I am trying to derive the following result: $$ I=\frac{2}{\pi}\int_0^\infty dr\, r^2 j_l(kr)j_l(k'r)= \frac{1}{k^2}\delta(k-k')\, , $$ where $l$ is an integer.
I am aware that it can be done as shown in the page 5 of this pdf, but I was hoping there was another way to obtain it.
For example, I have tried to express the spherical Bessel functions according to
this equation:
$$
j_l (x)=(-1)^l (x)^l\left(\frac{1}{x}\frac{d}{dx}\right)^l \frac{\sin x}{x}\, ,
$$
and then Taylor expanded the sinc function so I obtained the following expression with no derivatives in it:
$$
I=\frac{2}{\pi}\sum_{n, n'}\frac{(-1)^{2l+n+n'}}{(2n+1)(2n'+1)(2n-l)!(2n'-l)!}\int_0^\infty dr\, r^2 (kr)^{2n-l}(k'r)^{2n'-l}
$$
which looks pretty awful in my eyes...
I am no mathematician so I'm probably very far from attacking this problem in the easiest way. Can anybody help me out? Any hints? Thanks in advance.
