§14.15 Uniform Asymptotic Approximations

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

14.15.1		$${\mathsf{P}}_{\nu }^{-\mu }\left(\pm x\right)={\left(\frac{1\mp x}{1\pm x}\right)}^{\mu /2}\left(\sum _{j=0}^{J-1}\frac{{\left(\nu +1\right)}_j{\left(-\nu \right)}_j}{j!\mathrm{\Gamma }\left(j+1+\mu \right)}{\left(\frac{1\mp x}{2}\right)}^j+O\left(\frac{1}{\mathrm{\Gamma }\left(J+1+\mu \right)}\right)\right)$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $x$: real variable, $\mu$: general order, $\nu$: general degree and $J$: nonnegative integer Referenced by: §14.15(i) Permalink: http://dlmf.nist.gov/14.15.E1 Encodings: TeX, pMML, png See also: Annotations for §14.15(i), §14.15 and Ch.14

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

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

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

14.15.2		$$P_{\nu }^{-\mu }\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\mu +1\right)}{\left(\frac{2\mu u}{\pi }\right)}^{1/2}{K}_{\nu +\frac{1}{2}}\left(\mu u\right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree and $u$ Referenced by: §14.15(i) Permalink: http://dlmf.nist.gov/14.15.E2 Encodings: TeX, pMML, png See also: Annotations for §14.15(i), §14.15 and Ch.14

14.15.3		$${\mathit{Q}}_{\nu }^{\mu }\left(x\right)=\frac{1}{{\mu }^{\nu +(1/2)}}{\left(\frac{\pi u}{2}\right)}^{1/2}{I}_{\nu +\frac{1}{2}}\left(\mu u\right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree and $u$ Referenced by: §14.15(i) Permalink: http://dlmf.nist.gov/14.15.E3 Encodings: TeX, pMML, png See also: Annotations for §14.15(i), §14.15 and Ch.14

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).

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

As $\mu \to \mathrm{\infty }$,

14.15.4		$${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\mu +1\right)}{\left(1-{\alpha }^2\right)}^{-\mu /2}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{(\nu /2)+(1/4)}{\left(\frac{p}{x}\right)}^{1/2}{\mathrm{e}}^{-\mu \rho }\left(1+O\left(\frac{1}{\mu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\mathrm{e}$: base of natural logarithm, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$, $p$ and $\rho$ Referenced by: §14.15(ii) Permalink: http://dlmf.nist.gov/14.15.E4 Encodings: TeX, pMML, png See also: Annotations for §14.15(ii), §14.15 and Ch.14

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

14.15.5
ⓘ 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: $\ln z$: 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 ${\mathsf{P}}_{\nu }^{-\mu }\left(-x\right)$ is obtained by replacing ${\mathrm{e}}^{-\mu \rho }$ by ${\mathrm{e}}^{\mu \rho }$ on the right-hand side of (14.15.4). Approximations for ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$ and ${\mathsf{Q}}_{\nu }^{\mp \mu }\left(x\right)$ can then be achieved via (14.9.7), (14.9.9), and (14.9.10).

Next,

14.15.8		$$P_{\nu }^{-\mu }\left(x\right)=\begin{array}{l}{\left(\frac{2\mu }{\pi }\right)}^{1/2}\frac{1}{\mathrm{\Gamma }\left(\mu +1\right)}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{(\nu /2)+(1/4)}{\left(1-{\alpha }^2\right)}^{-\mu /2}\\ \times {\left(\frac{{\alpha }^2+{\eta }^2}{{\alpha }^2\left(x^2-1\right)+1}\right)}^{1/4}{K}_{\nu +\frac{1}{2}}\left(\mu \eta \right)\left(1+O\left(\frac{1}{\mu }\right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$ and $\eta$ Permalink: http://dlmf.nist.gov/14.15.E8 Encodings: TeX, pMML, png See also: Annotations for §14.15(ii), §14.15 and Ch.14

14.15.9		$${\mathit{Q}}_{\nu }^{\mu }\left(x\right)=\begin{array}{l}{\left(\frac{\pi }{2}\right)}^{1/2}{\left(\frac{\mathrm{e}}{\mu }\right)}^{\nu +(1/2)}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{\mu /2}{\left(1-{\alpha }^2\right)}^{-(\nu /2)-(1/4)}\\ \times {\left(\frac{{\alpha }^2+{\eta }^2}{{\alpha }^2\left(x^2-1\right)+1}\right)}^{1/4}{I}_{\nu +\frac{1}{2}}\left(\mu \eta \right)\left(1+O\left(\frac{1}{\mu }\right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$ and $\eta$ Permalink: http://dlmf.nist.gov/14.15.E9 Encodings: TeX, pMML, png See also: Annotations for §14.15(ii), §14.15 and Ch.14

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

14.15.10		$$\begin{array}{l}\alpha \ln \left({\left({\alpha }^2+{\eta }^2\right)}^{1/2}+\alpha \right)-\alpha \ln \eta -{\left({\alpha }^2+{\eta }^2\right)}^{1/2}\\ =\begin{array}{l}\frac{1}{2}\ln \left(\frac{\left(1+{\alpha }^2\right)x^2+1-{\alpha }^2-2x{\left({\alpha }^2{x}^2-{\alpha }^2+1\right)}^{1/2}}{\left(x^2-1\right)\left(1-{\alpha }^2\right)}\right)\\ +\frac{1}{2}\alpha \ln \left(\frac{{\alpha }^2\left(2x^2-1\right)+1+2\alpha x{\left({\alpha }^2{x}^2-{\alpha }^2+1\right)}^{1/2}}{1-{\alpha }^2}\right).\end{array}\end{array}$$
ⓘ Symbols: $\ln z$: 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 is mapped one-to-one to the interval , with the points $x=1$ and $x=\mathrm{\infty }$ corresponding to $\eta =\mathrm{\infty }$ and $\eta =0$, respectively. For asymptotic expansions and explicit error bounds, see Dunster (2003b).

For $\nu \to \mathrm{\infty }$ and fixed $\mu$ ($\ge 0$),

14.15.11		${\mathsf{P}}_{\nu }^{-\mu }\left(\cos \theta \right)$	$={\displaystyle \frac{1}{{\nu }^{\mu }}}{\left({\displaystyle \frac{\theta }{\sin \theta }}\right)}^{1/2}\left(J_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)+O\left({\displaystyle \frac{1}{\nu }}\right)\mathrm{env}{J}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, : Ferrers function of the first kind, $\cos z$: cosine function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and $\theta$: angle Permalink: http://dlmf.nist.gov/14.15.E11 Encodings: TeX, pMML, png See also: Annotations for §14.15(iii), §14.15 and Ch.14
14.15.12		${\mathsf{Q}}_{\nu }^{-\mu }\left(\cos \theta \right)$	$=-{\displaystyle \frac{\pi }{2{\nu }^{\mu }}}{\left({\displaystyle \frac{\theta }{\sin \theta }}\right)}^{1/2}\left(Y_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)+O\left({\displaystyle \frac{1}{\nu }}\right)\mathrm{env}{Y}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)\right),$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and $\theta$: angle Permalink: http://dlmf.nist.gov/14.15.E12 Encodings: TeX, pMML, png See also: Annotations for §14.15(iii), §14.15 and Ch.14

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

Next,

14.15.13		$P_{\nu }^{-\mu }\left(\cosh \xi \right)$	$={\displaystyle \frac{1}{{\nu }^{\mu }}}{\left({\displaystyle \frac{\xi }{\sinh \xi }}\right)}^{1/2}{I}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\xi \right)\left(1+O\left({\displaystyle \frac{1}{\nu }}\right)\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mu$: general order, $\nu$: general degree and $\xi$: variable Permalink: http://dlmf.nist.gov/14.15.E13 Encodings: TeX, pMML, png See also: Annotations for §14.15(iii), §14.15 and Ch.14
14.15.14		${\mathit{Q}}_{\nu }^{\mu }\left(\cosh \xi \right)$	$={\displaystyle \frac{{\nu }^{\mu }}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}}{\left({\displaystyle \frac{\xi }{\sinh \xi }}\right)}^{1/2}{K}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\xi \right)\left(1+O\left({\displaystyle \frac{1}{\nu }}\right)\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mu$: general order, $\nu$: general degree and $\xi$: variable Permalink: http://dlmf.nist.gov/14.15.E14 Encodings: TeX, pMML, png See also: Annotations for §14.15(iii), §14.15 and Ch.14

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

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 $n\to \mathrm{\infty }$ with $\theta$ fixed.

As $\nu \to \mathrm{\infty }$,

14.15.15		$${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)=\beta {\left(\frac{y-{\alpha }^2}{1-{\alpha }^2-x^2}\right)}^{1/4}\left(J_{\mu }\left(\left(\nu +\frac{1}{2}\right)y^{1/2}\right)+O\left(\frac{1}{\nu }\right)\mathrm{env}{J}_{\mu }\left(\left(\nu +\frac{1}{2}\right)y^{1/2}\right)\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, : Ferrers function of the first kind, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\beta$, $y$ and $\alpha$ Referenced by: §14.15(iv) Permalink: http://dlmf.nist.gov/14.15.E15 Encodings: TeX, pMML, png See also: Annotations for §14.15(iv), §14.15 and Ch.14

14.15.16		$${\mathsf{Q}}_{\nu }^{-\mu }\left(x\right)=-\frac{\pi \beta }{2}{\left(\frac{y-{\alpha }^2}{1-{\alpha }^2-x^2}\right)}^{1/4}\left(Y_{\mu }\left(\left(\nu +\frac{1}{2}\right)y^{1/2}\right)+O\left(\frac{1}{\nu }\right)\mathrm{env}{Y}_{\mu }\left(\left(\nu +\frac{1}{2}\right)y^{1/2}\right)\right),$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\beta$, $y$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E16 Encodings: TeX, pMML, png See also: Annotations for §14.15(iv), §14.15 and Ch.14

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		$$P_{\nu }^{-\mu }\left(x\right)=\beta {\left(\frac{{\alpha }^2-y}{x^2-1+{\alpha }^2}\right)}^{1/4}{I}_{\mu }\left(\left(\nu +\frac{1}{2}\right){|y|}^{1/2}\right)\left(1+O\left(\frac{1}{\nu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\beta$, $y$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E17 Encodings: TeX, pMML, png See also: Annotations for §14.15(iv), §14.15 and Ch.14

14.15.18		$${\mathit{Q}}_{\nu }^{\mu }\left(x\right)=\frac{1}{\beta \mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{{\alpha }^2-y}{x^2-1+{\alpha }^2}\right)}^{1/4}{K}_{\mu }\left(\left(\nu +\frac{1}{2}\right){|y|}^{1/2}\right)\left(1+O\left(\frac{1}{\nu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\beta$, $y$ and $\alpha$ Referenced by: §14.15(iv) Permalink: http://dlmf.nist.gov/14.15.E18 Encodings: TeX, pMML, png See also: Annotations for §14.15(iv), §14.15 and Ch.14

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

14.15.19
ⓘ 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		$$\beta ={\mathrm{e}}^{\mu }{\left(\frac{\nu -\mu +\frac{1}{2}}{\nu +\mu +\frac{1}{2}}\right)}^{(\nu /2)+(1/4)}{\left({\left(\nu +\frac{1}{2}\right)}^2-{\mu }^2\right)}^{-\mu /2},$$
ⓘ 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		$${\left(y-{\alpha }^2\right)}^{1/2}-\alpha \arctan \left(\frac{{\left(y-{\alpha }^2\right)}^{1/2}}{\alpha }\right)=\arccos \left(\frac{x}{{\left(1-{\alpha }^2\right)}^{1/2}}\right)-\frac{\alpha }{2}\arccos \left(\frac{\left(1+{\alpha }^2\right)x^2-1+{\alpha }^2}{\left(1-{\alpha }^2\right)\left(1-x^2\right)}\right),$$
$x\le {\left(1-{\alpha }^2\right)}^{1/2}$, $y\ge {\alpha }^2$,
ⓘ Symbols: $\arccos z$: arccosine function, $\arctan z$: 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		$$\begin{array}{l}{\left({\alpha }^2-y\right)}^{1/2}+\frac{1}{2}\alpha \ln |y|-\alpha \ln \left({\left({\alpha }^2-y\right)}^{1/2}+\alpha \right)\\ =\ln \left(\frac{x+{\left(x^2-1+{\alpha }^2\right)}^{1/2}}{{\left(1-{\alpha }^2\right)}^{1/2}}\right)+\frac{\alpha }{2}\ln \left(\frac{\left(1-{\alpha }^2\right)\left|1-x^2\right|}{\left(1+{\alpha }^2\right)x^2-1+{\alpha }^2+2\alpha x{\left(x^2-1+{\alpha }^2\right)}^{1/2}}\right),\end{array}$$
$x\ge {\left(1-{\alpha }^2\right)}^{1/2}$, $y\le {\alpha }^2$,
ⓘ Symbols: $\ln z$: 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 $x=\mathrm{\infty }$ are mapped to $y={\alpha }^2$, $y=0$, and $y=-\mathrm{\infty }$, respectively. The interval is mapped one-to-one to the interval , 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).

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

14.15.23		$\mathrm{env}U(-c,x)$
		$\mathrm{env}\overline{U}(-c,x)$
ⓘ Symbols: $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: 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 ${(U(-c,x))}^2$ and was rewritten as $U^2(-c,x)$. The other three terms were treated in similar fashion. Modification (effective with 1.0.12): Originally this equation used $f(x)$ to represent both $U(-c,x)$ and $\overline{U}(-c,x)$. 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 $U(-c,x)=\overline{U}(-c,x)$.

As $\nu \to \mathrm{\infty }$,

14.15.24		$${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)=\begin{array}{l}\frac{1}{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu )/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2-{\alpha }^2}{x^2-a^2}\right)}^{1/4}\\ \times \left(U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )+O\left({\nu }^{-2/3}\right)\mathrm{env}U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$, $U(a,z)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$, $\zeta$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E24 Encodings: TeX, pMML, png See also: Annotations for §14.15(v), §14.15 and Ch.14

14.15.25		$${\mathsf{Q}}_{\nu }^{-\mu }\left(x\right)=\begin{array}{l}\frac{\pi }{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu +2)/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2-{\alpha }^2}{x^2-a^2}\right)}^{1/4}\\ \times \left(\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )+O\left({\nu }^{-2/3}\right)\mathrm{env}\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$, $\overline{U}(a,x)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$, $\zeta$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E25 Encodings: TeX, pMML, png See also: Annotations for §14.15(v), §14.15 and Ch.14

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

14.15.26		$a$	$={\displaystyle \frac{{\left(\left(\nu +\mu +\frac{1}{2}\right)\left|\nu -\mu +\frac{1}{2}\right|\right)}^{1/2}}{\nu +\frac{1}{2}}},$
		$\alpha$	$={\left({\displaystyle \frac{2\left|\nu -\mu +\frac{1}{2}\right|}{\nu +\frac{1}{2}}}\right)}^{1/2},$
ⓘ 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		$$\begin{array}{l}\frac{1}{2}\zeta {\left({\zeta }^2-{\alpha }^2\right)}^{1/2}-\frac{1}{2}{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)\\ ={\left(1-a^2\right)}^{1/2}\mathrm{arctanh}\left(\frac{1}{x}{\left(\frac{x^2-a^2}{1-a^2}\right)}^{1/2}\right)-\mathrm{arccosh}\left(\frac{x}{a}\right),\end{array}$$
, ,
ⓘ Symbols: $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $x$: real variable, $a$, $\zeta$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E27 Encodings: TeX, pMML, png See also: Annotations for §14.15(v), §14.15 and Ch.14

and

14.15.28		$$\frac{1}{2}{\alpha }^2\arcsin \left(\frac{\zeta }{\alpha }\right)+\frac{1}{2}\zeta {\left({\alpha }^2-{\zeta }^2\right)}^{1/2}=\arcsin \left(\frac{x}{a}\right)-{\left(1-a^2\right)}^{1/2}\arctan \left(x{\left(\frac{1-a^2}{a^2-x^2}\right)}^{1/2}\right),$$
$-a\le x\le a$, $-\alpha \le \zeta \le \alpha$,
ⓘ Symbols: $\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable, $a$, $\zeta$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E28 Encodings: TeX, pMML, png See also: Annotations for §14.15(v), §14.15 and Ch.14

when $a>0$, and

14.15.29		$${\zeta }^2=-\ln \left(1-x^2\right),$$
,
ⓘ Symbols: $\ln z$: 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 is mapped one-to-one to the interval , with the points $x=-a$, $x=a$, and $x=1$ corresponding to $\zeta =-\alpha$, $\zeta =\alpha$, and $\zeta =\mathrm{\infty }$, respectively. When $a=0$ the interval is mapped one-to-one to the interval , with the points $x=-1$, $0$, and $1$ corresponding to $\zeta =-\mathrm{\infty }$, $0$, and $\mathrm{\infty }$, respectively.

Next, as $\nu \to \mathrm{\infty }$,

14.15.30		$${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)=\begin{array}{l}\frac{1}{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu )/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2+{\alpha }^2}{x^2+a^2}\right)}^{1/4}\\ \times U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\left(1+O\left({\nu }^{-1}\ln \nu \right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\ln z$: principal branch of logarithm function, $U(a,z)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$, $\zeta$ and $\alpha$ Referenced by: §14.15(v) Permalink: http://dlmf.nist.gov/14.15.E30 Encodings: TeX, pMML, png See also: Annotations for §14.15(v), §14.15 and Ch.14

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		$$\begin{array}{l}\frac{1}{2}\zeta {\left({\zeta }^2+{\alpha }^2\right)}^{1/2}+\frac{1}{2}{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)\\ ={\left(1+a^2\right)}^{1/2}\mathrm{arctanh}\left(x{\left(\frac{1+a^2}{x^2+a^2}\right)}^{1/2}\right)-\mathrm{arcsinh}\left(\frac{x}{a}\right),\end{array}$$
, ,
ⓘ Symbols: $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $x$: real variable, $a$, $\zeta$ and $\alpha$ Permalink: http://dlmf.nist.gov/14.15.E31 Encodings: TeX, pMML, png See also: Annotations for §14.15(v), §14.15 and Ch.14

when $a>0$, which maps the interval one-to-one to the interval : the points $x=-1$ and $x=1$ correspond to $\zeta =-\mathrm{\infty }$ and $\zeta =\mathrm{\infty }$, 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 ${\mathsf{Q}}_{\nu }^{\mp \mu }\left(x\right)$, 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).