Chebyshev nodes - Revision history

77.242.185.19: /* Definition */

2014-11-22T15:44:31Z

Definition

← Older revision		Revision as of 17:44, 22 November 2014
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	~~{{Use dmy dates\|date=June 2013}}~~		Hello, I'm Jayne, a 21 year old from Kilmux, Great Britain.<br>My hobbies include (but are not limited to) Skydiving, Home Movies and watching Supernatural.<br><br>Also visit my web-site; [http://69.89.27.245/~kickbox6/socialecorner/groups/taking-the-mystery-out-of-online-shopping/ 4inkjets coupons and Discounts]
	~~In mathematics~~, an '~~''asymptotic formula''' for a quantity (function or expression) depending on natural numbers, or on a variable taking real numbers as values, is a function of natural numbers~~, ~~or of~~ a ~~real variable~~, ~~whose values are nearly equal to the values of the former when both are evaluated for the same large values of the variable.~~
	~~An asymptotic formula for a quantity is a function which is asymptotically equivalent to the former.~~

	More generally, an asymptotic formula is "a statement of equality between two functions which is not a true equality but which means the ratio of the two functions approaches 1 as the variable approaches some value, usually infinity".<~~ref~~>~~{{Cite web\|url=http://www.answers.com/topic/asymptotic-formula\|title=Sci-Tech Dictionary: asymptotic formula\|accessdate=13 May 2010}}</ref>~~

	~~==Definition==~~
	~~Let ''P~~(~~n)'' be a quantity or function depending on ''n'' which is a natural number. A function ''F(n)'' of ''n'' is an asymptotic formula for ''P(n)'' if ''P(n)'' is asymptotically equivalent~~ to~~''F(n~~)'', ~~that is, if~~

	~~:<math>\lim_{n\rightarrow \infty}\frac{P(n)}{F(n)}=1~~.<~~/math~~>

	~~This is symbolically denoted by~~

	:<~~math~~>~~P(n) \sim F(n)\,</math>~~

	~~==Examples==~~

	~~===Prime number theorem===~~

	~~For a real number ''x'', let π (''x'') denote the number of prime numbers less than or equal to ''x''. The classical [[prime number theorem]] gives an asymptotic formula for &pi~~; ~~(''x''):~~

	~~:<math> \pi(x)\sim \frac{x}{\log(x)}.</math>~~

	~~===Stirling's formula===~~
	[~~[Image:Stirling's Approximation.svg\|thumb\|right\|300px\|Stirling's approximation approaches the factorial function as ''n'' increases.]]~~

	~~[[Stirling's approximation]] is a well-known asymptotic formula for the [[factorial]] function:~~

	:~~<math>n!=1\times 2\times\ldots \times n<~~/~~math>.~~

	~~The asymptotic formula is~~

	~~:<math>n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.<~~/~~math>~~

	~~===Asymptotic formula for the partition function===~~
	For a positive integer ''n'', the [[Partition function (number theory)\|partition function]] ''P''(''n''), sometimes also denoted ''p''(''n''), gives the number of ways of writing the integer ''n'' as a sum of positive integers, where the order of addends is not considered significant.~~<ref name="Wolfram">Weisstein, Eric W~~. ~~"Partition Function P." From MathWorld--A Wolfram Web Resource~~. ~~http:~~//~~mathworld.wolfram.com~~/~~PartitionFunctionP.html<~~/~~ref> Thus, for example, ''P''(4) = 5. [[G.H. Hardy]] and [[Srinivasa Ramanujan]] in 1918 obtained~~ the ~~following asymptotic formula for ''P''(''n''):<ref name="Wolfram"/>~~

	~~:<math>P(n)\sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{2n/3}}.</math>~~

	~~===Asymptotic formula for Airy function===~~
	~~The [[Airy function]] Ai(x) which is a solution~~ of ~~the differential equation~~

	~~:<math> y''~~-~~xy=0\,<~~/~~math>~~

	and ~~which has many applications in physics, has the following asymptotic formula:~~

	~~:<math> \mathrm{Ai}(x) \sim \frac{e^{-\frac{2}{3}x^{3/2}}}{2\sqrt{\pi}x^{1/4}}.</math>~~

	~~==See also==~~
	* [[Asymptotic analysis]]

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Asymptotic Formula}}~~
	~~[[Category:Asymptotic analysis]~~]

en>Kri: Rephrased sentence

2013-12-28T14:18:51Z

Rephrased sentence

← Older revision		Revision as of 16:18, 28 December 2013
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			{{Use dmy dates\|date=June 2013}}
			In mathematics, an '''asymptotic formula''' for a quantity (function or expression) depending on natural numbers, or on a variable taking real numbers as values, is a function of natural numbers, or of a real variable, whose values are nearly equal to the values of the former when both are evaluated for the same large values of the variable.
			An asymptotic formula for a quantity is a function which is asymptotically equivalent to the former.

			More generally, an asymptotic formula is "a statement of equality between two functions which is not a true equality but which means the ratio of the two functions approaches 1 as the variable approaches some value, usually infinity".<ref>{{Cite web\|url=http://www.answers.com/topic/asymptotic-formula\|title=Sci-Tech Dictionary: asymptotic formula\|accessdate=13 May 2010}}</ref>

	~~Satisfied to meet you! My own name~~ is ~~Eusebio Ledbetter~~. It'~~s not a common place but what I desire doing~~ is ~~[http~~://~~Search.Un.org/search?ie~~=~~utf8&site~~=~~un_org&output~~=~~xml_no_dtd~~&~~client=UN_Website_en~~&~~num~~=~~10&lr~~=~~lang_en&proxystylesheet~~=~~UN_Website_en&oe~~=~~utf8&q~~=~~bottle+counter&Submit~~=~~Go bottle counter~~] ~~tops collecting and now My personal have time to have~~ a ~~look at on new things. Software developing is how As~~ well ~~as support my family~~. ~~My house~~ is ~~thus in Vermont~~. I'~~ve been working on my website for~~ the ~~some time now~~. ~~Inspection it out here:~~ http://~~circuspartypanama~~.com<br><br>~~Visit my site; Clash of clans hack tool~~ ([~~http~~://~~circuspartypanama~~.~~com circuspartypanama.com~~])		==Definition==
			Let ''P(n)'' be a quantity or function depending on ''n'' which is a natural number. A function ''F(n)'' of ''n'' is an asymptotic formula for ''P(n)'' if ''P(n)'' is asymptotically equivalent to''F(n)'', that is, if

			:<math>\lim_{n\rightarrow \infty}\frac{P(n)}{F(n)}=1.</math>

			This is symbolically denoted by

			:<math>P(n) \sim F(n)\,</math>

			==Examples==

			===Prime number theorem===

			For a real number ''x'', let π (''x'') denote the number of prime numbers less than or equal to ''x''. The classical [[prime number theorem]] gives an asymptotic formula for π (''x''):

			:<math> \pi(x)\sim \frac{x}{\log(x)}.</math>

			===Stirling's formula===
			[[Image:Stirling's Approximation.svg\|thumb\|right\|300px\|Stirling's approximation approaches the factorial function as ''n'' increases.]]

			[[Stirling's approximation]] is a well-known asymptotic formula for the [[factorial]] function:

			:<math>n!=1\times 2\times\ldots \times n</math>.

			The asymptotic formula is

			:<math>n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.</math>

			===Asymptotic formula for the partition function===
			For a positive integer ''n'', the [[Partition function (number theory)\|partition function]] ''P''(''n''), sometimes also denoted ''p''(''n''), gives the number of ways of writing the integer ''n'' as a sum of positive integers, where the order of addends is not considered significant.<ref name="Wolfram">Weisstein, Eric W. "Partition Function P." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PartitionFunctionP.html</ref> Thus, for example, ''P''(4) = 5. [[G.H. Hardy]] and [[Srinivasa Ramanujan]] in 1918 obtained the following asymptotic formula for ''P''(''n''):<ref name="Wolfram"/>

			:<math>P(n)\sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{2n/3}}.</math>

			===Asymptotic formula for Airy function===
			The [[Airy function]] Ai(x) which is a solution of the differential equation

			:<math> y''-xy=0\,</math>

			and which has many applications in physics, has the following asymptotic formula:

			:<math> \mathrm{Ai}(x) \sim \frac{e^{-\frac{2}{3}x^{3/2}}}{2\sqrt{\pi}x^{1/4}}.</math>

			==See also==
			* [[Asymptotic analysis]]

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Asymptotic Formula}}
			[[Category:Asymptotic analysis]]

en>Helpful Pixie Bot: ISBNs (Build KH)

2012-05-11T22:10:47Z

ISBNs (Build KH)

New page

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