| Preface Errata 1 Introduction and Overview 1.1 Structure of the Book 1.2 Modem Applications of Conformal Mapping Electromagnetics 8, Vibrating Membranes & Acoustics 8, Transverse Vibrations & Buckling of Plates 9, Elasticity 10, Heat Transfer 11, Fluid Flow 12, Other Areas 12 1.3 Growth in Scope of Applications 2 Basic Mathematical Concepts 2.1 Transformation of Coordinates 2.2 Transformation by Means of Complex Functions 2.3 Analytic Functions 2.4 Conformality and Uniqueness 3 A Selection of Mapping Functions 3.1 Elementary Transformations 3.1.1 Linear Transformation 3.1.2 Power Transformation 3.1.3 Inverse or Reciprocal Transformation 3.1.4 Logarithmic and Exponential Transformations 3.1.5 Hyperbolic Transformation 3.1.6 Interpretation of Elementary Transformation 3.2 Composite Transformations 3.2.1 Bilinear Transformation 3.2.2 Joukowski Transformation 3.2.3 Trigonometric and Hyperbolic Transformations 3.2.4 Maxwelrs Transformation 3.3 Schwarz-Christoffel Transformation for Polygons 3.3.1 The Basic Method 3.3.2 Trigons 3.3.3 Maxwell's Transformation Revisited 3.3.4 Joukowski's Transformation Revisited 3.3.5 Mapping the Interior of a Rectangle The Rectangle 77, The Square 81, Weierstrass Formulation for Rectangle and Triangle 82, Application 83, The Inverse and Further Notes 84 3.3.6 Mapping the Regions Exterior to Rectangles and Between Rectangles 3.3.7 Polygons with Rounded Corners 3.4 Exploiting Symmetry 4 Numerical Methods 4.1 Methods of Approximation 4.2 Series Approximations 4.2.1 Mapping the Interior of the Unit Circle 4.2.2 Method of Simultaneous Equations (Kantorovich) 4.2.3 Mapping Exterior Regions 4.2.4 Successive Approximations (Fornberg) 4.3 Variational Methods 4.3.1 Carrying out the Minimization 4.3.2 Orthogonalization 4.3.3 Minimizing the Perimeter, Orthogonal Polynomials 4.3.4 Example of Minimum Perimeter Method 4.3.5 Minimizing the Area, Comparison of Methods 4.3.6 Additional Extremum Principles 4.3.7 Additional Approximation Methods 4.4 Integral Equation Methods 4.4.1 Method of Lichtenstein and Gershgorin 4.4.2 Method of Theodorsen and Garrick 4.4.3 Method of Symm 4.4.4 Comments on Integral Equation Methods 4.5 Numerical Determination of Schwarz-Christoffel Transformation 4.5.1 The Parameter Problem 4.5.2 Averting Singularities at Infinity 4.5.3 Method of Trefethen …… 5 Mathematical Models 6 Nonplanar Fields and Nonuniform Media 7 Static Fields in Electricity and Magnetinsm 8 Transmission Lines and Waveguides 9 Vibrating Membranes and Acoustics 10 Transverse Vibrations and Buckling of Plates 11 Stresse and Strains in an Elastic Medium 12 Steady State Heat Conduction in Doubly Connected Regions 13 Transient Heat Transter in Isotropic and Anisotropic Media 14 Fluid Flow 15 Conformal Mapping and Other Methods Appendices |
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