# 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 numberβs *period representation*. An *algebraic domain* is a subset of ${\mathrm{\beta \x84\x9d}}^{n}$ 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 $\mathrm{\pi \x9d\x92\xab}$.

## 1 Examples

###### Example 1.

The transcendental number^{} $\mathrm{{\rm O}\x80}$ is a period since we can write

$\mathrm{{\rm O}\x80}={\displaystyle \underset{{x}^{2}+{y}^{2}\beta \x89\u20ac1}{\beta \x88\xab}}\pi \x9d\x91\x91x\beta \x81\u2019\pi \x9d\x91\x91y.$ |

###### Example 2.

Any algebraic number^{} $\mathrm{\Xi \pm}$ is a period since we use the somewhat definition that integration over a 0-dimensional space is taken to mean evaluation:

$\mathrm{\Xi \pm}={\displaystyle {\beta \x88\xab}_{\{\mathrm{\Xi \pm}\}}}x$ |

###### Example 3.

The logarithms of algebraic numbers are periods:

$\mathrm{log}\beta \x81\u2018\mathrm{\Xi \pm}={\displaystyle {\beta \x88\xab}_{1}^{\mathrm{\Xi \pm}}}{\displaystyle \frac{1}{x}}\beta \x81\u2019\pi \x9d\x91\x91x$ |

## 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:

$\mathrm{\beta \x84\u20ac}\beta \x8a\x8a\mathrm{\beta \x84\x9a}\beta \x8a\x8a\stackrel{{\rm B}\u2015}{\mathrm{\beta \x84\x9a}}\beta \x8a\x8a\mathrm{\pi \x9d\x92\xab}\beta \x8a\x8a\mathrm{\beta \x84\x82},$ |

where $\stackrel{{\rm B}\u2015}{\mathrm{\beta \x84\x9a}}$ denotes the set of algebraic numbers. The periods promise to prove an interesting and important set of numbers in that nebulous between $\stackrel{{\rm B}\u2015}{\mathrm{\beta \x84\x9a}}$ and $\mathrm{\beta \x84\x82}$.

## 4 References

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

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 |