# characteristic function

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## characteristic function

[‚kar·ik·tə′ris·tik ′fəŋk·shən]
(mathematics)
The function χA defined for any subset A of a set by setting χA (x) = 1 if x is in A and χA = 0 if x is not in A. Also known as indicator function.
(physics)
A function, such as the point characteristic function or the principal function, which is the integral of some property of an optical or mechanical system over time or over the path followed by the system, and whose value for a path actually followed by a system is a maximum or a minimum with respect to nearby paths with the same end points.
(statistics)
A function that uniquely defines a probability distribution; it is equal to √(2π) times the Fourier transform of the frequency function of the distribution.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Characteristic Function

in mathematics:

(1) An eigenfunction.

(2) The characteristic, or indicator, function of a set A is a function f(x) that is defined on some set E containing A and that assumes the value f(x) = 1 if x is in A and the value f(x) = 0 if x is not in A.

(3) In probability theory, the characteristic function fx(t) of a random variable X is the mathematical expectation of the quantity exp (it X). For a random variable with probability density function PX(x), this definition yields the formula

For example, for a random variable having a normal distribution with parameters a and σ, the characteristic function is

The characteristic function has several noteworthy properties. To every random variable X there corresponds a definite characteristic function. The probability distribution for X is uniquely determined by fX(t). When independent random variables are added, the corresponding characteristic functions are multiplied. If the concept of closeness is suitably defined, to random variables with close distributions there correspond characteristic functions that differ little from each other, and, conversely, to close characteristic functions there correspond random variables with close distributions. These properties underlie the applications of characteristic functions, particularly applications to the derivation of the limit theorems of probability theory.

A mathematical apparatus more or less equivalent to that of the characteristic function was first used by P. Laplace in 1812, but the full power of the characteristic function method was demonstrated in 1911 by A. M. Liapunov, who used the method in obtaining the theorem that bears his name.

The concept of characteristic function can be generalized to finite and infinite systems of random variables—that is, to random vectors and stochastic processes. The theory of characteristic functions has much in common with the theory of Fourier integrals.

### REFERENCES

Gnedenko, B. V. Kursteorii veroiatnostei, 5th ed. Moscow, 1969.
Prokhorov, Iu. V., and Iu. A. Rozanov. Teoriia veroiatnostei, 2nd ed. Moscow, 1973.

## characteristic function

(mathematics)
The characteristic function of set returns True if its argument is an element of the set and False otherwise.
References in periodicals archive ?
A map [lambda] : F(P) [right arrow] [Z.sup.n.sub.2] is called a characteristic function if it satisfies the non-singularity condition : whenever the intersection [mathematical expression not reproducible] is non-empty, the set [mathematical expression not reproducible] forms a basis for [Z.sup.n.sub.2].
Since a SVNS is characterized by three functions independently, this paper introduced ([alpha], [beta], [gamma])-equalities corresponding to characteristic functions of SVNS.
Except for some special cases, the S[alpha]S distribution does not have a closed-form expression for the PDF, so it is usually described by its characteristic function
As long as you like, you can choose the other general characteristic functions; for example, [phi](x) = [e.sup.a(x)] - 1 and [phi](x) = sin(a(x)/M), to obtain the corresponding theorems.
These conditions avoid the loss of super-additivity of a class of general characteristic functions. Furthermore, if random influences are taken into account, the considered problem is quite involved and this is one of our future research works.
Some derivations employ characteristic functions in a variety of ways, since the characteristic function of a sum of independent random variables is the product of each summand's characteristic function and the inverse transform is not intractable ([11, pages 188-189], [12-14], [15, pages 362-363], [16, 17]).
This procedure determines characteristic functions of extracted stages.
Generally, the synthesis of vibrating mechanical systems comes down to the prime factorization of the characteristic function (decomposition into partial fractions, decomposition into continued fractions).
Characteristic functions for Sturm-Liouville problems with nonlocal boundary conditions.
Messer draws heavily from Barth's "ethics of creation" and pairs this approach with the Aristotelian/Thomist emphasis upon teleology and essentialism, especially as teleology and essentialism apply to human beings and their characteristic functions as beings of a particular kind.

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