A Second Course in Complex Analysis

Copyright

Copyright © 1967, 1995 by William A. Veech

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2008, is an unabridged and slightly corrected republication of the work originally published by W. A. Benjamin, Inc., New York, in 1967.

Library of Congress Cataloging-in-Publication Data

Veech, William A.

A second course in complex analysis / William A. Veech. – Dover ed. p. cm.

Originally published: New York : W. A. Benjamin, 1967.

Includes bibliographical references and index.

eISBN-13: 978-0-486-15193-9

   1. Functions of complex variables. I. Title.

QA331.7.V44 2008

515′.9–dc22

2007034715

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

Preface

It would be possible to guess with a fair degree of accuracy the contents of a book entitled A First Course in Complex Analysis. Not so however with one entitled A. Second Course in Complex Analysis, and this is one of the pleasant aspects of the subject. With a working knowledge of the elements the curious student may proceed in a number of directions, any one of which will be richly rewarding. In the following we have chosen one of these directions.

This book is intended for the student, graduate or undergraduate, who (a) has had previous experience with complex analysis, (b) is well grounded in (plane) point-set topology, and (c) has sufficient background to understand not only a mathematical proof but also the development of a mathematical theory.

Our goal has been to present an integrated theory, and the tone, if not the pace, of the presentation is that of a modern research paper. It appears in retrospect that our goal has been more successfully approached with the closely related Chapters 1–4 than it has with Chapters 5 and 6 which are of a different character. For an undergraduate course we would, therefore, suggest the first four chapters be taken as a one-semester unit, while at a quarter system school the entire book could be covered in the second two terms of a three-term sequence. In a graduate course, where it should be possible to move rapidly through the first two chapters, much of the book can be covered in one semester.

It is my hope and intention that this book will be consulted as frequently by students engaged in independent study as by students actually enrolled in a second course in complex analysis. Indeed, the choice of topics and manner of presentation has been greatly influenced by my own experience with such independent study.

The topics herein being for the most part classical, we have not attempted to cite chapter and verse of the literature for specific results. The bibliography contains many books which have directly influenced this work, and foremost among these are the books of Carathéodory. The geometric flavor of Chapters 2–4 is one manifestation of the Carathéodory influence.

There are a number of people to whom thanks are due in connection with this work, and not the least of these are the many undergraduates at Princeton University who, over a period of three years, were subjected to the material at various stages of its development. I would also like to thank: Robert C. Gunning, who suggested such a text might be useful and who specifically suggested the inclusion of the Prime Number Theorem; the Mathematics Department of Princeton University, not only for its delightful atmosphere, but also for affording me the opportunity to teach the relevant course; the secretarial staff at Princeton, and particularly Patricia Clark, who did such excellent work in typing the manuscript; and finally my wife, Kay, who with patience and encouragement was behind me all the way.

WILLIAM A. VEECH

Berkeley, California

March 1967

Contents

PREFACE

CHAPTER 1   ANALYTIC CONTINUATION

1.  THE EXPONENTIAL FUNCTION AND THE LOGARITHM

2.  CONTINUATION SEQUENCES

3.  CONTINUATION ALONG AN ARC

4.  GERMS

5.  EXISTENCE OF CONTINUATIONS

6.  THE WINDING NUMBER

7.  THE ARGUMENT PRINCIPLE

8.  THE MONODROMY THEOREM

9.  COMPOSITION OF GERMS

10.  COMPOSITION OF CONTINUATIONS

11.  COVERING SURFACES

CHAPTER 2   GEOMETRIC CONSIDERATIONS

1.  COMPLEX PROJECTIVE SPACE

2.  LINEAR TRANSFORMATIONS

3.  FRACTIONAL LINEAR TRANSFORMATIONS

4.  PROPERTIES OF FRACTIONAL LINEAR TRANSFORMATIONS

5.  SYMMETRY

6.  SCHWARZ’S LEMMA

7.  NON-EUCLIDEAN GEOMETRY

8.  THE SCHWARZ REFLECTION PRINCIPLE

CHAPTER 3   THE MAPPING THEOREMS OF RIEMANN AND KOEBE

1.  ANALYTIC EQUIVALENCE

2.  LOCAL UNIFORM CONVERGENCE

3.  A THEOREM OP HUBWITZ

4.  IMPLICATIONS OP POINTWISE CONVERGENCE

5.  IMPLICATIONS OF CONVERGENCE ON A SUBSET

6.  APPROXIMATELY LINEAR FUNCTIONS–ANOTHER APPLICATION OF SCHWARZ’S LEMMA

7.  A UNIFORMIZATION THEOREM

8.  A CLOSER LOOK AT THE COVERING

9.  BOUNDARY BEHAVIOR

10.  LINDELÖF’S LEMMA

11.  FACTS FROM TOPOLOGY

12.  CONTINUITY AT THE BOUNDARY

13.  A THEOREM OF FEJER

CHAPTER 4   THE MODULAR FUNCTION

1.  EXCEPTIONAL VALUES

2.  THE MODULAR CONFIGURATION

3.  THE LANDAU RADIUS

4.  SCHOTTKY’S THEOREM

5.  NORMAL FAMILIES

6.  MONTEL’S THEOREM

7.  PICARD’S SECOND THEOREM

8.  THE KOEBE-FABER DISTORTION THEOREM

9.  BLOCH’S THEOREM

CHAPTER 5   THE HADAMARD PRODUCT THEOREM

1.  INFINITE PBODUCTS

2.  PBODUCTS OF FUNCTIONS

3.  THE WEIERSTRASS PRODUCT THEOREM

4.  FUNCTIONS OF FINITE ORDER

5.  EXPONENT OF CONVERGENCE

6.  CANONICAL PRODUCTS

7.  THE BOREL-CARATHÉODORY LEMMA–ANOTHER FORM OF SCHWARZ’S LEMMA

8.  A LEMMA OF H. CARTAN

9.  THE HADAMARD PRODUCT THEOREM

10.  THE GAMMA FUNCTION

11.  STANDARD FORMULAS

12.  THE INTEGRAL REPRESENTATION OF Γ(z)

CHAPTER 6   THE PRIME NUMBER THEOREM

1.  DIRICHLET SERIES

2.  NUMBER-THEORETIC FUNCTIONS

3.  STATEMENT OF THE PRIME NUMBER THEOREM

4.  THE RIEMANN ZETA FUNCTION

5.  ANALYTIC CONTINUATION OF ζ(s)

6.  RIEMANN’S FUNCTIONAL EQUATION

7.  THE ZEROS OF ζ(s) IN THE CRITICAL STRIP

8.  ζ(s) FOR Re s = l

9.  INTEGRAL REPRESENTATION OF DIRICHLET SERIES

10.  INTEGRAL-THEORETIC LEMMAS

11.  WEAK LIMITS

12.  A TAUBERIAN THEOREM

BIBLIOGRAPHY

INDEX

chapter 1

Analytic Continuation

1. The Exponential Function and the Logarithm

The exponential function is defined by the power series

which converges for each w. This function’s usefulness stems from its being a solution to the differential equation

as one sees through term-by-term differentiation.

To illustrate the importance of (1.2) we derive the relation

as follows. Set up the function h(w) = e−wew and differentiate using (1.2). The result is h−(w) = ′h(w) + h(w) = 0, meaning that h is constant. To calculate the constant set w = 0 and observe directly from (1.1) that h(0) = 1. Thus h(w) = 1, and (1.3) is obtained. Notice we have also proved ew ≠ 0 for any w.

By a similar application of (1.2) one sees for fixed z that, as a function of w, ez+we−w ≡ ez. In view of (1.3) the formula

holds for all w. Letting z vary, it also holds for all z.

Equations (1.3) and (1.4) are basic to what follows.

Lemma 1.1. If w = u + iv, where u and v are real numbers, then

Furthermore, |ew| = 1 if and only if u = 0; i.e., if and only if w = iv.

Proof. We have ew = eueiv, and therefore

We compute separately the factors on the right of (1.6). First |eiv|:

If n is an integer, then because v is real, where denotes complex conjugation. Substituting w = iv into (1.1) we see that eiv = e−iv. Thus

As for |eu|, we claim if u > 0 that

and

The first inequality is immediate from (1.1) while the second is a consequence of the first. Since e° = 1, (1.5) and the second statement of our lemma now follow from (1.6)–(1.9).

The logarithm will be introduced as the inverse function to the exponential function. Before we make this introduction, it is necessary (a) to determine the range of ew and (b) to devise a procedure for computing the inverse function. Toward (a) and (b) we establish the following statements :

(A) If z ≠ 0, then z = ew for some w.

(B) Let U be an open disk which does not contain zero. If z0, W0 are such that z0 U and ew⁰ = z0, it is possible to define

on U an analytic function φ having the properties

eφ(z) = z, z U

and

φ(z0) = W0.

We will observe that (B) implies (A), and then we will prove (B). Let z be a nonzero complex number. We assume for the moment that z does not lie on the negative real axis. There is, as the reader can show, an open disk U containing z and 1 but not zero. According to (B), z belongs to the range of ew because 1 does. Thus all nonzero complex numbers, except possibly those which are real and negative, belong to the range of the exponential. If z is real and negative, let U be an open disk which contains z but not 0. U contains a point z0 which is not real, and by what we have just seen, z0 belongs to the range. Invoking (B) once more, z must belong to the range of ew, and (A) is proved.

Now to prove (B). Suppose z0 = ew⁰, and let U be an open disk which contains z0 but does not contain zero. For each z U let γs be the line segment from z0 to z parametrized by z(t) = z0 + t(z – z0), 0 ≤ t ≤ 1. Define

Of course φ is well defined; one can compute directly (Problem 4) that φ is analytic with derivative 1/z. Alternatively, one can call upon the following fact from the elementary theory:

Theorem. Suppose f is analytic on a disk U, and let z0 be a fixed point of U. For each z U join z0 to z by a differentiable arc γs. The function

is analytic on U and satisfies F′(z) = f(z).

Continuing with (1.10) we set up an auxiliary function, h, defined by

Using the relation φ′(z) = l/z, we find

Thus h is constant on U. To determine which constant, set z = z0. Since φ(zo) = w0, we have h(z0) = (ew⁰/z0) = 1. It follows that eφ(z) = z for all z, and (B) is established.

Having satisfied ourselves that the equation ew = z, z ≠ 0, has at least one solution w for each z, we will now look for all solutions. The problem becomes simpler if we first find all solutions to

For if w1 and w2 are solutions to ew = z, then w2 – W1 is a solution to (1.11). Therefore knowing all solutions to (1.11) and one solution to ew = z, one knows all solutions to the latter.

According to Lemma 1.1 the solutions to (1.11) have the form w = iv for certain real numbers v. Let E be the set of real numbers

E = {t| eit = 1}.

E is closed because eit is continuous in t Furthermore E contains together with any pair of numbers the sum and difference of that pair.

We will show that E contains a positive number. By statement (A) there exists a number w such that ew = –1. Lemma 1.1 tells us that w = is for some real number s. Since of course s ≠ O, one of the numbers 2s or –2s is positive. Whichever it is, let t be that number. Since eis = –1 = e−is, we have eit = eiseis = 1, meaning t E.

Because E contains positive numbers, the number

T = inf{t E | t > 0}

is well defined. By definition T ≤ 0, and since E is closed, T E¹. Our object is to show T > 0.

If there is a δ > 0 such that the interval 0 < t < contains no elements of E, then by definition T ≥ δ. The existence of such a δ is a consequence of the uniqueness theorem for analytic functions (Problem 7); we include the following direct proof.

Lemma 1.2. There exists δ < 0 such that if 0 < |w| ≥ δ, then ew ≠ 1.

Proof. By (1.1) ew – 1 = wh(w), where h is the power series h(w) = . Now h is continuous, and h(0) = 1. Therefore, there exists δ > 0 such that if |w| ≤ δ, then |h(w)| ≥ . Clearly, if 0 < |w| ≥ δ, wh(w) ≠ 0, and a fortiori ew ≠ 1. The lemma is proved.

As we have remarked before Lemma 1.2, it follows that T > 0. If ζ is a complex number of absolute value 1, we shall denote by Eζ the set

Eζ = {s| eis = ζ}.

If ζ = 1, we continue to write E = E1. Notice if s is one solution to eis = ζ, then

Lemma 1.3. The set E is generated by T in the sense that

Proof. Naturally nT E for each integer n. If t is an arbitrary element of E, we will use the definition of T to show that t = nT for some n. To this end let n be an integer (positive, negative, or zero) such that nT ≤ t < (n + 1)T. The number t – nT belongs to E, and by our choice of n, 0 ≤ t – nT < T. Since T is the smallest positive element of E, we have t – nT = 0 or t = nT. This completes the proof.

Equations (1.12) and (1.13) combine to imply for each number ζ of absolute value 1 the existence of exactly one element of Eζ in the interval [0, T). What is the same, the function z(t) = eit maps [0, T) in a one-to-one fashion onto the unit circle.

Similarly, for any real number v0, z(t) = eit maps [v0 – T/2, v0 + T/2) in a one-to-one fashion onto the unit circle.

Denote by γ the unit circle with parametrization z(t) = eit, 0 ≤ t ≤ T. The length of γ is

Define π to be the real number T/2. By definition the unit circle has length 2π.

REMARE. Throughout this book we are assuming the Cauchy integral formula, at least for circles. If f is analytic inside and on a neighborhood of a circle γ, then

for points z inside γ. The number π which enters here is the same as the number π defined above. To see this let us recall here in outline the proof of the integral formula.

The first step is to prove for points z inside 7 that

which, since z ∉ γ, is the same as

The problem then is to compute

To do this one shows that φ is constant as a function of z (Problem 8). Then letting z = z0 be the center of γ, one explicitly evaluates φ(z0). For the latter, γ can be parametrized by ζ(t) = z0 + Re²πit, 0 ≤ t ≤ 1, where R is the radius. Since dζ = 2 πiRe²πit dt and ζ – z0 = Re²πit, we have

and the integral formula follows.

In order to define log z we shall introduce two functions, one old and one new.

For real numbers r > 0 define In r, the natural logarithm of r, by the integral

over the real interval from 1 to r. Recall from statement (B) above, eIn r = r for each r > 0.

If z ≠ 0, arg z, the argument of z, is defined to be any real number t such that eit = (z/|z|). The upshot of our earlier discussion is that for any real number v0 there is a unique value of arg z in the interval [v0 – π, V0 + π). Note that arg z = arg (z/|z|).

REMARK. If v0 is real, consider the function z(t) = eit on the open interval (v0 – π, v0 + π). The image of this interval contains all points of the unit circle save one, ei(v⁰ − π) = −eiv⁰. It is an elementary problem in point set topology to show that the mapping eit → t is continuous on the set {eit | v0 – π < t < v0 + π]. The same function is not continuous on the full unit circle.

Lemma 1.4. Let U be an open disk which does not contain zero. If z0 U, and if v0 is a prescribed value of arg z0, the function h defined by

h(z) = arg z, v0 – π ≤ arg z < v0 + π

is continuous on U.

Proof. We write arg z as the composition of arg w, |w| = 1, with w(z) = Since w(z) is continuous on U, the lemma will follow from the remark preceding it if we can show that w(U) ⊆ {eit | v0 – π < t < v0 + π}.

Suppose to the contrary that w(z) = (z/|z|) = –eiv⁰ for some z U. Since by definition (z0/|z0|) = eiv⁰ it follows that z = – (|z|/|z0|)z0 is a positive multiple of –z0. U is a disk, and therefore the line segment joining z0 to z belongs to U. But zero lies on this segment contradicting the assumption that 0 E U. Thus –eiv⁰ w(U). The lemma is proved.

To define log z, let U be a disk which does not contain zero. Fix z0 U and a value v0 of arg z0. Define log z as

Both In |z| and arg z are continuous on U, the latter by Lemma 1.4. Therefore log z is also continuous. Furthermore log z is an inverse function to the exponential:

We now establish the relationship between equations (1.10) and (1.14).

Theorem 1.1. Equation (1.14) defines an analytic function on U. In fact if w0 = log z0, the function φ defined by equation (1.10) is equal to log z.

Proof. Since elog z = z = eφ(z), there exists an integer-valued function n(z) such that

φ(z) = log z + 2πin(z), z U.

What is more, n(z) is continuous on U because both and log z are. A continuous integer-valued function on a disk reduces to a constant; the constant can be evaluated by setting z = z0. Here by definition φ(z0) = log z0 and n(z0) = 0. Thus n(z) = 0, and the theorem is proved.

REMARK. By virtue of Theorem 1.1 we have (d/dz) log z = (1/z).

For later applications we will need the Taylor series of log(l – z) about zero. To be precise, define for |z|

log(l – z) = In|l – z| + i arg(l – z), –π ≤ arg(l – z) < π.

Being the composition of two analytic functions log(l – z) is analytic. By the chain rule

Another function with the same derivative for |z| < 1 is the series − Therefore log(l − z) = − for some constant c. Setting z = 0, we find c = 0. Thus we have obtained

for that version of log(l – z) with log 1 = 0.

Problems

1.   Let f be analytic on a disk U with f(z) ≠ 0, z U. If z0 U, and if w0 is such that ew⁰ = f(z0), there exists on U an analytic function ψ such that ψ(z0) = w0 and eψ(z) = f(z). (HINT: Let w = f(z), dw = f′(z) dz, and proceed in analogy with (1.10).)

2.   Using the fact that (d/dz) log z = (1/z), prove that

3.   If α is complex, define (1 + z)α, |z| < 1, by

(a) If k is a nonnegative integer, then prove (dk/dzk)(l + z)α = α(α – 1) . . . (α – k + 1)(1 + z)α−k, where (1 + z)α−k, is defined by (1.16) with α = α – k.

(b) Define

and use Taylor’s theorem to prove

(c) If α = (p/q) for integers p and q, show using (1.16) that {(1 +z)α}q = (1 +z)p, the latter being defined in the usual sense.

4.   Let U be a disk of