A simple proof of the modular identity for theta functions
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- by Wim Couwenberg
- Proc. Amer. Math. Soc. 131 (2003), 3305-3307
- DOI: https://doi.org/10.1090/S0002-9939-03-06902-8
- Published electronically: February 12, 2003
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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, DOI 10.1090/pspum/049.2/1013167
Bibliographic 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
- 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
Dedicated: To A.C.M. van Rooij on the occasion of his 65th birthday