Spherical Bessel Function of the First Kind

$\begin{figure}\begin{center}\BoxedEPSF{SphericalBesselj.epsf}\end{center}\end{figure}$

$\begin{eqnarray*} j_n(x) &\equiv& \sqrt{\pi\over 2x} J_{n+1/2}(x)\\ &=& 2^nx^... ...\ &=& (-1)^nx^n\left({d\over x\, dx}\right)^n {\sin x\over x}. \end{eqnarray*}$

The first few functions are

$\begin{eqnarray*} j_0(x) &=& {\sin x\over x}\\ j_1(x) &=& {\sin x\over x^2} -... ...t({{3\over x^3} - {1\over x}}\right)\sin x - {3\over x^2}\cos x. \end{eqnarray*}$

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.

Arfken, G. ``Spherical Bessel Functions.'' §11.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 622-636, 1985.