en>Zyxw: add category and link

2015-01-10T05:23:53Z

add category and link

78.146.112.194: /* Simulation argument */ deleted two word to make the article make more sense

2013-11-24T21:29:19Z

Simulation argument: deleted two word to make the article make more sense

← Older revision		Revision as of 23:29, 24 November 2013
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	I'~~m Loyd and I live in~~ a ~~seaside city in northern United States~~, ~~Baltimore~~. I'~~m 37 and I~~'~~m will soon finish my study at Neuroscience~~.<br><br>~~Here is my site~~ ... [~~http~~://~~Pocketbook~~.su/~~forum~~/~~skoraya~~-~~pomoshch/fifa~~-~~coin~~-~~generator~~-~~0 fifa 15 coin Generator~~]		The '''omega constant''' is a [[mathematical constant]] defined by

			:<math>\Omega\,e^{\Omega}=1.\,</math>

			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... {{OEIS\|id=A030178}}. It has properties that

			:<math> e^{-\Omega}=\Omega,\,</math>

			or equivalently,

			:<math> \ln \Omega = - \Omega.\,</math>

			One can calculate Ω [[iterative method\|iteratively]], by starting with an initial guess Ω<sub>0</sub>, and considering the [[sequence]]

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

			This sequence will [[limit of a sequence\|converge]] towards Ω as ''n''→∞. This convergence is due to the fact that Ω is an [[Fixed point (mathematics)\|attractive fixed point]] of the function ''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>

			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

			:<math> \Omega=\frac{1}{\displaystyle \int_{-\infty}^{+\infty}\frac{\,dt}{(e^t-t)^2+\pi^2}}-1 .</math>

			==Irrationality and transcendence==

			Ω can be proven [[irrational number\|irrational]] from the fact that [[e (mathematical constant)\|e]] is [[transcendental number\|transcendental]]; if Ω were rational, then there would exist integers ''p'' and ''q'' such that

			: <math> \frac{p}{q} = \Omega </math>

			so that

			: <math> 1 = \frac{p e^{\left( \frac{p}{q} \right)}}{q}</math>

			<!-- extra blank line for legibility; these two displays crowd each other -->

			: <math> e = \left( \frac{q}{p} \right)^{\left( \frac{q}{p} \right)} = \sqrt[p]{\frac{q^q}{p^q}} </math>

			and ''e'' would therefore be [[Algebraic number\|algebraic]] of degree ''p''. However ''e'' is transcendental, so Ω must be irrational.

			Ω is in fact [[transcendental number\|transcendental]] as the direct consequence of [[Lindemann–Weierstrass theorem]]. If Ω were algebraic, exp(Ω) would be transcendental and so would be exp<sup>−1</sup>(Ω). But this contradicts the assumption that it was algebraic.

			==See also==

			* [[Lambert W function]]

			==External links==
			* {{MathWorld\|urlname=OmegaConstant\|title=Omega Constant}}

			[[Category:Transcendental numbers]]
			[[Category:Mathematical constants\|Omega]]
			[[Category:Articles containing proofs]]

en>Dexbot: r2.7.1) (Robot: Adding fa:نیک باستروم

2012-06-25T09:32:50Z

r2.7.1) (Robot: Adding fa:نیک باستروم

New page

I'm Loyd and I live in a seaside city in northern United States, Baltimore. I'm 37 and I'm will soon finish my study at Neuroscience.<br><br>Here is my site ... [http://Pocketbook.su/forum/skoraya-pomoshch/fifa-coin-generator-0 fifa 15 coin Generator]

Nick Bostrom - Revision history

en>Zyxw: add category and link

78.146.112.194: /* Simulation argument */ deleted two word to make the article make more sense

en>Dexbot: r2.7.1) (Robot: Adding fa:نیک باستروم