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| | 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.
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| An asymptotic formula for a quantity is a function which is asymptotically equivalent to the former.
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| 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>
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| ==Definition==
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| 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
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| :<math>\lim_{n\rightarrow \infty}\frac{P(n)}{F(n)}=1.</math>
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| This is symbolically denoted by
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| :<math>P(n) \sim F(n)\,</math>
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| ==Examples==
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| ===Prime number theorem===
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| 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''):
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| :<math> \pi(x)\sim \frac{x}{\log(x)}.</math>
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| ===Stirling's formula===
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| [[Image:Stirling's Approximation.svg|thumb|right|300px|Stirling's approximation approaches the factorial function as ''n'' increases.]] | |
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| [[Stirling's approximation]] is a well-known asymptotic formula for the [[factorial]] function:
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| :<math>n!=1\times 2\times\ldots \times n</math>. | |
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| The asymptotic formula is
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| :<math>n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.</math>
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| ===Asymptotic formula for the partition function===
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| 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"/>
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| :<math>P(n)\sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{2n/3}}.</math>
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| ===Asymptotic formula for Airy function===
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| The [[Airy function]] Ai(x) which is a solution of the differential equation
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| :<math> y''-xy=0\,</math>
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| and which has many applications in physics, has the following asymptotic formula: | |
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| :<math> \mathrm{Ai}(x) \sim \frac{e^{-\frac{2}{3}x^{3/2}}}{2\sqrt{\pi}x^{1/4}}.</math>
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| ==See also==
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| * [[Asymptotic analysis]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Asymptotic Formula}}
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| [[Category:Asymptotic analysis]]
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Hello, I'm Jayne, a 21 year old from Kilmux, Great Britain.
My hobbies include (but are not limited to) Skydiving, Home Movies and watching Supernatural.
Also visit my web-site; 4inkjets coupons and Discounts