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value of one omega value of one omega
The omega constant is a mathematical constant defined by 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 0.5671432904097838729999686622... It has properties that or equivalently, One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence This sequence will converge towards Ω as n→∞. This convergence is due to the fact that 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).A beautiful identity due to Victor Adamchik is given by the relationship «»[1]==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 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.….….….….…..if you like my advise just click on approve button ...and please type for any other quary if you have,,,,,,,,,,,,,,,,,,,
The omega constant is a mathematical constant defined by
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 0.5671432904097838729999686622... It has properties that
or equivalently,
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
This sequence will converge towards Ω as n→∞. This convergence is due to the fact that 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).
A beautiful identity due to Victor Adamchik is given by the relationship
«»[1]==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 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.
….….….….…..if you like my advise just click on approve button ...and please type for any other quary if you have,,,,,,,,,,,,,,,,,,,
if you satisfy with my answe so please just click on approve button ...and if you have any other question or dought keep asking
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