When I calculate the Fourier transform of the function $f(t) = \mathrm e^{-|t|/\tau} \text{ with } \tau >0$ in Mathematica once via the function FourierTransform and once by hand, I get different results. And with "by hand" I mean letting Mathematica calculate the integral $ft_2(\omega) = \int_\mathbb{R} f(t) \mathrm e^{-2 \pi \mathrm i \omega t} \, \mathrm dt$
My input is:
f[t_] = Exp[-Abs[t]/\[Tau]]; ft1[\[Omega]_] = FourierTransform[f[t], t, \[Omega]] ft2[\[Omega]_] = Integrate[f[t]*Exp[-2*Pi*I*\[Omega]*t], {t, -Infinity, Infinity}, Assumptions -> {{\[Omega], \[Tau]} \[Element] Reals, \[Tau] > 0}] \[Omega] = 0.123; \[Tau] = 0.456; ft1[\[Omega]] ft2[\[Omega]]
And the generated output is:
$ft_1(\omega) = \sqrt{\frac{2}{\pi}} \cdot \frac{\tau}{1+\tau^2\omega^2}$ $ft_2(\omega) = \frac{2 \tau}{(-\mathrm i + 2 \pi \tau \omega)(\mathrm i + 2 \pi \tau \omega)} = \frac{2 \tau}{1 + 4 \pi^2 \omega^2 \tau^2}$ and
0.362694 0.811248+ 0. I
As you can see, we get different values for $\omega = 0.123$ and $\tau = 0.456$. I am most certain that there is some error with the integral, as the result from FourierTransform can also be obtained from rule 207 in the List from Wikipedia.