Real-Analyticity of Generalized Sine Functions with Two Parameters

TL;DR

This paper characterizes the maximal interval of real-analyticity for generalized sine functions with two parameters, identifying specific conditions on the parameter p and calculating the Taylor series radius.

Contribution

It establishes the exact conditions under which the generalized sine functions are analytic at certain points and determines the radius of convergence for their Taylor series expansions.

Findings

Analyticity at specific points occurs iff p=m/(m-1) for some integer m>1.

The maximal interval of real-analyticity is identified for these functions.

The radius of convergence of the Taylor series at key points is determined.

Abstract

We identify the maximal real interval on which $\sin_{p,n}$ is real-analytic for any real number $p>1$ and any integer $n>1$. We achieve this by first proving that $\sin_{p,n}$ is analytic at $(1/2){\pi}_{p,n}$ iff $p=m/(m-1)$ for some integer $m>1$, in which case we determine the radius of convergence of the Taylor series at $(1/2){\pi}_{p,n}$.