§13.20 Uniform Asymptotic Approximations for Large $\mu$

When $\mu \to \mathrm{\infty }$ in the sector , with $\kappa (\in ℂ)$ fixed

13.20.1		$$M_{\kappa ,\mu }\left(z\right)=z^{\mu +\frac{1}{2}}\left(1+O\left({\mu }^{-1}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: 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		$$W_{\kappa ,\mu }\left(x\right)={\pi }^{-\frac{1}{2}}\mathrm{\Gamma }\left(\kappa +\mu \right){\left(\frac{1}{4}x\right)}^{\frac{1}{2}-\mu }\left(1+O\left({\mu }^{-1}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable Referenced by: §13.20(i) Permalink: http://dlmf.nist.gov/13.20.E2 Encodings: TeX, pMML, png See also: Annotations for §13.20(i), §13.20 and Ch.13

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).

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

13.20.4		$$M_{\kappa ,\mu }\left(x\right)=\sqrt{\frac{2\mu x}{X}}{\left(\frac{4{\mu }^{2}x}{2{\mu }^2-\kappa x+\mu X}\right)}^{\mu }{\left(\frac{2(\mu -\kappa )}{X+x-2\kappa }\right)}^{\kappa }{\mathrm{e}}^{\frac{1}{2}X-\mu }\left(1+O\left(\frac{1}{\mu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $X$ Permalink: http://dlmf.nist.gov/13.20.E4 Encodings: TeX, pMML, png See also: Annotations for §13.20(ii), §13.20 and Ch.13

13.20.5		$$W_{\kappa ,\mu }\left(x\right)=\sqrt{\frac{x}{X}}{\left(\frac{2{\mu }^2-\kappa x+\mu X}{(\mu -\kappa )x}\right)}^{\mu }{\left(\frac{X+x-2\kappa }{2}\right)}^{\kappa }{\mathrm{e}}^{-\frac{1}{2}X-\kappa }\left(1+O\left(\frac{1}{\mu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $X$ Permalink: http://dlmf.nist.gov/13.20.E5 Encodings: TeX, pMML, png See also: Annotations for §13.20(ii), §13.20 and Ch.13

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

Let

13.20.6		$\alpha$	$=\sqrt{2|\kappa -\mu |/\mu },$
ⓘ 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		$X$	$=\sqrt{|x^2-4\kappa x+4{\mu }^2|},$
ⓘ 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		$\mathrm{\Phi }(\kappa ,\mu ,x)$	$={\left({\displaystyle \frac{{\mu }^2{\zeta }^2-2\kappa \mu +2{\mu }^2}{x^2-4\kappa x+4{\mu }^2}}\right)}^{\frac{1}{4}}\sqrt{x},$
ⓘ Defines: $\mathrm{\Phi }(\kappa ,\mu ,x)$: 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

13.20.9		$$\zeta \sqrt{{\zeta }^2+{\alpha }^2}+{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\mu }^2-{\kappa }^2}}\right)-2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\mu }^2-{\kappa }^2}}\right).$$
ⓘ Symbols: $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$ Permalink: http://dlmf.nist.gov/13.20.E9 Encodings: TeX, pMML, png See also: Annotations for §13.20(iii), §13.20 and Ch.13

(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: $\ln z$: 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 $(0,\mathrm{\infty })$ is mapped one-to-one onto the $\zeta$-interval $(-\mathrm{\infty },\mathrm{\infty })$, with $x=0$ and $\mathrm{\infty }$ corresponding to $\zeta =-\mathrm{\infty }$ and $\mathrm{\infty }$, respectively.) Then as $\mu \to \mathrm{\infty }$

13.20.11		$$W_{\kappa ,\mu }\left(x\right)={\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)U(\mu -\kappa ,\zeta \sqrt{2\mu })\left(1+O\left({\mu }^{-1}\ln \mu \right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function Referenced by: §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E11 Encodings: TeX, pMML, png See also: Annotations for §13.20(iii), §13.20 and Ch.13

13.20.12		$$M_{\kappa ,\mu }\left(x\right)=\begin{array}{l}{\left(8\mu \right)}^{\frac{1}{4}}{\left(\frac{2\mu }{\mathrm{e}}\right)}^{2\mu }{\left(\frac{\mathrm{e}}{\kappa +\mu }\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\\ \times U(\mu -\kappa ,-\zeta \sqrt{2\mu })\left(1+O\left({\mu }^{-1}\ln \mu \right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function Referenced by: §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E12 Encodings: TeX, pMML, png See also: Annotations for §13.20(iii), §13.20 and Ch.13

uniformly with respect to $x\in (0,\mathrm{\infty })$ 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.

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

13.20.13		$\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left({\displaystyle \frac{\zeta }{\alpha }}\right)$	$={\displaystyle \frac{X}{\mu }}-{\displaystyle \frac{2\kappa }{\mu }}\ln \left({\displaystyle \frac{X+x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}}\right)-2\ln \left({\displaystyle \frac{\kappa x-\mu X-2{\mu }^2}{x\sqrt{{\kappa }^2-{\mu }^2}}}\right),$
$x\ge 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,
ⓘ Symbols: $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$ Permalink: http://dlmf.nist.gov/13.20.E13 Encodings: TeX, pMML, png See also: Annotations for §13.20(iv), §13.20 and Ch.13
13.20.14		$\zeta \sqrt{{\alpha }^2-{\zeta }^2}+{\alpha }^2\arcsin \left({\displaystyle \frac{\zeta }{\alpha }}\right)$	$={\displaystyle \frac{X}{\mu }}+{\displaystyle \frac{2\kappa }{\mu }}\arctan \left({\displaystyle \frac{x-2\kappa }{X}}\right)-2\arctan \left({\displaystyle \frac{\kappa x-2{\mu }^2}{\mu X}}\right),$
$2\kappa -2\sqrt{{\kappa }^2-{\mu }^2}\le x\le 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,
ⓘ Symbols: $\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable, $\alpha$ and $X$ Permalink: http://dlmf.nist.gov/13.20.E14 Encodings: TeX, pMML, png See also: Annotations for §13.20(iv), §13.20 and Ch.13
13.20.15		$-\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(-{\displaystyle \frac{\zeta }{\alpha }}\right)$	$=-{\displaystyle \frac{X}{\mu }}+{\displaystyle \frac{2\kappa }{\mu }}\ln \left({\displaystyle \frac{2\kappa -X-x}{2\sqrt{{\kappa }^2-{\mu }^2}}}\right)+2\ln \left({\displaystyle \frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\kappa }^2-{\mu }^2}}}\right),$
,
ⓘ Symbols: $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$ Permalink: http://dlmf.nist.gov/13.20.E15 Encodings: TeX, pMML, png See also: Annotations for §13.20(iv), §13.20 and Ch.13

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

13.20.16		$$W_{\kappa ,\mu }\left(x\right)=\begin{array}{l}{\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\\ \times \left(U(\mu -\kappa ,\zeta \sqrt{2\mu })+\mathrm{env}U(\mu -\kappa ,\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function Referenced by: §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E16 Encodings: TeX, pMML, png See also: Annotations for §13.20(iv), §13.20 and Ch.13

13.20.17		$$M_{\kappa ,\mu }\left(x\right)=\begin{array}{l}{\left(8\mu \right)}^{\frac{1}{4}}{\left(\frac{2\mu }{\mathrm{e}}\right)}^{2\mu }{\left(\frac{\mathrm{e}}{\kappa +\mu }\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\\ \times \left(U(\mu -\kappa ,-\zeta \sqrt{2\mu })+\mathrm{env}\overline{U}(\mu -\kappa ,\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function Referenced by: §13.20(iv), §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E17 Encodings: TeX, pMML, png See also: Annotations for §13.20(iv), §13.20 and Ch.13

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

Also,

13.20.18		$$W_{\kappa ,\mu }\left(x\right)=\begin{array}{l}{\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\\ \times \left(U(\mu -\kappa ,\zeta \sqrt{2\mu })+\mathrm{env}\overline{U}(\mu -\kappa ,-\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function Referenced by: §13.20(iv), §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E18 Encodings: TeX, pMML, png See also: Annotations for §13.20(iv), §13.20 and Ch.13

13.20.19		$$M_{\kappa ,\mu }\left(x\right)=\begin{array}{l}{\left(8\mu \right)}^{\frac{1}{4}}{\left(\frac{2\mu }{\mathrm{e}}\right)}^{2\mu }{\left(\frac{\mathrm{e}}{\kappa +\mu }\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\\ \times \left(U(\mu -\kappa ,-\zeta \sqrt{2\mu })+\mathrm{env}U(\mu -\kappa ,-\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function Referenced by: §13.20(iv) Permalink: http://dlmf.nist.gov/13.20.E19 Encodings: TeX, pMML, png See also: Annotations for §13.20(iv), §13.20 and Ch.13

uniformly with respect to $\zeta \in (-\mathrm{\infty },0]$ 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 $-(1-\delta )\mu \le \kappa \le \mu /\delta$. Similarly for (13.20.12), (13.20.17), and (13.20.19).

For uniform approximations valid when $\mu$ is large, $x/\mathrm{i}\in (0,\mathrm{\infty })$, 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).