Asymptotic expansion

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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.

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, \varphi_{n+1}(x) = o(\varphi_n(x)) \ (x \rightarrow L). 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 is a formal series \sum_{n=0}^\infty a_n \varphi_{n}(x) if

f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = O(\varphi_{N}(x)) \ (x \rightarrow L).

or

f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = o(\varphi_{N-1}(x)) \ (x \rightarrow L).

If one or the other holds for all N, then we write

f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \ (x \rightarrow L).

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.

[edit] Examples of asymptotic expansions

\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)
xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \rightarrow \infty)
\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}}
where B2m are Bernoulli numbers and s^{\overline{2m-1}} 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 N > | s | .
\sqrt{\pi}x e^{x^2}{\rm erfc}(x) = 1+\sum_{n=1}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}.

[edit] 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

\frac{1}{1-w}=\sum_{n=0}^\infty w^n.

The expression on the left is valid on the entire complex plane w\ne 1, while the right hand side converges only for | w | < 1. Multiplying by e − w / t and integrating both sides yields

\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.

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 u = w / t, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion

e^{-1/t}\; \operatorname{Ei}\left(\frac{1}{t}\right) = \sum _{n=0}^\infty n! \; t^{n+1}.

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 \operatorname{Ei}(1/t). Substituting x = − 1 / t and noting that \operatorname{Ei}(x)=-E_1(-x) results in the asymptotic expansion given earlier in this article.

[edit] References

  • [Bleistein, N. and Handlesman, R.], Asymptotic Expansions of Integrals, Dover, New York, 1975
  • [Erdelyi, A.|A. Erdelyi], Asymptotic Expansions, Dover, New York, 1955
  • 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, 1963