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

[mike3]

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

Thought I'd comment on the idea for real bases less than \( \eta = e^{\frac{1}{e}} \). I suspect it could, but haven't tried. If we wanted to try and recover the regular iteration for these bases, one could modify the Cauchy integral equation so as to have the upper and lower part of the contour as parallel to the real axis and at distance each equal to the (magnitude of the) period. I suspect then one needs two grids of sample points, one on the imaginary axis and one on the real axis, to achieve the integration, so this would require a significant modification to the existing program.

Kouznetsov mentioned about this here:
http://math.eretrandre.org/tetrationforu...d-248.html

On the other hand, it may also be possible to generate the merged Kneser solution (complex-valued at the real axis but (apparently) analytically compatible with the solution for \( b > \eta \)) using a modified Cauchy integral equation where the axis the contour envelope is skew, going diagonally from the lower left part of the plane to the upper right, on which the function would behave as one asymptotically approaching the fixed points, like for other bases. This would require less modification, since we can still make do with only one grid.

Such a contour would look something like this:
(the graph in the background is for base \( b = \sqrt{2} \), obtained via your method, and the dotted line is the one on which the sampling nodes would be put)