08/20/2009, 01:44 PM
(This post was last modified: 08/20/2009, 06:12 PM by sheldonison.)

(08/20/2009, 10:28 AM)bo198214 Wrote: Though currently I wonder whether these arbitrary close singularities indeed imply that the function is not analytic in any point.Henryk, I finally caught up with your formula for the singularities -- its very helpful. I have a pretty good intuitive feel for the singularities for small values of k.

I mean there is a theorem that if a holomorphic function sequence converges locally uniformly (i.e. for each point there is a neighborhood where it converges uniformly) then the limit is again a holomorphic function (which is not true for just differentiable functions).

However I dont think that the inverse statement is also true, that if a function sequence does not converge locally uniformly that then resulting function can not be holomorphic.

For example a sequence of non-continuous functions can have a continuous function as a limit. Also Jay showed that the singularities gets milder with increasing n. So there maybe a very little tiny hope that the resulting function is analytic despite.

It appears there are about 500,000 or so n=3 singularities in the critical strip used by the base change equation (from 5.016 to 6.330). I would assume the n=4 singularities would be super-exponentially denser yet.

k=1, 4.8688+0.5713i

k=2, 5.0732+0.4586i

k=3, 5.1734+0.4068i

k=500,000 6.3301+0.0706i

In looking at the singularities for small values of k, It seems that the function becomes undefined (or multi-valued?), once passing the neighborhood of the singularity. Each singularity is associated with a particular increment of the windings. Even ignoring the singularities associated with very large values of k, values of n>4, (which approach arbitrarily close to the real axis), can we continue the function for smaller of k, where n=4 as opposed to n=infinity?

Typo correction: Actually I used n=3 in Henryk's equation, but to see the singularities in the "f" base change equation below requires using n=4.

I have not yet made the leap to understanding the singularities associated with larger values of k, and how they change the behavior of f, but I hope to do so.

- Sheldon