Heaviside step function: Difference between revisions

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The '''Heaviside step function''', or the '''unit step function''', usually denoted by {{mvar|H}} or {{mvar|θ}} (but sometimes {{mvar|u}}, {{math|'''1'''}} or {{math|{{not a typo|𝟙}}}}), is a [[step function]] named after [[Oliver Heaviside]], the value of which is [[0 (number)|zero]] for negative arguments and [[1 (number)|one]] for nonnegativepositive arguments.<ref name="Zhang ZhouDifferent 2021conventions pp.concerning the value 9–46">{{cite book math| last=Zhang | first=Weihong | last2=Zhou | first2=Ying | title=The Feature-Driven Method for Structural Optimization | chapter=Level-set functions and parametric functions | publisher=Elsevier | year=2021 | doi=10.1016/b978-''H''(0-12-821330-8.00002-x | pages=9–46 | quote=Heaviside function, also called the Heaviside step function, is a discontinuous function. As)}} illustratedare in Figuse. 2.13, it values zero for negative input and one for nonnegative input.}}</ref> It is an example of the general class of step functions, all of which can be represented as [[Linearlinear combination|linear combinations]]s of translations of this one.

The function was originally developed in [[operational calculus]] for the solution of [[differential equation]]s, where it represents a signal that switches on at a specified time and stays switched on indefinitely. [[Oliver Heaviside]], who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as {{math|'''1'''}}.

TheTaking the convention that {{math|''H''(0) {{=}} 0}}, the Heaviside function may be defined as:

* a [[piecewise function]]: <math display="block">H(x) := \begin{cases} 1, & x \gegeq 0 \\ 0, & x < 0 \end{cases}</math>

* using the [[Iverson bracket]] notation: <math display="block">H(x) := [x \gegeq 0]</math>

* an [[indicator function]]: <math display="block">H(x) := \mathbf{1}_{x \geq 0}=\mathbf 1_{\mathbb R_+}(x)</math>

* the derivative of the [[ramp function]]: <math display="block">H(x) := \frac{d}{dx} \max \{ x, 0 \}\quad \mbox{for } x \ne 0</math>

The [[Dirac delta function]] is the [[derivative]] of the Heaviside function :

<math display="block">\delta(x)= \frac{d}{dx} H(x).</math>

Hence the Heaviside function can be considered to be the [[integral]] of the Dirac delta function. This is sometimes written as

<math display="block">H(x) := \int_{-\infty}^x \delta(s)\,ds</math>

although this expansion may not hold (or even make sense) for {{math|''x'' {{=}} 0}}, depending on which formalism one uses to give meaning to integrals involving {{mvar|δ}}. In this context, the Heaviside function is the [[cumulative distribution function]] of a [[random variable]] which is [[almost surely]] 0. (See [[constantConstant random variable]].)

although this expansion may not hold (or even make sense) for {{math|''x'' {{=}} 0}}, depending on which formalism one uses to give meaning to integrals involving {{mvar|δ}}. In this context, the Heaviside function is the [[cumulative distribution function]] of a [[random variable]] which is [[almost surely]] 0. (See [[constant random variable]].)

In operational calculus, useful answers seldom depend on which value is used for {{math|''H''(0)}}, since {{mvar|H}} is mostly used as a [[Distribution (mathematics)|distribution]]. However, the choice may have some important consequences in functional analysis and game theory, where more general forms of continuity are considered. Some common choices can be seen [[#Zero argument|below]].

Approximations to the Heaviside step function are of use in [[biochemistry]] and [[neuroscience]], where [[logistic function|logistic]] approximations of step functions (such as the [[Hill equation (biochemistry)|Hill]] and the [[Michaelis–Menten kinetics|Michaelis–Menten equations]]) may be used to approximate binary cellular switches in response to chemical signals.