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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. The concept has been promoted by Maxim Kontsevich and Don Zagier. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
A number is an abstract idea used in counting and measuring. ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
This article or section does not cite its references or sources. ...
Maxim Kontsevich (Russian: МакÑим Концевич) (born August 25, 1964) is a Russian mathematician. ...
Don Bernhard Zagier (1951 - ) is an American mathematician. ...
Definition Kontsevich and Zagier define a period as - a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients, over domains in
 given by polynomial inequalities with rational coefficients. In this definition, "rational" can be exchanged for "algebraic" without changing the meaning, since irrational algebraic numbers and functions are themselves expressible as integrals of rational functions over rational domains. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ...
In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...
In mathematics, a series is a sum of a sequence of terms. ...
In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...
In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
The feasible regions of linear programming are defined by a set of inequalities. ...
The set of periods is denoted by  and places in the number hierarchy as
 where  denotes the algebraic numbers.
Sums and products of periods remain periods, so the periods form an algebra.
Purpose of the classification The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the class of transcendental numbers is excessively broad; in particular, the set of transcendental numbers is not countable, and its members are generally not computable. The set of all periods is countable and all periods are computable. In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...
In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ...
A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ...
Besides the algebraic numbers, the following numbers are known to be periods:
The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...
In mathematics, a zeta constant is a number obtained by plugging an integer into the Riemann zeta function. ...
In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...
In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ...
The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...
Extensions Some mathematical constants notably seem absent from the set of periods; in particular, it is not known whether Euler's number e and Euler-Mascheroni constant γ belong to . The mathematical constant e (occasionally called Eulers number after the Swiss mathematician Leonhard Euler, or Napiers constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function. ...
The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ...
The periods can be extended to the 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 functions. If Euler's constant is added as a new period, then according to Kontsevich and Zagier "all classical constants are periods in the appropriate sense". The exponential function is one of the most important functions in mathematics. ...
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred to...
Conjectures Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods". A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ...
It is conjectured that, if a period is given by two different integrals, then either integral can be transformed into the other using only the linearity of integrals, changes of variables, and the Newton-Leibniz formula. In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...
A useful property of algebraic numbers is that equality between two algebraic expressions can be determined algorithmically. It is conjectured that this is also possible for periods.
References - Kontsevich and Zagier. "Periods." Preprint, May 2001.
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