## Math 642-600 Current Assignment — Spring 2014

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

- Read 10.1-10.3.
- Do the following problems.
- Problem 6(a,c), page 329 (§ 7.2).
- Problem 11(a,c), page 330 (§ 7.2).
- This refers to Problem 7 on the midterm. Show that every f
satisfying the conditions of the problem is band limited, with
Ω ≤ ρ.
- 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.)
- 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.
- Use Laplace's method directly (don't simply apply the formula!)
and Watson's lemma to derive an asymptotic formula for the integral
$I(x) = \int_{-1}^1 (2+t^2)^{-x}dt$, $x\to \infty$.

Updated 4/24/2014 (fjn)