Complex Analysis by Stein and Shakarchi

[micromass]

• Author: Elias Stein, Rami Shakarchi
• Title: Complex Analysis
• Amazon Link: https://www.amazon.com/dp/0691113858/?tag=pfamazon01-20
• Prerequisities: Fourier Analysis by Stein, Shakarchi
• Level: Undergrad

Table of Contents:

[LIST]
[*] Foreword
[*] Introduction
[*] Preliminaries to Complex Analysis
[LIST]
[*] Complex numbers and the complex plane
[LIST]
[*] Basic properties
[*] Convergence
[*] Sets in the complex plane
[/LIST]
[*] Functions on the complex plane
[LIST]
[*] Continuous functions
[*] Holomorphic functions
[*] Power series
[/LIST]
[*] Integration along curves
[*] Exercises
[/LIST]
[*] Cauchy's Theorem and Its Applications
[LIST]
[*] Goursat's theorem
[*] Local existence of primitives and Cauchy's theorem in a disc
[*] Evaluation of some integrals
[*] Cauchy's integral formulas
[*] Further applications
[LIST]
[*] Morera's theorem
[*] Sequences of holomorphic functions
[*] Holomorphic functions defined in terms of integrals
[*] Schwarz reflection principle
[*] Runge's approximation theorem
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Meromorphic Function and the Logarithm
[LIST]
[*] Zeros and Poles
[*] The residue formula
[LIST]
[*] Examples
[/LIST]
[*] Singularities and meromorphic functions
[*] The argument principle and applications
[*] Homotopies and simply connected domains
[*] The complex logarithm
[*] Fourier series and harmonic functions
[*] Exercises
[*] Problems
[/LIST]
[*] The Fourier Transform
[LIST]
[*] The class F
[*] Action of the Fourier transform on F
[*] Paley-Wiener theorem
[*] Exercises
[*] Problems
[/LIST]
[*] Entire Functions
[LIST]
[*] Jensen's formula
[*] Functions of finite order
[*] Infinite products
[LIST]
[*] Generalities
[*] Example: the product formula for the sine function
[/LIST]
[*] Weierstrass infinite products
[*] Hadamard's factorization theorem
[*] Exercises
[*] Problems
[/LIST]
[*] The Gamma and Zeta Functions
[LIST]
[*] The gamma function
[LIST]
[*] Analytic continuation
[*] Further properties of \Gamma
[/LIST]
[*] The zeta function
[LIST]
[*] Functional equation and analytic continuation
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] The Zeta Function and Prime Number Theorem
[LIST]
[*] Zeros of the zeta function
[LIST]
[*] Estimates for 1/\zeta(s)
[/LIST]
[*] Reduction to the functions \psi and \psi_1
[LIST]
[*] Proofs of the asymptotics for \psi_1
[/LIST]
[*] Note on interchanging double sums
[*] Exercises
[*] Problems
[/LIST]
[*] Conformal Mappins
[LIST]
[*] Conformal equivalence and examples
[LIST]
[*] The disc and upper half-plane
[*] Further examples
[*] The Dirichlet problem in a strip
[/LIST]
[*] The Schwarz lemma; automorphisms of the disc and upper half-plane
[LIST]
[*] Automorphisms of the disc
[*] Automorphisms of the upper half-plane
[/LIST]
[*] The Riemann mapping theorem
[LIST]
[*] Necessary conditions and statement of the theorem
[*] Montel's theorem
[*] Proof of the Riemann mapping theorem
[/LIST]
[*] Conformal mappings onto polygons
[LIST]
[*] Some examples
[*] The Schwarz-Christoffel integral
[*] Boundary behavior
[*] The mapping formula
[*] Return to elliptic integrals
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] An Introduction to Elliptic Functions
[LIST]
[*] Elliptic functions
[LIST]
[*] Liouville's theorems
[*] The Weierstrass P function
[/LIST]
[*] The modular character of elliptic functions and Eisenstein series
[LIST]
[*] Eisenstein series
[*] Eisenstein series and divisor functions
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Applications of Theta Functions
[LIST]
[*] Product formula for the Jacobi theta function
[LIST]
[*] Further transformation laws
[/LIST]
[*] Generating functions
[*] The theorems about sums of squares
[LIST]
[*] The two-squares theorem
[*] The four-squares theorem
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Appendix: Asymptotics
[LIST]
[*] Bessel functions
[*] Laplace method; Stirling's formula
[*] The Airy function
[*] The partition function
[*] Problems
[/LIST]
[*] Simple Connectivity and Jordan Curve Theorem
[LIST]
[*] Equivalent formulations of simple connectivity
[*] The Jordan curve theorem
[LIST]
[*] Proof of a general form of Cauchy's theorem
[/LIST]
[/LIST]
[*] Notes and References
[*] Bilbiography
[*] Symbol Glossary
[*] Index
[/LIST]

 

[mathwonk]

I hope someone with more knowledge of this book will comment. I have very little, only having seen it briefly while supervising a graduate prep course in complex analysis before prelims. My impression then was that I was disappointed in it, given the name of the extremely eminent first author. Everything in it is excellent, but it seemed inadequate for students to learn from alone, or to get a really good feel for many sides of the subject. It was apparently notes from a lecture series possibly given only once.

The pedigree of the main author guarantees that the viewpoint is excellent, that there are interesting topics authoritatively treated that are not found elsewhere, the proofs are elegant, and that the insights offered are deep. The manner in which it was written though suggest it was not actually written by that main author, and that it may have been a bit of an experiment, and hence may not have received the repeated polishing that leads to a really outstanding text.

I would suggest that, maybe like the similarly generated physics books of Feynman, this is an outstanding series that everyone can benefit from and that is unique in its point of view, and that less qualified authors could not begin to imitate. However it may be hard to learn thoroughly the subject from only these books.

A remark: in the introduction there are three key principles of complex analysis mentioned briefly for the reader, namely contour integration, regularity, and analytic continuation. The hint of an idea about contour integration that is given says that for certain appropriate contours the integral of a holomorphic function is zero.

In my estimation this is not worth much to the reader since as stated it is true of all functions. I would suggest instead saying that for holomorphic functions, that contour integration is a "deformation invariant", i.e. if a contour is changed continuously the integral does not change.

Then, as a corollary, for integrals that can be continuously deformed to a point, the integral is the same as the integral over a point, namely zero. This last case is the usual homotopy version of the Cauchy theorem, but the main point is the deformation invariance, not the special case that comes out zero.

And now that I have tried to improve something from the book, you can be your own judge as to which version is more suitable for a text, maybe their simpler less informative one.

So I will guess that everyone could benefit from reading these books, but that one might well want some other standard sources around too.