A function f(z) is said to be doubly periodic if it has two periods ω_1 and ω_2 whose ratio ω_2/ω_1 is not real. A doubly periodic function that is analytic (except at poles) and that has no singularities other than poles in the finite part of the plane is called an elliptic function. The periods ω_1 and ω_2 play the same part in the theory of elliptic functions as does the single period in the case of the trigonometric functions. Jacobi proved that if a univariate single-valued function is doubly periodic, then the ratio of periods cannot be real, as well as the impossibility for a single-valued univariate function to have more than two distinct periods.

elliptic function | half-period ratio | periodic function | triply periodic function