Omega constant

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

<math>\Omega e^\Omega = 1.</math>

It is the value of Template:Math, where Template:Mvar is [[Lambert W function|Lambert's Template:Mvar function]]. The name is derived from the alternate name for Lambert's Template:Mvar function, the omega function. The numerical value of Template:Math is given by

Template:Math (sequence A030178 in the OEIS).

Template:Math (sequence A030797 in the OEIS).

Properties

Fixed point representation

The defining identity can be expressed, for example, as

<math>\ln \left(\tfrac{1}{\Omega} \right)=\Omega.</math>

or

<math>-\ln(\Omega)=\Omega</math>

as well as

<math>e^{-\Omega}= \Omega.</math>

Computation

One can calculate Template:Math iteratively, by starting with an initial guess Template:Math, and considering the sequence

<math>\Omega_{n+1}=e^{-\Omega_n}.</math>

This sequence will converge to Template:Math as Template:Mvar approaches infinity. This is because Template:Math is an attractive fixed point of the function Template:Math.

It is much more efficient to use the iteration

<math>\Omega_{n+1}=\frac{1+\Omega_n}{1+e^{\Omega_n}},</math>

because the function

<math>f(x)=\frac{1+x}{1+e^x},</math>

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, Template:Math can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also Template:Section link).

<math>\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}}.</math>

Integral representations

An identity due to Victor AdamchikTemplate:Cn is given by the relationship

<math>\int_{-\infty}^\infty\frac{dt}{(e^t-t)^2+\pi^2} = \frac{1}{1+\Omega}.</math>

Other relations due to Mező<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite arXiv.</ref> and Kalugin-Jeffrey-Corless<ref>Template:Cite arXiv.</ref> are:

<math>\Omega=\frac{1}{\pi}\operatorname{Re}\int_0^\pi\log\left(\frac{e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}\right) dt,</math>

<math>\Omega=\frac{1}{\pi}\int_0^\pi\log\left(1+\frac{\sin t}{t}e^{t\cot t}\right)dt.</math>

The latter two identities can be extended to other values of the Template:Mvar function (see also Template:Section link).

Transcendence

The constant Template:Math is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Template:Math is algebraic. By the theorem, Template:Math is transcendental, but Template:Math, which is a contradiction. Therefore, it must be transcendental.<ref name="Mezo">Template:Cite journal</ref>

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:OmegaConstant%7COmegaConstant.html}} |title = Omega Constant |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Template:Citation