Book:Eberhard Freitag/Complex Analysis
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Eberhard Freitag and Rolf Busam: Complex Analysis
Published $\text {2005}$, 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
- 5.4 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