---
tag: 09C4JP7Y0M
kind: object
name: Heaviside step function
aliases: [unit-step-function]
msc: [26A06, 44A10]
uses: [05XT5AVJBK]
sources:
  - url: https://en.wikipedia.org/wiki/Heaviside_step_function
    locator: Formulation
  - url: https://en.wikipedia.org/wiki/Heaviside_step_function
    locator: Unilateral Laplace transform
same_as:
  wikidata: Q322339
  wikipedia: 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|05XT5AVJBK]] 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|0KPHENZ5S7]].
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.