# Double-periodic function

A single-valued analytic function with only isolated singularities on the entire finite complex -plane, and such that there exists two numbers whose quotient is not a real number and which are periods of , i.e. are such that the identity

is valid. (If is real and rational, is a simply-periodic function; if is real and irrational, .) All numbers of the form where are integers are also periods of . All periods of a given double-periodic function form a discrete Abelian group with respect to addition, known as the period group (or the period module), a basis of which (a period basis) is constituted by two primitive periods , . All remaining periods of this double-periodic function may be represented in the form where are integers. Analytic functions of one complex variable with more than two primitive periods do not exist, except for constants.

The points of the form where are integers form the period lattice (subdividing the entire -plane into period parallelograms). Points (numbers) , for which

are said to be congruent (comparable with respect to the period module). At congruent points the double-periodic function assumes the same value, so that it is sufficient to study the behaviour of in some basic period parallelogram. This is usually the set of points

i.e. the parallelogram with vertices

A non-constant double-periodic function that is regular in the entire basic period parallelogram does not exist. Meromorphic double-periodic functions are called elliptic functions (cf. Elliptic function). The generalization of the concept of an elliptic function to include functions of complex variables are called Abelian functions (cf. Abelian function).

#### References

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