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Heaviside step function

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Indicator function of positive numbers

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Heaviside step
The Heaviside step function, using the half-maximum convention
General information
General definition	$H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}$ ^{[dubious –discuss]}
Fields of application	Operational calculus

TheHeaviside step function, or theunit step function, usually denoted byH orθ (but sometimesu,1 or𝟙), is astep function named afterOliver Heaviside, the value of which iszero for negative arguments andone for positive arguments. Different conventions concerning the valueH(0) are in use. It is an example of the general class of step functions, all of which can be represented aslinear combinations of translations of this one.

The function was originally developed inoperational calculus for the solution ofdifferential equations, 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 as1.

Formulation

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Taking the convention thatH(0) = 1, the Heaviside function may be defined as:

Apiecewise function: $H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}$
Using theIverson bracket notation: $H(x):=[x\geq 0]$
Anindicator function: $H(x):=\mathbf {1} _{x\geq 0}=\mathbf {1} _{\mathbb {R} _{+}}(x)$

For the alternative convention thatH(0) =⁠1/2⁠, it may be expressed as:

Apiecewise function: $H(x):={\begin{cases}1,&x>0\\{\frac {1}{2}},&x=0\\0,&x<0\end{cases}}$
Alinear transformation of thesign function: $H(x):={\frac {1}{2}}\left({\mbox{sgn}}\,x+1\right)$
Thearithmetic mean of twoIverson brackets: $H(x):={\frac {[x\geq 0]+[x>0]}{2}}$
Aone-sided limit of thetwo-argument arctangent: $H(x)=:\lim _{\epsilon \to 0^{+}}{\frac {{\mbox{atan2}}(\epsilon ,-x)}{\pi }}$
Ahyperfunction: $H(x)=:\left(1-{\frac {1}{2\pi i}}\log z,\ -{\frac {1}{2\pi i}}\log z\right)$ Or equivalently: $H(x)=:\left(-{\frac {\log -z}{2\pi i}},-{\frac {\log -z}{2\pi i}}\right),$ wherelogz is theprincipal value of the complex logarithm ofz.

Other definitions which are undefined atH(0) include:

Apiecewise function: $H(x):={\begin{cases}1,&x>0\\0,&x<0\end{cases}}$
The derivative of theramp function: $H(x):={\frac {d}{dx}}\max\{x,0\}\quad {\mbox{for }}x\neq 0$
Expressed in terms of theabsolute value function, such as: $H(x)={\frac {x+|x|}{2x}}$

Relationship with Dirac delta

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TheDirac delta function is theweak derivative of the Heaviside function: $\delta (x)={\frac {d}{dx}}\ H(x),$ Hence the Heaviside function can be considered to be theintegral of the Dirac delta function. This is sometimes written as: $H(x):=\int _{-\infty }^{x}\delta (s)\,ds,$ although this expansion may not hold (or even make sense) forx = 0, depending on which formalism one uses to give meaning to integrals involvingδ. In this context, the Heaviside function is thecumulative distribution function of arandom variable which isalmost surely 0. (SeeConstant random variable.)

Analytic approximations

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Approximations to the Heaviside step function are of use inbiochemistry andneuroscience, wherelogistic approximations of step functions (such as theHill and theMichaelis–Menten equations) may be used to approximate binary cellular switches in response to chemical signals.

For asmooth approximation to the step function, one can use thelogistic function: $H(x)\approx {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh kx={\frac {1}{1+e^{-2kx}}},$ where a largerk corresponds to a sharper transition atx = 0.

If we takeH(0) =⁠1/2⁠, equality holds in the limit: $H(x)=\lim _{k\to \infty }{\tfrac {1}{2}}(1+\tanh kx)=\lim _{k\to \infty }{\frac {1}{1+e^{-2kx}}}.$

A set of functions that successively approach the step function — ${\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh(kx)={\frac {1}{1+e^{-2kx}}}$
approaches the step function ask → ∞.

{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh(kx)={\frac {1}{1+e^{-2kx}}}} — ${\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh(kx)={\frac {1}{1+e^{-2kx}}}$
approaches the step function ask → ∞.

There aremany other smooth, analytic approximations to the step function.^[1] Among the possibilities are: ${\begin{aligned}H(x)&=\lim _{k\to \infty }\left({\tfrac {1}{2}}+{\tfrac {1}{\pi }}\arctan kx\right)\\H(x)&=\lim _{k\to \infty }\left({\tfrac {1}{2}}+{\tfrac {1}{2}}\operatorname {erf} kx\right)\end{aligned}}$ These limits holdpointwise and in the sense ofdistributions. 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, thenconvergence holds in the sense of distributions too.)

One could also use a scaled and shiftedSigmoid function.

In general, anycumulative distribution function of acontinuous probability distribution that is peaked around zero and has a parameter that controls forvariance can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations arecumulative distribution functions of common probability distributions: thelogistic,Cauchy andnormal distributions, respectively.

Non-Analytic approximations

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Approximations to the Heaviside step function could be made throughSmooth transition function like $1\leq m\to \infty$ : ${\begin{aligned}f(x)&={\begin{cases}{\displaystyle {\frac {1}{2}}\left(1+\tanh \left(m{\frac {2x}{1-x^{2}}}\right)\right)},&|x|<1\\\\1,&x\geq 1\\0,&x\leq -1\end{cases}}\end{aligned}}$

Integral representations

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Often anintegral representation of the Heaviside step function is useful: ${\begin{aligned}H(x)&=\lim _{\varepsilon \to 0^{+}}-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau +i\varepsilon }}e^{-ix\tau }d\tau \\&=\lim _{\varepsilon \to 0^{+}}\ {\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau -i\varepsilon }}e^{ix\tau }d\tau ,\end{aligned}}$ where the second representation is easy to deduce from the first, given that the step function is real and thus is its owncomplex conjugate.

Zero argument

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SinceH is usually used inintegration, and the value of a function at a single point does not affect its integral, it rarely matters what particular value is chosen ofH(0). Indeed whenH is considered as adistribution or an element ofL^∞ (seeL^p space) it does not even make sense to talk of a value at zero, since such objects are only definedalmost everywhere. If using some analytic approximation (as in theexamples above) then often whatever happens to be the relevant limit at zero is used.

There exist various reasons for choosing a particular value.

H(0) =⁠1/2⁠ is often used since thegraph then hasrotational symmetry; put another way,H −⁠1/2⁠ is then anodd function. In this case the following relation with thesign function holds for allx: $H(x)={\tfrac {1}{2}}(1+\operatorname {sgn} x).$ Also, $\forall x,\ H(x)+H(-x)=1$ .

H(0) = 1 is used whenH needs to beright-continuous. For instancecumulative distribution functions are usually taken to be right continuous, as are functions integrated against inLebesgue–Stieltjes integration. In this caseH is theindicator function of aclosed semi-infinite interval: $H(x)=\mathbf {1} _{[0,\infty )}(x).$ The corresponding probability distribution is thedegenerate distribution.

H(0) = 0 is used whenH needs to beleft-continuous. In this caseH is an indicator function of anopen semi-infinite interval: $H(x)=\mathbf {1} _{(0,\infty )}(x).$
In functional-analysis contexts fromoptimization andgame theory, it is often useful to define the Heaviside function as aset-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,H(0) = [0,1].

Discrete form

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An alternative form of the unit step, defined instead as a function $H:\mathbb {Z} \rightarrow \mathbb {R}$ (that is, taking in a discrete variablen), is: $H[n]={\begin{cases}0,&n<0,\\1,&n\geq 0,\end{cases}}$ Or using the half-maximum convention:^[2] $H[n]={\begin{cases}0,&n<0,\\{\tfrac {1}{2}},&n=0,\\1,&n>0,\end{cases}}$ wheren is aninteger. Ifn is an integer, thenn < 0 must imply thatn ≤ −1, whilen > 0 must imply that the function attains unity atn = 1. Therefore the "step function" exhibits ramp-like behavior over the domain of[−1, 1], and cannot authentically be a step function, using the half-maximum convention.

Unlike the continuous case, the definition ofH[0] is significant.

The discrete-time unit impulse is the first difference of the discrete-time step: $\delta [n]=H[n]-H[n-1].$ This function is the cumulative summation of theKronecker delta: $H[n]=\sum _{k=-\infty }^{n}\delta [k],$ where ${\textstyle \delta [k]=\delta _{k,0}}$ is thediscrete unit impulse function.

Antiderivative and derivative

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Theramp function is anantiderivative of the Heaviside step function: $\int _{-\infty }^{x}H(\xi )\,d\xi =xH(x)=\max\{0,x\}\,.$ Thedistributional derivative of the Heaviside step function is theDirac delta function: ${\frac {dH(x)}{dx}}=\delta (x)\,.$

Fourier transform

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TheFourier transform of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have ${\hat {H}}(s)=\lim _{N\to \infty }\int _{-N}^{N}e^{-2\pi ixs}H(x)\,dx={\frac {1}{2}}\left(\delta (s)-{\frac {i}{\pi }}\operatorname {p.v.} {\frac {1}{s}}\right).$ Herep.v.⁠1/s⁠ is thedistribution that takes a test functionφ to theCauchy principal value of $\textstyle \int _{-\infty }^{\infty }{\frac {\varphi (s)}{s}}\,ds$ . The limit appearing in the integral is also taken in the sense of (tempered) distributions.

Unilateral Laplace transform

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TheLaplace transform of the Heaviside step function is ameromorphic function. Using the unilateral Laplace transform we have: ${\begin{aligned}{\hat {H}}(s)&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}H(x)\,dx\\&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}\,dx\\&={\frac {1}{s}}\end{aligned}}$ When thebilateral transform is used, the integral can be split in two parts and the result will be the same.

References

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^Weisstein, Eric W."Heaviside Step Function".MathWorld.
^Bracewell, Ronald Newbold (2000).The Fourier transform and its applications (3rd ed.). New York: McGraw-Hill. p. 61.ISBN 0-07-303938-1.

External links

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Wikimedia Commons has media related toHeaviside function.

Digital Library of Mathematical Functions, NIST,[1].
Berg, Ernst Julius (1936). "Unit function".Heaviside's Operational Calculus, as applied to Engineering and Physics.McGraw-Hill Education. p. 5.
Calvert, James B. (2002)."Heaviside, Laplace, and the Inversion Integral".University of Denver.
Davies, Brian (2002). "Heaviside step function".Integral Transforms and their Applications (3rd ed.). Springer. p. 28.
Duff, George F. D.; Naylor, D. (1966). "Heaviside unit function".Differential Equations of Applied Mathematics.John Wiley & Sons. p. 42.