Omega Constant

$$W(1)\equiv 0.5671432904\ldots,$$ (1)

where $W(x)$ is Lambert's W-Function. It is available in Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) using the function ProductLog[1]. $W(1)$ can be considered a sort of ``Golden Ratio'' for exponentials since

$$\mathop{\rm exp}\nolimits [-W(1)]=W(1),$$ (2)

giving

$$\ln\left[{1\over W(1)}\right]=W(1).$$ (3)

See also Golden Ratio, Lambert's W-Function

References

Plouffe, S. ``The Omega Constant or $W(1)$.'' http://www.lacim.uqam.ca/piDATA/omega.txt.