Find Inverse Fourier Transform

Question

Find the inverse Fourier Transform of

$$ { F(\omega)=\frac{1}{2\pi(a+j\omega)^2} \ } $$ using the convolution theorem. Hint: the Fourier Transform of $e^{-at} u(t)=\frac{1}{\sqrt{2\pi}(a+j\omega)} $

Answer 1

The convolution theorem gives us: $$\mathcal{F}^{-1}[\mathcal{F}(f)\cdot\mathcal{F}(g)]= f*g$$

From the hint: $$F(\omega) = \mathcal{F}(e^{-at}u(t))\cdot\mathcal{F}(e^{-at}u(t))$$ and you seek $\mathcal{F}^{-1}[F(\omega)]$.

Can you finish it?