Generalized Sine Functions, Complexified

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.

10 pages, to appear in Math. Mag