period

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 ${ℝ}^n$ given by inequalities with algebraic coefficients. A complex number is defined to be a period if both its real and imaginary parts are. The set of all complex periods is denoted by $𝒫$.

1 Examples

Example 1.

The transcendental number $\pi$ is a period since we can write

$\pi ={\displaystyle \underset{x^2+y^2\le 1}{\int }}𝑑x𝑑y.$

Example 2.

Any algebraic number $\alpha$ is a period since we use the somewhat definition that integration over a 0-dimensional space is taken to mean evaluation:

$\alpha ={\displaystyle {\int }_{\{\alpha \}}}x$

Example 3.

The logarithms of algebraic numbers are periods:

$\log \alpha ={\displaystyle {\int }_1^{\alpha }}{\displaystyle \frac{1}{x}}𝑑x$

2 Non-periods

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

3 Inclusion

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

$\mathbb{Z}⊊\mathbb{Q}⊊\overline{\mathbb{Q}}⊊𝒫⊊ℂ,$

where $\overline{\mathbb{Q}}$ denotes the set of algebraic numbers. The periods promise to prove an interesting and important set of numbers in that nebulous between $\overline{\mathbb{Q}}$ and $ℂ$.

4 References

Kontsevich and Zagier. Periods. 2001. Available on line at http://www.ihes.fr/PREPRINTS/M01/M01-22.ps.gz.