Change of base formula for Tetration

[jaydfox]

@Jay for publishing you have to provide proofs, thats for sure.

The only item I haven't proven is that the power series for the fractional iteration of e^z-1 converges for some radius of convergence greater than 0. Some sources say it only converges for integer iterations, others say it has non-zero radius of convergence for fractional iterations.

So either the proof exists somewhere, or the disproof exists somewhere, or everybody's wrong so far. I just need to figure out which the case is. If I can find or provide a proof that fractional iteration of e^z-1 converges and is unique, then combined with my change of base formula, I've got "the" unique solution to tetration for all bases greater than eta.

I'm also working on a method for bases between 1 and eta. Bases in that range have three partitions, so each solution will need a formula.

The tricky ranges are \( e^{-e^2}\ \le\ b\ <\ 1 \), where the solution oscillates with period 2 (not period 1!) but converges, and \( 0\ <\ b\ <\ e^{-e^2} \), where the solution oscillates with period 2 and does not converge. Other than a sine function as part of generating the oscillations, I don't have much of a good starting point for those bases.