DLMF: Untitled Document

14.10.1

{\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,

â“˜

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.10
Permalink:: http://dlmf.nist.gov/14.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

… â–º

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

also satisfies (14.10.1)–(14.10.5). â–º

14.10.6

{P^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}P^{\mu+1}_{% \nu}\left(x\right)}-(\nu-\mu)(\nu+\mu+1)P^{\mu}_{\nu}\left(x\right)=0,

â“˜

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.10, §14.14
Permalink:: http://dlmf.nist.gov/14.10.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

… â–º

Q^{\mu}_{\nu}\left(x\right)

also satisfies (14.10.6) and (14.10.7). In addition,

P^{\mu}_{\nu}\left(x\right)

and

Q^{\mu}_{\nu}\left(x\right)

satisfy (14.10.3)–(14.10.5).

§13.20 Uniform Asymptotic Approximations for Large $\mu$

… â–º

§13.20(iii) Large $\mu$ , $-(1-\delta)\mu\leq\kappa\leq\mu$

… â–º(a) In the case

-\mu<\kappa<\mu

… â–ºwhen

\mu<\kappa

, and by (13.20.10) when

\mu=\kappa

. … â–º

§13.20(v) Large $\mu$ , Other Expansions

…

… â–º

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: â–º

(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.

â–º

(b)

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

â–ºFor all other values of the parameters

P^{\mu}_{\nu}\left(x\pm i0\right)

has no zeros in the interval

(-\infty,-1)

. â–ºFor complex zeros of

P^{\mu}_{\nu}\left(z\right)

see Hobson (1931, §§233, 234, and 238).

… â–º

13.25.1

M_{\kappa,\mu}\left(z\right)M_{\kappa,-\mu-1}\left(z\right)+\frac{(\frac{1}{2}% +\mu+\kappa)(\frac{1}{2}+\mu-\kappa)}{4\mu(1+\mu)(1+2\mu)^{2}}M_{\kappa,\mu+1}% \left(z\right)M_{\kappa,-\mu}\left(z\right)=1.

â“˜

Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Referenced by:: §13.25
Permalink:: http://dlmf.nist.gov/13.25.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.25 and Ch.13

â–ºFor integral representations, integrals, and series containing products of

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

see Erdélyi et al. (1953a, §6.15.3).

… â–º

§14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$

… â–º

\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{\mu}_{\nu}\left(x\right),

â–º

\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{-\mu}_{\nu}\left(x\right),

… â–º

§14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$

… â–º

§14.9(iii) Connections Between $P^{\pm\mu}_{\nu}\left(x\right)$ , $P^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\boldsymbol{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)$

…

… â–º â–º â–º â–º

$x$ , $y$ , $\tau$	real variables.
…
$\mu$ , $\nu$	general order and degree, respectively.
…

… â–ºThe main functions treated in this chapter are the Legendre functions

\mathsf{P}_{\nu}\left(x\right)

,

\mathsf{Q}_{\nu}\left(x\right)

,

P_{\nu}\left(z\right)

,

Q_{\nu}\left(z\right)

; Ferrers functions

\mathsf{P}^{\mu}_{\nu}\left(x\right)

,

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

(also known as the Legendre functions on the cut); associated Legendre functions

P^{\mu}_{\nu}\left(z\right)

,

Q^{\mu}_{\nu}\left(z\right)

,

\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)

; conical functions

\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

,

\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

,

\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

,

P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

,

Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

(also known as Mehler functions). … â–ºAmong other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote

\mathsf{P}^{\mu}_{\nu}\left(x\right)

and

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

by

\mathrm{P}_{\nu}^{\mu}(x)

and

\mathrm{Q}_{\nu}^{\mu}(x)

, respectively. Magnus et al. (1966) denotes

\mathsf{P}^{\mu}_{\nu}\left(x\right)

,

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

,

P^{\mu}_{\nu}\left(z\right)

, and

Q^{\mu}_{\nu}\left(z\right)

by

P_{\nu}^{\mu}(x)

,

Q_{\nu}^{\mu}(x)

,

\mathfrak{P}_{\nu}^{\mu}(z)

, and

\mathfrak{Q}_{\nu}^{\mu}(z)

, respectively. Hobson (1931) denotes both

\mathsf{P}^{\mu}_{\nu}\left(x\right)

and

P^{\mu}_{\nu}\left(x\right)

by

P^{\mu}_{\nu}\left(x\right)

; similarly for

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

and

Q^{\mu}_{\nu}\left(x\right)

.

… â–º

13.15.2

2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-(z+2% \mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,

â“˜

Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Referenced by:: §13.15(i)
Permalink:: http://dlmf.nist.gov/13.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

â–º

13.15.3

(\kappa-\mu-\tfrac{1}{2})M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)+% (1+2\mu)\sqrt{z}M_{\kappa,\mu}\left(z\right)-(\kappa+\mu+\tfrac{1}{2})M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,

â“˜

Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

… â–º

13.15.5

2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)-2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1% }{2},\mu-\frac{1}{2}}\left(z\right)-(\kappa-\mu-\tfrac{1}{2})\sqrt{z}M_{\kappa% -\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,

â“˜

Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

â–º

13.15.6

2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)+(z-2% \mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2})\sqrt{z}M_{% \kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,

â“˜

Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

â–º

13.15.7

2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-2\mu(% 1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}M_{\kappa% +\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0.

â“˜

Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

…

… â–ºThroughout this section we assume that

\mu

and

\nu

are real, and when they are not integers we write … â–º

(a)

$\mu\leq 0$ .

… â–º

(c)

$\mu>0$ , $n<m$ , and $m-n$ is odd.

… â–º

(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.

â–º

(b)

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

…

… â–º

See accompanying text — Figure 14.4.7: $\mathsf{P}^{-\mu}_{0}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

â–º

… â–º

…

… â–ºFrom (13.14.2) and (13.14.3)

M_{\kappa,\mu}\left(z\right)

has the same zeros as

M\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)

and

W_{\kappa,\mu}\left(z\right)

has the same zeros as

U\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)

, hence the results given in §13.9 can be adopted. â–ºAsymptotic approximations to the zeros when the parameters

\kappa

and/or

\mu

are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if

\mu(\geq 0)

is fixed and

\kappa(>0)

is large, then the

r

th positive zero

\phi_{r}

of

M_{\kappa,\mu}\left(z\right)

is given by â–º

13.22.1

\phi_{r}=\frac{j_{2\mu,r}^{2}}{4\kappa}+j_{2\mu,r}O\left(\kappa^{-\frac{3}{2}}% \right),

â“˜

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi_{r}$ : positive zeros and $j_{b,r}$ : positive zero of Bessel
Referenced by:: §13.22, §13.22
Permalink:: http://dlmf.nist.gov/13.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.22 and Ch.13

â–ºwhere

j_{2\mu,r}

is the

r

th positive zero of the Bessel function

J_{2\mu}\left(x\right)

(§10.21(i)). …

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1—10 of 162 matching pages

1: 14.10 Recurrence Relations and Derivatives

2: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20(iii) Large $\mu$ , $-(1-\delta)\mu\leq\kappa\leq\mu$

§13.20(v) Large $\mu$ , Other Expansions

3: 14.27 Zeros

4: 13.25 Products

5: 14.9 Connection Formulas

§14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$

§14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$

§14.9(iii) Connections Between $P^{\pm\mu}_{\nu}\left(x\right)$ , $P^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\boldsymbol{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)$

6: 14.1 Special Notation

7: 13.15 Recurrence Relations and Derivatives

8: 14.16 Zeros

9: 14.4 Graphics

10: 13.22 Zeros

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1—10 of 162 matching pages

1: 14.10 Recurrence Relations and Derivatives

2: 13.20 Uniform Asymptotic Approximations for Large Î¼

§13.20 Uniform Asymptotic Approximations for Large Î¼

§13.20(iii) Large Î¼ , − ( 1 − Î´ ) â¢ Î¼ ≤ Îº ≤ Î¼

§13.20(v) Large Î¼ , Other Expansions

3: 14.27 Zeros

4: 13.25 Products

5: 14.9 Connection Formulas

§14.9(i) Connections Between ð–¯ Î½ ± Î¼ â¡ ( x ) , ð–¯ − Î½ − 1 ± Î¼ â¡ ( x ) , ð–° Î½ ± Î¼ â¡ ( x ) , ð–° − Î½ − 1 Î¼ â¡ ( x )

§14.9(ii) Connections Between ð–¯ Î½ ± Î¼ â¡ ( ± x ) , ð–° Î½ − Î¼ â¡ ( ± x ) , ð–° Î½ Î¼ â¡ ( x )

§14.9(iii) Connections Between P Î½ ± Î¼ â¡ ( x ) , P − Î½ − 1 ± Î¼ â¡ ( x ) , ð‘¸ Î½ ± Î¼ â¡ ( x ) , ð‘¸ − Î½ − 1 Î¼ â¡ ( x )

6: 14.1 Special Notation

7: 13.15 Recurrence Relations and Derivatives

8: 14.16 Zeros

9: 14.4 Graphics

10: 13.22 Zeros

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2: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20(iii) Large $\mu$ , $-(1-\delta)\mu\leq\kappa\leq\mu$

§13.20(v) Large $\mu$ , Other Expansions

§14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$

§14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$

§14.9(iii) Connections Between $P^{\pm\mu}_{\nu}\left(x\right)$ , $P^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\boldsymbol{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)$