A difficult integral for help

[xylai]

$$\int^{2a}_{0}dpJ[0,b\sqrt{p}]J[0,b\sqrt{2a-p}]$$

where a and b are constant, and J[0,x] is Bessel function.

 

[benorin]

$$\int^{2a}_{0}dpJ[0,b\sqrt{p}]J[0,b\sqrt{2a-p}]$$

where a and b are constant, and J[0,x] is Bessel function.

Expand out both Bessel functions as a series and multiply them according to Cauchy's rule, namely

$$\left(\sum_{k=0}^{\infty}a_{k} x^{k}\right) \left(\sum_{j=0}^{\infty}b_{j} x^{j}\right) = \sum_{k=0}^{\infty}\sum_{j=0}^{k}a_{j}b_{k-j} x^k$$​

and pass the integral through to the inner most sum (the second time I've blatantly ignored convergence issues, perfering to hand-wave such until I get a result) and make the change of variables $p=2au$, you should get it from there... post your result so I can compare/check my work.

Ben