30 Spheroidal Wave Functions Properties 30.5 Functions of the Second Kind 30.7 Graphics

§30.6 Functions of Complex Argument

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Defines:: $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument and $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument
Keywords:: of complex argument, spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §§3.6, 3.63).
Permalink:: http://dlmf.nist.gov/30.6
See also:: Annotations for Ch.30

The solutions

30.6.1		$\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right),$
30.6.1		$\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right),$
ⓘ Symbols: $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.6.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.6 and Ch.30

of (30.2.1) with $\mu=m$ and $\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)$ are real when $z\in(1,\infty)$ , and their principal values (§4.2(i)) are obtained by analytic continuation to $\mathbb{C}\setminus(-\infty,1]$ .

Relations to Associated Legendre Functions

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See also:: Annotations for §30.6 and Ch.30

30.6.2	$\displaystyle\mathit{Ps}^{m}_{n}\left(z,0\right)$	$\displaystyle=P^{m}_{n}\left(z\right),$
30.6.2	$\displaystyle\mathit{Qs}^{m}_{n}\left(z,0\right)$	$\displaystyle=Q^{m}_{n}\left(z\right);$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer and $n\geq m$ : integer degree Permalink: http://dlmf.nist.gov/30.6.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.6, §30.6 and Ch.30

compare §14.3(ii).

Wronskian

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See also:: Annotations for §30.6 and Ch.30

30.6.3		$\mathscr{W}\left\{\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right),\mathit{Qs}^{m}% _{n}\left(z,\gamma^{2}\right)\right\}=\frac{(-1)^{m}(n+m)!}{(1-z^{2})(n-m)!}A_% {n}^{m}(\gamma^{2})A_{n}^{-m}(\gamma^{2}),$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $!$ : factorial (as in $n!$ ), $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.6.E3 Encodings: TeX, pMML, png See also: Annotations for §30.6, §30.6 and Ch.30

with $A_{n}^{\pm m}(\gamma^{2})$ as in (30.11.4).

Values on $(-1,1)$

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Keywords:: spheroidal differential equation, spheroidal wave functions, with complex parameter, with complex parameters
See also:: Annotations for §30.6 and Ch.30

30.6.4	$\displaystyle\mathit{Ps}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)$	$\displaystyle=(\mp\mathrm{i})^{m}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.6.E4 Encodings: TeX, pMML, png See also: Annotations for §30.6, §30.6 and Ch.30
30.6.5	$\displaystyle\mathit{Qs}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)$	$\displaystyle={(\mp\mathrm{i})^{m}\left(\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}% \right)\mp\tfrac{1}{2}\mathrm{i}\pi\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right% )\right)}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : 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 and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.6.E5 Encodings: TeX, pMML, png See also: Annotations for §30.6, §30.6 and Ch.30

For further properties see Arscott (1964b).

For results for Equation (30.2.1) with complex parameters see Meixner and Schäfke (1954).

§30.6 Functions of Complex Argument

Relations to Associated Legendre Functions

Wronskian

Values on (-1,1)

Values on $(-1,1)$