SpheroidalS2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

(Subscript[S, ν,μ]^((2)))(γ,z) is the radial spheroidal function of the second kind with variable z and parameters ν,  μ,  γ.  It is defined as the normalizable solution w(z)==(Subscript[S, ν,μ]^((2)))(γ,z) of the wave differential equation (1-(z^2)) w''(z)-2z w'(z)+(λ+(γ^2)(1-(z^2))-(μ^2)/(1-(z^2))) w(z)==0 with parameter λ equal to spheroidal eigenvalue λ==(Subscript[λ, ν,μ])(γ).  The parameter ν enumerates the spheroidal eigenvalues in such a manner that in the limit (γ->0), the eigenvalues are (Subscript[λ, ν,μ])(0)==ν (μ+1) and (lim_(γ->0)) (Subscript[S, ν,μ]^((2)))(γ,z/γ)==(Subscript[y, ν])(z), where (Subscript[y, ν])(z) is the spherical Bessel function of the second kind. The radial spheroidal functions are normalized according to the Meixner-Schäfke normalization scheme, meaning (lim_(z->∞)) (Subscript[S, ν,μ]^((2)))(γ,z/γ)/(Subscript[y, ν])(z)==1. (Subscript[S, ν,μ]^((2)))(γ,z) is an analytical function in the variables ν, μ, γ and z.

Specific values

General characteristics

Domain and analyticity

(Subscript[S, ν,μ]^((2)))(γ,z) is an analytical function of ν, μ, γ, z which is defined in (ℂ^4).

Symmetries and periodicities

Parity

(Subscript[S, ν,μ]^((2)))(γ,z) is an even function with respect to μ.

Mirror symmetry

Periodicity

No periodicity

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Differentiation

Low-order differentiation

With respect to z

Integration

Operations

Limit operation

Representations through equivalent functions

Theorems

History

Copyright