How to compute this integral in general case?
$$t(x)=\int\_{-\infty}^{\infty}\frac{\exp(ixy)}{1+y^{2q}}dy$$


Mathematica can compute it when q is known. For example,for q=1 this integral is
$$\exp(-{\left|x\right|})\pi$$ 


But even in this case, I don't really know how to get this result.



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.)


