For functions that are transformed between _time_ and _frequency domains_, the convention for [Fourier transform](Fourier%20transform.md)s is as follows. Recall that the general definition of the Fourier transform is as follows: [$\mathcal{F}[f(x)]_k=\int dx \, f(x) e^{-ixk}.$](Fourier%20transform.md#^7b1c33) The Fourier transform from the time to the frequency domain is expressed as $\widetilde{f}(\omega) = \int_{-\infty}^{\infty}dt \, f(t) e^{i\omega t}$ and its inverse is $f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega \, \widetilde{f}(\omega) e^{-i\omega t}$ ^7f3052 Be mindful of the sign convention used here. Here we switch the signs in the exponentials for the transform and inverse transform. --- # Recommended reading The convention used here for the [Time-frequency Fourier transform](Time-frequency%20Fourier%20transform.md) is presented in the following texts: * [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md), pg. 288 #MathematicalFoundations/Analysis/FourierAnalysis/Integrals #MathematicalFoundations/SignalProcessing #MathematicalFoundations/Analysis/Functions/Functionals/Integrals/IntegralTransforms