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# Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes, or), is a discontinuous function named after Oliver Heaviside (1850–1925), whose value is zero for negative argument and one for positive argument. [1]

55 relations: Almost everywhere, Almost surely, Antiderivative, Cauchy distribution, Cauchy principal value, Closed set, Continuous function, Cumulative distribution function, Degenerate distribution, Differential equation, Dirac delta function, Distribution (mathematics), Dominated convergence theorem, Even and odd functions, Fourier transform, Function (mathematics), George F. D. Duff, Graph of a function, Hyperfunction, Indicator function, Integer, Integral, Iverson bracket, John Wiley & Sons, Kronecker delta, Laplace transform, Laplacian of the indicator, Lebesgue–Stieltjes integration, Linear combination, Logistic distribution, Logistic function, Lp space, Macaulay brackets, McGraw-Hill Education, Multivalued function, Negative number, New York City, Normal distribution, Oliver Heaviside, Open set, Operational calculus, Pointwise, Probability distribution, Ramp function, Random variable, Rectangular function, Sign function, Smoothness, Step function, Step response, ... Expand index (5 more) »

## 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.

## Almost surely

In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one.

## Antiderivative

In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

## Cauchy distribution

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.

## Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF, also cumulative density function) 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. In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

## Degenerate distribution

In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension.

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Dirac delta function

In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.

## Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.

## Dominated convergence theorem

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm.

## Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## George F. D. Duff

George Francis Denton Duff (July 29, 1926 – March 2, 2001) was a Canadian mathematician who did research in partial differential equations and wave phenomena.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Hyperfunction

In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order.

## Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Iverson bracket

In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.

## Laplace transform

In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace.

## Laplacian of the indicator

In mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can be viewed as the surface delta prime function.

## Lebesgue–Stieltjes integration

In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

## Logistic distribution

In probability theory and statistics, the logistic distribution is a continuous probability distribution.

## Logistic function

A logistic function or logistic curve is a common "S" shape (sigmoid curve), with equation: where.

## Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

## Macaulay brackets

Macaulay brackets are a notation used to describe the ramp function A popular alternative transcription uses angle brackets, viz. \langle x \rangle.

## McGraw-Hill Education

McGraw-Hill Education (MHE) is a learning science company and one of the "big three" educational publishers that provides customized educational content, software, and services for pre-K through postgraduate education.

## Multivalued function

In mathematics, a multivalued function from a domain to a codomain is a heterogeneous relation.

## Negative number

In mathematics, a negative number is a real number that is less than zero.

## New York City

The City of New York, often called New York City (NYC) or simply New York, is the most populous city in the United States.

## Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.

## Oliver Heaviside

Oliver Heaviside FRS (18 May 1850 – 3 February 1925) was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations (equivalent to Laplace transforms), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Operational calculus

Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.

## 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 &mdash; operations defined on functions by applying the operations to function values separately for each point in the domain of definition.

## Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

## Ramp function

The ramp function is a unary real function, whose graph is shaped like a ramp.

## Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

## Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as: 0 & \mbox |t| > \frac \\ \frac & \mbox |t|.

## Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Step function

In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals.

## Step response

The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions.

## Trigonometric integral

In mathematics, the trigonometric integrals are a family of integrals involving trigonometric functions.

## University of Denver

The University of Denver (DU) is a research coeducational, four-year university in Denver, Colorado.

## Variance

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

## 1

1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.

## References

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