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§34.12 Physical Applications… ► , and symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).
§14.31(ii) Conical Functions►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)). … ►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)). …
§29.19(i) Lamé Functions►Simply-periodic Lamé functions ( noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
… ►Bessel functions of the first kind, , arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation , or by the Helmholtz equation . ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … …
… ►The next four differential equations apply to the complete case of and in the form (see (19.16.20) and (19.16.23)). … ►and , with , satisfies Laplace’s equation: …
… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates , which are related to Cartesian coordinates by …
An Introduction to Linear Difference Equations.
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Laplace approximations for hypergeometric functions with matrix argument.
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Laplace approximation for Bessel functions of matrix argument.
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