§30.6 Functions of Complex Argument

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

The solutions

30.6.1

$\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right),$

$\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right),$

Symbols:: $\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : spheroidal wave function of complex argument, $\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : 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

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

¶ Relations to Associated Legendre Functions

30.6.2

$\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,0\right)=\mathop{P^{{m}}_{{n}}\/}\nolimits\!\left(z\right),$

$\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,0\right)=\mathop{Q^{{m}}_{{n}}\/}\nolimits\!\left(z\right);$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : spheroidal wave function of complex argument, $\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : 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

compare §14.3(ii).

¶ Wronskian

30.6.3 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right),\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)\right\}=\frac{(-1)^{m}(n+m)!}{(1-z^{2})(n-m)!}A_{n}^{m}(\gamma^{2})A_{n}^{{-m}}(\gamma^{2}),$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $!$ : $n!$ : factorial, $\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : spheroidal wave function of complex argument, $\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : 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

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

¶ Values on $(-1,1)$

Keywords:: spheroidal differential equation, spheroidal wave functions

30.6.4 $\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x\pm i0,\gamma^{2}\right)=(\mp i)^{m}\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right),$

Symbols:: $\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : spheroidal wave function of complex argument, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : 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

30.6.5 $\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x\pm i0,\gamma^{2}\right)={(\mp i)^{m}\left(\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)\mp\tfrac{1}{2}i\pi\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)\right)}.$

Symbols:: $\mathop{\mathit{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : spheroidal wave function of complex argument, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : 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 and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.6.E5
Encodings:: TeX, pMML, png

For further properties see Arscott (1964b).

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