On the Generalization of Radamacher's Proof to $\vartheta_1(z,\tau)$

TL;DR

This paper extends Rademacher's classical proof of the eta function's transformation law to the Jacobi theta function, employing residue calculus to establish a broader generalization in modular form theory.

Contribution

It introduces a novel generalization of Rademacher's proof technique from the eta function to the Jacobi theta function using residue calculus.

Findings

Established a new transformation law for the Jacobi theta function.

Extended Rademacher's proof method to a broader class of modular forms.

Provided a residue calculus-based approach for such generalizations.

Abstract

In this paper, we generalize Rademacher's proof of the transformation law of the eta function to the Jacobi theta function using Residue calculus.