Bessel Function Fourier Expansion

Let $n\geq {1/2}$ and $\alpha_1$, $\alpha_2$, ...be 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

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

has Coefficients found as follows:

$$\int_0^1 xf(x)J_n(x\alpha_l)\,dx = \sum_{r=1}^\infty A_r \int_0^1 xJ_n(x\alpha_r)J_n(x\alpha_l)\, dx.$$ (2)

But Orthogonality of Bessel Function Roots gives

$$\int_0^1 xJ_n(x\alpha_l)J_n(x\alpha_r)\,dx = {\textstyle{1\over 2}}\delta_{l,r} {J_{n+1}}^2(\alpha_r)$$ (3)

(Bowman 1958, p. 108), so

$$\int_0^1 xf(x)J_n(x\alpha_l)\,dx = {\textstyle{1\over 2}}\sum_{r=1}^\infty A_r \delta_{l,r} {J_{n+1}}^2(x\alpha_r)$$

$$= {\textstyle{1\over 2}}A_l {J_{n+1}}^2(\alpha_l),$$ (4)

and the Coefficients are given by

$$A_l = {2\over {J_{n+1}}^2(\alpha_l)} \int_0^1 xf(x)J_n(x\alpha_l)\,dx.$$ (5)

References

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