Complex Analysis

ISBN-10: 1441972870

ISBN-13: 9781441972873

Edition: 3rd 2010

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Book details

List price: $49.99
Edition: 3rd
Copyright year: 2010
Publisher: Springer London, Limited
Publication date: 8/6/2010
Binding: Hardcover
Pages: 328
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 1.606
Language: English

Preface to the Third Edition
Preface to the Second Edition
The Complex Numbers
Introduction
The Field of Complex Numbers
The Complex Plane
The Solution of the Cubic Equation
Topological Aspects of the Complex Plane
Stereographic Projection; The Point at Infinity
Exercises
Functions of the Complex Variable z
Introduction
Analytic Polynomials
Power Series
Differentiability and Uniqueness of Power Series
Exercises
Analytic Functions
Analyticity and the Cauchy-Riemann Equations
The Functions e<sup>z</sup>, sin z, cos z
Exercises
Line Integrals and Entire Functions
Introduction
Properties of the Line Integral
The Closed Curve Theorem for Entire Functions
Exercises
Properties of Entire Functions
The Cauchy Integral Formula and Taylor Expansion for Entire Functions
Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem
Newton's Method and Its Application to Polynomial Equations
Exercises
Properties of Analytic Functions
Introduction
The Power Series Representation for Functions Analytic in a Disc
Analytic in an Arbitrary Open Set
The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points
Exercises
Further Properties of Analytic Functions
The Open Mapping Theorem; Schwarz' Lemma
The Converse of Cauchy's Theorem: Morera's Theorem; The Schwarz Reflection Principle and Analytic Arcs
Exercises
Simply Connected Domains
The General Cauchy Closed Curve Theorem
The Analytic Function log z
Exercises
Isolated Singularities of an Analytic Function
Classification of Isolated Singularities; Riemann's Principle and the Casorati-Weierstrass Theorem
Laurent Expansions
Exercises
The Residue Theorem
Winding Numbers and the Cauchy Residue Theorem
Applications of the Residue Theorem
Exercises
Applications of the Residue Theorem to the Evaluation of Integrals and Sums
Introduction
Evaluation of Definite Integrals by Contour Integral Techniques
Application of Contour Integral Methods to Evaluation and Estimation of Sums
Exercises
Further Contour Integral Techniques
Shifting the Contour of Integration
An Entire Function Bounded in Every Direction
Exercises
Introduction to Conformal Mapping
Conformal Equivalence
Special Mappings
Schwarz-Christoffel Transformations
Exercises
The Riemann Mapping Theorem
Conformal Mapping and Hydrodynamics
The Riemann Mapping Theorem
Mapping Properties of Analytic Functions on Closed Domains
Exercises
Maximum-Modulus Theorems for Unbounded Domains
A General Maximum-Modulus Theorem
The Phragm�n-Lindel�f Theorem
Exercises
Harmonic Functions
Poisson Formulae and the Dirichlet Problem
Liouville Theorems for Re f; Zeroes of Entire Functions of Finite Order
Exercises
Different Forms of Analytic Functions
Introduction
Infinite Products
Analytic Functions Defined by Definite Integrals
Analytic Functions Defined by Dirichlet Series
Exercises
Analytic Continuation; The Gamma and Zeta Functions
Introduction
Power Series
Analytic Continuation of Dirichlet Series
The Gamma and Zeta Functions
Exercises
Applications to Other Areas of Mathematics
Introduction
A Variation Problem
The Fourier Uniqueness Theorem
An Infinite System of Equations
Applications to Number Theory
An Analytic Proof of The Prime Number Theorem
Exercises
Answers
References
Appendices
Index
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