The Green's function satisfies the non homogeneous Bessel equation can be written as $xg''+g'+\left(k^2x-\frac{m^2}{x}\right)g=-\delta(x-\xi)$ where $m\geq0$ and an integer. The boundary conditions are $\lim_{x\to 0}|g\left(x|\xi\right)|<\infty$, and $g\left(L|\xi\right)=0$.
The Fourier-Bessel series representation of $g$ can be written as $g\left(x|\xi\right)=\sum_{n=1}^{\infty}G_n\left(\xi\right)J_m(k_{nm}x)$. Where $k_{nm}$ is the $n^{th}$ root of $J_m\left(k_{nm}x\right)$. Now if we substitute the second equation into the first one with the Fourier-Bessel representation of $\delta(x-\xi)$ then the $G_n(\xi)$ is expressed as
$\left(k^2-k_{nm}^2\right)G_n(\xi)=-\frac{2k_{nm}^2J_m(k_{nm}\xi)}{L^2\left[J_{m+1}\left(k_{nm}L\right)\right]^2}$
How the last result is obtained after substitution? Please help to find it out.