A real number x is a period if it is expressible as the integral of an (with algebraic coefficients) over an algebraic domain, and this integral is absolutely convergent. This is called the number’s period representation. An algebraic domain is a subset of ℝn given by inequalitiesMathworldPlanetmath with algebraic coefficients. A complex numberMathworldPlanetmathPlanetmath is defined to be a period if both its real and imaginary partsMathworldPlanetmath are. The set of all complex periods is denoted by 𝒫.

1 Examples

Example 1.

The transcendental numberMathworldPlanetmath Ο€ is a period since we can write

Example 2.

Any algebraic numberMathworldPlanetmath Ξ± is a period since we use the somewhat definition that integration over a 0-dimensional space is taken to mean evaluation:

Example 3.

The logarithms of algebraic numbers are periods:


2 Non-periods

It is by no means trivial to find complex non-periods, though their existence is clear by a counting argumentMathworldPlanetmath: The set of complex numbers is uncountable, whereas the set of periods is countableMathworldPlanetmath, as there are only countably many algebraic domains to choose and countably many algebraic functionsMathworldPlanetmath over which to integrate.

3 Inclusion

With the existence of a non-period, we have the following chain of set inclusions:


where β„šΒ― denotes the set of algebraic numbers. The periods promise to prove an interesting and important set of numbers in that nebulous between β„šΒ― and β„‚.

4 References

Kontsevich and Zagier. Periods. 2001. Available on line at \url

Title period
Canonical name Period
Date of creation 2013-03-22 13:55:43
Last modified on 2013-03-22 13:55:43
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Definition
Classification msc 11F67
Defines period representation
Defines algebraic domain