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) |

(2) |

(3) |

(4) |

(5) |

(6) |

**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.

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1999-05-25