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| {{dablink|For a more frequently used sense of the word "period" in mathematics, see [[Periodic function]].}}
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| In [[mathematics]], a '''period''' is a [[number]] that can be expressed as an [[integral]] of an [[algebraic function]] over an algebraic domain. Sums and products of periods remain periods, so the periods form a [[ring (mathematics)|ring]].
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| {{harvs|txt|first=Maxim|last= Kontsevich|authorlink=Maxim Kontsevich|first2=Don |last2=Zagier|author2-link=Don Zagier|year=2001}} gave a survey of periods and introduced some conjectures about them.
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| ==Definition==
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| A real number is called a period if it is the difference of volumes of regions of Euclidean space given by [[polynomial]] [[inequality (mathematics)|inequalities]] with rational coefficients. More generally a complex number is called a period if its real and imaginary parts are periods.
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| The values of [[absolutely convergent]] integrals of [[rational function]]s with algebraic coefficients, over domains in <math>\mathbb{R}^n</math> given by [[polynomial]] [[inequality (mathematics)|inequalities]] with algebraic coefficients are also periods, since integrals and irrational algebraic numbers are expressible in terms of areas of suitable domains.
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| ==Examples==
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| Besides the algebraic numbers, the following numbers are known to be periods:
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| * The [[natural logarithm]] of any algebraic number
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| * [[π]]
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| * [[Elliptic integral]]s with rational arguments
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| * All [[zeta constant]]s (the [[Riemann zeta function]] of an integer) and [[multiple zeta value]]s
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| * Special values of [[hypergeometric function]]s at algebraic arguments
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| * [[gamma function|Γ]](''p''/''q'')<sup>''q''</sup> for natural numbers ''p'' and ''q''.
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| An example of real number that is not a period is given by [[Chaitin's constant Ω]]. Currently there are no natural examples of [[computable number]]s that have been proved not to be periods, though it is easy to construct artificial examples using [[Cantor's diagonal argument]]. Plausible candidates for numbers that are not periods include ''[[e (mathematical constant)|e]]'', 1/π, and [[Euler–Mascheroni constant γ]].
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| ==Purpose of the classification==
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| The periods are intended to bridge the gap between the [[algebraic number]]s and the [[transcendental numbers]]. The class of algebraic numbers is too narrow to include many common [[mathematical constant]]s, while the set of transcendental numbers is not [[countable]], and its members are not generally [[computable number|computable]]. The set of all periods is countable, and all periods are computable, and in particular [[definable number|definable]].
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| ==Conjectures==
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| Many of the constants known to be periods are also given by integrals of [[transcendental function]]s. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods".
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| Kontsevich and Zagier conjectured that, if a period is given by two different integrals, then each integral can be transformed into the other using only the linearity of integrals, [[change of variables|changes of variables]], and the [[fundamental theorem of calculus|Newton–Leibniz formula]]
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| : <math> \int_a^b f'(x) \, dx = f(b) - f(a). </math>
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| A useful property of algebraic numbers is that equality between two algebraic expressions can be determined algorithmically. The conjecture of Kontsevich and Zagier implies that this is also possible for periods.
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| It is not expected that [[Euler's number]] ''e'' and [[Euler–Mascheroni constant]] γ are periods. The periods can be extended to ''exponential periods'' by permitting the product of an algebraic function and the [[exponential function]] of an algebraic function as an integrand. This extension includes all algebraic powers of ''e'', the gamma function of rational arguments, and values of [[Bessel function]]s. If, further, Euler's constant is added as a new period, then according to Kontsevich and Zagier "all classical constants are periods in the appropriate sense".
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| ==References==
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| *{{Citation | last1=Belkale | first1=Prakash | last2=Brosnan | first2=Patrick | title=Periods and Igusa local zeta functions | doi=10.1155/S107379280313142X | mr=2012522 | year=2003 | journal=International Mathematics Research Notices | issn=1073-7928 | issue=49 | pages=2655–2670}}
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| *{{Citation | last1=Kontsevich | first1=Maxim | last2=Zagier | first2=Don | editor1-last=Engquist | editor1-first=Björn | editor2-last=Schmid | editor2-first=Wilfried | title=Mathematics unlimited—2001 and beyond | url=http://www.ihes.fr/~maxim/TEXTS/Periods.pdf | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-66913-5 | mr=1852188 | year=2001 | chapter=Periods | pages=771–808}}
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| *{{Citation | last1=Waldschmidt | first1=Michel | title=Transcendence of periods: the state of the art | doi=10.4310/PAMQ.2006.v2.n2.a3 | mr=2251476 | year=2006 | journal=Pure and Applied Mathematics Quarterly | issn=1558-8599 | volume=2 | issue=2 | pages=435–463}}
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| ==External links==
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| * [http://planetmath.org/encyclopedia/Period2.html PlanetMath: Period]
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| [[Category:Mathematical constants]]
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| [[Category:Algebraic geometry]]
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| [[Category:Integral calculus]]
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| {{Number Systems}}
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