Math 673Assignment #6Applications of Fourier TransformsDue March 9, 2005Name1. The Cauchy problem for the Klein-Gordon equation isutt− c2uxx+ a2u = 0 for − ∞ < x < ∞, , t > 0u(x, 0) = f(x), ut(x, 0) = g(x) for − ∞ < x < ∞.Find a formula that represents the solution. Explain whyZ∞−∞|u(x, t)|2dx =Z∞−∞Èf(ξ) cos(tpc2ξ2+ a2) +ˆg(ξ)pc2ξ2+ a2sin(tpc2ξ2+ a2)!2dξ.Conclude that, for any time tZ∞−∞|u(x, t)|2dx ≤ 4Z∞−∞|f(x)|2dx + 4t2Z∞−∞|g(x)|2dx.2. For 0 < α < 1, show thatFcxα−1=r2πΓ(α)|ξ|αcos³απ2´.Hint: Replace cos xξ by its sum of exponentials, use Cauchy’s theorem and the contoursRiρiRρorRρRρ−i−ias appropriate.3. Use the previous problem to concludeFcx−p=rπ2|ξ|p−1Γ(p)sec³pπ2´for 0 < p < 1.4. FindFsµx√a2− x2H(a − x)¶.Not so much a hint as a potentially useful cryptic remark: A hint from a previous homeworkassignment may prove useful.5. Find Fc(H(b − x)). Use it to evaluateZ∞0sin bxx(x2+ a2)dx.6. Prove that Fc{f0(x)} = ξFsf −q2πf(0) and that Fs{f0(x)} = −ξFcf.7. Find Fc{e−αx2} for α > 0. Use that result and properties of the derivatives of Fourier sine andcosine transforms to find