A simple proof of the modular identity for theta functions

Abstract

The modular identity arises in the theory of theta functions in one complex variable. It states a relation between theta functions for parameters τ \tau and − 1 / τ -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.