### Canonical Structures in potential Theory - S.S. Vinogradov, P. D. Smith, E.D. Vinogradova

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1 Laplace’s Equation
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
1.1 Laplace’s equation in curvilinear coordinates
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
1.2 Solutions of Laplace’s equation: separation of variables
1.3 Formulation of potential theory for structures with edges
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
1.4 Dual equations: a classification of solution methods
1.5 Abel’s integral equation and Abel integral transforms
1.6 Abel-type integral representations of hypergeometric functions
1.7 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
2.4.1 Type A triple series equations
2.4.2 Type B triple series equations
2.4 Symmetric triple series equations involving Jacobi polynomials
2.5 Relationships between series and integral equations
2.6 Dual integral equations involving Bessel functions
2.7 Nonsymmetrical triple series equations
2.8 Coupled series equations
2.9 A class of integro-series equations
3 Electrostatic Potential Theory for Open Spherical Shells
3.1 The open conducting spherical shell
3.2.1 Approximate analytical formulae for capacitance
3.2 A symmetrical pair of open spherical caps and the spherical
barrel
3.3 An asymmetrical pair of spherical caps and the asymmetric
barrel
3.4 The method of inversion
3.5 Electrostatic fields in a spherical electronic lens
3.6 Frozen magnetic fields inside superconducting shells
3.7 Screening number of superconducting shells
4 Electrostatic Potential Theory for Open Spheroidal Shells
4.1 Formulation of mixed boundary value problems in spheroidal
geometry
4.2 The prolate spheroidal conductor with one hole
4.3 The prolate spheroidal conductor with a longitudinal slot
4.4 The prolate spheroidal conductor with two circular holes
4.5 The oblate spheroidal conductor with a longitudinal slot
4.6 The oblate spheroidal conductor with two circular holes
4.7.1 Open spheroidal shells
4.7.2 Spheroidal condensors
4.7 Capacitance of spheroidal conductors
5 Charged Toroidal Shells
5.1 Formulation of mixed boundary value problems in toroidal geometry
5.2 The open charged toroidal segment
5.3 The toroidal shell with two transversal slots
5.4 The toroidal shell with two longitudinal slots
5.5 Capacitance of toroidal conductors
5.6.1 The toroidal shell with one azimuthal cut
5.6 Anopentoroidal shell with azimuthal cuts
5.6.2 The toroidal shell with multiple cuts
5.6.3 Limiting cases
6 Potential Theory for Conical Structures with Edges
6.1 Non-coplanar oppositely charged infinite strips
6.2 Electrostatic fields of a charged axisymmetric finite open conical
conductor
6.3 The slotted hollow spindle
6.4 A spherical shell with an azimuthal slot
7 Two-dimensional Potential Theory
7.1 The circular arc
7.2 Axially slotted open circular cylinders
7.3 Electrostatic potential of systems of charged thin strips
7.4 Axially-slotted elliptic cylinders
7.5 Slotted cylinders of arbitrary profile
8 More Complicated Structures
8.1 Rigorous solution methods for charged flat plates
8.2.1 The spherically-curved elliptic plate
8.2 The charged elliptic plate
8.3 Polygonal plates
8.4 The finite strip
8.5 Coupled charged conductors: the spherical cap and circular disc
A Notation
B Special Functions
B.1 The Gamma function
B.2 Hypergeometric functions
B.3.1 The associated Legendre polynomials
B.3.2 The Legendre polynomials
B.3 Orthogonal polynomials: Jacobi polynomials, Legendre polynomials
B.4.1 Ordinary Legendre functions
B.4 Associated Legend refunctions
B.4.2 Conical functions
B.4.3 Associated Legendre functions of integer order
B.5.1 Spherical Bessel functions
B.5.2 Modified Bessel functions
B.5 Bessel functions
B.6 The incomplete scalar product
C Elements of Functional Analysis
C.1 Hilbert spaces
C.2 Operators
C.3 The Fredholm alternative and regularisation
D Transforms and Integration of Series
D.1 Fourier and Hankel transforms
D.2 Integration of series