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
Famous quotes containing the word constant:
“I should say that the most prominent scientific men of our country, and perhaps of this age, are either serving the arts and not pure science, or are performing faithful but quite subordinate labors in particular departments. They make no steady and systematic approaches to the central fact.... There is wanting constant and accurate observation with enough of theory to direct and discipline it. But, above all, there is wanting genius.”
—Henry David Thoreau (1817–1862)