ABSTRACT

I1 98 3 Integrals of a Real Variable of course, heuristic, but in following sections we shall place the method on a firm foundation.

The similarity of the approximations (1 1.03) and (1 1.05) to (7.02) and (7.03) attracts attention. From the standpoint of complex-variable theory (Chapter 4), Laplace's method and the method of stationary phase can be regarded as special cases of the same general procedure. This is reflected in the analysis: the proofs of $13 below resemble those of $7 in many ways. 11.3 The case in which stationary points are absent is an exercise in integration by parts. Sincepl(t) is of constant sign in [a, b] we may take v = p(t) as new integration variable. Then (1 1.01) becomes

I(x) = el""f(v) dv, l:; where f(v) = q(t)/pl(t). This is a Fourier integral, and the asymptotic analysis of $5 is directly applicable. In particular, iff (0) is continuous and f '(v) is sectionally continuous, that is, ifpl(t) and q(t) are continuous andpW(t) and qt(t) are sectionally continuous in [a, b], then

1 This confirms the predictions of $1 1.1 in this case. In other cases, the range of integration can be subdivided in such a way that the 1 only stationary point in each subrange is located at one of the endpoints, and without

loss of generality we may suppose that this is the left endpoint. Before proceeding to these cases we establish a number of preliminary results.