# Characteristic function

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

$\mathbf{1}_A\colon X \to \{0, 1\},$
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.
$\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}$