SpheroidalEigenvalue

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

λ==(Subscript[λ, ν,μ])(γ) is the spheroidal eigenvalue of degree ν and order μ of the corresponding Sturm-Liouville 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. 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). (Subscript[λ, ν,μ])(γ)  is an analytical function of the variables ν, μ and γ.

Specific values

Specialized values

For fixed ν , μ

For fixed ν , λ

For fixed λ

General characteristics

Domain and analyticity

(Subscript[λ, ν,μ])(γ) is an analytical function of ν, μ and γ which is defined in (ℂ^3).

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

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