The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
Ω e Ω = 1. {\displaystyle \Omega e^{\Omega }=1.}
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 numerical value of Ω is given by
Ω = 0.567143290409783872999968662210... (sequence A030178 in the OEIS). 1/Ω = 1.763222834351896710225201776951... (sequence A030797 in the OEIS).
Properties
Fixed point representation The defining identity can be expressed, for example, as
ln ( 1 Ω ) = Ω . {\displaystyle \ln \left({\tfrac {1}{\Omega }}\right)=\Omega .}
or
− ln ( Ω ) = Ω {\displaystyle -\ln(\Omega )=\Omega }
as well as
e − Ω = Ω . {\displaystyle e^{-\Omega }=\Omega .}
Computation One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
Ω n + 1 = e − Ω n . {\displaystyle \Omega _{n+1}=e^{-\Omega _{n}}.}
This sequence will converge to Ω as n approaches infinity. This is because Ω is an attractive fixed point of the function e−x. It is much more efficient to use the iteration
Ω n + 1 = 1 + Ω n 1 + e Ω n , {\displaystyle \Omega _{n+1}={\frac {1+\Omega _{n}}{1+e^{\Omega _{n}}}},}
because the function
f ( x ) = 1 + x 1 + e x , {\displaystyle f(x)={\frac {1+x}{1+e^{x}}},}
in addition to having the same fixed point, also has a derivative that vanishes there. This guarantees quadratic convergence; that is, the number of correct digits is roughly doubled with each iteration. Using Halley's method, Ω can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also Lambert W function § Numerical evaluation).
Ω j + 1 = Ω j − Ω j e Ω j − 1 e Ω j ( Ω j + 1 ) − ( Ω j + 2 ) ( Ω j e Ω j − 1 ) 2 Ω j + 2 . {\displaystyle \Omega _{j+1}=\Omega _{j}-{\frac {\Omega _{j}e^{\Omega _{j}}-1}{e^{\Omega _{j}}(\Omega _{j}+1)-{\frac {(\Omega _{j}+2)(\Omega _{j}e^{\Omega _{j}}-1)}{2\Omega _{j}+2}}}}.}
Integral representations An identity due to Victor Adamchik is given by the relationship
∫ − ∞ ∞ d t ( e t − t ) 2 + π 2 = 1 1 + Ω . {\displaystyle \int _{-\infty }^{\infty }{\frac {dt}{(e^{t}-t)^{2}+\pi ^{2}}}={\frac {1}{1+\Omega }}.}
Other relations due to Mező and Kalugin-Jeffrey-Corless are:
Ω = 1 π Re ∫ 0 π log ( e e i t − e − i t e e i t − e i t ) d t , {\displaystyle \Omega ={\frac {1}{\pi }}\operatorname {Re} \int _{0}^{\pi }\log \left({\frac {e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}}\right)dt,}
Ω = 1 π ∫ 0 π log ( 1 + sin t t e t cot t ) d t . {\displaystyle \Omega ={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt.}
The latter two identities can be extended to other values of the W function (see also Lambert W function § Representations).
Transcendence The constant Ω is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Ω is algebraic. By the theorem, e−Ω is transcendental, but Ω = e−Ω, which is a contradiction. Therefore, it must be transcendental.
References
External links Weisstein, Eric W. "Omega Constant". MathWorld. "Omega constant (1,000,000 digits)", Darkside communication group (in Japan), retrieved 2017-12-25