# §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)

where

 30.11.2 $\displaystyle{\cal C}_{\nu}^{(1)}$ $\displaystyle=\mathop{J_{\nu}\/}\nolimits,$ $\displaystyle{\cal C}_{\nu}^{(2)}$ $\displaystyle=\mathop{Y_{\nu}\/}\nolimits,$ $\displaystyle{\cal C}_{\nu}^{(3)}$ $\displaystyle=\mathop{{H^{(1)}_{\nu}}\/}\nolimits,$ $\displaystyle{\cal C}_{\nu}^{(4)}$ $\displaystyle=\mathop{{H^{(2)}_{\nu}}\/}\nolimits,$ Defines: ${\cal C}_{\nu}^{(j)}$: a Bessel function (locally) Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{Y_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the second kind, $\mathop{{H^{(1)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function) and $\mathop{{H^{(2)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

 30.11.3 $\mathop{S^{m(j)}_{n}\/}\nolimits\!\left(z,\gamma\right)=\frac{(1-z^{-2})^{% \frac{1}{2}m}}{A_{n}^{-m}(\gamma^{2})}\sum_{2k\geq m-n}a^{-m}_{n,k}(\gamma^{2}% )\psi_{n+2k}^{(j)}(\gamma z).$ Defines: $\mathop{S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{% \gamma}\right)$: 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)

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)

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

### Connection Formulas

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

## §30.11(iii) Asymptotic Behavior

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

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

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

## §30.11(iv) Wronskian

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

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

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

where

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

or

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

## §30.11(vi) Integral Representations

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

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

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).