Generalized Sine Functions, Complexified

TL;DR

This paper explores the extension of generalized sine functions into the complex plane, identifying their conformal properties and applications in complex analysis and geometry.

Contribution

It introduces the complexification of generalized sine functions and characterizes their conformal mappings from polygons to the complex plane with slits.

Findings

Generalized sine functions can be extended as complex analytic functions.

The natural domain for these functions is identified as a conformal equivalence from polygons to slit complex planes.

Applications include geometric and analytic insights into these functions.

Abstract

Generalized sine and cosine functions, $\sin_{n}$ and $\cos_{n}$, that parametrize the generalized unit circle $x^n+y^n=1$ are, much like their classical circular counterparts, extendable as complex analytic functions. In this article, we identify the natural domain on which $\sin_{n}$ is a conformal equivalence from a polygon to the complex plane with $n$ slits. We also give some geometric and analytic applications.