Legendre equation
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1: 14.21 Definitions and Basic Properties
2: 14.2 Differential Equations
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§14.2(i) Legendre’s Equation
… ►§14.2(ii) Associated Legendre Equation
… ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations , , , , . … ►§14.2(iii) Numerically Satisfactory Solutions
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14.2.7
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3: 14.29 Generalizations
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►For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).
4: 18.8 Differential Equations
5: 14.31 Other Applications
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►The conical functions appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)).
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§14.31(iii) Miscellaneous
►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …6: 19.4 Derivatives and Differential Equations
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§19.4(ii) Differential Equations
…7: 30.2 Differential Equations
8: 14.3 Definitions and Hypergeometric Representations
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14.3.8
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9: Howard S. Cohl
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►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and -series.
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