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| In [[mathematics]], an '''asymptotic expansion''', '''asymptotic series''' or '''Poincaré expansion''' (after [[Henri Poincaré]]) is a [[formal series]] of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.
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| If φ<sub>''n''</sub> is a sequence of [[continuous function]]s on some domain, and if ''L'' is a (possibly infinite) [[limit point]] of the domain, then the sequence constitutes an '''asymptotic scale''' if for every ''n'',
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| <math>\varphi_{n+1}(x) = o(\varphi_n(x)) \ (x \rightarrow L)</math>. If ''f'' is a continuous function on the domain of the asymptotic scale, then ''f'' has an asymptotic expansion of order ''N'' with respect to the scale as a formal series <math>\sum_{n=0}^\infty a_n \varphi_{n}(x)</math> if | |
| :<math>f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = O(\varphi_{N}(x)) \ (x \rightarrow L)</math>
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| or
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| :<math>f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = o(\varphi_{N-1}(x)) \ (x \rightarrow L).</math>
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| If one or the other holds for all ''N'', then we write
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| :<math> f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \ (x \rightarrow L).</math>
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| See [[asymptotic analysis]] and [[big O notation]] for the notation.
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| The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the [[Euler–Maclaurin summation formula]] and integral transforms such as the [[Laplace transform|Laplace]] and [[Mellin transform|Mellin]] transforms. Repeated [[integration by parts]] will often lead to an asymptotic expansion.
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| Since a ''[[Convergence (mathematics)|convergent]]'' [[Taylor series]] fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a ''non-convergent'' series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as '''superasymptotics'''.<ref>{{cite journal|first=John P.|last= Boyd|title= The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series |journal=Acta Applicandae Mathematicae|volume=56|issue=1|pages=1–98|year=1999|doi=10.1023/A:1006145903624}}</ref> The error is then typically of the form <math>\sim\exp\left(-\frac{c}{\epsilon}\right)</math> where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as [[Borel resummation]] to the divergent tail. Such methods are often referred to as '''hyperasymptotic approximations'''.
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| ==Examples of asymptotic expansions==
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| * [[Gamma function]]
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| ::<math>\frac{e^x}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots | |
| \ (x \rightarrow \infty)</math>
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| * [[Exponential integral]]
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| ::<math>xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \rightarrow \infty) </math>
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| * [[Riemann zeta function]]
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| ::<math>\zeta(s) \sim \sum_{n=1}^{N-1}n^{-s} + \frac{N^{1-s}}{s-1} +
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| N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^{\overline{2m-1}}}{(2m)! N^{2m-1}}</math><br />where <math>B_{2m}</math> are [[Bernoulli numbers]] and <math>s^{\overline{2m-1}}</math> is a [[rising factorial]]. This expansion is valid for all complex ''s'' and is often used to compute the zeta function by using a large enough value of ''N'', for instance <math>N > |s|</math>.
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| * [[Error function]]
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| ::<math> \sqrt{\pi}x e^{x^2}{\rm erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}} \ (x \rightarrow \infty).</math> | |
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| ==Detailed example==
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| Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its [[domain of convergence]]. Thus, for example, one may start with the ordinary series
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| :<math>\frac{1}{1-w}=\sum_{n=0}^\infty w^n.</math>
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| The expression on the left is valid on the entire [[complex plane]] <math>w\ne 1</math>, while the right hand side converges only for <math>|w|< 1</math>. Multiplying by <math>e^{-w/t}</math> and integrating both sides yields
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| :<math>\int_0^\infty \frac{e^{-w/t}}{1-w}\, dw
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| = \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n\, du,</math>
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| after the substitution <math>u=w/t</math> on the right hand side. The integral on the left hand side, understood as a [[Cauchy principal value]], can be expressed in terms of the [[exponential integral]]. The integral on the right hand side may be recognized as the [[gamma function]]. Evaluating both, one obtains the asymptotic expansion
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| :<math>e^{-1/t}\; \operatorname{Ei}\left(\frac{1}{t}\right) = \sum _{n=0}^\infty n! \; t^{n+1}. </math>
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| Here, the right hand side is clearly not convergent for any non-zero value of ''t''. However, by truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of <math>\operatorname{Ei}(1/t)</math> for sufficiently small ''t''. Substituting <math>x=-1/t</math> and noting that <math>\operatorname{Ei}(x)=-E_1(-x)</math> results in the asymptotic expansion given earlier in this article.
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| ==References==
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| <references/>
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| * Bleistein, N. and Handlesman, R., ''Asymptotic Expansions of Integrals'', Dover, New York, 1975.
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| * Copson, E. T., ''Asymptotic Expansions'', Cambridge University Press, 1965.
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| * [[Arthur Erdélyi|A. Erdélyi]], ''Asymptotic Expansions'', Dover, New York, 1955.
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| * [[G. H. Hardy|Hardy, G. H.]], ''Divergent Series'', Oxford University Press, 1949.
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| * Paris, R. B. and Kaminsky, D., ''Asymptotics and Mellin-Barnes Integrals'', Cambridge University Press, 2001.
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| * [[E. T. Whittaker|Whittaker, E. T.]] and Watson, G. N., ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1963.
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| ==External links==
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| * {{springer|title=Asymptotic expansion|id=p/a013670}}
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| [[Category:Mathematical analysis]]
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| [[Category:Asymptotic analysis]]
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| [[Category:Mathematical series]]
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The actual amend delivers a bunch of notable enhancements, arc of which could turn out to be the new Dynasty War Manner. In this mode, you can claiming combating dynasties and reduce utter rewards aloft her beat.
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