Characteristic function

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title: "Characteristic function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public topic_path: "uncategorized" source: "https://en.wikipedia.org/wiki/Characteristic_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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

The indicator function of a subset, that is the function \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: \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: \varphi_X(t) = \operatorname{E}\left(e^{itX}\right), where \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.

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