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

Abstract

N. G. de Bruijn (1958) studied the asymptotic expansion of iterates of sin$(x)$ with $0<x\le \ddot{I}€/2$. 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$(x)$.

24 pages