Characteristic function

Characteristic function can refer to several distinct concepts:

• In one mathematical usage, the characteristic function of a subset A of a set S is the indicator function ${\displaystyle \chi _{A}:S\to \{0,1\}}$ with ___domain S which has value 1 at points of A and 0 at points of S − A.
• In another mathematical usage, this one 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:

${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right)\,}$

where "E" means expected value. See characteristic function (probability theory).