Fundamental Pair Of Periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Definition A fundamental pair of periods is a pair of complex numbers \omega_1,\omega_2 \in \Complex such that their ratio \omega_2 / \omega_1 is not real. If considered as vectors in \R^2, the two are linearly independent. The lattice generated by \omega_1 and \omega_2 is :\Lambda = \left\. This lattice is also sometimes denoted as \Lambda(\omega_1, \omega_2) to make clear that it depends on \omega_1 and \omega_2. It is also sometimes denoted by \Omega\vphantom or \Omega(\omega_1, \omega_2), or simply by (\omega_1, \omega_2). The two generators \omega_1 and \omega_2 are called the ''lattice basis''. The parallelogram with vertices (0, \omega_1, \omega_1+\omega_2, \omega_2) is called the ''fundamental parallelogram''. While a fundamental pair gen ...
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