Is there any formula to calculate the Fourier Transform of a product of a exponential function and a rectangular windows?
i.e. Formally, calculating $$F(x(t))=F[(e^{-at})\Pi((t-T/2)/T]$$
Where
$\Pi$ is the rectangular window
$T$ a real positive number
I suppose that, given the formulas about the exponential decay and considering a windows as a combination of two Heaviside step function, these formulas can be applied to calculate this Fourier Trasform, but I have some doubts about how to proceed.
$$\begin{align*}\mathcal{F}\left\{x(t)\right\} & = \mathcal{F}\left\{e^{-at}\dfrac1{T}\Pi\left(t-\dfrac{T}{2}\right)\right\}\\ \\ &= \int_{-\infty}^{\infty}e^{-at}\dfrac1{T}\Pi\left(t-\dfrac{T}{2}\right)e^{-2\pi i st} \mathrm{d}t \\ \\ &= \dfrac1{T} \int_{0}^{T} e^{-at}e^{-2\pi i st} \mathrm{d}t \\ \\ &= \dfrac1{T} \int_{0}^{T} e^{-(a+2\pi is) t} \mathrm{d}t \\ \\ &= \dfrac1{T} \cdot \dfrac{-1}{a + 2\pi i s}e^{-(a+2\pi is) t} \biggr|_0^T\\ \\ &= \dfrac1{T} \cdot \dfrac{-1}{a + 2\pi i s}\left[e^{-(a+2\pi is) T} - 1 \right]\\ \\ \mbox{and if you don't want to stop there...}\\ \\ &= e^{-(a+2\pi is) \frac{T}{2}} \dfrac{\left[e^{(a+2\pi is) \frac{T}{2}} - e^{-(a+2\pi is) \frac{T}{2}} \right]}{2\left(a + 2\pi i s\right)\frac{T}{2}}\\ \\ &= e^{-i(2\pi s-ia) \frac{T}{2}} \dfrac{\left[e^{i(2\pi s - ia) \frac{T}{2}} - e^{-i(2\pi s-ia) \frac{T}{2}} \right]}{2i\left(2\pi s -ia\right)\frac{T}{2}}\\ \\ &= e^{-i(2\pi s-ia) \frac{T}{2}} \mathrm{sinc}\left(\left[2\pi s -ia\right]\frac{T}{2}\right)\\ \end{align*}$$
Where $\mathrm{sinc}()$ is the non-normalized $\mathrm{sinc}()$ function.