# §30.6 Functions of Complex Argument

The solutions

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

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

 30.6.2 $\displaystyle\mathit{Ps}^{m}_{n}\left(z,0\right)$ $\displaystyle=P^{m}_{n}\left(z\right),$ $\displaystyle\mathit{Qs}^{m}_{n}\left(z,0\right)$ $\displaystyle=Q^{m}_{n}\left(z\right);$

compare §14.3(ii).

## Wronskian

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

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

## Values on $(-1,1)$

 30.6.4 $\displaystyle\mathit{Ps}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)$ $\displaystyle=(\mp\mathrm{i})^{m}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$ 30.6.5 $\displaystyle\mathit{Qs}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)$ $\displaystyle={(\mp\mathrm{i})^{m}\left(\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}% \right)\mp\tfrac{1}{2}\mathrm{i}\pi\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right% )\right)}.$

For further properties see Arscott (1964b).

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