Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

General Berndt-Type Integrals and Series Associated with Jacobi Elliptic Functions

Ce Xu, Jianqiang Zhao

Abstract

In this paper, we prove two structural theorems on the general Berndt-type integrals with the denominator having arbitrary positive degrees by contour integrations involving hyperbolic and trigonometric functions, and hyperbolic sums associated with Jacobi elliptic functions. We first establish explicit relations between these integrals and four classes of hyperbolic sums. Then, using our previous results on hyperbolic series and applying the matrix method from linear algebra, we compute explicitly several general hyperbolic sums and their higher derivatives. These enable us to express two families of general Berndt-type integrals as polynomials in $Γ^4(1/4)$ and $π^{-1}$ with rational coefficients, where $Γ$ is the Euler gamma function. At the end of the paper, we provide some conjectures of general Berndt-type integrals.

General Berndt-Type Integrals and Series Associated with Jacobi Elliptic Functions

Abstract

In this paper, we prove two structural theorems on the general Berndt-type integrals with the denominator having arbitrary positive degrees by contour integrations involving hyperbolic and trigonometric functions, and hyperbolic sums associated with Jacobi elliptic functions. We first establish explicit relations between these integrals and four classes of hyperbolic sums. Then, using our previous results on hyperbolic series and applying the matrix method from linear algebra, we compute explicitly several general hyperbolic sums and their higher derivatives. These enable us to express two families of general Berndt-type integrals as polynomials in and with rational coefficients, where is the Euler gamma function. At the end of the paper, we provide some conjectures of general Berndt-type integrals.
Paper Structure (5 sections, 23 theorems, 155 equations)

This paper contains 5 sections, 23 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Notation and Some Preliminary Results
  3. Berndt-Type Integrals via Hyperbolic Series
  4. Explicit evaluations of Hyperbolic Sums
  5. General Berndt-Type Integrals

Key Result

Theorem 1.1

Let $X=\Gamma^4(1/4)$ and $Y=\pi^{-1}$. For all integers $m\geq 1$ and $p\geq [m/2]$, the Berndt-type integrals where the degrees of $X$ have the same parity as that of $m$ and are between $2p-m+2$ and $2p+m$, inclusive, while the degrees of $Y$ are between $2p-m+2$ and $2p+3m-2$, inclusive.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • proof
  • ...and 36 more