Fourier transform of $te^{-t^2}$?

Question

How can I find the Fourier transform of:

$$f(t) = te^{-t^2}$$

Answer 1

Hint: If using the Fourier transform $\hat{f}(\xi) = \int\limits_{-\infty}^{\infty}f(x)e^{-2\pi i x\xi}dx$,
then $x^nf(x) $ has the Fourier transform $\left(\frac{i}{2\pi}\right)^n\frac{d^n\hat{f}(\xi)}{d\xi}$,
and $f(x) = e^{-\alpha x^2}$ has the Fourier transform $\hat{f}(\xi) = \sqrt{\frac{\pi}{\alpha}}e^{-(\pi\xi)^2/\alpha}$.

What happens if you try to combine them?