Heaviside step function: Difference between revisions

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{{Short description|Indicator function whose value is zero for negative numbers and one forof positive numbers}}

{{refimprove|date=December 2012}}

{{Infobox mathematical function

[[Image:Dirac distribution CDF.svg|325px|thumb|The Heaviside step function, using the half-maximum convention]]

| 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]] (1850–1925), 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 function]]sfunctions, 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, and represented the function as {{math|'''1'''}}.

==Formulation==

The Heaviside function may be defined as:

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 > 0 \\ 0, & x \le 0 \end{cases}</math>

* anA [[indicatorpiecewise function]]: <math display="block">H(x) := \mathbfbegin{1cases}_{ 1, & x >\geq 0 \\ 0, & x < 0 \end{cases}</math>

* the derivative ofUsing the [[rampIverson functionbracket]] notation: <math display="block">H(x) := \frac{d}{dx} \max \{ [x, 0 \}\quad \mbox{for } x \negeq 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:

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

* A [[piecewise function]]: <math display="block">\deltaH(x) := \begin{cases} 1, & x > 0 \\ \frac{d1}{dx2}, & H(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 \to 0^{+}} \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:

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

* A [[piecewise function]]: <math display="block">H(x) := \int_begin{-\inftycases}^ 1, & x > 0 \delta(s)\ 0,ds & 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>

* Expressed in terms of the [[absolute value]] function, such as:<math display="block"> H(x) = \frac{x + |x|}{2x}</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]].)

==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]].)

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]].

== 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.

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}}.

== Analytic approximations ==

[[File:Step function approximation.png|alt=A set of functions that successively approach the step function|thumb|500x500px|<math>\frac{1}{2} + \frac{1}{2} \tanh(kx) = \frac{1}{1+e^{-2kx}}</math> approaches the step function as <math>k \rightarrow \infty</math>.]]

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>

<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>

[[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'' → ∞}}.|none]]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}

<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]].)

\end{align}</math>

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.

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]].)

== Non-Analytic approximations ==

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.

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}

<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]].

\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 [[Integral|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 + \tfrac12\sgn (x).</math>Also, <math> \forall x, \ H(x) + H(-x) = 1</math>.

* {{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'' ≤ &minus;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|&minus;1, 1}}, and cannot authentically be a step function, using the half-maximum convention.

An alternative form of the unit step, defined instead as a function <math>H: \mathbb{Z} \to \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'' ≤ &minus;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|&minus;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="inline"> \delta[k] = \delta_{k,0} </math> is the [[degenerate distribution|discrete unit impulse function]].

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>

<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.

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|{{intmath|int|−∞|∞}} {{sfrac|''φ''(''s'')|''s''}} ''ds''}}. 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}

<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 [[Laplace transform#Bilateral Laplace transform|bilateral transform]] is used, the integral can be split in two parts and the result will be the same.

&= \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.

==Other expressions==

The Heaviside step function can be represented as a [[hyperfunction]] as

<math display="block">H(x) = \left(1-\frac{1}{2\pi i}\log(z),\ -\frac{1}{2\pi i}\log(z)\right).</math>

where <math>\log(z)</math> is the [[Complex logarithm#Principal value|principal value of the complex logarithm]] of {{mvar|z}}.

It can also be expressed for <math>x \ne 0</math> in terms of the [[absolute value]] function as

<math display="block"> H(x) = \frac{x + |x|}{2x} \,.</math>

==See also==

{{Div col|colwidth=25em}}

* [[Gamma function]]

* [[Dirac delta function]]

* [[Indicator function]]

* [[RectangularIverson functionbracket]]

* [[Step response]]

* [[Sign function]]

* [[Negative number]]

* [[Laplace transform]]

* [[Iverson bracket]]

* [[Laplacian of the indicator]]

* [[List of mathematical functions]]

* [[Macaulay brackets]]

* [[Negative number]]

* [[Rectangular function]]

* [[Sign function]]

* [[Sine integral]]

* [[DiracStep delta functionresponse]]

{{Div col end}}

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[[Category:Special functions]]

[[Category:Generalized functions]]

[[Category:Schwartz distributions]]