I was looking at the documentation here. In the "Details and Options" section they mention that $f(t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} F(\omega)e^{-i \omega t} d\omega$. I interpreted this to mean
InverseFourierTransform[expr,$\omega$,t] = $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} $ expr $ e^{-i \omega t} d\omega$
Is this correct? I tried to type in the following two expressions to confirm this but I was not getting the same results.
InverseFourierTransform[1/(1 + I*\[Omega]), \[Omega], -t]
(1/Sqrt[(2*Pi)]) Integrate[(1/(1 + I*\[Omega]))*
Exp[-I*\[Omega]*t], {\[Omega], -Infinity, Infinity}]
What I was expecting was for both of the above expressions to give me E^-t Sqrt[2 \[Pi]] HeavisideTheta[t]
but that isn't what happened. I tried switching Integrate[] to NIntegrate[] but then the integral won't evaluate at all.
Am I misinterpreting what InverseFourierTransform[] does? Or is there an error in my other integral? Thank you very much in advance.
-t
as the time variable? TryInverseFourierTransform[1/(1 + I*w), w, t]
The mapping is fromw
(frequency) tot
(time) and not fromw
to-t
. The Fourier transforms mapst
tow
and nott
to-w
. Try to stick to the definitions unless you want results which will not make too much sense. $\endgroup$