I am wondering if there exists a closed form formula for the Fourier transform of the function $f(x) = \frac{e^x}{1 + e^x} \prod(x / a)$, $a \in \mathbb{R}_{>0}$, where $\prod(x / a) = 1$ if $|x| < a$, 0 else (i.e., the rectangular function). I have tried computing the Fourier transform of both and convolving them in the Fourier domain. However, I've not been able to obtain any closed form solution. Any ideas?
$(F\sigma)(\omega) = \sqrt{\frac{\pi}{2}} ( \delta(\omega) - i \ \text{csch}(\pi \omega) )$
$(F\prod)(\omega) = \frac{1}{\sqrt{2\pi}} a \ \text{sinc}(\frac{a \omega}{2})$
Using Only Mathematica 13.1:
$$\mathcal{F}_x\left[\sigma (x) \Pi \left(\frac{x}{a}\right)\right](\omega )=\\-\frac{i e^{\frac{i a \omega }{2}}}{\sqrt{2 \pi } \omega }-i \sqrt{\frac{\pi }{2}} \text{csch}(\pi \omega )+\frac{i e^{-\frac{1}{2} i a \omega } \, _2F_1\left(1,-i \omega ;1-i \omega ;-e^{a/2}\right)}{\sqrt{2 \pi } \omega }+\frac{i e^{\frac{i a \omega }{2}} \, _2F_1\left(1,i \omega ;1+i \omega ;-e^{a/2}\right)}{\sqrt{2 \pi } \omega }$$
MMA code:
FourierTransform[LogisticSigmoid[x] UnitBox[x/a] // FunctionExpand, x, \[Omega], Assumptions -> a > 0] // Expand(*-((I E^((I a \[Omega])/2))/(Sqrt[2 \[Pi]] \[Omega])) - I Sqrt[\[Pi]/2] Csch[\[Pi] \[Omega]] + ( I E^(-(1/2) I a \[Omega]) Hypergeometric2F1[1, -I \[Omega], 1 - I \[Omega], -E^(a/2)])/( Sqrt[2 \[Pi]] \[Omega]) + ( I E^((I a \[Omega])/2) Hypergeometric2F1[1, I \[Omega], 1 + I \[Omega], -E^(a/2)])/( Sqrt[2 \[Pi]] \[Omega])*)