# §13.21 Uniform Asymptotic Approximations for Large $\kappa$

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

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

When $\kappa\to\infty$ through positive real values with $\mu$ ($\geq 0$) fixed

 13.21.1 $M_{\kappa,\mu}\left(x\right)=\sqrt{x}\Gamma\left(2\mu+1\right)\kappa^{-\mu}\*% \left(J_{2\mu}\left(2\sqrt{x\kappa}\right)+\mathrm{env}\mskip-2.0mu J_{2\mu}% \left(2\sqrt{x\kappa}\right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$
 13.21.2 $W_{\kappa,\mu}\left(x\right)=\sqrt{x}\Gamma\left(\kappa+\tfrac{1}{2}\right)\*% \left(\sin\left(\kappa\pi-\mu\pi\right)J_{2\mu}\left(2\sqrt{x\kappa}\right)-% \cos\left(\kappa\pi-\mu\pi\right)Y_{2\mu}\left(2\sqrt{x\kappa}\right)+\mathrm{% env}\mskip-2.0mu Y_{2\mu}\left(2\sqrt{x\kappa}\right)O\left(\kappa^{-\frac{1}{% 2}}\right)\right),$
 13.21.3 $W_{-\kappa,\mu}\left(xe^{-\pi\mathrm{i}}\right)=\frac{\pi\sqrt{x}}{\Gamma\left% (\kappa+\tfrac{1}{2}\right)}e^{\mu\pi\mathrm{i}}\*\left({H^{(1)}_{2\mu}}\left(% 2\sqrt{x\kappa}\right)+\mathrm{env}\mskip-2.0mu Y_{2\mu}\left(2\sqrt{x\kappa}% \right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$
 13.21.4 $W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=\frac{\pi\sqrt{x}}{\Gamma\left(% \kappa+\tfrac{1}{2}\right)}e^{-\mu\pi\mathrm{i}}\*\left({H^{(2)}_{2\mu}}\left(% 2\sqrt{x\kappa}\right)+\mathrm{env}\mskip-2.0mu Y_{2\mu}\left(2\sqrt{x\kappa}% \right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$

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

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

 13.21.5 $2\sqrt{\zeta}=\sqrt{x+x^{2}}+\ln\left(\sqrt{x}+\sqrt{1+x}\right).$ ⓘ Symbols: $\ln\NVar{z}$: 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 13

Then

 13.21.6 $M_{-\kappa,\mu}\left(4\kappa x\right)=\frac{2\Gamma\left(2\mu+1\right)}{\kappa% ^{\mu-\frac{1}{2}}}\left(\frac{x\zeta}{1+x}\right)^{\frac{1}{4}}I_{2\mu}\left(% 4\kappa\zeta^{\frac{1}{2}}\right){\left(1+O\left(\kappa^{-1}\right)\right)},$
 13.21.7 $W_{-\kappa,\mu}\left(4\kappa x\right)=\frac{\sqrt{8/\pi}e^{\kappa}}{\kappa^{% \kappa-\frac{1}{2}}}\left(\frac{x\zeta}{1+x}\right)^{\frac{1}{4}}K_{2\mu}\left% (4\kappa\zeta^{\frac{1}{2}}\right){\left(1+O\left(\kappa^{-1}\right)\right)},$

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

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).

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

Let

 13.21.8 $c(\kappa,\mu)=e^{\mu\pi\mathrm{i}}\sqrt{\tfrac{1}{2}\pi}\left(\frac{\kappa-\mu% }{\kappa+\mu}\right)^{\frac{1}{2}\mu}\left(\frac{e^{2}}{\kappa^{2}-\mu^{2}}% \right)^{\frac{1}{2}\kappa},$ ⓘ Defines: $c(\kappa,\mu)$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{e}$: base of exponential function Permalink: http://dlmf.nist.gov/13.21.E8 Encodings: TeX, pMML, png See also: Annotations for 13.21(ii), 13.21 and 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 13
 13.21.10 $\Psi(\kappa,\mu,x)=\left(\frac{4\mu^{2}-\kappa\zeta}{x^{2}-4\kappa x+4\mu^{2}}% \right)^{\frac{1}{4}}\sqrt{x},$ ⓘ Defines: $\Psi(\kappa,\mu,x)$: 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 13

with the variable $\zeta$ defined implicitly by

 13.21.11 $\sqrt{4\mu^{2}-\kappa\zeta}-\mu\ln\left(\frac{2\mu+\sqrt{4\mu^{2}-\kappa\zeta}% }{2\mu-\sqrt{4\mu^{2}-\kappa\zeta}}\right)=\tfrac{1}{2}X+\mu\ln\left(\frac{x% \sqrt{\kappa^{2}-\mu^{2}}}{2\mu^{2}-\kappa x+\mu X}\right)+\kappa\ln\left(% \frac{2\sqrt{\kappa^{2}-\mu^{2}}}{2\kappa-x-X}\right),$ $0, ⓘ Symbols: $\ln\NVar{z}$: 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 13

and

 13.21.12 $\sqrt{\kappa\zeta-4\mu^{2}}-2\mu\operatorname{arctan}\left(\frac{\sqrt{\kappa% \zeta-4\mu^{2}}}{2\mu}\right)=\tfrac{1}{2}(X-\pi\mu)-\mu\operatorname{arctan}% \left(\frac{x\kappa-2\mu^{2}}{\mu X}\right)+\kappa\operatorname{arcsin}\left(% \frac{X}{2\sqrt{\kappa^{2}-\mu^{2}}}\right),$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}\leq x<2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$.

Then as $\kappa\to\infty$

 13.21.13 $\displaystyle M_{\kappa,\mu}\left(x\right)$ $\displaystyle=\Gamma\left(2\mu+1\right)\*\left(\frac{e^{2}}{\kappa^{2}-\mu^{2}% }\right)^{\frac{1}{2}\mu}\*\left(\frac{\kappa-\mu}{\kappa+\mu}\right)^{\frac{1% }{2}\kappa}\*\Psi(\kappa,\mu,x)\*\left(J_{2\mu}\left(\sqrt{\zeta\kappa}\right)% +\mathrm{env}\mskip-2.0mu J_{2\mu}\left(\sqrt{\zeta\kappa}\right)O\left(\kappa% ^{-1}\right)\right),$ 13.21.14 $\displaystyle W_{\kappa,\mu}\left(x\right)$ $\displaystyle=\frac{e^{-\mu\pi\mathrm{i}}}{\pi}\Gamma\left(\kappa+\mu+\tfrac{1% }{2}\right)\*\Gamma\left(\kappa-\mu+\tfrac{1}{2}\right)\*c(\kappa,\mu)\Psi(% \kappa,\mu,x)\*\left(\sin\left(\kappa\pi-\mu\pi\right)J_{2\mu}\left(\sqrt{% \zeta\kappa}\right)-\cos\left(\kappa\pi-\mu\pi\right)Y_{2\mu}\left(\sqrt{\zeta% \kappa}\right)+\mathrm{env}\mskip-2.0mu Y_{2\mu}\left(\sqrt{\zeta\kappa}\right% )O\left(\kappa^{-1}\right)\right),$
 13.21.15 $W_{-\kappa,\mu}\left(xe^{-\pi\mathrm{i}}\right)=c(\kappa,\mu)\Psi(\kappa,\mu,x% )\left({H^{(1)}_{2\mu}}\left(\sqrt{\zeta\kappa}\right)+\mathrm{env}\mskip-2.0% mu Y_{2\mu}\left(\sqrt{\zeta\kappa}\right)O\left(\kappa^{-1}\right)\right),$
 13.21.16 $W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=e^{-2\mu\pi\mathrm{i}}c(\kappa,% \mu)\Psi(\kappa,\mu,x)\left({H^{(2)}_{2\mu}}\left(\sqrt{\zeta\kappa}\right)+% \mathrm{env}\mskip-2.0mu Y_{2\mu}\left(\sqrt{\zeta\kappa}\right)O\left(\kappa^% {-1}\right)\right),$

uniformly with respect to $\mu\in[0,(1-\delta)\kappa]$ and $x\in\left(0,(1-\delta)(2\kappa+2\sqrt{\kappa^{2}-\mu^{2}})\right]$, where $\delta$ again denotes an arbitrary small positive constant. For the functions $J_{2\mu}$, $Y_{2\mu}$, ${H^{(1)}_{2\mu}}$, and ${H^{(2)}_{2\mu}}$ 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 $x$.

## §13.21(iii) Large $\kappa$, $0\leq\mu\leq(1-\delta)\kappa$ (Continued)

Let

 13.21.17 $\widehat{c}(\kappa,\mu)=\sqrt{2\pi}\kappa^{\frac{1}{6}}\left(\frac{\kappa-\mu}% {\kappa+\mu}\right)^{\frac{1}{2}\mu}\left(\frac{e^{2}}{\kappa^{2}-\mu^{2}}% \right)^{\frac{1}{2}\kappa},$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{e}$: base of exponential function Permalink: http://dlmf.nist.gov/13.21.E17 Encodings: TeX, pMML, png See also: Annotations for 13.21(iii), 13.21 and 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 13
 13.21.19 $\widehat{\Psi}(\kappa,\mu,x)=\left(\frac{\widehat{\zeta}}{x^{2}-4\kappa x+4\mu% ^{2}}\right)^{\frac{1}{4}}\sqrt{2x},$ ⓘ 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 13

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

 13.21.20 $\widehat{\zeta}=-\left(\frac{3}{2\kappa}\left(-\frac{1}{2}X+2\mu\operatorname{% arctan}\left(\frac{x\kappa-x\sqrt{\kappa^{2}-\mu^{2}}-2\mu^{2}}{\mu X}\right)+% \kappa\operatorname{arccos}\left(\frac{x-2\kappa}{2\sqrt{\kappa^{2}-\mu^{2}}}% \right)\right)\right)^{2/3},$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}, ⓘ Symbols: $\operatorname{arccos}\NVar{z}$: arccosine function, $\operatorname{arctan}\NVar{z}$: 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 13

and

 13.21.21 $\widehat{\zeta}=\left(\frac{3}{2\kappa}\left(\frac{1}{2}X+\mu\ln\left(\frac{x% \sqrt{\kappa^{2}-\mu^{2}}}{\kappa x-2\mu^{2}-\mu X}\right)+\kappa\ln\left(% \frac{2\sqrt{\kappa^{2}-\mu^{2}}}{x-2\kappa+X}\right)\right)\right)^{2/3},$ $x\geq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$. ⓘ Symbols: $\ln\NVar{z}$: 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 13

Then as $\kappa\to\infty$

 13.21.22 $M_{\kappa,\mu}\left(x\right)=\frac{1}{2\pi}\*\Gamma\left(2\mu+1\right)\*\Gamma% \left(\kappa-\mu+\tfrac{1}{2}\right)\*\widehat{c}(\kappa,\mu)\*\widehat{\Psi}(% \kappa,\mu,x)\*\left(\sin\left(\kappa\pi-\mu\pi\right)\mathrm{Ai}\left(\kappa^% {\frac{2}{3}}\widehat{\zeta}\right)+\cos\left(\kappa\pi-\mu\pi\right)\mathrm{% Bi}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)+\mathrm{envBi}\left(\kappa% ^{\frac{2}{3}}\widehat{\zeta}\right)O\left(\kappa^{-1}\right)\right),$
 13.21.23 $W_{\kappa,\mu}\left(x\right)=\sqrt{2\pi}\kappa^{\frac{1}{6}}\*\left(\frac{% \kappa+\mu}{\kappa-\mu}\right)^{\frac{1}{2}\mu}\*\left(\frac{\kappa^{2}-\mu^{2% }}{e^{2}}\right)^{\frac{1}{2}\kappa}\*\widehat{\Psi}(\kappa,\mu,x)\*\left(% \mathrm{Ai}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)+\mathrm{envAi}% \left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)O\left(\kappa^{-1}\right)% \right),$
 13.21.24 $W_{-\kappa,\mu}\left(xe^{-\pi\mathrm{i}}\right)=e^{(\kappa-\frac{1}{6})\pi% \mathrm{i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(\kappa,\mu,x)\left(\mathrm{Ai% }\left(\kappa^{\frac{2}{3}}\widehat{\zeta}e^{-\frac{2}{3}\pi\mathrm{i}}\right)% +\mathrm{envBi}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)O\left(\kappa^{% -1}\right)\right),$
 13.21.25 $W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=e^{-(\kappa-\frac{1}{6})\pi% \mathrm{i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(\kappa,\mu,x)\left(\mathrm{Ai% }\left(\kappa^{\frac{2}{3}}\widehat{\zeta}e^{\frac{2}{3}\pi\mathrm{i}}\right)+% \mathrm{envBi}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)O\left(\kappa^{-% 1}\right)\right),$

uniformly with respect to $\mu\in[0,(1-\delta)\kappa]$ and $x\in\left[(1+\delta)(2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}),\infty\right)$. 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 $x$.

## §13.21(iv) Large $\kappa$, Other Expansions

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 $x\in[\delta,\infty)$ 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).