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

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

Provided that $\mu-\nu\notin\mathbb{Z}$ the corresponding expansions for $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mp\mu}_{\nu}\left(x\right)$ 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 $P^{-\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\mu+1\right)}\left(\frac{2% \mu u}{\pi}\right)^{1/2}K_{\nu+\frac{1}{2}}\left(\mu u\right)\*\left(1+O\left(% \frac{1}{\mu}\right)\right),$
 14.15.3 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{1}{\mu^{\nu+(1/2)}}\left(\frac{% \pi u}{2}\right)^{1/2}I_{\nu+\frac{1}{2}}\left(\mu u\right)\left(1+O\left(% \frac{1}{\mu}\right)\right),$

where $u$ is given by (14.12.10). Here $I$ and $K$ 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 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\mu+1\right)}\left(% 1-\alpha^{2}\right)^{-\mu/2}\left(\frac{1-\alpha}{1+\alpha}\right)^{(\nu/2)+(1% /4)}\*\left(\frac{p}{x}\right)^{1/2}e^{-\mu\rho}\left(1+O\left(\frac{1}{\mu}% \right)\right),$

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 and Ch.14
 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), §14.15 and Ch.14

and

 14.15.7 $\rho=\frac{1}{2}\ln\left(\frac{1+p}{1-p}\right)+\frac{1}{2}\alpha\ln\left(% \frac{1-\alpha p}{1+\alpha p}\right).$ ⓘ Symbols: $\ln\NVar{z}$: 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), §14.15 and Ch.14

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

Next,

 14.15.8 $P^{-\mu}_{\nu}\left(x\right)=\left(\frac{2\mu}{\pi}\right)^{1/2}\frac{1}{% \Gamma\left(\mu+1\right)}\left(\frac{1-\alpha}{1+\alpha}\right)^{(\nu/2)+(1/4)% }\*\left(1-\alpha^{2}\right)^{-\mu/2}\left(\frac{\alpha^{2}+\eta^{2}}{\alpha^{% 2}\left(x^{2}-1\right)+1}\right)^{1/4}\*K_{\nu+\frac{1}{2}}\left(\mu\eta\right% )\left(1+O\left(\frac{1}{\mu}\right)\right),$
 14.15.9 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(\frac{\pi}{2}\right)^{1/2}\left% (\frac{e}{\mu}\right)^{\nu+(1/2)}\left(\frac{1-\alpha}{1+\alpha}\right)^{\mu/2% }\*\left(1-\alpha^{2}\right)^{-(\nu/2)-(1/4)}\left(\frac{\alpha^{2}+\eta^{2}}{% \alpha^{2}\left(x^{2}-1\right)+1}\right)^{1/4}\*I_{\nu+\frac{1}{2}}\left(\mu% \eta\right)\left(1+O\left(\frac{1}{\mu}\right)\right),$

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 $\alpha\ln\left(\left(\alpha^{2}+\eta^{2}\right)^{1/2}+\alpha\right)-\alpha\ln% \eta-\left(\alpha^{2}+\eta^{2}\right)^{1/2}=\frac{1}{2}\ln\left(\frac{\left(1+% \alpha^{2}\right)x^{2}+1-\alpha^{2}-2x\left(\alpha^{2}x^{2}-\alpha^{2}+1\right% )^{1/2}}{\left(x^{2}-1\right)\left(1-\alpha^{2}\right)}\right)+\frac{1}{2}% \alpha\ln\left(\frac{\alpha^{2}\left(2x^{2}-1\right)+1+2\alpha x\left(\alpha^{% 2}x^{2}-\alpha^{2}+1\right)^{1/2}}{1-\alpha^{2}}\right).$ ⓘ Symbols: $\ln\NVar{z}$: 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), §14.15 and Ch.14

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

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

Next,

 14.15.13 $\displaystyle P^{-\mu}_{\nu}\left(\cosh\xi\right)$ $\displaystyle=\frac{1}{\nu^{\mu}}\left(\frac{\xi}{\sinh\xi}\right)^{1/2}I_{\mu% }\left(\left(\nu+\tfrac{1}{2}\right)\xi\right)\*\left(1+O\left(\frac{1}{\nu}% \right)\right),$ 14.15.14 $\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(\cosh\xi\right)$ $\displaystyle=\frac{\nu^{\mu}}{\Gamma\left(\nu+\mu+1\right)}\left(\frac{\xi}{% \sinh\xi}\right)^{1/2}\*K_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)\xi\right)\*% \left(1+O\left(\frac{1}{\nu}\right)\right),$

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

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

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 and Ch.14
 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 natural logarithm, $\mu$: 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), §14.15 and Ch.14

and the variable $y$ is defined implicitly by

 14.15.21 $\left(y-\alpha^{2}\right)^{1/2}-\alpha\operatorname{arctan}\left(\frac{\left(y% -\alpha^{2}\right)^{1/2}}{\alpha}\right)=\operatorname{arccos}\left(\frac{x}{% \left(1-\alpha^{2}\right)^{1/2}}\right)-\frac{\alpha}{2}\operatorname{arccos}% \left(\frac{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}}{\left(1-\alpha^{2}% \right)\left(1-x^{2}\right)}\right),$ $x\leq\left(1-\alpha^{2}\right)^{1/2}$, $y\geq\alpha^{2}$, ⓘ Symbols: $\operatorname{arccos}\NVar{z}$: arccosine function, $\operatorname{arctan}\NVar{z}$: arctangent function, $x$: real variable, $y$ and $\alpha$ Referenced by: §14.15(iv) Permalink: http://dlmf.nist.gov/14.15.E21 Encodings: TeX, pMML, png See also: Annotations for §14.15(iv), §14.15 and Ch.14

and

 14.15.22 ${\left(\alpha^{2}-y\right)^{1/2}+\tfrac{1}{2}\alpha\ln|y|-\alpha\ln\left(\left% (\alpha^{2}-y\right)^{1/2}+\alpha\right)}={\ln\left(\frac{x+\left(x^{2}-1+% \alpha^{2}\right)^{1/2}}{\left(1-\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}% {2}\ln\left(\frac{\left(1-\alpha^{2}\right)\left|1-x^{2}\right|}{\left(1+% \alpha^{2}\right)x^{2}-1+\alpha^{2}+2\alpha x\left(x^{2}-1+\alpha^{2}\right)^{% 1/2}}\right)},$ $x\geq\left(1-\alpha^{2}\right)^{1/2}$, $y\leq\alpha^{2}$, ⓘ Symbols: $\ln\NVar{z}$: 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), §14.15 and Ch.14

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 $U\left(-c,x\right)$, $\overline{U}\left(-c,x\right)$, which are defined in §12.2. For $U\left(-c,x\right)$ or $\overline{U}\left(-c,x\right)$, with $c$ and $x$ nonnegative,

 14.15.23 $\displaystyle\mathrm{env}\mskip-1.0muU\left(-c,x\right)$ $\displaystyle=\begin{cases}\left({U}^{2}\left(-c,x\right)+{\overline{U}}^{2}% \left(-c,x\right)\right)^{1/2},&0\leq x\leq X_{c},\\ \sqrt{2}U\left(-c,x\right),&X_{c}\leq x<\infty,\end{cases}$ $\displaystyle\mathrm{env}\mskip-1.0mu\overline{U}\left(-c,x\right)$ $\displaystyle=\begin{cases}\left({U}^{2}\left(-c,x\right)+{\overline{U}}^{2}% \left(-c,x\right)\right)^{1/2},&0\leq x\leq X_{c},\\ \sqrt{2}\ \overline{U}\left(-c,x\right),&X_{c}\leq x<\infty,\end{cases}$ ⓘ Symbols: $\mathrm{env}\mskip-1.0muU\left(\NVar{c},\NVar{x}\right)$: envelope of parabolic cylinder function $U\left(\NVar{c},\NVar{x}\right)$, $\mathrm{env}\mskip-1.0mu\overline{U}\left(\NVar{c},\NVar{x}\right)$: envelope of parabolic cylinder function $\overline{U}\left(\NVar{c},\NVar{x}\right)$, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $\overline{U}\left(\NVar{a},\NVar{x}\right)$: parabolic cylinder function, $x$: real variable and $X_{c}$: root Referenced by: Erratum (V1.0.12) for Equation (14.15.23), Erratum (V1.0.14) for Equation (14.15.23) 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 $(U\left(-c,x\right))^{2}$ and was rewritten as ${U}^{2}\left(-c,x\right)$. The other three terms were treated in similar fashion. Modification (effective with 1.0.12): Originally this equation used $f(x)$ to represent both $U\left(-c,x\right)$ and $\overline{U}\left(-c,x\right)$. 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. See also: Annotations for §14.15(v), §14.15 and Ch.14

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

As $\nu\to\infty$,

 14.15.24 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\left(\nu+\frac{1}{2}\right)^{1% /4}2^{(\nu+\mu)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)% }\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(U\left(\mu% -\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+O\left(\nu^{-2/3}% \right)\mathrm{env}\mskip-1.0muU\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)% ^{1/2}\zeta\right)\right),$
 14.15.25 $\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\frac{\pi}{\left(\nu+\frac{1}{2}\right)^% {1/4}2^{(\nu+\mu+2)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}% \right)}\*\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(% \overline{U}\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+O% \left(\nu^{-2/3}\right)\mathrm{env}\mskip-1.0mu\overline{U}\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\right),$

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), §14.15 and Ch.14

and the variable $\zeta$ is defined implicitly by

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

and

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

when $a>0$, and

 14.15.29 $\zeta^{2}=-\ln\left(1-x^{2}\right),$ $-1, ⓘ Symbols: $\ln\NVar{z}$: 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), §14.15 and Ch.14

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

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 $\frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^{1/2}+\frac{1}{2}\alpha^{2}% \operatorname{arcsinh}\left(\frac{\zeta}{\alpha}\right)=\left(1+a^{2}\right)^{% 1/2}\operatorname{arctanh}\left(x\left(\frac{1+a^{2}}{x^{2}+a^{2}}\right)^{1/2% }\right)-\operatorname{arcsinh}\left(\frac{x}{a}\right),$ $-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 $\mathsf{Q}^{\mp\mu}_{\nu}\left(x\right)$, 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).