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Diffstat (limited to 'content/know/concept/heaviside-step-function/index.pdc')
| -rw-r--r-- | content/know/concept/heaviside-step-function/index.pdc | 27 |
1 files changed, 19 insertions, 8 deletions
diff --git a/content/know/concept/heaviside-step-function/index.pdc b/content/know/concept/heaviside-step-function/index.pdc index 0471acf..dbbca6f 100644 --- a/content/know/concept/heaviside-step-function/index.pdc +++ b/content/know/concept/heaviside-step-function/index.pdc @@ -50,7 +50,23 @@ $$\begin{aligned} \end{aligned}$$ The [Fourier transform](/know/concept/fourier-transform/) -of $\Theta(t)$ is noteworthy. +of $\Theta(t)$ is as follows, +where $\pv{}$ is the Cauchy principal value, +$A$ and $s$ are constants from the FT's definition, +and $\mathrm{sgn}$ is the signum function: + +$$\begin{aligned} + \boxed{ + \tilde{\Theta}(\omega) + = \frac{A}{|s|} \Big( \pi \delta(\omega) + i \: \mathrm{sgn}(s) \pv{\frac{1}{\omega}} \Big) + } +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-fourier"/> +<label for="proof-fourier">Proof</label> +<div class="hidden"> +<label for="proof-fourier">Proof.</label> In this case, it is easiest to use $\Theta(0) = 1/2$, such that the Heaviside step function can be expressed using the signum function $\mathrm{sgn}(t)$: @@ -77,15 +93,10 @@ $$\begin{aligned} &= A \pi \delta(s \omega) + \frac{A}{2} \pv{\int_{-\infty}^\infty \mathrm{sgn}(t) \exp(i s \omega t) \dd{t}} = \frac{A}{|s|} \pi \delta(\omega) + i \frac{A}{s} \pv{\frac{1}{\omega}} \end{aligned}$$ +</div> +</div> The use of $\pv{}$ without an integral is an abuse of notation, and means that this result only makes sense when wrapped in an integral. Formally, $\pv{\{1 / \omega\}}$ is a [Schwartz distribution](/know/concept/schwartz-distribution/). -We thus have: -$$\begin{aligned} - \boxed{ - \tilde{\Theta}(\omega) - = \frac{A}{|s|} \Big( \pi \delta(\omega) + i \: \mathrm{sgn}(s) \pv{\frac{1}{\omega}} \Big) - } -\end{aligned}$$ |