Generalization of Siegel's Method to theta 1

TL;DR

This paper provides a new proof of the transformation law for the theta function  under the full modular group, extending Siegel's method to this specific theta function.

Contribution

It introduces a novel proof technique for the transformation law of , generalizing Siegel's method to this case.

Findings

New proof of  transformation law using Siegel's method

Extension of Siegel's approach to theta 

Deeper understanding of modular transformations of theta functions

Abstract

We present a new proof of the transformation law of $\vartheta_1$ under the action of the generator of the full modular group $\Gamma$ using Siegel's method.