(09/10/2018, 03:00 PM)sheldonison Wrote:So I don't understand exactly what this wseries and xpoint is supposed to mean, I've never seen anyone use that notation. If it exists then what is relevant is a typical Taylor series (or some kind of functional series) just with a standard sum from n_0 to infinity for the 1 branch and that's it, nothing fancy, no fixed points or computational approximations.(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 W1 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
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?
How do I cite this document and does it say what I think it says?

09/10/2018, 09:30 PM
(This post was last modified: 09/15/2018, 01:47 AM by sheldonison.)
(09/10/2018, 06:50 PM)Chenjesu Wrote: So I don't understand exactly what this wseries and xpoint is supposed to mean, I've never seen anyone use that notation. If it exists then what is relevant is a typical Taylor series (or some kind of functional series) just with a standard sum from n_0 to infinity for the 1 branch and that's it, nothing fancy, no fixed points or computational approximations. wseries is just the Taylor series from this paper[45] ; we can express LambertW as follows where for negative z the +sqrt is the (1) branch the Op asked about. I'm not sure what nonstandard notation I'm using; this is just function composition ... Post#28 has the first 16 terms of the wseries Taylor series. The recursive formula for the coefficients was provided in post#30. wseries has the same a_3 to a_oo Taylor series coefficients as my xfixed series from post#27 which is before I discovered Corless's paper. My best guesss is that perhaps Chenjesu just views this as too complicated a solution, and he is looking for a simpler series. The simpler LambertW Taylor series for the main branch at z=0 only has a radius of convergence of 1/e, and won't work for the fixed points of any base>exp(1/e). So if that is Chenjesu's complaint, then yes, this is a more complicated series, but it is much more powerful since it gives the both the main branch and the (1) branch, and since it converges for a fairly large subset of the complex plane. Since the (1) branch is only real valued at the real axis from 1/e to 0, and it has a really complicated singularity at 0, so there is no hope of getting any series in x centered at x=0. Since the (1) branch also has a square root branch at 1/e, that requires a square root term in the composition, so that can't be a simple series either. There probably aren't any other rational x,W(x) pairings besides at 1/e. The approach from Corless's paper has rational coefficients, and a square root in the substitution and is recentered so that z=1/e is mapped to zero; that might be the best that we can do for the (1) branch.
 Sheldon

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