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 , where is Lambert's function. The name is derived from the alternate name for Lambert's function, the ''omega function''. The numerical value of is given by

: .
: .

Properties

Fixed point representation

The defining identity can be expressed, for example, as
:$\ln(\tfrac)=\Omega.$
or
:$-\ln(\Omega)=\Omega$
or
:$e^=\Omega.$

Computation

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

:$\Omega_=e^.$

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

It is much more efficient to use the iteration

:$\Omega_=\frac,$

because the function

:$f(x)=\frac,$

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

:$\Omega_=\Omega_j-\frac.$

Integral representations

An identity due to Victor Adamchik is given by the relationship
:$\int_^\infty\frac = \frac.$

Other relations due to Mező
and Kalugin-Jeffrey-Corless.
are:
:$\Omega=\frac\operatorname\int_0^\pi\log\left(\frac\right) dt,$
:$\Omega=\frac\int_0^\pi\log\left(1+\frace^\right)dt.$
The latter two identities can be extended to other values of the function (see also ).

References

External links

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{{Irrational number

Omega
Articles containing proofs
Real transcendental numbers