Complex Analysis

This Wikibook aims to be a comprehensive text on complex analysis, the study of complex numbers and complex functions.

Contributors include Samuel Cormier-Iijima, Tachyon01, Mathmensch, and Fephisto. Add your name here if you contribute!

See also: Calculus/Complex_analysis

Table of ContentsEdit

Chapter 1: Complex NumbersEdit

• 1.1 Introduction
• 1.2 The Topology of the Complex Plane and Stereographic Projection

Chapter 2: Complex FunctionsEdit

• 2.1 Complex Functions
• 2.2 Limits and Continuous Functions
• 2.3 Complex Derivatives
• 2.4 Holomorphic and Harmonic Functions

Chapter 3: Elementary FunctionsEdit

• 3.1 Polynomial Functions
• 3.2 Exponential and Trigonometric Functions
• 3.3 Logarithmic Functions
• 3.4 Inverse Trigonometric Functions

Chapter 4: Residue TheoryEdit

• The Basics
• Partial Fractions
• A More "Complex" Solution
• Some Consequences

ContentsEdit

1. Complex numbers
2. Complex differentiable, holomorphic, Cauchy–Riemann equations
3. Function series, power series, Euler's formula, polar form, argument
4. Contour integrals
5. Cauchy's theorem for star-shaped domains, Cauchy's integral formula, Montel's theorem
6. Differentiability and analyticity of holomorphic functions, Cauchy's differentiation formulae, Morera's theorem
7. Identity theorem, Liouville-type theorems, Riemann's theorem
8. Extremum principles, open mapping theorem, Schwarz' lemma
9. Types of singularities, meromorphic functions, Laurent series, Casorati–Weierstrass theorem
10. Chains, cycles, winding numbers, nullhomologous, the general Cauchy theorem
11. The residue theorem, argument principle, Rouché's theorem, Hurwitz' theorem
12. Exercises

AppendixEdit

1. Picard's little theorem
2. Picard's great theorem
3. The Riemann mapping theorem

ReferencesEdit

In EnglishEdit

• Alder, Michael D..An Introduction to Complex Analysis for Engineers. , 1997.
• Ahlfors, Lars Valerian. Complex Analysis: An Introduction to the theory of analytic functions of one complex variable. 2.ed. New York: McGraw-Will, 1966. ISBN 000700656X
• Boas, Ralph Philip. An invitation to complex analysis. 1.ed. New York: Random House, 1988. ISBN 0394350766
• Conway, John B.. Functions of One Complex Variable: Graduate Texts in Mathematics. 2.ed. Berlin: Springer, 1978. v. 1. ISBN 0387903283
• Henrici, Peter. Applied and Computational Complex Analysis. Wiley, 1974. v. 1. ISBN 0471372447
• Knopp, K.. Theory of Functions. New York: Dover, 1945. v. 2.
• Lang, Serge. Complex Analysis: Graduate Texts in Mathematics. New York: Springer, 1993.
• Rudin, Walter. "Real and Complex Analysis". 3.ed. McGraw Hill. ISBN 0070542341

In other languagesEdit

• Ávila, Geraldo Severo de Souza. Variáveis Complexas e Aplicações. 2.ed. Rio de Janeiro: LTC, 2000.
• Cartan, Henri Paul. Théorie Élémentaire Des Fonctions Analytiques d'Une Ou Plusieurs Variables Complexes. 2.ed. Paris: Hermann, 1967.
• Churchill, R. V.. Variáveis Complexas e suas Aplicações. São Paulo: Mac-Graw-Hill do Brasil e Editora da Universidade de São Paulo, 1975.
• Fernandez, Cecília S., Bernardes Jr., Nilson C.. Introdução às Funções de uma Variável Complexa. Rio de Janeiro: , 2006. ISBN 8585818336
• Lima, Elon Lages. Curso de Análise. Rio de Janeiro: IMPA, 1989. v. 2.
• Lins Neto, Alcides. Funções de uma variável complexa. Rio de Janeiro: IMPA, 1996.