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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 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]] |
Revision as of 16:18, 28 December 2013
30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. 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".[1]
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 toF(n), that is, if
This is symbolically denoted by
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):
Stirling's formula
Stirling's approximation is a well-known asymptotic formula for the factorial function:
The asymptotic formula is
Asymptotic formula for the partition function
For a positive integer n, the 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.[2] Thus, for example, P(4) = 5. G.H. Hardy and Srinivasa Ramanujan in 1918 obtained the following asymptotic formula for P(n):[2]
Asymptotic formula for Airy function
The Airy function Ai(x) which is a solution of the differential equation
and which has many applications in physics, has the following asymptotic formula:
See also
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ Template:Cite web
- ↑ 2.0 2.1 Weisstein, Eric W. "Partition Function P." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PartitionFunctionP.html