Complex Analysis

ISBN-10: 0691113858
ISBN-13: 9780691113852
Edition: 2003
List price: $99.95 Buy it from $95.31
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Description: With this second volume in the series, we enter the world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The book includes coverage of: the main properties of holomorphic functions; and an  More...

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Book details

List price: $99.95
Copyright year: 2003
Publisher: Princeton University Press
Publication date: 4/27/2003
Binding: Hardcover
Pages: 400
Size: 6.50" wide x 9.50" long x 1.25" tall
Weight: 2.046
Language: English

With this second volume in the series, we enter the world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The book includes coverage of: the main properties of holomorphic functions; and an introduction to elliptic functions.

Foreword
Introduction
Preliminaries to Complex Analysis
Complex numbers and the complex plane
Basic properties
Convergence
Sets in the complex plane
Functions on the complex plane
Continuous functions
Holomorphic functions
Power series
Integration along curves
Exercises
Cauchy's Theorem and Its Applications
Goursat's theorem
Local existence of primitives and Cauchy's theorem in a disc
Evaluation of some integrals
Cauchy's integral formulas
Further applications
Morera's theorem
Sequences of holomorphic functions
Holomorphic functions defined in terms of integrals
Schwarz reflection principle
Runge's approximation theorem
Exercises
Problems
Meromorphic Functions and the Logarithm
Zeros and poles
The residue formula
Examples
Singularities and meromorphic functions
The argument principle and applications
Homotopies and simply connected domains
The complex logarithm
Fourier series and harmonic functions
Exercises
Problems
The Fourier Transform
The class F
Action of the Fourier transform on F
Paley-Wiener theorem
Exercises
Problems
Entire Functions
Jensen's formula
Functions of finite order
Infinite products
Generalities
Example: the product formula for the sine function
Weierstrass infinite products
Hadamard's factorization theorem
Exercises
Problems
The Gamma and Zeta Functions
The gamma function
Analytic continuation
Further properties of T
The zeta function
Functional equation and analytic continuation
Exercises
Problems
The Zeta Function and Prime Number Theorem
Zeros of the zeta function
Estimates for 1/s(s)
Reduction to the functions v and v1
Proof of the asymptotics for v1
Note on interchanging double sums
Exercises
Problems
Conformal Mappings
Conformal equivalence and examples
The disc and upper half-plane
Further examples
The Dirichlet problem in a strip
The Schwarz lemma; automorphisms of the disc and upper half-plane
Automorphisms of the disc
Automorphisms of the upper half-plane
The Riemann mapping theorem
Necessary conditions and statement of the theorem
Montel's theorem
Proof of the Riemann mapping theorem
Conformal mappings onto polygons
Some examples
The Schwarz-Christoffel integral
Boundary behavior
The mapping formula
Return to elliptic integrals
Exercises
Problems
An Introduction to Elliptic Functions
Elliptic functions
Liouville's theorems
The Weierstrass p function
The modular character of elliptic functions and Eisenstein series
Eisenstein series
Eisenstein series and divisor functions
Exercises
Problems
Applications of Theta Functions
Product formula for the Jacobi theta function
Further transformation laws
Generating functions
The theorems about sums of squares
The two-squares theorem
The four-squares theorem
Exercises
Problems
Asymptotics
Bessel functions
Laplace's method; Stirling's formula
The Airy function
The partition function
Problems
Simple Connectivity and Jord

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