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 small 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.

\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}}.

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.

  • [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
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