Fundamental parallelogram (complex analysis)

In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined.

Although the concept of a two-dimensional lattice is quite simple, there is a considerable amount of specialized notation and language concerning the lattice that occurs in mathematical literature. This article attempts to review this notation, as well as to present some theorems that are specific to the two-dimensional case.

The fundamental pair of periods is a pair of complex numbers ${\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }$ such that their ratio ω₂/ω₁ is not real. In other words, considered as vectors in ${\displaystyle \mathbb {R} ^{2}}$, the two are not collinear. The lattice generated by ω₁ and ω₂ is

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