Certain functions, capable of expansion only as a divergent series,
may nevertheless be calculated with great accuracy by taking the
sum of a suitable number of terms. The theory of such asymptotic
expansions is of great importance in many branches of pure and
applied mathematics and in theoretical physics. Solutions of
ordinary differential equations are frequently obtained in the form
of a definite integral or contour integral, and this tract is
concerned with the asymptotic representation of a function of a
real or complex variable defined in this way. After a preliminary
account of the properties of asymptotic series, the standard
methods of deriving the asymptotic expansion of an integral are
explained in detail and illustrated by the expansions of various
special functions. These methods include integration by parts,
Laplace's approximation, Watson's lemma on Laplace transforms, the
method of steepest descents, and the saddle-point method. The last
two chapters deal with Airy's integral and uniform asymptotic
expansions.
General
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