Fourier transform of a Gaussian over a square window

Question

I am trying to evaluate an integral of the form $$ G_T(\omega) = \int_{-T/2}^{T/2} \mathrm{d}t ~ e^{-\sigma^2 t^2} e^{i\omega t} $$ and I am looking for a closed-form expression. I know this integral can be rearranged into a convolution between a Gaussian and a sinc function, and an "incomplete" expression for that integral can be found here. I am however rather confused by the complex erf function in that answer, and I really need some clarifications.

Answer 1

If, after @Sal's comment, you still need to be conviced that $$G_T(\omega)=\int_{-\frac T2}^{+\frac T2} e^{-\sigma^2 t^2} e^{i\omega t}\,dt$$ $$G_T(\omega)=i\,\frac{ \sqrt{\pi } }{2 \sigma }\,e^{-\frac{\omega ^2}{4 \sigma ^2}}\left(\text{erfi}\left(\frac{\omega -i \sigma ^2 T}{2 \sigma }\right)-\text{erfi}\left(\frac{\omega +i \sigma ^2 T}{2 \sigma }\right)\right)$$ is not "intimidating", expand it as a Taylor series around $T=0$ and let $k=\frac{\omega ^2}{ \sigma ^2}$ $$G_T(\omega)=T\Bigg[1+\sum_{n=1}^\infty(-1)^n\, \frac{\sigma^{2n}}{4^n\,(2 n+1)!} a_n\,T^{2n}\Bigg]$$ The first $a_n$ are $$\left( \begin{array}{cc} n & a_n \\ 1 & k+2 \\ 2 & k^2+12 k+12 \\ 3 & k^3+30 k^2+180 k+120 \\ 4 & k^4+56 k^3+840 k^2+3360 k+1680 \\ 5 & k^5+90 k^4+2520 k^3+25200 k^2+75600 k+30240 \\ \end{array} \right)$$