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

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

Published $\text {1991}$, Springer Verlag

ISBN 978-0387971957

Subject Matter

• Complex Analysis

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