Generalized Trigonometric Functions over Associative Algebras

TL;DR

This paper extends the concept of trigonometric functions to arbitrary associative algebras, exploring their properties, polar forms, and logarithms, and proposing new formulas and open questions in the field.

Contribution

It generalizes the notion of polar form and trigonometric functions from complex numbers to associative algebras, providing new formulas for logarithms and opening avenues for further research.

Findings

Generalized polar forms for associative algebras

Elegant formulas for logarithms over algebras

Identification of open questions in algebraic trigonometry

Abstract

Extending the work of Freese, we further develop the theory of generalized trigonometric functions. In particular, we study to what extent the notion of polar form for the complex numbers may be generalized to arbitrary associative algebras, and how the general trigonometric functions may be used to give particularly elegant formulas for the logarithm over an algebra. Finally, we close with an array of open questions relating to this line of inquiry.