7. Consider the general Laplace integral

where we suppose that h' (a) = h'' (a) = 0, h''' (a) < 0, and h' (t) < 0 for t ∈ (a, b], so that h(t) is maximised at the point of inflexion t = a. Show that

where f(t) ～ f0(t ? a)λ as t → a, with λ > ?1 and f0 a constant.

8. (FOR STUDENTS TAKING MA4AM. NOT FOR STUDENTS TAKING MA3AM.)

Derive the two-term asymptotic expansion

as x → ∞, as follows. First show that h(t) = ? sin2 t is maximised at t = 0, determine an expansion of h(t) around this point up to terms of O(t4), and hence show that

Then Taylor expand the quartic term up to terms of O(t 4 ) (valid since 0 < t < δ ? 1) to yield

Finally, expand the range of integration to ([0,∞) (incurring only exponentially small errors), and then proceed in the usual way. . .

9. (FOR STUDENTS TAKING MA4AM. NOT FOR STUDENTS TAKING MA3AM.)

Use Laplace’s method to show that

as x → ∞.

10. (FOR STUDENTS TAKING MA4AM. NOT FOR STUDENTS TAKING MA3AM.)

Use Laplace’s method to show that

as x → ∞.