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Neumann Series (Bessel Function)

A series of the form

\sum_{n=0}^\infty a_nJ_{\nu+n}(z),
\end{displaymath} (1)

where $\nu$ is a Real and $J_{\nu+n}(z)$ is a Bessel Function of the First Kind. Special cases are
z^\nu =2^\nu \Gamma({\textstyle{1\over 2}}\nu+1) \sum_{n=0}^...
... {({\textstyle{1\over 2}}z)^{\nu/2+n}\over n!} J_{\nu/2+n}(z),
\end{displaymath} (2)

where $\Gamma(z)$ is the Gamma Function, and
\sum_{n=0}^\infty b_nz^{\nu+n}=\sum_{n=0}^\infty a_n\left({{\textstyle{1\over 2}}z}\right)^{(\nu+n)/2}J_{(\nu+n)/2}(z),
\end{displaymath} (3)

a_n\equiv \sum_{m=0}^{\left\lfloor{n/2}\right\rfloor } {2^{\...
...{1\over 2}}\nu+{\textstyle{1\over 2}}n-m+1)\over m!} b_{n-2m},
\end{displaymath} (4)

and $\left\lfloor{x}\right\rfloor $ is the Floor Function.

See also Kapteyn Series


Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

© 1996-9 Eric W. Weisstein