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                            276                       Approximation of Integrals

                                                     9.5 Laplace’s Method

                            Laplace’s method provides a futher generalization of the results of the previous section to
                            the case in which the integral under consideration is not exactly of the form

                                                          T
                                                           h(t)exp(−nt) dt
                                                        0
                            or


                                                       T 1
                                                                    2
                                                         h(t)exp(−nt /2) dt.
                                                      −T 0
                              Consider an integral of the form
                                                         b
                                                         h(y)exp{ng(y)} dy.
                                                       a
                            Then, by changing the variable of integration, we can rewrite this integral in a form in
                            which Corollary 9.1 may be applied. It is worth noting that Theorem 9.14 below may be
                            generalized to the case in which g and h both depend on n;however, we will not consider
                            such a generalization here.


                            Theorem 9.14. Consider the integral
                                                 b

                                           I n =   h(y)exp{ng(y)} dy, −∞ ≤ a < b ≤∞
                                                 a
                            where
                               (i) g is three-times differentiable on (a, b)
                                                                               2
                               (ii) h is twice-differentiable on (a, b) and h(y) = O(exp(dy )) as |y|→∞, for some
                                  constant d
                              (iii) g is maximized at y = ˆ y, where a < ˆ y < b


                              (iv) g (ˆ y) = 0, |g (y)| > 0 for all y  = ˆ y, and g (ˆ y) < 0.

                            If |h(ˆ y)| > 0, then
                                                       √
                                                         (2π)h(ˆ y)     −1
                                         I n = exp{ng(ˆ y)}    1  [1 + O(n )] as n →∞.

                                                       [−ng (ˆ y)] 2
                            If h(ˆ y) = 0, then
                                                                 1

                                                I n = exp{ng(ˆ y)}O   as n →∞.
                                                                  3
                                                                n 2
                            Proof. Note that
                                     b                              b
                                     h(y)exp{ng(y)} dy = exp{ng(ˆ y)}  h(y)exp{−n[g(ˆ y) − g(y)]} dy.
                                   a                              a
                            Consider the change-of-variable

                                                                            1
                                                  u = sgn(ˆ y − y){2[g(ˆ y) − g(y)]} 2
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