Basic Complex Analysis by by J. E. Marsden and M.J. Hoffman

[Greg Bernhardt]

• Author: by J. E. Marsden and M.J. Hoffman
• Title: Basic Complex Analysis
• Amazon Link: https://www.amazon.com/dp/071672877X/?tag=pfamazon01-20
• Prerequisities:

Table of Contents:

[LIST]
[*] Analytic Functions
[LIST]
[*] Introduction to Complex Numbers
[*] Properties of Complex Numbers
[*] Some Elementary Functions 
[*] Continuous Functions
[*] Basic Properties of Analytic Functions
[*] Differentiation of the Elementary Functions
[/LIST]
[*] Cauchy's Theorem
[LIST]
[*] Contour Integrals
[*] Cauchy's Theorem—A First Look
[*] A Closer Look at Cauchy's Theorem
[*] Cauchy's Integral Formula
[*] Maximum Modulus Theorem and Harmonic Functions
[/LIST]
[*] Series Representation of Analytic Functions
[LIST]
[*] Convergent Series of Analytic Functions
[*] Power Series and Taylor's Theorem
[*] Laurent Series and Classification of Singularities
[/LIST]
[*] Calculus of Residues
[LIST]
[*] Calculation of Residues
[*] Residue Theorem
[*] Evaluation of Definite Integrals
[*] Evaluation of Infinite Series and Partial-Fraction Expansions
[/LIST]
[*] Conformal Mappings
[LIST]
[*] Basic Theory of Conformal Mappings
[*] Fractional Linear and Schwarz-Christoffel Transformations
[*] Applications of Conformal Mappings to Laplace's Equation, Heat Conduction, Electrostatics, and Hydrodynamics
[/LIST]
[*] Further Development of the Theory
[LIST]
[*] Analytic Continuation and Elementary Riemann Surfaces
[*] Rouche's Theorem and Principle of the Argument
[*] Mapping Properties of Analytic Functions
[/LIST]
[*] Asymptotic Methods
[LIST]
[*] Infinite Products and the Gamma Function
[*] Asymptotic Expansions and the Method of Steepest Descent
[*] Stirling's Formula and Bessel Functions
[/LIST]
[*] Laplace Transform and Applications
[LIST]
[*] Basic Properties of Laplace Transforms 
[*] Complex Inversion Formula
[*] Application of Laplace Transforms to Ordinary Differential Equations
[/LIST]
[*] Answers to Odd-Numbered Exercises
[*] Index 
[/LIST]

 

[jackmell]

I've tried over the last 5 years, many intro Complex Analysis textbooks and this one is by far the best one I've used. It's not overly-complicated, not too many proofs, and is a pleasant read compared to other textbooks which are difficult to follow. It presents the topics in a very accessible way that I believe the student can follow without difficulty.

I would highly recommend this text as a second book for anyone taking the subject for the first time. I use my often.