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Bessel Function Fourier Expansion

Let $n\geq {1/2}$ and $\alpha_1$, $\alpha_2$, the Positive Roots of $J_n(x)=0$. An expansion of a function in the interval (0,1) in terms of Bessel Functions of the First Kind

f(x)=\sum_{l=1}^\infty A_r J_n(x\alpha_r),
\end{displaymath} (1)

has Coefficients found as follows:
\int_0^1 xf(x)J_n(x\alpha_l)\,dx = \sum_{r=1}^\infty A_r \int_0^1
xJ_n(x\alpha_r)J_n(x\alpha_l)\, dx.
\end{displaymath} (2)

But Orthogonality of Bessel Function Roots gives
\int_0^1 xJ_n(x\alpha_l)J_n(x\alpha_r)\,dx = {\textstyle{1\over 2}}\delta_{l,r} {J_{n+1}}^2(\alpha_r)
\end{displaymath} (3)

(Bowman 1958, p. 108), so
$\displaystyle \int_0^1 xf(x)J_n(x\alpha_l)\,dx$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sum_{r=1}^\infty A_r \delta_{l,r} {J_{n+1}}^2(x\alpha_r)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}A_l {J_{n+1}}^2(\alpha_l),$ (4)

and the Coefficients are given by
A_l = {2\over {J_{n+1}}^2(\alpha_l)} \int_0^1 xf(x)J_n(x\alpha_l)\,dx.
\end{displaymath} (5)


Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958.

© 1996-9 Eric W. Weisstein