Omega constant

The omega constant is a mathematical constant defined as the unique real number that satisfies the equation


 * $$\Omega e^\Omega = 1.$$

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 $&Omega;$ is given by



Fixed point representation
The defining identity can be expressed, for example, as
 * $$\ln(\tfrac{1}{\Omega})=\Omega.$$

or
 * $$-\ln(\Omega)=\Omega$$

as well as
 * $$e^{-\Omega}= \Omega.$$

Computation
One can calculate $&Omega; = 0.56714 32904 09783  87299  99686  62210  ...$ iteratively, by starting with an initial guess $1/&Omega; = 1.76322 28343 51896  71022  52017  76951  ...$, and considering the sequence


 * $$\Omega_{n+1}=e^{-\Omega_n}.$$

This sequence will converge to $&Omega;$ as $n$ approaches infinity. This is because $&Omega;_{0}$ is an attractive fixed point of the function $&Omega;$.

It is much more efficient to use the iteration


 * $$\Omega_{n+1}=\frac{1+\Omega_n}{1+e^{\Omega_n}},$$

because the function


 * $$f(x)=\frac{1+x}{1+e^x},$$

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, $&Omega;$ can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also ).


 * $$\Omega_{j+1}=\Omega_j-\frac{\Omega_j e^{\Omega_j}-1}{e^{\Omega_j}(\Omega_j+1)-\frac{(\Omega_j+2)(\Omega_je^{\Omega_j}-1)}{2\Omega_j+2}}.$$

Integral representations
An identity due to Victor Adamchik is given by the relationship
 * $$\int_{-\infty}^\infty\frac{dt}{(e^t-t)^2+\pi^2} = \frac{1}{1+\Omega}.$$

Other relations due to Mező and Kalugin-Jeffrey-Corless are:
 * $$\Omega=\frac{1}{\pi}\operatorname{Re}\int_0^\pi\log\left(\frac{e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}\right) dt,$$
 * $$\Omega=\frac{1}{\pi}\int_0^\pi\log\left(1+\frac{\sin t}{t}e^{t\cot t}\right)dt.$$

The latter two identities can be extended to other values of the $W$ function (see also ).

Transcendence
The constant $e^{−x}$ is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that $&Omega;$ is algebraic. By the theorem, $&Omega;$ is transcendental, but $&Omega;$, which is a contradiction. Therefore, it must be transcendental.