Omega Constant

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