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In mathematics, specifically algebraic geometry, a period or algebraic period [1] is a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known mathematical constants such as the number π .
A nonzero constant P for which this is the case is called a period of the function. If there exists a least positive [2] constant P with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period.
Given a mapping f from a set X into itself, :, a point x in X is called periodic point if there exists an n >0 so that = where f n is the n th iterate of f.The smallest positive integer n satisfying the above is called the prime period or least period of the point x.
The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions.
A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a 1, a 2, a 3, ... satisfying . a n+p = a n. for all values of n. [1] [2] [3] If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.
The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication , the order of an element a of a group, is thus the smallest positive integer m such that a m = e , where e denotes the identity element of the group, and a m ...
The period of c / k , for c coprime to k, equals the period of 1 / k . If k = 2 a ·5 b n where n > 1 and n is not divisible by 2 or 5, then the length of the transient of 1 / k is max( a , b ), and the period equals r , where r is the multiplicative order of 10 mod n, that is the smallest integer such that 10 r ≡ 1 (mod ...
In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. [1] A function f {\displaystyle f} is quasiperiodic with quasiperiod ω {\displaystyle \omega } if f ( z + ω ) = g ( z , f ( z ) ) {\displaystyle f(z+\omega )=g(z,f(z))} , where g {\displaystyle g} is a " simpler " function than f ...