 2k  F (x, k) + 5 − k 2 E (x, k)

GU (241)(7b)

GU(241)(7c)

2.587 1.

2. 3. 4.

%  p+2 1 dx (b + cos x) p 2 sin xΔ + 2(2p + 3)bk (b + cos x) (p + 2)k2 Δ  (b + cos x)p+1 dx  2 2 2 2 −(p + 1) k − k + 6b k Δ  (b + cos x)p dx  2 2 2 2 +(2p + 1)b k − k + b k Δ &  (b + cos x)p−1 dx    2 + p 1 − b2 k  + k 2 b2 Δ   ik p = −2, b = ±1, b = k For p = n a natural number, this integral can be reduced to the following three integrals:  1 b + cos x dx = b F (x, k) + arcsin (k sin x) Δ k  2 2 2 2 (b + cos x) b k −k 1 2b dx = arcsin (k sin x) F (x, k) + 2 E (x, k) + Δ k2 k k

  b 1 cos x dx dx  , = 2 Π x, 2 ,k +  (b + cos x) Δ b −1 b −1 1 − b2 − sin2 x Δ where 

(b + cos x) Δ

p+3

dx

2.589

Elliptic and pseudo-elliptic integrals

 5. 2.588 1.

2.

 √ 1 − b2 Δ + k k  2 + k 2 b2 sin x cos x dx 1   = <   ln √  1 − b2 − sin2 x Δ 1 − b2 Δ − k k  2 + k 2 b2 sin x 2 (1 − b2 ) k  2 + k 2 b2

% 2 dx 1 −k  sin xΔ   2 = n −1 (b + cos x) Δ (n − 1) (1 − b2 ) k  + b2 k 2 (b + cos x)    dx −(2n − 3) 1 − 2k 2 + 2b2 k 2 b n−1 (b + cos x) Δ   2  dx −(n − 2) 2k − 1 − 6b2 k 2 n−2 (b + cos x) Δ &   dx dx 2 2 − (4n − 10)bk + (n − 3)k n−3 n−4 (b + cos x) Δ (b + cos x) Δ   ik n = 1, b = ±1, b = ± k This integral can be reduced to the following integrals: ⎡   2   1 dx dx ⎣ −k sin xΔ − 1 − 2k 2 + 2b2 k 2 b   = 2 2 2  2 2 b + cos x (b + cos x) Δ (b + cos x) Δ (1 − b ) k + b k ⎤   2 b + cos x (b + cos x) dx − k 2 dx⎦ + 2bk 2 Δ Δ



 3.

⎡

dx 3

(b + cos x) Δ

=

(see 2.587 2, 3, 4)

1 −k  sin xΔ ⎣  2 2 2 (1 − b2 ) k  + b2 k 2 (b + cos x)    dx −3b 1 − 2k2 + 2k 2 b2 2 (b + cos x) Δ ⎤    2 dx − 2bk 2 F (x, k)⎦ − 2k − 1 − 6b2 k 2 (b + cos x) Δ 2

(see 2.588 2 and 2.587 4) 2.589 1.



p+3

(c + tan x) Δ

⎡

dx

=

 p p+2 1 dx ⎣ (c + tan x) Δ + 2(2n + 3)ck 2 (c + tan x) 2 2 cos x Δ (p + 2)k   (c + tan x)p+1 dx 2 2 2 −(p + 1) 1 + k + 6c k Δ  (c + tan x)p dx  2 2 +(2p + 1)c 1 + k + 2c2 k  Δ ⎤  p−1    dx ⎦ (c + tan x) 2 − p 1 + c2 1 + k  c 2 Δ [p = −2]

For p = n a natural number, this integral can be reduced to the following three integrals:

199

200

Trigonometric Functions

 2.  3. 4.

5.

2

where  √  1 + c2 k  2 + 1 + c2 Δ sin x cos x dx 1   = < √  ln   c2 − (1 + c2 ) sin2 x Δ 1 + c2 k  2 − 1 + c2 Δ 2 (1 + c2 ) 1 + c2 k  2 

1.

⎡ dx 1 Δ ⎣−  = n n−1 (c + tan x) Δ (n − 1) (1 + c2 ) 1 + k  2 c2 (c + tan x) cos2 x   dx 2 2 +(2n − 3)c 1 + k  + 2c2 k  n−1 (c + tan x) Δ   dx 2 2 −(n − 2) 1 + k  + 6c2 k  n−2 (c + tan x) Δ ⎤   dx dx 2 2 ⎦ − (n − 3)k  + (4n − 10)ck n−3 n−4 (c + tan x) Δ (c + tan x) Δ

This integral can be reduced to the integrals: ⎡  dx 1 −Δ ⎣  = 2 2 2 2  (c + tan x) cos2 x (c + tan x) Δ (1 + c ) 1 + k c   dx 2 2 +c 1 + k + 2c2 k  (c + tan x) Δ ⎤   2 c + tan x (c + tan x) 2 2 dx + k  dx⎦ − 2ck Δ Δ 

3.

1 c + tan x Δ + k dx = c F (x, k) +  ln Δ 2k Δ − k

(c + tan x) 1 1 c Δ + k dx =  2 tan xΔ + c2 F (x, k) −  2 E (x, k) +  ln Δ k Δ − k k k

 c 1 + c2 dx 1 = Π x, − F (x, k) + ,k (c + tan x) Δ 1 + c2 c (1 + c2 ) c2 sin x cos x dx  , −  2 c − (1 + c2 ) sin2 x Δ

2.591

2.

2.591

⎡ dx 3

(c + tan x) Δ

=

(see 2.589 2, 3, 4)

1 −Δ ⎣  2 2 2  2 (1 + 1+k c (c + tan x) cos2 x   dx 2 2 +3c 1 + k  + 2c2 k  2 (c + tan x) Δ ⎤   dx 2 2 2 + 2ck F (x, k)⎦ − 1 + k  + 6c2 k  (c + tan x) Δ c2 )

(see 2.591 2 and 2.589 4)

2.593

2.592 1.

Elliptic and pseudo-elliptic integrals

201

 

n a + sin2 x dx Pn = Δ The recursion formula Pn+1 =

  n  a + sin2 x sin x cos xΔ + (2n + 2) 1 + k 2 + 3ak2 Pn+1        − (2n + 1) 1 + 2a 1 + k 2 + 3a2 k 2 Pn + 2na(1 + a) 1 + k 2 a Pn−1

1 (2n + 3)k 2

reduces this integral (for n an integer) to the integrals: 2.

P1

3.

P0

4.

P−1 =

5. 6.

(see 2.584 1 and 2.584 4)

 

1 dx 1  = Π x, , k 2 a a a + sin x Δ

(see 2.584 1)



For a = 0  dx sin2 xΔ  dx Tn =  n h + g sin2 x Δ

(see 2.584 70)

H (124)a

can be calculated by means of the recursion formula:      1 −g 2 sin x cos xΔ 2 2 Tn−3 =  n−1 + 2(n − 2) g 1 + k + 3hk Tn−2 2 (2n − 5)k 2 h + g sin x 



− (2n − 3) g + 2hg 1 + k 2.593 1.

2

2



2 2

+ 3h k





Tn−1 + 2(n − 1)h(g + h) g + hk

2



 Tn

n   b + cos2 x dx Δ The recursion formula ⎧ ⎨  n  1 b + cos2 x sin x sin xΔ − (2n + 2) 1 − 2k 2 − 3bk 2 Qn+1 Qn+2 = 2 (2n + 3)k ⎩ Qn =

⎫ ⎬      + (2n + 1) k2 + 2b k 2 − k 2 − 3b2 k 2 n− 2nb(1 − b) k 2 − k 2 b Qn−1 ⎭ 

reduces this integral (for n an integer) to the integrals: 2.

Q1

(see 2.584 1 and 2.584 6)

3.

Q0

(see 2.584 1)

202

Trigonometric Functions

 4.

5. 2.594 1.

Q−1 =

2.594



1 1 dx = Π x, − ,k (b + cos2 x) Δ b+1 b+1

For b = 0  dx cos2 xΔ

(see 2.584 72)

H (123)

 

n c + tan2 x dx Rn = Δ The recursion formula ⎧ ⎨ c + tan2 xn tan xΔ    1 2 2 − (2n + 2) 1 + k Rn+2 = − 3ck Rn+1 (2n + 3)k  2 ⎩ cos2 x

⎫ ⎬ , +      + (2n − 1) 1 − 2c 1 + k 2 + 3c2 k 2 Rn + 2nc(1 − c) 1 − k 2 c Rn−1 ⎭

reduces this integral (for n an integer) to the integrals: 2.

R1

3.

R0

4.

R−1 =

(see 2.584 1 and 2.584 90) 

(see 2.584 1)

1 1−c 1 dx   F (x, k) + Π x, ,k = c−1 c(1 − c) c c + tan2 x Δ

For c = 0, see 2.582 5. 2.595



Integrals of the type



< R sin x, cos x, 1 − p2 sin2 x dx for p2 > 1.

Notation: α = arcsin (p sin x).



1. 2.

3.



Basic formulas

 2  dx 1 1  p >1 = F α, 2 p p 1 − p2 sin x



 < 1 1 p2 − 1 F α, 1 − p2 sin2 x dx = p E α, − p p p   2 p >1

  2  r2 1 dx 1  Π α, p >1 , =   2 2 2 p p p 1 − r2 sin x 1 − p2 sin x 

To evaluate integrals of the form

R sin x, cos x,

<

1−

p2

k with p;

(2)

k



2

with 1 − p2 ;

BY (283.03) BY (283.02)

sin x dx for p2 > 1, we may use formulas 2

2.583 and 2.584, making the following modifications in them. We replace (1)

BY (283.00)

2.597

(3) (4)

Elliptic and pseudo-elliptic integrals

1  1 F α, p ; p



1 1 p2 − 1 E (x, k) with p E α, F α, − . p p p

F (x, k) with

For example (see 2.584 15): 2.596 ⎡ 

 4 2 2 sin2 x x dx 1 − p − 2 1 4p cos sin x cos x 10 ⎣  p E α, + = 1. 3p2 3p4 p 1 − p2 sin2 x ⎤



1 ⎦ 2 − 5p2 + 3p4 1 1 p2 − 1 F α, F α, · − + p p 3p4 p p 



1 1 sin x cos x 1 − p2 sin2 x p2 − 1 4p2 − 2 − F α, E α, = + 3p2 3p3 p 3p3 p

2.

3.

203



p2 > 1



For example (see 2.583 36):



   < 1 1 1 − p2 sin2 x 1 1 p2 − 1 2 sin2 x + dx = tan x F F α, α, 1 − p − p E α, − cos2 x p p p p 

p  < 1 1 = p F α, − E α, + tan x 1 − p2 sin2 x p p  2  p >1 For example (see 2.584 37): 

  dx 1 sin x cos x −1 1 p2 − 1 p2 <  F α, = p E α, − − 2 2  2 3 2 p − 1 p p p 1 − p 2 1 − 1 − p2 sin x

p sin x 2 sin x cos x 1 1 1 p p  E α, + F α, = 2 − 2 2 p − 1 1 − p2 sin x p p p −1 p   2 p >1



< 2 2 2.597 Integrals of the form R sin x, cos x, 1 + p sin x dx   1 + p2 sin x Notation: α = arcsin  1 + p2 sin2 x Basic formulas    dx p 1  1. F α,  = 1 + p2 1 + p2 1 + p2 sin2 x    <  sin x cos x p 2 2 2 2. 1 + p sin x dx = 1 + p E α,  − p2  2 1+p 1 + p2 sin2 x 

BY (282.00)

BY (282.03)

204

Trigonometric Functions

2.598

   1 + p2 sin2 x dx p 1 2 3. Π α, r ,  BY (282.02) = 1 + (p2 − r2 p2 − r2 ) sin2 x 1 + p2 1 + p2    p cos x sin x dx 1  4. = − arcsin  p 1 + p2 1 + p2 sin2 x

 < cos x dx 1 2 2  5. = ln p sin x + 1 + p sin x p 1 + p2 sin2 x   1 + p2 sin2 x − cos x dx 1   ln 6. = 2 sin x 1 + p2 sin2 x 1 + p2 sin2 x + cos x    1 + p2 sin2 x + 1 + p2 sin x dx 1  7. ln  =   2 1 + p2 cos x 1 + p2 sin2 x 1 + p2 sin2 x − 1 + p2 sin x    1 + p2 sin2 x + 1 + p2 1 tan x dx    ln = 8.  2 1 + p2 1 + p2 sin2 x 1 + p2 sin2 x − 1 + p2   cot x dx 1 1 − 1 + p2 sin2 x   9. = ln 2 1 + 1 + p2 sin2 x 1 + p2 sin2 x  "  2.598 To calculate integrals of the form R sin x, cos x, 1 + p2 sin2 x dx, we may use formulas 2.583 and 2.584, making the following modifications in them. We replace (1)

k2 with −p2 ;

(2)

k  with 1 + p2 ;

(3)

F (x, k) with √ 1

(4)

 2 E (x, k) with 1 + p E α, √ p

(5)

1 k

(6)

1 k

2

1+p2

F

α, √ p

; 2

1+p

1+p2

− p2 √sin x2cos x2 ; 1+p sin x

p cos x arcsin √ ; 1+p2   arcsin (k sin x) with p1 ln p sin x + 1 + p2 sin2 x .

ln (k cos x + Δ) with

1 p

For example (see 2.584 90):  1.

⎡ < 1 ⎣  tan x 1 + p2 sin2 x = 2) 2 2 (1 + p 1 + p sin x ⎤    sin x cos x p ⎦ − 1 + p2 E α,  + p2  1 + p2 1 + p2 sin2 x   p tan x 1 E α,  + = − 1 + p2 1 + p2 1 + p2 sin2 x tan2 x dx

For example (see 2.584 37):

2.611

Elliptic and pseudo-elliptic integrals

 2.

<

dx 1+

p2

1

3 = 1 + p2 E sin x 2



p



α,  1 + p2

   Integrals of the form R sin x, cos x, a2 sin2 x − 1 dx   a cos x . Notation: α = arcsin  a2 − 1

2.599

205



a2 > 1



Basic formulas:  √    2  a2 − 1 dx 1  a >1 1. = − F α, 2 2 a a a sin x − 1  √   √    2−1 2−1 a a 1 − a E α, 2. a2 sin2 x − 1 dx = F α, a a a  2  a >1     √  r 2 a2 − 1 a2 − 1 dx 1 Π α, 2 2 , 3. =   a (r2 − 1) a (r − 1) a 1 − r2 sin2 x a2 sin2 x − 1  2  a > 1, r2 > 1   2  sin x dx α  a >1 4. =− a a2 sin2 x − 1    2  cos x dx 1   5. a >1 = ln a sin x + a2 sin2 x − 1 2 2 a a sin x − 1   2  dx cos x  a >1 6. = − arctan  sin x a2 sin2 x − 1 a2 sin2 x − 1  √  a2 − 1 sin x + a2 sin2 x − 1 1 dx   ln √ = √ 7. 2 a2 − 2 cos x a2 sin2 x − 1 a2 − 1 sin x − a2 sin2 x − 1  2  a >1  √  a2 − 1 + a2 sin2 x − 1 tan x dx 1   8. ln √ = √ 2 a2 − 1 a2 sin2 x − 1 a2 − 1 − a2 sin2 x − 1  2  a >1

  2  1 cot x dx  a >1 = − arcsin 9. a sin x a2 sin2 x − 1    2.611 To calculate integrals of the type R sin x, cos x, a2 sin2 x − 1 dx for a2 > 1,

BY (285.00)a

BY (285.06)a

BY (285.02)a

we may use

formulas 2.583 and 2.584. In doing so, we should follow the procedure outlined below: (1)

In the right members of these formulas, the following functions should be replaced with integrals equal to them:

206

Trigonometric Functions

2.611

 F (x, k)

should be replaced with

E (x, k)

should be replaced with

1 − ln (k cos x + Δ) k

should be replaced with

1 arcsin (k sin x) k

should be replaced with

1 Δ − cos x ln 2 Δ + cos x

should be replaced with

1 Δ + k  sin x ln 2k  Δ − k  sin x

should be replaced with

1 Δ + k ln 2k  Δ − k 

should be replaced with

1 1−Δ ln 2 1+Δ

should be replaced with



(2)

dx Δ Δdx

     

sin x dx Δ cos x dx Δ dx Δ sin x dx Δ cos x tan x dx Δ cot x dx Δ

 2 Then, on both sides of the equations, we should replace Δ with i a2 sin2 x − 1, k with a and k 2 with 1 − a .

(3)

Both sides of theresulting equations should be multiplied by i, as a result of which only real functions a2 > 1 should appear on both sides of the equations.

(4)

The integrals on the right sides of the equations should be replaced with their values found from formulas 2.599.

Examples: 1.

We rewrite equation 2.584 4 in the form     sin2 x dx 1 1   dx = 2 − 2 i a2 sin2 x − 1 dx, a i a2 sin2 x − 1 i a2 sin2 x − 1 a from which we get  √       2−1 a sin2 x dx 1 dx 1   a2 sin2 x − 1 dx = − E α, = 2 + a a a a2 sin2 x − 1 a2 sin2 x − 1  2  a >1

2.

We rewrite equation 2.584 58 as follows:      2a4 a2 − 2 sin2 x − 3a2 − 5 a2 dx < < 5 = − 3 sin x cos x 2 a2 sin2 x − 1 a2 sin2 x − 1 i5 3 (1 − a2 ) i3    1 2a2 − 4 dx  − i a2 sin2 x − 1 dx − 3 (1 − a2 ) i a2 sin2 x − 1 3 (1 − a2 )2

2.613

Elliptic and pseudo-elliptic integrals

207

from which we obtain      2a4 a2 − 2 sin2 x − 3a2 − 5 a2 1 dx < < = sin x cos x + 2   5 3 2 3 (1 − a2 ) a a2 sin2 x − 1 a2 sin2 x − 1 3 (1 − a2 )     √  √  2  2   a2 − 1 a2 − 1 2 − 2a a − 2 E α, × a − 3 F α, a a  2  a >1 3.

We rewrite equation 2.584 71 in the form    dx cot x dx tan x dx    = + , 2 2 2 2 sin x cos xi a sin x − 1 i a sin x − 1 i a2 sin2 x − 1 from which we obtain  √

 1 a2 − 1 + a2 sin2 x − 1 dx 1   ln √ = √ − arcsin a sin x 2 a2 − 1 sin x cos x a2 sin2 x − 1 a2 − 1 − a2 sin2 x − 1   2 a >1

√  "  Integrals of the form R x, cos x, 1 − k 2 cos2 x dx.  sin   π To find integrals of the form R sin x, cos x, 1 − k 2 cos2 x dx, we make the substitution x = −y, 2 which yields

   <  R sin x, cos x, 1 − k 2 cos2 x dx = − R cos y, sin y, 1 − k 2 sin2 y dy.

 < 2 2 The integrals R cos y, sin y, 1 − k sin y dy are found from formulas 2.583 and 2.584. As a 2.612

result of the use of these formulas (where it is assumed that the original integral can be reduced only to integrals of the first and second Legendre forms), when we replace the functions F (x, k) and E (x, k) with the corresponding integrals, we obtain an expression of the form  <  dy 1 − k 2 sin2 y dy − B −g (cos y, sin y) − A  1 − k 2 sin2 y Returning now to the original variable x, we obtain       dx R sin x, cos x, 1 − k 2 cos2 x dx = −g (sin x, cos x) − A √ 1 − k 2 cos2 x dx −B 1 − k 2 cos2 x The integrals appearing in this expression are found from the formulas



 sin x dx √ 1. = F arcsin √ ,k 1 − k 2 cos2 x 1 − k2 cos2 x



  sin x k 2 sin x cos x 2 2 2. 1 − k cos x dx = E arcsin √ ,k − √ 1 − k 2 cos2 x 1 − k 2 cos2 x 

  [p > 1]. R sin x, cos x, 1 − p2 cos2 x dx    To find integrals of the type R sin x, cos x, 1 − p2 cos2 x dx, where [p > 1], we proceed as in

2.613

Integrals of the form

section 2.612. Here, we use the formulas

208

Trigonometric Functions



1 1  = − F arcsin (p cos x) , [p > 1] p p 1 − p2 cos2 x



  1 p2 − 1 1 2 2 F arcsin (p cos x) , 1 − p cos x dx = − p E arcsin (p cos x) , p p p

 1. 2.

2.614

dx



  R sin x, cos x, 1 + p2 cos2 x dx.    To find integrals of the type R sin x, cos x, 1 + p2 cos2 x dx, we need to make the substitution π x = − y. This yields 2

   <  R sin x, cos x, 1 + p2 cos2 x dx = − R cos y, sin y, 1 + p2 sin2 y dy.

 < To calculate the integrals − R cos y, sin y, 1 + p2 sin2 y dy, we need to use first what was said

2.614

Integrals of the form

in 2.598 and 2.612 and then, after returning to the variable x, the formulas    1 dx p  = 1. F x,  1 + p2 cos2 x 1 + p2 1 + p2      p 2. 1 + p2 cos2 x dx = 1 + p2 E x,  1 + p2 

  R sin x, cos x, a2 cos2 x − 1 dx [a > 1] .    To find integrals of the type R sin x, cos x, a2 cos2 x − 1 dx, we need to make the substitution π x = − y. This yields 2

   <  2 2 2 2 R sin x, cos x, a cos x − 1 dx = − R cos y, sin y, a sin y − 1 dy  "  To calculate the integrals − R cos y, sin y, a2 sin2 y − 1 dy, we use what was said in 2.611 and then, after returning to the variable x, we use the formulas  

√ 2  a sin x a −1 1 dx √ = F arcsin √ , 1. a a a2 cos2 x − 1 a2 − 1 2.615

2.

Integrals of the form

 

[a > 1] 

√ 2 a sin x a −1 2 2 a cos x − 1 dx = a E arcsin √ , a a2 − 1  

√ 2 a sin x 1 a −1 [a > 1] − F arcsin √ , a a a2 − 1 



2.616

Elliptic and pseudo-elliptic integrals

 11

2.616



Integrals of the form R sin x, cos x,   1 − p2 sin x . Notation: α = arcsin  1 − p2 sin2 x  1.

<

1−

p2

dx 1 < F =    1 − p2 1 − p2 sin2 x 1 − q 2 sin2 x

209

< 2 2 sin x, 1 − q sin x dx. 2

 α,

.

 q 2 − p2 1 − p2 + 0 < p2 < q 2 < 1,

0<x≤

π, 2 BY (284.00)

2.

3.

4.

5.

  tan2 x dx tan x 1 − q 2 sin2 x <   = (1 − q 2 ) 1 − p2 sin2 x . 1 − p2 sin2 x 1 − q 2 sin2 x   q 2 − p2 1  − E α, 1 − p2 (1 − q 2 ) 1 − p2 + π, 0 < p2 < q 2 < 1, 0 < x ≤ BY (284.07) 2  tan4 x dx <   1 − p2 sin2 x 1 − q 2 sin2 x  .  . %  & 2 − p2 2 − p2     q q 1 = 2 2 − p2 − q 2 E α, − 1 − q 2 F α, 3 × 2 1 − p2 1 − p2 3 (1 − q 2 ) (1 − p2 ) 2 .   2p2 + q 2 − 3 + sin2 x 4 − 3p2 − 2q 2 + p2 q 2 sin x 1 − q 2 sin2 x + 2 cos2 x 1 − p2 sin2 x 3 (1 − p2 ) (1 − q 2 ) + π, 0 < p2 < q 2 < 1, 0 < x ≤ BY (284.07) 2  sin2 x dx <  3 1 − p2 sin2 x 1 − q 2 sin2 x  .  .    1 − p2 q 2 − p2 q 2 − p2 1  E α, = F α, − (1 − q 2 ) (q 2 − p2 ) 1 − p2 1 − p2 (q 2 − p2 ) 1 − p2 sin x cos x < −   1 − p2 sin2 x 1 − q 2 sin2 x (1 − q 2 ) + π, BY (284.06) 0 < p2 < q 2 < 1, 0 < x ≤ 2  cos2 x dx < 3   1 − p2 sin2 x 1 − q 2 sin2 x  .  .    1 − p2 q 2 − p2 q 2 − p2 1 − q2  = 2 E α, F α, − q − p2 1 − p2 1 − p2 (q 2 − p2 ) 1 − p2 , + π 0 < p2 < q 2 < 1, 0 < x ≤ BY (284.05) 2

210

Trigonometric Functions

2.617

 6.

7.

cos4 x dx 5   1 − p2 sin2 x 1 − q 2 sin2 x  3 ⎡    .  2 2 2 − p2 2 + p2 − 3q 2 1 − q 2 1−p q ⎣ F α, = 2 2 1 − p2 3 (q 2 − p2 ) (1 − p2 )  . ⎤    1 − p2 sin x cos x 1 − q 2 sin2 x 2q 2 − p2 − 1 q 2 − p2 ⎦ < +2 E α, + 3 1 − p2 1 − p2 1 − p2 sin2 x 3 (q 2 − p2 ) + , π BY (284.05) 0 < p2 < q 2 < 1, 0 < x ≤ 2 .  .   1 − q 2 sin2 x q 2 − p2 dx 1 E α, = 1 − p2 1 − p2 sin2 x 1 − p2 sin2 x 1 − p2 + π, 0 < p2 < q 2 < 1, 0 < x ≤ 2 <

BY (284.01)

8.

/  .    0 2 2 − p2 0 1 − p2 sin2 x 1 − p q sin x cos x q 2 − p2 1 < dx = E α, −    3 2 2 2 2 2 1 − q 1 − p 1 − q 2 1 − q sin x 1 − p sin2 x 1 − q 2 sin2 x + π, . 0 < p2 < q 2 < 1, 0 < x ≤ 2 

9.

. dx 2 2 1 + (p r − p2 − r2 ) sin2 x

.



BY (284.04)



1 − p2 sin2 x q 2 − p2 1 2  Π α, r , = 1 − p2 1 − q 2 sin2 x 1 − p2 + π, 0 < p2 < q 2 < 1, 0 < x ≤ . 2 BY (284.02)

.√ 2.617

Notation: α = arcsin 

1.

b2 + c2 − b sin x − c cos x √ , 2 b 2 + c2

dx √ a + b sin x + c cos x 2 F (α, r) = − √ a + b 2 + c2   0 < b2 + c2 < a,

. r=

√ 2 b 2 + c2 √ . a + b 2 + c2

b b arcsin √ − π ≤ x < arcsin √ 2 2 2 b +c b + c2



BY (294.00) √ 2 = −√ F (α, r) 4 2 b + c2 

  b a b − arccos − √ 0 < |a| < b2 + c2 , arcsin √ ≤ x < arcsin √ b 2 + c2 b 2 + c2 b 2 + c2 BY (293.00)

2.618

Elliptic and pseudo-elliptic integrals

 2.

√ 2b 2c √ sin x dx √ {2 E (α, r) − F (α, r)} + 2 a + b sin x + c cos x = −< b + c2 3 4 a + b sin x + c cos x (b2 + c2 )

   b a b − arccos − √ ≤ x < arcsin √ 0 < |a| < b2 + c2 , arcsin √ b 2 + c2 b 2 + c2 b 2 + c2 BY (293.05)



3. 4.

211

√ (b cos x − c sin x) dx √ = 2 a + b sin x + c cos x a + b sin x + c cos x  √ 2 b + c2 + b sin x + c cos x √ dx a + b sin x + c cos x √   < 2 + c2  2 a − b = −2 a + b2 + c2 E (α, r) +  F (α, r) √ a + b 2 + c2    b b 2 2 0 < b + c < a, arcsin √ − π ≤ x < arcsin √ b 2 + c2 b 2 + c2 BY (294.04)

√  4 = −2 2 b2 + c2 E (α, r) 

  b a b 2 2 0 < |a| < b + c , arcsin √ − arccos − √ ≤ x < arcsin √ b 2 + c2 b 2 + c2 b 2 + c2

BY (293.01)

 5.

√

a + b sin x + c cos x dx <  = −2 a + b2 + c2 E (α, r)   0 < b2 + c2 < a,

 b b arcsin √ − π ≤ x < arcsin √ b 2 + c2 b 2 + c2 BY (294.01) √ √  2 + c2 − a √  2 b 4 √ = −2 2 b2 + c2 E (α, r) + F (α, r) 4 2 b + c2 

  −a b b − arccos √ 0 < |a| < b2 + c2 , arcsin √ ≤ x < arcsin √ b 2 + c2 b 2 + c2 b 2 + c2 BY (293.03)



    √ 1 R sin ax, cos ax, cos 2ax dx = R sin t, cos t, 1 − 2 sin2 t dt a where the substitution √ t = ax has been used. Notation: α = arcsin   2 sin ax √ The integrals R sin ax, cos ax, cos 2ax dx are special cases of the integrals 2.595. for (p = 2).

2.618

Integrals of the form

We give some formulas:

 dx 1 1 √ 1. = √ F α, √ a 2 cos 2ax 2

 2 cos ax 1 1 √ 2. dx = √ E α, √ cos 2ax a 2 2

+ π, 0 < ax ≤ 4 + π, 0 < ax ≤ 4

212

Trigonometric Functions

 3.

√

1 2 tan x √ dx √ E α, √ = − cos 2ax 2 a a cos ax cos 2ax 2

2.619

+ π, 0 < ax ≤ 4  √

√

  2 ax + 1 sin ax √ 6 cos 2 1 1 2 2 dx √ E α, √ F α, √ = cos 2ax 4. − − a 3a 3a,cos3 ax cos4 ax cos 2ax 2 2+ π 0<x≤ 4 √



 2 √ 1 2 1 tan ax dx 1 1 √ E α, √ = 5. − √ F α, √ − tan ax cos 2ax a a cos 2ax 2 a 2 2 + π, 0<x≤ 2

 tan4 ax dx 1 1 sin ax √ √ 6. = √ F α, √ cos 2ax − 3a cos3 ax cos 2ax 3a 2 2 + π, 0 < ax ≤ 4

 + 1 π, 1 dx  √ = √ Π α, r2 , √ 0 < ax ≤ 7. 2 4 1 − 2r2 sin ax cos 2ax a 2 2

√

 1 2 1 1 dx sin 2ax √ = √ F α, √ E α, √ 8. − + √ 3 a a 2 2 2 a cos 2ax cos 2ax + π, 0 < ax ≤ 4

 + sin 2ax sin2 ax dx 1 1 π, √ = √ 9. − √ E α, √ 0 < ax ≤ 4 2a cos 2ax a 2 2 cos3 2ax

 + 1 1 dx π, sin 2ax √ = √ F α, √ 10. 0 < ax ≤ + √ 4 2 3a 2 cos5 2ax 3a cos3 2ax √



 √ 1 2 1 1 E α, √ cos 2ax dx = 11. − √ F α, √ a 2 a 2 2 + π, 0 < ax ≤ 4 √



 √ √ cos 2ax 2 1 1 1 dx = 12. F α, √ + tan ax cos 2ax − E α, √ cos2 ax a a 2 2 + π, 0<x≤ 4     √   1 2.619 Integrals of the form R sin ax, cos ax, − cos 2ax dx = R sin x, cos x, 2 sin2 x − 1 dx a  √ 2 cos ax Notation: α = arcsin    The integrals R sin x, cos x, 2 sin2 x − 1 dx are special cases of the integrals 2.599 and 2.611 for (a = 2). We give some formulas:

2.621

Elliptic and pseudo-elliptic integrals

 1.  2.  3.  4.  5.

 6.  7.  8.  9.  10.  11.



1 dx 1 √ = − √ F α, √ − cos 2ax a 2 2 



 2 1 cos ax dx 1 1 √ = √ E α, √ − F α, √ − cos 2ax 2 2 a 2 

 4 √ 1 1 cos ax dx 1 5 1 √ sin 2ax − cos 2ax = √ 3 F α, √ − E α, √ − 2 12a − cos 2ax 2 2 3a 2 √

√ 1 2 dx 1 √ E α, √ = cot ax − cos 2ax − a a sin2 ax − cos 2ax 2 



 dx 2 1 1 √ = √ F α, √ − 6 E α, √ sin4 ax − cos 2ax 3a 2 2 2 √ 1 cos ax  2 6 sin ax + 1 − cos 2ax + 3a sin3 ax 



 √ cot2 ax dx 1 1 1 1 √ = √ F α, √ − 2 E α, √ + cot ax − cos 2ax a − cos 2ax a 2 2 2

dx 1 1 √ = − √ Π α, r2 , √ 2 2 (1 − 2r cos ax) − cos 2ax a 2 2 



 dx 1 1 1 sin 2ax √ = √ F α, √ − 2 E α, √ + √ 3 a − cos 2ax a 2 2 2 − cos 2ax

2 cos ax dx 1 sin 2ax 1 √ = √ − √ E α, √ 3 2a − cos 2ax a 2 2 − cos 2ax

1 1 dx sin 2ax √ = − √ F α, √ − √ 5 3a 2 2 − cos 2ax 3a − cos3 2ax 



 √ 1 1 1 − cos 2ax dx = √ F α, √ − 2 E α, √ a 2 2 2 

2.621

213

Integrals of the form 

 √ R sin ax, cos ax, sin 2ax dx.

2 sin ax . 1 + sin ax + cos ax √

 1 dx 2 √ F α, √ = a 2 sin 2ax ⎡ √

 1+i 1 2 ⎣1 + i sin ax dx √ Π α, ,√ = a 2 2 2 sin 2ax

Notation: α = arcsin

1.

2.

BY (287.50)

⎤





1−i 1 1−i 1 1 ⎦ Π α, ,√ + + F α, √ − 2 E α, √ 2 2 2 2 2

 3.

√ 



 2 sin ax dx 1 1 √ = F α, √ − E α, √ a 2 2 (1 + sin ax + cos ax) sin 2ax

BY (287.57) BY (287.54)

214

Trigonometric Functions

 4.

 5.  6.

 7.  8.

√

 2 √ sin ax dx 1 √ tan ax − E α, √ = a 2 (1 − sin ax + cos ax) sin 2ax + π, ax = 2 √

1 2 (1 + cos ax) dx √ E α, √ = a 2 (1 + sin ax + cos ax) sin 2ax √ 



√ 2 (1 + cos ax) dx 1 1 √ = F α, √ − E α, √ + tan ax a 2 2 (1 − sin ax + cos ax) sin 2ax + π, ax = 2 √



 2 1 (1 − sin ax + cos ax) dx 1 √ − F α, √ = 2 E α, √ a 2 2 (1 + sin ax + cos ax) sin 2ax √

2 (1 + sin ax + cos ax) dx 1 √ Π α, r2 , √ = . 2 a 2 [1 + cos ax + (1 − 2r ) sin ax] sin 2ax

2.631

BY (287.55) BY (287.51)

BY (287.56) BY (287.53)

BY (287.52)

2.63–2.65 Products of trigonometric functions and powers 2.631

⎡

 xr sinp x cosq x dx =

1.

1 ⎣(p + q)xr sinp+1 x cosq−1 x (p + q)2  p r−1 q +rx sin x cos x − r(r − 1) xr−2 sinp x cosq x dx

⎤   − rp xr−1 sinp−1 x cosq−1 x dx + (q − 1)(p + q) xr sinp x cosq−2 x dx⎦ ⎡ 1 ⎣ − (p + q)xr sinp−1 x cosq+1 x = (p + q)2  +rxr−1 sinp x cosq x − r(r − 1) xr−2 sinp x cosq x dx  + rq

r−1

x

p−1

sin

x cos

q−1

⎤  p−2 r q x dx + (p − 1)(p + q) x sin x cos x dx⎦ GU (331)(1)

 xm sinn x dx =

2.

 3.

xm cosn x dx =

m−1

n−1

sin x {m sin x − nx cos x} 2 n  n−1 m(m − 1) + xm−2 sinn x dx xm sinn−2 x dx − n n2 x

xm−1 cosn−1 x {m cos x + nx sin x} n2  n−1 m(m − 1) m n−2 x cos + x dx − xm−2 cosn x dx n n2

2.633

Trigonometric functions and powers



 n

4.

215

x sin

2m

2m xn+1 x dx = m 22m (n + 1)

 m−1 2m (−1)m  k (−1) + 2m−1 xn cos(2m − 2k)x dx k 2 k=0

(see 2.633 2)  xn sin2m+1 x dx =

5.

(−1) 22m



 xn cos2m x dx =

6.

m

m 

(−1)k

k=0

2m + 1 k

TI 333

 xn sin(2m − 2k + 1)x dx (see 2.633 1)



TI 333

n+1

2m x m 22m (n + 1) m−1  2m  1 + 2m−1 xn cos(2m − 2k)x dx k 2 k=0

(see 2.633 2)  xn cos2m+1 x dx =

7.

m 

1 22m

k=0

2m + 1 k

TI 333

 xn cos(2m − 2k + 1)x dx (see 2.633 2)

2.632

 xμ−1 sin βx dx =

1.  2.

xμ−1 cos βx dx = 

4. 2.633 1. 2.8

3.

i i −μ −μ (iβ) γ(μ, iβx) − (−iβ) γ (μ, −iβx) 2 2

[Re μ > −1, x > 0] ET I 317(2)    

 πi πi 1 (μ − 1) Γ(μ, −iax) + exp (1 − μ) Γ(μ, iax) xμ−1 sin ax dx = − μ exp 2a 2 2 [Re μ < 1, a > 0, x > 0] ET I 317(3)

 3.

TI 333

 1 −μ −μ (iβ) γ(μ, iβx) + (−iβ) γ (μ, −iβx) 2

[Re μ > 0, x > 0]     π π 1 Γ(μ, −iax) + exp −iμ Γ(μ, iax) xμ−1 cos ax dx = − μ exp iμ 2a 2 2

ET I 319(22) ET I 319(23)

1 k! cos ax + kπ x sin ax dx = − 2 k ak+1 k=0

 n  n xn−k  1 n k! sin ax + kπ x cos ax dx = 2 k ak+1 k=0   n  n−1 2n−2k   x2n−2k−1 2n k+1 x k cos x + sin x x sin x dx = (2n)! (−1) (−1) (2n − 2k)! (2n − 2k − 1)! 

n

n 

 n xn−k

k=0



k=0

TI (487)

TI (486)

216

Trigonometric Functions

 n 2n−2k  x2n−2k+1 k x cos x + sin x x sin x dx = (2n + 1)! (−1) (−1) (2n − 2k + 1)! (2n − 2k)! k=0 k=0   n  n−1 2n−2k 2n−2k−1   x x x2n cos x dx = (2n)! sin x + cos x (−1)k (−1)k (2n − 2k)! (2n − 2k − 1)! k=0 k=0   n  n   x2n−2k+1 x2n−2k 2n+1 k x sin x + cos x cos x dx = (2n + 1)! (−1) (2n − 2k + 1)! (2n − 2k)! 



4.

5.

6.

2.634

2n+1

n 

k+1

k=0

k=0

2.634  1.

n/2

(n+1)/2

(2k) (2k−1) (x) sin mx  (x) cos mx  k Pn k−1 Pn Pn (x) sin mx dx = − (−1) + (−1) m m2k m m2k−1 k=0

 2.

k=1

n/2

Pn (x) cos mx dx =

sin mx  m

k=0

(2k) Pn (x) (−1)k m2k

+

cos mx m

(n+1)/2



k=1

(−1)k−1

(2k−1)

Pn (x) m2k−1

(k)

In formulas 2.634, Pn (x) is any nth -degree polynomial, and Pn (x) is its k th derivative with respect to x. 2.635 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Notation: z1 = a + bx.  1 b z1 sin kx dx = − z1 cos kx + 2 sin kx k k  1 b z1 cos kx dx = z1 sin kx + 2 cos kx k k

 1 2b2 2bz1 2 2 z1 sin kx dx = − z1 cos kx + 2 sin kx k k2 k

 1 2b2 2bz1 z12 cos kx dx = z12 − 2 sin kx + 2 cos kx k k k



2  z1 6b 2b2 3b 3 2 2 − z1 cos kx + 2 z1 − 2 sin kx z1 sin kx dx = k k2 k k



 2 z1 6b 2b2 3b z13 cos kx dx = z12 − 2 sin kx + 2 z12 − 2 cos kx k k k k



 2 4 1 12b 2 24b 6b2 4bz1 4 4 2 cos kx + 2 z1 − 2 sin kx z1 sin kx dx = − z1 − 2 z1 + 4 k k k k k



 2 4 2 1 12b 2 24b 6b 4bz1 4 4 2 sin kx + 2 z1 − 2 cos kx z1 cos kx dx = z1 − 2 z1 + 4 k k k k k



 5b 12b2 24b4 20b2 120b4 z1 sin kx − cos kx z15 sin kx dx = 2 z14 − 2 z12 + 4 z14 − 2 z12 + k k k k k k4



 5b 12b2 2 24b4 20b2 2 120b4 z1 5 4 4 cos kx + sin kx z1 cos kx dx = 2 z1 − 2 z1 + 4 z1 − 2 z1 + k k k k k k4

2.637

Trigonometric functions and powers

217



6bz1 20b2 2 120b4 4 sin kx dx = 2 z1 − 2 z1 + sin kx k k k4

30b2 360b4 2 720b6 1 z − cos kx − z16 − 2 z14 + k k k4 1 k6

 6bz1 20b2 2 120b4 6 4 z1 cos kx dx = 2 z1 − 2 z1 + cos kx k k k4

30b2 360b4 2 720b6 1 z − + sin kx z16 − 2 z14 + k k k4 1 k6

 11.

12.

2.636

z16

 xn sin2 x dx =

1.

xn+1 2(n + ⎫ ⎧1) n/2

(n−1)/2

⎬  (−1)k+1 xn−2k−1 n! ⎨  (−1)k+1 xn−2k sin 2x + cos 2x + ⎭ 4 ⎩ 22k (n − 2k)! 22k+1 (n − 2k − 1)! k=0

GU (333)(2e)

 xn cos2 x dx =

2.

k=0

xn+1 2(n + ⎫ ⎧1) n/2

(n−1)/2

⎬  (−1)k+1 xn−2k−1 n! ⎨  (−1)k+1 xn−2k sin 2x + cos 2x − ⎭ 4 ⎩ 22k (n − 2k)! 22k+1 (n − 2k − 1)! k=0

k=0

GU (333)(3e)

 3. 4. 5. 6.

x 1 x2 − sin 2x − cos 2x 4 4 8

 3 x 1 x 1 x2 sin2 x dx = − cos 2x − x2 − sin 2x 6 4 4 2  x 1 x2 + sin 2x + cos 2x x cos2 x dx = 4 4 8

 3 x 1 x 1 x2 cos2 x dx = + cos 2x + x2 − sin 2x 6 4 4 2 x sin2 x dx =

2.637  1.11

MZ 241

MZ 245

⎧

⎨ n/2

 (−1)k xn−2k cos 3x n! 3 n − 3 cos x x sin x dx = 4 ⎩ (n − 2k)! 32k+1 k=0

(n−1)/2

−



k=0

xn−2k−1 (−1)k (n − 2k − 1)!



⎫

⎬ sin 3x − 3 sin x ⎭ 32k+2 GU(333)(2f)

218

Trigonometric Functions

 2.

⎧

n/2

n! ⎨  (−1)k xn−2k sin 3x xn cos3 x dx = + 3 sin x 4 ⎩ (n − 2k)! 32k+1 k=0



[(n−1)/2]

+

k=0

(−1)k

n−2k−1

x (n − 2k − 1)!



⎫

⎬ cos 3x + 3 cos x ⎭ 32k+2 GU(333)(3f)

 3. 4. 5. 6. 2.638

1 3 x 3 sin x − sin 3x − x cos x + cos 3x 4 36 4 12

2

 3 2 3 x 1 x 3 3 2 x sin x dx = − x + + sin 3x cos x + cos 3x + x sin x − 4 2 12 54 2 18  1 3 x 3 cos 3x + x sin x + sin 3x x cos3 x dx = cos x + 4 36 4 12

2

 3 2 3 1 x x 3 x2 cos3 x dx = x − − cos 3x sin x + sin 3x + x cos x + 4 2 12 54 2 18 x sin3 x dx =



1.

 2.

3.6

4.6

2.638

sinq x sinq−1 x [(p − 2) sin x + qx cos x] dx = − xp (p − 1)(p − 2)xp−1   q2 q(q − 1) sinq x dx sinq−2 x dx − + p−2 (p − 1)(p − 2) x (p − 1)(p − 2) xp−2 [p = 1, p = 2]

cosq x cosq−1 x [(p − 2) cos x − qx sin x] dx = − p p−1 x (p − 1)(p  − 2)x  2 q q q(q − 1) cos x dx cosq−2 x dx − + p−2 (p − 1)(p − 2) x (p − 1)(p − 2) xp−2 [p = 1, p = 2]   cos x dx sin x dx sin x 1 =− + p p−1 x (p − 1)x p−1 xp−1  sin x cos x 1 sin x dx =− − − p−1 p−2 (p − 1)x (p − 1)(p − 2)x (p − 1)(p − 2) xp−2 (p > 2)   cos x 1 cos x dx sin x dx =− − xp (p − 1)xp−1 p−1 xp−1  cos x sin x 1 cos x dx =− + − (p − 1)xp−1 (p − 1)(p − 2)xp−2 (p − 1)(p − 2) xp−2 (p > 2)

MZ 241

MZ 245, 246

TI (496)

TI (495)

TI (492)

TI (491)

2.641

Trigonometric functions and powers

2.639  1.

219

⎧ n−2 sin x dx (−1)n+1 ⎨  (−1)k (2k + 1)! = cos x x2n x(2n − 1)! ⎩ x2k+1 k=0 ⎫ n−1 ⎬ (−1)n+1  (−1)k+1 (2k)! ci(x) + sin x + ⎭ (2n − 1)! x2k k=0

 2.

⎧ ⎨ n−1  (−1)k+1 (2k)!

sin x (−1)n+1 dx = x2n+1 x(2n)! ⎩ +

n−1  k=0

 3.

cos x dx (−1)n+1 dx = x2n x(2n − 1)! ⎩ n−2  k=0

4.

cos x

⎫ ⎬ (−1)n (−1)k+1 (2k + 1)! si(x) sin x + ⎭ x2k+1 (2n)! k=0

⎧ ⎨ n−1  (−1)k+1 (2k)!

−



x2k

GU (333)(6b)a

x2k

GU (333)(6b)a

cos x

⎫ ⎬ (−1)k (2k + 1)! (−1)n si(x) sin x + ⎭ (2n − 1)! x2k+1 k=0

GU (333)(7b)

⎧ ⎨ n−1  (−1)k+1 (2k + 1)!

cos x dx (−1)n+1 = x2n+1 x(2n)! ⎩

cos x x2k+1 ⎫ n−1 ⎬ (−1)n  (−1)k+1 (2k)! − ci(x) sin x + ⎭ x2k (2n)! k=0

k=0

GU (333)(7b)

2.641 1. 2. 3. 4. 5.

  1 ka sin kx ka dx = si(u) − sin ci(u) cos a + bx b b b    cos kx 1 ka ka dx = ci(u) + sin si(u) cos a + bx b b b   sin kx k cos kx 1 sin kx + dx dx = − (a + bx)2 b a + bx b a + bx   k sin kx 1 cos kx cos kx − dx dx = − (a + bx)2 b a + bx b a + bx   sin kx k2 sin kx k cos kx − dx = − − (a + bx)3 2b(a + bx)2 2b2 (a + bx) 2b2



  k u = (a + bx) b   k u = (a + bx) b (see 2.641 2) (see 2.641 1) sin kx dx a + bx (see 2.641 1)

220

Trigonometric Functions

 6.  7.

 8.  9.

 10.

 11.

 12.

2.642 1.

2.

4.



cos kx dx a + bx (see 2.641 2)

sin kx sin kx k cos kx dx = − − 2 (a + bx)4 3b(a + bx)3 6b (a + bx)2 2 3  k k sin kx cos kx − dx + 2 6b (a + bx) 6b3 a + bx (see 2.641 2)  k3 cos kx cos kx k sin kx k cos kx sin kx + 3 dx dx = − + 2 + 3 4 3 2 (a + bx) 3b(a + bx) 6b (a + bx) 6b (a + bx) 6b a + bx (see 2.641 1) 2

sin kx k cos kx sin kx dx = − − 5 4 (a + bx) 4b(a + bx) 12b2 (a + bx)3  k 2 sin kx k 3 cos kx k4 sin kx dx + + 3 2 4 4 24b (a + bx) 24b (a + bx) 24b a + bx (see 2.641 1) cos kx k sin kx cos kx dx = − + 5 4 (a + bx) 4b(a + bx) 12b2 (a + bx)3  2 k4 k cos kx k 3 sin kx cos kx + dx + − 24b3 (a + bx)2 24b4 (a + bx) 24b4 a + bx (see 2.641 2) sin kx sin kx k cos kx k 2 sin kx k 3 cos kx dx = − − + + (a + bx)6 5b(a + bx)5 20b2 (a +bx)4 60b3 (a + bx)3 120b4 (a + bx)2 k5 k 4 sin kx cos kx + dx − 120b5 (a + bx) 120b5 a + bx (see 2.641 2) cos kx cos kx k sin kx k 2 cos kx dx = − + + 6 5 2 4 (a + bx) 5b(a + bx) 20b (a + bx) 60b3 (a + bx)3 3 4 k5 k sin kx k cos kx sin kx − dx − − 4 2 5 120b (a + bx) 120b (a + bx) 120b5 a + bx (see 2.641 1)





m−1 ln x 2m (−1)m  k + (−1) ci[(2m − 2k)x] k 22m 22m−1 k=0

 m 2m + 1 sin2m+1 x (−1)m  k dx = 2m (−1) si[(2m − 2k + 1)x] k x 2



sin2m x dx = x



2m m



k=0

m−1  2m

ln x 1 + ci[(2m − 2k)x] k 22m 22m−1 k=0

 m 1  2m + 1 cos2m+1 x dx = 2m ci[(2m − 2k + 1)x] k x 2

 3.

k2 cos kx k sin kx cos kx − dx = − + (a + bx)3 2b(a + bx)2 2b2 (a + bx) 2b2

2.642

cos2m x dx = x

2m m

k=0

2.643

Trigonometric functions and powers

 5.

7.

8.

2.643



1. 

2.

3.4

2m 1 2m m 2 x 

m−1 2m cos(2m − 2k)x (−1)m  + (2m − 2k) si[(2m − 2k)x] (−1)k+1 + 2m−1 k 2 x

sin2m x dx = − x2



k=0



m 2m + 1 (−1)m  k+1 (−1) k x2 22m k=0

 sin(2m − 2k + 1)x × − (2m − 2k + 1) ci[(2m − 2k + 1)x] x

 2m 2m 1 cos x dx = − 2 2m m 2 x x  m−1  2m cos(2m − 2k)x 1 + (2m − 2k) si[(2m − 2k)x] − 2m−1 k 2 x k=0 ⎧

 m cos2m+1 x 1  2m + 1 ⎨ cos(2m − 2k + 1)x = − ⎩ k x2 22m x k=0 ⎫ ⎬ + (2m − 2k + 1) si[(2m − 2k + 1)x] ⎭ 

6.

221

2m+1

sin

x

dx =

xp dx xp−1 [p sin x + (q − 2)x cos x] q − 2 =− + q sin x q−1 (q − 1)(q − 2) sinq−1 x



xp dx p(p − 1) + sinq−2 x (q − 1)(q − 2)



xp−2 dx sinq−2 x

xp dx xp−1 [p cos x − (q − 2)x sin x] =− q cos x (q − 1)(q − 2) cosq−1 x  p−2 p(p − 1) dx q−2 xp dx x + + q − 1 cosq−2 x (q − 1)(q − 2) cosq−2 x  2k−1   ∞ −1 xn  xn k+1 2 2 dx = + B2k xn+2k (−1) sin x n (n + 2k)(2k)! k=1

[|x| < π,  4.

xn 

5.8  6.

n−1 n 2 1 −1 dx = − n − [1 + (−1)n ] (−1) 2 Bn ln x − sin x nx n!

xn dx = cos x

∞  k=0

∞  k=1 k= n 2

n > 0] TU (333)(8b)  2n  2 2 −1 B2k x2k−n (−1)k (2k − n) · (2k)!

[n > 1, |x| > π] , + π |x| < , n > 0 2

|E2k |xn+2k+1 (n + 2k + 1)(2k)!

GU (333)(9b) GU (333)(10b)

∞  1 |E2k |x2k−n+1 dx n |En−1 | = [1 − (−1) ln x + ] n x cos x 2 (n − 1)! (2k − n + 1) · (2k)! k=0 k= n−1 2

+ π, |x| < 2

GU (333)(11b)

222

Trigonometric Functions

 7.  8.

2.644

∞  22k xn+2k−1 n xn dx n n−1 = −x x B2k cot x + + n (−1)k 2 n−1 (n + 2k − 1)(2k)! sin x k=1 [|x| < π, n > 1]

GU (333)(8c)

n

xn

n+1 cot x 2 n n dx =− n + Bn+1 ln x − [1 − (−1)n ] (−1) 2 2 n+1 x (n + 1)x (n + 1)! sin x ∞ n  (−1)k (2x)2k − n+1 B2k 2 (2k − n − 1)(2k)! k=1 k= n+1 2

 9.

xn dx = xn tan x + n cos2 x

 10.

xn

∞  k=1

(−1)k

2

 2k



22k − 1 xn+2k−1 B2k (n + 2k − 1) · (2k)! + n > 1,



1.

 2.

 3.

GU (333)(9c)

|x| <

π, 2

GU (333)(10c)

 n+1 tan x 2n n  n+1 dx = 2 − [1 − (−1)n ] (−1) 2 − 1 Bn+1 ln x 2 n cos x x  (n + 1)! ∞ n  (−1)k 22k − 1 (2x)2k − n+1 B2k x (2k − n − 1)(2k)! k=1 k= n+1 2

2.644

[|x| < π]

+ π, |x| < 2

GU (333)(11c)

n−1  (2n − 2)(2n − 4) . . . (2n − 2k + 2) sin x + (2n − 2k)x cos x x dx = − 2n (2n − 1)(2n − 3) . . . (2n − 2k + 3) (2n − 2k + 1)(2n − 2k) sin2n−2k+1 x sin x k=0 n−1 2 (n − 1)! + (ln sin x − x cot x) (2n − 1)!! n−1  (2n − 1)(2n − 3) . . . (2n − 2k + 1) x dx sin x + (2n − 2k − 1)x cos x =− 2n+1 2n(2n − 2) . . . (2n − 2k + 2) (2n − 2k)(2n − 2k − 1) sin2n−2k x sin x k=0  (2n − 1)!! x dx + 2n n! sin x (see 2.644 5) n−1  (2n − 2)(2n − 4) . . . (2n − 2k + 2) x dx (2n − 2k)x sin x − cos x = 2n cos x (2n − 1)(2n − 3) . . . (2n − 2k + 3) (2n − 2k + 1)(2n − 2k) cos2n−2k+1 x k=0 n−1

+  4.

2

(n − 1)! (x tan x + ln cos x) (2n − 1)!!

n−1  (2n − 1)(2n − 3) . . . (2n − 2k + 1) x dx (2n − 2k + 1)x sin x − cos x = 2n+1 cos x 2n(2n − 2) . . . (2n − 2k + 2) (2n − 2k)(2n − 2k − 1) cos2n−2k x k=0  (2n − 1)!! x dx + 2n n! cos x (see 2.644 6)

2.645

Trigonometric functions and powers

 5.  6.  7.  8.  9.  10.  11.  12.  13.

 14.

223

 2k−1  ∞  −1 x dx k+1 2 2 =x+ B2k x2k+1 (−1) sin x (2k + 1)! k=1

x dx = cos x

∞  k=0

|E2k |x2k+2 (2k + 2)(2k)!

x dx = −x cot x + ln sin x sin2 x x dx = x tan x + ln cos x cos2 x  x dx sin x + x cos x 1 x dx (see 2.644 5) = − + 2 sin x sin3 x 2 sin2 x  x sin x − cos x 1 x dx x dx = + (see 2.644 6) cos3 x 2 cos2 x 2 cos x 2 x dx x cos x 1 2 =− − − x cot x + ln (sin x) 3 sin4 x 3 sin3 x 6 sin2 x 3 x sin x 1 2 2 x dx = − + x tan x − ln (cos x) cos4 x 3 cos3 x 6 cos2 x 3 3  3 x dx x dx x cos x 1 3x cos x 3 + =− − − − 8 sin x 8 sin x sin5 x 4 sin4 x 12 sin3 x 8 sin2 x (see 2.644 5)  x dx cos x

x sin x 1 3x sin x 3 3 x dx = − + − + 5 4 3 2 cos x 4 cos x 12 cos x 8 cos x 8 cos x 8

(see 2.644 6) 2.645



1.

  x k m dx = (−1) x k cosn x p sin

3.

m



k=0



2.

2m

m  sin2m+1 x dx = (−1)k n k cos x m

xp dx cosn−2k x

(see 2.643 2)



xp sin x dx (see 2.645 3) cosn−2k x k=0   xp p sin x dx xp−1 = − dx xp cosn x (n − 1) cosn−1 x n − 1 cosn−1 x xp

[n > 1]  4.

5. 6.

m 

xp dx k sinn−2k x k=0  m   xp cos x  cos2m+1 x k m dx = dx xp (−1) k sinn x sinn−2k x k=0  p−1  cos x dx xp x p xp n = − + sin x (n − 1) sinn−1 x n − 1 sinn−1 x xp

cos2m x dx = sinn x

m 

(−1)k

(see 2.643 2)

GU (333)(12)

(see 2.643 1)

(see 2.645 6) [n > 1]

(see 2.643 1)

GU (333)(13)

224

Trigonometric Functions

2.646

 7. 8. 2.646

x x cos x x + ln tan dx = − sin x 2 sin2 x   x x π x sin x dx = − ln tan + 2 cos x cos x 2 4

 p

1.

x tan x dx =

∞ 

(−1)

k=1

 xp cot x dx =

2.

(−1)k

k=0

 3.

∞ 



 22k−1 − 1 B2k xp+2k (p + 2k) · (2k)!

k+1 2

2k

22k B2k xp+2k (p + 2k)(2k)!

xp tan2 x dx = x tan x + ln cos x −

x2 2

x cot2 x dx = −x cot x + ln sin x −

x2 2

 4. 2.647



1.  2.  3.  4.  5.  6.  7.  8.  9.  10.

+ p ≥ −1, [p ≥ 1,

|x| <

π, 2

|x| < π]

 xn n xn cos x dx xn−1 dx = − + m m−1 (m − 1)b (a + b sin x)m−1 (a + b sin x) (m − 1)b (a + b sin x) [m = 1]  xn sin x dx xn n xn−1 dx = − m m−1 (m − 1)b (a + b cos x)m−1 (a + b cos x) (m − 1)b (a + b cos x) [m = 1]   π x π x x dx = −x tan − + 2 ln cos − 1 + sin x 4 2 4 2 π x π x x dx = x cot − − + 2 ln sin 1 − sin x 4 2 4 2 x x x dx = x tan + 2 ln cos 1 + cos x 2 2 x x x dx = −x cot + 2 ln cos 1 − cos x 2 2 x π x cos x x + tan − dx = − 2 1 + sin x 2 4 (1 + sin x)  x π x x cos x 2 dx = 1 − sin x + tan 2 + 4 (1 − sin x) x sin x

dx =

2

dx = −

x sin x (1 − cos x)

GU (333)(13d)

MZ 247

MZ 247 PE (329) PE (330) PE (331) PE (332)

x x − tan 1 + cos x 2

2

(1 + cos x)

GU (333)(12d)

x x − cot 1 − cos x 2

MZ 247a

2.654

2.648 1. 2.

Trigonometric functions and powers

225



x x + sin x dx = x tan 1 + cos x 2  x x − sin x dx = −x cot 1 − cos x 2  x2 dx

GU (333)(16)

x sin x + cos x GU (333)(17) 2 = b [(ax − b) sin x + (a + bx) cos x] [(ax − b) sin x + (a + bx) cos x]  tan x dx 2.651 GU (333)(18) 2 = a [a + (ax + b) tan x]

 [a + (ax + b) tan x] cos(x − t) x dx = cosec 2t x ln − L(x + t) + L(x − t) 2.652 cos(x + t) cos(x − t) cos(x + t) !π !, + ! ! t = nπ; |x| < ! − |t0 |! , 2 where t0 is the  value of the argument t, which is reduced by multiples of the argument π to lie in the LO III 288 interval − π2 , π2 . 2.653  √ √  sin x √ dx = 2π S 1. x (cf. 8.251 21) x  √ √  cos x √ dx = 2π C 2. x (cf. 8.251 3) x 2.649

2.654 1. 2. 3. 4. 5. 6. 7. 8.

 √ Notation: Δ = 1 − k 2 sin2 x, k  = 1 − k 2 :  xΔ x sin x cos x 1 dx = − 2 + 2 E (x, k) Δ k k     k2 x sin3 x cos x 2k 2 + 5 1   dx = − 4 F (x, k) + E (x, k) − 4 3 3 − Δ2 x + k 2 sin x cos x Δ 4 Δ 9k 9k 9k   , 3 2 2  k x sin x cos x 7k − 5 1 +  2 2 2 3 Δ x − k dx = − 4 F (x, k) + E (x, k) − − 3k sin x cos x Δ Δ 9k 9k 4 9k 4  1 x cos x x sin x dx +  2 arcsin (k sin x) dx = −  2 Δ3 k Δ kk  x sin x 1 x cos x dx + ln (k cos x + Δ) = Δ3 Δ k  1 x sin x cos x dx x = 2 − 2 F (x, k) 3 Δ k Δ k  3 x sin x cos x dx 1 2 − k 2 sin2 x − 4 [E (x, k) + F (x, k)] =x 3 Δ k4 Δ k   x sin x cos3 x dx k2 sin2 x + k 2 − 2 k 2 1 + = x F (x, k) + 4 E (x, k) 3 4 4 Δ k Δ k k

226

Trigonometric Functions

2.655

2.655 Integrals containing sin x2 and cos x2 In integrals containing sin x2 and cos x2 , it is expedient to make the substitution x2 = u.   xp−1 p−1 p 2 2 cos x + xp−2 cos x2 dx 1. x sin x dx = − 2 2   xp−1 p−1 2. xp cos x2 dx = sin x2 − xp−2 sin x2 dx 2 2 ⎧    r ⎨ xn−4k+3 cos x2 xn−4k+1 sin x2 n 2 k − 3. x sin x dx = (n − 1)!! (−1) ⎩ 22k−1 (n − 4k + 3)!! 22k (n − 4k + 1)!! k=1 ⎫  ⎬ (−1)r xn−4r sin x2 dx + 2r ⎭ 2 (n − 4r − 1)!! + : n ;, GU (336)(4a) r= 4 ⎧    r ⎨ xn−4k+3 sin x2 xn−4k+1 cos x2 + 2k 4. xn cos x2 dx = (n − 1)!! (−1)k−1 2k−1 ⎩ 2 (n − 4k + 3)!! 2 (n − 4k + 1)!! k=1 ⎫  ⎬ r (−1) xn−4r cos x2 dx + 2r ⎭ 2 (n − 4r − 1)!! + : n ;, GU (336)(5a) r= 4  cos2 x 5. x sin x2 dx = − 2  sin2 x 6. x cos x2 dx = − 2   x 1 π C (x) 7. x2 sin x2 dx = − cos x2 + 2 2 2   x 1 π 2 2 2 S (x) 8. x cos x dx = sin x − 2 2 2  x2 1 9. x3 sin x2 dx = − cos x2 + sin x2 2 2  x2 1 sin x2 + cos x2 10. x3 cos x2 dx = 2 2

2.661

Trigonometric functions and exponentials

227

2.66 Combinations of trigonometric functions and exponentials 

2.661

e

ax

1 sin x cos x dx = 2 a + (p + q)2 p

q

eax sinp x cosq−1 x [a cos x + (p + q) sin x]

   − pa eax sinp−1 x cosq−1 x dx + (q − 1)(p + q) eax sinp x cosq−2 x dx

=

a2

1 + (p + q)2

eax sinp−1 x cosq x [a sin x − (p + q) cos x]

 + qa e

ax

sin

p−1

x cos

q−1

TI (523)

  p−2 ax q x dx + (p − 1)(p + q) e sin x cos x dx TI (524)

=

a2

1 + (p + q)2

  eax sinp−1 x cosq−1 x a sin x cos x + q sin2 x − p cos2 x

   p p−2 ax q−2 ax q x dx + p(p − 1) e sin x cos x dx + q(q − 1) e sin x cos TI (525)

1 = 2 a + (p + q)2

  eax sinp−1 x cosq−1 x a sin x cos x + q sin2 x − p cos2 x

 +q(q − 1) eax sinp−2 x cosq−2 x dx   − (q − p)(p + q − 1) eax sinp−2 x cosq x dx TI (526)

=

1 a2 + (p + q)2

   eax sinp−1 x cosq−1 x a sin x cos x + q sin2 x − p cos2 x

 +p(p − 1) eax sinp−2 x cosq−2 x dx   p ax q−2 + (q − p)(p + q − 1) e sin x cos x dx



GU (334)(1a)

For p = m and q = n even integers, the integral eax sinm x cosn x dx can be reduced by means of  these formulas to the integral eax dx. However, when only m or only n is even, they can be reduced to

228

Trigonometric Functions

 integrals of the form

 e

ax

2.662  1. eax sinn bx dx =

 eax cosn bx dx =

2.

2.662

n

cos x dx or 1 a2 + n2 b2

eax sinm x dx, respectively.



(a sin bx − nb cos bx) eax sinn−1 bx   n−2 2 ax + n(n − 1)b e sin bx dx 1 a2 + n2 b2



(a cos bx + nb sin bx) eax cosn−1 bx   2 ax n−2 + n(n − 1)b e cos bx dx

 eax sin2m bx dx

3.

=

m−1  k=0

(2m)!b2k eax sin2m−2k−1 bx (2m − 2k)! [a2 + (2m)2 b2 ] [a2 + (2m − 2)2 b2 ] · · · [a2 + (2m − 2k)2 b2 ]

(2m)!b2m eax × [a sin bx − (2m − 2k)b cos bx] + 2 2 2 2 [a + (2m) b ] [a + (2m − 2)2 b2 ] · · · [a2 + 4b2 ] a

ax

m ax  2m e 2m 1 e k + 2m−1 (−1) (a cos 2bkx + 2bk sin 2bkx) = 2m 2 m 2 a 2 m − k a + 4b2 k 2 k=1



eax sin2m+1 bx dx

4.

m 

(2m + 1)!b2k eax sin2m−2k bx [a sin bx − (2m − 2k + 1)b cos bx] (2m − 2k + 1)! [a2 + (2m + 1)2 b2 ] [a2 + (2m − 1)2 b2 ] · · · [a2 + (2m − 2k + 1)2 b2 ] k=0

m 2m + 1 eax  (−1)k = 2m [a sin(2k + 1)bx − (2k + 1)b cos(2k + 1)bx] 2 a2 + (2k + 1)2 b2 m − k

=

k=0

 eax cos2m bx dx =

5.8

m−1  k=0

(2m)!b2m eax + 2 [a + (2m)2 b2 ] [a2 + (2m − 2)2 b2 ] · · · [a2 + 4b2 ] a

ax

m 2m e 2m 1 eax  + [a cos 2kbx + 2kb sin 2kbx] = m 22m a 22m−1 m − k a2 + 4b2 k 2 k=1

 6.

(2m)!b2k eax cos2m−2k−1 bx [a cos bx + (2m − 2k)b sin bx] (2m − 2k)! [a2 + (2m)2 b2 ] [a2 + (2m − 2)2 b2 ] · · · [a2 + (2m − 2k)2 b2 ]

eax cos2m+1 bx dx m 

(2m + 1)!b2k eax cos2m−2k bx (2m − 2k + 1)! [a2 + (2m − 1)2 b2 ] · · · [a2 + (2m − 2k + 1)2 b2 ] k=0

m 1 eax  2m + 1 = 2m [a cos(2k + 1)bx + (2k + 1)b sin(2k + 1)bx] 2 m − k a + (2k + 1)2 b2 2

=

k=0

2.663  eax (a sin bx − b cos bx) 1. eax sin bx dx = a2 + b 2

2.666

Trigonometric functions and exponentials

229

 2.

3. 4.

2.664

eax sin bx (a sin bx − 2b cos bx) 2b2 eax + 4b2 + a2 (4b2 + a2 ) a eax  a eax − 2 cos 2bx + b sin 2bx = 2a a + 4b2 2  ax e (a cos bx + b sin bx) eax cos bx dx = a2 + b 2  eax cos bx (a cos bx + 2b sin bx) 2b2 eax eax cos2 bx dx = + 2 2 4b + a (4b2 + a2 ) a eax  a eax + 2 cos 2bx + b sin 2bx = 2a a + 4b2 2 eax sin2 bx dx =



1.

e

 eax a sin(b + c)x − (b + c) cos(b + c)x sin bx cos cx dx = 2 a2 + (b + c)2  a sin(b − c)x − (b − c) cos(b − c)x + a2 + (b − c)2 GU (334)(6b)



 eax a cos cx + c sin cx a cos(2b + c)x + (2b + c) sin(2b + c)x eax sin2 bx cos cx dx = − 2 4 a2 + c 2 a2 + (2b + c)2 a cos(2b − c)x + (2b − c) sin(2b − c)x − a2 + (2b − c)2



 eax a sin bx − b cos bx a sin(b + 2c)x − (b + 2c) cos(b + 2c)x eax sin bx cos2 cx dx = + 2 4 a2 + b 2 a2 + (b + 2c)2 a sin(b − 2c)x − (b − 2c) cos(b − 2c)x + a2 + (b − 2c)2

2.

3.

ax

GU (334)(6c)

GU (334)(6d)

2.665   eax dx eax [a sin bx + (p − 2)b cos bx] eax dx a2 + (p − 2)2 b2 1. = − + p p−1 2 sin bx (p − 1)(p − 2)b (p − 1)(p − 2)b2 sin bx sinp−2 bx  ax  eax [a cos bx − (p − 2)b sin bx] a2 + (p − 2)2 b2 e dx eax dx = − + 2. p 2 p−1 2 cos bx (p − 1)(p − 2)b cos bx (p − 1)(p − 2)b cosp−2 bx

TI (530)a TI (529)a

By successive for p a natural number, we obtain integrals of the form  applications  2.665  ax  ofaxformulas eax dx eax dx e dx e dx , , , which are not expressible in terms of a finite combination , sin bx cos bx cos2 bx sin2 bx of elementary functions. 2.666    eax a ax p p−1 ax p−1 tan 1. e tan x dx = x− x dx − eax tanp−2 x dx TI (527) e tan p−1 p−1    a eax cotp−1 x + 2. eax cotp x dx = − TI (528) eax cotp−1 x dx − eax cotp−2 x dx p−1 p−1   eax tan x 1 eax dx − (see remark following 2.665) 3. eax tan x dx = a a cos2 x

230

Trigonometric Functions

 eax (a tan x − 1) − a eax tan x dx e tan x dx = a   eax cot x 1 eax dx ax + e cot x dx = a a sin2 x   eax ax 2 (a cot x + 1) + a eax cot x dx e cot x dx = − a

2.667

 4. 5. 6.

ax

2

(see 2.666 3) (see remark following 2.665)

(see 2.666 5)  Integrals of type

R (x, eax , sin bx, cos cx) dx

Notation: sin t = − √

b ; a2 + b 2

a cos t = √ . a2 + b 2

2.667   xp eax p p ax (a sin bx − b cos bx) − 2 xp−1 eax (a sin bx − b cos bx) dx 1. x e sin bx dx = 2 a + b2 a + b2  p xp eax sin(bx + t) − √ xp−1 eax sin(bx + t) dx = √ 2 2 2 a +b a + b2   xp eax p (a cos bx + b sin bx) − xp−1 eax (a cos bx + b sin bx) dx 2. xp eax cos bx dx = 2 a + b2 a2 + b2  p xp eax cos(bx + t) − √ xp−1 eax cos(bx + t) dx =√ 2 2 2 a +b a + b2  n+1  (−1)k+1 n!xn−k+1 3. xn eax sin bx dx = eax sin(bx + kt) 2 2 k/2 k=1 (n − k + 1)! (a + b )  n+1  (−1)k+1 n!xn−k+1 n ax ax 4. x e cos bx dx = e cos(bx + kt) 2 2 k/2 k=1 (n − k + 1)! (a + b ) 



  eax a2 − b 2 2ab 5. xeax sin bx dx = 2 ax − sin bx − bx − cos bx a + b2 a2 + b 2 a2 + b 2 



  eax a2 − b 2 2ab ax 6. xe cos bx dx = 2 ax − 2 cos bx + bx − 2 sin bx a + b2 a + b2 a + b2 ⎧%  2  2  &  ⎨ 2 2 ax 2 a 2a a − b − 3b e 7. x2 eax sin bx dx = 2 x+ sin bx ax2 − 2 a + b2 ⎩ a2 + b 2 (a2 + b2 ) ⎫ %  2 & ⎬ 2 2b 3a − b 4ab − bx2 − 2 x + cos bx 2 ⎭ a + b2 (a2 + b2 )

TI 355

2.672

Trigonometric and hyperbolic functions

 8.

231

⎧%    & ⎨ 2 a2 − b 2 2a a2 − 3b2 e 2 ax 2 cos bx x e cos bx dx = 2 x+ ax − 2 a + b2 ⎩ a2 + b 2 (a2 + b2 ) ⎫ %  2 & ⎬ 2 2b 3a − b 4ab + bx2 − 2 x+ sin bx 2 2 ⎭ a +b (a2 + b2 ) ax

GU (335), MZ 274-275

2.67 Combinations of trigonometric and hyperbolic functions 2.671  1. sinh(ax + b) sin(cx + d) dx =

2.

3.

4.

a cosh(ax + b) sin(cx + d) a2 + c 2 c − 2 sinh(ax + b) cos(cx + d) a + c2  a cosh(ax + b) cos(cx + d) sinh(ax + b) cos(cx + d) dx = 2 a + c2 c + 2 sinh(ax + b) sin(cx + d) a + c2  a sinh(ax + b) sin(cx + d) cosh(ax + b) sin(cx + d) dx = 2 a + c2 c − 2 cosh(ax + b) cos(cx + d) a + c2  a sinh(ax + b) cos(cx + d) cosh(ax + b) cos(cx + d) dx = 2 a + c2 c + 2 cosh(ax + b) sin(cx + d) a + c2 GU (354)(1)

2.672 1. 2. 3. 4.



1 (cosh x sin x − sinh x cos x) 2  1 sinh x cos x dx = (cosh x cos x + sinh x sin x) 2  1 cosh x sin x dx = (sinh x sin x − cosh x cos x) 2  1 cosh x cos x dx = (sinh x cos x + cosh x sin x) 2 sinh x sin x dx =

232

2.673

Trigonometric Functions

2.673

 sinh2m (ax + b) sin2n (cx + d) dx





n−1

2n 2n (−1)m 2m (−1)m+n 2m  (−1)k = 2m+2n x + 2m+2n−1 sin[(2n − 2k)(cx + d)] m n m 2 2 (2n − 2k)c k k=0    2n m−1 n−1 (−1)j+k 2m j k (−1)n   + 2m+2n−2 2 2 2 (2m − 2j) a + (2n − 2k)2 c2 j=0

1.

k=0

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2 − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(3a)



sinh2m (ax + b) sin2n−1 (cx + d) dx

n−1

2n − 1 (−1)m+n 2m  (−1)k = 2m+2n−2 cos[(2n − 2k − 1)(cx + d)] m 2 (2n − 2k − 1)c k k=0    2n−1 m−1 n−1 (−1)j+k 2m j k (−1)n−1   + 2m+2n−3 2 (2m − 2j)2 a2 + (2n − 2k − 1)2 c2 j=0

2.

k=0

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)]} −(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] cos[(2n − 2k − 1)(cx + d)] GU (354)(3b)

 3.

sinh2m−1 (ax + b) sin2n (cx + d) dx  2n  =

n

22m+2n−2

2m−1 j



(−1)j

j=0

(2m − 2j − 1)a

n

+



m−1 

(−1)

22m+2n−3

m−1  n−1  j=0 k=0

cosh[(2m − 2j − 1)(ax + d)]

(−1)j+k



2m−1 j



2n k



(2m − 2j − 1)2 a2 + (2n − 2k)2 c2

× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2n − 2k)c sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(3c)

2.673

Trigonometric and hyperbolic functions

233

 sinh2m−1 (ax + b) sin2n−1 (cx + d) dx

4.

n−1 m−1  n−1 

(−1) = 2m−2n−4 2

j=0 k=0

(−1)j+k



2m−1 j



2n−1 k



(2m − 2j − 1)2 a2 + (2n − 2k − 1)2 c2

× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)]} −(2n − 2k − 1)c sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)] GU (354)(3d)

 sinh2m (ax + b) cos2n (cx + d) dx

5.

m

=







 2n 

m−1 

(−1)j



2m j



2m 2n n sinh[(2m − 2j)(ax + b)] x + 2m+2n−1 m n 2 (2m − 2j)a j=0   n−1  2n   (−1)m 2m m k + 2m+2n−1 sin[(2n − 2k)(cx + d)] 2 (2n − 2k)c k=0    2n j 2m m−1 n−1 (−1)   j k 1 + 2m+2n−2 2 a2 + (2n − 2k)2 c2 2 (2m − 2j) j=0 (−1) 22m+2n

k=0

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2 − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(4a)

 6.

sinh2m (ax + b) cos2n−1 (cx + d) dx   n−1  2n−1   (−1)m 2m m k = 2m+2n−2 sin[(2n − 2k − 2)(cx + d)] 2 (2n − 2k − 1)c k=0    2−1 j 2m m−1 n−1 (−1)   j k 1 + 2m+2n−3 2 (2m − 2j)2 a2 + (2n − 2k − 1)2 c2 j=0 k=0

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k − 1)(cx + d)]} +(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)] GU (354)(4a)

234

Trigonometric Functions

2.673

 sinh2m−1 (ax + b) cos2n (cx + d) dx

7.

 2n  =

n

22m+2n−2 +



2m−1 j



m−1 

(−1)j

j=0

(2m − 2j − 1)a

1 22m−2n−3

m−1  n−1  j=0 k=0

cosh[(2m − 2j − 1)(ax + d)]

(−1)j



2m j



2n k



(2m − 2j − 1)2 a2 + (2n − 2k)2 c2

× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2n − 2k)c sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(4b)

 sinh2m−1 (ax + b) cos2n−1 (cx + d) dx

8.

=

m−1  n−1 

1 22m+2n−4

j=0 k=0

(−1)j



2m−1 j



2n−1 k



(2m − 2j − 1)2 a2 + (2n − 2k − 1)2 c2

× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)]} +(2n − 2k − 1)c sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)] GU (354)(4b)

 9.

cosh2m (ax + b) sin2n (cx + d) dx  2m   2n 

  m−1    (−1)k 2n (−1)n 2m m k sin[(2n − 2k)(cx + d)] = x + 2m+2n−1 2 (2n − 2k)c  k=0  2n  m−1 2m  j n + 2m+2n−1 sinh[(2m − 2j)(ax + b)] 2 (2m − 2j)a j=0    2n k 2m m−1 n−1 (−1) n   j k (−1) + 2m+2n−2 2 a2 + (2n − 2k)2 c2 2 (2m − 2j) j=0 m n 22m+2n

k=0

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2n − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(5a)

2.673

Trigonometric and hyperbolic functions

235

 cosh2m−1 (ax + b) sin2n (cx + d) dx

10.

 2n  =

m−1 

n

22m+2n−2

j=0

n

+

(−1)

22m+2n−3



2m−1 j



(2m − 2j − 1)a

m−1  n−1  j=0 k=0

sinh[(2m − 2j − 1)(ax + b)]

(−1)k



2m−1 j



2n k



(2m − 2j − 1)2 a2 + (2n − 2k)2 c2

× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2n − 2k)c cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(5a)

 cosh2m (ax + b) sin2n−1 (cx + d) dx

11.

  n−1    (−1)k+1 2n−1 (−1)n−1 2m m k cos[(2n − 2k − 1)(cx + d)] = 22m+2n−2 (2n − 2k − 1)c k=0    2n−1 m−1 n−1 (−1)k 2m j k (−1)n−1   + 2m+2n−3 2 (2m − 2j)2 a2 + (2n − 2k − 1)2 c2 j=0 k=0

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)]} −(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] cos[(2n − 2k − 1)(cx + d)] GU (354)(5b)

 12.

cosh2m−1 (ax + b) sin2n−1 (cx + d) dx n−1 m−1  n−1 

(−1) = 2m+2n−4 2

j=0 k=0

(−1)k



2m−1 j



2n−1 k



(2m − 2j − 1)2 a2 + (2n − 2k − 1)2 c2

× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)]} −(2n − 2k − 1)c cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)] GU (354)(5b)

236

Trigonometric Functions

2.673

 cosh2m (ax + b) cos2n (cx + d) dx  2m   2n 

13.

m n 22m+2n

=

 2m  x+

 2n  +

+

n 22m+2n−1



m−1  j=0

2m j

k=0

(2m − 2j)a

m−1  n−1 

1 22m+2n−2

j=0 k=0

2

n−1 

m 22m+2n−1

k

(2n − 2k)c

sin[(2n − 2k)(cx + d)]

sinh[(2m − 2j)(ax + b)] 

2m j



2n k



(2m − 2j)2 a2 + (2n − 2k)2 c2

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2n − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(6)

 cosh2m−1 (ax + b) cos2n (cx + d) dx

14.

 2n  =

n

22m+2n−2 +



m−1  j=0

2m−1 j



(2m − 2j − 1)a

m−1  n−1 

1 22m+2n−3

j=0 k=0

sinh[(2m − 2j − 1)(ax + b)] 

2m−1 j



2n k



(2m − 2j − 1)2 a2 + (2n − 2k)2 c2

× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]} +(2n − 2k)c cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)] GU (354)(6)

 15.

cosh2m (ax + b) cos2n−1 (cx + d) dx  2m  =

m 22m+2n−2

+

n−1  k=0

1 22m+2n−3

 2n−1  k

(2n − 2k − 1)c

m−1  n−1  j=0 k=0

sin[(2n − 2k − 1)(cx + d)]    2m j

2n−1 k

(2m − 2j)2 a2 + (2n − 2k − 1)2 c2

× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos [(2n − 2k − 1)(cx + d)]} +(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)] GU (354)(6)

2.711

The logarithm

237

 cosh2m−1 (ax + b) cos2n−1 (cx + d) dx

16.

=

m−1  n−1 

1 22m+2n−4

j=0 k=0



2m−1 j



2n−1 k



(2m − 2j − 1)2 a2 + (2n − 2k − 1)2 c2

× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)]} +(2n − 2k − 1)c cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)] GU (354)(6)

2.674

 eax sinh bx sin cx dx =

1.

 eax sinh bx cos cx dx =

2.

 eax cosh bx sin cx dx =

3.

 eax cosh bx cos cx dx =

4.

e(a+b)x [(a + b) sin cx − c cos cx] 2 [(a + b)2 + c2 ] e(a−b)x − [(a − b) sin cx − c cos cx] 2 [(a − b)2 + c2 ] e(a+b)x [(a + b) cos cx + c sin cx] 2 [(a + b)2 + c2 ] (a−b)x e − [(a − b) cos cx + c sin cx] 2 [(a − b)2 + c2 ] e(a+b)x [(a + b) sin cx − c cos cx] 2 [(a + b)2 + c2 ] e(a−b)x + [(a − b) sin cx − c cos cx] 2 [(a − b)2 + c2 ] e(a+b)x [(a + b) cos cx + c sin cx] 2 [(a + b)2 + c2 ] e(a−b)x + [(a − b) cos cx + c sin cx] 2 [(a − b)2 + c2 ] MZ 379

2.7 Logarithms and Inverse-Hyperbolic Functions 2.71 The logarithm 

2.711



ln x dx = x ln x − m m

m

lnm−1 x dx

x  (−1)k (m + 1)m(m − 1) · · · (m − k + 1) lnm−k x m+1 m

=

k=0

(m > 0)

TI (603)

238

Logarithms and Inverse-Hyperbolic Functions

2.721

2.72–2.73 Combinations of logarithms and algebraic functions 2.721

 xn lnm x dx =

1.

m xn+1 lnm x − n+1 n+1

 xn lnm−1 x dx

(see 2.722)

For n = −1  m ln x dx lnm+1 x 2. = x m+1 For n = −1 and m = −1  dx = ln (ln x) 3. x ln x  m xn+1  lnm−k x (−1)k (m + 1)m(m − 1) · · · (m − k + 1) 2.722 xn lnm x dx = m+1 (n + 1)k+1

TI (604)

k=0

2.723 1. 2. 3. 2.724

 ln x 1 x ln x dx = x − n + 1 (n + 1)2  2   ln x 2 ln x 2 − xn ln2 x dx = xn+1 + n + 1 (n + 1)2 (n + 1)3  3   ln x 3 ln2 x 6 ln x 6 − xn ln3 x dx = xn+1 + − n + 1 (n + 1)2 (n + 1)3 (n + 1)4 n



1.

2. 2.725 1. 2.





n+1

xn dx xn+1 n+1 m =− m−1 + m − 1 (ln x) (m − 1) (ln x)



TI 375

TI 375

xn dx m−1

(ln x)

For m = 1  n   x dx = li xn+1 ln x    1 (a + bx)m+1 dx (a + bx)m+1 ln x − (m + 1)b x  m  m  m−k k k+1    b x 1 m m+1 m+1 k a (a + bx) ln x − (a + bx) ln x dx = −a (m + 1)b (k + 1)2 

(a + bx)m ln x dx =

k=0

For m = −1, see 2.727 2. 2.726 

  (a + bx)2 a2 1 − 1. (a + bx) ln x dx = ln x − ax + bx2 2b 2b 4

  b2 x3 1  abx2 2 3 3 2 2. (a + bx) ln x dx = (a + bx) − a ln x − a x + + 3b 2 9

TI 374

2.731

Combinations of logarithms and algebraic functions



239



 1  3 2 2 1 2 3 1 3 4 4 4 3 (a + bx) − a ln x − a x + a bx + ab x + b x (a + bx) ln x dx = 4b 4 3 16 3

3. 2.727 8

1.

2.8 3. 4. 5.

2.728 1. 2.9 2.729

   1 dx ln x dx ln x = + − (a + bx)m b(m − 1) (a + bx)m−1 x(a + bx)m−1 For m = 1   1 1 ln(a + bx) dx ln x dx = ln x ln(a + bx) − (see 2.728 2) a + bx b b x  1 x ln x dx ln x + ln =− 2 (a + bx) b(a + bx) ab a + bx  1 x ln x dx ln x 1 + ln =− + (a + bx)3 2b(a + bx)2 2ab(a + bx) 2a2 b a + bx  

 √ √ (a + bx)1/2 − a1/2 ln x dx 2 √ = [a > 0] (ln x − 2) a + bx − 2 a ln x1/2  a + bx b   √ √ a + bx 2 (ln x − 2) a + bx + 2 −a arctan [a < 0] = b −a 

  m+1  dx 1 x xm+1 ln(a + bx) − b m+1 a + bx

 bx bx ln(a + bx) = ln a ln x + Φ − , 2, 1 [a > 0] x a a 

xm ln(a + bx) dx =



  m+1 1 (−a)m+1 1  (−1)k xm−k+2 ak−1 m+1 x ln(a + bx) dx = − ln(a + bx) + x m+1 bm+1 m+1 (m − k + 2)bk−1 k=1     ax 1 2 a2 1 x2 − x ln(a + bx) dx = x − 2 ln(a + bx) − 2 b 2 2 b     3 3 ax2 a2 x 1 3 a 1 x 2 − + 2 x ln(a + bx) dx = x + 3 ln(a + bx) − 3 b 3 3 2b b     ax3 a2 x2 1 4 a4 a3 x 1 x4 x3 ln(a + bx) dx = − + − x − 4 ln(a + bx) − 4 b 4 4 3b 2b2 b3 ⎧ ⎨     1 x x2n ln x2 + a2 dx = x2n+1 ln x2 + a2 + (−1)n 2a2n+1 arctan 2n + 1 ⎩ a ⎫ n  (−1)n−k 2n−2k 2k+1 ⎬ a x −2 ⎭ 2k + 1 m

1.  2.  3.  4.  2.731

TI 376

k=0

240

Logarithms and Inverse-Hyperbolic Functions





x2n+1 ln x2 + a

2.7327

 2

dx =

2.732

⎧ ⎨

   1 x2n+2 + (−1)n a2n+2 ln x2 + a2 2n + 2 ⎩ ⎫ n+1 ⎬  (−1)n−k a2n−2k+2 x2k + ⎭ k k=1

2.733



1.  2.  3.  4.  5.  2.734

    x ln x2 + a2 dx = x ln x2 + a2 − 2x + 2a arctan a        2 1 x2 + a2 ln x2 + a2 − x2 x ln x + a2 dx = 2     2 3  2 1 3  2 x 2 2 2 2 3 x ln x + a dx = x ln x + a − x + 2a x − 2a arctan 3 3 a       x4  1  4 + a2 x2 x − a4 ln x2 + a2 − x3 ln x2 + a2 dx = 4 2     2 5 2 2 3  2 1 5  2 x 4 2 2 4 5 x ln x + a dx = x ln x + a − x + a x − 2a x + 2a arctan 5 5 3 a ! ! x2n ln !x2 − a2 ! dx

DW DW DW DW DW

 ! ! n  ! 2 ! !x + a! 1 2! 2n+1 2n−2k 2k+1 ! ! ! a −2 ln x − a + a ln ! x x x − a! 2k + 1 k=0    1 ! 2 ! ! n+1  2n+2  ! 2 1 2n+1 2 2n+2 2 2n−2k+2 2k 2.735 x a x ln !x − a ! − ln !x − a ! dx = −a x 2n + 2 k 1 = 2n + 1



2n+1

k=1

2.736



1.  2.  3.  4.  5.

! ! ! !x + a! ! ! ! ! ln !x2 − a2 ! dx = x ln !x2 − a2 ! − 2x + a ln !! x − a! ! ! !  !  1  2 x − a2 ln !x2 − a2 ! − x2 x ln !x2 − a2 ! dx = 2 ! !

! 2 ! 2 !x + a! ! ! 2 3 1 2 2! 3 2! 2 3 ! ! ! ! x ln x − a dx = x ln x − a − x − 2a x + a ln ! 3 3 x − a! 

! 2 ! ! x4  ! 2 1  4 3 2! 4 2! 2 2 ! ! x − a ln x − a − −a x x ln x − a dx = 4 2 ! !

! 2 ! 2 !x + a! ! ! 2 5 2 2 3 1 4 2! 5 2! 4 5 ! ! ! ! x ln x − a dx = x ln x − a − x − a x − 2a x + a ln ! 5 5 3 x − a!

DW DW DW DW DW

2.74 Inverse hyperbolic functions 2.741  x  x 1. arcsinh dx = x arcsinh − x2 + a2 a a

DW

2.813

Arcsines and arccosines

 2.

arccosh

 3. 4. 2.742 1. 2.

x  x dx = x arccosh − x2 − a2 a a x  2 = x arccosh + x − a2 a

241

+ , x arccosh > 0 a + , x arccosh < 0 a

 x a  x dx = x arctanh + ln a2 − x2 a a 2   x a  x arccoth dx = x arccoth + ln x2 − a2 a a 2 arctanh





x2 a2 + 2 4

DW

DW DW



x x 2 − x + a2 a 4 2

 x a2 x x x 2 − x arccosh dx = x − a2 arccosh − a 4

a 4 22 x a2 x x 2 = − x − a2 arccosh + 2 4 a 4 x x arcsinh dx = a

DW

arcsinh

DW

+ , x arccosh > 0 a + , x arccosh < 0 a DW

2.8 Inverse Trigonometric Functions 2.81 Arcsines and arccosines   n/2

n   x n x n−2k arcsin dx = x (−1)k · (2k)! arcsin 2k a a k=0

(n+1)/2

   n x n−2k+1 k−1 2 2 + a −x (−1) · (2k − 1)! arcsin 2k − 1 a k=1   n/2

n   x n x n−2k 2.812 arccos dx = x (−1)k · (2k)! arccos 2k a a k=0

(n+1)/2

   n x n−2k+1 k 2 2 + a −x (−1) · (2k − 1)! arccos 2k − 1 a 2.811

k=1

2.813     x x + a2 − x2 arcsin dx = sign(a) x arcsin 1.11 a |a|

2    x 2 x x arcsin − 2x 2.9 dx = x arcsin + 2 a2 − x2 arcsin a |a| |a| ⎡

3

2    x 3 x x 3. arcsin dx = sign(a) ⎣x arcsin + 3 a2 − x2 arcsin a |a| |a| ⎤  x − 6 a2 − x2 ⎦ − 6x arcsin |a|

242

Inverse Trigonometric Functions

2.814

2.814  x x  1. arccos dx = x arccos − a2 − x2 a a     2 x x 2 x 2. arccos dx = x arccos − 2 a2 − x2 arccos − 2x a a a       3 3 x x 2 x x arccos 3. dx = x arccos − 3 a2 − x2 arccos − 6x arccos + 6 a2 − x2 a a a a

2.82 The arcsecant, the arccosecant, the arctangent, and the arccotangent 2.821     x a x 1. arccosec dx = arcsin dx = x arcsin + a ln x + x2 − a2 a x 2   a = x arcsin − a ln x + x2 − a2 x  2.



  a a x arcsec dx = arccos dx = x arccos − a ln x + x2 − a2 a x x   a = x arccos − a ln x + x2 − a2 x

+ a π, 0 < arcsin < x 2 , + π a − < arcsin < 0 2 x DW

+ a π, 0 < arccos < x 2 , + π a − < arccos < 0 2 x DW

2.822   x a  x 1.8 arctan dx = x arctan − ln a2 + x2 a a 2   x a  x 2. arccot dx = x arccot − ln a2 + x2 a a 2   1 2 x ax x 9 x + a2 arctan − 3. x arctan dx = a 2 a 2  2  ax πx 1 2 x x + − x + a2 arctan 4.9 x arccot dx = a 2 4 2 a   ax2  1 x x 1 5.9 x2 arctan dx = x3 arctan + a3 ln x2 + a2 − a 3 a 6 6    1 ax2 πx3 x x 1 + 6.9 x2 arccot dx = − x3 arctan − a3 ln x2 + a2 + a 3 a 6 6 6

2.83 Combinations of arcsine or arccosine and algebraic functions 

DW DW

 n+1 dx xn+1 x 1 x x √ arcsin − 2.831 x arcsin dx = (see 2.263 1, 2.264, 2.27) 2 a n + 1 a n + 1 − x2   an+1 n+1 dx x x 1 x x √ arccos + 2.832 xn arccos dx = (see 2.263 1, 2.264, 2.27) 2 a n+1 a n+1 a − x2   arccos x arcsin x dx and dx) cannot be expressed as a 1. For n = −1, these integrals (that is, x x finite combination of elementary functions. n

2.838

Arcsine or arccosine and algebraic functions

 2. 2.8339



1.  2.  3.  4.  5.  6. 2.834



π 1 arccos x dx = − ln − x 2 x



243

arcsin x dx x





x2 a2 x 2 x 2 − + a −x arcsin 2 4 |a| 4    1 2 πx2 x 2 x x 2 2 − sign(a) 2x − a arcsin + a −x x arccos dx = a 4 4 |a| 4  3   x x 1 2 x arcsin + x + 2a2 a2 − x2 x2 arcsin dx = sign(a) a 3 |a| 9  3  3  x πx x 1 2 x 2 2 2 2 − sign(a) arcsin + x + 2a a −x x arccos dx = a 6 3 |a| 9  4 

 x 3a4 1  x x − + x 2x2 + 3a2 a2 − x2 x3 arcsin dx = sign(a) arcsin a 4 32 |a| 32 % &  4 4 4    8x − 3a πx x 1 x x3 arccos dx = − sign(a) arcsin + x 2x2 + 3a2 a2 − x2 a 8 32 |a| 32

x x arcsin dx = sign(a) a

√

a2 − x2 x √  1 x 1 a + a2 − x2 x 1 arccos dx = − arccos − ln 2. 2 x a x a a x .   2  arcsin x 2 (a − b)(1 − x) arcsin x 2.835 − √ a > b2 dx = − arctan 2 2 2 (a + bx) b(a + bx) b a − b (a + b)(1 + x)     2 (a + b)(1 + x) + (b − a)(1 − x) 1 arcsin x  − √ =− a < b2 ln  b(a + bx) b b2 − a2 (a + b)(1 + x) − (b − a)(1 − x) √  1 c + 1x arcsin x x arcsin x √ √ + dx = − [c > −1] arctan 2.8368 2 2) 2 2 2c (1 + cx 2c c + 1 1 (1 + cx )  √ −x 1 − x2 + x −(c + 1) 1 arcsin x  +  ln √ [c < −1] =− 2 2c (1 + cx ) 4c −(c + 1) 1 − x2 − x −(c + 1) 1.

1 x 1 a+ 1 x arcsin dx = − arcsin − ln 2 x a x a a

2.837   x arcsin x √ dx = x − 1 − x2 arcsin x 1. 2 1−x  x arcsin x x x2 1 2 √ 2. − dx = 1 − x2 arcsin x + (arcsin x) 4 2 4 1 − x2  3  2x 1  2 x arcsin x x3 √ + − x +2 3. dx = 1 − x2 arcsin x 9 3 3 1 − x2 2.838   arcsin x x arcsin x 1  < 1. dx = √ + ln 1 − x2 2 2 3 1−x (1 − x2 )

244

Inverse Trigonometric Functions

 2.

2.841

arcsin x 1 1−x x arcsin x < dx = √ + ln 2 2 1+x 3 1−x (1 − x2 )

2.84 Combinations of the arcsecant and arccosecant with powers of x 2.841



1.

x arcsec

 2.

 3.



  a 1 2 a arccos dx = x arccos − a x2 − a2 x 2 x  a 1 2 2 2 x arccos + a x − a = 2 x

x dx = a

+ a π, 0 < arccos < x 2, +π a < arccos < π 2 x DW

  a 1 x a a  2 a3  2 3 2 2 2 ln x + x − a x arcsec dx = arccos dx = x arccos − x x − a − a x 3 x 2 2 + a π, 0 < arccos < x 2

  1 a a  2 a3  3 2 2 2 = ln x + x − z x arccos + x x − a + 3 x 2 2 +π , a < arccos < π 2 x 

DW



  a 1 2 x a x arcsin + a x2 − a2 x arccosec dx = arcsin dx = a x 2 x  a 1 2 2 2 x arcsin − a x − a = 2 x

+ a π, 0 < arcsin < x 2 , + π a − < arcsin < 0 2 x DW

2.85 Combinations of the arctangent and arccotangent with algebraic functions 

2.851 2.852

3. 2.853 1. 2.

xn+1 x a x dx = arctan − a n+1 a n+1

xn arccot

xn+1 x a x dx = arccot + a n+1 a n+1



1.

2.

xn arctan





xn+1 dx a2 + x2 xn+1 dx a2 + x2

For n = −1  arctan x dx cannot be expressed as a finite combination of elementary functions. x   arccot x π arctan x dx = ln x − dx x 2 x 

 1 2 x ax x dx = x + a2 arctan − a 2 a 2   1 2 x ax x x + a2 arccot + x arccot dx = a 2 a 2 x arctan

2.859

Arctangent and arccotangent with algebraic functions

245



 ax2 x3 x a3  2 x dx = arctan + ln x + a2 − a 3 a 6 6  3 3   x x a ax2 πx3 x ln x2 + a2 + + x2 arccot dx = − arctan − 4.9 a 3 a 6 6 6  1 x 1 a2 + x2 x 1 ln arctan dx = − arctan − 2.854 2 a x x2 

a 2a  x arctan x 1 β − αx α + βx √ arctan x 2.855 dx = − ln (α + βx)2 α2 + β 2 α + βx 1 + x2 2.856      1 ln 1 + x2 dx  x arctan x 1 2 1. − arctan x ln 1 + x dx = 1 + x2 2 2 1 + x2  2  1 x arctan x 1  2 2. dx = x arctan x − ln 1 + x2 − (arctan x) 2 1+x 2 2   3  1 x arctan x x arctan x 1 2 x + 1 + x arctan x − 3. dx = − dx 2 1+x 2 2 1 + x2 3.

x2 arctan

9





(see 2.8511)

x 1 2 − x arctan x + (arctan x) 3 2

TI (689) TI (405)

 x arctan x 1 2  dx = − x2 + ln 1 + x2 + 2 1+x 6 3 & %  n  x (2n − 2k)!!(2n − 1)!! arctan x dx 1 (2n − 1)!! arctan x arctan x 2.857 + n+1 = (2n)!!(2n − 2k + 1)!! (1 + x2 )n−k+1 2 (2)!! (1 + x2 ) k=1 n 1 (2n − 1)!!(2n − 2k)!! 1 + 2 (2n)!!(2n − 2k + 1)!!(n − k + 1) (1 + x2 )n−k+1 k=1 √   √ 2 x arctan x x √ 2.858 dx = − 1 − x2 arctan x + 2 arctan √ − arcsin x 1 − x2 1 − x2   arctan x a + bx2 x arctan x 1 < 2.859 [a < b] dx = √ − √ arctan 2 b − a√ 3 a b−a a a + bx √ (a + bx2 ) x arctan x a + bx2 − a − b 1 √ = √ + √ [a > b] ln √ 2a a − b a a + bx2 a + bx2 + a − b 4.

4

3

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3–4 Definite Integrals of Elementary Functions 3.0 Introduction 3.01 Theorems of a general nature 3.011 Suppose that f (x) is integrable† over the largest of the intervals (p, q), (p, r), (r, q). Then (depending on the relative positions of the points p, q, and r) it is also integrable over the other two intervals, and we have    q

r

f (x) dx = p

q

f (x) dx + p

f (x) dx.

FI II 126

r

3.012 The first mean-value theorem. Suppose (1) that f (x) is continuous and that g(x) is integrable over the interval (p, q), (2) that m ≤ f (x) ≤ M , and (3) that g(x) does not change sign anywhere in the interval (p, q). Then, there exists at least one point ξ (with p ≤ ξ ≤ q) such that  q  q f (x)g(x) dx = f (ξ) g(x) dx. FI II 132 p

p

3.013 The second mean-value theorem. If f (x) is monotonic and non-negative throughout the interval (p, q), where p
q

 ξ f (x)g(x) dx = f (p) g(x) dx p

Under the conditions of Theorem 3.013 1, if f (x) is nondecreasing, then  q  q f (x)g(x) dx = f (q) g(x) dx [p ≤ ξ ≤ q] . 2. p

ξ

If f (x) is monotonic in the interval (p, q), where p
q

function f (x) is said to be integrable over the interval (p, q), if the integral

f (x) dx exists. Here, we usually mean p

the existence of the integral in the sense of Riemann. When it is a matter of the existence of the integral in the sense of Stieltjes or Lebesgue, etc., we shall speak of integrability in the sense of Stieltjes or Lebesgue.

247

248

Introduction



q

3.

 ξ  q f (x)g(x) dx = f (p) g(x) dx + f (q) g(x) dx

p

or



p

q

4.

3.020

 ξ  q f (x)g(x) dx = A g(x) dx + B g(x) dx

p

[p ≤ ξ ≤ q] ,

ξ

p

[p ≤ ξ ≤ q] ,

ξ

where A and B are any two numbers satisfying the conditions A ≥ f (p + 0) A ≤ f (p + 0)

B ≤ f (q − 0) B ≥ f (q − 0)

and and

[if f decreases], [if f increases].

In particular,  q  ξ  q 5. f (x)g(x) dx = f (p + 0) g(x) dx + f (q − 0) g(x) dx p

p

FI II 138

ξ

3.02 Change of variable in a definite integral 



β

3.020

ψ

f (x) dx = α

f [g(t)]g  (t) dt;

x = g(t).

ϕ

This formula is valid under the following conditions: 1.

f (x) is continuous on some interval A ≤ x ≤ B containing the original limits of integration α and β.

2.

The equalities α = g(ϕ) and β = g(ψ) hold.

3.

g(t) and its derivative g  (t) are continuous on the interval ϕ ≤ t ≤ ψ.

4.

As t varies from ϕ to ψ, the function g(t) always varies in the same direction from g(ϕ) = α to g(ψ) = β.∗ 

3.021

β

The integral

f (x) dx can be transformed into another integral with given limits ϕ and ψ by α

means of the linear substitution

αψ − βϕ β−α t+ : ψ−ϕ ψ−ϕ

 β  β−α αψ − βϕ β−α ψ t+ f (x) dx = f dt ψ−ϕ ϕ ψ−ϕ ψ−ϕ α x=

1.

In particular, for ϕ = 0 and ψ = 1, ∗ If this last condition is not satisfied, the interval ϕ ≤ t ≤ ψ should be partitioned into subintervals throughout each of which the condition is satisfied:  β  ϕ1  ϕ2  ψ f (x) dx = f [g(t)]g  (t) dt + f [g(t)]g  (t) dt + · · · + f [g(t)]g  (t) dt. α

ϕ

ϕ1

ϕn−1

3.033

General formulas



β

2.

 1 f (x) dx = (β − α) f ((β − α)t + α) dt

α

0

For ϕ = 0 and ψ = ∞,   β f (x) dx = (β − α) 3. α

3.022 1.

249



∞

f 0

α + βt 1+t



dt (1 + t)2

The following formulas also hold:  β  β f (x) dx = f (α + β − x) dx α β

 2.



α β

f (β − x) dx

f (x) dx = 0



0



α

3.

α

f (x) dx = −α

f (−x) dx −α

3.03 General formulas 3.031 1.

Suppose that a function f (x) is integrable over the interval (−p, p) and satisfies the relation f (−x) = f (x) on that interval. (A function satisfying the latter condition is called an even function.) Then,  p  p f (x) dx = 2 f (x) dx. FI II 159 −p

2.

0

Suppose that f (x) is a function that is integrable on the interval (−p, p) and satisfies the relation f (−x) = −f (x) on that interval. (A function satisfying the latter condition is called an odd function). Then,  p f (x) dx = 0. FI II 159 −p

3.032  1. 0



π 2

π 2

f (sin x) dx =

f (cos x) dx, 0

where f (x) is a function that is integrable on the interval (0, 1).  2π  π  2. f (p cos x + q sin x) dx = 2 f p2 + q 2 cos x dx, 0

0

   where f (x) is integrable on the interval − p2 + q 2 , p2 + q 2 .  π2  π2   3. f (sin 2x) cos x dx = f cos2 x cos x dx, 0

FI II 159

FI II 160

0

where f (x) is integrable on the interval (0, 1). 3.033 1. If f (x + π) = f (x) and f (−x) = f (x), then

FI II 161

250

Introduction



∞

f (x) 0

sin x dx = x



3.034 π 2

f (x) dx

LO V 277(3)

0

If f (x + π) = −f (x) and f (−x) = f (x), then  π2  ∞ sin x dx = f (x) f (x) cos x dx x 0 0

2.

LO V 279(4)

In formulas  ∞3.033, it is assumed that the integrals in the left members of the formulas exist. q f (px) − f (qx) 3.034 dx = [f (0) − f (+∞)] ln , x p 0 if f (x) is continuous for x ≥0 and if there exists a finite limit f (+∞) = lim f (x). x→+∞

3.035 1. 2.

    f α + exi + f α + e−xi 2π dx = f (α + p) [|p| < 1] 2 1 + 2p cos x + p 1 − p2 0  π      1 − p cos x f α + exi + f α + e−xi dx = π {f (α + p) + f (α)} 2 1 − 2p cos x + p 0 

 3. 0

FI II 633

π

π



 −xi



−f α+e f α+e 1 − 2p cos x + p2

[|p| < 1]

 xi sin x dx =

LA 230(16)

BE 169

π {f (α + p) − f (α)} π [|p| < 1]

BE 169

In formulas 3.035, it is assumed that the function f is analytic in the closed unit circle with its center at the point α. 3.036

 π  π    2  sin2 x 1.11 f f sin2 x dx p <1 dx = 2 1 + 2p cos x + p 0 0 π 2

  2 sin x f = dx p ≥1 2 p 0  2.

 π F (n) (cos x) sin2n x dx = (2n − 1)!!

0

3.037

LA 228(6) π

F (cos x) cos nx dx

B 174

0

If f is analytic in the circle of radius r and if f [r (cos x + i sin x)] = f1 (r, x) + if2 (r, x),

then 

1. 2. 3.

∞

f1 (r, x) π  −p  f re dx = 2 2 2p 0 p +x  ∞  x dx π   −p  − f (0) f re f2 (r, x) 2 = 2 p +x 2 0  ∞ f2 (r, x) π dx = [f (r) − f (0)] x 2 0

LA 230(19) LA 230(20) LA 230(21)

3.045

Improper integrals



251

∞

  π  f2 (r, x) dx = 2 f (r) − f re−p 2 + x2 ) x (p 2p 0  ∞  ∞   x dx √ 3.038 F qx + p 1 + x2 = F (p cosh x + q sinh x) sinh x dx 2 −∞ 1 + x −∞  ∞   F  sign p · p2 − q 2 cosh x sinh2 x dx = 2q 4.

LA 230(22)

0

[If F is a function with a continuous derivative in the interval (−∞, ∞), all these integrals converge.]

3.04 Improper integrals 3.041 Suppose that a function f (x) is defined on an interval (p, +∞) and that it is integrable over an arbitrary finite subinterval of the form (p, P ). Then, by definition  P  +∞ f (x) dx = lim f (x) dx, P →+∞ p p  +∞ f (x) dx exists or that it converges. if this limit exists. If it does exist, we say that the integral p

Otherwise, we say that the integral diverges. 3.042 Suppose that a function f (x) is bounded and integrable in an arbitrary interval (p, q − η) (for 0 < η < q − p) but is unbounded in every interval (q − η, q) to the left of the point q. The point q is then called a singular point. Then, by definition,  q−η  q f (x) dx = lim f (x) dx, η→0 p p  q f (x) dx exists or that it converges. if this limit exists. In this case, we say that the integral p

3.043 If not only the integral of f (x) but also the integral of |f (x)| exists, we say that the integral of f (x) converges absolutely.  +∞ 3.044 The integral f (x) dx converges absolutely if there exists a number α > 1 such that the limit p

lim {xα |f (x)|}

x→+∞

exists. On the other hand, if 

lim {x|f (x)|} = L > 0,

x→+∞ +∞

|f (x)| dx diverges.

the integral p

3.045



q

Suppose that the upper limit q of the integral

f (x) dx is a singular point. Then, this integral p

converges absolutely if there exists a number α <1 such that the limit lim [(q − x)α |f (x)|] x→q

exists. On the other hand, if 

lim [(q − x)|f (x)|] = L > 0,

x→q

q

f (x) dx diverges.

the integral p

252

Introduction

3.046

3.046 Suppose that the functions f (x) and g(x) are defined on the interval (p, +∞), that f (x) is integrable over every finite interval of the form (p, P ), that the integral  P f (x) dx p

is a bounded function of P , that g(x) is monotonic, and that g(x) →0 as x → +∞. Then, the integral  +∞ f (x)g(x) dx p

converges.

FI II 577

3.05 The principal values of improper integrals 3.051 Suppose that a function f (x) has a singular point r somewhere inside the interval (p, q), that f (x) is defined at r, and that f (x) is integrable over every portion of this interval that does not contain the point r. Then, by definition 

 r−η  q  q f (x) dx = lim f (x) dx + f (x) dx . η→0 η  →0

p

r+η 

p

Here, the limit must exist for independent modes of approach of η and η  to zero. If this limit does not exist but the limit 

 r−η  q f (x) dx + f (x) dx lim η→0 p r+η "q does exist, we say that this latter limit is the principal value of the improper integral p f (x) dx, and we  q say that the integral f (x) dx exists in the sense of principal values. FI II 603 p

3.052 Suppose that the function f (x) is continuous over the interval (p, q) and vanishes at only one point r inside this interval. Suppose that the first derivative f  (x) exists in a neighborhood of the point r. Suppose that f  (r) = 0 and that the second derivative f  (r) exists at the point r itself. Then,  q dx FI II 605 p f (x) diverges, but exists in the sense of principal values. 3.053 A divergent integral of a positive function cannot exist in the sense of principal values. 3.054 Suppose that the function f (x) has no singular points in the interval (−∞, +∞). Then, by definition  Q  +∞ f (x) dx = lim f (x) dx. P →−∞ Q→+∞

−∞

P

Here, the limit must exist for independent approach of P and Q to ±∞. If this limit does not exist but the limit  +P lim f (x) dx P →+∞ −P

does exist, this last limit is called the principal value of the improper integral  +∞ f (x) dx.

FI II 607

−∞

3.055 The principal value of an improper integral of an even function exists only when this integral FI II 607 converges (in the ordinary sense).

3.112

Rational functions

253

3.1–3.2 Power and Algebraic Functions 3.11 Rational functions 

∞

1. −∞

r2

p + qx π (p − qr cos λ) (principal value) dx = 2 + 2rx cos λ + x r sin λ (see also 3.194 8 and 3.252 1 and 2) 

3.11211

Integrals of the form

BI (22)(14)

∞

gn (x) dx , where h −∞ n (x)hn (−x) gn (x) = b0 x2n−2 + b1 x2n−4 + · · · bn−1 , hn (x) = a0 xn + a1 xn−1 + · · · an

[All roots of hn (x) lie in the upper half-plane.]  ∞ πi Mn gn (x) dx = , 1. h (x)h (−x) a n 0 Δn −∞ n where ! ! !a1 a3 a5 0 !! ! !a0 a2 a4 0 !! ! ! 0 a1 a3 0 !!, Δn = ! ! .. ! . .. !. ! ! ! !0 0 0 an !  2.

∞

g1 (x) dx

−∞ h1 (x)h1 (−x)

3.8

4.11



∞



∞

b1 a2 a1

b2 a4 a3

···

.. 0

0

.

! bn−1 !! 0 !! 0 !!. ! ! ! an !

πib0 a0 a1

g3 (x) dx = πi −∞ h3 (x)h3 (−x) ∞

g4 (x) dx

−∞ h4 (x)h4 (−x)  ∞ g5 (x) dx

6.

! ! b0 ! ! a0 ! ! Mn = ! 0 ! .. !. ! !0

JE

−b0 + aa0 2b1 g2 (x) dx = πi a0 a1 −∞ h2 (x)h2 (−x)

 5.

=

JE

−∞ h5 (x)h5 (−x)

a0 a1 b 2 a3 a0 (a0 a3 − a1 a2 )

−a2 b0 + a0 b1 −

a0 b 3 (a0 a3 − a1 a2 ) a4 a0 (a0 a23 + a21 a4 − a1 a2 a3 )

JE

b0 (−a1 a4 + a2 a3 ) − a0 a3 b1 + a0 a1 b2 + = πi = πi

JE

M5 , a0 Δ5

where

  M5 = b0 −a0 a4 a5 + a1 a24 + a22 a5 − a2 a3 a4 + a0 b1 (−a2 a5 + a3 a4 )  a0 b 4  −a0 a1 a5 + a0 a23 + a21 a4 − a1 a2 a3 , + a0 b2 (a0 a5 − a1 a4 ) + a0 b3 (−a0 a3 + a1 a2 ) + a5 Δ5 = a20 a25 − 2a0 a1 a4 a5 − a0 a2 a3 a5 + a0 a23 a4 + a21 a24 + a1 a22 a5 − a1 a2 a3 a4

JE

254

Power and Algebraic Functions

3.121

3.12 Products of rational functions and expressions that can be reduced to square roots of first- and second-degree polynomials 3.121



1

1. 0

 2. 3.

∞

 sin kλ 1 dx √ = 2 cosec λ 1 − 2x cos λ + x2 x 2k − 1

BI (10)(17)

k=1

1

dx 1 π  [0 < p < q] = q − px q(q − p) x(1 − x) 0   1 π 11∓r 1+r dx 1∓x =± ∓ arctan 2 1 − 2rx + r 1 ± x 4r r 1 ± r 1−r 0

BI (10)(9)

LI (14)(5, 16)

3.13–3.17 Expressions that can be reduced to square roots of third- and fourthdegree polynomials and their products with rational functions 

Notation: In 3.131–3.137 we set: α = arcsin 



c−u , b−u

.

(a − c)(b − u) , (b − c)(a − u) .  (a − c)(u − b) a−u , λ = arcsin , κ = arcsin (a − b)(u − c) a−b    u−a a−c a−b μ = arcsin , ν = arcsin , p= , u−b u−c a−c γ = arcsin

3.131 1. 2. 3. 4. 5. 6. 7.



u−c , b−c

a−c , β = arcsin a−u

δ = arcsin

 q=

b−c . a−c

u

dx 2  =√ F (α, p) a−c (a − x)(b − x)(c − x) −∞  c dx 2  =√ F (β, p) a−c (a − x)(b − x)(c − x) u  u dx 2  =√ F (γ, q) a−c (a − x)(b − x)(x − c) c  b dx 2  =√ F (δ, q) a−c (a − x)(b − x)(x − c) u  u dx 2  =√ F (κ, p) a−c (a − x)(x − b)(x − c) b  a dx 2  =√ F (λ, p) a−c (a − x)(x − b)(x − c) u  u dx 2  =√ F (μ, q) a −c (x − a)(x − b)(x − c) a

[a > b > c ≥ u]

BY (231.00)

[a > b > c > u]

BY (232.00)

[a > b ≥ u > c]

BY (233.00)

[a > b > u ≥ c]

BY (234.00)

[a ≥ u > b > c]

BY (235.00)

[a > u ≥ b > c]

BY (236.00)

[u > a > b > c]

BY (237.00)

3.133

Square roots of polynomials



∞

8. u

3.132



c

1. u



dx 2  =√ F (ν, q) a −c (x − a)(x − b)(x − c)

x dx

2  =√ [c F (β, p) + (a − c) E (β, p)] − 2 a−c (a − x)(b − x)(c − x)



c

b

u



BY (232.19)

√ 2a =√ F (γ, q) − 2 a − c E (γ, q) a−c (a − x)(b − x)(x − c) x dx

BY (233.17)

    2 =√ (b − a) Π δ, q 2 , q + a F (δ, q) a−c (a − x)(b − x)(x − c) x dx



b

x dx (a − x)(x − b)(x − c)

2 =√ a−c



BY (234.16)

   (b − c) Π κ, p2 , p + c F (κ, p) [a ≥ u > b > c]

a

5. u



(a − u)(c − u) b−u

[a > b > u ≥ c] u

4. 



BY (238.00)

[a > b ≥ u > c]

3. 

[u ≥ a > b > c]

[a > b > c > u] u

2. 

255

BY (235.16)

√ x dx 2c  =√ F (λ, p) + 2 a − c E (λ, p) a−c (a − x)(x − b)(x − c) [a > u ≥ b > c]

u

6.



a

BY (236.16)

  2 = √ a(a − b) Π(μ, 1, q) + b2 F (μ, q) b a−c (x − a)(x − b)(x − c) x dx

[u > a > b > c] 3.133  1.

BY (237.16)

u

dx 2  √ = [F (α, p) − E (α, p)] 3 (a − b) a − c (a − x) (b − x)(c − x) −∞



[a > b > c ≥ u] c

2. u



dx (a −

x)3 (b

− x)(c − x)

=

2 2 √ [F (β, p) − E (β, p)] + a−c (a − b) a − c



BY (231.08)

c−u (a − u)(b − u)

[a > b > c > u] BY (232.13)  u dx (b − u)(u − c) 2 2  √ = E (γ, q) − 3 (a − b)(a − c) a−u (a − b) a − c (a − x) (b − x)(x − c) c BY (233.09) [a > b ≥ u > c]  b dx 2  √ = BY (234.05) E (δ, q) [a > b > u ≥ c] 3 (a − b) a − c (a − x) (b − x)(x − c) u .  u dx u−b 2 2  √ = [F (κ, p) − E (κ, p)] + 3 a − b (a − u)(u − c) (a − b) a − c (a − x) (x − b)(x − c) b 

3.

4.

5.

[a > u > b > c]

BY (235.04)

256

Power and Algebraic Functions



∞

6. u

8.

. dx

2 2  √ = E (ν, q) + 3 a − b (b − a) a − c (x − a) (x − b)(x − c)

u−b (u − a)(u − c)

[u > a > b > c] √ 2 dx 2 a−c  √ E (α, p) − = F (α, p) 3 (a − b) a − c (a − x)(b − x) (c − x) (a − b)(b −∞  − c) c−u 2 − b − c (a − u)(b − u) [a > b > c ≥ u] √  c 2 dx 2 a−c  √ E (β, p) − = F (β, p) (a − b)(b − c) (a − b) a − c (a − x)(b − x)3 (c − x) u

BY (238.05)

[a > b > c > u] √ u dx 2 2 a−c  √ E (γ, q) = F (γ, q) − (a − b)(b − c) (a − x)(b − x)3 (x − c) (b − c) a − c  c 2 (a − u)(u − c) + (a − b)(b − c) b−u [a > b > u > c] √  a dx 2 2 a−c  √ E (λ, p) = F (λ, p) − (a − b)(b − c) (a − x)(x − b)3 (x − c) (a − b) a − c  u 2 (a − u)(u − c) + (a − b)(b − c) u−b [a > u > b > c] √  u 2 dx 2 a−c  √ E (μ, q) − = F (μ, q) 3 (a − b)(b − c) (b − c) a−c (x − a)(x − b) (x − c) a

BY (232.14)

[u > a > b > c] √ 2 dx 2 a−c  √ E (ν, q) − = F (ν, q) 3 (a − b)(b − c) (b − c) a−c (x − a)(x − b) (x − c) u  u−a 2 − a − b (u − b)(u − c) [u ≥ a > b > c] .  u dx b−u 2 2  √ = E (α, p) + 3 b − c (a − u)(c − u) (c − b) a − c (a − x)(b − x)(c − x) −∞

BY (237.12)

 7.

3.133

u

 9.

10.

11.

 12.

13.

b

14. u

 15. b

BY (233.10)

BY (236.09)

∞

[a > b > c > u] 

BY (231.09)

u



dx (a − x)(b − x)(x −



c)3

dx (a − x)(x − b)(x −

c)3

=

=

2 2 √ [F (δ, q) − E (δ, q)] + b−c (b − c) a − c

.

BY (238.04)

BY (231.10)

b−u (a − u)(u − c)

[a > b > u > c]

BY (234.04)

[a ≥ u > b > c]

BY (235.01)

2 √ E (κ, p) (b − c) a − c

3.134

Square roots of polynomials



a

16. u



dx

2 2  √ = E (λ, p) − (b − c)(a − c) (b − c) a − c (a − x)(x − b)(x − c)3





a

dx

2 2 √ [F (μ, q) − E (μ, q)] + a−c (b − c) a − c

=

(x − a)(x − b)(x − c)3

[u > a > b > c] ∞

18. u

=

3(a − +

c

2.



u



u

3.

dx (a − x)5 (b − x)(c − x)



b

4. u



=

dx (a −

x)5 (b

(a −

x)5 (b

c



=

− x)(x − c)

dx − x)(x − c)

=

5. b

u



b)2

u−a (u − b)(u − c) BY (237.13)

BY (238.03)

2  [(3a − b − 2c) F (α, p) − 2(2a − b − c) E (α, p)] 3 (a − c) .

2 3(a − c)(a − b)

(c − u)(b − u) (a − u)3 [a > b > c ≥ u]

BY (231.08)

2  [(3a − b − 2c) F (β, p) − 2(2a − b − c) E (β, p)] 3(a− b)2 (a − c)3  2 4a2 − 3ab − 2ac + bc − u(3a − 2b − c) c−u + 3(a − b)(a − c)2 (a − u)3 (b − u) BY (232.13) [a > b > c > u] 2  [2(2a − b − c) E (γ, q) − (a − b) F (γ, q)] 3(a − (a − c)3  . 2 5a2 − 3ab − 3ac + bc − 2u(2a − b − c) (b − u)(u − c) − 3(a − b)2 (a − c)2 (a − u)3 [a > b ≥ u > c] BY (233.09) b)3

3(a − −



BY (236.10)

u

dx  5 (a − x) (b − x)(c − x) −∞





dx 2  √ [F (ν, q) − E (ν, q)] = 3 (b − c) a − c (x − a)(x − b)(x − c) [u ≥ a > b > c]

3.134  1.

(a − u)(u − b) u−c

[a > u ≥ b > c] u

17. 

257

b)2

2  [2(2a − b − c) E (δ, q) − (a − b) F (δ, q)] 3 (a − c) .

2 3(a − b)(a − c)

(b − u)(u − c) (a − u)3 [a > b > u ≥ c]

BY (234.05)

dx (a −

x)5 (x

− b)(x − c) =

2  [(3a − b − 2c) F (κ, p) − 2(2a − b − c) E (κ, p)] (a − c)3 3(a −  . 2 4a2 − 2ab − 3ac + bc − u(3a − b − 2c) u−b + 3(a − b)2 (a − c) (a − u)3 (u − c) BY (235.04) [a > u > b > c] b)2

258

Power and Algebraic Functions



∞

6. u

 7.

3.134

dx 2   = [2(2a − b − c) E (ν, q) − (a − b) F (ν, q)] 5 2 (x − a) (x − b)(x − c) 3(a − b) (a − c)3  2 . 2 4a − 2ab − 3ac + bc + u(b + 2c − 3a) u−b + 3(a − b)2 (a − c) (u − a)3 (u − c) BY (238.05) [u > a > b > c]

u

dx 2  √ = 2 (b − c)2 a − c 5 3(a − b) (a − x)(b − x) (c − x) −∞



× [2(a − c)(a + c − 2b) E (α, p) + (b − c)(3b − a − 2c) F (α, p)]   2 3ab − ac + 2bc − 4b2 − u(2a − 3b + c) c−u − 3(a − b)(b − c)2 (a − u)(b − u)3 [a > b > c ≥ u] BY (231.09) c

8. u





dx (a − x)(b −

u



c

a

u

 a

3(a −

b)2 (b

2 √ − c)2 a − c

dx (a − x)(b −

x)5 (x

− c)

=

3(a −

b)2 (b

2 √ − c)2 a − c

× [(a − b)(2a − 3b + c) F (γ, q) + 2(a − c)(2b − a − c) E (γ, q)]  . 2 3ab + 3bc − ac − 5b2 − 2u(a − 2b + c) (a − u)(u − c) + 3(a − b)2 (b − c)2 (b − u)3 [a > b > u > c] BY (233.10)

10.

11.

− x)

=

× [(b − c)(3b − a − 2c) F (β, p) + 2(a − c)(a − 2b + c) E (β, p)] . (a − u)(c − u) 2 + 3(a − b)(b − c) (b − u)3 [a > b > c > u] BY (232.14)

9.



x)5 (c

dx 2  √ = 2 (b − c)2 a − c 5 3(a − b) (a − x)(x − b) (x − c) × [(b − c)(3b − 2c − a) F (λ, p) + 2(a − c)(a + c − 2b) E (λ, p)]  . 2 3ab + 3bc − ac − 5b2 + 2u(2b − a − c) (a − u)(u − c) + 2 2 3(a − b) (b − c) (u − b)3 [a > u > b > c] BY (236.09)

u



dx (x − a)(x − b)5 (x − c)

=

2 √ 3(a − b)2 (b − c)2 a − c × [(a − b)(2a + c − 3b) F (μ, q) + 2(a − c)(2b − a − c) E (μ, q)] . (u − a)(u − c) 2 + 3(a − b)(b − c) (u − b)3 [u > a > b > c] BY (237.12)

3.134

Square roots of polynomials



∞

12. u

 13.

14.

15.

16.

17.

18.

259

dx 2  √ = 2 (b − c)2 a − c 5 3(a − b) (x − a)(x − b) (x − c) × [(a − b)(2a + c − 3b) F (ν, q) + 2(a − c)(2b − c − a) E (ν, q)]   2 3bc + 2ab − ac − 4b2 + u(3b − a − 2c) u−a − 3(a − b)2 (b − c) (u − b)3 (u − c) BY (238.04) [u ≥ a > b > c]

u

dx 2   = [2(a + b − 2c) E (α, p) − (b − c) F (α, p)] 5 2 (a − x)(b − x)(c − x) 3(b − c) (a − c)3 −∞  . 2 ab − 3ac − 2bc + 4c2 + u(2a + b − 3c) b−u + 2 3(a − c)(b − c) (a − u)(c − u)3 [a > b > c > u] By (231.10)  b dx 2   = [(2a + b − 3c) F (δ, q) − 2(a + b − 2c) E (δ, q)] 5 2 (a − x)(b − x)(x − c) 3(b − c) (a − c)3 u  . 2 ab − 3ac − 2bc + 4c2 + u(2a + b − 3c) b−u + 2 3(b − c) (a − c) (a − u)(u − c)3 [a > b > u > c] BY (234.04)  u dx 2   = [2(a + b − 2c) E (κ, p) − (b − c) F (κ, p)] 5 2 (a − c)3 (a − x)(x − b)(x − c) 3(b − c) b . 2 (a − u)(u − b) + 3(a − c)(b − c) (u − c)3 [a ≥ u > b > c] BY (235.20)  a dx 2   = [2(a + b − 2c) E (λ, p) − (b − c) F (λ, p)] 5 2 (a − x)(x − b)(x − c) 3(b − c) (a − c)3 u  . 2 ab − 3ac − 3bc + 5c2 + 2u(a + b − 2c) (a − u)(u − b) − 2 2 3(b − c) (a − c) (u − c)3 [a > u ≥ b > c] BY (236.10)  u dx 2   = [(2a + b − 3c) F (μ, q) − 2(a + b − 2c) E (μ, q)] 5 2 (x − a)(x − b)(x − c) 3(b − c) (a − c)3 a  2 4c2 − ab − 2ac − bc + u(3a + 2b − 5c) u−a + 2 3(b − c)(a − c) (u − b)(u − c)3 [u > a > b > c] BY (237.13)  ∞ dx 2   = [(2a + b − 3c) F (ν, q) − 2(a + b − 2c) E (ν, q)] 5 2 3 (x − a)(x − b)(x − c) 3(b − c) (a − c) u . (u − a)(u − b) 2 + 3(a − c)(b − c) (u − c)3 [u ≥ a > b > c] BY (238.03)

260

3.135 1.6

2.

3.

4.

5.

6.

7.

Power and Algebraic Functions



u

3.135

dx 2  √ = [(b − c) F (α, p) − (2a − b − c) E (α, p)] 3 (c − x)3 (a − b)(b − c)2 a − c (a − x)(b − x) −∞ 2(b + c − 2u)  + 2 (b − c) (a − u)(b − u)(c − u) BY (231.13) [a > b > c > u]  a dx 2  √ = [(b − c) F (λ, p) − 2(2a − b − c) E (λ, p)] 3 3 (a − b)(b − c)2 a − c  (a − x)(x − b) (x − c) u a−u 2(a − b − c + u) + (a − b)(b − c)(a − c) (u − b)(u − c) BY (236.15) [a > u > b > c]  u dx 2  √ = [(2a − b − c) E (μ, q) − 2(a − b) F (μ, q)] 3 3 (a − b)(b − c)2  a−c (x − a)(x − b) (x − c) a u−a 2 + (a − c)(b − c) (u − b)(u − c) BY (236.14) [u > a > b > c]  ∞ dx 2  √ = [(2a − b − c) E (ν, q) − 2(a − b) F (ν, q)] 3 3 (a − b)(b − c)2  a−c (x − a)(x − b) (x − c) u u−a 2 − (a − b)(b − c) (u − b)(u − c) BY (238.13) [u ≥ a > b > c]  u dx 2   = [(2b − a − c) E (α, p) − (b − c) F (α, p)] 3 3 (a − x) (b − x)(c − x) (a − b)(b − c) . (a − c)3 −∞ b−u 2 + (b − c)(a − c) (a − u)(c − u) BY(231.12) [a > b > c > u]  b dx 2   = [(a − b) F (δ, q) + (2b − a − c) E (δ, q)] 3 3 (a − x) (b − x)(x − c) (b − c)(a − b) . (a − c)3 u b−u 2 + (b − c)(a − c) (a − u)(u − c) BY (234.03) [a > b > u > c]  u dx 2   = [(b − c) F (κ, p) − (2b − a − c) E (κ, p)] 3 3 (a − x) (x − b)(x − c) (a − b)(b − c) . (a − c)3 b u−b 2 + (a − b)(a − c) (a − u)(u − c) BY (235.15) [a > u > b > c]

3.136

Square roots of polynomials



∞

8. u

dx 2   = [(a + c − 2b) E (ν, q) − (a − b) F (ν, q)] 3 3 (x − a) (x − b)(x − c) (a − b)(b − c) . (a − c)3 +

9.

10.

11.

12.

3.136 1.



u



u

261

2 (a − b)(a − c)

u−b (u − a)(u − c) [u > a > b > c]

BY (238.14)

dx 2  √ = [(a + b − 2c) E (α, p) − 2(b − c) F (α, p)] 3 3 (b − c)(a − b)2  a−c (a − x) (b − x) (c − x) −∞ c−u 2 − (a − b)(b − c) (a − u)(b − u) BY (231.11) [a > b > c ≥ u]  c dx 2  √ = [(a + b − 2c) E (β, p) − 2(b − c) F (β, p)] 2 (b − c) a − c 3 3 (a − b) (a − x) (b − x) (c − x) u  c−u 2 + (a − b)(a − c) (a − u)(b − u) BY (232.15) [a > b > c > u]  u dx 2  √ = [(a − b) F (γ, q) − (a + b − 2c) E (γ, q)] 2 (b − c) a − c 3 3 (a − b) (a − x) (b − x) (x − c) c   2 2 2 a + b − ac − bc − u(a + b − 2c) u−c + (a − b)2 (b − c)(a − c) (a − u)(b − u) BY (233.11) [a > b > u > c]  ∞ dx 2  √ = [(a − b) F (ν, q) − (a + b − 2c) E (ν, q)] 2 3 3 (x − a) (x − b) (x − c) (a − b) (b − c) a − c u 2u − a − b  + 2 (a − b) (u − a)(u − b)(u − c) BY (238.15) [u > a > b > c]

dx  3 (a − x) (b − x)3 (c − x)3 −∞ =

2

 (a − − c)2 (a − c)3     × (b − c)(a + b − 2c) F (α, p) − 2 c2 + a2 + b2 − ab − ac − bc E (α, p) b)2 (b

+

2[c(a − c) + b(a − b) − u(2a − c − b)]  (a − b)(a − c)(b − c)2 (a − u)(b − u)(c − u) [a > b > c > u]

BY (231.14)

262

Power and Algebraic Functions



∞

2. u

dx  3 (x − a) (x − b)3 (x − c)3 =

1.6

2.

2

 (a − − c)2 (a − c)3     × (a − b)(2a − b − c) F (ν, q) − 2 a2 + b2 + c2 − ab − ac − bc E (ν, q) b)2 (b

+

3.137

3.137

2[u(a + b − 2c) − a(a − c) − b(b − c)]  (a − b)2 (a − c)(b − c) (u − a)(u − b)(u − c) [u > a > b > c]

BY (238.16)

  dx 2 a−r  √ , p − F (α, p) = Π α, a−c (a − r) a − c −∞ (r − x) (a − x)(b − x)(c − x)



u



c

[a > b > c ≥ u]

BY (231.15)

dx 2(c − b)  √ = (r − x) (a − x)(b − x)(c − x) (r − b)(r − c) a − c 2 r−b √ ,p + × Π β, F (β, p) r−c (r − b) a − c [a > b > c > u, r = 0]

 u dx 2 b−c  √ ,q = Π γ, r−c (r − c) a − c c (r − x) (a − x)(b − x)(x − c) u

3.



[a > b ≥ u > c, b

4. u

5.

6.8



u



u

r = c]

dx 2  √ = (r − x) (a − x)(b − x)(x − c) (r− a)(r − b)

 a−c r−a , q + (r − b) F (δ, q) × (b − a) Π δ, q 2 r−b [a > b > u ≥ c, r = b]

BY (232.17)

BY (233.02)

BY (234.18)

dx 2  √ = (c − r)(b − r) b (x − r) (a − x)(x − b)(x − c)

  a−c 2c− r , p + (b − r) F (κ, p) × (c − b) Π κ, p b−r [a ≥ u > b > c, r = b] BY (235.17)

 a dx 2 a−b  √ ,p = Π λ, a −r (a − r) a − c u (x − r) (a − x)(x − b)(x − c)

7. a

[a > u ≥ b > c,

r = a]

dx 2  √ = (x − r) (x − a)(x − b)(x − c) (b− r)(a − r)

 a−c b−r , q + (a − p) F (μ, q) × (b − a) Π μ, a−b [u > a > b > c, r = a]

BY (236.02)

BY (237.17)

3.139

Square roots of polynomials



∞

8. u

  2 r−c  √ , q − F (ν, q) = Π ν, a−c (r − c) a − c (x − r) (x − a)(x − b)(x − c) dx

[u ≥ a > b > c] 3.138



u

1.



0

 2.

3.

4.

5.

dx x(1 − x) (1 − k 2 x)

  √ = 2 F arcsin u, k

[0 < u < 1]

  √ dx [0 < u < 1]  2  = 2 F arccos u, k u x(1 − x) k  + k 2 x √

 1 dx 1−u < , k [0 < u < 1] = 2 F arcsin   k u x(1 − x) x − k  2  u   √ dx < [0 < u < 1]   = 2 F arctan u, k 2 0 x(1 + x) 1 + k  x  u   √ dx <    = F 2 arctan u, k  2 0 x 1 + x2 + 2 k  − k 2 x 

1

1

6. u



u

7. a



a

8. u

263

<

dx <   =F x k 2 (1 + x2 ) + 2 (1 + k 2 ) x

π 2

[0 < u < 1] √ − 2 arctan u, k

[0 < u < 1]  

dx u−α p+m−α 1  , = √ F 2 arctan p p 2p (x − α) [(x − m)2 + n2 ]

dx 1  =√ F 2 2 p (α − x) [(x − m) + n ]



2 arccot

BY (238.06)

PE (532), JA

PE(533)

PE (534)

PE (535)

JA

JA

[α < u] , 

α−u p−m+α , p 2p [u < α] ,

 where p = (m − α)2 + n2 .

√ 1− 3−u √ , 3.139 Notation α = arccos 1+ 3−u √ 3+1−u , γ = arccos √ 3−1+u  u dx 1 √ F (α, sin 75◦ ) = √ 1. 4 3 3 −∞ 1 − x  1 dx 1 √ F (β, sin 75◦ ) 2. = √ 4 3 3 1 − x u

√ 3−1+u β = arccos √ , 3+1−u √ u−1− 3 √ . δ = arccos u−1+ 3 H 66 (285)

H 65 (284)

264

3. 4. 5. 6. 7. 8. 9. 10. 11.

Power and Algebraic Functions



u



∞

3.139

dx 1 √ F (γ, sin 15◦ ) H 65 (283) = √ 4 3 3 x −1 1  ∞ dx 1 √ F (δ, sin 15◦ ) H 65 (282) = √ 4 3 3 x −1 u

3  1 dx 1 1 √ √ √ = MO 9 Γ 3 3 3 2π 3 2 1−x 0 √ 3  1 x dx 2 1 3 √ = √ MO 9 Γ 3 3 π 4 3 1−x 0  1   1 √ 4 1 − x3 dx = 27 F (β, sin 75◦ ) − 2u 1 − u3 BY (244.01) 5 u √  1  1 √ 1 x dx 2 1 − u3 4 √ = 3− 4 − 3 4 F (β, sin 75◦ ) + 2 3 E (β, sin 75◦ ) − √ BY (244.05) 3+1−u 1 − x3 u √  1 m  2(m − 2) 1 xm−3 dx x dx 2um−2 1 − u3 √ √ + = BY (244.07) 2m − 1 2m − 1 u 1 − x3 1 − x3 u √  u  1 √ 1 x dx 2 u3 − 1 4 −4 ◦ ◦ 4 √ BY (240.05) = 3 + 3 F (γ, sin 15 ) − 2 3 E (γ, sin 15 ) + √ 3−1+u x3 − 1 1 √  u dx 1 + u + u2 1 2 ◦ ◦ √ √ √  √ [F (α, sin 75 = √ ) − 2 E (α, sin 75 )] + 4 3 27 3 1+ 3−u 1−u −∞ (1 − x) 1 − x [u = 1]

12. u

BY (246.06)

√ dx 1 + u + u2 2 1 ◦ ◦ √ √ √ [F (δ, sin 15 ) − 2 E (δ, sin 15 )] + √  = √ 4 27 3 u−1+ 3 u−1 (x − 1) x3 − 1

[u = 1] √ u (1 − x) dx 2− 3 [F (α, sin 75◦ ) − E (α, sin 75◦ )] = √ √  2 √ 4 27 −∞ 1 + 3−x 1 − x3 √  1 (1 − x) dx 2− 3 √ [F (β, sin 75◦ ) − E (β, sin 75◦ )] = 4 √  2 √ 27 u 1+ 3−x 1 − x3  √ √ √  u 2 3−2 2− 3 (x − 1) dx u3 − 1 √ √ − 4 E (γ, sin 15◦ ) = √  2 √ u2 − 2u − 2 3 27 1 1+ 3−x x3 − 1 √  √  √  ∞ 2 2− 3 2− 3 (x − 1) dx u3 − 1 √ √ − E (δ, sin 15◦ ) = √  2 √ 4 2 − 2u − 2 u 3 27 3 u 1+ 3−x x −1 & √ % √ √  u (1 − x) dx 2 + 3 2 4 3 1 − u3 − E (α, sin 75◦ ) = √ √  2 √ 4 u2 − 2u − 2 27 −∞ 1 − 3−x 1 − x3 √  u (x − 1) dx 2+ 3 √ [F (γ, sin 15◦ ) − E (γ, sin 15◦ )] = 4 √  2 √ 3 27 1 1− 3−x x −1

BY (242.03)

 13.

14.

15.

16.

17.

18.

BY (246.07)

BY (244.04)

BY (240.08)

BY (242.07)

BY (246.08)

BY (240.04)

3.141

Square roots of polynomials

√ (x − 1) dx 2+ 3 [F (δ, sin 15◦ ) − E (δ, sin 15◦ )] = √ √  2 √ 4 3 27 u 1− 3−x x −1  2   u x + x + 1 dx 1 E (α, sin 75◦ ) = √ √  2 √ 4 3 −∞ 1 + 3−x 1 − x3  2   1 x + x + 1 dx 1 E (β, sin 75◦ ) = √ √ 2 √  4 3 3 u x−1+ 3 1−x  2   u x + x + 1 dx 1 E (γ, sin 15◦ ) = √ √ 2 √ 4 3 3 1 3+x−1 x −1  2   ∞ x + x + 1 dx 1 E (δ, sin 15◦ ) = √ √ 2 √  4 3 3 u x−1+ 3 x −1 √  u (x − 1) dx 2+ 3 4 ◦ √ E (γ, sin 15 ) − √ F (γ, sin 15◦ ) = √ 4 4 2 3 27 √ 27 1 (x + x + 1) x − 1  √ 2 − 3 2(u − 1) 3 + 1 − u √ √ − √ 3 3 − 1 + u u3 − 1 

19.

20.

21.

22.

23.

24.

265

∞

BY (242.05)

BY (246.01)

BY (244.02)

BY (240.01)

BY (242.01)

BY (240.09)

√ 2  u   1 + 3 − x dx 1 ,√ + Π α, p2 , sin 75◦ = √ √ √ 2 4 3 −∞ 1 + 3 − x − 4 3p2 (1 − x) 1 − x3 

25.



1

26. u



u

27. 1



u

3.141

√  2   1 + 3 − x dx 1 ,√ + Π β, p2 , sin 75◦ = √ √ √ 2 4 3 1 + 3 − x − 4 3p2 (1 − x) 1 − x3

BY (244.03)

√  2   1 − 3 − x dx 1 ,√ + Π γ, p2 , sin 15◦ = √ √ √ 2 4 3 1 − 3 − x − 4 3p2 (x − 1) x3 − 1

BY (240.02)

√ 2   1 − 3 − x dx 1 ,√ + Π δ, p2 , sin 15◦ = √ √ √ 2 4 3 1 − 3 − x − 4 3p2 (x − 1) x3 − 1

BY (242.02)

∞

28.

BY (246.02)



Notation: In 3.141 and 3.142 we set:    a−c c−u u−c , β = arcsin , γ = arcsin α = arcsin a−u b−u b−c . .  (a − c)(b − u) (a − c)(u − b) a−u , κ = arcsin , λ = arcsin , δ = arcsin (b − c)(a − u) (a − b)(u − c) a−b     u−a a−c a−b b−c μ = arcsin , ν = arcsin , p= , q= . u−b u−c a−c a−c

266

Power and Algebraic Functions

 √ a−x (a − u)(c − u) dx = 2 a − c [F (β, p) − E (β, p)] + 2 (b − x)(c − x) b−u

 c 1. u

 u c

√ a−x dx = 2 a − c E (γ, q) (b − x)(x − c)

u

√ a−x dx = 2 a − c E (δ, q) − 2 (b − x)(x − c)

2.  b 3.

 u 4. b

 a u

 u 6. a

7. u

 u. 8. c

u

BY (233.01)

(b − u)(u − c) a−u

[a > b > u ≥ c]  √ a−x (a − u)(u − b) dx = 2 a − c [F (κ, p) − E (κ, p)] + 2 (x − b)(x − c) u−c

BY (234.06)

[a ≥ u > b > c]

BY (235.07)

√ x−a dx = −2 a − c E (μ, q) + 2 (x − b)(x − c)



[a > u ≥ b > c]

BY (236.04)

(u − a)(u − c) u−b

[a > b > c > u]

BY (237.03)

BY (232.07)

√ b−x 2(a − b) dx = 2 a − c E (γ, q) − √ F (γ, q) (a − x)(x − c) a−c BY (233.04)

[a > b > u ≥ c]  √ x−b (a − u)(u − b) 2(b − c) dx = 2 a − c E (κ, p) − √ F (κ, p) − 2 (a − x)(x − c) u−c a−c

BY (234.07)

10. b

 a. 11. u

 u. a

[a > b ≥ u > c]

[a > b ≥ u > c]  √ b−x (b − u)(u − c) 2(a − b) dx = 2 a − c E (δ, q) − √ F (δ, q) − 2 (a − x)(x − c) a−u a−c

 u.

12.

BY (232.06)

[u > a > b > c]  √ 2(b − c) b−x (a − u)(c − u) dx = √ F (β, p) − 2 a − c E (β, p) + 2 (a − x)(c − x) b−u a−c

 c.

9.



[a > b > c > u]

√ a−x dx = 2 a − c [F (λ, p) − E (λ, p)] (x − b)(x − c)

5.

 b.

3.141

[a ≥ u > b > c]

BY (235.06)

√ x−b 2(b − c) dx = 2 a − c E (λ, p) − √ F (λ, p) (a − x)(x − c) a−c [a > u ≥ b > c]  √ 2(a − b) x−b (u − a)(u − c) dx = √ F (μ, q) − 2 a − c E (μ, q) + 2 (x − a)(x − c) u−b a−c [u > a > b > c]

BY (236.03)

BY (237.04)

3.141

Square roots of polynomials

 c

√ c−x dx = −2 a − c E (β, p) + 2 (a − x)(b − x)

13. u

 u 14. c

 b 15. u

16. b

 a 17. u

 u 18. a

 c 19. u

 u c

 b

21.11

22.

(a − u)(c − u) b−u [a > b > c > u]

BY (232.08)

√ x−c dx = 2 a − c [F (γ, q) − E (γ, q)] (a − x)(b − x)

[a > b ≥ u > c]  √ x−c (b − u)(u − c) dx = 2 a − c [F (δ, q) − E (δ, q)] + 2 (a − x)(b − x) a−u

 u

20.



267

√ x−c dx = 2 a − c E (κ, p) − 2 (a − x)(x − b) √ x−c dx = 2 a − c E (λ, p) (a − x)(x − b)



[a > b > u ≥ c]

BY (233.03)

BY (234.08)

(a − u)(u − b) u−c [a ≥ u > b > c]

BY (235.07)

[a > u ≥ b > c]

BY (236.01)

 √ x−c (u − a)(u − c) dx = 2 a − c [F (μ, q) − E (μ, q)] + 2 (x − a)(x − b) u−b [u > a > b > c]

BY (237.05)

2√ (b − x)(c − x) dx = a − c [(2a − b − c) E (β, p) − (b − c) F (β, p)] a−x 3  (a − u)(c − u) 2 + (2b − 2a + c − u) 3 b−u [a > b > c > u]

BY (232.11)

2√ (x − c)(b − x) dx = a − c [(2a − b − c) E (γ, q) − 2(a − b) F (γ, q)] a−x 3 2 − (a − u)(b − u)(u − c) 3 [a > b ≥ u > c]

BY (233.06)

2√ (x − c)(b − x) dx = a − c [2(b − a) F (δ, q) + (2a − b − c) E (δ, q)] a−x 3  u (b − u)(u − c) 2 + (b + c − a − u) 3 a−u [a > b > u ≥ c]  u 2√ (x − b)(x − c) dx = a − c [(2a − b − c) E (κ, p) − (b − c) F (κ, p)] a−x 3  b (a − u)(u − b) 2 + (b + 2c − 2a − u) 3 u−c [a ≥ u > b > c]

BY (234.11)

BY (235.10)

268

Power and Algebraic Functions

 a 23.

11 u

 u 24. a

 c 25. u

 u 26. c

 b 27.

28.

29.

30.11

3.141

2√ (x − b)(x − c) dx = a − c [(2a − b − c) E (λ, p) − (b − c) F (λ, p)] a−x 3  2 + (a − u)(u − b)(u − c) 3 [a > u ≥ b > c]

BY (236.07)

2√ (x − b)(x − c) dx = a − c [2(a − b) F (μ, q) + (b + c − 2a) E (μ, q)] x−a 3  (u − a)(u − b) 2 + (u + 2a − 2b − c) 3 u−c [u > a > b > c]

BY (237.08)

2√ (a − x)(c − x) dx = a − c [(2b − a − c) E (β, p) − (b − c) F (β, p)] b−x 3  (a − u)(c − u) 2 + (a + c − b − u) 3 b−u [a > b > c > u]

BY (232.10)

2√ (a − x)(x − c) dx = a − c [(2b − a − c) E (γ, q) + (a − b) F (γ, q)] b−x 3  2 − (a − u)(b − u)(u − c) 3 [a > b ≥ u > c]

BY (233.05)

2√ (a − x)(x − c) dx = a − c [(a − b) F (δ, q) + (2b − a − c) E (δ, q)] b−x 3  u (b − u)(u − c) 2 + (2a + c − 2b − u) 3 a−u [a > b > u ≥ c]   u 2√ (a − x)(x − c) dx = a − c [(b − c) F (κ, p) + (a + c − 2b) E (κ, p)] x−b 3  b (a − u)(u − b) 2 + (2b − a − 2c + u) 3 u−c [a ≥ u > b > c]  a 2√ (a − x)(x − c) dx = a − c [(a + c − 2b) E (λ, p) + (b − c) F (λ, p)] x−b 3 u 2 − (a − u)(u − b)(u − c) 3 [a > u ≥ b > c]  u 2√ (x − a)(x − c) dx = a − c [(a + c − 2b) E (μ, q) − (a − b) F (μ, q)] x − b 3  a (u − a)(u − c) 2 + (u + b − a − c) 3 u−b [u > a > b > c]

BY (234.10)

BY (235.11)

BY (236.06)

BY (237.06)

3.142

Square roots of polynomials

 c 31. u

 u 32. c

 b 33.

34.

35.

36.

3.142

2√ (a − x)(b − x) dx = a − c [2(b − c) F (β, p) + (2c − a − b) E (β, p)] c−x 3  (a − u)(c − u) 2 + (a + 2b − 2c − u) 3 b−u [a > b > c > u]

BY (232.09)

2√ (a − x)(b − x) dx = a − c [(a + b − 2c) E (γ, q) − (a − b) F (γ, q)] x−c 3  2 + (a − u)(b − u)(u − c) 3 [a > b ≥ u > c]

BY (233.07)

2√ (a − x)(b − x) dx = a − c [(a + b − 2c) E (δ, q) − (a − b) F (δ, q)] x − c 3  u (b − u)(u − c) 2 + (2c − 2a − b + u) 3 a−u [a > b > u ≥ c]  u 2√ (a − x)(x − b) dx = a − c [(a + b − 2c) E (κ, p) − 2(b − c) F (κ, p)] x − c 3  b (a − u)(u − b) 2 + (u + c − a − b) 3 u−c [a ≥ u > b > c]  a 2√ (a − x)(x − b) dx = a − c [(a + b − 2c) E (λ, p) − 2(b − c) F (λ, p)] x−c 3 u  2 − (a − u)(u − b)(u − c) 3 [a > u ≥ b > c]  u 2√ (x − a)(x − b) dx = a − c [(a + b − 2c) E (μ, q) − (a − b) F (μ, q)] x−c 3  a (u − a)(u − c) 2 + (u + 2c − a − 2b) 3 u−b [u > a > b > c] 

u



1. −∞

 b 2. u

+2

3. b

BY (234.09)

BY (235.09)

BY (236.05)

BY (237.07)

. √ a−x b−u 2(a − c) 2 2 a−c E (α, p) + dx = √ F (α, p) − (b − x)(c − x)3 b−c b−c (a − u)(c − u) a−c

[a > b > c > u] √ a−x 2 a−c a−b √ E (δ, q) F (δ, q) − dx = 2 3 (b − x)(x − c) b−c (b − c) .a − c

 u

269

a−c b−c

BY (231.05)

b−u (a − u)(u − c)

[a > b > u > c] √ 2 a−x 2 a−c E (κ, p) − √ dx = F (κ, p) (x − b)(x − c)3 b−c a−c [a ≥ u > b > c]

BY (234.13)

BY (235.12)

270

Power and Algebraic Functions

 √ 2 a−x (a − u)(u − b) 2 a−c 2 E (λ, p) − √ dx = F (λ, p) − (x − b)(x − c)3 b−c b−c u−c a−c

 a 4. u

 u 5. a

BY (236.12) [a > u ≥ b > c] √  x−a u−a 2(a − b) 2 a−c √ E (μ, q) − dx = F (μ, q) − 2 3 (x − b)(x − c) b−c (u − b)(u − c) (b − c) a − c

[u > a > b > c] √ x−a 2(a − b) 2 a−c √ E (ν, q) − dx = F (ν, q) (x − b)(x − c)3 b−c (b − c) a − c

BY (237.10)

[u ≥ a > b > c] √  2 a−c a−b a−x c−u dx = E (α, p) − 2 3 (b − x) (c − x) b−c b − c (a − u)(b − u)

BY (238.09)

[a > b > c ≥ u] √ a−x 2 a−c dx = E (β, p) [a > b > c > u] 3 (b − x) (c − x) b−c u  √  u 2 a−c 2 a−x (a − u)(u − c) dx = [F (γ, q) − E (γ, q)] + 3 (x − c) (b − x) b − c b − c b−u c

BY (231.03)



∞

6. u



u



7. −∞

 c 8. 9.

3.142

 a 10. u

 u 11. a



u



[a > b > u > c]  √ 2 a−c 2 a−x (a − u)(u − c) dx = E (λ, p) + (x − b)3 (x − c) c−b b−c u−b

BY (233.15)

[a > u > b > c]

BY (236.11)

√ 2 a−c x−a dx = [F (μ, q) − E (μ, q)] (x − b)3 (x − c) b−c

∞

12.

u

−∞

 c. u

 u. 15. c

 b. u

2 b−x dx = √ E (α, p) (a − x)3 (c − x) a−c

2 b−x 2(a − b) dx = √ E (β, p) − (a − x)3 (c − x) a−c a−c

14.

16.

[u > a > b > c] √  2 a−c x−a u−a dx = [F (ν, q) − E (ν, q)] + 2 3 (x − b) (x − c) b−c (u − b)(u − c)

.

13.

BY (232.01)



[u ≥ a > b > c]

BY (238.10)

[a > b > c ≥ u]

BY (231.01)

c−u (a − u)(b − u)

[a > b > c > u]  2 b−x (b − u)(u − c) 2 dx = √ [F (γ, q) − E (γ, q)] + (a − x)3 (x − c) a−c a−u a−c

2 b−x dx = √ [F (δ, q) − E (δ, q)] (a − x)3 (x − c) a−c

BY (237.09)

BY (232.05)

[a > b ≥ u > c]

BY (233.13)

[a > b > u ≥ c]

BY (234.15)

3.142

Square roots of polynomials

 u. 17.

18.

.

u

19. −∞

 b. 20. u

21. b

22. u

 u. 23. a

25.

∞

.

[a > b > c > u] b−u (a − u)(u − c) [a > b > u > c]

BY (234.14)

x−b 2 dx = √ [F (κ, p) − E (κ, p)] (a − x)(x − c)3 a−c

[a > u ≥ b > c] x−b 2 b−c dx = √ E (μ, q) − 2 (x − a)(x − c)3 a−c a−c



u−a (u − b)(u − c) BY (237.11)

x−b 2 dx = √ E (ν, q) [u ≥ a > b > c] 3 (x − a)(x − c) a −c u √  u  2 a−c 2(b − c) c−x √ dx = E (α, p) − F (α, p) 3 (b − x) (a − x) a − b (a − b) a − c −∞ √

u

 u 27. c

 b 28. u

[a > b > c ≥ u]

2 a−c 2(b − c) c−x √ dx = E (β, p) − F (β, p) − 2 3 (a − x) (b − x) a−b (a − b) a − c

26.

BY (235.03)

BY (236.14)

[u > a > b > c]

.

 c

BY (238.07)

BY (231.04)

[a ≥ u > b > c]  x−b (a − u)(u − b) 2 2 √ dx = [F (λ, p) − E (λ, p)] + (a − x)(x − c)3 a−c u−c a−c

 a.



BY (235.08)

[u > a > b > c] . b−x b−u 2 dx = √ [F (α, p) − E (α, p)] + 2 (a − x)(c − x)3 (a − u)(c − u) a−c

b−x 2 dx = − √ E (δ, q) + 2 (a − x)(x − c)3 a−c

 u.

24.

.

u−b (a − u)(u − c) b [a > u > b > c] .  ∞. 2 x−b u−b dx = √ [F (ν, q) − E (ν, q)] + 2 3 (x − a) (x − c) (u − a)(u − c) a−c u 

2 x−b dx = − √ E (κ, p) + 2 (a − x)3 (x − c) a−c

271

BY (238.01)



BY (231.07)

c−u (a − u)(b − u)

[a > b > c > u] BY (232.03)  2 a−c 2 x−c (b − u)(u − c) 2 dx = E (γ, q) − √ F (γ, q) − 3 (a − x) (b − x) a−b a−b a−u a−c √

[a > b ≥ u > c]

BY (233.14)

x−c 2 a−c 2 dx = E (δ, q) − √ F (δ, q) (a − x)3 (b − x) a−b a−c [a > b > u ≥ c]

BY (234.20)

√

272

Power and Algebraic Functions

 u 29. b

√ x−c 2(b − c) 2 a−c √ dx = E (κ, p) F (κ, p) − (a − x)3 (x − b) a−b (a − b) . a−c +2

a−c a−b

u−b (a − u)(u − c)

[a > u > b > c] 

∞

30. u



√ 2 2(a − c) x−c 2 a−c √ dx = E (ν, q) + F (ν, q) − (x − a)3 (x − b) a−b a−b a−c



u

31. −∞

 c 32. u

 u 33. c

 a 34. u

36.

3.143 1.

6

2.

3.144

1.

.

BY (235.13)

u−b (u − a)(u − c)

[u > a > b > c] √  c−x c−u 2 a−c [F (α, p) − E (α, p)] + 2 dx = 3 (a − x)(b − x) a−b (a − u)(b − u)

BY (238.08)

[a > b > c ≥ u]

BY (231.06)

[a > b > c > u]  2 x−c (a − u)(u − c) 2 a−c E (γ, q) + dx = − 3 (a − x)(b − x) a−b a−b b−u

BY (232.04)

[a > b > u > c]  √ 2 x−c (a − u)(u − c) 2 a−c [F (λ, p) − E (λ, p)] + dx = (a − x)(x − b)3 a−b a−b u−b

BY (233.16)

√ c−x 2 a−c [F (β, p) − E (β, p)] dx = (a − x)(b − x)3 a−b √

[a > u > b > c] √ x−c 2 a−c E (μ, q) [u > a > b > c] dx = (x − a)(x − b)3 a−b a √   ∞ b−c x−c u−a 2 a−c E (ν, q) − 2 dx = 3 (x − a)(x − b) a−b a − b (u − b)(u − c) u

BY (236.13)

[u ≥ a > b > c]

BY (238.11)

 u 35.

3.143



 √  √ 2 (1 − u) √ 4 arctan ,2 2 2−1 (1 + u) u  √   ∞ dx 2 u2 − 1 1 √ , = F arccos 2 4 2 u + 1 2 1 + x u 

1

dx 1 √ = F 2 1 + x4



1+

1 Notation: α = arcsin √ . 2 u −u+1  √   ∞ dx 3  = F α, 2 2 x(x − 1) (x − x + 1) u

[u ≥ 1]

BY (237.01)

H 66 (286)

H 66 (287)

BY (261.50)

3.144

Square roots of polynomials



∞

2. u



∞

3. u



∞

4. u

 5.

6.



2(2u − 1)

dx

273

 = − 4E x3 (x − 1)3 (x2 − x + 1) u(u − 1) (u2 − u + 1)

√  3 α, 2

BY (261.54) [u > 1] &  √  √  3 3 2u − 1 α, − E α, +  2 2 2 u(u − 1) (u2 − u − 1)

% 

(2x − 1)2 dx  =4 F x3 (x − 1)3 (x2 − x + 1)

[u > 1]  √ & √  3 3 α, − E α, 2 2

% 

dx 4 < = F 3 3 x(x − 1) (x2 − x + 1)

[u ≥ 1]

√  3 α, 2



∞

(2x − 1)2 dx < [u > 1] = 4E 3 u x(x − 1) (x2 − x + 1)  √   √   ∞. x(x − 1) 3 3 4 1 α, − F α, 3 dx = 3 E 2 2 3 2 (x − x + 1) u 

∞

7. u



∞

8. u



u

∞

10. u



. dx (2x − 1)2

∞

9.



dx (2x − 1)2

∞

11. u



(2x −

1)2

BY (261.56)

BY (261.52) BY (261.51)

BY (261.53) [u > 1]  √ &  √  1 3 3 u(u − 1) α, − E α, + 2 2 2(2u − 1) u2 − u + 1

% 

1 x(x − 1) = F x2 − x + 1 3

x2 − x + 1 =E x(x − 1)



[u > 1]   √ 3 3 u(u − 1) α, − 2 2(2u − 1) u2 − u + 1

dx 4  = E 2 3 x(x − 1) (x − x + 1)

dx 40  E = 5 5 2 3 x (x − 1) (x − x + 1)

dx 44 < F = 27 5 x(x − 1) (x2 − x + 1)

BY (261.57)

[u > 1] BY (261.58)  √   √  1 2 3 3 u(u − 1) α, − F α, − 2 3 2 2u − 1 u2 − u + 1



[u > 1] BY (261.55)  √    2 √  2(2u − 1) 9u − 9u − 1 3 3 4 α, − F α, −  2 3 2 3 u3 (u − 1)3 (u2 − u + 1)





[u > 1] BY (261.54)  √   √  2(2u − 1) u(u − 1) 3 3 56 < α, − E α, + 2 27 2 3 9 (u2 − u + 1) [u > 1]

BY (261.52)

274

Power and Algebraic Functions



∞

12. u

3.145



u

1. α



u

2.

 √   √  3 3 16 1  E α, F α, = − 4 2 27 2 27 2 (2x − 1) x(x − 1) (x − x + 1)   8 5u2 − 5u + 2 u(u − 1) − 3 9(2u − 1) u2 − u + 1 [u > 1] dx

dx

1  =√ F 2 2 pq (x − α)(x − β) [(x − m) + n ] 



.

q(u − α) 1 2 arctan , p(u − β) 2 [β < α < u]

.

BY (261.55)

(p + q)2 + (α − β)2 pq



dx (α − x)(x − β) [(x − m)2 + n2 ]

.  q(α − u) 1 −(p − q)2 + (α − β)2 2 arccot , p(u − β) 2 pq [β < u < α] . .    β dx q(β − u) 1 (p + q)2 + (α − β)2 1  , = √ F 2 arctan pq p(α − u) 2 pq (x − α)(x − β) [(x − m)2 + n2 ] u [u < β < α] β

1 =√ F pq

3.

3.145



.

where (m − α)2 + n2 = p2 , and (m − β)2 + n2 = q 2 .∗ 4.

Set 2

2

2

2

(m1 − m) + (n1 + n) = p2 , (m1 − m) + (n1 − n) = p21 , . 2 (p + p1 ) − 4n2 cot α = 2; 4n2 − (p − p1 )

then  u

dx 2  ,= p+p F + 1 m−n tan α 2 [(x − m)2 + n2 ] (xm1 ) + n21



√ u − m 2 pp1 , α + arctan n p + p1 [m − n tan α < u < m + n cot α]

3.146



1

1. 0

 2. 0

1

1 dx π 1√ √ = + 2K 4 1+x 8 4 1 − x4 x2 dx π √ = 4 4 1+x 8 1−x

√  2 2

BI (13)(6)

BI (13)(7)

∗ Formulas 3.145 are not valid for α + β = 2m. In this case, we make the substitution x − m = z, which leads to one of the formulas in 3.152.

3.147

Square roots of polynomials



1

3. 0

1√ x4 dx π √ =− + 2K 4 1+x 8 4 1 − x4

√  2 2

BI (13)(8)

. 3.147

Notation: In 3.147–3.151 we set: α = arcsin . β = arcsin . δ = arcsin

275

(a − c)(d − u) , (a − d)(c − u) .

(a − c)(u − d) , (c − d)(a − u)

γ = arcsin

(b − d)(u − c) , (b − c)(u − d)

κ = arcsin

.

(b − d)(c − u) , (c − d)(b − u)

(a − c)(b − u) , (b − c)(a − u)

. (a − c)(u − b) (b − d)(a − u) , μ = arcsin , λ = arcsin (a − b)(u − c) (a − b)(u − d) . . . (b − d)(u − a) (b − c)(a − d) (a − b)(c − d) , q= , r= . ν = arcsin (a − d)(u − b) (a − c)(b − d) (a − c)(b − d) .



d

1. u



dx 2  = F (α, q) (a − x)(b − x)(c − x)(d − x) (a − c)(b − d) [a > b > c > d > u]

u

2.



d





u

dx (a − x)(b − x)(c − x)(x − d)



c

u

F (β, r)

=

2 (a − c)(b − d)

dx (a − x)(b − x)(x − c)(x − d)

=

2 (a − c)(b − d)



dx (a − x)(b − x)(x − c)(x − d)

=

2 (a − c)(b − d)

u

b



dx (a − x)(x − b)(x − c)(x − d)

=

2 (a − c)(b − d)

a

u

BY (253.00)

F (δ, q) BY (254.00)

F (κ, q) BY (255.00)

F (λ, r) [a ≥ u > b > c > d]

7.11

BY (254.00)

F (γ, r)

[a > b > u ≥ c > d]

6. 

(a − c)(b − d)

[a > b ≥ u > c > d] b

5. 

2

[a > b > c > u ≥ d] u

4.



(a − x)(b − x)(c − x)(x − d)

=

[a > b > c ≥ u > d] c

3. 

dx

BY (251.00)

BY (256.00)

dx 2  = F (μ, r) (a − x)(x − b)(x − c)(x − d) (a − c)(b − d) [a > u ≥ b > c > d]

BY (257.00)

276

Power and Algebraic Functions



u

8. a



dx (x − a)(x − b)(x − c)(x − d)

=

2 (a − c)(b − d)

F (ν, q) [u > a > b > c > d]

3.148 1.



d

8 u





a−d  , q + c F (α, q) = (d − c) Π α, a−c (a − x)(b − x)(c − x)(d − x) (a − c)(b − d) x dx

2

[a > b > c > d > u] BY (251.03)



x dx 2 d−c  , r + a F (β, r) = (d − a) Π β, a−c (a − x)(b − x)(c − x)(x − d) (a − c)(b − d)

c

[a > b > c ≥ u > d] BY (252.11)



x dx 2 c−d  , r + b F (γ, r) = (c − b) Π γ, b−d (a − x)(b − x)(c − x)(x − d) (a − c)(b − d)

d

3. u



BY (258.00)

u

2. 

3.148

u

4. c

[a > b > c > u ≥ d] BY (253.11)



x dx 2 b−c  , q + d F (δ, q) = (c − d) Π δ, b−d (a − x)(b − x)(x − c)(x − d) (a − c)(b − d)

[a > b ≥ u > c > d] BY (254.10)



b x dx 2 b−c  , q + a F (κ, q) = (b − a) Π κ, a−c (a − x)(b − x)(x − c)(x − d) (a − c)(b − d) u

 5.



u

[a > b > u ≥ c > d] BY (255.17)



x dx 2 a−b  , r + c F (λ, r) = (b − c) Π λ, a−c (a − x)(x − b)(x − c)(x − d) (a − c)(b − d)

a

[a ≥ u > b > c > d] BY (256.11)



x dx 2 b−a  , r + d F (μ, r) = (a − d) Π μ, b−d (a − x)(x − b)(x − c)(x − d) (a − c)(b − d)

6.8 b

 7.

u



u

8. a

[a > u ≥ b > c > d] BY (257.11)



x dx 2 a−d  , q + b F (ν, q) = (a − b) Π ν, b−d (x − a)(x − b)(x − c)(x − d) (a − c)(b − d) [u > a > b > c > d]

3.149



d

1. u

dx  x (a − x)(b − x)(c − x)(d − x) 2 =  cd (a − c)(b − d)

BY (258.11)



c(a − d) , q + d F (α, q) (c − d) Π α, d(a − c) [a > b > c > d > u] BY (251.04)

3.149

Square roots of polynomials



u

2. d

dx  x (a − x)(b − x)(c − x)(x − d) =



c

3. u

2  ad (a − c)(b − d)



u

4.



a(d − c) , r + d F (β, r) (a − d) Π β, d(a − c) [a > b > c ≥ u > d] BY (252.12)

dx  x (a − x)(b − x)(c − x)(x − d) =

277

2  bc (a − c)(b − d)



b(c − d) , r + c F (γ, r) (b − c) Π γ, c(b − d) [a > b > c > u ≥ d] BY (253.12)

dx  x (a − x)(b − x)(x − c)(x − d)



d(b − c) =  , q + c F (δ, q) (d − c) Π δ, c(b − d) cd (a − c)(b − d) [a > b ≥ u > c > d] BY (254.11)

c

2



b

5. u

dx  x (a − x)(b − x)(x − c)(x − d) =



u

6.



2 a(b − c)  , q + b F (κ, q) × (a − b) Π κ, b(a − c) ab (a − c)(b − d) [a > b > u ≥ c > d] BY (255.18)

dx  x (a − x)(x − b)(x − c)(x − d)



c(a − b) =  , r + b F (λ, r) × (c − b) Π λ, b(a − c) bc (a − c)(b − d) [a ≥ u > b > c > d] BY (256.12)

b

2



a

7. u

 8. a

u

dx  x (a − x)(x − b)(x − c)(x − d)



2 d(b − a)  = , r + a F (μ, r) × (d − a) Π μ, a(b − d) ad (a − c)(b − d) [a > u ≥ b > c > d] BY (257.12)

dx  x (x − a)(x − b)(x − c)(x − d) =

2  ab (a − c)(b − d)



b(a − d) , q + a F (ν, q) (b − a) Π ν, a(b − d) [u > a > b > c > d] BY (258.12)

278

Power and Algebraic Functions

3.151  1.

d

u

dx  (p − x) (a − x)(b − x)(c − x)(d − x) =



u

2. d

dx  (p − x) (a − x)(b − x)(c − x)(x − d) =



c

3. u



dx 2   = (p − x) (a − x)(b − x)(c − x)(x − d) (p− b)(p − c) (a − c)(b − d)

 (c − d)(p − b) × (c − b) Π γ, , r + (p − c) F (γ, r) (b − d)(p − c) [a > b > c > u ≥ d, p = c] BY (253.39)

b

dx  (p − x) (a − x)(b − x)(x − c)(x − d)

5. u

=

 b

2  (p− a)(p − d) (a − c)(b − d)

 (d − c)(p − a) × (d − a) Π β, , r + (p − d) F (β, r) (a − c)(p − d) [a > b > c ≥ u > d, p = d] BY (252.39)

dx 2   = (p − x) (a − x)(b − x)(x − c)(x − d) (p− c)(p − d) (a − c)(b − d)

 (b − c)(p − d) × (c − d) Π δ, , q + (p − c) F (δ, q) (b − d)(p − c) [a > b ≥ u > c > d, p = c] BY (254.39)

c

6.

2  (p− c)(p − d) (a − c)(b − d)

 (a − d)(p − c) × (d − c) Π α, , q + (p − d) F (α, q) (a − c)(p − d) [a > b > c > d > u, p = d] BY (251.39)

u

4.



3.151

u

2  (p− a)(p − b) (a − c)(b − d)

 (b − c)(p − a) × (b − a) Π κ, , q + (p − b) F (κ, q) (a − c)(p − b) [a > b > u ≥ c > d, p = b] BY (255.38)

dx  (x − p) (a − x)(x − b)(x − c)(x − d) =

2  (b− p)(p − c) (a − c)(b − d)

 (a − b)(p − c) × (b − c) Π λ, , r + (p − b) F (λ, r) (a − c)(p − b) [a ≥ u > b > c > d, p = b] BY (256.39)

3.152

Square roots of polynomials



a

7. u

dx  (p − x) (a − x)(x − b)(x − c)(x − d) =



u

8. a

2  (p− a)(p − d) (a − c)(b − d)

 (b − a)(p − d) × (a − d) Π μ, , r + (p − a) F (μ, r) (b − d)(p − a) [a > u ≥ b > c > d, p = a] BY (257.39)

dx  (p − x) (x − a)(x − b)(x − c)(x − d) =

3.152

Notation: In 3.152–3.163 we set: 

279

2  (p− a)(p − b) (a − c)(b − d)

 (a − d)(p − b) × (a − b) Π ν, , q + (p − a) F (ν, q) (b − d)(p − a) [u > a > b > c > d, p = a] BY (258.39) u α = arctan , b

β = arccot

u a

 b a2 + b 2 u ε = arccos , ξ = arcsin δ = arccos , , b u a2 + u 2   a b2 − u2 a u2 − b 2 ζ = arcsin , κ = arcsin , b a2 − u 2 u a2 − b 2  √ u 2 − a2 a2 − b 2 a , q = , μ = arcsin , ν = arcsin u2 − b 2 u a b a b r= √ , s= √ , t= . 2 2 2 2 a a +b a +b

a2 + b 2 , a2 + u 2 u η = arcsin , b  2 a − u2 λ = arcsin , a2 − b 2

γ = arcsin



u

1.

∞

2. u



u

3.

[a > b > 0]

H 62(258), BY (221.00)

dx 1  = F (β, q) 2 2 2 2 a (x + a ) (x + b )

[a > b > 0]

H 63 (259), BY (222.00)



0



b

4.



u



u

5. 

∞

6. u

(x2

+

a2 ) (x2

+

b2 )

dx (x2

+

a2 ) (b2

+

a2 ) (b2

−

1 =√ F (γ, r) 2 a + b2

[b ≥ u > 0]

H 63 (260)

−

x2 )

1 =√ F (δ, r) 2 a + b2

[b > u ≥ 0]

H 63 (261), BY (213.00)

1 =√ F (ε, s) 2 a + b2

[u > b > 0]

H 63 (262), BY (211.00)

[u > b > 0]

H 63 (263), BY (212.00)

dx (x2

+

a2 ) (x2

=

x2 )

dx (x2



b

dx

1 F (α, q) a



0



u b

−

b2 )

dx 1  =√ F (ξ, s) 2 2 2 2 2 a + b2 (x + a ) (x − b )

280

Power and Algebraic Functions



u

7.

=

1 F (η, t) a

[a > b ≥ u > 0]

H 63 (264), BY (219.00)

=

1 F (ζ, t) a

[a > b > u ≥ 0]

H 63 (265), BY (220.00)

=

1 F (κ, q) a

[a ≥ u > b > 0]

H 63 (266), BY (217.00)

dx 1  = F (λ, q) 2 2 2 2 a (a − x ) (x − b )

[a > u ≥ b > 0]

H 63 (257), BY (218.00)

1 F (μ, t) a

[u > a > b > 0]

H 63 (268), BY (216.00)

dx 1  = F (ν, t) a (x2 − a2 ) (x2 − b2 )

[u ≥ a > b > 0]

H 64(269), BY (215.00)



0



b

8.



u



u

9. a

10. u



u

11. 





∞

u

3.153



u

1.



0



u

2.



0





u

u

u

5. b

6. u

−

x2 ) (x2

−

b2 )

dx (x2

−

a2 ) (x2

−

b2 )



x2 dx (x2

+

a2 ) (x2

=

+

b2 )

x2 dx (a2 + x2 ) (b2 − x2 )

=u =



a2 + u 2 − a E (α, q) b2 + u2

a2

+

b2

[u > 0,

a > b]

a2 E (γ, r) − √ F (γ, r) − u a2 + b 2



x dx +

x2 ) (b2

−

x2 )

=





 

b

a

8. u

b 2 − u2 a2 + u 2 BY (214.05)

a a2 + b2 E (δ, r) − √ F (δ, r) 2 a + b2 BY (213.06)

 b2 1 2 =√ F (ε, s) − a2 + b2 E (ε, s) + (u + a2 ) (u2 − b2 ) u a2 + b 2 (a2 + x2 ) (x2 − b2 ) x2 dx

x2 dx (a2 − x2 ) (b2 − x2 ) x2 dx

(a2



BY (221.09)

2

−

x2 ) (b2

−

x2 )

= a {F (η, t) − E (η, t)}  = a {F (ζ, t) − E (ζ, t)} + u

[u > b > 0]

BY (211.09)

[a > b ≥ u > 0]

BY (219.05)

b2 − u2 a2 − u 2 [a > b > u ≥ 0]

u

7. 

dx (a2

(a2

0



−

x2 )

2

b



−

x2 ) (b2

[b > u ≥ 0]

4. 

−

x2 )

[b ≥ u > 0] b

3. 

−

x2 ) (b2 dx

a

12.

(a2

(a2

b



dx

3.153

x2 dx (a2 − x2 ) (x2 − b2 )

= a E (κ, q) −

x2 dx  = a E (λ, q) (a2 − x2 ) (x2 − b2 )

BY (220.06)

1 2 (a − u2 ) (u2 − b2 ) u [a ≥ u > b > 0]

BY (217.05)

[a > u ≥ b > 0]

BY (218.06)

3.154

Square roots of polynomials

 9.

u

6



a



1

10.



0

3.154



u

1. 0



x2 dx (1 +

u

b

3.

u

4.





u

5.

b

6. u

u

7. b



1 k2





a

8. u



x2 ) (b2

+

−

x2 )

4

x dx (a2

+

x2 ) (x2

−

b2 )

x4 dx (a2

−

x2 ) (b2

−

x2 )

x4 dx (a2



u > 0]

 2     1 2a − b2 a2 F (γ, r) − 2 a4 − b4 E (γ, r) = √ (a2 + x2 ) (b2 − x2 ) 3 a2 + b2   b2 − u2 u 2 2 2 − 2b − a + u 3 a2 + u 2 [a ≥ u > 0]

(a2

0



=

BY (216.06) BI (14)(9)



a2 + u 2 b2 + u2

BY (221.09)

x4 dx

x4 dx

b



+

k 2 x2 )

[u > a > b > 0]  π  1 + k2 2 −E , 1−k 2 4

   u 2 a  2  2 a + b2 E (α, q) − b2 F (α, q) + u − 2a2 − b2 = 3 3 (x2 + a2 ) (x2 + b2 )

u



x2 ) (1



u 2 − a2 u2 − b2

x4 dx

0



(x2 − a2 ) (x2 − b2 )

= a {F (μ, t) − E (μ, t)} + u

[a > b,

2.





x2 dx

281

−

x2 ) (b2

−

x2 )

 2     1 2a − b2 a2 F (δ, r) − 2 a4 − b4 E (δ, r) = √ 2 2 3 a +b u 2 + (a + u2 ) (b2 − u2 ) 3 [b > u ≥ 0]

BY (213.06)

 2     1 2b − a2 b2 F (ε, s) + 2 a4 − b4 E (ε, s) = √ 2 2 3 a +b 2b2 − 2a2 + u2  2 + (u + a2 ) (u2 − b2 ) 3u [u > b > 0]

BY (211.09)

=

=

4

x dx (a2

−

x2 ) (x2

−

b2 )

BY (214.05)

=

    u a  2 2a + b2 F (η, t) − 2 a2 + b2 E (η, t) + (a2 − u2 ) (b2 − u2 ) 3 3 [a > b ≥ u > 0]

BY (219.05)

    a  2 2a + b2 F (ζ, t) − 2 a2 + b2 E (ζ, t) 3   b2 − u2 u 2 2 2 + u + a + 2b 3 a2 − u 2 [a > b > u ≥ 0]

BY (220.06)

  a  2 2 a + b2 E (κ, q) − b2 F (κ, q) 3 u2 + 2a2 + 2b2  2 (a − u2 ) (u2 − b2 ) − 3u [a ≥ u > b > 0]

BY (217.05)

  u x dx a  2  2 a + b2 E (λ, q) − b2 F (λ, q) + = (a2 − u2 ) (u2 − b2 ) 2 2 2 2 3 3 (a − x ) (x − b ) 4

[a > u ≥ b > 0]

BY (218.06)

282

Power and Algebraic Functions



u

9.



a

3.155 1.

 a u

2.

3.

4.

5.9

6.

7.

8.

x4 dx (x2

−

a2 ) (x2

−

b2 )

=

    a  2 2a + b2 F (μ, t) − 2 a2 + b2 E (μ, t) 3   u 2 − a2 u 2 2 2 + u + 2a + b 3 u2 − b 2 [u > a > b > 0]

(a2 − x2 ) (x2 − b2 ) dx =

3.155

BY (216.06)

  u a  2 a + b2 E (λ, q) − 2b2 F (λ, q) − (a2 − u2 ) (u2 − b2 ) 3 3

[a > u ≥ b > 0]  u     a  2 a + b2 E (μ, t) − a2 − b2 F (μ, t) (x2 − a2 ) (x2 − b2 ) dx = 3  a  u 2 − a2 u 2 2 2 + u − a − 2b 3 u2 − b 2 [u > a > b > 0]  u    a 2 2b F (α, q) − a2 + b2 E (α, q) (x2 + a2 ) (x2 + b2 ) dx = 3  0  a2 + u 2 u 2 2 2 + u + a + 2b 3 b2 + u2 [a > b, u > 0]  u      1 (a2 + x2 ) (b2 − x2 ) dx = a2 + b2 a2 F (γ, r) − a2 − b2 E (γ, r) 3  0  b2 − u2 u 2 2 2 + u + 2a − b 3 a2 + u 2 [a ≥ u > 0]  b     1 2 (a2 + x2 ) (b2 − x2 ) dx = a + b2 a2 F (δ, r) + b2 − a2 E (δ, r) 3 u u 2 (a + u2 ) (b2 − u2 ) + 3 [b > u ≥ 0]  u     1 (a2 + x2 ) (x2 − b2 ) dx = a2 + b2 b2 − a2 E (ε, s) − b2 F (ε, s) 3 b u 2 + a2 − b 2  2 (a + u2 ) (u2 − b2 ) + 3u [u > b > 0]  u     a  2 a + b2 E (η, t) − a2 − b2 F (η, t) (a2 − x2 ) (b2 − x2 ) dx = 3 0 u 2 (a − u2 ) (b2 − u2 ) + 3 [a > b ≥ u > 0]  b     a  2 a + b2 E (ζ, t) − a2 − b2 F (ζ, t) (a2 − x2 ) (b2 − x2 ) dx = 3  u  b2 − u2 u + u2 − 2a2 − b2 3 a2 − u 2 [a > b > u ≥ 0]

BY (218.11)

BY (216.10)

BY (221.08)

BY (214.12)

BY (213.13)

BY (211.08)

BY (219.11)

BY (220.05)

3.156

3.156



∞

6 u



x2



x2

b

dx 1  √ = a2 b 2 a2 + b 2 (x2 + a2 ) (x2 − b2 )

u > 0]



a2 + b

 2

4.

x2

u



dx (x2

+

a2 ) (x2

−

b2 )

=



1 √ a2 + b a2 b 2  a2 + b 2 u2 − b 2 1 − 2 b u a2 + u 2

 2

u

6.

x2

b



a

7. u

x2

dx 1  = 2 E (κ, q) 2 2 2 2 ab (a − x ) (x − b ) 

dx (a2

−

x2 ) (x2

−

b2 )

=

BY (213.09)

BY (211.11)

 E (ξ, s) − b2 F (ξ, s)

[u ≥ b > 0]   b dx b2 − u2 1 1  = 2 {F (ζ, t) − E (ζ, t)} + 2 2 2 2 2 2 ab b u a2 − u 2 (a − x ) (b − x ) u x 

BY (222.04)

 E (ε, s) − b2 F (ε, s) [u > b > 0]

∞

BY (217.09)

b2 + u2 1 − 2 E (β, q) 2 2 a +u ab

 2    dx 1  √ a F (δ, r) − a2 + b2 E (δ, r) = 2 2 2 2 2 2 2 2 (x + a ) (b − x ) a b a + b 1  2 + 2 2 (a + u2 ) (b2 − u2 ) a b u [b > u > 0]

u

3. 

1  = 2 2 2 2 2 2 ub x (x + a ) (x + b )

b

u



dx

[a ≥ b,

2.

5.

283

 u   a  2 a + b2 E (κ, q) − 2b2 F (κ, q) (a2 − x2 ) (x2 − b2 ) dx = 3 b u 2 − a2 − b 2  2 (a − u2 ) (u2 − b2 ) + 3u [a ≥ u > b > 0]

9.

1.

Square roots of polynomials

BY (212.06)

[a > b > u > 0]

BY (220.09)

[a ≥ u > b > 0]

BY (217.01)

1 1  2 E (λ, q) − 2 2 (a − u2 ) (u2 − b2 ) 2 ab a b u

[a > u ≥ b > 0]  u dx u 2 − a2 1 1  = 2 {F (μ, t) − E (μ, t)} + 2 2 ab a u u2 − b 2 (x2 − a2 ) (x2 − b2 ) a x

BY (218.12)

[u > a > b > 0]

BY (216.09)

[u ≥ a > b > 0]

BY (215.07)

 8.



∞

9. u

x2



dx (x2

−

a2 ) (x2

−

b2 )

=

1 {F (ν, t) − E (ν, t)} ab2

284

3.157

Power and Algebraic Functions



u

1. 0

3.

5.

6.

8.

9.

10.

11.

∞

b

[u ≥ b > 0] BY (212.12)

u dx b2 1  Π η, , t [a > b ≥ u > 0; p = b] = BY (219.02) 2 ap p (a2 − x2 ) (b2 − x2 ) 0 (p − x )  b dx 1  = 2 ) (p − b2 ) 2 2 2 2 2 a (p − a (a − x ) (b − x ) u (p − x )       2 2  2    b p − a × b − a2 Π ζ, 2 , t + p − b2 F (ζ, t) a (p − b2 )   a > b > u ≥ 0; p = b2 BY (220.13)        u   p a2 − b 2 dx 1 2 2  b Π κ, 2 , q + p − b F (κ, q) = 2 ap (p − b2 ) a (p − b2 ) (a2 − x2 ) (x2 − b2 ) b (p − x )   a ≥ u > b > 0; p = b2 BY (217.12)

 a a2 − b 2 dx 1  Π λ, 2 ,q = 2 2 a (a − p) a −p (a2 − x2 ) (x2 − b2 ) u (x − p)   a > u ≥ b > 0; p = a2 BY (218.02)  u dx  2 2 (x − a2 ) (x2 − b2 ) a (p − x ) 

 2    p − b2 1 2 2 a Π μ, F (μ, t) − b , t + p − a = a (p − a2 ) (p − b2 )  p − a2  u > a > b > 0; p = a2 , p = b2 BY (216.12) 

7.



 b2 p + b2 Π α, , q + F (α, q) p p

[b ≥ u > 0, p = 0] BY (214.13)a

2 dx 1 b  √ ,r = Π δ, 2 2 2 2 2 2 2 2 2 b −p (p − b ) a + b (a + x ) (b − x ) u (p − x )   b > u ≥ 0, p = b2 BY (213.02) 

 u   dx 1 p 2  √ F (ε, s) = , s + p − b b2 Π ε, 2 p − b2 (a2 + x2 ) (x2 − b2 ) p (p − b2 ) a2 + b2  b (p − x )  u > b > 0, p = b2 BY (211.14)



 ∞ dx 1 a2 + p  √ = , s − F (ξ, s) Π ξ, 2 2 a + b2 (a2 + p) a2 + b2 (a2 + x2 ) (x2 − b2 ) u (x − p) 

4.

[p = 0] BY (221.13)



2 dx 1 a +p  =− , q − F (β, q) BY (222.11) Π β, 2 + p) 2 2 2 2 2 a (a a2 (x + a ) (x + b ) u (p − x )        u b 2 p + a2 dx 1 2  √ , r + p F (γ, r) = a Π γ, 2 p (a2 + b2 ) p (p + a2 ) a2 + b2 (a2 + x2 ) (b2 − x2 ) 0 (p − x ) 

2.

dx 1  = a (p + b2 ) (p − x2 ) (x2 + a2 ) (x2 + b2 )

3.157

3.158

Square roots of polynomials



∞

12.

(x2 − p)

u



dx (x2 − a2 ) (x2 − b2 )

=

285

 1   p Π ν, 2 , t − F (ν, t) ap a [u ≥ a > b > 0;

3.158  1.

u

0



0

5. b

6. u

<

dx

=

3

(a2 + x2 ) (b2 − x2 )

a2

1 √ E (γ, r) a2 + b 2

u > 0]

BY (221.06)

u ≥ 0]

BY (222.03)

[b ≥ u > 0]

dx

1 u < = √ E (δ, r) − 2 2 2 2 2 a (a + b2 ) 3 a a +b (a2 + x2 ) (b2 − x2 )

u

<

b



dx

=

3

(a2 + x2 ) (x2 − b2 )

BY (214.01)a

b2 − u2 a2 + u 2

1 1 √ {F (ε, s) − E (ε, s)} + 2 2 2 2 (a + b2 ) u a a +b



[u > b > 0] ∞

8. u

<

0

dx (a2 + x2 ) (b2 −

= 3 x2 )

10. u

u2 − b 2 u 2 + a2 BY (211.05)

BY (212.03)

1 u √ {F (γ, r) − E (γ, r)} +  b 2 a2 + b 2 b2 (a2 + u2 ) (b2 − u2 ) [b > u > 0]

∞

BY (213.08)

dx 1 < = √ {F (ξ, s) − E (ξ, s)} 2 3 a a2 + b 2 (a2 + x2 ) (x2 − b2 ) [u ≥ b > 0]

u

9.



(x2 + b2 )

u 1 {F (α, q) − E (α, q)} +  a (a2 − b2 ) a2 (u2 + a2 ) (u2 + b2 )

[b > u ≥ 0]

7.



=

BY (222.05)

dx 1 < {F (β, q) − E (β, q)} = 2 a (a − b2 ) 3 (a2 + x2 ) (x2 + b2 )

0



3 a2 )

[a > b, u

BY (221.05)

u ≥ 0]

[a > b;

u



dx (x2 +

4.



u > 0]

 2  dx 1 u < a E (β, q) − b2 F (β, q) −  = 2 2 2 2 2 ab (a − b ) 3 b (a + u2 ) (b2 + u2 ) (x2 + a2 ) (x2 + b2 )

<

∞

BY (215.12)

 2  1 a E (α, q) − b2 F (α, q) 2 2 (a − b )

[a > b, u

3.



3

ab2

[a > b; ∞

u



=

(x2 + a2 ) (x2 + b2 )

2.



dx

<

p = 0]

BY (214.10)

dx 1 u < − √ =  E (ξ, s) 2 2 + b2 2 2 2 2 2 3 b a b (a + u ) (u − b ) 2 2 2 2 (a + x ) (x − b ) [u ≥ b > 0]

BY (212.04)

286

Power and Algebraic Functions



u

11.

<

0



(a2

b

12.

<

u



u

13.

(a2 −

<

∞

14. u



u

15.

a

16. u



<

<

∞

18. u



u

1.

= (b2 − x2 )

dx 3

(a2 − x2 ) (x2 − b2 )

1 E (ζ, t) a (a2 − b2 )

1 = a (a2 − b2 )

dx

= 3

(a2 − x2 ) (b2 − x2 )

b2 − u2 a2 − u 2



[a > b ≥ u > 0]

BY (219.07)

[a > b > u ≥ 0]

BY (220.10)



a F (κ, q) − E (κ, q) + u



u2 − b 2 a2 − u 2



[a > u > b > 0]   a u2 − b 2 E (ν, t) − u u 2 − a2

BY (217.10)



1 1 F (η, t) − 2 2 ab2 b (a − b2 )

dx (x2 − a2 ) (x2 −

= 3 b2 )



[u > a > b > 0]   a E (η, t) − u [a > b > u > 0]

b2 F (λ, q) − a2 E (λ, q) + au

a2 − u 2 b2 − u2 

BY (215.04)



a2 − u 2 u2 − b 2



x2 dx (x2 + a2 ) (x2 +

= 3 b2 )

∞

u

BY (218.04)

[u > a > b > 0]   1 b 2 u 2 − a2 − 2 F (ν, t) a E (ν, t) − 2 2 u u −b ab

BY (216.11)

[u ≥ a > b > 0]

BY (215.06)



a {F (α, q) − E (α, q)} a2 − b 2 [a > b,

2.

BY (219.06)

1 a E (μ, t) − 2 F (μ, t) b2 (a2 − b2 ) ab

dx 1 < = 2 2 b (a − b2 ) 3 (x2 − a2 ) (x2 − b2 )

<

0



3 x2 )

a E (η, t) − u

dx 1 < = 2 2 ab (a − b2 ) 3 (a2 − x2 ) (x2 − b2 )

a

3.159

−

x2 )



[a > u > b > 0] u

17.



(b2



1 = 2 2 a (a − b2 )

dx 1 < = 2 a (b − a2 ) 3 (x2 − a2 ) (x2 − b2 )

0



−

3 x2 )

dx

b



dx

3.159

u > 0]

BY (221.12)

x2 dx a u < = 2 {F (β, q) − E (β, q)} +  2 2 + u2 ) (b2 + u2 ) a − b 3 (a (x2 + a2 ) (x2 + b2 ) [a > b,

u ≥ 0]

BY (222.10)

3.159

Square roots of polynomials



u

3. 0



u

7. b

<

∞

u

9.

x2 dx

1 u =√ {F (δ, r) − E (δ, r)} + 2 2 + b2 a + b2 3 a (a2 + x2 ) (b2 − x2 )

0

u

u

11.

b

u

u

BY (213.07)

x2 dx (a2 + x2 ) (b2 −

3 x2 )

[u > b > 0]

BY (211.13)

[u ≥ b > 0]

BY (212.01)

1 u −√ = E (γ, r) 2 2 2 2 2 a + b2 (a + u ) (b − u ) BY (214.07)

x dx 1 u < =√ {F (ξ, s) − E (ξ, s)} +  2 2 2 2 3 a +b (a + u ) (u2 − b2 ) (a2 + x2 ) (x2 − b2 )

<

x2 dx

=

3

(a2 − x2 ) (b2 − x2 )

a2

1 − b2





a E (η, t) − u

[u > b > 0]  b2 − u2 1 − F (η, t) 2 2 a −u a [a > b ≥ u > 0]

12.

b

b2 − u2 a2 + u 2

2

0

13.

BY (214.04)

[b > u > 0] ∞

10.





2

x2 dx 1 < =√ E (ξ, s) 2 + b2 3 a (a2 + x2 ) (x2 − b2 )

<

BY (222.07)

1 =√ {F (γ, r) − E (γ, r)} 2 a + b2 (b2 − x2 )

[b > u ≥ 0]  x dx u2 − b 2 1 a < =√ E (ε, s) − 2 2 2 2 u (a + b ) u2 + a2 3 a +b (a2 + x2 ) (x2 − b2 )

u



3 x2 )

2

8.



u ≥ 0]

[b ≥ u > 0] b

BY (221.11)

x dx

<

u



2

(a2 +

6.



u > 0]

 2  x2 dx 1 < a E (β, q) − b2 F (β, q) = 2 2 a (a − b ) 3 (x2 + a2 ) (x2 + b2 )

0



 2  1 u a E (α, q) − b2 F (α, q) −  2 2 2 −b ) (a + u ) (b2 + u2 )

[a > b, u

5.



3

a (a2

[a > b, ∞

u



=

(x2 + a2 ) (x2 + b2 )

4.



x2 dx

<

287

<

x2 dx (a2 −

<

3 x2 )

(b2 − x2 )

x2 dx (a2 −

3 x2 )

=

(x2 − b2 )

=

BY (212.10)

BY (219.04)

a 1 E (ζ, t) − F (ζ, t) a2 − b 2 a

1 a (a2 − b2 )

[a > b > u ≥ 0]   3 2 − b2 u a b2 F (κ, q) − a2 E (κ, q) + u a2 − u 2



[a > u > b > 0]

BY (220.08)

BY (217.06)

288

Power and Algebraic Functions



∞

14. u



u

15.

x2 dx

< 3 (x2 − a2 ) (x2 − b2 )

<

0



a

16. u



u

17.

= 3

(a2 − x2 ) (b2 − x2 )

<

∞

18. u

2

x dx

= 3

(x2 − a2 ) (x2 − b2 ) x2 dx

< 3 (x2 − a2 ) (x2 − b2 )

a2

   1 a u2 − b 2 − E (ν, t) + F (ν, t) u u 2 − a2 a  

1 2 a − b2

x2 dx 1 < = 2 a − b2 3 (a2 − x2 ) (x2 − b2 )

a



x2 dx

a = 2 a − b2

3.161

u

[u > a > b > 0] 

a2 − u 2 − a E (η, t) b2 − u2

[a > b > u > 0]   a2 − u 2 a F (λ, q) − a E (λ, q) + u u2 − b 2



a E (μ, t) − b2

1 = 2 a − b2



b2 a E (ν, t) − u





∞

1. u

BY (218.07)

[u > a > b > 0]

BY (216.01)

u 2 − a2 u2 − b 2



2.

3.

4.

BY (215.11)

    a2 b2 − u2 2a2 + b2 1   2 2 2  = 3 4 2 a + b E (β, q) − b F (β, q) + 3a b 3a2 b4 u3 x4 (x2 + a2 ) (x2 + b2 ) dx

[a > b, 

BY (219.12)

[a > u > b > 0]

[u ≥ a > b > 0] 3.161

BY (215.09)

u > 0]

BY (222.04)

 2 2     dx 1  √ a 2a − b2 F (δ, r) − 2 a4 − b4 E (δ, r) = 4 4 2 2 2 2 2 2 (x + a ) (b − x ) 3a b a + b u  a2 b2 + 2u2 a2 − b2  2 + (b − u2 ) (a2 + u2 ) 3a4 b4 u3 [b > u > 0] BY (213.09) √    u a2 + b 2 dx 2b2 − a2 2 a2 − b 2  √ = F (ε, s) + E (ε, s) 4 3 a4 b 4 (x2 + a2 ) (x2 − b2 ) 3a4 b2 a2 + b2 b x  1 + 2 2 3 (u2 + a2 ) (u2 − b2 ) 3a b u [u > b > 0] BY (211.11)  ∞   4     dx 1  √ 2 a − b4 E (ξ, s) + b2 2b2 − a2 F (ξ, s) = 4 4 2 2 4 2 2 2 2 (x + a ) (x − b ) 3a b a +b u x  a2 b2 + u2 2a2 − b2 u2 − b 2 − 2 4 3 3a b u u 2 + a2 [u ≥ b > 0] BY (212.06) b

x4

3.162

Square roots of polynomials



b

5. u

 6.

7.

⎧ ⎨ 

    dx 1  2a2 + b2 F (ζ, t) − 2 a2 + b2 E (ζ, t) = 3 4 x4 (a2 − x2 ) (b2 − x2 ) 3a b ⎩ ⎫  2    2 2 2 2 2 2 2a + b u + a b a b − u ⎬ + u3 a2 − u 2 ⎭ [a > b > u > 0]

  dx 1    = 3 4 2 a2 + b2 E (κ, q) − b2 F (κ, q) 2 2 2 2 3a b (a − x ) (x − b ) b  1 + 2 2 3 (a2 − u2 ) (u2 − b2 ) 3a b u [a ≥ u > b > 0] ⎧  a  dx 1 ⎨   = 3 4 2 a2 + b2 E (λ, q) − b2 F (λ, q) 4 (a2 − x2 ) (x2 − b2 ) 3a b ⎩ u x ⎫   ⎬ 2 a2 + b 2 u 2 + a 2 b 2  2 2 ) (u2 − b2 ) − (a − u ⎭ au3 [a > u ≥ b > 0] u

8.

x4

a

⎧     1 ⎨  2 2a + b2 F (μ, t) − 2 a2 + b2 E (μ, t) = 3 4 3a b ⎩ ⎫  2    a + 2b2 u2 + a2 b2 b2 u2 − a2 ⎬ + au3 u2 − b 2 ⎭

⎧  ∞    dx 1 ⎨ 2  2a + b2 F (ν, t) − 2 a2 + b2 E (ν, t) = 3 4 4 2 2 2 2 ⎩ (x − a ) (x − b ) 3a b u x ⎫ ⎬ ab2  2 + 3 (u − a2 ) (u2 − b2 ) ⎭ u



1. 0

u

<

BY (217.14)

BY (218.12)

dx  2 (x − a2 ) (x2 − b2 )

[u ≥ a > b > 0] 3.162

BY (220.09)

u

x4



9.

289

dx 5

(x2 + a2 ) (x2 + b2 )

=

[u > a > b > 0]

BY (216.09)

BY (215.07)

 2     3a − b2 F (α, q) − 2 2a2 − b2 E (α, q)  −    u a2 4a2 − 3b2 + u2 3a2 − 2b2 < + 3 3a4 (a2 − b2 ) (u2 + a2 ) (u2 + b2 ) [a > b, u > 0] BY (221.06) 1

3a3

(a2

2 b2 )

290

Power and Algebraic Functions

 2.

3.

4.

5.

6.

7.

8.

∞

3.162

 2     dx 1 < 3a − b2 F (β, q) − 2 2a2 − b2 E (β, q) = 2 3 2 2 5 u (x2 + a2 ) (x2 + b2 ) 3a (a − b ) . u2 + b 2 u + 2 2 2 3a (a − b ) (a2 + u2 )3 [a > b, u ≥ 0] BY (222.03)  2   u 2 2 2 a 2a − 4b dx 3b − a < = 2 F (α, q) + 2 E (α, q) 2 2 2 5 3ab (a − b ) . 3b4 (a2 − b2 ) 0 (x2 + a2 ) (x2 + b2 ) u 2 + a2 u + 2 2 2 3b (a − b ) (u2 + b2 )3 [a > b, u > 0] BY (221.05)  ∞  2 2     dx 1 < a − 2b2 E (β, q) + b2 3b2 − a2 F (β, q) = 2 2a 4 2 2 5 3ab  (a − b ) u (x2 + a2 ) (x2 + b2 )    u b2 3a2 − 4b2 + u2 2a2 − 3b2 < − 3 3b4 (a2 − b2 ) (u2 + a2 ) (u2 + b2 ) [a > b, u ≥ 0] BY (222.05)  u   2   dx 1 < < 2 b + 2a2 E (γ, r) − a2 F (γ, r) = 5 3 0 (a2 + x2 ) (b2 − x2 ) 3a4 (a2 + b2 ) . b2 − u2 u + 2 2 2 3a (a + b ) (a2 + u2 )3 [b ≥ u > 0] BY (214.15)  b  2   dx 1 < < 4a + 2b2 E (δ, r) − a2 F (δ, r) = 5 3 u (a2 + x2 ) (b2 − x2 ) 3a4 (a2 + b2 )    .  2 2 u a 5a + 3b2 + u2 4a2 + 2b2 b2 − u2 − 2 3 4 2 2 3a (a + b ) (a2 + u2 ) [b > u > 0] BY (213.08)  u      dx 1 < < 3a2 + 2b2 F (ε, s) − 4a2 + 2b2 E (ε, s) = 5 3 b (a2 + x3 ) (x2 − b2 ) 3a4 (a2 + b2 )   .  2  3a + b2 u2 + 2 2a2 + b2 a2 u2 − b 2 + 2 3 3a2 (a2 + b2 ) u (u2 + a2 ) [u > b > 0] BY (211.05)  ∞  2     dx 1 < < 3a + 2b2 F (ξ, s) − 4a2 + 2b2 E (ξ, s) = 5 3 u (a2 + x2 ) (x2 − b2 ) 3a4 (a2 + b2 ) . u2 − b 2 u + 2 2 3a (a + b2 ) (a2 + u2 )3 [u > b > 0] BY (212.03)

3.162

Square roots of polynomials

 9.

u

<

0

 10.

11.

12.

13.

14.

15.

dx (a2 + x2 ) (b2 −

= 5 x2 )

<



1 3 b2 )

291

    2a2 + 3b2 F (γ, r) − 2a2 + 4b2 E (γ, r)

3b4 (a2 +      u 3a3 + 4b2 b2 − 2a2 + 3b2 u2 < + 3 3b4 (a2 + b2 ) (a2 + u2 ) (b2 − u2 ) [b > u > 0]

BY (214.10)

∞

 2   dx 1 < < 2a + 4b2 E (ξ, s) − b2 F (ξ, s) = 5 3 u (a2 + x2 ) (x2 − b2 ) 3b4 (a2 + b2 )     2  u 3a + 4b2 b2 − 2a2 + 3b2 u2 < + 3 3b4 (a2 + b2 ) (a2 + u2 ) (u2 − b2 ) [u > b > 0]    u 2a 2b2 − a2 dx 2a2 − 3b2 < F (η, t) + = 2 E (η, t) 3ab4 (a2 − b2 ) 5 3b4 (a2 − b2 )  0 (a2 − x2 ) (b2 − x2 )     2 u 3a − 5b2 b2 − 2 a2 − 2b2 u2 a2 − u 2 + 2 b2 − u2 3b4 (a2 − b2 ) (b2 − u2 ) [a > b > a > 0]    a 2a a2 − 2b2 dx 3b2 − a2 < = 2 F (λ, q) + 2 E (λ, q) 5 3ab2 (a2 − b2 )  3b4 (a2− b2 ) u (a2 − x2 ) (x2 − b2 )  u 2 2b2 − a2 u2 + 3a2 − 5b2 b2 a2 − u 2 + 2 u2 − b 2 3b4 (a2 − b2 ) (u2 − b2 ) [a > u > b > 0]    u 2a 2b2 − a2 dx 2a2 − 3b2 < F (μ, t) + = 2 E (μ, t) 3ab4 (a2 − b2 ) 5 3b4 (a2 − b2 ) a (x2 − a2 ) (x2 − b2 )  u 2 − a2 u + 2 2 2 2 2 3b (a − b ) (u − b ) u2 − b2 [u > a > b > 0]  2   ∞ 4b − 2a2 a dx 2a2 − 3b2 < F (ν, t) = E (ν, t) + 2 3ab4 (a2 − b2 ) 5 3b4 (a2 − b2) u (x2 − a2 ) (x2 − b2 )    3b2 − a2 u2 − 4b2 − 2a2 b2 u2 − a2 − 2 u2 − b 2 3b2 u (a2 − b2 ) (u2 − b2 ) [u ≥ a > b > 0]  u     dx 1 < 4a2 − 2b2 E (η, t) − a2 − b2 F (η, t) = 2 3 2 − b2 ) 5 0 (a2 − x2 ) (b2 − x2 ) 3a (a   2     u 5a − 3b2 a2 − 4a2 − 2b2 u2 b2 − u2 − a (a2 − u2 ) a2 − u 2 [a > b ≥ u > 0]

BY (212.04)

BY (219.06)

BY (218.04)

BY (216.11)

BY (215.06)

BY (219.07)

292

Power and Algebraic Functions



b

16.

<

u



u

17.

dx

<

∞

18. u

3.163



u

1.

5

(a2 − x2 ) (b2 − x2 )

b



=

dx

=

5

(a2 − x2 ) (x2 − b2 )

  2 2a2 − b2

1 F (ζ, t) (a2 − b2 ) −  b2 − u2 u + 2 2 2 2 2 3a (a − b ) (a − u ) a2 − u2 [a > b > u ≥ 0] 3a3

dx

< (x2 +

2. u



u

3.

 2     3a − b2 F (κ, q) − 4a2 − 2b2 E (κ, q)    −   2 2a2 − b2 a2 + b2 − 3a2 u2 u2 − b2 , + 2 a2 − u 2 3a2 u (a2 − b2 ) (a2 − u2 ) [a > u > b > 0] BY (217.10)

3 a2 )

= (x2 +

3 b2 )

∞

4. u



1 ab2

dx

(a2

a2

−

(a2

2 b2 )

−

b2 )



  a2 + b2 E (α, q) − 2b2 F (α, q) u

(a2 + u2 ) (b2 + u2 ) [a > b, u > 0]

3

3

(x2 + a2 ) (b3 − x2 )

1

<

= a2 b 2 +



BY (220.10)

2 b2 )

(a2

 2   dx 1 < a + b2 E (β, q) − 2b2 F (β, q) = 2 2 2 2 3 3 ab (a − b ) (x2 + a2 ) (x2 + b2 ) u  − 2 2 2 b (a − b ) (a2 + u2 ) (b2 + u2 ) [a > b, u ≥ 0]

<

0

3a3

1

3a3

− ∞

E (ζ, r) −

 2     dx 1 < 4a − 2b2 E (ν, t) − a2 − b2 F (ν, t) = 2 3 2 2 5 (x2 − a2 ) (x2 − b2 ) 3a (a − b )    4a2 − 2b2 a2 + b2 − 3a2 u2 u2 − b2 + 2 u 2 − a2 3a2 u (a2 − b2 ) (u2 − a2 ) [u > a > b > 0] BY (215.04)

0



2 b2 )

(a2

3.163

b2

(a2 + b2 )

(a2

+

b2 )

3



BY (221.07)

BY (222.12)

 2    a F (γ, r) − a2 − b2 E (γ, r) u

(a2 + u2 ) (b2 − u2 ) [b > u > 0]

dx b 2 − a2 1 < < = E (ξ, s) − < F (ξ, s) 3 3 3 3 (x2 + a2 ) (x2 − b2 ) a2 b2 (a2 + b2 ) a2 (a2 + b2 ) u  + 2 2 2 b (a + b ) (u2 + a2 ) (u2 − b2 ) [u > b > 0]

BY (214.15)

BY (212.05)

3.165

Square roots of polynomials



u

5.

<

0



∞

6. u

dx

=

3

3

(a2 − x2 ) (b2 − x2 )

293

a2 + b 2 1 F (η, t) − 2 E (η, t) ab2 (a2 − b2 ) ab2(a2− b2 )  4  a + b 4 − a2 + b 2 u 2 u  + 2 a2 b2 (a2 − b2 ) (a2 − u2 ) (b2 − u2 ) [a > b > u > 0] 2

BY (279.08)

2

a +b dx 1 < F (ν, t) − = 2 2 2 E (ν, t) 2 2 ab (a − b ) 3 3 ab (a2 − b2 ) (x2 − a2 ) (x2 − b2 ) 1  + u (a2 − b2 ) (u2 − a2 ) (u2 − b2 ) [u > a > b > 0]

BY (215.10)

. 2 1 u2 − ρρ (ρ − ρ) , r= . 3.164 Notation: α = arccos 2 − u + ρρ 2 ρρ  ∞ dx 1  = √ F (α, r) BY (225.00) 1. ρρ (x2 + ρ2 ) (x2 + ρ2 ) u   ∞ 2u (u2 + ρ2 ) (u2 + ρ2 ) 1 x2 dx  = − E (α, r) 2. 2 2 2√ 2 2 2 2 2 4 2 2 (x + ρ ) (x + ρ ) (ρ + ρ) (u − ρ ρ ) (ρ + ρ) ρρ u (x − ρρ) 

BY (225.03) ∞

3. u



∞

4. u

(x2

+ ρρ)

2



2

x dx (x2

+

ρ2 ) (x2

x2 dx

+

ρ2 )

=−

√ 4 ρρ

1 (ρ − ρ)

2

√ [F (α, r) − E (α, r)] ρρ

BY (225.07)

1 < F (α, r) =− 2 E (α, r) + 2√ 2 2 3 3 (ρ − ρ ) (x2 + ρ2 ) (x2 + ρ2 )  2 (ρ − ρ) ρρ 2u u − ρρ  − 2 2 (ρ + ρ) (u + ρρ) (u2 + ρ2 ) (u2 + ρ2 ) BY (225.05)



∞

5. u

 6.

7.

8.



2 x2 − ρρ dx

√ 4 ρρ

< =− 2 [F (α, r) − E (α, r)] 3 3 (ρ − ρ) (x2 + ρ2 ) (x2 + ρ2 )   2u u2 − ρρ  + (u2 + ρρ) (u2 + ρ2 ) (u2 + ρ2 )

 ∞ (x2 + ρ2 ) (x2 + ρ2 )

1 dx = √ E (α, r) ρρ + ρρ) u   √  ∞ 2 2 x2 −  dx 4  ( + )  = − F (α, r) E (α, r) + √ 2 2 2 2 (x2 + 2 ) (x2 + 2 ) ( − ) ( − )  u (x + )  2 2  ∞   x +  dx 1 + , = √ Π α, p2 , r 2  u (x2 + ) − 4p2 x2 (x2 + 2 ) (x2 + 2 ) (x2

2

BY (225.06) BY(225.01)

BY (225.08)

BY (225.02)

294

Power and Algebraic Functions

3.165

√ a2 − b 2 u 2 − a2 √ . 3.165 Notation: α = arccos 2 , r= 2 u +a a 2 √  a dx 2 √ √ 1. = √ 4 2 2 4 x + 2b x + a a 2⎡+ a2 + b2 u ⎤  <   √ √ 2 − b2 ) 2 2 2 a 2 (a a 2+ a −b a−u ⎦ √ √ , √ × F ⎣arctan a+u a2 + b 2 a 2 + a2 − b 2 

[a > b, ∞

2. u

dx 1 √ F (α, r) = 2a x4 + 2b2 x2 + a4



a > u ≥ 0]

a2 > b2 > −∞,

BY (264.00)

a2 > 0,

u≥0



BY (263.00, 266.00)

√ ∞ dx u4 + 2b2 u2 + a4 1 √ = 3 [F (α, r) − 2 E (α, r)] + 2 x4 + 2b2 x2 + a4 2a a2 u (u2 + a2 ) u x [a > b > 0, u > 0]  ∞ x2 dx 1 [F (α, r) − E (α, r)] = √ 2 2 2 2 4a (a − b2 ) x4 + 2b2 x2 + a4 u (x + a )  2 a > b2 > −∞, a2 > 0, 

3.

4.

BY (263.06)

u≥0



BY (263.03, 266.05)

√ ∞ 1 x2 dx u u4 + 2b2 u2 + a4 − E (α, r) = 2√ 4 2 + b2 ) (u4 − a4 ) 2 + b2 ) 2 2 2 2 4 2 (a 4a (a x + 2b x + a u (x − a )  2  a > b2 > −∞, u2 > a2 > 0

 5.



BY (263.05, 266.02) ∞

6. u

 7.

8.

9.

∞

2

x dx 1 a < E (α, r) − F (α, r) = 4 4 2 2 (a − b ) − b2 ) 3  2 4a (a  (x4 + 2b2 x2 + a4 ) u u − a2 √ − 2 (a2 + b2 ) (u2 + a2 ) u4 + 2b2 u2 + a4  a2 > b2 > −∞, a2 > 0, u ≥ 0

BY (263.08, 266.03)

 2 2 x − a2 dx

u a u 2 − a2 < √ = 2 [F (α, r) − E (α, r)] + 2 2 2 4 + 2b2 u2 + a4 a − b u + a 3 u u (x4 + 2b2 x2 + a4 ) ! 2 !  !b ! < a2 , u ≥ 0 BY (266.08)  2   ∞ 2 x + a2 dx a a2 − b 2 u 2 − a2 u < = 2 E (α, r) − · 2 ·√ 2 2 2 2 4 a +b a +b u +a 3 u + 2b2 u2 + a4 u (x2 + 2b2 x2 + a4 ) ! 2 !  !b ! < a2 , u ≥ 0 BY (266.06)a  2   ∞ 2 x − a2 dx a a2 + b 2 F (α, r) = 2 E (α, r) − √ 2 2 2 2 a −b 2a (a2 − b2 ) x4 + 2b2 x2 + a4 u (x + a )  2  a > b2 > −∞, a2 > 0, u ≥ 0 BY (263.04, 266.07)

3.166

Square roots of polynomials



∞

10.

√ x4 + 2b2 x2 + a4 (x2

u

 11.

+

2 a2 )

√ ∞ x4 + 2b2 x2 + a4 (x2 −

u

2 a2 )

dx =



1 E (α, r) 2a

295

a2 > b2 > −∞,

a2 > 0,

u≥0



BY (263.01, 266.01)

dx =

u  4 1 [F (α, r) − E (α, r)] + 4 u + 2b2 u2 + a4 2a u − a4

[a > b > 0,   2 ∞  x2 + a2 dx 1  + ,√ Π α, p2 , r = 2 2a u (x2 + a2 ) − 4a2 p2 x2 x4 + 2b2 x2 + a4

u > a]

BY (263)

[a > b > 0,

u ≥ 0]

BY (263.02)

 12.

3.166

Notation:

α = arccos

u2 − 1 , u2 + 1

β = arctan

  √ 1−u 1+ 2 , 1+u

1 − u2 1 , γ = arccos u, δ = arccos , ε = arccos u 1 + u2 √ < √ √ √ 2 4 , q =2 3 2−4=2 2 2 − 1 ≈ 0.985171 r= 2 

∞

1. u



∞

2. u

dx 1 √ = F (α, r) 4 2 x +1

[u ≥ 0]

H (287), BY (263.50)

√ dx u4 + 1 1 √ = [F (α, r) − 2 E (α, r)] + 2 u (u2 + 1) x2 x4 + 1

[u > 0]   ∞ u u2 − 1 x2 dx 1 1 √ √ = E (α, r) − F (α, r) − 4 4 2 4 2 (u2 + 1) u4 + 1 u (x + 1) x + 1

BY (263.57)

 3.

 4. 5. 6.

7. 8.

∞

2

1 x dx = [F (α, r) − E (α, r)] 2√ 4 4 + 1) x + 1 u √  ∞ u u4 + 1 1 x2 dx = − E (α, r) 2√ 4 2 2 (u4 − 1) 4 x +1 u (x − 1) √  ∞√ 4 u u4 + 1 x +1 1 2 dx = 2 [F (α, r) − E (α, r)] + u4 − 1 2 u (x − 1) (x2

2  ∞  2 x − 1 dx 1 = E (α, r) − F (α, r) 2√ 4 2 2 x +1 u (x + 1)  ∞√ 4 x + 1 dx 1 = E (α, r) 2 2 (x2 + 1) u

[u ≥ 0]

BY (263.59)

[u ≥ 0]

BY (263.53)

[u > 1]

BY (263.55)

[u > 1]

BY (263.58)

[u ≥ 0]

BY (263.54)

[u ≥ 0]

BY (263.51)

296

Power and Algebraic Functions



∞

9. u

 10. 11. 12. 13.

1

14. u



16. 17. 18.8

19. 20. 21.3 22.

H 66(288) BY (264.50)

[0 ≤ u < 1]

BY (264.51)

[0 ≤ u < 1] √ √ (1 + x)2 dx 3 2−4 3 2+4 √  √ E (β, q) − F (β, q) = 2 4 2 2 x −x 2+1 x +1

BY (264.55)

[0 ≤ u < 1]

BY (264.56)

1

[u < 1]

H 66 (290), BY (259.75)

[u > 1]

H 66 (289), BY (260.75)

[u < 1] [u = 0] BY (259.76)

√ x dx 1 4 1 √ u −1 [u > 1] = √ F (δ, r) − 2 E (δ, r) + 4 u 2 x −1 1  1 4 x dx u 1 √ = √ F (γ, r) + 1 − u4 [u < 1] 3 3 2 1 − x4 u  u 4 x dx 1  1 √ [u > 1] = √ F (δ, r) + u u4 − 1 3 3 2 x4 − 1 1  √    √   u 1+ 1− 3 u dx 2+ 3 1 √  ,   F arccos = √ 4 2 3 1+ 1+ 3 u x (1 + x3 ) 0 

23. 0

BY (263.52)

[0 ≤ u < 1]

dx 1 √ = √ F (γ, r) 4 2 1 − x u

2  1 dx 1 1 √ = √ Γ 4 4 4 2π 1−x 0  u dx 1 √ = √ F (δ, r) 2 x4 − 1 1  1 2 √ x dx 1 √ = 2 E (γ, r) − √ F (γ, r) 4 1−x u

22 1 3 =√ Γ 4 2π 

[u ≥ 0]

u

dx 1 √ = F (ε, r) 4 2 x +1 0  1  √ dx √ = 2 − 2 F (β, q) x4 + 1 u √   1  2  √ x + x 2 + 1 dx √  √ = 2 + 2 E (β, q) 2 x4 + 1 u x −x 2+1  1 (1 − x)2 dx 1 √  √ = √ [F (β, q) − E (β, q)] 2 4 2 x +1 u x −x 2+1 

15.

 2 2  x + 1 dx 1  + ,√ = Π α, p2 , r 2 2 2 2 2 4 (x + 1) − 4p x x +1

3.166

u

u

2



dx x (1 −

x3 )

1 = √ F 4 3

[u > 0] √   √  1− 1+ 3 u 2− 3 √  , arccos 2 1+ 3−1 u





[1 ≥ u > 0]

BY (260.77) BY (259.76) BY (260.77)

BY (260.50)

BY (259.50)

3.167

Square roots of polynomials

. 3.167

Notation: In 3.167 and 3.168 we set: . β = arcsin . δ = arcsin

α = arcsin

297

(a − c)(d − u) , (a − d)(c − u) .

(a − c)(u − d) , (c − d)(a − u)

γ = arcsin

(b − d)(u − c) , (b − c)(u − d)

κ = arcsin

.

(b − d)(c − u) , (c − d)(b − u)

(a − c)(b − u) , (b − c)(a − u)

. (a − c)(u − b) (b − d)(a − u) , μ = arcsin , λ = arcsin (a − b)(u − c) (a − b)(u − d) . . . (b − d)(u − a) (b − c)(a − d) (a − b)(c − d) ν = arcsin , q= , r= . (a − d)(u − b) (a − c)(b − d) (a − c)(b − d) .

 d.

2(c − d) d−x dx =  (a − x)(b − x)(c − x) (a − c)(b − d)

1. u

 u. 2. d

3. u

 u. 4. c

5. u

b

 a. 7. u

BY (251.05)

[a > b > c > u ≥ d]

2(c − d) x−d b−c  dx = ,q Π δ, (a − x)(b − x)(x − c) b−d (a − c)(b − d)

BY (253.14)

[a > b ≥ u > c > d] BY (254.02)



2 x−d b−c dx =  , q + (a − d) F (κ, q) (b − a) Π κ, (a − x)(b − x)(x − c) a−c (a − c)(b − d)

 u. 6.

[a > b > c > d > u]



2(d − a) x−d d−c dx =  , r − F (β, r) Π β, (a − x)(b − x)(c − x) a−c (a − c)(b − d)

[a > b > c ≥ u > d] BY (252.14)



2 x−d c−d dx =  , r + (b − d) F (γ, r) (c − b) Π γ, (a − x)(b − x)(c − x) b−d (a − c)(b − d)

 c.

 b.



a−d , q − F (α, q) Π α, a−c

[a > b > u ≥ c > d] BY (255.20)



2 x−d a−b dx =  , r + (c − d) F (λ, r) (b − c) Π λ, (a − x)(x − b)(x − c) a−c (a − c)(b − d) [a ≥ u > b > c > d]

x−d 2(a − d) b−a dx =  ,r Π μ, (a − x)(x − b)(x − c) b−d (a − c)(b − d)

BY (256.13)

[a > u ≥ b > c > d]

BY (257.02)

298

Power and Algebraic Functions

 u. 8. a

9. u

 u 10. d

 c 11. u

 u 12. c

13. u

14. b

 a 15. u

 u 16. a

 d. 17. u

 u. d

 a−d , q + (b − d) F (ν, q) (a − b) Π ν, b−d

BY (258.14)

[a > b > c > d > u] BY (251.02)

  2 c−x d−c dx =  , r − (a − c) F (β, r) (a − d) Π β, (a − x)(b − x)(x − d) a−c (a − c)(b − d) [a > b > c ≥ u > d]

  2(b − c) c−x c−d dx =  , r − F (γ, r) Π γ, (a − x)(b − x)(x − d) b−d (a − c)(b − d)

BY (252.13)

[a > b > c > u ≥ d]

  2(c − d) x−c b−c dx =  , q − F (δ, q) Π δ, (a − x)(b − x)(x − d) b−d (a − c)(b − d)

BY (253.13)

[a > b ≥ u > c > d] BY (254.12)

  2 x−c b−c dx =  , q + (a − c) F (κ, q) (b − a) Π κ, (a − x)(b − x)(x − d) a−c (a − c)(b − d)

 u

18.

[u > a > b > c > d]

2(c − d) c−x a−d dx =  ,q Π α, (a − x)(b − x)(d − x) a−c (a − c)(b − d)

 d

 b

2 x−d dx =  (x − a)(x − b)(x − c) (a − c)(b − d)

3.167

[a > b > u ≥ c > d]

2(b − c) x−c a−b dx =  ,r Π λ, (a − x)(x − b)(x − d) a−c (a − c)(b − d)

BY (259.19)

[a ≥ u > b > c > d] BY (256.02)

  2 x−c b−a dx =  , r + (d − c) F (μ, r) (a − d) Π μ, (a − x)(x − b)(x − d) b−d (a − c)(b − d) [a > u ≥ b > c > d] BY (257.13)

  2 x−c a−d  dx = , q + (b − c) F (ν, q) (a − b) Π ν, (x − a)(x − b)(x − d) b−d (a − c)(b − d) [u > a > b > c > d] BY (258.13)

  2 b−x a−d dx =  , q + (b − c) F (α, q) (c − d) Π α, (a − x)(c − x)(d − x) a−c (a − c)(b − d) [a > b > c > d > u] BY (251.07)

  2 b−x d−c dx =  , r − (a − b) F (β, r) (a − d) Π β, (a − x)(c − x)(x − d) a−c (a − c)(b − d) [a > b > c ≥ u > d]

BY (252.15)

3.167

Square roots of polynomials

 c.

2(b − c) b−x c−d dx =  ,r Π γ, (a − x)(c − x)(x − d) b−d (a − c)(b − d)

19. u

 u. 20. c

 b. 21. u

BY (254.14)

[a > b > u ≥ c > d]

  2(b − c) x−b a−b dx =  , r − F (λ, r) Π λ, (a − x)(x − c)(x − d) a−c (a − c)(b − d)

BY (255.21)

22. b

[a ≥ u > b > c > d] BY (256.15)

  2 x−b b−a dx =  , r − (b − d) F (μ, r) (a − d) Π μ, (a − x)(x − c)(x − d) b−d (a − c)(b − d)

 a. 23.8 u

 u. 24. a

25. u

 u 26. d

27. u

 u

29. u

BY (257.15)

[a > b > c > d > u]

2(a − d) a−x d−c dx =  ,r Π β, (b − x)(c − x)(x − d) a−c (a − c)(b − d)

BY (251.06)

[a > b > c ≥ u > d] BY (252.02)

  2 a−x c−d dx =  , r + (a − b) F (γ, r) (b − c) Π γ, (b − x)(c − x)(x − d) b−d (a − c)(b − d)

 c

 b

[a > u ≥ b > c > d]

2(a − b) x−b a−d dx =  ,q Π ν, (x − a)(x − c)(x − d) b−d (a − c)(b − d)

[u > a > b > c > d] BY (258.02)

  2 a−x a−d dx =  , q + (a − c) F (α, q) (c − d) Π α, (b − x)(c − x)(d − x) a−c (a − c)(b − d)

 d

c

[a > b > c > u ≥ d] BY (253.02)

  2 b−x b−c dx =  , q + (b − d) F (δ, q) (d − c) Π δ, (a − x)(x − c)(x − d) b−d (a − c)(b − d)

[a > b ≥ u > c > d]

  b−x 2(a − b) b−c dx =  , q − F (κ, q) Π κ, (a − x)(x − c)(x − d) a−c (a − c)(b − d)

 u.

28.

299

[a > b > c > u ≥ d] BY (253.15)

  a−x 2 b−c dx =  , q + (a − d) F (δ, q) (d − c) Π δ, (b − x)(x − c)(x − d) b−d (a − c)(b − d)

[a > b ≥ u > c > d]

2(a − b) a−x b−c dx =  ,q Π κ, (b − x)(x − c)(x − d) a−c (a − c)(b − d)

BY (254.13)

[a > b > u ≥ c > d]

BY (255.02)



300

Power and Algebraic Functions

 u 30. b

 a 31. u

 u 32. a

3.168

  a−x 2 a−b dx =  , r + (a − c) F (λ, r) (c − b) Π λ, (x − b)(x − c)(x − d) a−c (a − c)(b − d) [a ≥ u > b > c > d]

  2(d − a) a−x b−a dx =  , r − F (μ, r) Π μ, (x − b)(x − c)(x − d) b−d (a − c)(b − d)

BY (256.14)

[a > u ≥ b > c > d]

  2(a − b) x−a a−d dx =  , q − F (ν, q) Π ν, (x − b)(x − c)(x − d) b−d (a − c)(b − d)

BY (257.14)

[u > a > b > c > d] 3.168

 c

1. u

 u 2. c

 b 3.

4.

%

c−x 2 dx = (a − x)(b − x)(x − d)3 d−a x−c 2 dx = 3 (a − x)(b − x)(x − d) a−d

x−c 2 dx = (a − x)(b − x)(x − d)3 a−d



a−c E (γ, r) − b−d

.

BY (258.15)

(a − u)(c − u) (b − u)(u − d)

&

[a > b > c > u > d]

BY (253.06)

a−c [F (δ, q) − E (δ, q)] b−d [a > b ≥ u > c > d]



.

BY (254.04)

2 a−c (b − u)(u − c) [F (κ, q) − E (κ, q)] + b − d b − d (a − u)(u − d) u BY (255.09) [a > b > u ≥ c > d] . % &  u c − d (a − u)(u − b) x−c a−c 2 E (λ, r) − dx = 3 (a − x)(x − b)(x − d) a − d b − d b − d (u − c)(u − d) b  a

5. u

 u 6. a

 c. 7. u

[a ≥ u > b > c > d]

BY (256.06)

[a > u ≥ b > c > d]

BY (257.01)

x−c a−c 2 [F (ν, q) − E (ν, q)] dx = (x − a)(x − b)(x − d)3 a−d . b−d 2 (u − a)(u − c) + a − d (u − b)(u − d) [u > a > b > c > d]

BY (258.10)

x−c 2 dx = 3 (a − x)(x − b)(x − d) a−d





a−c E (μ, r) b−d

b−x 2  dx = 3 (a − x)(c − x)(x − d) (a − d)(c − d) (a − c)(b − d) × [(b − c)(a − d) F (γ, r) − (a − c)(b − d) E (γ, r)] . (a − u)(c − u) 2(b − d) + (a − d)(c − d) (b − u)(u − d) [a > b > c > u > d]

BY (253.03)

3.168

Square roots of polynomials

 u. 8. c

301

b−x 2  dx = (a − x)(x − c)(x − d)3 (a − d)(c − d) (a − c)(b − d) × [(a − c)(b − d) E (δ, q) − (a − b)(c − d) F (δ, q)]

 b. 9. u

[a > b ≥ u > c > d] b−x 2  dx = 3 (a − x)(x − c)(x − d) (a − d)(c − d) (a − c)(b − d)

 u. 10. b

11.

12.

13.

14.

BY (254.15)

× [(a − c)(b − d) E (κ, q) − (a − b)(c − d) F (κ, q)] . (b − u)(u − c) 2 − c − d (a − u)(u − d) [a > b > u ≥ c > d]

BY (255.06)

x−b 2  dx = (a − x)(x − c)(x − d)3 (a − d)(c − d) (a − c)(b − d)

× [(a − c)(b − d) E (λ, r) − (a − d)(b − c) F (λ, r)] . (a − u)(u − b) 2 − a − d (u − c)(u − d) BY (256.03) [a ≥ u > b > c > d]   a. (a − c)(b − d) x−b E (μ, r) dx = 2 3 (a − x)(x − c)(x − d) (a − d)(c − d) u 2(b − c)  − F (μ, r) (c − d) (a − c)(b − d) [a > u ≥ b > c > d] BY (257.09)  u. x−b dx (x − a)(x − c)(x − d)3 a . 2(b − d) 2(a − b) (u − a)(u − c)  = + F (ν, q) (a − d)(c − d) (u − b)(u − d) (a − d) (a − c)(b − d)  (a − c)(b − d) E (ν, q) +2 (a − d)(c − d) [u > a > b > c > d] BY (258.09) .   c 2 a−x a−c (a − u)(c − u) 2 [F (γ, r) − E (γ, r)] + dx = 3 (b − x)(c − x)(x − d) c − d b − d c − d (b − u)(u − d) u [a > b > c > u > d] BY (253.04)   u a−x a−c 2 E (δ, q) dx = (b − x)(x − c)(x − d)3 c−d b−d c [a > b ≥ u > c > d]

BY (254.01)

302

Power and Algebraic Functions

 b 15. u

a−x 2 dx = 3 (b − x)(x − c)(x − d) c−d

 u 16. b

 a 17. u

18.

a−x 2 dx = 3 (x − b)(x − c)(x − d) c−d a−x 2 dx = (x − b)(x − c)(x − d)3 c−d







a−c 2(a − d) E (κ, q) − b−d (b − d)(c − d)

3.168

.

(b − u)(u − c) (a − u)(u − d)

BY (255.08) [a > b > u ≥ c > d] . a−c (a − u)(u − b) 2 [F (λ, r) − E (λ, r)] + b−d b − d (u − c)(u − d) BY (256.05) [a ≥ u > b > c > d]

a−c [F (μ, r) − E (μ, r)] b−d

[a > u ≥ b > c > d] .   u 2 x−a a−c (u − a)(u − c) −2 E (ν, q) + dx = 3 (x − b)(x − c)(x − d) c − d b − d c − d (u − b)(u − d) a  d.

19. u

 u. 20. d

[u > a > b > c > d] d−x 2 dx = (a − x)(b − x)(c − x)3 b−c

x−d −2 dx = 3 (a − x)(b − x)(c − x) b−c





BY (257.06)

BY (258.05)

b−d [F (α, q) − E (α, q)] a−c [a > b > c > d > u] . 2 b−d (b − u)(u − d) E (β, r) + a−c b − c (a − u)(c − u)

BY (251.01)

BY (252.06) [a > b > c ≥ u > d] .  2 x−d b−d (b − u)(u − d) 2 [F (κ, q) − E (κ, q)] + dx = 3 (a − x)(b − x)(x − c) b − c a − c b − c (a − u)(u − c) u BY (255.05) [a > b > u > c > d]   u. x−d b−d 2 E (λ, r) dx = 3 (a − x)(x − b)(x − c) b − c a −c b

 b. 21.

22.

 a. 23. u

x−d 2 dx = 3 (a − x)(x − b)(x − c) b−c



[a ≥ u > b > c > d] BY (256.01) . 2(c − d) b−d (a − u)(u − b) E (μ, r) − a−c (a − c)(b − c) (u − c)(u − d)

BY (257.06) [a > u ≥ b > c > d] .  2 x−d b−d (u − a)(u − d) 2 [F (ν, q) − E (ν, q)] + dx = 3 (x − a)(x − b)(x − c) b − c a − c a − c (u − b)(u − c) a BY (258.06) [u > a > b > c > d]   a. 2 b−x b−d dx = E (α, q) 3 (d − x) (a − x)(c − x) c − d a −c u

 u. 24.

25.

[a > b > c > d > u]

BY (251.01)

3.168

Square roots of polynomials

. b−d (b − u)(u − d) 2 [F (β, r) − E (β, r)] + a−c c − d (a − u)(c − u) d BY (252.03) [a > b > c > u > d] . .   b 2 2 b−x b−d (b − u)(u − d) dx = E (κ, q) + 3 (a − x)(x − c) (x − d) d−c a−c c − d (a − u)(u − c) u  u.

26.

27.

 u. 28. b

30.

32.

33.

34.

35.

[a > b > u > c > d] 2 x−b dx = 3 (a − x)(x − c) (x − d) c−d



BY (255.03)

b−d [F (λ, r) − E (λ, r)] a−c



BY (258.03) [u > a > b > c > d]  2 (a − c)(b − d) a−b 2 a−x  dx = E (α, q) − F (α, q) 3 (b − x)(c − x) (d − x) (b − c)(c − d) b − c (a − c)(b − d) u [a > b > c > d > u] BY (251.08)   u (a − c)(b − d) 2(a − d) a−x  dx = E (β, r) F (β, r) − 2 3 (x − d) (b − x)(c − x) (b − c)(c − d) (c − d) (a − c)(b − d) d . a−c (b − u)(u − d) +2 (b − c)(c − d) (a − u)(c − u) BY (252.04) [a > b > c > u > d] .  b 2(a − b) a−x (a − c)(b − d)  dx = E (κ, q) F (κ, q) − 2 (b − x)(x − c)3 (x − d) (b − c)(c − d) (b − c) (a − c)(b − c) u . 2(a − c) (b − u)(u − d) + (b − c)(c − d) (a − u)(u − c) BY (255.04) [a > b > u > c > d]   u 2 (a − c)(b − d) 2(a − d) a−x  dx = E (λ, r) − F (λ, r) 3 (x − d) (x − b)(x − c) (b − c)(c − d) (c − d) (a − c)(b − d) b [a ≥ u > b > c > d] BY (256.09)   a 2 (a − c)(b − d) 2(a − d) a−x  dx = E (μ, r) − F (μ, r) 3 (x − b)(x − c) (x − d) (b − c)(c − d) (c − d) (a − c)(b − d) u . 2 (a − u)(u − b) − b − c (u − c)(u − d) BY (257.04) [a > u ≥ b > c > d]

 d 31.



b−x 2 dx = 3 (a − x)(c − x) (x − d) c−d

[a ≥ u > b > c > d] BY (256.08) . 2 2 x−b b−d (a − u)(u − b) dx = [F (μ, r) − E (μ, r)] + 3 (a − x)(x − c) (x − d) c−d a−c a − c (u − c)(u − d) u BY (257.03) [a > u ≥ b > c > d] . .   u 2 2(b − c) x−b b−d (u − a)(u − d) dx = E (ν, q) − 3 (x − d) (x − a)(x − c) c − d a − c (a − c)(c − d) (u − b)(u − c) a  a.

29.

303

304

Power and Algebraic Functions

 2 (a − c)(b − d) x−a 2(a − b)  dx = E (ν, q) − F (ν, q) (x − b)(x − c)3 (x − d) (b − c)(c − d) (a − c)(b − d) (b − c) a . 2 (u − a)(u − d) − c − d (u − b)(u − c) BY (258.04) [u > a > b > c > d] .   d 2 (a − c)(b − d) 2(c − d) d−x  dx = E (α, q) − F (α, q) 3 (a − x)(b − x) (c − x) (a − b)(b − c) (b − c) (a − c)(b − d) u . 2 (a − u)(d − u) − a − b (b − u)(c − u) BY (251.11) [a > b > c > d > u] .   u 2 (a − c)(b − d) 2(a − d) x−d  dx = E (β, r) − F (β, r) 3 (a − x)(b − x) (c − x) (a − b)(b − c) (a − b) (a − c)(b − d) d . 2 (c − u)(u − d) + b − c (a − u)(b − u) BY (252.07) [a > b > c ≥ u > d] .   c 2 (a − c)(b − d) 2(a − d) x−d  dx = E (γ, r) − F (γ, r) 3 (a − x)(b − x) (c − x) (a − b)(b − c) (a − b) (a − c)(b − d) u BY (253.07) [a > b > c > u ≥ d] .   u 2 (a − c)(b − d) 2(c − d) x−d  dx = E (δ, q) F (δ, q) − (a − x)(b − x)3 (x − c) (a − b)(b − c) (b − c) (a − c)(b − d) c . 2(b − d) (a − u)(u − c) + (a − b)(b − c) (b − u)(u − d) BY (254.05) [a > b > u > c > d] .   a 2 (a − c)(b − d) 2(a − d) x−d  dx = E (μ, r) F (μ, r) − (a − x)(x − b)3 (x − c) (a − b)(b − c) (a − b) (a − c)(b − d) u . 2(b − d) (a − u)(u − c) + (a − b)(b − c) (u − b)(u − d) [a > u > b > c > d] BY (257.07) .   u 2 (a − c)(b − d) 2(c − d) x−d  dx = E (ν, q) − F (ν, q) 3 (x − c) (x − a)(x − b) (a − b)(b − c) (b − c) (a − c)(b − d) a BY (258.07) [u > a > b > c > d] .   d c−x a−c (a − u)(d − u) 2 2(b − c) dx = E (α, q) − 3 (d − x) (a − x)(b − x) a − b b − d (a − b)(b − d) (b − u)(c − u) u [a > b > c > d > u]  u

36.

37.

38.

39.

40.

41.

42.

43.

3.168

3.168

Square roots of polynomials

 u 44. d

 c 45. u

 u 46. c

47.

48.

49.

50.

x−c 2 dx = 3 (a − x)(b − x) (x − d) b−a

a−c [F (γ, r) − E (γ, r)] b−d



[a > b > c > u ≥ d] . a−c (a − u)(u − c) 2 E (δ, q) + b−d a − b (b − u)(u − d)

BY (254.08)

[u > a > b > c > d] BY (258.01) . 2 2 a−x a−c (a − u)(d − u) dx = [F (α, q) − E (α, q)] + 3 (c − x)(d − x) (b − x) b − c b − d b − d (b − u)(c − u) u BY (251.12) [a > b > c > d > u] .   u 2 2(a − b) a−x a−c (u − d)(c − u) dx = E (β, r) − 3 (b − x) (c − x)(x − d) b−c b−d (b − c)(b − d) (a − u)(b − u) d  d



 c



u

53.



c−x 2 dx = (a − x)(b − x)3 (x − d) a−b

BY (254.08) [a > b ≥ u > c > d] .   a 2 2 x−c a−c (a − u)(u − c) dx = [F (μ, r) − E (μ, r)] + 3 (x − d) (a − x)(x − b) a − b b − d a − b (u − b)(u − d) u BY (257.10) [a > u ≥ b > c > d]   u 2 x−c a−c dx = E (ν, q) 3 (x − a)(x − b) (x − d) a−b b−d a

51.

52.

. a−c (c − u)(u − d) 2 [F (β, r) − E (β, r)] + b−d b − d (a − u)(b − u) BY (252.10) [a > b > c ≥ u > d]



c−x 2 dx = 3 (a − x)(b − x) (x − d) a−b

305

2 a−x dx = (b − x)3 (c − x)(x − d) b−c

[a > b > c ≥ u > d]

BY (252.09)

a−c E (γ, r) b−d

[a > b > c > u ≥ d] BY (253.01) .   u 2 2 a−x a−c (a − u)(u − c) dx = [F (δ, q) − E (δ, q)] + 3 (x − c)(x − d) (b − x) b − c b − d b − c (b − u)(u − d) c BY (254.06) [a > b > u > c > d] .   a 2 2 a−x a−c (a − u)(u − c) dx = E (μ, r) + 3 (x − c)(x − d) (x − b) c − b b − d b − c (u − b)(u − d) u  u

54. a

2 x−a dx = (x − b)3 (x − c)(x − d) b−c



[a > u > b > c > d]

BY (257.08)

a−c [F (ν, q) − E (ν, q)] b−d [u > a > b > c > d]

BY (258.08)

306

Power and Algebraic Functions

 d. 55. u

d−x 2 dx = 3 (a − x) (b − x)(c − x) b−a

d

58.

59.

61.

62.

63.

64.

(b − u)(d − u) (a − u)(c − u)

(a −

2 x−d dx = − x)(c − x) a−b



x)3 (b

BY (251.09)

b−d [F (β, q) − E (β, q)] a−c



[a > b > u ≥ c > d] BY (255.01) .  x−d b−d (u − b)(u − d) 2 2 dx = [F (λ, r) − E (λ, r)] + 3 (x − b)(x − c) (a − x) a − b a − c a − b (a − u)(u − c) b BY (256.10) [a > u > b > c > d]   d 2 (a − c)(b − d) c−x 2(c − d)  dx = E (α, q) F (α, q) − 3 (a − x) (b − x)(d − x) (a − b)(a − d) (a − d) (a − c)(b u . − d) 2(a − c) (b − u)(d − u) + (a − b)(a − d) (a − u)(c − u) [a > b > c > d > u] BY (251.15)   u 2 (a − c)(b − d) 2(b − c) c−x  dx = E (β, r) − F (β, r) 3 (b − x)(x − d) (a − x) (a − b)(a − d) (a − b) (a − c)(b − d) d [a > b > c ≥ u > d] BY (252.08)   c 2 (a − c)(b − d) c−x 2(b − c)  dx = E (γ, r) − F (γ, r) 3 (b − x)(x − d) (a − x) (a − b)(a − d) (a − b) (a − c)(b − d) u . 2 (c − u)(u − d) − a − d (a − u)(b − u) BY (253.10) [a > b > c > u ≥ d]   u 2 (a − c)(b − d) 2(c − d) x−c  dx = E (δ, q) − F (δ, q) 3 (a − x) (b − x)(x − d) (a − b)(a − d) (a − d) (a − c)(b − d) c . 2 (b − u)(u − c) − a − b (a − u)(u − d) BY (254.09) [a > b ≥ u > c > d]  u.

60.

.

[a > b > c ≥ u > d] BY (252.05) . x−d b−d (c − u)(u − d) 2 2 dx = [F (γ, r) − E (γ, r)] + 3 (a − x) (b − x)(c − x) a−b a−c a − c (a − u)(b − u) u BY (253.05) [a > b > c > u ≥ d] . .   u 2 2(a − d) x−d b−d (b − u)(u − c) dx = E (δ, q) − 3 (a − x) (b − x)(x − c) a−b a−c (a − b)(a − c) (a − u)(u − d) c BY (254.03) [a > b ≥ u > c > d] .   b 2 x−d b−d dx = E (κ, q) 3 (b − x)(x − c) (a − x) a − b a −c u  c.

57.

b−d 2 E (α, q) + a−c a−b

[a > b > c > d > u]

 u. 56.



3.168

3.169

Square roots of polynomials

307

 2 (a − c)(b − d) x−c 2(c − d)  dx = E (κ, q) − F (κ, q) (a − x)3 (b − x)(x − d) (a − b)(a − d) (a − d) (a − c)(b − d) u [a > b > u ≥ c > d] BY (255.10)   u 2 (a − c)(b − d) x−c 2(b − c)  dx = E (λ, r) F (λ, r) − 3 (x − b)(x − d) (a − x) (a − b)(a − d) (a − b) (a − c)(b b . − d) 2(a − c) (u − b)(u − d) + (a − b)(a − d) (a − u)(u − c) BY (256.07) [a > u > b > c > d] .   d. 2 2 b−x b−d (b − u)(d − u) dx = [F (α, q) − E (α, q)] + 3 (c − x)(d − x) (a − x) a − d a − c a − d (a − u)(c − u) u BY (251.13) [a > b > c > d > u]   u. 2 b−x b−d dx = E (β, r) 3 (c − x)(x − d) (a − x) a − d a −c d  b

65.

66.

67.

68.

 c.

2 b−x dx = (a − x)3 (c − x)(x − d) a−d

69. u

 u. 70. c

 b. 71. u

 u. 72. b

(a −



x−b −2 dx = − c)(x − d) a−d

x)3 (x

[a > b > c > u ≥ d] BY (253.08) . 2 b−d (b − u)(u − c) [F (δ, q) − E (δ, q)] + a−c a − c (a − u)(u − d) BY (254.07) [a > b ≥ u > c > d]



2 b−x dx = (a − x)3 (x − c)(x − d) a−d

b−x 2 dx = 3 (a − x) (x − c)(x − d) a−d

[a > b > c ≥ u > d] BY (252.01) . 2(a − b) b−d (c − u)(u − d) E (γ, r) − a−c (a − c)(a − d) (a − u)(b − u)



b−d [F (κ, q) − E (κ, q)] a−c



[a > b > u ≥ c > d] . b−d (u − b)(u − d) 2 E (λ, r) + a−c a − d (a − u)(u − c) [a ≥ u > b > c > d]

3.169

Notation: In 3.169–3.172, we set: 

a β = arctan , u

BY (256.04)

 b a2 + b 2 u ε = arccos , ξ = arcsin δ = arccos , , b u a2 + u 2   a b2 − u2 a u2 − b 2 ζ = arcsin , κ = arcsin , b a2 − u 2 u a2 − b 2  √ a2 − b 2 u 2 − a2 a , q = , μ = arcsin , ν = arcsin u2 − b 2 u a b a b r= √ , s= √ t= . 2 2 2 2 a a +b a +b

a2 + b 2 , a2 + u 2 u η = arcsin , b  2 a − u2 λ = arcsin , a2 − b 2

u γ = arcsin b

u α = arctan , b

BY (255.07)

308

Power and Algebraic Functions

 u 1. 0

 u 2.6 0

 u 3. 0

 b 4. 5.

x2 + a2 dx = a {F (α, q) − E (α, q)} + u x2 + b2



x2 + b2 b2 F (α, q) − a E (α, q) + u dx = 2 2 x +a a  x2 + a2 dx = a2 + b2 E (γ, r) − u 2 2 b −x



a2 + u 2 b2 + u2



6. 0

u > 0]

BY (221.03)

[a > b,

u > 0]

BY (221.04)

a2 + u 2 b2 + u2

b 2 − u2 a2 + u 2

[b ≥ u > 0]

 b2 − x2 dx = a2 + b2 {F (γ, r) − E (γ, r)} + u 2 2 a +x



[u > b > 0]

BY (214.11)

 b2 − x2 dx = a2 + b2 {F (δ, r) − E (δ, r)} [b > u ≥ 0] a2 + x2 u  u 2  x − b2 1 2 dx = (a + u2 ) (u2 − b2 ) − a2 + b2 E (ε, s) 2 2 a +x u b  u 0

b2 − x2 a2 − b 2 F (η, t) dx = a E (η, t) − a2 − x2 a

u

b2 − x2 a2 − b 2 F (ζ, t) − u dx = a E (ζ, t) − 2 2 a −x a

9.  b 10.

 u 11. b

 a 12. u

 u 13. a

 u 14. 0



BY (211.03)

b2 − u2 a2 + u 2 [b ≥ u > 0]

 b

8.

[a > b,

 a2 + x2 dx = a2 + b2 E (δ, r) [b > u ≥ 0] BY (213.01), ZH 64 (273) 2 2 b −x u  u 2  a + x2 1 2 dx = a2 + b2 {F (ε, s) − E (ε, s)} + (u + a2 ) (u2 − b2 ) 2 2 x −b u b  u

7.

3.169

BY (214.03) BY (213.03)

[u > b > 0]

BY (211.04)

[a > b ≥ u > 0]

BY (219.03)

b2 − u2 a2 − u 2 [a > b > u ≥ 0]

BY (220.04)

x2 − b2 1 2 b2 F (κ, q) − dx = a E (κ, q) − (a − u2 ) (u2 − b2 ) a2 − x2 a u x2 − b2 b2 F (λ, q) dx = a E (λ, q) − a2 − x2 a

[a ≥ u > b > 0]

BY (217.04)

[a > u ≥ b > 0]

BY (218.03)

 x2 − b2 u 2 − a2 a2 − b 2 F (μ, t) − a E (μ, t) + μ dx = x2 − a2 a u2 − b 2

a2 − x2 dx = a E (η, t) b2 − x2

[u > a > b > 0]

BY (216.03)

[a > b ≥ u > 0]

H 64 (276), BY (219.01)

3.171

Square roots of polynomials

309

   a2 − x2 u b2 − u2 dx = a E (ζ, t) − [a > b > u ≥ 0] b2 − x2 a a2 − u 2 u  u 2 a − x2 1 2 dx = a {F (κ, q) − E (κ, q)} + (a − u2 ) (u2 − b2 ) 2 2 x −b u b  b

15.

16.

 a 17. 18. 3.171 1. 2.

a2 − x2 dx = a {F (λ, q) − E (λ, q)} x2 − b2 u   u 2 x − a2 u 2 − a2 dx = u − a E (μ, t) 2 2 x −b u2 − b 2 a

[a ≥ u > b > 0]

BY (217.03)

[a > u ≥ b > 0]

BY (218.09)

[u > a > b > 0]

BY (216.04)



√ a2 + x2 a2 + b 2 = E (ε, s) [u > b > 0] 2 2 x −b b2 b   √  ∞ dx a2 + x2 u2 − b 2 a2 + b 2 a2 = E (ξ, s) − 2 2 2 2 2 x x −b b b u a2 + u 2 u 



u

b

3. u



u

4. b



a

5. u



u

6. a



dx x2

dx x2

dx x2

dx x2

∞

7. u



∞

8. u



b

9. u



dx x2

[u ≥ b > 0]  a2 − x2 b2 − u2 a2 − b 2 a a2 = F (ζ, t) − 2 E (ζ, t) + 2 2 2 2 b −x ab b b u a2 − u 2

 

dx x2

a2 − x2 a 1 = 2 E (κ, q) − F (κ, q) x2 − b2 b a a2 − x2 a 1 = 2 E (λ, q) − f (λ, q) − x2 − b2 b a



dx x2

dx x2

BY (220.03)





BY (212.09)

[a > b > u > 0]

BY (220.12)

[a ≥ u > b > 0]

BY (217.11)

 (a2 − u2 ) (u2 − b2 ) b2 u

x2 − a2 a a2 − b 2 1 = E (μ, t) − F (μ, t) − x2 − b2 b2 ab2 u



BY (211.01), ZH 64 (274)

a x2 + a2 1 a2 F (β, q) − = E (β, q) + x2 + b2 a b2 b2 u

x2 + b2 1 1 = {F (β, q) − E (β, q)} + 2 2 x +a a u







[a > u ≥ b > 0]

BY (218.10)

u 2 − a2 u2 − b 2 [u > a > b > 0]

BY (216.08)

b2 + u2 a2 + u 2 [a > b,

u > 0]

BY (222.08)

u > 0]

BY (222.09)

b2 + u2 a2 + u 2

[a > b,  √ (b2 − u2 ) (a2 + u2 ) b2 − x2 a2 + b 2 − = E (δ, r) 2 2 2 a +x a u a2

[b > u > 0]

BY (213.10)

310

Power and Algebraic Functions

11.

12.

13. 14. 15. 16.

17. 18.



√ x2 − b2 a2 + b 2 = {F (ε, s) − E (ε, s)} [a > b > 0] 2 2 a +x a2 b   √  ∞ dx x2 − b2 a2 + b 2 1 u2 − b 2 = {F (ξ, s) − E (ξ, s)} + x2 a2 + x2 a2 u a2 + u 2 u [u ≥ b > 0]   √  b (b2 − u2 ) (a2 + u2 ) dx a2 + x2 a2 + b 2 = {F (δ, r) − E (δ, r)} + 2 b2 − x2 b2 b2 u u x [b > u > 0]   ∞ dx x2 − a2 a a2 − b 2 = E (ν, t) − F (ν, t) [u ≥ a > b > 0] x2 x2 − b2 b2 ab2 u    b dx b2 − x2 1 b 2 − u2 1 [a > b > u > 0] = − E (ζ, t) 2 2 2 2 − u2 x a − x u a a u   u dx x2 − b2 1 [a ≥ u > b > 0] = {F (κ, q) − E (κ, q)} 2 2 2 a −x a b x    a (a2 − u2 ) (u2 − b2 ) dx x2 − b2 1 {F (λ, q) − E (λ, q)} + = 2 u2 − x2 a a2 u u x [a > u ≥ b > 0]    u 1 u 2 − a2 dx x2 − b2 1 = E (μ, t) − [u > a > b > 0] 2 2 2 x −a a u u2 − b 2 a x   ∞ dx x2 − b2 1 [u ≥ a > b > 0] = E (ν, t) x2 x2 − a2 a u 

10.

3.172

u

dx x2

BY (211.07)

BY (212.11)

BY (213.05) BY (215.08)

BY (220.11)

BY (217.08)

BY (218.08) BY (216.07)

BY (215.01), ZH 65 (281)

3.172

 u.

1.

3

∞

.

2. u

 u. 3. ∞

.

4. u

 u. 5. 0

+

3 a2 )

x2 + a2 (x2

0



x2 + b2 (x2

+

3 b2 )

x2 + a2 (x2 + b2 )

3

a2 − b 2 u 1  E (α, q) − 2 2 2 a a (a + u ) (b2 + u2 )

dx =

(x2 + a2 )

0



x2 + b2

[a > b,

u > 0]

BY (221.10)

1 E (β, q) a

[a > b,

u ≥ 0]

H 64 (271)

a E (α, q) b2

[a > b,

u > 0]

H 64 (270)

u ≥ 0]

BY (222.06)

dx =

dx =

dx =

u a a2 − b 2  E (β, q) − 2 2 b2 b2 (a + u ) (b2 + u2 )

[a > b, √ b2 − x2 a2 + b 2 1 dx = E (γ, r) − √ F (γ, r) 3 2 2 2 2 a a + b2 (a + x )

[b ≥ u > 0]

BY (214.08)

3.172

Square roots of polynomials

 b. 6. u

 √ a2 + b 2 1 u b2 − u2 E (δ, r) − √ F (δ, r) − 2 3 dx = a2 a a2 + u 2 a2 + b 2 (a2 + x2 ) b2 − x2

 u. 7. b



∞

[b > u ≥ 0]  √ x2 − b2 a2 + b 2 b2 1 u2 − b 2 E (ε, s) − √ F (ε, s) − 3 dx = a2 u u 2 + a2 a2 a2 + b 2 (a2 + x2 )

.

8.  u. 9.

∞

10. u

 u. 11.

14.

dx =

BY (211.06)

a2 + b 2 b2 E (ξ, s) − √ F (ξ, s) 2 2 a a a2 + b 2

a2 √ F (γ, r) − b 2 a2 + b 2

√

[u ≥ b > 0]

BY (212.08)

 2  a + b2 u

a2 + b 2 E (γ, r) +  b2 b2 (a2 + u2 ) (b2 − u2 )

b2 − x2 −

3 x2 )

dx =

1 a

 F (η, t) − E (η, t) +

u a



[u > b > 0]  b2 − u2 a2 − u 2 [a > b ≥ u > 0]

 b.

13.

3 x2 )

dx =

BY (213.04)

BY (214.09) [b > u > 0]  2  √ a + b2 u x2 + a2 a2 + b 2 1 √  dx = F (ξ, s) − E (ξ, s) + 3 b2 a2 + b 2 b2 (a2 + u2 ) (u2 − b2 ) (x2 − b2 )

(a2

0

3 x2 )

x2 + a2

.

√

x2 − b2

(b2 −

0

12.

[u > b > 0]

(a2 +

u



311

b2 − x2

BY (212.07)

BY (219.09)

1 {F (ζ, t) − E (ζ, t)} [a > b > u ≥ 0] a − u   u. 2 x − b2 1 u2 − b 2 1 [a > u > b > 0] dx = − E (κ, q) 3 2 − u2 2 2 u a a (a − x ) b   ∞. 2 1 u2 − b 2 x − b2 1 3 dx = a [F (ν, t) − E (ν, t)] + u u 2 − a2 (x2 − a2 ) u

BY (217.07)

[u > a > b > 0]

BY (215.05)

(a2

 u. 15. 0

 a. 16. u

 u. 17. a

3 x2 )

a2 − x2 3

(b2 − x2 )

a2 − x2 (x2 −

3 b2 )

x2 − a2 3

(x2 − b2 )

dx =

dx =

a u [F (η, t) − E (η, t)] + 2 2 b b 

a2 − u 2 a − 2 E (λ, q) 2 2 u −b b

dx =

u b2

dx =

a [F (μ, t) − E (μ, t)] b2



BY (220.07)

a2 − u 2 b2 − u2 [a > b > u > 0]

BY (219.10)

[a > u > b > 0]

BY (218.05)

[u > a > b > 0]

BY (216.05)

312

Power and Algebraic Functions



∞

.

18. u

x2 − a2

a 1 3 dx = b2 [F (ν, t) − E (ν, t)] + u 2 2 (x − b )



u 2 − a2 u2 − b 2 [u ≥ a > b > 0]

3.173



1

1. u



u

2. 1

3.174

dx x2

dx x2





x2 + 1 √ = 2E x2 − 1



√  2 1 arccos , u 2

√   1− 1+ 3 u √  , β = arccos 1+ 3−1 u u

1. 0

2.

3.

4.

3.175

BY (215.03)

%   √  √ & √ x2 + 1 √ 2 2 1 − u4 = 2 F arccos u, − E arccos u, + 2 1−x 2 2 u

Notation: In 3.174 and 3.175, we take:



3.173

. dx √  2   1+ 1+ 3 x

[u < 1]

BY (259.77)

[u > 1]

BY (260.76)

√   1+ 1− 3 u √  ,  1+ 1+ 3 u   √ √ 2+ 3 2− 3 , q= . p= 2 2 α = arccos

1 1 − x + x2 = √ E (α, p) 4 x(1 + x) 3

[u > 0]

BY (260.51)

. 1 dx 1 + x + x2 = √ E (β, q) [1 ≥ u > 0] BY (259.51)  √  2 4 x(1 − x) 3 0 1+ 3−1 x  √  √    √  u 2 2+ 3 1+ 1− 3 u dx x(1 + x) 1 2− 3 √  √ √  F (α, p) − E (α, p) + − √ 4 4 2 1 − x + x2 27 3 1+ 1+ 3 u 27 0 1−x+x  u(1 + u) × 1 − u + u2 [u > 0] BY (260.54)  √  √    √  u 2 2− 3 1− 1+ 3 u dx x(1 − x) 4 2+ 3 √ √  √ E (β, q) − √ F (β, q) − 4 4 2 2 1 + x + x 1 + x + x 27 27 3 3−1 u 1+ 0  u(1 − u) × 1 + u + u2 [1 ≥ u > 0] BY (259.55) 

u



u

1. 0

 2. 0

u

dx 1+x

dx 1−x





 u (1 − u + u2 ) x 2 1 √     [F (α, p) − 2 E (α, p)] + √ √ = √ 4 3 1+x 27 3 1+u 1+ 1+ 3 u [u > 0]  u (1 + u + u2 ) x 2 1 √ √  √   [F (β, q) − 2 E (β, q)] + = √ 4 1 − x3 27 3 1−u 1+ 3−1 u [0 < u < 1]

BY (260.55)

BY (259.52)

3.183

Fourth roots of polynomials

313

3.18 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions 3.181 1.

2.

3.182

. &  % 

1 1 4 4(a − u)(u − b)  = a−b 2 E √ ,√ + E arccos 4 (a − b)2 2 2 (a − x)(x − b) b . & % 

1 4(a − u)(u − b) 1 ,√ − K √ + F arccos 4 (a − b)2 2 2 [a ≥ u > b] BY (271.05) %     u a − b − 2 (u − a)(u − b) 1 dx a−b   F arccos ,√ 4 2 2 (x − a)(x − b) a − b + 2 (u − a)(u − b) a &   a − b − 2 (u − a)(u − b) 1  − 2 E arccos ,√ 2 a − b + 2 (u − a)(u − b)  4 2(2u − a − b) (u − a)(u − b)  + a − b + 2 (u − a)(u − b) BY (272.05) [u > a > b] 

u



u

1. b

√

dx

.  & %

1 4(a − u)(u − b) 1 2  + F arccos =√ ,√ K √ 4 (a − b)2 a−b 2 2 [(a − x)(x − b)]3 dx

[a ≥ u > b]    √ u a − b − 2 (u − a)(u − b) 1 dx 2   ,√ =√ F arccos 4 a−b 2 [(x − a)(x − b)]3 a − b + 2 (u − a)(u − b) a

BY (271.01)

 2.

[u > a > b] 3.183

Notation: In 3.183–3.186 we set: 1

, α = arccos √ 4 u2 + 1

2. 3.

β = arccos

 4

1−

u2 ,

√ 1 − u2 − 1 √ γ = arccos . 1 + u2 − 1





 √ dx 1 1 2u √ = 2 F α, √ − 2 E α, √ + √ 4 4 2 2 2 x +1 u2 + 1 0 [u > 0] 



  u √ dx 1 1 √ = 2 2 E β, √ − F β, √ [0 < u ≤ 1] 4 2 2 2 1−x 0 √



 u dx 1 2u 4 u2 − 1 1 √ √ √ √ − 2 E γ, + = F γ, 4 2 2 x2 − 1 1 + u2 − 1 1 [u > 1] 

1.

BY (272.00)

u

BY (273.55) BY (271.55)

BY (272.55)

314

3.184 1.

2.

3.185

Power and Algebraic Functions

√ 



 < x2 dx 2 2 1 1 2u 4 3 √ √ √ = (1 − u2 ) 2 E β, − F β, − 4 5 5 2 2 1 − x2 0 [0 < u ≤ 1] √ √



 u 1 dx u2 − 1 1 1 1 − u2 − 1 √ √ √ √ F γ, = E γ, · − − 2 4 x2 − 1 2 u 2 2 1 + u2 − 1 1 x 



u

u

1. 0



u

2. 0

4.

5.



u

0



u

7. 0

1. 2.

dx

< = 3 4 (1 − x2 )

√



1 √ α, 2



1 2 F β, √ 2

[u > 1]

BY (272.54)

[u > 0]

BY (273.50)

[0 < u ≤ 1]

BY (271.51)

[u > 1]

BY (272.50)

[0 < u ≤ 1]

BY (271.54)

[u > 0] 



 √ x2 dx 1 1 2u < = 2 2 F α, √ − 2 E α, √ + √ 4 2+1 5 4 2 2 u (x2 + 1)

2

x dx 1 < = √ F 7 4 3 2 (x2 + 1)



1 α, √ 2



u − < 3 4 6 (u2 + 1)

 3. 1

BY (273.54)

[u > 0]

BY (273.56)

[u > 0]

BY (273.53)

√

√ 1 + x2 + 1 1 √ √ dx = 2 2 E α, [u > 0] 4 2 2 x2 + 1 0 (x + 1) √



   u √ dx 1 u 4 1 − u2 1 √ √  √ − E β, √ + = 2 F β, √ 2 2 1 + 1 − x2 4 1 − x2 1 + 1 − u2 0 

BY (271.59)

u

6.

3.186

√ dx < = 2F 3 4 (x2 + 1)



dx 1 < = F γ, √ 2 1 4 (x2 − 1)3 √

 u 1 x2 dx 2 2 2  4 < F β, √ = − u 1 − u2 3 3 2 0 4 (1 − x2 )3



 u √ √ dx 1 1 < √ √ − 2 F α, = 2 2 E α, 2 2 0 4 (x2 + 1)5 

3.

3.184

u

u

[0 < u ≤ 1] 



 dx 1 1 1 √  √ = F γ, √ − E γ, √ 2 2 2 x2 + 2 x2 − 1 4 x2 − 1 [u > 1]

BY (273.51)

BY (271.58)

BY (272.53)

3.194

Powers of x and binomials



√ √ 



 √ 1 − 1 − x2 dx 1 1 2u 4 1 − u2 √ √ · < = 2 2 E β, √ − F β, √ − 2 2 1 + 1 − x2 4 (1 − x2 )3 1 + 1 − u2

u

4. 0



315

u

2

5.



1

x dx =E √ < 3 4 x2 + 2 x2 − 1 (x2 − 1)



1 γ, √ 2



[0 < u ≤ 1]

BY (271.57)

[u > 1]

BY (272.51)

3.19–3.23 Combinations of powers of x and powers of binomials of the form (α+βx) 3.191



u

xν−1 (u − x)μ−1 dx = uμ+ν−1 B(μ, ν)

1.

[Re μ > 0,

Re ν > 0]

ET II 185(7)

0



∞

2.

x−ν (x − u)μ−1 dx = uμ−ν B(ν − μ, μ)

u





1 ν−1

3.

x

(1 − x)

μ−1

[Re ν > Re μ > 0]

1

xμ−1 (1 − x)ν−1 dx = B(μ, ν)

dx =

0

0

[Re μ > 0, 3.192



1

1. 0



1

2. 0



1

3. 0



[−1 < p < 0]

BI (4)(6)

1

(x − 1)p− 2

n

0

∞

u

3. 0

n!nν+n ν(ν + 1)(ν + 2) . . . (ν + n)

xμ−1 dx uμ = 2 F 1 (ν, μ; 1 + μ; −βu) ν (1 + βx) μ

2.6 



dx = π sec pπ x

xν−1 (n − x)n dx =



BI (3)(4)

(1 − x)p dx = −π cosec pπ xp+1

3.193

0



BI (3)(5)



u

FI II 774(1)

[−1 < p < 0]

1



p2 < 1

Re ν > 0]

xp dx = −π cosec pπ (1 − x)p+1

∞

1.



xp dx = pπ cosec pπ (1 − x)p

4.

3.194

ET II 201(6)

xμ−1 dx uμ−ν = ν ν (1 + βx) β (ν − μ)

F 2 1 ν, ν − μ;

− 12 < p <

1 2



[Re ν > 0]

EH I 2

[|arg(1 + βu)| < π,

ν − μ + 1;

−

1 βu



[Re ν > Re μ] ∞

BI (23)(7)

Re μ > 0] ET I 310(20)

ET I 310(21)

μ−1

x dx = β −μ B(μ, ν − μ) (1 + βx)ν

[|arg β| < π,

Re ν > Re μ > 0] FI II 775a, ET I 310(19)

316

Power and Algebraic Functions

 4.

∞

11 0

 0

μ−1

μ

u x dx = 1 + βx μ

μ−1 n

cosec(μπ)

[|arg β| < π,

0 < Re μ < n + 1]

2 F 1 (1, μ; 1

+ μ; −βu)

[|arg(1 + uβ)| < π,

Re μ > 0] ET I 308(5)

∞

6. 0





ET I 308(6) u

5. 

xμ−1 dx π = (−1)n μ n+1 (1 + βx) β

3.195

∞

7.

xμ−1 dx (1 − μ)π = cosec μπ (1 + βx)2 βμ

BI (16)(4)

1

xm dx 1

(a + bx)n+ 2

0

[0 < Re μ < 2]

= 2m+1 m!

(2n − 2m − 3)!! am−n+ 2 (2n − 1)!! bm+1



m < n − 12 ,

a > 0,

b>0

 BI (21)(2)



1

xn−1 dx = 2−n (1 + x)m

8. 0



∞

3.19511 0

∞  k=0



m − n − 1 (−2)−k k n+k

(1 + x)p−1 1 − a−p dx = (a + x)p+1 p(a − 1) ln a = a−1 =1

BI (3)(1)

[p = 0,

a > 0,

a = 1]

[p = 0,

a > 0,

a = 1]

[a = 1] LI (19)(6)

3.196



u

(x + β)ν (u − x)μ−1 dx =

1. 0



∞

2.



β ν uμ u 2 F 1 1, −ν; 1 + μ; − μ β !  ! ! ! !arg u ! < π ! β!

(x + β)−ν (x − u)μ−1 dx = (u + β)μ−ν B(ν − μ, μ)

u

! ! ! ! !arg u ! < π, ! β!

ET II 185(8)

 Re ν > Re μ > 0 ET II 201(7)



b

(x − a)μ−1 (b − x)ν−1 dx = (b − a)μ+ν−1 B(μ, ν)

3.

[b > a,

Re μ > 0,

Re ν > 0]

a

 4. 5.

∞



EH I 10(13)

ν

b dx π = − cosec νπ ν (a − bx)(x − 1) b b − a 1

ν  1 b dx π cosec νπ = ν b a−b −∞ (a − bx)(1 − x)

[a < b,

b > 0,

[a > b > 0,

0 < ν < 1]

0 < ν < 1]

LI (23)(5)

LI (24)(10)

3.197

3.197

Powers of x and binomials



∞

1. 0



∞

2.11 u



1

3.

317



γ xν−1 (β + x)−μ (x + γ)− dx = β −μ γ ν− B(ν, μ − ν + ) 2 F 1 μ, ν; μ + ; 1 − β [|arg β| < π, |arg γ| < π, Re ν > 0, Re μ > Re(ν − )] ET II 233(9)

β x−λ (x + β)ν (x − u)μ−1 dx = u−λ (β + u)μ+ν B(λ − μ − ν, μ) 2 F 1 λ, μ; λ − μ; − ! ! !  u ! !β ! ! ! u !arg ! < π or ! ! < 1, 0 < Re μ < Re(λ − ν) ET II 201(8) !u! ! β!

xλ−1 (1 − x)μ−1 (1 − βx)−ν dx = B(λ, μ) 2 F 1 (ν, λ; λ + μ; β)

0



1

4.

Re μ > 0,

[Re μ > 0,

Re ν > 0,

∞

[|arg α| < π, ∞

  xλ−ν (x − 1)ν−μ−1 (αx − 1)−λ dx = α−λ B(μ, ν − μ) 2 F 1 ν, μ; λ; α−1

1

|arg(α − 1)| < π]

EH I 115(6)

[Re μ > 0]  u xν−1 (x + α)λ (u − x)μ−1 dx = αλ uμ+ν−1 B(μ, ν) 2 F 1 −λ, ν; μ + ν; − α  u ! +! ! ! < π, Re μ > 0, arg ! ! α

BI 19(5)

[1 + Re ν > Re λ > Re μ, ∞

7. 0

u

8. 0



− Re(μ + ν) > Re λ > 0] EH I 60(12), ET I 310(23)

6.



a > −1]

xλ−1 (1 + x)ν (1 + αx)μ dx = B(λ, −μ − ν − λ) 2 F 1 (−μ, λ; −μ − ν; 1 − α)

0



WH

BI(5)4, EH I 10(11)

5.



|β| < 1]

xμ−1 (1 − x)ν−1 (1 + ax)−μ−ν dx = (1 + a)−μ B(μ, ν)

0



[Re λ > 0,

xμ− 2 (x + a)−μ (x + b)−μ dx = 1

  √ 1−2μ Γ μ − 1 √ √ 2 π a+ b Γ(μ)

, Re ν > 0 ET II 186(9)

∞

9.

xλ−1 (1 + x)−μ+ν (x + β)−ν dx = B(μ − λ, λ) 2 F 1 (ν, μ − λ; μ; 1 − β)

0

[Re μ > Re λ > 0]  10. 11.

1

EH I 205

π xq−1 dx = cosec qπ [0 < q < 1, p > −1] BI (5)(1) q (1 + px) (1 − x) (1 + p)q 0  √     1 1 2 Γ p + 12 Γ(1 − p) xp− 2 dx √ sin (2p − 1) arctan q 2p  √  √ cos (arctan q) = p p π (2p − 1) sin arctan q 0 (1 − x) (1 + qx)   1 BI (11)(1) − 2 < p < 1, q > 0

318

Power and Algebraic Functions



1

12. 0

3.198

   √ 1−2p  √ 1−2p 1 Γ p + 12 Γ(1 − p) 1 − q − 1+ q xp− 2 dx √ = √ (1 − x)p (1 − qx)p (2p − 1) q π  1  − 2 < p < 1, 0 < q < 1



1

3.198

xμ−1 (1 − x)ν−1 [ax + b(1 − x) + c]−(μ+ν) dx = (a + c)−μ (b + c)−ν B(μ, ν)

0

[a ≥ 0,



BI (11)(2)

b

3.199

b ≥ 0,

c > 0,

Re μ > 0,

Re ν > 0]

FI II 787

(x − a)μ−1 (b − x)ν−1 (x − c)−μ−ν dx = (b − a)μ+ν−1 (b − c)−μ (a − c)−ν B(μ, ν)

a

[Re μ > 0,

Re ν > 0,

c < a < b] EH I 10(14)



1

3.211

xλ−1 (1 − x)μ−1 (1 − ux)− (1 − vx)−σ dx = B(μ, λ) F 1 ((λ, , σ, λ + μ; u, v))

0

[Re λ > 0, Re μ > 0] EH I 231(5)

    q a + b (1 + ax)−p + (1 + bx)−p xq−1 dx = 2(ab)− 2 B(q, p − q) cos q arccos √ 2 ab 0 BI (19)(9) [p > q > 0]

   ∞   q a + b (1 + ax)−p − (1 + bx)−p xq−1 dx = −2i(ab)− 2 B(q, p − q) sin q arccos √ 2 ab 0 BI (19)(10) [p > q > 0]  1   (1 + x)μ−1 (1 − x)ν−1 + (1 + x)ν−1 (1 − x)μ−1 dx = 2μ+ν−1 B(μ, ν) 

3.212

3.213

3.214

∞

0

[Re μ > 0,

Re ν > 0] LI(1)(15), EH I 10(10)



1

3.215

 μ μ−1  a x (1 − ax)ν−1 + (1 − a)ν xν−1 [1 − (1 − a)x]μ−1 dx = B(μ, ν)

0

[Re μ > 0,

Re ν > 0,

|a| < 1] BI (1)(16)

3.216



1. 0

 2. 3.217 3.218 3.219 3.221

1

xμ−1 + xν−1 dx = B(μ, ν) (1 + x)μ+ν

[Re μ > 0,

Re ν > 0]

∞

xμ−1 + xν−1 dx = B(μ, ν) [Re μ > 0, Re ν > 0] (1 + x)μ+ν 1   ∞ p p−1 b x (1 + bx)p−1 − dx = π cot pπ [0 < p < 1, b > 0] (1 + bx)p bp−1 xp 0 ∞ 2p−1 x − (a + x)2p−1 dx = π cot pπ [p < 1] (cf. 3.217) x)p xp  0 ∞ (a + xν xμ − dx = ψ(μ + 1) − ψ(ν + 1) ν+1 (x + 1) (x + 1)μ+1 0 [Re μ > −1, Re ν > −1]



1. a

FI II 775

∞

(x − a)p−1 dx = π(a − b)p−1 cosec pπ x−b

[a > b,

0 < p < 1]

FI II 775

BI(18)(13) BI (18)(7)

BI (19)(13)

LI (24)(8)

3.226

Powers of x and binomials

 2. 3.222

(a − x)p−1 dx = −π(b − a)p−1 cosec pπ x−b −∞



a

1

1. 0



319

xμ−1 dx = β(μ) 1+x

∞

2. 0

[a < b,

0 < p < 1]

LI (24)(8)

[Re μ > 0]

xμ−1 dx = π cosec(μπ)aμ−1 x+a = −π cot(μπ)(−a)μ−1

WH

for a > 0

FI II 718, FI II 737

for a < 0

BI(18)(2), ET II 249(28)

[0 < Re μ < 1] 3.223



∞

1. 0



∞

2. 0

3. 3.224

3.225

 π  μ−1 xμ−1 dx = β cosec(μπ) + αμ−1 cot(μπ) (β + x)(α − x) α+β [|arg β| < π,

0 < Re μ < 2]

α > 0,

ET I 309(7)

0 < Re μ < 2] ET I 309(8)



∞



∞

−b [a > b > 0, 0 < Re μ < 2] ET I 309(9) b−a 0

  ∞ γ − β μ−1 δ − β μ−1 (x + β)xμ−1 dx = π cosec(μπ) γ δ + (x + γ)(x + δ) γ−δ δ−γ 0 [|arg γ| < π, |arg δ| < π, 0 < Re μ < 1] ET I 309(10)

1. 1



∞

2. 1



∞

3. 0

3.226

 π  μ−1 xμ−1 dx = β − γ μ−1 cosec(μπ) (β + x)(γ + x) γ−β [|arg β| < π, |arg γ| < π,



1

1. 0

 2.

0

1

μ−1

a x dx = π cot(μπ) (a − x)(b − x)

μ−1

μ−1

(x − 1)p−1 dx = (1 − p)π cosec pπ x2

[−1 < p < 1]

BI (23)(8)

(x − 1)1−p 1 dx = p(1 − p)π cosec pπ x3 2

[0 < p < 1]

BI (23)(1)

xp dx π = p(1 − p) cosec pπ 3 (1 + x) 2

[−1 < p < 2]

BI (16)(5)

xn dx (2n)!! √ =2 (2n + 1)!! 1−x

BI (8)(1)

1

xn− 2 dx (2n − 1)!! √ π. = (2n)!! 1−x

BI (8)(2)

320

3.227

Power and Algebraic Functions



∞

1. 0



∞

2. 0

3.228 1.

2.

3.

3.227



xν−1 (β + x)1−μ γ 1−μ ν−1 dx = β γ B(ν, μ − ν) 2 F 1 μ − 1, ν; μ; 1 − γ+x β [|arg β| < π, |arg γ| < π, 0 < Re ν < Re μ]

ET II 217(9)

x− (β − x)−σ dx = πγ − (β − γ)−σ cosec(π) I 1−γ/β (σ, ) γ+x [|arg β| < π, |arg γ| < π, − Re σ < Re  < 1]

ET II 217(10)



ν  a−c (x − a)ν (b − x)−ν dx = π cosec(νπ) 1 − for c < a x−c b−c a 

ν  c−a = π cosec(νπ) 1 − cos(νπ) for a < c < b 

ν b − c c−a = π cosec(νπ) 1 − for c > b c−b ET II 250(31) [|Re ν| < 1] ! !  b ν−1 (x − a)ν−1 (b − x)−ν π cosec(νπ) !! a − c !! dx = for c < a or c > b; !b−c! x−c b−c a π(c − a)ν−1 =− cot(νπ) for a < c < b (b − c)ν [0 < Re ν < 1] ET II 250(32)  b (x − a)ν−1 (b − x)μ−1 dx x−c a

(b − a)μ+ν−1 b−a = B(μ, ν) 2 F 1 1, μ; μ + ν; for c < a or c > b; b−c b−c 

b

= π(c − a)ν−1 (b − c)μ−1 cot μπ − (b − a)μ+ν−2 B(μ − 1, ν)

b−c × 2 F 1 2 − μ − ν, 1; 2 − μ; for a < c < b b−a [Re μ > 0, Re ν > 0, μ + ν = 1, μ = 1, 2, . . .] ET II 250(33)  4. 0

 5.

0

1

π(a − b)ν−1 (1 − x)ν−1 x−ν dx = cosec(νπ) a − bx aν

∞

[0 < Re ν < 1,

0 < b < a]

 c xν−1 (x + a)1−μ dx = a1−μ (−c)ν−1 B(μ − ν, ν) 2 F 1 μ − 1, ν; μ; 1 + x−c a a1−μ−ν ν−1 1−μ B(μ − ν − 1, ν) = πc (a + c) cot[(μ − ν)π] −

a + c a × 2 F 1 2 − μ, 1; 2 − μ + ν; a+c [a > 0, 0 < Re ν < Re μ]

BI (5)(8)

for c < 0;

for c > 0 ET II 251(34)

3.237

Powers of x and binomials



∞

6.

xν−1

0

−n

(γ + x) x+β

⎡

⎤ ν−1 n−1

j  π β (1 − ν)j γ − β ⎦ ⎣1 − γ dx = sin πν (γ − β)n β j! γ j=0 ν−1

[|arg β| < π, 

1

3.229 0

3.231



1

1. 0



1

2.11 0



1

3. 0



1

4. 0



1

5. 0



xp−1 + x−p dx = π cosec pπ 1+x



1 xp − x−p dx = − π cot pπ x−1 p



xp − x−p 1 dx = − π cosec pπ 1+x p



xμ−1 − xν−1 dx = ψ(ν) − ψ(μ) 1−x

∞

−x 1−x

p−1

x

0



∞

3.232 0



∞

3.233 0



1



1.11 0



1



2. 0



3.23610

3.237

0 < Re ν < n]

AS 256 (6.1.22)

  xμ−1 dx π cosec μπ a b = − (1 − x)μ (1 + ax)(1 + bx) a−b (1 + a)μ (1 + b)μ [0 < Re μ < 1]

xp−1 − x−p dx = π cot pπ 1−x

6.

3.235

|arg γ| < π,

p2 < 1 p2 < 1 p2 < 1 p2 < 1

BI (5)(7)



BI (4)(4)



BI (4)(1)



BI (4)(3)



BI (4)(2)

[Re μ > 0,

Re ν > 0] FI II 815, BI(4)(5)



3.234

321

q−1

dx = π (cot pπ − cot qπ)

(c + ax)−μ − (c + bx)−μ b dx = c−μ ln x a

1 − (1 + x)−ν 1+x

xq−1 x−q − 1 − ax a − x xq−1 x−q + 1 + ax a + x





[p > 0,

q > 0]

[Re μ > −1;

a > 0;

FI II 718

b > 0;

c > 0] BI (18)(14)

dx = ψ(ν) + C x

[Re ν > 0]

EH I 17, WH

dx = πa−q cot qπ

[0 < q < 1,

a > 0]

BI (5)(11)

dx = πa−q cosec qπ

[0 < q < 1,

a > 0]

BI (5)(10)

∞

(1 + x)μ − 1 dx = ψ(ν) − ψ(ν − μ) [Re ν > Re μ > 0] (1 + x)ν x

0 1 μ 1−μ x 2 dx μ (1 − a)−μ − (1 + a)−μ  √ Γ 1+ Γ μ+1 = 2 2 2 2aμ π 0 [(1 − x) (1 − a x)] 2 [−2 < μ < 1, |a| < 1]   u 2  n+1 ∞  u Γ dx = ln  2 2 (−1)n+1 [|arg u| < π] x + u u+1 n n=0 2 Γ 2

BI (18)(5)

BI (12)(32) ET II 216(1)

322

3.238

Power and Algebraic Functions



∞

ν−1

νπ ν−1 |x| dx = −π cot |u| sign u 2 −∞ x − u

1.

3.238

[0 < Re ν < 1 u real,

u = 0] ET II 249(29)



∞

ν−1

2. −∞

νπ ν−1 |x| sign x dx = π tan |u| x−u 2

[0 < Re ν < 1 u real,

u = 0] ET II 249(30)



b

3.

(b − x)μ−1 (x − a)ν−1 |x − u|

a

μ+ν

dx =

(b − a)μ+ν−1 Γ(μ) Γ(ν) μ ν |a − u| |b − u| Γ(μ + ν) [Re μ > 0, Re ν > 0, 0 < u < a < b and 0 < a < b < u]

MO 7

3.24–3.27 Powers of x, of binomials of the form α + βxp and of polynomials in x 3.241



1

1. 0



xμ−1 dx 1 = β p 1+x p

∞

2. 0

 4.

0 ∞

 5. 6.10

3.242 1.

pπ xp−1 dx π = cot 1 − xq q q

xμ−1 dx n+1

(p + qxν )

0



μ ν−μ , ν ν

p > 0]

WH, BI (2)(13)

[Re ν > Re μ > 0] ET I 309(15)a, BI (17)(10)

∞

PV

11

[Re μ > 0,

μπ 1 xμ−1 dx π = B = cosec 1 + xν ν ν ν

 3.11



μ p

1 = n+1 νp

[p < q]  μ/ν  μ   Γ ν Γ 1 + n − μν p q Γ(1 + n) + μ 0 < < n + 1, ν

BI (17)(11)

p = 0,

,

BI (17)(22)a ∞

(p − q)π (p − q)π [p < 2q] = cosec 2 q )2 q q (1 + x 0   

p   b b−u x x 1/p − G(x) = sign du = (b − a)F c b−a c a where ⎧ ⎪ x≤0  1 ⎨−1 sign(x − t) dt = 2x − 1 0 < x < 1 F (x) = ⎪ 0 ⎩ 1 x≥1 xp−1 dx

  (2n − 2m − 1) π (2m + 1)π x2m dx = sin t cosec t cosec 4n 2n + 2x cos t + 1 n 2n 2n −∞ x   2 m < n, t < π 2



q = 0

BI (17)(18)

∞

FI II 642

3.247

Powers of x and binomials and polynomials

 2.

∞



11 0



∞

3.24311 0

3.244



1

1. 0

 2. 3. 4.

x2 x4 + 2ax2 + 1

c

x2 + 1 xb + 1

323



dx 1 1 −1/2−c 1/2−c =2 (1 + a) B c− , x2 2 2



xμ−1 dx (1 + x2ν ) (1 + x3ν ) 



π 4π 2π = 8 cosec(2ρ) + 12 cosec(3ρ) − 8 cosec 2ρ − + 8 cosec 2ρ − 48ν 3 3  ,  π π cosec ρ + sec(ρ) −3 cosec ρ − 6 6 μπ , [0 < Re μ < 5 Re ν] ET I 312(34) where ρ = 6ν

pπ xp−1 + xq−p−1 π dx = cosec 1 + xq q q

[q > p > 0]

BI (2)(14)

pπ xp−1 − xq−p−1 π [q > p > 0] dx = cot q 1−x q q 0  1 ν−1  μ , x − xμ−1 1+ dx = C + ψ [Re μ > Re ν > 0] 1 − xν ν ν 0



   ∞ 2m 2n + 1 2m + 1 x − x2n π π − cot π dx = cot 2l l 2l 2l −∞ 1 − x 1

[m < l, 

∞

n < l]

3.246

BI (2)(17)

FI II 640

 ν−μ  x − xν (1 + x)−μ dx =

ν B(ν, μ − ν) ν−μ+1 0 [Re μ > Re ν > 0]  ∞ qπ pπ (p + q)π 1 − xq p−1 π cosec cosec x dx = sin r r r r r 0 1−x [p + q < r, p > 0]

3.245

BI (2)(16)





BI (16)(13)

ET I 331(33), BI (17)(12)

 dx can be transformed by the substitution x = et Integrals of the form f x ± x , x ± x , . . . x  1  ∞  1+p  1−p −1 −t x or x = e . For example, instead of +x dx, we should seek to evaluate sech px dx 0 0  1 n−m−1  ∞ x + xn+m−1 −1 and, instead of dx, we should seek to evaluate cosh mx (cosh nx − cos a) dx n 2n 0 1 + 2x cos a + x 0 (see 3.514 2). 3.247  1 α−1 ∞  x (1 − x)n−1 ξk dx = (n − 1)! 1.11 b 1 − ξx (α + kb)(α + kb + 1) . . . (α + kb + n − 1) 0 p

−p

q

−q

k=0

 2. 0

[b > 0, ∞

(1 − x ) x 1 − xnp p

ν−1

dx =

|ξ| < 1]

AD (6704)

π

πν π (p + ν)π sin cosec cosec np n np np [0 < Re ν < (n − 1)p]

ET I 311(33)

324

3.248

Power and Algebraic Functions



∞

1. 0



1

2. 0

 3. 4.3

6.∗

3.249 1.0 2.9 3. 4. 5.

xμ−1 dx 1 √ = B ν 1 + xν



μ 1 μ , − ν 2 ν

3.248

[Re ν > Re 2μ > 0]

BI (21)(9)

x2n+1 dx (2n)!! √ = (2n + 1)!! 1 − x2

BI (8)(14)

x2n dx (2n − 1)!! π √ = 2 (2n)!! 2 1−x 0  ∞ dx π √ = 2 ) 4 + 3x2 3 (1 + x −∞ 1

BI (8)(13)

⎧   ⎪ 2 b ⎪ ⎪ √ −1 arctan ⎪ ⎪ ⎪ a b−a ⎪  ∞ ⎨ dx 2 = √    2 2 ⎪ a −∞ 1 + x ⎪ b + ax2 √ √

⎪ ⎪ ⎪ 1 a+ a−b ⎪ ⎪ √ √ ln √ ⎩ a−b a− a−b 

if a < b if a = b if a > b

∞

dx π (2n − 3)!! = 2n−1 2 + a2 )n 2 · (2n − 2)!! a (x 0  a  2 n− 12 (2n − 1)!! π. a − x2 dx = a2n 2(2n)!! 0 n  1 1 − x2 dx = 2n+1 Q n (a) n+1 −1 (a − x)

 1 μ μ+1 x dx 1 = β 2 2 2 0 1+x  1  μ−1   1 − x2 dx = 22μ−2 B(μ, μ) = 12 B 12 , μ

FI II 743 FI II 156

EH II 181(31)

[Re μ > −1]

BI (2)(7)

[Re μ > 0]

FI II 784

[p > 0]

BI (7)(7)

0

 6. 7. 8.11 3.251

1



√ p−1 x dx =

2 p(p + 1) 0

 1 1 1 1 1 μ −ν ,1 − (1 − x ) dx = B μ μ ν 0 

−n/2

 ∞ π(n − 1) n−1 x2 n Γ dx = 1+ n−1 2 Γ 2 −∞ 

1. 0

 2. 0

1

1−

 ν−1 1 μ ,ν xμ−1 1 − xλ dx = B λ λ

[Re μ > 0,

|ν| > 1]

[n > 1]

[Re μ > 0,

Re ν > 0,

λ > 0] FI II 787

∞

μ

μ

 ν−1 1 ,1 − ν − xμ−1 1 + x2 dx = B 2 2 2



Re μ > 0,

   Re ν + 12 μ < 1

3.252

Powers of x and binomials and polynomials



∞

3.

μ−1

x

ν−1

(x − 1) p

1



∞

0 ∞

0

x2m+1 dx m!(n − m − 2)! n = 2(n − 1)!am+1 cn−m−1 (ax2 + c) xμ+1 (1 +

0 1

7. 0



x2m dx (2m − 1)!!(2n − 2m − 3)!!π √ n = 2 · (2n − 2)!!am cn−m−1 ac (ax2 + c)

∞

6. 

2 x2 )

dx =

μπ 4 sin μπ 2

xq+p−1 (1 − xq )

−p q

dx =

0 1

9.

q

x p −1 (1 − xq )

1 −p



1

−p q

xp−1 (1 − xq ) ∞

11.



∞

1. 0

 2. 3.

μ−1

x 0

3.252

c > 0,

n > m + 1]

∞



μ−1 2

GU (141)(8b) WH



pπ pπ cosec 2 q q

[Re μ > 1]

LI (3)(11)

[q > p]

BI (9)(22)

[p > 1,

dx =

pπ π cosec q q

[q > p > 0]

p −ν

(1 + βx )

n > m + 1 ≥ 1]

[ac > 0,

π π cosec q p

0



[a > 0,

dx =

0

10.

Re μ < p − p Re ν]

[−2 < Re μ < 2]

1 μ−1 β 2 = −4 + 2 4 (1 + x ) xμ dx

1

8. 

Re ν > 0,

GU (141)(8a)

5. 

[p > 0,

ET I 311(32)

4. 



μ 1 dx = B 1 − ν − , ν p p

325



μ μ 1 − μp ,ν − dx = β B p p p [|arg β| < π, p > 0,

q > 0]

BI (9)(23)a

BI (9)(20)

0 < Re μ < p Re ν]

BI (17)(20), EH I 10(16)

  1 dx (−1)n−1 ∂ n−1 b √ arccot √ n = (n − 1)! ∂cn−1 (ax2 + 2bx + c) ac − b2 ac − b2   a > 0, ac > b2

 dx (2n − 3)!!πan−1 a > 0, = n 2 + 2bx + c) n− 12 (ax 2 −∞ (2n − 2)!! (ac − b )

  ∞ 1 dx (−2)n ∂ n √ √ = n+ 32 (2n + 1)!! ∂cn c ( ac + b) 0 (ax2 + 2bx + c)  a ≥ 0,

ac > b2

c > 0,



GW (131)(4) GW (131)(5)

√  b > − ac GW (213)(4)

326

Power and Algebraic Functions



∞

4. 0



x dx n (ax2 + 2bx + c)   b (−1)n ∂ n−2 b 1 − = arccot √ for ac > b2 ; (n − 1)! ∂cn−2 2 (ac − b2 ) 2 (ac − b2 ) 32 ac − b2   √ b b + b2 − ac 1 (−1)n ∂ n−2 √ + ln for b2 > ac > 0; = (n − 1)! ∂cn−2 2 (ac − b2 ) 4 (b2 − ac) 32 b − b2 − ac an−2 = for ac = b2 2(n − 1)(2n − 1)b2n−2 [a > 0, b > 0, n ≥ 2] GW (141)(5)

∞

5. −∞



x dx (2n − 3)!!πban−2 n =− (2n−1) + 2bx + c) (2n − 2)!! (ac − b2 ) 2



ac > b2 ,

a > 0,

n≥2

 GW (141)(6)

m

−∞

(ax2

m

xn dx n+ 32

(ax2 + 2bx + c)

0

n−m−1 m

x dx b (−1) πa n = n− 1 + 2bx + c) (2n − 2)!! (ac − b2 ) 2

k [m/2]   m ac − b2 (2k − 1)!!(2n − 2k − 3)!! × 2k b2 k=0   ac > b2 , 0 ≤ m ≤ 2n − 2

∞

7.



(ax2

∞

6.



3.252

=

n! √ √ n+1 (2n + 1)!! c ( ac + b) 

a ≥ 0,

c > 0,

GW (141)(17)

√  b > − ac GW (213)(5a)

∞

8.

xn+1 dx (ax2

0

n+ 32

=

+ 2bx + c)

n! √ √ n+1 (2n + 1)!! a ( ac + b) 

a > 0,

c ≥ 0,

√  b > − ac GW (213)(5b)



∞

9.

10.6 0

x

dx n+1

(ax2 + 2bx + c)

0



n+ 12

=

1 22n+ 2

(2n − 1)!!π √ n+ 1 √ (b + ac) 2 n! a



a > 0,

c > 0,

b+

 √ ac > 0 LI (21)(19)

∞

μ−1

x dx ν− 12 ν =2 2 (1 + 2x cos t + x )



1 1 1 2 −ν 2 −ν (sin t) tΓ ν + B(μ, 2ν − μ) P μ−ν− 1 (cos t) 2 2 [0 < t < π, 0 < Re μ < Re 2ν]

ET I 310(22)

3.255

Powers of x and binomials and polynomials



∞

11.

327

 μ− 12 −ν−1  μ 1 + 2βx + x2 x dx = 2−μ β 2 − 1 2 Γ(1 − μ) B(ν − 2μ + 1, −ν) P μν−μ (β)

0

Re(2μ − ν) < 1,

[Re ν < 0,

|arg (β ± 1)| < π] EH I 160(33)

1 2 −μ

= −π cosec νπ C ν (β)    −2 < Re 12 − μ < Re ν < 0,

|arg (β ± 1)| < π



EH I 178(24)



∞

12. 0

xμ−1 dx = −πaμ−2 cosec t cosec(μπ) sin[(μ − 1)t] x2 + 2ax cos t + a2 [a > 0, 0 < |t| < π,

0 < Re μ < 2] FI II 738, BI(20)(3)



∞

13.

xμ−1 dx (x2

0

2 a2 )

+ 2ax cos t +

=

πaμ−4 cosec μπ cosec3 t 2 × {(μ − 1) sin t cos [(μ − 2)t] − sin[(μ − 1)t]} [a > 0,



∞

14. 0

0 < |t| < π,

xμ−1 dx √ = π cosec(μπ) P μ−1 (cos t) 1 + 2x cos t + x2

0 < Re μ < 4] [−π < t < π,

LI(20)(8)a, ET I 309(13)

0 < Re μ < 1] ET I 310(17)



1

3.253 −1

(1 + x)2μ−1 (1 − x)2ν−1 μ+ν

(1 + x2 )

dx = 2μ+ν−2 B(μ, ν)

3.254  u  ν xλ−1 (u − x)μ−1 x2 + β 2 dx 1. 0 2ν λ+μ−1

=β u



∞

2.6



1 0

Re ν > 0]

FI II 787



λ λ + 1 λ + μ λ + μ + 1 −u2 ; , ; 2 B(λ, μ) 3 F 2 −ν, , 2 2 2 β 

2 u Re ET II 186(10) > 0, λ > 0, Re μ > 0 β

 −λ  ν x (x − u)μ−1 x2 + β 2 dx

u

3.255

[Re μ > 0,

1

1

xμ+ 2 (1 − x)μ− 2 μ+1

(c + 2bx − ax2 )

Γ(μ) Γ(λ − μ − 2ν) = uμ−λ+2ν Γ(λ − 2ν)

1+λ−μ λ 1+λ β2 λ−μ − ν, − ν; − ν, − ν; − 2 × 3 F 2 −ν, 2 2 2 u  2

β |u| > |β| and Re ET II 202(9) > 0, 0 < Re μ < Re(λ − 2ν) u   √ Γ μ + 12 π dx =  √ √ 2 μ+ 12 √ Γ(μ + 1) a+ c + 2b − a + c c + 2b − a   √ √ 2 c + 2b − a + c > 0, c + 2b − a > 0, Re μ > − 12 a+ BI (14)(2)

328

Power and Algebraic Functions

3.256



1

1.

(1 − x2 )

0

 2. 0



xp−1 + xq−1

1

p+q 2

xp−1 − xq−1 (1 −

∞

9

3.257

0

x2 )

p+q 2

%

b ax + x



1 dx = cos 2

dx =

1 sin 2





 q−p q+p p q π sec π B , 4 4 2 2

q−p π cosec 4

+c

∞

1.

  √ π Γ p + 12

2.

∞

3. 0



5.



p + q < 2]

BI (8)(25)

BI (8)(26)

[p > 0,

q > 0,

p + q < 2]

a > 0,

b < 0,

c > 0,

p > − 12



a > 0,

b > 0,

c > −4ab,

n nan+1 x2 + a2 − x dx = 2 n −1

n ≥ 2]

a

GW (215)(5)

[n ≥ 2]

GW (214)(7)

[n ≥ 2]

GW (214)(6a)

[a > 0, ∞

0

7.



∞

6. 

BI (20)(4)

p > − 12

dx n [n ≥ 2] √  n = n−1 2 2 2 a (n − 1) x + x + a 0  ∞  n n · m!am+n+1 xm x2 + a2 − x dx = (n − m − 1)(n − m + 1) . . . (m + n + 1) 0 



n−1 n+1     a2 1 b − b 2 − a2 b − b 2 − a2 − 2(n − 1) 2(n + 1)

[0 < a ≤ b, √ √  n+1 n−1  ∞  n b2 + 1 − b b2 + 1 − b 2 + x + 1 − x dx = 2(n − 1) 2(n + 1) b 

4.



1 2acp+ 2

 n  x − x2 − a2 dx =

b

[p > 0, q > 0, p q q+p π B , 4 2 2

dx

  Γ(p + 1) 1 B p + 12 , 12 = 1 2 p+ a (4ab + x) 2 



&−p−1

2

=

3.258

3.256

GW (214)(6)

x dx n · m! √  n = 2 2 (n − m − 1)(n − m + 1) . . . (m + n + 1)an−m−1 x+ x +a m

[a > 0, ∞

0 ≤ m ≤ n − 2]

GW (214)(5a)

0 ≤ m ≤ n − 2]

GW (214)(5)

 n  n · (n − m − 2)!(2m + 1)!am+n+1 (x − a)m x − x2 − a2 dx = 2m (n + m + 1)! [a > 0,

n ≥ m + 2]

GH (215)(6)

3.264

3.259 1.

Powers of x and binomials and polynomials



1 p−1

6

x 0

(1 − x)

n−1

329

∞ k  l b Γ(p + km) (1 + bx ) dx = (n − 1)! k Γ(p + n + km) m l

k=0

[|b| < 1 unless l = 0, 1, 2, . . . ;  2.

u

p, n, p + ml > 0]

BI (1)(14)

λ

xν−1 (u − x)μ−1 (xm + β m ) dx

11 0

3.11

= β mλ uμ+ν−1 B (μ, ν)

ν+m−1 μ+ν μ+ν+1 μ + ν + m − 1 −um ν ν+1 ,..., ; , ,..., ; m × m+1 F m −λ, , m m m m ! m β !  m ! ! u π !< Re μ > 0, Re ν > 0, !!arg ET II 186(11) β ! m



 ∞ λ λ β 1 λ −μ −ν ,μ + ν − ; μ + ν; 1 − xλ−1 (1 + αxp ) (1 + βxp ) dx = α−λ/p B ν, F 2 1 p p p p α 0 [|arg α| < π, |arg β| < π, p > 0, 0 < Re λ < 2 Re(μ + ν)] ET I 312(35)

3.261 1.11

 PV 0

 2. 3.

4.

1

∞

(1 − x cos t) xμ−1 dx  cos kt = 1 − 2x cos t + x2 μ+k

[Re μ > 0,

BI (6)(9)

 2  (xν + x−ν ) dx π sin νt ν < 1, t = (2n + 1)π = BI (6)(8) 2 1 + 2x cos t + x sin t sin νπ 0   1  1+p x + x1−p dx π (p sin t cos pt − cos t sin pt) = 2 )2 2 sin3 t sin pπ (1 + 2x cos t + x 0  2  p < 1, t = (2n + 1)π BI (6)(18)

 1 μ−1 x dx π cosec t cosec μπ a sin t · = μ sin t − μ arctan 2 2 μ 2 (1 − x) (1 + 2a cos t + a ) 2 1 + a cos t 0 1 + 2ax cos t + a x 1

[a > 0, 

0 < Re μ < 1]

BI (6)(21)

∞

(1 − p)π x−p dx π [−2 < p < 1] = cosec LI (18)(3) 3 1 + x 3 3 0 ∞ , + ν π x dx νπ νπ = + β ν cosec − 2γ ν cosec(νπ) γβ ν−1 sec 2 + β2) 2 + γ2) (x + γ) (x 2 (β 2 2 0 [Re β > 0, |arg γ| < π, −1 < Re ν < 2, ν = 0] ET II 216(7)

3.262 3.263

3.264

t = 2nπ]

k=0



1. 0

∞

π ap−2 + bp−2 cos xp−1 dx = (a2 + x2 ) (b2 − x2 ) 2 a2 + b 2

pπ 2

cosec

pπ 2 [0 < p < 4,

a > 0,

b > 0] BI (19)(14)

330

Power and Algebraic Functions



∞

2. 0



∞

3. 0



∞

4. 0



∞

5.

dx

π 2 = 2 2 2 (b + x ) (a + b + x ) dx

π 3 = 4 2 2 (b + x ) (a + b + x ) dx 4

∞

6. 0

∞

7. 0

 3.265

3.266

=

  ⎛



μ=

1 2



|arg γ| < π,

0 < Re μ < 4]

1 1 1 − − 3/2 1/2 2 a2 b1/2 2a (a + b) a (a + b)

ET I 309(4)

 MC

2 3 1 2 − − − 5/2 3/2 1/2 2 3 a3 b1/2 4a (a + b) a (a + b) a (a + b)



π⎝ 2 5 3 − − 7/2 5/2 2 4 a4 b1/2 8a (a + b) 4a (a + b) ⎞ 1 2 ⎠ − − 3/2 1/2 3 4 a (a + b) a (a + b)

dx n (b + x2 ) (a + b + x2 )



a+b π 1 1 1 1 3 − n; = − B n− , 2 F 1 1 − n, 1; n−1/2 2 an b1/2 2 2 2 a 2a (a + b)

=



μ

π γ 2 −1 − β 2 −1 μπ xμ−1 dx = cosec 2 2 (β + x ) (γ + x ) 2 β − γ

2 1 π 1 √ −√ = 2(γ − β) γ β [|arg β| < π,

(b + x2 ) (a + b + x2 )

0



μ

3.265

π 1 π − n−1/2 n 2 an b1/2 2a (a + b)

  n−1  12 j a j j! a+b j=0

AS 263 (6.6.3.2)

a + b > 0]  π β γ x2 dx π = − = 2 2 2 2 2 2 2 2 (x + α ) (x + β ) (x + γ ) 2α (β − γ ) β + α γ + α 2(α + β)(α + γ)(β + γ) 

[n > 0,

1 − xμ−1 dx = ψ(μ) + C [Re μ > 0] FI II 796, WH, ET I 16(13) 1−x 0 = ψ(1 − μ) + C − π cot(μπ) [Re μ > 0] EH I 16(15)a

  ∞ π β (xν − aν ) dx aν = ln β ν cosec(νπ) − aν cot(νπ) − (x − a)(β + x) a + β π a 0 [|arg β| < π, |Re ν| < 1, ν = 0] 1

ET II 216(8)

3.267 1. 2.

  Γ n + 13 x3n dx 2π   √ √ = 3 3 3 Γ 13 Γ(n + 1) 1 − x3 0    1 3n−1 (n − 1)! Γ 23 x dx   √ = 3 3 Γ n + 23 1 − x3 0 

1

BI (9)(6)

BI (9)(7)

3.272

3.

∗

Powers of x and binomials and polynomials



1

0

3.268



1

    Γ n − 13 Γ 23 x3n−2 dx   √ = 3 3 Γ n + 13 1 − x3

1. 0

 2. 3.

331

1 pxp−1 − 1 − x 1 − xp

dx = ln p

BI (5)(14)

1 − xμ ν−1 x dx = ψ(μ + ν) − ψ(ν) [Re ν > 0, 0 1−x 

 1 n  n xμ−1 n−k √ − dx = nC + ψ μ + 1−x 1− nx n 0 1

Re μ > 0]

BI (2)(3)

k=1

[Re μ > 0] 3.269



1

1. 0

 2. 3. 3.271



∞

0



∞

2. 0

4.



pπ xp − x−p 1 π x dx = − cosec 2 1 + x p 2 2 0



 1 μ ν+1 μ+1 x − xν 1 1 ψ ψ dx = − 2 2 2 2 2 0 1−x 1

1.

3.



pπ 1 xp − x−p π − x dx = cot 2 1−x 2 2 p

π xp − xq dx = x−1 x+a 1+a



π xp − ap xp − 1 dx = x−a x−1 a−1

1 a2p − 1 − ap ln a sin(2pπ) π

p2 < 1



BI (4)(12)



BI (4)(8)

[Re μ > −1,

ap − cos pπ aq − cos qπ − sin pπ sin qπ

p2 < 1

BI (13)(10)

Re ν > −1]

BI (2)(9)





p2 < 1,

q 2 < 1,

  1 p2 < 4

a>0



BI (19)(2)

BI (19)(3)



1 p p 2 (a − 1) cot pπ − (a + 1) ln a π 0  2  p <1 BI (18)(9)

  ∞ p ap+q − 1 ap − aq sin pπ x − ap 1 − x−p q π x dx = + x − a 1 − x a − 1 sin[(p + q)π] sin[(q − p)π] sin qπ 0   (p + q)2 < 1, (p − q)2 < 1 

∞



∞

π xp − ap x−p − 1 dx = x−a x−1 a−1

BI (19)(4)

5. 0

3.272



1. 0

1

2 xp − x−p dx = 2 (1 − 2pπ cot 2pπ) 1−x



0 < p2 <

1 4



BI (16)(3)

1

xn−1 + xn− 2 − 2x2n−1 dx = 2 ln 2 1−x

BI (8)(8)

332

Power and Algebraic Functions



1

2. 0

3.273



1

1. 0

2

3.273

1

xn−1 + xn− 3 + xn− 3 − 3x3n−1 dx = 3 ln 3 1−x

BI (8)(9)

 (k − 1)!ak−1 sin kt sin t − an xn sin[(n + 1)t] + an+1 xn+1 sin nt p−1 (1 − x) dx = Γ(p) 1 − 2ax cos t + a2 x2 Γ(p + k) n

k=1

[p > 0] 

1

2. 0

BI (6)(13)

 (k − 1)!ak−1 cos kt cos t − ax − an xn cos[(n + 1)t] + an+1 xn+1 cos nt p−1 (1 − x) dx = Γ(p) 1 − 2ax cos t + a2 x2 Γ(p + k) n

k=1

[p > 0] 

x 0



1

0



 cos kt 1 − x cos t − xn+1 cos[(n + 1)t] + xn+2 cos nt dx = 1 − 2x cos t + x2 k+1 n

∞

0

1

0

1 1 − xn dx = n+1 (1 + x)n+1 1 − x 2

∞

3. 0

3.275 1. 2.

3. 4.

n  k=1

BI (20)(13)

2k k

BI (5)(3)

π qπ xq − 1 dx = tan xp − x−p x 2p 2p

[p > q]

BI (18)(6)



xn−1 pxnp−1 − dx = p ln p [p > 0] 1−x 1 − x1/p 0



 1 n−1 n nx xmn−1 n−k 1 − ψ m + dx = C + 1 − xn 1−x n n 0 k=1

 1 p−1 x qxpq−1 − dx = ln q [q > 0] 1−x 1 − xq 0

 ∞ 1 dx 1 = 0. − 2n 2m 1 + x 1 + x x 0 

1

3.276

% 

1.

π μπ (μ + 1)π xμ−1 (1 − x) π cosec dx = sin cosec 1 − xn n n n n [0 < Re μ < n − 1]

2. 

BI (6)(11)

k=0

1. 

BI (6)(12)

k=1

4. 3.274

 sin kt sin t − xn sin[(n + 1)t] + xn+1 sin nt dx = 2 1 − 2x cos t + x k+1 n

1

3.

BI (6)(14)

10 0

∞

b ax + x

BI (13)(9)

BI (5)(13)

BI (5)(12) BI (18)(17)

&−p−1

2 +c x2

dx 1 = 2|b|

  B p + 12 , 12 1 p+ 2

(2a (b + |b|) + c)



a > 0,

c > −4ac,

p > − 12



3.278

Powers of x and binomials and polynomials

 2.

∞

10 0

3.277



∞

11

1.

&−p−1  

%

2 B p + 12 , 12 b b +c dx = a+ 2 ax + 1 x x p+ (4ab + c) 2  a > 0,

xμ−1

0



∞ xμ−1

2.

√

1 + x2 + β √ 1 + x2

+ 

0

ν

b > 0,

c > −4ac,

p > − 12



 ν+μ μ μ ν+ μ dx = 2 2 −1 β 2 − 1 2 4 Γ Γ(1 − μ − ν) P μ −12 (β) 2 2 [Re β > −1, 0 < Re μ < 1 − Re ν] ET I 310(25)

,ν β 2 + x2 + x

β 2 + x2

333

dx =



β μ+ν−1 1−μ−ν B μ, 2μ 2 [Re β > 0,

0 < Re μ < 1 − Re ν] ET I 311(28)

√    ν ∞ μ−1 Γ μ2 Γ(1 − μ − ν) cos t ± i sin t 1 + x2 x 1−μ μ−1 2 2 √ dx = 2 sin t Γ(−ν) 1 + x2 0   μ+1 1 i 1 − μ+1 −2 2 2 −ν 2 × π Q − μ+1 −ν (cos t) ∓ π P μ−1 (cos t) 2 2 2 [Re μ > 0] ET I 311 (27) + ,ν  ∞ xμ−1 (β 2 − 1) (x2 + 1) + β √ dx x2 + 1 0   μ−1  1 − μ μ−1 2 2 − 1 iπ(μ−1) Γ μ2 Γ(1 − μ − ν)  2 = √ e 2 Q −2μ+1 −ν (β) β −1 Γ(−ν) 4 π 2 [Re β > 1, Re ν < 0, Re μ < 1 − Re ν] ET I 311(26) 

3.

4.



∞

5.

(x − u)μ−1

u

6.

7.

∞ xμ−1

x−

[|arg u| < π, 3.2788

ET II 202(10)

√ √ ν  −ν ,

x2 − 1 + x − x2 − 1 1−μ+ν 1−μ−ν √ , dx = 2−μ B 2 2 x2 − 1 1 ET I 311(29) [Re μ < 1 + Re ν] +√ ,  √  √ √ 2ν 2ν  u (u − x)μ−1 < x+2+ x + x+2− x 1 2μ + 1 1 2 −μ  dx = 2 π[u(u + 2)]μ− 2 P ν− 1 (u+ 2 2 x(x + 2) 0 1) 

+

√ 2ν √ 1   2μ−1 12 −μ x+1− x−1 1 2ν+ 2 √ dx = √ e(μ− 2 )πi u2 − 1 4 Q ν− 1 (u) 2 2 π x −1 [|arg(u − 1)| < π, 0 < Re μ < 1 + Re ν]



1. 0

∞



xp 1 + x2p

q

dx =0 1 − x2

[pq > 1]

Re μ > 0]

ET II 186(12)

334

Exponential Functions

3.310

3.3–3.4 Exponential Functions 3.31 Exponential functions 

∞

11

3.310

e−px dx =

0

3.311



∞

1. 0



∞

2. 0

 3.11

1 p

[Re p > 0]

dx ln 2 = px 1+e p

LO III 284a

e−μx dx = β(μ) 1 + e−x

[Re μ > 0]

EH I 20(3), ET I 144(7)

∞

pπ e−px π cosec dx = −qx |q| q −∞ 1 + e [q > p > 0 or 0 > p > q]



∞

4. 0



∞

0

∞

 ak e−qx dx = −px 1 − ae q + kp

[0 < a < 1]

1 − eνx dx = ψ(ν) + C + π cot(πν) ex − 1

[Re ν < 1]

e

0

− e−νx dx = ψ(ν) + C 1 − e−x

9.

11.

Re ν > 0]

(cf. 3.231 5)

[b > 0,

0 < Re μ < 1]

[|arg b| < π,

ET I 120(14)a

0 < Re μ < 1] ET I 120(15)a

∞

∞

0

0

WH, EH I 16(14)

BI (27)(8)

12. 

EH I 16(16)

∞

−px

−qx

pπ e −e π cot dx = −(p+q)x p + q p +q 0 1−e 



  ∞ px r−q r−p e − eqx 1 dx = ψ − ψ rx − esx r−s r−s r−s 0 e 

13.∗

[Re μ > 0,

e−μx dx = πbμ−1 cot(μπ) −x b − e −∞  ∞ −μx e dx = πbμ−1 cosec(μπ) −x −∞ b + e 

10.11

[Re ν > 0]

− e−νx dx = ψ(ν) − ψ(μ) 1 − e−x

e

0

8.

(cf. 3.265)

∞ −μx

7. 

BI (27)(7)

∞ −x

6. 

BI (28)(7)

k=0

5. 

(cf. 3.241 2)

∞

 c

c  ln b ln a a −b 1 dx = −ψ c ψ c x x c −d ln d ln d ln dc x

−px

[p > 0,

q > 0]

[r > s, r > p, r > q]

GW (311)(16c)

GW (311)(16)

x

−qx

e +e π cosec dx = −(p+q)x p+q 1+e



πp p+q



[c > a > 0,

b > 0,

d > 0] GW (311)(16a)

3.317

3.312

Exponential functions



∞

1.

 ν−1 x e−μx dx = β B(βμ, ν) 1 − e− β

335

[Re β > 0,

Re ν > 0,

Re μ > 0]

0



LI(25)(13), EH I 11(24) ∞

2.

 −1    1 − e−x 1 − e−αx 1 − e−βx e−px dx = ψ(p + α) + ψ(p + β) − ψ(p + α + β) − ψ(p)

0

[Re p > 0,  3.

∞

11

Re p > − Re α,

Re p > − Re β,

Re p > − Re(α + β)]

 ν−1  − −μx 1 − e−x 1 − βe−x e dx = B(μ, ν) 2 F 1 (, μ; μ + ν; β)

0

[Re μ > 0,

3.313 1.

7

 

2.7 3.314

3.315 1.

|arg(1 − β)| < π]



EH I 116(15)

[0 < Re μ < 1]

∞

e−μx dx = B(μ, ν − μ) [0 < Re μ < Re ν] −x )ν −∞ (1 + e 

  ∞ e−μx dx ν   = γ exp β μ − B(γμ, ν − γμ) β/γ + e−x/γ ν γ −∞ e  

ν > Re μ > 0, |Im β| < π Re γ Re γ

ET I 120(21)

∞

  e−μx dx = exp[γ(μ − ) − βν] B(μ, ν +  − μ) 2 F 1 ν, μ; ν + ; 1 − eν−β β + e−x )ν (eγ + e−x ) (e −∞ [|Im β| < π, |Im γ| < π, 0 < Re μ < Re(ν + )] ET I 121(22)   π β μ−1 − γ μ−1 e−μx dx = cosec(μπ) −x ) (γ + e−x ) γ−β −∞ (β + e [|arg β| < π, |arg γ| < π,

 2.

Re ν > 0,

∞

e−μx dx = π cot πμ −x −∞ 1 − e

PV

ET I 145(15)

∞

 3.316

∞

β = γ,

0 < Re μ < 2]

ET I 120(18)

ν

(1 + e−x ) − 1 dx = ψ(μ) − ψ(μ − ν) −x )μ −∞ (1 + e

[Re μ > Re ν > 0]

(cf. 3.235) BI (28)(8)

3.317



∞



1. −∞  ∞

2. −∞



1 1 − μ 1 + e−x (1 + e−x )

dx = C + ψ(μ)

1 1 ν − μ (1 + e−x ) (1 + e−x )

[Re μ > 0]

(cf. 3.233)

BI (28)(10)

dx = ψ(μ) − ψ(ν)

[Re μ > 0,

Re ν > 0]

(cf. 3.219) BI (28)(11)

336

3.318 1.

2.7

Exponential Functions

3.318

√ √  −ν  −ν β + 1 − e−x + β − 1 − e−x √ e−μx dx 1 − e−x 0  (μ−ν)/2 2μ+1 e(μ−ν)πi β 2 − 1 Γ(μ) Q ν−μ μ−1 (β) = Γ(ν) [Re μ > 0] ET I 145(18)  ∞    ν 1 √ e−u 1 − e−2x − e−x 1 − e−2u e−μx dx 1 − e−2x u  √ 1 u − 1 (μ+ν) √ 2− 2 (μ+ν) πe− 2 (μ+ν) Γ(μ) Γ(ν + 1) P − 21 (μ−ν) 1 − e−2u 2 = Γ[(μ + ν + 1)/2] [u > 0, Re μ > 0, Re ν > −1] ET I 145(19)



∞

3.32–3.34 Exponentials of more complicated arguments 3.321 1.

√  u ∞  2 π π (−1)k u2k+1 Φ(u) = erf(u) = e−x dx = 2 2 k!(2k + 1) 0 k=0 ∞ k 2k+1 2  2 u = e−u (2k + 1)!!

√

11

k=0

(cf. 8.25) √ 2 2 π Φ(qu) [q > 0] e−q x dx = 2q 0 √  ∞ 2 2 π [q > 0] e−q x dx = 2q 0  u , 2 2 2 2 1 + xe−q x dx = 2 1 − e−q u 2q 0 √   u π 1 2 −q 2 x2 −q 2 u2 Φ(qu) − que x e dx = 3 2q 2 0  u ,   2 2 2 2 1 + x3 e−q x dx = 4 1 − 1 + q 2 u2 e−q u 2q 0  √

  u 2 2 2 2 3 1 3 π Φ(qu) − + q 2 u2 que−q u x4 e−q x dx = 5 2q 4 2 0

 2. 3. 4.∗ 5.∗ 6.∗ 7.∗

3.322  1.11

∞

u

FI II 624



    x2 u βγ 2 − γx dx = πβe exp − 1−Φ γ β+ √ 4β 2 β

[Re β > 0]

∞   ,  +  x2 − γx dx = πβ exp βγ 2 1 − Φ γ β exp − 4β 0

ET I 146(21)

[Re β > 0]

NT 27(1)a

 2.

AD 6.700

u



3.326

Exponentials of more complicated arguments

 3.

11

PV

∞

e

±iλx2

0

3.323 1.

11

2.10 3.11

1 dx = 2



π ±πi/4 e λ

337

[λ > 0]

PBM 343 (2.3.15(2))

  √ π q2 /4 1 e exp −qx − x dx = 1−Φ 1+ q 2 2 1 2 √  ∞     q π Re p2 > 0 exp −p2 x2 ± qx dx = exp 2 4p p −∞ 4

 ∞ 4 γ   3 γ γ exp −β 2 x4 − 2γ 2 x2 dx = 2− 2 e 2β2 K 14 β 2β 2 0 + π |arg β| < , 4 

∞



2



BI (29)(4) BI (28)(1)

|arg γ| <

π, 4 ET I 147(34)a

3.324 1. 2.11

.

 β β − γx dx = K1 exp − βγ 4x γ 0 %

2n &  ∞ 1 b 1 exp − x − dx = Γ x n 2n −∞





∞

3.325 0

3.326 1.



∞

8 0



∞

2.10 0

3.

∗



∞



∞



b exp −ax − 2 x



2

1 dx = 2

1 exp (−x ) dx = Γ μ



1 μ

xm exp (−βxn ) dx =

Γ(γ) nβ γ

μ



4.∗

γ=

(x − a) exp (−β(x − b) ) dx =

(x − a) exp (−β(x − b)n ) dx = 0

5.∗



∞

u

ET I 146(25)

[a > 0,

b > 0]

FI II 644

[Re μ > 0]

n

u

Re γ > 0]

[b ≥ 0]

 √ π exp −2 ab a

Γ

0



[Re β ≥ 0,

m+1 n

[Re β > 0,

2

n n , β(−b) nβ 2/n

2

 − (a − b)



2

Γ

BI (26)(4)

Re m > 0, 1

n n , β(−b) nβ 1/n

[Re n > 0,



Re n > 0]



Re β > 0,

|arg b| < π]

− Γ n , β(u − b) 2/n  1 nβ    Γ n , β(−b)n − Γ n1 , β(u − b)n −(a − b) nβ 1/n [Re n > 0, Re β > 0, |arg b| < π, |arg(u − b)| < π] Γ

(x − a) exp (−β(x − b)n ) dx =

Γ

n

n , β(−b)

2

n n , β(−b) 2/n nβ

 − (a − b)

n

Γ

1

n , β(u − nβ 1/n

[Re n > 0,

b)n



Re β > 0,

|arg(u − b)| < π]

338

Exponential Functions

Exponentials of exponentials  ∞ 1 3.327 exp (−aenx ) dx = − Ei(−a) n 0 ∞ 3.328 exp (−ex ) eμx dx = Γ(μ) −∞ % &   ∞ b a exp (−ceax ) b exp −cebx 3.329 − dx = e−c ln −ax −bx 1 − e 1 − e a 0 3.331  1.

∞

3.327

[n ≥ 1,

Re a ≥ 0,

a = 0]

LI (26)(5)

[Re μ > 0] [a > 0,

NH 145(14)

b > 0,

c > 0]

BI (27)(12)

  exp −βe−x − μx dx = β −μ γ(μ, β)

[Re μ > 0]

ET I 147(36)

exp (−βex − μx) dx = β μ Γ(−μ, β)

[Re β > 0]

ET I 147(37)

0



∞

2. 0



∞

3.11

 ν−1   μ+ν β 1 − e−x exp βe−x − μx dx = B(μ, ν)β − 2 e 2 M ν−μ , ν+μ−1 (β) 2

0

2

[Re μ > 0,



∞

4.

 ν−1 μ−1 β 1 − e−x exp (−βex − μx) dx = Γ(ν)β 2 e− 2 W

0

1−μ−2ν 2

∞

3.332

[Re μ > 0,

1.

3

2.3

3.∗ 3.33411



ET I 147(39)

Re ν > 0,

|arg(1 − λ)| < π]

ET I 147(40)

∞

e−μx dx = Γ(μ) ζ(μ) −x ) − 1 −∞ exp (e  ∞   e−μx dx = 1 − 21−μ Γ(μ) ζ(μ) −x )+1 −∞ exp (e = ln 2 

Re ν > 0]

 ν−1  −   1 − e−x 1 − λe−x exp βe−x − μx dx = B(μ, ν) Φ1 (μ, , ν, λ, β)

0

3.333

ET I 147(38)

(β) , −μ 2

[Re β > 0, 

Re ν > 0]

∞



[Re μ > 1] [Re μ > 0,

ET I 121(24)

μ = 1]

[μ = 1] ET I 121(25)



tanh(x) 1 7 ζ(3) − 2 dx = 2 3 x π2 x cosh (x) 0    ∞ β ν−1 β ν−1 − μx dx = Γ(μ − ν + 1)e 2 β 2 W ν−2μ−1 ,− ν (β) (ex − 1) exp − x 2 2 e −1 0 [Re β > 0, Re μ > Re ν − 1] ET I 137(41)

Exponentials of hyperbolic functions  ∞  νx  e + e−νx cos νπ exp (−β sinh x) dx = −π [Eν (β) + Y ν (β)] 3.335 0

[Re β > 0]

EH II 35(34)

3.338

3.336

Exponentials of more complicated arguments



∞

1. 0



∞

2. 0



exp (−νx − β sinh x) dx = π cosec νπ [Jν (β) − J ν (β)] + π π |arg β| < and |arg β| = for Re ν > 0; 2 2 exp (nx − β sinh x) dx =

, ν is not an integer

3. 0

WA 341(2)

1 [S n (β) − π En (β) − π Y n (β)] 2 [Re β > 0;

∞

339

exp (−nx − β sinh x) dx =

n = 0, 1, 2, . . .]

WA 342(6)

1 (−1)n+1 [S n (β) + π En (β) + π Y n (β)] 2 [Re β > 0; n = 0, 1, 2, . . .] EH II 84(47)

3.337



∞

1. −∞  ∞

2.

exp (−νx + iβ cosh x) dx = iπe −∞  ∞

3. −∞

+ π, |arg β| < 2

exp (−αx − β cosh x) dx = 2 K α (β) iνπ 2

exp (−νx − iβ cosh x) dx = −iπe−

H (1) ν (β) iνπ 2

H (2) ν (β)

WA 201(7)

[0 < arg z < π]

EH II 21(27)

[−π < arg z < 0]

EH II 21(30)

Exponentials of trigonometric functions and logarithms 3.338



π

1.

  {exp i [(ν − 1)x − β sin x] − exp i [(ν + 1)x − β sin x]} dx = 2π Jν (β) + i Eν (β)

0



exp [±i (νx − β sin x)] dx = π [Jν (β) ± i Eν (β)]

2.

[Re β > 0]

EH II 36

[Re β > 0]

EH II 35(32)

π

0

 3.

10

4.6

5.∗

∞  γ J k (kβ) 1 exp [−γ (x − β sin x)] dx = + 2 [Re γ > 0] γ γ 2 + k2 0 k=1   a + b sin x + c cos x  π exp 2π 1 + p sin x + q cos x dx =  e−α I 0 (β), 2 − q2 1 + p sin x + q cos x 1 − p −π . a2 − b 2 − c 2 bp + cq − a ; β = α2 − ; with α = 2 2 1−p −q 1 − p2 − q 2 % ∞ &  π/4  tan2n x √ exp − dx = ln 2 1 n+ 2 0 n=1 ∞

WA 619(4)

 2  p + q2 < 1

340

Exponential Functions



3.339

π

6

3.339

exp (z cos x) dx = π I 0 (z) 0

exp (−p tan x) dx = ci(p) sin p − si(p) cos(p)  1 ∞  pk−1 exp (−px ln x) dx = x−px dx = kk 0

3.341 

BI (277)(2)a

π 2

[p > 0]

BI (271)(2)a

0 1

3.34211 0

BI (29)(1)

k=1

3.35 Combinations of exponentials and rational functions 3.351



u

1.8

xn e−μx dx =

0

n! μn+1

− e−uμ

n  n! k=0

uk

k! μn−k+1

= μ−n−1 γ(n + 1, μu) [u > 0,

Re μ > 0, n = 0, 1, 2, . . .] ET I 134(5)



∞

2.11

xn e−μx dx = e−uμ

u

n  n! k=0

uk

k! μn−k+1

= μ−n−1 Γ(n + 1, μu) [u > 0,



Re μ > 0, n = 0, 1, 2, . . .] ET I 33(4)

∞

3.

xn e−μx dx = n!μ−n−1

[Re μ > 0]

ET I 133(3)

0



∞ −px

e

4.

xn+1

u



e

x

1

6. 7.9 8.11 9.7

= (−1)n+1

n−1 pn Ei(−pu) e−pu  (−1)k pk uk + n n! u n(n − 1) . . . (n − k) k=0

[p > 0] ∞ −μx

5. 

dx

dx

= − Ei(−μ)

NT 21(3)

[Re μ > 0]

BI (104)(10)

u

ex dx = li (eu ) = Ei(u) [u < 0] −∞ x  u 1 1 xe−μx dx = 2 − 2 e−μu (1 + μu) [u > 0] μ μ 0  u   2 1 [u > 0] x2 e−μx dx = 3 − 3 e−μu 2 + 2μu + μ2 u2 μ μ 0  u   6 1 x3 e−μx dx = 4 − 4 e−μu 6 + 6μu + 3μ2 u2 + μ3 u3 μ μ 0 [u > 0]

3.352  1.

u −μx

e dx = eμβ [Ei(−μu − μβ) − Ei(−μβ)] x+β

0



∞ −μx

2. u

e dx = −eβμ Ei(−μu − μβ) x+β

[u ≥ 0,

|arg β| < π]

[u ≥ 0,

|arg(u + β)| < π,

ET II 217(12)

Re μ > 0]

ET I 134(6), JA

3.354

Exponentials and rational functions



v −μx

3. u



e dx = eαμ {Ei[−(α + v)μ] − Ei[−(α + u)μ]} x+α e dx = −eβμ Ei(−μβ) x+β



∞ −px



∞ −μx

Re μ > 0]

ET II 217(11)

e

ET II 251(37)

e

0



3.353

[|arg β| < π,

dx = e−pa Ei(pa − pu) a − x u   p > 0, a < u; for a > u, one should replace Ei(pa − pu) in this formula with Ei(pa − pu)

6.8

7.

Re μ > 0]

∞ −μx

0

5.

[−α < n, and − α > v,

ET I 134 (7)

4. 7

341

dx = e−μa Ei(aμ) a−x [a < 0,

BI (91)(4)

∞

eipx dx = iπeiap x − a −∞



Re μ > 0]

∞

1. u

[p > 0]

ET II 251(38)

 (k − 1)!(−μ)n−k−1 e−μx dx (−μ)n−1 βμ −uμ e Ei[−(u + β)μ] = e − (x + β)n (n − 1)!(u + β)k (n − 1)! k=1 [n ≥ 2, |arg(u + β)| < π, n−1

Re μ > 0] ET I 134(10)



∞

2.7 0



n−1  e−μx dx 1 (−μ)n−1 βμ e Ei(−βμ) = (k − 1)!(−μ)n−k−1 β −k − n (x + β) (n − 1)! (n − 1)! k=1 [n ≥ 2, |arg β| < π,

Re μ > 0]

ET I 134(9), BI (92)(2) ∞

3. 0

e−px dx 1 = peαp Ei(−ap) + (a + x)2 a

[p > 0,

a > 0] LI (281)(28), LI (281)(29)

 4. 0



1

xex e dx = − 1. (1 + x)2 2

∞

5.7 0

BI (80)(6)

 xn e−μx dx = (−1)n−1 β n eβμ Ei(−βμ) + (k − 1)!(−β)n−k μ−k x+β k=1 [|arg β| < π, n

Re μ > 0] BI (91)(3)a, LET I 135(11)

3.354



∞ −μx

e dx 1 = [ci(βμ) sin βμ − si(βμ) cos βμ] 2 2 β +x β

1. 0

 2. 3.7

[Re β > 0,

Re μ > 0]

BI (91)(7)

[Re β > 0,

Re μ > 0]

BI (91)(8)

∞

xe−μx dx = − ci(βμ) cos βμ − si(βμ) sin βμ β 2 + x2 0  ∞ −μx  e dx 1  −βμ e = Ei(βμ) − eβμ Ei(−βμ) 2 2 2β 0 β −x

[|arg (±β)| < π,

Re μ > 0]

BI (91)(14)

342

Exponential Functions

 4.

3.355

∞

 xe−μx dx 1  −βμ e = Ei(βμ) + eβμ Ei(−βμ) 2 2 2  0 β − x |arg (±β)| < π, Re μ > 0; for β > 0 one should replace Ei(βμ) in this formula with Ei(βμ)

 5.8

BI (91)(15) ∞

e−ipx dx π = e−|ap| 2 + x2 a a −∞



∞

1.  0

−μx

xe

dx

(β 2

2 x2 )

+

e

−px

dx

−

2 x2 )

(a2

0

∞

4.3 0



=

p real]

ET I 118(1)a

1 {ci(βμ) sin βμ − si(βμ) cos βμ} − βμ [ci(βμ) cos βμ + si(βμ) sin βμ] 2β 3

=

1 {−βμ [ci(βμ) sin βμ − si(βμ) cos βμ]} 2β 2 [Re β > 0,

∞

3.3 

+

2 x2 )

[a = 0,

LI (92)(6) ∞

2. 

e−μx dx (β 2

0

3.356

3.355

∞

1. 0

−px

xe

dx

(a2

2 x2 )

−

=

=

Re μ > 0]

 1  (ap − 1)eap Ei(−ap) + (1 + ap)e−ap Ei(ap) 3 4a   2 Im a > 0,

p>0

  1  −2 + ap e−ap Ei(ap) − eap Ei(−ap) 2 4a   2 Im a > 0,

p>0





BI (92)(7)

BI (92)(8)

LI (92)(9)

x2n+1 e−px dx = (−1)n−1 a2n [ci(ap) cos ap + si(ap) sin ap] a2 + x2 n k−1  1  (2n − 2k + 1)! −a2 p2 + 2n p k=1

[p > 0] 

∞

2. 0

BI (91)(12)

2n −px

x e 1 dx = (−1)n a2n−1 [ci(ap) sin ap − si(ap) cos ap] + 2n−1 2 2 a +x p [p > 0]



∞

3. 0

  x2n+1 e−px 1 1 dx = a2n eap Ei(−ap) + e−ap Ei(ap) − 2n 2 2 a −x 2 p

4. 0

∞

k−1  (2n − 2k)! −a2 p2

k=1

BI (91)(11)

n 

k−1  (2n − 2k + 1)! a2 p2

k=1

[p > 0] 

n 

  x2n e−px 1 1 dx = a2n−1 e−ap Ei(ap) − eap Ei(−ap) − 2n−1 2 2 a −x 2 p [p > 0]

BI (91)(17) n 

k−1  (2n − 2k)! a2 p2

k=1

BI (91)(16)

3.358

Exponentials and rational functions

3.357  1.

∞

a3

0

 0



a3

∞

a3

0

 6. 0

∞

0



0

a > 0]

 x e dx 1  −ap e = Ei(ap) − eap Ei(−ap) − 2 ci(ap) sin ap + 2 si(ap) cos ap 4 4 a −x 4a

[p > 0,

BI (92)(23)

a > 0]

BI (91)(18)

BI (91)(19)

2 −px

[p > 0, ∞

BI (92)(22)

 e−px 1  dx = 3 e−ap Ei(ap) − eap Ei(−ap) + 2 ci(ap) sin ap − 2 si(ap) cos ap 4 4 a −x 4a

∞

0

4.

a > 0]

 xe−px dx 1  = 2 eap Ei(−ap) + e−ap Ei(ap) − 2 ci(ap) cos ap − 2 si(ap) sin ap 4 4 a −x 4a

3. 

a > 0]

∞

0

BI (92)(21)

x e dx 1 = {ci(aμ) (cos aμ − sin aμ) a3 − a2 x + ax2 − x3 2  + si(aμ) (cos aμ + sin aμ) + e−aμ Ei(aμ)

[p > 0,

2. 

a > 0]

2 −μx

[Re μ > 0, 3.358  1.

BI (92)(20)

xe−μx dx 1 {− ci(aμ) (sin aμ + cos aμ) = − a2 x + ax2 − x3 2a  − si(aμ) (sin aμ − cos aμ) + e−aμ Ei(aμ) [Re μ > 0,

∞

a > 0]

e−μx dx 1 = 2 {ci(aμ) (sin aμ − cos aμ) 2 2 3 − a x + ax − x 2a  − si(aμ) (sin aμ + cos aμ) + e−aμ Ei(aμ) [Re μ > 0,

5.

BI (92)(19)

x e dx 1 = {− ci(aμ) (sin aμ + cos aμ) 2 2 3 + a x + ax + x 2 − si(aμ) (sin aμ − cos aμ) − eaμ Ei(−aμ)} [Re μ > 0,

∞

0

a > 0]

2 −μx

a3

4.

BI (92)(18)

xe dx 1 {ci(aμ) (sin aμ − cos aμ) = a3 + a2 x + ax2 + x3 2a − si(aμ) (sin aμ + cos aμ) − eaμ Ei(−aμ)} [Re μ > 0,

0

a > 0]

−μx

∞

3.



e−μx dx 1 = 2 {ci(aμ) (sin aμ + cos aμ) + a2 x + ax2 + x3 2a + si(aμ) (sin aμ − cos aμ) − eaμ Ei(−aμ)} [Re μ > 0,

∞

2.



343

a > 0]

BI (91)(20)

 x3 e−px dx 1  ap e Ei(−ap) + e−ap Ei(ap) + 2 ci(ap) cos ap + 2 si(ap) sin ap = 4 4 a −x 4 [p > 0,

a > 0]

BI (91)(21)

344

Exponential Functions



∞

5. 0

3.359

  x4n e−px 1 dx = a4n−3 e−ap Ei(ap) − eap Ei(−ap) + 2 ci(ap) sin ap − 2 si(ap) cos ap 4 4 a −x 4 n k−1  1  (4n − 4k)! a4 p4 − 4n−3 p k=1



[p > 0, ∞

6. 0

a > 0]

BI (91)(22)

  e 1 dx = a4n−2 eap Ei(−ap) + e−ap Ei(ap) − 2 ci(ap) cos ap − 2 si(ap) sin ap a4 − x4 4 n k−1  1  (4n − 4k + 1)! a4 p4 − 4n−2 p

4n+1 −px

x

k=1



[p > 0, ∞

7. 0

a > 0]

BI (91)(23)

  x4n+2 e−px 1 dx = a4n−1 e−ap Ei(ap) − eap Ei(−ap) − 2 ci(ap) sin ap + 2 si(ap) cos ap 4 4 a −x 4 n k−1  1  (4n − 4k + 2)! a4 p4 − 4n−1 p k=1



[p > 0, ∞

8. 0

a > 0]

BI (91)(24)

  x4n+3 e−px 1 dx = a4n eap Ei(−ap) + e−ap Ei(ap) + 2 ci(ap) cos ap + 2 si(ap) sin ap 4 4 a −x 4 n k−1  1  (4n − 4k + 3)! a4 p4 − 4n p k=1

[p > 0,  3.359

a > 0]

BI (91)(25)

∞

(i − x)n e−ipx dx = (−1)n−1 2πpe−p Ln−1 (2p) n 2 −∞ (i + x) i + x =0

for p > 0; for p < 0. ET I 118(2)

3.36–3.37 Combinations of exponentials and algebraic functions 3.361 1.8 2.8 3.8 3.362



u −qx



π √ Φ ( qu) q 0   ∞ −qx e π √ dx = q x 0   ∞ −qx e π q √ dx = e q 1 + x −1 

1. 1

 2.

0

e √ dx = x

∞ −μx

e dx √ = x−1

∞ −μx

e dx √ = x+β

 

[q > 0] [q > 0]

BI(98)(10)

[q > 0]

BI (104)(16)

π −μ e μ

[Re μ > 0]

 , π βμ + 1−Φ e βμ μ

[Re μ > 0,

BI (104)(11)a

|arg β| < π]

ET I 135(18)

3.371

3.363

Exponentials and algebraic functions



∞√

x − u −μx e dx = x

1. u



∞

u



2

1. 0

 2. 3. 3.365

e−μx dx π √ √ = √ [1 − Φ ( uμ)] u x x−u

e−px dx  = πe−p I 0 (p) x(2 − x)

e2x dx √ = π I 0 (2) 2 −1 1 − x  ∞ −px  ap ap e dx  = e 2 K0 2 x(x + a) 0 

u

0



∞

u

2u

1. 0



∞

2. 0



Re μ ≥ 0]

[p > 0]

ET I 136(26)

GW (312)(7a)

BI (277)(2)a

[a > 0,

p > 0]

[u > 0,

Re μ > 0]

ET I 136(28)

xe−μx dx √ = u K 1 (uμ) x2 − u2

[u > 0,

Re μ > 0]

ET I 136(29)

(u − x)e−μx dx √ = πue−uμ I 1 (uμ) 2ux − x2

[Re μ > 0]

(x + β)e−μx dx  = βeβμ K 1 (βμ) x2 + 2βx

[Re μ > 0,

xe−μx dx πu √ [L1 (μu) − I 1 (μu)] + u = 2 2 2 u −x

2.



[u > 0,

ET I 136(23)

1

1.

3.366

√ π −uμ √ e − π u [1 − Φ ( uμ)] μ [Re μ > 0]

2. 3.364



345

GW (312)(8a)

ET I 136(31)

|arg β| < π]

, + xe−μx dx βπ π  [H1 (βμ) − Y 1 (βμ)] − β = |arg β| < , Re μ > 0 2 2 x2 + β 2 0  

 ∞  u exp 2μ cos2 2t e−μx dx −v cos t √ 3.367 K 0 (v)e dv = t − sin t 2 sin t 0 (1 + cos t + x) x + 2x 0 [Re μ > 0]  ∞ e−μx dx π 1  [H1 (βμ) − Y 1 (βμ)] − 2 2 3.368 = 2βμ β μ x2 + β 2 0 x+ + , π |arg β| < , Re μ > 0 2  ∞ −μx e dx 2 √ √ aμ 11  3.369 = √ − 2 πμe (1 − Φ ( aμ)) [|arg a| < π, Re μ > 0] a (x + a)3 0  ∞ √ 1 3 1 2n − 1 −n− 1 2 μ 3.37111 xn− 2 e−μx dx = π · · . . . 2 2 2 0 √ = π2−n μ−n−1/2 (2n − 1)!! [n ≥ 0]

ET I 136(30)

∞

3.

[Re μ > 0]

ET I 136(27)

ET I 136(33)

ET I 136(32) ET I 135(20)

ET I 135(17)

346

Exponential Functions



3.372

∞

1 1 (2n − 1)!! p xn− 2 (2 + x)n− 2 e−px dx = e K n (p) [p > 0, n = 0, 1, 2, . . .] pn 0 ∞ + n  n ,   x + x2 + β 2 + x − x2 + β 2 e−μx dx = 2β n+1 O n (βμ)

3.372 3.373

GW (312)(8)

0

3.374 1.

2.

[Re μ > 0]

WA 05(1)

√  n x + 1 + x2 1 √ e−μx dx = [S n (μ) − π En (μ) − π Y n (μ)] 2 2 1+x 0 [Re μ > 0] √ n  ∞ x − 1 + x2 1 √ e−μx dx = − [S n (μ) + π En (μ) + π Y n (μ)] 2 2 1+x 0 [Re μ > 0]

ET I 37(35)



∞

ET I 137(36)

3.38–3.39 Combinations of exponentials and arbitrary powers 3.381  u 1. xν−1 e−μx dx = μ−ν γ(ν, μu)

[Re ν > 0]

EH I 266(22), EH II 133(1)

0



u

2.

p−1 −x

x

e

dx =

0

∞ 

(−1)k

k=0

= e−u

∞  k=0



up+k k!(p + k)

up+k p(p + 1) . . . (p + k) AD 6.705

∞

3.8

xν−1 e−μx dx = μ−ν Γ(ν, μu)

[u > 0,

Re μ > 0]

u



EH I 256(21), EH II 133(2) ∞

4. 0



∞

5. 0



xν−1 e−μx dx =

u  ∞

7.

ν u e dx = u− 2 e− 2 W − ν , (1−ν) (u) ν 2 2 x

xk−1 eiμx dx =



FI II 779

0 < Re ν < 1]

EH I 12(32)

[u > 0]

Γ(k) (−iμ)k

WH

[0 < Re(k) < 1,

μ = 0] GH2 62 (313.14)

u

xm e−βx dx = n

0

9.∗

Re ν > 0]



− ν  q xν−1 e−(p+iq)x dx = Γ(ν) p2 + q 2 2 exp −iν arctan p [p > 0, Re ν > 0 and p = 0,

0

8.∗

[Re μ > 0,

∞ −x

6.



1 Γ(ν) μν

∞

u

xm e−βx dx = n

n

γ (v, βu ) nβ v Γ (v, βun ) nβ v

v= v=

m+1 n m+1 n

[u > 0, [u > 0,

Re v > 0, Re v > 0,

Re n > 0, Re n > 0,

Re β > 0] Re β > 0]

3.383

10.

∗

Exponentials and arbitrary powers



∞

xm e−βx dx = n

0

11.∗

γ (v, βun ) + Γ (v, βun ) nβ v

v=   ∞ 2m −βx2n x e dx = 2 −∞

347

m+1 n ∞

[u > 0,

x2m e−βx

2n

Re v > 0,

dx =

0

Re β > 0]

See also 3.326 1

2 (γ (v, βun ) + Γ (v, βun )) Γ(v) = nβ v nβ v

2m + 1 2n

v=

Re n > 0,

[u > 0,

3.382  u (u − x)ν e−μx dx = (−μ)−ν−1 e−uμ γ(ν + 1, −uμ) 1.6

Re v > 0,

[Re ν > −1,

Re n > 0,

u > 0]

Re β > 0]

ET I 137(6)

0



∞

2.

(x − u)ν e−μx dx = μ−ν−1 e−uμ Γ(ν + 1)

[u > 0,

Re ν > −1,

Re μ > 0]

u



ET I 137(5), ET II 202(11) ∞

3.

[Re μ > 0]

(x + β)ν e−μx dx = μ−ν−1 eβμ Γ(ν + 1, βμ)

[|arg β| < π,

ν

2

0



μ

(1 + x)−ν e−μx dx = μ 2 −1 e 2 W − ν , (1−ν) (μ)

∞

4.

WH

2

Re μ > 0]

0



ET I 137(4), ET II 233(10) u

5.

(a + x)μ−1 e−x dx = ea [γ(μ, a + u) − γ(μ, a)]

[Re μ > 0]

EH II 139

0



∞

6.

(β + ix)−ν e−ipx dx = 0

−∞

= 

[for p > 0] ν−1 βp

e 2π(−p) Γ(ν)

[for p < 0] [Re ν > 0,

∞

7. −∞

(β − ix)−ν e−ipx dx =

Re β > 0]

ET I 118(4)

Re β > 0]

ET I 118(3)

[Re μ > 0, Re ν > 0]



βu βu exp Γ(μ) I μ− 12 2 2

ET II 187(14)

ν−1 −βp

e 2πp Γ(ν)

[for p > 0]

=0

[for p < 0] [Re ν > 0,

3.383  u xν−1 (u − x)μ−1 eβx dx = B(μ, ν)uμ+ν−1 1 F 1 (ν; μ + ν; βu) 1.11 0



u

xμ−1 (u − x)μ−1 eβx dx =

2.11 0

√

π



u β

μ− 12

[Re μ > 0] 1



μ− ∞ 2 βu u 1 βu xμ−1 (x − u)μ−1 e−βx dx = √ Γ(μ) exp − K μ− 12 β 2 2 π u

ET II 187(13)

 3.

[Re μ > 0,

Re βu > 0]

ET II 202(12)

348

Exponential Functions

 4.

∞

11

ν−1

x

μ−1 −βx

(x − u)

e

dx = β

− μ+ν 2

u

μ+ν−2 2

u

 5.11

6.

3.384



βu Γ(μ) exp − W 2

ν−μ 1−μ−ν 2 , 2

[Re μ > 0,

(βu)

Re βu > 0]

ET II 202(13)

∞

e−px xq−1 (1 + ax)−ν dx 0 % & p q−ν  p   p ν  p q L−q Lν−q π2 −ν a a − [ν = ±1, ±2, . . .] = q p Γ(ν) sin [π(q − ν)] a sin(πν) Γ(1 − q) a sin(πq) Γ(1 − ν) Γ(q) = q [ν = 0] p [Re q > 0, Re p > 0, Re a > 0]  ∞  βμ 1 1 1 xν−1 (x + β)−ν+ 2 e−μx dx = 2ν− 2 Γ(ν)μ− 2 e 2 D 1−2ν 2βμ 0

[|arg β| < π, 

∞

7.

Re ν > 0,

xν−1 (x + β)−ν− 2 e−μx dx = 2ν Γ(ν)β − 2 e 1

1

βμ 2

Re μ ≥ 0,

D −2ν



0

μ = 0]

ET I 39(20), EH II 119(2)a

2βμ

[|arg β| < π,

Re ν > 0,

Re μ ≥ 0]

ET I 139(21), EH II 119(1)a



∞

8.

ν−1

x 0



(x + β)

e

1 dx = √ π



ν− 12 βμ βμ β e 2 Γ(ν) K 12 −ν μ 2 [|arg β| < π,

Re μ > 0,

Re ν > 0]

ET II 233(11), EH II 19(16)a, EH II 82(22)a ∞

9. u



ν−1 −μx

(x − u)ν e−μx dx = uν Γ(ν + 1) Γ(−ν, uμ) x

[u > 0,

Re ν > −1,

Re μ > 0] ET I 138(8)

∞

10. 0

xν−1 e−μx dx = β ν−1 eβμ Γ(ν) Γ(1 − ν, βμ) x+β

[|arg β| < π,

Re μ > 0,

Re ν > 0] EH II 137(3)

3.384



1

1. −1



(1 − x)ν−1 (1 + x)μ−1 e−ipx dx = 2μ+ν−1 B(μ, ν)eip 1 F 1 (μ; ν + μ; −2ip) [Re ν > 0,

v

2.

(x − u)2μ−1 (v − x)2ν−1 e−px dx

u

ET I 119(13)



u+v p exp −p M μ−ν,μ+ν− 12 (vp − up) 2 [v > u > 0, Re μ > 0, Re ν > 0] ET I 139(23)

μ+ν−1 −μ−ν

= B(2μ, 2ν)(v − u)

Re μ > 0]

3.385

Exponentials and arbitrary powers



∞

3.

(x + β)2ν−1 (x − u)2−1 e−μx dx

  (β − u)μ (u + β)ν+−1 = exp Γ(2) W ν−,ν+− 12 (uμ + βμ) μν+ 2 [u > 0, |arg(β + u)| < π, Re μ > 0, Re  > 0] ET I 139(22)

u



∞

4.

∞

5. u



(x + β) (x − u)

1 1 1 √ (x − u)ν−1 (x + u)−ν+ 2 e−μx dx = √ 2ν− 2 Γ(ν) D 1−2ν (2 uμ) μ [u > 0, Re μ > 0,

u

1 1 1 √ (x − u)ν−1 (x + u)−ν− 2 e−μx dx = √ 2ν− 2 Γ(ν) D −2ν (2 uμ) u [u > 0, Re μ ≥ 0,

∞

−μ

(β − ix)

−∞

(γ − ix)−ν e−ipx dx =

2πe−βp pμ+ν−1 1 F 1 (ν; μ + ν; (β − γ)p) Γ(μ + ν)

=0

∞

8.6

=

∞

6 −∞

Re γ > 0,

[for p > 0]

Re(μ + ν) > 1]

(β + ix)−μ (γ + ix)−ν e−ipx dx = 0

−∞

9.

Re ν > 0]

[for p < 0] [Re β > 0,



Re ν > 0]

ET I 139(19)

7.6



ET I 139(17)

ET I 139(18) ∞

6.



  (β + u)μ 1 − (β+u)μ 2 e dx = νπ cosec(νπ)e k2ν μ 2 [ν = 0, u > 0, |arg(u + β)| < π, Re μ > 0, Re ν < 1]

−ν −μx

ν

u



349

ET I 119(10)

[for p > 0]

2πeγp (−p)μ+ν−1 [for p < 0] 1 F 1 [μ; μ + ν; (β − γ)p] Γ(μ + ν) [Re β > 0, Re γ > 0, Re(μ + ν) > 1] ET I 19(11)

(β + ix)−2μ (γ − ix)−2ν e−ipx dx



β−γ pμ+ν−1 exp p W ν−μ, 12 −ν−μ (βp + γp) [for p > 0] Γ(2ν) 2

β−γ (−p)μ+ν−1 exp p W μ−ν, 12 −ν−μ (−βp − γp) [for p < 0] = 2π(β + γ)−μ−ν Γ(2μ) 2   Re β > 0, Re γ > 0, Re(μ + ν) > 12 ET I 19(12) = 2π(β + γ)−μ−ν

 11

3.385

1

xν−1 (1 − x)λ−1 (1 − βx)− e−μx dx = B(ν, λ)Φ1 (ν, , λ + ν, −μ, β)

0

[Re λ > 0,

Re ν > 0,

|arg(1 − β)| < π]

ET I 39(24)

350

Exponential Functions

3.386

3.386 

∞

(ix)ν0

n -

νk

(βk + ix)

e−ipx dx = 2πe−β0 p β0ν0

k=1

1. −∞

β0 − ix

%

Re ν0 > −1,

n 

Re βk > 0,

n -



∞

n -

νk

(βk + ix)

Re νk < 1,

π arg ix = sign x, 2

Re νk < 1,

π arg ix = sign x, 2

=0

β0 + ix

%

−∞

Re ν0 > −1,

Re βk > 0,

n  k=0

3.387



1

1.6



−1

 2.

1

6



−1

1 − x2

ν−1

e−μx dx =

 2 ν−1 iμx

1−x

e

√

π

√ dx = π

ν− 12 2 Γ(ν) I ν− 12 (μ) μ

+ Re ν > 0,



∞

1



∞

4.

ET I 118(8)

ν− 12 2 Γ(ν) J ν− 12 (μ) μ

 2 ν−1 −μx 1 x −1 e dx = √ π

ν− 12 2 Γ(ν) K ν− 12 (μ) μ

& p>0

ET I 119(9)

π, 2

WA 172(2)a

|arg μ| <

[Re ν > 0] 3.

p>0

e−ipx dx

k=1

2.

&

k=1

k=0

(ix)ν0

νk

(β0 + βk )

+ π |arg μ| < , 2

WA 25(3), WA 48(4)a

, Re ν > 0

WA 190(4)a

 2 ν−1 iμx x −1 e dx

1

√ ν− 12 π 2 (1) [Im μ > 0, Re ν > 0] EH II 83(28)a Γ(ν) H 1 −ν (μ) 2 2 μ

ν− 12 √ π 2 (2) [Im μ < 0, Re ν > 0] EH II 83(29)a = −i Γ(ν) H 1 −ν (−μ) − 2 2 μ √ ν− 12  u + ,  2  π 2u 2 ν−1 μx u −x e dx = Γ(ν) I ν− 12 (uμ) + Lν− 12 (uμ) 2 μ 0 =i

5.

[u > 0, 

∞

6. u

 2 ν−1 −μx 1 x − u2 e dx = √ π



2u μ

Re ν > 0]

ET II 188(20)a

ν− 12 Γ(ν) K ν− 12 (uμ) [u > 0,

Re μ > 0,

Re ν > 0] ET II 203(17)a

3.389

Exponentials and arbitrary powers

 7.

∞

 2 ν−1 −μx x + u2 e dx =

11 0

351

√ ν− 12 + , π 2u Γ(ν) Hν− 12 (uμ) − Y ν− 12 (uμ) 2 μ [|arg u| < π,

3.388



2u

1.

 ν−1 −μx √ 2ux − x2 e dx = π

0



2u μ

ν− 12



∞

0

 ν−1 −μx 1 2βx + x2 e dx = √ π



2β μ

ν− 12

ET I 138(10)

e−uμ Γ(ν) I ν− 12 (uμ) [u > 0,

2.

Re μ > 0]

Re ν > 0]

ET I 138(14)

eβμ Γ(ν) K ν− 12 (βμ) [|arg β| < π,

Re ν > 0,

Re μ > 0] ET I 138(13)



∞

3. 0



∞

4.

√

μ  2 ν−1 −μx i πe (2) x + ix e dx = − ν− 1 Γ(ν) H ν− 1 2 2 2μ 2  2 ν−1 −μx x − ix e dx =

0

3.389 1.

2.7

iμ 2

√ iμ i πe− 2 ν− 12

2μ

(1)

Γ(ν) H ν− 1

[Re μ > 0,

Re ν > 0]

ET I 138(15)

[Re μ > 0,

Re ν > 0]

ET I 138(16)

μ

2

2



 −1 μx μ2 u2 1 1 x2ν−1 u2 − x2 e dx = B(ν, )u2ν+2−2 1 F 2 ν; , ν + ; 2 2 4 0

1 3 1 μ + B ν + ,  u2ν+2−1 1 F 2 ν + ; , ν +  + 2 2 2 2 [Re  > 0, Re ν > 0] !

 ∞  2  μ2 u2 !! 1 − ν u2ν+2−2 2ν−1 2 −1 −μx 31 √ u +x x e dx = G 2 π Γ(1 − ) 13 +4 ! 1 −  − ν, 0, 12 0 π |arg u| < , Re μ > 0, 2



u

√  1 −ν u  + , ν−1 μx 1 π μ 2 u2ν + x u2 − x2 e dx = uν+ 2 Γ(ν) I ν+ 12 (μu) + Lν+ 12 (μu) 2ν 2 2 0

 3.7



[Re ν > 0] ∞

4. u

 5.



ET II 188(21)

, Re ν > 0 ET II 234(15)a

ET II 188(19)a

ν−1 −μx  1 √ −1 1 1 π x x2 − u2 e dx = 2ν− 2 μ 2 −ν uν+ 2 Γ(ν) K ν+ 12 (uμ) [Re(uμ) > 0]

∞

1 μ2 u2 ; 2 4

ET II 203(16)a

−ν −ipx

(ix) e dx = πβ −ν−1 e−|p|β 2 + x2 β −∞

+ |ν| < 1,

Re β > 0,

arg ix =

, π sign x 2

ET I 118(5)

352

Exponential Functions



∞

6. 0



∞

7. 0

 8.



xν e−μx 1 (ν − 1)π ν−1 dx = Γ(ν)β exp iμβ + i β 2 + x2 2 2

 (ν − 1)π × Γ(1 − ν, iβμ) + exp −iβμ − i Γ (1 − ν, −iβμ) 2 [Re β > 0, Re μ > 0, Re ν > −1] xν−1 e−μx dx = π cosec(νπ)Vν (2μ, 0) 1 + x2

(β + ix)−ν e−ipx π dx = (β + γ)−ν e−pγ 2 + x2 γ γ −∞ [Re ν > −1,

Re ν > 0]

ET I 138(9)

p > 0,

Re β > 0,

Re γ > 0]

ET I 118(6)

Re ν > −1]

ET I 118(7)

∞

(β − ix)−ν e−ipx π dx = (β + γ)−ν eγp 2 + x2 γ γ −∞ [p < 0, 

∞

3.391

 

0

3.392

[Re μ > 0,

ET II 218(22)

∞

 9.6

3.391



∞

1. 0

 2. 0

x + 2β +

Re β > 0,

Re γ > 0,

 √ 2ν  √ 2ν −μx ν x − x + 2β − x dx = 2ν+1 β ν eβμ K ν (βμ) e μ ET I 140(30) [|arg β| < π, Re μ > 0]

 ν  1 ν x + 1 + x2 e−μx dx = S 1,ν (μ) + S 0,ν (μ) μ μ [Re μ > 0]

∞



ET I 140(25)

ν 1 ν 1 + x2 − x e−μx dx = S 1,ν (μ) − S 0,ν (μ) μ μ

[Re μ > 0] ET I 140(26) √   ν ∞ x + 1 + x2 √ 3. e−μx dx = π cosec νπ [J−ν (μ) − J −ν (μ)] 1 + x2 0 ET I 140(27), EH II 35(33) [Re μ > 0] √  ν  ∞ 1 + x2 − x √ 4. e−μx dx = S 0,ν (μ) − ν S −1,ν (μ) [Re μ > 0] ET I 140(28) 1 + x2 0  2ν   ∞ x + x2 + 4β 2  e−μx dx 3.393 x3 + 4β 2 x 0   μπ 3  = 2ν+3/2 2ν J ν+1/4 (βμ) Y ν−1/4 (βμ) − J ν−1/4 (βμ) Y ν+1/4 (βμ) 2 β ET I 140(33) [Re β > 0, Re μ > 0] √ ν+1/2  ∞   2   √ 1+ 1+x √ −2iμ 3.394 e−μx dx = 2 Γ(−ν) D ν 2iμ D ν xν+1 1 + x2 0 [Re μ ≥ 0, Re ν < 0] ET I 140(32) 

3.411

3.395 1.

2.

Rational functions of powers and exponentials

353

√

ν √ −ν x2 − 1 + x x2 − 1 + x + √ e−μx dx = 2 K ν (μ) 2−1 x 1 [Re μ > 0] √ √      2ν 2ν  ∞ μ μ x + x2 − 1 + x − x2 − 1 2μ  K ν+1/4 K ν−1/4 e−μx dx = π 2 2 x (x2 − 1) 1 

∞

[Re μ > 0] √ √     ν −ν  ∞ x + x2 + 1 + cos νπ x + x2 + 1 √ e−μx dx = −π [Eν (μ) + Y ν (μ)] 2+1 x 0 [Re μ > 0]

3.

ET I 140(29)

ET I 140(34)

EH II 35(34)

3.41–3.44 Combinations of rational functions of powers and exponentials 3.411 1. 2. 3. 4. 5. 6.8



∞

1 xν−1 dx = ν Γ(ν) ζ(ν) [Re μ > 0, Re ν > 1] FI II 792a μx e −1 μ 0 2n  ∞ 2n−1 2π x dx B2n n−1 = (−1) [n = 1, 2, . . .] FI II 721a px e −1 p 4n 0  ∞ ν−1  1  x dx = ν 1 − 21−ν Γ(ν) ζ(ν) [Re μ > 0, Re ν > 0] FI II 792a, WH μx + 1 e μ 0

 ∞ 2n−1  2π 2n |B2n | x dx  1−2n = 1−2 [n = 1, 2, . . .] BI(83)(2), EH I 39(25) epx + 1 p 4n 0  ln 2 x dx π2 = BI (104)(5) −x 1−e 12 0  ∞ ν−1 −μx ∞  x e dx = Γ(ν) (μ + n)−ν β n = Γ(ν) Φ(β, ν, μ) 1 − βe−x 0 n=0 [Re μ > 0 and either |β| ≤ 1,

 7.

∞

11 0



xν−1 e−μx 1 μ dx = ν Γ(ν) ζ ν, 1 − e−βx β β

β = 1,

Re ν > 0; or β = 1, [Re β > 0,

Re ν > 1]

Re μ > 0,

EH I 27(3)

Re ν > 1] ET I 144(10)



∞

8. 0



∞

9. 0

 10.

0

∞

xn−1 e−px dx = (n − 1)! 1 + ex

∞  k=1

(−1)k−1 (p + k)n

[p > −1;

n = 1, 2, . . .]

BI (83)(9)

π2 xe−x dx = −1 ex − 1 6

(cf. 4.231 3)

BI (82)(1)

π2 xe−2x dx = 1 − e−x + 1 12

(cf. 4.251 6)

BI (82)(2)

354

Exponential Functions



∞

11. 0



∞

12.11 0

 13.11

14.7

xe−3x π2 3 dx = − −x e +1 12 4 2n−1  (−1)k−1 xe−(2n−1)x π2 + dx = − 1 + ex 12 k2

∞

∞

15.7 0

17.

 18.

(cf. 4.251 6)

BI (82)(5)

(cf. 4.251 5)

BI (82)(4)

x2 e−nx dx = 2 1 + e−x

∞  k=n

[n = 1, 2, . . .]

(cf. 4.261 12)

k=1

(−1)n+k = (−1)n+1 k3



BI (82)(9)



n−1 

(−1)k 3 ζ(3) + 2 2 k3 k=1 [n = 1, 2, . . .]

(cf. 4.261 11) LI (82)(10)

∞

  x e dx = π 3 csc3 μπ 2 − sin2 μπ −x −∞ 1 + e  ∞ 3 −nx n−1  1 x e π4 − 6 dx = −x 15 k4 0 1−e

16.

BI (82)(3)

2n

k=n



(cf. 4.251 5)

k=1

xe−2nx π 2  (−1)k + dx = 1 + ex 12 k2 0 k=1    ∞ 2 −nx ∞ n−1   1 x e 1 dx = 2 = 2 ζ(3) − −x k3 k3 0 1−e 

3.411

2 −μx

[0 < Re μ < 1] (cf. 4.262 5)

k=1

∞

11 0

∞

 (−1)n+k x3 e−nx dx = 6 = (−1)n+1 −x 1+e k4 k=n

ET I 120(17)a



 (−1)k 7 4 π +6 120 k4 n−1

BI (82)(12)



k=1

(cf. 4.262 4) 

∞

19.9 0



∞

20.9 0



 n dx =− e−px e−x − 1 x

(−1)k

n

k=0

k

ln(p + n − k)

∞

∞

0

 1 1 − e−mx dx = (n − 1)! 1 − ex kn m

xn−1

(cf. 4.272 11)

LI (83)(8)

∞

 qk 1 xp−1 dx = p Γ(p) = Γ(p)r−p Φ(q, p, 1) rx e −q qr kp k=1

[p > 0,

−1 < q < 1]

r > 0,

BI (83)(5) ∞

μx

xe dx = πβ μ−1 cosec(μπ) [ln β − π cot(μπ)] x −∞ β + e



LI (89)(15)

k=1

22.7

23.

LI (89)(10)

n   n dx  k n (p + n − k) ln(p + n − k) e−px e−x − 1 = (−1) k x2

0



LI (82)(13)

k=0

21. 

n 

[|arg β| < π,

0 < Re μ < 1] BI (101)(5), ET I 120(16)a

∞

24. −∞

xeμx dx = −1

eνx

π ν

cosec

μπ 2 ν

[Re ν > Re μ > 0]

(cf. 4.254 2) LI (101)(3)

3.414

Rational functions of powers and exponentials



∞

25.

x 0

 26. 27.

28. 29.

(cf. 4.231 4)

BI (82)(6)

∞

1 − e−x −x 2π 2 e dx = −3x 1+e 27 0 %  &   ∞ Γ μ2 + 1 √ 1 − e−μx dx  π  = ln 1 + ex x Γ μ+1 0 2      ∞ −νx Γ ν2 Γ μ+1 e − e−μx dx 2 = ln  μ   ν+1  e−x + 1 x Γ 2 Γ 2 0  ∞ px + qx pπ qπ , e − e dx = ln cot tan rx x 2r 2r −∞ 1 + e x

 30.

π2 1 + e−x dx = −1 ex − 1 3

355

LI (82)(7)a

[Re μ > −1] [Re μ > 0, [|r| > |p|,

BI (93)(4)

Re ν > 0] |r| > |q|,

BI (93)(6)

rp > 0,

rq > 0] BI (103)(3)

∞

+

qπ ,

pπ e − e dx = ln sin cosec rx 1 − e x r r −∞ px

qx

[|r| > |p|,

|r| > |q|,

rp > 0,

rq > 0] BI (103)(4)



∞ −qx

e

31. 0

 32. 3.412 3.413

+ e(q−p)x x dx = 1 − e−px



π qπ cosec p p

2 [0 < q < p]

∞ −px

pπ − e(p−q)x dx = ln cot [0 < p < q] −qx + 1 e x 2q 0   ∞ a + be−px dx a+b p a + be−qx = ln − qx px −px −qx ce + g + he ce + g + he x c+g+h q 0 [p > 0, q > 0]



BI (82)(8)

e

∞

1. 0

  1 − e−βx (1 − e−γx ) e−μx dx Γ(μ) Γ (β + γ + μ) = ln 1 − e−x x Γ(μ + β) Γ(μ + γ) [Re μ > 0, Re μ > − Re β, Re μ > − Re γ, Re μ > − Re(β + γ)]

BI (93)(7)

BI (96)(7)

(cf. 4.267 25) BI (93)(13)

 ∞

1−e qπ dx = ln cosec qx (q−2p)x x 2p e −e

2. 0



e

0

3.414 0

[0 < q < p]

BI (95)(6)

∞ −px

3.



 (q−p)x 2

∞

− e−qx 1 + e−(2n+1)x dx 1 + e−x x

 q(q + 2)(q + 4) · · · (q + 2n)(p + 1)(p + 3) · · · (p + 2n − 1) = ln p(p + 2)(p + 4) · · · (p + 2n)(q + 1)(q + 3) · · · (q + 2n − 1) [Re p > −2n, Re q > −2n] (cf. 4.267 14) BI (93)(11)

    1 − e−βx (1 − e−γx ) 1 − e−δx e−μx dx Γ(μ) Γ(μ + β + γ) Γ(μ + β + δ) Γ(μ + γ + δ) = ln 1 − e−x x Γ(μ + β) Γ(μ + γ) Γ(μ + δ) Γ(μ + β + γ + δ) [2 Re μ > |Re β| + |Re γ|+| Re δ|] (cf. 4.267 31) BI (93)(14), ET I 145(17)

356

3.415

Exponential Functions



∞

1. 0



3.415

 

βμ 1 βμ x dx π = − ψ ln − (x2 + β 2 ) (eμx − 1) 2 2π βμ 2π [Re β > 0,

Re μ > 0] BI (97)(20), EH I 18(27)

∞

2.11

x dx (x2

0

+

2 β2)

(e2πx

1 1 1  ψ (β) − 2+ 3 8β 4β 4β − 1) ∞ 1  |B2k+2 | ∼ 4β 4 β 2k =−

k=0

[asymptotic expansion for Re β > 0] 

∞

3.11 0



∞

4.8 0

3.416



∞

1. 0



∞

2. 0

 3.

∞

8 0

BI(97)(22), EH I 22(12)



 1 βμ 1 βμ x dx = + ψ − ln 2 2 μx (x + β ) (e + 1) 2 2π 2 2π

1 x dx 1  1 = ψ β+ − 2 4β 2 4β 2 (x2 + β 2 ) (e2πx + 1)

[Re β > 0,

Re μ > 0]

[Re β > 0,

Re μ > 0]

1 2n − 1 dx (1 + ix)2n − (1 − ix)2n = i e2πx − 1 2 2n + 1

[n = 1, 2, . . .]

BI (88)(4)

1 (1 + ix)2n − (1 − ix)2n dx = πx i e +1 2n + 1

[n = 1, 2, . . .]

BI (87)(1)

, 1 + (1 + ix)2n−1 − (1 − ix)2n−1 dx 2n = 1 − 2 B 2n i eπx + 1 2n [n = 1, 2, . . .]

3.417 1. 2. 3.418 1.6 2.6 3.



∞

b x dx π ln = 2 ex + b2 e−x a 2ab a −∞  ∞ x dx π2 = 2 x 2 −x 4ab −∞ a e − b e 

1 1 x dx = ψ − x + e−x − 1 e 3 3 0 

 ∞ 1 1 xe−x dx = ψ − x + e−x − 1 e 6 3 0  ln 2 π x dx = ln 2 x + 2e−x − 2 e 8 0 

BI (87)(2)

∞

[ab > 0]

(cf. 4.231 8)

(cf. 4.231 10)

 2 2 π = 1.1719536193 . . . 3  5 2 π = 0.3118211319 . . . 6

BI (101)(1)

LI (101)(2)

LI (88)(1) LI (88)(2)

BI (104)(7)

3.421

3.419

Rational functions of powers and exponentials



∞

2

(ln β) x dx = x −x ) 2(β − 1) −∞ (β + e ) (1 + e

1.

357

[|arg β| < π]

(cf. 4.232 2) BI (101)(16)

 2.

∞

x dx

−∞ (β



2

+

ex ) (1

−

e−x )

=

π + (ln β) 2(β + 1)

2

+

[|arg β| < π]

2

BI (101)(17)

,

π 2 + (ln β) ln β x2 dx = x −x ) 3(β + 1) −∞ (β + e ) (1 − e

3.

∞

(cf. 4.232 3)

[|arg β| < π]

(cf. 4.261 4) BI (102)(6)



+

2

π 2 + (ln β) x3 dx = x −x ) 4(β + 1) −∞ (β + e ) (1 − e

4.

∞

,2 [|arg β| < π]

(cf. 4.262 3) BI (102)(9)



+

2

π 2 + (ln β) x4 dx = x −x ) 15(β + 1) −∞ (β + e ) (1 − e

5.

∞

,2

, + 2 7π 2 + 3 (ln β) ln β (cf. 4.263 1)

 6.

11

2

π 2 + (ln β) x5 dx = x −x ) 6(β + 1) −∞ (β + e ) (1 − e

 7.

+

∞

+

,2

− 4π 2 + (ln β) (x − ln β) x dx = x −x ) 6(β − 1) −∞ (β − e ) (1 − e ∞

BI (102)(10)

, + 2 3π 2 + (ln β)

2

(cf. 4.264 3)

,

BI (102)(11)

ln β [|arg β| < π]

(cf. 4.257 4) BI (102)(7)

3.421



∞

1. 0

 −νx n  −ρx m dx e e −1 − 1 e−μx 2 x =

n  k=0

(−1)k

m n  m (−1)l k l l=0

× {(m − l)ρ + (n − k)ν + μ} ln [(m − l)ρ + (n − k)ν + μ] [Re ν > 0,  2. 0

∞

Re μ > 0,

Re ρ > 0]

BI (89)(17)

n n  n   dx 1  (ρ + kν + 1)2 1 − e−νx 1 − e−ρx e−x 3 = (−1)k k x 2 k=0 n n 1 (kν + 1)2 ln(kν + 1) × ln(ρ + kν + 1) + (−1)k−1 k 2 k=1

[n ≥ 2,

Re ν > 0,

Re ρ > 0]

BI (89)(31)

358

Exponential Functions

3.422

    π β μ−1 ln β − γ μ−1 ln γ π 2 β μ−1 − γ μ−1 cos μπ xe−μx dx = + −x ) (γ + e−x ) (β − γ) sin μπ (γ − β) sin2 μπ −∞ (β + e [|arg β| < π, |arg γ| < π, β = γ. 0 < Re μ < 2]

 3.

∞



∞

4. 0



∞

5. 0

ET I 120(19)

 −px   (p + s + 1)(q + r + 1) dx e = ln − e−qx e−rx − e−sx e−x x (p + r + 1)(q + s + 1) [p + s > −1, p + r > −1, q > p] (cf. 4.267 24)

BI (89)(11)

    dx 1 − e−px 1 − e−qx 1 − e−rx e−x x = (p + q + 1) ln(p + q + 1) +(p + r + 1) ln(p + r + 1) + (q + r + 1) ln(q + r + 1) −(p + 1) ln(p + 1) − (q + 1) ln(q + 1) − (r + 1) ln(r + 1) −(p + q + r) ln(p + q + r) [p > 0,



3.423  1.

(cf. 4.268 3)



∞

2

∞

xν−1 e−μx 2

(ex − 1)

0

dx = Γ(ν) [ζ(ν − 1) − ζ(ν)]

q −px

x e (1 −

0 ∞

4.7

dx

2 ae−px )

xν−1 e−μx

=

∞ Γ(q + 1)  ak

apq+1

k=1

−∞ (β

+

2 ex )

[a < 1,

kq

[Re ν > 0, xex dx

ET I 313(10)

Re ν > 2]

q > −1,

ET I 313(11)

p > 0]

BI (85)(13)

dx = Γ(ν) [Φ(β; ν − 1; μ) − (μ − 1) Φ(β; ν; μ)]

2

∞

BI (102)(8)a

dx = Γ(ν) [ζ(ν − 1, μ + 2) − (μ + 1) ζ(ν, μ + 2)]

(1 − βe−x )

0



[Re ν > 2]

[Re μ > −2, ∞

3.8 

xν−1 (ex − 1)

6



BI (89)(14)

∞

0

5.

r > 0]

−π 2 x(x − a)eμx dx = cosec2 μπ [(eαμ + 1) ln μ − 2π cot μπ (eαμ − 1)] x −x ) ea − 1 −∞ (β − e ) (1 − e [a > 0, |arg β| < π, |Re μ| < 1] (cf. 4.257 5)

3.422

2.

q > 0,

=

1 ln β β

Re μ > 0,

|arg(1 − β)| < π] [|arg β| < π]

(cf. 9.550)

ET I 313(12)

(cf. 4.231 5) BI (101)(10)

6.∗



t

x5 0

e−x (1 −

2 e−x )

∞ 

 e−kt  5 y + 5y 4 + 20y 3 + 60y 2 + 120y + 120 5 k k=1 ∞ 5 −t/2   e−kt  4 t e −5 = 120 ζ(5) − y + 4y 3 + 12y 2 + 24y + 24 2 sinh(t/2) k5 k=1 y = kt

dx = 120 ζ(5) −

3.427

Rational functions of powers and exponentials

3.424  1.7

∞

(1 + a)ex − a (1 − ex )

0



∞

2.

(1 + a)ex + a (1 + ex )

0

 3.

∞

∞

−∞

(a2 ex

1.

7

2



∞

4.

6.

n = 1, 2, . . .]

BI (85)(14)

π2 2ab

[ab > 0]

BI (102)(3)a

dx =

b π ln ab a

[ab > 0]

BI (102)(1)

x2 dx =

2 2 π −2 3

BI (85)(7)

∞

+

BI(101)(13), LI(101)(13)

=−

p+1 e−x )

(ex − ae−x ) x2 dx +

2 ex )

(1 +

p p , ln a B 2 2

1 ap+1

[a > 0,

p > 0]

∞

∞

BI (102)(5)

2

2 e−x )

(ex − ae−x ) x2 dx 2

n > 0]

=

(ln a) a−1

=

π 2 + (ln a) a+1

BI (102)(12) 2

2

+ ex ) (1 − e−x )

BI (102)(13)



e−x e−μx + −x dx = ψ(μ) [Re μ > 0] (cf. 4.281 4) x e −1 0

 ∞ 1 1 − (cf. 4.281 1) e−x dx = C −x 1 − e x 0

 ∞ 1 e−2x 1 1 π − dx = ln −x 2 1+e x 2 4 0

−μx

 ∞ 1 1 1 e 1 1 − + x dx = ln Γ(μ) − μ − ln μ + μ − ln(2π) 2 x e − 1 x 2 2 0 

5.

∞

(a2 ex

−∞ (a

3.427  1.

3.

[a > −1,

dx =

 2 x  a e − e−x x2 dx

−∞ (a



2.7

BI (85)(15)

∞

−∞

2.

2 2x 2 −x b e )

(ex − 1)

2.7

1.

+

n = 1, 2, . . .]

  √

 π Γ n − 12 1 xex dx a −C−ψ n− = 2 ln 2 2 2x n 4a2n−1 b Γ(n) 2b 2 −∞ (a + b e ) [ab > 0,





3.426

−

e − e−x + 2

0

3.425

2 ex

∞  (−1)k (a + k)n

[a > −1,

k=1

2 2x b2 e−x )

∞ x

5.

e−ax xn dx = n!

a2 ex − b2 e−x

4. 

2

e−ax xn dx = n! ζ(n, a)

a2 ex + b2 e−x

−∞ (a



2

359





[Re μ > 0]

1 −2x dx 1 1 e = − ln π − x 2 e +1 x 2 0

−x  ∞ μx e e −1 dx = − ln Γ(μ) − ln sin(πμ) + ln π −μ −x 1 − e x 0 [Re μ < 1]

WH BI (94)(1) BI (94)(5)

WH BI (94)(6)

EH I 21(6)

360

Exponential Functions

7. 8.

10.

∞

(cf. 4.281 5)

BI (94)(3)

[Re μ > 0, n = 1, 2, . . .]

−x −μx e 1−e dx = ln Γ(μ + 1) [Re μ > −1] μ− −x 1 − e x 0

 ∞ Γ(μ + ν + 1) e−μx − e−(μ+ν)x dx = ln νe−x − x e −1 x Γ(μ + 1) 0 

9.



e−νx e−μx − dx = ln μ − ψ(ν) 1 − e−x x 0

 ∞ n e−μx − e−x dx = n ψ(nμ + n) − n ln n x 1 − e−x/n 0 

3.428



BI (94)(4)

∞

∞

11.

[Re μ > −1,

, + −1 (1 − ex ) + x−1 − 1 e−xz dx = ψ(z) − ln z

Re ν > 0]

[Re z > 0]

WH

BI (94)(8) EH I 18(24)

0

3.428  1.

∞

0



∞



1 1 −x 1 e−1 − e−μνx dx −μx = ln Γ(μν) − ν ln μ − e − νe μ μ 1 − e−x x μ

2. 0

n−1 n−1 e(1−μ)x e−nμx + + + −x x/n 2 1−e 1 − e−x 1−e



[Re μ > 0,

e−x

Re ν > 0] BI (94)(18)

n−1 1 dx = ln 2π − nμ + ln n x 2 2

[Re μ > 0, n = 1, 2, . . .]



n−1 ∞  n e(1−μ)x e−x k n−1 − dx = + 1 − ln Γ μ − nμ − 2 1 − e−x x n 1 − ex/n 0 k=0

BI (94)(14)

 3.



∞



4. 0

5.

6.

−νx

−μνx

x

μx

e e e e − − + 1 − ex 1 − eμx 1 − ex 1 − eμx



[Re μ > 0,

n = 1, 2, . . .]

BI (94)(13)

dx = ν ln μ x

[Re μ > 0, Re ν > 0] LI (94)(15) 

1 dx μe−μx μ + 1 −μx − + aμ − + (1 − aμ)e−x e x −μx e −1 1−e 2 x 0

1 μ−1 ln(2π) + − aμ ln μ = 2 2 [Re μ > 0] BI (94)(16)

  ∞  −νx −μνx −μx 1 dx e μ−1 e (μ − 1)e μ − 1 −μ e x = ln(2π) + − μν ln μ − − − 1 − e−x 1 − e−μx 1 − e−μx 2 x 2 2 0 [Re μ > 0, Re ν > 0] (cf. 4.267 37) BI (94)(17) 

∞



∞

7. 0

 3.429 0



  (1 − e−νx ) (1 − e−μx ) dx −x = ln B(μ, ν) 1−e − 1 − e−x x [Re μ > 0,

∞

 −x  dx = ψ(μ) e − (1 + x)−μ x

[Re μ > 0]

Re ν > 0]

BI (94)(12) NH 184(7)

3.437

3.431

Rational functions of powers and exponentials



∞

1. 0



∞

2. 0

361



1 −1 Γ(ν + 3) e−μx − 1 + μx − μ2 x2 xν−1 dx = 2 ν(ν + 1)(ν + 2)μν [Re μ > 0, −2 > Re ν > −3] LI (90)(5)





  −px  1 −2 1 1 −1 −x dx = −1 + p + x − x (x + 2) 1 − e ln 1 + e 2 2 p [Re p > 0]

3.432



∞

1. 0



n n   n 1 xν−1 e−mx e−x − 1 dx = Γ(ν) (−1)k k (n + m − k)ν k=0

[n = 0, 1, . . . , Re ν > 0] ∞

2.

+  ν−1 , xν−1 e−x − e−μx 1 − e−x dx = Γ(ν) −

0

∞

3.433

% xp−1 e−x +

0

n 

(−1)k

k=1

xk−1 (k − 1)!

LI (90)(10)

Γ(μ) Γ(μ + ν) [Re μ > 0,



ET I 144(6)

Re ν > 0]

LI (81)(14)

& dx = Γ(p)

[−n < p < −n + 1,

n = 0, 1, . . .] FI II 805

3.434



∞ −νx

e

1.

xρ+1

0



∞ −μx

e

0



∞

1. 0



dx =

μρ − ν ρ Γ(1 − ρ) ρ

[Re μ > 0,

Re ν > 0,

Re ρ < 1] BI (90)(6)

2. 3.435

− e−μx



ν − e−νx dx = ln x μ

(x + 1)e−x − e− 2

x

[Re μ > 0,

Re ν > 0]

 dx = 1 − ln 2 x

LI (89)(19)

∞

 1 1 − e−μx dx = ln (βμ) + C − eβμ Ei(−βμ) [|arg β| < π, Re μ > 0] β 0 x(x + β)

 ∞ 1 dx −x −e =C 3. 1+x x 0

 ∞ dx a 1 = ln − C [a > 0, Re μ > 0] 4. e−μx − 1 + ax x μ 0   ∞ −npx e e−mpx − e−mqx dx − e−nqx m − [p > 0, q > 0] 3.436 = (q − p) ln 2 n m x n  0 ∞ −px dx 1−e = p ln p − p [p > 0] 3.437 pe−x − x x 0 2.11

FI II 634

ET II 217 (18) FI II 7 95, 802 BI (92)(10)

BI (89)(28) BI (89)(24)

362

3.438 1. 2.7

Exponential Functions



4.







dx ln 2 − 1 = x 2 0   ∞ 2 2 −px p −x p dx p p2 11 1−e e − − 2− = ln p − p3 3 6 2x x x x 6 36 0 1 1 + 2 x

e−x −

1 −x e 2 x

BI (89)(19)

[p > 0]

dx 1 = 1 − ln 2 e−x − e−2x − e−2x x x 0



  ∞ x 1 dx 1 −x x + 2  −px = p− e − e− 2 p− e + (ln p − 1) 2 2x x 2 0 

3.

∞

3.438

BI (89)(33)

∞

BI (89)(25)

BI (89)(22) [p > 0]

 ∞   dx r 1  −mpx = p ln p − q ln q − (p − q) 1 + ln e − e−mqx (p − q)e−rx + mx x m 0 [p > 0, q > 0, r > 0] LI(89)(26), LI(89)(27)  ∞   dx (p − r)e−qx + (r − q)e−px + (q − p)e−rx = (r − q)p ln p + (p − r)q ln q + (q − p)r ln r x2 0 [p > 0, q > 0, r > 0] (cf. 4.268 6) BI (89)(18)

 3.439

3.441

3.442



∞

1. 0

 2. 3. 3.443

∞



  1 dx x+2 q+1 = −1 + q + 1 − e−x e−qx 1− ln 2x x 2 q



e−x − 1 1 + x 1+x 0

 ∞ 1 −px − e 1 + a2 x2 0 

∞

1. 0



∞

2. 0

[q > 0] dx =C−1 x

n

 0

∞

[p > 0]

BI (92)(11)



dx p2 3 = ln p − p2 [p > 0] x 2 4 n n 1 dx = (−1)k−1 (q + kp)2 ln(q + kp) k 2

BI (89)(32)

k=2

[n > 2, 3.

BI (92)(16)

dx a = −C + ln x p

e−x p2 p 1 − e−px − + 2 x x2

(1 − e−px ) e−qx x3

BI (89)(23)

q > 0,

pn + q > 0]

(cf. 4.268 4)

 2 dx 1 − e−px e−qx 2 = (2p + q) ln(2p + q) − 2(p + q) ln(p + q) + q ln q x [q > 0, 2p > −q]

BI (89)(30)

(cf. 4.268 2) BI (89)(13)

3.456

Powers and algebraic functions of exponentials

363

3.45 Combinations of powers and algebraic functions of exponentials 3.451 1. 2. 3.452

√ 4 4 −x − ln 2 xe 1 − e dx = 3 3 0

 ∞  π 1 + ln 2 xe−x 1 − e−2x dx = 4 2 0 



∞

BI (99)(1)

(cf. 4.241 9)

2. 3. 4. 5. 3.453



∞

1. 0



[a2 ex

0



∞

11 0



∞

2. 0

3.455 1.

2.

3.456



FI II 643a,BI(99)(4) BI (99)(5) BI (99)(6) BI (99)(8) BI (99)(7)



dx b xex 2π √ ln 1 + = a2 ex − (a2 − b2 ) ex − 1 ab a

∞

2. 3.454

b xex dx 2π √ arctan = ab a − (a2 + b2 )] ex − 1

xe−2nx dx (2n − 1)!! π √ = 2x (2n)!! 2 e −1

 ln 2 +

xe−(2n−1)x dx (2n − 2)!! √ =− (2n − 1)!! e2x − 1



 0

2n  (−1)k k=1

ln 2 +

k=1

(cf. 4.298 17)

BI (99)(16)

[ab > 0]

(cf. 4.298 18)

BI (99)(17)

 LI (99)(10)

k 2n−1 

[ab > 0]

(−1)k k

 LI (99)(9)

∞

x2 ex dx < = 8π ln 2 3 0 (ex − 1)    ∞ x3 ex dx π2 2 < = 24π (ln 2) + 12 3 0 (ex − 1)

1.

BI (99)(2)

∞

x dx √ x = 2π ln 2 e −1 0

  ∞ 2 x dx π2 2 √ x = 4π (ln 2) + 12 e −1 0  ∞ −x xe dx π √ x = [2 ln 2 − 1] 2 e −1 0  ∞ −x xe dx √ = 1 − ln 2 e2x − 1 0

 ∞ −2x 7 xe dx 3 √ x = π ln 2 − 4 12 e −1 0

1.

1.

−x

∞

  x dx π π √ √ √ = ln 3 + 3 3x 3 3 3 3 e −1

BI (99)(11)

BI (99)(12)

BI (99)(13)

364

Exponential Functions



  π π < = √ ln 3 − √ 2 3 3 3 3 3 (e3x − 1)

∞

x dx

2. 0

3.457



∞

1. 0

 2. 3. 3.458

(cf. 4.244 3)

BI (99)(14)

 n−1/2 (2n − 1)!! π [C + ψ(n + 1) + 2 ln 2] xe−x 1 − e−2x dx = 4 · (2n)!! (cf. 4.241 5)

∞

BI (99)(3)

x

xe dx

2 [ln(4a) − 3C − 2 ψ(2n) − ψ(n)] (2n + 1)an+1/2 −∞ (a +  ∞ μ μ x dx 1 , ln a [a > 0, Re μ > 0] = − B μ 2 x −x ) 2aμ 2 2 −∞ (a e + e n+3/2 ex )



ln 2

=

p−1

xex (ex − 1)

1.7 0

 2.

3.457

BI (101)(12)

BI (101)(14)

% & ∞  (−1)k−1 1 ln 2 + dx = p p+k+1

BI (104)(4)

k=0

∞

xex dx

−∞ (a

+

ν+1 ex )

1 [ln a − C − ψ(ν)] νaν % & ν−1 1 1 = ν ln a − νa k =

[a > 0] [a > 0,

ν = 1, 2, . . .]

k=1

BI (101)(11)

3.46–3.48 Combinations of exponentials of more complicated arguments and powers 3.461



∞ −p2 x2

e

1.

x2n

u

√ (−1)n 2n−1 p2n−1 π [1 − Φ(pu)] dx = (2n − 1)!! 2 2 n−1 e−p u  (−1)k 2k+1 (pu)2k + 2n−1 2u (2n − 1)(2n − 3) · · · (2n − 2k − 1) k=0



∞

2.

x2n e−px dx = 2

0



∞

3.

x2n+1 e−px dx = 2

0



∞

6.∗

∞

π p

[p > 0,

n = 0, 1, . . .]

[p > 0]

FI II 743 BI (81)(7)

(2a)2k n! (2n − 1)!! √  k π (−1) 2n (2k)!(n − k)! k=0

2 dx 1 √ √ = e−μu − μπ [1 − Φ (u μ)] 2 x u u  ∞   exp −a x2 + b2 dx = b K 1 (ab)

0

NT 21(4)

n

2

−∞



[p > 0]

n! 2pn+1

(x + ai)2n e−x dx =

4. 5.11

(2n − 1)!! 2(2p)n



e−μx

2

BI (100)(12)

+ π |arg μ| < , 2 [Re a > 0,

, u>0

Re b > 0]

ET I 135(19)a

3.462

7.

∗

Exponentials of complicated arguments and powers



∞

0

8.∗



9.∗

[Re a > 0, 2 3  12b 3b x4 exp −a x2 + b2 dx = 3 K 2 (ab) + 2 K 1 (ab) a a

Re b > 0]

[Re a > 0, 3 4  90b 15b x6 exp −a x2 + b2 dx = 4 K 3 (ab) + 3 K 2 (ab) a a

Re b > 0]

∞

[Re a > 0,

Re b > 0]

γ √ 2β [Re β > 0,

Re ν > 0]

0

3.462



  2b b2 K 0 (ab) x2 exp −a x2 + b2 dx = 2 K 1 (ab) + a a

∞

0



∞

1.





xν−1 e−βx

2

−γx

dx = (2β)−ν/2 Γ(ν) exp

0



2.8

3.11

4.

∞

xn e−px

2

−∞ ∞

xe−μx

5.11

2

+2qx

−2νx

dx =

dx =

0



7.11

8. 9.∗



γ2 8β



D −ν

EH II 119(3)a, ET I 313(13)



n−1





2 π d qeq /p [p > 0] BI (100)(8) n−1 p dq −∞

k  n n/2

 2 p π q 1 [p > 0] LI (100)(8) = n!eq /p p p (n − 2k)!(k)! 4q 2 k=0



 ∞ √ −ν−1 q q2 ν −β 2 x2 −iqx − ν2 √ (ix) e dx = 2 πβ exp − 2 D ν 8β β 2 −∞ , + π ET I 121(23) Re β 2 > 0, Re ν > −1, arg ix = sign x 2  ∞   √ xn exp −(x − β)2 dx = (2i)−n π H n (iβ) EH II 195(31)



6.

365

∞

1 2n−1 p

ν 1 − 2μ 2μ 





 ν π νμ2 e 1 − erf √ μ μ



2

+ π |arg ν| < , 2

, Re μ > 0

q π exp [Re p > 0] p p −∞ 

   ∞ 2 ν π 2ν 2 + μ νμ2 ν e x2 e−μx −2νx dx = − 2 + 1 − erf √ 2μ μ5 4 μ 0 , + π |arg ν| < , Re μ > 0 2

2   ∞ 2 2 ν π 1 ν x2 e−μx +2νx dx = [|arg ν| < π, Re μ > 0] 1+2 eμ 2μ μ μ −∞

 ∞ 1 e±a −βxn ±a e dx = Γ [Re β > 0, Re n > 0] 1/n n nβ 0 xe−px

2

+2qx

dx =

q p

ET I 146(31)a BI (100)(7)

ET I 146(32) BI (100)(8)a

366

10.

Exponential Functions

∗



∞

0

11.∗ 12.∗ 13.∗

14.∗

15.∗ 16.∗

17.∗

18.∗

19.∗ 20.∗ 21.∗ 22.∗ 23.∗



∞

(x − a)e−β(x−a) dx = eaβ

(1 − aβ) β2

3.462

[Re β > 0]

(1 − aβ) [Re β > 0] β2 0 !

 !   ∞ ! ! am e±pb/a pb !arg b ! < π (ax ± b)m e−px dx = Γ m + 1, ± p > 0, ! pm+1 a a ! 0

 ∞ am e±pb/a pb (ax ± b)m e−px dx = Γ m + 1, pu ± pm+1 a u ! 

!  ! ! b ! ! ±u !<π p > 0, !arg a 



  u am e±pb/a pb pb (ax ± b)m e−px dx = Γ m + 1, ± − Γ m + 1, pu ± pm+1 a a 0 ! 

!  ! ! b ± u !! < π u > 0, p > 0, !!arg a ! 

 !  ∞ −px n−1 ±pb/a ! b !! e e p pb dx = Γ −n + 1, ± p > 0, !!arg !<π n n (ax ± b) a a a 0

 ∞ e−px pn−1 e±pb/a pb dx = Γ −n + 1, pu ± n an a u (ax ± b) ! 

!  ! ! b ± u !! < π u > 0, p > 0, !!arg a 



  u −px n−1 ±pb/a e e p pb pb dx = Γ −n + 1, ± − Γ −n + 1, pu ± n n (ax ± b) a a a 0 ! 

!  ! ! b ± u !! < π u > 0, p > 0, !!arg a      a k

j

k  ∞ b Γ j+1 k ,β −b x−a x−a exp −β dx = b b kβ (j+1)/k 0 +  a , arg − > 0, Re b > 0, Re β > 0, Re k > 0 b  ∞ −βxn e Γ(z, βun ) 1−m [u > 0, Re β > 0, Re n > 0, Re z > 0] dx = z= xm nβ z n u   √  ∞ exp −a x + b2 √ dx = K 0 (ab) [Re a > 0, Re b > 0] x2 + b2 0 √    ∞ 2 x exp −a x + b2 b √ dx = K 1 (ab) [Re a > 0, Re b > 0] 2 + b2 a x 0   √  ∞ 4 x exp −a x + b2 3b2 √ dx = 2 K 1 (ab) [Re a > 0, Re b > 0] 2 2 a x +b 0   √  ∞ 6 x exp −a x + b2 15b3 √ dx = 3 K 3 (ab) [Re a > 0, Re b > 0] a x2 + b2 0 (x − a)e−β(x+a) dx = e−aβ

3.471

24.

∗

Exponentials of complicated arguments and powers



∞

0

367

  √ n x2n exp −a x + b2 b √ dx = (2n − 1)!! K n (ab) a x2 + b2

[Re a > 0, Re b > 0]   2

2

2 exp −px a p a p 1 √ dx = exp 25.∗ K0 [Re a > 0, Re b > 0] 2 2 2 2 2 a +x 0  ∞ dx 2 1 e−x − e−x = C 3.463 x 2 0 ∞  dx √ √ √  −μx2 −νx2 e 3.464 −e = π ν− μ [Re μ > 0, Re ν > 0] 2 x  0 ∞   2 π μ+β 1 + 2βx2 e−μx dx = 3.465 [Re μ > 0] 2 μ3 0 3.466  ∞ −μ2 x2 + e π β 2 μ2 Re β > 0, |arg μ| < e dx = [1 − Φ(βμ)] 1. 2 2 2β 0 x +β √  ∞ 2 −μ2 x2 + x e π πβ μ2 β 2 − e 2. dx = [1 − Φ(βμ)] Re β > 0, |arg μ| < x2 + β 2 2μ 2 0  1 x2 ∞  e −1 1 3. dx = 2 x k!(2k − 1) 0 



∞

∞

3.467

2 e−x −

0

3.468 1. 2. 3.469



∞

k=1

1 1 + x2



∞



e−μx

4

−2νx2

dx =

1 4



2ν exp μ



 dx 4 3 e−x − e−x = C x 4 0  ∞ 4 2 dx 1 = C e−x − e−x x 4 0

3.471  1.

u

u

x 0

π, 4 π, 4

NT 19(13)

ET II 217(16)

FI II 683

BI (92)(12)

ν2 2μ

[u > 0] [Re μ > 0,

K 14

ν2 2μ

NT 33(17)

a > 0]

NT 19(11)

[Re μ ≥ 0]

ET I 146(23) BI (89)(7) BI (89)(6)





β β dx 1 exp − exp − = x x2 β u ν−1

2.

ET I 136(24)a

∞

0



FI II 645

2

0

3.

dx 1 =− C x 2

π e−x dx 2 √ = [1 − Φ(u)] √ 2 − u2 x 4u x u 2   ∞ −μx2 xe dx 1 π a2 μ √ √ e = [1 − Φ (a μ)] 2 + x2 2 μ a 0

1. 2.



BI (89)(5)

μ−1 − β x

(u − x)

e

dx = β

ν−1 2

u

ET II 188(22) 2μ+ν−1 2



β β exp − Γ(μ) W 1−2μ−ν , ν 2 2 2u u [Re μ > 0, Re β > 0,

u > 0] ET II 187(18)

368

Exponential Functions



u

3.

−μ−1

x

μ−1 − β x

(u − x)

e

dx = β

0



u

4. 0



−μ μ−1

u



β Γ(μ) exp − u

β β 1 1 x−2μ (u − x)μ−1 e− x dx = √ β 2 −μ e− 2u πu

∞

5.

β

xν−1 (x − u)μ−1 e x dx = B(1 − μ − ν, μ)uμ+ν−1

∞

6.

 β

x−2μ (x − u)μ−1 e x dx =

u





[Re μ > 0, u > 0]

β Γ(μ) K μ− 12 2u [u > 0, Re β > 0,

u



3.471

π 1 −μ β 2 Γ(μ) exp u



ET II 187(16)

Re μ > 0]



β 1 F 1 1 − μ − ν; 1 − ν; u [0 < Re μ < Re(1 − ν),

β 2u



I μ− 12

β 2u

ET II 187(17)

u > 0] ET II 203(15)



[Re μ > 0,

∞

7.

β

xν−1 (x + γ)μ−1 e− x

0

u > 0] ET II 202(14)

β ν−1 ν−1 β dx = β 2 γ 2 +μ Γ(1 − μ − ν)e 2γ W ν−1 +μ,− ν 2 2 γ [|arg γ| < π, Re(1 − μ) > Re ν > 0] ET II 234(13)a

 8.

u

 μ−1 − β x−2μ u2 − x2 e x

0

μ− 12

2 β 1 μ− 32 dx = √ u Γ(μ) K μ− 12 u π β [Re β > 0, u > 0,

Re μ > 0] ET II 188(23)a

ν2 ∞   β ν−1 − β −γx x x e dx = 2 K ν 2 βγ γ 0

 9.



∞

10.

 xν−1 exp

0

 11.

iμ exp 2

x−

 2

β x

β2 x+ x



Re γ > 0] ET II 82(23)a, LET I 146(29)

iνπ

dx = 2β ν e 2 K −ν (βμ)     Im μ > 0, Im β 2 μ < 0; note that K −ν ≡ K ν dx = iπβ ν e−

iνπ 2

EH II 82(24)

(1)

H −ν (βμ)



Im μ > 0,

   Im β 2 μ > 0 EH II 21(33)



∞  μ ν μ2 xν−1 exp −x − K −ν (μ) dx = 2 4x 2+ 0 , π WA 203(15) |arg μ| < , Re μ2 > 0; note that K −ν ≡ K ν 2

 ∞ ν−1 − β x e x β ν−1 β γ dx = γ e Γ(1 − ν) Γ ν, [|arg γ| < π, Re β > 0, Re ν < 1] x+γ γ 0 

13.



ν−1

x 0

12.

∞

iμ 2



[Re β > 0,

ET II 218(19)

3.475

Exponentials of complicated arguments and powers

  exp 1 − x1 − xν dx = ψ(ν) [Re ν > 0] x(1 − x) 0   ∞ 1 π −2√βγ e x− 2 e−γx−β/x dx = [Re β ≥ 0, γ 0  ∞ √ √ ∂n  1 xn− 2 e−px−q/x dx = (−1)n π n p−1/2 e−2 pq ∂p 0 [Re p > 0, 

14. 15. 16.

369

1

BI (80)(7)

Re γ > 0]

ET 245 (5.6.1)

Re q > 0] PBM 344 (2.3.16(2))

3.472



∞

1. 0



∞

2.

   a 2 1 π √ [exp (−2 aμ) − 1] exp − 2 − 1 e−μx dx = x 2 μ 

x2 exp −

0

4. 5.

1 a − μx2 dx = x2 4



[Re μ > 0,

Re a > 0]

ET I 146(30)

[Re μ > 0,

Re a > 0]

ET I 146(26)

[Re μ > 0,

a > 0]

π √ √ (1 + 2 aμ) exp (−2 aμ) μ3

 dx a 1 π √ 2 exp (−2 aμ) exp − 2 − μx = 2 x x 2 a 0  

  ∞ dx aπ 1 1 (1 + a)e−1/a exp − = x2 + 2 4 2a x x 2 0   ∞ π ∂ n −2√pq −n−1/2 −px−q/x n x e dx = (−1) e p ∂q n 0 

3.





∞

[a > 0] [Re p > 0,

ET I 146(28)a BI (98)(14)

Re q > 0] PBM 344 (2.3.16(3))



∞

3.473

(2m − 1)!! √ π 2m n

exp (−xn ) x(m+1/2)n−1 dx =

0

3.474



1

1. 0



1



2. 0

3.475 1.7 2. 3.

n exp (1 − x−n ) xnp − n 1−x 1−x 





n dx 1 k−1 = ψ p+ x n n

n exp (1 − x−n ) exp 1 − − 1 − xn 1−x

∞

 n exp −x2 −

k=1

 1 x

[p > 0] dx = − ln n x

 dx 1 1 = − nC n+1 2 x 2 1+x 0   ∞  n dx 1 = −2−n C exp −x2 − 2 1 + x x 0  ∞   dx   n = 1 − 2−n C exp −x2 − e−x x 0 

BI (98)(6)

BI (80)(8) BI (80)(9)

[n ∈ Z]

BI (92)(14) BI (92)(13) BI (89)(8)

370

3.476

Exponential Functions



∞

[exp (−νxp ) − exp (−μxp )]

1. 0



∞

[exp (−xp ) − exp (−xq )]

2. 0

3.477 1.

10

2.8



3.476

1 μ dx = ln x p ν

[Re μ > 0,

p−q dx = C x pq

[p > 0,

Re ν > 0]

BI (89)(3)

q > 0]

∞

e−a|x| dx = e−au γ(0, −au) − eau γ(0, au) [Re a > 0, Im u = 0, −∞ x − u  ∞ sign x exp (−a|x|) dx = − [exp (a|u|) Ei (−a|u|) − exp (−a|u|) Ei (a|u|)] x−u −∞

BI (89)(9)

arg u = 0]

[a > 0] 3.478



∞

1.

xν−1 exp (−μxp ) dx =

0



∞

2. 0

 3.

[Re μ > 0,

(u − x)

μ−1

n

∞

4. 0

∞

1. 0



∞

2.11

  2 xν−1 exp −βxp − γx−p dx = p

0 < Re ν < −p for p < 0]



ET I 313(18, 19)

μ+ν−1

nF n

ET II 187(15)

ν 2p   γ K νp 2 βγ β

Re γ > 0]

ET I 313(17)

Re ν > 0]

ET I 313(14)

Re ν > 0]

EH II 83(30)

 √  12 −ν xν−1 exp −β 1 + x β 2 √ Γ(ν) K 12 −ν (β) dx = √ π 2 1+x ν−1

x

0

3.481

p > 0]

BI(81)(8)a, ET I 313(15, 16)

[Re β > 0, 

Re ν > 0,

ν ν+1 ν+n−1 , ,..., ; n n n

μ+ν μ+ν+1 μ+ν+n−1 , ,..., ; βun n n n [Re μ > 0, Re ν > 0, n = 2, 3, . . .]

exp (βx ) dx = B(μ, ν)u

0

3.479

ET II 251(36)



u

x





ν p



ν 1 − νp ν−1 p x [1 − exp (−μx )] dx = − μ Γ |p| p [Re μ > 0 and − p < Re ν < 0 for p > 0, ν−1

11

1 − νp μ Γ p

MC

∞

1. −∞

[Re β > 0,   √ √ 1−ν   exp iμ 1 + x2 ν π μ 2 (1) √ dx = i Γ H 1−ν (μ) 2 2 2 2 2 1+x [Im μ > 0,

xex exp (−μex ) dx = −

1 (C + ln μ) μ

[Re μ > 0]

BI (100)(13)

3.511

Hyperbolic functions



∞

2. −∞

3.482



∞

1.3



xe exp −μe x

2x



1 dx = − [C + ln(4μ)] 4

[Re μ > 0]

BI (100)(14)

[Re β > 0] ∞

2. 0



π μ

1 [S n (β) − π En (β) − π Y n (β)] 2

exp (nx − β sinh x) dx =

0





371

ET I 168(11)

1 exp (−nx − β sinh x) dx = (−1)n+1 [S n (β) + π En (β) + π Y n (β)] 2 [Re β > 0]

∞

3.

exp (−νx − β sinh x) dx =

0

ET I 168(12)

π [Jν (β) − J ν (β)] sin νπ

[Re β > 0] ⎧

iνπ ⎪ ⎪  ∞ K ν (a) for a > 0, ⎨2 exp − exp (ν arcsinh x − iax) 2

√ 3.483 dx = iνπ ⎪ 1 + x2 −∞ ⎪ K ν (−a) for a < 0 ⎩2 exp 2 

qx

px  dx a q a = (ea − 1) ln [p > 0, 3.484 − 1+ 1+ qx px x p 0 π/2   πe [1 − Φ(1)] 3.485 exp − tan2 x dx = 2  1 0 1 ∞  3.4866 x−x dx = e−x ln x dx = k −k = 1.2912859970627 . . . 

0

3.487 1.

∗



π/4

% exp −

0

q > 0]

BI (89)(34)

FI II 483

k=1

& ∞  tan2k+1 x k=0

[|Re ν| < 1] ET I 122(32)

∞

0

ET I 168(13)

k+

1 2

dx = ln 2

3.5 Hyperbolic Functions 3.51 Hyperbolic functions 3.511



∞

1. 0



∞

2. 0



∞

3. 0

 4.

0

∞

π dx = cosh ax 2a

[a > 0]

π aπ sinh ax dx = tan sinh bx 2b 2b

[b > |a|]

BI (27)(10)a

[b > |a|]

GW (351)(3b)

[b > |a|]

BI (4)(14)a

π aπ 1 sinh ax dx = sec − β cosh bx 2b 2b b π aπ cosh ax dx = sec cosh bx 2b 2b



a+b 2b



372

Hyperbolic Functions



∞

5. 0



∞

6. 0



∞

7. 0



∞

8.11 0

 9. 10. 3.512

sin aπ π sinh ax cosh bx c dx = bπ sinh cx 2c cos aπ c + cos c

[c > |a| + |b|]

BI (27)(11)

bπ π cos aπ cosh ax cosh bx 2c cos 2c dx = bπ cosh cx c cos aπ c + cos c

[c > |a| + |b|]

BI (27)(5)a

bπ sinh ax sinh bx π sin aπ 2c sin 2c dx = bπ cosh cx c cos aπ c + cos c

[c > |a| + |b|]

BI (27)(6)a

dx =1 cosh2 x

BI (98)(25)

∞



sinh2 ax dx = 1 − aπ cot aπ 2 −∞ sinh x  ∞ sinh ax sinh bx aπ aπ dx = 2 sec 2 2b 2b cosh bx 0 

∞

1. 0



∞

2. 0

3.512

1 sinh x dx = B coshν x 2



μ+1 ν−μ , 2 2



BI (16)(3)a

[b > |a|]



β β cosh 2βx 4ν−1 B ν + ,ν − dx = a a a cosh2ν ax μ

a2 < 1

BI (27)(16)a

[Re (ν ± β) > 0,

a > 0,

β > 0]

LI(27)(17)a, EH I 11(26)

[Re μ > −1,

Re(μ − ν) < 0] EH I 11(23)

3.513 1. 2.

√ 1 dx a + b + a2 + b 2 √ =√ ln a2 + b 2 a + b − a2 + b 2 0 a + b sinh x √  ∞ dx b 2 − a2 2 =√ arctan 2 2 a√ +b b −a 0 a + b cosh x 1 a + b + a2 − b 2 √ =√ ln a2 − b 2 a + b − a2 − b 2 

√ ∞ dx 2 b 2 − a2 =√ arctan a√ +b b 2 − a2 0 a sinh x + b cosh x 1 a + b + a2 − b 2 √ =√ ln a2 − b 2 a + b − a2 − b 2

 3.

∞

[ab = 0]  

b 2 > a2 b 2 < a2

GW (351)(8)

  GW (351)(7)

 2  b > a2  2  a > b2 GW (351)(9)

3.516

Hyperbolic functions



∞

4. 0

373

& % √ 2 b 2 − a2 − c 2 dx =√ + π arctan a + b cosh x + c sinh x a+b+c b 2 − a2 − c 2 ⎧ ⎡ ⎤ =0 for (b − a)(a + b + c) > 0 ⎪ ⎪ ⎪ ⎢ ⎨|| = 1 for (b − a)(a + b + c) < 0 ⎥ ⎢ ⎥ 2 2 2 ⎢when b > a + c ; and ⎥ ⎪ ⎣ ⎦ =1 for a < b + c ⎪ ⎪ ⎩  = −1 for a > b + c √ 1 a + b + c + a2 − b 2 + c 2 √ =√ ln a2 − b2 + c2 a + b + c − a2 − b2 +c2  b2 < a2 + c2 , a2 = b2 1 a+c = ln c a [a = b = 0, c = 0] 2(a − b) = c(a − b − c)  2  b = a2 + c2 , c(a − b − c) < 0 GW (351)(6)

3.514



∞

1. 0



∞

2. 0



[0 < t < π,

cosh ax dx 2

(cosh x + cos t)

0

∞

a > 0]

BI (27)(22)a

a(πt ) cosh ax − cos t1 πt2 π sin b 2 − dx = cos t1 cosh bx − cos t2 b sin t2 sin ab π b sin t2

[0 < |a| < b, ∞

3. 

dx t = cosec t cosh ax + cos t a

=

sinh ax sinh bx

π (− cos t sin at + a sin t cos at) sin3 t sin aπ



0 < a2 < 1,

bt bπ bπ sin cosec t cosec 2 a a a

0 < t2 < π]

0


[0 < |b| < a, 0 < t < π] (cosh ax + cos t)  √  ∞ 2 cosh x dx = − ln 2 3.515 1− √ cosh 2x −∞ 3.516  ∞  dx dx 1 ∞ 1. √ √   μ = μ = Q μ−1 (z) 2 2 2 z + z + z − 1 cosh x z − 1 cosh x 0 −∞ 4.

0

2

dx =

BI (6)(20)a

BI (6)(18)a BI (27)(27)a

BI (21)(12)a

[Re μ > −1] For a suitable choice of a single-valued branch of the integrand, this formula is valid for arbitrary values of z in the z-plane cut from −1 to +1 provided μ <0. If μ > 0, this formula ceases to be valid for points at which the denominator vanishes. CO, WH  ∞ dx 1. EH II 181(32)  n+1 = Q n (β)  0 β + β 2 − 1 cosh x

374

Hyperbolic Functions



∞

2. 0



∞

3. 0

3.517

cosh γx dx e−iγπ Γ(ν − γ + 1) Q γν (β) =  ν+1  Γ(ν + 1) β + β 2 − 1 cosh x [Re (ν ± γ) > −1, μ −iμπ

2μ

  Γ(ν − 2μ + 1) Γ μ + 12

2 e sinh x dx μ  ν+1 = √  π (β 2 − 1) 2 Γ(ν + 1) β + β 2 − 1 cosh x

ν = −1, −2, −3, . . .] EH I 157(12)

Q μν−μ (β)

[Re(ν − 2μ + 1) > 0,

Re(ν + 1) > 0] EH I 155(2)

3.517



∞

1.

ν+ 12

2. 0

3.518





a

cosh γ +

∞



∞

∞

∞

7 0

 5.6 0

ν+ 12

=

1

sinh2μ+1 x dx

∞

Re(ν + γ + 1) > 0] EH I 156(11)



πΓ 2 −ν P νγ (cosh a) 2 sinhν a

sinh2μ x dx

ν+1

0

4.

x dx





Re ν < 12 ,

a>0



EH I 156(8)

  2μ e−iμπ Γ(ν − 2μ + 1) Γ μ + 12 Q μν−μ (cosh a) =√ Γ(ν + 1) π sinhμ a [Re(ν + 1) > 0, Re(ν − 2μ + 1) > 0, a > 0] EH I 155(3)a

  μ−ν = 2μ β 2 − 1 2 Γ(ν − 2μ) Γ(μ + 1) P μ−ν (β) μ

(β + cosh x) [Re(ν − μ) > Re μ > −1,

3. 



ν+1

10 0

− ν Γ(ν + γ + 1) Γ(ν − γ) P −ν π 2 γ (β)   β −1 2 2 Γ ν + 12 [Re(ν − γ) > 0,

(cosh a + sinh a cosh x)

0

2.

1 2

(cosh a − cosh x)

1.



 =

(β + cosh x)

0



  cosh γ + 12 x dx

β does not lie on the ray (−∞, +1) of the real axis]

sinh2μ−1 x cosh x dx 1  ν = a−μ B(μ, ν − μ) 2 1 + a sinh2 x ν−1

sinhμ−1 x (cosh x + 1)  (β + cosh x)

a > 0]

EH I 11(22)

B

1 1 1 × 2 F 1 ,  + 2 − μ − ν; 2 − μ − ν; − β 2 2 2 [Re μ > 0, Re( − μ − ν) > −2, |arg(1 + β)| < π]

EH I 115(11)

dx

[Re ν > Re μ > 0,

EH I 155(1)

=2

μ+ν−ρ

1 μ,  + 2 − μ − ν 2





μ 1−β = 2−(2−μ−ν+) 2 F 1 , 2 − μ − ν + ; 1 +  − ; 2 2  μ × B 2 − μ − ν + , −1 + ν + 2 [β ∈ (−∞, −1) , Re(2 + ) Re(μ + ν), Re(2ν + μ) > 2] EH I 115(10) ν−1

sinhμ−1 x (cosh x − 1)  (β + cosh x)

dx

3.522

Hyperbolic and algebraic functions



∞

6.7 0



sinhμ−1 x coshν−1 x   dx = cosh2 x − β

π/2

3.519 0

2F 1

, 1 +  −

[β ∈ (1, ∞) ,

375



μ ν μ+ν μ+ν ;1 +  − ;β 2B ,1 +  − 2 2 2 2 Re μ > 0, 2 Re(1 + ) > Re(μ + ν)] EH I 115(9)

∞

 1 pkπ sinh [(r − p)] tan x dx = π sin sinh (r tan x) kπ + r r



p2 < r2



BI (274)(13)

k=1

3.52–3.53 Combinations of hyperbolic functions and algebraic functions 3.521



∞

1. 0



∞

2. 0



∞

3. 1



∞

1



∞

0

4. 5. 6. 7. 8. 9.

LI III 225(103a), BI(84)(1)a

∞

 dx = −2 Ei[−(2k + 1)a] x sinh ax

[a > 0]

LI (104)(14)

[a > 0]

LI (104)(13)

∞

 dx =2 (−1)k+1 Ei[−(2k + 1)a] x cosh ax

∞

 (−1)k π x dx = + π (b2 + x2 ) sinh ax 2ab ab + kπ

∞

(b2

0



π x dx = 2G = π ln 2 − 4 L = 1.831931188 . . . cosh x 4

[a > 0,

b > 0]

k=1

2. 3.

GW (352)(2b)

k=0

1. 

[a > 0]

k=0

4. 3.522

π2 x dx = 2 sinh ax 4a

x dx 1 = − β(b + 1) 2 + x ) sinh πx 2b

∞ 2π  dx (−1)k−1 = 2 2 b 2ab + (2k − 1)π 0 (b + x ) cosh ax k=1

 ∞ 1 1 dx = β b + 2 2 b 2 0 (b + x ) cosh πx  ∞ 1 x dx = ln 2 − 2 ) sinh πx (1 + x 2 0  ∞ π dx =2− 2 2 0 (1 + x ) cosh πx  ∞ x dx π = −1 2 ) sinh πx (1 + x 2 0 2  ∞ dx πx = ln 2 2 0 (1 + x ) cosh 2  ∞ , √ x dx 1 + √ = 2 + 1 −2 π + 2 ln πx 2 2 0 (1 + x ) sinh 4

[b > 0]

BI(97)(16), GW(352)(8)

∞

[a > 0, [b > 0]

b > 0]

BI (97)(5)

BI (97)(4) BI (97)(7) BI (97)(1) BI (97)(8) BI (97)(2) BI (97)(9)

376

Hyperbolic Functions



∞

10. 0

3.523 1. 2.



√ , dx 1 + √ π − 2 ln = 2 + 1 πx (1 + x2 ) cosh 4 2

BI (97)(3)

∞

2β − 1 xβ−1 dx = β−1 β Γ(β) ζ(β) 2 a 0 sinh ax  ∞ 2n−1 2n 2 − 1  π 2n x dx = |B2n | 2n a 0 sinh ax 

3.523

∞

3. 0



[Re β > 1, [a > 0,

a > 0]

WH

n = 1, 2, . . .] WH, GW(352)(2a)



2 xβ−1 1 dx = Γ(β) Φ −1, β, cosh ax (2a)β 2

β ∞  2 2 k Γ(β) (−1) = (2a)β 2k + 1 k=0



[Re β > 0, ∞

4. 0



∞

5. 0



∞

6. 0



∞

7. 0



∞

8. 0



∞

9. 0



∞

10. 0



∞

11. 0



∞

0



∞

1.

[a > 0]

π3 x2 dx = cosh x 8

(cf. 4.261 6)

BI (84)(3)

π4 x3 dx = sinh x 8

(cf. 4.262 1 and 2)

BI (84)(5)

2.11 0

BI(84)(12)a, GW(352)(1a)

5 5 x4 dx = π cosh x 32

BI (84)(7)

x5 π6 dx = sinh x 4

BI (84)(8)

61 7 x6 dx = π cosh x 128

BI (84)(9)

17 8 x7 dx = π sinh x 16

BI (84)(10)

∞

x1/2 dx √  1 = π (−1)k cosh x (2k + 1)3/2

BI (98)(7)a

∞ √  dx (−1)k π = 2 x1/2 cosh x (2k + 1)1/2 k=0

xμ−1

0



EH I 35, ET I 322(1)

k=0

12. 3.524

a > 0]

 π 2n+1 x2n dx = |E2n | cosh ax 2a

sinh βx Γ(μ) dx = sinh γx (2γ)μ

BI (98)(25)a



 

 1 β 1 β ζ μ, 1− − ζ μ, 1+ 2 γ 2 γ [Re γ > |Re β|, Re μ > −1] ET I 323(10)

∞

x2m

2m



aπ

π d sinh ax tan dx = sinh bx 2b da2m 2b

[b > |a|]

BI (112)(20)a

3.524

Hyperbolic and algebraic functions



∞

3. 0

4.11 5.

6. 7.

∞

 sinh ax dx = Γ(1 − p) p sinh bx x

k=0

377

1 1 − [b(2k + 1) − a]1−p [b(2k + 1) + a]1−p



[b > |a|, p < 1] BI (131)(2)a  aπ sinh ax π d sec dx = [b > |a|] x2m+1 BI (112)(18)a 2m+1 cosh bx 2b da 2b 0



 

  ∞ Γ(μ) cosh βx 1 β 1 β dx = xμ−1 ζ μ, 1 − + ζ μ, 1 + sinh γx (2γ)μ 2 γ 2 γ 0 [Re γ > |Re β|, Re μ > 1] 

∞



∞

2m+1

ET I 323(12)

 π d cosh ax aπ dx = x2m sec [b > |a|] cosh bx 2b da2m 2b 0

  ∞ ∞  1 cosh ax dx 1 k · = Γ(1 − p) (−1) + cosh bx xp [b(2k + 1) − a]1−p [b(2k + 1) + a]1−p 0 2m

BI(112)(17)

k=0



[b > |a|, ∞

8.

x2m+1

0



9.8 10.

∞

∞

11.

x2

x6

0



13.

[b > |a|]

π3 sinh ax aπ aπ dx = 3 sin sec3 [b > |a|] sinh bx 4b 2b 2b 0  ∞ π sinh ax aπ aπ 5 aπ  dx = 8 sec · 2 + sin2 x4 · sin sinh bx 2b 2b 2b 2b 0 

12.

aπ π d2m+1  cosh ax tan dx = 2m+1 sinh bx 2b da 2b

p < 1]

π

aπ 7

sinh ax dx = 16 sec sinh bx 2b 2b

[b > |a|]  aπ aπ aπ + 2 cos4 sin 45 − 30 cos2 2b 2b 2b [b > |a|]

π2 aπ sinh ax aπ dx = 2 sin sec2 [b > |a|] cosh bx 4b 2b 2b 0  ∞ π aπ 4 sinh ax aπ aπ  dx = sec · 6 − cos2 x3 sin cosh bx 2b 2b 2b 2b 0 ∞

14.

x5

0



∞

15.

x7

0

 16.

∞

x 0

BI (112)(19)a BI (84)(18)

BI (82)(17)a

BI (82)(21)a

∞

x



BI(131)(1)a

BI (84)(15)a

π

aπ 6

[b > |a|]  aπ aπ aπ 120 − 60 cos2 + cos4 sin 2b 2b 2b

π

aπ 8

BI (82)(18)a [b > |a|]  aπ aπ aπ aπ 5040 − 4200 cos2 + 546 cos4 − cos6 sin 2b 2b 2b 2b

sinh ax dx = sec cosh bx 2b 2b sinh ax dx = sec cosh bx 2b 2b

π aπ 2 cosh ax dx = sec sinh bx 2b 2b

BI (82)(14)a

[b > |a|]

BI (82)(22)a

[b > |a|]

BI (84)(16)a

378

Hyperbolic Functions

 17. 18.

π aπ 4  cosh ax aπ 1 + 2 sin2 dx = 2 sec [b > |a|] sinh bx 2b 2b 2b 0  ∞ π aπ 6  cosh ax aπ aπ 15 − 15 cos2 dx = 8 sec + 2 cos4 x5 sinh bx 2b 2b 2b 2b 0 

∞

∞

19.

x3

x7

0

21.



cosh ax dx = 16 sec sinh bx 2b 2b

[b > |a|] 315 − 420 cos2

BI (82)(19)a

aπ

aπ aπ + 126 cos4 − 4 cos6 2b 2b 2b

∞

0



∞

23. 0



∞

1. 0



∞

2. 0



∞

3. 0



BI(82)(23)a

∞

22.

3.525

aπ 8 

π

BI (82)(15)a

[b > |a|]  cosh ax aπ π3 aπ dx = 3 2 sec3 − sec [b > |a|] x2 cosh bx 8b 2b 2b 0  ∞ π aπ 5  cosh ax aπ aπ dx = sec + cos4 x4 24 − 20 cos2 cosh bx 2b 2b 2b 2b 0 

20.

3.525

x6

π

aπ 7 

cosh ax dx = sec cosh bx 2b 2b  aπ

BI (84)(17)a

[b > |a|] 720 − 840 cos2

BI (82)(16)a

aπ

aπ aπ + 182 cos4 − cos6 2b 2b 2b

π

sinh ax dx · = ln tan + cosh bx x 4b 4

[b > |a|]

BI (82)(20)a

[b > |a|]

BI (95)(3)a

dx 1 sinh ax a · = − cos a + sin a ln [2 (1 + cos a)] sinh πx 1 + x2 2 2 dx 1 1 − sin a sinh ax π · = sin a + cos a ln π 2 sinh 2 x 1 + x 2 2 1 + sin a

[π ≥ |a|]

BI (97)(10)a

[π ≥ 2|a|]

BI (97)(11)a

1 cosh ax x dx 1 · = (a sin a − 1) + cos a ln [2 (1 + cos a)] sinh πx 1 + x2 2 2 [π > |a|]

∞

4. 0

1 1 + sin a cosh ax x dx π · = cos a − 1 + sin a ln sinh π2 x 1 + x2 2 2 1 − sin a

+π

> |a|

2 ∞ sinh ax x dx a+π a π · = −2 sin + sin a − cos a ln tan 2 2 2 4 0 cosh πx 1 + x

BI (97)(12)a

, BI (97)(13)a

 5.

 6. 0

[π > |a|] ∞

GW (352)(12)

dx a+π cosh ax a π · = 2 cos − cos a − sin a ln tan cosh πx 1 + x2 2 2 4 [π > |a|]

GW (352)(11)

3.527

Hyperbolic and algebraic functions



∞

7. 0



∞

0



∞

0

∞

 cos π x dx cosh ax π b · 2 + π = sinh bx c + x2 2bc bc + kπ

[b > |a|]

BI (97)(19)

k(b−a)

 sinh ax cosh bx dx 1 (a + b + c)π (b + c − a)π · = ln tan cot cosh cx x 2 4c 4c [c > |a| + |b|]

∞

2. 0



BI (97)(18)

k=1

1. 

[b ≥ |a|]

k=1

8. 3.526

∞ k(b−a) dx sinh ax π  sin b π · 2 = sinh bx c + x2 c bc + kπ

379

∞

3. 0

1 a sinh2 ax dx · = ln sec π sinh bx x 2 b Γ(μ) xμ−1 dx = sinh βx cosh γx (2γ)μ

BI (93)(10)a

[b > |2a|]

BI (95)(5)a



 

 1 β 1 β Φ −1, μ, 1+ + Φ −1, μ, 1− 2 γ 2 γ [Re γ > |Re β|, Re μ > 0] ET I 323(11)

3.527



∞

1. 0

 2. 3.6

5. 6. 7. 8. 9.

[Re a > 0,

Re μ > 2]

BI (86)(7)a

m = 1, 2, . . .]

BI(86)(5)a

∞

x2m π 2m |B2m | dx = 2 a2m+1 0 sinh ax  ∞ μ−1  x 4  1 − 22−μ Γ(μ) ζ(μ − 1) dx = 2 μ (2a) 0 cosh ax 1 = 2 ln 2 a

 4.

xμ−1 4 Γ(μ) ζ(μ − 1) dx = (2a)μ sinh2 ax

[a > 0,

[Re a > 0,

Re μ > 0,

[Re a > 0,

μ = 2]

μ = 2]

BI (86)(6)a ∞

x dx ln 2 [a = 0] = 2 2 a cosh ax 0  2m   ∞ 2 − 2 π 2m x2m dx = |B2m | [a > 0, m = 1, 2, . . .] 2 (2a)2m a 0 cosh ax  ∞ ∞ 2 Γ(μ)  sinh ax (−1)k dx = xμ−1 [Re μ > 1, a > 0] aμ (2k + 1)μ−1 cosh2 ax 0 k=0  ∞ x sinh ax π [a > 0] dx = 2 2a cosh2 ax 0  ∞ sinh ax 2m + 1  π 2m+1 x2m+1 |E2m | [a > 0, m = 0, 1, . . .] dx = 2 a 2a cosh ax 0  ∞ cosh ax 22m+1 − 1 x2m+1 (2m + 1)! ζ(2m + 1) dx = a2 (2a)2m sinh2 ax 0 [a = 0,

m = 1, 2, . . .]

LO III 396

BI(86)(2)a

BI (86)(15)a

BI (86)(8)a BI (86)(12)a

BI (86)(13)a

380

Hyperbolic Functions



10.

11

11.8 12. 13. 14.11 15.10 16.∗ 3.528

cosh ax 22m − 1  π 2m |B2m | dx = a a sinh2 ax 0 √  ∞ x sinh ax π Γ(μ)   dx = 2μ+1 2 Γ μ+ 1 4μa cosh ax 0 2  ∞ 2 x dx π2 = 2 3 −∞ sinh x  ∞ cosh ax π2 x2 dx = 2a3 sinh2 ax 0  ∞ sinh x dx = 4G x2 cosh2 x 0  ∞ tanh x2 dx = ln 2 cosh x 0  ∞  cosh ax 2 Γ (μ) ζ (μ − 1)  1 − 21−μ xμ−1 = 2 μ a sinh ax 0 

∞

∞

1. 0



∞

2. 0

3.529



∞

1. 0

 2. 3. 3.531

∞

0



∞

2.10 0



4.

(1 + xi)2n − (1 − xi)2n dx = (−1)n+1 2|E2n | + 2 i sinh πx 2

1 1 − sinh x x



m = 1, 2, . . .]

[μ > 0,

a > 0]

BI (86)(14)a LI (86)(9)

BI (102)(2)a

[a > 0]

BI (86)(11)a

[a = 0]

BI (86)(10)a BI (93)(17)a

BI (87)(8)

[n = 0, 1, . . .]

dx = − ln 2 x

aπ cosh ax − 1 dx · = − ln cos sinh bx x 2b 0

 ∞ a b dx − = (b − a) ln 2 sinh ax sinh bx x 0 

[a > 0,

(1 + xi)2n−1 − (1 − xi)2n−1 dx = 2 i sinh πx 2

∞

1.7

3.

x2m

3.528

BI (87)(7)

BI (94)(10)a

[b > |a|]

GW (352)(66) BI (94)(11)a

 π , 4 +π x dx =√ ln 2 − L = 1.1719536193 . . . 2 cosh x − 1 3 3 3 [see 8.26 for L(x)] t ln 2 − L(t) x dx = cosh 2x + cos 2t sin 2t

LI (88)(1) LO III 402

∞

t π 2 − t2 x2 dx = · 3 sin t 0 cosh x + cos t  2    ∞ 4 t π − t2 7π 2 − 3t2 x dx = 15 sin t 0 cosh x + cos t

[0 < t < π]

BI (88)(3)a

[0 < t < π]

BI (88)(4)a

3.533

Hyperbolic and algebraic functions



∞

 sin 2kaπ x2m dx = 2(2m)! cosec 2aπ cosh x − cos 2aπ k 2m+1 k=1   = 2 22m−1 − 1 π 2m |B2m |

∞

5.3 0



1 2

a=

a =

1 2





xμ−1 dx cosh x − cos t

0

   i Γ(μ)  −it  −it e Φ e , μ, 1 − eit Φ eit , μ, 1 sin t   = 2 − 23−μ Γ(μ) ζ(μ − 1)

[μ = 2,

t = π]

= 2 ln 2

[μ = 2,

t = π]

[Re μ > 0,

=

∞

7. 0





0 < a < 1,

BI (88)(5)a ∞

6.3





381

0 < t < 2π,

t = π] ET I 323(5)

∞

xμ dx sin kt 2 Γ(μ + 1)  = (−1)k−1 μ+1 cosh x + cos t sin t k

[μ > −1,

0 < t < π]

BII (96)(14)a

k=1

u

8. 0

1 x dx = cosec 2t [L(θ + t) − L(θ − t) − 2 L(t)] cosh 2x − cos 2t 2 [θ = arctan (tanh u cot t) ,

t = nπ] LO III 402

3.532



∞

1.11 0

 2. 0

3.533



k=0

u

∞

0

0

0

k

[t = mπ] x

3.

b−a b+a





 π t (π − t) x cosh x dx = cosec t ln 2 − L −L cosh 2x − cos 2t 2 2 2

∞

2.6 



[a > 0, b > 0, n > −1] GW (352)(5)







θ+t 1 θ−t x cosh x dx ψ+t = cosec t L −L +L π− cosh 2x − cos 2t 2 2 2

2

 ψ−t t π−t +L − 2L − 2L 2 2   2 u t ψ u t θ LO III 288a tan = tanh cot , tan = coth cot ; t = nπ 2 2 2 2 2 2

1. 

∞

2n!  xn dx 1 = a cosh x + b sinh x a+b (2k + 1)n+1

∞

sinh ax dx 2

(cosh ax − cos t)

x3

=

π−t cosec t a2 

sinh x dx 2

(cosh x + cos t)

=

t π2 − t sin t

[a > 0,

LO III 403

0 < t < π]

(cf. 3.5141) BI (88)(11)a

 2 [0 < t < π]

(cf. 3.531 3) BI (88)(13)

382

Hyperbolic Functions



∞

4.11

x2m+1

0

sinh x dx 2

(cosh x − cos 2aπ)

= 2(2m + 1)! cosec 2aπ 

= 2(2m + 1) 2

3.534 ∞  sin 2kaπ k=1

2m−1



k 2m+1

 − 1 π 2m |B2m |



0 < a < 1, a=

1 2

a =

1 2



 BI (88)(14)

3.534

 1 π I 1 (a) 1 − x2 cosh ax dx = WA 94(9) 1. 2a 0  1 cosh ax π √ 2. dx = I 0 (a) WA 94(9) 2 2 1−x 0  1 π x arcsin (tanh a) dx √ = √ [a > 0] 3.535 · BI (80)(11) · 2 sinh ax sinh a cosh 2a − cosh 2ax 2 2a 0 3.536  ∞ π2 x2 11 dx = BI (98)(7) 1. 2 12 0 cosh x √   ∞ 2 ∞ x tanh x2 dx π (−1)k √ 2. BI (98)(8) = 2 2 2k + 1 cosh x 0 k=0



 ∞ νπ sin μπ 1−μ−ν 1−μ+ν xμ−1 2 sin 2 Γ(μ) Γ 3. sinh (ν arcsinh x) √ dx = ×Γ 2μ π 2 2 1 + x2 0 

∞

νπ cos μπ xμ−1 2 cos 2 cosh (ν arccosh x) √ dx = 2μ π 1 + x2

4. 0

ET I 324(14) [−1 < Re μ < 1 − |Re ν|]



1−μ−ν 1−μ+ν Γ(μ) Γ ×Γ 2 2

[0 < Re μ < 1 − |Re ν|]

ET I 324(15)

3.54 Combinations of hyperbolic functions and exponentials 3.541



∞

1.

e−μx sinhν βx dx =

0



∞

2. 0



e−μx e−x

4. 0

0

μ ν − ,ν + 1 2β 2

[Re β > 0,

Re ν > −1, Re μ > Re βν] EH I 11(25), ET I 163(5)

[Re (μ + b ± β) > 0] ∞

−∞  ∞

5.

2ν+1 β

B





 1 1 μ+β 1 μ−β sinh βx dx = + + e−μx ψ −ψ sinh bx 2b 2 2b 2 2b

3.



1

∞

π μπ sinh μx dx = tan sinh βx 2β β

1 π aπ sinh ax dx = − cot sinh x a 2 2

e−px dx (cosh px)

2q+1

=

1 22q−2 B(q, q) − p 2qp

[Re β > 2|Re μ|] [0 < a < 2] [p > 0,

q > 0]

EH I 16(14)a BI (18)(6) BI (4)(3) LI (27)(19)

3.545

Hyperbolic functions and exponentials

6. 7. 8. 9. 10. 3.542



μ+1 e 2 0  ∞ μ 1 e−μx tanh x dx = β − 2 μ 0  ∞ −μx  e μ −1 dx = μ β 2 2 0 cosh x  ∞ sinh μx 1 e−μx dx = (1 − ln 2) 2 μ cosh μx 0  ∞ 1 π pπ sinh px dx = − cot e−qx sinh qx p 2q 2q 0 



∞

−μx

∞

1.

e 0



∞

2. 0

−μx

dx =β cosh x

383

e−μx (cosh x − cosh u)

ν−1

ET I 163(7)

[Re μ > 0]

ET I 163(9)

[Re μ > 0]

ET I 163(8)

[Re μ > 0]

LI (27)(15)

[0 < p < 2q]

BI (27)(9)a

μ − ν, 2ν + 1 β   1 Re β > 0, Re ν > − , Re μ > Re βν ET I 163(6) 2  1 1 2 iπν 2 −ν e Γ(ν) sinhν− 2 u Q μ− dx = −i 1 (cosh u) 2 π [Re ν > 0, Re μ > Re ν − 1]

1 (cosh βx − 1) dx = ν B 2 β ν

[Re μ > −1]



EH I 155(4), ET I 164(23)

3.543 1.



∞

  iπeitb e−ibx dx =− cosh πb − e−2itb sinh πb cosh t −∞ sinh x + sinh t [t > 0]



∞

2. 0

 3. 4.

[Re μ > −1,

∞

∞

1. 0

 2.

0

BI (6)(10)a

∞

∞

u



t = 2nπ]

k=1

3.544 3.545

∞

 sin kt e−μx dx = 2 cosec t cosh x − cos t μ+k

 cos kt 1 − e−x cos t −(μ−1)x e dx = 2 μ+k 0 cosh x − cos t k=0

 ∞ epx − cos t 1 1 dx = t cosec t + 2 p 1 + cos t 0 (cosh px + cos t) 

ET I 121(30)

∞

    exp − n + 12 x  dx = Q n (cosh u) , 2 (cosh x − cosh u)

[Re μ > 0,

t = 2nπ]

[p > 0]

BI (6)(9)a

BI (27)(26)a

[u > 0]

EH II 181(33)

π aπ 1 sinh ax dx = cosec − epx + 1 2p p 2a

[p > a,

p > 0]

BI (27)(3)

1 π aπ sinh ax dx = − cot px e −1 2a 2p p

[p > a,

p > 0]

BI (27)(9)

384

3.546 1. 2. 3. 4.

Hyperbolic Functions



√ a 1 π a2 √ exp √ Φ 2 β 4β 2 β 0   ∞ 2 2 a 1 π exp e−βx cosh ax dx = 2 β 4β 0

  ∞ 1 π a2 2 −βx2 −1 e sinh ax dx = exp 4 β β 0

  ∞ 2 1 π a2 +1 e−βx cosh2 ax dx = exp 4 β β 0 

3.547  1.

∞

e−βx sinh ax dx = 2

∞

exp (−β sinh x) sinh γx dx =

0



ET I166(38)a

[Re β > 0]

FI II 720a

[Re β > 0]

ET I 166(40)

[Re β > 0]

ET I 166(41)

[Re β > 0] exp (−β cosh x) sinh γx sinh x dx =

0 ∞

exp (−β sinh x) cosh γx dx =

3. 0



[Re β > 0]

γπ π π cot [J γ (β) − Jγ (β)] − [Eγ (β) + Y γ (β)] = γ S −1,γ (β) 2 2 2

∞

2. 

3.546

WA 341(5), ET I 168(14)a

γ K γ (β) β

πγ π π tan [Jγ (β) − J γ (β)] − [Eγ (β) + Y γ (β)] = S 0,γ (β) 2 2 2 [Re β > 0, γ not an integer] ET I 168(16)a, WA 341(4), EH II 84(50)

∞

4.

exp (−β cosh x) cosh γx dx = K γ (β)

[Re β > 0]

ET I 168(16)a, WA 201(5)

[Re β > 0]

ET I 168(7), EH II 85(51)

0



∞

5.

exp (−β sinh x) sinh γx cosh x dx = 0



γ S 0,γ (β) β

∞

6.

exp (−β sinh x) sinh[(2n + 1)x] cosh x dx = O 2n+1 (β) 0



[Re β > 0] ∞

7.

exp (−β sinh x) cosh γx cosh x dx = 0



exp (−β sinh x) cosh 2nx cosh x dx = O 2n (β) 0



∞

9. 0



11.

[Re β > 0]

∞

8.

10.11

1 S 1,γ (β) β

ET I 167(5)

∞

1 exp (−β cosh x) sinh2ν x dx = √ π

[Re β > 0]

ν 2 1 Γ ν+ K ν (β) β 2  Re β > 0,

ET I 168(6)

Re ν > − 12



1 ν β Γ (μ − ν) W −μ,ν− 12 (4β) 2 0 [Re β > 0, Re μ > Re ν]

 ∞   β2 exp − sinh x sinhν−1 x coshν x dx = −π D ν βeiπ/4 D ν βe−iπ/4 2 0 + π, Re ν > 0, |arg β| ≤ 4

EH II 82(20)

exp [−2 (β coth x + μx)] sinh2ν x dx =

EH II 120(10)

3.548

Hyperbolic functions and exponentials

 12.

13.

14.

, 1 3 + exp (2νx − 2β sinh x) √ dx = π β J ν+ 14 (β) J ν− 14 (β) + Y ν+ 14 (β) Y ν− 14 (β) 2 sinh x 0 [Re β > 0] EH I 169(20)  ∞ + , exp (−2νx − 2β sinh x) 1 3 √ dx = π β J ν+ 14 (β) Y ν− 14 (β) − J ν− 14 (β) Y ν+ 14 (β) 2 sinh x 0 ET I 169(21) [Re β > 0]   ∞ exp (−2β sinh x) sinh 2νx 1 π 3 β  νπi (1) (2) √ e H 1 +ν (β) H 1 −ν (β) dx = 2 4i 2 sinh x 0 2 (1) (2) −νπi −e H 1 −ν (β) H 1 +ν (β) ∞

2



∞

15. 0



17.8

18.

19.

20.

∞

2

[Re β > 0]



ET I 170(24)

exp (−2β sinh x) cosh 2νx 1 π 3 β  νπi (1) (2) √ dx = e H 1 +ν (β) H 1 −ν (β) 2 2 4 2 sinh x  (1) (2) −νπi +e H 1 −ν (β) H 1 +ν (β) 2

16.

385

exp (−2β cosh x) cosh 2νx √ dx = cosh x

2

[Re β > 0]



ET I 170(25)

β K 1 (β) K ν− 14 (β) π ν+ 4 0 ET I 170(26) [Re β > 0]   ∞ exp [−2β (cosh x − 1)] cosh 2νx β 2β √ · e K ν+ 14 (β) K ν− 14 (β) dx = π cosh x 0 [Re β > 0] ET I 170(27)        ∞ cos ν + 14 π exp (−2νx − 2β sinh x) + sin ν + 14 π exp (2νx − 2β sinh x) √ dx sinh x 0 + ,  1 = π 3 β J 14 +ν (β) J 14 −ν (β) + Y 14 +ν (β) Y 14 −ν (β) 2 ET I 169(22) [Re β > 0]        ∞ 1 1 sin ν + 4 π exp (−2νx − 2β sinh x) − cos ν + 4 π exp (2νx − 2β sinh x) √ dx sinh x 0 + ,  1 = π 3 β J 14 +ν (β) Y 14 −ν (β) − J 14 −ν (β) Y 14 +ν (β) 2 ET I 169(23) [Re β > 0]  ∞ exp [−β(cosh x − 1)] cosh νx sinh x  dx = eβ K ν (β) cosh x (cosh x − 1) 0 [Re β > 0]

3.548  1.

∞

e

−μx4

0

 2.

0

∞

π sinh ax dx = 4

e−μx cosh ax2 dx = 4



2

π 4



a exp 2μ a exp 2μ



a2 8μ a2 8μ



I 14

I − 14

a2 [Re μ > 0, 8μ 2

a 8μ [Re μ > 0,

ET I 169(19)

a ≥ 0]

ET I 166(42)

a > 0]

ET I 166(43)

386

3.549

Hyperbolic Functions



∞

1.

e−βx sinh [(2n + 1) arcsinh x] dx = O 2n+1 (β)

3.549

[Re β > 0]

(cf. 3.547 6)

0



ET I 167(5) ∞

2.

e−βx cosh (2n arcsinh x) dx = O 2n (β)

[Re β > 0]

(cf. 3.547 8)

0



ET I 168(6) ∞

3.

e−βx sinh (ν arcsinh x) dx =

ν S 0,ν (β) β

[Re β > 0]

(cf. 3.5475)

e−βx cosh (ν arcsinh x) dx =

1 S 1,ν (β) β

[Re β > 0]

(cf. 3.547 7)

0



∞

4. 0

ET I 168(7)

A number of other integrals containing hyperbolic functions and exponentials, depending on arcsinh x or arccosh x, can be found by first making the substitution x = sinh t or x = cosh t.

3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers 3.551  1.

∞

xμ−1 e−βx sinh γx dx =

0



xμ−1 e−βx cosh γx dx =

0

∞

3. 0

  1 Γ(μ) (β − γ)−μ + (β + γ)−μ 2 [Re μ > 0,

∞

4.

ET I 164(19)



β μ−1 −βx 1−μ −μ x e coth x dx = Γ(μ) 2 ζ μ, −β 2

xn e−(p+mq)x sinhm qx dx = 2−m n!

0

 5.11 

1 −βx

e

x

x e

x ∞

ET I 164(21)

m < p + qm] LI (81)(4)

BI (80)(4)

sinh γx dx =

1 β+γ ln 2 β−γ

[Re β > |Re γ|]

ET I 163(12)

cosh γx dx =

1 [− Ei(γ − β) − Ei(−γ − β)] 2

[Re β > |Re γ|]

ET I 164(15)

∞ −βx

1

0

 1 β+γ sinh γx dx = + Ei(γ − β) − Ei(−γ − β) ln 2 β−γ

e

7. 

q > 0,

[β > γ]

0

8.6

k

(−1)k (p + 2kq)n+1 [p > 0,

Re β > 0]



∞ −βx

6. 

m   m k=0

0

Re β > |Re γ|]



[Re μ > 1, 

Re β > |Re γ|] ET I 164(18)

∞

2.



  1 Γ(μ) (β − γ)−μ − (β + γ)−μ 2 [Re β > −1,

xe−x coth x dx =

π2 −1 4

BI (82)(6)

3.554

Hyperbolic functions, exponentials, and powers



∞

9. 0



∞

10.6

 β Γ 4 β dx −βx

= ln + 2 ln e tanh x β 1 x 4 + Γ 4 2 xe−x coth(x/2) dx =

0

3.552 1. 2. 3.



∞

ET I 164(16)

[Re μ > 1, [a > 0,

Re β > −1]

m = 1, 2, . . .]

[Re μ > 0,

ET I 164(20) EH I 38(24)a

μ = 1]

[if μ = 1] EH I 32(5)

∞

4. 0



[Re β > 0]

π2 −1 3

  xμ−1 e−βx 1 1−μ dx = 2 Γ(μ) ζ μ, (β + 1) sinh x 2 0  ∞ 2m−1 −ax  1 x e π 2m dx = |B2m | sinh ax 2m a 0  ∞ μ−1 −x   x e dx = 21−μ 1 − 21−μ Γ(μ) ζ(μ) cosh x 0 = ln 2



387

∞

5. 0

 π 2m x2m−1 e−ax 1 − 21−2m dx = |B2m | cosh ax 2m a

[a > 0,

m = 1, 2, . . .]

EH I 39(25)a

∞

 x2 e−2nx 1 dx = 4 sinh x (2k + 1)3

[n = 0, 1, 2, . . .]

(cf. 4.261 13)

k=n

BI(84)(4)



∞

6.11 0

π4 x3 e−2nx dx = − 12 sinh x 8

n  k=1

1 (2k − 1)4

[n = 0, 1, . . .]

(cf. 4.262 6) BI (84)(6)

3.553



∞

1. 0



∞

2.11 0

3.554

2. 3.

[a < 1]

BI (95)(7)

sinh2 x2 e−x dx 1 4 · = ln cosh x x 2 π

(cf. 4.267 2)

BI (95)(4)

 β+3 Γ 4 dx β

− ln = 2 ln [Re β > 0] e−βx (1 − sech x) β + 1 x 4 0 Γ 4



 ∞ β+1 1 β − cosech x dx = ψ [Re β > 0] e−βx − ln x 2 2 0 &    ∞% sinh 12 − β x dx = 2 ln Γ(β) − ln π + ln (sin πβ) − (1 − 2β)e−x x sinh x 0 2 

1.11

sinh2 ax e−x dx 1 = ln (aπ cosec aπ) sinh x x 2

∞

[0 < Re β < 1]

ET I 164(17)

ET I 163(10)

EH I 21(7)

388

Hyperbolic Functions



β 1 1 β − coth x dx = ψ [Re β > 0] e − ln + x 2 2 β 0 

 ∞ dx 1 sinh qx −x = 2 ln Γ q + + 2qe − + ln cos πq − ln π sinh x2 x 2 0   2 q < 12

 ∞ β xμ−1 e−βx (coth x − 1) dx = 21−μ Γ(μ) ζ μ, + 1 2 0 

4. 5.

6.

3.555

∞

−βx



[Re β > 0; 3.555



∞

1. 0

1 sinh2 ax dx = ln · 1 − epx x 4



p 2aπ sin 2aπ p

ET I 163(11)

WH

Re μ > 1]

ET I 164(22)

[0 < 2|a| < p]

(cf. 3.545 2) BI (93)(15)



∞

2. 0

3.556



1 sinh2 ax dx · = − ln (aπ cot aπ) x e +1 x 4

∞

1.

x −∞



∞

2. 0

3.557



∞

1.

π2 pπ 1 − epx dx = − tan2 sinh x 2 2

1 − e−px 1 − e−(p+1)x · dx = 2p ln 2 sinh x x

1 2



(cf. 3.545 1)

[p < 1]

(cf. 4.255 3)

[p > −1]

n

[p > −1, ∞

BI (93)(9)

BI (101)(4)

BI (95)(8)

dx e−px − e−qx · cosh x − cos m π x n

k=1

2.

a<

 

Γ n+q+k Γ p+k  m n−1  2n 2n km  = 2 cosec π π ln  (−1)k−1 sin n+p+k q+k n n Γ Γ k=1  2n  2n n−1

2 m  Γ n+q−k Γ p+k n n km  π π ln  (−1)k−1 sin = 2 cosec n+p−k q+k n n Γ Γ

0





[m + n odd]

[m + n even]

n

q > −1]

BI (96)(1)

2

dx (1 − e−x ) · m cosh x + cos π x 0 n   n+k+1 2  k+2   k 

 m n−1  Γ Γ 2n Γ 2n km k−1 π π × ln   2n2  (−1) sin = 2 cosec    k+1 n+k n n Γ 2n Γ 2n Γ n+k+2 k=1 2n n−1   n−k+1 2  k+2   k 

2 m  Γ Γ n Γ n km k−1 n π π × ln   = 2 cosec (−1) sin   n−k   n−k+2  k+1 2 n n Γ n Γ n Γ k=1 n

[m + n odd]

[m + n even] BI (96)(2)

3.558

Hyperbolic functions, exponentials, and powers



∞

3.



∞

4. 0



∞

5. 0



  e−px sin m m dx −x nπ π− e tan · m 2n cosh x + cos n π x



Γ p+n+k n−1 m  2n km  = tan π ln(2n) + 2 π ln (−1)k−1 sin 2n n Γ p+k k=1 2n  n−1 p+n−k

2 m Γ  n km  π ln n + 2 π ln (−1)k−1 sin = tan 2n n Γ p+k k=1 n

0

dx 1 + e−x a · 1−p = 2 sec Γ(p) cosh x + cos a x 2

−x 2

0 ∞

7.



BI (88)(8)

 cos kaπ e−x − cos aπ dx = 2 · (2m + 1)! cosh x − cos aπ k 2m+2

∞

∞

5.

x

n−1  n−k 1 − (−1)n e−nx nπ 2 + 4 dx = (−1)k 2 2 x 3 k cosh 2 k=1

LI (85)(1)

x2

∞

6.

0

∞

LI (85)(6)

k=1

n−1  n−k 1 + (−1)n e−nx dx = 6n ζ(3) − 8 k3 cosh2 x2 k=1

LI (85)(4)

x3

n−1  n−k 1 − e−nx 4 4 nπ dx = − 24 2 x 15 k4 sinh 2 k=1

BI (85)(9)

x3

n−1  1 + (−1)n e−nx n−k 7 4 nπ dx = + 24 (−1)k 4 2 x 30 k cosh 2 k=1

BI (85)(8)

0



BI (85)(5)

x2

0

7.

BI (88)(6)

BI (85)(3)

k=1



LI (96)(5)a

2

n−1  n−k 1 − e−nx dx = 8n ζ(3) − 8 x k3 0 sinh2 k=1 2  ∞ ∞ n−1 −2nx   n−k 1 2 x1 − e x e − 8 dx = 8n 2 (2k − 1)3 (2k − 1)3 sinh x 0



LI (96)(5)

n−1  n−k 1 − e−nx 2nπ 2 − 4 dx = 2 x 3 k2 sinh 2 k=1

∞

0

4.

BI (96)(3)

x

∞

2. 

2

[m + n even]

∞

x2m+1

0

3.

[p > 0]   1 cos k − 2 λ (−1)k−1 k q+1

[m + n odd]

k=1

1. 

k=1

  1 k−1 cos k − 2 a (−1) kp

a π e − cos a dx = |a|π − − cosh x − cos a 2 3

0

3.558

∞ 

[q > −1] −x

x 

∞ 

x 2

xq e cosh Γ(q + 1) dx = cosh x + cos λ cos λ2 k=1

∞

6.

389

390

Trigonometric Functions



%

∞

3.559

e

−x

0



2.

1 1 dx = a − + ln Γ(a) − ln(2π) x 2 2

[a > 0] BI (96)(6)

e

0

1.

&

∞ −2x

3.561 3.562

1 (1 − e−x ) (1 − ax) − xe−x (2−a)x e a− + 2 4 sinh2 x2

3.559

tanh x2 π dx = 2 ln √ x cosh x 2 2

BI (93)(18)





 γ γ − D −2μ √ D −2μ − √ 2β 2β 0   1 Re μ > − 2 , Re β > 0 ET I 166(44) 2 

  ∞ 2 γ γ 1 γ x2μ−1 e−βx cosh γx dx = Γ(2μ)(2β)−μ exp + D −2μ √ D −2μ − √ 2 8β 2β 2β 0 

∞



∞

3.

x2μ−1 e−βx sinh γx dx = 2

xe−βx sinh γx dx = 2

0



∞

4.

−βx2

xe 0

γ 4β

γ cosh γx dx = 4β

1 Γ(2μ)(2β)−μ exp 2

 

π exp β π exp β



2

γ 4β γ2 4β

γ2 8β



[Re μ > 0,



[Re β > 0]

Φ

γ √ 2 β

+

Re β > 0]

ET I 166(45)

BI(81)(12)a,ET I 165(34)

1 2β

[Re β > 0]   √



∞ π 2β + γ 2 γ2 γ γ 2 −βx2 √ √ x e sinh γx dx = exp + 2 Φ 2 4β 4β 8β β 2 β 0

ET I 166(35)

 5.



∞

6. 0

 √  2

π 2β + γ 2 γ 2 −βx2 √ x e cosh γx dx = exp 2 4β 8β β

[Re β > 0]

ET I 166(36)

[Re β > 0]

ET I 166(37)

3.6–4.1 Trigonometric Functions 3.61 Rational functions of sines and cosines and trigonometric functions of multiple angles 3.611



2π

n

(1 − cos x) sin nx dx = 0

1.

BI (68)(10)

0



2π

n

(1 − cos x) cos nx dx = (−1)n

2. 0

 3.

π

 n

(cos t + i sin t cos x) dx = 0

π 2n−1

π

(cos t + i sin t cos x) 0

BI (68)(11) −n−1

dx = π P n (cos t)

EH I 158(23)a

3.613

3.612

Rational functions of sines and cosines



π

1.6 0



391

sin nx cos mx dx = 0 sin x =π

for n > m,

if m + n is odd and positive

=0

for n > m,

if m + n is even

for n ≤ m;

LI (64)(3) π

2. 0

sin nx dx = 0 sin x =π

for n even for n odd BI (64)(1, 2)



π sin(2n − 1)x dx = sin x 2 0

 π/2 1 1 (−1)k−1 sin 2nx dx = 2 1 − + − · · · + sin x 3 5 2n − 1 0

 π/2  π (−1)n−1 sin 2nx sin 2nx 1 1 dx = 2 dx = (−1)n−1 4 1 − + − · · · + cos x cos x 3 5 2n − 1 0 0  π2  π cos(2n + 1)x cos(2n + 1)x dx = 2 dx = (−1)n π cos x cos x 0 0  π/2 π sin 2nx cos x dx = sin x 2 0

3. 4. 5. 6. 7. 3.613



π/2

π

1.6 0

 2.

π

6 0

 3. 0

π cos nx dx =√ 1 + a cos x 1 − a2

√

1 − a2 − 1 a

cos nx dx πan = 1 − 2a cos x + a2 1 − a2 π = 2 (a − 1) an

n

  

a2 < 1,

n≥0

a2 < 1,

n≥0

a2 > 1,

n≥0



FI II 145

GW (332)(21b)

GW (332)(22a)

GW (332)(22b)

LI (45)(17)

BI (64)(12)

  BI (65)(3)

π

sin nx sin x dx π = an−1 2 1 − 2a cos x + a 2 π = n+1 2a

 

a2 < 1,

n≥1

a2 > 1,

n≥1

  BI(65)(4), GW(332)(34a)

392

Trigonometric Functions

 4.

π

10 0



π

5. 0

 6. 7.

8.

cos nx cos x dx π 1 + a2 n−1 · = a 1 − 2a cos x + a2 2 1 − a2 2 π a +1 = n+1 · 2 2a a −1 πa = 1 − a2 π = a (a2 − 1) cos(2n − 1)x dx = 1 − 2a cos 2x + a2

3.614

   

a2 < 1,

n≥1

a2 > 1,

n≥1

n = 0,

a2 < 1

n = 0,

a2 > 1

π 0

cos 2nx cos x dx =0 1 − 2a cos 2x + a2



a2 = 1

π

0

0

π

11.6 0



π

12. 0

 13.

0



= 1

= 1 <1 >1

 



BI (65)(9, 10) BI (65)(12)

BI (65)(6, 7)

 

cos(2n − 1)x cos x dx π a = · 1 − 2a cos 2x + a2 2 1−a 1 π = · 2 (a − 1)an n−1

 

a2 < 1 a2 > 1

  BI (65)(11)

π

10.





BI (65)(8) π

9.





BI(65)(5), GW(332)(34b)



 2 cos(2n − 1)x cos 2x dx =0 a 2 1 − 2a cos 2x + a 0  π  π sin 2nx sin x dx sin(2n − 1)x sin 2x dx = =0 2 1 − 2a cos 2x + a 1 − 2a cos 2x + a2 0 0  2 a  π  2 sin(2n − 1)x sin x dx π an−1 a = · 2 1 − 2a cos 2x + a 2 1 + a 0  2 1 π a = · 2 (1 + a)an 



π

sin nx − a sin(n − 1)x sin mx dx = 0 1 − 2a cos x + a2 π = am−n 2 cos nx − a cos(n − 1)x π  |m|−n a cos mx dx = − 1 1 − 2a cos x + a2 2

for m < n for m ≥ n  2  a <1



sin nx − a sin[(n + 1)x] dx = 0 1 − 2a cos x + a2



cos nx − a cos[(n + 1)x] dx = πan 1 − 2a cos x + a2



a2 < 1 a2 < 1 a2 < 1

  

LI (65)(13)

BI (65)(14) BI (68)(13) BI (68)(14)

3.616

Rational functions of sines and cosines



π

7

3.614

0

sin x sin px · dx · a2 − 2ab cos x + b2 1 − 2ap cos px + a2p πbp−1 = p+1 2a (1 − bp ) πap−1 = 2b (bp − a2p )

393

[0 < b ≤ a ≤ 1, 

0 < a ≤ 1,

p = 1, 2, 3, . . .]  p = 1, 2, 3, . . .

a2 < b,

BI (66)(9)

3.615 1. 2. 3.

3.616



2n √  2  1 − a2 a <1 BI (47)(27) a 0   



 π 1 cos x sin 2nx dx s s π 2n sin 2n arctan arccos = − tan 2 b 2 2 2a2 0 1 + (a + b sin x)   



 π 1 cos x cos(2n + 1)x dx s s π 2n+1 cos arccos (2n + 1) arctan = tan 2 b 2 2 2a2 1 + (a + b sin x) 0 <   2 where s = − 1 + b2 − a2 + (1 + b2 − a2 ) + 4a2 BI (65)(21, 22) 

π/2



π

1.



cos 2nx dx (−1)n π √ = 1 − a2 sin2 x 2 1 − a2

1 − 2a cos x + a2

n

dx = π

0



π

2.10 0

dx 1 n = 2 (1 − 2a cos x + a2 )

 0

1−

n  2  n k=0

2π

k

BI (63)(1)

dx n (1 − 2a cos x + a2 ) n−1  (n + k − 1)! a2 k π = n 2 (1 − a2 ) (k!) (n − k − 1)! 1 − a2 =



a2k

(a2

π n − 1)

k=0 n−1 

(n + k − 1)! 2

k=0

 

1 k

(k!) (n − k − 1)! (a2 − 1)

a2 < 1 a2 > 1

 

BI (331)(63) π

3.



1 − 2a cos x + a2

n

cos nx dx = (−1)n πan

BI (63)(2)

0

 4.

π



0

1 − 2a cos x + a2

n

cos mx dx  n 1 2π  = 1 − 2a cos x + a2 cos mx dx 2 0 =0  n−m = π(−a)m 1 + a2

[n < m] [(n−m)/2] 



k=0

2k

n n − k a k m+k 1 + a2

[n ≥ m]

GW (332)(35a)

 5. 0

2π

sin nx dx m =0 (1 − 2a cos 2x + a2 )

GW (332)(32a)

394

Trigonometric Functions



π

6. 0



π

7. 0

 0

3.61710 0

[a = 0,

cos nx dx cos nx dx 1 2π = m 2 0 (1 − 2a cos x + a2 )m (1 − 2a cos x + a2 )





k m−1 2m − k − 2 1 − a2 a2m+n−2 π  m + n − 1 = 2m−1 k m−1 a2 (1 − a2 ) k=0



m−1  m+n−1 k 2m − k − 2  2 π a −1 = 2m−1 n 2 k m − 1 a (a − 1) k=0 cos 2nx dx  n+1 = 2 2 a cos x + b2 sin2 x

π

dx n+1/2

(1 − 2a cos x + a2 )



2n n



 2 n



π/2



dx

n+1/2 1 − k 2 sin2 x where the Fn (k) satisfies the recurrence relation k dFn (k) , n = 0, 1, 2, . . . Fn+1 (k) = Fn (k) + 2n + 1 dk and  π/2 dx F0 (k) = K(k) ≡  1/2 0 1 − k 2 sin2 x is the complete elliptic integral of the first kind. Introducing the complete elliptic integral of the second kind  π/2  1/2 1 − k 2 sin2 x dx E(k) = Fn (k) =

0

0

d K(k) E(k) K(k) = − , dk k (1 − k 2 ) k combined with the recurrence relation lead to

d E(k) E(k) − K(k) = dk k

d F 0 (k) dk E(k) E(k) = K(k) + − K(k) = , 1 − k2 1 − k2

E(k) E(k) k d + F2 (k) = 2 1−k 3 dk 1 − k 2

  4 − 2k 2 1 = E(k) − K(k) 3 (1 − k 2 ) 1 − k2

F1 (k) = F 0 (k) + k

 

a2 < 1 a2 > 1

 

GW (332)(31)

b2 − a π (2ab)2n+1

[a > 0, b > 0]    2 |a| 2 = , |a| = 1 2n+1 Fn |1 + a| |1 + a|

with

the derivatives

±1] GW (332)(32c)



π/2

8.



  1 sin x dx 1 1 − m = 2(m − 1)a (1 − a)2m−2 (1 + a)2m−2 (1 − 2a cos 2x + a2 )

3.617

GW (332)(30b)

3.623

Powers of trigonometric functions

395

3.62 Powers of trigonometric functions 3.621 1. 2. 3. 4. 5.

6.∗ 7.∗ 3.622 1. 2. 3.





μ μ , 2 2 0 0  2  π/2  π/2 1 1 Γ sin3/2 x dx = cos3/2 x dx = √ 4 6 2π 0 0  π/2  π/2 (2m − 1)!! π sin2m x dx = cos2m x dx = (2m)!! 2 0 0  π/2  π/2 (2m)!! sin2m+1 x dx = cos2m+1 x dx = (2m + 1)!! 0 0  π/2  μ ν 1 , [Re μ > 0, sinμ−1 x cosν−1 x dx = B 2 2 2 0 π/2

π/2

sinμ−1 x dx =

cosμ−1 x dx = 2μ−2 B

FI II 789

FI II 151

FI II 151

Re ν > 0]

LO V 113(50), LO V 122, FI II 788





2 3 2 sin x dx = Γ π 4 0   1 2  π/2 Γ 4 dx √ √ = 2 2π sin x 0 



π/2 √

π/2

μπ π sec 2 2 0

 π/4 μ+1 1 tanμ x dx = β 2 2 0  π/4 n−1 π  (−1)k tan2n x dx = (−1)n + 4 2n − 2k − 1 0 tan±μ x dx =

[|Re μ| < 1]

BI (42)(1)

[Re μ > −1]

BI (34)(1)

BI (34)(2)

k=0



tan2n+1 x dx = (−1)n 0

3.623



ln 2  (−1)k + 2 2n − 2k n−1

π/4

4.11



π/2

0

cotμ−1 x sin2ν−2 x dx = 0

π/4

2.6

tanμ x sin2 x dx =

1+μ β 4

tanμ x cos2 x dx =

1−μ β 4

0

 3.6

0

μ 1 μ B ,ν − 2 2 2 [0 < Re μ < 2 Re ν] BI(42)(6), BI(45)(22)

π/2

tanμ−1 x cos2ν−2 x dx =

1.



BI (34)(3)

k=0

π/4



μ+1 2 μ+1 2

−

1 4

[Re μ > −1]

BI (34)(4)

+

1 4

[Re μ > −1]

BI (34)(5)



396

3.624 1. 2.3

3.11 4.8 5.

6.6 3.625

Trigonometric Functions



π/4

1 sinp x dx = [p > −1] p+2 cos x p+1 0     1  π/2  π/2 1 μ 1 Γ 2 + 14 Γ(1 − μ) sinμ− 2 x cosμ− 2 x   dx = dx = cos2μ−1 x 2 Γ 54 − μ2 sin2μ−1 x 0 0   1 − 2 < Re μ < 1  π/4 1 (2n)!! cosn− 2 (2x) dx = π 2 2n+1 2n+1 cos (x) 2 (n!) 0  π/4 cosμ 2x [Re μ > −1] dx = 22μ B(μ + 1, μ + 1) 2(μ+1) x cos 0    π/4 2μ−2 Γ μ − 12 Γ(1 − μ) sin x 1−2μ √ dx = 2 B(2μ − 1, 1 − μ) = cosμ 2x 2 π 0 1  2 < Re μ < 1

2  π/2 sin ax aπ 1 − sin πa [2a β(a) − 1] , [a > 0] dx = sin x 2 2 0 

π/4

1. 0



π/4

2. 0



π/4

3. 0



π/4

4.8 0

3.626

3.624



π/4

1. 0



π/4

2. 0

 0

3.62811 0

sin2n x cosp 2x dx = cos2p+2n+2 x

π 2

1 2

  B n + 12 , p + 1

[p > −1]

(cf. 3.251 1)

LI (55)(12) BI (38)(3)

BI (35)(1)

BI (35)(4)

BI (35)(2) BI (35)(3)

1

(2n − 2)!!(2m − 1)!! sin2n−1 x cosm− 2 2x dx = cos2n+2m x (2n + 2m − 1)!!

BI (38)(6)

1

(2n − 1)!!(2m − 1)!! π sin2n x cosm− 2 2x dx = · cos2n+2m+1 x (2n + 2m)!! 2

BI (38)(7)

sin2n−1 x √ (2n − 2)!! cos 2x dx = 2n+2 cos x (2n + 1)!!

(cf. 3.251 1)

BI (38)(4)

sin2n x √ (2n − 1)!! π · cos 2x dx = 2n+3 cos x (2n + 2)!! 2

(cf. 3.251 1)

BI (38)(5)

  Γ(μ) Γ 12 − μ cotμ x μπ √ dx = sin μ μ sin x 2 π 0 2  −1 < Re μ < 12     1 sec2p x sin2p−1 x dx = √ Γ(p) Γ 12 − p 0 < p < 12 2 π

π/2

3.627 

(n − 1)! Γ(p + 1) sin2n−1 x cosp 2x dx = · cos2p+2n+1 x 2 Γ(p + n + 1) (n − 1)! 1 = = B(n, p + 1) 2(p + n)(p + n − 1) · · · (p + 1) 2 [p > −1] (cf. 3.251 1)

GW (331)(34b)

tanμ x dx = cosμ x



π/2

BI (55)(12)a WA 691

3.631

Powers of trigonometric functions

397

3.63 Powers of trigonometric functions and trigonometric functions of linear functions 3.631  1.

π

0



2ν−1 ν B

π/2

2 sinν−2 x sin νx dx =

2.7 0



π

3.6

νπ 1 cos 1−ν 2

sinν x sin νx dx = 2−ν π sin

0



π sin aπ 2

ν+a+1 ν−a+1 , 2 2



sinν−1 x sin ax dx =

νπ 2

[Re ν > 0]

LO V 121(67a), WA 337a

[Re ν > 1]

GW(332)(16d), FI I 152

[Re ν > −1]

LO V 121(69)

π

sinn x sin 2mx dx = 0

4.

GW (332)(11a)

0





π

π/2

sin2n x sin(2m + 1)x dx =

5.

sin2n x sin(2m + 1)x dx

0

0

(−1)m 2n+1 n!(2n − 1)!! (2n − 2m − 1)!!(2m + 2n + 1)!! (−1)n 2n+1 n!(2m − 2n − 1)!!(2n − 1)!! = (2m + 2n + 1)!! =

[m ≤ n] ∗ [m ≥ n] ∗ GW (332)(11b)





π

π/2

sin2n+1 x sin(2m + 1)x dx = 2

sin2n+1 x sin(2m + 1)x dx

(−1)m π 2n + 1 = 2n+1 n−m 2

6. 0

0

=0 

[n ≥ m] [n < m] BI(40)(12), GW(332)(11c)

π

sinn x cos(2m + 1)x dx = 0

7.

GW (332)(12a)

0



π



sinν−1 x cos ax dx =

8. 0

2ν−1 ν B

π cos aπ 2

ν+a+1 ν−a+1 , 2 2 [Re ν > 0]



π/2

0



2ν ν B

π/2

sinν−2 x cos νx dx =

10. 0 ∗ In



cosν−1 x cos ax dx =

9.

LO V 121(68)a, WA 337a

π

ν+a+1 ν−a+1 , 2 2

νπ 1 sin ν−1 2

3.631.5, for m = n we should set (2n − 2m − 1)!! = 1

[Re ν > 0]

GW (332)(9c)

[Re ν > 1]

GW(332)(16b), FI II 15 2

398

Trigonometric Functions

 11. 12.

3.632

π

π νπ [Re ν > −1] cos ν 2 2 0

 π  π/2 2n (−1)m sin2n x cos 2mx dx = 2 sin2n x cos 2mx dx = π n−m 22n 0 0 sinν x cos νx dx =

LO V 121(70)a

[n ≥ m]

=0 

[n < m] BI(40)(16), GW(332)(12b)

π

sin2n+1 x cos 2mx dx  π/2 =2 sin2n+1 x cos 2mx dx =

13.7 0

(−1)m 2n+1 n!(2n + 1)!! (2m − 2n − 3)!!(2m + 2n + 1)!! 0 (−1)n+1 2n+1 n!(2m − 2n + 3)!!(2n + 1)!! = (2m + 2n + 1)!!

[n ≥ m − 1] [n < m − 1] GW (332)(12c)



π/2

cosν−2 x sin νx dx =

14. 0



1 ν−1



π

[Re ν > 1]

cos x sin nx dx = 1 − (−1) m

15.



π/2

0 π

cosn x sin mx dx =

17.11

cosm x sin nx dx

FI II 153

π

n

2n+1

k

0 π

cosm x cos ax dx =

18.6 0



π/2

0r−1  

2n+1 k k=1   1 + (−1)m+n

0





n  2k

1

cosn x sin nx dx =

16. 



 (m + n − 2k − 2)!! m!(n − m − 2)!! m! +s = 1 − (−1)m+n (m − k)! (m + n)!! (m + n)!! k=0 ⎧ ⎡ ⎤  ⎪ ⎨2 if n − m = 4l + 2 > 0 m if m ≤ n ⎢ ⎥ s = 1 if n − m = 2l + 1 > 0 ⎣r = ⎦ GW (332)(13a) ⎪ n if m ≥ n ⎩ 0 if n − m = 4l or n − m < 0

0



m+n

GW(332)(16c), FI II 152

if m ≤ n and n − m = 2k otherwise

m



a+m (−1) sin aπ a+m ;1 − ; −1 2 F 1 −m, − 2m (m + a) 2 2



[a = 0, ±1, ±2, . . .]

GW (332)(15a)

WA 313

π/2

cosν−2 x cos νx dx = 0

19.

[Re ν > 1]

GW(332)(16a), FI II 152

[Re n > −1]

LO V 122(78), FI II 153

0



π/2

cosn x cos nx dx =

20.10 0

3.632



1. 0

π

π 2n+1

  p+a   , + π Γ 2 Γ p−a 2 − x dx = 2p−1 Γ(p) sinp−1 x cos a 2 Γ(p − a) Γ(p + a)  2  p < a2

BI (62)(11)

3.634

Powers of trigonometric functions



π 2

2. −π 2

3.10

+  π , dx = cosν−1 x sin a x + 2

2ν−1 ν B

399

π sin aπ 2

ν+a+1 ν−a+1 , 2 2

[Re ν > 0]  π/2 n−1  (−1)k 2k n − 1

cosp x sin[(p + 2n)x] dx = (−1)n−1 p+k+1 k 0

WA 337a

k=0

[n > 0] 

π

4. −π

  cosn−1 x cos[m(x − a)] dx = 1 − (−1)n+m = =



π/2

cosp+q−2 x cos[(p − q)x] dx =

5. 0



π 2

−π 2

LI (41)(12)

cosn−1 x cos[m(x − a)] dx

[1 − (−1)n+m ] π cos ma

n+m+1 n−m+1 , 2n−1 n B 2 2 [n ≥ m] LO V 123(80), LO V 139(94a)

π 2p+q−1 (p + q − 1) B(p, q) [p + q > 1]

3.633



π/2



cosp−1 x sin ax sin x dx =

1. 0



2p+1 p(p + 1) B 

π/2

cosn x sin nx sin 2mx dx =

2. 0



cosn x cos nx cos 2mx dx =

π/2

cosn−1 x cos[(n + 1)x] cos 2mx dx =

3. 0



π/2

cosp+q x cos px cos qx dx =

4. 0



π/2

cosp+q x sin px sin qx dx =

5.6 0

aπ

p+a p−a + 1, +1 2 2

π 2p+q+2

π 2p+q+2



π 2n+1

 1+

∞   k=1

n−1 m−1



LO V 150(110)

π

n

2n+2

m

π/2

0

WH

[n > m − 1]

BI (42)(19, 20)

BI (42)(21)

1 (p + q + 1) B(p + 1, q + 1)



[p + q > −1] GW (332)(10c)   Γ(p + q + 1) p  q π = p+q+2 −1 k k 2 Γ(p + 1) Γ(q + 1) [p + q > −1]

3.634



π/2

sinμ−1 x cosν−1 x sin(μ + ν)x dx = sin

1. 0

BI (42)(16)

μπ B(μ, ν) 2 [Re μ > 0,

Re ν > 0] BI(42)(23), FI II 814a

400

Trigonometric Functions



π/2

sinμ−1 x cosν−1 x cos(μ + ν)x dx = cos

2. 0

3.635

μπ B(μ, ν) 2 [Re μ > 0,

Re ν > 0] BI(42)(24), FI II 814a



π/2

cosp+n−1 x sin px cos[(n + 1)x] sin x dx =

3. 0

π 2p+n+1

Γ(p + n) n! Γ(p) [p > −n]

3.635



 μ  μ+1 1 cos 2x tan x dx = BI (34)(7) ψ −ψ [Re μ > 0] 4 2 2 0  π/2 ∞   n Γ(p + n − k) π cosp+2n x sin px tan x dx = p+2n+1 k 2 Γ(p) (n − k)! 0 k=0 Γ(p + 2n) pπ = p+2+n+1 2 Γ(n + 1) Γ(p + n + 1) [p > −2n] BI (42)(22)  π/2 π cosn−1 x sin[(n + 1)x] cot x dx = BI (45)(18) 2 0 

π/4

μ−1

1. 2.7

3. 3.636



π/2

1.

tan±μ x sin 2x dx =

0



π/2

2. 

π/2

11 0

μπ μπ cosec 2 2

tan±μ x cos 2x dx = ∓

0

3.

BI (42)(15)

tan2μ x dx = cos x



π/2 0

μπ μπ sec 2 2

[0 < Re μ < 2]

BI (45)(20)a

[|Re μ| < 1]

  Γ μ + 12 Γ(−μ) cot2μ x √ dx = sin x 2 π 

− 12 < Re μ < 1

BI (45)(21)



(cf. 3.251 1) BI (45)(13, 14)

3.637



π/2

tanp x sinq−2 x sin qx dx = − cos

1. 0

(p + q)π B(p + q − 1, 1 − p) 2 [p + q > 1 > p]



π/2

tanp x sinq−2 x cos qx dx = sin

2. 0

(p + q)π B(p + q − 1, 1 − p) 2 [p + q > 1 > p]



π/2

cotp x cosq−2 x sin qx dx = cos

3. 0

GW (332)(15d)

GW (332)(15b)

pπ B(p + q − 1, 1 − p) 2 [p + q > 1 > p]

GW (332)(15c)

3.642

Trigonometric functions: powers and rational functions



π/2

cotp x cosq−2 x cos qx dx = sin

4. 0

401

pπ B(p + q − 1, 1 − p) 2 [p + q > 1 > p]

3.638



π/4

1.

1 cosμ+ 2

0



π/4

2. 0



sin2μ x dx

π/2

3. 0

2x cos x

=



π sec μπ 2 

μ− 12

|Re μ| <

1 2



GW (332)(15a)

(cf. 3.192 2) BI (38)(8)

 1

Γ μ + 2 Γ(1 − μ) sin 2x dx 2 √ = · sin cosμ 2x cos x 2μ − 1 π



π cosp−1 x sin px dx = sin x 2



2μ − 1 π 4  1  − 2 < Re μ < 1 [p > 0]

BI (38)(17) GW(332)(17), BI(45)(5)

3.64–3.65 Powers and rational functions of trigonometric functions 3.641



π/2

1. 0

sinp−1 x cos−p x dx = a cos x + b sin x



π/2

0

π cosec pπ sin−p x cosp−1 x dx = a sin x + b cos x a1−p bp [ab > 0,



π/2

2.

sin

1−p

x cos x 3

(sin x + cos x)

0



p

dx = 0

π/2

p

sin x cos

1−p

x 3

(sin x + cos x)

dx =

0 < p < 1]

(1 − p)p π cosec pπ 2 [−1 < p < 2]

3.642 1.

2. 3.



GW (331)(62)

BI(48)(5)

π/2

sin2μ−1 x cos2ν−1 x dx 1 [Re μ > 0, Re ν > 0] BI (48)(28)  μ+ν = 2μ 2ν B(μ, ν) 2 2a b 0 a2 sin x + b2 cos2 x    π/2 B n2 , n2 sinn−1 x cosn−1 x dx [ab > 0] GW (331)(59a)  n = 2(ab)n a2 cos2 x + b2 sin2 x 0  π/2  sin2n x dx sin2n x dx 1 π =   n+1  2 0 a2 cos2 x + b2 sin2 x n+1 0 a2 cos2 x + b2 sin2 x   π/2 cos2n x dx cos2n x dx 1 π (2n − 1)!!π = =    n+1 = n+1 n+1 2 2 2 2 2 2 2 2 2 2 n!ab2n+1 0 0 a sin x + b cos x a sin x + b cos x 

4. 0

π/2

n 

cosp+2n x cos px dx  n+1 = π a2 cos2 x + b2 sin2 x k=0

2n − k n



[ab > 0] GW (331)(58)

p+k−1 bp−1 k (2a)2n−k+1 (a + b)p+k [a > 0, b > 0, p > −2n − 1] GW (332)(30)

402

3.643

Trigonometric Functions



π/2

1. 0



π/2

2. 0

3.644

3.643



cosp x cos px dx π (1 + a)p−1 = · 1 − 2a cos 2x + a2 2p+1 1−a

a2 < 1,

p > −1



m−1 m−k−1 β 2n sin2n x cosμ x cos βx (−1)n π(1 − a)2n−2m+1   dx = m k l 22m−β−1 (1 + a)2m+β+1 (1 − 2a cos 2x + a2 ) k=0 l=0

2m − k − l − 2 × (a − 1)k l m − 1(−2)  2  a < 1, β = 2m − 2n − μ − 2, μ > −1

GW (332)(33c)

GW (332)(33)

ν−1

k

2 k 2  m + 1 − 2ν m + 1 − 2ν p − q2 p − q2 sinm x m−2 p 1. dx = 2 , B A + q 2 ν=1 −4q 2 2 2 −q 2 0 p + q cos x . ⎧   ⎪ πp q2 ⎪ ⎪ if m = 2k + 2 ⎪ ⎨ q 2 1 − 1 − p2   k ≥ 1, q = 0, p2 − q 2 ≥ 0 where A = ⎪ ⎪ 1 p+q ⎪ ⎪ if m = 2k + 1 ⎩ ln q p−q

 π m−1 m+1 sinm x dx = 2m−1 B , 2. [m ≥ 2] 2 2 0 1 + cos x

 π m−1 m+1 sinm x m−1 dx = 2 , B 3. [m ≥ 2] 2 2 0 1 − cos x .    π pπ sin2 x q2 dx = 2 1 − 1 − 2 4. q p 0 p + q cos x

 π p sin3 x 1 p+q p2 dx = 2 2 + 5. 1 − 2 ln q q q p−q 0 p + q cos x

k  π n  a+b cosn x dx π k (2n − 2k − 1)!!(2k − 1)!! √ (−1) 3.645 n+1 = n (n − k)!k! a−b 2 (a + b)n a2 − b2 k=0 0 (a + b cos x)  2  2 a >b LI (64)(16)

3.646



π



π/2

1. 0



π/2

2. 0

 3.647 0

π/2

cosn x sin nx sin 2x π dx = 2 1 − 2a cos 2x + a 4a



1+a 2

n

1 − n 2





a2 < 1



BI (50)(6)

∞ 1 − a cos 2nx π  m k π m a + m+1 cos x cos mx dx = 2 m+2 kn 1 − 2a cos 2nx + a 2 2 k=1  2  a <1

π cosp x cos px dx ap−1 = · 2 2 2 2b (a + b)p sin x + b cos x

a2

[p > −1,

a > 0,

b > 0]

LI (50)(7)

BI (47)(20)

3.652

3.648

Trigonometric functions: powers and rational functions



π/4

1. 0

403

tanl x dx 1 + cos m n π sin 2x





 n−1 n+l+k m  l+k km 1 cosec π π ψ (−1)k−1 sin −ψ 2n n n 2n 2n k=0 n−1 



 2 n+l−k m  l+k km 1 π ψ (−1)k−1 sin = cosec π −ψ n n n n n =

[m + n is odd]

[m + n is even]

k=0

[l is a natural number] 

π/2

2. 0

3.649



π/2

1. 0



π/2

2. 0

3.651 1. 2. 3.652

tan±μ x dx = π cosec t sin μt cosec(μπ) 1 + cos t sin 2x 

μ  μπ 1−a tan±μ x sin 2x dx π cosec = 1 − 1 ∓ 2a cos 2x + a2 4a 2  1 + a μ  μπ a−1 π cosec = 1+ 4a 2 a+1 μπ tan±μ x (1 ∓ a cos 2x) π dx = sec 2 1 ∓ 2a cos 2x + a 4 2 μπ π = sec 4 2



|Re μ| < 1,

 

a2 < 1 a2 > 1

π/2

0

π/2

2. 0



0

BI (50)(3)

BI (50)(4)

tanμ x dx = (sin x + cos x) sin x



π/2 0

[Re μ > −1]

BI (36)(3)

[Re μ > −1]

BI (36)(4)a

cotμ x dx = π cosec μπ (sin x + cos x) cos x

tanμ x dx = (sin x − cos x) sin x



π/2 0

π/2

μ+ 12 x

cot dx = (sin x + cos x) cos x

 0

π/2

BI (49)(1)

cotμ x dx = −π cot μπ (cos x − sin x) cos x [0 < Re μ < 1]

3.

BI (47)(4)



[0 < Re μ < 1]





π/4

1. 

t2 < π 2

[−2 < Re μ < 1] 

μ   2  1−a a <1 1+  1 + a μ   2  a−1 a >1 1− a+1 [|Re μ| < 1]





 1 μ+2 μ+1 tanμ x dx = ψ −ψ 1 + sin x cos x 3 3 3 0 



  π/4 μ 1 μ+2 μ+1 tan x dx = β +β 1 − sin x cos x 3 3 3 0 



BI (36)(5)

BI (49)(2)

μ− 12 x

tan dx = π sec μπ (sin x + cos x) cos x   |Re μ| < 12

BI (61)(1, 2)

404

3.653

Trigonometric Functions



π/2

1. 0



π/2

2.11 0



tan1−2μ x dx = a2 cos2 x + b2 sin2 x 

μ

tan x dx = 1 − a sin2 x

π/2

0

π/2 0

3.653

cot1−2μ x dx π = 2μ 2−2μ 2 2 2 2 2a b sin μπ a sin x + b cos x [0 < Re μ < 1]

μ

π sec cot x dx =  2 1 − a cos x 2 (1 −

GW (331)(59b)

μπ 2 a)μ+1

[|Re μ| < 1, π/2 + π , μπ tan±μ x dx π cosec t sec cos − t μ = 2 2 2 1 − cos2 t sin2 2x 0  |Re μ| < 1,

a < 1]

BI (49)(6)

 3.

t2 < π 2



BI(49)(7), BI(47)(21)

4.

5.



π/2



π/2

+ π , tan±μ x sin 2x μπ sin − t μ dx = π cosec 2t cosec 2 2 1 − cos2 t sin2 2x 0   |Re μ| < 1, t2 < π 2 BI (47)(22)a  π/2  π/2 + μπ , π μπ tanμ x sin2 x dx cotμ x cos2 x dx = = cosec 2t sec cos − (μ + 1)t 2 1 − cos2 t sin2 2x 1 − cos2 t sin2 2x 2 0 0 2 |Re μ| < 1, t2 < π 2 BI(47)(23)a, BI(49)(10)

6. 0

3.654



π/2

1.

tanμ x cos2 x dx = 1 − cos2 t sin2 2x

tanμ+1 x cos2 x dx (1 + cos t sin 2x)

0

2



π/2 0



+ μπ , μπ cotμ x sin2 x dx π cosec 2t sec cos − (μ − 1)t = 2 2 1 − cos2 t sin2 2x 2 |Re μ| < 1, t2 < π 2 BI(47)(24)a, BI(49)(9)

π/2

= 0

cotμ+1 x sin2 x dx 2

(1 + cos t sin 2x)

=

π (μ sin t cos μt − cos t sin μt) 2 sin μπ sin3 t   |Re μ| < 1, t2 < π 2 BI(48)(3), BI(49)(22)



π/2

2.

2

(sin x + cos x)

0



π/2

3. 0

 3.655 0

tan±μ x dx

π/2

=

μπ sin μπ

μπ tan±(μ−1)x dx π = ± cot 2 2 2 2 cos x − sin x tan2μ−1 x dx   = 1 − 2a cos t1 sin2 x + cos t2 cos2 x + a2



π/2

[0 < Re μ < 1]

BI (56)(9)a

[0 < Re μ < 2]

BI (45)(27, 29)

cot2μ−1 x dx   1 − 2a cos t1 cos2 x + cos t2 sin2 x + a2 0 π cosec μπ = 2 )μ (1 − 2a cos t + a2 ) 1 − μ (1 − 2a cos t + a 2 1   0 < Re μ < 1, t21 < π 2 , t22 < π 2 BI (50)(18)

3.662

3.656

Trigonometric functions: powers of linear functions



π/4

1. 0



π/2

2. 0

405





μ+1 μ+2 tanμ x dx 1 = −ψ −ψ 1 − sin2 x cos2 x 12

6

6



 μ+4 μ+5 μ+2 μ+1 +ψ +ψ + 2ψ − 2ψ 6 6 3 3 [Re μ > −1] (cf. 3.651 1 and 2) LI (36)(10) tanμ−1 x cos2 x dx = 1 − sin2 x cos2 x



π/2 0

π cotμ−1 x sin2 x dx μπ = √ cosec cosec 2 2 6 1 − sin x cos x 4 3 [0 < Re μ < 4]



2+μ π 6



LI (47)(26)

3.66 Forms containing powers of linear functions of trigonometric functions 3.661



2π

1.

(a sin x + b cos x)

2n+1

(a sin x + b cos x)

2n

dx = 0

BI (68)(9)

0



2π

2. 0



π

3.

n

(a + b cos x) dx = 0

 n (2n − 1)!! · 2π a2 + b2 (2n)!!

dx =

1 2



BI (68)(8)

n  n (a + b cos x) dx = π a2 − b2 2 P n

2π



0 n/2

√

a 2 a − b2



k π  (−1)k (2n − 2k)! n−2k  2 a a − b2 2n k!(n − k)!(n − 2k)! k=0  2  a > b2 GW (332)(37a)

 2π a dx 1 π √ = = n+1 P n 2 − b2 2 0 (a + b cos x)n+1 2 2 2 a (a − b )

k n  a+b π (2n − 2k − 1)!!(2k − 1)!! √ = · (n − k)!k! a−b 2n (a + b)n a2 − b2 =

 4. 0

π

dx n+1

(a + b cos x)

k=0

3.662



π/2



π/2

μ

(sec x − 1) sin x dx =

1. 0

[a > |b|]

μ

(cosec x − 1) cos x dx = μπ cosec μπ 0

[|Re μ| < 1] 

π/2

μ

(cosec x − 1) sin 2x dx = (1 − μ)μπ cosec μπ

2. 0



π/2

 μ

(sec x − 1) tan x dx =

3. 0

 0

π/2

[−1 < Re μ < 2]

BI (55)(13) BI (48)(7)

μ

(cosec x − 1) cot x dx = −π cosec μπ 0

π/4

μ

(cot x − 1)

4.

GW(332)(38), LI(64)(14)

π dx = − cosec μπ sin 2x 2

[−1 < Re μ < 0]

BI (46)(4,6)

[−1 < Re μ < 0]

BI (38)(22)a

406

Trigonometric Functions



π/4

dx = μπ cosec μπ cos2 x

μ

(cot x − 1)

5. 0

3.663



u

ν− 12

(cos x − cos u)

1.

 cos ax dx =

0



ν−1

(cos x − cos u)

2.

[|Re μ| < 1]

BI (38)(11)a



π 1 sinν u Γ ν + P −ν 1 (cos u) 2 2  a− 2 Re ν > − 12 ; a > 0, √

u

3.663

cos[(ν + β)x] dx =

0

0


EH I 159(27), ET I 22(28) 2ν−1

π Γ(β + 1) Γ(ν) Γ(2ν) sin u ν   C β (cos u) 1 ν 2 Γ(β + 2ν) Γ ν + 2 [Re ν > 0, Re β > −1, 0 < u < π] EH I 178(23)

3.664



π

 z+

1. 0



q  z 2 − 1 cos x dx = π P q (z) + Re z > 0,

π



2. 0

z+

√

  π, arg z + z 2 − 1 cos x = arg z for x = 2

dx q = π P q−1 (z) z 2 − 1 cos x + Re z > 0,



π

 z+

3.

q  z 2 − 1 cos x cos nx dx =

0

 4.

SM 482

  π, arg z + z 2 − 1 cos x = arg z for x = 2

WH

π P nq (z) (q + 1)(q + 2) · · · (q + n)    π Re z > 0, arg z + z 2 − 1 cos x = arg z for x = , 2  z lies outside the interval (−1, 1) of the real axis WH, SM 483(15)

π



μ  z + z 2 − 1 cos x sin2ν−1 x dx

0

2

22ν−1 Γ(μ + 1) [Γ(ν)] C νμ (z) Γ(2ν + μ)  √ 1 1−ν π Γ(ν) Γ(2ν) Γ(μ + 1) ν π 2 ν 2 −ν   = z − 1 4 2 Γ(ν) P μ+ν− C μ (z) = 2 1 (z) 1 2 2 Γ(2ν + μ) Γ ν + 2 EH I 155(6)a, EH I 178(22) [Re ν > 0]  2π + ,    ν ν−1 dx β + β 2 − 1 cos(a − x) γ + γ 2 − 1 cos x 0    = 2π P ν βγ − β 2 − 1 γ 2 − 1 cos a =

5.

[Re β > 0,

Re γ > 0]

EH I 157(18)

3.667

3.665

Trigonometric functions: powers of linear functions



π

1. 0



π

2. 0

μ μ sinμ−1 x dx 2μ−1 , B μ =  μ 2 2 (a + b cos x) (a2 − b2 )

[Re μ > 0,

3.666 π

(β + cos x)

1.

μ−ν− 12

0



π

2.6 0



4.

5.10

3.667

FI II 790a

EH I 81(9)

Re ν > − 12

√   π μ+ν (cosh β + sinh β cos x) sin−2ν x dx = ν sinhν (β) Γ 12 − ν P νμ (cosh β) 2   Re ν < 12

π

1

μ

0



|a| < 1]

  μ  1 2ν+ 2 e−iμπ β 2 − 1 2 Γ ν + 12 Q μν− 1 (β) 2   sin2ν x dx = Γ ν + μ+ 12  Re ν + μ + 12 > 0,

(cos t + i sin t cos x) sin2ν−1 x dx = 2ν− 2

3.

0 < b < a]

  1  sin2μ−1 x dx 2 1 1 ν = B μ, 2 F ν, ν − μ + 2 ; μ + 2 ; a 2 (1 + 2a cos x + a ) [Re μ > 0,



407

2π

√

1

EH I 155(5)a

EH I 156(7)

−ν

2 π sin 2 −ν t Γ(ν) P μ+ν− 1 (cos t) 2   Re ν > 0, t2 < π 2 1



i3m 2π Γ(ν + 1) cos ma P m ν (cos t) Γ(ν + m + 1) 0 + , π 0
EH I 158(23)

ν

[cos t + i sin t cos(a − x)] cos mx dx =



π/4

1.

2μ

(cos x + sin x)

0



sinμ−1 2x dx

π/4

2.

EH I 159(26)

√

=

π Γ(μ)   2μ+1 Γ μ + 12

sinμ x dx μ+1

(cos x − sin x)

0

EH I 159(25)

cos x

= −π cosec μπ

[Re μ > 0] [−1 < Re μ < 0]

BI (37)(1)

(cf. 3.192 2) BI (37)(16)



π/4

3. 0



π/4

4. 0

 5.

0

π/4

μ

π (cos x − sin x) dx = − cosec μπ sinμ x sin 2x 2

[−1 < Re μ < 0]

sinμ x dx π = cosec μπ μ 2 (cos x − sin x) sin 2x

[0 < Re μ < 1]

sinμ x dx = μπ cosec μπ μ (cos x − sin x) cos2 x

[|Re μ| < 1]

BI (35)(27)

LI (37)(20)a

BI (37)(17)

408

Trigonometric Functions



π/4

sinμ x dx

6.

μ−1

(cos x − sin x)

0



π/2

sinμ−1 x cosν−1 x

7.

μ+ν

(sin x + cos x)

0

3.668





π 4

1. −π 4



v

2. u

 0



2.∗

π

√

0

dx = a ± b cos x

[Re μ > 0,

BI(35)(24), BI(37)(18)

Re ν > 0]

BI (48)(8)

π 2 sin (π cos2 t)  μ−1 1 − 2a cos u + a2 sin x dx π · = μ · 2 2 1 − 2a cos x + a sin μπ (1 − 2a cos v + a )   0 < Re μ < 1, a2 < 1



π√

0



[|Re μ| < 1]

dx = B(μ, ν)

sinp−1 x cosq−p−1 x dx = q (a cos x + b sin x)

a ± b cos x dx =

1.

1−μ μπ cosec μπ 2

dx =

μ−1

π/2

=

cos 2t

(cos u − cos x) μ (cos x − cos v)

3.669 3.670

cos x + sin x cos x − sin x

cos3 x

3.668



π/2

√

−π/2



π/2 0

FI II 788

BI (73)(2)

sinq−p−1 x cosp−1 x B(p, q − p) q dx = aq−p bp (a sin x + b cos x) [q > p > 0, ab > 0]

√ a ± b cos x dx = 2 a + b K

π/2

dx 2 √ =√ E a+b −π/2 a ± b sin x





2b a+b

BI (331)(9)



[a > b > 0]  2b a+b [a > b > 0]

3.67 Square roots of expressions containing trigonometric functions 3.671



π/2



 α+1 β+1 α+1 1 α+β+2 2 1 2 2 , ,− ; ;k sin x cos x 1 − k sin x dx = B F 2 2 2 2 2 2 [α > −1, β > −1, |k| < 1] α

1. 0

β

GW (331)(93)



π/2

2. 0

sinα x cosβ x 1  dx = B 2 2 2 1 − k sin x



α+1 β+1 , 2 2



F



α+1 1 α+β+2 2 , ; ;k 2 2 2 [α > −1, β > −1,

|k| < 1] GW (331)(92)

 3. 0

π



∞ 

(2j − 1)!! (2n + 2j − 1)!! 2j k 22j j!(n + j)! 1 − k 2 sin x j=0

j ∞ 2 k2 (2n − 1)!!π  [(2j − 1)!!] √ = 2n 1 − k 2 j=0 22j j!(n + j)! k 2 − 1 2n

sin

x dx

2

=

π 2n

 

k2 < 1 k2 <



1 2

 LI (67)(2)

3.676

4.∗

Trigonometric functions and square roots





π√

a + b cos x dx =



√

√ a + b sin x dx = 2 a + bE



−π/2

0

5.∗

π/2

π

√

0

dx = a ± b cos x



π/2

dx 2 √ K = a + b a ± b sin x −π/2



409

2b a+b



[a > b]  2b a+b [a > b]

3.672



π/4

1. 0

 0



π 2

3.673 u



1.8

dx sinn x (2n − 1)!! · π = n+1 cos x (2n)!! sin x (cos x − sin x)

√ dx π − 2u √ = 2K sin 4 sin x − sin u

π 2



0



π

2.



0



π

3.8 0

3.675 1. 2. 3.676 1. 2.

dx 1 − (p2 /2) (1 − cos 2x)

= K(p),



=2 1 − 2p cos x + p2 2 = p

BI (74)(11)

[1 > p > 0]



p2 ≤ 1 p2 ≥ 1

BI (67)(5)

 

√

√  2 p 2 p cos x dx 1 1+p  K = − (1 + p)E 2 p 1+p 1+p 1+p 1 − 2p cos x + p   2 p <1 

BI (67)(6)

2

BI (67)(7)

π

WH

FI II 684, WH

π/2

sin x dx 1  = arctan p 2 2 p 0 1 + p sin x  π/2 < tan2 x 1 − p2 sin2 x dx = ∞ 0

BI (39)(6)



sin x dx

  sin n + 12 x dx π  = P n (cos u) 2 2 (cos u − cos x) u    u cos n + 12 x dx π  = P n (cos u) 2 2 (cos x − cos u) 0 

BI (39)(5)

π/4

2.

3.674

dx sinn x (2n)!! · =2· cosn+1 x (2n + 1)!! cos x (cos x − sin x)

BI (60)(5)

BI (53)(8)

410

Trigonometric Functions



π/2

3. 0

3.677 1.

2. 3.678

dx

1  = K 2 2 2 2 p p cos x + q sin x



p2 − q 2 p

3.677

 [0 < q < p]

√  √  √ 1 sin2 x dx 2 2  −√ K = 2E 2 2 2 2 0 1 + sin x % √   √ &  π/2 √ cos2 x dx 2 2  = 2 K −E 2 2 2 0 1 + sin x 



π/2

dx  = ln 2 sec1/2 2x − 1 tan x 0  π/4  tan2 x dx 1  = 1 − k 2 − E(k) + K(k) 2 2 2 0 1 − k sin 2x     u π cos 2x − cos 2u π2 2 dx = (1 − cos u) u < cos 2x + 1 2 4 0  π/4 n− 12 √ (cos x − sin x) (2n − 1)!! π cosec x dx = n+1 cos x (2n)!! 0  π/4 n− 1 √ (cos x − sin x) 2 (2n − 1)!!(2m − 1)!! tanm x cosec x dx = π n+1 x cos (2n + 2m)!! 0

1. 2. 3. 4. 5. 3.679



FI II 165

BI (60)(2)

BI (60)(3)

π/4

π/2

1. 0

BI (38)(23)

BI (39)(2)

LI (74)(6)

BI (38)(24)

BI (38)(25)

dx cos2 x · 1 − cos2 β cos2 x 1 − k 2 sin2 x π  1    ∗  − K E (β, k = ) − E F (β, k ) + K F (β, k ) sin β cos β 1 − k 2 sin2 β 2 MO 138



π/2

2. 0

2

sin x dx   · 1 − 1 − k 2 sin2 β sin2 x 1 − k 2 sin2 x   π 1    ∗  − K E (β, k = ) − E F (β, k ) + K F (β, k ) k 2 sin β cos β 1 − k 2 sin2 β 2 MO 138

 3. 0 ∗ In

π/2

sin x K E (β, k) − E F (β, k) dx  = · 2 2 sin β sin x k 2 sin β cos β 1 − k 2 sin2 β 1 − k 2 sin2 x 2

1−

3.679, k =

√

k2

1 − k2 .

MO 138

3.683

Trigonometric functions: various powers

411

3.68 Various forms of powers of trigonometric functions 3.681



π/2

1. 0



π/2

2. 0

 3.

4.8

  sin2μ−1 x cos2ν−1 x dx 1 = B(μ, ν) F , μ; μ + ν; k 2   2 2 1 − k 2 sin x 2μ−1

[Re μ > 0,

Re ν > 0]

EH I 115(7)

[Re μ > 0,

Re ν > 0]

EH I 10(20)

2ν−1

sin x cos x dx B(μ, ν)  μ+ν = μ 2 2 (1 − k 2 ) 1 − k 2 sin x

π/2

sinμ x dx   μ −1 0 cosμ−3 x 1 − k 2 sin2 x 2     

Γ μ+1 Γ 2 − μ2 1 + (μ − 3)k + k 2 1 − (μ − 3)k + k 2 2 =  − (1 + k)μ−3 (1 − k)μ−3 k 3 π(μ − 1)(μ − 3)(μ − 5) [−1 < Re μ < 4] BI (54)(10)

 π/2  1−μ sinμ+1 x dx μ (1 − k)−μ − (1 + k)−μ √ Γ Γ 1+ μ+1 =   2 2 2kμ π 2 2 μ 2 0 cos x 1 − k sin x [−2 < Re μ < 1] 

π/2

3.682 0

sinμ x cosν x 1 B  dx = 2a (a − b cos2 x)



μ+1 ν+1 , 2 2



F

BI (61)(5)

μ+ν b ν+1 , ; + 1; 2 2 a [Re μ > −1, Re ν > −1,

a > |b| ≥ 0] GW (331)(64)

3.683



π/4

(sinn 2x − 1) tan

1. 0

2.

3.

π 4

 + x dx =

11 2 k n

π/4

(cosn 2x − 1) cot x dx = −

0

k=1

1 = − [C + ψ(n + 1)] 2 [n ≥ 0] BI(34)(8), BI(35)(11)   π/4 π/4 π + x dx = (sinμ 2x − 1) cosecμ 2x tan (cosμ 2x − 1) secμ 2x cot x dx 4 0 0 1 = [C + ψ(1 − μ)] 2 [Re μ < 1] BI (35)(20)   π4 π/4  π  2μ   + x dx = sin 2x − 1 cosecμ 2x tan cos2μ 2x − 1 secμ 2x cot x dx 4 0 0 1 π =− + cot μπ 2μ 2 BI (35)(21)





π/4

π/4

(1 − secμ 2x) cot x dx =

4. 0

(1 − cosecμ 2x) tan 0

1 + x dx = [C + ψ(1 − μ)] 4 2

π

[Re μ < 1]

BI (35)(13)

412

Trigonometric Functions



π/4

3.684 0

(cotμ x − 1) dx = (cos x − sin x) sin x



π/2 0

3.684

(tanμ x − 1) dx = −C − ψ(1 − μ) (sin x − cos x) cos x

[Re μ < 1] BI (37)(9)

3.685



π/4

1. 0



π/2

2. 0

 π/4 π  μ−1  μ−1   + x dx = sin cos 2x − sinν−1 2x tan 2x − cosν−1 2x cot x dx 4 0 1 = [ψ(ν) − ψ(μ)] 2 [Re μ > 0, Re ν > 0] BI(34)(9), BI(35)(12)  μ−1  dx sin = x − sinν−1 x cos x



π/2

0



 μ ,  dx 1 + ν cosμ−1 x − cosν−1 x = ψ −ψ sin x 2 2 2 [Re μ > 0,



π/2

(sinμ x − cosecμ x)

3. 0

dx = cos x



π/2

(cosμ x − secμ x) 0

Re ν > 0]

π μπ dx = − tan sin x 2 2 [|Re μ| < 1]



π/4

(sinμ 2x − cosecμ 2x) cot

4. 0

π 4

+ x dx =



BI (46)(2)

BI (46)(1, 3)

π/4

(cosμ 2x − secμ 2x) tan x dx 0

π 1 − cosec μπ 2μ 2 [|Re μ| < 1] BI (35)(19, 22)   π/4 π/4 π + x dx = 5. (sinμ 2x − cosecμ 2x) tan (cosμ 2x − secμ 2x) cot x dx 4 0 0 π 1 + cot μπ =− 2μ 2 [|Re μ| < 1] BI (35)(14)  π/4   μ−1  π + x dx sin 6. 2x + cosecμ 2x cot 4 0  π/4  μ−1  π = cos 2x + secμ 2x tan x dx = cosec μπ 4 0 [0 < Re μ < 1] BI (35)(18, 8)  π/4  π/4 π  μ−1  μ−1   π μ + x dx = sin cos 7. 2x − cosec 2x tan 2x − secμ 2x cot x dx = cot μπ 4 2 0 0 [0 < Re μ < 1] BI(35)(7), LI(34)(10)  π/2  π/2 π tan x dx cot x dx = = 3.686 BI(47)(28), BI(49)(14) μ μ μ μ cos x + sec x sin x + cosec x 4μ 0 0 3.687    π/2 μ−1  π/2 μ ν π cos ν−μ sin x + sinν−1 x cosμ−1 x + cosν−1 x 4

1. dx = , B dx = ν+μ cosμ+ν−1 x 2 2 sinμ+ν−1 x 0 0 π 2 cos 4 [Re μ > 0, Re ν > 0, Re(μ + ν) < 2] =

BI (46)(7)

3.688

Trigonometric functions: various powers



π/2

2. 0

sinμ−1 x − sinν−1 x dx = cosμ+ν−1 x



π/2

0

413

   sin ν−μ cosμ−1 x − cosν−1 x μ ν 4 π   dx = , B 2 2 sinμ+ν−1 x 2 sin ν+μ 4 π [Re μ > 0, Re ν > 0, Re(μ + ν) < 4] BI(46)(8)



π/2

3. 0

sinμ x + sinν x cot x dx = sinμ+ν x + 1



π 2

0



μ−ν π π cosμ x + cosν x tan x dx = sec · cosμ+ν x + 1 μ+ν μ+ν 2 [Re μ > 0, Re ν > 0]



BI (49)(15)a, BI (47)(29)



π/2

4. 0

sinμ x − sinν x cot x dx = sinμ+ν x − 1



π 2

0



μ−ν π π cosμ x − cosν x tan x dx = tan · cosμ+ν x − 1 μ+ν μ+ν 2 [Re μ > 0, Re ν > 0]



BI(149)(16)a, BI(47)(30)

 5. 6. 3.688

μ π π cosμ x + secμ x tan x dx = sec · ν ν cos x + sec x 2ν ν 2 0  π/2 μ π π cosμ x − secμ x tan x dx = tan · ν ν cos x − sec x 2ν ν 2 0 

π/2

π/4

1. 0



π/4

2. 0



[|Re ν| > |Re μ|]

BI (49)(12)

[|Re ν| > |Re μ|]

BI (49)(13)

tanν x − tanμ x dx · = ψ(μ) − ψ(ν) cos x − sin x sin x

[Re μ > 0,

BI (37)(10)

tanμ x − tan1−μ x dx · = π cot μπ cos x − sin x sin x

[0 < Re μ < 1]

π/4

(tanμ x + cotμ x) dx =

3. 0



μπ π sec 2 2

π/4

(tanμ x − cotμ x) tan x dx =

4. 0



π/4

5. 0



π/4

6. 0



π/4

7. 0



π/4

8. 0



π/4

9. 0

 10.

0

π/4

π μπ 1 − cosec μ 2 2

Re ν > 0]

[|Re μ| < 1]

BI (37)(11)

BI (35)(9)

[0 < Re μ < 2]

BI (35)(15)

π μπ tanμ−1 x − cotμ−1 x dx = cot cos 2x 2 2

[|Re μ| < 2]

BI (35)(10)

1 π μπ tanμ x − cotμ x tan x dx = − + cot cos 2x μ 2 2

[−2 < Re μ < 0]

BI (35)(23)

tanμ x + cotμ x dx = π cosec t cosec μπ sin μt 1 + cos t sin 2x

[t = nπ,

tanμ−1 x + cotμ x dx = π cosec μπ (sin x + cos x) cos x

[0 < Re μ < 1]

BI (37)(3)

tanμ x − cotμ x 1 dx = −π cosec μπ + (sin x + cos x) cos x μ

[0 < Re μ < 1]

BI (37)(4)

tanν x − cotμ x dx = ψ(1 − μ) − ψ(1 + ν) (cos x − sin x) cos x

[Re μ < 1,

|Re μ| < 1]

Re ν > −1]

BI (36)(6)

BI (37)(5)

414

Trigonometric Functions



π/4

11. 0



π/4

12. 0

 13. 14.

3.689

tanμ−1 x − cotμ x dx = π cot μπ (cos x − sin x) cos x

[0 < Re μ < 1]

BI (37)(7)

1 tanμ x − cotμ x dx = π cot μπ − (cos x − sin x) cos x μ

[0 < Re μ < 1]

BI (37)(8)

π/4

dx π 1 · = [Re μ = 0] BI (37)(12) μ tan x + cot x sin 2x 8μ 0 √  π/2  π/2 1 1 π Γ(ν) dx dx   = = 2ν+1 ν · ν · μ μ μ μ 2 μ Γ ν + 12 (tan x + cot x) tan x (tan x + cot x) sin 2x 0 0 μ

[ν > 0] 

π/4

(tanμ x − cotμ x) (tanν x − cotν x) dx =

15. 0

BI(49)(25), BI(49)(26)

μπ 2

sin νπ 2π sin 2 cos μπ + cos νπ [|Re μ| < 1,



π/4

(tanμ x + cotμ x) (tanν x + cotν x) dx =

16. 0

π/4

17. 0

π/4

18. 0



π/4

19. 0



π/2

20. 3.689



π/2

1. 0

2.

|Re ν| < 1]

BI (35)(25)

[0 < Re ν < 1]

BI (37)(14)

dx π νπ tanν x + cotν x · = sec tanμ x + cotμ x sin 2x 4μ 2μ

[0 < Re ν < 1]

BI (37)(13)

[μ > 0,

BI (49)(29)

ν

(1 + tan x) − 1 μ+ν

dx = ψ(μ + ν) − ψ(μ) sin x cos x

ν > 0]

π μπ μt (sinμ x + cosecμ x) cot x dx = cosec t cosec sin sinν x − 2 cos t + cosecν x ν ν ν [μ < ν]



BI (35)(16)

dx π νπ tanν x − cotν x · = tan tanμ x − cotμ x sin 2x 4μ 2μ

(1 + tan x)

0

|Re ν| < 1]

sin μπ (tanμ x − cotμ x) (tanν x + cotν x) dx = −π cos 2x cos μπ + cos νπ [|Re μ| < 1,



BI (35)(17)

νπ 2π cos μπ 2 cos 2 cos μπ + cos νπ

[|Re μ| < 1, 

|Re ν| < 1]

LI (50)(14)

π μt2 t2 sinμ x − 2 cos t1 + cosecμ x μπ · cot x dx = cosec t2 cosec sin − cosec t2 cos t1 ν νx sin x + 2 cos t + cosec ν ν ν ν 2 0 [(ν > μ > 0) or (ν < μ < 0) or (μ > 0, ν < 0, and μ + ν < 0) or (μ < 0, ν > 0, and μ + ν > 0)] π/2

BI (50)(15)

3.691

Trigonometric functions: complicated arguments

415

3.69–3.71 Trigonometric functions of more complicated arguments 3.691



∞

1.



2

sin ax



 dx =

0



0

1

2.

  sin ax2 dx =

0



3. 4.

∞





1 cos ax dx = 2 2

1

  cos ax2 dx =

∞

5. 0



∞

6.

[a > 0]

π √  S a 2a

  1 sin ax2 cos 2bx dx = 2  cos ax2 sin 2bx dx =

0



∞

7. 0



FI II 743a, ET I 64(7)a

[a > 0]

cos ax2 cos 2bx dx =

1 2



ET I 8(5)a

[a > 0, b > 0]  2



2 2 b b π π b 1 π − sin cos + cos = 2a a a 2 a a 4 [a > 0, b > 0]

ET I 82(1)a

ET I 82(18), BI(70)(13) GW(334)(5a)





 b b π b2 b2 sin C √ − cos S √ 2a a a a a



 b2 π b2 + sin cos 2a a a

[a > 0,

b > 0]

[a > 0,

b > 0]

ET I 83(3)a

GW(334)(5a), BI(70)(14), ET I 24(7) ∞

8. 0

10.

π 2a

√  π C a [a > 0] 2a 0 

  ∞  2 b b π b2 b2 sin ax sin 2bx dx = cos C √ + sin S √ 2a a a a a 0 

9.





∞



∞

  (cos ax + sin ax) sin b2 x2 dx  2  2   a  a a a 1 π  = cos sin 1 + 2C − 1 − 2S 2b 2 2b 4b2 2b 4b2 [a > 0, b > 0] ET I 85(22)

  (cos ax + sin ax) cos b2 x2 dx 0  2  2   a  a a a 1 π  = 1 + 2C sin cos + 1 − 2S 2b 2 2b 4b2 2b 4b2 [a > 0, b > 0] ET I 25(21) 2

√  ∞   b + c2 2bc π π sin 2 cos sin a2 x2 sin 2bx sin 2cx dx = − 2 2a a a 4 0

11. 0

  sin a2 x2 cos 2bx cos 2cx dx =

√

2bc π cos 2 cos 2a a



[a > 0, b 2 + c2 π + 2 a 4 [a > 0,

b > 0,

c > 0]

ET I 84(15)

b > 0,

c > 0]

ET I 84(21)

416

Trigonometric Functions



∞

12.



2 2

cos a x 0



∞

13. 0

15. 16. 17.

2

√ b + c2 π 2bc π sin 2 sin sin 2bx sin 2cx dx = − 2a a a2 4 

    1 π sin ax2 cos bx2 dx = 4 2 1 π = 4 2

[a > 0, b > 0,

1 1 √ +√ [a > b > 0] a − b

a+b 1 1 √ −√ [b > a > 0] b+a b−a



 2 2  1 π π − sin ax − sin2 bx2 dx = [a > 0, 8 b a 0 

  ∞  2 2  1 π π 2 2 + cos ax − sin bx dx = [a > 0, 8 b a 0 

  ∞  2 2  1 π π 2 2 − cos ax − cos bx dx = [a > 0, 8 a b 0 

  ∞ √  4 2  1  π π 8− 2 − sin ax − sin4 bx2 x = 64 b a 0 

14.



3.692



∞

18. 0



∞

19. 0



∞

20.

 4 2  1 cos ax − sin4 bx2 dx = 8

b > 0]

BI (178)(1)

b > 0]

BI (178)(3)

b > 0]

BI (178)(5)

[a > 0, b > 0] 



 π π π π 1 + − + a b 32 2a 2b

BI (178)(2)

[a > 0, 

  √  4 2  1 π π 8+ 2 − cos ax − cos4 bx2 dx = 64 a b

b > 0]

BI (178)(4)

[a > 0,

b > 0]

BI (178)(6)



∞

sin2n ax2 dx =

0





cos2n ax2 dx = ∞

BI (177)(5, 6)

0 ∞

21.

2n+1

sin

 2 ax dx =

0

1 22n+1

n 

n+k

(−1)

k=0

2n + 1 k



π 2(2n − 2k + 1)a [a > 0]



∞

22.

  cos2n+1 ax2 dx =

0

1 22n+1

n  k=0

2n + 1 k





∞

1.

     sin a − x2 + cos a − x2 dx =

0

 2.



∞

cos 0

x2 π − 2 8



cos ax dx =

π cos 2



BI (70)(9)

π 2(2n − 2k + 1)a [a > 0]

3.692

ET I 25(19)

BI (177)(21)



∞

c > 0]

π sin a a

a2 π − 2 8

BI(177)(7)a, BI(70)(10)

GW(333)(30c), BI(178)(7)a

[a > 0]

ET I 24(8)

3.695

Trigonometric functions: complicated arguments



∞

3. 0



∞

4. 0



∞

5.

   1 sin a 1 − x2 cos bx dx = − 2    1 cos a 1 − x2 cos bx dx = 2 sin ax2 +

0

2

b a









b2 π π cos a + + a 4a 4

2

b π π sin a + + a 4a 4

∞

cos 2bx dx =

417

cos ax2 +

0

2

b a

[a > 0]

ET I 23(2)

[a > 0] cos 2bx dx =

1 2



ET I 24(10)

π 2a

[a > 0] 6.8

3.693



∞



∞

BI (70)(19, 20)

∞ + ,    π  cos x2 − 1 − cos x2 + 1 dx =   4n+1 [(2n)!]2 n + 1 −∞ n=0 2 2

1.

  sin ax2 + 2bx dx =



0



∞

2.

  cos ax2 + 2bx dx =

0



π 2a

2

2



b b b2 1 b2 1 − S2 − C2 cos − sin a 2 a a 2 a

[a > 0] 2





b2 b π b2 1 b2 1 − C2 − S2 cos + sin 2a a 2 a a 2 a [a > 0]

3.694 1.

2.

3.695

BI (70)(3)

BI (70)(4)



2

2

 b b b2 1 1 π cos − C2 − S2 sin c + cos c 2a a 2 0  a 2

a 2

2  b b b2 1 1 π sin − S2 − C2 + sin c − cos c 2a a 2 a 2 a [a > 0] GW (334)(4a) 





  ∞  2  b2 b2 b2 1 1 π cos − C2 − S2 cos ax + 2bx + c dx = cos c − sin c 2a a 2 a

2 a

0   b2 b2 b2 1 1 π sin − S2 − C2 + cos c + sin c 2a a 2 a 2 a [a > 0] GW (334)(4b) 

∞



∞

1.

  sin ax2 + 2bx + c dx =



3 3

sin a x 0

 2. 0

∞



π sin(bx) dx = 6a



  π cos a3 x3 cos(bx) dx = 6a

b 3a



b 3a





J 13  J 13

2b 3a





2b 3a

b 3a





b 3a

 + J − 13



2b 3a



b 3a



   √ 3 b 2b − K 13 π 3a 3a

[a > 0, b > 0] ET I 83(5)    √    3 b b 2b 2b + J − 13 + K 13 3a 3a π 3a 3a [a > 0,

b > 0]

ET I 24(11)

418

3.696

Trigonometric Functions



∞



4

sin ax

1.





2

sin bx



0



∞

2. 0



∞

3. 0



∞

4. 0



    π sin ax4 cos bx2 dx = − 4     π cos ax4 sin bx2 dx = 4     π cos ax4 cos bx2 dx = 4

∞

3.697

sin 0

3.698





∞

sin

1. 0





∞

2.8

sin 0





∞

3.

cos 0





∞

4.

π dx = − 4

cos 0

a2 x a2 x2 2

a x2 a2 x2









2

a x2







b sin 2a

b sin 2a

b cos 2a



b cos 2a







  1 cos b2 x2 dx = 4b 

2 2

sin b x



1 dx = 4b

  1 cos b2 x2 dx = 4b

b2 π − 8a 8



[a > 0, b > 0] 2

b J − 14 8a

[a > 0, 2

b b 3 − π J 14 8a 8 8a

b > 0]

ET I 84(19)

[a > 0, 2

b J − 14 8a

b > 0]

ET I 83(4), ET I 25(24)

[a > 0,

b > 0]

ET I 25(25)

b2 π − 8a 8



[a > 0,

b > 0]

1. 2. 3.

ET I 83(6)

 π sin 2ab − cos 2ab + e−2ab 2

 



[a > 0,

b > 0]

 π sin 2ab + cos 2ab − e−2ab 2

ET I 24(13)

[a > 0,

b > 0]

ET I 84(12)

b > 0]

ET I 24(14)

 π cos 2ab − sin 2ab + e−2ab 2

√

2π b2 (cos 2ab + sin 2ab) [a > 0, b > 0] sin a x + 2 dx = x 4a 0 √

 ∞ 2π b2 2 2 (cos 2ab − sin 2ab) [a > 0, b > 0] cos a x + 2 dx = x 4a 0 √



 ∞  ∞ 2π b2 b2 2 2 2 2 sin a x − 2ab + 2 dx = cos a x − 2ab + 2 dx = x x 4a 0 0 [a > 0, b > 0] 

∞

ET I 83(9)

 π sin 2ab + cos 2ab + e−2ab 2

[a > 0, 3.699

ET I 83(2)



2

 √ aπ sin(bx) dx = √ J 1 2a b 2 b   1 sin b2 x2 dx = 4b

2

b2 b 3 1 − π J4 8a 8 8a





3.696



2 2

BI (70)(27)

BI (70)(28)

BI(179)(11, 12)a, ET I 83(6)

3.715

Trigonometric functions: complicated arguments

√

2π −2ab b2 e 4. sin a x − 2 dx = [a > 0, x 4a 0 √

 ∞ 2π −2ab b2 2 2 e 5. cos a x − 2 dx = [a > 0, x 4a 0  u     πau 3.711 sin a u2 − x2 cos bx dx = √ J 1 u a2 + b 2 2 a2 + b 2 0 

∞

419



2 2

b > 0]

GW (334)(9b)a

b > 0]

GW (334)(9b)a

[a > 0,

b > 0,

u > 0] ET I 27(37)

3.712 

 ∞

1.

sin (axp ) dx =

0



∞

2.

cos (axp ) dx =

Γ



π sin 2p 1

pa p  π Γ p1 cos 2p 1

[a > 0,

p > 1]

EH I 13(40)

[a > 0,

p > 1]

EH I 13(39)

pa p

0

3.713

1 p

∞

1. 0

∞

1  (−b)k − kq+1 a p Γ sin (ax + bx ) dx = p k! p



q

k=0

kq + 1 p

 k(q − p) + 1 π sin 2p [a > 0, b > 0, p > 0,





q > 0] BI (70)(7)



∞

2.

∞

cos (axp + bxq ) dx =

0

1  (−b)k −(kq+1)/p a Γ p k! k=0







kq + 1 k(q − p) + 1 π cos p 2p [a > 0, b > 0, p > 0,



q > 0] BI (70)(8)

3.714  1.

∞

cos (z sinh x) dx = K 0 (z)

[Re z > 0]

WA 202(14)

[Re z > 0]

MO 36

[Re z > 0]

MO 37

0



∞

2.

sin (z cosh x) dx = 0



∞

3. 0

 4. 5.

3.715

π J 0 (z) 2

cos (z cosh x) dx = −

π Y 0 (z) 2

∞

μπ K μ (z) [Re z > 0, 2 0 μ

 π √ 2 1 cos (z cosh x) sin2μ x dx = π Γ μ+ I μ (z) z 2 0  Re z > 0, cos (z sinh x) cosh μx dx = cos



1.

π

sin (z sin x) sin ax dx = sin aπ s 0,a (z) = sin aπ 0

∞  k=1

(12

−

|Re μ| < 1]

Re μ > − 12

WA 202(13)



WH

(−1)k−1 z 2k−1 − a2 ) . . . [(2k − 1)2 − a2 ]

a2 ) (32

[a > 0]

WA 338(13)

420

Trigonometric Functions



π

2.



π/2

3.

sin (z sin x) sin 2x dx = 0



π

sin (z sin x) sin nx dx  π/2 π = [1 − (−1)n ] sin (z sin x) sin nx dx = [1 − (−1)n ] J n (z) 2 0 [n = 0, ±1, ±2, . . .] WA 30(6), GW(334)(153a)

0



1 sin (z sin x) sin nx dx = 2

3.715

−π

2 (sin z − z cos z) z2

LI (43)(14)

π

4.

sin (z sin x) cos ax dx = (1 + cos aπ) s 0,a (z) 0

= (1 + cos aπ)

∞  k=1



(−1)k−1 z 2k−1 (12 − a2 ) (32 − a2 ) . . . [(2k − 1)2 − a2 ] [a > 0]

WA 338(14)

π

5.

sin (z sin x) cos[(2n + 1)x] dx = 0

GW (334)(53b)

0



π

cos (z sin x) sin ax dx = −a (1 − cos aπ) s −1,a (z)   ∞  1 (−1)k−1 z 2k = −a (1 − cos aπ) − 2 + a a2 (22 − a2 ) (42 − a2 ) . . . [(2k)2 − a2 ]

6. 0

k=1



[a > 0]

WA 338(12)

π

7.

cos (z sin x) sin 2nx dx = 0

GW (334)(54a)

0



π

cos (z sin x) cos ax dx = −a sin aπ s −1,a (z)   ∞  1 (−1)k−1 z 2k = −a sin aπ − 2 + a a2 (22 − a2 ) (42 − a2 ) . . . [(2k)2 − a2 ]

8. 0

k=1

 9.

π

cos (z sin x) cos nx dx =

1 2



[a > 0]

WA 338(11)

π

cos (z sin x) cos nx dx  π/2 π n = [1 + (−1) ] cos (z sin x) cos nx dx = [1 + (−1)n ] J n (z) 2 0

0

−π

GW (334)(54b)



π/2

cos (z sin x) cos2n x dx =

10.8 0

 11.

π/2

sin (z cos x) sin 2x dx = 0

π (2n − 1)!! J n (z) 2 zn

2 (sin z − z cos z) z2

[n = 0, 1, 2, . . .]

FI II 486, WA 35a

LI (43)(15)

3.716

Trigonometric functions: complicated arguments



π/2

12.8 0

421

aπ aπ π s 0,a (z) = cosec [Ja (z) − J−a (z)] sin (z cos x) cos ax dx = cos 2 4 2 π aπ = − sec [Ea (z) + E−a (z)] 4 4 ∞ (−1)k−1 z 2k−1 aπ  = cos 2 (12 − a2 ) (32 − a2 ) . . . [(2k − 1)2 − a2 ] k=1



π

13.

sin (z cos x) cos nx dx = 0



1 2

[a > 0]



π

sin (z cos x) cos nx dx = π sin −π

π/2

sin (z cos x) cos[(2n + 1)x] dx = (−1)n

14. 0



π/2

15.11

sin (a cos x) tan x dx = si(a) + 0



π/2 2ν

16.

sin (z cos x) sin 0



π/2

17.7 0

WA 339

nπ J n (z) 2

π J 2n+1 (z) 2

π 2

GW (334)(55b)

WA 30(8)

[a > 0]

BI (43)(17)

√ ν π 2 1 x dx = Γ ν+ Hν (z) 2 z 2   Re ν > − 12

WA 358(1)

aπ s −1,a (z) cos (z cos x) cos ax dx = −a sin 2 aπ aπ π π [Ja (z) + J−a (z)] = cosec [Ea (z) − E−a (z)] = sec 4 2  4 2  ∞  aπ (−1)k−1 z 2k 1 = −a sin − 2+ 2 a a2 (22 − a2 ) (42 − a2 ) . . . [(2k)2 − a2 ] k=1

 18.

π

cos (z cos x) cos nx dx = 0

 19. 20.

21.

3.716

1 2



[a > 0] π

cos (z cos x) cos nx dx = π cos −π

WA 339

nπ J n (z) 2

π/2

π J 2n (z) 2 0

√ ν  π/2 π 2 1 2ν cos (z cos x) sin x dx = Γ ν+ J ν (z) 2 z 2 0   Re ν > − 12 μ

 π √ 2 1 2μ cos (z cos x) sin x dx = π Γ μ+ J μ (z) z 2 0   Re μ > − 12 cos (z cos x) cos 2nx dx = (−1)n ·



sin (a tan x) dx =

 1  −a e Ei(a) − ea Ei(−a) 2

[a > 0]

cos (a tan x) dx =

π −a e 2

[a ≥ 0]

π/2

1. 0

 2.

0

GW (334)(56b)

π/2

(cf. 3.723 1)

WA 30(9)

WA 35, WH

WH

BI (43)(1)

BI (43)(2)

422

Trigonometric Functions



π/2

3.

sin (a tan x) sin 2x dx =

aπ −a e 2

[a ≥ 0]

BI (43)(7)

cos (a tan x) sin2 x dx =

1 − a −a πe 4

[a ≥ 0]

BI (43)(8)

cos (a tan x) cos2 x dx =

1 + a −a πe 4

[a ≥ 0]

BI (43)(9)

[a > 0]

BI (43)(5)

0



π/2

4. 0



π/2

5. 0



π/2

6.

sin (a tan x) tan x dx = 0



π −a e 2

π/2

cos (a tan x) tan x dx = −

7.

3.717

0

 1  −a e Ei(a) + ea Ei(−a) 2 [a > 0]



π/2

sin (a tan x) sin2 x tan x dx =

8. 0

 

[a ≥ 0]

(cf. 3.742 1)

BI (43)(3)

cos2 (a tan x) dx =

 π 1 + e−2a 4

[a ≥ 0]

(cf. 3.742 3)

BI (43)(4)

π/2

0



π/2

sin2 (a tan x) cot2 x dx =

11. 0



π/2

12. 0



 π  −2a e + 2a − 1 4

[a ≥ 0]

BI (43)(19)

  dx =C 1 − sec2 x cos (tan x) tan x

π/2

13.

BI (43)(11)

 π 1 − e−2a 4

0

10.

[a > 0]

BI (43)(6)

sin2 (a tan x) dx =

π/2

9.

2 − a −a πe 4

(cf. 3.723 5)

sin (a cot x) sin 2x dx = 0

BI (51)(14)

aπ −a e 2

[a ≥ 0]

(cf. 3.716 3)

In general, formulas 3.716 remain valid if we replace tan x in the argument of the sine or cosine with cot x if we also replace sin x with cos x, cos x with sin x, hence tan x with cot x, cot x with tan x, sec x with cosec x, and cosec x with sec x in the factors. Analogously,  π/2  π/2 π dx dx = = sin a [a ≥ 0] sin (a cosec x) sin (a cot x) sin (a sec x) sin (a tan x) 3.717 cos x sin x 2 0 0 BI (52)(11, 12)

3.718



π/2

1.

sin 0



π 2

 p − a tan x tanp−1 x dx = 0

π/2

cos

π p − a tan x tanp x dx = e−a 2 2  2  p < 1, p = 0, a ≥ 0

π

BI (44)(5, 6)

π/2

sin (a tan x − νx) sinν−2 x dx = 0

2.

[Re ν > 0,

a > 0]

NH 157(15)

0

 3.

π/2

sin (n tan x + νx) 0

π cosν−1 x dx = sin x 2

[Re ν > 0]

BI (51)(15)

3.722

Trigonometric and rational functions



π/2

cos (a tan x − νx) cosν−2 x dx =

4. 0

πe−a aν−1 Γ(ν)

423

[Re ν > 1,

a > 0] LO V 153(112), NT 157(14)



π/2

5.

cos (a tan x + νx) cosν x dx = 2−ν−1 πe−a

[Re ν > −1,

a ≥ 0]

BI (44)(4)

0



π/2

cos (a tan x − γx) cosν x dx =

6. 0



π/2

7. 0

ν (2a) πa 2 W γ2 ,− ν+1 2   · ν γ+ν +1 22 Γ 1+ 2  a > 0, Re ν > −1,

sin nx − sin (nx − a tan x) cosn−1 x dx = sin x

 ν+γ

= −1, −2, . . . 2

 π/2 [n = 0, −a π (1 − e ) [n = 1,

EH I 274(13)a

a > 0] , a ≥ 0] LO V 153(114)

3.719 1.



π

6

sin (νx − z sin x) dx = π Eν (z)

WA 336(2)

cos (nx − z sin x) dx = π J n (z)

WH

cos (νx − z sin x) dx = π Jν (z)

WA 336(1)

0



π

2. 0

 3.

π

0

3.72–3.74 Combinations of trigonometric and rational functions 3.721



∞

1. 0



∞

2. 1

 3.

∞

8 1

3.722



∞

1. 0

 2.11 3.

π sin(ax) dx = sign a x 2

FI II 645

sin(ax) dx = − si(a) x

BI 203(1)

cos(ax) dx = − ci(a) x

BI 203(5)

sin(ax) dx = ci(aβ) sin(aβ) − cos(aβ) si(aβ) x+β

[|arg β| < π,

a > 0] BI(16)(1), FI II 646a

∞

sin(ax) dx = πeiaβ −∞ x + β  ∞ cos(ax) dx = − sin(aβ) si(aβ) − cos(aβ) ci(aβ) x+β 0

[a > 0,

Im β > 0]

[|arg β| < π,

a > 0] ET I 8(7), BI(160)(2)

424

Trigonometric Functions

 4.

8

5.10

7.10

∞

∞

0

0

4. 5.11 6. 7. 8.

9. 10. 11.

[a > 0,

Im β > 0]

 sin(ax) 1  −aβ e dx = Ei(aβ) − eaβ Ei(−aβ) 2 2 β +x 2β

[a > 0,

β > 0]

cos(ax) π −aβ e dx = 2 2 β +x 2β

[a ≥ 0,

Re β > 0]

ET I 65(14), BI(160)(3)

FI II 741, 750, ET I 8(11), WH ∞

3. 

β not real and positive] ET I 8(8), BI(161)(2)a

2. 

Im β > 0]

∞

0



β not real and positive] FI II 646, BI(161)(1)

cos(ax) dx = −iπeiaβ −∞ β − x

3.723  1.11

Im β > 0]

∞

sin(ax) dx = −πeiaβ [a > 0, −∞ β − x  ∞ cos(ax) dx = − cos(aβ) ci(aβ) + sin(aβ) [si(aβ) + π] β−x 0 [a > 0, 

8.11

∞

cos(ax) dx = −iπeiaβ [a > 0, −∞ x + β  ∞ sin(ax) dx = sin(βa) ci(βa) − cos(βa) [si(βa) + π] β−x 0 [a > 0, 

6.8

3.723

x sin(ax) π dx = e−aβ β 2 + x2 2

[a > 0,

Re β > 0] FI II 741, 750, ET I 65(15), WH

∞

x sin(ax) dx = πe−aβ [a > 0, Re β > 0] BI (202)(10) 2 2 −∞ β + x  ∞  x cos(ax) 1 dx = − e−aβ Ei(aβ) + eaβ Ei(−aβ) [a > 0, β > 0] BI (160)(6) 2 2 β +x 2 0  ∞ sin[a(b − x)] π dx = e−ac sin(ab) [a > 0, b > 0, c > 0] LI (202)(9) 2 + x2 c c −∞  ∞ cos[a(b − x)] π dx = e−ac cos(ab) [a > 0, b > 0, c > 0] LI (202)(11)a 2 + x2 c c −∞  ∞  π , sin(ax) 1+ sin(aβ) ci(aβ) − cos(aβ) si(aβ) + dx = 2 2 β 2 0 β −x BI (161)(3) [|arg β| < π, a > 0]  ∞ cos(ax) π sin(ab) [a > 0, b > 0] dx = BI(161)(5), ET I 9(15) 2 − x2 b 2b 0  ∞ x sin(ax) π [a > 0] dx = − cos(ab) FI II 647, ET II 252(45) 2 − x2 b 2 0  ∞ + x cos(ax) π, dx = cos(aβ) ci(aβ) + sin(aβ) si(aβ) + β 2 + x2 2 0 BI (161)(6) [|arg β| < π, a > 0]

3.726

Trigonometric and rational functions

 12. 3.724 1.

2.

3.

3.725

425

∞

cos(ab) − 1 sin(ax) dx = π x(x − b) b −∞

[a > 0,

b > 0]

 √ 2 cq − b  sin(aq) + c cos(aq) πe−a p−q p − q2   a > 0, p > q 2    ∞ √ b + cx b − cq −a p−q 2  cos(ax) dx = cos(aq) + c sin(aq) πe 2 p − q2 −∞ p + 2qx + x   a > 0, p > q 2  ∞ cos[(b − 1)t] − x cos(bt) cos(ax) dx = πe−a sin t sin (bt + a cos t) 1 − 2x cos t + x2 −∞   a > 0, t2 < π 2 

∞

b + cx sin(ax) dx = 2 −∞ p + 2qx + x



∞

1. 0



∞

2. 0



∞

3. 0

ET II 252(44)



 π  sin(ax) dx = 1 − e−aβ 2 2 2 x (β + x ) 2β

[Re β > 0,

π sin(ax) dx = 2 (1 − cos(ab)) 2 2 x (b − x ) 2b

[a > 0]

π −βb sin(ax) cos(bx) dx = e sinh(aβ) x (x2 + β 2 ) 2β 2 π π = − 2 e−aβ cosh(bβ) + 2 2β 2β

a > 0]

BI (202)(12)

BI (202)(13)

BI (202)(14)

BI (172)(1) BI (172)(4)

[0 < a < b] [a > b > 0] ET I 19(4)

3.726



∞

1.11 0

 2.7 0

x sin(ax) dx ± b2 x + bx2 ± x3  1 + −ab π , =± e Ei(ab) − eab Ei(−ab) − 2 ci(ab) sin(ab) + 2 cos(ab) si(ab) + 4b 2 πe−ab − π cos(ab) + 4b [a > 0, b > 0; if the lower sign is taken, then the integral is a principal value integral]

b3

ET I 65(21)a, BI(176)(10, 13) ∞

x2 sin(ax) dx b3 ± b2 x + bx2 ± x3 =

[a > 0,

b > 0;

 1 + ab π , e Ei(−ab) − e−ab Ei(ab) + 2 ci(ab) sin(ab) − 2 cos(ab) si(ab) + 4 2   −ab + cos(ab) ±π e

if the lower sign is taken, then the integral is a principal value integral] ET I 66(22), BI(176)(11, 14)

426

3.727

Trigonometric Functions



∞

1. 0



∞

2.8 0



∞

3. 0

5. 6.11

√



cos(ax) ab π 2 ab ab dx = exp − √ cos √ + sin √ b4 + x4 4b3 2 2 2 [a > 0, +  π sin(ax) 1 dx = 3 2 sin(ab) ci(ab) − 2 cos(ab) si(ab) + 4 4 b −x 4b 2 , −ab ab Ei(ab) − e Ei(−ab) +e

b > 0]

BI(160)(25)a, ET I 9(19)

[a > 0,

b > 0]

BI (161)(12)

 cos(ax) π  dx = 3 e−ab + sin(ab) b4 − x4 4b [a > 0,



∞

7. 0



∞

0



11. 12.7

b > 0]

(cf. 3.723 5 and 3.723 11)

BI (161)(16)

∞

BI (160)(23)a BI (161)(13)

BI (161)(17)

√



ab x2 cos(ax) ab π 2 ab √ √ √ exp − − sin dx = cos b4 + x4 4b 2 2 2 [a > 0,

8.11

10.

(cf. 3.723 2 and 3.723 9)

∞

[a > 0,

9.

b > 0]

x sin(ax) ab π ab √ sin √ [a > 0, b > 0] dx = exp − 4 + x4 2 b 2b 2 2 0  ∞  x sin(ax) π  dx = 2 e−ab − cos(ab) [a > 0, b > 0] 4 4 b −x 4b 0   ∞  x cos(ax) 1 π dx = 2 cos(ab) ci(ab) + 2 sin(ab) si(ab) + b4 − x4 4b2 2 0  −ab ab − e Ei(ab) − e Ei(−ab) 

4.

3.727

b > 0]

x2 sin(ax) dx 1+ 2 sin(ab) ci(ab) = 4 4 b −x 4b  , π −2 cos(ab) si(ab) + − e−ab Ei(ab) + eab Ei(−ab) 2 [a > 0, b > 0]

 x cos(ax) π  sin(ab) − e−ab [a > 0, dx = 4 4 b −x 4b 0

 ∞ 3 ab x sin(ax) ab π √ exp − cos √ [a > 0, dx = 4 + x4 b 2 2 2 0  ∞ 3  x sin(ax) −π  −ab e dx = − cos(ab) [a > 0, 4 4 b −x 4 0  ∞ 3  π x cos(ax) dx 1 + = 2 cos(ab) ci(ab) + 2 sin(ab) si(ab) + 4 4 b −x 4 2 0 , −ab ab +e Ei(ab) + e Ei(−ab)

BI (160)(26)a

BI (161)(14)

2

[a > 0,

b > 0]

BI (161)(18)

b > 0]

BI (160)(24)

b > 0]

BI (161)(15)

b > 0]

BI(161)(19)

3.729

Trigonometric and rational functions



∞

13.

3

(x2 + b2 )

0



∞

14.



x3 sin ax 4

(x2 + b2 )

0

3.728

x3 sin ax

∞

1. 0

dx =

 πe−ab  3a − ba2 16b

dx =

 πe−ab a  3 + 3ab − a2 b2 3 96b

  π βe−aγ − γe−aβ cos(ax) dx = (β 2 + x2 ) (γ 2 + x2 ) 2βγ (β 2 − γ 2 )

  ∞ π e−aβ − e−aγ x sin(ax) dx = 2 2 2 2 2 (γ 2 − β 2 ) 0 (β + x ) (γ + x )

[a > 0,

427

Re β > 0,

Re γ > 0] BI (175)(1)

 2.

  ∞ π βe−aβ − γe−aγ x2 cos(ax) dx = 2 2 2 2 2 (β 2 − γ 2 ) 0 (β + x ) (γ + x )

[a > 0,

Re β > 0,

Re γ > 0] BI (174)(1)

 3.

  ∞ π β 2 e−aβ − γ 2 e−aγ x3 sin(ax) dx = 2 2 2 2 2 (β 2 − γ 2 ) 0 (β + x ) (γ + x )

[a > 0,

Re β > 0,

Re γ > 0] BI (175)(2)

 4.



(b2

0 ∞

6. 0

 7. 8. 9. 3.729

π (b sin(ac) − c sin(ab)) cos(ax) dx = 2 2 2 − x ) (c − x ) 2bc (b2 − c2 )

π (cos(ab) − cos(ac)) x sin(ax) dx = (b2 − x2 ) (c2 − x2 ) 2 (b2 − c2 )

∞

1. ∞

2.

+

x sin(ax) dx 2

∞

3.

cos(px) 0

 4.

0

=

2 x2 )

(b2 + x2 )

0



cos(ax) dx (b2

0



Re γ > 0]

[a > 0,

b > 0,

c > 0]

[a > 0]

BI (175)(3) BI (174)(3)

∞

x2 cos(ax) dx π (c sin(ac) − b sin(ab)) = 2 2 2 2 2 (b2 − c2 ) 0 (b − x ) (c − x )  2   ∞ π b cos(ab) − c2 cos(ac) x3 sin(ax) dx = 2 2 2 2 2 (b2 − c2 ) 0 (b − x ) (c − x )  ∞ π e−ac − cos ba x sin ax dx = 2 2 2 2 2 a2 + c 2 0 (b − x ) (c + x ) 

Re β > 0,

BI (174)(2) ∞

5. 

[a > 0,

∞

π (1 + ab)e−ab 4b3

=

π −ab ae 4b

1 − x2 (1 + x2 )

x3 sin(ax) dx (b2 + x2 )

2

=

2

dx =

[a > 0,

b > 0,

c > 0]

BI (175)(4)

[a > 0,

b > 0,

c > 0]

BI (174)(4)

[a > 0,

c > 0,

b real]

[a > 0,

b > 0]

BI (170)(7)

[a > 0,

b > 0]

BI (170)(3)

πp −p e 2

π (2 − ab)e−ab 4

BI (43)(10)a

[a > 0,

b > 0]

BI (170)(4)

428

3.731 1.

Trigonometric Functions

√ √ Notation: 2A2 = b4 + c2 + b2 , 2B 2 = b4 + c2 − b2 ,  ∞ cos(ax) dx π e−aA (B cos(aB) + A sin(aB)) √ = 2 2 2 2 2c b 4 + c2 0 (x + b ) + c 

2.

∞

x sin(ax) dx

=

2 b2 )

π −aA e sin(aB) 2c

(x2 + + c2   ∞ 2 2 x + b cos(ax) dx 0

3.

3.731

(x2 +

0



∞

4.



x x2 + b (x2

0

3.732  1.

∞



0



∞

∞



3. 0



∞

4.

+

c2

BI (176)(3)

[a > 0,

b > 0,

c > 0]

BI (176)(1)

[a > 0,

b > 0,

c > 0]

BI (176)(4)

[a > 0,

b > 0,

c > 0]

BI (176)(2)

π e−aA (A cos(aB) − B sin(aB)) √ 2 b 4 + c2

=

π −aA e cos(aB) 2

ET I 65(16)

[a > 0,



∞

1. 0



∞

2. 0

 3. 0

|Im γ| < Re β]

γ+x γ−x − 2 sin(ax) dx = πe−aβ cos(aγ) β 2 + (γ + x)2 β + (γ − x)2 [a > 0, Re β > 0,



ET I 8(13)

γ + iβ is not real] LI (175)(17)



γ+x γ−x + 2 cos(ax) dx = πe−aβ sin(aγ) 2 + (γ + x) β + (γ − x)2 [a > 0,

3.733

γ + iβ is not real]

1 1 π + 2 cos(ax) dx = e−aβ cos(aγ) 2 2 + (γ − x) β + (γ + x) β 

β2

0

sin(ax) dx

2 b2 )

c > 0]



β2

0

+

 2

+ c2

=

b > 0,

 1 1 π − sin(ax) dx = e−aβ sin(aγ) β 2 + (γ − x)2 β 2 + (γ + x)2 β [a > 0, Re β > 0,



2. 

2 b2 )

[a > 0,

|Im a| < Re β]

LI (176)(21)

cos(ax) dx π sin (t + ab sin t) = 3 exp (−ab cos t) x4 + 2b2 x2 cos 2t + b4 2b sin 2t + a > 0, b > 0,

|t| <

π, 2

x sin(ax) dx π sin (ab sin t) = 2 exp (−ab cos t) x4 + 2b2 x2 cos 2t + b4 2b sin 2t+ a > 0,

|t| <

π, 2

b > 0,

BI (176)(7)

BI(176)(5), ET I 66(23) ∞

x4

sin (t − ab sin t) x2 cos(ax) dx π exp (−ab cos t) = + 2b2 x2 cos 2t + b4 2b sin 2t + a > 0,

b > 0,

|t| <

π, 2

BI (176)(8)

3.736

Trigonometric and rational functions

 4.

5.

3.734

429

∞

x3 sin(ax) dx sin (2t − ab sin t) π = exp (−ab cos t) 4 2 2 4 2 sin 2t 0 x + 2b x cos 2t + b + π, a > 0, b > 0, |t| < 2    ∞ π sin(ax) dx sin (2t + ab sin t) = 4 1 − exp (−ab cos t) 4 2 2 4 2b sin 2t 0 x (x + 2b x cos 2t + b ) + π, a > 0, b > 0, |t| < 2



 π sin(ax) dx ab ab 1. = 4 1 − exp − √ cos √ [a > 0, b > 0] 4 4 2b 2 2 0 x (b + x )  ∞  π  sin(ax) dx = 4 2 − e−ab − cos(ab) 2. [a > 0, b > 0] 4 4 4b 0 x (b − x )    ∞ sin(ax) dx π 1 −ab 3.735 = 4 1 − e (2 + ab) [a > 0, b > 0] 2 2 2 2b 2 0 x (b + x ) 3.736  ∞  π  cos(ax) dx 1. = 5 sin(ab) + (2 + ab)e−ab 2 2 4 4 8b 0 (b + x ) (b − x ) 



∞

(b2

0

∞

3. 0



∞

4.

∞

5.

6. 0

BI (172)(10)

WH, BI (172)(22)

[a > 0,

b > 0]

BI (176)(5)

[a > 0,

b > 0]

BI (174)(5)

[a > 0,

b > 0]

BI (175)(6)

[a > 0,

b > 0]

BI (174)(6)

[a > 0,

b > 0]

BI (175)(7)

[a > 0,

b > 0]

BI (174)(7)

 π  x sin(ax) dx = 4 (1 + ab)e−ab − cos(ab) 2 4 4 + x ) (b − x ) 8b

 π  x cos(ax) dx = 3 sin(ab) − abe−ab (b2 + x2 ) (b4 − x4 ) 8b 2

 π  x3 sin(ax) dx = 2 (1 − ab)e−ab − cos(ab) 2 4 4 + x ) (b − x ) 8b

 π  x cos(ax) dx = sin(ab) + (ab − 2)e−ab 2 4 4 + x ) (b − x ) 8b 4

(b2

0



BI (172)(7)

(b2

0



BI (176)(22)

∞

2. 

BI (176)(6)

∞

 π x sin(ax) dx = (ab − 3)e−ab − cos(ab) 2 4 4 + x ) (b − x ) 8 5

(b2

430

3.737

Trigonometric Functions



∞

1.8 0

 2.

3.

3.737

 (2n − k − 2)!(2ab)k cos(ax) dx πe−ab = n (2b)2n−1 (n − 1)! k!(n − k − 1)! (b2 + x2 ) −ab√p   k=0 n−1 n−1 e d (−1) π = 2n−1 √ 2b (n − 1)! dpn−1 p p=1  n−1 −abp  n−1 e d π (−1) = 2n−1 2b (n − 1)! dpn−1 (1 + p)n p=1 [a > 0, b > 0] n−1

GW(333)(67b), WA 209, WA 192

n−1 πae−aβ  (2n − k − 2)!(2aβ)k 2 2 n+1 22n n!β 2n−1 k!(n − k − 1)! 0 (x + β ) k=0 π −aβ = e [n = 0, β ≥ 0] 2 [a > 0, Re β > 0] GW (333)(66c)    ∞ sin(ax) dx π e−aβ = 1 − n Fn (aβ) n+1 2n+2 2 2 2β 2 n! 0  x (β + x )   a > 0, Re β > 0, F0 (z) = 1, F1 (z) = z + 2, . . . , Fn (z) = (z + 2n)Fn−1 (z) − zFn−1 (z)



∞

x sin(ax) dx (b2

0 ∞

5. ∞

6.

+

4 x2 )

=

x sin ax n+1

+ β2)

πa (1 + ab)e−ab 16b3

[a > 0,

b > 0]

BI(170)(5), ET I 67(35)a

 πa  3 + 3ab + a2 b2 e−ab [a > 0, b > 0] BI(170)(6), ET I 67(35)a 96b5  πe−aβ dx = 2n 2n−1 (2n − 3)!!(2 − βa) 2 n!β 2n−2 & n−1  (2n − k − 2)!2k (βa)k−1   k(k + 1) − 2(k + 1)βa + β 2 a2 − k!(n − k − 1)!

=

3

(x2

0

+

3 x2 )

x sin(ax) dx (b2

0



=

GW (333)(66e) ∞

4. 

x sin(ax) dx

k=1

3.738



∞

1. 0

  n xm−1 sin(ax) πβ m−2n  (2k − 1)π dx = − exp −aβ sin x2n + β 2n 2n 2n k=1

 (2k − 1)mπ (2k − 1)π + aβ cos × cos 2n 2n + [m is even] ,

 2. 0

∞

a > 0,

|arg β| <

π , 2n

  n xm−1 cos(ax) πβ m−2n  (2k − 1)π dx = exp −aβ sin x2n + β 2n 2n 2n

k=1  (2k − 1)mπ (2k − 1)π + aβ cos × sin 2n 2n , + π , 0 < m < 2n + 1 [m is odd] , a > 0, |arg β| < 2n

, 0 < m ≤ 2n

ET I 67(38)

BI(160)(29)a, ET I 10(29)

3.741

3.739

Trigonometric and rational functions



∞

1.

sin(ax) dx x (x2 + 22 ) (x2 + 42 ) . . . (x2 + 4n2 )

% n−1

&  π(−1)n k 2n 2(k−n)a n 2n = (−1) + (−1) 2 e k n (2n)!22n+1

0

k=0



[a > 0, ∞

2.

(x2

0

+

12 ) (x2

n ≥ 0]

LI(174)(8)

cos(ax) dx + 32 ) . . . [x2 + (2n + 1)2 ]

n π  (−1)n k 2n + 1 (−1) = e(2k−2n−1)a k (2n + 1)! 22n+1 =



431

[a ≥ 0,

n ≥ 0]

[a = 0,

n ≥ 0]

k=0

π2−2n−1 2

(2n + 1) (n!)

BI(175)(8) ∞

3.

(x2

0

+

12 ) (x2

x sin(ax) dx + 32 ) . . . [x2 + (2n + 1)2 ]

n  π(−1)n k 2n + 1 (−1) = (2n − 2k + 1)e(2k−2n−1)a k (2n + 1)!22n+1 k=0

[a > 0, n ≥ 0]

n ∞ 2n π21−2n  cos ax dx k = (−1) k e−2ak 2 2 2 2 2 2 n−k (2n)! 0 (x + 2 ) (x + 4 ) . . . (x + 4n )

LI (174)(9)

 4.

k=1

3.741



∞

1. 0



∞

2. 0

 3. 0

sin(ax) sin(bx) 1 dx = ln x 4 sin(ax) cos(bx) π dx = x 2 π = 4 =0



a+b a−b

[n ≥ 1,

a ≥ 0]

[a > 0,

b > 0,

2 a = b]

FI II 647

[a > b ≥ 0] [a = b > 0] [b > a ≥ 0] FI II 645

∞

sin(ax) sin(bx) aπ dx = x2 2 bπ = 2

[0 < a ≤ b] [0 < b ≤ a] BI (157)(1)

432

Trigonometric Functions

3.742  1.

∞

0



b ≥ a ≥ 0]

[a > 0,

b > 0,

Re β > 0]

[β > 0,

a ≥ b ≥ 0]

[β > 0,

b ≥ a ≥ 0] BI(163)(1)a, GW(333)(71c)

∞

  x cos(ax) cos(bx) 1 dx = − eaβ ebβ Ei [− β(a + b)] + e−bβ Ei [β(b − a)] 2 2 β +x 4   1 − e−aβ ebβ Ei [β(a − b)] + e−bβ Ei [β(a + b)] 4 =∞

[a = b] [a = b] BI (163)(2)

∞

5. 0

x sin(ax) cos(bx) π dx = e−aβ cosh(bβ) x2 + β 2 2 π −2aβ = e 4 π = − e−bβ sinh(aβ) 2

[0 < b < a] [0 < b = a] [0 < a < b] BI (162)(4)

∞

6. 0

0

[β > 0,

BI (162)(3)

0

7.

a ≥ b ≥ 0]

, cos(ax) cos(bx) π + −|a−b|β e dx = + e−(a+b)β 2 2 β +x 4β π −aβ = e cosh bβ 2β π −bβ e = cosh aβ 2β

4.



[β > 0,

∞

0



Re β > 0]

BI(162)(1)a, GW(333)(71a)

3.



b > 0,

 sin(ax) cos(bx) 1 −aβ  bβ e e Ei [β(a − b)] + e−bβ Ei [β(a + b)] dx = 2 2 β +x 4β   1 − eaβ ebβ Ei [− β(a + b)] + e−bβ Ei [β(b − a)] 4β

0



[a > 0,

∞

2.



sin(ax) sin(bx) π  −|a−b|β −(a+b)β dx = − e e β 2 + x2 4β π −aβ e sinh bβ = 2β π −bβ e = sinh aβ 2β

3.742

sin(ax) sin(bx) π dx = − cos(ap) sin(bp) 2 2 p −x 2p π = − sin(2ap) 4p π = − sin(ap) cos(bp) 2p

[a > b > 0] [a = b > 0] [b > a > 0] BI (166)(1)

∞

sin(ax) cos(bx) π x dx = − cos(ap) cos(bp) p2 − x2 2 π = − cos(2ap) 4 π = sin(ap) sin(bp) 2

[a > b > 0] [a = b > 0] [b > a > 0] BI (166)(2)

3.747

Trigonometric and rational functions



∞

8. 0

cos(ax) cos(bx) π sin(ap) cos(bp) dx = p2 − x2 2p π = sin(2ap) 4p π = cos(ap) sin(bp) 2p

433

[a > b > 0] [a = b > 0] [b > a > 0] BI (166)(3)

3.743



∞

1. 0



∞

2. 0



∞

3. 0

 4. 5.6

Re β > 0]

ET I 80(21)

x dx sin(ax) π sinh(aβ) · 2 =− · 2 cos(bx) x + β 2 cosh(bβ)

[0 < a < b,

Re β > 0]

ET I 81(30)

x dx cos(ax) π cosh(aβ) · = · sin(bx) x2 + β 2 2 sinh(bβ)

[0 < a < b,

Re β > 0]

ET I 23(37)

[0 < a < b,

Re β > 0]

ET I 23(36)

dx cos(ax) π cosh(aβ) · 2 · = 2 cos(bx) x + β 2β cosh(bβ) 0  ∞ dx sin(ax) · 2 PV =0 sin x b − x2 0 π = sin(a − 1)b b ∞

sin(ax) cos(bx) 0 ∞ sin(ax) cos(bx) 0

3.7443 3.7453 

xn+1

0



∞

2. 0



n dx -

∞

1.

3.747

[0 < a < b,

∞



3.746

dx sin(ax) π sinh(aβ) · · = sin(bx) x2 + β 2 2β sinh(bβ)

π/2

π/2

0 ∞

3. 0



dx π sinh(aβ) = · x (x2 + β 2 ) 2β 2 cosh(bβ) dx · =0 x (c2 − x2 )

Re β > 0]

ET I 82(32)

[0 < a < b,

c > 0]

ET I 82(31)

%

n π sin (ak x) = ak 2

a0 >

k=1

k=1

⎡ ⎣a >

n 

& ak ,

ak > 0

k=1 n  k=1

|ak | +

m 

FI II 646

⎤ |bj |⎦

WH

j=1

& % ∞  π m 1  7 xm 22k−1 − 1 dx = + ζ(2k) = 2πG − ζ(3) sin x 2 m 42k−1 (m + 2k) 2

x dx = sin x

 π/2 π



2 0

 − x dx = 2G cos x

π x dx = (x2 + b2 ) sin(ax) 2 sinh(ab)

[m = 2]

LI (206)(2)

BI(204)(18), BI(206)(1), GW(333)(32)

[b > 0]

GW (333)(79c)

π

x tan x dx = −π ln 2 0

[0 < a < b,

k=1

2.

4.

[b real, b/π ∈ Z]

·

k=1

0



if 1 ≤ a ≤ 2

n m n sin(ax) π dx sin (a x) cos (b x) = ak k j xn+1 2 j=1

1.7



k=0

if 0 ≤ a ≤ 1

BI (218)(4)

434

Trigonometric Functions



3.748

π/2

x tan x dx = ∞

5.

BI (205)(2)

0



π/4

x tan x dx = −

6. 0



π/2

7.

x cot x dx = 0

 8. 9. 10. 11. 3.748 1.

2.

3.

1 π ln 2 + G = 0.1857845358 . . . 8 2

BI (204)(1)

π ln 2 2

FI II 623

π/4

1 π ln 2 + G = 0.7301810584 . . . 8 2 0   π/2  π 1 π π π − x tan x dx = − x tan x dx = ln 2 2 2 0 2 2 0  ∞ π dx = [a > 0] tan ax x 2 0  π/2 π x cot x dx = ln 2 cos 2x 4 0 x cot x dx =

BI (204)(2)

GW(333)(33b), BI(218)(12) LO V 279(5)

BI (206)(12)

 ∞  1  π m  4k − 1 ζ(2k) 2 4 42k−1 (m + 2k) 0 k=1    π/2 ∞  π p 1  1 p −2 ζ(2k) x cot x dx = 2 p 4k (p + 2k) 0 k=1    π/4 ∞  ζ(2k) 1  π m 2 m − x cot x dx = 2 4 m 42k−1 (m + 2k) 0 

π/4

xm tan x dx =

LI (204)(5)

LI (205)(7)

LI (204)(6)

k=1

3.749



∞

1. 0

 2. 3.

x tan(ax) dx π = 2ab x2 + b2 e +1

[a > 0,

b > 0]

GW (333)(79a)

∞

x cot(ax) dx π [a > 0, b > 0] = 2ab 2 + b2 x e −1 0  ∞  ∞  ∞ x tan(ax) dx x cot(ax) dx x cosec(ax) dx = = =∞ 2 2 2 2 b −x b −x b2 − x2 0 0 0

GW (333)(79b) BI (161)(7, 8, 9)

3.75 Combinations of trigonometric and algebraic functions 3.751



∞

1. 0

 2.9 0

∞

sin(ax) dx √ = x+β cos(ax) dx √ = x+β





  , π + cos(aβ) − sin(aβ) + 2 C aβ sin(aβ) − 2 S aβ cos(aβ) 2a ET I 65(12)a [a > 0, |arg β| < π]  ,  π + aβ cos(aβ) − 2 S aβ sin(aβ) cos(aβ) + sin(aβ) − 2 C 2a

[a > 0,

|arg β| < π]

ET I 8(9)a

3.755

Trigonometric and algebraic functions



∞

3. u  ∞

4. u

3.752 1.



1

8

sin(ax) √ dx = x−u cos(ax) √ dx = x−u

 



1

1

1.8 0

 2. 3. 4. 5. 3.754

u > 0]

ET I 65(13)

π [cos(au) − sin(au)] 2a

[a > 0,

u > 0]

ET I 8(10)

π (−1)k a2k+1 = H1 (a) (2k − 1)!!(2k + 3)!! 2a [a > 0]

0



[a > 0,

k=0

2. 3.753

π [sin(au) + cos(au)] 2a

∞   sin(ax) 1 − x2 dx =

0

KU 65(6)a

∞

sin(ax) dx  (−1)k a2k+1 π √ = 2 = 2 H0 (a) 2 1−x [(2k + 1)!!] k=0

∞

0 ∞

2. 0

 0

3.755 ∞

0

2. 0

WA 30(7)a

[a > 0]

WA 200(14) WA 200(15)

[a > 0]

WA 30(6)

sin(ax) dx π  = [I 0 (aβ) − L0 (aβ)] 2 2 2 β +x

[a > 0,

Re β > 0]

cos(ax) dx  = K 0 (aβ) β 2 + x2

[a > 0,

Re β > 0]

x sin(ax) < dx = a K 0 (aβ) 3 2 2 (β + x ) <

1. 

BI (149)(9)

ET I 66(26)

WA 191(1), GW(333)(78a) ∞

3.



[a > 0]

1

1. 

BI (149)(6)

 π J 1 (a) cos(ax) 1 − x2 dx = 2a

cos(ax) dx π √ = J 0 (a) 2 2 1−x 0  ∞ sin(ax) dx π √ = J 0 (a) 2 2 x −1 1  ∞ cos(ax) π √ dx = − Y 0 (a) 2−1 2 x 1  1 x sin(ax) π √ dx = J 1 (a) 2 2 1−x 0 

435

∞

<

x2 + β 2 − β sin(ax) dx  = x2 + β 2 x2 + β 2 + β cos(ax) dx  = x2 + β 2

[a > 0,





π −aβ e 2a

[a > 0]

π −aβ e 2a

[a > 0,

Re β > 0]

ET I 66(27)

ET I 66(31)

Re β > 0]

ET I 10(25)

436

3.756

Trigonometric Functions



∞

1. 0

 2.

x

n 2 −1

cos(ax)

0

3.757



∞

0

2.

n -

ak > 0,

a>

%

cos (ak x) dx = 0

∞

11 0

sin(ax) √ dx = x cos(ax) √ dx = x

 

n 

& ak

k=2

ak > 0,

a>

k=1

1.11 

%

n sin(ax) sin (ak x) dx = 0 n x 2 −1 k=2

∞

3.756

n 

ET I 80(22)

& ak

ET I 22(26)

k=1

π 2a

[a > 0]

BI (177)(1)

π 2a

[a > 0]

BI (177)(2)

3.76–3.77 Combinations of trigonometric functions and powers 3.761



1

xμ−1 sin(ax) dx =

1. 0

−i [ 1 F 1 (μ; μ + 1; ia) − 2μ

1F 1

(μ; μ + 1; −ia)] [a > 0,



u

∞

3. 1



μ = 0] ET I 68(2)a

∞

2.8 

Re μ > −1,

 π i  − π iμ e 2 Γ(μ, iu) − e 2 iμ Γ(μ, −iu) xμ−1 sin x dx = 2

sin(ax) a2n−1 dx = x2n (2n − 1)!

%2n−1  (2n − k − 1)! k=1

a2n−k

[Re μ < 1]

&

 π + (−1)n ci(a) sin a + (k − 1) 2 [a > 0]

∞

4.

xμ−1 sin(ax) dx =

0

EH II 149(2)

μπ 2

π sec Γ(μ) μπ = μ sin aμ 2 2a Γ(1 − μ)

[a > 0;

LI (203)(15)

0 < |Re μ| < 1] FI II 809a, BI(150)(1)



π

m/2

m! (−1)n+1  (nπ)m−2k (−1)k nm+1 (m − 2k)! k=0= > = > m! m−2 m m/2

2 −1 −(−1) nm+1

xm sin(nx) dx =

5.10 0

GW(333)(6)



1

xμ−1 cos(ax) dx =

6.8 0



1 [ 1 F 1 (μ; μ + 1; ia) + 2μ

1F 1

(μ; μ + 1; −ia)] [a > 0,

∞

7. u

Re μ > 0]

ET I 11(2)

 π 1  − π iμ e 2 Γ(μ, iu)s + e 2 iμ Γ(μ, −iu) xμ−1 cos x dx = 2 [Re μ < 1]

EH II 149(1)

3.763

Trigonometric functions and powers



∞

8. 1



437

% 2n &  cos(ax) a2n  (2n − k)! π + (−1)n+1 ci(a) dx = cos a + (k − 1) x2n+1 (2n)! a2n−k+1 2 k=1

[a > 0] ∞

9.8

xμ−1 cos(ax) dx =

0

μπ 2

π cosec Γ(μ) μπ = μ cos μ a 2 2a Γ(1 − μ)

[a > 0,

LI (203)(16)

0 < Re μ < 1] FI II 809a, BI(150)(2)



π

(−1)n nm+1

(m−1)/2



m! (nπ)m−2k−1 (m − 2k − 1)! k=0 (m+1)/2 2[(m + 1)/2] − m +(−1) · m! nm+1

xm cos(nx) dx =

10. 0

(−1)k

GW (333)(7)



m/2

π/2



m

11.

x cos x dx = 0

(−1)k

k=0

 :m;  π m−2k m! − m m! + (−1) m/2 2 (m − 2k)! 2 2 GW (333)(9c)



2nπ

xm cos kx dx = −

12. 0

3.762



m−1  j=0

∞

1.

j! k j+1

xμ−1 sin(ax) sin(bx) dx =

0



m j+1 π (2nπ)m−j cos j 2

BI (226)(2)

, + μπ 1 −μ cos Γ(μ) |b − a| − (b + a)−μ 2 2 [a > 0, b > 0,

a = b,

−2 < Re μ < 1]

(for μ = 0, see 3.741 1, for μ = −1, see 3.741 3) 

BI(149)(7), ET I 321(40) ∞

2.

[a > 0, ∞

3.

, μπ 1 −μ sin Γ(μ) (a + b)−μ + |a − b| sign(a − b) 2 2 b > 0, |Re μ| < 1] (for μ = 0 see 3.741 2) BI(159)(8)a, ET I 321(41)

xμ−1 sin(ax) cos(bx) dx =

0



+

xμ−1 cos(ax) cos(bx) dx =

0

+ , μπ 1 −μ cos Γ(μ) (a + b)−μ + |a − b| 2 2 [a > 0, b > 0,

0 < Re μ < 1] ET I 20(17)

3.763



1. 0

∞

 νπ sin(ax) sin(bx) sin(cx) 1 ν−1 cos Γ(1 − ν) (c + a − b) dx = − (c + a + b)ν−1 xν 4 2  ν−1 ν−1 − |c − a + b| sign(a − b − c) + |c − a − b| sign(a + b − c) [c > 0,

0 < Re ν < 4,

ν = 1, 2, 3,

a ≥ b > 0]

GW(333)(26a)a, ET I 79(13)

438

Trigonometric Functions



∞

2. 0



∞

3. 0



∞

4. 0

3.764  1. 2.

sin(ax) sin(bx) sin(cx) dx = 0 x π = 8 π = 4

[c < a − b and c > a + b] [c = a − b and c = a + b] [a − b < c < a + b] [a ≥ b > 0,

3.765  1.10

∞

c > 0]

FI II 645

sin(ax) sin(bx) sin(cx) 1 dx = (c + a + b) ln(c + a + b) 2 x 4 1 1 − (c + a − b) ln(c + a − b) − |c − a − b| ln |c − a − b| 4 4 1 × sign(a + b − c) + |c − a + b| ln |c − a + b| sign(a − b − c) 4 [a ≥ b > 0, c > 0] BI(157)(8)a, ET I 79(11) sin(ax) sin(bx) sin(cx) πbc dx = x3 2 πbc π(a − b − c)2 − = 2 8

[0 < c < a − b and c > a + b] [a − b < c < a + b] [a ≥ b > 0,

 pπ Γ(1 + p) cos b + 2 0  ∞  1 πp xp cos(ax + b) dx = − p+1 Γ(1 + p) sin b + a 2 0 ∞

3.764

xp sin(ax + b) dx =

1

ap+1

c > 0]

BI(157)(20), ET I 79(12)

[a > 0,

−1 < p < 0]

GW (333)(30a)

[a > 0,

−1 < p < 0]

GW (333)(30b)

sin ax dx xν (x + b)



1 πν ν 3 ν Γ(−1 − ν) 1 F 2 1; 1 + , + ; − a2 b2 sign(a) 2 2 2 2 4

ν 1 2 2 π cosec(πν) sin(ab) ν πν ν − − a Γ(−ν) 1 F 2 1; 1 + , 1 + ; − a b sign(a) sin bν 2 2 4 2 [Im a = 0, −1 < Re b < 2, arg b = π] MC

0

= a1+ν b cos



∞

2. 0

 Γ(1 − ν)  iaβ cos(ax) dx = e Γ(ν, iaβ) + e−iaβ Γ(ν, −iaβ) ν ν x (x + β) 2β [a > 0, |Re ν| < 1,

|arg β| < π] ET II 221(52)

3.766  1.10 0

∞

xμ−1 sin ax dx 1 + x2



πμ 3 − μ 4 − μ a2 πμ π = −a2−μ Γ(μ − 2) 1 F 2 1; , ; + sec sinh(a) sign(a) sin 2 2 4 2 2 2 MC [Im a = 0, −1 < Re μ < 3]

3.768

Trigonometric functions and powers



∞

xμ−1 cos(ax) μπ π cosh a dx = cosec 2 1+x 2 2 1 μπ + cos Γ(μ) {exp [−a + iπ(1 − μ)] γ(1 − μ, −a) − ea γ(1 − μ, a)} 2 2 [a > 0, 0 < Re μ < 3] ET I 319(24)

∞

x2μ+1 sin(ax) dx π = − b2μ sec(μπ) sinh(ab) x2 + b2 2 sin(μπ) + Γ(2μ) [ 1 F 1 (1; 1 − 2μ; ab) + 1 F 1 (1; 1 − 2μ; −ab)] 2a2μ   a > 0, − 32 < Re μ < 12 ET II 220(39)

2. 0

 3.9 0



∞

4.9 0



 1 x2μ+1 cos(ax) dx π 2(μ+ 12 ) b cosec μ + = − π cosh(ab) x2 + b2 2  2   

 cos μ + 12 π 1 Γ 2 μ+ + 1 F 1 1; 1 − 2 μ + 1 2  2a2(μ+ 2 )

1 + 1 F 1 1; 1 − 2 μ + ; −ab 2  a > 0, −1 < Re μ <

3.767 

∞ xβ−1

1.

∞ xβ

2.



cos ax −

∞ xβ−1

3.

βπ 2



sin ax −

∞ xβ

4.



cos ax − x2 − b2

0

1 2





; ab



ET II 221(56)

π dx = − γ β−2 e−aγ 2

[a > 0,

Re γ > 0,

0 < Re β < 2] BI (160)(20)

dx =

βπ 2

βπ 2

1 2



x2 − b2

0



βπ 2

γ 2 + x2

0



 sin ax − γ 2 + x2

0



439

π β−1 −aγ γ e 2

[a > 0,

Re γ > 0,

|Re β| < 1] BI (160)(21)

dx =



π β−2 πβ b cos ab − 2 2

[a > 0,

b > 0,

0 < Re β < 2] BI (161)(11)



π β−1 πβ dx = − b sin ab − 2 2

[a > 0,

b > 0,

|β| < 1] GW (333)(82)

3.768 1. 2. 3.11



 Γ(μ) μπ [a > 0, 0 < Re μ < 1] sin au + aμ 2 u  ∞  Γ(μ) μπ [a > 0, 0 < Re μ < 1] (x − u)μ−1 cos(ax) dx = μ cos au + a 2 u  1 1 Γ(ν + 1) (1 − x)ν sin(ax) dx = − C ν (a) = a−ν−1/2 s ν+1/2,1/2 (a) a aν+1 0 [a > 0, Re ν > −1] ∞

(x − u)μ−1 sin(ax) dx =

ET II 203(19) ET II 204(24)

ET I 11(3)a

440

Trigonometric Functions

3.768

Here C ν (a) is the Young’s function given by: C ν (a) = 

1 ν 2a

Γ(ν + 1)

[ 1 F 1 (1; ν + 1; ia) +

1F 1

(1 − x)ν cos(ax) dx = 0



[a > 0, u

xν−1 (u − x)μ−1 sin(ax) dx =

5. 0

u

μ+ν−1

2i



u

xν−1 (u − x)μ−1 cos(ax) dx =

6. 0



Re μ > 0,

1F 1

Re ν > −1,

ET I 11(3)a

(ν; μ + ν; −iau)] ν = 0]

ET II 189(26)

uμ+ν−1 B(μ, ν) [ 1 F 1 (ν; μ + ν; iau) + 1 F 1 (ν; μ + ν; −iau)] 2 [a > 0, Re μ > 0, Re ν > 0] ET II 189(32)

u

7. 0



Re ν > −1]

B(μ, ν) [ 1 F 1 (ν; μ + ν; iau) − [a > 0,

9.

∞  (−1)n aν+2n . Γ(ν + 2n + 1) n=0



 i i −ν−1 a exp (νπ − 2a) γ (ν + 1, −ia) 2 2    i − exp − (νπ − 2a) γ (ν + 1, ia) 2  2 n ∞  −a = Γ(ν + 1) Γ(ν + 2 + 2n) n=0

1

4.3

8.

(1; ν + 1; −ia)] =

 au √  u μ−1/2 au Γ(μ) J μ−1/2 xμ−1 (u − x)μ−1 sin(ax) dx = π sin a 2 2 [Re μ > 0]

ET II 189(25)

∞

xμ−1 (x − u)μ−1 sin(ax) dx u √  μ−1/2  au  au , + au π u au J 1/2−μ − sin Y 1/2−μ = Γ(μ) cos 2 a 2 2 2 2 a > 0, 0 < Re μ < 12 ET II 203(20)  u  au √  u μ− 12 au Γ(μ) J μ− 12 xμ−1 (u − x)μ−1 cos(ax) dx = π cos a 2 2 0 ET II 189(31) [Re μ > 0] √ ∞ +  au  au , π  u μ− 12 au au J 12 −μ Y 12 −μ xμ−1 (x − u)μ−1 cos(ax) dx = − Γ(μ) sin − cos 2 a 2 2 2 2 u   1 ET II 204(25) a > 0, 0 < Re μ < 2  1 i xν−1 (1 − x)μ−1 sin(ax) dx = − B(μ, ν) [ 1 F 1 (ν; ν + μ; ia) − 1 F 1 (ν; ν + μ; −ia)] 2 0 [Re μ > 0, Re ν > −1, ν = 0]

 10.

11.3

ET I 68 (5)a, ET I 317(5)



1

xν−1 (1 − x)μ−1 cos(ax) dx =

12.3 0

1 B(μ, ν) [ 1 F 1 (ν; ν + μ; ia) + 2

1F 1

[Re μ > 0,

(ν; ν + μ; −ia)] Re ν > 0]

ET I 11(5)

3.771

Trigonometric functions and powers



√

1

xμ (1 − x)μ sin(2ax) dx =

13. 0



1

xμ (1 − x)μ cos(2ax) dx =

14. 0

π

1 (2a)μ+ 2

Γ(μ + 1) J μ+ 12 (a) sin a [a > 0,

√ π 1

(2a)μ+ 2



∞

1. 0

 0

  πiaν−1 e−aβ (β + ix)−ν − (β − ix)−ν sin(ax) dx = − Γ(ν) [a > 0,

  πa x (β + ix)−ν + (β − ix)−ν sin(ax) dx = −

0

(ν − 1 − aβ) −aβ e Γ(ν) [a > 0, Re β > 0,

Re ν > 0]

  (−1) (2n)!πaν−2n−1 e−aβ Lν−2n−1 x2n (β + ix)−ν + (β − ix)−ν cos(ax) dx = (aβ) 2n Γ(ν) [a > 0, Re β > 0, 0 ≤ 2n < Re ν] n

ET I 13(20) ∞

0

∞

7. 0

0

ν−2

ET I 70(17)

6.

1.

Re ν > 0]

∞

0



Re β > 0,

  (−1)n i (2n)!πaν−2n−1 e−aβ Lν−2n−1 x2n (β − ix)−ν − (β + ix)−ν sin(ax) dx = (aβ) 2n Γ(ν) [a > 0, Re β > 0, 0 ≤ 2n < Re ν]

5.

3.771

Re ν > 0]

ET I 70(16)

0



Re β > 0,

∞

4.



ET I 11(4)

ET I 13(19) ∞

3.



Re μ > −1]

  πaν−1 e−aβ (β + ix)−ν + (β − ix)−ν cos(ax) dx = Γ(ν) [a > 0,



ET I 68(4)

ET I 70(15) ∞

2.



Re μ > −1]

Γ(μ + 1) J μ+ 12 (a) cos a [a > 0,

3.769

441

∞

+

−ν

x2n+1 (β + ix)−ν + (β − ix)

,

(−1)n+1 (2n + 1)!πaν−2n−2 e−aβ Lν−2n−2 (aβ) 2n+1 Γ(ν) [a > 0, Re β > 0, −1 ≤ 2n + 1 < Re ν] ET I 70(18)

sin(ax) dx =

+ , (−1)n+1 −ν (2n + 1)!πaν−2n−2 e−aβ Lν−2n−2 x2n+1 (β + ix)−ν − (β − ix) (aβ) cos(ax) dx = 2n+1 Γ(ν) [a > 0, Re β > 0, 0 ≤ 2n < Re ν − 1] ET I 13(21)

√ ν  2  1 π 2β 1 2 ν− 2 β +x sin(ax) dx = Γ ν+ [I −ν (aβ) − Lν (aβ)] 2 a 2   a > 0, Re β > 0, Re ν < 12 , ν = − 12 , − 32 , − 52 , . . . EH II 38a, ET I 68(6)

442

Trigonometric Functions



∞

2. 0

 3.

 2 ν− 12 1 β + x2 cos(ax) dx = √ π

2β a

ν



1 cos(πν) Γ ν + K −ν (aβ)  2 a > 0, Re β > 0,

 μ−1 x2ν−1 u2 − x2 sin(ax) dx

a 2μ+2ν−1 = u B μ, ν + 2

1 3 1F 2 ν + ; , μ + ν + 2 2  Re μ > 0, Re ν > − 12

 u  2  a2 u 2 1 2μ+2ν−2 1 2ν−1 2 μ−1 u −x x cos(ax) dx = u B(μ, ν) 1 F 2 ν; , μ + ν; − 2 2 4 0 

∞

5.7

ν− 12  x x2 + β 2

0



u

6.



u2 − x2

ν− 12

∞

7.

 2 ν− 12 x − u2



u2 − x2

ν− 12

0

∞

9.

10. 0



1 a2 u 2 ;− 2 4



ET II 189(29)

ν

√ ν π 2u 1 Γ ν+ Hν (au) 2 a 2  a > 0,

√ ν π 2u 1 cos(ax) dx = Γ ν+ J ν (au) 2 a 2  a > 0,

 2 ν− 12 x − u2

u





u > 0,

Re ν > − 12



ET I 69(7), WA 358(1)a

|Re ν| <

1 2



EH II 81(12)a, ET I 69(8), WA 187(3)a

u

8.



1 2

√ ν π 2u 1 sin(ax) dx = Γ ν+ J −ν (au) 2 a 2  a > 0, u > 0,

u





[Re μ > 0, Re ν > 0] ET II 190(35)

2β 1 1 sin(ax) dx = √ β cos νπ Γ ν + K ν+1 (aβ) 2 π a

ν √ 2β 1 1  K ν+1 (aβ) = πβ a Γ 2 −ν [a > 0, Re β > 0, Re ν < 0] ET I 69(11)

sin(ax) dx =

0



1 2

Re ν <

WA 191(1)a, GW(333)(78)a u

0

4.



3.771

u > 0,

√ ν π 2u 1 cos(ax) dx = − Γ ν+ Y −ν (au) 2 a 2  a > 0, u > 0,

Re ν > − 12

 ET I 11(8)

|Re ν| <

1 2



WA 187(4)a, EH II 82(13)a, ET I 11(9)

u

ν− 12  x u2 − x2

√ ν 2u π 1 u sin(ax) dx = Γ ν+ J ν+1 (au) 2 a 2  a > 0, u > 0,

Re ν > − 12

 ET I 69(9)

3.772

Trigonometric functions and powers



∞

11.



x x −u 2

 1 2 ν− 2

u



√

π u sin(ax) dx = 2

  x u2 − x2

ν− 12

∞

13.



x x −u

u



2

 2 ν−1/2

 2 ν−1/2 x + 2βx sin(ax) dx =

∞

 2 ν−1/2 x + 2βx cos(ax) dx

0





1 Γ ν+ Y −ν−1 (au) 2 a > 0, u > 0,

− 12 < Re ν < 0



ν+1

√ ν πu 2u 1 cos(ax) dx Γ ν+ J −ν−1 (au) 2 a   2 a > 0, u > 0, 0 < Re ν < 12

∞

1.

2.

ν

u s (ν−1)ν+1 (au) ν a

−1

√ ν 2u π 1 1 1 2ν+1 u u − Γ ν+ = ν+ Hν+1 (au) 2 2 a 2   2 a > 0, u > 0, Re ν > − 12 ET I 12(10)

cos(ax) dx = −

0

3.772

2u a

ET I 69(10) u

12.7





443

ET I 12(11)

√ ν π 2β 1 Γ ν+ [J −ν (aβ) cos(aβ) + Y −ν (aβ) sin(aβ)] 2 a  2  a > 0, |arg β| < π, 12 > Re ν > − 32 ET I 69(12)

√ ν π 2β 1 Γ ν+ [Y −ν (aβ) cos(aβ) − J −ν (aβ) sin(aβ)] 2 a 2  a > 0, |Re ν| < 12 ET I 12(13) ν

 2u  ν−1/2 √ 2u 1 2ux − x2 sin(ax) dx = π Γ ν+ sin(au) J ν (au) a 2 0   a > 0, u > 0, Re ν > − 12 0

=−

3.



ET I 69(13)a

∞

4. 2u



2u

5.

√ ν  2 ν−1/2 π 2β 1 x − 2ux sin(ax) dx = Γ ν+ [J −ν (au) cos(au) − Y −ν (au) sin(au)] 2 a  2 a > 0, u > 0, |Re ν| < 12 ET I 70(14)  ν−1/2 √ 2ux − x2 cos(ax) dx = π

0





2u a

ν



1 Γ ν+ J ν (au) cos(au) 2  a > 0, u > 0, Re ν > − 12 ET I 12(4)

∞

6.

 2 ν−1/2 x − 2ux cos(ax) dx

2u

=−

√ ν π 2u 1 Γ ν+ [J −ν (au) sin(au) + Y −ν (au) cos(au)] 2 a  2 a > 0, u > 0, |Re ν| < 12 ET I 12(12)

444

3.773 1.

Trigonometric Functions



∞

x2ν

8

μ+1

(x2 + β 2 )

0

3.773

sin(ax) dx

1 2ν−2μ 3 β 2 a2 β a B (1 + ν, μ − ν) 1 F 2 ν + 1; ν + 1 − μ, ; 2√ 2 4

β 2 a2 πa2μ−2ν+1 Γ(ν − μ) 3   + 1 F 2 μ + 1; μ − ν + , μ − ν + 1; 4μ−ν+1 Γ μ − ν + 32 2 4 !   √ 2 2 ! −ν + 1 π a β ! 2 β 2ν−2μ−1 G 21 = ! 13 2 Γ(μ + 1) 4 ! μ − ν + 12 , 12 , 0 [a > 0, Re β > 0, −1 < Re ν < Re μ + 1] ET I 71(28)a, ET II 234(17)

=

 2.

∞

8

x2m+1 sin(ax) (z +

0

n+1 x2 )

dx =

(−1)n+m π dn  m −a√z · z e n! 2 dz n [a > 0,

√ ∞ 2m+1 x sin(ax) dx (−1)m+1 π d2m+1 n   = [a K n (aβ)] n+ 1 2n β n Γ n + 12 da2m+1 0 (β 2 + x2 ) 2 [a > 0,

0 ≤ m ≤ n,

|arg z| < π] ET I 68(39)

 3.

 0

∞

5.





x cos(ax) dx 1 2ν−2μ−1 1 1 1 β 2 a2 1 1 B ν + ,μ − ν + 1F 2 ν + ; ν − μ + , ; μ+1 = 2 β 2 2 2 2 2 4 (x2 + β 2 ) 

√ 2μ−2ν+1  1 Γ ν−μ− 2 πa 3 β 2 a2 ; + μ + 1; μ − ν + 1, μ − ν + 1F 2 4μ−ν+1 Γ(μ − ν +1) 2 4 !  √ 2 2 ! −ν + 1 π a β ! 2 β 2ν−2μ−1 G 21 = ! 13 2 Γ(μ + 1) 4 ! μ − ν + 12 , 0, 12   a > 0, Re β > 0, − 12 < Re ν < Re μ + 1 ET I 14(29)a, ET II 235(19) 2ν

x2m cos(ax) dx n+1

(z + x2 )

0



−1 ≤ m ≤ n] ET I 67(37)

∞

4.



Re β > 0,

∞

6.7 0

= (−1)m+n

√ 1 dn  π · n z m− 2 e−a z 2 · n! dz [a > 0,

√ x cos(ax) dx π (−1) d2m   = {an K n (aβ)} · 1 n+ da2m 2n β n Γ n + 12 (β 2 + x2 ) 2  a > 0, 2m

n + 1 > m ≥ 0,

|arg z| < π] ET I 10(28)

m

Re β > 0,

0≤m
1 2



ET I 14(28)

3.774



1. 0

∞

  sin(ax) dx π i i νπ I ν (ab) + Jν (iab) − Jν (−iab) sin √ √  ν = ν b sin(νπ) 2 2 2 x2 + b2 x + x2 + b2 [a > 0, b > 0, Re ν > −1] ET I 70(19)

3.775

Trigonometric functions and powers



∞

2. 0

3.

  1 cos(ax) dx π 1 νπ Jν (iab) + Jν (−iab) − cos I ν (ab) √ √  ν = ν b sin(νπ) 2 2 2 x2 + b2 x + x2 + b2 [a > 0, b > 0, Re ν > −1]

 ν  



 ∞ x + x2 + β 2 aβ aβ aπ ν  β I 14 − ν2 sin(ax) dx = K 14 + ν2 2 2 2 2 2 x (x + β ) 0  a > 0, Re β > 0, 

∞

4. 0

445



ν 



x2 + β 2 − x aβ aβ aπ ν  β I − 14 + ν2 cos(ax) dx = K − 14 − ν2 2 2 2 2 2 x (x + β )  a > 0, Re β > 0,

ET I 12(15)

Re ν <

3 2



ET I 71(23)

Re ν > − 32



ET I 12(17)

5.

 ν  

 ∞ β + x2 + β 2 3 ν 1 2  Γ − sin(ax) dx = W ν2 , 14 (aβ) M − ν2 , 14 (aβ) 1 β a 4 2 xν+ 2 x2 + β 2 0   a > 0, Re β > 0, Re ν < 32

6.

 ν 

 ∞ β + x2 + β 2 1 ν 1 √  − Γ cos(ax) dx = W ν2 ,− 14 (aβ) M − ν2 ,− 14 (aβ) 1 4 2 β 2a xν+ 2 β 2 + x2 0   a > 0, Re β > 0, Re ν < 12

ET I 71(27)

ET I 12(18)

3.775 

∞



ν  ν x2 + β 2 + x − x2 + β 2 − x νπ  K ν (aβ) sin(ax) dx = 2β ν sin 2 2 2 x +β [a > 0, Re β > 0, |Re ν| < 1]



ν  ν x2 + β 2 + x + x2 + β 2 − x νπ  K ν (aβ) cos(ax) dx = 2β ν cos 2 2 2 x +β [a > 0, Re β > 0, |Re ν| < 1]

1. 0

 2. 0

∞

ET I 70(20)

ET I 13(22)

3.

√ √ ν  ν  ∞ + x + x2 − u2 + x − x2 − u2 νπ νπ , √ − Y ν (au) sin sin(ax) dx = πuν J ν (au) cos 2 2 x2 − u2 u [a > 0, u > 0, |Re ν| < 1]

4.

√ √ ν  ν  ∞ + x + x2 − u2 + x − x2 − u2 νπ νπ , √ + J ν (au) sin cos(ax) dx = −πuν Y ν (au) cos 2 2 x2 − u2 u [a > 0, u > 0, |Re ν| < 1]

ET I 70(22)

ET I 13(25)

446

Trigonometric Functions

3.776

√ √ ν  ν x + i u2 − x2 + x − i u2 − x2 π νπ √ [Jν (au) − J−ν (au)] sin(ax) dx = uν cosec 2 − x2 2 2 u 0 ET I 70(21) [a > 0, u > 0] √ √ ν  ν  u x + i u2 − x2 + x − i u2 − x2 π νπ √ [Jν (au) + J−ν (au)] cos(ax) dx = uν sec 2 − x2 2 2 u 0 [a > 0, u > 0, |Re ν| < 1] 

5.

6.

u



ET I 13(24)

√ √  ν  ν ∞ x + x2 − u2 + x − x2 − u2  sin(ax) dx 2 − u2 ) x (x u   au  au  au  au , π 3 ν+ =− Y 1/4−ν/2 + J 1/4−ν/2 Y 1/4+ν/2 au J 1/4+ν/2 2 2 2 2   2 ET I 71(25) a > 0, u > 0, |Re ν| < 32

 7.6

 8.

∞

6 u



∞

9. 0



∞

10. 0



2u

11. 0

3.776  1.

∞

0

 2.

0

∞

√ √  ν  ν x + x2 − u2 + x − x2 − u2  cos(ax) dx 2 − u2 ) x (x   au  au  au  au , π 3 ν+ =− Y −1/4−ν/2 + J −1/4−ν/2 Y −1/4+ν/2 au J −1/4+ν/2 2 2 2 2  2 ET I 13(26) a > 0, u > 0, |Re ν| < 32 ν  ν    x + β + x2 + 2βx + x + β − x2 + 2βx  sin(ax) dx x2 + 2βx   + νπ νπ , = πβ ν Y ν (βa) sin βa − + J ν (βa) cos βa − 2 2 [a > 0, |arg β| < π, |Re ν| < 1] ET I 71(26) ν  ν    x + β + x2 + 2βx + x + β − x2 + 2βx  cos(ax) dx x2 + 2βx +   νπ νπ , = πβ ν J ν (βa) sin βa − − Y ν (βa) cos βa − 2 2 [a > 0, |arg β| < π, |Re ν| < 1] ET I 13(23) √ 4ν √ 4ν √ √ 2u + x + i 2u − x + 2u + x − i 2u − x √ cos(ax) dx 4u2 x − x3  a 2ν 3/2 J ν−1/4 (au) J −ν−1/4 (au) = (4u) π 2 ET I 14(27) [a > 0, u > 0] a2 (b + x)2 + p(p + 1) a sin(ax) dx = p p+2 (b + x) b

[a > 0,

b > 0,

p > 0]

BI (170)(1)

a2 (b + x)2 + p(p + 1) p cos(ax) dx = p+1 (b + x)p+2 b

[a > 0,

b > 0,

p > 0]

BI (170)(2)

3.784

Rational and trigonometric functions

447

3.78–3.81 Rational functions of x and of trigonometric functions 3.781 1. 2. 3.782



1. 0



∞

u

1 − cos x dx − x

∞

2. 0

 3. 3.783 1. 2. 3.784

∞

u

(cf. 3.784 4 and 3.781 2)

cos x dx = C + ln u x

1 − cos ax aπ dx = x2 2

BI (173)(8)

[u > 0]

GW (333)(31)

[a ≥ 0]

BI (158)(1)

[a > 0,

b real,



∞

∞



∞

2. 0



∞

3. 0



∞

4. 0



∞

5. 0



∞

6. 0



b = 0]

ET II 253(48)



 cos x − 1 1 3 dx 1 = C− + 2 x 2(1 + x) x 2 4 0

 ∞ dx 1 = −C cos x − 2 1 + x x 0 

BI (173)(7)

dx = −C x

sin ab 1 − cos ax dx = π x(x − b) b −∞

0

8.



dx =1−C x

∞

1.

7.



sin x 1 − x 1+x 0

 ∞ 1 cos x − 1+x 0 

BI (173)(19) EH I 17, BI(273)(21)

b cos ax − cos bx dx = ln x a

[a > 0,

b > 0]

FI II 635, GW(333)(20)

a sin bx − b sin ax a dx = ab ln 2 x b

[a > 0,

b > 0]

FI II 647

cos ax − cos bx (b − a)π dx = x2 2

[a ≥ 0,

b ≥ 0]

BI(158)(12), FI II 645

sin x − x cos x dx = 1 x2 cos ax − cos bx 1 dx = x(x + β) β

BI (158)(3)



 b ci(aβ) cos aβ + si(aβ) sin aβ − ci(bβ) cos bβ − si(bβ) sin bβ + ln a [a > 0, b > 0, |arg β| < π] ET II 221(49)

cos ax + x sin ax dx = πe−a 1 + x2

[a > 0]

GW (333)(73)

∞

sin ax − ax cos ax π dx = a2 sign a 3 x 4 0    ∞ π (b − a)β + e−bβ − e−aβ cos ax − cos bx dx = x2 (x2 + β 2 ) 2β 3 0

LI (158)(5)

[a > 0,

b > 0,

|arg β| < π] BI(173)(20)a, ET II 222(59)

448

Trigonometric Functions



∞

9.10 0

n  π cos mx √ = e−m sin u sinh φ (cos β sin u cosh φ + sin β cos u sinh φ) 1 + a2 T n (x) 2n 1 + a2 k=1

[u = (2k − 1) π/ (2n) , 

∞

3.785 0

3.786



∞

1. 0



∞

2.11 0

 3.

∞

11 0

3.785

 1 ak cos bk x dx = − ak ln bk x n

n

k=1

k=1

φ = arcsinh(1/a), %

β = m cos u cosh φ, & n  ak = 0 bk > 0,

0 < |a| < 1] FI II 649

k=1

(1 − cos ax) sin bx b b 2 − a2 a a+b dx = ln + ln 2 x 2 b2 2 a−b (1 − cos ax) cos bx dx = ln x



|a2 − b2 | b

(1 − cos ax) cos bx π dx = (a − b) x2 2 =0

[a > 0,

b > 0]

[a > 0,

b > 0,

ET I 81(29)

a = b]

FI II 647

[a < b ≤ 0] [0 < a ≤ b] ET I 20(16)

3.787



∞

1. 0

π (cos a − cos nax) sin mx dx = (cos a − 1) x 2 π = cos a 2

[m > na > 0] [na > m] BI(155)(7)



∞

2. 0



sin2 ax − sin2 bx 1 a dx = ln x 2 b

[a > 0,

b > 0]

GW (333)(20b)

∞

x3 − sin3 x 13 π dx = 5 x 32 0   ∞ 3 − 4 sin2 ax sin2 ax 1 dx = ln 2 4. x 2 0

 π/2 1 π − cot x dx = ln 3.788 x 2 0 π/2 2 4x cos x + (π − x)x dx = π 2 ln 2 3.789 sin x 0 3.791  π/2 x dx = ln 2 1. 1 + sin x 0  π x cos x dx = π ln 2 − 4G 2. 0 1 + sin x  π/2 x cos x dx = π ln 2 − 2G 3. 1 + sin x 0 3.

BI (158)(6)

[a real, a = 0]

HBI (155)(6)

GW (333)(61)a LI (206)(10)

GW (333)(55a) GW (333)(55c)

GW (333)(55b)

3.792

Rational and trigonometric functions



π

449

π

   π/2  π − x cos x 2 − x cos x dx = 2 dx = π ln 2 + 4G = 5.8414484669 . . . 1 − sin x 1 − sin x 0

2

4. 0

BI(207)(3), GW(333)(56c)



π/2

5. 0

 6. 7.

x2 dx π2 =− + π ln 2 + 4G = 3.3740473667 . . . 1 − cos x 4

BI (207)(3)

π

x2 dx = 4π ln 2 0 1 − cos x    π/2 p+1 ∞  π p+1  π p x dx 1 2  =− − ζ(2k) + (p + 1) 1 − cos x 2 2 p 42k−1 (p + 2k) 0

BI (219)(1)

k=1



[p > 0] π/2

8. 0

 9. 10. 11. 12.

3.

π x dx = − ln 2 1 + cos x 2

GW (333)(55a)

π/2

π x sin x dx = ln 2 + 2G 1 − cos x 2 0  π x sin x dx = 2π ln 2 0 1 − cos x  π/2  π π x − sin x x − sin x dx = + dx = 2 1 − cos x 2 1 − cos x 0 0  π/2 π x sin x dx = − ln 2 + 2G 1 + cos x 2 0

3.792  1. 2.

LI (207)(4)

GW (333)(56a) GW (333)(56b) GW (333)(57a) GW (333)(55b)

 2 dx 2π a = 2 2 1−a −π 1 − 2a cos x + a  π/2 ∞  x cos x dx a2k π k ln(1 + a) − = (−1) 1 + 2a sin x + a2 2a (2k + 1)2 0 k=0  2 a  π  2 x sin x dx π a = ln(1 + a) 2 a

0 1 − 2a cos x + a  2 1 π a = ln 1 + a a 

π

2π

0

5. 0

<1



FI II 485



< 1,  <1

LI (241)(2)

a = 0



BI (221)(2)

4.



<1

2π

x sin x dx 2π ln(1 − a) = 1 − 2a cos x + a2 a

1 2π ln 1 − = a a %



a2 < 1,  2  a >1 n−1 

a = 0

&



  −n x sin nx dx a−k − ak 2π n ln(1 − a) + = − a a 1 − 2a cos x + a2 1 − a2 n−k k=1  2  a < 1, a = 0

BI (223)(4)

BI (223)(5)

450

Trigonometric Functions



∞

6. 0

!  ! π !! 1 + a !! sin x dx −1 = · 1 − 2a cos x + a2 x 4a ! 1 − a !

3.792

[a real,

a = 0,

a = 1] GW (333)(62b)



∞

7.8 0

 0

∞

9. 0



∞

10.3 0



∞

11. 0



∞

12. 0



∞

13. 0

 0

0

sin x cos bx π dx = a[b] · 1 − 2a cos x + a2 x 2(1 − a) π π = ab + ab−1 2(1 − a) 4 [0 < a < 1,

[b = 0, 1, 2, . . .] [b = 1, 2, 3, . . .] ; b > 0] ; (for b = 0, see 3.792 6)

(1 − a cos x) sin bx dx π 1 − a[b]+1 = · · 1 − 2a cos x + a2 x 2 1−a π 1 − ab πab = · + 2 1−a 4

ET I 19(5)

[b = 1, 2, 3, . . .] [b = 1, 2, 3, . . .] [0 < a < 1,

b > 0]

ET I 82(33)

−bβ

1 + ae 1 dx π = 2 2 2 2 1 − 2a cos bx + a β + x 2β (1 − a ) 1 − ae−bβ





a2 < 1,

b≥0

1 dx sin bβ aπ = 1 − 2a cos bx + a2 β 2 − x2 β (1 − a2 ) 1 − 2a cos bβ + a2  2 a < 1,

b>0

e−βbc − ac sin bcx x dx π = 1 − 2a cos bx + a2 β 2 + x2 2 (1 − ae−bβ ) (1 − aebβ )  2 a < 1,

b > 0,



1 sin bx x dx π = 1 − 2a cos bx + a2 β 2 + x2 2 ebβ − a 1 π = 2a aebβ − 1



a2 < 1,

b>0

a2 > 1,

b>0

BI (192)(1)



BI (193)(1)

c>0



BI (192)(8)

 

a − cos βbc sin bcx x dx π = 2 2 2 1 − 2a cos bx + a β − x 2 1 − 2a cos βb + a2 c





 2

a2 < 1,

b > 0,

c>0



1 − a sin βbc + 2ac+1 sin βb cos bcx dx π = 1 − 2a cos bx + a2 β 2 − x2 2β (1 − a2 ) 1 − 2a cos βb + a2  2  a < 1, b > 0, c > 0

∞

1 − a cos bx dx π e = 1 − 2a cos bx + a2 1 + x2 2 eb − a

0

16.

[0 < a < 1]

∞

15. 

[b = 1, 2, . . .] ;

BI (192)(2) ∞

14. 

[b = 0, 1, 2, . . .]

ET I 81(26) ∞

8.



sin bx dx π 1 + a − 2a[b]+1 = · 1 − 2a cos x + a2 x 2 (1 − a2 ) (1 − a) π 1 + a − ab − ab+1 = 2 (1 − a2 ) (1 − a)

b



a2 < 1,

b>0



BI (193)(5)

BI (193)(9) FI II 719

3.794

Rational and trigonometric functions



∞

17. 0



  π eβ−βb + aeβb cos bx dx · = 1 − 2a cos x + a2 x2 + β 2 2β (1 − a2 ) (eβ − a)

451

[0 ≤ b < 1,

|a| < 1,

Re β > 0] ET I 21(21)

∞

18. 0

sin bx sin x dx · 1 − 2a cos x + a2 x2 + β 2

π sinh bβ [0 ≤ b < 1] 2β eβ − a + , π = am eβ(m+1−b) − e(1−b)β 4β (aeβ − 1) , + π m −(m+1−b)β −(1−b)β [m ≤ b ≤ m + 1] a e − e − 4β (ae−β − 1) [0 < a < 1, Re β > 0] ET I 81(27) =



∞

19. 0



∞

20.

(cos x − a) cos bx dx π cosh βb · = 1 − 2a cos x + a2 x2 + β 2 2β (eβ − a) sin x (1 − 2a cos 2x +

0

n+1 a2 )

dx = x



= 0

|a| < 1,

Re β > 0] ET I 21(23)

∞

0



[0 ≤ b < 1,

∞

tan x (1 − 2a cos 2x + tan x

n+1 a2 )

(1 − 2a cos 4x + a2 )

n+1

dx x n  2  n π dx = a2k 2n+1 2 k x 2 (1 − a ) k=0 BI (187)(14)

3.793



2π

1.3 0



k=1

[|a| < 1] 2π

2. 0

3.794 1.3

2.

3.3



% & n  sin nx − a sin[(n + 1)x] 1 n x dx = −2πa ln(1 − a) + 1 − 2a cos x + a2 kak 

cos nx − a cos[(n + 1)x] x dx = 2πan 1 − 2a cos x + a2 ∞

a2 < 1



BI (223)(9) BI (223)(13)

 a2k+1 x dx π2 4 = + 2 2 (1 − a2 ) (1 − a2 ) (2k + 1)2 0 1 + a + 2a cos x k=0  2  a <1 ⎡ √ √  n  n  2π 1 − a2 − 1 − 1 − a2 2π x sin nx n 1+ ⎣ dx = √ (∓1) 1 ± a cos x an 1 − a2 0 √ √    ⎤  √ n−1  (∓1)k 1 + 1 − a2 k − 1 − 1 − a2 k 2 1±a ⎦ √ × ln √ + ak 1 + a + 1 − a k=0 n − k  2  a <1 BI (223)(2)   √ n  2π  2  x cos nx 2π 2 1 − 1 − a2 dx = √ a <1 BI (223)(3) 2 1 ± a cos x ∓a 1−a 0 π

452

Trigonometric Functions

√ x sin x dx π a + a2 − b 2 = ln b 2(a − b) 0 a + b cos x √  2π 2π a + a2 − b2 x sin x dx = ln a + b cos x b 2(a + b) 0  ∞ dx π sin x · = √ x 2 a2 − b 2 0 a ± b cos 2x 

4. 5. 6.

3.795

π

=0

[a > |b| > 0]

GW (333)(53a)

[a > |b| > 0]

GW (333)(53b)

 

a2 > b 2 a2 < b 2

  BI (181)(1)

    ∞ 2 b + c2 + x2 x sin ax − b2 − c2 − x2 c sinh ac dx = π 3.795 2 2 2 2 −∞ [x + (b − c) ] [x + (b + c) ] (cos ax + cosh ac) 2π = ab e +1 [a > 0] 3.796  π/2 π cos x ± sin x x dx = ∓ ln 2 − G 1. cos x ∓ sin x 4 0  π/4 π 1 cos x − sin x x dx = ln 2 − G 2. cos x + sin x 4 2 0 3.797 1. 2. 3. 3.798



1 π2 π π − x tan x tan x dx = ln 2 + − + ln 2 4 2 32 4 8 0   π/4  π π 1 4 − x tan x dx = − ln 2 + G cos 2x 8 2 0  π/4 π π 1 4 − x tan x dx = ln 2 + G cos 2x 8 2 0 

π/4

∞

1.8 0

π

tan x dx π · = √ 2 a + b cos 2x x 2 a − b2 =0

 2.8 0

[c > b > 0] [b > c > 0] BI (202)(18)

BI (207)(8, 9)

BI (204)(23)

BI (204)(8)

BI (204)(19)

BI (204)(20)

[0 < b < a] [0 < a < b] BI (181)(2)

∞

dx π tan x · = √ 2 a + b cos 4x x 2 a − b2 =0

[0 < b < a] [0 < a < b] BI (181)(3)

3.799



1. 0

π/2

x dx 2

(sin x + a cos x)

=

ln a a π − 2 1+a 2 1 + a2

[a > 0]

BI (208)(5)

3.812

Rational and trigonometric functions



π/4

2. 

x dx 2

(cos x + a sin x)

0

=

453

1−a 1 1+a π ln √ + · 1 + a2 4 (1 + a) (1 + a2 ) 2 [a > 0]

π

3.

a cos x + b

2 2x

(a + b cos x)

0

dx =

2(a − b) 2π √ ln b a + a2 − b 2

BI (204)(24)

[a > |b| > 0]

GW (333)(58a)

3.811 

π

1. 0



t1 x dx t1 + t2 t1 − t2 1 + tan 2 sin x · = π cosec cosec ln t2 1 − cos t1 cos x 1 − cos t2 cos x 2 2 1 + tan 2 (cf. 3.794 4)

π/2

2. 0



π/4

3. 0



π/4

4. 0



π/4

5. 0

3.812



1. 0

2. 3. 4.11

π

π x dx = ln 2 + G (cos x ± sin x) sin x 4

BI (208))(16, 17)

π x dx = − ln 2 + G (cos x + sin x) sin x 8

BI (204)(29)

π x dx = ln 2 (cos x + sin x) cos x 8

BI (204)(28)

sin x x dx π π 1 = − ln 2 + − ln 2 2 sin x + cos x cos x 8 4 2

BI (204)(30)

 π x sin x dx b = √ arctan 2 a + b cos x a√ ab √ π a + −b √ = √ ln √ 2 −ab a − −b

√  π/2 π 1+ 1+a x sin 2x dx = ln 1 + a cos2 x a 2 0   √  π/2 π 2 1+a− 1+a x sin 2x dx = ln a 2 1 + a sin2 x 0  π 2 π x dx = √ 2 − cos2 x a 2a a2 − 1 0 =0 = divergent ! ! π !1 + a! π x sin x dx ! ! = ln 2 2 2a ! 1 − a ! 0 a − cos x

[a > 0,

b > 0]

[a > −b > 0] GW (333)(60a)

[a > −1,

a = 0]

BI (207)(10)

[a > −1,

a = 0]

BI (207)(2)

 

a2 > 1



principal value for 0 < a2 < 1



[a = 0] BI (219)(10)

 5.7

BI (222)(5)

[0 < a < 1]

divergent if a = 0 BI (219)(13)

454

Trigonometric Functions

 6.

π

11 0

   x sin 2x dx = π ln 4 1 − a2 a2 − cos2 x , +   = 2π ln 2 1 − a2 + a a2 − 1 = divergent

3.813



principal value for 0 ≤ a2 < 1  2  a >1



[|a| = 1] BI (219)(19)



π/2

7. 0

 8. 9. 10. 11. 12.∗ 13.∗

BI (207)(1)

π

x sin x dx = π(π − 2t) cosec 2t 2 2 0 1 − cos t sin x  π ∞  x cos x dx sin(2k + 1)t = 4 cosec t 2 2 (2k + 1)2 0 cos t − cos x k=0  π π x sin x dx = (π − 2t) cot t 2 2 2 0 tan t + cos x  ∞ t x (a cos x + b) sin x dx = 2aπ ln cos + πbt tan t 2 2x 2 cot t + cos 0 % &   π x sin x cos x a−1 dx = −π ln 2 + ln 1 + a a − sin2 x 0  π/2   √ ln a − sin2 x dx = −π ln 2 + iπ ln arccos a 0

14.∗

∞  x sin x dx sin(2k + 1)t = −2 cosec t 2 2 (2k + 1)2 cos t − sin x k=0



π/2

PV

BI (219)(12)

BI (219)(17)

BI (219)(14) BI (219)(18)

[a > 1]

[0 < a < 1]

! ! ln !a − sin2 x! dx = −π ln 2

[0 < a < 1]

! ! ln !a − cos2 x! dx = −π ln 2

[0 < a < 1]

0

15.∗



π/2

PV 0

3.813



1. 0



π

x dx 1 = 2 2 2 2 4 a cos x + b sin x

∞

2. 0



0

0

x dx π2 = 2ab a2 cos2 x + b2 sin2 x [a > 0,

b > 0]



GW (333)(36)



GW(333)(81), ET II 222(63)

sin x dx π = 2 2 2 2ab sin x + b cos x

[ab > 0]

sin2 x dx π = 2 2 2 2 2b(a + b) x a cos x + b sin x

[a > 0,

x

0



2π

β γ 2 1 π sinh(2aδ) dx   − − = · 2 2 2 2 γ β sinh(2aδ) β 2 sin2 ax + γ 2 cos2 ax x2 + δ 2 4δ β sinh (aδ) ! − γ cosh ! (aδ)  ! ! β !arg ! < π, Re δ > 0, a > 0 ! γ!

∞

3. 4.



∞

 

a2

BI (181)(8)

b > 0]

BI (181)(11)

3.814

Rational and trigonometric functions

 5.

455

π/2

π x sin 2x dx a+b = 2 [a > 0, b > 0, a = b] GW (333)(52a) ln a − b2 2b a2 cos2 x + b2 sin2 x  π x sin 2x dx a+b 2π [a > 0, b > 0, a = b] GW (333)(52b) ln = 2 2 2 cos2 x + b2 sin2 x a − b 2a a 0  ∞ π sin 2x dx = [a > 0, b > 0] BI (182)(3) · 2 2 2 2 x a(a + b) 0 a cos x + b sin x    ∞ β−γ sin 2ax π x dx −2aδ   − e = · 2 2 2 2 2 2 β+γ 2 β 2 sinh2 (aδ) − γ 2 cosh2 (aδ) 0 β sin ax + γ cos ax x + δ ! !  ! ! β ! ! a > 0, !arg ! < π, Re δ > 0 γ 0

6. 7. 8.



ET II 222(64), GW(333)(80) ∞

9. 0



∞

11. 0



b > 0]

BI (182)(7)a

dx π sin x cos2 x · = 2 2 2 2a(a + b) cos x + b sin x x

[a > 0,

b > 0]

BI (182)(4)

π 2 sin3 x dx = · · 2 2 2 2 2b a + b a cos x + b sin x x

[a > 0,

b > 0]

BI (182)(1)

a2

0

3.814

[a > 0,

∞

10. 

dx π (1 − cos x) sin x · = 2b(a + b) a2 cos2 x + b2 sin2 x x

π/2

1. 0



π/4

2. 0

 

5.

dx π tan x = 2 2 2ab x + b sin x x

b > 0]

BI (181)(9)

[a > 0,

b > 0]

LI (208)(20)

[a > 0,

b > 0]

GW (333)(59)

[a > 0,

b > 0]

BI (182)(6)

π tan x dx = · 2ab a2 cos2 2x + b2 sin2 2x x

[a > 0,

b > 0]

BI (181)(10)a

dx π 1 sin2 2x tan x · = · 2 2 2 2 2b a + b a cos 2x + b sin 2x x

[a > 0,

b > 0]

BI (182)(2)a

π 1 cos2 2x tan x dx = · · 2a a + b a2 cos2 2x + b2 sin2 2x x

[a > 0,

b > 0]

BI (182)(5)a

cos2

π/2

x cot x dx a+b π = 2 ln 2 2 2 2a b cos x + b sin x 0    π/2  π  π π 1 2 − x tan x dx 2 − x tan x dx = a2 cos2 x + b2 sin2 x 2 0 a2 cos2 x + b2 sin2 x 0 a+b π = 2 ln 2b a a2



∞

6. ∞

7. 0



∞

8. 0

 9.

0

2

a2

0



BI (204)(30)

[a > 0,

a2

0

BI (206)(9)

π π 1 x tan x dx = − ln 2 + − ln 2 (sin x + cos x) cos x 8 4 2

∞

3. 4.

(1 − x cot x) dx π = 2 4 sin x

∞

π sin x tan x dx = · 2 2 2 2b(a + b) cos x + b sin x x

456

Trigonometric Functions



∞

10. 0

 11. 12. 13. 14. 15. 3.815

dx sin2 x cos x π a · =− 2 2 2 2 2 8b a + b2 a cos 2x + b sin 2x x cos 4x

∞

sin x 2 cos2 x + b2 sin2 x a 0  ∞ sin x cos x 2 2 2 2 0 a cos x + b sin x  ∞ sin x cos2 x 2 2 2 2 0 a cos x + b sin x  ∞ sin3 x 2 2 2 2 0 a cos x + b sin x  ∞ 1 − cos x 2 2 2 2 0 a cos x + b sin x 

π/2

1. 0



π/2

2. 0

3.815

[a > 0,

b > 0]

BI (186)(12)a

·

dx π b 2 − a2 = · x cos 2x 2ab b2 + a2

[a > 0,

b > 0]

BI (186)(4)a

·

π b dx = · x cos 2x 2a a2 + b2

[a > 0,

b > 0]

BI (186)(7)a

·

π b2 dx = · 2 x cos 2x 2ab a + b2

[a > 0,

b > 0]

BI (186)(8)a

·

π a dx =− · 2 x cos 2x 2b a + b2

[a > 0,

b > 0]

BI (186)(10)

·

dx π = x sin x 2ab

[a > 0,

b > 0]

BI (186)(3)a

x sin 2x dx π   = ln 2 2 a − b 1 + a sin x 1 + b sin x

√ √  1+ 1+b 1+a √ ·√ 1+ 1+a 1+b [a > 0, b > 0]

 √ √ 1+ 1+n 1+a π x sin 2x dx   √ = ln a + ab + b 1+ 1+a 1 + a sin2 x (1 + b cos2 x) [a > 0, b > 0]

(cf. 3.812 3) BI (208)(22)

(cf. 3.812 2 and 3) BI (208)(24)

√ π/2 π 1+ 1+a x sin 2x dx √ = ln (1 + a cos2 x) (1 + b cos2 x) a−b 1+ 1+b 0

 3.

[a > 0,

b > 0]

(cf. 3.812 2) BI (208)(23)



π/2

4. 0

t1 cos x sin 2x dx 2π 2   = ln cos2 t1 − cos2 t2 cos t2 1 − sin2 t1 cos2 x 1 − sin2 t2 cos2 x 2 [−π < t1 < π,

−π < t2 < π] BI (208)(21)

3.816 1.



π

2

2.

(a2

0

 3.

dx = π 2

(a2 − cos2 x)   π 2 a − 1 − sin2 x cos x 0

7

√

x2 sin 2x

11 0

π

−

cos2

2

x)

a2 − 1 − a a (a2 − 1)

! ! π !! 1 − a !! x dx = ln ! 2 1 + a! 2

[a > 1] 

a2 > 1

LI (220)(9)



(cf. 3.812 5)

   √ √ a cos 2x − sin2 x 2  2 x dx = −2π ln 2 −a + a a + 1 2 a + sin+ x ,  √ √ a < −1 and a > 0. When a > 0, can write a a + 1 as a(a + 1).

BI (220)(12)

LI (220)(10)

3.818

Rational and trigonometric functions

 4.11 0

π

   √ √ a cos 2x + sin2 x 2  2 x dx = 2π ln 2 a − a a + 1 2 a +− sin x ,  √ √ a < 0 and a > 1. When a > 1, can write a a + 1 as a(a + 1).

457

(cf. 3.812 6) LI (220)(11)

3.817



∞

1. 0



∞

2. 0



∞

3. 0



∞

4. 0



∞

5. 0



∞

6. 0



∞

7. 0



∞

8. 0

3.818



∞

1. 0



∞

2. 0



∞

3. 0



∞

4. 0

 5. 0

∞

sin x π a2 + b 2 dx = · 3 3 ·   2 x 4 a b a2 cos2 x + b2 sin2 x

[ab > 0]

BI (181)(12)

π sin x cos x dx = 3  2 · 2 2 2 2 x 4a b a cos x + b sin x

[ab > 0]

BI (182)(8)

π sin3 x dx =  2 · 2 x 4ab3 a2 cos2 x + b2 sin x

[ab > 0]

BI (181)(15)

π sin x cos2 x dx = 3  2 · 2 2 2 2 x 4a b a cos x + b sin x

[ab > 0]

BI (182)(9)

π a2 + b 2 tan x dx = · 3 3 ·   2 x 4 a b a2 cos2 x + b2 sin2 x

[ab > 0]

BI (181)(13)

π a2 + b 2 tan x dx = ·   2 x 4 a3 b 3 a2 cos2 2x + b2 sin2 2x

[ab > 0]

BI (181)(14)

π sin2 x tan x dx =  2 · 2 x 4ab3 a2 cos2 x + b2 sin x

[ab > 0]

BI (182)(11)

tan x cos2 2x π dx = 3  2 · 2 2 2 2 x 4a b a cos 2x + b sin 2x

[ab > 0]

BI (182)(10)

π 3a4 + 2a2 b2 + 3b4 sin x dx = · ·   3 x 16 a5 b 5 a2 cos2 x + b2 sin2 x [ab > 0]

BI (181)(16)

[ab > 0]

BI (182)(13)

[ab > 0]

BI (182)(14)

π 3a2 + b2 dx = · ·   3 x 16 a3 b 5 a2 cos2 x + b2 sin2 x

[ab > 0]

LI (181)(19)

π 3a2 + b2 sin3 x cos x dx = · ·   3 x 64 a3 b 5 a2 cos2 2x + b2 sin2 2x

[ab > 0]

BI (182)(17)

2

π a + 3b sin x cos x dx = ·  3 · 2 x 16 a5 b 3 a2 cos2 x + b2 sin x

2

π a2 + 3b2 sin x cos2 x dx = · ·   3 x 16 a5 b 3 a2 cos2 x + b2 sin2 x sin3 x

458

Trigonometric Functions



∞

6. 0



∞

7. 0



∞

8. 0



∞

9. 0

3.819



∞

1. 0



∞

0

0

∞

4. 0



∞

5. 0



∞

6. 0



∞

7. 0



∞

8. 0

 9. 0

π 3a2 + b2 sin2 x tan x dx = · ·   3 x 16 a3 b 5 a2 cos2 x + b2 sin2 x

[ab > 0]

BI (181)(17)

[ab > 0]

BI (182)(16)

π 3a4 + 2a2 b2 + 3b4 tan x dx = · ·   3 x 16 a5 b 5 a2 cos2 2x + b2 sin2 2x 2

2

π a + 3b tan x cos 2x dx = ·  3 · 2 x 16 a5 b 3 a2 cos2 2x + b2 sin 2x

[ab > 0]

BI (181)(18)

[ab > 0]

BI (182)(15)

2

π 5a6 + 3a4 b2 + 3a2 b4 + 5b6 sin x dx = · ·   4 x 32 a7 b 7 a2 cos2 x + b2 sin2 x

∞

BI (181)(20)

π a4 + 2a2 b2 + 5b4 sin x cos x dx = · ·   4 x 32 a7 b 5 a2 cos2 x + b2 sin2 x [ab > 0]

∞

3.



tan x π 3a4 + 2a2 b2 + 3b4 dx = ·   3 x 16 a5 b 5 a2 cos2 x + b2 sin2 x

[ab > 0]

2.



3.819

BI (182)(18)

π a4 + 2a2 b2 + 5b4 sin x cos2 x dx = · ·   4 x 32 a7 b 5 a2 cos2 x + b2 sin2 x [ab > 0]

BI (182)(19)

[ab > 0]

BI (181)(23)

[ab > 0]

BI (182)(26)

π a2 + 5b2 sin x cos3 x dx = · ·   4 x 32 a7 b 3 a2 cos2 x + b2 sin2 x

[ab > 0]

BI (182)(23)

π a2 + b 2 sin3 x cos2 x dx = · 5 5 ·   4 x 32 a b a2 cos2 x + b2 sin2 x

[ab > 0]

BI (182)(27)

π a2 + 5b2 sin x cos4 x dx = · ·   4 x 32 a7 b 3 a2 cos2 x + b2 sin2 x

[ab > 0]

BI (182)(24)

[ab > 0]

BI (181)(24)

3

4

2 2

π 5a + a b + b sin x dx = ·  4 · 2 x 32 a5 b 7 a2 cos2 x + b2 sin x 3

2

π a +b sin x cos x dx = · 5 5  4 · 2 x 32 a b a2 cos2 x + b2 sin x

sin5 x

4

2

π 5a2 + b2 dx = ·  4 · 2 x 32 a3 b 7 a2 cos2 x + b2 sin x

3.821

Powers and trigonometric functions



∞

10. 0



∞

11. 0



∞

12. 0



∞

13. 0



∞

14. 0



∞

15. 0



∞

16. 0

459

sin3 x cos x π 5a4 + 2a2 b2 + b4 dx = · ·   4 x 128 a5 b 7 a2 cos2 2x + b2 sin2 2x π 5a2 + b2 sin5 x cos3 x dx = · ·   4 x 512 a3 b 7 a2 cos2 2x + b2 sin2 2x

[ab > 0]

BI (182)(22)

[ab > 0]

BI (182)(30)

π 5a4 + 2a2 b2 + b4 sin2 x tan x dx = ·  4 · 2 x 32 a5 b 7 a2 cos2 x + b2 sin x 4

2

sin x tan x π 5a + b dx = ·  4 · 2 2 2 2 x 32 a3 b 7 a cos x + b sin x

[ab > 0]

BI (182)(21)

[ab > 0]

BI (182)(29)

2

π a4 + 2a2 b2 + 5b4 cos2 2x tan x dx = · ·   4 x 32 a7 b 5 a2 cos2 2x + b2 sin2 2x 3

2

π a +b sin 4x tan x dx = · 5 5  4 · 2 2 2 2 x 8 a b a cos 2x + b sin 2x

[ab > 0]

BI (182)(29)

[ab > 0]

BI (182)(28)

[ab > 0]

BI (182)(25)

2

cos4 2x tan x π a2 + 5b2 dx = · ·   4 x 32 a7 b 3 a2 cos2 2x + b2 sin2 2x

3.82–3.83 Powers of trigonometric functions combined with other powers 3.821  π Γ(p + 1) π2 x sinp x dx = p+1 +  1. ,2 p 2 0 +1 Γ 2  rπ 2 (2m − 1)!! 2 π · r 2. x sinn x dx = 2 (2m)!! 0 (2m)!! r = (−1)r+1 π (2m + 1)!!

[p > −1]

BI(218)(7), LO V 121(71)

[n = 2m] [n = 2m + 1] [r is a natural number]



π/2

x cosn x dx =

3.11 0

=  4. 5.

1 π (n − 1)!! − n−1 2 (n)!! 2

n

m−1 

1 π 2 (n − 1)!! − n−2 8 (n)!! 2

k=0,m−k odd m−1  k=0

n k

k

1 (n − 2k)2

1 (n − 2k)2

GW (333)(8c)

[n = 2m] [n = 2m − 1] GW (333)(9b)

π (2m − 1)!! 2 (2m)!! 0  sπ  (2m − 1)!! π2  2 s − r2 x cos2m x dx = 2 (2m)!! rπ π

x cos2m x dx =

2

BI (218)(10) BI (226)(3)

460

Trigonometric Functions



  √ p p Γ p2 sinp x π dx = ·  p+1  = 2p−2 B , x 2 Γ 2 2 2 [p is a fraction with odd numerator and denominator]

∞

6. 0



∞

7. 0



∞

8. 0



∞

9. 0



∞

10. 0



∞

11. 0



3.822

∞

LO V 278, FI II 808

(2n − 1)!! π sin2n+1 x dx = · x (2n)!! 2

BI (151)(4)

sin2n x dx = ∞ x

BI (151)(3)

sin2 ax aπ dx = 2 x 2

[a > 0]

LO V 307, 312, FI II 632

sin2m ax (2m − 3)!! aπ · dx = x2 (2m − 2)!! 2

[a > 0]

GW (333)(14b)

a2 π sin2m+1 ax (2m − 3)!! (2m + 1) dx = 3 x (2m)!! 4

[a > 0]

GW (333)(14d)

sinp x dx xm

12.

 ∞ p−1 sin x p cos x dx = m−1 m−1 0 x  ∞ p−2  ∞ p sin x sin x p2 p(p − 1) dx − dx = (m − 1)(m − 2) 0 xm−2 (m − 1)(m − 2) 0 xm−2

0

[p > m − 1 > 0] [p > m − 1 > 1] GW (333)(17)



∞

2n

sin px √ dx = ∞ x

13. 0



∞

2n+1

14.

sin 0

3.822



1 dx px √ = 2n 2 x

π/2

xp cosm x dx = −

1. 0

BI (177)(5)



π  (−1)k 2p n



k=0

p(p − 1) m2



2n + 1 n+k+1

√

π/2

xp−2 cosm x dx + 0

1 2k + 1

m−1 m



π/2

xp cosm−2 x dx 0

[m > 1, 

∞

2.

x−1/2 cos2n+1 (px) dx =

0



∞

3.823 0

xμ−1 sin2 ax dx = −

1 22n



π 2p

n  k=0

Γ(μ) cos μπ 2 2μ+1 aμ

2n + 1 n+k+1



BI (177)(7)

p > 1]

1 √ 2k + 1 [a > 0,

GW (333)(9a) BI (177)(8)

−2 < Re μ < 0] ET I 319(15), GW(333)(19c)a

3.824 1. 2.



∞

 sin2 ax π  1 − e−2aβ dx = 2 + β2 x 4β 0  ∞  cos2 ax π  1 + e−2aβ dx = 2 2 4β 0 x +β

[a > 0,

Re β > 0]

BI (160)(10)

[a > 0,

Re β > 0]

BI (160)(11)

3.824

Powers and trigonometric functions



∞

3.7

2m

sin 0

dx (−1)m π x 2 = 2m+1 · 2 a +x 2 2

 2

2m

sinh

2m

a−2

m 

461

k

(−1)

k=0

2m k



sinh[2(m − k)a]

BI (160)(12) [a > 0]

 ∞ 2m+1  2m + 1 −2ka dx (−1)m−1 (2m+1)a sin2m+1 x 2 = (−1)k Ei[(2k − 2m − 1)a] e e 2 2m+2 k a +x 2 a 0 k=0 

2m+1  −(2m+1)a k−1 2m + 1 2ka (−1) +e e Ei[(2m + 1 − 2k)a] k



4.7

k=0

[a > 0]

2m + 1 2ka x dx π −(2m+1)a m+k sin2m+1 x 2 = e (−1) e k a + x2 22m+1 0 k=0 + π, |arg a| < , 2



 ∞ m 2m 2m dx π π  2m cos x 2 = 2m+1 e−2ka + 2m m+k a + x2 2 a m 2 0 

5.7

6.7

BI (160)(14)

m 

∞

m = 0, 1, 2, . . .

k=1

[a > 0] 

∞

7.

cos2m+1 x

0

a2

dx π = 2m+1 2 +x 2 a

m  k=1

BI (160)(16)



2m + 1 e−(2k+1)a m+k+1 [a > 0]



∞

8.

cos2m+1 x

0

BI (160)(17)

2m+1 x dx e−(2m+1)a  2m + 1 2ka = − e Ei[(2m − 2k + 1)a] k a2 + x2 22m+2 k=0

2m+1 e(2m+1)a  2m + 1 −2ka Ei[(2k − 2m − 1)a] − 2m+2 e k 2 k=0



BI (160)(18) ∞

9. 0

 10.

0

∞

2

cos ax π sin 2ab dx = 2 2 b −x 4b

[a > 0,

b > 0]

  sin2 ax cos2 bx π 1 2(b−a)β 1 −2(a+b)β −2bβ −2aβ dx = +e − e −e 1− e β 2 + x2 8β 2 2   π 1 − e−4aβ = 16β  π 1 2(a−b)β 1 −2(a+b)β −2bβ −2aβ = +e − e −e 1− e 8β 2 2 [a > 0, b > 0] , (cf. 3.824 1

BI (161)(10)

[a > b] [a = b] [a < b] and 3)

BI (162)(6)

462

Trigonometric Functions



∞

11. 0

3.825

, x sin 2ax cos2 bx π + −2aβ −2(a+b)β 2(b−a)β 2e dx = + e + e β 2 + x2 8  π  −4aβ = e + 2e−2aβ 8 , π + −2aβ + e−2(a+b)β − e2(a−b)β = 2e 8

[a > 0] [a = b] [a < b] LI (162)(5)

3.825



∞

1. 0

  π b − c + ce−2ab − be−2ac sin2 ax dx = (b2 + x2 ) (c2 + x2 ) 4bc (b2 − c2 ) 

 ∞ π b − c + be−2ac − ce−2ab cos2 ax dx = 2 2 2 2 4bc (b2 − c2 ) 0 (b + x ) (c + x )

 2.



∞

3.3 0



(b2

π (c sin 2ab − b sin 2ac) sin2 ax dx = − x2 ) (c2 − x2 ) 4bc (b2 − c2 )

(b2

π (b sin 2ac − c sin 2ab) cos ax dx = 2 2 2 − x ) (c − x ) 4bc (b2 − c2 )

[a > 0,

b > 0,

c > 0]

BI (174)(15)

[a > 0,

b > 0,

c > 0]

BI (175)(14)

[a > 0,

b > 0,

c > 0,

b = c] LI (174)(16)

∞

4.3 0

2

[a > 0,

b > 0,

c > 0,

b = c] LI (175)(15)

3.826 1. 2. 3.827

   π sin2 ax dx 1 −2ab = 2 2a − 1−e 2 2 2 4b b 0 x (b + x )

 ∞ π sin2 ax dx 1 = sin 2ab 2a − 2 2 2 4b2 b 0 x (b − x ) 



∞

∞

1.8 0

sin3 ax 3 − 3ν−1 ν−1 νπ a Γ(1 − ν) dx = cos ν x 4 2

[a > 0,

b > 0]

BI (172)(13)

[a > 0,

b > 0]

BII (172)(14)

[a < Re ν < 4, ν = 1, 2, 3] GW (333)(19f)



∞

2.8 0



∞

3. 0



∞

4.8 0



∞

5. 0

 6.

0

∞

3

π sin ax dx = x 4

LO V 277

sin3 ax 3 dx = a ln 3 2 x 4

BI (156)(2)

sin3 ax 3 dx = a2 π x3 8 sin4 ax aπ dx = 2 x 4 sin4 ax dx = a2 ln 2 x3

BI(156)(7)a,LO V 312

[a > 0]

BI (156)(3) BI (156)(8)

3.828

Powers and trigonometric functions



∞

7. 0



∞

8. 0



∞

9. 0



∞

10. 0



∞

11. 0



∞

12. 0



∞

13. 0



∞

14. 0



∞

15. 0

3.828 1.8 2.8 3.8

sin4 ax a3 π dx = x4 3

[a > 0]

sin5 ax 5 a (3 ln 3 − ln 5) dx = x2 16 sin5 ax 5 2 a π dx = 3 x 32

[a > 0]

sin5 ax 5 3 a (25 ln 5 − 27 ln 3) dx = x4 96

∞

0

BI (156)(12)

sin5 ax 115 4 a π dx = x5 384

[a > 0]

BI(156)(13), LO V 312

sin6 ax 3 aπ dx = 2 x 16

[a > 0]

BI (156)(5)

sin6 ax 3 2 a (8 ln 2 − 3 ln 3) dx = x3 16

BI (156)(10)

sin6 ax 1 4 a (27 ln 3 − 32 ln 2) dx = 5 x 16

BI (156)(14)

sin6 ax 11 5 a π dx = x6 40

sin2 ax cos 2bx 1 dx = π max(0, a − b) 2 x 2

∞

π sin 2ax cos2 bx dx = x 2 3 = π 8 π = 4

0

6. 0

1 4a2 − b2 sin2 ax cos bx dx = ln x 4 b2

∞

5.8 

BI (156)(9)

[a > 0]

LO V 312

FI II 647 BI (157)(1)

BI (151)(10)

4.8 

BI(156)(11), LO V 312 BI (156)(4)

In 3.828 1–21 the restrictions a > 0, b > 0, c > 0 apply. ! !  ∞ sin ax sin bx 1 !! a + b !! [a =

b] dx = ln ! x 2 a − b! 0  ∞ dx 1 sin ax sin bx 2 = π min(a, b) x 2 0  ∞ 2 π sin ax sin bx dx = [b < 2a] x 4 0 π = [b = 2a] 8 =0 [b > 2a] 

463

[2a = b]

BI (151)(12)

[a > b] [a = b] [a < b] BI (151)(9)

464

Trigonometric Functions



∞

7.8 0

3.828

sin2 ax sin bx sin cx π (|b − 2a − c| − |2a − b − c| + 2c) dx = 2 x 16 [a > 0, 0 < c ≤ b]

! ! ∞ 1 !! (b + c)2 (2a − b + c)(2a + b − c) !! sin2 ax sin bx sin cx dx = ln ! x 8 (b − c)2 (2a + b + c)(2a − b − c) ! 0 [b = c, 2a + c = b, 2a + b = c,

BI(157)(9)a, ET I 79(15)

 8.8



∞

9. 0

sin2 ax sin2 bx π dx = a x2 4 π = b 4

2a = b + c]

LI (152)(2)

[0 ≤ a ≤ b] [0 ≤ b ≤ a] BI (157)(3)



∞

10.8 0

 11.

∞

8 0



∞

12. 0



∞

13. 0

 14.10 0

∞

  sin ax sin bx 1 dx = π min a2 , b2 [3 max(a, b) − min(a, b)] 4 x 6 2

2

BI (157)(27)

sin2 ax cos2 bx 1 dx = π [a + max(0, a − b)] 2 x 4 sin3 ax sin 3bx a3 π dx = 4 x 2  π  3 8a − 9(a − b)3 = 16  9bπ  2 a − b2 = 8 sin3 ax cos bx dx = 0 x π =− 16 π =− 8 π = 16 π = 4

BI (157)(6)

[b > a] [a ≤ 3b ≤ 3a]

BI (157)(28)

[3b ≤ a]

LI (157)(28)

[b > 3a] [b = 3a] [3a > b > a] [b = a] [a > b] ⎛

[a > 0,

b > 0]

BI (151)(15)

sin3 ax cos 3bx 3 ⎝ a ln 81 − 2(a − 3b) ln(a − 3b) + 2(a − b) ln(a − b) dx = 2 x 16 ⎞ + 2(a + b) ln(a + b) − 2(a + 3b) ln(a + 3)⎠ [Im a = 0,

Im b = 0]

MC

3.828

Powers and trigonometric functions



∞

15. 0



 sin3 ax cos bx π 2 3a − b2 dx = 3 x 8 πb2 = 4 π = (3a − b)2 16 =0

[b < a] [a = b] [a < b < 3a] [3a < b] [a > 0,

∞

16. 0

 sin3 ax sin bx bπ  2 9a − b2 dx = 4 x 24  π  24a3 − (3a − b)3 = 48 πa3 = 2

465

b > 0]

BI(157)(19), ET I 19(10)

[0 < b ≤ a] [0 < a ≤ b ≤ 3a] [0 < 3a ≤ b] ET I 79(16)



∞

17. 0



∞

18.8 0



∞

19.11 0

3

2

π sin ax sin bx dx = x 8 5π = 32 3π = 16 3π = 32 =0

[2b > 3a] [2b = 3a] [3a > 2b > a] [2b = a] [a > 2b]

[a > 0, b > 0] ! ! ! (2a + b)3 (b − 2a)3 (2a + 3b)(3b − 2a) ! 1 sin ax cos3 bx ! dx = ln !! ! x 16 9b8 [2a = b, 2a = 3b]

BI (151)(14)

2

2

2

BI (151)(13)

2

sin ax sin bx sin cx dx x π = 4 sign(c) − 2 sign(2b + c) + 2 sign(2b − c) + sign(2a − 2b + c) − sign(2a − 2b − c) 32

+ 2 sign(2a − c) + sign(2a + 2b + c) − sign(2a + 2b − c) − 2 sign(2a + c)

 20. 0

[Im a = 0, ∞

Im b = 0,

Im c = 0]

MC

sin2 ax sin2 bx sin 2cx dx x2 a−b−c a+b+c a+b−c ln 4(a − b − c)2 − ln 4(a + b + c)2 + ln 4(a + b − c)2 = 16 16 16 a−b+c a+c a−c ln 4(a − b + c)2 + ln 4(a + c)2 − ln 4(a − c)2 − 16 8 8 b+c b−c 1 ln 4(b + c)2 − ln 4(b − c)2 − c ln 2c + 8 8 2 [a > 0, b > 0, c > 0] BI (157)(10)

466

Trigonometric Functions



∞

21.8 0

sin2 ax sin3 bx 3b2 π dx = x3 16 a2 π = 12 6b2 − (3b − 2a)2 π = 32 a2 π = 4

3.829

[2a > 3b] [2a = 3b] [3b > 2a > b] [b ≥ 2a] BI (157)(18)

3.829 1. 2. 3.831 1. 2. 3. 4.



[(n−1)/2]   xn − sinn x π k n (n − 2k)n+1 dx = (−1) k xn+2 2n (n + 1)! 0 k=0

 ∞  ∞     dx dx mπ 2m 2m−1 2m 1 − cos 1 − cos x 2 = 2m x 2 = m x x 2 0 0



∞

(2n − 1)!! b sin2n ax − sin2n bx dx = ln [ab > 0, x (2n)!! a 0    ∞ cos2n ax − cos2n bx (2n − 1)!! b dx = 1 − [ab > 0, ln x (2n)!! a 0  ∞ b cos2m+1 ax − cos2m+1 bx dx = ln [ab > 0, x a 0

 ∞ 1 cosm ax cos max − cosm bx cos mbx b dx = 1 − m ln x 2 a 0



π/2

1.

x cos 0

p−1

x sin ax dx =

π 2p+1

Γ(p)

 p+a+1 



− ψ p−a+1 2 2   p−a+1   p+a+1 Γ Γ 2 2

ψ

0

0

FI II 651

m = 0, 1, . . .]

FI II

m = 0, 1, . . .]

LI (155)(8)

−(p + 1) < a < p + 1]

,  dx (−1) π + −2a 2m 1 − e sin2m+1 x sin 2mx 2 = − 1 sinh a a + x2 22m+1 a

sin2m−1 x sin[(2m − 1)x]

0

4.

n = 0, 1, . . .]

m

[a > 0, ∞

3. 

FI II 651

BI (205)(6) ∞

2.3 

n = 1, 2, . . .]



[p > 0, 

BI (158)(7, 8)

∞

[ab > 0, 3.832

GW (333)(63)

sin2m−1 x sin[(2m + 1)x]

BI (162)(17)

2m−1 π dx (−1) 1 − e−2a = a2 + x2 22m a [a > 0,

∞

m = 0, 1. . . .]

m+1

a2

m = 1, 2, . . .]

BI (162)(11)

2m−1 dx (−1)m−1 π −2a  e 1 − e−2a = 2 2m +x 2 a [a > 0,

m = 1, 2, . . .]

BI (162)(12)

3.832

Powers and trigonometric functions

 5.

6.3

7.

8.

9.

10. 11.

12.

13.

14.

15. 16.3 17.

18.3

∞

dx (−1)m π −3(2m+1)a e = sinh2m+1 a a2 + x2 2a 0 [a > 0]  ∞ + m  ,  x dx (−1) π a  −2a 2m −2a 1 − e sin2m x sin[(2m − 1)x] 2 = e − 1 + e a + x2 22m+1 0 [a ≥ 0, m = 0, 1, . . .]  ∞ , +  x dx (−1)m π  −2a 2m 1 − e sin2m x sin(2mx) 2 = − 1 a + x2 22m+1 0 [a > 0, m = 0, 1, . . .]  ∞ m  2m x dx (−1) π sin2m x sin[(2m + 2)x] 2 = 2m+1 e−2a 1 − e−2a 2 a +x 2 0 [a > 0, m = 0, 1, . . .]  ∞ m x dx (−1) π −4ma e sin2m x sin 4mx 2 = sinh2m a 2 a + x 2 0 [a > 0, m = 1, 2, . . .]  ∞ dx (2m − 3)!! π · [m = 1, 2, . . .] sin2m x cos x 2 = x (2m)!! 2 0  ∞ ,  dx (−1)m π + −2a 2m−1 1 − e sin2m x cos[(2m − 1)x] 2 = − 1 sinh a a + x2 22m a 0 [a > 0, m = 1, 2, . . .]  ∞ 2m dx (−1)m π  sin2m x cos(2mx) 2 = 2m+1 1 − e−2a 2 a +x 2 a 0 [a > 0, m = 0, 1, . . .]  ∞ m  2m dx (−1) π sin2m x cos[(2m + 2)x] 2 = 2m+1 e−2a 1 − e−2a 2 a +x 2 a 0 [a > 0, m = 0, 1, . . .]  ∞ m dx (−1) π −4ma e sin2m x cos 4mx 2 = sinh2m a 2 a + x 2a 0 [a > 0, m = 0, 1, . . .]  ∞ (2m − 1)!! π dx = · [m = 0, 1, . . .] sin2m+1 x cos x x (2m + 2)!! 2 0  ∞ dx (2m − 3)!! π · [m = 1, 2, . . .] sin2m+1 x cos x 3 = x (2m)!! 2 0  ∞ ,  x dx (−1)m π + −2a 2m−1 1 − e sin2m−1 x cos[(2m − 1)x] 2 = − 1 a + x2 22m 0 [m = 1, 2, . . . , a > 0]  ∞ +  , m−1   π a  x dx (−1) −2a 2m+1 −a e 1 − e sin2m+1 x cos 2mx 2 = − 1 − e a + x2 22m+2 0 [m = 0, 1, . . . , a ≥ 0]

467

sin2m+1 x sin[3(2m + 1)x]

BI (162)(18)

BI (162)(13)

BI (162)(14)

BI (162)(15)

BI (162)(16) GW (333)(15a)

BI (162)(25)

BI (162)(26)

BI (162)(27)

BI (162)(28) GW (333)(15) GW (333)(15b)

BI (162)(23)

BI (162)(29)

468

Trigonometric Functions



∞

19.

sin2m−1 x cos[(2m + 1)x]

0



3.832

2m−1 x dx (−1)m π −2a  1 − e−2a = e a2 + x2 22m [m = 1, 2, . . . ,

∞

20.

sin2m+1 x cos[2(2m + 1)x]

0

∞

21. 0



cosm x sin mx

x dx (−1)m−1 π −2(2m+1)a e = sinh2m+1 a 2 +x 2

22.

cosn sx sin nsx

0

 0

x dx π = n+1 2 +x 2

a2

cosn sx sin nsx

cosm−1 x sin[(m + 1)x]

cosm x sin[(m + 1)x]

0

∞

26.3

cosm x sin[(m − 1)x]

0



27.11 28. 29. 30.

Re a > 0,

n ≥ 0]

∞

BI (166)(10)

 m−1 x dx π = m e−2a 1 + e−2a 2 +x 2 m = 1, 2, . . .]

x dx π = m cosh a 2 +x 2

+

a2

[m = 0, 1, . . . , , m−1 1 + e−2a −1

a > 0]

BI (163)(10)

[m = 0, 1, . . . ,

a ≥ 0]

BI (163)(7)

31. 0

cosm x sin(3mx)

[m = 1, 2, . . . , ∞

cosm x cos[(m − 1)x]

BI (163)(6)

 m x dx π = m+1 e−a 1 + e−2a a2 + x2 2

x dx π = e−3ma coshm a [a > 0, m = 1, 2, . . .] 2 + x2 a 2 0  ∞ n dx π  cosn sx cos nxs 2 = n+1 1 + e−2as [n = 0, 1, . . .] 2 a +x 2 a 0  ∞ dx π cosn as sin nas cosn sx cos nsx 2 = [n = 0, 1, . . .] a − x2 2a 0  ∞  m−1 dx π cosm−1 x cos[(m + 1)x] 2 = m e−2a 1 + e−2a 2 a +x 2 a 0 

BI (163)(9)

a2

[a > 0,

25. 

BI (162)(8)

 x dx π  −n 2 − cosn as cos nas = a2 − x2 2

0

∞

BI (162)(30)

+ , n 1 + e−2as − 1

[n = 0, 1, . . .] ∞

24. 

k=1

a > 0]

 m  −2ka e Ei(2ka) − e2ka Ei(−2ka) k

[s > 0, ∞

23. 

x dx 1 = m+1 + x2 2 a

a2

m  

[a > 0] ∞

BI (162)(24)

a2

[m = 0, 1, . . . , 

a > 0]

a2

dx π = m+1 e 2 +x 2 a

+ a

1 + e−2a

m

 − 1 − e−2a

[m = 0, 1, . . . ,

BI (163)(11) BI (163)(16)

a > 0] ,

BI (163)(14)

a > 0]

BI (163)(15)

3.834

Powers and trigonometric functions



∞

32.

cosm x cos[(m + 1)x]

0



∞

33. 0



∞

34.

cos2m x cos 2nx sin x

cosp ax sin bx cos x

0 ∞

36.

dx x= x



∞

cos2m−1 x cos 2nx sin

0

∞

37. 0

π dx = x 2

cosp ax sin pax cos x

0



[m = 0, 1, . . . ,

a > 0]

BI (163)(17)

[p > q − 1 > 0]

[p > q − 1 > 1]

π dx x = 2m+1 x 2



2m m+n



GW (333)(18)

BI (152)(5, 6) ∞

35. 



∞ sinp−1 x dx p p + 1 ∞ sinp+1 x = dx − dx q q−1 x q−1 0 x  q−1 0 xq−1 ∞ p(p − 1) dx = sinp−2 x cos x q−2 (q − 1)(q − 2) 0 x ∞ (p + 1)2 dx − sinp x cos x q−2 (q − 1)(q − 2) 0 x

0



 m dx π = m+1 e−a 1 + e−2a a2 + x2 2 a 

sinp x cos x

469



dx x2

n -

k=1



[b > ap,

π dx = p+1 (2p − 1) x 2

π cospk ak x sin bx sin x = 2

p > −1]

[p > −1] % b>

n 

BI (153)(12) BI (153)(2)

& ak pk ,

ak > 0,

pk > 0

k=1

BI (157)(15)

3.833 1.



∞

10

2m+1

sin 0

x cos

2n

dx = x x



∞

sin2m+1 x cos2n−1 x

0

(2m − 1)!!(2n − 1)!! dx = π x 2m+n+1 (m + n)! BI (151)(24, 25)



1 1 1 = B m + ,n + 2 2 2

GW (333)(24)



∞

2. 0

3.834



1. 0

∞

sin2m+1 2x cos2n−1 2x cos2 x

π (2m − 1)!!(2n − 1)!! dx = · x 2 (2m + 2n)!!

LI (152)(4)

! !2m−1 sin2m+1 x dx (−1)m π(1 + a)4m !! 1 − a !! = · !1 + a! 1 − 2a cos x + a2 x 22m+2 a2m+1

k  2m 1  4a k m− 2 − (−1) k (1 + a)2 k=0

[|a| = 1]

GW (333)(62a)

470

Trigonometric Functions



∞

2. 0

sin2m+1 x cosn x dx p · 2 x (1 − 2a cos x + a ) =

3.835  1.

∞

0



∞

2. 0

3.836  1.

∞

2.

∞

n  (−1)k (2m + 2n − 2k + 1)!!(2m + 2k − 1)!! n!π 2n+1 (2m + n + 1)!(1 + a)2p k!(n − k)! k=0

4a 3 × F m + n − k + , p; 2m + n + 2; 2 (1 + a)2 [a = ±1] GW (333)(62)

π b2m−1 cos2m x cos 2mx sin x dx = · 2 x 2 a(a + b)2m a2 cos2 x + b2 sin x

[ab > 0]

BI (182)(31)a

π b2m−1 cos2m−1 x cos 2mx sin x dx = · x 2a (a + b)2m a2 cos2 x + b2 sin2 x

[ab > 0]

LI (182)(32)a

[m ≥ n]

LI (159)(12)



0



3.835



11 0

sin x x sin x x

n

n

π sin mx dx = x 2 nπ cos mx dx = n 2

12 (m+n)   k=0

(−1)k (n + m − 2k)n−1 k!(n − k)!

[0 ≤ m < n] [m ≥ n ≥ 2]

=0 π = 4

[m = n = 1] GI(159)(14), ET I 20(11)



∞

3. 0



∞



4.8 0

sin x x sin x x

n−1 sin nx cos x

n

π dx = x 2 ⎡

[n ≥ 1]

BI (159)(20)

⎤ 

12 n(1+a)   n π⎢ sin(anx) 1 ⎥ (n + an − 2k)n ⎦ dx = ⎣1 − n−1 (−1)k k x 2 2 n! k=0

[all real a, n ≥ 1] 5.10

I n (b) =



∞



6.11 0

3.837



π/2

1. 0

 2.

0

π/4

2 π

ET I 20(11)



n r

n  −1  sin x (n − b − 2k)n−1 cos bx dx = n 2n−1 n! (−1)k k x 0 k=0 where 0 ≤ b < n, n ≥ 1, r = (n − b)/2, and r is the largest integer contained in r



sin x x

∞

LO V 340(14)

n cos anx dx = 0

[a ≤ −1 or a ≥ 1,

x2 dx = π ln 2 sin2 x π x2 dx π2 + ln 2 + G = 0.8435118417 . . . = − 2 16 4 sin x

n ≥ 2;

for n = 1 see 3.741 2]

BI (206)(9)

BI (204)(10)

3.839

Powers and trigonometric functions



471

π/4

π2 π x2 dx = + ln 2 − G 2 cos x 16 4 0    π/4 p+1 ∞  π p+1  π p 1 1  x 1 dx = − − ζ(2k) + (p + 1) 4 4 p 2 42k−1 (p + 2k) sin2 x 0

3. 4.

GW (333)(35a)

k=1

[p > 0] 

π/2

5. 0



π/2

6. 0



∞

7. 0



∞

8. 0



1

9. 0



π

10.3 0

 11.3

0

3.838



π

BI (206)(7)

3 x3 cos x π3 + π ln 2 dx = − 3 16 2 sin x

BI (206)(8)

cos 2nx dx sin2n+1 x m = 0 cos x x

1 x dx = cosec a · ln sec a cos ax cos[a(1 − x)] a

 m−1 , n> 2  m−2 , n> 2 + π, a< 2

1 x sin(2n + 1)x dx = π 2 sin x 2

[n = 0, 1, 2, . . .]

 x sin 2nx dx = −4 (2k − 1)−2 sin x

[n = 1, 2, 3, . . .]

π/2

π/4

2. 0

4.

m>0

BI (180)(16)

 m>0

BI (180)(17)

BI (149)(20)

n

πp x cosp−1 x π sec dx = 2p 2 sinp+1 x π 1 x sinp−1 x dx = − β p+1 cos x 4p 2p

[p < 1]

p+1 2

BI (206)(13)a

[p > −1]

m−1 π 1  (−1)k−1 x sin2m−1 x dx = (1 − cos mπ) + cos2m+1 x 8m 2m 2m − 2k − 1 0 k=0 % &  π/4 m−1  (−1)k−1 1 π x sin2m x dx = + (−1)m−1 ln 2 + cos2m+2 x 2(2m + 1) 2 m−k 0

3.



k=1

0



π2 x2 cos x dx = − + 4G = 1.1964612764 . . . 2 4 sin x

cos 2nx dx sin2n x m = 0 cos x x

1. 

LI (204)(14)

LI (204)(15)

π/4

BI (204)(17)

BI (204)(16)

k=0

3.839 1.



π/4

11

x tan2 x dx =

1 π π2 − − ln 2 4 32 2

BI (204)(3)

x tan3 x dx =

1 π 1 π − + ln 2 − G 4 2 8 2

BI (204)(7)

0



π/4

2. 0

 3.

0

π/4

1 π π2 x2 tan x dx = ln 2 − + cos2 x 2 4 16

(cf. 3.839 1)

BI (204)(13)

472

Trigonometric Functions



π/4

4. 0



1 x2 tan2 x dx = cos2 x 3

π π2 π +G 1 − ln 2 − + 4 2 16

π/2

Γ(p + 1) ·+  ,2 p 0 +1 Γ 2

 π/2 p−1 p+1 p+1 2 π p − B , x sin x cot x dx = 2p p 2 2 0 x cosp x tan x dx =

5.

6.



∞

7.

sin2n x tan x

0

 8. 9. 10.

∞

π

2p+1 p



∞

11 0

BI (204)(12)

[p > −1]

BI (205)(3)

[p > −1]

BI (206)(11)

π dx = [s > −1] x 2 0

2n  ∞ sin x cos[(2n − 1)x] 22n − 1 2n−1 · ·2 dx = (−1)n−1 π|B2n | cos x x (2n)! 0  ∞  2  rπ dx π sec tanhr pq tanr px 2 = r <1 2 q +x 2q 2 0 coss rx tan qx

GW (333)(16) BI (151)(26)

BI (180)(15) BI (160)(19)

 √ 1 − k2 sin2 x, 1 − k2 cos2 x, and similar expressions

√ Notation: k  = 1 − k 2 3.841  ∞  dx = E(k) sin x 1 − k 2 sin2 x 1. x 0  ∞  dx = E(k) 2. sin x 1 − k 2 cos2 x x 0  ∞  dx = E(k) 3. tan x 1 − k 2 sin2 x x 0  ∞  dx = E(k) 4. tan x 1 − k 2 cos2 x x 0

1.

(cf. 3.839 2)

π (2n − 1)!! dx = · x 2 (2n)!!

3.84 Integrals containing

3.842

3.841

BI (154)(8) BI (154)(20) BI (154)(9) BI (154)(21)

dx sin x  2 1 + sin x x ∞ tan x dx  = · 2 x

0 ∞ 1 + sin x  ∞ 1 sin x tan x 1 dx dx √ √ = = =√ K √ ≈ 1.3110287771 2 2 1 + cos2 x x 1 + cos2 x x 0 0 BI (183)(4, 5, 9, 10)

 2. u

π 2



x cos x dx sin x − sin u 2

2

=

π ln (1 + cos u) 2

BI (226)(4)

3.844

Integrals containing



∞

3. 0

 1 − k 2 sin2 xf unction]squareroot and similar expressions

 ∞ dx tan x   = 2 2 2 2 x 1 − k sin x 0 ∞ 1 − k sin x sin x √ = 1 − k 2 cos2 x 0 sin x

473

dx x  ∞ tan x dx dx √ = = K(k) 2 cos2 x x x 1 − k 0 BI (183)(12, 13, 21, 22)



π/2

4. 0



π/2

5. 0

 6.

0

7. 3.843 1. 2. 3.11

α



 dx = E(k) tan x 1 − k 2 sin2 2x x 0  ∞  dx = E(k) tan x 1 − k 2 cos2 2x x 0

 ∞  ∞ 1 dx 1 tan x tan x dx  √ = =√ K √ ≈ 1.3110287771 2 2 1 + cos2 2x x 0 0 1 + sin2 2x x ∞



∞

1. 0



4. 5. 6.

BI (214)(1)

LO III 284

LO III 284

∞

0

3.

x sin x cos x 1 √ dx = 2 [π − 2 E(k)] 2 2 2k 1 − k cos x

π sin2 α2 x sin x dx  = cos2 α cos2 x sin2 α − sin2 x

4.

2.

BI (211)(1)

 cos α + 1 − sin2 α sin2 β π ln  β 2 cos β cos2 α2 x sin x dx   =   0 1 − sin2 α sin2 x sin2 β − sin2 x 2 cos α 1 − sin2 α sin2 β



3.844

x sin x cos x 1  dx = 2 [−πk  + 2 E(k)] 2 2 2k 1 − k sin x

tan x dx  = 2 1 − k2 sin 2x x

 0

BI (154)(10) BI (154)(22)

BI (183)(6, 11) ∞

√

tan x dx = K(k) 2 2 1 − k cos 2x x

1 sin x cos x dx √ = 2 [K(k) − E(k)] 2 2 k 1 − k cos x x

BI (183)(14, 23)

BI (185)(20)

∞

1 sin x cos2 x dx √ = 2 [K(k) − E(k)] · 2 2 x k 1 − k cos x 0  ∞     sin x cos3 x 1  dx √ = 4 2 + k 2 K(k) − 2 1 + k2 E(k) · 2 2 3k 1 − k cos x x 0  ∞ 4     1  sin x cos x dx √ = 4 2 + k 2 K(k) − 2 1 + k2 E(k) · 2 2 x 3k 1 − k cos x 0  ∞   1  sin3 x cos x dx √ = 4 1 + k 2 E(k) − 2k 2 K(k) · 3k 1 − k 2 cos2 x x 0  ∞ 3 2   1  sin x cos x dx √ = 4 1 + k 2 E(k) − 2k 2 K(k) · 2 2 3k 1 − k cos x x 0

BI (185)(21) BI (185)(22) BI (185)(23)

BI (185)(24)

BI (185)(25)

474

Trigonometric Functions

 7. 8. 3.845 1.11

2.11

3.11

∞

 sin2 x tan x 1  dx √ = 2 E(k) − k2 K(k) · 2 2 k 1 − k cos x x 0  ∞ 4     1  sin x tan x dx √ = 4 2 + 3k 2 k 2 K(k) − 2 k 2 − k 2 E(k) · 2 2 x 3k 1 − k cos x 0  √ & % √  1 sin x cos x 2 2 dx √ √ = 2 E − K ≈ 0.5990701174 · 2 x 2 2 2 1 + cos x 0  √ & % √   ∞ sin x cos2 x dx √ 2 2 1 √ = 2 E · − K ≈ 0.5990701174 2 x 2 2 2 1 + cos x 0  √ & % √   ∞ sin2 x tan x dx √ 2 2 √ = 2 K −E ≈ 0.7119586598 · 2 x 2 2 1 + cos x 0 

3.846  1.

3.845

BI (184)(16) BI (184)(18)

∞

BI (185)(6)

BI (185)(7)

BU (184)(8)

∞

 1  sin x cos x dx  = 2 E(k) − k2 K(k) BI (185)(9) · 2 2 x k 0 1 − k sin x  ∞  1  sin x cos2 x dx  = 2 E(k) − k2 K(k) 2. BI (185)(10) · 2 2 x k 0 1 − k sin x  ∞     1  sin x cos3 x dx  = 4 2 − 3k2 k 2 K(k) − 2 k 2 − k 2 E(k) 3. BI (185)(11) · 2 2 x 3k 0 1 − k sin x  ∞     1  sin x cos4 x dx  = 4 2 − 3k2 k 2 K(k) − 2 k 2 − k 2 E(k) 4. BI (185)(12) · 2 2 x 3k 0 1 − k sin x  ∞   1  sin3 x cos x dx  = 4 1 + k2 E(k) − 2k 2 K(k) 5. BI (185)(13) · 2 2 x 3k 0 1 − k sin x  ∞   1  sin3 x cos2 x dx  = 4 1 + k2 E(k) − 2k 2 K(k) 6. BI (185)(14) · 2 3k 0 1 − k2 sin x x  ∞ 1 sin2 x tan x dx  = 2 [K(k) − E(k)] 7. BI (184)(9) · 2 k 0 1 − k2 sin x x  ∞     1  sin4 x tan x dx  = 4 2 + k2 K(k) − 2 1 + k 2 E(k) 8. BI (184)(11) · 2 3k 0 1 − k2 sin x x % √   √ &  ∞  ∞ sin x cos x sin x cos2 x dx √ 2 2 dx 11   = = 2 K 3.847 · · −E ≈ 0.7119586598 2 2 x x 2 2 0 0 1 + sin x 1 + sin x BI (185)(3, 4)

3.848 1. 2.



∞

sin3 x cos x 1 dx  = 2 [K(k) − E(k)] · 2 2 x 4k 0 1 − k sin 2x  ∞  1  cos2 2x tan x dx  = 2 E(k) − k2 K(k) · k 0 1 − k2 sin2 2x x

BI (185)(15)

BI (184)(12)

3.852

Trigonometric functions with powers

 3. 4. 5. 6. 7. 3.849 1.11

2.11

3.11

475

∞

    cos4 2x tan x 1  dx  = 4 2 − 3k2 k 2 K(k) − 2 k 2 − k 2 E(k) · 2 3k 0 1 − k2 sin 2x x  ∞ 2   4  sin 4x tan x dx  = 4 1 + k2 E(k) − 2k 2 K(k) · 2 3k 0 1 − k2 sin 2x x  ∞ 3  sin x cos x 1  dx √ = 2 E(k) − k2 K(k) · 2 2 4k 1 − k cos 2x x 0  ∞ 2 1 cos 2x tan x dx √ = 2 [K(k) − E(k)] · 2 2 x k 1 − k cos 2x 0  ∞     1  cos4 2x tan x dx √ = 4 2 + k2 K(k) − 2 1 + k 2 E(k) · 2 2 3k 1 − k cos 2x x 0  √ & % √  1 sin3 x cos x 2 2 dx √ = √ K −E ≈ 0.1779896649 · 2 x 2 2 2 2 1 + cos 2x 0 √   √ & √ %  ∞ sin3 x cos x 2 2 2 dx  = 2E −K ≈ 0.1497675293 · 2 x 8 2 2 0 1 + sin 2x  √ & % √   ∞ cos2 2x tan x dx √ 2 2  = 2 K −E ≈ 0.7119586598 · 2 x 2 2 0 1 + sin 2x 

BI (184)(13)

BI (184)(17)

BI (185)(26) BI (184)(19) BI (184)(20)

∞

BI (185)(8)

BI (185)(5)

BI (184)(7)

3.85–3.88 Trigonometric functions of more complicated arguments combined with powers 3.851



∞

5. 0

  dx bπ sin ax2 cos(bx) 2 = x 2

2







√ b π b b √ √ + S −C + aπ sin 4a 4 2 a 2 a [a > 0, b > 0] , (cf. 3.691 7) ET I 23(3)a

3.852



∞

1. 0



∞

2. 0

   sin ax2 aπ dx = x2 2

 √    dx 1 π √  sin ax2 cos bx2 2 = a+b+ a−b x 2 2 1√ πa = 2 √ 1 π √ a+b− b−a = 2 2

[a ≥ 0] [a > b > 0] [b = a ≥ 0]

[b > a > 0] ,  3. 0

∞

  √ sin2 a2 x2 2 π 3 a dx = x4 3

BI (177)(10)a

[a ≥ 0]

(cf. 3.852 1)

BI (177)(23)

GW (333)(19e)

476

Trigonometric Functions

   √ sin3 a2 x2 a π dx = 3 3 − x2 4 2 0   ∞  2  1 π 2 2 dx sin x − x cos x = x4 3 2 0

 ∞ dx 1 1 =− C cos2 x − 2 1+x x 2 0 

4.

10

5. 6. 3.853



∞

∞

1. 0

2.

3.

4.

3.854

3.853



Im a2 = 0



MC BI (178)(8) BI (173)(22)

    ,  sin ax2 π +√ π √  √ π √  C S dx = 2 sin aβ 2 + aβ − 2 cos aβ 2 + aβ − sin aβ 2 2 2 β +x 2β 4 4

ET II 219(33)a [a > 0, Re β > 0]  +    2 √ √  √ √ , cos ax2 π π π 2 2 cos C S aβ − dx = 2 cos aβ + aβ − 2 sin aβ + aβ β 2 + x2 2β 4 4 0 ET II 221(51)a [a > 0, Re β > 0]  2  ∞ 2 x sin ax dx β 2 + x2 0   π √  √ π √ , βπ +  2  √ sin aβ − 2 sin aβ 2 + C S aβ + 2 cos aβ 2 + aβ = 2  4 4 1 π − 2 2a ET II 219(32)a [a > 0, Re β > 0]  2   ∞ 2    √   √ x cos ax βπ 1 π π − cos aβ 2 − 2 cos aβ 2 + C dx = aβ 2 2 β +x 2 2a  2 4 0  √ √  π S aβ − 2 sin aβ 2 + 4 ET II 221(50)a [a > 0, Re β > 0]



∞



∞

1.



     cos ax2 − sin ax2

0

dx πe−ab √ = x4 + b4 2b3 2

2

[a > 0,

b > 0] LI (178)(11)a, BI (168)(25)



∞

2. 0



     x2 dx πe−ab √ cos ax2 + sin ax2 = x4 + b4 2b 2

2

∞

     cos ax2 + sin ax2

πe−ab √ = 2 4 2b3 (x4 + b4 )

∞

     cos ax2 − sin ax2

πe−ab √ 4 2b

3. 0

 4. 0

[a > 0, b > 0] 2

x2 dx

x4 dx (x4 +

2 b4 )

2

=

LI (178)(12)



1 a+ 2 2b

[a > 0,

1 −a 2b2

b > 0]

LI (178)(14)

[a > 0,

b > 0]

BI (178)(15)

3.856

3.855 1. 2. 3. 4.

Trigonometric functions with powers

  



sin ax2 aβ aβ 1 aπ  I 14 dx = K 14 [a > 0, 2 4 2 2 2 2 β +x 0   



 ∞ cos ax2 aβ aβ 1 aπ  1 1 I −4 dx = K4 [a > 0, 2 4 2 2 2 2 β +x 0    

2  u sin a2 x2 a2 a π3 √ dx = 2 [a > 0] J 14 4 2 u2 u4 − x4 0    2 2

2 2

 ∞ sin a2 x2 a u a u a π3 √ J 14 dx = − Y 14 4 4 4 2 2 2 x −u u 

6.

∞

Re β > 0]

ET I 66(28)

Re β > 0]

ET I 9(22)

ET I 66(29)

[a > 0]    2 2

 2 2 2 cos a x a u a π3 √ dx = J − 14 4 4 4 2 2 u −x 0   2 2 2 2

 ∞ cos a x a2 u 2 a u a π3 √ J − 14 dx = − Y − 14 4 4 4 2 2 2 x −u u 

5.

477

3.856 

∞



1. 0



∞



2. 0

 3. 0

ET I 66(30)

u

∞



β 4 + x4 + x2  β 4 + x4

β 4 + x4 + x2  β 4 + x4

β 4 + x4 − x2  β 4 + x4

ν 

2 2

sin a x



a dx = 2

ν   a cos a2 x2 dx = 2

ν   a cos a2 x2 dx = 2



π 2ν β I 14 − ν2 2



a2 β 2 2

ET I 9(23)

ET I 10(24)



K 14 + ν2

 3 Re ν < , 2

a2 β 2 2

|arg β| <



π 4

 ET I 71(23)



2 2

2 2

a β a β π 2ν β I − 14 − ν2 K − 14 + ν2 2 2 2   π 3 ET I 12(16) Re ν < , |arg β| < 2 4



2 2

2 2

a β a β π 2ν β I − 14 + ν2 K − 14 − ν2 2 2 2   π 3 Re ν > − , |arg β| < 2 4 ET I 12(17)

 2 2 2 2

∞ sin a2 x2 dx sinh a 2β a β < = √ K0   2 2 2β 0 β 4 + x4 x2 + β 4 + x4 + π, |arg β| < 4  2 2 2 2

 ∞ a2 β 2 cos a x dx sinh a β  = √ 2 K1 4 3   2 2 2β 0 x2 + β 4 + x4 β 4 + x4 + π, |arg β| < 4 

4.

5.



ET I 66(32)

ET I 10(27)

478

Trigonometric Functions



∞

6. 0

3.857  1.

∞

0



∞

2. 0

3.858  1.

<

β 4 + x4 + x2   a2 β 2 π  sin a2 x2 dx = √ e− 2 I 0 2 2 β 4 + x4

x2 R1 R2

x2 R1 R2







3.857

a2 β 2 2 + π, |arg β| < 4

  1 R2 − R1 sin ax2 dx = √ K 0 (ac) sin ab R2 + 2 b  R1 < < 2 2 2 2 R1 = c + (b − x ) , R2 = c2 + (b + x2 ) ,   1 R2 + R1 cos ax2 dx = √ K 0 (ac) cos ab R2 − R 2 b 1  < < 2 2 2 2 R1 = c + (b − x ) , R2 = c2 + (b + x2 ) ,

ET I 67(33)

 a > 0,

c>0

ET I 67(34)

 a > 0,

c>0

ET I 10(26)

√  2 √ ν  ν   x + x4 − u4 + x2 − x4 − u4 √ sin a2 x2 dx 4 4 u x − u  2 2

2 2

2 2

2 2  a u a u a u a u a π 3 2ν u =− J 14 + ν2 Y 14 − ν2 + J 14 − ν2 Y 14 + ν2 4 a 2 2 2 2  Re ν < 32 ET I 71(25) √ ν  ν  ∞ 2 √ 4   x + x − u4 + x2 − x4 − u4 √ 2. cos a2 x2 dx 4 4 x −u u   2 2

2 2

2 2

2 2  a u a u a u a u a π 3 2ν 1 ν 1 ν 1 ν 1 ν u =− J−4+2 Y −4−2 + J−4−2 Y −4+2 4 a 2 2  2 2  Re ν < 32 ET I 13(26)   ∞  n dx 1 1 = − nC 3.859 BI (173)(24) cos x2 − n+1 2 x 2 1+x 0 3.861

√ m− 1  ∞ n+1   2  dx 1 πa 2 2n+1 k−1 2n + 1 ax sin = ± 2n−m+ 1 (−1) 1. (2k − 1)m− 2 2m n + k x 2 (2m − 1)!! 2 0 k=1   the + sign is taken when m ≡ 0 (mod 4) or m ≡ 1 (mod 4), BI (177)(19)a the − sign is taken when m ≡ 2 (mod 4) or m ≡ 3 (mod 4)

√ m− 1  ∞ n    dx 1 2n πa 2 2. sin2n ax2 2m = ± 2n−2m+1 (−1)k k m− 2 n+k x 2 (2m − 1)!! 0 k=1   the + sign is taken when m ≡ 0 (mod 4) or m ≡ 3 (mod 4), BI (177)(18)a, LI (177)(18) the − sign is taken when m ≡ 2 (mod 4) or m ≡ 1 (mod 4)

 ∞   2√   2 √  sin2 x n cos ax n + sin ax n 3.862 dx x2 0 √ n n   √ n− 12 π √ n − 2k + a n = (−1)k k (2n − 1)!! 2 k=0   √ a> n>0 BI (178)(9) ∞

3.866

3.863

Trigonometric functions with powers



∞

1. 0



∞

2. 0

3.864 1. 2.



    π x cos ax4 sin 2bx2 dx = − 8



2

    π x2 cos ax4 cos 2bx2 dx = − 8



479

 2 2

2 

2

b b b b3 π b π 1 3 − − sin J + cos J −4 3 4 a 2a 8 2a 2a 8 2a ET I 25(22) [a > 0, b > 0]  2 





b2 b2 b π b2 π b3 3 1 + + sin J + cos J −4 −4 a3 2a 8 2a 2a 8 2a

 √  √ dx π b sin ax = Y 0 2 ab + K 0 2 ab x x 2 0  ∞  √  √ dx π b = − Y 0 2 ab + K 0 2 ab cos cos ax x x 2 0

[a > 0,

b > 0]

ET I 25(23)

[a > 0,

b > 0]

WA 204(3)a

[a > 0,

b > 0]

∞

sin

WA 204(4)a, ET I 24 (12)

3.865



u



1. 0



∞

2. u

u2 − x2 x2μ

μ−1

(x − u) x2μ

sin

μ−1

sin

√ μ− 12 a 3 π 2 a dx = uμ− 2 Γ(μ) J 12 −μ x 2 a u [a > 0, u > 0,

a dx = x



0 < Re μ < 1] ET II 189(30)

π 1 −μ a a 2 Γ(μ) sin J 1 u 2u μ− 2

a 2u [a > 0,

u > 0,

Re μ > 0] ET II 203(21)

 u 3. 0



∞

4. u

μ−1 √ μ− 12 a u2 − x2 3 π 2 a dx = − cos Γ(μ)uμ− 2 Y 12 −μ 2μ x x 2 a u [a > 0, u > 0,

(x − u) x2μ

μ−1

cos

a dx = x



0 < Re μ < 1] ET II 190(36)

π 1 −μ a a 2 Γ(μ) cos J 1 u 2u μ− 2

a 2u [a > 0,

u > 0,

Re μ > 0] ET II 204(26)

3.866



1.

μ−1

x 0

  b2 π sin sin a2 x dx = x 4

μ b μπ [J μ (2ab) − J −μ (2ab) + I −μ (2ab) − I μ (2ab)] cosec a 2 [a > 0, b > 0, |Re μ| < 1] ET I 322(42)

μ ∞  2  b2 π b μπ μ−1 [J μ (2ab) + J −μ (2ab) + I μ (2ab) − I −μ (2ab)] x sin cos a x dx = sec x 4 a 2 0 [a > 0, b > 0, |Re μ| < 1]

 2.

∞

ET I 322(43)

480

Trigonometric Functions



∞

3.

μ−1

x 0

  b2 π cos cos a2 x dx = x 4

3.867

μ b μπ [J −μ (2ab) − J μ (2ab) + I −μ (2ab) − I μ (2ab)] cosec a 2 [a > 0, b > 0, |Re μ| < 1] ET I 322(44)

3.867



1

1. 0

 2. 0

3.868



1

cos ax − cos xa 1 dx = 1 − x2 2 cos ax + cos xa 1 dx = 1 + x2 2

∞

1. 0

3. 4.

b2 sin a x + x



2



b2 2 cos a x + x 0

 ∞ b2 sin a2 x − x 0

 ∞ b2 2 cos a x − x 0 

2.



3.869  1.

∞

∞

0



∞

2. 0



b sin ax − x





∞

0



∞

0

cos ax − cos xa π dx = sin a 1 − x2 2 [a > 0]

GW (334)(7a)

[a > 0]

GW (334)(7b)

cos ax + cos xa π dx = e−a 1 + x2 2

dx = π J 0 (2ab) x

[a > 0,

b > 0] GW (334)(11a), WA 200(16)

dx = −π Y 0 (2ab) x

[a > 0,

b > 0]

GW (334)(11a)

dx =0 x

[a > 0,

b > 0]

GW (334)(11b)

dx = 2 K 0 (2ab) x

[a > 0,

b > 0]

GW (334)(11b)

[a > 0,

b > 0,



b x dx π = exp −αβ − β 2 + x2 2 β





dx b b π exp −aβ − cos ax − = x β 2 + x2 2β β

Re β > 0] ET II 220(42)

[a > 0,

b > 0,

Re β > 0] ET II 222(58)

3.871



∞

1. 0

 2. 0

∞



 + b2 μπ μπ , − Y μ (2ab) sin xμ−1 sin a x + dx = πbμ J μ (2ab) cos x 2 2 [a > 0, b > 0, Re μ < 1] 

 + b2 μπ μπ , μ−1 + Y μ (2ab) cos x cos a x + dx = −πbμ J μ (2ab) sin x 2 2 [a > 0, b > 0, |Re μ| < 1]



 ∞ b2 μπ xμ−1 sin a x − dx = 2bμ K μ (2ab) sin x 2 0

ET I 321(35)

 3.

ET I 319(17)

[a > 0,

b > 0,

|Re μ| < 1] ET I 319(16)

3.874

Trigonometric functions with powers



∞

4.

μ−1

x 0

481



 b2 μπ cos a x − dx = 2bμ K μ (2ab) cos x 2 [a > 0,

b > 0,

|Re μ| < 1] ET I 321(36)

3.872 1.

2.

3.873 1.

2.

3.874



 

 dx 1 1 sin a x + sin a x − x x 1 − x2  0

 

 dx 1 ∞ 1 π 1 = sin a x + = − sin 2a sin a x − 2 0 x x 1 − x2 4 [a ≥ 0] BI (149)(15), GW (334)(8a) 

 

  1 dx 1 1 cos a x + cos a x − 2 x x 1 + 0 

 

  ∞x dx 1 1 π 1 = cos a x + = e−2a cos a x − 2 0 x x 1 + x2 4 [a ≥ 0] GW (334)(8b) 

1

√  π  a2 2 2 dx √ sin(2ab) + cos(2ab) + e−2ab cos b x = 2 2 x x 4 2a 0 [a > 0, √  ∞ 2  dx π  a cos(2ab) − sin(2ab) + e−2ab cos 2 cos b2 x2 2 = √ x x 4 2a 0 [a > 0, 

∞

sin



∞

1. 0



∞

2. 0

4. 5. 6.



√  π π b2 dx 2 2 cos 2ab + cos a x + 2 = x x2 2b 4

∞

2

[a > 0,

ET I 24(15)

b > 0]

ET I 24(16)

b > 0] BI (179)(6)a, GW(334)(10a)

[a > 0,

b > 0] GI (179)(8)a, GW(334)(10a)



√ dx π b 2 2 sin a x − 2 = − √ e−2ab 2 x x 2 2b 0

√  ∞ 2 dx π b cos a2 x2 − 2 = √ e−2ab 2 x x 2 2b 0 √

2  ∞ dx 2π b sin ax − = 2 x x 4b 0 √

2  ∞ dx 2π b cos ax − = 2 x x 4b 0 

3.



√  π π b2 dx sin 2ab + sin a2 x2 + 2 = x x2 2b 4

b > 0]

[a ≥ 0,

b > 0]

GW (335)(10b)

[a ≥ 0,

b > 0]

GW (334)(10b)

[a > 0,

b > 0]

BI (179)(13)a

[a > 0,

b > 0]

BI (179)(14)a

482

3.875

Trigonometric Functions



∞

1. u



∞

2. u



∞

3.6 3.876



1. 0

2.

  √   x sin p x2 − u2 π 2 + u2 cosh ab exp −p cos bx dx = a x2 + a2 2 [0 < b < p]   √   x sin p x2 − u2 π −ap 2 − a2 e cos bx dx = cos b u a2 + x2 − u2 2   √ sin p a2 + x2 (a2 + x2 )

0

∞

3.875

3/2

cos bx dx =

πp −ab e 2a

  √   sin p x2 + a2 π √ cos bx dx = J 0 a p2 − b2 2 x2 + a2 =0

  √  ∞   cos p x2 + a2 π √ cos bx dx = − Y 0 a p2 − b2 2  x2 + a2 0 = K 0 a b2 − p2

ET I 27(39)

[0 < b < p,

a > 0]

ET I 27(38)

[0 < p < b,

a > 0]

ET I 26(29)

[0 < b < p] [b > p > 0] [a > 0]

ET I 26(30)

[0 < b < p] [b > p > 0]

[a > 0] √   ∞   cos p x2 + a2 π −bc 2 − c2 e cos bx dx = cos p a x2 + c2 2c 0

ET I 26(34)

 3.

[c > 0, b > p] √ √     ∞ sin p x2 + a2 π e−bc sin p a2 − c2 √ √ cos bx dx = [c = a] 2 2 2 2 2c a2 − c 2 0 (x + c ) x + a p π [c = a] = e−ba 2 a [b > p, c > 0]   √  ∞ cos p x2 + a2 π −ab e cos bx dx = [b > p > 0; a > 0] 2 + a2 x 2a 0   √  ∞ x cos p x2 + a2 π sin bx dx = e−ab [a > 0, b > p > 0] 2 2 x +a 2 0   √  u   cos p u2 − x2 π √ cos bx dx = J 0 u b2 + p2 2 u2 − x2 0 √    ∞   2 2 cos p x − u √ [0 < b < |p|] cos bx dx = K 0 u p2 − b2 x2 − u2 u   π = − Y 0 u b2 − p2 [b > |p|] 2

ET I 26(33)

 4.

5.6 6.6 7. 8.

ET I 26(31)a ET I 27(35)a

ET I 85(29)a

ET I 28(42)

ET I 28(43)

3.881

3.877

Trigonometric functions with powers



u

1. 0

483

   √ , , + u  + u  sin p u2 − x2 π3 p < J 14 cos bx dx = b2 + p2 − b J 14 b2 + p2 + b 8 2 2 3 4 (u2 − x2 )

[b > 0, p > 0] ET I 27(40)  √   ∞ +u  , +u  ,   sin p x2 − u2 π3 p < J 14 cos bx dx = − b − b2 − p2 Z 14 b + b2 − p2 8 2 2 3 4 u (x2 − u2 )

 2.

[b > p > 0] ET I 27(41)  √   u , , + u  + u  cos p u2 − x2 π3 p < J − 14 cos bx dx = p2 + b2 − b J − 14 p2 + b2 + b 8 2 2 3 4 0 (u2 − x2 )

 3.

4.

[u > 0, p > 0] ET I 28(44)  √    ∞ +u  , +u  ,   cos p x2 − u2 π3 p < J − 14 b − b2 − p2 Y 14 b + b2 − p2 cos bx dx = − 8 2 2 3 4 u (x2 − u2 ) [b > p > 0]

3.878 1.

2.

3.

   √  2  2     sin p x4 + a4 a a 1  π 3 2 2 2 2 2 √ cos bx dx = b J − 14 J 14 p− p −b p+ p −b 2 2 2 2 x4 + a4 0 [p > b > 0] ET I 26(32) √    ∞ cos p x4 + a4 √ cos bx2 dx x4 + a4 0   2  2     a a 1  π 3 2 2 2 2 b J − 14 =− Y 14 p− p −b p+ p −b 2 2 2 2 [a > 0, p > b > 0] ET I 27(36)    √     u  cos p u4 − x4 u2  2 u2   2 π 3 1 2 2 2 √ cos bx dx = b J − 14 p + b − p J − 14 p +b +p 2 2 2 2 u4 − x4 0 

∞



∞

3.879 0

3.881



sin axp

π dx = x 2p

π/2

1.

x sin (a tan x) dx = 0

 2. 3.

ET I 28(45)

[p > 0,

b > 0]

ET I 28(46)

[a > 0,

p > 0]

GW (334)(6)

 π −a  C + ln 2a − e2a Ei(−2a) e 4 [a > 0]

BI (205)(9)

[a > 0]

BI (151)(6)

[a > 0]

BI (151)(19)

∞

 π dx = 1 − e−a x 2 0  ∞  π dx = 1 − e−a sin (a tan x) cos x x 2 0 sin (a tan x)

484

Trigonometric Functions



∞

4.

cos (a tan x) sin x 0



π dx = e−a x 2

∞

5. 

∞



[a > 0]

BI (152)(11)

cos (a tan x) sin3 x

1 − a −a dx = πe x 4

[a > 0]

BI (151)(23)

[a > 0]

BI (152)(13)

∞

sin (a tan x) tan 0



π/2

8.

π x dx = − Ei(−a) sin 2x 4

[a > 0]

BI (206)(15)

sin (a cot x)

x dx 1 − e−a π = 2a sin2 x

[a > 0]

LI (206)(14)

π/2

9. 0



π/2

10. 0



1 + a −a x dx cos2 x = πe 2 x 4

cos (a tan x) 0



  π x cos (a tan x) tan x dx = − e−a C + ln 2a + e2a Ei(−2a) 4

∞

11.

cos (a tan x) tan x 0



∞

12.

π dx = e−a x 2

cos (a tan x) sin2 x tan x

0



∞

13.



∞



17.

[a > 0]

BI (151)(22)

[a > 0]

BI (152)(13)

[a > 0]

BI (152)(10)

[a > 0]

BI (180)(6)

sin (a tan 2x) cos2 2x tan x

1 + a −a dx = πe x 4

∞

∞

0

   x dx π exp (−a tanh b) − e−a sin a tan2 x 2 = b + x2 2 [a > 0,

∞

2.

  cos a tan2 x cos x

0

 3. 0

BI (152)(15)

π dx = e−a x 2

π dx = e−a x 2 0  ∞  π dx = 1 − e−a sin (a tan 2x) tan x cot 2x x 2 0



[a > 0]

cos (a tan 2x) tan x

sin (a tan 2x) tan x tan 2x

3.882  1.

BI (151)(21)

BI (152)(9)

0

16.

[a > 0]

[a > 0]

0

15.

BI (205)(10)

π dx = e−a x 2

∞

14.

1 − a −a dx = πe x 16

[a > 0]

sin (a tan x) tan2 x

0



BI (151)(20)

1 + a −a dx = πe x 2

0

7.

[a > 0]

sin (a tan x) sin 2x 0

6.

3.882

b2

BI (160)(22)

 dx π  cosh b exp (−a tanh b) − e−a sinh b = 2 +x 2b [a > 0,

∞

b > 0]

b > 0]

BI (163)(3)

b > 0]

BI (191)(10)



 x dx π exp (−a tanh b) cos a tan2 x cosec 2x 2 = b + x2 2 sinh 2b [a > 0,

3.892

Trigonometric functions and exponentials

 4.

5.11

6.

∞

 −a    x dx π e cosh b − exp (−a tanh b) sinh b cos a tan2 x tan x 2 = 2 b + x 2 cosh b 0 [a > 0, b > 0]  ∞    x dx π coth b exp (−a tanh b) − e−a cos a tan2 x cot x 2 = 2 b +x 2 0 [a > 0, b > 0]  ∞    x dx π coth 2b exp (−a tanh b) − e−a cos a tan2 x cot 2x 2 = 2 b +x 2 0 [a > 0, b > 0]

3.883  1.

1

cos (a ln x)

0



xμ−1 sin (β ln x) dx = − 0



3.88411

BI (163)(4)

BI (163)(5)

BI (191)(11)

dx aπ = (1 + x)2 2 sinh aπ

1

2. 3.

485

BI (404)(4)

β β 2 + μ2

[Re μ > |Im β|]

ET I 319(19)

1

μ [Re μ > |Im β|] + μ2 0   ∞ +     , sin a |x| sign x dx = π exp −a |−b| + exp −a |b| x−b −∞ [a > 0, Im b = 0] xμ−1 cos (β ln x) dx =

ET I 321(38)

β2

ET II 253(46)

3.89–3.91 Trigonometric functions and exponentials 3.891  1.

2π

[m = n; or m = n = 0]

eimx sin nx dx = 0

0

[m = n = 0]

= πi 

2π

[m = n]

eimx cos nx dx = 0

2. 0

=π

[m = n = 0]

= 2π

[m = n = 0]

3.892  π 1.11 eiβx sinν−1 x dx = 0



π

2ν−1 ν B

πeiβ 2

ν+β+1 ν−β+1 , 2 2 [Re ν > −1]

π 2

2. −π 2



eiβx cosν−1 x dx = 2ν−1 ν B

NH 158, EH I 12(29)

π

ν+β+1 ν−β+1 , 2 2 [Re ν > −1]

GW (335)(19)

486

Trigonometric Functions



π/2

ei2βx sin2μ x cos2ν x dx =

3.6 0

3.893



   exp iπ β − ν − 12 B (β − μ − ν, 2ν + 1)    × F (−2μ, β − μ − ν; 1 + β − μ + ν; −1) + exp iπ μ + 12 1

22μ+2ν+1

 × B(β − μ − ν, 2μ + 1) F (−2ν, β − μ − ν; 1 + β + μ − ν; −1)   Re μ > − 12 , Re ν > − 12 EH I 80(6)



π

ei2βx sin2μ x cos2ν x dx =

4. 0

π exp [iπ(β − ν)] F (−2ν, β − μ − ν; 1 + β + μ − ν; −1) 4μ+ν (2μ + 1) B(1 − β + μ + ν, 1 + β + μ − ν) EH I 80(8)



π/2

0

3.893



π

ei(μ+ν)x sinμ−1 x cosν−1 x dx = eiμ 2 B(μ, ν) 

1 1 1 iμ π F (1 − ν, 1; μ + 1; −1) + F (1 − μ, 1; ν + 1; −1) = μ+ν−1 e 2 2 μ ν [Re μ > 0, Re ν > 0] EH I 80(7)

5.

∞

1.8

e−px sin(qx + λ) dx =

0



∞

2.8

e−px cos(qx + λ) dx =

0



∞

3. 

[Re p > 0]

BI (261)(3)

1 (p cos λ − q sin λ) p2 + q 2

[Re p > 0]

BI (261)(4)

e−x cos t cos (t − x sin t) dx = 1

0 ∞ −βx

e

4.8

1 (q cos λ + p sin λ) + q2

p2

0

sin ax dx = Re sin bx

BI (261)(7)





 1 β β a+b b−a −i −i ψ −ψ 2bi 2b 2b 2b 2b [Re β > 0,



∞ −2px

e

5.8 0

 6.

1 sin[(2n + 1)x] dx = + sin x 2p

e

0



k=1

p p2 + k 2

 sin 2nx 1 dx = 2p sin x p2 + (2k + 1)2

∞ −px

8

n 

∞

[Re p > 0]

BI (267)(15)

[Re p > 0]

GW (335)(15c)

n−1

 (−1)k (2k + 1) 2n + 1 n + (−1) 2 p2 + (2n + 1)2 p2 + (2k + 1)2 n−1

e−px cos[(2n + 1)x] tan x dx =

0

k=0

[p > 0] 

3.895

,ν +  2π Γ(ν + 1) P m ν (β) β + β 2 − 1 cos x einx dx = Γ(ν + m + 1) −π [Re β > 0]



1. 0

GW (335)(15)

k=0

7.

3.894

b = 0]

LI (267)(16)

π

∞

e−βx sin2m x dx =

ET I 157(15)

(2m)! β(β 2 + 22 ) (β 2 + 42 ) · · · [β 2 + (2m)2 ] [Re β > 0]

FI II 615, WA 620a

3.895

Trigonometric functions and exponentials

 2.

π

10

e−px sin2m x dx =

0

 3.

π/2

10

=

(2m)! 2 + 22 ) (p2 + 42 ) · · · [p2 + (2m)2 ] p (p  %  2   2   2  & 2 2 2 2 2 2 p p p p · · · p + 2 + 2 + (2m − 2) pπ p × 1 − e− 2 1 + + + ··· + 2! 4! (2m)! BI (270)(4)

∞

4.

e−βx sin2m+1 x dx =

0

 5.10 6.

(β 2

+

12 ) (β 2

(2m + 1)! + 32 ) · · · [β 2 + (2m + 1)2 ] [Re β > 0]

π

e−px sin2m+1 x dx =

0



GW (335)(4a)

e−px sin2m x dx

0



(2m)! (1 − e−pπ ) p (p2 + 22 ) (p2 + 42 ) · · · [p2 + (2m)2 ]

487

π/2

8

FI II 615, WA 620a

−pπ

(p2

+

) (2m + 1)! (1 + e 2 2 + 3 ) · · · [p + (2m + 1)2 ]

12 ) (p2

GW (335)(4b)

e−px sin2m+1 x dx

0

=



(2m + 1)! +%32 ) · · · [p2 + (2m + 1)2 ] +  2     & p + 12 p2 + 32 · · · p2 + (2m − 1)2 −pπ p2 + 12 2 × 1 − pe + ··· + 1+ 3! (2m + 1)! (p2

12 ) (p2

BI (270)(5) ∞

7.

e−px cos2m x dx =

0

(2m)! p (p2

 +

× 1+

22 ) · · · [p2 2

p p2 + 2!



+ (2m)2 ]      p2 p2 + 22 · · · p2 + (2m − 2)2 p2 + 22 + ··· + 4! (2m)! [p > 0]



π/2

8.10

e−px cos2m x dx

0

=

(2m)! p (p2

 +

22 ) · · · [p2

× −e−p 2

π

 9.7

BI (262)(3)

+ (2m)2 ]       p2 p2 + 22 p2 p2 + 22 · · · p2 + (2m − 2)2 p2 + + ··· + +1+ 2! 4! (2m)! BI (270)(6)

∞

e−px cos2m+1 x dx

0

=

(2m + 1)!p 2 + 12 ) (p2 + 32 ) · · · [p2 + (2m + 1)2 ] (p  2  2       p + 12 p2 + 32 p + 12 p2 + 32 · · · p2 + (2m − 1)2 p2 + 12 + + ··· + × 1+ 3! 5! (2m + 1)! [p > 0]

BI (262)(4)

488

Trigonometric Functions



π/2

10.11

3.896

e−px cos2m+1 x dx

0

=



∞

11.8

e−βx sinn ax

0



16. 3.896 1. 2. 3.

4.

BI (270)(7)



−n−2

dx =

1 2 e 4 (1∓1+2n)πi a(n + 1) ⎧   −1 ⎫ ⎨ b+na+iβ −1 ⎬ b+na−iβ 2a 2a × ± (−1)n ⎩ ⎭ n+1 n+1

b > 0,

a2 + 2m2 a (a2 + 4m2 )

  a a2 + m2 + n2 e cos mx cos nx dx = 2 (a + (m − n)2 ) (a2 + (m + n)2 ) 0    ∞ m a2 + m2 − n2 −ax e sin mx cos nx dx = 2 (a + (m − n)2 ) (a2 + (m + n)2 ) 0  ∞ 2m [a > 0] e−ax sin2 mx dx = a (a2 + 4m2 ) 0  ∞ 2amn e−ax sin mx sin nx dx = 2 2 ] [a2 + (m + n)2 ] [a + (m − n) 0 

15.

sin bx cos bx

e−ax cos2 mx dx =

0

14.

[a > 0, ∞

12. 13.

(2m + 1)! 2 + 12 ) (p2 + 32 ) · · · [p2 + (2m + 1)2 ] (p %  2    &  p + 1 p2 + 32 · · · p2 + (2m − 1)2 p2 + 12 −p π × e 2 +p 1+ + ··· + 3! (2m + 1)!



∞

∞

−ax

2

x2

DW61 (861.06)

DW61 (861.15)

DW61 (861.14) DW61 (861.10) DW61 (861.13)

√

p2 π − 4q e 2 sin pλ q −∞ √  ∞ p2 2 2 π − 4q e 2 cos pλ e−q x cos[p(x + λ)] dx = q −∞ 2



 ∞ 1 3 b2 b b −ax2 e sin bx dx = exp − ; ; 1F 1 2a 0 4a

2 2 4a 2 b 3 b = 1 F 1 1; ; − 2a 2 4a 2 k−1 ∞ 1 b  b = − [a > 0] 2a (2k − 1)!! 2a k=1 2

  ∞ b 1 π −βx2 exp − e cos bx dx = [Re β > 0] 2 β 4β 0

e−q

Re β > 0]

sin[p(x + λ)] dx =

BI (269)(2) BI (269)(3)

ET I 73(18) FI II 720

BI (263)(2)

3.911

3.897 1.8

2.

Trigonometric functions and exponentials



 γ − ib π (γ − ib)2 √ exp 1−Φ β 2 β 0

 4β (γ + ib)2 γ + ib √ − exp 1−Φ 4β 2 β [Re β > 0] 

   ∞ 2  2 (γ − ib) γ − ib 1 π √ exp e−βx −γx cos bx dx = 1−Φ 4 β 4β 2 β 0

  γ + ib (γ + ib)2 √ + exp 1−Φ 4β 2 β [Re β > 0] 

3.898  1.

∞

e−βx

2

∞

e

−βx2

0



∞

2.

−γx

sin bx dx = −

i 4



∞

3.8

e−βx cos ax cos bx dx = 2

e−px sin2 ax dx = 2

0

1.

7

2.



1 sin ax sin bx dx = 4

0

3.899

489

1 4





1 4

π β



π β

e

− (a−b) 4β

e−

2

−e

(a−b)2 4β

a2 π 1 − e− p p

− (a+b) 4β

+ e−

2

(a+b) 4β

ET I 74(27)

ET I 15(16)



[Re β > 0]  2

BI (263)(4)

[Re β > 0]

BI (263)(5)

[Re p > 0]

BI (263)(6)

& √ % n sin[(2n + 1)x] π 1  −( kp )2 dx = + e [p > 0] sin x p 2 0 k=1 & √ %  ∞ −p2 x2 2n 2 e cos[(4n + 1)x] π 1  k −( k ) dx = + (−1) e p cos x p 2 0 

∞ p2 x2

e

BI (267)(17)

k=0



∞

3. 0

<

[p > 0]





2 π 2 ∞ e−px dx p k 1  k a exp − 1 − 2a cos x + a2 = 1 − a2 2 + 4p k=1 <  π 2  ∞ p 1  −k k + = 2 a exp − a −1 2 4p

BI (267)(18)

 

a2 < 1,

p>0

a2 > 1,

p>0

 

EI (266)(1)

LI (266)(1)

k=1

3.911  1. 2. 3.11

∞

1 π sin ax dx = − aπ βx + 1 e 2a 2β sinh 0 β

 ∞ πa sin ax π 1 dx = coth − βx − 1 e 2β β 2a 0  ∞ sin ax x/2 1 e dx = π tanh(aπ) x 2 0 e −1

[a > 0,

Re β > 0]

BI (264)(1)

[a > 0,

Re β > 0]

BI (264)(2), WH

[a > 0]

ET I 73(13)

490

Trigonometric Functions



∞

4. 0



sin ax − eγx

eβx

[a > 0] 



 β + ia β − ia 1 dx = ψ −ψ 2i(β − γ) β−γ β−γ

0 ∞

0

i sin ax dx = [ψ(β + ia) − ψ(β − ia)] eβx (e−x − 1) 2

1.

a > 0]

ET I 15(10)



1−e

 −γx ν−1



eiβx cosν x β 2 eix + ν 2 e

 −ix μ

2.11

∞

 μ−ν−u a2 ; 1 + ; πb2ν 2 F 1 −ν, − u+μ+ν 2 2 b2

= u + μ +ν u + ν − μ ,1 + 2μ (μ + 1) B 1 − 2 2  u−μ−ν μ−ν+u b2 2ν πa 2 F 1 −ν, 2 ; 1 + 2 ; a2

= μ+ν−u u+μ−ν μ ,1 + 2 (μ + 1) B 1 + 2 2 [Re μ > −1] e−β

√

γ 2 +x2

cos bx dx = 

0

 2. 0

|ν| > |β|]

EH I 81(11)a

ν  e−iux cosμ x a2 eix + b2 e−ix dx

−π 2

3.914  1.

 2 π 2 F 1 −μ, β2 − ν2 − μ2 ; 1 + β2 + ν2 − μ2 ; βν 2

dx = ν μ β ν μ β 2ν (ν + 1) B 1 + + − , 1 − + + 2 2 2 2 2 2 [Re ν > −1,

π 2

ET 73(15)





 1 β − ia β + ia cos ax dx = B ν, + B ν, 2γ γ γ [Re β > 0, Re γ > 0, Re ν > 0,

−βx

−π 2



[Re β > −1]

ET I 73(17)

e

π 2

GW (335)(8)

a > 0]

0



Re γ > 0]





  ν−1 i β − ia β + ia e−βx 1 − e−γx sin ax dx = − B ν, − B ν, 2γ γ γ [Re β > 0, Re γ > 0, Re ν > 0,

∞

2.

3.913

BI (264)(8)

[Re β > 0, ∞

6.



n−1

∞

0

3.912  1.

 sin ax −nx a 1 π π − + − e dx = 1 − e−x 2 2a e2πa − 1 a2 + k 2 k=1

5. 

3.912

∞

γ 2 + x2 e−β

βγ β 2 + b2

√

γ 2 +x2



for a2 < b2

for b2 < a2





ET I 122(31)a

  K 1 γ β 2 + b2

2 2

cos bx dx =



β γ K 0 (γA) + A2



[Re β > 0, 2



Re γ > 0]

2β γ γ − K 1 (γA) A3 A , +  A = β 2 + b2

ET I 16(26)

3.915

Trigonometric functions and exponentials



491

√ 2 2  2  γ + x2 e−β γ +x cos bx dx 0



3βγ 2 4β 3 γ 2 6βγ 8β 3 γ β3γ3 = − 2 + + K 0 (γA) + − 3 + K 1 (γA) 5 A A4 A A3 ,A +  A = β 2 + b2  ∞ −β √γ 2 +x2   e  [Re β > 0, Re γ > 0, b > 0] cos bx dx = K 0 γ β 2 + b2 γ 2 + x2 0

3.

4.



∞

∞

5.

1 3/2

β (γ 2 + x2 )

0





+

1 2 γ + x2

∞

xe−β

√

γ 2 +x2

sin bx dx =

0

8.

9.

10. 3.915

e−β

√

γ 2 +x2

cos bx dx =

  1  2 β + b2 K 1 γ β 2 + b2 βγ (6.726(4))

6. 7.

ET I 16(27)





∞



π

  bβγ 2 K 2 γ β 2 + b2 2 2 β +b

ET I 175(35)

√ 2 2  x γ 2 + x2 e−β γ +x sin bx dx 0



bγ 2 4bβ 2 γ 2 2bγ 8bβ 2 γ bβ 2 γ 3 = − 2 + + K 0 (γA) + − 3 + K 1 (γA) 5 A A4 A A A3 , +  A = β 2 + b2

 ∞ √ 2 2  2  12bβγ 2 24bβ 3 γ 2 bβ 3 γ 4 γ + x2 e−β γ +x x sin bx dx = − + + K 0 (γA) 4 A6 A4 0 A

24bβγ 48bβ 3 γ 3bβγ 3 8bβ 3 γ 3 + − + − + K 1 (γA) 7 A5 A A3 , A5 +  A = β 2 + b2  ∞ −β √γ 2 +x2   xe γb  sin bx dx =  K 1 γ β 2 + b2 ET I 75(36) γ 2 + x2 β 2 + b2 0   ∞   √ 2 2 1 b 1 + 2 (6.726(3)) e−β γ +x x sin bx dx = K 0 γ β 2 + b2 2 3/2 γ +x β 0 β (γ 2 + x2 )

ea cos x sin x dx =

1. 0



2 sinh a a

GW (337)(15c)

π

eiβ cos x cos nx dx = in π J n (β)

2.

EH II 81(2)

0

ν

2 1 e cos Γ ν+ J ν (β) β 2 −π 2 ν

 π √ 2 1 e±β cos x sin2ν x dx = π Γ ν+ I ν (β) β 2 0 

3. 4.

3

π 2

iβ sin x

2ν

√ x dx = π

 

Re ν > − 12 Re ν > − 12

 

EH II 81(6)

GW (337)(15b)

492

Trigonometric Functions

 5.

π

e

iβ cos x

2ν

sin

√ x dx = π

0

3.916



π/2

1.

e−p

2

x√ tan x sin 2 cos x

sin 2x

0



π/2

2. 0

ν

2 1 Γ ν+ J ν (β) β 2

3.916



Re ν > − 12



WA 34(2), WA 60(6)

 2  2 1 1 dx = C (p) − + S (p) − 2 2

exp (−p tan x) dx 1 = − eap Ei(−ap) sin 2x + a cos 2x + a 2

[p > 0] ,

NT 33(18)a

(cf. 3552 4 and 6) BI (273)(11)



π/2

3. 0

1 exp (−p cot x) dx = − e−ap Ei(ap) sin 2x + a cos 2x − a 2

[p > 0] ,

(cf. 3.552 4 and 6) BI (273)(12)



π/2

4.

(1 −

0

 1 exp (−p tan x) sin 2x dx = − e−ap Ei(ap) + eap Ei(−ap) 2 2 2 − 2a cos 2x − (1 + a ) cos 2x 4

a2 )

[p > 0] 

π/2

5.

(1 −

0

BI (273)(13)

 1 exp (−p cot x) sin 2x dx = − e−ap Ei(ap) + eap Ei(−ap) 2 2 2 + 2a cos 2x − (1 + a ) cos 2x 4

a2 )

[p > 0] 3.917 1.

2.

3.918

BI (273)(14)





√ π 1 1 e−2β cot x cosν−1/2 x sin−(ν+1) x sin β − ν − Γ ν + x dx = J ν (β) 2 2(2β)ν 2 0   WA 186(7) Re ν > − 12 



√  π/2 π 1 1 e−2β cot x cosν−1/2 x sin−(ν+1)x cos β − ν − Γ ν+ x dx = Y ν (β) ν 2 2(2β) 2 0   Re ν > − 12 WA 186(8) 

π/2



π/2

1. 0



π/2

2. 0

 cosμ x i γ(β−μx)−2β cot x π iγ (ε) (2β)−μ Γ(μ + 1) H μ+ 1 (β) dx = e 2μ+2 2 2 2β sin x   ε = 1, 2, γ = (−1)ε+1 , Re β > 0, Re μ > −1 cosμ x sin(β − μx) −2β cot x 1 dx = e 2 sin2μ+2 x



π (2β)−μ Γ(μ + 1) J μ+ 12 (β) 2β [Re β > 0,

 3. 0

π/2

cosμ x cos(β − μx) −2β cot x 1 e dx = − 2μ+2 2 sin x

GW (337)(16)



Re μ > −1]

WH

π (2β)−μ Γ(μ + 1) Y μ+ 12 (β) 2β [Re β > 0,

Re μ > −1]

GW (337)(17b)

3.922

3.919

Trigonometric functions and exponentials



π/2

1. 0



π/2

2. 0

493

sin 2nx 2n − 1 dx = (−1)n−1 · 2n+2 4(2n + 1) sin x exp (2π cot x) − 1

BI (275)(6), LI (275)(6)

dx sin 2nx n n−1 2n+2 exp (π cot x) − 1 = (−1) 2n + 1 sin x

BI (275)(7), LI (275)(7)

3.92 Trigonometric functions of more complicated arguments combined with exponentials 3.9216





∞

1.

e

−γx

cos ax (cos γx − sin γx) dx = 2

0

[a > 0,   1 π exp − tan2n x = − 1 n 2 0 n=1 % & % ∞ &  π/2  π/2 ∞  1 1 π 2n 2n sin x = cos x = exp − exp − n n 4 0 0 n=1 n=1 

2.10

3.10 3.922 1.

2.

2

γ π exp − 8a 2a Reγ ≥ |Im γ|]

 . π β 2 + a2 − β e−βx e−βx sin ax2 dx = 8 β 2 + a2 0 −∞

√ 1 a π arctan =  sin 4 2 2 2 β 2 β +a [Re β > 0, a > 0] .    ∞  2 π β 2 + a2 + β 1 ∞ −βx2 e−βx cos ax2 dx = e cos ax2 dx = 2 −∞ 8 β 2 + a2 0

√ 1 a π arctan =  cos 4 2 2 2 β 2 β +a [Re β > 0, a > 0] 

ET I 26(28)

∞ π/4 -

∞

2

1 sin ax2 dx = 2



∞

2

FI II 750, BI (263)(8)

FI II 750, BI (263)(9)

[In formulas 3.922 3 and 4, a > 0, b > 0, Re β > 0, and     1  2 1  2 b2 , B= A= β + a2 + β , C= β + a2 − β . 2 2 4 (a + β ) 2 2 If a is complex, then Re β > |Im a|.]   ∞ 2 π 1 e−βx sin ax2 cos bx dx = − e−Aβ (B sin Aa − C cos Aa) 3. 2 + a2 2 β 0

 √ 1 a ab2 π βb2 arctan − exp − =  sin 4 (β 2 + a2 ) 2 β 4 (β 2 + a2 ) 2 4 β 2 + a2 LI (263)(10), GW (337)(5)

494

Trigonometric Functions



∞

4.

e

−βx2

0

3.923



π e−Aβ (B cos Aa + C sin Aa) β 2 + a2

 √ 1 a ab2 π βb2 arctan − exp − =  cos 4 (β 2 + a2 ) 2 β 4 (β 2 + a2 ) 2 4 β 2 + a2

1 cos ax cos bx dx = 2 2

LI (263)(11), GW (337)(5)

3.923 1.



     exp − ax2 + 2bx + c sin px2 + 2qx + r dx −∞     √ a b2 − ac − aq 2 − 2bpq + cp2 π  = 4 exp a2 + p2 a2  + p2  2    p p q − pr − b2 p − 2abq + a2 r 1 arctan − × sin 2 a a2 + p2

 2.

∞

[a > 0]

GW (337)(3), BI (296)(6)

∞

     exp − ax2 + 2bx + c cos px2 + 2qx + r dx −∞     √ a b2 − ac − aq 2 − 2bpq + cp2 π =  exp 4 2 a2 + p2 a2 + p  2    p p q − pr − b2 p − 2abq + a2 r 1 arctan − × cos 2 a a2 + p2 [a > 0]

3.924



∞

e

1.

−βx4

0



∞

2.

e−βx cos bx2 dx = 4

0

3.925



∞

2

e

1.

p −x 2

0

 2.

∞

e 0

p2 −x 2

.

2

2

b b b 1 exp − I4 2β 8β 8β

.

2

2

b b b exp − I − 14 2β 8β 8β

π sin bx dx = 4 2

π 4

1 sin 2a x dx = 2



∞

2 2

cos 2a2 x2 dx =

1 2

2

e

p −x 2

−∞



∞

e −∞

p2 −x 2

GW (337)(3), BI (269)(7)

[Re β > 0,

b > 0]

ET 73(22)

[Re β > 0,

b > 0]

ET I 15(12)

√ π −2ap e sin 2a x dx = (cos 2ap + sin 2ap) 4a 2 2

[a > 0, b > 0] √ π −2ap e cos 2a2 x2 dx = (cos 2ap − sin 2ap) 4a [a > 0,

b > 0]

BI (268)(12)

BI (268)(13)

3.932

Trigonometric and exponential functions

3.926

1.

Notation:

495

. a2 + β 2 + β a2 + β 2 − β , v= u= 2 2   ∞ √ γ 2 π 1 √ √ e−(βx + x2 ) sin ax2 dx = e−2u γ [v cos (2v γ) + u sin (2v γ)] 2 2 2 a +β 0 

∞

2.

e−(βx

2

.

+ xγ2 )

0

cos ax2 dx =

1 2

[Re β > 0,

 a2

∞

3.927

p

e− x sin2

0

3.928



∞

1. 0



BI (268)(14)

√ π √ √ e−2u γ [u cos (2v γ) − v sin (2v γ)] 2 +β

[Re β > 0, 

Re γ > 0]

2a p p2 a dx = a arctan + ln 2 x p 4 p + 4a2

[a > 0,

Re γ > 0]

BI (268)(15)

p > 0]

LI (268)(4)





√ π −2rs cos(A+B) q2 b2 e exp − p2 x2 + 2 sin {A + 2rs sin(A + B)} sin a2 x2 + 2 dx = x x 2r BI (268)(22)

∞

2. 0





√ π −2rs cos(A+B) q2 b2 2 2 2 2 e exp − p x + 2 cos {A + 2rs sin(A + B)} cos a x + 2 dx = x x 2r

BI (268)(23)



∞

3.929

+ ,  √  2 e−x cos p x + pe−x sin px dx = 1

0

Notation: For the formulas in 3.928: a2 + p2 > 0, r = B=

LI (268)(3)

  a2 1 4 a4 + p4 , s = 4 b4 + q 4 , A = arctan 2 , and 2 p

1 b2 arctan 2 . 2 q

3.93 Trigonometric and exponential functions of trigonometric functions 3.931



π/2

1. 0



π

2.

e−p cos x sin (p sin x) dx = Ei(−p) − ci(p)

 e−p cos x sin (p sin x) dx = −

−π

0



π/2

3. π/2

4. 0

3.932



e−p cos x cos (p sin x) dx =

1 2



π

ep cos x sin (p sin x) sin mx dx =

1. 0

e−p cos x sin (p sin x) dx = −2 shi(p)

GW (337)(11b)

e−p cos x cos (p sin x) dx = − si(p)

0



0

NT 13(27)

2π

NT 13(26)

e−p cos x cos (p sin x) dx = π

GW (337)(11a)

0

1 2



2π

ep cos x sin (p sin x) sin mx dx = 0

π pm · 2 m! BI (277)(7), GW (337)(13a)

496

Trigonometric Functions



π

ep cos x cos (p sin x) cos mx dx =

2. 0

1 2



3.933

2π

ep cos x cos (p sin x) cos mx dx = 0

π pm · 2 m!

BI (277)(8), GW (337)(13b)



π

ep cos x sin (p sin x) cosec x dx = π sinh p

3.933

BI (278)(1)

0

3.934



π

1.

ep cos x sin (p sin x) tan

x dx = π (1 − ep ) 2

BI (271)(8)

ep cos x sin (p sin x) cot

x dx = π (ep − 1) 2

BI (272)(5)

0



π

2. 0



ep cos x cos (p sin x) 0

3.936



 p2k+1 sin 2nx dx = π sin x (2k + 1)! n−1

π

3.935



2π

π

ep cos x cos (p sin x − mx) dx = 2

1.

[p > 0]

LI (278)(3)

k=0

0

ep cos x cos (p sin x − mx) dx = 0

2πpm m! BI (277)(9), GW (337)(14a)



2π

2.

ep sin x sin (p cos x + mx) dx =

mπ 2πpm sin m! 2

[p > 0]

GW (337)(14b)

ep sin x cos (p cos x + mx) dx =

mπ 2πpm cos m! 2

[p > 0]

GW (337)(14b)

0



2π

3. 0



2π

ecos x sin (mx − sin x) dx = 0

4.

WH

0

 5.

π

eβ cos x cos (ax + β sin x) dx = β −a sin(aπ) γ(a, β)

EH II 137(2)

0

3.937 Notation: In formulas 3.937 1 and 2, (b−p)2 +(a+q)2 > 0, m = 0, 1, 2, . . . , A = p2 −q 2 +a2 −b2 , B = 2(pq + ab), C = p2 + q 2 − a2 − b2 , and D = 2(ap + bq).  1.11 0

2π

exp (p cos x + q sin x) sin (a cos x + b sin x − mx) dx  √ √  − m  = iπ (b − p)2 + (a + q)2 2 (A + iB)m/2 I m C − iD − (A − iB)m/2 I m C + iD GW (337)(9b)



2π

exp (p cos x + q sin x) cos (a cos x + b sin x − mx) dx √ √   − m  m m = π (b − p)2 + (a + q)2 2 (A + iB) 2 I m C − iD + (A − iB) 2 I m C + iD

2. 0

 3. 0

2π



GW (337)(9a)

m q 2π  2 p + q2 sin m arctan exp (p cos x + q sin x) sin (q cos x − p sin x + mx) dx = m! 2 p



GW (337)(12)

3.944

Trigonometric and exponentials functions and powers



2π

exp (p cos x + q sin x) cos (q cos x − p sin x + mx) dx =

4. 0

497



m q 2π  2 p + q2 cos m arctan m! 2 p GW (337)(12)

3.938



π

er(cos px+cos qx) sin (r sin px) sin (r sin qx) dx =

1. 0



∞ 1 π r(p+q)k 2 Γ(pk + 1) Γ(qk + 1) k=1

π

er(cos px+cos qx) cos (r sin px) cos (r sin qx) dx =

2. 0

π 2

 2+

∞  k=1

r(p+q)k Γ(pk + 1) Γ(qk + 1)



BI (277)(14)

BI (277)(15)

3.939



π

eq cos x

1. 0



∞ sin rx π  (pq)kr sin (q sin x) dx = 1 − 2pr cos rx + p2r 2pr Γ(kr + 1) k=1

%

π

eq cos x

2.3 0

1 − pr cos rx π 2+ cos (q sin x) dx = 1 − 2pr cos rx + p2r 2

[r > 0, ∞  k=1

0 < p < 1] & (pq)kr Γ(kr + 1)

[r > 0, 0 < p < 1] 



q−1 cos (p sin 2x) dx π exp p = 2q q+1 cos2 x + q 2 sin2 x

π/2 p cos 2x

e

3. 0

BI (278)(15)

BI (278)(16)

BI (273)(8)

3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers 3.941  1.

∞

e−px sin qx

q dx = arctan x p

e−px cos qx

dx =∞ x

e−px cos px

 √ x dx π = exp −bp 2 b4 + x4 4b2

0



∞

2. 0

3.942  1.

∞

0



∞

2.

e−px cos px

0



3.943 0

∞

x dx π = 2 e−bp sin bp 4 −x 4b

b4

e−βx (1 − cos ax)

1 a2 + β 2 dx = ln x 2 β2

[p > 0]

BI (365)(1) BI (365)(2)

[p > 0,

b > 0]

BI (386)(6)a

[p > 0,

b > 0]

BI (386)(7)a

[Re β > 0]

BI (367)(6)

3.944  u i i −μ −μ xμ−1 e−βx sin δx dx = (β + iδ) γ [μ, (β + iδ) u] − (β − iδ) γ [μ, (β − iδ) u] 1. 2 2 0 [Re μ > −1]

ET I 318(8)

498

Trigonometric Functions



∞

2.

xμ−1 e−βx sin δx dx =

u



xμ−1 e−βx cos δx dx =

0

xμ−1 e−βx cos δx dx =

u

∞

5.11

xμ−1 e−βx sin δx dx =

0



∞

6. 0

 0

xμ−1 e−βx cos δx dx =

0

1 1 −μ −μ (β + iδ) Γ [μ, (β + iδ) u] + (β − iδ) Γ [μ, (β − iδ) u] 2 2 Γ(μ) (β 2

Γ(μ) (δ 2 + β 2 ) μ2

0

11.

FI II 812, BI (361)(9)



δ cos μ arctan β

Re β > |Im δ|]

[Re μ > 0,

xμ−1 exp (−ax cos t) sin (ax sin t) dx = Γ(μ)a−μ sin(μt) + Re μ > −1,

xμ−1 exp (−ax cos t) cos (ax sin t) dx = Γ(μ)a−μ cos(μt) + Re μ > −1,

xp−1 e−qx sin (qx tan t) dx =

1 Γ(p) cosp t sin pt qp

xp−1 e−qx cos (qx tan t) dx =

1 Γ(p) cosp (t) cos pt qp

0



μ/2

+ δ2)

[Re β > |Im δ|] ET I 320(29)

δ sin μ arctan β [Re μ > −1, Re β > |Im δ|]

a > 0,

|t| <

π, 2

a > 0,

|t| <

π, 2 EH I 13(35)

∞

9. 10.

ET I 320(28)

EH I 13(36) ∞

8.



1 1 −μ −μ (β + iδ) γ [μ, (β + iδ) u] + (β − iδ) γ [μ, (β − iδ) u] 2 2

FI II 812, BI (361)(10) ∞

7.



ET I 318(9)

[Re μ > 0] ∞

4. 

i i −μ −μ (β + iδ) Γ [μ, (β + iδ) u] − (β − iδ) Γ [μ, (β − iδ) u] 2 2 [Re β > |Im δ|]

u

3. 

3.944

∞

+ π |t| < , 2

, q>0

LO V 288(16)

+ , π |t| < , q > 0 LO V 288(15) 2

n+1 

2k+1  ∞ β n+1 b n −βx k x e sin bx dx = n! (−1) 2 + b2 2k + 1 β β 0 0≤2k≤n

n b n ∂ = (−1) ∂β n b2 + β 2 [Re β > 0, b > 0] GW (336)(3), ET I 72(3)

3.947

Trigonometric and exponentials functions and powers



∞

12.

∞

13.



n+1 

2k β b k n+1 (−1) 2k β 2 + b2 β 0≤2k≤n+1

n β ∂ n = (−1) ∂β n b2 + β 2 [Re β > 0, b > 0] GW (336)(4), ET I 14(5)

xn e−βx cos bx dx = n!

0



 xn−1/2 e−βx sin bx dx = (−1)n

0

14.

499

n

⎛ <

⎞ β 2 + b2 − β ⎠  β 2 + b2

π d ⎝ 2 dβ n

[Re β > 0, ⎛ < ⎞   ∞ 2 + b2 + β n β π d ⎝ ⎠  xn−1/2 e−βx cos bx dx = (−1)n 2 dβ n β 2 + b2 0

b > 0]

ET I 72(6)

[Re β > 0,

b > 0]

ET I 15(6)

3.945  1.

∞



∞



∞

 −βx  dx e sin ax − e−γx sin bx r x  0 

   2   r−1  b a 2 r−1 sin (r − 1) arctan sin (r − 1) arctan = Γ(1 − r) b + γ − a2 + β 2 2 γ 2 β [Re β > 0, Re γ > 0, r < 2, r = 1] BI(371)(6)

2.

 −βx  dx e cos ax − e−γx cos bx r x 0 

   2  r−1  a b 2 r−1 cos (r − 1) arctan cos (r − 1) arctan = Γ(1 − r) a + β − b2 + γ 2 2 β 2 γ [Re β > 0, Re γ > 0, r < 2, r = 1] BI (371)(7)

3. 0

   −βx  dx 1 a2 + γ 2 γ β β γ ln 2 arccot − arccot ae sin bx − be−γx sin ax 2 = ab + x 2 b + β2 a a b b [Re β > 0,

3.946



∞

1.

e−px sin2m+1 ax

0

Re γ > 0]



m 2m + 1 (−1)m  dx (2m − 2k + 1)a = 2m (−1)k arctan k x 2 p k=0

[m = 0, 1, . . . , 

∞

2. 0

e−px sin2m ax

m+1 m−1 

(−1) dx = x 22m

(−1)k

k=0



2m k





1. 0

∞

e−βx sin γx sin ax

1 β 2 + (a + γ)2 dx = ln 2 x 4 β + (a − γ)2

p > 0]

  1 ln p2 + (2m − 2k)2 a2 − 2m 2 [m = 1, 2, . . . ,

3.947

BI (368)(22)

[Re β > |Im γ|,

p > 0]

a > 0]

GW (336)(9a)



2m m

ln p

GW (336)(9b)

BI (365)(5)

500

Trigonometric Functions

 2.

∞

11

e 0

 3.

∞

11 0

3.948



−px

3.948





|a + b| |a − b| dx |a + b| |a − b| arctan arctan sin ax sin bx 2 = − x 2 p

2 p 2 2 p + (a − b) p + ln 4 p2 + (a + b)2 [p > 0, for p = 0 see 3.741 3] BI (368)(1), FI II 744

e−px sin ax cos bx

a+b a−b dx = arctan + arctan x p p [a ≥ 0,

∞

1.11

e−βx (sin ax − sin bx)

0

p > 0]

a br dx = arctan − arctan x β β [Re β > 0] ,



∞

1 b2 + β 2 dx = ln 2 x 2 a + β2

e−βx (cos ax − cos bx)

0

e−βx (cos ax − cos bx)

0

0

2

2

a dx β a +β b = ln 2 + b arctan − a arctan 2 2 x 2 b +β β β

∞

0

  dx 2b p p + 4a 2a − b arctan − ln 2 e−βx sin2 ax − sin2 bx 2 = a arctan x p p 4 p + 4b2

1.

2.8

2

∞

BI (368)(25)

  dx 2b p p2 + 4a2 2a + b arctan + ln 2 e−βx cos2 ax − cos2 bx 2 = −a arctan x p p 4 p + 4b2 [p > 0]



BI (368)(20) 2

[p > 0]

5.

3.949

(cf. 3.951 3)

[Re p > 0] ∞

4. 

[Re β > 0] ,

BI (367)(8), FI II 748a ∞

3. 

(cf. 3.951 2) BI (367)(7)

2. 

GW (336)(10b)

BI (368)(26)

1 a+b+c 1 a+b−c 1 a−b+c dx = − arctan + arctan + arctan x 4 p 4 p 4 p 0 −a + b + c 1 + arctan 4 p [p > 0] BI (365)(11)    ∞ 1 b 1 1 2pb dx π = arctan − arctan 2 e−px sin2 ax sin bx +s 2 2 x 2 p 2 2 p %+ 4a− b 2 0 & 1 for p2 + 4a2 − b2 < 0 s= 0 for p2 + 4a2 − b2 ≥ 0

 3. 0

∞

e−px sin ax sin bx sin cx

e−px sin2 ax cos bx

1 dx = ln x 8

 2   p + (2a + b)2 p2 + (2a − b)2

BI (365)(8)

2

(p2 + b2 )

[p > 0]

BI (365)(9)

3.951

Trigonometric and exponentials functions and powers

 4.

∞

8

e

−px

0



  1 a 1 1 2pa dx π = arctan + arctan 2 sin ax cos bx +s 2 − a2 x 2 p 2 2 p %+ 4b 2 & 2 1 for p + 4b2 − a2 < 0 s= 0 for p2 + 4b2 − a2 ≥ 0 2

BI (365)(10) ∞

5.

e−px sin2 ax sin bx sin cx

0

3.951



∞

1. 0



e

0 ∞ −γx

e

3. 0

 4.11

∞ −γx

e

0

 5. 6.

[Re β > 0,

Re γ ≥ 0]

BI (367)(3)

1 b2 + β 2 − e−βx cos bx dx = ln 2 x 2 b + γ2

[Re β > 0,

Re γ ≥ 0]

BI (367)(4)

Re γ > 0]

BI (368)(21)a

b − e−βx b b2 + β 2 b sin bx dx = ln 2 + β arctan − γ arctan 2 2 x 2 b +γ β γ [Re β > 0, 2

1 x bπ π cos bx dx = 2 − 2 cosech2 [Re β > 0] −1 2b 2β β 0

 ∞ 1 1 1 − cos bx dx = ln b − [ψ(ib) + ψ(−ib)] x−1 e x 2 0

7. 0



9.

(β − γ)b − e−βx sin bx dx = arctan 2 x b + βγ

∞

∞

ET I 15(18)

1 − cos ax dx a 1 1−e · = + ln 2πx e −1 x 4 2 a

[b > 0]

ET I 15(9)

[a > 0]

BI (387)(10)

−a

∞

 −βx  dx 1 a2 + γ 2 = ln e − e−γx cos ax [Re β > 0, x 2 β2 0



 ∞ 1 √ π cos px − e−px dx 1 √ = 4 exp − bp 2 sin bp 2 b4 + x4 x 2b 2 2 0 

∞

10. 0

 11.

0

BI (365)(15)

FI II 745

eβx



8.

dx 1 p2 + (b + c)2 = ln 2 x 8 p + (b − c)2 ,  2 + 2 2 2 p p + (2a − b + c) + (2a + b − c) 1 + ln 2 16 [p + (2a + b + c)2 ] [p2 + (2a − b − c)2 ] [p > 0]

√   dx = ln 2 1 − e−x cos x x

∞ −γx

2. 

501

∞



1 cos x − ex − 1 x

Re γ > 0]

[p > 0]



BI (367)(10)

BI (390)(6)

dx = C

NT 65(8)

dx a a2 + q 2 e−qx a sin ax = ln + q arctan − a ae−px − x x 2 p2 q [p > 0,

q > 0]

BI (368)(24)

502

Trigonometric Functions

  2m π x2m sin bx 1 m ∂ dx = (−1) coth bπ − [b > 0] ex − 1 ∂b2m 2 2b 0    ∞ 2m+1 2m+1 π 1 x cos bx m ∂ dx = (−1) coth bπ − ex − 1 ∂b2m+1 2 2b 0 

12. 13.

3.952



∞

∞

14. 0



∞

15. 0



∞

16. 0



∞

17. 0

2m x2m sin bx dx m ∂ = (−1) ∂b2m e(2n+1)cx − e(2n−1)cx

%

[b > 0] bπ  b π tanh − 2 4c 2c b + (2k − 1)2 c2

2m x2m sin bx dx m ∂ = (−1) ∂b2m e(2n−2)cx

n

%

[b > 0] b bπ  π tanh − 4c 2c b2 + (2k − 1)2 c2 n

∞

18.

[b > 0]  b bπ 1 π coth − − 4c 2c 2b b2 + (2k)2 c2 n−1

∞

19. 0

x2m+1 cos bx dx ∂ 2m+1 = (−1)m 2m+1 2ncx (2n−2)cx ∂b e −e

%

[b > 0,

c > 0]

 bπ 1 b π coth − − 2 4c 2c 2b b + (2k)2 c2 n−1

0 ∞

21. 0

3.952



∞

1.

bπ 2p



0

GW (336)(14c)

c > 0]

GW (336)(14d)

m 

cosh cos ax − cos bx dx 1 1 b2 + (2k − 1)2 p2 = ln − ln 2 aπ (2m−1)px x 2 cosh 2 a + (2k − 1)2 p2 −e k=1 2p GW (336)(16a)

bπ m−1 cos ax − cos bx dx 1 a sinh 2p 1  b2 + 4k 2 p2 = ln − ln 2 2 b sinh aπ 2 a + 4k 2 p2 e2mpx − e(2m−2)px x 2p k=1

1 (b + 1) sinh[(b − 1)π] sin x sin bx dx = ln · x 1−e x 4 (b − 1) sinh[(b + 1)π]

∞



b2 = 1

1 2aπ sin2 ax dx = ln · 1 − ex x 4 sinh 2aπ

xe−p

x2

xe−p

x2

2

sin ax dx =

0

2.

&

k=1

[p > 0] ∞

20. 

GW (336)(14b)

&

k=1

[p > 0]



GW (336)(14a)

e(2m+1)px

0



&

k=1

[b > 0, 

GW (336)(15b)

&

k=1

x2m+1 cos bx dx ∂ 2m+1 = (−1)m 2m+1 (2n+1)cx (2n−1)cx ∂b e −e %

GW (336)(15a)

2

cos ax dx =

GW (336)(16b)



LO V 305

LO V 306, BI (387)(5)



√ a π a2 exp − 4p3 4p2

BI (362)(1)

2k+1 ∞ 1 a  (−1)k k! a − 2p2 4p3 (2k + 1)! p k=0

[a > 0]

BI (362)(2)

3.953

Trigonometric and exponentials functions and powers



∞

3.

x2 e−p

2

x2

sin ax dx =

0

4. 5. 6.3

2k+1 ∞ a 2p2 − a2  (−1)k k! a + 4p4 8p5 (2k + 1)! p k=0

[a > 0]

2 √ 2p − a 2 2 a x2 e−p x cos ax dx = π exp − 2 5 8p 4p 0

 ∞ 2 3 √ 6ap − a a2 3 −p2 x2 x e sin ax dx = π exp − 2 16p7 4p 0



2k √   ∞ ∞ a a π a (−1)k dx π −p2 x2 = e sin ax = Φ x 2p k!(2k + 1) 2p 2 2p 0 

503

∞

2

BI (362)(4)

2

BI (362)(5) BI (362)(6)

BI (365)(21)

k=0



∞

7.

x

e

sin γx dx =

0



∞

8.10

xμ−1 e−βx cos ax dx = 2

0

10.

3.953 1.

2.

γe− 4β 2β

μ+1 2

Γ

1+μ 2





μ 3 γ2 ; ; 1 − F 1 1 2 2 4β

1 −μ/2  μ −a2 /4β β e Γ 2 2

ET I 318(10) [Re β > 0, Re μ > −1]

μ 1 1 a2 + ; ; 1F 1 − 2 2 2 4β [Re β > 0, Re μ > 0, a > 0]





√ a π a2 2n −β 2 x2 n √ x e cos ax dx = (−1) n+1 2n+1 exp − 2 D 2n 2 √β 8β

β 2 0 a π a2 exp − 2 H 2n = (−1)n 2n+1 (2β) 4β 2β + , π |arg β| < , a > 0 4



√  ∞ 2 2 2 a π a 2n+1 −β x n √ x e sin ax dx = (−1) n+ 3 exp − 2 D 2n+1 8β β 2 2 √2 β 2n+2 0

a π a2 exp − 2 H 2n+1 = (−1)n 2n+2 (2β) 4β + 2β , π |arg β| < , a > 0 4 

9.



γ2

μ−1 −βx2

ET I 320(30)

∞



∞



∞

WH, ET I 15(13)

WH, ET I 74(23)

xμ−1 e−γx−βx sin ax dx 0 2









γ − a2 iaγ γ − ia γ + ia i iaγ =− − exp Γ(μ) exp − D −μ √ D −μ √ μ exp 8β 4β 4β 2β 2β 2(2β) 2 [Re μ > −1, Re β > 0, a > 0] ET I 318(11) 2

xμ−1 e−γx−βx cos ax dx 0

2







iaγ γ − a2 γ − ia γ + ia 1 iaγ = + exp Γ(μ) exp − D −μ √ D −μ √ μ exp 8β 4β 4β 2β 2β 2(2β) 2 [Re μ > 0, Re β > 0, a > 0] ET I 16(18) 2

504

Trigonometric Functions

   √ γ − ia i π (γ − ia)2 √ xe sin ax dx =  (γ − ia) exp − 1−Φ 4β 2 β 8 β3 0 

   (γ + ia)2 γ + ia √ − (γ + ia) exp − 1−Φ 4β 2 β [Re β > 0, a > 0]

  √  ∞ 2 π γ − ia (γ − ia)2 √ xe−γx−βx cos ax dx = −  (γ − ia) exp 1−Φ 4β 2 β 8 β3 0

  (γ + ia)2 γ + ia 1 √ + (γ + ia) exp + 1−Φ 4β 2β 2 β [Re β > 0, a > 0] 

3.

4.

3.954

∞

3.954  1.11

∞

−γx−βx2

e−βx sin ax 2

0

ET I 74(28)

ET I 16(17)



    x dx π βγ 2 a a −γa γa √ √ e = − Φ γ β − Φ γ β + − e 2 sinh aγ + e γ 2 + x2 4 2 β 2 β [Re β > 0, Re γ > 0, a > 0] ET I 74(26)a



∞

2.11 0



    dx π βγ 2 a a −βx2 −γa γa e e cos ax 2 = Φ γ β− √ −e Φ γ β+ √ 2 cosh aγ − e γ + x2 4γ 2 β 2 β [Re β > 0, Re γ > 0, a > 0] ET I 15(15)





 π − β2 π e 4 D ν (β) dx = cos βx − ν [Re ν > −1] EH II 120(4) 2 2 0  ∞ 2 dx √ e − 1 3.956 e−x (2x cos x − sin x) sin x 2 = π BI (369)(19) x 2e 0 3.957 

∞ −β 2 1. xμ−1 exp sin ax dx 4x 0



   πi √  √ μ i i i μπ K μ βe−πi/4 a = μ β μ a− 2 exp − μπ K μ βe 4 a − exp 2 4 4 [Re β > 0, Re μ < 1, a > 0] ET I 318(12) ∞

3.955



∞

2.

xν e−

xμ−1 exp

0

3.958



x2 2

∞

−∞



cos ax dx



    √ √ μ i 1 i μπ K μ βe−πi/4 a = μ β μ a− 2 exp − μπ K μ βeπi/4 a + exp 2 4 4 [Re β > 0, Re μ < 1, a > 0] ET I 320(32)a

n −(ax2 +bx+c)

x e

1.

−β 2 4x

2

n/2

n   b − p2 −1 π n! exp −c ak sin(px + q) dx = − 2a a 4a (n − 2k)!k! k=0

n−2k  n − 2k

pb π −q+ j × bn−2k−j pj sin j 2a 2 j=0

[a > 0]

GW (37)(1b)

3.963

Trigonometric and exponentials functions and powers



∞

n −(ax2 +bx+c)

2.

x e −∞

n 

2

n/2

 b − p2 π n! exp −c ak a 4a (n − 2k)!k! k=0

n−2k  n − 2k

pb π −q+ j × pj cos j 2a 2 j=0

cos(px + q) dx =

505

−1 2a

[a > 0] 

∞

3.959

xe−p

2

x2

tan ax dx =

0

GW (337)(1a)

2 2

√ ∞ a π a k k (−1) k exp − 2 p3 p k=1

[p > 0] 3.961



∞

1. 0



BI (362)(15)

    x dx aγ exp −β γ 2 + x2 sin ax  = K 1 γ a2 + β 2 γ 2 + x2 a2 + β 2 [Re β > 0, Re γ > 0,

a > 0] ET I 75(36)

∞

2.

+

exp −β



0

,

dx γ 2 + x2 cos ax  γ 2 + x2

  = K 0 γ a2 + β 2 [Re β > 0,

Re γ > 0,

a > 0] ET I 17(27)

3.962 

∞

<

1. 0

     2 + β2 a exp −γ γ 2 + x2 − γ exp −β γ 2 + x2 a π  < sin ax dx =  2 2 2 2 γ +x β + a2 β + a2 + β 2 [Re β > 0,

Re γ > 0,

a > 0] ET I 75(38)

<      ∞ x exp −β γ 2 + x2 2 2 , +  π β+ a +β 2 + β2 <  exp −γ a cos ax dx =   2 a2 + β 2 0 γ 2 + x2 γ 2 + x2 − γ [Re β > 0, Re γ > 0, a > 0]

2.

ET I 17(29)

3.963



1.

∞

e− tan

2

√ sin x dx π = cos2 x x 2

x

0



π/2

2.

e−p tan x

x dx 1 = [ci(p) sin p − cos p si(p)] 2 cos x p

xe− tan

x

sin 4x

xe− tan

x

sin3 2x

0



π/2

3.8

2

0

 4.

8 0

π/2

2

BI (391)(1)

[p > 0]

(cf. 3.339)

BI (396)(3)

3√ dx =− π 8 cos x 2

BI (396)(5)

√ dx =2 π 8 cos x

BI (396)(6)

506

3.964

Trigonometric Functions



π/2

1.

xe−p tan x

0



π/2

2. 

π/2

xe−p tan

xp

2

0

3.965 1. 2. 3.966



∞

1 p − cos x dx = 4 cos x cot x 4 2

x

0

3.8

p sin x − cos x dx = − sin p si(p) − ci(p) cos p cos3 x

xe−p tan

2



π p

1 + 2p − 2 cos2 x dx = 6 cos x cot x 8p



π p





∞

xe−βx sin ax2 sin βx dx =

β 4

[p > 0]

LI (396)(4)

[p > 0]

BI (396)(7)

[p > 0]

BI (396)(8)

+ π |arg β| < , 4

π − β2 e 2a 2a3 0   ∞ π − β2 β −βx 2 xe cos ax cos βx dx = e 2a 4 2a3 0

1.

3.964

[a > 0,

  xe−px cos 2x2 + px dx = 0

, a>0

Re β > |Im β|]

ET I 84(17) ET 26(27)

[p > 0]

BI (361)(16)

[p > 0]

BI (361)(17)

[p > 0]

BI (361)(18)

0



√   1 p π exp − p2 xe−px cos 2x2 − px dx = 8 4 0  ∞      x2 e−px sin 2x2 + px + cos 2x2 + px dx = 0 

2. 3.

∞

0



 1 π 2 − p2 exp − p2 16 4 0

 ∞ 1   1 e 4a Γ(μ) μπ D −μ √ xμ−1 e−x cos x + ax2 dx = cos μ 4 a (2a) 2 0

 4. 5.3



∞

∞

6.6

     x2 e−px sin 2x2 − px − cos 2x2 − px dx =

  e xμ−1 e−x sin x + ax2 dx =

0

1 4a

√

Γ(μ) μπ D −μ sin μ 4 (2a) 2



[Re μ > 0,

1 √ a [Re μ > −1,

3.967



∞

1.

β2

e− x2 sin a2 x2

0

 2.

∞

e 0

2 −β x2

√ √ π −√2aβ dx e = sin 2aβ x2 2β √

cos a2 x2

dx π − e = x2 2β

[Re β > 0,

a > 0]

a > 0]

BI (361)(19)

ET I 321(37)

ET I 319(18)

a > 0] ET I 75(30)a, BI(369)(3)a

√

√ 2aβ cos 2aβ

[Re β > 0,

a > 0] BI (369)(4), ET I 16(20)

3.972

Trigonometric and exponentials functions and powers



∞

3.

2 −βx2

x e 0

√

π

cos ax dx = < cos 3 4 4 (a2 + β 2 ) 2



3 a arctan 2 β

507



[Re β > 0] 3.968  1.

∞

e

−βx2

0



∞

2.

π sin ax dx = − 8

e−βx cos ax4 dx = 2

0



4

π 8



ET I 14(3)a

2

 2

2

2

 β β β β β π π J 14 cos + + Y 14 sin + a 8a 8a 8 8a 8a 8

ET I 75(34) [Re β > 0, a > 0] 2

 2

2

2

 β β β β π β π + J 14 sin − Y 14 cos + a 8a 8a 8 8a 8a 8

[Re β > 0,

a > 0]

ET I 16(24)

3.969  √ ∞      π −p2 x4 +q 2 x2 3 3 2px cos 2pqx + q sin 2pqx dx = e 1. 2 0  ∞     2 4 2 2  2. e−p x +q x 2px sin 2pqx3 − q cos 2pqx3 dx = 0

BI (363)(7) BI (363)(8)

0

  3.971 Notation: In formulas 3.971 1 and 2, p ≥ 0, q ≥ 0, r = 4 a2 + p2 , s = 4 b2 + q 2 , A = arctan ap , and B = arctan qb .



 ∞    dx 1 ∞ dx q b q b 2 2 2 2 exp −px − 2 sin ax + 2 = exp −px − 2 sin ax + 2 1. 2 x x x 2√ −∞ x x x2 0 π exp [−2rs cos(A + B)] sin [A + 2rs sin(A + B)] = 2s 

BI (369)(16, 17)

∞

2. 0





  dx 1 ∞ dx q b q b 2 2 cos ax exp −px2 − 2 cos ax2 + 2 = exp −px − + 2 2 2 x x x 2√ −∞ x x x2 π exp [−2rs cos(A + B)] cos [A + 2rs sin(A + B)] = 2s 

BI (369)(15, 18)

3.972  1.

2.

, +  dx exp −β γ 4 + x4 sin ax2  4 + x4 γ 0   2   2  γ γ aπ I 1/4 = β 2 + a2 − β K 1/4 8 2 + 4 π Re β > 0, |arg γ| < , 4  ∞ +  , dx exp −β γ 4 + x4 cos ax2  4 4 0 γ + x  2   2  γ γ aπ I −1/4 = β 2 + a2 − β K 1/4 8 2 + 4 π Re β > 0, |arg γ| < , 4 ∞

 β 2 + a2 + β , a>0 ET I 75(37)



 β 2 + a2 + β , a>0 ET I 17(28)



508

Trigonometric Functions

3.973  1.

∞

exp (p cos ax) sin (p sin ax)

0



π dx = (ep − 1) x 2

∞

2.

exp (p cos ax) sin (p sin ax + bx) 0



exp (p cos ax) cos (p sin ax + bx) 0

a > 0]

  x dx π = exp −cb + pe−ac 2 +x 2 [a > 0, b > 0,

WH, FI II 725

c2

c > 0,

exp (p cos x) sin (p sin x + nx) 0 ∞

5.

exp (p cos x) sin (p sin x) cos nx 0

  dx π exp −cb + pe−ac = c2 + x2 2c [a > 0, b > 0,

c > 0,

π dx = ep x 2

[p > 0]

∞ pn π π  pk dx = · + x n! 4 2 k! k=n+1

∞

6.

exp (p cos x) cos (p sin x) sin nx 0

LI (366)(3)

n−1 π  pk pn π dx = + x 2 k! n! 4 k=0

[p > 0] 3.974



p > 0]

BI (366)(2)

[p > 0] 

p > 0]

BI (372)(4) ∞

4. 

[p > 0,

BI (372)(3) ∞

3.



3.973

∞

1. 0

LI (366)(4)

   π ep − exp pe−ab dx exp (p cos ax) sin (p sin ax) cosec ax 2 = b + x2 2b sinh ab [a > 0, b > 0, p > 0]    π ep − exp pe−ab x dx [1 − exp (p cos ax) cos (p sin ax)] cosec ax 2 = b + x2 2 sinh ab

BI (391)(4)

∞

[a > 0, b > 0, p > 0]    π ep − exp pe−ab − ab dx exp (p cos ax) sin (p sin ax + ax) cosec ax 2 = b + x2 2b sinh ab

BI (391)(5)

∞

[a > 0, b > 0, p > 0]    ∞ π ep − exp pe−ab − ab x dx exp (p cos ax) cos (p sin ax + ax) cosec ax 2 = b + x2 2 sinh ab 0

BI (391)(6)

 2. 0

 3. 0

 4.



[a > 0, ∞

5.

exp (p cos ax) sin (p sin ax) 0

 6.

exp (p cos ax) cos (p sin ax) 0

p > 0]

BI (391)(7)

x dx π = [1 − exp (p cos ab) cos (p sin ab)] 2 −x 2

b2

[p > 0, ∞

b > 0,

a > 0]

BI (378)(1)

dx π exp (p cos ab) sin (p sin ab) = b2 − x2 2b [a > 0,

b > 0,

p > 0]

BI (378)(2)

3.981

Trigonometric and hyperbolic functions



∞

7.

exp (p cos ax) sin (p sin ax) tan ax 0

    dx π · tanh ab exp pe−ab − ep = b2 + x2 2b [a > 0,



∞

8.

exp (p cos ax) sin (p sin ax) cot ax 0

b2

∞

9.

exp (p cos ax) sin (p sin ax) cosec ax 0

∞

10.

[1 − exp (p cos ax) cos (p sin ax)] cosec ax

0

3.975 

∞ sin

1.

 β arctan γx

(γ 2 + x2 )

0

β 2

·

b > 0,

b > 0,

∞

2.

sin (β arctan x) (1 + x2 )

0



∞

3.976 0

β 2

·

BI (372)(14)

p > 0]

BI (372)(15)

p > 0]

BI (391)(12)

x dx π = − exp (p cos ab) sin (p sin ab) cosec ab b2 − x2 2 [a > 0, b > 0, p > 0] BI (391)(13)

1 1 dx γ 1−β = ζ(β, γ) − β − −1 2 4γ 2(β − 1)

e2πx

[Re β > 1,



p > 0]

dx π cosec ab [ep − exp (p cos ab) cos (p sin ab)] = b2 − x2 2b [a > 0,



b > 0,

   dx π coth ab ep − exp pe−ab = 2 +x 2b [a > 0,



509

1 ζ(β) dx = − β +1 2(β − 1) 2

e2πx

Re γ > 0]

[Re β > 1]

WH, ET I 26(7) EH I 33(13)

 β− 12 −px2 e−p 1 + x2 e cos [2px + (2β − 1) arctan x] dx = β sin πβ Γ(β) 2p [Re β > 0, p > 0]

WH

3.98–3.99 Combinations of trigonometric and hyperbolic functions 3.981



∞

1. 0



∞

2. 0



[Re β > 0,

a > 0]

∞

0





 π aπ i β + ai β − ai sin ax dx = − tanh − ψ −ψ cosh βx 2β 2β 2β 4β 4β [Re β > 0, a > 0] cos ax π aπ dx = sech cosh βx 2β 2β

∞

4.

sin ax 0

[Re β > 0,

all real a]

5.

∞

cos ax 0

BI (264)(14)





 sinh aπ π β + γ + ia β + γ − ia sinh βx i γ dx = + ψ − ψ βπ sinh γx 2γ cosh aπ 2γ 2γ 2γ γ + cos γ [|Re β| < Re γ,



BI (264)(16)

GW (335)(12), ET I 88(1)

3. 

π aπ sin ax dx = tanh sinh βx 2β 2β

πβ γ

sin π sinh βx dx = βπ sinh γx 2γ cosh aπ γ + cos γ

[|Re β| < Re γ]

a > 0]

ET I 88(5) BI (265)(7)

510

6.

7.

Trigonometric Functions



∞



∞

3.982

βπ aπ π sin 2γ sinh 2γ sinh βx dx = sin ax [|Re β| < Re γ, a > 0] BI (265)(2) βπ cosh γx γ cosh aπ 0 γ + cos γ ⎡







  ∞ 3γ − β + ia 1 ⎣ 3γ − β − ia 3γ + β − ia sinh βx dx = ψ cos ax +ψ −ψ cosh γx 4γ 4γ 4γ 4γ 0 ⎤

2π sin πβ 3γ + β + ia γ ⎦ −ψ + πβ 4γ cos γ + cosh πa γ

[|Re β| < Re γ, 8.

9.



∞

cos ax 0



βπ aπ π cos 2γ cosh 2γ cosh βx dx = βπ cosh γx γ cosh aπ γ + cos γ

π/2

cos2m x cosh βx dx =

11.11 0



π/2

0

2.

a > 0]

[|Re β| < Re γ,

all real a]

(2m + 1)! cosh (β 2 + 12 ) (β 2 + 32 ) . . . [β 2 + (2m + 1)2 ]

0

WA 620a

∞

γ

3.11

BI (265)(6)

WA 620a

πβ 2

aπ cos ax dx = [Re β > 0, aπ 2 0 cosh βx 2β 2 sinh 2β  βπ βπ aπ aπ  ∞ cosh − β cos sinh π a sin 2γ 2γ 2γ 2γ sinh βx  sin ax dx = 2 βπ aπ cosh γx 0 γ 2 cosh − cos 

ET I 88(6)

(2m)! sinh πβ 2 β (β 2 + 22 ) . . . [β 2 + (2m)2 ]

cos2m+1 x cosh βx dx = 

[|Re β| < Re γ,

[β = 0]

12.11

1.

ET I 31(13)

sinh π cosh βx dx = · [|Re β| < Re γ, a > 0] BI (265)(4) πβ sinh γx 2γ cosh πa 0 γ + cos γ ⎡





 ∞ 3γ + β + ia i ⎣ 3γ + β − ai 3γ − β + ia cosh βx dx = ψ sin ax −ψ +ψ cosh γx 4γ 4γ 4γ 4γ 0 ⎤

2πi sinh πa 3γ − β − ai γ ⎦ −ψ − βπ 4γ cosh aπ + cos γ γ sin ax

10.

3.982

a > 0]

πa γ

∞

sin2 x cos ax π dx = 2 4 sinh hx

a > 0]

BI (264)(16)

γ

[|Re β| < 2 Re γ, a > 0]  a−2 + = I(a) 1 − e−π(a−2)

a+2 2a − 1 − e−πa 1 − e−π(a+2)  1 I(0) = (π coth π − 1) , 2

I(±2) =

ET I 88(9)

1 π + (coth 2π − coth π) 4 2



3.984

Trigonometric and hyperbolic functions

3.983 

∞

1.6 0

 a c π sin arccosh β b cos ax dx = √ b cosh βx + c β c2 − b2 sinh aπ β  a π sinh β arccos cb = √ β b2 − c2 sinh aπ β

511

[c > b > 0] [b > |c| > 0] [Re β > 0,



∞

2. 0



3.3 4.

5.

aγ β



cos ax dx π sinh = cosh βx + cos γ β sin γ sinh aπ β

a > 0]

π Re β < Im βγ,

GW (335)(13a)

a>0



BI (267)(3)

∞

sin ab cos ax dx = −π coth aπ [a > 0, b > 0] ET I 30(8) sinh b 0 cosh x − cosh b π  ∞ cos ax dx 2 ET I 30(9) [a > 0] < = < 2 2 0 1 + 2 cosh πx 1 + 2 cosh πa 3 3  + , + , + , + , β β a a  ∞ π sin (π − δ) sinh (π + δ) − sin (π + δ) sinh (π − δ) γ γ γ γ sin ax sinh βx  dx = 2πβ 2πa cosh γx + cos δ 0 γ sin δ cosh − cos γ

[π Re γ > |Re γδ|, 

∞

6. 0

γ

|Re β| < Re γ,

a > 0]

BI (267)(2)

 + , + , + , + , β β a a π cos (π − b) cosh (π + b) − cos (π + b) cosh (π − b) γ γ γ γ cos ax cosh βx  dx = 2πβ 2πa cosh γx + cos b γ sin b cosh γ − cos γ [|Re β| < Re γ, 0 < b < π, a < 0] BI (267)(6)



∞

7. 0

ai cos ax dx aπ Q ν (β)  ν+1 = Γ(ν + 1 − ai)e  Γ(ν + 1) β + β 2 − 1 cosh x

[Re ν > −1,

|arg(β + 1)| < π,

a > 0]

ET I 30(10)

3.984 1.

6

2.6

 lim

∞

 lim

∞

c↑1 0

c↑1 0  ∞ 0 ∞

4. 0

 5.

0

[|b| ≤ π,

sinh ab cos ax cosh cx dx = −π cot b cosh x + cos b sinh aπ

[0 < |b| < π,

sin ax sinh x2 π sinh aβ dx = cosh x + cos β 2 sin β2 cosh aπ

3.8 

sin ax sinh cx cosh ab dx = π cosh x + cos b sinh aπ

∞

π cos aγ cos ax cosh β2 x β dx = cosh βx + cosh γ 2β cosh γ2 cosh aπ β sin ax sinh βx aπ dx = 2 cosh 2βx + cos 2ax 4 (a + β 2 )

a real]

[Reβ < π, 

a real] a > 0]

!  ! π Re β > !Im βγ !

[a > 0,

Re β > 0]

BI (267)(1) BI (267)(5) ET I 80(10)

ET I 31(16)

BI (267)(7)

512

Trigonometric Functions



∞

6. 0



∞

7.8 0

βπ cos ax cosh βx dx = cosh 2βx + cos 2ax 4 (a2 + β 2 )

3.985

[Re β > 0,

a > 0]

BI (267)(8)

sinh2μ−1 x cosh2−2ν+1 x 1   dx = B(μ, ν − μ) 2 F 1 (, μ; ν; β) 2 cosh2 x − β sinh2 x [Re ν > Re μ > 0]

3.985  1.

∞

0

2ν−2 cos ax dx = Γ coshν βx β Γ(ν)



ν ai + 2 2β



ν ai − Γ 2 2β

[Re β > 0,

Re ν > 0,

EH I 115(12)

a > 0] ET I 30(5)

 2.

3.

3.986

n−1 -

∞



a2 cos ax dx 4n−1 πa + k2 = aπ 2n 2 2 4β cosh βx 0 2(2n − 1)!β sinh 2β k=1   2     πa a + 22 β 2 a2 + 42 β 2 · · · a2 + (2n − 2)2 β 2 = aπ 2(2n − 1)!β 2n sinh 2β [n ≥ 2, a > 0] % &

 ∞ n 2 2k − 1 cos ax dx a2 π22n−1 + = 2n+1 2 βx (2n)!β cosh aπ k=1 4β 2 0 cosh 2β  2      π a + β 2 a2 + 32 β 2 · · · a2 + (2n − 1)2 β 2 = aπ 2(2n)!β 2n+1 cosh 2β [Re β > 0, n = 0, 1, . . . , all real a]



∞

1. 0



∞

2. 0



∞

3. 0



∞

4.3 0

∞

[|Im(β + γ)| < Re δ]

BI (264)(19)

π sinh πα sin αx cos βx γ  dx = απ sinh γx 2γ cosh γ + cosh βπ γ

[|Im(α + β)| < Re γ]

LI (264)(20)

γπ cosh βπ π cos βx cos γx 2δ cosh 2δ dx = · γπ cosh δx δ cosh βπ δ + cosh δ

[|Im(β + γ)| < Re δ]

BI (264)(21)

β−1 β coth β − 1 sin2 βx β + = dx = 2 2β π (e − 1) 2π 2π sinh πx

2. 0

∞

EH I 44(3)

sin ax (1 − tanh βx) dx =

π 1 − a 2β sinh απ 2β

[Re β > 0]

ET I 88(4)a

sin ax (coth βx − 1) dx =

aπ 1 π coth − 2β 2β a

[Re β > 0]

ET I 88(3)

0



ET I 30(4)

γπ sinh βπ π sin βx sin γx 2δ sinh 2δ dx = · cosh δx δ cosh βδ π + cosh γδ π

[|Im β| < π] 3.987  1.

ET I 30(3)

3.991

3.988

Trigonometric and hyperbolic functions



π/2

1. 0

513

cos ax sinh (2b cos x) π√ √ dx = πb I α2 + 14 (b) I − a2 + 14 (b) 2 cos x [a > 0]



π/2

2. 0

ET I 37(66)

cos ax cosh (2b cos x) π√ √ dx = πb I a2 − 14 (b) I − a2 − 14 (b) 2 cos x [a > 0]



∞

3. 0

3.989



∞

1. 0



∞

2. 0



∞

3. 0

 4. 5.

π P − 12 +ia (cos b) cos ax dx √ √ = 2 cosh aπ cosh x cos b 2

[a > 0,

b > 0]

ET I 30(7)

[a > 0,

b > 0]

ET I 93(44)

[a > 0,

b > 0]

ET I 93(45)

2

sin a πx sin bx π πb2 πb dx = sin 2 cosech sinh ax 2a 4a 2a 2

ET I 37(67)

2

2

πb cos a πx sin bx π cosh πb a − cos 4a2 dx = sinh ax 2a sinh πb 2a a2 π

sin xπ cos ax π cos 4 − √2 dx = cosh x 2 cosh aπ 2 2

1

ET I 36(54)

2

2 a π 1 cos xπ cos ax π sin 4 + √2 dx = · cosh x 2 cosh aπ 0 2    ∞ ∞ + 2    2 ,   sin πax cos bx dx = − exp − k + 12 b sin k + 12 πa cosh πx 0 k=0 %  %  & 2 & ∞ k + 12 π b k + 12 b2 1  π − + exp − +√ sin a 4 4πa a a

∞

ET I 36(55)

k=0

ET I 36(56) [a > 0, b > 0] & %  



∞ 2 ∞  cos πax2 cos bx 1 1 dx = (−1)k exp − k + πa b cos k + cosh πx 2 2 0 k=0 % %   & 2 & ∞ k + 12 π b k + 12 1  π b2 +√ cos − + exp − a 4 4πa a a



 6.

k=0

[a > 0, 3.991 1. 2.11



π a a2 1 + sin πx sin ax coth πx dx = tanh sin 2 2 4 4π 0 

  ∞ a π a2 1 + cos πx2 sin ax coth πx dx = tanh 1 − cos 2 2 4 4π 0



∞

2

b > 0]

ET I 36(57)

ET I 93(42) ET I 93(43)

514

3.992

Trigonometric Functions



3.992

a2 − √ 4π sin πx cos ax

dx = − 3 + 1. a 2 0 4 cosh √ − 2 1 + 2 cosh √ πx 3 3  π a2  ∞ sin 2 − 12 4π cos πx cos ax

dx = 1 − 2. a 2 0 4 cosh √ − 2 1 + 2 cosh √ πx 3 3 √  ∞ 2 sin x + cos x2 π sin2 a + cos a2 √ √ · cos ax dx = 3.993 2 cosh ( πx) cosh ( πa) 0 3.994  ∞ sin (2a cosh x) cos bx π√ + √ dx = − 1. aπ J 1 + ib (a) Y 1 − ib (a) + J 1 − ib (a) Y 4 2 4 2 4 2 4 cosh x 0 



∞

[a > 0,



∞

ib 1 4+ 2

, (a)



∞

b > 0]

ET I 37(63)

ET I 93(47)

, cos (2a sinh x) sin bx i√ √ dx = − πa I − 1 − ib (a) K − 1 + ib (a) − I − 1 + ib (a) K − 1 − ib (a) 4 2 4 2 4 2 4 2 2 sinh x 0 [a > 0, b > 0] ET I 93(48) √ +  ∞ , sin (2a sinh x) cos bx πa √ I 1 − ib (a) K 1 + ib (a) + I 1 + ib (a) K 1 − ib (a) dx = 4 2 4 2 4 2 4 2 2 sinh x 0 [a > 0, b > 0] ET I 37(64) √ +  ∞ , cos (2a sinh x) cos bx πa √ I − 1 − ib (a) K − 1 + ib (a) + I − 1 + ib (a) K − 1 − ib (a) dx = 4 2 4 2 4 2 4 2 2 sinh x 0

7.

+

[a > 0, sin (a cosh x) sin (a sinh x)

0

3.995

ET I 37(58)

[a > 0, b > 0] + , sin (2a sinh x) sin bx i√ √ dx = − πa I 1 − ib (a) K − 1 + ib (a) − I 1 + ib (a) K 1 − ib (a) 4 2 4 2 4 2 4 2 2 sinh x

0

6.

ET I 37(61)

∞

3.

5.

ET I 37(60)

[a > 0, b > 0] ET I 37(62) + , cos (2a cosh x) cos bx π√ √ dx = − aπ J − 1 + ib (a) Y − 1 − ib (a) + J − 1 − ib (a) Y − 1 + ib (a) 4 2 4 2 4 2 4 2 4 cosh x

0

4.

π 12

∞

2. 

cos

2



π/2

1.

  sin 2a cos2 x cosh (a sin 2x) b2 cos2 x + c2 sin2 x

0

 2. 0

π dx = sin a sinh x 2

π/2



 cos 2a cos x cosh (a sin 2x)

dx =

2

b2

cos2

x+

c2

2

sin x

dx =

b > 0]

[a > 0]

ET I 37(65) BI (264)(22)

2ac π sin 2bc b+c [b > 0,

c > 0]

BI (273)(9)

[b > 0,

c > 0]

BI (273)(10)

2ac π cos 2bc b+c

3.997

3.996

Trigonometric and hyperbolic functions



∞

sin (a sinh x) sinh βx dx = sin

1. 0

 2. 3. 4. 5. 3.997

βπ K β (a) 2

[|Re β| < 1,

a > 0]

EH II 82(26)

a > 0]

WA 202(13)

∞

βπ K β (a) [|Re β| < 1, 2 0  π/2  π cos (a sin x) cosh (β cos x) dx = J 0 a2 − β 2 2 0  ∞   π sin a cosh x − 12 βπ cosh βx dx = J β (a) [|Re β| < 1, 2 0  ∞   π cos a cosh x − 12 βπ cosh βx dx = − Y β (a) [|Re β| < 1, 2 0 cos (a sinh x) cosh βx dx = cos



515

π/2

sinν x sinh (β cos x) dx =

1. 0

MO 40

a > 0]

WA 199(12)

a > 0]

WA 199(13)

√ ν2 ν+1 π 2 Γ L ν2 (β) 2 β 2

[Re ν > −1] ν



π √ 2 2 ν+1 sinν x cosh (β cos x) dx = π Γ I ν2 (β) β 2 0

EH II 38(53)

[Re ν > −1]

WH

 2.



π/2

3. 0

 4. 0

π/2

∞ √  dx (−1)k √ √ = 2π 2k + 1 cosh (tan x) cos x sin 2x k=0

BI (276)(13)

∞

Γ(q)  dx tanq x sin kλ = (−1)k−1 q cosh (tan x) + cos λ sin 2x sin λ k k=1

[q > 0]

BI (275)(20)

516

Trigonometric Functions

4.111

4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers 4.111



∞

1. 0

∂ 2m sin ax π · x2m dx = (−1)m · 2m sinh βx 2β ∂a



aπ tanh 2β [Re β > 0]



∞

2. 0

cos ax π ∂ 2m+1 · x2m+1 dx = (−1)m sinh βx 2β ∂a2m+1

GW (336)(17a)



aπ tanh 2β [Re β > 0]



∞

3. 0



∞

4. 0

∂ 2m+1 sin ax π · x2m+1 dx = (−1)m+1 · 2m+1 cosh βx 2β ∂a

∂ 2m cos ax π · x2m dx = (−1)m · 2m cosh βx 2β ∂a



(cf. 3.981 1)



GW (336)(17b)



1 cosh aπ 2β [Re β > 0]

1 cosh aπ 2β

(cf. 3.981 1)

(cf. 3.981 3) GW (336)(18b)



[Re β > 0]

(cf. 3.981 3) GW (336)(18a)



∞

5.

x 0

 6.

7.

aπ β

sinh π2 sin 2ax dx = · 2 cosh βx 4β cosh2 aπ β

[Re β > 0,

a > 0]

BI (364)(6)a

[Re β > 0,

a > 0]

BI (364)(1)a

[Re β > 0,

a > 0]

∞

π2 cos 2ax 1 dx = · 2 sinh βx 4β cosh2 aπ 0 β

 ∞ πa sin ax dx π = 2 arctan exp − 2β 2 0 cosh βx x x

BI (387)(1), ET I 89(13), LI (298)(17)

4.112 1.

2.



∞

 2  cos ax 2β 3 x + β2 πx dx = cosh3 aβ 0 cosh 2β  ∞  cos ax  6β 4 x x2 + 4β 2 πx dx = cosh4 aβ 0 sinh 2β

[Re β > 0,

a > 0]

ET I 32(19)

[Re β > 0,

a > 0]

ET I 32(20)

4.113

Trigonometric and hyperbolic functions and powers

4.113  1.

∞

0

dx sin ax 1 πe−aβ · 2 =− 2 − 2 sinh πx x + β 2β β sin πβ   1  + 2 2 F 1 1, −β; 1 − β; −e−a + 2β ∞  (−1)k e−ak πe−aβ 1 − − = 2 2β 2β sin πβ k2 − β 2 k=1

[Re β > 0,  2.

3. 4.

5.

6.

7.

2F 1

517

  1, β; 1 + β : −e−a

β = 0, 1, 2, . . . ,

a > 0]

ET I 90(18)

m−1  dx 1  (−1)k e−ka (−1)m e−ma  sin ax (−1)m ae−ma · 2 + + ln 1 + e−a = 2 2m 2m m−k 2m 0 sinh πx x + m k=1   1 dm−1 (1 + z)m−1 + ln(1 + z) 2m! dz m−1 z z=e−a [a > 0] ET I 89(17)  ∞  ∞  sin ax sin ax dx dx a 1 a · = = − cosh a+sinh a ln 2 cosh GW (336)(21b) 2 2 sinh πx 1 + x 2 sinh πx 1 + x 2 2 0 −∞  ∞  sin ax dx dx 1 ∞ sin ax π · · = = sinh a − cosh a arctan (sinh a) π π 2 2 2 −∞ sinh x 1 + x 2 0 sinh x 1 + x 2 2 ∞

GW (336)(21a)

√ √ sin ax 2 dx π −a sinh a 2 cosh a + 2 √ π · 1 + x2 = − √2 e + √2 ln 2 cosh a − √2 + 2 cosh a arctan 2 sinh a 0 sinh x 4 [a > 0] LI (389)(1) √

 ∞ 1 sin ax x dx π −a sinh a 2 cosh a + 2 √ √ √ √ √ e ln − · = + 2 cosh a arctan π 2 2 2 2 cosh a − 2 2 sinh a 0 cosh x 1 + x 4 BI (388)(1) [a > 0]  ∞   x dx cos ax 1 a · = − + e−a + cosh a ln 1 + e−a 2 sinh πx 1 + x 2 2 0 

∞



∞

8. 0



∞

9.11 0

 10.11 0

[a > 0]

BI (389)(14), ET I 32(24)

 −a  π −a cos ax x dx + e −1 π · 1 + x2 = 2 sinh a arctan e 2 sinh x 2 [a > 0]

BI (389)(11)

∞

 (−1)k e−(k+1/2)a cos ax dx πe−aβ · 2 − =  2 cosh πx x + β 2 2β cos(βπ) k + 1 − β2 k=0

∞

 −aβ

2

(−1)m e aβ + dx cos ax · =   2 cosh πx x2 + m + 1 2β 2 2

[Re β > 0,

 1 2

−

a > 0]

ET I 32(26)

∞ 

(−1)k e−(k+1/2)a  2 k + 12 − β 2 k=0 [Re β > 0,

a > 0]

ET I 32(25)

518

Trigonometric Functions



∞

11. 0

 12. 13.

4.114

 a  a  dx cos ax a  · = 2 cosh − ea arctan e− 2 + e−a arctan e 2 cosh πx 1 + x2 2 [a > 0]

ET I 32(21)

∞

  dx cos ax · [a > 0] = ae−a + cosh a ln 1 + e−2a BI (388)(6) π 2 0 cosh 2 x 1 + x √

 ∞ 1 dx cos ax π −a 2 sinh a cosh a 2 cosh a + 2 √ · arctan √ =√ e + √ − √ ln π 2 2 2 cosh a − 2 2 2 2 sinh a 0 cosh 4 x 1 + x [a > 0]

4.114



∞

1. 0



∞

2. 0

4.115



∞

1. 0

 0

4.

βπ aπ cos ax sinh βx 1 cosh 2γ + sin 2γ dx = ln βπ x cosh γx 2 cosh aπ 2γ − sin 2γ

[|Re β| < Re γ,

a > 0]

[|Re β| < Re γ]

BI (387)(6)a

ET I 33(34)

∞

π e−ab sin bβ  x sin ax sinh βx ke−ak sin kβ dx = + · (−1)k 2 2 x + b sinh πx 2 sin bπ k 2 − b2 k=1 [0 < Re β < π,

a > 0,

b > 0] BI (389)(23)

∞

2.

3.



βπ aπ sin ax sinh βx dx = arctan tan tanh x sinh γx 2γ 2γ

BI (388)(5)



∞



∞



 1 x sin ax sinh βx 1 · dx = e−a (a sin β − β cos β) − sinh a sin β ln 1 + 2e−a cos β + e−2a 2 x + 1 sinh πx 2 2 sin β + cosh a cos β arctan a e + cos β [|Re β| < π, a > 0] LI (389)(10)

x sin ax sinh βx · dx x2 + 1 sinh π x 0 2

cos β cosh a + sin β 1 π − sin β cosh a arctan = e−a sin β + cos β sinh a ln 2 2 cosh a+ − sin β sinh a , π BI (389)(8) |Re β| < , a > 0 2  ∞ ∞ π e−ab sin bβ  cos ax sinh βx e−ak sin kβ dx = · + · (−1)k 2 2 sinh πx 2b sin bπ k 2 − b2 0 x +b k=1 [0 < Re β < π, a > 0, b > 0]

5. 0

BI (389)(22)

 1 cos ax sinh βx 1 · dx = e−a (a sin β − β cos β) + cosh a sin β ln 1 + 2e−a cos β + e−2a 2 x + 1 sinh πx 2 2 sin β − sinh a cos β arctan a e + cos β [|Re β| < π, a > 0, b > 0] BI (389)(20)a 

4.115

Trigonometric and hyperbolic functions and powers



∞

6. 0

 0

∞

8. 0



∞

0



11.



 β β  sin ax sinh βx β a −a dx = e 2 a sin − β cos · − sinh sin ln 1 + 2e−a cos β + e−2a 1 cosh πx 2 2 2 2 x2 + β sin β a 4 + cosh cos arctan 2 2 1 + e−a cos β [|Re β| < π, a > 0] ET I 91(26) ∞

−ak 1 sin ax cosh γx cos kγ π e−aβ cos βγ  k−1 e dx = · + · − (−1) 2 2 2 2 x +β sinh πx 2β 2β sin βπ k − β2 k=1 [0 ≤ Re β, |Re γ| < π,

a > 0] BI (389)(21)

9.

10.

cosh a + sin β cos β cos ax sinh βx π 1 · + sinh a sin β arctan dx = e−a sin β − cosh a cos β ln x2 + 1 sinh π x 2 2 cosh a − sin β sinh a 2 + , π |Re β| < , a > 0, b > 0 2 BI (389)(18)

∞

7.



519

  1 sin ax cosh βx 1 · dx = − e−a (a cos β + β sin β) + sinh a cos β ln 1 + 2e−a cos β + e−2a 2 x + 1 sinh πx 2 2 sin β + cosh a sin β arctan a e + cos β [|Re β| < π, a > 0] ET I 91(25), LI (389)(9)

∞

cosh a + sin β cos β sin ax cosh βx π −a 1 · π dx = − 2 e cos β+ 2 sinh a sin β ln cosh a − sin β +cosh a cos β arctan sinh a 2+1 x 0 sinh x 2 + , π |Re β| < , a > 0 BI (389)(7) 2  ∞ ∞ −ak π e−ab cos bβ  x cos ax cosh βx cos kβ k ke dx = · + · (−1) 2 2 2 x +b sinh πx 2 sin bπ k − b2 0 k=1



[|Re β| < π, ∞

12. 0

 13. 0

∞

a > 0]

BI (389)(24)

1 x cos ax cosh βx · dx = e−a (a cos β + β sin β) x2 + 1 sinh πx 2   1 1 − + cosh a cos β ln 1 + 2e−a cos β + e−2a 2 2 sin β + sinh a sin β arctan a e + cos β [|Re β| < π, a > 0]

BI (389)(19)

x cos ax cosh βx cosh a + sin β π 1 · dx = −1 + e−a cos β + cosh a sin β ln x2 + 1 sinh π x 2 2 cosh a − sin β cos β 2 + sinh a cos β arctan sinh a + , π |Re β| < , a > 0 2

BI (389)(17)

520

Trigonometric Functions



∞

14. 0

4.116 1.



4.116

cos ax cosh βx e−2a sin 2β −a −a · dx = ae cos β + βe sin β + sinh a sin β arctan x2 + 1 cosh π x 1 + e−2a cos 2β   1 2 + cosh a cos β ln 1 + 2e−2a cos 2β + e−4a 2 , + π |Re β| < , a > 0 ET I 34(37) 2

∞

6

x cos 2ax tanh x dx

the integral is divergent

BI (364)(2)

[Re β > 0,

BI (387)(8)

0



∞

2.

cos ax tanh βx 0

4.117



∞

1. 0

 2. 3. 4. 5.

aπ dx = ln coth x 4β

sin ax πx dx = a cosh a − sinh a ln (2 sinh a) tanh 1 + x2 2 [a > 0]

BI (388)(3)

∞

π sin ax πx a dx = − ea + sinh a ln coth + 2 cosh a arctan (ea ) tanh 2 4 2 2 0 1+x  ∞   sin ax a [a > 0] coth πx dx = e−a − sinh a ln 1 − e−a 2 2 0 1+x  ∞ a sin ax π [a > 0] coth x dx = sinh a ln coth 2 1 + x 2 2 0  ∞   x cos ax π tanh x dx = −ae−a − cosh a ln 1 − e−2a 2 1+x 2 0 

∞

0

 0

∞

8. 0



∞

9. 0

BI (389)(6)

BI (388)(7)

x cos ax π π a tanh x dx − ea + cosh a ln coth + 2 sinh a arctan (ea ) 1 + x2 4 2 2   x cos ax a 1 coth πx dx = − e−a − − cosh a ln 1 − e−a 2 1+x 2 2 a x cos ax π coth x dx = −1 + cosh a ln coth 2 1+x 2 2

∞

BI (388)(8) BI (389)(15)a, ET I 33(31)a

[a > 0]

BI (389)(12)

  π x cos ax π a coth x dx = −2 + e−a + cosh a ln coth + 2 sinh a arctan e−a 2 1+x 4 2 2 [a > 0]



BI (389)(5)

[a > 0] ∞

7. 

BI (388)(4)

[a > 0]

6.



1 x sin ax 1 1 π πa coth πa − 1 dx = 2 1 2 sinh 2 πa 2 2

0 ∞ cosh x pπ 1 − cos px dx · = ln cosh 4.119 sinh qx x 2q 0 4.1188

a > 0]

BI (389)(13)

ET I 89(14) BI (387)(2)a

4.123

Trigonometric and hyperbolic functions and powers

521

4.121 

∞

1. 0



∞

2. 0

sin ax − sin bx dx · = 2 arctan cosh βx x

bπ aπ − exp 2β 2β (a + b)π 1 + exp 2β

exp

bπ cos ax − cos bx dx 2β · = ln aπ sinh βx x cosh 2β

[Re β > 0]

GW (336)(19b)

[Re β > 0]

GW (336)(19a)

cosh

4.122 γπ sinh cos βx sin γx dx 2δ · = arctan βπ cosh δx x 0 cosh 2δ  ∞ 1 cosh 2aπ + cos βπ cosh βx dx 2 · = ln sin ax sinh x x 4 1 + cos βπ 0 

1.6

2. 4.123



∞

∞

1. 0



∞

2. 0



∞

3. 0



∞

4. 0



∞

5. 0

[Re δ > |Im β| + |Im γ|]

[|Re β| < 1]

ET I 93(46)a

BI (387)(7)

x dx sin x 1 1 · = arctan − cosh ax + cos x x2 − π 2 a a

BI (390)(1)

x dx sin x a 1 · 2 = − arctan 2 2 cosh ax − cos x x − π 1+a a

BI (390)(2)

x dx sin 2x 1 1 + 2a2 1 · 2 · = − arctan 2 cosh 2ax − cos 2x x − π 2a 1 + a2 a

BI (390)(4)

x dx cosh ax sin x −1 · = cosh 2ax − cos 2x x2 − π 2 2a (1 + a2 )

LI (390)(3)

dx cos ax πe−aγ · 2 = 2 cosh πx + cos πβ x + γ 2γ (cos γπ + cos βπ)  ∞  e−(2k+1−β)a e−(2k+1+β)a 1 − 2 + sinh βπ γ 2 − (2k + 1 − β)2 γ − (2k + 1 + β)2 k=0

[0 < Re β < 1, 

∞

6. 0

 7. 0

∞

Re γ > 0,

a > 0]

ET I 33(27)

 sin ax sinh bx Γ(p) a  (−1)k xp−1 dx = p arctan p sin cos 2ax + cosh 2bx b (2k + 1)p (a2 + b2 ) 2 k=0 ∞

sin ax2



sin sinh 1 ∂ ϑ1 (z | q) · x dx = cos πx + cosh πx 4 ∂z πx 2

πx 2



[p > 0]

BI (364)(8)

z=0,q=e−2a

[a > 0]

ET I 93(49)

522

4.124 1. 2. 4.125

Trigonometric Functions

4.124

  √  cos px cosh q 1 − x2 π √ dx = J 0 p2 − q 2 2 1 − x2 0    ∞  dx π u 2 2 cos ax cosh β (u − x ) · √ = J0  2 u2 − x2 a2 − β 2 u 



1

∞

1. 0

MO (40)

ET I 34(38)

  (−1)n−1 a2n−1 π dx a2 = sinh (a sin x) cos (a cos x) sin x sin 2nx 1+ x (2n − 1)! 8 2n(2n + 1) LI (367)(14)

  ∞ (−1)n−1 a2(n−1) π dx a2 = cosh (a sin x) cos (a cos x) sin x cos(2n − 1)x 1− x [2(n − 1)]! 8 2n(2n − 1) 0

 2.

LI (367)(15)



∞

3.

sinh (a sin x) cos (a cos x) cos x cos 2nx 0

∞ (−1)n a2n+1 3π dx π  (−1)k a2k+1 = + x 2 (2k + 1)! (2n + 1)! 8 k=n+1

(−1)n−1 a2n−1 π + (2n − 1)! 8 LI (367)(21)

4.126  1.

∞

sin (a cos bx) sinh (a sin bx)

0



[b > 0] ∞

2.

sin (a cos bx) cosh (a sin bx) 0



cos (a cos bx) sinh (a sin bx) 0

BI (381)(2)

dx π cos (a cos bc) sinh (a sin bc) = c2 − x2 2c [b > 0,

∞

3. 

x dx π = [cos (a cos bc) cosh (a sin bc) − 1] 2 −x 2

c2

c > 0]

x dx π = [a cos bc − sin (a cos bc) cosh (a sin bc)] 2 −x 2

c2

[b > 0] ∞

4.

cos (a cos bx) cosh (a sin bx) 0

BI (381)(1)

BI (381)(4)

dx π = − sin (a cos bc) sinh (a sin bc) c2 − x2 2c [b > 0]

BI (381)(3)

4.13 Combinations of trigonometric and hyperbolic functions and exponentials 4.131  1. 0

∞

⎫   ⎧ β−νγ−ai β−νγ+ai ⎨ ⎬ Γ Γ 2γ 2γ i Γ(ν + 1)  −  sin ax sinhν γxe−βx dx = − ν+2 2 γ ⎩ Γ β+νγ−ai + 1 Γ β+γν+ai +1 ⎭ 2γ 2γ [Re ν > −2,

Re γ > 0,

|Re(γν)| < Re β]

ET I 91(30)a

4.133

Trigonometric and hyperbolic functions and exponentials



∞

2. 0

⎫   ⎧ β−νγ−ai β−νγ+ai ⎬ ⎨ Γ Γ 2γ 2γ Γ(ν + 1)  −  cos ax sinhν γxe−βx dx = ν+2 2 γ ⎩ Γ β+γν−ai + 1 Γ β+νγ+ai +1 ⎭ 2γ 2γ [Re ν > −1,



∞

3.

e−βx

0



∞

4. 0

4.132



∞

1. 0



∞

2. 0

e−x

Re γ > 0,

∞  2a sin ax dx = 2 2 sinh γx a + [β + (2k − 1)γ] k=1 



 β + γ + ia β + γ − ia 1 = ψ −ψ 2γi 2γ 2γ

|Re(γν)| < Re β]

∞

3. 0

[Re β > |Re γ|]

π aπ 1 sin ax dx = coth − sinh x 2 2 a

∞

4. 0

sinh 2πa sin ax sinh βx a π γ dx = − + · 2πβ eγx − 1 2 (a2 + β 2 ) 2γ cosh 2πa γ − cos γ

  β a a β i +i +1 −ψ −i +1 + ψ 2γ γ γ γ γ [Re γ > |Re β|, a > 0]



∞

0

a π sin ax cosh βx dx = − · eγx + 1 2 (a2 + β 2 ) γ

1.



∞

11 0

 2.11 0

∞

BI (265)(5)a, ET I 92(34)

βπ sinh aπ γ cos γ 2βπ cosh 2aπ γ − cos γ

ET I 92(35)

sin 2πβ β π cos ax sinh βx γ dx = − · eγx − 1 2 (a2 + β 2 ) 2γ cosh 2aπ − cos 2βπ γ γ LI (265)(8)

πβ πa β π sin γ cosh γ cos ax sinh βx dx = − + 2βπ eγx + 1 2 (a2 + β 2 ) γ cosh 2aπ γ − cos γ

[Re γ > |Re β|] 4.133

ET I 92(33)

sinh 2πa a π sin ax cosh βx γ dx = − + · eγx − 1 2 (a2 + β 2 ) 2γ cosh 2πa − cos 2πβ γ γ

[Re γ > |Re β|] 5.

ET I 91(28)

ET I 91(29)

[Re γ > |Re β|] 

ET I 34(40)a

BI (264)(9)a

[Re γ > |Re β|] 

523

ET I 34(39)

2

   x √ sin ax sinh βx exp − dx = πγ exp γ β 2 − a2 sin(2aβγ) 4γ [Re γ > 0] 2

   x √ cos ax cosh βx exp − dx = πγ exp γ β 2 − a2 cos(2aβγ) 4γ [Re γ > 0]

ET I 92(37)

ET I 35(41)

524

4.134

Trigonometric Functions



∞

1.

e−βx (cosh x − cos x) dx = 2

0



∞

2.

e−βx (cosh x − cos x) dx = 2

 

0

4.135



∞

sin ax cosh 2γxe

1. 0



∞

2.

1 π cosh β 4β

[Re β > 0]

ME 24

1 π sinh β 4β

[Re β > 0]

ME 24

. −βx2

2

1 dx = 2

4





aγ 2 a π2 βγ 2 1 arctan exp − + sin a2 + β 2 a2 + β 2 a2 + β 2 2 β

. cos ax2 cosh 2γxe−βx dx = 2

0

1 2

4.134

4

[Re β > 0] LI (268)(7)



aγ 2 a π2 βγ 2 1 arctan exp − + cos a2 + β 2 a2 + β 2 a2 + β 2 2 β [Re β > 0]

4.136



∞

1. 0

LI (268)(8)

√

  4 1 2π 1 sinh2 x + sin x2 e−βx dx = √ I 14 cosh 8β 8β 4 β

[Re β > 0] √

∞  4 1 2π 1 sinh2 x − sin x2 e−βx dx = √ I 14 sinh 8β 8β 4 β 0

ME 24

[Re β > 0] √

∞  4 1 2π 1 cosh2 x + cos x2 e−βx dx = √ I − 14 cosh 8β 8β 4 β 0

ME 24

[Re β > 0] √

∞  4 1 2π 1 cosh2 x − cos x2 e−βx dx = √ I − 14 sinh 8β 8β β 4 0

ME 24

[Re β > 0]

ME 24

 2.

 3.

 4.

4.137



∞

2

2 −βx4

sin 2x sinh 2x e

1. 0



[Re β > 0]



4 1 1 π π 1 − sin 2x2 cosh 2x2 e−βx dx =  J cos 4 β 4 128β 2 4 β

MI 32

[Re β > 0]



4 1 1 π −π 1 − cos 2x2 sinh 2x2 e−βx dx =  J sin 4 β 4 128β 2 4 β

MI 32

∞

[Re β > 0]

MI 32

0

3. 0

dx =  J − 14 4 128β 2





1 1 π + cos β β 4

∞

2. 

π

4.141

Trigonometric and hyperbolic functions, exponentials, and powers



∞

4.

2 −βx4

2

cos 2x cosh 2x e 0

π

dx =  J − 14 4 128β 2





1 1 π + sin β β 4 [Re β > 0]

4.138



∞

1. 0







1 1 cos β β

[Re β > 0]



 2  −βx4 1 1 π 2 2 2 sin 2x cosh 2x − cos 2x sinh 2x e dx =  J 14 sin 4 2 β β 32β

MI 32

[Re β > 0]



 2  −βx4 1 1 π 2 2 2 cos 2x cosh 2x + sin 2x sinh 2x e dx =  J − 14 cos 4 2 β β 32β

MI 32

∞

[Re β > 0]



 2  −βx4 1 1 π 2 2 2 cos 2x cosh 2x − sin 2x sinh 2x e dx =  J − 14 sin 4 2 β β 32β

MI 32

∞

0

3. 0



 2  4 π sin 2x cosh 2x2 + cos 2x2 sinh 2x2 e−βx dx =  J1 4 32β 2 4

MI 32

∞

2. 

525

4. 0

[Re β > 0]

MI 32

4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 4.141



∞

−βx2

xe

1. 0



∞

2.

1 cosh x sin x dx = 4

xe−βx sinh x cos x dx = 2

0



∞

3.

x2 e−βx cosh x cos x dx = 2

0

 4. 0

∞

1 4

x2 e−βx sinh x sin x dx = 2





1 4 1 4

π β3



1 1 + sin cos 2β 2β [Re β > 0]

MI 32

[Re β > 0]

1 1 π 1 − sin cos β3 2β β 2β

MI 32

[Re β > 0]

1 1 π 1 + cos sin β3 2β β 2β

MI 32

[Re β > 0]

MI 32



1 π 1 − sin cos β3 2β 2β





526

4.142

Trigonometric Functions



∞

1.

xe−βx (sinh x + sin x) dx = 2

0

 2. 3.

∞

1 2

 

4.142

π 1 cosh β3 4β

[Re β > 0]

π 1 [Re β > 0] sinh 3 β 4β 0

  ∞ 1 1 1 π 1 2 −βx2 + sinh x e (cosh x + cos x) dx = cosh 2 β3 4β 2β 4β 0 

∞

4.

xe−βx (sinh x − sin x) dx = 2

1 2

x2 e−βx (cosh x − cos x) dx = 2

0

1 2



[Re β > 0]

1 1 π 1 + cosh sinh β3 4β 2β 4β [Re β > 0]

4.143



∞

−βx2

xe

1. 0



∞

2.

1 (cosh x sin x + sinh x cos x) dx = 2β

xe−βx (cosh x sin x − sinh x cos x) dx = 2

0



∞

4.144

e−x sinh x2 cos ax 2

0

4.145



∞

1.

−βx2

xe 0



∞

2.

dx = x2



1 2β





a cosh (2ax sin t) sin (2ax cos t) dx = 2

xe−βx sinh (2ax sin t) cos (2ax cos t) dx = 2

0

a 2



∞

1.8

e−βx sinh ax sin bx dx = 2

0

 2.8 0

∞

e−βx cosh ax cos bx dx = 2

1 2



1 2

π exp β





π exp β



a −b 4β 2



MI 32

MI 32

ET I 35(44)



2 a2 π a sin 2t exp − cos 2t cos t − β3 β β [Re β > 0] BI (363)(5)



2 2 a π a sin 2t exp − cos 2t sin t − β3 β β



2

ME 24

1 π sin β 2β



a2 − b 2 4β

ME 24

[Re β > 0]

[Re β > 0] 4.14610

ME 24

1 π cos β 2β

[Re β > 0] 

 a πa − 1−Φ √ 4 8 [a > 0]

π − a2 e 8 2

ME 24

sin

ab 2β [Re β > 0]

cos

ab 2β [Re β > 0]

BI (363)(6)

4.212

Logarithmic functions



∞

3.

−βx2

xe 0



∞

4.

a cosh ax sin ax dx = 4β

xe−βx sinh ax cos ax dx = 2

0



∞

5.8

a 4β

x2 e−βx cosh ax sin ax dx = 2

0



∞

6.8

x2 e−βx cosh ax cos ax dx = 2

0

1 4

1 4









π β

π β

π β3



527

a2 a2 + sin cos 2β 2β



2

cos



2

a a − sin 2β 2β



[Re β > 0]

[Re β > 0]

a2 a2 a2 + cos sin 2β β 2β

[Re β > 0]

2 2 a a2 π a − sin cos β3 2β β 2β [Re β > 0]

4.2–4.4 Logarithmic Functions 4.21 Logarithmic functions 4.211  1.

∞

dx = −∞ 1 e ln x  u dx = li u 0 ln x

2. 4.212



1

1.7 0



1

2. 0

 3.

1

7

1

4. 5.

1

8

1

6. 7. 1

BI (31)(4)

dx = −ea Ei(−a) a − ln x

[a > 0]

BI (31)(5)

[a ≥ 0]

BI (31)(14)

[a > 0]

BI (31)(16)

dx

1 = − + e−a Ei(a) a (a + ln x) dx

e

1 + ea Ei(−a) a

2

=

2

= 1 + (1 − a)e−a Ei(a)

[a ≥ 0]

BI (31)(15)

2

= 1 + (1 + a)ea Ei(−a)

[a > 0]

BI (31)(17)

2

=

ln x dx ln x dx (a − ln x)

0



[a > 0]

(a + ln x)

0



dx = e−a Ei(a) a + ln x

(a − ln x)

0



FI III 653, FI II 606

2

0



BI (33)(9)

ln x dx (1 + ln x)

e −1 2

BI (33)(10)

528

Logarithmic Functions

 8.

1

7 0

4.213

n−1  dx 1 1 e−a Ei(a) − (n − k − 1)!ak−n n = (n − 1)! (n − 1)! (a + ln x) k=1

[a ≥ 0] 

1

9. 0

BI (31))(22)

n−1 dx (−1)n a (−1)n−1  e = Ei(−a) + (n − k − 1)!(−a)k−n n (n − 1)! (n − 1)! (a − ln x) k=1

[a > 0, 

n odd]

BI (31)(23)

m

(ln x)

dx, it is convenient to make the substitution x = e−t . n l [an + (ln x) ] Results 4.212 3, 4.212 5, and 4.212 8 [for n > 1] and 4.213 6, 4.213 8 below are divergent but may be considered to be valid if defined as follows: 

n−1   a  a f (z) dz f (z) d 1 dz PV n = (n − 1)! dz0 0 (z − z0 ) 0 z − z0 where a > z0 > 0, n = 1, 2, 3, . . . and PV indicates the Cauchy principal value. 4.213  1 dx 1 [a > 0] 1. BI (31)(6) 2 = a [ci(a) sin a − si(a) cos a] 2 0 a + (ln x)  1  dx 1  −a 2.7 Ei(a) − ea Ei(−a) [a > 0] , (cf. 4.212 1 and 2) 2 = 2a e 2 0 a − (ln x)

In integrals of the form

BI (31)(8)



1

3. 0



ln x dx a2 + (ln x)

1

4.7

= ci(a) cos a + si(a) sin a

2

=−

ln x dx a2 − (ln x)

0

2

 1  −a e Ei(a) + ea Ei(−a) 2

[a > 0] [a > 0] ,

BI (31)(7)

(cf. 4.212 1 and 2) BI (31)(9)



1

+

5. 0

dx 2

a2 + (ln x)

,2 =

1 1 [ci(a) sin a − si(a) cos a] − 2 [ci(a) cos a + si(a) sin a] 3 2a 2a [a > 0]



1

6.8

+

0



1

+

7. 0

dx 2

a2 − (ln x) ln x dx

2

a2 + (ln x)

,2

,2 =

is divergent 1 1 [ci(a) sin a − si(a) cos a] − 2 2a 2a [a > 0]

 8.8 0

1

+

ln x dx 2

a2 − (ln x)

,2

LI (31)(18)

is divergent

BI (31)(19)

4.221

4.214

Logarithms of more complicated arguments



1

1. 0



dx a4

− (ln x)

4

=−

 1  a e Ei(−a) − e−a Ei(a) − 2 ci(a) sin a + 2 si(a) cos a 4a3 [a > 0]

1

2.

ln x dx a4

0

− (ln x)

4

=−

1

3.

2

(ln x) dx a4 − (ln x)

0

4

=−



1

3

(ln x) dx a4

0

− (ln x)

4

=−



1

1. 0

 2.

3. 4.

4.216 1.

2.∗



μ−1 1 dx = Γ(μ) ln x

BI (31)(13)

[Re μ > 0]

FI II 778

[Re μ < 1]

BI (31)(1)

1

dx π

μ = cosec μπ Γ(μ) 1 0 ln x √  1 π 1 ln dx = x 2 0  1 √ dx  = π 1 0 ln x



BI (31)(12)

 1 a e Ei(−a) + e−a Ei(a) + 2 ci(a) cos a + 2 si(a) sin a 4 [a > 0]

4.215

BI (31)(11)

 1  a e Ei(−a) − e−a Ei(a) + 2 ci(a) sin a − 2 si(a) cos a 4a [a > 0]

4.7

BI (31)(10)

 1  a e Ei(−a) + e−a Ei(a) − 2 ci(a) cos a − 2 si(a) sin a 2 4a [a > 0]



529

BI (32)(1) BI (32)(3)

1/e

dx < = K 0 (1) 2 0 (ln x) − 1 √  1/e π dx √ = e − ln x − 1 0

GW (32)(2)

4.22 Logarithms of more complicated arguments 4.221



1

1.

ln x ln(1 − x) dx = 2 −

π2 6

BI (30)(7)

ln x ln(1 + x) dx = 2 −

π2 − 2 ln 2 12

BI (30)(8)

0

 2.

0

1

530

Logarithmic Functions



∞

1

3.

ln 0

4.222



∞

ln 0

2. 3. 4.

 ln(1 + k) 1 − ax dx =− ak 1 − a ln x k

BI (31)(3)

a2 + x2 dx = (a − b)π b2 + x2

[a > 0,

b > 0]

a2 + x2 aa dx = π(b − a) + π ln [a > 0, b > 0] b2 + x2 bb 0

 ∞ b2 ln x ln 1 + 2 dx = πb (ln b − 1) [b > 0] x 0 

  ∞   1 + ab b2 2 2 ln(1 + ab) − b ln 1 + a x ln 1 + 2 dx = 2π x a 0 ∞

5. 0





ln 1 +

0

∞

7.

[a > 0,

2

  b ln a2 + x2 ln 1 + 2 x

∞

6. 

GW (322)(20)

∞

ln x ln



2

a x2

ln a2 +

0

8.∗

[a < 1]

k=1

1. 

4.222

1 x2

ln 1 +



ln 1 +

2

b x2

BI (33)(2)

BI (33)(3)

dx = 2π [(a + b) ln(a + b) − a ln a − b]

2

b x2

b > 0]

BI (33)(1)

[a > 0,

b > 0]

BI (33)(4)

dx = 2π [(a + b) ln(a + b) − a ln a − b ln b]



b > 0]  1 + ab ln(1 + ab) − b ln b dx = 2π a 

[a > 0,

BI (33)(5)

[a > 0, b > 0] BI (33)(7) &

b b−m ∞   (k − 1)! b! (−1)b−m−1 1/a 1 ln (1 + ax) xb e−x dx = e Ei − + (b − m)! ab−m a (−a)b−m−k 0 m=0 %



k=1

[b > 0, 4.223



∞

1. 0



∞

2. 0



∞

3. 0

4.224



  π2 ln 1 + e−x dx = 12

BI (256)(10)

  π2 ln 1 − e−x dx = − 6

BI (256)(11)

  t2 π2 − ln 1 + 2e−x cos t + e−2x dx = 6 2

u

ln sin x dx = L

1. 0



π −u −L 2 2

π

π/4

ln sin x dx = −

2. 0

an integer]

1 π ln 2 − G 4 2

[|t| < π]

BI (256)(18)

LO III 186(15)

BI (285)(1)

4.225

Logarithms of more complicated arguments



π/2

3.

ln sin x dx = 0



1 2



π

ln sin x dx = − 0

531

π ln 2 2

FI II 629,643

u

ln cos x dx = − L(u)

4.

LO III 184(10)

0



π/4

ln cos x dx = −

5. 0



1 π ln 2 + G 4 2

BI (286)(1)

π/2

π ln 2 2 0    π/2 π π2 2 2 (ln sin x) dx = (ln 2) + 2 12 0    π/2 π π2 2 2 (ln cos x) dx = (ln 2) + 2 12 0 √  π a + a2 − b 2 ln (a + b cos x) dx = π ln 2 0  π ln (1 ± sin x) dx = −π ln 2 ± 4G ln cos x dx = −

6. 7. 8. 9.8 10.

BI 306(1)

BI (305)(19)

BI (306)(14)

[a ≥ |b| > 0]

GW (322)(15) GW (322)(16a)

0



∞ k  k  b (−1)n+1 π a [a > 0] 11. ln (1 + a sin x) dx = ln + 2G + 2 2 2 k n=1 2n − 1 0 k=1 π = − ln 2 + 2G [a = 1] 2   √  π  2  1 + 1 − a2 12. a ≤1 ln (1 + a cos x) dx = π ln 2 0   ⎧ √ 2 ⎪  π ⎨2π ln 1 + 1 − a for a2 ≤ 1 2 2 12 (1) ln (1 + a cos x) dx = ⎪ 0 ⎩ π a2 for a2 ≥ 1 2 ln 4

2k+1  π/2 ∞ 2    2a 22k (k!) 13. ln 1 + 2a sin x + a2 dx = (2k + 1) · (2k + 1)!! 1 + a2 0 k=0  2  a ≤1  nπ   14.11 ln a2 − 2ab cos x + b2 dx = 2nπ ln [max (|a|, |b|)] π/2

7

b=

1−a 1+a

BI (330)(1)

BI (308)(24)

0

 15.8

[ab > 0] nπ

  ln 1 − 2a cos x + a2 dx = 0

0

= nπ ln a2 4.225



π/4

ln (cos x − sin x) dx = −

1. 0

1 π ln 2 − G 8 2

 

a2 ≤ 1 a2 ≥ 1

FI II 142, 163, 688

 

GW (322)(9b)

532

Logarithmic Functions



π/4

2.

ln (cos x + sin x) dx = 0



1 2



π/2

ln (cos x + sin x) dx = − 0

2π

3.

ln (1 + a sin x + b cos x) dx = 2π ln

1+

0



2π

4.

4.226

√

1 − a2 − b 2 2 

1 π ln 2 + G 8 2

a2 + b 2 < 1

  ln 1 + a2 + b2 + 2a sin x + 2b cos x dx = 0



  = 2π ln a2 + b2

0



GW (322)(9a)



BI (332)(2)

a2 + b 2 ≤ 1 a2 + b 2 ≥ 1

  BI (322)(3)

4.226



π/2

1.

 2 ln a2 − sin2 x dx = −2π ln 2

0

= 2π ln

a+

 √ a2 − 1 = 2π (arccosh a − ln 2) 2

a2 ≤ 1



[a > 1] FI II 644, 687

2.

3.

4.



π/2



u





π/2      1 π  ln 1 + a sin2 x dx = ln 1 + a sin2 x dx = ln 1 + a cos2 x dx 2 0 0 √ 0  1 π  1+ 1+a 2 = ln 1 + a cos x dx = π ln 2 0 2 BI (308)(15), GW(322)(12) [a ≥ −1]

 u   1 π α sin α − ln 2 ln 1 − sin2 α sin2 x dx = (π − 2θ) ln cot + 2u ln 2 2 2 0 π − 2u + L(θ + u) − L(θ − u) + L 2 + π π, cot θ = cos α tan u; −π ≤ α ≤ π, − ≤ u ≤ LO III 287 2 2 %  &   π/2   2  α β β 1 2 2 2 2 2 α + cos4 + sin cos2 ln 1 − cos x sin α − sin β sin x dx = π ln cos 2 2 2 2 2 0

5. 0

LO III 283 [α > β > 0]   2

π sin x π −α+u +L −α−u ln 1 − dx = −u ln sin2 α − L 2 2 sin α , + π 2 π − ≤ u ≤ , |sin u| ≤ |sin α| 2 2 LO III 287

 6. 0

π/2

  1 ln a2 cos2 x + b2 sin2 x dx = 2

 0

π

  a+b ln a2 cos2 x + b2 sin2 x dx = π ln 2 [a > 0,

b > 0]

GW (322)(13)

4.227

Logarithms of more complicated arguments



π/2

7. 0

4.227



t 1 + sin π−t 1 + sin t cos2 x 2 dx = π ln = π ln cot ln 2 t 1 − sin t cos x 4 cos 2 + π, |t| < 2

u

ln tan x dx = L(u) + L

1. 0



π/4

2. 0



BI (286)(11)

4 ln (a tan x) dx =

0

4.

LO III 283

LO III 186(16)

 π ln tan x dx = − π2 ln tan x dx = −G

π/2

3. 

π −u −L 2 2

π

533

π/4

7

π ln a 2

n

[a > 0] n

(ln tan x) dx = n!(−1) 0

∞ 

(−1)k (2k + 1)n+1

1  π n+1 |En | 2 2 k=0

=

BI (307)(2)

[n even] BI (286)(21)



π/2

5.7

(ln tan x)

2n

dx = 2(2n)!

0



∞  k=0

π/2

6.

(ln tan x)

2n+1

k

(−1) = (2k + 1)2n+1

 π 2n+1 2

|E2n |

BI (307)(15)

dx = 0

BI (307)(14)

π3 16

BI (286)(16)

5 5 π 64

BI (286)(19)

0



π/4

7.

2

(ln tan x) dx = 0



π/4

8.

4

(ln tan x) dx = 0



π/4

9.

ln (1 + tan x) dx =

π ln 2 8

BI (287)(1)

ln (1 + tan x) dx =

π ln 2 + G 4

BI (308)(9)

ln (1 − tan x) dx =

π ln 2 − G 8

BI (287)(2)

0



π/2

10. 0



π/4

11. 0



π/2

2

(ln (1 − tan x)) dx =

12.11 0



π/4

13. 14. 0

BI (308)(10)

ln (1 + cot x) dx =

π ln 2 + G 8

BI (287)(3)

ln (cot x − 1) dx =

π ln 2 8

BI (287)(4)

0



π ln 2 − 2G 2

π/4

534

Logarithmic Functions



π/4

15.

ln (tan x + cot x) dx = 0



π/4

1 2



2

(ln (cot x − tan x)) dx =

16.11 0

4.228

π/2

ln (tan x + cot x) dx = 0

1 2



π/2

π ln 2 2

2

(ln (cot x − tan x)) dx = 0

BI (287)(5), BI (308)(11)

π ln 2 2 BI (287)(6), BI (308)(12)



π/2

17. 0

  1 ln a2 + b2 tan2 x dx = 2



π

  ln a2 + b2 tan2 x dx = π ln(a + b)

0

[a > 0, 4.228 1. 2.

b > 0]

GW (322)(17)





  t π−t π ln sin t sin x + 1 − cos2 t sin2 x dx = ln 2 − 2 L LO III 290 − 2L 2 2 2 0  u  π  1 1 − t − ϕ ln cos t + L(u + ϕ) − L(u − ϕ) − L(ϕ) ln cos x + cos2 x − cos2 t dx = − 2 2 2 0   sin u π cos ϕ = 0≤u≤t≤ sin t 2 

π/2



t

LO III 290

3. 4.

π  − t ln cos t ln cos x + cos2 x − cos2 t dx = − 2 0 & %    u t sin u + sin t cos x sin2 u − sin2 x t 2 2  sin u + 1 ln dx = π ln tan sin u + tan 2 2 0 sin u − sin t cos x sin2 u − sin2 x 

[t > 0, u > 0] √ ∞ π/4 √ π (−1)k  ln cot x dx = 2 (2k + 1)3 0 k=0  π/4 ∞ √  dx (−1)k √ √ = π 2k + 1 ln cot x 0 k=0  π/4 √  √ √ 1 1 π/2 √ π ln tan x + cot x dx = ln tan x + cot x dx = ln 2 + G 2 8 2 0 0

LO III 285

LO III 283

 5.

6.

7.

BI (297)(9)

BI (304)(24)

BI (287)(7), BI (308)(22)



π/4

ln2

8. 0

√ √ 1 cot x − tan x dx = 2



π/2

ln2 0

√ √ π cot x − tan x dx = ln 2 − G 4 BI (287)(8), BI (308)(23)

4.229 1. 2.11



1 ln ln dx = −C x 0  1  ∞ −u dx e

= PV du ≈ −0.154479 PV 1 0 0 ln u ln ln x



1

FI II 807

BI (31)(2)

4.231

Logarithms and rational functions



√ dx 1  ln ln = − (C + 2 ln 2) π x 1 0 ln x



μ−1  1 1 1 ln ln dx = ψ(μ) Γ(μ) ln x x 0

 3.

4.11

5.7

535

1

BI (32)(4)

[Re μ > 0]

BI (30)(10)

If the integrand contains (ln ln x1 ), it is convenient to make the substitution ln x1 = u so that x = e−u .  1 ln (a + ln x) dx = ln a − e−a Ei(a) [a > 0] BI (30)(5) 0



1

ln (a − ln x) dx = ln a − ea Ei(−a)

6. 0



π/2

7. π/4

π ln ln tan x dx = ln 2

[a > 0]

BI (30)(6)

    Γ 34 √   2π Γ 14

BI (308)(28)

4.23 Combinations of logarithms and rational functions 4.231



1

1. 0



1

2. 0



1

3. 0



1

4. 0



π2 ln x dx = − 1+x 12

FI II 483a

π2 ln x dx = − 1−x 6

FI II 714

π2 x ln x dx = 1 − 1−x 6

BI (108)(7)

π2 1+x ln x dx = 1 − 1−x 3

BI (108)(9)

∞

5.11 0

 6.

0

 7.

1

∞

0

∞

9. 0



0

 √ 

 Γ n − 12 π 1 dx a −C−ψ n− ln x 2 2 ln n = 4(n − 1)!a2n−1 b 2b 2 (a + b2 x2 ) ln x dx a π ln = 2 2 +b x 2ab b

[ab > 0]

ln px π ln pq dx = q 2 + x2 2q

[p > 0,

ln x dx π2 = − a2 − b2 x2 4ab

[ab > 0]

a2

0

10.

BI (139)(1)

BI (111)(1)

[a > 0, ∞

8. 

[0 < a]

ln x dx = − ln 2 (1 + x)2

7



ln x dx ln a = 2 (x + a) a

∞

b > 0]

LI (139)(3) BI (135)(6)

q > 0]

BI (135)(4)

536

Logarithmic Functions

 11. 12. 13. 14. 15. 16.

a

ln x dx π ln a G − = 2 + a2 x 4a a 0  1  ∞ ln x ln x dx = − dx = −G 2 1 + x 1 + x2 0 1  1 ln x dx π2 = − 2 8 0 1−x  1 x ln x π2 dx = − 2 48 0 1+x  1 x ln x π2 dx = − 2 24 0 1−x  1 n 1 − x2n+2 (n + 1)π 2  n − k + 1 + ln x dx = − 2 8 (2k − 1)2 (1 − x2 ) 0 

ln x

0



BI (111)(2)

1 − xn+1 (n + 1)π 2  n − k + 1 + dx = − (1 − x)2 6 k2

BI (111)(3)

π2 x ln x dx = −1 + 1+x 2

1

π2 (1 − x) ln x dx = 1 − 1+x 6

υ

ln uυ (u + υ)2 ln x dx = ln (x + u)(x + υ) 2(υ − u) 4uυ

1. u



BI (111)(5)

n−k+1 1 + (−1)n xn+1 (n + 1)π 2  − dx = − (−1)k 2 (1 + x) 12 k2

1

0

4.232

GW (324)(7b)

k=1

0

20.∗

BI (108)(11)

n

1

ln x



FI II 482, 614

k=1

18. 

GW (324)(7b)

n

1

0

19.∗

[a > 0]

k=1

17. 

4.232

∞

2. 0

2

BI (145)(32)

2

ln x dx (ln β) − (ln γ) = (x + β)(x + γ) 2(β − γ)

[|arg β| < π,

|arg γ| < π] ET II 218(24)



∞

3. 0

4.233 1.

3

2.3 3.11 4.3

2

π 2 + (ln a) ln x dx = x+ax−1 2(a + 1)

[a > 0]

  1 ln x dx 2 2π 2  −ψ = = −0.7813024129 . . . 2 1 + x + x 9 3 3 0    1 1 ln x dx 1 2π 2  − ψ = = −1.17195361934 . . . 2 1 − x + x 3 3 3 0    1 1 x ln x dx 1 7π 2  − ψ = − = −0.15766014917 . . . 2 9 6 3 0 1+x+x    1 1 x ln x dx 1 5π 2  −ψ = = −0.3118211319 . . . 2 6 6 3 0 1−x+x 

BI (140)(10)

1

LI (113)(1)

LI (113)(2)

LI (113)(2)

LI (113)(4)

4.236

Logarithms and rational functions



∞

5. 0

4.234



∞

1.11

2

1

2.

x ln x dx

∞

3. ∞

4. 1

5. 0

 6. 7.

3.11 4. 4.236

2

ln x dx = −

BI (142)(2)a

π 2

BI (142)(1)a

BI (112)(21)

∞

[a > 0, ∞

(a2

0

2.

BI (111)(4)

π2 x2 ln x dx √  =−  2 4 (1 − x ) (1 + x ) 16 2 + 2

8.

1.



x2 ln x dx aπ b = ln + b2 x2 ) (1 + x2 ) 2b (b2 − a2 ) a

[ab > 0]

π (1 − x)xn−2 π2 [n > 1] dx = − 2 tan2 2n 1 − x 4n 2n 0        ∞ 1 − x2 xm−1 π 2 sin m+1 π sin nπ n ln x dx = − 2 2  mπ  2  m+2  1 − x2n 4n sin 2n sin 0 2n π    ∞ π 1 − x2 xn−3 π2 [n > 2] ln x dx = − 2 tan2 2n 1−x 4n n 0  1 xm−1 + xn−m−1 π2   ln x dx = − [n > m] 1 − xn n2 sin2 m 0 nπ 1

1. 0

 2. 0

BI (317)(16)a

b > 0]

LI (140)(12) LI (140)(12), BI (317)(15)a

∞

ln x



GW (324)(13c)

1 = − ln 2 4

bπ a ln x dx = ln [ab > 0] 2 + b2 x2 ) (1 + x2 ) 2 − a2 ) (a 2a (b b 0

 ∞ ln x 1 dx π ln a + b ln b · = 2 2 1 + b2 x2 2 (1 − a2 b2 ) a 0 x +a 

4.235

0 < t < π]

BI (144)(18)a

ln x dx = 0

1 − x2

[a > 0,

G π − 2 8

=

2

(1 + x2 )

0



1 + x2 (1 − x2 )

0



2 x2 )

(1 +

0



ln x dx (1 + x2 )

1



ln x dx t ln a = x2 + 2xa cos t + a2 a sin t

537

1

1 + (p − 1) ln x x ln x + 1−x (1 − x)2

1 x ln x + 1 − x (1 − x)2



dx =

LI (135)(12)

BI (135)(11)

BI (108)(15)

xp−1 dx = −1 + ψ  (p) [p > 0]



BI (135)(10)

π2 −1 6

BI (111)(6)a, GW (326)(13) GW (326)(13a)

538

Logarithmic Functions

4.241

4.24 Combinations of logarithms and algebraic functions 4.241 1.

2.

3.

  2n x2n ln x (2n − 1)!! π  (−1)k−1 √ · − ln 2 dx = (2n)!! 2 k 1 − x2 0 k=1    1 2n+1 2n+1  (−1)k x ln x (2n)!! √ dx = ln 2 + (2n + 1)!! k 1 − x2 0 k=1    1 2n  1 (2n − 1)!! π  (−1)k−1 2n 2 · − − ln 2 x 1 − x ln x dx = (2n + 2)!! 2 k 2n + 2 0 

1

BI (118)(5)a

BI (118)(5)a

k=1



1

x2n+1

4.



1 − x2 ln x dx =

0

(2n)!! (2n + 3)!!

 ln 2 +

2n+1  k=1

1 (−1)k − k 2n + 3

LI (117)(4), GW (324)(53a)



BI (117)(5), GW (324)(53b)

 5.

< 1 (2n − 1)!! 2n−1 π [ψ(n + 1) + C + ln 4] ln x · (1 − x2 ) dx = − 4 · (2n)!! 0



.

6. 0

 7. 8. 9. 10.

1 2 √ln x dx = − π ln 2 − 1 G 4 2 1 − x2

BI (145)(1)

1

ln x dx π √ = − ln 2 2 2 1 − x 0  ∞ ln x dx √ = 1 − ln 2 2 x2 − 1 1 x  1 π π 1 − x2 ln x dx = − − ln 2 8 4 0  1  x 1 − x2 ln x dx = 13 ln 2 − 49 0



1

11. 0

4.242

BI (117)(3)

 0

 2. 0

b

BI (144)(17)

BI (117)(1), GW (324)(53c)

BI (117)(2)

√

 2 1 2π  =− Γ 2 8 4 x (1 − x ) ln x dx

∞

1.

FI II 614, 643

ln x dx

1  K = 2 2 2 2 2a (a + x ) (x + b )



+

x2 ) (b2

√  a2 − b 2 ln ab a 

ln x dx (a2

GW (324)(54a)

−

x2 )

1 = √ K 2 2 a + b2



b √ 2 a + b2



[a > b > 0] ln ab − [a > 0,

π K 2

BY (800.04)



a √ 2 a + b2

b > 0]



BY (800.02)

4.247

Logarithms and algebraic functions



∞

3. b



 b a π  = √ K √ ln ab + K √ 2 2 a2 + b 2 a2 + b 2 a2 + b 2 (x2 + a2 ) (x2 − b2 ) ln x dx

1

[a > 0, b > 0] √ &

b b ln x dx a2 − b 2 1 π  K = ln ab − K 2a a 2 a (a2 − x2 ) (b2 − x2 ) 0 %

 4.

539

[a > b > 0] √  a ln x dx a2 − b 2 1  K ln ab = 2a a (a2 − x2 ) (x2 − b2 ) b %

√ &  ∞ b ln x dx a2 − b 2 1 π  K = ln ab + K 2a a 2 a (x2 − a2 ) (x2 − b2 ) a

BY (800.06)

BY (800.01)

 5.

6.

[a > b > 0] 

1

4.243 0

4.244



1

1. 0

3. 4.245 1.

2.

ln x dx

< 2 3 x (1 − x2 )

1

 3 1 1 =− Γ 8 3

GW (324)(54b)

ln x dx π √ =− √ 3 3 3 1 − x3

BI (118)(7) BI (118)(8)

  2n x4n+1 ln x (2n − 1)!! π  (−1)k−1 √ · − ln 2 dx = (2n)!! 8 k 1 − x4 0 k=1    1 4n+3 2n+1  (−1)k x ln x (2n)!! √ ln 2 + dx = 4 · (2n + 1)!! k 1 − x4 0 

BY (800.05) GW (324)(56b)



π ln 3 + √ 3 3 0

 1 x ln x dx π π √ − ln 3 < = √ 3 3 3 3 0 3 (1 − x3 )2 

2.

x ln x π √ dx = − ln 2 4 8 1−x

BY (800.03)

1

GW (324)(56a)

GW (324)(56c)

k=1



1

4.246 0



1−x

2

n−

% & 1 n  1 (2n − 1)!! π 2 ln x dx = − · 2 ln 2 + (2n)!! 4 k

GW (324)(55)

k=1

4.247





1

1.6 0

 2.6 0

1

1 1 , ln x 2n 2n √ dx = − π n 2n 2 1−x 8n sin 2n

1 1 , πB ln x dx 2n 2n  = − π n xn−1 (1 − x2 ) 8 sin 2n πB

[n > 1]

GW (324)(54c)a

GW (324)(54)

540

Logarithmic Functions

4.251

4.25 Combinations of logarithms and powers 4.251



∞

1. 0



∞

2. 0



1

3.10 0



1

4. 0



ln x π dx = πaμ−1 cot μπ ln a − a−x sin2 μπ

1

0

0 < Re μ < 1]

ET I 314(5)

xμ−1 ln x dx = − ψ  (μ) = − ζ(2, μ) 1−x

[Re μ > 0]

BI (108)(8)

π 2  (−1)k−1 x2n dx = − + 1+x 12 k2 2n

BI (108)(4)

2n−1  (−1)k π2 x2n−1 dx = + ln x 1+x 12 k2

BI (108)(5)

k=1

∞

 μ−1   π xμ−1 ln x dx = β ln β − γ μ−1 ln γ − π cot μπ β μ−1 − γ μ−1 (x + β)(x + γ) (γ − β) sin μπ [|arg β| < π, |arg γ| < π, 0 < Re μ < 2, μ = 1] BI (140)(9)a, ET 314(6)

∞

  π xμ−1 ln x dx π − β μ−1 (sin μπ ln β − π cos μπ) = 2 (x + β)(x − 1) (β + 1) sin μπ [|arg β| < π, 0 < Re μ < 2,

2. 0

μ = 1] BI (140)(11)

∞

3. 0



[a > 0,

GW (324)(6), ET I 314(3)

0



BI (135)(1)



[Re μ > 0]

1.



0 < Re μ < 1]

k=1

11



[|arg β| < π,

xμ−1 ln x dx = β  (μ) x+1

0

4.252

x

ln x 



μ−1

1

5.11

6.

πβ μ−1 xμ−1 ln x dx = (ln β − π cot μπ) β+x sin μπ

∞

4.6 0

xp−1 ln x pπ π2 cosec2 dx = − 2 1−x 4 2 xμ−1 ln x (1 − μ)aμ−2 π dx = 2 (x + a) sin μπ

[0 < p < 2] ln a − π cot μπ +

1 μ−1 [|arg a| < π

(see also 4.254 2)

0 < Re μ < 2

(μ = 1)] GW (324)(13b)

4.253



1

ν−1

xμ−1 (1 − xr )

1.8 0

ln x dx =

, μ 1 μ + μ , ν ψ + ν B − ψ r2 r r r [Re μ > 0,

Re ν > 0,

r > 0]

GW (324)(3b)a, BI (107)(5)a

 2. 0

1

xp−1 π ln x dx = − cosec pπ p+1 (1 − x) p

[0 < p < 1]

bi (319)(10)a

4.255

Logarithms and powers



∞

3. u



(x − u)μ−1 ln x dx = uμ−λ B(λ − μ, μ) [ln u + ψ(λ) − ψ(λ − μ)] xλ

∞

4.11

ln x 0



∞

5.

x a2 + x2

p

[0 < Re μ < Re λ] ln a  p p dx = pB , x 2a 2 2

(x − 1)p−1 ln x dx =

1



∞

6.7

ln x 0



ln x 0

π cosec πp p

dx n+ 12

(a + x)

=



2 (2n − 1)a

n− 12

[a > 0,

p > 0]

ln a + 2 ln 2 − 2

[Re μ > 0,



1

1. 0



xp−1 ln x 1 dx = − 2 ψ  1 − xq q

∞

2. 0

 3.

4.3

5.

6. 4.255 1.

2. 3.

∞



p q

xp−1 ln x π2 dx = − pπ q 1−x q 2 sin2 q

π2 p−1 0 π q 2 sin2 q

 1 p−1 x ln x 1  p dx = 2 β 1 + xq q q 0 pπ  ∞ p−1 cos x ln x π2 q dx = − 2 pπ q 2 1 + x q 0 sin q  1 q−1 2 x ln x π dx = − 2 2q 1 − x 8q 0 ln x dx = − 1 xp

xq

n−1  k=1

1 2k − 1

1

[p > 0,

BI (289)(12)a

a = 0,

|arg a| < π] NT 68(7)

n = 1, 2, . . .]

q > 0]

BI (142)(5)

GW (324)(5)

[0 < p < q]

BI (135)(8)

[p < 1,

p + q > 1]

BI (140)(2)

[p > 0,

q > 0]

[0 < p < q]

GW (324)(7)

BI (135)(7)

[q > 0]

BI (108)(12)

[p > 1]

BI (108)(13)

[p > 1]

BI (108)(14)

[p < 1]

BI (140)(3)



 2 π sin 2p 1 − x2 xp−2 π  ln x dx = − π 1 + x2p 2p 0 cos2 2p   2

 1 1 + x2 xp−2 π π 2 ln x dx = − sec 2p 1 − x 2p 2p 0  ∞  pπ 1 − xp π2 tan2 ln x dx = 2 1−x 4 2 0 

BI (140)(6)



[|arg a| < π, 4.254

ET II 203(18)

[−1 < p < 0]

dx 1 = (ln a − C − ψ(μ)) (a + x)μ+1 μaμ

∞

7.7

541

542

Logarithmic Functions



1

4.256

ln 0

4.257 

∞

1. 0

4.256



 μ  xμ−1 dx μ+m 1 1 μ m < , = 2B ψ −ψ x n n n n n n n−m (1 − xn ) [Re μ > 0]

LI (118)(12)

+ , x γ ν ν ν dx π γ ln + π (β − γ ) cot νπ β β = (x + β)(x + γ) sin νπ(γ − β) xν ln

[|arg β| < π, 

∞

2.

ln 0



∞

x q



p

q 2p

x + x2p

xp 2p q + x2p



ln

4.



x q

r

| Re ν| < 1] ET II 219(30)

dx =0 x

dx =0 2 + x2 q 0 , + 2 2  ∞ + (ln a) ln a 4π dx x = ln x ln a (x − 1)(x − a) 6(a − 1) 0

3.

|arg γ|< π,

[q > 0]

BI (140)(4)a

[q > 0]

BI (140)(4)a

[a > 0]

(for a = 1 see 4.261 5) BI (141)(5)

∞

5.

ln x ln 0

xp dx π 2 [(ap + 1) ln a − 2π (ap − 1) cot pπ] x = a (x − 1)(x − a) (a − 1) sin2 pπ   2 p < 1, a > 0

BI (141)(6)

4.26–4.27 Combinations involving powers of the logarithm and other powers 4.261 1.

7

2. 3.

4. 5. 6. 7. 8.11

  t π 2 − t2 dx [0 ≤ t ≤ π] (ln x) = 1 + 2x cos t + x2 6 sin t 0  1  2 2 1 ∞ (ln x) dx 10π 3 (ln x) dx √ = = 2 2 0 x2 − x + 1 81 3 0 x −x+1  1  2 2 1 ∞ (ln x) dx 8π 3 (ln x) dx √ = = 2 2 0 x2 + x + 1 81 3 0 x +x+1 , + 2 2  ∞ + (ln a) ln a π dx 2 = [a > 0] (ln x) (x − 1)(x + a) 3(1 + a) 0  ∞ dx 2 2 (ln x) = π2 2 (1 − x) 3 0  1 3 dx π 2 (ln x) = 2 1 + x 16 0 √  1  2 2 1 ∞ 3 2 3 2 1+x 2 1+x π (ln x) dx = (ln x) dx = 1 + x4 2 0 1 + x4 64 0 √  1 8 3π 3 + 351 ζ(3) 2 1−x (ln x) dx = 1 − x6 486 0 

1

2

BI (113)(7)

GW (324)(16c)

GW (324)(16b)

BI (141)(1) BI (139)(4) BI (109)(3) BI (109)(5), BI (135)(13)

4.261

Powers and logarithms

  dx π π2 2 = (ln 2) + 2 12 1 − x2 0    ∞ 2 3 μ−1 π 2 − sin μπ 2 x dx = (ln x) [0 < Re μ < 1] 1+x sin3 μπ 0    1 ∞ n n   (−1)n+k (−1)k 3 2 x dx n =2 ζ(3) + 2 (ln x) = (−1) 1+x (k + 1)3 2 k3 0 

9. 10. 11.7

543

1

2 (ln x) √

k=n



∞

1

12.7

2

(ln x) 0



1

2

(ln x) 0



ET I 315(10)

k=1



  1 xn dx 1 =2 = 2 ζ(3) − 3 1−x (k + 1) k3 n

k=n

13.11

BI (118)(13)



[n = 0, 1, . . .]

BI (109)(1)

[n = 0, 1, . . .]

BI (109)(2)

k=1

∞

n

k=n

k=1

  x2n dx 1 1 7 ζ(3) − 2 = 2 = 1 − x2 (2k + 1)3 4 (2k − 1)3 [n = 0, 1, . . .]

∞

14.

(ln x) 0

2

x2

BI (109)(4)

 xp−1 dx π sin(1 − p)t  2 = π − t2 + 2π cot pπ [π cot pπ + t cot(1 − p)t] + 2x cos t + 1 sin t sin pπ [0 < t < π, 0 < p < 2, p = 1]

⎧ &2 ⎫ % 2n 2n ⎨ π2  ⎬ 1 2n k k  x dx (−1) (−1) (2n − 1)!! 2 π + + ln 2 (ln x) √ = + ⎭ 2 · (2n)!! ⎩ 12 k2 k 1 − x2 0 k=1 k=1 ⎧ &2 ⎫ %2n+1  1 2n+1 ⎬ ⎨ 2n+1 2 k k   dx (−1) (−1) (2n)!! π 2 x − + ln 2 (ln x) √ = + − ⎭ (2n + 1)!! ⎩ 12 k2 k 1 − x2 0

GW (324)(17)

 15.

16.

k=1



GW (324)(60a)

k=1

GW (324)(60b) 1

17.7

  2 2 (ln x) xμ−1 (1 − x)ν−1 dx = B(μ, ν) [ψ(μ) − ψ(ν + μ)] + ψ  (μ) − ψ  (μ + ν)

0

[Re μ > 0, 

1

18.

2

(ln x) 0



1

19. 0

20.

1

2

(ln x) 0

0

LI (111)(8)

 1 + (−1)n xn+1 n−k+1 3 dx = (n + 1) ζ(3) − 2 (−1)k−1 2 (1 + x) 2 k3

1

2

1 − x2n+2

LI (111)(7)

2

(1 − x2 )

 n−k+1 7 (n + 1) ζ(3) − 2 4 (2k − 1)3 n

dx =

k=1

q−1

(ln x) xp−1 (1 − xr )

21.

n−k+1 k3

ET I 315(11)

k=1

7



k=1

Re ν > 0]

n

2

(ln x) 

1 − xn+1 dx = 2(n + 1) ζ(3) − 2 (1 − x)2

n 

[n = 0, 1, . . .] LI (111)(9)

+ p ,2  p p p 1 p ,q ψ +q + ψ +q dx = 3 B − ψ −ψ r r r r r r [p > 0,

q > 0,

r > 0]

GW (324)(8a)

544

4.262

Logarithmic Functions



1

1.

3

(ln x) 0



1

2.

3

(ln x) 0

3. 4.

7 4 dx =− π 1+x 120

BI (109)(9)

π4 dx =− 1−x 15

BI (109)(11)

,2 + 2 π 2 + (ln a) dx 3 = (ln x) (x + a)(x − 1) 4(a + 1) 0 % &  1 n−1 n 4  (−1)k 3 x dx n+1 7π = (−1) −6 (ln x) 1+x 120 (k + 1)4 0 

4.262

∞

[a > 0]

BI (141)(2)

[n = 1, 2, . . .]

BI (109)(10)

 π4 xn dx 1 =− +6 1−x 15 (k + 1)4

[n = 1, 2, . . .]

BI (109)(12)

 x2n dx 1 π4 +6 = − 2 1−x 16 (2k + 1)4

[n = 1, 2, . . .]

BI (109)(14)

k=0



n−1

1

5.

3

(ln x) 0



k=0

n−1

1

6.

3

(ln x) 0



k=0

3

(ln x) 0



 1 + (−1)n xn+1 n−k+1 7(n + 1)π 4 + 6 dx = − (−1)k−1 (1 + x)2 120 k4

BI (111)(10)

n

1

3

(ln x) 0

k=1

1

9.

3

(ln x) 0

4.263 

BI (111)(11)

k=1

8. 

 n−k+1 1 − xn+1 (n + 1)π 4 + 6 dx = − (1 − x)2 15 k4 n

1

7.

∞

1.8 0

1 − x2n+2 2

(1 − x2 )

 n−k+1 (n + 1)π 4 +6 16 (2k − 1)4 n

dx = −

,+ , + 2 2 7π 2 + 3 (ln a) ln a π 2 + (ln a) dx 4 = (ln x) (x − 1)(x + a) 15(1 + a) [a > 0]



1

2.

4

(ln x) 0



1

3. 0



1

1.

dx 5π = 2 1+x 64

BI (109)(17)

   t π 2 − t2 7π 2 − 3t2 dx (ln x) = 1 + 2x cos t + x2 30 sin t 4

5

(ln x) 0

 2.

1

5

(ln x) 0

BI (141)(3)

5

[|t| < π] 4.264

BI (111)(12)

k=1

BI (113)(8)

31π 6 dx =− 1+x 252

BI (109)(20)

8π 6 dx =− 1−x 63

BI (109)(21)

4.267

Powers and logarithms



∞

3. 0

545

,2 + , + 2 2 3π 2 + (ln a) π 2 + (ln a) dx 5 = (ln x) (x − 1)(x + a) 6(1 + a) [a > 0]



1

4.265

6

(ln x) 0

4.266  1.

1

7

(ln x)

0



1

2. 0

4.267  1.

1

0



1

2. 0



1

3.8 0

 4. 5. 6.11 7. 8. 9. 10.

7

(ln x)

BI (141)(4)

7

dx 61π = 2 1+x 256

BI (109)(25)

127π 8 dx =− 1+x 240

BI (109)(28)

8π 8 dx =− 1−x 15

BI (109)(29)

2 1 − x dx = ln 1 + x ln x π

BI (127)(3)

π (1 − x)2 dx = ln 2 1 + x ln x 4

BI (128)(2)

(1 − x)2 dx · mx 2 ln x +x 1 + 2x cos n   n+k+1 2  k+2   k 

n−1  Γ 2n Γ 2n Γ kmπ 1  mπ  = (−1)k sin ln   2n2     k+1 n+k n sin n k=1 Γ 2n Γ 2n Γ n+k+2 2n   n−k+1 2  k+2   k 

12 (n−1)

  Γ n Γ n Γ kmπ 1 k n   (−1) sin = ln   2  n−k   n−k+2  k+1 n sin mπ Γ n Γ n Γ n k=1 n [m < n]

1 ln 2 1−x dx · =− · 2 ln x 1 + x 1 + x 2 0 √  1 2 x 2 2 1−x dx · = ln · 2 ln x π 0 1+x 1+x  1 ∞   p dx ln(1 + k) [p ≥ 1] = (1 − x)p (−1)k k ln x 0 k=1

 1 dx 1 − xp −p = ln Γ(p + 1) 1−x ln x 0  1 p−1 p x − xq−1 dx = ln [p > 0, ln x q 0      1 p−1 Γ q Γ p+1 dx x − xq−1 2  · = ln  2p   q+1 [p > 0, ln x 1+x Γ 2 Γ 2 0  1 p−1   1 ∞ xp−1 − x−p pπ x − x−p dx = dx = ln tan 2 0 (1 + x) ln x 2 0 (1 + x) ln x

[m + n is odd]

[m + n is even] BI (130)(3)

1

BI (130)(16) BI (130)(17)

BI (123)(2)

GW (326)(10)

q > 0]

FI II 647

q > 0]

FI II 186

[0 < p < 1]

FI II 816

546

Logarithmic Functions

 11. 12.

13.

1

p+r dx = ln ln x r+q 0  1 p ∞ q  n + k − 1

x −x dx p+k = ak ln n k q+k 0 (1 − ax) x ln x k=0  1 p+q+1 dx = ln (xp − 1) (xq − 1) ln x (p + 1)(q + 1) 0 (xp − xq ) xr−1



4.267

1

14. 0

xp − xq 1 + x2n+1 · 1+x x ln x

[r > 0, 

p > 0,

1

15. 0

q > 0]

q > 0,

a2 < 1

[p > −1,

q > −1,

LI (123)(5)



  q   + n + 1 Γ q+1 Γ 2 Γ + n Γ p+1 2 2 2



 dx = ln  p+1 q+1 q p +n+1 Γ +n Γ Γ Γ 2 2 2 2

Γ(q + 1) Γ(p + r + 1) xp − xq 1 − xr · dx = ln 1−x ln x Γ(p + 1) Γ(q + r + 1) [p > −1, q > −1,



q p+r Γ Γ 2r 1 p−1 q−1 x −x 2r

 dx = ln r ) ln x q + r p (1 + x 0 Γ Γ 2r 2r  1 2p−2q q−1 1−x x dx qπ = ln tan 2p 1 + x ln x 4p 0  ∞ p−1  pπ qπ x − xq−1 dx = ln tan cot r 2r 2r 0 (1 + x ) ln x

q > 0]

p + r > −1,

BI (130)(15)

p + q > −1]

p

[p > 0, 

p > 0,

GW (324)(19b)

BI (127)(7)

q + r > −1]

GW (324)(23)

 16.

17. 18.

[p > 0,

q > 0,

[0 < q < p] [0 < p < r,

pπ ⎞  ∞ p−1 sin x − xq−1 r ⎠ dx = ln ⎝ r sin qπ 0 (1 − x ) ln x r

20.

21.

BI (128)(6)

0 < q < r]

[0 < p < r,

0 < q < r]

  q Γ p+2 xp−1 − xq−1 1 − x2 2n Γ 2n dx = ln  q+2   p  · [p > 0, q > 0] 1 − x2n ln x Γ 2n Γ 2n 0     q+2 p+4n+2 q  1 p−1 Γ p+4n+4 4(2n+1) Γ 4(2n+1) Γ 4(2n+1) Γ 4(2n+1) x − xq−1 1 + x2    dx = ln  2(2n+1) ln x p+2 p 0 1+x Γ q+4n+4 Γ Γ q+4n+2 Γ 

22. 0

BI (143)(4)

1

4(2n+1)



GW (324)(21)

GW (324)(22), BI (143)(2)

⎛

19.

r > 0]

∞

xp−1 − xq−1 1 + x2(2n+1)

4(2n+1)

4(2n+1)

BI (128)(11)

4(2n+1)

BI (128)(7) [p > 0, q > 0]

 pπ (p + 2)π qπ (q + 2)π 1 + x2 dx = ln tan · tan · cot · cot · ln x 4(2n + 1) 4(2n + 1) 4(2n + 1) 4(2n + 1) [0 < p < 4n, 0 < q < 4n] BI (143)(5)

4.267

Powers and logarithms



∞

23. 0



pπ · sin (q+2)π sin 2n xp−1 − xq−1 1 − x2 2n dx = ln (p + 2)π qπ 1 − x2n ln x · sin sin 2n 2n [0 < p < 2n,

1

(1 − xp ) (1 − xq )

24. 1

(1 − xp ) (1 − xq )

25. 0



0 < q < 2n]

BI (143)(6)

r−1

x

0



547

(p + q + r)r dx = ln ln x (p + r)(q + r)

[p > 0,

Γ(p + r) Γ(q + r) xr−1 dx = ln (1 − x) ln x Γ(p + q + r) Γ(r) [r > 0, r + p > 0,

1

q > 0,

r + q > 0,

r > 0]

BI (123)(8)

r + p + q > 0]

(p + q + 1)(q + r + 1)(r + p + 1) dx = ln ln x (p + q + r + 1)(p + 1)(q + 1)(r + 1) r > −1, p + q > −1, p + r > −1, q + r > −1,

FI II 815a

(1 − xp ) (1 − xq ) (1 − xr )

26. 0



[p > −1,

q > −1,

p + q + r > −1] GW (324)(19c)

1

Γ(p + 1) Γ(q + 1) Γ(r + 1) Γ(p + q + r + 1) dx = ln (1 − x) ln x Γ(p + q + 1) Γ(p + r + 1) Γ(q + r + 1) r > −1, p + q > −1, p + r > −1, q + r > −1, p + q + r > −1]

(1 − xp ) (1 − xq ) (1 − xr )

27. 0



[p > −1,

q > −1,

FI II 815 1

(1 − xp ) (1 − xq ) (1 − xr )

28. 0



1

29. 0

(p + q + s)(p + r + s)(q + r + s)s xs−1 dx = ln ln x (p + s)(q + s)(r + s)(p + q + r + s) [p > 0, q > 0, r > 0,

BI (123)(10)

   q+s  Γ p+s Γ r dx x p q r  = ln  s   p+q+s (1 − x ) (1 − x ) (1 − xr ) ln x Γ r Γ r s−1

[p > 0, 

∞

30.

(1 − xp ) (1 − xq )

0



s−1

dx x =2 p+q+2s (1 − x ) ln x

1

(1 − xp ) (1 − xq ) (1 − xr )

31. 0



1

0



q > 0,

r > 0,

s > 0] GW (324)(23a)

1

s−1

dx x (1 − xp ) (1 − xq ) p+q+2s (1 − x )

ln x 0 (p + s)π sπ cosec = 2 ln sin p + q + 2s p + q + 2s [s > 0, s + p > 0, s + p + q > 0] GW (324)(23b)a

Γ(p + s) Γ(q + s) Γ(r + s) Γ(p + q + r + s) xs−1 dx = ln (1 − x) ln x Γ(p + q + s)4 Γ(p + r + s) Γ(q + r + s) Γ(s) [p > 0, q > 0, r > 0, s > 0] ∗ BI (127)(11)

   q+s   r+s   p+q+r+s  Γ p+s Γ Γ Γ xs−1 dx t = ln  p+q+s  t q+r+s  t  p+r+s  t  s  (1 − x ) (1 − x ) (1 − x ) t (1 − x ) ln x Γ Γ Γ t Γ t t t [p > 0, q > 0, r > 0, s > 0, t > 0] ∗ GW (324)(23b) p

32.

s > 0]

q

r

∗ In 4.267.31 the restrictions can be somewhat weakened by writing, for example, s > 0, p + s > 0, q + s > 0, r + s > 0, p + q + s > 0, p + r + s > 0, q + r + s > 0, p + q + r + s > 0, in 4.267 31 and 32.

548

33. 34. 35.

36.

Logarithmic Functions

1



1

0



39.6

xp − xp+q −q 1−x



dx Γ(p + q + 1) = ln [p > −1, p + q > −1] BI (127)(19) ln x Γ(p + 1) 0   1 μ dx x −x − x(μ − 1) = ln Γ(μ) [Re μ > 0] WH, BI (127)(18) x − 1 x ln x 0   1 dx (1 − xp ) (1 − xq ) = − ln {B(p, q)} 1−x− 1−x x ln x 0 [p > 0, q > 0] BI (130)(18)   1 p−1 pq−1 x x 1 dx 1 − + = q ln p − q q 1−x 1−x x(1 − x) x (1 − x ) ln x 0

37.

38.



4.268

xq−1 xpq−1 p − 1 p−1 p − 1 p−1 − x − x − 1 − x 1 − xp 1 − xp 2



[p > 0]

dx 1−p 1 = ln(2π) + pq − ln x 2 2

(1 − x ) (1 − x ) − (1 − x) dx = ln B(p, q) x(1 − x) ln x 0

 1 n  n n dx p = (x − 1) (−1)n−k ln(pk + 1) n−k ln x 0 1

p

q





BI (130)(20)

ln p

[p > 0,

q > 0]

BI (130)(22)

[p > 0,

q > 0]

GW (324)(24)

[n > 0,

pn > −1]

2

k=0

GW (324)(19d), BI (123)(12)a



1

40.6 0



1

p n

(1 − x ) dx = 1 − x ln x

(−1)k−1 ln Γ[(n − k)p + 1]

[n > 1,

pn > −1]

[n > 0,

q > 0,

BI (127)(12)

k=0

n  dx ln[q + (n − k)p] = (−1)k k ln x n

n

(xp − 1) xq−1

41.

n 

0

k=0

pn > −q] BI (123)(12)



1

n

(1 − xp ) xq−1

42.6 0

dx = (1 − x) ln x

n 

(−1)k−1 ln Γ[(n − k)p + q]

k=0

[n > 1,

q > 0,

pn > −q] BI (127)(13)



1

n

m

(xp − 1) (xq − 1)

43.10 0

 m m n x dx ln[r + (m − k)q + (n − j)p] = (−1)j (−1)k j k ln x j=0 r−1

k=0

[n ≥ 0, 4.268  1. 0

n 

1

(xp − xq ) (1 − xr ) (ln x)

2

m ≥ 0,

n + m > 0,

r > 0,

pn + qm + r > 0]

BI (123)(16)

dx = (p + 1) ln(p + 1) − (q + 1) ln(q + 1) −(p + r + 1) ln(p + r + 1) + (q + r + 1) ln(q + r + 1) [p > −1,

q > −1,

p + r > −1,

q + r > −1]

GW (324)(26)

4.269

Powers and logarithms



1

2

(xp − xq )

2. 0



dx 2

(ln x)

= (2p + 1) ln(2p + 1) + (2q + 1) ln(2q + 1) − 2(p + q + 1) ln(p + q + 1) 

1

(1 − xp ) (1 − xq ) (1 − xr )

3.

549

0

p > − 12 ,

q > − 12



GW (324)(26a)

dx (ln x)

2

= (p + q + 1) ln(p + q + 1) + (q + r + 1) ln(q + r + 1) + (p + r + 1) ln(p + r + 1) −(p + 1) ln(p + 1) − (q + 1) ln(q + 1) − (r + 1) ln(r + 1) − (p + q + r) ln(p + q + r) [p > −1,

q > −1,

r > −1,

p + q > −1,

p + r > −1,

q + r > −1,

p + q + r > 0] BI (124)(4)

 4.

5.

 dx 1 n k n (1 − xp ) xq−1 = (−1) (pk + q)2 ln(pk + q) 2 k 2 (ln x) 0 k=0 + q, q > 0, p > − n ⎛ ⎞ 

 1 n m    dx p n q m r−1 j n ⎠ k m ⎝ (1 − x ) (1 − x ) x (−1) (−1) 2 = j k (ln x) 0 j=0 n 

1

BI (124)(14)

k=0

×[(m − k)q + (n − j)p + r] ln[(m − k)q + (n − j)p + r] [r > 0, 

1

6.



mq + r > 0,

(q − r)xp−1 + (r − p)xq−1 + (p − q)xr−1

0



np + r > 0,

mq + np + r > 0]

BI (124)(8)

dx (ln x)

2

= (q − r)p ln p + (r − p)q ln q + (p − q)r ln r  7. 0

4.269 1. 2.11

1



[p > 0,

q > 0,

r > 0]

BI (124)(9)

xp−1 xq−1 xr−1 + + + (p − q)(p − r)(p − s) (q − p)(q − r)(q − s) (r − p)(r − q)(r − s)  xs−1 dx p2 ln p q 2 ln q 1 + + = 2 (s − p)(s − q)(s − r) (ln x) 2 (p − q)(p − r)(p − s) (q − p)(q − r)(q − s)  2 2 s ln s r ln r + + (r − p)(r − q)(r − s) (s − p)(s − q)(s − r) [p > 0, q > 0, r > 0, s > 0] BI (124)(16)

√   1 ∞ π (−1)k 1 dx  ln = x 1 + x2 2 (2k + 1)3 0 k=0  1 ∞ √  dx (−1)k  √ = π  2k + 1 1 0 k=0 1 + x2 ln x

BI (115)(33)

BI (133)(2)

550

3. 4.

5.

6.

7.

4.271

Logarithmic Functions

  1 1 1 π ln xp−1 dx = [p > 0] x 2 p3 0   1 p−1 x π  [p > 0] dx = p 1 0 ln x  1 n √  sin t − xn sin[(n + 1)t] + xn+1 sin nt sin kt dx  √ · π = 2 1 − 2x cos t + x k 1 0 k=1 ln x [|t| < π]  1  cos kt √ n−1 cos t − x − xn−1 cos nt + xn cos[(n − 1)t] dx  √ · π = 2 1 − 2x cos t + x k 1 0 k=1 ln x [|t| < π]  v dx  =π [uv > 0] x v u x · ln ln u x 

1

2n

(ln x)

1. 0



1

2.

22n − 1 dx = · (2n)! ζ(2n + 1) 1+x 22n

2n−1

(ln x) 0



1

3.

2n−1

(ln x) 0

 4.

4.271

1

GW (324)(1c)

BI (133)(1)

BI (133)(5)

BI (133)(6) BI (145)(37)

BI (110)(1)

1 − 22n−1 2n dx = π |B2n | 1+x 2n

[n = 1, 2, . . .]

BI (110)(2)

1 dx = − 22n−2 π 2n |B2n | 1−x n

[n = 1, 2, . . .]

BI (110)(5), GW(324)(9a)

dx = ei(p−1)π Γ(p) ζ(p) [p > 1] 1−x  1 ∞  dx (−1)k n n (ln x) = (−1) n! 1 + x2 (2k + 1)n+1 0 k=0  1  dx dx 1 ∞ π 2n+1 2n 2n (ln x) = (ln x) = 2n+2 |E2n | 2 2 1+x 2 0 1+x 2 0  ∞ 2n+1 (ln x) dx = 0 [|b| < 2] 2 0 1 + bx + x  1 dx 22n+1 − 1 2n (ln x) = · (2n)! ζ(2n + 1) [n = 1, 2, . . .] 1 − x2 22n+1 0  ∞ dx 2n (ln x) =0 1 − x2 0   1 dx dx 1 ∞ 1 − 22n 2n 2n−1 2n−1 π |B2n | (ln x) = (ln x) = 2 2 1−x 2 0 1−x 4n 0 p−1

(ln x)

GW (324)(9b)

0

5.

6. 7. 8. 9. 10.

[n = 1, 2, . . .]

BI (110)(11)

GW (324)(10)a

BI (135)(2) BI (110)(12) BI (312)(7)a

BI (290)(17)a, BI(312)(6)a

4.272

Powers and logarithms



1

11.

2n−1

(ln x) 0



1

12.

2n

(ln x) 

2

dx =

22n − 1 2n π |B2n | 2

[n = 1, 2, . . .]

BI (290)(19)a

[n = 1, 2, . . .]

BI (296)(17)a

∞

1

2n+1

(ln x) 0



1 + x2 (1 − x2 )

0

13.

x dx 1 = − π 2n |B2n | 2 1−x 4n

551

 cos 2akπ (cos 2aπ − x) dx = −(2n + 1)! 2 1 − 2x cos 2aπ + x k 2n+2 k=1

∞

14.6 0

[a is not an integer]   x dx d n ν−2 sin(ν − 1)t (ln x) 2 = −π cosec t n a a + 2ax cos t + x2 dν sin νπ [a > 0, 0 < Re ν < 2, ν−1

LI (113)(10)

n

0 < |t| < π] ET I 315(12)



1 p−1 p 1 n x (n) (ln x) dx = − ψ q n+1 1 − x q q 0

 1 p−1 p 1 n x (ln x) dx = n+1 β (n) 1 + xq q q 0

 15. 16.3 4.272

[p > 0,

q > 0]

GW (324)(9)

[p > 0,

q > 0]

GW (324)(10)

 q−1 1 dx ∞ 1 ln  sin kt x = cosec t Γ(q) (−1)k−1 q [|t| < π, q < 1] 2 k 0 1 + 2x cos t + x k=1   

q−1  1 ∞ 1  (1 + x) dx t 1 k−1 cos k − 2 t = sec · Γ(q) (−1) ln x 1 + 2x cos t + x2 2 kq 0 k=1   |t| < π, q < 12  1  μ ∞ 1 xν−1 dx ak sin kt Γ(μ + 1)  = ln x 1 − 2ax cos t + x2 a2 a sin t (ν + k − 1)μ+1 0 

1.

2.

3.9

k=1

[a > 0, 

1

4. 0



[r > 0, ∞

1

7.

−π < t < π]

BI (140)(14)a



r−1 ∞  cos λ − px pk−1 cos kλ 1 q−1 x dx = Γ(r) ln x 1 + p2 x2 − 2px cos λ (q + k − 1)r

(ln x)

p

dx = Γ(1 + p) x2



μ−1 1 1 xν−1 dx = μ Γ(μ) ln x ν 0 

n− 12  1 (2n − 1)!! π 1 ν−1 x dx = ln x (2ν)n ν 0 

Re ν > 0,

LI (130)(5)

k=1

5. 6.

Re μ > 0,

LI (130)(1)

q > 0]

[p > −1]

BI (113)(11) BI (149)(1)

1

[Re μ > 0, [Re ν > 0]

Re ν > 0]

BI (107)(3)

BI (107)(2)

552

Logarithmic Functions



1

8.11 0

4.272



n−1 ν−1

1 x 1  1 −3 1 dx = Γ 3 − p n − q n −3 ln x 1+x n

[Re ν > 0] BI (110)(4)

n−1 1 xν−1 1 dx = (n − 1)! ζ(n, ν) [Re ν > 0] BI (110)(7) ln x 1−x 0

μ−1

 1 n  (−1)k n(n − 1) . . . (n − k + 1) 1 nx (x − 1)n a + ln xa−1 dx = Γ(μ) x x−1 (a + n − k)μ−1 k! 0 

9. 10.

k=0

[Re μ > 0]

n−1 1 1 ln x 0

μ−1  1 1 ln x 0  11.

12.

1 − xm dx = (n − 1)! 1−x

m  k=1

LI (110)(10)

1 kn

LI (110)(9)

 ν  xν−1 dx 1 1 = Γ(μ) = μ Γ(μ) ζ μ, 2 μ 1−x (ν + 2k) 2 2 ∞

k=0

[Re μ > 0, Re ν > 0]



∞ p 1 q  1 x − x−q 1 1 dx = Γ(p + 1) − ln 1 − x2 x (2k + q − 1)p+1 (2k − q − 1)p+1 0 k=1   p > −1, q 2 < 1

r−1 p−1  1 ∞  −s 1 x dx 1 = Γ(r) ln q )s k x (p + kq)r (1 + x 0 k=0 [p > 0, q > 0, r > 0,

BI (110)(13)

 13.

14.

LI (326)(12)a

0 < s < r + 2] GW (324)(11)

n m  1  m 1 1 m (1 + xq ) xp−1 dx = n! ln n+1 k x (p + kq) 0

 15.

k=0

[p > 0,

m n 1   m (−1)k 1 m (1 − xq ) xp−1 dx = n! ln k (p + kq)n+1 x 0

q > 0]

BI (107)(6)

[p > 0,

q > 0]

BI (107)(7)

[p > 0,

q > 0,

 16.

k=0

p−1 q−1 ∞ 1  x dx ak 1 1 = Γ(p) ln x 1 − axq aq p kp 0

 17.

a < 1]

LI (110)(8)

k=1



1

18. 0

 19. 0

1



2− n1  p−1  1 1 1 n 1 q−1 Γ q 1− n − p1− n x dx = −x ln x n−1 n [q > p > 0]

2n−1 p

∞ k x − x−p q−1 |B2k | 1  2pπ 1 x dx = 2n ln q x 1−x p q 2k · (2k − 2n)! k=n + q, p< 2

BI (133)(4)

LI (110)(16)

4.282

Rational functions of ln x and powers



 x p−1  v q−1 dx  v p+q−1 ln ln = B(p, q) ln u x x u u 1 √  √ q qπ e  x dx = √ q e 0 x − (1 + ln x)

4.273

4.274 4.275



1

v

%

1 ln x

1. 0



1

2.

553

 x−

0

&

q−1 −x

p−1

1 1 − ln x

q 

(1 − x)

q−1

dx =

[p > 0,

q > 0,

uv > 0]

[q > 0]

BI (145)(36)

BI (145)(4)

Γ(q) [Γ(p + q) − Γ(p)] Γ(p + q) [p > 0,

dx = − ψ(q) x ln x

q > 0]

[q > 0]

BI (107)(8) BI (126)(5)

4.28 Combinations of rational functions of ln x and powers 4.281



1



1. 0



∞

2. 1

 3. 4. 5. 6. 7.

 1 1 + dx = C ln x 1 − x

BI (127)(15)

1 dx = li(p) x2 (ln p − ln x) p

LA 281(30)

1

xp−1 dx = ±e∓pq Ei (±pq) [p > 0, q > 0] 0 q ± ln x   1 1 xμ−1 + dx = − ψ(μ) [Re μ > 0] ln x 1 − x 0   1  p−1 x xq−1 + dx = ln p − ψ(q) [p > 0, q > 0] ln x 1−x 0   1 1 ln 2 dx 1 = + 2 1−x 2x ln x ln x 2 0   1 1 ln 2π 1 (1 − x) (1 + q ln x) + x ln x q−1 dx = − q − ln Γ(q) + x q− + 2 2 (1 − x) ln x 2 2 0 [q > 0]

4.282 1. 2. 3. 4.



1

1 1 dx = − C 1 − x 4 2 + (ln x) 0

 1 2a + π 1 dx 1 β · = 2 1 + x2 2 2a 4π 0 a + (ln x)  1 1 dx 4−π 2 1 + x2 = 4π 2 0 π + (ln x)

 1 ln x dx 1 1 − ln 2 2 · 1 − x2 = 2 2 2 0 π + (ln x) ln x

4π 2

2

·

LI (144)(11,12) WH

BI (127)(17)

LI (130)(19)

BI (128)(15)

BI (129)(1)

+ π, a>− 2

BI (129)(9)

BI (129)(6)

BI (129)(10)

554

Logarithmic Functions

 5. 6. 7. 8. 9.10 10. 11.

 a , π x dx 1+π + ln + ψ [a > 0] = 2 1 − x2 2 2 2a a π 0 a + (ln x)

 1 ln x x dx 1 1 − C · = 2 1 − x2 2 2 2 0 π + (ln x)  1 1 dx ln 2 2 · 1 + x2 = 4π 2 0 π + 4 (ln x)  1 ln x dx 2−π · = 2 2 2 1−x 16 0 π + 4 (ln x)  1 , √ 1 dx 1 + √ · = 2 − 1 π + 2 ln 2 1 + x2 2 8π 2 0 π + 16 (ln x)  1 √ 1 ln x 1 dx π √ √ + + ln · = − 2 − 1 2 1 − x2 2 32 2 16 16 2 0 π + 16 (ln x) 2k−2  1 ∞ 2π π2  ln x dx =− 4 |B2k | + ,2 1 − x a a 2 0 k=1 a2 + (ln x) 

1

+

0



1

13. 0

1. 2. 3. 4. 5. 6. 7. 8.

ln x

1

12.

4.283

4.283

·

ln x 2

a2 + (ln x)

,2

∞  π 2k−2 x dx π2  = − |B | 2k 1 − x2 4a4 a

BI (129)(13)

BI (129)(7)

BI (129)(11)

BI (129)(8)

BI (129)(12)

BI (129)(4)

BI (129)(16)

k=1

∞ dx xp − x−p sin kpπ 2π  = (−1)k−1 2 2 2 x − 1 q + (ln x) q 2q + kπ



p2 < 1



BI (132)(13)a

k=1

dx x−1 −x = ln 2 − 1 ln x ln x 0

 1 1 1 1 dx ln 2π + − = −1 ln x 1 − x 2 ln x 2 0

 1 1 x x dx ln 2π + + = ln x 1 − x 2 x ln x 2 0 &  1% x 1 1 dx = C − 2 − (1 − x)2 2 (ln x) 0

 1 1 1 dx ln 2 − 1 1 − = + 2 1 − x 2 ln x 2 ln x 2 0

 1 dx 1 1 1+x ln 2π + · − ln x = ln x 2 1 − x ln x 2 0   1 dx 1 −x = −C 1 − ln x x ln x 0 &  1% q q x −1 2 − ln x dx = q ln q − q x (ln x) 0 

BI (129)(14)

1



BI (132)(17)a

BI (127)(20)

BI (127)(23)

GW (326)(8a)

BI (128)(14)

BI (127)(22)

GW (326)(11a)

[q > 0]

BI (126)(2)

4.291

Logarithmic functions and powers

10.

 x+

 dx a 1 = ln + C [a > 0, q > 0] a ln x − 1 x ln x q 0  p−1

 1 1 1+x x 1 ln 2π + dx = − ln Γ(p) + p − ln p − p + ln x 2(1 − x) ln x 2 2 0 

9.

1

1 p−1−

 11.

1 + 1−x

0



1

% −

12. 0



1

0



1



14. 0



1

%

1. 0



1

%

2.

2

(ln x)

p−

1 2

1 q− 2





+

1 1 − 2 ln x



xp−1

dx = ln x



[p > 0]

1 ln 2π − p ln p + p − 2 2 [p > 0]

BI (127)(25)

[p > 0]



  2p−1  dx 1 1 1 3 = − p (ln p − 1) x −1 x + 1− 2 ln x ln x 2 −x ln x

p−1

x

r−1

pq−1

rq−1

px rx + − p 1−x 1 − xr



[p > 0]

1

0

GW (326)(8)

BI (132)(23)a



1 dx ln 2π = (p − r) − q − ln Γ(q) + ln x 2 2



[q > 0]

BI (132)(13)

[q > 0]

BI (126)(3)

xq − 1

q

−

q x (ln x)

3

−

&

q2 2x (ln x)

2

−

11 q3 q3 ln q − q 3 dx = 6 ln x 6 36 [q > 0]



GW (326)(9)

& 3 2 q2 q2 3 − 2 − 2 ln x dx = 2 ln q − 4 q x (ln x) x (ln x)

x (ln x)

4.285

BI (126)(8)

(p − 2)xp − (p − 1)xp−1 3 dx = − ψ(p) + p − 2 (1 − x) 2

xq − 1

4

0



&

1



13.

4.284

555

BI (126)(4)

n−1  xp−1 dx pn−1 −pq 1 e = Ei(pq) − (n − k − 1)!(pq)k−1 n n−1 (n − 1)! (n − 1)!q (q + ln x) k=1



[p > 0,

q < 0]

BI (125)(21)

n

xa (ln x) dx

, we should make the substitution x = et or x = e−t and m l [b ± (ln x) ] then seek the resulting integrals in 3.351–3.356. In integrals of the form

4.29–4.32 Combinations of logarithmic functions of more complicated arguments and powers 4.291



1. 0

1

π2 ln(1 + x) dx = x 12

FI II 483

556

Logarithmic Functions



1

2. 0

 3. 4.

5. 6. 7.7 8. 9. 10. 11. 12. 13.

4.291

π2 ln(1 − x) dx = − x 6

1 ln(1 − x) π2 2 dx = (ln 2) − x 2 12 0  1  1 x dx π2 2 = (ln 2) − ln 1 − 2 x 2 12 0  1 ln 1 + x 2 2 dx = 1 (ln 2)2 − π 1−x 2 12 0  1 ln(1 + x) 1 2 dx = (ln 2) 1+x 2 0  a  ∞  ln(1 + ax) ln u du π  2 ln 1 + a − dx = 2 2 1 + x 4 0 0 1+u  1 ln(1 + x) π dx = ln 2 2 1+x 8 0  ∞ ln(1 + x) π dx = ln 2 + G 1 + x2 4 0  1 ln(1 − x) π dx = ln 2 − G 2 1+x 8 0  ∞ ln(x − 1) π dx = ln 2 1 + x2 8 1  1 π2 1 ln(1 + x) 2 dx = − (ln 2) 12 2 0 x(1 + x)  ∞ π2 ln(1 + x) dx = . x(1 + x) 6 0

FI II 714

1/2

BI (145)(2)

BI (114)(18)

BI (115)(1)

BI (114)(14)a

[a > 0]

GI II (2209)

FI II 157 BI (136)(1)

BI (114)(17) BI (144)(4)

BI (144)(4)

BI (141)(9)a



1

14. 0

a+b 2 ln 2 ln(1 + x) 1 ln + 2 dx = (ax + b)2 a(a − b) b b − a2 1 = 2 (1 − ln 2) 2a

[a = b,

ab > 0]

[a = b] LI (114)(5)a



∞

15. 0

 16.

1

ln(a + x) 0

 17.

a ln ln(1 + x) b dx = (ax + b)2 a(a − b) √ dx 1 = √ arccot a ln[(1 + a)a] 2 a+x 2 a

[ab > 0]

BI (139)(5)

[a > 0]

BI (114)(20)

∞

dx a ln a − b ln b = (b + x)2 b(a − b)  a   ln(1 + ax) 1 dx = arctan a ln 1 + a2 2 1 + x 2 0 ln(a + x)

[a > 0,

b > 0,

a = b]

LI (139)(6)

0

18.

GI II (2195)

4.291

Logarithmic functions and powers

 19. 20.

√ ln(1 + ax) 1 dx = √ arctan a ln(1 + a) [a > 0] 2 1 + ax 2 a 0    1 1 ln(ax + b) 1 (a + b) ln(a + b) − b ln b − a ln 2 dx = 2 a−b 2 0 (1 + x) 

21. 22.

1

[a > 0,

[a > 0, 1

23.

0

BI (139)(9)

[a > 0]



1 dx 1 1 a+b 1−x 1 ln(a + b) − ln b − ln a = 2 2 2 2 (ax + b) (bx + a) a −b a−b ab a b  4 ln 2 + 2 b − a2   a > 0, b > 0, a2 = b2

∞

26. 0

[a > 0, 1−x

2

ln(1 + ax) 0

(1 +

∞

ln(a + x) 0

LI (114)(11)

LI (114)(13)

b 1−x dx 1 ln · = 2 2 2 2 (ax + b) (bx + a) ab (a − b ) a

1

28.



2

ln(1 + x)



b > 0]

LI (114)(12)

2

ln(1 + x)

27.

BI (139)(8)

+π  ,  1 + x2 dx 1 2 ln 1 + a − 2 arctan a · ln a · = a2 + x2 1 + a2 x2 2a (1 + a2 ) 2

1

25.



BI (114)(22)

b > 0]

ln(1 + x) 0



a = b]

23 1 + x2 1 dx = − ln 2 + (1 + x)4 3 72

1

24. 

b > 0,

ln(1 + x) 0



BI (114)(21)

∞

ln(ax + b) 1 [a ln a − b ln b] [a > 0, dx = 2 (1 + x) a − b 0

 ∞ x dx 1 aπ a2 ln(a + x) 2 = 2 (a2 + b2 ) ln b + 2b + b2 ln a (b2 + x2 ) 0 

dx =

2 x2 )

b2 − x2 (b2 +

b > 0]

2

dx =

2 x2 )

LI (139)(14) 2

a a 1 (1 + a) 1 π ln(1 + a) − · ln 2 − · 2 2 2 1+a 2 1+a 4 1 + a2

a2

1 + b2

a ln

bπ b − a 2



[a > −1]

BI (114)(23)

[a > 0,

b > 0]

BI (139)(11)

[a > 0,

∞ x dx 1 aπ a2 2 + 2 ln a ln (a − x) ln b − 2 = a2 + b 2 2b b (b2 + x2 ) 0

b > 0]

BI (139)(12)

[a > 0,

b > 0]

BI (139)(10)

 29. 0

∞

ln2 (a − x)

b2 − x2 2

(b2 + x2 )

dx =

2 a2 + b 2

a ln

a bπ − b 2



 30.

557

558

4.292

Logarithmic Functions



1

1. 0



1

2. 0

3.



a



1

ln (1 ± x) π √ dx = − ln 2 ± 2G 2 2 1−x x ln (1 ± x) π √ dx = −1 ± 2 2 1−x

ln(1 + bx) 1+ √ dx = π ln 2 2 a −x −a

4. 0

4.292

GW (325)(20) GW (325)(22c)

√ 1 − a2 b 2 2

  1 0 ≤ |b| ≤ a

√

√ x ln(1 + ax) 1 − a2 π 1 − 1 − a2 √ + arcsin a dx = −1 + · 2 2 a 1−x √ a   π a2 − 1 = −1 + + ln a + a2 − 1 2a a

BI (145)(16, 17)a, GW (325)(21e)

[|a| ≤ 1] [a ≥ 1] GW (325)(22)



1

5. 0

4.293



π2 1 ln(1 + ax) 1 2 √ − (arccos a) dx = arcsin a (π − arcsin a) = 2 8 2 x 1 − x2 [|a| ≤ 1]

1

xμ−1 ln(1 + x) dx =

1. 0



∞

2.6

1 [ln 2 − β(μ + 1)] μ

∞

3. 4.

5.

6.11 7.

BI (106)(4)a ET I 315(17)

−1 [β(−μ) + ln 2] μ

[Re μ < 0]

xμ−1 ln(1 + x) dx =

π μ sin μπ

[−1 < Re μ < 0]

0



[Re μ > −1]

xμ−1 ln(1 + x) dx =

1



BI (120)(4), GW (325)(21a)

1  (−1)k−1 2n k 0 k=1 % &  1 2n+1  (−1)k 1 2n ln 4 + x ln(1 + x) dx = 2n + 1 k 0 k=1 % &  1 n (−1)n · 4 π  (−1)k 2 ln 2 n− 12 + − x ln(1 + x) dx = 2n + 1 2n + 1 4 2k + 1 0 k=0  ∞ π [−1 < Re μ < 0] xμ−1 ln |1 − x| dx = cot(μπ) μ 0

GW (325)(3)a

2n

1

x2n−1 ln(1 + x) dx =

GW (325)(2b)

GW (325)(2c)

GW (325)(2f)

BI (134)(4), ET I 315(18)



1

xμ−1 ln(1 − x) dx = −

8. 0

 9.7 1

∞

xμ−1 ln(x − 1) dx =

1 1 [ψ(μ + 1) − ψ(1)] = − [ψ(μ + 1) + C] μ μ [Re μ > −1]

ET I 316(19)

[Re μ < 0]

ET I 316(20)

1 [π cot(μπ) + ψ(μ + 1) + C] μ

4.294

Logarithmic functions and powers



∞

10.

xμ−1 ln(1 + γx) dx =

0



∞

0 1

12. 0



1

13. 0



[−1 < Re μ < 0,

|arg γ| < π] BI (134)(3)

11.11 

π μγ μ sin μπ

559

π xμ−1 ln(1 + x) dx = − [C + ψ(1 − μ)] 1+x sin μπ

[−1 < Re μ < 1]

ET I 316(21)

2μ − 1 ln(1 + x) ln 2 + dx = − (1 + x)μ+1 2μ μ 2μ μ2

BI (114)(6)

xμ−1 ln(1 − x) dx = B(μ, ν) [ψ(ν) − ψ(μ + ν)] (1 − x)1−ν

∞

14. 0

[Re μ > 0,

Re ν > 0]

xμ−1 ln(γ + x) dx = γ μ−ν B(μ, ν − μ) [ψ(ν) − ψ(ν − μ) + ln γ] (γ + x)ν [0 < Re μ < Re ν]

4.294



1

1.

ln(1 + x) 0



1

2.

ln(1 + x) 0



ln(1 + x) 0



1

ln(1 + x) 0 1

ln(1 + x) 0 1

ln(1 − x) 0

 1  1  (−1)k−1 1 − x2n dx = 2 ln 2 · − 1+x 2k + 1 j=1 j k

BI (114)(8)

j n−1 2n  1  (−1)j  (−1)k−1 1 − x2n dx = 2 ln 2 · + 1−x 2k + 1 i=1 j k

BI (114)(9)

1

ln(1 − x) 0 ∞

1

j n  (−1)j  1 1 − (−1)n xn dx = 1−x j k j=1

BI (114)(15)

j n  11 1 − xn dx = − 1−x j k j=1

BI (114)(16)

ln2 (1 − x)xp dx =

k=1

2π cot pπ p+1

[−2 < p < −1]

BI (134)(13)a

 (−1)k n! (ln 2) n! r+1 + 2 (r + 1)n+1 (n − k)!(r + 1)k+1 n

n

0

n−k

LI (106)(34)a

k=0

1

n

[ln(1 − x)] (1 − x)r dx = (−1)n 0

k=1

BI (114)(10)

[ln(1 + x)] (1 + x)r dx = (−1)n−1

9.

10.

k=1

j n 2n+1   (−1)j  1 (−1)k−1 1 − x2n+1 dx = 2 ln 2 + 1−x 2k + 1 j k j=1

0



j

k=1

8. 

2n

k=1

7. 

BI (114)(7)

k=1

k=0

6. 

j n 2n+1   1 1 (−1)k−1 1 + x2n+1 dx = 2 ln 2 − 1+x 2k + 1 j k j=1

k=0

5. 

BI (114)(2)

k=0

4. 

[0 < p < 1]

n−1

1

ET I 316(23)

π (p − 1)xp−1 − px−p dx = 2 ln 2 − x sin pπ

k=0

3.

ET I 316(122)

n! (r + 1)n+1

[r > −1]

BI (106)(35)a

560

Logarithmic Functions



1

11.

ln

0



1

12.

2n

(ln x) 0

 13.

1 1 − x2

1

6 0

4.295



n x2q−1 dx =

n! ζ(n + 1, q + 1) 2

∞

  ln μx2 + β

0

[−1 < q < 0]

BI (311)(15)a

 dx  π 2n+2 =− |B2n+2 | ln 1 − x2 x 2(n + 1)(2n + 1)

 m ∞    Γ(m + 1) 1 ln 1 − x2 dx = − ln x n(2n + 1)m+1 n=1

1.

4.295

√  dx π ln = μγ + β √ γ + x2 γ

BI (309)(5)a

[m + 1 > 0,

[Re β > 0,

n + 1 > 0]

Re μ > 0,

|arg γ| < π] ET II 218(27)

 2. 3. 4. 5. 6. 7.

 dx  π ln 1 + x2 2 = − ln 2 x 2 0  ∞  dx  ln 1 + x2 2 = π x 0  ∞   dx 2a  π + ln a ln 1 + x2 = (a + x)2 1 + a2 2a 0  1  dx  π ln 1 + x2 = ln 2 − G 2 1 + x 2 0  ∞  dx  π ln 1 + x2 = ln 2 + G 2 1+x 2 1  ∞   2 ag + bc dx π ln ln a + b2 x2 2 = 2 x2 c + g cg g 0 

8.

1

9. 10. 11. 12. 13. 14.

GW (325)(4c)

[a > 0]

BI (319)(6)a

BI (114)(24) BI (114)(5)

[a > 0,

b > 0,

c > 0,

g > 0] BI (136)(11-14)a

∞

  ln a2 + b2 x2

0

c2

bc dx π = − arctan 2 2 −g x cg ag

   ∞ ln 1 + p2 x2 − ln 1 + q 2 x2 dx = π(p − q) x2 0  1 1 + a2 x2 dx 2 ln = − (arctan a) 2 1 − x2 1 + a 0  1  dx  π2 =− ln 1 − x2 x 12 0  ∞   dx ln2 1 − x2 2 = 0 x 0  1  dx  π ln 1 − x2 = ln 2 − G 2 1+x 4 0  ∞  2  dx π ln x − 1 = ln 2 + G 2 1 + x 4 1 

GW (325)(2g)

[a > 0,

b > 0,

c > 0,

g > 0] BI (136)(15)a



[p > 0,

q > 0]

FI II 645

BI (115)(2)

BI (142)(9)a

GW (325)(17) BI (144)(6)

4.295

Logarithmic functions and powers



∞

15.

  ln2 a2 − x2

0



 dx π  = ln a2 + b2 b2 + x2 b

561

[b > 0]

BI (136)(16)

∞

  b2 − x2 2bπ ln2 a2 − x2 dx = − 2 [b > 0] 2 a + b2 (b2 + x2 ) 0    1   1 π2 1 dx 2 = − (ln 2) ln 1 + x2 x (1 + x2 ) 2 12 2 0  ∞   π2 dx = ln 1 + x2 x (1 + x2 ) 12 0  1   dx ln cos2 t + x2 sin2 t = −t2 1 − x2 0

 ∞   a b2 c c a2 g a dx 2 ln b + π + 2 ln + 2 ln ln a2 + b2 x2 = (c + gx)2 cg a2 g 2 + b 2 c 2 b g g b2 c b 0 [a > 0, b > 0, c > 0, g

16. 17. 18. 19. 20.

BI (136)(20)

BI (114)(25) BI (141)(9)

BI (114)(27)a

> 0] BI (139)(16)a

 21.

1

  ln a2 + b2 x2

0

 22.

∞

11 0



∞

23. 0



∞

24. 0



∞

25. 0

 26. 0

∞

dx (c + gx)2   2a b2 a cb2 − ga2 a2 + b2 2 c c+g ln a + 2 2 arccot + ln ln = − 2 c(c + g) a g + b 2 c2 b b b2 (c + g) a2 g c [a > 0, b > 0, c > 0, g > 0] BI (114)(28)a

    ∞  2 ln 1 + p2 x2 ln p + x2 q + pr π ln dx = dx = r2 + q 2 x2 q 2 + r2 x2 qr r 0 [qr > 0,   ln 1 + a2 x2 dx π = 2 2 2 2 2 2 2 2 b +c x d + g x b g − c 2 d2 a > 0, b > 0,



  ln 1 + a2 x2 x2 dx π = 2 2 2 2 2 2 2 2 b +c x d + g x b g − c 2 d2 a > 0, b > 0,



  ln a2 + b2 x2

dx 2

(c2 + g 2 x2 )

=

π 2c3 g

p > 0]

FI II 745a, BI (318)(1)a, BI (318)(4)a



 g ad ab c ln 1 + − ln 1 + d g b c  c > 0, d > 0, g > 0, b2 g 2 = c2 d2

BI (141)(10)



 b ab ad d ln 1 + − ln 1 + c c g g  c > 0, d > 0, g > 0, b2 g 2 = c2 d2

BI (141)(11)



bc ag + bc − ln g ag + bc [a > 0, b > 0,



  2 x dx bc π ag + bc 2 2 + ln a + b x ln 2 = 2cg 3 g ag + bc (c2 + g 2 x2 ) [a > 0, b > 0,

c > 0,

g > 0] GW (325)(18a)

2

c > 0,

g > 0] GW (325)(18b)

562

Logarithmic Functions



1

27. 0



1

28. 0

 29. 30.6 31. 32. 33.

34.

35. 36.

  π ln 1 + ax2 1 − x2 dx = 2

4.295

√ √ 

1+ 1+a 11− 1+a √ + ln 2 21+ 1+a

[a > 0] √ √ 

  π 1+ 1+a 11− 1+a 2 2 √ − ln 1 + a − ax 1 − x dx = ln 2 2 21+ 1+a √

[a > 0]

BI (117)(7)

 2    dx 1 + 1 − a2 a <1 ln 1 − a2 x2 √ = π ln 2 2 1−x 0  1    dx π 2 ln 1 − a2 x2 √ = − arccos |a| − 2 x 1 − x2 0  1   π dx k = ln K(k) − K (k  ) ln 1 − x2  2 2 2 k 2 (1 − x ) (1 − k x ) 0  1   dx π 1 2 ± 2k K(k)− K (k  ) = ln √ ln 1 ± kx2  BI (120)(8), 2 2 2 2 8 k (1 − x ) (1 − k x ) 0    1 ln 1 − k 2 x2  dx = ln k  K(k) (1 − x2 ) (1 − k 2 x2 ) 0 .  1     1 − k 2 x2 ln 1 − k 2 x2 dx = 2 − k 2 K(k) − (2 − ln k  ) E(k) 2 1−x 0 .  1   1  1 − x2 2 2 ln 1 − k 2 x2 dx = 2 1 + k  − k  ln k  K(k) − (2 − ln k ) E(k) 2 2 1−k x k 0 √  1   a2 − b 2 dx 2π √ √ ln 1 − x2 =√ ln a2 − b 2 a + a2 − b 2 (a + bx) 1 − x2 −1 1

[a > 0, 1 q p    +1 +ψ +1 ln 1 − x2 pxp−1 − qxq−1 dx = ψ 2 2 0

b > 0,

 37.8

√ 1   dx 1+ 1+a 2 √ ln 1 + ax = π ln 2 1 − x2 0  1 , +    μ 1 +1 ln 1 + x2 xμ−1 dx = ln 2 − β μ 2 0  ∞   π ln 1 + x2 xμ−1 dx = μπ 0 μ sin 2 

38. 39. 40.

BI (117)(6)

[p > −2,

a = b]

q > −2]

[a ≥ −1] [Re μ > −2]

BI (119)(1)

LI (120)(11)

BI (120)(12)

BI (120)(14)

BI (119)(27)

BI (119)(3)

BI (119)(7)

BI (145)(15)

BI (106)(15) GW (325)(21b)

BI (106)(12)

[−2 < Re μ < 0] BI (311)(4)a, ET I 315(15)

4.298

Logarithmic functions and powers



∞

41. 0

 xμ−1 dx  ln 1 + x2 1+x π = sin μπ

4.296  1. 2. 3. 4.





 1−μ 2−μ μπ μπ β β ln 2 − (1 − μ) sin − (2 − μ) cos 2 2 2 2 [−2 < Re μ < 1] ET I 316(25)

 dx  π2 t2 = − ln 1 + 2x cos t + x2 x 6 2 0  ∞    2  dx ln a − 2ax cos t + x2 = π ln 1 + 2a|sin t| + a2 2 1+x −∞  ∞   2π cos μt [|t| < π, −1 < Re μ < 0] ln 1 + 2x cos t + x2 xμ−1 dx = μ sin μπ 0  2 

 ∞ 2 a − b2 cos t x + 2ax cos t + a2 x dx 1 2 ln = π − πt + π arctan 2 x2 − 2ax cos t + a2 x2 + b2 2 (a + b2 ) sin t + 2ab 0 1

[a > 0, 4.297  1.

1

0



[a > 0, ln

0



1

3.

ln 0



5.11 6. 7. 8. 9.

BI (145)(28) ET I 316(27)

0 < t < π]

dx ax + b =0 bx + a (1 + x)2

b > 0]

[ab > 0]

1 − x dx π = ln 2 x 1 + x2 8

BI (115)(16) BI (139)(23) BI (115)(5)

1

1 + x dx =G BI (115)(17) 1 − x 1 + x2 0

2  ∞ 1+x π2 dx = ln BI (141)(13) 1−x x (1 + x2 ) 2 0  v 1  v 2 v + x dx = ln ln [uv > 0] BI (145)(33) u+x x 2 u u  ∞ b ln(1 + ax) − a ln(1 + bx) b [a > 0, b > 0] dx = ab ln FI II 647 2 x a 0  1 dx 1 + ax √ ln = π arcsin a [|a| ≤ 1] GW (325)(21c), BI (122)(2) 1 − ax x 1 − x2 0

 v 1 + ax dx u π   = F arcsin av, ln 1 − ax v v (x2 − u2 ) (v 2 − x2 ) u ln

 10.8

b > 0,

BI (114)(34)

  dx ax + b 1 a+b − a ln a − b ln b ln = (a + b) ln bx + a (1 + x)2 a−b 2

∞

2.

4.

563

PV 0

1

! ! !a + y ! π2 !  dy ln !! = ! a − y y 1 − y2 2

[|av| < 1] [0 < a ≤ 1]

BI (145)(35)

564

4.298  1.

Logarithmic Functions

4.298

ln

ln 2 1 1 + x2 x2n−1 1 dx = + 2− β(2n + 1) x 1+x 2n 4n 2n

BI (137)(1)

ln

ln 2 1 1 + x2 x2n 1 dx = + 2− β(2n + 1) x 1+x 2n 4n 2n

BI (137)(3)

ln

ln 2 1 1 + x2 x2n−1 1 dx = + 2− β(2n + 1) x 1−x 2n 4n 2n

BI (137)(2)

ln

ln 2 1 1 + x2 x2n 1 dx = − − 2+ β(2n + 1) x 1−x 2n 4n 2n

BI (137)(4)

∞

0



∞

2. 0



∞

3. 0



∞

4. 0

 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

∞

1 1 + x2 x2n−1 ln 2 1 + 2− β(2n + 1) dx = 2 x 1+x 2n 4n 2n 0    1 n−1  1 (−1)k 1 + x2 2n 1 nπ x dx = + ln 2 − +2 (−1) ln x 2n + 1 2 2n + 1 2n − 2k − 1 0 k=0    1 n−1  (−1)k 1 + x2 2n−1 1 1 n+1 n+1 x + (−1) (−1) ln dx = ln 2 + ln 2 − x 2n 2n k 0 k=1  1 1 + x2 dx π ln = ln 2 2 x 1 + x 2 0  ∞ 2 dx 1+x ln = π ln 2 x 1 + x2 0  ∞ 1 + x2 dx ln =0 x 1 − x2 0  1 1 − x2 dx π ln = ln 2 2 x 1 + x 4 0  ∞ 2 dx 1+x 3π ln 2 ln = 2 x+1 1+x 8 1  1 1 + x2 dx 3π ln 2 − G ln = 2 x + 1 1 + x 8 0  ∞ 1 + x2 dx 3π ln 2 + G ln = 2 x − 1 1 + x 8 1  1 1 + x2 dx 3π ln 2 ln = 2 1 − x 1 + x 8 0  ∞ 1 + x2 x dx π2 ln = x2 1 + x2 12 0  ∞ 2 2 2 dx ag + bc a +b x π ln [a > 0, b > 0, c > 0, ln = 2 2 2 2 x c +g x cg c 0 ln

 18.

BI (294)(8)

BI (294)(9)a

BI (115)(7) BI (137)(8) BI (137)(9) BI (115)(9) BI (144)(8) BI (115)(18) BI (144)(9) BI (115)(19) BI (138)(3)

g > 0]

BI (138)(6, 7, 9, 10)a ∞

ln 0

BI (137)(10)

ag dx a2 + b2 x2 1 arctan = x2 c2 − g 2 x2 cg bc

[a > 0, b > 0, c > 0, g > 0] BI (138)(8, 11)a

4.311

Logarithmic functions and powers

 19. 20. 21.

23. 4.299 1.

∞

1 + x2 x2 dx π = (ln 4 − 1) 2 2 2 x 4 (1 + x ) 0

  1 2 1−x ln2 1 − x2 dx = π x2 0  1  1 ∞ 1 + 2x cos t + x2 dx t2 1 + 2x cos t + x2 dx = = − ln ln (1 + x)2 x 2 0 (1 + x)2 x 2 0 ln

 22.

BI (139)(21)

FI II 643a

[|t| < π] ∞

1 + 2x cos t + x2 p−1 2π (1 − cos pt) x dx = − 2 (1 + x) p sin pπ 0

 1 π−t dx 1 + x2 sin t √ ln = π ln cot 1 − x2 sin t 1 − x2 4 0 ln

BI (115)(23), BI (134)(15)

[0 < |p| < 1,

|t| < π]

[|t| < π]

  (x + 1) x + a2 dx 2 = (ln a) [a > 0] 2 (x + a) x 0    1 √ (1 − ax) 1 + ax2 dx 1 ln = √ arctan a ln(1 + a) 2 2 1 + ax 2 a (1 − ax2 ) 0 



∞



1

3.

ln 0

 4. 0

11

2. 3. 4. 5. 6.8

  1 − a2 x2 1 + ax2 (1 − 

1

ln

4.311

2 ax2 )

(x + 1) x + a (x + a)2

2

BI (134)(14)

[a > 0]

BI (115)(25)

√ dx 1 = √ arctan a ln(1 + a) 1 + ax2 a [a > 0]



BI (115)(26)

2

xμ−1 dx =

π (aμ − 1) μ sin μπ

  π cosec πn ln (1 + xn ) dx = xn n−1 0  ∞   dx 2π ln 1 + x3 = √ ln 3 2 1 − x + x 3 0  ∞  dx  π2 π √ ln 3 − ln 1 + x3 = 1 + x3 9 3 0  ∞  x dx  π2 π √ ln 3 + ln 1 + x3 = 1 + x3 9 3 0  ∞  1−x  2 ln 1 + x3 dx = − π 2 3 1+x 9 0 √ !  ∞! 3! ! !1 − x ! dx = − π 3 ! a3 ! x3 6a2 0 

BI (134)(17)

GW (325)(21d)

ln

2.

1.

565

[a > 0,

Re μ > 0]

BI (134)(16)

∞

n = 2, 3, . . . LI (136)(8) LI (136)(6) LI (136)(7) BI (136)(9)

566

4.312

Logarithmic Functions



∞

1.

ln

π2 1 + x3 dx π √ ln 3 + = x3 1 + x3 9 3

BI (138)(12)

ln

π2 1 + x3 x dx π = √ ln 3 − 3 3 x 1+x 9 3

BI (138)(13)

0



∞

2. 0

4.313 1. 2.



∞

 dx  ln x ln 1 + a2 x2 2 = πa (1 − ln a) [a > 0] x 0  

 ∞    dx  b b ln 1 + c2 x2 ln a2 + b2 x2 2 = 2π c + ln(b + ac) − ln b − c ln c x a a 0 [a > 0, b > 0, c > 0] 

∞

3. 0

 

ln x ln

b a2 + 2bx + x2 dx = 2π ln a arcsin 2 2 a − 2bx + x x a

[a ≥ |b|]

BI (134)(25)

[a > 0]

BI (141)(7)

[a > 0]

LI (141)(8)

∞

0 ∞

7.

(ln a) x ln x − x − a dx = 2 (x + a) x 2(a − 1) 2

ln2 (1 − x)

0



ln(1 + ax) 0



π 2 + (ln a) x ln x − x − a dx = (x + a)2 x 1+a ∞

1

1.11

 ak p + k xp−1 − xq−1 dx = (−1)k+1 ln ln x k q+k k=1

∞

2. 0



(q − 1)x 1 1 + − (1 + x)2 x + 1 (1 + x)q



[|a| < 1,

3.

x ln x + 1 − x 2

ln(1 + x)dx = ln

4 π

x (ln x)  

 1 ln 1 − x2 dx q+π q π π  = − ln Γ ln 2q + ln − 1 + 2 q π 2q π 0 x q 2 + (ln x) 0

4.

1

p > 0,

q > 0]

BI (123)(18)

dx = ln Γ(q) x ln(1 + x) [q > 0]



BI (134)(24)

2

ln(1 + x) 

b > 0]

BI (134)(22, 23)a

[a > 0,

0



a + bc > 0]

1 + a2 x2 dx bb = π(a − b) + π ln 1 + b2 x2 x2 aa

∞

6.

[a > 0, ln x ln

0

5.

BI (134)(18)

BI (134)(20, 21)a

 

  a + bc a b2 dx 2 2 2 ln(a + bc) − ln a − c ln 1 + c x ln a + 2 = 2π x x2 b b

∞

4.

4.314

4.312

[q > 0]

BI (143)(7) BI (126)(12)

LI (327)(12)a

4.317

4.315

Logarithmic functions and powers



1

1.

ln(1 + x) (ln x)

n−1

ln(1 + x) (ln x)

2n

ln(1 − x) (ln x)

n−1

ln(1 − x) (ln x)

2n

0



1

2. 0



1

3. 0



1

4. 0

4.316



1

1. 0

 2. 0

4.317 1. 2.

3.

4.

5. 6. 7. 8. 9.

567



dx 1 n−1 = (−1) (n − 1)! 1 − n ζ(n + 1) x 2

BI (116)(3)

22n+1 − 1 dx = π 2n+2 |B2n+2 | x (2n + 1)(2n + 2)

BI (116)(1)

dx = (−1)n (n − 1)! ζ(n + 1) x

BI (116)(4)

22n dx =− π 2n+2 |B2n+2 | x (n + 1)(2n + 1)

BI (116)(2)



p ∞  1 dx ak 1 = − p+1 Γ(p + 1) ln (1 − axr ) ln x x r k p+2 k=1

1



  1 ln 1 − 2ax cos t + a2 x2 ln x

p

[p > −1, dx = −2 Γ(p + 1) x

∞  k=1

a < 1,

r > 0]

ak cos kt k p+2

√ 1 + x2 + a dx √ ln √ = π arcsin a [|a| < 1] 2−a 1 + x2 1 + x 0 √  1 √ 1 1 − a2 x2 − x 1 − a2 dx 2 = (arcsin a) ln 1−x x 2 0 v−t √  1 cos dx 1 + cos t 1 − x2 2 √ = π cot t ln v+t 1 − cos t 1 − x2 x2 + tan2 v 0 sin 2   √  1 π2 x + 1 − x2 x dx √ ln2 = 2 2 x − 1 − x2 1 − x 0  1   √ √ dx π 1 = ln(4k) K(k) + K (k  ) ln 1 + kx + 1 − kx  2 2 2 4 8 (1 − x ) (1 − k x ) 0  1   √ √ 3 1 dx = ln(4k) K(k) + π K (k  ) ln 1 + kx − 1 − kx  2 2 2 4 8 (1 − x ) (1 − k x ) 0  1    dx π 1 = ln k K(k) + K (k  ) ln 1 + 1 − k 2 x2  2 4 (1 − x2 ) (1 − k 2 x2 ) 0  1    dx 3 1 = ln k K(k) − π K (k  ) ln 1 − 1 − k 2 x2  2 ) (1 − k 2 x2 ) 2 4 (1 − x 0 √  1  2  1 + p 1 − x2 dx √ = π arcsin p p <1 ln 2 1 − x 1−p 1−x 0 

BI (116)(7) LI (116)(8)

∞

BI (142)(11)

BI (115)(32)

BI (115)(30)

BI (115)(31)

BI (121)(8)

BI (121)(9)

BI (121)(6)

BI (121)(7)

BI (115)(29)

568

Logarithmic Functions

√ dx 1 + q 1 − k2 x2  √ = π F (arcsin q, k  ) ln 2 2 2 1−q 1−k x (1 − x ) (1 − k 2 x2 ) 0  2  q <1 ! √  ∞ !! ! π2 ! 1 + 2 1 + x2 ! dx √ √ ln ! = ! ! 1 − 2 1 + x2 ! 1 + x2 3 −∞

 10.

11.10 4.318

4.318



1

1

1. 0



BI (122)(15)

  q ln q  ln (1 − xq ) dx q  q = π ln Γ +1 − + ln −1 2 2π 2 2π 2π 1 + (ln x) x %

∞

ln (1 + xr )

2. 0

xq − xp (p − r)xp − (q − r)xq + 2 ln x (ln x)

&

[q > 0]

BI (126)(11)

 pπ qπ dx cot = r ln tan xr+1 2r 2r [p < r,

q < r]

BI (143)(9)

In integrals containing ln (a + bxr ), it is useful to make the substitution xr = t and then to seek the resulting integral in the tables. For example,  ∞  1 ∞ p −1 π xp−1 ln (1 + xr ) dx = t r ln(1 + t) dt = (see 4.293 3) pπ r 0 0 p sin r 4.319    ∞  dx  1 −2aπx ln 2aπ + a (ln a − 1) − ln Γ(a + 1) ln 1 − e = −π 1. 1 + x2 2 0 [a > 0] 

 ∞  dx  1 ln 1 + e−2aπx = π ln Γ(2a) − ln Γ(a) + a (1 − ln a) − 2a − ln 2 1 + x2 2 0

 2.



∞

3.

ln 0

[a > 0]  b > −1, a

a p a + be−px dx = ln ln a + be−qx x a+b q



BI (354)(6)

BI (354)(7)

pq > 0 FI II 635, BI (354)(1)

4.321



∞

x ln cosh x dx = 0

1. −∞  ∞

2.

ln cosh x −∞

4.322



1.11

dx =0 1 − x2

π

x ln sin x dx = 0

 2.

0

∞

BI (358)(2)a

1 2



π

x ln cos2 x dx = − 0

ln sin2 ax π 1 − e−2ab ln dx = b2 + x2 b 2

BI (138)(20)a

π2 ln 2 2

BI (432)(1, 2) FI II 643

[a > 0,

b > 0]

GW (338)(28b)

4.324

Logarithmic functions and powers



∞

3. 0



∞

4. 0



∞

5.11 0



∞

6. 0



ln cos2 ax π 1 + e−2ab ln dx = b2 + x2 b 2

[a > 0,

b > 0]

GW (338)(28a)

ln sin2 ax π2 + aπ dx = − b2 − x2 2b

[a > 0,

b > 0]

BI (418)(1)

ln cos2 ax dx = ∞ b2 − x2

BI (418)(2)

ln cos2 x dx = −π x2

FI II 686

π/4

7.7 0



% & ∞  π μ  ζ(2k) 1 2 ln sin xxμ−1 dx = − ln 2 + − 2μ 4 μ 42k−1 (μ + 2k) k=1

%

π/2

ln sin xxμ−1 dx = −

8.7 0



[Re μ > 0] &

LI (425)(1)

[Re μ > 0] & ζ(2k) 42k−1 (μ + 2k)

LI (430)(1)

[Re μ > 0]

LI (430)(2)

 ζ(2k) 1  π μ 1 −2 μ 2 μ 4k (μ + 2k) ∞

k=1

%

π/2

ln (1 − cos x) xμ−1 dx = −

9. 0



569

∞

10.

  ln 1 ± 2p cos βx + p2

0

1  π μ 2 − μ 2 μ

∞  k=1

 dx π  = ln 1 ± pe−βq q 2 + x2 q  π  = ln p ± e−βq q

 

p2 < 1 p2 > 1

  FI II 718a

4.323



π

x ln tan2 x dx = 0

1.11

BI (432)(3)

0

 2. 3. 4.324

∞

ln tan2 ax π dx = ln tanh ab 2 2 b +x b 0

2  ∞ 1 + tan x π2 dx = ln 1 − tan x x 2 0 



∞

ln

1. 0

 2.

∞

ln 0

1 + sin x 1 − sin x

2

[a > 0,

b > 0]

GW (338)(26)

dx = π2 x

q2 1 + 2a cos px + a2 dx = ln(1 + a) ln 2 1 + 2a cos qx + a2 x

p 2 1 q = ln 1 + ln 2 a p

GW (338)(28c)

GW (338)(25)

[−1 < a ≤ 1] [a < −1 or a ≥ 1] GW (338)(27)

570

Logarithmic Functions



∞

3.

  ln a2 sin2 px + b2 cos2 px

0

4.325

π dx = [ln (a sinh cp + b cosh cp) − cp] c2 + x2 c [a > 0, b > 0, c > 0,

p > 0] GW (338)(29)

4.325



1

1.3

ln ln 0



∞  1 dx 1 ln k 2 = −C ln 2 + = −C ln 2 + 0.159868905 · · · = − (ln 2) (−1)k x 1+x k 2 k=2

GW (325)(25a)



∞  1 dx (−1)k −ikλ e ln ln = (C + ln k) GW (325)(26) iλ x x+e k 0 k=1



  1  ∞ 1 dx 1 dx 1 1 π ln − C ln ln = ln ln x = ψ + ln 2π = x (1 + x)2 (1 + x)2 2 2 2 2 0 1 

2.

3.

√  

 ∞ 2π Γ 34 1 dx dx π   ln ln = ln ln x = ln x 1 + x2 1 + x2 2 Γ 14 0 1 √  

 1  ∞ 3 2π Γ 23 1 dx dx π   ln ln = ln ln x = √ ln x 1 + x + x2 1 + x + x2 Γ 13 3 0 1 

  1  ∞ 1 1 dx dx 2π 5 √ ln 2π − ln Γ ln ln = ln ln x = 2 2 x 1 − x + x 1 − x + x 6 6 3 0 1 

4. 5. 6.

1

BI (147)(7)

1

BI (148)(1)

BI (148)(2)

BI (148)(5)



1 t

 1  ∞ + (2π) Γ 1 dx dx π 2 2π

ln ln ln = ln ln x = 2 2 1 t x 1 + 2x cos t + x 1 + 2x cos t + x 2 sin t 0 1 − Γ 2 2π



t/π

7.

BI (147)(9)

 8. 0

 9.

1

1 1 ln ln xμ−1 dx = − (C + ln μ) x μ

∞

ln ln x 1

[Re μ > 0]

BI (147)(1)

xn−2 dx 1 + x2 + x4 + · · · + x2n−2



n+k π π π kπ 2n

tan ln 2π + ln = (−1)k−1 sin k 2n 2n n n k=1 Γ 2n

n−1 n−k Γ 2 π π  kπ π n

tan ln π + ln (−1)k−1 sin = k 2n 2n n n k=1 Γ n n−1 

Γ

[n is even]

[n is odd] BI (148)(4)

4.331

Logarithms and exponentials



1

10.

ln ln 0



1 x



dx  (1 + x2 )

ln

1 x

4.326



1

ln (a − ln x) xμ−1 dx = 0

4.327

∞ √  (−1)k+1 √ π [ln(2k + 1) + 2 ln 2 + C] 2k + 1 k=0

1 [ln a − eaμ Ei(−aμ)] μ

2μ−1  1 1 2 e ln 2 ln 1 − 1 x dx = − [Ei(−μ)] x ln x 2 0 

1.

1

, + 2 ln a2 + (ln x)

0

  2 Γ 2a+3π dx π π 4π  2a+π  + ln = π ln 2 1+x 2 2 Γ 4π

  1 + , dx 2 Γ a+3π π 2 2 4π  a+π  + ln π ln a + 4 (ln x) = π ln 2 1+x 2 Γ 4π 0

 2.

BI (147)(4)

1

1.

2.

dx √ (1 + x2 ) ln x

μ−1  1 x dx π  [Re μ > 0] ln ln = − (C + ln 4μ) x μ 1 0 ln x

μ−1  1 1 1 1 ln ln xν−1 dx = μ Γ(μ) [ψ(μ) − ln(ν)] ln x x ν 0 

12.

ln ln x 1

=

11.

∞

=

571



∞

3. 0

BI (147)(3)

[Re μ > 0,

Re ν > 0]

[Re μ > 0,

a > 0]

BI (147)(2)

BI (107)(23)

[Re μ > 0]

BI (145)(5)

+ π, a>− 2

BI (147)(10)

[a > −π] , + 2 2 ln a2 + (ln x) xμ−1 dx = [− cos aμ ci(aμ) − sin aμ si(aμ) + ln a] μ [a > 0,

Re μ > 0]

BI (147)(16)a

GW (325)(28)

If the integrand contains a logarithm whose argument also contains a logarithm, for example, if the integrand contains ln ln x1 , it is useful to make the substitution ln x = t and then seek the transformed integral in the tables.

4.33–4.34 Combinations of logarithms and exponentials 4.331



∞

1.

e−μx ln x dx = −

1 (C + ln μ) μ

[Re μ > 0]

BI (256)(2)

e−μx ln x dx = −

1 Ei(−μ) μ

[Re μ > 0]

BI (260)(5)

0

 2.

1

∞

572

Logarithmic Functions



1

eμx ln x dx = −

3. 0

4.332 1. 2.

1 μ



−1 dx x

1 μx

0

e

  1 2π 5 ln x dx =√ ln 2π − ln Γ x −x −1 6 3 6 0 e +e %   &  ∞ 2 √ Γ 3 π ln x dx   2π = √ ln x −x +1 Γ 13 3 0 e +e 

4.332

[μ = 0]

GW (324)(81a)

∞

(cf. 4.325 6)

BI (257)(6)

(cf. 4.325 5)

BI (257)(7)a, LI (260)(3)

 2 π 1 [Re μ > 0] e−μx ln x dx = − (C + ln 4μ) 4 μ 0  ∞  ∞ ln x dx C + ln 4k kπ 1 π √ sin 4.334 = (−1)k 2 2 x −x 2 3 3 +1+e k 0 e 

∞

4.333

BI (256)(8), FI II 807a BI (357)(13)

k=1

4.335  1. 2. 3.7

  1 π2 2 + (C + ln μ) [Re μ > 0] μ 6 0  √   ∞ π π2 2 2 −x2 e (ln x) dx = (C + 2 ln 2) + 8 2 0    ∞ 1 π2 3 3 (C + ln μ) − ψ  (1) e−μx (ln x) dx = − (C + ln μ) + μ 2 0 ∞

4.336 1.7

 0 ∞

2. 4.337  1.

4.7 5.∗

FI II 808 MI 26

e dx = −0.154479567 ln x

e−μx dx 2

∞

ET I 149(13)

∞ −x

π 2 + (ln x)

0

3.

2

PV 

2.

e−μx (ln x) dx =

= ν  (μ) − eμ

BI (260)(9)

[Re μ > 0]

MI 26

 1 ln β − eμβ Ei(−βμ) [|arg β| < π, Re μ > 0] BI (256)(3) μ 0

 ∞ 1 μ μ e−μx ln(1 + βx)dx = − e β Ei − ET I 148(4) [|arg β| < π, Re μ > 0] μ β 0  ∞  1 ln a − e−aμ Ei(aμ) e−μx ln |a − x| dx = [a > 0, Re μ > 0] BI (256)(4) μ 0 ! !  ∞ ! β !   ! dx = 1 e−βμ Ei(βμ) e−μx ln !! [Re μ > 0] MI 26 ! β−x μ 0 %

ζ−μ

ζ−μ−k &  ∞ ζ   (−1)ζ−μ−1 1/a ζ! 1 1 ζ −x ln(1 + ax)x e dx = e Ei − (k − 1)! − + (ζ − μ)! aζ−μ a a 0 μ=0 e−μx ln(β + x)dx =

k=1

4.338  1. 0

∞

  2 e−μx ln β 2 + x2 dx = [ln β − ci(βμ) cos(βμ) − si(βμ) sin(βμ)] μ [Re β > 0,

Re μ > 0]

BI (256)(6)

4.352

Logarithms, exponentials, and powers



∞

2. 0

4.341

1. 2. 3.11

   2 2 ln β − eβμ Ei(−βμ) − eβμ Ei(βμ) e−μx ln2 x2 − β 2 dx = μ

[Im β > 0, Re μ > 0] ! !  ∞ !x + 1!   ! dx = 1 e−μ (ln 2μ + γ) − eμ Ei(−2μ) e−μx ln !! ! x−1 μ 0 [Re μ > 0] √ √  ∞ x + ai + x − ai π √ [H0 (aμ) − Y 0 (aμ)] dx = e−μx ln 4μ 2a 0 [a > 0, Re μ > 0]

4.339

4.342

573

BI (256)(5)

MI 27

ET I 149(20)

  1 1 + ln 2 − 2 β(2n + 1) e ln (sinh x) dx = 2n n 0     ∞ μ 1 1 e−μx ln (cosh x) dx = β − [Re μ > 0] μ 2 μ 0   ∞  μ  1 1 μ −μx e [ln (sinh x) − ln x] dx = ln − − ψ μ 2 μ 2 0

ET I 165(32)

[Re μ > 0]

ET I 165(33)





∞

π

4.343

−2nx

BI (256)(17)

    eμ cos x ln 2μ sin2 x + C dx = −π K 0 (μ)

WA 95(16)

0

4.35–4.36 Combinations of logarithms, exponentials, and powers 4.351



1

1.

(1 − x)e−x ln x dx =

0

 2.

0



1

e

∞

1. 0



∞

2. 0

 3. 0

BI (352)(2)

∞ −μx

1



BI (352)(1)

  1 eμx μx2 + 2x ln x dx = 2 [(1 − μ)eμ − 1] μ

3. 4.352

1−e e

∞

1 ln x 2 dx = eμ [Ei(−μ)] 1+x 2

xν−1 e−μx ln x dx =

1 Γ(ν) [ψ(ν) − ln μ] μν

[Re μ > 0]

[Re μ > 0,

  1 n! 1 1 xn e−μx ln x dx = n+1 1 + + + · · · + − C − ln μ μ 2 3 n

NT 32(10)

Re ν > 0] BI (353)(3), ET I 315(10)a

[Re μ > 0] 

 √ (2n − 1)!! 1 1 1 n− 12 −μx x e ln x dx = π 2 1 + + + ··· + − C − ln 4μ 1 3 5 2n − 1 2n μn+ 2 [Re μ > 0]

ET I 148(7)

ET I 148(10)

574

Logarithmic Functions



∞

4.

4.353

xμ−1 e−x ln x dx = Γ (μ)

[Re μ > 0]

GW (324)(83a)

(x − ν)xν−1 e−x ln x dx = Γ(ν)

[Re ν > 0]

GW (324)(84)

[Re μ > 0]

BI (357)(2)

0

4.353



∞

1. 0



∞

2. 0





 (2n − 1)!! π 1 n− 12 −μx e ln x dx = μx − n − x 2 (2μ)n μ

1

(μx + n + 1)xn eμx ln x dx = eμ

3. 0

n 

(−1)k−1

k=0

n! n! + (−1)n n+1 k+1 (n − k)!μ μ [μ = 0]

4.354 1.



∞

6 0

GW (324)(82)

∞

 (−1)k−1 xν−1 ln x dx = Γ(ν) [ψ(ν) − ln k] ex + 1 kν k=1 1 2 = − (ln 2) 2

[Re ν > 0] [for ν = 1] GW (324)(86a)



∞

2.7

∞

3. ∞

4.

0



∞

1. 0 ∞

2. 0

 3. 0

+ 1)

∞

(−1) (k − 1) [ψ(ν) − ln k] kν k

xν−1 ln x dx = Γ(ν) x2n−1 ln x dx =

xν−1 ln x n Γ(ν) n dx = (−1) x (n − 1)! (e + 1)

2 −μx2

∞  k=1

(x − 2n)ex − 2n

x e 

2

(ex + 1) ∞

∞  k=2

2

5.

4.355

+ 1)

dx = Γ(ν)

(x − ν)ex − ν

0



ln x 2

(ex

0



x

(ex

0



ν−1

(−1)k−1 kν

[Re ν > 1]

GW (324)(86b)

[Re ν > 0]

GW (324)(87a)

[n = 1, 2, . . .]

GW (324)(87b)

22n−1 − 1 2n π |B2n | 2n

∞  k=n

(−1)k (k − 1)! [ψ(ν) − ln k] (k − n)!k ν

1 (2 − ln 4μ − C) ln x dx = 8μ



π μ

  2 ν 1 + x μx2 − νx − 1 e−μx +2νx ln x dx = 4μ 4μ  2  2 (n − 1)! μx − n x2n−1 e−μx ln x dx = 4μn



[Re ν > 0]

GW (324)(86c)

[Re μ > 0]

BI (357)(1)a

π exp μ



ν2 μ



 ν 1+Φ √ μ

[Re μ > 0]

BI (358)(1)

[Re μ > 0]

BI (353)(4)

4.356

Logarithms, exponentials, and powers



∞

4. 0

  2 (2n − 1)!! 2μx2 − 2n − 1 x2n e−μx ln x dx = 2(2μ)n



575

π μ [Re μ > 0]

4.356 1. 2.



BI (353)(5)

+  x a , dx + ln x = 2 ln a K 0 (2μ) exp −μ [a > 0, Re μ > 0] GW (324)(91) a x x 0

 ∞   1 b exp −ax − ln x 2ax2 − (2n + 1)x − 2b xn− 2 dx x 0 n  ∞ b 2 π −2√ab  (n + k)! e =2  √ k a a k=0 (n − k)!(2k)!! 2 ab 

∞

∞

3.

exp −ax −

0

b x



[a > 0,

b > 0]

BI (357)(4)

 dx  ln x 2ax2 + (2n − 1)x − 2b n+ 3 x 2 ∞ a n π √  2 e−2 ab =2 b a

(n + k − 1)!  √ k k=0 (n − k − 1)!(2k)!! 2 ab

[a > 0,

b > 0]

BI (357)(11)

[a > 0,

b > 0]

GW (324)(92c)

For n = 0: 

 ∞ π −2√ab b 2ax2 − x − 2b √ e dx = 2 5. exp −ax − ln x x a x x 0 [a > 0,

b > 0]

For n = 12 :  ∞  √  ax2 − b a ln x exp −ax − dx = 2 K 4. 0 2 ab x x2 0

BI (357)(7), GW(324)(92a)

For n = −1: √ 

 ∞ 1 + 2 ab π −2√ab b 2ax2 − 3x − 2b √ e dx = exp −ax − 6. ln x x a a x 0 [a > 0, b > 0]  7.9 0

∞

exp −ax −

b x



ln x a −

b x2



LI (357)(6), GW (324)(92b)

 √ dx = K 0 2 ab [a > 0,

b > 0]

576

8.

9

9.9

Logarithmic Functions



∞



∞









4.357

  3 ln x 2ax2 − (2n + 1)x − 2b xn− 2 dx 0 (2n+1)/4  √ b K n+ 12 2 ab =4 a n2  n b π −2√ab  (n + k)! e =2  √ k a a k=0 (n − k)!(2k)!! 2 ab [n = 0, 1, . . . , a > 0, b > 0]

 ∞   dx  b exp −ax − ln ax2 − b cos (α ln x) + αx sin (α ln x) 2 x x 0  √   = 2 cos α ln b/a K iα 2 ab

10.9

b exp −ax − x

exp −ax −

0

b x

b > 0, −∞ < α < ∞]  dx ax2 − b sin (α ln x) − αx cos (α ln x) 2 x   √  = 2 sin α ln b/a K iα 2 ab [a > 0,

ln x





[a > 0, b > 0, −∞ < α < ∞]  



α/2 ∞  √ b b α b q xα ln x a − − 2 exp −ax − K α 2 ab dx = 2 x x x a 0 

11.9

[a > 0, 4.357



∞

1. 0

3.

4.358 1.

1 + x4 exp − 2ax2



−∞ < α < ∞]

√ 2a3 π 1 + ax2 − x4 dx = − √ ln x 2 x 2ae

[a > 0] √

∞ 2a3 π 1 + x4 x4 + ax2 − 1 √ [a > 0] exp − dx = ln x 2 4 2ax x 2ae 0 √

 ∞ 1 + x4 (1 + a) 2a3 π x4 + 3ax − 1 √ exp − dx = ln x 2ax2 x6 2ae 0

BI (357)(8)

[a > 0]

BI (357)(10)



 2.



b > 0,



∞

6 1

 0

3.9 0

m

dx =

 ∂ m  −ν μ Γ(ν, μ) ∂ν m

[m = 0, 1, . . . ,

Re μ > 0,

Re ν > 0] MI 26

∞

2. 

xν−1 e−μx (ln x)

BI (357)(9)

∞

 Γ(ν)  2 2 xν−1 e−μx (ln x) dx = ν [ψ(ν) − ln μ] + ζ(2, ν) μ [Re μ > 0, Re ν > 0]   Γ(ν) 3 3 xν−1 e−μx (ln x) dx = ν [ψ(ν) − ln μ] + 3 ζ(2, ν) [ψ(ν) − ln μ] − 2 ζ(3, ν) μ [Re μ > 0,

Re ν > 0]

MI 26

MI 26

4.364

Logarithms, exponentials, and powers

 4.

∞

7

ν−1 −μx

x

e

0

Γ(ν) (ln x) dx = ν

577

4

4

[ψ(ν) − ln μ] + 6 ζ(2, ν) [ψ(ν) − ln μ]

2

 −8 ζ(3, ν) [ψ(ν) − ln μ] + 3 [ζ(2, ν)] + 6 ζ(4, ν) 2



[Re μ > 0, ∞

5.3

 ∂ n  −ν μ Γ(ν) n ∂ν

xν−1 e−μx (ln x) dx = n

0

4.359



∞

1.

e−μx

0

[n = 0, 1, 2, . . .]

xp−1 − xq−1 1 dx = [λ(μ, p − 1) − λ(μ, q − 1)] ln x μ [Re μ > 0,



1

eμx

2.11 0

4.361



∞

1. 

∞

2.



(x + 1)e−μx

+

2

k=0

[p > 0,

q > 0]

∞

2.

∞

1. 0

 2.

[Re μ > 0]

MI 27

2

e−μx ln(2x − 1)

1

e−μx ln(a + x)

 0

[Re μ > 0]

ET I 148(8)

μ(x + a) ln(x + a) − 2 dx x+a  ∞ 1 μ(x − a) ln2 (x − a) − 4 2 dx = (ln a) = e−μx ln2 (a − x) 4 0 x−a [Re μ > 0, a > 0] BI (354)(4, 5)

x(1 − x)(2 − x)e−(1−x) ln(1 − x) dx =

∞

BI (352)(5)a

1 +  μ ,2 dx = Ei − x 2 2

2

0

1.

BI (352)(9)

, = eμ − ν(μ)

e−μx dx

1



MI 27

MI 27

xex ln(1 − x) dx = 1 − e 

q > 0]

[Re μ > 0]

0

4.364

μk p + k ln k! q+k

p > 0,

1

1.

4.363

∞ 

dx = ν  (μ) − ν  (μ)

x π 2 + (ln x)

0

4.362

xp−1 − xq−1 dx = ln x

π 2 + (ln x)

0

Re ν > 0]

e−μx ln[(x + a)(x + b)]

1−e 4e

BI (352)(4)

dx = e(a+b)μ {Ei(−aμ) Ei(−bμ) − ln(ab) Ei[−(a + b)μ]} x+a+b [a > 0, b > 0, Re μ > 0] BI (354)(11)

578

Logarithmic Functions



∞

2.

e

−μx

ln(x + a + b)

0

1 1 + x+a x+b

4.365

dx = (1 + ln a ln b) ln(a + b) + e−(a+b)μ {Ei(−αμ) Ei(−bμ)} + (1 − ln(ab)) Ei[−(a + b)μ] [a > 0,



∞

4.365

 e−x −

0

4.366 1. 2. 3.



x (1 + x)p+1 ln(1 + x)

dx = ln p x

b > 0,

[p > 0]

∞

−μx

[Re μ > 0] ; ∞

BI (354)(12)

BI (354)(15)



x2 dx 2 2 = [ci(aμ)] + [si(aμ)] e ln 1 + 2 [Re μ > 0] a x 0 ! !  ∞ ! x2 ! dx = Ei(aμ) Ei(−aμ) [Re μ > 0] e−μx ln !!1 − 2 !! a x 0 ! !  ∞ ! 1 + x2 ! −μx2 ! dx = 1 [cosh μ sinh(iμ) − sinh μ cosh(iμ)] ! xe ln ! 2 1−x ! μ 0 



NT 32(11)a ME 18

(cf. 4.339)

MI 27



x2 + 2β eβμ √ K 0 (βμ) [|arg β| < π, Re μ > 0] ET I 149(19) dx = 4μ 2β   0 2u +  2   2 , x2 4u2 − x2 2 dx π −2u2 μ π √ e Y 2iu e−μx ln = μ − (C − ln 2) J 0 0 2iu μ u4 2 2 4u2 − x2 0 ET I 149(21)a [Re μ > 0]

4.367 4.368 4.369

Re μ > 0]



∞

1.

xe−μx ln



∞

x+

xν−1 e−μx [ψ(ν) − ln x] dx =

0

2.

2

xn e−μx

 ln x −

0

1 2

Γ(ν) ln μ μν

2 ψ(n + 1) −

1 2

[Re ν > 0]

 ψ  (n + 1) dx =

n! μn+1

ET I 149(12)

  2 1  1 ln μ − ψ(n + 1) + ψ (n + 1) 2 2 [Re μ > 0]

MI 26

4.37 Combinations of logarithms and hyperbolic functions 4.371



1. 0

ln x dx = π ln cosh x

%√

 & 2π Γ 34   Γ 14



π+t t/π (2π) Γ ∞ π ln x dx 2π

= ln π − t cosh x + cos t sin t 0 Γ 2π

 2.

∞

LI (260)(1)a



t2 < π 2



BI (257)(7)a

4.374

Logarithms and hyperbolic functions



∞

3. 0

4.372



ln x dx =ψ cosh2 x

∞

1.

ln x 1



579



1 + ln π = ln π − 2 ln 2 − C 2

BI (257)(4)a

  n−1 π mπ π sinh mx kmπ Γ n+k  dx = tan ln 2π + ln  2n (−1)k−1 sin k sinh nx 2n 2n n n Γ 2n k=1 n−1   2 π mπ π kmπ Γ n−k k−1 n   = tan ln π + ln (−1) sin 2n 2n n n Γ nk k=1

[m + n is odd]

[m + n is even] BI (148)(3)a

∞

2.

ln x 1

cosh mx dx cosh nx   n π (2k − 1)mπ Γ 2n+2k−1 π ln 2π k−1 4n   ln + (−1) cos = 2n cos mπ n 2n Γ 2k−1 2n 4n k=1 n−1  2n−2k+1  2 π ln π π (2k − 1)mπ Γ 2n   2k−1 = ln (−1)k−1 cos mπ + 2n cos 2n n 2n Γ 2n k=1

[m + n is odd]

[m + n is even] BI (148)(6)a

4.373



∞

1. 0

 2. 3.

∞

∞

4. 0

6.

4.374

+ b > 0,

  ln 1 + x2

dx 4 πx = 2 ln cosh π 0 2    a+5  2   ∞   2 Γ 6 sinh π 6 Γ a+4 2 3 πx 6  dx = 2 sin ln  a+1   a+2 ln a + x sinh πx 3 Γ 6 Γ 6 0 

5.

% &     ln a2 + x2 2 Γ 2ab+3π 2b π 4π  − ln  dx = 2 ln cosh bx b π Γ 2ab+π 4π

  ln 1 + x2



π dx 2 a −ψ = ln + a π 2a sinh2 ax



π+a π

[a > −1] .



[a > 0] π  ∞  cosh 2 x  2π − 4  π dx = ln 1 + x2 2 π 0 x sinh 2 π √  ∞ x cosh √   16 8 2 √ 4π dx = 4 2 − + ln ln 1 + x2 2+1 π π 0 x sinh2 4 

1. 0

∞

  ln cos2 t + e−2x sin2 t

dx = −2t2 sinh x

a>−

π, . 2b

BI (258)(11)a

BI (258)(1)a

BI (258)(12)

BI (258)(5)

BI (258)(3)

BI (258)(2)

BI (259)(10)a

580

Logarithmic Functions



∞

2.



ln a + be 0

−2x



4.375

  a+b dx 2 ln(a + b) − a ln a − b ln 2 = (b − a) 2 cosh2 x [a > 0,

4.375



∞

1.11

ln cosh 0



ln coth x 0

4.376



∞

1. 0



5. 6.

BI (259)(11)

π dx = ln 2 cosh x 2

BI (259)(16)

∞

√  (−1)k+1 ln x √ √ dx = 2 π {ln(2k + 1) + 2 ln 2 + C} x cosh x 2k + 1

∞

ln x 0

4.

BI (147)(4)

∞  (−1)k+1 (μ + 1) cosh x − x sinh x μ dx = 2 Γ(μ + 1) x 2 (2k + 1)μ+1 cosh x k=0

[Re μ > −1]

(n + 1) cosh x − x sinh x n (−1) (n) 1 ln x β x dx = 2 2n 2 cosh x 0  ∞  2n n sinh 2ax − ax 2n−1 1 π ln 2x dx = − |B2n | x 2 n a sinh ax 0



∞



∞

[n = 1, 2, . . .]

∞

7. 0



ln x

BI (356)(14)

[n = 1, 2, . . . , a > 0]  (2n + 1) cosh ax − ax sinh ax 2n π 2n+1 ln |E2n | x dx = − 2 2a cosh ax

BI (356)(15)

ln x 0

 9.6

[a > 0] BI (356)(11) ⎧2   π 2n  2n−1 ⎪ −1 |B2n | n = 1, 2, . . . ⎪ ⎨a 2 2a

2ax sinh ax − (2n + 1) cosh ax 2n x dx = 1 ⎪ cosh3 ax ⎪ ⎩ a

n=0 [a > 0]

∞

ln x 0

BI (356)(9)a

2n+1

∞

8.6

BI (356)(10)

n

−1 ax cosh ax − (2n + 1) sinh ax 2n 2 (2n)! ζ(2n + 1) x dx = 2 2 2n+1 (2a) sinh ax 0  ∞  π 2n ax cosh ax − 2n sinh ax 2n−1 22n−1 − 1 |B2n | ln x dx = x 2 2n a sinh ax 0 

LI (259)(14)

k=0

2.

3.

π x dx = G − ln 2 2 cosh x 4

∞

2.

a + b > 0]

2ax cosh ax − (2n + 1) sinh ax 2n 1  π 2n |B2n | x dx = 3 a a sinh ax [a > 0,

BI (356)(2)

n = 1, 2, . . .]

BI (356)(6)a

4.382

Logarithms and trigonometric functions

 10.

ln x 0



x

t   2 − 6 cos2 2 2 x2 dx = π − t t 2 3 sin t (cosh x + cos t)

x sinh x − 6 sinh2

∞

581

[0 < t < π]

BI (356)(16)a

∞

 cosh πx + πx sinh πx dx  ln 1 + x2 =4−π x2 cosh2 πx 0  ∞  cosh πx + πx sinh πx dx  12. ln 1 + 4x2 = 4 ln 2 x2 cosh2 πx 0    ∞ ax − n 1 − e−2ax 2n−1 1  π 2n 4.377 ln 2x dx = |B2n | x 2n a sinh2 ax 0 [n = 1, 2, . . .] 11.

BI (356)(12) BI (356)(13)

LI (356)(8)a

4.38–4.41 Logarithms and trigonometric functions 4.381 1. 2. 3. 4. 4.382 1. 2.10



1

1 ln x sin ax dx = − [C + ln a − ci(a)] a 0  1 π, 1+ ln x cos ax dx = − si(a) + a 2 0  2π 1 ln x sin nx dx = − [C + ln(2nπ) − ci(2nπ)] n 0  2π 1+ π, ln x cos nx dx = − si(2nπ) + n 2 0

[a > 0]

GW (338)(2a)

[a > 0]

BI (284)(2)

GW (338)(1a)

GW (338)(1b)

! ! !x + a! ! sin bx dx = π sin ab ! [a < 0, b > 0] ln ! x − a! b 0 !  ∞ ! +   , !x + a! ! cos bx dx = 2 cos(ab) si(ab) + π − sin(ab) ci(ab) ! ln ! ! x−a b 2 0





∞

∞

3.

ln 0





∞

4.

ln 0

 5.

x2 + x + a2 2π exp −b sin bx dx = x2 − x + a2 b

 a2 −

1 4

b > 0]

[a > 0,

b > 0,

ET I 18(9)

c > 0] FI III 648a, BI (337)(5)

 sin

b 2

[b > 0] ∞

ln 0

 a2 + x2 π  −bc e cos cx dx = − e−ac 2 2 b +x c

[a > 0,

ET I 77(11)

(x + β)2 + γ 2 2π −γb e sin bx dx = sin βb (x − β)2 + γ 2 b

[Re γ > 0,

ET I 77(12)

|Im β| ≤ Re γ,

b > 0] ET I 77(13)

582

Logarithmic Functions

4.383  1.

2. 4.384

∞

  β ln 1 + e−βx cos bx dx = 2 − 2b

π

πb 0 2b sinh β

 ∞   πb β π coth ln 1 − e−βx cos bx dx = 2 − 2b 2b β 0 

4.383

[Re β > 0,

b > 0]

ET I 18(13)

[Re β > 0,

b > 0]

ET I 18(14)

1

1.

ln (sin πx) sin 2nπx dx = 0

GW (338)(3a)

0





1

2.7

1/2

ln (sin πx) sin(2n + 1)πx dx = 2

ln (sin πx) sin(2n + 1)πx dx % & n  1 1 2 −2 ln 2 − = (2n + 1)π 2n + 1 2k − 1

0

0

k=1

GW (338)(3b)





1

3.6

1/2

ln (sin πx) cos 2nπx dx = 2 0

ln (sin πx) cos 2nπx dx 0

= − ln 2 1 =− 2n  4.

[n = 0] [n > 0] GW (338)(3c)

1

ln (sin πx) cos(2n + 1)πx dx = 0

GW (338)(3d)

0



π/2

ln sin x sin x dx = ln 2 − 1

5.

BI (305)(4)

0

 6.

π/2

ln sin x cos x dx = −1 ⎧ ⎪  π/2 ⎨− π , for n > 0 4n ln sin x cos 2nx dx = π ⎪ 0 ⎩− ln 2, for n = 0 2  π π cos 2mn ln sin x cos[2m(x − n)] dx = − 2m 0  π/2 π ln sin x sin2 x dx = (1 − ln 4) 8 0  π/2 π ln sin x cos2 x dx = − (1 + ln 4) 8 0  π/2 1 ln sin x sin x cos2 x dx = (ln 8 − 4) 9 0  π/2 π2 ln sin x tan x dx = − 24 0

BI (305)(5)

0

7.

8. 9. 10. 11. 12.

LI (305)(6)

LI (330)(8)

BI (305)(7)

BI (305)(8)

BI (305)(9)

BI (305)(11)

4.384

Logarithms and trigonometric functions





π/2

13. 

π

0

ln sin x 0

18.



p2 < 1



a2 < 1 a2 > 1

FI II 484

  BI (331)(8)

ln sin bx 0

17.



dx π 1 − a2 = ln 1 − 2a cos x + a2 1 − a2 2 a2 − 1 π ln = 2 a −1 2a2

π

16. 



ln (1 + p cos x) dx = π arcsin p cos x

π

15.



BI (305)(16, 17)

0

14. 

π/2

ln sin 2x cos x dx = 2 (ln 2 − 1)

ln sin 2x sin x dx = 0

583

dx π 1−a = ln 2 2 1 − 2a cos x + a 1−a 2

2b



a2 < 1



BI (331)(10)

 2  dx π 1 + a2b a <1 = ln 2 2 1 − 2a cos x + a 1−a 2 0  π/2  π dx 1 dx ln sin x = ln sin x 2 1 − 2a cos 2x + a 2 1 − 2a cos 2x + a2 0 0 1−a π ln = 2 (1 − a2 ) 2 a−1 π ln = 2 (a2 − 1) 2a π

ln cos bx



ln sin bx

dx π 1−a = ln 1 − 2a cos 2x + a2 1 − a2 2

ln cos bx

dx π 1 + ab = ln 2 2 1 − 2a cos 2x + a 1−a 2

0 π

20. 0



π/2

21. 0



1+p ln cos x dx π ln = 2 2 1 − 2p cos 2x + p 2 (1 − p ) 2 p+1 π ln = 2 (p2 − 1) 2p

π

ln sin x 0

ln sin bx 0

24.

a2 > 1

 

   

a2 < 1 a2 < 1 p2 < 1 p2 > 1



BI (331)(18)



BI (331)(21)

 

cos x dx = 1 − 2a cos 2x + a2





a2 < 1 a2 > 1

  LI (331)(9)

π

ln cos bx 0



cos x dx =0 1 − 2a cos 2x + a2 [0 < a < 1]

π

ln sin x 0

b

 aπ ln 2 cos x dx π 1 + a2  = ln 1 − a2 − 2 2 1 − 2a cos x + a 2a 1 − a 1 − a2 2 2 π a +1 a −1 π ln 2 ln = − 2a a2 − 1 a2 a (a2 − 1)

π

23. 



a2 < 1

BI (321)(8)

22.





BI (321)(1), BI (331)(13) π

19. 

BI (331)(11)

π ln 2 cos2 x dx π 1+a ln(1 − a) − = 1 − 2a cos 2x + a2 4a 1 − a 2(1 − a) π a+1 a−1 π ln 2 = ln − 4a a − 1 a 2a(a − 1)

BI (331)(19, 22)

[0 < a < 1] [a > 1] BI (331)(16)

584

Logarithmic Functions



π/2

25.

ln sin x 0

cos 2x dx 1 = 1 − 2a cos 2x + a2 2



4.385

π

cos 2x dx ln sin x 1 − 2a cos 2x + a2 

2 1+a π 2 ln(1 − a) − a = ln 2 2a (1 − a2 ) 2  1 + a2 a − 1 π ln − ln 2 = 2a (a2 − 1) 2 a 0

 2  a <1  2  a >1

BI (321)(2), BI (331)(15), LI (321))(2)



π/2

26. 0

 1 + a2 cos 2x dx π 2 ln(1 + a) − a ln 2 ln cos x = 1 − 2a cos 2x + a2 2a (1 − a2 ) 2  1 + a2 1 + a π ln − ln 2 = 2a (a2 − 1) 2 a

 

a2 < 1 a2 > 1

  BI (321)(9)

4.385 1. 2.

3.

√ π a2 − b 2 dx √ =√ ln sin x ln [a > 0, a > b] 2 2 a + b cos x a −b a + a2 − b 2 0  π/2  π/2 dx dx ln sin x ln cos x 2 = 2 (a sin x ± b cos x) 0 0 (a cos x ± b sin

x) 1 a bπ = ∓a ln − b (a2 + b2 ) b 2 [a > 0, b > 0]  π/2  π/2 b ln sin x dx ln cos x dx π ln = = 2 sin2 x + b2 cos2 x 2 sin2 x + a2 cos2 x 2ab a + b a b 0 0 

π

[a > 0, 

π/2

ln sin x 

4. 0



sin 2x dx

2 = a sin2 x + b cos2 x =

π/2

ln cos x 

0

π/2

5. 0



2 b sin x + a cos2 x

a2 sin2 x − b2 cos2 x ln sin x  2 dx = a2 sin2 x + b2 cos2 x

 0

π/2

ln cos x 

ln sin x 



sin x 2

dx =

π/2

0

b > 0]

BI (319)(3, 7), LI (319)(3)

a2 cos2 x − b2 sin2 x 2 dx a2 cos2 x + b2 sin2 x

π 2b(a + b)

cos x ln cos x π √ dx = − ln 2 8 1 + cos2 x

1 + sin x  π/2 3  π/2 sin x ln sin x cos3 x ln cos x ln 2 − 1  √ dx = dx = 2 2 4 1 + cos x 0 0 1 + sin x 0

2.

π/2

BI (317)(4, 10)

sin 2x dx

[a > 0,

1.

b > 0]

a 1 ln 2b(b − a) b

=

4.386

BI (319)(1,6)a

2

[a > 0, 

BI (331)(6)

b > 0]

LI (319)(2, 8)

BI (322)(1, 6)

BI (322)(2, 7)

4.387

Logarithms and trigonometric functions



π/2

3. 0



π/2

4. 0

4.387 1.

2.

3.

4.

5.

6.



ln sin x 

dx 1−

k2

π 1 = − K(k) ln k − K (k  ) 2 4 sin x 2

k π ln cos x dx 1  = K(k) ln − K (k  ) 2 2 2 k 4 1 − k sin x 

π/2

585

BI (322)(3)

BI (322)(9)

π/2

ln sin x sinμ x cosν x dx =

ln cos x cosμ x sinν x dx 0 0





 μ+1 ν+1 μ+1 μ+ν+2 1 , = B ψ −ψ 4 2 2 2 2 [Re μ > −1, Re ν > −1] GW (338)(6c) μ √

    π/2 πΓ 2 ψ μ −ψ μ+1 ln sin x sinμ−1 x dx = μ+1 2 2 0 4Γ 2 [Re μ > 0] ν √  

  π/2 πΓ ν+1 ν 2 ψ −ψ ln sin x cosν−1 x dx = ν+1 2 2 0 4Γ 2

GW (338)(6a)

[Re ν > 0]    π/2 2n (2n − 1)!! π  (−1)k+1 2n − ln 2 ln sin x sin x dx = (2n)!! 2 k 0 k=1 2n+1   π/2  (−1)k (2n)!! 2n+1 + ln 2 ln sin x sin x dx = (2n + 1)!! k 0 k=1 % n &  π/2 (2n − 1)!! π  1 2n + ln 4 ln sin x cos x dx = − (2n)!! 4 k 0 k=1 (2n − 1)!! π [C + ψ(n + 1) + ln 4] =− (2n)!! 4

GW (338)(6b) FI II 811

BI (305)(13)

BI (305)(14)



π/2

1 (2n)!! (2n + 1)!! 2k + 1 k=0 

 1 (2n)!! 3 =− ψ n+ −ψ 2(2n + 1)!! 2 2

ln sin x cos2n+1 x dx = −

7. 0

n 

GW (338)(7b)



π/2

ln cos x sin2n x dx = −

8. 0

(2n − 1)!! π {C + 2 ln 2 + ψ(n + 1)} 2n+1 · n! 2 BI (306)(8)

586

Logarithmic Functions



π/2

9. 0



(2n − 1)!! π ln cos x cos2n x dx = − 2n n! 2

ln cos x cos2n x dx = 0

ln 2 +

2n  (−1)k k=1



k

%

π/2

10.



4.388

2n−1  (−1)k 2n−1 (n − 1)! ln 2 + (2n − 1)!! k

BI (306)(10)

&

k=1

BI (306)(9)

4.388 1.

2.

3.

4.

5. 6.

% & n−1  1 (−1)k sin2n x 1 nπ ln 2 + (−1) + ln sin x 2n+2 dx = cos x 2n + 1 2 4 2n − 2k − 1 0 k=0 % &  π/4 n−1  (−1)k 1 sin2n−1 x n − ln 2 + (−1) ln 2 + ln sin x 2n+1 dx = cos x 4n n−k 0 k=1 % &  π/4 n  1 1 (−1)k−1 sin2n x n+1 π + − ln 2 + (−1) ln cos x 2n+2 dx = cos x 2n + 1 2 4 2n − 2k + 1 0 k=0 % &  π/4 n−1  (−1)k 1 sin2n−1 x n − ln 2 + (−1) ln 2 + ln cos x 2n+1 dx = cos x 4n n−k 0 k=0  π/2 π pπ sinp−1 x [0 < p < 2] ln sin x p+1 dx = − cosec cos x 2p 2 0  π/2  2  1 π pπ dx = sec p <1 ln sin x p−1 4 p − 1 2 tan x sin 2x 0 

4.389  1.

π/4

(2n − 1)!! π (2n)!! 4n + 2

π

ln sin x sin2n 2x cos 2x dx = −

0



π/4

ln sin x cosn 2x sin 2x dx = −

2. 0



ln cos x cosμ−1 2x tan 2x dx = 0



π/2

0



π/2

ln cos x cosμ−1 x sin x dx = −

5.3

0

ln cos x cosp x cos px dx =

−π 2

 0

BI (310)(3)

1 μ2 BI (306)(11)

π [C + ψ(p + 1) − 2 ln 2] 2p+1

π/2

ln cos x cosp−1 x sin px sin x dx =

6.

BI (310)(4)

BI (286)(2)

[Re μ > 0] π 2

BI (288)(11)

1 β(μ) 4(1 − μ)

ln sin x sinμ−1 x cos x dx =

4.

BI (288)(10)

BI (285)(2)

[Re μ > 0] 

LI (288)(2)

BI (330)(9)

1 {C + ψ(n + 2) + ln 2} 4(n + 1)

π/4

3.

BI (288)(1)

π 2p+2

[p > −1]   1 C + ψ(p) − − 2 ln 2 p [p > 0]

BI (306)(12)

4.394

4.391

Logarithms and trigonometric functions



π/4



π/4

n

0

n

(ln sin 2x) sinp−1 2x tan

(ln cos 2x) cosp−1 2x tan x dx =

1.

587

0

[p > 0] 

π/4

n

(ln sin 2x) sinp−1 2x tan

2. 

π/4

2n−1

(ln cos 2x)

tan x dx =

0



π/4

4.

2n

(ln cos 2x)

tan x dx =

0

4.392



π/4

1. 0



BI (286)(10), BI (285)(18)

(−1)n n! + x dx = ζ(n + 1, p) 4 2

π

0

3.

1 − x dx = β (n) (p) 4 2

π

1 − 22n−1 2n π |B2n | 4n

BI (285)(17)

[n = 1, 2, . . .]

BI (286)(7)

22n 1 (2n)! ζ(2n + 1) 22n+1

BI (286)(8)

% & n−1  (−1)k−1 1 1 sin2n x n+1 π dx = − ln 2 + +2 ln (sin x cos x) (−1) cos2n+2 x 2n + 1 2 2n + 1 2n − 2k − 1 k=0

%

π/4

2.

ln (sin x cos x) 0

 (−1)k 1 sin2n−1 x 1 n n dx = + (−1) (−1) ln 2 − ln 2 + cos2n+1 x 2n 2n k n−1

&

BI (294)(8)

k=1

BI (294)(9)

4.393



π/2

1.

ln tan x sin x dx = ln 2

BI (307)(3)

ln tan x cos x dx = − ln 2

BI (307)(4)

0



π/2

2. 0



π/2

3.

 ln tan x sin2 x dx = −

0



π/4

0

sin x ln cot 0



π/2

1. 0

π 4

BI (307)(5, 6)

π2 ln tan x dx = − cos 2x 8

π/2

5. 4.394

ln tan x cos2 x dx = 0

4. 

π/2

GW (338)(10b)a

x dx = ln 2 2

LO III 290

1−a ln tan x dx π ln = 2 2 1 − 2a cos 2x + a 2 (1 − a ) 1 + a a−1 π ln = 2 2 (a − 1) a + 1

 

a2 < 1 a2 > 1

  BI (321)(15)

 2. 0

π/2

ln tan x cos 2x dx π 1+a 1−a = ln 2 2 1 − 2a cos 2x + a 4a 1 − a 1+a π a2 + 1 a − 1 ln = 4a a2 − 1 a + 1 2

 

a2 < 1 a2 > 1

  BI (321)(16)

588

Logarithmic Functions



π

3. 0



π

4. 0



ln tan bx dx π 1 − ab = ln 1 − 2a cos 2x + a2 1 − a2 1 + ab

[0 < a < 1,

ln tan bx cos x dx =0 1 − 2a cos 2x + a2

[0 < a < 1]

ln tan x

arcsin a cos 2x dx =− (π + arcsin a) 1 − a sin 2x 4a



ln tan x

π cos 2x dx = − arcsin a 2 2 4a 1 − a sin 2x



π/4

5. 0



π/4

6. 0

 7.

8.

4.395

a2 ≤ 1 a2 < 1

b > 0]

 

BI (331)(24) BI (331)(25)

BI (291)(2,3)

BI (291)(9)

  π π cos 2x dx 2 = − arcsinh a = − ln a + 1 + a 2 4a 4a 1 + a2 sin 2x 0  2  a <1 BI (291)(10)  u sin x ln cot x    2 dx = cosec 2α π ln 2 + L(ϕ − α) − L(ϕ + α) − L π − 2α 2 2 2 2 0 1 − cos α sin x [tan ϕ = cot α cos u; 0 < u < π] π/4

ln tan x

LO III 290



π/4

9. 0

4.395



π/2

1. 0



π/4

2. u

4.396

2.

3.

ln tan x dx  = − ln k  K(k) 2 2 1 − k sin x



√ sin υ + 1 − cos2 u cos2 υ ln tan x sin 4x dx π cos2 υ ln =−  2  2 2 sin u sin υ sin u (1 + sin υ) sin u + tan2 υ sin2 2x sin 2x − sin2 u + π, π LO III 285a 0
π/2

π/2

ln tan x cosq−1 x cos[(q + 1)x] dx = −

4. 0

 5.

π/4

n

p

(ln tan x) tan x dx = 0

LO III 290a

BI (322)(11)

  a 2 Γ 2 [a > 0, Re μ > 0] ln (a tan x) sinμ−1 2x dx = 2μ−2 ln a Γ(a) 0  

 √  π/2 2μ − 1 π Γ u − 12 2(μ−1)x ln tan x cos dx = − C+ψ + ln 4 4 Γ(μ) 2 0   Re μ > 12  π/2 π ln tan x cosq−1 x cot x sin[(q + 1)x] dx = − [C + ψ(q + 1)] 2 0 

1.

π , + π ln tan x sin 2x dx − t − − t ln 2 = cosec 2t L 2 2 1 − cos2 t sin2 2x

1 2n+1

B

(n)

π 2q

p+1 2

LI (307)(8)

BI (307)(9)

[q > −1]

BI (307)(11)

[q > 0]

BI (307)(10)

[p > −1]

LI (286)(22)



4.397

Logarithms and trigonometric functions

 6. 7.

589

1 − 22n 2n dx = π |B2n | [n = 1, 2, . . .] cos 2x 2n 0 & %  π/4 n (−1)n+1 π 2  (−1)k 2n+1 + ln tan x tan x dx = 4 12 k2 0 π/2

(ln tan x)

2n−1

BI (312)(6)

GW (338)(8a)

k=1

4.397



ln (1 + p sin x)

 π2 1 dx = − arccos p2 sin x 8 2



ln (1 + p cos x)

π2 1 dx 2 = − (arccos p) cos x 8 2



π/2

1. 0



π/2

2. 0



π

3.

ln (1 + p cos x) 0

 4.

5.

6.

dx = π arcsin p cos x

=

1 2



2π 0

π = − an n π =− n na 

8.

π

p2 < 1 p2 < 1



BI (313)(1)



BI (313)(8)



BI (331)(1)



− α − α ln sin α L cos x ln (1 + cos α cos x) 2 dx = 1 − cos2 α cos2 x sin α cos α 0 + 0<α< π  π/2 − α + (π − α) ln sin α L cos x ln (1 − cos α cos x) 2 dx = 1 − cos2 α cos2 x sin α cos α 0 + 0<α<  π   ln 1 − 2a cos x + a2 cos nx dx π/2

0

7.



p2 < 1

π, 2

LO III 291

π, 2

LO III 291

  ln 1 − 2a cos x + a2 cos nx dx  2  a < 1 BI (330)(11), BI (332)(5)  2  a >1 GW (338)(13a)

    1 2π  ln 1 − 2a cos x + a2 sin nx sin x dx = ln 1 − 2a cos x + a2 sin nx sin x dx 2 0 0

π an+1 an−1 = − 2 n + 1 n− 1  BI (330)(10), BI (332)(4) a2 > 1  2π  π     1 ln 1 − 2a cos x + a2 sin nx sin x dx = ln 1 − 2a cos x + a2 cos nx cos x dx 2 0 0

π an+1 an−1 =− + 2 n+1 n−1 π

BI (330)(12), BI (332)(6)

590

Logarithmic Functions



π

9.

4.398

  ln 1 − 2a cos 2x + a2 cos(2n − 1)x dx = 0



  ln 1 − 2a cos 2x + a2 sin 2nx sin x dx = 0



a2 < 1



BI (330)(15)

0



π

10. 0



π

11. 0



π

12.

  π ln 1 − 2a cos 2x + a2 sin(2n − 1)x sin x dx = 2



  ln 1 − 2a cos 2x + a2 cos 2nx cos x dx = 0

0



π

13. 0



  π ln 1 − 2a cos 2x + a2 cos(2n − 1)x cos x dx = − 2

π/2

14. 0

  aπ ln 1 + 2a cos 2x + a2 sin2 x dx = − 4 π π ln a2 − = 4 4a

a2 < 1



BI (330)(13)

an an−1 − n n−1   2 a <1 

a2 < 1



BI (330)(14)



BI (330)(16)



an−1 an + n n−1   2 a <1  

a2 < 1 a2 > 1



BI (330)(17)

  BI (309)(22), LI (309)(22)



π/2

15. 0



π

16. 0

4.398



  aπ ln 1 + 2a cos 2x + a2 cos2 x dx = 4 π π ln a2 + = 4 4a

  ln 1 − 2a cos x + a2 2π ln(1 − ab) dx = 1 − 2b cos x + b2 1 − b2

π

1.

ln 0



ln

 3.

1. 0

 BI (309)(23), LI (309)(23)



a2 ≤ 1,

a2 < 1

m

b2 < 1





n m/n a 1 − 2a cos x + a a cos mx dx = 2π − 2 1 − 2a cos nx + a m m

n −m/n a−m a − = 2π m m

BI (331)(26)

BI (330)(18)

 

a2 ≤ 1 a2 ≥ 1

 

2

ln



a2 > 1



BI (332)(9) π

0

4.399



2

0



a2 < 1

2n+1 1 + 2a cos x + a2 n 2πa sin(2n + 1)x dx = (−1) 1 − 2a cos x + a2 2n + 1 

2π

2.



π/2

1 + 2a cos 2x + a cot x dx = 0 1 + 2a cos 2nx + a2 

 π ln 1 + a sin x sin2 x dx = 2 2

BI (331)(5), LI(331)(5)

√ √

1+ 1+a 11− 1+a √ − ln 2 21+ 1+a [a > −1]

BI (309)(14)

4.413

Logarithms and trigonometric functions



π/2

2. 0



π/2

3. 0

4.411  1. 2.

  π ln 1 + a sin2 x cos2 x dx = 2 

√ √

1+ 1+a 11− 1+a √ + ln 2 21+ 1+a

 ln 1 − cos β cos x π 1 + sin α dx = − ln 2 2 1 − cos α cos x sin α sin α + sin β 2

591

[a > −1]

BI (309)(15)

2

+ π 0<β< , 2

0<α<

π, 2

LO III 285

 2  dx 1 + sin x = λ2 λ < π2 BI (331)(2) 1 + cos λ sin x sin x 0  π/2  π/2  π/2 q p + q sin ax dx p + q cos ax dx p + q tan ax dx = = = π arcsin ln ln ln p − q sin ax sin ax p − q cos ax cos ax p − q tan ax tan ax p 0 0 0 [p > q > 0] π

ln

FI II 695a, BI (315)(5, 13,17)a



π/2

3. 0

α−β cos 1 + cos β cos x 2π cos x 2 ln dx = ln α+β 1 − cos2 α cos2 x 1 − cos β cos x sin 2α sin 2 +

0<α≤β<

4.412



π/4

1.

ln tan 0



π/4

2.

ln tan 0



π/4

3.

ln tan 0



π/4

4.

ln tan 0



π/4

5.

ln tan 0

4.413  1.

π/2

π 4 π 4 π 4 π 4 π 4

±x ±x

π, 2

LO III 284

π2 dx =± sin 2x 8

BI (293)(1)

dx π2 =± tan 2x 16

BI (293)(2)

2n ± x (ln tan x)

22n+2 − 1 dx =± π 2n+2 |B2n+2 | sin 2x 4(n + 1)(2n + 1)

2n−1 ± x (ln tan x) n−1 ± x (ln sin 2x)

  ln p2 + q 2 tan2 x

0

BI (294)(24)

1 − 22n+1 dx = ± 2n+2 (2n)! ζ(2n + 1) sin 2x 2 n

BI (294)(25)

dx (−1)n−1 = (n − 1)! ζ(n + 1) tan 2x 2

LI (294)(20)

π ap + bq dx = ln ab a a2 sin2 x + b2 cos2 x [a > 0,

b > 0,

p > 0,

q > 0] BI (318)(1–4)a

 2. 0

π/2

  ln 1 + q 2 tan2 x

p2

dx 1 2 2 2 2 sin x + r cos x s sin x + t2 cos2 x

 p2 − r2 qr qt π t2 − s2 ln 1 + ln 1 + = 2 2 + p t − s2 r2 pr p st s [q > 0, p > 0, r > 0, s > 0, t > 0] BI (320)(18) 2

592

Logarithmic Functions



π/2

  ln 1 + q 2 tan2 x

dx sin2 x p2 sin2 x + r2 cos2 x s2 sin2 x + t2 cos2 x

 t qr qt π r ln 1 + ln 1 + = 2 2 − p t − s2 r2 s p p s [q > 0, p > 0, r > 0, s > 0, t > 0] BI (320)(20)

π/2

  ln 1 + q 2 tan2 x

dx cos2 x 2 2 2 2 2 2 p sin x + r cos x s sin x + t2 cos2 x

 p qr qt π s ln 1 + = 2 2 − ln 1 + p t − s2 r2 r p t s [q > 0, p > 0, r > 0, s > 0, t > 0] BI (320)(21)

3. 0

 4. 0

 5. 0

4.414



π



ln tan rx dx π 1 − p2r = ln 1 − 2p cos x + p2 1 − p2 1 + p2r

π/2

1. 0



4.414

π/2

2. 0

p2 < 1



BI (331)(12)

  dx ln 1 − k 2 sin2 x  = ln k  K(k) 1 − k 2 sin2 x

BI (323)(1)

    sin2 x dx 1  ln 1 − k 2 sin2 x  = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) k 1 − k 2 sin2 x BI (323)(3)



π/2

3. 0

,   cos x  1 + 2 2 ln 1 − k 2 sin2 x 1 − k2 sin2 x = 2 1 + k  − k  ln k  K(k) − (2 − ln k ) E(k) dx k 2

BI (323)(6)



π/2

4. 0

  ln 1 − k 2 sin2 x <

  1  =  2 k 2 − 2 K(k) + (2 + ln k ) E(k)  3 k 1 − k 2 sin2 x dx

BI (323)(9)



π/2

5. 0

  sin2 x ln 1 − k 2 sin2 x dx

<



3 1 − k 2 sin2 x =

,  1 +  2 2  (2 + ln k K(k) ) E(k) − 1 + k + k ln k k2 k 2 BI (323)(10)



π/2

6. 0

,   cos2 x dx 1 + 2   1 + k K(k) − (2 + ln k ln 1 − k 2 sin2 x < = + ln k ) E(k) 3 k2 1 − k 2 sin2 x BI (323)(16)

 7. 0

π/2

   2 ln 1 − k 2 sin2 x 1 − k 2 sin2 x dx = 1 + k  K(k) − (2 − ln k ) E(k)

BI (324)(18)

4.416

Logarithms and trigonometric functions



π/2

8. 0

593

    1 2 −2 + 11k 2 − 6k 4 + 3k  ln k  K(k) ln 1 − k 2 sin2 x sin2 x 1 − k 2 sin2 x dx = 2 9k      + 2 − 10k2 − 3 1 − 2k 2 ln k  E(k) BI (324)(20)

 π/2    1 2 ln 1 − k 2 sin2 x cos2 x 1 − k 2 sin2 x dx = 2 2 + 7k 2 − 3k 4 − 3k  ln k  K(k) 9k 0      2 2  − 2 + 8k − 3 1 + k ln k E(k)

 9.

BI (324)(21), LI (324)(21)



π/2

10. 0

    sin x cos x dx 2   1−2n [1 + (2n − 1) ln k ln 1 − k 2 sin2 x < = ] k − 1 2n+1 (2n − 1)2 k 2 1 − k 2 sin2 x BI (324)(17)

4.415  1. 2. 4.416



1. 0

2.

3.

4.7



π  π ln 4a + C − 2a 2 0    ∞ π 1 π ln 4a + C − ln x cos ax2 dx = − 4 2a 2 0 ∞

ln x sin ax2 dx = −

 π/2 cos x ln

1+

1 4

 sin2 β − cos2 β tan2 α sin2 x 1 − sin2 α cos2 x

[a > 0]

GW (338)(19)

[a > 0]

GW (338)(19)

dx

= cosec 2α {(2α + 2γ − π) ln cos β + 2 L(α) − 2 L(γ) + L(α + γ) − L(α − γ)}   sin α π cos γ = ; 0<α<β< LO III 291 sin β 2    π/2 cos x ln 1 − sin2 β − cos2 β tan2 α sin2 x dx 1 − sin2 α cos2 x 0 = cosec 2α {(π + 2α − 2γ) ln cos β + 2 L(α) + 2 L(γ) − L(α + γ) + L(α − γ)}   π sin α ; 0<α<β< cos γ = LO III 291 sin β 2    π/2 ln sin x + sin2 x − sin2 β dx 1 − cos2 α cos2 x β  

tan β 1 + sin α π  = − cosec α arctan ln sin β + ln sin α 2 sin α + 1 − cos2 α cos2 β + π, 0 < α < π, 0 < β < LO III 285 2  π/4  n−1 ln tan x (ln cos 2x) tan 2x dx = 12 (−1)n (n − 1)! 1 − 2−(n+1) ζ(n + 1) 0

BI (287)(20)

594

Logarithmic Functions

4.421

4.42–4.43 Combinations of logarithms, trigonometric functions, and powers 4.421



∞

ln x sin ax

1. 0



π dx = − (C + ln a) x 2

∞

2.

ln ax sin bx 0



∞

ln ax cos bx 0

, x dx π −bβ  π + bβ    −bβ   e e = ln (aβ ) − Ei (−bβ ) + e Ei (bβ ) β 2 + x2 2 4 [β  = β sign β; a > 0,

∞

ln ax sin bx 0

,   β  dx π π + bβ  e Ei (−bβ  ) − e−bβ Ei (bβ  ) = e−bβ ln (aβ  ) + 2 2 β +x 2 4 [β  = β sign β; a > 0,

ln ax cos bx 0

x dx π = {− si(bc) sin bc + cos bc [ln ac − ci(bc)]} 2 −c 2

∞

1.

ln x sin axxμ−1 dx =

0



∞

2. 0

4.423



∞

1. 0



0

4.424



1.

∞

BI (422)(6)

1 C + ln ab 2

BI (411)(5)

0 < Re μ < 1]

[a > 0,

b > 0]

GW (338)(21a)

b > 0]

GW (338)(21b)

BI (411)(6)

cos ax − cos bx π dx = [(a − b) (C − 1) + a ln a − b ln b] 2 x 2 sin ax aπ (C + ln 2a − 1) dx = − 2 x 2

2

(ln x) sin ax 0

c > 0]

2

ln x 0

b > 0,

[a > 0,

[a > 0, ∞

3.

BI (422)(5)

π μπ , Γ(μ) μπ + ψ(μ) − ln a + cot sin aμ 2 2 2

a cos ax − cos bx dx = ln ln x x b ln x



c > 0]

[a > 0, |Re μ| < 1] + π μπ , Γ(μ) μπ ψ(μ) − ln a − tan ln x cos axxμ−1 dx = μ cos a 2 2 2

∞

2.

b > 0,

dx π {sin bc [ci(bc) − ln ac] − cos bc si(bc)} = x2 − c2 2c [a > 0,



b > 0]

x2

[a > 0, ∞

5.

4.422

b > 0]

ET I 17(3), NT 27(11)a

4. 

FI II 810a

ET I 76(5), NT 27(10)a

3.



[a > 0]

[a > 0]

GW (338)(20b)

[a > 0]

ET I 77(9), FI II 810a

π π dx π3 2 = C2 + + πC ln a + (ln a) x 2 24 2

4.429

Logarithms, trigonometric functions, and powers

 2.

∞

6

2

(ln x) sin axxμ−1 dx =

0

4.425  1.

∞

ln(1 + x) cos ax

0

 2. 3.

∞



b+x b−x



Γ(μ) μπ +  μπ − 2 ψ(μ) ln a sin ψ (μ) + ψ 2 (μ) + π ψ(μ) cot aμ 2 2  μπ 1 2 − π ln a cot + (ln a) − π 2 2 4 [a > 0, 0 < Re μ < 1] ET I 77(10)

 1 dx 2 2 = [si(a)] + [ci(a)] x 2

dx = −2π si(ab) x 0  ∞  a   dx = −π Ei − ln 1 + b2 x2 sin ax x b 0 ln2

cos ax

[a > 0]

ET I 18(8)

[a ≥ 0,

b > 0]

[a > 0,

b > 0]

ET I 18(11)

GW (338)(24), ET I 77(14)



1

4. 0

4.426  1.11

  1 pπ dx π = 2+ coth ln 1 − x2 cos (p ln x) x 2p 2p 2

∞

x ln

0

LI (309)(1)a

 b2 + x2 π  sin ax dx = 2 (1 + ac)e−ac − (1 + ab)e−ab 2 2 c +x a

[b ≥ 0, ∞ +  ap  ap , b2 x2 + p2 dx = π Ei − ln 2 2 sin ax − Ei − 2 c x +p x c b 0 [b > 0,

 2.

595

c ≥ 0,

a > 0]

c > 0,

p > 0,

GW (338)(23)

a > 0] ET I 77(15)



∞

4.427

 ln x +

0

4.428  1. 2.

3.

∞





sin ax π π β 2 + x2  dx = K 0 (aβ) + ln(β) [I 0 (aβ) − L(aβ)] 2 2 2 2 β +x [Re β > 0, a > 0]

cos bx dx = πb ln 2 − aπ [a > 0, b > 0] x2 0  ∞     cos bx π ln 4 cos2 ax 2 dx = cosh(bc) ln 1 + e−2ac 2 x +c c 0 + π, a < b < 2a < c  ∞     sin bx dx = π ln 1 + e−2a sinh b − π ln 2 1 − e−b ln cos2 ax 2) x (1 + x 0 

ln cos2 ax

[a > 0, ∞

4. 0

ln cos2 ax

 0

1

ET I 22(29)

ET I 22(30)

ET I 82(36)

    cos bx dx = −π ln 1 + e−2a cosh b + b + e−b π ln 2 − aπ x2 (1 + x2 ) [a > 0,

4.429

b > 0]

ET I 77(16)

π (1 + x)x sin (ln x) dx = ln x 4

b > 0]

ET I 22(31) BI (326)(2)a

596

4.431

Logarithmic Functions



∞

1.

ln (2 ± 2 cos x)

0



∞

2.

ln (2 ± 2 cos x)

0



∞

3. 0

4.431

sin bx x dx = − π sinh(bc) ln (1 ± e−c ) x2 + c2 [b > 0,

c > 0]

ET I 22(32)

[b > 0,

c > 0]

ET I 22(32)

  cos bx π dx = cosh(bc) ln 1 ± e−c 2 2 x +c c

 sin bx  π  (−a)k dx = − [1 + sign(b − k)] ln 1 + 2a cos x + a2 x 2 k [b]

k=1

[0 < a < 1, 

∞

4. 0



ET I 82(25)

b

 cos bx  π  ak π  −c sinh[c(b − k)] cosh(bc) + ln 1 − 2a cos x + a dx = ln 1 − ae x2 + c2 c c k



2

k=1

[|a| < 1, 4.432

b > 0]

∞

1. 0



 sin x dx = ln 1 − k sin x  2 2 1 − k sin x x 2



2

0

∞

b > 0,

  ln 1 − k 2 cos2 x √

c > 0]

ET I 22(33)

sin x dx = ln k  K(k) 2 2 1 − k cos x x BI ((412, 414))(4)



π/2

2. 0

  sin x cos x ln 1 − k 2 sin2 x  x dx 1 − k 2 sin2 x    1  = 2 πk  (1 − ln k  ) + 2 − k 2 K(k) − (4 − ln k  ) E(k) k BI (426)(3)



π/2

3. 0



BI (426)(6) ∞

4. 0

 5. 0

    sin x cos x  1  ln 1 − k 2 cos2 x √ x dx = 2 −π − 2 − k 2 K(k) + (4 − ln k  ) E(k) k 1 − k 2 cos2 x

   sin x cos x dx 1  2 2 − k 2 − k  ln k  K(k) − (2 − ln k ) E(k) = 2 ln 1 − k 2 sin2 x  k 1 − k 2 sin2 x x BI (412)(5)

∞

   sin x cos x dx  1  = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 cos2 x √ 2 2 k 1 − k cos x x BI (414)(5)

4.432

Logarithms, trigonometric functions, and powers



∞

6. 0

597

 ∞    dx sin x sin x dx = ln 1 ± k sin x  ln 1 ± k cos2 x √ 2 2 2 2 x 1 − k cos x x 1 − k sin x 0 ∞   dx tan x ln 1 ± k sin2 x  = 2 2 1 − k sin x x 0 ∞   tan x dx = ln 1 ± k cos2 x √ 2 cos2 x x 1 − k 0 ∞   dx tan x ln 1 ± k sin2 2x  = 2 2 1 − k sin 2x x 0 ∞   tan x dx = ln 1 ± k 2 cos2 2x √ 2 2 1 − k cos 2x x 0 π 1 2 (1 ± k) K(k) − K (k  ) = ln √ 2 8 k 

2

BI (413)(1–6), BI (415)(1–6)



∞

7. 0

    1  sin x dx = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 sin2 x  2 k 1 − k 2 sin x x 3

BI (412)(6)



  ln 1 − k 2 cos2 x √

∞

   sin x cos2 x dx 1  2 2 − k 2 − k  ln k  K(k) − (2 − ln k ) E(k) = 2 ln 1 − k 2 sin2 x  k 1 − k 2 sin2 x x

0



BI (414)(6)a

9. 0



BI (412)(7) ∞

10. 0



 1  sin3 x dx 2 2 − k 2 − k  ln k  K(k) − (2 − ln k ) E(k) = 2 k 1 − k 2 cos2 x x

∞

8.

∞

11. 0

   sin x cos2 x dx  1  = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 cos2 x √ k 1 − k 2 cos2 x x   tan x dx = ln 1 − k 2 sin2 x  2 1 − k 2 sin x x



BI (414)(7) ∞

  ln 1 − k 2 cos2 x √

0

tan x dx = ln k  K(k) 2 2 1 − k cos x x BI ((412, 414))(9)



∞

   sin2 x tan x dx  1  = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 sin2 x  k 1 − k 2 sin2 x x

∞

   sin x tan x dx 1  2 2 − k 2 − k  ln k  K(k) − (2 − ln k ) E(k) = 2 ln 1 − k 2 cos2 x √ k 1 − k 2 cos2 x x

12. 0

BI (412)(8)

 13. 0

2

BI (414)(8)

 14. 0

∞

  ln 1 − k 2 sin2 x <

sin2 x



∞

  dx sin x ln 1 − k 2 cos2 x < 3 x 0 (1 − k 2 cos2 x)   1  = 2 k 2 − 2 K(k) + (2 + ln k ) E(k)  k

dx =  3 x 1 − k 2 sin2 x

BI ((412, 414))(13)

598

Logarithmic Functions



π/2

15. 0

4.432

   1  sin x cos x  π  (1 + ln k ln 1 − k 2 sin2 x < x dx = ) − (2 + ln k ) K(k) 3 k2 k 1 − k 2 sin2 x BI (426)(9)

16.

17.



π/2



∞

  1 sin x cos x ln 1 − k 2 cos2 x < x dx = 2 {−π + (2 + ln k  ) K(k)} BI (426)(15) k 3 0 (1 − k 2 cos2 x)  ∞  ∞     dx dx sin x cos x sin3 x 2 2 2 2 < = ln 1 − k sin x < ln 1 − k cos x  3 x x 3 0 0 (1 − k 2 cos2 x) 1 − k 2 sin2 x   1  = 2 2 − k2 + ln k  K(k) − (2 + ln k ) E(k) k BI (412)(14), BI(414)(15)

18. 0

  ln 1 − k 2 sin2 x <

3

sin x

dx  3 x 1 − k 2 sin2 x  =

∞

  dx sin x cos x ln 1 − k 2 cos2 x < 3 x 0 (1 − k2 cos2 x)    1 2 (2 + ln k ) E(k) − 2 − k 2 + k  ln k  K(k) = 2 k2 k



∞

  dx sin x cos x = ln 1 − k 2 sin2 x <  3 x 1 − k 2 sin2 x

∞

  dx sin x tan x ln 1 − k 2 sin2 x <  3 x 1 − k 2 sin2 x  =

19. 0





BI (412)(15), BI(414)(14) ∞

  dx sin2 x tan x ln 1 − k 2 cos2 x < 3 x 0 (1 − k 2 cos2 x)   1  = 2 2 − k2 + ln k  K(k) − (2 + ln k ) E(k) k

2

BI (412)(16), BI(414)(17)

20. 0

2

∞

  dx sin x cos2 x ln 1 − k 2 cos2 x < x 3 0 (1 − k 2 cos2 x)    1  2 2  (2 + ln k K(k) = ) E(k) − 2 − k + k ln k k2 k 2



∞

21. 0

 22. 0

∞

  ln 1 − k 2 sin2 x <

BI (412)(17), BI(414)(16) ∞

  dx tan x ln 1 − k 2 cos2 x < x 3 0 (1 − k 2 cos2 x)   1  = 2 k 2 − 2 K(k) + (2 + ln k ) E(k)  k

dx 3 x = 2 1 − k 2 sin x tan x



  dx = ln 1 − k 2 sin2 x 1 − k 2 sin2 x sin x x



BI ((412, 414))(18)

  dx 1 − k 2 cos2 x sin x ln 1 − k 2 cos2 x x 0   2  = 2 − k K(k) − (2 − ln k ) E(k) ∞

BI ((412, 414))(1)

4.511

Inverse trigonometric functions

 23.

  ln 1 − k 2 sin2 x 1 − k 2 sin2 x sin x cos x · x dx 0   1  3  2 4 2  = (1 − 3 ln k ) + 22k + 6k − 3k ln k 3πk K(k) 27k 2

599

π/2

  − 2 − k 2 (14 − 6 ln k  ) E(k) BI (426)(1)



  ln 1 − k 2 cos2 x 1 − k 2 cos2 x sin x cos x · x dx     1  2 2 −3π − 22k  + 6k 4 − 3k ln k  K(k) + 2 − k 2 (14 − 6 ln k  ) E(k) = 2 27k

π/2

24. 0



∞

25. 0



∞

26. 0

  dx = ln 1 − k 2 sin2 x 1 − k 2 sin2 x tan x x



BI (426)(2)

  dx ln 1 − k 2 cos2 x 1 − k2 cos2 x tan x x 0   2  = 2 − k K(k) − (2 − ln k ) E(k) 

∞

((412,414))(2) ∞

    sin x tan x dx dx = ln sin2 x + k cos2 x √ ln sin2 x + k  cos2 x √ 2 2 2 2 1 − k cos x x 1 − k cos x x 0 ∞  2  tan x dx  2 ln sin 2x + k cos 2x √ = 2 cos2 2x x 1 − k 0 ⎡ √ 3 ⎤ k ⎥ 2 1 ⎢ = ln ⎣ ⎦ K(k) 2 1 + k BI (415)(19–21)

4.44 Combinations of logarithms, trigonometric functions, and exponentials 4.441 1.



∞

7

e

−qx

0



∞

2.

   p  2 1 p 2 sin px ln x dx = 2 q arctan − pC − ln p − q p + q2 q c

e−qx cos px ln x dx = −

0



1 p2 + q 2



[q > 0,

p > 0]   q  2 p 2 ln p + q + p arctan + qC 2 q

BI (467)(1)

[q > 0]

BI (467)(2)

π/2 −p tan x

e

4.442 0

1 ln cos x dx 1 2 2 = − [ci(p)] + [si(p)] sin x cos x 2 2

[Re p > 0]

NT 32(11)

4.5 Inverse Trigonometric Functions 4.51 Inverse trigonometric functions 

4.511

∞

arccot px arccot qx dx = 0

π 2



 1 p q 1 ln 1 + + ln 1 + p q q p [p > 0, q > 0]

BI (77)(8)

600

Inverse Trigonometric Functions



4.512

π

4.512

arctan (cos x) dx = 0

BI (345)(1)

0

4.52 Combinations of arcsines, arccosines, and powers 4.521



1

1. 0



1

2. 0

4. 5.

6. 7. 8. 4.522 1. 2. 3. 4. 5. 6. 7.

FI II 614, 623

arccos x π dx = ∓ ln 2 + 2G 1±x 2

BI (231)(7, 8)

√ 2 1+q x π √ ln arcsin x dx = 1 + qx2 2q 1 + 1 + q 0   1 x π 1 + 1 − p2  arcsin x dx = ln 1 − p2 x2 2p2 2 1 − p2 0  1 ∞  sin[(2k + 1)λ] dx arccos x 2 = 2 cosec λ 2 (2k + 1)2 sin λ − x 0 k=0 √  1 π 1+ 1+q dx √ = ln arcsin x x (1 + qx2 ) 2 1+q 0 √  1 x π 1+q−1 arcsin x 2 dx = 4q 1+q (1 + qx2 ) 0 √  1 x π 1+q−1 arccos x dx = 2 )2 4q 1+q (1 + qx 0 

3.

π arcsin x dx = ln 2 x 2

1

[q > −1] 

p2 < 1



LI (231)(3)

BI (231)(10)

[q > −1]

BI (235)(10)

[q > −1]

BI (234)(2)

[q > −1]

BI (234)(4)

    1 3 2 2 π + k  K(k) − 2 1 + k E(k) x 1 − k 2 x2 arccos x dx = 2 9k 2 0    1   1 3 3 2 2 x 1 − k 2 x2 arcsin x dx = 2 − πk  − k  K(k) + 2 1 + k  E(k) 9k 2 0    1   1 3 2 2 2  2 2 E(k) π + k K(k) − 2 1 + k x k + k x arcsin x dx = 2 9k 2 0  1 , x arcsin x 1 + π √ dx = 2 − k  + E(k) k 2 1 − k 2 x2 0  1 , x arccos x 1 +π √ − E(k) dx = 2 k 2 1 − k 2 x2 0  1 , x arcsin x 1 +π  − E(k) dx = 2 k 2 0 k  2 + k 2 x2  1 , x arccos x 1 + π  dx = 2 − k  + E(k) k 2 0 k  2 + k 2 x2 

BI (231)(1)

1

BI (236)(9)

BI (236)(1)

BI(236)(5)

BI (237)(1)

BI (240)(1)

BI (238)(1)

BI (241)(1)

4.531

Arctangents, arccotangents, and powers



1

8. 0



1

9.

∞ x arcsin x dx 2  sin[(2k + 1)λ] √ = sin λ (2k + 1)2 (x2 − cos2 λ) 1 − x2 k=0



0



1

10.



0

4.523 1. 2. 3. 4. 5. 6. 4.524

x arcsin kx (1 −

x2 ) (1

−

k 2 x2 )

x arccos kx (1 −

x2 ) (1

−

k 2 x2 )

dx = − dx =

π ln k  2k

π ln(1 + k) 2k

  π 2n n! 1 − 2n + 1 2 (2n + 1)!! 0    1 π (2n − 1)!! 2n−1 x arcsin x dx = 1− 4n 2n n! 0  1 2n n! x2n arccos x dx = (2n + 1)(2n + 1)!! 0  1 π (2n − 1)!! x2n−1 arccos x dx = 4n 2n n! 0  1  n 2n n! 1 − x2 arccos x dx = π (2n + 1)!! −1  1  n− 12 π 2 (2n − 1)!! 1 − x2 arccos x dx = 2 2n n! −1 

1

1.

2

(arcsin x)

x2

0



BI (243)(11)

BI (239)(1)

BI (242)(1)

1

x2n arcsin x dx =



601

1

dx √ = π ln 2 1 − x2

0

dx

2

(arccos x) √

2.

1 − x2

3 = π ln 2

BI (229)(1)

BI (229)(2)

BI (229)(4)

BI (229)(5)

BI (254)(2)

BI (254)(3)

BI (243)(13)

BI (244)(9)

4.53–4.54 Combinations of arctangents, arccotangents, and powers 4.531



1

1. 0



arctan x dx = x

∞

0

 4.

0

1

arccot x dx = G x

FI II 482, BI (253)(8) BI (248)(6, 7)

π arccot x dx = − ln 2 + G x(1 + x) 8

BI (235)(11)

0



1

∞

π arccot x dx = ± ln 2 + G 1±x 4

2. 3.



∞

arctan x dx = −G. 1 − x2

BI (248)(2)

602

Inverse Trigonometric Functions



1

5.

arctan qx 0

 6. 0

7. 0



π 1 arctan x dx = ln 2 + G 2 x (1 + x ) 8 2

∞

8. 0



∞

9. 0



10.11 11. 12.

13.

4.532

x arctan x π2 dx = 4 1+x 16

BI (248)(3)

x arctan x π dx = − ln 2 1 − x4 8

BI (248)(4)

1

xp arctan x dx = 0



3. 4. 5. 4.533

BI (235)(12)

x arccot x π dx = ln 2 4 1−x 8 0  ∞  ∞ arccot x arccot x √ √ dx = dx = 2G 2 x 1 + x 1 + x2 0 0  1 √ arctan x π  √ dx = ln 1 + 2 2 2 0 x 1−x ⎤ ⎡  1 π 1 ⎣ π x arctan x dx ⎦ <  = k2 F 4 , k − <    2 2 2 2   0 (1 + x ) 1 + k x 2 2 1+k 

∞

p , +π 1 −β +1 2(p + 1) 2 2

pπ π cosec 2(p + 1) 2 0  1 +  , π p 1 +β +1 xp arccot x dx = 2(p + 1) 2 2 0  ∞ pπ π cosec xp arccot x dx = − 2(p + 1) 2 0 √

 ∞ 2q xp π3 Γ(q) dx   = 2q+2 arctan x 2p 1 + x x 2 p Γ q + 12 0 

∞

1. 0

 2.

0

BI (234)(10)

∞

1. 2.

q dx 1 1+q p 1 arccot q = ln + 2 arctan q + 2 2 2 2 2 (1 + px) 2p +q (1 + p) p +q 1+p [p > −1]

1

BI (243)(7)

2

arccot qx 

q dx 1 (1 + p)2 q2 − p arctan q = ln + (1 + px)2 2 p2 + q 2 1 + q2 (1 + p) (p2 + q 2 ) [p > −1]

1

4.532

1

xp arctan x dx =

(1 − x arccot x) dx =

π

π 4

dx π − arctan x = − ln 2 + G 4 1−x 8

BI (248)(12) BI (251)(3, 10)

FI II 694

BI (294)(14)

[p > −2]

BI (229)(7)

[−1 > p > −2]

BI (246)(1)

[p > −1]

BI (229)(8)

[−1 < p < 0]

BI (246)(2)

[q > 0]

BI (250)(10)

BI (246)(3)

BI (232)(2)

4.535

Arctangents, arccotangents, and powers



π

1 + x dx 1 π − arctan x = ln 2 + G 2 4 1−x1+x 8 2 0

 1 dx π 1 4. = − ln 2 x arccot x − arctan x 2 x 1 − x 4 0  ∞  ∞ dx x dx π2 2 2 √ + 4G 4.534 (arctan x) = (arccot x) √ =− 4 x2 1 + x2 1 + x2 0 0 4.535  1   1 arctan px dx = 2 arctan p ln 1 + p2 1. 2 2p 0 1+p x 

 1   1 π 1 arccot px dx = 2 + arccot p ln 1 + p2 [p > 0] 2. 2x 1 + p p 4 2 0  ∞  π arctan qx q ln pq − pq [p > 0, q > 0] 3. dx = − (p + x)2 1 + p2 q 2 2 0

 ∞ arccot qx q π 4. dx = ln pq + [p > 0, q > 0] (p + x)2 1 + p2 q 2 2pq 0  ∞ x arccot px π 1 + pq [p > 0, q > 0] 5. dx = ln 2 2 q +x 2 pq 0  ∞ x arccot px dx π 1 + p2 q 2 6. = ln [p > 0, q > 0] 2 2 x −q 4 p2 q 2 0  ∞ π arctan px dx = ln(1 + p) [p ≥ 0] 7. 2) x (1 + x 2 0  ∞  π  arctan px dx = ln 1 + p2 [p ≥ 0] 8. 2 4 0 x (1 − x )  ∞ π dx = 2 ln(1 + pq) 9. arctan qx [p > 0, q ≥ 0] 2 + x2 ) x (p 2p 0  ∞ π p2 + q 2 dx = ln 10. arctan qx [p ≥ 0] x (1 − p2 x2 ) 4 p2 0  ∞ x arctan qx πq [p > 0, q ≥ 0] 11. dx = 2 + x2 )2 4p(1 + pq) (p 0  ∞ x arccot qx π [p > 0, q ≥ 0] 12. 2 dx = 4p2 (1 + pq) 2 2 (p + x ) 0  1  arctan qx π  √ 13. dx = ln q + 1 + q 2 2 2 0 x 1−x

⎧ π 1 + |αβ| ⎪  ∞ ⎨ ln sign(α) for β = γ x arctan(αx) dx 2 2 1 + |αγ| = β − γπα 14.9 2 2 2 2 ⎪ −∞ (x + β ) (x + γ ) ⎩ for β = γ 2|β| (1 + |αβ|) 3.

1

for α, β, γ real

603

BI (235)(25)

BI (232)(1)

BI (251)(9, 17)

BI (231)(19)

BI (231)(24) BI (249)(1) BI (249)(8) BI (248)(9) BI (248)(10) FI II 745 BI (250)(6) BI (250)(3) BI (250)(6) BI (252)(12)a

BI (252)(20)a

BI (244)(11)

604

Inverse Trigonometric Functions



π 1 + |α/γ| ∞ ln sign(α) x arctan (α/x) dx 2 2 1 + |α/β| = β − γπα 2 2 2 2 ⎪ −∞ (x + β ) (x + γ ) ⎩ 2 2β (|β| + |α|) ⎧ ⎪ ⎨

 15.9

4.536



∞

1. 0

 0

∞

3. 0

4.537 1.

8

2.



(β = γ)

 π  1 + 1 + q2 dx 1 π  arctan qx arcsin x 2 = qπ ln  + ln q + 1 + q 2 − − arctan q x 2 2 2 1 + q2 π p arctan px − arctan qx dx = ln x 2 q

[p > 0,

q > 0]

FI II 635

arctan px arctan qx π (p + q)p+q dx = ln 2 x 2 pp q q

[p > 0,

q > 0]

FI II 745

1

1

 arctan 1 − x2

  arctan tan λ 1 − k 2 x2

.

0

 4.

1

  arctan tan λ 1 − k 2 x2

0

.

[p > 0]

  1 − k 2 x2 π π 2 sin2 λ E (λ, k) − cot λ 1 − dx = 1 − k 1 − x2 2 2

√  1 arctan tan λ 1 − k 2 x2 π  dx = F (λ, k) 2 2 2 2 (1 − x ) (1 − k x ) 0



∞

arctan x2

BI (245)(10)

BI (245)(12)

 ∞ dx dx = arctan x3 2 1+x 1 + x2  0 0 ∞ ∞ dx dx π2 = arccot x2 = arccot x3 = 2 2 1+x 1+x 8 0 0  ∞  2 1−x π √ 2. arctan x2 dx = 2−1 2 x 2 0  ∞     4.539 xs−1 arctan ae−x dx = 2−s−1 Γ(s)a Φ −a2 , s + 1, 12 0

∞  p sin qx x dx π  4.541 arctan = ln 1 + pe−q [p > −eq ] 2 1 + p cos qx 1 + x 2 0 1.

BI (245)(9)

, 1 − x2 π + 2 dx = 2 E (λ, k) − k  F (γ, k) 2 2 1−k x 2k

< π − 2 cot γ 1 − 1 − k 2 sin2 γ 2k





4.538

β = γ)





 π − 4λ dx π π + 4λ = ln cos cosec 1 − x2 cos2 λ 2 cos λ 8 8 0  1 dx     1 2 π ln p + arctan p 1 − x2 = 1 + p 1 − x2 2 0 

3.

5.

(α, β, γ real;

BI (230)(7) ∞

2. 

4.536

BI (245)(11) BI (245)(13)

BI (252)(10, 11) BI (252)(18, 19) BI (244)(10)a

ET I 222(47) BI (341)(14)a

4.573

Inverse and direct trigonometric functions

605

4.55 Combinations of inverse trigonometric functions and exponentials 4.551



1

1.9

(arcsin x) e−bx dx =

0



1

2.

∞

3.9 0



∞

4.9 0

ET I 160(1)

π 1 [L0 (b) − I 0 (b) + b L1 (b) − b I 1 (b)] + 2b2 b

x (arcsin x) e−bx dx =

0



π πe−b [I 0 (b) − L0 (b)] − 2b 2b

ET I 161(2)

 x −bx 1 arctan e dx = [ci(ab) sin(ab) − si(ab) cos(ab)] a b

[Re b > 0]  + , x −bx 1 π arccot e − ci(ab) sin(ab) + si(ab) cos(ab) dx = a b 2

ET I 161(3)

[Re b > 0]

ET I 161(4)

x  

1 1 1 q dx = ln 2π ln Γ(q) − q − ln q + q − e2πx − 1 2 2 2 0 [q > 0]

 ∞ 2 dx arccot x − e−px = C + ln p [p > 0] π x 0 

4.552

4.553

∞ arctan

WH NT 66(12)

4.56 A combination of the arctangent and a hyperbolic function 

∞

arctan e−x 1 dx = 2q 2 cosh px −∞

4.561

√ Π(x) π 3 Γ(q)   dx = 2q 4p Γ q + 12 −∞ cosh px [q > 0]



∞

LI (282)(10)

4.57 Combinations of inverse and direct trigonometric functions  4.571 4.572 4.573 1. 2.

π/2

sin x dx π arcsin (k sin x)  = − ln k  2 2k 0 1 −

k 2 sin x  ∞ 2 arccot x − cos px dx = C + ln p π 0



BI (344)(2)

[p > 0]

p π  [p > 0, 1 − e− q 2p 0



  ∞ p p 1 p −p q q arccot qx cos px dx = e Ei − e Ei − 2p q q 0 ∞

arccot qx sin px dx =

 3.

[p > 0, ∞

arccot rx 0

NT 66(12)

1±q sin px dx π ln =± p 1 ± 2q cos px + q 2 2pq 1 ± qe− r q±1 π ln =± p 2pq q ± e− r

 

q > 0]

BI (347)(1)a

q > 0]

BI (347)(2)a

p2 < 1,

r > 0,

p>0

q 2 > 1,

r > 0,

p>0



 BI (347)(10)

606

Inverse Trigonometric Functions



∞

4. 0

4.574  1.



1 tan x dx r π arccot px 2 = 2 ln 1 + tanh 2r q p q cos2 x + r2 sin2 x [p > 0,

∞

arctan

0



∞

2.7

arctan 0



arctan 0

 4.

arctan 0

4.575



π

1.

arctan 0

3.

4.576 1. 2. 4.577

sin(bx) dx =

π −ab e sinh(ab) b

[Re a > 0,



[a > 0,

2 x2

cos(bx) dx =

√ −b a2 +c2

π

arctan



π

π −b e sin b b

π n p sin x sin nx dx = p 1 − p cos x 2n



π/2

1. 0

BI (347)(9)

ET I 87(8)

π p sin x sin nx cos x dx = 1 − p cos x 4

ET I 29(7)

sinh(ab) [b > 0]

ET I 87(9)

[b > 0]

ET I 29(8)



p2 < 1



BI (345)(4)



dx π 1+p p sin x = ln 1 − p cos x sin x 2 1−p 0  π  dx π  p sin x = − ln 1 − p2 arctan 1 − p cos x tan x 2 0 arctan

r > 0]

b > 0]

b > 0]

pn−1 pn+1 + n+1 n−1 0  2  p <1

 π π pn+1 pn−1 p sin x cos nx sin x dx = − arctan 1 − p cos x 4 n+1 n−1 0  2  p <1 

2.



2ax π sin(bx) dx = e x2 + c2 b



∞

2a x

q > 0,

 1  −ab a cos(bx) dx = e Ei(ab) − eab Ei(−ab) x 2b 

∞

3.

4.574

 

p2 < 1 p2 < 1

 

BI (345)(5)

BI (345)(6)

BI(346)(1) BI(346)(3)

  sin2 x dx arctan tan λ 1 − k 2 sin2 x  1 − k 2 sin+2 x  ,  π = 2 F (λ, k) − E (λ, k) + cot λ 1 − 1 − k 2 sin2 λ 2k BI (344)(4)

 2. 0

π/2

  cos2 x dx arctan tan λ 1 − k 2 sin2 x  1 − k+2 sin2 x  , π 2 = 2 E (λ, k) − k F (λ, k) + cot λ 1 − k 2 sin2 λ − 1 2k BI (344)(5)

4.601

Change of variables in multiple integrals

607

4.58 A combination involving an inverse and a direct trigonometric function and a power 

∞

4.58110

dx = x

arctan x cos px 0



∞

arctan 0

dx π x cos x = − Ei(−p) p x 2 [Re(p) > 0]

ET I 29(3), NT 25(13)

4.59 Combinations of inverse trigonometric functions and logarithms 4.591  1.

1

0



1 arcsin x ln x dx = 2 − ln 2 − π 2

BI (339)(1)

arccos x ln x dx = ln 2 − 2

BI (339)(2)

1

2. 0



∞

1

4.592

arccos x 0

4.593



 (2k − 1)!! ln(2k + 2) dx =− ln x 2k k! 2k + 1

1

1.

arctan x ln x dx = 0



π 1 1 ln 2 − + π 2 2 4 48

1

arccot x ln x dx = −

2. 0



BI (339)(8)

k=0

1

4.594

n−1

arctan x (ln x)

BI (339)(3)

1 2 π 1 π − − ln 2 48 4 2

(ln x + n) dx =

0

 −n  n! 2 − 1 ζ(n + 1) n+1 (−2)

BI (339)(4)

BI (339)(7)

4.6 Multiple Integrals 4.60 Change of variables in multiple integrals 4.601    1. f (x, y) dx dy = f [ϕ(u, υ), ψ(u, υ)] |Δ| du dυ (σ  )

(σ)

where x = ϕ(u, υ), y = ψ(u, υ), and Δ =

2.

D(ϕ, ψ) ∂ϕ ∂ψ ∂ψ ∂ϕ − ≡ is the Jacobian determinant ∂u ∂υ ∂u ∂υ D(u, υ)

of the functions ϕ and ψ.   f (x, y, z) dx dy dz = f [ϕ(u, υ, w), ψ(u, υ, w), χ(u, υ, w)|Δ| du dυ dw] (V  )

(V )

where x = ϕ(u, υ, w), y = ψ(u, υ, w), and z = χ(u, υ, w) and where ! ∂ϕ ∂ϕ ∂ϕ ! ! ! ∂υ ∂w ! ! ∂u ∂ψ D(ϕ, ψ, χ) ∂ψ ! ≡ Δ = !! ∂ψ ∂u ∂υ ∂w ! ! ∂χ ∂χ ∂χ ! D(u, υ, w) ∂u

∂υ

∂w

is the Jacobian determinant of the functions ϕ, ψ, and χ. Here, we assume, both in (4.601 1) and in (4.601 2) that

608

Multiple Integrals

(a)

the functions ϕ, ψ, and χ and also their first partial derivatives are continuous in the region of integration; the Jacobian does not change sign in this region; there exists a one-to-one correspondence between the old variables x, y,z and the new ones u, υ,w in the region of integration; when we change from the variables x, y,z to the variables u, υ,w, the region V (resp. σ) is mapped into the region V  (resp. σ  ).

(b) (c) (d) 4.602

Transformation to polar coordinates: x = r cos ϕ,

4.603

4.602

y = r sin ϕ;

D(x, y) =r D(r, ϕ)

Transformation to spherical coordinates: x = r sin θ cos ϕ,

y = r sin θ sin ϕ,

z = r cos θ,

D(x, y, z) = r2 sin θ D(r, θ, ϕ)

4.61 Change of the order of integration and change of variables 4.611 

α

1.

 x  dx f (x, y) dy =

0



0

α

2.

 dx

0



α

f (x, y) dx

0

y



β αx



β

f (x, y) dy = 0

α

dy

α

dy

f (x, y) dx α βy

0

4.612 

R

1. 0

 2. 0

√ R2 −x2

 dx

 f (x, y) dy =

dy 0

0

2p

√

 dx

 √R2 −y2

R

q/p



2px−x2

f (x, y) dy = 0

f (x, y) dx 0



q

dy 0

+ √ , p 1+ 1−(y/q)2 + √ , p 1− 1−(y/q)2

f (x, y) dx

4.613

4.613

Change of the order of integration and change of variables



α

1.

 dx

0



β/(β+x)



β/(β+α)

f (x, y) dy = 0

0

 +



α

 dx

0



1



2α

3.

δ β+γ

 dx



3α−x

α

 0

R



√ 2 αy

f (x, y) dy = dy f (x, y) dx+  03α 0 3α−y + dy f (x, y) dx α

4.

f (x, y) dx 0

 αβ  y/β f (x, y) dy = dy f (x, y) dx 0  0δ  (δ−y)/γ + dy f (x, y) dx αβ 0  , a > 0, β > 0, γ > 0

x2 /4α

0

β(1−y)/y

dy

δ−νx βx

 α=

f (x, y) dx 0

β/(β+α)

2.

α

dy

 x+2R  dx √ f (x, y) dy = R2 −x2

0

 R dy √ f (x, y) dx 0 R2 −y 2  2R  R + dy f (x, y) dx R 0  3R  R + dy f (x, y) dx R

2R

y−2R

609

610

Multiple Integrals





π/2

4.614

4.615 0



2R

4.616

 dx

4.617 α



π/2 0

ϕ2 (x)



 dx

β

f (x, y)dy = ϕ1 (x)

 +

f (r cos ϕ, r sin ϕ) r dr 0

ϕ2 (x)



f (x, y)dy −

 α dx

0

2R cos ϕ

dϕ

0

0

ϕ1 (x)

 dx

β

ϕ1 (x)



 dx

α

f (x, y)dy −

0

0

ϕ2 (x)

f (x, y)dy 0

f (x, y) dy 0

[ϕ1 (x) ≤ ϕ2 (x) for α ≤ x ≤ β]  1 f (x, y) dy = dx f [x, zϕ(x)] ϕ(x)dz [y = zϕ(x)] 0 0 0  1 0 ϕ(γz) =γ dz f (γz, y) dy [x = γz] 0 0  x1  y1  x1  1 dx f (x, y) dy = dx (y1 − y0 ) f [x, y0 + (y1 − y0 ) t] dt



4.619

f (r cos ϕ, r sin ϕ) r dr 0

f (x, y) dy =

0

4.618

R

dϕ



2R−x2

0

β



π/2 0

√

 dx

0





0

f (x, y) dy = 0

r 2R

f (r, ϕ) dϕ

0

√ R2 −x2

 dx

arccos

dr

0

R



2R

f (r, ϕ) dr =

0





2R cos ϕ

dϕ

4.614

γ

 dx

x0

ϕ(x)

y0



γ

x0

0

[y = y0 + (y1 − y0 ) t]

4.62 Double and triple integrals with constant limits 4.620 1.

General formulas  π  ∞  π sign p  dω f  (p cosh x + q cos ω sinh x) sinh x dx = −  f sign p p2 − q 2 p2 − q 2 0 0   2 2 p >q , lim f (x) = 0 x→+∞

LO III 389

4.621

Double and triple integrals with constant limits





2π

2.

∞

dω 0

611

f  [p cosh x + (q cos ω + r sin ω) sinh x] sinh x dx   2π sign p = − f sign p p2 − q 2 − r2 p2 − q 2 − r2   2 2 2 lim f (x) = 0 LO III 390 p >q +r ,

0

x→+∞

 π

π

3. 0

0





  2π sign p dx dy  p − q cos x + r cot y = −  f sign p p2 − q 2 − r2 2 f sin x sin y sin x sin y p2 − q2 − r2  lim f (x) = 0 p2 > q 2 + r2 , x→+∞



∞

4.

 dx

−∞

LO III 280 ∞

f  (p cosh x cosh y + q sinh x cosh y + r sinh y) cosh y dy

−∞

  2π sign p = − f sign p p2 − q 2 − r2 p2 − q 2 − r2   lim f (x) = 0 LO III 390 p2 > q 2 + r2 , x→+∞

 5.

 π  ∞  ∞  dx f (p cosh x + q cos ω sinh x) sinh2 x sin ω dω = 2 f sign p p2 − q 2 cosh x sinh2 x dx 0 0  0  lim f (x) = 0 LO III 391 x→+∞

 6.

∞

 dx

0



2π

π

f [p cosh x + (q cos ω + r sin ω) sin θ sinh x] sinh2 x sin θ dθ  ∞   f sign p p2 − q 2 − r2 cosh x sinh2 x dx =4 0  p2 > q 2 + r2 , lim f (x) = 0 LO III 390

dω 0

0

x→+∞

 7. 0

 ∞ dx



2π

dω 0

0

π

f {p cosh x + [(q cos ω + r sin ω) sin θ + s cosh θ] sinh x} sinh2 x sin θ dθ  ∞   f sign p p2 − q 2 − r2 − s2 cosh x sinh2 x dx = 4π 0  2 2 2 2 p >q +r +s , lim f (x) = 0 LO III 391 x→+∞

4.621 1. 2. 3.

 1 − k 2 sin2 x sin2 y π dx dy = √ 2 2 1 − k sin y 2 1 − k2 0 0   π/2  π/2 cos y 1 − k 2 sin2 x sin2 y dx dy = K(k) 1 − k 2 sin2 y 0 0  π/2  π/2 sin α sin y dx dy πα  = 2 2 2 2 0 0 1 − sin α sin x sin y 

π/2  π/2

sin y

LO I 252(90)

LO I 252(91)

LO I 253

612

Multiple Integrals

4.622

 π π

π

1. 0

0

0

 π π

dx dy dz = 4π K 2 1 − cos x cos y cos z

√  2 2

4.622

MO 137

 √ π dx dy dz = 3π K 2 sin MO 137 12 0 0 0 3 − cos y cos z − cos x cos z − cos x cos y  π π π + + √ √ √ , √ , √ √ dx dy dz = 4π 18 + 12 2 − 10 3 − 7 6 K 2 2 − 3 3− 2 0 0 0 3 − cos x − cos y − cos z

2. 3.

π

MO 137



∞ ∞

4.6233 0 π 

0 2π

  π ϕ a2 x2 + b2 y 2 dx dy = 2ab



∞

  ϕ x2 x dx

0

4.624

f (α cos θ + β sin θ cos ψ + γ sin θ sin ψ) sin θ dθ dψ  π  1 = 2π f (R cos p) sin p dp = 2π f (Rt) dt −1 +0 ,  R = α2 + β 2 + γ 2  a  b    −3/2 4.6258 pl (a, b) = dx dy x2 + y 2 + 1 P l 1/ x2 + y 2 + 1 0

0

0

0

Then, for even and odd subscripts:   l+k l−1 (−1)l−k−1 22k 2l+2k  l+k l−k−1 ab 1  √ • p2l (a, b) = 2l 2 2 2k l(2l + 1)2 a + b + 1 k=0 (2k + 1) k    2j k  j 1 1 1 ×(2l + 2k + 1) + k−j+1 22j (a2 + b2 + 1)j (a2 + 1)k−j+1 (b2 + 1) j=0



l  l+k+1 2l + 2k + 1 (−1)l+k l 1 • p2l+1 (a, b) = 2l+1 k k l+k 2 (2l + 1) 22k k=0  b a a b 1 1 √ √ arctan−1 √ + arctan−1 √ × k k 2+1 2+1 2+1 2+1 2 2 b b a a (b + 1) (a + 1)  ⎫ k ⎬ 2j−1  2 1 1 1  · +ab + j k−j+1 k−j+1 ⎭ 2j (a2 + b2 + 1) (a2 + 1) (b2 + 1) j=1 j j

4.63–4.64 Multiple integrals

 x 1 (x − t)n−1 f (t) dt, (n − 1)! p p p p where f (t) is continuous on the interval [p, q] and p ≤ x ≤ q. 

4.631



x

dtn−1



tn−1

dtn−2 . . .

t1

f (t) dt =

FI II 692

4.635

Multiple integrals

4.632

 

 ···

1.

dx1 dx2 · · · dxn =

x1 ≥0,x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤h

613

hn n! [the volume of an n-dimensional simplex]

√ πn Rn ··· dx1 dx2 · · · dxn =  n +1 Γ x21 +x22 +···+x2n ≤R2 2  

2.

FI III 472



[the volume of an n-dimensional sphere] FI III 473

 



dx1 dx2 · · · dxn

π (n+1)/2

[n > 1] n+1 1 − − − · · · − x2n Γ x21 +x22 +···+x2n ≤1 2   half-area of the surface of an (n + 1)-dimensional sphere x21 + x22 + · · · + x2n+1 = 1    8 4.634 x1p1 −1 xp22 −1 · · · xnpn −1 dx1 dx2 . . . dxn ··· 

x1 q1

4.635 1.



···

4.633

x21

 2α≥0,...,xn ≥0 α α x1 ≥0,x 1 2 x n n + q2 +···+( x ≤1 qn )

 



 ···

f

 2α≥0,...,xn ≥0 α  α x1 ≥0,x 1 2 x1 x n n + q2 +···+( x ≥1 q qn )

FI III 474





p1 p2 pn Γ Γ . . . Γ q1p1 q2p2 . . . qnpn α1 α2 αn

= p1 p2 pn α1 α2 . . . αn Γ + + ··· + +1 α1 α2 αn [αi > 0, pi > 0, qi > 0, i = 1, 2, . . . , n] FI III 477

2

8

1

=

x22

x1 q1



α1 +

x2 q2



α2 + ··· +

xn qn

αn 

2

×x1p1 −1 xp22 −1 · · · xnpn −1 dx1 dx2 · · · dxn



p1 p2 pn  ∞ Γ Γ . . . Γ p1 p2 pn q1p1 q2p2 . . . qnpn α1 α2 αn

= f (x)x α1 + α2 +···+ αn −1 dx p1 p2 pn α1 α2 · · · αn 1 Γ + + ··· + α1 α2 αn under the assumption that the integral on the right converges absolutely.

FI III 487

614

Multiple Integrals

  2.



x1 q1



 ···

8

f

≥0,··· ,xn ≥0  2α α x1 ≥0,x 1 2 x n αn ≤1 + q2 +···+( x qn )

x1 q1



α1 +

x2 q2

4.636



α2 + ··· +

xn qn

αn 

2

×x1p1 −1 x2p2 −1 · · · xnpn −1 dx1 dx2 · · · dxn



p1 p2 pn  1 Γ Γ ···Γ p1 p2 pn q1p1 q2p2 . . . qnpn α1 α2 αn

= f (x)x α1 + α2 +···+ αn −1 dx p1 p2 pn α1 α2 . . . αn 0 Γ + + ··· + α1 α2 αn under the assumptions that the one-dimensional integral on the right converges absolutely and that the numbers qi , αi , and pi are positive. FI III 479 In particular,    ···

3.

xp11 −1 xp22 −1 . . . xnpn −1 e−q(x1 +x2 +···+xn ) dx1 dx2 . . . dxn

x1 ≥0,x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤1

=  

 ···

4.8

x1 ≥0,x2 ≥0,··· ,xn ≥0 α α n x1 1 +x2 2 +···+xα n ≤1

4.636 1.

8

 

 ···

x1 ≥0,x2 ≥0,...,xn ≥0 α α α x1 1 +x2 2 +···+xn n ≥1

 Γ (p1 ) Γ (p2 ) . . . Γ (pn ) 1 p1 +p2 +···+pn −1 −qx x e dx Γ (p1 + p2 + · · · + pn ) 0 [n > 0, p1 > 0, p2 > 0, . . . , pn > 0]

xp11 −1 x2p2 −1 . . . xnpn −1 α2 αn μ dx1 dx2 . . . dxn 1 (1 − xα 1 − x2 − · · · − xn )



p1 p2 pn Γ Γ ...Γ Γ(1 − μ) α1 α2 αn

= p1 p2 pn α1 α2 . . . αn Γ 1−μ+ + + ··· + α1 α2 αn [p1 > 0, p2 > 0, . . . , pn > 0, μ < 1] FI III 480

xp11 −1 xp22 −1 . . . xnpn −1 α2 αn μ dx1 dx2 . . . dxn 1 (xα 1 + x2 + · · · + xn )





p2 pn p1 Γ ...Γ Γ 1 α α2 αn 1



= p1 p2 pn p2 pn p1 α1 α2 . . . αn μ − − − ··· − Γ + + ··· + α1 α2 αn α1 α2   αn p1 p2 pn p1 > 0, p2 > 0, . . . , pn > 0; μ > + + ··· + FI III 488 α1 α2 αn

4.638

Multiple integrals

 

 ···

2.8

x1 ≥0,x2 ≥0,··· ,xn ≥0 α α n x1 1 +x2 2 +···+xα n ≤1

  3.

8

 ···

x1 ≥0,x2 ≥0,...,xn ≥0 α α α x1 1 +x2 2 +···+xn n ≤1

615

xp11 −1 xp22 −1 · · · xnpn −1 α2 αn μ dx1 dx2 . . . dxn 1 (xα 1 + x2 + · · · + xn )





p1 p2 pn Γ Γ ...Γ 1 α1 α2 αn



= p1 p1 p2 pn p2 pn α1 α2 . . . αn + + ··· + −μ Γ + + ··· + α1 α2 α1 α2  αn αn p1 p2 pn μ< + + ··· + FI III 480 α1 α2 αn . α2 αn 1 1 − xα 1 − x2 − · · · − xn x1p1 −1 xp22 −1 . . . xnpn −1 dx1 dx2 . . . dxn α1 α2 n 1 + x1 + x2 + · · · + xα n =

√ Γ π 2

p1 α1





⎫ ⎧ m p1 pn m+1 ⎪ ⎪ ⎪ ⎪ Γ ...Γ Γ ⎬ 1 ⎨ Γ 2 α2 αn 2

−

, m+1 m+2 ⎪ α1 α2 . . . αn Γ(m) ⎪ ⎪ ⎪ ⎩Γ ⎭ Γ 2 2

p1 p2 pn + + ··· + . α1 α2 αn    f (x1 + x2 + · · · + xn ) ···

where m = 4.6378

x1 ≥0, x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤1

=

FI III 480

xp11 −1 x2p2 −1 . . . xnpn −1 dx1 dx2 . . . dxn (q1 x1 + q2 x2 + · · · + qn xn + r)

Γ (p1 ) Γ (p2 ) . . . Γ (pn ) Γ (p1 p2 + . . . pn )



1

f (x) 0

p1 +p2 +···+pn

xp1 p2 +···pn −1 p p dx, (q1 x + r) (q2 x + r) 2 . . . (qn x + r) n [q1 ≥ 0, q2 ≥ 0, . . . , qn ≥ 0; r > 0] p1

where f (x) is continuous on the interval (0, 1). 4.638    ∞ p1 −1 p2 −1 ∞ ∞ x1 x2 · · · xnpn −1 e−(q1 x1 +q2 x2 +···+qn xn ) ... dx1 dx2 . . . dxn 1. s (r0 + r1 x1 + r2 x2 + · · · + rn xn ) 0 0 0  ∞ er0 x xs−1 dx Γ (p1 ) Γ (p2 ) . . . Γ (pn ) = p1 p2 pn Γ(s) 0 (q1 r1 x) (q1 r2 x) . . . (qn rn x) where pi , qi , ri , and s are positive. This result is also valid for r0 = 0, provided p1 +p2 +· · ·+pn > s.  ∞ ∞  ∞ x1p1 −1 xp22 −1 . . . xnpn −1 2. ... s dx1 dx2 . . . dxn 0 0 0 (r0 + r1 x1 + r2 x2 + · · · + rn xn ) Γ (p1 ) Γ (p2 ) . . . Γ (pn ) Γ (sp1 p2 − · · · − pn ) = r1p1 r2p2 · · · rnpn r0s−p1 −p2 −···−pn Γ(s) [pi > 0, ri > 0, s > 0]  ∞  ∞ ∞ p1 −1 p2 −1 pn −1 x1 x2 . . . xn ... 3.8 q1 q2 q s dx1 dx2 . . . dxn · · + (rn x n ) n ] 0 0 0 [1 + (r1 x1 ) + (r2 x2 ) + ·

p1 p2 pn p1 p2 pn Γ s− − − ··· − Γ Γ ...Γ q1 q2 qn q1 q2 qn = q1 q2 . . . qn r1p1 q1 r2p2 q2 . . . rnpn qn Γ(s) [pi > 0, qi > 0, ri > 0, s > 0]

616

4.639

Multiple Integrals

 

 ···

1.

(p1 x1 + p2 x2 + · · · + pn xn )

x21 +x22 +···+x2n ≤1

 

2m

dx1 dx2 . . . dxn

√  m πn (2m − 1)!! p21 + p22 + · · · + p2n  = n 2m +m+1 Γ 2 FI III 482

 ···

2.

4.639

(p1 x1 + p2 x2 + · · · + pn xn )

2m+1

dx1 dx2 . . . dxn = 0

FI III 483

x21 +x22 +···+x2n ≤1

4.641

 

 ···

1.11

ep1 x1 +p2 x2 +···+pn xn dx1 dx2 . . . dxn

x21 +x22 +···+x2n ≤1

∞ √  = πn k=0

 

 ···

2.

ep1 x1 p2 x2 +···p2n x2n dx1 dx2 . . . dx2n =

x21 +x22 +···+x22n ≤1

k! Γ

n 2

1 +k+1



p21 + p22 + · · · + p2n 4

 (2π)n I n p21 + p22 + · · · + p22n

k

FI III 483

n/2

(p21 + p22 + · · · + p22n )

FI III 483a

√  <

2 π n R n−1 2 2 2 4.642 ··· f x1 + x2 + · · · + xn dx1 dx2 . . . dxn =  n x f (x) dx, 0 Γ 2 2 2 2 x1 +x2 +···+xn ≤R 2 where f (x) is a function that is continuous on the interval (0, R).  1 1  1 p −1 p −1 p −1 4.643 ... f (x1 x2 · · · xn ) (1 − x1 ) 1 (1 − x2 ) 2 . . . (1 − xn ) n  

0

0



FI III 485

0

×xp21 xp31 +p2 · · · xpn1 +p2 +···+pn−1 dx1 dx2 . . . dxn  Γ (p1 ) Γ (p2 ) . . . Γ (pn ) 1 = f (x)(1 − x)p1 +p2 +···+pn −1 dx Γ (p1 + p2 + · · · + pn ) 0 under the assumption that the integral on the right converges absolutely. n−1 ?   @A  B dx1 dx2 . . . dxn−1 ··· 4.644 f (p1 x1 + p2 x2 + · · · + pn xn ) |xn | x21 +x22 +···+x2n =1    dx1 dx2 · · · dxn−1 f (p1 x1 + p2 x2 + · · · + pn xn ) < =2 ··· 1 − x21 − x22 − · · · − x2n−1 x21 +x22 +···+x2n−1 ≤1 √

 π < 2 π n−1 2 2 2

= f p1 + p2 + · · · + pn cos x sinn−2 x dx n−1 0 Γ 2     where f (x) is continuous on the interval − p21 + p22 + · · · + p2n , p21 + p22 + · · · + p2n .

FI III 488

[n ≥ 3]

FI III 489

4.648

Multiple integrals

617

4.645 Suppose that two functions f (x1 , x2 , . . . , xn ) and g (x1 , x2 , . . . , xn ) are continuous in a closed, bounded region D and that the smallest and greatest values of the function g in D are m and M , respectively. Let ϕ(u) denote a function that is continuous for m ≤ u ≤M. We denote by ψ(u) the integral    1. ψ(u) = ··· f (x1 , x2 , . . . , xn ) dx1 dx2 . . . dxn , m≤g(x1 ,x2 ,...,xn )≤u

2.

over that portion of the region D on which the inequality m ≤ g (x1 , x2 , . . . , xn ) ≤ u is satisfied. Then    ··· f (x1 , x2 , . . . , xn ) ϕ [g (x1 , x2 , . . . , xn )] dx1 dx2 . . . dxn m≤g(x1 ,x2 ,...,xn )≤M

 = (S)

M

 ϕ(u)d ψ(u) = (R)

m

M

ϕ(u) m

d ψ(u) du du

where the middle integral must be understood in the sense of Stieltjes. If the derivative dψ du exists and is continuous, the Riemann integral on the right exists.  M " +∞ M may be +∞ in formulas 4.645 2, in which case m should be understood to mean lim . M →+∞

 

 ···

4.6468

x1 ≥0,x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤1

xp11 −1 xp22 −1 . . . xnpn −1 r dx1 dx2 . . . dxn (q1 x1 + q2 x2 + · · · + qn xn )

 ∞ xr−1 dx Γ (p1 ) Γ (p2 ) . . . Γ (pn ) = p1 Γ (p1 + p2 + · · · + pn − r + 1) Γ(r) 0 (1 + q1 x) (1 + q2 x)p2 · · · (1 + qn x)pn p2 > 0, . . . , pn > 0, q1 > 0, q2 > 0, . . . , qn > 0, p1 + p2 + · · · + pn > r > 0]

= [p1 > 0,  



 ···

4.647

exp

0≤x21 +x22 +···+x2n ≤1

p1 x1 + p2 x2 + · · · + pn xn  x21 + x22 + · · · + x2n =



∞ ∞

4.6488 0

0

m



∞

··· 0

FI III 493

 dx1 dx2 . . . dxn √ 2 πn n

n (p21 + p22 + · · · + p2n ) 4

 exp − x1 + x2 + · · · + xn +

λn+1 x1 x2 . . . xn 1

− 12

I n2 −1

<

p21 + p22 + · · · + p2n FI III 495

 −1

2

−1

n

×xc1n+1 x2n+1 . . . xnn+1 n 1 =√ (2π) 2 e−(n+1)λ n+1

−1

dx1 dx2 · · · dxn

FI III 496

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5 Indefinite Integrals of Special Functions 5.1 Elliptic Integrals and Functions Notation: k  =

√ 1 − k 2 (cf. 8.1).

5.11 Complete elliptic integrals 5.111

 K(k)k2p+3 dk =

1.

2.

1 (2p + 3)2

 + , 2 4(p + 1)2 K(k)k2p+1 dk + k 2p+2 E(k) − (2p + 3) K(k)k

⎧   ⎨ 1 E(k)k2p+3 dk = 2 4(p + 1)2 E(k)k2p+1 dk 4p + 16p + 15 ⎩ − E(k)k2p+2

+

BY (610.04)

⎫ ⎬ 2 2 (2p + 3)k  − 2 − k 2p+2 k  K(k) ⎭ ,

BY (611.04)

5.112  1.

⎡ ⎤ ∞ 2  πk ⎣ [(2j)!] k 2j ⎦ K(k) dk = 1+ 4 4j 2 j=1 (2j + 1)2 (j!) ⎡

 2.6

E(k) dk = 

3.

∞ 

2

BY (610.00)

⎤

2j

[(2j)!] k πk ⎣ ⎦ 1− 2 − 1) 24j (j!)4 2 (4j j=1

K(k)k dk = E(k) − k K(k) 2

BY (611.00)

BY (610.01)

 4. 5.

,  1 + 2 1 + k 2 E(k) − k K(k) 3  ,   1 + 2 4 + k 2 E(k) − k  4 + 3k 2 K(k) K(k)k3 dk = 9 E(k)k dk =

619

BY (611.01) BY (610.02)

620

Elliptic Integrals and Functions

5.113

 6. 7. 8.

,   1 + 2 4 + k 2 + 9k 4 E(k) − k  4 + 3k 2 K(k) BY 611.02) 45  ,   1 + 2 64 + 16k2 + 9k 4 E(k) − k  64 + 48k 2 + 45k 4 K(k) K(k)k5 dk = BY (610.03) 225  ,   1 + 2 64 + 16k 2 + 9k 4 + 225k 6 E(k) − k  64 + 48k 2 + 45k 4 K(k) E(k)k5 dk = 1575 E(k)k3 dk =

BY (611.03)

 9. 10. 11. 12. 13. 5.113

E(k) K(k) dk = − 2 k k  + , E(k) 1 2 k dk = K(k) − 2 E(k) k2 k  E(k) dk = k K(k) k 2  ,  E(k) 1 +  2 2 2 k dk = − 2 E(k) + k K(k) k4 9k 3  k E(k) dk = K(k) − E(k) k 2 

dk = − E(k) k + , dk 2 2 E(k) − k K(k) = 2 E(k) − k  K(k) k   dk  2 = −k  K(k) 1 + k 2 K(k) − E(k) k , dk 1+ 2 E(k) − k  K(k) [K(k) − E(k)] 2 = k k + , dk 1 2 E(k) − k K(k) 2  2 = [K(k) − E(k)] k k k + , dk  E(k) 2 1 + k2 E(k) − k  K(k) 4 =  kk k 2 [K(k) − E(k)]

1.  2.  3.  4.  5.  6. 



5.114 5.115 1. 2. 3.

k K(k) dk E(k) −



k 2

1 2 =  2 k K(k) − E(k) K(k)

 π 2  , r2 , k k dk = k 2 − r2 Π , r , k − K(k) + E(k) 2 2  + π ,  π 2  K(k) − Π , r2 , k k dk = k2 K(k) − k 2 − r2 Π ,r ,k 2 2   π   π 2  2 E(k) 2 2 , r ,r ,k Π + Π , k k dk = k − r 2 2 2 k Π

π

BY (612.05) BY (612.02) BY (612.01) BY (612.03) BY (612.04)

BY (612.06) BY (612.09) BY (612.12) BY (612.07)

BY (612.13)

BY (612.11)

BY (612.14) BY (612.15) BY (612.16)

5.124

Elliptic integrals

621

5.12 Elliptic integrals 

2 + F (x, k) dx [F (x, k)] π,  = 0<x≤ 2 2 0 1 − k 2 sin2 x  x 2  [E (x, k)] 5.12211 E (x, k) 1 − k 2 sin2 x dx = 2 0 5.123  x 1 F (x, k) sin x dx = − cos x F (x, k) + arcsin (k sin x) 1. k 0 .

 x 1 1 − k 2 sin2 x 1 1 2. F (x, k) cos x dx = sin x F (x, k) + arccosh − arccosh 2   k k k k 0 x

5.121

5.124 1.

2.

3.∗



,  1+ k sin x 1 − k 2 sin2 x+arcsin (k sin x) 2k 0 ⎡  x  1 ⎣ k cos x 1 − k 2 sin2 x E (x, k) cos x dx = sin x E (x, k) + 2k 0 ⎤ .

2 2 1 1 − k sin x 2 2 ⎦ − k  arccosh − k + k  arccosh k k 2

5.∗

BY (630.11)

BY (630.21)

a

x E(x) dx π = 2√ 2 2 2 2 2 4 (k + k x ) a − x



BY (630.12)

BY (630.22)

√ a 1 − a2

a2 E (λ, k)



1−a

 2

F (λ, k)



+ 2 + 2 3/2 3/2 (k 2 + k 2 a2 ) k (k 2 + k 2 a2 ) (k 2 + k 2 a2 )

 a λ = arcsin √ [0 < a < 1, 0 < k < 1] k = 1 − k2 k 2 + k 2 a2   √  a F (φ, k) x E(x) dx π a2 E (φ, k) a 1 − a2 + √ = + √ 3/2 2 2 2 4 k 2 (k 2 − a2 ) k 2 k 2 − a2 a2 − x2 0 (k − x ) k 2 (k 2 − a2 ) a [0 < a < k < 1] φ = arcsin k  π/2 E (x, k ) sin x cos x dx   0 1 − k 2 cosh2 v sin2 x 1 − k 2 sin2 x 

tanh v 1 π tanh v π  − [F (φ, k) − E (φ, k)] = 2 E(k ) arctanh − k sinh v cosh v 2 2

k  tanh v  2 φ = arcsin [0 < tanh v < k < 1] k = 1−k k 0

4.∗

BY (630.32)

x

E (x, k) sin x dx = − cos x E (x, k)+



BY (630.01)

622

6.∗

Elliptic Integrals and Functions



5.124

π/2

E (x, k) sin x cos x dx   2 2 ψ sin2 x 1 − k cos 1 − k 2 sin2 x 

  tan ψ π tan ψ 1 π 2 2 1 − 1 − k cos ψ = 2 − E (β, k) +  E(k) arctan k sin ψ cos ψ k 2 2 1 − k 2 cos2 ψ

+  π, tan ψ 0 < k < 1, 0 < ψ < k = 1 − k2 β = arctan k 2  π/2 E (x, k ) sin x cos x dx   0 1 + k 2 sinh2 μ sin2 x 1 − k 2 sin2 x   < −1 π 2  2 = 2 E(k ) arctanh (k tanh μ) − F (φ, k) − E (φ, k) + tanh μ 1 + k sinh μ k sinh μ cosh 2 μ<

 π − coth μ 1 − 1 + k2 sinh2 μ 2  [0 < k < 1, 0 < tanh μ < 1] φ = arcsin(tanh μ) k  = 1 − k 2 0

7.∗

∗



π/2

8.

0

F (x, k ) sin x cos x dx   1 + k 2 sinh2 μ sin2 x 1 − k 2 sin2 x

+ , −1 π  F (φ, k) K(k ) arctanh (k tanh μ) − k 2 sinh μ cosh 2 μ  2 [0 < k < 1, 0 < tanh μ < 1] φ = arcsin(tanh μ) k = 1 − k =

9.∗



π/2

0

F (x, k ) sin x cos x dx   1 − k 2 cosh2 ν sin2 x 1 − k 2 sin2 x

 

tanh ν 1 π  F (φ, k) ) arctanh K(k − k 2

sinh ν cosh ν k 2  tanh ν φ = arcsin [0 < k < 1, 0 < tanh ν < 1] k = 1 − k2 k =

10.∗



π/2

0

11.∗





 tan ψ 1 π F (β, k) = 2 − K(k  ) arctanh k 2 k sin

ψ cos ψ  tan ψ β = arctan [0 < k < 1, 0 < ψ < 1] k = 1 − k2 k



b

ln a

F (x, k) sin x cos x dx   2 1 − k cos2 ψ sin2 x 1 − k 2 sin2 x

+x −x



x2 dx (x2 − a2 ) (b2 − x2 )

=

π 2  2  − ( − a2 ) (2 − b2 ) + πβ [F (φ, k) − E (φ, k)] 

β a [0 < a < b < ] φ = arcsin k=  b

5.131

5.125

Jacobian elliptic functions



x

1.

  Π x, α2 , k sin x dx

⎡. ⎤ 2 2 k − α 1 = − cos x Π x, α2 , k + √ arctan ⎣ sin x⎦ 1 − k 2 sin2 x k 2 − α2 ⎡. ⎤ 2 2   α −k 1 = − cos x Π x, α2 , k + √ arctanh ⎣ sin x⎦ 1 − k 2 sin2 x α2 − k 2

0



 2.

623



  2 α < k2   2 α > k2 BY (630.13)

x

    Π x, α2 , k cos x dx = sin x Π x, α2 , k − f − f0

0

%     &  2 1 − α2 α2 − k 2 + 1 − α2 sin2 x 2k 2 − α2 − α2 k 2  f=  arctan  2 (1 − α2 ) (α2 − k 2 ) 2α2 (1 − α2 ) (α2 − k 2 ) cos x 1 − k2 sin2 x  for 1 − α2 α2 − k 2 > 0; %        2 2 α2 − 1 α2 − k 2 + 1 − α2 sin x α2 + α2 k 2 − 2k 2 1 =  ln 1 − α2 sin2 x 2 (α2 − 1) (α2 − k 2 ) &   2α2 (α2 − 1) (α2 − k 2 ) cos x 1 − k 2 sin2 x + 1 − α2 sin2 x    for 1 − α2 α2 − k 2 < 0, f0 is the value of f at x = 0 BY (630.23) where

1

Integration with respect to the modulus   2 5.126 F (x, k)k dk = E (x, k) − k F (x, k) + 1 − k 2 sin2 x − 1 cot x BY (613.01)   , +   1  2 1 + k 2 E (x, k) − k  F (x, k) + 5.127 E (x, k)k dk = 1 − k 2 sin2 x − 1 cot x BY (613.02) 3          5.128 Π x, r2 , k k dk = k 2 − r2 Π x, r2 , k − F (x, k) + E (x, k) + 1 − k 2 sin2 x − 1 cot x BY (613.03)

5.13 Jacobian elliptic functions 5.131  1.

⎡    1 ⎣ sn m+1 u cn u dn u + (m + 2) 1 + k2 sn m u du = sn m+2 u du m+1 ⎤  − (m + 3)k2 sn m+4 u du⎦ SI 259, PE(567)

624

Elliptic Integrals and Functions

⎡

 cn m u du =

2.

1 ⎣ − cn m+1 u sn u dn u (m + 1)k 2 

+ (m + 2) 1 − 2k

 2



 cn m+2 u du + (m + 3)k 2

⎤ cn m+4 u du⎦ PE (568)

⎡

 dn m u du =

3.

5.132

1 ⎣k 2 dn m+1 u sn u cn u (m + 1)k 2 

+ (m + 2) 2 − k

 2



⎤  dn m+2 u du − (m + 3) dn m+4 u du⎦ PE (569)

By using formulas 5.131, we can reduce the integrals (for m = 1) dn m u du to the integrals 5.132, 5.133 and 5.134. 5.132  sn u du = ln 1. sn u cn u + dn u dn u − cn u = ln sn u  1 du k  sn u + dn u =  ln 2. cn u k cn u  1 du k  sn u − cn u =  arctan  3. dn u k k sn u + cn u 1 cn u =  arccos k dn u 1 cn u + ik sn u =  ln ik dn u 1 k  sn u =  arcsin k dn u "

5.133 1.

2.

"

sn m u du,

"

cn m u du, and

H 87(164) SI 266(4) SI 266(5) H 88(166) JA SI 266(6) JA



1 ln (dn u − k cn u) k dn u − k 2 cn u 1 = arccosh 2 k 1−k

dn u − cn u 1 = arcsinh k ; k 1 − k2 1 = − ln (dn u + k cn u) k  1 cn u du = arccos (dn u) ; k i = ln (dn u − ik sn u) ; k 1 = arcsin (k sn u) k sn u du =

H 87(161) JA JA SI 365(1) H 87(162) SI 265(2)a, ZH 87(162) JA

5.137

Jacobian elliptic functions

625

 3.

dn u du = arcsin (sn u) ; = am u = i ln (cn u − i sn u)

5.134 1. 2. 3.

H 87(163) SI 266(3), ZH 87(163)



1 [u − E (am u, k)] k2  , 1 + 2 cn 2 u du = 2 E (am u, k) − k u k  dn 2 u du = E (am u, k) sn 2 u du =

5.135  1 dn u + k  sn u du =  ln 1. cn u k cn u 1 dn u + k  =  ln 2k dn u − k   i sn u ik − k cn u du =  ln 2. dn u kk dn u 1 k cn u =  arccot kk k  1 − dn u cn u du = ln 3. sn u sn u 1 1 − dn u = ln 2 1 + dn u  1 1 − k sn u cn u du = − ln 4. dn u k dn u 1 + k sn u 1 ln = 2k 1 − k sn u  1 1 + sn u dn u du = ln 5. cn u 2 1 − sn u 1 + sn u = ln cn u  1 1 − cn u dn u du = ln 6. sn u 2 1 + cn u

PE (564) PE (565) PE (566)

SI 266(7) H 88(167) SI 266(8)

SI 266(10) H 88(168) SI 266(9)

H 88(172) JA H 87(170)

5.136  1 1. sn u cn u du = − 2 dn u k  2. sn u dn u du = − cn u  3. cn u dn u du = sn u 5.137 1.



1 dn u sn u du = 2 2 cn u k  cn u

H 88(173)

626

Elliptic Integrals and Functions

 2.  3.  4.  5.  6.

5.138

1 cn u sn u du = − 2 dn 2 u k  dn u dn u cn u du = − sn 2 u sn u sn u cn u du = dn 2 u dn u cn u dn u du = − sn 2 u sn u sn u dn u du = cn 2 u cn u

H 88(175) H 88(174) H 88(177) H 88(176) H 88(178)

5.138  sn u cn u du = ln 1. sn u dn u dn u  1 sn u dn u du = 2 ln 2.  cn u dn u cn u k  sn u dn u du = ln 3. sn u cn u cn u

H 88(183) H 88(182) H 88(184)

5.139  cn u dn u du = ln sn u 1.11 sn u  1 sn u dn u du = ln 2. cn u cn u  1 sn u cn u du = − 2 ln dn u 3. dn u k

H 88(179) H 88(180) H 88(181)

5.14 Weierstrass elliptic functions The invariants g1 and g2 used below are defined in 8.161. 5.141  ℘(u) du = − ζ(u)

1. 

2. 3. 4.8

1  1 ℘ (u) + g2 u 6 12  1 3 1 ℘ (u) − g2 ζ(u) + g3 u ℘3 (u) du = 120 20 10    1 du σ(u − v) =  2u ζ(v) + ln ℘(u) − ℘(v) ℘ (v) σ(u + v) ℘2 (u) du =

 5.

H 120(192) H 120(193)

[℘(v) = e1 , e2 , e3 ]

(see 8.162) H 120(194)

  au αδ − βγ α℘(u) + β σ(u + v) du = + 2  − 2u ζ(v) ln γ℘(u) + δ γ γ ℘ (v) σ(u − v) −1

where v = ℘



−δ γ

H 120(195)

5.221

The exponential integral function and powers

627

5.2 The Exponential Integral Function 5.21 The exponential integral function 



∞

5.211

Ei(−βx) Ei(−γx) dx = x

1 1 + β γ



Ei[−(β + γ)x]

−x Ei(−βx) Ei(−γx) −

e−γx e−βx Ei(−γx) − Ei(−βx) β γ [Re(β + γ) > 0]

NT 53(2)

5.22 Combinations of the exponential integral function and powers 5.221



∞

1. x



k=0

∞

2. x

3.∗ 4.∗

   n−1 1 Ei[−a(x + b)] (−1)n Ei[−a(x + b)] e−ab  (−1)n−k−1 ∞ e−ax + dx = n − dx k+1 xn+1 x bn n n bn−k x x Ei[−a(x + b)] dx = x2



1 1 + x b

[a > 0,

Ei[−a(x + b)] −

b > 0]

NT 52(3)

e−ab Ei(−ax) b [a > 0, b > 0]



NT 52(4)

1 xe−ax x2 Ei(−ax) + 2 e−ax + [a > 0] 2 2a 2a  ∞ n!e−ax  (ax)k xn+1 Ei(−ax) + xn Ei(−ax) dx = n+1 (n + 1)an+1 k! x Ei(−ax) dx =

k=0

5.∗

6.∗

7.∗

8.∗





[a > 0] x Ei(−ax)e−bx dx =

[a > 0,   2 Ei(−ax)e−ax − Ei(−2ax) Ei2 (−ax) dx = x Ei2 (−ax) + a

 x Ei2 (−ax) dx =



x2 2 Ei (−ax) + 2

Ei(−ax) dx = u Ei(−au) + 

0



1 x + a2 a



b > 0]

[a > 0] Ei(−ax)e−ax −

1 1 Ei(−2ax) + 2 e−2ax a2 a

[a > 0] u

0

9.∗

1 1 x 1 e−(a+b)x Ei [−(a + b)x] − 2 Ei(−ax)e−bx − Ei(−ax)e−bx − 2 b b b b(a + b)

∞

e

−au

a

−1

[a > 0]

2

 x  x a + b2 b2 ab a2 Ei − dx = ln a − ln b − x Ei − ln(a + b) − a b 2 2 2 2 [a > 0,

b > 0]

628

10.

The Sine Integral and the Cosine Integral

∗



∞

0

5.231

   x  x   ab  2 2  3 3 3 3 2 a + b ln(a + b) − a ln a − b ln b − a − ab + b x Ei − Ei − dx = a b 3 a+b 2

[a > 0,

b > 0]

5.23 Combinations of the exponential integral and the exponential 5.231



x

ex Ei(−x) dx = − ln x − C + ex Ei(−x)

1.

ET II 308(11)

0

 1.

x

e−βx Ei(−αx) dx = −

0

1 β



 β e−βx Ei(−αx) + ln 1 + − Ei[−(α + β)x] α

ET II 308(12)

5.3 The Sine Integral and the Cosine Integral 5.31



1.

cos αx ci(βx) dx = 

sin αx ci(βx) dx = −

2. 5.32



1.

cos αx si(βx) dx = 



1.



cos αx si(βx) si(αx + βx) − si(αx − βx) + α 2α

NT 49(1) NT 49(2)

NT 49(3) NT 49(4)

1 (si(αx + βx) + si(αx − βx)) ci(αx) ci(βx) dx = x ci(αx) ci(βx) + 2α 1 1 1 + (si(αx + βx) + si(βx − αx)) − sin αx ci(βx) − sin βx ci(αx) 2β α β NT 53(5)

1 (si(αx + βx) + si(αx − βx)) 2β 1 1 1 − (si(αx + βx) + si(βx + αx)) + cos αx si(βx) + cos βx si(αx) 2α α β

si(αx) si(βx) dx = x si(αx) si(βx) −

2.

 3.

cos αx ci(βx) ci(αx + βx) + ci(αx − βx) + α 2α

sin αx si(βx) ci(αx + βx) − ci(αx − βx) + α 2α

sin αx si(βx) dx = −

2. 5.33

sin αx ci(βx) si(αx + βx) + si(αx − βx) − α 2α

NT 54(6)

1 cos αx ci(βx) α



1 1 1 1 1 − sin βx si(αx) − + − ci(αx + βx) − ci(αx − βx) β 2α 2β 2α 2β

si(αx) ci(βx) dx = x si(αx) ci(βx) +

NT 54(10)

5.54

Combinations of the exponential integral and the exponential

5.34



∞

si[a(x + b)]

1. x



∞

2.

ci[a(x + b)] x

dx = x2 dx = x2





1 1 + x b 1 1 + x b

si[a(x + b)] −

cos ab si(ax) + sin ab ci(ax) b [a > 0,

ci[a(x + b)] +

629

b > 0]

NT 52(6)

sin ab si(ax) − cos ab ci(ax) b [a > 0,

b > 0]

NT 52(5)

5.4 The Probability Integral and Fresnel Integrals 

5.41

11

5.42 5.43

e−α x Φ(αx) dx = x Φ(αx) + √ α π  cos2 αx2 √ S (αx) dx = x S (αx) + α 2π  sin2 αx2 C (αx) dx = x C (αx) − √ α 2π 2

2

NT 12(20)a NT 12(22)a NT 12(21)a

5.5 Bessel Functions Notation: Z and Z denote any of J, N , H (1) , H (2) . In formulae 5.52–5.56, Z p (x) and Zp (x) are arbitrary Bessel functions of the first, second, or third kinds.  ∞  J p+2k+1 (x) JA, MO 30 5.51 J p (x) dx = 2 k=0

5.52



1.  2.11 5.5310

xp+1 Z p (x) dx = xp+1 Z p+1 (x)

WA 132(1)

x−p Z p+1 (x) dx = −x−p Z p (x)

WA 132(2)

  

  p2 − q 2 α2 − β 2 x − Z p (αx) Zq (βx) dx x = αx Z p+1 (αx) Zq (βx) − βx Z p (αx) Zq+1 (βx) − (p − q) Z p (αx) Zq (βx) = βx Z p (αx) Zq−1 (βx) − αx Z p−1 (αx) Zq (βx) + (p − q) Z p (αx) Zq (βx) JA, MO 30, WA 134(7)

5.54 1.



10

αx Z p+1 (αx) Zp (βx) − βx Z p (αx) Zp+1 (βx) α2 − β 2 βx Z p (αx) Zp−1 (βx) − αx Z p−1 (αx) Zp (βx) = α2 − β 2

x Z p (αx) Zp (βx) dx =

WA 134(8)

 2.

2

x [Z p (αx)] dx =

 x2  2 [Z p (αx)] − Z p−1 (αx) Z p+1 (αx) 2

WA 135(11)

630

3.

Bessel Functions



∗

x Z p (ax) Zp (ax) dx =  10

5.55

5.55

 x4  2 Z p (ax) Zp (ax) − Z p−1 (ax) Zp+1 (ax) − Z p+1 (ax) Zp−1 (ax) 4

Z p (αx) Zq+1 (αx) − Z p+1 (αx) Zq (αx) Z p (αx) Zq (αx) 1 Z p (αx) Zq (αx) dx = αx + x p2 − q 2 p+q Z p−1 (αx) Zq (αx) − Z p (αx) Zq−1 (αx) Z p (αx) Zq (αx) = αx − p2 − q 2 p+q WA 135(13)

5.56



1.

Z 1 (x) dx = − Z 0 (x)

JA

x Z 0 (x) dx = x Z 1 (x)

JA

 2.

6–7 Definite Integrals of Special Functions 6.1 Elliptic Integrals and Functions Notation: k  =

√ 1 − k 2 (cf. 8.1).

6.11 Forms containing F (x, k) 

π/2

6.111

F (x, k) cot x dx = 0

6.112



π/2

1. 0



F (x, k)

2 π sin x cos x 1 √ − K(k) ln K(k ) dx = 4k 1 − k sin2 x (1 − k) k 16k

F (x, k)

sin x cos x 1 dx = − 2 ln k  K(k) 2k 1 − k 2 sin2 x

0



π/2

3. 0

6.113



π/2

1.

F (x, k )

0



π/2

2.

F (x, k) 0

BI (350)(1)

√ (1 + k) k π sin x cos x 1 K(k) ln + K(k ) F (x, k) dx = 4k 2 16k 1 + k sin2 x

π/2

2.

π 1 K(k  ) + ln k K(k) 4 2

2 sin x cos x dx 1 √ K(k  ) ln = 4(1 − k) (1 + k) k cos2 x + k sin2 x

BI (350)(6)

BI (350)(7)

BI (350)(2)a, BY(802.12)a

BI (350)(5)

sin x cos x dx · 1 − k 2 sin2 t sin2 x 1 − k 2 sin2 x =−

, + 1 π  F (t, k) tan t) − K(k) arctan (k k 2 sin t cos t 2 BI (350)(12)

 6.114

6.115

 dx 1 K(k) K 1 − tan2 u cot2 v =    2 cos u sin v u sin2 x − sin2 u sin2 v − sin2 x   2 BI (351)(9) k = 1 − cot2 u · cot2 v √  1 (1 + k) k π x dx 1 K(k) ln + K(k ) F (arcsin x, k) = 2 1 + kx 4k 2 16k 0 (cf. 6.112 2) BI (466)(1) v

F (x, k) <

631

632

Elliptic Integrals and Functions

6.121

This and similar formulas can be obtained from formulas 6.111–6.113 by means of the substitution x = arcsin t.

6.12 Forms containing E (x, k) 

 1  sin x cos x 2 1 + k  K(k) − (2 + ln k ) E(k) dx = 2 BI (350)(4) 2 2 2k 1 − k sin x 0 π/2 dx 1 6.122 E (x, k)  BI (350)(10), BY (630.02) = {E(k) K(k) − ln k } 2 2 0 1 − k 2 sin x  π/2 sin x cos x dx 6.123 E (x, k) · 2 2 2 1 − k sin t sin x 0 1 − k 2 sin2 x +  ,  π 1 π E(k) arctan (k tan t) − E (t, k) + cot t 1 − 1 − k 2 sin2 t =− 2 k sin t cos t 2 2 BI (350)(13) ⎛. ⎞  v 2 dx 1 ⎝ 1 − tg u ⎠ E(k) K 6.124 E (x, k) < =   2 cos u sin v tg 2 v u sin2 x − sin2 u sin2 v − sin2 x ⎞ ⎛. k 2 sin v sin2 2u ⎠ + K⎝ 1− 2 cos u sin2 2v  2  k = 1 − cot2 u cot2 v BI (351)(10) π/2

6.121

E (x, k)

6.13 Integration of elliptic integrals with respect to the modulus 

x 1 − cos x = tan sin x 2 0 1 sin2 x + 1 − cos x 6.132 E (x, k)k dk = 3 sin x  0 1  2    1 + r sin x x − r2 Π x, r2 , 0 6.133 Π x, r , k k dk = tan − r ln 2 1 − r sin x 0 1

6.131

F (x, k)k dk =

BY (616.03) BY (616.04) BY (616.05)

6.14–6.15 Complete elliptic integrals 6.141



1

1.

K(k) dk = 2G

FI II 755

π2 4

BY (615.03)

0



1

2.

K(k  ) dk =

0





π dk = π ln 2 − 2G 2 k 0 √ 

 1 2 dk 1 4 1 2 7 Γ 6.143 K(k)  = K = k 2 16π 4 0  1 2 π dk = 6.144 K(k) 1 + k 8 0 6.142

1

K(k) −

BY (615.05) BY (615.08) BY (615.09)

6.161

The theta function



, 1 + 4 dk 2 = 24 (ln 2) − π 2 K(k  ) − ln k k 12 0  1 1 2 n 2 6.146 n k K(k) dk = (n − 1) k n−2 K(k) dk + 1 0 0  1  1 6.147 n k n K(k  ) dk = (n − 1) k n−2 E(k) dk 

633

1

6.145

0

6.148



E(k) dk =

1 +G 2

E(k  ) dk =

π2 8

0



1

2. 0

3.

∗



1

0

6.149



1



0 1

2. 

E(k) −

(see 6.152)

BY (615.11)

BY (615.02) BY (615.04)

π dk π = π ln 2 − 2G + 1 − 2 k 2

(E(k  ) − 1)

0

3.∗

[n > 1]

E(k) dk = 1 1+k

1. 

BY (615.12)

0

1

1.

BY (615.13)

BY (615.06)

dk = 2 ln 2 − 1 k

BY (615.07)

1

E(k) dk = 1 1+k



 1 4 dx  E(x) π 2 π 2 a − x K(x) − √ + x = − ln √ 3 4 4 e 1 − x2 0 x ⎡ ⎤ 0

4.∗

6.151

6.152 6.1536 6.154

√  ⎢ 2 dk 1⎢ 2 4K + E(k)  = ⎢ ⎢ k 8⎣ 2

⎥ ⎥ π2 √ ⎥ ⎥ 0 2 ⎦ K2 2  1  1 (n + 2) k n E(k  ) dk = (n + 1) k n K(k  ) dk 0 0

 a 1+a 1 K(k)k dk π √ = √ ln 4 1 − a2 1−a k  2 a2 − k 2 0 π/2 E (p sin x) π sin x dx =  2 sin2 x 1 − p 2 1 − p2 0



1

BY (615.10)

[n > 1]

(see 6.147)

[0 < a < 1] 

p2 > 1



BY (615.14) LO I 252 FI II 489

6.16 The theta function 6.161  1.

∞

0

 2. 0

∞

      s xs−1 ϑ2 0 | ix2 dx = 2s 1 − 2−s π − 2 Γ 12 s ζ(s)       s xs−1 ϑ3 0 | ix2 − 1 dx = π − 2 Γ 12 s ζ(s)

[Re s > 2]

ET I 339(20)

[Re s > 2]

ET I 339(21)

634

Elliptic Integrals and Functions



∞

3. 0



6.162

       1 xs−1 1 − ϑ4 0 | ix2 dx = 1 − 21−s π − 2 s Γ 12 s ζ(s) [Re s > 2]

∞

4. 0

ET I 339(22)

           1 xs−1 ϑ4 0 | ix2 + ϑ2 0 | ix2 − ϑ3 0 | ix2 dx = − (2s − 1) 21−s − 1 π − 2 s Γ 12 s ζ(s) ET I 339(24)

6.162



∞

1.11

e−ax ϑ4



0



∞

2.

e−ax ϑ1



0



∞

3.11

e−ax ϑ2



0



∞

4.11 0

6.16310



∞

1. 0



!

! iπx  √  √  l ! dx = √ cosh b a cosech l a ! l2 a

[Re a > 0, !

 √  √  bπ !! iπx l dx = − √ sinh b a sech l a ! 2 2l l a

|b| ≤ l]

ET I 224(1)a

[Re a > 0, !

 √  √  (l + b)π !! iπx l dx = − √ sinh b a sech l a ! l2 2l a

|b| ≤ l]

ET I 224(2)a

[Re a > 0, !

 √  √  (l + b)π !! iπx l dx = √ cosh b a cosech l a ! 2 2l l a

|b| ≤ l]

ET I 224(3)a

[Re a > 0,

|b| ≤ l]

ET I 224(4)a

√ √ 1  √ √  √  e−(a−μ)x ϑ3 (π μx |iπx ) dx = √ coth a + μ + coth a − μ 2 a [Re a > 0]

∞

2.10 0



e−ax ϑ3



bπ 2l

∞

ϑ3 (iπkx | iπx) e−(k

2

+l2 )x

dx =

ET I 224(7)a

sinh 2l l (cosh 2l − cos 2k)

       1 ϑ4 0 | ie2x + ϑ2 0 | ie2x − ϑ3 0 | ie2x e 2 x cos(ax) dx 0      1 1 1 1  1 +ia = − 1 1 − 2 2 −ia π − 4 − 2 ia Γ 14 + 12 ia ζ 12 + ia 22 2 ET I 61(11) [a > 0]  ∞     1 6.165 e 2 x ϑ3 0 | ie2x − 1 cos(ax) dx 0  +  − 1 ia− 1  1   , 2 2 1 1 1 2 4 Γ = 1 + a π ζ ia + + ia + 4 2 4 2 1 + 4a2 [a > 0] ET I 61(12) 6.16411

6.165

Generalized elliptic integrals

635

6.1710 Generalized elliptic integrals 1.

Set



π

Ωj (k) ≡

 −(j+ 12 ) 1 − k 2 cos φ dφ,

0

j! (4m + 2j)! π αm (j) = (64)m (2j)! (2m + j)!



1 m!

2 ,

π λ= 2



(2j + 1)k 2 , 1 − k2

then

⎡

  π 1 1 −j 4m 2 −1 ⎣ 1−k erf λ + (2j + 1) αm (j)k = 1+ 2 Ωj (k) = (2j + 1)k 2 2 2k m=0







 2 1 1 13 2 2 −λ2 −2 √ λe × erf λ − − (2j + 1) 16 + 2 + 4 1+ λ 3 12 k k π ⎤





 2 2 4 2 λe−λ + . . .⎦ × erf λ − √ 1 + λ2 + λ4 3 15 π 

∞ 

while for large λ  lim Ωj (k) =

j→∞

2.

 −j π k2 1 − k2 (2j + 1) 



  1 1 1 4 13 −1 −2 × 1 + (2j + 1) + 1 + 2 − (2j + 1) 1+ + ... 2 2k 3 16k 2 16k 4

Set  Rμ (k, α, δ) =

π

cos2α−1 (θ/2) sin2δ−2α−1 (θ/2) dθ

, μ+ 1 [1 − k 2 cos θ] 2 0 < k < 1, Re δ > Re α > 0, Re μ > −1/2,   (−1)ν 2ν μ + 12 ν Γ(α) Γ (δ − α + ν) Mν (μ, α, δ) = , ν! Γ(δ + ν) with (λ)ν = Γ(λ + ν)/ Γ(λ), and   ν 2 μ + 12 ν Γ(α + ν) Γ (δ − α) , Wν (μ, α, δ) = ν! Γ(δ + ν) 0

then: • for small k: 

Rμ (k, α, δ) = 1 − k

1  2 −(μ+ 2 )

−(μ+ 12 )  = 1 + k2

∞   ν=0 ∞  ν=0



ν  k2 / 1 − k2 Mν (μ, α, δ) ν  k2 / 1 + k2 W ν (μ, α, δ),

636

The Exponential Integral Function and Functions Generated by It

6.211

• for k 2 close to 1: Rμ (k, α, δ)     2 α−δ  δ−α−μ− 12   2k 1 − k2 = Γ(δ − α) Γ μ + α − δ + 12 Γ μ + 12 +      μ+ 12 , × Γ δ − α − μ − 12 Γ(α) Γ δ − μ − 12 2k 2     Re μ + α − δ + 12 not an integer , +   1 = 2μ+ 2 k 2μ+1 Γ μ + 12 Γ(1 − α) ×

∞   n=0

   α−δ+μ−n+ 1  2 Γ (δ − α + n) Γ(1 − α + n) Γ α − δ + μ − n + 12 n! 2k 2 / 1 − k 2   α − δ + μ + 12 = m, with m a non-negative integer

6.2–6.3 The Exponential Integral Function and Functions Generated by It 6.21 The logarithm integral 

1

li(x) dx = − ln 2

6.211

BI (79)(5)

0

6.212



1

1. 0



1

2. 0





1 li x dx = 0 x 1 li(x)xp−1 dx = − ln(p + 1) p

1

3.

li(x) 0



∞

4.

li(x) 1

6.213 1. 2. 3. 4. 5.

dx 1 = ln(1 − q) xq+1 q dx 1 = − ln(q − 1) xq+1 q



1 π 1  a ln a − li sin (a ln x) dx = x 1 + a2 2 0  ∞

 1 π 1 + a ln a li sin (a ln x) dx = − 2 x 1+a 2 1  1

 1 π 1 a li ln a + cos (a ln x) dx = − x 1 + a2 2 0  ∞

1 π 1  ln a − a li cos (a ln x) dx = x 1 + a2 2 1    1 ln 1 + a2 dx = li(x) sin (a ln x) x 2a 0 

BI (255)(1)

[p > −1]

BI (255)(2)

[q < 1]

BI (255)(3)

[q > 1]

BI (255)(4)

[a > 0]

BI (475)(1)

[a > 0]

BI (475)(9)

[a > 0]

BI (475)(2)

[a > 0]

BI (475)(10)

[a > 0]

BI(479)(1), ET I 98(20)a

1

6.216

The logarithm integral

 6. 7. 8. 9. 10. 11.

1

arctan a dx =− x a 0  1 dx 1  π li(x) sin (a ln x) 2 = a ln a + [a > 0] x 1 + a2 2 0  ∞ dx 1 π − a ln a [a > 0] li(x) sin (a ln x) 2 = x 1 + a2 2 1  1 π dx 1  ln a − a [a > 0] li(x) cos (a ln x) 2 = x 1 + a2 2 0  ∞ π dx 1  ln a + a [a > 0] li(x) cos (a ln x) 2 = − x 1 + a2 2 1

  1  a  a 1 p−1 2 2 ln (1 + p) + a − p arctan li(x) sin (a ln x) x dx = 2 a + p2 2 1+p 0 li(x) cos (a ln x)



1

li(x) cos (a ln x) xp−1 dx = −

12. 0

6.214 1. 2. 6.215

p  a + ln (1 + p)2 + a 1+p 2

li





1

xp−1 dx = −2 1 ln x

li(x) 0

  2

BI (479)(3) BI (479)(13)

BI (479)(4) BI (479)(14)

BI (477)(1)

[p > 0]

BI (477)(2)

[0 < p < 1]

BI (340)(1)

[0 < p < 1]

BI (340)(9)

1

π √ arcsinh p = −2 p



 π √ ln p+ p+1 p

[p > 0]

BI (444)(3)

[1 > p > 0]

BI (444)(4)

ax 1 = − Γ(p) x p

[0 < p ≤ 1]

BI (444)(1)

dx π Γ(p) =− 2 x sin pπ

[0 < p < 1]

BI (444)(2)

dx . = −2 1 xp+1 ln x

 p−1 1 li(x) ln x 0  p−1  1 1 li(x) ln x 0 



li(x) .

1

2.

2.

[p > 0]

BI (479)(2)

1

0

1.

1 a2 + p2

a arctan



p−1 1 1 dx = −π cot pπ · Γ(p) ln x x 0  ∞

1 π p−1 Γ(p) li dx = − (ln x) x sin pπ 1 

1.

6.216

637



π √ arcsin p p

638

The Exponential Integral Function and Functions Generated by It

6.221

6.22–6.23 The exponential integral function 

6.221 6.222

6.223

1 − eαp NT 11(7) 0 α

∞ 1 1 ln q ln p + − Ei(−px) Ei(−qx) dx = ln(p + q) − p q p q 0 [p > 0, q > 0] FI II 653, NT 53(3)  ∞ Γ(μ) Ei(−βx)xμ−1 dx = − μ [Re β ≥ 0, Re μ > 0] μβ 0 p

Ei(αx) dx = p Ei(αp) +

NT 55(7), ET I 325(10)

6.224



∞

1.

Ei(−βx)e−μx dx = −

0



μ 1 ln 1 + μ β

= −1/β 

[Re(β + μ) ≥ 0,

μ > 0]

[μ = 0] FI II 652, NT 48(8)

∞

2.

Ei(ax)e−μx dx = −

0

μ

1 ln −1 μ a

[a > 0,

Re μ > 0,

μ > a]

ET I 178(23)a, BI (283)(3)

6.225



∞

1.



Ei −x

2



e

−μx2

0



∞

2. 0

6.226 1. 2. 3.

  2  px2 π √ arcsin p Ei −x e dx = − p

∞

∞

4. 0



1. 0

∞

[Re μ > 0]

BI (283)(5), ET I 178(25)a

[1 > p > 0]

NT 59(9)a



1 2 √ Ei − e−μx dx = − K 0 ( μ) 4x μ 0  ∞ 2

a 2 √ Ei e−μx dx = − K 0 (a μ) 4x μ 0

  ∞ π 1 √ −μx2 Ei (− μ) Ei − 2 e dx = 4x μ 0 



6.227

   π π √ √ arcsinh μ = − ln dx = − μ+ 1+μ μ μ



1 Ei − 2 4x

e

−μx2 +

Ei(−x)e−μx x dx =

[Re μ > 0]

MI 34

[a > 0,

MI 34

Re μ > 0]

[Re μ > 0]

MI 34

1  π √ √ √ √ 2 4x dx = [cos μ ci μ − sin μ si μ] μ

1 1 − 2 ln(1 + μ) μ(μ + 1) μ

[Re μ > 0]

MI 34

[Re μ > 0]

MI 34

6.241

The sine integral and cosine integral functions



∞



2. 0

 e−ax Ei(ax) eax Ei(−ax) − dx = 0 x−b x+b = π 2 e−ab

[a > 0,

b < 0]

[a > 0,

b > 0]

639

ET II 253(1)a

6.228



∞

1.

Ei(−x)ex xν−1 dx = −

0



∞

2.

Ei(−βx)e

−μx ν−1

x

0

6.231

1. 2. 6.233

[0 < Re ν < 1]



Γ(ν) μ dx = − 2 F 1 1, ν; ν + 1; ν(β + μ)ν β+μ [|arg β| < π, Re(β + μ) > 0,

ET II 308(13)

Re ν > 0]





√ dx 1 1 √ √ √ √ Ei − 2 exp −μx2 + 2 = 2 π (cos μ si μ − sin μ ci μ) 2 4x 4x x 0 [Re μ > 0]  ∞   −x  −μx 1 [a < 1, Re μ > 0] Ei(−a) − Ei −e e dx = γ(μ, a) μ − ln a 

6.229

6.232

π Γ(ν) sin νπ

∞



b2  ∞ ln 1 + 2 a Ei(−ax) sin bx dx = − 2b 0  ∞ b 1 Ei(−ax) cos bx dx = − arctan b a 0 

∞

1.

Ei(−x)e−μx sin βx dx = −

0



∞

2.

Ei(−x)e−μx cos βx dx = −

0

1 β 2 + μ2

β2

1 + μ2



MI 34 MI 34

[a > 0,

b > 0]

BI (473)(1)a

[a > 0,

b > 0]

BI (473)(2)a

 β  β ln (1 + μ)2 + β 2 − μ arctan 2 1+μ [Re μ > |Im β|]  μ  β ln (1 + μ)2 + β 2 + β arctan 2 1+μ [Re μ > |Im β|]



ET II 308(14)





BI (473)(7)a

BI (473)(8)a

∞

Ei(−x) ln x dx = C + 1

6.234

NT 56(10)

0

6.24–6.26 The sine integral and cosine integral functions 6.241



∞

1.

si(px) si(qx) dx =

π 2p

[p ≥ q]

BI II 653, NT 54(8)

ci(px) ci(qx) dx =

π 2p

[p ≥ q]

FI II 653, NT 54(7)

0

 2.

0

∞

640

The Exponential Integral Function and Functions Generated by It



∞

3. 0

 2 2

2 p − q2 p+q 1 1 ln ln si(px) ci(qx) dx = + 4q p−q 4p q4 1 = ln 2 q

6.242

[p = q] [p = q] FI II 653, NT 54(10, 12)



∞

6.242 0

6.243



 1 ci(ax) 2 2 dx = − [si(aβ)] + [ci(aβ)] β+x 2

2. 6.244



∞

1.8 2.

x dx π = − ci(pq) q 2 − x2 2

[p > 0,

q > 0]

BI (255)(6)

[p > 0,

q > 0]

BI (255)(7)

[p > 0,

q > 0]

BI (255)(8)

[a > 0,

0 < Re μ < 1]

∞

ci(px) 0



∞

1.

q2

dx π Ei(−pq) = 2 +x 2q

dx π si(pq) = q 2 − x2 2q

si(ax)xμ−1 dx = −

0



ET II 253(2)

si(px)

0

2.

[a > 0]

BI (255)(6)

ci(px)

6.246

ET II 253(3)

q > 0]

∞

1. 

b > 0]

[p > 0,

0



[a > 0,

x dx π = Ei(−pq) q 2 + x2 2

∞

8

6.245

ET II 224(1)

si(px) 0



|arg β| < π]

∞

si (a|x|) sign x dx = π ci (a|b|) −∞ x − b  ∞ ci (a|x|) dx = −π sign b · si (a|b|) −∞ x − b

1.

[a > 0,

Γ(μ) μπ sin μ μa 2

NT 56(9), ET I 325(12)a ∞

2.

ci(ax)xμ−1 dx = −

0

Γ(μ) μπ cos μaμ 2

[a > 0,

0 < Re μ < 1] NT 56(8), ET I 325(13)a

6.247

∞

μ 1 arctan μ β 0 .  ∞ 1 μ2 −μx ci(βx)e dx = − ln 1 + 2 μ β 0

1.

2. 6.248 1.





8

si(βx)e−μx dx = −

∞

si(x)e 0

−μx2



 1 π x dx = Φ −1 √ 4μ 2 μ

[Re μ > 0]

NT 49(12), ET I 177(18)

[Re μ > 0]

NT 49(11), ET I 178(19)a

[Re μ > 0]

MI 34

6.253

The sine integral and cosine integral functions



1 π Ei − ci(x)e [Re μ > 0] μ 4μ 0 

2  2

2   ∞+  2  π , −μx μ2 μ π 1 1 si x + e S dx = + C − − 2 μ 4 2 4 2 0

 2.

6.249

6.251  1. 2.

641

∞

−μx2

1 dx = 4





1 2 √ si e−μx dx = kei (2 μ) x μ 0  ∞

1 2 √ ci e−μx dx = − ker (2 μ) x μ 0

6.252  1.

[Re μ > 0]

ME 26

[Re μ > 0]

MI 34

[Re μ > 0]

MI 34

∞

∞

0



π 2p π =− 4p

sin px si(qx) dx = −

 

=0 

MI 34

∞

2.6

cos px si(qx) dx = −

0

=

1 ln 4p



1 q

p+q p−q

p2 > q 2 p2 = q 2 p2 < q 2

   FI II 652, NT 50(8)

2



p = 0,

p2 = q 2



[p = 0] FI II 652, NT 50(10)



∞

3.

sin px ci(qx) dx = −

0



1 ln 4p



p2 −1 q2

2



=0

[p = 0]

π 2p π =− 4p



0

cos px ci(qx) dx = −

 

=0

6.253 0

p2 = q 2



FI II 652, NT 50(9) ∞

4.



p = 0,

∞



m

m+1

p2 > q

 2

p2 = q 2 p2 < q 2

  FI II 654, NT 50(7)



π r +r si(ax) sin bx dx = − 2 2 1 − 2r cos x + r 4b(1  − r) (1 −mr ) m+1  π 2 + 2r − r − r =− 4b(1 − r) (1 − r2 ) πrm+1 =− 2 2b(1  − r) (1 − r ) π 1 + r − rm+1 =− 2b(1 − r) (1 − r2 )

[b = a − m] [b = a + m] [a − m − 1 < b < a − m] [a + m < b < a + m + 1] ET I 97(10)

642

The Exponential Integral Function and Functions Generated by It

6.254  1.∗

∞

0



 1 1 dx 1 = ci(x) sin x L2 − L2 − x 2 2 2 2

 where L2 (x) is the Euler dilogarithm defined as L2 (z) = −

2.11

log(1 − t) dt and this in turn can t 0 be expressed as L2 (z) = Φ(z, 2, 1) in terms of the Lerch function defined in 9.550, with z real.  ∞+ π a π, dx = ln H(a − b) si(ax) + cos bx · 2 x 2 b 0 [a > 0, b > 0, H(x) is the Heaviside step function] ET I 41(11)

6.255  1.

∞

[cos ax ci (a|x|) + sin (a|x|) si (a|x|)]

−∞



z

dx = −π [sign b cos ab si (a|b|) − sin ab ci (a|b|)] x−b [a > 0]

∞

2. −∞

[sin ax ci (a|x|) − sign x cos ax si (a|x|)]

6.256  1.

∞

0



∞

 2  π si (x) + ci2 (x) cos ax dx = ln(1 + a) a 2

[si(x) cos x − ci(x) sin x] dx =

0

3.∗



∞

si2 (x) cos(ax) dx =

0

4.∗ 6.257



∞

6.258  1.

∞

0

 2. 0

ET II 253(5)

[a > 0]

π 2

π log(1 + a) 2a

π log(1 + a) 2a 0  ∞   √ π a sin bx dx = − J 0 2 ab si x 2b 0 ci2 (x) cos(ax) dx =

ET II 253(4)

dx = −π [sin (a|b|) si (a|b|) + cos ab ci (a|b|)] x−b [a > 0]

2.∗

6.254

[0 ≤ a ≤ 2] [0 ≤ a ≤ 2] [b > 0]

+ π, dx si(ax) + sin bx 2 2 x + c2  π  −bc e = [Ei(bc) − Ei(−ac)] + ebc [Ei(−ac) − Ei(−bc)] 4c π = e−bc [Ei(ac) − Ei(−ac)] 4c

ET I 42(18)

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0] BI (460)(1)

∞

+ π, x dx si(ax) + cos bx 2 2 x + c2  π  −bc e =− [Ei(bc) − Ei(−ac)] + ebc [Ei(−bc) − Ei(−ac)] 4 π = e−bc [Ei(−ac) − Ei(ac)] 4

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0]

BI (460)(2, 5)

6.262

6.259

The sine integral and cosine integral functions



∞

si(ax) sin bx

1. 0



dx π Ei(−ac) sinh(bc) = x2 + c2 2c π = e−cb [Ei(−bc) + Ei(bc) − Ei(−ac) − Ei(ac)] 4c π + Ei(−bc) sinh(bc) 2c

ci(ax) sin bx 0

∞

ci(ax) cos bx 0

4.∗

x dx π = − sinh(bc) Ei(−ac) 2 +c 2 π π = − sinh(bc) Ei(−bc) + e−bc [Ei(−bc) + Ei(bc) 2 4 − Ei(−ac) − Ei(ac)]

x2

dx x2 + c2 π cosh bc Ei(−ac) = 2c  π  −bc = e [Ei(ac) + Ei(−ac) − Ei(bc)] + ebc Ei(−bc) 4c

5.∗

[0 < a ≤ b,

c > 0]

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0]

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0]

BI (460)(4), ET I 41(15) ∞

[ci(x) sin x − Si(x) cos x] sin x

0



c > 0]

BI (460)(3)a, ET I 97(15)a

3.



[0 < b ≤ a,

ET I 96(8) ∞

2.



643

2 x dx 1 Ei(a)e−a − Ei(−a)ea = 2 +x 8

a2

[a real] ∞

[ci(x) sin x − Si(x) cos x]

2

0

2 π  x dx π 3 e−|a| sinh(a) − Ei(a)e−a − Ei(−a)ea = 2 +x 8a 8|a|

a2

[a real] 6.261



∞

1.

−px

si(bx) cos axe 0

  a p2 + (a + b)2 1 2bp ln dx = − + p arctan 2 2 (a2 + p2 ) 2 p2 + (a − b)2 b − a2 − p2 [a > 0,



∞

2.

si(βx) cos axe−μx dx = −

0



1.

∞

−μx

ci(bx) sin axe 0

p > 0]

ET I 40(8)

Re μ > |Im β|]

ET I 40(9)

μ − ai μ + ai arctan β β − 2(μ + ai) 2(μ − ai)

arctan

[a > 0, 6.262

b > 0,

1 dx = 2 2 (a + μ2 )



  2 2 μ + b2 − a2 + 4a2 μ2 2aμ a μ arctan 2 − ln μ + b 2 − a2 2 b4 [a > 0, b > 0, Re μ > 0] ET I 98(16)a

644

The Exponential Integral Function and Functions Generated by It



∞

2.

ci(bx) cos axe−px dx =

0

3.

⎧ ⎨p

−1 ln 2 (a2 + p2 ) ⎩ 2

, + 2 b2 + p2 − a2 + 4a2 p2 b4

[a > 0, b > 0,     2 (μ − ai)2 (μ + ai)  ∞ ln 1 + − ln 1 + β2 β2 − ci(βx) cos axe−μx dx = 4(μ + ai) 4(μ − ai) 0 [a > 0,

6.263

⎫ ⎬

+ a arctan

2ap b2 + p2 − a2 ⎭

Re p > 0]

Re μ > |Im β|]

ET I 41(16)

ET I 41(17)

6.263 π − μ ln μ [ci(x) cos x + si(x) sin x] e dx = 2 1 + μ2 0 π  ∞ − μ + ln μ −μx [si(x) cos x − ci(x) sin x] e dx = 2 1 + μ2 0    ∞ ln 1 + μ2 [sin x − x ci(x)] e−μx dx = 2μ2 0 

1.

2. 3. 6.264



∞

−μx

−

[Re μ > 0]

ME 26a, ET I 178(21)a

[Re μ > 0]

ME 26a, ET I 178(20)a

[Re μ > 0]

ME 26

∞

si(x) ln x dx = C + 1

1.

NT 46(10)

0



∞

2.

ci(x) ln x dx = 0

π 2

NT 56(11)

6.27 The hyperbolic sine integral and hyperbolic cosine integral functions 6.271



∞

μ+1 1 1 ln = arccoth μ 2μ μ − 1 μ 0  ∞   1 ln μ2 − 1 2.11 chi(x)e−μx dx = − 2μ 0

  ∞ 2 1 1 π Ei 6.27211 chi(x)e−px dx = 4 p 4p 0 6.273  ∞ ln μ [cosh x shi(x) − sinh x chi(x)] e−μx dx = 2 1.11 μ −1 0  ∞ μ ln μ 2.11 [cosh x chi(x) + sinh x shi(x)] e−μx dx = 1 − μ2 0 1.

shi(x)e−μx dx =

[Re μ > 1]

MI 34

[Re μ > 1]

MI 34

[p > 0]

MI 35

[Re μ > 0]

MI 35

[Re μ > 2]

MI 35

6.284

The probability integral



∞

11

6.274

[cosh x shi(x) − sinh x chi(x)] e

−μx2

0





1 π 4μ 1 e Ei − μ 4μ [Re μ > 0]

MI 35

  ln μ2 − 1 −μx [x chi(x) − sinh x] e dx = − [Re μ > 1] 2μ2

 0 ∞ 1 1 π 1 −μx2 [cosh x chi(x) + sinh x shi(x)] e x dx = exp Ei − 8 μ3 4μ 4μ 0 [Re μ > 0] 

6.275 6.276

6.277

1 dx = 4

645



∞

∞

1.

[chi(x) + ci(x)] e

−μx

0



∞

2.

  ln μ4 − 1 dx = − 2μ

[chi(x) − ci(x)] e−μx dx =

0

μ2 + 1 1 ln 2 2μ μ − 1

MI 35

MI 35

[Re μ > 1]

MI 34

[Re μ > 1]

MI 35

6.28–6.31 The probability integral 6.281 1.



∞

6

[1 − Φ(px)] x

2q−1

0

  Γ q + 12 √ dx = 2 πqp2q

[Re q > 0,

Re p > 0] NT 56(12), ET II 306(1)a



∞

2.6 0





b 2b b 1 − Φ atα ± α dt = √ t π a

1−α 2α

+

, K 1+α (2ab) ± K 1−α (2ab) e±2ab 2α

2α

[a > 0, 6.282



∞

1.

Φ(qt)e−pt dt =

0



  2

p p 1 1−Φ exp p 2q 4q 2

b > 0,

+ Re p > 0,

α = 0]

|arg q| <

π, 4

MO 175, EH II 148(11)

∞

2. 0



 

 1 1 (μ + 1)2 μ+1 1 1 exp Φ x+ −Φ e−μx+ 4 dx = 1−Φ 2 2 (μ + 1)(μ + 2) 4 2

ME 27

6.283 1. 2.

  √  √  α 1 √ eβx 1 − Φ αx dx = −1 β α−β 0  ∞ √ √  −pt q 1 √ Φ qt e dt = p p +q 0 

∞

[Re α > 0,

Re β < Re α]

[Re p > 0,

Re(q + p) > 0]

ET II 307(5)

EH II 148(12)



∞

6.284 0



 √ q 1 √ 1−Φ e−px dx = e−q p p 2 x

+ Re p > 0,

|arg q| <

π, 4

EF 147(235), EH II 148(13)

646

6.285

The Exponential Integral Function and Functions Generated by It



∞

1.

[1 − Φ(x)] e−μ

2

x2

dx =

0



∞

2.

Φ(iat)e

−a2 t2 −st

0

arctan μ √ πμ

−1 √ exp dt = 2ai π

6.285

[Re μ > 0]

s2 4a2



s2 Ei − 2 4a + Re s > 0,

MI 37



|arg a| <

π, 4 EH II 148(14)a

6.286



ν+1 Γ ∞ 2 2 2 [1 − Φ(βx)] eμ x xν−1 dx = √ πνβ ν 0



 1.



∞

%

√ 1−Φ

2. 0

2x 2

2F 1

ν ν+1 ν μ2 , ; + 1; 2 2 2 2 β Re2 β > Re μ2 ,

Re ν > 0

 ET II 306(2)

& e

x2 2

xν−1 dx = 2 2 −1 sec ν

νπ  ν Γ 2 2 [0 < Re ν < 1]

6.287



∞

1.

Φ(βx)e−μx x dx = 2

0



∞

2.



β  2μ μ + β 2

[1 − Φ(βx)] e−μx x dx = 2

0

1 2μ



β 1−  μ + β2



Re μ > − Re β 2 ,

ET I 325(9)

Re μ > 0



ME 27a, ET I 176(4)



Re μ > − Re β 2 ,

Re μ > 0



NT 49(14), ET I 177(9)





  r 1 r 1 A B Q(rA) Q(rB) dr = − 3.∗ I= exp α arctan + β arctan B = A 2 σ2 4 2π αB βA −∞ σ 1 1 1 B=A = − α arctan 4 π  α     ∞ 2 x σ 2 A2 σ2 B 2 1 1 Q(x) = √ , α= e−t /2 dt = , β= , 1 − erf √ 2 2 2 1+σ A 1 + σ2 B 2 2π x 2  ∞   2 ai 6.288 a > 0, Re μ > Re a2 Φ(iax)e−μx x dx =  MI 37a 2 2μ μ − a 0 6.289  ∞ + 2 2 2 β π, 2 2 Re 1. Φ(βx)e(β −μ )x x dx = μ > Re β , |arg μ| < 2μ (μ2 − β 2 ) 4 0 

 2. 0

∞

ET I 176(5) ∞

[1 − Φ(βx)] e(β

2

−μ2 )x2

x dx =

1 2μ(μ + β)

+ Re2 μ > Re β 2 ,

arg μ <

π, 4 ET I 177(10)

6.297

The probability integral

√ b−a √ [Re μ > −a > 0, 2(μ + a) μ + b 0 2    ∞ μ μ i 1 −(μx+x2 ) + Ei − x dx = √ Φ(ix)e [Re μ > 0] 4 π μ 4  0 ∞ 2 2 arctan μ 1 1 [1 − Φ(x)] e−μ x x2 dx = √ − μ3 μ2 (μ2++ 1) 2 π 0 π, |arg μ| < 4 √  ∞  μ+1+1 1 −μx2 dx = ln √ Φ(x)e = arccoth μ + 1 x 2 μ+1−1 0 [Re μ > 0]

 3.

6.291 6.292

6.293 6.294



∞

∞

1. 0



∞

2. 0

6.295

647

√ 2 Φ b − ax e−(a+μ)x x dx =

  2 2 β 1 exp(−2βμ) 1−Φ e−μ x x dx = x 2μ2

+ π |arg β| < , 4

  2 2 dx 1 = − Ei(−2μ) 1−Φ e−μ x x x

+ π, |arg μ| < 4

b > a]

ME 27

MI 37

MI 37

MI 37a

|arg μ| <

π, 4 ET I 177(11) MI 37

 

1 1 1 2 2 1. 1−Φ exp −μ x + 2 dx = √ [sin 2μ ci(2μ) − cos 2μ si(2μ)] x x πμ 0 + π, |arg μ| < 4 

 ∞ 1 1 1 π [H1 (2μ) − Y 1 (2μ)] − 2 2. 1−Φ exp −μ2 x2 + 2 x dx = x x 2μ μ 0 + π, |arg μ| < 4 

 ∞ 1 dx π 1 = [H0 (2μ) − Y 0 (2μ)] 3. 1−Φ exp −μ2 x2 + 2 x x x 2 0 + π, |arg μ| < 4   

   ∞   2 2 2 a2 2 a 1 −aμ√2 ax · e− 2x2 e−μ x x dx = 6.296 e x + a2 1 − Φ √ − π 2μ4 2x 0 , + π |arg μ| < , a > 0 4 6.297

  ∞ 2 2 1 β √  exp [−2 (βγ + β μ)] 1. 1 − Φ γx + e(γ −μ)x x dx = √ √ x 2 μ μ+γ 0 

 2. 0

∞

∞

[Re β > 0, Re μ > 0] 

 2     b + 2ax e−bμ 1−Φ exp − μ2 − a2 x2 + ab x dx = 2x 2μ(μ + a) [a > 0,

b > 0,

Re μ > 0]

MI 37

MI 37

MI 37

MI 38a

ET I 177(12)a

MI 38

648

The Exponential Integral Function and Functions Generated by It



∞

3. 0

6.298







    2 b + 2ax2 b − 2ax2 1 −ab + 1−Φ 1−Φ e eab e−μx x dx = exp −b a2 + μ 2x 2x μ

MI 38 [a > 0, b > 0, Re μ > 0]



 ∞ 2 2 b − 2ax2 b + 2ax2 1 √ exp (−b μ) 2 cosh ab − e−ab Φ − eab Φ e−(μ−a )x x dx = 2 2x 2x μ − a 0 MI 38 [a > 0, b > 0, Re μ > 0]  ∞ + ,     1 2 exp 12 a2 K ν a2 cosh(2νt) exp (a cosh t) [1 − Φ (a cosh t)] dt = 2 cos(νπ) 0  Re a > 0, − 12 < Re ν < 12

 6.298

6.299



b2 1 [1 − Φ(ax)] sin bx dx = [a > 0, b > 0] 1 − e− 4a2 b  0 √ √   ∞ b + a2 + a 2b a 2b 1 2 √ + 2 arctan ln Φ(ax) sin bx dx = √ b − a2 4 2πb b + a2 − a 2b 0 [a > 0, b > 0]

6.311 6.312

∞

6.313 

α ⎞ 12 ,− 12 + 1  √  1 α2 + β 2 2 − α sin(βx) 1 − Φ αx dx = − ⎝ 2 2 2 ⎠ β α +β

∞

⎛ √  cos(βx) 1 − Φ αx dx = ⎝

0

[Re α > |Im β|]  2.



0



∞

1. 0

ET I 96(3)

ET II 307(6)

α ⎞ + ,− 12 1 2 2 2 2 ⎠ α + β + α α2 + β 2 1 2

[Re α > |Im β|] 6.314

ET I 96(4)

⎛

∞

1.

ET II 308(10)

ET II 307(7)

  , + , + 1 1 a sin(bx) 1 − Φ dx = b−1 exp −(2ab) 2 cos (2ab) 2 x 

[Re a > 0, b > 0]  

 ∞ + , + , 1 1 a cos(bx) 1 − Φ dx = −b−1 exp −(2ab) 2 sin (2ab) 2 x 0

ET II 307(8)

 2.

[Re a > 0, 6.315



∞

1.

ν−1

x 0

 2. 0

∞

b > 0]

ET II 307(9)

 

Γ 1 + 12 ν β ν+1 ν 3 ν+3 β2 , + 1; , ; − sin(βx) [1 − Φ(αx)] dx = √ F 2 2 2 2 2 2 4α2 π(ν + 1)αν+1 1

xν−1 cos(βx) [1 − Φ(αx)] dx =

Γ 2 + 12 ν √ πναν



2F 2

Re ν > −1]

ν ν+1 1 ν β2 , ; , + 1; − 2 2 2 2 2 4α [Re α > 0,

[Re α > 0,

Re ν > 0]

ET II 307(3)

ET II 307(4)

6.323

Fresnel integrals



∞

3. 0



∞

4.





 1 b2 1 b2 [1 − Φ(ax)] cos bx · x dx = 2 exp − 2 − 2 1 − exp − 2 2a 4a b 4a 

[Φ(ax) − Φ(bx)] cos px

0



∞

5. 0

649



2

1 dx p = Ei − 2 x 2 4b

 √  1 1 x− 2 Φ a x sin bx dx = √ 2 2πb



− Ei

[a > 0,

 p2 4a2

b > 0]

ET I 40(5)

[a > 0, b > 0, p > 0] & % √ & √ b + a 2b + a2 a 2b √ ln + 2 arctan 2 b − a2 b − a 2b + a



%

[a > 0,   



 ∞ 1 2 x b π b2 x 2 2 √ √ e e 1−Φ 1−Φ sin bx dx = 2 2 2 0 [b > 0] √  ∞ 2 2 i π − b22 e 4a e−a x Φ(iax) sin bx dx = [b > 0] a 2 0 ∞  2 2 2 1 − e−p − √ (1 − Φ(p)) [1 − Φ(x)] si(2px) dx = πp π 0 [p > 0]

b > 0]

ET I 40(6)

ET I 96(3)

 6.316

6.3176 6.318

ET I 96(5) ET I 96(2)

NT 61(13)a

6.32 Fresnel integrals 6.321 

∞



1. 0



∞

2. 0

6.322 1. 2.



 1 − S (px) x2q−1 dx = 2  1 − C (px) x2q−1 dx = 2

√   2q + 1 π 2 Γ q + 12 sin 4 √ 2q 4 πqp √   2q + 1 π 2 Γ q + 12 cos 4 √ 2q 4 πqp

  p  p2 1 p2 −C cos + sin 4 2 2 4 0

   ∞  2 p 1 p 1 p2 −S C (t)e−pt dt = cos − sin p 4 2 2 4 0 

∞

S (t)e−pt dt =

1 p

6.323  1. 0

∞

√ S t e−pt dx =





0 < Re q < 32 ,

p>0

0 < Re q < 32 ,

p>0

 p  1 −S 2 2    p 1 −C 2 2





NT 56(14)a

NT 56(13)a



MO 173a MO 172a



12 p2 + 1 − p  2p p2 + 1

EF 122(58)a

650

The Gamma Function and Functions Generated by It



∞

2.

√ C t e−pt dt =

0

6.324 1. 2. 6.325

12  p2 + 1 + p  2p p2 + 1

 1 1 + sin p2 − cos p2 − S (x) sin 2px dx = 2 4p 0   ∞ 1 1 − sin p2 − cos p2 − C (x) sin 2px dx = 2 4p 0 

∞



∞

1. 0



∞

2.

6.324

EF 122(58)a



√ π −5 2 2 S (x) sin b x dx = b =0

[p > 0]

NT 61(11)a

0 < b2 < 1   2 b >1



ET I 98(21)a

√

0

NT 61(12)a



2 2

C (x) cos b2 x2 dx =

[p > 0]



π −5 2 2 b

0 < b2 < 1   2 b >1

=0



ET I 42(22)

6.326



∞



∞



1. 0

 2. 0

  π 1/2 1 1 + sin p2 − cos p2 − S (x) si(2px) dx = (S (p) + C (p) − 1) − 2 8 4p [p > 0]   1 π 1/2 1 − sin p2 − cos p2 − C (x) si(2px) dx = (S (p) − C (p)) − 2 8 4p [p > 0]

NT 61(15)a

NT 61(14)a

6.4 The Gamma Function and Functions Generated by It 6.41 The gamma function 

∞

11

6.411

−∞

Γ(α + x) Γ(β − x) dx = −iπ21−α−β Γ(α + β) [Re(α + β) < 1 and either Im α < 0 < Im β or Im β < 0 < Im α ] ET II 297(1)

= iπ2

1−α−β

Γ(α + β) [Re(α + β) < 1,

Im α < 0,

Im β < 0] ET II 297(2)

=0 [Re(α + β) < 1,

Im α > 0,

Im β > 0] ET II 297(3)

6.415

The gamma function



i∞

6.412 −i∞

Γ(α + s) Γ(β + s) Γ(γ − s) Γ(δ − s) ds = 2πi

651

Γ(α + γ) Γ(α + δ) Γ(β + γ) Γ(β + δ) Γ(α + β + γ + δ) [Re α, Re β, Re γ, Re δ > 0] ET II 302(32)

6.413



∞

1.

√ 2

|Γ(a + ix) Γ(b + ix)| dx =

0

 2.

    √ !2 1 1 ! Γ b − a − π Γ(a) Γ a + Γ(a + ix) 2 2 ! !   ! Γ(b + ix) ! dx = 2 Γ(b) Γ b − 12 Γ(b − a)

1.



2. 3.

5.

6.

b > 0]



0
[Im α = 0,

1 2



ET II 302(28)

ET II 297(4) ∞

α+β−2

ET II 297(5)

ET II 299(18)

∞

±2π 2 i Γ(α + β − γ − δ − 1) Γ(γ + x) Γ(δ + x) dx = sin[π(γ − δ)] Γ(α − γ) Γ(α − δ) Γ(β − γ) Γ(β − δ) −∞ Γ(α + x) Γ(β + x)

[Re(α + β − γ − δ) > 1, Im γ < 0, Im δ < 0. In the numerator, we take the plus sign if Im γ > Im δ and the minus sign if Im γ < Im δ.] ET II 300(19)  1   ∞ π exp ± 2 π(δ − γ)i Γ(α − β − γ + x + 1) dx     = Γ(β + γ − 1) Γ 12 (α + β) Γ 12 (γ − δ + 1) −∞ Γ(α + x) Γ(β − x) Γ(γ + x) [Re(β + γ) > 1, δ = α − β − γ + 1, Im δ = 0. The sign is plus in the argument if the ET II 300(20) exponential for Im δ > 0 and minus for Im δ < 0.]  ∞ Γ(α + β + γ + δ − 3) dx = Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) −∞ [Re(α + β + γ + δ) > 3]

6.415

ET II 302(27)

Re(α − β) < −1]

2 dx = [Re(α + β) > 1] Γ(α + x) Γ(β − x) Γ(α + β − 1) −∞  ∞ Γ(γ + x) Γ(δ + x) dx = 0 Γ(α + x) Γ(β + x) −∞ [Re(α + β − γ − δ) > 1, Im γ, Im δ > 0] 

4.

∞

Γ(α + x) dx = 0 −∞ Γ(β + x)



[a > 0,

∞ !!

0

6.414

    π Γ(a) Γ a + 12 Γ(b) Γ b + 12 Γ(a + b)   2 Γ a + b + 12



−∞

1. −∞

R(x) dx Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x)

ET II 300(21)

 1 Γ(α + β + γ + δ − 3) R(t) dt = Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) 0 [Re(α + β + γ + δ) > 3, R(x + 1) = R(x)] ET II 301(24)

652

The Gamma Function and Functions Generated by It



  R(t) cos 12 π(2t + α − β) dt R(x) dx



= 0 α+β γ+δ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ Γ Γ Γ(α + δ − 1) 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2, R(x + 1) = −R(x)]

 2.

6.421

1

∞

ET II 301(25)

6.42 Combinations of the gamma function, the exponential, and powers 6.421



∞

1. −∞

Γ(α + x) Γ(β − x) exp [2(πn + θ)xi] dx = 2πi Γ(α + β)(2 cos θ)−α−β exp[(β − α)iθ]

% Re(α + β) < 1,

 2.

3.

4.

6.422

−

π π <θ< , 2 2

× [ηn (β) exp(2nπβi) − ηn (−α) exp(−2nπαi)]  &   0 if 12 − n Im ξ > 0     n an integer, ηn (ξ) = sign 12 − n if 12 − n Im ξ < 0 ET II 298(7)

∞

eπicx dx =0 −∞ Γ(α + x) Γ(β − x) Γ(γ + kx) Γ(δ − kx) [Re(α + β + γ + δ) > 2, c and k are real,



∞



i∞

|c| > |k| + 1]

ET II 301(26)

Γ(α + x) exp[(2πn + π − 2θ)xi] dx −∞ Γ(β + x)   (2 cos θ)β−α−1 = 2πi sign n + 12 exp[−(2πn + π − θ)αi + θi(β − 1)] Γ(β − α) + ,   π π Re(β − α) > 0, − < θ < , n is an integer, n + 12 Im α < 0 ET II 298(8) 2 2  ∞ Γ(α + x) exp[(2πn + π − 2θ)xi] dx = 0 −∞ Γ(β + x) + ,   π π Re(β − α) > 0, − < θ < , n is an integer, n + 12 Im α > 0 ET II 297(6) 2 2

1. −i∞

    Γ(s − k − λ) Γ λ + μ − s + 12 Γ λ − μ − s + 12 z s ds 



Re(k + λ) < 0,

1

   z − k − μ Γ 12 − k + μ z λ e 2 W k,μ (z)  Re λ > |Re μ| − 12 , |arg z| < 32 π ET II 302(29)

= 2πi Γ

2

γ+i∞

2. γ−i∞

Γ(α + s) Γ(−s) Γ(1 − c − s)xs ds = 2πi Γ(α) Γ(α − c + 1)Ψ(α, c; x)   − Re α < γ < min (0, 1 − Re c) , − 32 π < arg x < 32 π

EH I 256(5)

6.422

The gamma function, the exponential, and powers



γ+i∞

3.

Γ(−s) Γ(β + s)ts ds = 2πi Γ(β)(1 + t)−β

653

[0 > γ > Re(1 − β),

|arg t| < π]

γ−i∞





∞i

4.

Γ −∞i



t−p 2

i∞

5.

Γ(s) Γ −i∞

 6.

EH I 256, BU 75

√ t−p−2 1 2 2 z t dt = 2πie 4 z Γ(−p) D p (z) Γ(−t)   |arg z| < 34 π, p is not a positive integer

2ν +

1 4

  − s Γ 12 ν −

1 4

 −s



z2 2

WH

s ds

    1 1 1 3 2 = 2πi · 2 4 − 2 ν z − 2 e 4 z Γ 12 ν + 14 Γ 12 ν − 14 D ν (z)   |arg z| < 34 π, ν = 12 , − 12 , − 32 , . . . EH II 120

c+i∞

3 c−i∞



1



 1 −s  1   −1 Γ 2 ν + 12 s Γ 1 + 12 ν − 12 s ds = 4πi J ν (x) 2x [x > 0, − Re ν < c < 1]

−c+i∞

7. −c−i∞

 ν+2s 1 Γ(−ν − s) Γ(−s) − 12 iz ds = −2π 2 e 2 iνπ H (1) ν (z) + π |arg(−iz)| < , 2

EH II 21(34)

, 0 < Re ν < c EH II 83(34)



−c+i∞

8. −c−i∞

Γ(−ν − s) Γ(−s)

 1 ν+2s 1 ds = 2π 2 e− 2 iνπ H (2) ν (z) 2 iz + π |arg(iz)| < , 2

, 0 < Re ν < c EH II 83(35)

9. 10.

 1 ν+2s  i∞ x ds = 2πi J ν (x) Γ(−s) 2 [x > 0, Re ν > 0] EH II 83(36) Γ(ν + s + 1) −i∞  i∞   5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (−2iz)s ds = −π 2 e−i(z−νπ) sec(νπ)(2z)−ν H (1) ν (z) −i∞   |arg(−iz)| < 32 π, 2ν = ±1, ±3 . . . EH II 83(37)

 11.

  5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (2iz)s ds = π 2 ei(z−νπ) sec(νπ)(2z)−ν H (2) ν (z) −i∞  |arg(iz)| < 32 π, 2ν = ±1, i∞

 ±3 . . . EH II 84(38)



i∞

12.

Γ(s) Γ −i∞

1 2

   3 3 1 − s − ν Γ 12 − s + ν (2z)s ds = 2 2 π 2 iz 2 ez sec(νπ) K ν (z)  |arg z| < 32 π, 2ν = ±1,

 ±3, . . . EH II 84(39)



− 12 +i∞

13. − 12 −i∞

Γ(−s) 2s x ds = 4π s Γ(1 + s)



∞

2x

J 0 (t) dt t

[x > 0]

MO 41

654

The Gamma Function and Functions Generated by It

 14.

6.423

i∞

Γ(α + s) Γ(β + s) Γ(−s) Γ(α) Γ(β) (−z)s ds = 2πi F (α, β; γ; z) Γ(γ + s) Γ(γ) −i∞

[For arg(−z) < π, the path of integration must separate the poles of the integrand at the points s = 0, 1, 2, 3, . . . from the poles s = −α − n and s = −β − n (for n = 0, 1, 2, . . . ).] 

15.

16.

δ+i∞

Γ(α + s) Γ(−s) 2πi Γ(α) (−z)s ds = 1 F 1 (α; γ; z) Γ(γ + s) Γ(γ) δ−i∞ + π , π − < arg(−z) < , 0 > δ > − Re α, γ = 0, 1, 2, . . . 2 2 & %   2  i∞ +  1  1 , Γ 12 − s 1 z s ds = 2πiz 2 2π −1 K 0 4z 4 − Y 0 4z 4 Γ(s) −i∞

EH I 62(15), EH I 256(4)

[z > 0]     i∞ Γ λ + μ − s + 12 Γ λ − μ − s + 12 s z z ds = 2πiz λ e− 2 W k,μ (z) Γ(λ − k − s + 1) −i∞ +

ET II 303(33)

 17.

Re λ > |Re μ| − 12 ,

 18.



Γ(k − λ + s) Γ λ + μ − s +   Γ μ − λ + s + 12 + −i∞ i∞



 1 2

z s ds = 2πi

Re(k − λ) > 0,

m -



i∞

19. −i∞

j=1 q j=m+1

Γ (bj − s)

n -

Γ (1 − aj + s)

j=1

Γ (1 − bj + s)

p j=n+1

Γ (aj − s)

|arg z| <

ET II 302(30)

 1

Γ k + μ + 2 λ −z z e 2 M k,μ (z) Γ(2μ + 1) Re(λ + μ) > − 12 ,

π, 2

|arg z| <

π, 2

ET II 302(31)

!

! a1 , . . . , ap ! z z s ds = 2πi G pq mn ! b1 , . . . , bq  p + q < 2(m + n); Re ak < 1,

  |arg z| < m + n − 12 p − 12 q π;  k = 1, . . . , n; Re bj > 0, j = 1, . . . , m ET II 303(34)

6.423



∞

1.

e−αx

0

 2. 3.

∞

  dx = ν e−α Γ(1 + x)

  dx = eβα ν e−α , β Γ(x + β + 1) 0  ∞   xm dx = μ e−α , m Γ(m + 1) e−αx Γ(x + 1) 0

MI 39, EH III 222(16)

e−αx

MI 39, EH III 222(16)

[Re m > −1]

MI 39, EH III 222(17)

6.433

Gamma functions and trigonometric functions

 4.

6.424

∞

655

  xm dx = enα μ e−α , m, n Γ(m + 1) MI 39, EH III 222(17) Γ(x + n + 1) 0  α+β−2 θ   1  ∞ 2 cos 1 R(x) exp[(2πn + θ)xi] dx 2 = exp θ(β − α)i R(t) exp(2πnti) dt Γ(α + x) Γ(β − x) Γ(α + β − 1) 2 −∞ 0 [Re(α + β) > 1, −π < θ < π, n is an integer, R(x + 1) = R(x)] ET II 299(16) e−αx

6.43 Combinations of the gamma function and trigonometric functions 6.431 

 −∞

1. −∞

sin rx dx = Γ(p + x) Γ(q − x)

2 cos

r p+q−2 r(q − p) sin 2 2 Γ(p + q − 1)

=0

[|r| > π] [r is real;

 r p+q−2 r(q − p) 2 cos cos cos rx dx 2 2 = Γ(p + q − 1) −∞ Γ(p + x) Γ(q − x)

 2.

[|r| < π]

Re(p + q) > 1]

dx sin(mπx) =0 sin(πx) Γ(α + x) Γ(β − x) −∞

[m is an even integer] α+β−2

2 Γ(α + β − 1)

[m is an odd integer] [Re(α + β) > 1]

ET II 300(22)

, +π (β − α) cos cos πx dx

2

= α+β γ+δ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ 2Γ Γ Γ(α + δ − 1) 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]

ET II 301(23)

 2.

ET II 298(11, 12)

, +π (β − α) sin sin πx dx

2

= α+β γ+δ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ 2Γ Γ Γ(α + δ − 1) 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]

 1.

MO 10a, ET II 299(13, 14)

∞

= 6.433

MO 10a, ET II 298(9, 10)

[|r| > π] [r is real;

6.432

Re(p + q) > 1]

∞

=0



[|r| < π]

∞

∞

656

The Gamma Function and Functions Generated by It

6.441

6.44 The logarithm of the gamma function∗ 6.441



p+1

1.

ln Γ(x) dx = p





1

2.

1 ln 2π + p ln p − p 2 1

ln Γ(1 − x) dx =

ln Γ(x) dx = 0



0 1

3.

ln Γ(x + q) dx = 0



z

FI II 784

1 ln 2π 2

1 ln 2π + q ln q − q 2

FI II 783

[q ≥ 0]

NH 89(17), ET II 304(40)

z(z + 1) z ln 2π − + z ln Γ(z + 1) − ln G(z + 1), 2 2

∞

 z z k z(z + 1) Cz 2 -  z2 2 where G(z + 1) = (2π) exp − 1+ − exp −z + 2 2 k 2k

4.

ln Γ(x + 1) dx = 0

WH

k=1



n

5.

ln Γ(α + x) dx = 0

n−1  k=0

1 1 (a + k) ln(a + k) − na + n ln(2π) − n(n − 1) 2 2 [a ≥ 0;



1

6.442

[a > 0; 

ET II 304(41)

exp(2πnxi) ln Γ(a + x) dx = (2πni)−1 [ln a − exp(−2πnai) Ei(2πnai)]

0

6.443

n = 1, 2, . . .]

n = ±1, ±2, . . .]

ET II 304(38)

1

1 [ln(2πn) + C] NH 203(5), ET II 304(42) 2πn 0  

  1 1 1 1 π 1 + 2 1 + + ··· + ln Γ(x) sin(2n + 1)πx dx = ln + (2n + 1)π 2 3 2n − 1 2n + 1 0

1.

ln Γ(x) sin 2πnx dx =

2.

ET II 305(43)



1

3.

ln Γ(x) cos 2πnx dx = 0

 4.

1

8 0



1

5.

1 4n

2 ln Γ(x) cos(2n + 1)πx dx = 2 π

NH 203(6), ET II 305(44)

%

∞

 ln k 1 (C + ln 2π) + 2 2 2 (2n + 1) 4k − (2n + 1)2

& NH 203(6)

k=2

sin(2πnx) ln Γ(a + x) dx = −(2πn)−1 [ln a + cos(2πna) ci(2πna) − sin(2πna) si(2πna)]

0

[a > 0;  6.

1

n = 1, 2, . . .]

ET II 304(36)

cos(2πnx) ln Γ(a + x) dx = −(2πn)−1 [sin(2πna) ci(2πna) + cos(2πna) si(2πna)]

0

[a > 0;

n = 1, 2, . . .]

ET II 304(37)

∗ Here, we are violating our usual order of presentation of the formulas in order to make it easier to examine the integrals involving the gamma function.

6.457

The incomplete gamma function

657

6.45 The incomplete gamma function 6.451 1. 2. 6.452



∞

1 Γ(β)(1 + α)−β α 0    ∞ 1 1 e−αx Γ(β, x) dx = Γ(β) 1 − α (α + 1)β 0 

∞

1. 0

e−αx γ(β, x) dx =

[β > 0]

MI 39

[β > 0]

MI 39



2 x2 1 e−μx γ ν, 2 dx = 2−ν−1 Γ(2ν)e(aμ) D −2ν (2aμ) 8a μ  |arg a| <

π , 4

1 Re ν > − , 2

 Re μ > 0 ET I 179(36)



∞

2.

e−μx γ

0





1 x2 , 4 8a2



√   2 2 a dx = √ e(aμ) K 14 a2 μ2 μ 3 4

+ π |arg a| < , 4

, Re μ > 0

+ ,  a 1 1 π √ |arg a| < , Re μ > 0 dx = 2a 2 ν μ 2 ν−1 K ν (2 μa) e−μx Γ ν, x 2

2 0 ∞   √ 1 1 α α −βx − 2 ν ν − 2 ν−1 e γ ν, α x dx = 2 α β Γ(ν) exp D −ν √ 8β 2β 0 [Re β > 0, Re ν > 0]

6.453 6.454

ET I 179(35)

∞

ET I 179(32)

ET II 309(19), MI 39a

6.455



∞

1.



αν Γ(μ + ν) β 1, μ + ν; μ + 1; F 1 2 μ(α + β)μ+ν α+β [Re(α + β) > 0, Re μ > 0, Re(μ + ν) > 0]

ET II 309(16)



αν Γ(μ + ν) α γ(ν, αx) dx = 2 F 1 1, μ + ν; ν + 1; ν(α + β)μ+ν α+β [Re(α + β) > 0, Re β > 0, Re(μ + ν) > 0]

ET II 308(15)

xμ−1 e−βx Γ(ν, αx) dx =

0



∞

2.

x 0

6.456 1. 2. 6.457 1. 2.

μ−1 −βx

e



√ √ γ (2ν, α) 1 1 e−αx (4x)ν− 2 γ ν, dx = π 1 4x αν+ 2 0

√ √  ∞ π Γ (2ν, α) 1 −αx ν− 12 e (4x) Γ ν, dx = 1 4x αν+ 2 0 

∞



√ √ γ (2ν + 1, α) 1 √ γ ν + 1, e dx = π 1 4x x αν+ 2 0

√  ∞ √ Γ (2ν + 1, α) (4x)ν 1 e−αx √ Γ ν + 1, dx = π 1 4x x αν+ 2 0 

∞

ν −αx (4x)

MI 39a MI 39a

MI 39 MI 39

658

The Gamma Function and Functions Generated by It



∞

6.458

1−2ν

x



2

exp αx

0



6.458

2

    3  1 b b 2 −ν ν−1 2 sin(bx) Γ ν, αx dx = π 2 α Γ 2 − ν exp D 2ν−2 1 8α  (2α) 2  3π , 0 < Re ν < 1 |arg α| < 2 ET II 309(18)

6.46–6.47 The function ψ(x) 

x

6.461

ψ(t) dt = ln Γ(x) 1

6.462

1

ψ(α + x) dx = ln α 0 ∞ x−α [C + ψ(1 + x)] = −π cosec(πα) ζ(α) 0  1 e2πnxi ψ(α + x) dx = e−2πnαi Ei(2πnαi)

6.463 6.464

[α > 0]

ET II 305(1)

[1 < Re α < 2]

ET II 305(6)

[α > 0;

n = ±i, ±2, . . .]

ET II 305(2)

0

6.465 1.



1

8 0

% & ∞  ln k 2 C + ln 2π + 2 ψ(x) sin πx dx = − π 4k 2 − 1 k=2

(see 6.443 4)



1

2. 0



1 ψ(x) sin(2πnx) dx = − π 2 ∞

6.466

[n = 1, 2, . . .]

NH 204 ET II 305(3)

 −1 [ψ(α + ix) − ψ(α − ix)] sin xy dx = iπe−αy 1 − e−y

0

[α > 0, 6.467  1.

y > 0]

ET I 96(1)

1

sin(2πnx) ψ(α + x) dx = sin(2πnα) ci(2πnα) + cos(2πnα) si(2πnα)

0



[α ≥ 0;

n = 1, 2, . . .]

ET II 305(4)

1

cos(2πnx) ψ(α + x) dx = sin(2πnα) si(2πnα) − cos(2πnα) ci(2πnα)

2. 0

[α > 0; 

1

6.468 0

6.469  1.

1 ψ(x) sin2 πx dx = − [C + ln(2π)] 2

1

ψ(x) sin πx cos πx dx = −

0

 2.8

1

ET II 305(5) NH 204

π 4

n 1 − n2 1 n−1 = ln 2 n+1

ψ(x) sin πx sin(nπx) dx = 0

n = 1, 2, . . .]

NH 204

[n is even] [n > 1 is odd] NH 204(8)a

6.511

6.471

Bessel functions



∞

1.

x−α [ln x − ψ(1 + x)] dx = π cosec(πα) ζ(α)

659

[0 < Re α < 1]

ET II 306(7)

0



∞

2.

  x−α [ln(1 + x) − ψ(1 + x)] dx = π cosec(πα) ζ(α) − (α − 1)−1

0



[0 < Re α < 1] ∞

[ψ(x + 1) − ln x] cos(2πxy) dx =

3. 0

6.472



∞

1.

ET II 306(8)

1 [ψ(y + 1) − ln y] 2

ET II 306(12)

    x−α (1 + x)−1 − ψ  (1 + x) dx = −πα cosec(πα) ζ(1 + α) − α−1

0



[|Re α| < 1] ∞

2.

ET II 306(9)

  x−α x−1 − ψ  (1 + x) dx = −πα cosec(πα) ζ(1 + α)

0

[−2 < Re α < 0] 

∞

6.473

x−α ψ (n) (1 + x) dx = (−1)n−1

0

π Γ(α + n) ζ(α + n) Γ(α) sin πα [n = 1, 2, . . . ;

ET II 306(10)

0 < Re α < 1] ET II 306(11)

6.5–6.7 Bessel Functions 6.51 Bessel functions 6.511



∞

J ν (bx) dx =

1. 0



∞

2. 0

1 b

 νπ 1 Y ν (bx) dx = − tan b 2

[Re ν > −1,

b > 0]

[|Re ν| < 1,

b > 0]

ET II 22(3)

WA 432(7), ET II 96(1)



a

3.

J ν (x) dx = 2 0



a

J 12 (t) dt = 2 S

0

J − 12 (t) dt = 2 C

0

6.

ET II 333(1)

WA 599(4)

√  a

a

J 0 (x) dx = a J 0 (a) + 0

[Re ν > −1]

√  a

a

5. 

J ν+2k+1 (a)

k=0

4. 

∞ 

WA 599(3)

πa [J 1 (a) H0 (a) − J 0 (a) H1 (a)] 2 [a > 0]

ET II 7(2)

660

Bessel Functions



6.512

a

J 1 (x) dx = 1 − J 0 (a)

7.

[a > 0]

ET II 18(1)

0



∞

8.

J 0 (x) dx = 1 − a J 0 (a) +

a



πa [J 0 (a) H1 (a) − J 1 (a) H0 (a)] 2 [a > 0]

ET II 7(3)

[a > 0]

ET II 18(2)

∞

9.

J 1 (x) dx = J 0 (a) a



b

10.

Y ν (x) dx = 2 a

 11.

a

I ν (x) dx = 2  

∞

∞

0

6.512 1.11

ET II 339(46)

∞ 

[Re ν > −1]

(−1)n I ν+2n+1 (a)

ET II 364(1)

n=0

K 0 (ax) =

π 2a

[a > 0]

K 20 (ax) =

π2 4a

[a > 0]

0

13.∗

[Y ν+2n+1 (b) − Y ν+2n+1 (a)]

n=0

0

12.∗

∞ 



μ+ν+1

 ∞ Γ b2 μ+ν+1 ν−μ+1 2 ν −ν−1

F , ; ν + 1; 2 J μ (ax) J ν (bx) dx = b a μ−ν+1 2 2 a 0 Γ(ν + 1) Γ 2 [a > 0, b > 0, Re(μ + ν) > −1, b < a. For a > b, the positions of μ and ν should be reversed.] 

∞

2.7 0





β2 Γ(ν) β F ν, −n; ν − n; 2 J ν+n (αt) J ν−n−1 (βt) dt = ν−n α n! Γ(ν − n) α 1 = (−1)n 2α =0

0

β αν 1 = 2β =0

0

[0 < β = α] [0 < α < β]

∞

MO 50

ν−1

J ν (αx) J ν−1 (βx) dx =

4.

[0 < β < α]

[Re(ν) > 0] ∞

3.8



ET II 48(6)

ν−n−1

[β < α] [β = α] [β > α]

[Re ν > 0] WA 444(8), KU (40)a

2 2b J ν+2n+1 (ax) J ν (bx) dx = bν a−ν−1 P (ν,0) 1− 2 [Re ν > −1 − n, 0 < b < a] n a =0

[Re ν > −1 − n,

0 < a < b] ET II 47(5)

6.513

Bessel functions



∞

5.

661



1 2a

J ν+n (ax) Y ν−n (ax) dx = (−1)n+1

0

Re ν > − 12 ,

a > 0,

 n = 0, 1, 2, . . . ET II 347(57)



∞ b2 b−1 ln 1 − 2 J 1 (bx) Y 0 (ax) dx = − π a 0  a ∞  2 J ν (x) J ν+1 (x) dx = [J ν+n+1 (a)] 

6. 7.

0

 8.

9

9.∗ 10.∗ 6.513

ET II 21(31)

[Re ν > −1]

ET II 338(37)

n=0 ∞

1 δ(b − a) a 0

 ∞ b2 1 ln 1 + 2 K 0 (ax) J 1 (bx) = 2b a 0

 ∞ b2 1 K 0 (ax) I 1 (bx) = − ln 1 − 2 2b a 0

[n = 0, 1, . . .]

k J n (ka) J n (kb) dk =

JAC 110

[a > 0,

b > 0]

[a > 0,

b > 0]



1 + ν + 2μ ∞ 2 2

[J μ (ax)] J ν (bx) dx = a2μ b−2μ−1 1 + ν − 2μ 2 0 [Γ(μ + 1)] Γ 2 ⎡ ⎛

 1.

[0 < b < a]

Γ

1− ⎢ ⎜ 1 − ν + 2μ 1 + ν + 2μ ⎜ , ; μ + 1; ×⎢ ⎣F ⎝ 2 2 [Re ν + Re 2μ > −1, 

∞

2. 0

b−1 Γ [J μ (ax)] K ν (bx) dx = 2 2



2μ + ν + 1 2



∞

3. 0

 4. 0

∞



⎞⎤2 4a2 1 − 2 ⎟⎥ b ⎟⎥ ⎠⎦ 2

0 < 2a < b]

ET II 52(33)

 &2



% 2μ − ν + 1 4a2 −μ 1+ 2 Γ P 1 ν− 1 2 2 2 b [2 Re μ > |Re ν| − 1, Re b > 2|Im a|] ET II 138(18)



ν + 2μ + 1     eμπi Γ 4a2 4a2 2 −μ −μ

P 1 ν− 1 I μ (ax) K μ (ax) J ν (bx) dx = 1 + 2 Q 1 ν− 1 1+ 2 2 2 2 2 ν − 2μ + 1 b b bΓ 2 [Re a > 0, b > 0, Re ν > −1, Re(ν + 2μ) > −1] ET II 65(20)  νπ π sec J μ (ax) J −μ (ax) K ν (bx) dx = P μ1 ν− 1 2 2 2b 2



   2 4a 4a2 1+ 2 1 + 2 P −μ 1 1 2 ν− 2 b b [|Re ν| < 1, Re b > 2|Im a|] ET II 138(21)

662

Bessel Functions

6.514



1 + ν + 2μ %  &2  ∞ Γ e 4a2 2 2 −μ

[K μ (ax)] J ν (bx) dx = 1+ 2 Q 1 ν− 1 2 2 1 + ν − 2μ b 0 bΓ 2     Re a > 0, b > 0, Re 12 ν ± μ > − 12 2μπi

5.



z

J μ (x) J ν (z − x) dx = 2

6. 0



∞ 

(−1)k J μ+ν+2k+1 (z)

[Re μ > −1,

ET II 66(28)

Re ν > −1]

WA 414(2)

k=0 z

7.

J μ (x) J −μ (z − x) dx = sin z

[−1 < Re μ < 1]

WA 415(4)

J μ (x) J 1−μ (z − x) dx = J 0 (z) − cos(z)

[−1 < Re μ < 2]

WA 415(4)

0

 8.

z

0

9.

∗



∞

1 b

b 2 = arcsin πb 2a

J 20 (ax) J 1 (bx) =

0

6.514



∞

Jν

1.

a x

0



∞

2.

Jν

Jν

∞

0

 ET II 57(9)

b > 0,

− 12 < Re ν <

3 2



+ 1 √ , + √ , 1 1 1 K ν (bx) dx = b−1 e 2 i(ν+1)π K 2ν 2e 4 iπ ab + b−1 e− 2 i(ν+1)π K 2ν 2e− 4 πi ab x   a > 0, Re b > 0, |Re ν| < 52

a x

J ν (bx) dx = −

 √ π  √ , 2b−1 + K 2ν 2 ab − Y 2ν 2 ab π 2  a > 0, b > 0,

|Re ν| <

1 2

 ET II 62(37)a

∞

5.

Yν 0

a x

 √ Y ν (bx) dx = −b−1 J 2ν 2 ab



a > 0,

b > 0,

|Re ν| <

1 2

 ET II 110(14)

∞

Yν 0

Re ν > − 12

a

Yν

6.

b > 0,

ET II 141(31)

4.



a > 0,

  √ 2 √  −1 2ab Y 2ν 2 ab + K 2ν Y ν (bx) dx = b x π  a > 0,

0





ET II 110(12) ∞

3.



[2a > b > 0]

a

0



 √ J ν (bx) dx = b−1 J 2ν 2 ab

[b > 2a > 0]

a

 1 √  √ 1 1 1 K ν (bx) dx = −b−1 e 2 νπi K 2ν 2e 4 πi ab − b−1 e− 2 νπi K 2ν 2e− 4 πi ab x   a > 0, Re b > 0, |Re ν| < 52 ET II 143(37)

6.516

Bessel functions



∞

7.

Kν

a x

0



∞

Kν

a x

0



∞

1.

Jμ

a

0



∞

2. 0







 √  √  3νπ 3νπ sin ker2ν 2 ab + cos kei2ν 2 ab 2 2   Re a > 0, b > 0, |Re ν| < 12 ET II 113(28)

8. 6.515

Y ν (bx) dx = −2b

−1

663

∞

3. 0

x

 √ K ν (bx) dx = πb−1 K 2ν 2 ab

Yμ

a x

[Re a > 0,

Re b > 0]

 √  √ K 0 (bx) dx = −2b−1 J 2μ 2 ab K 2μ 2 ab

[a > 0, Re b > 0] +  a ,2  1 √  √ 1 K 0 (bx) dx = 2πb−1 K 2μ 2e 4 πi ab K 2μ 2e− 4 πi ab Kμ x H (1) μ

2

a x



H (2) μ

ET II 146(54)

2

a x



ET II 143(42)

[Re a > 0, Re b > 0] ET II 147(59)   √ √ J 0 (bx) dx = 16π −2 b−1 cos μπ K 2μ 2eπi/4 a b K 2μ 2e−πi/4 a b , + π |arg a| < , b > 0, |Re μ| < 14 4 ET II 17(36)

6.516



∞

1.

 √  J 2ν a x J ν (bx) dx = b−1 J ν



0



∞

2.

a2 4b

 √  J 2ν a x Y ν (bx) dx = −b−1 Hν

0

 3. 0

4.10

5.

∞



 √  π J 2ν a x K ν (bx) dx = b−1 I ν 2







2

a 4b

b > 0,

Re ν > − 12

 ET II 58(16)

2



a 4b



a > 0,



− Lν

2

a 4b

a > 0,

 

b > 0,

Re ν > − 12



ET II 111(18)

 Re b > 0, Re ν > − 12 ET II 144(45) 2

2

 ∞  √  a a 1 1 J −ν Y 2ν a x J ν (bx) dx = J ν cot(2πν) − cosec(2πν) b 4b 2b 4b 0

 a4 3 3 23ν−3 a2−2ν bν−2  1 , − ν; Γ ν − 1; − F 1 2 2 2 2 64b2 π 3/2 [a > 0, b > 0] MC  ∞  √  Y 2ν a x Y ν (bx) dx 0 2

2

2   a a a b−1 = sec(νπ) J −ν + cosec(νπ) H−ν − 2 cot(2νπ) Hν 2 4b 4b 4b   a > 0, b > 0, |Re ν| < 12 ET II 111(19)

664

Bessel Functions

⎡ 2

2

−1  √  a a πb ⎣ cosec(2νπ) L−ν Y 2ν a x K ν (bx) dx = − cot(2νπ) Lν 2 4b 4b 0 ⎤ 2

2

a a ⎦ sec(νπ) Kν − tan(νπ) I ν − 4b π 4b   ET II 144(46) Re b > 0, |Re ν| < 12  2

2   ∞  √  a a 1 K 2ν a x J ν (bx) dx = πb−1 sec(νπ) H−ν − Y −ν 4 4b 4b 0   Re a > 0, b > 0, Re ν > − 12 

6.

7.



∞

ET II 70(22) ∞

 √  K 2ν a x Y ν (bx) dx  2

2

2  a a a 1 −1 = − πb sec(νπ) J −ν − cosec(νπ) H−ν + 2 cosec(2νπ) Hν 4 4b 4b 4b   Re a > 0, b > 0, |Re ν| < 12 ET II 114(34)

∞

 √  K 2ν a x K ν (bx) dx =

∞

 √  πb−1 I 2ν a x K ν (bx) dx = 2

8. 0

 9. 0

 10. 0

 6.517

z

J0

0



0



J 2ν (2z sin x) dx =

π 2 J (z) 2 ν



π/2

2.



π 2 J (z) 2 ν

0



 2  π Jν (z) + Nν2 (z) 8 cos νπ

J 2ν (2z cos x) dx =

π/2

1.

MO 48 2

K 2ν (2z sinh x) dx = 

2

2

2 

 a a a π Kν + L−ν − Lν 4b 2 sin(νπ) 4b 4b   1 Re b > 0, |Re ν| < 2 ET II 147(63) 2   2

a a Iν + Lν 4b 4b   Re b > 0, Re ν > − 12 ET II 147(60)

πb−1 4 cos(νπ)

z 2 − x2 dx = sin z

∞

6.518 6.519

6.517

0

− 12 < Re ν <

Re z > 0,

Re ν > − 12 Re ν > − 12

1 2





MO 45

WH



WA 42(1)a

6.52 Bessel functions combined with x and x2 6.521



1.

1

β J ν−1 (β) J ν (α) − α J ν−1 (α) J ν (β) α2 − β 2 α J ν (β) J ν (α) − β J ν (α) J ν (β) = β 2 − α2

x J ν (αx) J ν (βx) dx = 0

[α = β,

ν > −1]

[α = β,

ν > −1] WH

Bessel functions combined with x and x2

6.522

 2.

∞

10

x K ν (ax) J ν (bx) dx = 0

bν aν (b2 + a2 )

665

[Re a > 0,

b > 0,

Re ν > −1] ET II 63(2)

  ∞ π(ab)−ν a2ν − b2ν x K ν (ax) K ν (bx) dx = 2 sin(νπ) (a2 − b2 ) 0

 3.

[|Re ν| < 1,

Re(a + b) > 0] ET II 145(48)

  ν a −1  λ x J ν (λx) K ν (μx) dx = μ2 + λ2 + λa J ν+1 (λa) K ν (μa) − μa J ν (λa) K ν+1 (μa) μ 0

 4.



5.∗

[Re ν > −1] ∞

x K 1 (ax) =

π 2a2

[a > 0]

x K 20 (ax) =

1 2a2

[a > 0]

0



6.∗

∞

0

7.



∗

∞

b a (a2 + b2 )

x K 1 (ax) J 1 (bx) = 0



8.∗

∞

x K 0 (ax) I 0 (bx) = 0



9.∗

∞

x K 1 (ax) I 1 (bx) = 0

10.

∗



∞



∞

12.∗



∞

[a > b > 0]

b a (a2 − b2 )

[a > b > 0]

π 2a3

[a > 0]

x2 K 1 (ax) =

2 a3

[a > 0]

0

x2 K 0 (ax) J 1 (bx) =

0

13.∗



∞

x2 K 1 (ax) J 0 (bx) =

0

14.

∗



∞

x2 K 0 (ax) I 1 (bx) =

0

15.∗



∞

x2 K 1 (ax) I 0 (bx) =

0

6.522

Notation: 1 = 

1.8 0

∞

b > 0]

1 − b2

a2

x2 K 0 (ax) =

0

11.∗

[a > 0,

2b 2

[a > 0,

2

[a > b > 0]

2

[a > b > 0]

2

[a > b > 0]

(a2 + b2 ) 2a (a2

+ b2 ) 2b

(a2

− b2 ) 2a

(a2

− b2 )

ET II 367(26)

b > 0]

, ,   1 + 1 + (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2

    2 x [J μ (ax)] K ν (bx) dx = Γ μ + 12 ν + 1 Γ μ − 12 ν + 1 b−2 + − 1  1 , −μ + 1 ,  2 −2 2 2 −2 2 1 + 4a P 1 + 4a × 1 + 4a2 b−2 2 P −μ b b 1 1 − ν ν 2

[Re b > 2|Im a|,

2

2 Re μ > |Re ν| − 2]

ET II 138(19)

666

Bessel Functions



∞

2. 0



∞

3.11 0



6.522

  2e2μπi Γ 1 + 12 ν + μ x [K μ (ax)] J ν (bx) dx =  1   b 4a2 + b2 2 Γ 12 ν − μ   2 b−2 ) Q −μ 2 b−2 ) (1 + 4a (1 + 4a × Q −μ 1 1 2ν 2 ν−1     b > 0, Re a > 0, Re 12 ν ± μ > −1 2

ν ν −ν x K 0 (ax) J ν (bx) J ν (cx) dx = r1−1 r2−1 (r2 − r1 ) (r2 − r1 ) = ν 2 1 2 , 2 (2 − 1 ) +   2 2 2 2 r1 = a + (b − c) , r2 = a + (b + c) , c > 0, Re ν > −1,

ET II 66(27)a

Re a > |Im b|

,

ET II 63(6) ∞

4.10

− 1  x I 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 b2 + 2a2 c2 + 2b2 c2 2

0

[Re b > Re a,

5.10

c > 0]

ET II 16(27)

alternatively, with a and c interchanged  ∞ 1 x I 0 (cx) K 0 (bx) J 0 (ax) dx = 2 [Re b > Re c, a > 0]  − 21 0 2  ∞ − 1  x J 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 c2 + 2a2 b2 + 2b2 c2 2 0

alternatively, with a and b interchanged  ∞ 1 x J 0 (bx) K 0 (ax) J 0 (cx) dx = 2 2 − 21 0  ∞ x J 0 (ax) Y 0 (ax) J 0 (bx) dx = 0

6.



c > 0]

[Re a > |Im b|,

c > 0]

− 12

[0 < 2a < b < ∞] ET II 15(21)

∞

7. 0





 − 1 3+ν 3−ν x J μ (ax) J μ+1 (ax) K ν (bx) dx = Γ μ + Γ μ+ b−2 1 + 4a2 b−2 2 2 2 + , 1 1 + , 1 1 2 ν− 2 2 ν− 2 ×P −μ 1 + 4a2 b−2 P −μ−1 1 + 4a2 b−2 [Re b > 2|Im a|,



9.

0

ET II 138(20)

x K μ− 12 (ax) K μ+ 12 (ax) J ν (bx) dx   +  1 , −μ− 12 + 1 , 2e2μπi Γ 12 ν + μ + 1 −μ+ 12  2 −2 2 2 −2 2 1 + 4a 1 + 4a =− Q b b Q 1 1 1 1 1    ν− 2 ν− 2 b Γ 12 ν − μ b2 + 4a2 2 2 2  b > 0, Re a > 0, Re ν > −1, |Re μ| < 1 + 12 Re ν ET II 67(29)a

0



2 Re μ > |Re ν| − 3]

∞

8.

8

ET II 15(25)

[0 < b < 2a]

  = −2π −1 b−1 b2 − 4a2

0

[Re b > |Im a|,

∞

 − 1 x I 12 ν (ax) K 12 ν (ax) J ν (bx) dx = b−1 b2 + 4a2 2 [b > 0,

Re a > 0,

Re ν > −1] ET II 65(16)

Bessel functions combined with x and x2

6.522



∞

10.

x J 12 ν (ax) Y

0

1 2ν

(ax) J ν (bx) dx =0   = −2π −1 b−1 b2 − 4a2



− 12

[a > 0,

Re ν > −1,

0 < b < 2a]

[a > 0,

Re ν > −1,

2a < b < ∞] ET II 55(48)

∞

11.8

x J 12 (ν+n) (ax) J 12 (ν−n) (ax) J ν (bx) dx

0

= 2π

−1 −1

b



4a − b 2

 1 2 −2

Tn

b 2a



=0  0

∞

8 0

 14.

8

Re ν > −1,

0 < b < 2a]

[a > 0,

Re ν > −1,

2a < b]

  x I 12 (ν−μ) (ax) K 12 (ν+μ) (ax) J ν (bx) dx = 2−μ a−μ b−1 b2 + 4a2 [b > 0,

13.

[a > 0,

ET II 52(32) ∞

12.



667

Re a > 0,

Re ν > −1,

− 12

+   1 ,μ b + b2 + 4a2 2

Re(ν − μ) > −2] μ

ν

ET II 66(23)

ν−μ

μ−ν

(cos ψ) (sin ϕ) (sin ψ) (cos ϕ)   x J μ (xa sin ϕ) K ν−μ (ax cos ϕ cos ψ) J ν (xa sin ψ) dx = 2 2 2 a 1 − sin ϕ ,sin ψ + π π ET II 64(10) a > 0, 0 < ϕ < , 0 < ψ < , Re μ > −1, Re ν > −1 2 2

∞

x J μ (xa sin ϕ cos ψ) J ν−μ (ax) J ν (xa cos ϕ sin ψ) dx 0 μ

15.10

16.10

17.11

−ν

−μ

−1

= −2π −1 a−2 sin(μπ) (sin ϕ) (sin ψ) (cos ϕ) (cos ψ) [cos(ϕ + ψ) cos(ϕ − ψ)] + , π a > 0, 0 < ϕ < , 0 < ψ < 12 π, Re ν > −1 ET II 54(39) 2    ∞ 23ν (abc)ν Γ ν + 12 ν+1 x J ν (bx) K ν (ax) J ν (cx) dx = √ 2ν+1 π (22 − 21 ) 0 [Re a > |Im b|, c > 0]    ∞ 1 3ν ν 2 (abc) Γ ν + 2 xν+1 I ν (cx) K ν (bx) J ν (ax) dx = √ 2ν+1 π (22 − 21 ) 0 [Re b > |Im a| + |Im c|]  ∞ tν−μ−ρ+1 J μ (ct) J ν (bt) K ρ (at) dt 0  μ−ν+ρ−1  1 1+2ν−2ρ  2 1 − x2 22 − x2 x 21+ν−μ−ρ dx = μ ν ρ μ−ν c b a Γ (μ − ν + ρ) 0 (b2 − x2 ) , ,   1 + 1 + 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0] ν

668

Bessel Functions

 18.

∞

11

6.523

tμ−ν+ρ+1 J μ (ct) J ν (bt) K ρ (at) dt  ν−μ−ρ−1  1 1+2μ+2ρ  2 1 − x2 22 − x2 x 21+μ−ν+ρ aρ dx = μ ν ν−μ c b Γ (ν − μ − ρ) 0 (c2 − x2 ) , , + +   1  1  1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0]

0



∞

6.523 0

+ −1  2 −1 , b   ln x 2π −1 K 0 (ax) − Y 0 (ax) K 0 (bx) dx = 2π −1 a2 + b2 + b − a2 a [Re b > |Im a|, Re(a + b) > 0] ET II 145(50)

6.524



∞

1. 0





x J 2ν (ax) J ν (bx) Y ν (bx) dx = 0



= −(2πab)−1 ∞

2.

2

x [J 0 (ax) K 0 (bx)] dx = 0

1 π − arcsin 8ab 4ab



b −a b 2 + a2 2

2

0 < a < b,

Re ν > − 12

0 < b < a,

Re ν > − 12

1.10

 ET II 352(14)



[a > 0, 6.525



b > 0]

ET II 373(9)

, ,   1 + 1 + (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2  ∞ ,− 32  + 2 2  a + b2 + c2 − 4a2 c2 x2 J 1 (ax) K 0 (bx) J 0 (cx) dx = 2a a2 + b2 − c2 Notation: 1 =

0

[c > 0,

Re b ≥ |Im a|,

Re a > 0] ET II 15(26)

2.10

alternatively, with a and b interchanged    ∞ 2b a2 + b2 − c2 2 x J 1 (bx) K 0 (ax) J 0 (cx) dx = [Re a > |Im b|, Re b > 0, 3 (22 − 21 ) 0  ∞ ,− 32  + 2 2  a + b2 + c2 − 4a2 b2 x2 I 0 (ax) K 1 (bx) J 0 (cx) dx = 2b b2 + c2 − a2

c > 0]

0



∞

3.10

x2 I 0 (cx) K 0 (bx) J 0 (ax) dx =

  2b a2 + b2 − c2

0

6.526



1. 0

∞



2

x J 12 ν ax



−1

J ν (bx) dx = (2a)

(22 − 21 ) J 12 ν

b2 4a

3

[Re b > |Re a|,

c > 0]

[Re a > |Im b|,

c > 0]

ET II 16(28)

[a > 0,

b > 0,

Re ν > −1]

ET II 56(1)

Bessel functions combined with x and x2

6.527



∞

2. 0

  x J 12 ν ax2 Y ν (bx) dx −1

= (4a)



∞

0

∞

4.

xY 0



2

x J 12 ν ax

3.





 1 2ν



xY 0

1 2ν

b2 4a

− tan



 νπ

∞

6. 0

∞

7. 0

 ax2 J ν (bx) dx = −(2a)−1 H 12 ν

 1 2ν

J 12 ν 2 [a > 0,



ET II 140(27)

2

b 4a

[a > 0,

∞

8. 0

2

2    2 b b π x K 12 ν ax J ν (bx) dx = I 12 ν − L 12 ν 4a 4a 4a [Re a > 0,   π ⎣ cosec(νπ) L− 12 ν x K 12 ν ax2 Y ν (bx) dx = 4a

  x K 12 ν ax2 K ν (bx) dx

2. 3.



∞

Re ν > −1]

b > 0,

ET II 68(9)

⎡

π = 8a

1.

Re ν > −1]

Re b > 0,

2

2

2    νπ  νπ b b b π H− 12 ν J − 12 ν cos − sin − H 12 ν 4a sin(νπ) 2 4a 2 4a 4a [a > 0, Re b > 0, |Re ν| < 1] ET II 141(28)

 νπ 2

I 12 ν



b2 4a

b2 4a





− cot(νπ) L 12 ν

 νπ 1 K 12 ν − sec π 2

[Re a > 0,

6.527

Re ν > −1]

 ax2 K ν (bx) dx

− tan



2 

 νπ b2 b H− 12 ν + sec 4a 2 4a b > 0, Re ν > −1] ET II 109(9)

2

2   b b π  νπ H− 1 ν K ν (bx) dx = − Y − 12 ν 2 4a 4a 8a cos 2 [a > 0, Re b > 0,

=





ET II 61(35) ∞

5.



 Y

669

b > 0,

b2 4a





⎤

b2 ⎦ 4a

|Re ν| < 1]

ET II 112(25)

2

2

2 

  b b b νπ 1 1 1 K 2ν sec + π cosec(νπ) L− 2 ν − L2ν 2 4a 4a 4a [Re a > 0, |Re ν| < 1] ET II 146(52)

    1 x2 J 2ν (2ax) J ν− 12 x2 dx = a J ν+ 12 a2 2 0  ∞     1 x2 J 2ν (2ax) J ν+ 12 x2 dx = a J ν− 12 a2 2 0  ∞     1 x2 J 2ν (2ax) Y ν+ 12 x2 dx = − a Hν− 12 a2 2 0





a > 0,

Re ν > − 12

[a > 0,

Re ν > −2]

ET II 355(35)

[a > 0,

Re ν > −2]

ET II 355(36)

ET II 355(33)

670

Bessel Functions

 6.528 0

6.529



∞

1. 0

 2.



∞

x K 14 ν

x2 4



I 14 ν

x2 4



J ν (bx) dx = K 14 ν

x2 4

 √   √  2a 1 x J ν 2 ax K ν 2 ax J ν (bx) dx = b−2 e− b 2

6.528





b2 I 14 ν 4 [b > 0,

ν > −1]

[Re a > 0,

b > 0,

MO 183a

Re ν > −1] ET II 70(23)

a

    x J λ (2a) I λ (2x) J μ 2 a2 − x2 I μ 2 a2 − x2 dx

0

=

a2λ+2μ+2 2 Γ(λ + 1) Γ(μ + 1) Γ(λ + μ + 2)

λ+μ+1 λ+μ+3 ; λ + 1, μ + 1, λ + μ + 1, ; −a4 × 1F 4 2 2 [Re λ > −1, Re μ > −1] ET II 376(31)

6.53–6.54 Combinations of Bessel functions and rational functions 6.531 1.10



∞



πν a2 b 2 1 2−ν 2+ν = −π J ν (ab) cot(πν) cosec(πν) − π J −ν (ab) cosec2 (πν) + cot , ; − 1; F 1 2 ν 2 2 2 4

2 2 a b 3−ν 3+ν ab πν , ; − + 2 tan 1 F 2 1; ν −1 2 2 4 2 [Re ν < 1, arg a = π, b > 0] MC  ∞   Y ν (bx) 2 dx = π cot(νπ) [Y ν (ab) + Eν (ab)] + Jν (ab) + 2 [cot(νπ)] [Jν (ab) − J ν (ab)] x−a 0 [b > 0, a > 0, |Re ν| < 1] 0

2.

Y ν (bx) dx x+a



ET II 98(9) ∞

3. 0

, + 1 1 π2 K ν (bx) 2 dx = [cosec(νπ)] I ν (ab) + I −ν (ab) − e− 2 iνπ Jν (iab) − e 2 iνπ J−ν (iab) x+a 2 [Re b > 0, |arg a| < π, |Re ν| < 1] ET II 128(5)

6.532



1.11 0

∞

 νπ , 1+ , J ν (x) i+ π S 0,ν (ia) − e−iνπ/2 K ν (a) = i s 0,ν (ia) + sec I ν (a) dx = 2 2 x +a a a 2 2 [Re a > 0,

Re ν > −1]

6.536

Bessel functions and rational functions



∞

2. 0

⎡

 νπ Y ν (x) π 1 ⎣ 1 tan I ν (ab) − K ν (ab) − dx = νπ x2 + a2 2a 2 a cos 2  ⎤ νπ

b sin 2 2 3−ν 3+ν a b ⎦ 2 , ; + 1 F 2 1; 1 − ν2 2 2 4 [b > 0,



∞

3. 0

∞

4. 0



∞

5. 0



∞

6. 0

2. 3.11

3b

4.

Re a > 0,

|Re ν| < 1]

ET II 99(13)

 νπ  νπ   Y ν (bx) π  J ν (ab) + tan tan [Jν (ab) − J ν (ab)] − Eν (ab) − Y ν (ab) dx = 2 2 x −a 2a 2 2 [b > 0, a > 0, |Re ν| < 1] ET II 101(21)



6.533  1.

671

x J 0 (ax) dx = K 0 (ak) x2 + k 2

[a > 0,

Re k > 0]

WA 466(5)

Y 0 (ax) K 0 (ak) dx = − x2 + k 2 k

[a > 0,

Re k > 0]

WA 466(6)

J 0 (ax) π [I 0 (ak) − L0 (ak)] dx = 2 2 x +k 2k

[a > 0,

Re k > 0]

WA 467(7)

z

J p+q (z) dx = x p 0

 z 1 1 J p+q (z) J p (x) J q (z − x) dx = + x z−x p q z 0  ∞ + a, dx b [J 0 (ax) − 1] J 1 (bx) 2 = − 1 + 2 ln x 4 b 0 a2 =− 4b J p (x) J q (z − x)

[Re p > 0,

Re q > −1]

WA 415(3)

[Re p > 0,

Re q > 0]

WA 415(5)

[0 < b < a] [0 < a < b] ET II 21(28)a

⎧

b 1 1 b2 ⎪ ⎪  ∞ , ; 2, − 1 [0 < b < a] ⎨ F 2 1 dx 2a 2 2 a2 = [J 0 (ax) − 1] J 1 (bx) 2 ⎪ x 0 ⎪2 E b −1 [0 < a < b] ⎩ π a2  ∞ dx =0 [0 < a < b] [1 − J 0 (ax)] J 0 (bx) x 0 a = ln [0 < b < a] b

ET II 14(16)

 6.534 6.535 6.536

∞

x3 J 0 (x) 1 1 dx = K 0 (a) − π Y 0 (a) 4 4 2 4 0 ∞ x − a x 2 [J ν (x)] dx = I ν (a) K ν (a) 2 2 0 ∞x 3+ a x J 0 (bx) dx = ker(ab) x4 + a4 0

[a > 0]

ET II 340(5)

[Re a > 0, 

b > 0,

Re ν > −1]

|arg a| < 14 π

ET II 342(26)

 ET II 8(9), MO 46a

672

Bessel Functions



∞

6.537 0

6.538



∞

1. 0



x2 J 0 (bx) 1 dx = − 2 kei(ab) 4 4 x +a a

+ b > 0,

|arg a| <

6.537

π, 4

MO 46a

%  √   √ & dx a+b 2i ab 2i ab J 1 (ax) J 1 (bx) 2 = E −K x π |b − a| |b − a| [a > 0,

∞

2.8

x−1 J ν+2n+1 (x) J ν+2m+1 (x) dx = 0

0

b > 0]

ET II 21(30)

[m = n with m, n integers, ν > −1]

= (4n + 2ν + 2)−1

[m = n,

ν > −1] EH II 64

6.539 1. 2.

  π Y ν (b) Y ν (a) − 2 = 2 J ν (b) J ν (a) a x [J ν (x)]    b J ν (b) dx π J ν (a) − = 2 2 Y ν (a) Y ν (b) a x [Y ν (x)] 

b

dx

[J ν (x) = 0 for x ∈ [a, b]]

ET II 338(41)

[Y ν (x) = 0 for x ∈ [a, b]] ET II 339(49)

 3. a

6.541



b



J ν (a) Y ν (b) π dx = ln x J ν (x) Y ν (x) 2 J ν (b) Y ν (a)

∞

x J ν (ax) J ν (bx)

1. 0

 2.8 0

 ET II 339(50)

dx = I ν (bc) K ν (ac) x2 + c2 = I ν (ac) K ν (bc)

[0 < b < a,

Re c > 0,

Re ν > −1]

[0 < a < b,

Re c > 0,

Re ν > −1] ET II 49(10)

∞

dx x1−2n J ν (ax) J ν (bx) 2 x + c2  2 2 p n−1−p  2 2 k &

n % ν n−1   a c /4 b c /4 π 1 1 b = − 2 I ν (bc) K ν (ac) − c 2 a sin(πν) p=0 p! Γ(1 − ν + p) k! Γ(1 − ν + k) k=0

=

−

1 c2

n %

    & ν n−1  a2 c2 /4 p n−1−p  b2 c2 /4 k 1 b I ν (bc) K ν (ac) − 2ν a p!(1 − ν)p k!(1 + ν)k p=0

[0 < b < a]

k=0

[n = 1, 2, . . . ,

Re ν > n − 1,

Re c > 0,

0 < b < a]

6.544

Bessel functions and rational functions

 3.

∞

8 0

673

xα−1 1  c 2ρ−α J (cx) J (cx) dx = μ ν ρ 2 2 (x2 + z 2 )   (μ + ν + α)/2 − ρ, 1 + 2ρ − α ×Γ (μ⎛− ν − α)/2 + ρ + 1, (μ + ν − α)/2 + ρ + 1, (ν − μ − α)/2 + ρ + 1 α μ+ν+α μ−ν−α 1−α + ρ, 1 − + ρ, ρ; ρ + 1 − ,ρ + 1 + , 2 2 2 2 ⎞ μ+ν−α ν − μ − α 2 2 ⎠ z α−2ρ  cz μ+ν ρ+1+ ,ρ + 1 + ;c z + , 2 2 2 2 ⎛   ρ − (α + μ + ν)/2, (α + μ + ν) /2 ⎝1 + μ + ν , 1 + μ + ν Γ 3F 4 ρ, μ + 1, ν + 1 2 2 ⎞ α+μ+ν α+μ+ν ;1 − ρ + , μ + 1, ν + 1, μ + ν + 1; c2 z 2 ⎠ 2 2 × 3F 4 ⎝

    a1 , . . . , ap Γ (a1 ) . . . Γ (ap ) , c > 0, Re z > 0, Re(α + μ + ν) > 0; Re(α − 2ρ) > 1 Γ = b1 , . . . , bq Γ (b1 ) . . . Γ (bq ) ν  ∞ J ν (ax) Y ν (bx) − J ν (bx) Y ν (ax) π b   6.542 [0 < b < a] ET II 352(16) dx = − 2 2 2 a 0 x [J ν (bx)] + [Y ν (bx)]  

    ∞ 1 1 x dx 6.543 J μ (bx) cos (ν − μ)π J ν (ax) − sin (ν − μ)π Y ν (ax) = I μ (br) K ν (ar) 2 2 2 x + r2 0 [Re r > 0, 6.544



∞

Jν

1.

√

√    x dx 2 a 2 a 1 2 K 2ν √ =− Yν − Y 2ν √ x b x2 a π b  b a > 0, b > 0,

∞

2. 0



√

 a  x dx 2 a 1 √ J Jν = Jν 2ν x b x2 a b

∞

3.

Jν

4.

|Re ν| <

1 2

 EI II 357(47)



a > 0,

b > 0,

Re ν > − 12

 ET II 57(10)

√

2 a − 1 iπ 1 − 1 iνπ e 2 K 2ν √ e 4 a b  Re b > 0, a > 0, |Re ν| < 12

√

√    x dx 2 a 2 a 2 π Jν = K 2ν √ + Y 2ν √ x b x2 aπ 2 b  b a > 0, b > 0,

ET II 142(32)

a

|Re ν| <

1 2

 ET II 62(38)

√

√

  ∞ a  x dx 2 a 1 iπ 2 a − 1 iπ 4 1 i(ν+1)π − 12 i(ν+1)π 2 4 4 √ √ e e Kν Yν = K K + e e 2ν 2ν x b x2 a b b 0   Re b > 0, a > 0, |Re ν| < 12

 5.

∞

Yν 0

√

 x dx 2 a 1 iπ 1 1 iνπ = e2 K 2ν √ e 4 + Kν x b x2 a b

a

0



Re μ > |Re ν| − 2]

a

0



a ≥ b > 0,

ET II 143(38)

674

Bessel Functions



∞

6.

Kν 0



6.551

 √

√   x dx 1 i 1 νπi πi 2 a − 12 νπi − 14 πi 2 a 2 4 √ √ Jν = K 2ν e K 2ν e e −e x b x2 a b b   Re a > 0, b > 0, |Re ν| < 52

a

ET II 70(19)

∞

7.

Kν 0



√

√    x dx 3 2 a 2 a 3 2 Yν πν kei2ν √ πν ker2ν √ − cos = sin x b x2 a 2 2 b  b  Re a > 0, b > 0, |Re ν| < 52

a

ET II 113(29)

√

∞ a  x dx 2 a π √ Kν K Kν = 2ν 2 x b x a b 0

 8.

[Re a > 0,

Re b > 0]

ET II 146(55)

6.55 Combinations of Bessel functions and algebraic functions 6.55110



1 1/2

1.

x 0



∞

2. 1

  √ −3/2 Γ 34 + 12 ν   J ν (xy) dx = 2y Γ 14 + 12 ν    +y −1/2 ν − 12 J ν (y) S −1/2,ν−1 (y) − J ν−1 (y) S 1/2,ν (y)   y > 0, Re ν > − 32

    x1/2 J ν (xy) dx = y −1/2 J ν−1 (y) S 1/2,ν (y) + 12 − ν J ν (y) S −1/2,ν−1 (y) [y > 0]

6.552



∞

1.

J ν (xy) 0



dx 1/2 a2 )

(x2 +

= I ν/2

1

2 ay



K ν/2

1

2 ay



ET II 22(2)

[Re a > 0,

y > 0,

Re ν > −1]

ET II 23(11), WA 477(3), MO 44 ∞

2.

Y ν (xy) 0



ET II 21(1)

dx (x2

+

1/2 a2 )

=−

          1 sec 12 νπ K ν/2 12 ay K ν/2 12 ay + π sin 12 νπ I ν/2 12 ay π [y > 0, Re a > 0, |Re ν| < 1] ET II 100(18)

∞

3.

K ν (xy) 0

dx (x2

+

 π sec 12 νπ 8 2

1/2 a2 )

=

 

J ν/2

1

2 ay

2

  + Y ν/2 12 ay

[Re a > 0,

2 

Re y > 0,

|Re ν| < 1] ET II 128(6)



1

4.

J ν (xy) 0

 5.

dx (1 −

1

Y 0 (xy) 0

 6.

dx (1 −

∞

J ν (xy) 1

1/2 x2 )

1/2 x2 )

dx 1/2

(x2 − 1)

=

 2 π J ν/2 12 y 2

[y > 0,

=

    π J 0 12 y Y 0 12 y 2

[y > 0]

ET II 102(26)a

[y > 0]

ET II 24(23)a

=−

    π J ν/2 12 y Y ν/2 12 y 2

Re ν > −1]

ET II 24(22)a

6.561

Bessel functions and powers



∞

7.

Y ν (xy) 1

dx 1/2

(x2 − 1)

=

675

 2   2  π  J ν/2 12 y − Y ν/2 12 y 4 [y > 0]



∞

6.553

x−1/2 I ν (x) K ν (x) K μ (2x) dx =

0

Γ

1 4

ET II 102(27)

       + 12 μ Γ 14 − 12 μ Γ 14 + ν + 12 μ Γ 14 + ν − 12 μ 3    4 Γ 4 + ν + 12 μ Γ 34 + ν − 12 μ  |Re μ| < 12 , 2 Re ν > |Re μ| − 12 ET II 372(2)

6.554



∞

1.

x J 0 (xy) 0



x J 0 (xy) 0

dx (1 −

∞

3.

x J 0 (xy) 1



x J 0 (xy) 0

5.

∞

11

[y > 0]

ET II 7(6)a

= a−1 e−ay

[y > 0,

Re a > 0]

 1 ν √ 2a  π  dx = J ν (ak) K ν (ak) 2ν (2k) Γ ν + 12  a > 0,

|arg k| >

3/2 a2 )

ν+1/2

(x4 + 4k 4 )

0

ET II 7(4)

= y −1 cos y

− 1)

xν+1 J ν (ax)

Re a > 0]

ET II 7(5)a

1/2

+

[y > 0,

= y −1 sin y

dx (x2

= y −1 e−ay

[y > 0]

1/2 x2 )

dx (x2

∞

4. 

1/2

(a2 + x2 )

1

2. 

dx

ET II 7(7)a

π 4,

Re ν > − 12



WA 473(1)



∞

6.555 0



∞

6.556 0

 a x1/2 J 2ν−1 ax1/2 Y ν (xy) dx = − 2 Hν−1 2y



2

a 4y

a > 0,

+  a a 1/2 , dx π √ Y ν/2 J ν a x2 + 1 = − J ν/2 2 2 2 x2 + 1

y > 0,

Re ν > − 12



ET II 111(17)

[Re ν > −1,

a > 0]

MO 46

6.56–6.58 Combinations of Bessel functions and powers 6.561



1

1. 0



1

2. 0

 3. 0

1

  1 xν J ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [J ν (a) Hν−1 (a) − Hν (a) J ν−1 (a)]   Re ν > − 12   1 xν Y ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [Y ν (a) Hν−1 (a) − Hν (a) Y ν−1 (a)]   Re ν > − 12   1 xν I ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [I ν (a) Lν−1 (a) − Lν (a) I ν−1 (a)]   Re ν > − 12

ET II 333(2)a

ET II 338(43)a

ET II 364(2)a

676

Bessel Functions



1

4. 0



1

5.

6.561

  1 xν K ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [K ν (a) Lν−1 (a) + Lν (a) K ν−1 (a)]   Re ν > − 12 xν+1 J ν (ax) dx = a−1 J ν+1 (a)

ET II 367(21)a

[Re ν > −1]

ET II 333(3)a

0



1

6.

xν+1 Y ν (ax) dx = a−1 Y ν+1 (a) + 2ν+1 a−ν−2 π −1 Γ(ν + 1)

0



1

7.

xν+1 I ν (ax) dx = a−1 I ν+1 (a)

[Re ν > −1]

ET II 339(44)a

[Re ν > −1]

ET II 365(3)a

[Re ν > −1]

ET II 367(22)a

0



1

8.

xν+1 K ν (ax) dx = 2ν a−ν−2 Γ(ν + 1) − a−1 K ν+1 (a)

0



1

x1−ν J ν (ax) dx =

9. 0



1

x1−ν Y ν (ax) dx =

10. 0



1

11.

a

ν−2

2ν−1

Γ(ν)

− a−1 J ν−1 (a)

aν−2 cot(νπ) − a−1 Y ν−1 (a) 2ν−1 Γ(ν)

x1−ν I ν (ax) dx = a−1 I ν−1 (a) −

0



1

12.

ET II 333(4)a

[Re ν < 1]

ET II 339(45)a

aν−2 Γ(ν)

ET II 365(4)a

2ν−1

x1−ν K ν (ax) dx = 2−ν aν−2 Γ(1 − ν) − a−1 K ν−1 (a)

0



1

xμ J ν (ax) dx =

13.7 0

2μ Γ

[Re ν < 1]

 ν+μ+1 

aμ+1 Γ

2  + a−μ {(μ + ν − 1) J ν (a) S μ−1,ν−1 (a) − J ν−1 (a) S μ,ν (a)}  ν−μ+1 2

1 1  1 ∞ μ μ −μ−1 Γ  2 + 2 ν + 2 μ x J ν (ax) dx = 2 a Γ 12 + 12 ν − 12 μ 0

 14.



∞

15. 0

 16. 0

∞

ET II 367(23)a

[a > 0, 

Re(μ + ν) > −1]

− Re ν − 1 < Re μ < 12 ,

  1  −μ−1 Γ 12 + 12 ν + 12 μ μ μ   x Y ν (ax) dx = 2 cot 2 (ν + 1 − μ)π a Γ 12 + 12 ν − 12 μ |Re ν| − 1 < μ < 12 , xμ K ν (ax) dx = 2μ−1 a−μ−1 Γ





1+μ+ν 1+μ−ν Γ 2 2 [Re (μ + 1 ± ν) > 0,

ET II 22(8)a

a>0



EH II 49(19)

a>0

 ET II 97(3)a

Re a > 0] EH II 51(27)

6.564

Bessel functions and powers



∞

17. 0



∞

18. 0

  Γ 12 q + 12 J ν (ax)  dx = ν−q q−ν+1  xν−q 2 a Γ ν − 12 q + 12 Γ Y ν (x) dx = xν−μ

1 2



677

−1 < Re q < Re ν −

1 2



WA 428(1), KU 144(5)

     + 12 μ Γ 12 + 12 μ − ν sin 12 μ − ν π 2ν−μ π   |Re ν| < Re(1 + μ − ν) < 32

WA 430(5)





1

x2m+n+1/2 K n+1/2 (αx) dx =

19. 0

6.562



∞

xμ Y ν (bx)

0

∞

2. 0



STR

k=0

1.



n π  (n + k)! γ(2m + n − k + 1, α) 2 k!(n − k)! α2m+n+3/2 2k

       dx = (2a)μ π −1 sin 12 π(μ − ν) Γ 12 (μ + ν + 1) Γ 12 (1 + μ − ν) S −μ,ν (ab) x+a        −2 cos 12 π (μ − ν) Γ 1 + 12 μ + 12 ν Γ 1 + 12 μ − 12 ν S −μ−1,ν (ab)   ET II 98(8) b > 0, |arg a| < π, Re (μ ± ν) > −1, Re μ < 32

πk ν xν J ν (ax) dx = [H−ν (ak) − Y −ν (ak)] x+k 2 cos νπ



− 12 < Re ν < 32 ,

a > 0,

|arg k| < π



WA 479(7) ∞

3.

dx x+a

    μ − ν a2 b 2 μ+ν μ−2 ,1 − ; Γ 12 (μ + ν) Γ 12 (μ − ν) b−μ 1 F 2 1; 1 − =2 2 2 4

    3 − μ + ν a2 b 2 3 − μ − ν μ−3 1−μ 1 1 , ; Γ 2 (μ − ν − 1) Γ 2 (μ + ν − 1) ab −2 1 F 2 1; 2 2 4

xμ K ν (bx)

0

−πaμ cosec[π(μ − ν)] {K ν (ab) + π cos(μπ) cosec[π(ν + μ)] I ν (ab)} [Re b > 0, 

∞

6.563

x−1 J ν (bx)

0

6.564



1.

∞

ν+1

x 0

|arg a| < π,

Re μ > |Re ν| − 1]

ET II 127(4)

dx πa−μ−1 = (x + a)1+μ sin[( ⎧ + ν − μ)π] Γ(μ + 1)  ν+2m ∞ ⎨ (−1)m 12 ab Γ( + ν + 2m) × ⎩ m! Γ(ν + m + 1) Γ ( + ν − μ + 2m) m=0 ⎫ μ+1−+m ⎬ 1 ∞ 1  Γ(μ + m + 1) sin 2 ( + ν − μ − m)π 2 ab     − 1 m! Γ (μ + ν −  + m + 3) Γ 12 (μ − ν −  + m + 3) ⎭ 2 m=0   ET II 23(10), WA 479 b > 0, |arg a| < π, Re( + ν) > 0, Re( − μ) < 52

dx J ν (bx) √ = 2 x + a2



2 ν+ 1 a 2 K ν+ 12 (ab) πb



Re a > 0,

b > 0,

−1 < Re ν <

1 2



ET II 23(15)

678

Bessel Functions



∞

2.

1−ν

x 0

dx J ν (bx) √ = 2 x + a2



6.565

, π 1 −ν + a2 I ν− 12 (ab) − Lν− 12 (ab) 2b  Re a > 0,

b > 0,

Re ν > − 12



ET II 23(16)

6.565



∞

1. 0



∞

2.





 −ν− 12 ab ab Γ(ν + 1) Iν x−ν x2 + a2 J ν (bx) dx = 2ν a−2ν bν Kν Γ(2ν + 1)  2 2  Re a > 0, b > 0, Re ν > − 12  −ν− 12 xν+1 x2 + a2

0



∞

3.

 −ν− 32 xν+1 x2 + a2

0



WA 477(4), ET II 23(17)



Re a > 0,

√ bν π  J ν (bx) dx = ν+1 ab  2 ae Γ ν + 32 [Re a > 0,

b > 0,

Re ν > − 12



ET II 24(18)

b > 0,

Re ν > −1] ET II 24(19)

∞

4.

J ν (bx)xν+1 (x2 +

0



√ ν−1 πb  J ν (bx) dx = ν ab  2 e Γ ν + 12

∞

5. 0

μ+1 a2 )

dx =

aν−μ bμ K ν−μ (ab) Γ(μ + 1)    −1 < Re ν < Re 2μ + 32 ,

2μ

a > 0,

b>0



MO 43

 μ xν+1 x2 + a2 Y ν (bx) dx = 2ν−1 π −1 a2μ+2 (1 + μ)−1 Γ(ν)b−ν

a2 b 2 −1 × 1 F 2 1; 1 − ν, 2 + μ; − 2μ aμ+ν+1 [sin(νπ)] 4 × Γ(μ + 1)b−1−μ [I μ+ν+1 (ab) − 2 cos(μπ) K μ+ν+1 (ab)] [b > 0,



ET II 100(19)

 μ 2μ a1+μ−ν b−1−μ π I −1−μ+ν (ab) cot[π(μ − ν)] cosec(πμ) x1−ν x2 + a2 Y ν (bx) dx = Γ(−μ) 2μ a1+μ−ν b−1−μ π − I 1+μ−ν (ab) cosec[π(μ − ν)] cosec(πν) Γ(−μ)

2−1−ν a2+2μ bν a2 b 2 + cos(πν) Γ(−μ) 1 F 2 1; 2 + μ, 1 + ν; 4 (1 + μ)π 2 MC Re ν < 1, Re(ν − 2μ) > −3, arg a = π, b > 0

∞

 μ x1+ν x2 + a2 K ν (bx) dx = 2ν Γ(ν + 1)aν+μ+1 b−1−μ S μ−ν,μ+ν+1 (ab)

0

7.

−1 < Re ν < −2 Re μ]

∞

6.10



Re a > 0,

0

[Re a > 0,

Re b > 0,

Re ν > −1] ET II 128(8)

6.567

Bessel functions and powers

 8.

∞

11

μ+1

(x2 + k 2 )

0

6.566



x−1 J ν (ax)

∞

1.

xμ Y ν (bx)

0



∞

2.

x2

xν+1 J ν (ax)

0

 0

WA 477(1)

dx = 2μ−2 π −1 b1−μ + a2 ,  +π    (μ − ν + 1) Γ 12 μ + 12 ν − 12 Γ 12 μ − 12 ν − 12 × cos 2

μ + 1 − ν a2 b 2 μ+1+ν ,2 − ; × 1 F 2 1; 2 − 2 2 4 , + , + π π 1 μ−1 cosec (μ + ν + 1) cot (μ − ν + 1) I ν (ab) − πa 2 2 , +π 2 −aμ−1 cosec (μ − ν + 1) K ν (ab) 2  b > 0, Re a > 0, |Re ν| − 1 < Re μ < 52 ET II 100(17)

x2

dx = bν K ν (ab) + b2



a > 0,

Re b > 0,

−1 < Re ν <

3 2



xν K ν (ax)

2 ν−1

x2

dx π b [H−ν (ab) − Y −ν (ab)] = 2 +b 4 cos νπ 

a > 0,

Re b > 0,

Re ν > − 12



WA 468(9) ∞

4.

x−ν K ν (ax)

0



    aν k +ν−2μ−2 Γ 12  + 12 ν Γ μ + 1 − 12  − 12 ν dx = 2ν+1 Γ(μ + 1) Γ(ν + 1)

+ν +ν a2 k 2 ; − μ, ν + 1; × 1F 2 2  2  4 a2μ+2− Γ 12 ν + 12  − μ − 1

+ 1 1 22μ+3− Γ μ + 2 + ν −  2 2

ν +  a2 k 2 ν− × 1 F 2 μ + 1; μ + 2 + ,μ + 2 − ; 2 2 4   a > 0, − Re ν < Re  < 2 Re μ + 72 , Re k > 0

EH II 96(58) ∞

3.



679

2

dx π [Hν (ab) − Y ν (ab)] = ν+1 x2 + b2 4b cos νπ  a > 0,

Re b > 0,

Re ν <

1 2



WA 468(10) ∞

5. 0

x−ν J ν (ax)

x2

dx π = ν+1 [I ν (ab) − Lν (ab)] 2 +b 2b



a > 0,

Re b > 0,

Re ν > − 52



WA 468(11)

6.567



1. 0

1

 μ xν+1 1 − x2 J ν (bx) dx = 2μ Γ(μ + 1)b−(μ+1) J ν+μ+1 (b) [b > 0,

Re ν > −1,

Re μ > −1] ET II 26(33)a

680

Bessel Functions



1

2.

 μ xν+1 1 − x2 Y ν (bx) dx

0

  = b−(μ+1) 2μ Γ(μ + 1) Y μ+ν+1 (b) + 2ν+1 π −1 Γ(ν + 1) S μ−ν,μ+ν+1 (b) [b > 0,

 3. 4.

6.567

Re μ > −1,

Re ν > −1]

 μ 21−ν S ν+μ,μ−ν+1 (b) [b > 0, Re μ > −1] x1−ν 1 − x2 J ν (bx) dx = bμ+1 Γ(ν) 0   1  μ x1−ν 1 − x2 Y ν (bx) dx = b−(μ+1) 21−ν π −1 cos(νπ) Γ (1 − ν)

ET II 103(35)a

1

ET II 25(31)a

 × S μ+ν,μ−ν+1 (b) − 2 cosec(νπ) Γ(μ + 1) J μ−ν+1 (b)

0

μ

[b > 0, 

1 1−ν

5.

x



 2 μ

1−x

K ν (bx) dx = 2

0

−ν−2 ν

−1

b (μ + 1)

Re μ > −1,

Γ(−ν) 1 F 2

Re ν < 1]

b2 1; ν + 1, μ + 2; 4

ET II 104(37)a



+π2μ−1 b−(μ+1) cosec (νπ) Γ(μ + 1) I μ−ν+1 (b)  6. 7.

8.

9.

10.

11.

12. 13.

1



[Re μ > −1,

Re ν < 1]

π Hν− 12 (b) [b > 0] 2b 0   1 + , π dx cosec(νπ) cos(νπ) J ν+ 12 (b) − H−ν− 12 (b) x1+ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν > −1]    1 + ,  π dx cot(νπ) Hν− 12 (b) − Y ν− 12 (b) − J ν− 12 (b) x1−ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν < 1]  2  1  ν− 12   √ b xν 1 − x2 J ν (bx) dx = 2ν−1 πb−ν Γ ν + 12 J ν 2 0   b > 0, Re ν > − 12



 1   1 √ −ν b b 1 ν 2 ν− 2 ν−1 x 1−x Y ν (bx) dx = 2 πb Γ ν + Jν Yν 2 2 2 0   b > 0, Re ν > − 12



 1  ν− 12 √ b b 1 xν 1 − x2 K ν (bx) dx = 2ν−1 πb−ν Γ ν + Iν Kν 2 2 2 0   Re ν > − 12

 2  1   1 √ −ν b 1 ν 2 ν− 2 −ν−1 x 1−x I ν (bx) dx = 2 πb Γ ν + Iν 2 2 0

 1 ν−1 1   1 b −ν− 2 − ν sin b xν+1 1 − x2 J ν (bx) dx = 2−ν √ Γ 2 π 0   b > 0, |Re ν| < 12 x1−ν J ν (bx) √

dx = 1 − x2

ET II 129(12)a ET II 24(24)a

ET II 102(28)a

ET II 102(30)a

ET II 24(25)a

ET II 102(31)a

ET II 129(10)a ET II 365(5)a

ET II 25(27)a

6.571

Bessel functions and powers









√ −ν b b b b 1 x x −1 Y ν (bx) dx = 2 πb Γ ν + Jν J −ν −Yν Y −ν 2 2 2 2 2 1   |Re ν| < 12 , b > 0 ET II 103(32)a



  ∞ 2  ν− 12 b 2ν−1 1 xν x2 − 1 K ν (bx) dx = √ b−ν Γ ν + Kν 2 2 π 1   Re b > 0, Re ν > − 12 ET II 129(11)a





 ∞  −ν− 12 √ 1 b b − ν Jν x−ν x2 − 1 J ν (bx) dx = −2−ν−1 πbν Γ Yν 2 2 2 1   b > 0, |Re ν| < 12 ET II 25(26)a

 ∞  ν− 12 1 2ν + ν cos b x−ν+1 x2 − 1 J ν (bx) dx = √ b−ν−1 Γ 2 π 1   b > 0, |Re ν| < 12 ET II 25(28) 

14.

15.

16.

17.8

6.568



∞

∞

1.

ν



2

ν− 12

xν Y ν (bx)

0



xμ Y ν (bx)

0



ν−2

dx π = aν−1 J ν (ab) x2 − a2 2



a > 0,

b > 0,

− 12 < Re ν <

5 2



ET II 101(22) ∞

2.

+π

, (μ − ν + 1)

dx π = aμ−1 J ν (ab) + 2μ π −1 aμ−1 cos x2 − a2 2



2 μ−ν+1 μ+ν+1 ×Γ Γ S −μ,ν (ab) 2  2  a > 0, b > 0, |Re ν| − 1 < Re μ < 52

ET II (101)(25)

1

xλ (1 − x)μ−1 J ν (ax) dx

6.569 0

=

6.571

681



∞

1.

+

0

 2. 0

∞

+

x2 + a2

 12

Γ(μ) Γ(1 + λ + ν)2−ν aν Γ(ν + 1) Γ(1 + λ + μ + ν)

λ+1+ν λ+2+ν λ+1+μ+ν λ+2+μ+ν a2 , ; ν + 1, , ;− × 2F 3 2 2 2 2 4 [Re μ > 0, Re(λ + ν) > −1] ET II 193(56)a





,μ ab ab dx ± x J ν (bx) √ = aμ I 12 (ν∓μ) K 12 (ν±μ) 2 2 x2 + a 2  Re a > 0, b > 0, Re ν > −1, Re μ < 32

ET II 26(38)

,μ 1 dx x2 + a2 2 − x Y ν (bx) √ 2 2  x + a



 ab ab ab ab = aμ cot(νπ) I 12 (μ+ν) K 12 (μ−ν) − cosec(νπ) I 12 (μ−ν) K 12 (μ+ν) 2  2 2 2  Re a > 0, b > 0, Re μ > − 32 , |Re ν| < 1 ET II 104(40)

682

Bessel Functions



∞

3.

+

6.572



∞

−μ

x

1.

,μ dx + x K ν (bx) √ 2 2 x +  a





 ab ab ab ab π2 μ a cosec(νπ) J 12 (ν−μ) = Y − 12 (ν+μ) − Y 12 (ν−μ) J − 12 (ν+μ) 4 2 2 2 2 [Re a > 0, Re b > 0] ET II 130(15)

x2 + a2

0

+

2

 12

x +a

2

 12

,μ +a

0



6.572

  Γ 1+ν−μ dx 2 W 12 μ, 12 ν (ab) M − 12 μ, 12 ν (ab) J ν (bx) √ = ab Γ(ν + 1) x2 + a2 [Re a > 0, b > 0, Re(ν − μ) > −1] ET II 26(40)

∞

2. 0

+ ,μ 1 dx x−μ x2 + a2 2 + a K ν (bx) √ 2 x + a2

 1+ν−μ   1−ν−μ  Γ 2 2 W 12 μ, 12 ν (iab) W 12 μ, 12 ν (−iab) = 2ab Re b > 0, Re μ + |Re ν| < 1] ET II 130(18), BU 87(6a) Γ

[Re a > 0, 

∞

3.

x−μ

+

x2 + a2

 12

−a

,μ

0

6.573



∞

1.

xν−M +1 J ν (bx)

0

k -

dx + a2   

Γ 1+ν+μ ν−μ 1 2 tan π M 12 μ, 12 ν (ab) = − W − 12 μ, 12 ν (ab) ab Γ(ν + 1) 2

 ν−μ π W 12 μ, 12 ν (ab) + sec 2   Re a > 0, b > 0, |Re ν| < 12 + 12 Re μ ET II 105(42)

Y ν (bx) √

x2

J μi (ai x) dx = 0

i=1 %

ai > 0,

k 

M=

k 

μi &

i=1

ai < b < ∞,

−1 < Re ν < Re M +

1 2k

−

1 2

ET II 54(42)

i=1

 2. 0

∞

xν−M −1 J ν (bx) %

k -

J μi (ai x) dx = 2ν−M −1 b−ν Γ(ν)

i=1

ai > 0,

k  i=1

ai < b < ∞,

k -

aμi i , Γ (1 + μi ) i=1 &

0 < Re ν < Re M + 12 k +

3 2

M=

k 

μi

i=1

WA 460(16)a, ET II 54(43)

6.576

6.574

Bessel functions and powers

683



ν+μ−λ+1  ∞ α Γ 2

J ν (αt) J μ (βt)t−λ dt = −ν + μ + λ + 1 0 2λ β ν−λ+1 Γ Γ(ν + 1) 2

ν+μ−λ+1 ν−μ−λ+1 α2 , ; ν + 1; 2 ×F 2 2 β [Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < α < β] ν

1.8

2.

WA 439(2)a, MO 49

If we reverse the positions of ν and μ and at the same time reverse the positions of α and β, the function on the right-hand side of this equation will change. Thus, the right-hand side represents α α that is not analytic at = 1. a function of β β For α = β, we have the following equation:

ν+μ−λ+1  ∞ αλ−1 Γ(λ) Γ 2





J ν (αt) J μ (αt)t−λ dt = −ν + μ + λ + 1 ν + μ + λ +1 ν−μ+λ+1 0 λ 2 Γ Γ Γ 2 2 2 [Re(ν + μ + 1) > Re λ > 0, α > 0] MO 49, WA 441(2)a

3.∗

If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right-hand side of equation 6.574 1 (or 6.574 3) vanishes. The cases in which the hypergeometric function F in 6.574 3 (or 6.574 1) can be reduced to an elementary function are then especially important.

μ+ν−λ+1  ∞ βν Γ 2

J ν (αt) J μ (βt)t−λ dt = ν − μ + λ+1 0 2λ αμ−λ+1 Γ Γ(ν + 1) 2

ν + μ − λ + 1 −ν + μ − λ + 1 β2 , ; μ + 1; 2 ×F 2 2 α [Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < β < α] MO 50, WA 440(3)a If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right-hand side of equation 6.754 1 (or 6.574 3) vanishes. The cases in which the hypergeometric function F in 6.754 3 (or 6.574 1) can be reduced to an elementary function are then especially important.

6.575  1.11

∞

J ν+1 (αt) J μ (βt)tμ−ν dt = 0

0

=  2. 0

∞



 2 ν−μ

[α < β]

α −β β Γ(ν − μ + 1) 2

2ν−μ αν+1 √

μ

[α ≥ β] [Re(ν + 1) > Re μ > −1]

MO 51

J ν (x) J μ (x) π Γ(ν + μ)      dx = ν+μ  xν+μ 2 Γ ν + μ + 12 Γ ν + 12 Γ μ + 12 [Re(ν + μ) > 0]

KU 147(17), WA 434(1)

684

6.576

Bessel Functions



∞

1.

xμ−ν+1 J μ (x) K ν (x) dx =

0

2.11

1 2

6.576

Γ(μ − ν + 1)

[Re μ > −1,

Re(μ − ν) > −1]



1−λ ν ν  ∞ a b Γ ν+ 2

x−λ J ν (ax) J ν (bx) dx = 1+λ 0 λ 2ν−λ+1 2 (a + b) Γ(ν + 1) Γ 2

1 4ab 1−λ , ν + ; 2ν + 1; ×F ν + 2 2 (a + b)2 [a > 0, b > 0, 2 Re ν + 1 > Re λ > −1]

ET II 370(47)

ET II 47(4)





ν−λ+μ+1 ν−λ−μ+1  ∞ b Γ Γ 2 2 x−λ K μ (ax) J ν (bx) dx = λ+1 ν−λ+1 2 a Γ(1 + ν) 0

ν−λ+μ+1 ν−λ−μ+1 b2 , ; ν + 1; − 2 ×F 2 2 a [Re (a ± ib) > 0, Re(ν − λ + 1) > |Re μ|] EH II 52(31), ET II 63(4), WA 449(1) ν

3.



∞

4.

−λ

x 0



∞

5.

−λ

x 0



∞

6.





1−λ+μ+ν 1−λ−μ+ν 2−2−λ a−ν+λ−1 bν Γ K μ (ax) K ν (bx) dx = Γ 2 Γ(1 − λ)

2

1−λ+μ−ν 1−λ−μ−ν ×Γ Γ 2 2

1−λ+μ+ν 1−λ−μ+ν b2 , ; 1 − λ; 1 − 2 ×F 2 2 a [Re a + b > 0, Re λ < 1 − |Re μ| − |Re ν|] ET II 145(49), EH II 93(36)    − 12 λ + 12 μ + 12 ν Γ 12 − 12 λ − 12 μ + 12 ν K μ (ax) I ν (bx) dx = 2λ+1 Γ(ν + 1)a−λ+ν+1

2 1 1 − λ + 12 μ + 12 ν, 12 − 12 λ − 12 μ + 12 ν; ν + 1; ab 2 ×F 2 2 [Re (ν + 1 − λ ± μ) > 0, a > b] EH II 93(35) bν Γ

[a > b,  7.

0

∞

2

 π(ν − μ − λ) ∞ −λ 2 sin x K μ (ax) I ν (bx) dx π 2 0 Re λ > −1, Re (ν − λ + 1 ± μ) > 0] (see 6.576 5)

x−λ Y μ (ax) J ν (bx) dx =

0

8

1

xμ+ν+1 J μ (ax) K ν (bx) dx = 2μ+ν aμ bν

EH II 93(37)

Γ(μ + ν + 1) μ+ν+1

(a2 + b2 )

[Re μ > |Re ν| − 1,

Re b > |Im a|]

ET 137(16), EH II 93(36)

6.578

Bessel functions and powers

6.577  1.8

∞

[a > 0,  2.

∞

dx = (−1)n cν−μ+2n I μ (ac) K ν (bc) x2 + c2 Re c > 0, 2 + Re μ − 2n > Re ν > −1 − n, n ≥ 0 an integer]

ET II 49(13)

dx = (−1)n cμ−ν+2n I ν (bc) K μ (ac) + c2 Re ν − 2n + 2 > Re μ > −n − 1, n ≥ 0 an integer]

ET II 49(15)

xν−μ+1+2n J μ (ax) J ν (bx)

0

8

b > a,

xμ−ν+1+2n J μ (ax) J ν (bx)

0

[b > 0, 6.578



∞

1. 0



∞

2.

∞

3. 0



∞

4. 0



a > b,

x2

 2−1 aλ bμ c−λ−μ− Γ λ+μ+ν+ 2

x−1 J λ (ax) J μ (bx) J ν (cx) dx = λ+μ−ν+ Γ(λ + 1) Γ(μ + 1) Γ 1 − 2

λ+μ−ν+ λ+μ+ν+ a2 b 2 , ; λ + 1, μ + 1; 2 , 2 ×F4 2 2 c c   5 Re(λ + μ + ν + ) > 0, Re  < , a > 0, b > 0, c > 0, c > a + b ET II 351(9) 2 x−1 J λ (ax) J μ (bx) K ν (cx) dx





+λ+μ−ν +λ+μ+ν 2−2 aλ bμ c−−λ−μ Γ = Γ Γ(λ + 1) Γ(μ + 1) 2 2

+λ+μ−ν +λ+μ+ν a2 b2 , ; λ + 1, μ + 1; − 2 , − 2 ×F4 2 2 c c [Re( + λ + μ) > |Re ν|, Re c > |Im a| + |Im b|] ET II 373(8)

0



685

∞

5.

xλ−μ−ν+1 J ν (ax) J μ (bx) J λ (cx) dx = 0  Re λ > −1, Re(λ − μ − ν) < 12 ,



c > b > 0,

0
2λ−μ−ν−1 aν bμ Γ(λ) xλ−μ−ν−1 J ν (ax) J μ (bx) J λ (cx) dx = λ c Γ(μ + 1) Γ(ν + 1)  Re λ > 0, Re(λ − μ − ν) < 52 , c > b > 0,

0
x1+μ Y μ (ax) J ν (bx) J ν (cx) dx = 0

0 < a < c − b]

[0 < b < c,



ET II 53(36)

ET II 53(37)

0



ET II 352(13) ∞

6.11 0

 7.11 0

 − 1 μ− 1 μ+ 1 1 1 xμ+1 K μ (ax) J ν (bx) J ν (cx) dx = √ aμ b−μ−1 c−μ−1 e−(μ+ 2 )πi u2 − 1 2 4 Q ν− 12 (u) 2 2π   Re a > |Im b| + |Im c|, Re ν > −1, Re(μ + ν) > −1 2bcu = a2 + b2 + c2 , WA 452(2), ET II 64(12)

∞

 − 1 μ− 1 μ+ 1 1 1 1 xμ+1 I ν (ax) K μ (bx) J ν (cx) dx = √ a−μ−1 bμ c−μ−1 e−(μ− 2 ν+ 4 )πi v 2 + 1 2 4 Q ν− 12 (iv), 2 2π [Re b > |Re a| + |Im c|; Re ν > −1, Re(μ + ν) > −1] ET II 66(22) 2acv = b2 − a2 + c2

686

Bessel Functions

 8.

∞

x1−μ J μ (ax) J ν (bx) J ν (cx) dx  1 1 2 −μ μ− 1 2 −μ a (bc)μ−1 (sinh u) 2 sin[(μ − ν)π]e(μ− 2 )πi Q ν− = 1 (cosh u) 3 2 π 1 1 −μ μ− 12 μ−1 2 −μ (sin v) P ν− 1 (cos v) = √ a (bc) 2 2π

11 0

[a > b + c] [|b − c| < a < b + c] [0 < a < |b − c|]

=0

 9.

6.578

2bc cosh u = a − b − c , 2bc cos v = b + c − a ,  ∞ J ν (ax) J ν (bx) J ν (cx)x1−ν dx = 0 2

0

2

2

2

2

ν−1

2

b > 0,

c > 0;

Re ν > −1, Re μ > − 12



[0 < c ≤ |a − b| or c ≥ a + b]

2ν−1

Δ 2     [|a − b| < c < a + b] (abc)ν Γ ν + 12 Γ 12    a > 0, b > 0, c > 0; Re ν > − 12 Δ = 14 [c2 − (a − b)2 ] [(a + b)2 − c2 ], =

10.11

(Δ > 0 is equal to the area of a triangle whose sides are a, b, and c.) √ ν  ∞ πc Γ(ν + μ + 1) Γ(ν − μ + 1) −ν− 12 ν+1 x K μ (ax) K μ (bx) J ν (cx) dx = P μ− 1 (u) 1 ν+ 1 2 0 23/2 (ab)ν+1 (u2 − 1) 2 4   2 2 2 2abu = a + b + c , Re(a + b) > |Im c|, Re (ν ± μ) > −1, Re ν > −1 

∞

11.11 0



∞

12.8

ET II 67(30)

1 ν+ 1 (ab)−ν−1 cν e−(ν+ 2 )πi Q μ− 21 (u) 2 xν+1 K μ (ax) I μ (bx) J ν (cx) dx = 2abu = a2 + b2 + c2 √ 1 ν+ 1 2π (u2 − 1) 2 4 [Re a > |Re b| + |Im c|; Re ν > −1, Re(μ + ν) > −1] ET II 66(24)



2

xν+1 [J ν (ax)] Y ν (bx) dx = 0

0

−ν− 12 23ν+1 a2ν b−ν−1  2 1  b − 4a2 = √ πΓ 2 −ν



0 < b < 2a,

|Re ν| <

1 2

0 < 2a < b,

|Re ν| <

1 2

 

ET II 109(3)



∞

13.

xν+1 J ν (ax) Y ν (ax) J ν (bx) dx

0



=0 −ν− 12 23ν+1 a2ν b−ν−1  2 1  b − 4a2 =− √ πΓ 2 −ν



a > 0,

|Re ν| < 12 ,

a > 0,

2a < b < ∞,

0 < b < 2a |Re ν| <



1 2



ET II 55(49)



∞

14.

xν+1 J μ (xa sin ψ) J ν (xa sin ϕ) K μ (xa cos ϕ cos ψ) dx

0 ν

= +

tan 12 α = tan ψ cos ϕ,

a > 0,

π > ϕ > 0, 2

2ν Γ(μ + ν + 1) (sin ϕ) aν+2 0<ψ<

π , 2

 2ν+1 cos α2

2ν+2

(cos ψ) Re ν > −1,

P −μ ν (cos α) , Re(μ + ν) > −1 ET II 64(11)

6.581

Bessel functions and powers



∞

15.

ν+1

x

J ν (ax) K ν (bx) J ν (cx) dx =

0



∞

16.8

  23ν (abc)ν Γ ν + 12

,ν+ 12 √ + 2 2 π (a + b2 + c2 ) − 4a2 c2  Re b > |Im a|, 3ν

xν+1 I ν (ax) K ν (bx) J ν (cx) dx =

0

687

ν



1 2

c > 0,

Re ν > − 12



ET II 63(8)



2 (abc) Γ ν + ,ν+ 12 √ + 2 2 π (b − a2 + c2 ) + 4a2 c2  Re b > |Re a| + |Im c|;

Re ν > − 12



ET II 65(18)

6.579  1.

∞

x2ν+1 J ν (ax) Y ν (ax) J ν (bx) Y ν (bx) dx

a2ν Γ(3ν + 1) 3 a2 1  1   = 3 F ν + 2 , 3ν + 1; 2ν + 2 ; b2 2πb4ν+2   Γ 2 − ν Γ 1 2ν + 2 EH II 94(45), ET II 352(15) 0 < a < b, − 3 < Re ν < 12

0



∞

2. 0



∞

3.

x2ν+1 J ν (ax) K ν (ax) J ν (bx) K ν (bx) dx      

2ν−3 a2ν Γ ν+1 Γ ν + 12 Γ 3ν+1 a4 3ν + 1 2 2 √ 4ν+2 ; 2ν + 1; 1 − 4 = F ν + 12 , 2 b πb Γ(ν + 1)   0 < a < b, Re ν > − 13 ET II 373(10) 4

x1−2ν [J ν (ax)] dx =

0



∞

4.

2

Γ(ν) Γ(2ν) 2   2π Γ ν + 12 Γ(3ν) 2

x1−2ν [J ν (ax)] [J ν (bx)] dx =

0

[Re ν > 0]

a2ν−1 Γ(ν)   F  2πb Γ ν + 12 Γ 2ν + 12

ET II 342(25)

1 a2 1 ν, − ν; 2ν + ; 2 2 2 b ET II 351(10)

6.581



a λ−1

x

1. 0



∞  (−1)m Γ(λ + μ + m) Γ(λ + m) J λ+μ+ν+2m (a) J μ (x) J ν (a − x) dx = 2 m! Γ(λ) Γ(μ + m + 1) m=0 [Re(λ + μ) > 0, Re ν > −1] λ

ET II 354(25) a

2.8

xλ−1 (a − x)−1 J μ (x) J ν (a − x) dx

0

=

∞ 2λ  (−1)m Γ(λ + μ + m) Γ(λ + m) (λ + μ + ν + 2m) J λ+μ+ν+2m (a) aν m=0 m! Γ(λ) Γ(μ + m + 1)

[Re(λ + μ) > 0, 

a

0

ET II 354(27)

    Γ μ + 12 Γ ν + 12 μ+ν+ 1 2 J a x (a − x) J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 1)   Re μ > − 12 , Re ν > − 12 μ

3.

Re ν > 0]

ν

ET II 354(28), EH II 46(6)

688

Bessel Functions



a

x (a − x) μ

4.

ν+1

0

 5.

a

    Γ μ + 12 Γ ν + 32 μ+ν+ 3 2 J a J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 2)   Re ν > −1, Re μ > − 12 

xμ (a − x)−μ−1 J μ (x) J ν (a − x) dx =

0



∞

6.582

−μ

xμ−1 |x − b|

0



∞

6.583

6.582

ET II 354(29)

 1

2μ Γ μ + 2 Γ(ν − μ) μ √ a J ν (a) π Γ(μ + ν + 1)   Re ν > Re μ > − 12

ET II 355(30)

  1 K μ (|x − b|) K ν (x) dx = √ (2b)−μ Γ 12 − μ Γ(μ + ν) Γ(μ − ν) K ν (b) π   b > 0, Re μ < 12 , Re μ > |Re ν|

xμ−1 (x + b)−μ K μ (x + b) K ν (x) dx =

√

0

ET II 374(14)

π Γ(μ + ν) Γ(μ − ν)   K ν (b) 2μ bμ Γ μ + 12 [|arg b| < π, Re μ > |Re ν|] ET II 374(15)

6.584  1.8

∞ x−1

+

 , πi H (1) axeπi H (1) ν (ax) − e ν m+1

(x2 − r2 ) m = 0, 1, 2, . . . ,

0



∞

2.8 0



∞

3. 0

 4. 0

∞



m + , d πi (1) −2 r H (ar) ν 2m m! r dr  Im r > 0, a > 0, |Re ν| < Re  < 2m + 72

dx =

WA 465

  x−1 1 1 cos ( − ν)π J ν (ax) + sin ( − ν)π Y ν (ax) m+1 dx 2 2 (x2 + k 2 )

m  −2  d (−1)m+1 k K ν (ak) = m 2 · m! k dk   m = 0, 1, 2, . . . , Re k > 0, a > 0, |Re ν| < Re  < 2m + 72 WA 466(2) x1−ν dx am K ν+m (ak) {cos νπ J ν (ax) − sin νπ Y ν (ax)} = m+1 2m · m!k ν+m (x2 + k 2 )  m = 0, 1, 2, . . . , Re k > 0, a > 0, −2m − 

 1

3 2

< Re ν < 1



WA 466(3)

      x−1 − 12 ν − μ π J ν (ax) + sin 12  − 12 ν − μ π Y ν (ax) μ+1 dx (x2 + k 2 ) ⎡  ν  

1 ak Γ 12  + 12 ν +ν +ν a2 k 2 πk−2μ−2 2 ⎣ 1  ; − μ, ν + 1; = 1F 2 2 sin νπ · Γ(μ + 1) Γ(ν + 1) Γ 2  + 12 ν − μ 2 2 4 ⎤  1 −ν  1 

Γ 2  − 12 ν −ν −ν a2 k 2 ⎦ 2 ak   ; − μ, 1 − ν; − 1F 2 2 2 4 Γ(1 − ν) Γ 12  − 12 ν − μ   WA 407(1) a > 0, Re k > 0, |Re ν| < Re  < 2 Re μ + 72

cos

2

6.591

Powers and Bessel functions of complicated arguments



∞

5.8 0

689

⎤⎧ ⎡ ⎛ ⎞ ⎤ ⎨  1 ⎣ J μj (bn x)⎦ cos ⎣ ⎝ + μj − ν ⎠ π ⎦ J ν (ax) ⎩ 2 j=1 j ⎫ ⎡ ⎛ ⎞ ⎤ ⎬ x−1  1 + sin ⎣ ⎝ + μj − ν ⎠ π ⎦ Y ν (ax) dx ⎭ x2 + k 2 2 j ⎡ ⎤ n = −⎣ I μj (bn k)⎦ K ν (ak)k −2 ⎡

n -

j=1

⎡ ⎣Re k > 0,

a>



⎛

|Re bj |,

Re ⎝ +

j



⎞

⎤

μj ⎠ > |Re ν|⎦

WA 472(9)

j

6.59 Combinations of powers and Bessel functions of more complicated arguments 6.591



∞

1. 0



1

x2ν+ 2 J ν+ 12

a x

K ν (bx) dx =

√ √ √ 1 2πb−ν−1 aν+ 2 J 1+2ν 2ab K 1+2ν 2ab [a > 0,

∞

0

1

x2ν+ 2 Y ν+ 12

a x

K ν (bx) dx =

√ √ √ 1 2πb−ν−1 aν+ 2 Y 2ν+1 2ab K 2ν+1 2ab [a > 0,

0

∞

4. 0

1

x2ν+ 2 K ν+ 12

a

x−2ν+ 2 J ν− 12 1

x

K ν (bx) dx =

√

∞

5.

−2ν+ 12

x 0

 6. 0

Re ν > −1]

2πb−ν−1 aν+ 2 1

 1 √  1 √ K 2ν+1 e 4 iπ 2ab K 2ν+1 e− 4 iπ 2ab

[Re a > 0, Re b > 0] ET II 146(56)  √ √ 1 K ν (bx) dx = 2πbν−1 a 2 −ν K 2ν−1 2ab x + √ √ , 2ab + cos(νπ) Y 2ν−1 2ab × sin(νπ) J 2ν−1

a

[a > 0, 

Re b > 0,

ET II 143(41) ∞

3. 

Re ν > −1] ET II 142(35)

2.



Re b > 0,

Y ν− 12

Re b > 0,

x−2ν+ 2 J 12 −ν 1

ET II 142(34)

 √ π ν−1 1 −ν K ν (bx) dx = − b a 2 sec(νπ) K 2ν−1 2ab x 2 √ √ , + 2ab − J 1−2ν 2ab × J 2ν−1

a

[a > 0, ∞

Re ν < 1]

a x

Re ν < 1]

ET II 143(40)

J ν (bx) dx

  1 1 = − i cosec(2νπ)bν−1 a 2 −ν e2νπi J 1−2ν (u) J 2ν−1 (v) − e−2νπi J 2ν−1 (u) J 1−2ν (v) , + 2 1 1  1  1 ET II 58(12) u = 12 ab 2 e 4 πi , v = 12 ab 2 e− 4 πi , a > 0, b > 0, − 12 < Re ν < 3

690

Bessel Functions



∞

7. 0

x−2ν+ 2 K ν− 12 1

a x

Y ν (bx) dx =

6.592

√ √ √ 1 2πbν−1 a 2 −ν Y 2ν−1 2ab K 2ν−1 2ab   b > 0, Re a > 0, Re ν > 16 ET II 113(30)

 

∞ aν− bν Γ 12 μ + 12  − 12 ν b −1   x J μ (ax) J ν dx = 2ν−+1 x 2 Γ(ν + 1) Γ 12 μ + 12 ν − 12  + 1 0

ν+μ− a2 b 2 ν−μ− + 1, + 1; × 0 F 3 ν + 1, 16 2 2 aμ bμ+ Γ 12 ν − 12 μ − 12    + 2μ++1 2 Γ(μ + 1) Γ 12 μ + 12 ν + 12  + 1

ν+μ+ a2 b 2 μ−ν+ + 1, + 1; × 0 F 3 μ + 1, 2    2  16  3 a > 0, b > 0, − Re μ + 2 < Re  < Re ν + 32

 8.

6.592



∞

1.

x (1 − x) λ

μ−1

0

 2.10

1

WA 480(1)

   √  Γ(μ) Γ λ + 1 + 12 ν −ν ν   Y ν a x dx = 2 a cot(νπ) Γ(1 + ν) Γ λ + 1 + μ + 12 ν

a2 1 1 × 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + 2 ν; −   4 Γ(μ) Γ λ + 1 − 12 ν ν −ν   −2 a cosec(νπ) Γ(1 − ν) Γ λ + 1 + μ − 12 ν

a2 × 1 F 2 λ − 12 ν + 1; 1 − ν, λ + 1 + μ − 12 ν; − 4  Re λ > −1 + 12 |Re ν|, Re μ > 0 ET II 197(76)a

 √  xλ (1 − x)μ−1 K ν a x dx

0

=2

+2

−ν−1 −ν

a

 

Γ(ν) Γ(μ) Γ λ + 1 − 12 ν a2 1 1   1 F 2 λ + 1 − 2 ν; 1 − ν, λ + 1 + μ − 2 ν; 4 Γ λ + 1 + μ − 12 ν

 

Γ(−ν) Γ λ + 1 + 12 ν Γ(μ) 1 a2 1   a 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + ν; 2 4 Γ λ + 1 + μ + 12 ν

−1−ν ν

2ν−1 = ν Γ(μ) G 21 13 a



a2 4

!ν

! −λ !2 ! ν, 0, ν − λ − μ 2 OB 159 (3.16)



Re λ > −1 + 12 |Re ν|,

Re μ > 0



ET II 198(87)a

6.592

Powers and Bessel functions of complicated arguments

 3.

∞

11

x (x − 1) λ

μ−1

1



∞

4.

1

5.

x− 2 (1 − x)− 2 1

1

0



1

6.

x− 2 (1 − x)− 2 1

1

0

 7.

1

x− 2 (1 − x)− 2

0

8. 9.



a2 4

!

!0 ! ! −μ, λ + 1 ν, λ − 1 ν Γ(μ) 2 2   a > 0, 0 < Re μ < 34 − Re λ

 √  xλ (x − 1)μ−1 K ν a x dx = Γ(μ)22λ−1 a−2λ G 30 13

1



 √  J ν a x dx = 22λ a−2λ G 20 13

1

1

691



ET II 205(36)a

!

a !! 0 4 ! −μ, 12 ν + λ, − 12 ν + λ 2

[Re a > 0, Re μ > 0]  2  √  1 a J ν a x dx = π J 1 ν [Re ν > −1] 2 2  2  √  1 a I ν a x dx = π I 1 [Re ν > −1] ν 2 2

+   a , a  √  1 a 1 νπ I ν2 + I − ν2 K ν2 K ν a x dx = π sec 2 2 2 2 2

ET II 209(60)a ET II 194(59)a

ET II 197(79)

[|Re ν| < 1] ET II 198(85)a  , + 2   √ 1 1 a x− 2 (x − 1)− 2 K ν a x dx = K ν2 [Re a > 0] ET II 208(56)a 2 1

 1 + +  a ,2  a ,2   √  − 12 − 12 x (1 − x) Y ν a x dx = π cot(νπ) J ν2 − cosec(νπ) J − ν2 2 2 0 

∞



∞

[|Re ν| < 1]

10. 1



a > 0,

0 < Re μ <

1 2

Re ν +

3 4



ET II 205(34)a ∞

11. 1



 √  1 x− 2 ν (x − 1)μ−1 J ν a x dx = Γ(μ)2μ a−μ J ν−μ (a) 

ET II 195(68)a

 √  1 x− 2 ν (x − 1)μ−1 J −ν a x dx = Γ(μ)2μ a−μ [cos(νπ) J ν−μ (a) − sin(νπ) Y ν−μ (a)]   a > 0, 0 < Re μ < 12 Re ν + 34 ET II 205(35)a

∞

12.

 √  1 x− 2 ν (x − 1)μ−1 K ν a x dx = Γ(μ)2μ a−μ K ν−μ (a)

1



[Re a > 0, ∞

13. 1



 √  1 x− 2 ν (x − 1)μ−1 Y ν a x dx = 2μ a−μ Y ν−μ (a) Γ(μ)  a > 0,

Re μ > 0]

0 < Re μ <

1 2

ET II 209(59)a

Re ν +

3 4



ET II 206(40)a ∞

14.

 √  1 (1) a x dx = 2μ a−μ H ν−μ (a) Γ(μ) x− 2 ν (x − 1)μ−1 H (1) ν

1

 15.

[Re μ > 0, ∞

Im a > 0]

ET II 206(45)a

Im a < 0]

ET II 207(48)a

 √  1 (2) a x dx = 2μ a−μ H ν−μ (a) Γ(μ) x− 2 ν (x − 1)μ−1 H (2) ν

1

[Re μ > 0,

692

Bessel Functions



1

16. 0

 17. 0

1

6.593

 √  1 22−ν a−μ s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 J ν a x dx = Γ(ν) [Re μ > 0]

ET II 194(64)a

 √  1 22−ν a−μ cot(νπ) s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 Y ν a x dx = Γ(ν) −2μ a−μ cosec(νπ) J μ−ν (a) Γ(μ) [Re μ > 0,

2

  √   a 1 b > 0, Re ν > − 12 x J 2ν−1 a x J ν (bx) dx = ab−2 J ν−1 2 4b 0  2

2   ∞  √  √ a a πa x J 2ν−1 a x K ν (bx) dx = 2 I ν−1 − Lν−1 4b 4b 4b 0   Re b > 0, Re ν > − 12

6.593  1. 2.

Re ν < 1]

6.594  1.

∞√

∞

0



 √   √  √ 1 x I 2ν−1 a x J 2ν−1 a x K ν (bx) dx = π2−ν a2ν−1 b−2ν− 2 J ν− 12 ν

[Re b > 0, ∞

2.



a2 2b

ET II 196(75)a

ET II 58(15)

ET II 144(44)



Re ν > 0]

ET II 148(65)

 √   √  xν I 2ν−1 a x Y 2ν−1 a x K ν (bx) dx

0

√ −ν−1 2ν−1 −2ν− 1 2 cosec(νπ) π2 a b  2

2

2  a a a × H 12 −ν + cos(νπ) J ν− 12 + sin(νπ) Y ν− 12 2b 2b 2b [Re b > 0, Re ν > 0] ET II 148(66)  ∞  √   √  xν J 2ν−1 a x K 2ν−1 a x K ν (bx) dx 0  2

2  a a 2 −ν−2 2ν−1 −2ν− 12 =π 2 a b cosec(νπ) H 12 −ν − Y 12 −ν 2b 2b [Re b > 0, Re ν > 0] ET II 148(67) =

3.

6.595  1.

∞

ν+1

x

J ν (cx)

0

n -

zi−μi J μi (ai zi ) dx = 0 i=1 % ai > 0,

Re bi > 0,

< zi = x2 + b2i  &  n n   1 1 n+ > Re ν > −1 ai < c; Re μi − 2 2 i=1 i=1 EH II 52(33), ET II 60(26)

 2. 0

∞

xν−1 J ν (cx)

<  −μi  bi J μi (ai bi ) zi−μi J μi (ai zi ) dx = 2ν−1 Γ(ν)c−ν zi = x2 + b2i i=1 % i=1  &  n n   1 3 n+ > Re ν > 0 ai < c, Re μi + ai > 0, Re bi > 0, 2 2 i=1 i=1 n -

n -

EH II 52(34), ET II 60(27)

6.596

6.596

Powers and Bessel functions of complicated arguments



∞

1. 0

693

  x2μ+1 2μ Γ(μ + 1) J ν α x2 + z 2  dx = μ+1 ν−μ−1 J ν−μ−1 (αz) ν α z (x2 + z 2 ) 

 1 1 ν− α > 0, Re > Re μ > −1 2 4

 √  ∞ α α J ν α t2 + 1 π √ Y ν2 [Re ν > −1, dt = − J ν2 2 2 2 t2 + 1 0  ∞   x2μ+1 2μ Γ(μ + 1) K ν α x2 + z 2  dx = K ν−μ−1 (αz) ν αμ+1 z ν−μ−1 (x2 + z 2 ) 0

WA 457(5)

 2. 3.

 4.8

∞

J ν (βx) 0

 √  J μ−1 α x2 + z 2 (x2 + z 2 )

1 1 2 μ+ 2

[α > 0, xν+1 dx =

αμ−1 z ν K ν (βz) 2μ−1 Γ(μ) [α < β,

α > 0]

MO 46

Re μ > −1]

WA 457(6)

Re(μ + 2) > Re ν > −1]

 √  ∞ J μ α x2 + z 2 ν−1 2ν−1 Γ(ν) J μ (αz) J ν (βx)  x dx = μ βν zμ (x2 + z 2 ) 0 [Re(μ + 2) > Re ν > 0,

ET II 59(19)

 5.8

β > α > 0] WA 459(12)

 √  ∞ J μ α x2 + z 2 ν+1 J ν (βx)  dx μ x (x2 + z 2 ) 0

 6.6

=0

μ−ν−1     βν α2 − β 2 = μ J μ−ν−1 z α2 − β 2 α z

8.8

[α > β > 0]

[Re μ > Re ν > −1] WA 415(1)   √  μ−ν−1  ∞   K μ α x2 + z 2 ν+1 α2 + β 2 βν 2 + β2 z J ν (βx)  x dx = K α μ−ν−1 μ αμ z (x2 + z 2 ) 0 + π, α > 0, β > 0, Re ν > −1, |arg z| < KU 151(31), WA 416(2) 2 

 ν  ∞     μ 1 π u 2 2 − 2 ν+1 2 2 x −y J ν (ux) K μ v x − y x dx = exp −iπ μ − ν − · μ 2 2 v 0 &μ−ν−1 %√   2 2 u +v (2) · H μ−ν−1 y u2 + v 2 y + , 1α  1α  2 1 Re μ < 1, Re ν > −1, u > 0, v > 0, y > 0; x − y 2 2 = e 2 απi y 2 − n2 2 if x < y 

7.8

[0 < α < β]

694

Bessel Functions

 9.

∞

8

6.597

   − μ v x2 + y 2 x2 + y 2 2 xν+1 dx J ν (ux) H (2) μ

0

uν = μ v ⎡ ⎣ Re μ > Re ν > −1, 

u > 0,

arg v 2 − u

 2 σ

%√ &μ−ν−1   v 2 − u2 (2) H μ−ν−1 y v 2 − u2 y [u < v]

v > 0,

y > 0; ,

arg

 v 2 − u2 = 0, for v > u ⎤

= −πσ for v < u, where σ =

μ−ν−1 ⎦ 1 or σ = 2 2 MO 43



∞

10.8 0



∞

11.8 0

    xν−1 2ν−1 Γ(ν) J μ (αz) J μ (γz) J ν (βx) J μ α x2 + z 2 J μ γ x2 + z 2 dx = μ β ν zμ  zμ (x2 + z 2 )   α > 0; β > α + γ; γ > 0, Re 2μ + 52 > Re ν > 0 WA 459(14) n n   <  −μ  −nμ x J μ (αk x) J ν (βt)tν−1 J μ αk t2 + x2 dt = 2ν−1 β −ν Γ(ν) (t2 + x2 ) k=1 k=1 % &

n 1 1 x > 0, α1 > 0, α2 > 0, . . . , αn > 0, β > αk ; Re nμ + n + > Re ν > 0 2 2 k=1

MO 43

√      a2 + x2 2ν−2 Γ ν − 12 8 √ Hν (2a) 12. x dx = Re ν > 12 WA 457(8) ν ν+1 2 2 2a π (a + x ) 0  ∞ +  1 ,  − 1 μ  −1 6.597 tν+1 J μ b t2 + y 2 2 t2 + y 2 2 t2 + β 2 J ν (at) dt 0 +  1 ,  − 1 μ = β ν J μ b y 2 − β 2 2 y 2 − β 2 2 K ν (aβ) 

∞J2 μ

[a ≥ b, 

1

6.598

Re β > 0,

−1 < Re ν < 2 + Re μ]

EH II 95(56)

  √   √  − 1 (ν+μ+1)  μ ν x 2 (1 − x) 2 J μ a x J ν b 1 − x dx = 2aμ bν a2 + b2 2 J ν+μ+1 a2 + b 2

0

[Re ν > −1,

Re μ > −1]

EH II 46a

6.61 Combinations of Bessel functions and exponentials 6.611  1. 0

∞

e−αx J ν (βx) dx =

β −ν

+ ,ν α2 + β 2 − α  α2 + β 2

[Re ν > −1,

Re (α ± iβ) > 0] EH II 49(18), WA 422(8)

6.611

Bessel functions and exponentials



∞

2. 0

− 1  e−αx Y ν (βx) dx = α2 + β 2 2 cosec(νπ)

+ + ,−ν ,ν   1 1 × β ν α2 + β 2 2 + α cos(νπ) − β −ν α2 + β 2 2 + α [Re α > 0,



∞

3.

695

e−αx K ν (βx) dx =

0

β > 0,

sin(νθ) π β sin(νπ) sin θ

|Re ν| < 1]

 α cos θ = ; β

MO 179, ET II 105(1)

θ→

π 2

 for β → ∞

ET II 131(22)  −ν  ν   π cosec(νπ) −ν  =  α2 − β 2 + α β α + α2 − β 2 − β ν 2 α2 − β 2 [|Re ν| < 1, Re(α + β) > 0] ET I 197(24), MO 180



∞

4.8 0

+ ,ν  β −ν α − α2 − β 2  e−αx I ν (βx) dx = α2 − β 2

[Re ν > −1,

Re α > |Re β|] MO 180, ET I 195(1)

5.

⎧ ⎡ 2ν ⎤⎫   ⎪ ⎪ 2 + β2 ⎨ ⎬ α α + i ⎥ ⎢ 1± cos(νπ) − ⎦ ⎣ 2ν ⎪ ⎪ sin(νπ) b ⎩ ⎭

 ν  ∞ α2 + β 2 − α  e−αx H (1,2) (βx) dx = ν β ν α2 + β 2 0

(2) [−1 < Re ν < 1; a plus sign corresponds to the function H (1) ν , a minus sign to the function H ν .]

⎧ ⎡ ⎤⎫ . 2 ⎬ ⎨ ∞ α α 1 2i (1) ⎦ e−αx H 0 (βx) dx =  1 − ln ⎣ + 1 + ⎭ π β β α2 + β 2 ⎩ 0

MO 180, ET I188(54, 55)

 6.

7.

[Re α > |Im β|] ⎧ ⎡ ⎤⎫ . 2 ⎬  ∞ ⎨ α ⎦ α 1 2i (2) e−αx H 0 (βx) dx =  1 + ln ⎣ + 1 + 2 2 ⎭ ⎩ π β β α +β 0

MO 180, ET I 188(53)

[Re α > |Im β|]

MO 180, ET I 188(53)

[Re α > |Im β|]

MO 47, ET I 187(44)



∞

8. 0

 9.11 0

∞

e−αx Y 0 (βx) dx =

π



−2 α2 + β 2

ln

α+



α2 + β 2 β

α β

arccos e−αx K 0 (βx) dx =  β 2 − α2 1 = ln 2 α − β2

[Re(α + β) > 0] WA 424, ET II 131(22) 

. α + β

α2 −1 β2

 [Re(α + β) > 0] MO 48

696

Bessel Functions

 10.10



b

∞

α dα a

dk J 1 (kα)e

−k|β|



b

=

0

 1− 

a

|β| α2 + β 2

6.612

 dα (see 3.241 6)

6.612  1.

∞

e−2αx J 0 (x) Y 0 (x) dx =

+  − 1 , K α α2 + 1 2 1

π (α2 + 1) 2  ∞ 1 , 1 + e−2αx I 0 (x) K 0 (x) dx = K 1 − α2 2 2 % 0

1 & 1 2 1 K 1− 2 = 2α α

[Re α > 0]

ET II 347(58)

0

2.



∞

3. 0

 4.

6.

[1 < α < ∞]



−αx

2



2



ET II 370(48)

2

α2 + β 2 + γ 1 e−αx J ν (βx) J ν (γx) dx = √ Q ν− 12 2βγ  π γβ Re (α ± iβ ± iγ) > 0, γ > 0,

∞

2β

Re ν > − 12



WA 426(2), ET II 50(17)



K  α2 + 4β 2 α2 + 4β 2



   2   ∞ 2α + β 2 K √ 2β 2 − 2 α2 + β 2 E √ 2β 2 α +β α +β  e−2αx J12 (βx) dx = 2 2 2 πβ α +β 0  ∞ r2 e−3x I l (x) I m (x) I n (x) dx = r1 g + 2 + r3 π g 0 where √

11 3−1 2 1 g= Γ Γ2 96π 3 24 24 e

0

5.

[0 < α < 1]

and

[J 0 (βx)] dx =

π

MO 178

WA 428(3)

6.614

Bessel functions and exponentials

(lmn) 000 100 110 111 200 210 211 220 221 222 300 310 311 320 321 322 330 331 332 333 400 410 411 420 421 422 430 431 

∞

11

6.613

e

r1 1 1 5/12 − 1/8 10/3 3/8 − 2/3 73/36 − 15/16 5/8 35/2 − 79/36 − 11/4 319/48 − 125/36 35/16 50/3 − 35/3 35/9 − 35/16 994/9 − 515/16 − 9/2 12907/120 − 229/16 35/3 2641/48 − 1505/36 −xz

0

6.614



∞

1.

e 0

−αx

J

ν+ 12

r2 0 0 − 1/2 3/4 2 − 9/4 2 − 29/6 21/8 − 27/20 21 − 85/6 21/2 − 119/8 269/30 − 213/40 − 1046/25 148/5 − 1012/105 1587/280 542/3 − 879/8 357/5 − 13903/10 1251/40 − 1024/35 − 28049/200 118051/1050 x2 2

dx =

r3 0 − 1/3 0 0 −2 1/3 0 0 0 0 −13 4 − 2/3 − 1/3 0 0 0 0 0 0 −92 115/3 −12 −6 1 0 1/3 0

(lmn) 432 433 440 441 442 443 444 500 510 511 520 521 522 530 531 532 533 540 541 542 543 544 550 551 552 553 554 555

r1 525/32 − 595/72 6025/36 − 29175/224 2975/48 − 539/32 77/8 9287/12 − 189029/180 275/4 2897/16 − 937/12 509/8 3589/18 − 1329/8 2555/36 − 2233/48 18471/32 − 1390/3 7777/32 − 5621/72 1155/32 197045/108 − 12023/8 1683/2 − 5159/16 24563/312 − 9251/208

697

r2 − 4617/112 8809/420 − 620161/1470 131379/400 − 31231/200 119271/2800 − 186003/7700 3005/2 − 138331/50 5751/10 − 15123/20 27059/30 − 4209/28 − 1993883/3075 297981/700 − 187777/1050 164399/1400 − 28493109/19600 286274/245 − 1715589/2800 4550057/23100 − 560001/6160 − 101441689/22050 18569853/4900 − 5718309/2695 2504541/3080 − 1527851/77000 12099711/107800

  π  πi Γ(ν + 1) √ D −ν−1 ze 4 i D −ν−1 ze− 4 π

r3 0 0 0 0 0 0 0 − 2077/3 348 −150 − 229/3 24 0 0 − 4/3 0 0 − 1/3 0 0 0 0 0 0 0 0 0 0

[Re ν > −1]

 2

2 

  √  β β π β β2 J ν β x dx = exp − I 12 (ν−1) − I 12 (ν+1) 4 α3 8α 8α 8α 2 1 = e−β /4α α

MO 122

[ν = 0] MO 178

 2. 0



1 β   β e− 2 α Γ(ν + 1) M 12 ,ν e−αx Y 2ν 2 βx dx = √ cot(νπ) − cosec(νπ) W Γ(2ν + 1) α αβ

 β 1 2 ,nu α

[Re α > 0,

∞   β e Γ(ν + 1) M − 12 ,ν e−αx I 2ν 2 βx dx = √ Γ(2ν + 1) α αβ 0

|Re ν| < 1]

ET I 188(50)a

[Re α > 0,

Re ν > −1]

ET I 197(20)a

 3.

∞

1 β 2 α

698

Bessel Functions



∞

4. 0

6.615



1 β   β e2 α e−αx K 2ν 2 βx dx = √ Γ(ν + 1) Γ(1 − ν) W − 12 ,ν α 2 αβ

[Re α > 0, |Re ν| < 1] ET I 199(37)a



  √  β2 β2 β2 π β 5. e−αx K 1 β x dx = exp MO 181 K1 − K0 3 8 α 8α 8α 8α 0



 ∞  √   √  2βγ 1 β2 + γ2 6.615 e−αx J ν 2β x J ν 2γ x dx = I ν exp − [Re ν > −1] α α α 0 



∞



MO 178

6.616 1. 2. 3.



∞



∞

  , +   1 e−αx J 0 β x2 + 2γx dx =  exp γ α − α2 + β 2 MO 179 α2 + β 2 0  ∞     1 e−αx J 0 β x2 − 1 dx =  exp − α2 + β 2 MO 179 α2 + β 2 1 √  ∞ 2 2   eiα r +x (1) itx 2 2 e H 0 r α − t dt = −2i √ 2 2 −∞ + , r + x 0 ≤ arg α2 − t2 < π, 0 ≤ arg α < π; r and x are real MO 49

4.

e −∞



1

5.3 −1



∞

1. 0



√ 2 2   e−iα r +x 2 2 r α − t dt = 2i √ 2 2 +  r +x −π < arg α2 − t2 ≤ 0,

, r and x are real

−π < arg α ≤ 0,

[a > 0, ∞

0



(2) H0

∞

2. 0

b > 0]

∞ +  ,  P n (x) e−xy J 0 y 1 − x2 /(α + y) dy = n! n+1 α n=0

K q−p (2z sinh x) e(p+q)x dx =

π2 [J p (z) Y q (z) − J q (z) Y p (z)] 4 sin[(p − q)π] [Re z > 0, −1 < Re(p − q) < 1] MO 44

 π ∂ Y p (z) ∂ J p (z) −2px − Y p (z) K 0 (2z sinh x) e dx = − J p (z) 4 ∂p ∂p [Re z > 0]

6.618



1. 0

MO 49

    −1/2 e−ax I 0 b 1 − x2 dx = 2 a2 + b2 sinh a2 + b2

6.8 6.617

−itx

∞

2

√ 2 β π β2 e−αx J ν (βx) dx = √ exp − I 12 ν 8α 8α 2 α

[Re α > 0,

MO 44

β > 0,

Re ν > −1] WA 432(5), ET II 29(8)

6.621

Bessel functions, exponentials, and powers



∞

2.

e

−αx2

0

699

2

2 

 √  νπ β β π β2 νπ 1 I 12 ν K 12 ν Y ν (βx) dx = − √ exp − tan + sec 8α 2 8α π 2 8α 2 α [Re α > 0, β > 0, |Re ν| < 1]

2

2

∞  νπ √π 2 β β 1 √ exp e−αx K ν (βx) dx = sec K 12 ν 4 2 8α 8α α 0 [Re α > 0,

WA 432(6), ET II 106(3)

 3.



∞

4. 0



∞

5.

|Re ν| < 1] EH II 51(28), ET II 132(24)

2

2

√ β β π −αx2 e I ν (βx) dx = √ exp I 12 ν 8α 8α 2 α

[Re ν > −1,

Re α > 0]

EH II 92(27)

e−αx J μ (βx) J ν (βx) dx 2

  Γ μ+ν+1 2 =2 α β Γ(μ + 1) Γ(ν + 1)

ν+μ+1 ν+μ+2 ν+μ+1 β2 , , ; μ + 1, ν + 1, ν + μ + 1; − × 3F 3 2 2 2 α [Re(ν + μ) > −1, Re α > 0] EH II 50(21)a

0

−ν−μ−1 − ν+μ+1 2

ν+μ

6.62–6.63 Combinations of Bessel functions, exponentials, and powers 6.621

Notation: ,  1 + (a + ρ)2 + z 2 − (a − ρ)2 + z 2 , 1 = 2 

1.

∞

e−αx J ν (βx)xμ−1 dx

0



=

ν



Γ(ν + μ) F

αμ Γ(ν + 1) 

=

β 2α

2 =

,  1 + (a + ρ)2 + z 2 + (a − ρ)2 + z 2 2

ν+μ ν+μ+1 β2 , ; ν + 1; − 2 2 2 α



WA 421(2)

ν

1 −μ

Γ(ν + μ) ν−μ+1 ν−μ β2 β2 2 , + 1; ν + 1; − 2 F 1+ 2 αμ Γ(ν + 1) α 2 2 α

β 2α

WA 421(3)

 ν =<

β 2

Γ(ν + μ)

ν+μ 1−μ+ν β , ; ν + 1; 2 2 2 α + β2

F Γ(ν + 1) [Re(ν + μ) > 0,

ν+μ

(α2 + β 2 )



2

Re (α + iβ) > 0,

+  − 1 μ  1,  2 2 −2 = α2 + β 2 2 Γ(ν + μ) P −ν μ−1 α α + β [α > 0,

β > 0,



Re (α − iβ) > 0] WA 421(3)

Re(ν + μ) > 0] ET II 29(6)

700

Bessel Functions



∞

2.

e−αx Y ν (βx)xμ−1 dx

6.621

 ν

0

β 2

= cot νπ <



Γ(ν + μ) F ν+μ

(α2 + β 2 )

 −ν β 2

− cosec νπ <

Γ(ν + 1)



Γ(μ − ν) μ−ν

(α2 + β 2 )

ν+μ ν−μ+1 β2 , ; ν + 1; 2 2 2 α + β2

F

Γ(1 − ν)



μ−ν 1−ν−μ β2 , ; 1 − ν; 2 2 2 α + β2

[Re μ ≥ |Re ν|, +   − 1 μ  1, 2 2 2 −2 = − Γ(ν + μ) β 2 + α2 2 Q −ν + β α α μ−1 π [α > 0, β > 0,



Re (α ± iβ) > 0] WA 421(4)

Re μ > |Re ν|] ET II 105(2)



∞

3.

μ−1 −αx

x

e

0



∞

4.



√ 1 α−β π(2β)ν Γ(μ + ν) Γ(μ − ν) 1   K ν (βx) dx = F μ + ν, ν + ; μ + ; (α + β)μ+ν 2 2 α+β Γ μ + 12 [Re μ > |Re ν|, Re(α + β) > 0] ⎡ 

m+1 −αx

x

e

m+1 −ν

J ν (βx) dx = (−1)

β

0

 5.10

e−zx J 1 (ax) J 1/2 (ρx) x−3/2 dx

8.10

9.10



ET II 131(23)a, EH II 50(26)

⎦

Re ν > −m − 2]

ET II 28(3)



 < 1 a2 arcsin ρ2 − 21 − ρ +z 2 2 [arg a > 0, arg ρ > 0, arg z > 0]     ∞ < 2 1 −zx −1/2 2 2 e J 1 (ax) J 1/2 (ρx) x dx = ρ − ρ − 1 a πρ 0 [arg a > 0, arg ρ > 0, arg z > 0]    ∞ 2 1 a2 − 21 1 e−zx J 1 (ax) J 1/2 (ρx) x1/2 dx = a πρ 22 − 21 0 [arg a > 0, arg ρ > 0, arg z > 0]    ∞ 2 21 ρ2 − 21 e−zx J 1 (ax) J 3/2 (ρx) x1/2 dx = π ρ3/2 a (22 − 21 ) 0 [arg a > 0, arg ρ > 0, arg z > 0]

   ∞ < 1 1 1 2 2 − 2 e−zx J 1 (ax) J 3/2 (ρx) x−3/2 dx = √ arcsin a a −  1 1 a 2π ρ3/2 a 0 [arg a > 0, arg ρ > 0, arg z > 0] =

7.10

α2 + β 2 − α  α2 + β 2 [β > 0,

∞

0

6.10

dm+1 ⎣ dαm+1

ν ⎤

1 a

2 πρ

1 2

<

a2 − 21 +

6.621

Bessel functions, exponentials, and powers



∞

10.10

e 0



∞

11.10

0

13.10

14.10

15.10

16.10

17.10

J 1 (ax) J 5/2 (ρx) x

% < & 2 1  z 1 2a 1 2 dx = √ − 3a arcsin 1 a2 − 21 +  2 2 a 2π ρ5/2 a a − 1 [arg a > 0, arg ρ > 0, arg z > 0]

e−zx J 1 (ax) J 5/2 (ρx) x−3/2 dx

[arg a > 0, ∞

12.10



−1/2

⎡ 2

7a 1 5a2 21 1 ⎣ 41 1 2 2  =√ −a z − − 8 4 8 2π ρ5/2 a a2 − 21 ⎤

< 4   1 3 2 2 1 2 2 3a ⎦ 1 2  + 22 1 a2 − 21 + arcsin a z + a ρ − − 2 1 a 2 2 8

0



−zx

701

∞

arg ρ > 0,

arg z > 0]

e−zx J 1 (ax) J 5/2 (ρx) x−5/2 dx ⎧ +  ,  2  2  5/2 2 5/2 ⎨ − ρ −  2 ρ 1 1 ρ2 z2 3a 1 1 2 + za arcsin − − =√ 15 a 8 2 2 2π ρ5/2 a ⎩ ⎫  2  < ρ 3a2 z2 21 z 3 a2 1 ⎬ 2 2 − + − + z1 a − 1 +  2 8 6 4 3 a2 − 21 ⎭ 



[arg a > 0,

arg ρ > 0,

arg z > 0]

22 − ρ2 2 2 3/2 a ρ 2 π (2 − 21 ) 42 0 [arg a > 0, arg ρ > 0, arg z > 0] %    2 3/2 &  ∞ ρ − 21 ρ2 − 21 2 ρ3/2 2 −zx −1/2 − + e J 2 (ax) J 3/2 (ρx) x dx = π a2 3 ρ 3ρ3 0 [arg a > 0, arg ρ > 0, arg z > 0]  ∞ e−zx J 3 (ax) J 1/2 (ρx) x−1/2 dx 0 

  <  2   2 2 1 2 2 2 2 2 2 2 = ρ 3a − 4ρ + 12z − ρ − 1 122 − 16ρ + 41 − 3a πρ 3a3 [arg a > 0, arg ρ > 0, arg z > 0]  ∞ e−zx J 3 (ax) J 3/2 (ρx) x1/2 dx 0  %     2 3/2 &  ρ − 21 ρ2 − 21 a 22 − a2 2 3/2 4 2 ρ − + − 2 = π a3 3 ρ 3ρ2 (2 − 21 ) 32 [arg a > 0, arg ρ > 0, arg z > 0] %<   &   2   ∞ 2 2 4 3/2 2ρ −  4ρ −  2 ρ 1 1 e−zx J 3 (ax) J 3/2 (ρx) x−1/2 dx = 22 − ρ2 − 8z π 3a3 ρ4 0 [arg a > 0, arg ρ > 0, arg z > 0] e−zx J 2 (ax) J 3/2 (ρx) x1/2 dx =

702

Bessel Functions

 18.

∞

10 0



∞

19.10 0

6.622

e−zx J 3 (ax) J 3/2 (ρx) x−3/2 dx 

 2  < 42 24ρ 821 a2 41 2 ρ3/2 4 2 2 2 2 − 2 ρ − + − + − + 4z − ρ a = 1 π 3a3 5 ρ 5 5ρ ρ 5ρ3 [arg a > 0, arg ρ > 0, arg z > 0] e−zx J 3 (ax) J 3/2 (ρx) x−5/2 dx ⎧ 

2 ρ3/2 ⎨ 2 4 2 4z 3 =− ρ − z + a π 3a3 ⎩ 5 3   < 32 2 12 2 4 2 241 a4 21 a2 21 61 2 2 2 + + + 2 − ρ a + ρ − 1 − 2 + 2 + 15⎫ 5 3 5ρ 16ρ4 24ρ4 30ρ4

ρ ⎬ a6 − arcsin 16ρ3 2 ⎭ [arg a > 0,

6.622



∞

1. 0

 2.

3.8

  dx = ln 2α J 0 (x) − e−αx x

arg ρ > 0,

arg z > 0]

[α > 0]

NT 66(13)

∞ i(u+x)

e π (1) J 0 (x) dx = i H 0 (u) u+x 2 0 μ− 12   ∞ 2 −(μ− 12 )πi Q ν− 12 (cosh α) −x cosh α μ−1 e e I ν (x)x dx = 1 π 0 sinhμ− 2 α [Re(μ + ν) > 0,

MO 44

Re (cosh α) > 1] WA 388(6)a

6.623



∞

1.

e 0

−αx

  (2β)ν Γ ν + 12 J ν (βx)x dx = √ ν+ 1 π (α2 + β 2 ) 2



ν

Re ν > − 12 ,

Re α > |Im β|

 WA 422(5)

  ∞ 2α(2β)ν Γ ν + 32 e−αx J ν (βx)xν+1 dx = √ ν+ 3 0 π (α2 + β 2 ) 2

 2.



∞

3. 0

e−αx J ν (βx)

dx = x

[Re ν > −1,

WA 422(6)

 ν α2 + β 2 − α νβ ν [Re ν > 0;

6.624  1. 0

∞

Re α > |Im β|]

Re α > |Im β|]

(cf. 6.611 1)

⎫ ⎤ .

2 ⎬ α α 1 α  xe−αx K 0 (βx) dx = 2 ln ⎣ + − 1⎦ − 1 2 ⎭ α − β ⎩ α2 − β 2 β β ⎧ ⎨

WA 422(7)

⎡

MO 181

6.625

Bessel functions, exponentials, and powers

 2.

∞√

−αx

xe

0



∞

3.

e−tz(z

2

 K ± 12 (βx) dx = −1/2

−1)

π 1 2β α + β

∞

4.

e−tz(z

2

e−tz(z

2

−1/2

−1)

(z 2

I −μ (t)tν dt =

0



∞

5.

−1 2

−1)

I μ (t)tν dt =

0



∞

6.

MO 181

Γ(ν − μ + 1)

K μ (t)tν dt =

0



− 12 (ν+1)

− 1)

Γ(−ν − μ) 1

(z 2 − 1) 2

ν

P μν (z)

Γ(ν + μ + 1) − 12 (ν+1)

(z 2 − 1)

eiμπ Q μν (z)

∞

7. 0

[Re (ν ± μ) > −1]

EH II 57(7)

[Re(ν + μ) < 0]

EH II 57(8)

[Re(ν + μ) > −1]

EH II 57(9)

P −μ ν (z)

e−t cos θ J μ (t sin θ) tν dt = Γ(ν + μ + 1) P −μ ν (cos θ)

0





Re(ν + μ) > −1,





EH II 57(10)

 ∞  (2b) Γ ν + 12  J ν (bx)xν 1 √ dx = 1 πx e −1 π 2 2 2 ν+ 2 n=1 (n π + b )

1 λ−ν−1

x

1.

0 ≤ θ < 12 π

ν

[Re ν > 0, 6.625

703

μ−1 ±iαx

(1 − x)

e

0

|Im b| < π]

WA 423(9)

2−ν αν Γ(λ) Γ(μ) 1 J ν (αx) dx = 2 F 2 λ, ν + ; λ + μ, 2ν + 1; ±2iα Γ(λ + μ) Γ(ν + 1) 2

[Re λ > 0, Re μ > 0] ET II 194(58)a  

1 (2α)ν Γ(μ) Γ ν + 12 1 ν μ−1 ±iαx x (1 − x) e J ν (αx) dx = √ 1 F 1 ν + ; μ + 2ν + 1; ±2iα 2 π Γ(μ + 2ν + 1) 0   Re μ > 0, Re ν > − 12 ET II 194(57)a  

 1 (2α)ν Γ ν + 12 Γ(μ) 1 ν μ−1 ±αx x (1 − x) e J ν (αx) dx = √ 1 F 1 ν + ; μ + 2ν + 1; ±2α 2 π Γ(μ + 2ν + 1)  0  Re μ > 0, Re ν > − 12 

2.

3.

BU 9(16a), ET II 197(77)a

 1 ν 1 α Γ (λ + ν) Γ(μ) λ−1 μ−1 ±αx x (1 − x) e I ν (αx) dx = 2 Γ(ν + 1) Γ(λ + μ + ν) 0

1 × 2 F 2 ν + , λ + ν; 2ν + 1, μ + λ + ν; ±2α 2 [Re μ > 0, Re(λ + ν) > 0] ET II 197(78)a

 4.





1

xμ−κ (1 − x)2κ−1 I μ−κ

5. 0

1 x 1 1 Γ(2κ) xz e− 2 xz dx = √ e 2 z −κ− 2 M κ,u (z) 2 π Γ(1 + 2μ)     Re κ − 12 − μ < 0, Re κ > 0 BU 129(14a)

704

Bessel Functions



∞

6.

−λ

x

μ−1 −αx

(x − 1)

1



∞

7.

e

6.626

!1

! − λ, 0 (2α)λ Γ(μ) 21 2 ! √ I ν (αx) dx = G 23 2α ! π  −μ, ν − λ,1−ν − λ  0 < Re μ < 2 + Re λ, Re α > 0

√ x−λ (x − 1)μ−1 e−αx K ν (αx) dx = Γ(μ) π(2α)λ G 30 23

1



∞

8. 1



∞

9.

! 1

! 0, − λ 2 ! 2α ! −μ, ν − λ, −ν − λ

[Re μ > 0, Re α > 0]  (2α) Γ 2 − μ + ν Γ(μ) −ν μ−1 −αx √ x (x − 1) e I ν (αx) dx = π Γ(1 − μ + 2ν)

1 − μ + ν; 1 − μ + 2ν; −2α × 1F 1 2  0 < Re μ < 12 + Re ν, Re α > 0 ν−μ

1

[Re μ > 0, ∞

10.

x−μ− 2 (x−1)μ−1 e−αx K ν (αx) dx = 1

ET II 207(50)a

ET II 208(55)a

ET II 207(49)a

√ 1 1 π Γ(μ)(2α)− 2 μ− 2 e−α W − 12 μ,ν− 12 μ (2α)

x−ν (x − 1)μ−1 e−αx K ν (αx) dx =

1





Re α > 0]

ET II 208(53)a

√ 1 π Γ(μ)(2α)− 2 e−α W −μ,ν (2α)

1

[Re μ > 0,

ET II 207(51)a Re α > 0] 1     −1/2 −ax   −1/2 2 sinh a − a a2 + b2 1 − x2 xe I 1 b 1 − x2 dx = sinh a2 + b2 b −1

 11.3

[a > 0, 6.626  1.11

∞

xλ−1 e−αx J μ (βx) J ν (γx) dx =

0



∞

2.

e 0

−2αx

ν+μ

J ν (βx) J μ (βx)x

∞ β μ γ ν −ν−μ −λ−μ−ν  Γ(λ + μ + ν + 2m) 2 α Γ(ν + 1) m! Γ(μ + m + 1) m=0

m 2 γ β2 × F −m, −μ − m; ν + 1; 2 − 2 β 4α [Re(λ + μ + ν) > 0, Re (α ± iβ ± iγ) > 1] EH II 48(15)

  Γ ν + μ + 12 β ν+μ √ dx = π3  π2 cosν+μ ϕ cos(ν − μ)ϕ  dϕ × ν+μ 2 2 2 2 + β 2 cos2 ϕ α 0 (α + β cos ϕ)   Re α > |Im β|, Re(ν + μ) > − 12

 3.

K e−2αx J 0 (βx) J 1 (βx)x dx =

√

β α2 +β 2

−E

√

WA 427(1)



β α2 +β 2

 α2 + β 2 

 ∞ β β α 1 1 −2αx e I 0 (βx) I 1 (βx)x dx = E − K 2 − β2 2πβ α α α α 0 0

4.

∞

b > 0]

2πβ

[Re α > Re β]

WA 427(2)

WA 428(5)

6.628

5.

10

Bessel functions, exponentials, and powers



∞



∞

1  a μ−ν−2n−1  ρ ν xν−μ+2n e−zx J μ (αx) J ν (ρx) dx = √ a  π 2 0   ∞ Γ ν +n+q+ 1  ν − μ + n + 12 q 1 2    ×  Γ μ − ν − n + 12 q=0 q! Γ ν + q + 12

q  1 /ρ dx z2 √ ×a−2q x2ν+2q ρ2 + 2 1 − x2 0 + , 1 − x   μ > ν + 2n, n = 0, 1, . . . , ν > − 12 where 1 = 12 (a + ρ)2 + z 2 − (a − ρ)2 + z 2

6.627 0

6.628



∞

1. 0

2.

3.

705

x−1/2 −x πea K ν (a) e K ν (x) dx = √ x+a a cos(νπ)



|arg a| < π,

|Re ν| <

1 2



e−x cos β J −ν (x sin β) xμ dx = Γ(μ − ν + 1) P νμ (cos β) + π 0<β< , 2

ET II 368(29)

Re(μ − ν) > −1

,

WA 424(3), WH



∞



∞

sin μπ Γ (μ − ν + 1) sin(μ + ν)π π 0 , + 1 1 ν × Q μ (cos β + 0 · i) e 2 νπi + Q νμ (cos β − 0 · i) e− 2 νπi + π, Re(μ + ν) > −1, 0 < β < WA 424(4) 2

 1 u xu π ixu B(2ν, 2μ − 2ν + 1) e 2 e 2 (1 − x)2ν−1 xμ−ν J μ−ν dx = 22(ν−μ) e 2 (μ−ν)i 1 M ν,μ (u) 2 Γ(μ − ν + 1) uν+ 2 0 MO 118a

4.8 0



e−x cos β Y ν (x sin β) xμ dx = −

e−x cosh α I ν (x sinh α) xμ dx = Γ(ν + μ + 1) P −ν μ (cosh α)  Re(μ + ν) > −1,

|Im α| < 12 π

WA 423(1) ∞

5.

e−x cosh α K ν (x sinh α) xμ dx =

0

sin μπ Γ(μ − ν + 1) Q νμ (cosh α) sin(ν + μ)π [Re(μ + 1) > |Re ν|]





∞

6.

e−x cosh α I ν (x)xμ−1 dx =

0

WA 423(2)

ν− 12 μ− 12

(cosh α) Q cos νπ  sin(μ + ν)π π (sinh α)μ− 12 2 [Re(μ + ν) > 0,

Re (cosh α) > 1] WA 424(6)

 7.



∞

e 0

−x cosh α

μ−1

K ν (x)x

dx =

1

−μ

2 P ν− 1 (cosh α) π 2 Γ(μ − ν) Γ(μ + ν) μ− 1 2 (sinh α) 2 [Re μ > |Re ν|,

Re (cosh α) > −1] WA 424(7)

706

Bessel Functions



∞

8

6.629

x−1/2 e−xα cos ϕ cos ψ J μ (αx sin ϕ) J ν (αx sin ψ) dx

0

  1 = Γ μ + ν + 12 α− 2 P −μ (cos ϕ) P −ν 1 (cos ψ) ν− 12  μ− 2 π 1 π 0 < ϕ < , 0 < ψ < , Re(μ + ν) > − ET II 50(19) 2 2 2

 α > 0, 6.631



∞

1.

6.629

μ −αx2

x e



1

1 1 2ν + 2μ + 2 1 2ν+1 α 2 (μ+ν+1) Γ(ν +

βν Γ

J ν (βx) dx =

0

1

1)

1F 1

ν+μ+1 β2 ; ν + 1; − 2 4α

1 2

BU 8(15)



2

β β2 = exp − M 12 μ, 12 ν 1 μ 8α 4α 2 βα Γ(ν + 1) [Re α > 0, 1 2ν + 2μ +

Γ



Re(μ + ν) > −1]

EH II 50(22), ET II 30(14), BU 14(13b)



∞

2.

xμ e−αx Y ν (βx) dx 2



ν−μ β2 = −α π exp − β sec 2 8α   

2

1 1 1 Γ 2 + 2μ + 2ν ν−μ β sin π M 12 μ, 12 ν × +W Γ(1 + ν) 2 4α

0



− 12 μ −1

Re μ > |Re ν| − 1,

[Re α > 0, 

∞

3.

xμ e−αx K ν (βx) dx = 2

0



∞

4.11

1 − 1 μ −1 α 2 β Γ 2

xν+1 e−αx J ν (βx) dx = 2

0



∞

5. 0



∞

6.

7. 0

∞

1+ν+μ 2



β (2α)ν+1

xν+1 e±iαx J ν (βx) dx = 2

xe−αx J ν (βx) dx = 2

√ πβ 8α

3 2

β > 0]

[Re μ > |Re ν| − 1]

2

[Re α > 0,

exp −



 2

β 8α

I 12 ν− 12

ET II 106(4)

ET II 132(25)

WA 431(4), ET II 29(10)

[Re α > 0, Re ν > 0] 

 ν+1 β2 π− exp ±i 2  4α α > 0, −1 < Re ν < 12 ,





Re ν > −1]



 β2 ν−1 −αx2 ν−1 −ν x e J ν (βx) dx = 2 β 1 − γ ν, 4α ν

1 1 2 μ, 2 ν

β2 4α

2



2

β 1−ν+μ β Γ exp W − 12 μ, 12 ν 2 8α 4α

βν β exp − ν+1 (2α) 4α

0







2

β 8α

− I 12 ν+ 12

[Re α > 0,

 2

ET II 30(11)

β>0



ET II 30(12)

β 8α

Re ν > −2]

ET II 29(9)

6.633

Bessel functions, exponentials, and powers

 8.

1

n+1 −αx2

x

e

0



∞

9.

& % n  1 α −α I n (2αx) dx = I r (2α) e −e 4α r=−n %

x1−n e−αx I n (2αx) dx = 2

1



∞

10. 0

707

[n = 0, 1, . . .] &

ET II 365(8)a

[n = 1, 2, . . .]

ET II 367(20)a

n−1  1 eα − e−α I r (2α) 4α r=1−n

 √  2 1 n! e−x x2n+μ+1 J μ 2x z dx = e−z z 2 μ Lμn (z) 2

[n = 0, 1, . . . ;

n + Re μ > −1] BU 135(5)



∞

6.632

+  1 ,  − 1 1 x− 2 exp − x2 + a2 − 2ax cos ϕ 2 x2 + a2 − 2ax cos ϕ 2 K ν (x) dx

0

= πa− 2 sec(νπ) P ν− 12 (− cos ϕ) K ν (a)   |arg a| + |Re ϕ| < π, |Re ν| < 12 ET II 368(32) 1

6.633



∞

1.

λ+1 −αx2

x

e

0

2.

3.

4.

 

m μ+ν+λ+2 ∞  2 Γ m + 12 ν + 12 μ + 12 λ + 1 β μ γ ν α− β2 J μ (βx) J ν (γx) dx = ν+μ+1 − 2 Γ(ν + 1) m=0 m! Γ(m + μ + 1) 4α

γ2 × F −m, −μ − m; ν + 1; 2 β [Re α > 0, Re(μ + ν + λ) > −2, β > 0, γ > 0] EH II 49(20)a, ET II 51(24)a





αβ 1 α2 + β 2 e J p (αx) J p (βx)x dx = 2 exp − Ip 2 42 22 0 + , π Re p > −1, |arg | < , α > 0, β > 0 KU 146(16)a, WA 433(1) 4

 ∞ 1 1 − 3 ν− 1 1 2ν+1 −αx2 2 2 x e J ν (x) Y ν (x) dx = − √ α exp − W 12 ν, 12 ν 2α α 2 π 0   Re α > 0, Re ν > − 12 ET II 347(59) 2



 ∞ 2 β − γ2 βγ 1 exp xe−αx I ν (βx) J ν (γx) dx = Jν 2α 4α 2α 0 

∞



∞

5. 0

−2 x2

[Re α > 0, xλ−1 e−αx J μ (βx) J ν (βx) dx

Re ν > −1]

ET II 63(1)

2

 + 12 μ + 12 ν =2 α β Γ(μ + 1) Γ(ν + 1)   ν μ 1 ν μ ν+μ+λ β2 + + , + + 1, ; μ + 1, ν + 1, μ + ν + 1; − × 3F 3 2 2 2 2 2 2 α [Re(ν + λ + μ) > 0, Re α > 0] WA 434, EH II 50(21) −ν−μ−1 − 12 (ν+λ+μ) ν+μ Γ

1

2λ

708

Bessel Functions



∞

6.634

x2

xe− 2a [I ν (x) + I −ν (x)] K ν (x) dx = aea K ν (a)

6.634

[Re a > 0,

−1 < Re ν < 1]

0

ET II 371(49)

6.635  1.

∞

x−1 e− x J ν (βx) dx = 2 J ν α

  2αβ K ν 2αβ

0

[Re α > 0,



∞

2.

x−1 e− x

α

β > 0]

ET II 30(15)

  Y ν (βx) dx = 2 Y ν 2αβ K ν 2αβ

0



∞

3.

ET II 106(5) [Re α > 0, β > 0]



1 + , + , 12    √ √ α 2 x−1 e− x −βx J ν (γx) dx = 2 J ν 2α β2 + γ2 − β 2α β2 + γ2 + β Kν

0

[Re α > 0,

Re β > 0,

γ > 0] ET II 30(16)



∞

6.636 0

√  1 1  1 √  1 1 1 2  − 12 −α x x e J ν (βx) dx = √ Γ ν + 12 D −ν− 12 2− 2 αe 4 πi β − 2 D −ν− 12 2− 2 αe− 4 πi β − 2 πβ   Re α > 0, β > 0, Re ν > − 12

ET II 30(17)

6.637  1.

∞

+  2 − 1 1 ,  β + x2 2 exp −α β 2 + x2 2 J ν (γx) dx

+

+ , , 1 1 1  2 1  2 β α + γ 2 2 − α K 12 ν β α + γ2 2 + α = I 12 ν 2 2 [Re α > 0, Re β > 0, γ > 0, Re ν > −1] ET II 31(20)

∞

+  2 − 1 1 ,  β + x2 2 exp −α β 2 + x2 2 Y ν (γx) dx

+  νπ , 1 1  2 2 2 β α +γ K1 = − sec +α 2 2ν 2 

+ + ,  νπ ,

1 1 1  2 1  2 1 K 12 ν β α + γ 2 2 + α + sin β α + γ2 2 − α × I 12 ν π 2 2 2 [Re α > 0, Re β > 0, γ > 0, |Re ν| < 1] ET II 106(6)

∞

+  2 − 1 1 ,  x + β 2 2 exp −α x2 + β 2 2 K ν (γx) dx



 νπ  2  2 1 , 1 , 1 + 1 + 1 2 2 2 2 K 12 ν β α+ α −γ β α− α −γ = sec K 12 ν 2 2 2 2 [Re α > 0, Re β > 0, Re(γ + β) > 0, |Re ν| < 1] ET II 132(26)

0

 2. 0

 3. 0

6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers 

6.641 0

∞√

−αx

xe



2

J ± 14 x



2

2   √ α α πα dx = H∓ 14 − Y ∓ 14 4 4 4

MI 42

6.645

6.642

Bessel functions of complicated arguments, exponentials, and powers



∞

1.10

x−1 e−αx Y ν

0



∞

2. 0

6.643



∞

x

1. 0

2.

3.



∞



∞



 √   √  2 dx = 2 K ν 2 a Y ν 2 a x



√  √  2 x−1 e−αx H (1,2) α Kν α dx = H (1,2) ν ν x

μ− 12 −αx

e

709

J 2ν



[Re a > 0]

MC MI 44, EH II 91(26)

  2

√  Γ μ + ν + 12 − β2 −μ β 2α e 2β x dx = α M μ,ν β Γ(2ν + 1) α     Re μ + ν + 12 > 0 

 √  Γ μ+ν+ 1 xμ− 2 e−αx I 2ν 2β x dx = Γ(2ν + 1)

1 2

BU 14(13a), MI 42a





2



β2 β β −1 e 2α α−μ M −μ,ν α 0     Re μ + ν + 12 > 0 MI 45    

 ∞ 1 1 2  √  Γ μ + ν + 2 Γ μ − ν + 2 β2 −μ 1 β e 2α α W −μ,ν xμ− 2 e−αx K 2ν 2β x dx = 2β 0     α Re μ + ν + 12 > 0 , (cf. 6.631 3)

4.

 √  β2 1 xn+ 2 ν e−αx J ν 2β x dx = n!β ν e− α α−n−ν−1 Lνn

0



 exp − β  √  8α 1 π x− 2 e−αx Y 2ν β x dx = − α cos(νπ)





2

β α

MI 47a



[n + ν > −1]

MO 178a

2

2  β β 1 sin(νπ) I ν + Kν 8α π 8α 0   |Re ν| < 12 MI 44 1 1



 ∞ m− 2  √  1 1 1 Γ(m + 1) 1 2 6. x 2 m e−αx K m 2 x dx = e 2α W − 12 (m+1),− 12 m MI 48a 2α α α 0



 ∞  √  a2 b 1 a2 β −βx  6.644 e J 2ν 2a x J ν (bx) dx = exp − 2 Jν 2 2 + b2 2 β + b β β + b2 0   Re β > 0, b > 0, Re ν > − 12 

∞

5.

2



ET II 58(17)

6.645



∞

1. 1

 2. 1

∞

         2 − 1 1 1 x − 1 2 e−αx J ν β x2 − 1 dx = I 12 ν α2 + β 2 + α α2 + β 2 − α K 12 ν 2 2      2 − 1 ν− 1  12 ν −αx 2 ν 2 β α + β 2 2 4 K ν+ 12 x −1 e J ν β x2 − 1 dx = α2 + β 2 π

MO 179a

MO 179a

710

Bessel Functions



1

3.3



−1

1 − x2

−1/2

6.646

   2 cosh a2 + b2 − cosh a e−ax I 1 b 1 − x2 dx = b [a > 0,

6.646 

∞

1. 1

2.



x−1 x+1

12 ν

b > 0]

   ν   exp − α2 + β 2 β −αx 2   e J ν β x − 1 dx = α2 + β 2 α + α2 + β 2

[Re ν > −1] EF 89(52), MO 179    ν

1ν  ∞ exp − α2 − β 2 x − 1 2 −αx   2 β   e I ν β x − 1 dx = x+1 α2 − β 2 α + α2 − β 2 1 

∞

3.7 b



[Re ν > −1,

ν/2

α > β]

MO 180

+  1/2 ,  Γ(ν + 1)  ν −bx x e K ν a t2 − b 2 Γ(−ν, bx) − y ν ebs Γ(−ν, by) dt = ν 2sa 1/2  2 [Re(p + a) > 0, |Re(ν)| < 1] . where x = p − s, y = p + s, s = p − a2

e−pt

t−b t+b

ME 39a

6.647



∞

1.

x−λ− 2 (β + x) 1

λ− 12

e−αx K 2μ

+ , x(β + x) dx

0

=

   1 1 αβ  1 e 2 Γ 2 − λ + μ Γ 12 − λ − μ W λ,μ (z1 ) W λ,μ (z2 ) β   z1 = 12 β α + α2 − 1 ,

 |arg β| < π,  2.

∞

(α + x)− 2 x− 2 e−x cosh t K ν 1

1

, + x(α + x) dx

Re α > −1,

  z2 = 12 β α − α2 − 1  Re λ + |Re μ| < 12 ET II 377(37)





 νπ 1 1 t 1 −t 1 α cosh t 2 e αe K 12 ν αe = sec K 12 ν 2 2 4 4 [−1 < Re ν < 1] ET II 377(36)  α , +  1 1 3.11 xλ− 2 (α − x)−λ− 2 e−x sinh t I 2μ x(α − x) dx 0 1  1 



1 t 1 −t −(α/2) sinh t 2 Γ 2 + λ + μ Γ 2 − λ + μ αe M −λ,μ αe =e M λ,μ 2 2 2 α [Γ(2μ + 1)]   Re μ > |Re λ| − 12 ET II 377(32)

 ∞ ν +  12 , α + βex 2 2 α dx = 2 K ν+ (α) K ν− (β) 6.648 ex K + β + 2αβ cosh x 2ν αex + β −∞ [Re α > 0, Re β > 0] ET II 379(45) 0

6.651

6.649

Bessel and exponential functions and powers



∞

1.

K μ−ν (2z sinh x) e(ν+μ)x dx =

0

 0

∞

3.

J ν+μ (2x sinh t) e(ν−μ)t dt = K ν (x) I μ (x)  Re(ν − μ) < 32 , Y ν−μ (2x sinh t) e−(ν+μ)t dt =

0



π2 [J ν (z) Y μ (z) − J μ (z) Y ν (z)] 4 sin[(ν − μ)π] [Re z > 0, −1 < Re(ν − μ) < 1] MO 44

∞

2.



711

∞

4.

K 0 (2z sinh x) e−2νx dx = −

0

π 4

Re(ν + μ) > −1,

x>0



EH II 97(68)

1 {I μ (x) K ν (x) − cos[(ν − μ)π] I ν (x) K μ (x)} sin[π(μ − ν)]   |Re(ν − μ)| < 1, Re(ν + μ) > − 12 , x > 0 EH II 97(73)

J ν (z)

∂ Y ν (z) ∂ J ν (z) − Y ν (z) ∂ν ∂ν



6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers 6.651



∞

1.

xλ+ 2 e− 4 α 1

1

2

x2

Iμ

0

1

4α

2 2

x

3 1 = √ 2λ+1 β −λ− 2 2π



J ν (βx) dx 2 !

β !! 1 − μ, 1 + μ 21 G 23 2α2 ! h, 12 , k + π |arg α| < , 4



∞

2. 0

h= β > 0,

3. 0

∞

+ 12 λ + 12 ν,

k=

3 4

− 32 − Re(2μ + ν) < Re λ < 0

+ 12 λ − 12 ν , ET II 68(8)

  1 1 2 2 xλ+ 2 e− 4 α x K μ 14 α2 x2 J ν (βx) dx  !

π λ+1 −λ− 3 12 β 2 !! 1 − μ, 1 + μ 2 = 2 β G 23 2 2α2 ! h, 12 , k +



3 4

x2μ−ν+1 e− 4 αx I μ 1

2

1

2 4 αx



h= |arg α| <

π , 4

3 4

+ 12 λ + 12 ν,

k=

Re (λ + ν ± 2μ) > − 32

3

,4

+ 12 λ − 12 ν ET II 69(15)

J ν (βx) dx



β ν−2μ−1 1 1 1 β2 μ−ν+ 12 − 12   1F 1 =2 +μ + μ; − μ + ν; − (πα) Γ 2 2 2 2α Γ 12 − μ + ν   ET II 68(6) Re α > 0, β > 0, Re ν > 2 Re μ + 12 > − 12

712

Bessel Functions



∞

4.

x2μ+ν+1 e− 4 α 1

2

x2



∞

2

x



J ν (βx) dx

2 2 αx



β2 3 ;− 2 2 2α



ET II 69(13)

K ν (βx) dx

2

2

  β β 2 −μ− 32 − 12 μ− 12 ν− 14 1 √ β α Γ(2μ + ν + 1) Γ μ + 2 exp = W k,m 8α 4α π 2k = −3μ − ν − 12 , 2m = μ + ν + 12  Re α > 0, Re μ > − 12 , Re (2μ + ν) > −1 ET II 146(53)

0



2 2

1

x2μ+ν+1 e− 2 αx I μ 1

4α

√ μ −2μ−2ν−2 ν Γ (1 + 2μ + ν)  1 F 1 1 + 2μ + ν; μ + ν +  = π2 α β Γ μ + ν + 32   |arg α| < 14 π, Re ν > −1, Re(2μ + ν) > −1, β > 0

0

5.

1

Kμ

6.652

μ− 12

∞

6.

− 14 αx2

xe 0

J 12 ν

1

2

4 βx







 1  2 βγ 2 αγ 2 2 −2 J ν (γx) dx = 2 α + β exp − 2 J 12 ν α + β2 α2 + β 2 [γ > 0, Re α > |Im β|, Re ν > −1] ET II 56(2)



∞

7. 0



∞

8.

xe− 4 αx I 12 ν 1

2

x1−ν e− 4 α 1

2

x2

1

 2 J ν (βx) dx = 4 αx

Iν

0



∞

9.

x−ν−1 e− 4 α 1

2

x2

1



1 πα 2

 2 2 J ν (βx) dx = 4α x

I ν+1

1

0

4α

2 2

x







− 12



β2 β −1 exp − 2α [Re α > 0,

β > 0,

Re ν > −1] ET II 67(3)

2 β ν−1 π α 

J ν (βx) dx =





β β2 exp − 2 D −2ν α  4α |arg α| < 14 π, β > 0,

Re ν > − 12



ET II 67(1)





β 2 ν β2 β exp − 2 D −2ν−3 π 4α α  |arg α| < 14 π, Re ν > −1,

β>0



ET II 67(2)



∞

6.652



2ν −

x e 0

6.653



∞

1. 0

x2 8

+αx

Iν

x2 8



2

α   Γ(4ν + 1) e 2 W 3 1 α2 dx = 4ν 2 Γ(ν + 1) αν+1 − 2 ν, 2 ν   Re ν + 14 > 0





 ab dx 1  2 1 2 a +b = 2 I ν (a) K ν (b) exp − x − Iν 2 2x x x = 2 K ν (a) I ν (b) [Re ν > −1]

 2. 0

∞

MI 45

[0 < a < b] [0 < b < a] WA 482(2)a, EH II 53(37), WA 482(3)a

   zw dx  1  2 1 z + w2 K ν = 2 K ν (z) K ν (w) exp − x − 2 2x x x [|arg z| < π, |arg w| < π, arg(z + w)] < 14 π

WA 483(1), EH II 53(36)

6.662

Bessel, hyperbolic, and exponential functions

6.655



√  √  1 β2 x e Kν ME 39 dx = 4πα− 2 K 2ν β α 8x 0



∞  − 1 α2 x α2 β √ x β 2 + x2 2 exp − 2 Jν J ν (γx) dx = γ −1 e−βγ J 2ν (2α γ) 2 2 2 β +x β +x 0   Re β > 0, γ > 0, Re ν > − 12

 6.654

713

∞

2

− 12 − β 8x −αx

ET II 58(14)

6.656



∞

1.

+ , 1 e−(ξ−z) cosh t J 2ν 2(zξ) 2 sinh t dt = I ν (z) K ν (ξ)

0





Re ν > − 12 ,

Re(ξ − z) > 0

 EH II 98(78)

∞

2.

+

,

e−(ξ+z) cosh t K 2ν 2(zξ) 2 sinh t dt = 1

0

1 K ν (z) K ν (ξ) sec(νπ) 2  |Re ν| < 12 ,

 2  1 1 Re z 2 + ξ 2 ≥ 0 EH II 98(79)

6.66 Combinations of Bessel, hyperbolic, and exponential functions Bessel and hyperbolic functions 6.661



∞

sinh(ax) K ν (bx) dx =

1. 0



∞

2. 0

π cosec 2

 νπ  2

   sin ν arcsin ab √ b 2 − a2 [Re b > |Re a|,

   π cos ν arcsin ab  νπ cosh(ax) K ν (bx) dx =  2 b2 − a2 cos 2

|Re ν| < 2] ET II 133(32)

[Re b > |Re a|,

|Re ν| < 1] ET II 134(33)

6.662

Notation: ,  1 + 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 

1.

10 0

∞

K(k) cosh(βx) K 0 (αx) J 0 (γx) dx = √ u+v

2 =

,  1 + (b + c)2 + a2 + (b − c)2 + a2 2



< 2 2 2 2 2 2 (α + β + γ ) − 4α β + α2 − β 2 − γ 2 

< 1 2 2 2 2 2 2 (α + β + γ ) − 4α β − α2 + β 2 + γ 2 v= 2

1 u= 2

k 2 = v(u + v)−1

[Re α > |Re β|,

γ > 0]

ET II 15(23)

714

Bessel Functions

6.663

alternatively, with a = γ, b = β, c = α,  ∞ K(k) cosh(bx) K 0 (cx) J 0 (ax) dx =  2 2 − 21 0 k2 =  2.

∞

10 0

22 − c2 , 22 − 21

[Re c > |Re b|,

a > 0]

  K(k) sn u dn u sinh(βx) K 1 (αx) J 0 (γx) dx = a u E(k) − K(k) E(u) + cn u

+ −1 ,1  2  2 2 2 2 2 2 2 2 2 2 2 α +β +γ cn u = 2γ − 4α β −α +β +γ −1

1 k = 2 2

,− 12   + 2   2 2 2 2 2 2 2 2 α +β +γ − 4α β 1− α −β −γ [Re α > |Re β|,

γ > 0]

ET II 15(24)

alternatively, with a = γ, b = β, c = α,    ∞ K(k) sn u dn u −1 sinh(bx) K 1 (cx) J 0 (ax) dx = c u E(k) − K(k) E(u) + cn u 0 2 2 2 − c a  , k2 = 22 [Re c > |Re b|, cn 2 u = 2 2 − c 2 2 − 21 6.663



∞

1.

K ν±μ (2z cosh t) cosh [(μ ∓ ν) t] dt =

0

 2.

Y μ+ν (2z cosh t) cosh[(μ − ν)t] dt =

0



0

∞

5. 0



1.

π [J μ (z) Y ν (z) + J ν (z) Y μ (z)] 4 EH II 97(65)

1 J μ+ν (2z sinh t) cosh[(μ − ν)t] dt = [I ν (z) K μ (z) + I μ (z) K ν (z)] 2  Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,

z>0

1 J μ+ν (2z sinh t) sinh[(μ − ν)t] dt = [I ν (z) K μ (z) − I μ (z) K ν (z)] 2  Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,

z>0

∞

J 0 (2z sinh t) sinh(2νt) dt = 0

EH II 96(64)

[z > 0] ∞

4.

6.664

π [J μ (z) J ν (z) − Y μ (z) Y ν (z)] 4

J μ+ν (2z cosh t) cosh[(μ − ν)t] dt = −

0



WA 484(1), EH II 54(39)

[z > 0] ∞

3. 

1 K μ (z) K ν (z) 2 [Re z > 0]

∞

a > 0]

sin(νπ) 2 [K ν (z)] π



|Re ν| < 34 ,

z>0







EH II 97(71)

EH II 97(72)

EH II 97(69)

6.668

Bessel, hyperbolic, and exponential functions



∞

715

  cos(νπ) 2 [K ν (z)] |Re ν| < 34 , z > 0 EH II 97(70) π 0    ∞ 1 ∂ K ν (z) ∂ I ν (z) 1 2 − K ν (z) Y 0 (2z sinh t) sinh(2νt) dt = I ν (z) − cos(νπ) [K ν (z)] π ∂ν ∂ν π 0   |Re ν| < 34 , z > 0 EH II 97(75)  ∞ 2   π Jν2 (z) + Nν2 (z) K 0 (2z sinh t) cosh 2νt dt = [Re z > 0] MO 44 8 0



 ∞ 1 1 1 Γ +μ−ν Γ − μ − ν W ν,μ (iz) W ν,μ (−iz) K 2μ (z sinh 2t) coth2ν t dt = 4z 2 2+ 0 , π |arg z| ≤ , |Re μ| + Re ν < 12 2

2. 3.

4. 5.

Y 0 (2z sinh t) cosh(2νt) dt = −

MO 119



∞

6.

cosh(2μx) K 2ν (2a cosh x) dx = 0



1 K μ+ν (a) K μ−ν (a) 2

∞

6.665

sech x cosh(2λx) I 2μ (a sech x) dx = 0

[Re a > 0]     Γ 12 + λ + μ Γ 12 − λ + μ

ET II 378(42)

M λ,μ (a) M −λ,μ (a) 2 2a [Γ(2μ  + 1)]  |Re λ| − Re μ < 12 ET II 378(43)

Bessel, hyperbolic, and algebraic functions  ∞ ∞ 2 xν+1 sinh(αx) cosech(πx) J ν (βx) dx = (−1)n−1 nν+1 sin(nα) K ν (nβ) 6.666 π n=1 0 [|Re α| < π, Re ν > −1] ET II 41(3), WA 469(12)

6.667 1.

3

2.

 √



a2 − x2 sinh t I 2ν (x) 1 t 1 −t π √ ae I ν ae dx = I ν 2 2 a2 − x2 0 2  Re ν > − 12 ET II 365(10) √   a          cosh a2 − x2 sinh t K 2ν (x) π2 √ cosec(νπ) I −ν aet I −ν ae−t − I ν aet I ν ae−t dx = 2 2 4 a −x 0   |Re ν| < 12 ET II 367(25) 

a

cosh

Exponential, hyperbolic, and Bessel functions 6.668

Notation: ,  1 + 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2

2 =

,  1 + (b + c)2 + a2 + (b − c)2 + a2 2

716

Bessel Functions

 1.

10

2.10

6.669

∞



∞

0



∞

2. 0

4.

1

−1

e−αx sinh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 − r1 ) 2 (r2 + r1 ) 2 0   r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α,  ∞ 1 e−cx sinh(bx) J 0 (ax) dx = 2 2 − 21 0 [Re c > |Re b|, a > 0]  ∞ 1 1 −1 e−αx cosh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 − r1 ) 2 (r2 + r1 ) 2 0   r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α,  ∞ 2 e−cx cosh(bx) J 0 (ax) dx = 2 2 − 21 0 [Re c > |Re b|, a > 0]

1.

3.

6.669

1

ET II 12(52)

ET II 12(54)

   2λ , + 1 Γ 12 − λ + μ 1 x M −λ,μ α2 + β 2 2 − β e−β cosh x J 2μ (α sinh x) dx = coth 2 α Γ(2μ + 1) , + 1 × W λ,μ α2 + β 2 2 + β   Re β > |Re α|, Re(μ − λ) > − 12 BU 86(5b)a, ET II 363(34)  2λ 1 x e−β cosh x Y 2μ (α sinh x) dx coth 2   sec[(μ + λ)π] W λ,μ α2 + β 2 + β W −λ,μ α2 + β 2 − β =− α     tan[(μ + λ)π] Γ 12 − λ + μ W λ,μ α2 + β 2 + β M −λ,μ α2 + β 2 − β − α Γ(2μ + 1)   Re β > |Re α|, Re λ < 12 − |Re μ| ET II 363(35)

 2ν √ 1 1 x e− 2 (a1 a2 )t cosh x coth K 2μ (t a1 a2 sinh x) dx 2 0     Γ 12 + μ − ν Γ 12 − μ − ν W ν,μ (a1 t) W ν,μ (a2 t) = √ 2t a1 a2   + √ √ 2, 1 ± 2μ , Re t ( a1 + a2 ) > 0 BU 85(4a) Re ν < Re 2    ∞ +  x ,2ν Γ 12 + μ − ν √ − 12 (a1 a2 )t cosh x coth W ν,μ (a1 t) M ν,μ (a2 t) e I 2μ (t a1 a2 sinh x) dx = √ 2 0   t a1 a2 Γ(1 + 2μ)  1 BU 86(5c) Re 2 + μ − ν > 0, Re μ > 0, a1 > a2 

∞



∞

5.

e −∞

2νs− x−y 2 tanh s

    √

Γ 12 + μ + ν Γ 12 + μ − ν xy ds = I 2μ M ν,μ (x) M −ν,μ (y) √ 2 cosh s cosh s xy [Γ(1 + 2μ)]     Re ±ν + 12 + μ > 0 BU 83(3a)a

6.671

Bessel and trigonometric functions



∞

6.

e −∞

2νs− x+y 2 tanh s

717

    √

Γ 12 + μ + ν Γ 12 + μ − ν xy ds = J 2μ M ν,μ (x) M ν,μ (y) √ 2 cosh s cosh s xy [Γ(1 + 2μ)]     Re ∓ν + 12 + μ > 0 BU 84(3b)a

6.67–6.68 Combinations of Bessel and trigonometric functions 6.671 

∞

1. 0



 β sin ν arcsin α  J ν (αx) sin βx dx = α2 − β 2 = ∞ or 0

[β = α]

αν cos νπ 2  ν =  2 2 β − α β + β 2 − α2

[β > α]



∞

2. 0

[β < α]

β cos ν arcsin α  J ν (αx) cos βx dx = α2 − β 2

[Re ν > −2]



[β < α]

= ∞ or 0

[β = α]

−α sin  ν =  β 2 − α2 β + β 2 − α2 ν



0

[β > α] [Re ν > −1]

Y ν (ax) sin(bx) dx    νπ  − 1 b a2 − b2 2 sin ν arcsin = cot 2 a  νπ  − 1 1 b 2 − a2 2 = cosec 2 2  + +  1 ,ν  1 ,−ν  2  2 −ν 2 2 ν 2 2 × a cos(νπ) b − b − a −a b− b −a

0

4.

νπ 2

WA 444(5)

∞

3.



WA 444(4)

[0 < b < a, |Re ν| < 2]

[0 < a < b,

|Re ν| < 2] ET I 103(33)

∞

Y ν (ax) cos(bx) dx     tan νπ b 2 = 1 cos ν arcsin a (a2 − b2 ) 2

+  νπ  1    1 ,ν − b2 − a2 2 a−ν b − b2 − a2 2 + cot(νπ) = − sin 2  +  1 ,−ν  ν +a b − b2 − a2 2 cosec(νπ)

[0 < b < a,

|Re ν| < 1]

[0 < a < b,

|Re ν| < 1] ET I 47(29)

718

Bessel Functions



6.671

∞

5.

K ν (ax) sin(bx) dx

 νπ  ,ν + ,ν   1 + 2 1 1 1 −ν 2 2 −2 2 2 2 2 2 b +a a +b = πa cosec +b − b +a −b 4 2 [Re a > 0, b > 0, |Re ν| < 2, ν = 0] ET I 105(48)

0



∞

6.

K ν (ax) cos(bx) dx

 + +  νπ  2  2  1  1 ,ν  1 ,−ν π 2 2 −2 −ν 2 2 ν 2 2 b+ b +a b +a = sec +a b+ b +a a 4 2 [Re a > 0, b > 0, |Re ν| < 1] ET I 49(40)

0



∞

7.

J 0 (ax) sin(bx) dx = 0 0



1 = √ 2 b − a2

0

∞

9. 0

1 J 0 (ax) cos(bx) dx = √ 2 a − b2

∞

10. 0

[a = b]

=0

[0 < a < b]



b 1 J 2n+1 (ax) sin(bx) dx = (−1)n √ T 2n+1 a a2 − b 2

∞

11.



b 1 J 2n (ax) cos(bx) dx = (−1)n √ T 2n 2 2 a a −b

 12.

ET I 99(2)

[0 < b < a]

ET I 43(2)

b

2 arcsin a √ π a2 − b 2 % &  1 b2 b 2 − ln −1 = √ π b 2 − a2 a a2

[0 < b < a] [0 < a < b] ET I 103(31)

∞

Y 0 (ax) cos(bx) dx = 0 0

[0 < b < a]

[0 < a < b]

Y 0 (ax) sin(bx) dx = 0

ET I 43(1)

[0 < a < b]

=0 

[0 < b < a]

=∞

=0 

[0 < a < b] ET I 99(1)

∞

8.



[0 < b < a]

1 = −√ 2 b − a2

[0 < b < a] [0 < a < b] ET I 47(28)

6.672

Bessel and trigonometric functions



∞

13. 0

 0



K 0 (βx) sin αx dx =  ln α2 + β 2

. α2 +1 β2

α + β



[α > 0, ∞

14.8 6.672



1

719

π K 0 (βx) cos αx dx =  2 α2 + β 2

β > 0]

WA 425(11)a, MO 48

[α > 0]

WA 425(10)a, MO 48

∞

J ν (ax) J ν (bx) sin(cx) dx

1. 0

=0





b +a −c 1 = √ P ν− 12 2 ab 2ab2

b + a2 − c 2 cos(νπ) Q ν− 12 − =− √ 2ab π ab 

2

2

∞

2.

J ν (x) J −ν (x) cos(bx) dx = 0

2

1 P 1 2 ν− 2



1 2 b −1 2

[Re ν > −1,

0 < c < b − a,

[Re ν > −1,

b − a < c < b + a,

[Re ν > −1,

b + a < c,

∞

0

∞

4. 0



   π2 K ν (ax) K ν (bx) cos(cx) dx = √ sec(νπ) P ν− 12 a2 + b2 + c2 (2ab)−1 4 ab  Re(a + b) > 0, c > 0, 1 K ν (ax) I ν (bx) cos(cx) dx = √ Q ν− 12 2 ab

2

0



|Re ν| <

1 2



ET I 50(51)

2

a2 + b 2 + c 2ab  Re a > |Re b|,

c > 0,

Re ν > − 12



  1 P ν− 12 1 − 2a2 2   1 = cos(νπ) Q ν− 12 2a2 − 1 π

Re ν > −1]

[0 < a < 1, [a > 1,

Re ν > −1] ET II 343(30)

∞

6.

2

  1 Q ν− 12 1 − 2a2 π   1 = − sin(νπ) Q ν− 12 2a2 − 1 π

cos(2ax) [J ν (x)] dx = 0

7.

[0 < b < 2] [2 < b]

sin(2ax) [J ν (x)] dx =



ET I 102(27)

ET I 49(47) ∞

5.



0 < a < b]

ET I 46(21)

3.



0 < a < b]



=0 

0 < a < b]

 

Re ν > − 12  Re ν > − 12

0 < a < 1, a > 1,



ET II 344(32) ∞

sin(2ax) J 0 (x) Y 0 (x) dx = 0 0

=−

+ 1 , K 1 − a−2 2 πa

[0 < a < 1] [a > 1] ET II 348(60)

720

Bessel Functions



∞

8. 0





1

K 0 (ax) I 0 (bx) cos(cx) dx =  K c2 + (a + b)2





√ 2 ab c2 + (a + b)2

[Re a > |Re b|, ∞

9.

1 K(a) π

1 1 =− K πa a

cos(2ax) J 0 (x) Y 0 (x) dx = −

0



6.673

c > 0]

ET I 49(46)

[0 < a < 1] [a > 1] ET II 348(61)

∞

10.

2

1 K 1 − a2 π   2 1 = K 1− 2 πa a

cos(2ax) [Y 0 (x)] dx = 0





[0 < a < 1] [a > 1] ET II 348(62)

6.673  1.

∞

0

+  νπ  νπ , J ν (ax) cos − Y ν (ax) sin sin(bx) dx 2 2 =0 =



2aν

1 √ b 2 − a2

+

 b + b 2 − a2

 1 ,ν

+

 + b − b 2 − a2

2

  1 ,ν 2

[0 < b < a,

|Re ν| < 2]

[0 < a < b,

|Re ν| < 2] ET I 104(39)

∞

2. 0

+  νπ  νπ , Y ν (ax) cos + J ν (ax) sin cos(bx) dx 2 2 =0 =−

2aν

1 √ b 2 − a2

+

 b + b2 − a

+

 ,ν 2

 + b − b2 − a

1 2

  ,ν 2 1 2

[0 < b < a,

|Re ν| < 1]

[0 < a < b,

|Re ν| < 1] ET I 48(32)

3.∗



ea − 1 a

π/2

[cos x I 0 (a cos x) + I 1 (a cos x)] dx = 0

6.674



a

sin(a − x) J ν (x) dx = a J ν+1 (a) − 2ν

1. 0



(−1)n J ν+2n+2 (a)

n=0

[Re ν > −1] a

cos(a − x) J ν (x) dx = a J ν (a) − 2ν

2. 0



∞ 

∞ 

(−1)n J ν+2n+1 (a)

n=0

a

%

[Re ν > −1]

sin(a − x) J 2n (x) dx = a J 2n+1 (a) + (−1)n 2n cos a − J 0 (a) − 2

3. 0

ET II 334(12)

ET II 336(23) n 

&

(−1)m J 2m (a)

m=1

[n = 0, 1, 2, . . .]

ET II 334(10)

6.676

Bessel and trigonometric functions



%

a

cos(a − x) J 2n (x) dx = a J 2n (a) − (−1)n 2n sin a − 2

4. 0

5.

n−1 

721

& (−1)m J 2m+1 (a)

m=0

ET II 335(21) [n = 0, 1, 2, . . .] % &  a n  sin(a − x) J 2n+1 (x) dx = a J 2n+2 (a) + (−1)n (2n + 1) sin a − 2 (−1)m J 2m+1 (a) 0



m=0

[n = 0, 1, 2, . . .] %

a

ET II 334(11)

cos(a − x) J 2n+1 (x) dx = a J 2n+1 (a) + (−1) (2n + 1) cos a − J 0 (a) − 2 n

6. 0



n 

&

m

(−1) J 2m (a)

m=1

[n = 0, 1, 2, . . .]

ET II 336(22)

z

7.

sin(z − x) J 0 (x) dx = z J 1 (z)

WA 415(2)

cos(z − x) J 0 (x) dx = z J 0 (z)

WA 415(1)

0

 8.

z

0

6.675



∞

1. 0



2 2

2 

2

√   √  a a a νπ νπ a a π − − J ν a x sin(bx) dx = cos J 12 ν− 12 − sin J 12 ν+ 12 3 8b 4 8b 8b 4 8b 2 4b [a > 0, b > 0, Re ν > −4] ET I 110(23)

∞

2.

 √  J ν a x cos(bx) dx

0

=− 

∞

3. 0



∞

4. 0

6.676



∞

1. 0



∞

2. 0

 3. 0

∞

2

2 

2

√  2 a a a a π νπ νπ a 1 1 1 1 − − sin J + cos J 3 ν− 2 ν+ 2 2 2 8b 4 8b 8b 4 8b 4b 2 [a > 0, b > 0, Re ν > −2] ET I 53(22)a

 √  1 J 0 a x sin(bx) dx = cos b  √  1 J 0 a x cos(bx) dx = sin b



a2 4b a2 4b

[a > 0,

b > 0]

ET I 110(22)

[a > 0,

b > 0]

ET I 53(21)



 √   √  1 J ν a x J ν b x sin(cx) dx = J ν c

 √   √  1 J ν a x J ν b x cos(cx) dx = J ν c





 √   √  1 K0 J 0 a x K 0 a x sin(bx) dx = 2b

ab 2c

ab 2c



cos



sin

2

a 2b

a2 + b 2 νπ − 4c 2 [a > 0, b > 0,

c > 0,

ET I 111(29)a



a2 + b 2 νπ − 4c 2 [a > 0, b > 0,

Re ν > −2]

c > 0,

Re ν > −1] ET I 54(27)

[Re a > 0,

b > 0]

ET I 111(31)

722

Bessel Functions



∞

4.

J0 0

6.

 a , √  √  π + a I0 − L0 ax K 0 ax cos(bx) dx = 4b 2b 2b

[Re a > 0, b > 0]   √ √  √   a 1 Re a > 0, b > 0 K0 ax Y 0 ax cos(bx) dx = − K 0 2b 2b 0  ∞ √  a  a , √ 1 1 π2 + H0 −Y0 K0 axe 4 πi K 0 axe− 4 πi cos(bx) dx = 8b 2b 2b 0 

5.

6.677

ET I 54(29)

∞

ET I 54(30)

[Re a > 0, b > 0] 6.677



∞

1.

  J 0 b x2 − a2 sin(cx) dx = 0

 √  cos a c2 − b2 √ = c2 − b 2

a

√  ∞   exp −a b2 − c2 2 2 √ J 0 b x − a cos(cx) dx = 2 2 a b √− c  − sin a c2 − b2 √ = c2 − b 2



∞

3.6

  cos z J 0 α x2 + z 2 cos βx dx = 

0



0

5. 0

ET I 113(47)

[0 < c < b] [0 < b < c]

α2 − β 2

α2 − β 2

[0 < β < α,

z > 0]

[0 < α < β,

z > 0] MO 47a

∞

4.



[0 < b < c]

ET I 57(48)a

=0 

[0 < c < b]



 2.

ET I 54(31)

    1 Y 0 α x2 + z 2 cos βx dx =  sin z α2 − β 2 α2 − β 2   1 = − exp −z β 2 − α2 β 2 − α2

[0 < β < α,

z > 0]

[0 < α < β,

z > 0] MO 47a

∞

+  ,   π K 0 α x2 + β 2 cos(γx) dx =  exp −β α2 + γ 2 2 α2 + γ 2 [Re α > 0, Re β > 0,

  √ a   sin a b2 + c2 √ J 0 b a2 − x2 cos(cx) dx = b 2 + c2 0   √  ∞   cosh a b2 − c2 √ J 0 b x2 − a2 cos(cx) dx = b 2 − c2 0

γ > 0] ET I 56(43)

 6. 7.

=0

[b > 0]

MO 48a, ET I 57(47)

[0 < c < b,

a > 0]

[0 < b < c,

a > 0] ET I 57(49)

6.681

Bessel and trigonometric functions



∞

8. 0



∞

9. 0



∞

6.678 0

6.679



    i exp −iβ α2 + γ 2 (2)  H 0 α β 2 − x2 cos(γx) dx = α2 + γ 2 +  −π < arg β 2 − x2 ≤ 0,

α > 0,

γ>0



+  √  π  √ , 1 π sin K 0 2 x + Y 0 2 x sin(bx) dx = 2 2b b

,

,

ET I 59(64) ∞

[a > 0, ∞

ET I 115(58)

[a > 0, +  x , 2 2 cos(bx) dx = − cosh(πb) [K ib (a)] Y 0 2a sinh 2 π

b > 0]

ET I 59(62)

[a > 0,   π 2 2 cos(bx) dx = [J ib (a)] + [Y ib (a)] 4

b > 0]

ET I 59(65)

J 0 2a sinh ∞

6. 0



∞

7.

 x ,

+ K 0 2a sinh

0

2

cos(bx) dx = [I ib (a) + I −ib (a)] K ib (a)

 x , 2

2

[Re a > 0, 

π 2

cos(2μx) J 2ν (2a cos x) dx =

1. 0

 2.

π 2

cos(2μx) Y 2ν (2a cos x) dx = 0

ET I 59(63)

b > 0]

+

0

6.681

ET I 111(34)

ET I 115(59)

5. 

ET I 58(58)

[b > 0]

+  x , π cos(bx) dx = − [J ν+ib (a) Y ν−ib (a) + J ν−ib (a) Y ν+ib (a)] J 2ν 2a cosh 2 2 0  ∞ +  x , 2 2 sin(bx) dx = sinh(πb) [K ib (a)] J 0 2a sinh 2 π 0 

ET I 59(59)

+  x , cos(bx) dx = I ν−ib (a) K ν+ib (a) + I ν+ib (a) K ν−ib (a) J 2ν 2a sinh 2   a > 0, b > 0, Re ν > − 12

0

4.

γ>0

∞

2.

3.

α > 0,

+  x , sin(bx) dx = −i [I ν−ib (a) K ν+ib (a) − I ν+ib (a) K ν−ib (a)] J 2ν 2b sinh 2 [a > 0, b > 0, Re ν > −1]

0



    exp iβ α2 + γ 2 (1)  H 0 α β 2 − x2 cos(γx) dx = −i α2 + γ 2 +  π > arg β 2 − x2 ≥ 0,

∞

1.



723

π J ν+μ (a) J ν−μ (a) 2



Re ν > − 12

b > 0]



ET I 59(66)

ET II 361(23)

π [cot(2νπ) J ν+μ (a) J ν−μ (a) − cosec(2νπ) J μ−ν (a) J −μ−ν (a)] 2   |Re ν| < 12 ET II 361(24)

724

Bessel Functions



π 2

3.

cos(2μx) I 2ν (2a cos x) dx = 0



π 2

4.

cos(νx) K ν (2a cos x) dx = 0



π I ν−μ (a) I ν+μ (a) 2

π I 0 (a) K ν (a) 2

6.682



Re ν > − 12



ET I 59(61)

[Re ν < 1]

WA 484(3)

π

J 0 (2z cos x) cos 2nx dx = (−1)n πJn2 (z).

5.

MO 45

0



π

J 0 (2z sin x) cos 2nx dx = πJn2 (z).

6.

WA 43(3), MO 45

0



π 2

7.

cos(2nπ) Y 0 (2a sin x) dx = 0



π J n (a) Y n (a) 2

[n = 0, 1, 2, . . .]

ET II 360(16)

π

8.

sin(2μx) J 2ν (2a sin x) dx = π sin(μπ) J ν−μ (a) J ν+μ (a) 0



[Re ν > −1]

9.

cos(2μx) J 2ν (2a sin x) dx = π cos(μπ) J ν−μ (a) J ν+μ (a)   Re ν > − 12

0



π 2

10. π 2

11.

ET II 360(14)

J ν+μ (2z cos x) cos[(ν − μ)x] dx =

π J ν (z) J μ (z) 2

[Re(ν + μ) > −1]

cos[(μ − ν)x] I μ+ν (2a cos x) dx =

π I μ (a) I ν (a) 2

[Re(μ + ν) > −1]

0



ET II 360(13)

π

0

MO 42

WA 484(2), ET II 378(39)



π 2

12.

cos[(μ − ν)x] K μ+ν (2a cos x) dx =

0

π2 cosec[(μ + ν)π] [I −μ (a) I −ν (a) − I μ (a) I ν (a)] 4 [|Re(μ + ν)| < 1]

 13.8

π 2

K ν−m (2a cos x) cos[(m + ν)x] dx = (−1)m

0

π I m (a) K ν (a) 2 [|Re(ν − m)| < 1]

6.682 1.

7



ET II 378(40)

WA 485(4)



π 2

π J ν (x) 2x 0 [ν may be zero, a natural number, one half, or a natural number plus one half; x > 0] J ν− 12 (x sin t) sin

 2.

π 2

ν

ν+ 12

J ν (z sin x) sin x cos 0

t dt =

2ν

x dx = 2

ν−1



z √ 1 πΓ ν + z −ν Jν2 2 2   Re ν > − 12

MO 42a

MO 42a

6.683

Bessel and trigonometric functions

6.683  1. 0



π 2

2. 0

 0

[Re ν > Re μ > −1]  z1ν z2μ J ν+μ+1 z12 + z22 < J ν (z1 sin x) J μ (z2 cos x) sinν+1 x cosμ+1 x dx = ν+μ+1 (z12 + z22 )

π 2

4.

J μ (z sin θ) (sin θ)

1−μ

2ν+1

(cos θ)

dθ =

0

Re μ > −1]

WA 410(1)

k=0

Re μ > −1]

(see also 6.513 6)

J μ (z sin θ) (sin θ)

dθ =

μ+1

(cos θ)

0 π 2

6.

J μ (a sin θ) (sin θ)

WA 407(2)

Hμ− 12 (z)  2z π

1−μ

2+1

WA 407(3)

dθ = 2 Γ( + 1)a−−1 J +μ+1 (a)

0

[Re  > −1,

Re μ > −1] WA 406(1),



π 2

7.

ν

J ν (2z sin θ) (sin θ) (cos θ) 0

2ν

WA 414(1)

s μ+ν,ν−μ+1 (z) 2μ−1 z ν+1 Γ(μ) [Re ν > −1]

π 2

5. 

WA 407(4)

∞     1 J ν z cos2 x J μ z sin2 x sin x cos x dx = (−1)k J ν+μ+2k+1 (z) z

[Re ν > −1,



Γ

[Re ν > −1, π 2

3.





μ−ν 2 2

J μ (z) J ν (z sin x) I μ (z cos x) tanν+1 x dx = μ+ν +1 Γ 2  z ν

π 2

725

EH II 46(5)

dθ

    ∞ 1  (−1)m z ν+2m Γ ν + m + 12 Γ ν + 12 = 2 m=0 m! Γ(ν + m + 1) Γ(2ν + m + 1)   √ 1 2 = z −ν π Γ ν + 12 [J ν (z)] 2

  Re ν > − 12 EH II 47(10)



π 2

8.

ν+1

J ν (z sin θ) (sin θ) 0

 9. 0

π 2

−2ν

(cos θ)

z ν−1 dθ = 2−ν √ Γ π

    2ν+1 2ν+1 J ν z sin2 θ J ν z cos2 θ (sin θ) (cos θ)





1 − ν sin z 2   −1 < Re ν < 12   Γ 12 + ν J 2ν+ 12 (z) dθ = 2ν+ 3 √ 2 Γ(ν + 1) 2 z   1 Re ν > − 2

EH II 68(39)

WA 409(1)

726

Bessel Functions





π 2

2





2



2μ+1

J μ z sin θ J ν z cos θ sin

10. 0

6.684 

π

1.8

(sin x)

2ν

0



π

2.

(sin x)

2ν

0

θ cos

2ν+1

6.684

    Γ μ + 12 Γ ν + 12 J μ+ν+ 12 (z) √ θ dθ = √ 2 π Γ(μ + ν + 1) 2z   WA 417(1) Re μ > − 12 , Re ν > − 12



Jν α2 + β 2 − 2αβ cos x √ 1 J ν (α) J ν (β)  ν dx = 2ν π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x   Re ν > − 12 

Yν α2 + β 2 − 2αβ cos x √ 1 J ν (α) Y ν (β) ν  ν dx = 2 π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x 



π 2

6.685

sec x cos(2λx) K 2μ (a sec x) dx = 0

6.686



∞



2

sin ax

1. 0



|α| < |β|,

π W λ,μ (a) W −λ,μ (a) 2a

Re ν > − 12

[Re a > 0]

ET II 362(27)



ET II 362(28) ET II 378(41)

2 2

√ b b ν+1 π √ 1 − π J 2ν J ν (bx) dx = − sin 8a 4 8a 2 a

[a > 0, b > 0, Re ν > −3] ET II 34(13)



√ ∞   b2 b2 ν+1 π − π J 12 ν cos ax2 J ν (bx) dx = √ cos 8a 4 8a 2 a 0 [a > 0, b > 0, Re ν > −1]

 2.



ET II 38(38) ∞

3.

√  νπ π = − √ sec 2 4 a



2

2   2 b b b2 3ν + 1 b ν−1 − π J 12 ν + π Y 12 ν × cos − sin 8a 4 8a 8a 4 8a [a > 0, b > 0, −3 < Re ν < 3] ET II 107(7)

0



∞

4.

 0

  cos ax2 Y ν (bx) dx

√  νπ π = √ sec 4 a 2



2

2  2 b b b2 3ν + 1 b ν−1 − π J 12 ν + π Y 12 ν × sin + cos 8a 4 8a 8a 4 8a [a > 0, b > 0, −1 < Re ν < 1] ET II 107(8)

0

5.

  sin ax2 Y ν (bx) dx

∞

  b2 1 sin ax2 J 1 (bx) dx = sin b 4a

[a > 0,

b > 0]

ET II 19(16)

6.693

Bessel and trigonometric functions and powers



b2 6. cos ax 8a 0 2

 ∞   b 1 cos 7. sin2 ax2 J 1 (bx) dx = 2b 8a 0 2

 ∞  π  π x 6.687 cos K 2ν xei 4 K 2ν xe−i 4 dx 2a 0 

∞



2



2 J 1 (bx) dx = sin2 b

= 6.688



π 2

1. 0



π J ν (μz sin t) cos (μx cos t) dt = J ν2 2

Γ

1 4

[a > 0,

b > 0]

ET II 20(20)

[a > 0,

b > 0]

ET II 19(17)

  √  π + ν Γ 14 − ν π √ W 14 ,ν aei 2 W 8 a  a > 0, |Re ν| < 14

 √

x2 + z 2 + x μ 2

 √

 J ν2

x2 + z 2 − x μ 2

 1 4 ,ν

ae−i 2

π



ET II 372(1)



[Re ν > −1,

π 2

2. 0



727

π 2

3. 0

MO 46 Re z > 0] + ,   − 1 ν− 1 1 1√ ν+1 (sin x) cos (β cos x) J ν (α sin x) dx = 2− 2 παν α2 + β 2 2 4 J ν+ 12 α2 + β 2 2

[Re ν > −1] +  , π cos [(z − ζ) cos θ] J 2ν 2 zζ sin θ dθ = J ν (z) J ν (ζ) 2   Re ν > − 12

ET II 361(19)

EH II 47(8)

6.69–6.74 Combinations of Bessel and trigonometric functions and powers 

∞

6.691

x sin(bx) K 0 (ax) dx = 0

6.692



∞

1. 0



∞

0



1. 0

[Re a > 0,

b > 0]

ET I 105(47)

   − 1 3 1 x K ν (ax) I ν (bx) sin(cx) dx = − (ab)− 2 c u2 − 1 2 Q 1ν− 1 (u), u = (2ab)−1 a2 + b2 + c2 2 2   Re a > |Re b|, c > 0, Re ν > − 32 ET I 106(54)

2.

6.693

− 3 πb  2 a + b2 2 2

∞

 − 1     3 π x K ν (ax) K ν (bx) sin(cx) dx = (ab)− 2 c u2 − 1 2 Γ 32 + ν Γ 32 − ν P −1 (u) ν− 12 4     Re(a + b) > 0, c > 0, |Re ν| < 32 u = (2ab)−1 a2 + b2 + c2 ET I 107(61)



β dx 1 = sin ν arcsin J ν (αx) sin βx x ν α αν sin νπ 2 ν =   2 ν β + β − α2

[β ≤ α] [β ≥ α] [Re ν > −1]

WA 443(2)

728

Bessel Functions

 2.

∞

8 0





β dx 1 = cos ν arcsin J ν (αx) cos βx x ν α αν cos νπ 2 ν =   2 ν β + β − α2

6.693

[β ≤ α] [β ≥ α]

[Re ν > 0] WA 443(3)

∞

3.

Y ν (ax) sin(bx) 0

dx x

   νπ 1 b = − tan sin ν arcsin ν 2 a

[0 < b < a, |Re ν| < 1]  +  νπ + 1 ,ν  2   1 ,−ν  1 −ν 2 2 2 ν 2 2 sec −a b− b −a = a cos(νπ) b − b − a 2ν 2 [0 < a < b, |Re ν| < 1]



ET I 103(35) ∞

4.

dx 2 x√       b cos ν arcsin ab a2 − b2 sin ν arcsin ab − = ν 2− 1 √ ν (ν 2 − 1)    νπ −aν cos 2 b + ν b2 − a2 = √ ν  ν (ν 2 − 1) b + b2 − a2

J ν (ax) sin(bx) 0

 0

dx J ν (ax) cos(bx) 2 x       a cos (ν + 1) arcsin ab a cos (ν − 1) arcsin ab + = 2ν(ν − 1)  2ν(ν + 1)   aν sin νπ aν+2 sin νπ 2 2 = √ √ ν−1 − ν+1   2ν(ν − 1) b + b2 − a2 2ν(ν + 1) b + b2 − a2

∞

J 0 (αx) sin x 0

7.

8. 9.

[0 < a < b,

Re ν > 0]

[0 < b < a,

Re ν > 1]

[0 < a < b,

Re ν > 1] ET I 44(6)

6.



Re ν > 0]

ET I 99(6) ∞

5.



[0 < b < a,

dx π = x 2 = arccosecα

[0 < α < 1] [α > 1] WH

∞

dx π = x 2 0 = arcsin β π =− 2  ∞ dx = ln 2α [J 0 (x) − cos αx] x 0  z ∞ 2 dx = J ν (x) sin(z − x) (−1)k J ν+2k+1 (z) x ν 0 J 0 (x) sin βx

k=0

[β > 1]  2  β <1 [β < −1] NT 66(13)

[Re ν > 0]

WA 416(4)

6.697

Bessel and trigonometric functions and powers



z

J ν (x) cos(z − x)

10. 0

729

∞ 1 dx 2 = J ν (z) + (−1)k J ν+2k (z) x ν ν k=1

[Re ν > 0]  6.69410 0

6.695



∞

J 1 (ax) sin(bx) dx x 



 4a b b 1 b2 b2 = b− 1+ 2 E + 1− 2 K 2 3π% 4a 2a  4a 2a



2 −1  & 2a 2a 2b 4a 1 b2 = b− K 1+ 2 E − 1− 2 3π 4a b b2 b

∞

1. 0



∞

2. 0



[0 ≤ b ≤ 2a] ET I 102(22) [0 ≤ 2a ≤ b]

sin αx sinh αβ K 0 (βu) J 0 (ux) dx = 2 2 β +x β

[α > 0,

Re β > 0,

u > α]

cos αx π e−αβ I 0 (βu) J (ux) dx = 0 β 2 + x2 2 β

[α > 0,

Re β > 0,

−α < u < α]

MO 46

MO 46 ∞

3.

x2

0



WA 416(5)

2

x π sin(αx) J 0 (γx) dx = e−αβ I 0 (γβ) 2 +β 2

[α > 0,

Re β > 0,

0 < γ < α] ET II 10(36)

∞

4. 0

x cos(αx) J 0 (γx) dx = cosh(αβ) K 0 (βγ) x2 + β 2

[α > 0,

Re β > 0,

α < γ] ET II 11(45)



∞

6.696 0

[1 − cos(αx)] J 0 (βx)

dx = arccosh x



α β

=0

[0 < β < α] [0 < α < β] ET II 11(43)

6.697  1.

∞

sin[α(x + β)] J 0 (x) dx = 2 x+β −∞



α

0

cos βu √ du 1 − u2



∞

0



∞

3. 0

 4. 5.

∞

WA 463(2)

[1 ≤ α < ∞]

WA 463(1), ET II 345(42)

sin(x + t) π J 0 (t) dt = J 0 (x) x+t 2

[x > 0]

WA 475(4)

cos(x + t) π J 0 (t) dt = − Y 0 (x) x+t 2

[x > 0]

WA 475(5)

= π J 0 (β) 2.

[0 ≤ α ≤ 1]

|x| sin[α(x + β)] J 0 (bx) dx = 0 x −∞ + β  ∞ ,2 + ,2 sin[α(x + β)] + J n+ 12 (x) dx = π J n+ 12 (β) x+β −∞

[0 ≤ α < b] [2 ≤ α < ∞,

WA 464(5), ET II 345(43)a

n = 0, 1, . . .] ET II 346(45)

730

Bessel Functions

 6.

7.

6.698

∞

sin[α(x + β)] J n+ 12 (x) J −n− 12 (x) dx = π J n+ 12 (β) J −n− 12 (β) x+β −∞ [2 ≤ α < ∞,

n = 0, 1, . . .]

√ <2  ∞ π a Γ(μ + ν) J μ+ν− 12 [a(z − ζ)] J μ [a(z + x)] J ν [a(ζ + x)]   · dx =  1 1 1 μ ν (ζ + x) Γ μ+ 2 Γ ν+ 2 (z − ζ)μ+ν− 2 −∞ (z + x) [Re(μ + ν) > 0]

6.698



1.

∞√

0

 x J ν+ 14 (ax) J −ν+ 14 (ax) sin(bx) dx =

  b  2 cos 2ν arccos 2a √ πb 4a2 − b2

=0  2.

∞√

0

 x J ν− 14 (ax) J −ν− 14 (ax) cos(bx) dx =

ET II 346(46)

WA 463(3)

[0 < b < 2a] [0 < 2a < b]





b 2 cos 2ν arccos 2a √ πb 4a2 − b2

ET I 102(26)

 [0 < b < 2a]

=0

[0 < 2a < b] ET I 46(24)

 3.

∞√

0

x I 14 −ν

√  2ν 



1 1 π −2ν b + a2 + b2 √ ax K 14 +ν ax sin(bx) dx = a 2 2 2b  a2 + b 2  Re a > 0, b > 0, Re ν < 54 ET I 106(56)

 4.

∞√

0

x I − 14 −ν

√

 2ν 



1 1 π −2ν b + a2 + b2 √ ax K − 14 +ν ax cos(bx) dx = a 2 2 2b a2 + b 2  Re a > 0, b > 0, Re ν < 34 ET I 50(49)

6.699



1.

∞

λ

0

 2+λ+ν 



2 + λ + ν 2 + λ − ν 3 b2 , ; ; 2 2 2 2 a  0 < b < a, − Re ν − 1 < 1 + Re λ < 32  ν

 1+λ+ν 1 Γ (ν + λ + 1) a b−(ν+λ+1) sin π = 2 Γ(ν + 1) 2

2+λ+ν 1+λ+ν a2 , ; ν + 1; 2 ×F 2 2  b  0 < a < b, − Re ν − 1 < 1 + Re λ < 32

x J ν (ax) sin(bx) dx = 2

1+λ −(2+λ)

a

b

Γ

 2  F Γ ν−λ 2 

ET I 100(11)

6.699

Bessel and trigonometric functions and powers



∞

2. 0



xλ J ν (ax) cos(bx) dx  

2λ a−(1+λ) Γ 1+λ+ν 1 + λ + ν 1 + λ − ν 1 b2 2  ν−λ+1  F , ; ; 2 = 2 2 2 a Γ 2   0 < b < a, − Re ν < 1 + Re λ < 32    a ν −(ν+1+λ)

b Γ (1 + λ + ν) cos π2 (1 + λ + ν) 1+λ+ν 2+λ+ν a2 2 F , ; ν + 1; 2 = Γ(ν + 1) 2 2 b   0 < a < b, − Re ν < 1 + Re λ < 32  2+λ−μ Γ 2λ b Γ 2+μ+λ 2 2

∞

a2+λ

0



∞

4. 0



ET I 45(13)



xλ K μ (ax) sin(bx)dx =

3.

∞

5. 0



F

∞

6. 0

 7. 0

2+μ+λ 2+λ−μ 3 b2 , ; ;− 2 2 2 2 a [Re (−λ ± μ) < 2, Re a > 0, b > 0] ET I 106(50)





μ+λ+1 1+λ−μ λ λ−1 −λ−1 x K μ (ax) cos(bx) dx = 2 a Γ Γ 2 2

μ+λ+1 1+λ−μ 1 b2 , ; ;− 2 ×F 2 2 2 a [Re (−λ ± μ) < 1, Re a > 0, −ν− 12 √ ν ν 2 π2 b a − b2   x sin(ax) J ν (bx) dx = Γ 12 − ν ν

=0 

731



 

b > 0]

0 < b < a,

−1 < Re ν <

1 2

0 < a < b,

−1 < Re ν <

1 2

ET I 49(42)

  ET II 32(4)



 −ν− 12 1 sin(νπ) √ + ν b ν a2 − b 2 Γ 2 π

ν  −ν− 12 1 b + ν b 2 − a2 = 2ν √ Γ 2 π

xν cos(ax) J ν (bx) dx = −2ν

 

0 < b < a,

|Re ν| <

1 2

0 < a < b,

|Re ν| <

1 2

 

ET II 36(29) ∞

xν+1 sin(ax) J ν (bx) dx



−ν− 32 sin(νπ) ν 3  2 a − b2 b Γ ν+ = −21+ν a √ 2 π

−ν− 32 21+ν ν 3  2 b − a2 = − √ ab Γ ν + 2 π

 

0 < b < a,

− 32 < Re ν < − 12

0 < a < b,

− 32 < Re ν < − 12

 

ET II 32(3) 3  2  ∞ 2 −ν− 2 √ ν+1 1+ν ν a − b  πab x cos(ax) J ν (bx) dx = 2 Γ − 12 − ν 0

 8.

=0

 

0 < b < a,

−1 < Re ν < − 12

0 < a < b,

−1 < Re ν < − 12

 

ET II 36(28)

732

Bessel Functions



1

xν sin(ax) J ν (ax) dx =

9. 0

 10.

11.

6.711

1 [sin aJν (a) − cos a J ν+1 (a)] 2ν + 1 [Re ν > −1]

ET II 334(9)a

1

1 [cos aJν (a) + sin a J ν+1 (a)] 2ν + 1 0   Re γ > − 12

 ∞  − 3 −ν √ 3 1+ν ν + ν b b 2 + a2 2 x K ν (ax) sin(bx) dx = π(2a) Γ 2 0  Re a > 0, b > 0, xν cos(ax) J ν (ax) dx =



∞

12. 0



xμ K μ (ax) cos(bx) dx =

1√ 1 π(2a)μ Γ μ + 2 2



Re ν > − 32



ET I 105(49)



−μ− 12 b 2 + a2  Re a > 0,

b > 0,

Re μ > − 12



ET I 49(41) ∞

13.



xν Y ν−1 (ax) sin(bx) dx = 0

√ −ν− 12 2ν πaν−1 b  2 1  b − a2 = Γ 2 −ν

0





ET II 335(20)



0 < b < a,

|Re ν| <

1 2

0 < a < b,

|Re ν| <

1 2

  ET I 104(36)

∞

14.

xν Y ν (ax) cos(bx) dx = 0

 −ν− 12 √ ν b 2 − a2   = −2 πa Γ 12 − ν

0

ν

 0 < b < a,

|Re ν| <

1 2

 0 < a < b,

|Re ν| <

1 2

  ET I 47(30)

6.711



∞

1.

[0 < c < b − a,

xν−μ J μ (ax) J ν (bx) sin(cx) dx = 0

−1 < Re ν < 1 + Re μ]

0



ET I 103(28) ∞

2. 0



xν−μ+1 J μ (ax) J ν (bx) cos(cx) dx = 0 [0 < c < b − a,

∞

3.

[0 < a, 4.

∞

b > 0,

−1 < Re ν < Re μ]

c Γ(ν) Γ(μ + 1) 0 < c < b − a, 0 < Re ν < Re μ + 3]

ET I 47(25)

xν−μ−2 J μ (ax) J ν (bx) sin(cx) dx = 2ν−μ−1 aμ b−ν

0



a > 0,

0 < b,

Γ() Γ(μ + 1) 0 < c < b − a, 0 < Re  < Re μ + 2]

ET I 103(29)

x−μ−1 J μ (ax) J  (bx) cos(cx) dx = 2−μ−1 b− aμ

0

[b > 0,

a > 0,

ET I 47(26)

6.713

Bessel and trigonometric functions and powers



∞

5.

x 0

 6.10 7.10 6.712 1.

2.

1−2ν

∞

733

 

Γ 32 − ν a 3 3 2   F − ν, − 2ν; 2 − ν; a sin(2ax) J ν (x) Y ν (x) dx = − 2 2 2 Γ 2ν − 12 Γ(2 − ν)  0 < Re ν < 32 , 0 < a < 1 ET II 348(63)



ρ2 a2 2z 2 Γ (ν) aμ ρ−ν − − μ−ν+3 2 Γ (μ + 1) ν − 1 μ + 1 3 0   2  ∞ μ −ν 2 ρ a Γ (ν) a ρ − − 2z 2 cos (zx)xν−μ−3 J μ (ax) J ν (ρx) dx = μ−ν+3 2 Γ (μ + 1) ν − 1 μ +1 0 √ −ν− 12 π(2a)ν  2 1  b + 2ab Γ 2 −ν 0   b > 0, −1 < Re ν < 12 √  ∞ −ν− 12 π(2a)ν  2 ν  b + 2ab x [Y ν (ax) cos(ax) − J ν (ax) sin(ax)] cos(bx) dx = −  1 Γ 2 −ν 0 



∞

xν [J ν (ax) cos(ax) + Y ν (ax) sin(ax)] sin(bx) dx =

ET I 104(40)

ET I 48(35) ∞

3.

xν [J ν (ax) cos(ax) − Y ν (ax) sin(ax)] sin(bx) dx

0



=0

√ −ν− 12 2ν πbν  2  b − 2ab = 1 Γ 2 −ν





arg sin (zx)xν−μ−4 J μ (ax) J ν (ρx) dx = z



−1 < Re ν <  −1 < Re ν < 12

0 < b < 2a, 2a < b,

1 2



ET I 104(41) ∞

4.

xν [J ν (ax) sin(ax) + Y ν (ax) cos(ax)] cos(bx) dx

0

=0

√ −ν− 12 π(2a)ν  2  b − 2ab = − 1 Γ 2 −ν

 0 < b < 2a,  0 < 2a < b,

|Re ν| <

1 2

|Re ν| <

1 2

 

ET I 48(33)

6.713



∞

1.

  2 2 x1−2ν sin(2ax) [J ν (x)] − [Y ν (x)] dx

0

=

 2. 0

∞

sin(2νπ) Γ

3

  

− ν Γ 32 − 2ν a 3 3 F − ν, − 2ν; 2 − ν; a2 π Γ(2 − ν) 2 2   0 < Re ν < 34 , 0 < a < 1 ET II 348(64) 2

x2−2ν sin(2ax) [J ν (x) J ν−1 (x) − Y ν (x) Y ν−1 (x)] dx    

sin(2νπ) Γ 32 − ν Γ 52 − 2ν a 3 5 =− F − ν, − 2ν; 2 − ν; a2 π Γ(2 − ν) 2 2  1 5 < Re ν < , 0 < a <1 ET II 348(65) 2 4

734

Bessel Functions



∞

3. 0

6.714



∞

1. 0

 2.

6.714

x2−2ν sin(2ax) [J ν (x) Y ν−1 (x) + Y ν (x) J ν−1 (x)] dx  

Γ 32 − ν a 3 5 2  F − ν, − 2ν; 2 − ν; a =−  2 2 Γ 2ν − 32 Γ(2 − ν)  1 5 < Re ν < , 0 < a <1 ET II 349(66) 2 2

2

sin(2ax) [xν J ν (x)] dx  

a−2ν Γ 12 + ν 1 1 + ν, ; 1 − ν; a2 = √ F 2 2 π Γ(1 − ν) 2

a−4ν−1 Γ 12 + ν 1 1 1  F + ν, + 2ν; 1 + ν; 2 = 2 2 a 2 Γ (1 + ν) Γ 12 − 2ν

 

|Re ν| <

0 < a < 1, a > 1,

|Re ν| <

1 2

1 2





ET II 343(31) ∞

2

cos(2ax) [xν J ν (x)] dx 0

a−2ν Γ(ν) 1 1 2   F ν + , ; 1 − ν; a = √ 2 2 2 π Γ 12 −   1ν

Γ(−ν) Γ 2 + 2ν 1 1 2   F + ν, + 2ν; 1 + ν; a + 2 2 2π Γ 12 − ν  

sin(νπ)a−4ν−1 Γ 12 + 2ν 1 1 1   + ν, + 2ν; 1 + ν; 2 =− F 2 2 a Γ(1 + ν) Γ 12 − ν

 0 < a < 1,  a > 1,

− 14 < Re ν <

− 14 < Re ν <

1 2

1 2





ET II 344(33)

6.715



∞

1. 0



∞

2. 0

6.716



xν π sin(x + β) J ν (x) dx = sec(νπ)β ν J −ν (β) x+β 2 xν π cos(x + β) J ν (x) dx = − sec(νπ)β ν Y −ν (β) x+β 2

a

xλ sin(a − x) J ν (x) dx = 2aλ+1

1. 0

 0



|arg β| < π,

|Re ν| <

1 2

|arg β| < π,

|Re ν| <

1 2





ET II 340(8)

ET II 340(9)

∞  (−1)n Γ(ν − λ + 2n) Γ(ν + λ + 1) (ν + 2n + 1) J ν+2n+1 (a) Γ(ν − λ) Γ(ν + λ + 3 + 2n) n=0

[Re(λ + ν) > −1] a

xλ cos(a − x) J ν (x) dx =

2.



ET II 335(16)

λ+1

J ν (a) a + 2aλ+1 λ+ν+1 ∞  (−1)n Γ (ν − λ + 2n − 1) Γ(ν + λ + 1) (ν + 2n) J ν+2n (a) × Γ(ν − λ) Γ(ν + λ + 2n + 2) n=1 [Re(λ + ν) > −1]

ET II 335(26)

6.721

Bessel and trigonometric functions and powers



∞

sin[a(x + β)] J ν+2n (x) dx = πβ −ν J ν+2n (β) ν −∞ x (x + β)  1 ≤ a < ∞, n = 0, 1, 2, . . . ;

6.717

6.718

735



∞

1. 0



∞

2.

∞

3.



x1−ν π sin(αx) J ν (γx) dx = β −ν e−αβ I ν (βγ) 2 +β 2

x2

0



xν+1 cos(αx) J ν (γx) dx = β ν cosh(αβ) K ν (βγ) + β2  0 < α ≤ γ, Re β > 0,



−1 < Re ν <

x2

0



xν sin(αx) J ν (γx) dx = β ν−1 sinh(αβ) K ν (βγ) x2 + β 2  0 < α ≤ γ, Re β > 0,

Re ν > − 32

−1 < Re ν <

0 < γ ≤ α,

Re β > 0,

1 2

ET II 345(44)

3 2





ET II 33(8)

ET II 37(33)

Re ν > − 12



ET II 33(9) ∞

4.

−ν

x π cos(αx) J ν (γx) dx = β −ν−1 e−αβ I ν (βγ) 2 +β 2 

x2

0

0 < γ ≤ α, Re β > 0, Re ν > − 32



ET II 37(34)

6.719



α

1.6 0

 2. 0

∞      sin(βx) √ J ν (x) dx = π (−1)n J 2n+1 (αβ) J 12 ν+n+ 12 12 α J 12 ν−n− 12 12 α 2 2 α −x n=0

[Re ν > −2] α

+

 cos(βx) π √ J ν (x) dx = J 0 (αβ) J 12 ν 12 α 2 2 2 α −x

, 2

+π

∞ 

(−1)n J 2n (αβ) J 12 ν+n



1.

∞√

0

 2.

∞√

0

 3.

∞√

0

 4.

0

∞√

√   x J 14 a2 x2 sin(bx) dx = 2−3/2 a−2 πb J 14



b2 4a2

√   x J − 14 a2 x2 cos(bx) dx = 2−3/2 a−2 πb J − 14

xY

 1 4

√  a2 x2 sin(bx) dx = −2−3/2 πba−2 H 14 

2 2

x Y − 14 a x



cos(bx) dx = −2

−3/2

1  1  1 2 α J 2 ν−n 2 α

n=1

[Re ν > −1] 6.721

ET II 335(17)





ET II 336(27)



[b > 0]

2

ET I 108(1)

[b > 0]

2

ET I 51(1)

b 4a2 b 4a2

√ πba−2 H− 14

b2 4a2

ET I 108(7)

ET I 52(7)

736

Bessel Functions

 2

2  √ b b −2 3 x K 14 a x sin(bx) dx = 2 π ba I 14 − L 14 2 4a 4a2 0 + , π |arg a| < , b > 0 4  2

2   ∞ √  2 2 √ b b 1 x K − 14 a x cos(bx) dx = 2−5/2 π 3 ba−2 I − 14 − L −4 2 2 4a 4a 0 

5.

6.

6.722

∞√



2 2



−5/2

[b > 0] 6.722



∞√

1. 0

ET I 109(11)

ET I 52(10)

  2

2

√     Γ 58 − ν b b  5  W ν, 1 1 x K 18 +ν a2 x2 I 18 −ν a2 x2 sin(bx) dx = 2πb−3/2 M −ν, 8 2 2 8 8a 8a Γ 4   π 5 Re ν < , |arg a| < , b > 0 8 4 ET I 109(13)

 2.10

∞√

    x J − 18 −ν a2 x2 J − 18 +ν a2 x2 cos(bx) dx    4  √ Γ 14 b π 3 3 3 7 3    2F 3 = 3/4 3/2  3   5 − ν, + ν; , , ; − 8 8 8 4 8 4a 2 a Γ 4 Γ 8 − ν Γ 58 + ν   4  b 1 1 7 9 1 1 2b cos(πν) 2 F 3 − ν, + ν; , , ; − − 2 a π 2 2 2 8 8 4a   4  5/2 b 11 3 13 b ν 2 sin(πν) 2 F 3 1 − ν, 1 + ν; , , ; − − 4 15a π 8 2 8 4a  2  a > 0, Im b = 0 MC

∞√

    x J 18 −ν a2 x2 J 18 +ν a2 x2 sin(bx) dx 

0

 3.

 2 πi/2

2 πi/2

b e b e 2 −3/2 πi/8 = b W ν, 18 e W −ν, 18 2 π 8a 8a2  ⎤ 2 −πi/2

πi b e b 2 e− 2 ⎦ 1 + e−iπ/8 W ν, 18 W −ν, 8 8a2 8a2

0

[b > 0]



∞√

4. 0

    x K 18 −ν a2 x2 I − 18 −ν a2 x2 cos(bx) dx =

 6.723 0

∞

√

2πb

−3/2 Γ

ET I 108(6)

3  2

2

b b 8−ν 1 1 M W ν,− 8 −ν,− 8 3 2 2 8a 8a Γ  4  3 Re ν < 8 , b > 0 ET I 52(12)

          1 x J ν x2 sin(νπ) J ν x2 − cos(νπ) Y ν x2 J 4ν (4ax) dx = J ν a2 J −ν a2 4 [a > 0, Re ν > −1] ET II 375(20)

6.726

Bessel and trigonometric functions and powers

6.724  1.

∞

x2λ J 2ν

0



∞

2. 0

x2λ J 2ν

a x

a x

737

sin(bx) dx

√ 2ν πa Γ(λ − ν + 1)b2ν−2λ−1 1 a2 b 2

0 F 3 2ν + 1, ν − λ, ν − λ + ; = 1 2 16 42ν−λ Γ(2ν + 1) Γ ν − λ + 2

3 a2 b 2 a2λ+2 Γ(ν − λ − 1)b , λ − ν + 2, λ + ν + 2; + 2λ+3 0F 3 2 Γ(ν + λ + 2) 2 16  − 54 < Re λ < Re ν, a > 0, b > 0 ET I 109(15) cos(bx) dx

 

√ 2ν 2ν−2λ−1 Γ λ − ν + 12 a2 b 2 1 =4 πa b 0 F 3 2ν + 1, ν − λ + , ν − λ; Γ(2ν + 1) Γ(ν − λ) 2 16  

1 1 3 3 a2 b 2 −λ−1 2λ+1 Γ ν − λ − 2

,λ − ν + ,ν + λ + ; +4 a 0F 3 3 2 2 2 16 Γ ν+λ+ 2  3  − 4 < Re λ < Re ν − 12 , a > 0, b > 0 ET I 53(14) λ−2ν

6.725



∞

1. 0



∞

2. 0



∞

3.

 2 2

 √  a a νπ π sin(bx) π √ sin − − J ν a x dx = − J ν2 b 8b 4 4 8b x [Re ν > −3,  √  cos(bx) √ J ν a x dx = x



π cos b



2

a νπ π − − 8b 4 4

 √  1 x 2 ν J ν a x sin(bx) dx = 2−ν aν b−ν−1 cos



0



∞

4.

 √  1 x 2 ν J ν a x cos(bx) dx = 2−ν b−ν−1 aν sin

0







2

a > 0,

b > 0] ET I 110(27)



a 8b [Re ν > −1,

J 12 ν

a > 0,

b > 0] ET I 54(25)



a2 νπ − 4b 2 −2 < Re ν < 12 , 2

a > 0,

b>0



ET I 110(28)



a νπ − 4b 2 −1 < Re ν < 12 ,

a > 0,

b>0



ET I 54(26)

6.726  1. 0

∞

  − 1 ν  x x2 + b2 2 J ν a x2 + b2 sin(cx) dx     1 ν− 34 π −ν −ν+ 3  2 2 c a − c2 2 a b = J ν− 32 b a2 − c2 2 =0

 0 < c < a,  0 < a < c,

Re ν >

1 2

Re ν >

1 2

 

ET I 111(37)

738

Bessel Functions



∞

2. 0

   2 − 1 ν x + b2 2 J ν a x2 + b2 cos(cx) dx     1 ν− 1 π −ν −ν+ 1  2 2 a b a − c2 2 4 J ν− 12 b a2 − c2 = 2 =0



Re ν > − 12

b > 0,

Re ν > − 12

 

ET I 55(37)

∞

   2 ∓ 1 ν x + b2 2 K ν a x2 + b2 cos(cx) dx    ± 1 ν− 1 π ∓ν 1 ∓ν  2 a b2 a + c2 2 4 K ±ν− 12 b a2 + c2 = 2 [Re a > 0, Re b > 0, c is real] ET I 56(45)

∞

   2 − 1 ν x + a2 2 Y ν b x2 + a2 cos(cx) dx      1 ν− 1 aπ (ab)−ν b2 − c2 2 4 Y ν− 12 a b2 − c2 = 2     1 ν− 1 2a (ab)−ν c2 − b2 2 4 K ν− 12 a c2 − b2 =− π

11 0



b > 0,

  1ν  x x2 + b2 2 K ±ν a x2 + b2 sin(cx) dx    − 1 ν− 3 π ν ν+ 3  2 a b 2 c a + c2 2 4 K −ν− 32 b a2 + c2 = 2 [Re a > 0, Re b > 0, c > 0] ET I 113(45)

0

4.

 0 < c < a,  0 < a < c,

∞

3.



6.727

5. 0

 0 < c < b,

a > 0,

Re ν > − 12

 0 < b < c,

a > 0,

Re ν > − 12

 

ET I 56(41)

6.727



a

1.9 0



  + a  + a  , , π cos(cx) √ b 2 + c2 + c J ν b a2 − x2 dx = J 12 ν b2 + c2 − c J 12 ν 2 2 2 a2 − x2 [Re ν > −1, c > 0, a > 0] ET I 113(48)

∞

  +a  , +a  ,   sin(cx) π √ J ν b x2 − a2 dx = J 12 ν c − c2 + b2 J − 12 ν c + c2 + b 2 2 2 2 x2 − a2 [0 < b < c, a > 0, Re ν > −1]

∞

  +a  , +a  ,   cos(cx) π √ J ν b x2 − a2 dx = − J 12 ν c − c2 − b2 Y − 12 ν c + c2 − b 2 2 2 2 x2 − a2 [0 < b < c, a > 0, Re ν > −1]

2. a



ET I 113(49)

3. a

 4.8 0

a



a2 − x2



1 2ν

 a2 − x2 dx = cos x I ν

√ 2ν+1

ET I 58(54) 2ν+1

πa   Γ ν + 32   Re ν > − 12

WA 409(2)

6.731

6.728

Bessel and trigonometric functions and powers



∞

1.

∞

2.

∞

3. 0



∞

4. 0



  x cos ax2 J ν (bx) dx

 2 2

2 

2

√ b b b νπ b νπ πb = 3/2 cos − − J 12 ν+ 12 + sin J 12 ν− 12 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −2] ET II 38(39)

0



  x sin ax2 J ν (bx) dx

 2 2

2 

2

√ b b b νπ b νπ πb = 3/2 cos − − J 12 ν− 12 − sin J 12 ν+ 12 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −4] ET II 34(14)

0



∞

5.

  β2 1 cos J 0 (βx) sin αx2 x dx = 2α 4α

[α > 0,

β > 0]

MO 47

  β2 1 sin J 0 (βx) cos αx2 x dx = 2α 4α

[α > 0,

β > 0]

MO 47

b2 νπ − 4a 2  a > 0,

b > 0,

  xν+1 sin ax2 J ν (bx) dx =

0



739

∞

6. 0

  xν+1 cos ax2 J ν (bx) dx =

bν cos ν+1 2 aν+1

bν 2ν+1 aν+1



sin

−2 < Re ν <



ET II 34(15)



b2 νπ − 4a 2 

1 2

a > 0,

b > 0,

−1 < Re ν <

1 2



ET II 38(40)

6.729



∞

1. 0



∞

2. 0

  1 cos x sin ax2 J ν (bx) J ν (cx) dx = 2a

  1 sin x cos ax2 J ν (bx) J ν (cx) dx = 2a







b 2 + c2 bc νπ − Jν 4a 2 2a [a > 0, b > 0,



c > 0,

Re ν > −2] ET II 51(26)



b 2 + c2 bc νπ − Jν 4a 2 2a [a > 0, b > 0,

c > 0,

Re ν > −1] ET II 51(27)

6.731



1.11 0

∞

    x sin ax2 J ν bx2 J 2ν (2cx) dx





ac2 bc2 1 sin 2 J = √ ν 2 2 2 2 b 2 − a2 b −2a b −2a

ac bc 1 cos = √ Jν 2 − b2 2 − b2 2 2 a a 2 a −b

[0 < a < b,

Re ν > −1]

[0 < b < a,

Re ν > −1] ET II 356(41)a

740

Bessel Functions

 2.

∞

10

6.732

    x cos ax2 J ν bx2 J 2ν (2cx) dx





ac2 bc2 1 = √ cos 2 Jν 2 2 2 2 b 2 − a2 b −2a b −2 a

ac bc 1 = √ sin Jν a2 − b 2 a2 − b 2 2 a2 − b 2

0

 

0 < a < b,

Re ν > − 12

0 < b < a,

Re ν > − 12

 

ET II 356(42)a



∞

6.732

x2 cos

0

6.733



∞

1.

sin 0



∞

2. 0

x2 Y 1 (x) K 1 (x) dx = −a3 K 0 (a) 2a

[a > 0]

ET II 371(52)

a √  √  dx = π J0 [sin x J 0 (x) + cos x Y 0 (x)] a Y0 a 2x x

[a > 0] a √  √  dx = π J0 [sin x Y 0 (x) − cos x J 0 (x)] cos a Y0 a 2x x

ET II 346(51)

ET II 347(52) [a > 0] a √ √   πa J1 3. x sin a K1 a [a > 0] ET II 368(34) K 0 (x) dx = 2x 2 0  ∞ a √  √  πa Y1 K 0 (x) dx = − 4. x cos a K1 a [a > 0] ET II 369(35) 2x 2 0  ∞  √  dx 6.734 cos a x K ν (bx) √ x 0





 a a π a a = √ sec(νπ) D ν− 12 √ D −ν− 12 − √ + D ν− 12 − √ D −ν− 12 √ 2 b 2b 2b 2b  2b 1 Re b > 0, |Re ν| < 2 ET II 132(27) 6.735  ∞    √  √ 1. [a > 0] x1/4 sin 2a x J − 14 (x) dx = πa3/2 J 34 a2 ET II 341(10)



∞

0



∞

2. 0



∞

3. 0



∞

4. 0

6.736  1.11

∞

0

 2. 0

∞

   √  √ x1/4 cos 2a x J 14 (x) dx = πa3/2 J − 34 a2

[a > 0]

ET II 341(12)

   √  √ x1/4 sin 2a x J 34 (x) dx = πa3/2 J − 14 a2

[a > 0]

ET II 341(11)

   √  √ x1/4 cos 2a x J − 34 (x) dx = πa3/2 J 14 a2

[a > 0]

ET II 341(13)

   ,  √  √ + π  2 π J 0 a − sin a2 − Y 0 a2 x−1/2 sin x cos 4a x J 0 (x) dx = −2−3/2 π cos a2 − 4 4 ET II 341(18) [a > 0]  +     ,   √ √ π π J 0 a2 + cos a2 − Y 0 a2 x−1/2 cos x cos 4a x J 0 (x) dx = −2−3/2 π sin a2 − 4 4

[a > 0]

ET II 342(22)

6.737

Bessel and trigonometric functions and powers



∞

3.

−1/2

x

 √  sin x sin 4a x J 0 (x) dx =



0



∞

4.

 √  x−1/2 cos x sin 4a x J 0 (x) dx =

0



741

 π π  2 cos a2 + J0 a 2 4 [a > 0]    π π cos a2 − J 0 a2 2 4

ET II 341(16)

[a > 0] ET II 342(20) ∞    ,  √  √ + π  2 π J 0 a − cos a2 − Y 0 a2 x−1/2 sin x cos 4a x Y 0 (x) dx = 2−3/2 π 3 sin a2 − 4 4 0

 5.



[a > 0] ∞

6.



∞

1. 0



∞

2. 0

4.

5.

  √         sin a x2 + b2 b b π 2 2 2 2 √ J ν (cx) dx = J 12 ν J − 12 ν a− a −c a+ a −c 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 35(19)   √         cos a x2 + b2 b b π √ a − a2 − c2 Y − 12 ν a + a2 − c 2 J ν (cx) dx = − J 12 ν 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 39(44)

  √ , , + a  + a  cos b a2 − x2 π √ J ν (cx) dx = J 12 ν b2 + c2 − b J 12 ν b 2 + c2 + b 2 2 2 a2 − x2 0 [c > 0, Re ν > −1] ET II 39(47)  √ √  a cos a2 − x2 πa2ν+1

xν+1 √ I ν (x) dx = ET II 365(9) [Re ν > −1] 3 a2 − x2 0 ν+1 2 Γ ν+ 2   √  ∞ 2 2 sin a b + x √ xν+1 J ν (cx) dx b2 + x2 0      π 1 +ν ν  2 2 − 14 − 12 ν 0 < c < a, Re b > 0, −1 < Re ν < 12 b 2 c a −c = J −ν− 12 b a2 −c2 2   =0 0 < a < c, Re b > 0, −1 < Re ν < 12 

3.

 √  x−1/2 cos x cos 4a x Y 0 (x) dx

   , √ + π  2 π = −2−3/2 π 3 cos a2 − J 0 a + sin a2 − Y 0 a2 4 4 ET II 347(56) [a > 0]

0

6.737

ET II 347(55)

a

ET II 35(20)

742

Bessel Functions



∞

6.

ν+1 cos

x 0

6.738

  √    − 1 − 1 ν a x2 + b2 π 1 +ν ν  2 √ b 2 c a − c2 4 2 Y −ν− 12 b a2 − c2 J ν (cx) dx = − 2 x2 + b2   =

0 < c < a,

−1 < Re ν <

Re b > 0,

  − 1 − 1 ν 2 1 +ν ν  2 b 2 c c − a2 4 2 K ν+ 12 b c2 − a2 π  0 < a < c,

1 2

1 −1 < Re ν < 2

Re b > 0,





ET II 39(45)

6.738



a ν+1

x

1. 0



     − 1 ν− 3 π ν+ 3  2 2 a 2 b 1 + b2 2 4 J ν+ 32 a 1 + b2 sin b a − x J ν (x) dx = 2

ET II 335(19) [Re ν > −1]   xν+1 cos a x2 + b2 J ν (cx) dx      , − 1 ν− 3 + π ν+ 3 ν  2 ab 2 c a − c2 2 4 cos(πν) J ν+ 32 b a2 − c2 − sin (πν) Y ν+ 32 b a2 − c2 = 2   0 < c < a, Re b > 0, −1 < Re ν < − 12

∞

2. 0

=0

 0 < a < c,

−1 < Re ν < − 12

Re b > 0,



ET II 39(43)



t

6.739

x 0

6.741 1.

2.

−1/2 cos



 √   √    √t    √  b t−x t 2 2 2 2 √ J 2ν a x dx = π J ν a + b + b Jν a +b −b 2 2 t−x   1 Re ν > − 2 EH II 47(7)

a a cos (μ arccos x) π √ J 12 (ν−μ) J ν (ax) dx = J 12 (μ+ν) 2 2 2 1 − x2 0 [Re(μ + ν) > −1,   1 a  a cos [(ν + 1) arccos x] π √ cos J ν+ 12 J ν (ax) dx = a 2 2 1 − x2 0 

3. 0

1

1

cos [(ν − 1) arccos x] √ J ν (ax) dx = 1 − x2



a

π sin J ν− 12 a 2

[Re ν > −1, a 2 [Re ν > 0,

a > 0]

a > 0]

a > 0]

ET II 41(54)

ET II 40(53)

ET II 40(52)a

6.75 Combinations of Bessel, trigonometric, and exponential functions and powers 6.751

Notation: 1 =

, ,   1 + 1 + (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2

6.752

Combinations of Bessel, trigonometric, and exponential functions and powers

<  1 1 1 ax dx = √ √ e sin(bx) I 0 b + b 2 + a2 2 2b b2 + a2 0 [Re a > 0,

 ∞ 1 1 a 1  ax dx = √ √ e− 2 ax cos(bx) I 0 √ 2 2 2 2b a + b b + a2 + b2 0 

1.

2.

∞

− 12 ax

743



b > 0]

ET I 105(44)

[Re a > 0,

 3.

< ∞

10

e−bx cos(ax) J 0 (cx) dx =

0

b > 0] 1/2 2 (b2 + c2 − a2 ) + 4a2 b2 + b2 + c2 − a2 √ < 2 2 (b2 + c2 − a2 ) + 4a2 b2

ET I 48(38)

[c > 0] alternatively, with a and b interchanged,   ∞ 2 − b 2 −ax e cos(bx) J 0 (cx) dx = 2 2 2 2 − 1 0 6.752 1.



∞

10

e 0

 2.10 0

∞

−ax

dx = arcsin J 0 (bx) sin(cx) x

 

ET II 11(46)

[c > 0]

2c



 a2 + (c + b)2 + a2 + (c − b)2

= arcsin

c 2



[Re a > |Im b|, c > 0] ET I 101(17)  b − b2 − 21 b dx = (1 − r) = , e−ax J 1 (cx) sin(bx) x c c   c2 a2 − , c > 0 b2 = 1 − r2 r2 ET II 19(15)

Notation: For integrals 6.752 3–6.752 5 we define the auxiliary functions ,  1 + (a + ρ)2 + z 2 − (a − ρ)2 + z 2 2 ,  1 + 2 (a) ≡ 1 (a, ρ, z) = (a + ρ)2 + z 2 + (a − ρ)2 + z 2 2

1 (a) ≡ 1 (a, ρ, z) =

when a ≥ 0, ρ ≥ 0, and z ≥ 0.   ∞ √ π 3.10 e−zx J ν+1/2 (ax) J ν+1 (ρx) x dx 2 0

  a ρ2 − 21 2ν+2 1  =a ρ 2  (2 − 2 ) 2 1  ρ − 1 1 2 ν+1 2 2  − a ρ = aν+1/2 2ν+2 2 2 2 2 − 1 2 −ν−3/2 −ν−1

[Re z > |Im a| + |Im ρ|]

744

4.

10

5.10

6.753 1.

8

2.

Bessel Functions

6.753

  ∞ π dx e−zx J ν+1/2 (ax) J ν (ρx) √ 2 0 x

 1/2 1 1 1 ν+1/2 ν  ρ d =a 2ν 2 2   2 1 − a /2 0 2  a/2   dx ν > − 12 , Re z > |Im a| + |Im ρ| = a−ν−1/2 ρν x2ν √ 2 1−x 0  2  

2 − a2 a −  ∞ 2 − 2  a 2 1 a ρ dx −zx + arcsin e sin(ax) J 1 (ρx) 2 = x 2aρ 2 2 0 [Re z > |Im a| + |Im ρ|] 

∞



∞

 ϕ ν sin (xa sin ψ) −xa cos ϕ cos ψ e J ν (xa sin ϕ) dx = ν −1 tan sin(νψ) x 2 0 + π π, Re ν > −1, a > 0, 0 < ϕ < , 0 < ψ < ET II 33(10) 2 2  ∞  ϕ ν cos (xa sin ψ) −xa cos ϕ cos ψ e J ν (xa sin ϕ) dx = ν −1 tan cos(νψ) x 0 + 2 π, Re ν > 0, a > 0, 0 < ϕ, ψ < 2 ET II 38(35)

3.8 0

  2(2a) Γ(ν + 32 )R−2ν−3 b cos(ν + 32 )ϕ + s sin(ν + 32 )ϕ xν+1 e−sx sin(bx) J ν (ax) dx = − √ π  ν

Re s > |Im a| + |Im b|,   2  2 2 ϕ = arg s + a − b − 2ibs

Re ν > − 32 , 2  R4 = s2 + a2 − b2 + 4b2 s2 , 

∞

4.8 0

xν+1 e−sx cos(bx) J ν (ax) dx =

  2(2a)ν √ Γ(ν + 32 )R−2ν−3 s cos(ν + 32 )ϕ − b sin(ν + 32 )ϕ , π  Re ν > −1,

2  R4 = s2 + a2 − b2 + 4b2 s2 ,  5.10 0

∞

Re s > |Im a| + |Im b|,   2  2 2 ϕ = arg s + a − b − 2ibs

xν e−ax cos ϕ cos ψ sin (ax sin ψ) J ν (ax sin ϕ) dx   1   −ν− 12  ν νΓ ν + 2 √ a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ =2 sin ν + 12 β π + , π π β a > 0, 0 < ϕ < , 0 < ψ < , Re ν > −1 ET II 34(12) tan = tan ψ cos ϕ 2 2 2

6.755

Combinations of Bessel, trigonometric, and exponential functions and powers



∞

6. 0

6.754 1. 2.

xν e−ax cos ϕ cos ψ cos (ax sin ψ) J ν (ax sin ϕ) dx   1   −ν− 12  ν νΓ ν + 2 √ =2 a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ cos ν + 12 β π   π 1 β a > 0, 0 < ϕ, ψ < , Re ν > − ET II 38(37) tan = tan ψ cos ϕ 2 2 2

2

√ b π − b2 8 e e I0 ET I 108(9) [b > 0] 3/2 8 2 0   2

2

2

2

  ∞  2  2 a a π π a a 1 π −ax − + e cos x J 0 x dx = J0 cos −Y0 cos 4 2 16 16 4 16 16 4 0 



∞

∞

3. 0

−x2

  sin(bx) I 0 x2 dx =

    1 e−ax sin x2 J 0 x2 dx = 4



[a > 0] MI 42 2

 2

2

2

 a a π π a a π − + J0 sin −Y0 sin 2 16 16 4 16 16 4 [a > 0]

6.755



∞

1. 0

 2.

3.

4.

6.

7.

 ν−1  √  2 x−ν e−x sin 4a x I ν (x) dx = 23/2 a e−a W

1 3 1 1 2 − 2 ν, 2 − 2 ν

[a > 0,

MI 42

 2 2a Re ν > 0]

ET II 366(14)

∞

   √  2 1 3 x−ν− 2 e−x cos 4a x I ν (x) dx = 2 2 ν−1 aν−1 e−a W − 32 ν, 12 ν 2a2 0   a > 0, Re ν > − 12  ∞  ν−1 Γ  3 − 2ν  2    √  −ν x 3/2  ea W 3 ν− 1 , 1 − 1 ν 2a2 x e sin 4a x K ν (x) dx = 2 a π 21 2 2 2 2 Γ 2 +ν 0   a > 0, 0 < Re ν < 34    ∞    √  Γ 12 − 2ν a2 3 −ν− 12 x ν−1 ν−1 1  e W 3 ν,− 1 ν 2a2 x e cos 4a x K ν (x) dx = 2 2 πa 2 2 Γ 2 + ν 0  a > 0, − 12 < Re ν < 14

ET II 366(16)

ET II 369(38)

ET II 369(42)



√ ∞  √  3 1 πa Γ( + ν) Γ( − ν) 3 2   ,  + ; −2a x− 2 e−x sin 4a x K ν (x) dx =  + ν,  − ν; 2F 2 2 2 2−2 Γ  + 12 0 ET II 369(39) [Re  > |Re ν|]

√  ∞  √  π Γ( + ν) Γ( − ν) 1 1 −1 −x 2   x e cos 4a x K ν (x) dx = 2 F 2  + ν,  − ν; ,  + ; −2a 2 2 2 Γ  + 12 0 ET II 370(43) [Re  > |Re ν|]  ∞     √ 2 1 x−1/2 e−x cos 4a x I 0 (x) dx = √ e−a K 0 a2 2π 0 [a > 0] ET II 366(15) 

5.

745

746

Bessel Functions



∞

8.



√  e cos 4a x K 0 (x) dx =

−1/2 x

x



0



∞

9. 0

6.756  1.

∞

∞

2.

∞

3.

1

1

∞

0

6.757  1.

∞

 √  cos a x J ν (bx) dx



 a ia 1 1 ia Γ ν+ =√ D −ν− 12 √ D −ν− 12 √ + D −ν− 12 − √ 2 2πb b b  b  a > 0, b > 0, Re ν > − 12 ET II 39(42)

√ −1/2 −a x

e

√

x−1/2 e−a

x

 √  1 sin a x J 0 (bx) dx = a I 14 2b

 √  a I 1 cos a x J 0 (bx) dx = 2b − 4



a2 4b





a2 4b



K 14 + π |arg a| < , 4 2

2

a a K 14 4b 4b + π |arg a| < , 4

=2

∞

, b>0

ET II 12(49)

ET I 193(26)

     e−bx cos a 1 − e−x J ν ae−x dx

0

∞

=

J ν (a)  Γ(ν − b + 2n) Γ(ν + b) + (ν + 2n) J ν+2n (a) 2(−1)n ν + b n=0 Γ(ν − b + 1) Γ(ν + b + 2n + 1) [Re b > − Re ν]



ET II 11(40)

∞  (−1)n Γ(ν − b + 2n + 1) Γ (ν + b) (ν + 2n − 1) J ν+2n+1 (a) Γ(ν − b + 1) Γ(ν + b + 2n + 2) n=0

[Re b > − Re ν]

2.

, b>0

     e−bx sin a 1 − e−x J ν ae−x dx

0



ET II 369(41)

√ x

x− 2 e−a

x



ET II 369(40)

 √  sin a x J ν (bx) dx





a ia 1 i ia Γ ν+ =√ D −ν− 12 √ − D −ν− 12 − √ D −ν− 12 √ 2 2πb b b b [a > 0, b > 0, Re ν > −1] ET II 34(17)

0

4.

[a > 0]

√ x

x− 2 e−a

0



  π a2 e K 0 a2 2

   √  2 1 x−1/2 e−x cos 4a x K 0 (x) dx = √ π 3/2 e−a I 0 a2 2

0



6.756

π 2

6.758 −π 2

ei(μ−ν)θ (cos θ)

ν+μ

ET I 193(27)

(λz)−ν−μ J ν+μ (λz) dθ < = π(2az)−μ (2bz)−ν J μ (az) J ν (bz); λ = 2 cos θ (a2 eiθ + b2 e−iθ ) < λ = 2 cos θ (a2 eiθ + b2 e−iθ ) [Re(ν + μ) > −1] EH II 48(12)

6.775

Bessel functions and the logarithm, or arctangent

747

6.76 Combinations of Bessel, trigonometric, and hyperbolic functions 

∞

6.761 0

 √   −x  J 2ν 2 b2 − a2 √ dx = cosh x cos (2a sinh x) J ν (be ) J ν be 2 b 2 − a2 x

=0 

∞

6.762

=0 =− ∞

6.763 0

Re ν > −1]

[0 < b < a,

Re ν > −1]

     cosh x sin (2a sinh x) J ν (bex ) Y ν be−x − Y ν (bex ) J ν be−x dx

0



[0 < a < b,

 +   −1/2 1/2 , 2 cos(νπ) a2 − b2 K 2ν 2 a2 − b2 π

  cosh x cos (2a sinh x) Y ν (bex ) Y ν be−x dx +  −1/2 1/2 , 1 = − b 2 − a2 J 2ν 2 b2 − a2 2 +   −1/2 1/2 , 2 = cos(νπ) a2 − b2 K 2ν 2 a2 − b2 π



ET II 359(10)

0 < a < b,

|Re ν| <

1 2

0 < b < a,

|Re ν| <

1 2

 

ET II 360(12)

[0 < a < b,

|Re ν| < 1]

[0 < b < a,

|Re ν| < 1] ET II 360(11)

6.77 Combinations of Bessel functions and the logarithm, or arctangent 

∞

6.771

μ+ 12

x 0

 



 1 3  2μ− 2 Γ μ+ν μ+ν 3 ν−μ 1 a2 2 + 4 + + ln x J ν (ax) dx =  ν−μ 1  μ+ 3 ψ +ψ − ln 2 4 2 4 4 Γ 2 +4 a 2   3 a > 0, − Re ν − 2 < Re μ < 0 ET II 32(25)

6.772



∞

1 ln x J 0 (ax) dx = − [ln(2a) + C] WA 430(4)a, a 0  ∞ , 1 + a +C 2. ln x J 1 (ax) dx = − ln a 2 0  ∞   2 3. ln a2 + x2 J 1 (bx) dx = [K 0 (ab) + ln a] b 0  ∞  2 4. J 1 (tx) ln 1 + t4 dt = ker x x 0 √  



  ∞  ln x + x2 + a2 1 2 ab ab ab √ K0 6.773 J 0 (bx) dx = + ln a I 0 K0 2 2 2 2 2 2 x +a 0 [a > 0, b > 0]

 ∞ √ 2 ab x + a2 + x dx 6.774 ln √ = K 20 J 0 (bx) √ [Re a > 0, b > 0] 2 + a2 − x 2 + a2 2 x x 0 ∞ +  ,   1  6.775 x ln 1 + a2 + x2 − ln x J 0 (bx) dx = 2 1 − e−ab b 0 [Re a > 0, b > 0] 1.

ET II 10(27) ET II 19(11) ET II 19(12) MO 46

ET II 10(28) ET II 10(29)

ET II 12(55)

748

Bessel Functions

  2 1 − a K 1 (ab) 6.776 J 0 (bx) dx = b b 0  ∞ 2 6.777 J 1 (tx) arctan t2 dt = − kei x x 0 

∞



a2 x ln 1 + 2 x

6.776



[Re a > 0,

b > 0]

ET II 10(30) MO 46

6.78 Combinations of Bessel and other special functions 

∞

6.781 0

1 si(ax) J 0 (bx) dx = − arcsin b



b a

=0

[0 < b < a] [0 < a < b] ET II 13(6)

6.782  ∞  √  e−z − 1 1. Ei(−x) J 0 2 zx dx = z 0  ∞  √  sin z 2. si(x) J 0 2 zx dx = − z 0  ∞  √  cos z − 1 3. ci(x) J 0 2 zx dx = z 0  ∞  √  dx Ei(−z) − C − ln z √ 4. Ei(−x) J 1 2 zx √ = x z 0  ∞ π  √  dx − si(z) 5. si(x) J 1 2 zx √ = − 2 √ x z 0  ∞  √  dx ci(z) − C − ln z √ 6. ci(z) J 1 2 zx √ = x z 0  ∞  √  C + ln z − e2 Ei(−z) 7. Ei(−x) Y 0 2 zx dx = πz 0 6.783  1. 2. 3.

NT 60(4) NT 60(6) NT 60(5) NT 60(7) NT 60(9) NT 60(8) NT 63(5)

2

  b 2 x si a2 x2 J 0 (bx) dx = − 2 sin [a > 0] 2 b 4a 0  2   ∞  2 2 b 2 x ci a x J 0 (bx) dx = 2 1 − cos [a > 0] 2 b 4a 0 2

  2

 ∞   b b 1 ci a2 x2 J 0 (bx) dx = + ln + 2C ci b 4a2 4a2 0 

∞

∞

4. 0

6.784  1. 0

∞

 2

  b 1 π 2 2 si a x J 1 (bx) dx = − − si b 4a2 2 

xν+1 [1 − Φ(ax)] J ν (bx) dx = a−ν

ET II 13(7)a ET II 13(8)a

[a > 0]

ET II 13(8)a

[a > 0]

ET II 20(25)a

 

2

Γ ν + 32 b b2 1 1 1 1 exp − M 2 2 2 ν+ 2 , 2 ν+ 2 b2 Γ(ν + 2) 8a 4a + , π |arg a| < , b > 0, Re ν > −1 4 ET II 92(22)

6.792

Integration of Bessel functions



∞

2.

 x [1 − Φ(ax)] J ν (bx) dx = ν

0

749

 

2

1 b 2 a 2 −ν Γ ν + 12 b2   1 1 1 1 M exp − 2 ν− 4 , 2 ν+ 4 π b3/2 Γ ν + 32 8a2 4a2   1 π |arg a| < , Re ν > − , b > 0 4 2 ET II 92(23)





  ∞ exp a2 − x    2x a π 5/2 2 2 sec(νπ) [J ν (a)] + [Y ν (a)] 6.785 1−Φ √ K ν (x) dx = x 4 2x 0   Re a > 0, |Re ν| < 12 ET II 370(46)  ∞   6.786 xν−2μ+2n+2 ex2 Γ μ, x2 Y ν (bx) dx 0 3  3  2

2

b b nΓ 2 −μ+ν +n Γ 2 −μ+n exp = (−1) W μ− 12 ν−n−1, 12 ν b Γ(1 − μ) 8 4  n is an integer, b > 0, Re(ν − μ + n) > − 32 , Re(−μ + n) > − 32 , Re ν < 12 − 2n 

ET II 108(2) ∞

6.787 0

1

xν+2n− 2 J ν (bx) dx = 0 B(a + x, a − x)



π ≤ b < ∞,

−1 < Re ν < 2a − 2n −

7 2



ET II 92(21)

6.79 Integration of Bessel functions with respect to the order 6.791



∞

1. −∞  ∞

2. −∞  ∞

3. −∞

K ix+iy (a) K ix+iz (b) dx = π K iy−iz (a + b)

[|arg a| + |arg b| < π]

J ν−x (a) J μ+x (a) dx = J μ+ν (2a)

[Re(μ + ν) > 1]

ET II 382(21) ET II 379(1)

J κ+x (a) J λ−x (a) J μ+x (a) J ν−x (a) dx =

Γ(κ + λ + μ + ν + 1) Γ (κ +⎛ λ + 1) Γ(λ + μ + 1) Γ(μ + ν + 1) Γ (ν + κ + 1) × 4F 5 ⎝

κ+λ+μ+ν κ+λ+μ+ν+1 κ+λ+μ+ν+1 κ+λ+μ+ν , , + 1, + 1; 2 2 2 2 ⎞

κ + λ + μ + ν + 1, κ + λ + 1, λ + μ + 1, μ + ν + 1, ν + κ + 1; −4a2 ⎠ [Re(κ + λ + μ + ν) > −1] 6.792



∞

1. −∞

ET II 379(3)

eπx K ix+iy (a) K ix+iz (b) dx = πe−πz K i(y−z) (a − b) [a > b > 0]

ET II 382(22)

750

Bessel Functions





∞

2.

e −∞



ix

K ν+ix (α) K ν−ix (β) dx = π

αeρ + β α + βeρ

6.793

ν K 2ν

 α2 + β 2 + 2αβ cosh  [|arg α| + |arg β| + |Im | < π] ET II 382(23)

∞

3. −∞

e(π−γ)x K ix+iy (a) K ix+iz (b) dx = πe−βy−αz K iy−iz (c)

[0 < γ < π,

a > 0,

b > 0,

c > 0,

α, β, γ—the angles of the triangle with sides a, b, c] ET II 382(24), EH II 55(44)a

2ν ∞ h (2) (2) (2) e−cxi H ν−ix (a) H ν+ix (b) dx = 2i H 2ν (hk) k −∞ < <

 4.11

h= 

∞

5. −∞

ae 2 c + be− 2 c , 1

1

k=

ae− 2 c + be 2 c 1

1

[a, b > 0,

c is real]

ET II 380(11)

a−μ−x b−ν+x ecxi J μ+x (a) J ν−x (b) dx % & 12 μ+ 12 ν  

+ , +c c  ,1/2  2 cos 2c 2 − 12 ci 2 12 ci = exp (ν − μ)i J μ+ν 2 cos +b e a e 1 1 2 2 a2 e− 2 ci + b2 e 2 ci [a > 0, b > 0, |c| < π, Re(μ + ν) > 1] =0 [a > 0,

b > 0,

|c| ≥ π,

Re(μ + ν) > 1]

EH II 54(41), ET II 379(2)

6.793 1.



∞

e−cxi [J ν−ix (a) Y ν+ix (b) + Y ν−ix (a) J ν+ix (b)] dx = −2 −∞ < < 1 1 1 1 k = ae− 2 c + be 2 c h = ae 2 c + be− 2 c ,



∞

2.

e

−cxi

−∞



∞

−∞

6.794



∞

1. 2.



α, β, γ ∈ R,

ae 2 c + be− 2 c , 1

1

k=

α, β > 0,

ae− 2 c + be 2 c 1

1

σ = α2 + β 2 − 2αβ cosh γ,

K ix (a) K ix (b) cosh[(π − ϕ)x] dx =

∞

cosh 0

Im c = 0]

ET II 380(9)

[a, b > 0,

Im c = 0]

ET II 380(10)

 eiγx sech(πx) [J −ix (α) J ix (β) − J ix (α) J −ix (β)] dx = 2i H(σ) sign(β − α) J 0 σ 1/2

0



[a, b > 0,

2ν h [J ν−ix (a) J ν+ix (b) − Y ν−ix (a) Y ν+ix (b)] dx = 2 Y 2ν (hk) k < < h=

3.10

2ν h J 2ν (hk) k

π π x K ix (a) dx = 2 2

 H(σ) the Heaviside step function

 π K0 a2 + b2 − 2ab cos ϕ 2 [a > 0]

EH II 55(42) ET II 382(19)

6.794

Integration of Bessel functions



∞

3.

cosh(x) K ix+ν (a) K −ix+ν (a) dx = 0

5. 6.

+   , π K 2ν 2a cos 2 2

[2|arg a| + |Re | < π] π x J ix (a) dx = 2 sin a sech [a > 0] 2 −∞  ∞ π x J ix (a) dx = −2i cos a cosech [a > 0] 2 −∞  ∞   2 2 sech(πx) [J ix (a)] + [Y ix (a)] dx = − Y 0 (2a) − E0 (2a) 

4.

751

ET II 383(28)

∞

ET II 380(6) ET II 380(7)

0



∞

7. 0



∞

8.

x tanh(πx) K ix (β) K ix (α) dx = 0

9.

10.

π πa x K ix (a) dx = x sinh 2 2

[a > 0]

ET II 380(12)

[a > 0]

ET II 382(20)

π  exp(−β − α) αβ 2 α+β

[|arg β| < π, |arg α| < π] ET II 175(4)

π α α2 x sinh(πx) K 2ix (α) K ix (β) dx = 5/2 √ exp −β − 8β 2 β 0 + π, β > 0, |arg α| < ET II 175(5) 4  ∞ x sinh(πx) π2 I n (β) K n (α) K ix (α) K ix (β) dx = [0 < β < α; n = 0, 1, 2, . . .] 2 2 x +n 2 0 2 π I n (α) K n (β) [0 < α < β; n = 0, 1, 2, . . .] = 2 

∞



∞

11. 0

3/2



 γ α β αβ π2 exp − + + 2 x sinh(πx) K ix (α) K ix (β) K ix (γ) dx = 4 2 +β α γ π |arg α| + |arg β| < , 2

, γ>0 ET II 176(9)

%

&  ∞ π (α + β) γ 2 + 4αβ π2 γ √ x K 12 ix (α) K 12 ix (β) K ix (γ) dx =  exp − x sinh 2 2 αβ 2 γ 2 + 4αβ 0 [|arg α| + |arg β| < π, γ > 0]

 12.

ET II 176(8)

 13. 0

ET II 176(10) ∞

x sinh(πx) K 12 ix+λ (α) K 12 ix−λ (α) K ix (γ) dx = 0 2

π γ 22λ+1  α2λ z z = γ 2 − 4α2

=

[0 < γ < 2α] , + 2λ (γ + z) + (γ − z)2λ [0 < 2α < γ]

ET II 176(11)

752

Bessel Functions

6.795  1.

∞

cos(bx) K ix (a) dx =

0

 0

∞

3.

x sin(ax) K ix (b) dx = 0 −∞

4. −∞



x sin

0

1. 2. 3. 4.



∞

−1/2 1 1 − a2 4

πb sinh a exp (−b cosh a) 2

[|a| < 1] , b>0

ET II 175(1)

[0 < a < b;

n = 0, 1, . . .]

[0 < b < a;

n = 0, 1, . . .]



1 b2 π 3/2 b πx K 12 ix (a) K ix (b) dx = √ exp −a − 2 8a 2a + π |arg a| < , 2

ET II 382(25)

, b>0

ET II 175(6)

1

e 2 πx cos(bx) J ix (a) dx = −i exp (ia cosh b) [a > 0, −∞ sinh(πx)

 ∞ 1 π πx K ix (a) dx = cos (a sinh b) cos(bx) cosh 2 2 0

 ∞ 1 π πx K ix (a) dx = sin (a sinh b) sin(bx) sinh 2 2 0    ∞ b π2 2 cos(bx) cosh(πx) [K ix (a)] dx = − Y 0 2a sinh 4 2 0

b > 0]

ET II 380(8) EH II 55(47) EH II 55(48)

[a > 0, 

 ∞ b π2 2 J 0 2a sinh sin(bx) sinh(πx) [K ix (a)] dx = 4 2 0

b > 0]

ET II 383(27)

[a > 0,

b > 0]

ET II 382(26)

 5.

ET II 380(4)

+ π |Im a| < , 2

sin[(ν + ix)π] K ν+ix (a) K ν−ix (b) dx = π 2 I n (a) K n+2ν (b) n + ν + ix = π 2 K n+2ν (a) I n (b)

∞

5.

6.796

, a>0 EH II 55(46), ET II 175(2)

J x (ax) J −x (ax) cos(πx) dx =



+ π |Im b| < , 2

π −a cosh b e 2

∞

2. 

6.795

6.797  1.

∞

0

(2)

(2)

xeπx sinh(πx) Γ(ν + ix) Γ(ν − ix) H ix (a) H ix (b) dx   √ = i2ν π Γ 12 + ν (ab)ν (a + b)−ν K ν (a + b) [a > 0,

 2. 0

∞

b > 0,

Re ν > 0]

ET II 381(14)

iπ 3/2 2ν (2) (2)  (b−a)−ν H (2) xeπx sinh(πx) cosh(πx) Γ(ν +ix) Γ(ν −ix) H ix (a) H ix (b) dx =  1 ν (b−a) Γ − ν 2   0 < a < b, 0 < Re ν < 12 ET II 381(15)

6.812

Struve functions



∞

3.

πx

xe

sinh(πx) Γ

0

ν + ix 2

753



ν − ix (2) (2) Γ H ix (a) H ix (b) dx 2

  − 1 ν a2 + b 2 = iπ22−ν (ab)ν a2 + b2 2 H (2) ν [a > 0,

 4.

∞

11 0

b > 0,

Re ν > 0]

ET II 381(16)

  x sinh(πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx = 2λ−1 π 3/2 (ab)λ (a + b)−λ Γ λ + 12 K λ (a + b) [|arg a| < π,

Re λ > 0,

b > 0] ET II 176(12)



∞

5.

x sinh(2πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx =

0



∞

6. 0



∞

7. 0

λ



5 2

ab 2λ π 1  K λ (|b − a|) |b − a| Γ 2 − λ  a > 0, 0 < Re λ < 12 , b > 0



ET II 176(13)

    ab √ x sinh(πx) Γ λ + 12 ix Γ λ − 12 ix K ix (a) K ix (b) dx = 2π 2 a2 + b 2 K 2λ 2 a2 + b 2 + , π |arg a| < , Re λ > 0, b > 0 2 x tanh(πx) K ix (a) K ix (b) 1  3 1   3 1  dx = 2 Γ 4 + 2 ix Γ 4 − 2 ix





ET II 177(14)

  πab 2 + b2 exp − a a2 + b 2 + π |arg a| < , 2

, b>0 ,

ET II 177(15)

6.8 Functions Generated by Bessel Functions 6.81 Struve functions 6.811



∞

1.

Hν (bx) dx = −

0





∞

2.

Hν 0



Hν−1 0

a x



∞

3.

2

cot

 νπ  2

b

Hν (bx) dx = −

 √ J 2ν 2a b



b

a > 0,

b > 0,

b > 0]

ET II 158(1)

Re ν > − 32



ET II 170(37)

2

a x

[−2 < Re ν < 0,

Hν (bx)

 √ 1 dx = − √ J 2ν−1 2a b x a b



a > 0,

b > 0,

Re ν > − 12



ET II 170(38)

6.812



1. 0

∞

H1 (bx) dx π [I 1 (ab) − L1 (ab)] = 2 2 x +a 2a

[Re a > 0,

b > 0]

ET II 158(6)

754

Functions Generated by Bessel Functions



∞

2. 0

6.813

 

b cot νπ Hν (bx) π 3 − ν 3 + ν a2 b 2 2  ; ; dx = − (ab) + 1; L F ν 2 1 νπ x2 + a2 1 − ν2 2 2 2 2a sin 2 [Re a > 0, b > 0, |Re ν| < 2] ET II 159(7)

6.813



∞

s−1

x

1. 0



∞

2.

 

2s−1 Γ s+ν s+ν 2  π Hν (ax) dx = s  1 tan 1 2

  a Γ 2ν − 2s + 1 3 , 1 − Re ν a > 0, −1 − Re ν < Re s < min 2

x−ν−1 Hν (x) dx =

0



∞

3.

x−μ−ν Hμ (x) Hν (x) dx =

0



Re ν > − 32



1

√ 2−μ−ν π Γ(μ + ν)      Γ μ + 12 Γ ν + 12 Γ μ + ν + 12

xν+1 Hν (ax) dx =

1 Hν+1 (a) a

x1−ν Hν (ax) dx =

aν−1 1   √ 1 − a Hν−1 (a) ν−1 πΓ ν + 2 2

0 1

5. 0

ET II 383(2)



[Re(μ + ν) > 0]

4. 



2−ν−1 π Γ(ν + 1)

WA 429(2), ET I 335(52)



a > 0,

WA 435(2), ET II 384(8)

Re ν > − 32



[a > 0] 6.814  1.

∞

(x2 +

0

6.815



1.

1

1−μ a2 )

dx =

ET II 159(8)

 √  1 x 2 ν (1 − x)μ−1 Hν a x dx = 2μ a−μ Γ(μ) Hμ+ν (a)

0

2.

ET II 158(3)a

2μ−1 πaμ+ν b−μ [I −μ−ν (ab) − Lμ+ν (ab)] Γ(1 − μ) cos[(μ + ν)π]   Re a > 0, b > 0, Re ν > − 32 , Re(μ + ν) < 12 , Re(2μ + ν) < 32

xν+1 Hν (bx)

ET II 158(2)a



 Re ν > − 32 , Re μ > 0 ET II 199(88)a

 1  √  1 3 3 a2 B(λ, μ)aν+1 3   , ν + , λ + μ; − xλ− 2 ν− 2 (1 − x)μ−1 Hν a x dx = ν √ 1, λ; 2F 3 2 2 4 2 π Γ ν + 32 0 [Re λ > 0,

Re μ > 0]

ET II 199(89)a

6.82 Combinations of Struve functions, exponentials, and powers 6.821  1.6 0

∞

 −n− 12  1 1  e−αx H−n− 12 (βx) dx = (−1)n β n+ 2 α + α2 + β 2 2 α + β2 [Re α > |Im β|]

ET I 206(6)

6.831

Struve and trigonometric functions

 2.

∞

6 0



 −n− 12  1 1  e−αx L−n− 12 (βx) dx = β n+ 2 α + α2 − β 2 2 α − β2 √

∞

755

2 e−αx H0 (βx) dx = π

α2 +β 2 +β α

ln

[Re α > |Re β|]



ET I 208(26)

 [Re α > |Im β|] ET II 205(1) α2 + β 2  β  ∞ arcsin α 2 −αx 4. e L0 (βx) dx =  [Re α > |Re β|] ET II 207(18) π α2 + β 2 0   ∞ a  a , + a a π cosec(νπ) sinh I ν+ 12 − cosh I −ν− 12 6.822 e(ν+1)x Hν (a sinh x) dx = a 2 2 2 2 0 [Re a > 0, −2 < Re ν < 0] 3.

0

ET II 385(11)

6.823



∞

λ −αx

x e

1. 0





λ+ν+3 3 3 b2 bν+1 Γ(λ + ν + 2) λ+ν

3 F 2 1, + 1, ; ,ν + ;− 2 Hν (bx) dx = √ 3 2 2 2 2 a 2ν aλ+ν+2 π Γ ν + 2 [Re a > 0, b > 0, Re(λ + ν) > −2] ET II 161(19)

 ν

∞

2.

ν −αx

x e 0

  β

Γ(2ν + 1) α (2β)ν Γ ν + 12 β −ν− 12 Lν (βx) dx = P −ν− 1 2ν+1 −  √  2 1 1   2 α π ν+ π α − β2 α β 2 − α2 2 4 2   Re α > |Re β|, Re ν > − 12 ET I 209(35)a

6.824  1.

∞

ν −at

t e

 √ L2ν 2 t dt =

0

 2. 6.825

∞

√ t dt =

1 a2ν+1

1 a

e Φ

1 √ a

MI 51





1 1 1 1  − 2ν, ea γ MI 51 2ν+1 2 a Γ 2 − 2ν a 0  

 ∞ β ν+1 Γ 12 + 2s + ν2 2 2 3 β2 ν+s+1 3   ; , ν + ; − xs−1 e−α x Hν (βx) dx = ν+1 √ ν+s+1 1, 2F 2 2 2 2 4α2 2 πα Γ ν + 32+ 0 π, Re s > − Re ν − 1, |arg α| < 4

tν e−at L−2ν

1

ET I 335(51)a, ET II 162(20)

6.83 Combinations of Struve and trigonometric functions 

6.831

∞

x−ν sin(ax) Hν (bx) dx = 0

0

=

√

 π2

−ν −ν

b

ν− 12 b 2 − a2   Γ ν + 12

 0 < b < a,

Re ν > − 12

 0 < a < b,

Re ν > − 12

  ET II 162(21)

756

Functions Generated by Bessel Functions



∞√

6.832 0

6.832

2

√  2 2 √ a a −3/2 x sin(ax) H 14 b x dx = −2 π 2 Y 14 b 4b2 [a > 0]

ET I 109(14)

6.84–6.85 Combinations of Struve and Bessel functions 

∞

6.841



Hν−1 (ax) Y ν (bx) dx = −aν−1 b−ν



0

=0 

∞

6.842 0

6.843  1.

∞

0



∞

2. 0

4 K [H0 (ax) − Y 0 (ax)] J 0 (bx) dx = π(a + b)

 √  1 J 2ν a x Hν (bx) dx = − Y ν b



a2 4b



|Re ν| <

1 2

0 < a < b,

|Re ν| <

1 2

|a − b| a+b [a > 0,



 √  2ν Γ(ν + 1) S −ν−1,ν K 2ν 2a x Hν (bx) dx = πb

0 < b < a,



a > 0,

  ET II 114(36)

b > 0] b > 0,

ET II 15(22)

−1 < Re ν <

5 4



ET II 164(10)

2

a b

[Re a > 0,

b > 0,

Re ν > −1] ET II 168(27)



∞

6.844 0

6.845  1.

∞

0





   √   √   √  μ−ν μ−ν π J μ a x − sin π Y μ a x K μ a x Hν (bx) dx cos 2 2 2

2

a a 1 = 2 W 12 ν, 12 μ W − 12 ν, 12 μ a 2b 2b + , π |arg a| < , b > 0, Re ν > |Re μ| − 2 ET II 169(35) 4 + a  a ,  √ 4 cos(νπ) K 2ν 2 ab H−ν − Y −ν J ν (bx) dx = x x πb  |arg a| < π,

b > 0,

|Re ν| <

1 2



ET II 73(7)

∞

2. 0

 2

2    √  √  a a 1 2 K 2ν 2a b − Y 2ν 2a b J −ν + sin(νπ) Hν Hν (bx) dx = x x b π  a > 0, b > 0, − 32 < Re ν < 0 ∞

 6.846 0



∞

6.847 0





ET II 170(39)

2

 √   √  2 a 1 K 2ν 2a x + Y 2ν 2a x Hν (bx) dx = J ν π b  b a > 0,

b > 0,

|Re ν| <

, dx + νπ π νπ J ν (ax) + sin Hν (ax) 2 [I ν (ak) − Lν (ak)] = cos 2 2 2 x +k 2k  a > 0, Re k > 0,

1 2

 ET II 169(30)

− 12 < Re ν < 2



ET II 384(5)a, WA 467(8)

6.852

Combinations of Struve and Bessel functions

6.848  1.

∞

x [I ν (ax) − L−ν (ax)] J ν (bx) dx =

0



2  a ν−1 1 cos(νπ) 2 2 π b a + b Re a > 0,

757

b > 0,

−1 < Re ν < − 12



ET II 74(12) ∞

2.

x [H−ν (ax) − Y −ν (ax)] J ν (bx) dx = 2

0

cos(νπ) ν−1 1 b aν π a+ b |arg a| < π,

− 12 < Re ν,

b>0



ET II 73(5)

6.849  1.

∞

x K ν (ax) Hν (bx) dx = a−ν−1 bν+1

0



∞

2. 0

6.851  1. 0

a2



1 + b2

Re a > 0,

Re ν > − 32

  2μ + (z − b)2μ 2 −μ−1 −2μ (z + b) sec(μπ), x [K μ (ax)] H0 (bx) dx = −2 πa bz    z = 4a2 + b2 Re a > 0, b > 0, |Re μ| < 32

+ ,2 + x J 12 ν (ax) − Y

∞

,2  1 (ax) Hν (bx) dx 2ν =0 1 4 √ = 2 πb b − 4a2



b > 0,

 



ET II 164(12)

ET II 166(18)

0 < b < 2a,

− 32 < Re ν < 0

0 < 2a < b,

− 32 < Re ν < 0

 

ET II 164(7) ∞

2.

  2 2 xν+1 [J ν (ax)] − [Y ν (ax)] Hν (bx) dx

0



=0 −ν− 12 23ν+2 a2ν b−ν−1  2 1  b − 4a2 = √ πΓ 2 −ν



0 < b < 2a,

− 34 < Re ν < 0

0 < 2a < b,

− 34 < Re ν < 0

 

ET II 163(6)

6.852  1.

∞

x1−μ−ν J ν (x) Hμ (x) dx =

0

 2.

(2ν − 1)2−μ−ν     (μ + ν − 1) Γ μ + 12 Γ ν + 12 Re ν > 12 ,

Re(μ + ν) > 1

 ET II 383(4)

∞

xμ−ν+1 Y μ (ax) Hν (bx) dx

0

=0 =

ν−μ−1 21+μ−ν aμ b−ν  2 b − a2 Γ(ν − μ)

 

0 < b < a,

Re(ν − μ) > 0,

− 32 < Re μ <

1 2

0 < a < b,

Re(ν − μ) > 0,

− 32 < Re μ <

1 2

 

ET II 163(3)

758

Functions Generated by Bessel Functions



∞

3.

μ+ν+1

x 0

6.853  1.

∞



b2 2μ+ν+1 bν+1 3 3 3 K μ (ax) Hν (bx) dx = √ μ+2ν+3 Γ μ + ν + F 1, μ + ν + ; ; − 2 2 2 2 a πa  Re a > 0, b > 0, Re ν > − 32 , Re(μ + ν) > − 32 ET II 165(13)

x1−μ [sin (μπ) J μ+ν (ax) + cos(μπ) Y μ+ν (ax)] Hν (bx) dx

0



=0

 μ−1 b ν b 2 − a2 = μ−1 μ+ν 2 a Γ(μ) 



0 < b < a,

1 < Re μ < 32 ,

Re ν > − 32 ,

Re(ν − μ) <

1 2

0 < a < b,

1 < Re μ < 32 ,

Re ν > − 32 ,

Re(ν − μ) <

1 2

 

ET II 163(4) ∞

2.

1

xλ+ 2 [I μ (ax) − L−μ (ax)] J ν (bx) dx

! ⎞ !1+μ μ μ ! , 1 − , 1 + 2 2 1 cos(μπ) 3 ⎜ b2 ! 2 ⎟ b−λ− 2 G 22 = 2λ+ 2 ⎠ 33 ⎝ 2 ! 3 π a ! + λ+ν , 1+μ , 3 + λ−ν ! 2 2 2 4  4 Re(μ + ν + λ) > − 32 , − Re ν − 52 < Re(λ − μ) < 1 ET II 76(21)

0

 

6.853

∞

3.

Re a > 0,

b > 0,

⎛

1

xλ+ 2 [Hμ (ax) − Y μ (ax)] J ν (bx) dx

0

! ⎞ !1−μ μ μ ! , 1 − , 1 + 2 2 1 cos(μπ) 3 ⎜b ! 2 ⎟ b−λ− 2 G 23 = 2λ+ 2 ⎠ 33 ⎝ 2 ! 3 π2 a ! + λ+ν , 1−μ , 3 + λ−ν ! 2 2 4 2 4  |arg a| < π, Re(λ + μ) < 1, Re(λ + ν) + 32 > |Re μ| ET II 73(6) ⎛

2

 b > 0,  4.

∞√

0

 0

∞

6.

μ−ν+1

∞

xH

1. 0



1 b2 a b 2 1 μ−ν+1   x [I μ (ax) − Lμ (ax)] J ν (bx) dx = √ F 1, ; ν − μ + ; − 2 2 2 a π Γ ν − μ + 12   1 −1 < 2 Re μ + 1 < Re ν + 2 , Re a > 0, b > 0 ET II 74(13) x



1 2

μ−ν+1 μ−1 ν−2μ−1

0

6.854

|Re ν| <

ET II 74(11)

∞

5.



 , 2 ν− 1 −ν 1 a 2b √ x I ν− 12 (ax) − Lν− 12 (ax) J ν (bx) dx = 2 π a + b2  Re a > 0, b > 0, +

 1 2ν

1 b2 2μ−ν+1 a−μ−1 bν−1 1    F 1, + μ; + ν; − 2 [I μ (ax) − L−μ (ax)] J ν (bx) dx =  1 2 2 a Γ 2 − μ Γ 12 + ν   1 Re a > 0, Re ν > − 2 , Re μ > −1, b > 0 ET II 75(18)

2

ax



K ν (bx) dx =

Γ

1

2ν + 1 1 1− 2 2 ν aπ



S

− 12 ν−1, 12 ν

b2 4a [a > 0,

Re b > 0,

Re ν > −2] ET II 150(75)

6.857

Combinations of Struve and Bessel functions



∞



2

x H 12 ν ax

2. 0



1 J ν (bx) dx = − Y 2a

1 2ν

b2 4a





a > 0,

759

b > 0,

−2 < Re ν <

3 2



ET II 73(3)

6.855



∞

1.

2ν+ 12

x 0



ET II 76(22)

+ a  a ,  √ 4 dx H−ν−1 − Y −ν−1 J ν (bx) = − √ cos(νπ) K −2ν−1 2 ab x x x π ab   |arg a| < π, b > 0, |Re ν| < 12

∞

2. 0



1 + a  a , √ √ 3 aν+ 2 2 2ab K 2ν+1 2ab I ν+ 12 − Lν+ 12 J ν (bx) dx = 2 √ ν+1 J 2ν+1 x x πb  Re a > 0, b > 0, −1 < Re ν < 12

ET II 74(8) ∞

3.

1

x2ν+ 2

+ a  a , Hν+ 12 − Y ν+ 12 J ν (bx) dx x x

√ √ 1 1 1 = −25/2 π −3/2 aν+ 2 b−ν−1 sin(νπ) K 2ν+1 2abe 4 πi K 2ν+1 2abe− 4 πi   |arg a| < π, b > 0, −1 < Re ν < − 16 ET II 74(9)

0



∞

6.856 0

2

 √   √  1 a x Y ν a x K ν a x Hν (bx) dx = 2 exp − 2b 2b+ b > 0,

|arg a| <

π , 4

Re ν > − 32

,

ET II 169(32)

6.857





∞

1.

x exp 0

a2 x2 8



K 12 ν

k = 14 ν,  2. 0

∞

a2 x2 8

m=

1 2

Hν (bx) dx

2

2

 νπ 1

b b 2 − ν −1 ν −1 2 2 Γ − ν exp =√ a b cos W k,m 2 2 2a2  a2 π  + 14 ν |arg a| < 34 π, b > 0, − 32 < Re ν < 0 ET II 167(24)





1 2 2 1 a x Hν (bx) dx xσ−2 exp − a2 x2 K μ 2 2     √ + μ Γ ν+σ −μ π −ν−σ ν+1 Γ ν+σ 2 2       b = ν+2 a 2 Γ 32 Γ ν + 32 Γ ν+σ 2

ν+σ 3 3 ν+σ b2 ν+σ + μ, − μ; , ν + , ;− 2 × 3 F 3 1, 2 2 2 2 , 2 4a + π b > 0, |arg a| < , Re(σ + ν) > 2|Re μ| ET II 167(23) 4

760

Functions Generated by Bessel Functions

6.861

6.86 Lommel functions 6.861



∞

1.

λ−1

x

S μ,ν (x) dx =

Γ

1

2 (1

0

6.862



u

1.

       + λ + μ) Γ 12 (1 − λ − μ) Γ 12 (1 + μ + ν) Γ 12 (1 + μ − ν)     22−λ−μ Γ 12 (ν − λ) + 1 Γ 1 − 12 (λ + ν)   ET II 385(17) − Re μ < Re λ + 1 < 52

 √  1 1 xλ− 2 μ− 2 (u − x)σ−1 s μ,ν a x dx

0

aμ+1 uλ+σ Γ(λ + 1) (μ − ν + 1) (μ + ν + 1) Γ(λ + σ + 1)

a2 u μ−ν+3 μ+ν+3 , , λ + σ + 1; − × 2 F 3 1, 1 + λ; 2 2 4 [Re λ > −1, Re σ > 0] ET II 199(92)   1 1  ∞  √   √  B μ, 12 (1 − λ − ν) − μ u 2 μ+ 2 ν 1 ν μ−1 2 x (x − u) s λ,ν a x dx = S λ+μ,μ+ν a u μ a u   √ ! ! !arg a u ! < π, 0 < 2 Re μ < 1 − Re(λ + ν) ET II 211(71) = Γ(σ)

2.

 6.863

∞√

0

 6.864 0

6.865

xe−αx s μ, 14



x2 2

∞



2

√ α 3 dx = 2−2μ−1 α Γ 2μ + S −μ−1, 14 2  2  Re α > 0, Re μ > − 34

ET I 209(38)

exp[(μ + 1)x] s μ,ν (a sinh x) dx = 2μ−2 π cosec(μπ) Γ() Γ(σ) + a a a  a , × I Iσ − I − I −σ 2 2 2 2 2 = μ + ν + 1, 2σ = μ − ν + 1 [a > 0, −2 < Re μ < 0] ET II 386(22)    ∞√ μ−ν 1 B 14 − μ+ν 2 , 4 − 2 sinh x cosh(νx) S μ, 12 (a cosh x) dx = S μ+ 12 ,ν (a) √ μ+ 3 a2 2 0  |arg a| < π, Re μ + |Re ν| < 12 ET II 388(31)

6.866



∞

1.

x−μ−1 cos(ax) s μ,ν (x) dx

0

=0 =2  2. 0

∞

μ− 12

√ πΓ



μ+ν+1 2



 1 μ+ 1 μ− 1 μ−ν+1  1 − a2 2 4 P ν− 12 (a) Γ 2 2

[a > 1] [0 < a < 1] ET II 386(18)



 1 μ− 1 μ− 1 √ μ+ν μ−ν  2 −μ −μ− 12 a − 1 2 4 P ν− 12 (a) x sin(ax) S μ,ν (x) dx = 2 πΓ 1− Γ 1− 2 2 2 [a > 1, Re μ < 1 − |Re ν|]

ET II 387(23)

6.871

6.867

Thomson functions



761

π/2

1.

cos(2μx) S 2μ−1,2ν (a cos x) dx 0

a a a  a , π22μ−3 a2μ cosec(2νπ) + J μ+ν Y μ−ν − J μ−ν Y μ+ν Γ(1 − μ − ν) Γ (1 − μ + ν) 2 2 2 2 [Re μ > −2, |Re ν| < 1] ET II 388(29)  π/2 a a cos [(μ + 1) x] s μ,ν (a cos x) dx = 2μ−2 π Γ() Γ(σ) J  Jσ 2 2 0 2 = μ + ν + 1, 2σ = μ − ν + 1 [Re μ > −2] ET II 386(21) =

2.



π/2

6.868 0

6.869



∞

1.

1−μ−ν

x 0



 π  π cos(2μx) π22μ−1 S 2μ,2ν (a sec x) dx = W μ,ν aei 2 W μ,ν ae−i 2 cos x a [|arg a| < π, Re μ < 1]

√ ν−1  12 (μ+ν−1) μ+ν−1 πa Γ(1 − μ − ν)  2   a J ν (ax) S μ,−μ−2ν (x) dx = − 1 P μ+ν (a) 1 2μ+2ν Γ ν + 2   a > 1, Re ν > − 12 , Re(μ + ν) < 1 ET II 388(28)

∞

2. 0

x−μ J ν (ax) s ν+μ,−ν+μ+1 (x) dx  μ = 2ν−1 Γ(ν)a−ν 1 − a2 =0



ET II 388(30)

 

0 < a < 1, 1 < a,

Re μ > −1,

Re μ > −1,

−1e < Re ν <  −1 < Re ν < 32

3 2



ET II 388(28)

∞

3. 0





2

 2 b 1 1 1 Γ μ + ν + 1 Γ μ − ν + 1 S −μ−1, 12 ν x K ν (bx) s μ, 12 ν ax dx = 4a 2 2 4a   Re μ > 12 |Re ν| − 2, a > 0, Re b > 0 ET II 151(78)

6.87 Thomson functions 6.871

1/2  β4 + 1 + β2  e−βx ber x dx = 2 (β 4 + 1) 0 1/2   ∞ β4 + 1 − β2  e−βx bei x dx = 2 (β 4 + 1) 0 

1.

2.

∞

ME 40

ME 40

762

6.872

Functions Generated by Bessel Functions



∞

1. 0



∞

2. 0

 3. 4. 5. 6. 7.

3. 4.

⎡

MI 49

1 3ν + π 2β 4 ⎤



1 3ν + 6 ⎦ 1 − J 12 (ν+1) + π sin 2β 2β 4

 √  1 e−βx beiν 2 x dx = 2β



π⎣ J 12 (ν−1) β



1 2β





sin

 √  1 1 e−βx ber 2 x dx = cos β β 0  ∞  √  1 1 e−βx bei 2 x dx = sin β β 0    ∞   √ 1 1 1 1 1 e−βx ker 2 x dx = − cos ci + sin si 2β β β β β 0    ∞  √  1 1 1 1 1 −βx e kei 2 x dx = − sin ci − cos si 2β β β β β 0



 ∞  √   √  2 3νπ 2 1 Jν + e−βx berν 2 x beiν 2 x dx = sin 2β β β 2 0

0

2.



MI 49

6.873

1.

⎡



1 1 3νπ π⎣ J 12 (ν−1) + cos β 2β 2β 4 ⎤



1 1 3ν + 6 ⎦ + π − J 12 (ν+1) cos 2β 2β 4

 √  1 e−βx berν 2 x dx = 2β

∞



6.874

∞



 2 √   √  2 1 berν 2 x + bei2ν 2 x e−βx dx = I ν β β

ME 40 ME 40 MI 50 MI 50

[Re ν > −1]

MI 49

[Re ν > −1]

ME 40





  √ 1 1 3π 3νπ e π √ ber2ν 2 2x dx = Jν − + cos β β β 4 2 x 0   Re ν > − 12



  ∞ −βx  √ 1 1 3π 3νπ e π √ bei2ν 2 2x dx = Jν − + sin β β β 4 2 x 0   Re ν > − 12

 ∞ √  −βx ν 1 3νπ 2−ν 2 + x berν x e dx = 1+ν cos [Re ν > −1] β 4β 4 0

 ∞ √  −βx ν 1 3νπ 2−ν + x 2 beiν x e dx = 1+ν sin [Re ν > −1] β 4β 4 0 

6.872

∞ −βx

MI 49

MI 49 ME 40 ME 40

6.921

6.875 1. 2. 6.876 1. 2.

Mathieu, hyperbolic, and trigonometric functions

763

     √   √  1 π 1 1 1 e ker 2 x − ln x ber 2 x dx = ln β cos + sin 2 β β 4 β 0      ∞     √ √ π 1 1 1 1 e−βx kei 2 x − ln x bei 2 x dx = ln β sin − cos 2 β β 4 β 0 



∞

∞

−βx

1 arctan a2 2a 0  ∞  1 ln (1 + a4 ) x ker x J 1 (ax) dx = 2a 0 x kei x J 1 (ax) dx = −

MI 50 MI 50

[a > 0]

ET II 21(32)

[a > 0]

ET II 21(33)

6.9 Mathieu Functions (m)

Notation: k 2 = q. For definition of the coefficients Ap

(m)

and Bp

, see section 8.6.

6.91 Mathieu functions 6.911



2π

1.

[m = p]

cem (z, q) cep (z, q) dz = 0

MA

0



2π

2.

∞ + ,2 ,2 +  (2n) (2n) 2 A2r [ce2n (z, q)] dz = 2π A0 +π =π

0



2π

3.

2

[ce2n+1 (z, q)] dz = π 0



MA

r=1 ∞ + ,2  (2n+1) A2r+1 =π

MA

r=0 2π

4.

[m = p]

sem (z, q) sep (z, q) dz = 0

MA

0



2π

5.

2

[se2n+1 (z, q)] dz = π 0



MA

r=0 2π

6.

2

[se2n+2 (z, q)] dz = π 0



∞ + ,2  (2n+1) =π B2r+1 ∞ + ,2  (2n+2) B2r+2 =π

MA

r=0 2π

7.

sem (z, q) cep (z, q) dz = 0

[m = 1, 2, . . . ;

p = 1, 2, . . .]

MA

0

6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions 6.921



(2n)

π

cosh (2k cos u sinh z) ce2n (u, q) du =

1. 0

πA0   (−1)n Ce2n (z, −q) ce2n π2 , q [q > 0]

MA

764

Mathieu Functions



(2n)

π

2.

6.922

cosh (2k sin u cosh z) ce2n (u, q) du = 0

πA0 (−1)n Ce2n (z, −q) ce2n (0, q) [q > 0]



(2n+1)

π

3.

MA

sinh (2k sin u cosh z) se2n+1 (u, q) du = 0

πkB1 (−1)n Ce2n+1 (z, −q) se2n+1 (0, q) [q > 0]



(2n+1)

π

4.

MA

sinh (2k cos u sinh z) ce2n+1 (u, q) du = 0

πkA1   (−1)n+1 Se2n+1 (z, −q) ce2n+1 π2 , q [q > 0]



(2n+1)

π

5.

sinh (2k sin u sin z) se2n+1 (u, q) du = 0

MA

πkB1 se2n+1 (z, q) se2n+1 (0, q) [q > 0]

6.922



(2n+1)

π

πA1 Ce2n+1 (z, q) 2 ce2n+1 (0, q)

cos u cosh z cos (2k sin u sinh z) ce2n+1 (u, q) du =

1. 0



[q > 0] π

2.

sin u sinh z cos (2k cos u cosh z) se2n+1 (u, q) du = 0



MA

π

3.

sin u sinh z sin (2k cos u cosh z) se2n+2 (u, q) du = 0

MA

πB1 (2n+1)  π Se2n+1 (z, q) ,q 2 se2n+1 2 [q > 0] (2n+2) πkB2   − 2 se2n+2 π2 , q

Se2n+2 (z, q)

[q > 0] 

cos u cosh z sin (2k sin u sinh z) se2n+2 (u, q) du = 0



MA

(2n+2)

π

4.

πkB2 Se2n+2 (z, q) 2 se2n+2 (0, q) [q > 0]

π

5.

sin u cosh z cosh (2k cos u sinh z) se2n+1 (u, q) du =

πB1

MA

(2n+1)

2 se2n+1

0

π 2

,q

(−1)n Ce2n+1 (z, −q)

[q > 0] 

π

6.

cos u sinh z cosh (2k sin u cosh z) ce2n+1 (u, q) du = 0

MA

(2n+1) πA1

2 ce2n+1 (0, q)

(−1)n Se2n+1 (z, −q)

[q > 0]  7.

π

sin u cosh z sinh (2k cos u sinh z) se2n+2 (u, q) du = 0

MA

MA

(2n+2) πkB2  π (−1)n+1  ,q 2 se2n+2

Se2n+2 (z, −q)

2

[q > 0]

MA

6.924

Mathieu, hyperbolic, and trigonometric functions



(2n+2)

π

8.

cos u sinh z sinh (2k sin u cosh z) se2n+2 (u, q) du = 0

πkB2 (−1)n Se2n+2 (z, −q) 2 se2n+2 (0, q) [q > 0]

6.923



∞

1.

[q > 0] ∞

2.

cos (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −

0

 3.

[q > 0] ∞

4.

cos (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −

0

∞

5.

πA0   Ce2n (z, q) 2 ce2n 12 π, q

cos (2k cosh z cosh u) Ce2n (u, q) du =

(2n) πA0 1  − 2 ce2n 2 π, q

[q > 0] ∞

6. 0



∞

kπA1   Fey2n+1 (z, q) 2 ce2n+1 12 π, q

cos (2k cosh z cosh u) Ce2n+1 (u, q) du =

(2n+1) kπA1   2 ce2n+1 12 π, q

[q > 0] ∞

8. 0



cos (2k cos u cos z) ce2n (u, q) du = 0

πA0 1

ce2n

2 π, q

 ce2n (z, q) [q > 0]

 0

MA

(2n+1)

π

sin (2k cos u cos z) ce2n+1 (u, q) du = −

2.

MA

(2n)

π

1.

MA

Ce2n+1 (z, q)

[q > 0] 6.924

MA

(2n+1)

sin (2k cosh z cosh u) Ce2n+1 (u, q) du = 0



MA

Fey2n (z, q)

[q > 0] 7.

MA

(2n)

sin (2k cosh z cosh u) Ce2n (u, q) du = 0



MA

kπB2 (2n+2)   Se2n+2 (z, q) 4 se2n+2 12 π, q

[q > 0] 

MA

kπB2 (2n+2)   Gey2n+2 (z, q) 4 se2n+2 12 π, q

sin (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −

0



MA

πB1 (2n+1)   Gey2n+1 (z, q) 4 se2n+1 12 π, q

[q > 0] ∞

MA

πB1 (2n+1)   Se2n+1 (z, q) 4 se2n+1 12 π, q

sin (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −

0



765

πkA1   ce2n+1 (z, q) ce2n+1 12 π, q [q > 0]

MA

766

Mathieu Functions



(2n)

π

3.

6.925

cos (2k cos u cosh z) ce2n (u, q) du = 0

πA0 1

ce2n

2 π, q

 Ce2n (z, q) [q > 0]



(2n)

π

4.

cos (2k sin u sinh z) ce2n (u, q) du = 0

πA0 Ce2n (z, q) ce2n (0, q) [q > 0]



sin (2k cos u cosh z) ce2n+1 (u, q) du = − 0

πkA1   Ce2n+1 (z, q) ce2n+1 12 π, q [q > 0]



π

6.

sin (2k sin u sinh z) se2n+1 (u, q) du = 0

(2n+1) πkB1 se2n+1 (0, q)

Notation: z1 = 2k 

MA

Se2n+1 (z, q) [q > 0]

6.925

MA

(2n+1)

π

5.

MA

MA

 cosh2 ξ − sin2 η, and tan α = tanh ξ tan η

2π

sin [z1 cos(θ − α)] ce2n (θ, q) dθ = 0.

1.

MA

0



(2n)

2π

cos [z1 cos(θ − α)] ce2n (θ, q) dθ =

2. 0



2πA0   Ce2n (ξ, q) ce2n (η, q) ce2n (0, q) ce2n 12 π, q (2n+1)

2π

sin [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = −

3.

MA

0

2πkA1   Ce2n+1 (ξ, q) ce2n+1 (η, q) ce2n+1 (0, q) ce2n+1 12 π, q MA



2π

cos [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = 0

4.

MA

0



(2n+1)

2π

sin [z1 cos(θ − α)] se2n+1 (θ, q) dθ =

5. 0

2πkB1   Se2n+1 (ξ, q) se2n+1 (η, q) se2n+1 (0, q) se2n+1 12 π, q MA



2π

6.

cos [z1 cos(θ − α)] se2n+1 (θ, q) dθ = 0

MA

sin [z1 cos(θ − α)] se2n+2 (θ, q) dθ = 0

MA

0



2π

7. 0



2π

cos [z1 cos(θ − α)] se2n+2 (θ, q) dθ =

8. 0

2πk 2 B2 (2n+2)   Se2n+2 (ξ, q) se2n+2 (η, q) se2n+2 (0, q) se2n+2 12 π, q MA



(2n+2)

π

sin u sin z sin (2k cos u cos z) se2n+2 (u, q) du = −

6.926 0

πkB2   se2n+2 (z, q) 2 se2n+2 π2 , q [q > 0]

MA

6.941

Eigenfunctions of Helmholtz equation

767

6.93 Combinations of Mathieu and Bessel functions 6.931

,2 + (2n) π A0  π ce2n (z, q) 0 ,q ce2n (0, q) ce2n 2 + ,2 (2n)  2π   2π A0 1/2  π Fey2n (z, q) ce2n (u, q) du = Y 0 k [2 (cos 2u + cosh 2z)] 0 ,q ce2n (0, q) ce2n 2 

1.

2.

π

  1/2 ce2n (u, q) du = J 0 k [2 (cos 2u + cos 2z)]

MA

MA

6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems Notation: Particular solutions of the Helmholtz equation in three-dimensional infinite space ∇2 Ψ + k 2 Ψ = 0 in Cartesian (x, y, z), spherical (r, θ, φ), and cylindrical (ρ, z, φ) coordinates are Ψkx ky kz (x, y, z) ∝ ei(kx x+ky y+kz z) with k 2 = kx2 + ky2 + kz2  k imφ Z l+1/2 (kr) P m Ψlm (r, θ, φ) ∝ e l (cos θ) r   Ψmkz (ρ, z, φ) ∝ ei(mφ+kz z) Z l+1/2 ρ k 2 − kz2 with P m l (cos θ) the associated Legendre function, Z is any Bessel function, m = 0, 1, . . . , l; l ∈ N, r2 = ρ2 + z 2 , ρ = r sin θ, z = r cos θ, φ = arccot(x/y), and kt2 = k 2 − kz2 . 6.941



k

1.

e −k



∞

2. −∞

 3. 0

∞

iρz

    z 2πk m p l−m 2 2 dp = i J l+1/2 (kr) P m J m ρ k − ρ Pl l k r r

e−iρz J l+1/2 (kr) P m l

z r

 dz = im−l

[ρ > 0, l ≥ m ≥ 0]  ρ  2πr J m ρ k 2 − ρ2 P m l k k

[ρ > 0, l ≥ m ≥ 0]   x J m (ρkt ) cos kx x + m arcsin dx ρ  <  kx (−1)m 2 2 cos y kt − kx + m arccos = 2 2 kt kt − kx =0

 

kx2 < kt2 kx2 > kt2

 

768

Mathieu Functions

  x Y m (ρkt ) cos kx x + m arcsin dx 0 ρ <  m kx (−1) 2 2 sin y kt − kx + m arccos = 2 2 kt kt − kx  <

 |kx | (−1)m 2 2 = exp −y kx − kt − m sign (kx ) arccosh kt kx2 − kt2 

 ∞ z kz 2πr (j)   2 (j) −ikz z m−l 2 Pm ρ e H H l+1/2 (kr) P m dx = i k − k l m l z r k k −∞

 4.

5.

∞

7.

8. 9.

10.

[ρ > 0]

 

kx2 < kt2 kx2 > kt2

 

 The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0. 

 ∞   z kz 2πk (j) (j) m ikz z l−m 2 2 H l+1/2 (kr) P m H m ρ k − kz P l dkz = i e l k r r −∞   The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0. 

 ∞     2  kz 2πr m z −ikz z m−l e J m ρ k 2 − kz2 P m kz < k2 J l+1/2 (kr) P l dz = i l r k k −∞   2 =0 kz > k2 

 k   z kz 2πk m ikz z l−m 2 2 J l+1/2 (kr) P m J m ρ k − kz P l dkz = i e l k r r −k 

 ∞     2  kz 2πr m z m −ikz z m−l 2 2 Y m ρ k − kz P l kz < k2 Y l+1/2 (kr) P l dz = i e r k k −∞ 

   2  kz 2r K m ρ kz2 − k 2 P m kz > k2 = −2im−l l kπ k

 k   kz Y m ρ k 2 − kz2 P m il−m eikz z dkz l k −k

 ∞     m kz 4 1 − cos kz z + 2 π(m − l) P l K m ρ kz2 − k 2 eikz z dkz π k k  z 2πk Y l+1/2 (kr) P m = l r r 

6.

6.941

7.113

Associated Legendre functions

769

7.1–7.2 Associated Legendre Functions 7.11 Associated Legendre functions 

1

P ν (x) dx = sin ϕ P −1 ν (cos ϕ)

7.111

MO 90

cos ϕ

7.112



1

1. −1

= 

2 (n + m)! 2n + 1 (n − m)!

[n = k] SM III 185, WH

1

2. −1



[n = k]

m Pm n (x) P k (x) dx = 0

m m Qm n (x) P k (x) dx = (−1)

1 − (−1)n+k (n + m)! (k − n)(k + n + 1)(n − m)!

1

3. −1

P ν (x) P σ (x) dx 2π sin π(σ − ν) + 4 sin(πν) sin(πσ) [ψ(ν + 1) − ψ(σ + 1)] π 2 (σ − ν) (σ + ν + 1) 2  2 π − 2 (sin πν) ψ (ν + 1)

= 1 2 π ν+ 2

=



EH I 171(18)

1

4. −1

Q ν (x) Q σ (x) dx =

=

[σ + ν + 1 = 0] [σ = ν]

EH I 170(7)

EH I 170(9)a

[ψ(ν + 1) − ψ(σ + 1)] [1 + cos(πσ) cos(νπ)] − π2 sin π(ν − σ) (σ − ν) (σ + ν + 1) [σ + ν + 1 = 0; ν, σ = −1, −2, −3, . . .] 1 2 2π



+

− ψ (ν + 1) 1 + (cos νπ)

2

EH I 170(11)

,

2ν + 1 [ν = σ,

ν = −1, −2, −3, . . .] EH I 170(12)



1

5. −1

P ν (x) Q σ (x) dx =

1 − cos π(σ − ν) − 2π −1 sin(πν) cos(πσ) [ψ(ν + 1) − ψ(σ + 1)] (ν − σ) (ν + σ + 1) [Re ν > 0, Re σ > 0,

=−

sin(2νπ) ψ  (ν + 1) π(2ν + 1)

σ = ν]

EH I 170(13)

[Re ν > 0,

σ = ν]

EH I 171(14)

7.113

Notation: 

    Γ 12 + ν2 Γ 1 + σ2  A = 1 σ  Γ 2 + 2 Γ 1 + ν2

1

P ν (x) P σ (x) dx =

1. 0

πν πσ −1 sin πν A sin πσ 2 cos 2 − A 2 cos 2 1 2 π(σ − ν)(σ + ν + 1)

EH I 171(15)

770

Associated Legendre Functions



1

2.

Q ν (x) Q σ (x) dx =

ψ(ν + 1) − ψ(σ + 1) −



1

P ν (x) Q σ (x) dx = 0

7.114



−1 A−1 cos π(ν−σ) 2 (σ − ν)(σ + ν + 1)

∞

1.

+ ,    π(σ−ν) −1 A − A−1 sin π(σ+ν) A + A sin 2 2 (σ − ν)(σ + ν + 1) [Re ν > 0, Re σ > 0]

0

3.

π 2

7.114

P ν (x) Q σ (x) dx = 1

1 (σ − ν)(σ + ν + 1)

[Re ν > 0,

Re σ > 0]

[Re(σ − ν) > 0,

EH I 171(16) EH I 171(17)

Re(σ + ν) > −1] ET II 324(19)



∞

2.

Q ν (x) Q σ (x) dx = 1



∞

3.

ψ  (ν + 1) 2ν + 1

2

[Q ν (x)] dx = 1



∞

7.115

Q ν (x) dx = 1

ψ(σ + 1) − ψ(ν + 1) (σ − ν)(σ + ν + 1) [Re(ν + σ) > −1;

1 ν(ν + 1)



σ, ν = −1, −2, −3, . . .]

Re ν > − 12



EH I 170(5)

EH I 170(6)

[Re ν > 0]

ET II 324(18)

7.12–7.13 Combinations of associated Legendre functions and powers 

1

7.121

x P ν (x) dx = cos ϕ

7.122



1

1. 0



  − sin ϕ sin ϕPν (cos ϕ) + cos ϕ P 1ν (cos ϕ) (ν − 1)(ν + 2)

MO 90

2

[P m 1 (n + m)! n (x)] dx = 1 − x2 2m (n − m)!

1

[P μν (x)]

2. 0

2

dx Γ(1 + μ + ν) =− 2 1−x 2μ Γ(1 − μ + ν)

[0 < m ≤ n] [Re μ < 0,

MO 74

ν + μ is a positive integer] EH I 172(26)





1

3.

2 P n−ν (x) ν

0

dx n! =− 2 1−x 2(n − ν) Γ(1 − n + 2ν)

[n = 0, 1, 2, . . . ;

Re ν > n] ET II 315(9)



1

7.123 −1



k Pm n (x) P n (x)

dx =0 1 − x2

[0 ≤ m ≤ n,

0 ≤ k ≤ n;

m = k] MO 74

1

7.124 −1

 1m  2 1m m k z − 1 2 Qm xk (z − x)−1 1 − x2 2 P m n (x) dx = (−2) n (z) · z [m ≤ n; k = 0, 1, . . . , n − m; z is in the complex plane with a cut along the interval (−1, 1) on the real axis] ET II 279(26)

7.128

Associated Legendre functions and powers





1

7.125 −1

1 − x2

 12 m

771

m m m −3/2 (k + m)!(l + m)! (n + m)!(s − m)! Pm k (x) P l (x) P n (x) dx = (−1) π  (k − m)!(l − m)!(n  − m)!(s  −  k)!   Γ m + 12 Γ t − k + 12 Γ t − l + 12 Γ t − n + 12   × (s − l)!(s − n)! Γ s + 32 [2s = k + l + n + m and 2t = k + l − n − m are both even l ≥ m, m ≤ k − l − m ≤ n ≤ k + l + m]

ET II 280(32)

7.126  1.

√

1



P ν (x)xσ dx =

0



1

xσ P m ν (x) dx =

2. 0

> −1]

EH I 171(23)

ET II 313(2)

 

π 1/2 22μ−1 Γ 1+σ ν−μ+1 μ+ν μ 3+σ−μ 2   3+σ−μ  3 F 2  ,− , 1 − ; 1 − μ, ;1 x 2 2 2 2 Γ Γ 1−μ 0 2 2 [Re σ > −1, Re μ < 2] ET II 313(3)  ∞ 1   μ xμ−1 Q ν (ax) dx = eμπi Γ(μ)a−μ a2 − 1 2 Q −μ ν (a) 

4.



m+ν+1 m−ν m 3+σ+m , , + 1; m + 1, ;1 2 2 2 2 [Re σ > −1; m = 0, 1, 2, . . .]

× 3F 2

3.

π2−σ−1 Γ(1 + σ)  

[Re σ Γ 1 + 12 σ − 12 ν Γ 12 σ + 12 ν + 32   (−1)m π 1/2 2−2m−1 Γ 1+σ Γ(1 + m + ν) 2 1 1  3 σ m Γ 2 + 2 m Γ 2 + 2 + 2 Γ(1 − m + ν)

1

σ

P μν (x) dx =

1

[|arg(a − 1)| < π,



−1

7.128  1.

1

Re(ν − μ) > −1]

ET II 325(26)

2

1

7.127

Re μ > 0,

(1 + x)σ P ν (x) dx = 1

2σ+1 [Γ(σ + 1)] Γ(σ + ν + 2) Γ(1 + σ − ν)

μ− 1

[Re σ > −1]

ET II 316(15)

μ− 3

(1 − x)− 2 μ (1 + x) 2 2 (z + x) 2 P μν (x) dx −1   1 −1/2 Γ μ − 12 (z − 1)μ− 2 (z + 1) = − 1/2 2μπi π e% Γ (μ + ν)&Γ(μ − ν % − 1)  % %

1/2

1/2 &

1/2 &

1/2 & 1 + z 1+z 1+z 1+z μ−1 μ μ μ−1 × Qν Q −ν−1 + Qν Q −ν−1 2 2 2 2 1

[− 12 < Re μ < 1, z is in the complex plane with a cut along the interval (−1, 1) of the real axis] 

ET II 317(20) 1

2. −1

1

(1 − x)− 2 μ (1 + x) 2 1

μ− 12

μ− 1

(z + x) 2 P μν (x) dx % %  

1/2 &

1/2 & 2e−2μπi Γ 12 + μ 1+z 1+z μ μ μ = 1/2 Q −ν−1 (z − 1) Q ν 2 2 π Γ(μ − ν) Γ (μ + ν + 1) [− 12 < Re μ < 1, z is in the complex plane with a cut along the interval (−1, 1) of the real axis] ET II 316(18)

772

Associated Legendre Functions



4

1

7.129 −1

7.131  1.

∞

7.129

P ν (x) P λ (x)(1 + x)λ+ν dx =

1

(x − 1)− 2 μ (x + 1) 2 1

μ− 12

2λ+ν+1 [Γ(λ + ν + 1)] 2

[Γ(λ + 1) Γ(ν + 1)] Γ(2λ + 2ν + 2) [Re(ν + λ + 1) > 0]

(z + x)

μ− 12

EH I 172(30)

P μν (x) dx

 %

1/2 &2 1+z − ν) Γ (1 − μ + ν) μ μ   Pν =π (z − 1) 2 Γ 12 − μ [Re(μ + ν) < 0, Re(μ − ν) < 1, |arg(z + 1)| < π] ET II 321(6)

1

1/2 Γ(−μ

 2.

∞

1

(x − 1)− 2 μ (x + 1) 2 1

μ− 12

(z + x)

μ− 32

P μν (x) dx

% %

1/2 &

1/2 & 1 −1/2 1+z 1+z π 1/2 Γ(1 − μ − ν) Γ (2 − μ + ν) (z − 1)μ− 2 (z + 1) μ μ−1   = Pν Pν 2 2 Γ 32 − μ [Re μ < 1, Re(μ + ν) < 1, Re(μ − ν) < 2, |arg(1 + z)| < π] ET II 321(7) 1

7.132 1.

2.

    π2μ Γ λ + 12 μ Γ λ − 12 μ         1−x Γ λ + 12 ν + 12 Γ λ − 12 ν Γ − 12 μ + 12 ν + 1 Γ − 12 μ − 12 ν + 12 −1 [2 Re λ > |Re μ|] ET II 316(16)        ∞  2 λ−1 μ 2μ−1 Γ λ − 12 μ Γ 1 − λ + 12 ν Γ 12 − λ − 12 ν      x −1 P n (x) dx =  Γ 1 − 12 μ + 12 ν Γ 12 − 12 μ − 12 ν Γ 1 − λ − 12 μ 1 [Re λ > Re μ, Re(1 − 2λ − ν) > 0, Re(2 − 2λ + ν) > 0] ET II 320(2) 

 3.

1



∞

9

 2 λ−1

P μν (x) dx =

 2 λ−1 μ Γ x −1 Q ν (x) dx = eμπi

1



1

4.

 − 1 μ xσ 1 − x2 2 P μν (x) dx =

0



1

5. 0

 6. 0

1

1 2

       + 12 ν + 12 μ Γ 1 − λ + 12 ν Γ λ + 12 μ Γ λ − 12 μ     22−μ Γ 1 + 12 ν − 12 μ Γ 12 + λ + 12 ν [|Re μ| < 2 Re λ < Re ν + 2] μ−1

2 Γ  1 1 Γ 1 + 2σ − 2ν

1

 

ET II 324(23)



1 1 2 + 2 σ Γ 1 + 2 σ − 12 μ Γ 12 σ + 12 ν − 12 μ

+

3 2



EH I 172(24) [Re μ < 1, Re σ > −1]    1 1  1m (−1)m 2−m−1 Γ 2 + 2 σ Γ 1 + 2 σ Γ (1 + m + ν)     xσ 1 − x2 2 P m ν (x) dx = Γ(1 − m + ν) Γ 1 + 12 σ + 12 m − 12 ν Γ 32 + 12 σ + 12 m + 12 ν [Re σ > −1, m is a positive integer] EH I 172(25), ET II 313(4)

1



 2 η

1−x

P μν (x) dx =

    2μ−1 Γ 1 + η − 12 μ Γ 12 + 12 σ   Γ(1 − μ) Γ 32 + η + 12 σ − 12 μ

ν−μ+1 μ+ν μ 3+σ−μ ,− , 1 + η − ; 1 − μ, + η; 1 × 3F 2 2 2   2 2  Re η − 12 μ > −1, Re σ > −1 ET II 314(6)

7.135

Associated Legendre functions and powers



∞

7.

−ρ

x



x −1 2

− 12 μ

P μν (x) dx

=

[Re μ < 1,



∞

1.

[|arg(u − 1)| < π,

∞

2.



[|arg(u − 1)| < π,

∞

1. 1



∞

2. 1

7.135



1

1. −1

2.

0 < Re μ < 1 + Re ν]

MO 90a

 2  1λ  1 λ+ 1 μ μ−1 x − 1 2 Q −λ dx = Γ(μ)eμπi u2 − 1 2 2 Q ν−λ−μ (u) ν (x)(x − u)

u

7.134

ET II 320(3)

 1μ Q ν (x)(x − u)μ−1 dx = Γ(μ)eμπi u2 − 1 2 Q −μ ν (u)

u



 ρ+μ+ν   ρ+μ−ν−1  Γ 2 √2 π Γ(ρ) Re(ρ + μ + ν) > 0, Re(ρ + μ − ν) > 1]

2ρ+μ−2 Γ

1

7.133

773

0 < Re μ < 1 + Re(ν − λ)]

 1μ 2λ+μ Γ(λ) Γ(−λ − μ − ν) Γ(1 − λ − μ + ν) (x − 1)λ−1 x2 − 1 2 P μν (x) dx = Γ(1 − μ + ν) Γ(−μ − ν) Γ(1 − λ − μ) [Re λ > 0, Re(λ + μ + ν) < 0, Re(λ + μ − ν) < 1]

ET II 204(30)

ET II 321(4)

 − 1 μ 2λ−μ sin πν Γ(λ − μ) Γ(−λ + μ − ν) Γ(1 − λ + μ + ν) (x − 1)λ−1 x2 − 1 2 P μν (x) dx = − π Γ(1 − λ) [Re(λ − μ) > 0, Re(μ − λ − ν) > 0, Re(μ − λ + ν) > −1] ET II 321(5)



1 − x2

− 12 μ

 − 1 μ (z − x)−1 P μμ+n (x) dx = 2e−iμπ z 2 − 1 2 Q μμ+n (z)

[n = 0, 1, 2, . . . , Re μ + n > −1, z is in the complex plane with a cut along the interval (−1, 1) ET II 316(17) of the real axis.]  ∞  μ/2 −ρ (x − 1)λ−1 x2 − 1 (x + z) P μν (x) dx 1

=

2λ+μ−ρ Γ(λ − ρ) Γ(ρ − λ − μ − ν) Γ (ρ − λ − μ + ν + 1) Γ(1 − μ + ν) Γ(−μ − ν) Γ (1 + ρ − λ − μ)

1+z × 3 F 2 ρ, ρ − λ − μ − ν, ρ − λ − μ + ν + 1; ρ − λ + 1, ρ − λ − μ + 1; 2

Γ(ρ − λ) Γ(λ) μ 1+z λ−ρ 2 (z + 1) + λ, −μ − ν, 1 − μ + ν; 1 − μ, 1 − ρ + λ; 3F 2 Γ(ρ) Γ(1 − μ) 2 [Re λ > 0, Re(ρ − λ − μ − ν) > 0, Re(ρ − λ − μ + ν + 1) > 0, |arg(z + 1)| < π] ET II 322(9)

774

Associated Legendre Functions



∞

3.

7.136

 −μ/2 −ρ (x − 1)λ−1 x2 − 1 (x + z) P μν (x) dx

1

=−

sin(νπ) Γ(λ − μ − ρ) Γ (ρ − λ + μ − ν) Γ(ρ − λ + μ + ν + 1) 2ρ−λ+μ π Γ (1 + ρ − λ)

1+z × 3 F 2 ρ, ρ − λ + μ − ν, ρ − λ + μ + ν + 1; 1 + ρ − λ, 1 + ρ − λ + μ; 2 Γ(λ − μ) Γ (ρ − λ + μ) λ−ρ−μ (z + 1) + Γ(ρ) Γ(1 − μ)

1+z × 3 F 2 λ − μ, −ν, ν + 1; 1 + λ − μ − ρ, 1 − μ; 2 [Re(λ − μ) > 0, Re (ρ − λ + μ − ν) > 0, Re(ρ − λ + μ + ν + 1) > 0, |arg(z + 1)| < π] ET II 322(10)

7.136  1.

1



1 − x2

λ−1 

μ/2

P ν (ax) dx



π2μ Γ(λ) μ+ν 1−μ+ν 1 2  1 1    , ; + λ; a − 2F 1 2 2 2 Γ 2 + λ Γ 2 − 2 μ − 12 ν Γ 1 − 12 μ + 12 ν ET II 318(31) [Re λ > 0, −1 < a < 1]  ∞  2 λ−1  2 2 μ/2 μ x −1 a x −1 P ν (ax) dx 1     Γ(λ) Γ 1 − λ − 12 μ + 12 ν Γ 12 − λ − 12 μ − 12 ν     = Γ 1 − 12 μ + 12 ν Γ 12 − 12 ν − 12 μ Γ(1 − λ − μ)

1−μ+ν μ−ν 1 ,1 − λ − ; 1 − λ − μ; 1 − 2 ×2μ−1 aμ−ν−1 2 F 1 2 2 a [Re a > 0, Re λ > 0, Re(ν − μ − 2λ) > −2, Re(2λ + μ + ν) < 1] ET II 325(25) −1

=

2.

1 − a2 x2



∞

3.

1

1

7.137  1.

∞

 μ+ν+1 

  μ−2 μπi −μ−ν−1 Γ(λ) Γ 1 − λ + μ+ν 2 e a 2   3 Γ ν+2

μ+ν+1 μ+ν 3 −2 × 2F 1 ,1 − λ + ;ν + ;a 2 2 2 [|arg(a − 1)| < π, Re λ > 0, Re(2λ − μ − ν) < 2] ET II 325(27)

 2 λ−1  2 2 − 1 μ Γ x −1 a x − 1 2 Q μν (ax) dx =

2

x− 2 μ− 2 (x − 1)−μ− 2 (1 + ax) 2 μ Q μν (1 + 2ax) dx 1

1

1

1

1

  2. 1

∞

1

|arg a| < π,

x− 2 μ− 2 (x − 1)−μ− 2 (1 + ax) 2 Q μν (1 + 2ax) dx 1

1

3

,2  1μ  μ + 1/2 2 − μ a Q (1 + a) ν 2  Re μ < 12 , Re(μ + ν) > −1 ET II 325(28)

= π −1/2 e−μπi Γ

1

μ

, + ,  1 1 −1/2 μ+1 +  1/2 (1 + a) Q μν (1 + a)1/2 = −π −1/2 e−μπi Γ −μ − 12 a 2 μ+ 2 1 + a2 Qν   |arg a| < π, Re μ < − 12 , Re(μ + ν + 2) > 0 ET II 326(29)

7.137

Associated Legendre functions and powers



1

3. 0



1

4.

,2   1  + 1 1 1 1 x− 2 μ− 2 (1 − x)−μ− 2 (1 + ax) 2 μ P μν (1 + 2ax) dx = π 1/2 Γ 12 − μ a 2 μ P μν (1 + a)1/2   Re μ < 12 , |arg a| < π ET II 319(32) 1

x− 2 μ− 2 (1 − x)−μ− 2 (1 + ax) 2 P μν (1 + 2ax) dx 1

1

3

μ

+ ,     1 1 1/2 (1 + a) P μν (1 + a)2 = π 1/2 Γ − 12 − μ a 2 μ+ 2 P μ+1 ν   ET II 319(33) Re μ < − 12 , |arg a| < π

0



1

5.

x 2 μ− 2 (1 − x)μ− 2 (1 + ax)− 2 μ P μν (1 + 2ax) dx 1

1

1

1

0

= π 1/2 Γ 

1

1

1

− 12 μ

3

x 2 μ− 2 (1 − x)μ− 2 (1 + ax)

6. 0

=



1

7.

μ

1 2

 + , + ,  1 1 1 1/2  −1/2 1/2 P 1−μ (1 + a) P μν (1 + a)1/2 π Γ μ − 12 a 2 − 2 μ (1 + a) ν 2 + , + , 1/2 +(μ + ν)(1 − μ + ν) P −μ (1 + a) P μν (1 + a)1/2 ν   Re μ > 12 , |arg a| < π ET II 319(35)

x− 2 − 2 (1 − x)−μ− 2 (1 + ax) 2 μ Q μν (1 + 2ax) dx 1

1

1

= π 1/2 Γ 1

8.

μ

1

3

=



y

0

1

1 2

+ , + ,  1 − μ a 2 μ P μν (1 + a)1/2 Q μν (1 + a)1/2   Re μ < 12 , |arg a| < π ET II 320(38)

x− 2 − 2 (1 − x)−μ− 2 (1 + ax) 2 Q μν (1 + 2ax) dx

0

9.

+ , + ,  1 1/2 (1 + a) + μ a− 2 μ P μν (1 + a)1/2 P −μ ν   Re μ > − 12 , |arg a| < π ET II 319(34)

P μν (1 + 2ax) dx

0



775

μ

 1 1/2  −1/2 12 μ+ 12 π Γ −μ − 12 (1 + a) a 2 + , + , + , + , 1/2 1/2 (1 + a) Q μν (1 + a) + P μν (1 + a)1/2 Q μ+1 (1 + a)1/2 × P μ+1 ν ν   Re μ < − 12 , |arg a| < π ET II 320(39)

  − 12 λ λ (y − x)μ−1 x 1 + 12 γx P ν (1 + γx) dx

 12 μ− 12 λ 12 μ  2 1 P λ−μ (1 + γy) y 1 + γy ν γ 2 [Re λ < 1, Re μ > 0, |arg γy| < π] ET II 193(52)

= Γ(μ)  10. 0

y

 − 1 λ 1 (y − x)μ−1 xσ+ 2 λ−1 1 + 12 γx 2 P λν (1 + γx) dx  γ − 12 λ

Γ(σ) Γ(μ)y σ+μ−1 1 γy = 2 −ν, 1 + ν, σ; 1 − λ, σ + μ; − 3F 2 Γ(1 − λ) Γ(σ + μ) 2 [Re σ > 0, Re μ > 0, |γy| < 1] ET II 193(53)

776

Associated Legendre Functions



y

11.

(y − x)μ−1 [x(1 − x)]− 2 λ P λν (1 − 2x) dx = Γ(μ)[y(1 − y)] 2 μ− 2 λ P λ−μ (1 − 2y) ν 1

1

0

 12.

7.138

1

[Re λ < 1,

Re μ > 0,

0 < y < 1] ET II 193(54)

y

(y − x)μ−1 xσ+ 2 λ−1 (1 − x)− 2 λ P λν (1 − 2x) dx 1

1

0

=



∞

7.138

−μ−ν−2

(a + x)

Pμ

0

a−x a+x



Pν

Γ(μ) Γ(σ)y σ+μ−1 3 F 2 (−ν, 1 + ν, σ; 1 − λ, σ + μ; y) Γ (σ + μ) Γ(1 − λ) [Re σ > 0, Re μ > 0, 0 < y < 1] ET II 193(155)

a−x a+x



a−μ−ν−1 [Γ(μ + ν + 1)]

4

dx =

2

[Γ(μ + 1) Γ(ν + 1)] Γ(2μ + 2ν + 2) [|arg a| < π, Re(μ + ν) > −1] ET II 326(3)

7.14 Combinations of associated Legendre functions, exponentials, and powers 7.141



∞

 1μ e−ax (x − 1)λ−1 x2 − 1 2 P μν (x) dx =

∞

  e−ax (x − 1)λ−1 x2 − 1

1. 1



[Re a > 0,

2.

1 2μ

∞

3.

e

−ax

(x − 1)

λ−1

1



∞

4. 1



Re λ > 0]

ET II 323(13)

Q μν (x) dx

!

! 1 + μ, 1 Γ(ν + μ + 1)eμπi −λ−μ −a 22 ! a e G 23 2a ! = λ + μ, ν + 1, −ν 2 Γ(ν − μ + 1) [Re a > 0, Re λ > 0, Re(λ + μ) > 0] ET II 325(24)

1



!

! 1 + μ, 1 a−λ−μ e−a 31 G 23 2a !! λ + μ, −ν, 1 + ν Γ(1 − μ + ν) Γ(−μ − ν)

!

! 1, 1 − μ  2 − 12 μ μ −1 μ−λ −a 31 ! x −1 P ν (x) dx = −π sin(νπ)a e G 23 2a ! λ − μ, 1 + ν, −ν [Re a > 0, Re(λ − μ) > 0] ET II 323(15)

!

! 1 − μ, 1  2 − 12 μ μ 1 μπi μ−λ −a 22 −ax λ−1 ! x −1 e (x − 1) Q ν (x) dx = e a e G 23 2a ! λ − μ, ν + 1, −ν 2 [Re a > 0, Re λ > 0, Re(λ − μ) > 0]

ET II 323(14) ∞

5. 1

 − 1 μ 1 e−ax x2 − 1 2 P μν (x) dx = 21/2 π −1/2 aμ− 2 K ν+ 12 (a) [Re a > 0,

Re μ < 1] ET II 323(11), MO 90



∞

7.142 1

e− 2 ax 1



x+1 x−1

12 μ

P μν− 1 (x) dx = 2

2 W μ,ν (a) a



Re μ < 1,

ν−

1 2



= 0, ±1, ±2, . . . BU 79(34), MO 118

7.146

7.143

Associated Legendre functions, exponentials, and powers



∞

1

[x(1 + x)]

1.

− 12 μ −βx

e

P μν (1

0



∞

2. 0

7.144  1.

∞

β μ− 2 1 + 2x) dx = √ e 2 β K ν+ 12 π

∞

1



∞

1. 0



∞

2. 0



Re β > 0]

ET I 179(1)

[Re μ < 1,

Re β > 0]

ET I 179(2)

1

sin(νπ) Γ(ν + μ + 1) E (−ν, ν + 1, λ + μ; μ + 1 : 2β) λ+μ Γ(ν − μ + 1) 2β sin(μπ)  sin [(μ + ν) π] − 1−μ λ E (ν − μ + 1, −ν − μ, λ : 1 − μ : 2β) 2 β sin(μπ) [Re β > 0, Re λ > 0, Re (λ + μ) > 0] ET I 181(16) sin(νπ) E (−ν, ν + 1, λ − μ : 1 − μ : 2β) sin(μπ) sin[(μ − ν)π] − 1+μ λ E (μ + ν + 1, μ − ν, λ : 1 + μ : 2β) 2 β sin(μπ) [Re β > 0, Re λ > 0, Re(λ − μ)] > 0 ET I 181(17)

e−βx xλ− 2 μ−1 (x + 2) 2 μ Q μν (1 + x) dx = − 1

0

7.145

[Re μ < 1,

e−βx xλ+ 2 μ−1 (x + 2) 2 μ Q μν (1 + x) dx =





β 2



1μ 1 e2β 1 2 −βx μ W μ,ν+ 12 (β) e P ν (1 + 2x) dx = 1+ x β

0

2.

1

2β λ−μ

  1 e−βx eβ Pν W ν+ 12 ,0 (β) W −ν− 12 ,0 (β) − 1 dx = 1+x (1 + x)2 β

x−1 e−βx Q − 12

[Re β > 0]    2  2   1 1 π2 −2 β β 1 + 2x dx = + Y0 J0 8 2 2 [Re β > 0]

∞

3. 0



∞

1. 0

 2. 0

x− 2 μ e−βx P μν 1

√  β 1 5 1 + x dx = 2μ β 2 μ− 4 e 2 W

1 1 1 1 2 μ+ 4 , 2 ν+ 4

ET II 327(5)

√  β 1 1 3 e−βx x− 2 μ √ 1 + x dx = 2μ β 2 μ− 4 e 2 W P μν 1+x

Re ν > −1]

ET II 327(6)

(β)

[Re μ < 1, ∞

ET I 180(6)

  1 2 x−1 e−ax Q ν 1 + 2x−2 dx = [Γ(ν + 1)] a−1 W −ν− 12 ,0 (ai) W −ν− 12 ,0 (−ai) 2 [Re a > 0,

7.146

777

1 1 1 1 2 μ+ 4 , 2 ν+ 4

Re β > 0]

ET I 180(7)

(β)

[Re μ < 1, Re β > 0]

ET I 180(8)a

778

Associated Legendre Functions



∞√

3.

−βx

xe

P 1/4 ν



1+

x2



P −1/4 ν

0

7.147





  1 1 1 π (1) (2) 2 H 1 β H ν+ 1 β 1 + x dx = 2 2 2β ν+ 2 2 2 [Re β > 0]



∞

7.147

 1ν xλ−1 x2 + a2 2 e−βx P μν

0

%

x (x2

ET I 180(9)

& 1/2

+ a2 )

dx

2−ν−2 aλ+ν 32 = G π Γ(−μ − ν) 24

!  ! 1 − λ , 1−λ ! 2 2 ! 1 λ+μ+ν ! 0, 2 , − 2 , − λ−μ+ν 2 Re β > 0, Re λ > 0] ET II 327(7) 

a2 β 2 4

[a > 0,

1 1 1 1 1 1 1−x y P μν (x) dx = 2ν y 2 μ+ν− 2 e 2 y W 7.148 (1 − x)− 2 μ (1 + x) 2 μ+ν−1 exp − 1+x −1 [Re y > 0]  ∞ +   2 −1/2 1/2 , 2 2 2 α + β + 2αβx 7.149 exp − α + β + 2αβx P ν (x) dx 

1 1 1 2 μ−ν− 2 , 2 μ

(y)

ET II 317(21)

1

= 2π −1 (αβ)−1/2 K ν+ 12 (α) K ν+ 12 (β) [Re α > 0,

Re β > 0]

ET II 323(16)

7.15 Combinations of associated Legendre and hyperbolic functions 7.151



∞

1.

(sinh x)

α−1

P −μ ν

(sinh x)

α−1

Q μν

0



∞

2. 0

      2−1−μ Γ 12 α + 12 μ Γ 12 ν − 12 α + 1 Γ 12 − 12 α − 12 ν       (cosh x) dx = Γ 12 μ + 12 ν + 1 Γ 12 + 12 μ − 12 ν Γ 1 + 12 μ − 12 α [Re(α + μ) > 0, Re(ν − α + 2) > 0, Re(1 − α − ν) > 0] EH I 172(28)

    eiμπ 2μ−α Γ 12 + 12 ν + 12 μ Γ 1 + 12 ν − 12 α     (cosh x) dx = Γ 1 + 12 ν − 12 μ Γ 12 + 12 ν + 12 α     × Γ 12 α + 12 μ Γ 12 α − 12 μ [Re (α ± μ) > 0,



∞

7.152

e 0

−αx

sinh

2μ

Re(ν − α + 2) > 0]

EH I 172(29)

     1  −2μ   1  Γ 2μ + 12 Γ(α − n − μ) Γ α + n − μ + 12   cosh 2 x dx = μ √ 2 x P 2n 1 4 π Γ(α  + n + μ + 1) Γ α − n + μ +1 2 Re α > n + Re μ, Re μ > − 4 ET I 181(15)

7.162

Associated Legendre functions, powers, and trigonometric functions

779

7.16 Combinations of associated Legendre functions, powers, and trigonometric functions 7.161



1

1.

 − 1 μ xλ−1 1 − x2 2 sin(ax) P μν (x) dx

0

=

 2.

1

π 1/2 2μ−λ−1 Γ (λ + 1) a  Γ 3+λ−μ+ν Γ 1 + λ−μ−ν 2 2

1+λ λ 3 λ−μ−ν 3+λ−μ+ν a2 ,1 + ; ,1 + , ;− × 2F 3 2 2 2 2 2 4 [Re λ > −1, Re μ < 1] ET II 314(7) 

 − 1 μ xλ−1 1 − x2 2 cos(ax) P μν (x) dx

0

=



∞

3.

 2 1μ x − 1 2 sin(ax) P μν (x) dx =

0

π 1/2 2μ−λ Γ(λ)



1+λ−μ−ν λ−μ+ν Γ 1+ Γ 2 2

λ λ+1 1 1+λ−μ−ν λ−μ+ν a2 , ; , ,1 + ;− × 2F 3 2 2 2 2 2 4 [Re λ > 0, Re μ < 1] ET II 314(8) 2μ π 1/2 a−μ− 2   S 1 1 (a) 1 − 2 μ − 12 ν Γ 1 − 12 μ + 12 ν μ+ 2 ,ν+ 2  a > 0, Re μ < 32 , Re(μ + ν) < 1 1

Γ

1 2

ET II 320(1)

7.162



∞

2 −2

P ν 2x a

1. a



3. 0

πa − 1 sin(bx) dx = − 4 cos(νπ)

 2  2  ab ab − J −ν− 12 J ν+ 12 2 2 [a > 0, b > 0, −1 < Re ν < 0]

  P ν 2x2 a−2 − 1 cos(bx) dx







 ab ab ab ab π = − a J ν+ 12 J −ν− 12 − Y ν+ 12 Y −ν− 12 4 2 2 2 2 [a > 0, b > 0, −1 < Re ν < 0] ET II 326(2)

a





ET II 326(1) ∞

2.

∞

+ ,2   2  2 −1/2  x +2 x + 1 dx = 2−1/2 π −1 a sin(νπ) K ν+ 12 2−1/2 a sin(ax) P −1 ν

[a > 0, −2 < Re ν < 1] ET I 98(22) ∞    −1/2  x2 + 2 sin(ax) Q 1ν x2 + 1 dx = −2−3/2 πa K ν+ 12 2−1/2 a I ν+ 12 2−1/2 a 0   ET 98(23) a > 0, Re ν > − 32

 4.



780

Associated Legendre Functions

√ 

2   a 2 sin(νπ) K ν+ 12 √ cos(ax) P ν 1 + x2 dx = − π 2 0 [a > 0,



 ∞   a a π 2 I ν+ 12 √ cos(ax) Q ν 1 + x dx = √ K ν+ 12 √ 2 2 2 0 [a > 0,  1     a a π cos(ax) P ν 2x2 − 1 dx = J ν+ 12 J −ν− 12 2 2 2 0 

5.

6.

7.

7.163

∞

−1 < Re ν < 0]

ET I 42(23)

Re ν > −1]

ET I 42(24)

[a > 0] 7.163



∞

1. a

 2. 0

1

 1  2  1 ν− 1  1 π νπ −ν  + x − a2 2 4 sin(bx) P 02 ax−1 dx = b−ν− 2 cos ab − 2 4   a > 0, |Re ν| < 12

7.164 ∞

1.

 ,2 2 x2 x1/2 sin(bx) P −1/4 1 + a dx = ν

x1/2 sin(bx) P −1/4 ν



0

∞

3.

x1/2 sin(bx) P −1/4 ν

0

 4. 0

Γ

−1 < Re ν < 0]

ET II 327(4)

2  b  K ν+ 1 2 2a ν  Re a > 0, b > 0, − 54 < Re ν < 14

2 −1 −1/2 b πa  1 Γ 4− 4 +ν 

5

ET II 327(8) ∞

2.



<

+

0



ET I 98(24)

  1 x−1 cos(ax) P ν 2x−2 − 1 dx = − π cosec(νπ) 1 F 1 ((ν + 1; 1; ai)) 1 F 1 (ν + 1; 1; −ai) 2 [a > 0,



ET I 42(25)

∞

x1/2 sin(bx) P 1/4 ν





 −1/4

1 + a2 x2 Q ν−1



1 + a2 x2 dx   π − 1 πi 



4 Γ ν + 54 b b 2e I ν+ 12 = K ν+ 12   1 3 2a 2a 2   ab Γ ν + 4 ET II 328(9) Re a > 0, b > 0, Re ν > − 54

 dx −1/4 1 + a2 x2 P ν−1 1 + a2 x2 √ 1 + a2 x2



b b a−2 b1/2 5  5  K ν− 12 =√ K ν+ 12 2a 2a   2π Γ 4 + ν Γ 4 − ν 5 ET II 328(10) Re a > 0, b > 0, − 4 < Re ν < 54

  dx 2 x2 √ 1 + a2 x2 P −3/4 1 + a ν 1 + a2 x2 2  b a−2 b1/2 7  3  K ν+ 1 =√ 2 2a 2π Γ 4 + ν Γ 4 − ν   Re a > 0, b > 0, − 74 < Re ν < 34 ET II 328(11)

7.171

Associated Legendre function and probability integral



∞

5.

1/2

x

+ cos(bx)

P 1/4 ν



1+

a2 x2

,2

0

 0

∞

7.

  x1/2 cos(bx) P 1/4 1 + a2 x2 Q 1/4 1 + a2 x2 dx ν ν   π 1 πi 



4 Γ ν + 34 b b 2e   = I ν+ 12 K ν+ 12 1/2 Γ ν + 5 2a 2a ab 4   ET II 328(13) Re a > 0, b > 0, Re ν > − 34 x1/2 cos(bx) P −1/4 ν

0



∞

8. 0



 −1/2 2  a−1 πb b 2   1  K ν+ 1 dx =  3 2 2a Γ 4 + ν Γ − 4 − ν  Re a > 0, b > 0, − 34 < Re ν < − 14 ET II 328(12)

∞

6.



781

x1/2 cos(bx) P 1/4 ν

  dx 1 + a2 x2 P 3/4 1 + a2 x2 √ ν 1 + a2 x2 2  b a−2 b1/2 5  1  K ν+ 1 =√ 2 2a 2π Γ + ν Γ − ν 4 4   Re a > 0, b > 0, − 54 < Re ν < 14 ET II 328(14)

  dx 1/4 1 + a2 x2 P ν−1 1 + a2 x2 √ 1 + a2 x2



b b a−2 b1/2 1 3  3  K ν− 1 =√ K ν+ 2 2 2a 2a 2π Γ   4 +ν Γ 4 −ν ET II 329(15) Re a > 0, b > 0, |Re ν| < 34

∞

7.165

cos(ax) P ν (cosh x) dx









1 + ν − iα ν + iα ν − iα Γ Γ − Γ − 2 2 2 [a > 0, −1 < Re ν < 0] ET II 329(18)      π 2−μ π Γ 12 α + 12 μ Γ 12 α − 12 μ −μ α−1        7.166 P ν (cos ϕ) sin ϕ dϕ =  1 1 Γ 2 + 2 α + 12 ν Γ 12 α − 12 ν Γ 12 μ + 12 ν + 1 Γ 12 μ − 12 ν + 12 0 [Re (α ± μ) > 0] MO 90, EH I 172(27)    η  a η 1 η 2 Γ(μ − η) Γ η + sin(a − x) dx −η 2 (sin a) √ = P −μ 7.167 P −μ (cos x) P [cos(a − x)] ν ν ν (cos a) sin x sin x  π Γ(η + μ + 1) 0 Re μ > Re η > − 12 ET II 329(16) 0

=−

sin(νπ) Γ 4π 2

1 + ν + iα 2

7.17 A combination of an associated Legendre function and the probability integral 

7.171 1

∞

 2  − 1 μ  x − 1 2 exp a2 x2 [1 − Φ(ax)] P μν (x) dx



3 a2 1+μ+ν μ−ν −1 μ−1 =π 2 Γ Γ aμ− 2 e 2 W 2 2 [Re a > 0,

Re μ < 1,

Re (μ + ν) > −1,

1 1 1 1 4 − 2 μ, 4 + 2 ν

 2 a

Re(μ − ν) > 0] ET II 324(17)

782

Associated Legendre Functions

7.181

7.18 Combinations of associated Legendre and Bessel functions 7.181  1.

∞

1



∞

2. 1

7.182



∞

         P ν− 12 (x)x1/2 Y ν (ax) dx = 2−1/2 a−1 cos 12 a J ν 12 a − sin 12 a Y ν 12 a   a > 0, Re ν < 12         1  cos 12 a Y ν 12 a + sin 12 a J ν 12 a P ν− 12 (x)x1/2 J ν (ax) dx = − √ 2a   |Re ν| < 12

ν

x

1.



x −1 2

 12 λ− 12

1



∞

3.

0



1

1

∞

6.

∞

7.

1

    π μ− 1 a 2 J 12 −μ 12 a J ν+ 12 12 a 2 [Re μ < 1, Re(μ − ν) < 2]

 1 −μ

x2 − 1

− 12 μ

P μν− 1 (x) K ν (ax) dx = (2π)−1/2 aμ− 2 1

2

− 1 μ 1  xμ+ 2 x2 − 1 2 P μν− 1 (x) K ν (ax) dx =



− 1 μ 3  xμ− 2 x2 − 1 2 P μν− 1 (x) K ν (ax) dx =



2

1

8.



2

1



+ a a a  a , 1 = 2−3/2 π 1/2 aμ− 2 J ν J μ− 12 −Yν Y μ− 12 2 2 2 2  1 − 4 < Re μ < 1, a > 0, Re(2μ − ν) > − 12 ET II 349(67)a

ET II 337(33)a ∞

x2





+ a a  a  a , 1 = −2−3/2 π 1/2 aμ− 2 J μ− 12 Yν + Y μ− 12 Jν 2 2 2  1  2 1 − 4 < Re μ < 1, a > 0, |Re ν| < 2 + 2 Re μ ET II 344(37)a

 − 1 μ 1 x 2 −μ 1 − x2 2 P μν (x) J ν+ 12 (ax) dx =

5. 

3 2

 − 1 μ 1 x 2 −μ x2 − 1 2 P μν− 1 (x) Y ν (ax) dx 2

1



Re(2λ + ν) <

 − 1 μ 1 x 2 −μ x2 − 1 2 P μν− 1 (x) J ν (ax) dx 2

1

4.

  2λ+ν a−λ Γ 12 + ν = S λ−ν,λ+ν (a) π 1/2 Γ(1 −λ) a > 0, Re ν < 52 ,

ET II 344(36)a

ET II 345(38)a ∞

2.



P λ−1 (x) J ν (ax) dx λ

ET II 108(3)a

∞

− 1 μ 1  xμ− 2 x2 − 1 2 P μν− 3 (x) K ν (ax) dx = 2



    K ν 12 a K μ− 12 12 a

[Re μ < 1,

Re a > 0]

ET II 135(5)a

π −3/2 − 1 a a e 2 W μ,ν (a) 2 [Re μ < 1,

Re a > 0]

ET II 135(3)a

π −1/2 − 1 a a e 2 W μ−1,ν (a) 2 [Re μ < 1,

Re a > 0]

ET II 135(4)a

π −1 − 1 a a e 2 W μ− 12 ,ν− 12 (a) 2 [Re μ < 1]

ET II 135(6)a

7.182

Associated Legendre and Bessel functions



∞

+  a ,2   1 ν− 1 1 −ν  2  2x − 1 K ν (ax) dx = π −1/2 a−ν 2ν−1 K μ+ 12 x1/2 x2 − 1 2 4 P μ2 2   1 Re ν > − 2 , Re a > 0 ET II 136(11)a

∞

  1 ν− 1 1 −ν  2  2x − 1 Y ν (ax) dx x1/2 x2 − 1 2 4 P μ2 + a a a  a , = π 1/2 2ν−2 a−ν J μ+ 12 J −μ− 12 − Y μ+ 12 Y −μ− 12 2 2 2  2  ET II 108(5)a Re ν > − 12 , a > 0, Re ν + |2 Re μ + 1| < 32

∞

  1 ν− 1 1 −ν  2  2x − 1 J ν (ax) dx x1/2 x2 − 1 2 4 P μ2

9. 1

 10. 1

 11. 1

 

783

∞

12.

+  a ,2 +  a ,2  J μ+ 12 − J −μ− 12 2 2  < 2 Re μ < 12 − Re ν ET II 345(39)a

= −2ν−2 a−ν π 1/2 sec(μπ) Re ν > − 12 ,

Re ν −

a > 0,

3 2

  − 1 ν  x x2 − 1 2 P νμ 2x2 − 1 K ν (ax) dx = 2−ν aν−1 K μ+1 (a)

1



[Re a > 0, ∞

13.



x x2 + a

1 2 2ν



P νμ 1 + 2x2 a

 −2

ET II 136(10)a

K ν (xy) dx = 2−ν ay −ν−1 S 2ν,2μ+1 (ay)

0



Re ν < 1]

[Re a > 0,

Re y > 0,

Re ν < 1] ET II 135(7)

∞

14. 0

1ν       x x2 + a2 2 (μ − ν) P νμ 1 + 2x2 a−2 + (μ + ν) P ν−μ 1 + 2x2 a−2 K ν (xy) dx = 21−ν μy −ν−2 S 2ν+1,2μ (ay) [Re a > 0,



∞

15. 0

Re y > 0,

Re ν < 1]

ET II 136(8)

 1 ν−1  ν      P μ 1 + 2x2 a−2 + P ν−μ 1 + 2x2 a−2 K ν (xy) dx = 21−ν y −ν S 2ν−1,2μ (ay) x x2 + a2 2 [Re a > 0,

Re y > 0,

Re ν < 1] ET II 136(9)



∞

16. 0

 17. 0

 − 1 ν− 1 −ν− 1 x1/2 x2 + 2 2 4 P μ 2

+  , 2 1 y −1/2 2 2 −ν π −1/2 K μ+ 12 2−1/2 y  2      x + 1 J ν (xy) dx = 3 1   Γ3 ν + μ + 2 Γ ν − μ + 2 1 − 2 − Re ν < Re μ < Re ν + 2 , y > 0 ET II 44(1)

∞

 − 1 ν− 1 ν+ 1   x1/2 x2 + 2 2 4 Q μ 2 x2 + 1 J ν (xy) dx

  1 1 = 2−ν− 2 π 1/2 e(ν+ 2 )πi y ν K μ+ 12 2−1/2 y I μ+ 12 2−1/2 y   ET II 46(12) Re ν > −1, Re(2μ + ν) > − 52 , y > 0

784

Associated Legendre Functions



∞

7.183

 − 1 μ− 1 μ+ 1 x1−μ 1 + a2 x2 2 4 Q ν− 12 (±iax) J ν (xy) dx 2

0

= i(2π)1/2 eiπ(μ∓ 2 ν∓ 4 ) a−1 y μ−1 I ν 1

 3 −4 − 7.184



1 2

1

Re ν < Re μ < 1 + Re ν,

y > 0,

1

   y K μ 12 a−1 y  Re a > 0 ET II 46(11) 2a

−1

∞

   1 μ− 1 − 1 −μ    x1/2 x2 − 1 2 4 P − 21 +ν x−1 J ν (xa) dx = 21/2 a−1−μ π −1/2 cos a + 12 (ν − μ)π 2   |Re μ| < 12 , Re ν > −1, a > 0

∞

  x−ν x2 − 1

1. 1



7.183

ET II 44(2)a

2.

1 1 4−2ν

ν− 12

Pμ



 2x−2 − 1 K ν (ax) dx

1



= π 1/2 2−ν a−2+ν W μ+ 12 ,ν− 12 (a) W −μ− 12 ,ν− 12 (a)   Re ν < 32 , a > 0 ET II 370(45)a

∞

3. 0

 4. 0



1

  1 ν+ 1 −ν− 1   xν 1 − x2 2 4 P μ 2 2x−2 − 1 J ν (xy) dx 3  1  ν+ 12 ν Γ 2 + μ + ν Γ 2 + ν − μ =2 y    1/2 Γ 3 + ν 2 (2π) 2



3 3 × 1 F 1 ν + μ + ; 2ν + 2; iy 1 F 1 ν + μ + ; 2ν + 2; −iy 2 2   y > 0, − 32 − Re ν < Re μ < Re ν + 12 ET II 45(3)

∞

  1 − 1 ν 1 −ν   1 + 2a2 x−2 K ν (xy) dx x−ν x2 + a2 4 2 Q μ2   2 1 = ie−iπν π 1/2 2−ν−1 a−ν− 2 y ν−2 Γ 32 + μ − ν W −μ− 12 ,ν− 12 (iay) W −μ− 12 ,ν− 12 (−iay)   Re a > 0, Re y > 0, Re μ > − 32 , Re(μ − ν) > − 32 ET II 137(13)

∞

   1 − 1 ν 1 −ν  1 + 2x−2 J ν (ax) dx x−ν x2 + 1 4 2 Q μ2

5. 0

 6.



2 1 + 2 J ν (ax) dx x  3 2  1  iπν − 12 ν −ν−2 Γ 2 +μ+ν Γ 2 +ν−μ = −ie π 2 a % & cos(μπ) sin(μπ)  W μ+ 1 ,ν+ 1 (a)  1 (a) + M 1 × W −μ− 12 ,ν+ 12 (a) 2 2 Γ(2 + 2ν) μ+ 2 ,ν+ 2 Γ ν + μ + 32   3 1 a > 0, Re(μ + ν) > − 2 , Re(μ − ν) < 2 ET II 46(14)

  1 + ν ν+ 1 xν 1 + x2 4 2 Q μ 2

0

=2

  Γ 32 + μ − ν M μ+ 12 ,ν− 12 (a) W −μ− 12 ,ν− 12 (a) Γ(2ν)   a > 0, 0 < Re ν < Re μ + 32 ET II 47(15)a

−iνπ 1/2 π −ν −ν−2 ie

a

7.187

Associated Legendre and Bessel functions



∞

7.

  1 − 1 ν 12 −ν   1 + 2a2 x−2 K ν (xy) dx x−ν x2 + a2 4 2 Q − 1 2

0

−iπν 3/2 −ν−3

= ie

π

2

a

1 2 −ν

y

ν−1

2

[Γ(1 − ν)] ×

+

+  ay ,2  J ν− 12 + Y ν− 12 2 2 Re y > 0, Re ν < 1] ET II 136(12)

[Re a > 0, 

785

 ay ,2

+    1/2 ,  1  a2 + x2 x−1 J ν (xy) dx = 2−1/2 πy −1 exp − a2 − 14 y Jν 2y 0   ET II 46(10) Re ν > − 12 , y > 0

 ∞ 2   1 − x −ν−1 −2 x 1 + x2 Pν J 0 (xy) dx = y 2ν [2ν Γ(ν + 1)] K 0 (y) 1 + x2 0 ET II 13(10) [Re ν > 0]

7.185

7.186 7.187  1.

∞

∞

x1/2 Q ν− 12

x P νμ

0



∞

2.



+

[Re ν < 1, Re y > 0]  ,2  y ,2 + 1 + a2 x2 J 0 (xy) dx = 2π −2 y −1 a−1 cos(λπ) K λ 2a  Re a > 0, |Re λ| < 14 ,

0

− 12 ν

x Pμ





− 12 ν

1 + a2 x2 Q μ

0

∞

x P μσ− 1 2



∞

6.

x P μσ− 1 2

0

2

=y Re a > 0,

 0

∞

 y y +σ−μ W μ,σ M −μ,σ Γ(1 + 2σ) a a ET II 14(15) Re σ > − 14 , Re μ < 1

−2 μπi Γ

e

y > 0,

1 2

 2 x2 J (xy) dx 1 + a2 x2 P −μ 1 + a 1 0 σ− 2



7.

ET II 137(15)

1 + a2 x2 J ν (xy) dx

  1 + a2 x2 Q μσ− 1 1 + a2 x2 J 0 (xy) dx



Re y > 0]



 



  1 y y y −1 e− 2 νπi Γ 1 + μ + 12 ν   1 1 = K I μ+ 2 μ+ 2 2a 2a a Γ 1 + μ − 12 ν   3 Re a > 0, y > 0, Re μ > − 4 , Re ν > −1 ET II 47(16)

0

5.

y>0

−1/2 ν   x 1 + x2 Pμ 1 + x2 K ν (xy) dx = y −1/2 S ν− 12 ,μ+ 12 (y) [Re ν < 1,

∞

4.



ET II 137(14)

ET II 13(11) ∞

3. 

 1 + x2 K ν (xy) dx = y −3/2 S ν+ 12 ,μ+ 12 (y)

x P λ− 12

0



= 2π −1 y −2 cos(σπ) W μ,σ Re a > 0,

y > 0,

y

a |Re σ| < 14

W −μ,σ

y a

ET II 14(14)

  2 y +  y  y , x P μσ− 1 1 + a2 x2 J 0 (xy) dx = −iπ −1 y −2 W μ,σ W μ,σ eπi − W μ,σ e−πi 2 a a a   ET II 14(13) Re a > 0, y > 0, |Re σ| < 14 , Re μ < 1

786

Associated Legendre Functions



∞

8. 0



∞

9.

1 1  −1/2 − 12 − 12 ν   − ν x 1 + a2 x2 Pμ 1 + a2 x2 P μ2 2 1 + a2 x2 J ν (xy) dx +  y ,2 K μ+ 12 2a     = πa2 Γ ν2 + μ + 32 Γ ν2 − μ + 12  ET II 46(9) Re a > 0, y > 0, − 54 < Re μ < 14 

 −1ν

x Pμ 2

1 + a2 x2

2

0



∞

10.

+  y ,2 −1 2 K μ+ 12 2a y     J ν (xy) dx = 1 1 πa Γ 1 + μ + ν Γ ν − μ 2 2   Re a > 0, y > 0, − 34 < Re μ < − 14 , Re ν > −1





∞

1. 0



∞

2.

∞

3.

Re a > 0,

y y K μ+ 12 2a K μ+ 32 2a     = πa2 Γ 2 + 12 ν + μ Γ 12 ν − μ ET II 45(8) y > 0, − 74 < Re μ < − 14

  − 1 μ  a y μ−2 e−ay √ x a2 + x2 2 P −ν J ν (xy) dx = μ−1 Γ(μ + ν) a2 + x2  Re a > 0, y > 0, Re ν > −1,  1ν xν+1 x2 + a2 2 P ν

0



ET II 45(7)

 −1/2 − 12 ν   − 12 ν x 1 + a2 x2 Pμ 1 + a2 x2 P μ+1 1 + a2 x2 J ν (xy) dx

0

7.188

7.188



x2 + 2a2 √ 2a x2 + a2

 − 1 ν x1−ν x2 + a2 2 P ν−1



0

2

J ν (xy) dx =

2

x + 2a √ 2a x2 + a2

Re μ >

1 2



 ya ,2 (2a)ν+1 y −ν−1 + K ν+ 12 π Γ(−ν) 2 [Re a > 0, −1 < Re ν < 0,

ET II 45(4)

y > 0] ET II 45(5)



 ay y (2a) I ν− 12 K ν− 12 Γ(ν) 2 2 [Re a > 0, y > 0, 0 < Re ν < 1] 1−ν ν−1

J ν (xy) dx =

 ay

ET II 45(6)

7.189



∞

μ −x

(a + x) e

1.

P −2μ ν

0

 2.

∞

−μ −x

(x + a) 0

e



2x 1+ a

P −2μ ν

I μ (x) dx = 0 − 12 < Re μ < 0,

 − 12 + Re μ < Re ν < − 12 − Re μ

ET II 366(18)



2x 1+ I μ (x) dx a     2μ−1 Γ μ + ν + 12 Γ μ − ν − 12 ea = W 12 −μ, 12 +ν (2a) π 1/2 Γ (2μ + ν + 1) Γ(2μ! − ν) ! |arg a| < π, Re μ > !Re ν + 12 ! ET II 367(19)

7.192

Associated Legendre functions and functions generated by Bessel functions



∞

3.

−μ x

x

e

P 2μ ν

0





2x 1+ K μ (x + a) dx a

    = π −1/2 2μ−1 cos(μπ) Γ μ + ν + 12 Γ μ − ν + 12 W 12 −μ, 12 +ν (2a) ! !  |arg a| < π, Re μ > !Re ν + 12 ! ET II 373(11)

∞

4.

− 12 μ

x

−1/2 −x

(x + a)

e

0



787

P μν− 1 2

a−x a+x



K ν (a + x) dx =

π −1μ a 2 Γ(μ, 2a) 2

[a > 0, ∞

5.

(sinh x) 0

μ+1

(cosh x)

−2μ− 32

Re μ < 1]

ET II 374(12)

P −μ ν [cosh(2x)] I μ− 12 (a sech x) dx 1

=

2μ− 2 Γ(μ − ν) Γ (μ + ν + 1)

M ν+ 12 ,μ (a) M −ν− 12 ,μ (a) 3 2 π 1/2 aμ+ 2 [Γ (μ + 1)] [Re μ > Re ν, Re μ > − Re ν − 1] ET II 378(44)

7.19 Combinations of associated Legendre functions and functions generated by Bessel functions 7.191



∞

1.

  2   1 2  = 2−ν−2 π 1/2 a cosec(μπ) cos(νπ) Y ν 12 a − Jν 2a   −1 < Re μ < 0, Re ν < 12 ET II 384(6)

a



∞

2. 0

7.192



1. 0

 − 1 − 1 ν ν+ 1   x1/2 x2 − a2 4 2 P μ 2 2x2 a−2 − 1 [Hν (x) − Y ν (x)] dx

1

 −1/4−ν/2 ν+1/2  2 −2  2x a − 1 [I −ν (x) − Lν (x)] dx x1/2 x2 − a2 Pμ     2   2 = 2−ν−1 π 1/2 a cosec(2μπ) cos(νπ) I ν 12 a − I −ν 12 a   −1 < Re μ < 0, Re ν < 12 ET II 385(15)

 (ν−μ−2)/4 (μ−ν+2)/2 x(ν−μ−1)/2 1 − x2 P ν−1/2 (x) S μ,ν (ax) dx





μ+ν+3 μ − 3ν + 3 μ−ν μ−3/2 1/2 −(ν−μ−1)/2 π =2 π a Γ Γ cos 4 4 2  1  1  1   1  × J ν 2 a Y −(μ−ν+1)/2 2 a − Y ν 2 a J −(μ−ν+1)/2 2 a [Re(μ − ν) < 0,

a > 0,

|Re(μ + ν)| < 1,

Re(μ − 3ν) < 1]

ET II 387(24)a

788

Associated Legendre Functions



∞

2.

 −β/2 β x1/2 x2 − 1 P ν (x) S μ,1/2 (ax) dx

1

 1 Γ β−μ−ν − 2 4  S μ−β+1,ν+1/2 (a) −μ  Re(μ + ν − β) < − 12 , Re(μ − ν − β) < 12 ET II 387(25)a

2−3/2+β−μ aβ−1 Γ

7.193 1.



∞



∞



β−μ+ν 2  π 1/2 Γ 12

=  Re β < 1,

a > 0,

7.193

+

1 4

 1/4−ν/2 ν−1/2  −2  x−ν x2 − 1 P μ/2−ν/2 2x − 1 S μ,ν (ax) dx 1     2μ−ν aν−2 π 1/2 Γ 3ν−μ−1 2  1+ν−μ  = W ρ,σ aeiπ/2 W ρ,σ ae−iπ/2 Γ 2   ρ = 12 (μ + 1 − ν), σ = ν − 12 , Re(μ − ν) < 0, a > 0, Re ν < 32 , Re(3ν − μ) > 1 ET II 387(27)a

2.

 −ν/2 ν  2  x x2 − 1 P λ 2x − 1 S μ,ν (ax) dx

1

   + λ Γ ν−μ−1 −λ 2 2   1−μ+ν   1−μ−ν S μ−ν+1,2λ+1 (a) Γ 2Γ 2 2 Re(μ − ν + λ) < −1, Re(μ − ν + λ) < 0] ET II 387(26)a =

[Re ν < 1,

a > 0,

aν−1 Γ

 ν−μ+1

7.21 Integration of associated Legendre functions with respect to the order 7.211



1 1 cosec θ [0 < θ < π] 2 2 0

 ∞ 1 θ [0 < θ < π] 2. P x (cos θ) dx = cosec 2 −∞  ∞   1 7.212 x−1 tanh(πx) P − 12 +ix (cosh a) dx = 2e− 2 a K e−a 

∞

1.

P −x− 12 (cos θ) dx =

ET II 329(19) ET II 329(20)

0

[a > 0] x tanh(πx) P − 12 +ix (cosh b) dx = Q a− 12 (cosh b) [Re a > 0] 2 2 0 ∞ a + x 1 sinh(πx) cos(ax) P − 12 +ix (b) dx =  2 (b + cosh a) 0 [a > 0, |b| < 1] 

7.213 7.214  7.215

0

∞

∞

cos(bx) P μ− 1 +ix (cosh a) dx = 0

π

2

= Γ

1 2

ET II 330(22) ET II 387(23)

ET I 42(27)

[0 < a < b] μ

2 (sinh a)  μ+ 1 − μ (cosh a − cosh b) 2

[0 < b < a] ET II 330(21)

7.221

Integration of associated Legendre functions



∞

7.216

cos(bx) Γ(μ + ix) Γ(μ − ix) P

0

1 2 −μ − 12 +ix

789

π

μ− 1

Γ(μ) (sinh a) 2 (cosh a) dx = μ (cosh a + cosh b) [a > 0, b > 0, Re μ > 0] 2

ET II 330(24)





1 1 1 1 2 −ν − ix Γ 2ν − + ix P ν+ix−1 (cos θ) I ν− 12 +ix (a) K ν− 12 +ix (b) dx ν − + ix Γ 2 2 2 −∞ ν √ ab ν− 1 = 2π (sin θ) 2 K ν (ω) ω , + 1/2  ET II 383(29) ω = a2 + b2 + 2ab cos θ

7.217  1.

∞



∞

2. 0



∞

3.

2(ab)1/2 −ikR (2) (2) e xeπx tanh(πx) P − 12 +ix (− cos θ) H ix (ka) H ix (kb) dx = − ; πR  2  1/2 R = a + b2 − 2ab cos θ [a > 0, b > 0, 0 < θ < π, Im k ≤ 0] 1

−ν

(2)

(2)

2 xeπx sinh(πx) Γ (ν + ix) Γ(ν − ix) P − (− cos θ) H ix (a) H ix (b) dx 1 +ix 2

0

ν− 12



= i(2π)1/2 (sin θ) 1/2  R = a2 + b2 − 2ab cos θ 

∞

4. 0

[a > 0,

b > 0,

ET II 381(17)

0 < θ < π,

Re ν > 0]

ab R

ν H (2) ν (R) ET II 381 (18)

λ 1  2  1 λ− 1 π 1/2 ab 2 −λ √ β − 1 2 4 K λ (z) x sinh(πx) Γ(λ+ix) Γ(λ−ix) K ix (a) K ix (b) P − (β) dx = 1 +ix 2 z 2 , +  π 2 2 z = a + b + 2abβ |arg a| < , |arg(β − 1)| < π, Re λ > 0 ET II 177(16) 2

7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions 7.221  1.

1

−1

 2.6

[m = n]

P n (x) P m (x) dx = 0 2 = 2n + 1

1

[m = n]

1 2n + 1 =0

[m = n]

P n (x) P m (x) dx = 0

=  3.

(−1)

2π

P 2n (cos ϕ) dϕ = 2π 0

[n − m is even,

2m+n−1 (m 

WH, EH I 170(8, 10)



1 2 (m+n−1)

m!n!  2    ! − n) (n + m + 1) n2 ! m−1 2

2n −2n 2 n

m = n]

[n is even, m is odd] WH

2 .

MO 70, EH II 183(50)

790

7.222

Associated Legendre Functions



1

1. −1



[m < n] 4

−1

(1 + x)m+n P m (x) P n (x) dx =

2m+n+1 [(m + n)!]

ET II 277(15)

2

(m!n!) (2m + 2n + 1)!

1

3. −1



xm P n (x) dx = 0

1

2. 

7.222

(1 + x)m−n−1 P m (x) P n (x) dx = 0 

1

4. −1

1 − x2

n

P 2m (x) dx =

[m > n]

2n2 (n − m)(2m + 2n + 1)





1 −1

1 − x2

n−1

ET II 278(16)

P 2m (x) dx

[m < n] 

1

x2 P n+1 (x) P n−1 (x) dx =

5. 0

 7.223 7.224

1 2 {P n (x) P n−1 (x) − P n−1 (x) P n (z)} dx = − −x n [z belongs to the complex plane with a discontinuity along the interval from −1 to +1.] 1

−1



1

2. −1



1

3. −1



1

4. −1



1

5. −1



1

6. −1



1

7. −1



1

8. −1

2.

WH

1

1.

1.

n(n + 1) (2n − 1)(2n + 1)(2n + 3)

−1 z



7.225

WH

(z − x)−1 P n (x) dx = 2 Q n (z)

ET II 277(7)

x(z − x)−1 P 0 (x) dx = 2 Q 1 (z)

ET II 277(8)

xn+1 (z − x)−1 P n (x) dx = 2z n+1 Q n (z) −

2

2n+1 (n!) (2n + 1)!

ET II 277(9)

xm (z − x)−1 P n (x) dx = 2z m Q n (z)

[m ≤ n]

ET II 277(10)a

(z − x)−1 P m (x) P n (x) dx = 2 P m (z) Q n (z)

[m ≤ n]

ET II 278(18)a

(z − x)−1 P n (x) P n+1 (x) dx = 2 P n+1 (z) Q n (z) −

2 n+1

x(z − x)−1 P m (x) P n (x) dx = 2z P m (z) Q n (z) x(z − x)−1 [P n (x)] dx = 2z P n (z) Q n (z) − 2

[m < n]

2 2n + 1



−1 1 (x − t) P n (t) dt = n + (1 + x)−1/2 [T n (x) + T n+1 (x)] 2 −1

−1  1 1 −1/2 −1/2 (t − x) P P n (t) dt = n + (1 − x)−1/2 [T n (x) − T n+1 (x)] 2 x 

x

WH

−1/2

ET II 278(19)

ET II 278(21)

ET II 278(20)

EH II 187(43)

EH II 187(44)

7.232

Legendre polynomials and powers



1

3. −1



1

4. −1

5.10

7.226 1.

2. 3.

1 2

(1 − x)−1/2 P n (x) dx =



−1/2

(cosh 2p − x)

23/2 2n + 1

EH II 183(49)

√ 2 2 exp[−(2n + 1)p] P n (x) dx = 2n + 1 [p > 0]

P  (z) dz  = P  (x) Q  (y) 2 (xy − z) − (x2 − 1) (y 2 − 1) −1 = P  (y) Q  (x)

1

 7.227 0

(1 < x ≤ y) (1 < y ≤ x)



1

 2 −1/2

ET II 276(4)

ET II 276(5)

[|p| < 1] 1/2 ,−2n−1

MO 71

+  a + a2 + 1    2 2 −1/2 2 x a +x P n 1 − 2x dx = 2n + 1 

[Re a > 0]  1   1 −μ/2 Γ(1 + μ) P l (x)(z − x)−μ−1 dx = z 2 − 1 e−iπμ Q μl (z) 2 −1

7.2286

WH

1

%   &2 Γ 12 + m 1−x P 2m (x) dx = m! −1      1 −1/2  Γ 12 + m Γ 32 + m x 1 − x2 P 2m+1 (x) dx = m!(m + 1)! −1  1  −m−3/2 2 (−p)m (1 + p)−m−1/2 1 + px2 P 2m (x) dx = 2m + 1 −1 

791

[l = 0, 1, 2, . . . ,

ET II 278(23)

|arg(z − 1)| < π]

7.23 Combinations of Legendre polynomials and powers 7.231 1. 2.6

    (−1)m Γ m − 12 λ Γ 12 + 12 λ     x P 2m (x) dx = [Re λ > −1] 2 Γ − 12 λ Γ m + 32 + 12 λ 0      1 (−1)m Γ m + 12 − 12 λ Γ 1 + 12 λ λ     x P 2m+1 (x) dx = 2 Γ 12 − 12 λ Γ m + 2 + 12 λ 0



1

λ

[Re λ > −2] 7.232



EH II 183(51)

EH II 183(52)

1

1. −1

(1 − x)a−1 P m (x) P n (x) dx =

2a Γ(a) Γ(n − a + 1) 4 F 3 (−m, m + 1, a, a; 1, a + n + 1, a − n; 1) Γ(1 − a) Γ(n + a + 1) [Re a > 0] ET II 278(17)

792

Associated Legendre Functions



1

2. −1

2a+b−1 Γ(a) Γ(b) 3 F 2 (−n, 1 + n, a; 1, a + b; 1) Γ(a + b)

(1 − x)a−1 (1 + x)b−1 P n (x) dx =

[Re a > 0,



1

(1 − x)μ−1 P n (1 − γx) dx =

3. 0





(1 − x)μ−1 xν−1 P n (1 − γx) dx = 0

[Re μ > 0]

Γ(μ) Γ(ν) 1 3 F 2 −n, n + 1, ν; 1, μ + ν; γ Γ(μ + ν) 2 [Re μ > 0,



1

7.233

  x2μ−1 P n 1 − 2x2 dx =

0

Re b > 0]

ET II 276(6)

Γ(μ)n! P (μ,−μ) (1 − γ) Γ(μ + n + 1) n

1

4.

7.233



Re ν > 0]

ET II 190(37)a

ET II 190(38)

2

(−1)n [Γ(μ)] 2 Γ(μ + n + 1) Γ(μ − n) [Re μ > 0]

ET II 278(22)

7.24 Combinations of Legendre polynomials and other elementary functions 

∞

7.241 0



∞



Pn e

7.242 0

7.243



n a

1 d e P n (1 − x)e−ax dx = e−a an a da a

n 1 1 d n =a 1+ 2 da an+1



 −x

[Re a > 0] (a − 1)(a − 2) · · · (a − n + 1) e−ax dx = (a + n)(a + n − 2) · · · (a − n + 2) [n ≥ 2, Re a > 0]

P 2n (cosh x) e

−ax

0



[Re a > 2n]  2  2   2  2 2 a a −2 a − 4 · · · a − (2n)2 −ax P 2n+1 (cosh x) e dx = 2 (a − 1) (a2 − 32 ) · · · [a2 − (2n + 1)2 ]

ET I 171(6)

[Re a > 2n + 1]  2  2   2  2 2 a +1 a + 3 · · · a + (2n − 1)2 −ax P 2n (cos x) e dx = a (a2 + 22 ) (a2 + 42 ) · · · [a2 + (2n)2 ]

ET I 171(7)

∞

[Re a > 0]  2   2   2 2 2 a + 4 · · · a + (2n)2 a a +2 −ax P 2n+1 (cos x) e dx = 2 (a + 12 ) (a2 + 32 ) · · · [a2 + (2n + 1)2 ]

ET I 171(4)

∞

0

3. 0

 4. 0



    a2 − 12 a2 − 32 · · · a2 − (2n − 1)2 dx = a (a2 − 22 ) (a2 − 42 ) · · · [a2 − (2n)2 ]

∞

2. 

ET I 171(3)



∞

1.

ET I 171(2)



1

5.11 −1

eixα P n (x) dx = in

[Re a > 0] 2π J 1 (α) α n+ 2

[n = 0, 1, 2, . . . ,

ET I 171(5)

a > 0] GH2 24 (171.10)

7.249

Legendre polynomials and elementary functions

7.244  1. 2.

793

 a ,2   π+ J n+ 12 P n 1 − 2x2 sin ax dx = [a > 0] 2 2 0  1 a a   π J −n− 12 P n 1 − 2x2 cos ax dx = (−1)n J n+ 12 2 2 2 0 1

ET I 94(2)

[a > 0] 7.245  1.

2π

P 2m+1 (cos θ) cos θ dθ =

0



π

2.

P m (cos θ) sin nθ dθ = 0

π 24m+1



2m m



2m + 2 m+1

ET I 38(1)

MO 70, EH II 183(5)

2(n − m + 1)(n − m + 3) · · · (n + m − 1) (n − m)(n − m + 2) · · · (n + m)

[n ≤ m or n + m is even]

=0 

2π

3.10 0



  √ 3 n+1 2 π Γ n + 2 P1 P 2n+1 (sin α sin φ) sin φ dφ = (−1) (cos α) (2n + 1) Γ (n + 2) 2n+1   α = 12 (2n + 1)π, n an integer

−1

cos(αx) P n (x) dx = 0



= (−1)v

7.246 7.247 7.248

[n is odd] 2π 1 (α) J α 2v+ 2

[n = 2v is even] GH2 24 (171.10a)

  2 sin(2n + 1)θ P n 1 − 2 sin2 x sin2 θ sin x dx = 0 (2n + 1) sin θ 1 π dx 3 (a) J P 2n+1 (x) sin ax √ = (−1)n+1 2a 2n+ 2 x 0



π

1

1.



−1



MO 71

[a > 0]

1

−1



ET I 94(1)

+  −1/2 1/2 , P n (x) dx = π(ab)−1/2 J n+ 12 (aλ) J n+ 12 (bλ) a2 + b2 − 2abx sin λ a2 + b2 − 2abx [a > 0,

2.

b > 0]

ET II 277(11)

+  −1/2 1/2 , P n (x) dx = −π(ab)−1/2 J n+ 12 (aλ) Y n+ 12 (bλ) a2 + b2 − 2abx cos λ a2 + b2 − 2abx [0 ≤ a ≤ b]

7.249  1.

MO 71

1

4.



[n > m and n + m is odd]

ET II 277(12)

1

−1

P n (x) arcsin x dx = 0

=π

⎧ ⎪ ⎪ ⎨

⎫2 ⎪ ⎪ ⎬

(n − 2)!!

⎪ 1 n+1 ⎪ ⎪ ⎩ 2 2 (n+1) ⎭ !⎪ 2

[n is even]

[n is odd] WH

794

2.

Associated Legendre Functions

n t−1  1 2πr 2 P n (x) = x + x − 1 cos t t=0 t

7.251

[t > n]

7.25 Combinations of Legendre polynomials and Bessel functions 7.251



1

1.

  x P n 1 − 2x2 Y ν (xy) dx = π −1 y −1 [S 2n+1 (y) + π Y 2n+1 (y)]

0

[n = 0, 1, . . . ;

y > 0,

ν > 0] ET II 108(1)

  1   i 2 −1 n+1 x P n 1 − 2x K 0 (xy) dx = y K 2n+1 (y) + S 2n+1 (iy) (−1) 2 0

 2.



1

3.

  x P n 1 − 2x2 J 0 (xy) dx = y −1 J 2n+1 (y)

0



1

4.

  2 x P n 1 − 2x2 [J 0 (ax)] dx =

0



1

5.

ET II 134(1)

[y > 0]

ET II 13(1)

  1 2 2 [J n (a)] + [J n+1 (a)] 2(2n + 1)

  x P n 1 − 2x2 J 0 (ax) Y 0 (ax) dx =

0

[y > 0]

ET II 338(39)a

1 [J n (a) Y n (a) + J n+1 (a) Y n+1 (a)] 2(2n + 1) ET II 339(48)a

 6.

1

  x2 P n 1 − 2x2 J 1 (xy) dx = y −1 (2n + 1)−1 [(n + 1) J 2n+2 (y) − n J 2n (y)]

0

[y > 0] ET II 20(23)    2 1   2−ν−1 aν Γ 12 μ + 12 ν     7. xμ−1 P n 2x2 − 1 J ν (ax) dx = Γ(ν + 1) Γ 12 μ + 12 ν + n + 1 Γ 12 + 12 ν − n 0

μ+ν μ+ν μ+ν μ+ν a2 , ; ν + 1, + n + 1, − n; − × 2F 3 2 2 2 2 4 [a > 0, Re(μ + ν) > 0] ET II 337(32)a  1 e−a [I n (a) + I n+1 (a)] 7.252 e−ax P n (1 − 2x) I 0 (ax) dx = 2n + 1 0 [a > 0] ET II 366(11)a  π/2 7.253 sin(2x) P n (cos 2x) J 0 (a sin x) dx = a−1 J 2n+1 (a) ET II 361(20) 0 1   7.254 x P n 1 − 2x2 [I 0 (ax) − L0 (ax)] dx = (−1)n [I 2n+1 (a) − L2n+1 (a)] 

0

[a > 0]

ET II 385(14)a

Gegenbauer polynomials Cnν (x) and powers

7.313

795

7.3–7.4 Orthogonal Polynomials 7.31 Combinations of Gegenbauer polynomials Cnν (x) and powers 7.311



1

1.



−1

3.

4.

ν− 12



C νn (x) dx = 0

Re ν > − 12

n > 0,



      1 ν Γ(2ν + n) Γ(2ρ + n + 1) Γ ν + 12 Γ ρ + 12 2 ν− 2 1−x x C n (x) dx = 2n+1 Γ(2ν) Γ(2ρ + 1)n! Γ(n + ν + ρ + 1) 0   Re ρ > − 12 , Re ν > − 12    1  1 2β+ν+ 2 Γ(β + 1) Γ ν + 12 Γ(2ν + n) Γ β − ν + ν ν− 12 β     (1 − x) (1 + x) C n (x) dx = n! Γ(2ν) Γ β − ν − n + 32 Γ β + ν + n + 32 −1   Re β > −1, Re ν > − 12  1 2α+β+1 Γ(α + 1) Γ(β + 1) Γ (n + 2ν) β (1 − x)α (1 + x) C νn (x) dx = −1

n! Γ(2ν) Γ(α + β + 2) 1 × 3 F 2 −n, n + 2ν, α + 1; ν + , α + β + 2; 1 2 [Re α > −1, Re β > −1] 

2.

1 − x2

ET II 280(1)

1

n+2ρ

ET II 280(2) 3 2



ET II 280(3)

ET II 281(4)

7.312 In the following integrals, z belongs to the complex plane with a cut along the interval of the real axis from −1 to 1.  1 3   1 ν  1 ν− 1 ν− 12 π 1/2 2 2 −ν −(ν− 12 )πi m  2 m −1 2 ν− 2 e 1−x 1. x (z − x) C n (x) dx = z z − 1 2 4 Q n+ν− 1 (z) 2 Γ(ν) −1   1 m ≤ n, Re ν > − 2 ET II 281(5)  1 3  ν− 12 ν  1 ν− 1 ν− 12 π 1/2 2 2 −ν −(ν− 12 )πi n+1  2 e z − 1 2 4 Q n+ν− 2. xn+1 (z − x)−1 1 − x2 C n (x) dx = z 1 (z) 2 Γ(ν) −1 π21−2ν−n n! − Γ(ν) Γ(ν +n + 1)  ET II 281(6) Re ν > − 12  1 3  ν− 12 ν  1 ν− 1 π 1/2 2 2 −ν −(ν− 12 )πi  2 ν− 12 e z − 1 2 4 C νm (z) Q n+ν− 3.6 (z−x)−1 1 − x2 C m (x) C νn (x) dx = 1 (z) 2 Γ(ν) −1   1 m ≤ n, Re ν > − 2 ET II 283(17) 7.313



1

1.



−1

1 − x2

ν− 12



C νm (x) C νn (x) dx = 0

m = n,

Re ν > − 12



ET II 282(12), MO 98a, EH I 177(16)



1

2. −1



1 − x2

ν− 12

2

[C νn (x)] dx =

π2

1−2ν

Γ(2ν + n)

n!(n + ν) [Γ(ν)]

2



Re ν > − 12



ET II 281(8), MO 98a, EH I 177(17)

796

7.314 1.

2.

Orthogonal Polynomials

7.314

  π 1/2 Γ ν − 12 Γ(2ν + n) (1 − x) (1 + x) dx = n! Γ(ν) Γ(2ν) −1   Re ν > 12  1  1 2  23ν− 2 [Γ(2ν + n)] Γ 2n + ν + 12 1 2 (1 − x)ν− 2 (1 + x)2ν−1 [C νn (x)] dx =   2 (n!) Γ(2ν) Γ 3ν + 2n + 12 −1 

1

ν− 32

ν− 12

2 [C νn (x)]

[Re ν > 0]  3.

4.

5.

6.

7.

1

3

1

ET II 281(9)

ET II 282(10)

2

(1 − x)3ν+2n− 2 (1 + x)ν− 2 [C νn (x)] dx −1   2     π 1/2 Γ ν + 12 Γ ν + 2n + 12 Γ (2ν + 2n) Γ 3ν + 2n − 12 =    2   1 1 n + Γ(2ν) 22ν+2n n! Γ ν + Γ 2ν + 2n + 2 2   ET II 282(11) Re ν > 16  1 1 3 (1 − x)ν− 2 (1 + x)ν+m−n− 2 C νm (x) C νn (x) dx −1     Γ ν − 12 + m − n Γ 12 − ν + m − n 22−2ν−m+n π 3/2 Γ(2ν + n) m     = (−1)  2  Γ 12 − ν − n Γ 12 + m − n m!(n − m)! [Γ(ν)] Γ 12 + ν + m   Re ν > − 12 ; n ≥ m ET II 282(13)a  1 ν− 1 (1 − x)2ν−1 (1 + x) 2 C νm (x) C νn (x) dx −1       1 23ν− 2 Γ ν + 12 Γ (2ν + m) Γ(2ν + n) Γ ν + 12 + m + n Γ 12 − ν + n − m       = m!n! Γ(2ν) Γ 12 − ν Γ ν + 12 + n − m Γ 3ν + 12 + m + n ET II 282(14) [Re ν > 0]  1 1 3 (1 − x)ν− 2 (1 + x)3ν+m+n− 2 C νm (x) C νn (x) dx −1    2     24ν+m+n−1 Γ ν + 12 Γ(2ν + m + n) Γ ν + m + n + 12 Γ 3ν + m + n − 12    =  Γ(2ν + n) Γ(4ν + 2m + 2n) Γ ν + m + 12 Γ ν + n + 12 Γ (2ν + m)   Re ν > 16 ET II 282(15)  1 ν− 1 (1 − x)α (1 + x) 2 C μm (x) C νn (x) dx −1     1 2α+ν+ 2 Γ(α + 1) Γ ν + 12 Γ ν − α + n − 12 Γ(2μ + m) Γ (2ν + n)     = 1 3 Γ(2μ) Γ(2ν) m!n!

Γ ν−α− 2 Γ ν−α+n+ 2 1 3 3 3 × 4 F 3 −m, m + 2μ, α + 1, α − ν + ; μ + , ν + α + n + , α − ν − n + ; 1 2  2 2 2  Re α > −1, Re ν > − 12 ET II 283(16) 

1

7.315 −1



1−x

2

 12 ν−1

C ν2n (ax) dx

  1  π 1/2 Γ 12 ν ν  2 2a2 − 1 = 1 1 Cn Γ 2ν + 2 [Re ν > 0]

ET II 283(19)

Gegenbauer polynomials Cnν (x) and elementary functions

7.323





1

7.316 −1

7.317



1 − x2

1

2

C νn (cos α cos β + x sin α sin β) dx =

μ−1 λ− 12

(1 − x)

1.

ν−1

x

  Γ (2λ + n) Γ λ + 12 Γ(μ) (α,β)  P n (1 − γ)  (1 − γx) dx = Γ(2λ) Γ λ + μ + n + 12   Re λ > −1, λ = 0, − 12 , Re μ > 0 β = λ − μ − 12

α = λ + μ − 12 , 

1

(1 − x)μ−1 xν−1 C λn (1 − γx) dx = 0



1 2ν

7.318

x



1−x

7.319



C νn

1

(1 − x)

μ−1 ν−1

1.

x

C λ2n





2

1/2

1

2. 0

ET II 191(40)a

ET II 283(21)



+ n) Γ(μ) Γ(ν) 1 2 3 F 2 −n, n + λ, ν; , μ + ν; γ n! Γ(λ) Γ(μ + ν) 2

n Γ(λ

dx = (−1)

0



γ 1 × 3 F 2 −n, n + 2λ, ν; λ + , μ + ν; 2 2 [2λ = 0, −1, −2, . . . , Re μ > 0, Re ν > 0]





γx

ET II 190(39)a

Γ(2λ + n) Γ(μ) Γ(ν) n! Γ(2λ) Γ(μ + ν)

  Γ(2ν + n) Γ ν + 12 Γ(σ) (α,β)  P n (1 − y),  1 − x y dx = 2 Γ(2ν) Γ n + ν + σ + 12  α = ν + σ − 12 , Re ν > − 12 , Re σ > 0 β = ν − σ − 12

 2 σ−1

0

22ν−1 n! [Γ(ν)] C νn (cos α) C νn (cos β) Γ(2ν + n) [Re ν > 0] ET II 283(20)

C λn

0

2.

797

[Re μ > 0, Re ν > 0] ET II 191(41)a   1  n (−1) 2γ Γ(μ) Γ(λ + n + 1) Γ ν + 2   (1 − x)μ−1 xν−1 C λ2n+1 γx1/2 dx = 1 n! Γ(λ) Γ μ + ν + 2

1 2 1 3 × 3 F 2 −n, n + λ + 1, ν + 2 ; 2 , μ + ν + ; γ  2 Re μ > 0, Re ν > − 12 ET II 191(42)

7.32 Combinations of Gegenbauer polynomials Cnν (x) and elementary functions 

1

7.321



−1



ν− 12

0



π21−ν in Γ(2ν + n) −ν a J ν+n (a) n! Γ(ν)   Re ν > − 12 ET II 281(7), MO 99a x  ν π Γ(2ν + n) a − 1 e−bx dx = (−1)n C νn e−ab I ν+n (ab) a n! Γ(ν) 2b   Re ν > − 12 ET I 171(9)

eiax C νn (x) dx =

1

[x(2a − x)]ν− 2

7.322 7.323

2a

1 − x2

π

C νn (cos ϕ) (sin ϕ)

1. 0

2ν

dϕ = 0 = 2−2ν π Γ(2ν + 1) [Γ(1 + ν)]

[n = 1, 2, 3, . . .] −2

[n = 0] EH I 177(18)

798

Complete System of Orthogonal Step Functions

 2.

π

11

C νn (cos ψ cos ψ  + sin ψ sin ψ  cos ϕ) (sin ϕ)

2ν−1

7.324

dϕ

0

= 22ν−1 n! [Γ(ν)] C νn (cos ψ) C νn (cos ψ  ) [Γ(2ν + n)] 2

[Re ν > 0] 7.324



1

1.



1 − x2

ν− 12

C ν2n+1 (x) sin ax dx = (−1)n π

0

 2.

1



1 − x2

ν− 12

C ν2n (x) cos ax dx =

0

EH I 177(20)

Γ(2n + 2ν + 1) J 2n+ν+1 (a) (2n + 1)! Γ(ν)(2a)ν   Re ν > − 12 , a > 0

(−1)n π Γ(2n + 2ν) J ν+2n (a) (2n)! Γ(ν)(2a)ν  Re ν > − 12 ,

−1

a>0



ET I 94(4)

ET I 38(3)a

7.325∗ Complete System of Orthogonal Step Functions Let sj (x) = (−1) 2jx for j ∈ N and cj (x) = (−1) 2jx+1/2 for j ∈ 0 + N where z denotes the integer part of z. Thus, cj (z) and sj (z) have minimal period j −1 and manifest even and odd symmetry about x = 1/2, respectively, and so are the discrete analogues of cos 2πjx and sin 2πjx. Furthermore, for j ∈ N let j denote its odd part: the quotient of j by its highest power-of-two factor. Then for all j and k ∈ N, if (j, k) denotes their highest common factor and [j, k] denotes their lowest common multiple:   1 (j,k) if j/j = k/k sj (x)sk (x) dx = [j,k] 1. 0 otherwise 0   1 if j/j = k/k (−1)(j+k)/2+1 (j,k) [j,k] 2. cj (x)ck (x) dx = 0 otherwise 0

7.33 Combinations of the polynomials C νn (x) and Bessel functions; Integration of Gegenbauer functions with respect to the index 7.331



∞

2n+1−ν

x

1. 1



x −1 2

ν−2n− 12

C ν−2n 2n



1 J ν (xy) dx x −1

−1

= (−1)n 22n−ν+1 y −ν+2n−1 [(2n)!] Γ(2ν − 2n) [Γ(ν − 2n)] cos y   y > 0, 2n − 12 < Re ν < 2n + 12 ET II 44(10)a

7.334

7.332

Gegenbauer functions and Bessel functions



∞

1. 0

799

+  − 1 ν− 3 ν+ 12 + 2 −1/2 , 1/2 , x + β2 xν+1 x2 + β 2 2 4 C 2n+1 β J ν+ 32 +2n x2 + β 2 a J ν (xy) dx %

1/2 & +   2   , ν+ 12 y2 n 1/2 −1/2 12 −ν ν 2 −1/2 2 2 1/2 C 2n+1 a y a −y sin β a − y = (−1) 2 π 1− 2 a [0 < y < a] =0 [a < y < ∞]



[a > 0,

Re ν > −1]

Re β > 0,

ET II 59(23) ∞

2. 0

+  − 1 ν− 3 ν+ 1 +  −1/2 , 1/2 , J ν+ 12 +2n x2 + β 2 xν+1 x2 + β 2 2 4 C 2n 2 β x2 + β 2 a J ν (xy) dx %

1/2 & +   2   , ν+ 12 y2 n 1/2 −1/2 12 −ν ν 2 −1/2 2 2 1/2 C 2n a y a −y cos β a − y = (−1) 2 π 1− 2 a [0 < y < a] =0 [a < y < ∞]

[a > 0,

Re ν > −1]

Re β > 0,

ET II 59(24)

7.333



π

(sin x)

1.

ν+1

ν+ 12

cos (a cos θ cos x) C n



0 n

= (−1) 2

(cos x) J ν (a sin θ sin x) dx

1/2 2π ν+ 1 ν (sin θ) C n 2 (cos θ) J ν+ 12 +n (a) a

=0 

[n = 0, 2, 4, . . .] [n = 1, 3, 5, . . .]

[Re ν > −1] π

2.

(sin x)

ν+1

ν+ 12

sin (a cos θ cos x) C n

WA 414(2)a

(cos x) J ν (a sin θ sin x) dx

0

=0 = (−1)

n−1 2



2π a

[n = 0, 2, 4, . . .]

1/2 ν

ν+ 12

(sin θ) C n

(cos θ) J ν+ 12 +n (a)

[n = 1, 3, 5, . . .]

[Re ν > −1] 7.334



π

1.

(sin x)

2ν

0

 2.

π

(sin x) 0

2ν

J ν (ω) π Γ(2ν + n) J ν+n (α) J ν+n (β) C νn (cos x) dx = ν−1 , ων 2 n! Γ(ν) αν βν   1/2 ω = α2 + β 2 − 2αβ cos x n = 0, 1, 2, . . . ; C νn (cos x)

Y ν (ω) π Γ(2ν + n) J ν+n (α) Y ν+n (β) dx = ν−1 , ων 2 n! Γ(ν) αν βν  2   1/2 ω = α + β 2 − 2αβ cos x |α| < |β|,

WA 414(3)a

Re ν > − 12

Re ν −

1 2





ET II 362(29)

ET II 362(30)

800

Complete System of Orthogonal Step Functions

7.335

Integration of Gegenbauer functions with respect to the index  c+i∞ −ν  −1 7.335 [sin(απ)] tα C να (z) dα = −2i 1 + 2tz + t2 c−i∞

[−2 < Re ν < c < 0,

EH I 178(25)

1 sech(πx) ν − + ix K ν− 12 +ix (a) I ν− 12 +ix (b) C ν− 1 +ix (− cos ϕ) dx 2 2 −∞ 2−ν+1 (ab)ν −ν ω K ν (ω) = Γ(ν)  ω = a2 + b2 − 2ab cos ϕ EH II 55(45)

 7.336

|arg (z ± 1)| < π]

∞



7.34 Combinations of Chebyshev polynomials and powers 

1

7.341 −1



 −1 2 [T n (x)] dx = 1 − 4n2 − 1 1

7.342 −1

7.343



1

1. −1

ET II 271(6)

, +  1/2  1/2 1 − z2 U n x 1 − y2 + yz dx =

T n (x) T m (x) √

2 U n (y) U n (z) n+1 [|y| < 1, |z| < 1]

dx =0 1 − x2 π = 2 =π

ET II 275(34)

[m = n] [m = n = 0] [m = n = 0] MO 104



1

2. −1

 1 − x2 U n (x) U m (x) dx = 0 =

7.344



1

1. −1



1

2. −1

7.345



1

1. −1



1

2. −1

π 2

[m = n]

ET II 274(28)

[m = n]

ET II 274(27), MO 105a

 −1/2 (y − x)−1 1 − y 2 T n (y) dy = π U n−1 (x)

[n = 1, 2, . . .]

EH II 187(47)

 1/2 (y − x)−1 1 − y 2 U n−1 (y) dy = −π T n (x)

[n = 1, 2, . . .]

EH II 187(48)

(1 − x)−1/2 (1 + x)m−n− 2 T m (x) T n (x) dx = 0

[m > n]

ET II 272(10)

3

(1 − x)−1/2 (1 + x)m+n− 2 T m (x) T n (x) dx = 3

π(2m + 2n − 2)! − 1)!(2n − 1)!

2m+n (2m

[m + n = 0]

ET II 272(11)

7.349

Chebyshev polynomials and powers



1

3. −1



1

4. −1

5.



1



1

π(2m + 2n + 2)! + 1)!(2n + 1)!

3

(1 − x)1/2 (1 + x)m+n+ 2 U m (x) U n (x) dx =

2m+n+2 (2m

1

(1 − x)1/2 (1 + x)m−n− 2 U m (x) U n (x) dx = 0

[m > n]

801

ET II 274(31)

ET II 274(30)

25/2 (m + 1)(n + 1)   (1 − x)(1 + x)1/2 U m (x) U n (x) dx =  m + n + 32 m + n + 52 [1 − 4(m − n)2 ] −1 ET II 274(29)

6.



1

7.

−1/2

(1 − x)α−1 T m (x) T n (x) dx   1 π 1/2 2α− 2 Γ(α) Γ n − α + 12 1 1 1  1   = −m, m, α, α + ; , α + n + , α − n + F 4 3 1 2 2 2 Γ 2 −α Γ α+n+ 2 [Re α > 0] ET II

(1 + x) −1

(1 + x)

1/2

−1



1

xs−1 T n (x) √

0

7.347



−1



−1

dx π  = s 1 1 2 s2 B 2 + 2 s + 12 n, 12 + 12 s − 12 n 1−x [Re s > 0] β

(1 − x)α (1 + x) T n (x) dx =

1 ;1 2



275(32)

ET II 324(2)

2α+β+2n+1 (n!) Γ(α + 1) Γ (β + 1)

(2n)! Γ(α + β + 2) 1 × 3 F 2 −n, n, α + 1; , α + β + 2; 1 2 [Re α > −1, Re β > −1]

ET II 271(2)

2

1

2.

272(12)

2

1

1.



(1 − x)α−1 U m (x) U n (x) dx

  1 π 1/2 2α− 2 (m + 1) (n + 1) Γ(α) Γ n − α + 32 3  3  = Γ 2 −α Γ 2 +α+n 3 1 3 × 4 F 3 −m, m + 2, α, α − ; , α + n + , α − n − 2 2 2 [Re α > 0] ET II

7.346

1 ;1 2

β

(1 − x)α (1 + x) U n (x) dx =

2α+β+2n+2 [(n + 1)!] Γ(α + 1) Γ (β + 1)

(2n + 2)! Γ(α + β + 2) 3 × 3 F 2 −n, n + 1, α + 1; , α + β + 2; 1 2 ET II 273(22)



1

7.348  7.349

−1 1 

−1



1 − x2

1 − x2

−1/2

−1/2

  U 2n (xz) dx = π P n 2z 2 − 1

[|z| < 1]

  1 T n 1 − x2 y dx = π [P n (1 − y) + P n−1 (1 − y)] 2

ET II 275(33) ET II 222(14)

802

Complete System of Orthogonal Step Functions

7.351

7.35 Combinations of Chebyshev polynomials and elementary functions 

1

7.351 0

   − 1 2a x−1/2 1 − x2 2 e− x T n (x) dx = π 1/2 D n− 12 2a1/2 D −n− 12 2a1/2 [Re a > 0]

7.352 

∞

1. 0



∞

2.

+  −1/2 , x U n a a2 + x2



a−n a+1 −n−1 −2 dx = ζ n + 1, 1 n+1 2n 2 (a2 + x2 ) 2 (eπx + 1) [Re a > 0]

+  −1/2 , x U n a a2 + x2 (a2 + x2 )

0

1 2 n+1

(e2πx − 1)

dx =

2.

ET II 276(40)





 +  ,  1 a+1 a+3 2 2 −1/2 1−2n πx T n a a + x ζ n, dx = 2 − ζ n, 2 4 4 0

a + 1 = 21−n Φ −1, n, 2 [Re a > 0] ET II 273(19)

−2 

 ∞ + ,  2  1   2 1 a+1 2 −2n 2 −1/2 −1 1−n πx a +x Tn a a + x ζ n + 1, cosh dx = π n2 2 2 0 ∞

 2 − 1 n a + x2 2 sech



[Re a > 0] 7.354

ET II 275(39)

a−n−1 a−n 1 ζ(n + 1, a) − − 2 4 2n [Re a > 0]

7.353  1.

ET II 272(13)



+

1

1.

sin(xyz) cos −1



1

sin(xyz) sin −1

1 − y2

1/2 , z T 2n+1 (x) dx = (−1)n π T 2n+1 (y) J 2n+1 (x)

+ 1/2  1/2 , 1/2  1 − y2 1 − x2 z U 2n+1 (x) dx = (−1)n π 1 − y 2 U 2n+1 (y) J 2n+2 (z) ET II 274(25)

1

3.

cos(xyz) cos −1



1/2 

ET II 271(4)

2. 

1 − x2

ET II 273(20)

1

4.

cos(xyz) sin −1

+ 1/2  1/2 , 1 − y2 1 − x2 z T 2n (x) dx = (−1)n π T 2n (y) J 2n (z) +

1 − x2

1/2 

1 − y2

ET II 271(5)

1/2 , 1/2  z U 2n (x) dx = (−1)n π 1 − y 2 U 2n (y) J 2n+1 (z) ET II 274(24)

7.355



1

1. 0

 2.

0

1

π dx T 2n+1 (x) sin ax √ = (−1)n J 2n+1 (a) 2 1 − x2 T 2n (x) cos ax √

π dx = (−1)n J 2n (a) 2 2 1−x

[a > 0]

ET I 94(3)a

[a > 0]

ET I 38(2)a

7.374

Hermite polynomials

803

7.36 Combinations of Chebyshev polynomials and Bessel functions 

1

7.361



 2 −1/2

1−x

0



∞

7.362

1 T n (x) J ν (xy) dx = π J 12 (ν+n) 2

 2 − 1 x − 1 2 Tn

1



1 π W K 2μ (ax) dx = x 2a



1 2 n,μ



1 1 y J 12 (ν−n) y 2 2 [y > 0, Re ν > −n − 1]

ET II 42(1)

(a) W − 12 n,μ (a) [Re a > 0]

ET II 366(17)a

7.37–7.38 Hermite polynomials 

x

7.371 

0

H n (y) dy = [2(n + 1)]−1 [H n+1 (x) − H n+1 (0)] 1

7.372



−1

7.373



1.

x

1 − t2

α− 12

H 2n

EH II 194(27)

 1



√  (−1)n π 1/2 (2n)! Γ α + 2 Lα n (x) xt dx = Γ(n + α + 1)   Re a > − 12

e−y H n (y) dy = H n−1 (0) − e−x H n−1 (x) 2

2

EH II 195(34)

[see 8.956]

EH II 194(26)

0



∞

2. −∞

7.374  1.

∞

−∞



∞

2.11 −∞

e−x H 2m (xy) dx = 2

m √ (2m)!  2 y −1 π m!

e−x H n (x) H m (x) dx = 0 2

EH II 195(28)

[m = n]

√ = 2n · n! π

SM II 567

[m = n]

2 m+n−1 m n e−2x H m (x) H n (x) dx = (−1) 2 + 2  2 2 Γ



m+n+1 2

[m + n is even]

=0 

[m + n is odd] ET II 289(10)a

∞

3. −∞  ∞

4. −∞  ∞

5.

e−x H m (ax) H n (x) dx = 0 2

e−x H 2m+n (ax) H n (x) dx = 2

e −∞



SM II 568



∞

6. −∞

−2α2 x2

[m < n]

ET II 290(20)a

m √ n (2m + n)!  2 a − 1 an π2 m!

ET II 291(21)a

m+n−1 2

−m−n−1



 m+n 2



m+n+1 Γ

2



1 − 2α α α2 1−m−n ; 2 × 2 F 1 −m, n; 2 2α − 1   Re α2 > 0, α2 = 12 , m + n is even

H m (x) H n (x) dx = 2

e−(x−y) H n (x) dx = π 1/2 y n 2n 2

2

ET II 289(12)a

ET II 288(2)a, EH II 195(31)

804

Complete System of Orthogonal Step Functions



∞

7. −∞



∞

8. −∞



∞

7.375

  2 −2y 2 e−(x−y) H m (x) H n (x) dx = 2n π 1/2 m!y n−m Ln−m m %

 n 2 e−(x−y) H n (αx) dx = π 1/2 1 − α2 2 H n

αy

[m ≤ n] &

BU 148(15), ET II 289(13)a ET II 290(17)a

1/2

(1 − α2 )

e−(x−y) H m (αx) H n (αx) dx 2

9. −∞



min(m,n)

= π 1/2

k=0



m 2k k! k

% &



m+n  n  αy −k 1 − α2 2 H m+n−2k 1/2 k (1 − α2 ) ET II 291(26)a



∞

e−

10.

(x−y)2 2u

−∞

7.375  1.

∞

+ , n H n (x) dx = (2πu)1/2 (1 − 2u) 2 H n y(1 − 2u)−1/2   0 ≤ u < 12

e−2x H k (x) H m (x) H n (x) dx = π −1 2 2 (m+n+k−1) Γ(s − k) Γ(s − m) Γ(s − n) 2

−∞

1

2s = k + m + n + 1 

∞

EH II 195(30)

[k + m + n is even]

ET II 290(14)a

m+n+k

e−x H k (x) H m (x) H n (x) dx = 2

2. −∞

2 2 π 1/2 k!m!n! , (s − k)!(s − m)!(s − n)! 2s = m + n + k

[k + m + n is even] ET II 290(15)a

7.376



∞

eixy e−

1. −∞



∞

2.

x2 2

H n (x) dx = (2π)1/2 e−

y2 2

H n (y)in

e−2αx xν H 2n (x) dx = (−1)n 22n− 2 − 2 ν 2

3

1

Γ

0



∞

3.

e−2αx xν H 2n+1 (x) dx = (−1)n 22n− 2 ν 2

1

Γ

0

MO 165a

  ν+1  

Γ n + 12 ν+1 1 1 2 ; ; F −n, √ 1 (ν+1) 2 2 2α πα 2

ν 2

[Re α > 0, Re ν > −1] BU 150(18a)   

+ 1 Γ n + 32 3 1 ν F −n, + 1; ; √ 1 ν+1 2 2 2α 2 πα [Re α > 0,



∞

−∞

7.378 0

∞

BU 150(18b)

e−x H m (x + y) H n (x + z) dx = 2n π 1/2 m!z n−m Ln−m (−2yz) m 2

7.3778 

Re ν > −2]

[m ≤ n]

xα−1 e−βx H n (x) dx = 2n

n

 2 n! Γ(α + n − 2m)

m=0

m!(n − 2m)!

[Re α > 0, if n is even;

ET II 292(30)a

(−1)m 2−2m β 2m−α−n

Re α > −1, if n is odd;

Re β > 0]

ET I 172(11)a

7.385

7.379



∞

−∞  ∞

2. −∞



xe−x H 2m+1 (xy) dx = π 1/2 2

1.

ν

−∞

7.382

m (2m + 1)!  2 y y −1 m!

EH II 195(28)

2

∞

∞

805

xn e−x H n (xy) dx = π 1/2 n!Pn (y)

7.381 

Hermite polynomials

(x ± ic) e

−x2

H n (x) dx = 2

EH II 195(29)

n−1−ν 1/2 Γ

π

 n−ν  2

Γ(−ν)

  exp ± 12 π(ν + n)i

ET II 288(3)a [c > 0] +  √ ,   2 2 1 −1 −x x−1 x2 + a2 e H 2n+1 (x) dx = (−2)n π 1/2 a−2 2ν n! − (2n + 1)!e 2 a D −2n−2 a 2

0

ET II 288(4)a

7.383



∞

1.

e−xp H 2n+1

√  3 x dx = (−1)n 2n (2n + 1)!!π 1/2 (p − 1)n p−n− 2

0



∞

2. 0



∞

3. 0



∞

4.

[Re p > 0] EF 151(261)a, ET I 172(12)a   √ (2n + 1)! (b − α)n e−(b−βx) H 2n+1 (α − β)x dx = (−1)n π α − β 3 n! (b − β)n+ 2 [Re(b − β) > 0]   √ (2n)! (b − α)n 1 √ e−(b−β)x H 2n (α − β)x dx = (−1)n π n! (b − β)n+ 12 x

ET I 172(15)a

[Re(b − β) > 0]

ET I 172(16)a

xa− 2 n−1 e−bx H n 1

0

√    x dx = 2n Γ(a)b−a 2 F 1 − 12 n, 12 − 12 n; 1 − a; b  Re a > 12 n, if n is even, Re a > 12 n − 12 , if n is odd, If a is even, only the first 1 +

:n; 2

terms are kept in the series for

2F 1

⎦

ET I 172(14)a



∞

5.

x−1/2 e−px H 2n

0



∞

7.384 0

√  1 x dx = (−1)n 2n (2n − 1)!!π 1/2 (p − 1)n p−n− 2



1. 0

∞

MO 177a

⎛ ⎞   √

√  n   α+ x a− x 1 2π α ⎠ √ e−bx H n 1 − λ−2 b−1 2 H n ⎝ < + Hn dx = λ λ b x 2 λ −1 b

[Re b > 0] 7.385

Re b > 0, ⎤

  1 + , √ (2n)! Γ b + e−bx n 2n 2 √ x Ln (s) π s (1 − e−x ) dx = (−1) 2 H 2n Γ(n + b + 1) n e −1   Re b > − 12

ET I 173(17)a

ET I 174(23)a

806

Complete System of Orthogonal Step Functions



∞

2.

+√ √ , √ (2n + 1)! Γ(b) b  Ln (s) s 1 − e−x dx = (−1)n 22n πs  Γ n + b + 32

e−bx H 2n+1

0



∞

7.386

x−

n+1 2

7.386

q2

e− 4x H n



0

q √ 2 x



[Re b > 0] e−px dx = 2n π 1/2 p

n−1 2

e−q

ET I 174(24)a

√ p

EF 129(117)

7.387  ∞ √ 2 1 1 2 1. e−x sinh 2βx H 2n+1 (x) dx = 2n− 2 π 1/2 β 2n+1 e 2 β 0  ∞ √ 2 1 2 2. e−x cosh 2βx H 2n (x) dx = 2n−1 π 1/2 β 2n e 2 β

ET II 289(7)a ET II 289(8)a

0

7.388  1.

∞

e−x sin 2

√

1 1 2 2βx H 2n+1 (x) dx = (−1)n 2n− 2 π 1/2 β 2n+1 e− 2 β

√

 n+ 12 − 1 β 2 2βx H 2n+1 (ax) dx = (−1)n 2−1 π 1/2 a2 − 1 e 2 H 2n+1

ET II 288(5)a

0



∞

2.

e

−x2

sin



ET II 290(18)a ∞

3.

√ 2 1 2 e−x cos 2βx H 2n (x) dx = (−1)n 2n−1 π 1/2 β 2n e− 2 β

0



∞

4.

e

−x2

0



ET II 289(6)a

% & √   aβ −1 1/2 2 n − 12 β 2 1−a cos 2βx H 2n (ax) dx = 2 π e H 2n √ 1/2 2 (a2 − 1) ET II 290(19)a

∞

5.

e−y [H n (y)] cos 2

2

√



2βy dy = π 1/2 2n−1 n!e−

β2 2

  Ln β 2

EH II 195(33)

0



∞

6.11

√ 2 b2 e−x sin(bx) H n (x) H n+2m+1 (x) dx = 2n−1 (−1)m πn!b2m+1 e− 4 L2m+1 n



0



∞

7.

 e−x cos(bx) H n (x) H n+2m (x) dx = 2n− 2 2

1

0





√ 1/2 2 (a2 − 1)

0



aβ

π

7.389 0

[b > 0] b2 π n!(−1)m b2m e− 4 L2m n 2



2

b 2

b2 2



[b > 0] + , (2n)! n 1/2 2 dx = 2−n (−1)n π (cos x) H 2n a (1 − sec x) 2 [H n (a)] (n!)



ET I 39(11)a

ET I 39(11)a ET II 292(31)

7.39 Jacobi polynomials 7.391  1.

1

−1

β

(1 − x)α (1 + x) P (α,β) (x) P (α,β) (x) dx n m =0

[m = n,

Re α > −1,

Re β > −1]

[m = n,

Re α > −1,

Re β > −1]

α+β+1

=

Γ(α + n + 1) Γ (β + n + 1) 2 n! (α + β + 1 + 2n) Γ(α + β + n + 1)

ET II 285(5, 9)

7.391

Jacobi polynomials



1

2. −1

σ

(1 − x)ρ (1 + x) P (α,β) (x) dx = n

807

2ρ+σ+1 Γ(ρ + 1) Γ(σ + 1) Γ(n + 1 + α) n! Γ(ρ + σ + 2) Γ(1 + α) × 3 F 2 (−n, α + β + n + 1, ρ + 1; α + 1, ρ + σ + 2; 1)



[Re ρ > −1, 1

3.6 −1



1

−1

5.

(1 − x)ρ (1 + x)β P (α,β) (x) dx = n

7.

Re σ > −1]

ET II 284(1)

2β+ρ+1 Γ(ρ + 1) Γ(β + n + 1) Γ(α − ρ + n) n! Γ(α − ρ) Γ(β + ρ + n + 2) [Re ρ > −1,

Re β > −1]

ET II 284(2)

,2 + 2α+β Γ(α + n + 1) Γ(β + n + 1) (1 − x)α−1 (1 + x)β P (α,β) (x) dx = n n!α Γ(α + β + n + 1) −1 1

ET II 285(6) [Re α > 0, Re β > −1]  2 1 + ,2 24α+β+1 Γ α + 12 [Γ(α + n + 1)] Γ(β + 2n + 1) (1 − x)2α (1 + x)β P (α,β) (x) dx = √ n 2 π (n!) Γ(α + 1) Γ(2α + β + 2n + 2) −1   Re α > − 12 , Re β > −1 ET II 285(7)  1 β (1 − x)ρ (1 + x) P (α,β) (x) P (ρ,β) (x) dx n n



 6.

ET II 284(3)

2α+σ+1 Γ(σ + 1) Γ(α + 1) Γ(σ − β + 1) n! Γ(σ − β − n + 1) Γ(α + σ + n + 2) [Re α > −1,

4. 

(1 − x)α (1 + x)σ P (α,β) (x) dx = n

Re σ > −1]

−1

= 

1

8. −1



(1 − x)ρ−1 (1 + x)β P (α,β) (x) P (ρ,β) (x) dx = n n

2ρ+β Γ(α + n + 1) Γ(β + n + 1) Γ(ρ) n! Γ(α + 1) Γ(ρ + β + n + 1) [Re β > −1,

1

9.7 −1

−1

ET II 286(11)

σ

1

10.6

Re ρ > 0]

(1 − x)α (1 + x) P (α,β) (x) P (α,σ) (x) dx n m =



2ρ+β+1 Γ(ρ + n + 1) Γ(β + n + 1) Γ (α + β + 2n + 1) n! Γ(β + ρ + 2n + 2) Γ(α + β + n + 1) [Re ρ > −1, Re β > −1] ET II285(10)

2α+σ+1 Γ(α + n + 1) Γ (α + β + m + n + 1) Γ(σ + m + 1) Γ(σ − β + 1) m!(n − m)! Γ (α + β + n + 1) Γ(α + σ + m + n + 2) Γ (α − β + m − n + 1) [Re α > −1, Re σ > −1] ET II 286(12)

β

(1 − x)ρ (1 + x) P (α,β) (x) P (ρ,β) (x) dx n m

2β+ρ+1 Γ(α + β + m + n + 1) Γ (β + n + 1) Γ(ρ + m + 1) Γ (α − ρ − m + n) m!(n − m)! Γ(α + β + n + 1) Γ (β + ρ + m + n + 2) Γ(α − ρ) [Re β > −1, Re ρ > −1] ET II 287(16)  x , 1 + (α+1,β+1) (α+1,β+1) P n−1 (1 − y)α (1 + y)β P (α,β) (y) dy = (0) − (1 − x)α+1 (1 + x)β+1 P n−1 (x) n 2n 0 =

11.

EH II 173(38)

808

7.392

Complete System of Orthogonal Step Functions



7.392

1

xλ−1 (1 − x)μ−1 P (α,β) (1 − γx) dx n

1. 0

= 

Γ(α + n + 1) Γ(λ) Γ(μ) 1 γ −n, n + α + β + 1, λ; α + 1, λ + μ; 3F 2 n! Γ (α + 1) Γ(λ + μ) 2 [Re λ > 0, Re μ > 0] ET II 192(46)a

1

xλ−1 (1 − x)μ−1 P (α,β) (γx − 1) dx n

2. 0

+ n + 1) Γ(λ) Γ(μ) 1 3 F 2 −n, n + α + β + 1, λ; β + 1, λ + μ; γ a n! Γ (β + 1) Γ(λ + μ) 2 [Re λ > 0, Re μ > 0] ET II 192(47)a

n Γ(β

= (−1) 

1

xα (1 − x)μ−1 P (α,β) (1 − γx) dx = n

3. 0



Γ(α + n + 1) Γ(μ) (α+μ,β−μ) P (1 − γ) Γ(α + μ + n + 1) n [Re a > −1,

1

xβ (1 − x)μ−1 P (α,β) (γx − 1) dx = n

4. 0



1

1.



 2 ν

1−x

(ν,ν) sin bx P 2n+1 (x) dx

=



1



1 − x2

ν

(ν,ν)

cos bx P 2n (x) dx =

Re μ > 0]

ET II 191(44)a

√ (−1)n π Γ(2n + ν + 2) J 2n+ν+ 32 (b) 2 2 −ν (2n + 1)!bν+ 2 1

0

2.

ET II 191(43)a

Γ(β + n + 1) Γ(μ) (α−μ,β+μ) P (γ − 1) Γ(β + μ + n + 1) n [Re β > −1,

7.393

Re μ > 0]

(−1)n 2

ν− 12

1

[b > 0, Re ν > −1] √ π Γ(2n + ν + 1) J 2n+ν+ 12 (b)

ET I 94(5)

1

(2n)!bν+ 2

0

[b > 0,

Re ν > −1]

ET I 38(4)

7.41–7.42 Laguerre polynomials 7.411



t

Ln (x) dx = Ln (t) − Ln+1 (t)/(n + 1)

1. 0





t α α Lα n (x) dx = Ln (t) − Ln+1 (t) −

2. 0





t

Lα+1 n−1 (x) dx

3.

=

− Lα n (t)

+

0



n+α n

n+α n

MO 110



+

n+1+α n+1

EH II 189(16)a EH II 189(15)a

t

Lm (x) Ln (t − x) dx = Lm+n (t) − Lm+n+1 (t)

4.

EH II 191(31)

0

5.

2 ∞  t  Lk (x) dx = et − 1 k! 0

k=0

[t ≥ 0]

MO 110

7.414

7.412

Laguerre polynomials



1

(1 − x)μ−1 xα Lα n (ax) dx =

1. 0

809

Γ(α + n + 1) Γ(μ) α+μ L (a) Γ(α + μ + n + 1) n [Re α > −1,

Re μ > 0] EH II 191(30)a, BU 129(14c)



1

(1 − x)μ−1 xλ−1 Lα n (βx) dx =

2. 0

Γ(α + n + 1) Γ(λ) Γ(μ) 2 F 2 (−n, λ; α + 1, λ + μ : β) n! Γ(α + 1) Γ(λ + μ) [Re λ > 0,



1

0

7.414  1.11

∞

 α  −y Ln (y) − Lα e−x Lα n (x) dx = e n−1 (y)

y



∞

2.

e

−bx

0

 3.8

 2  b − (λ + μ)b + 2λμ (b − λ − μ)n Ln (λx) Ln (μx) dx = Pn bn+1 b(b − λ − μ)

α e−x xα Lα n (x) Lm (x) dx = 0

0

= 

EH II 191(29)

[Re b > 0] ∞

∞

Γ(α + n + 1) n!

ET I 175(34)

[m = n,

Re α > −1]

BU 115(8), ET II 293(3)

[m = n,

Re α > 0]

BU 115(8), ET II 292(2)

Γ(m + n + α + 1) (b − λ)n (b − μ)m bm+n+α+1  m!n!  0 b(b − λ − μ) × F −m, −n; −m − n − α, (b − λ)(b − μ) [Re α > −1, Re b > 0]      ∞ Γ(α + n + 1)2 Γ(α + m + 1) Γ α + 32 Γ m − 12 α −x α+1/2 α 9   4(1) . e x Ln (x) Lm (x) dx = n!m! Γ(α + 1) Γ − 12 0   × 3 F 2 −n, α + 32 , 32 ; α + 1, 32 − m; 1 4.



α e−bx xα Lα n (λx) Lm (μx) dx =

∞

e

5. 0



−bx

Lan (x) dx

n  a + m − 1 (b − 1)n−m

=

m=0 ∞

6.

m

bn−m+1

e−bx Ln (x) dx = (b − 1)n b−n−1

0



∞

7.

e−st tβ Lα n (t) dt =

Γ(β + 1) Γ(α + n + 1) −β−1 s F n! Γ(α + 1)

e−st tα Lα n (t) dt =

Γ(α + n + 1)(s − 1) n!sα+n+1

0

 8. 0

ET II 192(50)a

(m + n)! Γ(α + m + 1) Γ(β + n + 1) α+β+1 Lm+n (y) m!n! Γ(α + β + m + n + 2) [Re α > −1, Re β > −1] ET II 293(7)

β xα (1 − x)β Lα m (xy) Ln [(1 − x)y] dx =

7.413

Re μ > 0]

ET I 175(35)

[Re b > 0]

ET I 174(27)

[Re b > 0]

ET I 174(25)



1 −n, β + 1; α + 1; s [Re β > −1, Re s > 0] BU 119(4b), EH II 191(133)

∞

n

[Re α > −1,

Re s > 0] EH II 191(32), MO 176a

810

Complete System of Orthogonal Step Functions



∞

9.

e

−x α+β

x



β Lα m (x) Ln (x) dx

m+n

= (−1)

0



∞

10.6

e−bx x2a [Lan (x)] dx = 2

0



∞

11.

e−x xγ−1 Lμn (x) dx =

0



∞

12.

e−x(s+

0

a1 +a2 2

α+m (α + β)! n



β+n m



[Re(α + β) > −1]

    22a Γ a + 12 Γ n + 12

ET II 293(4)

2

2a+1 π (n!) b

2  2 1 1 Γ(a + n + 1) × F −n, a + ; − n; 1 − 2 2 b   1 Re a > − , Re b > 0 2

Γ(γ) Γ(1 + μ + n − γ) n! Γ(1 + μ − γ)

[Re γ > 0]

ET I 174(30) BU 120(4b)

) xμ+β Lμ (a x) Lμ (a x) dx 1 2 k k

⎤⎫ ⎧ ⎡  1+μ+β μ+β A2 ⎬ ⎨ F , 1 + ; 1 + μ; k 2 2 2 B d ⎣ Γ(1 + μ + β) Γ(1 + μ + k) ⎦ = k 1+μ 1+μ+β ⎭ ⎩ dh k!k! Γ (1 + μ) (1 − h) B 4a1 a2 h A = ; (1 − h)2 2

13.

7.415

h=0

a1 + a 2 1 + h B =s+ 1−h 2

 a1 + a2 BU 142(19) > 0, a1 > 0, a2 > 0, Re(μ + β) > −1 Re s + 2  2

  ∞ b1 a1 + a2 Γ(1 + μ + k) bk0 (μ,0) · P exp −x s + · xμ Lμk (a1 x) Lμk (a2 x) dx = k 1+μ+k 2 k! b 0 b2 b0 0 a1 + a2 a1 + a2 b0 = s + , b21 = b0 b2 + 2a1 a2 , b2 = s − 2 2 

 a1 + a 2 Re μ > −1, Re s + >0 2 BU 144(22)



1

Γ(α + n + 1) B(λ, μ) 2 F 2 (α + n + 1, λ; α + 1, λ + μ; −β) n! Γ(α + 1) 0 [Re λ > 0, Re μ > 0] ET II 193(51)a  



 ∞ 1/2   n+m iy iy (2π) 1 2 n−m − 2 √ √ i x dx = 7.416 xm−n exp − (x − y)2 Lm−n 2 H H n m n 2 n! 2 2 −∞ 7.415

(1 − x)μ−1 xλ−1 e−βx Lα n (βx) dx =

BU 149(15b), ET II 293(8)a

7.417  1.

∞

0

 2. 0

xν−2n−1 e−ax sin(bx) Lν−2n−1 (ax) dx = (−1)n i Γ(ν) 2n

b2n [(a − ib)−ν − (a + ib)−ν ] 2(2n)! [b > 0, Re a > 0, Re ν > 2n] ET I 95(12)

∞

−ν

−ν

[(a + ib) + (a − ib) ] 2(2n + 1)! [b > 0, Re a > 0, Re ν > 2n + 1]

xν−2n−2 e−ax sin(bx) Lν−2n−2 (ax) dx = (−1)n+1 Γ(ν) 2n+1

b

2n+1

ET I 95(13)

7.421

Laguerre polynomials



∞

3.

n+1 xν−2n e−ax cos(bx) L2n−1 Γ(ν) ν−2n (ax) dx = i(−1)

0



811

b2n−1 [(a − ib)−ν − (a + ib)−ν ] 2(2n − 1)! [b > 0, Re a > 0, Re ν > 2n − 1] ET I 39(12)

∞

4.

xν−2n−1 e−ax cos(bx) Lν−2n−1 (ax) dx = (−1)n Γ(ν) 2n

0

b2n [(a + ib)−ν + (a − ib)−ν ] 2(2n)! [b > 0, Re ν > 2n, Re a > 0] ET I 39(13)

7.418



∞

1. 0

   1 2 1  i 2 2 [D −n−1 (ib)] − [D −n−1 (−ib)] e− 2 x sin(bx) Ln x2 dx = (−1)n n! √ 2 2π

[b > 0]

2  ∞  2 b π −1 − 12 b2 −n − 12 x2 (n!) e e cos(bx) Ln x dx = 2 Hn √ 2 2 0 [b > 0] 



 ∞ 1 2 π 2n+1 − 1 b2 n+ 12 b2 n+ 12 2n+1 − 12 x2 2 x b x e sin(bx) Ln e Ln dx = 2 2 2 0 

2.

3.



∞

4.

n− 12

x2n e− 2 x cos(bx) Ln 2

1



0



∞

5.

xe− 2 x Lα n 2

1



0



∞

6.

e− 2 x Lα n 2

1



0



1 2 x Ln 2

1 2 −α

1 2 − 1 −α x Ln 2 2

1 2 x 2









1 2 x 2

ET I 95(14)



dx =

ET I 39(14)

[b > 0]

π 2n − 1 b2 n+ 12 1 2 2 b e b Ln 2 2

ET I 95(15)

[b > 0] ET I 39(16)



 π 1/2 1 1 2 1 2 − 12 y 2 α 2 −α y Ln y ye Ln sin(xy) dx = 2 2 2



 π 1/2 1 2 1 2 x cos(xy) dx = e− 2 y L α n 2 2





1 2 −α− 1 y Ln 2 2



ET II 294(11)

1 2 y 2



ET II 294(12)



∞

7.419 0

7.421



∞

1. 0

 2. 0

1

xn+2ν− 2 exp[−(1 + a)x] L2ν n (ax) K ν (x) dx    

π 1/2 Γ n + ν + 12 Γ n + 3ν + 12 1 1 1 F n + ν + , n + 3ν + ; 2ν + 1; − a = 1 2 2 2 2n+2ν+  2 n! Γ (2ν + 1)  1 1 Re a > −2, Re(n + ν) > − 2 , Re(n + 3ν) > − 2 ET II 370(44)

xe− 2 αx Ln 1

2



  1 2 βy 2 (α − β)n − 1 y2 2α βx J 0 (xy) dx = e L n 2 αn+1 2α(β − α) [y > 0,

∞

  2 2−2n−1 2n − 1 y2 y e 4 xe−x Ln x2 J 0 (xy) dx = n!

Re α > 0]

ET II 13(4)a ET II 13(5)

812

Hypergeometric Functions



∞

3.

2n+ν+1 − 12 x2

x

e

Lν+n n

0

7.422



1 2 1 2 2n+ν − 12 y 2 ν+n x J ν (xy) dx = y y e Ln 2 2

Re ν > −1]   ∞   y2 2 αy 2 xν+1 e−βx Lνn αx2 J ν (xy) dx = 2−ν−1 β −ν−n−1 (β − α)n y ν e− 4β Lνn 4β(α − β) 0 [y > 0,

 4.



∞

5.

e− 2q x xν+1 Lνn 2

1

0

6.∗



x2 q n+ν+1 − qy2 ν ν e 2 y Ln J ν (xy) dx = 2q(1 − q) (q − 1)n

∞



∞

1.

xν+1 e−βx

2

y 2

 y2 yν  Γ n + 1 + 12 ν (2β)−ν−1 e− 4β πn!     

2l  n  (−1)l Γ n − l + 12 Γ l + 12 αy 2 2α − β ν   × L 2l β 2β(2α − β) Γ l + 1 + 12 ν (n − l)! l=0 [y > 0,



∞

9

MO 183

+ 1  ,2 ν Ln2 αx2 J ν (xy) dx =

 2 σ  2 2 αx Ln αx J ν (xy) dx xν+1 e−αx Lν−σ m



∞

1.

e− 2 x Ln 1

2



0



∞

2.

e

− 12 x2

0

Ln

Re α > 0,

Re ν > −1,



y2 4α σ = 0,

y2

= (−1)m+n (2α)−ν−1 y ν e− 4α Lm−n−σ n [y > 0,

Re ν > −1]

Re β > 0,

0

7.423

ET II 43(5)

2

  2 1  y 2n+ν − 1 y2 xν+1 e−x Lνn x2 J ν (xy) dx = e 4 2n! 2

0

2.



[ν > 0]

 0

7.422



MO 183

n = 0,

ET II 43(7)



Ln−m+σ−ν m α = 1]

y2 4α



ET II 43(8)







 π 1/2 1 2 1 2 x 1 2 y √ √ x H 2n+1 y H 2n+1 e− 2 y Ln sin(xy) dx = 2 2 2 2 2 2 2

1 2 x H 2n 2



x √ 2 2

cos(xy) dx =

 π 1/2 2

e

− 12 y 2

Ln



1 2 y H 2n 2



ET II 294(13)a

y √ 2 2



ET II 294(14)a

7.5 Hypergeometric Functions 7.51 Combinations of hypergeometric functions and powers 

7.511 0

∞

Γ(a + s) Γ(b + s) Γ(c) Γ(−s) Γ(a) Γ(b) Γ(c + s) [c = 0, −1, −2, . . . , Re s < 0, Re(a + s) > 0,

F (a, b; c; −z)z −s−1 dx =

Re(b + s) > 0]

EH I 79(4)

7.512

Hypergeometric functions and powers

7.512 

1

1. 0



  α α Γ 1+ Γ(γ) Γ (α − γ + 1) Γ γ − − β 2 2   xα−γ (1 − x)γ−β−1 F (α, β; γ; x) dx = α α Γ (1 + α) Γ 1 + − β Γ γ − 2  + 2 , α Re α + 1 > Re γ > Re β, Re γ − − β > 0 ET II 398(1) 2

1

Γ(γ) Γ(ρ) Γ(β − γ + 1) Γ(γ − ρ + n) Γ(γ + n) Γ(γ − ρ) Γ(β − γ + ρ + 1) [n = 0, 1, 2 . . . ; Re ρ > 0, Re(β − γ) > n − 1]

ET II 398(2)

Γ(γ) Γ(ρ) Γ(β − ρ) Γ(γ − α − ρ) Γ(β) Γ(γ − α) Γ(γ − ρ) [Re ρ > 0, Re(β − ρ) > 0, Re(γ − α − ρ) > 0]

ET II 399(3)

Γ(γ) Γ(ρ) Γ(γ + ρ − α − β) Γ(γ + ρ − α) Γ(γ + ρ − β) [Re γ > 0, Re ρ > 0, Re(γ + ρ − α − β) > 0]

ET II 399(4)

Γ(ρ) Γ(σ) 3 F 2 (α, β, ρ; γ, ρ + σ; 1) Γ(ρ + σ) [Re ρ > 0, Re σ > 0, Re(γ + σ − α − β) > 0]

ET II 399(5)

xρ−1 (1 − x)β−γ−n F (−n, β; γ; x) dx =

2. 0



1

xρ−1 (1 − x)β−ρ−1 F (α, β; γ; x) dx =

3. 0



1

xγ−1 (1 − x)ρ−1 F (α, β; γ; x) dx =

4. 0



1

xρ−1 (1 − x)σ−1 F (α, β; γ; x) dx =

5. 0



1

6.10 0

 7.

813

 zx xλ−1 (1 − x)β−λ−1 F α, β; λ; dx = B(λ, β − λ)(1 − z/b)−α b

BU 9

1

xγ−1 (1 − x)δ−γ−1 F (α, β; γ; xz) F (δ − α, δ − β; δ − γ; (1 − x)ζ) dx

11 0

= [0 < Re γ < Re δ, 

1 γ−1

8.

x

(1 − x)

−1

(1 − xz)

0

 9.

1

−δ

Γ(γ) Γ(δ − γ) (1 − ζ)α+β−δ F (α, β; δ; z + ζ − zζ) Γ(δ) |arg(1 − z)| < π, |arg(1 − ζ)| < π] ET II 400(11)

  (1 − x)z F (α, β; γ; xz) F δ, β − γ; ; dx (1 − xz) Γ(γ) Γ() F (α + δ, β; γ + ; z) = Γ(γ + ) [Re γ > 0, Re  > 0, |arg(z − 1)| < π] ET II 400(12), Eh I 78(3)

xγ−1 (1 − x)ρ−1 (1 − zx)−σ F (α, β; γ; x) dx

0

=

Γ(γ) Γ(ρ) Γ(γ + ρ − α − β) (1 − z)−σ Γ(γ + ρ − α) Γ(γ + ρ − β) × 3 F 2 ρ, σ, γ + ρ − α − β; γ + ρ − α, γ + ρ − β;

[Re γ > 0,

Re ρ > 0,

Re (γ + ρ − α − β) > 0,

|arg(1 − z)| < π]

z z−1



ET II 399(6)

814

Hypergeometric Functions



∞

10.

7.513

Γ(γ) Γ(α − γ + σ) Γ (β − γ + σ) Γ(σ) Γ(α + β − γ + σ)

xγ−1 (x + z)−σ F (α, β; γ; −x) dx =

0

× F (α − γ + σ, β − γ + σ; α + β − γ + σ; 1 − z) [Re γ > 0, 

Re(α − γ + σ) > 0,

Re (β − γ + σ) > 0,

|arg z| < π]

ET II 400(10)

1

(1 − x)μ−1 xν−1 p F q (a1 , . . . , ap ; ν, b2 , . . . , bq ; ax) dx

11. 0

Γ(μ) Γ(ν) p F q (a1 , . . . , ap ; μ + ν, b2 , . . . , bq ; a) Γ(μ + ν) p ≤ q + 1; if p = q + 1, then |a| < 1] ET II 200(94) =

[Re μ > 0, 

Re ν > 0,

1

(1 − x)μ−1 xν−1 p F q (a1 , . . . , ap ; b1 , . . . , bq ; ax) dx

12. 0

Γ(μ) Γ(ν) p+1 F q+1 (ν, a1 , . . . , ap ; μ + ν, b1 , . . . , bq ; a) Γ(μ + ν) Re ν > 0, p ≤ q + 1, if p = q + 1, then |a| < 1] ET II 200(95) =

[Re μ > 0, 

1

7.513 0

  ν   s s 1  s xs−1 1 − x2 F −n, a; b; x2 dx = B ν + 1, 3 F 2 −n, a, ; b, ν + 1 + ; 1 2 2 2 2 [Re s > 0, Re ν > −1] ET I 336(4)

7.52 Combinations of hypergeometric functions and exponentials 

∞

7.521

e−st p F q (a1 , . . . , ap ; b1 , . . . , bq , t) dt =

0

7.522  1.11

∞

∞

2.6 0



∞

3. 0



p+1 F q



1, a1 , . . . , ap ; b1 , . . . , bq , s−1



[p ≤ q] e−λx xγ−1 2 F 1 (α, β; δ; −x) dx =

0



1 s

e−bx xa−1 F



EH I 192

Γ(δ)λ−γ E (α, β, γ : δ : λ) Γ(α) Γ(β)

1 1 x + ν, − ν; a; − 2 2 2



[Re λ > 0,

Re γ > 0]

EH I 205(10)

1 1 dx = 2a eb √ Γ(a)(2b) 2 −a K ν (b) π

ET I 212(1) [Re a > 0, Re b > 0] α+β− 12

b b b e−bx xγ−1 F (2α, 2β; γ; −λx) dx = Γ(γ)b−γ e 2λ W 12 −α−β,α−β λ λ [Re b > 0, Re γ > 0, |arg λ| < π] BU 78(30), ET I 212(4)

∞

4.6

e−xt tb−1 F (a, a − c + 1; b; −t) dt = xa−b Γ(b)Ψ(a, c; x)

0

 5.

[Re b > 0, ∞

e−x xs−1 p F q (a1 , . . . , ap , b1 , . . . , bq ; ax) dx = Γ(s)

p+1 F q

Re x > 0]

EH I 273(11)

(s, a1 , . . . , ap ; b1 , . . . , bq ; a)

0

[p < q,

Re s > 0]

ET I 337(11)

7.525

Hypergeometric functions and exponentials



∞

6.

β−1 −μx

x

e

2 F 2 (−n, n

−β

+ 1; 1, β; x) dx = Γ(β)μ

Pn

0



∞

7.

xβ−1 e−μx 2 F 2



0

2 1− μ



[Re μ > 0, Re β > 0]



 1 1 −n, n; β, ; x dx = Γ(β)μ−β cos 2n arcsin √ 2 μ [Re μ > 0,



∞

8.

xρn −1 e−μx

mF n

815

Re β > 0]

ET I 218(6)

ET I 218(7)

(a1 , . . . , am ; ρ1 , . . . , ρn ; λx) dx



λ , . . . , a ; ρ , . . . , ρ ; a m F n−1 1 m 1 n−1 μ Re μ > 0, if m < n; Re μ > Re λ, if m = n] ET I 219(16)a

0

= Γ (ρn ) μ−ρn

[m ≤ n; 

∞

9.

xσ−1 e−μx

mF n

Re ρn > 0,

(a1 , . . . , am ; ρ1 , . . . , ρn ; λx) dx

0



λ , . . . , a , σ; ρ , . . . , ρ ; a F m+1 n 1 m 1 n μ Re μ > 0, if m < n; Re μ > Re λ, if m = n] ET I 219(17) = Γ(σ)μ−σ

[m ≤ n, 

∞

7.523 1

Re σ > 0,

(x − 1)μ−1 x−μ− 2 e− 2 ax W 2μ+ 12 ,λ (ax) dx = Γ(μ)e− 2 a W μ+ 12 ,λ (a) 1

1

1

[Re μ > 0, 7.524



∞

1. 0



∞

2.

e−λx F



1 α, β; ; −x2 dx = λα+β−1 S 1−α−β,α−β (λ) 2

  e−st p F q a1 , . . . , ap ; b1 , . . . , bq ; t2 dx = s−1

0



∞

3.

e−st 0 F q



0

7.525



1. 0

∞

Re a > 0]

xσ−1 e−μx

1 2 q−1 t , ,..., , 1; q q q q q q



ET II 401(13) [Re λ > 0]

1 4 p+2 F q a1 , . . . , ap , 1, ; b1 , . . . , bq ; 2 2 s

[p < q]   dt = s−1 exp s−q

MO 176 MO 176

 a1 , . . . , am ; ρ1 , . . . , ρn ; (λx)k dx  k  kλ σ+k−1 σ σ+1 −σ = Γ(σ)μ m+k F n a1 , . . . , am , , ,..., ; ρ1 , . . . , ρn ; k k k μ  m + k ≤ n + 1, Re σ > 0; Re μ > 0, if m + k ≤ n;   2πi Re μ + kλe k > 0; r = 0, 1, . . . , k − 1 for m + k = n + 1

mF n



ET I 220(19)

816

Hypergeometric Functions



∞

2. 0

7.526

  xe−λx F α, β; 32 ; −x2 dx = λα+β−2 S 1−α−β,α−β (λ) [Re λ > 0]

7.526



γ+i∞

1.

st −b

e s γ−i∞





1 Γ(a + b − c + 1) b−1 t F a, b; a + b − c + 1; 1 − Ψ(a; c; t) dx = 2πi s Γ(b) Γ(b − c + 1)  Re b > 0, Re(b − c) > −1,

γ>

1 2



EH I 273(12)

∞

2. 0



ET II 401(14)

∞

3.

  t(x + y + t) −t γ−1 −α −a  e t (x + t) (y + t) F a, a ; γ; dt = Γ(γ)Ψ(a, c; x)Ψ (a , c; y) , (x + t)(y + t) [Re γ > 0, xy = 0] EH I 287(21) γ = a + a − c + 1 γ−1

x 0

−α

(x + y)

−β −x

(x + z)

e

  x(x + y + z) F α, β; γ; dx (x + y)(x + z) y+z

= Γ(γ)(zy)− 2 −μ e 2 W ν,μ (y) W λ,μ (z) 2ν = 1 − α + β − γ; 2λ = 1 + α − β − γ; 2μ = α + β − γ 1

[Re γ > 0,

|arg y| < π,

|arg z| < π] ET II 401(15)

7.527



∞

1.

 λ−1 −μx   1 − e−x e F α, β; γ; δe−x dx = B(μ, λ) 3 F 2 (α, β, μ; γ, μ + λ; δ)

0

[Re λ > 0, 

∞

2. 0



∞

0



∞

ET I 213(9)

 μ   B(α, μ + n + 1) B(α, β + n − α) 1 − e−x e−αx F −n, μ + β + n; β; e−x dx = B(α, β − α) [Re α > 0,

3.

4.

|arg(1 − δ)| < π]

Re μ > 0,

Re μ > −1]

 γ−1 −μx   Γ(μ) Γ(γ − α − β + μ) Γ(γ) 1 − e−x e F α, β; γ; 1 − e−x dx = Γ(γ − α + μ) Γ(γ − β + μ) [Re μ > 0, Re μ > Re(α + β − γ), Re γ > 0]

ET I 213(10)

ET I 213(11)

 γ−1 −μx    1 − e−x e F α, β; γ; δ 1 − e−x dx = B(μ, γ) F (α, β; μ + γ; δ)

0

[Re μ > 0,

Re γ > 0,

|arg(1 − δ)| < π]

ET I 213(12)

7.542

Hypergeometric and Bessel functions

817

7.53 Hypergeometric and trigonometric functions 7.531



∞

1. 0



μ

3 2 2 −α−β+1 −α−β α+β−2 K α−β c x sin μx F α, β; ; −c x πc μ dx = 2 2 Γ(α) Γ(β)  μ > 0, Re α > 12 ,

∞

2.

cos μx F 0

1 α, β; ; −c2 x2 2



μ

K α−β c Γ(α) Γ(β) [μ > 0, Re α > 0,

dx = 2−α−β+1 πc−α−β μα+β−1

Re β >

1 2



ET I 115(6)

Re β > 0,

c > 0] ET I 61(9)

7.54 Combinations of hypergeometric and Bessel functions 

∞

7.541

xα+β−2ν−1 (x + 1)−ν exz K ν [(x + 1)z] F (α, β; α + β − 2ν; −x) dx

0

= π − 2 cos(νπ) Γ 1

γ = α + β − 2ν



Re(α + β − 2ν) > 0,

1

   1 1 − α + ν Γ 12 − β + ν Γ(γ)(2z)− 2 − 2 γ W 12 γ, 12 (β−α) (2z)  1   1 Re 2 − α + ν > 0, Re 2 − β + ν > 0, |arg z| < 32 π 2

ET II 401(16)

7.542



∞

1.

  xσ−1 p F p−1 a1 , . . . , ap ; b1 , . . . , bp−1 ; −λx2 Y ν (xy) dx

!

y 2 !! b∗0 , . . . , b∗p+1 , 1 4λ ! h, k, a∗1 , . . . , a∗p , l 2λ 2 σ Γ (a1 ) . . . Γ (ap ) σ σ σ a∗j = aj − , j = 1, . . . , p; b∗0 = 1 − ; b∗j = bj − , 2 2 2 ν 1+ν ν j = 1, . . . , p − 1; h = , k = − , l = − 2 2 2   |arg λ| < π, Re σ > |Re ν|, Re aj > 12 Re σ − 34 , y > 0

0

=

 2.

Γ (b1 ) . . . Γ (bp−1 )



p+2,1

G p+2,p+3

ET II 118(53) ∞

  xσ−1 p F p a1 , . . . , ap ; b1 , . . . , bp ; −λx2 Y ν (xy) dx

0

!

y 2 !! b∗0 , . . . , b∗p , l 1 4λ ! h, k, a∗1 , . . . , a∗p , l 2λ 2 σ Γ (a1 ) . . . Γ (ap ) ν ν 1+ν σ bj ∗ = bj − ; j = 1, . . . , p; h = , k = − , l = − 2 2 2 2   Re λ > 0, Re σ > |Re ν|, Re aj > 12 Re σ − 34 , y > 0 =

b∗0 = 1 −

σ ; 2

a∗j = aj −

σ , 2

Γ (b1 ) . . . Γ (bp )



p+2,1 G p+2,p+3

ET II 119(54)

818

Hypergeometric Functions



∞

3.

7.542

  xσ−1 p F q a1 , . . . , ap ; b1 , . . . , bq ; −λx2 Y ν (xy) dx

, σ + ν σ − ν

= −π 2 y cos (σ − ν) Γ Γ 2 2 2

σ+ν σ−ν 4λ , ; b1 , . . . , bq ; − 2 × p+2 F q a1 , . . . , ap , 2 2 y [y > 0, p ≤ q − 1, Re σ > |Re ν|] ET II 119(55)

0

+π

−1 σ−1 −σ



∞

  xσ−1 p F q a1 , . . . , ap ; b1 , . . . , bq ; −λx2 K ν (xy) dx



σ+ν σ−ν σ+ν σ−ν 4λ σ−2 −σ , ; b1 , . . . , bq ; 2 =2 y Γ Γ p+2 F q a1 , . . . , ap , 2 2 2 2 y [Re y > 0, p ≤ q − 1, Re σ > |Re ν|] ET II 153(88)

∞

  x2ρ p F p a1 , . . . , ap ; b1 , . . . , bp ; −λx2 J ν (xy) dx

4. 0

 5. 0

 

∞

6.

x2ρ

m+1 F m

0

 

∞

7.

y > 0,

y > 0,



2!

y !! 1, b1 , . . . , bp 22ρ Γ (b1 ) . . . Γ (bp ) p+1,1 = 2ρ+1 G y Γ (a1 ) . . . Γ (ap ) p+1,p+2 4λ ! h, a1 , . . . , ap , k h = 12 + ρ + 12 ν, k = 12 + ρ − 12 ν 1 Re λ > 0, −1 − Re ν < 2 Re ρ < 2 + 2 Re ar , r = 1, . . . , p ET II 91(18)

 a1 , . . . , am+1 ; b1 , . . . , bm ; −λ2 x2 J ν (xy) dx

2 !

y !! 1, b1 , . . . , bm 22ρ Γ (b1 ) . . . Γ (bm ) y −2ρ−1 m+2,1 = G m+1,m+3 Γ (a1 ) . . . Γ (am+1 ) 4λ2 ! h, a1 , . . . , am+1 , k 1 h = 2 + ρ + 12 ν, k =12 + ρ − 12 ν, 1 Re λ > 0, Re(2ρ + ν) > −1, Re (ρ − ar ) < 4 ; r = 1, . . . , m + 1 ET II 91(19)

  xδ F α, β; γ; −λ2 x2 J ν (xy) dx

 

∞

8. 0

 9. 0

∞

! ⎞ ! ! 1 − α, 1 − β ! 1+δ−ν ⎠ !1+δ+ν , 0, 1 − γ, ! 2  2 −1 − Re ν − 2 min (Re α, Re β) < Re δ < − 12 ET II 82(9)

⎛ 2δ Γ(γ) −δ−1 22 ⎝ y 2 = y G 24 Γ(α) Γ(β) 4λ2

0

y > 0,

Re λ > 0,

! ⎞ ! 1, γ !   y Γ(γ) y 2 31 xδ F α, β; γ; −λ2 x2 J ν (xy) dx = G 24 ⎝ 2 !! 1 + δ + ν 1+δ−ν ⎠ Γ(α) Γ(β) 4λ ! , α, β, 2  2 y > 0, Re λ > 0, − Re ν − 1 < Re δ < 2 max (Re α, Re β) − 12 ET II 81(6) δ −δ−1

⎛

2

!

  2ν+1 Γ(γ) −ν−2 30 y 2 !! γ y xν+1 F α, β; γ; −λ2 x2 J ν (xy) dx = G 13 Γ(α) Γ(β) 4λ2 ! ν + 1, α, β  y > 0, Re λ > 0, −1 < Re ν < 2 max (Re α, Re β) − 32 ET II 81(5)

7.542

Hypergeometric and Bessel functions



∞

y   2ν−α−β+2 Γ(ν + 1) α+β−ν−2 y xν+1 F α, β; ν + 1; −λ2 x2 J ν (xy) dx = K α−β λα+β Γ(α) Γ(β) λ  ET II 81(3) y > 0, Re λ > 0, −1 < Re ν < 2 max (Re α, Re β) − 32

∞

y   xν+1 F α, β; ν + 1; −λ2 x2 K ν (xy) dx = 2ν+1 λ−α−β y α+β−ν−2 Γ(ν + 1) S 1−α−β,α−β λ [Re y > 0, Re λ > 0, Re ν > −1]

10. 0

 11. 0





∞

12.

ν+1

x 0



∞

13. 0



∞

14.



ET II 152(86)



 y 2 Γ β+ν+2 y β−1 λ−ν−β−1 2 β+ν 2 2 + 1; −λ x Jν (xy) dx = F α, β; K 12 (ν−β+1) 1 β−1 2 2λ π 2 Γ(α)   Γ(β)2  y > 0, −1 < Re ν < 2 max (Re α, Re β) − 32 ET II 81(4)

!

1   1 λ−σ−1 y − 2 Γ(γ) 41 y 2 !! 1 − p, γ − p, l xσ+ 2 F α, β; γ; −λ2 x2 Y ν (xy) dx = √ G 35 4λ2 ! h, k, α − p, β − p, l 2 Γ(α) Γ(β) 1 1 1 1 h = 4 + 2 ν, k = 4 − 2 ν, l = − 14 − 12 ν,  p = 12 + 12 σ  y > 0, Re λ > 0, Re σ > |Re ν| − 32 , Re σ < 2 Re α, Re σ < 2 Re β ET II 118(52) ν+2

x

F

0



819

y y 1 1 3 2ν y −ν−1 2 2 , − ν; ; −λ x Y ν (xy) dx = 1 K ν+1 1  Kν 2 2 2 2λ 2λ π 2 λ2 Γ 2 − ν  y > 0, Re λ > 0, − 32 < Re ν < − 12 ET II 117(49)

∞

15. 0



 y ,2 1 Γ(ν + 2) + 3  K xν+2 F 1, 2ν + ; ν + 2; −λ2 x2 Y ν (xy) dx = π − 2 2−ν λ−2ν−3  ν 2 2λ Γ 2ν + 32   1 y > 0, Re λ > 0, − 2 < Re ν < 12

ET II 117(50)



1 ∞ y 3 3 π 2 2−μ−ν−1 λ−μ−2ν−3 y μ+ν ν+2 2 2   x F 1, μ + ν + ; ; −λ x Y ν (xy) dx = K μ 2 2 λ Γ μ + ν + 32 0   3 1 3 y > 0, Re λ > 0, − 2 < Re ν < 2 , Re(2μ + ν) > − 2 ET II 118(51)

 16.



1 2 2 x F α − ν − , α; 2α; −λ x J ν (xy) dx 2  0   1 y +   y , i Γ 2 + α Γ 12 + α + ν y W 12 −α,− 12 −ν e−iπ − W 12 −α,− 12 −ν eiπ W 12 −α,− 12 −ν = 1−ν−2α 2α−1 ν+2 π2 λ y λ λ λ   ET II 80(1) y > 0, Re λ > 0, Re ν < − 12 , Re(α + ν) > − 12

 17.

 18. 0

∞

∞

2α+ν

x2α−ν F



1 ν + α − , α; 2α; −λ2 x2 J ν (xy) dx 2   y 22α−ν Γ 12 + α y ν−2 1 1 M = W α− ,ν− 2 2 λ2α−1 Γ(2ν) λ

y 1 1 2 −α,ν− 2

λ

ET II 80(2)

820

7.543

Confluent Hypergeometric Functions



∞

1.

−2α−1

x

F

0



∞

2.

x 0

1 4λ2 + α, 1 + α; 1 + 2α; − 2 J ν (xy) dx = λ−2α I 12 ν+α (λy) K 12 ν−α (λy) 2   x y > 0, Re λ > 0, Re ν > −1, Re α > − 12 ET II 81(7)

ν+1−4α

F

λ2 1 α, α + ; ν + 1; − 2 2 x 



∞

7.544 0

7.543



y > 0,

J ν (xy) dx



1 1 Γ(ν) ν 1−2α 2α−ν−1 = 2 λ λy K 2α−ν−1 λy y Iν Γ(2α) 2 2  3 Re λ > 0, Re α − 1 < Re ν < 4 Re α − 2 ET II 81(8)

  4x 1 xν+1 (1 + x)−2α F α, ν + ; 2ν + 1; J ν (xy) dx 2 (1 + x)2 Γ(ν + 1) Γ(ν − α + 1) 2ν−2α+1 2(α−ν−1) 2 y J ν (y) = Γ(α)   y > 0, −1 < Re ν < 2 Re α − 32 ET II 82(10)

7.6 Confluent Hypergeometric Functions 7.61 Combinations of confluent hypergeometric functions and powers 7.611



∞

1. 0



∞

2.

3

x−1 W k,μ (x) dx =

Γ

3 4

π 2 2k sec(μπ)    − 12 k + 12 μ Γ 34 − 12 k − 12 μ   |Re μ| < 12

x−1 M k,μ (x) W λ,μ (x) dx =

0

 3.

5.



Re μ > − 12 ,

Re(k − λ) > 0



BU 116(11), ET II 409(39) ∞

x−1 W k,μ (x) W λ,μ (x) dx

% & 1 1 1     − 1    = (k − λ) sin(2μπ) Γ 12 − k + μ Γ 12 − λ − μ Γ 2 − k − μ Γ 12 − λ + μ   BU 116(12), ET II 409(40) |Re μ| < 12  1  1  ∞ ψ 2 +μ−κ −ψ 2 −μ−κ π 2 dz     = {W κ,μ (z)} z sin 2πμ Γ 12 + μ − κ Γ 12 − μ − κ 0   BU 117(12a) |Re μ| < 12    ∞  1 ψ −κ 1 2 [W κ,0 (z)] dx =   2 BU 117(12b) 2 0 z Γ 12 − κ 0

4.

Γ(2μ + 1)   (k − λ) Γ 12 + μ − λ

ET II 406(22)

7.613

Confluent hypergeometric functions and powers



∞

6.

ρ−1

x 0



    Γ(ρ + 1) Γ 12 ρ + 12 + μ Γ 12 ρ + 12 − μ     W k,μ (x) W −k,μ (x) dx = 2 Γ 1 + 12 ρ + k Γ 1 + 12 ρ − k [Re ρ > 2|Re μ| − 1]

∞

7.11

=

∞

Γ(1 − μ + ν + ρ) Γ(1 + μ + ν + ρ) Γ(−2ν)     Γ 12 − λ − ν Γ 32 − k + ν + ρ

3 1 × 3 F 2 1 − μ + ν + ρ, 1 + μ + ν + ρ, − λ + ν; 1 + 2ν, − k + ν + ρ; 1 2 2 Γ(1 + μ − ν + ρ) Γ(1 − μ − ν + ρ) Γ(2ν)     + Γ 12 − λ + ν Γ 32 − k − ν + ρ

3 1 × 3 F 2 1 + μ − ν + ρ, 1 − μ − ν + ρ, − λ − ν; 1 − 2ν, − k − ν + ρ; 1 2 2 [|Re μ| + |Re ν| < Re ρ + 1] ET II 410(42)

tb−1 1 F 1 (a; c; −t) dt =

0



∞

2.

ET II 409(41)

xρ−1 W k,μ (x) W λ,ν (x) dx

0

7.612  1.

821

tb−1 Ψ(a, c; t) dt =

0

Γ(b) Γ(c) Γ(a − b) Γ(a) Γ(c − b)

[0 < Re b < Re a]

Γ(b) Γ(a − b) Γ(b − c + 1) Γ(a) Γ(a − c + 1)

EH I 285(10)

[0 < Re b < Re a

Re c < Re b + 1] EH I 285(11)

7.613



t

xγ−1 (t − x)c−γ−1 1 F 1 (a; γ; x) dx = tc−1

1. 0



BU 9(16)a, EH I 271(16) t

xβ−1 (t − x)γ−1 1 F 1 (t; β; x) dx =

2. 0



Γ(γ) Γ(c − γ) 1 F 1 (a; c; t) Γ(c) [Re c > Re γ > 0]



1

xλ−1 (1 − x)2μ−λ 1 F 1

3. 0

Γ(β) Γ(γ) β+γ−1 t 1 F 1 (t; β + γ; t) Γ(β + γ)

1 + μ − ν; λ; xz 2



[Re β > 0,

1

t

xβ−1 (t − x)δ−1 1 F 1 (t; β; x) 1 F 1 (γ; δ; t − x) dx = 0

0

Re(2μ − λ) > −1]

Γ(β) Γ(δ) β+δ−1 t 1 F 1 (t + γ; β + δ; t) Γ(β + δ) [Re β > 0, Re δ > 0] ET II 402(2), EH I 271(15)

t

1

1

xμ− 2 (t − x)ν− 2 M k,μ (x) M λ,ν (t − x) dx =

5.

1

BU 14(14)

4.



ET II 401(1)

dx = B(λ, 1 + 2μ − λ)e 2 z z − 2 −μ M ν,μ (z) [Re λ > 0,



Re γ > 0]

Γ(2μ + 1) Γ(2ν + 1) μ+ν t M k+λ,μ+ν+ 12 (t) Γ(2μ +2ν + 2)  1 1 Re μ > − , Re ν > − 2 2 BU 128(14), ET II 402(7)

822

Confluent Hypergeometric Functions



7.621

1

xβ−1 (1 − x)σ−β−1 1 F 1 (α; β; λx) 1 F 1 [σ − α; σ − β; μ(1 − x)] dx

6. 0

Γ(β) Γ(σ − β) λ e 1 F 1 (α; σ; μ − λ) Γ(σ) [0 < Re β < Re σ] ET II 402(3) =

7.62–7.63 Combinations of confluent hypergeometric functions and exponentials 7.621



∞

1.

e

 

Γ α + ν + 32 1 2 3 t M μ,ν (t) dt =  α+ν+ 32 F α + ν + 2 , −μ + ν + 2 ; 2ν + 1; 2s + 1 1 2 +s     Re α + μ + 32 > 0, Re s > 12

−st α

0



BU 118(1), MO 176a, EH I 270(12)a ∞

2. 0



e−st tμ− 2 M λ,μ (qt) dt = q μ+ 2 Γ(2μ + 1) s − 12 q 1

1

λ−μ− 12 

s + 12 q

−λ−μ− 12

 1 Re μ > − , 2

|Re q| Re s > 2



BU 119(4c), MO 176a, EH I 271(13)a



3.

4.

  

−α−μ− 32 1  ∞ Γ α + μ + 32 Γ α − μ + 32 q μ+ 2 1 −st α e t W λ,μ (qt) dt = s+ q Γ(α − λ + 2) 2 0

1 2s − q 3 × F α + μ + , μ − λ + ; α − λ + 2; 2 2  2s + q 

3 q EH I 271(14)a, BU 121(6), MO 176 Re α ± μ + > 0, Re s > − , q > 0 2 2  ∞   [|s| > |k|] e−st tb−1 1 F 1 (a; c; kt) dt = Γ(b)s−b F a, b; c; ks−1

0 k = Γ(b)(s − k)−b F c − a, b; c; [|s − k > |k||] k−s [Re b > 0, Re s > max (0, Re k)] EH I 269(5) 

∞

5.

 −a tc−1 1 F 1 (a; c; t)e−st dt = Γ(c)s−c 1 − s−1

[Re c > 0,

Re s > 1]

EH I 270(6)

0



∞

6.

Γ(b) Γ(b − c + 1) F (b, b − c + 1; a + b − c + 1; 1 − s) Γ(a + b − c + 1) [Re b > 0, Re c < Re b + 1,  Γ(b) Γ(b − c + 1) −b  = s F a, b; a + b − c + 1; 1 − s−1 Γ(a + b − c + 1)

tb−1 Ψ (a, c; t) e−st dt =

0

 7. 0

∞

e− 2 x xν−1 M κ,μ (bx) dx = b

1



|1 − s| < 1] 

Re s >

1 2



EH I 270(7)

Γ(1 + 2μ) Γ (κ − ν) Γ 2 + μ + ν ν     b Γ 12 + μ + κ Γ 12 + μ − ν   Re ν + 12 + μ > 0,

Re (κ − ν) > 0



BU 119(3)a, ET I 215(11)a

7.622

Confluent hypergeometric functions and exponentials



∞

8.

e

−sx

0

2 Γ(1 + 2μ)e−iπκ dx   = M κ,μ (x) x Γ 12 + μ + κ



s− s+

1 2 1 2

823

κ2 Q κμ− 1 (2s) 2  1  Re 2 + μ > 0,

Re s >

1 2

 BU 119(4a)



∞

9.

e−sx W κ,μ (x)

0

π dx  πμ = x cos 2

s− s+

κ 1 2

2 1 2

P κμ− 1 (2s) 2



Re

1 2

 ± μ > 0,

Re s > − 12

   ∞ Γ k + μ + 12 Γ 14 (2k + 6μ + 5) k+2μ−1 − 32 x     x e W k,μ (x) dx = k + 3μ + 12 Γ 14 (2μ − 2k + 3) 0 Re(k + μ) > − 12 ,



∞

11. 0



∞

12. 0



  Γ ν+ −μ Γ ν+ − 12 x ν−1 e x W κ,μ (x) dx = Γ (ν − κ + 1) 1 2

1 2

 Re ν +

Re(k + 3μ) > − 12



BU 122(8a), ET II 406(23)

 +μ 

 BU 121(7)



 10.



1 2

  ±μ >0

    Γ 12 + μ + ν Γ 12 − μ + ν 1 x ν−1    e2 x W κ,μ (x) dx = Γ (−κ − μ)  1 Γ 2 − μ − κ Γ 12 +μ − κ  Re ν + 12 ± μ > 0,

BU 122(8b)

Re (κ + ν) < 0



BU 122(8c)a

7.622



∞

1.

e−st tc−1 1 F 1 (a; c; t) 1 F 1 (α; c; λt) dt

0

  = Γ(c)(s − 1)−a (s − λ)−α sa+α−c F a, α; c; λ(s − 1)−1 (s − λ)−1 [Re c > 0,



∞

2.

Re s > Re λ + 1]

EH I 287(22)

e−t tρ 1 F 1 (a; c; t)Ψ (a ; c ; λt) dt

0

=C

 Γ(c) Γ(β) σ  λ F c − a, β; γ; 1 − λ−1 , Γ(γ) ρ = c − 1,

ρ = c + c − 2,

σ = −c,

β = c − c + 1,

σ = 1 − c − c ,

γ = c − a + a − c + 1,

β = c + c − 1,

γ = a − a + c,

C=

C=

Γ (a − a) , or Γ (a )

Γ (a − a − c + 1) Γ (a − c + 1) EH I 287(24)

824

Confluent Hypergeometric Functions



∞

3. 0

7.623

xν−1 e−bx M λ1 ,μ1 − 12 (a1 x) . . . M λn ,μn − 12 (an x) dx = aμ1 1 . . . aμnn (b + A)−ν−M Γ(ν + M )

an a1 ,..., × F A ν + M ; μ1 − λ1 , . . . , μn − λn ; 2μ1 , . . . , 2μn ; , b+A b+A A = 12 (a1 + · · · + an ) M = μ1 + · · · + μn ,  1 1 Re(ν + M ) > 0, Re b ± 2 a1 ± · · · + 2 an > 0 ET I 216(14)

7.623



∞

1.

e−x xc+n−1 (x + y)−1 1 F 1 (a; c; x) dx = (−1)n Γ(c) Γ(1 − a)y c+n−1 Ψ(c − a, c; y)

0

3.

t

−1

k−1

|arg y| < π]

EH I 285(16)

1 2 (t−x)

ET II 402(6)

1

   t Γ(λ) Γ 2 − k − λ + μ Γ 12 − k − λ − μ 1 1  1  x−k−λ−1 (t − x)λ−1 e 2 x W k,μ (x) dx = W k+λ,μ (t) tk+1 Γ 2 − k + μ Γ 2 − k − μ 0  Re λ > 0, Re(k + λ) < 12 − |Re μ|

 4.

n = 0, 1, 2, . . . ,



1 Γ(k) Γ(2μ + 1) 1 k− 1 2 2   t x (t − x) e M k,μ (x) dx = lμ 1 π t 2 Γ k + μ + 0 2   Re k > 0, Re μ > − 12 ET II 402(5)   k+λ−1  t 1 Γ(λ) Γ k + μ + 2 t 1   xk−1 (t − x)λ−1 e 2 (t−x) M k+λ,μ (x) dx = M k,μ (t) 1 Γ k+λ+ 0  μ+ 2  Re(k + μ) > − 12 , Re λ > 0 

2.

[− Re c < n < 1 − Re a,

ET II 405(21)



∞

5. 1



∞

6.11 1

 7.

  Γ(μ) Γ 12 − k − λ − μ − 1 μ 1 a μ−1 λ− 12 12 ax 1  (x − 1) x e W k,λ (ax) dx = a 2 e 2 W k+ 12 μ,λ+ 12 μ (a) Γ − k − λ 2   |arg(a)| < 32 π, 0 < Re μ < 12 − Re(k + λ) ET II 211(72)a (x − 1)μ−1 xμ− 2 e− 2 ax W 2μ+ 12 ,λ (ax) dx = Γ(μ)e− 2 a W μ+ 12 ,λ (a) 1

1

1

[Re μ > 0, ∞

Re a > 0]

ET II 211(74)a

Re a > 0]

ET II 211(73)a

(x − 1)μ−1 xk−μ−1 e− 2 ax W k,λ (ax) dx = Γ(μ)e− 2 a W k−μ,λ (a) 1

1

1

[Re μ > 0,

7.624

Confluent hypergeometric functions and exponentials

 8.

1

825

(1 − x)μ−1 xk−μ−1 e− 2 ax W k,λ (ax) dx 1

0

= Γ(μ)e− 2 a sec[(k − μ − λ)π]     Γ k − μ + λ + 12 × sin(μπ) M k−μ,λ (a) + cos[(k − λ)π] W k−μ,λ (a) Γ(2λ + 1)   0 < Re μ < Re k − |Re λ| + 12 ET II 200(93)a 1

7.624



∞

1.

+ 1 , 1 2σ 1 xρ−1 x 2 + (a + x) 2 e− 2 x M k,μ (x) dx

  ! ! 1 , 1, 1 − k + ρ −σ Γ(2μ + 1)aσ !2 23 = 1   G 34 a ! 1 ! 2 + μ + ρ, −σ, σ, 12 − μ + ρ π 2 Γ 12 + k + μ   1 |arg a| < π, Re(μ + ρ) > − 2 , Re(k − ρ − σ) > 0 ET II 403(8)

0



∞

2.

ρ−1

x 0



∞

3.

  ! ! 1 , 1, 1 − k + ρ + 1 , 1 2σ 1 1 ! − x − σ 32 2 x 2 + (a + x) 2 e 2 W k,μ (x) dx = −π 2 σa G 34 a ! 1 ! 2 + μ + ρ, 12 − μ + ρ, −σ, σ   |arg a| < π, Re ρ > |Re μ| − 12 ET II 406(24)

+ 1 , 1 2σ 1 xρ−1 x 2 + (a + x) 2 e− 2 x W k,μ (x) dx

  ! 1 ! 1 , 1, 1 + k + ρ σπ − 2 aσ ! 2    G 33 = − 1 34 a ! 1 ! 2 + μ + ρ, 12 − μ + ρ, −σ, σ Γ 2 − k + μ Γ 12 − k − μ   |arg a| < π, Re ρ > |Re μ| − 12 , Re(k + ρ + σ) < 0 ET II 406(25)

0



∞

4.

+ 1 , 1 1 2σ 1 xρ−1 (a + x)− 2 x 2 + (a + x) 2 e− 2 x M k,μ (x) dx

0

! 1 1

! 0, , − k − ρ Γ(2μ + 1)aσ 23 2 2 ! a   G 34 1 ! −σ, ρ + μ, ρ − μ, σ π 2 Γ 12 + k + μ  Re(ρ + μ) > − 12 , Re(k − ρ − σ) > − 12 ET II 403(9) =

 |arg a| < π, 

∞

5.

+ 1 , 1 1 2σ 1 xρ−1 (a + x)− 2 x 2 + (a + x) 2 e− 2 x W k,μ (x) dx

! 1 1

1 ! 0, , + k + ρ π − 2 aσ 33 2 2 !    = G 34 a ! −σ, ρ + μ, ρ − μ, σ Γ 2 − k + μ Γ 12 − k − μ   1 ET II 406(26) |arg a| < π, Re ρ > |Re μ| − 2 , Re(k + ρ + σ) < 12

0

1

 6.

∞

ρ−1

x 0

− 12

(a + x)

+

1 2

x + (a + x)

1 2

,2σ e

− 12 x

! 1 1

! 0, , − k + ρ 2 2 ! W k,μ (x) dx = π a a! −σ, ρ + μ, ρ − μ, σ  |arg a| < π, Re ρ > |Re μ| − 12 ET II 406(27) − 12

σ

32 G 34

826

7.625

Confluent Hypergeometric Functions



∞

1. 0

  xρ−1 exp − 12 (α + β)x M k,μ (αx) W λ,ν (βx) dx =



∞

2.

ρ−1

x 0



∞

3.

ρ−1

x 0

 0



Γ(1 + μ + ν + ρ) Γ(1 + μ − ν + ρ) μ+ 1 −μ−ρ− 1 2 3  α 2β Γ − λ + μ + ρ 2

1 3 α + k + μ, 1 + μ + ν + ρ, 1 + μ − ν + ρ; 2μ + 1, − λ + μ + ρ; − × 3F 2 2 2 β [Re α > 0, Re β > 0, Re (ρ + μ) > |Re ν| − 1] ET II 410(43)

  1 exp − (α + β)x W k,μ (αx) W λ,ν (βx) dx 2         −1 = β −ρ Γ 12 − k + μ Γ 12 − k − μ Γ 12 − λ + ν Γ 12 − λ − ν   ! β !! 12 + μ, 12 − μ, 1 + λ + ρ 33 × G 33 ! α ! 12 + ν + ρ, 12 − ν + ρ, −k [|Re μ| + |Re ν| < Re ρ + 1, Re(k + λ + ρ) < 0] ET II 410(44)a   !   1 β !! 12 + μ, 12 − ν, 1 − λ + ρ −ρ 22 exp − (α + β)x W k,μ (αx) W λ,ν (βx) dx = β G 33 ! 2 α ! 12 + ν + ρ, 12 − ν + ρ, k ET II 411(46)

∞

4.

7.626

1

  1 ρ−1 x exp − (α − β)x W k,μ (αx) W λ,ν (βx) dx 2   !      β !! 12 + μ, 12 − μ, 1 + λ + ρ −1 −ρ 23 1 1 Γ 2 −λ+ν Γ 2 −λ−ν =β ! G 33 α ! 12 + ν + ρ, 12 − ν + ρ, k [Re α > 0, |Re μ| + |Re ν| < Re ρ + 1] ET II 411(45)



1. 0

   k 1 1 − (ξ + η) exp − (ξ + η)x xc 1 F 1 (a; c; ξx) 1 F 1 (a; c; ηx) dx x 4 2 =0 a 2 = e−ξ [ 1 F 1 (a + 1; c; ξ)] ξ [where ξ and η are any two zeros of the function

 2. 1

7.625

∞



[ξ = η,

Re c > 0]

[ξ = η,

Re c > 0]

1F 1

(a; c; x)]

 1 k 1 − (ξ + η) e− 2 (ξ+η)x xc Ψ (a, c; ξx) Ψ(a, c; ηx) dx = 0 x 4 = −ξ −1 e−ξ [Ψ(a − 1, c; ξ)]2 [where ξ and η are any two zeros of the function Ψ(a, c; x)]

EH I 285

[ξ = η] ; [ξ = η] EH I 286

7.627

7.627

Confluent hypergeometric functions and exponentials



∞

1.

2λ−1

x

−μ− 12

(a + x)

e

1 2x

0



∞

2.

  Γ(2λ) Γ 12 − k + μ − 2λ λ−μ− 1 2 W

a W k,μ (a + x) dx = k+λ,μ−λ (a) 1 −k+μ Γ 2   1 ET II 411(50) |arg a| < π, 0 < 2 Re λ < − Re(k + μ) 2 −1x

x2λ−1 (a + x)−μ− 2 e− 2 x M k,μ2 (a + x) dx 1

1

  Γ(2λ) Γ(2μ + 1) Γ k + μ − 2λ + 12 λ−μ− 1 2 M   a = k−λ,μ−λ (a) 1 Γ(1 − 2λ + 2μ) Γ k+ μ + 2   Re λ > 0, Re(k + μ − 2λ) > − 12 ET II 405(20)

0



827

∞

3.

x2λ−1 (a + x)−μ− 2 e− 2 x W k,μ (a + x) dx = Γ(2λ)aλ−μ− 2 W k−λ,μ−λ (a) 1

1

1

0



[|arg a| < π, ∞

4.

Re λ > 0]

ET II 411(47)

xλ−1 (a + x)k−λ−1 e− 2 x W k,μ (a + x) dx = Γ(λ)ak−1 W k−λ,μ (a) 1

0



∞

5. 0



[|arg a| < π, Re λ > 0] ET II 411(48) !

! 0, 1 − k − σ 1 1 ! xρ−1 (a + x)−σ e− 2 x W k,μ (a + x) dx = Γ(ρ)aρ e 2 a G 30 a 23 ! −ρ, 1 + μ − σ, 1 − μ − σ 2 2 [|arg a| < π,

∞

6.

1

=

∞

7.

e

− 12 (a+x) (a

0

 8. 0

∞

ET II 411(49)

xρ−1 (a + x)−σ e 2 x W k,μ (a + x) dx

0



Re ρ > 0]

2

!

1 ! k − σ + 1, 0 Γ(ρ)aρ e− 2 a !  1  G 31 a 23 ! −ρ, 1 + μ − σ, 1 − μ − σ −k+μ Γ 2 −k−μ 2 2 [|arg a| < π, 0 < Re ρ < Re(σ − k)] ET II 412(51)

Γ + x)2κ−1 dx = W κ,μ (x) κ (ax) x

1 2

   − μ − κ Γ 12 + μ − κ W κ,μ (a) a Γ (1 − 2κ)  1   Re 2 ± μ − κ > 0

BU 126(7a)

dx (x + a)α 

Γ(1 + 2μ) Γ 12 + μ + γ Γ (κ − γ) 1 1 1  1  + μ − γ, − μ − γ; a = α, κ − γ; F 2 2 2 2 Γ 2 + μ − γ Γ  2 +  μ + κ1  1 Γ α + γ + 2 + μ Γ −γ − 2 − μ γ+ 1 +μ a 2 + Γ(α)

1 3 1 × 2 F 2 α + γ + μ + , κ + μ + ; 1 + 2μ, + μ + γ; a 2 2   2  Re γ + α + 12 + μ > 0, Re (γ − κ) < 0 BU 126(8)a

e− 2 x xγ+α−1 M κ,μ (x) 1

Γ

1

828

Confluent Hypergeometric Functions

 9. 0

7.628  1.

2.

7.628

1 dx n+1 n+μ+ 12 12 a = (−1) − μ + κ W −κ,μ (a) e x M κ,μ (x) a e Γ(1 + 2μ) Γ x+a 2 

  1 n > 0, Re κ − μ − BU 127(10a)a n = 0, 1, 2, . . . , Re μ + 1 + < n, |arg a| < π 2 2

∞

− 12 x n+μ+ 12



  2 1 1 1 e−st e−t t2c−2 1 F 1 a; c; t2 dt = 21−2c Γ(2c − 1)Ψ c − , a + ; s2 2 2 4 0   1 Re c > 2 , Re s > 0 EH I 270(11) 2

2

 ∞ 2 1 2 t as 1 t2ν−1 e− 2a t e−st M −3ν,ν dt = √ Γ(4ν + 1)a−ν s−4ν eas /8 K 2ν a 8 2 π 0   Re a > 0, Re ν > − 14 , Re s > 0 

∞

∞

3.

t2μ−1 e− 2a t e−st M λ,μ 1

2

0



2

t a

=2

7.629  1.8

∞

tk exp

0



∞

2. 0

7.631  1.

∞

a 2t

2.

∞

 xρ−1 exp

−3μ−λ

a t

1 2 (λ+μ−1)

λ−μ−1



as2 8

as2 4



Γ(4μ + 1)a s e W − 12 (λ+3μ), 12 (λ−μ)   Re a > 0, Re μ > − 14 , Re s > 0 ET I 215(13)

 √  √ 1 dt = 21−2k as−k− 2 S 2k,2μ 2 as  |arg a| < π, Re (k ± μ) > − 12 ,

Re s > 0



ET I 217(21)

Re s > 0]

      1  −1 α x − βx−1 W k,μ α−1 x W λ,ν βx−1 dx 2     −1 = β ρ Γ 12 − k + μ Γ 12 − k − μ ! β !! 1 + k, 1 − λ − ρ 41 × G 24 α ! 12 + μ, 12 − μ, 12 + ν − ρ,   3 |arg α| < 2 π, Re β > 0, Re(k + ρ) < −|Re ν| − 12

ET I 217(22)

1 2

−ν−ρ

ET II 412(55)



ρ−1

x 0

dt

 a a  √  √ 1 t−k exp − e−st W k,μ dt = 2 ask− 2 K 2μ 2 as 2t t [Re a > 0,

0



e−st W k,μ

ET I 215(12)



      1  −1 −1 α x − βx exp W k,μ α−1 x W λ,ν βx−1 dx 2         −1 = β ρ Γ 12 − k + μ Γ 12 − k − μ Γ 12 − λ + ν Γ 12 − λ − ν !

β !! 1 + k, 1 + λ − ρ × G 42 24 α ! 12 + μ, 12 − μ, 12 + ν − ρ, 12 − ν − ρ   3 3 |arg α| < 2 π, |arg β| < 2 π, Re(λ − ρ) < 12 − |Re μ|, Re(k + ρ) < 12 − |Re ν| ET II 412(57)

7.644

Confluent hypergeometric and trigonometric functions

829



      1  −1 −1 α x + βx 3. x exp W k,μ α−1 x W λ,ν βx−1 dx 2 0 !

β !! 1 − k, 1 − λ − ρ ρ 40 = β G 24 α ! 12 + μ, 12 − μ, 12 + ν − ρ, 12 − ν − ρ [Re α > 0, Re β > 0] ET II 412(54)

 ∞   μ− 12  1 7.632 e−st et − 1 exp − λet M k,μ λet − λ dt 2 0   Γ(2μ + 1) Γ 12 + k − μ + s W −k− 12 s,μ− 12 s (λ) = Γ(s + 1)   Re μ > − 12 , Re s > Re(μ − k) − 12 ET I 216(15) 

∞

ρ−1

7.64 Combinations of confluent hypergeometric and trigonometric functions 

∞

7.641

cos(ax) 1 F 1 (ν + 1; 1; ix) 1 F 1 (ν + 1; 1; −ix) dx

0



∞

7.64211 0

7.643



∞

4ν − 12 x2

sin(bx) 1 F 1

0

=0

[1 < a < ∞]

1 1 − 2ν; 2ν + 1; x2 2 2



dx =



π 4ν − 1 b2 b c 2 1F 1 2



2.

3.

4.

7.644



11



∞

∞

−μ− 12 − 12 x

x 0

e





b > 0,

ET II 402(4) EH I 285(12)

1 1 − 2ν; 1 + 2ν; b2 2 2  1

Re ν > − 4

1 2 1 2 1 2 π 2ν−1 − 1 b2 x b b x2ν−1 e− 4 x sin(bx) M 3ν,ν e 4 M 3ν,ν dx = 2 2 2 0   b > 0, Re ν > − 14 



 ∞ 1 2 1 2 π −2ν−1 1 b2 −2ν−1 14 x2 x b b x e cos(bx) W 3ν,ν e 4 W 3ν,ν dx = 2 2 2 0   1 Re ν < 4 , b > 0 



 ∞ 1 2 1 2 π −2ν 1 b2 −2ν 14 x2 x b b x e sin(bx) W 3ν−1,ν e 4 W 3ν−1,ν dx = 2 2 2 0   1 Re ν < 2 , b > 0 

1.

[0 < a < 1] ;

[−1 < Re ν < 0]     2 1 Γ(c) 1 2α−1 |y| cos(2xy) 1 F 1 a; c; −x2 dx = π 2 e−y Ψ c − 12 , a + 12 ; y 2 2 Γ(a)

x e

1.

  = −a−1 sin(νπ) P ν 2a−2 − 1



ET I 115(5)

ET I 116(10)

ET I 61(7)

ET I 116(9)

2

  Γ(3 − 2μ) a 1  exp − sin 2ax M k,μ (x) dx = π a W ρ,σ a2 , 2 Γ 2 +k+μ 2ρ = k − 3μ + 1, 2σ = k + μ − 1 [a > 0, Re(k + μ) > 0] ET II 403(10) 

1 2



1 2

k+μ−1

830

2.

3.

Confluent Hypergeometric Functions



∞



∞

7.651

 1 1 c Γ(1 + μ + ρ) Γ (1 − μ + ρ)   xρ−1 sin cx 2 e− 2 x W k,μ (x) dx = Γ 3 −k+ρ 0 2

c2 3 3 × 2 F 2 1 + μ + ρ, 1 − μ + ρ; , − k + ρ; − 2 2 4 [Re ρ > |Re μ| − 1] ET II 407(28)  ∞  1 1 xρ−1 sin cx 2 e 2 x W k,μ (x) dx 0   ! 1 π2 c2 !! 12 + μ − ρ, 12 − μ − ρ 22    G 23 = 1 ! 4 ! 12 , −k − ρ, 0 Γ 2 − k + μ Γ 12 − k − μ   c > 0, Re ρ > |Re μ| − 1, Re(k + ρ) < 12 ET II 407(29)

4.

ρ−1

x

 cos cx

1 2

e

0



∞

5.

− 12 x

1

   + μ + ρ Γ 12 − μ + ρ W k,μ (x) dx = Γ(1 − k + ρ)

1 1 1 c2 + μ + ρ, − μ + ρ; , 1 − k + ρ; − × 2F 2 2 2  4  2 Re ρ > |Re μ| − 12 ET II 407(30) Γ

2

 1 1 xρ−1 cos cx 2 e 2 x W k,μ (x) dx

 !  1 2 ! 1 + μ − ρ, 1 − μ − ρ π2 c ! 2    G 22 = 1 !2 23 4 ! 0, −k − ρ, 12 Γ 2 − k + μ Γ 12 − k − μ   c > 0, Re ρ > |Re μ| − 12 , Re(k + ρ) < 12 ET II 407(31)

0

7.65 Combinations of confluent hypergeometric functions and Bessel functions 7.651



∞

J ν (xy) M − 12 μ, 12 ν (ax) W

1. 0

 2. 0

1 1 2 μ, 2 ν

(ax) dx +  2  1 ,μ  2 − 1 Γ(ν + 1) 2 2  a + a a + y2 2 = ay −μ−1  1 1 + y 1 Γ 2 − 2μ + 2ν   y > 0, Re ν > −1, Re μ < 12 , Re a > 0 ET II 85(19)

∞

M k, 12 ν (−iax) M −k, 12 ν (−iax) J ν (xy) dx ae− 2 (ν+1)πi [Γ(1 + ν)] 1    y −1−2k Γ 2 + k + 12 ν Γ 12 − k + 12 ν  − 1 +  1 ,2k +  1 ,2k    × a2 − y 2 2 + a − a2 − y 2 2 a + a2 − y 2 2 1

=

=0

2



a > 0,

Re ν > −1,

|Re k| <

[0 < y < a] ;

1 4

[a < y < ∞] 

ET II 85(18)

7.661

Confluent hypergeometric functions, Bessel functions, and powers



∞

7.652 0

831

 +  + , , 1 1 M −μ, 12 ν a b2 + x2 2 − b W μ, 12 ν a b2 + x2 2 + b J ν (xy) dx + ,2μ 1 +  ay −2μ−1 Γ(1 + ν) a2 + y 2 2 + a 1 , 2 2 2 exp −b a + y =   1 Γ 12 + 12 ν − μ (A2 + Y 2 ) 2   ET II 87(29) y > 0, Re ν > −1, Re μ < 14 , Re a > 0, Re b > 0

7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers 7.661



∞

1.

x−1 W k,μ (ax) M −k,μ (ax) J 0 (xy) dx

% %

1 &

1 & 2 2 2 2 Γ(1 + 2μ) y y k k  P μ− 1 = e−ikπ  1 1+ 2 Q μ− 1 1+ 2 2 2 a a Γ 2 +μ+k   1 3 y > 0, Re a > 0, Re μ > − 2 , Re k < 4 ET II 18(44)

0



∞

2.

−1

x 0



% %

1 &

1 & 1 y2 2 y2 2 k −k W k,μ (ax) W −k,μ (ax) J 0 (xy) dx = π cos(μπ) P μ− 1 1+ 2 P μ− 1 1+ 2 2 2 2 a a   1 y > 0, Re a > 0, |Re μ| < 2 ET II 18(45)

∞

3.

x2μ−ν W k,μ (ax) M −k,μ (ax) J ν (xy) dx

0

Γ(2μ + 1)  = 22μ−ν+2k a2k y ν−2μ−2k−1  Γ ν − k − μ + 12

1 1 1 y2 − k, 1 − k, − k + μ; 1 − 2k, − k − μ + ν; − 2 × 3F 2 2 2 2 a   y > 0, Re μ > − 12 , Re a > 0, Re(2μ + 2k − ν) < 12 ET II 85(20)  4. 0

 5.

∞

x2ρ−ν W k,μ (iax) W k,μ (−iax) J ν (xy) dx  ! 



−1  1 1 y 2 !! 12 , 0, 12 − μ, 12 + μ 2ρ−ν ν−2ρ−1 − 12 24 −k+μ Γ −k−μ =2 y π Γ ! G 44 2 2 a2 ! ρ + 12 , −k, k, ρ − ν + 12   y > 0, Re a > 0, Re ρ > |Re μ| − 1, Re(2ρ + 2k − ν) < 12 ET II 86(23)a

∞

x2ρ−ν W k,μ (ax) M −k,μ (ax) J ν (xy) dx

0



y > 0,

Re a > 0,

 !  22ρ−ν Γ(2μ + 1) ν−2ρ−1 23 y 2 !! 12 , 0, 12 − μ, 12 + μ = 1 1 y ! G 44 a2 ! ρ + 12 , −k, k, ρ − ν + 12 π2 Γ 2 − k + μ  Re ρ > −1, Re(ρ + μ) > −1, Re(2e + 2k + ν) < 12 ET II 86(21)a

832

Confluent Hypergeometric Functions



∞

6.

x2ρ−ν W k,μ (ax) W −k,μ (ax) J ν (xy) dx

0

=

7.662



∞

1.

−1

x 0



∞

2. 0



∞

3.

M −μ, 14 ν

Γ(ρ + 1 + μ) Γ(ρ + 1 − μ) Γ(2ρ + 2) ν −ν−1 −2ρ−1     y 2 a Γ 32 + k + ρ Γ 32 − k + ρ Γ(1 + ν)

3 3 y2 3 × 4 F 3 ρ + 1, ρ + , ρ + 1 + μ, ρ + 1 − μ; + k + ρ, − k + ρ, 1 + ν; − 2 2 2 2 a [y > 0, Re ρ > |Re μ| − 1, Re a > 0] ET II 86(22)a

 







Γ 1 + 12 ν 1 2 1 2 1 2 1 2  I 1 ν−μ 1 x W μ, 14 ν x J ν (xy) dx =  1 1 y y K 4 ν+μ 2 2 4 4 Γ 2 + 4ν − μ 4

x−1 M α−β, 14 ν−γ

x−1 M k,0

0













[y > 0,

4. 0

Re ν > −1]

ET II 86(24)

1 2 1 2 x W α+β, 14 ν+γ x J ν (xy) dx 2 2  



Γ 1 + 12 ν − 2γ −2 1 2 1 2   y W α+γ, 14 ν+β y = y M α−γ, 14 ν−β 2 2 Γ 1 + 12 ν − 2β   ET II 86(25) y > 0, Re β < 18 , Re ν > −1, Re(ν − 4γ) > −2

   π iax2 M k,0 −iax2 K 0 (xy) dx = 16

 2  2 2  y2 y Jk + Yk 8a 8a [a > 0]

∞

7.662



ET II 152(83)



    iy iy 2 2 x−1 M k,μ iax2 M k,μ −iax2 K 0 (xy) dx = ay −2 [Γ(2μ + 1)] W −μ,k W −μ,k − 4a   4a a > 0, Re y > 0, Re μ > − 12

2

ET II 152(84)

7.663



∞

1.

2ρ

x 0



∞

2. 0

 3. 0

∞

2!

  y !! 1, b 22ρ Γ(b) 2 21 J ν (xy) dx = G 1 F 1 a; b; −λx Γ(a)y 2ρ+1 23 4λ ! 12 + ρ + 12 ν, a, 12 + ρ − 12ν  y > 0, −1 − Re ν < 2 Re ρ < 12 + 2 Re a, Re λ > 0 ET II 88(6)





1 1 2 2ν−a+ 2 Γ(a + 1) 2a−ν−1 − 1 y2 1 4 1 y xν+1 1 F 1 2a − ν; a + 1; − x2 J ν (xy) dx = e K y 1 a−ν− 2 2 4 π 2 Γ(2a − ν)   1 y > 0, Re ν > −1, Re(4a − 3ν) > 2 ET II 87(1)



1 y2 1+a+ν 1+a+ν ; − x2 J ν (xy) dx = y a−1 1 F 1 a; ;− xa 1 F 1 a; 2 2 2 2   y > 0, Re a > − 12 , Re(a + ν) > −1 ET II 87(2)

7.664

Confluent hypergeometric functions, Bessel functions, and powers



∞

4.

ν+1−2a

x

1F 1

0



∞

5.

833

1 2 a; 1 + ν − a; − x J ν (xy) dx 2

1 1 2 π 2 Γ(1 + ν − a) −2a+ν+ 1 2a−ν−1 − 1 y2 2y 2 y = e 4 I a− 12 4  Γ(a)  y > 0, Re a − 1 < Re ν < 4 Re a − 12 ET II 87(3)

   −1 2λ−2 − 1 y2 x 1 F 1 λ; 1; −x2 J 0 (xy) dx = 22λ−1 Γ(λ) y e 4

0



[y > 0, ∞

6.

  xν+1 1 F 1 a; b; −λx2 J ν (xy) dx

0

=



∞

7. 0

7.664



1

1

2

Γ(a)λ 2 a+ 2 ν

y

a−2 − y 8λ

e

xW

a

0

1 2 ν,μ

x

W − 12 ν,μ

a x

ET II 18(46)



y2 W k,μ , 2k = a − 2b + ν + 2, 2μ = a − ν − 1 4λ   y > 0, −1 < Re ν < 2 Re a − 12 , Re λ > 0 ET II 88(4)

  22b−2a−ν−1 Γ(b) −a 2a−2b+ν λ y x2b−ν−1 1 F 1 a; b; −λx2 J ν (xy) dx = Γ(a − b + ν + 1)

y2 × 1 F 1 a; 1 + a − b + ν; − 4λ     y > 0, 0 < Re b < 34 + Re a + 12 ν , Re λ > 0

∞

1.

21−a Γ(b)

Re λ > 0]

ET II 88(5)

+ , + , 1 1 1 1 K ν (xy) dx = 2ay −1 K 2μ (2ay) 2 e 4 iπ K 2μ (2ay) 2 e− 4 iπ

ET II 152(85) [Re y > 0, Re a > 0]



∞ 2 2 x W 12 ν,μ W − 12 ν,μ J ν (xy) dx x x 0    1  1   1      K 2μ 2y 2 = −4y −1 sin μ − 12 ν π J 2μ 2y 2 + cos μ − 12 ν π Y 2μ 2y 2

 2.

[y > 0,

4.

ET II 87(27)





2 2 W Y ν (xy) dx 1 1 ν,μ ν,μ − x x 2 0  2   1  1   1       cos μ − 12 ν π J 2μ 2y 2 − sin μ − 12 ν π Y 2μ 2y 2 K 2μ 2y 2 = 4y −1   y > 0, |Re μ| < 14 ET II 117(48)



 ∞  1  1 2 2 4 Γ(1 + 2μ)y −1  J 2μ 2y 2 K 2μ 2y 2 x W − 12 ν,μ M 12 ν,μ J ν (xy) dx =  1 1 x x Γ 2 + 2 ν + μ 0  y > 0, Re ν > −1, Re μ > − 14 

3.

Re (ν ± 2μ) > −1]

∞

xW

ET II 86(26)

834

Confluent Hypergeometric Functions

 5.

x W − 12 ν,μ

0

7.665



∞

1. 0





∞

∞

2.

ia x





ia W − 12 ν,μ − J ν (xy) dx x + , + ,     −1 1 1 K μ (2iay) 2 K μ (−2iay) 2 = 4ay −1 Γ 12 + μ + 12 ν Γ 12 − μ + 12 ν   y > 0, Re a > 0, |Re μ| < 12 , Re ν > −1 ET II 87(28)



 1 1 1 x M k,μ (x) dx x− 2 J ν ax 2 K 12 ν−μ 2 2

2

a a Γ(2μ + 1)   = M 12 (k+μ), 12 k+ 14 ν W 12 (k−μ), 12 k− 14 ν 1 2 2 a Γ k + ν + 1 2   1 1 ET II 405(18) a > 0, Re k > − 4 , Re μ > − 2 , Re ν > −1 + , 1 1  1 x 2 c+ 2 c −1 Ψ(a, c; x) 1 F 1 (a ; c ; −x) J c+c −2 2(xy) 2 dx

0

=



∞

7.666

7.665

1 1  Γ (c ) y 2 c+ 2 c −1 Ψ (c − a , c + c − a − a ; y) 1 F 1 (a ; a + a ; −y) Γ (a + a )  Re c > 0, 1 < Re (c + c ) < 2 Re (a + a ) + 12 EH I 287(23)

  + , 1 1 1 1 1 x 2 c− 2 1 F 1 a; c; −2x 2 Ψ a, c; 2x 2 J c−1 2(xy) 2 dx

0

= 2−c

, 1 c−2a 1 Γ(c) a− 1 c− 1 + y 2 2 1 + (1 + y) 2 (1 + y)− 2 Γ(a)   Re c > 2, Re(c − 2a) < 12 EH I 285(13)

7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers 7.671



∞

k− 32

x

1. 0

 2. 0

∞



 1 1 ax M k,ν (x) dx exp − (a + 1)x K ν 2 2 1   π 2 Γ(k) Γ(k + 2ν) −1  = k+ν  1 2 F 1 k, k + 2ν; 2ν + 1; −a a Γ k+ν+ 2 [Re a > 0, Re k > 0, Re(k + 2ν) > 0] ET II 405(17)



 3 1 1 ax W k,μ (x) dx x−k− 2 exp − (a − 1)x K μ 2 2   π Γ(−k) Γ(2μ − k) Γ(−2μ − k)      22k+1 ak−ν 2 F 1 −k, 2μ − k; −2k; 1 − a−1 = 1 Γ 2 − k Γ 12 + μ − k Γ 12 − μ − k [Re a > 0, Re k < 2 Re μ < − Re k] ET II 408(36)

7.672

7.672

Confluent hypergeometric functions, Bessel functions, exponentials, and powers



∞

1.

  2 1 x2ρ e− 2 ax M k,μ ax2 J ν (xy) dx

  ! Γ(2μ + 1) 2ρ −2ρ−1 21 y 2 !! 12 − μ, 12 + μ 2 y =  ! G 23 4a ! 12 + ρ + 12 ν, k, 12 + ρ − 12 ν Γ μ + k + 12     y > 0, −1 − Re 12 ν + μ < Re ρ < Re k − 14 , Re a > 0 ET II 83(10)

0



∞

2.

  2 1 x2ρ e− 2 ax W k,μ ax2 J ν (xy) dx

    Γ 1 + μ + 12 ν + ρ Γ 1 − μ + 12 ν + ρ 2−ν−1 − 1 ν−ρ− 1 ν 2y   = a 2 Γ(ν + 1) Γ 32 − k + 12 ν + ρ

y2 1 × 2 F 2 λ + μ, λ − μ; ν + 1, − k + λ; − , 2 4a     y > 0, Re a > 0, Re ρ ± μ + 12 ν > −1 ET II 85(16) λ = 1 + 12 ν + ρ

0



∞

3.

  2 1 x2ρ e 2 ax W k,μ ax2 J ν (xy) dx =

0

 

∞

4.

∞

5. 0

7.

e

y > 0,

M k,μ

22ρ y −2ρ−1    Γ 2 +μ − k! Γ 12 − μ − k  y 2 !! 12 − μ, 12 + μ 22 × G 23 ! 4a ! 12 + ρ + 12 ν, −k, 12 + ρ − 12 ν    |arg a| < π, −1 − Re 12 ν ± μ < Re ρ < − 14 − Re k ET II 85(17) 1

!

1 2 2λ y −1/2 Γ(2μ + 1) 31 y 2 !! −μ − λ, μ − λ, 1  G 34 x Y ν (xy) dx = 2 2 ! h, κ, −λ − 12 , l Γ 2 +k+μ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = − 14 − 12 ν  y > 0, Re(k − λ) > 0, Re (2λ + 2μ ± ν) > − 52 ET II 116(45)



1 1 2 1 2 x Y ν (xy) dx x2λ+ 2 e 4 x W k,μ 2



−1  2!

1 1 y !! −μ − λ, μ − λ, l 32 −k+μ Γ −k−μ = 2λ Γ y −1/2 , G 34 2 2 2 ! h, κ, − 12 − k − λ, l h = 14 + 12 ν, κ = 14 − 12 ν, l = − 14 − 12 ν  y > 0, Re(k + λ) < 0, Re (2λ ± 2μ ± ν) > − 52 ET II 117(47)

    1 2 1 y x−1/2 e− 2 x M 12 ν− 14 , 12 ν+ 14 x2 J ν (xy) dx = (2ν + 1)2−ν y ν−1 1 − Φ 2 0   y > 0, Re ν > − 12    ∞   1 2 1 Γ(ν + 2)y ν  y x−1 e− 2 x M 12 ν+ 12 , 12 ν+ 12 x2 J ν (xy)dx =  1 − Φ 3 ν 2 Γ ν+2 2 0 

6.

2λ+ 12 − 14 x2

x 0



835

∞

[y > 0, Re ν > −1]

ET II 82(1)

ET II 82(2)

836

Confluent Hypergeometric Functions



∞

8.

e

− 14 x2

0





1 2−k Γ(ν + 1) 2k−1 − 1 y2 y M k, 12 ν e 2 x2 J ν (xy) dx =  2 Γ k + 12 ν + 12  y > 0, Re ν > −1,

0

∞

10.

ν−2μ

x

e

1 2 4x



∞

11.

W k,±μ

xν−2μ e− 4 x W k,±μ 1

2

0

 y > 0, ∞

12.

2μ−ν − 14 x2

x

e

13.

∞

2μ−ν − 14 x2

x

e

M k,μ

Re(ν − 2μ) > −1] ET II 84(14)



1 2 x J ν (xy) dx 2

1 2 Γ(1 + ν − 2μ) 12 ( 12 +k−3μ+ν ) μ−k− 3 1 y2 2 4   = y , y e W α,β 2 2 Γ 12 + μ − k 1 2α = k + 3μ −ν − 12 , 2β = k1 − μ + ν + 2 1 Re ν > −1, Re(ν − 2μ) > −1, Re k − μ + 2 ν < − 4 ET II 84(15)

1 2 x J ν (xy) dx 2

1 1 3 1 2 1 2 Γ(2μ + 1)  2 2 ( 2 −k+3μ−ν ) y k−μ− 2 e− 4 y M α,β = 1 y 2 Γ 2 +k−μ+ν 1 1 + k + 3μ − ν, 2β = − + k − μ +ν 2α = 2    2  y > 0, − 12 < Re μ < Re k + 12 ν − 14 ET II 83(8)



1 2 x Y ν (xy) dx 2

M k,μ

0



Re ν > −1,



0







1 2 1 2 Γ(1 + ν − 2μ) β−μ k+μ− 3 − 1 y2 2 4 x J ν (xy) dx = 2 y y e M α,β 2 Γ(1 + 2β) 2 1 1 2α = 2 + k + ν − 3μ, 2β = 2 − k + ν − μ [y > 0,



1 2



1 ν−2μ − 14 x2 x e M k,μ x2 J ν (xy) dx 2

1 1 1 2 Γ(2μ + 1) k+μ− 32 − 14 y 2  = 2 2 ( 2 −k−3μ+ν )  y e W , y α,β 2 Γ μ + k + 12 1 2β = k + μ − ν − 12  2α = k − 3μ + ν + 2 , y > 0, −1 < Re ν < 2 Re(k + μ) − 12 ET II 83(9)

0



Re k <

ET II 83(7)

∞

9.



7.672

= π −1 2μ+β y k−μ− 2 e− 4 y Γ (2μ + 1)

    Γ(2μ − ν − 1) M α,β 12 y 2 × Γ 12 − k − μ cos[(ν − 2μ)π] Γ(2β + 1)   1 2 − sin[(ν + k − μ)π] W α,β 2 y 3



y > 0,

1

2

2α = 3μ − ν + k + 12 , 2β = μ − ν − k + 12 1 −1 < 2 Re μ < Re(2k + ν) + 2 , Re(2μ − ν) > −1 ET II 116(44)

7.673

Confluent hypergeometric functions, Bessel functions, exponentials, and powers



∞

14.

2μ+ν − 14 x2

x

e

M k,μ

0

837

1 2 x Y ν (xy) dx 2 = π −1 2μ+β y k−μ− 2 Γ(2μ + 1)   1 2  − 1 y2 1 Γ(2μ + ν + 1)  ×Γ 2 − μ − k e 4 cos(2μπ)  3 M α,β 2 y Γ μ+ν−k+ 2   1 2 + sin[(μ − k)π] W α,β 2 y 3

 

∞

15.

y > 0,

2α = 3μ + ν + k + 12 , 2β = μ + ν − k + 12 1 −1 < 2 Re μ < Re(2k − ν) + 2 , Re(2μ + ν) > −1 ET II 116(43)

  2 1 1 1 1 3 x2μ+ν e− 2 ax M k,μ ax2 K ν (xy) dx = 2μ−k− 2 a 4 − 2 (μ+ν+k) y k−μ− 2

0



2

y2 y W κ,m , 8a 4a 2m = μ + ν − k + 12 2κ = −3μ − ν − k − 12 , 1 Re a > 0, Re μ > − 2 , Re(2μ + ν) > −1 ET II 152(82) × Γ(2μ + 1) Γ(2μ + ν + 1) exp



7.673 1.



∞

10 0

Re y > 0,

 √ 1 1 e− 2 ax x 2 (μ−ν−1) M κ, 12 μ (ax) J ν 2 bx dx 1+μ κ−1 2 − 4 1 b b a− 2 (μ+1−ν) Γ(1 + μ)e− 2a = a

b × M 1 (κ−ν−1)+ 3 (1+μ), κ+ν − 1+μ 2 4 2 4 a 

ν−μ 3 Re(1 + μ) > 0, Re κ + >− , 2 4

 2. 0

∞

1

1+μ κ+ν − Γ 1+ 2 4  Im b = 0

BU 128(12)a

b  √ 1 1 1 Γ (ν + 1 ∓ μ) e 2a  a 12 (κ+1)+ 14 (1∓ν)  1±μ  e 2 ax x 2 (ν−1∓μ) W κ, 12 μ (ax) J ν 2 bx dx = a− 2 (ν+1∓μ) b Γ 2 −κ

b × W 12 (κ+1−ν)− 34 (1∓μ), 12 (κ+ν)+ 14 (1∓μ)

 a  ν∓μ 3 + κ < , Re ν > −1 BU 128(13) Re 2 4

838

Confluent Hypergeometric Functions

7.674  1.

∞

0

2.

7.674

  1 xρ−1 e− 2 κ J λ+ν ax1/2 J λ−ν ax1/2 W k,μ (x) dx  1 2λ  1    Γ 2 + λ + μ + ρ Γ 12 + λ − μ + ρ 2a = Γ (1 + λ + ν) Γ(1 + λ − ν) Γ(1 + λ − k + ρ)

× 4 F 4 1 + λ, 12 + λ, 12 + λ + μ + ρ, 12 + λ − μ + ρ; 1 + λ + ν,

1 + λ − ν, 1 + 2λ, 1 + λ − k + ρ; −a2   |Re μ| < Re(λ + ρ) + 12 ET II 409(37)  ∞   1 xρ−1 e− 2 κ I λ+ν ax1/2 K λ−ν ax1/2 W k,μ (x) dx 0 !

π −1/2 24 2 !! 0, 12 , 12 + μ − ρ, 12 − μ − ρ = G 45 a ! 2 λ, ν, −λ, −ν, k − ρ  |Re μ| < Re(λ + ρ) + 12 , |Re μ| < Re(ν + ρ) + 12 ET II 409(38)

Combinations of Struve functions and confluent hypergeometric functions 7.675  1.

∞

1

1

2

0



∞

2.

1

1

∞

3.

2λ+ 12

x 0

 y > 0,  4. 0

∞

2





ET II 171(42)



1 2 x Hν (xy) dx 2     Γ 74 + 12 ν + λ + μ Γ 74 + 12 ν + λ − μ 1 1 −λ− ν −1/2 ν+1 2 π     y = 24 Γ ν + 32 Γ 94 + λ − k − 12 ν

7 ν 3 3 9 ν y2 7 ν × 3 F 3 1, + + λ + μ, + + λ − μ; , ν + , + λ − k + ; − 4 2 2 2 4 2 2  4 2  Re(2λ + ν) > 2|Re μ| − 74 , y > 0 ET II 171(43)

x2λ+ 2 e− 4 x W k,μ

0



2!

1 2 y !! l, −μ − λ, mu − λ 2−λ Γ(2μ + 1)  G 22 x Hν (xy) dx = 1/2  1 34 2 2 ! l, k − λ − 12 , h, κ y Γ 2 +k+μ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = 34 + 12 ν   Re(2λ + 2μ + ν) > − 72 , Re(k − λ) > 0, y > 0, Re(2λ − 2k + ν) < − 12

x2λ+ 2 e− 4 x M k,μ

1 2 x Hν (xy) dx e W k,μ 2



−1  2!

1 1 y !! l, −μ − λ, μ − λ −k+μ Γ −k−μ = 2λ Γ y −1/2 G 23 34 2 2 2 ! l, −k − λ − 12 , h, κ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = 34 + 12 ν 7 Re(2λ + ν) > 2|Re μ| − 2 , Re (2k + 2λ + ν) < − 12 , Re(k + λ) < 0 ET II 172(46)a 1 2 4x



+  y ,   1 2 1 2 e 2 x W − 12 ν− 12 , 12 ν x2 Hν (xy) dx = 2−ν−1 y ν πe 4 y 1 − Φ 2 [y > 0,

Re ν > −1]

ET II 171(44)

7.681

Confluent hypergeometric functions and other special functions

839

7.68 Combinations of confluent hypergeometric functions and other special functions Combinations of confluent hypergeometric functions and associated Legendre functions 7.681



∞

1. 0

 1 x x−1/2 (a + x)μ e− 2 x P −2μ 1+2 M k,μ (x) dx ν a     1 sin(νπ) Γ(2μ + 1) Γ k − μ + ν + 12 Γ k − μ − ν − 12 e 2 a W ρ,σ (a), =− π Γ(k) ρ = 12 !− k + μ, ! σ = 12 + ν  1 |arg a| < π, Re μ > − , Re(k − μ) > !Re ν + 1 ! ET II 403(11) 2



∞

 1 x 1+2 M k,μ (x) dx x−1/2 (a + x)−μ e− 2 x P −2μ ν a    1 Γ(2μ + 1) Γ k + μ + ν + 12 Γ k + μ − ν − 12 e 2 a   W 12 −k−μ, 12 +ν (a) = Γ k + μ + 12 Γ(2μ + ν + 1) Γ(2μ − ν) ! !  |arg a| < π, Re μ > − 12 , Re(k + μ) > !Re ν + 12 ! ET II 403(12)

∞

 1 1 1 1 x W k,ν (x) dx x− 2 − 2 μ−ν (a + x) 2 μ e− 2 x P μk+ν− 3 1 + 2 2 a 1 1 1 1 Γ(1 − μ − 2ν)  a− 4 + 2 k− 2 ν e 2 a W ρ,σ (a) = 3 Γ 2 −k−μ−ν 2ρ = 12 + 2μ + ν − k, 2σ = k + 3ν − 32

2. 0



2

3. 0

[|arg a| < π, 



x− 2 − 2 μ−ν (a + x)− 2 μ e− 2 x P μk+μ+ν− 3 1 + 2 1

1

1

1

2

0

x a =

W k,ν (x) dx Γ

1 1 1 1 Γ(1 − μ − 2ν) 3  a− 2 + 2 k− 2 ν e 2 a W ρ,σ (a) − k − μ − ν 2 2ρ = 12 − k + ν, 2σ = k + 2μ + 3ν − 32

[|arg a| < π,

5. 0

Re(μ + 2ν) < 1] ET II 407(32)

∞

4.



Re μ < 1,

Re μ < 1,

Re(μ + 2ν) < 1] ET II 408(33)

∞



xμ− 4 k− 2 ν− 2 (a + x) 2 ν e− 2 x Q νμ−k+ 3 1 + 2 1

1

1

1

1

x

M k,ν (x) dx a   eνπi Γ(1 + 2μ − ν) Γ(1 + 2μ) Γ 52 − k + μ + ν 1 (κ+2μ−2ν+5) 1 a   = e 2 W ρ,σ (a) a4 2 Γ 12 + k + μ 2ρ = 12 − k − μ + 2ν, 2σ = k − 3μ − 32   |arg a| < π, Re μ > − 12 , Re(2μ − ν) > −1 2

ET II 404(14)

840

7.682

Confluent Hypergeometric Functions



∞

1.

x−1/2 e− 2 x P −2μ ν 1

 1+

0



∞

2. 0



∞

3. 0



∞

4.

 x 1/2 M k,μ (x) dx a  1    Γ(2μ + 1) Γ k + 12 ν Γ k − 12 ν − 12 e 2 a      W 3 −k, 1 + 1 ν (a) = 2μ 1/4  1 1 1 1 4 4 2  2 a Γ k + μ + 2 1Γ μ + 21ν + 2 Γ μ − 12 ν  |arg a| < π, Re k > 2 Re ν − 2 , Re k > − 2 Re ν ET II 404(13)

  x 1/2 1+ M k,μ (x) dx  1a  Γ 2 − ν Γ(1 + 2μ) Γ(k + μ + ν) 1 a 1   = e(1−k+μ−ν)πi 2μ−k−ν a 2 (k+μ−1) e 2 W ρ,σ (a), Γ k + μ + 12 ρ = 12 − k − 12 ν,  σ = μ + 12 ν  1 |arg a| < π, Re μ > − 2 , Re(k + μ + ν) > 0 ET II 404(15)

1−k+μ−ν x 2 (k+μ+ν)−1 (a + x)−1/2 e− 2 x Q k−μ−ν−1 1

1

 x 1/2 M k,μ (x) dx a   1 2(μ−ν)πi 2μ−2ν−1 12 (k+μ−1) 12 a Γ(2μ + 1) Γ(ν + 1) Γ k + μ − 2ν − 2 2 a e =e W ρ,σ (a), Γ k + μ + 12 2ρ = 1 − k + μ − 2ν, 2σ = k − μ − 2ν − 2  |arg a| < π, Re μ > − 12 , Re ν > −1, Re(k + μ − 2ν) > 12 ET II 404(16)

xν− 2 e− 2 x Q 2μ−2ν 2k−2ν−3 1

1

− 12 − 12 μ−ν − 12 x

x 0

e



1+

P μ2k+μ+2ν−3



 x 12 1+ W k,ν (x) dx a 2μ Γ(1 − μ − 2ν) − 1 + 1 k− 1 ν 1 a  a 2 2 2 e 2 W ρ,σ (a), = 3 Γ 2 −k−μ−ν 2ρ = 1 − k + μ + ν, 2σ = k + μ + 3ν − 2 [|arg a| < π,

 5.8 0

∞

7.682

Re μ < 1,

  x 1/2 − 12 − 12 μ−ν −1/2 − 12 x μ x (a + x) e P 2k+μ+2ν−2 1 + W k,ν (x) dx a μ 2 Γ(1 − μ − 2ν) − 1 + 1 k− 1 ν 1 a  a 2 2 2 e 2 W ρ,σ (a), 2ρ = μ + ν − k, = 3 Γ 2 −k−μ−ν [|arg a| < π, Re μ > 0,

Re(μ + 2ν) < 1] ET II 408(34)

2σ = k + μ + 3ν − 1 Re ν > 0]

ET II 408(35)

A combination of confluent hypergeometric functions and orthogonal polynomials  1 μ−α 1 8 e− 2 ax xα (1 − x) 2 −1 Lα 7.683 n (ax) M α− 1+α , μ−α−1 [a(1 − x)] dx 0

2

1

Γ(μ − α) Γ(1 + n + α) − 1+α a 2 M α+n, μ2 (a) Γ(1 + μ) n! Re(μ − α) > 0, n = 0, 1, 2, . . .] BU 129(14b) =

[Re a > −1,

7.711

Parabolic cylinder functions

841

A combination of hypergeometric and confluent hypergeometric functions

 ∞ λ ρ−1 − 12 x 7.684 x e M γ+ρ,β+ρ+ 12 (x) 2 F 1 α, β; γ; − dx x 0 Γ(α + β + 2ρ) Γ(2β + 2ρ) Γ(γ) 1 β+ρ− 1 1 λ 2 e2 λ2 W k,μ (λ); = Γ(β) Γ(β + γ + 2ρ) 1 1 k = 2 − α − 2 β − ρ, μ = 12 β + ρ [|arg λ| < π, Re(β + ρ) > 0, Re (α + β + 2ρ) > 0, Re γ > 0] ET II 405(19)

7.69 Integration of confluent hypergeometric functions with respect to the index   1 (aβ)1/2 exp − (α + β) 7.691 sech(πx) W ix,0 (α) W −ix,0 (β) dx = 2 α+β 2 −∞ i∞ 7.692 Γ(−a) Γ(c − a)Ψ(a, c; x)Ψ(c − a, c; y) da = 2πi Γ(c)Ψ(c, 2c; x + y) 

∞

ET II 414(61) EH I 285(15)

−i∞

7.693



∞

Γ(ix) Γ(2k + ix) W k+ix,k− 12 (α) W −k−ix,k− 12 (β) dx

1. −∞

= 2π

1/2

k

Γ(2k)(aβ) (α + β)

1 2 −2k

K

2k− 12

a+β 2



ET II 414(62)



i∞

2.

Γ −i∞

1 2

       + ν + μ + x Γ 12 + ν + μ − x Γ 12 + ν − μ + x Γ 12 + ν − μ − x × M μ+ix,ν (α) M μ−ix,ν (β) dx 1



2

2π(aβ)ν+ 2 [Γ(2ν + 1)] Γ(2ν + 2μ + 1) Γ(2ν − 2μ + 1) M 2μ,2ν+ 12 (α + β) = (α + β)2ν+1 Γ(4ν+ 2)  1 Re ν > |Re μ| − 2 ET II 413(59) ∞

7.69411 −∞

e−2ρxi Γ

1 2

   + ν + ix Γ 12 + ν − ix M ix,ν (α) M ix,ν (β) dx     1 2 = π αβ [Γ(2ν + 1)] sech ρ exp − (α + β) tanh ρ J 2ν αβ sech ρ  2  |Im ρ| < 12 π, Re ν > − 12

7.7 Parabolic Cylinder Functions 7.71 Parabolic cylinder functions 7.711



∞

1. −∞

D n (x) D m (x) dx = 0 = n!(2π)1/2

[m = n] [m = n] WH

842

Parabolic Cylinder Functions



%

& 1 1     ∓ 1 1   1  Γ 12 − 12 μ Γ − 12 ν Γ 2 − 2ν Γ −2μ [when the lower sign is taken, Re μ > Re ν] BU 11 117(13a), EH II 122(21)

∞

1

π2 2 (μ+ν+1) D μ (±t) D ν (t) dt = μ−ν

2. 0



7.721

∞

2

3.

[D ν (t)] dt = π

1/2 −3/2 ψ

2

0

1 2

   − 12 ν − ψ − 12 ν Γ(−ν)

BU 117(13b)a, EH II 122(22)a

7.72 Combinations of parabolic cylinder functions, powers, and exponentials 7.721



∞

e− 4 x (x − z)−1 D n (x) dx = ±ie∓nπi (2π)1/2 n!e− 4 z D −n−1 (∓iz) 1

1. −∞

2

1

2

[The upper or lower sign is taken accordingly as the imaginary part of z is positive or negative.] 

∞

2.

xν (x − 1)

1

1 1 2 μ− 2 ν−1

 

μ ν μ−ν (x − 1)2 a2 exp − D μ (ax) dx = 2μ−ν−2 a 2 − 2 −1 Γ D ν (a) 4 2 [Re(μ − ν) > 0]

7.722



∞

1. 0

 2. 3.11 7.723

3 2 1 1 1 e− 4 x xν D ν+1 (x) dx = 2− 2 − 2 ν Γ(ν + 1) sin (1 − ν)π 4

1/2 − 12 μ− 12 ν

Γ(μ) π 2  1 1 Γ 2 μ + 2 ν + 12 0

 ∞ 1 − 34 x2 ν − 12 ν πν e x D ν−1 (x) dx = 2 Γ(ν) sin 4 0



∞

1. 0



ET II 395(4)a

[Re ν > −1] ∞

WH

e− 4 x xμ−1 D −ν (x) dx = 1

2

WH

[Re μ > 0]

EH II 122(20)

[Re ν > −1]

ET II 395(2)

 π 1/2  −1 1 2 1 2 e− 4 x xν x2 + y 2 D ν (x) dx = Γ(ν + 1)y ν−1 e 4 y D −ν−1 (y) 2 [Re y > 0, Re ν > −1] EH II 121(18)a, ET II 396(6)a

∞

2.

 −1/2 1 2 1 2 e− 4 x xν−1 x2 + y 2 D ν (x) dx = y ν−1 Γ(ν)e 4 y D −ν (y)

0

 3. 0

[Re y > 0, 1

ET II 396(7)

 λ−1 a2 x2 Γ(λ) Γ(2ν) λ−1 a2 2 x2ν−1 1 − x2 e 4 D −2λ−2ν (ax) dx = e 4 D −2ν (a) Γ(2λ + 2ν) [Re λ > 0,



Re ν > 0]

∞

7.724 −∞

e−

(x−y)2 2μ

Re ν > 0]

+ , y2 1 2 1 e 4 x D ν (x) dx = (2πμ)1/2 (1 − μ) 2 ν e 4−4μ D ν y(1 − μ)−1/2

ET II 395(3)a

[0 < Re μ < 1] EH II 121(15)

7.731

7.725

Parabolic cylinder and hyperbolic functions



∞

1.

e

−pt

(2t)

ν−1 2

e

− 2t

0



∞

2.

e−pt (2t)

ν−1 2

0

4.

5.

6.

√  π 1/2 2t dt = 2

[Re ν > −1] ν p+1−1 √ pν p + 1

√

MO 175

MO 175 [Re ν > −1] √ 3  n  −n− 2  b + 12 e−bx D 2n+1 2x dx = (−2)n Γ n + 32 b − 12 0   Re b > − 12 ET I 210(3) 1



n

−n− 2  ∞ √ √ −1 −bx 1 1 1 x e D 2n 2x dx = (−2)n Γ n + b+ b− 2 2 2 0   1 Re b > − 2 ET I 210(5)

ν  ∞ < √  √ 1 1 < x− 2 (ν+1) e−sx D ν x dx = π 1 + 12 + 2s 1 0 4 +s   Re s > − 14 , Re ν < 1 ET I 210(7)

 ∞ ν + , 1−β− 1/2 2π β ν β ν+β+1 z−k 2 Γ(β) −zt −1+ β 1/2 − 2 2  (z + k) dt =  1 , ; ; e t D −ν 2(kt) F 2 2 2 z+k Γ 2 ν + 12 β + 12 0 , + z Re(z + k) > 0, Re > 0 k

 3.

 π 1/2 √p + 1 − 1ν+1 √ D −ν−2 2t dt = 2 (ν + 1)pν+1

e− 2 D −ν t

843

∞

EH II 121(11)



∞

eixy−

7.726

+ , +  1/2 , (1+λ)y 2 1 D ν x(1 − λ)1/2 dx = (2π)1/2 λ 2 ν e− 4λ D ν i λ−1 − 1 y

(1+λ)x2 4

−∞



∞

7.727 0



∞

7.728

e 2 x e−bx 1

μ+ 12

(ex − 1)

− ν2

(2t) 0

e−pt e

exp −

2 − q8t

a 1 − e−x

D ν−1

q √ 2t



D 2μ

dt =

EH II 121(16)

√  √  2 a √ dx = e−a 2b+μ Γ(b + μ) D −2b 2 a −x 1−e [Re a > 0, Re b > − Re μ]

 π 12 2

[Re λ > 0]

ET I 211(13)

p

1 2 ν−1

e

√ −q p

MO 175

7.73 Combinations of parabolic cylinder and hyperbolic functions 7.731



1. 0

∞

, +   3 2 cosh(2μx) exp − (a sinh x) D 2k (2a cosh x) dx = 2k− 2 π 1/2 a−1 W k,μ 2a2  2  Re a > 0

ET II 398(20)

844

Parabolic Cylinder Functions



∞

2. 0

7.741

+ ,   Γ(μ − k) Γ(−μ − k) 2 cosh(2μx) exp (a sinh x) D 2k (2a cosh x) dx = W k+ 12 ,μ 2a2 5 k+ 2 a Γ(−2k)  2 3π , Re k + |Re μ| < 0 |arg a| < 4 ET II 398(21)

7.74 Combinations of parabolic cylinder and trigonometric functions 7.741



∞

1. 0



∞

2.

    1 2 i √ 2 2 sin(bx) [D −n−1 (ix)] − [D −n−1 (−ix)] dx = (−1)n+1 π 2πe− 2 b Ln b2 n!  e− 4 x sin(bx) D 2n+1 (x) dx = (−1)n 1

2

0

 3. 4.

5.

7.742

∞



[b > 0]

ET I 115(3)

[b > 0]

ET I 115(1)

π 2n+1 − 1 b2 b e 2 2

π 2n − 1 b2 b e 2 [b > 0] 2 0  ∞ + , √    1 2 1 1 2 e− 4 x sin(bx) D 2ν− 12 (x) − D 2ν− 12 (−x) dx = 2π sin ν − 14 π b2ν− 2 e− 2 b 0   Re ν > 14 , b > 0 √  ∞ 1 1 1 2 + , 2 4 −2ν πb2ν− 2 e− 4 b − 12 x2    e cos(bx) D 2ν− 12 (x) + D 2ν− 12 (−x) dx = cosec ν + 14 π 0   Re ν > 14 , b > 0 

∞

1.

e− 4 x cos(bx) D 2n (x) dx = (−1)n 1

2

x2ρ−1 sin(ax)e−

∞

2.

x2ρ−1 sin(ax)e

x2 4

0

ET I 115(2)

ET I 61(4)

x2 4

0



1 Γ (2ρ + 1) D 2ν (x) dx = 2ν−ρ− 2 π 1/2 a Γ(ρ − ν + 1)

3 a2 1 × 2 F 2 ρ + , ρ + 1; , ρ − ν + 1; − 2 2 2  Re ρ > − 12   ! 2ρ−ν−2 22 a2 !! 12 − ρ, 1 − ρ D 2ν (x) dx = ! G Γ(−2ν) 23 2 ! −ρ − ν, 12 , 0  a > 0, Re ρ > − 12 ,

ET I 60(2)

ET II 396(8)

Re(ρ + ν) <

1 2



ET II 396(9)

 3. 0

∞

x2ρ−1 cos(ax)e−

x2 4



1 a2 2ν−ρ Γ(2ρ)π 1/2 1 1  ; , ρ − ν + ; − D 2ν (x) dx =  ρ, ρ + 2F 2 2 2 2 2 Γ ρ − ν + 12 [Re ρ > 0]

ET II 396(10)a

7.752

Parabolic cylinder and Bessel functions



∞

4.

2ρ−1

x

cos(ax)e

0

x2 4

2ρ−ν−2 22 D 2ν (x) dx = G Γ(−2ν) 23



a2 2

845

!  ! 1 − ρ, 1 − ρ !2 ! ! −ρ − ν, 0, 12  a > 0, Re ρ > 0,

Re(ρ + ν) <

1 2



ET II 396(11)



π/2

7.743

(cos x)

−μ−2

(sin x)

−ν

D ν (a sin x) D μ (a cos x) dx = −

0

 1 1/2 (1 + μ)−1 D μ+ν+1 (a) 2π

[Re ν < 1, 7.744



∞

1. 0



∞

2. 0

Re μ < −1]

ET II 397(19)

+ √  √ , √ sin(bx) D −ν− 12 2x − D −ν− 12 − 2x D ν− 12 2x dx

+ √ cos(bx) D −2ν− 12 2x + D −2ν− 12

√  ν √  1 1   −ν− 1 1 + 1 + b2 2 √ = − 2π sin 4 + 2 ν π b 1 + b2 [b > 0] ET I 115(4)  √ , √ 2x dx − 2x D 2ν− 12 √   2ν  √ π sin ν − 14 π 1 + 1 + b2 √ =− 1 1 + b2 b2ν+ 2 [b > 0] ET I 60(3)

7.75 Combinations of parabolic cylinder and Bessel functions 7.751 1. 2.

3.



+  y ,2 2 [D n (ax)] J 1 (xy) dx = (−1)n−1 y −1 D n [y > 0] a 0  ∞ y y D n+1 J 0 (xy) D n (ax) D n+1 (ax) dx = (−1)n y −1 D n a a 0   y > 0, |arg a| < 14 π  ∞ J 0 (xy) D ν (x) D ν+1 (x) dx = 2−1 y −1 [D ν (−y) D ν+1 (y) − D ν+1 (−y) D ν (y)] ∞

ET II 20(24)

ET II 17(42) ET II 397(17)a

0

7.752



∞

1. 0



ET II 76(1), MO 183 ∞

2. 0

 3. 0

1 2 1 2 1 xν e− 4 x D 2ν−1 (x) J ν (xy) dx = − sec(νπ)y ν−1 e− 4 y [D 2ν−1 (y) − D 2ν−1 (−y)] 2   y > 0, Re ν > − 12

∞

  1 2 1 1 2 xν e 4 x D 2ν−1 (x) J ν (xy) dx = 2 2 −ν π sin(νπ)y −ν Γ(2ν)e 4 y K ν 14 y 2   y > 0, − 12 < Re ν < 12 xν+1 e− 4 x D 2ν (x) J ν (xy) dx = 1

2

ET II 77(4)

1 2 1 sec(νπ)y ν−1 e− 4 y [D 2ν+1 (y) − D 2ν+1 (−y)] 2

[y > 0,

Re ν > −1]

ET II 78(13)

846

Parabolic Cylinder Functions



∞

4.

xν e− 4 x D 2ν+1 (x) J ν (xy) dx = 2

1

0



∞

5. 0

 0

0

∞

8.

xν+1 e− 4 x D 2ν+2 (x) J ν (xy) dx = − 12 sec(νπ)y ν e− 4 y [D 2ν+2 (y) + D 2ν+2 (−y)] 2

1

1

9.

10.

2

y > 0]

ET II 78(16)

  1 2 1 2 xν+1 e 4 x D 2ν+2 (x) J ν (xy) dx = π −1 sin(νπ) Γ(2ν + 3)y −ν−2 e 4 y K ν+1 14 y 2   y > 0, −1 < Re ν < − 56 xν e− 4 x D −2ν (x) J ν (xy) dx = 2−1/2 π 1/2 y −ν e− 4 y I ν  2

1

2

1

2

1

1



2

xν e 4 x D −2ν (x) J ν (xy) dx = y ν−1 e 4 y D −2ν (y)

0



ET II 77(5)

ET II 78(19) ∞

7. 

1 2 1 sec(νπ)e− 4 y y ν [D 2ν (y) + D 2ν (−y)] 2   y > 0, Re ν > − 12

[Re ν > −1, ∞

6.



7.752

1

4y

2



y > 0,

Re ν > − 12

Re ν > − 12 ,

y>0



ET II 77(8)



ET II 77(9), EH II 121(17) ∞

xν e 4 x D −2ν−2 (x) J ν (xy) dx = (2ν + 1)−1 y ν e 4 y D −2ν−1 (y) 0   y > 0, Re ν > − 12 ET II 77(10)  

1  ∞ 2μ− 2 Γ ν + 12 y ν 1 2 2 y2 1 ; ν − μ + 1; − xν e− 4 a x D 2μ (ax) J ν (xy) dx = ν + F 1 1 Γ(ν − μ + 1)a1+2ν  2 2a2 0  y > 0, |arg a| < 14 π, Re ν > − 12 

∞

11.

2

1

1

2

xν e 4 a

1

x2

D 2μ (ax) J ν (xy) dx =

Γ

0



∞

12.

1

 0

∞



1

2

x2

1

2

xν+1 e 4 a

x2

+ ν a2k 2m+μ y22 y2 1  μ+ 3 e 4a W k,m 4a2 Γ 2 −μ y 2 2k = 12 + μ − ν, 2m = 12 + μ + ν    y > 0, |arg a| < 14 π, − 12 < Re ν < Re 12 − 2μ

D 2μ (ax) J ν (xy) dx =

D 2μ (ax) J ν (xy) dx =



2

0

13.

ET II 77(11)







xν+1 e− 4 a

2

Γ

ET II 78(12)



3 y2 3 ν + ;ν − μ + ;− 2 2 2 2a  y > 0, |arg a| < 14 π, Re ν > −1

2μ Γ ν + 32 y ν   1F 1 Γ ν − μ + 32 a2ν+2  3 2



ET II 79(23) 1





+ ν 2 2 +m+μ a2k+1 y22 y2 4a W e k,m Γ(−μ)y μ+2 2a2 2k = μ − ν − 1, 2m = μ + ν + 1   y > 0, |arg a| < 34 π, −1 < Re ν < − 12 − 2 Re μ ET II 79(24)

7.754

Parabolic cylinder and Bessel functions



∞

14.

x

e

1 2 2 4a x

0



∞

15. 0



∞

16.

2λ− 2 μ π − 2 1

λ+ 12



1

22 G 23

D μ (ax) J ν (xy) dx = 3 Γ(−μ)y λ+ 2  y > 0, |arg a| < 34 π,

2

1

847

!  ! 1, 1 !2 ! 3 λ+ν ! 4 + 2 , − μ2 , 34 + λ−ν 2 Re μ < − Re λ < Re ν + 32 ET II 80(26) y2 2a2

2

1

xν+1 e 4 x D −2ν−1 (x) J ν (xy) dx = (2ν + 1)y ν−1 e 4 y D −2ν−2 (y)   y > 0, Re ν > − 12 xν+1 e− 4 x D −2ν−3 (x) J ν (xy) dx = 1

2

1 2 2−1/2 π 1/2 y −ν−2 e− 4 y

I ν+1

0



[y > 0, ∞

17.

2

1

1

1

4y

2

ET II 79(20)



Re ν > −1]

ET II 79(21)

2

xν+1 e 4 x D −2ν−3 (x) J ν (xy) dx = y ν e 4 y D −2ν−3 (y)

0



[y > 0, ∞

18.

ν

x e

1 2 2 4a x

0

7.753



∞

1. 0



−1

3 3 4 ν+ 4

D 12 ν− 12 (ax) Y ν (xy) dx = −π 2  y > 0,

xν− 2 e−(x+a) I ν− 12 (2ax) D ν (2x) dx = 2

1

a

−ν −1

y

Re ν > −1]

Γ(ν + 1)e

|arg a| < 34 π,

2. 0

xν− 2 e−(x+a) I ν− 32 (2ax) D ν (2x) dx = 2

3



∞

1.

y2 2a2



1 1 −1/2 π Γ(ν)aν− 2 D −ν (2a) 2

Re ν > 0]

ET II 397(12)

3 1 −1/2 π Γ(ν)aν− 2 D −ν (2a) 2

[Re a > 0, 7.754

ET II 79(22)



W − 12 ν− 12 , 12 ν  − 12 < Re ν < 23 ET II 115(39)

[Re a > 0, ∞

y2 4a2

Re ν > 1]

ET II 397(13)

xν e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν−1 (x) − D 2ν−1 (−x)} J ν (xy) dx 1

2

0

= ±y ν−1 e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν−1 (y) − D 2ν−1 (−y)}   y > 0, Re ν > − 12 ET II 76(2, 3) 1



∞

2.

2

xν e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν+1 (x) − D 2ν+1 (−x)} J ν (xy) dx 1

2

0

= ∓y ν e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν (y) + D 2ν (−y)}   y > 0, Re ν > − 12 ET II 77(6, 7) 1

 3.

∞

2

xν+1 e− 4 x {[1 ± 2 cos(νπ)] D 2ν (x) + D 2ν (−x)} J ν (xy) dx 1

2

0

= ±y ν−1 e− 4 y {[1 ± 2 cos(νπ)] D 2ν+1 (y) − D 2ν+1 (−y)} 1

2

[y > 0,

Re ν > −1]

ET II 78(14, 15)

848

Parabolic Cylinder Functions



∞

4.

7.755

xν+1 e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν+2 (x) + D 2ν+2 (−x)} J ν (xy) dx 1

2

0

= ±y ν e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν+2 (y) + D 2ν+2 (−y)} 1

2

[y > 0, 7.755



∞

1.

x−1/2 D ν

√  √  ax D −ν−1 ax J 0 (xy) dx

0

=2 

∞

 0

∞

4. 0

 0

%

4y 2 1+ 2 a

1/2 & P

1 1 2 ν− 4 1 4

%

4y 2 1+ 2 a

1/2 &

ET II 17(43)

 1  1 x1/2 D − 12 −ν ae 4 πi x1/2 D − 12 −ν ae− 4 πi x1/2 J ν (xy) dx

∞

a

5.

P

1 1 2 ν+ 4 − 14

ET II 78(17, 18)

,2ν  −1 + 2 −1/2   1/2 a + 2y = 2−ν π 1/2 y −ν−1 a2 + 2y Γ ν + 12 −a   y > 0, Re a > 0, Re ν > − 12 ET II 80(27)

0



πa

−1/2

[y > 0, Re a > 0]

2.

3.

−3/2

Re ν > −1]

∞

 1  1 D − 12 −ν ae 4 πi x−1/2 D − 12 −ν ae− 4 πi x−1/2 J ν (xy) dx

, +   −1 = 21/2 π 1/2 y −1 Γ ν + 12 exp −a(2y)1/2   y > 0, Re a > 0, e Re ν > − 12 ET II 80(28)a

  x1/2 D ν− 12 ax−1/2 D −ν− 12 ax−1/2 Y ν (xy) dx

 x1/2 D ν− 12 ax−1/2 D −ν− 12

+    , = y −3/2 exp −ay 1/2 sin ay 1/2 − 12 ν − 12 π   y > 0, |arg a| < 14 π ET II 115(40)  , + ax−1/2 K ν (xy) dx = 2−1 y −3/2 π exp −a(2y)1/2   Re y > 0, |arg a| < 14 π ET II 151(81)

Combinations of parabolic cylinder and Struve functions  ∞ 1 2 7.756 x−ν e− 4 x [D μ (x) − D μ (−x)] Hν (xy) dx 0  



23/2 Γ 1 μ + 12 μ+ν 1 1 1 1 1 y μπ 1 F 1 μ + ; μ + ν + 1; − y 2 sin = 1 2 2 2 2 2 2 Γ 2 μ + ν + 1  3 y > 0, Re(μ + ν) > − 2 , Re μ > −1 ET II 171(41)

7.773

Integration of a parabolic cylinder function

849

7.76 Combinations of parabolic cylinder functions and confluent hypergeometric functions 7.761



∞

1. 0



∞

2. 0



1 2 1 e 4 t t2c−1 D −ν (t) 1 F 1 a; c; − pt2 dt 2  

1 1 π 1/2 Γ(2c) Γ 12 ν − c + a 1  F a, c + ; a + + ν; 1 − p = c+ 1 ν  1   2 2 2 2 2 Γ 2 ν Γ a + 12 + 12 ν [|1 − p| < 1, Re c > 0, Re ν > 2 Re(c − a)] EH II 121(12)

1 2 1 e 4 t t2c−2 D −ν (t) 1 F 1 a; c; − pt2 dt 2  

1 π 1/2 Γ(2c − 1) Γ 12 ν + 12 − c + a 1     = c+ 1 ν− 1 F a, c − ; a + ν; 1 − p 2 2 Γ 12 + 12 ν Γ a + 12 ν 2 2 2  1 |1 − p| < 1, Re c > 2 , Re ν > 2 Re(c − a) − 1 EH II 121(13)

7.77 Integration of a parabolic cylinder function with respect to the index 

7.771 0

∞



1  π 1/2 β 2 cos a cos(ax) D x− 12 (β) D −x− 12 (β) dx = exp − 2 cos a 2 =0

  |a| < 12 π   |a| > 12 π ET II 298(22)

7.772 

− 12 +i∞

1. − 12 −i∞

⎡ ν  1  1 tan 12 ϕ ⎣ 4 iπ ξ −e e 4 iπ η D D ν −ν−1 cos 12 ϕ ⎤ ν   1  1 cot 12 ϕ dν D −ν−1 e 4 iπ ξ D ν −e 4 iπ η ⎦ + 1 sin νπ sin 2 ϕ     = −2i(2π)1/2 exp − 14 i ξ 2 − η 2 cos ϕ − 12 iξη sin ϕ

 − 12 +i∞  2. − 12 −i∞

ν

tan 12 ϕ cos 12 ϕ

EH II 125(7)

 1  1 dν D ν −e 4 iπ ζ D −ν−1 e 4 iπ η sin νπ + 1  + 1  , , iπ 1 ζ cos 2 ϕ + η sin 12 ϕ D −1 e 4 iπ η cos 12 ϕ − ζ sin 12 ϕ = −2i D 0 e 4 EH II 125(8)

7.773



c+i∞

1. c−i∞

D ν (z)tν Γ(−ν) dν = 2πie− 4 z 1

2

−zt− 12 t2

+ c < 0,

|arg t| <

π, 4

EH II 126(10)

850

Meijer’s and MacRobert’s Functions (G and E)



t−ν−1 dν sin(−νπ)   − 1  1 1 − t2  2 txy 2πi  2 x + i =  1/2 1 + t2 2 exp + y π 4 1 + t2 1 + t2 2   1 EH II 126(11) −1 < c < 0, |arg t| < π 2

c+i∞

2.

7.774

[D ν (x) D −ν−1 (iy) + D ν (−x) D −ν−1 (iy)] c−i∞



c+i∞

7.774 c−i∞

+ 1 + 1 , ,       (2)  D ν k 2 (1 + i)ξ D −ν−1 k 2 (1 + i)η Γ − 12 ν Γ 12 + 12 ν dν = 21/2 π 2 H 0 12 k ξ 2 + η 2 [−1 < c < 0,

Re ik ≥ 0]

EH II 125(9)

7.8 Meijer’s and MacRobert’s Functions (G and E) 7.81 Combinations of the functions G and E and the elementary functions 7.811 1.

!

!

! a1 , . . . , ap ! c1 , . . . , cσ μ,ν ηx !! dx G σ,τ ωx !! b1 , . . . , bq d, . . . , d τ ! 0

ω !! −b1 , . . . , −bm , c1 , . . . , cσ , −bm+1 , . . . , −bq 1 = G n+μ,m+ν η q+σ,p+τ η ! −a1 , . . . , −an , d1 , . . . , dτ , −an+1 , . . . , −ap subject to the following constraints



∞

m,n

G p,q

• • • • • • •

m, n, p, q, μ, ν, σ, τ are integers; 1≤n≤p −1 (j = 1, . . . , n; k = 1, . . . ,τ ) Re (aj + ck ) < 1   ω = 0, η = 0, |arg η| < m + n − 12 p − 12 q π, |arg ω| < μ + ν − 12 σ − 12 τ π The following must not be integers: bj − bk aj − a k dj − d k aj + d k

(j (j (j (j

= 1, . . . , m; k = 1, . . . , m; j = k) , = 1, . . . , n; k = 1, . . . , n; j = k) , = 1, . . . , μ; k = 1, . . . , μ; j = k) , = 1, . . . , n; k = 1, . . . , n) ;

• The following must not be positive integers: aj − b k c j − dk

(j = 1, . . . , n; k = 1, . . . , m) (j = 1, . . . , ν; k = 1, . . . , μ)

Formula 7.811 1 also holds for four sets of restrictions. See C. S. Meijer, Neue Integraldarstellungen f¨ ur Whittakersche Funktionen, Nederl. Akad. Wetensch. Proc. 44 (1941), 82–92. ET II 422(14) mn Hereafter, G m,n p,q will be written as G pq , and commas will only be inserted in entries like m,n+1 G p+1,q+1 , where their omission could cause ambiguity.

7.811

G and E and the elementary functions



1 ρ−1

2.

x

(1 − x)

σ−1

0

mn G pq

851

!

!

! a1 , . . . , ap ! 1 − ρ, a1 , . . . , ap m,n+1 ! αx ! dx = Γ(σ) G p+1,q+1 α !! b1 , . . . , bq b1 , . . . , bq , 1 − ρ − σ

where • • • • •

(p + q) <2(m + n)  |arg a| < m + n − 12 p − 12 q π Re (ρ + bj ) > 0, j = 1, . . . , m Re σ > 0 either   p + q ≤ 2(m + n), |arg α| ≤ m + n − 12 ρ − 12 q π, Re (ρ + bj ) > 0; j = 1, . . . , m; Re σ > 0, ⎡ ⎤

p q   1 ⎦ > −1, Re ⎣ aj − bj + (p − q) ρ − 2 2 j=1 j=1

or p
Re (p + bj ) > 0;

j = 1, . . . , m;

Re σ > 0 ET II 417(1)

!

!

! a1 , . . . , ap ! a1 , . . . , ap , ρ m+1,n ! ! x−ρ (x − 1)σ−1 G mn αx dx = Γ(σ) α G p+1,q+1 pq ! b1 , . . . , bq ! ρ − σ, b1 , . . . , bq 1 where

 3.

(or p ≤ q for |α| < 1) ,

∞

• • • • •

p + q < 2(m   + n) 1 |arg α| < m + n − 2 p − 12 q π Re (ρ − σ − aj ) > −1; j = 1, . . . , n Re σ > 0 either p + q ≤ 2(m + n),

  |arg α| ≤ m + n − 12 p − 12 q π,

Re (ρ − σ − aj ) > −1; j = 1, . . . , n; Re σ > 0, ⎡ ⎤

p q   1 ⎦ > −1, aj − bj + (q − p) ρ − σ + Re ⎣ 2 2 j=1 j=1 or q
(or q ≤ p for |α| > 1) ,

Re (ρ − σ − aj ) > −1;

Re σ > 0 ET II 417(2)

$n $m !

∞ ! a1 , . . . , ap Γ (bj + ρ) j=1 Γ (1 − aj − ρ) j=1 ρ−1 mn $p α−ρ x dx = $q G pq αx !! b , . . . , b Γ (1 − b − ρ) Γ (a + ρ) 1 q j j 0 j=m+1  j=n+1  p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, − min Re bj < Re ρ < 1 − max Re aj

 4.

j = 1, . . . , n;

1≤j≤m

1≤j≤n

ET II 418(3)a, ET I 337(14)

852

Meijer’s and MacRobert’s Functions (G and E)



∞

5.

ρ−1

x

−σ

(x + β)

mn G pq

0

7.812

!

!

! a1 , . . . , ap ! 1 − ρ, a1 , . . . , ap β ρ−σ m+1,n+1 ! ! αx ! dx = αβ ! G b1 , . . . , bq Γ(σ) p+1,q+1 σ − ρ, b1 , . . . , bq

where • • • • • •

p + q < 2(m   + n) |arg α| < m + n − 12 p − 12 q π |arg β| < π Re (ρ + bj ) > 0, j = 1, . . . , m Re (ρ − σ + aj ) < 1, j = 1, . . . , n either p ≤ q,

p + q ≤ 2(m + n),

  |arg α| ≤ m + n − 12 p − 12 q π,

|arg β| < π

Re (ρ = bj ) > 0, j = 1, . . . , m, Re (ρ − σ + aj ) < 1, j = 1, . . . , n, ⎤ ⎡

p q   1 ⎦ > 1, Re ⎣ aj − bj − (q − p) ρ − σ − 2 j=1 j=1 or p ≥ q,

p + q ≤ 2(m + n),



1 1 |arg α| ≤ m + n − p − q π, 2 2

|arg β| < π,

j = 1, . . . , m, Re (ρ − σ + aj ) < 1, j = 1, . . . , n, ⎡ ⎤

p q   1 ⎦>1 Re ⎣ aj − bj + (p − q) ρ − 2 j=1 j=1

Re (ρ + bj ) > 0,

ET II 418(4)

7.812



1. 0

1

 z xβ−1 (1 − x)γ−β−1 E a1 , . . . , ap : ρ1 , . . . , ρq ; m dx x = Γ(γ − β)mβ−γ E (a1 , . . . , ap+m : ρ1 , . . . , ρq+m : z) β+k−1 γ+k−1 , ρq+k = , k = 1, . . . , m ap+k = m m [Re γ > Re β > 0, m = 1, 2, . . .] ET II 414(2)

 2.

∞

xρ−1 (1 + x)−σ E [a1 , . . . , ap : ρ1 , . . . , ρq : (1 + x)z] dx

0

= Γ(ρ) E (a1 , . . . , ap , σ − ρ; ρ1 , . . . , ρq , σ; z) [Re σ > Re ρ > 0]

ET II 415(3)

7.815

G and E and the elementary functions



∞

3.

−β s−1

(1 + x)

x

mn G pq

0

853

!

!

! 1 − s, a1 , . . . , ap ax !! a1 , . . . , ap m,n+1 ! dx = Γ(β − s) G p+1,q+1 a ! 1 + x ! b1 , . . . , bq b1 , . . . , bq , 1 − β − min Re bk < Re s < Re β,

1 ≤ k ≤ m;

(p + q) < 2(m + n),    1 1 |arg a| < m + n − 2 p − 2 q π ET I 338(19)

7.813  1.

∞

−ρ −βx

x

e

mn G pq

0



!

!

! a1 , . . . , ap α !! ρ, a1 , . . . , ap m,n+1 ρ−1 ! αx ! dx = β G p+1,q b1 , . . . , bq β ! b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  |arg β| < 12 π, Re (bj − ρ) > −1, j = 1, . . . , m ET II 419(5)

∞

2. 0

! !



! 4α !! 0, 12 , a1 , . . . , ap m,n+2 2 ! a1 , . . . , ap −1/2 −1 e−βx G mn β αx dx = π G p+2,q pq ! b1 , . . . , bq β 2 ! b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  1 1 |arg β| < 2 π, Re bj > − 2 ; j = 1, . . . , m

ET II 419(6)

7.814  1.

∞

0

xβ−1 e−x E (a1 , . . . , ap : ρ1 , . . . , ρq : xz) dx    = π cosec(βπ) E a1 , . . . , ap : 1 − β, ρ1 , . . . , ρq : e±iπ z −z

2.

−β

  E a1 + β, . . . , ap + β : 1 + β, ρ1 + β, . . . , ρl + β : e±iπ z



[p ≥ q + 1, Re (ar + β) > 0, r = 1, . . . , p, |arg z| < π. The formula holds also for p < q + 1, ET II 415(4) provided the integral converges.]  ∞   xβ−1 e−x E a1 , . . . , ap : ρ1 , . . . , ρq : x−m z dx 0

 Re β > 0, 7.815  1. 0

∞

ap+k

1 1 1   = (2π) 2 − 2 m mβ− 2 E a1 , . . . , ap+m : ρ1 , . . . , ρq : m−m z  β+k−1 , k = 1, . . . , m; m = 1, 2, . . . = ET II 415(5) m

! !



! √ −1 m,n+1 4α ! 0, a1 , . . . , ap , 12 2 ! a1 , . . . , ap ! sin(cx) G mn πc αx dx = G p+2,q pq ! b1 , . . . , bq c2 ! b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  c > 0, Re bj > −1, j = 1, 2, . . . , m, Re aj < 12 , j = 1, . . . , n ET II 420(7)

854

Meijer’s and MacRobert’s Functions (G and E)



∞

2.

cos(cx) G mn pq

0

7.821

! !



! 4α !! 12 , a1 , . . . , ap , 0 m,n+1 2 ! a1 , . . . , ap 1/2 −1 αx ! dx = π c G p+2,q b1 , . . . , bq c2 ! b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  c > 0, Re bj > − 12 , j = 1, . . . , m, Re aj < 12 , j = 1, . . . , n ET II 420(8)

7.82 Combinations of the functions G and E and Bessel functions 7.821  1.

∞

0

!

!

! a1 , . . . , ap ! ρ − 1 ν, a1 , . . . , ap , ρ + 1 ν  √  m,n+1 2 2 ! ! x−ρ J ν 2 x G mn αx dx = α G p+2,q pq ! b1 , . . . , bq ! b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π  3 1 − 4 + max Re aj < Re ρ < 1 + 2 Re ν + min Re bj 1≤j≤n



∞

2.

 √  x−ρ Y ν 2 x G mn pq



1≤j≤m

ET II 420(9)

!

! a1 , . . . , ap αx !! dx b1 , . . . , bq

  ! ! ρ − 1 ν, ρ + 1 ν, a , . . . , a , ρ + 1 + 1 ν 1 p ! 2 2 2 2 = α! ! b1 , . . . , bq , ρ + 12 + 12 ν    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  − 34 + max Re aj < Re ρ < min Re bj + 12 |Re ν| + 1

0

m,n+2 G p+3,q+1

1≤j≤n



∞

3.

 √  x−ρ K ν 2 x G mn pq



0

1≤j≤m

ET II 420(10)

!

!

! a1 , . . . , ap ! ρ − 1 ν, ρ + 1 ν, a1 , . . . , ap 1 m,n+2 2 2 ! ! αx ! dx = G p+2,q α ! b1 , . . . , bq 2 b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  Re ρ < 1 − 12 |Re ν| + min Re bj 1≤j≤m

ET II 421(11)

7.822  1. 0

∞

!

!

! a1 , . . . , ap 4λ !! h, a1 , . . . , ap , k 22ρ m,n+1 ! λx ! dx = 2ρ+1 G p+2,q b1 , . . . , bq y y 2 ! b1 , . . . , bq h = 12 − ρ − 12 ν, k = 12 − ρ + 12 ν      p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, Re bj + ρ + 12 ν > − 12 ,  j = 1, 2, . . . , m, Re (aj + ρ) < 34 , j = 1, . . . , n, y > 0

2ρ

x

J ν (xy) G mn pq

2

ET II 91(20)

7.823

G and E and Bessel functions



∞

2.

1/2

x 0

Y ν (xy) G mn pq

855

!

! 2 ! a1 , . . . , ap λx ! dx b1 , . . . , bq



= (2λ)−1/2 y −1/2 G n+2,m q+1,p+3

!  y 2 !! 12 − b1 , . . . , 12 − bq , l ! 4λ ! h, k, 12 − a1 , . . . , 12 − ap , l

+ 12 ν, k = 14 − 12 ν, l = − 14 − 12 ν   p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, y > 0,    3 Re aj < 1, j = 1, . . . , n, Re bj ± 12 ν > − , j = 1, . . . , m 4 h=





∞

3.

1 4

ET II 119(56)

!

! 1/2 mn 2 ! a1 , . . . , ap x K ν (xy) G pq λx ! dx b1 , . . . , bq

!  y 2 !! 12 − b1 , . . . , 12 − bq =2 λ y ! 4λ ! h, k, 12 − a1 , . . . , 12 − ap h = 14 + 12 ν, k = 14 − 12 ν    Re y > 0, p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π ,  Re bj > 12 |Re ν| − 34 , j = 1, . . . , m

0

−3/2 −1/2 −1/2



n+2,m G q,p+2

ET II 153(90)

7.823



∞

1. 0

  xβ−1 J ν (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−2m z dx      = (2π)−m (2m)β−1 exp 12 π (β − ν − 1) i E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m ze−mπi     + exp − 12 π(β − ν − 1)i E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m zemπi , β + ν + 2k − 2 β − ν + 2k − 2 ap+k = , ap+m+k = , m = 1, 2, . . . , ; k = 1, . . . , m 2m 2m   Re(β + ν) > 0, Re (2ar m − β) > − 32 , r = 1, . . . , p ET II 415(7)

 2.

∞

  xβ−1 K ν (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−2m z dx

0

ap+k

= (2π)1−m 2β−2 mβ−1   × E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m z , β + ν + 2k − 2 β − ν + 2k − 2 , ap+m+k = , k = 1, 2, . . . , m = 2m 2m [Re β > |Re ν|, m = 1, 2, . . .] ET II 416(8)

856

Meijer’s and MacRobert’s Functions (G and E)

7.824



∞

1.

!

! 2 ! a1 , . . . , ap x1/2 Hν (xy) G mn λx dx pq ! b1 , . . . , bq

!  y 2 !! l, 12 − b1 , . . . , 12 − bq = (2λy) ! 4λ ! l, 12 − a1 , . . . , 12 − ap , h, k 1 ν 3 ν 1 ν h= + , k= − , l= + 4 2 4 2 4 2    p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, y > 0,   3 1  5 Re aj < min 1, 4 − 2 ν , j = 1, . . . , n, Re (2bj + ν) > − 2 , j = 1, . . . , m

0



−1/2



7.824

∞

2.

n+1,m+1 G q+1,p+3

ET II 172(47)

!

! a1 , . . . , ap  √  mn −ρ ! x Hν 2 x G pq αx ! dx b1 , . . . , bq

 ! !ρ − ! α! !ρ −

 − 12 ν, a1 , . . . , ap , ρ + 12 ν, ρ − 12 ν = − 12 ν, b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π , 

3 ν−1 1 3 max − , Re + max Re aj < Re ρ < min Re bj + 2 Re ν + 2 1≤j≤n 1≤j≤m 4 2

0

m+1,n+1 G p+3,q+1

1 2 1 2

ET II 421(12)

7.83 Combinations of the functions G and E and other special functions 

7.831

∞

−ρ

x 1

(x − 1)

σ−1

!

! a1 , . . . , ap ! F (k + σ − ρ, λ + σ − ρ; σ; 1 − αx ! dx b1!, . . . , bq

! a1 , . . . , ap , k + λ + σ − ρ, ρ m+2,n ! = Γ(σ) G p+2,q+2 α ! k, λ, b1 , . . . , bq x) G mn pq

where

ET II 421(13)

⎡ ⎤

p q   1 ⎦ 1 • Re ⎣ aj − bj + (q − p) k + >− 2 2 j=1 j=1 ⎡ ⎤

p q   1 ⎦ 1 ⎣ • Re aj − bj + (q − p) λ + >− 2 2 j=1 j=1 • either   p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, Re σ > 0, Re k ≥ Re λ > Re aj − 1, j = 1, . . . , n,

7.832

G and E and other special functions

857

or   p + q ≤ 2(m + n), |arg α| ≤ m + n − 12 p − 12 q π, Re σ > 0, Re k ≥ Re λ > Re aj − 1, j = 1, . . . , n, 

∞

7.832

  1 xβ−1 e− 2 x W κ,μ (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−m z dx

0

  1 1 1 = (2π) 2 − 2 m mβ+κ− 2 E a1 , . . . , ap+2m : ρ1 , . . . , ρq+m : m−m z ,

ap+k =

β + k + μ − 12 , m

ap+m+k =

β − μ + k − 12 β−κ+k , ρq+k = , k = 1, . . . , m m m   1 Re β > |Re μ| − 2 , m = 1, 2, . . . ET II 416(10)

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8–9 Special Functions 8.1 Elliptic Integrals and Functions 8.11 Elliptic integrals 8.110 1.

 Every integral of the form

  R x, P (x) dx, where P (x) is a third- or fourth-degree polyno-

mial, can be reduced to a linear combination of integrals leading to elementary functions and the following three integrals:  √   1 − k 2 x2 dx dx  √  , , dx, 2 2 2 2 2 1−x (1 − x ) (1 − k x ) (1 − nx ) (1 − x2 ) (1 − k 2 x2 ) which are called respectively elliptic integrals of the first, second, and third kind in the Legendre normal form. The results of this reduction for the more frequently encountered integrals are given√in formulas 3.13–3.17. The number k is called the modulus ∗ of these integrals; the number k = 1 − k 2 is called the complementary modulus, and the number n is called the parameter of BY (110.04) the integral of the third kind. 2.

3.11 4.∗

By means of the substitution x = sin ϕ, elliptic integrals can be reduced to the normal trigonometric forms    < dϕ dϕ 2 2  1 − k sin ϕ dϕ, , . BY (110.04)   2 2 2 1 − n sin ϕ 1 − k sin ϕ 1 − k 2 sin2 ϕ The results of reducing integrals of trigonometric functions to normal form are given in 2.58– 2.62. π in the 8.110 2 formuElliptic integrals from 0 to 1 in the 8.110 1 formulation (or from 0 to 2 lation) are called complete elliptic integrals. Take note that in mathematical software, and elsewhere, the notation for elliptic integrals is often modified by replacing the parameter k2 that is used here with k.

8.111 Notations: 1. ∗ The

Δϕ =

<

1 − k 2 sin2 ϕ;

k =

 1 − k2 ;

k2 < 1

quantity k is sometimes called the module of the functions.

859

860

Elliptic Integrals and Functions

8.112

2.

The elliptic integral of the first kind:  ϕ  sin ϕ dα dx   F (ϕ, k) = = 2 2 (1 − x ) (1 − k 2 x2 ) 0 0 1 − k 2 sin α

3.

The elliptic integral of the second kind:   ϕ 2 2 E (ϕ, k) = 1 − k sin α dα = 0

4.11

0

6.∗

√ 1 − k 2 x2 √ dx 1 − x2

FI II 135

The elliptic integral of the third kind:  ϕ  sin ϕ dα dx  Π(ϕ, n, k) = =   2 2 2 2 (1 − nx ) (1 − x2 ) (1 − k 2 x2 ) 0 0 1 − n sin α 1 − k sin α 

5.

sin ϕ



BY (110.04)

sin α dα x dx F (ϕ, k) − E (ϕ, k)   = = 2 2 k (1 − x2 ) (1 − k 2 x2 ) 0 0 1 − k 2 sin α  



 π/2 b dx i b π  F arcsin √ , arctan  = 2 2 2 2 2 2 2|a| a a +b +1 0 a + sin x a + sin x ϕ

2

sin ϕ

2

D(ϕ, k) =

[a and b are real] 8.112 1. 2. 3. 4. 5.

Complete elliptic integrals π K(k) = F , k = K  (k  ) 2 π , k = E  (k  ) E(k) = E 2 π , k = K (k ) K  (k) = F 2 π , k = E (k ) E  (k) = E 2 π K − E ,k = D=D 2 k2

In writing complete elliptic integrals, the modulus k, which acts as an independent variable, is often omitted, and we write     K (≡ K(k)) , K  ≡ K  (k) , E (≡ E(k)) , E  ≡ E  (k) . Series representations 8.113 1.

π K= 2



2

2

2 1 1 1·3 (2n − 1)!! 1 π 2 4 2n 2 F , ; 1; k k + k + ··· + k + . . . = 1+ 2 2·4 2n n! 2 2 2



FI II 487, WH 499

8.116

Elliptic integrals

π K= 1 + k

2.

861

 2

2

4

2n

2

2 1·3 (2n − 1)!! 1 1 − k 1 − k 1 − k + + ··· + + ... 1+ 2 1 + k 2·4 1 + k 2n n! 1 + k



DW

2

2

1 1·3 2 4 2 2 4 4 − K = ln  + ln  − ln  − k 2 + k 4 k 2 k 1·2 2·4 k 1·2 3·4

2

1·3·5 2 2 2 4 − − + ln  − k 6 + . . . 2·4·6 k 1·2 3·4 5·6

3.

DW

See also 8.197 1 and 8.197 2. 8.114 1.

2.

6

π E= 2



12 · 3 1 1 − 2 k2 − 2 2 k4 − · · · − 2 2 ·4

(1 + k  ) π E= 4

3.



1 1+ 2 2



(2n − 1)!! 2n n!

2



π 1 1 k 2n − . . . = F − , ; 1; k 2 2n − 1 2 2 2 WH 518, FI II 487





 2

1−k 1 + k

12 + 2 2 2 ·4



 4

1−k 1 + k

+ ··· +

1 1 1 1 ·3 2 4 4 − ln  − k 2 + 2 ln  − 2 k 1 · 2 2 ·4 k 1·2 3·4 2 1 12 · 32 · 5 2 4 − − + 2 2 ln  − k 6 + . . . 2 ·4 ·6 k 1·2 3·4 5·6

E= 1 +

2



(2n − 3)!! 2n n!

2

1 − k 1 + k

2n



+ ... DW

k 4

DW





2  2 2 (2n − 1)!! 1 1 2 1·3 n 2 2(n−1) 8.115 D = π + k + ··· + k + ... 1 2 3 2·4 2n − 1 2n n!     π2  arccos n1 1 − k 2 sin2 ϕ  2 − k 2  8.116 n dϕ = +R , where  2 − 1 1 − n2 sin2 ϕ 0 n n





1 1 k 2 k 4 6 1 1 R= + 1 + p+ −1 + p + 3 2 2  2 n 3 16 n  4 n

6 3 1 1 1 1 5 k 7 + 2 + 4 + p+ + − − 16 16 n 2 6 n 3 8

n n

 5 21 1 9 1 1 1 14 15k8 37 − + − 4 + p + + 4 + 6 + + ..., − 256 144 40n 2 8 n 3 24 20n 2 n n 3n 4 2 p = ln  , k = 4e−p , k2 = 1 − k 2 , n = 1 − n2 k

ZH 43(158)

ZH 44(163)

ZH 44(163)

862

Elliptic Integrals and Functions

8.117

Trigonometric series 8.117 1.

2.

8.118 1.

2.

For small values of k and ϕ, we may use the series

2 2 2·4 2 4 F (ϕ, k) = Kϕ − sin ϕ cos ϕ a0 + a1 sin ϕ + a2 sin ϕ + . . . , π 3 3·5  2 (2n − 1)!! 2 a0 = K − 1; an = an−1 − k 2n π 2n n!

2 2 2·4 2 4 b2 sin ϕ + . . . , E (ϕ, k) = Eϕ + sin ϕ cos ϕ b0 + b1 sin ϕ + π 3 3·5  2 (2n − 1)!! k 2n 2 b0 = 1 − E, bn = bn−1 − n π 2 n! 2n − 1

where

ZH 10(19)

where

ZH 27(86)

For k close to 1, we may use the series

2  2·4  2 4 a tan ϕ − . . . , − a1 tan ϕ + where 3 3·5 2  2 (2n − 1)!! 2 a0 = K − 1; an = an−1 − k 2n ZH 10(23) π 2n n! ϕ π 2 + E (ϕ, k) = (K − E ) ln tan π 2 2

,  2  2·4  1 + tan ϕ  2 4 b3 tan ϕ − . . . + + b1 − b2 tan ϕ + 1 − cos ϕ 1 − k 2 sin ϕ , cos ϕ 3 3·5 sin ϕ  ϕ π tan ϕ 2 + − F (ϕ, k) = K ln tan π 2 4 cos ϕ

where b0



a0

 2

(2n − 3)!! 2n − 1 2     = (K − E ) , bn = bn−1 − n−1 k 2n π 2 (n − 1)! 2n

ZH 27(90)

For the expansion of complete elliptic integrals in Legendre polynomials, see 8.928. 8.119 1.

Representation in the form of an infinite product: K(k) =

∞ π (1 + kn ) , 2 n=1

where < 2 1 − kn−1 < ; kn = 2 1 + 1 − kn−1 1−

See also 8.197.

k0 = k

FI II 166

8.125

Functional relations between elliptic integrals

863

8.12 Functional relations between elliptic integrals 8.121 1.

F (−ϕ, k) = − F (ϕ, k)

JA

2.

E (−ϕ, k) = − E (ϕ, k)

JA

3.

F (nπ ± ϕ, k) = 2n K(k) ± F (ϕ, k)

JA

4.

E (nπ ± ϕ, k) = 2n E(k) ± E (ϕ, k)

JA

8.122 E(k) K  (k) + E  (k) K(k) − K(k) K  (k) = 8.123 1. 2. 3. 4.

1 ∂F = 2 ∂k k



k sin ϕ cos ϕ E − k 2 F − k 1 − k 2 sin2 ϕ

π 2

FI II 691, 791

 MO 138, BY (710.07)

E(k) K(k) d K(k) = − dk kk2 k E−F ∂E = ∂k k E(k) − K(k) d E(k) = dk k

FI II 691 MO 138 FI II 690

8.124 1.

2.

8.125 1. 2.

3.

4.

The functions K and K satisfy the equation

 d 2 du kk − ku = 0. dk dk The functions E and E − K satisfy the equation

d du k 2 k + ku = 0. dk dk

WH 499, WH 502

WH



1 − k F ψ, [tan(ψ − ϕ) = k  tan ϕ] = (1 + k ) F (ϕ, k) 1 + k

1 − k 1 − k 2 E ψ, [E (ϕ, k) + k  F (ϕ, k)] − sin ψ =   1+k 1+k 1 + k [tan(ψ − ϕ) = k tan ϕ]  √    (1 + k) sin ϕ 2 k = (1 + k) F (ϕ, k) sin ψ = F ψ, 1+k 1 + k sin2 ϕ  √    < sin ϕ cos ϕ 2 k 1 2 2 2 1 − k sin ϕ E ψ, = 2E(ϕ, k) − k F (ϕ, k) + 2k 1+k 1+k 1 + k sin2 ϕ   (1 + k) sin ϕ sin ψ = 1 + k sin2 ϕ

MO 130

MO 131

MO 131

864

8.126 1. 2. 3.

4.

Elliptic Integrals and Functions

8.126

In particular,

1 − k 1 + k K(k) K =  1+k 2

1 − k 1 E [E(k) + k  K(k)] = 1 + k 1 + k  √  2 k = (1 + k) K(k) K 1+k  √   2 k 1  2 E(k) − k 2 K(k) E = 1+k 1+k

MO 130 MO 130

MO 130

MO 130

8.12711 k1

sin ϕ1

cos ϕ1

F (ϕ1 , k1 )

sin ϕ Δϕ

cos ϕ Δϕ

k  F (ϕ, k)

E (ϕ1 , k1 ) 1 + E (ϕ, k) − k

,

k k

k

k

−i tan ϕ

sec ϕ

−i F (ϕ, k)

ia [E (ϕ, k) − F (ϕ, k) − Δϕ tan ϕ]

1 k

k sin ϕ

Δϕ

k F (ϕ, k)

 1 E (ϕ, k) − k2 F (ϕ, k) k

1 k

−ik tan ϕ

Δϕ cos ϕ

−ik F (ϕ, k)

k ik

−ik sin ϕ Δϕ

1 Δϕ

−ik F (ϕ, k)

i

k2 sin ϕ cos ϕ Δϕ

 i  E (ϕ, k) − k2 F (ϕ, k) − Δϕ tan ϕ  k , 2 i+ ϕ cos ϕ E (ϕ, k) − F (ϕ, k) − k sinΔϕ k (see 8.111 1)

8.128 1. 2. 3.

In particular,

k K i  = k  K(k) k

  k K i  = k  K  (k  ) − i K(k) k

  1 K = k K(k) + i K  (k) k

MO 131

[Im(k) < 0]

MO 130

[Im(k) < 0]

MO 130

[Im(k) < 0]

MO 130

For integrals of elliptic integrals, see 6.11–6.15. For indefinite integrals of complete elliptic integrals, see 5.11. 8.129 Special values: √  √   2  √  1 dt π 2 2 1 1  √ 1. K sin =K = √ Γ MO 130 =K = 2 4 2 2 4 4 π 1 − t4 0 √ √ √ 2. K 2 − 1 = 2K 2−1 MO 130

8.130

3. 4. 5.∗ 6.∗ 7.∗

Elliptic functions

865

  π √ π = 3 K sin K  sin 12 12  √    π π 2− 2 √ = K = 2 K tan2 K  tan2 8 8 2+ 2 √  3−1 π √ = K sin 12 2 2  √ 2  π 3 + kK E= 12K 3  √ 2 π 3  kK E = + 4K 3

MO 130 MO 130

8.13 Elliptic functions 8.130 1.

Definition and general properties. A single-valued function f (z) of a complex variable, which is not a constant, is said to be elliptic if it has two periods 2ω1 and 2ω2 , that is f (z + 2mω1 + 2nω2 ) = f (z)

[m, n integers] .

The ratio of the periods of an analytic function cannot be a real number. For an elliptic function f (z), the z-plane can be partitioned into parallelograms—the period parallelograms—the vertices of which are the points z0 + 2mω1 + 2nω2 . At corresponding points of these parallelograms, the ZH 117, SI 299 function f (z) has the same value. 2.

Suppose that α is the angle between the sides a and b of one of the period parallelograms. Then, a , + a a ω1 a π cos α + i sin π cos α . τ= = eiα , q = eiπτ = e− b π sin α cos ω2 b b b

3.

The derivative of an elliptic function is also an elliptic function with the same periods. SM III 598

4.

A non-constant elliptic function has a finite number of poles in a period parallelogram: it can have no more than two simple and one second-order pole in such a parallelogram. Suppose that these poles lie at the points a1 , a2 , . . . , an and that their orders are α1 , α2 , . . . , αn . Suppose that the zeros of an analytic function that occur in a single parallelogram are b1 , b2 , . . . , bm and that the orders of the zeros are β1 , β2 , . . . , βm , respectively. Then, γ = α1 + α2 + · · · + αn = β1 + β2 + · · · + βm .

ZH 118

The number γ representing this sum is called the order of the elliptic function. 5.

The sum of the residues of an elliptic function with respect to all the poles belonging to a period parallelogram is equal to zero.

6.

The difference between the sum of all the zeros and the sum of all the poles of an elliptic function that are located in a period parallelogram is equal to one of its periods.

7.

Every two elliptic functions with the same periods are related by an algebraic relationship. GO II 151

866

8.7

Elliptic Integrals and Functions

8.141

A non-constant single-valued function which is not constant cannot have more than two periods. GO II 147

9.

An elliptic function of order γ assumes an arbitrary value γ times in a period parallelogram. SM 601, SI 301

8.14 Jacobian elliptic functions 8.141

Consider the upper limit ϕ of the integral  ϕ  u= 0

as a function of u. Using the notation

dα 1 − k 2 sin2 α

ϕ = am u we call this upper limit the amplitude. The quantity u is called the argument, and its dependence on ϕ is written u = arg ϕ. 8.142 The amplitude is an infinitely many-valued function of u and has a period of 4Ki. The branch points of the amplitude correspond to the values of the argument u = 2mK + (2n + 1)K i, where m and n are arbitrary integers (see also 8.151). 8.143

ZH 67–69

The first two of the following functions sn u = sin ϕ = sin am u, cn u = cos ϕ = cos am u, < dϕ dn u = Δϕ = 1 − k 2 sin2 ϕ = du

are called, respectively, the sine-amplitude and the cosine-amplitude while the third may be called the delta amplitude. All these elliptic functions were exhibited by Jacobi and they bear his name. SI 16 The Jacobian elliptic functions are doubly periodic functions and have two simple poles in a period ZH 69 parallelogram. 8.144  sn u dt  SI 21(23) 1. u= 2 (1 − t ) (1 − k 2 t2 ) 0  cn u dt  2. u= SI 21(23) 2 ) (k  2 + k 2 t2 ) (1 − t 1  dn u dt  3. u= SI 21(23) 2 (1 − t ) (t2 − k 2 ) 1 8.145 1.11

Power series representations: 1 + k 2 3 1 + 14k 2 + k 4 5 1 + 135k 2 + 135k 4 + k 6 7 u + u − u 3! 5! 7! 2 4 6 8 1 + 1228k + 5478k + 1228k + k 9 u − ... + 9! [|u| < |K |]

sn u = u −

ZH 81(97)

8.146

Jacobian elliptic functions

2.

cn u = 1 −

3.

dn u = 1−

1 2 1 + 4k 2 4 1 + 44k 2 + 16k 4 6 1 + 408k 2 + 912k 4 + 64k 6 8 u + u − u + u − ... 2! 4! 6! 8!  [|u| < |K |] ZH 81(98)

      k 2 2 k 2 4 + k 2 4 k 2 16 + 44k 2 + k 4 6 k 2 64 + 912k2 + 408k 4 + k 6 8 u + u − u + u − ... 2! 4! 6! 8!  [|u| < |K |] ZH 81(99)

      k2 3 k 2 4 + k 2 5 k 2 16 + 44k 2 + k 4 7 k 2 64 + 912k 2 + 408k 4 + k 6 9 u − u + u − ... =u− u + 3! 5! 7! 9!  [|u| < |K |] LA 380(4)

4.

am u

8.146

 πK  Representation as a trigonometric series or a product q = e− K = eπiτ ∗ ∞

1.

867

11

2π  q n− 2 πu sn u = sin(2n − 1) kK n=1 1 − q 2n−1 2K 1

∞

2.11

cn u =

2π  q n− 2 πu cos(2n − 1) kK n=1 1 + q 2n−1 2K

3.

dn u =

2π  q n π nπu + cos 2K K n=1 1 + q 2n K

4.11

am u =

WH 511a, ZH 84(108)

1

WH 511a, ZH 84(109)

∞

5.

6.

7.

8. 9.11

10.

∞  1 qn πu nπu +2 sin 2n 2K n1+q K n=1 % & ∞  π 1 q 2n−1 1 πu = sin(2n − 1) πu + 4 sn u 2K sin 2K 1 − q 2n−1 2K n=1 % & ∞ 2n−1  π 1 πu 1 n q =  (−1) cos(2n − 1) πu + 4 cn u 2k K cos 2K 1 + q 2n−1 2K n=1 % & ∞  π qn 1 nπu =  1+4 (−1)n cos 2n dn u 2k K 1+q K n=1 % & ∞ 2n  π πu sn u nπu n q =  tan +4 (−1) sin cn u 2k K 2K 1 + q 2n K n=1 1 ∞ sn u q n− 2 2π  πu =−  (−1)n sin(2n − 1) 2n−1 dn u kk K n=1 1+q 2K % & ∞  π q 2n cn u πnu πu = −4 sin cot 2n sn u 2K 2K 1+q K n=1

WH 511a, ZH 84(110)

WH 511a

LA 369(3)

LA 369(3)

LA 369(3)

LA 369(4)

LA 369(4)

LA 369(5)

! πu !! 1 ! expansions 1–22 are valid in every strip of the form !Im ! < π Im τ . The expansions 23–25 are valid in an 2K 2 arbitrary bounded portion of u. ∗ The

868

11.

12.

13.

14.

15.

16.

17.

18.

19.

Elliptic Integrals and Functions 1 ∞ 2π  q n− 2 cn u πu =− (−1)n cos(2n − 1) dn u kK n=1 1 − q 2n−1 2K % & ∞  π 1 q 2n−1 dn u πu = sin(2n − 1) πu − 4 sn u 2K sin 2K 1 + q 2n−1 2K n=1 % & ∞ 2n−1  π 1 dn u πu n q = (−1) cos(2n − 1) πu − 4 cn u 2K cos 2K 1 − q 2n−1 2K n=1 % & ∞  π πu qn cn u dn u nπu = cot −4 sin n sn u 2K 2K 1+q K n=1   ∞  π πu qn sn u dn u nπu = +4 tan sin cn u 2K 2K 1 + (−1)n q n K n=1

∞ 4π 2  q 2n−1 sn u cn u πu = 2 sin(2n − 1) dn u k K n=1 1 − q 2(2n−1) K % & ∞ n  π πu sn u nπu n q = +4 tan (−1) sin cn u dn u 2 (1 − k2 ) K 2K 1 − qn K n=1 % & ∞  (−1)n q n π πu cn u nπu = −4 cot sin n n sn u dn u 2K 2K 1 + (−1) q K n=1 % & ∞  π 1 q 2(2n−1) dn u πu = +4 sin(2n − 1) sn u cn u K sin πu K 1 − q 2(2n−1) K n=1

20.11 ln sn u = ln

∞  πu 1 qn nπu 2K + ln sin −4 sin2 n π 2K n 1 + q 2K n=1

21.

ln cn u = ln cos

22.

ln dn u = −8

∞  qn 1 nπu πu −4 sin2 n n 2K n 1 + (−1) q 2K n=1

∞ 

q 2n−1 1 πu sin2 (2n − 1) 2(2n−1) 2n − 1 2K 1 − q n=1

√ ∞ 4n 24q πu - 1 − 2q 2n cos πu K +q 23.11 sn u = √ sin πu 2n−1 2K n=1 1 − 2q cos K + q 4n−2 k √ √ ∞ 4n 2 k 4 q πu - 1 + 2q 2n cos πu K +q cos 24. cn u = √ 4n−2 2K n=1 1 − 2q 2n−1 cos πu k K +q 25.

26.

∞ 4n−2 √ 1 + 2q 2n−1 cos πu K +q  dn u = k πu 1 − 2q 2n−1 cos K + q 4n−2 n=1 % & 1 ∞ 2 πu  (2n + 1) π 2 2πq n+ 2 sin(2n + 1) 2K 1 + k2 3 sn u = − 3 3 2 2n+1 2k 2k 4K K (1 − q ) n=0 , +! u !! ! ! < Im τ !Im 2K

8.146

LA 369(5)

LA 369(6)

LA 369(6)

LA 369(7)

LA 369(7)

LA 369(7)

LA 369(8)

LA 369(8)

LA 369(8)

LA 369(2)

LA 369(2)

LA 369(2)

WH 508a, ZH 86(145)

WH 508a, ZH 86(146)

WH 508a, ZH 86(147)

MO 147

8.148

Jacobian elliptic functions

869

∞

27.

π2 K − E 2π 2  nq 2n cos nπu 1 2 πu K = + − 2 cosec sn 2 u 4K2 2K K K n=1 1 − q 2n  ! u !! 1 ! ! < Im τ !Im 2K 2

MO 148

8.147 1.

2.

3.

8.148

where

∞  π 1 π 2kK n=−∞ sin [u − (2n − 1)iK ] 2K ∞ (−1)n πi  cn u = 2kK n=−∞ sin π [u − (2n − 1)iK ] 2K ∞  (−1)n πi dn u = π 2K n=−∞ tan [u − (2n − 1)iK ] 2K

sn u =

MO 149

MO 150

MO150

The Weierstrass expansions of the functions sn u, cn u, dn u: C D B cn u = , dn u = , sn u = , A A A ∞ 

A=1−

(−1)n+1 an+1

n=1

C=

∞ 

(−1)n cn

n=0

u2n+2 (2n + 2)!

B=

2n

u (2n)!

D=

ZH 82–83(105,106,107) ∞ 

(−1)n bn

n=0 ∞ 

(−1)n dn

n=0

and

    a2 = 2k 2 , a3 = 8 k 2 + k 4 , a4 = 32 k 2 + k 6 + 68k 4 ,     a6 = 512 k 2 + k 10 + 3008 k 4 + k 8 + 5400k 6 , . . .

u2n+1 (2n + 1)! u2n (2n)!

    a5 = 128 k 2 + k 8 + 480 k 4 + k 6 ,

  b1 = 1 + k 2 , b2 = 1 + k 4 + 4k 2 , b3 = 1 + k 6 + 9 k 2 + k 4 ,       b4 = 1 + k 8 + 16 k 2 + k 6 − 6k 4 , b5 = 1 + k 10 + 25 k 2 + k 8 − 494 k 4 + k 6 ,     b6 = 1 + k 12 + 36 k 2 + k 10 − 5781 k 4 + k 8 − 12184k 6 , . . .

b0 = 1,

c0 = 1,

c1 = 1,

c2 = 1 + 2k 2 ,

c3 = 1 + 6k 2 + 8k 4 ,

c5 = 1 + 20k2 + 348k 4 + 448k 6 + 128k 8 , d0 = 1,

2

2

d1 = k , 2

4

d2 = 2k + k , 4

6

8

c4 = 1 + 12k2 + 60k 4 + 32k 6 ,

c6 = 1 + 30k2 + 2372k4 + 4600k6 + 2880k 8 + 512k 10 , 2

4

6

d3 = 8k + 6k + k , 10

d5 = 128k + 448k + 348k + 20k + k , d6 = 512k 2 + 2880k4 + 4600k 6 + 2372k 8 + 30k 10 + k 12 ,

...

2

4

4

8

d4 = 32k + 60k + 12k + k ,

...

870

Elliptic Integrals and Functions

8.151

8.15 Properties of Jacobian elliptic functions and functional relationships between them 8.151

The periods, zeros, poles, and residues of Jacobian elliptic functions:

1. Periods

Zeros

Poles

sn u

4mK + 2nK i

2mK + 2nK i

2mK + (2n + 1)K i

cn u

4mK + 2n (K + K i)

(2m + 1) K + 2nK i

2mK + (2n + 1)K i

Residues 1 (−1)m k (−1)m−1 ki

dn u

2mK + 4nK i

(2m + 1)K + (2n + 1)K i

2mK + (2n + 1)K i

(−1)n−1 i

SM 630, ZH 69–72

2. u∗ = u + K sn u∗ =

cn u dn u

cn u∗ = −k  dn u∗ = k 

sn u dn u

u + iK

u + K + iK

u + 2K

u + 2iK

u + 2K + 2iK

1 k sn u

1 dn u k cn u

− sn u

sn u

− sn u

ik k cn u

− cn u

− cn u

cn u

sn u cn u

dn u

− dn u

− dn u

−

1 dn u

i dn u k sn u

−i

cn u sn u

−

ik

SM 630

3. u∗ = 0

−u

sn u∗ = 0

− sn u

cn u∗ = 1

cn u

dn u∗ = 1

dn u

SI 19, SI 18(13), WH,

1 2K

1 √ 1 + k √ k √ 1 + k √ k

(K + iK ) √ √ 1+k+i 1−k √ 2k √ (1 − i) k  √ 2k √ √  √ k 1 + k − i 1 − k √ 2 1 2

WH

1  2 iK

u + 2mK + 2nK i

i √ k

(−1)m sn u

√ 1+k √ k

(−1)m+n cn u

√ 1+k

(−1)n dn u

WH

WH

dn (u1 , k1 )

ku

1 k

k sn(u, k)

dn(u, k)

cn(u, k)

iu

k

i

sn(u, k) cn(u, k)

1 cn(u, k)

dn(u, k) cn(u, k)

k u

i

k k

k

sn(u, k) dn(u, k)

cn(u, k) dn(u, k)

1 dn(u, k)

iku

i

k k

ik

sn(u, k) dn(u, k)

1 dn(u, k)

cn(u, k) dn(u, k)

sn(u, k) cn(u, k)

dn(u, k) cn(u, k)

1 cn(u, k)

cn(u, k) dn(u, k) 1 + k sn 2 (u, k)

1 − k sn 2 (u, k) 1 + k sn 2 (u, k)

1 − (1 + k  ) sn 2 (u, k) dn(u, k) √ dn(u, k) − k  √ 1 − k < 2(1+k ) × [1+dn(u,k)][k  +dn(u,k)]

1 − (1 − k  ) sn 2 (u, k) dn(u, k)  √ √ 1 + k1 dn(u, k) + k   [1 + dn(u, k)] [k + dn(u, k)]

ik u (1 + k)u (1 + k  ) u  √ 2 1 + k 2

1 k √ 2 k 1+k 1 − k 1 + k  √ 2 1 − k √ u 1 + k

ik

(1 + k) sn(u, k) 1 + k sn 2 (u, k) (1 + k  )

sn(u, k) cn(u, k) dn(u, k)

k 2 sn(u, k)d cn(u, k) √ k1 [1 + dn(u, k)] [k + dn(u, k)]

Jacobian elliptic functions

cn (u1 , k1 )

8.152

sn (u1 , k1 )

Transformation formulas

l1

8.152

u1

871

JA

872

Elliptic Integrals and Functions

8.153

8.153 1. 2. 3. 4. 5. 6. 7.11

8.11

9.11

sn (u, k  ) cn (u, k  ) 1 cn(iu, k) = cn (u, k  ) sn(iu, k) = i

SI 50(64) SI 50(65)

dn (u, k  ) cn (u, k  )   sn(u, k) = k−1 sn ku, k−1   cn(u, k) = dn ku, k−1   dn(u, k) = cn ku, k−1  √   2 , k 1 + k 2 −1/2 sn u 1 + k 1  √ sn(u, ik) = √ 1 + k 2 dn u 1 + k 2 , k (1 + k 2 )−1/2 dn(iu, k) =

SI 50(65)

  1/2  −1/2 sn u 1 + k 2 , k 1 + k2  cn(u, ik) = 1/2 −1/2 dn u (1 + k 2 ) , k (1 + k 2 ) 

dn(u, ik) =

dn u (1 +

1 1/2 k2 )

, k (1 + k 2 )

−1/2



Functional relations 8.154 1. 2. 3. 4. 5.

1 − cn 2u 1 + dn 2u cn 2u + dn 2u cn 2 u = 1 + dn 2u dn 2u + k2 cn 2u + k 2 dn 2 u = 1 + dn 2u sn 2 u =

sn 2 u + cn 2 u = 1 2

2

2

dn u + k sn u = 1

MO 146 MO 146 MO 146 SI 16(9) SI 16(9)

8.155 1. 2.

sn 2 u cn 2 u 1 − dn 2u = k2 1 + dn 2u dn 2 u 2 sn u dn 2 u 1 − cn 2u = 1 + cn 2u cn 2 u

MO 146 MO 146

8.156 1.

sn (u ± v) =

sn u cn v dn v ± sn v cn u dn u 1 − k 2 sn 2 u sn 2 v

SI 46(56)

8.160

The Weierstrass function ℘(u)

873

cn u cn v ∓ sn u sn v dn u dn v 1 − k 2 sn 2 u sn 2 v dn u dn v ∓ k 2 sn u sn v cn u cn v dn (u ± v) = 1 − k 2 sn 2 u sn 2 v cn (u ± v) =

2. 3. 8.157 1.

  1 1 − dn u u 1 − cn u =± =± 2 k 1 + cn u 1 + dn u   k cn u + dn u 1 − dn u u =± cn = ± 2 1 + dn u k dn u − cn u   cn u + dn u 1 − cn u u  = ±k dn = ± 2 1 + cn u dn u + cn u sn

2. 3.

SI 46(57) SI 46(58)

SI 47(61), SU 67(15) SI 48(62), SI 67(16) SI 48(63), SI 67(17)

8.158 1. 2. 3.8 8.159

d sn u = cn u dn u du d cn u = − sn u dn u du d dn u = −k2 dn u cn u du

2. 3.

SI 21(21) SI 21(21)

Jacobian elliptic functions are solutions of the following differential equations:  d sn u = (1 − sn 2 u) (1 − k 2 sn 2 u) du  d cn u = − (1 − cn 2 u) (k 2 + k 2 cn 2 u), du < d dn u = − (1 − dn 2 u) (dn 2 u − k  2 ) du

1.

SI 21(21)

SI 21(22) SI 21(22) SI 21(22)

For the indefinite integrals of Jacobi’s elliptic functions, see 5.13.

8.16 The Weierstrass function ℘(u) 8.160 1.

2.

The Weierstrass elliptic function ℘(u) is defined by    1 1 1 SI 307(6) ℘(u) = 2 + , 2 − 2 u (u − 2mω1 − 2nω2 ) (2mω1 + 2nω2 ) m,n # where the symbol means that the summation is made over all combinations of integers m and n except for the combination m = n = 0; 2ω1 and 2ω2 are the periods of the function ℘(u). Obviously,

ω1 ℘ (u + 2mω1 + 2nω2 ) = ℘(u) and Im

= 0, ω2

874

3.

Elliptic Integrals and Functions

8.161

 1 d ℘(u) = −2 3, du m,n (u − 2mω1 − 2nω2 ) where the summation is made over all integral values of m and n. The series 8.160 1 and 8.160 3 converge everywhere except at the poles, that is, at the points 2mω1 + 2nω2 (where m and n are integers).

4.

8.161

The function ℘(u) is a doubly periodic function and has one second-order pole in a period paralSI 306 lelogram. The function ℘(u) satisfies the differential equation 

1.

2.

d ℘(u) du where g2 = 60

2 = 4 ℘3 (u) − g2 ℘(u) − g3 , 

−4

(mω1 + nω2 )

;

g3 = 140

m,n

SI 142, 310, WH



−6

(mω1 + nω2 )

WH, SI 310

m,n

The functions g2 and g3 are called the invariants of the function ℘(u).  ∞  ∞ dz dz   8.162 u = , = 3 4 (z − e1 ) (z − e2 ) (z − e3 ) 4z − g2 z − g3 ℘(u) ℘(u) where e1 , e2 , and e3 are the roots of the equation 4z 3 − g2 z − g3 = 0; that is, g2 g3 e1 + e2 + e3 = 0, e1 e2 + e2 e3 + e3 e1 = − , e1 e2 e3 = SI 142, 143, 144 4 4 8.163 ℘ (ω1 ) = e1 , ℘ (ω1 ) + ω2 = e2 , ℘ (ω2 ) = e3 . Here, it is assumed that if e1 , e2 , and e3 lie on a straight line in the complex plane, e2 lies between e1 and e3 . 8.164 The number Δ = g23 − 27g32 is called the discriminant of the function ℘(u). If Δ > 0, all roots e1 , e2 , and e3 of the equation 4z 3 − g2 z − g3 = 0 (where g2 and g3 are real numbers) are real. In this case, the roots e1 , e2 , and e3 are numbered in such a way that e1 > e2 > e3 . 1.

If Δ > 0, then

2.

dz  , − g2 z − g3 g3 + g2 z − 4z 3 e1 −∞ where ω1 is real and ω2 is a purely imaginary number. Here, the values of the radical in the ω2 integrand are chosen in such a way that ω1 and will be positive. i If Δ < 0, the root e2 of the equation 4z 3 − g2 z − g3 = 0 is real, and the remaining two roots (e1 and e3 ) are complex conjugates. Suppose that e1 = α + iβ, and e3 = α − iβ. In this case, it is convenient to take  ∞  ∞ dz dz   and ω  = ω = 3 3 4z − g2 z − g3 4z − g2 z − g3 e1 e3 as basic semiperiods.



ω1 =

∞



dz

4z 3



,

e3

ω2 = i

In the first integral, the integration is taken over a path lying entirely in the upper half-plane and in the second over a path lying entirely in the lower half-plane. SI 151(21, 22)

8.169

8.165 1.

875

1 g3 u4 g 2 u6 g2 u2 3g2 g3 u8 + + 4 2 + ... + + 4 2 2 u 4·5 4·7 2 ·3·5 2 · 5 · 7 · 11

WH

Series representation: ℘(u) =

8.166

The Weierstrass function ℘(u)

Functional relations

℘ (u) = − ℘ (−u)  2 1 ℘ (u) − ℘ (v) 2. ℘(u + v) = − ℘(u) − ℘(v) + 4 ℘(u) − ℘(v)

g2 g3 2 8.167 ℘ (u; g2 , g3 ) = μ ℘ μu; 4 , μ μ6 1.

℘(u) = ℘(−u),

SI 163(32)

(the formula for homogeneity) SI 149(13)

The special case: μ = i. 1.

℘ (u; g2 , g3 ) = − ℘ (iu; g2 , −g3 )

8.168 An arbitrary elliptic function can be expressed in terms of the elliptic function ℘(u) having the same periods as the original function and its derivative ℘ (u). This expression is rational with respect to ℘(u) and linear with respect to ℘ (u). 8.169 A connection with the Jacobian elliptic functions. For Δ > 0 (see 8.164 1).

u cn 2 (u; k) 1. ℘ √ = e1 + (e1 − e3 ) 2 sn (u; k) e1 − e2 dn 2 (u; k) = e2 + (e1 − e3 ) 2 sn (u; k) 1 = e3 + (e1 − e3 ) 2 sn (u; k) SI 145(5), ZH 120(197–199)a

2.

3.

4. 5.

6.11

K iK ω1 = √ , ω2 = √ , e 1 − e3 e1 − e3 where   e2 − e 3 e1 − e2  , k = k= e1 − e 3 e1 − e3 For Δ < 0 (see 8.164 2)    1 + cn(2u; k) u ; = e2 + 9α2 + β 2 ℘  4 1 − cn(2u; k) 9α2 + β 2 K − iK K + iK ω =  , ω  =  , 4 2 9α2 + β 2 9α2 + β 2 where . . 1 1 3e2 3e2  −  +  k= ; k = 2 2 2 4 9α + β 2 4 9α2 + β 2

SI 154(29)

SI 145(7)

SI 147(12)

SI 153(28)

SI 147

For Δ = 0, all the roots e1 , e2 , and e3 are real, and if g2 g3 = 0, two of them are equal to each other. If e1 = e2 = e3 , then

876

7.

8.

Elliptic Integrals and Functions



3g3 9g3 9g3 2 ℘(u) = − coth u − g2 2g2 2g2 If e1 = e2 = e3 , then 1 9g3 3g3  < + ℘(u) = − 2g2 2g2 sin2 u 9g3

8.171

SI 148

SI 149

2g2

9.

If g2 = g3 = 0, then e1 = e2 = e3 = 0, and 1 ℘(u) = 2 u

SI 149

8.17 The functions ζ(u) and σ(u) 8.171 1. 2. 8.172 1.

2.

Definitions:



1 ℘(z) − 2 dz z 0

 u

 1 σ(u) = u exp ℘(z) − 2 dz z 0 1 ζ(u) = − u



u

Series and infinite-product representation    1 1 u 1 ζ(u) = + + + 2 u u − 2mω1 − 2nω2 2mω1 + 2nω2 (2mω1 − 2nω2 ) m,n  

- u2 u u σ(u) = u + 1− exp 2 2mω1 + 2nω2 2mω1 + 2nω2 2 (2mω1 + 2nω2 ) mn,

SI 181(45) SI 181(46)

SI 307(8)

SI 308(9)

8.173 g2 u 3 g3 u5 g 2 u7 3g2 g3 u9 − 2 − 4 2 2 − 4 − ··· · 3 · 5 2 · 5 · 7 2 · 3 · 5 · 7 2 · 5 · 7 · 9 · 11 g3 u7 g 2 u9 3g2 g3 u11 g2 u 5 − 3 − 9 22 − 7 2 2 − ··· 2. σ(u) = u − 4 2 · 3 · 5 2 · 3 · 5 · 7 2 · 3 · 5 · 7 2 · 3 · 5 · 7 · 11

∞ πu ζ (ω1 ) π πu π  ω2 8.174 ζ(u) = u+ cot + + nπ cot ω1 2ω1 2ω1 2ω1 n=1 2ω1 ω1

 πu ω2 + cot − nπ 2ω1 ω1 ∞ ζ (ω1 ) π πu 2π  q 2n πnu = u+ cot + sin ω1 2ω1 2ω1 ω1 n=1 1 − q 2n ω1 1.

ζ(u) = u −

22

SI 181(49) SI 181(49)

MO 154 MO 155

Functional relations and properties 8.175 ζ(u) = − ζ(−u), 8.176 1.

σ(u) = − σ(−u)

ζ (u + 2ω1 ) = ζ(u) + 2ζ (ω1 )

SI 181

SI 184(57)

8.181

Theta functions

877

2.

ζ (u + 2ω2 ) = ζ(u) + 2ζ (ω2 )

SI 184(57)

3.

σ (u + 2ω1 ) = − σ(u) exp {2 (u + ω1 ) ζ (ω1 )} .

SI 185(60)

4.

σ (u + 2ω2 ) = − σ(u) exp {2 (u + ω2 ) ζ (ω2 )} . π ω2 ζ (ω1 ) − ω1 ζ (ω2 ) = i 2

SI 185(60)

5.

SI 186(62)

8.177 1 ℘ (u) − ℘ (v) 2 ℘(u) − ℘(v)

1.

ζ(u + v) − ζ(u) − ζ(v) =

2.

℘(u) − ℘(v) = −

3.

ζ(u − v) + ζ(u + v) − 2ζ(u) =

SI 182(53)

σ(u − v) σ (u + v) σ 2 (u) σ 2 (v)

SI 183(54)

℘ (u) ℘(u) − ℘(v)

SI 182(51)

8.178 1.

ζ (u; ω1 , ω2 ) = t ζ (tu; tω1 , tω2 )

MO 154

2.8

σ (u; ω1 , ω2 ) = t−1 σ (tu; tω1 , tω2 )

MO 156

For the indefinite integrals of Weierstrass elliptic functions, see 5.14.

8.18–8.19 Theta functions 8.180 1.

Theta functions are defined as the sums (for |q| < 1) of the following series: ϑ4 (u) =

∞ 

2

(−1)n q n e2nui = 1 + 2

n=−∞

2.

3.

ϑ1 (u) = 11

ϑ2 (u) =

ϑ3 (u) =

2

(−1)n q n cos 2nu

∞ ∞  1 2 1 2 1  (−1)n q (n+ 2 ) e(2n+1)ui = 2 (−1)n+1 q (n− 2 ) sin(2n − 1)u i n=−∞ n=1 ∞ 

∞ 

2

1 q (n+ 2 ) e(2n+1)ui = 2

∞ 

2

q (n− 2 ) cos(2n − 1)u 1

WH

WH

n=1 2

q n e2nui = 1 + 2

n=−∞

WH

n=1

n=−∞

4.

∞ 

∞ 

2

q n cos 2nu

WH

n=1

The notations ϑ(u, q) and ϑ (u | τ ), where τ and q are related by q = eiπτ , are also used. Here, q is called the nome of the theta function and τ its parameter. 8.181 Representation of theta functions in terms of infinite products 1.

ϑ4 (u) =

∞  -

1 − 2q 2n−1 cos 2u + q 2(2n−1)



1 − q 2n



SI 200(9), ZH 90(9)

n=1

2.

ϑ3 (u) =

∞  n=1

1 + 2q 2n−1 cos 2u + q 2(2n−1)



1 − q 2n



SI 200(9), ZH 90(9)

878

Elliptic Integrals and Functions

8.182

∞

3. 4.8

-   √ 1 − 2q 2n cos 2u + q 4n 1 − q 2n ϑ1 (u) = 2 4 q sin u √ ϑ2 (u) = 2 4 q cos u

n=1 ∞ -



1 + 2q 2n cos 2u + q 4n



1 − q 2n



SI 200(9), ZH 90(9)

SI 200(0), ZH 90(9)

n=1

Functional relations and properties 8.182 Quasiperiodicity. Suppose that q = eπτ i (Im τ > 0). Then, theta functions that are periodic functions of u are called quasiperiodic functions of τ and u. This property follows from the equations 1.

ϑ4 (u + π) = ϑ4 (u)

SI 200(10)

2.

1 ϑ4 (u + τ π) = − e−2iu ϑ4 (u) q

SI 200(10)

3.

ϑ1 (u + π) = − ϑ1 (u)

SI 200(10)

4.

1 ϑ1 (u + τ π) = − e−2iu ϑ1 (u) q

SI 200(10)

5.

ϑ2 (u + π) = − ϑ2 (u)

SI 200(10)

6.

ϑ2 (u + τ π) =

7.

ϑ3 (u + π) = ϑ3 (u)

8.

ϑ3 (u + τ π) =

8.183  1. ϑ4 u +  2. ϑ1 u +  3. ϑ2 u +  4. ϑ3 u +  5. ϑ4 u +  6. ϑ1 u +  7. ϑ2 u +  8. ϑ3 u + 8.184



1 2π  1 2π  1 2π  1 2π

1 −2iu e ϑ2 (u) q

SI 200(10) SI 200(10)

1 −2iu e ϑ3 (u) q

SI 200(10)

= ϑ3 (u)

WH

= ϑ2 (u)

WH

= − ϑ1 (u)

WH

= ϑ4 (u)  −1/4 −iu 1 e ϑ1 (u) 2 πτ = iq  −1/4 −iu 1 e ϑ4 (u) 2 πτ = iq  −1/4 −iu 1 e ϑ3 (u) 2 πτ = q  −1/4 −iu 1 e ϑ2 (u) 2 πτ = q

WH WH WH WH WH

Even and odd theta functions

1.

ϑ1 (−u) = − ϑ1 (u)

WH

2.

ϑ2 (−u) = ϑ2 (u)

WH

3.

ϑ3 (−u) = ϑ3 (u)

WH

4.

ϑ4 (−u) = ϑ4 (u)

8.185 8.1867

ϑ44 (u)

ϑ42 (u)

WH

ϑ41 (u)

ϑ43 (u)

+ = + WH Considering the theta functions as functions of two independent variables u and τ , we have

8.192

Theta functions

879

∂ 2 ϑk (u | τ ) ∂ ϑk (u | τ ) =0 [k = 1, 2, 3, 4] +4 WH 2 ∂u ∂τ 8.187 We denote the partial derivatives of the theta functions with respect to u by a prime and consider them as functions of the single argument u. Then, πi

1.

ϑ1 (0) = ϑ2 (0) ϑ3 (0) ϑ4 (0)

2.

ϑ 1 (0) ϑ1 (0)

=

ϑ2 (0) ϑ2 (0)

ϑ3 (0)

+

ϑ3 (0)

WH

+

ϑ4 (0)

WH

ϑ4 (0)

8.188 ϑ1 (u) ϑ2 (u) ϑ3 (u) ϑ4 (0) = 12 ϑ1 (2u) ϑ2 (0) ϑ3 (0) ϑ4 (0) 8.189 The zeros of the theta functions: πτ π 1.8 ϑ4 (u) = 0 for u = 2m + (2n − 1) 2 2 πτ π 10 2. ϑ1 (u) = 0 for u = 2m + 2n 2 2 πτ π 3. ϑ2 (u) = 0 for u = (2m − 1) + 2n 2 2 πτ π 4. ϑ3 (u) = 0 for u = (2m − 1) + (2n − 1) 2 2

WH

SI 201 SI 201 SI 201

[m and n are integers or zero]

SI 201

For integrals of theta functions, see 6.16. 8.191 Connections with the Jacobian elliptic functions:  K K For τ = i K , i.e. for q = exp −π K ,

1.

2.

3.

 πu ϑ 1 1 1 H (u)  2K =√ sn u = √ πu k ϑ4 k Θ (u) 2K  πu   k  ϑ2 2K k  H 1 (u)  πu = cn u = k ϑ k Θ(u) 4 2K  πu √ ϑ3 √ Θ1 (u)  dn u = k   2K πu = k Θ(u) ϑ4 2K

SI 206(22), SI 209(35)

SI 207(23), SI 209(35)

SI 207(24), SI 209(35)

8.192 Series representation of H , H 1 , Θ, Θ1 .  the functions  In these formulas, q = exp −π KK . 1.

2.

3.

Θ(u) = ϑ4

H (u) = ϑ1

 πu 2K  πu

Θ1 (u) = ϑ3

2K

=1+2

2

(−1)n q n cos

n=1

=2

 πu 2K

∞ 

∞ 

<

(−1)n+1

4

q (2n+1) sin(2n − 1) 2

n=1

=1+2

∞ 

nπu K

2

q n cos

n=1

nπu K

SI 207(25), SI 212(42)

πu 2K

SI 207(25), SI 212(43)

SI 207(25), SI 212(45)

880

Elliptic Integrals and Functions

4.

H 1 (u) = ϑ2

8.193

 πu 2K

=2

∞   πu 4 q (2n−1)2 cos(2n − 1) 2K n=1

8.193

SI 207(25), SI 212(44)

Connections with the Weierstrass elliptic functions  √  √ ⎤2 ⎤2 ⎤2 ⎡ ⎡  √ H 1 u λ H  (0) Θ1 u λ H  (0) Θ u λ H  (0)  √ ⎦ λ = e3 + ⎣  √ ⎦ λ  √ ⎦ λ = e2 + ⎣ ℘(u) = e1 + ⎣ H 1 (0) H u λ Θ1 (0) H  u λ Θ(0) H  u λ ⎡

1.

SI 235(77,78)

 √  u λ H √ η1 u ζ(u) = + λ  √ ω1 H u λ 

H u √λ 2 η1 u 1 σ(u) = √ exp 2ω1 H  (0) λ

2.

3.

SI 234(73)

SI 234(72)

where λ = e1 − e 3 ; 8.194 1. 2.11

η1 = ζ (ω1 ) = −

ω1 λ H  (0) 3 H  (0)

SI 236

The connection with elliptic integrals: Θ (0) Θ (u) + SI 228(65) Θ(0) Θ(u)     u   sn a 1 Θ(u − a) Θ (a) dϕ 2 2 =u+ u + ln Π u, −k sin a, k = 2 2 2 cn a dn a Θ(a) 2 Θ(u + a) 0 1 − k sin a sn ϕ

E (u, k) = u − u

SI 228(65)

  q-series and products, q = exp −π KK % &2 ∞  2 π π 1+2 qn = K = Θ2 (K) 8.195 2 2 n=1 ∞  2 (−1)n+1 n2 q n  2 2π n=1 Θ (0) =K− 8.196 E = K − K ∞  Θ(0) K 2 1+2 (−1)n q n

(cf. 8.197 1)

SI 219

SI 230(67)

n=1

8.197 1.

2.

∞ 



2K = ϑ3 (0) π n=1  ∞  2n−1 2 1 kK ) ( = ϑ2 (0) q 2 = 2π 2 n=1 1+2

q

n2

=

(cf. 8.195)

WH

WH

8.199

3.

4.

5.

6.

Theta functions

881

4 ∞ 1 + q 2n √ 4 q =k 1 + q 2n−1 n=1

4 ∞ 1 − q 2n−1 = k 2n−1 1 + q n=1

2 ∞ √ K 1 − q 2n √ 24q =2 k 2n−1 1−q π n=1

∞ 2 √ K 1 − q 2n = 2 k 2n 1+q π n=1

SI 206(17, 18)

SI 206(19, 20)

WH

WH

8.198

1.

√ 1 1 − k √ = λ= 2 1 + k

∞ 

q (2n+1)

n=0

1+2

∞ 

q

2

4n

[for 0 < k < 1, we have 0 < λ < 12 ]

2

WH

n=1

The series 2.

q = λ + 2λ5 + 15λ9 + 150λ13 + 1707λ17 + . . .

WH

is used to determine q from the given modulus k. 8.19910

6.

Identities involving products of theta functions         ϑ1 (x, q) ϑ1 (y, q) = ϑ3 x + y, q 2 ϑ2 x − y, q 2 − ϑ2 x + y, q 2 ϑ3 x − y, q 2         ϑ1 (x, q) ϑ2 (y, q) = ϑ1 x + y, q 2 ϑ4 x − y, q 2 + ϑ4 x + y, q 2 ϑ1 x − y, q 2         ϑ2 (x, q) ϑ2 (y, q) = ϑ2 x + y, q 2 ϑ3 x − y, q 2 + ϑ3 x + y, q 2 ϑ2 x − y, q 2         ϑ3 (x, q) ϑ3 (y, q) = ϑ3 x + y, q 2 ϑ3 x − y, q 2 + ϑ2 x + y, q 2 ϑ2 x − y, q 2         ϑ3 (x, q) ϑ4 (y, q) = ϑ4 x + y, q 2 ϑ4 x − y, q 2 − ϑ1 x + y, q 2 ϑ1 x − y, q 2         ϑ4 (x, q) ϑ4 (y, q) = ϑ3 x + y, q 2 ϑ3 x − y, q 2 − ϑ2 x + y, q 2 ϑ2 x − y, q 2

7.

ϑ1 (x + y) ϑ1 (x − y) ϑ24 (0) = ϑ23 (x) ϑ22 (y) − ϑ22 (x) ϑ23 (y) = ϑ21 (x) ϑ24 (y) − ϑ24 (x) ϑ21 (y)

LW 8(1.4.16)

8.

ϑ2 (x + y) ϑ2 (x − y) ϑ24 (0)

=

ϑ22 (x) ϑ24 (y) − ϑ23 (x) ϑ21 (y)

LW 8(1.4.17)

9.

ϑ3 (x + y) ϑ3 (x − y) ϑ24 (0)

=

ϑ23 (x) ϑ24 (y) − ϑ22 (x) ϑ21 (y)

LW 8(1.4.18)

10.

ϑ4 (x + y) ϑ4 (x − y) ϑ24 (0) = ϑ24 (x) ϑ24 (y) − ϑ21 (x) ϑ21 (y)

1. 2. 3. 4. 5.

=

ϑ24 (x) ϑ22 (y) − ϑ21 (x) ϑ23 (y)

=

ϑ24 (x) ϑ23 (y) − ϑ21 (x) ϑ22 (y)

LW 8(1.4.8) LW 8(1.4.9) LW 8(1.4.10) LW 8(1.4.11) LW 8(1.4.12)

LW 8(1.4.15)

11.

ϑ4 (x + y) ϑ4 (x − y) ϑ24 (0)

ϑ24 (x) ϑ24 (y) − ϑ21 (x) ϑ21 (y)

LW 9(1.4.19)

12.

ϑ1 (x + y) ϑ1 (x − y) ϑ23 (0) = ϑ21 (x) ϑ23 (y) − ϑ23 (x) ϑ21 (y) = ϑ24 (x) ϑ22 (y) − ϑ22 (x) ϑ24 (y)

LW 9(1.4.23)

13.

ϑ2 (x + y) ϑ2 (x − y) ϑ23 (0) = ϑ22 (x) ϑ23 (y) − ϑ24 (x) ϑ21 (y) = ϑ23 (x) ϑ22 (y) − ϑ21 (x) ϑ24 (y)

LW 9(1.4.24)

14.

ϑ3 (x + y) ϑ3 (x − y) ϑ23 (0)

ϑ22 (x) ϑ22 (y) + ϑ24 (x) ϑ24 (y)

LW 9(1.4.25)

15.

ϑ4 (x + y) ϑ4 (x − y) ϑ23 (0) = ϑ21 (x) ϑ22 (y) + ϑ23 (x) ϑ24 (y) = ϑ22 (x) ϑ21 (y) + ϑ24 (x) ϑ23 (y)

LW 9(1.4.26)

=

=

ϑ23 (x) ϑ23 (y) − ϑ22 (x) ϑ22 (y)

LW 7(1.4.7)

ϑ21 (x) ϑ21 (y) + ϑ23 (x) ϑ23 (y)

=

=

882

Elliptic Integrals and Functions

16.

ϑ1 (x + y) ϑ1 (x − y) ϑ22 (0) = ϑ21 (x) ϑ22 (y) − ϑ22 (x) ϑ21 (y) = ϑ24 (x) ϑ23 (y) − ϑ23 (x) ϑ24 (y)

LW 9(1.4.30)

17.

ϑ2 (x+y) ϑ2 (x−y) ϑ22 (0)

ϑ23 (x) ϑ23 (y)−ϑ24 (x) ϑ24 (y)

LW 10(1.4.31)

18.

ϑ3 (x+y) ϑ3 (x−y) ϑ22 (0) = ϑ23 (x) ϑ22 (y)+ϑ24 (x) ϑ21 (y) = ϑ22 (x) ϑ23 (y)+ϑ21 (x) ϑ24 (y)

LW 10(1.4.32)

19.

ϑ4 (x+y) ϑ4 (x−y) ϑ22 (0)

LW 10(1.4.33)

=

=

ϑ24 (x) ϑ22 (y)+ϑ23 (x) ϑ21 (y)

=

=

ϑ21 (x) ϑ23 (y)+ϑ22 (x) ϑ24 (y)

20.

ϑ23 (x) ϑ23 (0)

ϑ22 (x) ϑ22 (0)

LW 11(1.4.49)

21.

ϑ24 (x) ϑ23 (0) = ϑ21 (x) ϑ22 (0) + ϑ23 (x) ϑ24 (0)

LW 11(1.4.50)

22.

ϑ24 (x) ϑ22 (0)

ϑ22 (x) ϑ24 (0)

LW 11(1.4.51)

23.

ϑ23 (x) ϑ22 (0) = ϑ21 (x) ϑ24 (0) + ϑ22 (x) ϑ23 (0)

LW 11(1.4.52)

24.8

ϑ43 (x) = ϑ42 (0) + ϑ44 (0)

LW 11(1.4.53)

8.199(2)10 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

=

ϑ21 (x) ϑ23 (0)

+

+

Derivatives of ratios of theta functions

d (ϑ1 / ϑ4 ) = ϑ24 (0) ϑ2 (x) ϑ3 (x)/ ϑ24 (x) dx d (ϑ2 / ϑ4 ) = − ϑ23 (0) ϑ1 (x) ϑ3 (x)/ ϑ24 (x) dx d (ϑ3 / ϑ4 ) = − ϑ22 (0) ϑ1 (x) ϑ2 (x)/ ϑ24 (x) dx d (ϑ1 / ϑ3 ) = ϑ23 (0) ϑ2 (x) ϑ4 (x)/ ϑ23 (x) dx d (ϑ2 / ϑ3 ) = − ϑ24 (0) ϑ1 (x) ϑ4 (x)/ ϑ23 (x) dx d (ϑ1 / ϑ2 ) = ϑ22 (0) ϑ3 (x) ϑ4 (x)/ ϑ22 (x) dx d (ϑ4 / ϑ1 ) = − ϑ24 (0) ϑ2 (x) ϑ3 (x)/ ϑ21 (x) dx d (ϑ4 / ϑ2 ) = ϑ23 (0) ϑ1 (x) ϑ3 (x)/ ϑ22 (x) dx d (ϑ4 / ϑ3 ) = ϑ22 (0) ϑ1 (x) ϑ2 (x)/ ϑ23 (x) dx d (ϑ3 / ϑ1 ) = − ϑ23 (0) ϑ2 (x) ϑ4 (x)/ ϑ21 (x) dx d (ϑ3 / ϑ2 ) = ϑ24 (0) ϑ1 (x) ϑ4 (x)/ ϑ22 (x) dx d (ϑ2 / ϑ1 ) = − ϑ22 (0) ϑ3 (x) ϑ4 (x)/ ϑ21 (x) dx

8.199(3)10 1.

=

ϑ24 (x) ϑ24 (0)

ϑ22 (x) ϑ22 (y)−ϑ21 (x) ϑ21 (y)

8.199(2)

Derivatives of theta functions

∞  q 2n d ln ϑ1 (u) = cot u + 4 sin 2u 2n du 1 − 2q cos 2u + q 4n n=1

LW 19(1.9.3) LW 19(1.9.6) LW 19(1.9.7) LW 19(1.9.8) LW 19(1.9.9) LW 19(1.9.10) LW 19(1.9.11) LW 20(1.9.12) LW 20(1.9.13) LW 20(1.9.14) LW 20(1.9.15) LW 20(1.9.16)

8.212

The exponential integral function Ei(x)

2.

∞  q 2n d ln ϑ2 (u) = − tan u − 4 sin 2u 2n du 1 + 2q cos 2u + q 4n n=1

3.

∞  q 2n−1 d ln ϑ3 (u) = −4 sin 2u du 1 + 2q 2n cos 2u + q 4n−2 n=1

4.

∞  q 2n−1 d ln ϑ4 (u) = 4 sin 2u du 1 − 2q 2n cos 2u + q 4n−2 n=1

5.

∞  d2 ln ϑ2 (u) = − sech2 {i(u + nπτ )} du2 n=−∞

883

8.2 The Exponential Integral Function and Functions Generated by It 8.21 The exponential integral function Ei(x) 8.211 1. 2.11

 Ei(x) = −



∞ −t

x

et dt = li (ex ) [x < 0] −x t −∞ t  −ε −t  ∞ −t   x t e e e dt + dt = PV dt Ei(x) = − lim ε→0+ t t −x ε −∞ t e

dt =

[x > 0] 3.7

Ei(x) =

1 2

{Ei(x + i0) + Ei(x − i0)}

8.212 1.8





2.7 3. 4. 5. 6. 7.8

e

[x > 0]

NT 11(1)

[x > 0]

NT 11(10)

0

  ∞ −t 1 e dt + 2 x 0 (x − t)    ∞ −t e dt 1 −x Ei(−x) = e − + 2 x 0 (x + t)  1 dt Ei (±x) = ±e±x x ± ln t 0  ∞ −xt e Ei (±xy) = ±e±xy dt 0 y∓t  ∞ −it e ±x dt Ei (±x) = −e t ± ix 0  1 y−1 t dt Ei(xy) = exy x + ln t 0 Ei(x) = ex

ET I 386

x −t

−1 dt t 0  x = C + e−x ln x + e−t ln t dt

Ei(−x) = C + ln x +

[x > 0]

[x > 0]

(cf. 8.211 1)

[x > 0]

(cf. 8.211 1)

[x > 0]

(cf. 8.211 1)

[Re y > 0, [x > 0]

x > 0]

LA 281(28)

NT 19(11) NT 23(2, 3)

LA 282(44)a

884

8.

The Exponential Integral Function and Functions Generated by It

= x−1 e 

9. 10. 11. 12. 13. 14. 15. 16.



1

ty−1 dt 0 %x − ln t  1 tx−1 −xy

Ei(−xy) = −e−xy

0

LA 282(45)a

& 2

(y − ln t)

dt − y −1

[x > 0,

y > 0]

LA 283(47)a

∞

dt 1 2 x − ln t t 1  ∞ dt 1 Ei(−x) = −e−x 2 x + ln t t 1  ∞ t cos t + x sin t Ei(−x) = −e−x dt t2 + x2 0  ∞ t cos t − x sin t Ei(−x) = −e−x dt t2 + x2 0  t 2 ∞ cos t arctan dt Ei(−x) = π 0 t x  2e−x ∞ x cos t − t sin t Ei(−x) = ln t dt π 0 t2 + x2  2ex ∞ x cos t + t sin t Ei(x) = 2 ln x − ln t dt π 0 t2 + x2  ∞ Ei(−x) = −x e−tx ln t dt Ei(x) = ex

8.213

[x > 0]

LA 283(48)

[x > 0]

LA 283(48)

[x > 0]

NT 23(6)

[x < 0]

NT 23(6)

[Re x > 0]

NT 25(13)

[x > 0]

NT 26(7)

[x > 0]

NT 27(8)

[x > 0]

NT 32(12)

1

See also 3.327, 3.881 8, 3.916 2 and 3, 4.326 1, 4.326 2, 4.331 2, 4.351 3, 4.425 3, 4.581. For integrals of the exponential integral function, see 6.22–6.23, 6.78. Series and asymptotic representations 8.213 1.

li(x) = C + ln (− ln x) +

∞ k  (ln x) k=1

2.

li(x) = C + ln ln x +

k · k!

∞ k  (ln x) k=1

k · k!

[0 < x < 1]

[x > 1]

NT 3(9)

NT 3(10)

8.214 1.

Ei(x) = C + ln(−x) +

∞  xk k · k!

[x < 0]

k=1

2.

Ei(x) = C + ln x +

∞  xk k · k!

[x > 0]

k=1

3.

Ei(x) − Ei(−x) = 2x

∞  k=0

x2k (2k + 1)(2k + 1)!

[x > 0]

NT 39(13)

8.219

The exponential integral function Ei(x)

8.2157

& % n ez  k! Ei(z) = + Rn (z) z zk

885

 −n−1 |Rn (z)| = O |z|

k=0

−n−1

[Re z ≤ 0] |arg(−z)| ≤ π − δ; δ > 0 small] |Rn (z)| ≤ (n + 1)!|z|

, 1 1 kn nx + Ei(nx) − Ei(−nx) = e + 3 3 , nx n2 x2 n x where x = x sign Re(x), kn = O(1), and n → ∞ NT 39(15) [z → ∞,

8.2167

8.217

Functional relations:  ∞    x sin t ex Ei (−x ) − e−x Ei (x ) = −2 NT 24(11) dt 2 2  0∞ t + x  x cos t 4 [x = x sign Re x] NT 27(9) = ln t dt − 2e−x ln x 2 2 π 0 t +x  ∞    t cos t 4 ∞ t sin t −x  ex Ei (−x ) + e−x Ei (x ) = −2 dt = 2e ln x − ln t dt 2 2 π 0 t2 + x2 0 t +x [x = x sign Re x] NT 24(10), NT 27(10)  

 ∞ 1 t x− x cos t 1 2 arctan Ei(−x) − Ei − dt = x π 0 t 1 + t2

1.

2.

3.

Ei(−αx) Ei(−βx) − ln(αβ) Ei[−(α + β)x] = e−(α+β)x

4.

 0

[Re x > 0] ∞ −tx

e

ln[(α + t)(β + t)] dt t+α+β

NT 25(14) NT 32(9)

See also 3.723 1 and 5, 3.742 2 and 4, 3.824 4, 4.573 2. • For a connection with a confluent hypergeometric function, see 9.237. • For integrals of the exponential integral function, see 5.21, 5.22, 5.23, 6.22, and 6.23. 8.218

Two numerical values:

1.

Ei(−1) = −0.219 383 934 395 520 273 665 . . .

NT 89

2.

Ei(1) = 1.895 117 816 355 936 755 478 . . .

NT 89

8.219∗ 1.∗ 2.∗ 3.∗

Definite integrals of exponential functions  ∞ π2 Ei2 (x)e−2x dx = 4 0  ∞ π2 Ei2 (−x)e2x dx = 4 0  ∞ Ei(x) Ei(−x) dx = 0 0

886

The Exponential Integral Function and Functions Generated by It

8.221

8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x 8.221 1.



+π , sinh t dt = −i + si(ix) t 2 0  x cosh t − 1 dt chi x = C + ln x + t 0 x

shi x =

2.11

(see 8.230 1)

EH II 146(17) EH II 146(18)

8.23 The sine integral and the cosine integral: si x and ci x 8.230 1.

10

2.10 8.231 1.

 si(x) = −

∞

π sin t dt = − + Si(x), where Si(x) = t 2 x  x  ∞ cos t cos t − 1 dt = C + ln x + dt ci(x) = − t t x 0  si(xy) = −

∞ x

 ci(xy) = −

2.

x

 si(x) = −

3.

∞

π/2

 0

x

sin t dt t

NT 11(3)

[ci(x) is also written Ci(x)]

NT 11(2)

sin ty dt t

NT 18(7)

cos ty dt t

NT 18(6)

e−x cos t cos (x sin t) dt

NT 13(26)

0

8.232

∞

si(x) = −

1.

π  (−1)k+1 x2k−1 + 2 (2k − 1)(2k − 1)!

NT 7(4)

k=1

2.7

∞ 

ci(x) = C + ln(x) +

(−1)k

k=1

x2k 2k(2k)!

NT 7(3)

8.233 1. ci(x) ± i si(x) = Ei (±ix)   2. ci(x) − ci xe±πi = ∓πi

8.234 7



2

NT 7(7)

π/2

Ei(−x) − ci(x) = 0

2.

NT 7(5)

si(x) + si(−x) = −π

3.

1.

NT 6a

e−x cos ϕ sin (s sin ϕ) dϕ



π/2

2

[ci(x)] + [si(x)] = −2 0

NT 13(27)

exp (−x tan ϕ) ln cos ϕ dϕ sin ϕ cos ϕ [Re x > 0]

(see also 4.366) NT 32(11)

See also 3.341, 3.351 1 and 2, 3.354 1 and 2, 3.721 2 and 3, 3.722 1, 3, 5 and 7, 3.723 8 and 11, 4.338 1, 4.366 1.

8.250

The probability integral, Fresnel integrals and error functions

887

8.235 lim (x si(x)) = 0,

1.

x→+∞

lim si(x) = −π,

2.

x→−∞

lim (x ci(x)) = 0

x→+∞

[ < 1]

lim ci(x) = ±πi

NT 38(5) NT 38(6)

x→−∞

• For integrals of the sine integral and cosine integral, see 6.24–6.26, 6.781, 6.782, and 6.783. • For indefinite integrals of the sine integral and cosine integral, see 5.3.

8.24 The logarithm integral li(x) 8.240



x

dt = Ei (ln x) [x < 1] 0 ln t  1−ε   x dt dt + li(x) = lim = Ei (ln x) [x > 1] ε→0 ln t 0 1+ε ln t      li exp −xe±πi = Ei −xe±iπ = Ei (x ∓ i0) = Ei(x) ± iπ = li (ex ) ± iπ

1.

li(x) =

2. 3.

JA JA

[x > 0]

JA, NT 2(6)

[x < 1]

LA 281(33)

Integral representations 8.241 1.



ln x t

li(x) = −∞

 li(x) = x

e 1 dt = x ln ln − t x

3.

∞

e−t ln t dt

− ln x

1

dt ln x + ln t 0  1 dt x +x = 2 ln x (ln x + ln t) 0  ∞ dt 1 =x 2 ln x − ln t t 1  x t a 1 dt li (ax ) = ln a −∞ t

2.



LA 280(22) LA 280(29)

[x < 1]

LA 280(30)

[x > 0]

For integrals of the logarithm integral, see 6.21

8.25 The probability integral Φ(x), the Fresnel integrals S (x) and C (x), the error function erf (x), and the complementary error function erfc(x) 8.250 1. 2.

11

Definition:

 x 2 2 Φ(x) = erf(x) = √ e−t dt π 0  x 2 S (x) = √ sin t2 dt 2π 0

(called the error function)

888

3. 4.11 5.∗

6.∗

7.∗ 8.∗

9.∗

The Exponential Integral Function and Functions Generated by It



2 C (x) = √ 2π

x

cos t2 dt 0

erfc(x) = 1 − erf(x) (called the complementary error function)  ∞ −(p+x)y  √  e sin a x dx 0 π(p + x)



a a 1 −a√p 1 a√p √ √ √ = − sinh (a p) + e Φ Φ √ − py + e √ + py 2 2 y 2 2 y 2

√

 ∞ −(p+x)y  √  p −a√p a a e 1 √ cos a x dx = √ exp − − py − e Φ √ − py πy a y 0 π(p + x) 4y

2 √ p √p a √ √ √ + e Φ √ + py − p cosh (a p) 2 2 y [Re p > 0, a, b are real]

2 √   p  p     p π exp −x2 Φ(p − x) dx = exp −x2 erf(p − x) dx = Φ √ 2 2 0 0  p  p     x2 exp −x2 Φ(p − x) dx = x2 exp −x2 erf(p − x) dx 0

2 2

√0  π p p x p − √ Φ − = Φ √ erf √ 4 2 2 2 2 2 √ √  (a+b)/ 2  (b+a)/ 2  2  √  2  √ √ Φ b Φ a exp −x 2 − x dx + exp −x 2 − x dx = π Φ(a) Φ(b) √ √ (b−a)/ 2

(a−b)/ 2

Integral representations 8.251 1. 2. 3.

1 Φ(x) = √ π



1 S (x) = √ 2π

2. 3.

2y Φ(xy) = √ π

x2 −t

e √ dt t

0



1 C (x) = √ 2π

8.252 1.

8.251

x2 0



x2 0



x 0

(see also 3.361 1)

sin t √ dt t cos t √ dt t

e−t

2 2

y

dt

 x   2y S (xy) = √ sin t2 y 2 dt 2π 0  x   2y C (xy) = √ cos t2 y 2 dt 2π 0



Re y 2 > 0



8.255

4.

5.7 6.8

The probability integral, Fresnel integrals and error functions

 ∞ −t2 y2   2 2 2 e ty dt Re y 2 > 0 Φ(xy) = 1 − √ e−x y √ π t2 + x2 0 2 2   2x −x2 y2 ∞ e−t y dt Re y 2 > 0 e =1− 2 2 π t +x 0

 y 4xie y2  ∞ 2 2   −y 2 = √ 4x Φ e−t y sin(ty) dt Re x2 > 0 −Φ 2xi 2xi π 0  ∞ y   y2 2 2 2 Φ e−t x −ty dt Re x2 > 0 = 1 − √ xe− 4 2x π 0

See also 3.322, 3.362 2, 3.363, 3.468, 3.897, 6.511 4 and 5. 8.2538 Series representations:

∞ x2k−1 2 −x2 3 2 2  11 erf(x) = √ e x F 1 1; ; x = √ (−1)k+1 1. 2 (2k − 1)(k − 1)! π π k=1 ∞ k 2k+1  2 2 x 2 = √ e−x (2k + 1)!! π k=0



2 5 3 1 2 7 5 1 2 2 3 2 2 x sin x F 1; , ; − x − x cos x F 1; , ; − x 2. S (x) = √ 4 4 4 3 4 4 4 2π ∞ k 4k+3  (−1) x 2 =√ 2π k=0 (2k + 1)!(4k + 3)   ∞ ∞   (−1)k 22k x4k+1 (−1)k 22k+1 x4k+3 2 2 2 − cos x sin x =√ (4k + 1)!! (4k + 3)!! 2π k=0 k=0



2 3 2 7 5 1 2 5 3 1 2 2 2 x sin x F 1; , ; − x − x cos x F 1; , ; − x 3. C (x) = √ 4 4 4 4 4 4 2π 3 ∞ k 4k+1  (−1) x 2 =√ 2π k=0 (2k)!(4k + 1)   ∞ ∞   (−1)k 22k+1 x4k+3 (−1)k 22k x4k+1 2 2 2 + cos x sin x =√ (4k + 3)!! (4k + 1)!! 2π k=0

889

NT 19(11)a NT 19(13)a

NT 28(3)a NT 27(1)a

NT 7(9)a NT 10(11)a

NT 8(14)a

NT 10(13)a

NT 8(13)a

NT 10(12)a

k=0

For the expansions in Bessel functions, see 8.515 2, 8.515 3. Asymptotic representations % n & 2  e−z  −2n−z k (2k − 1)!! 8 Φ(z) = 1 − √ (−1) + O |z| , 8.254 k πz (2z 2 ) k=0

[z → ∞, where |Rn | <

  Γ n + 12 |x|

n+ 12

cos

ϕ , 2

x = |x|eiϕ and ϕ2 < π 2

8.255 1.

|arg(−z)| ≤ π − δ;

1 1 S (x) = − √ cos x2 + O 2 2πx



1 x2

δ > 0 small] NT 37(10)a

[x → ∞]

MO 127a

890

The Exponential Integral Function and Functions Generated by It

1 1 sin x2 + O C (x) = + √ 2 2πx

2. 8.256 1. 2.

4.

5.11

1 x2



Functional relations: 

 z 2 z i 2 Φ √ =√ C (z) + i S (z) = eit dt 2 2π 0 i  z  √ 2 1 2 C (z) − i S (z) = √ Φ z i = √ e−it dt 2π 0 2i 

3.



  1 2 cos u C (u) + sin u S (u) = cos u + sin u2 + 2 2

2

8.256

[x → ∞]

MO 127a

  ∞ 2 e−2ut sin t2 dt π 0

[Re u ≥ 0]   ∞   2  1 2 2 cos u − sin u2 − cos u S (u) − sin u2 C (u) = e−2ut cos t2 dt 2 π 0

NT 28(6)a

[Re u ≥ 0]  2   2  2 √  π/2 exp −x tan ϕ sin ϕ2 cos ϕ 1 2 1 dϕ + S (x) − = C (x) − 2 2 π 0 sin 2ϕ

NT 28(5)a

(see also 6.322)

NT 33(18)a

• For a connection with a confluent hypergeometric function, see 9.236. • For a connection with a parabolic cylinder function, see 9.254. 8.257 1. 2.

   x S (x) − 12 = 0    lim x C (x) − 12 = 0

lim

x→+∞ x→+∞

[ < 1]

NT 38(11)

[ < 1]

NT 38(11)

1 2 1 lim C (x) = x→+∞ 2

3.

lim S (x) =

NT 38(12)a

x→+∞

4.

NT 38(12)a

• For integrals of the probability integral, see 6.28–6.31. • For integrals of Fresnel’s sine integral and cosine integral, see 6.32. 8.25810

Integrals involving the complementary error function

∞ 

2 1 1 − arccos erfc2 (x)e−βx dx = √ β + 2 arctan 1+β βπ 0

 1.

 2. 0

∞

x erfc2 (x)e−βx dx = 2

1 2β



 √ 4 arctan 1 + β √ 1− π 1+β

[β > 0]

[β > 0]

Lobachevskiy’s function Lfunction]Lobachevskiyfunction(ZdddZLZdddZ)(x)

8.262

 √ 4 arctan 1 + β √ x erfc (x)e 1− π 1+β 0  √  arctan 1 + β 1 1 − + 3 βπ (1 + β) (β 2 + 2β + 2) (1 + β) 2 [β > 0]    ∞ √  1 + 32 β 1 x erfc x e−βx dx = 2 1 − [β > 0] 3 β (1 + β) 2 0   √   ∞ √  −βx √ 1 arctan β 1 1 x erfc x e dx = √ − 3 2β(1 + β) π 2 β2 0 

3.

4. 5.11

891

∞

3

−βx2

2

1 dx = 2β 2



[β > 0] 8.259∗ 1.

2.

Integrals involving the error function and an exponential function    ∞ √  a p π −px2 Φ  [Re p > 0] , a, b real e Φ(a + bx) dx = p b2 + p −∞ 

  ∞ √  a p 2 π ab2 1 a2 p 2 −px Φ  − x e Φ(a + bx) dx = exp − 2 3/2 2p p b +p b2 + p −∞ p (b2 + p) 

∞

3.

x2n e−px Φ(a + bx) dx = (−1) 2

n

−∞

∂n ∂pn

%

π Φ p



[Re p > 0, √ & a p  b2 + p

a, b are real]

[n = 0, 1, . . . ,

Re p > 0,

a, b are real]

8.26 Lobachevskiy’s function L(x) 8.260

Definition:

 L(x) = −

x

ln cos t dt

LO III 184(10)

0

For integral representations of the function L(x), see also 3.531 8, 3.532 2, 3.533, and 4.224. 8.261 Representation in the form of a series: ∞ sin 2kx 1 L(x) = x ln 2 − (−1)k−1 LO III 185(11) 2 k2 k=1

8.262

Functional relationships:

1.

L(−x) = − L(x)

2.

L(π − x) = π ln 2 − L(x)

3.

L(π + x) = π ln 2 + L(x)  π π 1 π −x = x− ln 2 − L − 2x L(x) − L 2 4 2 2

4.

+ π π, − ≤x≤ 2 2

LO III 185(13) LO III 286

+ π, 0≤x< 4

LO III 286 LO III 186(14)

892

Euler’s Integrals of the First and Second Kinds

8.310

8.3 Euler’s Integrals of the First and Second Kinds and Functions Generated by Them 8.31 The gamma function (Euler’s integral of the second kind): Γ(z) 8.310 1.

Definition:  ∞ Γ(z) = e−t tz−1 dt

[Re z > 0]

(Euler)

FI II 777(6)

0

2.

Generalization:  1 Γ(z) = − (−t)z−1 e−t dt 2i sin πz C for z not an integer. The contour C is shown in the drawing:

WH

Γ(z) is an analytic function z with simple poles at the points z = −l (for l = 0, 1, 2,. . . ) to which (−1)l . Γ(z) satisfies the relation Γ(1) = 1. correspond to residues WH, MO 1 l! Integral representations 8.311 Γ(z) = 8.312 1.

∞



z−1



1

Γ(z) = 

1 ln t

∞

Γ(z) = xz 0

3.

4.

5.

2az ea Γ(z) = sin πz

Γ(z) =

(0+)

e2πiz − 1

0

2.



1

e−t tz−1 dt

MO 2

dt

[Re z > 0]

e−xt tz−1 dt

[Re z > 0,



∞

e−at

2



1 + t2

z− 12

1 2 sin πz

Γ(y) = xy e−iβy

bz 2 sin πz

FI II 779(8)

cos [2at + (2z − 1) arctan t] dt



[a > 0] ∞

WH

 z 2 e−t tz−1 1 + t2 2 {3 sin [t + z arccot(−t)] + sin [t + (z − 2) arccot(−t)]} dt

0

[arccot denotes an obtuse angle]



∞

  ty−1 exp −xte−iβ dt

+ x, y, β real,

Γ(z) =

Re x > 0]

0

0

6.

FI II 778



∞

−∞

ebti (it)z−1 dt

[b > 0,

x > 0,

y > 0,

0 < Re z < 1]

|β| <

π, 2

WH

MO 8

NH 154(3)

8.315

The gamma function (Euler’s integral of the second kind): Γ(z)

z  ∞ a2 + b 2   e−at cos(bt)tz−1 dt cos z arctan ab 0 √ z  ∞ a2 + b 2   = e−at sin(bt)tz−1 dt sin z arctan ab 0

893

√

7.

Γ(z) =



z

8.

NH 152(2)

[a > 0,

b ≥ 0,

[b > 0,

0 < Re z < 1]

Re z > 0]

∞

b cos(bt)tz−1 dt cos πz 0 2  ∞ bz = sin(bt)tz−1 dt πz sin 2 0

Γ(z) =

 9.

NH 152(1)a

∞

Γ(z) =

NH 152(5)

e−t (t − z)tz−1 ln t dt

0

 10.

∞

Γ(z) = −∞

NH 173(7)

[Re z > 0]

NH 145(14)

  exp zt − et dt 

∞

11.11 Γ(x) cos αx = λx

tx−1 e−λt cos α cos (λt sin α) dt

0

 12.

[Re z > 0]

∞

Γ(x) sin αx = λx

tx−1 e−λt cos α sin (λt sin α) dt

0

x > 0,

−

π, π <α< 2 2

WH

+ λ > 0,

x > 0,

−

π π, <α< 2 2

WH

⎡

13.

⎤ n k  −t kt e − (−1)  ∞⎢ k! ⎥ ⎢ ⎥ k=0 ⎢ ⎥ dt Γ(−z) = ⎢ ⎥ z+1 t 0 ⎣ ⎦

+ λ > 0,

8.313

8.314∗

Γ

z+1 v

1.11 2.8

= vu

z+1 v



∞

exp (−utv ) tz dt

[Re u > 0,

MO 2

Re v > 0,

Re z > −1]

0

JA, MO 7a



∞

Γ(z) = 1

8.315



[n = Re z]

e−t tz−1 dt +

∞ 

(−1)k k!(z + k) n=0

 i 1 = (−t)−z e−t dt Γ(z) 2π C  ∞ ebti 2πe−ab bz−1 dt = 2 Γ(z) −∞ (a + it)  ∞ −bti  e dt = 0 Re a > 0, z (a + it) −∞

[z → 0, in |arg z| < π]

[for the contour C, see 8.310 2]

b > 0,

Re z > 0,

|arg(a + it)| < 12 π



894

3.

Euler’s Integrals of the First and Second Kinds

ea 1 = a1−z Γ(z) π



8.321

π/2

cos (a tan θ − zθ) cosz−2 θ dθ

[Re z > 1]

NH 157(14)

0

See also 3.324 2, 3.326, 3.328, 3.381 4, 3.382 2, 3.389 2, 3.433, 3.434, 3.478 1, 3.551 1, 2, 3.827 1, 4.267 7, 4.272, 4.353 1, 4.369 1, 6.214, 6.223, 6.246, 6.281.

8.32 Representation of the gamma function as series and products 8.321 1.6

Representation in the form of a series: Γ(z + 1) =

∞ 

ck z k

 k=0 c0 = 1,

#n cn+1 =

k+1 sk+1 cn−k k=0 (−1)

n+1

 ;

s1 = C,

sn = ζ(n) for n ≥ 2,

|z| < 1 NH 40(1, 3)

2.11

∞ 

1 = dk z k Γ(z + 1) #n  k=0 (−1)k sk+1 dn−k ; d0 = 1, dn+1 = k=0 n+1

 s1 = C,

sn = ζ(n) for n ≥ 2

NH 41(4, 6)

Infinite-product representation ∞

1 - ez/k [Re z > 0] z 1 + kz k=1 z ∞  1 - 1 + k1 = [Re z > 0] z 1 + kz k=1 n nz - k = lim [Re z > 0] n→∞ z z+k k=1 ∞ <   2k 8.3237 Γ(z) = 2z z e−z B 2k−1 z, 12 k=1

1 z + Γ ∞ 2 2k √ 8.3247 Γ(1 + z) = 4z π k=1 8.32211 Γ(z) = e−Cz

8.325 1. 2.11 3.7



 ∞  Γ(α) Γ(β) γ γ = 1+ 1− Γ(α + γ) Γ(β − γ) α+k β+k k=0  

∞ eCx Γ(z + 1) x = 1− [z = 0, −1, −2, . . . ; ex/k Γ(z − x + 1) z+k k=1

 √ ∞ π z z     1 − = 1 + 2k − 1 2k Γ 1 + z2 Γ 12 − z2 k=1

SM 269 WH SM 267(130)

NH 98(12)

MO 3

NH 62(2)

Re z > 0,

Re(z − x) > 0]

MO 2

8.331

Functional relations involving the gamma function

895

8.326 2

1.

[Γ(x)] ! !

∞ ! Γ(x) !2 y2 Γ(2x) ! = = !! 1 + B(x + iy, x − iy) Γ(x − iy) ! (x + k)2 k=0

[x, y are real,

2.11

∞ xe−iCy - exp Γ(x + iy) = Γ(x) x + iy n=1 1 +

x = 0, −1, −2, . . .] LO V, NH 63(4)

 iy  n iy x+n

[x, y are real,

x = 0, −1, −2, . . .] MO 2

8.327 1.∗

Asymptotic representation for large arguments:

 √  −5  1 1 139 571 z− 12 −z + Γ(z) ∼ z e 2π 1 + − − +O z 12z 288z 2 51840z 3 2488320z 4 [|arg z| < π]

WH

2.∗

For z real and positive, the remainder of the series is less than the last term that is retained.  n n  n n √ √ n! ∼ 2πn or equivalently Γ(n + 1) ∼ 2πn e e [Stirling’s asymptotic formula for n  0] AS 6.1.38

3.∗

ln Γ(z) ∼



1 1 1 1 1 1 − z− + − + ... ln z − z + ln(2π) + 2 2 12z 360z 3 1260z 5 1680z 7 [z → ∞,

8.328 1. 2.

1

lim |Γ(x + iy)|e 2 |y| |y| 2 π

−x

|y|→∞

lim

|z|→∞

=

√ 2π

[x and y are real]

Γ(z + a) −a ln z e =1 Γ(z)

8.33 Functional relations involving the gamma function 8.331 1.

Γ(x + 1) = x Γ(x)

2.∗

Γ(x + a) = (x + a − 1) Γ(x + a − 1) =

3.∗

Γ(x + a + 1) (x + a)

Γ(x − a) = (x − a − 1) Γ(x − a − 1) =

Γ(x − a + 1) (x − a)

|arg z| < π]

AS 6.1.38

MO 6 MO 6

896

Euler’s Integrals of the First and Second Kinds

8.332

8.332 π y sinh πy ! 1 ! π !Γ + iy !2 = 2 cosh πy 2

|Γ(iy)| =

1. 2.

Γ(1 + ix) Γ(1 − ix) =

[y is real]

MO 3

[y is real]

πx sinh xπ

[x is real]   2π 2 x2 + y 2 Γ(1 + x + iy) Γ(1 − x + iy) Γ (1 + x − iy) Γ(1 − x − iy) = cosh 2yπ − cos 2xπ

3. 4.

LO V

[x and y are real] n

8.333 [Γ(n + 1)] = G(n + 1)

n -

LO V

kk ,

k=1

where n is a natural number and   ∞

 z z n z(z + 1) C 2 -  z2 2 1+ − z G(z + 1) = (2π) exp − exp −z + 2 2 n 2n n=1 8.334 1. 2. 3.

 z n , -+ 1 n  1 − = −z k Γ −z exp 2πki n k=1 k=1 1  1  π Γ 2 +x Γ 2 −x = cos πx π Γ(1 − x) Γ(x) = sin πx n -

WH

∞



[n = 2, 3, 3 . . .]

MO 2

FI II 430

Special cases 8.3357

Γ(nx) = (2π)

1−n 2

1

nnx− 2

n−1 k=0

1. 2. 3. 4.10



k Γ x+ n

  22x−1 Γ(2x) = √ Γ(x) Γ x + 12 π

[product theorem]

FI II 782a, WH

[doubling formula]

1     33x− 2 Γ(x) Γ x + 13 Γ x + 23 2π

n−1 - k k (2π)n−1 Γ Γ 1− = n n n k=1   ∞  Γ2 n − 12 1 1 1 25 1 1 2 2  1  = 4 + 16 + 256 + 1024 + 65536 + · · · = π − n=0 4 (n!) Γ

Γ(3x) =

WH

2

8.336



 ∞ yz + xi yz − xi z+1 Γ − e−tx sinz (ty) dt Γ(1 − z) = (2i) y Γ 1 + 2y 2y 0 [Re(yi) > 0, Re(x − yzi) > 0] NH 133(10)

8.339

Functional relations involving the gamma function

897

• For a connection with the psi function, see 8.361 1. • For a connection with the beta function, see 8.384 1. • For integrals of the gamma function, see 8.412 4, 8.414, 9.223, 9.242 3, 9.242 4. 8.337 1. 2.



2 Γ (x) < Γ(x) Γ (x)

[x > 0]

For x > 0, min Γ(1 + x) = 0.88560 . . . is attained when x = 0.46163 . . .

MO 1 JA

Particular values 8.338 1. 2. 3. 4.

Γ(1) = Γ(2) = 1   √ Γ 12 = π   √ Γ − 12 = −2 π    4 ∞ (4k − 1)2 (4k + 1)2 − 1 1 2 Γ = 16π 4 [(4k − 1)2 − 1] (4k + 1)2

MO 1a

k=1

5.

8 k=1

8.339 1. 2. 3. 4. 5.∗





3 k π 640 Γ = 6 √ 3 3 3

For n a natural number Γ(n) = (n − 1)! √   π Γ n + 12 = n (2n − 1)!! 2 √ n 1  π n 2 Γ 2 − n = (−1) (2n − 1)!!        2 1 Γ p+n+ 2 4p − 12 4p2 − 32 . . . 4p2 − (2n − 1)2   = 22n Γ p − n + 12 Γ(n + k) = (n + k − 1)! =

6.∗

WH

Γ(n + k + 1) (n + k)

[n + k ≥ 0, 1, . . .]

Γ(n − k) = (n − k − 1)! =

Γ(n − k + 1) (n − k)

[n − k ≥ 0, 1, . . .]

WA 221

898

Euler’s Integrals of the First and Second Kinds

8.341

8.34 The logarithm of the gamma function 8.341 1.

Integral representation:

−tz  ∞ 1 1 1 e 1 1 − + t dt ln Γ(z) = z − ln z − z + ln 2π + 2 2 2 t e −1 t 0 

2.11

3. 4.

5.

7.

WH

√ arctan 1 dt ln Γ(z) = z ln z − z − ln z + ln 2π + 2 2 e2πt − 1 0    w du Re z > 0 and arctan w = is taken over a rectangular path in the w-plane 2 0 1+u   ∞ −zt e dt − e−t −t [Re z > 0] ln Γ(z) = + (z − 1)e −t 1−e t 0   ∞ (1 + t)−z − (1 + t)−1 dt −t ln Γ(z) = (z − 1)e + ln(1 + t) t 0

ln Γ(x) =

ln π − ln sin πx 1 + 2 2



∞ 0



t z

[Re z > 0]   sinh 2 − x t dt − (1 − 2x)e−t t t sinh 2 1

 dt tz − t − t(z − 1) ln Γ(z) = t − 1 t ln t 0   ∞ −tz −t dt −e e −t ln Γ(z) = (z − 1)e + −t 1 − e t 0 

6.

[Re z > 0] ∞

1

WH

WH

WH

[0 < x < 1]

WH

[Re z > 0]

WH

[Re z > 0]

NH 187(7)

See also 3.427 9, 3.554 5. 8.342 Series representations: 1.11

ln Γ(z + 1)    ∞  1 πz 1 − ζ(2k + 1) 2k+1 1+z = z ln − ln + (1 − C) z + 2 sin πz 1−z 2k + 1 k=1 ∞  zk = −Cz + (−1)k ζ(k) k

[|z| < 1] NH 38(16, 12)

k=2

2.

ln Γ(1 + x) =

∞  x2n+1 πx 1 ln − Cx − ζ(2n + 1) 2 sin πx 2n + 1 n=1

[|x| < 1] 8.343 1.

∞  √ ln Γ(x) = ln 2π + n=1

NH 38(14)

 1 1 cos 2nπx + (C + ln 2nπ) sin 2nπx 2n nπ [0 < x < 1]

FI III 558

8.352

The incomplete gamma function

ln Γ(z) = z ln z − z −

2.

899

∞ ∞  √ m 1 1 1  ln z + ln 2π + 2 2 m=1 (m + 1)(m + 2) n=1 (z + n)m+1

[|arg z| < π]

MO 9

Asymptotic expansion for large values of |z|:

8.3447

 √ B2k 1 + Rn (z), ln Γ(z) = z ln z − z − ln z + ln 2π + 2 2k(2k − 1)z 2k−1 n−1

k=1

where |Rn (z)| <

2n(2n − 1)|z|

|B2n | 2n−1 cos2n−1

1 2

arg z



MO5

For integrals of ln Γ(x), see 6.44.

8.35 The incomplete gamma function 8.350 1.

Definition:  x e−t tα−1 dt γ(α, x) =

[Re α > 0]

EH II 133(1), NH 1(1)

0

 2.11

∞

Γ(α, x) =

e−t tα−1 dt

EH II 133(2), NH 2(2), LE 339

x

3.∗

Γ(z, 0) = Γ(z)

4.∗

Γ(a, ∞) = 0

5.

∗

γ(a, 0) = 0

8.351

x−α γ(α, x) is an analytic function with respect to α and x Γ(α)

1.

γ ∗ (α, x) =

2.

Another definition of Γ(α, x) that is also suitable for the case Re α ≤ 0: γ(α, x) =

xα xα −x e Φ (1, 1 + α; x) = Φ(a, 1 + a; −x) α α

EH II 133(5)

EH II 133(3)

3.

For fixed x, Γ(α, x) is an entire function of α. For non-integral α, Γ(α, x) is a multiple-valued function of x with a branch point at x = 0.

4.

A second definition of Γ(α, x): Γ(α, x) = xα e−x Ψ(1, 1 + α; x) = e−x Ψ(1 − α, 1 − α; x)

8.352

Special cases: %

1.

EH II 133(4)



γ(1 + n, x) = n! 1 − e

−x

n  xm m! m=0

& [n = 0, 1, . . .] EH II 136(17, 16), NH 6(11)

2.

Γ(1 + n, x) = n!e−x

n 

xm m! m=0

[n = 0, 1, . . .]

EH II 136(16, 18)

900

Euler’s Integrals of the First and Second Kinds

3.11

Γ(−n, x) =

8.353



 n  1 1 z k−n−1 (−1)n 1 Ei(−z) − ln(−z) + ln − − ln z − e−z n! 2 2 z (−n)k k=1

[n = 1, 2, . . .] n−1 

xm m! m=0 %

4.∗

Γ(n, x) = (n − 1)!e−x

5.∗

n−2  m! (−1)n+1 Γ(−n + 1, x) = (−1)m m+1 Γ(0, x) − e−z (n − 1)! x m=0

% 6.∗

γ(n, x) = (n − 1)! 1 − e−x −x

n−1 

xm m! m=0

&

[n = 2, 3, . . .]

&

[n = 1, 2, . . .]

n−1 

xm m! m=0 %

7.

∗

Γ(n, x) = (n − 1)!e

8.

∗

n−k−1  m! (−1)n−k Γ(0, x) − e−x Γ(−n + k, x) = (−1)m m+1 (n − k)! x m=0

[n = 1, 2, . . .] &

[n − k ≥ 1, 8.353 1.

Integral representations:  π γ(α, x) = xα cosec πα ex cos θ cos (αθ + x sin θ) dθ

[x = 0,

k = 0, 1, . . .]

Re α > 0,

α = 1, 2, . . .]

0

1

EH II 137(2)



∞

γ(α, x) = x 2 α

2.

 √ 1 e−t t 2 α−1 J α 2 xt dt

0

3.

Γ(α, x) =

ρ−x xα Γ(1 − α)



[Re α > 0]

EH II 138(4)

∞ −t −α

e t dt x+t

0

[Re α < 1,

x > 0] EH II 137(3), NH 19(12)

1 2α

−x 

2x e Γ(1 − α)  α −xy Γ(α, xy) = y e

4.

Γ(α, x) =

5.

∞

0 ∞

+ √ , 1 e−t t− 2 α K α 2 xt dt

[Re α < 1]

EH II 138(5)

e−ty (t + x)α−1 dt

0

[Re y > 0,

x > 0,

Re α > 1]

(See also 3.936 5, 3.944 1–4)

NH 19(10)

For integrals of the gamma function, see 6.45. 8.354 1.

Series representations: γ(α, x) =

∞  (−1)n xα+n n!(α + n) n=0

EH II 135(4)

8.356

2.

The incomplete gamma function

Γ(α, x) = Γ(α) −

∞  (−1)n xα+n n!(α + n) n=0

901

[α = 0, −1, −2, . . .] EH II 135(5), LE 340(2)

3.

Γ(α, x) − Γ (α, x + y) = γ(α, x + y) − γ(α, x) = e−x xα−1

∞  (−1)k [1 − e−y ek (y)] Γ(1 − α + k) k=0

xk Γ(1 − α) ek (x) =

4.

γ(α, x) = Γ(α)e−x x 2 α 1

∞ 

n  √  1 (−1)m x 2 n I n+α 2 x m! n=0 m=0

k  xm m! m=0

[x = 0,

[|y| < |x|] α = 0,

EH II 139(2)

−1, −2, . . .] EH II 139(3)

5.

8.355

Γ(α, x) = e−x xα

∞ 

Lα n (x) n +1 n=0

[x > 0]

Γ(α, x) γ(α, y) = e−x−y (xy)α

EH II 140(5)

∞ 

n! Γ(α) α Lα n (x) Ln (y) (n + 1) Γ(α + n + 1) n=0 [y > 0, x ≥ y,

α = 0, −1, . . .] EH II 139(4)

8.356 1.11

Functional relations: γ(α + 1, x) = α γ(α, x) − xα e−x α −x

EH II 134(2)

2.

Γ(α + 1, x) = α Γ(α, x) + x e

EH II 134(3)

3.

Γ(α, x) + γ(α, x) = Γ(α)

EH II 134(1)

4. 5.

d Γ(α, x) d γ(α, x) =− = xα−1 e−x dx dx n−1  Γ(α, x) xα+s Γ(α + n, x) = + e−x Γ(α + n) Γ(α) Γ(α + s + 1) s=0

6.11

Γ(α) Γ(α + n, x) − Γ(α + n) Γ(α, x) = Γ(α + n) γ(α, x) − Γ(α) γ(α + n, x)

7.∗

Γ(a + k, x) = (a + k − 1) Γ(a + k − 1, x) + xa+k−1 e−x  1  Γ(a + k + 1, x) − xa+k e−x = a+k

8.∗

Γ(a − k, x) = (a − k − 1) Γ(a − k − 1, x) + xa−k−1 e−x  1  Γ(a − k + 1, x) − xa−k e−x = a−k

9.∗

γ(a + k, x) = (a + k − 1) γ(a + k − 1, x) − xa+k−1 e−x  1  Γ(a + k + 1, x) + xa+k e−x = a+k

EH II 135(8) NH 4(3) NH 5

902

10.∗

8.357 1.

8.358

8.359 1. 2. 3. 4.

11

Euler’s Integrals of the First and Second Kinds

8.357

γ(a − k, x) = (a − k − 1) γ(a − k − 1, x) − xa−k−1 e−x  1  γ(a − k + 1, x) + xa−k e−x = a−k Asymptotic representation for large values of |x|: %M −1 &   (−1)m Γ(1 − α + m) −M α−1 −x + O |x| Γ(α, x) = x e xm Γ(1 − α) m=0   3π 3π < arg x < , M = 1, 2, . . . |x| → ∞, − 2 2

EH II 135(6), NH 37(7), LE 340(3)

Representation as a continued fraction: e−x xα Γ(α, x) = 1−α x+ 1 1+ 2−α x+ 2 1+ 3−α x+ 1 + ... Relationships with other functions:

EH II 136(13), NH 42(9)

Γ(0, x) = − Ei(−x)

1 Γ 0, ln = − li(x) x  √  √ Γ 12 , x2 = π − π Φ(x)  √  γ 12 , x2 = π Φ(x)

EH II 143(1) EH II 143(2) EH II 147(2) EH II 147(1)

8.36 The psi function ψ(x) 8.360 1. 8.361 1.8 2. 3. 4.

Definition: ψ(x) =

d ln Γ(x) dx

Integral representations:

 ∞ −t e e−zt d ln Γ(z) = − ψ(z) = dt dz t 1 − e−t 0

  ∞ dt 1 ψ(z) = e−t − z (1 + t) t 0  ∞ t dt 1 −2 ψ(z) = ln z − 2 2 2πt − 1) 2z 0 (t + z ) (e

 1 1 tz−1 − ψ(z) = dt − ln t 1 − t 0

[Re z > 0]

NH 183(1), WH

[Re z > 0]

NH 184(7), WH

[Re z > 0]

WH

[Re z > 0]

WH

8.363

The psi function ψ(x)

 5.

∞ −t

− e−zt dt − C, 1 − e−t

e

ψ(z) = 0

 6.

∞

ψ(z) =



(1 + t)−1 − (1 + t)−z

0



7. 8.

903

WH

 dt − C, t

−1 dt − C t − 1 0    ∞ 1 1 − ψ(z) = ln z + e−tz dt t 1 − e−t 0

[Re z > 0]

WH

1 z−1

ψ(z) =

t

FI II 796, WH

[Re z > 0]

MO 4

See also 3.244 3, 3.311 6, 3.317 1, 3.457, 3.458 2, 3.471 14, 4.253 1 and 6, 4.275 2, 4.281 4, 4.482 5. For integrals of the psi function, see 6.46, 6.47. Series representation 8.362 1.

2.

∞ 

1 1 − x+k k+1 k=0 ∞  1 1 = −C − + x x k(x + k) k=1

 ∞   1 1 − ln 1 + ψ(x) = ln x − x+k x+k ψ(x) = −C −

FI II 799(26), KU 26(1) FI II 495

MO 4

k=0

∞

3.

 π2 (x − 1) − (x − 1) ψ(x) = −C + 6

k=1



1 1 − k+1 x+k

k−1 

1 x+n n=0

NH 54(12)

8.363 1.

ψ(x + 1) = −C +

∞ 

(−1)k ζ(k)xk−1

NH 37(5)

k=2

2.

3.

∞  π x2 1 − cot πx − − C + [1 − ζ(2k + 1)] x2k 2x 2 1 − x2 k=1

∞  1 1 − ψ(x) − ψ(y) = y+k x+k

ψ(x + 1) =

NH 38(10)

k=0

(see also 3.219, 3.231 5, 3.311 7, 3.688 20, 4.253 1, 4.295 37)

4.

5.

∞ 

2yi y 2 + (x + k)2 k=0



∞  p 1 q − ψ = −C + q k + 1 p + kq ψ(x + iy) − ψ(x − iy) =

k=0

NH 99(3)

(see also 3.244 3)

NH 29(1)

904

Euler’s Integrals of the First and Second Kinds

8.364

6.8

[ q+1

 2 ]−1   p pπ kπ 2kpπ π +2 ln sin ψ cos = −C − ln(2q) − cot q 2 q q q k=1 [q = 2, 3, . . . , p = 1, 2, . . . , q − 1]

7.





∞ ∞   p p−1 1 ψ −ψ =q n−1 q q (p + kq) n=2

MO 4, EH I 19(29) NH 59(3)

k=0

ψ (n) (x) = (−1)n+1 n!

8.

∞  k=0

1 = (−1)n+1 n! ζ(n + 1, x) (x + k)n+1

NH 37(1)

Infinite-product representation 8.364 ψ(x)

∞ =x 1+

1.

e

2.

ey ψ(x)

1 1 e− x+k x+k k=0

∞ y Γ(x + y) y = 1+ e− x+k Γ(x) x+k

NH 65(12)

NH 65(11)

k=0

See also 8.37. • • • •

For a connection with Riemann’s zeta function, see 9.533 2. For a connection with the gamma function, see 4.325 12 and 4.352 1. For a connection with the beta function, see 4.253 1. For series of psi functions, see 8.403 2, 8.446, and 8.447 3 (Bessel functions), 8.761 (derivatives of associated Legendre functions with respect to the degree), 9.153, 9.154 (hypergeometric function), 9.237 (confluent hypergeometric function). • For integrals containing psi functions, see 6.46–6.47. 8.365 1. 2. 3.

Functional relations: 1 ψ(x + 1) = ψ(x) + x

x x+1 = 2 β(x) ψ −ψ 2 2 ψ(x + n) = ψ(x) +

n−1  k=0

4.

JA

(cf. 8.37 0)

1 x+k

GA 154(64)a

n  1 ψ(n + 1) = −C + k

MO 4

k=1

5. 6.

lim [ψ(z + n) − ln n] = 0

MO 3

n→∞

n−1 1 k ψ(nz) = ψ z+ + ln n n n k=0

[n = 2, 3, 4, . . .]

MO 3

8.367

7.

The psi function ψ(x)

ψ(x − n) = ψ(x) −

n  k=1

8. 9. 10. 8.366

1 x−k

ψ(1 − z) = ψ(z) + π cot πz     ψ 12 + z = ψ 12 − z + π tan πz     ψ 34 − n = ψ 14 + n + π

10. 11.

ψ  (n) =

2. 3. 4. 5. 6. 7. 8. 9.

GA 155(68)a JA

[n = 0,

±1,

±2, . . .]

Particular values ψ(1) = −C   ψ 12 = −C − 2 ln 2 = −1.963 510 026 . . . & % n  1 1  − ln 2 ψ 2 ± n = −C + 2 2k − 1 k=1   π ψ 14 = −C − − 3 ln 2 2 3 π ψ 4 = −C + − 3 ln 2 2 < 1 π 1 3 ψ 3 = −C − − ln 3 2 3 2 <   π 1 3 ψ 23 = −C + − ln 3 2 3 2 π2 = 1.644 934 066 848 . . . ψ  (1) = 6   π2 = 4.934 802 200 5 . . . ψ  12 = 2 ψ  (−n) = ∞

1.

905

(cf. 8.367 1) GA 155a JA GA 157a GA 157a GA 157a GA 157a JA JA

[n is a natural number]

JA

[n is a natural number]

JA

n   π2 1 −4 +n = 2 (2k − 1)2

[n is a natural number]

JA

n   π2 1 +4 −n = 2 (2k − 1)2

[n is a natural number]

JA

π2  1 − 6 k2 n−1

k=1

12.

ψ

1 2

k=1

13.

ψ

1 2

k=1

8.367 1. 2. 3.

Euler’s constant (also denoted by γ): C = − ψ(1) = 0.577 215 664 90 . . . & %n−1 1 − ln n C = lim n→∞ k k=1   1 C = lim ζ(x) − x→1+0 x−1

FI II 319, 795 FI II 801a

FI II 804

906

Euler’s Integrals of the First and Second Kinds

8.370

Integral representations:  ∞ 4. C=− e−t ln t dt

FI II 807

0

5. 6. 7. 8. 9. 10. 11. 12.

13.



1 ln ln FI II 807 dt t 0   1 1 1 + C= DW dt ln t 1 − t 0   ∞ dt 1 C=− MO 10 cos t − 1+t t 0   ∞ sin t 1 dt − C=1− MO 10 t 1 + t t 0   ∞ dt 1 −t C=− FI II 795, 802 e − 1 + t t 0    ∞ dt 1 C=− DW, MO 10 e−t − 2 1+t t 0   ∞ 1 1 − C= DW dt et − 1 tet 0  ∞ −t  1   e −t dt − dt 1−e C= FI II 802 t t 0 1 See also 8.361 5–8.361 7, 3.311 6, 3.435 3 and 4, 3.476 2, 3.481 1 and 2, 3.951 10, 4.283 9, 4.331 1, 4.421 1, 4.424 1, 4.553, 4.572, 6.234, 6.264 1, 6.468.  C=−

1

Asymptotic expansions C =

n−1  k=1

1 1 1 1 1 1 − ln n + + − + − + ... k 2n 12n2 120n4 252n6 240n8

[0 < θ < 1]

FI II 827

θ B2r+2 B2r 1 + ··· + 2r 2r+2 2r n 2 (r + 1) n

8.37 The function β(x) Definition: 

 x  x+1 1 β(x) = ψ −ψ 2 2 2 8.371 Integral representations:  1 x−1 t 3 dt β(x) = 1. 1 +t 0  ∞ −xt e 2. β(x) = dt −t 0 1+e

 ∞ −xt x+1 e dt 3. β = 2 0 cosh t 8.370

NH 16(13)

[Re x > 0]

WH

[Re x > 0]

MO 4

[Re x > −1]

See also 3.241 1, 3.251 7, 3.522 2 and 4, 3.623 2 and 3, 4.282 2, 4.389 3, 4.532 1 and 3.

8.377

The function β(x)

907

Series representation 8.372 1.7

β(x) =

∞  (−1)k k=0

2.7

β(x) =

∞  k=0

x+k 1 (x + 2k)(x + 2k + 1)

[−x ∈ N]

NH 37, 101(1)

[−x ∈ N]

NH 101(2)

∞

3.8

β(x) =

1 k! 1 2 x(x + 1) . . . (x + k) 2k k=0

[−x ∈ N]

[β has simple poles at x = −n with residue (−1)n ]

NH 246(7)

8.373 1.6

β(x + 1) = ln 2 +

∞ 

  (−1)k 1 − 2−k ζ(k + 1)xk

[|x| < 1]

NH 37(5)

k=1

2.

6

∞      π 1 1 − + 1 − 1 − 2−2k ζ(2k + 1) x2k β(x + 1) = ln 2 − 1 + − 2x 2 sin πx 1 − x2 k=1

[0 < |x| < 2;

x = ±1]

NH 38(11)

∞

8.374

 (−1)k dn β(x) = (−1)n n! n dx (x + k)n+1

[−x ∈ N]

NH 37(2)

k=0

8.375

1.

6

Representation in the form of a finite sum: q−1

 2  p (2k + 1)π p(2k + 1)π π ln sin β − cos = q 2 sin pπ q 2q q

k=0

2.

β(n) = (−1)n+1 ln 2 +

n−1  k=1

[q = 2, 3, . . . , p = 1, 2, 3, . . . , q − 1]

NH 23(9)

(−1)k+n+1 k

Functional relations

2n  x+k 8.376 (−1)k β = (2n + 1) β(x) 2n + 1 k=0 n    8.377 β 2k x = ψ (2n x) − ψ(x) − n ln 2 k=1

(see also 8.362 5–7)

NH 19 NH 20(10)

908

Euler’s Integrals of the First and Second Kinds

8.380

8.38 The beta function (Euler’s integral of the first kind): B(x, y) Integral representation 8.380 1.



1

B (x, y) = 0 

tx−1 (1 − t)y−1 dt∗ 1

=2  2.

4. 5. 6.

sin2x−1 ϕ cos2y−1 ϕ dϕ

[Re x > 0, t

B(x, y) = z (1 + z)

π/2

x 0

10. 11.

Re y > 0]

KU 10

Re y > 0]

FI II 775

Re y > 0]

MO 7

Re y > 0]

BI (1)(15)

(1 − t)y−1 dt (t + z)x+y [Re x > 0, Re y > 0,

dt Re y > 0]

BI (1)(15)

1 x−1

B (x, y) = z y (1 + z)x



9.

[Re x > 0,

 ∞ tx−1 t2x−1 B(x, y) = dt = 2 dt [Re x > 0, x+y 2 x+y 0 (1 + t) 0 (1 + t )  1 (1 + t)2x−1 (1 − t)2y−1 B(x, y) = 22−y−x dt [Re x > 0, x+y (1 + t2 ) −1  1 x−1  ∞ x−1 t + ty−1 t + ty−1 B(x, y) = dt = dt [Re x > 0, x+y (1 + t)x+y 0 (1 + t) 1  1+ , 1 x−1 B(x, y) = x+y−1 (1 − t)y−1 + (1 + t)y−1 (1 − t)x−1 (1 + t) 2 0

y

FI II 774(1)

0 ∞

0

8.

Re y > 0]

π/2

 7.

[Re x > 0,

0

B(x, y) = 2 

3.

 y−1 t2x−1 1 − t2 dt

cos2x−1 ϕ sin2y−1 ϕ x+y

(z + cos2 ϕ) [Re x > 0,

0 > z > −1,

Re(x + y) < 1]

NH 163(8)

0 > z > −1,

Re(x + y) < 1]

NH 163(8)

dϕ

Re y > 0,

See also 3.196 3, 3.198, 3.199, 3.215, 3.238 3, 3.251 1–3, 11, 3.253, 3.312 1, 3.512 1 and 2, 3.541 1, 3.542 1, 3.621 5, 3.623 1, 3.631 1, 8, 9, 3.632 2, 3.633 1, 4, 3.634 1, 2, 3.637, 3.642 1, 3.667 8, 3.681 2.  1  1   (1 − t)x−1 1 1 2 x−1 √ 1−t dt = 2x−1 B(x, x) = 2x−2 dt 2 2 t 0 0 See 8.384 4, 8.382 3, and also 3.621 1, 3.642 2, 3.665 1, 3.821 6, 3.839 6.  ∞ cosh 2yt B(x + y, x − y) = 41−x MO 9 dt [Re x > |Re y|, Re x > 0] cosh2x t 0  1 ,  y + y x−1 y−1 =z B x, (1 − tz ) t dt Re z > 0, Re > 0, Re x > 0 z z 0 FI II 787a

∗ This

equation is used as the definition of the function B(x, y).

8.384

8.381 1.

The beta function (Euler’s integral of the first kind): B(x, y)



1−x−y

dt 2π (a + b) = x y (x + y − 1) B(x, y) −∞ (a + it) (b − it) [a > 0,

 2.

∞

b > 0;

x and y are real,

x + y > 1]

MO 7

b > 0;

x and y are real,

x + y > 1]

MO 7

∞

dt =0 x (b − it)y (a − it) −∞ [a > 0,

3.

909

B(x + iy, x − iy) = 2

1−2x

−2iγy

αe



∞

e2iαyt dt 2x −∞ cosh (αt − γ) [y, α, γ are real,

α > 0;

Re x > 0] MI 8a

4.

For an integral representation of ln B(x, y), see 3.428 7.  2x+y−1 (x + y − 1) π/2 1 = cos[(x − y)t] cosx+y−2 t dt B (x, y) π 0  2x+y−2 (x + y − 1) π   = cos[(x − y)t] sinx+y−2 t dt π cos (x − y) π2 0  2x+y−2 (x + y − 1) π   = sin[(x − y)t] sinx+y−2 t dt π sin (x − y) π2 0

NH 158(5)a NH 159(8)a NH 159(9)a

Series representation 8.382 1.

2.

∞ 1 (y − 1) . . . (y − n) [y > 0] (−1)n y WH y n=0 n!(x + n)  





  ∞ √ 1 − 1 − 2−2k ζ(2k + 1) 2k+1 tan πx 1+x 1 1+x 1 2 , x ln B ln 2π + ln − ln + 2 2 2 x 1−x 2k + 1

B(x, y) =

k=0

[|x| < 2] 3. 8.383

B z,

1 2

=

∞  (2k − 1)!! k=1

2k k!

1 1 + z+k z

NH 39(17)

(see also 8.384 and 8.380 9)

WH

Infinite-product representation: (x + y + 1) B(x + 1, y + 1) =

∞ k(x + y + k) (x + k)(y + k)

[x,

y = −1,

−2, . . .]

MO 2

k=1

8.384

Functional relations involving the beta function: Γ(x) Γ(y) = B(y, x) Γ(x + y)

1.

B(x, y) =

2.

B(x, y) B(x + y, z) = B(y, z) B(y + z, x)

FI II 779 MO 6

910

3.

Bessel Functions and Functions Associated with Them ∞ 

B(x, y + k) = B(x − 1, y)

k=0

4.

8.391

B(x, x) = 21−2x B

1

WH



2, x

(see also 8.380 9 and 8.382 3) FI II 784

5. 6.

  π B(x, x) B x + 12 , x + 12 = 4x−1 2 x



n+m−1 n+m−1 1 =m =n n−1 m−1 B(n, m)

WH

[m and n are natural numbers]

For a connection with the psi function, see 4.253 1.

8.39 The incomplete beta function Bx (p, q) 

8.392 I x (p, q) =

x

tp−1 (1 − t)q−1 dt =

8.3917 Bx (p, q) =

0 Bx (p, q)

xp 2 F 1 (p, 1 − q; p + 1; x) p

ET I 373 ET II 429

B(p, q)

8.4–8.5 Bessel Functions and Functions Associated with Them 8.40 Definitions 8.401

Bessel functions Z ν (z) are solutions of the differential equation

ν2 d2 Z ν 1 dZν + 1 − 2 Zν = 0 + KU 37(1) dz 2 z dz z Special types of Bessel functions are what are called Bessel functions of the first kind J ν (z), Bessel functions of the second kind Y ν (z) (also called Neumann functions and often written N ν (z)), and Bessel (2) functions of the third kind H (1) ν (z) and H ν (z) (also called Hankel’s functions). ∞ z 2k zν  [|arg z| < π] (−1)k 2k KU 55(1) 8.402 J ν (z) = ν 2 2 k! Γ(ν + k + 1) k=0

8.403 1.

Y ν (z) =

1 [cos νπ J ν (z) − J −ν (z)] sin νπ

[for non-integer ν,

|arg z| < π] KU 41(3)

8.407

Definitions

911

z  (n − k − 1)!  z 2k−n − 2 k! 2 n−1

2.

π Y n (z) = 2 J n (z) ln

k=0

−

∞ 

(−1)k

k=0

 = 2 J n (z) ln

 z n+2k 1 [ψ(k + 1) + ψ(k + n + 1)] k! (k + n)! 2

z +C − 2

n−1  k=0

KU 43(10)

(n − k − 1)!  z 2k−n k!

2

&  n+2k % n+k n ∞ k  z n 1   1  1  (−1)k z2 1 − + − 2 n! k k! (k + n)! m m=1 m m=1 k=1 k=1 [n + 1 a natural number, |arg z| < π] KU 44, WA 75(3)a

8.404 1.

Y −n (z) = (−1)n Y n (z)

[n is a natural number]

KU 41(2)

2.

J −n (z) = (−1)n J n (z)

[n is a natural number]

KU 41(2)

8.4057 1.

H (1) ν (z) = J ν (z) + i Y ν (z)

KU 44(1)

2.

H (2) ν (z) = J ν (z) − i Y ν (z)

KU 44(1)

In all relationships that hold for an arbitrary Bessel function Z ν (z), that is, for the functions J ν (z), (2) Y ν (z), and linear combinations of them, for example, H (1) ν (z) and H ν (z), we shall write simply the letter Z instead of the letters J, Y , H (1) , and H (2) . Modified Bessel functions of imaginary argument I ν (z) and K ν (z) 8.406 1. 2.

 π  π I ν (z) = e− 2 νi J ν e 2 i z  3 3 I ν (z) = e 2 πνi J ν e− 2 πi z

+ π, −π < arg z ≤ 2 , +π < arg z ≤ π 2

WA 92 WA 92

For integer ν, I n (z) = i−n J n (iz)

3. 8.407 1.8 2.8

πi π νi (1)  1 πi e 2 H ν ze 2 2 −πi − π νi (2)  − 1 πi e 2 H −ν ze 2 K ν (z) = 2 K ν (z) =

KU 46(1)

 

−π < arg z ≤ 12 π − 12 π < arg z ≤ π

For the differential equation defining these functions, see 8.494.

 

WA 92(8)

912

Bessel Functions and Functions Associated with Them

8.411

8.41 Integral representations of the functions Jν (z) and Nν (z) 8.411 1.

11

2.

 1 π −niθ+iz sin θ e dθ J n (z) = 2π −π π 1 [n = 0, 1, 2, . . .] cos (nθ − z sin θ) dθ = π 0   1 π 2 π/2 J 2n (z) = cos 2nθ cos (z sin θ) dθ = cos 2nθ cos (z sin θ) dθ π 0 π 0

WH

[n an integer]

WA 30(7)



3.11

4.

5.

6.

7. 8.

9.

10. 11.

12.

π

1 sin(2n + 1)θ sin (z sin θ) dθ π 0  π/2 2 WA 30(6) [n an integer] = sin(2n + 1)θ sin (z sin θ) dθ π 0  z ν  π/2 2    J ν (z) = 2  sin2ν θ cos (z cos θ) dθ Γ ν + 12 Γ 12 0   Re ν > − 12 WH  z ν  π   2

J ν (z) = sin2ν θ cos (z cos θ) dθ Re ν > − 12   1 0 Γ ν + 12 Γ 2  z ν  π/2 2

cos (z sin θ) cos2ν θ dθ J ν (z) =   1 −π/2 1 Γ ν+2 Γ 2   KU 65(5), WA 35(4)a Re ν > − 12  z ν  π     2    J ν (z) =  e±iz cos ϕ sin2ν ϕ dϕ Re ν + 12 > 0 WH 1 1 Γ ν+2 Γ 2 0  z ν  1  ν− 12   2     1 − t2 J ν (z) = cos zt dt Re ν > − 12 KU 65(6), WH 1 1 Γ ν + 2 Γ 2 −1  x −ν  ∞  1  sin xt J ν (x) = 2  1 2   1  − 2 < Re ν < 12 , x > 0 MO 37 1 dt ν+ Γ 2 − ν Γ 2 1 (t2 − 1) 2  z ν  1  ν− 12   2  1 J ν (z) =  eizt 1 − t2 dt Re ν > − 12 WA 34(3) 1 Γ ν + 2 Γ 2 −1   2 ∞ νπ J ν (x) = sin x cosh t − WA 199(12) cosh νt dt π 0 2   π/2 cosν− 12 θ sin z − νθ + 1 θ 2 2ν+1 z ν  1 J ν (z) =  e−2z cot θ dθ 2ν+1 1 Γ ν+2 Γ 2 0 sin θ , +   π |arg z| < , Re ν + 12 > 0 WH 2 J 2n+1 (z) =

8.414

13.

10

14.

Integral representations of the functions Jν (z) and Nν (z)

1 J ν (z) = π

J ν (z) =

e



π

0

±νπi

π

sin νπ cos (νθ − z sin θ) dθ − π 

π

0



∞

913

e−νθ−z sinh θ dθ

0



[Re z > 0] ∞



WA 195(4)

cos (νθ + z sin θ) dθ − sin νπ e−νθ+z sinh θ dθ 0  π π for < |arg z| < π, with the upper sign taken for |arg z| > 2 2  π and the lower sign taken for |arg z| < − 2 WH

8.412 1. 2. 3.8

4.

5.7



 + z 1 π, t exp t− dt |arg z| < 2 t 2 −∞

 (0+) zν z2 J ν (z) = ν+1 t−ν−1 exp t − dt 2 πi −∞ 4t  ∞ z ν  (−1)k z 2k (0+) t −ν−k−1 J ν (z) = ν+1 et dt 2 πi 22k k! −∞ k=1  i∞  x ν+2t Γ(−t) 1 J ν (x) = dt [Re ν > 0, x > 0] 2πi −i∞ Γ(ν + t + 1) 2   z ν   (1+,−1−) Γ 12 − ν 2 ν− 12  1 2 t −1 J ν (z) = cos(zt) dt 2πi Γ 2 A  1 J ν (z) = 2πi



(0+)

−ν−1

WH, WA 195(2)

WA 195(1)

WA 195(1)

WA 214(7)

ν = 12 , 32 , . . . ; The pointA falls to the right of the point t = 1,  and arg(t − 1) = arg(t + 1) = 0 at the point A

6.8

J ν (z) =

1 2π



WH π+∞i

e−iz sin θ+iνθ dθ

[Re z > 0]

−π+∞i

The path of integration being taken around the semi-infinite strip y ≥ 0, −π ≤ x ≤ π. 

 ∞ Jν z2 + ζ 2 1 8 8.413 = eζ cos t cos (z sin t − νt) dt ν π(z + ζ)ν (z 2 − ζ 2 ) 2 0 ∞

− sin νπ



exp (−z sinh t − ζ cosh t − νt) dt

0

 1 1 − 2 +i∞ Γ(−t) 2t J 0 (t) dt = x dt 8.414 t 4π − 12 −i∞ t Γ(1 + t) 2x See 3.715 2, 9, 10, 13, 14, 19–21, 3.865 1, 2, 4, 3.996 4. 

[Re(z + ζ) > 0]

MO 40

[x > 0]

MO 41

∞

914

Bessel Functions and Functions Associated with Them

8.415

• For an integral representation of J 0 (z), see 3.714 2, 3.753 2, 3, and 4.124. • For an integral representation of J 1 (z), see 3.697, 3.711, 3.752 2, and 3.753 5. 8.415 1.

Y 0 (x) =



1 0

arcsin t 4 √ sin(xt) dt − 2 π 1 − t2



∞ 1

√   ln t + t2 − 1 √ sin(xt) dt t2 − 1 [x > 0]

 x −ν  ∞ cos xt Y ν (x) = −2  1 2   1  1 dt Γ 2 − ν Γ 2 1 (t2 − 1)ν+ 2

2.

Y ν (x) = −

3. 4.

4 π2

8

5.

6.

1 Y ν (z) = π

2 π 

− 12 < Re ν < 12 ,

x>0



KU 89(28)a, MO 38



0



MO 37

∞



cos x cosh t −

0 π

νπ 2

cosh νt dt

1 sin (z sin θ − νθ) dθ − π



∞



[−1 < Re ν < 1,

x > 0]

WA 199(13)

 eνt + e−νt cos νπ e−z sinh t dt

0

[Re z > 0] WA 197(1) & %  z ν  π/2 ∞ 2 2 2ν    Y ν (z) =  sin (z sin θ) cos θ dθ − e−z sinh θ cosh2ν θ dθ Γ ν + 12 Γ 12 0 0   Re ν > − 12 , Re z > 0 WA 181(5)a   1  π2 cosν− 2 θ cos z − νθ + 12 θ −2z cot θ 2ν+1 z ν    Y ν (z) = −  dθ e Γ ν + 12 Γ 12 0 sin2ν+1 θ , +   π |arg z| < , Re ν + 12 > 0 2 WA 186(8)

For an integral representation of Y 0 (z), see 3.714 3, 3.753 4, 3.864. See also 3.865 3.

8.42 Integral representations of the functions Hν(1) (z) and Hν(2) (z) 8.421 1.

2.

νπi  ∞ e− 2 eix cosh t−νt dt πi −∞ νπi  ∞ 2e− 2 = eix cosh t cosh νt dt πi 0

H (1) ν (x) =

[−1 < Re ν < 1,

x > 0]

WA 199(10)

[−1 < Re ν < 1,

x > 0]

WA 199(11)



∞ e e−ix cosh t−νt dt πi −∞ νπi  ∞ 2e 2 =− e−ix cosh t cosh νt dt πi 0

H (2) ν (x) = −

νπi 2

(1)

8.422

3.

4.

5.

6. 7.

8.

9.

10.

11. 8.422

H (1) ν (z)

2ν+1 iz ν    =−  Γ ν + 12 Γ 12

H (2) ν (z) =

2ν+1 iz ν    Γ ν + 12 Γ 12 

2.





π/2 0

π/2

915

t 1 cosν− 2 t ei(z−νt+ 2 ) exp (−2z cot t) dt sin2ν+1 t   Re ν > − 12 , Re z > 0

cos

te−i( sin2ν+1 t

ν− 12

0

z−νt+ 2t

WA 186(5)

) exp (−2z cot t) dt 

Re ν > − 12 ,

Re z > 0



WA 186(6)

 −ν  ∞  1  2i x2 eixt (1) 1 1  H ν (x) = − √ − < Re ν < , x > 0 WA 87(1) 1 dt 2 2 π Γ 2 − ν 1 (t2 − 1)ν+ 2  −ν  ∞  1  2i x2 e−ixt (2) 1 1  H ν (x) = √ − < Re ν < , x > 0 WA 187(2) 1 dt 2 2 π Γ 2 − ν 1 (t2 − 1)ν+ 2 

  1 1 i − 1 iνπ ∞ 2 e iz t + H (1) (z) = − exp t−ν−1 dt ν π 2 t 0 [0 < arg z < π; or arg z = 0 and − 1 < Re ν < 1] MO 38 

  1 z2 i − 1 iνπ ν ∞ 2 e ix t + H (1) (xz) = − z exp t−ν−1 dt ν π 2 t 0 , + π π 0 < arg z < , x > 0, Re ν > −1; or arg z = , x > 0 and − 1 < Re ν < 1 2 2 , +   xν exp i xz − π ν − π  ∞

ν− 12 1 2 it (1) 2 4   H ν (xz) = tν− 2 e−xt dt 1+ 1 πz 2z Γ ν+2 0  Re ν > − 12 , − 12 π < arg z < 32 π, x > 0  z ν  ∞ −2ie−iνπ 2  H (1) (z) = eiz cosh t sinh2ν t dt √ ν π Γ ν + 12  0 0 < arg z < π, Re ν > − 12 or arg z = 0 and − (1) H 0 (x)

i =− π

 √ exp i x2 + t2 √ dt x2 + t2 −∞



1 2

< Re ν <

1 2



MO 38

MO 39

MO 38

∞

[x > 0]

  z ν  (1+) − ν  ν− 12 2 (1) izt 2 2   H ν (z) = t e − 1 dt [−π < arg z < 2π] πi Γ 12 1+∞i   z ν   (−1−) Γ 12 − ν  ν− 12 (2) 2   H ν (z) = eizt t2 − 1 dt 1 πi Γ 2 −1+∞i [−2π < arg z < π] Γ

1.

(2)

Integral representations of the functions Hν (z) and Hν (z)

MO 38

1

The paths of integration are shown in the drawing.

WA 183(4)

916

Bessel Functions and Functions Associated with Them

8.423 1. 2.

H (1) ν (z)

1 =− π

H (2) ν (z)

1 =− π

 

−π+∞i

e−iz sin θ+iνθ dθ

8.423

[Re z > 0]

WA 197(2)a

[Re z > 0]

WA 197(3)a

−∞i −∞i

e−iz sin θ+iνθ dθ

π+∞i

The path of integration for 8.423 1 is shown in the left-hand drawing and for 8.423 2 in the right-hand drawing.

8.424 1.

H (1) ν (z) J ν (ζ) =

1 πi



γ+i∞

exp 0





1 zζ dt z2 + ζ 2 t− Iν 2 t t t

[γ > 0, 



1 zζ dt i γ−i∞ z2 + ζ 2 H (2) (z) J (ζ) = exp t − I ν ν ν π 0 2 t t t

Re ν > −1,

|ζ| < |z|]

MO 45

[γ > 0,

Re ν > −1,

|ζ| < |z|]

MO 45



2.

8.43 Integral representations of the functions Iν (z) and Kν (z) The function I ν (z) 8.431 1.

I ν (z) =

2.

I ν (z) =

3.

I ν (z) =

4.

I ν (z) =

 z ν  1  ν− 12 ±zt 2     1 − t2 e dt 1 1 Γ ν + 2 Γ 2 −1  z ν  1  ν− 12 2     1 − t2 cosh zt dt 1 1 Γ ν + 2 Γ 2 −1  z ν  π 2     e±z cos θ sin2ν θ dθ Γ ν + 12 Γ 12 0  z ν  π 2     cosh (z cos θ) sin2ν θ dθ Γ ν + 12 Γ 12 0



   Re ν + 12 > 0

WA 94(9)



   Re ν + 12 > 0

WA 94(9)



   Re ν + 12 > 0

WA 94(9)



   Re ν + 12 > 0

WA 94(9)

8.432

5.

Integral representations of the functions Iν (z) and Kν (z)

1 I ν (z) = π



π

e

z cos θ

0

sin νπ cos νθ dθ − π



∞

0

e−z cosh t−νt dt + π |arg z| ≤ , 2

917

, Re ν > 0

WA 201(4)

See also 3.383 2, 3.387 1, 3.471 6, 3.714 5. For an integral representation of I 0 (z) and I 1 (z), see 3.366 1, 3.534 3.856 6. The function K ν (z) 8.432 1.



∞

K ν (z) =

, + π |arg z| < or Re z = 0 and ν = 0 2

e−z cosh t cosh νt dt

0

2.

3.

4. 5.

 z ν  1   ∞ Γ 2  K ν (z) = 2  e−z cosh t sinh2ν t dt Γ ν + 12  0 Re ν > − 12 , Re z > 0; or Re z = 0 and −  z ν  1   ∞  ν− 12 Γ 2  K ν (z) = 2  e−zt t2 − 1 dt 1 Γ ν+2 1 +   Re ν + 12 > 0,

|arg z| <

 ∞ 1 K ν (x) = cos (x sinh t) cosh νt dt cos νπ 0 2    Γ ν + 12 (2z)ν ∞ cos xt dt 1 K ν (xz) = ν+ 12 xν Γ 2 0 (t2 + z 2 )

MO 39

1 2

< Re ν <

1 2



WA 190(5), WH

, π ; or Re z = 0 and ν = 0 2

[x > 0,

−1 < Re ν < 1]

+   Re ν + 12 ≥ 0,

WA 190(4)

WA 202(13)

x > 0,

|arg z| <

π, 2

WA 191(1)

6.11 7.7

K ν (z) =

 1  z ν

9.

e

2 ν

K ν (xz) =

+ π |arg z| < , 2

dt

tν+1 

 ∞ z x z2 K ν (xz) = exp − t+ t−ν−1 dt 2 0 2 t + 2

 8.

∞ −t−z 2 /4t

0

π xν e−xz   2z Γ ν + 12



∞

e 0

−xt ν− 12

t



|arg z| <

t 1+ 2z

, WA 203(15)

, π π or |arg z| = and Re ν < 1 4 4

MO 39

ν− 12 dt 

|arg z| < π,

Re z > 0,

Re ν > − 12 , x > 0

 MO 39

√  x ν  ∞ exp −x√t2 + z 2  π  √ K ν (xz) =  t2ν dt 2 + z2 2z Γ ν + 12 t 0 + Re ν > − 12 ,

Re z 2 > 0

Re



t2 + z 2 > 0,

, x>0

MO 39

918

Bessel Functions and Functions Associated with Them

8.433

See also 3.383 3, 3.387 3, 6, 3.388 2, 3.389 4, 3.391, 3.395 1, 3.471 9, 3.483, 3.547 2, 3.856, 3.871 3, 4, 7.141 5. √

 ∞   2x x 3 √ 8.433 K 13 cos t3 + xt dt KU 98(31), WA 211(2) =√ x 0 3 3 For an integral representation of K 0 (z), see 3.754 2, 3.864, 4.343, 4.356, 4.367.

8.44 Series representation The function J ν (z) 8.440 8.441 1.

J ν (z) =

∞  z ν 

2

k=0

[|arg z| < π]

WH 358 a

Special cases: J 0 (z) =

∞ 

(−1)k

k=0

2.

 z 2k (−1)k k! Γ(ν + k + 1) 2

z 2k 22k (k!)

J 1 (z) = − J 0 (z) =

2

∞

z  (−1)k z 2k 2 22k k!(k + 1)! k=0

3.

4.

J 13 (z) =

Γ

1 4 3

 3

z 2

∞  k=0

 √ 2k z 3 (−1) 2k 2 k! · 1 · 4 · 7 · · · · · (3k + 1) k

    √ 2k ∞  z 3 1 3 2 1+ J − 13 (z) =  2  (−1)k 2k z 2 k! · 2 · 5 · 8 · · · · · (3k − 1) Γ 3 k=1

For the expansion of J ν (z) in Laguerre polynomials, see 8.975 3. 8.442  μ+ν+2m ∞  (−1)m 12 z (μ + ν + m + 1)m 7 J ν (z) J μ (z) = 1. m! Γ(μ + m + 1) Γ(ν + m + 1) m=0

 2k  az ν bz μ b2 k az (−1) F −k, −ν − k; μ − 1; ∞ 2  2 2 a2 2.8 J ν (az) J μ (bz) = Γ(μ + 1) k! Γ(ν + k + 1) k=0

The function Y ν (z)

 ∞  z ν  z 2k 1 (−1)k 2k 8.44311 Y ν (z) = cos νπ sin νπ 2 2 k! Γ (ν + k + 1) k=0  ∞  z −ν  z 2k k − (−1) 2k 2 2 k! Γ(k − ν + 1) k=0 [ν = an integer] (cf. 8.403 1) For ν + 1 a natural number, see 8.403 2.; for ν a negative integer, see 8.404 1

MO 28

8.447

8.444 1.

2.11

Series representation

919

Special cases, ∞ k  z  (−1)k  z 2k  1 π Y 0 (z) = 2 J 0 (z) ln + C − 2 2 2 2 m (k!) m=1 k=1  z 2k−1   ∞ (−1)k+1 k−1 2 z   z  1 1 2 +2 π Y 1 (z) = 2 J 1 (z) ln + C − − − 2 z 2 k!(k − 1)! k m m=1

KU 44

k=2

The functions I ν (z) and K n (z) 8.445 I ν (z) =

∞ 

 z ν+2k 1 k! Γ(ν + k + 1) 2

k=0 n−1 

8.4468 K n (z) =

1 2

(−1)k

k=0

n+1

+(−1)

WH 372a

(n − k − 1)!  n−2k k! z2

∞  k=0

 z n+2k 

 1 1 z ln − ψ(k + 1) − ψ(n + k + 1) k!(n + k)! 2 2 2 2

 l 

n+l ∞  z n+2l  1  1 1 1 n+1 n 2 + = (−1) I n (z) ln z + C + (−1) 2 2 l!(n + l)! k k

l=0

+

k=1

WA 95(15)

k=1

n−1 1  (−1)l (n − l − 1)!  z 2l−n 2 l! 2 l=0

[n + 1 is a natural number] MO 29

8.447 1.

Special cases: I 0 (z) =

∞ 

I 1 (z) =

2 2

k=0

2.

 z 2k (k!)

I 0 (z)

=

∞  k=0

 z 2k+1 2

k!(k + 1)! ∞

3.

K 0 (z) = − ln

 z 2k z I 0 (z) + 2 ψ(k + 1) 2 22k (k!) k=0

WA 95(14)

920

Bessel Functions and Functions Associated with Them

8.451

8.45 Asymptotic expansions of Bessel functions 8.451 1.

For large values of |z| ∗ & %n−1      Γ ν + 2k + 12 π  (−1)k 2 π  + R1  J ±ν (z) = cos z ∓ ν − πz 2 4 (2z)2k (2k)! Γ ν − 2k + 12 k=0 & %n−1    Γ ν + 2k + 32 π  (−1)k π  + R2  − sin z ∓ ν − 2 4 (2z)2k+1 (2k + 1)! Γ ν − 2k − 12 k=0

[|arg z| < π]

2.



WA 222(1, 3)

& %n−1   Γ ν + 2k + 12 π  (−1)k π  + R1  sin z ∓ ν − Y ±ν (z) = 2 4 (2z)2k (2k)! Γ ν − 2k + 12 k=0 & %n−1    Γ ν + 2k + 32 π  (−1)k π  + R2  + cos z ∓ ν − 2 4 (2z)2k+1 (2k + 1)! Γ ν − 2k − 12 

11

(see 8.339 4)

2 πz



k=0

[|arg z| < π]

(see 8.339 4)

WA 222(2, 4, 5)



%n−1   &   2 i(z− π2 ν− π4 )  (−1)k Γ ν + k + 12 (−1)n Γ ν + n + 12  + θ1    e πz (2iz)k k! Γ ν − k + 12 (2iz)n k! Γ ν − n + 12 k=0   (see 8.339 4) WA 221(5) Re ν > − 12 , |arg z| < π



%n−1   &   Γ ν + k + 12 Γ ν + n + 12 2 −i(z− π2 ν− π4 )  1 1  + θ2    e πz (2iz)k k! Γ ν − k + 12 (2iz)n k! Γ ν − n + 12 k=0   (see 8.339 4) WA 221(6) Re ν > − 12 , |arg z| < π

11

H (1) ν (z)

4.11

H (2) ν (z)

5.

For indices of the form ν = 2n−1 (where n is a natural number), the series 8.451 terminate. In 2 this case, the closed formulas 8.46 are valid for all values.   ∞ ez  (−1)k Γ ν + k + 12   I ν (z) ∼ √ 2πz k=0 (2z)k k! Γ ν − k + 12   ∞     exp −z ± ν + 12 πi  1 Γ ν + k + 12   √ + (2z)k k! Γ ν − k + 12 2πz k=0

3.

=

=

[The + sign is taken for − 12 π < arg z < 32 π, the − sign for − 32 π < arg z < 12 π]∗  6.11

K ν (z) =

π −z e 2z

%n−1  k=0

&   Γ ν+k+ 2 Γ ν + n + 12 1  + θ3    (2z)k k! Γ ν − k + 12 (2z)n n! Γ ν − n + 12 

 1

(see 8.339 4)

(see 8.339 4)] WA 226(2,3)

WA 231, 245(9)

An estimate of the remainders of the asymptotic series in formulas 8.451: ∗ An

estimate of the remainders in formulas 8.451 is given in 8.451 7 and 8.451 8. contradiction that this condition contains at first glance is explained by the so-called Stokes phenomenon (see Watson, G.N., A Treatise on the Theory of Bessel Functions, 2nd Edition, Cambridge Univ. Press, 1944, page 201). ∗ The

8.452

7.

8.

Asymptotic expansions of Bessel functions

! !   ! ! Γ ν + 2n + 12 !  !!  |R1 | < ! 1 ! (2z)2n (2n)! Γ ν − 2n + 2 ! ! !   ! ! Γ ν + 2n + 32 !  !!  |R2 | < ! 1 ! (2z)2n+1 (2n + 1)! Γ ν − 2n − 2 ! For −

3 π < arg z < π, ν real, and n + 2 2

1 2

1 2

  ν 3 n≥ − 2 4

> |ν|

 1, |θ1 | < |sec (arg z)|, 3 π For − π < arg z < , ν real, and n + 2 2

  ν 1 n> − 2 4

WA 231

WA 231

WA 245

if Im z ≥ 0 if Im z ≤ 0

> |ν|

 1, |θ2 | < |sec (arg z)|,

921

WA 246

if Im z ≤ 0 if Im z ≥ 0

For ν real,

WA 245

 1 if Re z ≥ 0 |θ3 | < |cosec (arg z)|, if Re z < 0 Re θ3 ≥ 0,

if Re z ≥ 0

For ν and z real and n ≥ ν − 12 ,

WA 231

0 ≤ |θ3 | ≤ 1 In particular, it follows from 8.451 7 and 8.451 8 that for real positive values of z and ν, the errors |R1 | and |R2 | are less than the absolute value of the first discarded term. For values of |arg z| close to π, the series 8.451 1 and 8.451 2 may not be suitable for calculations. In particular, the error for |arg z| > π can be greater in absolute value than the first discarded term. “Approximation by tangents” 8.45211 For large values of the index (where the argument is less than the index). Suppose that x > 0 and ν > 0. Let us set ν/x = cosh α. Then, for large values of ν, the following expansions are valid:

 ν exp (ν tanh α − να) 5 1 1 √ ∼ coth α − coth3 α 1. Jν 1+ cosh α ν 8 24 2νπ tanh α

 9 1 231 1155 2 4 + 2 coth α − coth α + coth6 α + . . . ν 128 576 3456 WA 269(3)

922

Bessel Functions and Functions Associated with Them

 2.

Yν

8.453



ν 5 1 1 exp (να − ν tanh α) 3 π coth α − coth α 1− ∼− cosh α ν 8 24 ν tanh α

 2 9 1 231 1155 coth2 α − coth4 α + coth6 α + . . . + 2 ν 128 576 3456 WA 270(5)

8.453 For large values of the index (where the argument is greater than the index). Suppose that x > 0 and ν > 0. Let us set ν/x = cos β. Then, for large values of ν, the following expansions are valid: ⎧⎡ ⎛  ⎨ 2 1 ⎝ 9 cot2 β + 231 cot4 β ⎣1 − 1. J ν (ν sec β) ∼ νπ tan β ⎩ ν 2 128 576 ⎞ ⎤  1155 π + cot6 β ⎠ + . . .⎦ cos ν tan β − νβ − 3456 4 ⎤

   1 1 5 π ⎦ cot β + cot3 β − . . . sin ν tan β − νβ − + ν 8 24 4

2.

3.

4.

⎧⎡ ⎛  ⎨ 2 1 ⎝ 9 cot2 β + 231 cot4 β ⎣1 − Y ν (ν sec β) ∼ νπ tan β ⎩ ν 2 128 576 ⎞ ⎤  1155 π + cot6 β ⎠ + . . .⎦ sin ν tan β − νβ − 3456 4 ⎤ ⎤

 5 π ⎦ 1 1 cot β + cot3 β − . . .⎦ cos ν tan β − νβ − − ν 8 24 4  

exp νi (tan β − β) − π4 i 5 i 1 (1) 3 π cot β + cot β H ν (ν sec β) ∼ 1− ν 8 24 ν tan β

 2 9 1 231 1155 cot2 β + cot4 β + cot6 β + . . . − 2 ν 128 576 3456  

exp −νi (tan β − β) + π4 i 5 i 1 (2) 3  cot β + cot β H ν (ν sec β) ∼ 1+ π ν 8 24 2 ν tan β

 9 1 231 1155 cot2 β + cot4 β + cot6 β + . . . − 2 ν 128 576 3456

WA 271(4)

WA 271(5)

WA 271(1)

WA 271(2) 1

Formulas 8.453 are not valid when |x − ν| is of a size comparable to x 3 . For arbitrary small (and also large) values of |x − ν|, we may use the following formulas:

8.457

8.454

Asymptotic expansions of Bessel functions

923

Suppose that x > 0 and ν > 0, we set  w=

x2 − 1; ν2

Then,

 

 π 1 w3 (1) ν 3 +ν w− − arctan w i H 1 w +O 3 6 3 3 |ν|



 

  1 w3 π w (2) ν 3 (2) − arctan w i H 1 w +O 2. H ν (x) = √ exp − − ν w − MO 34 3 6 3 3 |ν| 3 ! !

√ !1! 1 The absolute value of the error O is then less than 24 2!! !!. |ν| ν 8.455 For x real and ν a natural number (ν = n), if n 1, the following approximations are valid:    3 2 2(n − x) 1 [2(n − x)] √ 1.7 K 13 J n (x) ≈ π 3x 3 x [n > x] (see also 8.433) 1.

w H (1) ν (x) = √ exp 3

1 2 ≈ e 3 πi 2





2 (n − x) (1) H1 3 3x



3

i [2(n − x)] 2 √ 3 x

WA 276(1)



[n > x] 1 ≈√ 3



2(x − n) 3x



%

J 13

3

{2(x − n)} 2 √ 3 x

&

% + J − 13

3

{2(x − n)} 2 √ 3 x

&

MO 34

(see also 8.441 3, 8.441 4) WA 276(2)

 2.

Y n (x) ≈

2(x − n) 3x

 J − 13

%

3

{2(x − n)} 2 √ 3 x

&

% − J 13

3

{2(x − n)} 2 √ 3 x

&

[x > n] An estimate of the error in formulas 8.455has not yet  been achieved. ∞ 1  Γ ν + k + (2k − 1)!! 2 2   8.45611 J 2ν (z) + Y 2ν (z) ≈ [|arg z| < π] πz 2k z 2k k! Γ ν − k + 12

WA 276(3)

(see also 8.479 1)

k=0

WA 250(5)

8.457

2 J 2ν (x) + J 2ν+1 (x) ≈ πx

[x  |ν|]

WA 223

924

Bessel Functions and Functions Associated with Them

8.461

8.46 Bessel functions of order equal to an integer plus one-half The function J ν (z) 8.461 1.11

 J n+ 12 (z) =

⎧ ⎪

n2  π  (−1)k (n + 2k)! 2 ⎨  sin z − n (2z)−2k πz ⎪ 2 (2k)!(n − 2k)! ⎩ k=0

⎫ n−1 ⎪

 2  ⎬  k (−1) (n + 2k + 1)! π −(2k+1) (2z) + cos z − n ⎪ 2 (2k + 1)!(n − 2k − 1)! ⎭ k=0

[n + 1 is a natural number]  2.

J −n− 12 (z) =

(cf. 8.451 1)

KU 59(6), WA 66(2)

⎧ ⎪

n2   π  (−1)k (n + 2k)! 2 ⎨ cos z + n πz ⎪ 2 (2k)!(n − 2k)!(2z)2k ⎩ k=0

⎫ n−1 ⎪

 2  ⎬ k (−1) (n + 2k + 1)! π − sin z + n 2 (2k + 1)!(n − 2k − 1)!(2z)2k+1 ⎪ ⎭ 

k=0

[n + 1 is a natural number] 8.462 1.

2.

1 J n+ 12 (z) = √ 2πz

 eiz

n  i−n+k−1 (n + k)! k=0

1 J −n− 12 (z) = √ 2πz



k!(n − k)!(2z)k

+ e−iz

(cf. 8.451 1)

n  (−i)−n+k−1 (n + k)! k=0



k!(n − k)!(2z)k [n + 1 is a natural number] 

eiz

KU 58(7), WA 67(5)

KU 59(6), WA 66(1)

n n   in+k (n + k)! (−i)n+k (n + k)! −iz + e k!(n − k)!(2z)k k!(n − k)!(2z)k k=0 k=0 [n + 1 is a natural number]

KU 59(7), WA 67(4)

8.463 1.

 n n+ 12

J n+ 12 (z) = (−1) z

 2. 8.464 1. 2.

J −n− 12 (z) = z

n+ 12



sin z 2 dn π (z dz)n z  n cos z d

2 π (z dz)n

Special cases:  2 sin z J 12 (z) = πz  2 cos z J − 12 (z) = πz

z

KU 58(4)

KU 58(5)

DW DW

8.469

Bessel functions of order equal to an integer plus one-half

925



3. 4. 5.8 6.

2 sin z − cos z πz z   2 cos z − sin z − J − 32 (z) = πz z  

3 2 3 J 52 (z) = − 1 sin z − cos z πz z2 z 



3 2 3 sin z + J − 52 (z) = − 1 cos z πz z z2 J 32 (z) =

DW

DW DW

DW

The function Y n+ 12 (z) 8.465 1.

Y n+ 12 (z) = (−1)n−1 J −n− 12 (z)

JA

2.

Y −n− 12 (z) = (−1)n J n+ 12 (z)

JA

(1,2)

The functions H n+ 1 (z), I n+ 12 (z), K n+ 12 (z) 2

8.466 1.

 (1) H n− 1 (z) 2

=

1 2 −n iz  (n + k − 1)! i e (−1)k πz k!(n − k − 1)! (2iz)k n−1

k=0

(cf. 8.451 3)



2.

n−1 

1 2 n −iz (n + k − 1)! i e (cf. 8.451 4) πz k!(n − k − 1)! (2iz)k k=0 % & n n   (−1)k (n + k)! (n + k)! 1 z n+1 −z e ± (−1) e I ±(n+ 1 ) (z) = √ 2 k!(n − k)!(2z)k k!(n − k)!(2z)k 2πz (2)

H n− 1 (z) = 2

8.467

k=0

 8.468 K n+ 12 (z) = 8.469 1. 2. 3. 4.

k=0

(cf. 8.451 5) π −z e 2z

n  k=0

(n + k)! k!(n − k)!(2z)k

(cf. 8.451 6)

KU 60a KU 60

Special cases: 

2 Y 12 (z) = − cos z πz  2 Y − 12 (z) = sin z πz  π −z e K ± 12 (z) = 2z  2 eiz (1) H 1 (z) = 2 πz i

WA 95(13)

MO 27

926

Bessel Functions and Functions Associated with Them

8.471



5. 6. 7.

2 e−iz 2 πz −i  2 iz (1) e H − 1 (z) = 2 πz  2 −iz (2) e H − 1 (z) = 2 πz (2)

H 1 (z) =

MO 27 MO 27 MO 27

8.47–8.48 Functional relations 8.4718 1.

Recursion formulas: z Z ν−1 (z) + z Z ν+1 (z) = 2ν Z ν (z)

KU 56(13), WA 56(1), WA 79(1), WA 88(3)

d Z ν (z) KU 56(12), WA 56(2), WA 79(2), We 88(4) dz Sonin and Nielsen, in their construction of the theory of Bessel functions, defined Bessel functions as analytic functions of z that satisfy the recursion relations 8.471. Z denotes J, N , H (1) , H (2) or any linear combination of these functions, the coefficients of which are independent of z and ν. 8.472 Consequences of the recursion formulas for Z defined as above: d 1. z Z ν (z) + ν Z ν (z) = z Z ν−1 (z) KU 56(11), WA 56(3), WA 79(3), WA 88(5) dz d 2. z Z ν (z) − ν Z ν (z) = −z Z ν+1 (z) KU 56(10), WA 56(4), WA 79(4), WA 88(6) dz

m d 3. (z ν Z ν (z)) = z ν−m Z ν−m (z) KU 56(8), WA 57(5), WA 89(9) z dz

m  −ν  d z Z ν (z) = (−1)m z −ν−m Z ν+m (z) 4. WA 89(10), Ku 55(5), WA 57(6) z dz 2.

Z ν−1 (z) − Z ν+1 (z) = 2

5.

Z −n (z) = (−1)n Z n (z)

8.473 1. 2. 3. 4. 5. 6.

[n is a natural number]

(cf. 8.404)

Special cases: 2 J 2 (z) = J 1 (z) − J 0 (z) z 2 Y 2 (z) = Y 1 (z) − Y 0 (z) z 2 (1,2) (1,2) (1,2) H 2 (z) = H 1 (z) − H 0 (z) z d J 0 (z) = − J 1 (z) dz d Y 0 (z) = − Y 1 (z) dz d (1,2) (1,2) H (z) = − H 1 (z) dz 0

8.4748 Each of the pairs of functions J ν (z) and J −ν (z) (for ν = 0, ±1, ±2,. . . ), J ν (z) and Y ν (z), and (2) H (1) ν (z) and H ν (z), which are solutions of equation 8.401, and also the pair I ν (z) and K ν (z) is a pair of linearly independent functions. The Wronskians of these pairs are, respectively,

8.478

Functional relations

927

2 4i 1 2 sin νπ, , − , − KU 52(10, 11, 12), WA 90(1, 4) πz πz πz z 8.4756 The functions J ν (z), and Y ν (z), H (1,2) (z), I ν (z), K ν (z), with the exception of J n (z) and I n (z), ν for n an integer are non-single-valued : z = 0 is a branch point for these functions. The branches of these functions that lie on opposite sides of the cut (−∞, 0) are connected by the relations 8.476   WA 90(1) 1. J ν emπi z = emνπi J ν (z)  mπi  −mνπi 2. Yν e z =e Y ν (z) + 2i sin mνπ cot νπ J ν (z) WA 90(3)  mπi  3. Y −ν e z = e−mνπi Y −ν (z) + 2i sin mνπ cosec νπ J ν (z) WA 90(4)  mπi  mνπi 4. Iν e z =e I ν (z) WA 95(17)   sin mνπ I ν (z) 5. K ν emπi z = e−mνπi K ν (z) − iπ [ν not an integer] WA 95(18) sin νπ  mπi  −νπi sin mνπ J ν (z) e 6. H (1) z = e−mνπi H (1) ν ν (z) − 2e sin νπ sin mνπ (2) sin(1 − m)νπ (1) H ν (z) − e−νπi H ν (z) = sin νπ sin νπ −

WA 95(5)

9.

 mπi  νπi sin mνπ J ν (z) e H (2) z = e−mνπi H (2) ν ν (z) + 2e sin νπ sin mνπ (1) sin(1 + m)νπ (2) H ν (z) + eνπi H ν (z) = sin νπ sin νπ [m an integer]   (2) eiπ z = − H −ν (z) = −e−iπν H (2) H (1) ν ν (z)   (1) e−iπ z = − H −ν (z) = −eiπν H (1) H (2) ν ν (z)

10.8

H (2) ν (z) = H ν (z)

7.

8.

(1)

WA 90(6) MO 26 MO 26 MO 26

8.477 1.

J ν (z) Y ν+1 (z) − J ν+1 (z) Y ν (z) = −

2.

I ν (z) K ν+1 (z) + I ν+1 (z) K ν (z) =

2 πz

WA 91(12)

1 z

WA 95(20)

See also 3.863. • For a connection with Legendre functions, see 8.722. • For a connection with the polynomials C λn (t), see 8.936 4. • For a connection with a confluent hypergeometric function, see 9.235. 8.478

For ν > 0 and x > 0, the product   x J 2ν (x) + Y 2ν (x) ,

considered as a function of x, decreases monotonically, if ν >

1 2

and increases monotonically if 0 < ν < 12 . MO 35

928

8.479 1.11 2.

Bessel Functions and Functions Associated with Them

 1 1 π 2 J ν (x) + Y 2ν (x) ≥ > 2 2 x −ν |J n (nz)| ≤ 1 √



x2

x≥ν≥

1 2

8.479



MO 35

! & %! ! z exp √1 − z 2 ! ! ! √ ! < 1, n a natural number ! ! 1 + 1 − z2 !

MO 35

Relations between Bessel functions of the first, second, and third kinds Y −ν (z) − Y ν (z) cos νπ = H (1) ν (z) − i Y ν (z) sin νπ  1 (2) = H (2) H (1) ν (z) + i Y ν (z) = ν (z) + H ν (z) 2

8.481 J ν (z) =

(cf. 8.403 1, 8.405)

WA 89(1), JA

(cf. 8.403 1, 8.405)

WA 89(3), JA

J ν (z) cos νπ − J −ν (z) = i J ν (z) − i H (1) 8.482 Y ν (z) = ν (z) sin νπ  i (1) H (2) = i H (2) ν (z) − i J ν (z) = ν (z) − H ν (z) 2 8.483 1. 2.

Y −ν (z) − e−νπi Y ν (z) J −ν (z) − e−νπi J ν (z) = = J ν (z) + i Y ν (z) i sin νπ sin νπ Y −ν (z) − eνπi Y ν (z) eνπi J ν (z) − J −ν (z) = = J ν (z) − i Y ν (z) H (2) (z) = ν i sin νπ sin νπ (cf. 8.405) H (1) ν (z) =

WA 89(5)

WA 89(6)

8.484 (1)

1.

H −ν (z) = eνπi H (1) ν (z)

2.

(2) H −ν (z)

8.4857

WA 89(7)

= e−νπi H (2) ν (z)

K ν (z) =

π I −ν (z) − I ν (z) 2 sin νπ

WA 89(7)

[ν not an integer]

(see also 8.407) WA 92(6)

8.486

Recursion formulas for the functions I ν (z) and K ν (z) and their consequences:

1.

z I ν−1 (z) − z I ν+1 (z) = 2ν I ν (z)

2.

I ν−1 (z) + I ν+1 (z) = 2

3.

z

4. 5.

d I ν (z) dz

d I ν (z) + ν I ν (z) = z I ν−1 (z) dz d z I ν (z) − ν I ν (z) = z I ν+1 (z) dz

m d {z ν I ν (z)} = z ν−m I ν−m (z) z dz

WA 93(1) WA 93(2) WA 93(3) WA 93(4) WA 93(5)

8.486(1)

Functional relations

6. 7.

d z dz

m



 z −ν I ν (z) = z −ν−m I ν+m (z)

I −n (z) = ln (z)

10. 11.

K ν−1 (z) + K ν+1 (z) = −2

12.

z

9.

13. 14. 15.

K −ν (z) = K ν (z)

17.

K 2 (z) =

19.

WA 93(8)

WA 93(7) WA 93(1)

d K ν (z) dz

WA 93(2)

d K ν (z) + ν K ν (z) = −z K ν−1 (z) dz d z K ν (z) − ν K ν (z) = −z K ν+1 (z) dz

m d {z ν K ν (z)} = (−1)m z ν−m K ν−m (z) z dz

m  −ν  d z K ν (z) = (−1)m z −ν−m K ν+m (z) z dz

16.

18.

WA 93(6)

[n a natural number]

2 I 2 (z) = − l1 (z) + I 0 (z) z d I 0 (z) = I 1 (z) dz z K ν−1 (z) − z K ν+1 (z) = −2ν K ν (z)

8.

929

WA 93(3) WA 93(4) WA 93(5) WA 93(6) WA 93(8)

2 K 1 (z) + K 0 (z) z

d K 0 (z) = − K 1 (z) dz ∞ , ∂ J ν (z) + z (z/2)ν+1  (z/2)n J n+1 (z) = ln − ψ(ν + 1) J ν (z) + ∂ν 2 Γ(ν + 1) n=0 n!(ν + n + 1)2

WA 93(7) LUKE 360

8.486(1)7 1.

2.

3.

4.

Differentiation with respect to order  ν+2k ∞ 1 1 ∂ J ν (z) ψ(ν + k + 1) k = J ν (z) ln z − z (−1) ∂ν 2 2 k! Γ(ν + k + 1) k=0  ν = n or n + 12 ,  −ν+2k ∞ 1 1 ψ(−ν + k + 1) ∂ J −ν (z) k = − J −ν (z) ln z + z (−1) ∂ν 2 2 k! Γ(−ν + k + 1) k=0  ν = n or n + 12 ,

 n integer

MS 3.1.3

 n integer

MS 3.1.3

∂ J ν (z) ∂ J −ν (z) ∂ Y ν (z) = cot πν − cosec πν − π cosec πν Y ν (z) ∂ν ∂ν ∂ν   ν = n or n + 12 , n integer MS 3.1.3

ν+2k  ∞   1 1 ψ(ν + k + 1) ∂ I ν (z) = I ν (z) ln z − z ν = n or n + 12 , n integer ∂ν 2 2 k! Γ(ν + k + 1) k=0

MS 3.1.3

930

5.

6.

7.

Bessel Functions and Functions Associated with Them

8.486(1)

  ∂ I −ν (z) ∂ I ν (z) ∂ K ν (z) 1 = −π cot πν K ν (z) + π cosec πν − ∂ν 2 ∂ν ∂ν   ν = n or n + 12 , n integer MS 3.1.3     n−1  1 z k−n J k (z) ∂ J ν (z) 1 n n 1 2 [n = 0, 1, . . .] = π (±1) Y n (z)±(±1) n! MS 3.2.3 ∂ν 2 2 k!(n − k) ν=±n k=0     n−1  1 z k−n Y k (z) ∂ Y ν (z) 1 n n 1 2 [n = 0, 1, . . .] = − π (±1) J n (z) ± (±1) n! ∂ν 2 2 k!(n − k) ν=±n k=0

MS 3.2.3

 8.

∂ I ν (z) ∂ν

 ν=±n

  n−1  (−1)k 1 z k−n I k (z) n+1 n1 2 = (−1) K n (z) ± (−1) n! 2 k!(n − k)

[n = 0, 1, . . .]

k=0

MS 3.2.3

 9.

∂ K ν (z) ∂ν 

10.

(−1)n

 11.11

12. 13. 14. 15. 16. 17. 18. 19.

 ν=±n

1 = ± n! 2

 ∂ I ν (z) ∂ν ν=n

∂ K ν (z) ∂ν

 = ν=n

1 n! 2

  n−1  1 z k−n K k (z) 2

k=0

[n = 0, 1, . . .]

k!(n − k)

k−n 1 z I k (z) n−1 (−1) 1  2 = − K n (z) + n! 2 k!(n − k)

k

k=0

n−1  k=0

 1 k−n K k (z) 2z k!(n − k)

[n = 0, 1, . . .]

AS 9.6.44

[n = 0, 1, . . .]

AS 9.6.45

Special cases   ∂ J ν (z) = 12 π Y 0 (z) ∂ν ν=0   ∂ Y ν (z) = − 12 π J 0 (z) ∂ν ν=0   ∂ I ν (z) = − K 0 (z) ∂ν ν=0   ∂ K ν (z) =0 ∂ν ν=0    −1/2 ∂ J ν (x) = 12 πx [sin x Ci(3x) − cos x Si(2x)] ∂ν ν= 12    −1/2 ∂ J ν (x) = 12 πx [cos x Ci(2x) + sin x Si(2x)] ∂ν ν=− 12    −1/2 ∂ Y ν (x) = 12 πx {cos x Ci(2x) + sin x [Si(2x) − π]} ∂ν ν= 12    −1/2 ∂ Y ν (x) = − 12 πx {sin x Ci(2x) − cos x [Si(2x) − π]} ∂ν ν=− 1 2

MS 3.2.3

MS 3.2.3 MS 3.2.3 MS 3.2.3 MS 3.2.3 MS 3.3.3

MS 3.3.3

MS 3.3.3

MS 3.3.3

8.491

Differential equations leading to Bessel functions

 20.  21. 8.487 1. 2. 3.

∂ I ν (x) ∂ν



∂ K ν (x) ∂ν

931

  = (2πx)−1/2 ex Ei(−2x) ∓ e−x Ei(2x)

ν=± 12



=∓ ν=± 12

MS 3.3.3

 π 12 ex Ei(−2x) 2x

MS 3.3.3

Continuity with respect to the order∗ : lim Y ν (z) = Y n (z)

[n an integer]

WA 76

lim H (1,2) (z) = H (1,2) (z) ν n

[n an integer]

WA 183

lim K ν (z) = K n (z)

[n an integer]

WA 92

ν→n ν→n ν→n

8.49 Differential equations leading to Bessel functions See also 8.401 8.491

1 d ν2 (zu ) + β 2 − 2 u = 0 1. z dz z   νγ 2    1 d  γ−1 2 (zu ) + βγz 2. − u=0 z dz z    2 α 2 − ν 2 γ 2 1 − 2α  u + βγz γ−1 + 3. u + u=0 z z2     4ν 2 γ 2 − 1  γ−1 2 4. u + βγz − u=0 4z 2

4ν 2 − 1 5. u + β 2 − u=0 4z 2

1 − 2α  α2 − ν 2  2 u + β + 6. u + u=0 z z2 7.



8. 9.

u + bz u = 0

1−ν  1u u + =0 z 4z

u +

11.

u + β 2 γ 2 z 2β−2 u = 0

∗ The

2

1 ν u + u + 4 z 2 − 2 u = 0 z z

1 1 ν2 u + u + 1− u=0 z 4z z

10.

JA

u = Z ν (βz γ )

JA

u = z α Z ν (βz γ )

JA

u=

√ z Z ν (βz γ )

JA

u=

√ z Z ν (βz)

JA

u = z α Z ν (βz)

JA

√ 1 u = z Z m+2

m



u = Z ν (βz)



 √ 2 b m+2 z 2 m+2 JA 111(5)

  u = Z ν z2 u = Zν

√  z

√  z  β 1 γz u = z 1/2 Z 2β ν

u = z2 Zν

WA 111(6) WA 111(7) WA 111(9)a WA 110(3)

continuity of the functions J ν (z) and I ν (z) follows directly from the series representations of these functions.

932

12.

8.492 1. 2. 8.493 1. 2. 8.494 1. 2. 3. 4.10 5. 6.

Bessel Functions and Functions Associated with Them

  z 2 u + (2α − 2βν + 1)zu + β 2 γ 2 z 2β + α(α − 2βν) u = 0

  u = z βν−α Z ν γz β

  u + e2z − ν 2 u = 0 u +

e

u = Z ν (ez )  u = z Z ν e1/z

−ν u=0 z4

2/z

2

2

1 ν tan z − 2 tan z u − + u=0 z z2 z

2

1 ν cot z   + 2 cot z u − u + − u=0 z z2 z u +



1 ν2 u + u − 1 + 2 u = 0 z z    ν 2 1 1   + u + u − u=0 z z 2z

1 1 − ν2 u = 0 u + u + 2 z 4

2ν + 1 2ν + 1  − k u − ku = 0 u + z 2z 1−ν  1u u − =0 z 4z u u ± √ = 0 z

u ± zu = 0

8.

ν(ν + 1) u − c + z2

9.

u −

10.

u − c2 z 2ν−2 u = 0

1.

2

u = cosec z Z ν (z)

JA

u = Z ν (iz) = C1 I ν (z) + C2 K ν (z)

JA

 √  u = Z ν 2i z

JA

 3 √ z Z 23 43 z 4 , √

 z Z 13

2 32 3z

u=0

2ν  u − c2 u = 0 z

√ −z ze 2 Z ν



,

u=

JA JA WA 111(8)

u=

 √ 3 z Z 23 43 iz 4

WA 111(10)

u=

 √ 3 z Z 13 23 iz 2

WA 111(10)







1  ν2 u + u + i− 2 u=0 z z 

WA 112(22)

JA

iz 2

ikz −ν 12 kx u=z e Zν 2 √  ν u = z2 Zν i z

u=

8.495

WA 112(22)

u = sec z Z ν (z)

u=

u +



WA 112(21)



u= 7.

8.492

√ z Z ν+ 12 (icz) 1

u = z ν+ 2 Z ν+ 12 (icz)  c √ u = z Z 2ν1 i z ν ν  √ u = Zν z i

WA 108(1) WA 109(3, 4) WA 109(5, 6)

JA

8.511

2. 3. 4.

Series of Bessel functions



u +

d2 dz 2 2

2.

3.

4.

2

1 ν i  ∓ 2i u − ± u=0 z z2 z

u = e±iz Z ν (z)

1 u + u + seiα u = 0 z

1  iα u + se + 2 u = 0 4z

8.496 1.



d dz 2 d2 dz 2

z4

d2 u dz 2

z

16 5

z 12

u=

− z2u = 0

d u dz 2 d2 u dz 2

u=

JA

√ i sze 2 α

JA

√ √ i z Z0 sze 2 α

JA

u = Z0



2

933

 √  1  √  Z 2 2 z + Z 2 2i z z WA 122(7)



 3   3 −7/10 5 5 5 u=z Z 56 3 z + Z 56 3 iz 5

8

− z5u = 0

WA 122(8)

− z6u = 0

    u = z −4 Z 10 2z −1/2 + Z 10 2iz −1/2

WA 122(9)

4

ν − 4ν 2 d4 u 2 d3 u 2ν 2 + 1 d2 u 2ν 2 + 1 du + + − + − 1 u = 0, dz 4 z dz 3 z 2 dz 2 z 3 dz z4 u = A1 J ν (z) + A2 Y ν (z) + A3 I ν (z) + A4 K ν (z), where A1 , A2 , A3 , A4 are constants

MO 29

8.51–8.52 Series of Bessel functions 8.511 1.

Generating functions for Bessel functions:

∞ ∞   k  1 1 t + (−t)−k J k (z) = exp J k (z)tk t− z = J 0 (z) + 2 t k=1

2.

k=−∞

 ∞   ∞

  1 k m exp t − t J k (z) t J m (z) z= t m=−∞

[|z| < |t|]

KU 119(12) WA 40

k=−∞

3.

exp (±iz sin ϕ) = J 0 (z) + 2 

4.

∞ 

J 2k (z) cos 2kϕ ± 2i

k=1

∞ 

∞ π  exp (iz cos ϕ) = (2k + 1)ik J k+ 12 (z) P k (cos ϕ) 2z k=0 ∞  = ik J k (z)eikϕ k=−∞

= J 0 (z) + 2

∞  k=1

ik J k (z) cos kϕ

J 2k+1 (z) sin(2k + 1)ϕ

KU 120(13)

k=0

WA 401(1) MO 27

MO 27

934

Bessel Functions and Functions Associated with Them

 5.

The series

i iz cos 2ϕ e π #



√

2z cos ϕ

e

8.512

∞

 1 1 dt = J 0 (z) + e 4 kπi J k (z) cos kϕ 2 2

−it2

−∞

MO 28

k=1

J k (z)

8.512 1.

J 0 (z) + 2

∞ 

J 2k (z) = 1

WA 44

k=1

2.

∞  (n + 2k)(n + k − 1)!

k!

k=0

3.



∞  (4k + 1)(2k − 1)!!

J 2k+ 12 (z) =

2k k!

k=0

J n+2k (z) =

8.513 Notation: In formulas 8.513

(p) Qk

=

 z n

[n = 1, 2, . . .]

2 2z π

k−1

 2  (−1)m



m=0

1.

∞ 

(2k)2p J 2k (z) =

k=1

2.

∞ 

p 

(2p)

Q2k z 2k

k m

(k − 2m)p

2k k! [p = 1, 2, 3, . . .]

WA 46(1)

[p = 0, 1, 2, 3, . . .]

WA 46(2)

k=0

(2k + 1)2p+1 J 2k+1 (z) =

k=0

3.

WA 45

p 

(2p+1)

Q2k+1 z 2k+1

k=0

In particular: ∞   1 z + z3 (2k + 1)3 J 2k+1 (z) = 2

WA 47(4)

k=0

4.

∞ 

(2k)2 J 2k (z) =

k=1

5.

∞ 

1 2 z 2

WA 47(4)

2k(2k + 1)(2k + 2) J 2k+1 (z) =

k=1

1 3 z 2

WA 47(4)

8.514 1.

∞  k=0

2.

sin z 2

WH

(−1)k J 2k (z) = cos z

WH

(−1)k J 2k+1 (z) =

J 0 (z) + 2

∞  k=1

3.

∞  k=1

(−1)k+1 (2k)2 J 2k (z) =

z sin z 2

WA 32(9)

8.518

4.

Series of Bessel functions ∞ 

(−1)k (2k + 1)2 J 2k+1 (z) =

k=0

5.

J 0 (z) + 2

∞ 

z cos z 2

935

WA 32(10)

J 2k (z) cos 2kθ = cos (z sin θ)

KU 120(14), WA 32

k=1

6.

∞ 

J 2k+1 (z) sin(2k + 1)θ =

k=0

7.

∞  k=0

8.515 1.

1 J 2k+1 (x) = 2



sin (z sin θ) 2

KU 120(15), WA 32

x

J 0 (t) dt

[x is real]

WA 638

0



k

ν ∞  z (−1)k tk 2z + t J ν+k (z) = J ν (z + t) k! 2z z+t

AD (9140)

k=0

2.

∞ 

  J 2k− 12 x2 = S (x)

MO 127a

  J 2k+ 12 x2 = C (x)

MO 127a

k=1

3.

∞  k=0

8.516

∞  (2n + 2k)(2n + k − 1)!

k!

k=0

The series

#

Ak J k (kx) and

#

1.

k=1

2.

∞ 

z J k (kz) = 2(1 − z) (−1)k J k (kz) = −

k=1

3.

∞ 

J 2k (2kz) =

k=1

WA 47

Ak J k (kx)

8.517 ∞ 

2n

J 2n+2k (2z sin θ) = (z sin θ)

z 2(1 + z)

z2 2 (1 − z 2 )

! & %! ! z exp √1 − z 2 ! ! ! √ !<1 ! ! 1 + 1 − z2 ! ! & %! ! z exp √1 − z 2 ! ! ! √ ! !<1 ! 1 + 1 − z2 ! ! & %! ! z exp √1 − z 2 ! ! ! √ !<1 ! ! 1 + 1 − z2 !

WA 615(1)

WA 622(1)

MO 58

8.518 1.11

∞  J  (kx) k

k=1

2.11

∞  k=1

3.

∞  k=1

k (−1)k−1

1 x + 2 4

[0 ≤ x < 1]

MO 58

J k (kx) 1 x = − k 2 4

[0 ≤ x < 1]

MO 58

1 2(1 − x)2

[0 ≤ x < 1]

MO 58

=

k J k (kx) =

936

4.

Bessel Functions and Functions Associated with Them ∞ 

(−1)k−1 J k (kx)k =

k=1

The series

#

1 2(1 + x)2

8.519

[0 ≤ x < 1]

MO 58

Ak J 0 (kx)

8.519 If, on the interval [0 ≤ x ≤ π], a function f (x) possesses a continuous derivative with respect to x that is of bounded variation, then ∞

1.

f (x) =

a0  + ak J 0 (kx) 2

[0 < x < π]

k=1

where

2. 3. 8.521 1.

 π/2  2 π a0 = 2f (0) + du uf  (u sin ϕ) dϕ π 0 0  π/2  2 π du uf  (u sin ϕ) cos nu dϕ an = π 0 0 Examples: ∞  k=1

2.

∞ 

n  1 1 1 √ J 0 (kx) = − + + 2 2 − 4m2 π 2 2 x x m=1

(−1)k+1 J 0 (kx) =

k=1

3.

∞  k=1

4.

WH

∞ 

1 2

1 J 0 {(2k − 1)x} (2k − 1)2

[2nπ < x < 2(n + 1)π]

[0 < x < π] π2 |x| − 8 2 π π2  2 x + x − π 2 − − π arccos = 8 2 x

MO 59

KU 124(12)

[−π < x < π]

KU 124

[π < x < 2π]

MO 59

  e−kz J 0 k x2 + y 2

k=1

  ∞ 1 1  1 1  = − + − r 2 (2kiπ + z)2 + x2 + y 2 (2kiπ − z)2 + x2 + y 2 k=1 ∞  z 1 1  1 B2k r2k−1 P 2k−1 = − + r 2 (2k)! r

[0 < r < 2π] MO 59

k=1

 where r = x2 + y 2 + z 2 and where the radical indicates the square root with a positive real part. In formula 8.521 4, the first equation holds when x and y are real and Re z > 0; the second equation holds when x, y, and z are all real.

8.523

Series of Bessel functions

The series

#

#

Ak Z 0 (kx) sin kx and

937

Ak Z 0 (kx) cos kx

8.522 1.

2.

3.

∞ 

 1 1 1  1   J 0 (kx) cos kxt = − + + √ + 2 2 − (2πl + tx)2 2 − (2πl − tx)2 2 x 1 − t x x k=1 l=1 l=1 m

n

 n    ∞ m ∞   1 1 1 1 1  − + J 0 (kx) sin kxt = − 2π l l 2πl (2πl + tx)2 − x2 k=1 l=1 l=1 l=m+1  ∞  1 1  − − 2 2 2πl (2πl − tx) − x l=n+1 m  n 1 x 1 1 1 + Y 0 (kx) cos kxt = − C + ln + π 4π 2π l l k=1 l=1 l=1  ∞  1 1  − − 2 2 2πl (2πl + tx) − x l=m+1⎧ ⎫ ∞ ⎨  1 1 ⎬ < − − ⎩ 2πl ⎭ 2 l=n+1 (2πl − tx) − x2

MO 59

MO 59

∞ 

MO 60

In formulas 8.522, x > 0, 0 ≤ t < 1, 2πm < x(1 − t) < 2(m + 1)π, 2nπ < x(1 + t) < 2(n + 1)π, m + 1 and n + 1 are natural numbers. 8.523 ∞ m n   1 1 1  < < 1. (−1)k J 0 (kx) cos kxt = − + + 2 2 2 k=1 l=1 l=1 x2 − [(2l − 1)π + tx] x2 − [(2l − 1)π − tx]

2.

∞  k=1

k

(−1) J 0 (kx) sin kxt

 n 1

m  1



⎧ ∞ ⎨ 

1 < 2 l=1 l=m+1 [(2l − 1)π + tx] − x2 ⎫ 1 1 ⎬ < − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2 1 2π

l l=1 ⎧ ∞ ⎨ 

−

l

+

⎩

MO 60

⎫ 1 ⎬ − 2lπ ⎭

MO 60

938

3.

Bessel Functions and Functions Associated with Them ∞ 

k

(−1) Y 0 (kx) cos kxt

k=1

8.524

m  n 1 x 1 1 1 + − C + ln + π 4π 2π l l l=1 l=1 ⎧ ⎫ ∞ ⎨  1 ⎬ 1 < − − ⎩ 2lπ ⎭ 2 l=m+1 [(2l − 1)π + tx] − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬ < − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2 MO 60

In formulas 8.523, x > 0, 0 ≤ t <1, (2m − 1)π < x(1 − t) < (2m + 1)π, (2n − 1)π < x(1 + t) < (2n + 1)π, m and n are natural numbers. 8.524 ∞ n   1 1  J 0 (kx) cos kxt = − + MO 60 1. 2 2 x − (2lπ − tx)2 k=1 l=m+1   ∞ m ∞    1 1 1   2. J 0 (kx) sin kxt + − 2lπ (2lπ − tx)2 − x2 l=1 (2lπ + tx)2 − x2 k=1 l=0   ∞ n  1 1 1 1  − + − 2 2 2lπ 2π l (2lπ − tx) − x l=n+1

l=1

MO 60

3.6

∞  k=1

Y 0 (kx) cos kxt

x 1 1 1  C + ln − + 2 − x2 π 4π 2π (2πl − tx) l=0   ∞  1 1  − − 2lπ (2lπ + tx)2 − x2 l=1   ∞  1 1  − − 2lπ (2lπ − tx)2 − x2 m 

−

n  l=1

1 l

l=n+1

MO 61

In formulas 8.524, x > 0, t > 1, 2mπ < x(t − 1) < 2(m + 1)π, 2nπ < x(t + 1) < 2(n + 1)π, m + 1 and n + 1 are natural numbers. 8.525 ∞ n   1 1 < 1. (−1)k J 0 (kx) cos kxt = − + MO 61 2 2 2 k=1 l=m+1 x − [(2l − 1)π − tx]

8.526

2.

Series of Bessel functions ∞  k=1

(−1)k J 0 (kx) sin kxt =

939

m 

n 1 1 1 < + 2π l 2 l=1 l=1 [(2l − 1)π − tx] − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬ < + − ⎩ 2lπ ⎭ 2 l=1 [(2l − 1)π + tx] − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬ < − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2

MO 61

3.

∞  k=1

n x 1 1 1 C + ln + π 4π 2π l l=1 m  1 < − 2 l=1 [(2l − 1)π − tx] − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬ < − − ⎩ 2lπ ⎭ 2 l=1 [(2l − 1)π + tx] − x2 ⎧ ⎫ ∞ ⎨  1 ⎬ 1 < − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2

(−1)k Y 0 (kx) cos kxt = −

MO 61

In formulas 8.525, x > 0, t > 1, (2m − 1)π < x(t − 1) < (2m + 1)π, (2n − 1)π < x(t + 1) < (2n + 1)π, m and n are natural numbers. 8.526   ∞ ∞  x π 1 π 1 1  C + ln + √ 1. K 0 (kx) cos kxt = + − 2 4π 2 2lπ 2x 1 + t2 x2 + (2lπ − tx)2 k=1 l=1   ∞ π 1 1  + − 2 2 2 2lπ x + (2lπ + tx) l=1

2.

⎧ ⎫ ∞ ∞ ⎨    1 x π 1 1 ⎬ < C + ln + (−1)k K 0 (kx) cos kxt = − ⎩ 2 4π 2 2lπ ⎭ 2 k=1 l=1 x2 + [(2l − 1)π − xt] ⎧ ⎫ ∞ 1 1 ⎬ π ⎨ < − + ⎩ 2 2lπ ⎭ 2 l=1 x2 + [(2l − 1)π + xt] [x > 0, t real] (see also 8.66)

MO 61

MO 62

940

Bessel Functions and Functions Associated with Them

8.530

8.53 Expansion in products of Bessel functions “Summation theorems”  8.530 Suppose that r > 0,  > 0, ϕ > 0, and R = r2 + 2 − 2r cos ϕ; that is, suppose that r, , and R are the sides of a triangle such that the angle between the sides r and  is equal to ϕ. Suppose also that 
0<ψ<

π , 2

e2iψ =

r − e−iϕ r − eiϕ

When these conditions are satisfied, we have the “summation theorem” for Bessel functions: 1.

eiνψ Z ν (mR) =

∞ 

J k (m) Z ν+k (mr)eikϕ

[m is an arbitrary complex number]

k=−∞

WA 394(6)

For Z ν = J ν and ν an integer, the restriction 
J 0 (mR) = J 0 (m) J 0 (mr) + 2

∞ 

J k (m) J k (mr) cos kϕ

MO 31

WA 391(1)

k=1

2.

(1,2)

H0

(1,2)

(mR) = J 0 (m) H 0

(mr) + 2

∞ 

(1,2)

J k (m) H k

(mr) cos kϕ

MO 31

k=1

3.

∞ z z  +2 cos 2kα J 0 (z sin α) = J 20 Jk2 2 2 k=1 

∞ 2π  1 (2k − 1)!! = J 2k+ 12 (z) P 2k (cos α) 2k + z 2 2k k!

MO 31

k=0

8.532

The term “summation theorem” is also applied to the formula ∞

1.

 Z ν (mR) J ν+k (m) Z ν+k (mr) ν = 2ν m−ν Γ(ν) (ν + k) C k (cos ϕ) ν R ν rν k=0

[ν = −1, −2, −3, . . . ; the conditions on r, , R, ϕ, and m are the same as in formula 8.530; for Z ν = J ν and ν an integer, formula 8.532 1 is valid for arbitrary r, , and ϕ]. WA 398(4) 8.533 Special cases: ∞

1.

πi  eimR (1) = √ (2k + 1) J k+ 12 (m) H k+ 1 (mr) P k (cos ϕ) 2 R 2 r

MO 31

k=0

∞

2.

πi  e−imR (2) =− √ (2k + 1) J k+ 12 (m) H k+ 1 (mr) P k (cos ϕ) 2 R 2 r k=0

MO 31

8.543

The zeros of Bessel functions

A degenerate addition theorem (r → ∞):  ∞ π  k im cos ϕ = i (2k + 1) J k+ 12 (m) P k (cos ϕ) e 2m k=0 ∞  = 2ν Γ(ν) (ν + k)ik (m)−ν J ν+k (m) C νk (cos ϕ)

941

8.534

WA 401(1)

[ν = 0, −1, −2, . . .]

WA 401(2)

k=0

8.535

The term “product theorem” is also applied to the formula

k ∞ + ,  1 − λ2 1 2 |1 − λ| < 1 Z ν+k (z) z Z ν (λz) = λν k! 2 k=0 For Z ν = J ν , it is valid for all values of λ and z. 8.536 ∞  (2n + 2k)(2n + k − 1)! 2 (2n)!  z 2n Jn+k (z) = [n > 0] 1. 2 k! (n!) 2

MO 32

WA 47(1)

k=0

∞  k Γ(n + k) 2 (2n)!  z 2n Jk (z) = 2 Γ(k − n + 1) (n!) 2 k=n

2.

2

3.

J02 (z) + 2

∞ 

[n > 0]

Jk2 (z) = 1

WA 47(2)

WA 41(3)

k=1

8.537 1.

∞ 

Z ν−k (t) J k (z) = Z ν (z + t)

[|z| < |t|]

WA 158(2)

k=−∞

2.

∞ 

J k (z) J n−k (z) = J n (2z)

WA 41

k=−∞

8.538 1.

∞ 

(−1)k J −ν+k (t) J k (z) = J −ν (z + t)

[|z| < |t|]

WA 159

Z ν+k (t) J k (z) = Z ν (t − z)

[|z| < |t|]

WA 159(5)

k=−∞

2.

∞  k=−∞

8.54 The zeros of Bessel functions 8.541 For arbitrary real ν, the function J ν (z) has infinitely many real zeros. For ν > −1, all its zeros WA 526, 530 are real. A Bessel function Z ν (z) has no multiple zeros except possibly the coordinate origin. WA 528 8.542 All zeros of the function Y 0 (z) with positive real parts are real. WA 531 8.543 If −(2s + 2) < ν < −(2s + 1), where s is a natural number or 0, then J ν (z) has exactly 4s + 2 complex roots, two of which are purely imaginary. If −(2s + 1) < ν < −2s, where s is a natural number, WA 532 then the function J ν (z) has exactly 4s complex zeros, none of which are purely imaginary.

942

Bessel Functions and Functions Associated with Them

8.544

8.544 If xν and xν are, respectively, the smallest positive zeros of the functions J ν (z) and J ν (z) for ν > 0, then xν > ν and xν > ν. Suppose also that yν is the smallest positive zero of the function Y ν (z). WA 534, 536 Then, xν < yν < xν . Suppose that zν,m (for m = 1, 2, 3, . . . ) are the zeros of the function z −ν J ν (z), numbered in order of the absolute value of their real parts. Here, we assume that ν = −1, −2, −3, . . . . Then, for arbitrary z  z ν

∞ z2 2 WA 550 1− 2 . J ν (z) = Γ(ν + 1) m=1 zν,m 8.5458 The number of zeros of the function z −ν J ν (z) that occur between the imaginary axis and the line on which   WA 497 Re z = m + 12 Re ν + 14 π, is exactly m. 8.546 For ν ≥0, the number of zeros of the function K ν (z) that occur in the region Re z <0, |arg z| < π WA 562 is equal to the even number closest to ν − 12 . 8.547 Large zeros of the functions J ν (z) cos α − Y ν (z) sin α, where ν and α are real numbers, are given by the asymptotic expansion

1 1 4ν 2 − 1   xν,m ∼ m + ν − π − α −  2 4 8 m + 12 ν − 14 π − α  2   KU 109(24), WA 558 4ν − 1 28ν 2 − 31 −  3 − . . .  384 m + 12 ν − 14 π − α 8.548 In particular, large zeros of the function J 0 (z) are given by the expansion 1 31 π 3779 x0,m ∼ (4m − 1) + − + − ... KU 109(25), WA 556 4 2π(4m − 1) 6π 3 (4m − 1)3 15π 5 (4m − 1)5 This series is suitable for calculating all (except the smallest x01 ) zeros of the function J 0 (z) correctly to at least five digits. 8.549 To calculate the roots xν,m of the function J ν (z) of smallest absolute value, we may use the identity ∞  1 429ν 5 + 7640ν 4 + 53752ν 3 + 185430ν 2 + 311387ν + 202738 . KU 112(27)a, WA 554 = 16 16 x 2 (ν + 1)8 (ν + 2)4 (ν + 3)2 (ν + 4)2 (ν + 5)(ν + 6) (ν + 7) (ν + 8) m=1 ν,m

8.55 Struve functions 8.550 1.

2. 8.551 1.

Definitions:  z 2m+ν+1 2    (−1) Hν (z) = 3 3 Γ m + 2 Γ ν +m+ 2 m=0  z 2m+ν+1 ∞  iπ   −iν π 2  2 H   Lν (z) = −ie ν ze 2 = 3 3 Γ m + 2 Γ ν +m+ 2 m=0 ∞ 

m

WA 358(2)

WA 360(11)

Integral representations:  ν  ν  1  π/2  ν− 12 2 z2 2 z2 2ν 2     Hν (z) = √ 1 − t sin zt dt = sin (z cos ϕ) (sin ϕ) dϕ √ π Γ ν + 12 0 π Γ ν + 12 0   WA 358(1) Re ν > − 12

8.554

Struve functions

 ν  π/2 2 z2 2ν   Lν (z) = √ sinh (z cos ϕ) (sin ϕ) dϕ π Γ ν + 12 0

2.

8.552

1.6

2.6

943



Re ν > − 12



WA 360(11)

Special cases:   z n−2m−1  n−1

 2 Γ m + 1 2 1 2   Hn (z) = [n = 1, 2, . . .] − En (z) π m=0 Γ n + 12 − m   z −n+2m+1  n−1

 2 Γ n − m − 1 2 1 2  H−n (z) = (−1)n+1 − E−n (z) π m=0 Γ m + 32

EH II 40(66), WA 337(1)

[n = 1, 2, . . .]

EH II 40(67), WA 337(2)

[n = 0, 1, . . .]

EH II 39(64)

[n = 0, 1, . . .]

EH II 39(65)

L−(n+ 1 ) (z) = I n+ 12 (z) [n = 0, 1, . . .] 2 √ 2 H 12 (z) = √ (1 − cos z) πz

1/2   z 1/2 2 2 cos z H 32 (z) = 1+ 2 − sin z + 2π z πz z

EH II 39(65)

 z  1 n 1  Γ m+ 2 2 Hn+ 12 (z) = Y n+ 12 (z) + π m=0 Γ(n + 1 − m)

−2m+n− 12

3.

n

4.

H−(n+ 1 ) (z) = (−1) J n+ 12 (z) 2

5. 6. 7. 8.553 1.

Functional relations:   Hν zeimπ = eiπ(ν+1)m Hν (z)

[m = 1, 2, 3, . . .]

d ν [z Hν (z)] = z ν Hν−1 (z) dz   −1  d  −ν z Hν (z) = 2−ν π −1/2 Γ ν + 32 − z −ν Hν+1 (z) dz  z ν   −1 Γ ν + 32 Hν−1 (z) + Hν+1 (z) = 2νz −1 Hν (z) + π −1/2 2  ν   −1  −1/2 z Γ ν + 32 Hν−1 (z) − Hν+1 (z) = 2 Hν (z) − π 2

2. 3. 4. 5.

Asymptotic representations:

 ξ −2m+ν−1  1 p−1 Γ m + 2  1  2 ν−2p−1   Hν (ξ) = Y ν (ξ) + + O |ξ| π m=0 Γ ν + 12 − m [|arg ξ| < π] For the asymptotic representation of Y ν (ξ), see 8.451 2.

EH II 39, WA 364(3)

WA 364(3)

WA 362(5) WA 358 WA 359 WA 359(5) WA 359(6)

8.554

EH II 39(63), WA 363(2)

944

8.555

Bessel Functions and Functions Associated with Them

8.555

The differential equation for Struve functions: 2 





z y + zy + z − ν 2

2



 ν+1 1 4 z2   √ y= π Γ ν + 12

WA 359(10)

8.56 Thomson functions and their generalizations berν (z), beiν (z), herν (z), heiν (z), kerν (z), keiν (z) 8.561 1. 2. 8.562 1. 2.

 3 berν (z) + i beiν (z) = J ν ze 4 πi  3 berν (z) − i beiν (z) = J ν ze− 4 πi .

WA 96(6) WA 96(6)

 3 herν (z) + i heiν (z) = H ν(1) ze 4 πi  3 herν (z) − i heiν (z) = H ν(1) ze− 4 πi

(see also 8.567)

WA 96(7)

(see also 8.567)

WA 96(7)

8.563 1. 2.

ber0 (z) ≡ ber(z); bei0 (z) ≡ bei(z) π π ker(z) ≡ − hei0 (z); kei(z) ≡ hei0 (z) 2 2

WA 96(8) WA 96(8)

For integral representations, see 6.251, 6.536, 6.537, 6.772 4, 6.777. Series representation 8.564 1.

ber(z) =

∞  (−1)k z 4k

2.

bei(z) =

WA 96(3)

2

k=0 ∞ 

24k [(2k)!]

(−1)k z 4k+2 2

24k+2 [(2k + 1)!]

∞ 2k   π z 4k 1 2 ker(z) = ln − C ber(z) + bei(z) + (−1)k 2 4k z 4 2 [(2k)!] m=1 m

WA 96(4)

k=0

3.

WA 96(9)a, DW

k=1

4.

kei(z) =

∞ 2k+1   1 π z 4k+2 2 ln − C bei(z) − ber(z) + (−1)k 2 z 4 24k+2 [(2k + 1)!] m=1 m

WA 96(10)a, DW

k=0

8.565 ber2ν (z) + bei2ν (z) =

∞  k=0

2ν+4k

(z/2) k! Γ(ν + k + 1) Γ(ν + 2k + 1)

WA 163(6)

8.570

Lommel functions

945

Asymptotic representation 8.566 1. 2. 3. 4.

eα(z) ber(z) = √ cos β(z) 2πz eα(z) sin β(z) bei(z) = √ 2πz  π α(−z) e ker(z) = cos β(−z) 2z  π α(−z) e kei(z) = sin β(−z) 2z

+ π, |arg z| < 4 + π, |arg z| < 4   5 |arg z| < π 4   5 |arg z| < π , 4

WA 227(1)

WA 227(1)

WA 227(2)

WA 227(2)

where z 1 25 13 √ − α(z) ∼ √ + √ − − ..., 4 3 128z 2 8z 2 384z 2 1 π 1 z 25 √ + ... β(z) ∼ √ − − √ − − 2 8 8z 2 16z 384z 3 2 2 8.567 1. 2.

Functional relations

 √ ker(z) + i kei(z) = K 0 z i  √ ker(z) − i kei(z) = K 0 z −i

(see 8.562)

WA 96(5), DW

(see 8.562)

WA 96(5), DW

For integrals of Thomson’s functions, see 6.87.

8.57 Lommel functions 8.570 1.

2.11

Definitions of the Lommel functions s μ,ν (z) and S μ,ν (z): (−1)m z μ+1+2m , + 2 [(μ + 1)2 − ν 2 ] [(μ + 3)2 − ν 2 ] . . . (μ + 2m + 1) − ν 2  2m+2  1    ∞  (−1)m z2 Γ 2 μ − 12 ν + 12 Γ 12 μ + 12 ν + 12 μ−1     =z Γ 12 μ − 12 ν + m + 32 Γ 12 μ + 12 ν + m + 32 m=0 [μ ± ν is not a negative odd integer]

s μ,ν (z) =

    S μ,ν (z) = s μ,ν (z) + 2μ−1 Γ 12 μ − 12 ν + 12 Γ 12 μ + 12 ν + 12     cos 12 (μ − ν)π J −ν (z) − cos 12 (μ + ν)π J ν (z) × sin νπ     = s μ,ν (z) + 2μ−1 Γ 12 μ − 12 ν + 12 Γ 12 μ + 12 ν + 12       × sin 12 (μ − ν)π J ν (z) − cos 12 (μ − ν)π Y ν (z)

EH II 40(69), WA 377(2)

EH II 40(71), WA 379(2)

EH II 41(71), WA 379(3)

946

Bessel Functions and Functions Associated with Them

8.571

Integral representations    z  z π μ μ 8.571 s μ,ν (z) = WA 378(9) Y ν (z) z J ν (z) dz − J ν (z) z Y ν (z) dz 2 0 0 8.572 s μ,ν (z)

 π/2  z 12 (1+ν+μ) 1 1 1 1 (1+ν−μ) ν+μ + μ− ν Γ J 12 (1+μ−ν) (z sin θ) (sin θ) 2 (cos θ) dθ = 2μ 2 2 2 2 0 EH II 42(86) [Re(ν + μ + 1) > 0] 8.573 Special cases: 1. 2. 3. 4. 5. 6. 8.574 1. 2.

S 1,2n (z) = zO2n (z) z O 2n+1 (z) S 0,2n+1 (z) = 2n + 1 1 S 2n (z) S −1,2n (z) = 4n 1 S 0,2n+1 (z) = S 2n+1 (z) 2

1 √ ν−1 S ν,ν (z) = Γ ν + π2 Hν (z) 2   √ S ν,ν (z) = [Hν (z) − Y ν (z)] 2ν−1 π Γ ν + 12

WA 382(1) WA 382(1) WA 382(2) WA 382(2) EH II 42(84) EH II 42(84)

Connections with other special functions: 1 sin(νπ) [s 0,ν (z) − ν s −1,ν (z)] π 1 Eν (z) = − [(1 + cos νπ) s 0,ν (z) + ν (1 − cos νπ) s −1,ν (z)] π Jν (z) =

EH II 41(82) EH II 42(83)

A connection with a hypergeometric function 3.

s μ,ν (z) =



z μ+1 μ − ν + 3 μ + ν + 3 z2 , ; − 1; 1F 2 (μ − ν + 1)(μ + ν + 1) 2 2 4 EH II 40(69), WA 378(10)

8.575 1. 2.8 3. 4. 5.8

Functional relations:   s μ+2,ν (z) = z μ+1 − (μ + 1)2 − ν 2 s μ,ν (z) ν s μ,ν (z) = (μ + ν − 1) s μ−1,ν−1 (z) s μ,ν (z) + z ν s μ,ν (z) = (μ − ν − 1) s μ−1,ν+1 (z) s μ,ν (z) − z  ν s μ,ν (z) = (μ + ν − 1) s μ−1,ν−1 (z) − (μ − ν − 1) s μ−1,ν+1 (z) 2 z 2 s μ,ν (z) = (μ + ν − 1) s μ−1,ν−1 (z) + (μ − ν − 1) s μ−1,ν+1 (z)

In formulas 8.575 1–5, s μ,ν (z) can be replaced with S μ,ν (z).

EH II 41(73), WA 380(1) EH II 41(74), WA 380(2) EH II 41(75), WA 380(3) EH II 41(76), WA 380(4) EH II 41(77), WA 380(5)

8.579

Lommel functions

947

8.576 Asymptotic expansion of S μ,ν (z). In the case in which μ±ν is not a positive odd integer, S μ,ν (z) has the following asymptotic expansion:



 ∞  1−μ+ν 1−μ−ν z −2m μ−1 m S μ,ν (z) ∼ z (−1) 2 2 m m 2 m=0

[|z| → ∞, |arg z| < π] WA 347, 352 The series terminates and is equal to S μ,ν (z) when μ ± ν is a positive odd integer. 8.577 Lommel functions satisfy the following differential equation:   z 2 w + zw + z 2 − ν 2 w = z μ+1 WA 377(1), EH II 40(68) 8.578 Lommel functions of two variables U ν (w, z) and V ν (w, z): Definition 1.

U ν (w, z) =

∞ 

(−1)m

 w ν+2m z

m=0

J ν+2m (z)

  1 z2 + νπ + U −ν+2 (w, z) w+ 2 w Particular values:

2.

V ν (w, z) = cos

3.

U 0 (z, z) = V 0 (z, z) =

4.

U 1 (z, z) = − V 1 (z, z) = sin z   n−1  (−1)n m U 2n (z, z) = (−1) ε2m J 2m (z) cos z − 2 m=0

5.

EH II 42(87), WA 591(5)

EH II 42(88), WA 591(6)

{J 0 (z) + cos z}

1 2

WA 591(9)

1 2

WA 591(10)

[n ≥ 1] ,

6.



(−1)n U 2n+1 (z, z) = 2

sin z −

n−1 

 (−1) ε2m+1 J 2m+1 (z)

m=0

V n (w, z) = (−1)n U n  w 1/2 2

8.

U ν (w, 0) =

9.

V −ν+2 (w, 0) =

8.579 1. 2. 3.

z2 ,z w

S ν− 32 , 12

Γ(ν − 1)  w 1/2 2

Γ(ν − 1)

WA 591(11)

m

[n ≥ 0] , 7.

εm

 2, m > 0, = 1, m = 0

εm

 2, m > 0, = 1, m = 0

WA 591(12)

w

WA 593(9)

2

S ν− 32 , 12

w 2

Functional relations:  z 2 ∂ U ν (w, z) = U ν−1 (w, z) + U ν+1 (w, z) 2 ∂w w  ∂ z 2 V ν (w, z) = V ν+1 (w, z) + 2 V ν−1 (w, z) ∂w w The function U ν (w, z) is a particular solution of the differential equation

WA 593(10)

WA 593(2) WA 593(4)

948

Bessel Functions and Functions Associated with Them

 w ν−2 ∂2U z2U 1 ∂U + − = J ν (z) ∂z 2 z ∂z w2 z 4.

The function V ν (w, z) is a particular solution of the differential equation  w −ν z2V ∂2V 1 ∂V + − = J −ν+2 (z) ∂z 2 z ∂z w2 z

8.580

WA 592(2)

WA 592(3)

8.58 Anger and Weber functions Jν (z) and Eν (z) 8.580 1.

Definitions: The Anger function Jν (z): Jν (z) =

2.

1 π

The Weber function Eν (z): Eν (z) =

8.581

1 π



π

cos (νθ − z sin θ) dθ

WA 336(1), EH II 35(32)

sin (νθ − z sin θ) dθ

WA 336(2), EH II 35(32)

0



π

0

Series representations:  2n ∞ (−1)n z2 νπ      Jν (z) = cos 2 n=0 Γ n + 1 + 12 ν Γ n + 1 − 12 ν  2n+1 ∞ (−1)n z2 νπ      + sin 2 n=0 Γ n + 32 + 12 ν Γ n + 32 − 12 ν

1.

EH II 36(36), WA 337(3)

 2n ∞ (−1)n z2 νπ      Eν (z) = sin 2 n=0 Γ n + 1 + 12 ν Γ n + 1 − 12 ν  2n+1 ∞ (−1)n z2 νπ      − cos 2 n=0 Γ n + 32 + 12 ν Γ n + 32 − 12 ν

2.

EH II 36(37), WA 338(4)

8.582

Functional relations:

1.6

2 Jν (z) = Jν−1 (z) − Jν+1 (z)

2.6

2 Eν (z)

3.

6

4.6

EH II 36(40), WA 340(2)

= Eν−1 (z) − Eν+1 (z)

Jν−1 (z) + Jν+1 (z) = 2νz

−1

EH II 36(41), WA 340(6) −1

Jν (z) − 2(πz)

sin(νπ)

Eν−1 (z) + Eν+1 (z) = 2νz −1 Eν (z) − 2(πz)−1 (1 − cos νπ)

EH II 36(42), WA 340(1) EH II 36(43), WA 340(5)

8.591

8.583 1.6

Neumann’s and Schl¨ afli’s polynomials

Asymptotic expansions: ⎡     p−1 1+ν Γ n + 1−ν sin νπ ⎣  n 2n Γ n+ 2 2  1−ν  z −2n Jν (z) = J ν (z) + (−1) 2 1+ν πz Γ Γ 2 2 n=0 ⎤     p−1 1 1    Γ n + 1 + ν Γ n + 1 − ν −2p 2     2 z −2n−1 + ν O |z|−2p−1 ⎦ −ν + O |z| (−1)n 22n 1 1 Γ 1 + ν Γ 1 − ν 2 2 n=0 [|arg z| < π]

2.

949

EH II 37(47), WA 344(1)

Eν (z) = − Y ν (z)

& % p−1     1+ν 1−ν  1 + cos(νπ)  −2p n 2n Γ n + 2  Γ n +  2 −2n − (−1) 2 + O |z| z πz Γ 1−ν Γ 1+ν 2 2 n=0 % p−1 &     1 1  ν −2n−1 ν (1 − cos νπ)  −2p−1 n 2n Γ n +1 + 2 ν  Γ n + 1 − 2  (−1) 2 + O |z| − z zπ Γ 1 + 12 ν Γ 1 − 12 ν n=0 WA344(2), EH II 37(48)

For the asymptotic expansion of J ν (z) and Y ν (z), see 8.451. 8.584 The Anger and Weber functions satisfy the differential equation

ν2  −1  y + z y + 1 − 2 y = f (ν, z), z z−ν where f (ν, z) = sin νπ for Jν (z) πz 2 1 and f (ν, z) = − 2 [z + ν + (z − ν) cos νπ] for Eν (z) πz

WA 341(9), EH II 37(44) EH II 37(45), WA 341(10)

8.59 Neumann’s and Schl¨ afli’s polynomials: O n (z) and S n (z) 8.590

Definition of Neumann’s polynomials

1.

n2  1  n(n − m − 1)!  z 2m−n−1 O n (z) = 4 m=0 m! 2

[n ≥ 1]

WA 299(2), EH II 33(6)

2.

O −n (z) = (−1)n O n (z)

[n ≥ 1]

WA 303(8)

3.

O 0 (z) =

4. 5.

1 z 1 O 1 (z) = 2 z 4 1 O 2 (z) = + 3 z z

WA 299(3), EH II 33(7) EH II 33(7) EH II 33(7)

In general, O n (z) is a polynomial in z −1 of degree n + 1. 8.591 Functional relations: 1.

O 0 (z) = − O 1 (z)

2.

2 O n (z)

= O n−1 (z) − O n+1 (z)

EH II 33(9), WA 301(3)

[n ≥ 1]

EH II 33(10), WA 301(2)

950

3.

4. 5. 8.592

Mathieu Functions

8.592

   π 2 (n − 1) O n+1 (z) + (n + 1) O n−1 (z) − 2z −1 n2 − 1 O n (z) = 2nz −1 sin n 2 [n ≥ 1] EH II 33(11), WA 301(1)  2  2  π nz O n−2 (z) − n − 1 O n (z) = (n − 1)z O n (z) + n sin n EH II 33(12), WA 303(4) 2    π 2 nz O n+1 (z) − n2 − 1 O n (z) = −(n + 1)z O n (z) + n sin n EH II 33(13), WA 303(5)a 2 The generating function: ∞  1 = J 0 (ξ)z −1 + 2 J n (ξ) O n (z) z−ξ n=1

[|ξ| < |z|]

The integral representation: √ √ n  n  ∞ u + u2 + z 2 + u − u2 + z 2 O n (z) = e−u du 2z n+1 0 See also 3.547 6, 8, 3.549 1, 2. 8.594 The inequality

EH II 32(1), WA 298(1)

8.593

−n−1

8.595

8.596 1. 2.

3. 8.597 1.

EH II 32(3), WA 305(1)

|O n (z)| ≤ 2n−1 n!|z| e 4 |z| [n > 1] EH II 33(8), WA 300(8) Neumann’s polynomial O n (z) satisfies the differential equation    π 2 d2 y dy  2 π 2 + z + 1 − n2 y = z cos n z 2 2 + 3z + n sin n EH II 33(14), WA 303(1) dz dz 2 2 Schl¨ afli’s polynomials S n (z). These are the functions that satisfy the formulas 1

2

S 0 (z) = 0

   1 π 2 S n (z) = 2zOn (z) − 2 cos n n 2 n

 2  (n − m − 1)! z 2m−n = m! 2 m=0

EH II 34(18), WA 312(2)

[n ≥ 1]

EH II 34(19), WA 312(3)

[n ≥ 1]

EH II 34(18)

S −n (z) = (−1)n+1 S n (z)

WA 313(6)

Functional relations: S n−1 (z) + S n+1 (z) = 4 O n (z)

WA 313(7)

Other functional relations may be obtained from 8.591 by replacing O n (z) with the expression for S n (z) given by 8.596 2.

8.6 Mathieu Functions 8.60 Mathieu’s equation  d2 y  + a − 2k 2 cos 2z y = 0, 2 dz

k2 = q

MA

(2n)

8.621

(2n+1)

(2n+1)

(2n+2)

Recursion relations for the coefficients A2r , A2r+1 , B2r+1 , B2r+2

951

8.61 Periodic Mathieu functions 8.610 In general, Mathieu’s equation 8.60 does not have periodic solutions. If k is a real number, there exist infinitely many eigenvalues a, not identically equal to zero, corresponding to the periodic solutions y(z) = y(2π + z). If k is nonzero, there are no other linearly independent periodic solutions. Periodic solutions of Mathieu’s equations are called Mathieu’s periodic functions or Mathieu functions of the first kind, or, more simply, Mathieu functions. 8.611 Mathieu’s equation has four series of distinct periodic solutions: 1.

ce2n (z, q) =

∞ 

(2n)

A2r cos 2rz

MA

r=0

2.

ce2n+1 (z, q) =

∞ 

(2n+1)

A2r+1 cos(2r + 1)z

MA

r=0

3.

se2n+1 (z, q) =

∞ 

(2n+1)

B2r+1 sin(2r + 1)z

MA

r=0

4.

se2n+2 (z, q) =

∞ 

(2n+2)

B2r+2 sin(2r + 2)z

MA

r=0

5.

8.612

The coefficients A and B depend on q. The eigenvalues a of the functions ce2n , ce2n+1 , se2n , se2n+1 are denoted by a2n , a2n+1 , b2n , b2n+1 . The solutions of Mathieu’s equation are normalized so that  2π y 2 dx = π

MO 65

0

8.613 1. 2. 3.

1 lim ce0 (x) = √ 2 lim cen (x) = cos nx

q→0

[n = 0]

q→0

lim sen (x) = sin nx

MO 65

q→0

(2n)

(2n+1)

(2n+1)

(2n+2)

8.62 Recursion relations for the coefficients A2r , A2r+1 , B2r+1 , B2r+2 8.621 1. 2. 3.

(2n)

(2n)

− qA2

=0  (2n) (2n) (2n) =0 (a − 4)A2 − q A4 + 2A0   (2n)  (2n) (2n) a − 4r2 A2r − q A2r+2 + A2r−2 = 0 aA0

MA MA

[r ≥ 2]

MA

952

Mathieu Functions

8.622

8.622 1. 2.

(2n+1)

(2n+1)

(a − 1 − q)A1 − qA3 =0    (2n+1) (2n+1) (2n+1) =0 a − (2r + 1)2 A2r+1 − q A2r+3 + A2r−1

MA

[r ≥ 1]

MA

8.623 1. 2.

(2n+1)

(2n+1)

− qB3 =0 (a − 1 + q)B1    (2n+1) (2n+1) (2n+1) =0 a − (2r + 1)2 B2r+1 − q B2r+3 + B2r−1

MA

[r ≥ 1]

MA

8.624 1. 2.11

(2n+2)

(2n+2)

− qB4 =0 (a − 4)B2    (2n+2) (2n+2) (2n+2) =0 a − 4r2 B2r − q B2r+2 + B2r−2

MA

[r ≥ 2]

MA

8.625 We can determine the coefficients A and B from equations 8.612, 8.613 and 8.621-8.624 pro(2n) vided a is known. Suppose, for example, that we need to determine the coefficients A2r for the function ce2n (z, q). From the recursion formulas, we have ! ! ! a −q 0 0 0 . . .!! ! !−2q a − 4 −q 0 0 . . .!! ! ! 0 −q a − 16 −q 0 . . .!! ! ST 1. ! =0 ! 0 0 −q a − 36 −q ! ! ! ! 0 0 0 −q a − 64 ! ! ! .. .. .. . . !! ! . . . . For given q in equation 8.625 1, we may determine the eigenvalues [|A0 | ≤ |A2 | ≤ |A4 | ≤ . . .]

2.

a = A0 , A2 , A4 , . . .

3.

If we now set a = A2n , we can determine the coefficients A2r from the recursion formulas 8.621 up to a proportionality coefficient. This coefficient is determined from the formula ∞ + + ,2  ,2 (2n) (2n) 2 A0 + = 1, MA A2r

(2n)

r=1

which follows from the conditions of normalization.

8.63 Mathieu functions with a purely imaginary argument 8.630 1.11

If, in equation 8.60, we replace z with iz, we arrive at the differential equation d2 y + (−a + 2q cosh 2z) y = 0 dz 2

We can find the solutions of this equation if we replace the argument z with iz in the functions cen (z, q) and sen (z, q). The functions obtained in this way are called associated Mathieu functions of the first kind and are denoted as follows:

8.652

1.

Mathieu functions for negative q

Ce2n (z, q),

Ce2n+1 (z, q),

Se2n+1 (z, q),

953

Se2n+2 (z, q)

8.631 1.

Ce2n (z, q) =

∞ 

(2n)

A2r cosh 2rz

MA

r=0

2.

Ce2n+1 (z, q) =

∞ 

(2n+1)

A2r+1 cosh(2r + 1)z

MA

r=0

3.

Se2n+1 (z, q) =

∞ 

(2n+1)

B2r+1 sinh(2r + 1)z

MA

r=0

4.

Se2n+2 (z, q) =

∞ 

(2n+2)

B2r+2 sinh(2r + 2)z

MA

r=0

8.64 Non-periodic solutions of Mathieu’s equation Along with each periodic solution of equation 8.60, there exists a second non-periodic solution that is linearly independent. The non-periodic solutions are denoted as follows: fe2n (z, q), fe2n+1 (z, q), ge2n+1 (z, q), ge2n+2 (z, q). Analogously, the second solutions of equation 8.630 1 are denoted by Fe2n (z, q), Fe2n+1 (z, q), Ge2n+1 (z, q), Ge2n+2 (z, q).

8.65 Mathieu functions for negative q 8.651

If we replace the argument z in equation 8.60 with±

π

2 d2 y + (a + 2q cos 2z) y = 0. dz 2 This equation has the following solutions: 8.652   1. ce2n (z, −q) = (−1)n ce2n 12 π − z, q   2. ce2n+1 (z, −q) = (−1)n se2n+1 12 π − z, q   3. se2n+1 (z, −q) = (−1)n ce2n+1 12 π − z, q   4. se2n+2 (z, −q) = (−1)n se2n+2 12 π − z, q   5. fe2n (z, −q) = (−1)n+1 fe2n 12 π − z, q   6. fe2n+1 (z, −q) = (−1)n ge2n+1 12 π − z, q   7. ge2n+1 (z, −q) = (−1)n fe2n+1 12 π − z, q   8. ge2n+2 (z, −q) = (−1)n ge2n+2 12 π − z, q

± z , we get the equation MA

MA MA MA MA MA MA MA MA

954

8.653

Mathieu Functions

Analogously, if we replace z with

3. 4. 5. 6.11 7.

11

8.11

π

i + z, q 2   n+1 Ce2n+1 (z, −q) = (−1) i Se2n+1 12 πi + z, q   Se2n+1 (z, −q) = (−1)n+1 i Ce2n+1 12 πi + z, q   Se2n+2 (z, −q) = (−1)n+1 Se2n+2 12 πi + z, q   Fe2n (z, −q) = (−1)n Fe2n 12 πi + z, q   Fe2n+1 (z, −q) = (−1)n+1 i Ge2n+1 12 πi + z, q   Ge2n+1 (z, −q) = (−1)n+1 i Fe2n+1 12 πi + z, q   Ge2n+2 (z, −q) = (−1)n+1 Ge2n+2 12 πi + z, q Ce2n (z, −q) = (−1)n Ce2n

2.

π i + z in equation 8.630 1, we get the equation 2

d2 y − (a + 2q cosh z) y = 0. dz 2

It has the following solutions: 8.654 1.

8.653

MA MA MA MA MA MA MA MA

8.66 Representation of Mathieu functions as series of Bessel functions 8.661 1.

ce2n (z, q) = =

2.

ce2n

π

2,q (2n) A0

ce2n (0, q) (2n) A0

ce2n+1 (z, q) = −



∞  (2n) (−1)r A2r J 2r (2k cos z)

r=0 ∞ 

(2n)

(−1)r A2r I 2r (2k sin z)

r=0

ce2n+1

π

2,q (2n+1) kA1



∞ 

(2n+1)

(−1)r A2r+1 J 2r+1 (2k cos z)

4.

∞  ce2n+1 (0, q) (2n+1) cot z (−1)r (2r + 1)A2r+1 I 2r+1 (2k sin z) kA1 (2n + 1) r=0 π  ∞  se2n+1 2 , q (2n+1) se2n+1 (z, q) = tan z (−1)r (2r + 1)B2r+1 J 2r+1 (2k cos z) (2n+1) kB1 r=0 ∞ se2n+1 (0, q)  (2n+1) = (−1)r B2r+1 I 2r+1 (2k sin z) (2n+1) kB1 r=0 π  ∞   − se2n+2 2 , q (2n+2) se2n+2 (z, q) = tan z (−1)r (2r + 2)B2r+2 J 2r+2 (2k cosz ) (2n+2) k 2 B2 r=0 ∞  se2n+2 (0, q) (2n+2) = cot z (−1)r (2r + 2)B2r+2 I 2r+2 (2k sin z) (2n+2) 2 k B2 r=0

8.662 1.

MA

MA

r=0

=

3.

MA

MA

MA MA

MA MA

∞

fe2n (z, q) = −

     π fe2n (0, q)  (2n) π  (−1)r A2r Im J r keiz Y r ke−iz 2 ce2n 2 , q r=0

MA

8.663

2.

Representation of Mathieu functions

fe2n+1 (z, q) =

955

πk fe2n+1 (0, q)   2 ce2n+1 π2 , q ∞           (2n+1) (−1)r A2r+1 Im J r keiz Y r+1 ke−iz + J r+1 keiz Y r ke−iz × r=0

MA

3.

πk ge2n+1 (0, q)   2 se2n+1 π2 , q ∞           (2n+1) (−1)r B2r+1 Re J r keiz Y r+1 ke−iz − J r+1 keiz Y r ke−iz ×

ge2n+1 (z, q) = −

r=0

MA

4.

πk 2 ge2n+2 (0, q)   2 se2n+2 12 π, q ∞           (−1)r Re J k keiz Y r+2 ke−iz − J r+2 keiz Y r ke−iz ×

ge2n+2 (z, q) = −

r=0

MA

The expansions of the functions Fen and Gen as series of the functions Y ν are denoted, respectively, by Feyn and Geyn , and the expansions of these functions as series of the functions K ν are denoted, respectively, by Fekn and Gekn . 8.663 ∞ ce2n (0, q)  (2n) 1. Fey2n (z, q) = A2r Y 2r (2k sinh z) (2n) A0 r=0 k 2 = q [|sinh z| > 1, Re z > 0] MA   ∞ ce2n π2 , q  r (2n) = (−1) A2r Y 2r (2k cosh z) (2n) A0 r=0 [|cosh z| > 1] MA   ∞   ce2n (0, q) ce2n π2 , q  (2n) = (−1)r A2r J r ke−z Y r (kez ) + ,2 (2n) r=0 A0 MA

956

2.

Mathieu Functions

Fey2n+1 (z, q) =

8.663

∞ ce2n+1 (0, q) coth z  (2n+1) (2r + 1)A2r+1 Y 2r+1 (2k sinh z) , kA1 (2n + 1) r=0

k 2 = q, =−

ce2n+1

π

2,q (2n+1) kA1



∞ 

[|sinh z| > 1,

Re z > 0] MA

r

(−1)

(2n+1) A2r+1

Y 2r+1 (2k cosh z)

r=0

[|cosh z| > 1] ce2n+1 (0, q) ce2n+1 =− ,2 + (2n+1) k A1 ×

∞ 

(2n+1)

(−1)r A2r+1



π

2,q



MA

     J r ke−z Y r+1 (kez ) + J r+1 ke−z Y r (kez )

r=0

MA ∞

3.

Gey2n+1 (z, q) =

se2n+1 (0, q)  (2n+1)

kB1

(2n+1)

B2r+1 Y 2r+1 (2k sinh z)

r=0

[|sinh z| > 1, =

se2n+1

π

2,q

(2n+1)

 tanh z

kB1

∞ 

Re z > 0] MA

(−1)r (2r +

(2n+1) 1)B2r+1

Y 2r+1 (2k cosh z)

r=0

[|cosh z| > 1] se2n+1 (0, q) se2n+1 = ,2 + (2n+1) k B1

π

2,q



∞ 

MA r

(−1)

(2n+1) B2r+1

r=0

      × J r ke−z Y r+1 (kez ) J r+1 ke−z Y r (kez ) MA

8.671

4.

The general theory

Gey2n+2 (z, q) =

se2n+2 (0, q)

coth z

(2n+2)

k 2 B2

957

∞  (2n+2) (2r + 2)B2r+2 Y 2r+2 (2k sinh z) r=0

[|sinh z| > 1, =−

se2n+2

π

2,q (2n+2) 2 k B2



Re z > 0] MA

∞  (2n+2) tanh z (−1)r (2r + 2)B2r+2 Y 2r+2 (2k cosh z) r=0

[|cosh z| > 1] MA

∞ se2n+2 (0, q) se2n+2 π ,q  (2 ) (2n+2) = (−1)r B2r+2 + ,2 (2n+2) r=0 k 2 B2

      × J r ke−z Y r+2 (kez ) − J r+2 ke−z Y r (kez ) MA

8.664 ∞

1.

Fek2n (z, q) =

ce2n (0, q)  (2n) πA0

(2n)

(−1)r A2r K 2r (−2ik sinh z)

r=0

k 2 = q,

[|sinh z| > 1,

Re z > 0] MA

2.

3.

4.

Fek2n+1 (z, q) =

Gek2n+1 (z, q) =

Gek2n+2 (z, q) =

ce2n+1 (0, q) (2n+1)

coth z

πkA1

se2n+1

r=0

k2 = q π

2,q (2n+1) πkB1

se2n+2

∞  (2n+1) (−1)r (2r + 1)A2r+1 K 2r+1 (−2ik sinh z)

π

Re z > 0] MA

 tanh z

∞  (2n+1) (2r + 1)B2r+1 K 2r+1 (−2ik cosh z)

MA

r=0



2,q (2n+2) 2 πk B2

[|sinh z| > 1,

tanh z

∞  (2n+2) (2r + 2)B2r+2 K 2r+2 (−2ik cosh z)

MA

r=0

8.67 The general theory If iμ is not an integer, the general solution of equation 8.60 can be found in the form 8.671 ∞ ∞   1. y = Aeμz c2r e2rzi + Be−μz c2r e−2rzi r=−∞

MA

r=−∞

The coefficients c2r can be determined from the homogeneous system of linear algebraic equations 2.11

c2r + ξ2r (c2r+2 + c2r−2 ) = 0, where

r = . . . , −2, −1, 0, 1, 2, . . . ,

MA

958

Associated Legendre Functions

ξ2r =

3.7

The condition that this ! !· · · ! !· ξ−4 1 ! !· 0 ξ−2 Δ (iμ) = !! 0 0 !· !· 0 0 ! !· · ·

8.700

q 2

(2r − iμ) − a

system be compatible yields an equation that μ must satisfy: ! · · · · · ·!! ξ−4 0 0 0 0 ·!! 1 ξ−2 0 0 0 ·!! =0 1 ξ0 0 0 ·!! ξ0 1 ξ2 0 ·!! 0 ξ2 · · · · · ·!

MA

4.

This equation can also be written in the form √

π a 2 cosh μπ = 1 − 2Δ(0) sin , where Δ(0) is the value that is assumed by the determinant 2 of the preceding article if we set μ = 0 in the expressions for ξ2r .

5.

If the pair (a, q) is such that |cosh μπ| < 1, then μ = iβ, Im β = 0, and the solution 8.671 1 is bounded on the real axis.

6.

If |cosh μπ| > 1, μ may be real or complex, and the solution 8.671 1 will not be bounded on the real axis.

7.

If cosh μπ = ±1, then iμ will be an integer. In this case, one of the solutions will be of period π or 2π (depending on whether n is even or odd). The second solution is non-periodic (see 8.61 and 8.64).

8.7–8.8 Associated Legendre Functions 8.70 Introduction 8.700 1.

An associated Legendre function is a solution of the differential equation     d2 u μ2 du 1 − z2 + ν(ν + 1) − − 2z u = 0, dz 2 dz 1 − z2

in which ν and μ are arbitrary complex constants. This equation is a special case of (Riemann’s) hypergeometric equation (see 9.151). The points +1, −1, ∞ are, in general, its singular points, specifically, its ordinary branch points. We are interested, on the one hand, in solutions of the equation that correspond to real values of the independent variable z that lie in the interval [−1, 1] and, on the other hand, in solutions corresponding to an arbitrary complex number z such that Re z > 1. These are multiple-valued in the z-plane. To separate these functions into single-valued branches, we make a cut along the real axis from −∞ to +1. We are also interested in those solutions of equation 8.700 1 for which ν or μ or both are integers. Of special significance is the case in which μ = 0. 8.701 In connection with this, we shall use the following notations: The letter z will denote an arbitrary complex variable; the letter x will denote a real variable that varies over the interval [−1, +1]. We shall sometimes set x = cos ϕ, where ϕ is a real number. We shall use the symbols P μν (z), Q μν (z) to denote those solutions of equation 8.700 1 that are singlevalued and regular for |z| <1 and, in particular, uniquely determined for z = x.

8.706

Introduction

959

We shall use the symbols P μν (z), Q μν (z) to denote those solutions of equation 8.700 1 that are singlevalued and regular forRe z > 1. When these functions cannot be unrestrictedly extended without violating their single-valuedness, we make a cut along the real axis to the left of the point z = 1. The values of the functions P μν (z) and Q μν (z) on the upper and lower boundaries of that portion of the cuts lying between the points −1 and +1 are denoted, respectively, by P μν (x ± i0) , Q μν (x ± i0) . The letters n and m denote natural numbers or zero. The letters ν and μ denote arbitrary complex numbers unless the contrary is stated. The upper index will be omitted when it is equal to zero. That is, we set P 0ν (z) = P ν (z), Q 0ν (z) = Q ν (z) The linearly independent functions

μ

z+1 2 1 1−z μ 8.702 P ν (z) = F −ν, ν + 1; 1 − μ; Γ(1 − μ) z − 1  2  z+1 = 0, if z is real and greater than 1 and MO 80, WH arg z−1  

 μ2 −ν−μ−1 eμπi Γ(ν + μ + 1) Γ 12  2 ν+μ+2 ν+μ+1 3 1   z , ; ν + ; 8.703 Q μν (z) = − 1 z F 2 2 2 z2 2ν+1 Γ ν + 32   2 [arg z − 1 = 0 when z is real and greater than 1; arg z = 0 when z is real and greater than zero] which are solutions of the differential equation 8.700 1, are called associated Legendre functions (or spherical functions) of the first and second kinds, respectively. They are uniquely defined, respectively, in the intervals |1 − z| < 2 and |z| > 1, with the portion of the real axis that lies between −∞ and +1 excluded. They can be extended by means of hypergeometric series to the entire z-plane where the above-mentioned cut was made. These expressions for P μν (z) and Q μν (z) lose their meaning when 1 − μ and ν + 32 are nonMO 80 positive integers, respectively. When z is a real number lying on the interval [−1, +1], so that (z = x = cos ϕ), we take the following functions as linearly+ independent solutions of the equation: , e 2 μπi P μν (cos ϕ + i0) + e− 2 μπi P μν (cos ϕ − i0) EH I 143(1)

μ2

1+x 1 1−x = F −ν, ν + 1; 1 − μ; EH I 143(6) Γ(1 − μ) 1 − x 2 , + 1 1 8.705 Q μν (x) = 12 e−μπi e− 2 μπi Q μν (x + i0) + e 2 μπi Q μν (x − i0) EH I 143(2)   π Γ (ν + μ + 1) −μ P (x) = (cf. 8.732 5) P μν (x) cos μπ − 2 sin μπ Γ(ν − μ + 1) ν If μ = ±m is an integer, the last equation loses its meaning. In this case, we get the following formulas by passing to the limit: 8.706   m dm m 1 − x2 2 Q (x) (cf. 8.752 1) EH I 149(7) 1. Qm ν (x) = (−1) dxm ν Γ(ν − m + 1) m Q (x) 2.11 Q −m EH I 144(18) ν (x) = Γ(ν + m + 1) ν 8.704 P μν (x) =

1 2

1

1

The functions Q μν (z) are not defined when ν + μ is equal to a negative integer. Therefore, we must exclude the cases when ν + μ = −1, −2, −3, . . . for these formulas. The functions

960

Associated Legendre Functions

8.707

P ±μ Q ±μ P ±μ Q ±μ ν (±z) , ν (±z) , −ν−1 (±z) , −ν−1 (±z) are linearly independent solutions of the differential equation for ν + μ = 0, ±1, ±2, . . . . 8.707 Nonetheless, two linearly independent solutions can always be found. Specifically, for ν ± μ not an integer, the differential equation 8.700 1 has the following solutions: 1.

P ±μ ν (±z) ,

P ±μ −ν−1 (±z) ,

Q ±μ ν (±z) ,

Q ±μ −ν−1 (±z)

respectively, for z = x = cos ϕ, 2.

P ±μ ν (±x) ,

P ±μ −ν−1 (±x) ,

Q ±μ ν (±x) ,

Q ±μ −ν−1 (±x) .

If ν ± μ is not an integer, the solutions 3.

P μν (z),

Q μν (z), respectively, and P μν (x),

Q μν (x)

are linearly independent. If ν ±μ is an integer but μ itself is not an integer, the following functions are linearly independent solutions of equation 8.700 1: 4.

μ P −μ ν (z), respectively, and P ν (x),

P μν (z),

P −μ ν (x).

If μ = ±m, ν = n, or ν = −n − 1, the following functions are linearly independent solutions of equation 8.700 1 for n ≥m: 5.

Pm n (z),

m Qm n (z), respectively, and P n (x),

Qm n (x),

and for n <m, the following functions will be linearly independent solutions 6.

P −m n (z),

−m Qm n (z), respectively, and P n (x),

Qm n (x).

8.71 Integral representations 8.711 1.



P −μ ν (z)

 μ− 12 μ  1 z2 − 1 2 1 − t2   = μ√ dt √   2 π Γ μ + 12 −1 z + t z 2 − 1 μ−ν



Re μ > − 12 ,

|arg (z ± 1)| < π

MO 88



2.

3.

4.



,ν  (ν + 1)(ν + 2) . . . (ν + m) π + z + z 2 − 1 cos ϕ cos mϕ dϕ π 0  π cos mϕ dϕ m ν(ν − 1) . . . (ν − m + 1) = (−1) √  ν+1 π 0 z + z 2 − 1 cos ϕ,  +  π π |arg z| < , arg z + z 2 − 1 cos ϕ = arg z for ϕ = (cf. 8.822 1) SM 483(15), WH 2 2   2  μ2 ∞ √ sinh2μ t dt eμπi Γ(ν + μ + 1)  z Q μν (z) = π μ  − 1 √  ν+μ+1 1 2 Γ μ + 2 Γ(ν − μ + 1) 0 z + z 2 − 1 cosh t [Re (ν ± μ) > −1, |arg (z ± 1)| < π] (cf. 8.822 2) MO 88 Pm ν (z) =

Q μν (z) =

eμπi Γ(ν + 1) Γ(ν − μ + 1)

 0

∞



√

cosh μt dt

ν+1 z + z 2 − 1 cosh t [Re(ν + μ) > −1, ν = −1, −2, −3, . . . ,

|arg (z ± 1)| < π]

WH, MO 88

8.714

Integral representations

 5. 8.712

8.713 1.

961

1 l(l − 1) l! =− (l − 2)! 2l + 1 2l + 1 −1  1  ν − μ eμπi Γ(ν + μ + 1)  2 z −1 2 1 − t2 (z − t)−ν−μ−1 dt Q μν (z) = ν+1 2 Γ(ν + 1) −1 [Re(ν + μ) > −1, Re μ > −1, |arg (z ± 1)| < π] (cf. 8.821 2) 1

P 2l (x) P 0l (x) dx = −

MO 88a, EH I 155(5)a



1      ∞ eμπi Γ μ + 1 π −(ν+ 12 )t μ   t dt cos ν + e dt 2 2 √ z2 − 1 2 Q μν (z) = − cos νπ μ+ 12 μ+ 12 2π 0 (z − cos t) 0 (z + cosh t)   Re μ > − 12 , Re(ν + μ) > −1, |arg (z ± 1)| < π

MO 89



2.

P −μ ν (z)

μ  ∞ z2 − 1 2 sinh2ν+1 t = ν dt 2 Γ(μ − ν) Γ(ν + 1) 0 (z + cosh t)ν+μ+1 [Re z > −1, |arg (z ± 1)| < π, Re(ν + 1) > 0, 

3.

P −μ ν (z)

=

8.714 1.

2.

  μ  ∞   cosh ν + 12 t dt 2 Γ μ + 12 z 2 − 1 2 π Γ(ν + μ + 1) Γ(μ − ν) 0 (z + cosh t)μ+ 12 [Re z > −1, |arg (z ± 1)| < π, Re(ν + μ) > −1,

 P μν

(cos ϕ) =

P −μ ν (cos ϕ) =

2 sinμ ϕ   π Γ 12 − μ

 0

ϕ

  cos ν + 12 t dt (cos t − cos ϕ) μ



μ+ 12

Γ(2μ + 1) sin ϕ 2μ Γ(μ + 1) Γ(ν + μ + 1) Γ(μ − ν)

0 < ϕ < π,

Re(μ − ν) > 0]

MO 89

Re(μ − ν) > 0]

MO 89

Re μ <

1 2



;

(cf. 8.823) MO 87



∞ 0

t

ν+μ

dt μ+ 1

(1 + 2t cos ϕ + t2 ) 2 [Re(ν + μ) > −1, Re(μ − ν) > 0] MO 89

μ

3.

4.

Q μν (cos ϕ) =

P μν (cos ϕ) =

1 Γ(ν + μ + 1) sin ϕ   2μ+1 Γ(ν − μ + 1) Γ μ + 12 &  ∞% sinh2μ t sinh2μ t × dt ν+μ+1 + ν+μ+1 (cos ϕ + i sin ϕ cosh t) (cos ϕ − i sin ϕ cosh t) 0   Re(ν + μ + 1) > 0, Re(ν − μ + 1) > 0, Re μ > − 12

MO 89

i Γ(ν + μ + 1) sinμ ϕ   2μ Γ (ν − μ + 1) Γ μ + 12 % &  ∞ sinh2μ t sinh2μ t × dt ν+μ+1 − ν+μ+1 (cos ϕ + i sin ϕ cosh t) (cos ϕ − i sin ϕ cosh t) 0   Re (ν ± μ + 1) > 0, Re μ > − 12

MO 89

962

Associated Legendre Functions

8.715 1.

P μν

  √  α cosh ν + 12 t dt 2 sinhμ α   (cosh α) = √ 1 π Γ 12 − μ 0 (cosh α − cosh t)μ+ 2 

2.

Q μν (cosh α) =

π eμπi sinhμ α   2 Γ 12 − μ



∞

α



8.715

1 2



α > 0,

Re μ <

α > 0,

Re μ < 12 ,

MO 87

e−(ν+ 2 )t dt 1

(cosh t − cosh α)

μ+ 12



Re(ν + μ) > −1



MO 87

See also 3.277 1, 4, 5, 7, 3.318, 3.516 3, 3.518 1, 2, 3.542 2, 3.663 1, 3.894, 3.988 3, 6.622 3, 6.628 1, 4–7, and also 8.742.

8.72 Asymptotic series for large values of |ν| 8.7216 1.

For real values of μ, |ν|  1, |ν|  |μ|, |arg ν| < π, we have: + ,  cos ν + k + 1  ϕ + π (2k − 1) + μπ  ∞ 1  Γ μ+k+ 2 2 2 4 2  P μν (cos ϕ) = √ Γ(ν + μ + 1)   1 k+ 12 3 π Γ μ − k + (2 sin ϕ) k! Γ ν + k + 2 k=0 2  5π π 3 ν + μ = −1, −2, −3, . . . ; ν = − 2 , − 52 , 72 . . . ; for < ϕ < 6 6 This series also converges for complex values of ν and μ. In the remaining cases, it is an asymptotic expansion for  |ν|  |μ|, |ν|  1, if ν > 0, μ > 0 and 0 < ε ≤ ϕ ≤ π − ε MO 92

2.6

√ Q μν (cos ϕ) = π Γ(ν + μ + 1) +  μπ , π   1 ∞ 1 cos ν + k + (2k − 1) + ϕ −  Γ μ+k+ 2 2 4 2  × (−1)k    1 k+ 12 3 Γ μ − k + (2 sin ϕ) k! Γ ν + k + 2 k=0 2  5 π ν + μ = −1, −2, −3, . . . ; ν = − 32 , − 52 , − 72 , . . . ; for < ϕ < π 6 6 This series also converges for complex values of ν and μ. In the remaining cases, it is an asymptotic expansion for  |ν|  |μ|, |ν|  1, if ν > 0, μ > 0, 0 < ε ≤ ϕ ≤ π − ϕ

3.

+  π cos ν + 12 ϕ − + 2 Γ(ν + μ + 1) μ 4   √ P ν (cos ϕ) = √ π Γ ν + 32 2 sin ϕ

μπ 2

,

  1 1+O ν  0 < ε ≤ ϕ ≤ π − ε,

EH I 147(6), MO 92

1 |ν|  ε For ν > 0, μ > 0 and ν > μ, it follows from formulas 8.721 1 and 8.721 2 that

 MO 92

8.724

Asymptotic series



4. 5.



1 ν+ ν (cos ϕ) = ϕ− 2 

 1 π cos ν + ν −μ Q μν (cos ϕ) = ϕ+ 2ν sin ϕ 2 −μ

P μν

2 cos νπ sin ϕ

963

 1 π μπ + +O √ 4 2 ν3

 1 π μπ + O √ 4 2 ν3  0 < ε ≤ ϕ ≤ π − ε;

1 ν ε

 MO 92

8.722 If ϕ is sufficiently close to 0 or π that νϕ or ν(π −ϕ) is small in comparison with 1, the asymptotic formulas 8.721 become unsuitable. In this case, the following asymptotic representation is applicable for μ ≤ 0, ν 1, and small values of ϕ:  

μ   ϕ η 1 ϕ 2 ϕ J μ+1 (η) − J J P −μ (cos ϕ) = J (η)+sin (η) + (η) +O sin4 1. ν+ cos μ μ+2 μ+3 ν 2 2 2 2η 6 2 where η = (2ν + 1) sin ϕ2 . In particular, it follows that  x 1. lim ν μ P −μ cos = J μ (x) ν ν→∞ ν 8.723 1.

2.

[x ≥ 0, μ ≥ 0]

MO 93

We can see how the functions P μν (z) and Q μν (z) behave for large |ν| and real values of z >   

Γ −ν − 12 e(μ−ν)α sinhμ α 1 3 1 1 F μ + , −μ + ; ν + ; Γ(−ν − μ) (e2α − 1)μ+ 12 2 2 2 1 − e2α  

 Γ ν + 12 1 1 1 e(ν+μ+1)α sinhμ α 1 F μ + , −μ + ; −ν + ; + Γ(ν − μ + 1) (e2α − 1)μ+ 12 2 2 2 1 − e2α   ν = ± 12 , ± 32 , ± 52 , . . . ; a > 12 ln 2

3 √ : 2 2

2μ P μν (cosh α) = √ π

√ Γ(ν + μ + 1) e−(ν+μ+1)α   Q μν (cosh α) = eμπi 2μ π sinhμ α μ+ 12 −2α Γ ν + 32 (1 − e )

1 3 1 1 × F μ + , −μ + ; ν + ; 2 2 2 1 − e2α μ + ν + 1 = 0, −1, −2, . . . ;

α>

1 2

 ln 2

MO 94

MO 94

See also 8.776. 8.724 For the inequalities in 8.776 1–4, ν and μ are arbitrary real numbers satisfying the inequalities ν ≥ 1, ν − μ + 1 > 0, and μ ≥ 0:  ! ±μ ! 1 8 Γ (ν ± μ + 1) !P ν (cos ϕ)! < 1. MO 91-92 νπ Γ(ν + 1) sinμ+ 12 ϕ  ! ±μ ! 1 2π Γ (ν ± μ + 1) ! ! Q ν (cos ϕ) < 2. MO 91-92 μ+ ν Γ(ν + 1) sin 12 ϕ ! ±μ ! 1 !P ν (cos ϕ)! < √2 Γ (ν ± μ + 1) 3. MO 91-92 νπ Γ(ν + 1) sinμ+ 12 ϕ

964

4. 5.8

Associated Legendre Functions

! ±μ ! !Q ν (cos ϕ)! <

8.725



1 π Γ (ν ± μ + 1) μ+ ν Γ(ν + 1) sin 12 ϕ   ! !√ ! ! Γ n + 12 2 ! ! ! ! m 2(m+n) /n sup ! t J m (t)! ! sin ϕ P n (cos ϕ)! < Γ(n − m + 1) 0
MO 91-92

[uniformly 0 ≤ m ≤ n] 8.72510 For fixed z and ν and Re μ → ∞, with z not on the real axis between −∞ and −1 and +∞ and +1, the following are asymptotic expansions in which the upper and lower signs are taken according to whether Im z is greater than or less than 0: ⎡



12 μ z + 1 Γ(ν + μ + 1) Γ(μ − ν) 1 1 μ ⎣ sin μπ F −ν, ν + 1; 1 + μ; + z 1. P ν (z) = π Γ(μ + 1) z−1 2 2 ⎤



μ sin νπ ∓iμπ z − 1 1 1 ⎦ − e F −ν, ν + 1; 1 + μ; − z sin μπ z+1 2 2

2.

Q μν (z) =



AS 8.10.1



1 1 −ν, ν + 1; 1 + μ; + z 2 2 ⎤



μ z − 1 1 1 − e∓iνπ F −ν, ν + 1; 1 + μ; − z ⎦ z+1 2 2 1 iμπ Γ(ν + μ + 1) e 2 Γ(μ + 1)

z+1 z−1

⎡

3.

12 μ

⎡

Q −μ ν (z) =

e−iμπ cosec [π(ν − μ)] ⎣ ∓iνπ e 2π Γ(1 + μ) −

z−1 z+1



− 12 μ F

Γ(μ − ν) ⎣ F



AS 8.10.2



z+1 z−1



− 12 μ F

−ν, ν + 1; 1 + μ;

⎤

1 1 ⎦ −ν, ν + 1; 1 + μ; + z 2 2

1 1 − z 2 2



AS 8.10.3

8.73–8.74 Functional relations 8.731 1.



z2 − 1

 d P μν (z) = (ν − μ + 1) P μν+1 (z) − (ν + 1)z P μν (z) dz (cf. 8.832 1, 8.914 2) EH I 161(10), MO 81

1(1)9 1(2)



 d P μν (z) = νz P μν (z) − (ν + μ) P μν−1 (z) dz   2  d P μν (z) = (ν + μ)(ν − μ + 1) z 2 − 1 P μ−1 z −1 (z) − μz P μν (z) ν dz z2 − 1

AS 8.5.4 AS 8.5.2

8.733

Functional relations

965

(2ν + 1)z P μν (z) = (ν − μ + 1) P μν+1 (z) + (ν + μ) P μν−1 (z) (cf. 8.832 2, 8.914 1)

2.

EH I 160(2), MO 81

P μ+2 (z) + 2(μ + 1) √ ν

3. 3(1)9 4.

z

(z) = (ν − μ)(ν + μ + 1) P μν (z) P μ+1 ν z2 − 1  −1/2   (ν − μ)z P μν (z) − (ν + μ) P μν−1 (z) P μ+1 (z) = z 2 − 1 ν  P μν+1 (z) − P μν−1 (z) = (2ν + 1) z 2 − 1 P μ−1 (z) ν

4(1)9 (ν − μ +

1) P μν+1 (z)

AS 8.5.1 EH I 160(3), MO 82

μ) P μν−1 (z)

4(2)9

= (2ν + 1)z − (ν +  1/2 μ−1 P μν+1 (z) = P μν−1 (z) + (2ν + 1) z 2 − 1 P ν (z)

5.

P μ−ν−1 (z)

=

MO 82, EH I 160(1)

P μν (z)

P μν (z)

AS 334(8.5.3) AS 334(8.5.5)

(cf. 8.820, 8.832 4) EH I 140(1), MO 82

8.732 1.

2.

10



z2 − 1

 d Q μν (z) = (ν − μ + 1) Q μν+1 (z) − (ν + 1)z Q μν (z) dz (cf. 8.832 3)

(2ν + 1)z

Q μν (z)

= (ν − μ +

1) Q μν+1 (z)

+ (ν +

MO 82

μ) Q μν−1 (z) (cf. 8.832 4)

3. 4. 5. 8.733 1.

MO 82

z (z) = (ν − μ)(ν + μ + 1) Q μν (z) Q μ+1 ν z2 − 1  Q μν−1 (z) − Q μν+1 (z) = −(2ν + 1) z 2 − 1 Q μ−1 (z) ν + , 1 π e−μπi Q μν (x ± i0) = e± 2 μπi Q μν (x) ∓ i P μν (x) 2 Q μ+2 (z) + 2(μ + 1) √ ν



1 − x2

 d P μν (x) = P μν (x) − (ν − μ + 1) P μν+1 (x) dx = −νx P μν (x) + (ν + μ) P μν−1 (x)  = − 1 − x2 P μ+1 (x) − μx P μν (x); ν  = (ν − μ + 1)(ν + μ) 1 − x2 P μ−1 (x) + μx P μν (x) ν

MO 82 MO 82a MO 83

(cf. 8.731 1)

MO 82

2.

(2ν +

1)x P μν (x)

= (ν − μ +

1) P μν+1 (x)

+ (ν +

μ) P μν−1 (x) (cf. 8.731 2)

MO 82

MO 82

4.

x P μ+2 (x) + 2(μ + 1) √ P μ+1 (x) + (ν − μ)(ν + μ + 1) P μν (x) = 0 ν ν 2 1−x (cf. 8.731 3)  P μν−1 (x) − P μν+1 (x) = (2ν + 1) 1 − x2 P μ−1 (x) (cf. 8.731 4) ν

5.

P μ−ν−1 (x)

3.11

=

P μν (x)

(cf. 8.731 5)

MO 82

966

8.734 1. 2. 3. 4. 5.

Associated Legendre Functions

 (z) = (ν − μ + 1) Q μν+1 (z) (ν + μ + 1)z Q νμ (z) + z 2 − 1 Q μ+1 ν  (ν + μ) Q μν−1 (z) + z 2 − 1 Q μ+1 (z) = (ν − μ)z Q μν (z) ν  Q μν−1 (z) − z Q μν (z) = −(ν − μ + 1) z 2 − 1 Q μ−1 (z) ν  μ z Q μν (z) − Q ν+1 (z) = −(ν + μ) z 2 − 1 Q μ−1 (z) ν  μ (ν + μ)(ν + μ + 1) Q ν−1 (z) + (2ν + 1) z 2 − 1 Q μ+1 (z) = (ν − μ)(ν − μ + 1) Q μν+1 (z) ν

8.735 1. 2. 3. 4. 5.



1 − x2 P μ+1 (x) = (ν − μ + 1) P μν+1 (x) ν  (ν − μ)x P μν (x) − (ν + μ) P μν−1 (x) = 1 − x2 P μ+1 (x) ν  P μν−1 (x) − x P μν (x) = (ν − μ + 1) 1 − x2 P μ−1 (x) ν  x P μν (x) − P μν+1 (x) = (ν + μ) 1 − x2 P μ−1 (x) ν (ν + μ + 1)x P μν (x) +

P −μ ν (z) =

2.

P μν (−z) = eνπi P μν (z) −

2 sin[(ν + μ)π]e−μπi Q μν (z) π 2 P μν (−z) = e−νπi P μν (z) − sin[(ν + μ)π]e−μπi Q μν (z) π

MO 83

6. 7.6

Q μν (z) sin[(ν + μ)π] − Q μ−ν−1 (z) sin[(ν − μ)π] = πeμπi cos νπ P μν (z)

2.

P μν (−x) = cos[(ν + μ)π] P μν (x) −

4.

MO 82

[Im z < 0]

MO 82

[Im z > 0]

MO 82

  Γ(ν − μ + 1) 2 μ μ = cos μπ P ν (x) − sin(μπ) Q ν (x) Γ(ν + μ + 1) π

1.

MO 83

(cf. 8.833 2)

Γ(ν − μ + 1) μ Q (z) Γ(ν + μ + 1) ν

P −μ ν (x)

MO 83

[Im z > 0]

Q μν (−z)

3.

MO 83

MO 83

Q μν (−z) = −e−νπi Q μν (z)

8.737

MO 82

(cf. 8.833 1)

5.

= −e

MO 82

[Im z < 0]

−2μπi Q −μ ν (z) = e

Q μν (z)

MO 82

MO 83

4.

νπi

MO 82

MO 83

  Γ(ν − μ + 1) 2 P μν (z) − e−μπi sin μπ Q μν (z) Γ(ν + μ + 1) π

1.

MO 82

MO 83

 (ν − μ)(ν − μ + 1) P μν+1 (x) = (ν + μ)(ν + μ + 1) P μν−1 (x) + (2ν + 1) 1 − x2 P μ+1 (x) ν

8.736

3.

8.734

2 sin[(ν + μ)π] Q μν (x) π π Q μν (−x) = − cos[(ν + μ)π] Q μν (x) − sin[(ν + μ)π] P μν (x) 2 sin[(ν + μ)π] π cos νπ cos μπ μ Q μ (x) − P (x) Q μ−ν−1 (x) = sin[(ν − μ)π] ν sin[(ν − μ)π] ν

MO 83

MO 84 MO 84 MO 83, EH I 144(15) MO 84

8.742

Functional relations

967

8.738

 

 √ 1 ν+1 −ν− 1 1. sin ϕ P −μ− 21 (cos ϕ) (i cot ϕ) = exp iπ μ − π Γ(ν + μ + 1) 2 2 2 + π, 0<ϕ< 2  

 √ 1 2 sin ϕ −ν− 12 exp iπ ν + Q 2.6 P μν (i cot ϕ) = 1 (cos ϕ − i0) π 4 Γ(−ν − μ) −μ− 2 + π, 0<ϕ< 2 √ π Γ(ν + μ + 1) −ν− 12 √ P −μ− 1 (coth α) 8.739 e−μπi Q μν (cosh α) = [Re (cosh α) > 0] 2 2 sinh α 8.741 2 sin μπ d P μν (x) d P −μ μ ν (x) 1. P −μ − P = (x) (x) ν ν dx dx π (1 − x2 )  ν+μ+1   ν+μ  Γ 2 +1 22μ Γ d Q μν (x) d P μν (x) μ μ 2   ν−μ   − Q ν (x) = 2. P ν (x) dx dx 1 − x2 Γ ν−μ+1 Γ 2 +1 2 11

8.742 1.

2.

3.

4.

Q μν

MO 83

MO 83 MO 83

MO 83

MO 83

   

 2 cosecμ ϕ ϕ cos ν + 12 t dt 2 μ μ   cos μπ P ν (cos ϕ) − sin μπ Q ν (cos ϕ) = π π Γ μ + 12 0 (cos t − cos ϕ) 12 −μ   Re μ > − 12 MO 88 

Γ(ν − μ + 1) 2 cos νπ P μν (cos ϕ) − sin νπ Q μν (cos ϕ) Γ(ν + μ + 1) π      2 cosecμ ϕ π cos ν + 12 (t − π) dt   = 1 −μ π Γ μ + 12 ϕ (cos ϕ − cos t) 2   1 Re μ > − 2 MO 88      π cos ν + 12 (t − π) dt 2 sinμ ϕ 2 μ μ   P ν (cos ϕ) cos (ν + μ) π − Q ν (cos ϕ) sin(ν + μ)π = μ+ 1 π π Γ 12 − μ ϕ (cos ϕ − cos t) 2   Re μ < 12 MO 88 Γ(ν − μ − 1) Γ(ν + μ + 1)

cos μπ P μν (cos ϕ) −

2 sin μπ Q μν (cos ϕ) π

 π sin2μ t dt 1 Γ(ν + μ + 1) sinμ ϕ   = μ√ 1 2 π Γ(ν − μ + 1) Γ μ + 2 0 (cos ϕ ± i sin ϕ cos t)ν−μ   Re μ > − 12 , 0 < ϕ < π MO 38

For integrals of Legendre functions, see 7.11–7.21.

968

Associated Legendre Functions

8.751

8.75 Special cases and particular values 8.751 1. 2.

3.8

m 

+ m + 1) 1 − x2 2 1−x F m − ν, m + ν + 1; m + 1; = (−1) MO 84 2m Γ(ν − m + 1)m! 2  m

Γ(ν + m + 1) z 2 − 1 2 1−z m F m − ν, m + ν + 1; m + 1; P ν (z) = MO 84 2m m! Γ(ν − m + 1) 2

3

eμπi Γ μ + n +  μ2 1/2 −n−μ−3/2 μ + n + 52 μ + n + 32 1 2  2 μ , ; n + 2; 2 Q n+ 1 (z) = F z −1 π z 3 2 2 2 z 2n+ 2 (n + 1)! Pm ν (x)

m Γ(ν

MO 84

8.752 1. 2.

3.

  m dm m 1 − x2 2 Pm P ν (x) WH, MO 84, EH I 148(6) ν (x) = (−1) dxm   1  m 1  m Γ(ν − m + 1) 2 −2 Pm P −m ... P ν (x)(dx)m ν (x) = (−1) ν (x) = 1 − x Γ(ν + m + 1) x x  2 − m2 P −m ν (z) = z − 1





z

4. 5.

P ν (z)(dz)m

... 1

[m ≥ 1]

HO 99a, MO 85, EH I 149(10)a

[m ≥ 1]

MO 85, EH I 149(8)

z

1

 2  m2 dm Qm Q (z) ν (z) = z − 1 dz m ν   2 − m2 m z Q −m (z) = (−1) − 1 ν

WH, MO 85, EH I 148(5)



∞

∞

...

z

Q ν (z)(dz)m

z

[m ≥ 1]

MO 85, EH I 149(9)

Special values of the indices 8.753 ϕ 1 cotμ Γ(1 − μ) 2

1.

P μ0 (cos ϕ) =

2.

P −1 ν (cos ϕ) = −

3.

Pm n (z) ≡ 0,

8.754 1. 2. 3.

MO 84

d P ν (cos ϕ) 1 ν(ν + 1) dϕ

Pm n (x) ≡ 0

MO 84

for m > n

MO 85



2 cosh να (cosh α) = π sinh α  2 1/2 cos νϕ P ν− 1 (cos ϕ) = 2 π sin ϕ  2 sin νϕ −1/2 P ν− 1 (cos ϕ) = 2 π sin ϕ ν 1/2 P ν− 1 2

MO 85 MO 85 MO 85

8.762

Derivatives with respect to the order

 1/2 Q ν− 1 2

4. 8.755

(cosh α) = i

ν sin ϕ 2

ν sinh α 1 −ν P ν (cosh α) = Γ(1 + ν) 2 1 Γ(1 + ν)

P −ν ν (cos ϕ) =

1. 2.

π e−να 2 sinh α

969

MO 85



MO 85 MO 85

Special values of Legendre functions 8.756

√ 2μ π   −ν−μ+1  Γ 2 +1 Γ 2   1 μ μ+1 2 sin 2 (ν + μ)π Γ ν+μ d P ν (0) 2 +1   = √ dx π Γ ν−μ+1 2   √ Γ ν+μ+1 1 μ μ−1 2  Q ν (0) = −2 π sin (ν + μ)π  ν−μ 2 Γ 2 +1   √ Γ ν+μ d Q μν (0) 1 μ 2 +1 = 2 π cos (ν + μ)π  ν−μ+1  dx 2 Γ 2 P μν (0) =

1. 2. 3. 4.

 ν−μ

MO 84

MO 84

MO84

MO 84

8.76 Derivatives with respect to the order

μ2  ∞ (−ν)(1 − ν) . . . (n − 1 − ν)(ν + 1)(ν + 2) . . . (ν + n) (μ + 1)(μ + 2) . . . (μ + n)1 · 2 . . . n n=1

n 1−x × [ψ(ν + n + 1) − ψ(ν − n + 1)] 2 [ν = 0, ±1, ±2, . . . ; Re μ > −1] MO 94

8.761

1 ∂P −μ ν (x) = ∂ν Γ(μ + 1)

8.762



1.  2.  3.

∂ P ν (cos ϕ) ∂ν



1−x 1+x



∂P −1 ν (cos ϕ) ∂ν ∂P −1 ν (cos ϕ) ∂ν

= 2 ln cos ν=0



= − tan ν=0



ν=1

ϕ 2

MO 94

ϕ ϕ ϕ − 2 cot ln cos 2 2 2

MO 94

ϕ ϕ ϕ 1 = − tan sin2 + sin ϕ ln cos 2 2 2 2

• For a connection with the polynomials C λn (x), see 8.936. • For a connection with a hypergeometric function, see 8.77.

MO 94

970

Associated Legendre Functions

8.771

8.77 Series representation For a representation in the form of a series, see 8.721. It is also possible to represent associated Legendre functions in the form of a series by expressing them in terms of a hypergeometric function. 8.771

μ

z+1 2 1−z 1 F −ν, ν + 1; 1 − μ; MO 15 1. P μν (z) = z−1 Γ(1 − μ) 2 μ  

ν+μ ν+μ+1 3 1 eμπi Γ(ν + μ + 1) Γ 12 z 2 − 1 2 μ 8   + 1, ;ν + ; 2 2. Q ν (z) = ν+1 F MO 15 2 z ν+μ+1 2 2 2 z Γ ν + 32 See also 8.702, 8.703, 8.704, 8.723, 8.751, 8.772. The analytic continuation for |z| > 1 The formulas are consequences of theorems on the analytic continuation of hypergeometric series (see 9.154 and 9.155): 8.772

 μ2 −ν−μ−1 ν+μ ν+μ+1 3 1 sin(ν + μ)π Γ(ν + μ + 1)  2 μ   z −1 z + 1, ;ν + ; 2 F 1. P ν (z) = ν+1 √ 2 2 2 z 2 π cos νπ Γ ν + 32

1 ν μ   2 Γ ν+2 μ−ν+1 μ−ν 1 1 , ; − ν; 2 z 2 − 1 2 z ν−μ F +√ 2 2 2 z π Γ(ν − μ + 1) [2ν = ±1, ±3, ±5, . . . ; |z| > 1; |arg (z ± 1)| < π] MO 85

2.

3.

P μν (z) =

P μν (z) =

  − ν+1

2 Γ −ν − 12 z 2 − 1 ν−μ+1 ν+μ+1 3 1 √ , ; ν + ; F 2 2 2 1 − z2 2ν+1  π Γ(−ν  − μ)

1 ν  2 ν 2 Γ ν+2 μ−ν μ+ν 1 1 +√ ,− ; − ν; z −1 2 F 2 2 ! 2 ! 1 − z2 π Γ(ν − μ + 1)   2ν = ±1, ±3, ±5; . . . ; !1 − z 2 ! > 1; |arg (z ± 1)| < π 1 Γ(1 − μ)

8.773 1.

2.

Q μν (z)

=e

Q μν (z) =

μπi



z−1 z+1

− μ2

z+1 2



ν F

z−1 −ν, −ν − μ; 1 − μ; z+1 !  ! !z − 1! !<1 ! !z + 1!



MO 86

√ − ν+1 ν+μ+1 ν−μ+1 π Γ(ν + μ + 1)  2 2  z −1  , ;ν + F 3 ν+1 2 2 2 Γ ν+2  ν + μ = −1, −2, −3, . . . ; |arg (z ± 1)| < π; 



z+1 z−1

μ2

3 1 ; 2 1 − z2 ! !  !1 − z 2 ! > 1

MO 86



1−z F −ν, ν + 1; 1 − μ; Γ(μ) 2

μ2 Γ(−μ) Γ(ν + μ + 1) z − 1 + F −ν, ν + 1; 1 + μ; Γ (ν − μ + 1) z+1 1 μπi e 2

MO 85

1−z 2

[|arg (z ± 1)| < π,

 |1 − z| < 2]

MO 86

8.777

Series representation

971



8.774

P μν

 

1 ϕ ν+ 12 1 3 sin ϕ Γ −ν − 12 −i(ν+1) π  ϕ 2 tan e + μ, − μ; ν + ; sin2 (i cot ϕ) = F 2 2 2 2 2 2π Γ(−ν − μ)1 

 ν+ 12 1 π ϕ 1 1 sin ϕ Γ ν + 2 ϕ eiν 2 cot + μ, − μ; − ν; sin2 F + 2π Γ(ν − μ + 1) 2 + 2 2 2 2 , π MO 86 2ν = ±1, ±3, ±5, . . . , 0 < ϕ < 2

8.775 1.

6

2.6

P μν (x) =

  

μ  (ν + μ) π Γ ν+μ+1 ν+μ+1 μ−ν 1 2 2  ν−μ  , ; ;x 1 − x2 2 F √ 2 2 2 πΓ 2 +1   ν+μ  

μ  ν + μ −ν +μ+1 3 2 2μ+1 sin 12 (ν + μ)π Γ 2 + 1   ν−μ+1  + 1, ; ;x + √ x 1 − x2 2 F 2 2 2 π Γ 2 2μ cos

1 2

   ν+μ+1 

√ μ  ν+μ+1 μ−ν 1 2 π sin 12 (ν + μ)π Γ μ 2 2 2   , ; ;x 1−x Q ν (x) = − 1−μ F 2 2 2 2 Γ ν−μ 1 2 + 1   ν+μ 

μ  √ cos 2 (ν + μ)π Γ 2 + 1  ν+μ μ−ν+1 3 2 μ 2 2   + 1, ; ;x F +2 π x 1−x 2 2 2 Γ ν−μ+1 2

MO 87

MO 87

8.776 1.

2.

For |z| 1  P μν (z)

=

Q μν (z) =

    

2ν Γ ν + 12 Γ −ν − 12 1 ν −ν−1 √ z + ν+1 √ z 1+O z2 π Γ(ν − μ + 1) 2 π Γ(−ν − μ) [2ν = ±1, ±3, ±5, . . . ,

√ e Γ(μ + ν + 1) −ν−1  z  π ν+1 1+O 2 Γ ν + 32 μπi



1 z2

|arg z| < π] MO 87



[2ν = −3, −5, −7, . . . ;

|arg z| < π] MO 87

√ 8.777 Set ζ = z + z 2 − 1. The variable ζ is uniquely defined by this equation on the entire z-plane in which a cut is made from −∞ to +1. Here, we are considering that branch of the variable ζ for which values of ζ exceeding 1 correspond to real values of z exceeding 1. In this case,   μ

2μ Γ −ν − 12 z 2 − 1 2 1 3 1 μ + μ, ν + μ + 1; ν + ; 1. P ν (z) = √ F 2 2 ζ2 π Γ(−ν − μ) ζ ν+μ+1 μ    

z2 − 1 2 1 1 1 2μ Γ ν + 12 + μ, μ − ν; − ν; +√ F 2 2 ζ2 π Γ(ν − μ + 1) ζ μ−ν [2ν = ±1, ±3, ±5, . . . ; |arg(z − 1)| < π] MO 86

2.

Q μν (z)

μ μπi

=2 e

μ 

√ Γ(ν + μ + 1) z 2 − 1 2 1 3 1   + μ, ν + μ + 1; ν + ; π F ζ ν+μ+1 2 2 ζ2 Γ ν + 32 [|arg(z − 1)| < π]

MO 86

972

Associated Legendre Functions

8.781

8.78 The zeros of associated Legendre functions 8.781 The function P −μ ν (cos ϕ), considered as a function of ν, has infinitely many zeros for μ ≥0. These are all simple and real. If a number ν0 is a zero of the function P −μ ν (cos ϕ), the number −ν0 − 1 is also MO 91 a zero of this function. μ 8.782 If ν and μ are both real and μ ≤0, or if ν and μ are integers, the function P ν (t) has no real zeros exceeding 1. If ν and μ are both real with ν < μ <0, the function P μν (t) has no real zeros exceeding 1 when sin μπ sin(μ − ν)π > 0, but does have one such zero when sin μπ sin(μ − ν)π <0. Finally, if μ ≤ ν, the function P μν (t) has no zeros exceeding 1 for μ even but does have one zero for μ odd. MO 91 8.783 If ν > − 32 and ν + μ + 1 > 0, the function Q μν (t) has no real zeros exceeding 1. 8.784 The function P − 12 +iλ (z) has infinitely many zeros for real λ. All these zeros are real and greater than unity. 8.785 For n a natural number, the function P n (x) has exactly n real zeros which lie in the closed interval −1, +1. 8.786 The function Q n (z) has no zeros for which |arg(z − 1)| < π if n is a natural number. The function MO 91 Q n (cos ϕ) has exactly n + 1 zeros in the interval 0 ≤ ϕ ≤ π. 8.787 The following approximate formula can be used to calculate the values of ν for which the equation P −μ ν (cos ϕ) = 0 holds for given small values of ϕ:  

 sin2 ϕ2 1 jμ 4μ2 − 1 4 ϕ ν+ =− MO 93 1− + O sin 1− . 2 2 sin ϕ2 6 jμ2 2 Here, jμ denotes an arbitrary nonzero root of the equation J μ (z) = 0 (for μ ≥ 0). If ϕ is close to π then, instead of this formula, we can use the following formulas:

2μ π−ϕ Γ(2μ + k + 1) [μ > 0, k = 0, 1, 2, . . .] MO 93 1. ν ≈μ+k+ Γ(μ) Γ(μ + 1) Γ(k + 1) 3 1  2. ν ≈k+ MO 93 [μ = 0, k = 0, 1, 2, . . .] 2 2 ln π−ϕ

8.79 Series of associated Legendre functions 8.791 1.

! ! !, +!   ! ! ! ! !t + t2 − 1! < !z + z 2 − 1!

∞

 1 = (2k + 1) P k (t) Q k (z) z−t k=0

2.

8.792

Here, t must lie inside an ellipse passing through the point z with foci at the points ±1. √ ∞  1 z − t + 1 − 2tz + t2 √ √ ln tk Q k (z) = z2 − 1 1 − 2tz + t2 k=0 [Re z > 1, |t| < 1] MO 78   ∞ 1 1 sin νπ  −β −β k − P −α (cos ϕ) P (cos ψ) = (−1) P −α ν ν k (cos ϕ) P k (cos ψ) π ν−k ν+k+1 k=0

[a ≥ 0,

β ≥ 0,

ν real,

−π < ϕ ± ψ < π]

MO 94

8.796

Series of associated Legendre functions

8.793 P −μ ν (cos ϕ) =

973



∞ 1 1 sin νπ  − (−1)k P −μ k (cos ϕ) π ν−k ν+k+1

[μ ≥ 0,

0 < ϕ < π]

k=0

MO 94

Addition theorems 8.794 1.11

P ν (cos ψ1 cos ψ2 + sin ψ1 sin ψ2 cos ϕ) = P ν (cos ψ1 ) P ν (cos ψ2 ) + 2 = P ν (cos ψ1 ) P ν (cos ψ2 ) + 2 [0 ≤ ψ1 < π,

2.

0 ≤ ψ2 < π,

∞  k=1 ∞  k=1

k (−1)k P −k ν (cos ψ1 ) P ν (cos ψ2 ) cos kϕ

Γ(ν − k + 1) k P (cos ψ1 ) P kν (cos ψ2 ) cos kϕ Γ(ν + k + 1) ν

ψ1 + ψ2 < π,

Q ν (cos ψ1 ) cos ψ2 + sin ψ1 sin ψ2 cos ϕ

8.795 1.

0 < ψ2 < π,

(cf. 8.814, 8.844 1)

∞ 

∞ < <  2 2 (−1)k P kν (z1 ) P −k P ν z1 z2 − z1 − 1 z2 − 1 cos ϕ = P ν (z1 ) P ν (z2 ) + 2 ν (z2 ) cos kϕ k=1

[Re z1 > 0, 2.

MO 90

(−1)k P −k ν (cos ψ1 ) Q ν k(cos ψ2 ) cos kϕ k=1 , 0 < ψ1 + ψ2 < π; ϕ real (cf. 8.844 3) MO 90

= P ν (cos ψ1 ) Q ν (cos ψ2 ) + 2 + π 0 < ψ1 < , 2

ϕ real]

|arg (z1 − 1)| < π,

Re z2 > 0,

|arg (z2 − 1)| < π]

MO 91

∞ < <  k Q ν x1 x2 − x21 − 1 x22 − 1 cos ϕ = P ν (x1 ) Q ν (x2 ) + 2 (−1)k P −k ν (x1 ) Q ν (x2 ) cos kϕ k=1

[1 < x1 < x2 , 3.

∞ < <  Q n x1 x2 + x21 + 1 x22 + 1 cosh α = k=n+1

ν = −1, −2, −3, . . . ,

MO 91

1 Q k (ix1 ) Q kn (ix2 ) e−kα (k − n − 1)!(k + n)! n [x1 > 0,

x2 > 0,

8.796 P ν (− cos ψ1 cos ψ2 − sin ψ1 sin ψ2 cos ϕ) = P ν (− cos ψ1 ) P ν (cos ψ2 ) + 2

∞ 

α > 0] (−1)k

k=1

× P −k ν

(− cos ψ1 ) P −k ν

[0 < ψ2 < ψ1 < π, See also 8.934 3.

ϕ real]

MO 91

Γ(ν + k + 1) Γ(ν − k + 1)

(cos ψ2 ) cos kϕ

ϕ real]

(cf. 8.844 2)

MO 91

974

Associated Legendre Functions

8.810

8.81 Associated Legendre functions with integer indices 8.810 For integer values of ν and μ, the differential equation 8.700 1. (with |ν| > |μ|) has a simple solution in the real domain, namely:   m dm m 1 − x2 2 P n (x). u = Pm n (x) = (−1) dxm m The functions P n (x) are called associated Legendre functions (or spherical functions) of the first kind. The number n is called the degree, and the number m is called the order of the function P m n (x). The sin mϑ P m functions {cos mϑ P m n (cos ϕ) , n (cos ϕ)}, which depend on the angles ϕ and ϑ, are also called Legendre functions of the first kind, or, more specifically, tesseral harmonics for m < n and sectoral harmonics for m = n. These last functions are periodic with respect to the angles ϕ and ϑ. Their periods are, respectively, π and 2π. They are single-valued and continuous everywhere on the surface of the unit sphere x21 + x22 + x23 = 1 (where x1 = sin ϕ cos ϑ, x2 = sin ϕ sin ϑ, x3 = cos ϕ), and they are solutions of the differential equation

1 ∂ ∂Y 1 ∂2Y + n(n + 1)Y = 0. sin ϕ + sin ϕ ∂ϕ ∂ϕ sin2 ϕ ∂ϑ2 8.811 The integral representation   ϕ   2 (−1)m (n + m)! m− 1 −m   sin (cos ϕ) = ϕ (cos t − cos ϕ) 2 cos n + 12 t dt MO 75 Pm n 1 Γ m + 2 (n − m)! π 0 8.812 The series representation:

m (−1)m (n + m)!  (n − m)(m + n + 1) 1 − x 2 2 1 − x Pm (x) = 1− n 2m m!(n − m)! 1!(m + 1) 2 

2 (n − m)(n − m + 1)(m + n + 1)(m + n + 2) 1 − x + − ... MO 73 2!(m + 1)(m + 2) 2

m (−1)m (2n − 1)!!  (n − m)(n − m − 1) n−m−2 x 1 − x2 2 xn−m − = (n − m)! 2(2n − 1)  (n − m)(n − m − 1)(n − m − 2)(n − m − 3) n−m−4 x + − ... MO 73 2 · 4(2n − 1)(2n − 3)

m m−n m−n+1 1 1 (−1)m (2n − 1)!!  , ; − n; 2 1 − x2 2 xn−m F = MO 73 (n − m)! 2 2 2 x 8.813 Special cases: 1. 2. 3. 4. 5. 6.

1/2  P 11 (x) = − 1 − x2 = − sin ϕ   1/2 P 12 (x) = −3 1 − x2 x = − 32 sin 2ϕ   P 22 (x) = 3 1 − x2 = 32 (1 − cos 2ϕ)  1/2  2  5x − 1 = − 38 (sin ϕ + 5 sin 3ϕ) P 13 (x) = − 32 1 − x2   P 23 (x) = 15 1 − x2 x = 15 4 (cos ϕ − cos 3ϕ)   3/2 P 33 (x) = −15 1 − x2 = − 15 4 (3 sin ϕ − sin 3ϕ)

MO 73 MO 73 MO 73 MO 73 MO 73 MO 73

8.820

Legendre functions

975

Functional relations For recursion formulas, see 8.731. 8.814 P n (cos ϕ1 cos ϕ2 + sin ϕ1 sin ϕ2 cos Θ) = P n (cos ϕ1 ) P n (cos ϕ2 ) + 2 [0 ≤ ϕ1 ≤ π, 8.815

n  (n − m)! m P n (cos ϕ1 ) P m n (cos ϕ2 ) cos mΘ (n + m)! m=1

0 ≤ ϕ2 ≤ π]

(“addition theorem”)

MO 74

If Y n1 (ϕ, ϑ) = A0 P n1 (cos ϕ) + Z n2 (ϕ, ϑ) = α0 P n2 (cos ϕ) +

n1  m=1 n2 

(am cos mϑ + bm sin mϑ) P m n1 (cos ϕ) , (αm cos mϑ + βm sin mϑ) P m n2 (cos ϕ) ,

m=1

then





2π

π

dϑ 



2π

sin ϕ dϕ Y n1 (ϕ, ϑ) Y n2 (ϕ, ϑ) = 0, 0

π

sin ϕ dϕ Y n (ϕ, ϑ) P n [cos ϕ cos ψ + sin ϕ sin ψ cos(ϑ − θ)] =

dϑ 0

0

0

n 

4π Y n (ψ, θ) 2n + 1

n! cos mϑ P m n (cos ϕ) (n + m)! m=1 For integrals of the functions, P m n (x), see 7.112 1, 7.122 1. n

8.816 (cos ϕ + i sin ϕ cos ϑ) = P n (cos ϕ) + 2

(−1)m

MO 75 MO 75

8.82–8.83 Legendre functions 8.820

The differential equation    d  2 du 1−z + ν(ν + 1)u = 0 (cf. 8.700 1), dz dz where the parameter ν can be an arbitrary number, has the following two linearly independent solutions:

1−z 1. P ν (z) = F −ν, ν + 1; 1; 2 1

Γ(ν + 1) Γ 2 −ν−1 ν + 2 ν + 1 2ν + 3 1   , ; ; 2 F SM 518(137) 2. Q ν (z) = ν+1 z 2 2 2 z 2 Γ ν + 32 The functions P ν (z) and Q ν (z) are called Legendre functions of the first and second kind respectively. If ν is not an integer, the function P ν (z) has singularities at z = −1 and z = ∞. However, if ν = n = 0, 1, 2, . . . , the function P ν (z) becomes the Legendre polynomial P n (z) (see 8.91) For ν = −n = −1, −2, . . . , we have P −n−1 (z) = P n (z). 3.

If ν = 0, 1, 2, . . . , the function Q ν (z) has singularities at the points z = ±1 and z = ∞. These points are branch points of the function. On the other hand, if ν = n = 0, 1, 2, . . . , the function Q n (z) is single-valued for |z| > 1 and regular for z = ∞.

976

4.

Associated Legendre Functions

In the right half-plane,



P ν (z) = 5.

1+z 2

ν



z−1 F −ν, −ν; 1; z+1

8.821

[Re z > 0]

The function P ν (z) is uniquely determined by equations 8.820 1 and 8.820 4 within a circle of radius 2 with its center at the point z = 1 in the right half-plane.

6.

For z = x = cos ϕ, a solution of equation 8.820 is the function  ϕ P ν (x) = P ν (cos ϕ) = F −ν, ν + 1; 1; sin2 ; 2 In general,

7.

P ν (z) = P −ν−1 (z) = P ν (x) = P −ν−1 (x), for z = x

8.

The function Q ν (z) for |z| > 1 is uniquely determined by equation 8.820 2 everywhere in the z-plane in which a cut is made from the point z = −∞ to the point z = 1. By means of a hypergeometric series, the function can be continued analytically inside the unit circle. On the cut (−1 ≤ x ≤ +1) of the real axis, the function Q ν (x) is determined by the equation

9.

Q ν (x) =

1 2

[Q ν (x + i0) + Q ν (x − i0)]

HO 52(53), WH

Integral representations 8.821 1.

2.

3.

ν  (1+,z+)  2 t −1 1 dt 2πi A 2ν (t − z)ν+1 Here, A is a point on the real axis to the right of the point t = 1 and to the right of z if z is real. At the point A, we set

P ν (z) =

arg(t − 1) = arg(t + 1) = 0 and [|arg(t − z)| < π] WH ν  (1−,1+)  2 t −1 1 Q ν (z) = dt 4i sin νπ A 2ν (z − t)ν+1 [ν is not an integer; the point A is at the end of the major axis of an ellipse to the right of t = 1 drawn in the t-plane with foci at the points ±1 and with a minor axis sufficiently small that the point z lies outside it. The contour begins at the point A, follows the path (1−, −1+), and returns to A; |arg z| ≤ π and |arg(z − t)| → arg z as t → 0 on the contour; arg(t + 1) = arg(t − 1) = 0 at the point A; z does not lie on the real axis between −1 and 1.] For ν = n an integer,  1  n 1 1 − t2 (z − t)−n−1 dt Q n (z) = n+1 2 −1

8.822 1.

1 P ν (z) = π

 0

π



z+

√

SM 517(134), WH



ν  z + z 2 − 1 cos ϕ dϕ 0 z 2 − 1 cos+ϕ    π, Re z > 0 and arg z + z 2 − 1 cos ϕ = arg z for ϕ = 2 dϕ

ν+1

1 = π

π



WH

8.827

Legendre functions

977



2.

∞ dϕ Q ν (z) = √  ν+1 , 2 0 z + z − 1 cosh ϕ  +  ,  Re ν > −1; if ν is not an integer, z + z 2 − 1 cosh ϕ for ϕ = 0 has its principal value

WH

2 8.823 P ν (cos θ) = π  n 8.824 Q n (z) = 2 n!



θ



  cos ν + 12 ϕ

dϕ 2 (cos ϕ − cos θ)  ∞  ∞ (dz)n+1 (t − z)n n ... =2 n+1 n+1 dt 2 2 z z % (z − 1) z (t − & 1)  n ∞ dt (−1)n dn  2 = z − 1 n+1 n 2 (2n − 1)!! dz z (t − 1) 

WH, MO 78

P n (t) 1 dt 2 −1 z − t See also 6.622 3, 8.842. 8.826 Fourier series:

2.

[Re z > 1]

1

8.825 Q n (z) =

1.

WH

0 ∞

[|arg(z − 1)| < π]

WH, MO 78

 n! 2n+2 1 n+1 sin(n + 3)ϕ P n (cos ϕ) = sin(n + 1)ϕ + π (2n + 1)!! 1 2n + 3  1 · 3(n + 1)(n + 2) + sin(n + 5)ϕ + . . . 1 · 2(2n + 3)(2n + 5) [0 < ϕ < π] ⎡ n! ⎣ cos(n + 1)ϕ + 1 n + 1 cos(n + 3)ϕ Q n (cos ϕ) = 2n+1 (2n + 1)!! 1 2n + 3 ⎤ 1 · 3 (n + 1)(n + 2) + cos(n + 5)ϕ + . . .⎦ 1 · 2 (2n + 3)(2n + 5) [0 < ϕ < π]

MO 79

MO 79

The expressions for Legendre functions in terms of a hypergeometric function (see 8.820) provide other series representations of these functions. Special cases and particular values 8.827 1. 2. 3. 4.

1 1+x ln = arctanh x 2 1−x x 1+x −1 Q 1 (x) = ln 2 1−x  1+x 3 1 2 3x − 1 ln − x Q 2 (x) = 4 1−x 2  1+x 5 2 2 1 3 5x − 3x ln − x + Q 3 (x) = 4 1−x 2 3 Q 0 (x) =

JA JA JA JA

978

5. 6.

Associated Legendre Functions

8.828

 1 + x 35 3 55 1  35x4 − 30x2 + 3 ln − x + x 16 1−x 8 24  63 1  1 + x 49 8 63x5 − 70x3 + 15x ln − x4 + x2 − Q 5 (x) = 16 1−x 8 8 15 Q 4 (x) =

JA JA

8.828 1.

P ν (1) = 1

MO 79



 ν+1 1 sin νπ ν 2. P ν (0) = − √ Γ Γ − 2 π3 2 2

 ν+1 1 ν 8.829 Q ν (0) = √ (1 − cos νπ) Γ Γ − 2 2 4 π

MO 79

MO 79

Functional relationships 8.831 1. 2.

3.

π [cos νπ P ν (x) − P ν (−x)] [ν = 0, ±1, ±2, . . .] 2 sin νπ 1 1+x − Wn−1 (x) Q n (x) = P n (x) ln [n = 0, 1, 2, . . .] , 2 1−x where n−1

 n 2   2(n − 2k) − 1 1 P n−2k−1 (x) = P k−1 (x) P n−k (x) Wn−1 (x) = (2k + 1)(n − k) k Q ν (x) =

k=0

MO 76

k=1

and 4. 5.

W−1 (x) ≡ 0 (see also 8.839)

∞  1 1 π − P ν (cos ϕ) (−1)k P k (cos ϕ) = ν−k ν+k+1 sin νπ

SM 516(131), MO 76

k=0

[ν not an integer; 6.

∞ 

(−1)k

k=0

1 1 − ν−k ν+k+1

P k (cos ϕ) P k (cos ψ) = [ν not an integer,

See also 8.521 4. 8.832  d  2 P ν (z) = (ν + 1) [P ν+1 (z) − z P ν (z)] 1. z −1 dz 2. (2ν + 1)z P ν (z) = (ν + 1) P ν+1 (z) + ν P ν−1 (z) 3. 4.



 d   Q ν (z) = (ν + 1) Q ν+1 (z) − z Q ν (z) dz (2ν + 1)z Q ν (z) Q ν+1 (z) + ν Q ν−1 (z) z2 − 1

0 ≤ ϕ < π]

MO 77

π P ν (cos ϕ) P ν (cos ψ) sin νπ

−π < ϕ + ψ < π,

−π < ϕ − ψ < π]

MO 77

WH WH WH WH

8.838

Legendre functions

979

8.833 [Im z < 0]

MO 77

[Im z > 0]

MO 77

3.

2 sin νπ Q ν (z) π 2 P ν (−z) = e−νπi P ν (z) − sin νπ Q ν (z) π Q ν (−z) = −e−νπi Q ν (z)

[Im z < 0]

MO 77

4.

Q ν (−z) = −eνπi Q ν (z)

[Im z > 0]

MO 77

1. 2.

P ν (−z) = eνπi P ν (z) −

8.834 1. 2.

πi P ν (x) 2 1 z+1 − W n−1 (z) Q n (z) = P n (z) ln 2 z−1 Q ν (x ± i0) = Q ν (x) ∓

(see 8.831 3)

MO 77

[sin νπ = 0]

MO 77

Q −ν−1 (cos ϕ) = Q ν (cos ϕ) − π cot νπ P ν (cos ϕ) [sin νπ = 0] π Q ν (− cos ϕ) = − cos νπ Q ν (cos ϕ) − sin νπ P ν (cos ϕ) 2

MO 77

8.835 1. Q ν (z) − Q −ν−1 (z) = π cot νπ P ν (z) 2. 3. 8.836 1. 2.

MO 77

  n z + 1 1 dn  2 z+1 1 z − 1 ln Q n (z) = n − P n (z) ln 2 n! dz n z−1 2 z−1   n 1 + x 1 dn  2 1+x 1 x − 1 ln Q n (x) = n − P n (x) ln n 2 n! dx 1−x 2 1−x

8.837  ϕ 1. P ν (x) = P ν (cos ϕ) = F −ν, ν + 1; 1; sin2 2 ν ν+1 tan νπ Γ(ν + 1) −ν−1 z  + 1, ;ν + 2. P ν (z) = ν+1 √ F 2 2 2 π Γ ν + 32  

1−ν ν 1 1 2ν Γ ν + 12 ν z F , − ; − ν; 2 +√ 2 2 2 z π Γ(ν + 1)

(cf. 8.820 6)

3 1 ; 2 z2

MO 77

MO 79 MO 79

MO 76

MO 78

See also 8.820. For integrals of Legendre functions, see 7.1–7.2. 8.838 Inequalities (0 ≤ ϕ ≤ π, ν > 1, and C0 is a number that does not depend on the values of ν or ϕ):  1 MO 78 1. |P ν (cos ϕ) − P ν+2 (cos ϕ)| ≤ 2C0 νπ  ! ! !Q ν (cos ϕ) − Q ν+2 (cos ϕ)! < C0 π 2. MO 78 ν With regard to the zeros of Legendre functions of the second kind, see 8.784, 8.785, and 8.786. For the expansion of Legendre functions in series of associated Legendre functions, see 8.794, 8.795, and 8.796.

980

8.839

Associated Legendre Functions

A differential equation leading to the functions Wn−1 (see 8.831 3):  d2 Wn−1  dWn−1 d Pν + (n + 1)nWn−1 = 2 1 − x2 − 2x 2 dx dx dx

8.839

MO 76

8.84 Conical functions 8.840

Let us set

ν = − 12 + iλ, where λ is a real parameter, in the defining differential equation 8.700 1 for associated Legendre functions. We then obtain the differential equation of the so-called conical functions. A conical function is a special case of the associated Legendre function. However, the Legendre functions P − 12 +iλ (x), Q − 12 +iλ (x) have certain peculiarities that make us distinguish them as a special class—the class of conical functions. The most important of these peculiarities is the following 8.841 The functions  2   4λ + 12 4λ2 + 32 ϕ 4λ2 + 12 2 ϕ P − 12 +iλ (cos ϕ) = 1 + + sin sin4 + . . . 2 2 2 2 2 2 4 2 are real for real values of ϕ. Also, P − 12 +iλ (x) ≡ P − 12 −iλ (x) MO 95 8.842 1. 2.6

Integral representations:  ∞  cosh λu du cos λu du 2 2 ϕ   P − 12 +iλ (cos ϕ) = = cosh λπ π 0 π 2 (cos u − cos ϕ) 2 (cos ϕ + cosh u) 0  ∞  ∞ cos λu du cos λu du   + Q − 12 ∓λi (cos ϕ) = ±i sinh λπ 2 (cosh u + cos ϕ) 2 (cosh u − cos ϕ) 0 0

MO 95

MO 95

Functional relations (See also 8.73) 8.843 P − 12 +iλ (− cos ϕ) =

, cosh λπ + Q − 12 +iλ (cos ϕ) + Q − 12 −iλ (cos ϕ) π

MO 95

8.844 1.

P − 12 +iλ (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ)

k ∞ (−1)k 22k P k  − 1 +iλ (cos ψ) P − 1 +iλ (cos ϑ) cos kϕ

2 2 = P − 12 +iλ (cos ψ) P − 12 +iλ (cos ϑ) + 2 (4λ2 + 12 ) (4λ2 + 32 ) · · · [4λ2 + (2k − 1)2 ] k=1 , + π (cf. 8.794 1) MO 95 0 < ϑ < , 0 < ψ < π, 0 < ψ + ϑ < π 2

2.

P − 12 +iλ (− cos ψ cos ϑ − sin ψ sin ϑ cos ϕ)

k ∞ (−1)k 22k P k  − 12 +iλ (cos ψ) P − 12 +iλ (− cos ϑ) cos kϕ = P − 12 +iλ (cos ψ) P − 12 +iλ (− cos ϑ) + 2 (4λ2 + 1) (4λ2 + 32 ) · · · [4λ2 + (2k − 1)2 ] , + k=1 π (cf. 8.796) MO 95 0 < ψ < < ϑ, ψ + ϑ < π 2

8.852

3.

Toroidal functions

981

Q − 12 +iλ (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ)

k ∞ (−1)k 22k P k  − 12 +iλ (cos ψ) Q − 12 +iλ (cos ϑ) cos kϕ , + = P − 12 +iλ (cos ψ) Q − 12 +iλ (cos ϑ) + 2 2 (4λ2 + 1) (4λ2 + 32 ) · · · 4λ2 + (2k − 1) k=1 + , π 0 < ψ < < ϑ, ψ + ϑ < π (cf. 8.794 2) MO 96 2

Regarding the zeros of conical functions, see 8.784.

8.85 Toroidal functions 8.850 1.

Solutions of the differential equation

m2 d2 u cosh η du 1 2 − + n + − u = 0, dη 2 sinh η dη 4 sinh2 η

are called toroidal functions. They are equivalent (under a coordinate transformation) to associated Legendre functions. In particular, the functions Qm MO 96 Pm n− 12 (cosh η) , n− 12 (sinh η) are solutions of equation 8.850 1. The following formulas, obtained from the formulas obtained earlier for associated Legendre functions, are valid for toroidal functions: 8.851 Integral representations:    π m Γ n + m + 12 (sinh η) sin2m ϕ dϕ m    1. P n− 1 (cosh η) =  √ 1 2 Γ n − m + 12 2m π Γ m + 12 0 (cosh η + sinh η cos ϕ)n+m+ 2    2π cos mϕ dϕ (−1)m Γ n + 12   = 2π Γ n − m + 12 0 (cosh η + sinh η cos ϕ)n+ 12 

2.

1 2



MO 96



∞

cosh mt dt m Γ n +  Qm 1 n− 12 (cosh η) = (−1) 1 Γ n − m + 2 0 (cosh η + sinh η cosh t)n+ 2   η  ln coth 2 Γ n + m + 12 n− 1   = (−1)m (cosh η − sinh η cosh t) 2 cosh mt dt 1 Γ n+ 2 0

[n ≥ m]

MO 96

8.852 1.

Functional relations:

√   π 2m Γ n + m + 12 1 sinhm ηe−(n+m+ 2 )η Γ(n + 1)   1 × F m + 2 , n + m + 12 ; n + 1; e−2η

m Qm n− 1 (cosh η) = (−1) 2

MO 96 ∗ Sometimes

called torus functions

982

2.

Orthogonal Polynomials

P −m (cosh η) = n− 1 2

8.853

m   1 2−2m  1 − e−2η e−(n+ 2 )η F m + 12 , n + m + 12 ; 2m + 1; 1 − e−2η Γ(m + 1) MO 96

8.853

An asymptotic representation P n− 12 (cosh η) for large values of n: 1 Γ(n)e(n− 2 )η   P n− 12 (cosh η) = √ 1 π Γ n + 2 % &  

2 Γ2 n + 12 1 1 η −2nη −2η × ln (4e ) e , n + ; n + 1; e +A+B , F πn! Γ(n) 2 2 where

2 (2n − 1)!! 1 1 · (2n − 1) −2η 1 1 · 3 · (2n − 1)(2n − 3) −4η 1 e e A=1+ 2 + 4 + · · · + 2n−2 e−2(n−1)η 2 1 · (n − 1) 2 1 · 2 · (n − 1)(n − 2) 2 (n − 1)!

  1 1   ∞ Γ k+ 2 Γ n+k+ Γ n + 12  2  un+k + uk − vn+k− 12 − vk− 12 e−2(n+k)η B= √ Γ(n + k + 1) Γ(k + 1) π 3 Γ(n) k=1

Here, ur =

r  1 s=1

s

,

vr− 12 =

r  s=1

2 2s − 1

[r is a natural number]

MO 97

8.9 Orthogonal Polynomials 8.90 Introduction 8.901 Suppose that w(x) is a nonnegative real function of a real variable x. Let (a, b) be a fixed interval on the x-axis. Let us suppose further that, for n = 0, 1, 2, . . . , the integral  b xn w(x) dx exists and that the integral

a



b

w(x) dx a

is positive. In this case, there exists a sequence of polynomials p0 (x), p1 (x), . . . , pn (x), . . . , that is uniquely determined by the following conditions: 1.

pn (x) is a polynomial of degree n and the coefficient of xn in this polynomial is positive.

2.

The polynomials p0 (x), p1 (x), . . . are orthonormal; that is,   b 0 for n = m, pn (x)pm (x)w(x) dx = 1 for n = m. a We say that the polynomials pn (x) constitute a system of orthogonal polynomials on the interval (a, b) with the weight function w(x).

8.910

8.902 1.

Legendre polynomials

983

If qn is the coefficient of xn in the polynomial pn (x), then n 

pk (x)pk (y) =

k=0

qn pn+1 (x)pn (y) − pn (x)pn+1 (y) qn+1 x−y

(Darboux–Christoffel formula) EH II 159(10)

2.11

n 

2

[pk (x)] =

k=0

qn  qn+1

 pn (x)pn+1 (x) − pn (x)pn+1 (x)

EH II 159(11)

8.903

Between any three consecutive orthogonal polynomials, there is a dependence pn (x) = (An x + Bn ) pn−1 (x) − Cn pn−2 (x) [n = 2, 3, 4, . . .] In this formula, An , Bn , and Cn are constants and qn qn qn−2 , Cn = 2 An = qn−1 qn−1 8.904 Examples of normalized systems of orthogonal polynomials: 

n+

 1 1/2 2 

Notation and name

P n (x) 1/2 (n + λ) n! 2λ Γ(λ) C λn (x) 2π Γ (2λ + n)  εn T n (x), ε0 = 1, εn = 2 for n = 1, 2, 3, . . . π −1/2 −n 2 2 π −1/4 (n!) H n (x)  1/2 Γ(n + 1) Γ(α + β + 1 + n)(α + β + 1 + 2n) P (α,β) (x) n Γ(α + 1 + n) Γ(β + 1 + n)2α+β+1  1/2 Γ(n + 1) (−1)n Lα n (x) Γ (α + n + 1)

MO 102

Interval

Weight

see 8.91

(−1, +1)

see 8.93

(−1, +1)

1  λ− 12 1 − x2

see 8.94

(−1, +1)

see 8.95

(−∞, ∞)

e−x

see 8.96

(−1, +1)

(1 − x)α (1 + x)β

see 8.97

(0, ∞)

xα e−x



1 − x2

−1/2 2

Cf. 7.221 1, 7.313, 7.343, 7.374 1, 7.391 1, 7.414 3.

8.91 Legendre polynomials 8.910 Definition. The Legendre polynomials P n (z) are polynomials satisfying equation 8.700 1 with μ = 0 and ν = n: that is, they satisfy the differential equation 1.

2.

 d2 u du + n(n + 1)u = 0 − 2z dz 2 dz This equation has a polynomial solution if, and only if, n is an integer. Thus, Legendre polynomials constitute a special type of associated Legendre function. 

1 − z2

Legendre polynomials of degree n are of the form n 1 dn  2 P n (z) = n z −1 n 2 n! dz

984

8.911

1.

Orthogonal Polynomials

8.911

Legendre polynomials written in expanded form:

n2  1  (−1)k (2n − 2k)! n−2k z P n (z) = n 2 k!(n − k)!(n − 2k)! k=0

(2n)! n(n − 1) n−2 n(n − 1)(n − 2)(n − 3) n−4 n z z = + − ... z − 2 2(2n − 1) 2 · 4(2n − 1)(2n − 3) 2n (n!)

(2n − 1)!! n 1 n 1−n 1 = z F − , ; − n; 2 n! 2 2 2 z HO 13, AD (9001), MO 69

2.

3.

2n(2n + 1) 2 2n(2n − 2)(2n + 1)(2n + 3) 4 n (2n − 1)!! z + z − ... P 2n (z) = (−1) 1− 2n n! 2! 4!

1 1 (2n − 1)!! = (−1)n F −n, n + ; ; z 2 n 2 n! 2 2

AD (9002), MO 69

+ 1)!! 2n(2n + 3) 3 2n(2n − 2)(2n + 3)(2n + 5) 5 z + z − ... P 2n+1 (z) = (−1) z− 2n n! 3! 5!

3 3 (2n + 1)!! = (−1)n z F −n, n + ; ; z 2 n 2 n! 2 2

n (2n

AD (9002), MO 69

⎛ 4.

P n (cos ϕ) =

1 n (2n − 1)!! ⎝ cos(n − 2)ϕ cos nϕ + 2n n! 1 2n − 1 +

1·3 n(n − 1) cos(n − 4)ϕ 1 · 2 (2n − 1)(2n − 3)

⎞ 1·3·5 n(n − 1)(n − 2) + cos(n − 6)ϕ − . . .⎠ 1 · 2 · 3 (2n − 1)(2n − 3)(2n − 5) WH

5.

(2n − 1)!! P 2n (cos ϕ) = (−1)n 2n n!

 2n n! (2n)2 2n sin2n−2 ϕ cos2 ϕ + · · · + (−1)n cos2n ϕ × sin ϕ − 2! (2n − 1)!! AD (9011)

6.

(2n + 1)!! cos ϕ P 2n+1 (cos ϕ) = (−1)n 2n n!

 2n n! (2n)2 2n 2n−2 2 n 2n sin cos ϕ × sin ϕ − ϕ cos ϕ + · · · + (−1) 3! (2n + 1)!! AD (9012)

7.

P n (z) =

n  k=0

k

(−1) (n + k)! (n − k)! (k!)

2

2k+1

  (1 − z)k + (−1)n (1 + z)k

WH

8.914

8.912

Legendre polynomials

985

Special cases:

1.

P 0 (x) = 1

JA

2.

P 1 (x) = x = cos ϕ  1 1 2 3x − 1 = (3 cos 2ϕ + 1) P 2 (x) = 2 4  1 1 3 5x − 3x = (5 cos 3ϕ + 3 cos ϕ) P 3 (x) = 2 8  1 1 35x4 − 30x2 + 3 = (35 cos 4ϕ + 20 cos 2ϕ + 9) P 4 (x) = 8 64  1 1 (63 cos 5ϕ + 35 cos 3ϕ + 30 cos ϕ) 63x5 − 70x3 + 15x = P 5 (x) = 8 128  1  1 (231 cos 6ϕ + 126 cos 4ϕ + 105 cos 2ϕ + 50) 231x6 − 315x4 + 105x2 − 5 = P 6 (x) = 16 512  1  P 7 (x) = 429x7 − 693x5 + 315x3 − 35x 16 1 (429 cos 7ϕ + 231 cos 5ϕ + 189 cos 3ϕ + 175 cos ϕ) = 1024  1  6435x8 − 12012x6 + 6930x4 − 1260x2 + 35 P 8 (x) = 128 1 (6435 cos 8ϕ − 3432 cos 6ϕ + 2772 cos 4ϕ − 2520 cos 2ϕ + 1225) = 16384

JA

3. 4. 5. 6. 7.10 8.

9.

8.913



π



  sin n + 12 t

2 (cos ϕ − cos t) See also 3.611 3, 3.661 3, 4.

2.7

dt

JA

WH

Schl¨ afli’s integral formula:  C

2 n t −1 dt, 2n (t − z)n+1

with C a simple contour containing z. 3.

JA

ϕ

1 P n (z) = 2πi 10

JA

Integral representations: 2 P n (cos ϕ) = π

1.

JA

Laplace integral formula: P n (z) =

1 π



π

,n + 1/2  cos ϕ dϕ x + x2 − 1

SA 175(9)

[|x| ≤ 1]

SA 180(19)

0

Functional relations 8.914 1.

Recurrence formulas: (n + 1) P n+1 (z) − (2n + 1)z P n (z) + n P n−1 (z) = 0

WH

986

Orthogonal Polynomials



2.

z2 − 1

8.915

 d Pn n(n + 1) = n [z P n (z) − P n−1 (z)] = [P n+1 (z) − P n−1 (z)] dz 2n + 1

WH

8.915 n 

1.10

(2k + 1) P k (x) P k (y) = (n + 1)

k=0

P n (x) P n+1 (y) − P n (y) P n+1 (x) y−x (Christoffel summation formula) MO 70

1(1)10 .

(y − x)

n 

  (2k + 1) P k (x) Q k (y) = 1 − (n + 1) P n+1 (x) Q n (y) − P n (x) Q n+1 (y)

k=0

AS 335(8.9.2) n−1

 2 

2.7

(2n − 4k − 1) P n−2k−1 (z) = P n (z)

(summation theorem)

MO 70

k=0 n−2

 2 

3.7

(2n − 4k − 3) P n−2k−2 (z) = z P n (z) − n P n (z)

SM 491(42), WH

k=0

4.

n

 2

10

(2n − 4k + 1)[k(2n − 2k + 1) − 2] P n−2k (z) = z 2 P n (z) − n(n − 1) P n (z)

WH

k=1



m  am−k ak an−k 2n + 2m − 4k + 1 P n+m−2k (z) = P n (z) P m (z) an+m−k 2n + 2m − 2k + 1 k=0  (2k − 1)!! , ak = k!

5.11

8.916 1. 2. 3. 4. 5.



1 1 (2n − 1)!! ∓inϕ ±2iϕ e , −n; − n; e F 2n n! 2 2  ϕ P n (cos ϕ) = F n + 1, −n; 1; sin2 2  ϕ P n (cos ϕ) = (−1)n F n + 1, −n; 1; cos2 2

1 1 1 n 2 P n (cos ϕ) = cos ϕ F − n, − n; 1; − tan ϕ 2 2 2  ϕ ϕ P n (cos ϕ) = cos2n F −n, −n; 1; − tan2 2 2 P n (cos ϕ) =

 m≤n

AD (9036)

MO 69 MO 69 WH HO 23 HO 23, 29, WH

See also 8.911 1, 8.911 2, 8.911 3. For a connection with other functions, see 8.936 3, 8.836, 8.962 2. • For integrals of Legendre polynomials, see 7.22–7.25. • For the zeros of Legendre polynomials, see 8.785.

8.918

8.917

Legendre polynomials

Inequalities:

1.

P 0 (x) < P 1 (x) < P 2 (x) < · · · < P n (x) < . . .

2.

For x > −1, P 0 (x) + P 1 (x) + · · · + P n (x) > 0.

3.

[P n (cos ϕ)] >

4.

√ n sin ϕ|P n (cos ϕ)| ≤ 1.

2

[x > 1]

MO 71 MO 71

sin(2n + 1)ϕ (2n + 1) sin ϕ

MO 71 MO 71

|P n (cos ϕ)| ≤ 1.

5. 6.

987

10

WH

Let n ≥ 2. The successive relative maxima of |P n (x)|, when x decreases from 1 to 0, form a decreasing sequence. More precisely, if μ1 , μ2 , . . . , μ n/2 denote these maxima corresponding to decreasing values of x, we have 1 > μ1 > μ2 > · · · > μ n/2

7.10 8.

10

SZ 162(7.3.1)

1/2

Let n ≥ 2. The successive relative maxima of (sin θ) π/2, form an increasing sequence.

|P n (cos θ)| when θ increases from 0 to SZ 163(7.3.2)

We have 1/2

(sin θ)

|P n (cos θ)| < (2/π)1/2 n−1/2

[0 ≤ qθ ≤ qπ]

SZ 163(7.3.8)

Here the constant (2/π)1/2 cannot be replaced by a smaller one. 9.10

1/2

max (sin θ)

0≤qθ≤qπ

1 |P n (cos θ)| ∼ = (2/π)1/2 n− 2

[n → ∞]

SZ 164(7.3.12)

10.10 Stieltjes’ first theorem: |P n (cos θ)| ≤

1/2 2 4 √ π n sin θ

[n = 1, 2, . . . , 0 < θ < π]

SA 197(8)

11.10 Stieltjes’ second theorem:

12.10 13.10 8.91810

4 |P n (x) − P n+2 (x)| < √ √ π n+2 ! ! √ ! d P n (x) ! 2 n ! ! ! dx ! < √π 1 − x2 12 2 |P n+1 (x) + P n (x)| < 6 (1 − x)−1/2 πn

SA 199(15)

[|x| < 1,

n = 1, 2, . . .]

SA 201(18)

[|x| < 1,

n = 0, 1, . . .]

SA 201(19)

Asymptotic approximations:

1.

[|x| ≤ 1]

P n (cos θ) =

2 πn sin ϕ

1/2



  1 π θ− + O n−3/2 2 4 [ε ≤ θ ≤ π − ε, 0 < ε < π/2m]

cos

n+

(Laplace’s formula)

SA 208(1)

988

Orthogonal Polynomials

8.921



1/2





 2 1 1 1 π 1 π cos θ sin n + 1− cos n + θ− + θ− πn sin θ 4n 2 4 8n 2 4  −5/2 +O n

2.

P n (cos θ) =

[ε ≤ θ ≤ π − ε,

0 < ε < π/2]

(Bonnet–Heine formula)

SA 208(2)

8.91910 Series of products of Legendre and Chebyshev polynomials  1.

i+j=n   1

1

2 −1

T n (x) P n (x) dx =

i,j=0

−1

P i (x) P j (x) P n (x) dx

8.92 Series of Legendre polynomials 8.921

The generating function: ∞  1 = tk P k (z) √ 1 − 2tz + t2 k=0 ∞  1 = P k (z) tk+1

! !, +  ! ! |t| < min !z ± z 2 − 1!

SM 489(31), WH

! !, +  ! ! |t| > max !z ± z 2 − 1!

MO 70

k=0

8.922 ∞

1.

z 2n =

 1 2n(2n − 2) . . . (2n − 2k + 2) P 0 (z) + P 2k (z) (4k + 1) 2n + 1 (2n + 1)(2n + 3) . . . (2n + 2k + 1)

MO 72

k=1

∞

2.

3.

 3 2n(2n − 2) . . . (2n − 2k + 2) P 1 (z) + P 2k+1 (z) (4k + 3) 2n + 3 (2n + 3)(2n + 5) . . . (2n + 2k + 3) k=1

2 ∞ (2k − 1)!! 1 π √ = (4k + 1) P 2k (x) [|x| < 1, (−1)!! ≡ 1] 2 2k k! 1 − x2

z 2n+1 =

MO 72

k=0

MO 72, LA 385(15)

4.

5.

√

∞ x π (2k − 1)!!(2k + 1)!! P 2k+1 (x) = (4k + 3) 2k+1 2 2 2 k!(k + 1)! 1−x k=0

 π 1 − x2 = 2 

6.10



∞

[|x| < 1, 

(−1)!! ≡ 1]

LA 385(17)

[|x| < 1,

(−1)!! ≡ 1]

LA 385(18)

1  (2k − 3)!!(2k − 1)!! − P 2k (x) (4k + 1) 2k+1 2 2 k!(k + 1)! k=1

∞ 

2 1−x 1 = P 0 (x) − 2 P n (x) 2 3 (2n − 1)(2n + 3) n=1

[−1 ≤ x ≤ 1]

8.925

7.10

8.923

Series of Legendre polynomials

1 − ρ2 (1 − 2ρx +

1/2 ρ2 )

=1+

∞ 

(2n + 1)ρn P n (x),

[|ρ| < 1,

|x| ≤ 1]

SA 170(4)

n=0

2 ∞ π  (2k − 1)!! arcsin x = [P 2k+1 (x) − P 2k−1 (x)] + πx/2 2 2k k! k=1

[|x| < 1, 8.924

989

(−1)!! ≡ 1]

WH

    ∞ 1 + cos nπ 1 + cos nπ  (4k + 5)n2 n2 − 22 . . . n2 − (2k)2 P P 2k+2 (cos θ) (cos θ) − 0 2 (n2 − 1) 2 (n2 − 12 ) (n2 − 32 ) . . . [n2 − (2k + 3)2 ] k=0 3 (1 − cos nπ) P 1 (cos θ) − 2 (n2 − 22 )     ∞ 1 − cos nπ  (4k + 3) n2 − 12 . . . n2 − (2k − 1)2 − P 2k+1 (cos θ) = cos nθ 2 (n2 − 22 ) (n2 − 42 ) . . . [n2 − (2k + 2)2 ]

−

1.

k=1

AD (9060.1)

∞ 

    (4k + 5)n2 n2 − 22 . . . n2 − (2k)2 P 2k+2 (cos θ) (n2 − 12 ) (n2 − 32 ) . . . [n2 − (2k + 3)2 ]

− sin nπ sin nπ P 0 (cos θ) − 2 (n2 − 1) 2 k=0 3 sin nπ P 1 (cos θ) + 2 (n2 − 22 )      ∞ sin nπ  (4k + 3) n2 − 12 n2 − 32 . . . n2 − (2k − 1)2 + P 2k+1 (cos θ) = sin nθ 2 (n2 − 22 ) (n2 − 42 ) . . . [n2 − (2k + 2)2 ]

2.

k=1

AD (9060.2)

3.

3

n/2

 2n−1 n! 2n−2k−1 (n − k − 1)!(2k − 3)!! P n (cos θ) − n P n−2k (cos θ) (2n − 4k + 1) (2n − 1)!! (2n − 2k + 1)!!k! k=1

= cos nθ AD (9061.1)

4.

∞ n  (2n + 2k − 1)!!(2k − 1)!! (2n + 4k + 3) (2n − 1)!!Pn−1 (cos θ) − P n+2k+1 (cos θ) 2n−1 (n − 1)! 2n+1 22k (n + k + 1)!(k + 1)! k=0 4 sin nθ = π

AD (9061.2)

8.925 1.

2.

3.

 2 (2k − 1)!! 4k − 1 2θ P 2k−1 (cos θ) = 1 − 22k (2k − 1)2 k! π k=1  2 ∞  4k + 1 (2k − 1)!! 1 2 sin θ P 2k (cos θ) = − 22k+1 (2k − 1)(k + 1) k! 2 π k=1  2 ∞  (2k − 1)!! k(4k − 1) 2 cot θ P 2k−1 (cos θ) = 22k−1 (2k − 1) k! π ∞ 

k=1

AD (9062.2)

AD (9062.3)

990

Orthogonal Polynomials

 2 ∞  4k + 1 (2k − 1)!!

4.

22k

k=1

8.926 1.

k!

P 2k (cos θ) =

8.926

2 −1 π sin θ

AD (9062.4)



∞  2 tan π−θ θ θ 1 4 P n (cos θ) = ln = − ln sin − ln 1 + sin n sin θ 2 2 n=1 ∞ 

1 + sin θ2 1 P n (cos θ) = ln −1 n+1 sin θ2 n=1

2.

8.927

AD (9063.2)

AD (9063.1)

∞ 

  1 cos k + 12 β P k (cos ϕ) =  2 (cos β − cos ϕ) k=0 =0

[0 ≤ β < ϕ < π] [0 < ϕ < β < π] MO 72

8.928 1.

∞ 3  (−1)n (4k + 1) [(2n − 1)!!]

23n

n=1

2.

∞ 

3

(n!)

P 2n (cos θ) =

4 K (sin θ) −1 π2

3

(−1)n+1

n=1

(4n + 1) [(2n − 1)!!]

3

(2n − 1)(2n + 2)23n (n!)

P 2n (cos θ) =

4 E (sin θ) 1 − π2 2

AD (9064.1)

AD (9064.2)

• For series of products of Bessel functions and Legendre polynomials, see 8.511 4, 8.531 3, 8.533 1, 8.533 2, and 8.534. • For series of products of Legendre and Chebyshev polynomials, see 8.919.

8.93 Gegenbauer polynomials Cnλ (t) 8.930 Definition. The polynomials C λn (t) of degree n are the coefficients of αn in the power-series expansion of the function ∞ −λ   1 − 2tα + α2 = C λn (t)αn WH n=0

Thus, the polynomials C λn (t) are a generalization of the Legendre polynomials. 1.10

C λ0 (t) = 1

2.10

C λ1 (t) = 2λt

3.10

C λ2 (t) = 2λ(λ + 1)t2 − λ   C λ3 (t) = 13 λ 4λ2 + 12λ + 8 t3 − 2λ(λ + 1)t     C λ4 (t) = 23 λ λ3 + 6λ2 + 11λ + 6 t4 − 2λ λ2 + 3λ + 2 t2 + 12 λ(λ + 1)   1 C λ5 (t) = 15 λ 4λ4 + 40λ3 + 140λ2 + 200λ + 96 t5     − 13 λ 4λ3 + 24λ2 + 44λ + 24 t3 + λ λ2 + 3λ + 2 t

4.10 5.11 6.10

Gegenbauer polynomials Cnλ (t)

8.934

7.10

8.931

C λ6 (t) =

991

 5  λ + 60λ4 + 340λ3 + 900λ2 + 1096λ + 480 t6   − 13 λ 2λ4 + 20λ3 + 70λ2 + 100λ + 48 t4     +λ λ3 + 6λ2 + 11λ + 6 t2 + 16 λ λ2 + 3λ + 2 1 45 λ

Integral representation:

 π   n  1 Γ(2λ + n) Γ 2λ+1 2 √ t + t2 − 1 cos ϕ sin2λ−1 ϕ dϕ = Γ(λ) π n! Γ(2λ) 0 See also 3.252 11, 3.663 2, 3.664 4. C λn (t)

MO 99

Functional relations 8.932 1.

2. 3. 8.933 1. 2. 3. 4.

Expressions in terms of hypergeometric functions:

∗ 1 1−t Γ(2λ + n) λ F 2λ + n, −n; λ + ; C n (t) = Γ(n + 1) Γ(2λ) 2 2

n 1 2 Γ(λ + n) n n 1−n t F − , ; 1 − λ − n; 2 = n! Γ(λ) 2 2 t

n 1 (−1) F −n, n + λ; ; t2 C λ2n (t) = (λ + n) B(λ, n + 1) 2

n 3 2 (−1) 2t λ F −n, n + λ + 1; ; t C 2n+1 (t) = B(λ, n + 1) 2

MO 97 MO 99 MO 99 MO 99

Recursion formulas: (n + 2) C λn+2 (t) = 2(λ + n + 1)t C λn+1 (t) − (2λ + n) C λn (t) + , λ+1 n C λn (t) = 2λ t C λ+1 (t) − C (t) n−1 n−2 + , λ+1 (2λ + n) C λn (t) = 2λ C λ+1 (t) − t C (t) n n−1   n C λn (t) = (2λ + n − 1)t C λn−1 (t) − 2λ 1 − t2 C λ+1 n−2 (t)

8.934

  1 −λ  λ+n− 12 , 1 − t2 2 dn + (−1)n Γ(2λ + n) Γ 2λ+1 2  2λ+1  1 − t2 = n n 2 n! dt Γ(2λ) Γ 2 + n

1.

C λn (t)

2.

C λn (cos ϕ) =

n  Γ(λ + k) Γ(λ + l) k,l=0 k+l=n

∗ Equation

k!l! [Γ(λ)]

2

cos(k − l)ϕ

8.932.1 defines the generalized functions C λ n (t), where the subscript n can be an arbitrary number.

Mo 98 WH WH WH

WH

MO 99

992

3.

Orthogonal Polynomials

8.935

C λn (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ) n 2 Γ(2λ − 1)  22k (n − k)! [Γ(λ + k)] (2λ + 2k − 1) sink ψ si nk ϑ = 2 Γ(2λ + n + k) [Γ(λ)] k=0 λ− 1



4.

ψ, ϑ, ϕ real;

lim Γ(λ) C λn (cos ϕ) =

λ→0

λ+k 2 × C λ+k (cos ϕ) n−k (cos ψ) C n−k (cos ϑ) C k  1 λ = 2 [“summation theorem”] (see also 8.794–8.796)

2 cos nϕ n

WH

MO 98

For orthogonality, see 8.904, 7.313. 8.935 Derivatives: 1.

2.11

Γ(λ + k) λ+k dk λ C n−k (t) C n (t) = 2k k dt Γ(λ) In particular,

MO 99

d C λn (t) = 2λ C λ+1 n−1 (t) dt

WH

For integrals of the polynomials C λn (x) see 7.31–7.33. 8.936 Connections with other functions:   1 λ  4 − 2 12 −λ Γ(2λ + n) Γ λ + 12 12 λ t −1 1. C n (t) = P λ+n− 1 (t) 2 Γ(2λ) Γ(n + 1) 4 m   2 −2 m!2m m dm P n (t) 1 m+ 1 m 1−t P n (t) 2. C n−m2 (t) = = (−1) m (2m − 1)!! dt (2m)! [m + 1 a natural number] 3.

C 1/2 n (t) = P n (t)

4.

J λ− 12 (r sin ϑ sin α) (r sin ϑ sin α)

5.

MO 98

MO 98, WH

−λ+ 12

λ

lim λ− 2 C n2 n

λ→∞

e−ir cos ϑ cos α ∞ λ λ √ Γ(λ)  −k Jλ+k (r) C k (cos ϑ) C k (cos α)  = 2  (λ + k)i Γ λ + 12 k=0 rλ C λk (1)

  n 2− 2 2 = H n (t) t λ n!

MO 99 MO 99a

See also 8.932. 8.937 Special cases and particular values: sin(n + 1)ϕ sin ϕ

1.

C 1n (cos ϕ) =

2.

C 00 (cos ϕ) = 1

MO 98

3.

C λ0 (t) ≡ 1

MO 98

MO 99

8.940

The Chebyshev polynomials

4.

C λn (1)

≡

2λ + n − 1 n

993

MO 98

A differential equation leading to the polynomials C λn (t): (2λ + 1)t  n(2λ + n) y − y=0 (cf. 9.174) y  + 2 t −1 t2 − 1 For series of products of Bessel functions and the polynomials C λn (x), see 8.532, 8.534. 8.93910 Differentiation and Rodrigues’ formulas and orthogonality relation

8.938

1. 2. 3. 4. 5.

WH

d λ C (t) = 2λ C λ+1 MS n−1 (t) dt n dm λ C (t) = 2m λ(λ + 1)(λ + 2) . . . (λ + m − 1) C λ+m MS n−m (t) dtm n d λ d C (t) = t C λn (t) − n C λn (t) MS dt n−1 dt d λ d C (t) = t C λn (t) + (2λ + n) C λn (t) MS dt n+1 dt  d λ  C (t) = (n + 2λ − 1) C λn−1 (t) − nt C λn (t) = (n + 2λ)t C λn (t) − (n + 1) C λn+1 (t) 1 − t2 dt n   = 2λ 1 − t2 C λ+1 n−1 (t)

5.3.2 5.3.2 5.3.2 5.3.2

MS 5.3.2

6.

, d + λ C n+1 (t) − C λn−1 (t) = 2(n + λ) C λn (t) dt

7.

C λn (t) =



MS 5.3.2

 1 −λ  1,  (−1)n 2λ(2λ + 1)(2λ + 2) . . . (2λ + n − 1) 1 − t2 2 dn +  2 n+λ− 2       1 − t dtn 2n n! λ + 12 λ + 32 . . . λ + n − 12 1     −λ 1,  (−1)n Γ λ + 12 Γ(n + 2λ) 1 − t2 2 dn +  2 n+λ− 2   1 − t = dtn 2n n! Γ(2λ) Γ n + λ + 12 [Rodrigues’ formula]

1

8. −1

λ− 12  C λn (t) C λm (t) 1 − t2 dt = 0 =

π2

MS 5.3.2

n = m 1−2λ

Γ(n + 2λ)

n!(λ + n) [Γ(λ)]

2

n=m [λ = 0]

[Orthogonality relation]

MS 5.3.2

8.94 The Chebyshev polynomials Tn (x) and Un (x) 8.940 1.

Definition Chebyshev’s polynomials of the first kind n  n ,   1 + T n (x) = cos (n arccos x) = x + i 1 − x2 + x − i 1 − x2 2 n    n n−4  2  n n−6  3 n n−2 1 − x2 + x 1 − x2 − 1 − x2 + . . . =x − x x 4 2 6 NA 66, 71

994

2.

Orthogonal Polynomials

8.941

Chebyshev’s polynomials of the second kind:  n+1  n+1    1 sin [(n + 1) arccos x] 2 2 x+i 1−x = √ U n (x) = − x−i 1−x 2 x] sin [arccos





2i 1 − x  2 n+1 n n + 1 n−4  n + 1 n−2  = x − 1 − x2 + x 1 − x2 − . . . x 1 5 3

Functional relations 8.941

Recursion formulas:

1.

T n+1 (x) − 2x T n (x) + T n−1 (x) = 0

2.

U n+1 (x) − 2x U n (x) + U n−1 (x) = 0

3.

T n (x) = U n (x) − x U n−1 (x)   1 − x2 U n−1 (x) = x T n (x) − T n+1 (x)

4.

NA 358

EH II 184(3) EH II 184(4)

For the orthogonality, see 7.343 and 8.904. 8.942 Relations with other functions:

1 1−x 1. T n (x) = F n, −n; ; 2 2 √ n− 12 1 − x2 dn  1 − x2 2. T n (x) = (−1)n n (2n − 1)!! dx n+ 12 (−1)n (n + 1) dn  1 − x2 3. U n (x) = √ n 1 − x2 (2n + 1)!! dx

MO 104

MO 104 EH II 185(15)

See also 8.962 3. 8.94310 Special cases 1.

T 0 (x) = 1

10.

U 0 (x) = 1

2.

T 1 (x) = x

11.

U 1 (x) = 2x

3.

T 2 (x) = 2x2 − 1

12.

U 2 (x) = 4x2 − 1

4.

T 3 (x) = 4x3 − 3x

13.

U 3 (x) = 8x3 − 4x

5.

T 4 (x) = 8x4 − 8x2 + 1

14.

U 4 (x) = 16x4 − 12x2 + 1

6.

T 5 (x) = 16x5 − 20x3 + 5x

15.

U 5 (x) = 32x5 − 32x3 + 6x

7.

T 6 (x) = 32x − 48x + 18x − 1

16.

U 6 (x) = 64x6 − 80x4 + 24x2 − 1

8.

T 7 (x) = 64x7 − 112x5 + 56x3 − 7x

17.

U 7 (x) = 128x7 − 192x5 + 80x3 − 8x

9.

T 8 (x) = 128x8 − 256x6 + 160x4 − 32x2 + 1

18.

U 8 (x) = 256x8 − 448x6 + 240x4 − 40x2 + 1

6

4

2

8.949

8.944 1.

The Chebyshev polynomials

Particular values: T n (1) = 1 n

2.

T n (−1) = (−1)

3.

n

T 2n (0) = (−1)

4.

T 2n+1 (0) = 0

8.945

995

5.

U 2n+1 (0) = 0

6.

U 2n (0) = (−1)n

The generating function: ∞

1.11

 1 − t2 = T 0 (x) + 2 T k (x)tk 2 1 − 2tx + t

[|t| < 1]

MO 104

[|t| < 1]

MO 104a, EH II 186(31)

k=1

∞

2.11

 1 = U k (x)tk 2 1 − 2tx + t k=0

8.946 Zeros. The polynomials T n (x) and U n (x) only have real simple zeros. All these zeros lie in the interval (−1, +1). √ 8.947 The functions T n (x) and 1 − x2 U n−1 (x) are two linearly independent solutions of the differential equation   d2 y dy + n2 y = 0. 1 − x2 −x NA 69(58) dx2 dx 8.948 Of all polynomials of degree n with leading coefficient equal to 1, the one that deviates the least from zero on the interval [−1, +1] is the polynomial 2−n+1 T n (x). 8.94910 Differentiation and Rodrigues’ formulas and orthogonality relations 1. 2. 3. 4. 5. 6.

d T n (x) = n U n−1 (x) MS dx dm T n (x) = 2m−1 Γ(m)n C m MS n−m (x) dxm  d  T n (x) = n [T n−1 (x) − x T n (x)] = n [x T n (x) − T n+1 (x)] 1 − x2 MS dx d U n (x) = 2 C 2n−1 (x) MS dx dm m+1 U n (x) = 2m m!Cn−m (x) MS dxm  d  U n (x) = (n + 1) U n−1 (x) − nx U n (x) = (n + 2)x U n (x) − (n + 1) U n+1 (x) 1 − x2 dx

5.7.2 5.7.2 5.7.2 5.7.2 5.7.2

MS 5.7.2

7.

8.

 c 1  1, (−1)n π 1/2 1 − x2 2 dn + 2 n− 2   1 − x T n (x) = [Rodrigues’ formula] dxn 2n+1 Γ n + 12 −1/2   1, (−1)n π 1/2 (n + 1) 1 − x2 dn + 2 n+ 2   1 − x U n (x) = dxn 2n+1 Γ n + 32 [Rodrigues’ formula]

MS 5.7.2

MS 5.7.2

996

Orthogonal Polynomials



1

9. −1



1

10. −1

8.950

−1/2  T m (x) T n (x) 1 − x2

⎧ ⎪ m = n ⎨0, dx = π/2, m = n = 0 ⎪ ⎩ π, m=n=0

−1/2  U m (x) U n (x) 1 − x2

 0, m = n dx = π/8, m = n

[Orthogonality relation]

MS 5.7.2

[Orthogonality relation]

MS 5.7.2

8.95 The Hermite polynomials H n (x) 8.950

Definition 2

H n (x) = (−1)n ex

1.

dn  −x2 e dxn

or H n (x) = 2n xn − 2n−1

2. 3.10

SM 567(14)

n n n xn−2 + 2n−2 · 1 · 3 · xn−4 − 2n−3 · 1 · 3 · 5 · xn−6 + . . . 2 4 6

MO 105a

H 0 (x) = 1

4.

10

H 1 (x) = 2x

5.

10

H 2 (x) = 4x2 − 2

6.10

H 3 (x) = 8x3 − 12x

7.10

H 4 (x) = 16x4 − 48x2 + 12

8.10

H 5 (x) = 32x5 − 160x3 + 120x

9.10

H 6 (x) = 64x6 − 480x4 + 720x2 − 120

10.10 H 7 (x) = 128x7 − 1344x5 + 3360x3 − 1680x 11.10 H 8 (x) = 256x8 − 3584x6 + 13440x4 − 13440x2 + 1680 8.951

The integral representation: 2n H n (x) = √ π



∞

(x + it)n e−t dt 2

MO 106a

−∞

Functional relations 8.952

Recursion formulas: d H n (x) = 2n H n−1 (x) dx H n+1 (x) = 2x H n (x) − 2n H n−1 (x)

1. 2.

SM 569(22) SM 570(23)

For the orthogonality, see 7.374 1 and 8.904. 3.

10

n H n (x) = −n H n−1 (x) + x H n (x)

MS 5.6.2

8.957

Hermite polynomials

H n (x) = 2x H n−1 (x) − H n−1 (x)

4.10 8.953

 (2n)!  Φ −n, 12 ; x2 n!   (2n + 1)! x Φ −n, 32 ; x2 H 2n+1 (x) = (−1)n 2 n!

2.

MS 5.6.2

The connection with other functions: H 2n (x) = (−1)n

1.

997

MO 106a MO 106a

• For a connection with the polynomials C λn (x), see 8.936 5. • For a connection with the Laguerre polynomials, see 8.972 2 and 8.972 3. • For a connection with functions of a parabolic cylinder, see 9.253. 8.954

Inequalities: √ n! e2x n/2

n/2!

1.10

|H n (x)| ≤ 2 2 − 2 

2.10

√ 2 |H n (x)| < k n!2n/2 ex /2 ,

8.955 1. 2. 8.956

n

n

k ≈ 1.086435

MO 106a SA 324

Asymptotic representation:

   √ 1 cos 4n + 1x + O √ H 2n (x) = (−1) 2 (2n − 1)!!e 4 n 

 √  √ 2 1 1 H 2n+1 (x) = (−1)n 2n+ 2 (2n − 1)!! 2n + 1ex /2 sin 4n + 3x + O √ 4 n x2 /2

n n

SM 579 SM 579

Special cases and particular values:

1.

H 0 (x) = 1

2.

H 1 (x) = 2x

3.

H 2 (x) = 4x2 − 2

4.

H 3 (x) = 8x3 − 12x

5.

H 4 (x) = 16x4 − 48x2 + 12

6.

H 2n (0) = (−1)n 2n (2n − 1)!!

7.

H 2n+1 (0) = 0

SM 570(24)

Series of Hermite polynomials 8.957 1.

The generating function: ∞ k    t H k (x) exp −t2 + 2tx = k!

SM 569(21)

k=0

∞

2.

 1 1 sinh 2x = H 2k+1 (x) e (2k + 1)! k=0

MO 106a

998

Orthogonal Polynomials

8.958

∞

3.

 1 1 cosh 2x = H 2k (x) e (2k)!

MO 106a

k=0

4.

∞ 

e sin 2x =

(−1)k

k=0

5.

e cos 2x =

∞ 

(−1)k

k=0

8.958

1.11

2.

1 H 2k+1 (x) (2k + 1)!

MO 106a

1 H 2k (x) (2k)!

MO 106a

“The summation theorem”:  r  n2 ⎛ r ⎞   2 ak ak xk ⎟ ⎜  r mk  ⎜ k=1 ⎟ ak k=1 ⎟=  Hn ⎜ H (x ) # mk k ⎜ n! a2k ⎟ ⎝ ⎠ m1 +m2 +···+mr =n k=1 mk ! A special case: n

2 2 H n (x + y) =

n   n k=0

8.959 1.

2. 3.

2.

k

 √  √ H n−k x 2 H k y 2

MO 107a

Hermite polynomials satisfy the differential equation d2 u n dun + 2nun = 0; − 2x SM 566(9) 2 dx dx A second solution of this differential equation is provided by the functions (A and B are arbitrary constants):   u2n = Ax Φ 12 − n; 32 ; x2 ,   u2n+1 = B Φ − 12 − n; 12 ; x2 MO 107

8.959(1)10 1.

MO 106a

Rodrigues’ formula and orthogonality relation

dn + −x2 , e dxn   ∞ 0 −x2 e H m (x) H n (x) dx = π 1/2 2n n! −∞ 2

H n (x) = (−1)n ex

[Rodrigues’ formula] for m = n for m = n

MS 5.6.2 MS 5.6.2

8.96 Jacobi’s polynomials 8.960 1.

Definition n   (−1)n −α −β d (1 − x) (1 − x)α+n (1 + x)β+n (1 + x) n n 2 n!



dx n n+β 1  n+α = n (x − 1)n−m (x + 1)m m n−m 2 m=0

P (α,β) (x) = n

EH II 169(10), CO EH II 169(2)

8.962

8.961

Jacobi’s polynomials

999

Functional relations:

1.11

P (α,α) (−x) = (−1)n P (α,α) (x) n n

2.

2(n + 1)(n + α + β + 1)(2n + α + β) P n+1 (x)   = (2n + α + β + 1) (2n + α + β)(2n + α + β + 2)x + α2 − β 2 P (α,β) (x) n

EH II 169(13) (α,β)

(α,β)

−2(n + α)(n + β)(2n + α + β + 2) P n−1 (x) EH II 169(11)

3.

 d (α,β)  P (2n + α + β) 1 − x2 (x) = n[(α − β) − (2n + α + β)x] P (α,β) (x) n dx n (α,β) +2(n + α)(n + β) P n−1 (x) EH II 170(15) m

4.11

d dxm

+ , 1 Γ(n + m + α + β + 1) (α+m,β+m) P (α,β) P n−m (x) = m (x) n 2 Γ(n + α + β + 1) [m = 1, 2, . . . , n]





EH II 170(17) (α,β)

EH II 173(32)

6.

n + 12 α + 12 β + 1 (1 − x) P (α+1,β) (x) = (n + α + 1) P (α,β) (x) − (n + 1) P n+1 (x) n n   (α,β) n + 12 α + 12 β + 1 (1 + x) P (α,β+1) (x) = (n + β + 1) P (α,β) (x) + (n + 1) P n+1 (x) n n

7.

(1 − x) P (α+1,β) (x) + (1 + x) P (α,β+1) (x) = 2 P (α,β) (x) n n n

EH II 173(34)

8.

(2n + α + β) P (α−1,β) (x) = (n + α + β) P (α,β) (x) − (n + β) P n−1 (x) n n

9.

(2n + α + β) P (α,β−1) (x) = (n + α + β) P (α,β) (x) + (n + α) P n−1 (x) n n

10.

P (α,β−1) (x) − P (α−1,β) (x) = P n−1 (x) n n

5.

8.962 1.

(α,β)

(α,β)

(α,β)

EH II 173(33)

EH II 173(35) EH II 173(36) EH II 173(37)

Connections with other functions:



(−1)n Γ(n + 1 + β) 1+x F n + α + β + 1, −n; 1 + β; n! Γ(1 + β)

2 Γ(n + 1 + α) 1−x = F n + α + β + 1, −n; 1 + α; n! Γ(1 + α) 2

n

Γ(n + 1 + α) 1 + x x−1 = −n, −n − β; α + 1; F n! Γ(1 + α) 2

x + 1

n Γ(n + 1 + β) x − 1 x+1 = F −n, −n − α; β + 1; n! Γ(1 + β) 2 x−1

P (α,β) (x) = n

2.

P n (x) = P (0,0) (x) n

3.

T n (x) =

CO, EH II 170(16) EH II 170(16) EH II 170(16) EH II 170(16) CO, EH II 179(3)

2

4.

22n (n!) (− 1 ,− 1 ) P n 2 2 (x) (2n)!   Γ(n + 2ν) Γ ν + 12 ν  P (ν−1/2,ν−1/2)  C n (x) = (x) n Γ(2ν) Γ n + ν + 12

CO, EH II 184(5)a

MO 108a, EH II 174(4)

1000

Orthogonal Polynomials

8.963

8.963 The generating function: ∞  P (α,β) (x)z n = 2α+β R−1 (1 − z + R)−α (1 + z + R)−β , n n=0

R=

 1 − 2xz + z 2

[|z| < 1] EH II 172(29)

8.964 The Jacobi polynomials constitute the unique rational solution of the differential (hypergeometric) equation   EH II 169(14) 1 − x2 y  + [β − α − (α + β + 2)x]y  + n(n + α + β + 1)y = 0. 8.965 Asymptotic representation (cos θ) = P (α,β) n       cos n + 12 (α + β + 1) θ − 12 α + 14 π [Im α = Im β = 0, 0 < θ < π] EH II 198(10) + O n−3/2 α+ 12  β+ 12 √  1 1 cos 2 θ πn sin 2 θ 8.966 A limit relationship: +  z ,  z −α J α (z) EH II 173(41) lim n−α P (α,β) cos = n n→∞ n 2 8.967 If α > −1 and β > −1, all the zeros of the polynomial P (α,β) (x) are simple, and they lie in the n interval (−1, 1).

8.97 The Laguerre polynomials 8.970

1 x −α dn  −x n+α  e x e x n! dx n

n  n + α xm m = (−1) n − m m! m=0

Lα n (x) =

1.

[Rodrigues’ formula]

EH II 188(5), MO 108 MO 109, EH II 188(7)

L0n (x) = Ln (x)

2. 3.

Definition.

10

4.10 5.10 6.10 7.10

Lα 0 (x) Lα 1 (x)

=1

= −x + α + 1  1 2 x − 2(α + 2)x + (α + 1)(α + 2) Lα 2 (x) = 2  1 α L3 (x) = − x3 − 3(α + 3)x2 + 3(α + 2)(α + 3)x − (α + 1)(α + 2)(α + 3) 6  1 Lα (x) = x4 − 4(α + 4)x3 + 6(α + 3) (α + 4) x2 − 4(α + 2)(α + 3)(α + 4)x 4 24  + (α + 1)(α + 2)(α + 3)(α + 4)

8.10

Lα 5 (x) = −

 1 x5 − 5(α + 5)x4 + 10(α + 4)(α + 5)x3 − 10(α + 3)(α + 4)(α + 5)x2 120 

+ 5(α + 2)(α + 3)(α + 4)(α + 5)x − (α + 1)(α + 2)(α + 3)(α + 4)(α + 5)

ET I 369

8.973

8.971 1. 2.11 3.

The Laguerre polynomials

1001

Functional relations:  d  α α L (x) − Lα n+1 (x) = Ln (x) dx n α n Lα d α n (x) − (n + α) Ln−1 (x) Ln (x) = − Lα+1 n−1 (x) = dx x d α x Lα (x) = n Lα n (x) − (n + α) Ln−1 (x) dx n α = (n + 1) Lα n+1 (x) − (n + α + 1 − x) Ln (x)

EH II 189(16) EH II 189(15), SM 575(42)a

EH II 189(12), MO 109 α x Lα+1 (x) = (n + α + 1) Lα n n (x) − (n + 1) Ln+1 (x)

4.

α = (n + α) Lα n−1 (x) − (n − x) Ln (x)

SM 575(43)a, EH II 190(23)

Lnα−1 (x)

5. 6.

(n +

=

Lα n (x)

1) Lα n+1 (x)

−

Lα n−1 (x)

− (2n + α + 1 −

SM 575(44)a, EH II 190(24)

x) Lα n (x)

+ (n +

α) Lα n−1 (x)

=0

[n = 1, 2, . . .] 7.10 8.

10

α (n + α) Lnα−1 (x) = (n + 1) Lα n+1 (x) − (n + 1 − x) Ln (x)

n Lα n (x)

= (2n + α − 1 −

x) Lα n−1 (x)

− (n + α −

MS 5.5.2

1) Lα n−2 (x) [n = 2, 3, . . .]

8.972 1. 2. 3. 8.973

Connections with other functions:

n+α α Ln (x) = Φ(−n, α + 1; x) n  2 x H 2n (x) = (−1)n 22n n! L−1/2 n  2 x H 2n+1 (x) = (−1)n 22n+1 n!x L1/2 n

MO 109, EH II 190(25, 24)

MS 5.5.2

MO 109, FI II 189(14) EH II 193(2), SM 576(47) EH II 193(3), SM 577(48)

Special cases:

1.

Lα 0 (x) = 1

EH II 188(6)

2.

Lα 1 (x) = α + 1 − x

n+α α Ln (0) = n

EH II 188(6)

3.

xn n!

4.

n L−n n (x) = (−1)

5.

L1 (x) = 1 − x

6.

L2 (x) = 1 − 2x +

x2 2

EH II 189(13) MO 109

MO 109

1002

8.974 1.

Orthogonal Polynomials

8.974

Finite sums: n 

 α  m! (n + 1)! α α α Lα Ln (x) Lα m (x) Lm (y) = n+1 (y) − Ln+1 (x) Ln (y) Γ(m + α + 1) Γ(n + α + 1)(x − y) m=0 EH II 188(9)

2.11

3.

n  Γ(α − β + m) β Ln−m (x) = Lβn (x) Γ(α − β)m! m=0 n 

MO 110, EH II 192(39)

α+1 Lα (x) m (x) = Ln

EH II 192(38)

β α+β+1 Lα (x + y) m (x) Ln−m (y) = Ln

EH II 192(41)

m=0

4.11

n  m=0

8.975 1.

2.

Arbitrary functions: (1 − z)−α−1 exp e

−xz

α

(1 + z) =

∞  xz = Lα (x)z n z − 1 n=0 n ∞ 

Lnα−n (x)z n

[|z| < 1]

EH II 189(17), MO 109

[|z| < 1]

MO 110, EH II 189(19)

[α > −1]

EH II 189(18), MO 109

n=0

3. 8.976 1.

∞   √  1 J α 2 xz ez (xz)− 2 α =

zn Lα n (x) Γ(n + α + 1) n=0

Other series of Laguerre polynomials:

√

1 α n xyz (xyz)− 2 α x+y Lα n (x) Ln (y)z = exp −z n! Iα 2 Γ(n + α + 1) 1−z 1−z 1−z n=0 ∞ 

[|z| < 1] 2.

∞  Lα n (x) = ex x−α Γ(α, x) n + 1 n=0

3.6

2 Lα n (x) =

Γ(n + α + 1)  22n n! n

k=0

4.6

α Lα n (x) Ln (y) =

[α > −1,

2n − 2k n−k



EH II 189(20)

x > 0]

EH II 215(19)

(2k)! 1 L2α (2x) k! Γ(α + k + 1) 2k

n α+2k Γ(1 + α + n)  Ln−k (x + y) (xy)k n! Γ(1 + α + k) k!

MO 110

MO 110, EH II 192(42)

k=0

8.977 1.

Summation theorems: Lnα1 +α2 +···+αk +k−1 (x1 + x2 + · · · + xk ) =



αk α2 1 Lα i1 (x1 ) Li2 (x2 ) · · · Lik (xk )

MO 110

i1 +i2 +···+i2 =n

2.

y Lα n (x + y) = e

∞  (−1)k k=0

k!

y k Lnα+k (x)

MO 110

8.982

8.978 1. 2. 3.

The Laguerre polynomials

1003

Limit relations and asymptotic behavior:

2x (α,β) Lα (x) = lim P 1 − n n β→∞ β +  x ,  √  1 = x− 2 α J α 2 x lim n−α Lα n n→∞ n + √  1 3 1 1 1 1 1 1 απ π , 2 x x− 2 α− 4 n 2 α− 4 cos 2 − Lα nx − + O n 2 α− 4 n (x) = √ e 2 4 π [Im α = 0,

EH II 191(35) EH II 191(36)

x > 0]

Laguerre polynomials satisfy the following differential equation: du d2 u + nu = 0 x 2 + (α − x + 1) dx dx 8.98011 Orthogonality relation   ∞ 0, m = n α  n+α  e−x xα Lα n (x) Lm (x) dx = Γ(1 + α) , m=n 0 n

EH II 199(1)

8.979

8.98110

EH II 188(10), SM 574(34)

MS 5.5.2

Behavior of relative maxima of |Lα n (x)|

1.

Let α be arbitrary and real. The sequence formed by the relative maxima of |Lα n (x)| and by the value of this function at x = 0, is decreasing for x < α + 12 , and increasing for x > α + 12 . The successive relative maxima of |Lα n (x)| form a decreasing sequence for x ≤ 0, and an increasing SZ 174(7.6.1) sequence for x ≥ 0.

2.

Let α be an arbitrary real number. The successive relative maxima of −x/2 α/2+ 4 x |Lα e−x/2 x(α+1)/2 |Lα n (x)| and e n (x)| 1

form an increasing sequence, provided x > x0 . In the first case ⎧ ⎨0 if α2 ≤ 1, 2 x0 = ⎩ α −1 if α2 > 1 2n + α + 1 In the second case,

 0 x0 =  2 1  12 α −4

if α2 ≤ q 14 , if α2 >

1 4

SZ 174(7.6.2)

In the first case, we take n so large that 2n + α + 1 > 0. 8.98210 1.

Asymptotic and limiting behavior of Lα n (x) Let α be arbitrary and real, c and w fixed positive constants, and let n → ∞. Then   −α/2− 14 α/2− 14 if cn−1 ≤ qx ≤ qω O n x Lα n (x) = O (nα ) if 0 ≤ qx ≤ qcn−1 These bounds are precise as regards their orders in n. For α ≥ q − 12 , both bounds hold in both intervals, that is,

1004

Orthogonal Polynomials

Lα n (x) = 2.

  1 1 x−α/2− 4 O nα/2− 4 , O (nα ) ,

0 < x ≤ qω,

Let α be arbitrary and real. Then for an arbitrary complex z  −α/2 lim n−α Lα J α 2z 1/2 , n (x) = z n→∞

uniformly if z is bounded.

8.982

α≥q−

1 2

SZ 175(7.6.4)

SZ 191(8.1.3)

9.114

Integral representations

1005

9.1 Hypergeometric Functions 9.10 Definition 9.100

A hypergeometric series is a series of the form α(α + 1)β(β + 1) 2 α(α + 1)(α + 2)β(β + 1)(β + 2) 3 α·β z+ z + z + ... F (α, β; γ; z) = 1 + γ·1 γ(γ + 1) · 1 · 2 γ(γ + 1)(γ + 2) · 1 · 2 · 3 9.101 A hypergeometric series terminates if α or β is equal to a negative integer or to zero. For γ = −n (n = 0, 1, 2, . . .), the hypergeometric series is indeterminate if neither α nor β is equal to −m (where m < n and m is a natural number). However, 1.

lim

γ→−n

F (α, β; γ; z) α(α + 1) . . . (α + n)β(β + 1) . . . (β + n) = Γ(γ) (n + 1)! ×z n+1 F (α + n + 1, β + n + 1; n + 2; z) EH I 62(16)

9.102 If we exclude these values of the parameters α, β, γ, a hypergeometric series converges in the unit circle |z| <1. F then has a branch point at z = 1. Then we have the following conditions for convergence on the unit circle: 1.

1 > Re(α + β − γ) ≥0. point z = 1.

2.

Re(α + β − γ) <0.

The series converges (absolutely) throughout the entire unit circle.

3.

Re(α + β − γ) ≥1.

The series diverges on the entire unit circle.

The series converges throughout the entire unit circle, except at the

FI II 410, WH

9.11 Integral representations

 1 1 9.111 F (α, β; γ; z) = tβ−1 (1 − t)γ−β−1 (1 − tz)−α dt [Re γ > Re β > 0] B(β, γ − β) 0  2π   z −n Γ(p)n! cos nt dt 9.1128 F p, n + p; n + 1; z 2 = 2π Γ(p + n) 0 (1 − 2z cos t + z 2 )p [n = 0, 1, 2, . . . ; p = 0, −1, −2, . . . ; |z| < 1]

WH

WH, MO 16

 ∞i 1 Γ(α + t) Γ(β + t) Γ(−t) Γ(γ) (−z)t dt Γ(α) Γ(β) 2πi −∞i Γ(γ + t) Here, |arg(−z)| < π and the path of integration are chosen in such a way that the poles of the functions Γ(α + t) and Γ(β + t) lie to the left of the path of integration and the poles of the function Γ(−t) lie to the right of it.

 p+m (−2)m (p + m) π p+m ;1 − ; −1 = cosm ϕ cos pϕ dϕ 9.114 F −m, − 2 2 sin pπ 0 [m + 1 is a natural number; p = 0, ±1, . . . ] EH I 80(8), MO 16

9.113 F (α, β; γ; z) =

See also 3.194 1, 2, 5, 3.196 1, 3.197 6, 9, 3.259 3, 3.312 3, 3.518 4–6, 3.665 2, 3.671 1, 2, 3.681 1, 3.984 7.

1006

Hypergeometric Functions

9.121

9.12 Representation of elementary functions in terms of a hypergeometric functions 9.121 1.8 2. 3. 4. 5. 6. 7. 8.

F (−n, β; β; −z) = (1 + z)

n n − 1 1 z2 (t + z)n + (t − z)n ; ; 2 = F − ,− 2 2 2 t 2tn   z z n = 1+ lim F −n, ω; 2ω; − ω→∞ t 2t

n − 1 n − 2 3 z2 (t + z)n − (t − z)n ,− ; ; 2 = F − 2 2 2 t 2nztn−1  z (t + z)n − tn = F 1 − n, 1; 2; − t nztn−1 ln(1 + z) F (1, 1; 2; −z) = z

1+z ln 1 3 2 1−z , 1; ; z = F 2 2 2z   z z = 1 + z lim F 1, k; 2; lim F 1, k; 1; k→∞ k→∞ k k  z z2 lim F 1, k; 3; = · · · = ez =1+z+ 2 k→∞ k n

9. 10.

lim F

k→∞ k →∞

lim F k→∞

k →∞

11. 12. 13. 14. 15. 16. 17.



lim F

k→∞ k →∞

2

1 z k, k ; ; 2 4kk 3 z2 k, k ; ; 2 4kk 

z

GA 127 IIIa GA 127 IV GA 127 V GA 127 VI GA 127 VII

−z

=

e +e 2

= cosh z

GA 127 IX

=

sinh z ez − e−z = 2z z

GA 127 X



=

sin z z

z2 1 = cos z k, k ; ; − 2 4kk k →∞

1 1 3 z , ; ; sin2 z = F 2 2 2 sin z

3 z 2 F 1, 1; ; sin z = 2 sin z cos z

3 1 z , 1; ; − tan2 z = F 2 2 tan z

n+1 n−1 3 sin nz 2 ,− ; ; sin z = F 2 2 2 n sin z

n+2 n−2 3 sin nz ,− ; ; sin2 z = F 2 2 2 n sin z cos z lim F k→∞

GA 127 II

GA 127 VIII



z2 3 k, k ; ; − 2 4kk 

EH I 101(4), GA 127 Ia

GA 127 XI



GA 127 XII

GA 127 XIII GA 127 XIV GA 127 XV GA 127 XVI GA 127 XVII

9.121

Elementary functions as hypergeometric function

18.

F

19.

F

20.

F

21.

F

22.

F

23.

F

24.

F

25.

F

26.

F

27.

F

28.

F

29.

F

30.

F

31.

F

32.

F



n−2 n−1 3 sin nz 2 ,− ; ; − tan z = − 2 2 2 n sin z cosn−1 z

n+2 n+1 3 sin nz cosn+1 z , ; ; − tan2 z = 2 2 2 n sin z

n n 1 , − ; ; sin2 z = cos nz 2 2 2

n+1 n−1 1 cos nz ,− ; ; sin2 z = 2 2 2 cos z

n n−1 1 cos nz ; ; − tan2 z = − ,− 2 2 2 cosn z

n+1 n 1 , ; ; − tan2 z = cos nz cosn z 2 2 2

1 1 , 1; 2; 4z(1 − z) = 2 1−z

1 , 1; 1; sin2 z = sec z 2

1 1 3 2 arcsin z , ; ;z = 2 2 2 z

1 3 arctan z , 1; ; −z 2 = 2 2 z

1 1 3 arcsinh z , ; ; −z 2 = 2 2 2 z

1+n 1−n 3 2 sin (n arcsin z) , ; ;z = 2 2 2 nz

n n 3 sin (n arcsin z) √ 1 + , 1 − ; ; z2 = 2 2 2 nz 1 − z 2

n n 1 2 ,− ; ;z = cos (n arcsin z) 2 2 2

1+n 1−n 1 2 cos (n arcsin z) √ , ; ;z = 2 2 2 1 − z2

1007

GA 127 XVIII GA 127 XIX EH I 101(11), GA 127 XX EH I 101(11), GA 127 XXI EH I 101(11), GA 127 XXII GA 127 XXIII



|z| ≤ 12 ;

|z(1 − z)| ≤

1 4



(cf. 9.121 13) (cf. 9.121 15) (cf. 9.121 26) (cf. 9.121 16) (cf. 9.121 17) (cf. 9.121 20) (cf. 9.121 21)

The representation of special functions in terms of a hypergeometric function: • for complete elliptic integrals, see 8.113 1 and 8.114 1; • for integrals of Bessel functions, see 6.574 1, 3, 6.576 2–5, 6.621 1–3; • for Legendre polynomials, see 8.911 and 8.916. (All these hypergeometric series terminate; that is, these series are finite sums); • for Legendre functions, see 8.820 and 8.837; • for associated Legendre functions, see 8.702, 8.703, 8.751, 8.77, 8.852, and 8.853; • for Chebyshev polynomials, see 8.942 1; • for Jacobi’s polynomials, see 8.962;

1008

Hypergeometric Functions

9.122

• for Gegenbauer polynomials, see 8.932; • for integrals of parabolic cylinder functions, see 7.725 6. 9.122 1.

Particular values: F (α, β; γ; 1) =

Γ(γ) Γ(γ − α − β) Γ(γ − α) Γ(γ − β)

[Re γ > Re(α + β)] GA 147(48), FI II 793

2.

3.

F (α, β; γ; 1) = F (−α, −β; γ − α − β; 1) 1 = F (−α, β; γ − α; 1) 1 = F (α, −β; γ − β; 1)

3 1 π = F 1, 1; ; 2 2 2

[Re γ > Re(α + β)]

GA 148(49)

[Re γ > Re(α + β)]

GA 148(50)

[Re γ > Re(α + β)]

GA 148(51)

(cf. 9.121 14)

9.13 Transformation formulas and the analytic continuation of functions defined by hypergeometric series 9.130 The series F (α, β; γ; z) defines an analytic function that, speaking generally, has singularities at the points z = 0, 1, and ∞. (In the general case, there are branch points.) We make a cut in the z-plane along the real axis from z = 1 to z = ∞; that is, we require that |arg(−z)| < π for |z| ≥1. Then, the series f (α, β; γ; z) will, in the cut plane, yield a single-valued analytic continuation, which we can obtain by means of the formulas below (provided γ + 1 is not a natural number and α − β and γ − α − β are not integers). These formulas make it possible to calculate the values of F in the given region, even in the case in which |z| > 1. There are other closely related transformation formulas that can also be used to get the analytic continuation when the corresponding relationships hold between α, β, γ. Transformation formulas 9.131 1.11



z z − 1

z −β = (1 − z) F β, γ − α; γ; z−1 −α

F (α, β; γ; z) = (1 − z)

F

α, γ − β; γ;

GA 218(91) GA 218(92)

= (1 − z)γ−α−β F (γ − α, γ − β; γ; z) 2.

F (α, β; γ; z) =

Γ(γ) Γ(γ − α − β) F (α, β; α + β − γ + 1; 1 − z) Γ(γ − α) Γ(γ − β) Γ(γ) Γ(α + β − γ) +(1 − z)γ−α−β F (γ − α, γ − β; γ − α − β + 1; 1 − z) Γ(α) Γ(β) EH I 94, MO 13

9.136

Transformation formulas for hypergeometric series

9.132 1.

1009



(1 − z)−α Γ(γ) Γ(β − α) 1 F (α, β; γ; z) = F α, γ − β; α − β + 1; Γ(β) Γ(γ − α) 1−z

1 −β Γ(γ) Γ(α − β) +(1 − z) F β, γ − α; β − α + 1; Γ(α) Γ(γ − β) 1−z

Γ(γ) Γ(β − α) 1 −α (−z) F α, α + 1 − γ; α + 1 − β; F (α, β; γ; z) = Γ(β) Γ(γ − α) z

Γ(γ) Γ(α − β) 1 (−z)−β F β, β + 1 − γ; β + 1 − α; + Γ(α) Γ(γ − β) z [|arg z| < π, α − β = ±m, m = 0, 1, 2, . . .]

2.11

    F 2α, 2β; α + β + 12 ; z = F α, β; α + β + 12 ; 4z(1 − z) |z| ≤ 12 ,

9.133 9.134

|z(1 − z)| ≤



2  z 1 α α+1 1. , ;β + ; 2 2 2 2−z

4z 1 −2α 2. F (2α, 2α + 1 − γ; γ; z) = (1 + z) F α, α + ; γ; 2 (1 + z)2



1 2 1 4z 3. = (1 + z)−2α F α, β; 2β; F α, α + − β; β + ; z 2 2 (1 + z)2



1 1 2 2 ϕ 9.135 F α, β; α + β + ; sin ϕ = F 2α, 2β; α + β + ; sin 2 2 % 2 

z −α F (α, β; 2β; z) = 1 − F 2

1 4



MO 13

GA 220(93)

WH



ϕ real; x = sin 2 2

MO 13, EH I 111(4)

GA 225(100) GA 225(101)

& √ 1− 2 1 <x< 2 2 MO 13

8

9.136

then

We set

√  Γ α + β + 12 π   , A=  1 Γ α + 2 Γ β + 12

 √  − Γ α + β + 12 2 π ; B= Γ(α) Γ(β)

1.





√

√ 1 1− z 1 3 1 1 = AF α, β; ; z + B z F α + , β + ; ; z F 2α, 2β; α + β + ; 2 2 2 2 2 2

2.



√

√ 1 3 1 1+ z 1 1 = AF α, β; ; z − B z F α + , β + ; ; z F 2α, 2β; α + β + ; 2 2 2 2 2 2 

3.

 1

α− 2 β− α + β − 12

 1 2

√ A zF







√

1 1+ z 3 α, β; ; z = F 2α − 1, 2β − 1; α + β − ; 2 2 2 √

1 1− z − F 2α − 1, 2β − 1; α + β − ; 2 2

GA 227(106)

GA 227(107)

GA 229(110)

1010

9.1377

Hypergeometric Functions

9.137

Gauss’ recursion functions:

1.

γ[γ − 1 − (2γ − α − β − 1)z] F (α, β; γ; z) + (γ − α)(γ − β)z F (α, β; γ + 1; z) + γ(γ − 1)(z − 1) F (α, β; γ − 1; z) = 0

2.

(2α − γ − αz + βz) F (α, β; γ; z) + (γ − α) F (α − 1, β; γ; z) + α(z − 1) F (α + 1, β; γ; z) = 0

3.

(2β − γ − βz + αz) F (α, β; γ; z) + (γ − β) F (α, β − 1; γ; z) + β(z − 1) F (α, β + 1; γ; z) = 0 γ F (α, β − 1; γ; z) − γ F (α − 1, β; γ; z) + (α − β)z F (α, β; γ + 1; z) = 0

4. 5.

γ(α − β) F (α, β; γ; z) − α(γ − β) F (α + 1, β; γ + 1; z) + β(γ − α) F (α, β + 1; γ + 1; z) = 0

6.

γ(γ + 1) F (α, β; γ; z) − γ(γ + 1) F (α, β; γ + 1; z) − αβz F (α + 1, β + 1; γ + 2; z) = 0

7.

γ F (α, β; γ; z) − (γ − α) F (α, β + 1; γ + 1; z) − α(1 − z) F (α + 1, β + 1; γ + 1; z) = 0

8.

γ F (α, β; γ; z) + (β − γ) F (α + 1, β; γ + 1; z) − β(1 − z) F (α + 1, β + 1; γ + 1; z) = 0

9.

γ(γ − βz − α) F (α, β; γ; z) − γ(γ − α) F (α − 1, β; γ; z) + αβz(1 − z) F (α + 1, β + 1; γ + 1; z) = 0

10.

γ(γ − αz − β) F (α, β; γ; z) − γ(γ − β) F (α, β − 1; γ; z) + αβz(1 − z) F (α + 1, β + 1; γ + 1; z) = 0

11.

γ F (α, β; γ; z) − γ F (α, β + 1; γ; z) + αz F (α + 1, β + 1; γ + 1; z) = 0

12.8

γ F (α, β; γ; z) − γ F (α + 1, β; γ; z) + βz F (α + 1, β + 1; γ + 1; z) = 0

13.

γ[α − (γ − β)z] F (α, β; γ; z) − αγ(1 − z) F (α + 1, β; γ; z) + (γ − α)(γ − β)z F (α, β; γ + 1; z) = 0

14.

γ[β − (γ − α)z] F (α, β; γ; z) − βγ(1 − z) F (α, β + 1; γ; z) + (γ − α)(γ − β)z F (α, β; γ + 1; z) = 0

8

15.

γ(γ + 1) F (α, β; γ; z) − γ(γ + 1) F (α, β + 1; γ + 1; z) + α(γ − β)z F (α + 1, β + 1; γ + 2; z) = 0

16.

γ(γ + 1) F (α, β; γ; z) − γ(γ + 1) F (α + 1, β; γ + 1; z) + β(γ − α)z F (α + 1, β + 1; γ + 2; z) = 0

17.

γ F (α, β; γ; z) − (γ − β) F (α, β; γ + 1; z) − β F (α, β + 1; γ + 1; z) = 0

8

8

18.

γ F (α, β; γ; z) − (γ − α) F (α, β; γ + 1; z) − α F (α + 1, β; γ + 1; z) = 0

MO 13–14

9.14 A generalized hypergeometric series The series 1.

pF q

(α1 , α2 , . . . , αp ; β1 , β2 , . . . , βq ; z) =

∞  (α1 )k (α2 )k . . . (αp )k z k (β1 )k (β2 )k . . . (βq )k k!

MO 14

k=0

is called a generalized hypergeometric series (see also 9.210). 2.

2 F 1 (α, β; γ; z)

≡ F (α, β; γ; z)

MO 15

For integral representations, see 3.254 2, 3.259 2, and 3.478 3.

9.15 The hypergeometric differential equation 9.151

A hypergeometric series is one of the solutions of the differential equation du d2 u − αβu = 0, z(1 − z) 2 + [γ − (α + β + 1)z] dz dz which is called the hypergeometric equation.

WH

9.153

The hypergeometric differential equation

1011

The solution of the hypergeometric differential equation 9.152 The hypergeometric differential equation 9.151 possesses two linearly independent solutions. These solutions have analytic continuations to the entire z-plane, except possibly for the three points 0, 1, and ∞. Generally speaking, the points z = 0, 1, ∞ are branch points of at least one of the branches of each solution of the hypergeometric differential equation. The ratio w(z) of two linearly independent solutions satisfies the differential equation  2 w 1 − a21 1 − a22 a2 + a22 − a23 − 1 w , = + + 1 2  −3  2 2 w w z (z − 1) z(z − 1) where a21 = (1 − γ)2 , a22 = (γ − α − β)2 , a23 = (α − β)2 . If α, β, γ are real, the function w(z) maps the upper (Im z > 0) or the lower (Im z <0) half-plane onto a curvilinear triangle whose angles are πa1 , πa2 , πa3 . The vertices of this triangle are the images of the points z = 0, z = 1, and z = ∞. 9.153 Within the unit circle |z| <1, the linearly independent solutions u1 (z) and u2 (z) of the hypergeometric differential equation are given by the following formulas: 1.

If γ is not an integer, u1 = F (α, β; γ; z), u2 = z 1−γ e F (α − γ + 1, β − γ + 1; 2 − γ; z)

2.

If γ = 1, then u1 = F (α, β; 1; z), u2 = F (α, β; 1; z) ln z +

∞  k=1

zk

(α)k (β)k (k!)

2

× {ψ(α + k) − ψ(α) + ψ(β + k) − ψ(β) − 2 ψ(k + 1) + 2 ψ(1)} (see 9.14 2)

3.

If γ = m + 1 (where m is a natural number), and if neither α nor β is a positive number not exceeding m, then u1 = F (α, β; m + 1; z), u2 = F (α, β; m + 1; z) ln z +

∞  k=1

 (k − 1)!(−m)k (α)k (β)k z {h(k) − h(0)} − z −k (1 + m)k (1 − α)k (1 − β)k m

k

k=1

(see 9.14 2)

where h(n) = ψ(α + n) + ψ(β + n) − ψ(m + 1 + n) − ψ(n + 1) 4.11

[n + 1 is a natural number]

Suppose that γ = m + 1 (where m is a natural number) and that α or β is equal to m + 1, where 0 ≤ m < m. Then, for example, for α = m + 1, we obtain u1 = F (1 + m , β; 1 + m; z) , u2 = z −m F (1 + m − m, β − m; 1 − m; z) In this case, u2 is a polynomial in z −1 .

1012

5.

Hypergeometric Functions

9.154

If γ = 1 − m (where m is a natural number) and if α and β are both different from the numbers 0, −1, −2, . . . , 1 − m, then u1 = z m F (α + m, β + m; 1 + m; z), u2 = z m F (α + m, β + m; 1 + m; z) ln z +

∞ 

zk

k=1

−

∞  k=1

(α + m)k (β + m)k ∗ {h (k) − h∗ (0)} (1 + m)k k!

(k − 1)!(−m)k z m−n (1 − α − m)k (1 − β − m)k (see 9.14 2)

where h∗ (n) = ψ(α + m + n) + ψ(β + m + n) − ψ(1 + m + n) − ψ(1 + n) We note that 1 1 1 + + ··· + (cf. 8.365 3) α α+1 α+n−1 and that, for α = −λ, where λ is a natural number or zero and n = λ + 1, λ + 2, . . . the expression ψ(α + n) − ψ(α) =

(α)k [ψ(α + n) − ψ(α)] in formulas 9.153 2–5 should be replaced with the expression (−1)λ λ!(n − λ − 1)! 6.

Suppose that γ = 1 − m (where m is a natural number) and that α or β is an integer (−m ), where m is one of the following numbers: 0, 1, . . . , m − 1. Suppose, for example, that α = −m . Then, u1 = F (−m , β; 1 − m; z) , u2 = F (−m + m, β + m; 1 + m; z) MO 18

7.

For γ =

1 2 (α

+ β + 1)   u1 = F α, β; 12 (α + β + 1); z ,   u2 = F α, β; 12 (α + β + 1); 1 − z

are two linearly independent solutions of the hypergeometric differential equation, provided α, β, MO 17–19 and γ are not zero or negative integers.

The analytic continuation of a solution that is regular at the point z = 0 9.154 Formulas 9.153 make possible the analytic continuation, by means of the hypergeometric series, of the function F (α, β; γ; z) defined inside the circle |z| <1 to the region |z| > 1, and |arg(−z)| < π. Here, it is assumed that α − β is not an integer. In the event that α − β is an integer (for example, if β = α + m, where m is a natural number), then, for |z| > 1, and |arg(−z)| < π we have:

9.155

1.

2.

The hypergeometric differential equation

1013

Γ(α) Γ(α + m) F (α, α + m; γ; z) Γ(γ) m−1 sin π(γ − α)  Γ(α + k) Γ(1 − γ + α + k) Γ(m − k) (−z)−α−k = π k! k=0  ∞  Γ(α + m + k) Γ(1 − γ + α + m + k) g(k)z −k + (−z)−α−m k!(k + m)! k=0 where g(n) = ln(−z) + π cot π(γ − α) + ψ(n + 1) + ψ(n + m + 1) − ψ(α + m + n) − ψ(1 − γ + α + m + n) For m = 0, we should set

m−1 

= 0.

k=0

9.155 This formula loses its meaning when α, γ, or α−γ +1 is equal to one of the numbers 0, −1, −2, . . . . In this last case, we have 1.

If α is a non-positive integer and γ is not an integer, F (α, α + m; γ; z) is a polynomial in z.

2.

Suppose that γ is a non-positive integer and that α is not an integer. We then set γ = −λ, where λ = 0, 1, 2, . . . . Then, Γ(α + λ + 1) Γ(α + λ + m + 1) λ+1 z F (α + λ + 1, α + λ + m + 1; λ + 2; z) Γ(λ + 2) is a solution of the hypergeometric equation that is regular at the point z = 0. This solution is equal to the right-hand member of formula 9.154 1 if we replace γ with λ in this equation and in formula 9.154 2.

3.

If α − γ + 1 is a non-positive integer and if α and γ are not themselves integers, we may use the formula F (α, α + m; γ; z) = (1 − z)

γ−2α−m

F (γ − α − m, γ − α; γ; z)

and apply formula 9.154 1 to its right-hand member, provided γ − α − m > 0. However, if α − γ − m ≤0, the right member of this expression is a polynomial taken to the (1 − z)th power. 4.

If α, β, and γ are integers, the hypergeometric differential equation always has a solution that is regular for z = 0 and that is of the form R1 (z) + ln(1 − z)R2 (z), where R1 (z) and R2 (z) are rational functions of z. To get a solution of this form, we need to apply formulas 9.137 1–9.137 3 to the function F (α, β; γ; z). However, if γ = −λ, where λ + 1 is a natural number, formulas 9.137 1 and 9.137 2 should be applied not to F (α, β; γ; z) but to the function z λ+1 F (α + λ + 1, β + λ + 1; λ + 2, z). By successive applications of these formulas, we can reduce the positive values of the parameters to the pair, unity and zero. Furthermore, we can obtain the desired form of the solution from the formulas −1 F (1, 1; 2; z) = −z ln(1 − z), F (0, β; γ; z) = F (α, 0; γ; z) = 1

MO 19–20

1014

Hypergeometric Functions

9.160

9.16 Riemann’s differential equation 9.160 1.11

The hypergeometric differential equation is a particular case of Riemann’s differential equation   1 − α − α 1 − β − β 1 − γ − γ  du d2 u + + + dz 2 z−b z−c dz ⎤ ⎡z − a    ββ γγ u αα (a − b)(a − c) (b − c)(b − a) (c − a)(c − b) ⎦ + =0 +⎣ z−a z−b z−c (z − a)(z − b)(z − c) WH 

The coefficients of this equation have poles at the points a, b, and c, and the numbers α, α ; β, β  ; γ, γ  are called the indices corresponding to these poles. The indices α, α ; β, β  ; γ, γ  are related by the following equation: α + α + β + β  + γ + γ  − 1 = 0 2. 3.

WH

The differential equations 9.160 1 are written diagramatically as follows: ⎧ ⎫ ⎨a b c ⎬ u=P α β γ z ⎩  ⎭ α β γ

The singular points of the equation appear in the first row in this scheme, the indices corresponding to WH them appear beneath them, and the independent variable appears in the fourth column. 9.161 The two following transformation formulas are valid for Riemann’s P -equation: ⎧ ⎫ ⎧ ⎫

k

l ⎨ a b c a b c ⎬ ⎨ ⎬ z−a z−c P α β γ z =P α+k β−k−1 γ+l z WH 1. ⎩  ⎭ ⎩  ⎭ z−b z−b α β γ α + k β − k − l γ + l ⎧ ⎫ ⎧ ⎫ ⎨a b c ⎬ ⎨a1 b1 c1 ⎬ 2. P α β γ z = P α β γ z1 WH ⎩  ⎭ ⎩  ⎭ α β γ α β γ The first of these formulas means that if

⎧ ⎨a u=P α ⎩  α

then the function



b β β

c γ γ

⎫ ⎬ z

⎭

,

k

l z−a z−c u z−b z−b satisfies a second-order differential equation having the same singular points as equation 9.161 2 and indices equal to α + k, α + k; β − k − l, β  − k − l; γ + l, γ  + l. The second transformation formula converts a differential equation with singularities at the points a,b, and c, indices α, α ; β, β  ; γ, γ  , and an independent variable z into a differential equation with the same indices, singular points a1 , b1 , and c1 , and independent variable z1 . The variable z1 is connected with the variable z by the fractional transformation u1 =

9.164

Riemann’s differential equation

1015

Az1 + B [AD − BC = 0] Cz1 + D The same transformation connects the points a1 , b1 , and c1 with the points a, b, and c. z=

WH, MO 20

9.162 By the successive application of the two transformation formulas 9.161 1 and 9.161 2, we can convert Riemann’s differential equation into the hypergeometric differential equation. Thus, the solution of Riemann’s differential equation can be expressed in terms of a hypergeometric function. For k = −α, l = −γ, and z1 = (z−a)(c−b) (z−b)(c−a) , we have 1.

⎧ ⎧ ⎫ b ⎨a b c ⎬ z − a α z − c γ ⎨ a 0 β+α+γ u= P α β γ z = P ⎩  ⎩  ⎭ z−b z−b α β γ α − α β + α + γ ⎧ ⎫

α

γ ⎨ 0 ∞ 1 ⎬ z−a z−c (z−a)(c−b) 0 β+α+γ 0 = P (z−b)(c−a) ⎭ ⎩  z−b z−b α − α β + α + γ γ − γ

c 0 γ − γ

⎫ ⎬ z

⎭

MO 23

2.

Thus, this solution can be expressed as a hypergeometric series as follows:

α

γ

z−a z−c   (z − a)(c − b) u= F α + β + γ, α + β + γ; 1 + α − α ; z−b z−b (z − b)(c − a)

If the constants a, b, c; α, α ; β, β  ; γ, γ  are permuted in a suitable manner, Riemann’s equation remains unchanged. Thus, we obtain a set of 24 solutions of differential equations having the following form WH, MO 23 (provided none of the differences α − α , β − β  , γ − γ  is an integer): 9.163



1.

u1

2.

u2

3.

u3

4.

u4

9.164 1.10 2. 3.

α

γ  z−c   (c − b)(z − a) = F α + β + γ, α + β + γ; 1 + α − α ; z−b (c − a)(z − b)

α

γ  z−a z−c (c − b)(z − a)     = F α + β + γ, α + β + γ; 1 + α − α; z−b z−b (c − a)(z − b)

α

γ   z−a z−c     (c − b)(z − a) = , α + β + γ ; 1 + α − α ; α + β + γ F z−b z−b (c − a)(z − b)

α

γ   z−a z−c (c − b)(z − a)      = F α + β + γ , α + β + γ; 1 + α − α; z−b z−b (c − a)(z − b)

z−a z−b

β

α  z−a   (a − c)(z − b) F β + γ + α, β + γ + α; 1 + β − β ; z−c (a − b)(z − c)

β 

α  z−b z−a (a − c)(z − b)     u6 = + γ + α, β + γ + α; 1 + β − β; β F z−c z−c (a − b)(z − c)

β

α  z−b z−a     (a − c)(z − b) u7 = F β + γ + α ,β + γ + α ;1 + β − β ; z−c z−c (a − b)(z − c) u5 =

z−b z−c

1016

Hypergeometric Functions

4. 9.165 1. 2. 3. 4.

u8 =

z−b z−c

β 

z−a z−c

α

 (a − c)(z − b)       F β + γ + α , β + α + γ ; 1 + β − β; (a − b)(z − c)



γ

β  z−c z−b   (b − a)(z − c) u9 = F γ + α + β, γ + α + β; 1 + γ − γ ; z−a z−a (b − c)(z − a)

γ 

β  z−c z−b (b − a)(z − c)     u10 = + α + β, γ + α + β; 1 + γ − γ; γ F z−a z−a (b − c)(z − a)

γ

β   z−c z−b     (b − a)(z − c) u11 = F γ + α + β ,γ + α + β ;1 + γ − γ ; z−a z−a (b − c)(z − a)

γ 

β   z−c z−b (b − a)(z − c)       u12 = F γ + α + β , γ + α + β ; 1 + γ − γ; z−a z−a (b − c)(z − a)

9.166



u13 =

2.

u14

3.

u15

4.

u16

9.167



z−a z−c

γ

α  z−a   (a − b)(z − c) F γ + β + α, γ + β + α; 1 + γ − γ ; z−b (a − c)(z − b)

γ 

α  z−c z−a (a − b)(z − c)     = + β + α, γ + β + α; 1 + γ − γ; γ F z−b z−b (a − c)(z − b)

γ

α  z−c z−a     (a − b)(z − c) = F γ + β + α ,γ + β + α ;1 + γ − γ ; z−b z−b (a − c)(z − b)  

γ

α  z−c z−a (a − b)(z − c)       = F γ + β + α , γ + β + α ; 1 + γ − γ; z−b z−b (a − c)(z − b)

1.

u17 =

2.

u18

3.

u19

4.

u20

9.168

α

β  z−b   (b − c)(z − a) + β; 1 + α − α ; α + γ + β, α + γ F z−c (b − a)(z − c)

α

β  z−a z−b (b − c)(z − a)     = F α + γ + β, α + γ + β; 1 + α − α; z−c z−c (b − a)(z − c)

α

β   z−a z−b     (b − c)(z − a) = F α + γ + β ,α + γ + β ;1 + α − α ; z−c z−c (b − a)(z − c)

α

β   z−a z−b (b − c)(z − a)       = F α + γ + β , α + γ + β ; 1 + α − α; z−c z−c (b − a)(z − c)

1.



z−c z−b

β

γ  z−c   (c − a)(z − b) + γ; 1 + β − β ; β + α + γ, β + α F z−a (c − b)(z − a)

β 

γ  z−b z−c (c − a)(z − b)     = F β + α + γ, β + α + γ; 1 + β − β; z−a z−a (c − b)(z − a)

1.

u21 =

2.

u22

z−b z−a

9.165

9.175

Second-order differential equations



3.

u23

4.

u24

β

γ   z−c     (c − a)(z − b) = F β + α + γ ,β + α + γ ;1 + β − β ; z−a (c − b)(z − a)

β 

γ   z−b z−c (c − a)(z − b)       = F β + α + γ , β + α + γ ; 1 + β − β; z−a z−a (c − b)(z − a) z−b z−a

1017

WH

9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme 9.171

The hypergeometric equation (see 9.151): ⎧ ⎫ ∞ 1 ⎨ 0 ⎬ 0 α 0 z WH u=P ⎩ ⎭ 1−γ β γ−α−β 9.172 The associated Legendre’s equation defining the functions P m n (z) for n and m integers (see 8.700 1): ⎧ ⎫ ∞ 1 ⎪ ⎪ ⎨ 0 ⎬ 1−z 1 1 WH 1. u=P m n + 1 m 2 2 ⎪ 2 ⎪ ⎩ 1 ⎭ 1 − 2 m −n − 2 m ⎫ ⎧ 0 ∞ 1 ⎪ ⎪ ⎪ ⎬ ⎨ 1 1 ⎪ 1 0 2m 2. u = P −2n WH 2 1 − z ⎪ ⎪ ⎪ ⎪ ⎭ ⎩n + 1 −1m 1 2 2 2

z2 m 9.173 The function P n 1 − 2 satisfies the equation 2n ⎧ 2 ⎫ ∞ 0 ⎨ 4n ⎬ 1 1 2 m n + 1 m z WH u=P 2 ⎩ 21 ⎭ − 2 m −n − 12 m The function J m (z) satisfies the limiting form of this equation obtained as n → ∞. 9.174 The equation defining the Gegenbauer polynomials C λn (z) (see 8.938): ⎧ ⎫ ∞ 1 ⎨ −1 ⎬ WH u = P 12 − λ n + 2λ 12 − λ z ⎩ ⎭ 0 −n 0 9.175 Bessel’s equation (see 8.401) is the limiting form of the equations: ⎧ ⎫ ∞ c ⎨ 0 ⎬ 1 n ic + ic z WH 1. u=P 2 ⎩ ⎭ −n −ic 12 − ic ⎧ ⎫ ∞ c ⎨ 0 ⎬ 1 n 0 z 2. u = eiz P WH 2 ⎩ ⎭ −n 32 − 2ic 2ic − 1 ⎧ ⎫ ∞ c2 ⎨ 0 ⎬ 1 1 2 n (c − n) 0 z 3. u=P WH 2 ⎩ 21 ⎭ − 2 n − 12 (c + n) n + 1 as c → ∞.

1018

Hypergeometric Functions

9.180

9.18 Hypergeometric functions of two variables 9.180 1.

 F 1 (α, β, β , γ; x, y) =

∞ ∞   (α)m+n (β)m (β  )n m n x y (γ)m+n m!n! m=0 n=0

[|x| < 1,

|y| < 1] EH I 224(6), AK 14(11)

2.

  F 2 (α, β, β , γ, γ ; x, y) =

∞ ∞  



(α)m+n (β)m (β )n m n x y (γ)m (γ  )n m!n! m=0 n=0 [|x| + |y| < 1]

3.

  F 3 (α, α , β, β , γ; x, y) =

∞ ∞  



EH I 224(7), AK 14(12)



(α)m (α )n (β)m (β )n m n x y (γ)m+n m!n! m=0 n=0 [|x| < 1,

|y| < 1] EH I 224(8), AK 14(13)

4.

 F 4 (α, β, γ, γ ; x, y) =

∞ ∞   (α)m+n (β)m+n m n x y (γ)m (γ  )n m!n! m=0 n=0

!√ ! √  ! x! + | y| < 1 EH I 224(9), AK 14(14)

9.181 for z:

The functions F 1 , F 2 , F 3 , and F 4 satisfy the following systems of partial differential equations

1.

System of equations for z = F 1 :

2.

∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, + y(1 − x) 2 ∂x ∂x ∂y ∂x ∂y ∂2z ∂2z ∂z ∂z + [γ − (α + β  + 1) y] − βx − αβ  z = 0 y(1 − y) 2 + x(1 − y) ∂y ∂x ∂y ∂x ∂x System of equations for z = F 2 :

3.

∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, − xy 2 ∂x ∂x ∂y ∂x ∂y ∂2z ∂2z ∂z ∂z + [γ  − (α + β  + 1) y] − βx − αβ  z = 0 y(1 − y) 2 − xy ∂y ∂x ∂y ∂y ∂x System of equations for z = F 3 :

x(1 − x)

x(1 − x)

EH I 233(9)

EH I 234(10)

∂z ∂2z ∂2z + [γ − (α + β + 1)x] − αβz = 0, + y 2 ∂x ∂x ∂y ∂x ∂2z ∂2z ∂z y(1 − y) 2 + x + [γ − (α + β  + 1) y] − α β  z = 0 ∂y ∂x ∂y ∂y x(1 − x)

EH I 234(11)

9.182

4.

Hypergeometric functions of two variables

1019

System of equations for z = F 4 : x(1 − x)

2 ∂z ∂z ∂2z ∂2z 2∂ z + [γ − (α + β + 1)x] − (α + β + 1)y − αβz = 0, − y − 2xy 2 2 ∂x ∂y ∂x ∂y ∂x ∂y

EH I 234(12)

y(1 − y)

2

2

2

∂z ∂ z ∂ z ∂ z ∂z + [γ  − (α + β + 1)y] − (α + β + 1)x − αβz = 0 − x2 2 − 2xy ∂y 2 ∂x ∂x ∂y ∂y ∂x AK 44

9.182 For certain relationships between the parameters and the argument, hypergeometric functions of two variables can be expressed in terms of hypergeometric functions of a single variable or in terms of elementary functions:

  −α  x−y EH I 238(1), AK 24(28) 1. F 1 (α, β, β , β + β ; x, y) = (1 − y) F α, β; β + β ; 1−y

y   −α   2. (α, β, β , β, γ ; x, y) = (1 − x) ; γ ; EH I 238(2), AK 23 α, β F2 F 1−x

xy  −β −β   3. EH I 238(3) F 2 (α, β, β , α, α; x, y) = (1 − x) (1 − y) F β, β ; α; (1 − x)(1 − y) 4. 5.

F 3 (α, γ − α, β, γ − β, γ; x, y) = (1 − y)

α+β−γ



F (α, β; γ; x + y − xy)

EH I 238(4), AK 25(35)



F 4 (α, γ + γ − α − 1, γ, γ ; x(1 − y), y(1 − x)) = F (α, γ + γ  − α − 1; γ; x) F (α, γ + γ  − α − 1; γ  ; y)

6. 7.

−y x (1 − x)β (1 − y)α , EH I 238(6) = F 4 α, β, α, β; − (1 − x)(1 − y) (1 − x)(1 − y) (1 − xy)

y x ,− = (1 − x)α (1 − y)α F (α, 1 + α − β; β; xy) F 4 α, β, β, β; − (1 − x)(1 − y) (1 − x)(1 − y)

8.

EH I 238(5)



F 4 α, β, 1 + α − β, β; −

y x ,− (1 − x)(1 − y) (1 − x)(1 − y)

EH I 238(7)



  x(1 − y) = (1 − y) F α, β; 1 + α − β; − 1−x α



9.



1 1 1 √ −2α F 4 α, α + , γ, ; x, y = (1 + y) F 2 2 2





EH I 238(8)

x 1 α, α + ; γ;  √ 2 2 1+ y   1 x 1 √ −2α + (1 − y) F α, α + ; γ;  √ 2 2 2 1− y AK 23

10.

 F 1 (α, β, β , γ; x, 1) =



Γ(γ) Γ (γ − α − β )  F (α, β : γ − β ; x) Γ(γ − α) Γ (γ − β  )

EH I 239(10), AK 22(23)

1020

11. 9.183 1.

Hypergeometric Functions

9.183

  F 1 (α, β, β , γ; x, x) = F (α, β + β ; γ; x)

EH I 239(11), AK 23(25)

Functional relations between hypergeometric functions of two variables:

y x  −β −β  , F 1 (α, β, β , γ; x, y) = (1 − x) (1 − y) F 1 γ − α, β, β , γ; x−1 y−1 = (1 − x)−α F 1 α, γ − β − β  , β  , γ;

y−x x , x−1 1−x

EH I 239(1)



EH I 239(2)



y y−x −α  , = (1 − y) F 1 α, β, γ − β − β , γ; y−1 y−1 



= (1 − x)γ−α−β (1 − y)−β F 1 γ − α, γ − β − β  , β  , γ; x,





= (1 − x)−β (1 − y)γ−α−β F 1 γ − α, β, γ − β − β  , γ;

x−y 1−y

x−y ,y x−1





EH I 239(3)

EH I 240(4)

EH I 240(5), AK 30(5)

2.8

  −α   F 2 (α, β, β , γ, γ ; x, y) = (1 − x) F 2 α, γ − β, β , γ, γ ;

= (1 − y)−α F 2 α, β, γ  − β  , γ, γ  ;

= (1 − x − y)−α F 2



y x , x−1 1−x



y x , 1−y y−1

EH I 240(6)



EH I 240(7)

y x , α, γ − β, γ  − β  , γ, γ  ; x+y−1 x+y−1



EH I 240(8), AK 32(6)

3.7



Γ(γ  ) Γ(β − α) x 1  −α  (−y) F 4 α, α + 1 − γ , γ, α + 1 − β; , F 4 (α, β, γ, γ ; x, y) = Γ (γ  − α) Γ(β) y y

 x 1 Γ(γ (Γ(α − β) β  (−y) F 4 β + 1 − γ , β, γ, β + 1 − α; , + Γ (γ  − β) Γ(α) y y EH I 240(9), AK 26(37)

9.185

9.184 1.

Hypergeometric functions of two variables

Integral representations: Double integrals of the Euler type  F 1 (α, β, β , γ; x, y) =

Γ(γ)  ) Γ (γ − β − β  ) Γ(β) Γ (β    −α uβ−1 v β −1 (1 − u − v)γ−β−β −1 (1 − ux − vy) du dv × u≥0,v≥0 u+v≤1

Re β  > 0,

[Re β > 0, 2.

  F 2 (α, β, β , γ, γ ; x, y) =

EH I 230(1), AK 28(1)

Γ(γ) Γ (γ  ) − β) Γ (γ  − β  )

 1 1    uβ−1 v β −1 (1 − u)γ−β−1 (1 − v)γ −β −1 (1 − ux − vy)−α du dv × 0

0

Re β  > 0,

Re (γ  − β  ) > 0]

Re (γ − β) > 0,

EH I 230(2), AK 28(2)

  F 3 (α, α , β, β , γ; x, y)

=

Γ(γ) Γ(β) Γ (β  ) Γ (γ

 ×

u≥0,v≥0 u+v≤1

− β − β)

uβ−1 v β



−1

(1 − u − v)−γ−β−β

Re β  > 0,

[Re β > 0, 4.

Re (γ − β − β  ) > 0]

Γ(β) Γ (β  ) Γ(γ

[Re β > 0, 3.

1021



−1



(1 − ux)−α (1 − vy)−α du dv

Re (γ − β − β  ) > 0]

EH I 230(3), AK 28(3)

 F 4 (α, β, γ, γ ; x(1 − y), y(1 − x))  1 1  Γ(γ) Γ (γ  ) uα−1 v β−1 (1 − u)γ−α−1 (1 − v)γ −β−1 = Γ(α) Γ(β) Γ(γ − α) Γ (γ  − β) 0 0

×(1 − ux)α−γ−γ [Re α > 0,



+1

(1 − vy)β−γ−γ

Re β > 0,



+1

(1 − ux − vy)γ+γ

Re (γ − α) > 0,



−α−β−1

Re (γ  − β) > 0]

du dv EH I 230(4)

9.185 Integral representations: Integrals of the Mellin–Barnes type The functions F 1 , F 2 , F 3 , and F 4 can be represented by means of double integrals of the following form:  i∞  i∞ Γ(γ) Ψ(s, t) Γ(−s) Γ(−t)(−x)s (−y)t ds dt F (x, y) = Γ(α) Γ(β)(2πi)2 −i∞ −i∞

1022

Confluent Hypergeometric Functions

9.201

Ψ(s, t)

F (x, y)

Γ(α + s + t) Γ(β + s) Γ (β  + t) Γ (β  ) Γ(γ + s + t)

F1 (α, β, β  , γ; x, y)

Γ(α + s + t) Γ(β + s) Γ (β  + t) Γ (γ  ) Γ (β  ) Γ(γ + s) Γ (γ  + t)

F2 (α, β, β  , γ, γ  ; x, y)

Γ(α + s) Γ (α + t) Γ(β + s) Γ (β  + t) Γ (α ) Γ (β  ) Γ(γ + s + t)

  F 3 (α, α , β, β , γ; x, y)

Γ(α + s + t) Γ(β + s + t) Γ (γ  ) Γ(γ + s) Γ (γ  + t) [α, α , β, β  may not be negative integers]

F4 (α, β, γ, γ  ; x, y) EH I 232(9–13), AK 41(33)

9.19 A hypergeometric function of several variables F A (α; β1 , . . . , βn ; γ1 , . . . , γn ; z1 , . . . , zn ) ∞ ∞  ∞   (α)m1 +···+mn (β1 )m1 · · · (βn )mn m1 m2 z1 z2 · · · znmn ... = (γ ) · · · (γ ) m ! · · · m ! 1 n 1 n m1 mn m =0m =0 m =0 1

2

n

ET I 385

9.2 Confluent Hypergeometric Functions 9.20 Introduction 9.20110 A confluent hypergeometric function is obtained by taking the limit as c → ∞ in the solution of Riemann’s differential equation ⎧ ⎫ ∞ c ⎨ 0 ⎬ WH u = P 12 + μ −c c − λ z ⎩1 ⎭ − μ 0 λ 2 9.202 The equation obtained by means of this limiting process is of the form 1 2

λ d2 u du 4 −μ 1. + + + WH u=0 dz 2 dz z z2 Equation 9.202 1 has the following two linearly independent solutions:   1 2. z 2 +μ e−z Φ 12 + μ − λ, 2μ + 1; z   1 3. z 2 −μ e−z Φ 12 − μ − λ, −2μ + 1; z which are defined for all values of μ = ± 12 , ± 22 , ± 32 , . . .

MO 111

9.214

The functions Φ(α, γ; z) and Ψ(α, γ; z)

1023

9.21 The functions Φ(α, γ; z) and Ψ(α, γ; z) 9.21010 1.

The series α(α + 1) z 2 α(α + 1)(α + 2) z 3 αz + + + ... γ 1! γ(γ + 1) 2! γ(γ + 1)(γ + 2) 3! is also called a confluent hypergeometric function. Φ(α, γ; z) = 1 +

A second notation: Φ(α, γ; z) =

1F 1

(α; γ; z).

Γ(1 − γ) Γ(γ − 1) 1−γ Φ(α, γ; z) + z Φ(α − γ + 1, 2 − γ; z) Γ(α − γ + 1) Γ(α)

2.

Ψ(α, γ; z) =

3.

Bateman’s function kν (x) is defined by  2 π/2 kν (x) = cos (x tan θ − νθ) dθ π 0

9.211

2.

3.

4.8

[x, ν real]

EH I 267

[0 < Re α < Re γ]

MO 114

Integral representation: 1

1.

EH I 257(7)

21−γ e 2 z Φ(α, γ; z) = B(α, γ − α)

Φ(α, γ; z) =

1

−1

1 Γ(α)

1

(1 − t)γ−α−1 (1 + t)α−1 e 2 zt dt

1 z 1−γ B(α, γ − α)

Φ(−ν, α + 1; z) =

Ψ(α, γ; z) =





z

et tα−1 (z − t)γ−α−1 dt 0

Γ(α + 1) z − α e z 2 Γ(α + ν + 1)



∞

e−t tν+ 2

0

α

[0 < Re α < Re γ]  √ J α 2 zt dt + Re(α + ν + 1) > 0,

MO 114

|arg z| <

π, 2 MO 115



∞

e−zt tα−1 (1 + t)γ−α−1 dt

[Re α > 0,

Re z > 0]

EH I 255(2)

0

Functional relations 9.212 Φ(α, γ; z) = ez Φ(γ − α, γ; −z) z Φ(α + 1, γ + 1; z) = Φ(α + 1, γ; z) − Φ(α, γ; z) γ

MO 112

3.

α Φ(α + 1, γ + 1; z) = (α − γ) Φ(α, γ + 1; z) + γ Φ(α, γ; z)

MO 112

4.

α Φ(α + 1, γ; z) = (z + 2a − γ) Φ(α, γ; z) + (γ − α) Φ(α − 1, γ; z)

MO 112

1. 2.

α dΦ = Φ(α + 1, γ + 1; z) dz γ

1 n+1 α + n Φ(α, γ; z) = z 9.214 lim Φ(α + n + 1, n + 2; z) γ→−n Γ(γ) n+1

MO 112

9.213

MO 112

[n = 0, 1, 2, . . .]

MO 112

1024

Confluent Hypergeometric Functions

9.215

9.21510 Φ(α, α; z) = ez

1.

MO 15

  π    1 Φ(α, 2α; 2z) = 2 exp 4 (1 − 2α)πi Γ α + 12 ez z 2 −α J α− 12 ze 2 i  z −p   Φ p + 12 , 2p + 1; 2iz = Γ(p + 1) eiz J p (z) 2 α− 12

2. 3.

1

MO 112 MO 15

For a representation of special functions in terms of a confluent hypergeometric function Φ(α, γ; z), see: • • • • • • 9.216 1.

for for for for for for

the probability integral, 9.236; integrals of Bessel functions, 6.631 1; Hermite polynomials, 8.953 and 8.959; Laguerre polynomials, 8.972 1; parabolic cylinder functions, 9.240; the Whittaker functions M λ,μ (z), 9.220 2 and 9.220 3.

The function Φ(α, γ; z) is a solution of the differential equation d2 F dF − αF = 0 + (γ − z) dz 2 dz This equation has two linearly independent solutions: z

2.

Φ(α, γ; z)

3.

z 1−γ Φ(α − γ + 1, 2 − γ; z)

MO 111

MO 112

9.22–9.23 The Whittaker functions Mλ,μ (z) and Wλ,μ (z) 9.220 1.

2. 3.11

If we make the change of variable u = e− 2 W in equation 9.202 1, we obtain the equation 1 2

d2 W 1 λ 4 −μ + + + − MO 115 W =0 dz 2 4 z z2 Equation 9.220 1 has the following two linearly independent solutions:   1 M λ,μ (z) = z μ+ 2 e−z/2 Φ μ − λ + 12 , 2μ + 1; z   1 M λ,−μ (z) = z −μ+ 2 e−z/2 Φ −μ − λ + 12 , −2μ + 1; z MO 115 z

To obtain solutions that are also suitable for 2μ = ±1, ±2, . . . , we introduce Whittaker’s function 4.

W λ,μ (z) =

Γ

Γ(−2μ) Γ(2μ) 1  M λ,μ (z) +  1  M λ,−μ (z) − μ − λ Γ 2 2 +μ−λ

WH

which, for 2μ approaching an integer, is also a solution of equation 9.220 1. For the functions M λ,μ (z) and W λ,μ (z), z = 0 is a branch point and z = ∞ is an essential singular point. Therefore, we shall examine these functions only for |arg z| < π. These functions W λ,μ (z) and W −λ,μ (−z) are linearly independent solutions of equation 9.220 1.

9.226

The Whittaker functions Mλ,μ (z) and Wλ,μ (z)

1025

Integral representations  1 1 1 1 1 z μ+ 2  (1 + t)μ−λ− 2 (1 − t)μ+λ− 2 e 2 zt dt, 1 1 2μ 2 B μ + λ + 2 , μ − λ + 2 −1 if the integral converges. See also 6.631 1 and 7.623 3. 9.222  ∞ 1 1 1 z μ+ 2 e−z/2 11  W λ,μ (z) =  e−zt tμ−λ− 2 (1 + t)μ+λ− 2 dt 1. 1 Γ μ−λ+ 2 0 +

9.221 M λ,μ (z) =



Re(μ − λ) > − 12 ,

WH

|arg z| <

π, 2 MO 118



μ+λ− 12 ∞ z λ e−z/2 t μ−λ− 12 −t  2. W λ,μ (z) =  t e dt 1 + z Γ μ − λ + 12 0   Re(μ − λ) > − 12 , |arg z| < π WH     z  i∞ Γ(u − λ) Γ −u − μ + 12 Γ −u + μ + 12 u e− 2     9.223 W λ,μ (z) = z du 2πi −i∞ Γ −λ + μ + 12 Γ −λ − μ + 12 [the path of integration is chosen in such a way that   the poles of the function Γ(u − λ) are separated from the poles of the functions Γ −u − μ + 12 and Γ −u + μ + 12 .] See also 7.142. MO 118  ∞  ∞ 1 1 9.224 W μ, 12 +μ (z) = z μ+1 e− 2 z (1 + t)2μ e−zt dt = z −μ e 2 z t2μ e−t dt [Re z > 0] WH 

0

9.225 1.

z

 W λ,μ (x) W −λ,μ (x) = −x

∞

tanh2λ

0

t {J 2μ (x sinh t) sin(μ − λ)π 2

+ Y 2μ (x sinh t) cos(μ − λ)π} dt



  (z1 z2 ) exp − 12 (z1 + z2 ) W κ,μ (z1 ) W λ,μ (z2 ) = Γ(1 − κ − λ) 

|Re μ| − Re λ < 12 ;

x>0



MO 119

μ+ 12

2.

∞

× 0

×F Θ=

t (z1 + z2 + t) , (z1 + t) (z2 + t)

− 12 +κ−μ

e−t t−κ−λ (z1 + t)

1 2

− 12 +λ−μ

(z2 + t)

 − κ + μ, 12 − λ + μ; 1 − κ − λ; Θ dt [z1 = 0,

z2 = 0,

|arg z1 | < π,

|arg z2 | < π,

Re(κ + λ) < 1] MO 119

See also 3.334, 3.381 6, 3.382 3, 3.383 4, 8, 3.384 3, 3.471 2. 9.226 Series representations   ∞  1 z 2k +μ M 0,μ (z) = z 2 1+ 24k k!(μ + 1)(μ + 2) . . . (μ + k) k=1

WH

1026

Confluent Hypergeometric Functions

9.227

Asymptotic representations For large values of |z| + ⎛ 2 , + 2  2 , + 2  2 , ⎞  ∞ μ2 − λ − 12 μ − λ − 32 . . . μ − λ − k + 12  ⎠ W λ,μ (z) ∼ e−z/2 z λ ⎝1 + k!z k

9.2277

k=1

[|arg z| ≤ π − α < π] 9.228

WH

For large values of |λ|

√ 1 1 1 M λ,μ (z) ∼ √ Γ(2μ + 1)λ−μ− 4 z 1/4 cos 2 λz − μπ − π 4 π

9.229

MO 118



1  √ 4z 4 −λ+λ ln λ π 1. W λ,μ ∼ − e sin 2 λz − λπ − λ 4  z 14 √ 2. W −λ,μ ∼ eλ−λ ln λ−2 λz 4λ Formulas 9.228 and 9.229 are applicable for √ |λ|  1, |λ|  |z|, |λ|  |μ|, z = 0, |arg z| < 3π 4 and |arg λ| <

MO 118 MO 118

π 2.

MO 118

Functional relations 9.231 1.

M

n+μ+ 12 ,μ

1 1 dn  n+2μ −z  z 2 −μ e 2 z z (z) = e (2μ + 1)(2μ + 2) . . . (2μ + n) dz n [n = 0, 1, 2, . . . ;

2μ = −1, −2, −3, . . .] MO 117

2.

z

− 12 −μ

− 12 −μ

M λ,μ (z) = (−z)

M −λ,μ (−z)

[2μ = −1, −2, −3, . . .]

WH

9.232 1.

W λ,μ (z) = W λ,−μ (z)

2.

W −λ,μ (−z) =

9.233 1.

M λ,μ (z) =

Γ(−2μ) Γ(2μ) 1  M −λ,μ (−z) +  1  M −λ,−μ (−z) Γ 2 −μ+λ Γ 2 +μ+λ   |arg(−z)| < 32 π

MO 116

WH

     Γ(2μ + 1) iπλ Γ(2μ + 1) 1    W λ,μ (z) W −λ,μ eiπ z +  1 e 1 exp iπ λ − μ − 2 Γ μ−λ+ 2 Γ μ+λ+  23  − 2 π < arg z < 12 π; 2μ = −1, −2, . . . MO 117

2.

M λ,μ (z) =

     Γ(2μ + 1) −iπλ Γ(2μ + 1) 1    W λ,μ (z) W −λ,μ e−iπ z +  1 e 1 exp −iπ λ − μ − 2 Γ μ−λ+ 2 Γ μ +λ + 2  − 12 π < arg z < 32 π; 2μ = −1, −2, . . . MO 117

9.237

The Whittaker functions Mλ,μ (z) and Wλ,μ (z)

9.234

Recursion formulas   √ W λ,μ (z) = z W λ− 12 ,μ− 12 (z) + 12 + μ − λ W λ−1,μ (z)   √ W λ,μ (z) = z W λ− 12 ,μ+ 12 (z) + 12 − μ − λ W λ−1,μ (z) + 2 ,    d W λ−1,μ (z) z W λ,μ (z) = λ − 12 z W λ,μ (z) − μ2 − λ − 12 dz  

  1−z d μ+ W λ,μ (z) − z W λ,μ (z) μ + 12 + λ 2 dz  

 d 1+z = μ+ W λ,μ+1 (z) + z W λ,μ+1 (z) μ + 2 dz

1. 2.

11

3. 4.

1027

WH WH WH

1 2

−λ



MO 117

5.

3 2

  d + λ + μ 12 + λ + μ z W λ,μ (z) = z(z + 2μ + 1) W λ+1,μ+1 (z) dz     + 12 z 2 + μ − λ − 12 z + 2μ2 + 2μ + 12 W λ+1,μ+1 (z) MO 117

Connections with other functions 9.235 z √ 1. M 0,μ (z) = 22μ Γ(μ + 1) z I μ 2  z z Kμ 2. W 0,μ (z) = π 2

MO 125a MO 125

9.236 2

1. 2. 3.

   ex 2x  Φ(x) = 1 − √ 2 W − 14 , 14 x2 = √ Φ 12 , 32 ; −x2 πx π √ z li(z) = − < W − 12 ,0 (− ln z) ln 12 Γ(α, x) = e−x Ψ(1 − α, 1 − α; x)

WH, MO 126 WH

EH I 266(21)

α

4.

γ(α, x) =

x Φ(α, α + 1; −x) α

EH I 266(22)

9.237 (−1)2μ z μ+ 2 e− 2 z 1    Γ 2 − μ − λ Γ 12 + μ − λ   ∞     Γ μ + k − λ + 12 k  z Ψ(k + 1) + Ψ(2μ + k + 1) − Ψ μ + k − λ + 12 − ln z × k!(2μ + k)! k=0    2μ−1  Γ(2μ − k) Γ k − μ − λ + 1 −2μ k 2 (−z) + (−z) k! k=0   |arg z| < 32 π; 2μ + 1 is a natural number MO 116 1

1.

W λ,μ (z) =

1

1028

Confluent Hypergeometric Functions

9.238

2.

Set λ − μ −

3.

W l+μ+ 12 ,μ (z) = (−1)l z μ+ 2 e− 2 z (2μ + 1)(2μ + 2) · · · (2μ + l) Φ(−l, 2μ + 1; z)

1 2

= l, where l + 1 is a natural number. Then 1

1

= (−1)l z μ+ 2 e− 2 z L2μ l (z) 1

1

MO 116

9.238 1.

  2−ν xν e−ix Φ 12 + ν, 1 + 2ν; 2ix Γ(ν + 1)   2−ν xν e−x Φ 12 + ν, 1 + 2ν; 2x I ν (x) = Γ(ν + 1)   √ K ν (x) = πe−x (2x)ν Ψ 12 + ν, 1 + 2ν; 2x J ν (x) =

2. 3.

EH I 265(9) EH I 265(10) EH I 265(13)

9.24–9.25 Parabolic cylinder functions Dp (z)

2

p 1 z 9.240 D p (z) = 2 4 + 2 W 14 + p2 ,− 14 z −1/2 2 ⎧ ⎫ ⎪ ⎪ √ ⎪ ⎪



√ ⎨ p z2 1 − p 3 z2 ⎬ π 2πz p 1 z2 −

Φ − , ; , ; = 22 e 4 −  p Φ ⎪ 1−p 2 2 2 2 2 2 ⎪ ⎪ ⎪ Γ − ⎩Γ ⎭ 2 2 MO 120a

are called parabolic cylinder functions. Integral representations 9.241 1.

1 π z2 1 D p (z) = √ 2p+ 2 e− 2 pi e 4 π



∞

xp e−2x

−∞

2

+2ixz

dx

[Re p > −1;

for x < 0,

arg xp = pπi] MO 122

2. 9.242 1.10 2.

3.

D p (z) =

2 − z4

e Γ(−p)



∞

e−xz−

x2 2

x−p−1 dx

[Re p < 0]

(cf. 3.462 1)

MO 122

0

 Γ(p + 1) − 1 z2 (0+) −zt− 1 t2 2 e 4 e (−t)−p−1 dt [|arg(−t)| ≤ π] 2πi ∞ p   (−1+) 1 1 2 1 1 (p−1) Γ 2 + 1 2 D p (z) = 2 e 4 z t (1 + t)− 2 p−1 (1 − t) 2 (p−1) dt iπ −∞ , + π |arg z| < ; |arg(1 + t)| ≤ π 4    1 − 1 z2 ∞i Γ 12 t − 12 p Γ(−t) √ t−p−2 t e 4 D p (z) = 2 z dt 2πi Γ(−p) −∞i   |arg z| < 34 π; p is not a positive integer D p (z) = −

WH

WH

WH

9.246

4.

Parabolic cylinder functions Dp (z)

1 − 1 z2 e 4 D p (z) = 2πi



(0−)

Γ

1

2t

∞

1029

 − 12 p Γ(−t) √ t−p−2 t 2 z dt Γ(−p)

[for all values of arg z; also, the contours encircle the poles of the function Γ(−t), but they do not encircle  the poles of the function Γ 12 t − 12 p ]. WH 9.243 ⎧ ⎨ ∞  π −1/2 √  √  2 2 cos  n+1 1 z − 1 n 2 zt n dt n e4 e−n(t−1) 1. D n (z) = (−1)μ ⎩ −∞ 2 sin ⎫  ∞+  0 ⎬ ,     √ √ 2 2 2 cos 1 cos zt n dt − + zt n dt e−n(t−1) e 2 n(1−t ) tn − e−n(t−1) ⎭ sin sin 0 −∞ 

2.

[n is a natural number]

WH

∞

2 cos 1 2 (2zt) dt D n (z) = (−1)μ 2n+2 (2π)−1/2 e 4 z tn e−2t sin :n; 0 , and the cosine or sine is chosen accordingly as n is even or odd] [n is a natural number, μ = 2

WH

9.244 1.

D −p−1 [(1 + i)z] =

e− 2

p−1 2

p+1

2.

D p [(1 + i)z] =

2 2   Γ − p2

Γ





iz 2 2

 p+1 

∞

∞

0

2

e−ix

2 2

(1 +

z

xp

1+ p x2 ) 2



dx



p−1

e− 2 z i

2

x

1

(x + 1) 2 p dx (x − 1)1+ 2

Re p > −1, Re p < 0;

   Re iz 2 ≥ 0    Re iz 2 ≥ 0

MO 122

MO 122

See also 3.383 6, 7, 3.384 2, 6, 3.966 5, 6. 9.245 2

 ∞ x sinh t + pπ t 1 1 p+ 12 10 √ sin D p (x) D −p−1 (x) = − √ coth 1. dt 2 2 π 0 sinh t

2.

 π   π  D p ze 4 i D p ze− 4 i =

1 Γ(−p)



∞ 0

[x is real, Re p < 0]

dt z p coth t exp − sinh 2t 2 sinh t , + π |arg z| < ; Re p < 0 4

MO 122

2

MO 122

See also 6.613. 9.246 Asymptotic expansions. If |z|  1 and |z|  |p|, then

2 p(p − 1) p(p − 1)(p − 2)(p − 3) − z4 p 1. D p (z) ∼ e z 1− + − ... 2z 2 2 · 4z 4   |arg z| < 34 π MO 121

2 p(p − 1) p(p − 1)(p − 2)(p − 3) 2.11 D p (z) ∼ e−z /4 z p 1 − + − ... 2z 2 ⎛ 2 · 4z 4 ⎞ √ 2π pπi z2 /4 −p−1 ⎝ (p + 1)(p + 2) (p + 1)(p + 2)(p + 3)(p + 4) e e 1+ − z + + . . .⎠ Γ(−p) 2z 2 2 · 4z 4 1  5 MO 121 4 π < arg z < 4 π

1030

3.

11

Confluent Hypergeometric Functions

9.247

p(p − 1) p(p − 1)(p − 2)(p − 3) D p (z) ∼ e z 1− + − ... 2z 2 2 · 4z 4 ⎛ ⎞ √ 2π −pπi z2 /4 −p−1 ⎝ (p + 1)(p + 2) (p + 1)(p + 2)(p + 3)(p + 4) − e 1+ e z + + . . .⎠ Γ(−p) 2z 2 2 · 4z 4  1  − 4 π > arg z > − 54 π MO 121 −z 2 /4 p



Functional relations 9.247 1. 2. 3. 9.248 1.

Recursion formulas: D p+1 (z) − z D p (z) + p D p−1 (z) = 0 d D p (z) + dz d D p (z) − dz

WH

1 z D p (z) − p D p−1 (z) = 0 2 1 z D p (z) + D p+1 (z) = 0 2

WH MO 121

Linear relations:

, Γ(p + 1) + π/2 √ D −p−1 (iz) + e−πpi/2 D −p−1 (−iz) e 2π √ 2π −π(p+1)i/2 −pπi =e e D p (−z) + D −p−1 (iz) Γ(−p) √ 2π π(p+1)i/2 e D −p−1 (−iz) = epπi D p (−z) + Γ(−p)

D p (z) =

MO 121

  ∞ i 2 cos xt −it2 /4 21+p/2 π exp − x +p 9.24910 D p [(1 + i)x] + D p [−(1 + i)x] = e dt Γ(−p) 2 2 tp+1 0 MO 122 [x real; −1 < Re p < 0] n  2 2 d [n = 0, 1, 2, . . .] 9.25110 D n (z) = (−1)n ez /4 n e−z /2 WH dz

p  k ∞    2   p a (bx − ay) b √ D p−k 9.252 D p (ax+by) = exp a2 + b 2 x D k a2 + b 2 y 2 2 k 4 a a +b k=0

[a > b > 0,

x > 0,

y > 0,

Re p ≥ 0]

Connections with other functions

n z2 z 9.25311 D n (z) = 2− 2 e− 4 H n √ 2 9.254  

 z2 z π 4 1. D −1 (z) = e 1−Φ √ 2 2   

 z2 z π 2 − z2 11 4 2 e 2. D −2 (z) = e −z 1−Φ √ 2 π 2

“summation theorem”

MO 124

MO 123a

MO 123

MO 123

9.262

9.255 1.

Confluent hypergeometric series of two variables

1031

Differential equations leading to parabolic cylinder functions:

d2 u 1 z2 + p+ − u=0 dz 2 2 4 The solutions are u = D p (z), D p (−z), D −p−1 (iz), and D −p−1 (−iz). (These four solutions are linearly dependent. See 9.248.)

2.

 d2 u  2 + z + λ u = 0, 2 dz

u = D − 1+iλ [±(1 + i)z] 2

EH II 118(12,13)a, MO 123 2

3.7

d u du + (p + 1)u = 0, +z 2 dz dz

u = e−

z2 4

D p (z)

MO 123

9.26 Confluent hypergeometric series of two variables 9.261 ∞  (α)m+n (β)m m n x y (γ)m+n m!n! m,n=0

1.6

Φ1 (α, β, γ, x, y) =

2.

Φ2 (β, β  , γ, x, y) =

3.

Φ3 (β, γ, x, y) =

[|x| < 1]

∞ 

(β)m (β  )m m n x y (γ)m+n m!n! m,n=0

EH I 225(20)

EH I 225(21)a, ET I 385

∞ 

(β)m xm y n (γ) m!n! m+n m,n=0

EH I 225(22)

The functions Φ1 , Φ2 , Φ3 satisfy the following systems of partial differential equations: 9.262 1.

z = Φ1 (α, β, γ, x, y)

EH I 235(23)

∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, + y(1 − x) ∂x2 ∂x ∂y ∂x ∂y ∂2z ∂z ∂z ∂2z y 2 +x + (γ − y) −x − αz = 0 ∂y ∂x ∂y ∂y ∂x

x(1 − x)

2.

z = Φ2 (β, β  , γ, x, y)

EH I 235(24)

∂z ∂2z ∂2z + (γ − x) − βz = 0, + y 2 ∂x ∂x ∂y ∂x ∂2z ∂z ∂2z y 2 +x + (γ − y) − βz = 0 ∂y ∂x ∂y ∂y x

1032

3.

Meijer’s G-Function

9.301

z = Φ3 (β, γ, x, y)

EH I 235(25)

∂z ∂2z ∂2z + (γ − x) − βz = 0, + y 2 ∂x ∂x ∂y ∂x ∂2z ∂z ∂2z y 2 +x +γ −z =0 ∂y ∂x ∂y ∂y x

9.3 Meijer’s G-Function 9.30 Definition m -

9.301

m,n G p,q

!

 ! a1 , . . . , ap 1 x !! = b1 , . . . , bq 2πi

j=1 q j=m+1

Γ (bj − s)

n -

Γ (1 − aj + s)

j=1

Γ (1 − bj + s)

p -

xs ds Γ (aj − s)

j=n+1

[0 ≤ m ≤ q, 0 ≤ n ≤ p, and the poles of Γ (bj − s) must not coincide with the poles of Γ (1 − ak + s) for any j and k (where j = 1, . . . , m; k = 1, . . . , n]). Besides 9.301, the following notations are also used: !

! ar mn G(x) EH I 207(1) , Gmn G pq x !! pq (x), bs 9.302 Three types of integration paths L in the right member of 9.301 can be exhibited: 1.

The path L runs from −∞ to +∞ in such a way that the poles of the functions Γ (1 − ak + s) lie to the left, and the poles of the functions Γ (bj − s) lie to the right of L (for j = 1, 2, . . . , m and k = 1, 2, . . . , n). In this case, the conditions under which the integral 9.301 converges are of the form   EH I 207(2) p + q < 2(m + n), |arg x| < m + n − 12 p − 12 q π.

2.

L is a loop, beginning and ending at +∞, that encircles the poles of the functions Γ (bj − s) (for j = 1, 2, . . . , m) once in the negative direction. All the poles of the functions Γ (1 − ak + s) must remain outside this loop. Then, the conditions under which the integral 9.301 converges are: q ≥ 1 and either p < q or p = q and |x| < 1.

3.

EH I 207(3)

L is a loop, beginning and ending at −∞, that encircles the poles of the functions Γ (1 − ak + s) (for k = 1, 2, . . . , n) once in the positive direction. All the poles of the functions Γ (bj − s) (for j = 1, 2, . . . , m) must remain outside this loop. The conditions under which the integral in 9.301 converges are EH I 207(4) p ≥ 1 and either p > q or p = q and |x| > 1. !

! ar The function G mn x !! is analytic with respect to x; it is symmetric with respect to the pq bs parameters a1 , . . . , an and also with respect to an+1 , . . . , ap ; b1 , . . . , bm ; bm+1 , . . . , bq . EH I 208

9.304

Functional relations

1033

9.30311 If no two bj (for j = 1, 2, . . . , n) differ by an integer, then, under the conditions that either p < q or p = q and |x| < 1, n m Γ (bj − bh ) Γ (1 + bh − aj ) !  m ! ar j=1 j=1 mn ! = xbh G pq x ! q p bs h=1 Γ (1 + bh − bj ) Γ (aj − bh ) j=n+1  × p F q−1 1 + bh − a1 , . . . , 1 + bh − ap ; j=m+1

. . . , ∗, . . . , 1 + bh − bq ;

1 + bh − b1 , . . .  p−m−n (−1) x EH I 208(5)

The prime by the product symbol denotes the omission of the product when j = h. The asterisk in the function p F q−1 denotes the omission of the hth parameter. 9.3047 If no two ak (for k = 1, 2, . . . , n) differ by an integer then, under the conditions that q < p or q = p and |x| > 1, n m  Γ (a − a ) Γ (bj − ah + 1) h j !  n ! a j=1 j=1 r mn = xah −1 G pq x !! p q bs h=1 Γ (aj − ah + 1) Γ (ah − bj ) j=m+1  × q F p−1 1 + b1 − ah , . . . , 1 + bq − ah ; j=n+1

. . . , ∗, . . . , 1 + ap − ah ;

1 + a1 − ah , . . .  (−1)q−m−n x−1 EH I 208(6)

9.31 Functional relations If one of the parameters aj (for j = 1, 2, . . . , n) coincides with one of the parameters bj (for j = m + 1, m + 2, . . . , q), the order of the G-function decreases. For example, !

!

! a1 , . . . , ap ! a2 , . . . , ap m,n−1 mn ! ! = G p−1,q−1 x ! 1. G pq x ! b1 , . . . , bq−1 , a1 b1 , . . . , bq−1 [n, p, q ≥ 1] An analogous relationship occurs when one of the parameters bj (for j = 1, 2, . . . , m) coincides with one of the aj (for j = n + 1, . . . , p). In this case, it is m and not n that decreases by one unit.

2.

The G-function with p > q can be transformed into the G-function with p < q by means of the relationships: !

!

! ! 1 − bs mn −1 ar ! EH I 209(9) = G nm x G pq x !! qp ! 1 − ar bs

1034

3.

4.

5.

Meijer’s G-Function

9.304

!

!

!

! ar ! a1 − 1, a2 , . . . , ap ! ar d mn mn mn ! ! x G pq x ! = G pq x ! + (a1 − 1) G pq x !! dx bs b1 , . . . , bq bs [n ≥ 1] !

!

! ap , 1 − r ! 1 − r, ap m+1,n ! = (−1)r G m,n+1 z G p+1,q+1 z !! p+1,q+1 ! bq , 1 0, bq !

!

! ap ! ap + k k mn mn ! ! z G pq z ! = G pq z ! bq + k bq

EH I 210(13)

[r = 0, 1, 2, . . .]

MS2 6 (1.2.2) MS2 7 (1.2.7)

9.32 A differential equation for the G-function mn G pq

!

! ar x !! satisfies the following linear q th -order differential equation: bs ⎡ ⎤



p q d d ⎣(−1)p−m−n x − aj + 1 − − bj ⎦ y = 0 x x dx dx j=1 j=1

[p ≤ q]

EH I 210(1)

9.33 Series of G-functions mn G pq

!

!

∞  ! a1 , . . . , ap ! a1 , . . . , ap 1 r b1 mn ! ! (1 − λ) G pq x ! λx ! =λ b1 , . . . , bq r! b1 + r, b2 , . . . , bq r=0 [|λ − 1| < 1, m ≥ 1, if m = 1 and p < q, λ may be arbitrary] !

∞  ! a1 , . . . , ap 1 bq r mn ! (λ − 1) G pq x ! =λ r! b1 , . . . , bq−1 , bq + r r=0 [m < q,

EH I 213(1)

|λ − 1| < 1] EH I 213(2)

!



r ∞  ! a1 − r, a2 , . . . , ap 1 1 a1 −1 mn =λ λ− G pq x !! r! λ b1 , . . . , bq r=0   1 n ≥ 1, Re λ > 2 , (if n = 1 and p > q, then λ may be arbitrary) =λ

ap −1

r !

∞  ! a1 , . . . , ap−1 , ap − r 1 1 ! − 1 G mn x pq ! b1 , . . . , bq r! λ r=0  n < p,

EH I 213(3)

Re γ >

1 2



EH I 213(4)

For integrals of the G-function, see 7.8.

9.34 Connections with other special functions

!

1 2 !! x J ν (x)x = 2 4 ! 12 ν + 12 μ, 12 μ − 12 ν !  1 2 !! 12 μ − 12 ν − 12 μ μ 20 x ! Y ν (x)x = 2 G 13 4 ! 21 μ − 12 ν, 12 μ + 12 ν, 12 μ − 12 ν −

1. 2.

μ

μ

10 G 02

EH I 219(44)

 1 2

EH I 219(46)

9.304

Functional relations

1035

!

1 2 !! x K ν (x)x = 2 4 ! 12 μ + 12 ν, 12 μ − 12 ν !1

! √ x 20 2 ! K ν (x) = e π G 12 2x ! ν, −ν !   !1 1 1 1 μ μ 11 2 ! 2 + 2ν + 2μ x ! Hν (x)x = 2 G 13 4 ! 12 + 12 ν + 12 μ, 12 μ − 12 ν, 12 μ + 12 ν !   !1 1 1 1 μ−1 31 2 ! 2 + 2μ   1−μ+ν  G 13  x ! S μ,ν (x) = 2 4 ! 12 + 12 μ, 12 ν, − 12 ν Γ Γ 1−μ−ν 2 2 !

! −a, −b Γ(c)x 12 ! x! G 2 F 1 (a, b; c; −x) = −1, −c Γ(a) Γ(b) 22 $q !

! 1 − a1 , . . . , 1 − ap j=1 Γ (bj ) 1,p ! −x ! G p F q (a1 , . . . , ap ; b1 , . . . , bq ; x) = $p 0, 1 − b1 , . . . , 1 − bq Γ (aj ) p,q+1 $j=1 !

q ! Γ (b ) 1, b1 , . . . , bq j 1 j=1 p,1 = $p G q+1,p − !! x a1 , . . . , ap j=1 Γ (aj )

3. 4. 5.

6. 7.7 8.

9.

μ

μ−1

20 G 02

 √ 1 2k xe 2 x 40 x2 W k,m (x) = √ G 24 4 2π

EH I 219(47)

EH I 219(49)

EH I 220(51)

EH I 220(55)

EH I 222(74)a

EH I 215(1)

!  ! 1 − 1 k, 3 − 1 k !4 2 4 2 !1 1 ! 2 + 2 m, 12 − 12 m, 12 m, − 12 m

EH I 221(70)

9.4 MacRobert’s E-Function 9.41 Representation by means of multiple integrals E (p; αr : q; s : x) =

Γ (αq+1 ) Γ (1 − α1 ) Γ (2 − α2 ) · · · Γ (q − αq ) q  ∞ p−q−1 -  −μ μ −αμ −1 × λμ (1 − λμ ) dλμ μ=1 0  ∞ × e−λp λpαp −1 0

 1+

ν=2

λq+2 λq+3 · · · λp (1 + λ1 ) · · · (1 + λq ) x

∞

q+ν e−λq+ν λq+ν

α

−1

dλq+ν

0

−αq+1 dλp

[|arg x| < π, p ≥ q + 1, αr and s are bounded by the condition that the integrals on the right be EH I 204(3) convergent.]

9.42 Functional relations 1.

α1 x E (α1 , . . . , αp : 1 , . . . , q : x) = x E (α1 + 1, α2 , . . . , αp : 1 , . . . , q : x) + E (α1 + 1, α2 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) EH I 205(7)

1036

2.

Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s)

9.511

(1 − 1) x E (α1 , . . . , αp : 1 , . . . , q : x) = x E (α1 , . . . , αp : 1 − 1, 2 , . . . , q : x) + E (α1 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) EH I 205(9)

3.

d E (α1 , . . . , αp : 1 , . . . , q : x) = x−2 E (α1 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) dx EH I 205(8)

9.5 Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s) 9.51 Definition and integral representations 9.511

ζ(z, q) =

1 Γ(z)



0

∞ z−1 −qt

t e dt; 1 − e−t

1 q 1−z +2 = q −z + 2 z−1



∞



2

q +t

 z 2 −2

0



 dt t sin z arctan q e2πt − 1

WH

[0 < q < 1,

Re z > 1] WH

 Γ(1 − z) (0+) (−θ)z−1 e−qθ dθ 2πi 1 − e−θ ∞ This equation is valid for all values of z, except for z = 1, 2, 3, . . . . It is assumed that the path of integration (see drawing below) does not pass through the points 2nπi (where n is a natural number).

9.512 ζ(z, q) = −

See also 4.251 4, 4.271 1, 4, 8, 4.272 9, 12, 4.294 11. 9.513  ∞ z−1 1 t 1. ζ(z) = dt [Re z > 0] (1 − 21−z ) Γ(z) 0 et + 1  ∞ z−1 t t e 2z dt [Re z > 1] 2. ζ(z) = z (2 − 1) Γ(z) 0 e2t − 1 & %  ∞ z ∞  1−z 2 z π2 1 −1 −k πt 11 t 2 + t2 t + 3. ζ(z) =  z  e dt z(z − 1) Γ 2 1 k=1  ∞  − z 2z−1 dt − 2z 1 + t2 2 sin (z arctan t) πt 4. ζ(z) = z−1 e +1 0

 −z/2 ∞ 1 z 2 2z−1 dt + z + t2 5. ζ(z) = z sin (z arctan 2t) 2πt 2 −1z−1 2 −1 0 4 e −1 See also 3.411 1, 3.523 1, 3.527 1, 3, 4.271 8.

WH WH

WH

WH

WH

9.532

Functional relations

1037

9.52 Representation as a series or as an infinite product 9.521 1.

2.

∞ 

1 [Re z > 1, (q + n)z n=0 % & ∞ ∞ 2 Γ(1 − z) zπ  sin 2πqn zπ  cos 2πqn ζ(z, q) = + cos sin (2π)1−z 2 n=1 n1−z 2 n=1 n1−z ζ(z, q) =

[Re z < 0, 3.8

ζ(z, q) =

q = 0, −1, −2, . . .]

WH

0 < q ≤ 1]

WH

∞  1 1 − − F n (z) , (q + n)z (1 − z)(N + q)z−1 n=0 N 

n=N

where

1 1 1 1 − − F n (z) = 1 − z (n + 1 + q)z−1 (n + q)z−1 (n + 1 + q)z  n+1 (t − n) dt =z (t + q)z+1 n

WH

9.522 1.

ζ(z) =

∞  1 z n n=1

[Re z > 1]

WH

2.

ζ(z) =

∞  1 1 (−1)n+1 z 1 − 21−z n=1 n

[Re z > 0]

WH

9.523 1.7

The following product and summation are taken over all primes p: ζ(z) =

p

2.

ln ζ(z) =

1 1 − p−z

∞  p k=1

1 kpkz

[Re z > 1]

WH

[Re z > 1]

WH

∞  ζ  (z) Λ(k) =− , [Re z > 1] ζ(z) kz k=1 where Λ(k) = 0 when k is not a power of a prime and Λ(k) = ln p when k is a power of a prime p.

9.52411

WH

9.53 Functional relations 9.531

9.532

ζ(−n, q) = −

∞  (−1)k−1 k=2

k

B n+2 (q) − B n+1 (q) = (n + 1)(n + 2) n+1 [n is a nonnegative integer]

see EH I 27 (11)

WH

∞

z k ζ(k, q) = ln

qz e−Cz Γ(q) z  − + Γ(z + q) q k(q + k) k=1

[|z| < q]

WH

1038

Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s)

9.533

9.533 1. 2. 3.

ζ(z, q) = −1 − z)

 1 lim ζ(z, q) − = − Ψ(q) z→1 z−1 

d 1 ζ(z, q) = ln Γ(q) − ln 2π dz 2 z=0 lim

9.534 ζ(z, 1) = ζ(z) 9.535   1 ζ z, 12 1. ζ(z) = z 2 −1 2.11

WH

z→1 Γ(1

2z Γ(1 − z) ζ(1 − z) sin

WH WH

[Re z > 1]  zπ 2

= π 1−z ζ(z)

WH WH

zπ = π z ζ(1 − z) WH 2

z z−1 z 1−z π − 2 ζ(z) = Γ 4. Γ WH π 2 ζ(1 − z) 2 2

 1 9.536 lim ζ(z) − =C z→1 z−1   (z − 1) Γ z2 + 1 1 √ ζ(z) = Ξ(−t) is an even function of t with real 9.537 Set z = 2 + it. Then, Ξ(t) = πz 2 JA coefficients in its expansion in powers of t . 3.

21−z Γ(z) ζ(z) cos

9.54 Singular points and zeros 9.5417 1.

z = 1 is the only singular point of the function ζ(z)

2.

The function ζ(z) has simple zeros at the points −2n, where n is a natural number. All other zeros of the function ζ(z) lie in the strip 0 ≤ Re z <1.

3.8

Riemann’s hypothesis: All zeros of the function ζ(z) lie on the straight line Re z = 12 . It has been shown that a countably infinite set of zeros of the zeta function lie on this line. The first WH 1,500,000,001 zeros lying in 0 < Im z < 545, 439, 823.215 are known to have Re z = 12 .

9.542

Particular values: 22m−1 π 2m |B2m | (2m)!

[m is a natural number]

WH

B2m 2m

[m is a natural number]

WH

[m is a natural number]

WH

1.

ζ(2m) =

2.

ζ(1 − 2m) = −

3.

ζ(−2m) = 0

4.

WH



ζ (0) =

− 12

ln 2π

WH

9.559

The Lerch function Φ(z, s, v)

1039

9.55 The Lerch function Φ(z, s, v) 9.550

Definition: Φ(z, s, v) =

∞ 

(v + n)−s z n

v = 0, −1, . . .]

[|z| < 1,

EH I 27(1)

n=0

Functional relations 9.551

Φ(z, s, v) = z m Φ(z, s, m + v) +

m−1 

(v + n)−s z n

[m = 1, 2, 3, . . . ,

v = 0, −1, −2, . . .]

n=0

EH I 27(1)

9.552 Φ(z, s, v) = iz

−v

 s−1

(2π)

Γ(1 − s) e

−iπ 2s

Φ e

−2πiv

ln z , 1 − s, 2πi

−e

iπ ( 2s −2v )

Φ e

2πiv

ln z , 1 − s, 1 − 2πi



EH I 29(7)

Series representation 9.553

∞ 

Φ(z, s, v) = z −v Γ(1 − s)

s−1 2πnvi

(− ln z + 2πni)

e

[0 < v ≤ 1,

Re s < 0,

n=−∞

 9.554

Φ(z, m, v) = z

−v

∞   n=0

n

[m = 2, 3, 4, . . . , 9.555 Φ(z, −m, v) =

m! zv

Integral representation 9.556

Φ(z, s, v) =

1 Γ(s)

ln



1 z

m−1

(ln z) (ln z) + ζ(m − n, v) n! (m − 1)!

−m−1 −

|arg (− ln z + 2πni)| ≤ π]

EH I 28(6)



∗ 1 Ψ(m) − Ψ(v) − ln ln z

|ln z| < 2π,

∞ r 1  Bm+r+1 (v) (ln z) z v r=0 r!(m + r + 1)

v = 0, −1, −2, . . .] [|ln z| < 2π]

 ∞ s−1 −(v−1)t t e t e dt 1 dt = −t t 1 − ze Γ(s) 0 e −z [Re v > 0, or |z| ≤ 1, z = 1, Re s > 0, or z = 1,

EH I 30(9) EH I 30(11)

∞ s−1 −vt 0

Re s > 1]

EH I 27(3)

Limit relationships 9.557 9.558

lim (1 − z)1−s Φ(z, s, v) = Γ(1 − s)

z→1

lim

[Re s < 1]

Φ(z, 1, v) =1 − z)

EH I 30(12) EH I 30(13)

z→1 − ln(1

A connection with a hypergeometric function 9.559 ∗ In

Φ(z, 1, v) = v −1 2 F 1 (1, v; 1 + v; z)

9.554 the prime on the symbol

#

[|z| < 1]

means that the term corresponding to n = m − 1 is omitted.

EH I 30(10)

1040

Bernoulli Numbers and Polynomials, Euler Numbers

9.561

9.56 The function ξ (s)

  Γ 12 s 1 9.561 ξ(s) = s(s − 1) 1 s ζ(s) 2 π2 9.562 ξ(1 − s) = ξ (s)

EH III 190(10) EH III 190(11)

9.6 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), λ(x, y) and Euler Polynomials 9.61 Bernoulli numbers 9.610

The numbers Bn , representing the coefficients of ∞  t tn = B n et − 1 n=0 n!

are called Bernoulli numbers. Thus, the function bers. 9.611 1.

[0 < |t| < 2π] ,

t is a generating function for the Bernoulli numet − 1 GE 48(57), FI II 520

Integral representations  ∞ 2n−1 x dx B2n = (−1)n−1 4n 2πx − 1 0 e 

[n = 1, 2, . . .]

B2n = (−1)n−1 π −2n

3.

B2n

Bn = lim

d

n

x→0 dxn

ex

2n

x −1

[n = 1, 2, . . .]



See also 3.523 2, 4.271 3. Properties and functional relations 9.6128

A symbolic notation: (B + α)[n] =

n   n k=0

in particular Bn = (B + 1)[n] =

k

Bk αn−k

n   n k=0

hence by recursion

(cf. 3.411 2, 4) FI II 721a

∞

x dx [n = 1, 2, . . .] 2 0 sinh x   2n(1 − 2n) ∞ 2n−2  = (−1)n−1 x ln 1 − e−2πx dx π 0

2.

4.∗

tn in the expansion of the function n!

k

Bk

[n ≥ 2]

[n ≥ 2]

9.622

Bernoulli polynomials

Bn = −n!

n−1  k=0

Bk k!(n + 1 − k)!

1041

[n ≥ 2]

9.613 9.614

All the Bernoulli numbers are rational numbers. Every number Bn can be represented in the form  1 Bn = Cn − , k+1 where Cn is an integer and the sum is taken over all k > 0 such that k + 1 is a prime and k is a divisor GE 64 of n. 1 11 9.615 All the Bernoulli numbers with odd index are equal to zero, except that B1 = − 2 ; that is, GE 52, FI II 521 B2n+1 = 0 for n a natural number. n−1  1 2n(2n − 1) . . . (2n − 2k + 2) 1 + − Bk/2 B2n = − [n ≥ 1] 2n + 1 2 (2k)! k=1; k even

(−1)n−1 (2n)! 9.616 B2n = ζ(2n) [n ≥ 0] 22n−1 π 2n 2(2n)! 1 9.6177 B2n = (−1)n−1

∞ 2n (2π) 1 1 − 2n p p=2

(cf. 9.542)

[n ≥ 1]

GE 56(79), FI II 721a

(cf. 9.523)

(where the product is taken over all primes p). • For a connection with Riemann’s zeta function, see 9.542. • For a connection with the Euler numbers, see 9.635. • For a table of values of the Bernoulli numbers, see 9.71 9.619

An inequality

! ! ! ! !(B − θ)[n] ! ≤ |Bn |

[0 < θ < 1]

9.62 Bernoulli polynomials 9.620

The Bernoulli polynomials B n (x) are defined by n   n Bk xn−k B n (x) = k

GE 51(62)

k=0

or symbolically, B n (x) = (B + x)[n] . 9.621 The generating function ∞  tn−1 ext = B (x) n et − 1 n=0 n! 9.622 1.

7

GE 52(68)

[0 < |t| < 2π]

(cf. 1.213)

GE 65(89)a

Series representation   ∞ n!  cos 2πkx − 12 πn B n (x) = −2 (2π)n kn k=1

[n > 1,

1 ≥ x ≥ 0;

n = 1,

1 > x > 0]

AS 805(23.1.16)

1042

Bernoulli Numbers and Polynomials, Euler Numbers

9.623

∞

2.7

B 2n−1 (x) = 2

(−1)n 2(2n − 1)!  sin 2kπx (2π)2n−1 k 2n−1 k=1

[n > 1,

1 ≥ x ≥ 0;

n = 1,

1 > x > 0]

AS 805(23.1.17)

∞

3.10

B 2n (x) =

(−1)n−1 2(2n)!  cos 2kπx (2π)2n k 2n

[0 ≤ x ≤ 1,

n = 1, 2, . . .]

GE 71

k=1

9.623

Functional relations and properties: B m+1 (n) = Bm+1 + (m + 1)

1.

n−1 

km

k=1

[n and m are natural numbers] 2.

B n (x + 1) − B n (x) = nxn−1

3.

B n (x)

4.

B n (1 − x) = (−1) B n (x)

5.

(see also 0.121)

GE 51(65) GE 65(90)

= n B n−1 (x)

[n = 1, 2, . . .]

GE 66

n

10

9.6247

n

GE 66 n−1

(−1) B n (−x) = B n (x) + nx m−1 

B n (mx) = mn−1

k=0

[n = 0, 1, . . .]



k Bn x + m [m = 1, 2, . . . n = 0, 1, . . .] ;

9.625

AS 804(23.1.9)

“summation theorem”

GE 67

For n odd, the differences

B n (x) − Bn vanish on the interval [0, 1] only at the points 0, 12 , and 1. They change sign at the point x = 12 . For n even, these differences vanish at the end points of the interval [0, 1]. Within this interval, they do not change sign, and their greatest absolute value occurs at the point x = 12 . 9.626 The polynomials B 2n (x) − B2n and B 2n+2 (x) − B2n+2 have opposite signs in the interval (0, 1). GE 87 9.627 Special cases: 1.

B 1 (x) = x −

2.

B 2 (x) = x − x +

3.

B 3 (x) = x −

4.

B 4 (x) = x4 − 2x + x −

GE 70

5.

B 5 (x) = x −

GE 70

9.628

1 2

GE 70

2 3

5

1 6

3 2 2x 3

+

5 4 2x

+

GE 70 1 2x 2

GE 70

1 30 5 3 1 3x − 6x

Particular values:

1.

B n (0) = Bn

2.

B 1 (1) = −B1 = 12 ,

B n (1) = Bn

[n = 1]

GE 76

9.640

The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y)

1043

9.63 Euler numbers 9.630

n

The numbers En , representing the coefficients of tn! in the expansion of the function ∞ +  1 tn π, = En |t| < , cosh t n=0 n! 2

are known as the Euler numbers. Thus, the function

1 cosh t

is a generating function for the Euler numbers. CE 330

9.631

A recursion formula (E + 1)[n] + (E − 1)[n] = 0

[n ≥ 1] ,

E0 = 1

CE 329

Properties of the Euler numbers 9.632 The Euler numbers are integers. 9.633 The Euler numbers of odd index are equal to zero; the signs of two adjacent numbers of even indices are opposite; that is, E2n+1 = 0, E4n > 0, E4n+2 < 0. CE 329 9.634 If α, βγ, . . . are the divisors of the number n − m, the difference E2n − E2m is divisible by those of the numbers 2α + 1, 2β + 1, 2γ + 1, . . . that are primes. 9.635 A connection with the Bernoulli numbers (symbolic notation):

3.6

    (4B − 1)[n] − (4B − 3)[n] + 4(−1)n+1 3n−1 − 1 B1 En−1 + 4(−1)n 3n−1 − 1 B1 = 2n n(E + 1)[n−1] [n ≥ 2] Bn = n n 2 (2 − 1)  [2n+1] B + 14 = −4−2n−1 (2n + 1)E2n [n ≥ 0]

4.

En−1 =

1.11 2.

(4B + 3)[n] − (4B + 1)[n] 2n

CE 330 CE 330 CE 341

[n ≥ 1]

For a table of values of the Euler numbers, see 9.72.

9.64 The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y) 9.640 1.



∞

ν(x) = 0

2.



xt dt Γ(t + 1) ∞

ν(x, α) = 0

 3.

∞

μ(x, β) = 0

4.

xα+t dt Γ(α + t + 1)

EH III 217(1)

xt tβ dt Γ(β + 1) Γ(t + 1)

EH III 217(2)

μ(x, β, α) = 

5.



λ(x, y) = 0

0 y

EH III 217(1)

∞

xα+t tβ dt Γ(β + 1) Γ(α + t + 1)

Γ(u + 1) du xu

EH III 217(2) MI 9

1044

Bernoulli Numbers and Polynomials, Euler Numbers

9.650

9.6510 Euler polynomials 9.650

The Euler polynomials are defined by

n−k n   n Ek 1 E n (x) = x− k 2k 2

AS 804 (23.1.7)

k=0

9.651

The generating function: ∞  tn 2ext = E n (x) t e + 1 n=0 n!

9.652 1.

AS 804 (23.1.1)

Series representation:   ∞ n!  sin (2k + 1)πx − 12 πn π n+1 (2k + 1)n+1

E n (x) = 4

k=0

[n > 0,

1 ≥ x ≥ 0,

n = 1,

1 > x > 0]

AS 804 (23.1.16)

∞

2.

10

(−1)n 4(2n − 1)!  cos(2k + 1)πx E 2n−1 (x) = π 2n (2k + 1)2n

[n = 1, 2, . . . ,

1 ≥ x ≥ 0]

k=0

AS 804 (23.1.17) ∞

3.

E 2n (x) =

(−1)n 4(2n)!  sin(2k + 1)πx π 2n+1 (2k + 1)2n+1 k=0

[n > 0,

9.653 1.

1 ≥ x ≥ 0,

n = 0,

1 > x > 0]

AS 804 (23.1.18)

Functional relations and properties: E m (n + 1) = 2

n 

(−1)n−k k m + (−1)n+1 E m (0),

[m and n are natural numbers]

k=1

AS 804 (23.1.4)

2.

En (x) = nEn−1 (x). n

[n = 1, 2, . . .]

AS 804 (23.1.5) AS 804 (23.1.6)

3.

E n (x + 1) + E n (x) = 2x

[n = 0, 1, . . .]

4.8



m−1  k n k E n (mx) = m (−1) En x − m

[n = 0, 1, . . . , m = 1, 3, . . .]

k=0

AS 804 (23.1.10)

5.

E n (mx) =



m−1  −2 k mn (−1)k Bn+1 x + n+1 m

[n = 0, 1, . . . , m = 2, 4, . . .]

k=0

AS 804 (23.1.10)

9.654

Special cases:

1.

E 1 (x) = x −

2.

E 2 (x) = x2 − x

1 2

9.655

Euler numbers

3.

E 3 (x) = x3 − 32 x2 +

4.

E 4 (x) = x4 − 2x3 + x

5.

E 5 (x) = x5 − 52 x4 + 52 x2 −

9.655 1.

1045

1 4

1 2

Particular values: E2n+1 = 0.

[n = 0, 1, . . .]   E n (0) = − E n (1) = −2(n + 1)−1 2n+1 − 1 Bn+1 [n = 1, 2, . . .] 1 −n E n 2 = 2 En [n = 0, 1, . . .] 1 2     E 2n−1 3 = − E 2n−1 3 = −(2n)−1 1 − 31−2n 22n − 1 B2n

2. 3. 4.

[n = 1, 2, . . .]

AS 805 (23.1.19) AS 805 (23.1.20) AS 805 (23.1.21)

AS 806 (23.1.22)

9.7 Constants 9.71 Bernoulli numbers • • • • • • • • • •

B0 = 1 B1 = − 1/2 B2 = 1/6 B4 = − 1/30 B6 = 1/42 B8 = − 1/30 B10 = 5/66 B12 = − 691/2730 B14 = 7/6 B16 = − 3617/510

• • • • • • • • •

B18 B20 B22 B24 B26 B28 B30 B32 B34

= 43 867/798 = − 174 611/330 = 854 513/138 = − 236 364 091/2730 = 8 553 103/6 = − 23 749 461 029/870 = 8 615 841 276 005/14 322 = − 7 709 321 041 217/510 = 2 577 687 858 367/6

• • • • •

E12 E14 E16 E18 E20

= 2 702 765 = −199 360 981 = 19 391 512 145 = −2 404 879 675 441 = 370 371 188 237 525

9.72 Euler numbers • • • • • •

E0 = 1 E2 = −1 E4 = 5 E6 = −61 E8 = 1385 E10 = −50 521

The Bernoulli and Euler numbers of odd index (with the exception of B 1 ) are equal to zero.

1046

Constants

9.740

9.73 Euler’s and Catalan’s constants Euler’s constant C = 0.577 215 664 901 532 860 606 512 . . .

(cf. 8.367)

Catalan’s constant ∞  (−1)k = 0.915 965 594 . . . G= (2k + 1)2 k=0

9.7410 Stirling numbers (m)

(m)

9.740 The Stirling number of the first kind Sn is defined by the requirement that (−1)n−m Sn is the number of permutations of n symbols which have exactly m cycles. AS 824 (23.1.3) 9.741 Generating functions: 1.

x(x − 1) · · · (x − n + 1) =

n 

Sn(m) xm

AS 824 (24.1.3)

m=0

2.

{ln(1 + x)}

m

∞ 

= m!

Sn(m)

n=m

9.742 1.8

2.

9.743 1.

2.

3.

xn n!

[|x| < 1]

AS 824 (24.1.3)

Recurrence relations: (m)

Sn+1 = Sn(m−1) − nSn(m) ; m r

Sn(m) =

n−r  k=m−r

Sn(0) = δ0n ;

Sn(1) = (−1)n−1 (n − 1)!;

n (r) (m+r) S S k n−k k

Sn(n) = 1

[n ≥ m ≥ 1]

AS 824 (24.1.3)

[n ≥ m ≥ r]

AS 824 (24.1.3)

Functional relations and properties   m  k  hm Γ hx + 1 x (m)  = hm x(x − h)(x − 2h) · · · (x − mh + h) =  x Sk h Γ h −m+1 k=1 &−1 

−1 % p x+m −1 k (m) [(x + 1)(x + 2) · · · (x + m)] = m! = (x + m) Sk m k=1 &−1 %   m  k  Γ hx + 1 x (m) −1 m = h + m Sk [(x + h)(x + 2h) · · · (x + mh)] = m  x h h Γ h +m+1 k=1

(m)

9.744 The Stirling number of the second kind Sn elements into m non-empty subsets.

is the number of ways of partitioning a set of n

9.748

9.745 1.

Stirling numbers

1047

Generating functions: xn =

n 

S(m) n x(x − 1) · · · (x − m + 1)

AS 824 (24.1.4)

m=0

2.

m

(ex − 1)

∞ 

= m!

S(m) n

n=m

3.

xn n!

[(1 − x)(1 − 2x) · · · (1 − mx)]

AS 824 (24.1.4) −1

∞ 

=



n−m S(m) n x

|x| < m−1



AS 824 (24.1.4)

n=m

9.746

Closed form expression: m 1  kn (−1)m−k k m! m

1.

S(m) = n

AS 824 (24.1.4)

k=0

9.747 1.8

2.

3.

Recurrence relations: (m)

Sn+1 = mS(m) + S(m−1) , n n m r

k

k=m−r

Sn(m) =



n−m 

(−1)k

k=0

9.7487

n

n−r 

S(m) = n

S(0) n = δ0n ,

(r)

(n) S(1) n = Sn = 1

(m−r)

Sn−k Sk

n−1+k n−m+k



[n ≥ m ≥ 1]

AS 825(24.1.4)

[n ≥ m ≥ r]

AS 825 (24.1.4)

2n − m (k) Sn−m+k n−m−k

Particular values:

AS 824 (24.1.3)

(m)

Stirling numbers of the first kind Sn m 1 2 3 4 5 6 7 8 9

(m)

S1

1

(m)

S2

-1 1

(m)

S3

2 -3 1

(m)

S4

-6 11 -6 1

(m)

S5

24 -50 35 -10 1

(m)

S6 -120 274 -225 85 -15 1

(m)

S7 720 -1764 1624 -735 175 -21 1

(m)

S8 -5040 13068 -13132 6769 -1960 332 -28 1

(m)

S9 40320 -109584 118121 -67284 22449 -4536 546 -36 1

1048

Constants

9.749

(m)

Stirling numbers of the second kind Sn m 1 2 3 4 5 6 7 8 9 9.7498 1.

(m)

S1

(m)

S2

1

1 1

(m)

S3

(m)

S4

1 3 1

1 7 6 1

(m)

S5

1 15 25 10 1

(m)

S6

1 31 90 65 15 1

(m)

S7

1 63 301 350 140 21 1

(m)

(m)

S8

S9

1 127 966 1701 1050 266 28 1

1 255 3025 7770 6951 2646 462 36 1 −m

Relationship between Stirling numbers of the first kind and derivatives of (ln x) dn dxn



1 lnm x

=

1 lnm x

:

n (k)  (−1)k (m)k Sn k=1

xn lnk x where (m)k = Γ(m + k)/ Γ(m),

[m, n are positive integers]

10 Vector Field Theory 10.1–10.8 Vectors, Vector Operators, and Integral Theorems 10.11 Products of vectors Let a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ), and c = (c1 , c2 , c2 ) be arbitrary vectors, and i, j, k be the set of orthogonal unit vectors in terms of which the components of a, b, and c are expressed. Two different products involving pairs of vectors are defined, namely, the scalar product, written a · b, and the vector product, written either a × b or a ∧ b. Their properties are as follows: 1. 2.

3. 4.

a · b = a1 b 1 + a2 b 2 + a3 b 3 ! ! !i j k !! ! a × b = !!a1 a2 a3 !! ! b1 b2 b3 ! ! ! ! a1 a2 a3 ! ! ! a × b · c = !! b1 b2 b3 !! ! c1 c2 c3 !

(scalar product) (vector product)

(triple scalar product)

a × (b × c) = (a · c) b − (a · b) c

(triple vector product)

10.12 Properties of scalar product 1.

a·b=b·a

2.

a × b · c = b × c · a = c × a · b = −a × c · b = −b × a · c = −c × b · a.

(commutative)

Note: a × b · c is also written [a, b, c]; thus (2) may also be written 3.

[a, b, c] = [b, c, a] = [c, a, b] = − [a, c, b] = − [b, a, c] = − [c, b, a]

10.13 Properties of vector product 1.

a × b = −b × a

2.

a × (b × c) = −a × (c × b) = − (b × c) × a

3.

a × (b × c) + b × (c × a) + c × (a × b) = 0

(anticommutative)

1049

1050

Vectors, Vector Operators, and Integral Theorems

10.14 Differentiation of vectors If a(t) = (a1 (t), a2 (t), a3 (t)), b(t) = (b1 (t), b2 (t), b3 (t)), c(t) = (c1 (t), c2 (t), c3 (t)), φ(t) is a scalar and all functions of t are differentiable, then 1. 2. 3. 4. 5. 6. 7.

da1 da2 da3 da = i+ j+ k dt dt dt dt d da db (a + b) = + dt dt dt dφ da d (φa) = a+φ dt dt dt da db d (a · b) = ·b+a· dt dt dt da db d (a × b) = ×b+a× dt dt dt da db dc d (a × b · c) = ×b·c+a× ·c+a×b· dt dt dt dt



db da dc d {a × (b × c)} = × (b × c) + a × ×c +a× b× dt dt dt dt

10.21 Operators grad, div, and curl In cartesian coordinates O {x1 , x2 , x3 }, in which system it is convenient to denote the triad of unit vectors by e1 , e2 , e3 , the vector operator ∇, called either “del” or “nabla,” has the form 1.

2.

3.

4.

∂ ∂ ∂ + e2 + e3 ∂x1 ∂x2 ∂x3 If Φ(x, y, z) is any differentiable scalar function, the gradient of Φ, written grad Φ, is

∇ ≡ e1

∂Φ ∂Φ ∂Φ e1 + e2 + e3 ∂x1 ∂x2 ∂x3 The divergence of the differentiable vector function f = (f1 , f2 , f3 ), written div f , is grad Φ ≡ ∇ Φ =

∂f1 ∂f2 ∂f3 + + ∂x1 ∂x2 ∂x3 The curl, or rotation, of the differentiable vector function f = (f1 , f2 , f3 ), written either curl f or rot f , is



∂f1 ∂f2 ∂f3 ∂f2 ∂f3 ∂f1 − − − e1 + e2 + e3 , curl f ≡ rot f ≡ ∇ ×f = ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 or equivalently, ! ! ! e1 e2 e2 !! ! ∂ ∂ ∂ ! curl f = !! ∂x1 ∂x ∂x3 ! 2 ! f1 f2 f3 ! div f ≡ ∇ ·f =

Properties of the operator ∇

1051

10.31 Properties of the operator ∇ Let Φ (x1 , x2 , x3 ), Ψ (x1 , x2 , x3 ) be any two differentiable scalar functions, f (x1 , x2 , x3 ), g (x1 , x2 , x3 ) any two differentiable vector functions, and a an arbitrary vector. Define the scalar operator ∇2 , called the Laplacian, by ∂2 ∂2 ∂2 + + ∇2 ≡ 2 2 ∂x1 ∂x2 ∂x23 Then, in terms of the operator ∇, we have the following: MF I 114 1.

∇(Φ + Ψ) = ∇ Φ + ∇ Ψ

2.

∇(ΦΨ) = Φ ∇ Ψ + Ψ ∇ Φ

3.

∇ (f · g) = (f · ∇) g + (g · ∇) f + f × (∇ ×g) + g × (∇ ×f )

4.

∇ · (Φf ) = Φ (∇ ·f ) + f · ∇ Φ

5.

∇ · (f × g) = g · (∇ ×f ) − f · (∇ ×g)

6.

∇ × (Φf ) = Φ (∇ ×f ) + (∇ Φ) × f

7.

∇ × (f × g) = f (∇ ·g) − g (∇ ·f ) + (g · ∇) f − (f · ∇) g

8.

∇ × (∇ ×f ) = ∇ (∇ ·f ) − ∇2 f

9.

∇ × (∇ Φ) ≡ 0

10.

∇ · (∇ ×f ) ≡ 0

11.10 ∇2 (ΦΨ) = Φ ∇2 Ψ + 2 (∇ Φ) · (∇ Ψ) + Ψ ∇2 Φ The equivalent results in terms of grad, div, and curl are as follows: 1.

grad(Φ + Ψ) = grad Φ + grad Ψ

2.

grad(ΦΨ) = Φ grad Ψ + Ψ grad Φ

3.

grad (f · g) = (f · grad) g + (g · grad) f + f × curl g + g × curl f

4.

div (Φf ) = Φ div f + f · grad Φ

5.

div (f × g) = g · curl f − f · curl g

6.

curl (Φf ) = Φ curl f + grad Φ × f

7.

curl (f × g) = f div g − g div f + (g · grad) f − (f · grad) g

8.

curl (curl f ) = grad (div f ) − ∇2 f

9.

curl (grad Φ) ≡ 0

10.

div (curl f ) ≡ 0

11.

∇2 (ΦΨ) = Φ ∇2 Ψ + 2 grad Φ · grad Ψ + Ψ ∇2 Φ

The expression (a · ∇) or, equivalently (a · grad), defined by ∂ ∂ ∂ + a2 + a3 , (a · ∇) ≡ a1 ∂x1 ∂x2 ∂x3 is the directional derivative operator in the direction of vector a.

1052

Vectors, Vector Operators, and Integral Theorems

10.411

10.41 Solenoidal fields A vector field f is said to be solenoidal if div f ≡ 0. We have the following representation: 10.411 Representation theorem for vector Helmholtz equation. If u is a solution of the scalar Helmholtz equation ∇2 u + λ2 u = 0, and m is a constant unit vector, then the vectors 1 X = curl (mu) , Y = curl X λ are independent solutions of the vector Helmholtz equation ∇2 H + λ2 H = 0 involving a solenoidal vector H. The general solution of the equation is 1 H = curl (mu) + curl curl (mu) . λ

10.51–10.61 Orthogonal curvilinear coordinates Consider a transformation from the cartesian coordinates O {x1 , x2 , x3 } to the general orthogonal curvilinear coordinates O {u1 , u2 , u3 }: x1 = x1 (u1 , u2 , u3 ) , x2 = x2 (u1 , u2 , u3 ) , x3 = x3 (u1 , u2 , u3 ) Then, 1.

2.

∂xi ∂xi ∂xi du1 + du2 + du3 ∂u1 ∂u2 ∂u3 and the length element dl may be determined from dxi =

(i = 1, 2, 3) ,

dl2 = g11 du21 + g22 du22 + g33 du23 + 2g23 du2 du3 + 2g31 du3 du1 + 2g12 du1 du2 , where

3.3

4.

∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 + + = gji , ∂ui ∂uj ∂ui ∂uj ∂ui ∂uj provided the Jacobian of the transformation ! ! ! ∂x1 ∂x2 ∂x3 ! ! ∂u1 ∂u1 ∂u1 ! ! ∂x1 ∂x2 ∂x3 ! J = ! ∂u ∂u2 !! 2 ! ∂x12 ∂u ∂x2 ∂x3 ! ! ∂u ∂u ∂u

gij =

3

3

gij = 0,

i = j,

3

does not vanish (see 14.313). 5.

Define the metrical coefficients √ √ √ h1 = g11 , h2 = g22 , h3 = g33 ; then the volume element dV in orthogonal curvilinear coordinates is

6.

dV = h1 h2 h3 du1 du2 du3 , and the surface elements of area dsi on the surfaces ui = constant, for i = 1, 2, 3, are

7.

ds1 = h2 h3 du2 du3 ,

ds2 = h1 h3 du1 du3 ,

ds3 = h1 h2 du1 du2

Denote by e1 , e2 , and e3 the triad of orthogonal unit vectors that are tangent to the u1 , u2 , and u3 coordinate lines through any given point P , and choose their sense so that they form a right-handed set in this order. Then in terms of this triad of vectors and the components fu1 , fu2 , and fu3 of f along the coordinate line,

10.613

8. 10.611 1. 2.3

3.

4.

Orthogonal curvilinear coordinates

f = fu1 e1 + fu2 e2 + fu3 e3

1053

MF I 115

∇ Φ, div f , curl f , and ∇2 in general orthogonal curvilinear coordinates. e1 ∂Φ e2 ∂Φ e3 ∂Φ + + h1 ∂u1 h2 ∂u2 h3 ∂u3

∂ 1 ∂ ∂ div f = (h2 h3 fu1 ) + (h3 h1 fu2 ) + (h1 h2 fu3 ) h1 h2 h3 ∂u1 ∂u2 ∂u3 ! ! ! h1 e 1 h2 e 2 h3 e 3 ! ! ∂ ! 1 ∂ ∂ ! ! curl f = ∂u1 ∂u2 ∂u3 ! ! h1 h2 h3 ! h1 fu1 h2 fu2 h3 fu3 !





∂ h2 h3 ∂ h3 h1 ∂ h1 h2 ∂ 1 ∂ ∂ 2 ∇ ≡ + + h1 h2 h3 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 ∂u3 h3 ∂u3 grad Φ =

MF I 21-31

10.612 Cylindrical polar coordinates. In terms of the coordinates O{r, φ, z}, that is, u1 = r, u2 = φ, u3 = z, where x1 = r cos φ, x2 = r sin φ, x3 = z for −π < φ ≤ π, it follows that 1.

h1 = 1,

h2 = r,

h3 = 1,

and 2. 3.

4.

5.

1 ∂Φ ∂Φ ∂Φ er + eφ + ez , ∂r r ∂φ ∂z ∂fz 1 ∂fφ 1 ∂ (rfr ) + + , div f = r ∂r r ∂φ ∂z ! ! !e reφ ez !! 1 !! ∂r ∂ ∂ ! curl f = ! ∂r ∂φ ∂z ! , r! fr rfφ fz !

1 ∂ ∂2 ∂ 1 ∂2 2 ∇ ≡ r + 2 2+ 2 r ∂r ∂r r ∂φ ∂z grad Φ =

MF I 116

10.613 Spherical polar coordinates. In terms of the coordinates O{r, θ, φ}, that is, u1 = r, u2 = θ, u3 = φ, where x1 = r sin θ cos φ, x2 = r sin θ sin φ, x3 = r cos θ, for 0 ≤ θ ≤ π, −π < φ ≤ π, we have 1.

h1 = 1,

h2 = r,

h3 = r sin θ,

and 2.10 3.

4.

5.

∂Φ 1 ∂Φ 1 ∂Φ er + eθ + eφ , ∂r r ∂θ r sin θ ∂φ ∂ 1 1 ∂fφ 1 ∂  2  r fr + (fθ sin θ) + , div f = 2 r ∂r r sin θ ∂θ r sin θ ∂φ ! ! ! er reθ r sin θeφ ! ! ! 1 ∂ ∂ !, !∂ curl f = 2 ∂r ∂θ ∂φ ! ! r sin θ ! fr rfθ r sin θfφ !



∂2 ∂ ∂ 1 ∂ 1 ∂ 1 ∇2 ≡ 2 r2 + 2 sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2

grad Φ =

MF I 116

1054

Vectors, Vector Operators, and Integral Theorems

10.614

Special Orthogonal Curvilinear Coordinates and their Metrical Coefficients h1 , h2 , h3 10.614 1. 2. 10.615 1. 2. 10.616 1.

Elliptic cylinder coordinates O{u1 , u2 , u3 }. < x1 = u1 u2 , x2 = (u21 − c2 ) (1 − u22 ), . . u21 − c2 u22 u21 − c2 u22 h1 = , h2 = , 2 2 u1 − c 1 − u22

10.617 1. 2. 10.618 1. 2. 10.619 1. 2.

h3 = 1

MF I 657

Parabolic cylinder coordinates O{u1 , u2 , u3 }.  1 2 u1 − u22 , 2 < h1 = u21 + u22 , x1 =

x2 = u1 u2 , x3 = u3 < h2 = u21 + u22 , h3 = 1

Conical coordinates O{u1 , u2 , u3 }. < < u1 u1 x1 = (a2 − u22 ) (a2 + u23 ), x2 = (b2 + u22 ) (b2 − u23 ), a b .

2.

x3 = u3

h1 = 1,

h2 = u 1

(a2

u22 + u23 , − u22 ) (b2 + u22 )

MF I 658

x3 =

u1 u2 u3 ab

with a2 + b2 = 1 . u22 + u23 h3 = u 1 2 (a + u23 ) (b2 − u23 )

Rotational parabolic coordinates O{u1 , u2 , u3 }. <  1 2 u1 − u22 x2 = u1 u2 1 − u23 , x3 = x1 = u1 u2 u3 , 2 < < u1 u2 h2 = u21 + u22 , h3 =  h1 = u21 + u22 , 1 − u23 Rotational prolate spheroidal coordinates O{u1 , u2 , u3 }. < < x1 = (u21 − a2 ) (1 − u22 ), x2 = (u21 − a2 ) (1 − u22 ) (1 − u23 ), x3 = u1 u2 . . . u21 − a2 u22 u21 − a2 u22 (u21 − a2 ) (1 − u22 ) h1 = , h = , h = 2 3 u21 − a2 1 − u22 1 − u23 Rotational oblate spheroidal coordinates O{u1 , u2 , u3 }. < < x2 = (u21 + a2 ) (1 − u22 ) (1 − u23 ), x3 = u1 u2 x1 = u3 (u21 + a2 ) (1 − u22 ), . . . u21 + a2 u22 u21 + a2 u22 (u21 + a2 ) (1 − u22 ) h1 = , h2 = , h3 = 2 2 2 u1 + a 1 − u2 1 − u23

MF I 659

MF I 660

MF I 661

MF I 662

10.713

10.620 1.

2.

Vector integral theorems

1055

Ellipsoidal coordinates O{u1 , u2 , u3 }. . . (u21 − a2 ) (u22 − a2 ) (u23 − a2 ) (u21 − b2 ) (u22 − b2 ) (u23 − b2 ) u1 u2 u3 x1 = , x , x3 = = 2 a2 (a2 − b2 ) b2 (b2 − a2 ) ab . . . (u21 − u22 ) (u21 − u23 ) (u22 − u21 ) (u22 − u23 ) (u23 − u21 ) (u23 − u22 ) , h , h h1 = = = 2 3 (u21 − a2 ) (u21 − b2 ) (u22 − a2 ) (u22 − b2 ) (u23 − a2 ) (u23 − b2 ) MF I 663

10.621 1.

2.

Paraboloidal coordinates O{u1 , u2 , u3 }. . . (u21 − a2 ) (u22 − a2 ) (u23 − a2 ) (u21 − b2 ) (u22 − b2 ) (u23 − b2 ) , x2 = , x1 = 2 2 a −b b 2 − a2   x3 = 12 u21 + u22 + u23 − a2 − b2 . . . (u21 − u22 ) (u21 − u23 ) (u23 − u21 ) (u23 − u22 ) (u23 − u21 ) (u23 − u22 ) h1 = , h , h = u = u 2 2 3 3 2 2 2 2 (u1 − a2 ) (u1 − b2 ) (u2 − a2 ) (u2 − b2 ) (u23 − a2 ) (u23 − b2 ) MF I 664

10.622 1. 2.

Bispherical coordinates O{u1 , u2 , u3 }.   1 − u22 (1 − u22 ) (1 − u23 ) x1 = au3 , x2 = a , u1 − u2 u1 − u2 a  h1 = , (u1 − u2 ) u21 − 1

 x3 =

a  , h2 = (u1 − u2 ) 1 − u22

u21 − 1 u1 − u2

h3 =

a u1 − u2

.

1 − u22 1 − u23

MF I 665

10.71–10.72 Vector integral theorems 10.711 Gauss’s divergence theorem. Let V be a volume bounded by a simple closed surface S and let f be a continuously differentiable vector field defined in V and on S. Then, if dS is the outward drawn vector element of area,   f · dS = div f dV KE 39 S

V

10.712 Green’s theorems. Let Φ and Ψ be scalar fields which, together with ∇2 Φ and ∇2 Ψ, are defined both in a volume V and on its surface S, which we assume to be simple and closed. Then, if ∂/∂n denotes differentiation along the outward drawn normal to S, we have 10.713 Green’s first theorem     ∂Ψ Φ ∇2 Ψ + grad Φ · grad Ψ dV dS = Φ KE 212 S ∂n V

1056

Vectors, Vector Operators, and Integral Theorems

Green’s second

 theorem    ∂Φ ∂Ψ −Ψ Φ ∇2 Ψ − Ψ ∇2 Φ dV Φ dS = ∂n ∂n S V 10.715 Special cases    2 (Φ grad Φ) · dS = 1. Φ ∇2 Φ + (grad Φ) dV S V   ∂Φ dS = 2. ∇2 Φ dV S ∂n V

10.714

10.714

10.716 3.

Green’s reciprocal theorem. If Φ and Ψ are harmonic, so that ∇2 Φ = ∇2 Ψ = 0, then   ∂Ψ ∂Φ dS = Ψ dS Φ S ∂n S ∂n

KE 215

MV 81

MM 105

10.717 Green’s representation theorem. If Φ and ∇2 Φ are defined within a volume V bounded by a simple closed surface S, and P is an interior point of V , then in three dimensions

   1 1 2 1 ∂Φ 1 ∂ 1 1 ∇ Φ dV + dS − Φ KE 219 4. Φ(P ) = − dS 4π V r 4π S r ∂n 4π S ∂n r

5.

6.

If Φ is harmonic within V , so that ∇2 Φ = 0, then the previous result becomes

  1 ∂Φ 1 ∂ 1 1 dS − Φ Φ(P ) = dS 4π S r ∂n 4π S ∂n r In the case of two dimensions, result (4) takes the form  1 Φ(p) = ∇2 Φ(q) ln |p − q| dS 2π S  1 ∂ 1 ∂ + Φ(q) ln |p − q| dq − Φ(q) dq ln |p − q| 2π C ∂nq 2π ∂nq MM 116

7.

where C is the boundary of the planar region S, and result (5) takes the form   ∂ 1 ∂ 1 Φ(q) ln |p − q| dq − ln |p − q| Φ(q) dq Φ(p) = 2π C ∂nq 2π C ∂nq

VL 280

10.718 Green’s representation theorem in Rn . If Φ is twice differentiable within a region Ω in Rn bounded by the surface Σ with outward drawn unit normal n, then for p ∈ Σ and n > 3     −1 ∇2 Φ(q) ∂ Φ(q) 1 1 ∂ 1 Φ(p) = dΩq + − Φ(q) dΣq , (n − 2)σn Ω |p − q|n−2 (n − 2)σn Σ |p − q|n−2 ∂nq ∂nq |p − q|n−2 where 2π n/2 σn = VL 279 Γ(n/2) is the area of the unit sphere in Rn . 10.719 Green’s theorem of the arithmetic mean. If Φ is harmonic in a sphere, then the value of Φ at KE 223 the center of the sphere is the arithmetic mean of its value on the surface. 10.720 Poisson’s integral in three dimensions. If Φ is harmonic in the interior of a spherical volume V of radius R and is continuous on the surface of the sphere on which, in terms of the spherical polar coordinates (r, θ, φ), it satisfies the boundary condition Φ (R, θ, φ) = f (θ, φ), then

10.811

where

Integral rate of change theorems

1057

   π π R R2 − r 2 f (θ , φ ) sin θ dθ dφ Φ(r, θ, φ) = , 3/2 4π 0 −π (r 2 + R2 − 2rR cos γ)

cos γ = cos θ cos θ + sin θ sin θ cos (φ − φ ) . KE 241 10.721 Poisson’s integral in two dimensions. If Φ is harmonic in the interior of a circular disk S of radius R and is continuous on the boundary of the disk on which, in terms of the polar coordinates (r, θ), it satisfies the boundary condition Φ(R, θ) = f (θ), then  π  2 R − r2 f (φ) dφ . Φ(r, θ) = 2 2 2π −π r + R − 2rR cos(θ − φ) 10.722 Stokes’ theorem. Let a simple closed curve C be spanned by a surface S. Define the positive normal n to S, and the positive sense of description of the curve C with line element dr, such that the positive sense of the contour C is clockwise when we look through the surface S in the direction of the normal. Then, if f is continuously differentiable vector field defined on S and C with vector element S = n dS, C  f · dr = curl f · dS, MM 143 C

S

where the line integral around C is taken in the positive sense. 10.723 Planar case of Stokes’ theorem. If a region R in the (x, y)-plane is bounded by a simple closed curve C, and f1 (x, y), f2 (x, y) are any two functions having continuous first derivatives in R and on C, then

C   ∂f2 ∂f1 − (f1 dx + f2 dy) = MM 143 dx dy, ∂x ∂y C R where the line integral is taken in the counterclockwise sense.

10.81 Integral rate of change theorems 10.811 Rate of change of volume integral bounded by a moving closed surface. Let f be a continuous scalar function of position and time t defined throughout the volume V (t), which is itself bounded by a simple closed surface S (t) moving with velocity v. Then the rate of change of the volume integral of f is given by    ∂f D dV + f dV = f v · dS, Dt V (t) V (t) ∂t S(t) where dS is the outward drawn vector element of area, and D ∂ ≡ + v · ∇. Dt ∂t By virtue of Gauss’s theorem, this also takes the form

  D Df + f div v dV. f dV = MV 88 Dt V (t) Dt V (t)

1058

Vectors, Vector Operators, and Integral Theorems

10.812

10.812 Rate of change of flux through a surface. Let q be a vector function that may also depend on the time t, and n be the unit outward drawn normal to the surface S that moves with velocity v. Defining the flux of q through S as  m = q · n dS, then

S

∂q Dm = + v div q + curl (q × v) · n dS. MV 90 Dt ∂t S 10.813 Rate of change of the circulation around a given moving curve. Let C be a closed curve, moving with velocity v, on which is defined a vector field q. Defining the circulation ζ of q around C by  q · dr, ζ= 

C

then Dζ = Dt

 C

∂q + (curl q) × v · dr. ∂t

MV 94

11 Algebraic Inequalities 11.1–11.3 General Algebraic Inequalities 11.11 Algebraic inequalities involving real numbers 11.111

Lagrange’s identity. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two sets of real numbers; then  n 2  n  n      2 ak b k = a2k b2k − BB 3 (ak bj − aj bk ) k=1

k=1

k=1

11.112 Cauchy–Schwarz–Buniakowsky inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of real numbers; then  n 2  n  n     2 2 ak b k ≤ ak bk . k=1

k=1

k=1

The equality holds if, and only if, the sequences a1 , a2 , . . . , an and b1 , b2 , . . . , bn are proportional. MT 30

11.113 Minkowski’s inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two sets of nonnegative real numbers, and let p > 1; then 1/p  n 1/p  n 1/p  n   p  p p (ak + bk ) ≤ ak + bk . k=1

k=1

k=1

The equality holds if, and only if, the sequences a1 , a2 , . . . , an and b1 , b2 , . . . , bn are proportional. MT 55

11.114 H¨ older’s inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two sets of nonnegative real numbers, and let p1 + 1q = 1, with p > 1; then 1/p  n 1/q  n n  p  q  ak bk ≥ ak b k . k=1

k=1

k=1

The equality holds if, and only if, the sequences ap1 , ap2 , . . . , apn and bq1 , bq2 , . . . , bqn are proportional. MT 50

11.115 Chebyshev’s inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be two arbitrary sets of real numbers such that either a1 ≥ a2 ≥ · · · ≥ an and b1 ≥ b2 ≥ · · · ≥ bn , or a1 ≤ a2 ≤ · · · ≤ an and b1 ≤ b2 ≤ · · · ≤ bn ; then



n a1 + a 2 + · · · + a n b1 + b2 + · · · + bn 1 ak b k . ≤ n n n k=1 The equality holds if, and only if, either a1 = a2 = · · · = an or b1 = b2 = · · · = bn . 1059

1060

General Algebraic Inequalities

11.116

11.116 Arithmetic-geometric inequality. Let a1 , a2 , . . . , an be any set of positive numbers, with arithmetic mean

a1 + a2 + · · · + an An = n and geometric mean 1/n

then An ≥ Gn or, equivalently,

Gn = (a1 a2 . . . an )

;



a1 + a2 + · · · + an 1/n . ≥ (a1 a2 . . . an ) n The equality holds only in the event that all of the numbers ai are equal. 11.117 Carleman’s inequality. If a1 , a2 , . . . , an is any finite set of non-negative numbers, then n  1/r (a1 a2 . . . ar ) ≤ e (a1 + a2 + · · · + an ) ,

BB 4

r=1

where e is the best possible constant in this inequality. The inequality is strict except for the trivial case MT 131 when ar = 0 for r = 1, 2, . . . , n. 11.118 An inequality involving absolute values. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be two arbitrary sets of real numbers; then n  p p p p {|ai − bj | + |bi − aj | − |ai − aj | − |bi − bj | } ≥ 0, 0 < p ≤ 2. i,j=1

11.21 Algebraic inequalities involving complex numbers If α, β are any two real numbers, the complex number z = α + iβ with real part α and imaginary part β has for its modulus |z| the nonnegative number  |z| = α2 + β 2 , and for its argument (amplitude)arg z the angle arg z = θ such that α β cos θ = and sin θ = , |z| |z| where −π < θ ≤ π. The complex number z = α−iβ is said to be the complex conjugate of z = α+iβ. If z = reiθ = r (cos θ + i sin θ) , then z n = rn einθ = rn (cos nθ + i sin nθ) , and, setting r = 1, we have de Moivre’s theorem n

(cos θ + i sin θ) = cos nθ + i sin nθ. It follows directly that, if z = eiθ , then



1 1 i 1 cos θ = z+ , sin α = − z− , 2 z 2 z and



1 1 i 1 r r sin rθ = − cos rθ = z + r , z − r . 2 z 2 z p/q iθ If w = z with p, q integral, and z = re , then the q roots of w0 , w1 , . . . , wq−1 of z are

11.313

Inequalities for sets of complex numbers

wk = r

p/q

1061





 pθ + 2kπ pθ + 2kπ cos + i sin , q q

with k = 0, 1, 2, . . . , q − 1. 11.2117 Simple properties and inequalities involving the modulus and the complex conjugate. If the real part of z is denoted by Re z and the imaginary part by Im z, then z + z = 2 Re z = 2α, z − z = 2 Im z = 2iβ, z = (z),

1 1 = , z z n

(z n ) = (z) , ! ! ! z1 ! |z1 | ! != ! z2 ! |z2 | , (z1 + z2 + · · · + zn ) = z1 + z2 + · · · + zn , z1 z2 · · · zn = z1 z2 · · · zn . 11.212

Inequalities for pairs of complex numbers.

(i)

|a + b| ≤ |a| + |b|

(ii)

|a − b| ≥ ||a| − |b||.

If a,b are any two complex numbers, then

(triangle inequality),

11.31 Inequalities for sets of complex numbers 11.311 Complex Cauchy–Schwarz–Buniakowsky inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of complex numbers; then !2  !  n  n n ! !   ! ! 2 2 ak b k ! ≤ |ak | |bk | . ! ! ! k=1 k=1 k=1 The equality holds if, and only if, the sequences a1 , a2 , . . . , an and b1 , b2 , . . . , bn are proportional. MT 42

11.312 Complex Minkowski inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of complex numbers, and let the real number p be such that p > 1; then  n 1/p  n 1/p  n 1/p    p p p |ak + bk | ≤ |ak | + |bk | . MT 56 k=1

11.313

k=1

k=1

Complex H¨ older inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of 1 complex numbers, and let the real numbers p,q be such that p > 1 and + 1q = 1; then p ! 1/p  n 1/p ! n  n ! !   ! ! p q |ak | |bk | ≥! ak b k ! . ! ! k=1 k=1 k=1 The equality holds if, and only if, the sequences p p p p p p |a1 | , |a2 | , . . . , |an | and |b1 | , |b2 | , . . . |bn | , MT 53 are proportional and arg ak bk is independent of k for k = 1, 2, . . . , n.

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12 Integral Inequalities 12.11 Mean Value Theorems 12.111 First mean value theorem Let f (x) and g(x) be two bounded functions integrable in [a, b], and let g(x) be of one sign in this interval. Then  b  b f (x)g(x) dx = f (ξ) g(x) dx, CA 105 a

with a ≤ ξ ≤ b.

a

12.112 Second mean value theorem (i)

Let f (x) be a bounded, monotonic decreasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then,  ξ  b f (x)g(x) dx = f (a) g(x) dx, a

a

with a ≤ ξ ≤ b. (ii)

Let f (x) be a bounded, monotonic increasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then,  b  b f (x)g(x) dx = f (b) g(x) dx, a

η

with a ≤ η ≤ b. (iii)

Let f (x) be bounded and monotonic in [a, b], and let g(x) be a bounded integrable function which experiences only a finite number of sign changes in [a, b]. Then,  b  ξ  b f (x)g(x) dx = f (a + 0) g(x) dx + f (b − 0) g(x) dx, CA 107 a

a

ξ

with a ≤ ξ ≤ b.

12.113 First mean value theorem for infinite integrals Let f (x) be bounded for" x ≥ a, and integrable in the arbitrary interval [a, b], and let g(x) be of one sign ∞ in x ≥ a and such that a g(x) dx is finite. Then, 1063

1064

Differentiation of Definite Integral Containing a Parameter



∞

 f (x)g(x) dx = μ

a

∞

g(x) dx,

CA 123

a

where m ≤ μ ≤ M and m, M are, respectively, the lower and upper bounds of f (x) for x ≥ a.

12.114 Second mean value theorem for infinite integrals Let f (x) be bounded and monotonic when x ≥ a, and g(x) be bounded and integrable in the" arbitrary ∞ interval [a, b] in which it experiences only a finite number of changes of sign. Then, provided a g(x) dx is finite,  ξ  ∞  ∞ f (x)g(x) dx = f (a + 0) g(x) dx + f (∞) g(x) dx, CA 123 with a ≤ ξ ≤ ∞.

a

a

ξ

12.21 Differentiation of Definite Integral Containing a Parameter 12.211 Differentiation when limits are finite Let φ(α) and ψ(α) be twice differentiable functions in some interval c ≤ α ≤ d, and let f (x, α) be both integrable with respect to x over the interval φ(α) ≤ x ≤ ψ(α) and differentiable with respect to α. Then,



  ψ(α) dψ d ψ(α) dφ ∂f dx. f (x, α) dx = FI II 680 f (ψ(α), α) − f (φ(α), α) + dα φ(α) dα dα φ(α) ∂α

12.212 Differentiation when a limit is infinite Let f (x, α) and ∂f /∂α both be integrable with respect to x over the semi-infinite region x ≥ a, b ≤ α < c. Then, if the integral  ∞ f (x, α) dx f (α) = a " ∞ ∂f exists for all b ≤ α ≤ c, and if a ∂α dx is uniformly convergent for α in [b, c], it follows that  ∞  ∂f d ∞ dx f (x, α) dx = dα a ∂α a

12.31 Integral Inequalities 12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals Let f (x) and g(x) be any two real integrable functions on [a, b]. Then,  2     b b b 2 2 f (x)g(x) dx ≤ f (x) dx g (x) dx , a

a

a

and the equality will hold if, and only if, f (x) = kg(x), with k real.

BB 21

12.312 H¨ older’s inequality for integrals p

q

Let f (x) andg(x) be any two real functions for which |f (x)| and |g(x)| are integrable on [a, b] with p > 1 and p1 + 1q = 1; then

Gram’s inequality for integrals





b

1/p 

b

f (x)g(x) dx ≤

1065

|f (x)| dx

a

1/q

b

p

q

|g(x)| dx

a

.

a

p

q

The equality holds if, and only if, α|f (x)| = β|g(x)| , where α and β are positive constants.

BB 21

12.313 Minkowski’s inequality for integrals p

p

Let f (x) and g(x)be any two real functions for which |f (x)| and |g(x)| are integrable on [a, b] for p > 0; then 1/p  1/p  1/p  b b b p p p |f (x) + g(x)| dx ≤ |f (x)| dx + |g(x)| dx . a

a

a

The equality holds if, and only if, f (x) = kg(x) for some real k ≥ 0.

BB 21

12.314 Chebyshev’s inequality for integrals Let f1 , f2 , . . . , fn be nonnegative integrable functions on [a, b] which are all either monotonic increasing or monotonic decreasing; then  b  b  b  b n−1 f1 (x) dx f2 (x) dx . . . fn (x) dx ≤ (b − a) f1 (x)f2 (x) . . . fn (x) dx MT 39 a

a

a

a

12.315 Young’s inequality for integrals Let f (x) be a real-valued continuous strictly monotonic increasing function on the interval [0, a], with f (0) = 0 and b ≤ f (a). Then  b  a f (x) dx + f −1 (y) dy, ab ≤ 0

0

where f −1 (y) denotes the function inverse to f (x). The equality holds if, and only if, b = f (a).

BB 15

12.316 Steffensen’s inequality for integrals Let f (x) be nonnegative and monotonic decreasing in [a, b], and g(x) be such that 0 ≤ g(x) ≤ 1 in [a, b]. Then  b  a+k  b f (x) dx ≤ f (x)g(x) dx ≤ f (x) dx, b−k a a "b where k = a g(x) dx. MT 107

12.317 Gram’s inequality for integrals Let f1 (x), f2 (x), . . . , fn (x) be real square integrable functions on [a, b]; then ! "b ! "b "b ! ! 2 f (x)f (x) dx · · · f (x)f (x) dx 1 2 1 n ! " a f1 (x) dx ! a" "ab ! b ! b 2 ! a f2 (x)f1 (x) dx ! f (x) dx · · · f (x)f (x) dx n a 2 a 2 ! ! ≥ 0. .. .. .. .. ! ! . ! ! . . . !" b ! "b "b 2 ! fn (x)f1 (x) dx f (x)f2 (x) dx · · · f (x) dx ! a a n a n

MT 47

1066

Convexity and Jensen’s Inequality

12.318 Ostrowski’s inequality for integrals Let f (x) be a monotonic function integrable on [a, b], and let f (a)f (b) ≥ 0, |f (a)| ≥ |f (b)|. Then, if g is a real function integrable on [a, b], ! ! ! ! ! b ! ξ ! ! ! ! ! ! ! f (x)g(x) dx! ≤ |f (a)| max ! g(x) dx!. ! a ! ! a≤ξ≤b! a

12.41 Convexity and Jensen’s Inequality A function f (x) is said to be convex on an interval [a, b] if for any two points x1 , x2 in [a, b]

x1 + x2 f (x1 ) + f (x2 ) f . ≤ 2 2 A function f (x) is said to be concave on an interval [a, b] if for any two points x1 , x2 in [a, b] the function −f (x) is convex in that interval. If the function f (x) possesses a second derivative in the interval [a, b], then a necessary and sufficient condition for it to be convex on that interval is that f  (x) ≥ 0 for all x in [a, b]. A function f (x) is said to be logarithmically convex on the interval [a, b] if f > 0 and log f (x) is concave on [a, b]. If f (x) and g(x) are logarithmically convex on the interval [a, b], then the functions f (x) + g(x) and MT 17 f (x)g(x) are also logarithmically convex on [a, b].

12.411 Jensen’s inequality Let f (x), p(x) be two functions defined for a ≤ x ≤ b such that α ≤ f (x) ≤ β and p(x) ≥ 0, with p(x) ≡ 0. Let φ(u) be a convex function defined on the interval α ≤ u ≤ β; then  "b " b f (x)p(x) dx φ (f ) p(x) dx a ≤ a"b . φ HL 151 "b p(x) dx p(x) dx a a

12.412 Carleman’s inequality for integrals If f (x) ≥ 0 and the integrals exist, then  x

 ∞  ∞ 1 exp f (t) dt dx ≤ e f (x) dx. x 0 0 0

12.51 Fourier Series and Related Inequalities The trigonometric Fourier series representation of the function f (x) integrable on [−π, π] is ∞ a0  + f (x) ∼ (an cos nx + bn sin nx) , 2 n=1 where the Fourier coefficients an and bn of f (x) are given by   1 π 1 π an = f (x) cos nx dx, bn = f (x) sin nx dx. 2π −π 2π −π (See 0.320–0.328 for convergence of Fourier series on (−l, l).)

TF 1

Generalized Fourier series

1067

12.511 Riemann-Lebesgue lemma If f (x) is integrable on [−π, π], then

 lim

π

t→∞ −π

and



π

lim

t→∞ −π

12.512 Dirichlet lemma  

in which sin n +

1 2



x

D

π 0

2 sin

1 2x

f (x) sin tx dx → 0

f (x) cos tx dx → 0.

TF 11

  sin n + 12 x π dx = , 2 2 sin 12 x

is called the Dirichlet kernel.

ZY 21

12.513 Parseval’s theorem for trigonometric Fourier series If f (x) is square integrable on [−π, π], then  ∞  1 π 2 a20   2 + ar + b2r = f (x) dx. 2 π −π r=1

Y 10

12.514 Integral representation of the nth partial sum If f (x) is integrable on [−π, π], then the nth partial sum n a0  + sn (x) = (ar cos rx + br sin rx) 2 r=1 has the following integral representation in terms of the Dirichlet kernel:    sin n + 12 t 1 π f (x − t) sn (x) = dt. π −π 2 sin 12 t

12.515 Generalized Fourier series ∞

Let the set of functions {φn }n=0 form an orthonormal set over [a, b], so that   b 1 for m = n, φm (x)φn (x) dx = 0 for m = n. a Then the generalized Fourier series representation of an integrable function f (x) on [a, b] is ∞  f (x) ∼ cn φn (x), n=0

where the generalized Fourier coefficients of f (x) are given by  b cn = f (x)φn (x) dx. a

Y 20

1068

Fourier Series and Related Inequalities

12.516 Bessel’s inequality for generalized Fourier series For any square integrable function defined on [a, b],  b ∞  c2n ≤ f 2 (x) dx, n=0

a

where the cn are the generalized Fourier coefficients of f (x).

12.517 Parseval’s theorem for generalized Fourier series ∞

If f (x) is a square integrable function defined on [a, b] and {φn (x)}n=0 is a complete orthonormal set of continuous functions defined on [a, b], then  b ∞  2 cn = f 2 (x) dx, n=0

a

where the cn are generalized Fourier coefficients of f (x).

13 Matrices and Related Results 13.11–13.12 Special Matrices 13.111 Diagonal matrix A square matrix A of the form

⎡ λ1 ⎢0 ⎢ ⎢ A=⎢0 ⎢ .. ⎣.

0 λ2 0 .. .

0 0 λ3

... ... ..

0 0 0

.

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 0 0 λn in which all entries away from the leading diagonal are zero.

13.112 Identity matrix and null matrix The identity matrix is a diagonal matrix I in which all entries in the leading diagonal are unity. The null matrix is all zeros.

13.113 Reducible and irreducible matrices The n × n matrix A = [aij ] is said to be reducible, if the indices 1, 2, . . . , n can be divided into two disjoint non-empty sets i1 , i2 , . . . , iμ ; j1 , j2 , . . . , jν with (μ + ν = n), such that aiα jβ = 0 (α = 1, 2, . . . , μ; β = 1, 2, . . . , ν) . GA 61 Otherwise, A will be said to be irreducible.

13.114 Equivalent matrices An m×n matrix A is equivalent to an m×n matrix B if, and only if, B = PAQ for suitable non-singular m × m and n × n matrices P and Q, respectively.

13.115 Transpose of a matrix If A = [aij ] is an m × n matrix with element aij in the ith row and the j th column, then the transpose AT of A is the n × m matrix AT = [bij ] with bij = aji , that is, the matrix derived from A by interchanging rows and columns.

1069

1070

Special Matrices

13.116 Adjoint matrix If A is an n × n matrix, then its adjoint, denoted by adj A, is the transpose of the matrix of cofactors Aij of A, so that adj A = [Aij ]

T

(see 14.13).

13.117 Inverse matrix If A = [aij ] is an n × n matrix with a nonsingular determinant |A|, then its inverse A−1 is given by adj A . A−1 = |A|

13.118 Trace of a matrix The trace of an n × n matrix A = [aij ], written tr A, is defined to be the sum of the terms on the leading diagonal, so that tr A = a11 + a22 + . . . + ann .

13.119 Symmetric matrix The n × n matrix A = [aij ] is symmetric if aij = aji for i, j = 1, 2, . . . , n.

13.120 Skew-symmetric matrix The n × n matrix A = [aij ] is skew-symmetric if aij = −aji for i, j = 1, 2, . . . , n.

13.121 Triangular matrices An n × n matrix A = [aij ] is of upper triangular type if aij = 0 for i > j and of lower triangular type if aij = 0 for j > i.

13.122 Orthogonal matrices A real n × n matrix A is orthogonal if, and only if, AAT = I.

13.123 Hermitian transpose of a matrix If A = [aij ] is an n × n matrix with complex elements, then its hermitian transpose AH is defined to be AH = [aji ] , with the bar denoting the complex conjugate operation.

13.124 Hermitian matrix An n × n matrix A is hermitian if A = AH , or equivalently, if A = AT , with the bar denoting the complex conjugate operation.

Diagonally dominant

1071

13.125 Unitary matrix An n × n matrix A is unitary if AAH = AH A = I.

13.126 Eigenvalues and eigenvectors If A is an n × n matrix, each eigenvector x corresponding to λ satisfies the equation AX = λx, while the eigenvalues λ satisfy the characteristic equation |A − λI| = 0

(see 15.61).

13.127 Nilpotent matrix An n × n matrix A is nilpotent if Ak = 0 for some k.

13.128 Idempotent matrix An n × n matrix A is idempotent if A2 = A.

13.129 Positive definite An n × n matrix A is positive definite if xT Ax > 0, for x = 0 an n element column vector.

13.130 Non-negative definite An n × n matrix A is non-negative definite if xT Ax ≥ 0, for x = 0 an n element column vector.

13.131 Diagonally dominant An n × n matrix A is diagonally dominant if |aii | >

# j=i

|aij | for all i.

13.21 Quadratic Forms A quadratic form involving the n real variables x1 , x2 , . . . , xn that are associated with the real n × n matrix A = [aij ] is the scalar expression n  n  Q(x1 , x2 , . . . , xn ) = aij xi xj . i=1 j=1

In terms of matrix notation, if x is the n × 1 column vector with real elements x1 , x2 , . . . , xn , and xT is the transpose of x, then Q(x) = xT Ax. Employing the inner product notation, this same quadratic form may also be written Q(x) ≡ (x, Ax) . If the n × n matrix A is hermitian, so that AT = A, where the bar denotes the complex conjugate operation, then the quadratic form associated with the hermitian matrix A and the vector x, which may have complex elements, is the real quadratic form

1072

Quadratic Forms

Q(x) = (x, Ax) . It is always possible to express an arbitrary quadratic form n  n  αij xi xj Q(x) = i=1 j=1

in the form Q(x) = (x, Ax) , where A = [aij ] is a symmetric matrix, by defining aii = αii for i = 1, 2, . . . , n and for i, j = 1, 2, . . . , n and i = j. aij = 12 (αij + αji )

13.211 Sylvester’s law of inertia When a quadratic form Q in n variables is reduced by a nonsingular linear transformation to the form 2 2 − yp+2 − . . . − yr2 , Q = y12 + y22 + . . . + yp2 − yp+1 the number p of positive squares appearing in the reduction is an invariant of the quadratic form Q, and it does not depend on the method of reduction itself. ML 377

13.212 Rank The rank of the quadratic form Q in the above canonical form is the total number r of squared terms ML 360 (both positive and negative) appearing in its reduced form.

13.213 Signature The signature of the quadratic form Q above is the number s of positive squared terms appearing in its ML 378 reduced form. It is sometimes also defined to be 2s − r.

13.214 Positive definite and semidefinite quadratic form The quadratic form Q(x) = (x, Ax) is said to be positive definite when Q(x) > 0 for x = 0. It is said ML 394 to be positive semidefinite if Q(x) ≥ 0 for x = 0.

13.215 Basic theorems on quadratic forms 1.

Two real quadratic forms are equivalent under the group of linear transformations if, and only if, they have the same rank and the same signature.

2.

A real quadratic form in n variables is positive definite if, and only if, its canonical form is Q = z12 + z22 . + . . . + zn2 .

3.

A real symmetric matrix A is positive definite if, and only if, there exists a real nonsingular matrix M such that A = MMT .

4.

Any real quadratic form in n variables may be reduced to the diagonal form

Basic theorems on quadratic forms

1073

Q = λ1 z12 + λ2 z22 + . . . + λn zn2 , λ1 ≥ λ2 ≥ . . . ≥ λn by a suitable orthogonal point-transformation. 5.

The quadratic form Q = (x, Ax) is positive definite if, and only if, every eigenvalue of A is positive; it is positive semidefinite if, and only if, all the eigenvalues of A are nonnegative, and it is indefinite if the eigenvalues of A are of both signs.

6.

The necessary conditions for an hermitian matrix A to be positive definite are aii > 0 for all i, aii aij > |aij |2 for i = j, the element of largest modulus must lie on the leading diagonal, |A| > 0.

(i) (ii) (iii) (iv) 7.

The quadratic form Q = (x, Ax) with A hermitian will be positive definite if all the principal minors in the top left-hand corner of A are positive, so that ! ! !a11 a12 a13 ! ! ! ! ! !a11 a12 ! ! > 0, !a21 a22 a23 ! > 0, . . . . a11 > 0, !! ML 353-379 ! ! ! a21 a22 !a31 a32 a33 !

13.31 Differentiation of Matrices If the n × m matrices A(t) and B(t) have elements that are differentiable functions of t, so that B(t) = [bij (t)] A(t) = [aij (t)] , then   d d A(t) = aij (t) 1. dt dt   d d d [A(t) ± B(t)] = aij (t) ± bij (t) 2. dt dt dt d d = A(t) ± B(t). dt dt 3. If the matrix product A(t)B(t) is defined, then



d d d [A(t)B(t)] = A(t) B(t) + A(t) B(t) . dt dt dt 4.

If the matrix product A(t)B(t) is defined, then

T

T d d d T [A(t)B(t)] = B(t) A(t) AT (t) + BT (t) . dt dt dt

5.

If the square matrix A is nonsingular, so that |A| = 0, then

d  −1  d −1 A A(t) A−1 (t) = −A (t) dt dt % &  T

T

A(τ ) dτ =

6. t0

aij (τ ) dτ t0

1074

The Matrix Exponential

13.41 The Matrix Exponential If A is a square matrix, and z is any complex number, then the matrix exponential eAz is defined to be ∞  1 r r An z n + ... = A z . eAz = I + Az + . . . + n! r! r=0

3.411 Basic properties 1.

e−Az 2.

eIz = Iez , eA(z1 +z2 ) = eAz1 · eAz2 , −1  = eAz , eAz · eBz = e(A+B)z

e0 = I,

[when A + B is defined and AB = BA]

dr  Az  e = Ar eAz = eAz Ar . dz r ML 340

3.

  B 0 If the square matrix A can be expressed in the form A = , with B and C square matrices, 0 C then  Bz  e 0 Az e = . 0 eCz

14 Determinants 14.11 Expansion of Second- and Third-Order Determinants 1.

2.

! !a11 ! !a21 ! !a11 ! !a21 ! !a31

! a12 !! = a11 a22 − a12 a21 . a22 ! ! a12 a13 !! a22 a23 !! = a11 a22 a33 − a11 a23 a32 + a12 a23 a31 − a12 a21 a33 + a13 a21 a32 − a13 a22 a31 . a32 a33 !

14.12 Basic Properties Let A = [aij ] and B = [bij ] be n × n matrices. Then the following results are true: 1.

If any two adjacent rows (or columns) of a square matrix are interchanged, then the sign of the associated determinant is changed.

2.

If any two rows (or columns) of a determinant are identical, the determinant is zero.

3.

A determinant is not changed in value if any multiple of a row (or column) is added to any other row (or column).

4.

|kA| = kn |A| ! T! !A ! = |A|

5.

for any scalar k. where AT is the transpose of A.

7.

|AB| = |A||B|. ! −1 ! !A ! = 1

8.

If the elements aij of A are functions of x, then

6.

|A|

when the inverse exists. n  d|A| daij = Aij dx dx i,j=1

(see 14.13).

14.13 Minors and Cofactors of a Determinant The minor Mij of the element aij in the nth -order determinant |A| associated with the square n × n matrix A is the (n − 1)th -order determinant derived from A by deletion of the ith row and j th column. The cofactor Aij of the element aij is defined to be Aij = (−1)i+j Mij . 1075

ML 20

1076

Principal Minors

14.14 Principal Minors A principal minor is one whose elements are situated symmetrically with respect to the leading diagonal ML 197 of A.

14.15* Laplace Expansion of a Determinant The nth -order determinant denoted by |A|, or det A, associated with the n × n matrix A = [aij ] may be expanded either by elements of the ith row as n  aij Aij , |A| = j=1

or by elements of the j th column as |A| =

n 

aij Aij ,

i=1

where Aij is the cofactor of element aij . The cofactors Aij satisfy the following n linear equations: n n   aij Akj = δik |A| , aij Aik = δjk |A| , j=1

i=1

 1 for i = j for i, j, k = 1, 2, . . . , n and δij = 0 for i = j.

ML 21

14.16 Jacobi’s Theorem Let Mr be an r-rowed minor of the nth -order determinant |A|, associated with the n × n matrix A = [aij ], in which the rows i1 , i2 , . . . , ir are represented together with the columns k1 , k2 , . . . , kr . Define the complementary minor to Mr to be the (n−k)-rowed minor obtained from |A| by deleting all the rows and columns associated with Mr , and the signed complementary minor M (r) to Mr to be M (r) = (−1)i1 +i2 +···+ir +k1 +k2 +···+kr × (complementary minor to Mr ). Then, if Δ is the matrix of cofactors given by ! ! ! A11 A12 · · · A1n ! ! ! ! A21 A22 · · · A2n ! ! ! Δ=! . .. .. ! , .. ! .. . . . !! ! !An1 An2 · · · Ann ! and Mr and Mr are corresponding r-rowed minors of |A| and Δ, it follows that Corollary. If |A| = 0, then

Mr = |A|r−1 M (r) . Apk Anq = Ank Apq .

ML 25

Cramer’s Rule

1077

14.17 Hadamard’s Theorem If |A| is an n × n determinant with elements aij that may be complex, then |A| = 0 if n  |aii | > |aij | . j=1,j=i

14.18 Hadamard’s Inequality Let A = [aij ] be an arbitrary n × n nonsingular matrix with real elements and determinant |A|. Then  n  n  2 2 |A| ≤ aik . i=1

k=1

This result is also true when A is hermitian. Deductions. 1.

ML 418

If M = max |aij |, then |A| ≤ M n nn/2 .

2.

ML 419

If the n × n matrix A = [aij ] is positive definite, then |A| ≤ a11 a22 . . . ann .

3.

If the real n × n matrix A is diagonally dominant, so that then |A| = 0.

BL 126

#n j=1

|aij | < |aii | for i = 1, 2, . . . , n,

14.21 Cramer’s Rule If the n linear equations a11 x1 a21 x1 .. .

+ +

a12 x2 a22 x2 .. .

+ ··· + ··· .. .

+ +

a1n xn a2n xn .. .

= =

b1 , b2 , .. .

an1 x1 + an2 x2 + · · · + ann xn = bn , have a nonsingular coefficient matrix A = [aij ], so that |A| = 0, then there is a unique solution A1j b1 + A2j b2 + · · · + Anj bj xj = |A| for j = 1, 2, . . . , n, where Aij is the cofactor of element aij in the coefficient matrix A. ML 134

1078

Some Special Determinants

14.31 Some Special Determinants 14.311 Vandermonde’s determinant (alternant) Third order.

! ! !1 1 1 !! ! !x1 x2 x3 ! = (x3 − x2 ) (x3 − x1 ) (x2 − x1 ) , ! ! 2 !x1 x22 x23 ! and, in general, the nth -order Vandermonde’s determinant is ! ! ! 1 1 ··· 1 !! ! ! x1 x2 ··· xn !! ! 2 2 ! x1 x · · · x2n !! = 2 (xj − xi ) , ! ! .. .. .. ! 1≤i<j≤n .. ! . ! . . . ! ! n−1 !x ! xn−1 · · · xn−1 n 1 2 where the right-hand side is the continued product of all the differences that can be formed from the 1 2 n(n − 1) pairs of numbers taken from x1 , x2 , . . . , xn , with the order of the differences taken in the ML 17 reverse order of the suffixes that are involved.

14.312 Circulants Second order.

! !x1 ! !x2

! x2 !! = (x1 + x2 ) (x1 − x2 ) . x1 !

Third order.

! ! !x1 x2 x3 ! ! !    !x3 x1 x2 ! = (x1 + x2 + x3 ) x1 + ωx2 + ω 2 x3 x1 + ω 2 x2 + ωx3 , ! ! !x2 x3 x1 ! where ω and ω 2 are the complex cube roots of 1. In general, the nth -order circulant determinant is ! ! ! x1 x2 x3 · · · xn !! ! ! xn x1 x2 · · · xn−1 !! n !  !xn−1 xn x1 · · · xn−2 ! -  x1 + x2 ωj + x3 ωj2 + · · · + xn ωjn−1 , ! != ! .. .. .. .. ! j=1 .. ! . . . . . !! ! ! x2 x3 x4 · · · x1 ! where ωj is an nth root of 1. The eigenvalues λ (see 15.61) of an n × n circulant matrix are λj = x1 + x2 ωj + x3 ωj2 + · · · + xn ωjn−1 , th where ωj is again an n root of 1.

ML 36

14.313 Jacobian determinant If f1 , f2 , . . . , fn are n real-valued functions which are differentiable with respect to x1 , x2 , . . . , xn , then the Jacobian Jf (x) of the fi with respect to the xj is the determinant ! ! ∂f ! 1 ∂f1 · · · ∂f1 ! ! ∂x1 ∂x2 ∂xn ! ! ∂f2 ∂f2 ∂f2 ! · · · ∂x ! ! ∂x ∂x2 n! . Jf (x) = !! . 1 . . . .. .. .. !! ! .. ! ∂fn ∂fn ∂fn !! ! ∂x · · · ∂x ∂x 1

2

n

Properties

1079

The notation ∂(f1 , f2 , . . . , fn ) ∂(x1 , x2 , . . . , xn ) is also used to denote the Jacobian Jf (x).

14.314 Hessian determinants ∂φ ∂φ ∂φ , ,..., of a function φ(x1 , x2 , . . . , xn ) with respect to x1 , x2 , ∂x1 ∂x2 ∂xn . . . , xn is called the Hessian H of φ, so that ! 2 ! 2 ∂2φ ∂2φ φ ! ! ∂ φ · · · ∂x∂1 ∂x ! ∂x21 ∂x1 ∂x2 ∂x1 ∂x3 n! ! ∂2φ ! ∂2φ ∂2φ ∂ 2 d2φ ! ! · · · ∂x ∂x2 ∂x3 ! ∂x2 ∂x1 ∂x22 2 ∂xn ! H=! . .. .. .. !! . .. ! . . . . . . ! ! 2 2 2 ! ∂2φ ∂ φ ∂ φ ∂ φ !! ! ∂x ∂x ··· ∂x ∂x ∂x ∂x ∂x2

The Jacobian of the derivatives

n

1

n

2

n

3

n

14.315 Wronskian determinants Let f1 , f2 , . . . , fn be n functions each n times differentiable with respect to x in some open interval (a, b). Then the Wronskian W (x) of f1 , f2 , . . . , fn is defined by ! ! ! f1 f2 ··· fn !! ! (1) (1) (1) ! f f2 ··· fn !! ! 1 ! (2) (2) (2) ! f2 ··· fn ! , W (x) = !! f1 .. .. !! .. ! .. . ! . . . ! ! ! (n−1) (n−1) (n−1) ! !f f · · · f n 1 2 dr fi (r) where fi = . dxr

14.316 Properties 1.

2.

dW follows from W (x) by replacing the last row of the determinant defining W (x) by the nth dx (n) (n) (n) derivatives f1 , f2 , . . . , fn . If constants k1 , k2 , . . . , kn exist, not all zero, such that k1 f1 + k2 f2 + · · · + kn fn = 0 for all x in (a, b), then W (x) = 0 for all x in (a, b).

3.

The vanishing of the Wronskian throughout (a, b) is necessary, but not sufficient, for the linear dependence of f1 , f2 , . . . , fn .

1080

Some Special Determinants

14.318

14.317 Gram-Kowalewski theorem on linear dependence A necessary and sufficient condition for n functions f1 , f2 , . . . , fn square integrable over a ≤ n ≤ b to be linearly dependent in this interval is the vanishing of the Gram determinant ! ! "b "b "b ! 2 f (x)f2 (x) dx · · · f1 (x)fn (x) dx!! ! " a f1 (x) dx a "1 a "b ! ! b b 2 ! a f2 (x)f1 (x) dx f (x) dx ··· f (x)fn (x) dx! a 2 a 2 !. ! G(f1 , f2 , . . . , fn ) = ! SA 2 (Theorem 3) .. .. .. .. ! . ! ! . . . ! !" b "b "b 2 ! fn (x)f1 (x) dx f (x)f2 (x) dx · · · f (x) dx ! a a n a n 14.318 If the n functions f1 , f2 , . . . , fn are square integrable over a ≤ n ≤ b, then the Gram determinant G (f1 , f2 , . . . , fn ) ≥ 0, and the equality sign holds only when the functions are linearly dependent in a ≤ n ≤ b. SA 4 (Corollary 1)

14.319 The rank of the matrix corresponding to the Gram determinant G(f1 , f2 , . . . , fn ) gives the maximum number of linearly independent functions f1 , f2 , . . . , fn in a ≤ x ≤ b. If the rank is r, then r of the functions are linearly independent, and the other n − r functions are linearly dependent on these. SA 3 (Theorem 4)

15 Norms 15.1–15.9 Vector Norms 15.11 General Properties The vector norm ||x|| of an n × 1 column vector x is a nonnegative number having the property that 1.

||x|| > 0 when x = 0 and ||x|| = 0 if, and only if, x = 0;

2.

||kx|| = |k|||x|| for any scalar k;

3.

||x + y|| ≤ ||x|| + ||y||.

15.21 Principal Vector Norms 15.211 The norm ||x||1 If x is a vector with complex components x1 , x2 , . . . , xn , then n  ||x||1 = |xr |.

VA 15

r=1

15.212 The norm ||x||2 (Euclidean or L2 norm) If x is a vector with complex components x1 , x2 , . . . , xn , then  n 1/2  2 ||x||2 = |xr | .

VA 8

r=1

15.213 The norm ||x||∞ If x is a vector with complex components x1 , x2 , . . . , xn , then ||x||∞ = max|xi |. i

1081

VA 15

1082

Matrix Norms

15.31 Matrix Norms 15.311 General properties The matrix norm ||A|| of a square matrix A is a nonnegative number associated with A having the properties that 1.

||A|| > 0 when A = 0 and ||A|| = 0 if, and only if, A = 0;

2.

||kA|| = |k| ||A|| for any scalar k;

3.

||A + B|| ≤ ||A|| + ||B||;

4.

||AB|| ≤ ||A|| ||B||.

VA 9

The matrix norm ||A|| associated with A = [aij ], and the vector norm ||x|| associated with the column vector x for which the matrix product Ax is defined, are said to be compatible if ||Ax|| ≤ ||A|| ||x||.

15.312 Induced norms When a vector z with norm ||z|| exists such that the maximum is attained in the expression ||A|| = max ||Az||, ||z||=1

then ||A|| is a matrix norm and is said to be the natural norm induced by, or subordinate to, the NO 428 vector norm ||z||.

15.313 Natural norm of unit matrix If I is the unit matrix, then for any natural norm ||I|| = 1.

NO 429

15.41 Principal Natural Norms The natural matrix norms induced on matrix A = [aij ] by the 1, 2, and ∞ vector norms are as follows:

15.411 Maximum absolute column sum norm ||A||1 = max j

n 

|aij |

NO 429

i=1

15.412 Spectral norm If AH denotes the Hermitian transpose of the square matrix A = [aij ], so that AH = [aji ] with a bar denoting the complex conjugate operation, then  ||A||2 = maximum eigenvalue of AH A, or, equivalently, ||Ax||2 . NO 429 ||A||2 = max ||x||2 =0 ||x||2

Deductions from Gerschgorin’s theorem (see 15.814)

1083

15.413 Maximum absolute row sum norm ||A||∞ = max i

n 

|aij |

NO 429

j=1

15.51 Spectral Radius of a Square Matrix Let A = [aij ] be an n×n matrix with elements that may be complex, and with eigenvalues λ1 , λ2 , . . . , λn . Then the spectral radius ρ(A) of A is the number ρ(A) = max |λi |. VA 9 1≤i≤n

15.511 Inequalities concerning matrix norms and the spectral radius 1.

||A||2 2 ≤ ||A||1 ||A||∞ .

2.

If A is any arbitrary n × n matrix with elements that may be complex, and the n × n matrix U is unitary, so that UH = U−1 , with H denoting the Hermitian transpose of A (see 13.123), then

NO 431

||AU|| = ||UA|| = ||A||. 3.

If A is any nonsingular n × n matrix with elements that may be complex with eigenvalues λ1 , λ2 , λn , then 1 ≤ |λ| ≤ ||A||. ||A−1 ||

4.

VA 16

For any square matrix A with spectral radius ρ(A) and any natural norm ||A||, ρ (A) ≤ ||A||.

5.

VA 15

NO 430

If the square matrix A is Hermitian, then ρ (A) = ||A||.

6.

If the square matrix A is Hermitian and Pm (x) is any polynomial of degree m with real coefficients, then ||Pm (A)|| = ρ (Pm (A)) .

7.

If A is any arbitrary n × n matrix with elements that may be complex, then the sequence of matrices A, A2 , A3 , . . . converges to the null matrix as n → ∞ if, and only if, ρ(A) < 1. NO 303

15.512 Deductions from Gerschgorin’s theorem (see 15.814) 1.

Let ⎛ A be any arbitrary n × n matrix with elements that may be complex; then ⎞ n n   |aij |, max |aij |⎠ . min ⎝ max 1≤i≤n

j=1

1≤j≤n

i=1

ρ(A) ≤ VA 17

1084

2.

Inequalities Involving Eigenvalues of Matrices

Let A be any arbitrary n×n matrix with elements , x2 , . . . , xn be any  thatmay  and x1  #nbe complex, n  |aij | j=1 |aij |xj set of n positive numbers; then ρ(A) ≤ min max , max xj . 1≤i≤n 1≤j≤n xi xi i=1 VA 18

15.61 Inequalities Involving Eigenvalues of Matrices The eigenvalues (characteristic values or latent roots) λ of an n × n matrix A = [aij ] are the solutions to the characteristic equation |A − λI| = 0. When expanded, the determinant |A − λI| is called the characteristic polynomial, and it has the form |A − λI| = (−1)n λn + cn−1 λn−1 + cn−2 λn−2 + · · · + c1 λ + c0 . The zeros of this polynomial satisfy the characteristic equation and so are the eigenvalues of A. In the characteristic polynomial the coefficients have the form cn−r = (−1)n−r It then follows that

(sum of all principal minors of |A| of order r) .

bn−1 = (−1)n (a11 + a22 + · · · + ann ) ,  bn−2 = (−1)n (aii ajj − aij aji ) , i<j

b0 = |A|. Since the sum of the elements of the leading diagonal of A is called the trace of A, written tr A, it follows that bn−1 = (−1)n tr A. ML 198

15.611 Cayley-Hamilton theorem Every square matrix A satisfies its characteristic equation, so that (−1)n An + cn−1 An−1 + cn−2 An−2 + · · · + c1 A + c0 I = 0.

ML 206

15.612 Corollaries 1.

If A is nonsingular, then its adjoint, denoted by adj A, is   adj A = − (−1)n An−1 + cn−1 An−2 + cn−2 An−3 + · · · + c2 A + c1 I .

2.

If A is nonsingular, then the characteristic polynomial of A−1 is

c1 n−1 c2 n−2 (−1)n λ λ + + ··· + (−1)n λn + . |A| |A| |A|

15.71 Inequalities for the Characteristic Polynomial The first group of inequalities that follow, which relate to the characteristic polynomial of an n × n matrix A whose elements may be complex, refer directly to the coefficients of the polynomial when written in the form

Named and unnamed inequalities

1085

P (λ) ≡ |λI − A| = λn + b1 λn−1 + b2 λn−2 + · · · + bn−1 λ + bn , and only implicitly to the coefficients aij of A that give rise to the bi .

15.711 Named and unnamed inequalities The first group of inequalities relating to the eigenvalues λ satisfying P (λ) = 0 are unnamed and are as follows: 1.

All the eigenvalues λ lie within or on the circle ||z|| ≤ r, where r is the positive root of . |bn | + |bn−1 |z + |bn−2 |z 2 + · · · + |b1 |z n−1 − z n = 0

2.

All the eigenvalues λ lie within the circle |z| < 1 + max|bi |.

MG 123

i

3.

When bn = 0 the eigenvalue λ of smallest modulus lies in the annulus R ≤ |z| ≤ R is the positive root of |bn | − |bn−1 |z − |bn−2 |z 2 − · · · − z n = 0.

4.

5.

MG 122

All the eigenvalues λ lie on or outside the circle   |bn | |z| = min . k (|bn | + |bk |)

R , where 21/n − 1 MG 126

MG 126

If the eigenvalues λ are ordered so that |λ1 | ≥ |λ2 | ≥ · · · ≥ |λp | > 1 ≥ |λp+1 | ≥ · · · ≥ |λn |, then 1

|z1 z2 . . . zp | ≤ N,

|zp | ≤ N p ,

where 2

2

2

N 2 = 1 + |b1 | + |b2 | + · · · + |bn | . 6.

MG 129

All the eigenvalues λ lie in or on the circle |z| ≤

n 

|bj |

1/j

.

MG 126

j=1

7.

8.

All the eigenvalues λ lie on the disk ! ! ! ! ! ! ! ! !z + b1 ! ≤ ! b1 ! + |b2 |1/2 + |b3 |1/3 + · · · + |bn |1/n . ! 2! !2! All the eigenvalues λ lie in the annulus m ≤ ||z|| ≤ M , where

+ , 2 2 m = max 0, min 1 − |bj |, |bn | 1≤j≤n−1

and

MG 145

1086

Inequalities for the Characteristic Polynomial

⎧ ⎫ n−1 ⎨ ⎬  2 2 M 2 = max 1 + |bj |, |bn | + 2 |bj | . ⎩ ⎭ j=1

The next group of inequalities are named theorems that apply to the explicit form of the characteristic MG 145 polynomial P (λ).

15.712 Parodi’s theorem The eigenvalues λ satisfying P (λ) = 0 lie in the union of the disks n  |z| ≤ 1, |z + b1 | ≤ |bj |.

MG 143

j=1

15.713 Corollary of Brauer’s theorem If |b1 | > 1 +

n 

|bj |,

j=2

then one and only one eigenvalue satisfying P (λ) = 0 lies on the disk n  |bj |. |z + b1 | ≤

MG 141

j=2

15.714 Ballieu’s theorem For any set μ = (μ1 , μ2 , . . . , μn ) of positive numbers, let μ0 = 0 and   μk + μn |bn−k | Mμ = max . 0≤k≤n−1 μk+1 Then all the eigenvalues satisfying P (λ) = 0 lie on the disk ||z|| ≤ Mμ .

MG 144

15.715 Routh-Hurwitz theorem Consider the characteristic equation |λI − A| = λn + b1 λn−1 + · · · + bn−1 λ + bn = 0 determining the n eigenvalues λ of the real n × n matrix A. Then the eigenvalues λ all have negative real parts if Δ2 > 0, ... , Δn > 0, Δ1 > 0, where ! ! ! b1 1 0 0 0 0 . . . 0 !! ! ! b3 b2 b1 1 0 0 . . . 0 !! ! ! b5 b4 b3 b2 b1 0 . . . 0 !!. Δk = ! GM 230 ! .. ! .. .. .. .. ! . ! . . . . ! ! !b2k−1 b2k−2 b2k−3 b2k−4 b2k−5 b2k−6 . . . bk !

Poincare’s separation theorem

1087

15.81–15.82 Named Theorems on Eigenvalues In the following theorems involving eigenvalue inequalities the elements aij of matrix A enter directly, and not in the form of the coefficients of the characteristic polynomial.

15.811 Schur’s inequalities If A = [aij ] is an n × n matrix with elements that may be complex, and eigenvalues λ1 , λ2 , . . . , λn , then 1.

n 

2

|λi | ≤

i=1

2.

n 

n 

n 

2

i,j=1 2

|Re λi | ≤

i=1

3.

|aij |

2

|Im λi | ≤

i=1

! n !  ! aij + aji !2 ! ! ! ! 2 i,j=1

! n !  ! aij − aji !2 ! ! ! ! 2 i,j=1

ML 309

15.812 Sturmian separation theorem Let Ar = [aij ] with i, j = 1, 2, . . . , r and r = 1, 2, . . . , N be a sequence of N symmetric matrices of increasing order. Then if λk (Ar ) for k = 1, 2, . . . , r denotes the k th eigenvalue of Ar , where the ordering is such that λ1 (Ar ) ≥ λ2 (Ar ) ≥ · · · ≥ λr (Ar ) , it follows that BL 115 λk+1 (Ai+1 ) ≤ λk (Ai ) ≤ λk (Ai+1 ) .

15.813 Poincare’s separation theorem

  Let yk , with k = 1, 2, . . . , K, be a set of orthonormal vectors so that the inner product (yk , yk ) = 1. Set K  x= uk y k , k=1

so that for any square matrix A for which the product Ax is defined, the quadratic form K    (x, Ax) = uk ul yk , Ayl . k,l=1

Then if

  bK = yk , Ayl for k, l = 1, 2, . . . , K,

it follows that λi (bK ) ≤ λi (A) λK−j (bK ) ≥ λN −j (A)

for i = 1, 2, . . . , K, for j = 0, 1, 2, . . . , K − 1.

BL 117

1088

Named Theorems on Eigenvalues

15.814 Gerschgorin’s theorem Let A = [aij ] be any arbitrary n × n matrix with elements that may be complex, and let n  |aij | for i = 1, 2, . . . , n. Λi ≡ j=1,i=j

Then all of the eigenvalues λi of A lie in the union of the n disks Γi , where Γi : |z − aii | ≤ Λi for i = 1, 2, . . . , n.

VA 16

15.815 Brauer’s theorem If in Gerschgorin’s theorem for a given m |ajj − amm | ≥ Λj + Λm for all j = m, then one and only one eigenvalue of A lies in the disk Γm .

MG 141

15.816 Perron’s theorem If μ = (μ1 , μ2 , . . . , μn ) is an arbitrary set of positive numbers, then all the eigenvalues λ of the n × n matrix A = [aij ] lie on the disk |z| ≤ Mμ , where n  μj Mμ = max |aij |. MG 141 1≤i≤n μ j=1 i

15.817 Frobenius theorem If A = [aij ] is a matrix with positive coefficients, so that aij > 0 for all i, j = 1, 2, . . . , n, then A has a positive eigenvalue λ0 , and all its eigenvalues lie on the disk MG 142 |z| ≤ λ0 .

15.818 Perron–Frobenius theorem If all elements aij of an irreducible matrix A are nonnegative, then R = min Mλ is a simple eigenvalue of A, and all the eigenvalues of A lie on the disk |z| ≤ R, where, if λ = (λ1 , λ2 , . . . , λn ) is a set of nonnegative numbers, not all zero, ⎧ ⎫ n ⎨ ⎬  Mλ = inf μ : μλi > |aij |λj , 1 ≤ i ≤ n ⎩ ⎭ j=1

and R = min Mλ . Furthermore, if A has exactly p eigenvalues (p ≤ n) on the circle |z| = R, then the set of all its GM 69 eigenvalues is invariant under rotations 2π/p about the origin.

15.819 Wielandt’s theorem If the n × n matrix A satisfies the conditions of the Perron–Frobenius theorem and if in the n × n matrix C = [cij ] |cij | ≤ aij , i, j = 1, 2, . . . , n, then any eigenvalue λ0 of C satisfies the inequality |λ0 | ≤ R. The equality sign holds only when there exists an n × n matrix D = [±δij ] such that δii = 1 for all i, δij = 0 for all i = j, and

Hermitian matrices and diophantine relations

C = (λ0 /R) DAD−1 .

1089

GM 69

15.820 Ostrowski’s theorem If A = [aij ] is a matrix with positive coefficients and λ0 is the positive eigenvalue in Frobenius’ theorem, then the n − 1 eigenvalues λj = λ0 satisfy the inequality M 2 − m2 |λj | ≤ λ0 2 , M + m2 where M = max aij , m = min aij for i, j = 1, 2, . . . , n. MG 145

15.821 First theorem due to Lyapunov In order that all the eigenvalues of the real n × n matrix A have negative real parts, it is necessary and sufficient that if V is an n × n matrix, the equation AT V + VA = −I has as a solution the matrix of coefficients V of some positive-definite quadratic form (x, Vx) (see 13.21). GM 224

15.822 Second theorem due to Lyapunov If all the eigenvalues of the real matrix A have negative real parts, then to an arbitrary negative-definite quadratic form (x, Wx) with x = x(t) there corresponds a positive-definite quadratic form (x, Vx) such that if one takes dx = Ax dt then (x, Vx) and (x, Wx) satisfy d (x, Vx) = (x, Wx) . dt Conversely, if for some negative-definite form (x, Wx) there exists a positive-definite form (x, Vx) connected to (x, Wx) by the preceding two equations, then all the eigenvalues of A have negative real parts (see 13.21, 13.31). GM 222

15.823 Hermitian matrices and diophantine relations involving circular functions of rational angles due to Calogero and Perelomov 1.

The off-diagonal Hermitian matrix A of rank n whose elements are given by

  (j − k) π ajk = (1 − δjk ) 1 + i cot , n has the integer eigenvalues λ(a) s = 2s − n − 1 for s = 1, 2, . . . , n, and the corresponding eigenvectors v (s) have the components

1090

Named Theorems on Eigenvalues

(s) vj

2.



2πisj = exp − n

for j = 1, 2, . . . , n.

The two off-diagonal Hermitian matrices B and C whose elements are defined by the formulas   (j − k) π , bjk = (1 − δjk ) sin−2 n   (j − k) π , cjk = (1 − δjk ) sin−4 n are related to the matrix A in (1) by the equations 1 2 A + 2A − σn(1) I , 2  1 2 B − 2 2 + σn(1) B − σn(2) I , C=− 6 B=

where I is the unit matrix and  1 2 n −1 , σn(1) = 3

σn(2) =

  1  2 n − 1 n2 + 11 . 45 (s)

The eigenvalues of B and C corresponding to the eigenvector vj (1) λ(b) s = σn − 2s(n − s) (2) λ(c) s = σn − 2s(n − s)

3.

in (1) have the form

for s = 1, 2, . . . , n, s(n − s) + 2 3

for s = 1, 2, . . . , n.

Together, the above two results imply the following diophantine summation rules: (a)

(b)

(c)

(d)





kπ 2skπ sin = n − 2s n n k=1



n−1  kπ 2skπ sin−2 cos = bs n n k=1



n−1  kπ 2skπ −4 sin cos = cs n n k=1

n−1  kπ −2p sin = σn(p) , n n−1 

for s = 1, 2, . . . , n − 1

cot

for s = 1, 2, . . . , n − 1, for s = 1, 2, . . . , n − 1,

k=1

(1)

(2)

with σn and σn as defined in (2), and σn(3) = σn(1)

2n4 + 23n2 + 191 , 315

bs = σn(1) − 2s(n − s),

σn(4) = σn(2)

3n4 + 10n2 + 227 315

2 cs = σn(2) − s(n − s)[s(n − s) + 2]. 3

Basic theorems

1091

15.91 Variational Principles 15.911 Rayleigh quotient If A is an Hermitian matrix, the Rayleigh quotient ρ(x) is the expression (x, Ax) . ρ (x) = (x, x)

NO 407

15.912 Basic theorems 1.

If the n × n matrix A is Hermitian and has eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn , then λ1 ≤ ρ ≤ λn , where ρ is the Rayleigh quotient for any x = 0, and λ1 = min x=0

2.

(x, Ax) (x, x)

and

λn = max x=0

(x, Ax) . (x, x)

NO 407

If the n × n matrix A is Hermitian and has eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn corresponding to the eigenvectors x1 , x2 , . . . , xn , respectively, and x = 0 is such that (x, x1 ) = (x, x2 ) = · · · = (x, xn ) = 0, then λj = min x

(x, Ax) , (x, x)

and λj ≤ 3.

(x, Ax) ≤ λn . (x, x)

NO 410

If the n × n matrix A is Hermitian, then the eigenvalue

(x, Ax) λr = max min , (x, x) where first the minimum over x is taken subject to (bi , x) = 0, i = 1, 2, . . . , r − 1, with the bi regarded as fixed vectors, and then the maximum over all possible bi . Also, the eigenvalue

(x, Ax) λr = min max , (x, x) where now the maximum over x is taken first subject to (bi , x) = 0, i = r + 1, r + 2, . . . , n for NO 414 fixed bi , and then the minimum over all possible bi .

4.

The (n − 1) eigenvalues λ1 , λ2 , . . . , λn−1 obtained from the (n − 1) × (n − 1) matrix derived from an Hermitian matrix A from which the last row and column have been omitted separate the n eigenvalues of A, so that λ1 < λ1 < λ2 < λ2 < · · · < λn−1 < λn

(see 15.812).

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16 Ordinary Differential Equations 16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations 16.11 First-Order Equations 16.111 Solution of a first-order equation Consider the real function f (t, x) that is defined and continuous in an open set D ⊂ R2 . Then a solution to the first-order differential equation dx = f (t, x) dt in the open interval I ⊂ R is a real function u(t) that is defined and is both continuous and differentiable in I, with the property that (i)

(t, u(t)) ∈ D for t ∈ I,

(ii)

du = f (t, u(t)) for t ∈ I. dt

16.112 Cauchy problem The Cauchy problem for the differential equation dx = f (t, x) dt is the problem of existence and uniqueness of the solution to this equation satisfying the initial condition u(t0 ) = x0 , where (t0 , u(t0 )) ∈ D, the open set defined above. The solution to the initial value problem may be expressed in the form of the integral equation  t f (τ, u(τ )) dτ (see 16.316). u(t) = x0 + t0

16.113 Approximate solution to an equation The real function φ(t) is said to be an approximate solution, to within the error , of the differential equation

1093

1094

Fundamental Inequalities and Related Results

dx = f (t, x) dt if φ is piecewise continuous, and for a given  > 0 and an open interval I ⊂ R, |φ (t) − f (t, φ(t))| ≤ , except at points of discontinuity of the derivative.

HU 3

16.114 Lipschitz continuity of a function The real function f (t, x) defined and continuous in some open set D ⊂ R2 is said to be Lipschitz continuous with respect to x for some constant k > 0 if, for all points (t, x1 ) and (t, x2 ) belonging to D |f (t, x1 ) − f (t, x2 )| ≤ k|x1 − x2 |. HU 5

16.21 Fundamental Inequalities and Related Results 16.211 Gronwall’s lemma Let the three piecewise continuous, non-negative functions u, v, and w be defined in the interval [0, a] and satisfy the inequality  t v(τ )w(τ ) dτ, w(t) ≤ u(t) + 0

except at points of discontinuity of the functions. Then, except at these same points,  t

 t u(τ )v(τ ) exp v (σ) dσ dτ. w(t) ≤ u(t) + 0

BB 135

τ

16.212 Comparison of approximate solutions of a differential equation Let f be a real function that is defined in an open set D ⊂ R2 , in which it is both continuous and Lipschitz continuous. In addition, let u1 and u2 be two approximate solutions of dx = f (t, x) dt in an open set I ⊂ R in the sense already defined, with |u1 (t) − f (t, u1 (t))| ≤ 1 , |u2 (t) − f (t, u2 (t))| ≤ 2 , except where the derivatives are discontinuous. Then, if for all t0 ∈ I |u1 (t0 ) − u2 (t0 )| ≤ δ, it follows that

1 + 2 |u1 (t) − u2 (t)| ≤ δ exp {|t − t0 |} + [exp {k|t − t0 |} − 1] . k

16.31 First-Order Systems 16.311 Solution of a system of equations The system of n first-order differential equations

HU 6

Lipschitz continuity of a vector

1095

dx1 = f1 (t, x1 , x2 , . . . , xn ) , dt dx2 = f2 (t, x1 , x2 , . . . , xn ) , dt .. . dxn = fn (t, x1 , x2 , . . . , xn ) , dt in which the functions f1 , f2 , . . . , fn are real and continuous in an open set D ⊂ Rn+1 , may be written in the concise matrix form dx = f (t, x) , dt where x and f are n × 1 column vectors. Its solution in the open interval I ⊂ R is the vector u(t) with elements u1 (t), u2 (t), . . . , un (t) with the property that (i)

(t, u(t)) ∈ D for t ∈ I,

(ii)

du = f (t, u(t)) for t ∈ I. dt

HU 24

16.312 Cauchy problem for a system The Cauchy problem for the system dx = f (t, x) dt is the problem of existence and uniqueness of the solution to this system satisfying the initial vector condition u (t0 ) = x0 , where (t0 , u(t0 ) ∈ D, the open set defined above in connection with the system. The solution to the initial value problem may be expressed in the form of the vector integral equation  t f (τ, u(τ )) dτ. u(t) = x0 + t0

16.313 Approximate solution to a system The real vector φ(t) is said to be an approximate vector solution, to within the order , of the system dx = f (t, x) , dt  if the elements of φ are piecewise continuous, and for a given  > 0 and open interval I ⊂ R, ||φ (t) − f (t, φ(t))|| ≤ , except at points of discontinuity of the derivative, where ||w|| denotes the supremum norm ||w|| = sup (|w1 |, |w2 |, . . . , |wn |) .

HU 25

16.314 Lipschitz continuity of a vector The real vector f (t, x) defined and continuous in some open set D ⊂ Rn is said to be Lipschitz continuous with respect to x for some constant k > 0 if, for all points (t, x1 ), (t, x2 ) belonging to D, ||f (t, x1 ) − f (t, x2 )|| ≤ k||x1 − x2 ||. HU 26

1096

First-Order Systems

16.315 Comparison of approximate solutions of a system Let f be a real vector defined in an open set D ⊂ R × Rn in which it is both continuous and Lipschitz continuous. In addition, let u1 and u2 be two approximate solutions of the system dx = f (t, x) dt in an open set I ⊂ R in the sense already defined, with |u1 (t) − f (t, u1 (t))| ≤ 1 , |u2 (t) − f (t, u2 (t))| ≤ 2 , except where the derivatives are discontinuous. Then, if for all t0 ∈ I ||u1 (t0 ) − u2 (t0 )|| ≤ δ, it follows that

1 + 2 ||u1 (t) − u2 (t)|| ≤ δ exp {k|t − t0 |} + [exp {k|t − t0 |} − 1] . k

HU 27

16.316 First-order linear differential equation The first-order linear differential equation when expressed in the canonical form dy + P (t)y = Q(t) dt has an integrating factor 

μ(t) = exp P (t) dt , and a general solution 1 y(t) = μ(t)

 t μ(ξ)Q(ξ) dξ , μ (t0 ) y0 + t0

where y0 = y(t0 ).

16.317 Linear systems of differential equations Consider the homogeneous system of linear differential equations dx = A(t)x, dt where x is an n×1 column vector and A(t) an n×n matrix. Then a fundamental system of solutions of this system is a set of n linearly independent solution vectors φ1 (t), φ2 (t), . . . , φn (t), The square matrix K(t) whose columns comprise the vectors φ1 (t), φ2 (t), . . . , φn (t) is called the fundamental matrix of the differential equation, and we have the representation  t

tr A(τ ) dτ . |K(t)| = |K (t0 )| exp t0

Using the fundamental matrix K(t) defined in terms of the homogeneous system, the unique solution to the inhomogeneous system dx = A(t)x + b(t), dt assuming the initial value x(t0 ) = x0 , is  t φ(t) = K(t)[K (t0 )]−1 x0 + K(t) [K(τ )]−1 b(τ ) dτ, HU 43 where b(t) is an n × 1 column vector.

t0

CL 69

Homogeneous differential equations

1097

16.41 Some Special Types of Elementary Differential Equations 16.411 Variables separable A first-order differential equation is said to be variables separable if it is of the form dy = M (x)N (y), dx or P (x)Q(y) dx + R(x)S(y) dy = 0. It may then be written in the form 1 M (x) dx − dy = 0, N (y) or P (x) S(y) dx + dy = 0, R(x) Q(y) provided R(x)Q(y) = 0.

16.412 Exact differential equations A differential equation M (x, y) dx + N (x, y) dy = 0 is said to be exact if there exists a function h(x, y) such that d [h(x, y)] = M (x, y) dx + N (x, y) dy.

IN 16

16.413 Conditions for an exact equation A necessary and sufficient condition that an equation of this form is exact is that the functions M (x, y) and N (x, y) together with their partial derivatives ∂M/∂y and ∂N/∂x exist and are continuous in a region in which ∂N ∂M = . IN 16 ∂y ∂x

16.414 Homogeneous differential equations A differential equation M (x, y) dx + N (x, y) dy = 0 is said to be algebraically homogeneous if, for arbitrary k, M (kx, ky) M (x, y) = . N (kx, ky) N (x, y) Setting y = sx, it may then be expressed in the form [M (1, s) + sN (1, s)] dx + xN (1) dx = 0, in which the variables s and x are separable.

IN 18

1098

Second-Order Equations

16.51 Second-Order Equations 16.511 Adjoint and self-adjoint equations The linear second-order differential equation du d2 u + c(x)u = 0 L(u) ≡ a(x) 2 + b(x) dx dx has associated with it the adjoint equation d2 d M (v) ≡ 2 [a(x)v] − [b(x)v] + c(x)v = 0. dx dx The equation L(u) = 0 is said to be self-adjoint if L(u) ≡ M (u). A linear self-adjoint second-order differential equation defined on [α, β] can always be expressed in the form

d du p(x) + q(x)u = 0, dx dx where p(x) and q(x) are continuous on [α, β] and p(x) > 0. The general equation L(u) = 0 can always be made self-adjoint and written in this form by multiplication by the factor    1 b(x) dx , exp a(x) a(x) when     b(x) c(x) b(x) p(x) = exp dx and q(x) = dx . exp a(x) a(x) a(x) In general, if dn u dn−1 u du L(u) = p0 n + p1 n−1 . . . + pn−1 + pn u, dx dx dx then its adjoint is dn dn−1 d [pn−1 v] + pn v. M (v) = (−1)n n [p0 v] + (−1)n−1 n−1 [p1 v] + . . . − HI 391 dx dx dx

16.512 Abel’s identity If p(x) and q(x) are continuous in [α, β] in which p(x) > 0, and u(x) and v(x) are suitably differentiable with

d du p(x) + q(x)u = 0, dx dx then the result

du dv −v p(x) u ≡ const. dx dx is known as Abel’s identity. More generally, if we consider the linear nth -order equation dn u dn−1 u du + pn = 0, p0 n + p1 n−1 + . . . + pn−1 dx dx dx and Δ is the Wronskian of a (fundamental) set of linearly independent solutions u1 , u2 , . . . , un , the Abel identity takes the form

 x p1 (x) dx , Δ = Δ0 exp − x0 p0 (x) where Δ0 is the value of Δ at x = x0 . IN 119

Solutions of the Riccati equation

1099

16.513 Lagrange identity If the linear nth -order equation L(u) = 0 is defined by dn u dn−1 u du + pn u, L(u) ≡ p0 n + p1 n−1 + . . . + pn−1 dx dx dx then the expression d vL(u) − uM (v) = {P (u, v)} , dx where M (v) is the adjoint of L(u), is called the Lagrange identity. The expression P (u, v), which is linear and homogeneous in dn−1 u dn−1 v du dv u, , . . . , n−1 and v, , . . . , n−1 , dx dx dx dx is then known as the bilinear concomitant. In the case of the second-order equation d2 u du + c(x)u = 0, L(u) = a(x) 2 + b(x) dx dx with adjoint M (v), the Lagrange identity becomes

d d du vL(u) − uM (v) = − (a(x)v)u + b(x)uv . IN 124 a(x)v dx dx dx

16.514 The Riccati equation The general Riccati equation has the form dz + a(x)z + b(x)z 2 + c(x) = 0, dx and an equation of this form results from the substitution   p(x) du dx z= u in the general self-adjoint equation

d du p(x) + q(x)u = 0.  dx  " xdx The further substitution v = u exp α a(x) dx in the Riccati equation then gives the more convenient form dv + r(x)v 2 + s(x) = 0, dx with  x

 x

r(x) = b(x) exp − a(x) dx and s(x) = c(x) exp a(x) dx . HI 273 α

α

16.515 Solutions of the Riccati equation If in the Riccati equation dv + r(x)v 2 + s(x) = 0, dx r(x) = 0, while r(x) and s(x) are continuous on the interval [α, β], then every solution v(x) may be expressed in the form 1 Au (x) + Bv  (x) , r(x) Au(x) + Bv(x) with A, B arbitrary constants, not both zero, and the prime denoting differentiation, while u and v are linearly independent solutions of

1100

Oscillation and Non-Oscillation Theorems for Second-Order Equations



1 dz d + s(x)z = 0. dx r(x) dx Conversely, if u(x) and v(x) are linearly independent solutions of this last equation and A and B are arbitrary constants, not both zero, the function 1 Au (x) + Bv  (x) r(x) Au(x) + Bv(x) is a solution of the Riccati equation wherever Au(x) = Bv(x) = 0. IN 24

16.516 Solution of a second-order linear differential equation A fundamental system of solutions of a homogeneous second-order linear differential equation in the canonical form d2 x dx + b(t)x = 0 + a(t) 2 dt dt is a system of two linearly independent solutions φ1 (t) and φ2 (t). The Wronskian of these solutions is ! ! !φ1 (t) φ2 (t)! ! ! = φ1 (t)φ2 (t) − φ2 (t)φ1 (t), W (t) = !  φ1 (t) φ2 (t)! and the solution to the inhomogeneous equation d2 x dx + b(t)x = f (t), + a(t) dt2 dt subject to the initial conditions x(t0 ) = x0 and x (t0 ) = x1 may be written  t φ1 (ξ)φ2 (t) − φ2 (ξ)φ1 (t) f (ξ) dξ, x(t) = c1 φ1 (t) + c2 φ2 (t) + W (ξ) t0 where the constants c1 and c2 are chosen such that x(t) satisfies the initial conditions. The linear combination c1 φ1 (t) + c2 φ2 (t) is known as the complementary function where c1 and c2 are arbitrary constants.

16.61–16.62 Oscillation and Non-Oscillation Theorems for SecondOrder Equations Equations whose solutions possess an infinite number of zeros in the interval (0, ∞) are said to have oscillatory solutions. The following theorems relate to such properties:

16.611 First basic comparison theorem If all solutions of the equation

are oscillatory, and if

d2 u + φ(x)u = 0 dx2 ψ(x) ≥ φ(x),

then all the solutions of d2 v + ψ(x)v = 0 dx2 are oscillatory, and conversely. That is, if ψ(x) ≥ φ(x) and some solutions v are non-oscillatory, then so also must some solutions u be non-oscillatory. BS 119

Szeg¨ o’s comparison theorem

1101

16.622 Second basic comparison theorem If all the solutions of the self-adjoint equation

d du p1 (x) + q1 (x)u = 0 dx dx are oscillatory as x → ∞, and if q2 (x) ≥ q1 (x), p2 (x) ≥ p1 (x) > 0, then all the solutions of the self-adjoint equation

d dv p2 (x) + q2 (x)v = 0 dx dx are oscillatory.

BS 120

16.623 Interlacing of zeros Let y1 (x) and y2 (x) be two linearly independent solutions of d2 y + F (x)y = 0, dx2 and suppose that y1 (x) has at least two zeros in the interval (a, b). Then if x1 and x2 are two consecutive HI 374 zeros of y1 (x), the function y2 (x) has one, and only one, zero in the interval (x1 , x2 ).

16.624 Sturm separation theorem Let u(x) and v(x) be two linearly independent solutions of the self-adjoint equation

d dy p(x) + q(x) = 0, dx dx in which p(x) > 0 and p(x), q(x) are continuous on [a, b]. Then, between any two consecutive zeros of IN 224 u(x) there will be one, and only one, zero of v(x).

16.625 Sturm comparison theorem Let p1 (x) ≥ p2 (x) > 0 and q1 (x) ≥ q2 (x) be continuous functions in the differential equations

d du p1 (x) + q1 (x)u = 0, dx dx

d dv p2 (x) + q2 (x)v = 0. dx dx Then between any two zeros of a non-trivial solution u(x) of the first equation there will be at least one zero of every non-trivial solution v(x) of the second equation. IN 228

16.626 Szeg¨ o’s comparison theorem Suppose, under the conditions of the Sturm comparison theorem, that p1 (x) ≡ p2 (x), q1 (x) ≡ q2 (x), and u(x) > 0, v(x) > 0 for a < x < b, together with

du dv v− u = 0. lim p1 (x) x→a dx dx Then, if u(b) = 0, there is a point ξ in (a, b) such that v(ξ) = 0. HI 379

1102

Oscillation and Non-Oscillation Theorems for Second-Order Equations

16.627 Picone’s identity Consider the equations



d du p1 (x) + q1 (x)u = 0, dx dx

d dv p2 (x) + q2 (x)v = 0, dx dx with p1 , p2 , q1 , and q2 positive and continuous for a < x < b, where q2 (x) > q1 (x) and p1 (x) > p2 (x). Then with a < α < β < b, Picone’s identity is

β  β 2

2  β  β u du dv du dv p2 du 2 −u = (q2 − q1 ) u ds+ (p1 − p2 ) ds+ ds. v p1 v − p2 u 2 v dx dx ds ds ds α α α v α IN 226

16.628 Sturm-Picone theorem Consider the self-adjoint equations



du p1 (x) + q1 (x)u = 0 dx and

d dv p2 (x) + q2 (x)v = 0. dx dx Let p1 , p2 , q1 , and q2 be positive and continuous for a < x < b, where q2 (x) > q1 (x) and p1 (x) > p2 (x). Then, if x1 and x2 is a pair of consecutive zeros of u(x) in (a, b), v(x) has at least one zero in the open IN 225 interval (a, b). d dx

16.629 Oscillation on the half line Consider the self-adjoint equation d dx



du p(x) + q(x)u = 0. dx

We then have the following results: (i)

Let p(x) > 0 and p, q be continuous on [0, ∞). If the two improper integrals  ∞  ∞ dx and q(x) dx 1 p(x) 1 diverge, then every solution u(x) has infinitely many zeros on the interval [1, ∞). Also, if the two integrals  1  1 dx = +∞ and q(x) dx = +∞, 0 p(x) 0 then every solution u(x) has infinitely many zeros on the interval (0, 1).

(ii)

(Moore’s theorem). Every non-trivial solution u(x) has at most a finite number of zeros on the interval [a, ∞) if the improper integral  ∞ dx p(x) a converges, and if

Kneser’s non-oscillation theorem

! x ! ! ! ! q(s) ds!! < M !

for

1103

a≤x<∞

a

with M > 0 a finite constant.

16.71 Two Related Comparison Theorems 16.711 Theorem 1 Consider the equations in the Sturm comparison theorem with the same assumptions on p(x) and q(x), and let u(x), v(x) be solutions such that u (x1 ) = v (x1 ) = 0, u (x) = v  (x1 ) > 0. Then if u(x) is increasing in [x1 , x2 ] and reaches a maximum at x2 , the function v(x) reaches a maximum HI 376 at some point x3 such that x1 < x3 < x2 .

16.712 Theorem 2 Consider the equation d2 y + F (x)y = 0, dx2 in which F (x) is continuous in (a, b) and such that 0 < m ≤ F (x) ≤ M. Then, if the solution y(x) has two successive zeros x1 , x2 , it follows that πM −1/2 ≤ x2 − x1 ≤ πm−1/2 .

16.81–16.82 Non-Oscillatory Solutions The real solution y(x) of d2 y + F (x)y = 0 dx2 is said to be non-oscillatory in the wide sense in (0, ∞) if there exists a finite number c such that the solution has no zeros in [c, ∞). HI 376

16.811 Kneser’s non-oscillation theorem Consider the equation d2 y + F (x)y = 0, dx2 and let   lim sup x2 F (x) = γ ∗ ,   lim inf x2 F (x) = γ∗ . Then the solution y(x) is non-oscillatory if γ ∗ < 14 , oscillatory if if either γ ∗ or γ∗ equals 14 .

1 4

< γ∗ and no conclusion can be drawn HI 461

1104

Some Growth Estimates for Solutions of Second-Order Equations

16.822 Comparison theorem for non-oscillation Consider the differential equations d2 y + F (x)y = 0, dx2

 f (x) = x

∞

F (s) ds,

x  ∞

d2 y + G(x)y = 0, g(x) = x G(s) ds, dx2 x where 0 < g(x) < f (x). Then if the first equation is non-oscillatory in the wide sense, so also is the HI 460 second.

16.823 Necessary and sufficient conditions for non-oscillation Consider the equation d2 y + F (x)y = 0. dx2  ∞

lim sup x F (s) ds = F ∗ ,

Then, if

x→∞

 lim inf x

x→∞

it follows that:

x ∞

F (s) ds = F∗ ,

x

(i)

a necessary condition that the solution y(x) be non-oscillatory is that F∗ ≤

(ii)

a sufficient condition that the solution y(x) be non-oscillatory is that F ∗ <

1 4 and 1 4.

F ∗ ≤ 1;

16.91 Some Growth Estimates for Solutions of Second-Order Equations 16.911 Strictly increasing and decreasing solutions Suppose that G(x) > 0 be continuous in (−∞, ∞) and such that xG(x) ∈ L(0, ∞). Then the equation d2 y − G(x)y = 0 has one, and only one, solution y+ (x) passing through the point (0, 1), which is positive dx2 and strictly monotonic decreasing for all x, and one and only one solution y− (x) through the point (0, 1), which is positive and strictly increasing for all x. The solution y+ (x) has the property that dy+ (x) 1/2 ∈ L2 (0, ∞). [G(x)] y+ (x) ∈ L2 (0, ∞) and dx 2 2 If, in addition, 0 < α ≤ G(x) ≤ β < ∞, then e−βx ≤ y+ (x) ≤ e−αx

for

x > 0.

HI 359

16.912 General result on dominant and subdominant solutions Consider the equations d2 y d2 Y − g(x)y = 0, − G(x)Y = 0, 2 dx dx2 where g and G are continuous on (0, ∞) with 0 < g(x) < G(x), and xg(x) ∈ L(0, ∞). In addition, let yα and Yα be the solutions of these respective equations corresponding to

A theorem due to Lyapunov

1105

yα (0) = Yα (0) = 1, yα (0) = Yα (0) = α for − ∞ < α < ∞. Let yω and Yω be determined, respectively, by yω (0) = Yω (0) = 0, yω (0) = Yω (0) = 1, and let y+ and Y+ be the subdominant solutions for which y+ (0) = Y+ (0) = 1 2 2  2   2   while y+ (x) , g(x) [y+ (x)] , Y+ (x) , and G(x) Y+ (x) belong to L(0, ∞). Then, if β and γ are such that y−β = y+ and Y−γ = Y+ , it follows that β < γ and yα (x) < Yα (x), yω (x) < Yω (x),

0 < x < ∞,

−γ ≤ α, HI 440

y+ (x) > Y+ (x).

16.913 Estimate of dominant solution Let G(x) be positive and continuous with continuous first- and second-order derivatives satisfying G(x)G (x) < 54 [G (x)] . Then there exists a dominant solution y(x) of the fundamental solutions Y0 (x) and Y1 (x) of d2 y − G(x)y = 0, dx2 determined by the initial conditions 2Y0 (0) = 0, Y1 (0)= 1,  Y0 (0) = 1, Y1 (0) = 0, such that  x

−1/4 1/2 y(x) < [G(x)] exp [G(ξ)] dξ , 2

0

and a positive constant C such that the normalized subdominant solution y+ (x), for which y+ (0) = 1   2 2 and y+ (x) ∈ L(0, ∞), G(x) [y+ (x)] ∈ L(0, ∞), satisfies  x

1/2 y+ (x) > CG(x)−1/4 exp − [G(ξ)] dξ . HI 443 0

16.914 A theorem due to Lyapunov Let y(x) be any solution of d2 y − G(x)y = 0 dx2 with G(x) positive and continuous in (0, ∞) with xG(x) ∈ L(0, ∞). Then  x

2 2 exp − [G(ξ) + 1] dξ < [y(x)] + [y  (x)] 0  x

< C exp [G(ξ) + 1] dξ , where C = [y(0)]2 + [y  (0)]2 .

0

HI 446

1106

Boundedness Theorems

16.92 Boundedness Theorems 16.9216 All solutions of the equation

are bounded, provided that "∞ (i) |φ(x)| dx < ∞,  ∞ |ψ(x)| dx < ∞ and (ii)

d2 u + (1 + φ(x) + ψ(x)) u = 0 dx2

ψ(x) → 0 as x → ∞.

BS 112

16.922 If all solutions of the equation d2 u + a(x)u = 0 dx2

are bounded, then all solutions of

are also bounded if

d2 u + (a(x) + b(x))u = 0 dx2  ∞ |b(x)| dx < ∞.

BS 112

16.923 If a(x) → ∞ monotonically as x → ∞, then all solutions of d2 u + a(x)u = 0 dx2

are bounded as x → ∞.

BS 113

16.924 Consider the equation d2 u + a(x)u = 0 dx2

in which





∞

x|a(x)| dx < ∞.

du exists, and the general solution is asymptotic to d0 + d1 x as x → ∞, where d0 and d1 dx may be zero, but not simultaneously. BS 114 Then lim

x→∞

16.9310 Growth of maxima of |y| Sonin’s theorem generalized by P´ olya may be stated as follows: Let y(x) satisfy the differential equation 

{k(x)y  } + φ(x)y = 0, where k(x) > 0, φ(x) > 0, and both functions k(x), φ(x) have a continuous derivative. Then the relative maxima of |y| form an increasing or decreasing sequence according as k(x)φ(x) is decreasing or increasing. SZ 164

17 Fourier, Laplace, and Mellin Transforms 17.1–17.4 Integral Transforms 17.11 Laplace transform The Laplace transform of the function f (x), denoted by F (s), is defined by the integral  ∞ F (s) = f (x)e−sx dx, Re s > 0. 0

The functions f (x) and F (s) are called a Laplace transform pair, and knowledge of either one enables the other to be recovered. If f is summable over all finite intervals, and there is a constant c for which  ∞ |f (x)|e−c|x| dx 0

is finite, then the Laplace transform exists when s = σ + iτ is such that σ ≥ c. Setting F (s) = L [f (x); s] to emphasize the nature of the transform, we have the symbolic inverse result f (x) = L−1 [F (s); x] .   The inversion of the Laplace transform is accomplished for analytic functions F (s) of order O s−k with k > 1 by means of the inversion integral  γ+i∞ 1 f (x) = F (s)esx ds, 2πi γ−i∞ SN 30 where γ is a real constant that exceeds the real part of all the singularities of F (s).

17.12 Basic properties of the Laplace transform 1.8

For a and b arbitrary constants, L [af (x) + bg(x)] = aF (s) + bG(s)

2.

(linearity)

If n > 0 is an integer and lim f (x)e−sx = 0, then for x > 0, x→∞ , + L f (n) (x); s = sn F (s) − sn−1 f (0) − sn−2 f (1) (0) − · · · − f (n−1) (0)

(transform of a derivative) SN 32

1107

1108

3.11

Integral Transforms

  "x If lim e−sx 0 f (ζ) dζ = 0, then x→∞

 L 0

x

 1 f (ξ) dξ; s = F (s) s

(transform of an integral)

  L e−ax f (x); s = F (s + a)

4.

SN 37

(shift theorem)

SU 143

The Laplace convolution f ∗ g of two functions f (x) and g(x) is defined by the integral  x f ∗ g(x) = f (x − ξ)g(ξ) dξ,

5.

0

and it has the property that f ∗ g = g ∗ f and f ∗ (g ∗ h) = (f ∗ g) ∗ h. In terms of the convolution operation L [f ∗ g(x); s] = F (s)G(s)

(convolution (Faltung) theorem).

SN 30

17.13 Table of Laplace transform pairs f (x)

F (s)

1

1

1/s

2

xn ,

n = 0, 1, 2, . . .

3

xν ,

ν > −1

4

xn− 2

5

x−1/2 (x + a)−1 ,

1

n! , sn+1 Γ(ν + 1) , sν+1   Γ n + 12 1

sn+ 2 |arg a| < π

,

Re s > 0

ET I 133(3)

Re s > 0

ET I 137(1)

Re s > 0

ET I 135(17)

 πa−1/2 eas erfc a1/2 s1/2 , Re s ≥ 0

ET I 136(25)

Re s > 0

ET I 142(14)

 x for 0 < x < 1 1 for x > 1

1 − e−s , s2

7

e−ax

1 , s+a

Re s > − Re a

ET I 143(1)

8

xe−ax

1 , (s + a)2

Re s > − Re a

ET I 144(2)

9a

e−ax − e−bx b−a

(s + a)−1 (s + b)−1 ,

6

Re s > {− Re a, − Re b}

AS 1022(29.3.12) continued on next page

Table of Laplace transform pairs

1109

continued from previous page

f (x) 9b11

F (s)

αe−ax + βe−bx + γe−cx (a − b)(b − c)(c − a)

(s + a)−1 (s + b)−1 (s + c)−1 , Re s > {− Re a, − Re b, − Re c}

a, b, c distinct , α = c − b, β = a − c, γ = b − a 1011

ae−ax − be−bx b−a

s(s + a)−1 (s + b)−1 , Re s > {− Re a, − Re b}

11 12 13

eax − 1 a eax − ax − 1 a2  ax 1 2 2  e − 2 a x − ax − 1 a3

AS 1022(29.3.13)

s−1 (s − a)−1 ,

Re s > Re a

s−2 (s − a)−1 ,

Re s > Re a

s−3 (s − a)−1 ,

Re s > Re a

14

(1 + ax)eax

s , (s − a)2

Re s > Re a

15

1 + (ax − 1)eax a2

s−1 (s − a)−2 ,

Re s > Re a

16

2 + ax + (ax − 2)eax a3

s−2 (s − a)−2 ,

Re s > Re a

17

xn eax ,

n!(s − a)−(n+1) ,

Re s > Re a

18 19 20 21 22

n = 0, 1, 2, . . .



 x + 12 ax2 eax

s , (s − a)3

Re s > Re a



 1 + 2ax + 12 a2 x2 eax

s2 , (s − a)3

Re s > Re a

(s − a)−4 ,

Re s > Re a

s , (s − a)4

Re s > Re a

s2 (s − a)−4 ,

Re s > Re a

1 3 ax 6x e

1

2 2x



 + 16 ax3 eax

 x + ax2 + 16 a2 x3 eax

continued on next page

1110

Integral Transforms

continued from previous page

f (x) 23

24 25



F (s)

 1 + 3ax + 32 a2 x2 + 16 a3 x3 eax

aeax − bebx a−b  1 ax 1 bx − be + ae a−b

26

xν−1 e−ax ,

27

xe−x

2

/(4a)

,

1 b

−

1 a

s3 (s − a)−4 ,

Re s > Re a

s(s − a)−1 (s − b)−1 ,

Re s > {Re a, Re b}

s−1 (s − a)−1 (s − b)−1 ,

Re s > {Re a, Re b}



Re ν > 0

Γ(ν)(s + a)−ν ,

Re a > 0

 2 2a − 2π 1/2 a3/2 seas erfc sa1/2

Re s > − Re a

ET I 144(3)

ET I 146(22)

28

exp (−aex ) ,

Re a > 0

as Γ (−s, a)

298

x1/2 e−a/(4x) ,

Re a ≥ 0

1 1/2 −3/2 s 2π

ET I 147(37)

 + , 1 + a1/2 s1/2 exp (−as)1/2 , Re s > 0

308

x−1/2 e−a/(4x) ,

Re a ≥ 0

, + π 1/2 s−1/2 exp (−as)1/2 , Re s > 0

318

x−3/2 e−a/(4x) ,

Re a > 0

ET I 146(26)

ET I 146(27)

+ , 2π 1/2 a−1/2 exp (−as)1/2 , Re s ≥ 0

ET I 146(28)

32

sin(ax)

−1  a s2 + a2 ,

Re s > |Im a|

ET I 150(1)

33

cos(ax)

−1  s s2 + a2 ,

Re s > |Im a|

ET I 154(3)

34

|sin(ax)|,

 πs −1  a s2 + a2 coth , 2a Re s > 0

ET I 150(2)

a>0

continued on next page

Table of Laplace transform pairs

1111

continued from previous page

f (x) 3511

36

|cos(ax)|,

a>0

1 − cos(ax) a2

F (s)  πs ,  2 −1 + s + a cosech , s + a2 2a Re s > 0  −1 s−1 s2 + a2 , Re s > |Im a|

37

38

ax − sin(ax) a3 sin(ax) − ax cos(ax) 2a3

AS 1022(29.3.19)

 −1 s−2 s2 + a2 , Re s > |Im a|

AS 1022(29.3.20)

Re s > |Im a|

AS 1022(29.3.21)

 2 −2 s + a2 ,

39

x sin(ax) 2a

−2  s s2 + a2 ,

40

sin(ax) + ax cos(ax) 2a

 −2 s2 s2 + a2 ,

Re s > |Im a|

Re s > |Im a| 41

x cos(ax)

cos(ax) − cos(bx) b 2 − a2 1

43  44 45

 1 − cos(ax) − 12 ax sin(ax) a4 1  1 b sin(bx) − a sin(ax) a2 − b 2 

4611

 2 2 2 a x − 1 + cos(ax) a4

 1 − cos(ax) + 12 ax sin(ax) a2

ET I 152(14)

AS 1023(29.3.23)

 2  −2 s − a2 s2 + a2 , Re s > |Im a|

42

ET I 155(44)

ET I 157(57)

−1  2 −1  s + b2 s s2 + a2 , Re s > {|Im a|, |Im b|}

AS 1023(29.3.25)

 −1 s−3 s2 + a2 ,

Re s > |Im a|

 −2 s−1 s2 + a2 ,

Re s > |Im a|

 2 −1  2 −1 s + a2 s + b2 , Re s > {|Im a|, |Im b|}  −2  2  2s + a2 , s−1 s2 + a2

Re s > |Im a| continued on next page

1112

Integral Transforms

continued from previous page

f (x) 47

a sin(ax) − b sin(bx) a2 − b 2

F (s)  −1  2 −1 s + b2 s2 s2 + a2 , Re s > {|Im a|, |Im b|}

48

sin(a + bx)

−1  (s sin a + b cos a) s2 + b2 ,

Re s > |Im b|

49

cos(a + bx)

−1  (s cos a − b sin a) s2 + b2 ,

Re s > |Im b|

1 50

a

 sinh(ax) − 1b sin(bx) a2 + b 2

 2 −1  2 −1 s − a2 s + b2 , Re s > {|Re a|, |Im b|}

51

cosh(ax) − cos(bx) a2 + b 2

−1  2 −1  s + b2 s s2 − a2 , Re s > {|Re a|, |Im b|}

52

a sinh(ax) + b sin(bx) a2 + b 2

 −1  2 −1 s + b2 s2 s2 − a2 , Re s > {|Re a|, |Im b|}

53

sin(ax) sin(bx)

−1  2 −1  s + (a + b)2 2abs s2 + (a − b)2 , Re s > {|Im a|, |Im b|}

54

cos(ax) cos(bx)

 −1  2 −1  s + (a + b)2 s s2 + a2 + b2 s2 + (a − b)2 , Re s > {|Im a|, |Im b|}

55

sin(ax) cos(bx)

 −1  2 −1  s + (a + b)2 a s2 + a2 − b2 s2 + (a − b)2 , Re s > {|Im a|, |Im b|}

56

sin2 (ax)

 −1 2a2 s−1 s2 + 4a2 ,

Re s > |Im a|

57

cos2 (ax)

 2   −1 s + 2a2 s−1 s2 + 4a2 ,

Re s > |Im a|

58

sin(ax) cos(ax)

−1  a s2 + 4a2 ,

Re s > |Im a|

e−ax sin(bx)

−1  b (s + a)2 + b2 ,

59

Re s > {− Re a,

|Im b|}

continued on next page

Table of Laplace transform pairs

1113

continued from previous page

f (x) 60

F (s) −1  (s + a) (s + a)2 + b2 ,

e−ax cos(bx)

Re s > {− Re a, 61

x−1 sin(ax)

arctan(a/s),

62

x−1 [1 − cos(ax)]

1 2

|Im b|}

Re s > |Im a|

ET I 152(16)

Re s > |Im a|

ET I 157(59)

  ln 1 + a2 /s2 ,

63

sinh(ax)

−1  a s2 − a2 ,

Re s > |Re a|

ET I 162(1)

64

cosh(ax)

−1  s s2 − a2 ,

Re s > |Re a|

ET I 162(2)

65

xν−1 sinh(ax),

Re ν > −1

1 2

  Γ(ν) (s − a)−ν − (s + a)−ν , Re s > |Re a|

66

xν−1 cosh(ax),

Re ν > 0

1 2

ET I 164(18)

  Γ(ν) (s − a)−ν + (s + a)−ν , Re s > |Re a|

ET I 164(19)

67

x sinh(ax)

−2  2as s2 − a2 ,

Re s > |Re a|

68

x cosh(ax)

 2  −2 s + a2 s2 − a2 ,

Re s > |Re a|

69

sinh(ax) − sin(ax)

 −1 2a3 s4 − a4 , Re s > {|Re a|, |Im a|}

70

cosh(ax) − cos(ax)

AS 1023(29.3.31)

−1  2a2 s s4 − a4 , Re s > {|Re a|, |Im a|}

AS 1023(29.3.32)

71

sinh(ax) + ax cosh(ax)

 −2 2as2 a2 − s2 ,

Re s > |Re a|

72

ax cosh(ax) − sinh(ax)

 −2 2a3 a2 − s2 ,

Re s > |Re a| continued on next page

1114

Integral Transforms

continued from previous page

f (x) 73

x sinh(ax) − cosh(ax) 1

74

a

 sinh(ax) − 1b sinh(bx) a2 − b 2

F (s)  −2  s a2 + 2a − s2 a2 − s2 ,

Re s > |Re a|

 2 −1  2 −1 a − s2 b − s2 , Re s > {|Re a|, |Re b|}

75

cosh(ax) − cosh(bx) a2 − b 2

−1  2 −1  s − b2 s s2 − a2 , Re s > {|Re a|, |Re b|}

76

a sinh(ax) − b sinh(bx) a2 − b 2

 −1  2 −1 s − b2 s2 s2 − a2 , Re s > {|Re a|, |Re b|}

77

sinh(a + bx)

−1  (b cosh a + s sinh a) s2 − b2 ,

Re s > |Re b|

78

cosh(a + bx)

−1  (s cosh a + b sinh a) s2 − b2 ,

Re s > |Re b|

79

sinh(ax) sinh(bx)

−1  2 −1  s − (a − b)2 2abs s2 − (a + b)2 , Re s > {|Re a|, |Re b|}

808

cosh(ax) cosh(bx)

 −1  2 −1  s − (a − b)2 s s2 − a2 − b2 s2 − (a + b)2 , Re s > {|Re a|, |Re b|}

81

sinh(ax) cosh(bx)

 −1  2 −1  s − (a − b)2 a s2 − a2 + b2 s2 − (a + b)2 , Re s > {|Re a|, |Re b|}

82

sinh2 (ax)

 −1 2a2 s−1 s2 − 4a2 ,

Re s > |Re a|

83

cosh2 (ax)

 2   −1 s − 2a2 s−1 s2 − 4a2 ,

Re s > |Re a|

84

sinh(ax) cosh(ax)

−1  a s2 − 4a2 ,

Re s > |Re a|

85

cosh(ax) − 1 a2

 −1 s−1 s2 − a2 ,

Re s > |Re a| continued on next page

Table of Laplace transform pairs

1115

continued from previous page

f (x) 86 87 88 89

F (s)

sinh(ax) − ax a3   cosh(ax) − 12 a2 x2 − 1 a4   1 − cosh(ax) + 12 ax sinh(ax) a4

 −1 s−2 s2 − a2 ,

Re s > |Re a|

 −1 s−3 s2 − a2 ,

Re s > |Re a|

 −2 s−1 s2 − a2 ,

Re s > |Re a|

 , + π 1/2 /4 (s − a)3/2 − (s + a)3/2 ,

x1/2 sinh(ax)

Re s > |Re a| −s−1 ln (Cs) ,

Re s > 0

ET I 148(1)

s−1 es/a Ei(−s/a),

Re s > 0

ET I 148(4)

Re s > 0

ET I 148(9)

90

ln x

91

ln(1 + ax),

92

x−1/2 ln x

− (π/s)

 0 for x < a H(x − a) = 1 for x > a

s−1 e−as ,

93

|arg a| < π

1/2

ln (4Cs) ,

a≥0

(Heaviside step function) 94

δ(x)

(Dirac delta function)

95

δ(x − a)

e−as ,

a≥0

96

δ  (x − a)

se−as ,

a≥0

 97

x

Si(x) ≡ 0

98

1 sin ξ dξ ≡ π + si(x) ξ 2

 Ci(x) ≡ ci(x) ≡ −

∞ x

998

erf

x 2a

cos ξ dξ ξ

1

s−1 arccot s,

Re s > 0

ET I 177(17)

  1 − s−1 ln 1 + s2 , 2

Re s > 0

ET I 178(19)

s−1 ea

2 2

s

erfc(as), Re s > 0, |arg a| < π/4

ET I 176(2)

continued on next page

1116

Integral Transforms

continued from previous page

f (x) 100

F (s)

 √  erf a x

 −1/2 as−1 s + a2 ,   Re s > 0, − Re a2

101

ET I 176(4)

,  − 1 + 1/2 s−1 s + a2 2 s + a2 −a ,

 √  erfc a x

Re s > 0 1028

erfc

a √ x



√ s

s−1 e−2a

, Re s > 0,

1038

104

105

J ν (ax),

x J ν (ax),

Re ν > −1

Re ν > −2

a−ν



ET I 177(9)

Re a > 0

ET I 177(11)

ν  −1/2 s2 + a2 s2 + a2 − s ,

Re s > |Im a|, ET I 182(1) , +   1/2 1/2 ,−ν s + s2 + a2 aν s + ν s2 + a2 −3/2  × s2 + a2 , +

Re s > |Im a|, + 1/2 ,−ν  aν ν −1 s + s2 + a2 ,

J ν (ax) x

ET I 182(2)

Re s ≥ |Im a| 106

 −(n+ 12 ) , 1 · 3 · 5 · · · (2n − 1)an s2 + a2

xn J n (ax)

Re s > |Im a| 107

xν J ν (ax),

Re ν > − 12

xν+1 J ν (ax),

Re ν > −1

I ν (ax),

Re ν > −1

ET I 182(7)

 −(ν+ 32 )   , 2ν+1 π −1/2 Γ ν + 32 aν s s2 + a2 Re s > |Im a|

1098

ET I 182(4)

  −(ν+ 12 )  , 2ν π −1/2 Γ ν + 12 aν s2 + a2 Re s > |Im a|,

108

ET I 182(5)

ET I 182(8)

+ ,ν   −1/2 s2 − a2 a−ν s − s2 − a2 , Re s > |Re a|

ET I 195(1)

continued on next page

Fourier transform

1117

continued from previous page

f (x) 110

xν I ν (ax),

F (s) Re ν > − 12

  −(ν+ 12 )  , 2ν π −1/2 Γ ν + 12 aν s2 − a2 Re s > |Re a|

111

xν+1 I ν (ax),

Re ν > −1

 −(ν+ 32 )   , 2ν+1 π −1/2 Γ ν + 32 aν s s2 − a2 Re s > |Re a|

112

x−1 I ν (ax),

Re ν > 0

ET I 195(6)

ET I 196(7)

+  1/2 ,−ν ν −1 aν s + s2 − a2 , Re s > |Re a|

ET I 195(4)

113

 sin 2a1/2 x1/2

(πa)1/2 s−3/2 e−a/s ,

Re s > 0

ET I 153(32)

114

 x−1/2 cos 2a1/2 x1/2

π 1/2 s−1/2 e−a/s ,

Re s > 0

ET I 158(67)

115

x−1 e−ax I 1 (ax)

+ ,+ ,−1 1/2 , (s + 2a)1/2 − s1/2 (s + 2a) + s1/2 Re s > |Re a|

116

J k (ax) x

k −1 a−k

+

s2 + a2

1/2

,k −s ,

Re s > |Im a|, k > −1 117

 x k− 12 J k− 12 (ax) 2a

AS 1024(29.3.52)

AS 1025(29.3.58)

 k Γ(k)π −1/2 s2 + a2 , Re s > |Im a|,

k>0

AS 1024(29.3.57)

118

J 0 (ax) − ax J 1 (ax)

 −3/2 s2 s2 + a2 ,

Re s > |Im a|

119

I 0 (ax) + ax I 1 (ax)

 −3/2 s2 s2 − a2 ,

Re s > |Im a|

17.21 Fourier transform The Fourier transform, also called the exponential or complex Fourier transform, of the function f (x), denoted by F (ξ), is defined by the integral  ∞ 1 F (ξ) = √ f (x)eiξx dx. 2π −∞ The functions f (x) and F (ξ) are called a Fourier transform pair, and knowledge of either one enables the other to be recovered. Setting F (ξ) = F [f (x); ξ] , to emphasize the nature of the transform, we have

1118

Integral Transforms

the symbolic inverse result f (x) = F −1 [F (ξ); x] . The inversion of the Fourier transform is accomplished by means of the inversion integral  ∞ 1 f (x) = √ F (ξ)e−iξx dξ. 2π −∞

17.22 Basic properties of the Fourier transform 1.

For a and b arbitrary constants, F [af (x) + bg(x)] = aF (ξ) + bG(ξ)

(linearity)

If n > 0 is an integer, and lim f (r) (x) = 0 for r = 0, 1, . . . , n − 1 with f (0) (x) ≡ f (x), then

2.

|x|→∞

+ , F f (n) (x); ξ = (−iξ)n F (ξ)

(transform of a derivative)

SN 27

The Fourier convolution f ∗ g of two functions f (x) and g(x) is defined by the integral  ∞ 1 f ∗ g(x) = √ f (x − ξ)g(ξ) dξ, 2π −∞ and it has the property f ∗ g = g ∗ f , and f ∗ (g ∗ h) = (f ∗ g) ∗ h. In terms of the convolution operation,

3.

F [f ∗ g(x); ξ] = F (ξ)G(ξ)

(convolution [Faltung] theorem).

SN 24

17.23 Table of Fourier transform pairs f (x)

F (ξ)

1

1

(2π)1/2 δ(ξ)

27

1 x

(π/2)

3

δ(x)

(2π)−1/2

SU 496

48

δ(ax + b),

a = 0

(2π)−1/2 eibξ/a

SU 517

a>0

(2/π)1/2 ξ −1 sin(aξ)

5 8

6

1/2

a, b ∈ R,

 1 |x| < a , 0 |x| > a  0 x<0 H(x) = 1 x>0

SU 496

i sign ξ

1 − √ + iξ 2π



π δ(ξ) 2

SU 50

SN 523 continued on next page

Table of Fourier transform pairs

1119

continued from previous page

f (x)

F (ξ)   (2/π)1/2 Γ(1 − a) sin 12 aπ

7

1 a, |x|

0 < Re a < 1

8

eiax ,

a∈R

(2π)1/2 δ(ξ + a)

SU 50

9

e−a|x| ,

a>0

a(2/π)1/2 a2 + ξ 2

SU 50

107

xe−a|x| ,

a>0

11

|x|e−a|x| ,

a>0

12

e−a|x| |x|

1/2

,

a>0

,

a>0

13

e−a

14

1 , 2 a + x2

Re a > 0

x , + x2

Re a > 0

157

2

a2

x2

|ξ|

2aiξ(2/π)1/2

2 , (a2 + ξ 2 )   (2/π)1/2 a2 − ξ 2

ξ>0

2

(a2 + ξ 2 ) +  1/2 ,1/2 a + a2 + ξ 2

SU 50

SU 51

1/2 −a|ξ|

(π/2)

e

SU 51

a 1/2 −a|ξ|

i sign ξ (π/2)



e



  sin ax2

17

  cos ax2

18

e−a|x| cos(bx),

a > 0,

b>0

19

e− 2 ax sin(bx),

a > 0,

b>0

209

sinh(ax) , sinh(bx)

|a| < |b|

(π/2) sin(πa/b) b [cosh (πξ/b) + cos(πa/b)]

219

cosh(ax) , sinh(bx)

|a| < |b|

i (π/2) sinh (πξ/b) b [cosh (πξ/b) + cos(πa/b)]

2

SU 50

SN 523

1/2

x (a2 + ξ 2 )  √ −1 2 2 a 2 e−ξ /4a

169

1

SN 523

1−a

1 cos (2a)1/2

ξ2 π + 4a 4

SN 523

2

ξ π 1 − cos SN 523 4a 4 (2a)1/2   1 1 a(2π)−1/2 2 + a + (b + ξ)2 a2 + (b − ξ)2

  1 −1/2 1 (ξ − b)2 ia exp − 2 2 a  1 (ξ + b)2 − exp − 2 a

1/2

SU 123

1/2

SU 123 continued on next page

1120

Integral Transforms

continued from previous page

f (x) 22

sin(ax) x

2311

x sinh x

247

xn sign x,

257

|x| ,

F (ξ)

 1/2 |ξ| < a, (π/2) 0 |ξ| > a  3 1/2 πξ 2π e

SN 523

2

SU 123

(2/π)1/2 (−iξ)−(1+n) n!

SU 506

(1 + eπξ ) n = 1, 2, . . .

ν

(2/π)1/2 Γ(ν + 1)|ξ|

−ν−1

cos [π(ν + 1)/2]

−1 < ν < 0, but not integral 267

SU506

i sign ξ(2/π)1/2 sin [(π/2) (ν + 1)] Γ(ν + 1)

ν

|x| sign x,

|ξ|

ν+1

−1 < ν < 0, but not integral ! ! e−ax ln !1 − e−x !,

27

SU 506

 π 1/2 cot (πa − iξπ) 2 a − iξ

ET I 121(26)

 π 1/2 csc (πa − iξπ) 2 a − iξ

ET I 121 (27)

−1 < Re a < 0   e−ax ln 1 + e−x ,

28

−1 < Re a < 0 In deriving results for the preceding table from ET I, account has been taken of the fact that the normalization factor 1/(2π)1/2 employed in our definition of F has not been used in those tables, and that there is a difference of sign between the exponents used in the definitions of the exponential Fourier transform.

17.24 Table of Fourier transform pairs for spherically symmetric functions

2

e−ar

311

e−ar r

 1 E(||k||) = f (||r||)e−ik·r dr (2π)3/2   21 ∞ E(k) = f (r) sin(kr)r dr πk 0  2a 2 π (a2 + k 2 )2  1 2 π (a2 + k 2 )2

411

1

(2π)3/2 δ(k)

1

 1 f (||r||) = E(||k||)eik·r dk (2π)3/2   21 ∞ f (r) = E(k) sin(kr)k dk πr 0

Basic properties of the Fourier sine and cosine transforms

1121

17.31 Fourier sine and cosine transforms The Fourier sine and cosine transforms of the function f (x), denoted by Fs (ξ) and Fc (ξ), respectively, are defined by the integrals   ∞   ∞ 2 2 f (x) sin(ξx) dx and Fc (ξ) = f (x) cos(ξx) dx. Fs (ξ) = π 0 π 0 The functions f (x) and Fs (ξ) are called a Fourier sine transform pair, and the functions f (x) and Fc (ξ) a Fourier cosine transform pair, and knowledge of either Fs (ξ) or Fc (ξ) enables f (x) to be recovered. Setting and Fc (ξ) = F c [f (x); ξ] , Fs (ξ) = F s [f (x); ξ] to emphasize the nature of the transforms, we have the symbolic inverses f (x) = F s −1 [Fs (ξ); x] and f (x) = F c −1 [Fc (ξ); x] . The inversion of the Fourier sine transform is accomplished by means of the inversion integral   ∞ 2 Fs (ξ) sin(ξx) dξ [x ≥ 0] f (x) = π 0 and the inversion of the Fourier cosine transform is accomplished by means of the inversion integral   ∞ 2 Fc (ξ) cos(ξx) dξ [x ≥ 0] . SN 17 f (x) = π 0

17.32 Basic properties of the Fourier sine and cosine transforms 1.

For a and b arbitrary constants, F s [af (x) + bg(x)] = aFs (ξ) + bGs (ξ) and

2.

F c [af (x) + bg(x)] = aFc (ξ) + bGc (ξ) (linearity) < If lim f (r−1) (x) = 0 and lim π2 f (r−1) (x) = ar−1 , then denoting the Fourier sine and cosine x→∞

x→∞

transforms of f (r) (x) by Fs (r) and Fc (r) , respectively, (i)

Fc (r) (ξ) = −ar−1 + ξFs (r−1) .

(ii)

Fs (r) (ξ) = −ξFc (r−1) (ξ), r−1  (−1)n a2r−2n−1 ξ 2n + (−1)r ξ 2n Fc (ξ), Fc (2r) (ξ) = −

(iii)

n=0 r−1



(iv)

Fc (2r+1) (ξ) = −

(v)

(ξ) = ξar−2 − ξ 2 Fs (r−2) (ξ), r  Fs (2r) (ξ) = − (−1)n ξ 2n−1 a2r−2n + (−1)r ξ 2r Fs (ξ),

(−1)n a2r−2n ξ 2n + (−1)r ξ 2r+1 Fs (ξ),

n=0

(vi)6

Fs

(r)

n=1

(vii)

Fs

(2r+1)

(ξ) = −

r   n=1

(−1)n ξ 2n−1 a2r−2n+1 + (−1)r+1 ξ 2r+1 Fc (ξ).

SN 28

1122

Integral Transforms

 1 ∞ Fs (ξ)Gs (ξ) cos(ξx) dξ = g(s) [f (s + x) + f (s − x)] ds, 2 0 0 ∞  ∞ 1 Fc (ξ)Gc (ξ) cos(ξx) dξ = g(s) [f (s + x) + f (|x − s|)] ds 2 0 0 (convolution (Faltung) theorem) 

3.

(i) (ii)

4.

(i) (ii)

∞

SN 24

If Fs (ξ) is the Fourier sine transform of f (x), then the Fourier sine transform of Fs (x) is f (ξ). If Fc (ξ) is the Fourier cosine transform of f (x), then the Fourier cosine transform of Fc (x) is f (ξ).

(v)

If f (x) is an odd function in (−∞, ∞), then the Fourier sine transform of f (x) in (0, ∞) is −iF (ξ). If f (x) is an even function in (−∞, ∞), then the Fourier cosine transform of f (x) in (0, ∞) is F (ξ). The Fourier sine transform of f (x/a) is aFs (aξ).

(vi) (vii)

The Fourier cosine transform of f (x/a) is aFc (aξ). F s [f (x); ξ] = Fs (|ξ|) sign ξ

(iii) (iv)

SU 45

17.33 Table of Fourier sine transforms f (x) 1

x−1

2

x−ν ,

Fs (ξ) 1/2

(π/2) 0 < Re ν < 2

,

(ξ > 0) ξ>0

ET I 64(3)

(2/π)1/2 ξ ν−1 Γ(1 − ν) cos (νπ/2) , ξ>0

ET I 68(1)

3

x−1/2

ξ −1/2 ,

ξ>0

ET I 64(6)

4

x−3/2

2ξ 1/2 ,

ξ>0

ET I 64(9)

(2/π)1/2 ξ −1 [1 − cos(aξ)] ,

ξ>0

ET I 63(1)

(2/π)1/2 Si(aξ),

ξ>0

ET I 64(4)

5 6 7

 1 0<xa  x−1 0 < x < a 0 x>a 1 , a−x

a>0

   (2/π)1/2 sin(aξ) Ci(aξ) − cos(aξ) 12 π + Si(aξ) , ξ>0

ET I 64(11)

continued on next page

Table of Fourier sine transforms

1123

continued from previous page

f (x) 87

x2

Fs (ξ)

1 , + a2

a>0

  (2π)−1/2 a−1 e−aξ Ei(aξ) − eaξ Ei(−aξ) ,

9

−3/2  x x2 + a2 ,

Re a > 0

(2/π)1/2 ξ K 0 (aξ),

10

 −1/2 x−1/2 x2 + a2 ,

Re a > 0

ξ 1/2 I 14

117

−ν− 32  x x2 + a2 , Re ν > −1,

12 13

x , 2 a + x2

−1

+

Re a > 0

15

x−1 e−ax ,

16

xν−1 e−ax ,

Re a > 0 Re a > 0

K 14

1

2 aξ



,

ξ>0

ET I 66(27)

ξ>0

ET I 66(28)

 π 1/2 2

e−aξ ,

ξ>0

ET I 65(15)

ξ>0

ET I 67(35)

  π/2  1 − e−aξ , 2 a

ξ (2/π)1/2 tan−1 , a

ξ>0

ET I 65(20)

ξ>0

ET I 72(2)

  −ν/2  ξ (2/π)1/2 Γ(ν) a2 + ξ 2 sin ν tan−1 , a Re ν > −1,

Re a > 0

17

e−ax ,

Re a > 0

18

xe−ax ,

Re a > 0

−ax2



 π/8a−1 ξe−aξ ,

 2 −1 x + a2 ,

x

2 aξ

ET I 65(14)

Re a > 0

2 x2 )

14

1

ξ>0

ξ ν+1 √  K ν (aξ),  2(2a)ν Γ ν + 32

x (a2

(ξ > 0)

19

xe

,

|arg a| < π/2

20

sin ax , x

a>0

21

sin ax , x2

a>0

 2/πξ , 2 a + ξ2 (2/π)1/2 2aξ (a2 + ξ 2 )

,

−ξ 2 ξ exp 4a ! ! !ξ + a! !, ! ln ! ξ − a!

−3/2

(2a)

2

1 (2π)1/2   1/2 ξ π2  1/2 a π2

ξ>0

ET I 72(7)

ξ>0

ET I 72(1)

ξ>0

ET I 72(3)

,

0<ξ
ξ>0

,

ET I 73(19)

ξ>0

ET I 78(1)

ξ>0

ET I 78(2)

continued on next page

1124

Integral Transforms

continued from previous page

22

23

sin

−1

x

f (x)

2

a x

,

sin

a>0

a2 x

,

a>0

Fs (ξ) (ξ > 0)   π 1/2 1 a ξ −1/2 J 1 2aξ 2 , 2 ξ>0 ET I 83(6)  π 1/2  2 1/2  Y 0 2aξ 1/2 + K 0 2aξ 1/2 2 π ET I 83(7)

24

2510

x−2 sin

27 28

29 30

31

2

a x

,

cosech(ax),

26



coth



1 ax − 1, 2

1 − x2

−1

sin2 (ax) , x

2

cos ax



Re a > 0

,

 π 1/2 2

 a−1 ξ 1/2 J 1 2aξ 1/2 ,

1/2

(π/2)

a−1 tanh

Re a > 0

a>0

a>0

1

2 πa

−1

ξ>0

ET I 83(8)

ξ>0

ET I 88(2)

 ξ ,

  (2π)1/2 a−1 coth πa−1 ξ − ξ,  (2/π)1/2 sin ξ 0

a>0

  sin ax2 ,



Re a > 0

sin(πx)

e−ax sin(bx), 2

a>0

ξ>0 0≤ξ≤π π<ξ

ET I 88(3) ET I 78(4)

    (2a)−1/2 exp − ξ 2 + b2 /(4a) sinh (bξ/2a) , ξ>0 ⎧ 1/2 −3/2 ⎪ 0 < ξ < 2a ⎨π 2 1/2 −5/2 ξ = 2a π 2 ⎪ ⎩ 0 2a < ξ ,   +  2 −1/2 a cos ξ /4a C (2πa)−1/2 ξ ,   + + sin ξ 2 /4a S (2πa)−1/2 ξ ,

ξ>0   ,  + 2 −1/2 sin ξ /4a C (2πa) a ξ ,  +  − cos ξ 2 /4a S (2πa)−1/2 ξ ,

ET I 78(7)

ET I 78(8)

ET I 82(1)

−1/2

ξ>0 continued on next page

Table of Fourier sine transforms

1125

continued from previous page

32 337 34 35 367 37

f (x) x arctan , a

2a arctan , x

Fs (ξ) a>0 Re a > 0

1/2 −1 −aξ

(π/2)

ξ

e

(ξ > 0)

,

(2π)−1/2 e−aξ sinh(aξ),

ξ>0

ET I 87(3)

ξ>0

ET I 87(8)

ξ>0

ET I 76(2)

ln x x ! ! !x + a! !, ! ln ! x − a!   ln 1 + a2 x2 , x

a>0

(2π)1/2 ξ −1 sin(aξ),

ξ>0

ET I 77(11)

a>0

−(2π)1/2 Ei (−ξ/a) ,

ξ>0

ET I 77(14)

J 0 (ax),

a>0

− (π/2)

1/2

 0 1/2

(2/π)

(C + ln ξ) ,

 2 −1/2 ξ − a2

0<ξ
38

39

Re ν > −2,

J ν (ax),

J 0 (ax) , x

a>0

a>0

   2  ξ −1 1/2 2 −1/2 a −ξ (2/π) sin ν sin a for 0 < ξ < a   aν cos 12 νπ + ,ν for a < ξ < ∞ 1/2 1/2 ξ + (ξ 2 − a2 ) (ξ 2 − a2 )

  (2/π)1/2 sin−1 aξ 1/2

ET I 99(3)

0<ξ
(π/2)

ET I 99(4)

407



x2 + b2

−1

a > 0, 41

(2/π)1/2 sinh(bξ) K 0 (ab)/b,

J 0 (ax), Re b > 0

−1  J 0 (ax), x x2 + b2 a > 0,

1/2 −bξ

(π/2) Re b > 0

e

0<ξ
ET I 100(12)

a<ξ<∞

ET I 100(13)

I 0 (ab),

In deriving results  for the preceding table from ET I, account has been taken of the fact that the normalization factor 2/π employed in our definition of Fs has not been used in those tables.

1126

Integral Transforms

17.34 Table of Fourier cosine transforms f (x) 1

2 3 4 5 69

7

x−ν ,

9

10

11

12

0 < Re ν < 1

xν−1 ,

x2 + a2

−1

sec

1

2 νπ



ξ ν−1 ,

sin(aξ) , ξ

ET I 10(1) ET I 7(1)

−(2/π)1/2 Ci(aξ),

ξ>0

ET I 8(3)

2ξ −1/2 C (aξ),

ξ>0

ET I 8(5)

ξ>0

ET I 8(6)

1 2

 − C (aξ) ,

(2/π)1/2 Γ(ν)ξ −ν cos

1

2 νπ



,

0<ν<1

ET I 10(1)

ξ>0

ET I 11(7)

ξ>0

ET I 11(7)

ξ>0

ET I 11(7)

1/2 −aξ

(π/2)

e

a

,

1

2,

Re a > 0

−ν− 12

,

1 Re a > 0, Re ν > − 2  ν a2 − x2 0<xa  0 −ν− 12  2 x − a2

e−ax ,

[Γ(ν)]

ξ>0

2ξ −1/2

Re a > 0

(x2 + a2 ) 

(2/π)1/2

0<ν<1

1 , x2 + a2 1

1/2

(π/2)

ξ>0

 1 0<xa  0 0<x a  x−1/2 0 < x < a 0 x>a  0 0<xa

11

8

Fc (ξ)

(π/2) 2 (1 + aξ)e−aξ , 2a3 ν √ ξ K ν (aξ) ,  2 2a Γ ν + 12

1

2ν Γ(ν + 1)(a/ξ)ν+ 2 J ν+ 12 (aξ), ξ>0

Re ν > −1 0<x
,

x>a 1 1 − < Re ν < 2 2 Re a > 0

−2−(ν+ 2 ) Γ 1

1 2

ET I 11(8)

 ν − ν (ξ/a) Y ν (aξ),

−1  (2/π)1/2 a a2 + ξ 2 ,

ξ>0

ET I 11(9)

ξ>0

ET I 14(1)

continued on next page

Table of Fourier cosine transforms

1127

continued from previous page

f (x) 13

147

Fc (ξ)

xe−ax ,

Re a > 0

ξ>0 ET I 15(7)     2 ξ 1/2 2 −ν/2 −1 (2/π) Γ(ν) a + ξ cos ν tan , a

xν−1 e−ax , Re a > 0,

15

  −2 (2/π)1/2 a2 − ξ 2 a2 + ξ 2 ,

x−1/2 e−ax ,

Re ν > a Re a > 0

ξ>0 

a2 + ξ 2

−1/2 +

a2 + ξ 2

1/2

ET I 15(7)

,1/2 +a

,

ξ>0 167 17 18

19

e−a

2

x2

,

−1 −x

x

e



2

sin ax

Re a > 0

2−1/2 |a|

−1/2

2 ξ2

a>0

a>0

a>0

217

sin2 (ax) , x2

a>0

227

e−bx sin(ax),

a > 0,

Re b > 0

x2 + a2

+

2 a2 )

−1/2

,

ξ>0

ξ>0 2   2

ξ 1 ξ √ cos + sin , 4a 4a 2 a

ξ>0 a>0

+  1/2 , , sin b x2 + a2 a>0

ET I 19(7)

ET I 23(1)

ET I 24(7) ξ>0 ⎧ 1/2 ⎪ ξa   1/2  a − 12 ξ ξ < 2a (π/2) ET I 19(8) 0 2a < ξ   a+ξ a−ξ (2π)−1/2 2 + , b + (a + ξ)2 b2 + (a − ξ)2

+  1/2 , sin b x2 + a2 (x2

ET I 15(11)



2   2

ξ 1 ξ √ cos − sin , 4a 4a 2 a

,

tan



ξ>0



sin(ax) , x



−1

,

(2π)

20

24

e

sin x

  cos ax2 ,

23

−1 −ξ 2 /4a2

ET I 14(4)

ET I 19(6)

1/2 −aξ

(b/a) (π/2)

e , ξ>0 ET I 26(29) +   1/2 , 0<ξ
b<ξ ET I 26(30) continued on next page

1128

Integral Transforms

continued from previous page

f (x) 25 26

27

28

1 − cos(ax) , x2   2 e−ax sin bx2 ,

e

−ax2



2

cos bx



a>0 Re a > |Im b|

Re a > |Im b|

,

sinh(ax) sinh(bx)

|Re a| < Re b

Fc (ξ)  1/2 (π/2) (a − ξ) ξ < a ET I 20(16) 0 a<ξ  −1/4     2−1/2 a2 + b2 exp −aξ 2 / 4 a2 + b2 +  −1 , × sin 12 arctan(b/a) − 14 bξ 2 a2 + b2 , ET I 23(5) ξ>0    2 2 a +b 2 exp −aξ / 4 a + b2 , +  −1 1 − 2 arctan(b/a) , × cos 14 bξ 2 a2 + b2 −1/2



 π 1/2 2

 2 −1/4

2



ξ>0 sin(πa/b) , b [cosh (πξ/b) + cos(πa/b)] ξ>0

29

cosh(ax) , cosh(bx)

|Re a| < Re b

sech(ax),

Re a > 0

a−1 (π/2)

1/2

32 337

 πx  x2 + a2 sech , 2a

a2 ln 1 + 2 , x 2

a + x2 ln , b2 + x2 

ET I 31(12)

sech (πξ/2a) , ξ>0

31

ET I 31(14)

(2π)1/2 cos(πa/2b) cosh (πξ/2b) , b [cosh (πξ/b) + cos(πa/b)] ξ>0

30

ET I 24(6)

ET I 30(1)

Re a > 0

2(2/π)1/2 a3 sech3 (aξ),

ξ>0

ET I 32(19)

Re a > 0

  (2π)1/2 ξ −1 1 − e−aξ ,

ξ>0

ET I 18(10)

  (2π)1/2 e−bξ − e−aξ ,

ξ>0

ET I 18(12)

a<ξ<∞

ET I 45(14)

Re a > 0, Re b > 0 34



x2 + b2

−1

1/2 −1 −bξ

J 0 (ax), a > 0,

(π/2) Re b > 0

b

e

I 0 (ab),

continued on next page

Mellin transform

1129

continued from previous page

f (x) 35

Fc (ξ)

−1  J 0 (ax), x x2 + b2 a > 0,

(2/π)1/2 cosh(bξ) K 0 (ab), Re b > 0

0<ξ
ET I 45(15)

In deriving results  for the preceding table from ET I, account has been taken of the fact that the normalization factor 2/π employed in our definition of Fc has not been used in those tables.

17.35 Relationships between transforms The following relationships exist between transforms, and they may be used to derive further transform pairs from among the results given in Sections 17.13–17.34. The appropriate sections of the main body of the tables may also be used to extend the list of transform pairs. 17.351 Fourier cosine transform and Laplace transform relationship 1 1 F c [f (x); ξ] = √ L [f (x); iξ] + √ L [f (x); −iξ] . 2π 2π 17.352 Fourier sine transform and Laplace transform relationship i i F s [f (x); ξ] = √ L [f (x); iξ] − √ L [f (x); −iξ] . 2π 2π 17.353 Exponential Fourier transform and Laplace transform relationship √ √ F [f (x); ξ] = 2πL [f (x); −iξ] + 2πL [f (−x); iξ] .

17.4110 Mellin transform The Mellin transform of the function f (x), denoted by f ∗ (s), is defined by the integral  ∞ f ∗ (s) = f (x)xs−1 dx. 0

The functions f (x) and f ∗ (s) are called a Mellin transform pair, and knowledge of either one enables the other to be recovered. The transform exists, provided the integral  ∞ |f (x)|xk−1 dx 0

is bounded for some k > 0, and then the inversion of the Mellin transform is accomplished by means of the inversion integral

1130

Integral Transforms

f (x) =

1 2πi



c+i∞

f ∗ (s)x−s ds,

c−i∞

where c > k. Setting f ∗ (s) = M [f (x); s] to denote the Mellin transform, we have the symbolic expression for the inverse result f (x) = M−1 [f ∗ (s); x] .

MS 397(6)

17.42 Basic properties of the Mellin transform 1.

For a and b arbitrary constants, M [af (x) + bg(x)] = af ∗ (s) + bg ∗ (s)

2.

x→0

(ii)

3.

r = 0, 1, . . . , n − 1,

If lim xs−r−1 f (r) (x) = 0, (i)

(linearity)

+ , M f (n) (x); s = (−1)n

Γ(s) f ∗ (s − n) Γ(s − n) (transform of a derivative)

SU 267 (4.2.3)

(transform of a derivative)

SU 267 (4.2.5)

+ , Γ(s + n) ∗ f (s) M xn f (n) (x); s = (−1)n Γ(s)

Denoting the nth repeated integral of f (x) by In [f (x)], where  x In [f (x)] = In−1 [f (u)] du, 0

(i)

M [In [f (x)] ; s] = (−1)n

Γ(s) f ∗ (s + n) Γ(n + s) (transform of an integral)

(ii)

M [In∞ [f (x)] ; s] =

Γ(s) f ∗ (s + n), Γ(s + n)

where



In∞ [f (x)] =

∞

∞ In−1 [f (u)] du

(transform of an integral)

SU 269 (4.2.15)

SU 269 (4.2.18)

x

4.

M [f (x)g(x); s] =

1 2πi



c+i∞

f ∗ (u)g ∗ (s − u) du

c−i∞

(Mellin convolution theorem)

SU 275(4.4.1)

Table of Mellin transforms

1131

17.43 Table of Mellin transforms f ∗ (s)

f (x) 1

e−x

Γ(s),

2

e−x

1 2

3

cos x

Γ(s) cos

1



4

sin x

Γ(s) sin

1



5

1 1−x

π cot(πs),

0 < Re s < 1

SU 521(M1)

6

1 1+x

π cosec(πs),

0 < Re s < 1

SU 521(M2)

7

(1 + xa )

2

Γ

Re s > 0

SU 521(M13)

Re s > 0

SU 521(M14)

,

0 < Re s < 1

SU 521(M15)

,

0 < Re s < 1

SU 521(M16)

1  2s , 2 πs

2 πs

Γ(s/a) Γ(b − s/a) , a Γ(b)

−b

0 < Re s < ab SU 521(M3)

8

9

10

11

T n (x) H(1 − x)  (1 − x2 )

Tn

Γ



 x−1 H(1 − x)  (1 − x2 )

P n (x) H(1 − x)

  P n x−1 H(1 − x)

1 2

+

1 2s

2−s π Γ(s)   , + 12 n Γ 12 + 12 s − 12 n

Re s > 0     2s−2 Γ 12 n + 12 s Γ 12 s − 12 n , Γ(s)

SU 521(M4)

Re s > n  1  1 Γ 2 s Γ 2 s + 12    , 2 Γ 12 s − 12 n + 12 Γ 12 s + 12 n + 1

SU 521(M5)

Re s > 0     2s−1 Γ 12 s + 12 n + 12 Γ 12 s − 12 n √ , π Γ(s + 1)

SU 521(M6)

Re s > n 12

1 + x cos φ 1 − 2x cos φ + x2

13

x sin φ , 1 − 2x cos φ + x2

−π < φ < π

SU 521(M7)

π cos(sφ) , sin(sπ)

0 < Re s < 1

SU 521(M11)

π sin(sφ) , sin(sπ)

0 < Re s < 1

SU 521(M12)

continued on next page

1132

Integral Transforms

continued from previous page

f ∗ (s)

f (x) 14

e−x cos φ cos (x sin φ) , 1 2π

15

− 12 π < φ < 12 π 16

x

J ν (x),

17

Y ν (x),

Re s > 0

SU 522(M17)

Γ(s) sin(sφ),

Re s > −1

SU 522(M18)

< φ < 12 π

e−x sin φ sin (x sin pφ) ,

−ν

Γ(s) cos(sφ),

ν>

− 12

ν∈R

  2s−ν−1 Γ 12 s   , 0 < Re s < 1 Γ ν − 12 s + 1     −2s−1 π −1 Γ 12 s + 12 ν Γ 12 s − 12 ν   × cos 12 s − 12 ν π, |ν| < Re s <

18

19

20

21

22

K ν (x),

Hν (x),

1 , a + xn |arg a| < π, 

1 + axh

−ν

2s−2 Γ

ν∈R

Re s > ν > 0     2s−1 tan 12 πs + 12 πν Γ 12 s + 12 ν   , Γ 12 ν − 12 s + 1   −1 − ν < Re s < min 32 , 1 − ν πn−1 cosec

24

arctan x

 πs n

SU 522(M21)

SU 522(M22)

a(s/n)−1 , 0 < Re s < n

n = 1, 2, 3, . . . ,

MS 453

h−1 a−s/h B (s/h, ν − (s/h))

,

h > 0, ln(1 + ax),

2s

SU 522(M20)

   + 12 ν Γ 12 s − 12 ν ,

ν∈R

h > 0, |arg a| < π   ν−1 1 − xh for 0 < x < 1 , 0 for x > 1

23

1

3 2

SU 522(M19)

0 < Re s < h Re ν h−1 B (ν, s/h)

MS 454 MS 454

Re ν > 0 |arg a| < π

πs−1 a−s cosec(πs),

−1 < Re s < 0

MS 454

− 12 πs−1 sec(πs/2),

−1 < Re s < 0

MS 454

continued on next page

Table of Mellin transforms

1133

continued from previous page

f ∗ (s)

f (x) −1 1 2 πs

25

arccot x

26

cosech(ax)

Re a > 0

  a−s 2 1 − 2−s Γ(s) ζ(s),

27

sech2 (ax),

Re a > 0

4a−s (1 − 22−s ) Γ(s)2−s ζ(s − 1),

28 2911

30

31

32

cosech2 (ax), 

x2 + b2

− 12 ν

Re a > 0 +  1/2 , J ν a x2 + b2

+  ⎧ 1  , 2 2 2ν 2 2 1/2 ⎪ a a b − x J − x ⎪ ν ⎨ for 0 < x < a ⎪ ⎪ for x > a ⎩0

0 < Re s < 1 Re s > 1

4a−s Γ(s)2−s ζ(s − 1), 2 2 s−1 a− 2 s b 2 s−ν Γ 1

1

1

1

MS 454

Re s > 2

MS 454

Re s > 2

MS 454

1  2 s J ν−s/2 (ab),

0 < Re s < 2 2 s−1 Γ

MS 454

3 2

+ Re ν

ET I 328

 1  − 1 s ν+ 1 s 2 a ,2 J ν+ 1 s (ab), 2s b 2 Re s > 0

MS 455

Re ν > −1 +  ⎧ 1   , − ν 2 2 2 2 1/2 2 ⎪ a − x J − x b a ⎪ ν ⎨ 1 −1 1 for 0 < x < a 21−ν [Γ(ν)] a 2 s−ν b−, 2 ν s ν−1+ 12 s, 12 s−ν (ab), ⎪ ⎪ for x > a ⎩0 Re s > 0 MS 455

α−s 2s−2 Γ

K ν (αx) − 1 ν βa2 ++ x2 2  1/2 , × K ν α βa2 + x2

 33

sec(πs/2),

Re (α, β) > 0

1

2s

   − 12 ν Γ 12 s + 12 ν , Re s > |Re ν|

  1 1 1 α− 2 s 2 2 s−1 β 2 s−ν Γ, 12 s K

MS 455

1 (αβ), ν− 2 s

Re s > 0

MS 455

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18 The z-Transform 18.1–18.3 Definition, Bilateral, and Unilateral z-Transforms 18.1 Definitions The z-transform converts a numerical sequence x[n] into a function of the complex variable z, and it takes two different forms. The bilateral or two-sided z-transform, denoted here by Zb {x[n]}, is used mainly in signal and image processing, while the unilateral or one-sided z-transform, denoted here by Zu {x[n]}, is used mainly in the analysis of discrete time systems and the solution of linear difference equations. ∞ The bilateral z-transform, Xb (z) of the sequence x[n] = {xn }n=−∞ is defined as ∞  Zb {x[n]} = xn z −n = Xb (z), n=−∞

∞

and the unilateral z-transform Xu (z) of the sequence x[n] = {xn }n=0 is defined as ∞  xn z −n = Xu (z), Zb {x[n]} = n=0

where each has its own domain of convergence (DOC). The series Xb (z) is a Laurent series, and Xu (z) is the principal part of the Laurent series for Xb (z). When xn = 0 for n < 0, the two z-transforms Xb (z) and Xu (z) are identical. In each case the sequence x[n] and its associated z-transform is a called a z-transform pair. The inverse z-transformation x[n] = Z −1 {X(z)} is given by  1 x[n] = X(z)z n−1 dz, 2πi Γ where X(z) is either Xb (z) or Xu (z), and Γ is a simple closed contour containing the origin and lying entirely within the domain of convergence of X(z). In many practical situations, the z-transform is either found by using a series expansion of X(z) in the inversion integral or, if X(z) = N (z)/D(z) where N (z) and D(z) are polynomials in z, by means of partial fractions and the use of an appropriate table of z-transform pairs. In order for the inverse z-transform to be unique, it is necessary to specify the domain of convergence, as can be seen by comparison of entries 3 and 4 of Table 18.2. Table 18.1 lists general properties of the bilateral z-transform, and Table 18.2 lists  some bilateral z-transform pairs. In what 0 for n < 0 follows, use is made of the unit integer function h(n) = , that is, a generalization of 1 for n ≥ 0  1 for n = k the Heaviside step function, and the unit integer pulse function Δ(n − k) = , that is, 0 for n = k a generalization of the delta function. 1135

1136

Definition, Bilateral, and Unilateral z-Transforms

18.2 Bilateral z-transform Table 18.1 General properties of the bilateral z-transform Xb (n) =

∞ 

xn z −n .

n=−∞

Term in sequence

z-Transform Xb (z)

1

αxn + βyn

αXb (z) + βYb (z)

2

xn−N

z −n Xb (z)

3

nxn

−z

4

z0n xn

dXb (z) dz

z Xb z0

DOC of Xb (z) scaled by |z0 | DOC of Xb (z) scaled by |z0 | to which it may be necessary to add or delete the origin and the point at infinity DOC of radius 1/R, where R is the radius of convergence of DOC of Xb (z) DOC of radius 1/R, where R is the radius of convergence of DOC of Xb (z)

5

nz0n xn

dXb (z/z0 ) −z dz

6

x−n

Xb (1/z)

7

nx−n

−z

8

x ¯n

9

Re xn

10

Im xn

Xb (z)   1 2 Xb (z) + Xb (z)   1 2i Xb (z) − Xb (z)

11

∞ 

Domain of Convergence Intersection of DOC’s of Xb (z) and Yb (z) with α, β constants DOC of Xb (z), to which it may be necessary to add or delete the origin or the point at infinity DOC of Xb (z), to which it may be necessary to add or delete the origin and the point at infinity

xk yn−k

dXb (1/z) dz

The same DOC as xn DOC contains the DOC of xn DOC contains the DOC of xn

Xb (z)Yb (z)

k=−∞

12

xn yn

1 2πi

 Xb (ξ)Yb Γ



z ξ −1 dξ ξ

∞ 

13

Parseval formula

1 xn y¯n = 2πi n=−∞

14

Initial value theorem for xn h(n)

x0 = lim Xb (z) z→∞



z Xb (ξ)Yb ξ −1 dξ ξ Γ



DOC contains the intersection of the DOCs of Xb (z) and Yb (z) (convolution theorem) DOC contains the DOCs of Xb (z) and Yb (z), with Γ inside the DOC and containing the origin (convolution theorem) DOC contains the intersection of DOCs of Xb (z) and Yb (z), with Γ inside the DOC and containing the origin

Bilateral z-transform

1137

Table 18.2 Basic bilateral z-transforms

1

Term in sequence

z-Transform Xb (z)

Domain of Convergence

Δ(n)

1

Converges for all z

2

Δ(n − N )

z −n

When N > 0 convergence is for all z except at the origin. When N < 0 convergence is for all z except at ∞

3

an h(n)

z z−a

|z| > |a|

4

an h(−n − 1)

z z−a

|z| < |a|

5

nan h(n)

az (z − a)2

|z| > a > 0

6

nan h(−n − 1)

az (z − a)2

|z| < a,

7

n2 an h(n)

az(z + a) (z − a)3

|z| > a > 0

bz az + az − 1 bz − 1   z 1 − (a/z)N z−a

|z| > max





a>0



8

1 1 + n an b

9

an h(n − N )

10

an h(n) sin Ωn

az sin Ω z 2 − 2az cos Ω + a2

|z| > a > 0

11

an h(n) cos Ωn

z(z − a cos Ω) z 2 − 2az cos Ω + a2

|z| > a > 0

12

ean h(n)

z z − ea

|z| > e−a

13

e−an h(n) sin Ωn

zea sin Ω z 2 e2a − 2zea cos Ω + 1

|z| > e−a

14

e−an h(n) cos Ωn

zea (zea − cos Ω) z 2 e2a − 2zea cos Ω + 1

|z| > e−a

h(n)

|z| > 0

1 1 |a| , |b|



1138

Definition, Bilateral, and Unilateral z-Transforms

18.3 Unilateral z-transform The relationship between the Laplace transform of a continuous function x(t) sampled at t = 0, T , 2T , ∞  x(nT )δ(t − nT ) follows from the result . . . and the unilateral z-transform of the function x ˆ(t) = 

∞

L{ˆ x(t)} = 0

=

∞ 

%

n=0 ∞ 

&

x(kT )δ(t − kT ) e−st dt

k=0

x(kT )e−ksT .

k=0

Setting z = esT , this becomes: L{ˆ x(t)} =

∞ 

x(kT )z −k = X(z),

k=0

showing that the unilateral z-transform Xu (z) can be considered to be the Laplace transform of a continuous function x(t) for t ≥ 0 sampled at t = 0, T , 2T , . . . . Table 18.3 lists some general properties of the unilateral z-transform, and Table 18.4 lists some unilateral z-transform pairs.

Unilateral z-transform

1139

Table 18.3 General properties of the unilateral z-transform Term in sequence

z-Transform Xu (z)

Domain of Convergence

1

αxn + βyn

αXu (z) + βYu (z)

Intersection of DOC’s of Xu (z) and Yu (z) with α, β constants

2

xn+k

z k Xu (z) − z k x0 − z k−1 x1 −z k−2 x2 − · · · − zxk−1

3

nxn

−z

4

z0n xn

Xu

5

nz0n xn

−z

6

x ¯n

Xu (z)

7

Re xn

1 2

8

∂ xn (α) ∂α

∂ Xu (z, α) ∂α

9

Initial value theorem

x0 = lim Xu (z)

10

Final value theorem

DOC of Xu (z), to which it may be necessary to add or delete the origin and the point at infinity DOC of Xb (z) scaled by |z0 |, to which it may be necessary to add or delete the origin and the point at infinity DOC of Xu (z) scaled by |z0 |, to which it may be necessary to add or delete the origin and the point at infinity

dXu (z) dz

z z0



dXu (z/z0 ) dz

The same DOC as xn

  Xu (z) + Xu (z)

DOC contains the DOC of xn Same DOC as xn (α)

z→∞

 lim xn = lim

n→∞

z→1

z−1 z



 Xu (z)

When Xu (z) = N (z)/D(z) with N (z), D(z) polynomials in z and the zeros of D(z) inside the unit circle |z| = 1 or at z = 1

1140

Definition, Bilateral, and Unilateral z-Transforms

Table 18.4 Basic unilateral z-transforms Term in sequence

z-Transform Xu (z)

Domain of Convergence

1

Δ(n)

1

Converges for all z

2

Δ(n − k)

z −k

Convergence for all z = 0

3

an h(n)

z z−a

|z| > |a|

4

nan h(n)

az (z − az)2

|z| > a > 0

5

n2 an h(n)

az(z + a) (z − a)3

|z| > a > 0

6

nan−1 h(n)

z (z − a)2

|z| > a > 0

7

(n − 1)an h(n)

z(2a − z) (z − a)2

|z| > a > 0

8

e−an h(n)

zea zea − 1

|z| > e−a

9

ne−an h(n)

10

n2 e−an h(n)

11

e−an h(n) sin Ωn

12

zea

|z| > e−a

2

(zea − 1)

zea (1 + zea ) 3

(zea − 1)

|z| > e−a

zea sin Ω − 2zea cos Ω + 1

|z| > e−a

e−an h(n) cos Ωn

zea (zea − cos Ω) z 2 e2a − 2zea cos Ω + 1

|z| > e−a

13

h(n) sinh an

z sinh a z 2 − 2z cosh a + 1

|z| > e−a

14

h(n) cosh an

z(z − cosh a) z 2 − 2z cosh a + 1

|z| > e−a

15

h(n)an−1 e−an sin Ωn

zea sin Ω z 2 e2a − 2zaea cos Ω + a2

|z| > e−a

16

h(n)an e−an cos Ωn

zea (zea − a cos Ω) z 2 − 2zaea cos Ω + a2

|z| > e−a

z 2 e2a

Bibliographic References Used in Preparation of Text (See the introduction for an explanation of the letters preceding each bibliographic reference.) AS AD AK BB BE BI BL BR BS BU

BY CA CE CL CO DW

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications, New York, 1972. Adams, E. P. and Hippisley, R. L., Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922. Appell, P. and Kamp´e de F´eriet, Fonctions hyperg´eometriques et hypersph´eriques, polynomes d’Hermite, Gauthier Villars, Paris, 1926. Beckenbach, E. F. and Bellman, R., Inequalities, 3rd printing. Springer–Verlag, Berlin, 1971. Bertrand, J., Traite de calcul diff´erentiel et de calcul int´egral, vol. 2, Calcul int´egral, int´egrales d´efinies et ind´efinies, Gauthier-Villars, Paris, 1870. Bierens de Haan, D., Nouvelles tables d’int´egrales d´efinies, Amsterdam, 1867. (Reprint) G. E. Stechert & Co., New York, 1939. Bellman, R., Introduction to Matrix Analysis, McGraw Hill, New York, 1960. Bromwich, T. I’A., An Introduction to the Theory of Infinite Series, Macmillan, London, 1908, 2nd edition, 1926.∗ Bellman, R., Stability Theory of Differential Equations, McGraw-Hill, New York, 1953. Buchholz, H., Die konfluente hypergeometrische Funktion mit besonderer Ber¨ ucksichtigung ihrer Anwendungen, Springer–Verlag, Berlin, 1953. Also an English edition: The confluent Hypergeometric Function, Springer–Verlag, Berlin, 1969. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer–Verlag, Berlin, 1954. Carslaw, H. S., Introduction to the Theory of Fourier’s Series and Integrals, Macmillan, London, 1930. Ces`aro, Z., Elementary Class Book of Algebraic Analysis and the Calculation of Infinite Limits, 1st ed. ONTI, Moscow and Leningrad, 1936. Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Wiley (Interscience), New York, 1953. Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1934.

∗ The Bibliographic Reference BR* refers to the 1908 edition of Bromwich T. I.’A., An Introduction to the Theory of Infinite Series; BR refers to the 1926 edition.

1141

1142

References

DW61 Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961. Efros, A. M. and Danilevskiy, A. M., Operatsionnoye ischisleniye i konturnyye integraly EF (Operational calculus and contour integrals). GNTIU, Khar’kov, 1937. Erd´elyi, A., et al., Higher Transcendental Functions, vols. I, II, and III. McGraw Hill, New York, EH 1953–1955. Erd´ elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. ET Euler, L., Introductio in Analysin Infinitorum, Bousquet, Lausanne, 1748. EU Fikhtengol’ts, G. M., Kurs differentsial’nogo i integral’nogo ischisleniya (Course in differential FI and integral calculus), vols. I, II, and III. Gostekhizdat, Moscow and Leningrad, 1947–1949. Also a German edition: Differential-und Integralrechnung I–III, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986–1987. Gauss, K. F., Werke, Bd. III. G¨ ottingen, 1876. GA Gel’fond, A. O., Ischisleniye konechnykh raznostey (Calculus of finite differences), part I. ONTI, GE Moscow and Leningrad, 1936. Gr¨ obner, W. and Hofreiter, N., Integraltafel, vol. 2, Bestimmte Integrale, Springer, Wien, 1961. GH2 Giunter, N. M. and Kuz’min, R. O. (eds.), Sbornik zadach po vysshey matematike (Collection of GI problems in higher mathematics), vols. I, II, and III. Gostekhizdat, Moscow and Leningrad, 1947. Gantmacher, F. R., Applications of the Theory of Matrices, translation by J. L. Brenner. Wiley GM (Interscience), New York, 1959. Goursat, E. J. B., Cours d’Analyse, vol. I, Gauthier–Villars, Paris, 1923. GO Gr¨ obner, W. et al., Integraltafel, Teil I, Unbestimmte Integrale, Akad. Verlag, Braunschweig, GU 1944. Gr¨ obner, W. and Hofreiter, N., Integraltafel, Teil II, Bestimmte Integrale, Springer–Verlag, Wien GW and Innsbruck, 1958. Hille, E., Lectures on Ordinary Differential Equations, Addison- Wesley, Reading, HI Massachusetts, 1969. Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities, Cambridge University Press, HL London, 2nd ed., 1952. Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University HO Press, London, 1931. Hurewicz, W., Lectures on Ordinary Differential Equations, MIT Press, Cambridge, HU Massachusetts, 1958. Ince, E. L., Ordinary Differential Equations, Dover, New York, 1944. IN Jahnke, E. and Emde, F., Tables of Functions with Formulas and Curves, Dover, New York, JA 1943. JAC Jackson, J. D., Classical Electrodynamics, Wiley, New York, 1975. James, H. M. et al. (eds.), Theory of Servomechanisms, McGraw Hill, New York, 1947. JE Jolley, L., Summation of Series, Chapman and Hall, London, 1925. JO Kellogg, O. D., Foundations of Potential Theory, Dover, New York, 1958. KE Krechmar, V. A., Zadachnik po algebre (Problem book in algebra), 2nd ed. Gostekhizdat, KR Moscow and Leningrad, 1950. Kuzmin, R. O., Besselevy funktsii (Bessel functions). ONTI, Moscow and Leningrad, 1935. KU Laska, W., Sammlung von Formeln der reinen und angewandten Mathematik, Friedrich Viewig LA und Sohn, Braunschweig, 1888–1894.

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Legendre, A. M., Exercises calcul int´egral, Paris, 1811. Lindeman, C. E., Examen des nouvelles tables d’int´egrales d´efinies de M. Bierens de Haan, Amsterdam, 1867, Norstedt, Stockholm, 1891. Lobachevskiy, N. I., Poloye sobraniye sochineniy (Complete works), vols. I, III, and V. LO Gostekhizdat, Moscow and Leningrad, 1946–1951. Luke, Y. L., Mathematical Functions and their Approximations, Academic Press, New York, LUKE 1975. Lawden, D. F., Elliptic Functions and Applications, Springer–Verlag, Berlin, 1989. LW McLachlan, N. W., Theory and Application of Mathieu Functions, Oxford University Press, MA London, 1947. Computation by Mathematica. MC McLachlan, N. W. and Humbert, P., Formulaire pour le calcul symbolique, L’Acad. des Sciences ME de Paris, Fasc. 100, 1950. Morse, M. P. and Feshbach, H., Methods of Theoretical Physics, vol. I, McGraw Hill, New York, MF 1953. Marden, M., Geometry of Polynomials, American Mathematical Society, Mathematical Survey MG 3, Providence, Rhode Island, 1966. McLachlan, N. W. et al., Suppl´ement au formulaire pour le calcul symbolique, L’Acad. des MI Sciences de Paris et al., Fasc. 113, 1950. Mirsky L., An Introduction to Linear Algebra, Oxford University Press, London, 1963. ML MM MacMillan, W. D., The Theory of the Potential, Dover, New York, 1958. Magnus, W. and Oberhettinger, F., Formeln und S¨ atze f¨ ur die speziellen Funktionen der MO mathematischen Physik, Springer–Verlag, Berlin, 1948. Magnus, W., Oberhettinger, F. and Soni, R. P., Formulas and Theorems for the Special MS Functions of Mathematical Physics, 3rd ed. Springer–Verlag, Berlin, 1966. MS2 Mathai, A. M. and Saxens, R. K. , Generalized Hypergeometrics Functions With Applications in Statistics and Physical Science, Springer–Verlag, Berlin, 1973. Mitrinovi´ c, D. S., Analytic Inequalities, Springer–Verlag, Berlin, 1970. MT Milne, E. A., Vectorial Mechanics, Methuen, London, 1948. MV Meyer Zur Capellen, W., Integraltafeln, Sammlung unbestimmer Integrale elementarer MZ Funktionen, Springer–Verlag, Berlin, 1950. Natanson, I. P., Konstruktivnaya teoriya funktsiy (Constructive theory of functions). NA Gostekhizdat, Moscow and Leningrad, 1949. Nielsen, N., Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906. NH Noble, B., Applied Linear Algebra, Prentice Hall, Englewood Cliffs, New Jersey, 1969. NO Nielsen, N., Theorie des Integrallogarithmus und verwandter Transcendenten, Teubner, Leipzig, NT 1906. Novoselov, S. I., Obratnyye trigonometricheskiye funktsii, posobive dlya uchiteley (Inverse NV trigonometric functions, textbook for students), 3rd ed. Uchpedgiz, Moscow and Leningrad, 1950. Oberhettinger, F., Tables of Bessel Transforms, Springer–Verlag, New York: 1972. OB PBM Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Gordan and Breach, New York, vols. I (1986), II (1986), III (1990). Peirce, B. O., A Short Table of Integrals, 3rd ed. Ginn, Boston, 1929. PE Sansone, G., Orthogonal Functions (Revised English Edition), Interscience, New York, 1959. SA

LE LI

1144

SI

SN SM ST STR SU SZ TF TI VA VL WA WH ZH ZY

References

Sikorskiy, Yu. S., Elementy teorii ellipticheskikh funktsiy s prilozheniyama k mekhanike (Elements of theory of elliptic functions with applications to mechanics). ONTI, Moscow and Leningrad, 1936. Sneddon, I. N., Fourier Transforms, 1st ed. McGraw Hill, New York, 1951. Smirnov, V. I., Kurs vysshey matematiki (A course of higher mathematics), vol. III, Part 2, 4th ed. Gostekhizdat, Moscow and Leningrad, 1949. Strutt, M. J. O., Lam´esche, Mathieusche und verwandte Funktionen in Physik and Technik, Springer–Verlag, Berlin, 1932. Stratton, J. C., Phys. Rev A, 43(3), pages 1381–1388, 1991. Sneddon, I. N., The Use of Integral Transforms, McGraw Hill, New York, 1972. Szeg¨o, G., Orthogonal Polynomials, Revised Edition, Colloquium Publications XXIII, American Mathematical Society, New York, 1959. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, 2nd ed. Oxford University Press, London, 1948. Timofeyev, A. F. Integrirovaniye funktsiy (Integration of functions), part I. GTTI, Moscow and Leningrad, 1933. Varga, R. S., Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1963. Vladimirov, V. S., Equations of Mathematical Physics, Dekker, New York, 1971. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, London, 1966. Whittaker, E. T. and Watson, G. N., Modern Analysis, 4th ed. Cambridge University Press, London, 1927, part II, 1934. Zhuravskiy, A. M., Spravochnik po ellipticheskim funktsiyam (Reference book on elliptic functions). Izd. Akad. Nauk. U.S.S.R., Moscow and Leningrad, 1941. Zygmund, A., Trigonometrical Series, 2nd ed. Chelsea, New York, 1952.

Classified Supplementary References (Prepared by Alan Jeffrey for the English language edition.)

General reference books 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Bromwich, T. I’A., An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926 (Reprinted 1942). Carlitz, L., “Generating Functions”, 1969, Fibonacci Quarterly, 7 (4): 359–393. Copson, E. T., An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, London, 1935. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Interscience Publishers, New York, 1953. Davis, H. T., Summation of Series, Trinity University Press, San Antonio, Texas, 1962. Erd´elyi, A. et al. Higher Transcendental Functions, vols. I to III, McGraw Hill, New York 1953–1955. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. Fletcher, A., Miller, J. C. P., and Rosenhead, L., An Index of Mathematical Tables, 2nd ed., Scientific Computing Service, London, 1962. Gr¨ obner, W. and Hofreiter, N., Integraltafel, I, II. Springer–Verlag, Wien and Innsbruck, 1949. Hardy, G. H., Littlewood, J. E., and P´ olya, G., Inequalities, 2nd ed., Cambridge University Press, London, 1952. Hartley, H. O. and Greenwood, J. A., Guide to Tables in Mathematical Statistics, Princeton University Press, Princeton, New Jersey, 1962. Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics, Cambridge University Press, London, 1956. Jolley, L. B. W., Summation of Series, Dover Publications, New York, 1962. Knopp, K., Theory and Application of Infinite Series, Blackie, London, 1946, Hafner, New York, 1948. Lebedev, N. N., Special Functions and their Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1965. Magnus, W. and Oberhettinger, F., Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea, New York, 1949. McBride, E. B., Obtaining Generating Functions, Springer–Verlag, Berlin, 1971. National Bureau of Standards, Handbook of Mathematical Functions, U.S. Government Printing Office, Washington, D.C., 1964. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vols. 1–5, Gordon and Breach, New York, 1986–1992. Truesdell, C. A Unified Theory of Special Functions, Princeton University Press, Princeton, New Jersey, 1948.

1145

1146

21. 22.

Supplemental References

Vein, R. and Dale, P., Determinants and Their Applications in Mathematical Physics, Springer–Verlag, New York, 1999. Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, 4th ed., Cambridge University Press, London, 1940.

Asymptotic expansions 1. 2. 3. 4. 5. 6. 7.

De Bruijn, N. G., Asymptotic Methods in Analysis, North-Holland Publishing Co., Amsterdam, 1958. Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, 3rd ed., Springer, New York, 1971. Copson, E. T., Asymptotic Expansions, Cambridge University Press, London, 1965. Erd´elyi, A., Asymptotic Expansions, Dover Publications, New York, 1956. Ford, W. B., Studies on Divergent Series and Summability, Macmillan, New York, 1916. Hardy, G. H., Divergent Series, Clarendon Press, Oxford, 1949. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.

Bessel functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Bickley, W. G., Bessel Functions and Formulae, Cambridge University Press, London, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vols. I and II. McGraw Hill, New York, 1954. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed., Macmillan, 1922. McLachlan, N. W., Bessel Functions for Engineers, 2nd ed., Oxford University Press, London, 1955. Luke, Y. L., Integrals of Bessel Functions, McGraw Hill, New York, 1962. Petiau, G., La th´eorie des fonctions de Bessel, Centre National de la Recherche Scientifique, Paris, 1955. Relton, F. E., Applied Bessel Functions, Blackie, London, 1946. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958. Wheelon, A. D., Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day, San Francisco, 1968.

Complex analysis 1. 2. 3. 4. 5. 6. 7.

Ahlfors, L. V., Complex Analysis, 3rd ed., McGraw Hill, New York, 1979. Ahlfors, L. V. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1971. Bieberbach, L., Conformal Mapping, Chelsea, New York, 1964. Henrici, P., Applied and Computational Complex Analysis, 3 vols, Wiley, New York, 1988, 1991, 1977. Hille, E., Analytic Function Theory, 2 vols. 2nd ed., Chelsea, New York, 1990, 1987. Kober, H., Dictionary of Conformal Representations, Dover Publications, New York, 1952. Titchmarsh, E. C., The Theory of Functions, 2nd ed., Oxford University Press, London, 1939. (Reprinted 1975).

Supplemental References

1147

Error function and Fresnel integrals 1. 2. 3. 4. 5.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Erd´elyi, A. et al., Tables of Integral Transforms, vol. I, McGraw Hill, New York, 1954. Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press, London, 1960. Tricomi, F. G., Funzioni ipergeometriche confluenti, Edizioni Cremonese, Turan, Italy, 1954. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.

Exponential integrals, gamma function and related functions 1. 2. 3. 4. 5. 6. 7. 8. 9.

Artin, E., The Gamma Function, Holt, Rinehart, and Winston, New York, 1964. Busbridge, I. W., The Mathematics of Radiative Transfer, Cambridge University Press, London, 1960. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II, McGraw Hill, New York, 1954. Hastings, Jr., C., Approximations for Digital Computers, Princeton University Press, Princeton, New Jersey, 1955. Kourganoff, V., Basic Methods in Transfer Problems, Oxford University Press, London, 1952. L¨ osch, F. and Schoblik, F., Die Fakult¨ at (Gammafunktion) und verwandte Funktionen, Teubner, Leipzig, 1951. Nielsen, N., Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906. Oberhettinger, F., Tabellen zur Fourier Transformation, Springer–Verlag, Berlin, 1957.

Hypergeometric and confluent hypergeometric functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Appell, P., Sur les Fonctions Hyperg´eometriques de Plusieures Variables, Gauthier-Villars, Paris, 1926. Bailey, W. N., Generalized Hypergeometric Functions, Cambridge University Press, London, 1935. Buchholz, H., Die konfluente hypergeometrische Funktion, Springer–Verlag, Berlin, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. I, McGraw Hill, New York, 1953. Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics, Cambridge University Press, London, 1956. Klein, F., Vorlesungen u ¨ber die hypergeometrische Funktion, Springer–Verlag, Berlin, 1933. N¨ orlund, N. E., Sur les Fonctions Hyperg´eometriques d’Ordre Superior, North–Holland, Copenhagen, 1956. Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press, London, 1960. Slater, L. J. Generalized Hypergeometric Functions, Cambridge University Press, London, 1966. Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd ed., National Bureau of Standards, Washington, D.C., 1952. Swanson, C. A. and Erd´elyi, A., Asymptotic Forms of Confluent Hypergeometric Functions, Memoir 25, American Mathematical Society, Providence, Rhode Island, 1957. Tricomi, F. G., Lezioni sulla funzioni ipergeometriche confluenti, Gheroni, Torino, 1952.

Integral transforms 1.

Bochner, S., Vorlesungen u ¨ber Fouriersche Integrale, Akad. Verlag, Leipzig, 1932. Reprint Chelsea, New York, 1948.

1148

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.

Supplemental References

Bochner, S. and Chandrasekharan, K., Fourier Transforms, Princeton University Press, Princeton, New Jersey, 1949. Campbell, G. and Foster, R., Fourier Integrals for Practical Applications, Van Nostrand, New York, 1948. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Oxford University Press, London, 1948. Doetsch, G., Theorie und Anwendung der Laplace-Transformation, Springer–Verlag, Berlin, 1937. (Reprinted by Dover Publications, New York, 1943) Doetsch, G., Theory and Application of the Laplace-Transform, Chelsea, New York, 1965. Doetsch, G., Handbuch der Physik, Mathematische Methoden II, 1st ed., Springer–Verlag, Berlin, 1955. Doetsch, G., Guide to the Applications of the Laplace and Z-Transforms, 2nd ed., Van Nostrand-Reinhold, London, 1971. Doetsch, G., Handbuch der Laplace-Transformation, Vols. I–IV, Birkh¨ auser Verlag, Basel, 1950–56. Doetsch, G., Kniess, H., and Voelker, D., Tabellen zur Laplace- Transformation, Springer–Verlag, Berlin, 1947. Erd´elyi, A., Operational Calculus and Generalized Functions, Holt, Rinehart and Winston, New York, 1962. Exton, H., Multiple Hypergeometric Functions and Applications, Horwood, Chichester, 1976. Exton, H., Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs, Horwood, Chichester, 1978. Hirschmann, J. J. and Widder, D. V., The Convolution Transformation, Princeton University Press, Princeton, New Jersey, 1955. Marichev, O. I., Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood Ltd., Chichester (1982). Oberhettinger, F., Tabellen zur Fourier Transformation, Springer–Verlag, Berlin (1957). Oberhettinger, F., Tables of Bessel Transforms, Springer–Verlag, New York (1972). Oberhettinger, F., Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973. Oberhettinger, F., Fourier Transforms of Distributions and Their Inverses, Academic Press, New York, 1973. Oberhettinger, F., Tables of Mellin Transforms, Springer–Verlag, Berlin, 1974. Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer–Verlag, Berlin, 1973. Oberhettinger, F. and Higgins, T. P., Tables of Lebedev, Mehler and Generalized Mehler Transforms, Math. Note No. 246, Boeing Scientific Research Laboratories, Seattle, Wash., 1961. Roberts, G. E. and Kaufman, H., Table of Laplace Transforms, McAinsh, Toronto, 1966. Sneddon, I. N., Fourier Transforms, McGraw Hill, New York, 1951. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, Oxford University Press, London, 1937. Van der Pol, B. and Bremmer, H., Operational Calculus Based on the Two Sided Laplace Transformation, Cambridge University Press, London, 1950. Widder, D. V., The Laplace Transform, Princeton University Press, Princeton, New Jersey, 1941. Wiener, N., The Fourier Integral and Certain of its Applications, Dover Publications, New York, 1951.

Jacobian and Weierstrass elliptic functions and related functions 1. 2. 3. 4.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer– Verlag, Berlin, 1954. Graeser, E., Einf¨ uhrung in die Theorie der Elliptischen Funktionen und deren Anwendungen, Oldenbourg, Munich, 1950. Hancock, H., Lectures on the Theory of Elliptic Functions, vol. I, Dover Publications, New York, 1958.

Supplemental References

5. 6. 7. 8. 9.

1149

Neville, E. H., Jacobian Elliptic Functions, Oxford University Press, London, 1944 (2nd ed. 1951). Oberhettinger, F. and Magnus, W., Anwendungen der Elliptischen Funktionen in Physik und Technik, Springer–Verlag, Berlin, 1949. Roberts, W. R. W., Elliptic and Hyperelliptic Integrals and Allied Theory, Cambridge University Press, London, 1938. Tannery, J. and Molk, J., El´ements de la Th´eorie des Fonctions Elliptiques, 4 volumes. Gauthier-Villars, Paris, 1893–1902. Tricomi, F. G., Elliptische Funktionen, Akad. Verlag, Leipzig, 1948.

Legendre and related functions 1. 2. 3. 4. 5. 6. 7.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. I, McGraw Hill, New York, 1953. Helfenstein, H., Ueber eine Spezielle Lam´esche Differentialgleichung, Brunner and Bodmer, Zurich, 1950 (Bibliography). Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, London, 1931. Reprinted by Chelsea, New York, 1955. Lense, J., Kugelfunktionen, Geest and Portig, Leipzig, 1950. MacRobert, T. M., Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications, Methuen, England, 1927. (Revised ed. 1947; reprinted Dover Publications, New York, 1948). Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd ed., National Bureau of Standards, Washington, D.C., 1952. Stratton, J. A., Morse, P. M., Chu, L. J. and Hunter, R. A., Elliptic Cylinder and Spheroidal Wave Functions Including Tables of Separation Constants and Coefficients, Wiley, New York, 1941.

Mathieu functions 1. 2. 3. 4.

Erd´elyi, A., Higher Transcendental Functions, vol. III, McGraw Hill, New York, 1955. McLachlan, N. W., Theory and Application of Mathieu Functions, Oxford University Press, London, 1947. Meixner, J. and Sch¨ afke, F. W., Mathieusche Funktionen und Sph¨ aroidfunktionen mit Anwendungen auf Physikalische und Technische Probleme, Springer–Verlag, Heidelberg, 1954. Strutt, M. J. O., Lam´esche, Mathieusche und verwandte Funktionen in Physik und Technik, Ergeb. Math. Grenzgeb. 1, 199–323 (1932). Reprint Edwards Bros., Ann Arbor, Michigan, 1944.

Orthogonal polynomials and functions 1. 2. 3. 4. 5. 6. 7.

Bibliography on Orthogonal Polynomials, Bulletin of National Research Council No. 103, Washington, D.C., 1940. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Interscience, New York, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954. Kaczmarz, St. and Steinhaus, H., Theorie der Orthogonalreihen, Chelsea, New York, 1951. Lorentz, G. G., Bernstein Polynomials, University of Toronto Press, Toronto, 1953. Sansone, G., Orthogonal Functions, Interscience, New York, 1959. Shohat, J. A. and Tamarkin, J. D., The Problem of Moments, American Mathematical Society, Providence, Rhode Island, 1943.

1150

Supplemental References

8.

Szeg¨ o, G., Orthogonal Polynomials, American Mathematical Society Colloquim Pub. No. 23, Providence, Rhode Island, 1959. Titchmarsh, E. C., Eigenfunction Expansions Associated with Second Order Differential Equations, Oxford University Press, London, part I (1946), part II (1958). Tricomi, F. G., Vorlesungen u ¨ber Orthogonalreihen, Springer–Verlag, Berlin, 1955.

9. 10.

Parabolic cylinder functions 1. 2.

Buchholz, H., Die konfluente hypergeometrische Funktion, Springer–Verlag, Berlin, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954.

Probability function 1. 2. 3.

Cramer, H., Mathematical Methods of Statistics, Princeton University Press, Princeton, New Jersey, 1951. Erd´elyi, A. et al., Higher Transcendental Functions, vols. I, II, and III. McGraw Hill, New York, 1953– 1955. Kendall, M. G. and Stuart, A., The Advanced Theory of Statistics, vol. I: Distribution Theory, Griffin, London, 1958.

Riemann zeta function 1. 2.

Titchmarsh, E. C., The Zeta Function of Riemann, Cambridge University Press, London, 1930. Titchmarsh, E. C., The Theory of the Riemann Zeta Function, Oxford University Press, London, 1951.

Struve functions 1. 2. 3.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed., Macmillan, London, 1922. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.

Index of Functions and Constants This index shows the occurrence of functions and constants used in the expressions within the text. The numbers refer to pages on which the function or constant appears.

Symbols

arccot function . . . . . . xxxi, xxxii, 51, 56–58, 64, 242, 244, 245, 263, 274, 279, 325, 326, 499, 556, 561, 599, 601–607, 625, 767, 892, 1115, 1133 arccoth function . . . . . . . . . xxxi, xxxii, 56, 60, 62, 75, 131–134, 172, 177, 178, 241, 644, 647 arcosech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 arcsec function . . . . . . . . . . . . . . . . . 61, 66, 99, 242, 244 arcsech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 arcsin function. . . . . . . . . . . . . . .xxxi, xxxii, 27, 56–61, 64, 66, 94, 97, 99, 116, 124–126, 133–138, 173, 179–182, 186–193, 195, 196, 198, 202–213, 225, 241–245, 254, 263, 265, 272, 275, 279, 297, 307, 382, 558, 563, 566–568, 583, 588, 589, 591, 600, 601, 604, 605, 607, 621, 622, 624, 625, 631, 632, 637, 638, 662, 668, 700–702, 713, 717, 718, 727, 728, 743, 744, 748, 755, 767, 768, 793, 815, 860, 914, 989, 1007 arcsinh function . . xxxi, xxxii, 54, 56, 60–62, 64, 94, 97, 126, 133, 135, 139, 240, 241, 371, 382, 386, 448, 588, 624, 637, 638, 1007 arctan function . . . . . . . . . . xxxi, xxxii, 27, 30, 49, 51, 52, 55–61, 63–67, 71–77, 79, 83–85, 87, 90, 97, 103, 104, 106, 114, 116, 117, 119, 126, 128–133, 147, 148, 171–175, 177, 178, 190, 205, 239, 240, 242–245, 254, 263, 272, 274, 279, 294, 295, 307, 317, 324, 329, 346, 363, 372, 373, 381, 393, 409, 453, 493–501, 507, 509, 516–521, 524, 556, 557, 560, 563–565, 593, 599–607, 612, 622–624, 631, 632, 637, 639, 640, 643, 644, 646–649, 748, 763, 860, 884, 885, 890–893, 898, 923, 1007, 1036, 1113, 1125, 1128, 1132 arctanh function . . . . . . . . . . . . . xxxi, xxxii, 56, 60–62, 64, 75, 79, 97, 125–128, 131, 132, 134, 172, 177, 178, 241, 621–623, 977

! and !! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see factorials (m) Sn . . . . . . . . . . . . see Stirling numbers, second kind (m) Zn . . . . . . . . . . . . . . . . . . . . . . . . see Bessel functions, Z ∇ . . . . . . . . . . . . . . . . . xliv, 767, 1050–1053, 1055–1057 β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see beta function δ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see delta function δij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Kronecker delta γ and Γ . . . . . . . . . . . . . . . . . . . . . . see gamma functions λ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1043 μ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1043 ν function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1043 Φ . . . . . . . . . . . see Lerch function and hypergeometric functions, confluent Ψ . . . . . . . . . . . see Euler function and hypergeometric functions, confluent Θ function . . . . . . . . . . . . . . . see Jacobi theta function ℘(x) . . . . . . . . . . . . . . . . . . . . . . see Weierstrass function ξ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1040 || · || . . . . . . . 1081–1083, 1085, 1086, 1095, 1096, 1120 || · ||1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081–1083 || · ||2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081–1083 || · ||∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081, 1083

A Airy function (Ai) . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii am function . . . . . . . . . . . . . . . . . . . xxxix, 625, 866, 867 Anger function (J) . . . . . xli, 339, 352, 371, 384, 421, 423, 444–446, 670, 671, 946, 948, 949, 992 arccos function . . xxxi, xxxii, 23, 56–60, 64, 99, 135, 139, 173, 179–183, 210, 211, 241–244, 263, 272, 279, 293–296, 307, 312, 313, 318, 393, 454, 511, 558, 562, 589, 600, 601, 607, 610, 624, 695, 730, 742, 767, 768, 861, 890, 936, 993, 994 arccosec function . . . . . . . . . . . . . . . . . . . . . 242, 244, 728 arccosh function . . . . . . xxxi, xxxii, 56, 60–62, 64, 97, 126, 133, 135, 137, 138, 241, 382, 386, 511, 532, 621, 624, 729, 768

1151

1152

INDEX OF FUNCTIONS AND CONSTANTS

associated Legendre functions first kind (P ). . . .xli, 326, 327, 333, 336, 374, 406, 407, 486, 660, 661, 665, 666, 686, 699, 703, 705, 706, 727, 755, 760, 761, 767–789, 792, 793, 797, 806–808, 810, 823, 831, 839, 840, 848, 958–972, 974, 975, 980–983, 992 second kind (Q) . . . . . . xli, 333, 336, 374, 383, 407, 511, 661, 662, 666, 685, 686, 700, 702, 703, 705, 727, 769–781, 783–785, 791, 795, 823, 831, 839, 840, 958–973, 981

B Bn (x) . . . . . . . . . . . . . . . . . . . see Bernoulli polynomials B(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Beta function Bateman function (k) . . . . . . . . . . . . . . . . xli, 349, 1023 bei(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions ber(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions Bernouli number (Bn ) . . xxxii, xxxiii, xxxix, 1–3, 8, 26, 42, 43, 46, 55, 145, 146, 148, 221–224, 353, 356, 376, 379–382, 387, 472, 550–552, 554, 560, 567, 574, 580, 581, 587, 589, 591, 764–766, 899, 906, 936, 1038–1045 Bernouli number (Bn∗ ) . . . . . . . . . . . . . . . . . . . . xxxiii, 62 Bernoulli polynomial (Bn (x)) . . . xxxii, xxxiii, xxxix, 46, 1037, 1041, 1042, 1045 Bessel functions In (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, xli, 13, 320, 339, 340, 345, 347, 350, 351, 368, 382, 385, 419, 435, 441, 444, 445, 470, 477–480, 491, 494, 496, 507, 513–515, 524, 595, 605, 616, 617, 660, 661, 663–681, 684–687, 689, 691, 692, 695–699, 702–716, 719, 720, 722–725, 727, 729, 730, 735, 736, 738, 741, 743, 745–747, 751–760, 762, 779–781, 783–787, 789, 794, 797, 800, 820, 832, 833, 838, 846, 847, 901, 911, 916, 917, 919, 920, 925–933, 943, 954, 1002, 1027, 1028, 1116, 1117, 1123, 1125, 1128 Jn (x) . . . . . . . . . . . xxxvii, xxxviii, xli, 13, 339, 350, 352, 371, 384, 385, 417–421, 423, 435, 440–443, 445, 446, 477–480, 482, 483, 491, 492, 507, 514, 515, 522, 524, 525, 578, 629, 642, 653, 659–694, 696–753, 756–759, 761–763, 767, 768, 777, 779, 780, 782–787, 792–794, 797–799, 802, 803, 808, 811, 812, 818–820, 830–838, 841, 845–848, 854, 855, 900, 910–914, 916, 918–931, 933–950, 954– 957, 963, 964, 972, 992, 1000, 1002–1004, 1017, 1023–1025, 1028, 1034, 1116, 1117, 1124–1129, 1132, 1133

Bessel functions (continued) Kn (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, xli, 2, 337, 339, 345–348, 350–353, 364–368, 370, 371, 384, 385, 417, 419, 435, 442, 444, 445, 477–482, 490, 491, 504, 505, 507, 514, 515, 529, 573, 575, 576, 578, 595, 638, 645, 648, 653, 654, 657, 660– 682, 684–696, 698–700, 702–716, 718–724, 726, 727, 729–732, 735, 736, 738, 740, 742, 745–753, 756–759, 761, 768, 776–787, 789, 794, 800, 803, 811, 814, 817–820, 828, 832–834, 837, 838, 841, 845, 846, 848, 854, 855, 900, 911, 917–920, 923, 925–933, 939, 942, 945, 955, 957, 1027, 1028, 1035, 1123–1126, 1129, 1132, 1133 Nn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii, 910 Yn (x) . . . . . . . . . . . . . xxxvii, xlii, 338, 339, 345, 346, 351–353, 371, 384, 385, 419, 435, 440, 442, 443, 445, 446, 477–480, 482, 483, 492, 507, 514, 515, 573, 578, 647, 654, 659–664, 666–682, 684–693, 695–700, 705–709, 711, 714–720, 722–724, 726, 728, 729, 732–736, 738, 740–742, 745, 747–752, 756–759, 761, 767, 768, 777, 779, 782, 783, 785, 787, 793, 794, 799, 817–819, 832, 833, 835–837, 847, 848, 854, 855, 910, 911, 914, 918–920, 922, 923, 925–931, 933, 937–939, 941–943, 945, 946, 949, 954–957, 975, 1025, 1034, 1124, 1126, 1132 Zn (x). . . . . . xlii, 483, 629, 630, 767, 910, 911, 926, 931–933, 937, 940, 941, 975 Zn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii, 629, 630 Hankel . . . . . . . . . . . . . . . . . . . . . . see Hankel function beta function (β) . . . . . . . . . . . 319, 322, 324, 325, 334, 371, 375, 383, 395, 396, 403, 432, 471, 553, 558, 562–564, 573, 586, 602, 904, 906, 907 Beta function (B) . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 6, 129, 175, 315–318, 320, 322–330, 332, 333, 335, 338, 347–349, 351, 359, 360, 364, 368, 370, 372, 374, 375, 382, 383, 395–397, 399–402, 407, 408, 411–413, 440, 442, 444, 460, 469, 472, 485, 486, 490, 512, 539–543, 548, 553, 559, 585, 705, 749, 754, 760, 801, 810, 813, 814, 816, 821, 894, 895, 908–910, 991, 1005, 1023, 1025 Bi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii bilateral z transform . . . . . . . . 1135–1137, 1139, 1159 binomial coefficients . . . . . . . . . . . . . . . . . . . . xxxiii, xliii, 1–6, 11, 12, 15, 22, 23, 25, 31, 33, 46, 84, 86, 88, 89, 100–103, 106, 110, 111, 114, 115, 119, 120, 140, 143, 148, 153, 157, 161, 173, 215, 220, 221, 223, 228, 232–238, 241, 316, 326, 329, 354, 357, 361, 362, 386, 393, 394, 397–402, 416, 431, 437, 459–461, 466, 469, 470, 478, 488, 498, 499, 504, 505, 545, 546, 548, 549, 552, 612, 808, 910, 934, 993, 994, 996, 998, 1003, 1023, 1030, 1040, 1041, 1044, 1046, 1047

INDEX OF FUNCTIONS AND CONSTANTS

C Cn (x) . . . . . . . . . . . . . . . . . see Gegenbauer polynomials C(x) . . see Fresnel sine integral and Young function Catalan constant (G) . . . . . . . . . xxxii, xl, 9, 375, 380, 433, 434, 448, 449, 452, 453, 470–472, 530–534, 536–538, 556, 558, 560, 563, 564, 580, 600–603, 632, 633, 1046 cd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Ce function . . . . . . . . . . . xxxviii, xl, 763–766, 953, 954 ce function . . . . . . . . . . . . xxxviii, xl, 763–767, 951–957 Chebyshev polynomials first kind (Tn (x)) . . . . . . . . . xli, 448, 667, 718, 790, 800–803, 983, 988, 993–996, 999, 1131 second kind (Un (x)) . . . . . . . . . xxxvii, xli, 800–802, 994–996 chi function . . . . . . . xxxvi, xl, 142–144, 644, 645, 886 ci function . . . . xxxv, xl, 219–221, 340–344, 423–426, 436, 437, 447, 495, 505, 506, 528, 529, 571, 572, 578, 581, 594, 595, 599, 605, 628, 629, 638–645, 647, 656, 658, 748, 762, 886, 887, 1115 Cin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi cn function . . . . . . . . . . xxxiii, xxxiv, xl, 623–626, 714, 866–873, 875, 879, 880 complex conjugate . . . . . . . . . . . . . . xliii, 293, 341, 342, 421, 422, 424–426, 511, 528, 529, 927, 931, 933, 1060, 1061, 1070, 1071, 1082, 1087, 1136, 1139 confluent hypergeometric functions . . . . . . . . . . . . . see hypergeometric functions, confluent constants Catalan . . . . . . . . . . . . . . . . . . . . see Catalan constant Euler . . . . . . . . . . . . . . . . . . . . . . . . . see Euler constant cos function . . . . . . . . . . xxix, xxxvi, xxxviii, 4, 13, 19, 20, 26–39, 41–52, 54–56, 64, 74–76, 78, 79, 126, 147, 151–237, 249, 250, 253, 254, 317, 318, 320, 322, 323, 326–329, 331–333, 338–345, 353, 358, 372, 373, 377–383, 385, 388–534, 537, 540–545, 550, 551, 561, 563, 565, 567–573, 576, 578, 579, 581–601, 604–608, 610–612, 616, 621–623, 628, 629, 631, 632, 634–644, 647–650, 652, 655, 656, 658, 659, 662–664, 667, 669, 671, 673–675, 677– 681, 686, 688, 689, 691, 692, 695, 703–707, 709, 711, 713, 715, 717–748, 750–752, 754, 756–770, 779–782, 784, 785, 787–789, 792–794, 797–800, 802, 806, 808, 811, 812, 815, 817, 818, 823, 825, 829–831, 833, 836, 837, 844, 845, 847–849, 854, 862–869, 877–880, 882–886, 888–894, 896, 898– 900, 904, 906–910, 912–918, 920, 922, 924, 925, 928, 930, 933–943, 945–951, 953, 954, 958–964, 966–981, 984–993, 997, 998, 1000, 1003, 1005– 1007, 1023, 1025, 1026, 1029, 1030, 1037, 1038, 1041, 1042, 1044, 1053, 1057, 1060, 1061, 1066,

1153

1067, 1090, 1110–1113, 1115, 1117, 1119–1122, 1124–1128, 1131, 1132, 1137, 1140 cosec function . . xxvii, xxix, 36–39, 43, 44, 49, 50, 64, 113–115, 126, 155, 156, 160, 225, 254, 315–323, 325, 327–332, 334, 335, 339, 349, 352, 354, 355, 358, 373, 381–383, 387, 388, 400, 401, 403–408, 410–414, 421, 422, 434, 437–439, 446, 453, 454, 471, 479, 480, 484, 496, 508, 509, 529, 540, 541, 547, 551, 565, 586, 588, 593, 600, 602, 604, 658, 659, 663, 664, 669, 670, 677–682, 689–692, 695, 697, 700, 713, 715, 717, 718, 723, 724, 755, 760, 761, 780, 787, 788, 844, 853, 869, 900, 921, 927, 929, 930, 932, 964, 967, 1131, 1132 cosech function . . . . . . 27, 43, 113–115, 126, 387, 501, 513, 634, 715, 751, 1111, 1124, 1133 cosh function . . . . . . . . . . . . . . xxviii, xxxvi, 27–36, 38, 42, 43, 45, 47, 48, 50–52, 64, 110–151, 231–237, 251, 323, 338, 339, 371–390, 407, 419, 425, 432, 433, 439, 448, 451, 452, 454, 455, 468, 482, 484, 485, 502, 504, 509–527, 568, 570, 573, 578–581, 595, 596, 605, 610, 611, 621, 622, 634, 643–645, 648, 686, 702, 705, 710, 713–717, 722, 723, 729, 735, 747, 750–752, 755, 760, 763–767, 778, 781, 787–789, 791, 792, 802, 806, 843, 844, 886, 888, 896, 906, 908, 909, 912–917, 921, 922, 952–958, 960–963, 967–969, 973, 977, 980–982, 998, 1006, 1043, 1112–1115, 1119, 1128, 1129, 1140 cosine integral (Ci) . . . . xxxv, xxxvi, 886, 930, 1115, 1122, 1126 cot function . . xxvii, xxviii, 28, 36, 37, 39, 42, 44, 46, 49, 52, 56, 64, 147, 157–161, 168, 174, 176–178, 185, 188, 189, 194, 195, 204–207, 213, 222–225, 229, 230, 274, 318–323, 330, 331, 334, 354, 355, 358, 372, 379, 381–384, 388, 395, 396, 400, 401, 403–406, 411–414, 422, 434, 448, 454, 455, 472, 484, 485, 492, 493, 496, 506, 509, 511, 532–534, 540, 542, 543, 546, 558, 559, 565, 567, 568, 587, 588, 590, 594, 595, 604, 606, 611, 623, 631, 632, 637, 663, 664, 669, 670, 676, 678–681, 690–692, 697, 700, 717, 723, 753, 754, 849, 867, 868, 876, 882, 903–905, 912, 914, 915, 922, 927, 929, 930, 932, 954, 967–969, 971, 979, 989, 1013, 1089, 1090, 1120 coth function . . . . . . . 28, 39, 40, 42, 44, 64, 110, 116, 118–120, 124, 130–134, 138, 145–148, 381, 384, 386–388, 485, 489, 502, 509–513, 520, 523, 580, 582, 595, 622, 634, 715, 716, 876, 921, 922, 956, 957, 967, 981, 1029, 1110, 1124 cs function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii curl . . . . . . . . . . . . . . . . . . . . . . . . . 1050–1053, 1057, 1058 cylinder function . . . . see parabolic cylinder function

1154

INDEX OF FUNCTIONS AND CONSTANTS

D dc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263–265 delta function (δ(x)) . . . . . . . . . . 661, 1115, 1118–1120 determinant . . . . . 1070, 1075–1077, 1084–1086, 1096, 1100 dilogarithm (L2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 div . . . . . . . . . . . . . . 1050, 1051, 1053, 1055, 1057, 1058 dn function . . . . . . . . . xxxiii, xxxiv, xl, 623–626, 714, 866–873, 875, 879, 880 double factorials . . . . . . . . . . . . . . . see factorial, double ds function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii

E En (x) . . . . . . . . . . . . . . . . . . . . . . . see Euler polynomials E(x) . . . . . . . . . . . . . . . . . . . . . . see MacRobert function elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 860 complete . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 860, 861 E . . . . . . . . . . . . . . . . xxxi, xl, 60, 135–139, 179–183, 185–195, 197, 198, 200, 202–214, 225, 255–262, 264–274, 280–283, 285–296, 300–315, 410, 604, 606, 621–623, 625, 632, 672, 777, 814, 852, 853, 855, 857, 860, 862–864, 880 complete . . . . . . . . . . xxxiv, xl, 313, 394, 408–410, 472–475, 562, 592, 593, 596–600, 619–622, 632, 633, 671, 696, 704, 714, 729, 860–865, 869, 880, 990 F . . . . . . . xxxi, 60, 61, 134–139, 179–183, 186–195, 197–200, 202–214, 225, 250, 251, 254–315, 407, 408, 410, 411, 470, 486, 563, 568, 602, 604, 606, 621–623, 631, 632, 654, 660, 661, 683, 684, 687, 699, 700, 704, 707, 730, 731, 733, 734, 758, 804, 809–820, 822, 823, 843, 849, 856, 860–864, 889, 918, 959, 963, 964, 968, 970, 971, 974–976, 979, 981, 982, 984, 986, 991, 994 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 complete . . . . . . . . . . . . . . xxxiv, xli, 274, 275, 313, 394, 408–410, 472–475, 538, 539, 562, 567, 585, 588, 592, 593, 596–600, 611, 612, 619, 620, 622, 631–633, 696, 704, 713, 714, 719, 720, 729, 756, 788, 860–870, 875, 879–881, 990 Π . . . . . . . xxxi, xxxiv, xxxix, 51, 52, 135, 137, 138, 180, 181, 183, 197, 198, 200–202, 204, 205, 210, 212–214, 255, 262, 263, 265, 276–279, 284, 285, 293, 295–300, 605, 620, 623, 632, 860, 880 erf . . . . . . . . . . . xxxvi, xl, 107–109, 336, 365, 635, 646, 887–889, 1115, 1116 erfc . . . . . . . xxxvi, xl, 887, 888, 890, 891, 1108, 1110, 1115, 1116 error functions . . . . . . . . . . . . . . . . . . . . . see erf and erfc

Euler constant (C ) . . . . xxxii, xxxv, xxxvi, xl, 3, 15, 321, 323, 330–332, 334, 335, 359, 361, 362, 364, 367, 369–371, 411, 412, 422, 447, 476, 478, 483, 484, 501, 534, 535, 538, 541, 553–555, 558, 559, 570–574, 578–581, 585, 586, 588, 593, 594, 599, 605, 628, 639, 644, 656, 658, 747, 748, 883, 884, 886, 894–896, 898, 903–906, 911, 919, 937–939, 944, 1037, 1038, 1046, 1125 Euler function (ψ) . . . . . . . . . . . . xxxix, 318, 321, 323, 330–332, 334, 335, 356, 359, 360, 364, 369, 382, 387, 388, 400, 403, 405, 411–414, 466, 486, 490, 501, 509, 510, 523, 535–538, 540–543, 553–555, 558, 559, 562, 570–574, 576–579, 585, 586, 588, 594, 595, 607, 617, 658, 659, 747, 769, 770, 820, 842, 902–907, 911, 919, 929, 969, 1011–1013 Euler number (En ) . . . . . . . . . . . . . . . . . . . . xxxii, xxxiii, xl, 8, 43, 145, 146, 221, 223, 376, 379, 380, 533, 550, 580, 1043–1045 Euler polynomial (En (x)) . . xxxii, xxxiii, 1044, 1045 exponential function (exp) . . . . . . . . . . . . . . . . . . . . . . 27, 108, 109, 143, 144, 215, 335–340, 345–349, 352, 364–371, 383–385, 390, 426, 428–430, 439, 440, 480, 482, 484–486, 488, 489, 492–498, 501–509, 513, 516, 521, 523, 524, 526, 574–576, 581, 617, 639, 645–649, 651, 652, 655–658, 693, 697–699, 706–710, 712, 713, 722, 723, 748–753, 759, 760, 768, 778, 781, 785, 791, 805, 810, 811, 815, 826, 828, 829, 831, 834, 837, 841–844, 848–850, 855, 876, 877, 879, 880, 886–888, 890–893, 895, 896, 913, 915–917, 920–923, 928, 933, 935, 967, 997, 1002, 1024–1026, 1029, 1030, 1066, 1090, 1094, 1096, 1098, 1099, 1105, 1110, 1119, 1123, 1124, 1128 exponential integral (En (x)) . . . . . . . . . . . . . . . . . . xxxv exponential integral (Ei(x)) . . . . . . . . . . . . . . . . . . xxxv, xl, 107, 109, 143, 144, 150, 151, 338, 340–344, 361, 370, 375, 386, 421, 422, 424–426, 432, 461, 468, 483, 484, 492, 495, 527–530, 535, 553, 555, 571–573, 577, 578, 594, 595, 605–607, 627, 628, 638–647, 649, 656, 658, 748, 883–887, 900, 902, 931, 1115, 1123, 1125

F

F . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Fourier transform F (x) . . . . . . . . . . . . . . . . . . see hypergeometric function

INDEX OF FUNCTIONS AND CONSTANTS

factorial ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii–xxxiv, xxxvii, xliii, 2–5, 8, 12, 13, 18, 19, 22, 23, 25–27, 33, 34, 42–44, 46, 49–51, 54, 55, 60–62, 66, 68, 77, 79, 85, 90, 92, 94, 106–109, 114, 115, 127, 128, 140–143, 145, 146, 148, 152, 155, 156, 163–167, 174, 215–219, 221–224, 228, 230, 241, 315, 316, 321, 323, 325, 326, 328–330, 332, 336, 340–342, 344, 346, 353, 354, 359, 361, 364, 365, 367, 379, 381, 382, 386, 389, 393, 396–398, 400–402, 405, 408, 417, 419, 430, 431, 436, 437, 441, 444, 466, 469, 470, 472, 486–488, 495–499, 502–505, 508, 510, 512, 517, 522, 528, 530, 531, 533, 535, 550– 552, 555, 559, 560, 567, 572–578, 580, 585–587, 591, 593, 601, 607, 612, 613, 616, 619, 627, 635, 636, 660, 672, 677, 687, 688, 698, 704, 705, 707, 709, 725, 769–771, 789–793, 795–801, 803–812, 840–842, 844, 860–862, 866, 867, 869, 884–886, 889, 892, 893, 895–897, 899–901, 904, 907, 909– 911, 913, 918–921, 923–925, 929, 930, 934–936, 940, 941, 944, 949, 950, 961, 962, 968, 973–975, 977, 979, 982–984, 986, 988–993, 995, 997–1002, 1005, 1010–1013, 1018, 1022, 1023, 1025–1027, 1031, 1034, 1038–1044, 1046, 1047, 1074, 1108, 1109, 1120 double (!!) . . . . . . . . . . . . . . . . . . . . . . . xliii, 23, 77, 79, 94, 110, 111, 113, 114, 127, 128, 146, 152, 155, 156, 174, 222, 226, 245, 250, 316, 319, 324–326, 336, 345, 346, 363, 364, 367, 369, 395–398, 401, 402, 405, 408–410, 420, 430, 435, 459, 460, 466, 467, 469, 470, 472, 478, 488, 531, 538, 539, 543, 551, 573–576, 585, 586, 601, 607, 616, 793, 805, 860–862, 889, 897, 909, 923, 934, 940, 974, 977, 982, 984, 986, 988–990, 992, 994, 997 Fe function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953–955 fe function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953–955 Fek function . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 955, 957 Fey function . . . . . . . . . . . . . . . . . . xl, 765, 767, 955, 956 Fourier transform . . . . . . xliv, 1117, 1118, 1121, 1122, 1129 cosine. . . . . . . . . . . . . . . .xliv, 1121, 1122, 1126–1129 sine. . . . . . . . . . . . . . . . . . . . . . . . xliv, 1121–1125, 1129 Fresnel integral cosine (C) . . . . . xxxvi, xl, 171, 225, 226, 415, 434, 475, 476, 492, 629, 641, 649, 650, 659, 887–890, 935, 1126 sine (S) . . xxxvi, xli, 170, 171, 225, 226, 415, 434, 475, 476, 492, 629, 641, 649, 650, 659, 887–890, 935, 1057, 1124

1155

G

a1 ,... Gpq nm (x | b1 ,... ) . . . . . . . . . . . . . . . see Meijer G function gamma function Γ(x) . . . . . xxxiv, xxxvii–xxxix, xliii, 6, 9, 68, 107– 109, 121–123, 163–167, 264, 296, 317, 318, 321, 322, 324, 326–333, 336–338, 346–355, 358–361, 365–368, 370, 374, 376, 377, 379–384, 386–390, 395, 396, 398–401, 406, 407, 411, 414, 419, 421, 423, 436–445, 459, 460, 462, 466, 472, 479, 486, 491, 492, 497–499, 503, 506, 509, 511, 512, 515, 521–523, 529, 535, 538, 539, 545–548, 550–553, 555, 560, 566–568, 570–574, 576–580, 585, 588, 594, 595, 602, 604, 605, 613–617, 632–640, 645, 646, 648–663, 665–668, 670, 672–688, 690–694, 696–700, 702–712, 715–717, 724–727, 730–734, 736–738, 741, 744–749, 752–761, 769–789, 791– 801, 803–853, 856, 864, 889, 892–902, 904, 909, 910, 912–921, 923, 929, 940–949, 959–964, 966– 975, 978, 979, 981–983, 991–993, 995, 999, 1002, 1003, 1005, 1008, 1009, 1013, 1019–1030, 1032, 1033, 1035–1040, 1043, 1046, 1048, 1056, 1108, 1110, 1113, 1116, 1117, 1119, 1120, 1122, 1123, 1126, 1127, 1130–1133 γ(x) . . 215, 335, 338, 340, 346, 347, 370, 440, 492, 496, 639, 657, 677, 706, 899, 902, 1027 incomplete (Γ(x, y)) . . . . . . . . xxxix, 215, 338, 340, 346–348, 352, 366, 368, 436, 438, 498, 576, 657, 658, 710, 749, 787, 899–902, 1002, 1027, 1110 incomplete (γ(x, y)) . . . . . . . . . xxxix, 439, 899–902 gd(x) . . . . . . . . . . . . . . . . . . see Gudermannian function Ge function . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953–955 ge function . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953, 955 Gegenbauer polynomial (Cn (x)) . . . . . . . xl, 327, 406, 795–800, 927, 940, 941, 969, 983, 990–993, 995, 997, 999, 1017 Gek function . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 955, 957 Gey function . . . . . . . . . . . . . . . . . . . . . . xl, 765, 955–957 grad . . . . . . . . . . . . . . . . . . 1050, 1051, 1053, 1055, 1056 Gudermannian (gd) . . . . . . . . . . . . . . . . . xl, 52, 53, 116

H H function . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 879, 880 Hn (x) . . . . . . . . . . . . . . . . . . . . see Hermite polynomials H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . see Struve function H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . see Hankel function H(x) . . . . . . . . . . . . . . . . . . . . . . . . see Heaviside function Hankel function (Hn (x)) . . . . . . . . . . . . . . . . . . . . xxxvii, xli, 339, 350, 351, 368, 370, 385, 492, 653, 663, 688, 691, 693–695, 698, 702, 709, 723, 750, 752, 753, 768, 778, 789, 850, 910, 911, 914–916, 920, 922, 923, 925–928, 931, 940, 944 Hen (x) . . . . . . . . . . . . . . . . . . . see Hermite polynomials

1156

INDEX OF FUNCTIONS AND CONSTANTS

Heaviside Function (H(x)) . . . . . xliv, 642, 750, 1115, 1118, 1131 heiν (z) . . . . . . . . . . . . . . . . . . . . . see Thomson functions herν (z) . . . . . . . . . . . . . . . . . . . . . see Thomson functions Hermite polynomials Hn (x) . . . . . . xxxvi, xxxvii, xli, 365, 503, 803–806, 810–812, 983, 992, 996–998, 1001, 1030 Hen (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi, xxxvii Hermitian . . . . . . . . . . . . . . xliv, 1070, 1071, 1082, 1083 hyperbolic cosine integral . . . . . . . . . . . . . . . . . . . see chi function sine integral . . . . . . . . . . . . . . . . . . . . . see shi function hypergeometric functions F . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, xl, 315–318, 320, 327, 329, 330, 335, 347–349, 351, 368, 370, 374, 375, 394, 398, 436, 438–440, 442, 444, 488, 490, 503, 512, 517, 639, 646, 648, 654, 657, 663, 670, 671, 673, 677–681, 683, 685, 688, 690, 699, 703, 704, 706, 707, 711, 712, 736, 737, 745, 749, 754, 755, 759, 760, 771–776, 779, 780, 784, 791, 792, 794–797, 801, 803, 805, 807–810, 813–818, 821–824, 826–835, 838, 841, 844, 846, 848, 849, 889, 910, 946, 982, 999, 1005–1013, 1015–1023, 1025, 1033, 1035, 1037, 1039 confluent (Φ) . . . . . . xxxix, 1022–1024, 1027, 1028, 1030–1032 confluent (Ψ) . . 816, 1023, 1027, 1028, 1038, 1039

I incomplete beta function I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 910 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 910, 1132 incomplete Gamma function . . see gamma function, incomplete inverse functions . . . . . . . . . . . . . . . . . . . . . . . . 1118, 1121

J Jacobi elliptic functions . . . see cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, sn Jacobi polynomial (pn (x)) . . . . . . xli, 998–1000, 1003 Jacobi theta function (Θ) . . . . xxxiv, xxxix, 879, 880 Jacobi zeta function (zn) . . . . . . . . . . . . . . . . . . . . . xxxiv

K kei(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions ker(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions K  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 k  . . . . . . . . . . . . . . . . . . . . . . . . . . xliv, 134, 135, 184–200, 204, 206, 225, 263, 410, 472–475, 562, 567, 568, 585, 588, 592, 593, 596–602, 604–606, 619–626, 631–633, 859–868, 870–873, 875, 879, 881, 1006 Kronecker delta . . . . . . . . xliv, 1046, 1047, 1076, 1088

L

L. . . . . . . . . . . . . . . . . . . . . . . . . . . . see Laplace transform L2 (x) . . . . . . . . . . . . . . . . . . . . . see dilogarithm function Ln (x) or Lα n (x) . . . . . . . . . . . see Laguerre polynomials L(x) . . . . . . . . . . . . . . . . . . . . see Lobachevskiy function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . see Struve function Laguerre function (Lα n (x)) . . . . 348, 441, 707, 709, 803–806, 808–812, 840, 901, 983, 1000–1004, 1028 polynomial (Ln (x)) . . xli, 344, 806, 808, 809, 811, 812, 844 Laplace transform . . . . . . xliv, 1107, 1108, 1129, 1138 Legendre functions first kind (Pn (x)) . . . . . xli, 93, 106, 327, 390, 405, 406, 409, 513, 612, 698, 707, 719, 769–772, 774, 776–782, 785, 786, 788–794, 801, 809, 815, 829, 933, 936, 940, 941, 959–961, 963–969, 972–990, 992, 999, 1017, 1131 second kind (Qn (x)) . . . . . . xli, 324, 373, 383, 696, 719, 769–771, 773, 777, 780, 785, 788, 790, 791, 959, 960, 965, 966, 968, 972, 973, 975–981, 986 Lerch function (Φ) . . . . . . . . . . . . . . . . xxxix, 642, 1039 li function . . . xxxv, xli, 238, 340, 527, 553, 636, 637, 883, 884, 887, 902, 1027 limit . . . . . . . xxxii, 6–8, 14, 21, 26, 53, 250–252, 511, 610, 611, 617, 635, 883, 887, 890, 894, 895, 904, 905, 931, 951, 963, 992, 1000, 1003–1006, 1023, 1038–1040, 1067, 1101, 1104, 1106–1108, 1118, 1121, 1130, 1136, 1139 ln function . . . . xxvii–xxix, xxxi, xxxii, xxxiv–xxxvii, 3, 9–11, 23, 26, 27, 43, 44, 46, 47, 49, 51–56, 61–67, 69–85, 87, 90, 94, 97, 99, 103, 104, 106, 113–120, 123–130, 133, 143, 145–148, 150, 155– 161, 167–172, 174–176, 178, 186–197, 199, 200, 204–207, 220–225, 237–245, 250, 316, 321, 324, 326, 330–332, 334, 338–340, 353–364, 369–373, 375, 376, 378–381, 383, 386–390, 395, 402, 410, 431, 433, 434, 438, 447–449, 451–457, 462–466, 470–473, 483–485, 495, 497, 499–502, 517–521, 527–607, 622–628, 631–633, 636–645, 647–649, 656, 658, 659, 661, 668, 671, 672, 695, 702, 718, 719, 728, 747, 748, 755, 763, 861, 862, 868, 880, 882–887, 891–893, 895, 898–900, 902–907, 909, 911, 914, 919, 929, 937–939, 944, 963, 969, 972, 977–979, 981, 982, 990, 1006, 1011–1013, 1026, 1027, 1037–1040, 1046, 1048, 1056, 1113, 1115, 1120, 1123, 1125, 1128, 1132, see log function Lobachevskiy function (L) . . . xli, 147, 225, 375, 380, 381, 530–534, 588, 589, 593, 891 log function . . . . . . . . . . . . . . . . 27, 642, see ln function

INDEX OF FUNCTIONS AND CONSTANTS

Lommel function (S) . . . . . . xxxvi, xli, 339, 346, 352, 371, 384, 386, 417, 670, 674, 676–678, 680, 681, 756, 758, 760, 761, 779, 782, 783, 785, 787, 788, 794, 815, 816, 819, 828, 945–947, 950, 1035 Lommel function (s) . . . . xli, 419–421, 439, 443, 670, 692, 725, 760, 761, 945, 946, 1133 Lommel function (U) . . . . . . . . . . . . . . . . . . . . . . xlii, 947 Lommel function (V) . . . . . . . . . . . . . . . . . xlii, 947, 948

M

M . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Mellin transform Mλ,μ (z) . . . . . . . . . . . . . . . . . . . see Whittaker functions MacRobert function (E) . . . . . . . . . . . . . . . . 1035, 1036 Mathieu functions Se . . . . . . . . . . . . . . . . . xxxviii, xli, 764–766, 953, 954 se . . . . . . . . . . . . . . . . . xxxviii, xli, 763–766, 951–957 max . . . . . 851, 854, 856, 987, 1066, 1081–1086, 1088, 1091 Meijer function (G) . . . . . xl, 351, 444, 654, 690, 691, 704, 711, 758, 776, 778, 817–819, 825–832, 835, 838, 844, 845, 847, 850–856, 1032–1035 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . xliv, 1130 min . . . . . . . . . . . . . . . . . . . . . . 851, 854, 856, 1085, 1091

N Nν (z) . . . . . . . . . . . . . . . . . . . . . . . see Bessel function, Y nc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii nd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Neumann function . . . . . . . . . . . see Bessel function, Y Neumann polynomial (On (x)) . . . . . . xxxvii, xli, 346, 384, 386, 946, 949, 950 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . see || · || and || · ||p ns function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii

O On (x) . . . . . . . . . . . . . . . . . . . see Neumann polynomials orthogonal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798

P Pn (x) . . . . . . . . see Jacobi polynomials and Legendre polynomials Pn (x) . . . . . . . . . . . see Legendre functions (first kind) Pnm (x) . . . . . see Legendre functions (associated, first kind) parabolic cylinder function (D) . . . . . . . . . . xxxviii, xl, 348, 349, 352, 365, 384, 390, 503, 504, 506, 653, 657, 658, 697, 708, 712, 740, 746, 802, 805, 811, 841–850, 1028–1031

1157

Phi function (Φ) . . . . . . . . . . . . . . . . . . . . . . xxxvi, xxxix, xl, 239, 336–338, 344, 345, 353, 354, 358, 364, 367, 371, 376, 379, 381, 384, 390, 489, 503, 504, 526, 574, 604, 629, 640, 645–649, 748, 749, 755, 781, 802, 835, 838, 887–891, 899, 902, 997, 998, 1001, 1050, 1051, 1056, 1057 Pochhammer symbol . . . . . . . . . . . . xliii, 321, 330, 635, 672, 705, 900, 918, 947, 1010–1012, 1018, 1022, 1031, 1048 polynomials . . . . . . . . . . . . . . . . . . . . . . see specific name principal value (PV). . .xliii, 322, 329, 335, 337, 433, 454, 528, 534, 563, 572, 883

Q Qn (x) . . . . . . . . see Legendre functions (second kind) Qm n (x) . . . see Legendre function (associated, second kind)

R root . . . . . . . . . . . . 15, 84, 104, 331, 539, 542, 553, 576 3 . . . . 72, 86–88, 264, 330, 331, 363, 364, 539, 570, 918 4. . . . . . . . .73, 78, 83, 105, 135, 136, 139, 210, 211, 263–265, 272, 295, 296, 312–315, 483, 493–495, 507, 524, 525, 868, 875, 878–881, 997 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894 rot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050

S Sn (x) . . . . . . . . . . . . . . . . . . . . . . see Schlafli polynomials (m) Sn . . . . . . . . . . . . . . . . see Stirling numbers, first kind s(x) . . . . . . . . . . . . . . . . . . . . . . . . . . see Lommel function S(x) . . . . . . . see Lommel function and Fresnel cosine integral sc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Schlafli polynomial (Sn (x)) . . . . . . . . . . . . xli, 949, 950 sd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Se(x) . . . . . . . . . . . . . . . . . . . . . . . . see Mathieu functions se(x) . . . . . . . . . . . . . . . . . . . . . . . . see Mathieu functions sec function . . . . . . . . . . . . . . . . . . 36, 39, 43, 44, 50, 52, 64, 113, 114, 155, 156, 315, 323, 328, 329, 371, 372, 377–379, 389, 395, 396, 400, 401, 403–405, 410–414, 421, 422, 436, 438, 439, 446, 471, 472, 479, 541, 551, 586, 646, 653, 661, 663, 664, 669, 670, 674, 682, 689, 691, 699, 706–708, 710, 713, 716, 718, 719, 726, 728, 734, 740, 749, 757, 761, 783, 806, 820, 825, 845, 846, 864, 921, 922, 932, 1007, 1126, 1132, 1133 sech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27, 43, 62, 113–115, 323, 387, 509, 634, 715, 750, 751, 787, 800, 802, 841, 883, 1128, 1133

1158

shi function . . xxxvi, xli, 142–144, 495, 644, 645, 886 Si function . . . . . . . . . xxxv, 643, 886, 930, 1115, 1122 si function . . . . . . . . xxxv, xli, 219–221, 340–344, 421, 423–426, 447, 495, 505, 506, 528, 529, 571, 572, 578, 581, 594, 595, 599, 605, 628, 629, 638–644, 647, 649, 650, 656, 658, 748, 762, 886, 887, 992, 1115 sigma function . . . . . . . . . . . . . . . . . . . . . xxxix, 876, 877 sign function . . xlv, 46, 177, 241, 243, 251, 322, 350, 351, 365, 370, 423, 437, 438, 447, 465, 485, 594, 596, 603, 604, 610, 611, 640, 642, 652, 750, 768, 885, 1118–1120, 1122 sin function . . . . . . . . . . . . . . . . xxvii, xxix, xxxi, xxxii, xxxiv–xxxviii, 4, 13, 19, 20, 23, 26–52, 55, 56, 64, 74–76, 79, 147, 151–237, 249, 250, 253, 254, 263–265, 317, 318, 321–323, 325–329, 331–333, 339–345, 348, 354, 355, 358, 359, 371–373, 375, 377–383, 385, 387–534, 537, 539–547, 550, 551, 554, 558, 559, 561–563, 565, 567–573, 576, 578, 579, 581–601, 604–606, 608, 610–612, 616, 621– 623, 628, 629, 631–633, 635–644, 647–651, 655, 656, 658, 659, 662–665, 667, 669, 671–675, 677, 678, 680, 684, 686, 688, 689, 691, 692, 695, 698, 703, 705, 706, 708, 709, 711, 713–715, 717–748, 751, 752, 754–756, 758–760, 762–770, 773, 774, 776, 777, 779–782, 784, 789, 793, 794, 797–800, 802, 806, 808, 810–812, 817, 820, 825, 829, 830, 833, 836, 837, 839, 842, 844–846, 848–850, 853, 859–869, 876–880, 882–893, 896, 898, 900, 904, 906–910, 912–918, 920, 922, 924, 925, 927, 928, 930, 933–940, 942, 943, 945–951, 954, 958, 959, 961–964, 966–981, 984–992, 994, 997, 998, 1000, 1005–1007, 1009, 1013, 1025, 1026, 1029, 1036– 1038, 1042, 1044, 1053, 1057, 1060, 1061, 1066, 1067, 1090, 1110–1113, 1115, 1117–1128, 1131, 1132, 1137, 1140 sinh function . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi, 27–36, 38, 40, 42, 43, 45, 47, 48, 50–52, 64, 110–151, 231–237, 251, 338, 339, 358, 371–390, 407, 419, 425, 432, 433, 438, 439, 448, 451, 452, 454, 455, 461, 466–468, 477, 484, 485, 489, 491, 496, 502, 504, 508–527, 570, 573, 578–582, 595, 596, 606, 610, 611, 621, 622, 634, 643–645, 664, 686, 698, 702, 704, 705, 710, 711, 713–716, 723, 729, 735, 747, 751–753, 755, 760, 763–766, 778, 787–789, 806, 843, 844, 886, 888, 896, 898, 913–915, 917, 943, 953, 955–957, 960–963, 967–969, 980, 981, 997, 1006, 1025, 1029, 1040, 1112–1115, 1119, 1120, 1124, 1125, 1128, 1140 sn function . . . . . . . . . xxxiii, xxxiv, xli, 623–626, 714, 866–873, 875, 879, 880 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859

INDEX OF FUNCTIONS AND CONSTANTS

square root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii, xxxiv, xxxvi, xxxviii, xliv, 2, 9–11, 14, 15, 23, 25, 26, 30, 37, 43, 44, 54–61, 63–67, 71–79, 83– 99, 103–109, 125–139, 158, 170–175, 177–184, 197, 199, 200, 202–214, 225, 226, 230, 239–245, 249, 251, 254–315, 317–319, 321, 324–328, 330, 333, 336, 337, 339, 344–353, 355, 359, 363–376, 380, 382–385, 390, 391, 393–396, 400–402, 404– 411, 414–419, 421, 425, 426, 428–430, 434–436, 440–446, 448, 451–454, 456, 457, 460, 472–479, 481–483, 485, 486, 488–497, 499, 501–507, 511, 513–515, 517, 518, 522–527, 529, 531, 532, 534, 535, 537–539, 542, 543, 545, 549–551, 553, 554, 556–558, 560, 562, 563, 565–568, 570–576, 578– 581, 584, 585, 588, 590–593, 595–606, 608–613, 615–617, 619, 621–623, 629, 631–635, 637–642, 644–651, 653, 657, 659, 661–668, 670, 672–675, 677, 678, 680–683, 685–763, 766–768, 771, 773, 777, 778, 780–782, 785–789, 791–794, 800–808, 810–812, 814, 815, 819, 828, 829, 837, 841, 843– 845, 848, 853, 854, 856, 859–866, 868, 870–876, 879–881, 887–891, 893–902, 905, 908, 909, 913– 915, 917, 918, 920–926, 928, 931–946, 950, 951, 958, 960–974, 976–983, 985, 987, 988, 990–998, 1000, 1002, 1003, 1007, 1009, 1018, 1019, 1023, 1026–1030, 1035, 1038, 1052, 1054, 1055, 1060, 1082, 1116–1121, 1123, 1125–1127, 1129, 1131 step function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 Stirling number first kind (Snm ) . . . . . . . . . . . . . . . . . . . xlv, 1046–1048 second kind (Sm n ) . . . . . . . . . . . . . . . . xlv, 1046–1048 Struve function H(x) . . xli, 345, 351, 421, 435, 442, 443, 573, 647, 659, 660, 663, 664, 669, 675, 677, 679, 680, 692, 694, 708, 722, 725, 735, 753–759, 787, 838, 848, 856, 942, 943, 946, 1035, 1132 modified (L(x)) . . . . . . xli, 345, 350, 351, 435, 441, 515, 595, 605, 663, 664, 669, 671, 675, 676, 678, 679, 692, 722, 736, 753–759, 787, 794, 942, 943

INDEX OF FUNCTIONS AND CONSTANTS

1159

T

U

Tn (x) . . . . . . . . . . . . . . . . . . see Chebyshev polynomials tan function . . . . . . . . . . . . . . xxvii–xxix, 27–30, 35, 36, 39, 40, 42, 44, 46, 50, 52, 53, 55, 56, 60, 64, 126, 151, 155–161, 167–178, 180, 181, 183–185, 188, 189, 194, 195, 199, 200, 202–207, 209, 212, 214, 222–225, 229, 230, 274, 322, 332, 339, 340, 355, 371, 375–379, 381, 382, 384, 388, 389, 393, 395, 396, 400, 403–405, 409–414, 421–423, 433, 434, 451–455, 457–459, 471–475, 483–486, 492, 493, 496, 498, 505, 506, 509, 515, 518, 532–535, 537, 541, 545, 546, 567–570, 579, 582, 586–589, 591– 594, 597–599, 604, 606, 622, 631, 632, 659, 664, 669–671, 682, 686, 699, 716, 717, 725, 728, 744, 745, 754, 766, 849, 862–865, 867–869, 883, 886, 890, 894, 905, 909, 922, 932, 954, 969, 971, 979, 986, 990, 1006, 1007, 1023, 1123, 1127, 1132 tanh function . . . . . . . . . 12, 27–29, 31, 39, 40, 42, 44, 51, 52, 62, 64, 110, 113–120, 123–126, 128–132, 134–137, 139, 145–148, 338, 380–383, 387, 390, 472, 484, 485, 489, 502, 509, 512, 513, 516, 518, 520, 569, 606, 621, 622, 716, 717, 751, 753, 766, 788, 789, 841, 921, 922, 956, 957, 1025, 1124 theta function (θ) . . . . xxxiv, 521, 633, 634, 877–883 Thomson functions bei(x) . . . . . . . . . . . . . . . . . . xxxix, 761–763, 944, 945 ber(x) . . . . . . . . . . . . . . . . . . xxxix, 761–763, 944, 945 hei(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 944 her(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 944 kei(x) . . . . . . xli, 641, 663, 672, 674, 748, 762, 763, 944, 945 ker(x) . . . . . . xli, 641, 663, 671, 674, 747, 762, 763, 944, 945 toroidal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see trace trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 transpose . . . . . . . . . . . . . . . xlv, 1069–1073, 1075, 1089

Un (x) . . . . . . . . . . . . . . . . . . see Chebyshev polynomials unilateral z transform . . . . . . . 1135, 1138–1140, 1159

W Wλ,μ (z) . . . . . . . . . . . . . . . . . . . see Whittaker functions Weber function (E) . . . . 338, 339, 346, 353, 371, 384, 421, 423, 670, 671, 751, 943, 946, 948, 949 Weierstrass function (℘) . . . . xxxix, xl, 626, 873–877, 880 Whittaker functions M . . . . . xli, 338, 348, 445, 654, 682, 697, 703, 705, 706, 709, 710, 715–717, 736, 748, 749, 784, 785, 787, 819–841, 1024–1027 W . . . . . . . . . xlii, 338, 346–349, 367, 368, 384, 423, 445, 635, 652, 654, 682, 697, 698, 704, 706, 707, 709, 710, 712, 715, 716, 726, 727, 736, 745, 749, 756, 759, 761, 776–778, 781, 782, 784–788, 803, 814–817, 819–841, 843, 844, 846, 847, 857, 979, 1024–1028, 1035

X Xb . . . . . . . . . . . . . . . . . . . . . . . . see bilateral z transform Xu . . . . . . . . . . . . . . . . . . . . . . see unilateral z transform

Y Y . . . . . . . . . . . . . . . . . . . . . . . . . . . see Bessel function, Y Young function (C) . . . . . . . . . xxxvi, xl, 417, 439, 440

Z zeta function (ζ) . . xxxix, 8, 338, 353, 354, 358, 359, 376, 377, 379–381, 386–389, 433, 434, 449, 471, 509, 540, 542, 543, 550, 552, 560, 567, 569, 576, 577, 580, 587, 591, 593, 607, 626, 633, 634, 658, 659, 802, 876, 877, 880, 894, 898, 903–905, 907, 909, 1036–1041, 1133 zn(x) . . . . . . . . . . . . . . . . . . . . . . see Jacobi zeta function

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Index of Concepts This index refers to concepts appearing in the text.

A

arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 242 arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 244 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 747 argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 of a complex number . . . . . . . . . . . . . . . . . . . . . . . . xliv arithmetic mean theorem . . . . . . . . . . . . . . . . . . . . . 1056 arithmetic progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 arithmetic-geometric inequality . . . . . . . . . . . . . . . 1060 arithmetic-geometric progression . . . . . . . . . . . . . . . . . 1 associated Legendre functions . . . 769, 788, 958, 972, 974 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 789 and Bessel functions . . . . . . . . . . . . . . . . . . . . 782, 787 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 778 and powers . . . . . . . . . . . . . . . . . . . . . . . . 770, 776, 779 and probability integral . . . . . . . . . . . . . . . . . . . . . . 781 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 789 and trigonometric functions . . . . . . . . . . . . . . . . . . 779 associated Mathieu functions . . . . . . . . . . . . . . . . . . . 952 asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 1146 asymptotic result . . . . . . . . . . 21, 356, 895, 1026, 1029 asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . 973, 975 adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 algebraic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059, 1060 algebraic functions . . . . . . . . . . . . . . . . . . . . . . . . . 82, 253 and arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and associated Legendre functions . . . . . . . . . . . 789 and Bessel functions . . . . . . . . . . . . . . . . . . . . 674, 715 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . 344, 363 and hyperbolic functions . . . . . . . . . . . 132, 375, 715 and logarithmic functions . . . . . . . . . . . . . . . . . . . . 538 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 789 and trigonometric functions . . . . . . . . . . . . . . . . . . 434 alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 analytic continuation . . . . . . . . . . . . . . . . . . . . . 970, 1012 Anger functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 angle of parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 anticommutative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 approximate solution . . . . . . . . . . . . . . . . . . . . 1093–1096 approximation by tangents . . . . . . . . . . . . . . . . . . . . . 921 arccosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 242 arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 244 arcsecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

B Ballieu theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 Bateman’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040, 1045 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 1040, 1041 Bessel functions. . .629, 659, 748, 749, 753, 910, 912, 914, 916–920, 924, 925, 928, 931, 933–937, 940, 941, 954, 1146, see Constant/Function index and algebraic functions . . . . . . . . . . . . . . . . . 674, 715 and arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 and associated Legendre functions . . . . . . 782, 787

1161

1162

INDEX OF CONCEPTS

Bessel functions (continued) and Chebyshev polynomials . . . . . . . . . . . . . . . . . 803 and exponentials . . . . 694, 699, 708, 711, 713, 715, 742, 834 and Gegenbauer functions . . . . . . . . . . . . . . . . . . . 798 and hyperbolic functions . . . . . . . . . . . 713, 715, 747 and hypergeometric functions . . . . . . . . . . . . . . . . 817 confluent . . . . . . . . . . . . . . . . . . . . . . . . . 830, 831, 834 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 794 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 854 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 767 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 854 and parabolic cylinder functions . . . . . . . . . . . . . 845 and powers . . . . . 664, 675, 689, 699, 708, 711, 727, 742, 831, 834 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 670 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 756 and trigonometric functions . . . 717, 727, 742, 747 generating functions . . . . . . . . . . . . . . . . . . . . . . . . . 933 imaginary arguments . . . . . . . . . . . . . . . . . . . . . . . . 911 Bessel inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 bilateral z-transform . . . . . . . . . . . . . . . . . . . . 1135, 1136 bilinear concomitant . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315, 322 Bonnet–Heine formula . . . . . . . . . . . . . . . . . . . . . . . . . 988 bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 boundedness theorems . . . . . . . . . . . . . . . . . . . . . . . . 1106 branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866, 1024 Brauer theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1086, 1088 Buniakowsky inequality . . . . . . . . . . . 1059, 1061, 1064

C Calogero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 Carleman inequality . . . . . . . . . . . . . . . . . . . . 1060, 1066 Catalan constant . . . . . . . . . . . . . . . . . . . xxxii, 1046, see Constant/Function index Cauchy principal value . . . . . . . . . . . . . . . . . . . . . . . . . 528 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . 1093, 1095 Cauchy–Schwarz–Buniakowsky inequality . . . . . 1059, 1061, 1064 Cayley–Hamilton theorem . . . . . . . . . . . . . . . . . . . . 1084 change of variables . . . . . . . . . . . . . . . . . . . 248, 607, 608 characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . 1071 characteristic polynomial. . . . . . . . . . . . . . . . . . . . . .1084 characteristic values . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 Chebyshev inequality . . . . . . . . . . . . . . . . . . . 1059, 1065

Chebyshev polynomials. . . . . . . . . . . . . . . . . . . .988, 993 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 803 and elementary functions . . . . . . . . . . . . . . . . . . . . 802 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 Christoffel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 Christoffel summation formula . . . . . . . . . . . . . . . . . 986 circle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 circulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 classification system . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi classified references . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145 column norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 comparison of approximate solutions . . . . 1094, 1096 comparison theorem . . . . . . . . . 1100, 1101, 1103, 1104 complementary error function. . .see error functions, complementary complementary modulus . . . . . . . . . . . . . . . . . . . . . . . 859 complete elliptic integrals . . . . . . . . . . . . . 619, 632, 859 complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146 complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii conditional convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 6 conditions, Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 confluent hypergeometric function . . . . . . . . . . . . . . see hypergeometric function, confluent conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980 constant of integration . . . . . . . . . . . . . . . . . . . . . . . . . . 63 constants . . . . . . . . . . . . . see Constant/Function index Catalan . . . . . . . . . . . . . . . . . . . . see Catalan constant Euler . . . . . . . . . . . . . . . . . . . . . . . . . see Euler constant continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902 continuity, Lipschitz . . . . . . . . . . . . . . . . . . . . 1094, 1095 converge absolutely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 conditionally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 uniformly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 convergence circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 19 convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 theorem . . . . . . . . . . . . 1108, 1118, 1122, 1130, 1136 coordinates, curvilinear . . . . . . . . . . . . . . . . . . . . . . . 1052 cosine and rational functions . . . . . . . . . . . . . . . . . . 171, 390 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628, 639, 886 hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 cosine-amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 Cramer’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050

INDEX OF CONCEPTS

1163

curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1052 cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 cylinder function . . . . see parabolic cylinder function

D Darboux–Christoffel formula . . . . . . . . . . . . . . . . . . . 983 de Moivre’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 decreasing solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 definite integrals . . . . . . . . . 247, see integrals, definite delta amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 derivative of a composite function . . . . . . . . . . . . . . . 22 determinants . . . . . . . . . . . . . . . . . . . . . . 1075, 1076, 1078 Gram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Vandermonde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 differential equations . . . . . . . . . . . . . . . . . . . . . 873, 874, 910, 931, 944, 947–950, 952, 958, 974, 975, 980, 981, 983, 993, 995, 998, 1000, 1003, 1011, 1013, 1015, 1024, 1031, 1034, 1093 adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 exact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 homogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 hypergeometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018, 1031 Riccati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014, 1022 second-order . . . . . . . . . . . . . . 1017, 1098, 1100, 1104 self-adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 special types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 variables separable . . . . . . . . . . . . . . . . . . . . . . . . . 1097 differentiation of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 1064 of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 dilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 diophantine relations. . . . . . . . . . . . . . . . . . . . . . . . . . 1089 directional derivative. . . . . . . . . . . . . . . . . . . . . . . . . . 1051 Dirichlet conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 DOC (domain of convergence) . . . . . . . . . . . . . . . . 1135 domain of convergence (DOC) . . . . . . . . . . . . . . . . 1135 dominant solutions . . . . . . . . . . . . . . . . . . . . . . 1104, 1105 double factorial symbol . . . . . . . . . . . . . . . . . . . . . . . . xliii double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 610, 1021 doubling formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 doubly-periodic function . . . . . . . . . . . . . . . . . . 866, 874

E eigenvalues . . . . . . . . . . . . . . . . . . . 951, 1071, 1084, 1087 eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 elementary functions . . . . . . . . . . . . . . . . . 25, 247, 1006 and Chebyshev polynomials . . . . . . . . . . . . . . . . . 802 and Gegenbauer polynomials . . . . . . . . . . . . . . . . 797 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 792 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 850 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 850 indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 elliptic functions . . . . . . . . . . . . . . . . 619, 631, 865, 1148 Jacobian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866, 870 order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 Weierstrass . . . . . . . . . . 626, see Weierstrass elliptic functions elliptic integrals . . . 104, 184, 619, 621, 631, 632, 859 complete . . . . . . . . . . . . . . . . . 394, 472–474, 632, 859 derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 394, 863, 865 functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . 863 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 equations differential . . . . . . . . . . . . . see differential equations first-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093, 1096 linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 special types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094, 1095 error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 887, 1147 complementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 essential singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 Euclidean norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 Euler constant . . . . . xxxii, see Constant/Function index dilogarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892, 908 numbers . . . . . . . . . . . . . . . . . . . . . . . . 1040, 1043, 1045 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044 substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 exact differential equations . . . . . . . . . . . . . . . . . . . . 1097 expansion of determinants. . . . . . . . . . . . . . .1075, 1076 expansions, asymptotic . . . . . . . . . . . . . . . . . . . . . . . 1146 expansions, Weierstrass . . . . . . . . . . . . . . . . . . . . . . . . 869 exponential integrals . . 627, 636, 638, 883, 885, 1147 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 106, 334 and algebraic functions . . . . . . . . . . . . . . . . . 344, 363 and associated Legendre functions . . . . . . . . . . . 776 and Bessel functions . . . . . 694, 699, 708, 711, 713, 715, 742, 834 and complicated arguments . . . . . . . . . . . . . . . . . . 336

1164

INDEX OF CONCEPTS

exponentials (continued) and exponential integrals . . . . . . . . . . . . . . . . . . . . 628 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 652 and hyperbolic functions . . . . . . 148, 338, 382, 386, 522, 525, 713, 715 and hypergeometric functions . . . . . . . . . . . . . . . . 814 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822, 834 and inverse trigonometric functions . . . . . . . . . . 605 and logarithmic functions . . . . . 339, 571, 573, 599 and parabolic cylinder functions . . . . . . . . . . . . . 842 and powers . . . . . 148, 346, 353, 363, 364, 386, 497, 525, 573, 699, 708, 711, 742, 754, 776, 834, 842 and rational functions . . . . . . . . . . . . . . 106, 340, 353 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 754 and trigonometric functions . . . 227, 339, 485, 493, 495, 497, 522, 525, 599, 742 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

F factorial symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 figures . . . . . . . . . . . . 608–610, 892, 913, 915, 916, 1036 final value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139 finite sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 first mean value theorem . . . . . . . . . . . . . . . . . . . . . . 1063 first-order equations . . . . . . . . . . . . . . . . . . . . 1093, 1096 first-order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix, xxxi, 82, 132, 247, 248, 274, 397, 410, 547, 656, 859, 867, 908, 920, 931, 981, 991, 1039, 1141 Fourier series . . . . . . . . . . . . . . . . . . . . 19, 46, 1066, 1067 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 1107, 1117 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121, 1129 properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126 exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129 sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121, 1129 properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118, 1120 fourth roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 fractional transformation . . . . . . . . . . . . . . . . . . . . . . 1014 Fresnel integrals . . . . . . . . . . . . . . . . 629, 649, 887, 1147 Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 functional series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

functions . . . . . . . . . . . . . see Constant/Function index inner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 outer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii fundamental inequalities . . . . . . . . . . . . . . . . . . . . . . 1094 fundamental system. . . . . . . . . . . . . . . . . . . . . . . . . . . 1100

G gamma functions . . . . . . . . . . 650, 892, 894, 895, 1147 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 and trigonometric functions . . . . . . . . . . . . . . . . . . 655 incomplete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657, 899 Gauss divergence theorem. . . . . . . . . . . . . . . . . . . . . 1055 Gegenbauer functions and Bessel functions . . . . . 798 Gegenbauer polynomials . . . . . . . . . . . . . . . . . . . . . . . 990 and elementary functions . . . . . . . . . . . . . . . . . . . . 797 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 general formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65, 249 generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . 635 generalized Fourier series . . . . . . . . . . . . . . . . 1067, 1068 generalized Legendre polynomials . . . . . . . . . . . . . . 990 generating functions Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1040 Bernoulli polynomials . . . . . . . . . . . . . . . . xxxii, 1041 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933 Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . 995 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 Euler polynomials . . . . . . . . . . . . . . . . . . . . xxxii, 1044 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . 997 Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 1000 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . 988 Neumann polynomials . . . . . . . . . . . . . . . . xxxvii, 950 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . 1046, 1047 geometric progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Gerschgorin theorem . . . . . . . . . . . . . . . . . . . . 1083, 1088 grad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 Gram determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 Gram inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 Gram–Kowalewski theorem . . . . . . . . . . . . . . . . . . . 1080 Green theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1055, 1056 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 growth estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1104 growth of maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Gudermannian (gd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

INDEX OF CONCEPTS

1165

H Hadamard’s inequality . . . . . . . . . . . . . . . . . . . . . . . . 1077 Hadamard’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 Hankel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 910, 925 Heaviside step function . . . . . . . . . . . . . . . . . . . . . . . . xliv Heine formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . 767, 1052 Hermite method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Hermite polynomials . . . . . . . . . . . . . . . . . 803, 996, 997 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . 1077, 1089 Hessian determinant . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 H¨ older inequality . . . . . . . . . . . . . . . . . 1059, 1061, 1064 homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 homogeneous differential equations . . . . . . . . . . . . 1097 hyperbolic amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 cosine integral . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 sine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 hyperbolic functions . . . . . . . . . . . . . . . . . . . 28, 110, 371 and algebraic functions . . . . . . . . . . . . . 132, 375, 715 and associated Legendre functions . . . . . . . . . . . 778 and Bessel functions . . . . . . . . . . . . . . . 713, 715, 747 and exponentials . . . . 148, 338, 382, 386, 522, 525, 713, 715 and inverse trigonometric functions . . . . . . . . . . 605 and linear functions . . . . . . . . . . . . . . . . . . . . . . . . . 120 and logarithmic functions . . . . . . . . . . . . . . . . . . . . 578 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 763 and parabolic cylinder functions . . . . . . . . . . . . . 843 and powers . . . . . . . . . . . . . . . 139, 148, 386, 516, 525 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 125 and trigonometric functions . . . 231, 509, 516, 522, 525, 747, 763 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 240 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110, 120 hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . . . . 1010 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005, 1008 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 hypergeometric functions . . . . . . . 812, 841, 946, 1005, 1006, 1039, 1147 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 817 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812 and trigonometric functions . . . . . . . . . . . . . . . . . . 817

hypergeometric functions (continued) confluent . . . . . . . . . . . . . . 820, 841, 1022, 1023, 1147 and Bessel functions . . . . . . . . . . . . . . 830, 831, 834 and exponentials . . . . . . . . . . . . . . . . . . . . . . 822, 834 and Legendre functions . . . . . . . . . . . . . . . . . . . . . 839 and parabolic cylinder functions . . . . . . . . . . . . 849 and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840 and powers . . . . . . . . . . . . . . . . . . . . . . . 820, 831, 834 and special functions . . . . . . . . . . . . . . . . . . . . . . . 839 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . 838 and trigonometric functions . . . . . . . . . . . . . . . . 829 several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022 two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018

I identities Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Picone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . 251, 252 incomplete beta functions . . . . . . . . . . . . . . . . . . . . . . 910 incomplete gamma function . . . . . . . . . . . . . . . . . . . . 657 increasing solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 indefinite integrals elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . 63 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 induced norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 inequalities . . . . . 950, 963, 979, 987, 997, 1041, 1061, 1083–1085, 1094 algebraic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059, 1060 Carleman . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060, 1066 for sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 Hadamard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063–1066 Schur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 infinite products . . . . . . . . . . . . . . . . . . . . . . . . . 6, 14, 862 initial value theorem . . . . . . . . . . . . . . . . . . . . 1136, 1139 inner function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi integer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 integer pulse function . . . . . . . . . . . . . . . . . . . . . . . . . 1135 integral differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 1064 formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063–1066 inversion . . . . . . . . . . . . . . . . . 1107, 1118, 1121, 1129 part (symbol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii representations . . . . . . 887, 888, 892, 898, 900, 902, 906, 908, 912, 914, 916, 942, 946, 950, 960, 974, 976, 980, 981, 985, 991, 996, 1005, 1021, 1023, 1025, 1028, 1035, 1036, 1039, 1040, 1067

1166

INDEX OF CONCEPTS

integral (continued) theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107, 1147 relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129 integrals definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 double. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610, 1021 elliptic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104, 859 Fresnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 improper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251, 252 indefinite . . . . . . . . . . . . . . . . . see indefinite integrals Mellin–Barnes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607, 612 pseudo-elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 integration constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 termwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 inverse z-transformation . . . . . . . . . . . . . . . . . . . . . . 1135 inverse hyperbolic functions . . . . . . . . . see hyperbolic functions inverse trigonometric functions . . . . . . . . . . . . . 599, see trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 605 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 and powers . . . . . . . . . . . . . . . . . . . . . . . . 600, 601, 607 and trigonometric functions . . . . . . . . . . . . . 605, 607 inversion integral . . . . . . . . . . . . 1107, 1118, 1121, 1129

J Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . 806, 998 Jacobi theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 Jacobian determinant . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Jacobian elliptic functions . . . . . . 866, 870, 879, 1148 Jacobian elliptic integrals . . . . . . . . . . . . . . . . . . . . . . 623 Jensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066

K Kneser’s non-oscillation theorem . . . . . . . . . . . . . . 1103 Kowalewski theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1080

L L2 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . 1059, 1099 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . 808, 1000 Laplace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Laplace integral formula . . . . . . . . . . . . . . . . . . . . . . . 985

Laplace transform . . . . . . . . . . . . . . . . . . . . . . 1107, 1129 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767, 1051 latent roots values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 least common factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 least common multiple . . . . . . . . . . . . . . . . . . . . . . . . . 798 Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . 975, 1149 and hypergeometric functions confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 associated . . . . . see associated Legendre functions special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 Legendre normal form . . . . . . . . . . . . . . . . . . . . . . . . . 859 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . 983, 988 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 794 and elementary functions . . . . . . . . . . . . . . . . . . . . 792 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 lemmas Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Gronwall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 Riemann–Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . 1067 letters, conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 linear dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 L∞ norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . 1094, 1095 Lobachevskiy’s “angle of parallelism” . . . . . . . . . . . . 51 Lobachevskiy’s function . . . . . . . . . . . . . . . . . . . . . . . . 891 logarithm integrals . . . . . . . . . . . . . . . . . . . . . . . . 636, 887 logarithms . . . . . . . . . . . . . . . . . . . . . . . . 53, 237, 527, 529 and algebraic functions . . . . . . . . . . . . . . . . . 238, 538 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 747 and exponentials . . . . . . . . . . . . . . 339, 571, 573, 599 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 656 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 578 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 and inverse trigonometric functions . . . . . . . . . . 607 and powers . . . . . . . . . . 540, 542, 553, 555, 573, 594 and rational functions . . . . . . . . . . . . . . . . . . 535, 553 and trigonometric functions . . . 339, 581, 594, 599 gamma functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . 760, 945 two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947 Lyapunov theorem . . . . . . . . . . . . . . . . . . . . . . 1089, 1105

M MacRobert functions . . . . . . . . . . . . . . . . . . . . . 850, 1035 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 854 and elementary functions . . . . . . . . . . . . . . . . . . . . 850 and special functions . . . . . . . . . . . . . . . . . . . . . . . . 856

INDEX OF CONCEPTS

Mathieu functions . . . . . 763, 950, 951, 953, 954, 1149 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 767 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 763 and trigonometric functions . . . . . . . . . . . . . . . . . . 763 imaginary argument . . . . . . . . . . . . . . . . . . . . . . . . . 952 matrix adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 cofactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 determinants . . . . . . . . . . . . . . . . . . . see determinants diagonal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . 1071 differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 Hermitian . . . . . . . . . . . . . . . . . . . . . . 1070, 1077, 1089 idempotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 irreducible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 principal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 nilpotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 non-negative definite . . . . . . . . . . . . . . . . . . . . . . . 1071 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082, 1083 null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 positive definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 reducible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 skew-symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 special . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069, 1070 triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 unitary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 mean value theorems . . . . . . . . . . . . . . . 247, 1063, 1064 Meijer functions. . . . . . . . . . . . . . . . . . . . . . . . . . 850, 1032 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 854 and elementary functions . . . . . . . . . . . . . . . . . . . . 850 and special functions . . . . . . . . . . . . . . . . . . . . . . . . 856 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . 1107, 1129 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131 Mellin–Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . 1021 metric coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 metrical coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054 Minkowski inequality . . . . . . . . . . . . . . 1059, 1061, 1065 modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632, 859, 860 multiple angle expansion . . . . . . . . . . . . . . . . . . . . . . . . 31 multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . 607, 612

1167

N named theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 natural norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 Neumann functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910 Neumann polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 949 nome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 non-oscillation . . . . . . . . . . . . . . . . . . . . 1100, 1103, 1104 normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 compatible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 Euclidean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 induced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082, 1083 natural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083 spectral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii

O one-sided z-transform . . . . . . . . . . . . . . . . . . . . . . . . . 1135 order of presentation . . . . . . . . . . . . . . . . . . . . . . . . . xxvii ordinary differential equations . . . . . . . . . . . . . . . . . 1093 orthogonal curvilinear coordinates . . . . . . . . . . . . . 1052 orthogonal polynomials . . . . . . . . . . . . . . 795, 982, 1149 oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100, 1102 Ostrogradskiy–Hermite method . . . . . . . . . . . . . . . . . 67 Ostrowski inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 Ostrowski theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 outer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi

P parabolic cylinder functions. . . .841, 849, 1028, 1150 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 845 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 843 and hypergeometric functions . . . . . . . . . . . . . . . . 849 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 848 and trigonometric functions . . . . . . . . . . . . . . . . . . 844 parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 parameter of the integral . . . . . . . . . . . . . . . . . . . . . . . 859 Parodi theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 Parseval formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 Parseval theorem . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 partial sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Perelomov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089

1168

periodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 951 periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865, 870 permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Perron theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . . 1088 Picone identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Pochhammer symbol . . . . . . . . . . . . . . . . . . . . . . . . . . xliii Poincare’s separation theorem . . . . . . . . . . . . . . . . . 1087 points, singular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 Poisson integral . . . . . . . . . . . . . . . . . . . . . . . . . 1056, 1057 poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865, 870, 874, 892 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 254, 313, 322 and hypergeometric functions confluent . . . . . . 840 characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 Chebyshev . . . . . . . . . . . see Chebyshev polynomials degree 3 or 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 Gegenbauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 Hermite . . . . . . . . . . . . . . . . see Hermite polynomials Jacobi . . . . . . . . . . . . . . . . . . . see Jacobi polynomials Laguerre . . . . . . . . . . . . . . . see Laguerre polynomials Legendre . . . . . . . . . . . . . . see Legendre polynomials orthogonal . . . . . . . . . . . . . . . . . . . . . . . . 795, 982, 1149 positive definite . . . . . . . . . . . . . . . . . . . . . . . . . 1071, 1072 positive semidefinite . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–18, 25 expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 363 and arccosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and arcsecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and associated Legendre functions . . 770, 776, 779 and Bessel functions . . . . . 664, 675, 689, 699, 708, 711, 727, 742, 831, 834 and binomials . . . . . . . . . . . . . . . . . . . . . . . . . . 315, 322 and Chebyshev polynomials . . . . . . . . . . . . . . . . . 800 and exponential integrals . . . . . . . . . . . . . . . . . . . . 627 and exponentials . . . . 148, 346, 353, 363, 364, 386, 497, 525, 573, 699, 708, 711, 742, 754, 776, 834, 842 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 652 and Gegenbauer polynomials . . . . . . . . . . . . . . . . 795 and hyperbolic functions . . 139, 148, 386, 516, 525 and hypergeometric functions . . . . . . . . . . . . . . . . 812 confluent . . . . . . . . . . . . . . . . . . . . . . . . . 820, 831, 834 and inverse trigonometric functions . . . . . 600, 601, 607 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 791 and logarithmic functions . . . . . 540, 542, 553, 555, 573, 594 and parabolic cylinder functions . . . . . . . . . . . . . 842

INDEX OF CONCEPTS

powers (continued) and rational functions . . . . . . . . . . . . . . 353, 401, 553 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 754 and trigonometric functions . . . 214, 397, 401, 405, 411, 436, 459, 475, 497, 516, 525, 594, 607, 727, 742, 779 binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 hyperbolic functions . . . . . . . . . . . . . . . . . . . . 110, 120 trigonometric functions . . . . . . . . . . . . . . . . . 151, 395 principal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi natural norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 252, 528 vector norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 probability function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 probability integrals . . . . . . . . . . . . . . . . . . 629, 645, 887 and associated Legendre functions . . . . . . . . . . . 781 problem, Cauchy . . . . . . . . . . . . . . . . . . . . . . . . 1093, 1095 product finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 infinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 14, 45 of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 8 pseudo-elliptic integrals . . . . . . . . . . . . . . . . . . . 105, 184 pulse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

Q q-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880 quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 quasiperiodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878

R radius of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 rate of change theorems . . . . . . . . . . . . . . . . . . . . . . . 1057 rational functions . . . . . . . . . . . . . . . . . . . . . . 66, 253, 254 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 789 and associated Legendre functions . . . . . . . . . . . 789 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 670 and cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171, 390 and exponentials . . . . . . . . . . . . . . . . . . . 106, 340, 353 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 125 and logarithmic functions . . . . . . . . . . . . . . . 535, 553 and powers . . . . . . . . . . . . . . . . . . . . . . . . 353, 401, 553 and sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171, 390 and trigonometric functions . . . . . . . . 401, 423, 447 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv reciprocal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 reciprocals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 12

INDEX OF CONCEPTS

1169

references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141 supplementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145 remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 representation theorem. . . . . . . . . . . . . . . . . . . . . . . .1056 residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870 Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Riemann differential equation . . . . . . . . . . . . . . . . . 1014 Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 1038 Riemann zeta functions . . . . . . . . . . . . . . . . . 1036, 1150 Riemann–Lebesgue lemma . . . . . . . . . . . . . . . . . . . . 1067 Rodrigues’ formula . . . . . . . . . . . . . 993, 995, 998, 1000 roots . . . . . . see square roots and Constant/Function index fourth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Routh–Hurwitz theorem . . . . . . . . . . . . . . . . . . . . . . 1086 row norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083

S saltus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 Schl¨ afli integral formula . . . . . . . . . . . . . . . . . . . . . . . . 985 Schl¨ afli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949 Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Schwarz inequality . . . . . . . . . . . . . . . . 1059, 1061, 1064 second mean value theorem . . . . . . . . . . . . . 1063, 1064 second-order equations . . . . . . 1017, 1098, 1100, 1104 self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098 semiconvergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 separation theorem . . . . . . . . . . . . . . . . . . . . . 1087, 1101 series . . . . . . . . . . . . . . . . . . . . . . . . . 860, see specific type alternating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 asymptotic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 diverge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Fourier . . . . . . . . . . . . . . . . . . . . . . . . 19, 46, 1066–1068 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 51 hypergeometric . . . . . . . . . . . . . . . . . . . . . . . 1005, 1008 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 of logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–18 rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 semiconvergent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 trigonometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46, 862 sign function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 signum function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv

sine and rational functions . . . . . . . . . . . . . . . . . . 171, 390 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628, 639, 886 hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 sine-amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . 958, 1038 solenoidal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 Sonin theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix and hypergeometric functions confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 856 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 856 indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 spectral radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083 spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974 square roots . . . 84, 88, 92, 94, 99, 103, 179, 184, 254 and cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 and sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 408 Steffensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv Stieltjes’ theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . 1046, 1048 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047, 1048 Stokes phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 920 Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Struve functions . . . . . . . . . . . . . . . . . . . . . 753, 942, 1150 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 756 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 and hypergeometric functions confluent . . . . . . 838 and parabolic cylinder functions . . . . . . . . . . . . . 848 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 and trigonometric functions . . . . . . . . . . . . . . . . . . 755 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . 1101 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . 1101 Sturm–Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . 1102 Sturmian separation theorem . . . . . . . . . . . . . . . . . 1087 subdominant solutions . . . . . . . . . . . . . . . . . . . . . . . . 1104 subordinate norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 substitutions, Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986 summation theorems . . . . . . 940, 986, 992, 998, 1002, 1030, 1042

1170

INDEX OF CONCEPTS

sums binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 powers of trigonometric functions . . . . . . . . . . . . . 37 products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 products of trigonometric functions . . . . . . . . . . . 38 reciprocals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 12 tangents of multiple angles . . . . . . . . . . . . . . . . . . . . 39 trigonometric and hyperbolic functions . . . . . . . . 36 supplementary references . . . . . . . . . . . . . . . . . . . . . 1145 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . 1072 symbol binomial coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . xliii factorial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii integral part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii Pochhammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii synonyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii system of equations . . . . . . . . . . . . . . . . . . . . . 1094, 1095 linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 Szeg¨ o comparison theorem . . . . . . . . . . . . . . . . . . . . 1101

T table usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi tangent approximation . . . . . . . . . . . . . . . . . . . . . . . . . 921 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 termwise integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 tests, convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 19 theorems addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973, 975 arithmetic mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 Ballieu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Brauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086, 1088 Cayley–Hamilton . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 comparison . . . . . . . . . . . . . . . 1100, 1101, 1103, 1104 convolution . . . . . . . . . 1108, 1118, 1122, 1130, 1136 de Moivre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 final value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139 Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 general nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Gerschgorin . . . . . . . . . . . . . . . . . . . . . . . . . . 1083, 1088 Gram–Kowalewski . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 Green . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055, 1056 Hadamard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 initial value . . . . . . . . . . . . . . . . . . . . . . . . . . 1136, 1139 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 Jacobi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076

theorems (continued) Kneser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089, 1105 mean value . . . . . . . . . . . . . . . . . . . . . . 247, 1063, 1064 named . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 non-oscillation . . . . . . . . . . . . . . . . . . 1100, 1103, 1104 oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100 Ostrowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 Parodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 Parseval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 Perron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Perron–Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Poincare’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 rate of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 Routh–Hurwitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 second-order equations . . . . . . . . . . . . . . . . . . . . . 1100 separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Sonin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Stieltjes’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Stokes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Sturm comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 Sturm separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 Sturm–Picone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Sturmian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 summation . . 940, 986, 992, 998, 1002, 1030, 1042 Szeg¨ o comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 vector integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 Wielandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633, 877 Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . 761, 944 total variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 transformation formulas. . . . . . . . . . . . . . . . . . . . . . .1008 transforms Fourier . . . . . . . . . . . . . . . . . . . . see Fourier transform fractional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 Hankel. . . . . . . . . . . . . . . . . . . . . see Hankel transform integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 Laplace . . . . . . . . . . . . . . . . . . . see Laplace transform Mellin . . . . . . . . . . . . . . . . . . . . . . see Mellin transform of a derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 triangle inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 trigonometric functions . . . . . . . . . . . 28, 151, 390, 415 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 434 and associated Legendre functions . . . . . . . . . . . 779 and Bessel functions . . . . . . . . . . . 717, 727, 742, 747

INDEX OF CONCEPTS

1171

trigonometric functions (continued) and exponentials . . . . 227, 339, 485, 493, 495, 497, 522, 525, 599, 742 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 655 and hyperbolic functions . . . . . . 231, 509, 516, 522, 525, 747, 763 and hypergeometric functions . . . . . . . . . . . . . . . . 817 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 and inverse trigonometric functions . . . . . 605, 607 and logarithmic functions . . . . . 339, 581, 594, 599 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 763 and parabolic cylinder functions . . . . . . . . . . . . . 844 and powers . . . . . 214, 395, 397, 401, 405, 411, 436, 459, 475, 497, 516, 525, 594, 607, 727, 742, 779 and rational functions . . . . . . . . . . . . . . 401, 423, 447 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . .408 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 755 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 241 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151, 459 trigonometric series . . . . . . . . . . . . . . . . . . . . . . . . 46, 862 triple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 triple vector product . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 two-sided z-transform . . . . . . . . . . . . . . . . . . . . . . . . . 1135

V Vandermonde determinant . . . . . . . . . . . . . . . . . . . . 1078 variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 variational principles . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 vector differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

W Weber functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 Weierstrass elliptic functions . . . . 626, 873, 880, 1148 Weierstrass expansions . . . . . . . . . . . . . . . . . . . . . . . . . 869 weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982 Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 Wielandt theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Wronskian determinant . . . . . . . . . . . . . . . . . . . . . . . 1079

Y Young inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065

U

Z

uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 unilateral z-transform . . . . . . . . . . . . . . . . . . . 1135, 1138 unit integer function . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 unit integer pulse function . . . . . . . . . . . . . . . . . . . . 1135 use of the tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi

zeros . . . . . . . . . . . . . . . . . . . . . . 865, 870, 879, 972, 1038 interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 simple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 z-transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

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