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$(x)$.

翻译：N. G. de Bruijn (1958) 研究了当 $0 < x \leq \pi/2$ 时正弦函数 sin$(x)$ 迭代的渐近展开。Bencherif 与 Robin (1994) 将此结果推广至在 0 处具有吸引不动点、且 $x > 0$ 充分小的递增解析函数 $f(x)$。Mavecha 与 Laohakosol (2013) 提出了一种用于显式推导所需参数的算法。我们回顾了他们的方法，首先在逻辑函数 $\ell(x)$、某一根式函数 $r(x)$ 上进行了测试，随后又应用于若干超越函数。在此过程中，我们展示了 $\ell(x)$ 与 $r(x)$ 为何是同类函数；这一结论同样适用于 sin$(x)$ 与 arcsinh$(x)$。