Half-Iterates and Delta Conjectures

Abstract

The vivid contrast between two competing algorithms for solving Abel's equation $g(\hat{I}¸(x))=g(x)+1$, given $\hat{I}¸(x)$, is easily sketched. EJ is faster and more efficient, but ML evaluates a limit characterizing the principal solution $g(x)$ directly. EJ finds $g(x)+\hat{I}´$, where $\hat{I}´$ is possibly nonzero but independent of $x$. If we were to know an exact expression for $\hat{I}´$, then the "intrinsicality" of ML would be subsumed by EJ. Filling this gap in our knowledge is the aim of this paper.

Two distinct earlier preprints are relevant. The first covers half-iterates of $x(1+x)$, sin$(x)$ & exp$(x/e)$ and appears at arXiv:2506.07625v1. The second covers half-iterates of $x$exp$(x)$, $x+1/x$ & arcsinh$(x)$ and appears at arXiv:2506.07625v2. Reading these will help to motivate the study of $\hat{I}´$ in the current paper. 9 pages; 2 figures