Master Complex Analysis with Serge Lang: Prerequisites & Techniques for Grads

[micromass]

• Author: Serge Lang
• Title: Complex Analysis
• Amazon Link: https://www.amazon.com/dp/0387985921/?tag=pfamazon01-20
• Prerequisities: Basic analysis
• Level: Grad

Table of Contents:

[LIST]
[*] Foreword
[*] Prerequisites
[*] Basic Theory
[LIST]
[*] Complex Numbers and Functions
[LIST]
[*] Definition
[*] Polar Form
[*] Complex Valued Functions
[*] Limits and Compact Sets
[LIST]
[*] Compact Sets
[/LIST]
[*] Complex Differentiability
[*] The Cauchy-Riemann Equations
[*] Angles Under Holomorphic Maps
[/LIST]
[*] Power Series
[LIST]
[*] Formal Power Series
[*] Convergent Power Series
[*] Relations Between Formal and Convergent Series
[LIST]
[*] Sums and Products
[*] Quotients
[*] Composition of Series
[/LIST]
[*] Analytic Functions
[*] Differentiation of Power Series
[*] The Inverse and Open Mapping Theorems
[*] The Local Maximum Modulus Principle
[/LIST]
[*] Cauchy's Theorem, First Part
[LIST]
[*] Holomorphic Functions on Connected Sets
[LIST]
[*] Appendix: Connectedness
[/LIST]
[*] Integrals Over Paths
[*] Local Primitive for a Holomorphic Function
[*] Another Description of the Integral Along a Path
[*] The Homotopy Form of Cauchy's Theorem
[*] Existence of Global Primitives. Definition of the Logarithm
[*] The Local Cauchy Formula
[/LIST]
[*] Winding Numbers and Cauchy's Theorem
[LIST]
[*] The Winding Number
[*] The Global Cauchy Theorem
[LIST]
[*] Dixon's Proof of Theorem 2.5 (Cauchy's Formula)
[/LIST]
[*] Artin's Proof
[/LIST]
[*] Applications of Cauchy's Integral Formula
[LIST]
[*] Uniform Limits of Analytic Functions
[*] Laurent Series
[*] Isolated Singularities
[LIST]
[*] Removable Singularities
[*] Poles
[*] Essential Singularities
[/LIST]
[/LIST]
[*] Calculus of Residues
[LIST]
[*] The Residue Formula
[LIST]
[*] Residues of Differentials
[/LIST]
[*] Evaluation of Definite Integrals
[LIST]
[*] Fourier Transforms
[*] Trigonometric Integrals
[*] Mellin Transforms
[/LIST]
[/LIST]
[*] Conformal Mappings
[LIST]
[*] Schwarz Lemma
[*] Analytic Automorphisms of the Disc
[*] The Upper Half Plane
[*] Other Examples
[*] Fractional Linear Transformations
[/LIST]
[*] Harmonic Functions
[LIST]
[*] Definition 
[LIST]
[*] Application: Perpendicularity
[*] Application: Flow Lines
[/LIST]
[*] Examples
[*] Basic Properties of Harmonic Functions
[*] The Poisson Formula
[LIST]
[*] 
The Poisson Integral as a Convolution
[/LIST]
[*] Construction of Harmonic Functions
[*] Appendix. Differentiating Under the Integral Sign
[/LIST]
[/LIST]
[*] Geometric Function Theory
[LIST]
[*] Schwarz Reflection
[LIST]
[*] Schwarz Reflection (by Complex Conjugation)
[*] Reflection Across Analytic Arcs
[*] Application of Schwarz Reflection
[/LIST]
[*] The Riemann Mapping Theorem
[LIST]
[*] Statement of the Theorem
[*] Compact Sets in Function Spaces
[*] Proof of the Riemann Mapping Theorem
[*] Behavior at the Boundary
[/LIST]
[*] Analytic Continuation Along Curves
[LIST]
[*] Continuation Along a Curve
[*] The Dilogarithm
[*] Application to Picard's Theorem
[/LIST]
[/LIST]
[*] Various Analytic Topics
[LIST]
[*] Applications of the Maximum Modulus Principle and Jensen's Formula
[LIST]
[*] Jensen's Formula
[*] The Picard-Borel Theorem
[*] Bounds by the Real Part, Borel-Caratheodory Theorem
[*] The Use of Three Circles and the Effect of Small Derivatives
[LIST]
[*] Hermite Interpolation Formula
[/LIST]
[*] Entire Functions with Rational Values
[*] The Phragmen-Lindelof and Hadamard Theorems
[/LIST]
[*] Entire and Meromorphic Functions
[LIST]
[*] Infinite Products
[*] Weierstrass Products
[*] Functions of Finite Order
[*] Meromorphic Functions, Mittag-Leffler Theorem
[/LIST]
[*] Elliptic Functions
[LIST]
[*] The Liouville Theorems
[*] The Weierstrass Function
[*] The Addition Theorem
[*] The Sigma and Zeta Functions
[/LIST]
[*] The Gamma and Zeta Functions
[LIST]
[*] The Differentiation Lemma
[*] The Gamma Function
[LIST]
[*] Weierstrass Product
[*] The Gauss Multiplication Formula (Distribution Relation)
[*] The (Other) Gauss Formula
[*] The Mellin Transform
[*] The Stirling Formula
[*] Proof of Stirling's Formula
[/LIST]
[*] The Lerch Formula
[*] Zeta Functions
[/LIST]
[*] The Prime Number Theorem
[LIST]
[*] Basic Analytic Properties of the Zeta Function
[*] The Main Lemma and its Application
[*] Proof of the Main Lemma
[/LIST]
[/LIST]
[*] Appendix
[LIST]
[*] Summation by Parts and Non-Absolute Convergence
[*] Difference Equations
[*] Analytic Differential Equations
[*] Fixed Points of a Fractional Linear Transformation
[*] Cauchy's Formula for C^\infty Functions
[*] Cauchy's Theorem for Locally Integrable Vector Fields
[*] More on Cauchy-Riemann
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

 

[mathwonk]

i love this book, but it was mostly too hard for my undergraduate class. it begins with a discussion pif formal and convergent powers dries, as does the book of cartan, which may be more accessible, and has excellent chapters near the end on isomorphisms of the complex plane and extended plane. highly recommended but may not be a good first book. for that i prefer the out of print book of frederick green leaf.