Conformal Mapping 9780486611372, 048661137X

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Conformal Mapping
 9780486611372, 048661137X

Table of contents :
Preface
Contents
Chapter 1. Harmonic Functions
1. Definitions and Preliminary Remarks
2. Elementary Properties of Harmonic Functions
3. Green's Formula
4. Applications of Green's Formula
5. The Green's Function and the Boundary Value Problem of the First Kind
6. The Poisson Formula
7. The Neumann Function and the Boundary Value Problem of the Second Kind
8. The Conjugate Harmonic Function
9. Multiply-connected Domains
10. The Harmonic Measure
11. Dependence of the Green's Function on the Domain
Chapter 2. Analytic Functions
1. Complex Numbers
2. The Complex Plane
3. Sequences and Series of Complex Numbers
4. Analytic Functions
5. The Cauchy-Riemann Equations
6. Power Series
7. Preliminary Study of Some Elementary Functions
Chapter 3. The Complex Integral Calculus
1. Complex Integration
2. Cauchy's Theorem
3. The Cauchy Integral Formula
4. The Taylor Series
5. Analytic Continuation
6. The Laurent Series
7. Liouville's Theorem
8. The Maximum Principle
9. The Residue Theorem
10. The Argument Principle
Chapter 4. Families of Analytic Functions
1. Equicontinuity and Uniform Boundedness
2. Normal Families of Analytic Functions
3. Extremal Problems
Chapter 5. Conformal Mapping of Simply-Connected Domains
1. Mapping Properties of Analytic Functions
2. The Linear Transformation
3. The Schwarz Lemma
4. The Riemann Mapping Theorem
5. The Symmetry Principle
6. The Schwarz-Christoffel Formula
7. Domains Bounded by Circular Arcs
8. Univalent Functions
9. Subordination
10. The Kernel Function
11. Conformal Mapping of Nearly Circular Domains
Chapter 6. Mapping Properties of Special Functions
1. The Rational Function of the Second Degree
2. Exponential and Trigonometric Functions
3. Elliptic Functions
4. Domains Bounded by Arcs of Confocal Conics
5. The Schwarzian s-functions
6. The Elliptic Modular Function
Chapter 7. Conformal Mapping of Multiply-Connected Domains
1. Canonical Domains
2. Characterization of Canonical Mappings by Extremal Problems
3. The Green's Function and the Dirichlet Problem
4. Area Problems
5. The Kernel Function and Orthonormal Sets
6. Bounded Functions
Index

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