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

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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Keywords:: Whittaker functions, asymptotic approximations for large parameters, large $\kappa$
Referenced by:: §13.22
Permalink:: http://dlmf.nist.gov/13.21
See also:: Annotations for Ch.13

§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 $\mu$

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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
See also:: Annotations for §13.21 and Ch.13

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 $M_{\kappa,\mu}\left(x\right)=\sqrt{x}\Gamma\left(2\mu+1\right)\kappa^{-\mu}\*% \left(J_{2\mu}\left(2\sqrt{x\kappa}\right)+\operatorname{env}\mskip-2.0muJ_{2% \mu}\left(2\sqrt{x\kappa}\right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\operatorname{env}\mskip-2.0muJ_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $J_{\NVar{\nu}}\left(\NVar{x}\right)$ and : 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 $W_{\kappa,\mu}\left(x\right)=\sqrt{x}\Gamma\left(\kappa+\tfrac{1}{2}\right)\*% \left(\sin\left(\kappa\pi-\mu\pi\right)J_{2\mu}\left(2\sqrt{x\kappa}\right)-% \cos\left(\kappa\pi-\mu\pi\right)Y_{2\mu}\left(2\sqrt{x\kappa}\right)+% \operatorname{env}\mskip-2.0muY_{2\mu}\left(2\sqrt{x\kappa}\right)O\left(% \kappa^{-\frac{1}{2}}\right)\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{env}\mskip-2.0muY_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\sin\NVar{z}$ : sine function and : 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 $W_{-\kappa,\mu}\left(xe^{-\pi\mathrm{i}}\right)=\frac{\pi\sqrt{x}}{\Gamma\left% (\kappa+\tfrac{1}{2}\right)}e^{\mu\pi\mathrm{i}}\*\left({H^{(1)}_{2\mu}}\left(% 2\sqrt{x\kappa}\right)+\operatorname{env}\mskip-2.0muY_{2\mu}\left(2\sqrt{x% \kappa}\right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{env}\mskip-2.0muY_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : 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 $W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=\frac{\pi\sqrt{x}}{\Gamma\left(% \kappa+\tfrac{1}{2}\right)}e^{-\mu\pi\mathrm{i}}\*\left({H^{(2)}_{2\mu}}\left(% 2\sqrt{x\kappa}\right)+\operatorname{env}\mskip-2.0muY_{2\mu}\left(2\sqrt{x% \kappa}\right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{env}\mskip-2.0muY_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : 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 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 $2\sqrt{\zeta}=\sqrt{x+x^{2}}+\ln\left(\sqrt{x}+\sqrt{1+x}\right).$

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and : 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 $M_{-\kappa,\mu}\left(4\kappa x\right)=\frac{2\Gamma\left(2\mu+1\right)}{\kappa% ^{\mu-\frac{1}{2}}}\left(\frac{x\zeta}{1+x}\right)^{\frac{1}{4}}I_{2\mu}\left(% 4\kappa\zeta^{\frac{1}{2}}\right){\left(1+O\left(\kappa^{-1}\right)\right)},$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and : 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 $W_{-\kappa,\mu}\left(4\kappa x\right)=\frac{\sqrt{8/\pi}e^{\kappa}}{\kappa^{% \kappa-\frac{1}{2}}}\left(\frac{x\zeta}{1+x}\right)^{\frac{1}{4}}K_{2\mu}\left% (4\kappa\zeta^{\frac{1}{2}}\right){\left(1+O\left(\kappa^{-1}\right)\right)},$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and : 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 $x\in(0,\infty)$ .

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 $\kappa$ , $0\leq\mu\leq(1-\delta)\kappa$

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Referenced by:: §13.21(i), §13.8(iii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/13.21.ii
See also:: Annotations for §13.21 and Ch.13

Let

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

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

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

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Defines:: $\Psi(\kappa,\mu,x)$ : function (locally)
Symbols:: : 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 $\sqrt{4\mu^{2}-\kappa\zeta}-\mu\ln\left(\frac{2\mu+\sqrt{4\mu^{2}-\kappa\zeta}% }{2\mu-\sqrt{4\mu^{2}-\kappa\zeta}}\right)=\tfrac{1}{2}X+\mu\ln\left(\frac{x% \sqrt{\kappa^{2}-\mu^{2}}}{2\mu^{2}-\kappa x+\mu X}\right)+\kappa\ln\left(% \frac{2\sqrt{\kappa^{2}-\mu^{2}}}{2\kappa-x-X}\right),$ $0<x\leq 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}$ ,

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : real variable and
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 $\sqrt{\kappa\zeta-4\mu^{2}}-2\mu\operatorname{arctan}\left(\frac{\sqrt{\kappa% \zeta-4\mu^{2}}}{2\mu}\right)=\tfrac{1}{2}(X-\pi\mu)-\mu\operatorname{arctan}% \left(\frac{x\kappa-2\mu^{2}}{\mu X}\right)+\kappa\operatorname{arcsin}\left(% \frac{X}{2\sqrt{\kappa^{2}-\mu^{2}}}\right),$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}\leq x<2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, : real variable and
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 $\kappa\to\infty$

13.21.13 $M_{\kappa,\mu}\left(x\right)=\Gamma\left(2\mu+1\right)\*\left(\frac{e^{2}}{% \kappa^{2}-\mu^{2}}\right)^{\frac{1}{2}\mu}\*\left(\frac{\kappa-\mu}{\kappa+% \mu}\right)^{\frac{1}{2}\kappa}\*\Psi(\kappa,\mu,x)\*\left(J_{2\mu}\left(\sqrt% {\zeta\kappa}\right)+\operatorname{env}\mskip-2.0muJ_{2\mu}\left(\sqrt{\zeta% \kappa}\right)O\left(\kappa^{-1}\right)\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\operatorname{env}\mskip-2.0muJ_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $J_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, : real variable and $\Psi(\kappa,\mu,x)$ : 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 $W_{\kappa,\mu}\left(x\right)=\frac{e^{-\mu\pi\mathrm{i}}}{\pi}\Gamma\left(% \kappa+\mu+\tfrac{1}{2}\right)\*\Gamma\left(\kappa-\mu+\tfrac{1}{2}\right)\*c(% \kappa,\mu)\Psi(\kappa,\mu,x)\*\left(\sin\left(\kappa\pi-\mu\pi\right)J_{2\mu}% \left(\sqrt{\zeta\kappa}\right)-\cos\left(\kappa\pi-\mu\pi\right)Y_{2\mu}\left% (\sqrt{\zeta\kappa}\right)+\operatorname{env}\mskip-2.0muY_{2\mu}\left(\sqrt{% \zeta\kappa}\right)O\left(\kappa^{-1}\right)\right),$

13.21.15 $W_{-\kappa,\mu}\left(xe^{-\pi\mathrm{i}}\right)=c(\kappa,\mu)\Psi(\kappa,\mu,x% )\left({H^{(1)}_{2\mu}}\left(\sqrt{\zeta\kappa}\right)+\operatorname{env}% \mskip-2.0muY_{2\mu}\left(\sqrt{\zeta\kappa}\right)O\left(\kappa^{-1}\right)% \right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{env}\mskip-2.0muY_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : real variable, $c(\kappa,\mu)$ : function and $\Psi(\kappa,\mu,x)$ : 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 $W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=e^{-2\mu\pi\mathrm{i}}c(\kappa,% \mu)\Psi(\kappa,\mu,x)\left({H^{(2)}_{2\mu}}\left(\sqrt{\zeta\kappa}\right)+% \operatorname{env}\mskip-2.0muY_{2\mu}\left(\sqrt{\zeta\kappa}\right)O\left(% \kappa^{-1}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{env}\mskip-2.0muY_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : real variable, $c(\kappa,\mu)$ : function and $\Psi(\kappa,\mu,x)$ : 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 $x\in\left(0,(1-\delta)(2\kappa+2\sqrt{\kappa^{2}-\mu^{2}})\right]$ , 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 .

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

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Referenced by:: §13.21(iv), §13.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/13.21.iii
See also:: Annotations for §13.21 and Ch.13

Let

13.21.17 $\widehat{c}(\kappa,\mu)=\sqrt{2\pi}\kappa^{\frac{1}{6}}\left(\frac{\kappa-\mu}% {\kappa+\mu}\right)^{\frac{1}{2}\mu}\left(\frac{e^{2}}{\kappa^{2}-\mu^{2}}% \right)^{\frac{1}{2}\kappa},$

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

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

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Defines:: $\widehat{\Psi}(\kappa,\mu,x)$ : function (locally)
Symbols:: : 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 $\widehat{\zeta}=-\left(\frac{3}{2\kappa}\left(-\frac{1}{2}X+2\mu\operatorname{% arctan}\left(\frac{x\kappa-x\sqrt{\kappa^{2}-\mu^{2}}-2\mu^{2}}{\mu X}\right)+% \kappa\operatorname{arccos}\left(\frac{x-2\kappa}{2\sqrt{\kappa^{2}-\mu^{2}}}% \right)\right)\right)^{2/3},$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}<x\leq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ ,

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Symbols:: $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, : real variable and
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 $\widehat{\zeta}=\left(\frac{3}{2\kappa}\left(\frac{1}{2}X+\mu\ln\left(\frac{x% \sqrt{\kappa^{2}-\mu^{2}}}{\kappa x-2\mu^{2}-\mu X}\right)+\kappa\ln\left(% \frac{2\sqrt{\kappa^{2}-\mu^{2}}}{x-2\kappa+X}\right)\right)\right)^{2/3},$ $x\geq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ .

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : real variable and
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 $\kappa\to\infty$

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{envBi}\left(\NVar{x}\right)$ : envelope of Airy function $\operatorname{Bi}\left(\NVar{x}\right)$ , $\sin\NVar{z}$ : sine function, : real variable, $\widehat{c}(\kappa,\mu)$ : function and $\widehat{\Psi}(\kappa,\mu,x)$ : 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 $W_{\kappa,\mu}\left(x\right)=\sqrt{2\pi}\kappa^{\frac{1}{6}}\*\left(\frac{% \kappa+\mu}{\kappa-\mu}\right)^{\frac{1}{2}\mu}\*\left(\frac{\kappa^{2}-\mu^{2% }}{e^{2}}\right)^{\frac{1}{2}\kappa}\*\widehat{\Psi}(\kappa,\mu,x)\*\left(% \operatorname{Ai}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)+% \operatorname{envAi}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)O\left(% \kappa^{-1}\right)\right),$

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{envAi}\left(\NVar{x}\right)$ : envelope of Airy function $\operatorname{Ai}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, : real variable and $\widehat{\Psi}(\kappa,\mu,x)$ : 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 $W_{-\kappa,\mu}\left(xe^{-\pi\mathrm{i}}\right)=e^{(\kappa-\frac{1}{6})\pi% \mathrm{i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(\kappa,\mu,x)\left(% \operatorname{Ai}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}e^{-\frac{2}{3}\pi% \mathrm{i}}\right)+\operatorname{envBi}\left(\kappa^{\frac{2}{3}}\widehat{% \zeta}\right)O\left(\kappa^{-1}\right)\right),$

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{envBi}\left(\NVar{x}\right)$ : envelope of Airy function $\operatorname{Bi}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : real variable, $\widehat{c}(\kappa,\mu)$ : function and $\widehat{\Psi}(\kappa,\mu,x)$ : 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 $W_{-\kappa,\mu}\left(xe^{\pi\mathrm{i}}\right)=e^{-(\kappa-\frac{1}{6})\pi% \mathrm{i}}\widehat{c}(\kappa,\mu)\widehat{\Psi}(\kappa,\mu,x)\left(% \operatorname{Ai}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}e^{\frac{2}{3}\pi% \mathrm{i}}\right)+\operatorname{envBi}\left(\kappa^{\frac{2}{3}}\widehat{% \zeta}\right)O\left(\kappa^{-1}\right)\right),$

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{envBi}\left(\NVar{x}\right)$ : envelope of Airy function $\operatorname{Bi}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : real variable, $\widehat{c}(\kappa,\mu)$ : function and $\widehat{\Psi}(\kappa,\mu,x)$ : 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 $x\in\left[(1+\delta)(2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}),\infty\right)$ . For the functions $\operatorname{Ai}$ and $\operatorname{Bi}$ see §9.2(i), and for the $\mathrm{env}$ functions associated with $\operatorname{Ai}$ and $\operatorname{Bi}$ 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 .

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

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Keywords:: Whittaker functions, asymptotic approximations for large parameters, large $\kappa$ , uniform
Referenced by:: §2.8(iii), §2.8(iv)
Permalink:: http://dlmf.nist.gov/13.21.iv
See also:: Annotations for §13.21 and Ch.13

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).

§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