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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 ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, an ...
, the value of which is
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
for negative arguments and
one 1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sp ...
for positive arguments. Different conventions concerning the value are in use. It is an example of the general class of step functions, all of which can be represented as
linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
s of translations of this one. The function was originally developed in
operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomia ...
for the solution of differential 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 as .


Formulation

Taking the convention that , the Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x \geq 0 \\ 0, & x < 0 \end * using the Iverson bracket notation: H(x) := \geq 0/math> * an
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
: H(x) := \mathbf_=\mathbf 1_(x) For the alternative convention that , it may be expressed as: * a piecewise function: H(x) := \begin 1, & x > 0 \\ \frac, & x = 0 \\ 0, & x < 0 \end * a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
of the
sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...
, H(x) := \frac \left(\mbox\, x + 1\right) * the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
of two Iverson brackets, H(x) := \frac * a one-sided limit of the two-argument arctangent H(x) =: \lim_ \frac * a hyperfunction H(x) =: \left(1-\frac\log z,\ -\frac\log z\right) or equivalently H(x) =: \left( -\frac, -\frac\right) where is the principal value of the complex logarithm of Other definitions which are undefined at include: * a piecewise function: H(x) := \begin 1, & x > 0 \\ 0, & x < 0 \end * the derivative of the ramp function: H(x) := \frac \max \\quad \mbox x \ne 0 * in terms of the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
function as H(x) = \frac


Relationship with Dirac delta

The
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
is the weak derivative of the Heaviside function: \delta(x)= \frac H(x). Hence the Heaviside function can be considered to be the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
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 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
0. (See Constant random variable.)


Analytic approximations

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, a ...
and
neuroscience Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions, and its disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, ...
, 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, and is usually applied to peaks which are above elevation compared to the relative landmass, though not as prominent as Mountain, mountains. Hills ...
and the Michaelis–Menten equations) may be used to approximate binary cellular switches in response to chemical signals. For a smooth approximation to the step function, one can use the
logistic function A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation f(x) = \frac where The logistic function has domain the real numbers, the limit as x \to -\infty is 0, and the limit as x \to +\infty is L. ...
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 (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...
and in the sense of distributions. In general, however,
pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of a continuous
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
that is peaked around zero and has a parameter that controls for
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
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.


Non-Analytic approximations

Approximations to the Heaviside step function could be made through Smooth transition function like 1 \leq m \to \infty : \beginf(x) &= \begin , & , x, < 1 \\ \\ 1, & x \geq 1 \\ 0, & x \leq -1 \end\end


Integral representations

Often an
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
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 In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
.


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 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 In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. 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 Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
; put another way, is then an odd function. In this case the following relation with the
sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...
holds for all : H(x) = \tfrac12(1 + \sgn x). Also, H(x) + H(-x) = 1 for all x. * is used when needs to be right-continuous. For instance
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
s are usually taken to be right continuous, as are functions integrated against in Lebesgue–Stieltjes integration. In this case is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of a closed semi-infinite interval: H(x) = \mathbf_(x). The corresponding probability distribution is the
degenerate distribution In probability theory, a degenerate distribution on a measure space (E, \mathcal, \mu) is a probability distribution whose support is a null set with respect to \mu. For instance, in the -dimensional space endowed with the Lebesgue measure, an ...
. * is used when needs to be left-continuous. 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), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...
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 A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathe ...
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 H : \mathbb \rarr \mathbb (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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. 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 In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
: 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 continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
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 In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
: \frac = \delta(x) \,.


Fourier transform

The
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
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 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 In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
of the Heaviside step function is a
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the 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.


See also

*
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
*
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
*
Indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
* Iverson bracket *
Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
* Laplacian of the indicator * List of mathematical functions * Macaulay brackets *
Negative number In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...
* Rectangular function *
Sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...
*
Sine integral In mathematics, trigonometric integrals are a indexed family, family of nonelementary integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operato ...
* Step response


References


External links

* Digital Library of Mathematical Functions, NIST

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