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.

• 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 (convex analysis) 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.

• In probability theory, the characteristic function (probability theory) 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 polynomial in linear algebra.

• The Euler characteristic, a topological invariant.

• The cooperative game in game theory.