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A simple proof of the modular identity for theta functions

Author: Wim Couwenberg
Journal: Proc. Amer. Math. Soc. 131 (2003), 3305-3307
MSC (2000): Primary 14K25
Published electronically: February 12, 2003
MathSciNet review: 1990617
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Abstract | References | Similar Articles | Additional Information

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.

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  • 1. Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR 0125252
  • 2. 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

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
Article copyright: © Copyright 2003 American Mathematical Society