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

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 4 (Translated from Russian) MR0444912 Zbl 0357.30002
[2]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 2 , Springer (1968) MR1535615 MR0173749 MR0011320 Zbl 0945.30001 Zbl 0135.12101 Zbl 55.0171.01 Zbl 51.0236.12 Zbl 48.1207.01
[3]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , 2 , Cambridge Univ. Press (1952) pp. Chapt. 20 MR1424469 MR0595076 MR0178117 MR1519757 Zbl 0951.30002 Zbl 0108.26903 Zbl 0105.26901 Zbl 53.0180.04 Zbl 47.0190.17 Zbl 45.0433.02 Zbl 33.0390.01
[4]	N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian) MR1054205 Zbl 0694.33001

References

[a1]	S. Lang, "Elliptic functions" , Addison-Wesley (1973) MR0409362 Zbl 0316.14001