Some evaluation of infinite series involving trigonometric and hyperbolic functions

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.