Characteristic function

(mathematics) The function χ_A defined for any subset A of a set by setting χ_A(x) = 1 if x is in A and χ_A = 0 if x is not in A. Also known as indicator function. eigenfunction
(physics) A function, such as the point characteristic function or the principal function, which is the integral of some property of an optical or mechanical system over time or over the path followed by the system, and whose value for a path actually followed by a system is a maximum or a minimum with respect to nearby paths with the same end points.
(statistics) A function that uniquely defines a probability distribution; it is equal to √(2π) times the Fourier transform of the frequency function of the distribution.

In mathematics, characteristic function can refer to any of several distinct concepts:

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

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:

• 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:

where E means expected value. This concept extends to multivariate distributions. For more see characteristic function (probability).

• The characteristic polynomial in linear algebra

• The Euler characteristic, a topological invariant

• The characteristic function in game theory

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