Book:Jerrold E. Marsden/Basic Complex Analysis/Third Edition

Subject Matter

 * Complex Analysis

Contents

 * Preface


 * 1 Analytic Functions
 * 1.1 Introduction to Complex Numbers
 * 1.2 Properties of Complex Numbers
 * 1.3 Some Elementary Functions
 * 1.4 Continuous Functions
 * 1.5 Basic Properties of Analytic Functions
 * 1.6 Differentiation of Elementary Functions


 * 2 Cauchy's Theorem
 * 2.1 Contour Integrals
 * 2.2 Cauchy's Theorem – A First Look
 * 2.3 A Closer Look at Cauchy's Theorem
 * 2.4 Cauchy's Integral Formula
 * 2.5 Maximum Modulus Theorem and Harmonic Functions


 * 3 Series Representations of Analytic Functions
 * 3.1 Convergent Series of Analytic Functions
 * 3.2 Power Series and Taylor's Theorem
 * 3.3 Laurent Series and Classification of Singularities


 * 4 Calculus of Residues
 * 4.1 Calculations of Residues
 * 4.2 Residue Theorem
 * 4.3 Evaluation of Definite Integrals
 * 4.4 Evaluation of Infinite Series and Partial-Fraction Expansions


 * 5 Conformal Mappings
 * 5.1 Basic Theory of Conformal Mappings
 * 5.2 Fractional Linear and Schwarz-Christoffel Transformations
 * 5.3 Applications of Conformal Mappings to Laplace's Equation, Heat Conduction, Electrostatics and Hydrodynamics


 * 6 Further Development of the Theory
 * 6.1 Analytic Continuation and Elementary Riemann Surfaces
 * 6.2 Rouché's Theorem and Principle of the Argument
 * 6.3 Mapping Properties of Analytic Functions


 * 7 Asymptotic Methods
 * 7.1 Infinite Products and the Gamma Function
 * 7.2 Asymptotic Expansions and the Method of Steepest Descent
 * 7.3 Stirling's Formula and Bessel Functions


 * 8 Laplace Transform and Applications
 * 8.1 Basic Properties of Laplace Transforms
 * 8.2 Complex Inversion Formula
 * 8.3 Application of Laplace Transforms to Ordinary Differential Equations


 * Answers to Odd-Numbered Exercises


 * Index