Asymptotic Series

An asymptotic series is a Series Expansion of a Function in a variable 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 . To form an asymptotic series of , written

 (1)

take
 (2)

where
 (3)

The asymptotic series is defined to have the properties
 (4)

 (5)

Therefore,
 (6)

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

References

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.