Francis J. Flanigan - Complex Variables

Copyright 1972 by Francis J. Flanigan.

All rights reserved.

This Dover edition, first published in 1983, is an unabridged and corrected republication of the work originally published in 1972 by Allyn and Bacon, Inc., Boston.

International Standard Book Number: 0-486-61388-7

Library of Congress Cataloging in Publication Data

Flanigan, Francis J.

Complex variables.

Reprint. Originally published: Boston : Allyn and Bacon, 1972.

Bibliography: p.

Includes index.

1. Functions of complex variables. 2. Harmonic functions. 3. Analytic functions. I. Title.

QA331.F62 1983 515.9 82-17732

Manufactured in the United States by Courier Corporation

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This book originated at the University of Pennsylvania in the Spring of 1968 as a set of lecture notes addressed to undergraduate math and science majors. It is intended for an introductory one-semester or quarter-and-a-half course with minimal prerequisites; it is neither a reference nor a handbook.

We approach complex analysis via real plane calculus, Greens Theorem and the Greens identities, determination by boundary values, harmonic functions, and steady-state temperatures. The conscientious student will compute many line integrals and directional derivatives as he works through the early chapters. The beautiful Cauchy theory for complex analytic functions is preceded by its harmonic counterpart.

The young student is likely to assume that an arbitrary differentiable function defined somewhere enjoys the remarkable properties of complex analytic functions. From the beginning we stress that

(i) the analytic f ( z ) = u ( x, y ) + i ( x, y ) is much better behaved than the arbitrary function encountered in freshman calculus or the first , course;

(ii) this is because u ( x, y ), ( x, y ) satisfy certain basic partial differential equations;

(iii) one can obtain much useful information about solutions of such equations without actually solving them.

In developing integration theory, we emphasize the analytic aspects at the expense of the topological or combinatorial. Thus, a complex function f ( z ) is defined to be analytic at a point if it is continuously complex differentiable in a neighborhood of that point. The Cauchy Integral Theorem is thereby an easy consequence of Greens Theorem and the Cauchy-Riemann equations. Goursats remarkable deepening of the Integral Theorem is discussed, but is not proved. On the other hand, we make much of the standard techniques of representing a function as an integral and then bounding that integral (the ML- inequality) or differentiating under the integral sign. The integral representation formulas (Greens Third Identity, the Poisson Integral, the Cauchy Formula) are the true heroes of these chapters.

The second half of the book () is motivated by two concerns: the integration of functions which possess singularities, and the behavior of analytic mappings w = f ( z ). Power series are developed first; thence flows the basic factorization from this comes all the rest. The book concludes with a discussion (no proof) of the Riemann Mapping Theorem.

The author recalls with pleasure many, many hours spent discussing complex analysis with Professor Jerry Kazdan at the University of Pennsylvania and nearby spots. Particular thanks are due Professor Kazdan and Professor Bob Hall for reading the manuscript and making many usable suggestions. Finally, the author is happy to record his gratitude to the staff of Allyn and Bacon for encouragement and prompt technical assistance over the months and miles.

F RANCIS J. F LANIGAN

Heres what well do in the first few chapters:

1. We examine the geography of the xy -plane. Some of this will be familiar from basic calculus (for example, distance between points), some may be new to you (for example, the important notion of domain). We must also consider curves in the plane.

2. We consider real-valued functions u ( x, y ) defined in the plane. We will examine the derivatives (partial derivatives, gradient, directional derivatives) and integrals (line integrals, double integrals) of these functions. Most of (1) above will be necessary for (2). All this happens in this chapter.

3. We next focus attention on a particular kind of real-valued function u ( x, y ), the so-called harmonic function (). These are very interesting in their own right, have beautiful physical interpretations, and point the way to complex analytic functions.

4. At last () we consider points ( x, y ) of the plane as complex numbers x + iy and we begin our study of complex-valued functions of a complex variable. This study occupies the rest of the book.

One disadvantage of this approach is the fact that complex numbers and complex analytic functions (our chief topic) do not appear until the third chapter. Admittedly, it would be possible to move directly from step (1) to step (4), making only brief reference to real-valued functions. On the other hand, the present route affords us

(i) a good look at some very worthwhile two-variable real calculus, and

(ii) an insight into the reasons behind some of the magical properties of complex analytic functions, which (as we will see) flow from (a) the natural properties of real-valued harmonic functions u ( x, y ) and (b) the fact that we can multiply and divide points in the plane. In the present approach the influences (a) and (b) will be considered separately before being combined.

One effect we hope for: You will learn to appreciate the difference between a complex analytic function (roughly, a complex-valued function f ( z ) having a complex derivative f ( z )) and the real functions y = f ( x ) which you differentiated in calculus. Dont be deceived by the similarity of the notations f ( z ), f ( x ). The complex analytic function f ( z ) turns out to be much more special, enjoying many beautiful properties not shared by the run-of-the-mill function from ordinary real calculus. The reason (see (a) above) is that f ( x ) is merely f ( x ), whereas the complex analytic function f ( z ) can be written as

where z = x + iy and u ( x, y ), ( x, y ) are each real-valued harmonic functions related to each other in a very strong way: the CauchyRiemann equations

In summary, the deceptively simple hypothesis that

forces a great deal of structure on f ( z ); moreover, this structure mirrors the structure of the harmonic u ( x, y ) and ( x, y ), functions of two real variables.