SpheroidalPS

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

(Subscript[PS, ν,μ])(γ,z) is the angular spheroidal function of the first kind with variable z and parameters ν,  μ,  γ.  It is defined as the normalizable solution w(z)==(Subscript[PS, ν,μ])(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 spherical limit (γ->0), the eigenvalues are (Subscript[λ, ν,μ])(0)==ν(ν+1) and (Subscript[PS, ν,μ])(0,z)==(Subscript[P, ν]^μ)(z), where (Subscript[P, ν]^μ)(z) is the Legendre function of the first kind. The angular spheroidal functions are normalized according to the Meixner-Schäfke normalization scheme, meaning (∫_(-1)^1)(Subscript[PS, (Subscript[n, 1]),m])(γ,η) (Subscript[PS, (Subscript[n, 2]),m])(γ,η)dη==2 (m+(Subscript[n, 1]))! /((2 (Subscript[n, 1])+1) ((Subscript[n, 1])-m)!)(Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2])])/;(Subscript[n, 1])∈ℕ∧(Subscript[n, 2])∈ℕ∧m∈ℕ∧(Subscript[n, 1])≥m∧(Subscript[n, 2])≥m. (Subscript[PS, ν,μ])(γ,z)  is an analytical function in the variables ν, μ, γ and z.

Specific values

Specialized values

For fixed ν , μ , z

For fixed ν , γ , z

For fixed ν , z

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Parity

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

Mirror symmetry

Periodicity

No periodicity

Series representations

Generalized power series

Expansions at γ==0

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

Expansions at z==0

Other series representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Differentiation

Low-order differentiation

With respect to z

Integration

Definite integration

Involving the direct function

Representations through equivalent functions

Theorems

History

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