I would like to find if possible a solution (closed form) for the following integral:
$$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin x)\,\mathrm dx$$
where $a,b$ are positive real constants and $j_{0}$ is the spherical bessel function of order zero ($j_0(z)=sinc(z)$). Anyone has a clever substituion or a known integral that might help me in finding the solution?
I would note that the j_0's are just sin y/y and so you can rewrite the numerators as proportional to exp[ibcos x] - exp[-ibcos x]. Then the whole integral will involve only complex exponentials and 1/(b cos x* b sin x). You can then pull out a and b such that you have factors of the form exp[i cos x]^b. A change of variable u = cos x will then convert this to simpler exponentials, and simplify the denominator; what results is straightforward to integrate.