09/10/2007, 06:58 AM
Nice graph!
I think Scott Draves once said that fractals are only half-way between math and art, the other half is coloring algorithms
Anyways, I wanted to give some pointers to info about the Lambert W-function. Mostly anything published by Corless/Jeffery et.al. or Galidakis is a great place to start. I also have done research on it, and one of the series expansions I found for it could only be found on OEIS, but I'm assuming it has to be published somewhere, so I'm including it below.
I'm not sure if series expansions are really what you need, it sounds like you need Maple or Mathematica, but if it helps, here are the two series expansions I know for the solution to \( b = h^{1/h} \):
found in (A000272) and using the Taylor-Puiseux conversion detailed in Abel Functional Equation, the other one is relatively easy to derive:
found in (A033917).
Andrew Robbins
I think Scott Draves once said that fractals are only half-way between math and art, the other half is coloring algorithms
Anyways, I wanted to give some pointers to info about the Lambert W-function. Mostly anything published by Corless/Jeffery et.al. or Galidakis is a great place to start. I also have done research on it, and one of the series expansions I found for it could only be found on OEIS, but I'm assuming it has to be published somewhere, so I'm including it below.
I'm not sure if series expansions are really what you need, it sounds like you need Maple or Mathematica, but if it helps, here are the two series expansions I know for the solution to \( b = h^{1/h} \):
\(
h = {}^{\infty}{b} = \sum^{\infty}_{k=0} \frac{\log(b)^k}{k!} (k+1)^{(k-1)}
\)
h = {}^{\infty}{b} = \sum^{\infty}_{k=0} \frac{\log(b)^k}{k!} (k+1)^{(k-1)}
\)
found in (A000272) and using the Taylor-Puiseux conversion detailed in Abel Functional Equation, the other one is relatively easy to derive:
\(
h = {}^{\infty}{b} = \sum^{\infty}_{k=0} \frac{(b - 1)^k}{k!} \sum^k_{j=0} \left[{k \atop j}\right] (j+1)^{(j-1)}
\)
h = {}^{\infty}{b} = \sum^{\infty}_{k=0} \frac{(b - 1)^k}{k!} \sum^k_{j=0} \left[{k \atop j}\right] (j+1)^{(j-1)}
\)
found in (A033917).
Andrew Robbins