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Diffstat (limited to 'latex/know/concept/fourier-transform/source.md')
| -rw-r--r-- | latex/know/concept/fourier-transform/source.md | 8 |
1 files changed, 3 insertions, 5 deletions
diff --git a/latex/know/concept/fourier-transform/source.md b/latex/know/concept/fourier-transform/source.md index 58830df..3e25980 100644 --- a/latex/know/concept/fourier-transform/source.md +++ b/latex/know/concept/fourier-transform/source.md @@ -63,10 +63,9 @@ on whether the analysis is for forward ($s > 0$) or backward-propagating ## Derivatives -The FT of a derivative has a very interesting property, let us take a -look. Below, after integrating by parts, we remove the boundary term by -assuming that $f(x)$ is localized, i.e. $f(x) \to 0$ for -$x \to \pm \infty$: +The FT of a derivative has a very interesting property. +Below, after integrating by parts, we remove the boundary term by +assuming that $f(x)$ is localized, i.e. $f(x) \to 0$ for $x \to \pm \infty$: $$\begin{aligned} \hat{\mathcal{F}}\{f'(x)\} @@ -75,7 +74,6 @@ $$\begin{aligned} &= A \big[ f(x) \exp(i s k x) \big]_{-\infty}^\infty - i s k A \int_{-\infty}^\infty f(x) \exp(i s k x) \dd{x} \\ &= (- i s k) \tilde{f}(k) - \qedhere \end{aligned}$$ Therefore, as long as $f(x)$ is localized, the FT eliminates derivatives |