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Spherical Bessel Function of the Second Kind


$\displaystyle n_n(x)$ $\textstyle \equiv$ $\displaystyle \sqrt{\pi\over 2x} Y_{n+1/2}(x)$  
  $\textstyle =$ $\displaystyle {(-1)^{n+1}\over 2^nx^{n+1}} \sum_{n=0}^\infty{(-1)^s(s-n)!\over s!(2s-2n)!} x^{2s}$  
  $\textstyle =$ $\displaystyle -{(2n-1)!!\over x^{n+1}}\left[{1-{{1\over 2}x^2\over 1!(1-2n)}+{\left({{1\over 2}x^2}\right)^2\over 2!(1-2n)(3-2n)}+\ldots}\right]$  
  $\textstyle =$ $\displaystyle (-1)^{n+1}\sqrt{\pi\over 2x}\, J_{-n-1/2}(x).$  

The first few functions are
$\displaystyle n_0(x)$ $\textstyle =$ $\displaystyle -{\cos x\over x}$  
$\displaystyle n_1(x)$ $\textstyle =$ $\displaystyle -{\cos x\over x^2} - {\sin x\over x}$  
$\displaystyle n_2(x)$ $\textstyle =$ $\displaystyle -\left({{3\over x^3} - {1\over x}}\right)\cos x - {3\over x^2}\sin x.$  

See also 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