30 Spheroidal Wave Functions Notation 30 Spheroidal Wave Functions 30.2 Differential Equations

§30.1 Special Notation

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Keywords:: notation, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.1
See also:: Annotations for Ch.30

(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, $-1<x<1$ .
$\gamma^{2}$	real parameter (positive, zero, or negative).
$m$	order, a nonnegative integer.
$n$	degree, an integer $n=m,m+1,m+2,\dots$ .
$k$	integer.
$\delta$	arbitrary small positive constant.

The main functions treated in this chapter are the eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ , $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$ , $\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right)$ , $\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right)$ , and $S^{m(j)}_{n}\left(z,\gamma\right)$ , $j=1,2,3,4$ . These notations are similar to those used in Arscott (1964b) and Erdélyi et al. (1955). Meixner and Schäfke (1954) use $\mathrm{ps}$ , $\mathrm{qs}$ , $\mathrm{Ps}$ , $\mathrm{Qs}$ for $\mathsf{Ps}$ , $\mathsf{Qs}$ , $\mathit{Ps}$ , $\mathit{Qs}$ , respectively.

Other Notations

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Keywords:: other notations, spheroidal coordinates, spheroidal wave functions
See also:: Annotations for §30.1 and Ch.30

Flammer (1957) and Abramowitz and Stegun (1964) use $\lambda_{mn}(\gamma)$ for $\lambda^{m}_{n}\left(\gamma^{2}\right)+\gamma^{2}$ , $R_{mn}^{(j)}(\gamma,z)$ for $S^{m(j)}_{n}\left(z,\gamma\right)$ , and

30.1.1	$\displaystyle S^{(1)}_{mn}(\gamma,x)$	$\displaystyle=d_{mn}(\gamma)\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$
30.1.1	$\displaystyle S^{(2)}_{mn}(\gamma,x)$	$\displaystyle=d_{mn}(\gamma)\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right),$
ⓘ Symbols: $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $d_{mn}(\gamma)$ : normalization constant and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.1.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.1, §30.1 and Ch.30

where $d_{mn}(\gamma)$ is a normalization constant determined by

30.1.2		$\displaystyle S^{(1)}_{mn}(\gamma,0)$	$\displaystyle=(-1)^{m}\mathsf{P}^{m}_{n}\left(0\right),$
	$n-m$ even,
		$\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}x}S^{(1)}_{mn}(\gamma,x)\right\|% _{x=0}$	$\displaystyle=(-1)^{m}\left.\frac{\mathrm{d}}{\mathrm{d}x}\mathsf{P}^{m}_{n}% \left(x\right)\right\|_{x=0},$
	$n-m$ odd.
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter A&S Ref: 21.7.13 21.7.14 Permalink: http://dlmf.nist.gov/30.1.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.1, §30.1 and Ch.30

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