Asymptotic Series

An asymptotic series is a Series Expansion of a Function in a variable $x$ which may converge or diverge (Erdelyi 1987, p. 1), but whose partial sums can be made an arbitrarily good approximation to a given function for large enough $x$. To form an asymptotic series $R(x)$ of $f(x)$, written

$$f(x)\sim R(x),$$ (1)

take

$$x^n R_n(x) = x^n[f(x)-S_n(x)],$$ (2)

where

$$S_n(x) \equiv a_0 + {a_1 \over x} + {a_2 \over x^2} + \ldots + {a_n \over x^n}.$$ (3)

The asymptotic series is defined to have the properties

$$\lim_{x \to \infty} x^n R_n(x) = 0 \qquad \hbox{for fixed $n$}$$ (4)

$$\lim_{n \to \infty} x^n R_n(x) = \infty \qquad \hbox{for fixed $x$}.$$ (5)

Therefore,

$$f(x) \approx \sum_{n=0}^\infty a_nx^{-n}$$ (6)

in the limit $x\to\infty$. If a function has an asymptotic expansion, the expansion is unique. The symbol $\sim$ is also used to mean directly Similar.

References

Asymptotic Series

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 15, 1972.

Arfken, G. ``Asymptotic of Semiconvergent Series.'' §5.10 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 339-346, 1985.

Bleistein, N. and Handelsman, R. A. Asymptotic Expansions of Integrals. New York: Dover, 1986.

Copson, E. T. Asymptotic Expansions. Cambridge, England: Cambridge University Press, 1965.

de Bruijn, N. G. Asymptotic Methods in Analysis, 2nd ed. New York: Dover, 1982.

Dingle, R. B. Asymptotic Expansions: Their Derivation and Interpretation. London: Academic Press, 1973.

Erdelyi, A. Asymptotic Expansions. New York: Dover, 1987.

Morse, P. M. and Feshbach, H. ``Asymptotic Series; Method of Steepest Descent.'' §4.6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 434-443, 1953.

Olver, F. W. J. Asymptotics and Special Functions. New York: Academic Press, 1974.

Wasow, W. R. Asymptotic Expansions for Ordinary Differential Equations. New York: Dover, 1987.