cv.complex variables - How can I calculate the characteristic function of these distributions? [previously: difficult integral]

If $q$ is a positive integer, then I think one can find this in any one of several undergraduate textbooks on complex analysis, where it's usually one of the standard examples to show the power of contour integration. I dimly remember something like this in Priestley's little OUP book, for instance. For arbitrary positive real values of $q$, I can't remember how this works I'm afraid.

(This is probably the sort of question which you could try out on fellow colleagues/students first, in my view.)