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

TL;DR

This paper compiles a chronological bibliography of various continued compositions, including square roots and powers, up to 2016, focusing on obscure primary sources and excluding well-studied forms like continued fractions.

Contribution

It provides a comprehensive, organized bibliography of lesser-known continued compositions, filling gaps in historical and mathematical references up to 2016.

Findings

Includes diverse continued compositions like roots, powers, cotangents, and logarithms.

Focuses on obscure primary sources not widely documented.

Excludes extensively studied forms such as continued fractions and exponentials.

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.