About the Project

spherical%20coordinates

AdvancedHelp

(0.003 seconds)

1—10 of 193 matching pages

1: 14.30 Spherical and Spheroidal Harmonics
§14.30(i) Definitions
As an example, Laplace’s equation 2 W = 0 in spherical coordinates1.5(ii)): … Here, in spherical coordinates, L 2 is the squared angular momentum operator: …
2: 10.73 Physical Applications
§10.73(ii) Spherical Bessel Functions
The functions 𝗃 n ( x ) , 𝗒 n ( x ) , 𝗁 n ( 1 ) ( x ) , and 𝗁 n ( 2 ) ( x ) arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates ρ , θ , ϕ 1.5(ii)): …With the spherical harmonic Y , m ( θ , ϕ ) defined as in §14.30(i), the solutions are of the form f = g ( k ρ ) Y , m ( θ , ϕ ) with g = 𝗃 , 𝗒 , 𝗁 ( 1 ) , or 𝗁 ( 2 ) , 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
§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 α , β , γ : …
§29.18(iii) Spherical and Ellipsoidal Harmonics
4: 10.55 Continued Fractions
§10.55 Continued Fractions
For continued fractions for 𝗃 n + 1 ( z ) / 𝗃 n ( z ) and 𝗂 n + 1 ( 1 ) ( z ) / 𝗂 n ( 1 ) ( z ) see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
5: 10.48 Graphs
§10.48 Graphs
See accompanying text
Figure 10.48.5: 𝗂 0 ( 1 ) ( x ) , 𝗂 0 ( 2 ) ( x ) , 𝗄 0 ( x ) , 0 x 4 . Magnify
See accompanying text
Figure 10.48.6: 𝗂 1 ( 1 ) ( x ) , 𝗂 1 ( 2 ) ( x ) , 𝗄 1 ( x ) , 0 x 4 . Magnify
See accompanying text
Figure 10.48.7: 𝗂 5 ( 1 ) ( x ) , 𝗂 5 ( 2 ) ( x ) , 𝗄 5 ( x ) , 0 x 8 . Magnify
6: 14.31 Other Applications
§14.31(i) Toroidal Functions
§14.31(ii) Conical Functions
The conical functions 𝖯 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates14.19(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
§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives
  • The main tables in Abramowitz and Stegun (1964, Chapter 10) give 𝗃 n ( x ) , 𝗒 n ( x ) n = 0 ( 1 ) 8 , x = 0 ( .1 ) 10 , 5–8S; 𝗃 n ( x ) , 𝗒 n ( x ) n = 0 ( 1 ) 20 ( 10 ) 50 , 100, x = 1 , 2 , 5 , 10 , 50 , 100 , 10S; 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , n = 0 , 1 , 2 , x = 0 ( .1 ) 5 , 4–9D; 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , n = 0 ( 1 ) 20 ( 10 ) 50 , 100, x = 1 , 2 , 5 , 10 , 50 , 100 , 10S. (For the notation see §10.1 and §10.47(ii).)

  • Zhang and Jin (1996, pp. 296–305) tabulates 𝗃 n ( x ) , 𝗃 n ( x ) , 𝗒 n ( x ) , 𝗒 n ( x ) , 𝗂 n ( 1 ) ( x ) , 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , 𝗄 n ( x ) , n = 0 ( 1 ) 10 ( 10 ) 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; x 𝗃 n ( x ) , ( x 𝗃 n ( x ) ) , x 𝗒 n ( x ) , ( x 𝗒 n ( x ) ) (Riccati–Bessel functions and their derivatives), n = 0 ( 1 ) 10 ( 10 ) 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; real and imaginary parts of 𝗃 n ( z ) , 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗒 n ( z ) , 𝗂 n ( 1 ) ( z ) , 𝗂 n ( 1 ) ( z ) , 𝗄 n ( z ) , 𝗄 n ( z ) , n = 0 ( 1 ) 15 , 20(10)50, 100, z = 4 + 2 i , 20 + 10 i , 8S. (For the notation replace j , y , i , k by 𝗃 , 𝗒 , 𝗂 ( 1 ) , 𝗄 , respectively.)

  • §10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions
  • Olver (1960) tabulates a n , m , 𝗃 n ( a n , m ) , b n , m , 𝗒 n ( b n , m ) , n = 1 ( 1 ) 20 , m = 1 ( 1 ) 50 , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n .

  • 8: 10.52 Limiting Forms
    §10.52 Limiting Forms
    10.52.1 𝗃 n ( z ) , 𝗂 n ( 1 ) ( z ) z n / ( 2 n + 1 ) !! ,
    𝗁 n ( 1 ) ( z ) i n 1 z 1 e i z ,
    9: 10.50 Wronskians and Cross-Products
    §10.50 Wronskians and Cross-Products
    𝒲 { 𝗁 n ( 1 ) ( z ) , 𝗁 n ( 2 ) ( z ) } = 2 i z 2 .
    𝒲 { 𝗂 n ( 1 ) ( z ) , 𝗂 n ( 2 ) ( z ) } = ( 1 ) n + 1 z 2 ,
    𝒲 { 𝗂 n ( 1 ) ( z ) , 𝗄 n ( z ) } = 𝒲 { 𝗂 n ( 2 ) ( z ) , 𝗄 n ( z ) } = 1 2 π z 2 .
    Results corresponding to (10.50.3) and (10.50.4) for 𝗂 n ( 1 ) ( z ) and 𝗂 n ( 2 ) ( z ) are obtainable via (10.47.12).
    10: 10.47 Definitions and Basic Properties
    Equation (10.47.1)
    Equation (10.47.2)
    §10.47(iv) Interrelations