# Book:Reinhold Remmert/Theory of Complex Functions/Second Edition

## Reinhold Remmert: Theory of Complex Functions (2nd Edition)

Published $1991$, Springer Verlag

ISBN 978-0387971957.

### Contents

Preface to the English Edition
Preface to the Second German Edition
Preface to the First German Edition
Historical Introduction
Chronological Table
Part A. Elements of Function Theory
Chapter 0. Complex Numbers and Continuous Functions
$\S$1. The Field $\C$ of complex numbers
$\S$2. Fundamental topological concepts
$\S$3. Convergent sequences of complex numbers
$\S$4. Convergent and absolutely convergent series
$\S$5. Continuous functions
$\S$6. Connected Spaces. Regions in $\C$
Chapter 1. Complex-Differential Calculus
$\S$1. Complex-differentiable functions
$\S$2. Complex and real differentiability
$\S$3. Holomorphic functions
$\S$4. Partial differentiation with respect to $x$, $y$, $z$ and $\bar z$
Chapter 2. Holomorphy and Conformality. Biholomorphic Mappings
$\S$1. Holomorphic functions and angle-preserving mappings
$\S$2. Biholomorphic mappings
$\S$3. Automorphisms of the upper half-plane and the unit disc
Chapter 3. Modes of Convergence in Function Theory
$\S$1. Uniform, locally uniform and compact convergence
$\S$2. Convergence criteria
$\S$3. Normal convergence of series
Chapter 4. Power Series
$\S$1. Convergence criteria
$\S$2. Examples of convergent power series
$\S$3. Holomorphy of power series
$\S$4. Structure of the algebra of convergent power series
Chapter 5. Elementary Transcendental Functions
$\S$1. The exponential and trigonometric functions
$\S$2. The epimorphism theorem for $\exp z$ and its consequences
$\S$3. Polar coordinates, roots of unity and natural boundaries
$\S$4. Logarithm functions
$\S$5. Discussion of logarithm functions
Part B. The Cauchy Theory
Chapter 6. Complex Integral Calculus
$\S$0. Integration over real intervals
$\S$1. Path integrals in $\C$
$\S$2. Properties of complex path integrals
$\S$3. Path independence of integrals. Primitives
Chapter 7. The Integral Theorem, Integral Formula and Power Series Development
$\S$1. The Cauchy Integral Theorem for star regions
$\S$2. Cauchy's Integral Formula for discs
$\S$3. The development of holomorphic functions into power series
$\S$4. Discussion of the representation theorem
$\S$5*. Special Taylor Series. Bernoulli numbers
Part C. Cauchy-Weierstrass-Riemann Function Theory
Chapter 8. Fundamental Theorems about Holomorphic Functions
$\S$1. The Identity Theorem
$\S$2. The concept of holomorphy
$\S$3. The Cauchy estimates and inequalities for Taylor coefficients
$\S$4. Convergence theorems of Weierstrass
$\S$5. The open mapping theorem and the maximum principle
Chapter 9. Miscellany
$\S$1. The fundamental theorem of algebra
$\S$2. Schwarz' lemma and the groups $\operatorname{Aut} \mathbb E$, $\operatorname{Aut} \mathbb H$
$\S$3. Holomorphic logarithms and holomorphic roots
$\S$4. Biholomorphic mappings. Local normal forms
$\S$5. General Cauchy theory
$\S$6*. Asymptotic power series developments
Chapter 10. Isolated Singularities. Meromorphic Functions
$\S$1. Isolated singularities
$\S$2*. Automorphisms of punctured domains
$\S$3. Meromorphic functions
Chapter 11. Convergent Series of Meromorphic Functions
$\S$1. General convergence theory
$\S$2. The partial fraction development of $\pi \cot \pi z$
$\S$3. The Euler formulas for $\sum_{\nu \ge 1} \nu^{-2n}$
$\S$4*. The Eisenstein theory of the trigonometric functions
Chapter 12. Laurent Series and Fourier Series
$\S$1. Holomorphic functions in annuli and Laurent series
$\S$2. Properties of Laurent series
$\S$3. Periodic holomorphic functions and Fourier series
$\S$4. The theta function
Chapter 13. The Residue Calculus
$\S$1. The residue theorem
$\S$2. Consequences of the residue theorem
Chapter 14. Definite Integrals and the Residue Calculus
$\S$1. Calculation of integrals
$\S$2. Further evaluation of integrals
$\S$3. Gauss sums
Short Biographies of Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass
Photograph of Riemann's gravestone
Literature
Symbol Index
Name Index
Subject Index
Portraits of famous mathematicians