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192  4. The method of multiple scales



               of   (This  procedure is a very neat way to obtain higher-order terms in the
               WKB approach.)
          Q4.16 WKB (exponential case). Consider  the equation





               where       for        and         Introduce       and
                      and hence find, completely, the first term of a uniformly valid asymp-
               totic expansion (which will be the counterpart of equation (4.47)).
          Q4.17 Eigenvalues. Use the method of multiple scales, in the WKB form, to find the
               leading approximation to the eigenvalues   of the  problem





                with                  and a  > 0  for          (You  should in-
                troduce       and                 Evaluate your results,  explicitly,
               for the  cases:  (a)       (b) a (X)  = 1 + X.  [Written in the  form
                                        it is  evident  that  this is the  problem of find-
               ing approximations to the large eigenvalues.]
          Q4.18 A turning-point problem I. Consider




                where           with f > 0 and  analytic  throughout the  given  domain.
               Show that the relevant scaling (cf.  §2.7) in the neighbourhood of the turning
               point (at x= 1) is            and then introduce the more useful fast
               scale                  and  use x as the slow scale. Show that
                                    where              = constant and Ai(X) is the
                (bounded) Airy function (a solution of      Determine h(x) and
               write down the first term of the asymptotic expansion of
          Q4.19 A turning-point problem II. Show that the equation




               where       and       has  turning points at x = 0  and at x = 1.  [Hint:
                write          Use the WKB approach (for     to find the first term
               in  each of the asymptotic  expansions  valid in x <0, 0 < x <1, x > 1. Also
               write down the leading  term in  the  asymptotic  expansions  valid  near x = 0
                and near x = 1.
          Q4.20 A higher-order turning-point. Show that the equation
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