PaperIII_72 - MATHEMATICAL TRIPOS Part III Friday 29 May...

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MATHEMATICAL TRIPOS Part III Friday, 29 May, 2015 1:30 pm to 4:30 pm PAPER 72 PERTURBATION AND STABILITY METHODS Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 (a) Find the first two nonzero terms in the expansion for all real roots of the quintic equation t5 − t3 + 3 = 0 ϵ ϵ as → 0. ϵ (b) Consider the integral 1 f (x) = (lnt) exp(ixt)dt 0 as x → ∞. By suitably deforming the integration contour, show that f (x) −ilnx − iγ + π/2 + exp(ix) + O(x−3 ) , x x x2 where γ is Euler’s constant. You are given that lnu exp(−u)du = −γ . 0 (c) Consider b I (λ) = exp(λφ(t))dt , −∞ where φ(t) = t3 − 2t2 + t and b is a real positive constant. Determine the first nonzero term in the expansion of I as λ → ∞, being careful to identify the behaviours found for different values of b. Part III, Paper 72
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3 2 (a) Consider the differential equation d2x t + x = 0 , subject to x = dx/dt = 1 when t = 0, where ≪ 1 and f (x) is a given function. ϵ Find the leading-order approximation to x(t; ) which is uniformly valid for t ≤ ϵ O(1/ ) when ϵ (i) f (x) = x2 − 1; (ii) f (x) = sin x.
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