THE METHOD OF STATIONARY PHASE 3 Now suppose that radiation is steadily being emitted from all points y on a surface S, with density c(y)dy where we assume that the c(y) all have the same phase we thus may as well assume that c(y) is real. Then for any x £ S, the radiation at x will be of the form e-ik'l k {.x) where the Ik(x) is an integral over S explicitly, r Jk\x-y\ h\x-y\ We wish to evaluate this integral asymptotically for large values of k. We thus are interested in an integral of the form a(y)eik^dy !• for large values of k. The evaluation of integrals of this type is known as the method of stationary phase. Observe that the major contribution will come from a neighborhood of the points where dj = 0. In fact we can break up a into a sum of pieces of small support by using a partition of unity. Suppose that dj 0 on supp a. Then we claim that integration by parts shows that the above integral is 0(k~N) for any TV , if a is a C°° function of y. To see this let £ be the vector field * ^ ay ay so that {«**-»*[2(|^)2]«**-*l*l2*** where |£| =^0. Let
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