14 Legendre and Related Functions Real Arguments 14.15 Uniform Asymptotic Approximations 14.17 Integrals

§14.16 Zeros

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Permalink:: http://dlmf.nist.gov/14.16
See also:: Annotations for Ch.14

§14.16(i) Notation
§14.16(ii) Interval
§14.16(iii) Interval $1<x<\infty$

§14.16(i) Notation

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Permalink:: http://dlmf.nist.gov/14.16.i
See also:: Annotations for §14.16 and Ch.14

Throughout this section we assume that $\mu$ and $\nu$ are real, and when they are not integers we write

14.16.1

$\mu=m+\delta_{\mu},$

$\nu=n+\delta_{\nu},$

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Symbols:: : nonnegative integer, : nonnegative integer, $\mu$ : general order, $\nu$ : general degree and $\delta_{\mu}$
Permalink:: http://dlmf.nist.gov/14.16.E1
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §14.16(i), §14.16 and Ch.14

where , $n\in\mathbb{Z}$ and $\delta_{\mu}$ , $\delta_{\nu}\in(0,1)$ . For all cases concerning $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $P^{\mu}_{\nu}\left(x\right)$ we assume that $\nu\geq-\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)).

§14.16(ii) Interval

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Keywords:: Ferrers functions, zeros
Notes:: See Hobson (1931, pp. 386–388 and 399–401).
Permalink:: http://dlmf.nist.gov/14.16.ii
See also:: Annotations for §14.16 and Ch.14

The number of zeros of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ in the interval is $\max(\lceil\nu-|\mu|\rceil,0)$ if any of the following sets of conditions hold:

(a)

$\mu\leq 0$ .
(b)

$\mu>0$ , $n\geq m$ , and $\delta_{\nu}>\delta_{\mu}$ .
(c)

$\mu>0$ , , and is odd.
(d)

$\nu=0,1,2,3,\dots$ .

The number of zeros of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ in the interval is $\max(\lceil\nu-|\mu|\rceil,0)+1$ if either of the following sets of conditions holds:

(a)

$\mu>0$ , , and $\delta_{\nu}\leq\delta_{\mu}$ .
(b)

$\mu>0$ , , and is even.

The zeros of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ in the interval interlace those of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ . $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ has $\max(\lceil\nu-|\mu|\rceil,0)+k$ zeros in the interval , where can take one of the values −1, 0, 1, 2, subject to $\max(\lceil\nu-|\mu|\rceil,0)+k$ being even or odd according as $\cos\left(\nu\pi\right)$ and $\cos\left(\mu\pi\right)$ have opposite signs or the same sign. In the special case $\mu=0$ and $\nu=n=0,1,2,3,\dots$ , $\mathsf{Q}_{n}\left(x\right)$ has zeros in the interval .

For uniform asymptotic approximations for the zeros of $\mathsf{P}^{-m}_{n}\left(x\right)$ in the interval when $n\to\infty$ with $(\geq 0)$ fixed, see Olver (1997b, p. 469).

§14.16(iii) Interval $1<x<\infty$

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Keywords:: associated Legendre functions, zeros
Notes:: See Hobson (1931, pp. 388–389 and 399).
Permalink:: http://dlmf.nist.gov/14.16.iii
See also:: Annotations for §14.16 and Ch.14

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

(a)

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

$\mu\leq\nu$ , $\mu\notin\mathbb{Z}$ , and $\lfloor\mu\rfloor$ is odd.

For all other values of $\mu$ and $\nu$ (with $\nu\geq-\frac{1}{2}$ ) $P^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$ .

$\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$ when $\nu>-1$ , and at most one zero in the interval $(1,\infty)$ when $\nu<-1$ .

14.15 Uniform Asymptotic Approximations 14.17 Integrals

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§14.16 Zeros

Contents

§14.16(i) Notation

§14.16(ii) Interval

§14.16(iii) Interval

§14.16(iii) Interval $1<x<\infty$