# §30.11 Radial Spheroidal Wave Functions

## §30.11(i) Definitions

Denote

 30.11.1 $\psi_{k}^{(j)}(z)=\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}{\cal C}_{k+\frac{1% }{2}}^{(j)}(z),$ $j=1,2,3,4$, ⓘ Defines: $\psi_{k}^{(j)}(z)$: spherical function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and ${\cal C}_{\nu}^{(j)}$: a Bessel function A&S Ref: 21.9.1 (in different form) Permalink: http://dlmf.nist.gov/30.11.E1 Encodings: TeX, pMML, png See also: Annotations for 30.11(i), 30.11 and 30

where

 30.11.2 $\displaystyle{\cal C}_{\nu}^{(1)}$ $\displaystyle=J_{\nu},$ $\displaystyle{\cal C}_{\nu}^{(2)}$ $\displaystyle=Y_{\nu},$ $\displaystyle{\cal C}_{\nu}^{(3)}$ $\displaystyle={H^{(1)}_{\nu}},$ $\displaystyle{\cal C}_{\nu}^{(4)}$ $\displaystyle={H^{(2)}_{\nu}},$ ⓘ Defines: ${\cal C}_{\nu}^{(j)}$: a Bessel function (locally) Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function) and ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function) Permalink: http://dlmf.nist.gov/30.11.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 30.11(i), 30.11 and 30

with $J_{\nu}$, $Y_{\nu}$, ${H^{(1)}_{\nu}}$, and ${H^{(2)}_{\nu}}$ as in §10.2(ii). Then solutions of (30.2.1) with $\mu=m$ and $\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)$ are given by

 30.11.3 $S^{m(j)}_{n}\left(z,\gamma\right)=\frac{(1-z^{-2})^{\frac{1}{2}m}}{A_{n}^{-m}(% \gamma^{2})}\sum_{2k\geq m-n}a^{-m}_{n,k}(\gamma^{2})\psi_{n+2k}^{(j)}(\gamma z).$ ⓘ Defines: $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$: radial spheroidal wave function Symbols: $z$: complex variable, $m$: nonnegative integer, $n\geq m$: integer degree, $\psi_{k}^{(j)}(z)$: spherical function, $A_{n}^{m}(\gamma^{2})$, $\gamma^{2}$: real parameter and $a^{m}_{n,k}(\gamma^{2})$: coefficients A&S Ref: 21.9.1 (in different form) Referenced by: §30.11(i), §30.16(iii) Permalink: http://dlmf.nist.gov/30.11.E3 Encodings: TeX, pMML, png See also: Annotations for 30.11(i), 30.11 and 30

Here $a^{-m}_{n,k}(\gamma^{2})$ is defined by (30.8.2) and (30.8.6), and

 30.11.4 $A_{n}^{\pm m}(\gamma^{2})=\sum_{2k\geq\mp m-n}(-1)^{k}a^{\pm m}_{n,k}(\gamma^{% 2})\quad(\neq 0).$ ⓘ Defines: $A_{n}^{m}(\gamma^{2})$ (locally) Symbols: $m$: nonnegative integer, $n\geq m$: integer degree, $\gamma^{2}$: real parameter and $a^{m}_{n,k}(\gamma^{2})$: coefficients Referenced by: §30.5, §30.6 Permalink: http://dlmf.nist.gov/30.11.E4 Encodings: TeX, pMML, png See also: Annotations for 30.11(i), 30.11 and 30

In (30.11.3) $z\neq 0$ when $j=1$, and $|z|>1$ when $j=2,3,4$.

### Connection Formulas

 30.11.5 $\displaystyle S^{m(3)}_{n}\left(z,\gamma\right)$ $\displaystyle=S^{m(1)}_{n}\left(z,\gamma\right)+\mathrm{i}S^{m(2)}_{n}\left(z,% \gamma\right),$ $\displaystyle S^{m(4)}_{n}\left(z,\gamma\right)$ $\displaystyle=S^{m(1)}_{n}\left(z,\gamma\right)-\mathrm{i}S^{m(2)}_{n}\left(z,% \gamma\right).$

## §30.11(iii) Asymptotic Behavior

For fixed $\gamma$, as $z\to\infty$ in the sector $|\operatorname{ph}z|\leq\pi-\delta$ ($<\pi$),

 30.11.6 $S^{m(j)}_{n}\left(z,\gamma\right)=\begin{cases}\psi_{n}^{(j)}(\gamma z)+O\left% (z^{-2}e^{|\Im z|}\right),&j=1,2,\\ \psi_{n}^{(j)}(\gamma z)\left(1+O\left(z^{-1}\right)\right),&j=3,4.\end{cases}$

For asymptotic expansions in negative powers of $z$ see Meixner and Schäfke (1954, p. 293).

## §30.11(iv) Wronskian

 30.11.7 $\mathscr{W}\left\{S^{m(1)}_{n}\left(z,\gamma\right),S^{m(2)}_{n}\left(z,\gamma% \right)\right\}=\frac{1}{\gamma(z^{2}-1)}.$

## §30.11(v) Connection with the $\mathit{Ps}$ and $\mathit{Qs}$ Functions

 30.11.8 $S^{m(1)}_{n}\left(z,\gamma\right)=K_{n}^{m}(\gamma)\mathit{Ps}^{m}_{n}\left(z,% \gamma^{2}\right),$
 30.11.9 $S^{m(2)}_{n}\left(z,\gamma\right)=\frac{(n-m)!}{(n+m)!}\frac{(-1)^{m+1}\mathit% {Qs}^{m}_{n}\left(z,\gamma^{2}\right)}{\gamma K_{n}^{m}(\gamma)A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})},$

where

 30.11.10 $K_{n}^{m}(\gamma)=\frac{\sqrt{\pi}}{2}\left(\frac{\gamma}{2}\right)^{m}\frac{(% -1)^{m}a_{n,\frac{1}{2}(m-n)}^{-m}(\gamma^{2})}{\Gamma\left(\frac{3}{2}+m% \right)A_{n}^{-m}(\gamma^{2})\mathsf{Ps}^{m}_{n}\left(0,\gamma^{2}\right)},$ $n-m$ even,

or

 30.11.11 $K_{n}^{m}(\gamma)=\frac{\sqrt{\pi}}{2}\left(\frac{\gamma}{2}\right)^{m+1}\*% \frac{(-1)^{m}a_{n,\frac{1}{2}(m-n+1)}^{-m}(\gamma^{2})}{\Gamma\left(\frac{5}{% 2}+m\right)A_{n}^{-m}(\gamma^{2})(\left.\ifrac{\mathrm{d}\mathsf{Ps}^{m}_{n}(z% ,\gamma^{2})}{\mathrm{d}z}\right|_{z=0})},$ $n-m$ odd.

## §30.11(vi) Integral Representations

When $z\in\mathbb{C}\setminus(-\infty,1]$

 30.11.12 $A_{n}^{-m}(\gamma^{2})S^{m(1)}_{n}\left(z,\gamma\right)=\frac{1}{2}{\mathrm{i}% ^{m+n}}\gamma^{m}\frac{(n-m)!}{(n+m)!}z^{m}(1-z^{-2})^{\frac{1}{2}m}\*\int_{-1% }^{1}e^{-\mathrm{i}\gamma zt}(1-t^{2})^{\frac{1}{2}m}\mathsf{Ps}^{m}_{n}\left(% t,\gamma^{2}\right)\mathrm{d}t.$

For further relations see Arscott (1964b, §8.6), Connett et al. (1993), Erdélyi et al. (1955, §16.13), Meixner and Schäfke (1954), and Meixner et al. (1980, §3.1).