# §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 $\mu-\nu\notin\Integer$ 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 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).

# §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
 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

and

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

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

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

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

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

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

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

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

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

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

 14.15.26 $\displaystyle a$ $\displaystyle=\frac{\left(\left(\nu+\mu+\frac{1}{2}\right)\left|\nu-\mu+\frac{% 1}{2}\right|\right)^{1/2}}{\nu+\frac{1}{2}},$ $\displaystyle\alpha$ $\displaystyle=\left(\frac{2\left|\nu-\mu+\frac{1}{2}\right|}{\nu+\frac{1}{2}}% \right)^{1/2},$

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: $\mathop{\ln\/}\nolimits z$: 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

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).