A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis

To my departed mathematical forebears: Fourier, Dirichlet, Lipschitz, Lindemann, Hilbert, Steinhaus, Rajchman, Zygmund-all of them real analysts. And of course to Hypatia, who knows why I dedicate my books to her.

Steven G. Krantz

A Handbook of Real Variables With Applications to Differential Equations and Fourier Analysis

Birkhauser Boston • Basel· Berlin

Sleven G. Krantz Departmenl of Mathematics Washington University SI. Louis. MO 63130-4899 U.S.A.

Library of Congress Cataloging.in·Publication Data Krantz. Steven G. (Steven George), 1951· A handbook of real variables: with applications to differential equations and Pontier analysis I Sleven Krantz. p. em. Includes bibliograpbical references and index. ISBN ()'8176-4329·X (alk. paper) -ISBN 3·7643-4329·X (all<. paper) I. Functions of real variables. 2. Mathematical analysis. I. TiUe. QA331.5.K7 2003 5I5'.8-
2003050248 CIP

AMS Subject CJassjfications: Prima:y: 26-00, 26-01; SecondaJy: 26A03, 26A06, 26A09, 26A15, 42-01.35-01 ISBN ()'8176·4329·X ISBN 3-7643-4329-X

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Contents Preface

xi

1 Basics

1 1 1 4

1.1 1.2 1.3 1.4

1.5 1.6

2

Sets . . Operations on Sets Functions .... Operations on Functions Number Systems .. 1.5.1 The Real Numbers Countable and Uncountable Sets

.

Sequences 2.1 Introduction to Sequences. . . . 2.1.1 The Definition and Convergence 2.1.2 The Cauchy Criterion . 2.1.3 Monotonicity.. 2.1.4 The Pinching Principle . . . . . . 2.1.5 Subsequences.. . . . . . . 2.1.6 The Bolzano-Weierstrass Theorem . . . . 2.2 Limsup and Liminf . . . . . 2.3 Some Special Sequences

5

6 6 9 11 11 11 12

13 14 14 15 15

17

3 Series

21

3.1

21 21 22 23 23 24 24 25 25

3.2

3.3

Introduction to Series 3.1.1 The Definition and Convergence . . 3.1.2 Partial Sums . Elementary Convergence Tests . . . . . . . 3.2.1 The Comparison Test. . 3.2.2 The Cauchy Condensation Test . 3.2.3 Geometric Series . . 3.2.4 The Root Test. . 3.2.5 The Ratio Test . 3.2.6 Root and Ratio Tests for Divergence Advanced Convergence Tests . . . . . . . . 3.3.1 Summation by Parts .•..

27

28 28 v

Contents

vi 3.3.2 Abel's Test . . . . . .... 3.3.3 Absolute and Conditional Convergence 3.3.4 Rearrangements of Series 3.4 Some Particular Series 3.4.1 The Series for e .... 3.4.2 Other Representations for e 3.4.3 Sums of Powers. . . . 3.5 Operations on Series . . . . . . . 3.5.1 Sums and SCalar Products of Series . . . . 3.5.2 Products of Series .. . . . . . • . . . 3.5.3 The Cauchy Product . . . . . 4

5

29 31 32 33 33 34 35 36 36 36

37

The Topology of the Real Line 4.1 Open and Closed Sets . . . . . . . 4.1.1 Open Sets . . . . . • . . . 4.1.2 Closed Sets . . . . 4.1.3 Characterization of Open and Closed Sets in Terms of Sequences . . . . . 4.1.4 Further Properties of Open and Closed Sets 4.2 Other Distinguished Points . . . . . 4.2.1 Interior Points and Isolated Points . . . 4.2.2 Accumulation Points . . . . . . . . . . . 4.3 Bounded Sets . . . . . . . . . . . 4.4 Compact Sets . . . 4.4.1 Introduction. . . .. . . . . . 4.4.2 The Heine-Borel Theorem. Part I . . . . . 4.4.3 The Heine-Borel Theorem, Part n . . . . . 4.5 The Cantor Set . 4.6 Connected and Disconnected Sets .. . . . 4.6.1 Connectivity 4.7 Perfect Sets . . .

39 39 39 40

Limits and the Continuity of Functions 5.1 Definitions and Basic Properties . . . . . . . . •. . . •. . • 5.1.1 Limits............. . .....•.....• 5.1.2 A Limit that Does Not Exist. . . . . . . . • . . 5.1.3 Uniqueness of Limits . . . . . . . . . . 5.1.4 Properties of Limits . 5.1.5 Characterization of Limits Using Sequences 5.2 Continuous Functions . .. • • . • . . . . . . . . . . . • . . . . 5.2.1 Continuity at a Point . . . . . . . . . . . . . . . 5.2.2 The Topological Approach to Continuity .. . . . . 5.3 Topological Properties and Continuity .. . . . • . . . . . . . 5.3.1 The Image ofa Function . 5.3.2 Uniform Continuity.. •..........

53

41 42 43 43 43 44

45 45 45 47 48

50 50 51

53 53 54 54

55 57 57 57 60

62 62 63

Contents

5.4

6

7

vii 5.3.3 Continuity and Connectedness . 5.3.4 The Intermediate Value Property . . . . . Classifying Discontinuities and Monotonicity . . . 5.4.1 Left and Right Limits. . . 5.4.2 Types of Discontinuities . . . 5.4.3 Monotonic Functions

65 65 66 66 66 67

The Derivative 6.1 The Concept of Derivative . . . . . . . . . 6.1.1 The Definition .. . . . . . . . 6.1.2 Properties of the Derivative. 6.1.3 The Weierstrass Nowhere Differentiable Function 6.1.4 The Chain Rule .. 6.2 The Mean Value Theorem and Applications . . . . 6.2.1 Local Maxima and Minima. 6.2.2 Fermat's Test . . . . . 6.2.3 Darboux's Theorem .. 6.2.4 The Mean Value Theorem . 6.2.5 Examples of the Mean Value Theorem. 6.3 Further Results on the Theory of Differentiation 6.3.1 L'Hopital's Rule 6.3.2 The Derivative of an Inverse Function 6.3.3 Higher-Order Derivatives . 6.3.4 Continuous Differentiability . . . . . The Integral 7.1 The Concept oflntegral . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Partitions. . . . . 7.1.2 Refinements of Partitions . . . . . . 7.1.3 Existence of the Riemann Integral 7.1.4 Integrability of Continuous Functions ..... 7.2 Properties of the Riemann Integral . . . . . . . 7.2.1 Existence Theorems . . . . . . . . . . 7.2.2 Inequalities for Integrals . . . . . . . . . . . . . 7.2.3 Preservation oflntegrable Functions under Composition 7.2.4 The Fundamental Theorem of Calculus . . . . 7.3 Further Results on the Riemann Integral . . . . 7.3.1 The Riemann-Stieltjes Integral. . . . . 7.3.2 Riemann's Lemma 7.4 Advanced Results on Integration Theory . . . . . 7.4.1 Existence of the Riemann-Stieltjes Integral 7.4.2 Integration by Parts . . . . . 7.4.3 Linearity Properties . 7.4.4 Bounded Variation

71 71 71 72 73 74 74 74 74 75 75

78 79 79 81

82 82 85

. . . . . .

. .

85 85 88 89 89 89 89 91 91 92

93 93 96

96 96 97 98 98

viii 8 Sequences and Series of Functions 8.1 Partial Sums and Pointwise Convergence. . . . . 8.1.1 Sequences of Functions. • . . . . . . . . 8.1.2 Uniform Convergence . 8.2 More on Uniform Convergence. 8.2.1 Commutation of Limits. . . 8.2.2 The Uniform Cauchy Condition 8.2.3 Limits of Derivatives 8.3 Series of Functions . . . . . . . 8.3.1 Series and Partial Sums. 8.3.2 Uniform Convergence of a Series 8.3.3 The Weierstrass M -Test .. . . . . 8.4 The Weierstrass Approximation Theorem . 8.4.1 Weierstrass's Main Result .

Contents

103 . . . . . . . . • . 103 103 . . . . . . . . . . • . . . . . . .

104 106 106 107

107

lOS lOS 109 110 111 III

.

9 Some Special Functions 9.1 Power Series . 9.1.1 Convergence......... 9.1.2 Interval of Convergence . . . . . . . . . . . • . 9.1.3 Real Analytic Functions 9.1.4 Multiplication of Real Analytic Functions . . 9.1.5 Division of Real AnalytiC Functions 9.2 More on Power Series: Convergence Issues . . . •. . 9.2.1 The Hadamard Formula . 9.2.2 The Derived Series . • . . . . . 9.2.3 Formula for the Coefficients of a Power Series 9.2.4 Taylor's Expansion . . . . . 9.3 The Exponential and Trigonometric Functions . . 9.3.1 The Series Definition. 9.3.2 The Trigonometric Functions . 9.3.3 Euler's Formula. . . 9.3.4 The Trigonometric Functions. . .•.. 9.4 Logarithms and Powers of Real Numbers 9.4.1 The Logarithmic Function 9.4.2 Characterization of the Logarithm 9.5 The Gamma Function and Stirling's Formula .. 9.5.1 Stirling's Formula • . .. . . . . . . . . . 9.6 An Introduction to Fourier Series .•. . . •. ..... 9.6.1 Trigonometric Series . . . . 9.6.2 Formula for the Fourier Coefficients . . 9.6.3 Bessel's Inequality .. . . . . • . 9.6.4 The Dirichlet Kernel . . . . .•.....•....•.

.

113

113 113 114 114 115 115 117 117 118 119

121 123 123 126 126 127

130 130 132

132 133 133 134 135 135 136

Contents 10 Advanced Topics 10.1 Metric Spaces . . . . . 10.1.1 The Concept of a Metric ... 10.1.2 Examples of Metric Spaces .. 10.1.3 Convergence in a Metric Space .. 10.1.4 The Cauchy Criterion . 10.1.5 Completeness. . . . . . 10.1.6 Isolated Points . . 10.2 Topology in a Metric Space. 10.2.1 Balls in a Metric Space 10.2.2 Accumulation Points 10.2.3 Compactness 10.3 The Baire Category Theorem . 10.3.1 Density . . . . 10.3.2 Closure . . 10.3.3 Baire's Theorem ... 10.4 The Ascoli-Arzela Theorem . 10.4.1 Equicontinuity . 10.4.2 Equiboundedness 10.4.3 The Ascoli-Arzela Theorem

ix

139 139 139 139 140 . .

141 142 142 144 144 145 146

147 147 148 149 150 150 151 lSI

11 Differential Equations 11.1 Picard's Existence and Uniqueness Theorem 11.1.1 The Form of a Differential Equation . . 11.1.2 Picard's Iteration Technique . 11.1.3 Some Illustrative Examples . 11.1.4 Estimation of the Picard Iterates . . . . . . . . . . 11.2 The Method of Characteristics 11.3 Power Series Methods 11.4 Fourier Analytic Methods. . 11.4.1 Remarks on Different Fourier Notations 11.4.2 The Dirichlet Problem on the Disc . . 11.4.3 The Poisson Integral . . 11.4.4 The Wave Equation . .

153

Glossary of Terms from Real Variable Theory

177

List of Notation

189

Guide to the Literature

193

Bibliography

197

Index

199

153 153 154 ISS

156 158 160 168 168 169 172

174

Preface The subject of real analysis dates to the mid-nineteenth century - the days of Riemann and Cauchy and Weierstrass. Real analysis grew up as a way to make the calculus rigorous. Today the two subjects are intertwined in most people's minds. Yet calculus is only the first step of a longjoumey, and real analysis is one of the first great triumphs along that road. In real analysis we learn the rigorous theories of sequences and series, and the profound new insights that these tools make possible. We learn of the completeness of the real number system, and how this property makes the real numbers the natural set of limit points for the rational numbers. We learn of compact sets and uniform convergence. The great classical examples, such as the Weierstrass nowhere-differentiable function and the Cantor set, are part of the bedrock of the subject. Of course complete and rigorous treatments of the derivative and the integral are essential parts of this process. The Weierstrass approximation theorem, the Riemann integral, the Cauchy property for sequences, and many other deep ideas round out the picture of a powerful set of tools. And the world of applications has come to appreciate, and to embrace, real analysis. Many engineering programs and many curricula in physics, economics and other disciplines have come to require real analysis. The theories of Fourier analysis, approximation theory, splines, control theory, systems science, differential equations, and functional analysis (to name a few) all depend decisively on real analysis. Yet real analysis remains a recondite subject. dreamed up and designed mainly for the theoretical mathematician. The purpose of this book is to acknowledge that there is a large audience of scientists and others who wish to use the fruits of real analysis, and who are not equipped to (or do not have the time to) stop and appreciate all the theory. We have created a handbook so that those who use real analysis in the field can quickly look up ideas, without becoming bogged down in long explanations and proofs. Surely it does the applied mathematical scientist little good to endeavor to look up an idea like "uniform convergence" in a standard real analysis text and to find that he first must learn about sequences, about the Cauchy condition, about the completeness of the real numbers, and several other ancillary ideas before he can get an answer to his question. We wish here to cut through that impasse and provide a quick and decisive treatment of each key topic. Thus this is a book of practice, not of theory. We provide cogent and incisive explanations of each important idea. with suitable cross-references as needed. We xi

xii

Preface

provide an ample number of examples. and there are some proofs. We make the treatment of each topic as self-contained as possible. We never refer the reader to outside sources for key ideas. This book is entirely self-contained. Of course we do provide references for further reading. but these are optional. There is no other book like the present one on the market today. There are a great many fine texts on real analysis, and these are listed in the Bibliography and the Guide to the Literature. The present book may be thought of as a concordance and a tour of the subject. Many engineers and other applied scientists will find that they need look no further than this volume for the real analysis ideas that they require. When further reading is necessary. suitable references may be found right here. The author is grateful to the many fine analysis teachers that he has had over the years. and also to Washington University for providing him the freedom to engage in academic pursuits of this kind. He thanks his editor, Ann Kostant, for helping him to develop the idea for this book and for making the publication process as painless and expeditious as possible. In addition. the author is very grateful for the careful work of his copyeditor. Avanti Athreya. All errors and malapropisms are the sole province of the author. He is always happy to hear of corrections and criticisms so that future editions may be made more accurate and useful. Steven G. Krantz Washington University. St. Louis. Missouri

A Handbook ofReal Variables

Chapter 1

Basics 1.1

Sets

Set theory is the bedrock of all of modem mathematics. A set is a collection of objects. We usually denote a set by an upper case roman letter. If S is a set and s is one of the objects in that set. then we say that s is an element of S and we write s e S. If t is /lot an element of S. then we write t rt S. Some of the sets that we study will be specified just by listing their elements: S = (2.4.6. 8). More often we shall use set-builder /lotatio/l: S = (x e JR : 4 < x 2 + 3 < 9). The collection of all objects /lot in the set S is called the compleme/lt of S and is denoted by'S. The complement of S must be understood in the context of some "universal set"-see Example I.I. If Sand T are sets and if each element of S is also an element of T. then we say that S is a subset of T and we write S CT. If S is /lot a subset of T. then we write SfJ:.T Example 1.1 Let S = (a. b. c. d. e). T = (a. c. e. g. i). and U = (a. c. d). Then

aeS. aeT. deS. drtT. UCS. UfJ:.T. If the universe is understood to be the standard 26 roman letters. then it follows that 'T = (b.d,J.h.j.k.I •...• x.y.z).

1.2

o

Operations on Sets

If Sand T are sets. then we let S n T denote the collection of all objects that are both in S and in T. We call S n T the i/ltersectio/l of Sand T In case S•• S2. S) • ... are sets. then the collection of all objects common to all the I

Chapter I: Basics

2 Sj. called the intersection of the Sj. is denoted by

or

nSj. j

If Sand T are sets. then we let S U T denote the collection of all objects that are either in S or in T or both. We call S U T the unioll of Sand T In case S. , S2. S3. . .. are sets. then the collection of all objects that lie in at least one of the Sj. called the ullion of the Sj. is denoted by or

USj. i

Figure 1.1 illustrates the concepts of intersection and union. by way of what is known as a Vellli diagram. Example /.2

Let S = {l. 2. 3.4. 5} and T = {2. 4. 6. 8. to}. Then S n T = {2.4}

and

o

S U T = {l. 2. 3, 4. 5. 6, 8.IO}.

If Sand T are sets. then we let

5 x T = {(so t) : s E 5 and t

E T} .

We call 5 x T the cartesian product of S and T Observe that 5 x T and T x S are distinct. Sometimes we will take the product of finitely many sets S•• S2 • ...• Sk. Thus 5.

X

S2

X ..• X

5k = {(SJ,S2•...• Sk) :Sj

E

5j forallj = I •...• k}

If Sand T are sets. then we let their set-theoretic difference be

S\ T

= (x e S : x

¢ T) .

=

If 5. T C lit the real numbers. then observe that C S lR \ 5 and S \ T Figure 1.2 illustrates the concept of set-theoretic difference.

= S n cT

Example J.3

Let 5 = (a. b. 1.2). T = {b. c. d. 2. 5}. and U = {a. Pl. Then

S\ T = {a. I}

and

T\5={c.d.5}.

Also S x U = {(a. a). (b. a). (I. a), (2. a). (a. 13). (b. 13). (1.13). (2. PH

1.2

3

Operations on Sets

T Two sets S and T

T Intersection

Union Figure l.l

T Two sets Sand T

T S\T Figure 1.2

4

Chapter 1: Basics and

ux

S = «a, a), (a, b), (a, 1), (a, 2), (/3, a), (~, b), (~, I), (/3,2»).

0

We conclude by noting that there is a distinguished set that will arise frequently in our work. That is the empty set 0. The empty set is the set with no elements.

1.3 Functions Let S and T be sets. Afimction ofS a unique element of T.

f

from S to T is a rule that assigns to each element

Example J.4 Let S = (l, 2, 3) and T = (a, b). The rule l---+a 2---+ a 3---+b

is a function, just because it assigns a unique element of T to each element of S. Notice that the function assigns the same element of T to each of 1 0 and 2 in S; that is allowed. We write f : S .... T if f is a function from S to T We call S the domain of { and we call T the range of {.

Example J.5 Let S

= (a, b,x) and T = {I,a, y). Define the function {by f:

{::~ X"" a

Then the domain of f is the set S itself. There are several choices for the range. Certainly the set (a, I) can be said to be the range. Also the entire 0 set T can be said to be the range. Many of our functions are given by formulas. If we write, for example, {(x) = .j2x + 5, then we mean that the function { assigns to each number x the number obtained by doubling x and adding 5 and then taking the principal (or positive) square roOI. In this situation, we understand the domain of f to be all numbers x for which the formula defining f makes sense - for this example, the domain is (x : x 2: -5/2). We understand the range of { to be any set containing all the values of {-for this example, the range could be taken to be (x : x > 0). If the function f has domain S and range T, and if for each element t E T there is some s E S such that res) = t, then we say that f is onto.

1.4

5

Operations on Functions

If the function g has domain S and range T, and if the only way that !(SI) can equal !(S2) is if SI = sz, then we say that 1 is one-Io-one.

Example 1.6 LetS = (-3,-2, -I,O,I,2,3}andletT = (O,I,4,9}.Letthefunction! be given by I(x) = x 2 Then the set of all values of I, applied to elements of S, is (0, 1,4, 9). Therefore 1 is onto. However, notice that [(-2) = 1(2) = 4. Therefore the function I is not one-te-one. 0

1.4 Operations on Functions Let

I

and g be functions with domain S and range T. We define

• If + g](x) = {(x) + g(x) • If • If

g](x)

g(x)

g](x) - I(x) . g(x)

[I]

•

= {(x) -

-

g

(x)

[(x) . =provldedthatg(x) ,!-O. g(x)

Example 1.7 Let I(x)

=x3 -

x and g(x)

=x4

Then

[I + g ](x) = x 3 - x [I - g ](x) = x 3

+ x4

x - x4

-

[/·g](x) = (x 3 -x) x 4 =x7 _x S

[gI] (x) = x3x-x

o

4

If I by

:S

-+ T and g : T -+ U, then we may consider the function g

0 {

defined

go I(x) = g(/(x». If I : S -+ T is both one-to-one and onto, then we may define a function I-I by the rule I-I (x) = y if and only if I(Y) = x. We call I-I the inverse of the function I. We have the essential properties

10

r

l (I)

=

1 for all 1 E

T

and

I-I

0

{(s) =

Example 1.8 Let I(x) = x 2 - 3x andg(x) = x 3 + \. Then

[0 g(x) = [x 3 + If - 3· [x 3 + I)

S

for all S E S.

6

Chapter I: Basics and go {(x) = [x 2 - 3x]3

+I.

The function / is not one-ta-one because /(0) = {(3) = O. But g : lR. -+ IR is both one-ta-one and onto. We may solve the equation gog-I(x) =x

to find that

or

o

1.5 Number Systems The most rudimentary number system is the natural numbers. These are the counting numbers 1.2.3•.... We denote the natural numbers by the symbol 1\1. Of the four standard arithmetic operations. the natural numbers are closed only under addition and multiplication. The integers comprise both the positive and negative whole numbers and also O. We denote the set of all integers by Z. Of the four standard arithmetic operations, the integers are closed under addition. subtraction. and multiplication. The ratio/lal numbers consist of all quotients of integers. Thus III / /I is a rational number provided that m. n E Z and n l' o. We denote the set of all rational numbers by Q. The set of rational numbers is closed under all four of the standard arithmetic operations. except that division by 0 is not allowed.

Example /.9 The number 4 is a natural number. Of course it is also an integer. Writing it as 4 = 4/1. we also see that this number is a rational number. The number -6 is an integer. It is /lot a natural number. Writing it as -6 = (-6)/1. we also see that this number is a rational number. The number 2/3 is /lot an integer. It is /lot a natural number. But it is a 0 rational number. The number system of greatest interest to us is the real number system. It contains the rational numbers. and has a number of other interesting properties as well. We explore the real numbers in the next subsection.

1.5.1

The Real Numbers

The rational numbers are afield. This means that there are operations of addition (+) and multiplication (x) that satisfy the usual laws of arithmetic. But in addition the field Q satisfies certain properties of the ordering «):

1.5

Number Systems

7

1. Ifx,y,z eQandy 0, and y > 0, then x . y > O.

Thus Q is an orderedfield. The real numbers will be an ordered field containing the rationals and satisfying an additional completeness properly. We formulate that property in terms of the concept of least upper bound.

Definition 1.1 Let S C JR. The set S is called bounded above if there is an element b E JR such that x :5 b for all XES. We call the element b an upper bound for the set S. Definition 1.2 Let S C JR. An element b E JR is called a least upper bound (or supremum) for S if b is an upper bound for S and there is no upper bound b' for S which is less than b. We write b = sup S = lub S. Example 1.10 Let S = {x e Q : 3 < x < 51. Then the number 9 is an upper bound for S. Also 7 is an upper bound. The least upper bound for S is 5. We write 5=lubS. By its very definition, if a least upper bound exists, then it is unique. Before we go on, let us record a companion notion for lower bounds:

Definition 1.3 Let S C lR. The set S is called bounded below if there is an element c E JR such that x > c for all xES. We call the element c a lower bound for the set

S. Definition 1.4 Let S c JR. An element c e JR is called a greatest lower bound (or infimum) for S if c is a lower bound for S and there is no lower bound c' for S which is greaterthan c. We write c = inf S = glb S. By its very definition, if a greatest lower bound exists, then it is unique. Example

J.n

LetS = (x e JR:O <x < 1) and T = (x eJR: 0 <x < 1). Then-l isa lower bound both for S and for T and 0 is the greatest lower bound for both sets. We write 0 = glb Sand 0 = glb S. Notice that 0 ¢ S while 0 e T. Also 5 is an upper bound both for S and for T. And 1 is the least upper bound for both sets. We write 1 = lub S and 1 = lub T Observe that 1 is 0 not in S and is not in T.

8

Chapter 1: Basics Now we have:

Theorem 1.1 There exists an ordered field lR which (i) contains iQ and (ii) has the property that any nonempty subset oflR which has an upper bound also has a least upper bound (that is also an element oflR). An equivalent. companion. statement is that if T is any set that is bounded below then T has a greatest lower bound (that is also an element of lR).

Example J.J2 It is known (see (KRA2. page 114]) that there is no rational number whose square is 2. (The details of this assertion are provided in Example 1.13.) Now let S = Ix E lR : x > 0 and x 2 < 2) . Of course S is bounded above (by 2. for example). So there is a least upper bound Ci. Of course Ci will be an element of lR. but Ci ¢ iQ. It can be shown that Ci 2 = 2 (see (KRA1. Section 2.5. Theorem 12). Thus the real number system contains numbers that are missing from the rational number system. 0 These are called the irratiollalllllmbers. It can also be shown that the number 7( • which represents the ratio of the circumference of a circle to its diameter. is not a rational number. But 7( does exist as a real number.

Example 1.13 Let US confirm that ..fi is IIOt a rational number. Suppose to the contrary that it is. So ..fi = p/q. with p and q integers. By division. we may suppose that p and q have no cornmon divisors. Thus

Multiplying this out gives 2q2 = p2 Since 2 divides the left side. we conclude that 2 divides the right side. So 2 divides p. We may write p = 2r for r an integer. Thus we have Simplifying gives

q2 = 2r2 Since 2 divides the right side. we conclude that 2 divides the left side. So 2 divides q. But we have shown now that 2 divides p and also that 2 divides q.

9

1.6 Countable and Uncountable Sets This contradicts the assumption that p and q have no common divisors. We conclude that .J2 cannot be rational. 0

It is considerably more difficult to prove that 7C is irrational. We cannot treat the mailer here. We shall learn below that the set of numbers IR \ Q (the irrational numbers) is much larger than Q itself. Thus "most" real numbers are not rational.

1.6

Countable and Uncountable Sets

Georg Cantor's theory of countable and uncountable sets, and more generally of many orders of infinity. is an integral part of any treatment of real analysis. What we give here is a summary. Complete treatments may be found in [KRAI. Section 1.8] and [KRA2. Section 5.8]. Two sets Sand T are said to have the same cardi/lality if there is a one-to-one. onto function tP : S --> T We write card S = card T. In this context we refer to such a function tP as a set-theoretic isomorphism. or just an isomorphism. The surprise is that some unlikely pairs of sets have the same cardinality. In particular. it is possible for S C T. S", T. and yet card S = cardT Example 1.14

Let A = [0••• "'I. B =

1%. &. #}. and C =

II. 2}. Then

is a set-theoretic isomorphism of A to B. This gives mathematical confirmation of the obvious fact that A and B have the same cardinality. On the other hand. it is impossible to construct a set-theoretic isomorphism from A to C. So A and C do /lot have the same cardinality. 0 Example US

Let S = [...• -6. -4. -2. O. 2. 4, 6•... } (the even integers) and let T = Z. Then obviously S C T but S ", T Yet tP(lI) = 11/2 is an isomorphism ofSto T. So card S=cardT 0

If two sets have the same cardinality. then we think of them as being the same size. For finite sets. this idea coincides with our intuition: two sets have the same cardinality if and only if they have the same (finite) number of elements. But for infinite sets. this says something new. If a set S has the same cardinality as N. the natural numbers. then we say that S is COUll/able.

10

Chapter 1: Basics

Example 1.16 Let S = (... , -6. -4. -2. O. 2. 4. 6•... ). Then the last example shows that S is countable. A similar argument shows that T = (.... -5, -3. -I, 1.3.5•... ) is countable. We will see below that the set IR of real numbers is not countable. We say that IR is uncountable. 0 Cantor's great insight was that the set IR of real numbers is in fact not countable (see (KRAI, Section 1.8) or (KRA2, Subsection 5.8.3) for a proof). If S is infinite and has cardinality different from the cardinality ofN, then we say that S is uncoulltable. We now list some of the key properties of countable sets: 1. If both Sand T are countable, then S U T. S nT, and S x T are at most countable. 2. If X is uncountable and Y ;:) X. then Y is uncountable. 3. If X is countable and Y

ex, then Y is at most countable.

Here we have used the phrase "at most countable" to mean either countably infinite or finite or empty. It is common to refer to a set of this type as "denumerable" (although many sources do not make this distinction very clearly). The word "countable" is reserved for infinite sets that have the same cardinality as the set N.

Example 1.17 The set iQl can be identified in a natural way with a subset of Z x Z. using 0 the map m/n ~ (m,n). It follows that iQl is countable.

Example U8 The set of all lines in the plane which contain (at least) two points having integer coefficients is countable. For each line may be identified with the 4tuple of integers coming from the coordinates of the two given points. And 0 the set of such 4-tuples is countable.

Example 1.19 The set Z x IR is uncountable. for it contains a copy of IR by way of the map IR 3 x ~ (0, x) e Z x IR.

o

Example 1.20 The set
+ iO e
o

Chapter 2

Sequences 2.1

Introduction to Sequences

2.1.1

The Definition and Convergence

Informally. a sequence is a list of numbers:

In more formal treatments, we say that a sequence on a set S is a function f from N to S. and we identify !(i) with aj. Such precision will not be required here. Example 2.1

Let i(j) =

Iii

This function defines the sequence III

12 ' 22 '

32 •....

We also write aj

1 = "':2 J

o

The primary property of a sequence is its convergence or its nonconvergence. We say that a sequence (a j) converges to a numerical limit l if. for every E > 0, there is a positive integer N such that i > N implies that la j - II < E. We write limj_co a j = e. Otherwise we say that the sequence diverges. Example 2.2

Consider the sequence 1,1/2,1/3• .... This sequence converges to O. To see this assertion, let E > O. Choose N so large that liN < E. Then, if i > N. it follows thatll/i - 01 = Iii < liN < E. Thus the sequence 0 converges to O.

11

Chapter 2: Sequences

12

Example 2.3 Consider the sequence -1.1. -1,1•.... This sequence does not converge. We commonly say that it diverges. To see this, let E = 1/2. Denote an element of the sequence by aj = (-I)j. Suppose that there were a limit £ and an N > 0 such that j > N implies that Ja j - £1 < E = 1/2. It follows that. for j > N, we have 2 = laj -aj+ll = I(aj

-e) + (£ -aj+I)1 -£1 + 1£ -aj+d

< laj <E+E =1.

This. of course, is a contradiction. It is not the case that 2 < 1. So the limit £ does not exist. 0

2.1.2

The Cauchy Criterion

It is convenient to be able to discuss the convergence or divergence of a sequence without making direct reference to the (putative) limit value £. This is the significance of the Cauchy criterion or Cauchy condition. Let (ajl be a sequence. We say that the sequence satisfies the Cauchy criterion if. for each E > 0, there is an N > 0 such that, whenever j, k > N. then la j - ak I < E. The Cauchy condition says. in effect. that the elements of the sequence are getting ever closer together (without making any statement about what point they may be getting close to). We sometimes say that a sequence satisfying this condition "is Cauchy."

Example 2.4 Let a j = 1/2 j This sequence is Cauchy. For let E > O. Choose N so large that 1/2N < E. Then, for k > j > N,

laj -akl < lajl < laNI = TN < E. Thus the sequence is Cauchy.

o

The significance of the Cauchy criterion is validated by the following result.

Proposition 2.1 Let {ajl bea Cauchy sequence. Then {ajl converges to an element of JR. Conversely, a convergent sequence in JR satisfies the Cauchy criterion. The proof of this result involves a careful investigation of the completeness of the real number system. We cannot treat the matter here.

In particular, it follows from the last proposition that any Cauchy sequence will /rave a limit ill JR.

2.1

Introduction to Sequences

13

The intuitive content of the Cauchy condition is that a Cauchy sequence gets close together and stays close together. With this thought in mind, we readily see that the sequence in Example 2.3 cannot be Cauchy, so it cannot converge.

2.1.3 Monotonicity Definition 2.1

Let (a i) be a sequence of real numbers. The sequence is said to be monotone increasing if al :5 a2 :5 .... It is monotone decreasing if al ::: a2 ::: .... The word "monotone" is used here primarily for reasons of tradition. In many contexts the word is redundant and we omit it. Example 2.5

Let ai = 1I./T Then the sequence (ai) is monotone decreasing. Let bi = (j - 1)lj. Then the sequence (bi) is monotone increasing. 0

Proposition 2.2 If (a i) is a monotone increasing sequence which is bounded above, that is, ai :5 M < 00 for all j. then (a i) is convergent. If (a i) is a monotone decreasing sequence which is bounded below. that is. ai ::: N > -00 for all j. then (a i) is conve~ent.

Corollmy 2.1 Let S be a set of real numbers which is bounded above and below. Let fJ be its supremum and ex i/s infimum. If E > 0 then there are s, t E S such that Is - fJl < E and It - exl < E. This fact can now be construed in the language of sequences:

Corollmy 2.2 Let S be a set of real numbers which is bounded above and below. Let fJ be its supremum and ex its infimum. There is a sequence (a i) c S and a sequence (b i) c S such that ai --+ ex and bi ...... fJ· Example 2.6

Let S = (x E IR : 0 < X < 1). Of course the infimum of S is 0, and the sequence ai = Ilj E S converges to O. Likewise, the supremum of S is I. and the sequence bi = (j - I)lj E S converges to I. 0

14

Chapter 2: Sequences

2.1.4 The Pinching Principle We next tum to one of the most useful results for calculating the limit of a sequence:

Proposition 2.3 [The Pinching Principle]

Let (aj}, (bj}, and (ej) be sequences of real numbers

satisfying

-

-

aj
for every j. If lim a' = lim Cj = j~oo J j-+oo

ex

for some real number ex, then .lim bj =ex. }-+oo

Example 2.7

Let a j = [sin ill j2. Observe that

1

1

J

I

--=2 < aj ::: -=2' The two sequences between which (aj) is pinched obviously tend to zero (see Example 2.1). Hence (aj) converges to O. 0

2.1.5 Subsequences Let (a)) be a given sequence. If O
are positive integers, then the function

is called a subsequence of the given sequence. We usually write the subsequence as or Sometimes a sequence will be divergent, but it will have a convergent subsequence. Example 2.8

Consider the sequence a) - (-1)), just as in Example 2.3. The subsequences 1,1,1, 1, ... and the subsequence -I, -I, -I, -I .... are both convergent. 0

2.2 Limsup and Liminf

15

A basic result about subsequences is this.

Proposition 2.4 If{aj} is a convergent sequence with limit l, then every subsequence converges tol. Conversely, if {b j} is a given sequence, and ifevery subsequence converges to some limit m, then the full sequence converges to the limit III. Exalllple 2.9 Let at = 1/2. Let a2 be chosen so that la21 < I, l-la21 < 1/2{1-lat 1>, and with randomly selected sign. Inductively. choose a j+1 such that laj+11 < 1. 1 -Iaj+ll < 1/2{1 -Iajl>. and with randomly selected sign. Then it easy to see that there is either a monotone increasing subsequence or a monotone decreasing subsequence of the aj. Since the full (divergent) sequence is bounded in absolute value by I. we conclude (Proposition 2.2) that this subsequence converges. 0

2.1.6 The Bolzano-Weierstrass Theorem The fundamental theorem about the existence of convergent subsequences is this:

Theorem 2. I Bolzano-Weierstrass Let {aj} be a bounded sequence. Then there is a convergent subsequence. Example 2.10 We know that the set Ql of rational numbers in the unit interval {O, I) is countable. Let them be enumerated as {at. a2, ... }. This sequence will be a quite chaotic subset of the unit interval. Nevertheless, the BolzanoWeierstrass theorem guarantees that there will be a convergent subsequence. Likewise, the sequence a j = sin j is a bounded sequence. If you write out the first ten or twenty terms (use your calculator), you will see that this too is a rather unpredictable sequence. But the theorem guarantees the existence of a convergent subsequence. 0

2.2 Limsup and Liminf Let {a j I be any sequence of real numbers. The limit supremum of this sequence is the greatest limit of al1 subsequences of the given sequence. More rigorously, for each j let

Then {A j I is a monotone decreasing sequence (since as j becomes large we are taking the supremum of a smaller set of numbers), so it has a limit. We define the

Chapter 2: Sequences

16 limit supremum of (a j } to be limsupaj = .lim Aj. ' .... 00

Note that the limit supremum may be ±oo. Likewise. the limit infimum of the given sequence is the least limit of all subsequences of the given sequence. In detail. let

Then (Bj} is a monotone increasing sequence (since as j becomes large we are taking the infimum of a smaller set of numbers). so it has a limit. We define the limit infimum of (a j} to be Iiminfaj = .lim Bj. ' ....00

Note that the limit infimum may be ±oo.

Example 2.11 Consider the sequence aj = (-I)j Then its limit supremum is 1 and its limit infimum is -1. It is less obvious. but true. that the limit supremum of the sequence (sin j} is 1 and the limit infimum of this sequence is -1. 0 The following result is now intuitively obvious, but worth noting explicitly.

Proposition 2.5 Let (aj} be a sequence and set Iimsupaj = f3 and Iiminfaj = a. Assume that a, f3 are finite real numbers. Let E > O. Then there are arbitrarily large j such that a + E > a j > a - E. Also there are arbitrarily large k such that f3 - E < ak < f3 + E. Compare Corollaries 2.1.2.2. Exwllple 2.12 Let aj = I sin j I. Then you can convince yourself that the limit supremum of (a j} is 1. And the limit infimum is 0- use your calculator. for instance. In the course of calculating. you will have produced elements of the sequence that are arbitrarily near to 0, and you will also have produced elements that are arbitrarily near to 1. Thus your calculations illustrate the proposition. 0 We conclude this brief consideration of lim sup and lim inf with a result that ties all the ideas together.

Proposition 2.6 Let (a j} be a sequence of real numbers. Let f3

= lim sup aj j-+oo

and a

=

Ii.m inf aj. J-+OO

2.3

17

Some Special Sequences

If (a jk I is any subsequence of the given sequence, then

Moreover, there is a subsequence (a j, I such that

and another sequence (a jm I such that lim ajm =

m_oo

p.

Again, compare with Corollary 2.2.

Example 2.13 Let

a j = ; . 7f

-

[the greatest integer not exceeding j . 7f] .

Of course every element of (ajl lies between 0 and 1. And none is equal to 0 or 1. You can use your calculator to convince yourself that there are elements of the sequence that are arbitrarily near to 0 and other elements that are arbitrarily near to 1. Thus you will see empirically that there is a subsequence converging to 0, and another subsequence converging to 1. 0

2.3 Some Special Sequences It is useful 10 have a collection of special sequences for comparison and study.

Example 2.14 Fix a real number J-. The sequence (J-j I is called a power sequence. If -1 < J- < 1. then the sequence converges to O. If JI, then the sequence is a constant sequence and converges to 1. If J- > I, then the sequence converges to +00. Finally, if J- ~ -1. then the sequence diverges. 0

=

For a > O. we define amln = (am)l/n •

where n is a positive integer and m E Z. Here the nib root (i.e., (1/ n)lh power) of a positive number is defined just like the square root was in Example 1.12. Thus we may talk about rational powers of a positive number. Next, if P E JR, then we may define a P = sup(a q : q E Q1. q < PI. Thus we can define any real power of a positive real number.

18

Chapter 2: Sequences

Lemma 2.1 If Ol > 1 is a real number and f3 > O. then 0lf3 > 1.

Example 2.15 Fix a real number Ol and consider the sequence (j"). If Ol > O. then it is easy to see that j" ~ +00; to verify this assertion. fix M > 0 and lake the number N to be the first integer after Mil". If Ol = O. then j" is a constant sequence. identically equal to 1. If Ol < O. then j" = I/r". The denominator of this last expression tends 0 to +00; hence the sequence j" tends to O. Example 2.16 The sequence (jl/j) converges to I. In fact, consider the expressions Olj = j Iii _ 1 > O. We have (by the Binomial Theorem-see Section 11.3) that j = (Olj

+ I)i ?

JJ:; .( .

1) (Oli)2.

Thus

0< Oli :: J2/{j - 1)

as long as j ? 2. It follows from Proposition 2.3 that Oli ~ 0 or j1/i -+ 1.

o

Example 2.17 LetOl be a positive real number. Then the sequence Oli/i converges to 1. To see this. first note that the case Ol = 1 is trivial. and the case Ol > I implies the case Ol < 1 (by taking reciprocals). So we concentrate on Ol > 1. But then we have 1 < Olili < jlli when j > Ol. Since j1/i tends to 1. PropoSition 2.3 applies and the argument 0 is complete. Example 2.18 Let A > 1 and let Ol be real. Then the sequence

{~~CI converges to O. To see this. fix an integer k > Ol and consider j > 2k. (Notice that k is fixed once and for all but j will be allowed to tend to +00 at the appropriate moment.) Writing A= 1 + IL. IL > O. we have that Ai = (IL+I)i > j(j -I)(j -2) ... (j -k+ 1) ILk .Ij-k. k(k - l)(k - 2) .. ·2 . I

2.3

19

Some Special Sequences Of course this comes from picking out the kth term of the binomial expansion for (J1. + 1)i. Notice that since j > 2k then each of the expressions j. (j - 1)•... (j - k + 1) in the numerator on the right exceeds j 12. Thus ·k

») > /

2 . k!

and

r o < -. <'

.a

),.J

2k ·k! "k

J • J1.

k

. J1.k

r=

k

·2k ·k! J1.

Since a - k < O. the right side tends to 0 as j ~

k

00.

o

Example 2.19

The sequence

{(I +~)j}~ ,

J=I

converges. In fact it is monotone increasing and bounded above. Use the Binomial Expansion to verify this assertion. The limit of the sequence is the number that we shall later call e (in honor of Leonard Euler. 17071783. who first studied it in detail). We shall study this sequence further in Section 3.4. 0 Example 2.20

The sequence

{ (l-~)j}~ J

J=\

converges to lie. where the definition of e is given in the last example. More generally. the sequence

converges to t! (here t! is defined as in the discussion following Example 2.14 above). 0

Chapter 3

Series 3.1

Introduction to Series 3.1.1 The Definition and Convergence A series is, informally speaking, an infinite sum. We write a series as

We think of the series as meaning 00

:~:>j =CI +C2+C3+···· j=1

The basic question about a series is, "Does the series converge?" That is. does the infinite sum have any meaning? Does it represent some finite real number?

Example 3.1 Consider the series

1

00

L3

j

j=1

Although we do not yet know the rigorous ideas connected with the convergence of series. we may think about this series heuristically. We may consider the "sum" of this series by adding together finitely many of its terms: N 1 SN=

L-·· 3J j=1

!O -

It is easy to calculate that SN = 3- N ). Thus the limit as N tends to 00 of SN is We intuit therefore that the sum of this series is The theory presented below will confirm this calculation. 0

!.

!.

21

22

Chapter 3: Series

3.1.2 Partial Sums With a view to answering the convergence question, we define the partial sum of the series to be

We say that the series converges if the sequence of partial sums (SN} converges to a finite limit.

Exnmple3.2 Let Cj = 2- j • Then the Nih partial sum of L~I Cj is

We readily see that lim SN

N-+oo

= N-+oo lim 1 -

TN

= 1.

Thus the limit of the partial sums exists and the series converges.

0

The series in the last example is commonly known as a geometric series.

Example 3.3 Let Cj = <-I)j. Then the sequence of partial sums is

-1. O. -1. 0•.... It is plain that this sequence has no limit. So the series converge.

Example 3.4 Let Cj = 1/j. Then observe that 1

S2=1+2

11I

[I] [I I]

S4=1+-+-+-=I+ -2 + -+2 3 4 3 4

[1] [1 I]

1

J >1+ -2 + -+=1+-+4 4 2:1

L j Cj

does not 0

23

3.2 Elementary Convergence Tests

1 1 1 1 111

~=1+-+-+-+-+-+-+-

2

3

4

5

678

= 1+ [~] + [~+~] + U+ ~+~ +4] > 1+ ~ + [~ + ~] + [~ + ~ + ~ + ~] 2448888

111 =1+ 2 + 2 + 2 etc.

We see that the sequence SI, S2, ... of partial sums is strictly increasing. and that it has a subsequence that tends to +00. Thus the sequence of partial sums does not tend to a finite limit, and the series diverges. 0 The series in the last example is commonly known as the harmonic series.

3.2

Elementary Convergence Tests

3.2.1

The Comparison Test

Proposition 3.1 Suppose that L~I cJ is a convergent series of nonnegative terms. If {bj} are real numbers. and iflbJI ~ cJ for every j, then the series ~I bj converges. Corollary 3.1 If L~I cj is as in the proposition, and if 0 ~ bj ~ cJ for every j. then the series L~I bJ converges. Example 3.5

The series L~l 2-J sin j is seen to converge by comparing it with the series L~I 2- j

0

Example 3.6

The series L~I In j /3 j is seen to converge by comparing it with the series

L~ll/2J.

0

24

Chapter 3: Series

3.2.2 The Cauchy Condensation Test Theorem 3.1 Cauchy Condensation Test Assume thaI CJ > C2 ~ ••• ~ cj ~ ••• O. The series

converges if and only if the series

converges.

Example3.?

We apply the Cauchy condensation test to the harmonic series 00

1

j=1

J

z:>·

It leads us to examine the series

Since the latter series diverges. the harmonic series diverges as wen.

0

Example 3.8

The series 001

L-:r

j=1 J

converges if r is a real number that exceeds 1 and diverges if r < 1. We leave the details as an exercise for the reader.

3.2.3 Geometric Series Proposition 3.2 LeI Ci be a fixed real number. The series

25

3.2 Elemenlary Convergence TeslS

is called a geometric series. It is useful to write SN

= I +a +a2 + ... +a N - 1 +a N :

hence

Thus SN

1- aN+! = ---.,.--

I-a It follows that the series converges if and only iflal < I. In this circumstance, the sum of the series (that is, the limit of the partial sums) is 1/(1 - a). Note that we already examined a particular geometric series in Example 3.1.

Example 3.9 Consider the series Lj(3.1)-j This is a geometric series. The partial sums are 1- 3.1-(N+I) SN = 1- 3.1 I The series converges to S= 31

21

3.2.4 The Root Test Theorem 3.2 Root Test Consider the series

If

then the series converges.

3.2.5 The Ratio Test Theorem 3.3 Ratio Test Consider the series

o

26

Chapter 3: Series

If Ii~ sup J-+OO

I I C '+1 _J_._ CJ

< 1,

then the series converges. Remark 3.1 If a series passes the Ratio Test, then it passes the Root Test (the converse is not true). Put another way. the Root Test is a better test than the Ratio Test, because it will give information whenever the Ratio Test does, and also in some circumstances when the Ratio Test does not. Why do we therefore learn the Ratio Test? The answer is that there are circumI stances when the Ratio Test is much easier to apply than the Root Test. Example 3.10

The series 00

2j

L"":'J.j" j=]

is easily studied using the Ratio Test (recall that j! Indeed Cj = 2 j jj! and Cj+ll Cj

=j

. (j - I) ... 2· I).

j

= 2 +l j(j + I)!. 2Jjj!

We can perform the division to see that

IC~;II =

j:I'

The lim sup of the last expression is O. By the Ratio Test, the series converges. Notice that in this example. while the Root Test applies in principle. it would be difficult to use in practice. 0 Exomple 3.11

We apply the Root Test to the series 00

·2

L~ 2J

j=l

Observe that ·2

J

CJ =-. 2J

27

3.2 Elementary Convergence Tests hence

As j ....

00,

we see that lim sup ICjl

II' J

1 = -2 < 1.

1-+00

By the Root Test, the series converges.

o

3.2.6 Root and Ratio Tests for Divergence It is natural to ask whether the Ratio and Root Tests can detect divergence. Neither test is necessary and sufficient: there are series which elude the analysis of both tests. We still have these useful results:

Theorem 3.4 Root Test for Divergence Consider the series

If

then the series diverges.

Theorem 3.5 Ratio Test for Divergence Consider the series

If there is an N > 0 such that

IC~;II:::

1 , forallj::: N

then the series diverges. In both the Root Test and the Ratio Test, if the lim sup is equal to 1, then no conclusion is possible. Example 3.12

Consider the series

00

L

j=1

.jf2

J

3j

Chapter 3: Series

28 Setting Cj = jil2 /3 j • we calculate that .lim )-+00

Icl lj =

l

.lim j l2/3 J-+OO

= +00.

We conclude. by Theorem 3.4. that the series diverges. Now consider the series

1

00

L":2J

j=1

Ifwesetcj = l/p.thenweseethat I · ICj Il/j .Im J-+OO

I·1m [.J/J·]2 I = j-+oo =I. J

The Root Test therefore gives us no information. However. one can use the Cauchy Condensation Test to see that the series converges. See also Example 3.8. 0 Example 3.13

Consider the series

00

.,

L 4J: J

j=1

Setting Cj = j !/4i. we calculate that .lim ICj+I/cjl ) .....00

= J-+OO .lim [j + IJ/4 = +00.

We conclude. by Theorem 3.5. that the series diverges. Now consider the series 00 I

L"7I

j=1

If we set Cj = I/j then we see that • Cj+l . j 11m - - = 11m - - = 1.

j-+oo

C

j

j-+oo j

+1

The Ratio Test therefore gives us no information. However, one can use the Cauchy Condensation Test to see that the series converges. See also Example 3.4. 0

3.3 Advanced Convergence Tests 3.3.1 Summation by Parts In this section we consider convergence tests for series which depend on cancellation among the terms of the series.

3.3 Advanced Convergence Tests

29

Proposition 3.3 [Summation by Parts) Let (aj lj.,o and (bj lj.,o be two sequences ofrenl or complex numbers. For N = 0, I, 2, ...• set N

AN =

L::>j j=o

(we adopt the convention that A_I = 0.) Then for any 0::: III

::: /I

<

00

it holds that

n

I:aj ·bj = [An ·bn -Am-I·bm]

j=m

n-I

+L

Aj . (bj - bj+I).

j=m

3.3.2

Abel's Test

Summation by parts may be used to derive the following test of Niels Henrik Abel (1802-1829).

Theorem 3.6 Abel Consider the series

Suppose that 1.

the partial sums AN = EJ=aaj form a bounded sequence;

2.

bo

~

bl ~ b2 ~ ..• ;

3. lim j ...."" b j = O.

Then the original series

converges. Example 3.14 [Alternating Series Testl As a first application of Abel's convergence test, we examine alternating series. Consider a series of the form

""

L(-l)j. bj j=1

(3.14.1)

Chapter 3: Series

30

withbl ~ iJ'l > b3 ~ .•. ~Oandbj ~ Oasj ~ 00. We set aj = (-I)j and apply Abel's test. We see immediately that all partial sums AN are either -lor O. In particular, this sequence of partial sums is bounded. And the b j's are monotone decreasing and tending to zero. By Abel's convergence test, the alternating series (3.14.1) converges. 0 Proposition 3.4 Let bl ~ iJ'l ~ ..• ~ 0 and assume that bj ~ O. Consider the alternating series Lj.,I(-I)jbj, as in the last example. It is convergent. Let S be its sum. Then the partial sums SN satisfy IS - SNI ::: bN+I·

Example 3.15 Consider the series

00 I L(-l)j-;· j=1 J

This series converges by Example 3.14. Then the partial sum SIOO -0.688172 is within 0.01 (in fact within 1/101) of the full sum S and the partial sum SIOOOO = -0.6930501 is within 0.0001 (in fact within 1/10001) ~~

0

Example 3./6 Next we examine a series which is important in the study of Fourier analysis. Consider the series

t si~j. J=I

(3.16.1)

J

L.!,

We already know that the series diverges. However, the expression sin j changes sign in a rather sporadic fashion. We might hope that the series (3.16.1) converges because of cancellation of the summands. We take aJ = sinj and bJ = I/j. Abel's test will apply if we can verify that the partial sums AN of the aj's are bounded. To see this, we use a trick: Observe that cos(j

+ 1/2) =

cos j . cos(l/2) - sin j . sin(l/2)

and cos(j - 1/2) = cos j . cos(l/2)

+ sin j

. sin(1/2).

Subtracting these equations and solving for sin j yields that . . Sin)

cosU - 1/2) - cos(j Sin (1/") _ = "_ ..

+ 1/2)

3.3 Advanced Convergence Tests

31

We conclude that AN

=

taj =t j=1

cos(j - 1/2).- cosU

+ 1/2).

2· slO(1/2)

j=1

Of course this sum collapses. and we see that A _ -cos(N + 1/2) + cos(1/2) N 2. sin(1/2) .

Thus 2 _ 1 IA N 1< - 2. sin(1/2) - sin(l/2)'

independent of N. Thus the hypotheses of Abel's test are verified and the series

tSi~j j=1

J

o

is seen to converge. Remark 3.2

It is interesting to notice that both the series

f., ~:-=. Isinjl ..:.

L

j=1

J

d f., sin2 j an L . j=1

J

diverge. The details of these assertions are left to the reader.

I

3.3.3 Absolute and Conditional Convergence We tum next to the topic of absolute and conditional convergence. A series of real or complex constants

is said to be absolutely convergent if

converges. We have:

Proposition 3.5 If the se£ies L:~=I

aj

is absolutely convergent, then it is convergent.

Chapter 3: Series

32

Definition 3.1 A series L~=I a} is said to be conditionally convergent if converges. but it does not converge absolutely.

L~=J a}

We see that absolutely convergent series are convergent. but the next example shows that the converse is not true.

Example 3.17 The series

(3.17.1)

converges by the Alternating Series Test. However. it is not absolutely convergent because the harmonic series 00

1

L-:J

}=I

diverges. Thus the series (3.17.1) is conditionally convergent. Remark 3.3

o

We know from Example 3.16 that the series

f: si~j }=I

J

converges. The terms of this series vary in sign in a fairly erratic fashion (calculate the first ten terms on your calculator). But the cancellation is very subtle- this series I is 'lOt an alternating series.

3.3.4 Rearrangements ofSeries There is a remarkable robustness result for absolutely convergent series that fails dramatically for conditionally convergent series. This result is enunciated in the next theorem. We first need a definition.

Definition 3.2 Let L~=I c} be a given series. Let(Pl 1j.,1 be a sequence in which every positive integer occurs once and only once (but not necessarily in the usual order). Then the series k

LC

Pi

}=I

is said to be a rearrangement of the given series.

3.4

Some Particular Series

33

Theorem 3.7 Weierstrass, Riemann If the series :L'=t aj of real numbers is absolutely convergent, then of course it is convergent; let the sum be e. Then every rearrangement of the series converges also

tot.

If the series :L'=t bj is conditionally convergent and if {3 is any real number or ±oo, then there is a rearrangement of the series such that its sequence ofpartial sums converges to {3.

Example 3.18 The series

f: (-~)j j=1

J

is conditionally convergent (because it is an alternating series). By Weierstrass's theorem. there will be a rearrangement of the series that converges to 5. How can we find it? First observe that the series consisting of all the positive terms of the series will diverge (exercise). Likewise. the series consisting of all the negative terms of the series will diverge. Thus we construct the desired rearrangement by using the following steps: • First select just enough positive terms to obtain a partial sum that is greater than 5. • Then add on enough negative terms so that the partial sum falls below 5. • Now add on enough positive terms so that the partial sum once again exceeds

5. • Again add on enough negative terms so that the partial sum falls below 5. Now continue in this fashion. Because the series of positive terms diverges. Steps I and 3 (and subsequent odd-numbered) steps are possible. Because the series of negative terms diverges, Steps 2 and 4 (and subsequent even-numbered steps) are possible. Because the series converges conditionally, the terms of the series tend to zero. So the partial sums we are constructing are gelling ever closer together. In sum, the construction yields a rearrangement that converges to

5.

0

3.4 Some Particular Series 3.4.1

The Series for e

We begin with a series that defines a special constant of mathematical analysis.

34

Definition 3.3

Chapter 3: Series The series 00

1

L"7j '=oJ.

•

J-

=

where j! = j . (j - 1) . (j - 2).·.1 for j ~ I and O! I, is convergent (by the Ratio Test, for instance). Its sum is denoted by the symbol e in honor of the Swiss mathematician Leonard Euler, who first studied it. Like the number 11:, to be considered later in this book. the number e is one that arises repeatedly in a variety of contexts in mathematics. It has many special properties. The first of these that we shall consider is that the definition that we have given for e is equivalent to another involving a sequence (this sequence was considered earlier in Examples 2.19, 2.20):

3.4.2 Other Representations for e Proposition 3.6 The limit

lim

n-+OO

(1+-nI)"

exists and equals e. Of course we have already noted this fact in Example 2.19. I! of the series The next result tells us how rapidly the partial sums AN = defining e converge to e. This is of theoretical interest, but can alsc5 be applied to determine the irrationality of e.

'L7=0

Proposition 3.7 With AN as above. we have that

o< e -

1

AN < -:-:---,-,.,. N N!

With some sharp theoretical work. the last estimate can be used to establish the following:

Theorem 3.8 Euler's number e is irrational. For a reference, see [RUD) or [KRAl).

3.4

35

Some Particular Series

3.4.3

Sums ofPowers

It is part of mathematical legend that Carl Friedrich Gauss (1777-1855) was given the task, as a child, to sum the integers from 1 to 100. The story is that he derived a remarkable formula and gave the correct answer in a few moments. Indeed it is said that he reasoned as follows: Let S = 1 + 2 + ... + 99 + 100. Then S= 1 + 2 + 3 S = 100 + 99 + 98

+ +

+ 98 + 99 + 100 + 3 + 2 + I.

Adding vertically, we find that

2S = .101 + 101 + 101 + ... + 101 + 101 + 101. , tOO times

Thus

2S = 100· 101 = 10100 and so

S = 5050. Precisely the same reasoning may be used to show that

=

SI,N

tj

= N(N/ I)

j=l

It is frequently of interest to sum up higher powers of j. Say that we wish to calculate N

Sk.N

=:2::/ j=1

for some positive integer k exceeding 1. We may proceed as follows: write

(j + 1)k+1 _

/+t =

[/+1 +

(k + I)

+,..+

(k

l + (k +;) . k l-I .?

•

= (k + I) . / + (k + I) . k 2 + (k

J

+2I). k 'J-+(k+l)'J+I-J·k+1

+ I) . k 2

.

/-1 + ...

P + (k + I) . j + I.

Summing from j = I to j = N yields

:2:: {(j + Il+l - l+l N

}

= (k + 1) . Sk,N

+

(k

+ I). k 2

. Sk-I,N

+ ...

j=1

+

(k

+ I) . k 2

S2,N

+ (k + I)· SI,N + N.

Chapter 3: Series

36 The sum on the left collapses to (N 1

[

+ l)k+l

-

1. We may solve for Sk.N and obtain (k + 1) . k

k I

Sk,N = k + I' (N + I) + - 1 - N - ... -

(k

+21)· k

. SZ.N - (k

2

+

. Sk-I,N

1

1) . SI.N .

We have succeeded in expressing Sk,N in terms of SI,N. S2,N •...• Sk-I.N. Thus we may inductively obtain formulas for Sk,N for any k. It turns out that

N(N + I) 2

S.,N = 1 + 2 + ... + N = S2.N

2

2

2

= 1 + 2 +... + N =

N(N

+ 1)(2N + 1) 6

2 3 3 N (N + 1)2 3 S3.N = 1 +2 + ... +N = 4 4

4 (•.N..:.....:.+....:I~)N:..:...::(2....:N_+.:....,:::I):..:.(3:..:N_2_+.:.....:..3N __I.. .:.)

4

S4N=1 +2 +· .. +N =•

30

3.5 Operations on Series Some operations on series, such as addition, subtraction, and scalar multiplication, are straightforward. Others. such as multiplication. entail subtleties.

3.5.1 Sums and Scalar Products ofSeries Proposition 3.8 Let

00

:L>j j=1

00

and

:L)j j=l

be convergent series ofreal or complex numbers; assume that the series sum to limits Ci and p respectively. Then (a) The series E~I (aj

+ hj) converges to the limit ex + p.

(b) If c is a constant, then the series E~. c . aj converges to c . ex.

3.5.2 Products ofSeries In order to keep our discussion of multiplication of series as straightforward as pos_ sible, we deal at first with absolutely convergent series. It is convenient in this discussion to begin our sum at j = 0 instead of j = I. If we wish to multiply

3.5

37

Operalions on Series

lhen we need 10 specify what the partial sums of the prodUCl series should be. An obvious necessary condition that we wish to impose is that if the first series converges to a and the second converges to {3. then the product series Ej.,o C j. whatever we define illO be. should converge to a . {3. The naive method for defining the summands of the product series is to let Cj = aj • bj. However. a glance at the producl of two partial sums of the given series shows that such a definition would be ignoring lhe distributivily of multiplication over addition.

3.5.3 The Cauchy Product Cauchy's idea was that the terms for the product series should be rn

Cm :;

La

j . bm - j •

j=O

This particular form for the summands can be easily motivated using power series considerations (which we shall provide laler on). For now we concentrale on confirming that lhis "Cauchy product" of two series really works.

Theorem 3.9 Cauchy Let E~ aj and Ej.,o b j be two absolutely convergent series which converge to limits a and 13 respectively. Define the series E:;;'=o Cm with summands rn

Crn

=

Laj ·b"'-i j=o

Then the series E:;;'=ocrn converges toa· {3. Example 3.19 Consider the Cauchy producl of the two condilionally convergenl series

f

(-I)j

.Jj + 1

j=O

and

f

(-l)j .

j=o.[J+1

Observe lhal Cm

=

(_1)0(_1)'"

..If.,Jm + 1

+

(_I)I(_I)m-1

.,fi.,(iii

+ ...

(-1)m(-I)O

+ -'-..;rm""+=;Ii'"-:..If"ff 1 m

= L(-1)m j=O

1

.,J(j + I) . (111 + 1-

j)

.

Chapter 3: Series

38

However. (j

+ 1)' (m + 1- j) ::: (m + 1)

Thus

Icml ~

m

L

(m

+ 1) =

(m

+ 1)2.

1 = l.

+1 We thus see that the terms of the series E:::'=o Cm do not tend to zero. so the '=om J-

series cannot converge.

0

Example 3.20

The series

00

A=

LZj=O

j

and j=o

are both absolutely convergent. We challenge the reader to calculate the Cauchy product and to verify that that product converges to 3. 0

Chapter 4

The Topology of the Real Line 4.1

Open and Closed Sets

4.1.1 Open Sets An open interval in JR is any set of the form (a, b) =

Ix e JR : a

<x <

bl.

A closed interval in JR is any set of the form

[a,bl=lx elR:a

~x ~b).

See Figure 4.1. c

c

a

b

An open interval

•

• a A closed interval

b

Figure 4.1

Observe that the intersection of any two open intervals is either empty (i.e., has no points in it) or is another open interval. The union of two open intervals is either another open interval (if the two component intervals overlap) or is just two disjoint open intervals. The key property of an open interval is this: If I is an open interval and x e I, then there is an fi > 0 such that (x -fi,X+fi) C I.

39

Chapter 4: The Topology of the Real Line

40

x-€ (

:

.

x

o

I

Figure 4.2

Thus any point in an open interval 1 has a little interval around it that still lies in I. See Figure 4.2. More generally. we call a set U c JR open if, whenever x E U, there is an E > 0 such that (x - E. X + E) cU. In JR. any open set U is the countable union of disjoint open intervals. See Figure 4.3.

--o_-eo>-------ooFigure 4.3

It may be noied that the union of any number (finite or infinite) of open sets is open. The intersection of finitely many (but. in general. not of infinitely many) open sets is open.

Example 4.1 Let U = (3,4) U (7,9). Then U is open. To illustrate this point. we take. for instance. the point x 8.88 E U. Then we may select E = 0.1 and see 0 that (x - E. X + €) = (8.78,8.98) C S.

=

4.1.2 Closed Sets A set E c JR is called closed if its complement C E is open. Unlike an open set. which is simply a union of intervals. a closed set can be rather complicated (see our discussion of the Cantor set below in Section 4.5). Figure 4.4 depicts a closed set.

•

•

•

•

•

I"

Figure 4.4

The intersection of any number (finite or infinite) of closed sets is closed. The union of finitely many (but. in general. not of infinitely many) closed sets is closed.

4.1

41

Open and Closed Sets

Example 4.2 Let E = [1,3) U {51. Then E is closed. To illustrate this point. we take x = 3.15 in the complement of E. Let ~ = 0.05. Then the interval (x -~. x +~) = (3.1. 3.2) lies entirely in the complement of E (illustrating that the complement of E is open, hence E is closed).

o 4.1.3

Characterization ofOpen and Closed Sets in Tenus ofSequences

Proposition 4.1 Let S c JR be a set. Then S is closed ifand only if each Cauchy sequence has a limit that is also an element of S.

{Sj

I in S

Example 4.3 The set E = [-2. 3) c JR is of course closed. If {a j I is any Cauchy sequence in E, then the sequence will have a limit in E - since the endpoints are included in the set, there is no possibility for the sequence to converge to an exterior point. 0 Note that it follows from the completeness of the real numbers that any Cauchy sequence whatever will have a limit in JR. The main point of this proposition is that. when the set S is closed, then a Cauchy sequence in S has its limit i/l S. Of course such a characterization cannot hold for open sets. For instance. let I = (0.1) and let aj = 1/(j + 1). Then aj e ( for each j, and the sequence certainly has a limit (namely, the point 0), yet that limit point is /lot in (. We may state an obverse to the last proposition, which is in fact trivially tautologically equivalent to it:

Proposition 4.2 Let U S;; JR be a set. Then U is open if, whenever (OJ I is a sequence in
42

Chapter 4: The Topology of the Real Line

4.1.4 Further Properties ofOpen and Closed Sets Let S c JR be a set. We call b e JR a boulldary poillt of S if every nontrivial neighborhood (b - E, b + E) contains both points of S and points ofJR \ S. We denote the set of boundary points of S by as. Refer to Figure 4.5. Boundary points

•

/~ s

•

Figure 4.5

A boundary point b might lie in S and might lie in the complement of S. The next example serves to illustrate the concept:

Example 4.5 Let S be the interval (0, I). Then no point of (0, 1) is in the boundary of S, since every point of (0, 1) has a neighborhood that lies inside (0, 1). Also no point of the complement of [0, 1] lies in the boundary of S for a similar reason. Indeed, the only candidates for elements of the boundary of S are 0 and 1. The point 0 is an element of the boundary since every neighborhood (O-E,O+E) contains the points (0, E) c S and the points (-E,O) C JR\S. A similar calculation shows that 1 lies in the boundary of S. Now consider the set T = [0, 1). Certainly there are no boundary points in (0, I), for the same reason as before. And there are no boundary points in JR \ [0, I), since that set is open. Thus the only candidates for elements of the boundary are 0 and 1. As before, these are both indeed boundary points for T. Notice that neither of the boundary points of S lie in S while both of the 0 boundary points of T lie in T.

Example 4.6 The boundary of the set Q is the entire real line. For if x is any element of JR then every interval (x - E, X + E) contains both rational numbers and 0 irrational numbers.

4.2

Other Distinguished Points

43

4.2 Other Distinguished Points 4.2.1 Interior Points and Isolated Points Definition 4.1 Let S S; IR. A point s E S is called an inlerior poinl of S if there is an E > 0 such that the interval (s - E, s + E) lies in S. A point I E S is called an isolaled poill/ of S if there is an E > 0 such that the intersection of the interval (I-E,I+E) with S isjustthe singleton (I}. See Figure 4.6.

Interior point

/•

Isolated point

"-. • Figure 4.6

By the definitions given here, an isolated point I of a set S c IR is a boundary point. For any interval (1- E,I +E) contains a point of S (namely I itself) and points of IR \ S (since t is isolated). A set consisting only of isolated points is called discrete. For instance, the integers Z c IR is a discrete set. Also the set (I, 1/2, 1/3, ... } S; IR is discrete.

Proposition 4.3 Let S C IR. Then each point of S is either an interior point or a boundary point. Example 4.7 Let S = (0, I]. Then the interior points of S are the elements of (0, I). The boundary points of S are the points 0 and I. The set S has no isolated points. Let T = (I, 1/2, 1/3, ... } U (0). Then the points 1,1/2,1/3, ... are isolated points of T The point 0 is nol isolated. Every element of T is a 0 boundary point, and there are no others. Remark 4.1 Observe that the interior points of a set S are elements of S- by their very definition. Also isolated points of S are elements of S. However. a boundary point of S mayor may not be an element of S. I

4.2.2 Accumulation Points Let S be a subset of IR. A point x is called an accumulalion point of S if every neighborhood of x contains infinitely many distinct elements of S.In particular, x is an accumulation point of S if it is the limit of an eventually nonconstant sequence in S.

Definition 4.2

44

Chapter 4: The Topology of the Real Line

Obviously a closed set contains all its accumulation points. If x is an accumulation point of S, then every open neighborhood of x contains infinitely many elements of S. Hence x is either a boundary point of S or an interior point of S; it cannot be an isolated point of S.

Example 4.8 Let S = (x E Q : 0 :5 x :5 I). Then every point of S is an accumulation point of S. Let T = (x E Z : 1 :5 x :5 10). Then no point of T is an accumulation point of T

Proposition 4.4 Let S be a subset of the real numbers. Then the boundary of S equals the boundary ofIR \ S. The next theorem allows us to use the concept of boundary to distinguish open sets from closed sets.

Theorem 4.1 A closed set conmins all of its boundary points. An open set contains none of its boundary points.

Example 4.9 Let E = (2.7) c IR. Then E is closed, and E contains its two boundary points 2, 7. The set S = (-4, 0) c IR is not closed. and it is missing one of its boundary points (namely. -4).

Proposition 4.5 Every nonisolared boundary point ofa set S is an accumulation point of the set S. Example 4.10 Consider the set S = [-1,2) U (3) U (5,7). The boundary points of S are (-1,2,3.5, 7). The nonisolaled boundary points are (-1,2,5, 7). We see plainly that each of these last four points is an accumulation point of S.

4.3 Bounded Sets Definition 4.3

A subset S of the real numbers is called bounded if there is a positive number M such that lsi :5 M for every element s of S.

4.4

Compact Sets

45

The next result is one of the great theorems of 19th century analysis. It is essentially a restatement of the Bolzano-Weierstrass Theorem of Subsection 2.16.

Theorem 4.2 Every bounded, intinite subset of lR has an accumulation point. Corollary 4.1 Let S C lR be a closed and bounded set. If(ajl is any sequence in S, then there is a Cauchy subsequence (ajtl that converges to an element of S.

Example 4.11 The set E = [4, 10) is a closed and bounded set. Let (a j I be a sequence in E. We may use the method of bisection to identify a convergent subsequence. Namely, write [4, 10) = [4,7) U [7, 10). One of these two subintervals will contain infinitely many elements of the sequence. Say that it is [4,7). Select an element ail that lies in [4,7). Now write [4,7) = [4,5.5) U [5.5,7). Again, one of these two subintervals will contain infinitely many elements of the sequence. Say that it is [5.5,7). Select an element ajz, with h > h. that lies in [5.5. 7). Continue to bisect and choose, at each stage selecting a subinterval that contains infinitely many elements of the sequence and an element ait that is further along in the sequence. In this manner we obtain the desired subsequence. It is clear that it converges because it lies in a telescoping list of closed intervals that are shrinking to a point (i.e., the limit point). 0

4.4 Compact Sets 4.4.1

Introduction

Compact sets are sets (usually infinite) which share many of the most important properties of finite sets. They play an important role in real analysis.

Dejitlitioll 4.4

A set S C lR is called compact if every sequence in S has a subsequence that converges to an element of s.

4.4.2

The Heine-Borel Theorem, Part I

Propositioll 4.6 A set in IR is compact ifand only ifit is closed and bounded. In the theory of topology, a different definition of compactness is used. It is equiv-

46

Chapter 4: The Topology of the Real Line

alent to the one just given in the context of the real line or more generally in metric spaces (see Chapter 10). We discuss it here.

Definition 4.5 Let S be a subset of the real numbers. A collection of open sets (00' }aeA (each 00' is an open set of real numbers) is called an open covering of S if

UOa 'JS. aeA

EXQmple 4.12 The collection C = (II). I)}~I is an open covering of the interval 1 = (0, I). Observe, however, that no subcollection of C covers 1. The collection V = (I/},I)}~I U «-liS. liS)} U (4/5. 615} is an open covering of the interval J = [0. 1]. However, not all the elements V are actually needed to cover J. In fact

(-115. 1/5) , 0/6, I) , (4/5.6/5)

o

cover the interval J-see Figure 4.7.

"·-(~·""OH----""(:--')

o

I

•

Figure 4.7

It is the special property displayed in this example that distinguishes compact sets from the point of view of topology. We need another definition:

Definition 4.6 If C is an open covering of a set S and if V is another open covering of S such that each element of V is also an element ofC. then we call Va subcovering ofC. We call V afinite subcovering if V has just finitely many elements. Example 4.13 The collection of intervals

C = () - I.} + l)}~1 is an open covering of the set S = [5.9]. The collection V= ((j -l,)+ I)}~s

4.4

47

Compact Sets is a subcovering. However, the collection

e=

(4,6), (5,7), (6, 8), (7,9), (8, 10»

o

is afinite subcovering.

4.4.3 The Heine-Borel Theorem, Part II Theorem 4.3 Heine-Borel A set S c lR is compact ifand only if every open covering C = (0" )"eA of S has a finite subcovering. Example 4.14 If A c B and both sets are nonempty, then A n B = A f= 0. A similar assertion holds when intersecting finitely many nonempty sets AI 2 Az ::> ... 2 Ak; it holds in this ciccumstance that n'=1 Aj = Ak. However, it is possible to have infinitely many nonempty nested sets with null intersection. An example is the sets Ij = (0, 1/j). Certainly Ij ::> Ij+! for all j, yet

n 00

Ij =0.

j=1

n, then

By contrast. if we take K j = [0, 1/

n 00

KJ = {O).

j=1

The next proposition shows that compact sets have the intuitively appealing property of the sets K j rather than the unsettling and nonintuitive property of the sets Ij. 0

Proposition 4.7 Let

KI ::>Kz2···::>Kj ... be nonempty compact sets ofreal numbers. Set

Then K. is compact and K. f= 0.

48

Chapter 4: The Topology of the Real Line

4.5 The Cantor Set In this section we describe the construction of a remarkable subset of lR with many pathological properties. We begin with the unit interval So = [0. I]. We extract from So its open middle third; thus S. = So \ 0/3.2/3). Observe that S. consists of two closed intervals of equal length 1/3. Now we construct S2 from S. by extracting from each of its two intervals the middle third: S2 = [0. 1/9] U[2/9.3/9] U[6/9.7/9] U[8/9. I]. Figure 4.8 shows S2·

-j--+---------+--+--+--!-

o

1 Figure 4.8. The sel S2.

Continuing in this fashion, we construct Sj+1 from Sj by extracting the open middle third from each of its component subintervals. We define the Cantor set C to be 00

C = nSj. j=1

Notice that each of the sets Sj is closed and bounded. hence compact. By Proposition 4.7 of the last section. C is therefore not empty. The set C is closed and bounded, hence compact. Proposition 4.8 The Canlor set C has zero length, in the sense thaI [0. I] \ C has length 1.

Idea of the Calculation: In the construction of SI. we removed from the unit interval one interval of length 3- 1• In constructing S2, we further removed two intervals of length 3-2 . In constructing Sj. we removed 2 j - 1 intervals of length 3- j Thus the total length of the intervals removed from the unit interval is 00

I)j-1 . r

i .

j=1

This last equals

~ f(~)j 3'=0 3 J-

The geometric series sums easily (see Subsection 3.2.3) and we find that the total length of the intervals removed is -1 (

3

1 ) -1 1-2/3 - .

4.5

49

The Cantor Set

Thus the Cantor set has length zero because its complement in the unit interval has length 1.

Proposition 4.9 The Cantor set is uncountable. In fact we can think of each element of the Cantor set as a limit of a sequence of intervals coming from the 5j (see the discussion below). This makes it possible to assign an address (consisting of a sequence of O's and I's-at each step we assign for the left interval and I for the right interval) to each element of the Cantor set. But there are uncountably many such addresses. The Cantor set is quite thin (it has zero length) but it is large in the sense that it has uncountably many elements. Also it is compact. The next result reveals a surprising property of this "thin" set:

o

Theorem 4.4 Let C be the Cantor set and define 5 = (x

+ Y:x

E C, Y E

Cl·

Then 5 = [0, 2). Idea of the Calculation: We sketch the argument. Since C c [0, 1), it is clear that 5 c [0,2). For the reverse inclusion, fix an element 1 E [0, 2). Our job is to find two element c and d in C such that c + d = 1. First observe that (x + Y : x E 51, Y E 51) = [0,2). Therefore there exist XI E 51 and YI E 51 such that XI + YI = 1. Similarly, (x + Y : x E 52, Y E Sz) = [0,2). Therefore there exist X2 E 52 and Y2 E 52 such thatx2 + Y2 = 1. Continuing in this fashion we may find for each i numbers x j and Yj such that Xj,Yj E 5j and Xj + Yj = t. Of course (Xj) C C and (Yj) C C, hence there are subsequences (Xj,) and (Yj,) which converge to real numbers c and d. Since C is compact, we can be sure that c E C and dEC. But the operation of addition respects limits; thus we may pass to the limit as k -> 00 in the equation

to obtain

c+d=t. Therefore [0,2) C (x

+ Y : x, Y E Cl. This completes the proof.

Observe that. whereas any open set is the union of open intervals, the existence of the Cantor set shows us that there is no such structure theorem for closed sets. In fact, closed intervals are atypically simple examples of closed sets.

SO

Chapter 4: The Topology of the Real Line

4.6 Connected and Disconnected Sets 4.6.1 Connectivity Let S be a set of real numbers. We say that S is disconnected if it is possible to find a pair of open sets U and V such that

un S i= 0, V n S i= 0, (U

and

n S) n (V n S) = 0,

s=(uns)u(vns).

If no such U and V exist, then we call S connected. See Figure 4.9.

A disconnected set

/

\ Figure 4.9

Example 4. J5

The set T = (x E IR : Ixl < I, x i= 01 is disconnected. For take U x < 01 and V = Ix : x > 01. Then U n T = Ix : -I < x < 0)

= (x :

i= 0

and

n T = Ix : 0 < x < II i= 0. Also (U n T) n (V n T) = 0. Clearly T = (U n T) u (V n T); hence T is V

0

disconnected. Example4.J6

The set X = (-1, I] is connected. To see this. suppose to the contrary that there exist open sets U and V such that U n X i= 0, V n X i= 0. (U n X) n (V n X) = 0. and

s= (UnX)U(V nX). Choose. a E U n X and b E V n X. Set a = sup (U

n (a, bJ).

Now (a,b] C X; hence U n(a.b] is disjoint from V. Thus a ::: b.ButCV is closed; hence a ¢ V. It follows that a < b.

4.7

Perfect Sets

51

If 0< e U then. because U is open. there exists an a e U such that 0< < a < b. This would mean that we chose 0< incorrectly. Hence 0< ¢ U. But 0< ¢ U and 0< ¢ V means 0< ¢ X. On the other other hand. 0< is the supremum of a subset of X (since a eX. b eX, and X is an interval). Since X is a closed interval. we conclude that 0< eX. This contradiction 0 shows that X must be connected. With small modifications. the discussion in the last example demonstrates that any closed interval is connected. Also we may similarly see that any open interval or half-open interval is connected. In fact the converse is true as well:

Theorem 4.5 If S is a connected subset ofJR, then S is an interval. The Cantor set is not connected; indeed it is disconnected in a special sense. Call a set S totally discollliected if. for each distinct xeS, yeS. there exist disjoint open setsU and V such that x e U,y e V. andS= (UnS)U(VnS).

Proposition 4.10 The Cantor set is toudly disconnected.

4.7 Perfect Sets A set S c JR is called perfect if it is nonempty. closed. and if every point of S is an accumulation point of S. The property of being perfect is a rather special one: it means that the set has no isolated points. Obviously a closed interval (a, bl is perfect. After all, a point x in the interior of the interval is surrounded by an entire open interval (x - ". x +,,) of elements of the interval; moreover a is the limit of elements from the right and b is the limit of elements from the left.

EXLlmple 4.17 The Cantor set. a totally disconnected set. is perfect. It is certainly closed. Now fix x e C. Then certainly x e St. Thus x is in one of the two intervals composing St. One (or perhaps both) of the endpoints of that interval does not equal x. Call that endpoint at. Likewise x e Sz· Therefore x lies in one of the intervals of Sz. Choose an endpoint az of that interval which does not equal x. Continuing in this fashion. we construct a sequence (ajl. Notice that each of the elements of this sequence lies in the Cantor set (why?). Finally. Ix - ajl :':: 3- j for each j. Therefore x is the limit of the sequence. We have thus proved that the Cantor set is perfect. 0

52

Chapter 4: The Topology of the Real Line

The fundamental theorem about perfect sets tells us that such a set must be rather large. We have

Theorem 4.6 A nonempty perfect set must be uncountJIble. Corollary 4.2 Ifa < b. then the closed interval [a. b) is uncountJIble.

We also have a new way of seeing that the Cantor set is uncountable. since it is perfect: Corollary 4.3

The Cantor set is uncountable.

Theorem 4.7 Cantor-Bendixon Any uncountable. compact set in lR. is the union ofa perfect set and a countable set. Example 4. J8 LetE = (I/j: j = 1.2.... }U{-I-l/j: j = I.2.... }U[-1.0). Then E is compact. Moreover. if we let A = [-I. OJ and B = {l/j : j = 1,2•... }U {-I - I/j : j = 1.2•... }. then A is perfect and B is countable and E = AUB. 0

Chapter 5

Limits and the Continuity of Functions 5.1

Definitions and Basic Properties

5.1.1

limits

Let E C IR be a set and let I be a real-valued function with domain E. Fix a point P E IR that is either in E or is an accumulation point of E. We say that I has limit t at P. and we write

Definition 5.1

lim

E?JX~P

with t a real number. if for each o < Ix - PI < 8 then

E

t.

I(x) =

> 0 there is a 8 > 0 such that when x E E and

I/(x) -

tl

<

E.

Example 5.1

Let E = R \ {O} and {(x) = x . sin(l/x) if x E E.

Then limx-oo I(x) Ix - 01 < 8. then

= O. To see this. let E

I/(x) - 01 =

> O. Choose 8

Ix . sin(l/x)1 < Ixl < 8 =

as desired. Thus the limit exists and equals O.

= E. If 0

<

E.

o

53

54

Chapter 5: Limits and the Continuity of Functions

5.1.2 A Limit that Does Not Exist Example 5.2 Let E = Rand

I if x is rational g(x) = { 0 if x is irrational.

[The function g is called the Diriclilet function.} Then limx-+ p g(x) does not exist for any point P of E. To see this. fix PeR Seeking a contradiction. assume that there is a limiting value £ for g at P. If this is so. then we take" = 1/2 and we can find a Ii> 0 such that 0 < Ix - PI < Ii implies Ig(x)

-£1

1

<" = 2'

(5.2.1)

If we take x to be rational. then (5.2.1) says that 1

11 -£1<2'

(5.2.2)

while if we take x irrational. then (5.2.1) says that

I

10 -£1<2'

(5.2.3)

But then the triangle inequality gives that

II - 01 = 1(1 - £) + (£ - 0)1

:::: 11 - £1 + 1£ - 01. which by (5.2.2) and (5.2.3) is

<1. This contradiction. that 1 < 1. allows US to conclude that the limit does not 0 exist at P.

5.1.3 Uniqueness ofLimits Proposition 5.1 Let f be a function with domain E. and let either PeE or P be an accumulation point of E.lflimx-+p f(x) = £ and limx-+p f(x) = 11/. then £ = 11/. The point of the last proposition is that if a limit is calculated by two different methods. the same answer will result. While of primarily philosophical interest now. this will be important information later.

5.1 Definitions and Basic Properties

55

This is a good time to observe that the limits lim f(x)

x_p

and lim f(P +h)

11-.0

are equal in the sense that if one limit exists. then so does the other. and they both have the same value. These really amount to two different ways to write the same thing.

5.1.4 Properties ofLimits In order to facilitate checking that certain limits exist. we now record some elementary properties of the limit. This requires that we first recall how functions are combined. Suppose that f and g are each functions which have domain E. We define the sUln or difference of f and g to be the function (f ± g)(x) = f(x) ± g(x).

the product of f and g to be the function (f· g)(x)

= f(x) . g(x).

and the quotient of f and g to be f) (x) = f(x) ( g g(x)

Notice that the quotient is only defined at points x for which g(x) :F O. See also Section 1.4. Now we have:

TheoremS.l Let f and g be functions with domain E and fix a point P that is either in E or is an accumulation point of E. Assume that i) lim f(x) = £ x_p

ii) lim g(x) x_p

= 11/.

Then a) lim (f ± g)(x) = £ ± 11/ x_p

b) Iim(f·g)(x)=£·m x_p

c) lim (f/g)(x) = £/11/ provided m :F O. x_p

56

Chapter 5: Limits and the Continuity of Functions Example 5.3

It is a simple matter to check that if !(x) = x. then lim f(x) = P

x-p

for every real P. (Indeed, for E > 0 we may take 8 is the constant function identically equal to a. then lim g(x) =

x-+p

= E.) Also if g(x) = Ct

Ct.

It then follows from parts a) and b) of the theorem that if f(x) is any polynomial function, then lim f(x) = f(P). x-+p

Moreover. if r(x) is any rational function (quotient of polynomials), then we may also use part c) of the theorem to conclude that lim r(x) = r(P)

x-+p

for all points P at which the rational function r(x) is defined.

o

Example 5.4

If t is a small positive real number. then 0 < sin t < t. This is true because sin t is the nearest distance from the point (cos t , sin t} to the x-axis while t is the distance from that point to the x-axis along an arc. See Figure 5.1. If

t

) '---- sin t

Figure 5.1

E

> O. we set 8 = E. We conclude that if 0 < It - 01 < 8. then

Isint -01 < It I < 8 =

E.

5.2 Continuous Functions

57

Since sin( -t) = - sin t. the same result holds when t is a negative number with small absolute value. Therefore lim sint = O. ' ....0

Since 2 COS

t = I-sin 2 t.

we may conclude from the preceding theorem that lim cost = 1. ' ....0

Now fix any real number P. We have lim sint = lim sin(P + h)

I-+P

h-+O

= lim sin P cosh + cos P sinh h....O

=sinp·l+cosP·O = sin P. We of course have used parts (a) and (b) of the theorem to commute the limit process with addition and multiplication. A similar argument shows that lim cost = COS P o ,.... P

5.1.5 Characterization ofLimits Using Sequences Proposition 5.2 Let f be a function with domain E and P be either an element of E or an accumulation point of E. Then lim f(x) = f. x.... P

if and only iffor any sequence (aj I

C E \ (PI satisfying

limj-+oo aj = p. it holds

that

5.2 Continuous Functions 5.2.1 Continuity at a Point Definition 5.2 Let E c IR be a set and let f be a real-valued function with domain E. Fix a point PEE. We say that f is continuous at P if lim f(x) = f(P).

x.... P

58

Chapter 5: Limits and the Continuity of Functions

Observe that, in the definition of continuity (as distinct from the definition of limit), we require that PeE. This is necessary because we are comparing the value f(P) with the value of the limit. Example 5.5

The function h(x) = {sin I/x

1

~f x # 0

If x = 0

is discontinuous at O. See Figure 5.2.

II

Figure 5.2

The reason is that lim h(x)

x-+o

does not exist. (Details of this assertion are left for you: notice that h(I/(j1l'» = 0 while h(2/[(4j + 1)11') = 1 fori = 1,2•....) The function _ sin I/x if # 0 k( ) x-I if x = 0

{x.

x

is also discontinuous at x = O. This time the Iimitlimx-+ok(x) exists (see Example 5.1); but the limit does not agree with k(O). Refer to Figure 5.3. However, the function _ k( ) x -

{X' sin I/x if x # 0 0

if x =0

is continuous at x = 0 because the limit at 0 exists and agrees with the value of the function there. See Figure 5.4. 0

5.2 Continuous Functions

59 , ••

,,

", ",

,

·.." ,

,,

,, ,

.,

.' ,· . , .• ,· ,.

.-

,,

.

,,

.,

, '

,,

,,

•

,,

.., ,,

,

.". Figure 5.3

.., ·, ,'. ,

. ..

..,.

•, ,

, .. ,, · •

,, . •

.

,

. .,•

,,•

,

, ,

,

••

.,. ,

,,•

".

'.

Figure 5.4 Theorem 5.2 Let I and g be functions with domain E and let P be a point of E. If I and g are continuous at P. then so are I ± g. I· g. and (provided g(P) # 0) Ilg.

Continuous functions may also be characterized using sequences: Proposition 5.3 Let I be a function with domain E and fix PEE. The function I is continuous at P if and only if, for every sequence {a j} C E satisfying lim j-co aj = p. it holds that

Chapter 5: Limits and the Continuity of Functions

60 Proposition 5.4

Letg have domain D and range E and let I have domain E and range H. Let P E D. Assume that g is continuous at P and that I is continuous at g(P). Then fog is continuous at P

Remark 5.1

It is not the case that if

e

lim g(x) =

x.... P

and

,lim ....,I(t) =

//I

then lim

x.... P

10 g(x) =

1/1.

A counterexample is given by the functions

g(x) = 0 f(x) =

{251fx=O. ~f x # 0

Notice that limx.... og(x) = 0 and lim,....o I(x) = 2, yet Iimx .... o I 0 g(x) = 5. The additional hypothesis that I be continuous at is necessary in order to guarantee that the limit of the composition will behave as expected. I

e

5.2.2

The Topological Approach to COlltilluity

Next we explore the topological approach to the concept of continuity. Whereas the analytic approach that we have been discussing so far considers continuity one point at a time, the topological approach considers all points simultaneously. Let us call a function continuous if it is continuous at every point of its domain.

Definition 5.3

Let numbers. We define

f be a function with domain E and let 0 be any set of real

r

l (0) = (x E E : f(x) EO}.

We sometimes refer to I-I (0) as the inverse image of 0 under f. See Figure 5.5.

Theorem 5.3 Let I be a function with domain E and range F The function I is continuous if and only if the inverse image of any open set in F under f is the intersection of E with an open set. In particular, if E is open, then I is continuous if and only if the inverse image of any open set under I is open.

5.2

61

Continuous Functions

(J

Figure 5.5

Since any open subset of the real numbers is a countable union of disjoint open intervals, then, in order to check that the inverse image under a function f of every open set is open, it is enough to check that the inverse image of any open I interval is open. This is frequently easy to do. as the next example shows.

Remark 5.2

Example 5.6 If [(x)

= x 2 , then the inverse image of an open interval (a, b) is

(-..Ib, -"fQ) U ("fQ, ..Ib) if a

> 0; is (-..Ib, .Jb) if a < 0, b ~ 0; and is " if a < b < O. Thus the function [ is continuous. Note that. by contrast. it is somewhat tedious to give an €-8 proof of the continuity of [(x) = x 2 0

Example 5.7 Let [ : IR ~ IR be a strictly increasing function (see Subsection 5.4.3). Assume that [ is continuous. Then it is obvious that [ takes the open interval (a. b) to the open interval (f(a). [(b». Likewise, " takes the interval (01, fJ) (with 01, fJ in the image of f) to the open interval (f-I (01). [-I (fJ». This we see immediately that [-I is continuous. It is rather tricky to check continuity of [-I from the original € - 8 definitions. 0

Corollmy 5.1

Let [ be a function with domain E. The function [ is continuous if and only if the inverse image of any closed set F under f is the intersection of E with some

closed set. In particular, if E is closed. then [ is continuous if and only if the inverse image of any closed set F under f is closed.

62

Chapter 5: Limits and the Continuity of Functions

5.3 Topological Properties and Continuity 5.3.1

The Image ofa Function

Definition 5.4

Let 1 be a function with domain E and let G be a subset of E. We

define

I(G) = {f(x) : x

e G).

The set f(G) is called the image of Gunder f. See Figure 5.6.

f(G)

Figure 5.6

Theorem 5.4 The image of a compact set under a continuous function is also compact. ExampleS.8 It is not the case that the continuous image of a closed set is closed. For instance, take f(x) = 1/(1 x 2 ) and E = 1R: then E is closed and 1 is continuous, but l(E) = (0, I) is not closed. It is also not the case that the continuous image of a bounded set is bounded. As an example, take I(x) I/x and E (0, I). Then E is 0 bounded and f continuous. but f(E) = (1,00) is unbounded.

+

=

=

Corollary 5.2 Let 1 be a function with compact domain K. Then there is a number L such that

I/(x)1

~

L

for all x e K. In fact we can prove an important strengthening of the corollary. Since f(K) is compact, It contains its supremum C and its infimum c. Therefore. there must be a number M e K such that f(M) = C and a number III e K such that 1(111) = c. In other words,j(III) ~ I(x) ~ I(M) for all x e K. We summarize:

5.3 Topological Properties and Continuity

63

Theorem 5.5 LeI I be a continuous function on a compacl sel K. Then there exisl numbers m and M in K such thaI {(m) ::: I(x) ::: I(M) for all x E K. We call m an absolute minimum for { on K and M an absolute maximum for { on K. Example 5.9 Notice thaI, in the last theorem. M and m need not be unique. For instance. the function sin x on the compact interval [0. 4rr I has an absolute minimum at 3rr /2 and 7rr /2. It has an absolute maximum atrr f2 and at 5rr /2. 0

5.3.2 Uniform Continuity Now we define a refined type of continuity:

Definition 5.5 Let I be a function with domain E. We say that I is uniformly continuous on E if. for any Ii > 0, there is a a > 0 such that whenever s, tEE and Is - tl < a.then I/(s) - {(t)1 < Ii. Observe that "uniform continuity" differs from "continuity" in that it treats all points of the domain simultaneously: the a > 0 that is chosen is independent of the points s, tEE. This difference is highlighted in the next example.

Example 5.10 Consider the function I(x) = x 2. Fix a point P E JR, P > 0, and let Ii > O. In order to guarantee that II (x) - {( P) I < Ii. we must have. for x > 0,

Ix 2 - p 2 1 < or

Ix - PI <

Ii

Ii

x+P

.

Since x will range over a neighborhood of P, we see that the required a in the definition of continuity cannot be larger than 1i/(2P). In fact the choice Ix - PI < a = 1i/(2P + 1) will do the job. Thus the choice of a depends not only on Ii (which we have come to expect) but also on P. In particular, f is not uniformly continuous on lR. This is a quantitative reflection of the fact that the graph of { becomes ever steeper as the variable moves to the right. Notice that the same calculation shows that the function I. with domain restricted [a. hI, 0 < a < h < 00, is uniformly continuous. See Figure 5.7. 0 Now the main result about uniform continuity is the following:

Chapter 5: Limits and the Continuity of Functions

64

a b

Figure 5.7 Theorem 5.6 Let f be a continuous function with compact domain K. Then

f

is uniformly con-

tinuous on K.

Example 5.11

The function f(x) = sin(l/x) is continuous on the domain E = (0,00) since it is the composition of continuous functions. However, it is not uniformly continuous. since

for j = I. 2, .... Thus. even though the arguments are becoming arbitrarily close together, the images of these arguments remain bounded apart. We conclude that f cannot be uniformly continuous. However, if f is considered as a function on any restricted interval of the form [a. bl. 0 < a < b < 00, then the preceding theorem tells us that f is uniformly continuous. 0 As an exercise. you should check that

( )_(x0 sin(l/x)

g x -

if x oF 0 if x =0

is uniformly continuous on any interval of the form [-N, N).

5.3

65

Topological Properties and Continuity

5.3.3 Contilluity aml Connectedness Last we note a connection between continuous functions and connectedness.

Theorem 5.7 Let f be a continuous function with domain an open interval I. Suppose that L is a connected subset of I. Then f(L) is connected. In other words, the image of an interval under a continuous function is also an interval.

Example 5.12 Let f be a continuous function on the interval la. bl. Let ()( = f(a) and f3 = f(b). Now choose a number y that lies between ()( and f3. Is there a number c E [a, bl such that f(c) = y? Because the continuous image of an interval is an interval, the answer is obviously "yes." Thus we have established the important intenllediate value property for continuous functions. We record this result formally in the next subsection. 0

5.3.4 The IntermedUlte Value Property Corollary 5.3

Let f be a continuous function whose domain contain the interval [a, bl. Let y be a number that lies between f(a) and f(b). Then there is a number c between a and b such that f(c) = y. See Figure 5.8.

(b,f(b))

Y=f(x) c b

(a,f(a))

'Y Figure 5.8

66

Chapter 5: Limits and the Continuity of Functions

5.4 Classifying Discontinuities and Monotonicity 5.4.1 Left and Right limits We begin by refining our notion of limit: Definition 5.6 Let f be a function with domain E. Fix a point PEE. We say that 1 has left limit f. at P, and write

lim I(x) = l,

x-+p-

if, for every E > O. there is a B > 0 such that whenever P - B < x < P and x E E, then it holds that I/(x) - II < Eo We say that 1 has right limit m at P. and write lim I(x) =m,

X--foP+

if. for every E > O. there is a B > 0 such that whenever P < x < P + B and x E E. then it holds that I/(x) -1111 < E This definition simply formalizes the notion of either letting x tend to P from the left only or from the right only. Example 5.13

Let

Then limx--+I- I(x) at 1 is 1(1) = o.

x2 I(x) = 0 { 2x-4

if 0 ~ x < 1 if x = 1 ifl<x<2.

= 1 while Iimx--+t+ I(x) = -2. The actual value of 1 0

5.4.2 Types ofDiscontinuities Let 1 be a function with domain E. Let P be in E and assume that 1 is discontinuous at P There are two ways in which this discontinuity can occur: I. If limx--+p- I(x) and limx--+p+ I(x) exist. but either do not equal each other or do not equal I(P). then we say that 1 has a discontinuity olthejirst kind (or sometimes a simple discontinuity) at P. II. If either limx--+p- does not exist or limx--+p+ does not exist. then we say that 1 has a discontinuity 01the second kind at P.

5.4

Classifying Discontinuities and Monotonicity

67

discontinuity of the first kind

discontinuity of the second kind Figure 5.9 See Figure 5.9.

Example 5.14 Define

f(x) = {sin(l/X)

o

~f x =f. 0 If x = 0

I if x> 0 g(x) = 0 if x = 0 { -I if x <0 h(x) = { 1 ifx is. irrat~onal

o

tf x

IS

ratIonal

Then f has a discontinuity of the second kind at 0 while g has a discontinuity of the first kind at O. The function h has a discontinuity of the second 0 kind at every point.

5.4.3 Monotonic Functions DejinilionS.7 Let f be a function whose domain contains an open interval (a. b). We say that f is monotonically increasing on (a. b) if, whenever a < s < t < b. it holds that f(s) ::: f(t). We say that f is monotonically decreasing on (a. b) if,

68

Chapter 5: Limits and the Continuity of Functions

monotonically increasing function

monotonically decreasing funClion

Figure 5.10 whenever a < s < t < b. it holds that I(s) ~ I(t). (See Figure 5.10.) Functions that are either monotonically increasing or monotonically decreasing are simply referred to as "monotonic:'

Example 5. J5 The function I(x) = sinx is monotonically increasing on the interval [-n"/2. rr/2]. and on all intervals of the form [(-1 +4k)rr /2. (l +4k)rr/2]. Also the function is monotonically decreasing on the interval [rr/2. 3rr/2]. and on all intervals of the form [(1 +4k)rr/2. (3 + 4k)rr/2]. 0 As with sequences. the word "monotonic" is superfluous in many contexts. But its use is traditional and occasionally convenient. Proposition 5.5

Let 1 be a monotonic function on an open interval (a. b). Then all of the discontinu-

ities of 1 are of the !irst kind. Corollary 5.4

Let 1 be a monotonic function on an interval (a. b). Then many discontinuities.

1 has at most countably

Theorem 5.8 Let 1 be a continuous function whose domain is a compact set K. Let 0 be any open set inR. Then I(K nO) has the form I(K) nu for some open setU C JR.

5.4 Classifying Discontinuities and Monotonicity

69

Suppose that f is a function on (a, b) such that a < s < t < b implies f(s) < f(t). Such a function is called strictly monotonically illcreasillg (strictly mOllotollically decreasing functions are defined similarly). It is clear that a strictly monotonically increasing (resp. strictly monotonically decreasing) function is one-to-one, and hence has an inverse. We summarize (see also Example 1.8): Theorem 5.9 Let f be a strictly monotone, continuous function with domain la, b). Then

exists and is continuous.

f- I

Chapter 6

The Derivative 6.1

The Concept of Derivative

6.1.1 The Definition Let 1 be a function with domain an open interval I. If x

E

I, then the quantity

I(t) - I(x) t-x measures the slope of the chord of the graph of 1 that connects the points (x, I(x» and (t, I(t». If we let t ...... x, then the limit of the quantity represented by this "Newton quotient" should represent the slope of the graph at tire point x. These considerations motivate the definition of the derivative: If 1 is a function with domain an open interval 1 and if x E I, then

Definition 6.1 the limit

. /(t) - I(x) I 1m • I_X t-x

when it exists, is called the derivative of 1 at x'lf the derivative of 1 at x exists, then we say that 1 is differelltiable at x. If 1 is differentiable at every x E I, then we say that 1 is differentiable 011 I. We write the derivative of 1 at x either as

1 (x) I

or

_d 1 or

dx

dl dx

Example 6.1 Consider the function I(x) = x 2 and x = 1. We endeavor to calculate the derivative: lim

I_x

I(t) - I(x) t -x

.

=hm

I-X

t 2 _x 2 t -x

•

=hm[t+xl=2x. I_X

Thus the derivative of I(x) = x 2 at the point x exists and is equal to 2x. 0

71

72

Chapter 6: The Derivative

6.1.2 Properties ofthe Derivative We begin our discussion of the derivative by establishing some basic properties and relating the notion of derivative to continuity.

Lemma 6.1 If

f is differentiable at a point x then f is continuous at x. In particular,

lim,....x f(t) = f(x). Thus all differentiable functions are continuous: differentiability is a stronger property than continuity.

Theorem 6.1 Assume that f and g are functions with domain an open interval I and that f and g are differentiable at x e I. Then f ± g. f g. and fig are differentiable at x (for fig we assume thatg(x) # 0). Moreover (a) (f

± g)'(x) = f'(x) ± g'(x);

(b) (f g)'(x) = f'(x)· g(x) (c) (f)' (x)

g

+

f(x), g'(x);

= g(x) . f'(x) - f(x) g'(x) g2(x)

Example 6.2 That f(x) = x is differentiable follows from lim f(t) - f(x) = lim t -x = l. I-+X

t -x

I_X

t-x

Hence !,(x) = 1 for all x.If g(x) == c is a constant function. then lim g(t) - g(x) I-+x

I - X

= lim c - c = 0; I-+x

t-

X

hence g' (x) == 0. It follows from the theorem that any polynomial function is differentiable. On the other hand. the function f (x) = Ix I is lIot di fferentiable at the point x = 0. This is because

.

' ....0-

while

ItI - 101 = hm . -t -

°

° =-1 1m ItI - 101 = I m -°= . x ,.... °

hm

t-

X

' .... 0-

I'

,....0+ t -

"

So the required limit does not exist,

t-

t0+ t -

1

o

6.1

The Concept of Derivative

6.1.3

73

The Weierstrass Nowhere Differelltiable Fllllctioll

Theorem 6.2 Weierstrass Define a function 1/1 with domain lR by the rule if n ::: x < if II ::: x <

X -II

1/I(x) =

{ 11+ I-x

II II

+ I and II is even

+ I and II is odd.

The graph of this function is exhibited in Figure 6.1.

neven

(n+1) odd

Figure 6.1

Then the function

!(x) =

f(~)i 1/1 (4ix)

i=1

4

is continuous at every real x and differentiable at no real x. This startling example of Weierstrass emphasizes the fact that continuity does not imply differentiability.

Example 6.3 The function

g(x)

=~

(:6/ 1/1 (4i x)

has the property that it is continuously differentiable. but not twice differentiable at any point. The function

gk(X)=~(41:kY 1/1 (4i x) has the property that it is k times continuously differentiable, but not (k + 1) times differentiable at any point. 0

Chapter 6: The Derivative

74

6.1.4 The Chain Rule Next we tum to the Chain Rule. Theorem 6.3 Let g be a differentiable function on an open interval I and let I be a differentiable function on an open interval chat contains the range ofg. Then fog is differentiable on the interval I and (f og)' (x) = !,(g(x»' g'(x) for each x

e

I.

Intuitively, if body :F moves Ct times as fast as body g, and if body 9 moves at velocity p, then:F moves at velocity Ct • p. Example 6.4

Let I(x) = x 3 and g(x) = sinx. Then 10 g(x) = sin 3 x. Thus we have, by the chain rule, that [sin3 xl'

= I'(g(x» . g'(x) = 3 sin2 x . cos x .

0

6.2 The Mean Value Theorem and Applications 6.2.1

Local Maxima and Minima

We begin this section with some remarks about local maxima and minima of functions. Definitwn 6.2 Let f be a function with domain (a, b). A point x e (a, b) is called a local minimum for I if there is a ~ > 0 such that f(t) ::: I(x) for all t e (x -~, x + ~). A point x e (a, b) is called a local maximum for I if there is a ~ > 0 such that I(t) :::: I(x) for alit e (x -~, x + ~). Local minima (plural of minimum) and local maxima (plural of maximum) are referred to collectively as local extrema.

6.2.2 Fennat's Test Propositwn 6.1 Let I be a function with domain (a, b). If I has a local extremum at x e (a, b), and if I is differentiable at x, then I'(x) = o. Example 6.5

Let !(x)=x+sinx.

6.2

75

The Mean Value Theorem and Applications Then / is differentiable on the entire real line, ('(x) = 1 + cosx. and /' vanishes at odd multiples of 1£. Yet, as a glance at the graph of / reveals, / has no local maxima nor minima. This result does nOl contradict the proposition. On the other hand. let g(x) = sinx. Then g has local (indeed global) maxima at points of the form x (4k + 1)1£/2, and g' vanishes at those points as well. Also g has local (indeed global) minima at points of the form x = (4k - 1)1£/2. and g' vanishes at those points. These results about the function g confirm the proposition. 0

6,2.3 Darboux's Theorem Before going on to mean value theorems, we provide a striking application of the proposition: Theorem 6.4 Darboux Let / be a differentiable function on an open interval I. Pick points s. 1 E 1 with s < 1 in 1 and suppose that I'(s) < p < {'(I). Then there is a point u between s and 1 such that I'(u) = p.

If /' were a continuous function. then the theorem would just be a special instance of the Intermediate Value Property of continuous functions (see Corollary 5.3). But derivatives need not be continuous. Example 6.6

Consider the function {(x) = {x 0

2

• sin(l/x)

if x '# 0 ifx = O.

Verify for yourself that /'(0) exists and vanishes but Iimx ... o /' (x) does not 0 exist. This example illustrates the significance of the theorem. Since /' will always satisfy the intermediate value property (even when it is not continuous), its discontinuities cannot be of the first kind. In other words: If / is a differentiable function on an open interval I. then the discontinuities of {' are all of the second kind.

6,2.4 The Mean Value Theorem Next we turn to the simplest form of the Mean Value Theorem. known as Rolle's theorem.

Chapter 6: The Derivative

76

Theorem 6.5 Rolle Let I be a continuous function on the closed interval [a, h] which is differentiable on (a, h). If I(a) = I(h) = 0, then there isapoint~ E (a, h) such that I'(n = o. See Figure 6.2.

-----------------

(~,f(m -~~-~-------------

IVc)

y

h

a Figure 6.2

Example 6.7

Let 11(X) = xe x sin x

+ sin2x.

=

=

Then h satisfies the hypotheses of Rolle's theorem with a 0 and h 1f. We can be sure, therefore, that there is a point ~ between 0 and 1f at which h' vanishes, even though it may be rather difficult to say exactly what that ~nt~

0

Example 6.8

Of course, the point ~ in Rolle's Theorem need not be unique. If I(x) = x 3 - x 2 - 2x on the interval [-1,2], then f(-I) = 1(2) = 0, and f' vanishes at two points of the interval (-1,2). 0

If you rotate the graph of a function satisfying the hypotheses of Rolle's Theorem, the result suggests that, for any continuous function I on an interval [a, h], differentiable on (a, h), we should be able to relate the slope of the chord connecting (a, I(a» and (h, I(h» with the value of I' at some interior point. That is the content of the Mean Value Theorem:

Theorem 6.6 Let f be a continuous function on the closed interval [a, h] that is differentiable on (a, h). There exists a point ~

E

(a, h) such that I(h) - I(a) = f'(~). h-a

6.2

77

The Mean Value Theorem and Applications

See Figure 6.3.

Figure 6.3

Example 6.9 Let I(x) = x sinx - x 2 on the interval [11", 21r). Observe that 1(11") and 1(211") = -411"2. Thus

= _11"2

1(211") - 1(11") = -311" 211" -11"

The Mean Value Theorem guarantees that there is a point ~ between 11" and 211" at which the derivative of I equals -311". It would be difficult to say concretely where that point is. 0

Corollary 6.1 If I is a differentiable function on the open interval f and if I' (x) = 0 for all x then I is a constant function.

E

f,

PROOF

If a < b are points of f, then the Mean Value Theorem tells us that there is a point ~ between a and b such that

o = f'(~) = f(b) -

I(a)

b-a

We conclude that I(a) = I(b). Since a and b were arbitrary points of f, we see that

I

is a constant function.

I

CoroUary 6.2 If I is differentiable on an open interval f and I' (x) > 0 for all x E f, then I is monotone increasing on f; that is, ifs < 1 are elements of f then I(s) ::: 1(1).

78

Chapter 6: The Derivative

If 1 is differentiable on an open interval 1 and /' (x) < 0 for all x e I, then / is monotone decreasing on I; that is, iss < 1 are elements of! then /($) ::: /(t). For the first assertion. let s < t be elements of I. According to the Mean Value Theorem. there is a point ~ between s and 1 such that

PROOF

0::: I'(n = /(1) - /(s) I-S

We see, therefore, that /(s) :s 1(1), so / is monotone increasing. The proof for monotone decreasing is similar.

I

6.2.5 Examples ofthe Mean Value Theorem Example 6.10 Let us verify that lim

x ....+oo

(.Jx + 5 -

-IX) = O.

Here the limit operation means thaI, for any € > 0, there is an N > 0 such that x > N implies that the expression in parentheses has absolute value less than €. Define /(x) = -IX for x > O. Then the expression in parentheses is just f(x + 5) - /(x). By the Mean Value Theorem. this equals

!'(n·5 for some x <

~ <

x

+ 5. But this last expression is 1 ~ -1/2 . 5. 2.

By the bounds on ~. this is < ~x-1/2 -2 .

Clearly, as x

~

+00. this expression tends to O.

o

A powerful tool in analysis is a generalization of the usual Mean Value Theorem that is due to Cauchy:

Theorem 6.7 Cauchy Let / and g be functions continuous on the interval [a. bl and differentiable on the interval (a, b). Then there is a point ~ e (a. b) such that f(b) - f(a)

f'(~)

g(") - g(o)

= g'(~)

6.3

Further Results on the Theory of Differentiation

79

Clearly the usual Mean Value Theorem is obtained from Cauchy's by taking g(x) to be the function x. We conclude this section by illustrating a typical application of the result. Example 6.11

Let f be a differentiable function on an interval 1 such that f' is differentiable at a point x E 1. Then (f(x . I1m

+ h) + f(x -

h.... o+

h

2

h) - 2f(x»

To see this, fix x and define :F(h) = f(x Q(h) = h 2 . Then (f(x

+ h) + f(x -

,, = (f ) (x).

+ h) +

h) - 2f(x»

h2

-

f(x - h) - 2f(x) and

:F(h) - :F(O) Q(h) - Q(O)

According to Cauchy's Mean Value Theorem, there is a ~ between 0 and h such that the last line equals

:F'(n Q'(~)

.

Writing this expression out gives f'(x+~)-f'(x-n

~

1 f'(x+~)-f'(x) = -. ~

2

1

+ 2'

f'(x -

n - f'(x) -~

,

and the last line tends, by the definition of the derivative, to the quantity (f')' (x). See also Subsection 6.3.3. 0

6.3 6.3.1

Further Results on the Theory of Differentiation L'Hopital's Rule

L'Hopital's Rule, actually due to his teacher J. Bernoulli (1667-1748), is a useful device for calculating limits, and a nice application of the Cauchy Mean Value Theorem. Here we present a special case of the theorem. Theorem 6.8 L'Hopital Suppose that f and g are differentiable functions on an open interval 1 and that p E 1. [flim".... p f(x) = lim".... p g(x) = 0 and jf

. f'(x) II m " .... p g'(x)

(6.8.1)

80

Chapter 6: The Derivative

exists and equals a real number i, then lim f(x) = i. x-+p g(x) Theorem 6.9 L'Hopitai Suppose that ! and g are differentiable functions on an open interval I and that p E I.lflim x... p !(x) = Iimx-+pg(x) = ±oo and if " !,(x) II m - -

(6.9.1)

x-+p g'(x)

exists and equals a real number i, then lim !(x) = i. x-+p g(x) Both Theorems 6.8 and 6.9 are valid for one-sided limits. Example 6.12 Let us calculate

" x-I II m - - . x-+l Inx We see that the hypotheses ofL'H6pital's rule are satisfied. Call the desired limit i. Then

(.D = I'1m -x-I x-+I

Inx

I'1m - 1 = 1. =x-+I l/x

o

Thus the limit we seek to calculate equals 1. Example 6.13 To calculate the limit

we set Inx A=ln [ XX ] =xlnx=I/x

and notice that Iimx...o A satisfies the hypotheses of the second version of L'Hopital's rule. Applying L'H6pital, we find that the limit of A is 0; hence 0 the original limit is 1.

6.3

81

Further Results on the Theory of Differentiation

6.3.2

The Derivative ofan Inverse Function

Proposition 6.2

Let 1 be an invertible function on an interval (a, b) with nonzero derivative at a point x E (a. b). Let X = f(x). Then (I-I)' (X) exists and equals Iff'(x). See Figure 6.4.

:•

· .··

graph off

..: •·· • graph off-I

.- _.- .'

..

'

Figure 6.4

Example 6.14 We know that the function f(x) = x k , k a positive integer, is one-to-one k I and differentiable on the interval (0,1). Moreover the derivative k x never vanishes on that interval. Therefore the Proposition applies and we l (X) x, that find, for X E (0, I) = 1((0, 1» and

(f

_I)'

r __ 1 _

(X) -

=

I I'(x) - k . x k -

_ 1 - k. Xl-Ilk

I

-'!'.xi- I -

k

In other words,

o

82

Chapter 6: The Derivative

6.3.3

Higher-Order Derivatives

If I is a differentiable function on an open interval I. then we may ask whether the function I' is differentiable. If it is, we denote its derivative by

I

"

or

I

(2)

2

2

d d I or dx 2 I or dx 2 '

and call it the second derivative of I. Likewise the derivative of the (k -l)th derivative. if it exists. is called the k th derivative and is denoted

I"""

dk dkl or t(k) or - lk o r - dx dx k

Observe that we cannot even consider whether exists in a neighborhood of that point.

I(k)

exists at a point unless

I(k-I)

Example 6.15

Let I(x) = x 2 lnx. Then !,(x) = 2xlnx + x. !,,(x)

= 21nx + 3, !,"(x) = x2 .

0

6.3.4 Continuous Differentiability If I is k times differentiable on an open interval 1 and if each of the derivatives 1(1), 1(2), •••• I(k) is continuous on I, then we say that I is k times cOlltitlUously differentiable on I. Obviously there is some redundancy in this definition since the continuity of I(J-I) follows from the existence of I(J). Thus only the continuity of the last derivative I(k) need be checked. Continuously differentiable functions are useful tools in analysis. We denote the class of k times continuously differentiable functions on 1 by Ck(l).

Example 6.16 For k = I. 2, ...• the function !k(x)

X k+ 1 = { -x k+1

if x :::: 0 ifx <0

will be k times continuously differentiable on IR but will fail to be k + 1 times differentiable at x = O. More dramatically. an analysis similar to the one we used on the Weierstrass nowhere differentiable function shows that the function 00 3J . J gdx) = 4J+Jk sin(4 x)

L

i=1

6.3

83

Further Results on the Theory of Differentiation is k times continuously differentiable on IR but will not be k + 1 times differentiable at any point (this function. with k = O. was Weierstrass's original example). 0

A more refined notion of smoothness or continuity of functions is that of Lipschitz or HOlder continuity. If I is a function on an open interval 1 and if Ct is a real number such that 0 < Ct < 1. then we say that I satisfies a lipschitz cOlldilioll of order Ct on 1 if there is a constant M such that for all s. lEI we have I/(s) - 1(1)1 ::: M . Is -

Ii a

Such a function is said to be of class LiPa(l). Clearly a function of class LiPa is uniformly continuous on I. For if E > 0 then we may take 8 = (E/M)I/a : then. if Is-rl <8.wehave I/(s) - 1(1)1::: M ·Is -Il

a < M· ElM =

E.

Example 6./7 Let I(x) = x 2 Then I is not in LiPI on the entire real line. For I(s) - 1(1) = Is

+ II.

s -I which grows without bound when s. I are large and positive. But I is LiPI on any bounded interval [a. b]. For. if s.1 E [a. bl. then

I

I(s) - 1(1) = s- r

Is + II < 2(lal + Ibl).

o

Example 6.18 When Ct > 1. the class LiPa contains only constant functions. In this instance the inequality

I/(s) - [(1)1 ::: M ·Is -Ila entails

I

I(s) - 1(1) < M ·Is

_ll a -

1

S -I

Because Ct -1 > O. letting s --+ I shows that If (I) exists for every tEl and equals O. It follows from Corollary 6.1 of the last section that I is constant on 1. 0 Instead of trying to extend the definition of Lipa(l) to Ct > 1. it is customary to define classes of functions Ck,a. for k = O. 1•... and 0 < Ct < 1. by the condition that I be of class C k on 1 and that Ilk) be an element of Lipa (I). We leave it as an exercise to verify that C k •a C C'·fJ if either k > eor both k = e and Ct ~ p.

84

Chapter 6: The Derivative

In more advanced studies in analysis, it is appropriate to replace LiPI (I), and more generally C k • I , with another space (invented by Antoni Zygmund, 1900-1992) defined in a more subtle fashion. In fact it uses the expression If(x + h) + f(x - h) - 2f(x)1 that we saw earlier in Example 6.11. See [KRA3] for further details on these matters.

Chapter 7

The Integral 7.1

The Concept of Integral

7.1.1

Partitions

The integral is a generalization of the summation process. That is the point of view that we shall take in this chapter. Let [a. b) be a closed interval in lit A finite. ordered set of points P = (xo. XI. X2 • •••• Xk-I. xd such that

Definition 7.1

a

=xo ::: XI < X2 ::: ••• ::: Xk-I

::: Xk

=b

is called a partition of [a. b). Refer to Figure 7.1. UP is a partition of [a. b). we let Ij denote the interval [Xj-I. X j). j = 1.2•...• k. Thesymbol6.j denotes thelengtlz of Ij. Th~mesh ofP. denoted by m(P). is defined to be maxj 6.j.

f-----Hf---I---f-----+I--I Xo

X2

X3

X4

XS

X6

JC]

Figure 7.1 The points of a partition need not be equally spaced. nor must they be distinct from each other. Example 7.1

The set P = to. 1. 1.9/8.2.5.21/4.23/4. 6} is a partition of the interval [0.6) with mesh 3 (because Is = [2.5). with length 3. is the longest interval in the partition). 0

85

Chapter 7: The Integral

86

Definition 7.2 Let [a. bI be an interval and let I be a function with domain [a. bI· UP = [xO.XI.X2 •.•.• Xk-I.Xkl is a partition of [a.bl and if. for each j. Sj is an (arbitrarily chosen) element of I j. then the corresponding Riemann sum is defined to be k

7?(f, P) =

L I(sj) 6j. j=1

See Figure 7.2.

y=f~)

.... v

----

Figure 7.2

Remark 7.1 In many applications. it is useful to choose Sj to be the right endpoint (or the left endpoint) of the intervallj. In a theoretical development, it is most

I

convenient to leave the S j unspecified.

Example 7.2 Let I(x) = x 2 - x and [a. bl = [1.41. Define the partition P = {I, 3/2. 2, 7/3.4) of this interval. Then a Riemann sum for this I and P is

2

7?(f. P) = (1 -1).

~ + ((7/4)2 -

(7/4»)

~

+ ( (7/3)2 - (7/3») . ~ + (32 - 3) . ~ 10103

864

o

Remark 7.2 We stress that the Riemann sum constructed in this last example is not the only one possible. Another, equally valid, Riemann sum would be

7.1

87

The Concept of Integral R(f, P) = (3/2)2 - 3/2) . ~

+ (22 - 2) . ~

+ (7/3)2_(7/3»). ~+ (42 -4). ~ 3 3 4841 =216

I

Definition 7.3 Let [a, b) be an interval and f a function with domain [a, b). We say that tlte Riemallll sums of f telld to a limit £ as m(P) telu/s to 0 if fat any IE > 0 there is a 8 > 0 such that ifP is any partition of [a, b) with m('P) < 8. then IR(f, P) - £1 < IE fat every choice of Sj E Ij (i.e., fat evety possible choice of

Riemann sum with mesh less than 8). Definition 7.4 A function f on a closed intetval [a, b) is said to be Riemalln illtegrable on [a, b) if the Riemann sums ofR(f, P) tend to a limit as m('P) tends to

zero. The value of the limit, when it exists, is called the Riemallll illtegral of f over [a, b) and is denoted by

l

b

f(x)dx.

Example 7.3

Let f(x) = x 2 For N a positive integer. consider the partition P = (0, I/N, 2/N, ... , [N -I)/N, I) of the interval [0, I). To keep this discussion simple. we will choose the point Sj to be the tight endpoint of the interval [(j - I)/ N , j / N) for each j (it turns out that, fat a continuous function f. this results in no loss of generality). The cOtresponding Riemann sum is

R(f, P)

=?=N(j)2 N

1 N

J=I

N =N3I ?=i J=I

Now we may use the formula that we discussed at the end of Section 3.4 to see that this last equals I N3'

As N

--? 00,

N(N

+ 1)(2N + I) 6

this last tends to 1/3. We conclude that

o

88

Chapter 7: The Integral

7.1.2 Refinements ofPartitions The basic idea in the theory of the Riemann integral is that refining the partition makes the Riemann sum more closely approximate the desired integral. Remark 7.3 We mention now a useful fact that will be formalized in later remarks. Suppose that I is Riemann integrable on [a. b]. with the value of the integral being I.. Let to > O. Then. as stated in the definition (with to/3 replacing to). there is a 0 > 0 such that if Q is a partition of [a. b] of mesh smaller than othen 1'R(f. Q}-I.I < to/3. It follows that ifP and P' are partitions of [a, b] of mesh smaller than 0, then

I'R(I. P} - 'R(f. pl}1

=: ,'R(f, P} -1.1 + II. -

'R(f. p'}1 <

to

to

2tO

3" + 3" = 3"

Note. however. that we may choose P' to equal the partition P. Also. for each j. we may choose the point S j where I is evaluated for the Riemann sum over P to be a point where I very nearly assumes its supremum on Ij. Then for each j we may choose the point s} where I is evaluated for the Riemann sum over pi to be a point where I very nearly assumes its infimum on Ij' It easily follows that when the mesh ofP is less than O. then

L (sup I- inf I) Aj < j

Ii

to.

(7.3.l)

Ii

Inequality (7.3.1) is a sort of Cauchy condition for the integral. This consequence of integrability will prove useful to us in some of the discussions in this and the next ~~. I

Definition 7.5 If P and Q are partitions of an interval [a. b]. then we say that Q is a refinement of P if the point set P is a subset of the point set Q. If P, pi are partitions of [a. b]. then their common rejillemellt is the union of all the points of P and P'. We record now a technical lemma that plays an implicit role in several of the results that follow:

Lemma 7.1 Let I be a function with domain the closed interval [a. b]. The Riemann integral

l

b

I(x}dx

exists if and only if for every to > 0 there is a 0 > 0 such that if P and P' are partitions of [a. b] with m(P} < 0 and 1Il(P'} < O. then their common refinement Q has the property that 1'R(f. P} - 'R(f. Q}I < to

89

7.2 Properties of the Riemann Integral and

In(f. p') - n(f. Q)I <

7.1.3

€.

Existence ofthe Riemann Integral

The most important. and perhaps the simplest. fact about the Riemann integral is that a large class of familiar functions is Riemann integrable. These include the continuous functions. the piecewise continuous functions. and more general classes of functions as well. The great classical result. which we can only touch on here. is that a function on an interval [a. b) is Riemann integrable if alld ollly if the set of its discontinuities has measure 0. 1 See [RUDI for all the details of this assertion.

7.1.4 Integrability ofContinuous Functions We now formalize the preceding discussion.

Theorem 7.1 Let f be a continuous function on a nonempty closed interval [a. b). Then f is Riemann integrable on [a. bl. That is to say. f(x)dx exists.

f:

Example 7.4 We can be sure that the integral

{I

X

2

10 e- dx exists (just because the integrand is continuous). even though this integral is impossible to compute by hand. 0 We next note an important fact about Riemann integrable functions. A Riemann integrable function on an interval [a. b) must be boullded. If it were not. then one could choose the points sj in the construction of n(f. P) so that f (s j) is arbitrarily large; then the Riemann sums would become arbitrarily large. and hence cannot converge.

7.2 Properties of the Riemann Integral 7.2.1

Existence Theorems

We begin this section with a few elementary properties of the integral that reflect its linear nature. IHere a set S has measure zero if, for any whose lengths is less than E.

E

> O. S can be covered by a union of intervals the sum of

Chapter 7: The Integral

90

Theorem 7.2 Let la, b] be a nonempty interval. let f and g be Riemann integrable functions on the interval. and let ex be a real number. Then [ ± g and ex [are integrable and we have 1.

1 1

[(x) ± g(x)dx =

[(x)dx ±

1

g(x)dx;

b

b

2.

1 1

b

b

b

ex [(x)dx = ex

f(x)dx.

Theorem 7.3 If c is a point of the interval la, b] and if f is Riemann integrable on both la. c] and Ie, b]. then f is integrable on la, b] and

1< Remark 7.4

1 b

[(x)dx

+

1 b

[(x)dx =

[(x)dx.

If we adopt the convention that

L"

[(x)dx =

-l

b

f(x)dx

(which is consistent with the way that the integral was defined in the first place), then Theorem 7.3 is true even when c is not an element of la, b]. For instance. suppose that c < a < b. Then, by Theorem 7.3.

1"

1 b

f(x)dx

+

1 b

{(x)dx =

[(x)dx

But this may be rearranged to read

1 b

f(x)dx

=

-1"

1 b

[(x)dx

+

1 b

[(x)dx

=[

f(x)dx

+

[(x)dx.

Example 7.5 Suppose that we know that

10

4

1 4

[(x)dx = 3

and

f(x)dx = -5.

Then we may conclude that

o

I

7.2 Properties of the Riemann Integral

91

7.2.2 Inequalities for Integrals One of the basic techniques of analysis is to perform estimates. Thus we require certain fundamental inequalities about integrals. These are recorded in the next theorem. Theorem 7.4 Let I and g be integrable functions on a nonempty interval la, b]. Then

1.

f

f(X)dxl

~

f

I/(x)ldx;

2. If I(x) < g(x) for all x

E

[a, b], then

l

b

I(x) dx

~

l

b

g(x)dx.

Example 7.6

We may estimate that

"/2 L"/2 x· sinxdx sinxdx = -. - 2 2 Lo < -1t

1t

0

Likewise

I. I

ezdx<1 lnx I.e -dx=-. lnx 1 x

I

x

2

o

Lemma 7.2 If f is a Riemann integrable function on la, b], and if tf> is a continuous function on a compact interval that contains the range of I, then tf> 0 I is Riemann integrable. Corollary 7.1 If f and g are Riemann integrable on [a, b], then so is the function I . g.

7.2.3 Preservation ofIntegrable Functions under Composition The following result is the so-called "change of variables formula." In some calculus books it is also referred to as the "u-substitution." This device is useful for transforming an integral into another (on a different domain) that may be easier to handle. Theorem 7.5 Let f be an integrable function on an interval la, b] of positive length. Let!/J be a continuously differentiable function from another interval [ao, P] of positive length into [a, b]. Assume that!/J is monotone increasing, one-to-one, and onto. Then

f

f(x)dx =

t

I(!/J(t». !/J'(t)dt.

92

Chapter 7: The Integral Example 7.7

=

Let f(x) sinx2 .2x on the interval [0.11']. Let 1{f(t) the theorem then,

= ../i. According to

fo" sin x 2 • 2x dx = fo" f (x) dx = fo"2 f(1{f(t}) ·1{f'(t)dt ,,2

1 sint·2../i·-dt 2../i

=

Lo

=

fo"2 sin t dr

=

-COS1l'2+cosO.

Example 7.8

=

Let f(x) e 1/ x /x 2 on the interval [-1,1]. Let 1{f(r) the theorem,

1

2 el/x

1

2

- 2 dx

x

1 1 1 1

= I/r. According to

=

f(x)dx

I

1/2

=

=

f(1{f(t»1{f' (r) dr

1

1/2

1

2-1

e'·r· -2d t r

1/2

= -

1

e'dr

= _e l / 2 +e.

7.2.4 The Funtwmental Theorem ofCalculus Theorem 7.6 Let f be an integrable function on the interval [a, b]. For x e [a. b] we define F(x) =

LX f(s)ds

If f is continuous at x e (a, b), then F'(x)

= f(x).

We conclude with this important interpretation of the fundamental theorem:

7.3 Further Results on the Riemann Integral

93

Corollary 7.2 If f is a continuous function on [a, b] and if G is any continuously differentiable function on [a, b] whose derivative equals f on (a, b), then

[

f(x)dx = G(b) - G(a).

Example 7.9

Let us calculate

L

x3

-d d x

sin(lnt)dt.

x2

It is useful to let G(t) be an antiderivative of the function sin(ln t). Then the expression (*) may be rewritten as

Of course this is something that we can calculate using the chain rule. The result is that

7.3 Further Results on the Riemann Integral 7.3.1

The Riemann-Stieltjes 1ntegral

Fix an interval [a, b] and a monotonically increasing function 01 on [a, b]. If P = (po, PI, ... , Pk) is a partition of [a, b], let 6.OIj = OI(Pj) - OI(Pj_I). Let f be a bounded function on [a, b] and define the upper Riemann swn of f with respect to 01 and the lower Riemann sum of f with respect to 01 as follows: k

U(f, P,OI) = 'L,Mj6.OIj j=1

and

k

.cU, P,OI) = 'L,mj6.OIj. j=1

Here the notation Mj denotes the supremum of f on the intervallj = [Pj-I, Pj] and mj denotes the infimum of f on I j. In the special case OI(X) = x, the Riemann sums discussed here have a form similar to the Riemann sums considered in the first two sections. Moreover,

.cu, P, 01) ::: nu, P) ::: U
94

Chapter 7: The Integral

Returning to general a, we define l'(f) = inf U(f. P. a)

and l.(f) = sup £(f. P. a).

Here the supremum and infimum are taken with respect to all partitions P of the interval la. b]. These are. respectively. the upper and lower integrals of f with respect to a on la. b]. By definition. it is always true that. for any partition P. (7.3.1.1)

It is natural to declare the integral to exist when the upper and lower integrals agree:

Definition 7.6 Let a be a monotone increasing function on the interval la. b] and let f be a bounded function on la. b]. We say that the Riema/llz-8tieltjes integral of f with respect to a exists if I"(f) = l.(f). When the integral exists, we denote it by

Notice that the definition of Riemann-5tieltjes integral is different from the definition of Riemann integral that we used in the preceding sections. It turns out that, when a(x) = x. then the two definitions are equivalent. In the present generality. it is easier to deal with upper and lower integrals in order to determine the existence of integrals. We now repeat an essential definition.

DejilliJion 7.7 Let P and Q be partitions of the interval la. b]. If each point ofP is also an element of Q. then we call Q a refinement of P. Notice that the refinement Q is obtained by adding points to P. The mesh of Q will be less than or equal to that ofP. The following lemma enables us to deal effectively with our new language: Lemma 7.3 LetP be a partition of the interval la. b] and f a function on la. b]. Fix a monotone increasing function a on la. b]. IfQ is a refinement ofP. then

U(f. Q. a)

s. U(f. P. a)

£(f. Q. a)

~

and £(f. P. a).

7.3

95

Further Results on the Riemann Integral Example 7.10 Let [a, b) = [0, 10) and let a(x) be the greatest integer function. That is, a(x) is the greatest integer that does not exceed x. So, for example, a(0.5) = 0, a(2) = 2, and a(-3/2) = -2. Certainly a is a monotone increasing function on [0, 10). Let I be any continuous function on [0, 10). We shall determine whether riO

10

Ida

exists and, if it does, calculate its value. Let 'P be a partition of [0, 10). By the lemma, it is to our advantage to assume that the mesh of'P is smaller than 1. Observe'that I::>a j equals the number of integers that lie in the interval Ij - that is, either 0 or 1. Let I io' Ih' ... I i10 be the intervals from the partition which do in fact contain integers (the first of these contains 0, the second contains I, and so on up to 10). Then to 10 U(f, 'P, a) = L Mj.l::>aj. = L Mj. and

£(f, 'P,a)

l=O

l=1

10

10

= Lmj.l::>aj. = Lmj. l=O

1=1

because any term in these sums corresponding to an interval not containing an integer must have I::>aj = O. Notice that I::>aio = 0 since a(O) = a(Pl)

= O.

Let E > O. Since I is uniformly continuous on [0, 10), we may choose a 8 > 0 such that Is - tl < 8 implies that I/(s) - l(t)1 < E/20. If m('P) < 8, then it follows thatl/(l) - Mj.1 < E/20 and I/(l) - mj.1 < E/20 for l = 0, I, ... 10. Therefore

U(/, 'P, a) <

L

10 (

E)

I(l) + 20

1=1

and 10 (

£(/, 'P,a) > L

E

I(l) - 20)

1=1

Rearranging the first of these inequalities leads to

U(f, 'P, a) <

(

10

)

10

)

{;. I(l)

+ 2:E

and

£(/, 'P,a) >

LI(l) ( l=l

-

E

2'

Chapter 7: The Integral

96

Thus, since 10 and 10 are trapped between U and £. we conclude that

1/

0 (/) -

1*(/)1 <

E.

We have seen that, if the partition is fine enough, then the upper and lower integrals of f with respect to a differ by at most E. It follows that J~O fda exists. Moreover, 10

r(/) - Lf(l) <

E

t=1

and

10

10 ( / )

-

L f(l)

<

E.

t=1

We conclude that 10

(

10

10

fda =

L

f(l).

t=1

o The example demonstrates that the language of the Riemann-Stieltjes integral allows us to think of the integral as a generalization of the summation process. This is frequently useful, for both philosophical and practical reasons.

7.3.2 Riemann's Lemma The next result, sometimes called Riemann's Lemma. is crucial to proving the existence of Riemann-5tieltjes integrals. Proposition 7.1 Leta be a monotone increasing function on la. b) and f a bounded function on the interval. The Riemann-Stieltjes integral of f with respect to a exists if and only if for every E > 0, there is a partition P such that

IU(/. P. a) - £(/, P, a») <

E.

(7.1.1)

We note in passing that the basic properties of the Riemann integral noted in Section 7.2 (Theorems 7.2 and 7.3) hold without change for the Riemann-Stieltjes integral.

7.4 Advanced Results on Integration Theory 7.4.1 Existence ofthe Riemann-Stieltjes Integral We now tum to enunciating the existence of certain Riemann-Stieltjes integrals.

97

7.4 Advanced Results on Integration Theory

Theorem 7.7 Let I be continuous on la, b] and assume that 01 is monotonically increasing. Then

l

b

IdOl

exists. Theorem 7.8 If 01 is a monotone increasing and continuous function on the interval la, b] and if I is monotonic on la, b] then I dOl exists.

It

7.4.2 Integration by Parts One of the useful features of Riemann-Stielljes integration is that it puts integration by parts into a very natural setting. We begin with a lemma:

Lemma 7.4 Let I be continuous on an interval la, b] and let g be monotone increasing and continuous on that interval. IfG is an antiderivative for g, then

l

b

lI b

I(x)g(x)dx =

dG

Theorem 7.9 Suppose that both I and g are continuous, monotone increasing functions on the interval la, b]. Let F be an antiderivative for f on la, b] and G an antiderivative for g on la, b]. Then we have

l

b

F dG = IF(b)· G(b) - F(a)· G(a)] -

f

G dF

Example 7.11 We may apply integration by parts to the integral I =

fo" x . cos x dx .

The result is I

= [x sinx]~ -

fo" sinx dx = -2.

o

Remark 7.S The integration by parts formula can be proved by applying summation by parts (Proposition 3.3) to the Riemann sums for the integral

t

Idg.

I

98

Chapter 7: The Integral

7.4.3 Linearity Properties We have already observed that the Riemann-5tieltjes integral

t

fda

is linear in I; that is,

and

l

·l

b

b C•

Ida = c

fda

where c is any constant and both I and g are Riemann-5tieltjes integrable with respect to a. We also would expect, from the very way that the integral is constructed, that it would be linear in the a entry. But we have not even defined the RiemannStieltjes integral for nonincreasing a. And what of a function a that is the difference of two monotone increasing functions? Such a function cenainly need not be monotone. Is it possible to identify which functions a can be decomposed as sums or differences of monotonic functions? It turns out that there is a satisfactory answer to these questions, and we should like to discuss these matters briefly.

7.4,4 Bounded Variation Definition 7.8 If a is a monotonically decreasing function on [a, b] and I is a function on [a, b] then we define

l

b

Ida =

-l

b

fd(-a)

when the right side exists. The definition exploits the simple observation that if a is monotone decreasing then -a is monotone increasing; hence the preceding theory applies to the function -a. Next we have Definition 7.9

Let a be a function on [a, b] that can be expressed as a(x) = a. (x) - a2(x) ,

where both a. and a2 are monotone increasing. Then for any f on [a, b], we define

l

b

fda =

l

b

fda.

-l

b

f da2,

99

7.4 Advanced Results on Integration Theory provided that both integrals on the right exist.

J:

Now. by the very way that we have formulated our definitions. I da is linear in both the I entry and the a entry. But the definitions are not satisfactory unless we can identify those a that can actually occur in the previous definition. This leads us to a new class of functions.

Definition 7.10

Let I be a function on the interval [a. b]. For x

E

[a. b]. we define

k

V/(x) = sup

L I/(pj) -

I(pj-I)I.

j=1

where the supremum is taken over all partitions P. with a = PO ::: ... ::: Pk = b. of the interval [a. x]. If VI 25 V/(b) < 00. then the function I is said to be of bounded variatioll on the interval [a. b]. In this circumstance the quantity VI(b) is called the total variatioll of I on [a. b]. A function of bounded variation has the property that its graph does unbounded total oscillation.

1I0t

have

Example 7.12

Define I(x) = sinx. with domain the interval [0.211]. Let us calculate VI. Let P be a partition of [0. 211]. Since adding points to the partition only makes the sum k

L I/(pj) -

I(pj-I)I

j=1

larger (by the triangle inequality). we may as well suppose that P = (po. PI. P2•...• Pk) contains the points 11/2.311/2. Say that PI, = 11/2 and PI2 = 311/2. Then t,

k

L I/(pj) j=1

{(pj-t)1

=

L I/(pj) -

I(pj-I)

I

j=1 /2

+

L

/I(pj) - I(pj-I)I

j=I,+1

k

+

L j=12+J

I/(pj) - {(pj-I)I·

Chapter 7: The Integral

100

However. [ is monotone increasing on the interval [0. rr/2] = [0, Pt,]. Therefore the first sum is just t,

LJ(Pj) - [(Pj-I)

= ((Pt.) -

[(po)

= [(rr/2) -

1(0)

= 1.

j=1

Similarly. ! is monotone on the intervals [rr/2,3rr/2] = [Pt" Ptz] and [3rr/2,27f] = [Ptz' Pk]. Thus the second and third sums equal I(pt.) [(Ptz) = 2 and !(Pk) - I(ptz) = 1 respectively. It follows that VI= VI(2rr) = 1+2+1 =4.

Of course VI(x) for any x E [0, 2rr] can be computed by similar means. In general. if I is a continuously differentiable function on an interval [a, b]. then VI(x) =

[1/'(t)ldt.

o

Lemma 7.5 Let I be a function of bounded variation on the interval[a, b]. Then the function VI is monotone increasing on [a, b]. Lemma 7.6 Let I be a function of bounded variation on the interval [a, b]. Then the function VI - { is monotone increasing on the interval [a, b]. Now we may combine the last two lemmas to obtain our main result:

Proposition 7.2 If a function I is of bounded variation on [a, b]. then I may be written as the difference of two monotone increasing functions. Namely.

1= VI-[Vf-/]· Conversely, the difference of two monotone increasing functions is a function of bounded variation. Now the main point of this discussion is the following theorem:

Theorem 7.10 If I is a continuous function on [a, b] and ifa is of bounded variation on [a. b], then the integral

7.4

Advanced Results on Integration Theory

101

exists. If g is of bounded variation on [a, hI and if f3 is a continuous function ofbounded variation on [a, hI. then the integral

exists. Both of these results follow by expressing the function of bounded variation as the difference of two monotone functions. as in Proposition 7.2.

ChapterS

Sequences and Series of Functions 8.1

Partial Sums and Pointwise Convergence

8.1.1 Sequences ofFunctions A sequence offllllc/ions is usually written

II (x), h(x),... We will generally assume that the functions

or

{Ii 1;1

Ii all have the same domain S.

Definition 8.1 A sequence of functions Iii )i=1 with domain S ~ lR is said to converge pointwise to a limit function f on S If, for each xES. the sequence of numbers {fj(x») converges to f(x). We write Iimj_OO Ii (x) = f(x).

Example8.l Define Ii(x) = x j with domain S = Ix : 0 :5 x :5 I). If 0 :5 x < I. then Ii(x) --+ O. However. fj(l) --+ 1. Therefore the sequence Ii converges to the function if 0:5x<1 f( x ) -lifx=1

_{O

o

See Figure 8.1.

Here are some basic questions that we must ask about a sequence of functions fj that converges to a function f on a domain S : 1. If the functions fj are continuous, then is f continuous? 2. If the functions Ii are integrable on an interval I, then is f integrable on I? If f is integrable on I. then does the sequence II fJ\x)dx converge to Itf(x)dx? 3. If the functions Ii are differentiable, then is f differentiable? If f is differenconverge to I'? tiable. then does the sequence

fi

103

104

Chapter 8: Sequences and Series of Functions

o

Figure 8.1

8.1.2

Uniform Convergence

We see from Example 8.1 that the answer to the first question of the last subsection is "no": Each of the Ij is continuous but I certainly is not. Ittums out that, in order to obtain a favorable answer to our questions, we must consider a stricter notion of convergence of functions. This motivates the next definition.

Definition 8.2

Let /j be a sequence of functions on a domain S. We say that the functions Ij converge uniformly to I if, given" > 0, there is an N > 0 such that for any j > N and any XES, it holds thatl/j(x) - I(x)1 < ". Notice that the special feature of uniform convergence is that the rate at which Ij(x) converges is independent of XES. In Example 8.1, /j(x) is converging very rapidly to zero for x near zero, but very slowly to zero for x near I (draw a sketch to help you understand this point). We shall establish this assertion rigorously in the next example.

Example 8.2 The sequence Ij(x) = x j does not converge uniformly to the limit function

I(x) =

{~

if 0 ::: x < I if x = I

on the domain S = [0, I]. In fact it does not even do so on the smaller domain [0, I). Again see Figure 8.1. To see :his notice that, no matter how large j is, the Mean Value Theorem shows that

Ij(l) - Ij(l - 1/(2j» =

2~ . IN)

8.1 Partial Sums and Pointwise Convergence

105

forsome~ between 1-1/(2j) and 1. But f;(x) = j .x}-I; hence If;(nl < j. and we conclude that If}(l) - !J(1- 1/(2j»\ <

or f}(l - 1/(2j» > f}(I) -

~

~ = ~. 2

2

In conclusion. no matter how large j, there will be values of x (namely x = 1 - 1/(2j) ) at which !J(x) is at least distance 1/2 from the limit O. We conclude that the convergence is not uniform. 0

Theorem 8.1 If !J are continuous functions on a set S and if !J converge unifonnly on S to a function f. then f is also continuous.

Next we tum our attention to integration. Example 8.3

Define functions }j(x) =

Then Iim}-+oo !J(x)

{!

ifx =0 ifO<x
= 0 for all x in the interval J = [0, 1]. However.

fo

I

f}(x)dx =

fo

II}

j dx = 1

for every j. Thus the !J converge to the integrable limit function f(x) but their integrals do not converge to the integral of f.

= 0, 0

Example 8.4

Let q., lJ2, ... be an enumeration of the rationals in the interval J = [0, I]. Define functions

ifx e {ql,q2,' .. ,qj) ifx It {m,q2, .... qj) Then the functions !J converge pointwise to the Dirichlet function f which is equal to 1 on the rationals and 0 on the irrationals. Each of the functions f} has integral 0 on I. But the function f is not integrable on I. 0 The last two examples show that something more than pointwise convergence is needed in order for the integral to respect the limit process.

106

Chapter 8: Sequences and Series of Functions

Theorem 8.2 Let Ii be integrable functions on a bounded interval [a, b] and suppose that Ii converge uniformly to a limit function f. Then f is integrable on [a. b] and

.lim )-+00

l

b

u

Ii(x)dx =

l

b

a

f(x)dx.

We have succeeded in answering questions 1 and 2 that were raised at the beginning of the section. In the next section we will answer question 3.

8.2 More on Uniform Convergence 8.2.1 Commutation oflimits In general. limits do not commute. Since the integral is defined with a limit. and since we saw in the last section that integrals do not always respect limits of functions. we know some concrete instances of the noncommutation of limits. The fact that continuity is defined with a limit. and that the limit of continuous functions need not be continuous. gives us further examples of limits that do not commute. Let us now tum to a situation in which limits do commute: Theorem 8.3 Fix a set S and a point s e S. Assume that the functions Ii converge uniformly on the domain S\ Is} roa limit function f. Suppose that each function f) (x) has a limit as x --+ s. Then f itself has a limit as x --+ sand

lim f(x) = .lim lim /j(x). )-+ooX-+3

X-i>.f

Because of the way that f is defined, we may rewrite this conclusion as lim .lim fl(x) = .Iim lim fl(x).

x-+s )-+00

/-+ooX-+3

In other words. the limits Iimx .... and lim)"'00 commute. Example8.S

Consider the limit lim lim

X-t- 1-

x) .

j-+co

This is easily seen to equal O. But lim lim xl

j-+oox-+l-

equals 1. The reason that these two limits are unequal is that the convergence of xl is not uniform (See Example 8.2).

8.2 More on Uniform Convergence

107

By contrast. the limit

sinjx --=-x-ur j-+oo j lim lim

can be calculated in any order (because the functions converge uniformly). 0 The limit is equal to zero.

8.2.2

The Uniform Cauchy Condition

In parallel with our notion of cauchy sequence of numbers. we have a concept of Cauchy sequence offunctions in the uniform sense: Definition 8.3 A sequence of functions Ii on a domain S is called a uniformly Cauchy sequence if. for each E > O. there is an N > 0 such that if j. k > N. then Ifj(x) -

A(x)1 <

E

for all XES.

Proposition 8.1 A sequence of functions Ii is unifonnly Cauchy on a domain S if and only if the sequence converges unifonnly to a limit function ! on the domain S.

We will use the last two results in our study of the limits of differentiable functions. First we consider an example. Example 8.6

Define the function

Ii(x) =

{~x2 x - Ij(4j)

if x :: 0 if 0 < x :: Ij(2j) if Ij(2j) < x < 00

We leave it as an exercise for you to check that the functions uniformly on the entire real line to the function !(x) =

{~

Ii converge

ifx :: 0 if x> 0

(draw a sketch to help you see this). Notice that each of the functions Ii is continuously differentiable on the entire real line, but ! is no/ differentiable ~~

0

8.2.3 Limits ofDerivatives It tums out that we must strengthen our convergence hypotheses if we want the limit process to respect differentiation. The basic result is the following:

108

Chapter 8: Sequences and Series of Functions

Theorem 8.4 Suppose that a sequence fj of differentiable functions on an open interval 1 converges pointwise to a limit function f. Suppose further that the differentiated sequence fj converges uniformly on 1 to a limit function g. Then the limit function f is differentiable on 1 and !,(x) = g(x) forall x E I.

Remark 8.1 A little additional effort shows that we need only assume in the theorem that the functions Ii converge at a single point xo in the domain. I Example 8.7

Consider the sequence

f j(x ) =

sinix . J

These functions converge uniformly to 0 on the entire real line. But their derivatives do not converge. Check for yourself to see that the key hypothesis of Theorem 8.4 fails for this example. Draw a sketch of hand f4 so that you can see what is going on. 0

8.3 Series of Functions 8.3.1

Series and Parlial Sums

Definition 8.4

The formal expression 00

LIi(x). j=1

where the Ii are functions on a common domain S. is called a series offunctiolls. For N = 1. 2. 3. . .. the expression N

SN(X) =

L fj(x) = fl (x) + h(x) + ... + fN(X) j=1

is called the Nih partial sum for the series. If

exists and is finite. we say that the series cOllverges at x. Otherwise we say that the series diverges at x. Notice that the question of convergence of a series of functions. which should be thought of as an additioll process. reduces to a question about the sequence ofpanial

8.3 Series of Functions

109

Sometimes. as in the next example, it is convenient to begin the series at some index other than j = 1.

SIll/IS.

Example 8.8

Consider the series 00

Lxi.

i=O

This is the geometric series from Subsection 3.2.3. It converges absolutely for IxI < I and diverges otherwise. By the formula for the partial sums of a geometric series. l-x N+ 1 SN(X) = --:--I-x

For Ixi < I. we see that

o 8.3.2

Uniform Convergence ofa Series

Definition 8.5

Let 00

LIi(x) i=1

be a series of functions on a domain S. Ifthe partial sums SN(X) converge uniformly on S to a limit function g(x), then we say that the series converges uniformly on S. Of course all of our results about uniform convergence of sequences of functions translate, via the sequence of partial sums of a series. to results about uniformly convergent series of functions. For example: (8) If

Ii are continuous functions on a domain S and if the series

converges uniformly on S to a limit function i,then i is also continuous onS. (b) If Ii are integrable functions on [a, bl and if

110

Chapter 8: Sequences and Series of Functions converges uniformly on [a, b] to a limit function grable on [a, b] and

l

b

a f(x)dx =

I. then I

is also inte-

f; lb co

a /j(x)dx.

Example 8.9 The series

fi e .1i . I

)=

2J

converges uniformly on any bounded interval [a, b]. The Weierstrass Mtest, discussed in the next subsection. provides a means for confirming this 0 assertion. Now we turn to an elegant test for uniform convergence that is due to Weierstrass.

8.3.3

The Weierstrass M-Test

Theorem 8.5 Weierstrass Let (!Jlj.,1 be functions on a common domain S. Assume that each I/jl is bounded on S by a constant Mj and that co

LMj <00. j=1

Then the series (8.5.1)

converges uniformly on the set S. Example 8.10 Let us consider the series co

?=r sin(2 x) . j

j

)=1

The sine terms oscillate so wildly that it would be difficult to calculate partial. sums for this series. However. noting that the jib summand /j(x) = 2- J sin(2 J x) is dominated in absolute value by 2- j, we see that the Weierstrass M-Test applies to this series. We conclude that the series converges uniformly on the entire real line.

8.4 The Weierstrass Approximation Theorem

III

By the above-noted property (a) of uniformly convergent series of continuous functions, we may conclude that the function f defined by our series is continuous. It is also 2rr-periodic: !(x + 2rr) = !(x) for every x, since this assertion is true for each summand. Since the continuous function! restricted to the compact interval [0, 2rr I is uniformly continuous (Theorem 5.6), we may conclude that! is uniformly continuous on the entire real line. However, it turns out that! is nowhere differentiable. The proof of this assertion follows lines similar to the treatment of nowhere differentiable 0 functions in Subsection 6.1.3. Exercise: Verify the assertion of Example 8.9.

o

8.4 The Weierstrass Approximation Theorem Karl Weierstrass (1815-1897) revolutionized analysis with his examples and theorems, and this section is devoted to one of his most striking results. We introduce it with a motivating discussion. It is natural to wonder whether the standard functions of calculus-sin x, cosx, and for instance-are actually polynomials of some very high degree. Since polynomials are so much easier to understand than these transcendental functions, an affirmative answer to this question would certainly simplify mathematics. Of course a moment's thought shows that this wish is impossible: a polynomial of degree k has at most k real roots. Since sine and cosine have infinitely many real roots, they cannot be polynomials. A polynomial of degree k has the property that if it is differentiated k + 1 times, then the final derivative is zero. Since no derivative of ever vanishes, we conclude that cannot be a polynomial. However, in calculus we learned of a formal procedure, called Taylor series, for associating polynomials with a given function!. In some instances these polynomials form a sequence that converges back to the original function. This might cause us to speculate that any reasonable function can be approximated in some fashion by polynomials. In fact, the theorem of Weierstrass gives a spectacular affirmation of this speculation:

ex,

ex

8.4.1

ex

Weierstrass's Main Result

Theorem 8.6 Weierstrass Let f be a continuous function on an interval [a, bl. Then there is a sequence of polynomials Pj (x) with the property that the sequence Pj converges uniformly on [a, bl to f. See Figure 8.2.

Let US consider some consequences of the theorem. A restatement of the theorem would be that, given a continuous function f on [a, bl and an (i > 0, there is a polynomial P such that I!(x) - p(x)1 < {i

112

Chapter 8: Sequences and Series of Functions

y =/(x)

Figure 8.2 for every x E [a. b]. If one were programming a computer to calculate values of a fairly wild function I. the theorem guarantees that. up to a given degree of accuracy, one could use a polynomial instead (which would in fact be much easier for the computer to handle). Advanced techniques can even tell what degree of polynomial is needed to achieve a given degree of accuracy. And notice this: Let 1 be the Weierstrass nowhere differentiable function. The theorem guarantees that, on any compact interval, 1 is the uniform limit of polynomials. Thus even the uniform limit of infinitely differentiable functions need not be differentiable-even at one point. This explains why the hypotheses of Theorem 8.4 need to be so stringent. Remark 8.2 If 1 is a given continuous function. then it is a mailer of great interest to actually produce the polynomial that will approximate 1 to a pre-specified degree of accuracy. There is a large theory built around this question. Certainly the Lagrange interpolation polynomials (see [BUB] or [ABR» will do the trick. An examination of the proof of the Weierstrass theorem that is presented in either [RUD] or [KRAI] will give another method of approximation. I Example 8.11

Let 1 be a continuously differentiable function on the interval [0. I]. Can we approximate it by polynomials p j so that p j --+ 1 uniformly and also --+ f' uniformly? The answer is "yes." For apply Weierstrass's theorem to find polynomials qj that converge uniformly to f'. Then integrate the qj to produce the desired polynomials Pj. We leave the details to the reader. If it is known that 1(1/2) = O. then we can produce polynomials Pi that perform the approximation described in the last two paragraphs and such that Pi (1/2) = O-just subtract a suitable constant from each polynomial. Again. details are left to the reader. 0

pi

Chapter 9

Some Special Functions 9.1 Power Series 9.1.1 Convergence A series of the form

00

2:>j(x -c)J J=O

is called a power series expanded about the point c. Our first task is to determine the nature of the set on which a power series converges.

Proposition 9.1 Assume that the power series 00

2::>J(x _c)J J=O

converges at the value x = d. Let r = )d - cl. Then the series converges uniformly and absolutely on compact subsets of I = Ix : Ix - cl < r}. Example 9.1 The power series

f:

(x -l)J

J=O (j

+ 1)2

obviously converges at x = 2, for then it simply reduces to the sum 00

f;

1 (j + 1)2'

The proposition therefore tells us that the series converges uniformly and absolutely on compact subsets of the interval (0.2). This assertion may also 0 be verified directly using the ratio test. 113

114

Chapter 9: Some Special Functions

9.1.2 Interval ofConvergence An immediate consequence of Proposition 9.1 is that the set on which the power series 00

Laj(x -c)j j=O

converges is an interval centered about c. We call this set the interval ofconvergellce. The series will converge absolutely and uniformly on compact subsets of the interval of convergence. The radius of the interval of convergence (called the radius of cOllvergellce) is defined to be half its length. Whether convergence holds at the endpoints of the interval will depend on the particular series. Let us use the notation C to denote the (maximal) open interval of convergence. It happens that if a power series converges at either of the endpoints of its interval of convergence. then the convergence is uniform up to that endpoint. This is a consequence of Abel's partial summation test. On the interval of convergence C. the power series defines a function f. Such a function is said to be real allalytic-see [KRPj for more on this subject. We will see more on this topic below. Example 9.2

The function I f(x) = I-x

has power series expansion about the origin given by 00

f(x) = Lx j j=O

This fact may be verified just by long division. The series converges neither at I nor at-I. In fact one may check by hand that f has a convergent power series expansion about any point xo in the real line except I. Thus f is real analytic 0 at all points except 1.

9.1.3 Real Analytic Functions A function f. with domain an open set U C IR and range either the real or the complex numbers. is called real allalytic if, for each c E U. the function f may be represented by a convergent power series on an interval of positive radius centered at c :

Definition 9.1

00

f(x) = Laj(x - c)j . j=O

9.1

Power Series

115

9.1.4 Multiplication ofReal Analytic FUllctions We need to know both the algebraic and the calculus properties of real analytic functions: are they continuous? differentiable? How does one add. subtract. multiply. or divide two real analytic functions?

Proposition 9.2 Let 00

00

~::>j(x - c)j and

L bj(x -

j=O

j=o

c)j

be two power series with intervals of convergence CI and C2. respectively. Let II (x) be the function defined by the firslSeries on C. and hex) the function defined by the second series on C2. Then, on their common domain C = CI n C2, it holds that

1. II (x) ± h(x) =

L:~(aj ±bj)(x -c)j;

2. II(x) h(x) = L:~=OL:j+k=m(aj ·bt}(x _c)m Example 9.3

The function I(x) = I/O - x)2 has power series expansion 00

I(x) =

L jx j - I j=1

while the function g(x) = 1- x is its own power series. Then we know that

1

-=(1-x)· 1- x

1

00

l .- I " 2=(1-x)'LJjx (1 - x) j=1

00

l . ". =LJx j=O

Observe that the product series has the same domain of convergence as the series for I. 0

9.1.5 Division ofReal Analytic Functions Next we tum to division of real analytic functions. If I and g are real analytic functions defined on a common open interval I. and if g does not vanish on I, then we would like 1/g to be a well-defined real analytic function (it certainly is a welldefinedjimctioll) and we would like to be able to calculate its power series expansion by formal long division. This is what the next result tells us:

Proposition 9.3 Let I and g be real analytic functions. both of which are defined on an open interval I. Assume that g does not vanish on I. Then the function

hex) = I(x) g(x)

116

Chapter 9: Some Special Functions

is real analytic on I. Moreover, if 1 is centered at the point c and if 00

!(x)

00

= l:>j(x -

c)j and g(x)

= Ebj(x -

j~

c)j •

j~

then the power series expansion ofh about c may be obtained by forma/long division of the latrer series into the fonner. Thar is, the zeroeth coefflcienr co of h is

co = ao/bo. the order one coefficienr CJ is

erc. Example 9.4 Ler

f(x) = I-x Then

and

g(x)

= (l_x)2 = 1-2x +x2

_1__ f(x) _ ~xj 1 - x - g(x) - 4--' . J~

Observe that rhe quotient series converges on (-I, 1). In practice it is often useful to calculate f / g by expanding g in a "geometric series." To illustrate this idea, we assume for simplicity that f and g are real analytic in a neighborhood of o. Then

f(x) = f(x) . _1_ g(x) g(x)

= f(x)

I

bo + btx + ... 1 1 = f(x)· bo· 1+(b./bo)x+ ... Now we use the fact that, for /3 small.

1 2 1_/3=1+/3+/3 + .... Setting /3 = -(bl/bo)x - ... and substituting the resulting expansion into OUT expression for f(x)/g(x) then yields a formula that can be multiplied out to give a power series expansion for f(x)/g(x).

9.2

117

More on Power Series: Convergence Issues Example 9.5 Let us redo the last example in light of these comments about using geometric series to perform long division of real analytic functions. Now f(x) l-x,g(x) (l-x)2. and

=

=

f(x) I-x g(x) = (1 - x)2

1 = (1 - x)· 1 _ (2x _ x2) = (l - x) .

[1 +

(2x - x 2) + (2x - x 2)2 + ... ]

=I+x+x2 +...

9.2

0

More on Power Series: Convergence Issues

9.2.1

The Hadamard Formulo.

We now introduce the Hadamard fomlllia for the radius of convergence of a power series. Lemma 9.1 For the power series 00

I>j(x-c)j, jdJ

define A and p by A = limsuplajll/i j-+oo

p=

{

if A = 00 if 0 < A < if A =0.

0 ooIIA

00

Then p is the radius ofconvergence of the power series aboutc. Example 9.6 Consider the power series 00

xj

L2

j

j=O

Thenaj =2- j and A = li~sup laili = li~supTI = )-+00

)-+00

1

2

118

Chapter 9: Some Special Functions It follows that p = 2 is the radius of convergence of the power series. The 0 series converges on the interval (-2.2).

Example 9.7 Consider the power series

f:

(x -',2)j

j=o

J.

Then aj = I/j!. Notice that the first j/2 terms of j! - j . (j - 1) (j - 2) ... 3 . 2 . 1 are of size at least j /2. Hence .

o ~ A = I~~:P lajlll}:s

(

1 ) Ilj (jf2)jf2

1

= (j/2)1/2

-+ O.

Hence A = O. Thus the radius of convergence of the power series is p = +00. Therefore the series converges on the entire real line. 0

Corollilry 9.1 The power series 00

I>j(x -c)j j=o

has radius ofconvergence p if and only if, when 0 < R < p, there exists a constant 0< C =CR such that for all j.

9.2.2 The Derived Series From the power series 00

:~:>j(x -c/ j=O

it is natural to create the derived series 00

Ljaj(x-c)H j=1

using term-by-term differentiation.

Proposition 9.4 The radius of convergence of the derived series is the same as the radius of convergence of the original power series.

9.2 More on Power Series: Convergence Issues

119

This result follows from the root test.

Proposition 9.5 Let f be a real analytic function defined on an open interval I. Then f is continuous and has continuous. real analytic derivatives of all ordet'S on I. In fact the derivatives of f are obtained by term-by-tenn differentiation of its series representation. Example 9.8

We know that 1

00

i; -=Ll 1-1 '=0

)-

hence (substituting 1 = x 2 ) _1--" _

I-x 2

~x2i

- ~

)=0

Now differentiating the series yields oo 2x 2'JX 2J'-1 = . (1- x 2 )2

L

J=o

Since the original series converges for 1 E (-1, I), we may be sure that the new series converges for x E (-1, 1). This assertion may also be checked with the ratio test. It can be verified (Example 9.9 below) that .

smx =

00

f;

(_I)ix2i+l

(2j + 1)1

Differentiating both sides yields 00

cosx =

L

j=O

(_I)ix2i

(2 ")1

J .

Both these series converge for all x.

9.2.3

o

Fonnula for the Coefficients ofa Power Series

We can now show that a real analytic function has a unique power series representation at any point.

Chapter 9: Some Special Functions

120

Corollary 9.2 Ifthe function { is represented by a conveTgent poWeT series on an interval ofpositive radius centeTed at c, 00

{(x) = I:>j(x - c)j , j=O

then the coefficients of the power series are related to the derivatives of the func-

tion by a'J -

{U)(c) .,

}.

Example 9.9 Let {(x) = sinx. Then {(O) =0

1'(0) = 1

/,,(0) = 0 {III(O) = -1 [(iv)(O) = 0

and so forth. It follows that •

sm x =

2j 1

(-l)j x + L .:..",:.:-.:-,.,..,.... j=O (2j + I)! 00

o

Finally, we note that integration of power series is as well behaved as differentiation.

Proposition 9.6 The power seTies 00

Laj(x -c)j j=o

and the seTies a' -._J_(x - c)j+! .~} + 1 J=V 00

L

obtained from term-by-teTm integration have the same radius of conveTgence, and the function F defined by F(x) =

L -.a'_J_(x 00

j=O}

+1

c)J+\

9.2 More on Power Series: Convergence Issues

121

on the common interval ofconvergence satisfies 00

F'(x) = I>j(x - c)j = f(x). j=O

Example 9.10 As we will see in a moment. the theory of power series is valid. without change. for a complex argument. Thus I

----::2

I +x

=

I

1-

'2 (IX)

~ 2' ~ , 2' = L.J(ix) ) = L.J(-I»)x ). j=1

j=O

Integrating both sides yields that ~ (-I)jx2j +l arctan x = L.J '=0 )-

2j + I

o

It is sometimes convenient to allow the variable in a power series to be a complex number. In this case we write 00

Laj(z-c)j j=O

where z is the complex argument. We now allow c and the aj 's to be complex numbers as well. Noting that the elementary facts about series hold for complex series as well as real series (you should check this for yourself). we see that the domain of convergence of a complex power series is a disc in the complex plane with radius p given as follows: Let A = lim sup lani l/n . n.... OO

Then p=

{

0

if A =

ool/A

ifO
00

These observations about complex power series will be useful in Section 9.3.

9.2.4 Taylor's Expansion We conclude this section with a consideration of Taylor series: Theorem 9.1 Let k be a nonnegative integer and suppose that f is a k + I-times continuously differentiable function on an open interval I = (a - E. a + E). Then. for x e I. k

•

. (x-a») f(x) = f()(a) ., j=O J.

L

+ Rk,u(X)

Chapter 9: Some Special Functions

122 where

Example 9.11 Consider the function f(x) = ex. Then all derivatives ofthe function fare also equal to ex Thus the Taylor series expansion about 0 for this f is xj

00

f(x) -

L -:r' J.

'=0 J-

Notice that we write - instead of = because, even though the series obviously converges (by the ratio test), we do not know that it converges to [ until we check the remainder term. Let us now perform that check. Now Rk.O(X) =

r e'·

10

(x _t)k

k!

dt.

We could actually evaluate this integral by integration by parts, but nothing so precise is required for our present purposes. Instead let us restrict attention to x lying in an interval [-A, A]. For such x, we may estimate IRk,O(x)1

~e

A •

r 10

(x - t)k

k!

dt = e

A

This expression clearly tends to 0 as k _ series converges to f as desired.

xk+t

(k + I)! 00,

~

e A Ak + 1

(k

+ I)!

uniformly in x. Thus the 0

Taylor's theorem allows us to associate with any infinitely differentiable function a formal expansion of the form 00

I>j(x-a)j j=o

with aj = [(j)(a)/jl. However, as already noted, there is no guarantee that this series will converge; even if it does converge, it may not converge back to f(x).

Example 9.12 Consider the function h(x)

=

o {e-l/x1

ifx=O if x oF O.

This function is infinitely differentiable at every point of the real line (including 0). However, all of its derivatives at x = 0 are equal to zero. Therefore the formal Taylor series expansion of h about a = 0 is 00

L 0 . (x j=o

O)i = O.

9.3 The Exponential and Trigonometric Functions

123

We see that the formal Taylor series expansion for" converges to the zero 0 function at every x. not to the original function" itself. In fact the theorem tells us that the Taylor expansion of a function f converges to f at a point x if and only if Rk.u(X) -+- O. We have the following more quantitative assertion. An infinitely differentiable function f on an interval I has Taylor series expansion about a E I that converges to f on a neighborhood J of a if and only if there are positive constants C. R such that. for every x E J and every k. it holds that

Example 9.13

Refer to Example 9.11. We can now see more easily that f(x) = eX is real analytic-i.e.• that the formal power series expansion for factually converges to f. Merely note (for instance) that. for x E [-2.2]. It
leXl :'0 leAl :'0

~~

for k large. Thus f is real analytic on [-2.2]. Similar estimates apply on 0. any other interval. The function" considered in Example 9.12 should 1I0t be thought of as an isolated exception. For instance. we know from calculus that the function f(x) = sinx has Taylor expansion that converges to f at every x. But then for € small the function g.(x) = f(x) +
9.3 The Exponential and Trigonometric Functions 9.3.1

The Series Definition

We begin by defining the exponential function: Definition 9.2

The power series 00

zJ

L"7i

J=O J.

converges. by the Ratio Test. for every complex value of z. The function defined thereby is called the exponential function and is written exp(z).

Chapter 9: Some Special Functions

124

Proposition 9.7 The function exp(z) satisfies exp(a + b) = exp(a)· exp(b) for any complex numbers a and b. This fact can be verified directly by multiplication ofpower series. We set e = exp(1). This is consistent with our earlier treatment of the number e in Section 2.3. The Proposition tells us that. for any positive integer k. we have ek

= 'e-. e·v..-e' = •exp(1) . exp(1) ... exp(1) = exp(k) . • • k times

k times

If m is another positive integer. then

(exp(kjm»m

=exp(k) = i

whence exp(kjm) = ifni We may extend this formula to negative rational exponents by using the fact that exp(a)· exp(-a) = 1. Thus. for any rational number q. exp(q) = e q .

Example 9./4 One may calculate from the series expansion for e. 00

e=

1

I:-=i' j=O J.

that e "" 2.718281828 .... It can be shown that e is irrational. indeed transcendental: that is. it is not the root of any polynomial with integer coefficients. 0 Now note that the function exp is monotone increasing and continuous. It follows that if we set. for any r E JR. e' = sup(eq : q

E

Q. q < r}

(this is a definition of the expression e') then eX = exp(x) for every real x. (You may find it useful to review the discussion of exponentiation in Sections 1.5.1 and 2.3; the presentation here parallels that one.) We will often adhere to custom and write eX instead of exp(x) when the argument of the function is real.

Pl'Oposition 9.8 The exponential function ex satisfies

9.3

The Exponential and Trigonometric Functions

125

1. eX> 0 foral/x;

2.

eO

= I;

3. (eX)' = eX;

4. e' is strictly increasing; 5. the graph of eX is asymptotic to the negative x -axis; 6. for each integer N > 0, there is a number CN such that eX > x> O.

CN

x N when

See Figure 9.1.

y =eX

Figure 9.1

All these assertions may be verified directly from the power series definition of the exponential.

Example 9.15 Let p(x) be any polynomial. Property 6 implies that there is a constant C such that Ip(x)1 ::: C . C for all real x. We will learn below that logarithmic functions grow more slowly than any polynomial. Thus there is a hierarchy logarithms < polynomials < exponentials for growth rates of functions. This is important information for qualitative analysis in the mathematical sciences.

126

Chapter 9: Some Special Functions

9.3.2

The Trigonometric Functions

Now we turn to the trigonometric functions. The definition of the trigonometric functions given in most calculus texts is unsatisfactory because (i) it relies too heavily on a picture and because (ii) the continual need to subtract off superfluous multiples of 211" is clumsy. We have nevertheless used the trigonometric functions in earlier chapters to illustrate various concepts. It is time now to give a rigorous definition of the trigonometric functions that is independent of these earlier considerations.

Definition 9.3

The power series 00

x2}+1

•

j;(-I») (2j + 1)! converges at every point of the real line (by the Ratio Test). The function that it defines is called the sine function and is usually written sinx. The power series 00

•

x2}

L(-I») (2 ')1 }=o J . converges at every point of the real line (by the Ratio Test). The function that it defines is called the cosine function and is usually written cos x. Example 9.16

Observe that 4 4 3 45 sin4= - - - + l!

3!

5!

-+ ....

Using the standard estimate for the error term of an alternating series. we thus see that sin 4 < O. Of course sin 0 = O. and one may perform a calculation like (*) to see that sin 2> O. We could. in principle, define 11" to be the first number a > 0 at which sin a = O. We will provide a more discursive discussion of the number 11" following Proposition 9.11. 0 You may recall that the power series that we use to define the sine and cosine functions are precisely the Taylor series expansions for the functions sine and cosine that were derived in your calculus text. But now we begin with the power series and must derive the properties of sine and cosine that we need from rhese series.

9.3.3 Euler's Formula In fact the most convenient way to achieve this goal is to proceed by way of the exponential function. (The point here is mainly one of convenience. It can be verified by direct manipulation of the power series that sin2 x + cos2 X = 1 and so forth, but

9.3

127

The Exponential and Trigonometric Functions

the algebra is extremely unpleasant.) The formula in the next proposition is due to Euler.

Proposition 9.9 Euler The exponential function and the functions sine and cosine are related by the formula (for x and y real and i 2 = -1) exp(x + iy) = If . (cos y + i sin y) . To verify this formula. just write out the power series expansions for all the relevant functions and multiply out both sides of the formula. Because of this formula. exp(iy) = cos y + i sin y.

(9.9.1)

We will usually write this as';'Y = cosy+i siny, where this expressiondejines what we mean by ';'Y. As a result.

= If . e iy = If . (cos y + i sin y).

If+iy

Example 9.17 Let us confirm formula (9.9.1). Now

i

y

= 1 + iy I! =

+ (iy)2 + (iy)3 + (;y)4 + ... 2!

(1 _2! + 4! y2

y4 _

3!

4!

+... ) + i

= cos y + i sin y.

(y _3! + ... ) y3

0

9.3.4 The TrigoIWmetri£ Functions Notice that e-iy = cos(-y)+i sin(-y) = cos y-; siny (we know from their power series expansions that the sine function is odd and the cosine function even). Then formula (9.9.1) tells us that cosy =

and

Proposition 9.10 For every real x, it holds that

2 e iy _ e- iy

siny = Now we may prove:

,;.y + e- iy

2i

(9.3.4.1)

(9.3.4.2)

Chapter 9: Some Special Functions

128

To see this. just use (9.3.4.1) and (9.3.4.2). We list several other properties of the sine and cosine functions that may be proved by similar methods.

Proposition 9.11 The funclions sine and cosine have the following properties:

1. sin(s + I) = sins cos 1 + cosssinl; 2. cos(s+t)=cosscost-sinssint;

3. cos(2s)

=cos2 s -

sin2 s;

4. sin(2s) = 2sinscoss;

S. sin(-s) = -sins; 6. cos(-s) =coss;

7. sin'(s)=coss; 8. cos'(s) = -sins. Example 9. J8

Let us prove part 1 of Proposition 9.11. Now

.

.

smscost+cosssml =

[eiS - e- is eit + e-i'l 2i

2

+

[is

+ e- is i' - e- it 2

.

eis it

eis e- it

e- is it

e-iJ.'e- it

=41+

4i

4i

4i

isi'

iSe- i,

+41-

4i

+

e-isi'

e-ise- it

4i

4i

2i

1

ei(s+t) _ e-i(s+,) 2i = sin(s + t). This confirms the formula. One important task in any course on the foundations of analysis is to define the number Tr and establish its basic properties. In a course on Euclidean geometry. the number Tr is defined to be the ratio of the circumference of a circle to its diameter. Such a definition is not useful for our purposes (however. it is consistent with the definition given here). Observe that cos 0 is the real part of i O• which is I. Thus if we set Ct

= infix> 0: cosx = 01

129

9.3 The Exponential and Trigonometric Functions

then a > 0 and. by the continuity of the cosine function. cos a = O. We define :rr = 2a. Applying Proposition 9.10 to the number a yields that sina = ±l. Since a is the first zero of cosine on the right half line, the cosine function must be positive on (O, a). But cosine is the derivative of sine. Thus the sine function is increasing on (0. a). Since sinO is the imaginary pan of eiO-which is O-we conclude that sina > 0 hence that sina = +1. Now we may apply parts 3 and 4 of Proposition 9.11 with s = a to conclude that sin:rr = 0 and cos:rr = -1. A similar calculation with s = :rr shows that sin 2:rr = 0 and cos 2:rr = 1. Next we may use parts 1 and 2 of Proposition 9.11 to calculate that sin{x + 2:rr) = sin x and cos{x + 2:rr) = cos x for all x. In other words. the sine and cosine functions are 2:rr-periodic.

Example 9.19 The business of calculating a decimal expansion for :rr would take us far afield. One approach would be to utilize the already noted fact that the sine function is strictly increasing on the interval [0. :rr/2]; hence its inverse function Sin-I: [0. I] ~ [O.:rr /2] is well-defined. Then one can determine (see Proposition 6.2) that ( Sin-I)' (x) =

1

.Jl- xi

.

By the fundamental theorem of calculus.

=1 °. 1

-=

:rr 1 Sin-I{l) dx. 2 ./1 -x2 By approximating the integral by its Riemann sums. one obtains an approximation to :rr/2 and hence to:rr itself. 0

Remark 9.1

Some sources use the notation arcsin instead of Sin-I

I

Let us for now observe that 22 24 26 cos2 = 1- - + - - - + _ ...

2!

4!

6!

16 64 =1-2+ + .... 24 720 As we noted in Chapter 3, since the series defining cos 2 is an alternating series with terms that strictly decrease to zero in magnitude, we may conclude that the last line is less than the sum of the first three terms: 2 cos 2 < -1+ < O.

3

130

Chapter 9: Some Special Functions

It follows that Ct = Tr /2 < 2 hence Tr < 4. A similar calculation of cos(3/2) would allow us to conclude that Tr > 3.

9.4 Logarithms and Powers of Real Numbers Since the exponential function exp(x) = ex is positive and strictly increasing. it is therefore a one-to-one function from IR to (0. 00). Thus it has a well-defined inverse function that we call the natural logarithm. We write this function as Inx (or sometimes log x).

9.4.1

The Logarithmic Function

Proposition 9.12 The narural logarithm function has the following properties: 1. (lnx)' = l/x; 2. lnx is strictly increasing;

3. In(1) = 0; 4. Ine = 1;

s.

the graph of the narurallogarithm function is asymptotic to the negative y axis.

6. In(s· t) = Ins + Int;

7.ln(s/t)=lns-Int. See Figure 9.2. These properties are all immediate from Proposition 9.8 and the definition of In.

Example 9.20 We have discussed earlier (part 6 of Proposition 9.8 and Example 9.15) that the exponential function grows faster than any polynomial. Now let p(x) = akxk +ak_lx k- 1+ak_2xk-2+ ... alx+ao be any polynomial with positive leading coefficient ak. Then it follows from our earlier considerations that. for some constant C. x ::: C . eP(x) for all real x. Using part 2 of Proposition 9.12. we may apply the logarithm to this inequality to obtain Inx ::: p(x) + InC This confirms the statement that we made in Example 9.15 about logarithms growing more slowly than polynomials.

9.4

Logarithms and Powers of Real Numbers

I3l

y = lnx

Figure 9.2

Proposition 9.13 If a and b are positive real numbers. then

Remark 9.2 We have discussed several different approaches to the exponentiation process. We proved the existence of nth roots. /I E N, as an illustration of the completeness of the real numbers (recall that we took the supremum of a certain set). We treated rational exponents by composing the usual arithmetic process of taking III Ih powers with the process of taking nth roots. Then. in Section 2.3, we passed to arbitrary powers by way of a limiting process. Proposition 9.13 gives us a unified and direct way to treat all exponentials at once. This unified approach will prove particularly advantageous when we wish to perform calculus operations on exponential functions (see the next proposition). I

Proposition 9.14 Fix a > O. The function !(x) = aX has the following properties:

2. {(O) = I;

3. if 0 < a < I. then { is decreasing and the graph of ! is asymptotic to the positive x-axis; 4. if 1 < a. then! is increasing and the graph of! is asymptotic to the negative

x-axis.

Chapter 9: Some Special Functions

132 Example 9.21

Let us differentiate the function g(x) = [sinx)COSx. It is convenient to write g(x) = ecosx.1n\sinx). Then it is clear that g'(x) = ecosx.ln\sinx) • [cosx .In[sinx))' =

sinx In[sinx) + cos x

. [_ ecosx.ln(smx).

COSX] .Sin X COSX]

= [sinx)cosx. [ -sinx In[sinx) +cosx' - . - . Sin X

o

9.4.2 Characterization ofthe Logarithm The logarithm function arises. among other places. in the context of probability and in the study of entropy. The reason is that the logarithm function is uniquely determined by the way that it interacts with the operation of multiplication:

Theorem 9.2 Let (x) be a continuously differentiable function with domilin the positive reills. and suppose satisfies the equality (5.1) = (5)

+ (1)

(9.2.1)

for all positive 5 and I. Then there is a constant C > 0 such thilt f(x) = C ·Inx

for all x. Observe that the natural logarithm function is then the unique continuously differentiable function that satisfies the condition (9.2.1) and whose derivative at 1 equals 1. That is the reason the natural logarithm function (rather than the common logarithm. or logarithm to the base 10) is singled out as the focus of our considerations in this section.

9.S

The Gamma Function and Stirling's Formula

Definition 9.4

For x > O. we define r(x)

=

10

00

e-'Ix-Idl.

Notice that. by Proposition 9.8 part 6. the integrand for fixed x is bounded by the function Ix-I if 0<1:;:1 {(I) = { CN' tx-N-1 if 1 < I < 00.

9.6

An Introduction to Fourier Series

133

We choose N so large that x - N -1 < -2. Then the function It can be shown, then, that the integral defining r converges.

f

is clearly integrable.

Proposition 9.15 For x > 0, we have

rex + 1) =

x

rex).

To verify this formula, integrate by parts.

Corollary 9.3 For II = 1,2, ... we have r(1I + 1) = II!. The corollary shows that the gamma function r is an extension of the factorial function from the positive integers to the positive real numbers.

9.5.1

Stirling's Formula

Theorem 9.3 Stirling The limit

r

{

II!

n~~ $e-nll,,+1/2

}

exists and equals I. In particular, the value of II! is asymptotically equal to

$

lI n+l/2

en

as II becomes large. Remark 9.3 Stirling's formula is important in calculating limits, because without the formula it is diffi;;ult to estimate the size of II! for large II. In this capacity, it plays an important role in probability theory, for instance, when one is examining the probable outcome of an event after a very large number of trials. I

Corollary 9.4 We have r(1/2) = .j1r.

9.6 An Introduction to Fourier Series In this section it will be convenient for us to work on the interval [0, 21t). We will perform arithmetic operations on this interval modulo 2lT : for example, 3lT 12+3lT12 is understood to be equivalent to IT because we subtract from the sum (3lT) the largest multiple of 2lT that it exceeds. When we refer to a function f as being continuous

134

Chapter 9: Some Special Functions

on [0. 211" 1. we require that it be right continuous at O. left continuous at 2rr, and that 1(0) = 1(2rr)·

9.6.1 Trigonometric Series If f is a (either real- or complex-valued) Riemann integrable function on this interval and if II

E

Z, then we define ~ 1 (1Jr . I(n) = 2rr fo l(t)e-tn1dt.

We call 1(11) the nih Fourier coefficient of I. The formal expression 00

L

S/(x) -

!(II)i

nx

n=-CO

is called the Fourier series of the function I In circumstances where the Fourier series converges to the function I. some of which we shall discuss below, the series provides a decomposition of I into simple component functions. This type of analysis is of importance in the theory of differential equations, in signal processing, and in scattering theory. There is a rich theory of Fourier series which is of interest in its own right. Example 9.22

Let us calculate the Fourier series of I(x) = x. Now 1(0) =

2~ fo1Jr t dt = 11" .

For II f= 0 we have 1(11) =

~ {1Jr I(t)e- int dt 211" fo

_ ~ {1Jr t . e-inl dt 211" fo

int e11Jr 1 fo1Jr e-inl = -·t·-- ---dt

(pans)

1

2rr

211" 0

-ill 0

-i II

- ....!....-O -in n

Therefore x=/(x)-1I"+

i

L _·i "=-00 n 00

nFO

nx

•

o

9.6 An Introduction to Fourier Series

135

We will see later. in Example 9.23. that this formula can be used to determine that L~I 1/11 2 = Tr 2 /6.

9.6.2

Formula/or the Fourier Coefficients Observe that. in case I has the special form (called a trigollometric polYllomial) K

L

I(x) =

aninI.

(9.6.2.1)

n=-K

then the coefficients an are given by an

1

= -1 2Tr 0

2 "

. dt I(t)e- Inl

(just perform the integrations on both sides. noting the simple formula f~" iiI dt = 0 when j # 0). Since functions of the form (9.6.2.1) are dense in the continuous functions. 1 we might hope that the coefficients 1<11) of the Fourier series of I will contain important information about I. The other theory that you know for decomposing a function into simple components is the theory of Taylor series. However. in order for a function to have a Taylor series it must be infinitely differentiable. Even then. as we have learned. the Taylor series of a function usually does not converge. and if it does converge then its limit may not be the original function. The Fourier series of I converges to I under fairly mild hypotheses on I. and thus provides a useful tool in analysis.

9.6.3 Bessel's Inequality The first result we shall enunciate about Fourier series gives a growth condition on the coefficients 1(11) : ~

Proposition 9.16 If 12 is integrable. then

Example 9.23 In fact it can be shown (this is Plancherel'slormllla) that

1See Subsection 10.3.1 for the notion of density. The assertion here is a variant of Theorem 8.6.

136

Chapter 9: Some Special Functions We may apply this formula to the result of Example 9.22 to find that

or

,,-=L..J n 6 1

00

1f2

2

1/=1

We will relearn this fact from a different point of view in Example 9.24. 0 Corollary 9.5 If f2 is integrable, then the Fourier coefficients 1(11) satisfy the property that

Definition 9.5 Let f be an integrable function on the interval (0, 21f). Let SN(X) denote the Nth partial sum ofthe Fourier series of f : N

L

SNf(x) =

{(n)i" X

n=-N

9.6.4 The Dirichlet K£rnel Since the coefficients of the Fourier series, at least for a square integrable function, tend to zero, we might hope that the Fourier series will converge. Of course the best circumstance would be that SN f -+ f in some sense. We now turn our attention to addressing this problem. Proposition 9.17 If f is integrable, then 2

SNf(x) = - 1

10 "

2Jr 0

DN(X - t)f(t)dt.

where (9.17.1)

9.6

137

An Introduction to Fourier Series

This fonnula is derived by noting that N

SNf(x) =

L

l(j)i jx

j=-N

f.

= L..J - 1 j=-N 2Jr

= _I [2" 21T

10

12n: !(t)e-'jl.. dt . e'jX.. 0

rt

ij(X-/l] f(t) dt .

L=-N

The whole problem. then. devolves upon calculating the sum in the brackets. But the upper half of the sum is a geometric series, and so is the lower half. Thus the series may be summed by the methods of Subsection 3.2.3. and the result is as in (9.17.1).

Remark 9.4 Note that. by a change of variable. the formula for SN presented in Proposition 9.17 can also be written as SN f(x) =

~ 12n: DN(I)!(X 21T 0

I) dl

provided we adhere to the convention of doing all arithmetic modulo multiples I of21T.

Lemma 9.2 For any N. it holds that -1

12n: DN(I)dl = I.

21T 0

To see this. integrate the original sum that defines DN. Next we claim that. for a large class of functions. the Fourier series converges back to the function at every point.

Theorem 9.4

Let f be a function on [0. 21T 1that satisfies a Lipschitz condition: there is a constant C > 0 !Ouch that ifs, I E [0. 21T 1. then

I!(s) - [(01 < C Is - II·

(9.4.1)

[Note that at 0 and 2Jr this condition is required to hold modulo 21T -see the remarks at the beginning of the section.} Then for every x E [0. 21T 1. it holds that SN!(X)-!(x)

Indeed. the convergence is uniform in x.

as N_oo.

Chapter 9: Some Special Functions

138 CoroUary 9.6 If feCI ([0, 21T]) then SN f -+

f uniformly.

In fact the proof of the theorem suffices to show that if f is a Riemann integrable function on [0, 21T] and if f is differentiable at x, then SN f(x) -+ f(x). Example 9.24

Let f(t) = t 2 - 2m,O ~ t < 21T. Then f(O) = f(21T) = 0 and f is Lipschitz modulo 2lr. A straightforward but tedious calculation shows that ~

f(/I) =

~

2 2" if

/I

/I

#0

21T 2

f(O)

=-3·

Applying Theorem 9.4 at the point x = 0 then yields that 21T 2

oo?

--+2" -, =0 3 L.J /1n=\

or 00

1T 2

1

L2"=-· 6

n=\/I

This is an important fact in analytic number theory. It tells us that the Riemann zeta function takes the value 1T 2 /6 at 2. 0 In fact the Riemann zeta function in general is defined to be 00

1

~(z) = "-:L.J II' n=1

for complex numbers z with real part exceeding 1. The sum we have been discussing is obviously just ~ (2). The behavior of the analytic function ~ has close links with the distribution of the prime numbers, and is the object of intense study. The celebrated Riemallll hypothesis is a conjecture about the zeros of the Riemann zeta function.

Chapter 10

Advanced Topics Part of the power of modem analysis is to look at things from an abstract point of view. This provides both unity and clarity. and also treats all dimensions at once. We shall endeavor to make these points clear as we proceed.

10.1 10.1.1

Metric Spaces The Concept ofa Metric

This section formalizes a general context in which we may do analysis any time we have a reasonable notion of calculating distance. Such a structure will be called a metric:

Definition 10.1

A metric space is a pair (X. p), where X is a set and p : X x X -+ {t e JR : t

~

O}

is a function satisfying 1. ForaIlx,y e X. p(x,y) =p(y.x);

2. p(x, y)

= 0 if and only ifx = y;

3. for all x, y, z e X, p(x, y) ::: p(x. z) + p(z, y). The function p is called a metric on X. Condition 3 is called the triangle inequality.

10.1.2 Examples ofMetric Spaces Example 10.1 The pair (JR, p). where p(x. y) = Ix - yl, is a metric space. Each of the properties required of a metric is in this case a restatement of familiar facts from the analysis of one dimension. 139

Chapter 10: Advanced Topics

140

The pair (]R3. p). where

is a metric space. Each of the properties required of a metric is in this case a restatement of familiar facts from the analysis of three dimensions. 0 The first example presented familiar metrics on two familiar spaces. Now we look at some new metrics. Example 10.2

The pair (]R2. p). where p(x. y) = max{lxl - yd.lx2 - nil. is a metric space. Only the triangle inequality is not trivial to verify; but that reduces. by consideration of several cases. to the triangle inequality of one variable. The pair (IR, p.). where p.(x. y) = 1 if x # y and = 0 otherwise. is a metric space. Checking the triangle inequality reduces to seeing that if x # y. then either x # z or y # z. 0 Example 10.3

Let X denote the space of continuous functions on the interval [0. 1]. If I. g e X. then let p(f. g) = sup I/(t) - g(t)l. IE(O.I)

Then the pair (X. p) is a metric space. The first two propenies of a metric are obvious. and the triangle inequality reduces to the triangle inequality for real numbers. We sometimes refer to p as the "uniform metric." This example is a dramatic new departure from the analysis we have done in the previous nine chapters. For X is a very large space- infinite dimensional in a cenain sense. Using the ideas that we are about to develop. it is nonetheless possible to study convergence. continuity. compactness. and other basic concepts of analysis in this more general context. We shall see 0 applications of these new techniques in later sections.

10.1.3 Convergence in a Metric Space Now we begin to develop the tools of analysis in metric spaces. Definition 10.2 Let (X. p) be a metric space. A sequence {Xj I of elements of X is said to converge to a point a e X if for each E > 0 there is an N > 0 such that if j > N. then p (x j • a) < E. We call a the limit of the sequence (x j ). We sometimes write Xj -+ a.

10.1

141

Metric Spaces

Compare this definition of convergence with the corresponding definition for convergence in the real line in Section 2.1. Notice that it is identical, except that the sense in which distance is measured is now more general.

Example 10.4 Let (X. p) be the metric space from Example 10.3. consisting of the continuous functions on the unit interval with the indicated metric function p. Then f = sinx is an element of this space. and so are the functions

f ·-

j

x 21+1

"(_I)l _ _

J-f=o

(2£+ I)!

Observe that the functions Ii are the partial sums for the Taylor series of sin x. We can check from simple estimates on the error term of Taylor's theorem that the functions Ii converge wliforlllly to f. (Note that uniform convergence is the right notion for this metric space.) Thus. in the language of metric spaces, fj ->- f in the metric space sense. 0

10.1.4 The Cauchy Criterion Definition 10.3 Let (X. p) be a metric space. A sequence (x j) of elements of X is said to be Cauchy if for each E > 0 there is an N > 0 such that if j. k > N. then p(Xj.Xk) < E. Now the Cauchy criterion and convergence are connected in the expected fashion:

Proposition 10.1 Let (Xj) be a conveTgent sequence, with limita. in the metric space (X. p). Then the sequence Ix j ) is Cauchy.

Example 10.5 The converse of the proposition is true in the real numbers (with the usual metric). as we proved in Section 2.1. However. it is not true in every metric space. For example. the rationals Q with the usual metric pes. t) = Is - tl is a metric space; but the sequence

3,3.1.3.14,3.141.3.1415,3.14159•...• while certainly Cauchy. does not converge to a rationalnulllber. Thus we are led to a definition:

0

142

Chapter 10: Advanced Topics

10.1.5 Completeness Definition 10.4 We say that a metric space (X. p) is complete if every Cauchy sequence converges to an element of the metric space. Thus the real numbers. with the usual metric. form a complete metric space. The rational numbers do not.

Example /0.6 Consider the metric space (X. p) from Examples 10.3 and 10.4 above. consisting of the continuous functions on the closed unit interval with the indicated uniform metric function p. If (gj I is a Cauchy sequence in this metric space. then each gj is a continuous function on the unit interval. and this sequence of continuous functions is Cauchy in the uniform sense (see Chapter 8). Therefore this sequence converges uniformly to a limit function g that must be continuous. We conclude that the metric space (X. p) is complete. 0

Example /0.7 Consider the metric space (X. p) consisting of the polynomials with domain the interval [0. 1). with the distance function p(f. g) = sUP'EIO.lllf(t) g(t)l. This metric space is not complete. For if h is any continuous function on [0. 1) that is not a polynomial. such as h(x) = sin x. then by the Weierstrass Approximation Theorem there is a sequence (p j I of polynomials that converges uniformly on [0. 1) to h. Thus this sequence (p j I will be Cauchy in the metric space. but it does not converge to an elemell1 of the metric space. We conclude that the metric space (X. p) is not complete. 0

10.1.6 Isolated Points Definition 10.5 Let (X. p) be a metric space and E a subset of X. A point PEE is called an isolated point of E if there is an r > Osuch that En B(P. r) = (PI. If a point of E is not isolated. then it is called nonisolated. We see that the notion of "isolated" has intuitive appeal: an isolated point is one that is spaced apart - at least distance r - from the other points of the space. A nonisolated point. by contrast. has neighbors that are arbitrarily close.

Example /0.8 Every point of the integers. with the usual metric. is isolated. because each integer has a ball of radius 1/2 about it that contains only that integer. By

10.1

Metric Spaces

143

contrast. no point of the interval [0. 1] is isolated. In the set

s = { 1. ~. ~ •... } U to},

o

every poi nt is isolated except O.

Definition 10.6 Let (X. p) be a metric space and f : X £ E JR. we say that the limit of f at P is £. and we write

~

JR. If P E X and

lim f(x) = £.

x.... P

iffor any E > 0 there is a 8 > 0 such that if 0 < p(x. P) < 8. then If(x) - £1 <

E.

Notice in this definition that we use p to measure distance in X-this is the natural notion of distance with which X comes equipped-but we use absolute values to measure distance in R The following lemma will prove useful. Lemma 10.1 Let (X. p) be a metric space and P EX. Let f be a function from X to JR. Then Iimx .... p f(x) = £ jf and only jf for every sequence (x j I C X satisfying x j ~ P. it holds that f(xj) ~ f(P).

Definition 10.7 Let (X. p) be a metric space and E a subset of X. Suppose that PEE. We say that a function f : E ~ JR is continuous at P if lim f(x) = f(P).

x .... P

Example 10.9

Let (X. p) be the space of continuous functions on the interval [0.1] equipped with the supremum metric as in Examples 10.3 and 10.4 above. Define the function F : X ~ JR by the formula F(f) =

£

f(t)dt.

Then F takes an element of X. namely a continuous function. to a real number. namely its integral over [0. 1]. We claim that F is continuous at every point of X. For fix a point f EX. If (fJ) is a sequence of elements of X converging in the metric space sense to the limit f. then (in the language of classical analysis as in Chapter 8) the fj are continuous functions converging uniformly to the continuous function f on the interval [0. 1]. But. by

144

Chapter 10: Advanced Topics Theorem 8.2. it follows that

f

fj(t)dt

~

f

f(t)dt.

However. this simply says that :F(/j) ~ :F(f). Using the lemma. we conclude that lim :F(g) = :F(f). g .... f Therefore :F is continuous at f. Since f E X was chosen arbitrarily. we conclude that the function :F is 0 continuous at every point of X. In the next section we shall develop some topological properties of metric spaces.

10.2 Topology in a Metric Space 10.2.1

Balls in a Metric Space

Fix a metric space (X. p). An open ball in the metric space is a set of the form B(P.r)

== Ix EX: p(x. P)

< r}.

where P E X and r > O. A set U ~ X is called open if for each r > Osuch that B(u.r) c U. We define a closed ball in the metric space (X. p) to be B(P.r)

= Ix EX: p(x. P)

1/ E

U there is an


A set E C X is called closed if its complement in X is open. Example 10.10

Consider the set of real numbers lR equipped with the metric p(s. t) = 1 if s ,p t and p(s. t) = 0 otherwise. Then each singleton U = {x) is an open set. For let P be a point of U. Then P = x. and the ball B(P.l/2) lies in

U. However. each singleton is also closed: the complement of the singleton U = {x} is the set S = lR \ {x}. If s E S. then B(s. 1/2) C S. as in the preceding paragraph. 0 Example 10.11

Let (X. p) be the metric space of continuous functions on the interval 10. 1]. equipped with the metric p(f. g) = sUPxelO.I) If(x) - g(x)l. Define U=

If EX:

f(lf2) > 5}.

10.2

Topology in a Metric Space

145

Then V is an open set in the metric space. To verify this. fix an element / E V. Let € = f(I/2) - 5 > O. We claim thatlhe metric ball 8(/, €) lies in V. For let g E 8(f, E). Then g(l/2) ~ f(I/2) -1/(1/2) - g(I/2)1 ~

/(1/2) - p(f, g)

> f(I/2) - €

=5. It follows that g E V. Since g E 8(/, €) was chosen arbitrarily, we may conclude that 8(/, €) C V. But this says that V is open. We may also conclude from this calculation that


o

is closed.

10.2.2 Accumulation Points Definition 10.8 Let (X, p) be a metric space and S C X. A point x E X is called an accumulation point of S if every 8(x, r) contains infinitely many distinct elements of S. Example 10.12 Let T = (0, I]. Then every point of T is an accumulation point. Let

S = { I.

~, ~, ... } U (01.

Then only the point 0 E S is an accumulation point.

o

Proposition 10.2 Let (X, p) bea metric space. A set S C X is closed ifand only ifevery accumulation point of S lies in S. Definition 10.9

Let (X, p) be a metric space. A subset S C X is said to be bounded if S lies in some ball 8(P, r).

Example 10.13 Consider the real numbers IR with the usual notion of distance. Then the sets (x € IR : xl - 3x + 7 = 0), {x E IR : x 2 < 7), (x E IR : 0 < x < I/xl

146

Chapter 10: Advanced Topics are all bounded. By contrast, the sets

{x E 1R: sinx = O},

{x E 1R: x 2 > 7},

{x E IR : x> l/x}

o

are all unbounded.

10,2.3 Compactness Definition 10.10 Let (X, p) be a metric space. A set S C X is said to be compact if every sequence in S has a subsequence that converges to an element of S. Example 10.14 In Chapter 4 we learned that, in the real number system, compact sets are closed and bounded, and conversely. Such is not the case in general metric spaces. As an example, consider the metric space (X, p) consisting of all continuous functions on the interval [0, 1], with the supremum metric, as in previous examples. Let

S

= l/j(x) = xi : j = 1,2, ... }.

This set is bounded since it lies in the ball 8(0, 2) (here 0 denotes the identically zero function). We claim that S contains no Cauchy sequences. This follows (see the discussion of uniform convergence in Chapter 8) because, no matter how large N is, if k > ; > N then we may write

Fix j. If x is sufficiently near to I, then Ixil > 3/4. But then we may pick k so large that Ix k - j I < 1/4. Thus

I/k(x) - fi(x)1 ~ 9/16. So there is no Cauchy subsequence. We may conclude (for vacuous reasons) that S is closed. But S is not compact. For, as just noted, the sequence {fi} consists of infinitely many distinct elements of S which do not have a convergent subsequence (indeed not even a Cauchy subsequence). 0 In spite of the last example, half of the Heine-Borel Theorem is true:

Proposition 10.3 Let (X, p) be a metric space and S a subset of X. If S ;s compact, then S ;s closed and bounded.

10.3

The Baire Category Theorem

147

Definition 10.11 Let S be a subset of a metric space (X, pl. A collection of open sets {Oa}aeA (each Oa is an open set in X) is called an open covering of S if

U Oa:> S. aeA

DejiJlition 10.12 If C is an open covering of a set S and if V is another open covering of S such that each element of 1) is also an element of C. then we call 1) a s/lbcovering of C. We call V afinite subcovering if 1) has just finitely many elements.

Theorem 10.1

A subset S of a metric space (X, p) is compact if and only if every open covering C = {Oa }aeA of S has a finite subcovering. Proposition 10.4 Let S be a compact subset ofa metric space (X, E is compact.

pl. If E is a closed subset ofS.then

Example 10.15 Let S = (0. 1). Define Uj = {x e lR : 1/j < x < II. Then the collection U = (Uj 1j..1 is an open covering of S. But there is no finite subcovering. so S is not compact. We observe also that S is bounded, but not closed; this gives a second reason why S is not compact. By contrast, the set T = [0, 1] is compact. First of all. it is closed and bounded. Second. the method of bisection can be used to see that any sequence in S has a convergent subsequence. Third. any open cover of T has a finite subcover. This is tricky to prove in general, but we can look at an example: Let Uo = (-0.1,0.1) U (0.9,1.1) and let Uj for j ~ 1 be as in the last paragraph. Then U = {Uj I~ certainly covers T. In addition, the collection

v=

{Uo, UI, ...• UII}

is a finite subcovering.

10.3

o

The Baire Category Theorem

10.3.1 Density Let (X. p) be a metric space and S C X a subset. A set E C S is said to be dense in S if every element of S is the limit of some sequence of elements of E.

Chapter 10: Advanced Topics

148 Example 10.16

The set of rational numbers Q is dense in any nonempty, open subset of the 0 reals JR equipped with the usual metric. Example 10.17 Let (X, p) be the metric space of continuous functions on the interval [0, 1] equipped with the supremum metric as usual. Let P C X be the polynomial functions. Then the Weierstrass Approximation Theorem tells us that P is dense in X. 0 Example 10.18 Consider the real numbers JR with the metric p(s, t) = 1 if s # t and p(s, t) = 0 otherwise. Then no proper subset E of JR is dense in JR. To see this, notice that if E were dense and were not all of JR, and if P E JR \ E, then p(P, e) > 1/2 for all e E E. So elements of E do not get close to P; thus E is not dense in JR. 0

10.3.2 Closure Definition 10.13 If (X, p) is a metric space and E C X then the closure of E is defined to be the union of E with the set of its accumulation points. Example 10.19 Let (X, p) be the set of real numbers with the usual metric and set E = Q() (-2, 2). Then the closure of E is [-2, 2]. Let (Y, (1) be the continuous functions on [0, I] equipped with the supremum metric as in Example 10.3. Take E C Y to be the polynomials. Then the closure of E is the set Y. 0 We note in passing that if B(P, r) is a ball in a metric space (X, p) then B(P, r) will contain but need not be equal to the closure of B(P, r). See Example 10.18. Definition 10.14 Let (X, p) be a metric space. We say that E C X is nowhere dense in X if the closure of E contains no ball B(x, r) for any x EX, r > O. Example 10.20 Let us consider the integers Z as a subset of the metric space JR equipped with the standard metric. Then the closure of Z is Z itself. And of course Z contains no metric balls. Therefore Z is nowhere dense in R 0

10.3 The Baire Category Theorem

149

Example 10.2/

Consider the metric space X of all continuous functions on the unit interval [0.11. equipped with the usual supremum metric. Fix k > O. and consider Ek

= {p(x) : p is a polynomial of degree not exceeding kl.

Then the closure of Ek is Ek itself (that is. the limit of a sequence of polynomials of degree not exceeding k is still a polynomial of degree not exceeding k). And Ek contains no metric balls. For if peEk and r > O. then p(x) + (r/2). xk+1 e B(p. r) but p(x) + (r/2)· xk+1 ¢ E. We recall. as noted in Example 10.17 above. that the set of all polynomials is dense in X; but if we restrict allention to polynomials of degree not exceeding a fixed integer k. then the resulting set is nowhere dense. 0

10.3.3 Haire's Theorem Theorem 10.2 Baire Let (X. p) be a complete metric space.

Then X cannot be written as the union of

countably many nowhere dense sets. Before we apply the Baire Category Theorem. let us formulate some restatements. or corollaries. of the theorem which follow immediately from the definitions.

Corollary 10.1 Let (X. p) be a complete metric space. Let YI. y2•... be countably many closed subsets of X. each of which contains no nontrivial open ball. Then Uj Yj also has

the property that it contains no nontrivial open ball.

Corollary 10.2 Let (X. p) be a complete mecric space. Let 0 .. 02 •... be countably many dense open subsets of X. Then j OJ is dense in X.

n

Note that the result of the second corollary follows from the first corollary by complementation. The set OJ. while dense, need not be open.

nj

Example 10.22

The metric space JR. equipped with the standard Euclidean metric. cannot be written as a countable union of nowhere dense sets. 0 By contrast. Q call be written as the union of the singletons {qj I where the qj represent an enumeration of the rationals. However. Q is not complete.

150

Chapter 10: Advanced Topics Example 10.23 Baire's theorem contains the fact that a perfect set of real numbers must be uncountable. For if P is perfect and countable, we may write P (PI, P2, ... ). Therefore 00

P = U(Pj}. j=1

But each of the singletons (Pj) is a nowhere dense set in the metric space P, and P is complete. (You should verify both these assertions for yourself.) This contradicts the Category Theorem, so P cannot be countable. 0 A set that can be written as a countable union of nowhere dense sets is said to be of first category. If a set is not of first category, then it is said to be of second category. The Baire Category Theorem says that a complete metric space must be of second category. We should think of a set of first category as being ''thin'' and a set of second category as being "fat" or "robust." (This is one of many ways that we have in mathematics of distinguishing "fat" sets. Countability and uncountability is another. Lebesgue's measure theory, not covered in this book, is a third.) One of the most striking applications of the Baire Category Theorem is the following result to the effect that "most" continuous functions are nowhere differentiable. This explodes the myth that most of us learn in calculus that a typical function is differentiable at all points except perhaps at a discrete set of bad points. Theorem 10.3 Let (X. p) be the metric space of continuous functions on the unit interval [0, 1] equipped with the metric p(f, g) =

sup I/(x) - g(x)l . .telO,I)

Define a subset of E of X as follows: lEE if !here exists one point at which differentiable. Then E is offirst category in the complete metric space (X, pl.

I

is

10.4 The Ascoli-Arzela Theorem 10,4.1 Equicontinuity Let :F = (fa laeA be a family, not necessarily countable, of functions on a metric space (X, pl. We say that the family :F is equicolltilluous on X if for every E > 0 there is a ~ > 0 such that when pes, t) < ~, then I/a(s) - la(t») < E. Notice that equicontinuity mandates not only uniform continuity of each fa, but also that the uniformity occur simultaneously, and at the same rate, for all the fa.

10.4 The Ascoli-Arzela Theorem

151

Example /0.24

Let (X. p) be the unit interval [0. I] with the usual Euclidean metric. Let F consist of all functions / on X that satisfy the Lipschitz condition

I/(s) - l(t)1 ::: 2 ·Is - tl for all s. t. Then F is an equicontinuous family of functions. For. if £ > 0, then we may take IS = £/2. Then if Is - tl < IS and I E F. we have

I/(s) - I(t») < 2 ·Is - tl < 2 . IS =

£ •

Observe. for instance. that the Mean Value Theorem tells us that sin x, cos x • 2x. x 2 are elements of F. 0

10.4.2 Equiboundedness If F is a family of functions on X, we call F eqllibollllded if there is a number M > 0

such that

I/(x») ::: M for all x E X and all I E F. For example. the functions !i(x) = sin jx on (0. 1] form an equibounded family.

10.4.3 The Ascoli-Arzela Theorem One of the cornerstones of classical analysis is the following result of Ascoli and Arzela: Theorem 10.4 Ascoli-Arzela Let (Y, 0") be a compact metric space. Let F bean equibounded. equicontinuous fam-

ily of functions on Y. Then there is a sequence Iii) to a continuous function on Y

C

F that converges uniformly

Let (X. p) be the metric space consisting of the continuous functions on the unit interval [0, 1], equipped with the usual supremum norm. Let F be a closed. equicontinuous, equibounded family of functions lying in X. Then the theorem says that F is a compact set in this metric space; for any infinite subset of F is guaranteed to have a convergent subsequence with limit in F. As a result. we may interpret the AscoliArzela theorem as identifying certain compact collections of continuous functions. Example 10.25

Refer to Example 10.24. The set F of functions on 10. 1] that are bounded by 2 and satisfy the Lipschitz condition

)/(s) - l(t)1

:s 21s - tl

152

Chapter 10: Advanced Topics forms an equibounded, equicontinuous family in the metric space (X, p) of continuous functions on the unit interval with the usual uniform metric. By the Ascoli-Arzela theorem, every sequence in :F has a convergent subse0 quence.

It is common in the theory of partial differential equations to derive the existence of a solution by first proving an a priori estimate for smooth functions and then extracting a solution in general, using the Ascoli-Arzela theorem, as the limit of smooth solutions.

Chapter 11

Differential Equations Differential equations are the heart and soul of analysis. Virtually any law of physics, engineering, biology. or chemistry can be expressed as a differential equation - and frequently as a first-order equation (Le.• an equation involving only first derivatives). Much of mathematical analysis has been developed in order to find techniques for solving differential equations. Most introductory books on differential equations devote themselves to elementary techniques for finding solutions to a very limited selection of equations. In the present book we take a different point of view. We explore certain central principles which have broad applicability to virtually any differential equation. These principles. in particular. illustrate some of the key ideas of the book.

11.1 11.1.1

Picard's Existence and Uniqueness Theorem The Form ofa Differential Equation

A fairly general first-order differential equation will have the form

dy

-

dx

(11.1.1.1)

= F(x.y).

Here F is a continuously differentiable function on some domain (a. b) x (c. d). We think of y as the dependent variable (that is. the function that we seek) and x as the independent variable. For technical reasons. we assume that the function F is bounded, (11.1.1.2) IF(x, y») =:: M. and in addition that F satisfies a Lipschitz condition: IF(x. s) - F(x. I») :::: C ·Is -

tl·

(11.1.1.3)

a

[In many treatments it is standard to assume that F is bounded and Flay is bounded. It is easy to see. using the Mean Value Theorem. that these two conditions imply (11.1.1.2). (11.1.\.3).]

153

154

Chapter 11: Differential Equations Example 11.1

Consider the equation dy dx

-=X

smy-y Inx.

2·

This equation fits the paradigm of equation (11.1.1.1) with F(x, y) = x 2 siny - y Inx, provided that 1 < x < 2 and 0 < y < 3 (for instance). 0 In fact the most standard, and physically appealing, setup for a first-order equation such as (11. 1.1.1) is to adjoin to it an initial condition. For us this condition will have the form y(XO) = yo. (11.1.1.4) Thus the problem we wish to solve is (11. 1.1.1) and (1 I. I.1.4) together. Picard's idea is to set up an iterative scheme for doing so. The most remarkable fact about Picard's technique is that it always works: As long as F satisfies the Lipschitz condition, then the problem will possess one and only one solution.

11.1.2 Picard's Iteration Technique While we will not actually give a complete proof that Picard's technique works. we will set it up and indicate the sequence of functions it produces; this sequence converges uniformly to the solution of our problem. Picard's approach is inspired by the fact that the differential equation (11.1.1.1) and initial condition (1 I. 1.1.4), taken together, are equivalent to the single integral equation y(x)

= YO +

1 x

F[r, y(r»dr (11.1.2.1) Xo We invite the reader to differentiate both sides of this equation, using the fundamental theorem of calculus. to derive the original differential equation (1 I. 1.1.1). Of course the initial condition (11.1.1.4) is built into (11.1.2.1). This integral equation inspires the iteration scheme that we now describe. We assume that XO E (a, b) and that YO E (c. d). We set x

YI(x) =yo+1 F(t.yo)dt. Xo

For x near to xo, this definition makes sense. Now we define }'2(x) =

LX F(t,YI(r»dr Xo

and. more generally. Yj+l(x) =

J~ F(r, Yj(r»dr

(11.1.2.2)

It turns out that the sequence of functions {YI. }'2 •••• } will converge uniformly on an interval of the form (xo - II, xo + II) C (a, b).

Il.l

155

Picard's Existence and Uniqueness Theorem

11.1.3 Some Illustrative Examples Picard's iteration method is best apprehended by way of some examples that show how the iterates arise and how they converge to a solution. We now proceed to de· velop such illustrations. Example 11.2

Consider the initial value problem y' = 2y,

y(O) = 1.

Of course this could easily be solved by the method of first order linear equations, or by separation of variables (see (SIK) for a description of these methods). Our purpose here is instead to illustrate how the Picard method works. First notice that the stated initial value problem is equivalent to the integral equation y(X) = I

+ fax 2y(t) dt .

Following the paradigm (11.1.2.1), we thus find that Yj+l(x) = 1 +

fax 2Yj(x)dx.

Using yo (x ) '" I, we then find that YI(X) = I

+ fax 2dt =

n(x) = I

+ fax 2(1 +2t)dt = I +2x +2x 2 ,

I +2x,

[X

2

2

n(x)=I+ Jo 2(1+2t+2t )dt=I+2x+2x +

4x3

T ·

In general, we find that Yj(x)

=

4x 3 (2x)j I +2x +2x + - + ... + -.,3 J. 2

j (2x)(

= L:-,(=0 e.

It is plain that these are the partial sums for the power series expansion of Y = e2x • We conclude that the solution of our initial value problem is y=e2x 0 Example 11.3

Let us use Picard's method to solve the initial value problem l=2x-y,

y(O) = I.

Chapter II: Differential Equations

156 The equivalent integral equation is

y(x) = I + foX [21 - y(t)1 dt and (11.1.2.2) tells us that

Taking yo(x) =: 1. we then find that

Yl (x) = I + foX (2t - I) dt = I + x 2 - x . Y2(x) = 1+ foX (2t-[I+t 2 -tl) dt 3x 2 =l+T- x

x3

-"3'

n(x) = 1+ foX (2t-[1+3t2/2-t-t3/31) dt 3x 2 x3 x4 =1+ T -x-T+ 4.3' Y4(X) = 1+ foX (21-[1+3t 2/2-t-t 3/2+t 4/4.31) dt 3x 2 x3 x4 xS =l+T-x-T+ 4.2 - 5.4.3 In general. we find that 2

3

4

.

'+1

J J Y'(x)=I_x+3x _3x +3x +"'+(_1) j 3x + (-1)j+l 2x J 2! 3! 4! j! (j + I)! j xl 2x j + 1 = [2x - 2) + 3· + (_I)j+l. I l=O e. (j + I).

L(_l)l,

Thus we see that the iterates Yj(x) converge to the solution y(x) [2x - 2) + 3e-x for the initial value problem. 0

11.1.4 Estimation ofthe Picard Iterates To get an idea of why the functions Yj converge uniformly. let us do some elementary estimations. Choose h > 0 so small that h . C < 1. where C is the constant from the Lipschitz condition in (11.1.1.3). We will assume in the following calculations that Ix - xol < h.

11.1 Picard's Existence and Uniqueness Theorem

IS7

Now we proceed with lhe iteration. Let Yo(t) be identically equal to the initial value Yo. Then IYo(t) - YI (1)1

= IYo -

YI (1)1

=

11:

: : 1:

F(I, Yo) dl

IF(I, Yo) Idl

::: M . Ix

- xol

:::M·h.

We have of course used the boundedness condition (11.1.1.2). Next we have IYI(x) - Y2(x)1 =

1:

F(I,Yo(t))dl -

f~ F(I,YI(I»dll

: : Jxor I

F(I, yo(t)) - F(t, YI (1»1 dl

:: L

C ·IYo(t) - YI (t)1 dl

:::c

M·h·h

= M· h· (Ch).

One can continue this procedure to find that 1Y2(x) - Y3(x)1 ::: M . C2 . h 3 = M . h . (Ch)2 .

and, more generally, IYj(x) - Yj+l (x)1 ::: M . cj . h j + 1 < M . h . (Ch)j

Now if 0 < M < N are integers, lhen IYM(X) - YN(x)1 ::: IYM(X) - YM+l (x)1

+ IYM+I (x) -

YM+2(X)1

+... + IYN-I (x) - YN(x)1 ::: M· h· ([ChjM + [ChjM+1 + ... [ChjN-I). Since IChl < I by design,the geometric series Lj[Chj j converges. As a result,the expression on the right ofour last display is as small as we please, for M and N large, just by lhe Cauchy criterion for convergent series. II follows that lhe sequence (yj) of approximate solutions converges uniformly to a function Y = y(x). In particular, Yis continuous. Furthermore, we know lhat Yj+l(x) =

f~ F(I, Yj(l»dl.

Chapter 11: Differential Equations

158

Letting ; ~ 00, and invoking the uniform convergence of the Yj, we may pass to the limit and find that y(x) =

f~ F(t, y(x»dt.

This says that y satisfies the integral equation that is equivalent to our original initial value problem. This equation also shows that y is continuously differentiable. Thus y is the function that we seek. It can be shown that this y is in fact the ullique solution to our initial value problem. We shall not provide the details of the proof of this assertion. In case F is not Lipschitz-say that F is only continuous, then it is still possible to show that a solution y exists. But it will no longer be unique.

11.2 The Method of Characteristics Characteristics are a device for solving partial differelltial equatiolls. The idea is to reduce the partial differential equation to a family of ordillary differelltial equatiollS (as in Section 11.1) along curves. Here we shall illustrate the idea with a few carefully chosen examples. Consider a first-order partial differential equation of the form av a(x, t)ax

av + b(x, t)= c(x, t)v + d(x, t). at

(11.2.1)

The idea is to think of the left-hand side as a directional derivative along a curve. To that end, we solve the auxiliary equations dx ds = a(x, t)

and

dt - =b(x,t). ds

(11.2.2)

What is going on here is that we have created a family of curves x = x(s), t = t(s) whose tangent vector (x'(s), t'(s» coincides with the direction of the vector (a, b), which is the "direction" along which the differential equation is operating. This device enables us to reduce the partial differential equation (11.2.1) to an ordinary differential equation that often can be solved by elementary methods. With this idea in mind, we see that the derivative of vex, t) along the described curves becomes dv ds

dv[x(s), t(s») ds av dt av dx = -ax -ds + ds av av =a·-+b·ax at =cv+d.

-=

at .

Here we have used the chain rule and the equations (11.2.1) and (11.2.2). We now illustrate with some simple examples.

(11.2.3)

159

11.2 The Method of Characteristics Example 1l.4

Consider the partial differential equation

av al

av ax

-+c·_=O.

This is the unidireclional wave equation. We impose the following initial conditions at I = 0: v(x,O) = G(x). Here G is some input functions. It is convenient to parameterize the "initial curve," or the curve along which the initial condition is specified, by X=T,

t=O,

V=G(T).

(11.4.1)

Now the characteristic equations, as indicated in (11.2.2) and (11.2.3), are

dx

dt

-=c.

-

ds

ds

= I,

dv =0.

ds

Of course we may easily solve these equations (taking into account (11.4.1) with s = 0). The result is X(S, T)

= CS + T,

I(S, T) = s,

v(s, T)

= G(T).

(11.4.2)

Ultimately we wish to express the solution v in terms of the given data G. With this thought in mind, we solve the first two equations for sand T as functions of x and I. Thus S

= I,

T

=

X

-ct

Finally, we substitute these simple formulas into the equation for v in (11.4.2) to obtain v(S, T) = G[T(X, t)) = G[x - cII.

This function v solves the original partial differential equation with initial 0 data. Example 11.5

Let us use the method of characteristics to solve the differential equation

au ax

au al

x-+I-=CU,

u(x, 1)

= {(x).

We begin by parameterizing the initial curve as

x

= T.

I

= 1.

u = {(T).

160

Chapter II: Differential Equations The characteristic equations are

dx -=x, ds

dt =1, ds

-

du - =cu. ds

Now we may solve these characteristic equations. keeping in mind the initial conditions at s = O. The result is

xes, ,)

= u'.

t(s. ,) = e'.

u(s. ,)

= !(,)e".

[Here we have used our knowledge from elementary ordi nary differential equations of finding exponential solutions of first order differential equations.) As usual. we solve the first two of these for sand, in terms of x and t. Thus x s = Int and

,= -. t

Inserting these into the equation for u gives

u(x,t)

=! (;:). t'

This is the solution to the original problem. Note in passing that the differential equation we have been analyzing may be said to have singular coefficients. since the vector of coefficients on the left-hand side vanishes at the origin. It results that the solution has a corresponding singularity. 0

11.3 Power Series Methods One of the techniques of broadest applicability in the subject of differential equations is that of power series, or real analytic functions. The philosophy is to guess that a given problem has a solution that may be represented by a power series. and then to endeavor to solve for the coefficients of that series. Along the way, one uses (at least tacitly) fundamental properties of these series -that they may be differentiated and integrated term by term, for instance. and that their intervals of convergence are preserved under standard operations.

Example IJ.6 Let p be an arbitrary real constant. Let us use a differential equation to derive the power series expansion for the function

y=(l+x)p. Of course the given y is a solution of the initial value problem (l

+ x) . y' =

py •

yeO) = 1.

11.3 Power Series Methods

161

We assume that the equation has a power series solution 00

j

y= Lajx =aO+aIX+a2X 2 + ... j=O

with positive radius of convergence R. Then 00

y' = L j . ajx j - I = al

+ 2a2X + 3a3x2 + ... ;

j=1 00

j xy' = L j . ajx = alx + 2a2x 2 + 3a3x3 + '"

;

j=1 00

py

=L

pajx

j

= pao + palX + pa2 x2 + ...

j=O

By the differential equation. the sum of the first two of these series equals the third. Thus 00

00

00

j=1

j=1

j=O

j j j Ljajx - I + Ljajx = Lpajx .

We immediately see two interesting anomalies: the powers of x on the lefthand side do not match up. so the two series cannot be immediately added. Also the summations do not all begin in the same place. We address these two concerns as follows. First. we can change the index of summation in the first sum on the left to obtain 00 00 00 j j j L<j + 1)aj+IX + L jajx = L pajx . j=o

j=1

j=O

Write out the first few terms of the sum we have changed. and the original sum. to see that they are just the same. Now every one of our series has x j in it. but they begin at different places. So we break off the extra terms as follows: 00

L(j + l)aj+lx j

00

+ Ljajx

j=1

j=1

j

00

j O O - Lpajx = -alx + paox (11.6.1) j=1

Notice that all we have done is to break off the zeroeth terms of the first and third series. and put them on the right. The three series on the left-hand side of (11.6.1) are begging to be put together: they have the same form. they all involve powers of x. and they all begin at the same index. Let us do so: 00

L[(j j=1

+ l)aj+1 + jaj -

j paj]x = -al

+ pao·

Chapter 11: Differential Equations

162

Now the powers of x that appear on the left are I. 2. . .. • and there are none of these on the right. We conclude that each of the coefficients on the left is zero; by the same reasoning. the coefficient (-at + pao) on the right (i.e.• the constant term) equals zero. So we have the equations I -al

(j

+ pao = 0

+ I)aj+l + (j -

p)aj = O.

Our initial condition tells us that ao = 1. Then our first equation implies p. The next equation. with j 1. says that that at

=

=

Hence a2 = (p - I)at/2 = (p - l)p/2. Continuing. we take p = 2 in the second equation to get

so a3 = (p - 2)a2/3 = (p - 2)(p - l)p/(3 . 2). We may proceed in this manner to obtain that aj

=

+ 1)

p(p - I)(p - 2)··· (p - j

., J.

Thus the power series expansion for our solution y is Y = I +px+

p(p - I) 2 p(p - l)(p - 2) 3 x + x + ...

+ p(p -

2!

3!

I)(p - 2)· .. (p - j

,

. J.

+ 1)

.

xl

+ ...

Since we knew in advance that the solution of our initial value problem was

+ x)P •

y = (1

we find that we have derived Isaac Newton's general binomial theorem (or binomial series): (1

+ x)P = 1+ px +

p(p -

I) x2 +

p(p - I)(p -

2!

+ p(p -

2\3 + ...

3!

I)(p - 2) ... (p - j

,

. J.

+ I)

.

xl

+ ...

.

0

IA set of equations like lIIis is called a recursion. 11 expresses a/s willi later indices in terms of a/s with earlier indices.

11.3 Power Series Methods

163

Example 11.7

Let us consider the differential equation

,

y = y.

Of course we know from elementary considerations that the solution to this equation is y = C . e'" • but let us pretend that we do not know this. Our goal is to instead use power series to derive the solution. We proceed by guessing that the equation has a solution given by a power series. and we proceed to solve for the coefficients of that power series. Our guess is a solution of the form y = ao +a\x +a2x2 +a3x3 + ....

Then

y' =a\ +202X +3a3x2 + ...

and we may substitute these two expressions into the differential equation. Thus a\ + 202x + 3a3x2 + ... = ao + a\x +a2x2 + ... Now the powers of x must match up (i.e.• the coefficients must be equal). We conclude that a\

=00

202 = al 3a3 = a2

and so forth. Let us take ao 10 be an unknown constant C. Then we see that al = C; C a2 = 2;

C a3 =3.2;

etc. In general.

C an =-. II!

In summary. our power series solution of the original differential equation

is

00

Y

c.

="Lj=fJ _xl ., = J.

xj c· "L -., = j=fJ J. 00

c· e".

Thus we have a new way. using power series. of discovering the general solution of the differential equation y' = y. 0

164

Chapter 11: Differential Equations

Example /1.8 Let us use the method of power series to solve the differential equation (l - x 2 )y" - 2xy' + p(p + I)y = o.

(11.8.1)

Here p is an arbitrary real constant. This is called Legendre's equation. We guess a solution of the form

and calculate 00

y'

= L,jajx j - t = a\ + 2a2x + 3a3x2 + ... j=1

and

00

y"

= L,j(j -

l)ajx j - 2 = 2a2 + 3·2· a3x

+ ...

j=2 It is most convenientto treat the differential equation in the form (11.8.1). We calculate 00

2 " =-L.-JJ' " .(. I) ajx j -x)' j=2

and

00

-2xy' = - L,2jajx j j=1

Substituting into the differential equation now yields 00

00

L,j(j -1)ajx j - 2 - L,j(j -I)ajx j j=2 j=2 00

- L,2jajx j j=l

00

+ p(p+ 1) L,ajx j

=

o.

j=O

We adjust the index of summation in the first sum so that it contains x j rather than x j - 2 and we break off spare terms and collect them on the right. The result is 00

L,(j + 2)(j j=2 00

+ I)a j+2xj 00

- L,j(j -I)ajx j - L,2jajx j j=2 j=2

00

+ p(p+ 1) L,QjX j j=2

= -2a2 - 6a3x + 2aIX - p(p + I)ao - p(p + I)a\x .

11.3

Power Series Methods

165

In other words, t [ ( j +2)(j + l)aj+2 - j(j - I)aj - 2jaj

+ pep + l)aj]X j

)=2

= - Za 2 - OO3x + Za\X - pep + I)ao - pep + l)alX . As a result,

[(j +2)(j

+ l)aj+2 -

j(j -I)aj -2jaj

+ p(p+ I)aj

1=0

fori = 2, 3, ... together with - Za 2 - pep + l)ao = 0

and

-003 +Za\ - pep + l)al = O. We have arrived at the recursion p(p+ I)

a2 =

1·2

.no,

(p - I)(p +2)

a3 =

. al

2·3

(p - j)(p + j + I) aj+2 = (j + 2)(j + I) . aj

t

fori = 2,3, ....

(11.8.2)

We recognize a familiar pattern: The coefficients ao and a\ are unspecified, so we set ao = A and a\ = B. Then we may proceed to solve for the rest of the coefficients. Now a2

a)= n.

=

"6=

=

2

(p - I)(p + 2)

2.3

3.4

4· 5

(p - 4)(p + 5)

5.6

(p - 5)(p + 6)

"4 _

as -

= 2

(p - 3)(p + 4)

6.7

pep + I) A.

(p - 2)(p + 3) a

....

as

=

=

8

"

pcp - 2)(p + I)(p + 3) . A

4!

•

_ (p - I)(p - 3)(p + 2)(p

a3 -

+ 4) . 8

5!

•

pcP - 2)(p - 4)(p

+ I)(p + 3)(p + 5)

6!

(p - I)(p - 3)(p - 5)(p + 2)(p + 4)(p

7!

A,

+ 6) . 8 •

166

Chapter II: Differential Equations and so forth. Putting these coefficient values into our supposed power series solution. we find that the general solution to our differential equation is y = A[1 _ pep + 1) x2 + pcp - 2)(p + l)(p + 3) x4 2! 4! _ pcp - 2)(p -4)(p + I)(p + 3)(p+ 5) x 6

+ _.. _J

6!

+ B[x -

(p - 1)(p + 2) 3

3!

x

+

(p - l)(p - 3)(p + 2)(p

5!

_ (p - l)(p - 3)(p - 5)(p + 2)(p + 4)(p + 6) x 7

+ 4)

5

x

+ - ... J.

7! We assure the reader that. when p is not an integer. then these are not familiar elementary transcendental functions. These are what we call Legendre/unctions. In the special circumstance that p is a positive even integer. the first function (that which is multiplied by A) terminates as a polynomial. In the special circumstance that p is a positive odd integer. the second function (that which is multiplied by B) terminates as a polynomial. These are called Legendre polynomials. and they play an important role in mathematical physics. representation theory. and interpolation theory. 0 Some differential equations have singularities. In the present context. this means that the higher order terms have coefficients that vanish to high degree. As a result. one must make a slightly more general guess as to the solution of the equation. This more general guess allows for a corresponding singularity to be built into the solution. Rather than develop the full theory ofFrobenius series. we merely give one example. Example 1/.9

We use the method of Frobenius series to solve the differential equation 2x 2 y"

+ x(2x + l)y' -

y = 0

(11.9.1)

about the regular singular point O. We guess a solution of the form 00

y =x

m

j

00

Lajx = Lajx m+ j=o j=O

and therefore calculate that 00

y' = L(m j=O

+ j)ajx m +j -

I

j

167

11.3 Power Series Methods

and

00

y" = L(m j=O

- l)ajx m+j - 2 .

+ j)(1II + j

Plugging these calculations into the differential equation yields 00

2

L(III + j)(1II + j j=O

00

l)ajx M + j +2 L(III + j)ajx M + j + 1 j=O

00

+ L(III + j)ajx + tn

00

j

j - Lajx tn + = O. j=o

j=O

We make the usual adjustments in the indices so that all powers of x are and we break off the dangling terms to put on the right-hand side of the equation. We obtain

x m +j, 00

2 L(III + j)(111

+j

- l)ajx M +j

j=1 00

+ 2 L(III + j

m j

- l)aj_lx +

00

+ L(III + j)ajx

tn

j

+ - Lajx m+

j=l

j=1

00

j

j=1

= -2111(111 - I)aox -lIIaox m +aox m tn

The result is [2(111

+ j)(1II + j

- I)aj + 2(111

+j

1

- l)aj_1 + (III + j)aj - aj = 0

for j = 1,2,3,... together with [-2m(1II - I) -

III

(11.9.2)

+ I]ao = O.

It is clearly not to our advantage to let no = O. Thus -2111(111 - 1) -

III

+ 1 = o.

This is the indicial equation. The roots of this quadratic equation are III = -1/2, I. We put each of these values into (11.9.2) and solve the resulting recursion. Now (11.9.2) says that (2m 2

+ 2l + 4mj - i -

For III = -1/2, this is

m - I)aj = (-2m - 2j

+ 2)aj_1

Chapter 11: Differential Equations

168 so al

= -aO,

a2

For 1/1 = I, we have aj

so

1

-2j = 3j + 2j2 aj -

2 al=-5ao,

1

= -2al = 2ao, etc.. 1

4 4 a2=-14al=35ao.

Thus we have found the linearly independent solutions

aox- I / 2 . (l-x and

aox· (1-

+ ~x2 - +... )

~x + ~x2 - + ... ). 5

35

The general solution of our differential equation is then

y = Ax- l12 (1- x + 4x2 -

+... ) + Bx· (1- ~x + ~x2 - +... ). 0

11.4 Fourier Analytic Methods An entire book could be wrillen about the applications of Fourier analysis to differential equations and to mathematical analysis. The subject of Fourier series was developed hand-in-hand with the analytical areas to which it is applied. In the present brief section we merely indicate a couple of examples.

11.4.1

Remarks on Different Fourier Notations

In Section 9.6, we found it convenient 10 define the Fourier coefficients of an integrable function on the interval [0, 21l') to be

~ = -1 1(11)

27r

L

2Jr

0

l(x)e-U1. ·' dx.

From the point of view of pure mathematics, this complex notation has pro 'ed to be useful, and it has become standardized. But, in applications, there are other Fourier paradigms. They are easily seen to be equivalent to the one we have already introduced. The reader who wants to be conversant in this subject should be aware of these different ways of writing the basic ideas of Fourier series. We will introduce one of them n,ow, and use it in the ensuing discussion. If 1 is integrable on the interval [-7r, 7r) (note that, by 21l'-periodicity, this is not essentially different from [0, 27r then we define the Fourier coefficients

n,

ao =

2~ f~ I(x)dx,

11.4

Fourier Analytic Methods an

169

= -1 [" [(x)cosllxdx 1r

forll~l.

_"

bn = -1 [ " [(x) sinllx dx 1r

forll~l.

_"

This new notation is not essentially different from the old. for ~

[(II)

1

2[an + ibn]

=

for II ~ 1. The change in normalization (Le.• whether the constant before the integral is 1/1r or 1/21r) is dictated by the useful fact that

_1 21r

f2" Ie-in, 12 dt = 1

10

in the theory from Section 9.6. and

1 2 -2 [" 1 dx = 1. 1r

.!.[" 1r

_"

2

I cos lit 1 dt = 1

-1 [ " I sin ntf dt = 1r

for II

_"

~ 1.

1 forn > 1

_"

in the theory that we are about to develop. It is clear that any statement (as in Section 9.6) that is formulated in the language of [(11) is easily translated into the language of an and bn • and vice versa. In the present discussion. we shall use an and bn • just because that is the custom. ~

11.4.2 The Dirichlet Problem on the Disc We now study the two-dimensional Laplace equation. which is

a2 u a2u /j. = ax 2 + a 2 = O. y This is probably the most important differential equation of mathematical physics. It describes a steady-state heat distribution. electrical fields, and many other important natural phenomena. It is useful to write this equation in polar coordinates. To do so. recall that

r 2 =x2 +l. x=rcos8. y=rsin8. Thus a ax a ay a a. a - =--+--=cos8-+sm8-

ar

ar ax

ar ay

ax

ay

a ax a ay a . a a = - - + - - =-rsm8-+rcos8a8 a8 ax a8 ay ax ay

-

170

Chapter 11: Differential Equations

We may solve these two equations for the unknowns a/ax and a/ay. The result is a -axa = cos8ar

sin8 a ---r a8

and

a = sin8- ay ar

-

a

cos8 a --. r a8

A tedious calculation reveals that

IJ.

a

2

a

2

= ax 2 + ay2 = ( cos 8 ara -

sin8 a ) (

-r-a8

a sin8 a ) cos 8 ar ----;- a8

cos8 a ) + ( sin8 ara - 7 a8

a2 = ar 2

1

a

(

a cos8 a ) sin8 ar --r-a8

1 (12

+ ;- ar + r 2 a8 2 •

Let us use the method of separation of variables. We will seek a solution w = w(r,8) = u(r)· v(8) of the Laplace equation. Using the polar form, we find that this leads to the equation

1 , (r) . v(8) + 2u(r) 1 u" (r) v(8) + -u . v"8 ( ) = 0. r r Thus

r 2 u"(r) +ru'(r) v"(8) ---'-u-(r-)--'--'- = --v-(8-) .

Since the left-hand side depends only on r, and the right-hand side only on 8, both sides must be constant Denote the common constant value by A. Then we have (11.4.2.1) v" +AV = 0 and

r 2 u" + ru' - AU = O.

(11.4.2.2)

If we demand that v be continuous and periodic, then we must demand that A > 0 and in fact that A = n2 for some nonnegative integer II. For II = 0, the only suitable solution is v constant, and for n > 0, the general solution (with A = n2 ) is

=

y = Acosn8 + B sin 118 • as you can verify directly. We set A = n2 in equation (11.4.1.2), and obtain

which is Euler's equidimensional equation. The change of variables x = e< transforms this equation to a linear equation with constant coefficients, and that can in tum be solved with standard techniques. The result is u

= A + Bin r

if II

= 0;

u=Arn+Br-n ifn= 1,2,3, ....

11.4 Fourier Analytic Methods

171

We are most interested in solutions u that are continuous at the origin; so we take B = 0 in all cases. The resulting solutions are 11=0.

w = a constant ao/2 ;

n = I, n = 2, n = 3,

w = real cosli +b. sinli); w = r2(a2cos2IJ +b2sin21i); w = r 3(a3 cos 31i + b3 sin 31i) ;

Of course any finite sum of solutions of Laplace's equation is also a solution. The same is true for infinite sums. Thus we are led to consider 1

w

00

= w(r, Ii) = Zao+ L(Gj cosjli + bj sin jli). j=O

On a formal level, lelling r _ 1- in this last expression gives 1

zao + L(aj cosjli +bj sinjli). 00

j=1

We draw all these ideas together under the following physical rubric. Consider a thin aluminum disc of radius 1, and imagine applying a heat distribution to the boundary of that disc. In polar coordinates, this distribution is specified by a function {(Ii). We seek to understand the steady-state heat distribution on the entire disc. So we seek a function w(r, Ii), continuous on the closure of the disc, which agrees with f on the boundary and which represents the steady-state distribution of heat on the interior. Some physical analysis shows that such a function w is the solution of the boundary value problem ~w=O,

wi aD =

f·

According to the calculations we previously performed, a natural approach to this problem is to expand the given function { in its sine/cosine series: f(li) =

~ao+ f'(ajCOSjli+bjSinjli) 2

t1

and then posit that the w we seek is 00

w(r,li) =

~ao+ Lrj(ajcosjli+bjsinjli). 2

j=1

This process is known as solving the Dirichlet problem on the disc with boundary data

f·

172

Chapter 11: Differential Equations

Example 11.10 Let us follow the paradigm just sketched to solve the Dirichlet problem on the disc with f(8) = 1 on the top half of the boundary. and f(8) = -Ion the bottom half of the boundary. It is straightforward to calculate that the Fourier series (sine/cosine series) expansion for this f is h) 4 (, 8 sin 38 ++ sin 58 + .. , ) f( u=-sm+

3

1r

5

The solution of the Dirichlet problem is therefore

r 3 sin 38 r S sin 58 ) 4( w(r.8) =;; rsin8 + 3 + + 5 + ....

0

11.4.3 The Poisson Integral We have presented a formal procedure with series for solving the Dirichlet problem. But in fact it is possible to produce a closed formula for this solution. as we will show. Referring back to our sine series expansion for !. and the resulting expansion for the solution of the Dirichlet problem. we recall that. for j > O.

Qj = -1 1T

1"

!@cosj¢d¢ and bj = -1 1r

-1r

1"

!(¢)sinj¢d¢.

-Jr

Thus

I

(11r_" 1"

00 w(r.8)=-2ao+Lrj [-

j=1

!(¢)COSj¢d¢]COSj8

+

[~

i:

!(¢)Sin j ¢d¢]sin j 8).

This. in tum. equals 00

1 1 Lr j -2QO+1r

J=I

1"

f(¢) (cos j¢ cos j8 + sinj¢ sin j8)d¢

-"

'1"

1 1 00 =-2ao+-LrJ 1r j=1

f(¢)[COSj(8-¢)]d¢.

-"

We finally simplify our expression to

w(r. 8) = -1

1"

1r_"

00. J cosj(8 -¢)] d¢. f(¢)[-1 + Lr

2

'I

1=

11.4 Fourier Analytic Methods

173

It behooves us, therefore, to calculate the sum inside the integral. For simplicity, we let a = B - tf>, and then we let

z

=ria = r(cosa + i sin a).

Likewise,

zn = rnina = rn(cosna + i sinna). Let Re z denote the real part of the complex number z. Then

1

[1

Jcosja = Re :2 + ?=zJ 00. 00 'J :2 + ?=r J=I

J=I

=Re

]

[_~+_1 2

I-z

-Re[ I+z

2(1- z)

= Re [(1

J

+ z)(I-

211- zl2

Z)]

1-lz12 - 211 - zl2 1- r 2

- "::"2(':":I-"::"2r-c-o-s-a-+-r-;2"") . Putting the result of this calculation into our original formula for w, we finally obtain the Poisson integral formula:

1"

2

1 1- r w(r,B) = 21r _" 1-2rcos(B -tf» +r 2f (tf»dtf>

Observe what this formula does for us: It expresses the solution of the Dirichlet problem with boundary data f as an explicit integral ofa universal expression (called a kernel) against that data function f. There is a great deal of information about wand its relation to f contained in this formula. As just one simple instance, we note that when r is set equal to 0 then we obtain w(O, B) = -2 1 7r

1"

f(tf»dtf>.

_"

This says that the value of the steady-state heat distribution at the origin is just the average value of f around the circular boundary.

174

Chapter 11: Differential Equations

11.4.4 The Wave Equation We consider the wave equation on the interval [0, 7f)

a 2 yxx = y"

(11.4.4.1)

with the boundary conditions y(O, t)

=0

and y(7f,t) =0.

This equation. along with its boundary conditions, is a mathematical model for a vibrating string with the ends (at x = 0 and x = 7f) pinned down. The function y(x. t) describes the ordinate of the point x on the string at time t. Physical considerations dictate that we also impose the initial conditions

aYI -0 at 1=0-

(11.4.4.2)

(indicating that the initial velocity of the string is 0) and y(x,O) = f(x)

(11.4.4.3)

(indicating that the initial configuration of the string is the graph of some function f). We solve the wave equation using a version of separation of variables. For convenience. we assume that the constant a = 1. We guess a solution of the form u(x, t) = u(x) . v(t). Putting this guess into the differential equation Uxx

= UII

gives ul/(x)v(t)

=

u(x)vl/(t).

We may obviously separate variables, in the sense that we may write ul/(x) u(x)

vl/(t)

=-v(t)

The left-hand side depends only on x while the right-hand side depends only on t. The only way this can be true is if ul/(x)

vl/(t)

---:...:.. = A = -v(t)u(x) for some constant A. But this gives rise to two second-order linear. ordinary differential equatiOns that we can solve explicitly: U"=AoU

(11.4.4.4)

vI/ = A • v.

(11.4.4.5)

11.4

Fourier Analytic Methods

175

Observe that this is the same constant A in both of these equations. Now, as we have already discussed, we want the initial configuration of the string to pass through the points (0,0) and (11,0). We can achieve these conditions by solving (11.4.4.4) with u(O) = 0 and u(1I) = O. But of course this is the eigenvalue problem that we discussed earlier. The problem has a nontrivial solution if and only if A = n2 for some positive integer II, and the corresponding eigenfunction is Un (x)

= sinnx.

For this same A. the general solution of (11.4.4.5) is

vet) = A sinnt + B COSIlt. If we impose the requirement that A = O. and we find the solution

Vi (0)

= 0, so that (11.4.4.2) is satisfied. then

vet) = B cosn/. This means that the solution we have found of our differential equation with boundary and initial conditions is

Yn(x, t) = sinnx cosnt.

(11.4.4.6)

In fact, any finite sum with coefficients (or linear combination) of these solutions will also be a solution:

Y=

al

sinxcost +a2sin2xcos2t + ... +aksinkxcoskt

This is called the "principle of superposition." Ignoring the rather delicate issue of convergence, we may claim that any infinite linear combination of the solutions (11.4.4.6) will also be a solution: 00

y= "L,bjsinjxcosj/.

(11.4.4.7)

j=1

Now we must examine the final condition (11.4.4.3). The mandate y(x, 0) = [(x) translates to 00

"L,bj sinjx

= y(x. 0) = {(x)

(11.4.4.8)

= y(x,O) = I(x)

(11.4.4.9)

j=1

or

00

"L,bjuj(x) j=1

Thus we demand that I have a valid Fourier series expansion. We know from our studies in Chapter 9 that such an expansion is valid for a rather broad class of functions [. Hence the wave equation is solvable in considerable generality.

Chapter 11: Differential Equations

176 We know that our eigenfunctions U j satisfy /I

urn

= -m2urn

/I

and

un =

-n 2Un·

Multiply the first equation by Un, the second by Um, and subtract. The result is " - Um"n"=( n 2-2 unum In) UnUm

or ' - "mUn']' = (2 [unum n -m 2)UnUm· We integrate both sides of this last equation from 0 to rr and use the fact that U j(O) = U j(rr) = 0 for every j. The result is

0= [unu:" - umu~l Thus or

"= (n o

2

m 2)

-

10" sin mx sin nx dx = 0

1"

10"0 Um(x)un(x)dx. -#m

(11.4.4.10)

-# m.

(11.4.4.11)

forn

um(x)un(x)dx = 0

for n

This is a standard fact from calculus. It played an important (tacit) role in Section 9.6, when we first learned about Fourier series. It is commonly referred to as an "orthogonality condition;' and is fundamental to the Fourier theory and the more general Sturm-Liouville theory. We now have discussed how the condition arises naturally from the differential equation. In view of the orthogonality condition (11.4.4.11), it is natural to integrate both sides of (11.4.4.9) against Uk (x). The result is [" I(x)· uk(x)dx = f"[f>jUj(X)]' uk(x)dx

Jo

Jo

J=I

= fbj j=1

rr

=

I

f" Uj(x)uk(x)dx Jo

bk '

The bk are the Fourier coefficients that we studied in Chapter 9. Certainly Fourier analysis has been one of the driving forces in the development of modem analysis: questions of sets of convergence for Fourier series led to Cantor's set theory. Other convergence questions led to Dirichlet's original definition of convergent series. Riemann's theory of the integral first occured in his classic paper on Fourier series. In tum, the tools of analysis shed much light on the fundamental questions of Fourier theory. In more modem times. Fourier analysis was an impetus for the development of functional analysis. pseudodifferential operators, and many of the other key ideas in the subject. It continues to enjoy a symbiotic relationship with many of the newest and most incisive ideas in mathematical analysis.

Glossary of Terms from Real Variable Theory absolutely convergent verges.

A series Lj Cj is absolutely convergent if Lj

ICjl con-

absolute maximum If f is a function with domain S and if there is a point M e S such that f(M) 2:: f(x) for all xeS. then the point M is called an absolute maximum for f absolute minimum If f is a function with domain S and if there is a point /II e S such that f(/II) ::s f(x) for all x e S.then the point //I is called an absolute minimum for f. accumulation point Let S be a set A point x is called an accumulation point of S if every neighborhood of x contains infinitely many distinct elements of S. accumulation point in a metric space Let (X. p) be a metric space and S a subset A point x e X is called an accumulation point of S if every B(x. r) contains infinitely many distinct elements of S. boundary point Let S be a sel. Then B is a boundary point of S if every nonempty neighborhood of B contains both points of S and points of c S. bounded above A set S is called bounded above if there is a number M such that s ::s M for every element s e S. bounded below A set S is called bounded below iflhere is a number N such that s > N for every element s e S. bounded set A set S is called bounded if there is a positive number K such that IsI ::s K for every element s e S. bounded set in a metric space Let (X. p) be a metric space. A subset S C X is said to be bounded if S lies in some ball B(P. r). bounded variation Let f be a function on the interval [a. b]. For x e [a. b] we define Vf(x) = sup L'=I If(pj) - f(pj-I)I. where the supremum is taken over all partitions P of the interval [a. x]. If Vf = Vf(b) < 00. then the function f is said to be of bounded variation on the interval [a. b]. 177

178 cardinality

Glossary of Terms from Real Variable Theory The size of a set, measured by set-theoretic isomorphism.

cartesian product given sets.

The collection of ordered pairs, or n-tuples, of objects from

Cauchy criterion We say that the sequence (a j I satisfies the Cauchy criterion if, for each € > 0, there is an N > 0 such that, whenever j, k > N, then la j - ak 1< €. Cauchy criterion in a metric space A sequence (x j} of elements of a metric space (X, p) is said to be Cauchy if, for each € > 0, there is an N > 0 such that if j,k > N,thenp(xj,xk) <E. characteristic curve tial equation. closed interval closed set

A curve that represents a flow induced by a partial differen-

A set of the form (a, h} = (x

E

IR : a::: X::: hI.

A set whose complement is open.

closure ofa set in a metric space Let (X, p) be a metric space and E ex. The closure of E is defined to be the union of E with the set of its accumulation points. We sometimes denote the closure of E by E. common refinement If P, P' are panitions of [a, h}, then their common refinement is the union of all the points of P and P'. commuting limits

For example, if lim .lim Yj(x) = .lim lim Yj(x). x-+c )-+00

j-+OOX-'C

compact set in a metric space Let (X, p) be a metric space. A set S C X is said to be compact if every sequence in S has a subsequence that converges to an element ofS. complement of a set

The collection of elements not in that set.

complete metric space A metric space (X, p) is called complete if every Cauchy sequence in X converges to an element of X. conditionally convergent A series Lj Cj is conditionally convergent if Lj Cj converges but L j Ie j 1diverges. connected continuous

If a set is not disconnected, then it is connected. A function f is continuous at P if Iimx .... p f(x) = f(P).

convergence of a sequence A sequence (a j I converges to a limit £ if, for every € > 0, there is a positive integer N such that j > N implies that la j - £1 < €. convergence of a sequence in a metric space A sequence (Xj} of elements of the metric space (X, p) is said to converge to a point IX E X iffor each € > 0 there is an N > Osuch that if j > N, then p(Xj,IX} < €. convergence ofa series A series Lj Cj converges if the sequence of panial sums (SN I converges to a finite limit.

Glossary of Terms from Real Variable Theory

179

convergence of a series of functions Let E· y)(x) be a series of functions. If the limit of partial sums IimN_oo SN(X) exists ~nd is finite, we say that the series E) y) (x) converges at x. converge pointwise A sequence of functions (fJ) converges pointwise to a limit function f on S if, for each xeS, the sequence of numbers (f) (x») converges to f(x). converge uniformly A sequence of functions (Ii) converges uniformly to f if, given € > 0, there is an N > 0 such that, for any; > N and any xeS. it holds that IfJ(x) - f(x)1 < €. cosine function

The power series function 00

.

x 2)

L(-1)l (2 ')1 .

)=0

countable set

J .

A set that has the same cardinality as the set of natural numbers.

covering of a set S

A collection of sets Ua such that Ua Ua 2 S.

derivative If f is a function with domain an open interval 1 and if x e I, then the limit Iimr-.< [(t~!(X). when it exists, is called the derivative of f at x. derived series From the power series E,!=o a) (x - c») we create. using formal differentiation, the derived series E'!=I ja)(x - c»)-I. differentiable at x.

If the derivative of f at x exists. then we say that f is differentiable

differential equation

An equation relating a function to some of its derivatives.

Dirichlet problem on the disc tor. disc in the complex plane

A boundary value problem for the Laplace opera-

A set of the form (z e

c: Iz -

cl <

rl.

disconnected A set S is disconnected if it is possible to find a pair of open sets U and V such that U n S # 13. V n S # 13. (U n S) n (V n S) = 13. and S = (U n S) u (V n S). discontinuity of the first kind If limx_p- f(x) and if Iimx _p+ f(x) exist. but either do not equal each other or do not equal f(P). then we say that f has a discontinuity of the first kind. discontinuity of the second kind If either Iim x _p- does not exist or Iimx _P+ does not exist, then we say that f has a discontinuity of the second kind at P divergence of a sequence sequence.

A sequence that does not converge is called a divergent

divergence of a series If the sequence of partial sums of a series does not converge. then the series diverges.

180

Glossary of Terms from Real Variable Theory

divergence of a series of functions If the sequence of partial sums of a series of functions does not converge. then the series of functions diverges. domain of a function eigenfunction

A characteristic function of a differential equation.

element of a set empty set

The set on which a function is defined.

An object in that set.

The set with no elements. The power series function exp(z) =

exponential function

L~O ~.

field A set with operations of addition. multiplication. SUbtraction. and division satisfying the usual laws of arithmetic. finite subcovering

A subcovering with finitely many elements.

6nite subcovering in a metric space Let S bea set in a metric space (X. p) and let C be an open covering of S. We call 1) a finite subcovering of C if 1) is a subcovering and if 1) has just finitely many elements. first category A set is of first category if it can be written as the countable union of nowhere dense sets. Fourier coefficient For an integrable function! on the interval [0.21£]. the numJo21r {(t)e-intdt. ber !
i!r

Fourier series The series S!(x) '" L~-oo !<1I)inx • formed with the Fourier coefficients of the function !. function A rule that assigns objects from one set (the domain) to the elements of another (the range). To each element of the domain there corresponds one and only one element of the range. geometric series

A series of the form

L j aj •

greatest integer function The function whose value x is equal to the greatest integer that does not exceed x. greatest lower bound bound greater than it. Hadamard formula and p by

A lower bound for a set such that there is no other lower For the power series ~ a j (x - c)i • Hadamard defines A A = lim sup lani l/n n_OO

if A = 00. ifO < A < 00. if A = O. harmonic series

The series L j 1/j .

181

Glossary of Terms from Real Variable Theory

image If 1 is a function and G a set, then the image I(G) is the set of f(x) such that x e G. infimum

See greatest lower bound.

initial condition f(xo) = YO· initial curve integers

For a first order differential equation. a condition of the form

A curve from which the characteristics emanate.

The positive and negative whole numbers and zero.

integral equation An equation that relates a function f to its integral. Often an integral equation is equivalent to some differential equation. or some initial value problem. interior point Let S be a set. A point s e S is called an interior point of S if there is an ~ > 0 such that the interval (s -~. s +~) lies in S. intersection

Those elements common to two or more given sets.

interval of convergence The interval centered at c on which a power series expanded about c converges. inverse function Given a function I. the inverse of to their corresponding domain elements.

f sends range elements of f

inverse image If f is a function and 0 a set. then the inverse image set of x such that I(x) e O.

f- I (0) is the

isolated point Let S be a set. A point t is called an isolated point of S if there is an ~ > 0 such that the intersection of the interval (t - ~. t + ~) with S is just the singleton {t}. isolated point in a metric space A point P of a set E in a metric space (X. p) is called isolated if there is an r > 0 such that En B(P. r) = {Pl· k-times continuously differentiable If I is k-times differentiable on an open interval I. and if each of the derivatives 1(1)./(2) • ...• I(k) is continuous on I. then we say that I is k-times continuously differentiable on I. Laplace equation

The partial differential equation t,. =

least upper bound bound less than it.

a2 u a2u ax 2 + ay2

= O.

An upper bound for a set such that there is no other upper

left limit The function Ion E has left limit £ at p. and we write Iim..... p- I(x) = £. if for every ~ > 0 there is a" > 0 such that whenever P - " < x < P and x e E. then it holds that I/(x) - £1 < ~.

182

Glossary of Terms from Real Variable Theory

limit of a function The function [ has limit £ at P if, for each EO > 0, there is a 8> Osuch that when x E EandO < lx-PI < 8,thenit holdsthatl/(x)-£I < EO. limit of a function on a metric space Let (X, p) be a metric space and [ a function on it. We say that [ has limit £ at P if for any EO > 0, there is a 8 > 0 such that if 0 < p(x, P) < 8, then I[(x) - £1 < EO. limit of a sequence the sequence.

If the sequence (a j) converges to £ then we call

ethe limit of

limit of a sequence in a metric space Ifthe sequence (x j) of elements of a metric space converges to CI then we call CI the limit of the sequence. The least limit of all subsequences of the given sequence.

limit infimum limit supremum

The greatest limit of all subsequences of the given sequence.

Lipschitz condition of order CI Let [ be a function. There is a constant M such that, for all s, 1 E 1, we have I/(s) - [(1)1 ::; M . Is - tl a . Here 0 < CI ::; 1. local extrema

Local maxima and local minima.

local maximum A point x E (a , b) is called a local maximum for [ if there is an 8 > 0 such that [(I) ::; [(x) for all t E (x - 8, x + 8). local minimum A point x E (a, b) is called a local minimum for [ if there is an 8> 0 such that I(t) ::: [(x) for all t E (x - 8, x + 8). lower bound

A number that is less than or equal to all elements of a given set.

lower Riemann integral Let [ be a function on the interval la, b]. Define the lower Riemann integral 1*(f) = sup£(f, P, CI), where the supremum is taken with respect to all partitions P of the interval la, b]. lower Riemann sum Fix an interval la, b] and a monotonically increasing function CI on la, b]. IfP (po, PI, .•• , Pkl is a partition of la, b], let 6.Clj CI(Pj)CI(Pj_I). Let [ be a bounded function on la, b] and define the lower Riemann sum of 1 with respect to CI as follows: £(f, P, CI) = L:~=I m j 6.CI j. Here 11/ j denotes the infimum of Ion lj.

=

=

mesh If P is a partition of la, b] we let lj denote the interval IXj-1> Xj), j = 1, 2, .•. ,k. The symbol 6. j denotes the length of I j. The mesh of P, denoted by 11/ (P), is defined to be max j 6. j. method of characteristics A method of reducing a partial differential equation to a family of ordinary differential equations along the characteristic curves. method of Frobenius A power series method for handling ordinary differential equations with singular coefficients. metric

The function p in the definition of a metric space.

metric space A metric space is a pair (X, p), where X is a set and p : X x X (t E lR : t ::: 0) is a function satisfying

--+

183

Glossary of Terms from Real Variable Theory 1. For all x, y E X, p(X, y) = p(y, x);

2. p(x, y) = 0 if and only if x = y;

3. for all x, y, Z E X, p(x, y) :::: p(x, z) + p(z, y). monotonically decreasing function The function f is monotonically decreasing on (a, b) if, whenever a < s < t < b, it holds that f(s) :::: [(t). monotone decreasing sequence al :::: a2

The sequence (aj} is monotone decreasing if

? ...

monotonically increasing function The function f is monotonically increasing on (a, b) if, whenever a < s < t < b, it holds that f(s) :::: [(t). monotonically increasing sequence

The sequence (a j} is monotone increasing if

a 1 :::: a2 :::: . . • .

natural logarithm function natural numbers neighborhood

The whole, or counting, numbers.

If x

nonisolated point

The inverse of the exponential function.

E

JR, then a neighborhood of x is an open set containing x.

A point that is not isolated.

nonisolated point in a metric space

A point in a metric space that is not isolated.

nowhere dense set in a metric space Let (X, p) be a metric space. The set E c X is nowhere dense in X if the closure of E contains no ball B(x, r) for any x EX, r > O. one-to-one A function that sends any two different domain values to two different range values. onto

A function that assumes all values in its range.

open interval

A set of the form (a, b) = (x E JR:a <x <

bl.

open covering A collection of open sets (Va laeA is called an open covering of S if UaeA Va :> S. open covering in a metric space Let S be a subset of a metric space (X, pl. A collection of open sets (ValaeA (each Va is an open set in X) is called an open covering of S jf UaeA Va :> S. A set with the property that whenever x E U, there is an € > 0 such that (x -€,x+€) C U.

open set

ordered field

A field equipped with an order relation.

ordinary differential equation An equation relating a function of one variable with some of its (ordinary) derivatives.

184

Glossary of Terms from Real Variable Theory

orthogonality condition An independence condition on functions. The orthogonality condition is modeled on the idea of perpendicularity of vectors. and is specified by an inner product. partial differential equation An equation relating a function of several variables with some of its (partial) derivatives. partial sum

The sum of finitely many terms of a series.

partial sum for a series of functions The expression SN(X) = LJ=I Yj(x) = YI (x) + n(x) + ... + YN(X) is called the Nth partial sum for the series Lj Yj(x). partial sum of a Fourier series If f is an integrable function on [0. 21f I and Ln !(n)e inx its Fourier series. then the Nth partial sum is N

SN f(x) =

L

!(Il)i nx

n=-N

partition Let [a. bl be a closed interval in JR. A finite. ordered set of points 'P = (xo. XI> X2.···. Xk_l. Xk) such that a = xo :5 XI :5 X2 :5 ... :5 Xk_1 :5 Xk = b is called a partition of [a. bl. perfect set A set S is perfect if it is nonempty, closed. and if every point of S is an accumulation point of S. Picard's iteration technique A recursive technique for producing a uniformly convergent sequence of functions that converges to a solution of the given first-order differential equation. Poisson integral formula

An explicit formula for solving the Dirichlet problem.

principle of superposition The idea that solutions of a differential equation may be combined linearly to obtain new solutions. power sequence

A sequence p.i) of powers.

power series expanded about the point c A series of the form L~ aj (x - c)j is called a power series expanded about the point c. radius of convergence range of a function

Half the length of the interval of convergence. The set in which a function takes its values.

rational function

A quotient of polynomials.

rational numbers disallowed).

The collection of quotients of integers (with division by zero

real analytic function A function f. with domain an open set U ~ JR and range either the real or the complex numbers. such that for each c e U. the function f may be represented by a convergent power series on an interval of positive radius centered at c : that is. f(x) = L~aj(x - c)j.

185

Glossary of Terms from Real Variable Theory rearrangement of a series

A series with its terms permuted.

refinement Let P and Q be partitions of the interval la. hI. If each point of P is also an element of Q. then we call Q a refinement of P. Riemann integrable A function I is Riemann integrable on la. hI if the Riemann sums of n(f. P) tend to a limit as the mesh of P tends to zero. Riemann integral exists.

The value of the limit of the Riemann sums. when that limit

Riemann-Stieltjes integral A version of the Riemann integral in which the lengths of segments of the partition are measured with a weight function 01. Riemann sum If I is a function on la. hI and P a partition with increment lengths I:J.j then the corresponding Riemann sum is defined to be n(/. P) = L'=I f(sj)l:J.j for points Sj in the intervals I j of the partition. right limit The function I on E has right limit £ at p. and we write Iim.<-> p+ I(x) = £. if for every € > 0 there is a Ii > 0 such that whenever P < x < P + Ii and x E E.then it holds thatl/(x) - £1 < €. same cardinality

1\vo sets with a set-theoretic isomorphism between them.

second category

A set is of second category if it is not of first category.

separation of variables method A method for solving differential equations in which the dependent and independent variables are isolated on separate sides of the equation. sequence on a set S

A list of numbers. or a function

f

from N to S.

sequence of functions A sequence whose terms are functions. usually written /l(x). !2(x)•... or (/j I~I' series

An infinite sum.

series offunctions The formal expression L'i=1 /j(x). where the Ij are functions on a common domain S. is called a series of functions. set-builder notation 1 <x2 +3<9}.

Specification of a set with the notation S =

set-theoretic difference not in the other.

A one-to-one. onto mapping.

A collection of objects.

simple discontinuity sine function

E lR :

Given two sets. the collection of objects in one set but

set-theoretic isomorphism set

Ix

See discontinuity of the first kind.

The power series function 00

.

x2j+l

~(-I») (2j + I)!

)=0

186

Glossary of Terms from Real Variable Theory

singleton

A set with one element.

Sturm·Liouville problem A theory of orthogonal functions. arising from certain differential equations. that is modeled on the theory of Fourier series. subcovering If C is a covering of a set S. and if 1) is another covering of S such that each element of 1) is also an element of C. then we call 1) a subcovering of C. subcovering in a metric space Let (X. p) be a metric space. If C is a covering of a set S ex. and if 1) is another covering of S such that each element of 1) is also an element of C, then we call 1) a subcovering of C. subsequence Let (a j) be a given sequence. If 0 < iI < h < ... are positive integers. then the function k t-+ a jt is called a subsequence of the given sequence. subset

A subcollection of objects in a given set.

summation by parts A summation procedure that is analogous to integration by parts: it switches the roles of addition and subtraction. supremum

See least upper bound.

totally disconnected A set S is totally disconnected if. for each distinct XES. yES. there exist disjoint open sets U and V such that x E U. Y E V. and S = (UnS)U(VnS).

The quantity VI (b) in the definition of bounded variation.

total variation

triangle inequality p(x. z)

On a metric space with melric P. the inequality p(x. y) <

+ p(z. y).

uncountable

An infinite set with cardinality at least as great as lit

uniform convergence of a sequence of functions A sequence of functions Ij converges uniformly to I if, given (i > 0, there is an N > 0 such that, for any j > N and any XES. it holds that Ih(x) - l(x)1 < (i. uniform convergence of a series of functions If the partial sums SN(X) of the series Ej Ij(x) converge uniformly on S to a limit function g(x), then we say that the series converges uniformly on S. uniformly Cauchy A sequence of functions Ij on a domain S is called uniformly Cauchy if, for each (i > O. there is an N > 0 such that if j. k > N then Ih(x) - fk(x)1 < (i for all XES. uniformly continuous A function I is uniformly continuous on a set E if for any (i > 0 there is a B > 0 such that whenever s. lEE and Is - II < B, then I/(s) - 1(1)1 < (i. union

Those elements in anyone of a collection of given sets.

upper bound

A number that is greater than or equal to all elements of a given set.

Glossary of Terms from Real Variable Theory

187

upper Riemann integral Let f be a function on the interval la, b]. Define /"(f) = inf U(f, P, or), where the infimum is taken with respect to all partitions P of the interval [a, b]. upper Riemann sum Fix an interval [a, b] and a monotonically increasing function or on [a, b). UP = {po, Ph ... , pkI is a partition of [a, b), let b.orj = or(Pj)or(Pj_I). Let f be a bounded function on la, b) and define the upper Riemann sum of f with respect to or as follows: U(f, P,or) = L~=I Mjb.orj. Here Mj denotes the supremum of f on / j. Venn diagram

A figure that displays sets as regions in the plane.

wave equation

The partial differential equation Uxx - Uu

=0

that describes a vibrating string and other physical waves.

List of Notation Symbol

Section

S E

Ix E IR : 1 <

S=

cs

seT S1=T SnT njSj

SUT UjSj

SxT SI x ... x Sk

S\T

o I

I:S-4T

S

T

f+g f-g f·g

2 X }

1.1 1.1 1.1 1.1 1.1 1.1

1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.4 1.4 1.4

Ilg fog

1.4 1.4

r

1.4 1.5

l

1\I

Z Ql

1.5

1.5

IR

1.5

<.~

1.6

cardS

1.7

Meaning

a set element of set-builder notation complement of the set S subset of not a subset of intersection of sets intersection of sets union of sets union of sets cartesian product of Sand T cartesian product of sets set-theoretic difference of Sand T the empty set a function a function domain of a function range of a function addition of functions subtraction of functions product of functions quotient of functions composition of functions inverse function the natural numbers the integers the rational numbers the real numbers ordering on numbers cardinality of the set S 189

190

List of Notation

Symbol

Section

a],a2. a3•••• or (aj}

l

limj_oo aj

=e

(ajt)

lim sup aj Iiminfaj (~.i)

e Lj..\ Cj SN

Lj I/j j LjCX Lj(-I)jbj LjCPi

r

n~oo

(I+'!'n

Sk.N

=L

lim

N

k

j=\ J. Lj[aj + bj] Lj c· aj Cn = aj' bn-j (a. b) [a. b]

LJ=I

U E

(O"J"eA

C.,/)

C Iim x _p [(x) r(x) rJ(O)

f(G) K m

M Iimx _p- {(x) Iimx_p+ I(x) I(t) - f(x) t-x f'(x). :Xf. ¥X

Meaning

2.l.l 2.1.1 2.1.1 2.1.5 2.2 2.2 2.3 2.3 3.l.l 3.1.2 3.1.2 3.2.3 3.3.2 3.3.4

a sequence limit of a sequence limit of a sequence subsequence limit supremum limit inferior power sequence Euler's number e a series partial sum the harmonic series a geometric series alternating series rearrangement of a series

3.4.2

expression for e

3.4.3 3.5.1 3.5.1 3.5.3 4.l.l 4.l.l 4.l.l 4.1.2 4.4.2 4.4.2 4.5 5.l.l 5.1.4 5.2.2. 5.3.1 5.3.1 5.3.1 5.3.1 5A.1 5.4.1

a sum of powers sum of series scalar product of series Cauchy product open interval closed interval an open set a closed set an open cover open covers the Cantor set the limit of a function a rational function inverse image of the set image of the set G a compact set an absolute minimum for the function an absolute maximum for the function left limit of 1 at P right limit of 1 at P

6.1.1

Newton quotient

6.l.l

the derivative of 1

°

List of Notation

191

Symbol

Section

1/F(x)

6.1.3

6.2.4

::2

f" (x), f(2) (x),

f,

~:{

Ij

D.j

m(P)

'R(/, P)

J:F(x)

UU, P, 0') £(/, P, 0') I*(f) I.U)

J:Q

7.3.1 7.3.1 7.4.4 8.1.1 8.1.1 8.3.1 8.3.1 8.3.3 9.1.1 9.1.1

9.2.1

Hadamard formula

9.2.4

Taylor's formula

9.2.4

remainder for Taylor's formula

7.1.1 7.2.4 7.3.1 7.3.1 7.3.1 7.3.1

I(x)dx

Ida

V(f) II (x), !2(x), ... or {fj(x») Iimj_oo /j(x) = I(x) L~I fj(x)

SN(X) Mj

00

.

Lj=oaj(x - c)J r

building block for Weierstrass function intermediate point for Mean Value Theorem higher derivatives a partition an interval in the partition length of an interval in the partition mesh of the partition P Riemann sum Riemann integral antiderivative of f upper Riemann sum lower Riemann sum upper integral of f lower integral of I Rieman-Stieltjes integral refinement of the partition P total variation of I sequence of functions pointwise convergence of functions a series of functions partial sum for a series of functions elements of the Weierstrass M -test power series radius of convergence

6.3.3 7.1.1 7.1.1 7.1.1 7.1.1 7.1.1

P

Meaning

A = IimsuPn_oo lanl 1/n

0 if A = 00, IfA ~fO < A <

p=

{ 00 k

If A .

I(x) = LIU)(a) j=o

00,

=0.

(x - a)j

., J.

+ Rk,o(X) x

Rk.o(X)

=

L 0

t
(x

t)k k!

dt

192

List of Notation

Symbol

Section

zi

00

Meaning

I>71 i=o J.

9.3.1

power series for exp(z)

eZ = exp(z) sinx cos x exp(x + iy) = e" (cos y + i sin y) Sin-I x [or arcsin x] Inx aX

9.3.1 9.3.2

9.3.2 9.3.3 9.3.4

f(x)

9.5

the exponential function the sine function the cosine function Euler's formula the inverse sine function the natural logarithm function general exponential function the gamma function

9.5.1

Stirling's formula

. f(t)e-In'dt

9.6.1

11 th

nx

9.6.1

Fourier series for

If(t)(2dt

9.6.3

Bessel's inequality

9.6.3

NIh

9.6.4

Dirichlet kernel

9.6.4

NIh

10.1.1 10.1.1 10.1.1 10.1.3 10.1.6 10.2.1 10.2.1 10.2.1 10.2.1 10.2.2 10.3.2

a metric space a metric triangle inequality xi converges to ex a linear functional on e[O, 1] open ball closed ball open set in a metric space closed set in a metric space open covering in a metric space closure of S

~

II!

.Jiiinn+ 1/2 en

f(II) = -1

21r

1

9.4.1

20

Fourier coefficient

0

00

L

Sf(x) -

9.4.1

!
f

n::::-OO

N

L

flo

1];;1 2 ::: 10

0

n=-N

N

SN f(x) =

L

nx 1(II)i

partial sum of Fourier series

n=-N

DN(t) =

sin(N + !)

SNf(x) =

.

I

sm 2t

~ 12nDN(X -

(X,p) p p(x, y) < p(x, z) Xi -+ ex

t)f(t)dt

+ p(z, y)

FU) B(P,r) B(P, r)

U

E C S

== Ix EX; p(x, P) < r} == Ix EX; p(x, P) ::: r}

partial sum of Fourier series

Guide to the Literature Classical Tracts on Real Analysis [BAR] R. G. Bartle. Introduction to Real Analysis, 3'd ed.• John Wiley & Sons, New York, 2000. [BOA] R. Boas, A Primer ofReal Functions, 4 th ed., revised and updated by Harold Boas, Mathematical Association of America, Washington, D.C., 1996. [BOR] E. Borel, Lefons sur les Fonctions de Variables Reel/es et les Developpements en Series de Polynomes. Gauthier-Villars, Paris. 1905. [BOU] N. Bourbaki, Fonctions d'une Variable Reel/e, Hermann. Paris, 1958. [BUC] R. C. Buck. Advanced Calculus. 2d ed., McGraw-Hili Book Company, New York,1965. [CAR] C. Carath60dory, Reel/e Funktionen. Chelsea. New York, 1946. [DEN] A. Denjoy. Introduction a la Theorie des Fonctions de Variables Reel/es, Hermann, Paris, 1937. [DIN] U. Dini, Grundlagen fUr eine Theorie der Funktionen einer veriinderlichen reel/en Grosse. Teubner. Leipzig, 1892.

[FED] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969. [GOF] C. Goffman, Introduction to Real Analysis. Harper & Row. New York, 1966. [GOU] E. Goursat. Cours d'Analyse Mathematique. 5'h ed., Gauthier-Villars, Paris. 1910. [GRA] L. M. Graves, The Theory ofFunctions ofReal Variables, 2 nd ed., McGrawHill, New Yotk, 1956. [HARl) G. H. Hardy, A Course ofPure Mathematics. lO'h ed., Cambridge University Press, 1958. [RAR2) G. H. Hardy, Integration ofFWlctiollS ofa Single Variable, 2nd ed., Cambridge University Press, Cambridge, 1928. 193

194

Guide to the Literature

[HLP] G. H. Hardy, J. E. Littlewood, and G. Polya.lnequalities. 2nd ed., Cambridge University Press, Cambridge. 1964. [HRO] G. H. Hardy and W. W. Rogosinski. Fourier Series, 3rd ed., Cambridge University Press, Cambridge, 1956. [HES] E. Hewitt and K. Stromberg. Real alld Abstract Analysis, Springer-Verlag, New York. 1965. [HOB] E. W. Hobson. The 17leory of Functions ofa Real Variable and the Theory ofFourier's Series, 3rd ed., Cambridge University Press. Cambridge. 1927. [JOL] L. B. W. Jolley. Summation ofSeries. Chapman & Hall, London. 1925. [KOF] A. N. Kolmogorov and S. V. Pomin. Elements of the Theory of Functions and Functional Analysis. Graylock Press. Rochester. NY, 1957. [LIT] J. E. Littlewood. Lectures on the Theory of Functiolls. Oxford University Press. Oxford, 1944. [NAT] I. P. Natanson. Theory of Functions of a Real Variable. Ungar. New York. 1960. [OLM] J. M. H. Olmsted, Real Variables. Appleton-Century-Crofts, New York. 1959. [OSG] W. F. Osgood. Functions ofReal Variables. Hafner. New York, 1947. [PIE] J. Pierpont, Lectures on the Theory of FWlctions of Real Variables. Ginn & Co.• Boston and New York. 1905. [ROY] H. Royden, Real Analysis. 3'd ed., Macmillan. New York. 1988. [RUD] W. Rudin. Principles ofMathematical Analysis. 3rd ed.• McGraw-Hili. New York, 1976. [SAK] S. Saks. Theory ofthe Integral. 2nd rev. ed.• Dover. New York. 1964. [STE] E. M. Stein. Singular Integrals and Differentiability Propenies ofFunctions. Princeton University Press, Princeton, 1970. [STW] E. M. Stein and G. Weiss. IlIIroduction to Fourier Analysis on Euclidean Space, Princeton University Press. Princeton. 1971. [SZE] G. Szego. Orthogonal Polynomials. American Mathematical Society. Providence. RI. 1939. [TIT] E. C. Titehmarsh. IlIIroduction to the Theory ofthe Fourier Integral. Oxford. The Clarendon Press. 1948. [WZY] R. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker. New York. 1977.

Guide to the Literature

195

[ZYG] A. Zygmund, Trigonometric Series, 2 nd ed., Cambridge University Press, Cambridge, 1968. Modern Treatments of Subjects in Real Analysis [BEN] J. Benedeno, Real Variables and Integration, Teubner, Stungart, 1976. [FOL] G. B. Folland, Real Analysis: Modern Techniques and their Applications, 2nd ed., John Wiley & Sons, New York, 1999. [GAR] R. Gariepy, Modern Real Analysis, Prindle, Weber, and Schmidt, Boston, 1995. [KRA] S. G. Krantz, Real Analysis and Foundations, CRC Press, Boca Raton, FL, 1992. [STEil] E. M. Stein, Hamwnic Analysis, Princeton University Press, Princeton, 1993. [NAS] B. Sz.-Nagy and 1. Szabados, Functions, Series, Operalors, North Holland, New York, 1983. Other References in Analysis [ABS] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, John Wiley & Sons, New York, 1972. [LIP] S. Lipschutz, SchaUIII's Outline ofthe Theory and Problems ofSet Theory and Related Topics, McGraw-Hill, New Yotk, 1964.

[RUDe] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hili Book Company, New York, 1987. [RUDF] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991. [SPI] M. Spiegel, Schaum's Outline ofthe Theory and Problems ofReal Variables, McGraw-Hill, New York, 1969. [ZWI] Zwillinger, et ai, CRC Standard Probability and Statistics Tables, CRC Press, Boca Raton, FL, 2000.

Bibliography [ABR] R. Abraham and J. Robbin, Transversal Mappings and Flows, Benjamin, New York,1967. [BOA] R. P. Boas. A Primer ofReal Functions. Carus Mathematical Monograph No. 13, John Wiley and Sons, Inc., New York, 1960. [BUCl R. C. Buck. Advanced Calculus. 2d ed., McGraw-Hill Book Company, New York, 1965. [BUB] P. Butzer and H. Berens, Semi-Groups ofOperators and Approximation, Springer-Verlag, Berlin and New York, 1967. [HOF] K. Hoffman. Analysis in Euclidean Space. Prentice-Hall, Inc., Englewood Cliffs, N.J.• 1962. [KAT] Y. Katznelson,Introduction 10 Harmonic Analysis, Wiley, New York, 1968. [KRAl] S. G. Krantz, Real Analysis and Foundations, CRC Press, Boca Raton, Florida,

1991. [KRA2] S. G. Krantz, Handbook ofLogic and ProofTeclllliques for Computer Science, Birkhliuser. Boston, 2002. [KRA3] S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Expositiones Math. 3(1983), 193-260. [KRP] S. G. Krantz and H. R. Parks, A Primer ofReal Analytic Functions, Birkhliuser, Boston, 2002. INIV] I. Niven. Irrational Numbers. Carus Mathematical Monograph No. II, John Wiley and Sons, Inc., New York, 1956. !ROO] W. Rudin. Principles of Mathenultical Analysis. 3d ed., McGraw-Hill Book Company, New York, 1976. [STR] K. Stromberg, An Introduction to Classical Real Analysis. Wadsworth Publishing, Inc., Belmont, Ca., 1981. 197

198

Bibliography

[81M] G. F. Simmons and S. G. Krantz, Differential Equations with Applications and Historical Notes, 3rd ed., McGraw-Hill, New York, 2004. [ZYGj A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge,

1959.

Index Abel's Convergence Test. 29 absolute convergence of series. 31 absolute maximum. 63 absolute minimum. 63 accumulation point of a set in a metric space. 145 addition of series. 36 Alternating Series Test. 29 Ascoli-Arzela theorem. 151 Baire category theorem. 149 Bessel's inequality. 135 boundary point. 42 bounded set. 44 bounded set in a metric space. 145 Cantor set. 48 Cauchy Condensation Test. 24 Mean Value theorem. 78 product of series. 37 sequences in a metric space. 143 chain rule. 74 change of variable. 91 characteristic curve. 158 characterization of connected subsets ofR51 closed ball in a metric space, 144 closure of a set in a metric space, 148 coefficients of a power series. 120 commOn refinement of partitions. 88 commuting limits. 106 compact set, 45 in a metric space. 146

comparison of the Root and Ratio Tests. 26 Comparison Test. 23 completeness of a metric space. 142 conditional convergence of series. 31 connected set. 50 continuity. 57 and closed sets. 61 and open sets. 60 and sequences. 59 of a function on a metric space. 143 under composition. 60 continuous functions are integrable. 89 continuous image(s) of a compact set. 62 of connected sets. 65 continuously differentiable. 82 convergence in a metric space. 140 of a sequence of functions. 103 cosine function. 126 counterexample to the convergence of Taylor series. 122 Darboux's theorem. 75 decomposition of a function of bounded variation. 100 density. 147 derivative, 71 of inverse function. 80 derived power series. 118 differentiable. 71

199

200 differential equations, 153 first order, 153 Dirichlet kernel, 136 Dirichlet problem on the disc, 171 disconnected set, 50 discontinuity of the first kind, 66 of the second kind, 66 eigenfunction, 175, 176 elementary operations on real analytic functions, 115 elementary properties of continuity, 59 of sine and cosine, 128 of the derivative, 72 of the exponential function, 124 of the integral, 90 equibounded family, 151 equicontinuous family, 150 Euler's equidimensional equation, 170 formula, 127 number e, 19, 34 exponential function(s), 123, 131 Fourier coefficient, 134 Fourier series, 134 function of bounded variation, 99 functional analysis, 176 Fundamental Theorem of Calculus, 92,93 gamma function, 132 genericity of nowhere differentiable functions, 150 geometric series, 24 harmonic series, 24 heat distribution on the disc, 171 Heine-Borel theorem, 47 image of a function, 62 initial condition, 154 initial curve, 159 integrable functions are bounded, 89

Index integral equation, 154 integratio!! by parts, 97 interior point, 43 Intermediate Value Theorem, 65 interval of convergence, 114 irrationality of e, 34 isolated point, 43 L'Hopital's Rule, 79 Laplace equation, 169 least upper bound, 7 left limit, 66 Legendre's equation, 164 length of a set, 48 limit of a function at a point, 53 a function on a metric space, 143 functions using sequences, 57 Riemann sums, 87 Lipschitz condition, 153 local maximum, 74 local minimum, 74 lower integral, 94 lower Riemann sum, 93 Mean Value theorem, 76 mesh of a partition, 85 method of bisection, 45, 147 characteristics, 158, 159 Frobenius, 166 metric space, 139 monotone decreasing function, 67 sequences, 13 monotone function, 67 monotone increasing function, 67 sequences, 13 natural logarithm function, 130 nowhere differentiable function, 73 number 1£ , 129 open ball ina metric space, 144 open covering, 46

201

Index open covering in a metric space, 147 open subcovering in a metric space, 147 orthogonality condition, 176 partial sum of a Fourier series, 136 partition, 85 perfect set, 51 Picard iterates, 154 iteration technique, 154 estimations of, 157 theorem, 153 Pinching Principle, 14 pointwise convergence of Fourier series, 137 Poisson integral formula, 172, 173 power sequences, 17 power series, 113 for solving a differential equation, 160 principle of superposition, 175 product of integrable functions, 91 pseudodifferential operators, 176 radius of convergence, 117 Ratio Test, 25, 27 rational and real exponents, 17 real analytic, 114 rearrangement of series, 32 refinement of a partition, 94 reversing the limits of integration, 90 Riemann integral, 87 lemma, 96 sum, 86 Riemann-Stieltjes integral, 93, 94 existence of, 96 right limit, 66 Rolle's theorem, 76 Root Test, 25, 27

scalar multiplication of series, 36 separation of variables, 174 method, 170 sequence jill, 18 sequence offunctions, 103 series offunctions, 108 set theory, 176 simple discontinuity, 66 sine function, 126 Stirling's formula, 133 strictly monotonically decreasing, 69 strictly monotonically increasing, 69 subcovering, 46 summation by parts, 29 Taylor's expansion, 121 term-by-Ierm integration of power series, 120 total variation, 99 totally disconnected set, 51 uniform continuity, 63 and compact sets, 64 uniform convergence, 104 uniformly Cauchy sequences offunctions, 107 uniqueness of limits, 54 upper bound, 7 upper integral, 94 upper Riemann sum, 93 value of 11", J29 vibrating string, 174 wave equation, J74 Weierstrass M-Test,IIO Approximation Theorem, 1II nowhere differentiable function, 73