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Real and complex behaviour of the base change function (was: The "cheta" function)
#31
Though currently I wonder whether these arbitrary close singularities indeed imply that the function is not analytic in any point.
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.
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#32
(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.
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.
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.

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
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#33
(08/20/2009, 01:44 PM)sheldonison Wrote: 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.

The singularities are branching points (like log has a branching point at 0). So properly there must be a cut associated with the singularity. I see whether I make a picture of the cuts associated with these singularities. But each , is multivalued, according along which path you continue the function to a particular point. (like the logarithm depends on how often winds the continuing path around the singularity at 0, the value of depends on how the path winds around any singularity. The values are only equal if you continue the function along two paths that you can deform into each other without crossing a singularity.)
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#34
(08/20/2009, 03:03 PM)bo198214 Wrote: The values are only equal if you continue the function along two paths that you can deform into each other without crossing a singularity.)
Or, interestingly enough, if you wind around non-trivial singularities in such a manner that the windings cancel out.

As you deform the paths to determine that they are the same, you may have to push one or more trivial singularities through a path. Each time you push a trivial singularity "through" a path, you keep of track of whether it crossed from the right or left side (relative to the direction of travel along the path from the starting point), and if the number of right and left singularities (corresponding to windings in opposite directions) are the same, then you can still manage to arrive at the same value. This is not necessarily true for the non-trivial singularities, and I suspect it is false.
~ Jay Daniel Fox
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#35
(08/21/2009, 09:11 PM)jaydfox Wrote: As you deform the paths to determine that they are the same, you may have to push one or more trivial singularities through a path. Each time you push a trivial singularity "through" a path, you keep of track of whether it crossed from the right or left side (relative to the direction of travel along the path from the starting point), and if the number of right and left singularities (corresponding to windings in opposite directions) are the same, then you can still manage to arrive at the same value.

Oh thats indeed interesting. So is the branch (say on the real axis to have a closed path) determined by the sum of the (oriented) winding numbers around each trivial singularity (assuming no windings around other singularities)?
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#36
(08/21/2009, 09:54 PM)bo198214 Wrote:
(08/21/2009, 09:11 PM)jaydfox Wrote: As you deform the paths to determine that they are the same, you may have to push one or more trivial singularities through a path. Each time you push a trivial singularity "through" a path, you keep of track of whether it crossed from the right or left side (relative to the direction of travel along the path from the starting point), and if the number of right and left singularities (corresponding to windings in opposite directions) are the same, then you can still manage to arrive at the same value.

Oh thats indeed interesting. So is the branch (say on the real axis to have a closed path) determined by the sum of the (oriented) winding numbers around each trivial singularity (assuming no windings around other singularities)?
Effectively, yes. I'm trying to test different types of paths to determine if there are exceptions, but as far as I can tell, it works approximately like this.

Given a starting and ending point, it is possible to create a path between them that starts and ends in the principal branch, even if it must sometimes leave the principal branch. I'm working on pictures of what I mean, so in the meantime, hopefully you can picture what I mean.

If this path is used to close the path we are interested in (the one that goes "through" the thicket of singularities), then we can simply count the number of windings around each singularity (most will be one winding), to determine which branch we are in for the first of the iterated logarithms. Care must be taken not to enclose a non-trivial singularity, but other than that, I'm not sure if there are any exceptions.

Here's the weird part: Unless we choose a path that "unwinds" the windings of the enclosed region, we will find that the image of the first of the iterated logarithms will simply go out along the imaginary axis, even as the real part sort of oscillates between very large and small values. The next logarithm will then give us some oscillating path that remains in the strip between 0*i and pi*i, such that the remaining iterated logarithms will converge on the real axis.

This has me concerned that the base-change formula is truly undefined for non-real values, because does it even make any sense to say that all complex values are somehow mapped back to a real number (which would be the result as n goes to infinity)? Worse yet, this mapping is determined by the path taken.
~ Jay Daniel Fox
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#37
(08/21/2009, 10:49 PM)jaydfox Wrote: Given a starting and ending point, it is possible to create a path between them that starts and ends in the principal branch, even if it must sometimes leave the principal branch. I'm working on pictures of what I mean, so in the meantime, hopefully you can picture what I mean.
Normally not, but in this case, yes Wink

Quote:This has me concerned that the base-change formula is truly undefined for non-real values,

I was thinking about a different aproach: I would say all singularities of all are bounded. So out there in the complex plane are points which are distant from all singularities of all . Perhaps one can define the value of each at by using a path that has all singularities to the right side, respectively.
Then one would show that converges uniformly in a neighborhood of . The limit function in the neighborhood of is necesarily again holomorphic there.
From *there* we continue the function to the real line. I guess has only singularities on the real line and at the fixed points of .
Then it would turn out that has an *asymptotic* power series development (for a certain sector of approach) at all points of the (upper) real axis. Which though of course has zero convergence radius.
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#38
(08/21/2009, 10:49 PM)jaydfox Wrote:
(08/21/2009, 09:54 PM)bo198214 Wrote:
(08/21/2009, 09:11 PM)jaydfox Wrote: As you deform the paths to determine that they are the same, you may have to push one or more trivial singularities through a path. Each time you push a trivial singularity "through" a path, you keep of track of whether it crossed from the right or left side (relative to the direction of travel along the path from the starting point), and if the number of right and left singularities (corresponding to windings in opposite directions) are the same, then you can still manage to arrive at the same value.

Oh thats indeed interesting. So is the branch (say on the real axis to have a closed path) determined by the sum of the (oriented) winding numbers around each trivial singularity (assuming no windings around other singularities)?



This has me concerned that the base-change formula is truly undefined for non-real values, because does it even make any sense to say that all complex values are somehow mapped back to a real number (which would be the result as n goes to infinity)? Worse yet, this mapping is determined by the path taken.

well C and R have the same cardinality so there are functions from C to R.

But those are not analytic functions.

But why map to reals only ?

why not map non-real complex numbers to non-real complex numbers ?


regards

tommy1729
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#39
(08/13/2009, 07:17 AM)bo198214 Wrote: Walker showed a similar convergence for , where dexp(x) = exp(x)-1.
He showed that the limit is infinitely differentiable on the real axis.
That means that he also wasnt clear about the complex behaviour otherwise he would have shown that the limit is holomorphic as a consequence of local uniform convergence.
But he could prove that local uniform (or compact) convergence only on the real axis, which does not suffice to imply holomorphy (because it could be that during the convergence non-real singularities get dense towads points on the real axis). I will persue this topic in the next days and have still some unexplored ideas at my hands.
Henryk, Do you have a reference for showing that the base change function is infinitely differentiable? Perhaps Walker's paper? I am now able to generate approximations for the Taylor series coefficients for the base change function. This would lead to a way of showing that the function is infinitely differentiable at the real axis, since all of the coefficients converge. But it would also lead to a way to show that the function is nowhere analytic, since the coefficients for large enough n eventually grow faster then any exponential. I wanted to read Walker's paper to see what his approach was, before posting my results.
- Sheldon
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#40
(08/13/2011, 10:32 AM)sheldonison Wrote: Henryk, Do you have a reference for showing that the base change function is infinitely differentiable? Perhaps Walker's paper?

There are two papers I know of:

Viana da Silva, M. (1988). The differentiability of the hairs of exp(Z). Proc. Am. Math. Soc., 103(4), 1179–1184.

Walker, P. L. (1991). Infinitely differentiable generalized logarithmic and exponential functions. Math. Comput., 57(196), 723–733.

I send them via e-mail to you.
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