§30.6 Functions of Complex Argument

The solutions

 30.6.1 $\mathop{\mathit{Ps}^{m}_{n}\/}\nolimits\!\left(z,\gamma^{2}\right),$ $\mathop{\mathit{Qs}^{m}_{n}\/}\nolimits\!\left(z,\gamma^{2}\right),$

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

compare §14.3(ii).

¶ Wronskian

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

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

¶ Values on $(-1,1)$

 30.6.4 $\displaystyle\mathop{\mathit{Ps}^{m}_{n}\/}\nolimits\!\left(x\pm i0,\gamma^{2}\right)$ $\displaystyle=(\mp i)^{m}\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,% \gamma^{2}\right),$ 30.6.5 $\displaystyle\mathop{\mathit{Qs}^{m}_{n}\/}\nolimits\!\left(x\pm i0,\gamma^{2}\right)$ $\displaystyle={(\mp i)^{m}\left(\mathop{\mathsf{Qs}^{m}_{n}\/}\nolimits\!\left% (x,\gamma^{2}\right)\mp\tfrac{1}{2}i\pi\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits% \!\left(x,\gamma^{2}\right)\right)}.$

For further properties see Arscott (1964b).

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