Asymptotic expansion
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In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion 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>\phi_{n+1}(x) = o(\phi(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 \phi_{n}(x)<math> such that
- <math>f(x) = \sum_{n=0}^N a_n \phi_{n}(x) + O(\phi_{N+1}(x)) \ (x \rightarrow L).<math>
In this case, we write
- <math> f(x) \sim \sum_{n=0}^\infty a_n \phi_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>
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, 1963it:sviluppo asintotico