Hankel Function of the First Kind

The Hankel functions of the first kind are defined as

(1)

where  is a Bessel function of the first kind and  is a Bessel function of the second kind. Hankel functions of the first kind is implemented in the Wolfram Language as HankelH1[n, z].

Hankel functions of the first kind can be represented as a contour integral over the upper half-plane using

(2)

The derivative of  is given by

(3)

The plots above show the structure of  in the complex plane.

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.