Mathematik-Online-Lexikon: Wichtige Fourier-Transformationen

	$\hat{f}(y)$
	$2\pi\delta(y)$
	$2\pi\mathrm{i}^n\delta^{(n)}(y),\quad n=1,2,3,\ldots$
$x^{-n}$	$\displaystyle\frac{\pi(-\mathrm{i})^n y^{n-1}\operatorname{sign}(y)}{(n-1)!}, \quad n=1,2,3,\ldots$
$\operatorname{sign}(x)$	$\displaystyle\frac{2}{\mathrm{i}y}$
$\vert x\vert=x\operatorname{sign}(x)$	$\displaystyle\frac{-2}{y^2}$
$x^n\operatorname{sign}(x)$	$\displaystyle\frac{2n!}{(\mathrm{i}y)^{n+1}}$
$\delta(x)$
$\delta^{(n)}(x)$	$(\mathrm{i}y)^n$
$\displaystyle e^{-a\vert x\vert}$	$\displaystyle\frac{2a}{a^2+y^2},\quad a>0$
$\displaystyle e^{-ax^2}$	$\displaystyle\sqrt{\frac{\pi}{a}}\,e^{-y^2/(4a)}$
$\cos(ax)$	$\pi(\delta(y+a)+\delta(y-a))$
$\sin(ax)$	$\mathrm{i}\pi(\delta(y+a)-\delta(y-a))$
$\cos(ax^2)$	$\displaystyle \sqrt{\frac{\pi}{a}} \cos\left(\frac{y^2}{4a}-\frac{\pi}{4}\right),\quad a>0$
$\sin(ax^2)$	$\displaystyle \sqrt{\frac{\pi}{a}} \cos\left(\frac{y^2}{4a}+\frac{\pi}{4}\right),\quad a>0$

	[Home] [Lexikon] [Aufgaben] [Tests] [Kurse] [Begleitmaterial] [Hinweise] [Mitwirkende] [Publikationen]
	Mathematik-Online-Lexikon:
	Wichtige Fourier-Transformationen