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Theta Functions in Complex Analysis and Number Theory

  • Hershel M. FarkasEmail author
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 17)

Summary

In these notes we try to demonstrate the utility of the theory of theta functions in combinatorial number theory and complex analysis. The main idea is to use identities among theta functions to deduce either useful number-theoretic information related to representations as sums of squares and triangular numbers, statements concerning congruences, or statements concerning partitions of sets of integers. In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk.

Key words

Theta functions triangular numbers partitions Riemann map Picard. 

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Copyright information

© Springer-Verlag New York 2008

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalem 91904Israel

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