Introductory Complex Analysis

545 Pages·1984·12.77 MB·English

by  Richard A. Silverman

INTRODUCTORY COMPLEX
ANALYSIS
RICHARD A. SILVERMAN
Based, in part, on material by A. I. Markushevich
DOVER PUBLICATIONS, INC.
NEW YORK
Copyright © 1972 by Dover Publications, Inc.
Copyright © 1967 by Richard A. Silverman.
All rights reserved.
This Dover edition, first published in 1972, is an unabridged and corrected republication of the
work originally published by Prentice-Hall, Inc., in 1967.
International Standard Book Number: 0-486-64686-6
Library of Congress Catalog Card Number: 72-84339
Manufactured in the United States by Courier Corporation 64686607
www.doverpublications.com
PREFACE
During the years 1963-1965, a substantial sector of my activities as editor of
Prentice-Hall’s series “Selected Russian Publications in the Mathematical
Sciences” was devoted to the translation and revision of Professor A. I.
Markushevich’s magnum opus “The Theory of Analytic Functions,” Moscow
(1950). The English edition takes the form of three volumes, collectively entitled
“Theory of Functions of a Complex Variable,” the first two published in 1965
and the third in 1967. Out of my strong interaction with the Russian original,
there emerged an English edition markedly different from its forebear, a fact
duly recognized by various reviewers.
Because of the grand scale of the master three-volume course, which runs to well
over a thousand pages, I decided to distill it down into a shorter one-volume
course, matching the usual one-year graduate course on complex analysis given
in the United States. The result is the present volume, called “Introductory
Complex Analysis” (hereinafter abbreviated as ICA), whose debt to the master
course (and to a brief course on complex analysis written by Markushevich
himself) is acknowledged by the credit line “based, in part, on material by A. I.
Markushevich” appearing on the title page.
The fact that ICA is in part the offspring of the master course has a pleasant and
useful side effect: An expanded discussion of material appearing in ICA can
often be found in the master course in precisely the same language. The
pedagogical assets of this “fringe benefit” should be obvious. Thus references to
Markushevich’s “Theory of Functions of a Complex Variable” (Prentice Hall,
Inc., Englewood Cliffs, New Jersey) are scattered throughout ICA, with “Volume
I” abbreviated as Ml (M for “Markushevich”), “Volume II” as M2 and “Volume
III” as M3. This feature also allows the student to pursue various advanced
topics whose beginnings are to be found in ICA, without any problem of
adjusting to new terminology and notation. References to Ml, M2 and M3 are
particularly prevalent in the problems. Thus the solutions of a number of
problems can in effect be “looked up” in the master course, but it is hoped that
the student will exercise restraint and try hard to solve such problems on his
own.
Having disposed of the parentage of ICA, let us now examine the traits of the
child. If complex analysis is regarded as differential and integral calculus in the
complex domain, then Chapters 1 and 2 are essentially “precalculus.”
Differentiation appears in Chapter 3, together with the basic notion of an analytic
function and a foretaste of conformal mapping. Chapters 4, 5 and 6 treat the
elementary functions in considerably more detail than in most books on this
level and, in my opinion, more in keeping with the needs of applied scientists.
Integration in the complex domain dominates Chapter 7 (on complex integrals
and Cauchy’s integral theorem) and Chapter 8 (on Cauchy’s integral formula and
its implications). Chapter 9 is devoted to the subject of complex series, as a
prelude to Chapter 10 on the key topic of power series. Chapter 11 on Laurent
series and Chapter 12 on the residue theorem and its implications (and manifold
applications) lie deep in the home territory of complex analysis. Chapter 13 is
devoted to harmonic functions, a subject too often slighted in first courses on
complex analysis. A substantial chunk of the theory and technique of infinite
product and partial fraction expansions is presented in Chapter 14.
The subject of conformal mapping is taken up again in Chapter 15, with
considerable space devoted to the Schwarz-Christoffel transformation which
figures so prominently in the applications. Finally, Chapter 16 deals with
analytic continuation and related topics, notably the symmetry principle and
Riemann surfaces.
The reference system is self-explanatory, with the prefix of each equation and
figure number referring to the chapter in which the equation or figure occurs. A
problem number unaccompanied by a section number always refers to a problem
at the end of the section where the reference is made. Otherwise the problem
number is preceded by the section number giving its proper address.
As in the case of all three volumes of the master course, ICA has benefited from
a careful reading by Dr. T. J. Rivlin, who made many helpful suggestions.
R. A. S.
CONTENTS
CHAPTER 1
COMPLEX NUMBERS, FUNCTIONS AND SEQUENCES
1. Introductory Remarks
2. Complex Numbers and Their Geometric Representation
3. Complex Algebra
4. Powers and Roots of Complex Numbers
5. Set Theory. Complex Functions
6. Complex Sequences
7. Proper and Improper Complex Numbers
8. Infinity and Stereographic Projection
CHAPTER 2
LIMITS AND CONTINUITY
9. More Set Theory. The Heine-Borel Theorem
10. The Limit of a Function at a Point
11. Continuous Functions
12. Curves and Domains
CHAPTER 3
DIFFERENTIATION. ANALYTIC FUNCTIONS
13. Derivatives. Rules for Differentiating Complex Functions
14. The Cauchy-Riemann Equations. Analytic Functions
15. Geometric Interpretation of Arg f′(z) and |f′(z)|. Conformal
Mapping
16. The Mapping
17. Conformal Mapping of the Extended Plane
CHAPTER 4
POLYNOMIALS AND RATIONAL FUNCTIONS
18. Polynomials. The Mapping w = P (z)
n
19. The Mapping w = (z – a)n
20. The Mapping w =
21. Rational Functions
22. The Mapping
CHAPTER 5
MÖBIUS TRANSFORMATIONS
23. The Group Property of Möbius Transformations
24. The Circle-Preserving Property of Möbius Transformations
25. Fixed Points of a Möbius Transformation. Invariance of the Cross
Ratio
26. Mapping of a Circle onto a Circle
27. Symmetry Transformations
28. Examples
CHAPTER 6
EXPONENTIALS AND LOGARITHMS
29. The Exponential
30. The Mapping w = ez
31. Some Functions Related to the Exponential
32. The Logarithm
33. The Function zα. Exponentials and Logarithms to an Arbitrary
Base
CHAPTER 7
COMPLEX INTEGRALS. CAUCHY’S INTEGRAL THEOREM
34. Rectifiable Curves. Complex Integrals
35. The Case of Smooth Curves
36. Cauchy’s Integral Theorem. The Key Lemma
37. Proof of Cauchy’s Integral Theorem
38. Application to the Evaluation of Definite Integrals
39. Cauchy’s Integral Theorem for a System of Contours
CHAPTER 8
CAUCHY’S INTEGRAL FORMULA AND ITS IMPLICATIONS
40. Indefinite Integrals
41. Cauchy’s Integral Formula
42. Morera’s Theorem. Cauchy’s Inequalities
CHAPTER 9
COMPLEX SERIES. UNIFORM CONVERGENCE
43. Complex Series
44. Uniformly Convergent Series and Sequences
45. Series and Sequences of Analytic Functions
CHAPTER 10
POWER SERIES
46. The Cauchy-Hadamard Theorem
47. Taylor Series. The Uniqueness Theorem for Power Series
48. Expansion of an Analytic Function in a Power Series
49. Liouville’s Theorem. The Uniqueness Theorem for Analytic
Functions
50. A-Points and Zeros
51. Weierstrass’ Double Series Theorem
52. Substitution of One Power Series into Another
53. Division of Power Series
CHAPTER 11
LAURENT SERIES. SINGULAR POINTS
54. Laurent Series
55. Laurent’s Theorem
56. Poles and Essential Singular Points
57. Behavior at an Essential Singular Point. Picard’s Theorem
58. Behavior at Infinity
CHAPTER 12
THE RESIDUE THEOREM AND ITS IMPLICATIONS
59. The Residue Theorem. Residues at Infinity
60. Jordan’s Lemma. Evaluation of Definite Integrals
61. The Argument Principle. The Theorems of Rouché and Hurwitz
62. Local Behavior of Analytic Mappings. The Maximum Modulus
Principle and Schwarz’s Lemma
CHAPTER 13
HARMONIC FUNCTIONS
63. Laplace’s Equation. Conjugate Harmonic Functions
64. Poisson’s Integral. Schwarz’s Formula
65. The Dirichlet Problem
CHAPTER 14
INFINITE PRODUCT AND PARTIAL FRACTION EXPANSIONS
66. Preliminary Results. Infinite Products
67. Weierstrass’ Theorem
68. Mittag-Leffier’s Theorem
69. The Gamma Function
70. Cauchy’s Theorem on Partial Fraction Expansions
CHAPTER 15
CONFORMAL MAPPING
71. General Principles of Conformai Mapping
72. Mapping of the Upper Half-Plane onto a Rectangle
73. The Schwarz-Christoffel Transformation
CHAPTER 16
ANALYTIC CONTINUATION
74. Elements and Chains
75. General and Complete Analytic Functions
76. Analytic Continuation Across an Arc
77. The Symmetry Principle
78. More on Singular Points
79. Riemann Surfaces
BIBLIOGRAPHY
INDEX