13 Confluent Hypergeometric Functions Whittaker Functions 13.19 Asymptotic Expansions for Large Argument 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

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Referenced by:: §13.22
Permalink:: http://dlmf.nist.gov/13.20
See also:: Annotations for Ch.13

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

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

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Keywords:: Whittaker functions, asymptotic approximations for large parameters, large $\mu$ , uniform
Notes:: For (13.20.2) use (13.14.3) and apply the method of steepest descents (§2.4(iv)) to the integral representation (13.4.14).
Referenced by:: §13.8(ii)
Permalink:: http://dlmf.nist.gov/13.20.i
See also:: Annotations for §13.20 and Ch.13

When $\mu\to\infty$ in the sector $|\operatorname{ph}\mu|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$ , with $\kappa(\in\mathbb{C})$ fixed

13.20.1 $M_{\kappa,\mu}\left(z\right)=z^{\mu+\frac{1}{2}}\left(1+O\left(\mu^{-1}\right)% \right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : 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 ; also

13.20.2 $W_{\kappa,\mu}\left(x\right)=\pi^{-\frac{1}{2}}\Gamma\left(\kappa+\mu\right)% \left(\tfrac{1}{4}x\right)^{\frac{1}{2}-\mu}\left(1+O\left(\mu^{-1}\right)% \right),$

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Symbols:: $O\left(\NVar{x}\right)$ : 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 and : 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 . For an extension of (13.20.1) to an asymptotic expansion, together with error bounds, see Olver (1997b, Chapter 10, Ex. 3.4).

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

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Notes:: See Olver (1997b, Chapter 7, §11.2).
Permalink:: http://dlmf.nist.gov/13.20.ii
See also:: Annotations for §13.20 and Ch.13

Let

13.20.3 $X=\sqrt{4\mu^{2}-4\kappa x+x^{2}}.$

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Defines:: (locally)
Symbols:: : 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 $\mu\to\infty$

13.20.4 $M_{\kappa,\mu}\left(x\right)=\sqrt{\frac{2\mu x}{X}}\*\left(\frac{4\mu^{2}x}{2% \mu^{2}-\kappa x+\mu X}\right)^{\mu}\*\left(\frac{2(\mu-\kappa)}{X+x-2\kappa}% \right)^{\kappa}\*e^{\frac{1}{2}X-\mu}\*\left(1+O\left(\frac{1}{\mu}\right)% \right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, : real variable and
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 $W_{\kappa,\mu}\left(x\right)=\sqrt{\frac{x}{X}}\*\left(\frac{2\mu^{2}-\kappa x% +\mu X}{(\mu-\kappa)x}\right)^{\mu}\*\left(\frac{X+x-2\kappa}{2}\right)^{% \kappa}\*e^{-\frac{1}{2}X-\kappa}\*\left(1+O\left(\frac{1}{\mu}\right)\right),$

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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, $\mathrm{e}$ : base of natural logarithm, : real variable and
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 $x\in(0,\infty)$ and $\kappa\in[0,(1-\delta)\mu]$ , where $\delta$ again denotes an arbitrary small positive constant.

§13.20(iii) Large $\mu$ , $-(1-\delta)\mu\leq\kappa\leq\mu$

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Referenced by:: §12.16, §13.20(iv), §13.8(ii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/13.20.iii
See also:: Annotations for §13.20 and Ch.13

Let

13.20.6 $\alpha=\sqrt{2|\kappa-\mu|/\mu},$

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

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

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Defines:: $\Phi(\kappa,\mu,x)$ : function (locally)
Symbols:: : 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 $-\mu<\kappa<\mu$

13.20.9 $\zeta\sqrt{\zeta^{2}+\alpha^{2}}+\alpha^{2}\operatorname{arcsinh}\left(\frac{% \zeta}{\alpha}\right)=\frac{X}{\mu}-\frac{2\kappa}{\mu}\ln\left(\frac{X+x-2% \kappa}{2\sqrt{\mu^{2}-\kappa^{2}}}\right)-2\ln\left(\frac{\mu X+2\mu^{2}-% \kappa x}{x\sqrt{\mu^{2}-\kappa^{2}}}\right).$

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Symbols:: $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable, $\alpha$ and
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 $\zeta=\pm\sqrt{\frac{x}{\mu}-2-2\ln\left(\frac{x}{2\mu}\right)},$

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and : 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 -interval $(0,\infty)$ is mapped one-to-one onto the $\zeta$ -interval $(-\infty,\infty)$ , with and $\infty$ corresponding to $\zeta=-\infty$ and $\infty$ , respectively.) Then as $\mu\to\infty$

13.20.11 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)\left(1+O\left(\mu^{-1}\ln\mu% \right)\right),$

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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, $\mathrm{e}$ : base of natural logarithm, $\ln\NVar{z}$ : principal branch of logarithm function, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real variable and $\Phi(\kappa,\mu,x)$ : 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 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)\left(1+O% \left(\mu^{-1}\ln\mu\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\ln\NVar{z}$ : principal branch of logarithm function, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real variable and $\Phi(\kappa,\mu,x)$ : 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 $x\in(0,\infty)$ and $\kappa\in[-(1-\delta)\mu,\mu]$ . For the parabolic cylinder function see §12.2.

These results are proved in Olver (1980b). This reference also supplies error bounds and corresponding approximations when , $\kappa$ , and $\mu$ are replaced by $\mathrm{i}x$ , $\mathrm{i}\kappa$ , and $\mathrm{i}\mu$ , respectively.

§13.20(iv) Large $\mu$ , $\mu\leq\kappa\leq\mu/\delta$

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Keywords:: Whittaker functions, asymptotic approximations for large parameters, imaginary $\kappa$ and/or $\mu$
Referenced by:: §12.16, §13.8(ii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/13.20.iv
See also:: Annotations for §13.20 and Ch.13

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

13.20.13 $\zeta\sqrt{\zeta^{2}-\alpha^{2}}-\alpha^{2}\operatorname{arccosh}\left(\frac{% \zeta}{\alpha}\right)=\frac{X}{\mu}-\frac{2\kappa}{\mu}\ln\left(\frac{X+x-2% \kappa}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)-2\ln\left(\frac{\kappa x-\mu X-2\mu% ^{2}}{x\sqrt{\kappa^{2}-\mu^{2}}}\right),$ $x\geq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ ,

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Symbols:: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable, $\alpha$ and
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 $\zeta\sqrt{\alpha^{2}-\zeta^{2}}+\alpha^{2}\operatorname{arcsin}\left(\frac{% \zeta}{\alpha}\right)=\frac{X}{\mu}+\frac{2\kappa}{\mu}\operatorname{arctan}% \left(\frac{x-2\kappa}{X}\right)-2\operatorname{arctan}\left(\frac{\kappa x-2% \mu^{2}}{\mu X}\right),$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}\leq x\leq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ ,

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Symbols:: $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, : real variable, $\alpha$ and
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 $-\zeta\sqrt{\zeta^{2}-\alpha^{2}}-\alpha^{2}\operatorname{arccosh}\left(-\frac% {\zeta}{\alpha}\right)=-\frac{X}{\mu}+\frac{2\kappa}{\mu}\ln\left(\frac{2% \kappa-X-x}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)+2\ln\left(\frac{\mu X+2\mu^{2}-% \kappa x}{x\sqrt{\kappa^{2}-\mu^{2}}}\right),$ $0<x\leq 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}$ ,

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Symbols:: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable, $\alpha$ and
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 $\mu<\kappa$ , and by (13.20.10) when $\mu=\kappa$ . (As in §13.20(iii) and $\infty$ correspond to $\zeta=-\infty$ and $\infty$ , respectively). Then as $\mu\to\infty$

13.20.16 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*\left(U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)+\mathrm{env}\mskip-1.0muU% \left(\mu-\kappa,\zeta\sqrt{2\mu}\right)O\left(\mu^{-\frac{2}{3}}\right)\right),$

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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, $\mathrm{env}\mskip-1.0muU\left(\NVar{c},\NVar{x}\right)$ : envelope of parabolic cylinder function $U\left(\NVar{c},\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real variable and $\Phi(\kappa,\mu,x)$ : 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 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*\left(U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+% \mathrm{env}\mskip-1.0mu\overline{U}\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)O% \left(\mu^{-\frac{2}{3}}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{env}\mskip-1.0mu\overline{U}\left(\NVar{c},\NVar{x}\right)$ : envelope of parabolic cylinder function $\overline{U}\left(\NVar{c},\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real variable and $\Phi(\kappa,\mu,x)$ : 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 $\zeta\in[0,\infty)$ and $\kappa\in[\mu,\mu/\delta]$ .

Also,

13.20.18 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*\left(U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)+\mathrm{env}\mskip-1.0mu% \overline{U}\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)O\left(\mu^{-\frac{2}{3}}% \right)\right),$

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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, $\mathrm{env}\mskip-1.0mu\overline{U}\left(\NVar{c},\NVar{x}\right)$ : envelope of parabolic cylinder function $\overline{U}\left(\NVar{c},\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real variable and $\Phi(\kappa,\mu,x)$ : 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 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*\left(U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+% \mathrm{env}\mskip-1.0muU\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)O\left(\mu^{% -\frac{2}{3}}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{env}\mskip-1.0muU\left(\NVar{c},\NVar{x}\right)$ : envelope of parabolic cylinder function $U\left(\NVar{c},\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : real variable and $\Phi(\kappa,\mu,x)$ : 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 $\zeta\in(-\infty,0]$ and $\kappa\in[\mu,\mu/\delta]$ .

For the parabolic cylinder functions and $\overline{U}$ see §12.2, and for the $\mathrm{env}$ functions associated with 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 , $\kappa$ , 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 $-(1-\delta)\mu\leq\kappa\leq\mu/\delta$ . Similarly for (13.20.12), (13.20.17), and (13.20.19).

§13.20(v) Large $\mu$ , Other Expansions

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Keywords:: Whittaker functions, asymptotic approximations for large parameters, large $\mu$
Referenced by:: §13.21(iv), §2.8(iii)
Permalink:: http://dlmf.nist.gov/13.20.v
See also:: Annotations for §13.20 and Ch.13

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