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Table of common Fourier transforms |
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The following table records some important Fourier transforms. ''F''(''s'') and ''G''(''s'') denote the Fourier transforms of ''f''(''t'') and ''g''(''t''), respectively. ''f'' and ''g'' may be integrable functions or tempered distributions.
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<td>''f''(''at'')</td>
<td>1/|''a''| ''F''(''s''/''a'')</td>
<td> </td>
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<td>''t'' ''f''(''t'')</td>
<td>1/(2π''i'') <em>F</em>'(-''s'')</td>
<td> </td>
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<td>''f''(''t'') ''g''(''t'')</td>
<td>(''F'' * ''G'')(-''s'')</td>
<td> </td>
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<td>1</td>
<td>δ(''s'')</td>
<td> </td>
</tr>
<tr>
<td>δ(''t''-''a'')</td>
<td>e<sup>-2π''ias''</td>
<td> </td>
</tr>
<tr>
<td>''t''<sup>''n''</sup></td>
<td>1/(-2π''i'')<sup>''n''</sup> &delta<sup>(''n'')</sup>(-''s'')</td>
<td>(''needs to be checked'') &delta<sup>(''n'')</sup>(''s'') is the ''n''-th distribution derivative of the Dirac delta</td>
</tr>
<tr>
<td>e<sup>2π''iat''</sup></td>
<td>δ(''s''-''a'')</td>
<td> </td>
</tr>
<tr>
<td>cos(2π''at'')</td>
<td>1/2 ( δ (''s'' - ''a'') + δ(''s'' + ''a'') )</td>
<td> </td>
</tr>
<tr>
<td>exp(-''a'' ''t''<sup>2</sup>)</td>
<td>(π/''a'')<sup>1/2</sup> exp(-π<sup>2</sup> ''s''<sup>2</sup> / ''a'')</td>
<td> </td>
</tr>
</table>
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