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Spherical Bessel Function of the First Kind


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}.

The first few functions are

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.

See also Poisson Integral Representation, Rayleigh's Formulas


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.

© 1996-9 Eric W. Weisstein