Modified Spherical Bessel Function

Solutions to the Modified Spherical Bessel Differential Equation, given by

$$i_n(x) \equiv \sqrt{\pi\over 2x}\,I_{n+1/2}(x)$$ (1)

$$i_0(x) = {\sinh x\over x}$$ (2)

$$k_n(x) \equiv \sqrt{2\pi\over x}\,K_{n+1/2}(x)$$ (3)

$$k_0(x) = {e^{-x}\over x},$$ (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.

References

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.