Conjectures

[jaydfox]

Super-logarithm Derivative
From my approximations, \( \text{slog}_e'(0) \approx 0.916 \), and \( {}^{i}{e} \approx 0.786 + i 0.916 \).

My first conjecture is that \( \text{Im}({}^{i}{e}) = \text{slog}_e'(0) \), and my second conjecture is that \( \text{slog}_e'(0) = 0.915965594177\cdots \) otherwise known as Catalan's constant.

In computing the solutions (for base e) to moderately large systems with your proposed slog, I think the first derivative is closer to 0.9159460564995..., which puts Catalan's constant out of the running (or puts your slog out of the running?).

Super-logarithm Equilibrium
My experiments with the super-logarithm have shown that there is a line in which the real part of the complex-valued superlog do not depend on the imaginary part of the input, in other words, my conjecture is that there exist functions f(b) and g(a, b) such that:

\( \text{slog}_b(f(b) + i a) = \text{slog}_b(f(b)) + i g(a, b) \)

Could you provide a bit more info on this one? Again using calculations based on your solution with base e, I've seen a line near z=0.5 (somewhere between 0.45 and 0.5), running almost a unit's length "up" and "down" in the imaginary direction, for which the slog has nearly fixed real part. But it's not quite exact, and as z approaches either primary fixed singularity, the slog eventually gets "sucked in". But maybe you had something else in mind?