Asymptotic Expansion
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An Asymptotic Expansion is a series expansion which partial sums are good approximation to a given function as the function arguments become large or tend to a specific point.
- AKA: Asymptotic Series, Poincaré Expansion.
- …
- Example(s):
- Counter-Example(s)
- See: Asymptotic Analysis, Henri Poincaré, Formal Series, Euler–Maclaurin Summation Formula, Laplace Transform, Mellin Transform, Integration by Parts, Convergence (Mathematics), Taylor Series, Borel Resummation.
References
2017a
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Asymptotic_expansion Retrieved:2017-5-21.
- 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. Deep investigations by Dingle [1] reveal that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. 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 and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a 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. The error is then typically of the form [math]\displaystyle{ \sim\exp\left(-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.
See asymptotic analysis, big O notation, and little o notation for the notation used in this article.
- 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. Deep investigations by Dingle [1] reveal that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. 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 and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a 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. The error is then typically of the form [math]\displaystyle{ \sim\exp\left(-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.
- ↑ R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press (1973).
2017b
- (Weisstein et al., 2017) ⇒ Bhatt, Bhuvanesh and Weisstein, Eric W. "Asymptotic Series." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AsymptoticSeries.html Retrieved:2017-5-21
- An asymptotic series is a series expansion of a function in a variable x which may converge or diverge (Erdélyi 1987, p. 1), but whose partial sums can be made an arbitrarily good approximation to a given function for large enough x. To form an asymptotic series R(x) of
- [math]\displaystyle{ f(x)∼R(x)\quad }[/math], (1)
- take
- [math]\displaystyle{ x^nR_n(x)=x^n[f(x)-S_n(x)]\quad }[/math], (2)
- where
- [math]\displaystyle{ S_n(x)=a_0+(a_1)/x+(a_2)/(x^2)+...+(a_n)/(x^n)\quad }[/math]. (3)
- The asymptotic series is defined to have the properties
- [math]\displaystyle{ \lim_(x\rightarrow \infty)x^nR_n(x)=0 }[/math] for fixed [math]\displaystyle{ n }[/math] (4)
- [math]\displaystyle{ \lim_(n\rightarrow\infty)x^nR_n(x)=infty }[/math] for fixed [math]\displaystyle{ x }[/math]. (5)
- Therefore,
- [math]\displaystyle{ f(x) \approx \sum_{n=0}^\infty a_n x^{-n} }[/math](6)
- in the limit [math]\displaystyle{ x\rightarrow\infty }[/math]. If a function has an asymptotic expansion, the expansion is unique. The symbol ∼ is also used to mean directly similar.
- Asymptotic series can be computed by doing the change of variable [math]\displaystyle{ x\rightarrow 1/x }[/math] and doing a series expansion about zero. Many mathematical operations can be performed on asymptotic series. For example, asymptotic series can be added, subtracted, multiplied, divided (as long as the constant term of the divisor is nonzero), and exponentiated, and the results are also asymptotic series (Gradshteyn and Ryzhik 2000, p. 20).
2017c
- (Enc. of Mathematics, 2021) ⇒ Asymptotic expansion. M.V. Fedoryuk (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Asymptotic_expansion&oldid=16660 Retrieved:2017-5-21