A chronology of continued square roots and other continued compositions, through the year 2016

Abstract

An infinite continued composition is an expression of the form \begin{equation*} \lim_{n\to\infty}t_0\circ t_1 \circ t_2 \circ \cdots \circ t_n(c)\;, \end{equation*} where the $t_i$ are maps from a set $D$ to itself, the initial value $c$ is a point in $D$, and the order of operations proceeds from right to left. This document is a bibliography, in chronological order through the year 2016, of selected continued compositions whose primary sources have typically been obscure. In particular, we include continued square roots: \begin{equation*} a_0+\sqrt{a_1+\sqrt{a_2+\sqrt{\ldots}}}\;, \end{equation*} as well as continued powers, continued cotangents, continued logarithms, and $f$-expansions. However, we do not include continued fractions, continued exponentials, or forms such as infinite sums and products in which the $t_i$ are linear functions, because the literature on these forms is extensive.