# §14.15 Uniform Asymptotic Approximations

## §14.15(i) Large $\mu$, Fixed $\nu$

For the interval $-1 with fixed $\nu$, real $\mu$, and arbitrary fixed values of the nonnegative integer $J$,

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

as $\mu\to\infty$, uniformly with respect to $x$. 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 $\mu-\nu\notin\mathbb{Z}$ 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 the following asymptotic approximations hold when $\mu\to\infty$, with $\nu$ ($\geq-\frac{1}{2}$) fixed, uniformly with respect to $x$:

 14.15.2 $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)=\frac{1}{\mathop{\Gamma\/}% \nolimits\!\left(\mu+1\right)}\left(\frac{2\mu u}{\pi}\right)^{1/2}\mathop{K_{% \nu+\frac{1}{2}}\/}\nolimits\!\left(\mu u\right)\*\left(1+\mathop{O\/}% \nolimits\!\left(\frac{1}{\mu}\right)\right),$
 14.15.3 $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)=\frac{1}{\mu^{% \nu+(1/2)}}\left(\frac{\pi u}{2}\right)^{1/2}\mathop{I_{\nu+\frac{1}{2}}\/}% \nolimits\!\left(\mu u\right)\left(1+\mathop{O\/}\nolimits\!\left(\frac{1}{\mu% }\right)\right),$

where $u$ 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). See also Temme (2015, Chapter 29).

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

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

As $\mu\to\infty$,

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

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 See also: Annotations for 14.15(ii)
 14.15.6 $p=\frac{x}{\left(\alpha^{2}x^{2}+1-\alpha^{2}\right)^{1/2}},$ Symbols: $x$: real variable, $\alpha$ and $p$ Permalink: http://dlmf.nist.gov/14.15.E6 Encodings: TeX, pMML, png See also: Annotations for 14.15(ii)

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: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\alpha$, $p$ and $\rho$ Permalink: http://dlmf.nist.gov/14.15.E7 Encodings: TeX, pMML, png See also: Annotations for 14.15(ii)

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

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: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $\eta$ Permalink: http://dlmf.nist.gov/14.15.E10 Encodings: TeX, pMML, png See also: Annotations for 14.15(ii)

The interval $1 is mapped one-to-one to the interval $0<\eta<\infty$, with the points $x=1$ 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 $\mu$

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

 14.15.11 $\displaystyle\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/% }\nolimits\theta\right)$ $\displaystyle=\frac{1}{\nu^{\mu}}\left(\frac{\theta}{\mathop{\sin\/}\nolimits% \theta}\right)^{1/2}\left(\mathop{J_{\mu}\/}\nolimits\!\left(\left(\nu+\tfrac{% 1}{2}\right)\theta\right)+\mathop{O\/}\nolimits\!\left(\frac{1}{\nu}\right)% \mathop{\mathrm{env}\mskip-2.0mu J_{\mu}\/}\nolimits\!\left(\left(\nu+\tfrac{1% }{2}\right)\theta\right)\right),$ 14.15.12 $\displaystyle\mathop{\mathsf{Q}^{-\mu}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/% }\nolimits\theta\right)$ $\displaystyle=-\frac{\pi}{2\nu^{\mu}}\left(\frac{\theta}{\mathop{\sin\/}% \nolimits\theta}\right)^{1/2}\left(\mathop{Y_{\mu}\/}\nolimits\!\left(\left(% \nu+\tfrac{1}{2}\right)\theta\right)+\mathop{O\/}\nolimits\!\left(\frac{1}{\nu% }\right)\mathop{\mathrm{env}\mskip-2.0mu Y_{\mu}\/}\nolimits\!\left(\left(\nu+% \tfrac{1}{2}\right)\theta\right)\right),$

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

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 Temme (2015, Chapter 29).

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

As $\nu\to\infty$,

 14.15.15 $\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)=\beta\left(\frac{y% -\alpha^{2}}{1-\alpha^{2}-x^{2}}\right)^{1/4}\*\left(\mathop{J_{\mu}\/}% \nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{1/2}\right)+\mathop{O\/}% \nolimits\!\left(\frac{1}{\nu}\right)\mathop{\mathrm{env}\mskip-2.0mu J_{\mu}% \/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{1/2}\right)\right),$
 14.15.16 $\mathop{\mathsf{Q}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)=-\frac{\pi\beta}{2% }\left(\frac{y-\alpha^{2}}{1-\alpha^{2}-x^{2}}\right)^{1/4}\left(\mathop{Y_{% \mu}\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{1/2}\right)+\mathop{O% \/}\nolimits\!\left(\frac{1}{\nu}\right)\mathop{\mathrm{env}\mskip-2.0mu Y_{% \mu}\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)y^{1/2}\right)\right),$

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

Next,

 14.15.17 $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)=\beta\left(\frac{\alpha^{2}% -y}{x^{2}-1+\alpha^{2}}\right)^{1/4}\mathop{I_{\mu}\/}\nolimits\!\left(\left(% \nu+\tfrac{1}{2}\right)|y|^{1/2}\right)\*\left(1+\mathop{O\/}\nolimits\!\left(% \frac{1}{\nu}\right)\right),$
 14.15.18 $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)=\frac{1}{\beta% \mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\left(\frac{\alpha^{2}-y}{x% ^{2}-1+\alpha^{2}}\right)^{1/4}\*\mathop{K_{\mu}\/}\nolimits\!\left(\left(\nu+% \tfrac{1}{2}\right)|y|^{1/2}\right)\left(1+\mathop{O\/}\nolimits\!\left(\frac{% 1}{\nu}\right)\right),$

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 See also: Annotations for 14.15(iv)
 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: $\mathrm{e}$: base of exponential function, $\mu$: general order, $\nu$: general degree and $\beta$ Permalink: http://dlmf.nist.gov/14.15.E20 Encodings: TeX, pMML, png See also: Annotations for 14.15(iv)

and the variable $y$ is defined implicitly by

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

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: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $x$: real variable, $y$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E22 Encodings: TeX, pMML, png See also: Annotations for 14.15(iv)

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

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

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

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 $\mathop{U\/}\nolimits\!\left(-c,x\right)$ or $\mathop{\overline{U}\/}\nolimits\!\left(-c,x\right)$, with $c$ and $x$ nonnegative,

 14.15.23 $\displaystyle\mathop{\mathrm{env}\mskip-1.0mu U\/}\nolimits\!\left(-c,x\right)$ $\displaystyle=\begin{cases}\left({\mathop{U\/}\nolimits^{2}}\!\left(-c,x\right% )+{\mathop{\overline{U}\/}\nolimits^{2}}\!\left(-c,x\right)\right)^{1/2},&0% \leq x\leq X_{c},\\ \sqrt{2}\mathop{U\/}\nolimits\!\left(-c,x\right),&X_{c}\leq x<\infty,\end{cases}$ $\displaystyle\mathop{\mathrm{env}\mskip-1.0mu \overline{U}\/}\nolimits\!\left(% -c,x\right)$ $\displaystyle=\begin{cases}\left({\mathop{U\/}\nolimits^{2}}\!\left(-c,x\right% )+{\mathop{\overline{U}\/}\nolimits^{2}}\!\left(-c,x\right)\right)^{1/2},&0% \leq x\leq X_{c},\\ \sqrt{2}\ \mathop{\overline{U}\/}\nolimits\!\left(-c,x\right),&X_{c}\leq x<% \infty,\end{cases}$ Symbols: $\mathop{\mathrm{env}\mskip-1.0mu U\/}\nolimits\!\left(\NVar{c},\NVar{x}\right)$: envelope of parabolic cylinder function $\mathop{U\/}\nolimits\!\left(\NVar{c},\NVar{x}\right)$, $\mathop{\mathrm{env}\mskip-1.0mu \overline{U}\/}\nolimits\!\left(\NVar{c},% \NVar{x}\right)$: envelope of parabolic cylinder function $\mathop{\overline{U}\/}\nolimits\!\left(\NVar{c},\NVar{x}\right)$, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $\mathop{\overline{U}\/}\nolimits\!\left(\NVar{a},\NVar{x}\right)$: parabolic cylinder function, $x$: real variable and $X_{c}$: root Referenced by: Other Changes, Other Changes Permalink: http://dlmf.nist.gov/14.15.E23 Encodings: TeX, TeX, pMML, pMML, png, png Clarification (effective with 1.0.14): Four terms were rewritten for improved clarity. The first of these appeared previously as $(\mathop{U\/}\nolimits\!\left(-c,x\right))^{2}$ and was rewritten as ${\mathop{U\/}\nolimits^{2}}\!\left(-c,x\right)$. The other three terms were treated in similar fashion. Reported 2016-11-22 Modification (effective with 1.0.12): Originally this equation used $f(x)$ to represent both $\mathop{U\/}\nolimits\!\left(-c,x\right)$ and $\mathop{\overline{U}\/}\nolimits\!\left(-c,x\right)$. This has been replaced by two equations giving explicit definitions for the two envelope functions. Some slight changes in wording were needed to make this clear to readers. Reported 2016-07-05 See also: Annotations for 14.15(v)

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 $\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)=\frac{1}{\left(\nu% +\frac{1}{2}\right)^{1/4}2^{(\nu+\mu)/2}\mathop{\Gamma\/}\nolimits\!\left(% \frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)}\left(\frac{\zeta^{2}-\alpha^% {2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(\mathop{U\/}\nolimits\!\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+\mathop{O\/}\nolimits\!% \left(\nu^{-2/3}\right)\mathop{\mathrm{env}\mskip-1.0mu U\/}\nolimits\!\left(% \mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\right),$
 14.15.25 $\mathop{\mathsf{Q}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)=\frac{\pi}{\left(% \nu+\frac{1}{2}\right)^{1/4}2^{(\nu+\mu+2)/2}\mathop{\Gamma\/}\nolimits\!\left% (\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)}\*\left(\frac{\zeta^{2}-% \alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(\mathop{\overline{U}\/}\nolimits% \!\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+\mathop{O\/% }\nolimits\!\left(\nu^{-2/3}\right)\mathop{\mathrm{env}\mskip-1.0mu \overline{% U}\/}\nolimits\!\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta% \right)\right),$

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

 14.15.26 $\displaystyle a$ $\displaystyle=\frac{\left(\left(\nu+\mu+\frac{1}{2}\right)\left|\nu-\mu+\frac{% 1}{2}\right|\right)^{1/2}}{\nu+\frac{1}{2}},$ $\displaystyle\alpha$ $\displaystyle=\left(\frac{2\left|\nu-\mu+\frac{1}{2}\right|}{\nu+\frac{1}{2}}% \right)^{1/2},$ Symbols: $\mu$: general order, $\nu$: general degree, $a$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E26 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 14.15(v)

and the variable $\zeta$ is defined implicitly by

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

and

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

when $a>0$, and

 14.15.29 $\zeta^{2}=-\mathop{\ln\/}\nolimits\!\left(1-x^{2}\right),$ $-1, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $x$: real variable and $\zeta$ Referenced by: §14.15(v) Permalink: http://dlmf.nist.gov/14.15.E29 Encodings: TeX, pMML, png See also: Annotations for 14.15(v)

when $a=0$. The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)).

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

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

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

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

when $a>0$, which maps the interval $-1 one-to-one to the interval $-\infty<\zeta<\infty$: the points $x=-1$ and $x=1$ correspond to $\zeta=-\infty$ and $\zeta=\infty$, respectively. When $a=0$ (14.15.29) again applies. (The inverse hyperbolic functions again take their principal values.)

Since (14.15.30) holds for negative $x$, corresponding approximations for $\mathop{\mathsf{Q}^{\mp\mu}_{\nu}\/}\nolimits\!\left(x\right)$, uniformly valid in the interval $-1, can be obtained from (14.9.9) and (14.9.10).

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