Fourier transform of $f(x) = \sin(x)e^{-x^2}$ I've tried looking at
$$\mathcal{F}\left(\sin(t)e^{-x^2}\right) = \dfrac{1}{\sqrt{\tau}}\int_{-\infty}^\infty \sin(x)e^{-x^2}e^{-iwx}\,\mathrm{d}x$$
but this seems like a dead end, as I don't know any easy ways to integrate $e^{-x^2}$ to make use of partial integration
I thought there might be some trick involving the derivatives of $f(x)$, as there is with $e^{-x^2}$, but I can't see any obvious solutions using that either, as $f(x ) = \dfrac{\cos(x)e^{-x^2}-f'(x)}{2x}$.
 A: Hints (although already indicated by Did): 
You may write $\sin x = \frac{e^{ix}-e^{-ix}}{2i}$ and then
use $$e^{-z^2}\int_{-\infty}^{\infty} e^{-x^2 - 2zx} dx =\int_{-\infty}^{\infty} e^{-(x+z)^2} dx = \sqrt{\pi}$$
valid for any complex number $z$.
A: Let us denote by FT the Fourier Transform $f \longrightarrow F$ defined by
$$F(\xi):=\int_{-\infty}^{+\infty}f(t)e^{-2 i \pi \xi t}dt$$
Let us write the initial function as the product:
$$f(t)g(t) \ \ \text{where} \ \ f(t):=\frac{\sin(t)}{t} \ \ \text{and} \ \  g(t):=te^{-t^2}$$
Let us take the convention $a=\frac{1}{2 \pi}.$ The Fourier Tranforms of $f$ and $g$ are:


*

*$F(\xi):=FT(f)(\xi)=\pi C_{[-a,a]} \ (\xi)$ where $C_I$ means "characteristic function of interval $I$" (classical FT of "sinc" function) and

*$G(\xi):=FT(g)(\xi)=-i \pi \xi e^{-(\pi \xi)^2} \ $ using a result given in (Fourier transform of $te^{-t^2}$?)
Using the convolution theorem, one gets:
$H(\xi):=(F\star G)(\xi)=\pi C_{[-a,a]}(\xi) \  \star \ -i\pi \xi e^{-(\pi \xi)^2}$
Let us differentiate $H$ by convolving $F'$ with $G$ under the form:
$H'(\xi):=(F'\star G)(\xi)=\pi \left(\delta(\xi+a)-\delta(\xi-a)\right) \star -i\pi \xi e^{-(\pi \xi)^2}$
Otherwise said :
$H'(\xi):=(F'\star G)(\xi)=-i \pi^2 \left( (\xi+a) e^{-\pi^2 (\xi+a)^2}-(\xi-a) e^{-\pi^2 (\xi-a)^2} \right)$
finally giving, by taking a primitive (with the help of Mathematica), the following Fourier transform:

$$H(\xi)= i \frac{\sqrt{\pi}}{2} (1-e^{2 \pi \xi})e^{-(\frac{1}{2}+\pi \xi)^2}=-i\sqrt{\pi}e^{-1/4}e^{-(\pi \xi)^2}\sinh(\pi \xi)$$

There is an interesting remark with the second expression of $H(\xi)$ : the Fourier transform of $\sin(t)e^{-t^2}$ bears (up to some constants) a striking similitude with the original function, a $\sinh$ taking the place of the $\sin$ function.
A: Refering to my other answer, since  $$\mathcal{F}\{\sin(x)\}=\frac{1}{2i}\mathcal{F}\{e^{ix}-e^{-ix}\}=\pi i[\delta(\omega+1)-\delta(\omega-1)]$$
the FT of $e^{-x^2}\sin(x)$ is $$\pi i [F(\omega+1)-F(w-1)]$$ where $$F(\omega)=\mathcal{F}\{e^{-x^2}\}$$
