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

Notes:: See Temme (1996b, §7.2).
Keywords:: Whittaker functions
Referenced by:: §13.29(ii), §2.11(vi), ¶ ‣ §3.10(ii)
Permalink:: http://dlmf.nist.gov/13.19

As $x\to\infty$

13.19.1 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)\sim\frac{\mathop{\Gamma\/% }\nolimits\!\left(1+2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}% +\mu-\kappa\right)}e^{{\frac{1}{2}x}}x^{{-\kappa}}\*\sum_{{s=0}}^{{\infty}}% \frac{\left(\frac{1}{2}-\mu+\kappa\right)_{{s}}\left(\frac{1}{2}+\mu+\kappa% \right)_{{s}}}{s!}x^{{-s}},$ $\mu-\kappa\neq-\tfrac{1}{2},-\tfrac{3}{2},\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\sim$ : asymptotic equality, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/13.19.E1
Encodings:: TeX, pMML, png

As $z\to\infty$

13.19.2 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\sim\frac{\mathop{\Gamma\/% }\nolimits\!\left(1+2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}% +\mu-\kappa\right)}e^{{\frac{1}{2}z}}z^{{-\kappa}}\*\sum_{{s=0}}^{{\infty}}% \frac{\left(\frac{1}{2}-\mu+\kappa\right)_{{s}}\left(\frac{1}{2}+\mu+\kappa% \right)_{{s}}}{s!}z^{{-s}}+\frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu% \right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+\kappa\right)}e^{{-% \frac{1}{2}z\pm(\frac{1}{2}+\mu-\kappa)\pi i}}z^{{\kappa}}\*\sum_{{s=0}}^{{% \infty}}\frac{\left(\frac{1}{2}+\mu-\kappa\right)_{{s}}\left(\frac{1}{2}-\mu-% \kappa\right)_{{s}}}{s!}(-z)^{{-s}},$ $-\tfrac{1}{2}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2% }\pi-\delta$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable and $\delta$ : small positive constant
Referenced by:: §13.14(iv)
Permalink:: http://dlmf.nist.gov/13.19.E2
Encodings:: TeX, pMML, png

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 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\sim e^{{-\frac{1}{2}z}}z^% {{\kappa}}\sum_{{s=0}}^{{\infty}}\frac{\left(\frac{1}{2}+\mu-\kappa\right)_{{s% }}\left(\frac{1}{2}-\mu-\kappa\right)_{{s}}}{s!}{(-z)^{{-s}}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{2}\pi-\delta$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable and $\delta$ : small positive constant
Referenced by:: §13.14(iv)
Permalink:: http://dlmf.nist.gov/13.19.E3
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 13.18 Relations to Other Functions 13.20 Uniform Asymptotic Approximations for Large $\mu$