In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

• The indicator function of a subset, that is the function 1 A : X → { 0 , 1 } , {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},} which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
• The characteristic function in convex analysis, closely related to the indicator function of a set: χ A ( x ) := { 0 , x ∈ A ; + ∞ , x ∉ A . {\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}
• 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: φ X ( t ) = E ⁡ ( e i t X ) , {\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),} where E {\displaystyle \operatorname {E} } denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.
• The characteristic function of a cooperative game in game theory.
• The characteristic polynomial in linear algebra.
• The characteristic state function in statistical mechanics.
• The Euler characteristic, a topological invariant.
• The receiver operating characteristic in statistical decision theory.
• The point characteristic function in statistics.