If $f: \mathbb R^n \to \mathbb C$ in $L^1(\mathbb R^n)$, we define $\hat f (\xi) = \int e^{-2\pi i \langle \xi, x \rangle} f(x) dx,$ for $\xi \in \mathbb R^n$.
I'm trying to find a function $f: \mathbb R \to \mathbb C$ such that
$$ \hat f (\omega) = \frac{1}{(1+i\omega)^2}. $$
However, I have no idea how to start this problem. Any hint?
One way is to guess, thinking about integrating by parts to get the right answer, but we can compute the inverse Fourier integral directly: $$ f(x) = \int_{-\infty}^{\infty} \frac{e^{2\pi i \omega x}}{(1+i\omega)^2} \, d\omega. $$ We can evaluate this using the Residue theorem by completing the contour in either the upper or lower half-plane with a large semicircle (and then Jordan's lemma implies the integral over the semicircle tends to zero, so it's just the sum of the residues). For $x<0$the lower half-plane contains no poles, so the integral is zero. For $x>0$, there's one double pole at $\omega = i$. One tedious computation using Laurent series later, we find the answer is $(2\pi)^2xe^{-2\pi x}$.