Analytic Continuation of the Doubly-periodic Barnes Zeta Function

TL;DR

This paper develops a method to extend the doubly-periodic Barnes zeta function analytically across the entire complex plane using complex integrals involving Hurwitz zeta and Jacobi theta functions, enabling explicit derivatives at non-positive integers.

Contribution

It provides the first explicit meromorphic extension of the doubly-periodic Barnes zeta function and formulas for its derivatives at non-positive integers.

Findings

Meromorphic extension to the entire complex plane.

Explicit formulas for derivatives at non-positive integers.

Representation involving Hurwitz zeta and Jacobi theta functions.

Abstract

The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorphic extension of the doubly periodic Barnes zeta function to the entire complex plane in terms of a real integral containing the Hurwitz zeta function and the first Jacobi theta function. These allow us to explicitly give expressions for the derivative at all non-positive integer points.