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 Kneser method question tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 01/28/2020, 09:02 PM It has been a while since I considered kneser’s method. So I will try to describe it and then ask my question. Correct me if I make mistake. Step one : We use the first upper fixpoint of exp : L. From L we compute the solution to  Exp(f(z))  = f(L z) By using the koenigs function. Step 2 : We find the value 1.  From this 1 we trace the positive reals. We notice the reals make a corner at every rotation L or equivalently at 1,e,e^e,.... First question : why at these values and not at say pi,exp(pi),exp(exp(pi)),... ? Second question : this indicates there are singularities at 1,e,e^e,.... What kind of singularities are they ??  The corners seem to suggest log or sqrt or such, but I am not sure. Third question : do we pick branches ? I thought we did not.  I can imagine that statistically such singularities or corners are expected because we need to respect the rotation. But why it is necessary, I have no formal explaination. So I guess that makes it question 4. We continue. First we take a log base L to solve  Exp(g(z)) = g(z) + 1 Notice we added a log on top of the singularities , right ? So we arrive at question 5 and 6 : Question 5 : did the log remove or simplify the singularities ? This is only possible with powers like log x^a = a log x I think. Question 6 : all singularities are still there right ?? Ok, so now we have a kind of Abel function g(z) but it does not map the positive reals to the positive reals. Notice also the positive reals values are now arranged 1-periodic within the function g(x). But often stacked on top of each other, hence unfortunately not describable by a Fourier series. So we need to map the positive reals to the positive reals kinda. We know that is possible with a somewhat unique analytic function from riemann’s mapping theorem. However the riemann mapping is mysterious to many. Apart from how it is done and closed form issues , error terms etc there is also this : Question 7 : How does the riemann mapping not destroy the functional equation ? Question 8 : The big question : How does the riemann mapping remove those singularities at 1,e,e^e,... ?? Question 9 : what happened to the other singularities ? I have the feeling I’m not the only wondering about these things. Regards Tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread Kneser method question - by tommy1729 - 01/28/2020, 09:02 PM RE: Kneser method question - by Gottfried - 01/28/2020, 10:19 PM RE: Kneser method question - by Daniel - 01/31/2020, 11:52 PM RE: Kneser method question - by sheldonison - 02/03/2020, 01:00 AM RE: Kneser method question - by Ember Edison - 02/07/2020, 11:41 PM excel plots for chistar - by sheldonison - 02/08/2020, 02:43 PM RE: excel plots for chistar - by Ember Edison - 02/08/2020, 04:14 PM RE: excel plots for chistar - by sheldonison - 02/09/2020, 10:27 AM RE: excel plots for chistar - by Ember Edison - 02/10/2020, 09:00 PM RE: excel plots for chistar - by sheldonison - 02/11/2020, 01:26 AM

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