Handbook of Conformal Mappings and Applications

Contents
Preface
Notations, Definitions, and Acronyms
Part I: Theory and Conformal Maps
	1 Introduction
		1.1 Historical Background
		1.2 Modern Developments
		1.3 In Retrospect
	2 Conformal Mapping
		2.1 Definitions
		2.2 Jordan Contour
		2.3 Metric Spaces
		2.4 Basic Theorems
		2.5 Harmonic Functions
		2.6 Univalent Functions
		2.7 Taylor Series Approximations
	3 Linear and Bilinear Transformations
		3.1 Definition of Certain Curves
		3.2 Bilinear Transformations
		3.3 Cross-Ratio
		3.4 Straight Lines and Circles
		3.5 Ellipses and Hyperbolas
	4 Algebraic Functions
		4.1 Polynomials
	5 Exponential Family of Functions
		5.1 Exponentials
		5.2 Specific Cases of Mappings
		5.3 Other Related Functions
		5.4 Complex Exponential Function
	6 Joukowski Airfoils
		6.1 Joukowski Maps
	7 Schwarz-Christoffel Transformation
		7.1 Schwarz-Christoffel Transformations
		7.2 Specific Transformations
		7.3 Regions Exterior to and between Rectangles
		7.4 Polygons with Round Corners
		7.5 Weierstrass Integral Equation
		7.6 Rectangles and Other Regions
		7.7 Inverse Schwarz-Christoffel Mapping
Part 2: Numerical Conformal Mapping
	8 Schwarz-Christoffel Integrals
		8.1. Parameter Problem
		8.2 Newton’s Method
		8.3 Numerical Computations
		8.4 Minimum Area Problem
		8.5 Numerical Methods for Minimum Area Problem
		8.6 Minimum Boundary Problem
		8.7 Numerical Parameter Problem
		8.8 Generalized Schwarz-Christoffel Parameter Problem
		8.9 Proofs
	9 Nearly Circular Regions
		9.1 Small Parameter Expansions
		9.2 Method of Infinite Systems
		9.3 Special Cases
		9.4 Exterior Regions
	10 Integral Equation Methods
		10.1 Neumann Kernel
		10.2 Interior Regions
		10.3 Exterior Regions
		10.4 Iterative Method
		10.5 Degenerate Kernel
		10.6. Szeg¨o Kernel
	11 Theodorsen’s Integral Equation
		11.1 Classical Iterative Method
		11.2 Convergence
		11.3 Theodorsen’s Method
		11.4 Integral Representation
		11.5 Starlike Regions
		11.6 Exterior Regions
		11.7 Trigonometric Interpolation
		11.8 Wegmann’s Method
		11.9 Newton’s Method
		11.10 Kantorovich’s Method
		11.11 Fornberg’s Method of Successive Approximation
		11.12 Lichtenstein-Gershgorin Method
		11.13 Theodorsen-Garrick Method
		11.14 Variational Methods
	12 Symm’s Integral Equation
		12.1. Symm’s Integral Equation
		12.2. Orthonormal Polynomial Method
		12.3 Modified ONP Method
		12.4. Lagrange Interpolation
		12.5. Spline Approximations
	13 Airfoils and Singularities
		13.1 Nearly Circular Approximations
		13.2 James’s Method
		13.3 Arbenz’s Integral Equation
		13.4 Boundary Corner
		13.5 Singularity Behavior
		13.6 Pole-Type Singularities
		13.7 Exterior Regions
	14 Doubly Connected Regions
		14.1 Annular Regions
		14.2 Area Theorem
		14.3 Source Density
		14.4 Dipole Distribution
		14.5 Gaier’s Variational Method
		14.6 Mapping onto Annulus
	15 Multiply Connected Regions
		15.1 Some Useful Results
		15.2 Dirichlet Problem
		15.3 Mikhlin’s Integral Equation
		15.4 Mayo’s Method
		15.5 Fast Poisson Solver
Part 3: Applications
	16 Grid Generation
		16.1 Computational Region
		16.2 Inlet Configurations
		16.3 Cascade Configurations
	17 Field Theories
		17.1 Mathematical Models
		17.2 Initial and Boundary Conditions
		17.3 Classification of Second-Order Equations
		17.4 Superposition Principle
		17.5 Parabolic Equations
		17.6 Hyperbolic Equations
		17.7 Elliptic Equations
		17.8 Poisson’s Equation
		17.9 Two-Dimensional Flows
	18 Fluid Flows
		18.1 Viscous Laminar Flows
		18.2 External Flows
		18.3 Ideal Fluid Flows
		18.4 Potential Flows
		18.5 Boundary Layer Flows
		18.6 Kirchhoff’s Flow Problem
		18.7 Streamlines in Fluid Flows
		18.8 Conformal Mapping of Flow Patterns
		18.9 Joukowski Maps
		18.10 Airfoils
	19 Heat Transfer
		19.1 Heat Flow
		19.2 Heat Transfer
		19.3 Conformal Transformations
		19.4 Poisson’s Integral Formulas
		19.5 Diffusion Equation
		19.6 High Temperature Effects
		19.7 Transient Problems
	20 Vibrations and Acoustics
		20.1 Wave Propagation and Dispersion
		20.2 Vibrations of Strings
		20.3 Vibrations of Membranes
		20.4 Acoustics
	21 Electromagnetic Field
		21.1 Electromagnetic Field
		21.2 Electrostatic Field
		21.3 Electric Field
		21.4 Electromagnetic Waves
		21.5 Electric Capacitors
		21.6 AC Circuits
		21.7 Laplace’s Equation
	22 Transmission Lines and Waveguides
		22.1 Maxwell’s Equations
		22.2 Wave Propagation
		22.3 Transmission Lines
		22.4 Conformal Mapping and Electric Transmission
		22.5 Conformal Mapping and Waveguides
		22.6 Helmholtz Equation and Rib-Shaped Waveguide
		22.7 Coplanar Waveguides
		22.8 Nonuniform Waveguides
	23 Elastic Medium
		23.1 Stress and Strain
		23.2 Stress Function
		23.3 Infinite Plate and Conformal Mapping
	24 Finite Element Method
		24.1 Weak Variational Form
		24.2 Galerkin Method
		24.3 Rayleigh-Ritz Method
		24.4 Linear Three-Node Triangular Elements
		24.5 Single Dependent Variable Problems
		24.6 Fluid Flows
		24.7 Free Axial Vibrations of an Elastic Rod
		24.8 Electric Potential
		24.9 Waveguide
		24.10 Motor Design
		24.11 Other Applications
	25 Computer Programs and Resources
		25.1 Numerical Methods
		25.2 Software
		25.3 Computer Codes
A Green’s Identities
B Cauchy’s P.-V. Integrals
	B.1 Numerical Evaluation
C Riemann Mapping Theorem
	C.1 Theorem and Proof
	C.2 Homogeneous Hilbert Problem
	C.3 Nonhomogeneous Hilbert Problem
	C.4 Riemann-Hilbert Problem
D Gudermannian
E Tables
F Elliptic Integrals
	F.1 Elliptic Integrals
	F.2 Jacobi’s Elliptic Functions
	F.3 Weierstrass Elliptic Function
	F.4 Jacobi’s Theta Functions
	F.5 Modular Function
	F.6 Weierstrass Zeta Function
	F.7 Weierstrass Sigma Function
G Gauss-Jacobi Rule
H Orthogonal Polynomials
I Special Finite Elements
J Schwarz Formula
Bibliography
Index