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

• As a synonym for indicator function, that is the function

$\mathbf {1} _{A}\colon X\to \{0,1\},$

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

• 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. This concept extends to multivariate distributions.

• 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.

• The characteristic polynomial in linear algebra.

• The Euler characteristic, a topological invariant.

• The cooperative game in game theory.

• The Receiver operating characteristic in statistical decision theory.