10/29/2013, 09:37 AM

(10/28/2013, 11:29 PM)tommy1729 Wrote: The only thing I can come up with is that :

1) we simply take the principal schroeder function near the fixpoint c.

2) in that domain we use analytic continuation ( monodromy ) so that by small radiuses recentered we can reach any value in the upper plane.

3) In that upper plane we find ( after many continuations ) the values equalling the reals between 0 and 1. ( and those are connected )

( for reasons yet to be explained !)

.....

10) we have sexp analytic near the positive real line.

....

And if I got It right , there is still alot to be explained !!

Tommy,

Is your question about using Kneser's method to get numerical results, or is it about Kneser's proof of a real valued sexp(z) solution? It is not practical to use Kneser's method to generate usable results for sexp(z), though it is a proof of the existence of such a real valued solution, even though numerically, Kneser's Riemann mapping is very difficult to work with. Also, Kneser's proof is hard to follow, and I am not good at math to recreate it.

On the other hand, I wrote a pari-gp program that is numerically equivalent to Kneser's method that is practical, though I cannot rigorously prove my method converges. But assuming my method converges, than I can show it converges to the same solution as would be given by Kneser's Riemann mapping solution. Obviously, I could explain my method, and its equivalence to Kneser's solution, but that would not be relevant to your questions.

- Sheldon