Omega Constant

Author

Eric W. Weisstein
August 13, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/OmegaConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/OmegaConstant.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

V6

Solve[Exp[-x]==x,x]

NSolve[Exp[-x]==x,x]

V7

Solve[Exp[-x]==x,x]

FindInstance[Exp[-x]==x,x]

Digits

N[ProductLog[1],20]

Nest

ProductLog[1.]

Table[Nest[Exp[-#]&,Random[Complex,{-1,1}2(I+1)],100],{10}]

Table[Nest[Exp[-#]&,Random[Real,{0,10}],100],{10}]

Table[Nest[(1/E)^#&,Random[Real,{0,10}],100],{10}]

-ProductLog[-Log[1/E]]/Log[1/E]

Integrals

(∫_(-∞)^∞)(1/(((((e^x)-x))^2)+(π^2)))dx

Plot[(1/((((Exp[x]-x))^2)+(π^2))),{x,-20,20},PlotStyle->Red,PlotRange->All]

NIntegrate[(1/((((Exp[x]-x))^2)+(π^2))),{x,-∞,∞}]

N[(1/(1+ProductLog[1])),20]

v5.2

Integrate[(1/((((Exp[x]-x))^2)+(π^2))),{x,-∞,∞}]//Timing

v6

Integrate[(1/((((Exp[x]-x))^2)+(π^2))),{x,-∞,∞}]//Timing

V7

Integrate[(1/((((Exp[x]-x))^2)+(π^2))),{x,-∞,∞}]//Timing