Complex Analysis

ISBN-10: 0070006571

ISBN-13: 9780070006577

Edition: 3rd 1979 (Revised)

Authors: Lars V. Ahlfors
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Book details

List price: $312.00
Edition: 3rd
Copyright year: 1979
Publisher: McGraw-Hill Education
Publication date: 1/1/1979
Binding: Hardcover
Pages: 352
Size: 6.50" wide x 9.50" long x 1.00" tall
Weight: 1.144

Complex Numbers
The Algebra of Complex Numbers
Arithmetic Operations
Square Roots
Justification
Conjugation, Absolute Value
Inequalities
The Geometric Representation of Complex Numbers
Geometric Addition and Multiplication
The Binomial Equation
Analytic Geometry
The Spherical Representation
Complex Functions
Introduction to the Concept of Analytic Function
Limits and Continuity
Analytic Functions
Polynomials
Rational Functions
Elementary Theory of Power Series
Sequences
Series
Uniform Coverages
Power Series
Abel's Limit Theorem
The Exponential and Trigonometric Functions
The Exponential
The Trigonometric Functions
The Periodicity
The Logarithm
Analytic Functions as Mappings
Elementary Point Set Topology
Sets and Elements
Metric Spaces
Connectedness
Compactness
Continuous Functions
Topological Spaces
Conformality
Arcs and Closed Curves
Analytic Functions in Regions
Conformal Mapping
Length and Area
Linear Transformations
The Linear Group
The Cross Ratio
Symmetry
Oriented Circles
Families of Circles
Elementary Conformal Mappings
The Use of Level Curves
A Survey of Elementary Mappings
Elementary Riemann Surfaces
Complex Integration
Fundamental Theorems
Line Integrals
Rectifiable Arcs
Line Integrals as Functions of Arcs
Cauchy's Theorem for a Rectangle
Cauchy's Theorem in a Disk
Cauchy's Integral Formula
The Index of a Point with Respect to a Closed Curve
The Integral Formula
Higher Derivatives
Local Properties of Analytical Functions
Removable Singularities. Taylor's Theorem
Zeros and Poles
The Local Mapping
The Maximum Principle
The General Form of Cauchy's Theorem
Chains and Cycles
Simple Connectivity
Homology
The General Statement of Cauchy's Theorem
Proof of Cauchy's Theorem
Locally Exact Differentials
Multiply Connected Regions
The Calculus of Residues
The Residue Theorem
The Argument Principle
Evaluation of Definite Integrals
Harmonic Functions
Definition and Basic Properties
The Mean-value Property
Poisson's Formula
Schwarz's Theorem
The Reflection Principle
Series and Product Developments
Power Series Expansions
Wierstrass's Theorem
The Taylor Series
The Laurent Series
Partial Fractions and Factorization
Partial Fractions
Infinite Products
Canonical Products
The Gamma Function
Stirling's Formula
Entire Functions
Jensen's Formula
Hadamard's Theorem
The Riemann Zeta Function
The Product Development
Extension of (s) to the Whole Plane
The Functional Equation
The Zeros of the Zeta Function
Normal Families
Equicontinuity
Normality and Compactness
Arzela's Theorem
Families of Analytic Functions
The Classical Definition
Conformal Mapping, Dirichlet's Problem
The Riemann Mapping Theorem
Statement and Proof
Boundary Behavior
Use of the Reflection Principle
Analytic Arcs
Conformal Mapping of Polygons
The Behavior at an Angle
The Schwarz-Christoffel Formula
Mapping on a Rectangle
The Triangle Functions of Schwarz
A Closer Look at Harmonic Functions
Functions with Mean-value Property
Harnack's Principle
The Dirichlet Problem
Subharmonic Functions
Solution of Dirichlet's Problem
Canonical Mappings of Multiply Connected Regions
Harmonic Measu
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