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

§30.1 Special Notation

ⓘ

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

ⓘ

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).

30 Spheroidal Wave Functions 30.2 Differential Equations

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§30.1 Special Notation

Other Notations

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!