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... (sequence A030178 in OEIS). 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 is an attractive fixed point of the function .
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
Read more about Omega Constant: Irrationality and Transcendence
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—Frederick Douglass (c.1817–1895)