§30.1 Special Notation

Keywords:: spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.1

(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 $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ and the spheroidal wave functions $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ , $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ , $\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ , $\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ , and $\mathop{S^{{m(j)}}_{{n}}\/}\nolimits\!\left(z,\gamma\right)$ , $j=1,2,3,4$ . 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 $\mathop{\mathsf{Ps}\/}\nolimits$ , $\mathop{\mathsf{Qs}\/}\nolimits$ , $\mathop{\mathit{Ps}\/}\nolimits$ , $\mathop{\mathit{Qs}\/}\nolimits$ , respectively.

¶ Other Notations

Keywords:: spheroidal coordinates, spheroidal wave functions

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

30.1.1

$S^{{(1)}}_{{mn}}(\gamma,x)=d_{{mn}}(\gamma)\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right),$

$S^{{(2)}}_{{mn}}(\gamma,x)=d_{{mn}}(\gamma)\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right),$

Symbols:: $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : 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
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where $d_{{mn}}(\gamma)$ is a normalization constant determined by

30.1.2

$S^{{(1)}}_{{mn}}(\gamma,0)=(-1)^{m}\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(0\right),$ $n-m$ even,

$\left.\frac{d}{dx}S^{{(1)}}_{{mn}}(\gamma,x)\right|_{{x=0}}=(-1)^{m}\left.\frac{d}{dx}\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)\right|_{{x=0}},$ $n-m$ odd.

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\frac{df}{dx}$ : 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
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For older notations see Abramowitz and Stegun (1964, §21.11) and Flammer (1957, pp. 14,15).