Characteristic function

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

• The indicator function, that is the function

${\mathbf{𝟏}}_A:X\to \{0,1\},$

which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.

• There is an indicator function for affine varieties over a finite field: given a finite set of functions $f_{\alpha }\in {\mathbb{𝔽}}_q[x_1,\dots ,x_n]$ let $V=\{x\in {\mathbb{𝔽}}_q^n:f_{\alpha }(x)=0\}$ be their vanishing locus. Then, the function $P(x)=\prod (1-f_{\alpha }(x{)}^{q-1})$ acts as an indicator function for $V$. If $x\in V$ then $P(x)=1$, otherwise, for some $f_{\alpha }$, we have $f_{\alpha }(x)\ne 0$, which implies that $f_{\alpha }(x{)}^{q-1}=1$, hence $P(x)=0$.
• The characteristic function in convex analysis, closely related to the indicator function of a set:

${\chi }_A(x):=\{\begin{array}{ll}0,& x\in A;\\ +\mathrm{\infty },& x\in ̸A.\end{array}$

• 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:

${\phi }_X(t)=\mathrm{E}\left(e^{itX}\right),$

where E means 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.