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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Heaviside step function</span></span>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Heaviside step</th></tr><tr><td colspan="2" class="infobox-image" style="padding-bottom:0.4em;"><span typeof="mw:File"></span><div class="infobox-caption">The Heaviside step function, using the half-maximum convention</div></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">General information</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">General definition</th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}</annotation>

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</math></span></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Fields of application</th><td class="infobox-data">Operational calculus</td></tr></tbody></table>

<p>The <b>Heaviside step function</b>, or the <b>unit step function</b>, usually denoted by <span class="texhtml mvar" style="font-style:italic;">H</span> or <span class="texhtml mvar" style="font-style:italic;">θ</span> (but sometimes <span class="texhtml mvar" style="font-style:italic;">u</span>, <span class="texhtml"><b>1</b></span> or <span class="texhtml">𝟙</span>), is a <a href="Step_function" title="Step function">step function</a> named after <a href="Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a>, the value of which is <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a> for negative arguments and <a href="1_(number)" class="mw-redirect" title="1 (number)">one</a> for positive arguments. Different conventions concerning the value <span class="texhtml"><i>H</i>(0)</span> are in use. It is an example of the general class of step functions, all of which can be represented as <a href="Linear_combination" title="Linear combination">linear combinations</a> of translations of this one.

</p><p>The function was originally developed in <a href="Operational_calculus" title="Operational calculus">operational calculus</a> for the solution of <a href="Differential_equation" title="Differential equation">differential equations</a>, 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 <span class="texhtml"><b>1</b></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2></div>

<p>Taking the convention that <span class="texhtml"><i>H</i>(0) = 1</span>, the Heaviside function may be defined as:

</p>

<ul><li>A <a href="Piecewise_function" title="Piecewise function">piecewise function</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}</annotation>

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</math></span></span></li>

<li>Using the <a href="Iverson_bracket" title="Iverson bracket">Iverson bracket</a> notation: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):=[x\geq 0]}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):=[x\geq 0]}</annotation>

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</math></span></span></li>

<li>An <a href="Indicator_function" title="Indicator function">indicator function</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):=\mathbf {1} _{x\geq 0}=\mathbf {1} _{\mathbb {R} _{+}}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):=\mathbf {1} _{x\geq 0}=\mathbf {1} _{\mathbb {R} _{+}}(x)}</annotation>

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<p>For the alternative convention that <span class="texhtml"><i>H</i>(0) = <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, it may be expressed as:

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<ul><li>A <a href="Piecewise_function" title="Piecewise function">piecewise function</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):={\begin{cases}1,&x>0\\{\frac {1}{2}},&x=0\\0,&x<0\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):={\begin{cases}1,&x>0\\{\frac {1}{2}},&x=0\\0,&x<0\end{cases}}}</annotation>

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</math></span></span></li>

<li>A <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> of the <a href="Sign_function" title="Sign function">sign function</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):={\frac {1}{2}}\left({\mbox{sgn}}\,x+1\right)}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):={\frac {1}{2}}\left({\mbox{sgn}}\,x+1\right)}</annotation>

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</math></span></span></li>

<li>The <a href="Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of two <a href="Iverson_bracket" title="Iverson bracket">Iverson brackets</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):={\frac {[x\geq 0]+[x>0]}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):={\frac {[x\geq 0]+[x>0]}{2}}}</annotation>

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<li>A <a href="One-sided_limit" title="One-sided limit">one-sided limit</a> of the <a href="Atan2" title="Atan2">two-argument arctangent</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=:\lim _{\epsilon \to 0^{+}}{\frac {{\mbox{atan2}}(\epsilon ,-x)}{\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)=:\lim _{\epsilon \to 0^{+}}{\frac {{\mbox{atan2}}(\epsilon ,-x)}{\pi }}}</annotation>

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<li>A <a href="Hyperfunction" title="Hyperfunction">hyperfunction</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=:\left(1-{\frac {1}{2\pi i}}\log z,\ -{\frac {1}{2\pi i}}\log z\right)}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)=:\left(1-{\frac {1}{2\pi i}}\log z,\ -{\frac {1}{2\pi i}}\log z\right)}</annotation>

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</math></span></span> Or equivalently: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=:\left(-{\frac {\log -z}{2\pi i}},-{\frac {\log -z}{2\pi i}}\right),}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)=:\left(-{\frac {\log -z}{2\pi i}},-{\frac {\log -z}{2\pi i}}\right),}</annotation>

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</math></span></span> where <span class="texhtml">log <i>z</i></span> is the <a href="Complex_logarithm#Principal_value" title="Complex logarithm">principal value of the complex logarithm</a> of <span class="texhtml mvar" style="font-style:italic;">z</span>.</li></ul>

<p>Other definitions which are undefined at <span class="texhtml"><i>H</i>(0)</span> include:

</p>

<ul><li>A <a href="Piecewise_function" title="Piecewise function">piecewise function</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):={\begin{cases}1,&x>0\\0,&x<0\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):={\begin{cases}1,&x>0\\0,&x<0\end{cases}}}</annotation>

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<li>The derivative of the <a href="Ramp_function" title="Ramp function">ramp function</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):={\frac {d}{dx}}\max\{x,0\}\quad {\mbox{for }}x\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):={\frac {d}{dx}}\max\{x,0\}\quad {\mbox{for }}x\neq 0}</annotation>

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<li>Expressed in terms of the <a href="Absolute_value" title="Absolute value">absolute value</a> function, such as:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)={\frac {x+|x|}{2x}}}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)={\frac {x+|x|}{2x}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Relationship_with_Dirac_delta">Relationship with Dirac delta</h2></div>

<p>The <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> is the <a href="Weak_derivative" title="Weak derivative">weak derivative</a> of the Heaviside function:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)={\frac {d}{dx}}\ H(x),}">

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<annotation encoding="application/x-tex">{\displaystyle \delta (x)={\frac {d}{dx}}\ H(x),}</annotation>

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</math></span></span>Hence the Heaviside function can be considered to be the <a href="Integral" title="Integral">integral</a> of the Dirac delta function. This is sometimes written as:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x):=\int _{-\infty }^{x}\delta (s)\,ds,}">

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<annotation encoding="application/x-tex">{\displaystyle H(x):=\int _{-\infty }^{x}\delta (s)\,ds,}</annotation>

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</math></span></span>although this expansion may not hold (or even make sense) for <span class="texhtml"><i>x</i> = 0</span>, depending on which formalism one uses to give meaning to integrals involving <span class="texhtml mvar" style="font-style:italic;">δ</span>. In this context, the Heaviside function is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of a <a href="Random_variable" title="Random variable">random variable</a> which is <a href="Almost_surely" title="Almost surely">almost surely</a> 0. (See <a href="Constant_random_variable" class="mw-redirect" title="Constant random variable">Constant random variable</a>.)

</p>

<div class="mw-heading mw-heading2"><h2 id="Analytic_approximations">Analytic approximations</h2></div>

<p>Approximations to the Heaviside step function are of use in <a href="Biochemistry" title="Biochemistry">biochemistry</a> and <a href="Neuroscience" title="Neuroscience">neuroscience</a>, where <a href="Logistic_function" title="Logistic function">logistic</a> approximations of step functions (such as the <a href="Hill_equation_(biochemistry)" title="Hill equation (biochemistry)">Hill</a> and the <a href="Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten equations</a>) may be used to approximate binary cellular switches in response to chemical signals.

</p><p>For a <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> approximation to the step function, one can use the <a href="Logistic_function" title="Logistic function">logistic function</a>:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)\approx {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh kx={\frac {1}{1+e^{-2kx}}},}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)\approx {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh kx={\frac {1}{1+e^{-2kx}}},}</annotation>

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</math></span></span>where a larger <span class="texhtml mvar" style="font-style:italic;">k</span> corresponds to a sharper transition at <span class="texhtml"><i>x</i> = 0</span>.

</p><p>If we take <span class="texhtml"><i>H</i>(0) = <span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, equality holds in the limit:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=\lim _{k\to \infty }{\tfrac {1}{2}}(1+\tanh kx)=\lim _{k\to \infty }{\frac {1}{1+e^{-2kx}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)=\lim _{k\to \infty }{\tfrac {1}{2}}(1+\tanh kx)=\lim _{k\to \infty }{\frac {1}{1+e^{-2kx}}}.}</annotation>

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<p>There are <a href="Sigmoid_function#Examples" title="Sigmoid function">many other smooth, analytic approximations</a> to the step function.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Among the possibilities are:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}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}}+{\tfrac {1}{2}}\operatorname {erf} kx\right)\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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}}+{\tfrac {1}{2}}\operatorname {erf} kx\right)\end{aligned}}}</annotation>

</semantics>

</math></span></span>These limits hold <a href="Pointwise" title="Pointwise">pointwise</a> and in the sense of <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. In general, however, <a href="Pointwise_convergence" title="Pointwise convergence">pointwise convergence</a> 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 <a href="Lebesgue_dominated_convergence_theorem" class="mw-redirect" title="Lebesgue dominated convergence theorem">convergence holds in the sense of distributions too</a>.)

</p><p>In general, any <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of a <a href="Continuous_distribution" class="mw-redirect" title="Continuous distribution">continuous</a> <a href="Probability_distribution" title="Probability distribution">probability distribution</a> that is peaked around zero and has a parameter that controls for <a href="Variance" title="Variance">variance</a> can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a> of common probability distributions: the <a href="Logistic_distribution" title="Logistic distribution">logistic</a>, <a href="Cauchy_distribution" title="Cauchy distribution">Cauchy</a> and <a href="Normal_distribution" title="Normal distribution">normal</a> distributions, respectively.

</p>

<div class="mw-heading mw-heading2"><h2 id="Non-Analytic_approximations">Non-Analytic approximations</h2></div>

<p>Approximations to the Heaviside step function could be made through <a href="Non-analytic_smooth_function#Smooth_transition_functions" title="Non-analytic smooth function">Smooth transition function</a> like <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq m\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq m\to \infty }</annotation>

</semantics>

</math></span><img src="./7d6d76c0b279d0cc75ebcbedab86104209085758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.239ex; height:2.343ex;" alt="{\displaystyle 1\leq m\to \infty }" loading="lazy"></span>:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}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{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Integral_representations">Integral representations</h2></div>

<p>Often an <a href="Integration_(mathematics)" class="mw-redirect" title="Integration (mathematics)">integral</a> representation of the Heaviside step function is useful:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}H(x)&=\lim _{\varepsilon \to 0^{+}}-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau +i\varepsilon }}e^{-ix\tau }d\tau \\&=\lim _{\varepsilon \to 0^{+}}\ {\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau -i\varepsilon }}e^{ix\tau }d\tau ,\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H(x)&=\lim _{\varepsilon \to 0^{+}}-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau +i\varepsilon }}e^{-ix\tau }d\tau \\&=\lim _{\varepsilon \to 0^{+}}\ {\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau -i\varepsilon }}e^{ix\tau }d\tau ,\end{aligned}}}</annotation>

</semantics>

</math></span></span>where the second representation is easy to deduce from the first, given that the step function is real and thus is its own <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Zero_argument">Zero argument</h2></div>

<p>Since <span class="texhtml mvar" style="font-style:italic;">H</span> is usually used in <a href="Integral" title="Integral">integration</a>, and the value of a function at a single point does not affect its integral, it rarely matters what particular value is chosen of <span class="texhtml"><i>H</i>(0)</span>. Indeed when <span class="texhtml mvar" style="font-style:italic;">H</span> is considered as a <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a> or an element of <span class="texhtml"><i>L</i><span style="padding-left:0.12em;"><sup>∞</sup></span></span> (see <a href="Lp_space" title="Lp space"><span class="texhtml"><i>L<span style="padding-left:0.12em;"><sup>p</sup></span></i></span> space</a>) it does not even make sense to talk of a value at zero, since such objects are only defined <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>. If using some analytic approximation (as in the <a href="#Analytic_approximations">examples above</a>) then often whatever happens to be the relevant limit at zero is used.

</p><p>There exist various reasons for choosing a particular value.

</p>

<ul><li><span class="texhtml"><i>H</i>(0) = <span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> is often used since the <a href="Graph_of_a_function" title="Graph of a function">graph</a> then has <a href="Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a>; put another way, <span class="texhtml"><i>H</i> − <span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> is then an <a href="Odd_function" class="mw-redirect" title="Odd function">odd function</a>. In this case the following relation with the <a href="Sign_function" title="Sign function">sign function</a> holds for all <span class="texhtml mvar" style="font-style:italic;">x</span>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)={\tfrac {1}{2}}(1+\operatorname {sgn} x).}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)={\tfrac {1}{2}}(1+\operatorname {sgn} x).}</annotation>

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</math></span></span>Also, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,\ H(x)+H(-x)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall x,\ H(x)+H(-x)=1}</annotation>

</semantics>

</math></span><img src="./65ea15d19a85824193d33d95f8cab5e589534229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.551ex; height:2.843ex;" alt="{\displaystyle \forall x,\ H(x)+H(-x)=1}" loading="lazy"></span>.</li></ul>

<ul><li><span class="texhtml"><i>H</i>(0) = 1</span> is used when <span class="texhtml mvar" style="font-style:italic;">H</span> needs to be <a href="Right-continuous" class="mw-redirect" title="Right-continuous">right-continuous</a>. For instance <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a> are usually taken to be right continuous, as are functions integrated against in <a href="Lebesgue%E2%80%93Stieltjes_integration" title="Lebesgue–Stieltjes integration">Lebesgue–Stieltjes integration</a>. In this case <span class="texhtml mvar" style="font-style:italic;">H</span> is the <a href="Indicator_function" title="Indicator function">indicator function</a> of a <a href="Closed_set" title="Closed set">closed</a> semi-infinite interval: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=\mathbf {1} _{[0,\infty )}(x).}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle H(x)=\mathbf {1} _{[0,\infty )}(x).}</annotation>

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</math></span></span> The corresponding probability distribution is the <a href="Degenerate_distribution" title="Degenerate distribution">degenerate distribution</a>.</li></ul>

<ul><li><span class="texhtml"><i>H</i>(0) = 0</span> is used when <span class="texhtml mvar" style="font-style:italic;">H</span> needs to be <a href="Left-continuous" class="mw-redirect" title="Left-continuous">left-continuous</a>. In this case <span class="texhtml mvar" style="font-style:italic;">H</span> is an indicator function of an <a href="Open_set" title="Open set">open</a> semi-infinite interval: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=\mathbf {1} _{(0,\infty )}(x).}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)=\mathbf {1} _{(0,\infty )}(x).}</annotation>

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</math></span></span></li>

<li>In functional-analysis contexts from <a href="Optimization" class="mw-redirect" title="Optimization">optimization</a> and <a href="Game_theory" title="Game theory">game theory</a>, it is often useful to define the Heaviside function as a <a href="Multivalued_function" title="Multivalued function">set-valued function</a> 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, <span class="texhtml"><i>H</i>(0) = [0,1]</span>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Discrete_form">Discrete form</h2></div>

<p>An alternative form of the unit step, defined instead as a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:\mathbb {Z} \rightarrow \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H:\mathbb {Z} \rightarrow \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./aff84464ec3d7759752023a65c881cd20cca880a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.843ex; height:2.176ex;" alt="{\displaystyle H:\mathbb {Z} \rightarrow \mathbb {R} }" loading="lazy"></span> (that is, taking in a discrete variable <span class="texhtml mvar" style="font-style:italic;">n</span>), is:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H[n]={\begin{cases}0,&n<0,\\1,&n\geq 0,\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle H[n]={\begin{cases}0,&n<0,\\1,&n\geq 0,\end{cases}}}</annotation>

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</math></span></span>Or using the half-maximum convention:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H[n]={\begin{cases}0,&n<0,\\{\tfrac {1}{2}},&n=0,\\1,&n>0,\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle H[n]={\begin{cases}0,&n<0,\\{\tfrac {1}{2}},&n=0,\\1,&n>0,\end{cases}}}</annotation>

</semantics>

</math></span></span>where <span class="texhtml mvar" style="font-style:italic;">n</span> is an <a href="Integer" title="Integer">integer</a>. If <span class="texhtml mvar" style="font-style:italic;">n</span> is an integer, then <span class="texhtml"><i>n</i> < 0</span> must imply that <span class="texhtml"><i>n</i> ≤ −1</span>, while <span class="texhtml"><i>n</i> > 0</span> must imply that the function attains unity at <span class="texhtml"><i>n</i> = 1</span>. Therefore the "step function" exhibits ramp-like behavior over the domain of <span class="texhtml">[−1, 1]</span>, and cannot authentically be a step function, using the half-maximum convention.

</p><p>Unlike the continuous case, the definition of <span class="texhtml"><i>H</i>[0]</span> is significant.

</p><p>The discrete-time unit impulse is the first difference of the discrete-time step:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta [n]=H[n]-H[n-1].}">

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<annotation encoding="application/x-tex">{\displaystyle \delta [n]=H[n]-H[n-1].}</annotation>

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</math></span></span>This function is the cumulative summation of the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H[n]=\sum _{k=-\infty }^{n}\delta [k],}">

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<annotation encoding="application/x-tex">{\displaystyle H[n]=\sum _{k=-\infty }^{n}\delta [k],}</annotation>

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</math></span></span>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta [k]=\delta _{k,0}}">

<msub>

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<annotation encoding="application/x-tex">{\textstyle \delta [k]=\delta _{k,0}}</annotation>

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</math></span><img src="./4bff2d5e266c1e63a7ee23f99482ce7998b0d8eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.053ex; height:3.009ex;" alt="{\textstyle \delta [k]=\delta _{k,0}}" loading="lazy"></span> is the <a href="Degenerate_distribution" title="Degenerate distribution">discrete unit impulse function</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Antiderivative_and_derivative">Antiderivative and derivative</h2></div>

<p>The <a href="Ramp_function" title="Ramp function">ramp function</a> is an <a href="Antiderivative" title="Antiderivative">antiderivative</a> of the Heaviside step function:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{x}H(\xi )\,d\xi =xH(x)=\max\{0,x\}\,.}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{x}H(\xi )\,d\xi =xH(x)=\max\{0,x\}\,.}</annotation>

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</math></span></span>The <a href="Distributional_derivative" class="mw-redirect" title="Distributional derivative">distributional derivative</a> of the Heaviside step function is the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dH(x)}{dx}}=\delta (x)\,.}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dH(x)}{dx}}=\delta (x)\,.}</annotation>

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</p>

<div class="mw-heading mw-heading2"><h2 id="Fourier_transform">Fourier transform</h2></div>

<p>The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have

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<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}(s)=\lim _{N\to \infty }\int _{-N}^{N}e^{-2\pi ixs}H(x)\,dx={\frac {1}{2}}\left(\delta (s)-{\frac {i}{\pi }}\operatorname {p.v.} {\frac {1}{s}}\right).}</annotation>

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</math></span></span>Here <span class="texhtml">p.v.<span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>s</i></span></span>⁠</span></span> is the <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a> that takes a test function <span class="texhtml mvar" style="font-style:italic;">φ</span> to the <a href="Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{-\infty }^{\infty }{\frac {\varphi (s)}{s}}\,ds}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{-\infty }^{\infty }{\frac {\varphi (s)}{s}}\,ds}</annotation>

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</math></span><img src="./8da42a2afa2dc00d93f8c4e58898d0d058f01b65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.293ex; height:4.176ex;" alt="{\displaystyle \textstyle \int _{-\infty }^{\infty }{\frac {\varphi (s)}{s}}\,ds}" loading="lazy"></span>. The limit appearing in the integral is also taken in the sense of (tempered) distributions.

</p>

<div class="mw-heading mw-heading2"><h2 id="Unilateral_Laplace_transform">Unilateral Laplace transform</h2></div>

<p>The <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> of the Heaviside step function is a <a href="Meromorphic_function" title="Meromorphic function">meromorphic function</a>. Using the unilateral Laplace transform we have:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\hat {H}}(s)&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}H(x)\,dx\\&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}\,dx\\&={\frac {1}{s}}\end{aligned}}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {H}}(s)&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}H(x)\,dx\\&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}\,dx\\&={\frac {1}{s}}\end{aligned}}}</annotation>

</semantics>

</math></span></span>When the <a href="Laplace_transform#Bilateral_Laplace_transform" title="Laplace transform">bilateral transform</a> is used, the integral can be split in two parts and the result will be the same.

</p>

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<ul><li><a href="Gamma_function" title="Gamma function">Gamma function</a></li>

<li><a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li>

<li><a href="Indicator_function" title="Indicator function">Indicator function</a></li>

<li><a href="Iverson_bracket" title="Iverson bracket">Iverson bracket</a></li>

<li><a href="Laplace_transform" title="Laplace transform">Laplace transform</a></li>

<li><a href="Laplacian_of_the_indicator" title="Laplacian of the indicator">Laplacian of the indicator</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of mathematical functions</a></li>

<li><a href="Macaulay_brackets" title="Macaulay brackets">Macaulay brackets</a></li>

<li><a href="Negative_number" title="Negative number">Negative number</a></li>

<li><a href="Rectangular_function" title="Rectangular function">Rectangular function</a></li>

<li><a href="Sign_function" title="Sign function">Sign function</a></li>

<li><a href="Sine_integral" class="mw-redirect" title="Sine integral">Sine integral</a></li>

<li><a href="Step_response" title="Step response">Step response</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HeavisideStepFunction.html">"Heaviside Step Function"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFBracewell2000" class="citation book cs1">Bracewell, Ronald Newbold (2000). <i>The Fourier transform and its applications</i> (3rd ed.). New York: McGraw-Hill. p. 61. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-303938-1</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Heaviside_function" class="extiw external" title="commons:Category:Heaviside function">Heaviside function</a></span>.</div></div>

</div>

<ul><li>Digital Library of Mathematical Functions, NIST, <a rel="nofollow" class="external autonumber" href="http://dlmf.nist.gov/1.16#iv">[1]</a>.</li>

<li><cite id="CITEREFBerg1936" class="citation book cs1">Berg, Ernst Julius (1936). "Unit function". <i>Heaviside's Operational Calculus, as applied to Engineering and Physics</i>. <a href="McGraw-Hill_Education" class="mw-redirect" title="McGraw-Hill Education">McGraw-Hill Education</a>. p. 5.</cite></li>

<li><cite id="CITEREFCalvert2002" class="citation web cs1">Calvert, James B. (2002). <a rel="nofollow" class="external text" href="http://mysite.du.edu/~jcalvert/math/laplace.htm">"Heaviside, Laplace, and the Inversion Integral"</a>. <a href="University_of_Denver" title="University of Denver">University of Denver</a>.</cite></li>

<li><cite id="CITEREFDavies2002" class="citation book cs1">Davies, Brian (2002). "Heaviside step function". <i>Integral Transforms and their Applications</i> (3rd ed.). Springer. p. 28.</cite></li>

<li><cite id="CITEREFDuffNaylor1966" class="citation book cs1"><a href="George_F._D._Duff" title="George F. D. Duff">Duff, George F. D.</a>; Naylor, D. (1966). "Heaviside unit function". <i>Differential Equations of Applied Mathematics</i>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. p. 42.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-06" href="https://en.wikipedia.org/wiki/?title=Heaviside_step_function&oldid=1304485200">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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