Book:Reinhold Remmert/Theory of Complex Functions/Second Edition
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Reinhold Remmert: Theory of Complex Functions (2nd Edition)
Published $\text {1991}$, Springer Verlag
- ISBN 978-0387971957
Subject Matter
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 0. Complex Numbers and Continuous Functions
- 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 1. Complex-Differential Calculus
- 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 2. Holomorphy and Conformality. Biholomorphic Mappings
- 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 3. Modes of Convergence in Function Theory
- 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 4. 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
- Chapter 5. Elementary Transcendental 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 6. Complex Integral Calculus
- 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
- Chapter 7. The Integral Theorem, Integral Formula and Power Series Development
- 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 8. Fundamental Theorems about Holomorphic Functions
- 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 9. Miscellany
- Chapter 10. Isolated Singularities. Meromorphic Functions
- $\S$1. Isolated singularities
- $\S$2*. Automorphisms of punctured domains
- $\S$3. Meromorphic functions
- Chapter 10. Isolated Singularities. 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 11. Convergent Series of Meromorphic 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 12. Laurent Series and Fourier Series
- Chapter 13. The Residue Calculus
- $\S$1. The residue theorem
- $\S$2. Consequences of the residue theorem
- Chapter 13. The Residue Calculus
- Chapter 14. Definite Integrals and the Residue Calculus
- $\S$1. Calculation of integrals
- $\S$2. Further evaluation of integrals
- $\S$3. Gauss sums
- Chapter 14. Definite Integrals and the Residue Calculus
- 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