Hankel Function of the First Kind

$$H_n^{(1)}(z) \equiv J_n(z)+iY_n(z),$$

where $J_n(z)$ is a Bessel Function of the First Kind and $Y_n(z)$ is a Bessel Function of the Second Kind. Hankel functions of the first kind can be represented as a Contour Integral using

$$H_n^{(1)}(z) = {1\over i\pi} \int_{0 {\rm\ [upper\ half\ plane]}}^\infty {e^{(z/2)(t-1/t)}\over t^{n+1}}\,dt.$$

See also Debye's Asymptotic Representation, Watson-Nicholson Formula, Weyrich's Formula

References

Arfken, G. ``Hankel Functions.'' §11.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 604-610, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 623-624, 1953.