4 INTRODUCTION Then r\e r = e so that faeik*dy = f afre^dy = fWa)eik*dy for some C00 function b of compact support. By repeating this operation we see that the integral is 0(k~N). Let us now assume that each critical point of / is non-degenerate. This means that the Hessian d / , i.e., the matrix \dy'dyJ/ is non-singular at each critical point. Let p be a critical point of . By a lemma of Morse, we can introduce coordinates z,, . . . , z„ about p so that = t(p) + G(*)/2 - z2 + zz 2 Z[ -r l+l + -„2]/2 where 6 is the index of the quadratic form d / . (See Appendix I to this chapter for a proof of Morse's lemma.) If m is the Jacobian matrix of the change of coordinates then, if supp a lies in the desired coordinate neighborhood, det jaeik*dy = eik^p) jaeikQ{z)l1 Notice that if Q is the matrix of Q(z) then at p we have dz. so that det dy dz (p) 8/9/
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