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 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.