# §13.20 Uniform Asymptotic Approximations for Large $\mu$

## §13.20(i) Large $\mu$, Fixed $\kappa$

When $\mu\to\infty$ in the sector $|\operatorname{ph}\mu|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$, with $\kappa(\in\mathbb{C})$ fixed

 13.20.1 $M_{\kappa,\mu}\left(z\right)=z^{\mu+\frac{1}{2}}\left(1+O\left(\mu^{-1}\right)% \right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function and $z$: complex variable Referenced by: §13.20(i) Permalink: http://dlmf.nist.gov/13.20.E1 Encodings: TeX, pMML, png See also: Annotations for §13.20(i), §13.20 and Ch.13

uniformly for bounded values of $|z|$; also

 13.20.2 $W_{\kappa,\mu}\left(x\right)=\pi^{-\frac{1}{2}}\Gamma\left(\kappa+\mu\right)% \left(\tfrac{1}{4}x\right)^{\frac{1}{2}-\mu}\left(1+O\left(\mu^{-1}\right)% \right),$

uniformly for bounded positive values of $x$. For an extension of (13.20.1) to an asymptotic expansion, together with error bounds, see Olver (1997b, Chapter 10, Ex. 3.4).

## §13.20(ii) Large $\mu$, $0\leq\kappa\leq(1-\delta)\mu$

Let

 13.20.3 $X=\sqrt{4\mu^{2}-4\kappa x+x^{2}}.$ ⓘ Defines: $X$ (locally) Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/13.20.E3 Encodings: TeX, pMML, png See also: Annotations for §13.20(ii), §13.20 and Ch.13

Then as $\mu\to\infty$

 13.20.4 $M_{\kappa,\mu}\left(x\right)=\sqrt{\frac{2\mu x}{X}}\*\left(\frac{4\mu^{2}x}{2% \mu^{2}-\kappa x+\mu X}\right)^{\mu}\*\left(\frac{2(\mu-\kappa)}{X+x-2\kappa}% \right)^{\kappa}\*e^{\frac{1}{2}X-\mu}\*\left(1+O\left(\frac{1}{\mu}\right)% \right),$
 13.20.5 $W_{\kappa,\mu}\left(x\right)=\sqrt{\frac{x}{X}}\*\left(\frac{2\mu^{2}-\kappa x% +\mu X}{(\mu-\kappa)x}\right)^{\mu}\*\left(\frac{X+x-2\kappa}{2}\right)^{% \kappa}\*e^{-\frac{1}{2}X-\kappa}\*\left(1+O\left(\frac{1}{\mu}\right)\right),$

uniformly with respect to $x\in(0,\infty)$ and $\kappa\in[0,(1-\delta)\mu]$, where $\delta$ again denotes an arbitrary small positive constant.

## §13.20(iii) Large $\mu$, $-(1-\delta)\mu\leq\kappa\leq\mu$

Let

 13.20.6 $\displaystyle\alpha$ $\displaystyle=\sqrt{2|\kappa-\mu|/\mu},$ ⓘ Defines: $\alpha$ (locally) Referenced by: §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E6 Encodings: TeX, pMML, png See also: Annotations for §13.20(iii), §13.20 and Ch.13 13.20.7 $\displaystyle X$ $\displaystyle=\sqrt{|x^{2}-4\kappa x+4\mu^{2}|},$ ⓘ Defines: $X$ (locally) Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/13.20.E7 Encodings: TeX, pMML, png See also: Annotations for §13.20(iii), §13.20 and Ch.13 13.20.8 $\displaystyle\Phi(\kappa,\mu,x)$ $\displaystyle=\left(\frac{\mu^{2}\zeta^{2}-2\kappa\mu+2\mu^{2}}{x^{2}-4\kappa x% +4\mu^{2}}\right)^{\frac{1}{4}}\sqrt{x},$ ⓘ Defines: $\Phi(\kappa,\mu,x)$: function (locally) Symbols: $x$: real variable Referenced by: §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E8 Encodings: TeX, pMML, png See also: Annotations for §13.20(iii), §13.20 and Ch.13

with the variable $\zeta$ defined implicitly as follows:

(a) In the case $-\mu<\kappa<\mu$

 13.20.9 $\zeta\sqrt{\zeta^{2}+\alpha^{2}}+\alpha^{2}\operatorname{arcsinh}\left(\frac{% \zeta}{\alpha}\right)=\frac{X}{\mu}-\frac{2\kappa}{\mu}\ln\left(\frac{X+x-2% \kappa}{2\sqrt{\mu^{2}-\kappa^{2}}}\right)-2\ln\left(\frac{\mu X+2\mu^{2}-% \kappa x}{x\sqrt{\mu^{2}-\kappa^{2}}}\right).$

(b) In the case $\mu=\kappa$

 13.20.10 $\zeta=\pm\sqrt{\frac{x}{\mu}-2-2\ln\left(\frac{x}{2\mu}\right)},$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function and $x$: real variable Referenced by: §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E10 Encodings: TeX, pMML, png See also: Annotations for §13.20(iii), §13.20 and Ch.13

the upper or lower sign being taken according as $x\gtrless 2\mu$.

(In both cases (a) and (b) the $x$-interval $(0,\infty)$ is mapped one-to-one onto the $\zeta$-interval $(-\infty,\infty)$, with $x=0$ and $\infty$ corresponding to $\zeta=-\infty$ and $\infty$, respectively.) Then as $\mu\to\infty$

 13.20.11 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)\left(1+O\left(\mu^{-1}\ln\mu% \right)\right),$
 13.20.12 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)\left(1+O% \left(\mu^{-1}\ln\mu\right)\right),$

uniformly with respect to $x\in(0,\infty)$ and $\kappa\in[-(1-\delta)\mu,\mu]$. For the parabolic cylinder function $U$ see §12.2.

These results are proved in Olver (1980b). This reference also supplies error bounds and corresponding approximations when $x$, $\kappa$, and $\mu$ are replaced by $\mathrm{i}x$, $\mathrm{i}\kappa$, and $\mathrm{i}\mu$, respectively.

## §13.20(iv) Large $\mu$, $\mu\leq\kappa\leq\mu/\delta$

Again define $\alpha$, $X$, and $\Phi(\kappa,\mu,x)$ by (13.20.6)–(13.20.8), but with $\zeta$ now defined by

 13.20.13 $\displaystyle\zeta\sqrt{\zeta^{2}-\alpha^{2}}-\alpha^{2}\operatorname{arccosh}% \left(\frac{\zeta}{\alpha}\right)$ $\displaystyle=\frac{X}{\mu}-\frac{2\kappa}{\mu}\ln\left(\frac{X+x-2\kappa}{2% \sqrt{\kappa^{2}-\mu^{2}}}\right)-2\ln\left(\frac{\kappa x-\mu X-2\mu^{2}}{x% \sqrt{\kappa^{2}-\mu^{2}}}\right),$ $x\geq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$, 13.20.14 $\displaystyle\zeta\sqrt{\alpha^{2}-\zeta^{2}}+\alpha^{2}\operatorname{arcsin}% \left(\frac{\zeta}{\alpha}\right)$ $\displaystyle=\frac{X}{\mu}+\frac{2\kappa}{\mu}\operatorname{arctan}\left(% \frac{x-2\kappa}{X}\right)-2\operatorname{arctan}\left(\frac{\kappa x-2\mu^{2}% }{\mu X}\right),$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}\leq x\leq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$, 13.20.15 $\displaystyle-\zeta\sqrt{\zeta^{2}-\alpha^{2}}-\alpha^{2}\operatorname{arccosh% }\left(-\frac{\zeta}{\alpha}\right)$ $\displaystyle=-\frac{X}{\mu}+\frac{2\kappa}{\mu}\ln\left(\frac{2\kappa-X-x}{2% \sqrt{\kappa^{2}-\mu^{2}}}\right)+2\ln\left(\frac{\mu X+2\mu^{2}-\kappa x}{x% \sqrt{\kappa^{2}-\mu^{2}}}\right),$ $0,

when $\mu<\kappa$, and by (13.20.10) when $\mu=\kappa$. (As in §13.20(iii) $x=0$ and $\infty$ correspond to $\zeta=-\infty$ and $\infty$, respectively). Then as $\mu\to\infty$

 13.20.16 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*\left(U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)+\mathrm{env}\mskip-1.0mu U% \left(\mu-\kappa,\zeta\sqrt{2\mu}\right)O\left(\mu^{-\frac{2}{3}}\right)\right),$
 13.20.17 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*\left(U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+% \mathrm{env}\mskip-1.0mu \overline{U}\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)O% \left(\mu^{-\frac{2}{3}}\right)\right),$

uniformly with respect to $\zeta\in[0,\infty)$ and $\kappa\in[\mu,\mu/\delta]$.

Also,

 13.20.18 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*\left(U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)+\mathrm{env}\mskip-1.0mu % \overline{U}\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)O\left(\mu^{-\frac{2}{3}}% \right)\right),$
 13.20.19 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*\left(U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+% \mathrm{env}\mskip-1.0mu U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)O\left(\mu^% {-\frac{2}{3}}\right)\right),$

uniformly with respect to $\zeta\in(-\infty,0]$ and $\kappa\in[\mu,\mu/\delta]$.

For the parabolic cylinder functions $U$ and $\overline{U}$ see §12.2, and for the $\mathrm{env}$ functions associated with $U$ and $\overline{U}$ see §14.15(v).

These results are proved in Olver (1980b). Equations (13.20.17) and (13.20.18) are simpler than (6.10) and (6.11) in this reference. Olver (1980b) also supplies error bounds and corresponding approximations when $x$, $\kappa$, and $\mu$ are replaced by $\mathrm{i}x$, $\mathrm{i}\kappa$, and $\mathrm{i}\mu$, respectively.

It should be noted that (13.20.11), (13.20.16), and (13.20.18) differ only in the common error terms. Hence without the error terms the approximation holds for $-(1-\delta)\mu\leq\kappa\leq\mu/\delta$. Similarly for (13.20.12), (13.20.17), and (13.20.19).

## §13.20(v) Large $\mu$, Other Expansions

For uniform approximations valid when $\mu$ is large, $x/\mathrm{i}\in(0,\infty)$, and $\kappa/\mathrm{i}\in[0,\mu/\delta]$, see Olver (1997b, pp. 401–403). These approximations are in terms of Airy functions.

For uniform approximations of $M_{\kappa,\mathrm{i}\mu}\left(z\right)$ and $W_{\kappa,\mathrm{i}\mu}\left(z\right)$, $\kappa$ and $\mu$ real, one or both large, see Dunster (2003a).