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

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