How do I cite this document and does it say what I think it says?

[Chenjesu]

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

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?

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.