A generalization of Iseki Formula and the transformation of $\theta_1(z, \tau)$

TL;DR

This paper generalizes Iseki's formula and employs it to establish the transformation law of the theta function (z, ), contributing to the understanding of its modular properties.

Contribution

It introduces a generalized form of Iseki's formula and applies it to derive the transformation law of (z, ), advancing the theory of theta functions.

Findings

Derived a generalized Iseki formula

Proved the transformation law of (z, )

Enhanced understanding of theta function transformations

Abstract

In this paper we give a generalization of Iseki's formula and use it to prove the transformation law of $\theta_1(z, \tau)$.