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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 .
The smallest positive integer n satisfying the above is called the prime period or least period of the point x. If every point in X is a periodic point with the same period n, then f is called periodic with period n (this is not to be confused with the notion of a periodic function). If there exist distinct n and m such that
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.
The following functions have period and take as their argument. The symbol ⌊ n ⌋ {\displaystyle \lfloor n\rfloor } is the floor function of n {\displaystyle n} and sgn {\displaystyle \operatorname {sgn} } is the sign function .
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.
In mathematics, an infinite periodic continued fraction is a simple continued fraction that can be placed in the form = + + + + + + + + + + + + + + where the initial block [;, …,] of k+1 partial denominators is followed by a block [+, +, …, +] of m partial denominators that repeats ad infinitum.
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 ...
If k 2 + 4 is a quadratic residue modulo p (where p > 2 and p does not divide k 2 + 4), then +, /, and / + can be expressed as integers modulo p, and thus Binet's formula can be expressed over integers modulo p, and thus the Pisano period divides the totient =, since any power (such as ) has period dividing (), as this is the order of the group ...