Overview

Functions defined by integrals often appear in the investigation of dynamical systems. For example, in chemistry, reaction rate coefficients take the form ^[1]

where, ? > 0, is inversely related to the temperature and the functions A( ?) and ?( x) are specified for a given type of reaction. This particular expression has been used to study the behaviors of K( ?) in the limits ? ? 0 and, ? ? ? ^[2].

Other integrals having this same structure also appear in a variety of situations arising in the natural and physical sciences, and applied mathematics. Part of the task of this chapter is to present results on functions defined by these kind of integrals and give formula for determining the leading term in their expansions when the relevant variable becomes large. We begin by briefly presenting the major results on the inverse-power set of expansion functions, i.e., { x ^{? n}: n = 0, 1, 2, }, and how they can be used to construct large- x representations for many of the functions defined by integrals. Next, we show that asymptotic series can often be obtained by the repeated use of integration by parts. Several examples are used to illustrate his procedure. In section 9.3, the general Laplace method is discussed along with Watson's lemma. Finally, we present the Euler-Maclaurin sum formula and show how it can be used to evaluate various sums. In all...