Asymptotic analysis: Working Note No. 2, Approximation of integrals

In this note we discuss the approximation of integrals that depend on a parameter. The basic tool is simple, namely, integration by parts. Of course, the power of the tool is evidenced in applications. The applications are many; they include Laplace integrals, generalized Laplace integrals, Fourier integrals, and Stokes' method of stationary phase for generalized Fourier integrals. 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''. The notation differs slightly from Working Note No. 1, for reasons that are mainly historical. The asymptotic analysis of integrals originated in complex analysis, where the (real or complex) parameter, usually denoted by x, is usually introduced in such a way that the interesting behavior of the integrals occurs when x [yields] [infinity] in some sector of the complex plane. As there is nothing sacred about notation, and historical precedent is as good a guide as any, we follow convention and denote the parameter by x, focusing on the behavior of integrals as x [yields] [infinity] along the real axis more » or, if x is complex, in some sector of the complex plane. The connection with the notation of Working Note No. 1 is readily established by identifying the small parameter [epsilon] with [vert bar]x[vert bar][sup [minus]1]. « less