# 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

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►Standard solutions: the associated Legendre functions
${P}_{\nu}^{\mu}\left(z\right)$, ${P}_{\nu}^{-\mu}\left(z\right)$, ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$, and ${\bm{Q}}_{-\nu -1}^{\mu}\left(z\right)$.
${P}_{\nu}^{\pm \mu}\left(z\right)$ and ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$ 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 ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$ 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 ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$ are real when $\nu $, $\mu \in \mathbb{R}$ and $z\in (1,\mathrm{\infty})$.
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►The generating function expansions (14.7.19) (with $\U0001d5af$ replaced by $P$) and (14.7.22) apply when $$; (14.7.21) (with $\U0001d5af$ replaced by $P$) applies when $|h|>\mathrm{max}\left|z\pm {\left({z}^{2}-1\right)}^{1/2}\right|$.