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

[sheldonison]

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 \( \text{wseries}=\text{xfixed}+1+\frac{x^2}{2} \) series in the previous post!  Duh, that seems obvious now!

\( f=\sqrt{2(\exp(x)-x-1)};\;\;\;\;\text{wseries}=f^{-1}(x)+1+\frac{x^2}{2} \)
\( W(z)=-\text{wseries}\left(\pm\sqrt{-2(\ln(-z)+1)}\right) \)

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.  

\( \frac{W(-\ln(\sqrt{2}))}{-\ln(\sqrt{2})}=4 \)
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)