Heaviside step function - Revision history

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2025-12-09T21:39:49Z

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	H(x) &= \lim_{k \to \infty}\left(\tfrac{1}{2} + \tfrac12\operatorname{erf} kx\right)		H(x) &= \lim_{k \to \infty}\left(\tfrac{1}{2} + \tfrac12\operatorname{erf} kx\right)
	\end{align}</math>These limits hold [[pointwise]] and in the sense of [[distribution (mathematics)\|distributions]]. In general, however, [[pointwise convergence]] need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then [[Lebesgue dominated convergence theorem\|convergence holds in the sense of distributions too]].)		\end{align}</math>These limits hold [[pointwise]] and in the sense of [[distribution (mathematics)\|distributions]]. In general, however, [[pointwise convergence]] need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then [[Lebesgue dominated convergence theorem\|convergence holds in the sense of distributions too]].)

			One could also use a scaled and shifted [[Sigmoid function]].

	In general, any [[cumulative distribution function]] of a [[continuous distribution\|continuous]] [[probability distribution]] that is peaked around zero and has a parameter that controls for [[variance]] can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are [[cumulative distribution function\|cumulative distribution functions]] of common probability distributions: the [[logistic distribution\|logistic]], [[Cauchy distribution\|Cauchy]] and [[normal distribution\|normal]] distributions, respectively.		In general, any [[cumulative distribution function]] of a [[continuous distribution\|continuous]] [[probability distribution]] that is peaked around zero and has a parameter that controls for [[variance]] can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are [[cumulative distribution function\|cumulative distribution functions]] of common probability distributions: the [[logistic distribution\|logistic]], [[Cauchy distribution\|Cauchy]] and [[normal distribution\|normal]] distributions, respectively.

imported>JJMC89 bot III: Removing :Category:Eponymous functions per Wikipedia:Categories for discussion/Log/2025 October 27#Eponyms in mathematics round 2

2025-11-09T22:42:14Z

Removing Category:Eponymous functions per Wikipedia:Categories for discussion/Log/2025 October 27#Eponyms in mathematics round 2

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	\end{align}</math>		\end{align}</math>

	These limits hold [[pointwise]] and in the sense of [[distribution (mathematics)\|distributions]]. In general, however, pointwise convergence need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then [[Lebesgue dominated convergence theorem\|convergence holds in the sense of distributions too]].)		These limits hold [[pointwise]] and in the sense of [[distribution (mathematics)\|distributions]]. In general, however, [[pointwise convergence]] need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then [[Lebesgue dominated convergence theorem\|convergence holds in the sense of distributions too]].)

	In general, any [[cumulative distribution function]] of a [[continuous distribution\|continuous]] [[probability distribution]] that is peaked around zero and has a parameter that controls for [[variance]] can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are [[cumulative distribution function\|cumulative distribution functions]] of common probability distributions: the [[logistic distribution\|logistic]], [[Cauchy distribution\|Cauchy]] and [[normal distribution\|normal]] distributions, respectively.		In general, any [[cumulative distribution function]] of a [[continuous distribution\|continuous]] [[probability distribution]] that is peaked around zero and has a parameter that controls for [[variance]] can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are [[cumulative distribution function\|cumulative distribution functions]] of common probability distributions: the [[logistic distribution\|logistic]], [[Cauchy distribution\|Cauchy]] and [[normal distribution\|normal]] distributions, respectively.
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	\end{align}</math>		\end{align}</math>

	where the second representation is easy to deduce from the first, given that the step function is real and thus is its own complex conjugate.		where the second representation is easy to deduce from the first, given that the step function is real and thus is its own [[complex conjugate]].

	== Zero argument ==		== Zero argument ==
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	There exist various reasons for choosing a particular value.		There exist various reasons for choosing a particular value.
	* {{math\|''H''(0) {{=}} {{sfrac\|1\|2}}}} is often used since the [[graph of a function\|graph]] then has rotational symmetry; put another way, {{math\|''H'' − {{sfrac\|1\|2}}}} is then an [[odd function]]. In this case the following relation with the [[sign function]] holds for all {{mvar\|x}}: <math display="block"> H(x) = \tfrac12(1 + \sgn x).</math>		* {{math\|''H''(0) {{=}} {{sfrac\|1\|2}}}} is often used since the [[graph of a function\|graph]] then has [[rotational symmetry]]; put another way, {{math\|''H'' − {{sfrac\|1\|2}}}} is then an [[odd function]]. In this case the following relation with the [[sign function]] holds for all {{mvar\|x}}: <math display="block"> H(x) = \tfrac12(1 + \sgn x).</math>
	Also, H(x) + H(-x) = 1 for all x.		Also, H(x) + H(-x) = 1 for all x.
	* {{math\|''H''(0) {{=}} 1}} is used when {{mvar\|H}} needs to be [[right-continuous]]. For instance [[cumulative distribution function]]s are usually taken to be right continuous, as are functions integrated against in [[Lebesgue–Stieltjes integration]]. In this case {{mvar\|H}} is the [[indicator function]] of a [[closed set\|closed]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{[0,\infty)}(x).</math> The corresponding probability distribution is the [[degenerate distribution]].		* {{math\|''H''(0) {{=}} 1}} is used when {{mvar\|H}} needs to be [[right-continuous]]. For instance [[cumulative distribution function]]s are usually taken to be right continuous, as are functions integrated against in [[Lebesgue–Stieltjes integration]]. In this case {{mvar\|H}} is the [[indicator function]] of a [[closed set\|closed]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{[0,\infty)}(x).</math> The corresponding probability distribution is the [[degenerate distribution]].

imported>Zvavybir: /* Formulation */ Added the piece-wise definition also to the other two kinds of Heaviside function

2025-04-25T22:18:03Z

Formulation: Added the piece-wise definition also to the other two kinds of Heaviside function

New page

{{Short description|Indicator function of positive numbers}}
{{refimprove|date=December 2012}}
{{Infobox mathematical function
| name = Heaviside step
| image = Dirac distribution CDF.svg
| imagesize = 325px
| caption = The Heaviside step function, using the half-maximum convention
| general_definition = <math display="block">H(x) := \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}</math>{{dubious|reason=This definition is not "general" because it adopts the H(0)=1 convention (hence disregarding other conventions).|date=August 2024}}
| fields_of_application = Operational calculus
}}

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 positive arguments. Different conventions concerning the value {{math|''H''(0)}} are in use. It is an example of the general class of step functions, all of which can be represented as [[linear combination]]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. Heaviside developed the operational calculus as a tool in the analysis of telegraphic communications and represented the function as {{math|'''1'''}}.

==Formulation==
Taking the convention that {{math|''H''(0) {{=}} 1}}, the Heaviside function may be defined as:
* a [[piecewise function]]: <math display="block">H(x) := \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}</math>
* using the [[Iverson bracket]] notation: <math display="block">H(x) := [x \geq 0]</math>
* an [[indicator function]]: <math display="block">H(x) := \mathbf{1}_{x \geq 0}=\mathbf 1_{\mathbb R_+}(x)</math>

For the alternative convention that {{math|''H''(0) {{=}} {{sfrac|1|2}}}}, it may be expressed as:
* a [[piecewise function]]: <math display="block">H(x) := \begin{cases} 1, & x > 0 \\ \frac{1}{2}, & x = 0 \\ 0, & x < 0 \end{cases}</math>
* a [[linear transformation]] of the [[sign function]], <math display="block">H(x) := \frac{1}{2} \left(\mbox{sgn}\, x + 1\right)</math>
* the [[arithmetic mean]] of two [[Iverson bracket]]s, <math display="block">H(x) := \frac{[x\geq 0] + [x>0]}{2}</math>
* a [[one-sided limit]] of the [[atan2|two-argument arctangent]] <math display="block">H(x) =: \lim_{\epsilon\to0^{+}} \frac{\mbox{atan2}(\epsilon,-x)}{\pi}</math>
* a [[hyperfunction]] <math display="block">H(x) =: \left(1-\frac{1}{2\pi i}\log z,\ -\frac{1}{2\pi i}\log z\right)</math> or equivalently <math display="block">H(x) =: \left( -\frac{\log -z}{2\pi i}, -\frac{\log -z}{2\pi i}\right)</math> where {{math|log ''z''}} is the [[Complex logarithm#Principal value|principal value of the complex logarithm]] of {{mvar|z}}

Other definitions which are undefined at {{math|''H''(0)}} include:
* a [[piecewise function]]: <math display="block">H(x) := \begin{cases} 1, & x > 0 \\ 0, & x < 0 \end{cases}</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>
* in terms of the [[absolute value]] function as
<math display="block"> H(x) = \frac{x + |x|}{2x}</math>

==Relationship with Dirac delta==
The [[Dirac delta function]] is the [[weak 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 [[Constant random variable]].)

== Analytic approximations ==
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.

[[File:Step function approximation.png|alt=A set of functions that successively approach the step function|thumb|500x500px|<math>\tfrac{1}{2} + \tfrac{1}{2} \tanh(kx) = \frac{1}{1+e^{-2kx}}</math><br>approaches the step function as {{math|''k'' → ∞}}.]]
For a [[Smooth function|smooth]] approximation to the step function, one can use the [[logistic function]]
<math display="block">H(x) \approx \tfrac{1}{2} + \tfrac{1}{2}\tanh kx = \frac{1}{1+e^{-2kx}},</math>

where a larger {{mvar|k}} corresponds to a sharper transition at {{math|''x'' {{=}} 0}}. If we take {{math|''H''(0) {{=}} {{sfrac|1|2}}}}, equality holds in the limit:
<math display="block">H(x)=\lim_{k \to \infty}\tfrac{1}{2}(1+\tanh kx)=\lim_{k \to \infty}\frac{1}{1+e^{-2kx}}.</math>

There are [[Sigmoid function#Examples|many other smooth, analytic approximations]] to the step function.<ref>{{MathWorld | urlname=HeavisideStepFunction | title=Heaviside Step Function}}</ref> Among the possibilities are:
<math display="block">\begin{align}
H(x) &= \lim_{k \to \infty} \left(\tfrac{1}{2} + \tfrac{1}{\pi}\arctan kx\right)\\
H(x) &= \lim_{k \to \infty}\left(\tfrac{1}{2} + \tfrac12\operatorname{erf} kx\right)
\end{align}</math>

These limits hold [[pointwise]] and in the sense of [[distribution (mathematics)|distributions]]. In general, however, pointwise convergence need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then [[Lebesgue dominated convergence theorem|convergence holds in the sense of distributions too]].)

In general, any [[cumulative distribution function]] of a [[continuous distribution|continuous]] [[probability distribution]] that is peaked around zero and has a parameter that controls for [[variance]] can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are [[cumulative distribution function|cumulative distribution functions]] of common probability distributions: the [[logistic distribution|logistic]], [[Cauchy distribution|Cauchy]] and [[normal distribution|normal]] distributions, respectively.

== Non-Analytic approximations ==
Approximations to the Heaviside step function could be made through [[Non-analytic_smooth_function#Smooth_transition_functions|Smooth transition function]] like <math> 1 \leq m \to \infty </math>:
<math display="block">\begin{align}f(x) &= \begin{cases}
{\displaystyle
\frac{1}{2}\left(1+\tanh\left(m\frac{2x}{1-x^2}\right)\right)}, & |x| < 1 \\
\\
1, & x \geq 1 \\
0, & x \leq -1
\end{cases}\end{align}</math>

==Integral representations==
Often an [[integration (mathematics)|integral]] representation of the Heaviside step function is useful:
<math display="block">\begin{align}
H(x)&=\lim_{ \varepsilon \to 0^+} -\frac{1}{2\pi i}\int_{-\infty}^\infty \frac{1}{\tau+i\varepsilon} e^{-i x \tau} d\tau \\
&=\lim_{ \varepsilon \to 0^+} \frac{1}{2\pi i}\int_{-\infty}^\infty \frac{1}{\tau-i\varepsilon} e^{i x \tau} d\tau.
\end{align}</math>

where the second representation is easy to deduce from the first, given that the step function is real and thus is its own complex conjugate.

== Zero argument ==
Since {{mvar|H}} is usually used in integration, and the value of a function at a single point does not affect its integral, it rarely matters what particular value is chosen of {{math|''H''(0)}}. Indeed when {{mvar|H}} is considered as a [[distribution (mathematics)|distribution]] or an element of {{math|''L''{{isup|∞}}}} (see [[Lp space|{{math|''L{{isup|p}}''}} space]]) it does not even make sense to talk of a value at zero, since such objects are only defined [[almost everywhere]]. If using some analytic approximation (as in the [[#Analytic approximations|examples above]]) then often whatever happens to be the relevant limit at zero is used.

There exist various reasons for choosing a particular value.
* {{math|''H''(0) {{=}} {{sfrac|1|2}}}} is often used since the [[graph of a function|graph]] then has rotational symmetry; put another way, {{math|''H'' − {{sfrac|1|2}}}} is then an [[odd function]]. In this case the following relation with the [[sign function]] holds for all {{mvar|x}}: <math display="block"> H(x) = \tfrac12(1 + \sgn x).</math>
Also, H(x) + H(-x) = 1 for all x.
* {{math|''H''(0) {{=}} 1}} is used when {{mvar|H}} needs to be [[right-continuous]]. For instance [[cumulative distribution function]]s are usually taken to be right continuous, as are functions integrated against in [[Lebesgue–Stieltjes integration]]. In this case {{mvar|H}} is the [[indicator function]] of a [[closed set|closed]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{[0,\infty)}(x).</math> The corresponding probability distribution is the [[degenerate distribution]].
* {{math|''H''(0) {{=}} 0}} is used when {{mvar|H}} needs to be [[left-continuous]]. In this case {{mvar|H}} is an indicator function of an [[open set|open]] semi-infinite interval: <math display="block"> H(x) = \mathbf{1}_{(0,\infty)}(x).</math>
* In functional-analysis contexts from optimization and game theory, it is often useful to define the Heaviside function as a [[Multivalued function|set-valued function]] to preserve the continuity of the limiting functions and ensure the existence of certain solutions. In these cases, the Heaviside function returns a whole interval of possible solutions, {{math|''H''(0) {{=}} [0,1]}}.

==Discrete form==

An alternative form of the unit step, defined instead as a function <math>H : \mathbb{Z} \rarr \mathbb{R}</math> (that is, taking in a discrete variable {{mvar|n}}), is:

<math display="block">H[n]=\begin{cases} 0, & n < 0, \\ 1, & n \ge 0, \end{cases} </math>

or using the half-maximum convention:<ref>{{cite book |last=Bracewell |first=Ronald Newbold |date=2000 |title=The Fourier transform and its applications |language=en |location=New York |publisher=McGraw-Hill |isbn=0-07-303938-1 |page=61 |edition=3rd}}</ref>

<math display="block">H[n]=\begin{cases} 0, & n < 0, \\ \tfrac12, & n = 0,\\ 1, & n > 0, \end{cases} </math>

where {{mvar|n}} is an [[integer]]. If {{mvar|n}} is an integer, then {{math|''n'' < 0}} must imply that {{math|''n'' ≤ −1}}, while {{math|''n'' > 0}} must imply that the function attains unity at {{math|1=''n'' = 1}}. Therefore the "step function" exhibits ramp-like behavior over the domain of {{closed-closed|−1, 1}}, and cannot authentically be a step function, using the half-maximum convention.

Unlike the continuous case, the definition of {{math|''H''[0]}} is significant.

The discrete-time unit impulse is the first difference of the discrete-time step

<math display="block"> \delta[n] = H[n] - H[n-1].</math>

This function is the cumulative summation of the [[Kronecker delta]]:

<math display="block"> H[n] = \sum_{k=-\infty}^{n} \delta[k] </math>

where

<math display="block"> \delta[k] = \delta_{k,0} </math>

is the [[degenerate distribution|discrete unit impulse function]].

== Antiderivative and derivative==
The [[ramp function]] is an [[antiderivative]] of the Heaviside step function:
<math display="block">\int_{-\infty}^{x} H(\xi)\,d\xi = x H(x) = \max\{0,x\} \,.</math>

The [[distributional derivative]] of the Heaviside step function is the [[Dirac delta function]]:
<math display="block"> \frac{d H(x)}{dx} = \delta(x) \,.</math>

== Fourier transform ==
The [[Fourier transform]] of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have
<math display="block">\hat{H}(s) = \lim_{N\to\infty}\int^N_{-N} e^{-2\pi i x s} H(x)\,dx = \frac{1}{2} \left( \delta(s) - \frac{i}{\pi} \operatorname{p.v.}\frac{1}{s} \right).</math>

Here {{math|p.v.{{sfrac|1|''s''}}}} is the [[distribution (mathematics)|distribution]] that takes a test function {{mvar|φ}} to the [[Cauchy principal value]] of <math>\textstyle\int_{-\infty}^\infty \frac{\varphi(s)}{s} \, ds</math>. The limit appearing in the integral is also taken in the sense of (tempered) distributions.

== Unilateral Laplace transform ==

The [[Laplace transform]] of the Heaviside step function is a [[meromorphic function]]. Using the unilateral Laplace transform we have:
<math display="block">\begin{align}
\hat{H}(s) &= \lim_{N\to\infty}\int^N_{0} e^{-sx} H(x)\,dx\\
&= \lim_{N\to\infty}\int^N_{0} e^{-sx} \,dx\\
&= \frac{1}{s} \end{align}</math>

When the bilateral transform is used, the integral can be split in two parts and the result will be the same.

==See also==
{{Div col|colwidth=25em}}
* [[Gamma function]]
* [[Dirac delta function]]
* [[Indicator function]]
* [[Iverson bracket]]
* [[Laplace transform]]
* [[Laplacian of the indicator]]
* [[List of mathematical functions]]
* [[Macaulay brackets]]
* [[Negative number]]
* [[Rectangular function]]
* [[Sign function]]
* [[Sine integral]]
* [[Step response]]
{{Div col end}}

==References==
{{Reflist}}
== External links ==
{{commons category|Heaviside function}}

* Digital Library of Mathematical Functions, NIST, [http://dlmf.nist.gov/1.16#iv].
*{{cite book |first=Ernst Julius |last=Berg |year=1936 |title=Heaviside's Operational Calculus, as applied to Engineering and Physics |chapter=Unit function |page=5 |publisher=[[McGraw-Hill Education]] }}
*{{cite web |first=James B. |last=Calvert |year=2002 |url=http://mysite.du.edu/~jcalvert/math/laplace.htm |title=Heaviside, Laplace, and the Inversion Integral |publisher=[[University of Denver]] }}
*{{cite book |first=Brian |last=Davies |year=2002 |title=Integral Transforms and their Applications |edition=3rd |page=28 |chapter=Heaviside step function |publisher=Springer }}
*{{cite book |first1=George F. D. |last1=Duff |author-link=George F. D. Duff |first2=D. |last2=Naylor |year=1966 |title=Differential Equations of Applied Mathematics |page=42 |chapter=Heaviside unit function |publisher=[[John Wiley & Sons]] }}

{{DEFAULTSORT:Heaviside Step Function}}
[[Category:Special functions]]
[[Category:Generalized functions]]
[[Category:Schwartz distributions]]