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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;

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

0.811248+ 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.

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You do know that FourierTransform[] has an unconventional normalization, don't you? Did you check the FourierParameters option? – J. M. Dec 3 '11 at 2:47
up vote 4 down vote accepted

Does FourierTransform[Exp[-Abs[t]/\[Tau]], t, \[Omega], FourierParameters -> {0, -2 Pi}] work for you? It does for me...

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Yup, that's it. Thank you very much! – Robert B. Warren Dec 3 '11 at 2:57
@J.M. what exactly are you doing with the FourierParameters there? I'm especially confused by the fact that the Wikipedia table… provides three separate columns for Fourier Transforms, one of which corresponds to Mathematica's interpretation .. – Vincent Tjeng Apr 1 '13 at 7:46
@Vincent, one way you can do the reconciliation of normalizations by your own would be to look at the general definition for FourierTransform[] in Mathematica's docs and compare accordingly... – J. M. Apr 2 '13 at 13:16
I did have a look through the documentation but wasn't sure.. will take another look through it! – Vincent Tjeng Apr 2 '13 at 13:46

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