Watson-Nicholson Formula

Let $H_\nu^{(\iota)}(x)$ be a Hankel Function of the First or Second Kind, let $x, \nu>0$, and define

$$w=\sqrt{\left({x\over\nu}\right)^2-1}.$$

Then

$$H_\nu^{(\iota)}(x)=3^{-1/2}w\mathop{\rm exp}\nolimits \{(-1)... ...}({\textstyle{1\over 3}}\nu w)+{\mathcal O}\vert\nu^{-1}\vert.$$

References

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