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In mathematics, characteristic function can refer to any of several distinct concepts: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

The most common and universal usage is as a synonym for indicator function, that is the function

$mathbf{1}_A: X to {0, 1}$

which for every subset A of X, has value 1 at points of A and 0 at points of X − A.

When applied to a natural number an effective procedure determines correctly if a natural number is or is not in the procedure's "set": "The characteristic function is the function that takes the value 1 for numbers in the set, and the value 0 for numbers not in the set" (cf Boolos-Burgess-Jeffrey (2002) p. 73).

The characteristic function in convex analysis:

$chi_{A} (x) := begin{cases} 0, & x in A; + infty, & x not in A. end{cases}$

The characteristic state function in statistical mechanics.

In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:

$varphi_X(t) = operatorname{E}left(e^{itX}right),$

where "E" means expected value. See characteristic function (probability theory).

The characteristic polynomial in linear algebra.

The Euler characteristic, a topological invariant.

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Categories: Disambiguation | Mathematical disambiguation In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. ... Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, a subdomain of optimization theory. ... The characteristic state function in statistical mechanics refers to a particular relationship between the partition function of an ensemble. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... Image File history File links Disambig_gray. ...

Results from FactBites:

Characteristic function - Wikipedia, the free encyclopedia (403 words)
	The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A.
	Characteristic functions are used in the most frequently seen proof of the central limit theorem.
	The characteristic function is closely related to the Fourier transform: the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f.

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