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

§14.27 Zeros

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

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

(a)

$\mu<0$ , $\mu\notin\Integer$ , $\nu\in\Integer$ , and $\mathop{\sin\/}\nolimits\!\left((\mu-\nu)\pi\right)$ and $\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)$ have opposite signs.
(b)

$\mu,\nu\in\Integer$ , $\mu+\nu<0$ , and $\nu$ is odd.

For all other values of the parameters $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$ .

For complex zeros of $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ see Hobson (1931, §§233, 234, and 238).

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