Approximation Of Integrals

1 1 Integration by Parts 1 2 Laplace Integrals 3 3 Laplace's Method 5 4 Fourier Integrals 8 5 Stokes' Method of Stationary Phase 11 References 16 Index 17 iv ASYMPTOTIC ANALYSIS Working Note #2 APPROXIMATION OF INTEGRALS by Marc Garbey and Hans G. Kaper Abstract In this note we discuss the approximation of integrals that depend on a parameter. The basic tool is simple, namely, integration by parts (Section 1). Of course, the power of the tool is evidenced in applications. The applications are many; they include Laplace integrals (Section 2), generalized Laplace integrals (Section 3), Fourier integrals (Section 4), and Stokes' method of stationary phase for generalized Fourier integrals (Section 5). These results illustrate beautifully Hardy's concept of applications of mathematics, that is, "certain regions of mathematical theory in which the notation and the ideas of the [method of integration by parts] may be used systematically with a great gain in clearness and simplicity" [G...