Omega constant
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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 are akin to those of the golden ratio, in that
or equivalently,
- log(1 / Ω) = Ω.
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
This sequence will converge towards Ω as n→∞.
[edit] 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
![e = \sqrt[p]{\frac{q^q}{p^q}}](../../../math/5/f/d/5fd7ff5ffff957d4bce919c317b2914b.png)
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, exp(Ω) would be transcendental and so would be exp−1(Ω). But this contradicts the assumption that it was algebraic.
[edit] See also
[edit] External links
- Weisstein, Eric W., Omega Constant at MathWorld.
