Green Eggs and HAM: Tetration for ALL bases, real and complex, now possible?

[sheldonison]

Hi.

I wanted to report to you some results I had trying out a new tetration method, or, well, actually a new twist on an old method. It's based on Kouznetsov's Cauchy integral method, only with a new and powerful method to solve the integral equation. It's actually just a new way of solving the integral equation in the Cauchy integral method.

What was the problem with Kouznetsov's method? While now it seems like it works for real bases greater than \( \eta \) and a lot of complex bases, it doesn't seem to work for the difficult challenge bases \( b = -1 \), \( b = 0.04 \), and \( b = e^{1/e} \).....

I'm on my cellphone... not computer. This method sounds very exciting! You should publish it. Biggest problem with Kouznetsov's method is finite rectangle in imag (z) and discreet sampling. Perhaps an infinite rectangle ?Riemann mapping? to a unit circle? Probably not the approach you're thinking about ...

Anyway, I would definitely be interested in details on your new ideas, and look forward to subsequent posts. Does it work for real bases less than \( \lt \exp(\frac{1}{e}) \)? Kouznetsov's method relies on limiting behavior at \( +/-\Im(\infty) \), whereas these bases are periodic in \( \Im(z) \).
- Sheldon