Characteristic function

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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
\mathbf{1}_A: X \to \{0, 1\}
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).
\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}
\varphi_X(t) = \operatorname{E}\left(e^{itX}\right)\,
where "E" means expected value. See characteristic function (probability theory).
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