Some mathematicians use the phrase "characteristic function" synonymously with "indicator function". The indicator function of a subset A of a set B is the function with ___domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A.
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:
If X is a vector-valued random variable, one takes the argument t to be a vector and tX to be a dot product.