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
$af(t)+bg(t)$	$aℱ\{f(t)\}+bℱ\{g(t)\}$	linearity
$f(t)*g(t)$	$ℱ\{f(t)\}ℱ\{g(t)\}$	convolution property
$f(t+\alpha )$	$F(s)\exp (-i\alpha s)$	time shift, where $F(s)=ℱ\{f(t)\}$
$f^{\prime }(t)$	$isℱ\{f(t)\}$	differentiation
$\overline{f(t)}$	$\overline{F(-s)}$	conjugation, where $F(s)=ℱ\{f(t)\}$
$f(\alpha t)$	${\displaystyle \frac{1}{|\alpha |}}F({\displaystyle \frac{s}{\alpha }})$	scaling, where $F(s)=ℱ\{f(t)\}$ with $\alpha \ne 0$