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The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
(1850–1925), the value of which is zero for negative arguments and
one 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, where it represents a signal that switches on at a specified time and stays switched on indefinitely.
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as . The Heaviside function may be defined as: * a
piecewise function In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
: H(x) := \begin 1, & x > 0 \\ 0, & x \le 0 \end * using the Iverson bracket notation: H(x) := >0/math> * an indicator function: H(x) := \mathbf_=\mathbf 1_(x) * the derivative of the ramp function: H(x) := \frac \max \\quad \mbox x \ne 0 The Dirac delta function is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the Heaviside function \delta(x)= \frac H(x) Hence the Heaviside function can be considered to be the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the Dirac delta function. This is sometimes written as H(x) := \int_^x \delta(s)\,ds although this expansion may not hold (or even make sense) for , depending on which formalism one uses to give meaning to integrals involving . In this context, the Heaviside function is the cumulative distribution function of a random variable which is
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
0. (See constant random variable.) In operational calculus, useful answers seldom depend on which value is used for , since is mostly used as a
distribution Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...
. 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
below Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred Below ...
. Approximations to the Heaviside step function are of use in
biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...
and
neuroscience Neuroscience is the science, scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, an ...
, where logistic approximations of step functions (such as the
Hill A hill is a landform that extends above the surrounding terrain. It often has a distinct summit. Terminology The distinction between a hill and a mountain is unclear and largely subjective, but a hill is universally considered to be not a ...
and the Michaelis–Menten equations) may be used to approximate binary cellular switches in response to chemical signals.


Analytic approximations

For a smooth approximation to the step function, one can use the logistic function H(x) \approx \tfrac + \tfrac\tanh kx = \frac, where a larger corresponds to a sharper transition at . If we take , equality holds in the limit: H(x)=\lim_\tfrac(1+\tanh kx)=\lim_\frac. There are many other smooth, analytic approximations to the step function. Among the possibilities are: \begin H(x) &= \lim_ \left(\tfrac + \tfrac\arctan kx\right)\\ H(x) &= \lim_\left(\tfrac + \tfrac12\operatorname kx\right) \end These limits hold
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
and in the sense of 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 convergence holds in the sense of distributions too.) In general, any cumulative distribution function of a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
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 functions of common probability distributions: the logistic, Cauchy and normal distributions, respectively.


Integral representations

Often an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
representation of the Heaviside step function is useful: \begin H(x)&=\lim_ -\frac\int_^\infty \frac e^ d\tau \\ &=\lim_ \frac\int_^\infty \frac e^ d\tau. \end 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 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 . Indeed when is considered as a
distribution Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...
or an element of (see 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 examples above) then often whatever happens to be the relevant limit at zero is used. There exist various reasons for choosing a particular value. * is often used since the graph then has rotational symmetry; put another way, is then an odd function. In this case the following relation with the sign function holds for all : H(x) = \tfrac12(1 + \sgn x). * is used when needs to be right-continuous. For instance cumulative distribution functions are usually taken to be right continuous, as are functions integrated against in Lebesgue–Stieltjes integration. In this case is the indicator function of a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
semi-infinite interval: H(x) = \mathbf_(x). The corresponding probability distribution is the degenerate distribution. * is used when needs to be
left-continuous In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
. In this case is an indicator function of an
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
semi-infinite interval: H(x) = \mathbf_(x). * In functional-analysis contexts from optimization and game theory, it is often useful to define the Heaviside function as a 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, .


Discrete form

An alternative form of the unit step, defined instead as a function (that is, taking in a discrete variable ), is: H \begin 0, & n < 0, \\ 1, & n \ge 0, \end or using the half-maximum convention: H \begin 0, & n < 0, \\ \tfrac12, & n = 0,\\ 1, & n > 0, \end where is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. If is an integer, then must imply that , while must imply that the function attains unity at . Therefore the "step function" exhibits ramp-like behavior over the domain of , and cannot authentically be a step function, using the half-maximum convention. Unlike the continuous case, the definition of is significant. The discrete-time unit impulse is the first difference of the discrete-time step \delta = H - H -1 This function is the cumulative summation of the Kronecker delta: H = \sum_^ \delta where \delta = \delta_ is the discrete unit impulse function.


Antiderivative and derivative

The ramp function is an
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
of the Heaviside step function: \int_^ H(\xi)\,d\xi = x H(x) = \max\ \,. The distributional derivative of the Heaviside step function is the Dirac delta function: \frac = \delta(x) \,.


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 \hat(s) = \lim_\int^N_ e^ H(x)\,dx = \frac \left( \delta(s) - \frac \operatorname\frac \right). Here is the
distribution Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...
that takes a test function to the Cauchy principal value of \textstyle\int_^\infty \frac \, 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: \begin \hat(s) &= \lim_\int^N_ e^ H(x)\,dx\\ &= \lim_\int^N_ e^ \,dx\\ &= \frac \end 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 H(x) = \left(1-\frac\log z,\ -\frac\log z\right). where is the principal value of the complex logarithm of . It can also be expressed for in terms of the absolute value function as H(x) = \frac \,.


See also

* 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 In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int_x^\infty\f ...
* Step response


References


External links

* Digital Library of Mathematical Functions, NIST

* * * * {{DEFAULTSORT:Heaviside Step Function Special functions Generalized functions