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

${\displaystyle \mathbf {1} _{A}:X\to \{0,1\}}$

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

• When applied to a natural number an effective procedure determines correctly if a natural number is or is not in the procedure's "set": "The characteristic function is the function that takes the value 1 for numbers in the set, and the value 0 for numbers not in the set" (cf Boolos-Burgess-Jeffrey (2002) p. 73).

• The characteristic function in convex analysis:

${\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}$

• 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:

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

where E means expected value. This concept extends to multivariate distributions. For more see characteristic function (probability).

• The characteristic polynomial in linear algebra

• The Euler characteristic, a topological invariant

• The characteristic function in game theory