Difficult integral

Gold Member
Hey guys, I am actually solving a physics problem, and at the last step of my solution, I encounter this integral:

$$\int_0^{k_F} k^2J_0(k|\vec{r}'-\vec{r}|)dk$$

The J is the 0th Bessel function of the 1st kind. I tried this on Wolfram alpha, and it gave me the integral in terms of other Bessel functions and Struve functions. I don't think my professor would have given me an integral that had such an elaborate solution (especially since I'm asked to graph this). Am I missing something basic here? Is there an easy way to evaluate this integral? If not, I probably did something wrong somewhere earlier.

vela
Staff Emeritus
Homework Helper
I don't know if this'll work, but perhaps you can use an integral representation of the Bessel function and then use some trick like switching the order of integration to get an answer.

hunt_mat
Homework Helper
You would perhaps differentiate w.r.t. r and see in it reveals anything that will make the integral easier.

Ray Vickson