THE METHOD OF STATIONARY PHASE 5 We are thus reduced to considering an integral of the form ( z j , . . . , zn). We claim that this integral has the asymptotic for large k. Here z expansion '??y/2gfr«-20/4[f(p) + 0(k -n/2-\ We prove this as follows: We can write/(z) = /(0) 4- 2 z iMz) where the^ are smooth, but, of course, not necessarily of compact support. If we could establish that feikQ{z)/2g(z)dz were well defined for suitable functions not having compact support, we could write ik / * 3z dz and this last integral is again of the type we are considering. Thus the highest order term in any asymptotic expansion will come from the constant term, and we are left with the task of evaluating the integral f elkQ^z''2dz. This is a product of one dimensional integrals, and we must evaluate the integral / Before evaluating this integral let us go to the problem of making sense of such integrals. We are thus interested in making sense of integrals of the form f el Qw*' h(z) dz. We can reduce the question of convergence of the multiple integral to that of the iterated integral, and hence to a problem in one real variable. Let h(t) be a C 2 function of the real variable t which is bounded and with bounded first two derivatives. Consider the integral „-*2/2 h(t)dt. We claim that this integral is uniformly convergent for Re X 0, |A| 1. Indeed, for 0 R S we have [S e^'2/2h(t)dt = -A"1 fS \{e-x,1/2)'Kt)dt J R J R I = -A-'rx'2/2(/i(0A) = _A-2e-x'2/2[A(M0A) - U/OWOA)' -\-2jSRe-x'2'2[{\/t){h{t)/t)']dt.) jSRe-Xll'2{h{t)/t)'dt +
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