Some evaluation of infinite series involving trigonometric and hyperbolic functions

TL;DR

This paper uses complex analysis techniques to evaluate and relate infinite series involving trigonometric and hyperbolic functions, providing closed-form expressions connected to Gamma functions and pi.

Contribution

It introduces new methods for evaluating complex infinite series involving hyperbolic and trigonometric functions using residue theorem and asymptotic analysis.

Findings

Closed-form evaluations of specific infinite series involving hyperbolic functions

Relations connecting series to Gamma functions and pi

New examples and consequences derived from the main results

Abstract

In this paper, by using the residue theorem and asymptotic formulas of trigonometric and hyperbolic functions at the poles, we establish many relations involving two or more infinite series of trigonometric and hyperbolic trigonometric functions. In particular, we evaluate in closed form certain classes of infinite series containing hyperbolic trigonometric functions, which are related to Gamma functions and \pi. Finally, some interesting new consequences and illustrative examples are considered.