§13.19 Asymptotic Expansions for Large Argument

As $x\to\infty$

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

As $z\to\infty$

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

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

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).