1. (Generalization of the standard method of stationary phase) Suppose that function (t) defined for t [a, b] satisfies the condition (c) = " (c) = ... =(-) (c) = 0, y) (c) #0, but (m) (t) #0 for any mp and t#ce [a, b]. Show that if c = a, the integral 1(z) = f* f(t)eize(t) dt has the asymptotic expansion I(x)~ f(a)eirv(a)tix/(2p) p! 11/PT(1/p) |(P)(a). as x . P Explain how the sign should be taken. How should the result be modified if c is an interior point of [a, b]? 2. Use the method of stationary phase, or its generalization, to find the leading-order approxi- mation of the integrals 1 (a) 1(x) = f' dt, (b) [ sin [2(t+ 6 - sinh t)] cost dt. Jo /0 3. (Method of steepest descent) Using their integral representations, obtain the leading- order approximation of the Airy function Ai(z) and the Bessel function J(x) (with v> 0) as x.