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

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 $\theta$-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 $\Gamma(1/4)$ and $\pi^{-1/2}$. Notably, this work establishes a connection between the integral and Barnes multiple zeta functions, providing a novel research pathway for solving related problems.