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 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, $\phi_{n+1}(x) = o(\phi(x)) \ (x \rightarrow L)$ . 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 $\sum_{n=0}^\infty a_n \phi_{n}(x)$ such that

$f(x) = \sum_{n=0}^N a_n \phi_{n}(x) + O(\phi_{N+1}(x)) \ (x \rightarrow L).$

In this case, we write

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

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

Examples of asymptotic expansions

Gamma function

$\frac{\exp(x)}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots \ (x \rightarrow \infty)$

Exponential integral

$x\exp(x)E_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \rightarrow \infty)$

Riemann zeta function

$\zeta(s) \sim \sum_{n=1}^{N-1}n^{-s} + \frac{N^{1-s}}{s-1} + N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^\overline{2m-1}}{(2m)! N^{2m-1}}$

where $B 2 m$ 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 $N > | s |$ .

Error function

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

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, 1963

Categories: Mathematical analysis | Asymptotic analysis | Mathematical series

Asymptotic expansion - Your Art History Reference Guide!

Examples of asymptotic expansions

References