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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
::<math>\mathbf{1}_A: X \to \{0, 1\}</math>
: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)|characteristic function]] in [[convex analysis]]:
::<math>\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}</math>
* The [[characteristic state function]] in [[statistical mechanics]]
* In [[probability theory]], the [[characteristic function (probability theory)|characteristic function]] of any [[probability distribution]] on the [[real number|real]] line is given by the following formula, where ''X'' is any [[random variable]] with the distribution in question:
::<math>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right)\,</math>
:where '''E''' means [[expected value]]. This concept extends to multivariate distributions. For more see [[characteristic function (probability)]].
* The [[characteristic polynomial]] in [[linearfunction algebra(probability)]].
* The [[Eulercharacteristic characteristicpolynomial]], ain [[topologicallinear invariant]]algebra
* The [[cooperativeEuler game|characteristic function]], ina [[gametopological theory]]invariant
* The [[cooperative game]] in game theory
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