09/10/2018, 03:00 PM
(This post was last modified: 09/10/2018, 09:26 PM by sheldonison.)
(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).
\( W(z)=-\text{wseries}\left(\pm\sqrt{-2(\ln(-z)+1)}\right) \)
\( a_0=1;\,a_1=1; \)
\( a_n=\frac{1}{n+1}\,\left(a_{n-1}-\sum_{k=2}^{n-1} k\,a_k\,a_{n+1-k}\right) \)
\( \text{wseries}=\sum_{n=0}^{\infty}a_n\,x^n;\;\;\;\text{wseries}=\text{xfixed}+1+\frac{x^2}{2};\;\;\; \) 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