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

$\mathbf {c} _{X}:S\to \{0,1\}$

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

• The characteristic function 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 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. See characteristic function (probability theory).

• The characteristic polynomial in linear algebra.

• The Euler characteristic, a topological invariant.

• The characteristic function in game theory.

In set theory, the characteristic function or indicator function of a subset A of a set X is the function, often denoted χ_A or I_A, from X to the set {0,1} which takes the value 1 on elements of A and 0 otherwise.

We can express elementary set-theoretic operations in terms of characteristic functions:

• Empty set: $\chi _{\emptyset }=0;\,$
• Intersection: $\chi _{A\cap B}=\min\{\chi _{A},\chi _{B}\}=\chi _{A}\cdot \chi _{B};\,$
• Union: $\chi _{A\cup B}=\max\{\chi _{A},\chi _{B}\}=\chi _{A}+\chi _{B}-\chi _{A}\cdot \chi _{B};\,$
• Symmetric difference: $\chi _{A\bigtriangleup B}=\chi _{A}+\chi _{B}{\pmod {2}};\,$
• Inclusion: $A\subseteq B\Leftrightarrow \chi _{A}\leq \chi _{B}.\,$