5. The Fourier Transform of a piecewise smooth function f(x), for which f(x)|dx exists and is...

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5. The Fourier Transform of a piecewise smooth function f(x), for which f(x)|dx exists and is finite, and the inverse Fourier Transform are given by ƒ(k) = F[ƒ(x)] = √/12/20 √/2² 1 ƒ(x)e¯ik²dx,_ƒ(x) = F¯¯¹[ƒ(k)] = √2/ƒ(k) e dk. (k)eikz -∞ (a) Show that the Fourier Transform of the function f(x) is a real parameter, is F[fa(x)] = = e-ika π 1+k² = = exp(-x - a), where a (b) Consider the partial differential equation - sin(t). ди Ət ди əx = 0, subject to the initial condition u(x, 0) = 4(x). Defining û (k, t) = F[u(x,t)] and ô(k) = F[y(x)], show that û(k, t) = ô(k)e¹k(1-cos(t)) (5) Using the result from part (a) find the solution u(x, t) of equation (5) with the initial condition (x) = exp(-|x|). 5. The Fourier Transform of a piecewise smooth function f(x), for which f(x)|dx exists and is finite, and the inverse Fourier Transform are given by ƒ(k) = F[ƒ(x)] = √/12/20 √/2² 1 ƒ(x)e¯ik²dx,_ƒ(x) = F¯¯¹[ƒ(k)] = √2/ƒ(k) e dk. (k)eikz -∞ (a) Show that the Fourier Transform of the function f(x) is a real parameter, is F[fa(x)] = = e-ika π 1+k² = = exp(-x - a), where a (b) Consider the partial differential equation - sin(t). ди Ət ди əx = 0, subject to the initial condition u(x, 0) = 4(x). Defining û (k, t) = F[u(x,t)] and ô(k) = F[y(x)], show that û(k, t) = ô(k)e¹k(1-cos(t)) (5) Using the result from part (a) find the solution u(x, t) of equation (5) with the initial condition (x) = exp(-|x|).

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a Showing the Fourier Transform of fax expxa Heres the derivation Ffax expxa expikx dx 2 expxa expik... View the full answer