Complex Analysis

by Gamelin, Theodore W.

• ISBN13: 9780387950693
• ISBN10: 0387950699
• Format: Paperback
• Copyright: 2001-05-01
• Publisher: Springer Verlag

Summary

The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis.

Table of Contents

Preface	vii
Introduction	xvii
FIRST PART
The Complex Plane and Elementary Functions	1	(32)
Complex Numbers	1	(4)
Polar Representation	5	(6)
Stereographic Projection	11	(4)
The Square and Square Root Functions	15	(4)
The Exponential Function	19	(2)
The Logarithm Function	21	(3)
Power Functions and Phase Factors	24	(5)
Trigonometric and Hyperbolic Functions	29	(4)
Analytic Functions	33	(37)
Review of Basic Analysis	33	(9)
Analytic Functions	42	(4)
The Cauchy-Riemann Equations	46	(5)
Inverse Mappings and the Jacobian	51	(3)
Harmonic Functions	54	(4)
Conformal Mappings	58	(5)
Fractional Linear Transformations	63	(7)
Line Integrals and Harmonic Functions	70	(32)
Line Integrals and Green's Theorem	70	(6)
Independence of Path	76	(7)
Harmonic Conjugates	83	(2)
The Mean Value Property	85	(2)
The Maximum Principle	87	(3)
Applications to Fluid Dynamics	90	(7)
Other Applications to Physics	97	(5)
Complex Integration and Analyticity	102	(28)
Complex Line Integrals	102	(5)
Fundamental Theorem of Calculus for Analytic Functions	107	(3)
Cauchy's Theorem	110	(3)
The Cauchy Integral Formula	113	(4)
Liouville's Theorem	117	(2)
Morera's Theorem	119	(4)
Goursat's Theorem	123	(1)
Complex Notation and Pompeiu's Formula	124	(6)
Power Series	130	(35)
Infinite Series	130	(3)
Sequences and Series of Functions	133	(5)
Power Series	138	(6)
Power Series Expansion of an Analytic Function	144	(5)
Power Series Expansion at Infinity	149	(2)
Manipulation of Power Series	151	(3)
The Zeros of an Analytic Function	154	(4)
Analytic Continuation	158	(7)
Laurent Series and Isolated Singularities	165	(30)
The Laurent Decomposition	165	(6)
Isolated Singularities of an Analytic Function	171	(7)
Isolated Singularity at Infinity	178	(1)
Partial Fractions Decomposition	179	(3)
Periodic Functions	182	(4)
Fourier Series	186	(9)
The Residue Calculus	195	(29)
The Residue Theorem	195	(4)
Integrals Featuring Rational Functions	199	(4)
Integrals of Trigonometric Functions	203	(3)
Integrands with Branch Points	206	(3)
Fractional Residues	209	(3)
Principal Values	212	(4)
Jordan's Lemma	216	(3)
Exterior Domains	219	(5)
SECOND PART
The Logarithmic Integral	224	(36)
The Argument Principle	224	(5)
Rouche's Theorem	229	(2)
Hurwitz's Theorem	231	(1)
Open Mapping and Inverse Function Theorems	232	(4)
Critical Points	236	(6)
Winding Numbers	242	(4)
The Jump Theorem for Cauchy Integrals	246	(6)
Simply Connected Domains	252	(8)
The Schwarz Lemma and Hyperbolic Geometry	260	(14)
The Schwarz Lemma	260	(3)
Conformal Self-Maps of the Unit Disk	263	(3)
Hyperbolic Geometry	266	(8)
Harmonic Functions and the Reflection Principle	274	(15)
The Poisson Integral Formula	274	(6)
Characterization of Harmonic Functions	280	(2)
The Schwarz Reflection Principle	282	(7)
Conformal Mapping	289	(26)
Mappings to the Unit Disk and Upper Half-Plane	289	(5)
The Riemann Mapping Theorem	294	(2)
The Schwarz-Christoffel Formula	296	(8)
Return to Fluid Dynamics	304	(2)
Compactness of Families of Functions	306	(5)
Proof of the Riemann Mapping Theorem	311	(4)
THIRD PART
Compact Families of Meromorphic Functions	315	(27)
Marty's Theorem	315	(5)
Theorems of Montel and Picard	320	(4)
Julia Sets	324	(9)
Connectedness of Julia Sets	333	(5)
The Mandelbrot Set	338	(4)
Approximation Theorems	342	(19)
Runge's Theorem	342	(6)
The Mittag-Leffler Theorem	348	(4)
Infinite Products	352	(6)
The Weierstrass Product Theorem	358	(3)
Some Special Functions	361	(29)
The Gamma Functions	361	(4)
Laplace Transforms	365	(5)
The Zeta Function	370	(6)
Dirichlet Series	376	(6)
The Prime Number Theorem	382	(8)
The Dirichlet Problem	390	(28)
Green's Formulae	390	(4)
Subharmonic Functions	394	(4)
Compactness of Families of Harmonic Functions	398	(4)
The Perron Method	402	(4)
The Riemann Mapping Theorem Revisited	406	(1)
Green's Function for Domains with Analytic Boundary	407	(6)
Green's Function for General Domains	413	(5)
Riemann Surfaces	418	(29)
Abstract Riemann Surfaces	418	(8)
Harmonic Functions on a Riemann Surface	426	(3)
Green's Function of a Surface	429	(5)
Symmetry of Green's Function	434	(2)
Bipolar Green's Function	436	(2)
The Uniformization Theorem	438	(3)
Covering Surfaces	441	(6)
Hints and Solutions for Selected Exercises	447	(22)
References	469	(2)
List of Symbols	471	(2)
Index	473