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# value of one omega

Gayatri Jayesh Bondriya
521 Points
6 years ago

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 ex.

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 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,,,,,,,,,,,,,,,,,,,

Gayatri Jayesh Bondriya
521 Points
6 years ago
if  you   satisfy  with   my  answe  so please   just click on approve  button  ...and  if  you  have  any  other  question   or  dought     keep asking