# §14.21(i) Associated Legendre Equation

 14.21.1 $\left(1-z^{2}\right)\frac{{d}^{2}w}{{dz}^{2}}-2z\frac{dw}{dz}+\left(\nu(\nu+1)% -\frac{\mu^{2}}{1-z^{2}}\right)w=0.$

Standard solutions: the associated Legendre functions $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$, $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(z\right)$, $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$, and $\mathop{\boldsymbol{Q}^{\mu}_{-\nu-1}\/}\nolimits\!\left(z\right)$. $\mathop{P^{\pm\mu}_{\nu}\/}\nolimits\!\left(z\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\infty$, which are branch points (or poles) of the functions, in general. When $z$ is complex $\mathop{P^{\pm\mu}_{\nu}\/}\nolimits\!\left(z\right)$, $\mathop{Q^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$, and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in(1,\infty)$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\infty,1]$; compare §4.2(i). The principal branches of $\mathop{P^{\pm\mu}_{\nu}\/}\nolimits\!\left(z\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$ are real when $\nu$, $\mu\in\Real$ and $z\in(1,\infty)$.

# §14.21(ii) Numerically Satisfactory Solutions

When $\realpart{\nu}\geq-\frac{1}{2}$ and $\realpart{\mu}\geq 0$, a numerically satisfactory pair of solutions of (14.21.1) in the half-plane $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{2}\pi$ is given by $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(z\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$.

# §14.21(iii) Properties

Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\infty)$ to the cut $z$-plane $\Complex\backslash(-\infty,1]$. This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). The generating function expansions (14.7.19) (with $\mathop{\mathsf{P}\/}\nolimits$ replaced by $\mathop{P\/}\nolimits$) and (14.7.22) apply when $|h|<\min\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$; (14.7.21) (with $\mathop{\mathsf{P}\/}\nolimits$ replaced by $\mathop{P\/}\nolimits$) applies when $|h|>\max\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$.