Problem: Solve the double integral $$I:=\int _{-\infty}^{\infty} \int _{-\infty}^{\infty} dx dy\ f(x,y),$$ where $f(x,y) = e^{-i a(x - y)}$ if $y > x$ and zero otherwise ($a$ is only a positive real constant).
So, I think I have
$$I=\int _{-\infty}^{\infty} dy \int_{-\infty}^{y} dx\ e^{-ia(x - y)}.$$
But this integral diverges. Is there some way to compute it?
Hint: let $u=(x+y)/2,v=(x-y)/2$, then $x=u+v,y=u-v$ and $dxdy=-2dudv$ $$I:=\int _{-\infty}^{\infty} \int _{-\infty}^{\infty} dx dy\ g(x-y)=-2\int _{-\infty}^{\infty} \left(\int _{-\infty}^{\infty} dv\ g(2v)\right)du$$ $$=-2\left(\int _{-\infty}^{\infty}du\right) \left(\int _{-\infty}^{\infty} dv\ g(2v)\right)$$