13 Confluent Hypergeometric Functions Whittaker Functions 13.20 Uniform Asymptotic Approximations for Large $\mu$ 13.22 Zeros

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

Keywords:: Whittaker functions
Referenced by:: §13.22
Permalink:: http://dlmf.nist.gov/13.21

§13.21(i) Large $\kappa$ , Fixed $\mu$
§13.21(ii) Large $\kappa$ , $0\leq\mu\leq(1-\delta)\kappa$
§13.21(iii) Large $\kappa$ , $0\leq\mu\leq(1-\delta)\kappa$ (Continued)
§13.21(iv) Large $\kappa$ , Other Expansions

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

Notes:: See Slater (1960, §4.4.3). The asymptotic approximations (13.21.2)–(13.21.4) follow from §13.21(ii) and (5.11.7).
Referenced by:: §13.8(iii), §2.8(iv)
Permalink:: http://dlmf.nist.gov/13.21.i

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 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\sqrt{x}\mathop{\Gamma\/}% \nolimits\!\left(2\mu+1\right)\kappa^{{-\mu}}\*\left(\mathop{J_{{2\mu}}\/}% \nolimits\!\left(2\sqrt{x\kappa}\right)+\mathop{\mathrm{env}J_{{2\mu}}\/}% \nolimits\!\left(2\sqrt{x\kappa}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-% \frac{1}{2}}}\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : real variable
Permalink:: http://dlmf.nist.gov/13.21.E1
Encodings:: TeX, pMML, png

13.21.2 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\sqrt{x}\mathop{\Gamma\/}% \nolimits\!\left(\kappa+\tfrac{1}{2}\right)\*\left(\mathop{\sin\/}\nolimits\!% \left(\kappa\pi-\mu\pi\right)\mathop{J_{{2\mu}}\/}\nolimits\!\left(2\sqrt{x% \kappa}\right)-\mathop{\cos\/}\nolimits\!\left(\kappa\pi-\mu\pi\right)\mathop{% Y_{{2\mu}}\/}\nolimits\!\left(2\sqrt{x\kappa}\right)+\mathop{\mathrm{env}Y_{{2% \mu}}\/}\nolimits\!\left(2\sqrt{x\kappa}\right)\mathop{O\/}\nolimits\!\left(% \kappa^{{-\frac{1}{2}}}\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and : real variable
Referenced by:: §13.21(i)
Permalink:: http://dlmf.nist.gov/13.21.E2
Encodings:: TeX, pMML, png

13.21.3 $\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(xe^{{-\pi i}}\right)=\frac{\pi% \sqrt{x}}{\mathop{\Gamma\/}\nolimits\!\left(\kappa+\tfrac{1}{2}\right)}e^{{\mu% \pi i}}\*\left(\mathop{{H^{{(1)}}_{{2\mu}}}\/}\nolimits\!\left(2\sqrt{x\kappa}% \right)+\mathop{\mathrm{env}Y_{{2\mu}}\/}\nolimits\!\left(2\sqrt{x\kappa}% \right)\mathop{O\/}\nolimits\!\left(\kappa^{{-\frac{1}{2}}}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function and : real variable
Permalink:: http://dlmf.nist.gov/13.21.E3
Encodings:: TeX, pMML, png

13.21.4 $\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(xe^{{\pi i}}\right)=\frac{\pi% \sqrt{x}}{\mathop{\Gamma\/}\nolimits\!\left(\kappa+\tfrac{1}{2}\right)}e^{{-% \mu\pi i}}\*\left(\mathop{{H^{{(2)}}_{{2\mu}}}\/}\nolimits\!\left(2\sqrt{x% \kappa}\right)+\mathop{\mathrm{env}Y_{{2\mu}}\/}\nolimits\!\left(2\sqrt{x% \kappa}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-\frac{1}{2}}}\right)% \right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function and : real variable
Referenced by:: §13.21(i)
Permalink:: http://dlmf.nist.gov/13.21.E4
Encodings:: TeX, pMML, png

uniformly with respect to $x\in(0,A]$ in each case, where 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 $2\sqrt{\zeta}=\sqrt{x+x^{2}}+\mathop{\ln\/}\nolimits\!\left(\sqrt{x}+\sqrt{1+x% }\right).$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : real variable
Permalink:: http://dlmf.nist.gov/13.21.E5
Encodings:: TeX, pMML, png

Then

13.21.6 $\mathop{M_{{-\kappa,\mu}}\/}\nolimits\!\left(4\kappa x\right)=\frac{2\mathop{% \Gamma\/}\nolimits\!\left(2\mu+1\right)}{\kappa^{{\mu-\frac{1}{2}}}}\left(% \frac{x\zeta}{1+x}\right)^{{\frac{1}{4}}}\mathop{I_{{2\mu}}\/}\nolimits\!\left% (4\kappa\zeta^{\frac{1}{2}}\right){\left(1+\mathop{O\/}\nolimits\!\left(\kappa% ^{{-1}}\right)\right)},$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : real variable
Referenced by:: §13.21(i)
Permalink:: http://dlmf.nist.gov/13.21.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : real variable
Referenced by:: §13.21(i)
Permalink:: http://dlmf.nist.gov/13.21.E7
Encodings:: TeX, pMML, png

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 see López (1999).

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

Referenced by:: §13.21(i), §13.8(iii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/13.21.ii

Let

13.21.8 $c(\kappa,\mu)=e^{{\mu\pi i}}\sqrt{\tfrac{1}{2}\pi}\left(\frac{\kappa-\mu}{% \kappa+\mu}\right)^{{\frac{1}{2}\mu}}\left(\frac{e^{2}}{\kappa^{2}-\mu^{2}}% \right)^{{\frac{1}{2}\kappa}},$

Symbols:: : base of exponential function and $c(\kappa,\mu)$ : function
Permalink:: http://dlmf.nist.gov/13.21.E8
Encodings:: TeX, pMML, png

13.21.9 $X=\sqrt{|x^{2}-4\kappa x+4\mu^{2}|},$

Symbols:: : real variable and
Permalink:: http://dlmf.nist.gov/13.21.E9
Encodings:: TeX, pMML, png

13.21.10 $\Psi(\kappa,\mu,x)=\left(\frac{4\mu^{2}-\kappa\zeta}{x^{2}-4\kappa x+4\mu^{2}}% \right)^{{\frac{1}{4}}}\sqrt{x},$

Symbols:: : real variable and $\Psi(\kappa,\mu,x)$ : function
Permalink:: http://dlmf.nist.gov/13.21.E10
Encodings:: TeX, pMML, png

with the variable $\zeta$ defined implicitly by

13.21.11 $\sqrt{4\mu^{2}-\kappa\zeta}-\mu\mathop{\ln\/}\nolimits\!\left(\frac{2\mu+\sqrt% {4\mu^{2}-\kappa\zeta}}{2\mu-\sqrt{4\mu^{2}-\kappa\zeta}}\right)=\tfrac{1}{2}X% +\mu\mathop{\ln\/}\nolimits\!\left(\frac{x\sqrt{\kappa^{2}-\mu^{2}}}{2\mu^{2}-% \kappa x+\mu X}\right)+\kappa\mathop{\ln\/}\nolimits\!\left(\frac{2\sqrt{% \kappa^{2}-\mu^{2}}}{2\kappa-x-X}\right),$ $0<x\leq 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable and
Permalink:: http://dlmf.nist.gov/13.21.E11
Encodings:: TeX, pMML, png

and

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

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, : real variable and
Permalink:: http://dlmf.nist.gov/13.21.E12
Encodings:: TeX, pMML, png

Then as $\kappa\to\infty$

13.21.13 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\mathop{\Gamma\/}% \nolimits\!\left(2\mu+1\right)\*\left(\frac{e^{2}}{\kappa^{2}-\mu^{2}}\right)^% {{\frac{1}{2}\mu}}\*\left(\frac{\kappa-\mu}{\kappa+\mu}\right)^{{\frac{1}{2}% \kappa}}\*\Psi(\kappa,\mu,x)\*\left(\mathop{J_{{2\mu}}\/}\nolimits\!\left(% \sqrt{\zeta\kappa}\right)+\mathop{\mathrm{env}J_{{2\mu}}\/}\nolimits\!\left(% \sqrt{\zeta\kappa}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-1}}\right)% \right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : real variable and $\Psi(\kappa,\mu,x)$ : function
Permalink:: http://dlmf.nist.gov/13.21.E13
Encodings:: TeX, pMML, png

13.21.14 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\frac{e^{{-\mu\pi i}}}{% \pi}\mathop{\Gamma\/}\nolimits\!\left(\kappa+\mu+\tfrac{1}{2}\right)\*\mathop{% \Gamma\/}\nolimits\!\left(\kappa-\mu+\tfrac{1}{2}\right)\*c(\kappa,\mu)\Psi(% \kappa,\mu,x)\*\left(\mathop{\sin\/}\nolimits\!\left(\kappa\pi-\mu\pi\right)% \mathop{J_{{2\mu}}\/}\nolimits\!\left(\sqrt{\zeta\kappa}\right)-\mathop{\cos\/% }\nolimits\!\left(\kappa\pi-\mu\pi\right)\mathop{Y_{{2\mu}}\/}\nolimits\!\left% (\sqrt{\zeta\kappa}\right)+\mathop{\mathrm{env}Y_{{2\mu}}\/}\nolimits\!\left(% \sqrt{\zeta\kappa}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-1}}\right)% \right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, $c(\kappa,\mu)$ : function and $\Psi(\kappa,\mu,x)$ : function
Permalink:: http://dlmf.nist.gov/13.21.E14
Encodings:: TeX, pMML, png

13.21.15 $\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(xe^{{-\pi i}}\right)=c(\kappa,\mu% )\Psi(\kappa,\mu,x)\left(\mathop{{H^{{(1)}}_{{2\mu}}}\/}\nolimits\!\left(\sqrt% {\zeta\kappa}\right)+\mathop{\mathrm{env}Y_{{2\mu}}\/}\nolimits\!\left(\sqrt{% \zeta\kappa}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-1}}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : real variable, $c(\kappa,\mu)$ : function and $\Psi(\kappa,\mu,x)$ : function
Permalink:: http://dlmf.nist.gov/13.21.E15
Encodings:: TeX, pMML, png

13.21.16 $\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(xe^{{\pi i}}\right)=e^{{-2\mu\pi i% }}c(\kappa,\mu)\Psi(\kappa,\mu,x)\left(\mathop{{H^{{(2)}}_{{2\mu}}}\/}% \nolimits\!\left(\sqrt{\zeta\kappa}\right)+\mathop{\mathrm{env}Y_{{2\mu}}\/}% \nolimits\!\left(\sqrt{\zeta\kappa}\right)\mathop{O\/}\nolimits\!\left(\kappa^% {{-1}}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : real variable, $c(\kappa,\mu)$ : function and $\Psi(\kappa,\mu,x)$ : function
Permalink:: http://dlmf.nist.gov/13.21.E16
Encodings:: TeX, pMML, png

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 $\mathop{J_{{2\mu}}\/}\nolimits$ , $\mathop{Y_{{2\mu}}\/}\nolimits$ , $\mathop{{H^{{(1)}}_{{2\mu}}}\/}\nolimits$ , and $\mathop{{H^{{(2)}}_{{2\mu}}}\/}\nolimits$ see §10.2(ii), and for the $\mathrm{env}$ functions associated with $\mathop{J_{{2\mu}}\/}\nolimits$ and $\mathop{Y_{{2\mu}}\/}\nolimits$ 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 .

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

Referenced by:: §13.21(iv), §13.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/13.21.iii

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:: : base of exponential function
Permalink:: http://dlmf.nist.gov/13.21.E17
Encodings:: TeX, pMML, png

13.21.18 $X=\sqrt{|x^{2}-4\kappa x+4\mu^{2}|},$

Symbols:: : real variable and
Permalink:: http://dlmf.nist.gov/13.21.E18
Encodings:: TeX, pMML, png

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:: : real variable
Permalink:: http://dlmf.nist.gov/13.21.E19
Encodings:: TeX, pMML, png

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

13.21.20 $\widehat{\zeta}=-\left(\frac{3}{2\kappa}\left(-\frac{1}{2}X+2\mu\mathop{% \mathrm{arctan}\/}\nolimits\!\left(\frac{x\kappa-x\sqrt{\kappa^{2}-\mu^{2}}-2% \mu^{2}}{\mu X}\right)+\kappa\mathop{\mathrm{arccos}\/}\nolimits\!\left(\frac{% x-2\kappa}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)\right)\right)^{{2/3}},$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}<x\leq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ ,

Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, : real variable and
Permalink:: http://dlmf.nist.gov/13.21.E20
Encodings:: TeX, pMML, png

and

13.21.21 $\widehat{\zeta}=\left(\frac{3}{2\kappa}\left(\frac{1}{2}X+\mu\mathop{\ln\/}% \nolimits\!\left(\frac{x\sqrt{\kappa^{2}-\mu^{2}}}{\kappa x-2\mu^{2}-\mu X}% \right)+\kappa\mathop{\ln\/}\nolimits\!\left(\frac{2\sqrt{\kappa^{2}-\mu^{2}}}% {x-2\kappa+X}\right)\right)\right)^{{2/3}},$ $x\geq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ .

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable and
Permalink:: http://dlmf.nist.gov/13.21.E21
Encodings:: TeX, pMML, png

Then as $\kappa\to\infty$

13.21.22 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\frac{1}{2\pi}\*\mathop{% \Gamma\/}\nolimits\!\left(2\mu+1\right)\*\mathop{\Gamma\/}\nolimits\!\left(% \kappa-\mu+\tfrac{1}{2}\right)\*\widehat{c}(\kappa,\mu)\*\widehat{\Psi}(\kappa% ,\mu,x)\*\left(\mathop{\sin\/}\nolimits\!\left(\kappa\pi-\mu\pi\right)\mathop{% \mathrm{Ai}\/}\nolimits\!\left(\kappa^{{\frac{2}{3}}}\widehat{\zeta}\right)+% \mathop{\cos\/}\nolimits\!\left(\kappa\pi-\mu\pi\right)\mathop{\mathrm{Bi}\/}% \nolimits\!\left(\kappa^{{\frac{2}{3}}}\widehat{\zeta}\right)+\mathop{\mathrm{% envBi}\/}\nolimits\!\left(\kappa^{{\frac{2}{3}}}\widehat{\zeta}\right)\mathop{% O\/}\nolimits\!\left(\kappa^{{-1}}\right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and : real variable
Permalink:: http://dlmf.nist.gov/13.21.E22
Encodings:: TeX, pMML, png

13.21.23 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\sqrt{2\pi}\kappa^{{\frac% {1}{6}}}\*\left(\frac{\kappa+\mu}{\kappa-\mu}\right)^{{\frac{1}{2}\mu}}\*\left% (\frac{\kappa^{2}-\mu^{2}}{e^{2}}\right)^{{\frac{1}{2}\kappa}}\*\widehat{\Psi}% (\kappa,\mu,x)\*\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(\kappa^{{\frac{2}% {3}}}\widehat{\zeta}\right)+\mathop{\mathrm{envAi}\/}\nolimits\!\left(\kappa^{% {\frac{2}{3}}}\widehat{\zeta}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-1}}% \right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function and : real variable
Permalink:: http://dlmf.nist.gov/13.21.E23
Encodings:: TeX, pMML, png

13.21.24 $\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(xe^{{-\pi i}}\right)=e^{{(\kappa-% \frac{1}{6})\pi i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(\kappa,\mu,x)\left(% \mathop{\mathrm{Ai}\/}\nolimits\!\left(\kappa^{{\frac{2}{3}}}\widehat{\zeta}e^% {{-\frac{2}{3}\pi i}}\right)+\mathop{\mathrm{envBi}\/}\nolimits\!\left(\kappa^% {{\frac{2}{3}}}\widehat{\zeta}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-1}% }\right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function and : real variable
Permalink:: http://dlmf.nist.gov/13.21.E24
Encodings:: TeX, pMML, png

13.21.25 $\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(xe^{{\pi i}}\right)=e^{{-(\kappa-% \frac{1}{6})\pi i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(\kappa,\mu,x)\left(% \mathop{\mathrm{Ai}\/}\nolimits\!\left(\kappa^{{\frac{2}{3}}}\widehat{\zeta}e^% {{\frac{2}{3}\pi i}}\right)+\mathop{\mathrm{envBi}\/}\nolimits\!\left(\kappa^{% {\frac{2}{3}}}\widehat{\zeta}\right)\mathop{O\/}\nolimits\!\left(\kappa^{{-1}}% \right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function and : real variable
Permalink:: http://dlmf.nist.gov/13.21.E25
Encodings:: TeX, pMML, png

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 $\mathop{\mathrm{Ai}\/}\nolimits$ and $\mathop{\mathrm{Bi}\/}\nolimits$ see §9.2(i), and for the $\mathrm{env}$ functions associated with $\mathop{\mathrm{Ai}\/}\nolimits$ and $\mathop{\mathrm{Bi}\/}\nolimits$ 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 .

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

Keywords:: Whittaker functions
Referenced by:: §2.8(iii), §2.8(iv)
Permalink:: http://dlmf.nist.gov/13.21.iv

For a uniform asymptotic expansion in terms of Airy functions for $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(4\kappa x\right)$ 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).

§13.21 Uniform Asymptotic Approximations for Large

Contents

§13.21(i) Large , Fixed

§13.21(ii) Large ,

§13.21(iii) Large , (Continued)

§13.21(iv) Large , Other Expansions

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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

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

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

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