Complex-type numbers and generalizations of the Euler identity

TL;DR

This paper explores various generalizations of Euler's formula, examining their properties through algebraic and geometric perspectives, and introduces new insights into the associated trigonometric functions.

Contribution

It presents novel generalizations of Euler's formula and analyzes their properties using algebraic and geometric frameworks.

Findings

New generalized Euler formulas introduced

Properties of generalized trigonometric functions characterized

Connections between algebraic and geometric formulations established

Abstract

We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural way in algebraic and geometric terms.