10/26/2021, 01:39 PM
(This post was last modified: 10/26/2021, 01:51 PM by sheldonison.)
Fascinating James,
I look forward to spending more time understanding the \(b=\sqrt{2}\) solutions, and how those solutions interact with the iterating functions "logbase" ... and how the two fixed points, 2 and 4 effect the solution, which has a much better chance of being analytic than base(e). Time is limited though during the working week. It would be interesting to see if this solution converges to the Schroeder solution as the logbase period gets larger, where the Schroeder solution is \(\frac{-2\pi i)}{\ln(\ln(2))}\approx17.143i\) periodic. My hunch is that the beta solution does not ... though how it behaves should be interesting. Perhaps we will see superexponential growth from the fixed point of four??? Maybe near \(\Im(z)=18i...19i+n\)
I look forward to spending more time understanding the \(b=\sqrt{2}\) solutions, and how those solutions interact with the iterating functions "logbase" ... and how the two fixed points, 2 and 4 effect the solution, which has a much better chance of being analytic than base(e). Time is limited though during the working week. It would be interesting to see if this solution converges to the Schroeder solution as the logbase period gets larger, where the Schroeder solution is \(\frac{-2\pi i)}{\ln(\ln(2))}\approx17.143i\) periodic. My hunch is that the beta solution does not ... though how it behaves should be interesting. Perhaps we will see superexponential growth from the fixed point of four??? Maybe near \(\Im(z)=18i...19i+n\)
- Sheldon