Beyond Sin and Cos

Abstract

The higher-order circular and hyperbolic functions deserve to be better known. Here we give their main properties in order to make them more accessible to teachers and students in calculus, linear algebra and differential equations courses. The study of these functions can be related to such diverse topics as the binomial theorem and the fast Fourier transform. Here, for each positive integer n, we define r functions FJ,(x), r = 0,1 .I., n-1. The cases a = 1 and a = -1 correspond, respectively, to what are usually known as generalized hyperbolic functions and generalized circular or trigonometric functions. We find it useful to retain the parameter a; the case a = 0 also gives something interesting. The functions considered here are elemnentary and can be a rich source for student projects and investigations.