spherical%20coordinates
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1: 14.30 Spherical and Spheroidal Harmonics
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§14.30(i) Definitions
… ► … ►As an example, Laplace’s equation in spherical coordinates (§1.5(ii)): … ►Here, in spherical coordinates, is the squared angular momentum operator: … ►2: 10.73 Physical Applications
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§10.73(ii) Spherical Bessel Functions
►The functions , , , and arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates (§1.5(ii)): …With the spherical harmonic defined as in §14.30(i), the solutions are of the form with , , , or , depending on the boundary conditions. Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …3: 29.18 Mathematical Applications
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§29.18(i) Sphero-Conal Coordinates
… ►(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). ►§29.18(ii) Ellipsoidal Coordinates
►The wave equation (29.18.1), when transformed to ellipsoidal coordinates : … ►§29.18(iii) Spherical and Ellipsoidal Harmonics
…4: 10.55 Continued Fractions
§10.55 Continued Fractions
►For continued fractions for and see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).5: 10.48 Graphs
6: 14.31 Other Applications
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§14.31(i) Toroidal Functions
… ►§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)). …7: 10.75 Tables
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§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives
►Zhang and Jin (1996, pp. 296–305) tabulates , , , , , , , , , 50, 100, , 5, 10, 25, 50, 100, 8S; , , , (Riccati–Bessel functions and their derivatives), , 50, 100, , 5, 10, 25, 50, 100, 8S; real and imaginary parts of , , , , , , , , , 20(10)50, 100, , , 8S. (For the notation replace by , , , , respectively.)
§10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions
… ►Olver (1960) tabulates , , , , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as .