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

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

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

• The characteristic state function in statistical mechanics.

• 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:

$\backslash varphi\_X(t)\; =\; \backslash operatorname\{E\}\backslash left(e^\{itX\}\backslash right)\backslash ,$

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

• The characteristic polynomial in linear algebra.

• The Euler characteristic, a topological invariant.

Charakteristische Funktion (Stochastik) | Funzione caratteristica | Funkcja charakterystyczna