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Modified Spherical Bessel Function

Solutions to the Modified Spherical Bessel Differential Equation, given by

$\displaystyle i_n(x)$ $\textstyle \equiv$ $\displaystyle \sqrt{\pi\over 2x}\,I_{n+1/2}(x)$ (1)
$\displaystyle i_0(x)$ $\textstyle =$ $\displaystyle {\sinh x\over x}$ (2)
$\displaystyle k_n(x)$ $\textstyle \equiv$ $\displaystyle \sqrt{2\pi\over x}\,K_{n+1/2}(x)$ (3)
$\displaystyle k_0(x)$ $\textstyle =$ $\displaystyle {e^{-x}\over x},$ (4)

where $I_n(x)$ is a Modified Bessel Function of the First Kind and $K_n(x)$ is a Modified Bessel Function of the Second Kind.


Abramowitz, M. and Stegun, C. A. (Eds.). ``Modified Spherical Bessel Functions.'' §10.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 443-445, 1972.

© 1996-9 Eric W. Weisstein