value of one omega

							
 
 
The omega constant is a mathematical constant defined by
	
It is the value of W(1) where W is Lambert's W function. The name is derived from the alternate name for Lambert's W function, the omega function.
The value of Ω is approximately 0.5671432904097838729999686622... It has properties that
	
or equivalently,
	
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
	
This sequence will converge towards Ω as n→∞. This convergence is due to the fact that Ω is an attractive fixed point of the function e−x.
It is much more efficient to use the iteration
	
because the function
	
has the same fixed point but features a zero derivative at this fixed point, therefore the convergence is quadratic (the number of correct digits is roughly doubled with each iteration).
A beautiful identity due to Victor Adamchik is given by the relationship
	
«»[1]==Irrationality and transcendence==
Ω can be proven irrational from the fact that e is transcendental; if Ω were rational, then there would exist integers p and q such that
	
so that
	
 
	
and e would therefore be algebraic of degree p. However e is transcendental, so Ω must be irrational.
Ω is in fact transcendental as the direct consequence of Lindemann–Weierstrass theorem. If Ω were algebraic, e-Ω would be transcendental; but Ω=exp(-Ω), so these cannot both be true.
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