# table of Fourier transforms

Below are tables of Fourier transforms (http://planetmath.org/FourierTransform); one lists some of the common properties, and the other lists some common examples.

## Properties

Original Transformed comment derivation
$af(t)+bg(t)$ $a\mathcal{F}\{f(t)\}+b\mathcal{F}\{g(t)\}$ linearity
$f(t)*g(t)$ $\mathcal{F}\{f(t)\}\mathcal{F}\{g(t)\}$ convolution property
$f(t+\alpha)$ $F(s)\exp(-i\alpha s)$ time shift, where $F(s)=\mathcal{F}\{f(t)\}$
$f^{\prime}(t)$ $is\mathcal{F}\{f(t)\}$ differentiation
$\overline{f(t)}$ $\overline{F(-s)}$ conjugation, where $F(s)=\mathcal{F}\{f(t)\}$
$f(\alpha t)$ $\displaystyle{\frac{1}{|\alpha|}F(\frac{s}{\alpha})}$ scaling, where $F(s)=\mathcal{F}\{f(t)\}$ with $\alpha\neq 0$

${{{{{{{}\end{center}\inner@par\@@section{subsubsection}{S0.SS0.SSSx2}{}{}{}{Examples}% \inner@par\begin{center}\begin{tabular}[]{|c|c|c|p{4cm}|c|}\hline\hlinef(t)&% \mathcal{F}\{f(t)\}&conditions&\tabularcell@hbox{explanation}&derivation\\ \hline\hline\delta(t)&1&&\tabularcell@hbox{Dirac delta function}&\\ \hline1&2\pi\delta(s)&&\tabularcell@hbox{}&\\ \hlinee^{iat}&2\pi\delta(s-\alpha)&a\in\mathbb{R}&\tabularcell@hbox{}&\\ \hline\cos(at)&\pi(\delta(s+a)+\delta(s-a))&a\in\mathbb{R}&% \tabularcell@hbox{}&\\ \hline\sin(at)&i\pi(\delta(s+a)-\delta(s-a))&a\in\mathbb{R}&% \tabularcell@hbox{}&\\ \hline\inner@par}\end{tabular}}}\end{center}\begin{flushright}\begin{tabular}% []{|ll|}\hline Title&table of Fourier transforms\\ Canonical name&TableOfFourierTransforms\\ Date of creation&2013-03-22 18:08:12\\ Last modified on&2013-03-22 18:08:12\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&7\\ Author&CWoo (3771)\\ Entry type&Feature\\ Classification&msc 42A38\\ Synonym&Groupoid Transforms\\ Defines&Fourier-Stieltjes generalization of FT\\ \hline}\end{tabular}}}\end{flushright}\end{document}$