What do sin$(x)$ and arcsinh$(x)$ have in Common?

TL;DR

This paper explores the asymptotic behavior of iterates of functions like sin(x) and arcsinh(x), revealing their shared properties and extending previous results to a broader class of functions.

Contribution

It reviews and tests an algorithm for deriving asymptotic parameters on various functions, highlighting the commonality between sin(x) and arcsinh(x).

Findings

sin(x) and arcsinh(x) share similar asymptotic properties.

The algorithm effectively derives parameters for transcendental functions.

Functions like the logistic and radical functions exhibit related behaviors.

Abstract

N. G. de Bruijn (1958) studied the asymptotic expansion of iterates of sin$(x)$ with $0 < x \leq \pi/2$. Bencherif & Robin (1994) generalized this result to increasing analytic functions $f(x)$ with an attractive fixed point at 0 and $x > 0$ suitably small. Mavecha & Laohakosol (2013) formulated an algorithm for explicitly deriving required parameters. We review their method, testing it initally on the logistic function $\ell(x)$, a certain radical function $r(x)$, and later on several transcendental functions. Along the way, we show how $\ell(x)$ and $r(x)$ are kindred functions; the same is also true for sin$(x)$ and arcsinh$(x)$.