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On a Class of Berndt-type Integrals and Related Barnes Multiple Zeta Functions

Xiang Chen, Ce Xu, Jianing Zhou

TL;DR

The paper addresses evaluating a class of Berndt-type integrals with denominators $[\cosh(2x)-\cos(2x)][\cosh x-\cos x]$ by converting them via contour integration into Ramanujan-type hyperbolic series, which are then analyzed using Fourier/MMaclaurin expansions of Jacobi elliptic functions and Ramanujan parameters $x,y,z$. The main contributions are explicit closed-form evaluations in terms of powers of $Γ(1/4)$ and $π$, together with a structural Barnes zeta-function framework that yields representations for $ζ_4$-values associated with these integrals. The work provides a unified approach linking Berndt-type hyperbolic integrals, Jacobi function theory, and Barnes multiple zeta functions, including concrete coefficient formulas and illustrative cases for higher-order parameters. This framework opens a systematic pathway for solving related Berndt-type problems and deepens connections between hyperbolic series and special zeta functions.

Abstract

This paper investigates a class of special Berndt-type integral calculations where the integrand contains only hyperbolic cosine functions. The research approach proceeds as follows: Firstly, through contour integration methods, we transform the integral into a Ramanujan-type hyperbolic infinite series. Subsequently, we introduce a $θ$-parameterized auxiliary function and apply the residue theorem from complex analysis to successfully simplify mixed-type denominators combining hyperbolic cosine and sine terms into a normalized Ramanujan-type hyperbolic infinite series with denominators containing only single hyperbolic function terms. For these simplified hyperbolic infinite series, we combine properties of Jacobi elliptic functions with composite analytical techniques involving Fourier series expansion and Maclaurin series expansion. This ultimately yields an explicit expression as a rational polynomial combination of $Γ(1/4)$ and $π^{-1/2}$. Notably, this work establishes a connection between the integral and Barnes multiple zeta functions, providing a novel research pathway for solving related problems.

On a Class of Berndt-type Integrals and Related Barnes Multiple Zeta Functions

TL;DR

The paper addresses evaluating a class of Berndt-type integrals with denominators by converting them via contour integration into Ramanujan-type hyperbolic series, which are then analyzed using Fourier/MMaclaurin expansions of Jacobi elliptic functions and Ramanujan parameters . The main contributions are explicit closed-form evaluations in terms of powers of and , together with a structural Barnes zeta-function framework that yields representations for -values associated with these integrals. The work provides a unified approach linking Berndt-type hyperbolic integrals, Jacobi function theory, and Barnes multiple zeta functions, including concrete coefficient formulas and illustrative cases for higher-order parameters. This framework opens a systematic pathway for solving related Berndt-type problems and deepens connections between hyperbolic series and special zeta functions.

Abstract

This paper investigates a class of special Berndt-type integral calculations where the integrand contains only hyperbolic cosine functions. The research approach proceeds as follows: Firstly, through contour integration methods, we transform the integral into a Ramanujan-type hyperbolic infinite series. Subsequently, we introduce a -parameterized auxiliary function and apply the residue theorem from complex analysis to successfully simplify mixed-type denominators combining hyperbolic cosine and sine terms into a normalized Ramanujan-type hyperbolic infinite series with denominators containing only single hyperbolic function terms. For these simplified hyperbolic infinite series, we combine properties of Jacobi elliptic functions with composite analytical techniques involving Fourier series expansion and Maclaurin series expansion. This ultimately yields an explicit expression as a rational polynomial combination of and . Notably, this work establishes a connection between the integral and Barnes multiple zeta functions, providing a novel research pathway for solving related problems.
Paper Structure (6 sections, 18 theorems, 115 equations)

This paper contains 6 sections, 18 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Some Definitions and Lemmas
  3. Berndt-Type Integrals via Hyperbolic Series
  4. Evaluations of Hyperbolic Summations via Jacobi Functions
  5. Evaluations of Berndt-type Integrals
  6. Berndt-type Integrals via Barnes Multiple Zeta Functions

Key Result

Lemma 2.1

(PB1998) Let $\xi(s)$ be a kernel function and let $r(s)$ be a function which is $O(s^{-2})$ at infinity. Then where $S$ denotes the set of poles of $r(s)$, and $O$ denotes the set of poles of $\xi(s)$ that are not poles of $r(s)$. Here, $\operatorname{Res}(r(s), s = \alpha)$ stands for the residue of $r(s)$ at $s = \alpha$. The kernel function $\xi(s)$ is meromorphic throughout the entire comple

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 3.1
  • ...and 27 more