§30.11 Radial Spheroidal Wave Functions

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Keywords:: radial, radial spheroidal wave functions, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.11
See also:: Annotations for Ch.30

§30.11(i) Definitions

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Keywords:: definitions, radial spheroidal wave functions
Notes:: See Arscott (1964b, §8.5).
Permalink:: http://dlmf.nist.gov/30.11.i
See also:: Annotations for §30.11 and Ch.30

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 Ch.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 Ch.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 Ch.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 Ch.30

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

Connection Formulas

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Keywords:: connection formulas, radial spheroidal wave functions
See also:: Annotations for §30.11(i), §30.11 and Ch.30

30.11.5	$\displaystyle S^{m(3)}_{n}\left(z,\gamma\right)$	$\displaystyle=S^{m(1)}_{n}\left(z,\gamma\right)+\mathrm{i}S^{m(2)}_{n}\left(z,% \gamma\right),$
30.11.5	$\displaystyle S^{m(4)}_{n}\left(z,\gamma\right)$	$\displaystyle=S^{m(1)}_{n}\left(z,\gamma\right)-\mathrm{i}S^{m(2)}_{n}\left(z,% \gamma\right).$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.11.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.11(i), §30.11(i), §30.11 and Ch.30

§30.11(ii) Graphics

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Keywords:: graphics, radial, radial spheroidal wave functions, spheroidal wave functions
Notes:: Figures 30.11.1–30.11.4 were produced at NIST with the aid of Maple procedures provided by the author.
Referenced by:: §30.1
Permalink:: http://dlmf.nist.gov/30.11.ii
See also:: Annotations for §30.11 and Ch.30

See accompanying text — Figure 30.11.1: $S^{0(1)}_{n}\left(x,2\right)$ , $n=0,1$ , $1\leq x\leq 10$ . Magnify

§30.11(iii) Asymptotic Behavior

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Keywords:: asymptotic behavior for large variable, radial spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §3.64).
Permalink:: http://dlmf.nist.gov/30.11.iii
See also:: Annotations for §30.11 and Ch.30

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

30.11.6		$S^{m(j)}_{n}\left(z,\gamma\right)=\begin{cases}\psi_{n}^{(j)}(\gamma z)+O\left% (z^{-2}e^{\|\Im z\|}\right),&j=1,2,\\ \psi_{n}^{(j)}(\gamma z)\left(1+O\left(z^{-1}\right)\right),&j=3,4.\end{cases}$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\psi_{k}^{(j)}(z)$ : spherical function and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.11.E6 Encodings: TeX, pMML, png See also: Annotations for §30.11(iii), §30.11 and Ch.30

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

§30.11(iv) Wronskian

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Keywords:: Wronskian, radial spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §3.65).
Permalink:: http://dlmf.nist.gov/30.11.iv
See also:: Annotations for §30.11 and Ch.30

30.11.7		$\mathscr{W}\left\{S^{m(1)}_{n}\left(z,\gamma\right),S^{m(2)}_{n}\left(z,\gamma% \right)\right\}=\frac{1}{\gamma(z^{2}-1)}.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.11.E7 Encodings: TeX, pMML, png See also: Annotations for §30.11(iv), §30.11 and Ch.30

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

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Keywords:: connection with spheroidal wave functions, radial spheroidal wave functions
Notes:: See Erdélyi et al. (1955, §16.11) and Meixner and Schäfke (1954, §3.66).
Referenced by:: §30.15(i), item
Permalink:: http://dlmf.nist.gov/30.11.v
See also:: Annotations for §30.11 and Ch.30

30.11.8		$S^{m(1)}_{n}\left(z,\gamma\right)=K_{n}^{m}(\gamma)\mathit{Ps}^{m}_{n}\left(z,% \gamma^{2}\right),$
ⓘ Symbols: $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $K_{n}^{m}(\gamma)$ : connection coefficient and $\gamma^{2}$ : real parameter A&S Ref: 21.10.1 (in different form) Permalink: http://dlmf.nist.gov/30.11.E8 Encodings: TeX, pMML, png See also: Annotations for §30.11(v), §30.11 and Ch.30

30.11.9		$S^{m(2)}_{n}\left(z,\gamma\right)=\frac{(n-m)!}{(n+m)!}\frac{(-1)^{m+1}\mathit% {Qs}^{m}_{n}\left(z,\gamma^{2}\right)}{\gamma K_{n}^{m}(\gamma)A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ , $K_{n}^{m}(\gamma)$ : connection coefficient and $\gamma^{2}$ : real parameter A&S Ref: 21.10.2 (in different form) Permalink: http://dlmf.nist.gov/30.11.E9 Encodings: TeX, pMML, png See also: Annotations for §30.11(v), §30.11 and Ch.30

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,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ , $K_{n}^{m}(\gamma)$ : connection coefficient, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : coefficients Referenced by: §30.16(iii) Permalink: http://dlmf.nist.gov/30.11.E10 Encodings: TeX, pMML, png See also: Annotations for §30.11(v), §30.11 and Ch.30

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.
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ , $K_{n}^{m}(\gamma)$ : connection coefficient, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : coefficients Referenced by: §30.16(iii) Permalink: http://dlmf.nist.gov/30.11.E11 Encodings: TeX, pMML, png See also: Annotations for §30.11(v), §30.11 and Ch.30

§30.11(vi) Integral Representations

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Keywords:: integral representation, radial spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §3.84).
Permalink:: http://dlmf.nist.gov/30.11.vi
See also:: Annotations for §30.11 and Ch.30

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

30.11.12		$A_{n}^{-m}(\gamma^{2})S^{m(1)}_{n}\left(z,\gamma\right)=\frac{1}{2}{\mathrm{i}% }^{m+n}\gamma^{m}\frac{(n-m)!}{(n+m)!}z^{m}(1-z^{-2})^{\frac{1}{2}m}\*\int_{-1% }^{1}e^{-\mathrm{i}\gamma zt}(1-t^{2})^{\frac{1}{2}m}\mathsf{Ps}^{m}_{n}\left(% t,\gamma^{2}\right)\,\mathrm{d}t.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral, $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.11.E12 Encodings: TeX, pMML, png See also: Annotations for §30.11(vi), §30.11 and Ch.30

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).

§30.11 Radial Spheroidal Wave Functions

Contents

§30.11(i) Definitions

Connection Formulas

§30.11(ii) Graphics

§30.11(iii) Asymptotic Behavior

§30.11(iv) Wronskian

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

§30.11(vi) Integral Representations

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§30.11 Radial Spheroidal Wave Functions

Contents

§30.11(i) Definitions

Connection Formulas

§30.11(ii) Graphics

§30.11(iii) Asymptotic Behavior

§30.11(iv) Wronskian

§30.11(v) Connection with the 𝑃𝑠 and 𝑄𝑠 Functions

§30.11(vi) Integral Representations

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