# P-function

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##### 1: 14.27 Zeros
###### §14.27 Zeros
$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P^{\mu}_{\nu}\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$. For complex zeros of $P^{\mu}_{\nu}\left(z\right)$ see Hobson (1931, §§233, 234, and 238).
##### 2: 14.21 Definitions and Basic Properties
Standard solutions: the associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $P^{-\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$, and $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(z\right)$. $P^{\pm\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ 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 $P^{\pm\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ 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 $P^{\pm\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ are real when $\nu$, $\mu\in\mathbb{R}$ and $z\in(1,\infty)$. … The generating function expansions (14.7.19) (with $\mathsf{P}$ replaced by $P$) and (14.7.22) apply when $|h|<\min\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$; (14.7.21) (with $\mathsf{P}$ replaced by $P$) applies when $|h|>\max\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$.