1. (Generalization of the standard method of stationary phase) Suppose that function (t) defined for t...

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1. (Generalization of the standard method of stationary phase) Suppose that function (t) defined for t€ [a, b] satisfies the condition (c) = " (c) = ... =(-¹) (c) = 0, y) (c) #0, but (m) (t) #0 for any m≥p 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) |₁(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 1³ (a) 1(x) = f' dt, (b) [₁ 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 J₂(x) (with v> 0) as x→∞. 1. (Generalization of the standard method of stationary phase) Suppose that function (t) defined for t€ [a, b] satisfies the condition (c) = " (c) = ... =(-¹) (c) = 0, y) (c) #0, but (m) (t) #0 for any m≥p 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) |₁(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 1³ (a) 1(x) = f' dt, (b) [₁ 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 J₂(x) (with v> 0) as x→∞.

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Q1 Generalization of the standard method of stationary phase Suppose that the function t defined for satisfies the condition But for any and Show that ... View the full answer