Solution to x * e^x = 1
The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
formula_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 Ω is given by
Ω = (sequence in the OEIS).
1/Ω = (sequence in the OEIS).
Properties.
Fixed point representation.
The defining identity can be expressed, for example, as
formula_2
or
formula_3
as well as
formula_4
Computation.
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
formula_5
This sequence will converge to Ω as n approaches infinity. This is because Ω is an attractive fixed point of the function "e"−"x".
It is much more efficient to use the iteration
formula_6
because the function
formula_7
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, Ω can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also ).
formula_8
Integral representations.
An identity due to Victor Adamchik is given by the relationship
formula_9
Other relations due to Mező
and Kalugin-Jeffrey-Corless
are:
formula_10
formula_11
The latter two identities can be extended to other values of the W function (see also ).
Transcendence.
The constant Ω is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Ω is algebraic. By the theorem, "e"−Ω is transcendental, but Ω = "e"−Ω, which is a contradiction. Therefore, it must be transcendental.
References.
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