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 and 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.

**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

Email:
w.couwenberg@chello.nl

DOI:
http://dx.doi.org/10.1090/S0002-9939-03-06902-8

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