# Double-periodic function

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A single-valued analytic function $f ( z)$ with only isolated singularities on the entire finite complex $z$- plane, and such that there exists two numbers $p _ {1} , p _ {2}$ whose quotient is not a real number and which are periods of $f ( z)$, i.e. $p _ {1} , p _ {2}$ are such that the identity

$$f ( z + p _ {1} ) = f ( z + p _ {2} ) = f ( z)$$

is valid. (If $p _ {1} / p _ {2}$ is real and rational, $f ( z)$ is a simply-periodic function; if $p _ {1} / p _ {2}$ is real and irrational, $f ( z) \equiv \textrm{ const }$.) All numbers of the form $mp _ {1} + np _ {2}$ where $m, n$ are integers are also periods of $f ( z)$. 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 $2 \omega _ {1} , 2 \omega _ {3}$, $\mathop{\rm Im} ( \omega _ {1} / \omega _ {3} ) \neq 0$. All remaining periods of this double-periodic function may be represented in the form $2 m \omega _ {1} + 2 n \omega _ {3}$ where $m , n$ 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 $2m \omega _ {1} + 2n \omega _ {3}$ where $m, n$ are integers form the period lattice (subdividing the entire $z$- plane into period parallelograms). Points (numbers) $z _ {1}$, $z _ {2}$ for which

$$z _ {1} - z _ {2} = 2m \omega _ {1} + 2n \omega _ {3} ,$$

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

$$\{ {z = z _ {0} + 2t _ {1} \omega _ {1} + 2t _ {3} \omega _ {3} } : { 0 \leq t _ {1} < 1 , 0 \leq t _ {3} < 1 } \} ,$$

i.e. the parallelogram with vertices

$$z _ {0,\ } z _ {0} + 2 \omega _ {1} , z _ {0} + 2 \omega _ {2} = \ z _ {0} + 2 \omega _ {1} + 2 \omega _ {3} ,\ z _ {0} + 2 \omega _ {3} .$$

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 $f ( z _ {1} \dots z _ {n} )$ of $n \geq 1$ complex variables are called Abelian functions (cf. Abelian function).

#### References

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