Spherical Hankel Function of the First Kind

The spherical Hankel function of the first kind  is defined by

			(1)
			(2)

where  is the Hankel function of the first kind and  and  are the spherical Bessel functions of the firstand second kinds.

It is implemented in the Wolfram Language as SphericalHankelH1[n, z].

Explicitly, the first few are

			(3)
			(4)
			(5)
			(6)

The derivative is given by

(7)

The plot above shows the real and imaginary parts of  on the real axis for , 1, ..., 5.

The plots above shows the real and imaginary parts of  in the complex plane.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437-442, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 623, 1985.