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base holomorphic tetration
#11
(11/07/2009, 08:21 AM)mike3 Wrote: So then is nth coefficient of mth power of g (truncated).
yes, while truncated is equal to untruncated.

Quote:I thought I also saw somethign ike . Note the positions of the super/subscripts are different.. would that mean the same thing or would that mean to raise the nth coefficient of f to the power m?
yes, the latter. Like one would read it, first take the index then take the power.
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#12
(11/06/2009, 11:29 PM)bo198214 Wrote: , , , .



This gives the following unsimplified coefficients of :


One can see that the coefficients are polynomials in with rational coefficients in .
One needs to investigate whether is analytic in with .

I just wanted to unify the variables: with we can write:
, , or shorter, setting and

, , .

The thing is now that Lambert has a singularity at , i.e. if approaches .

The question is whether this singularity gets compensated somehow by the infinite sum.

I want to further simplify the formula: with
,
where are polynomials in with coefficients that are rational functions in .
I hope i dint put errors somewhere;
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#13
And now, finally, the picture of regular tetration!

   

The red lines are , , , .
The blue lines are
And the green line is the limit .
In the range .

I computed the graphs with the powerseries development with 20 summands and 500 bits precision.

The same picture with x and y equally scaled:
   
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#14
And are the same g-coefficients as what are in the paper?
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#15
You sure that's actually the regular tetration against x or against the fixed point? Because the graph's x-coordinate looks to go way past if that scale is right.
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#16
(11/08/2009, 08:25 PM)mike3 Wrote: And are the same g-coefficients as what are in the paper?

yes.

(11/08/2009, 08:27 PM)mike3 Wrote: You sure that's actually the regular tetration against x or against the fixed point? Because the graph's x-coordinate looks to go way past if that scale is right.

No, thats this damn sage scale. As I wrote the x-axis starts at 1 (so does the y-axis). 1.22 is roughly the middle of . But sage just doesnt get it managed that at least two numbers are shown at every axis, sometimes there is not even one number at the scale.
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#17
(11/08/2009, 08:44 PM)bo198214 Wrote: No, thats this damn sage scale. As I wrote the x-axis starts at 1 (so does the y-axis). 1.22 is roughly the middle of . But sage just doesnt get it managed that at least two numbers are shown at every axis, sometimes there is not even one number at the scale.

Ah. I thought it started at 0... but I suppose that'd be wrong, as the regular iteration only goes down to and is complex-valued for . Oops, my bad... Smile
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#18
Actually some doubts are legitimate, as the convergence radius for bases near is too small than being able to compute the value at 1.

This is due to the fact that the non-integer iterates have a singularity at the upper fixed point. Thatswhy the convergence radius around the lower fixed point can be at most the distance to the upper fixed point.

In the following picture I show this distance from the lower to the upper fixed point (red) - which is the convergence radius - and compares it with the distance of the lower fixed point to 1 - which is the needed convergence radius (in dependency of b at the x-axis).

   

That means that for b right from the intersection of the both curves, the point 1 is not inside the convergence radius of the tetra-power (which is developed at the lower fixed point).
BUT, it seems that the divergent summation above that value is till precise enough.


   
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#19
So what would this indicate? You said "some doubts are legitimate".
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#20
(11/09/2009, 09:07 PM)mike3 Wrote: So what would this indicate? You said "some doubts are legitimate".

I just mean the powerseries convergence at the fixed point if it is near e.
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