How do I cite this document and does it say what I think it says? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: How do I cite this document and does it say what I think it says? (/showthread.php?tid=1199) Pages: 1 2 3 4 RE: How do I cite this document and does it say what I think it says? - Chenjesu - 09/10/2018 (09/10/2018, 03:00 PM)sheldonison Wrote: (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 0.0016exp(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 re-centered so that z=-1/e is mapped to zero; that might be the best that we can do for the (-1) branch.