Omega constant
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The Omega constant 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, viz., the Omega function.
The constant, Ω, is approximately 0.56714. It has properties that are akin to those of the golden ratio, in that
- <math> e^{-\Omega}=\Omega<math>
or equivalently,
- <math> \ln (1/\Omega) = \Omega.<math>
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
- <math> \Omega_{n+1}=e^{-\Omega_n}.<math>
This sequence will converge towards Ω as n→∞.
Irrationality
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
- <math> \frac{p}{q} = \Omega <math>
so that
- <math> 1 = \frac{p e^{\frac{p}{q}}}{q} <math>
- <math> e = \sqrt[p]{\frac{q^q}{p^q}} <math>
and e would therefore be algebraic of degree p. However e is transcendental, so Ω must be irrational.
See also
External link
Omega Constant -- from MathWorld (http://mathworld.wolfram.com/OmegaConstant.html)