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

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Jerrold E. Marsden and Michael J. Hoffman: Basic Complex Analysis (2nd Edition)

Published $\text {1999}$.

Subject Matter


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