# Omega constant

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

[itex] e^{-\Omega}=\Omega[itex]

or equivalently,

[itex] \ln (1/\Omega) = \Omega.[itex]

One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence

[itex] \Omega_{n+1}=e^{-\Omega_n}.[itex]

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

[itex] \frac{p}{q} = \Omega [itex]

so that

[itex] 1 = \frac{p e^{\frac{p}{q}}}{q} [itex]
[itex] e = \sqrt[p]{\frac{q^q}{p^q}} [itex]

and e would therefore be algebraic of degree p. However e is transcendental, so Ω must be irrational.

Omega Constant -- from MathWorld (http://mathworld.wolfram.com/OmegaConstant.html)

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