p-Trigonometric andp-Hyperbolic Functions in Complex Domain

Abstract

We study extension ofp-trigonometric functionssinpandcospand ofp-hyperbolic functionssinhpandcoshpto complex domain. Our aim is to answer the question under what conditions onpthese functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example,sin(z)=-i·sinh⁡i·z. In particular, we prove in the paper that forp=6,10,14,…thep-trigonometric andp-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 forp-trigonometric andp-hyperbolic functions.

Keplerian trigonometry

“…These generalizations are divided into two different lines of thought, depending on the possible fields of application. Some authors, like [7][8][9][10][13][14][15][16][17]22], taking the moves from the definition of the sine function as inverse of the arcsine function, introduced by the integral J (u) := u 0 (1 − t 2 ) −1/2 dt (where u is precisely the arc length of the circle), define the function sin p (x), where p ≥ 1, as the inverse of the function…”