ISBN 9789380108957,Classical Complex Analysis

Classical Complex Analysis

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Jones And Bartlett India

Publication Year 2011

ISBN 9789380108957

ISBN-10 9380108958


Number of Pages 430 Pages
Language (English)

Mathematics & science

Ideal for an introductory course in complex analysis at the advanced undergraduate or graduate level, this text has been developed over decades of teaching with an enthusiastic student reception. The first half of the book focuses in the core material. An early chapter on power series gives the reader concrete examples of analytic functions and a review of calculus. Mödius transformations are presented with emphasis on the geometric aspect and the Cauchy theorem is covered in the classical manner. The remaining chapters provide an elegant and solid overview of special topics such as Entire and Meromorphic Function, Analytic Continuation, Normal families, Conformal Mapping, and Harmonic Functions.

Key Features: 

Power Series approach gives students a chance to review calculus and discover complex analysis is a natural extension of calculus
Unique coverage of Phragmén-Lindelöf Theorem (§10.4), the Runge Approximation Theorem (§6.6), Conformal Mappings of Multiply-Connected Regions (§7.9), and Extensions of Theorems of MittagLeffler and Weierstrass (§7.3)
Also unique is the comparisons of Complex Analysis and Real analysis in Chapter 4,5, and 9
New, elementary proof of the Picard Theorems in Chapter 12
Generous exercise sets are a treasure trove of interesting and challenging problems
Over 60 illustrations help the reader visualize complex relationships

About the Author
Liang-shin Hahn-University of New Mexico, Albuquerque, New Mexico

Bernard Epstein-Emeritus, University of New Mexico, New Mexico

Table of Contents

Chapter 1: Complex Numbers: The Complex Field
Geometric Representation
The Riemann Sphere
Chapter 2: Power Series: Sequences
Some Terminology about the Topology of the Complex Plane
Uniform Convergence
Geometric Series
Circle of Convergence
Differentiation of Power Series
Some Elementary Functions
The Maximum and Minimum Modulus Principles
Chapter 3: Analytic Functions: The Cauchy-Riemann Differential Equations
Harmonic Functions
Geometric Significance of the Derivative
Möbius Transformations
Chapter 4: The Cauchy Theorem: Some Remarks on Curves
Line Integrals
The Cauchy-Goursat Theorem
The Cauchy Integral Formula
The Morera Theorem
Chapter 5: Singularities and Residues: Laurent Series
Isolated Singularities
Rational Functions
Evaluation of Real Integrals
The Argument Principle
Chapter 6: The Maximum Modulus Principle: The Maximum and Minimum Modulus Principles, Revisited
The Schwarz Lemma
The Three-Circle Theorem
A Maximum Theorem for an Unbounded Region
The Three-Line Theorem
The Phragmén-Lindelöf Theorems
Chapter 7: Entire and Meromorphic Functions: The Mittag-Leffler Theorem
A Theorem of Weierstrass
Extensions of Theorems of Mittag-Leffler and Weierstrass
Infinite Products
Blaschke Products
The Factorization of Entire Functions
The Jensen Formula
Entire Functions of Finite Order
The Runge Approximation
Chapter 8: Analytic Continuation: The Power-Series Method
Natural Boundaries
Multiple-Valued Functions
Riemann Surfaces
The Schwarz Symmetry Principle
The Monodromy Theorem
Law of Permanence
The Euler Gamma Function
Chapter 9: Normal Families: The Montel Selection Theorem
Univalent Functions
Chapter 10: Conformal Mapping: Classification of Regions
The Riemann Mapping Theorem Examples
Conformal Mappings of Multiply Connected Regions
Chapter 11: Harmonic Functions: Harmonic Conjugate
The Maximum and Minimum Principles for Harmonic Functions
The Poisson Integral Formula
The Dirichlet Problem
The Harnack Theorem
Green Functions
Chapter 12: The Picard Theorems: The Bloch Theorem
The Schottky Theorem
The Picard Theorems