Need help to solve a definit integral using Fourier transform. Hi I have some problem with solving the following definit integral.
$$
\int_{-\infty}^{\infty} e^{-\big(\dfrac{t}{\tau}\big)^2-\dfrac{i\omega t}{2}}\textrm{d}t
$$
My guess to solve it is by using Fourier transform. I need somehow get write it on the form
$$
F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}
$$
And after i get it on this form I can simply look up in a table what $F(\omega)$ is.

Hint. Alternatively, one may observe that, by parity of the integrand, your integral reduces to
$$
\int_{-\infty}^{\infty} e^{-\Big(\dfrac{t}{\tau}\Big)^2-\dfrac{i\omega t}{2}}\textrm{d}t=\int_{-\infty}^{\infty} e^{-\Big(\dfrac{t}{\tau}\Big)^2}\cos \Big(\dfrac{\omega t}{2} \Big)\:\textrm{d}t
$$ then one may use the standard result:
$$
\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x=\frac{\sqrt{\pi}}{2} e^{-\large\frac{a^2}{4}}.
$$
