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 φ_n 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, $\backslash varphi\_\{n+1\}(x)\; =\; o(\backslash varphi\_n(x))\; \backslash \; (x\; \backslash rightarrow\; L)$. 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 $\backslash sum\_\{n=0\}^\backslash infty\; a\_n\; \backslash varphi\_\{n\}(x)$ such that, for any fixed N,

$f(x)\; =\; \backslash sum\_\{n=0\}^N\; a\_n\; \backslash varphi\_\{n\}(x)\; +\; O(\backslash varphi\_\{N+1\}(x))\; \backslash \; (x\; \backslash rightarrow\; L).$

In this case, we write

$f(x)\; \backslash sim\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; \backslash varphi\_n(x)\; \backslash \; (x\; \backslash rightarrow\; L)$.

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.

Examples of asymptotic expansions

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• Gamma function

$\backslash frac\{e^x\}\{x^x\; \backslash sqrt\{2\backslash pi\; x\}\}\; \backslash Gamma(x+1)\; \backslash sim\; 1+\backslash frac\{1\}\{12x\}+\backslash frac\{1\}\{288x^2\}-\backslash frac\{139\}\{51840x^3\}-\backslash cdots$

\ (x \rightarrow \infty)

• Exponential integral

$xe^xE\_1(x)\; \backslash sim\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^nn!\}\{x^n\}\; \backslash \; (x\; \backslash rightarrow\; \backslash infty)$

• Riemann zeta function

$\backslash zeta(s)\; \backslash sim\; \backslash sum\_\{n=1\}^\{N-1\}n^\{-s\}\; +\; \backslash frac\{N^\{1-s\}\}\{s-1\}\; +$

N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^{\overline{2m-1}}}{(2m)! N^{2m-1}}
where $B\_\{2m\}$ are Bernoulli numbers and $s^\{\backslash overline\{2m-1\}\}$ 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 $N\; >\; |s|$.

• Error function

$\backslash sqrt\{\backslash pi\}x\; e^\{x^2\}\{\backslash rm\; erfc\}(x)\; =\; 1+\backslash sum\_\{n=1\}^\backslash infty\; (-1)^n\; \backslash frac\{(2n)!\}\{n!(2x)^\{2n\}\}.$

Detailed example

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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

$\backslash frac\{1\}\{1-w\}=\backslash sum\_\{n=0\}^\backslash infty\; w^n$

The expression on the left is valid on the entire complex plane $w\backslash ne\; 1$, while the right hand side converges only for . Multiplying by $e^\{-w/t\}$ and integrating both sides yields

$\backslash int\_0^\backslash infty\; \backslash frac\{e^\{-w/t\}\}\{1-w\}\; dw$

= \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

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

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 $\backslash operatorname\{Ei\}(1/t)$. Substituting $x=1/t$ and noting that $\backslash operatorname\{Ei\}(x)=-E\_1(-x)$ results in the asymptotic expansion given earlier in this article.

References

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• 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

mathematical analysis | asymptotic analysis | mathematical series

Asymptotische Folge | Sviluppo asintotico