Recently, I was fiddling around with computing Fourier transforms of different functions. At a certain point, I found out that $$\mathcal{F}_{x} [ \operatorname{sinc}(x)e^{-x} ] (t) = \int_{-\infty}^{\infty} \operatorname{sinc}(x)e^{-x}e^{-2 \pi i t x} dx = \pi . \tag{*}$$ It is therefore a constant value for all $t$.
I have two questions about this:
- How can one interpret this fact? Is there some sort of mathematical or physical reasoning behind this that allows for an intuitive understanding of $(*)$ ?
- Does $(*)$ bear any significance in mathematics, and if so, how and why?
Edit: it seems WA makes an error with the computation or calculates something differently. Either way, the results obtained above are probably wrong. I wonder what Wolfram computes in this case, and what the problem is.