Hankel Function

There are two types of functions known as Hankel functions. The more common one is a complex function (also called a Bessel function of the third kind, or Weber Function) which is a linear combination of Bessel functions of the first and second kinds.

Another type of Hankel function is defined by the contour integral

for , , , where  is a Hankel contour. The Riemann zeta function can be expressed in terms of  as

for  and , where  is the gamma function (Krantz 1999, p. 160).

REFERENCES:

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

Hankel, H. "Die Cylinderfunctionen erster und zweiter Art." Math. Ann. 1, 467-501, 1869.

Hankel, H. "Bestimmte Integrale mit Cylinderfunctionen." Math. Ann. 8, 453-470, 1875.

Krantz, S. G. "The Hankel Contour and Hankel Functions." §13.2.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 159, 1999.

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