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How do I cite this document and does it say what I think it says?
#21
Well I appreciate you taking the time to right about it. When I look at the formula you posted again though, I can understand how it applies to complex bases, but I still don't see how it addresses complex heights. You're still only picking an integer 2 to define the tetration, but I don't see how it tells me what something like  outside of maybe the sexp/slog formula that also doesn't seem to have an explicit representation.
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#22
Also, thanks James, for your comments.

For real bases, 1<b<exp(1/e), there is an attracting real valued fixed point so the wikipedia Schroder equation leads to the solution. For real bases>exp(1/e) there are no real valued fixed points, so then we use Kneser.  For real bases, 1<b<exp(1/e), lets L be the attracting fixed point, .  Then the has a formal solution centered around L, where is the derivative of b^z at the fixed point of L.  so the Taylor series for

For good convergence of , you may have to iterate a few times to get closer to the fixed point of L, but the equation works very well.
There is also a formal series for the inverse: 

This converts to an Abel equation,


Then the desired tetration solution of sexp(z) is:
- Sheldon
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#23
(08/21/2018, 02:33 AM)Chenjesu Wrote: Well I appreciate you taking the time to right about it. When I look at the formula you posted again though, I can understand how it applies to complex bases, but I still don't see how it addresses complex heights. You're still only picking an integer 2 to define the tetration, but I don't see how it tells me what something like  outside of maybe the sexp/slog formula that also doesn't seem to have an explicit representation.

I never worked too much on complex bases. My formula WILL NOT work for complex in your scenario. If you want to work with these bases, I developed a formal solution, but it is god awful, and it only works in the domain



There is no nice way to calculate the function , and I never proved the limiting expressions I had converged. Essentially I can prove that there is a dense subset of call this where is a holomorphic function in (on wildly different domains for each ). Then through some rough limiting arguments we can take where and and prove that (which is kind of where I wasn't sure how to continue). These are highly non unique though and branch cuts pop up wildly and in a crazy chaotic way.

Again though, this is in the hemisphere of knowledge, it's not proven per se in a modern sense.

Sadly, something like is a nightmare.  If you insist, I can write out the formula, but it's a nightmare, a god awful mess of a nightmare.
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#24
(08/21/2018, 02:33 AM)Chenjesu Wrote: Well I appreciate you taking the time to right about it. When I look at the formula you posted again though, I can understand how it applies to complex bases, but I still don't see how it addresses complex heights. You're still only picking an integer 2 to define the tetration, but I don't see how it tells me what something like  outside of maybe the sexp/slog formula that also doesn't seem to have an explicit representation.

Here are a few example bases with b^^(1/3) and b^^(i/3) using Kneser for real bases>exp(1/e), and Schroder for real bases<exp(1/e).  I also included b=i, for both Kneser and Schroder.  Schroder can only be used to calculate b^^z if there is an attracting fixed point for the limit of b^^n as n gets arbitrarily large.  Kneser is not real valued at the real axis for real bases<exp(1/e).  Rigorously extending Kneser to complex bases requires an application of perturbed fatou coordinates, which is the recent work of Shishikura in complex dynamics.  My fatou.gp program can compute tetration for complex bases.  Unfortunately, I only understand understand Shishikura's work at an high level (not in detail); whereas I do understand Kneser in detail.  One could use my program fatou.gp to calculate b^^(i/3) for real and complex valued bases and it would be an analytic function in b, with a singularity at b=exp(1/e).
Code:
base     algorithm            b^^(1/3)                                b^^(i/3)

1.25     Schroder           1.12034262083907             1.03239262899284+0.143429154452345i
sqrt(2)  Schroder           1.17241053375763             1.02664360767221+0.192822433840326i
2        kneser             1.30092658318488             0.998967851029509+0.292980086642819i
e        kneser             1.40302138219711             0.970673924797826+0.356232199460891i
3        kneser             1.43437813759630             0.961475968170544+0.373403025656933i
i        kneser     1.18928618189822+0.535109505735665i  0.719270646867979+0.215917090873838i 
i        Schroder   1.15607095286632+0.428615552193531i  0.712178499068194+0.193070761174410i 
- Sheldon
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#25
Is there by chance a taylor series for the -1 branch of the lambert w function that doesn't rely on a composite double sum with sterling numbers but looks instead as simply as the 0 branch?
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#26
Okay, I'll take over a week of silence to mean "no".
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#27
(08/23/2018, 08:35 PM)Chenjesu Wrote: Is there by chance a taylor series for the -1 branch of the lambert w function that doesn't rely on a composite double sum with sterling numbers but looks instead as simply as the 0 branch?

Personally, I rarely use the LambertW function but these links seem to be relevant to your question.  
See this math-stack question:  https://math.stackexchange.com/questions...ch/2614315
also see this tetration forum thread:  https://math.eretrandre.org/tetrationfor...70#pid7470


Since I don't personally use the LambertW function series formulation, I was hesitant to answer your question.  But one assumes you are interested in this equation for the fixed points.  Then you might also want to see this math-stack question.  I gave the 2nd answer, which gives a formal series for the approach I prefer to use for the two primary fixed points.  I needed both primary fixed points for my fatou.gp program.  The xfixed series is an alternative way to get both fixed point given that pari-gp's implementation of the LambertW function is very limited.  I use it along with Newton's method to get an answer to any desired accuracy.



Code:
{xfixed= x
+x^ 2* -1/6
+x^ 3*  1/36
+x^ 4* -1/270
+x^ 5*  1/4320
+x^ 6*  1/17010
+x^ 7* -139/5443200
+x^ 8*  1/204120
+x^ 9* -571/2351462400
+x^10* -281/1515591000
+x^11*  163879/2172751257600
+x^12* -5221/354648294000
+x^13*  5246819/10168475885568000
+x^14*  5459/7447614174000
+x^15* -534703531/1830325659402240000
+x^16*  91207079/1595278956070800000 + ...
- Sheldon
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#28
Chenjesu Wrote:Is there by chance a taylor series for the -1 branch of the lambert w function?
Sheldon: ... I don't personally use the LambertW function series much ... you might also want to see this math-stack question.  ... the 2nd answer ... gives a formal series ...

With a trivial amount of algebra, the LambertW function can also be expressed in terms of the series in the previous post!  Duh, that seems obvious now!




Using the positive square root in this series gives the Op's desired LambertW -1 branch.  For example with the 16 term wseries below, this equation for the upper fixed point is accurate to about 10^-19.  Using the negative square root gives the LambertW 0 branch and the lower fixed point.  


This seems cool enough that someone must have published it ... update with a little bit of searching, the xseries Taylor Series has been published by Corless, Jeffrey, and Donald Knuth http://www.apmaths.uwo.ca/~djeffrey/Offp...yKnuth.pdf in their 1997 paper "A sequence of Series for the Lambert W function"; see[48].

Code:
wseries= 1 + x + x^2/3 + x^3/36 - x^4/270 + x^5/4320 + x^6/17010 - 139*x^7/5443200 + x^8/204120 - 571*x^9/2351462400
- 281*x^10/1515591000 + 163879*x^11/2172751257600 - 5221*x^12/354648294000 + 5246819*x^13/10168475885568000
+ 5459*x^14/7447614174000 - 534703531*x^15/1830325659402240000 + 91207079*x^16/1595278956070800000 + O(x^17)
- Sheldon
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#29
I appreciate the work but the question was not limited to fixed points since the W function has a more general relationship to tetration. I looked on wikipedia and noticed that for some reason the Taylor series for the -1 branch is drastically more complicated, and so I was wondering if it has a simpler series representation.
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#30
(09/10/2018, 12:27 PM)Chenjesu Wrote: I appreciate the work but the question was not limited to fixed points since the W function has a more general relationship to tetration. I looked on wikipedia and noticed that for some reason the Taylor series for the -1 branch is drastically more complicated, and so I was wondering if it has a simpler series representation.

The nice thing about this particular implementation of LambertW for the W-1 and W0 branch pair, is that it has very nice convergence properties.  For example, this LambertW series converges for all z where 0.0016<abs(z)<84, plus many other points points with abs(z)<197.  Normally, this series would be used as a seed along with Newton's method.  The authors also give a closed form for the coefficients of the series in their paper (see below).




 relationship to my xfixed series in post#27

Anyway, -0.00069 at the limit of convergence is not zero, though it corresponds to an upper fixed point of ~13817 for b=1.00069.  So the question is how does the Lambert -1 branch singularity behave near z=0, and is there an asymptotic?
- Sheldon
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