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30 Spheroidal Wave FunctionsNotation

§30.1 Special Notation

(For other notation see Notation for the Special Functions.)

x real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, 1<x<1.
γ2 real parameter (positive, zero, or negative).
m order, a nonnegative integer.
n degree, an integer n=m,m+1,m+2,.
k integer.
δ arbitrary small positive constant.

The main functions treated in this chapter are the eigenvalues λnm(γ2) and the spheroidal wave functions 𝖯𝗌nm(x,γ2), 𝖰𝗌nm(x,γ2), 𝑃𝑠nm(z,γ2), 𝑄𝑠nm(z,γ2), and Snm(j)(z,γ), j=1,2,3,4. These notations are similar to those used in Arscott (1964b) and Erdélyi et al. (1955). Meixner and Schäfke (1954) use ps, qs, Ps, Qs for 𝖯𝗌, 𝖰𝗌, 𝑃𝑠, 𝑄𝑠, respectively.

Other Notations

Flammer (1957) and Abramowitz and Stegun (1964) use λmn(γ) for λnm(γ2)+γ2, Rmn(j)(γ,z) for Snm(j)(z,γ), and

30.1.1 Smn(1)(γ,x) =dmn(γ)𝖯𝗌nm(x,γ2),
Smn(2)(γ,x) =dmn(γ)𝖰𝗌nm(x,γ2),

where dmn(γ) is a normalization constant determined by

30.1.2 Smn(1)(γ,0) =(1)m𝖯nm(0),
nm even,
ddxSmn(1)(γ,x)|x=0 =(1)mddx𝖯nm(x)|x=0,
nm odd.

For older notations see Abramowitz and Stegun (1964, §21.11) and Flammer (1957, pp. 14,15).