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1: 34.12 Physical Applications
§34.12 Physical Applications
β–Ί 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
β–Ί
§14.31(ii) Conical Functions
β–ΊThe conical functions 𝖯 1 2 + i ⁒ Ο„ 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)). … β–ΊMany 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
β–Ί
§29.19(i) Lamé Functions
β–ΊSimply-periodic Lamé functions ( Ξ½ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
4: 10.73 Physical Applications
β–ΊBessel 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 . β–ΊLaplace’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
β–ΊThe 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)). … β–Ίand 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
β–ΊAs an example, Laplace’s equation 2 W = 0 in spherical coordinates (§1.5(ii)): …
7: 14.19 Toroidal (or Ring) Functions
β–ΊThis form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates ( Ξ· , ΞΈ , Ο• ) , which are related to Cartesian coordinates ( x , y , z ) by …
8: 1.14 Integral Transforms
β–Ί
1.14.19 β„’ ⁑ f ⁑ ( s ) 0 , ⁑ s .
β–Ί
1.14.21 β„’ ⁑ f ⁑ ( s a ) = β„’ ⁑ f a ⁑ ( s ) ,
β–Ί
1.14.27 β„’ ⁑ ( f ) ⁑ ( s ) = s ⁒ β„’ ⁑ ( f ) ⁑ ( s ) f ⁑ ( 0 + ) .
β–Ί
1.14.29 β„’ ⁑ ( f ( n ) ) ⁑ ( s ) = s n ⁒ β„’ ⁑ ( f ) ⁑ ( s ) s n 1 ⁒ f ⁑ ( 0 + ) s n 2 ⁒ f ⁑ ( 0 + ) β‹― f ( n 1 ) ⁑ ( 0 + ) .
β–Ί
1.14.31 β„’ ⁑ ( f g ) = β„’ ⁑ ( f ) ⁒ β„’ ⁑ ( g ) .
9: 2.5 Mellin Transform Methods
β–Ί
2.5.37 β„’ ⁑ h ⁑ ( ΞΆ ) = 0 h ⁑ ( t ) ⁒ e ΞΆ ⁒ t ⁒ d t .
β–Ί
2.5.38 ΞΆ ⁒ β„’ ⁑ h ⁑ ( ΞΆ ) = I 1 ⁑ ( x ) + I 2 ⁑ ( x ) ,
β–Ί
2.5.45 β„’ ⁑ h ⁑ ( ΞΆ ) = 0 e ΞΆ ⁒ t 1 + t ⁒ d t , ⁑ ΞΆ > 0 .
β–Ί
2.5.48 β„’ ⁑ h ⁑ ( ΞΆ ) ( ln ⁑ ΞΆ ) ⁒ k = 0 ΞΆ k k ! + k = 0 ψ ⁑ ( k + 1 ) ⁒ ΞΆ k k ! , ΞΆ 0 + .
β–Ί
2.5.49 β„’ ⁑ h ⁑ ( ΞΆ ) = e ΞΆ ⁒ E 1 ⁑ ( ΞΆ ) ;
10: 35.2 Laplace Transform
β–Ί
35.2.1 g ⁑ ( 𝐙 ) = 𝛀 etr ⁑ ( 𝐙 ⁒ 𝐗 ) ⁒ f ⁑ ( 𝐗 ) ⁒ d 𝐗 ,
β–Ί