§13.21 Uniform Asymptotic Approximations for Large $\kappa$

For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv).

When $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed

13.21.1		$$M_{\kappa ,\mu }\left(x\right)=\sqrt{x}\mathrm{\Gamma }\left(2\mu +1\right){\kappa }^{-\mu }\left(J_{2\mu }\left(2\sqrt{x\kappa }\right)+\mathrm{env}{J}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ and $x$: real variable Permalink: http://dlmf.nist.gov/13.21.E1 Encodings: TeX, pMML, png See also: Annotations for §13.21(i), §13.21 and Ch.13

13.21.2		$$W_{\kappa ,\mu }\left(x\right)=\sqrt{x}\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)(\begin{array}{l}\begin{array}{l}\sin \left(\kappa \pi -\mu \pi \right)J_{2\mu }\left(2\sqrt{x\kappa }\right)\\ -\cos \left(\kappa \pi -\mu \pi \right)Y_{2\mu }\left(2\sqrt{x\kappa }\right)\end{array}\\ +\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)),\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: 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, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\sin z$: sine function and $x$: real variable Referenced by: §13.21(i) Permalink: http://dlmf.nist.gov/13.21.E2 Encodings: TeX, pMML, png See also: Annotations for §13.21(i), §13.21 and Ch.13

13.21.3		$$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=\frac{\pi \sqrt{x}}{\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)}{\mathrm{e}}^{\mu \pi \mathrm{i}}\left(H_{2\mu }^{(1)}\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable Permalink: http://dlmf.nist.gov/13.21.E3 Encodings: TeX, pMML, png See also: Annotations for §13.21(i), §13.21 and Ch.13

13.21.4		$$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)=\frac{\pi \sqrt{x}}{\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)}{\mathrm{e}}^{-\mu \pi \mathrm{i}}\left(H_{2\mu }^{(2)}\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable Referenced by: §13.21(i) Permalink: http://dlmf.nist.gov/13.21.E4 Encodings: TeX, pMML, png See also: Annotations for §13.21(i), §13.21 and Ch.13

uniformly with respect to $x\in (0,A]$ in each case, where $A$ is an arbitrary positive constant.

Other types of approximations when $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed are as follows. Define

13.21.5		$$2\sqrt{\zeta }=\sqrt{x+x^2}+\ln \left(\sqrt{x}+\sqrt{1+x}\right).$$
ⓘ Symbols: $\ln z$: principal branch of logarithm function and $x$: real variable Permalink: http://dlmf.nist.gov/13.21.E5 Encodings: TeX, pMML, png See also: Annotations for §13.21(i), §13.21 and Ch.13

Then

13.21.6		$$M_{-\kappa ,\mu }\left(4\kappa x\right)=\frac{2\mathrm{\Gamma }\left(2\mu +1\right)}{{\kappa }^{\mu -\frac{1}{2}}}{\left(\frac{x\zeta }{1+x}\right)}^{\frac{1}{4}}{I}_{2\mu }\left(4\kappa {\zeta }^{\frac{1}{2}}\right)\left(1+O\left({\kappa }^{-1}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $x$: real variable Referenced by: §13.21(i) Permalink: http://dlmf.nist.gov/13.21.E6 Encodings: TeX, pMML, png See also: Annotations for §13.21(i), §13.21 and Ch.13

13.21.7		$$W_{-\kappa ,\mu }\left(4\kappa x\right)=\frac{\sqrt{8/\pi }{\mathrm{e}}^{\kappa }}{{\kappa }^{\kappa -\frac{1}{2}}}{\left(\frac{x\zeta }{1+x}\right)}^{\frac{1}{4}}{K}_{2\mu }\left(4\kappa {\zeta }^{\frac{1}{2}}\right)\left(1+O\left({\kappa }^{-1}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable Referenced by: §13.21(i) Permalink: http://dlmf.nist.gov/13.21.E7 Encodings: TeX, pMML, png See also: Annotations for §13.21(i), §13.21 and Ch.13

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

For (13.21.6), (13.21.7), and extensions to asymptotic expansions and error bounds, see Olver (1997b, Chapter 12, Exs. 12.4.5, 12.4.6). For extensions to complex values of $x$ see López (1999).

Let

13.21.8		$$c(\kappa ,\mu )={\mathrm{e}}^{\mu \pi \mathrm{i}}\sqrt{\frac{1}{2}\pi }{\left(\frac{\kappa -\mu }{\kappa +\mu }\right)}^{\frac{1}{2}\mu }{\left(\frac{{\mathrm{e}}^2}{{\kappa }^2-{\mu }^2}\right)}^{\frac{1}{2}\kappa },$$
ⓘ Defines: $c(\kappa ,\mu )$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit Permalink: http://dlmf.nist.gov/13.21.E8 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

13.21.9		$$X=\sqrt{|x^2-4\kappa x+4{\mu }^2|},$$
ⓘ Defines: $X$ (locally) Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/13.21.E9 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

13.21.10		$$\mathrm{\Psi }(\kappa ,\mu ,x)={\left(\frac{4{\mu }^2-\kappa \zeta }{x^2-4\kappa x+4{\mu }^2}\right)}^{\frac{1}{4}}\sqrt{x},$$
ⓘ Defines: $\mathrm{\Psi }(\kappa ,\mu ,x)$: function (locally) Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/13.21.E10 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

with the variable $\zeta$ defined implicitly by

13.21.11		$$\sqrt{4{\mu }^2-\kappa \zeta }-\mu \ln \left(\frac{2\mu +\sqrt{4{\mu }^2-\kappa \zeta }}{2\mu -\sqrt{4{\mu }^2-\kappa \zeta }}\right)=\frac{1}{2}X+\mu \ln \left(\frac{x\sqrt{{\kappa }^2-{\mu }^2}}{2{\mu }^2-\kappa x+\mu X}\right)+\kappa \ln \left(\frac{2\sqrt{{\kappa }^2-{\mu }^2}}{2\kappa -x-X}\right),$$
,
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $x$: real variable and $X$ Permalink: http://dlmf.nist.gov/13.21.E11 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

and

13.21.12		$$\begin{array}{l}\sqrt{\kappa \zeta -4{\mu }^2}-2\mu \arctan \left(\frac{\sqrt{\kappa \zeta -4{\mu }^2}}{2\mu }\right)\\ =\frac{1}{2}(X-\pi \mu )-\mu \arctan \left(\frac{x\kappa -2{\mu }^2}{\mu X}\right)+\kappa \arcsin \left(\frac{X}{2\sqrt{{\kappa }^2-{\mu }^2}}\right),\end{array}$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable and $X$ Permalink: http://dlmf.nist.gov/13.21.E12 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

Then as $\kappa \to \mathrm{\infty }$

13.21.13		$M_{\kappa ,\mu }\left(x\right)$	$=\mathrm{\Gamma }\left(2\mu +1\right){\left({\displaystyle \frac{{\mathrm{e}}^2}{{\kappa }^2-{\mu }^2}}\right)}^{\frac{1}{2}\mu }{\left({\displaystyle \frac{\kappa -\mu }{\kappa +\mu }}\right)}^{\frac{1}{2}\kappa }\mathrm{\Psi }(\kappa ,\mu ,x)\left(J_{2\mu }\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{J}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E13 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13
13.21.14		$W_{\kappa ,\mu }\left(x\right)$	$=\begin{array}{l}{\displaystyle \frac{{\mathrm{e}}^{-\mu \pi \mathrm{i}}}{\pi }}\mathrm{\Gamma }\left(\kappa +\mu +\frac{1}{2}\right)\\ \times \mathrm{\Gamma }\left(\kappa -\mu +\frac{1}{2}\right)c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\\ \times (\begin{array}{l}\begin{array}{l}\sin \left(\kappa \pi -\mu \pi \right)J_{2\mu }\left(\sqrt{\zeta \kappa }\right)\\ -\cos \left(\kappa \pi -\mu \pi \right)Y_{2\mu }\left(\sqrt{\zeta \kappa }\right)\end{array}\\ +\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)),\end{array}\end{array}$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: 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, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E14 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

13.21.15		$$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\left(H_{2\mu }^{(1)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E15 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

13.21.16		$$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)=\begin{array}{l}{\mathrm{e}}^{-2\mu \pi \mathrm{i}}c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\\ \times \left(H_{2\mu }^{(2)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),\end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E16 Encodings: TeX, pMML, png See also: Annotations for §13.21(ii), §13.21 and Ch.13

uniformly with respect to $\mu \in [0,(1-\delta )\kappa ]$ and $x\in (0,(1-\delta )(2\kappa +2\sqrt{{\kappa }^2-{\mu }^2})]$, where $\delta$ again denotes an arbitrary small positive constant. For the functions $J_{2\mu }$, $Y_{2\mu }$, $H_{2\mu }^{(1)}$, and $H_{2\mu }^{(2)}$ see §10.2(ii), and for the $\mathrm{env}$ functions associated with $J_{2\mu }$ and $Y_{2\mu }$ see §2.8(iv).

These approximations are proved in Dunster (1989). This reference also includes error bounds and extensions to asymptotic expansions and complex values of $x$.

Let

13.21.17		$$\widehat{c}(\kappa ,\mu )=\sqrt{2\pi }{\kappa }^{\frac{1}{6}}{\left(\frac{\kappa -\mu }{\kappa +\mu }\right)}^{\frac{1}{2}\mu }{\left(\frac{{\mathrm{e}}^2}{{\kappa }^2-{\mu }^2}\right)}^{\frac{1}{2}\kappa },$$
ⓘ Defines: $\widehat{c}(\kappa ,\mu )$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{e}$: base of natural logarithm Permalink: http://dlmf.nist.gov/13.21.E17 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

13.21.18		$$X=\sqrt{|x^2-4\kappa x+4{\mu }^2|},$$
ⓘ Defines: $X$ (locally) Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/13.21.E18 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

13.21.19		$$\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)={\left(\frac{\widehat{\zeta }}{x^2-4\kappa x+4{\mu }^2}\right)}^{\frac{1}{4}}\sqrt{2x},$$
ⓘ Defines: $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function (locally) Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/13.21.E19 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

and define the variable $\widehat{\zeta }$ implicitly by

13.21.20		$$\widehat{\zeta }=-{\left(\frac{3}{2\kappa }\left(-\frac{1}{2}X+2\mu \arctan \left(\frac{x\kappa -x\sqrt{{\kappa }^2-{\mu }^2}-2{\mu }^2}{\mu X}\right)+\kappa \arccos \left(\frac{x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)\right)\right)}^{2/3},$$
,
ⓘ Symbols: $\arccos z$: arccosine function, $\arctan z$: arctangent function, $x$: real variable and $X$ Permalink: http://dlmf.nist.gov/13.21.E20 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

and

13.21.21		$$\widehat{\zeta }={\left(\frac{3}{2\kappa }\left(\frac{1}{2}X+\mu \ln \left(\frac{x\sqrt{{\kappa }^2-{\mu }^2}}{\kappa x-2{\mu }^2-\mu X}\right)+\kappa \ln \left(\frac{2\sqrt{{\kappa }^2-{\mu }^2}}{x-2\kappa +X}\right)\right)\right)}^{2/3},$$
$x\ge 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$.
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $x$: real variable and $X$ Permalink: http://dlmf.nist.gov/13.21.E21 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

Then as $\kappa \to \mathrm{\infty }$

13.21.22		$$M_{\kappa ,\mu }\left(x\right)=\begin{array}{l}\frac{1}{2\pi }\mathrm{\Gamma }\left(2\mu +1\right)\mathrm{\Gamma }\left(\kappa -\mu +\frac{1}{2}\right)\widehat{c}(\kappa ,\mu )\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)\\ \times (\begin{array}{l}\sin \left(\kappa \pi -\mu \pi \right)\mathrm{Ai}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)\\ +\cos \left(\kappa \pi -\mu \pi \right)\mathrm{Bi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)+\mathrm{envBi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)O\left({\kappa }^{-1}\right)),\end{array}\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$, $\sin z$: sine function, $x$: real variable, $\widehat{c}(\kappa ,\mu )$: function and $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E22 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

13.21.23		$$W_{\kappa ,\mu }\left(x\right)=\begin{array}{l}\sqrt{2\pi }{\kappa }^{\frac{1}{6}}{\left(\frac{\kappa +\mu }{\kappa -\mu }\right)}^{\frac{1}{2}\mu }{\left(\frac{{\kappa }^2-{\mu }^2}{{\mathrm{e}}^2}\right)}^{\frac{1}{2}\kappa }\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)\\ \times \left(\mathrm{Ai}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)+\mathrm{envAi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)O\left({\kappa }^{-1}\right)\right),\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E23 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

13.21.24		$$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=\begin{array}{l}{\mathrm{e}}^{(\kappa -\frac{1}{6})\pi \mathrm{i}}\widehat{c}(\kappa ,\mu )\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)\\ \times \left(\mathrm{Ai}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }{\mathrm{e}}^{-\frac{2}{3}\pi \mathrm{i}}\right)+\mathrm{envBi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)O\left({\kappa }^{-1}\right)\right),\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\widehat{c}(\kappa ,\mu )$: function and $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E24 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

13.21.25		$$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)=\begin{array}{l}{\mathrm{e}}^{-(\kappa -\frac{1}{6})\pi \mathrm{i}}\widehat{c}(\kappa ,\mu )\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)\\ \times \left(\mathrm{Ai}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }{\mathrm{e}}^{\frac{2}{3}\pi \mathrm{i}}\right)+\mathrm{envBi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)O\left({\kappa }^{-1}\right)\right),\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\widehat{c}(\kappa ,\mu )$: function and $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function Permalink: http://dlmf.nist.gov/13.21.E25 Encodings: TeX, pMML, png See also: Annotations for §13.21(iii), §13.21 and Ch.13

uniformly with respect to $\mu \in [0,(1-\delta )\kappa ]$ and $x\in [(1+\delta )(2\kappa -2\sqrt{{\kappa }^2-{\mu }^2}),\mathrm{\infty })$. For the functions $\mathrm{Ai}$ and $\mathrm{Bi}$ see §9.2(i), and for the $\mathrm{env}$ functions associated with $\mathrm{Ai}$ and $\mathrm{Bi}$ see §2.8(iii).

These approximations are proved in Dunster (1989). This reference also includes error bounds and extensions to asymptotic expansions and complex values of $x$.

For a uniform asymptotic expansion in terms of Airy functions for $W_{\kappa ,\mu }\left(4\kappa x\right)$ when $\kappa$ is large and positive, $\mu$ is real with $|\mu |$ bounded, and $x\in [\delta ,\mathrm{\infty })$ see Olver (1997b, Chapter 11, Ex. 7.3). This expansion is simpler in form than the expansions of Dunster (1989) that correspond to the approximations given in §13.21(iii), but the conditions on $\mu$ are more restrictive.

For asymptotic expansions having double asymptotic properties see Skovgaard (1966).

See also §13.20(v).