Laplace equation
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1: 34.12 Physical Applications
§34.12 Physical Applications
… βΊ , and symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).2: 14.31 Other Applications
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βΊ
§14.31(ii) Conical Functions
βΊThe 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)). … βΊMany 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)). …3: 29.19 Physical Applications
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βΊ
§29.19(i) Lamé Functions
βΊSimply-periodic Lamé functions ( noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …4: 10.73 Physical Applications
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βΊBessel 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
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βΊLaplace’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.
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5: 19.18 Derivatives and Differential Equations
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βΊThe next four differential equations apply to the complete case of and in the form (see (19.16.20) and (19.16.23)).
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βΊand , with , satisfies Laplace’s equation:
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6: 14.30 Spherical and Spheroidal Harmonics
7: 14.19 Toroidal (or Ring) Functions
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βΊThis form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates
, which are related to Cartesian coordinates by
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