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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A simple proof of the modular identity for theta functions
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by Wim Couwenberg PDF
Proc. Amer. Math. Soc. 131 (2003), 3305-3307 Request permission

Abstract:

The modular identity arises in the theory of theta functions in one complex variable. It states a relation between theta functions for parameters $\tau$ and $-1/\tau$ situated in the complex upper half-plane. A standard proof uses Poisson summation and hence builds on results from Fourier theory. This paper presents a simple proof using only a uniqueness property and the heat equation.
References
  • Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR 0125252, DOI 10.1017/s0025557200044491
  • Leon Ehrenpreis, Fourier analysis, partial differential equations, and automorphic functions, Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 45–100. MR 1013167
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Additional Information
  • Wim Couwenberg
  • Affiliation: University Nijmegen, Toernooiveld-1, 6525 ED Nijmegen, The Netherlands
  • Email: w.couwenberg@chello.nl
  • Received by editor(s): July 6, 2001
  • Received by editor(s) in revised form: May 22, 2002
  • Published electronically: February 12, 2003

    • Dedicated: To A.C.M. van Rooij on the occasion of his 65th birthday
  • Communicated by: Juha M. Heinonen
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 3305-3307
  • MSC (2000): Primary 14K25
  • DOI: https://doi.org/10.1090/S0002-9939-03-06902-8
  • MathSciNet review: 1990617