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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