A doubly periodic function with periods 2ω_1 and 2ω_2 such that f(z + 2ω_1) = f(z + 2ω_2) = f(z), which is analytic and has no singularities except for poles in the finite part of the complex plane. The half-period ratio τ congruent ω_2/ω_1 must not be purely real, because if it is, the function reduces to a singly periodic function if τ is rational, and a constant if τ is irrational. ω_1 and ω_2 are labeled such that ℑ[τ] congruent ℑ[ω_2/ω_1]>0, where ℑ[z] is the imaginary part. A "cell" of an elliptic function is defined as a parallelogram region in the complex plane in which the function is not multi-valued.

doubly periodic function | elliptic curve | elliptic integral | half-period ratio | Jacobi elliptic functions | Jacobi theta functions | Liouville's elliptic function theorem | modular form | modular function | Neville theta functions | Weierstrass elliptic function