Spherical Hankel Function of the First Kind

$$h_n^{(1)}(x) \equiv \sqrt{\pi\over 2x} H_{n+1/2}^{(1)}(x) = j_n(x)+in_n(x),$$

where $H^{(1)}(x)$ is the Hankel Function of the First Kind and $j_n(x)$ and $n_n(x)$ are the Spherical Bessel Functions of the First and Second Kinds. Explicitly, the first few are

$$h_0^{(1)}(x) = {1\over x} (\sin x-i \cos x) = - {i\over x} e^{ix}$$

$$h_1^{(1)}(x) = e^{ix}\left({-{1\over x}- {i\over x^2}}\right)$$

$$h_2^{(1)}(x) = e^{ix}\left({{i\over x} - {3\over x^2} - {3i\over x^3}}\right).$$

References

Abramowitz, M. and Stegun, C. 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.