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Preface
Introduction to Complex Numbers
Properties of Complex Numbers
Some Elementary Functions
Continuous Functions
Analytic Functions
Differentiation of the Elementary Functions
Review Exercises for Chapter 1
Contour Integrals
Supplement: Riemann Sums
Cauchy's Theorem: Intuitive Version
Cauchy's Theorem: Precise Version
Supplement A: Integrals Along
Continuous Curves
Supplement B: Relationship of Cauchy's Theorem to the Jordan Curve Theorem
Cauchy's Integral Formula
Maximum Modulus Theorem and Harmonic Function
Review Exercises for Chapter 2
Convergent Series of Analytic Functions
Power Series and Taylor's Theorem
Laurent's Series and Classification of Singularities
Review Exercises for Chapter 3
Calculation of Residues
The Residue Theorem
Evaluation of Definite Integrals
Evaluation of Infinite Series and
Partial-Fraction Expansions
Review Exercises for Chapter 4
Basic Theory of Conformal Mappings
Fractional Linear and Schwarz-Christoffel Transformations
Applications of Conformal Mapping to Laplace's Equation, Heat
Conduction, Electrostatics, and Hydrodynamics
Review Exercises for Chapter 5
Analytic Continuation and Elementary Riemann Surfaces
Rouché's Theorem and the Principle of the Argument
Mapping Properties of Analytic
Functions
Supplement A: Normal Families and The Riemann Mapping Theorem
Supplement B: The Dynamics of Complex Analytic Mappings
Review Exercises for Chapter 6
Infinite Products and the Gamma Function
Asymptotic Expansions and the Method of Steepest Descent
Supplement: Bounded Variation and the Proof of the Stationary Phase Formula
Stirling's Formula and Bessel
Functions
Review Exercises for Chapter 7
Basic Properties of LaPlace Transforms
The Complex Inversion Formula
Application of Laplace Transforms to Ordinary Differential Equations
Supplement: The Fourier Transform and the Wave Equation
Review Exercises for Chapter 8
Index
Jerrold E. Marsden and Michael J. Hoffman
ISBN13: 978-0716728771Table of Contents
Preface
Introduction to Complex Numbers
Properties of Complex Numbers
Some Elementary Functions
Continuous Functions
Analytic Functions
Differentiation of the Elementary Functions
Review Exercises for Chapter 1
Contour Integrals
Supplement: Riemann Sums
Cauchy's Theorem: Intuitive Version
Cauchy's Theorem: Precise Version
Supplement A: Integrals Along
Continuous Curves
Supplement B: Relationship of Cauchy's Theorem to the Jordan Curve Theorem
Cauchy's Integral Formula
Maximum Modulus Theorem and Harmonic Function
Review Exercises for Chapter 2
Convergent Series of Analytic Functions
Power Series and Taylor's Theorem
Laurent's Series and Classification of Singularities
Review Exercises for Chapter 3
Calculation of Residues
The Residue Theorem
Evaluation of Definite Integrals
Evaluation of Infinite Series and
Partial-Fraction Expansions
Review Exercises for Chapter 4
Basic Theory of Conformal Mappings
Fractional Linear and Schwarz-Christoffel Transformations
Applications of Conformal Mapping to Laplace's Equation, Heat
Conduction, Electrostatics, and Hydrodynamics
Review Exercises for Chapter 5
Analytic Continuation and Elementary Riemann Surfaces
Rouché's Theorem and the Principle of the Argument
Mapping Properties of Analytic
Functions
Supplement A: Normal Families and The Riemann Mapping Theorem
Supplement B: The Dynamics of Complex Analytic Mappings
Review Exercises for Chapter 6
Infinite Products and the Gamma Function
Asymptotic Expansions and the Method of Steepest Descent
Supplement: Bounded Variation and the Proof of the Stationary Phase Formula
Stirling's Formula and Bessel
Functions
Review Exercises for Chapter 7
Basic Properties of LaPlace Transforms
The Complex Inversion Formula
Application of Laplace Transforms to Ordinary Differential Equations
Supplement: The Fourier Transform and the Wave Equation
Review Exercises for Chapter 8
Index