14 Legendre and Related Functions Real Arguments 14.14 Continued Fractions 14.16 Zeros

§14.15 Uniform Asymptotic Approximations

Keywords:: Ferrers functions, associated Legendre functions
Referenced by:: §14.26, §14.32
Permalink:: http://dlmf.nist.gov/14.15

§14.15(i) Large $\mu$ , Fixed $\nu$
§14.15(ii) Large $\mu$ , $0\leq\nu+\frac{1}{2}\leq(1-\delta)\mu$
§14.15(iii) Large $\nu$ , Fixed $\mu$
§14.15(iv) Large $\nu$ , $0\leq\mu\leq(1-\delta)(\nu+\frac{1}{2})$
§14.15(v) Large $\nu$ , $(\nu+\frac{1}{2})\delta\leq\mu\leq(\nu+\frac{1}{2})/\delta$

§14.15(i) Large $\mu$ , Fixed $\nu$

Notes:: (14.15.1) may be derived from (14.3.1) and §15.12(ii). For (14.15.2) and (14.15.3) see Dunster (2003b).
Keywords:: Ferrers functions, associated Legendre functions
Referenced by:: §14.26, §2.1(v), §2.8(iv)
Permalink:: http://dlmf.nist.gov/14.15.i

For the interval with fixed $\nu$ , real $\mu$ , and arbitrary fixed values of the nonnegative integer ,

14.15.1 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(\pm x\right)=\left(% \frac{1\mp x}{1\pm x}\right)^{{\mu/2}}\left(\sum_{{j=0}}^{{J-1}}\frac{\left(% \nu+1\right)_{{j}}\left(-\nu\right)_{{j}}}{j!\mathop{\Gamma\/}\nolimits\!\left% (j+1+\mu\right)}\left(\frac{1\mp x}{2}\right)^{j}+\mathop{O\/}\nolimits\!\left% (\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(J+1+\mu\right)}\right)\right)$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : real variable, $\mu$ : general order, $\nu$ : general degree and : nonnegative integer
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.15.E1
Encodings:: TeX, pMML, png

as $\mu\to\infty$ , uniformly with respect to . In other words, the convergent hypergeometric series expansions of $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(\pm x\right)$ are also generalized (and uniform) asymptotic expansions as $\mu\to\infty$ , with scale $\ifrac{1}{\mathop{\Gamma\/}\nolimits\!\left(j+1+\mu\right)}$ , $j=0,1,2,\dots$ ; compare §2.1(v).

Provided that $\mu-\nu\notin\Integer$ the corresponding expansions for $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{Q}^{{\mp\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ can be obtained from the connection formulas (14.9.7), (14.9.9), and (14.9.10).

For the interval $1<x<\infty$ the following asymptotic approximations hold when $\mu\to\infty$ , with $\nu$ ( $\geq-\frac{1}{2}$ ) fixed, uniformly with respect to :

14.15.2 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{\mathop{\Gamma% \/}\nolimits\!\left(\mu+1\right)}\left(\frac{2\mu u}{\pi}\right)^{{1/2}}% \mathop{K_{{\nu+\frac{1}{2}}}\/}\nolimits\!\left(\mu u\right)\*\left(1+\mathop% {O\/}\nolimits\!\left(\frac{1}{\mu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : real variable, $\mu$ : general order, $\nu$ : general degree and
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.15.E2
Encodings:: TeX, pMML, png

14.15.3 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{% \mu^{{\nu+(1/2)}}}\left(\frac{\pi u}{2}\right)^{{1/2}}\mathop{I_{{\nu+\frac{1}% {2}}}\/}\nolimits\!\left(\mu u\right)\left(1+\mathop{O\/}\nolimits\!\left(% \frac{1}{\mu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : real variable, $\mu$ : general order, $\nu$ : general degree and
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.15.E3
Encodings:: TeX, pMML, png

where is given by (14.12.10). Here $\mathop{I\/}\nolimits$ and $\mathop{K\/}\nolimits$ are the modified Bessel functions (§10.25(ii)).

For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000).

§14.15(ii) Large $\mu$ , $0\leq\nu+\frac{1}{2}\leq(1-\delta)\mu$

Notes:: See Dunster (2003b).
Keywords:: Ferrers functions, associated Legendre functions
Referenced by:: §14.26, §2.8(vi)
Permalink:: http://dlmf.nist.gov/14.15.ii

In this and subsequent subsections $\delta$ denotes an arbitrary constant such that $0<\delta<1$ .

As $\mu\to\infty$ ,

14.15.4 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{% \mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)}\left(1-\alpha^{2}\right)^{{-% \mu/2}}\left(\frac{1-\alpha}{1+\alpha}\right)^{{(\nu/2)+(1/4)}}\*\left(\frac{p% }{x}\right)^{{1/2}}e^{{-\mu\rho}}\left(1+\mathop{O\/}\nolimits\!\left(\frac{1}% {\mu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : base of exponential function, : real variable, $\mu$ : general order, $\nu$ : general degree, $\alpha$ , and $\rho$
Referenced by:: §14.15(ii)
Permalink:: http://dlmf.nist.gov/14.15.E4
Encodings:: TeX, pMML, png

uniformly with respect to $x\in(-1,1)$ and $\nu+\tfrac{1}{2}\in[0,(1-\delta)\mu]$ , where

14.15.5 $\alpha=\frac{\nu+\frac{1}{2}}{\mu}\,(<1),$

Symbols:: $\mu$ : general order, $\nu$ : general degree and $\alpha$
Referenced by:: §14.15(ii)
Permalink:: http://dlmf.nist.gov/14.15.E5
Encodings:: TeX, pMML, png

14.15.6 $p=\frac{x}{\left(\alpha^{2}x^{2}+1-\alpha^{2}\right)^{{1/2}}},$

Symbols:: : real variable, $\alpha$ and
Permalink:: http://dlmf.nist.gov/14.15.E6
Encodings:: TeX, pMML, png

and

14.15.7 $\rho=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{1+p}{1-p}\right)+\frac{1}% {2}\alpha\mathop{\ln\/}\nolimits\!\left(\frac{1-\alpha p}{1+\alpha p}\right).$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\alpha$ , and $\rho$
Permalink:: http://dlmf.nist.gov/14.15.E7
Encodings:: TeX, pMML, png

With the same conditions, the corresponding approximation for $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right)$ is obtained by replacing $e^{{-\mu\rho}}$ by $e^{{\mu\rho}}$ on the right-hand side of (14.15.4). Approximations for $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{Q}^{{\mp\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ can then be achieved via (14.9.7), (14.9.9), and (14.9.10).

Next,

14.15.8 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\frac{2\mu}{\pi}% \right)^{{1/2}}\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)}\left(% \frac{1-\alpha}{1+\alpha}\right)^{{(\nu/2)+(1/4)}}\*\left(1-\alpha^{2}\right)^% {{-\mu/2}}\left(\frac{\alpha^{2}+\eta^{2}}{\alpha^{2}\left(x^{2}-1\right)+1}% \right)^{{1/4}}\*\mathop{K_{{\nu+\frac{1}{2}}}\/}\nolimits\!\left(\mu\eta% \right)\left(1+\mathop{O\/}\nolimits\!\left(\frac{1}{\mu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : real variable, $\mu$ : general order, $\nu$ : general degree, $\alpha$ and $\eta$
Permalink:: http://dlmf.nist.gov/14.15.E8
Encodings:: TeX, pMML, png

14.15.9 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\frac% {\pi}{2}\right)^{{1/2}}\left(\frac{e}{\mu}\right)^{{\nu+(1/2)}}\left(\frac{1-% \alpha}{1+\alpha}\right)^{{\mu/2}}\*\left(1-\alpha^{2}\right)^{{-(\nu/2)-(1/4)% }}\left(\frac{\alpha^{2}+\eta^{2}}{\alpha^{2}\left(x^{2}-1\right)+1}\right)^{{% 1/4}}\*\mathop{I_{{\nu+\frac{1}{2}}}\/}\nolimits\!\left(\mu\eta\right)\left(1+% \mathop{O\/}\nolimits\!\left(\frac{1}{\mu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : real variable, $\mu$ : general order, $\nu$ : general degree, $\alpha$ and $\eta$
Permalink:: http://dlmf.nist.gov/14.15.E9
Encodings:: TeX, pMML, png

uniformly with respect to $x\in(1,\infty)$ and $\nu+\tfrac{1}{2}\in[0,(1-\delta)\mu]$ . Here $\alpha$ is again given by (14.15.5), and $\eta$ is defined implicitly by

14.15.10 $\alpha\mathop{\ln\/}\nolimits\!\left(\left(\alpha^{2}+\eta^{2}\right)^{{1/2}}+% \alpha\right)-\alpha\mathop{\ln\/}\nolimits\eta-\left(\alpha^{2}+\eta^{2}% \right)^{{1/2}}=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{\left(1+\alpha% ^{2}\right)x^{2}+1-\alpha^{2}-2x\left(\alpha^{2}x^{2}-\alpha^{2}+1\right)^{{1/% 2}}}{\left(x^{2}-1\right)\left(1-\alpha^{2}\right)}\right)+\frac{1}{2}\alpha% \mathop{\ln\/}\nolimits\!\left(\frac{\alpha^{2}\left(2x^{2}-1\right)+1+2\alpha x% \left(\alpha^{2}x^{2}-\alpha^{2}+1\right)^{{1/2}}}{1-\alpha^{2}}\right).$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable, $\alpha$ and $\eta$
Permalink:: http://dlmf.nist.gov/14.15.E10
Encodings:: TeX, pMML, png

The interval $1<x<\infty$ is mapped one-to-one to the interval $0<\eta<\infty$ , with the points and $x=\infty$ corresponding to $\eta=\infty$ and $\eta=0$ , respectively. For asymptotic expansions and explicit error bounds, see Dunster (2003b).

§14.15(iii) Large $\nu$ , Fixed $\mu$

Notes:: See Olver (1997b, pp. 463–469).
Keywords:: Ferrers functions, associated Legendre functions
Referenced by:: §14.26, §18.15(iii), §2.8(iv)
Permalink:: http://dlmf.nist.gov/14.15.iii

For $\nu\to\infty$ and fixed $\mu$ ( $\geq 0$ ),

14.15.11 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)=\frac{1}{\nu^{{\mu}}}\left(\frac{\theta}{\mathop{\sin\/% }\nolimits\theta}\right)^{{1/2}}\left(\mathop{J_{{\mu}}\/}\nolimits\!\left(% \left(\nu+\tfrac{1}{2}\right)\theta\right)+\mathop{O\/}\nolimits\!\left(\frac{% 1}{\nu}\right)\mathop{\mathrm{env}J_{{\mu}}\/}\nolimits\!\left(\left(\nu+% \tfrac{1}{2}\right)\theta\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mu$ : general order, $\nu$ : general degree and $\theta$ : angle
Permalink:: http://dlmf.nist.gov/14.15.E11
Encodings:: TeX, pMML, png

14.15.12 $\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)=-\frac{\pi}{2\nu^{{\mu}}}\left(\frac{\theta}{\mathop{% \sin\/}\nolimits\theta}\right)^{{1/2}}\left(\mathop{Y_{{\mu}}\/}\nolimits\!% \left(\left(\nu+\tfrac{1}{2}\right)\theta\right)+\mathop{O\/}\nolimits\!\left(% \frac{1}{\nu}\right)\mathop{\mathrm{env}Y_{{\mu}}\/}\nolimits\!\left(\left(\nu% +\tfrac{1}{2}\right)\theta\right)\right),$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mu$ : general order, $\nu$ : general degree and $\theta$ : angle
Permalink:: http://dlmf.nist.gov/14.15.E12
Encodings:: TeX, pMML, png

uniformly for $\theta\in(0,\pi-\delta]$ . For the Bessel functions $\mathop{J\/}\nolimits$ and $\mathop{Y\/}\nolimits$ see §10.2(ii), and for the $\mathop{\mathrm{env}\/}\nolimits$ functions associated with $\mathop{J\/}\nolimits$ and $\mathop{Y\/}\nolimits$ see §2.8(iv).

Next,

14.15.13 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi% \right)=\frac{1}{\nu^{{\mu}}}\left(\frac{\xi}{\mathop{\sinh\/}\nolimits\xi}% \right)^{{1/2}}\mathop{I_{{\mu}}\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}% \right)\xi\right)\*\left(1+\mathop{O\/}\nolimits\!\left(\frac{1}{\nu}\right)% \right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mu$ : general order, $\nu$ : general degree and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/14.15.E13
Encodings:: TeX, pMML, png

14.15.14 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)=\frac{\nu^{{\mu}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+% \mu+1\right)}\left(\frac{\xi}{\mathop{\sinh\/}\nolimits\xi}\right)^{{1/2}}\*% \mathop{K_{{\mu}}\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)\xi\right)\*% \left(1+\mathop{O\/}\nolimits\!\left(\frac{1}{\nu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mu$ : general order, $\nu$ : general degree and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/14.15.E14
Encodings:: TeX, pMML, png

uniformly for $\xi\in(0,\infty)$ .

For asymptotic expansions and explicit error bounds, see Olver (1997b, Chapter 12, §§12, 13) and Jones (2001). For convergent series expansions see Dunster (2004).

See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)$ as $n\to\infty$ with $\theta$ fixed.

§14.15(iv) Large $\nu$ , $0\leq\mu\leq(1-\delta)(\nu+\frac{1}{2})$

Notes:: See Boyd and Dunster (1986).
Referenced by:: §14.26, §2.8(vi)
Permalink:: http://dlmf.nist.gov/14.15.iv

As $\nu\to\infty$ ,

14.15.15 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\beta\left(% \frac{y-\alpha^{2}}{1-\alpha^{2}-x^{2}}\right)^{{1/4}}\*\left(\mathop{J_{{\mu}% }\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{{1/2}}\right)+\mathop{O\/% }\nolimits\!\left(\frac{1}{\nu}\right)\mathop{\mathrm{env}J_{{\mu}}\/}% \nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{{1/2}}\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable, $\mu$ : general order, $\nu$ : general degree, $\beta$ , and $\alpha$
Referenced by:: §14.15(iv)
Permalink:: http://dlmf.nist.gov/14.15.E15
Encodings:: TeX, pMML, png

14.15.16 $\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{\pi% \beta}{2}\left(\frac{y-\alpha^{2}}{1-\alpha^{2}-x^{2}}\right)^{{1/4}}\left(% \mathop{Y_{{\mu}}\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{{1/2}}% \right)+\mathop{O\/}\nolimits\!\left(\frac{1}{\nu}\right)\mathop{\mathrm{env}Y% _{{\mu}}\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{{1/2}}\right)% \right),$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable, $\mu$ : general order, $\nu$ : general degree, $\beta$ , and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E16
Encodings:: TeX, pMML, png

uniformly with respect to $x\in[0,1)$ and $\mu\in[0,(1-\delta)(\nu+\frac{1}{2})]$ . For $\alpha$ , $\beta$ , and see below.

Next,

14.15.17 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\beta\left(\frac{\alpha% ^{2}-y}{x^{2}-1+\alpha^{2}}\right)^{{1/4}}\mathop{I_{{\mu}}\/}\nolimits\!\left% (\left(\nu+\tfrac{1}{2}\right)|y|^{{1/2}}\right)\*\left(1+\mathop{O\/}% \nolimits\!\left(\frac{1}{\nu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : real variable, $\mu$ : general order, $\nu$ : general degree, $\beta$ , and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E17
Encodings:: TeX, pMML, png

14.15.18 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{% \beta\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\left(\frac{\alpha^{2}% -y}{x^{2}-1+\alpha^{2}}\right)^{{1/4}}\*\mathop{K_{{\mu}}\/}\nolimits\!\left(% \left(\nu+\tfrac{1}{2}\right)|y|^{{1/2}}\right)\left(1+\mathop{O\/}\nolimits\!% \left(\frac{1}{\nu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : real variable, $\mu$ : general order, $\nu$ : general degree, $\beta$ , and $\alpha$
Referenced by:: §14.15(iv)
Permalink:: http://dlmf.nist.gov/14.15.E18
Encodings:: TeX, pMML, png

uniformly with respect to $x\in(1,\infty)$ and $\mu\in[0,(1-\delta)(\nu+\frac{1}{2})]$ . In (14.15.15)–(14.15.18)

14.15.19 $\alpha=\frac{\mu}{\nu+\frac{1}{2}}\,(<1),$

Symbols:: $\mu$ : general order, $\nu$ : general degree and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E19
Encodings:: TeX, pMML, png

14.15.20 $\beta=e^{{\mu}}\left(\frac{\nu-\mu+\frac{1}{2}}{\nu+\mu+\frac{1}{2}}\right)^{{% (\nu/2)+(1/4)}}\left(\left(\nu+\tfrac{1}{2}\right)^{2}-\mu^{2}\right)^{{-\mu/2% }},$

Symbols:: : base of exponential function, $\mu$ : general order, $\nu$ : general degree and $\beta$
Permalink:: http://dlmf.nist.gov/14.15.E20
Encodings:: TeX, pMML, png

and the variable is defined implicitly by

14.15.21 $\left(y-\alpha^{2}\right)^{{1/2}}-\alpha\mathop{\mathrm{arctan}\/}\nolimits\!% \left(\frac{\left(y-\alpha^{2}\right)^{{1/2}}}{\alpha}\right)=\mathop{\mathrm{% arccos}\/}\nolimits\!\left(\frac{x}{\left(1-\alpha^{2}\right)^{{1/2}}}\right)-% \frac{\alpha}{2}\mathop{\mathrm{arccos}\/}\nolimits\!\left(\frac{\left(1+% \alpha^{2}\right)x^{2}-1+\alpha^{2}}{\left(1-\alpha^{2}\right)\left(1-x^{2}% \right)}\right),$ $x\leq\left(1-\alpha^{2}\right)^{{1/2}}$ , $y\geq\alpha^{2}$ ,

Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, : real variable, and $\alpha$
Referenced by:: §14.15(iv)
Permalink:: http://dlmf.nist.gov/14.15.E21
Encodings:: TeX, pMML, png

and

14.15.22 ${\left(\alpha^{2}-y\right)^{{1/2}}+\tfrac{1}{2}\alpha\mathop{\ln\/}\nolimits|y% |-\alpha\mathop{\ln\/}\nolimits\!\left(\left(\alpha^{2}-y\right)^{{1/2}}+% \alpha\right)}={\mathop{\ln\/}\nolimits\!\left(\frac{x+\left(x^{2}-1+\alpha^{2% }\right)^{{1/2}}}{\left(1-\alpha^{2}\right)^{{1/2}}}\right)+\frac{\alpha}{2}% \mathop{\ln\/}\nolimits\!\left(\frac{\left(1-\alpha^{2}\right)\left|1-x^{2}% \right|}{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}+2\alpha x\left(x^{2}-1+% \alpha^{2}\right)^{{1/2}}}\right)},$ $x\geq\left(1-\alpha^{2}\right)^{{1/2}}$ , $y\leq\alpha^{2}$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable, and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E22
Encodings:: TeX, pMML, png

where the inverse trigonometric functions take their principal values (§4.23(ii)). The points $x=\left(1-\alpha^{2}\right)^{{1/2}}$ , , and $x=\infty$ are mapped to $y=\alpha^{2}$ , , and $y=-\infty$ , respectively. The interval $0\leq x<\infty$ is mapped one-to-one to the interval $-\infty<y\leq y_{0}$ , where $y=y_{0}$ is the (positive) solution of (14.15.21) when .

For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986).

§14.15(v) Large $\nu$ , $(\nu+\frac{1}{2})\delta\leq\mu\leq(\nu+\frac{1}{2})/\delta$

Notes:: See Olver (1975b).
Keywords:: Ferrers functions, associated Legendre functions, parabolic cylinder functions
Referenced by:: §12.16, §13.20(iv), §2.8(vi), §2.8(vi)
Permalink:: http://dlmf.nist.gov/14.15.v

Here we introduce the envelopes of the parabolic cylinder functions $\mathop{U\/}\nolimits\!\left(-c,x\right)$ , $\mathop{\overline{U}\/}\nolimits\!\left(-c,x\right)$ , which are defined in §12.2. For $f(x)=\mathop{U\/}\nolimits\!\left(-c,x\right)$ or $\mathop{\overline{U}\/}\nolimits\!\left(-c,x\right)$ , with and nonnegative,

14.15.23 $\mathop{\mathrm{env}\/}\nolimits f(x)=\begin{cases}\left((\mathop{U\/}% \nolimits\!\left(-c,x\right))^{2}+(\mathop{\overline{U}\/}\nolimits\!\left(-c,% x\right))^{2}\right)^{{1/2}},&0\leq x\leq X_{c},\\ \sqrt{2}f(x),&X_{c}\leq x<\infty,\end{cases}$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function, : real variable and $X_{c}$ : root
Permalink:: http://dlmf.nist.gov/14.15.E23
Encodings:: TeX, pMML, png

where $x=X_{c}$ denotes the largest positive root of the equation $\mathop{U\/}\nolimits\!\left(-c,x\right)=\mathop{\overline{U}\/}\nolimits\!% \left(-c,x\right)$ .

As $\nu\to\infty$ ,

14.15.24 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{\left% (\nu+\frac{1}{2}\right)^{{1/4}}2^{{(\nu+\mu)/2}}\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)}\left(\frac{\zeta^{2}-% \alpha^{2}}{x^{2}-a^{2}}\right)^{{1/4}}\*\left(\mathop{U\/}\nolimits\!\left(% \mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{{1/2}}\zeta\right)+\mathop{O\/}% \nolimits\!\left(\nu^{{-2/3}}\right)\mathop{\mathrm{env}U\/}\nolimits\!\left(% \mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{{1/2}}\zeta\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : real variable, $\mu$ : general order, $\nu$ : general degree, , $\zeta$ and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E24
Encodings:: TeX, pMML, png

14.15.25 $\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\pi}{% \left(\nu+\frac{1}{2}\right)^{{1/4}}2^{{(\nu+\mu+2)/2}}\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)}\*\left(% \frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{{1/4}}\*\left(\mathop{% \overline{U}\/}\nolimits\!\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{{1/2% }}\zeta\right)+\mathop{O\/}\nolimits\!\left(\nu^{{-2/3}}\right)\mathop{\mathrm% {env}\overline{U}\/}\nolimits\!\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^% {{1/2}}\zeta\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function, : real variable, $\mu$ : general order, $\nu$ : general degree, , $\zeta$ and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E25
Encodings:: TeX, pMML, png

uniformly with respect to $x\in[0,1)$ and $\mu\in[\delta(\nu+\frac{1}{2}),\nu+\frac{1}{2}]$ . Here

14.15.26

$a=\frac{\left(\left(\nu+\mu+\frac{1}{2}\right)\left|\nu-\mu+\frac{1}{2}\right|% \right)^{{1/2}}}{\nu+\frac{1}{2}},$

$\alpha=\left(\frac{2\left|\nu-\mu+\frac{1}{2}\right|}{\nu+\frac{1}{2}}\right)^% {{1/2}},$

Symbols:: $\mu$ : general order, $\nu$ : general degree, and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E26
Encodings:: TeX, TeX, pMML, pMML, png, png

and the variable $\zeta$ is defined implicitly by

14.15.27 $\frac{1}{2}\zeta\left(\zeta^{2}-\alpha^{2}\right)^{{1/2}}-\frac{1}{2}\alpha^{2% }\mathop{\mathrm{arccosh}\/}\nolimits\!\left(\frac{\zeta}{\alpha}\right)=\left% (1-a^{2}\right)^{{1/2}}\mathop{\mathrm{arctanh}\/}\nolimits\!\left(\frac{1}{x}% \left(\frac{x^{2}-a^{2}}{1-a^{2}}\right)^{{1/2}}\right)-\mathop{\mathrm{% arccosh}\/}\nolimits\!\left(\frac{x}{a}\right),$ $a\leq x<1$ , $\alpha\leq\zeta<\infty$ ,

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, : real variable, , $\zeta$ and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E27
Encodings:: TeX, pMML, png

and

14.15.28 $\frac{1}{2}\alpha^{2}\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\frac{\zeta}{% \alpha}\right)+\frac{1}{2}\zeta\left(\alpha^{2}-\zeta^{2}\right)^{{1/2}}=% \mathop{\mathrm{arcsin}\/}\nolimits\!\left(\frac{x}{a}\right)-\left(1-a^{2}% \right)^{{1/2}}\mathop{\mathrm{arctan}\/}\nolimits\!\left(x\left(\frac{1-a^{2}% }{a^{2}-x^{2}}\right)^{{1/2}}\right),$ $-a\leq x\leq a$ , $-\alpha\leq\zeta\leq\alpha$ ,

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, : real variable, , $\zeta$ and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E28
Encodings:: TeX, pMML, png

when , and

14.15.29 $\zeta^{2}=-\mathop{\ln\/}\nolimits\!\left(1-x^{2}\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable and $\zeta$
Referenced by:: §14.15(v)
Permalink:: http://dlmf.nist.gov/14.15.E29
Encodings:: TeX, pMML, png

when . The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)).

When the interval $-a\leq x<1$ is mapped one-to-one to the interval $-\alpha\leq\zeta<\infty$ , with the points , , and corresponding to $\zeta=-\alpha$ , $\zeta=\alpha$ , and $\zeta=\infty$ , respectively. When the interval is mapped one-to-one to the interval $-\infty<\zeta<\infty$ , with the points , 0, and 1 corresponding to $\zeta=-\infty$ , 0, and $\infty$ , respectively.

Next, as $\nu\to\infty$ ,

14.15.30 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{\left% (\nu+\frac{1}{2}\right)^{{1/4}}2^{{(\nu+\mu)/2}}\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)}\left(\frac{\zeta^{2}+% \alpha^{2}}{x^{2}+a^{2}}\right)^{{1/4}}\*\mathop{U\/}\nolimits\!\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{{1/2}}\zeta\right)\left(1+\mathop{O\/}% \nolimits\!\left(\nu^{{-1}}\mathop{\ln\/}\nolimits\nu\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : real variable, $\mu$ : general order, $\nu$ : general degree, , $\zeta$ and $\alpha$
Referenced by:: §14.15(v)
Permalink:: http://dlmf.nist.gov/14.15.E30
Encodings:: TeX, pMML, png

uniformly with respect to $x\in(-1,1)$ and $\mu\in[\nu+\frac{1}{2},(1/\delta)(\nu+\frac{1}{2})]$ . Here $\zeta$ is defined implicitly by

14.15.31 $\frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^{{1/2}}+\frac{1}{2}\alpha^{2% }\mathop{\mathrm{arcsinh}\/}\nolimits\!\left(\frac{\zeta}{\alpha}\right)=\left% (1+a^{2}\right)^{{1/2}}\mathop{\mathrm{arctanh}\/}\nolimits\!\left(x\left(% \frac{1+a^{2}}{x^{2}+a^{2}}\right)^{{1/2}}\right)-\mathop{\mathrm{arcsinh}\/}% \nolimits\!\left(\frac{x}{a}\right),$

, $-\infty<\zeta<\infty$ ,

Symbols:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, : real variable, , $\zeta$ and $\alpha$
Permalink:: http://dlmf.nist.gov/14.15.E31
Encodings:: TeX, pMML, png

when , which maps the interval one-to-one to the interval $-\infty<\zeta<\infty$ : the points and correspond to $\zeta=-\infty$ and $\zeta=\infty$ , respectively. When (14.15.29) again applies. (The inverse hyperbolic functions again take their principal values.)

Since (14.15.30) holds for negative , corresponding approximations for $\mathop{\mathsf{Q}^{{\mp\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ , uniformly valid in the interval , can be obtained from (14.9.9) and (14.9.10).

For error bounds and other extensions see Olver (1975b).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 14.14 Continued Fractions 14.16 Zeros

§14.15 Uniform Asymptotic Approximations

Contents

§14.15(i) Large , Fixed

§14.15(ii) Large ,

§14.15(iii) Large , Fixed

§14.15(iv) Large ,

§14.15(v) Large ,

§14.15(i) Large $\mu$ , Fixed $\nu$

§14.15(ii) Large $\mu$ , $0\leq\nu+\frac{1}{2}\leq(1-\delta)\mu$

§14.15(iii) Large $\nu$ , Fixed $\mu$

§14.15(iv) Large $\nu$ , $0\leq\mu\leq(1-\delta)(\nu+\frac{1}{2})$

§14.15(v) Large $\nu$ , $(\nu+\frac{1}{2})\delta\leq\mu\leq(\nu+\frac{1}{2})/\delta$