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table of Fourier transforms
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(Feature)
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Below are tables of Fourier transforms; one lists some of the common properties, and the other lists some common examples.
| Original |
Transformed |
comment |
derivation |
| $af(t)+bg(t)$ |
$a\F{f(t)}+b\F{g(t)}$ |
linearity |
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| $f(t)*g(t)$ |
$\F{f(t)}\F{g(t)}$ |
convolution property |
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| $f(t+\alpha)$ |
$F(s)\exp(-i \alpha s)$ |
time shift, where $F(s)=\F{f(t)}$ |
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| $f'(t)$ |
$is \F{f(t)}$ |
differentiation |
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| $\overline{f(t)}$ |
$\overline{F(-s)}$ |
conjugation, where $F(s)=\F{f(t)}$ |
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| $f(\alpha t)$ |
$\displaystyle{\frac{1}{|\alpha|}F(\frac{s}{\alpha})}$ |
scaling, where $F(s)=\F{f(t)}$ with $\alpha\ne 0$ |
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| $f(t)$ |
$\F{f(t)}$ |
conditions |
explanation |
derivation |
| $\delta(t)$ |
$1$ |
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Dirac delta function |
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| $1$ |
$2\pi \delta(s)$ |
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| $e^{i a t}$ |
$2\pi \delta(s - \alpha)$ |
$a\in \mathbb{R}$ |
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| $\cos(at)$ |
$\pi (\delta(s+a) + \delta(s-a))$ |
$a\in \mathbb{R}$ |
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| $\sin(at)$ |
$i \pi (\delta(s+a) - \delta(s-a))$ |
$a\in \mathbb{R}$ |
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"table of Fourier transforms" is owned by CWoo. [ full author list (3) ]
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| Other names: |
Groupoid Transforms |
| Also defines: |
Fourier-Stieltjes generalization of FT |
This object's parent.
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Cross-references: Dirac delta function, scaling, conjugation, differentiation, convolution, derivation, properties
There is 1 reference to this entry.
This is version 4 of table of Fourier transforms, born on 2008-06-09, modified 2008-07-03.
Object id is 10689, canonical name is TableOfFourierTransforms.
Accessed 2363 times total.
Classification:
| AMS MSC: | 42A38 (Fourier analysis :: Fourier analysis in one variable :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type) |
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Pending Errata and Addenda
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