View source for Double-periodic function

A single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339301.png" /> with only isolated singularities on the entire finite complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339302.png" />-plane, and such that there exists two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339303.png" /> whose quotient is not a real number and which are periods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339304.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339305.png" /> are such that the identity

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is valid. (If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339307.png" /> is real and rational, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339308.png" /> is a simply-periodic function; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d0339309.png" /> is real and irrational, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393010.png" />.) All numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393011.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393012.png" /> are integers are also periods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393013.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393015.png" />. All remaining periods of this double-periodic function may be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393016.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393017.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393018.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393019.png" /> are integers form the period lattice (subdividing the entire <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393020.png" />-plane into period parallelograms). Points (numbers) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393022.png" /> for which

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are said to be congruent (comparable with respect to the period module). At congruent points the double-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393024.png" /> assumes the same value, so that it is sufficient to study the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393025.png" /> in some basic period parallelogram. This is usually the set of points

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i.e. the parallelogram with vertices

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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|Elliptic function]]). The generalization of the concept of an elliptic function to include functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033930/d03393029.png" /> complex variables are called Abelian functions (cf. [[Abelian function|Abelian function]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 4 (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''2''' , Springer (1968) {{MR|1535615}} {{MR|0173749}} {{MR|0011320}} {{ZBL|0945.30001}} {{ZBL|0135.12101}} {{ZBL|55.0171.01}} {{ZBL|51.0236.12}} {{ZBL|48.1207.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , '''2''' , Cambridge Univ. Press (1952) pp. Chapt. 20 {{MR|1424469}} {{MR|0595076}} {{MR|0178117}} {{MR|1519757}} {{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}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian) {{MR|1054205}} {{ZBL|0694.33001}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lang, "Elliptic functions" , Addison-Wesley (1973) {{MR|0409362}} {{ZBL|0316.14001}} </TD></TR></table>