Asymptotic expansion
From Free net encyclopedia
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.
If φn is a sequence of continuous functions 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, <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 an asymptotic expansion of f with respect to the scale is a formal series <math>\sum_{n=0}^\infty a_n \varphi_{n}(x)</math> such that, for any fixed N,
- <math>f(x) = \sum_{n=0}^N a_n \varphi_{n}(x) + O(\varphi_{N+1}(x)) \ (x \rightarrow L).</math>
In this case, we write
- <math> f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \ (x \rightarrow L)</math>.
See asymptotic analysis and big O notation for the notation.
The most common type of asymptotic expansion is a power series in either positive or negative terms. While a convergent Taylor series fits the definition as given, a non-convergent series is what is usually intended by the phrase. 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.
Examples of asymptotic expansions
- <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>
- <math>xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \rightarrow \infty) </math>
- <math>\zeta(s) \sim \sum_{n=1}^{N-1}n^{-s} + \frac{N^{1-s}}{s-1} +
N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^\overline{2m-1}}{(2m)! N^{2m-1}}</math> 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>.
- <math> \sqrt{\pi}x e^{x^2}{\rm erfc}(x) = 1+\sum_{n=1}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}.</math>
Detailed example
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
- <math>\frac{1}{1-w}=\sum_{n=0}^\infty w^n</math>
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
- <math>\int_0^\infty \frac{e^{-w/t}}{1-w} dw
= \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n du</math>
The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution <math>u=w/t</math>, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion
- <math>e^{-1/t}\; \operatorname{Ei}\left(\frac{1}{t}\right) = \sum _{n=0}^\infty n! \; t^{n+1} </math>
Here, the right hand side is clearly not convergent for any non-zero value of t. However, by keeping t small, and 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>. 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.
References
- Hardy, G. H., Divergent Series, Oxford University Press, 1949
- Paris, R. B. and Kaminsky, D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001
- Whittaker, E. and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963de:Asymptotische Folge