# Book:Eberhard Freitag/Complex Analysis

## Eberhard Freitag and Rolf Busam: Complex Analysis

Published $2005$, Springer

ISBN 978-3-540-25724-0.

## Subject Matter

Translation of Funktionentheorie 1 from 1993.

## Contents

1 Differential Calculus in the Complex Plane $\C$
1.1 Complex Numbers
1.2 Convergent Sequences and Series
1.3 Continuity
1.4 Complex Derivatives
1.5 The Cauchy–Riemann Differential Equations
2 Integral Calculus in the Complex Plane $\C$
2.1 Complex Line Integrals
2.2 The Cauchy Integral Theorem
2.3 The Cauchy Integral Formulas
3 Sequences and Series of Analytic Functions, the Residue Theorem
3.1 Uniform Approximation
3.2 Power Series
3.3 Mapping Properties for Analytic Functions
3.4 Singularities of Analytic Functions
3.5 Laurent Decomposition
3.6 The Residue Theorem
3.7 Applications of the Residue Theorem
4 Construction of Analytic Functions
4.1 The Gamma Function
4.2 The Weierstrass Product Formula
4.3 The Mittag–Leffler Partial Fraction Decomposition
4.4 The Riemann Mapping Theorem
5 Elliptic Functions
5.1 The Liouville Theorems
5.2 The Weierstrass ℘-function
5.3 The Field of Elliptic Functions
5.5 Elliptic Integrals
5.6 Abel’s Theorem
5.7 The Elliptic Modular Group
5.8 The Modular Function $j$
6 Elliptic Modular Forms
6.1 The Modular Group and Its Fundamental Region
6.2 The $k/12$-formula and the Injectivity of the $j$-function
6.3 The Algebra of Modular Forms
6.4 Modular Forms and Theta Series
6.5 Modular Forms for Congruence Groups
6.6 A Ring of Theta Functions
7 Analytic Number Theory
7.1 Sums of Four and Eight Squares
7.2 Dirichlet Series
7.3 Dirichlet Series with Functional Equations
7.4 The Riemann $\zeta$-function and Prime Numbers
7.5 The Analytic Continuation of the $\zeta$-function
7.6 A Tauberian Theorem
References
Symbolic Notations
Index