Editing Omega constant (section)

=== Computation ===
One can calculate {{math|&Omega;}} [[iterative method|iteratively]], by starting with an initial guess {{math|&Omega;<sub>0</sub>}}, and considering the [[sequence]]

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

This sequence will [[limit of a sequence|converge]] to {{math|&Omega;}} as {{mvar|n}} approaches infinity. This is because {{math|&Omega;}} is an [[Fixed point (mathematics)|attractive fixed point]] of the function {{math|''e''<sup>−''x''</sup>}}.

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]], {{math|&Omega;}} can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also {{section link|Lambert W function|Numerical evaluation}}).

:<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>