Book:Eberhard Freitag/Complex Analysis

Subject Matter
Translation of Funktionentheorie 1 from 1993.


 * Complex Analysis

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
 * 5.4 The Addition Theorem
 * 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