Characteristic function

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

• The indicator function, that is the function

[math]\displaystyle{ \mathbf{1}_A\colon X \to \{0, 1\}, }[/math]

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

• There is an indicator function for affine varieties over a finite field: given a finite set of functions [math]\displaystyle{ f_\alpha \in \mathbb{F}_q[x_1,\ldots,x_n] }[/math] let [math]\displaystyle{ V = \{ x \in \mathbb{F}_q^n : f_\alpha(x) = 0 \} }[/math] be their vanishing locus. Then, the function [math]\displaystyle{ P(x) = \prod(1 - f_\alpha(x)^{q-1}) }[/math] acts as an indicator function for [math]\displaystyle{ V }[/math]. If [math]\displaystyle{ x \in V }[/math] then [math]\displaystyle{ P(x) = 1 }[/math], otherwise, for some [math]\displaystyle{ f_\alpha }[/math], we have [math]\displaystyle{ f_\alpha(x) \neq 0 }[/math], which implies that [math]\displaystyle{ f_\alpha(x)^{q-1} = 1 }[/math], hence [math]\displaystyle{ P(x) = 0 }[/math].
• The characteristic function in convex analysis, closely related to the indicator function of a set:

[math]\displaystyle{ \chi_A (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases} }[/math]

• 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:

[math]\displaystyle{ \varphi_X(t) = \operatorname{E}\left(e^{itX}\right), }[/math]

where E means expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.

• The characteristic function of a cooperative game in game theory.
• The characteristic polynomial in linear algebra.
• The characteristic state function in statistical mechanics.
• The Euler characteristic, a topological invariant.
• The receiver operating characteristic in statistical decision theory.
• The point characteristic function in statistics.