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Change of base formula for Tetration
#16
bo198214 Wrote:@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 , where the solution oscillates with period 2 (not period 1!) but converges, and , 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.
~ Jay Daniel Fox
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Messages In This Thread
Change of base formula for Tetration - by jaydfox - 08/12/2007, 06:39 AM
RE: Change of base formula for Tetration - by jaydfox - 08/13/2007, 08:03 PM
RE: Parabolic Iteration - by jaydfox - 08/15/2007, 09:19 PM
RE: Parabolic Iteration - by bo198214 - 08/15/2007, 09:30 PM
RE: Parabolic Iteration - by jaydfox - 08/15/2007, 11:41 PM
RE: Parabolic Iteration - by bo198214 - 08/16/2007, 08:17 AM
RE: Parabolic Iteration - by jaydfox - 08/16/2007, 05:51 PM
RE: Parabolic Iteration - by bo198214 - 08/16/2007, 06:40 PM
RE: Parabolic Iteration - by jaydfox - 08/16/2007, 09:47 PM
RE: Parabolic Iteration - by bo198214 - 08/16/2007, 10:07 PM

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