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1: 34.12 Physical Applications
§34.12 Physical Applications
3 j , 6 j , and 9 j 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
§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 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)). …
3: 29.19 Physical Applications
§29.19(i) Lamé Functions
Simply-periodic Lamé functions ( ν noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
4: 10.73 Physical Applications
Bessel functions of the first kind, J n ( x ) , arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation 2 V = 0 , or by the Helmholtz equation ( 2 + k 2 ) ψ = 0 . 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. … …
5: 19.18 Derivatives and Differential Equations
The next four differential equations apply to the complete case of R F and R G in the form R - a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). … and U = R - a ( 1 2 , 1 2 ; z + i ρ , z - i ρ ) , with ρ = x 2 + y 2 , satisfies Laplace’s equation: …
6: 14.30 Spherical and Spheroidal Harmonics
As an example, Laplace’s equation 2 W = 0 in spherical coordinates (§1.5(ii)): …
7: 14.19 Toroidal (or Ring) Functions
This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates ( η , θ , ϕ ) , which are related to Cartesian coordinates ( x , y , z ) by …
8: 18.17 Integrals
§18.17(vi) Laplace Transforms
Jacobi
Laguerre
Hermite
18.17.40 0 e - a x L n ( α ) ( b x ) x z - 1 d x = Γ ( z + n ) n ! ( a - b ) n a - n - z F 1 2 ( - n , 1 + α - z 1 - n - z ; a a - b ) , a > 0 , z > 0 .
9: Bibliography B
  • P. M. Batchelder (1967) An Introduction to Linear Difference Equations. Dover Publications Inc., New York.
  • M. V. Berry (1980) Some Geometric Aspects of Wave Motion: Wavefront Dislocations, Diffraction Catastrophes, Diffractals. In Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Vol. 36, pp. 13–28.
  • P. Boalch (2005) From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.
  • R. W. Butler and A. T. A. Wood (2002) Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.
  • R. W. Butler and A. T. A. Wood (2003) Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.
  • 10: 15.4 Special Cases
    15.4.34 F ( 3 a , a ; 2 a ; e i π / 3 ) = π e i π a / 2 2 2 a Γ ( 1 2 + a ) 3 ( 3 a + 1 ) / 2 ( 1 Γ ( 1 3 + a ) Γ ( 2 3 ) + 1 Γ ( 2 3 + a ) Γ ( 1 3 ) ) ,