14 Legendre and Related Functions Complex Arguments 14.26 Uniform Asymptotic Expansions 14.28 Sums

§14.27 Zeros

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Keywords:: associated Legendre functions, zeros
Notes:: See Hobson (1931, pp. 391–399).
Permalink:: http://dlmf.nist.gov/14.27
See also:: Annotations for Ch.14

$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds:

(a)

$\mu<0$ , $\mu\notin\mathbb{Z}$ , $\nu\in\mathbb{Z}$ , and $\sin\left((\mu-\nu)\pi\right)$ and $\sin\left(\mu\pi\right)$ have opposite signs.
(b)

$\mu,\nu\in\mathbb{Z}$ , $\mu+\nu<0$ , and $\nu$ is odd.

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).

14.26 Uniform Asymptotic Expansions 14.28 Sums

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