Characteristic function

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

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

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

• In probability theory, the characteristic function (probability theory) 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 (convex analysis) 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.

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.