P-function

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1: 14.27 Zeros

§14.27 Zeros

► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ has no zeros in the interval $(-\mathrm{\infty },-1)$. ►For complex zeros of $P_{\nu }^{\mu }\left(z\right)$ see Hobson (1931, §§233, 234, and 238).

2: 14.21 Definitions and Basic Properties

… ►Standard solutions: the associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $P_{\nu }^{-\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{-\nu -1}^{\mu }\left(z\right)$. $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ 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 $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). The principal branches of $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are real when $\nu$, $\mu \in ℝ$ and $z\in (1,\mathrm{\infty })$. … ►The generating function expansions (14.7.19) (with $𝖯$ replaced by $P$) and (14.7.22) apply when ; (14.7.21) (with $𝖯$ replaced by $P$) applies when $|h|>\max \left|z\pm {\left(z^2-1\right)}^{1/2}\right|$.