Omega Constant

$\begin{displaymath} W(1)\equiv 0.5671432904\ldots, \end{displaymath}$

(1)

where

is Lambert's W-Function. It is available in Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) using the function ProductLog[1].

can be considered a sort of ``Golden Ratio'' for exponentials since

$\begin{displaymath} \mathop{\rm exp}\nolimits [-W(1)]=W(1), \end{displaymath}$

(2)

giving

$\begin{displaymath} \ln\left[{1\over W(1)}\right]=W(1). \end{displaymath}$

(3)

See also Golden Ratio, Lambert's W-Function

References

Plouffe, S. ``The Omega Constant or .'' http://www.lacim.uqam.ca/piDATA/omega.txt.