# Let ˆ t ky integraldisplay t xy e ikx dx ie let ˆ t

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Let ˆ T ( k,y ) = integraldisplay −∞ T ( x,y ) e ikx dx (i.e. let ˆ T be the Fourier transform of T with respect to the first variable). Explain why T ( x,y ) = 1 2 π integraldisplay −∞ ˆ T ( k,y ) e ikx dk. By using differentiation under the integral and the uniqueness of the Fourier transform, obtain a partial differential equation for ˆ T only involving partial differentiation with respect to y . Solve this differential equation, obtaining a result involving two unknown functions of k , call them A ( k ) and B ( k ). By setting y = 0 and y = 1 obtain A ( k ) and B ( k ) in terms of ˆ T ( k, 0) and ˆ T ( k, 1). Hence, find the function ˆ T ( k,y ). By exploiting symmetry, verify that T ( x,y ) is real. Q 10.36. (Q7(b), Paper II, 1995) [The first paragraph is routine. The second paragraph is heavily starred.] A linear system is such that that an input g ( t ) is related to an output f ( t ) by f ( t ) = integraldisplay t −∞ K ( t t ) g ( t ) dt . Let θ ( t ) be the step function defined by θ ( t ) = 0 for t < 0 and θ ( t ) = 1 for t 0. Suppose that g ( t ) = θ ( t ) e γt where γ > 0 and that f ( t ) = θ ( t ) e γt (1 e t ). Find K ( t ) for t > 0 assuming that K ( t ) = 0 for t < 0. Explain why you can drop the assumption K ( t ) = 0 for t< 0. Is it possible to find K ( t ) for t< 0 from the information given? Why? When g ( t ) = δ ( t ), use the expression previously found for K ( t ) to calcu- late f ( t ) both (i) by direct evaluation of the integral and (ii) by calculating 34

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the Fourier transform of g in this case and hence finding the Fourier transform of f . [Most students obtain different answers for (i) and (ii). If this happens to you, the object of the game is to discover what has gone wrong.] 35
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