Debye's Asymptotic Representation

An asymptotic expansion for a Hankel Function of the First Kind

$H_\nu^{(1)}(x)\sim{1\over\sqrt{\pi}} \mathop{\rm exp}\nolimits \{ix[\cos\alpha+(\alpha-\pi/2)\sin\alpha]\}$

$\times\left[{{e^{i\pi/4}\over X}+({\textstyle{1\over 8}}+{\textstyle{5\over 24... ...{385\over 3456}}\tan^4\alpha){3\cdot e^{5\pi i/4}\over 2^2 X^5}+\ldots}\right],$

where

$\begin{displaymath} {\nu\over x}=\sin\alpha, \end{displaymath}$

$\begin{displaymath} 1-{\nu\over x}>{3\over x}\nu^{1/2}, \end{displaymath}$

and

$\begin{displaymath} X\equiv \sqrt{-x\cos({\textstyle{1\over 2}}\alpha)}. \end{displaymath}$

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1475, 1980.

$H_\nu^{(1)}(x)\sim{1\over\sqrt{\pi}} \mathop{\rm exp}\nolimits \{ix[\cos\alpha+(\alpha-\pi/2)\sin\alpha]\}$
$\times\left[{{e^{i\pi/4}\over X}+({\textstyle{1\over 8}}+{\textstyle{5\over 24... ...{385\over 3456}}\tan^4\alpha){3\cdot e^{5\pi i/4}\over 2^2 X^5}+\ldots}\right],$