Period (number)

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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.

1 Definition
2 Purpose of the classification
3 Extensions
4 Conjectures
5 References
6 External links

[edit] 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 $\mathbb{R}^n$ 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.

The set of periods is denoted by $\mathcal{P}$ and places in the number hierarchy as

$\mathbb{Z} \subset \mathbb{Q} \subset \overline{\mathbb Q} \subset \mathcal{P} \subset \mathbb C$

where $\overline{\mathbb Q}$ denotes the algebraic numbers.

Sums and products of periods remain periods, so the periods form an algebra.

[edit] 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.

Besides the algebraic numbers, the following numbers are known to be periods:

The natural logarithm of any algebraic number
$π$
Elliptic integrals with rational arguments
All zeta constants (the Riemann zeta function of an integer) and multiple zeta value
Special values of hypergeometric functions at algebraic arguments
$Γ(p / q) q$ for natural numbers p and q, where Γ is the gamma function

[edit] 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 $\mathcal{P}$ .

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".

[edit] 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".

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.

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.