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©2001 CRC Press LLC To our children ©2001 CRC Press LLC Contents 1 Laplace’s Equation 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Laplace’s equation in curvilinear coordinates 1.1.1 Cartesian coordinates 1.1.2 Cylindrical polar coordinates 1.1.3 Spherical polar coordinates 1.1.4 Prolate spheroidal coordinates 1.1.5 Oblate spheroidal coordinates 1.1.6 Elliptic cylinder coordinates 1.1.7 Toroidal coordinates Solutions of Laplace’s equation: separation of variables 1.2.1 Cartesian coordinates 1.2.2 Cylindrical polar coordinates 1.2.3 Spherical polar coordinates 1.2.4 Prolate spheroidal coordinates 1.2.5 Oblate spheroidal coordinates 1.2.6 Elliptic cylinder coordinates 1.2.7 Toroidal coordinates Formulation of potential theory for structures with edges Dual equations: a classification of solution methods 1.4.1 The definition method 1.4.2 The substitution method 1.4.3 Noble’s multiplying factor method 1.4.4 The Abel integral transform method Abel’s integral equation and Abel integral transforms Abel-type integral representations of hypergeometric functions Dual equations and single- or double-layer surface potentials 2 Series and Integral Equations 2.1 Dual series equations involving Jacobi polynomials 2.2 Dual series equations involving trigonometrical functions 2.3 Dual series equations involving associated Legendre functions 2.4 Symmetric triple series equations involving Jacobi polynomials 2.4.1 Type A triple series equations 2.4.2 Type B triple series equations ©2001 CRC Press LLC 2.5 2.6 2.7 2.8 2.9 Relationships between series and integral equations Dual integral equations involving Bessel functions Nonsymmetrical triple series equations Coupled series equations A class of integro-series equations 3 Electrostatic Potential Theory for Open Spherical Shells 3.1 3.2 3.3 3.4 3.5 3.6 3.7