Math 642-600 Current Assignment — Spring 2014

Assignment 7 - Due Tuesday, 4/29/2014.

• Read 10.1-10.3.
• Do the following problems.
  1. Problem 6(a,c), page 329 (§ 7.2).
  2. Problem 11(a,c), page 330 (§ 7.2).
  3. This refers to Problem 7 on the midterm. Show that every f satisfying the conditions of the problem is band limited, with Ω ≤ ρ.
  4. Consider the one dimensional heat equation, u_t = u_xx, with u(x,0) = f(x), where − ∞ < x < ∞ and 0 ≤ t x <∞. By taking the Fourier transform in x, show that the solution u(x,t) is given by
     u(x,t) = ∫_R K(x−y,t)f(y)dy, K(ξ,t) = e^−ξ2/4t (4πt)^{− 1/2}.
     (The function K(x−y,t) is the one-dimensional heat kernel.)
  5. Consider the continuous piecewise linear function f(x) that is is 0 outside of [−2,2] and passes through the points (− 2,0), (−1,2), (0,-1), (1,2), and (2,0). Find the Fourier transform of f this way. First, find the distributional derivative f'', which is a linear combination of δ functions. Then, find the FT f''. Finally, use the FT to obtain the find Fourier transform of f.
  6. Use Laplace's method directly (don't simply apply the formula!) and Watson's lemma to derive an asymptotic formula for the integral $I(x) = \int_{-1}^1 (2+t^2)^{-x}dt$, $x\to \infty$.