9780387950693

Complex Analysis

by
  • ISBN13:

    9780387950693

  • ISBN10:

    0387950699

  • Format: Paperback
  • Copyright: 5/1/2001
  • Publisher: Springer Verlag

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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

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