Solution to x * e^x = 1
The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
![{\displaystyle \Omega e^{\Omega }=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5783bd22cc98558f94ad38a28bd4bd4b0172d3fa)
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
![{\displaystyle \ln({\tfrac {1}{\Omega }})=\Omega .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e39dc2c20c81f9efc47bf64b7a5b0bc0e5c043)
or
![{\displaystyle -\ln(\Omega )=\Omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/47d4470357842b5c12ac37e8b15db7df17aa3d00)
as well as
![{\displaystyle e^{-\Omega }=\Omega .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06d4afeae256cd5ace0c60c7658b279cb440a742)
Computation
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
![{\displaystyle \Omega _{n+1}=e^{-\Omega _{n}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acc355004a3330cc85b7488dd12b8c45a08f5580)
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
![{\displaystyle \Omega _{n+1}={\frac {1+\Omega _{n}}{1+e^{\Omega _{n}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/debd653e587ee7de4b1afe67317224b3d9a3dde7)
because the function
![{\displaystyle f(x)={\frac {1+x}{1+e^{x}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c0744e70cd54fe8b1d28afe6a2936aea5d5652)
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).
![{\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}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/446fa69dbf03ce30f828782e763dce492e969943)
Integral representations
An identity due to [citation needed]Victor Adamchik[citation needed] is given by the relationship
![{\displaystyle \int _{-\infty }^{\infty }{\frac {dt}{(e^{t}-t)^{2}+\pi ^{2}}}={\frac {1}{1+\Omega }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a247d3644ab29a12439d5d2dcab3ac5e36c41ca0)
Other relations due to Mező[1][2]
and Kalugin-Jeffrey-Corless[3]
are:
![{\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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7be38e0bb0040e89b03accf53bdd19f9867c0484)
![{\displaystyle \Omega ={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35b531281fbc5340ae10e230d4086b8f76bce639)
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.[4]
References
External links