13 Confluent Hypergeometric Functions Whittaker Functions 13.18 Relations to Other Functions 13.20 Uniform Asymptotic Approximations for Large

\mu

§13.19 Asymptotic Expansions for Large Argument

ⓘ

Keywords:: Whittaker functions, asymptotic expansions for large argument, error bounds, exponentially-improved
Notes:: See Temme (1996b, §7.2).
Referenced by:: §13.29(ii), §2.11(vi), §3.10(ii)
Permalink:: http://dlmf.nist.gov/13.19
See also:: Annotations for Ch.13

As $x\to\infty$

13.19.1		$M_{\kappa,\mu}\left(x\right)\sim\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}e^{\frac{1}{2}x}x^{-\kappa}\*\sum_{s=0}^{\infty}% \frac{{\left(\frac{1}{2}-\mu+\kappa\right)_{s}}{\left(\frac{1}{2}+\mu+\kappa% \right)_{s}}}{s!}x^{-s},$
$\mu-\kappa\neq-\tfrac{1}{2},-\tfrac{3}{2},\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/13.19.E1 Encodings: TeX, pMML, png See also: Annotations for §13.19 and Ch.13

As $z\to\infty$

13.19.2		$M_{\kappa,\mu}\left(z\right)\sim\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}e^{\frac{1}{2}z}z^{-\kappa}\\sum_{s=0}^{\infty}% \frac{{\left(\frac{1}{2}-\mu+\kappa\right)_{s}}{\left(\frac{1}{2}+\mu+\kappa% \right)_{s}}}{s!}z^{-s}+\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{% 2}+\mu+\kappa\right)}e^{-\frac{1}{2}z\pm(\frac{1}{2}+\mu-\kappa)\pi\mathrm{i}}% z^{\kappa}\\sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}% {\left(\frac{1}{2}-\mu-\kappa\right)_{s}}}{s!}(-z)^{-s},$
$-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $s$ : nonnegative integer, $z$ : complex variable and $\delta$ : small positive constant Referenced by: §13.14(iv) Permalink: http://dlmf.nist.gov/13.19.E2 Encodings: TeX, pMML, png See also: Annotations for §13.19 and Ch.13

provided that both $\mu\mp\kappa\neq-\tfrac{1}{2},-\tfrac{3}{2},\dots$ . Again, $\delta$ denotes an arbitrary small positive constant. Also,

13.19.3		$W_{\kappa,\mu}\left(z\right)\sim e^{-\frac{1}{2}z}z^{\kappa}\sum_{s=0}^{\infty% }\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}{\left(\frac{1}{2}-\mu-\kappa% \right)_{s}}}{s!}{(-z)^{-s}},$
$\|\operatorname{ph}z\|\leq\tfrac{3}{2}\pi-\delta$ .
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\operatorname{ph}$ : phase, $s$ : nonnegative integer, $z$ : complex variable and $\delta$ : small positive constant Referenced by: §13.14(iv) Permalink: http://dlmf.nist.gov/13.19.E3 Encodings: TeX, pMML, png See also: Annotations for §13.19 and Ch.13

Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3). See also Olver (1965).

For an asymptotic expansion of $W_{\kappa,\mu}\left(z\right)$ as $z\to\infty$ that is valid in the sector $|\operatorname{ph}z|\leq\pi-\delta$ and where the real parameters $\kappa$ , $\mu$ are subject to the growth conditions $\kappa=o\left(z\right)$ , $\mu=o\left(\sqrt{z}\right)$ , see Wong (1973a).

13.18 Relations to Other Functions 13.20 Uniform Asymptotic Approximations for Large

\mu

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