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[Update] Comparision of 5 methods of interpolation to continuous tetration
Ok some guts is needed to admit I do not fully understand and to say I find some things not perfectly well explained.

In particular because it sounds stupid and ungrateful , which I am not.

But it needs to be done.

For Kneser's solution alot of attention is going to the Riemann mapping but the " a priori " is not clear to me.

Maybe Im getting old and I am asking question that I have asked before or understood before so plz forgive me if so.

From my experience it is best to ask very specific questions so I will point to the post I find most confusing.

But first a silly question probably , what I will probably know right after I asked :p

About the schroeder equation

F( f(x) ) = q F(x).

Let c be the fixpoint of f(x).

Now it appears to me that F© must be either 0 or infinite.

What usefull stuff can be said if F© = oo ? Or is that completely useless ?

Second , Why do we prefer f ' © = 1 ?

I assume it is ONLY for the easy solvability of the Taylor series or limit formula for the " principal " schroeder function.

Having probably answered those questions somewhat myself , let continue with the MAIN question(s) and the principal schroeder function.

Since we have f © = 0 we thus have a Taylor series expanded at c.

However there is a limit radius of convergence.

And the real line is not included in the radius , so the trouble begins.

I was only able to find ONE POST adressing how to continue before the Riemann mapping ( all the others seemed to be copied or linked to that " mother post " )

It is still not clear to me what exactlty is mapped , what happened to the singularities and if all that does not loose the property of analyticity.

Also Im not sure what kind of " solution " we are suppose to end up with ?? A sexp that has no singularities for Re > 0 ?

What properties are claimed for the Kneser solution ?
Is it only the property of sexp being analytic near the real line for Re > 0 ?

The post that failed to enlighten me was this one :

- with respect to the poster of course -


It is said : we analyticly continue to ...


What happened to the singularities and limited radius ?

If you use Taylor series you CANNOT have a function that converges on the entire upper plane ; the Taylor series ALWAYS converges in a circle !!?

It is not said how the continuation is done , how we know it is possible , what series expansion we end up with etc etc

So what to make of that ?

I note that mapping a singularity or pole with exp or ln remains a problem ?

Then there follows a claim of simply connected which I find a bit handwaving ?? And what if it contains singularities ??

How is this different from Gottfriend's brown curve ?

I also note that the Riemann mapping may not change the functional equation.

Those 3 pics do not explain all that and perhaps a longer post should have been made.

With respect to the efforts though.

I hope I have sketched what I believe confuses most people about Kneser's method.




Messages In This Thread
RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/28/2013, 11:11 PM

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