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
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