# §13.19 Asymptotic Expansions for Large Argument

As $x\to\infty$

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

As $z\to\infty$

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

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

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).