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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
$\displaystyle h_0^{(1)}(x)$ $\textstyle =$ $\displaystyle {1\over x} (\sin x-i \cos x) = - {i\over x} e^{ix}$  
$\displaystyle h_1^{(1)}(x)$ $\textstyle =$ $\displaystyle e^{ix}\left({-{1\over x}- {i\over x^2}}\right)$  
$\displaystyle h_2^{(1)}(x)$ $\textstyle =$ $\displaystyle e^{ix}\left({{i\over x} - {3\over x^2} - {3i\over x^3}}\right).$  


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.

© 1996-9 Eric W. Weisstein