An analytic solution for a fourier transform

[jtceleron]

Homework Statement

the function is Exp[-w^2]/w^2, how to solve the Fourier transform analytically with Residue theorem?
It is better if there is more general results.
Mathematica can solve it analytically, but I need a human-soluable way.

Homework Equations

The Attempt at a Solution

My attempt follows the Fresnel integral, but there is still something wrong.

 

[vanhees71]

The integral is undefined as it stands, because there's the pole at $\omega=0$. You have to specify how to run around this pole first!