On generalized trigonometric functions and series of rational functions

Han Yu

Published 2016-10-15Version 1

Here we introduce a way to construct generlized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those generalized trigonometric functions have algebraic identities which generalizes the well known $\sin^2(x)+\cos^2(x)=1$. One application of the generalized trigonometric functions is to evaluating infinite series of rational functions. It is well known that we can express infinite series of rational functions with polygamma functions by using partial fraction decomposition. Here we consider the two sided sum $\lim_{k\to\infty}\sum_{n=-k}^{k}f(n)$ where $f$ is any rational function such that limit is finite. In this case we can use the residue theorem to have a finite expression by using cotangent functions. Using generalized trigonometric functions we obtain an alternative method of evaluating series of rational functions.