Heaviside step function

Description

The Heaviside step function is the function $H : \mathbb{R} \to \mathbb{R}$ with $H(x) = 0$ for $x < 0$ and $H(x) = 1$ for $x > 0$. Conventions for the value $H(0)$ differ: common choices are $0$, $\frac{1}{2}$ and $1$. For $a \ge 0$ and $s \in \mathbb{C}$ with $\operatorname{Re} s > 0$, the Laplace transform of $t \mapsto H(t - a)$ is $e^{-a s} / s$.

Remarks

The function $t \mapsto H(t - a)$ switches a signal on at the time $a$. For example, $f(t) H(t - a)$ is $0$ before $a$ and $f(t)$ after $a$. For $g : [0, \infty) \to \mathbb{C}$ with a Laplace transform $G$ and $a \ge 0$, the shifted function $t \mapsto H(t - a)\, g(t - a)$ has the transform $e^{-a s} G(s)$. In the sense of distributions, the derivative of $H$ is the Dirac delta. It is named after Oliver Heaviside.

Explanations

L1 human

The step function is off before time $0$ and on after it. It models a switch, such as a voltage turned on at one instant.