§13.20 Uniform Asymptotic Approximations for Large $\mu$

Referenced by:: §13.22
Permalink:: http://dlmf.nist.gov/13.20

§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 $\kappa$

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).
Keywords:: Whittaker functions
Referenced by:: §13.8(ii)
Permalink:: http://dlmf.nist.gov/13.20.i

When $\mu\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits\mu|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$ , with $\kappa(\in\Complex)$ fixed

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Referenced by:: §13.20(i)
Permalink:: http://dlmf.nist.gov/13.20.E1
Encodings:: TeX, pMML, png

uniformly for bounded values of $|z|$ ; also

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and $x$ : real variable
Referenced by:: §13.20(i)
Permalink:: http://dlmf.nist.gov/13.20.E2
Encodings:: TeX, pMML, png

uniformly for bounded positive values of $x$ . 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 $\mu$ , $0\leq\kappa\leq(1-\delta)\mu$

Notes:: See Olver (1997b, Chapter 7, §11.2).
Permalink:: http://dlmf.nist.gov/13.20.ii

Let

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

Symbols:: $x$ : real variable and $X$
Permalink:: http://dlmf.nist.gov/13.20.E3
Encodings:: TeX, pMML, png

Then as $\mu\to\infty$

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $x$ : real variable and $X$
Permalink:: http://dlmf.nist.gov/13.20.E4
Encodings:: TeX, pMML, png

13.20.5 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\sqrt{\frac{x}{X}}\*\left(\frac{2\mu^{2}-\kappa x+\mu X}{(\mu-\kappa)x}\right)^{\mu}\*\left(\frac{X+x-2\kappa}{2}\right)^{\kappa}\* e^{{-\frac{1}{2}X-\kappa}}\*\left(1+\mathop{O\/}\nolimits\!\left(\frac{1}{\mu}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $x$ : real variable and $X$
Permalink:: http://dlmf.nist.gov/13.20.E5
Encodings:: TeX, pMML, png

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

Referenced by:: §12.16, §13.20(iv), §13.8(ii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/13.20.iii

Let

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

Symbols:: $\alpha$
Referenced by:: §13.20(iv)
Permalink:: http://dlmf.nist.gov/13.20.E6
Encodings:: TeX, pMML, png

13.20.7 $X=\sqrt{|x^{2}-4\kappa x+4\mu^{2}|},$

Symbols:: $x$ : real variable and $X$
Permalink:: http://dlmf.nist.gov/13.20.E7
Encodings:: TeX, pMML, png

13.20.8 $\Phi(\kappa,\mu,x)=\left(\frac{\mu^{2}\zeta^{2}-2\kappa\mu+2\mu^{2}}{x^{2}-4\kappa x+4\mu^{2}}\right)^{{\frac{1}{4}}}\sqrt{x},$

Symbols:: $x$ : real variable and $\Phi(\kappa,\mu,x)$ : function
Referenced by:: §13.20(iv)
Permalink:: http://dlmf.nist.gov/13.20.E8
Encodings:: TeX, pMML, png

with the variable $\zeta$ defined implicitly as follows:

(a) In the case $-\mu<\kappa<\mu$

13.20.9 $\zeta\sqrt{\zeta^{2}+\alpha^{2}}+\alpha^{2}\mathop{\mathrm{arcsinh}\/}\nolimits\!\left(\frac{\zeta}{\alpha}\right)=\frac{X}{\mu}-\frac{2\kappa}{\mu}\mathop{\ln\/}\nolimits\!\left(\frac{X+x-2\kappa}{2\sqrt{\mu^{2}-\kappa^{2}}}\right)-2\mathop{\ln\/}\nolimits\!\left(\frac{\mu X+2\mu^{2}-\kappa x}{x\sqrt{\mu^{2}-\kappa^{2}}}\right).$

Symbols:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable, $\alpha$ and $X$
Permalink:: http://dlmf.nist.gov/13.20.E9
Encodings:: TeX, pMML, png

(b) In the case $\mu=\kappa$

13.20.10 $\zeta=\pm\sqrt{\frac{x}{\mu}-2-2\mathop{\ln\/}\nolimits\!\left(\frac{x}{2\mu}\right)},$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
Referenced by:: §13.20(iv)
Permalink:: http://dlmf.nist.gov/13.20.E10
Encodings:: TeX, pMML, png

the upper or lower sign being taken according as $x\gtrless 2\mu$ .

(In both cases (a) and (b) the $x$ -interval $(0,\infty)$ is mapped one-to-one onto the $\zeta$ -interval $(-\infty,\infty)$ , with $x=0$ and $\infty$ corresponding to $\zeta=-\infty$ and $\infty$ , respectively.) Then as $\mu\to\infty$

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $x$ : 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

13.20.12 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\left(8\mu\right)^{{\frac{1}{4}}}\*\left(\frac{2\mu}{e}\right)^{{2\mu}}\*\left(\frac{e}{\kappa+\mu}\right)^{{\frac{1}{2}(\kappa+\mu)}}\*\Phi(\kappa,\mu,x)\*\mathop{U\/}\nolimits\!\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)\left(1+\mathop{O\/}\nolimits\!\left(\mu^{{-1}}\mathop{\ln\/}\nolimits\mu\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $x$ : 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

uniformly with respect to $x\in(0,\infty)$ and $\kappa\in[-(1-\delta)\mu,\mu]$ . For the parabolic cylinder function $\mathop{U\/}\nolimits$ see §12.2.

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

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

Keywords:: Whittaker functions
Referenced by:: §12.16, §13.8(ii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/13.20.iv

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

13.20.13 $\zeta\sqrt{\zeta^{2}-\alpha^{2}}-\alpha^{2}\mathop{\mathrm{arccosh}\/}\nolimits\!\left(\frac{\zeta}{\alpha}\right)=\frac{X}{\mu}-\frac{2\kappa}{\mu}\mathop{\ln\/}\nolimits\!\left(\frac{X+x-2\kappa}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)-2\mathop{\ln\/}\nolimits\!\left(\frac{\kappa x-\mu X-2\mu^{2}}{x\sqrt{\kappa^{2}-\mu^{2}}}\right),$ $x\geq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ ,

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable, $\alpha$ and $X$
Permalink:: http://dlmf.nist.gov/13.20.E13
Encodings:: TeX, pMML, png

13.20.14 $\zeta\sqrt{\alpha^{2}-\zeta^{2}}+\alpha^{2}\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\frac{\zeta}{\alpha}\right)=\frac{X}{\mu}+\frac{2\kappa}{\mu}\mathop{\mathrm{arctan}\/}\nolimits\!\left(\frac{x-2\kappa}{X}\right)-2\mathop{\mathrm{arctan}\/}\nolimits\!\left(\frac{\kappa x-2\mu^{2}}{\mu X}\right),$ $2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}\leq x\leq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}$ ,

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $x$ : real variable, $\alpha$ and $X$
Permalink:: http://dlmf.nist.gov/13.20.E14
Encodings:: TeX, pMML, png

13.20.15 $-\zeta\sqrt{\zeta^{2}-\alpha^{2}}-\alpha^{2}\mathop{\mathrm{arccosh}\/}\nolimits\!\left(-\frac{\zeta}{\alpha}\right)=-\frac{X}{\mu}+\frac{2\kappa}{\mu}\mathop{\ln\/}\nolimits\!\left(\frac{2\kappa-X-x}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)+2\mathop{\ln\/}\nolimits\!\left(\frac{\mu X+2\mu^{2}-\kappa x}{x\sqrt{\kappa^{2}-\mu^{2}}}\right),$ $0<x\leq 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}$ ,

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable, $\alpha$ and $X$
Permalink:: http://dlmf.nist.gov/13.20.E15
Encodings:: TeX, pMML, png

when $\mu<\kappa$ , and by (13.20.10) when $\mu=\kappa$ . (As in §13.20(iii) $x=0$ and $\infty$ correspond to $\zeta=-\infty$ and $\infty$ , respectively). Then as $\mu\to\infty$

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $x$ : 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

13.20.17 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\left(8\mu\right)^{{\frac{1}{4}}}\*\left(\frac{2\mu}{e}\right)^{{2\mu}}\*\left(\frac{e}{\kappa+\mu}\right)^{{\frac{1}{2}(\kappa+\mu)}}\*\Phi(\kappa,\mu,x)\*\left(\mathop{U\/}\nolimits\!\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+\mathop{\mathrm{env}\overline{U}\/}\nolimits\!\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)\mathop{O\/}\nolimits\!\left(\mu^{{-\frac{2}{3}}}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $x$ : 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

uniformly with respect to $\zeta\in[0,\infty)$ and $\kappa\in[\mu,\mu/\delta]$ .

Also,

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $x$ : 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

13.20.19 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)=\left(8\mu\right)^{{\frac{1}{4}}}\*\left(\frac{2\mu}{e}\right)^{{2\mu}}\*\left(\frac{e}{\kappa+\mu}\right)^{{\frac{1}{2}(\kappa+\mu)}}\*\Phi(\kappa,\mu,x)\*\left(\mathop{U\/}\nolimits\!\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+\mathop{\mathrm{env}U\/}\nolimits\!\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)\mathop{O\/}\nolimits\!\left(\mu^{{-\frac{2}{3}}}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $x$ : 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

uniformly with respect to $\zeta\in(-\infty,0]$ and $\kappa\in[\mu,\mu/\delta]$ .

For the parabolic cylinder functions $\mathop{U\/}\nolimits$ and $\mathop{\overline{U}\/}\nolimits$ see §12.2, and for the $\mathrm{env}$ functions associated with $\mathop{U\/}\nolimits$ and $\mathop{\overline{U}\/}\nolimits$ 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 $x$ , $\kappa$ , and $\mu$ are replaced by $ix$ , $i\kappa$ , and $i\mu$ , 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

Keywords:: Whittaker functions
Referenced by:: §13.21(iv), §2.8(iii)
Permalink:: http://dlmf.nist.gov/13.20.v

For uniform approximations valid when $\mu$ is large, $x/i\in(0,\infty)$ , and $\kappa/i\in[0,\mu/\delta]$ , see Olver (1997b, pp. 401–403). These approximations are in terms of Airy functions.

For uniform approximations of $\mathop{M_{{\kappa,i\mu}}\/}\nolimits\!\left(z\right)$ and $\mathop{W_{{\kappa,i\mu}}\/}\nolimits\!\left(z\right)$ , $\kappa$ and $\mu$ real, one or both large, see Dunster (2003a).

§13.20 Uniform Asymptotic Approximations for Large

Contents

§13.20(i) Large , Fixed

§13.20(ii) Large ,

§13.20(iii) Large ,

§13.20(iv) Large ,

§13.20(v) Large , Other Expansions