From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- .../know/concept/heaviside-step-function/index.md | 96 ++++++++++++++++++++++ 1 file changed, 96 insertions(+) create mode 100644 source/know/concept/heaviside-step-function/index.md (limited to 'source/know/concept/heaviside-step-function') diff --git a/source/know/concept/heaviside-step-function/index.md b/source/know/concept/heaviside-step-function/index.md new file mode 100644 index 0000000..30b5f5d --- /dev/null +++ b/source/know/concept/heaviside-step-function/index.md @@ -0,0 +1,96 @@ +--- +title: "Heaviside step function" +date: 2021-02-25 +categories: +- Mathematics +- Physics +layout: "concept" +--- + +The **Heaviside step function** $\Theta(t)$, +is a discontinuous function used for enforcing causality +or for representing a signal switched on at $t = 0$. +It is defined as: + +$$\begin{aligned} + \boxed{ + \Theta(t) = + \begin{cases} + 0 & \mathrm{if}\: t < 0 \\ + 1 & \mathrm{if}\: t > 1 + \end{cases} + } +\end{aligned}$$ + +The value of $\Theta(t \!=\! 0)$ varies between definitions; +common choices are $0$, $1$ and $1/2$. +In practice, this rarely matters, and some authors even +change their definition on the fly for convenience. +For physicists, $\Theta(0) = 1$ is generally best, such that: + +$$\begin{aligned} + \boxed{ + \forall n \in \mathbb{R}: \Theta^n(t) = \Theta(t) + } +\end{aligned}$$ + +Unsurprisingly, the first-order derivative of $\Theta(t)$ is +the [Dirac delta function](/know/concept/dirac-delta-function/): + +$$\begin{aligned} + \boxed{ + \Theta'(t) = \delta(t) + } +\end{aligned}$$ + +The [Fourier transform](/know/concept/fourier-transform/) +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: + +$$\begin{aligned} + \boxed{ + \tilde{\Theta}(\omega) + = \frac{A}{|s|} \Big( \pi \delta(\omega) + i \: \mathrm{sgn}(s) \pv{\frac{1}{\omega}} \Big) + } +\end{aligned}$$ + +
+ + + +
+ +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/). + -- cgit v1.2.3