Spherical Hankel Function of the Second Kind

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

where $H^{(2)}(x)$ is the Hankel Function of the Second Kind and $j_n(x)$ and $n_n(x)$ are the Spherical Bessel Functions of the First and Second Kinds. Explicitly, the first is

$$h_0^{(2)}(x) = {1\over x} (\sin x+i \cos x) = {i\over x} e^{-ix}.$$

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.