Examples

Example 1 The transcendental number $\pi$ is a period since we can write

$\displaystyle \pi=\int\limits_{x^2+y^2\leq1}~dx~dy.$

Example 2 Any algebraic number $\alpha$ is a period since we use the somewhat natural definition that integration over a 0-dimensional space is taken to mean evaluation:

$\displaystyle \alpha=\int_{\{\alpha\}}x$

Example 3 The logarithms of algebraic numbers are periods:

$\displaystyle \log\alpha=\int_1^\alpha\frac{1}{x}~dx$

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.

Inclusion

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

$\displaystyle \mathbb{Z}\subsetneq \mathbb{Q}\subsetneq\overline {\mathbb{Q}}\subsetneq\mathcal {P}\subsetneq\mathbb{C},$

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 region between $\overline {\mathbb{Q}}$ and $\mathbb{C}$ .

References

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

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Also defines:

period representation, algebraic domain

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Cross-references: set inclusions, chain, algebraic functions, countable, uncountable, argument, clear, logarithms, mean, algebraic number, transcendental number, complex, imaginary parts, complex number, inequalities, subset, number's, absolutely convergent, coefficients, algebraic, integral, expressible, real number
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This is version 8 of period, born on 2003-09-03, modified 2004-09-25.
Object id is 4687, canonical name is Period2.
Accessed 5905 times total.

Classification:

AMS MSC:

11F67 (Number theory :: Discontinuous groups and automorphic forms :: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols)

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