Applied and Computational Complex Analysis - Vol 1: Power Series, Integration, Conformal Mapping, Location of Zeros 0471372447, 9780471372448

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APPLIED AND COMPUTATIONAL COMPLEX ANALYSIS VOLUME 1

Power Series-"Integration-Conformal Mapping -Location of Zeros

PETER HENRICI Professor of Mathematics Eidgeniissiscbe Technische Hochschule, Ziirlch

A WII.EY·INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, New York .

London . Sydney . Toronto

Copyright © 1974, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher.

Library of Congress Cataloging in Publication Data: Henrici, Peter, 1923Applied and computational complex analysis. (Pure and applied mathematics) "A Wiley-Interscience publication.'~ Bibliography: v. I, p. 1. Analytic functions. 2. Functions of complex variables. 3. Mathematical analysis. I. Title. QA331 .H453

1974

515'.9

73-19723

ISBN 0-471-37244-7 Printed in the United States nf America

10 9 8 7 6

~

4 J 2 I

Dedicated to the Memory of

HEINZ RUTISHAUSER

PREFACE

This book constitutes the first installment of a projected three-volume work that will present applications as well as the basic theory of analytic functions of one or several complex variables. Applications are made to other branches of mathematics, to science and engineering, and to computation. The algorithmic attitude toward mathematics-not to consider a problem solved unless an algorithm for constructing the solution has been found-prevails not only in the sections devoted to computation but forms one of the work's unifying themes. A short overview of the three volumes is in order. The first volume, after laying the necessary foundations in the theory of power series and of complex integration, discusses applications and basic theory (without the Riemann mapping theorem) of conformal mapping and the solution of algebraic and transcendental equations. The second volume will cover topics that are broadly connected with ordinary differential equations: special functions, integral transforms, asymptotics, and continued fractions. The third volume will center similarly around partial differential equations and will feature harmonic functions, the construction of conformal maps, the Bergman-Vekua theory of elliptic partial differential equations with analytic coefficients, and analytical . techniques for solving three-dimensional potential problems. In collecting all these topics under one cover, I have been guided by the idea that for today's applied mathematician it is not enough to specialize, however deeply, in any single narrowly restricted area. He should also be made aware as forcefully as possible of the light that radiates from the basic theories of mathematics into the neighboring fields of science. What I have tried to do here for complex analysis, should as part of an applied mathematics curriculum also be done, for instance, in real analysis and linear algebra. The prerequisites for Volume I include advanced calculus, matrix calculus, and a smattering of modern algebra. I have not attempted to hold the level of presentation to a uniform level; however, even in those places in which advanced notions are introduced (e.g., §2.1, 3.5, 4.6, 5.9, and 6.12) vii

1

viii

I'IU:t· A< 'E

the reader will find that the presentation i~ sclf-l'lllllallll'd ;111d. l11,pdull\. well motivated. It is felt that, with some omissions perhap~. 1h1, hulll- l'llllld form the basis of a well-rounded full-year course al lhe ~ettlur/flr~Hear graduate level of a modem, computation-oriented l'Urnculum 111 applied mathematics. A word about the layout of the contents of the presenl volume may he necessary. It is well known that there arc two essentially d1fferenl approaches to complex variable theory: Riemann's approach, hased on complex differentiability, and the approach of Weierstrass, hased on power series. Although the Weierslrassian example has been followed in a venerable series of classical texts (Whittaker and Watson [ 1927], firs I ed. 119031. Hurwitz and Courant [1929], Dienes [1931]), the "geometric" method of Riemann, brilliantly exposed, for example, by Ahlfors [1953], seems to have been preferred by most analysts to the "computational" method of Weierstrass, al least until recently, when under the influence of Bourbaki (Carlan [1961]) power series came back into fashion. Since this book is deliberately oriented toward computation, it was almost a foregone conclusion that the theory be based on power series. A computing machine cannot really deal with functions defined on a continuum, but it can readily handle sequences of power series coefficients and perform all kinds of algebraic operations on them. The elementary functions, and many of the important transcendental functions of mathematical physics, are most easily defined by power series. In my treatment of power series I make a sharp distinction between the purely algebraic aspects (Chapter 1) and the analytic aspects of the theory. In treating the analytic aspects I distinguish between properties of functions analytic at a point (Chapter 2) and functions analytic in a region (Chapter 3).

The emphasis on power series has several didactic advantages. First, it provides the student with the opportunity to renew his acquaintance with the basic notions of algebra (group, ring, field, isomorphism) which are illustrated here by interesting examples that are too elaborate to be treated in detail in the usual algebra course. Second, it teaches him to differentiate between the algebraic-computational and the analytic-functional content of the theory and to appreciate the increase in depth when proceeding to the latter. Third and most important, the early treatment of the basic facts on analytic continuation made possible by power series will impress the student at an early stage with the fundamental inner coherence of analytic functions which so completely sets them apart from more general types of function. After having described the overall trend of the book, I now wish to comment briefly on the treatment of certain details. The hypergeometric

PREFACE

ix

identities developed in § 1.6 are not immediately required for the theory that follows. They are inserted here as a source of illustrations and as an example of how the "formal" apparatus of power series can lead to concrete results. The formal treatment of the Lagrange-Biirmann expansion in Section 1.9, on the other hand, is indispensable to the theory of the inverse function in §2.4. Chapter 2 discusses power series with values in a Banach algebra, not only because this discussion can be carried through with absolutely no increase in complexity over the usual treatment of power series but also because it provides an excellent opportunity to introduce certain matrixvalued functions that are required in the theory of ordinary differential equations (Chapter 9). By using some simple notions from functional analysis we can once again relate complex analysis to another important branch of theoretical mathematics. In Chapter 3 the discussion of analytic continuation along an arc calls for the concept of homotopy, which is also required for a version of the Cauchy integral theorem. A nonstandard topic in Chapter 3 is the constructive treatment of Weierstrassian analytic continuation (§3.6). In the treatment of integration (Chapter 4) the local version of Cauchy's theorem becomes a triviality if analyticity is defined by power series. The fact that Goursat's generalization finds no place here does no irreparable damage because this result, although of great historical and intellectual interest, is seldom used in applied mathematics. In regard to nonlocal versions of Cauchy's theorem I have heeded the advice of my colleague and former teacher Albert Pfluger to present a homotopy version of the theorem before entering into homology. This suffices for a treatment of the Laurent series, which includes some applications to Bessel functions and Fourier series. In place of a formal treatment of homology theory, whose full power again is seldom required in applied mathematics, I use winding numbers to prove the Jordan curve theorem (for piecewise smooth curves). Thus in the theory of residues the concept of the "interior" of a Jordan curve with all its great intuitive appeal can be applied without embarrassment. In addition to the applications of residues to integration, I discuss the summation of infinite series by summatory functions, including the Plana summation formula. Chapter 5, on conformal mapping, after a brief discussion of elementary maps, features a careful study of the Joukowski map, a thorough understanding of which is essential to many applications. The treatment of Moebius transformations emphasizes symmetry rather than the cross-ratio. Added to technically motivated applications of conformal mapping to clcc1 rostatics and fluid dynamics is a treatment of Poisson's equation which includes a proof of Saint-Venant's isoperimetric inequality for

X

t•tu.tAt (n is a positive integer) expand the right member by the binomial formula and compare real and imaginary parts, thus proving Moivre's formula representing cosncp and sinncp as polynomials in coscp and sincp. 10. Characterize (in your own words or in a sketch) the sets of those points z in the complex plane that satisfy the following conditions: (a)

lz1I;

(d) Of is the same. We differentiate

and use the differential equation (1.4-8) to obtain

By the above P is proportional to Ba +b• but, because the zeroth coefficient of P equals I· 1-1 =0, P=O follows. • The method previously used to verify the binomial theorem now yields something more interesting. Calculating the nth coefficient of BaBb explicitly, we get

Ry the functional relation just established this equals

24

FORMAL POWER SI m > 0, prove that

L n

k-m

(

t>r

k n k -m ) ( n-k

= )

(

n 2n~m m) .

27

HYPERGEOMETRIC SERIES AND SUMS

§ 1.5.

FORMAL HYPERGEOMETRIC SERIES AND FINITE HYPERGEOMETRIC SUMS.

Let p,q be non-negative integers and let apa 2 , ••• ,aP; b 1,b 2 , ••• ,bq be elements of 5', subject to the condition that for all i and all non-negative integers n, b; + n + 0. The formal power series (1.5-1)

is called a generalized hypergeometric series. The following two notations are commonly used for F and appear indiscriminately in this book:

(1.5-la)

( 1.5-1 b)

The special case p = 2, q = 1 is called the classical (or Gaussian) hypergeometric series and is frequently denoted by F(a,b; c; x).

The generalized hypergeometric series (ghs) are of great importance in the theory of special functions of mathematical physics. Many such functions are defined as special cases of these series. Besides the Gaussian series, the case p = q= 1, called the confluent hypergeometric series, occurs frequently and is denoted by 1F 1 (a;

c; x)

or

x+b 2 0 >x 2 + · · ·)

Collecting coefficients of like powers, we find for the coefficient of xn (n=O, 1, ... )

( 1.6-4) This, in general, has no meaning; we have not defined convergence, and if we had we would not know whether it would take place here. Assume, however, that Q is a nonunit, i.e., that b0 = 0. Then, for n = 1, 2, ... , the series Q n begins at the earliest with the coefficient of x n, i.e.,

n>k. Consequently, the sums (1.6-4) terminate after the nth term and the fps

lo'OHMAI. POWER SERIES

obtained hy formally substituting Q into P cx1sh. Wl· denote it by PoQ or by P(Q) and call it the composition of I' with (j. If

we have, using the notation in ( 1.6-3), c = a 0 and

n= 1,2, ....

( 1.6-5)

Of course, as in a simple fps, we do not touch the convergence problem; no numerical value is assigned to the composed series for any value of x. Also, although the composition of a fps and a nonunit always exists, there may not always be a simple formula for its coefficients. One of our goals is to establish simple recurrence relations in certain cases. EXAMPLE)

To calculate the coefficients cO>c 1,c2 , ... , in the expansion

where Q: = yx- fx 2 andy is a complex constant, we have

_-y nX n - I

2

(

n~n-1 I

X n+l+ - I

(

22

n) y 2

n-2 X n+2

"'(n)2n

+ ... + ( - I ) 2" n x . Thus k 0, there exists an integer N=N(€) such that

llxn -xmll

Nand all m > N.

The sequence {xn} is called convergent if there exists an element x E.£ such that lim llxn -XII =0.

n-->00

It is clear that every convergent sequence is a Cauchy converse, however, need not hold. III. A normed linear space .£ is called complete if sequence of elements of.£ converges. A complete normed also called a Banach space. The space a:n is a Banach space under any of the norms for if the sequence of points

sequence. The every Cauchy linear space is defined above,

X (k) I X (k) 2

k=O, 1,2, ... ,

X (k)

n

is a Cauchy sequence the n sequences of complex numbers {x~>}, m = 1,2, ... ,n, are Cauchy sequences and therefore have limits x 1, ... ,xn by virtue of the completeness of G:. It is then easily seen that the sequence{xk} has the limit

x:=

These axioms evidently take care of the postulates (2) through (5) stated at the beginning of this section. However, we still cannot multiply. For

70

FUNCTIONS ANALYTIC AT A POINT

multiplication, an analogous set of axioms is required. IV. A set (1 of mathematical objects A,B, ... , is called an algebra if it is a linear space and if, in addition to addition and scalar multiplication, every ordered pair (A, B) of elements of 00

For any formal power series P we call radius of cOnvergence and denote by p(P) the supremum of all numbers p such that the series (2.2-3) converges for all Z satisfying liZ II ~p. Thus, by definition, if 0 < p(P) < oo, P(Z) converges for all Z such that IIZII < p(P) and diverges for some Z such that IIZII >p(P). It may or may not converge if IIZII =p(P). THEOREM 2.2a

(Cauchy-Hadamard formula)

If (2.2-4) is any formal power series, its radius of convergence is given by the formula

( 2.2-5)

78

Proof

FUNCTIONS ANALYTIC AT A POINT

We show that the number

a:= [ lim sup lanl 1/n]n---+00

1

satisfies a~p(P) and a';Pp(P). 1. Let a> 0. We show that P(Z) converges for every Z such that IJZIJ a. Then lanl 1/n >'T- 1 for an infinite number of nand lani>'T-n for an infinite number of n. Letting Z='TI, where I is the identity element ofoo an+ I

(2.2-7)

whenever the limit on the right exists. Proof

Let y: = lim

1..3!__ I·

n-->oo an+ I

Evidently y implies

>0. If y < oo, then for any (} > y there exists m such that n >m ..3!__ I(}-I. n-->oo

110

FUNCTIONS ANALYTIC AT A POINT

Similarly, if y > 0, we can show that for every (} such that 0 < (} < y lim sup lanll/n 0, the function defined by its sum P(Z) is continuous at all points Z such that IIZII < p(P).

Suppose now that P(Z) also converges for some Z 0 such that IIZ 0 11 =p(P). Is the function P(Z) still continuous at Z 0 ? In other words, does lim P(Z) = P(Z 0 )

(2.2-9)

Z-.Z 0

hold? Naturally, the approach of Z to Z 0 has to be restricted to those Z for which the series converges [which could include some Z=t=Z 0 such that IIZII =p(P)]. Disappointingly, there are examples that show that (2.2-9) does not always hold, not even if the approach of Z to Z 0 is restricted to Z such that IIZII 0 such that

IIP(Z) II< IIP(O) II= laol for all Z such that

OO such that lf(z)l < lf(O)I for lzl > p. LetS denote the compact set lz I.;;; p and let p. he the supremum of the values taken by lf(z)l on S. Then p. < oo, and by a the()rem of Weierstrass there exists z 0 such that lf(z 0 )1 = p.. The inequality lf(z)l < lf(z 0 )1 now holds for z E S and, since z fl. S implies lf(z)l < lf(O)I < lf(z0)1. for z fl. S. Because f is not constant, this contradicts the principle of the maximum. •

106

FUNCfiONS ANALYTIC AT A POINT

PROBLEMS I.

Show that in the algebra of 2 X 2 matrices over the field of complex numbers the matrix

has no square root. 2. For n-1,2, ... , let

Show that for

lzl sufficiently small

z sn ( z) = 1- z +

~ 00

(-

I)

m-n (

m-2 n- I

)

z

m

( I- z ) .

m-n+l

3.

Let the series F and G have radius of convergence infinity, let F be a nonunit, and let f(Z): =F(Z),

g(Z): = G(Z)

for all Z. Show that goj(Z) = GoF(Z)

for all Z.

§2.5.

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Here we apply the results of §§2.3 and 2.4 to the study of certain special functions well known in real analysis: the exponential function, the logarithm, and the general power function. Throughout this section the algebra (J3 is the algebra of complex numbers. Thus the functions to be considered are defined on certain subsets of the complex plane, and are complex valued.

I. The Exponential Function We saw in §2.2 that the exponential series has radius of convergence infinity. Thus we may define for all complex z a function "exp" by exp(z):=E 1 (z)=l+ ;,

+~~

+ .. ·.

(2.5-1)

107

ELEMENTARY TRANSCENDENTAL FUNCTIONS

This is called the exponential fundion and is also denoted by ez, for reasons to become apparent presently. By definition, the function exp is analytic at z=O; moreover, by Theorem 2.2d it is continuous at all points Evidently E 1(z)=Ez 0 such that e z is represented by a power series in h: = z- z 0 whenever Ihi< p. By the addition theorem

thus the desired expansion is 1 z h +-eo,+··· 1 z ,L2 e z =e z"+-eo 1! 2! '

and this actually holds for all complex h.

•

In the following we take for granted the elementary properties of the real sine and cosine functions as well as the representations y2 cosy= 1 - 2!

y4

y6

+ 4! - 6! + ... '

which by simple techniques of calculus can be shown to hold for all real values of y. (For a purely arithmetical account of these functions see the appendix of Whittaker and Watson [1927] or pp. 69-73 of Hurwitz and Courant [1929].) Setting z=iy in (2.5-1), wherey is real, we get 2

3

4

1 .y y .Y y e = +1----1-+-+···· 1! 2! 3! 4! ' '!Y.

ELEMENTARY TRANSCENDENTAL FUNCfiONS

109

v

Fig. Z.Sa. The number e;y = u+ iv.

hence

e;y=cosy + isiny.

(2.5-6)

By the geometric definition of the sine and cosine functions (see Fig. 2.5a) is a complex number with absolute value 1 and argument y-we have already referred to this fact in § 1.1. We note the special values e±iw = - 1, e ±iw/ 2 = ± i, and

eiv

(2.5-7) Ir z =X+~' the addition theorem yields

which shows that lezl=eRez, arge'=lmz. If z is any complex number, lhen by (2.5-7) (2.5-8)

A function f with the property that there exists a number p =1= 0 such that whenever z E D (f) then also z + p E D (f) and

f( z + p) = f( z) " ~:ailed periodic and p is called a period of f. Relation (2.5-8) proves the following theorem:

110

FUNCfiONS ANALYTIC AT A POINT

TIIEOREM 2.Sb

The function ez is periodic with period 2'1Ti. The Trigonometric Functions. By adding and subtracting (2.5-6) with the same relation in which y is replaced by - y we get the formulas of Euler, l . . ), cosy=2(e'Y+e-•Y

(2.5-9)

valid for all real y. We now define the functions cos and sin for all complex z by C03z: =

. 2I (e'Z. + e-•z),

(2.5-10) 1

.

.

sinz: = 2i (e•z_ e-'z). This definition is in agreement with the conventional definition for z real. By the two theorems just proved the functions cos and sin defined by (2.5-10) are analytic at every point in the complex plane and periodic with the real period 2'1T. The power series representations at z =0 are z2

z4

z6

z3

z5

z7

cosz = 1- - + - - - + ... 2! 4! 6! .

srnz=z--+---+··· 3! 5! 7! . Using the addition theorem of the exponential function, we show easily by the definitions (2.5-10) that the usual functional relations for the functions cos and sin also hold for complex values of the arguments. The functions cos and sin have no zeros in the complex plane except the real zeros z=(n+!)'TT and z=n'fT, respectively, where n=0,±1,±2, .... To prove this for the sine function note that sinz =0 if and only if e2 ;z = 1, i.e., if Re 2iz = 0 and Im 2iz is an integral multiple of 2'1T. It follows that z = n'TT. The proof for the cosine function is similar. The tangent and cotangent functions are defined for complex z by tanz: = srnz, cosz

cotz: = c~sz. smz

(2.5-11)

Again the usual functional relations hold. Both functions are periodic with the real period 'TT. It follow from Theorem 2.3d that tanz is analytic

ELEMENTARY TRANSCENDENTAL FUNCfiONS

111

whenever cosz*O; i.e., for z*(n+i)'IT, n=O, ± l,b+2, .... Similarly, cotz is analytic whenever sinz*O; i.e., for z *0, ±'IT,±2'11', .. .. The coefficients of the power series representing tanz and cotz near z = 0 can be expressed in terms of the Bernoulli numbers Bn, formally defined by (2.5-12) n=O

(see Problem 4, §1.2). By Theorem 2.3g we have for lzl sufficiently small, z*O, (2.5-13) The series 2::/=o(Bn/n!)zn+z/2 represents the function Z ( 2 ) z ez + 1 f(z):=2 ez-1 +l =2 ez-1'

which is easily seen to be even [i.e., to satisfy f(- z)= f(z)]. Hence the series contains only even powers of z. There follows B2n+1

n= 1,2, ... ,

=0,

and we have

z ez + 1 ~ B2n 2n 2 ez-1 = .ttC.J (2n)!z ' n=O

In view of . eliz + 1 . e 2IZ -1

cotz=1 we: now obtain

zcotz=

~ (-1)

oo

n=O

n

B2n

n

--(2z). (2n )!

I he identity 2

) tanz=cotz-2i(1+e4 IZ -1

(2.5-14)

FUNCfiONS ANALYTIC AT A POINT

112

finally yields

B

oo

tanz="" ( -) f-1_2n_ (22n -) )22nz2n-! £.J (2n )! n-1 (2.5-15)

The expansions (2.5-13), (2.5-14) and (2.5-15) have been established here only for lzl sufficiently small. We shall see in §3.3 that as a consequence of Theorem 3.3a they hold for lzl - al < 'TT. We call the

FUNCTIONS ANALYTIC AT A POINT

114

/UU+U

v

/

/

/

i(a+1t) ia

-; 7 7

Fig. 2.5b.

;--;-/-~1-)-a_J_)__/_/_/_/__ x The exponential function restricted.

function z = g(w) defined by (2.5-18) a simple branch of {logw} and denote it by logw (see Fig. 2.5b). The value of a which is used to determine the simple branch should be understood from the context. The simple branch determined by a= 0 is called the principal branch or principal value of the logarithm, and is denoted by Logw. Thus, for all w*O that are not real and negative, Logw = Loglwl + icp,

(2.5-19)

where cp is the unique value of argw satisfying -7r < cp < 7T. This use of the symbol Log is consistent with the notation Logu introduced earlier for positive real u. THEOREM 2.5d

A simple branch of {logw} is analytic at all points at which it is defined. Proof Let logw denote the simple branch defined by the number a and let w 0 be any point in the domain of definition T of logw. Let Z 0 = x 0 + iy 0 be the unique point in s that satisfies ezo= Wo. We know that ez is one-to-one on the set lz -z 0 1< p 1 : =min( a+ 7T-y 0,y 0 - a+ 7T). Because the coefficient of h in the expansion of ez in powers of h: = z- z 0 is not 0 (see the proof of Theorem 2.5a), there exists, by virtue of Theorem 2.4c, a number a 1 >0 such that ez assumes every wEN(w 0 ,a 1) at some zEN(z 0 ,p 1); moreover, the inverse function is analytic at w0 • But on N(w0,a 1) the inverse function agrees with logw. Hence logw is analytic at Wo.

•

It remains to determine the power series that represents logw near w 0 • We begin with the special case in which the simple branch is the principal branch and w 0 = 1. Then z 0 = 0 and since ez-ez 0 =ez -1 =E 1(z) -1

ELEMENTARY TRANSCENDENTAL FUNCTIONS

115

the problem is to invert the series E 1 - / . This was solved formally in § 1.7 and the reversion was found to be the logarithmic series

Thus by Theorem 2.4c, if k: = w- 1 has a sufficiently small modulus, we get Log( 1 + k) = L( k) = k- -!-k 2 + j-k 3 -

••••

(2.5-20)

To expand an arbitrary simple branch logw in powers of k: =w- w0 , where w0 is an arbitrary point of Ta, we note that

Hence by Theorem 2.5c logw=logw 0 +s, where sis a member of the set {log(l+k/w0)}. In addition to wETa we now assume that lw-w0 l