Double-periodic function

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

[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