P -Trigonometric and p -Hyperbolic Functions in Complex Domain

Abstract

We study extension of p -trigonometric functions s i n p and c o s p and of p -hyperbolic functions s i n h p and c o s h p to complex domain. Our aim is to answer the question under what conditions on p these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, s i n (z) = - i · sinh i · z. In particular, we prove in the paper that for p = 6,10,14,. the p -trigonometric and p -hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series for p -trigonometric and p -hyperbolic functions.