The omega constant is a mathematical constant defined by
Contents
- Fixed point representation
- Computation
- Integral representation
- Irrationality and transcendence
- References
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 value of Ω is approximately
Fixed point representation
The defining identity can be expressed, for example, as
or
or
In Mathematica
Plot[{x, Exp[-x], -Log[x]}, {x, 0, 1}]
Computation
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
init = 0.5;
FixedPoint[Exp[-#] &, init]
This sequence will converge towards Ω as n→∞. This convergence is because Ω is an attractive fixed point of the function e−x.
It is much more efficient to use the iteration
because the function
has the same fixed point but features a zero derivative at this fixed point, therefore the convergence is quadratic (the number of correct digits is roughly doubled with each iteration).
Integral representation
A beautiful identity due to Victor Adamchik[1] is given by the relationship
or
Irrationality and transcendence
Ω can be proven irrational from the fact that e is transcendental; if Ω were rational, then there would exist integers p and q such that
so that
and
The number e would therefore be algebraic of degree p. However e is transcendental, so Ω must be irrational.
Ω is in fact transcendental as the direct consequence of Lindemann–Weierstrass theorem. If Ω were algebraic, e−Ω would be transcendental; but Ω=exp(-Ω), so these cannot both be true.