Half-Iterates of $x$exp$(x)$, $x+1/x$ and arcsinh$(x)$

Abstract: Given $\theta(x)$, one method (EJ) for solving Abel's equation $g(\theta(x)) = g(x) \pm 1$ is significantly faster than a rival method (ML). On the other hand, ML evaluates a limit characterizing the principal solution $g(x)$ directly while EJ finds $g(x) + \delta$, where $\delta$ is possibly nonzero but independent of $x$. If an exact expression for $\delta$ is known, then the "intrinsicality" of ML carries over and relative quickness of EJ is preserved. We study $\delta$, as determined by $\theta$, and continue tangentially our earlier exploration of compositional square roots.