PHGN311 Homework #10
Due Friday, Nov. 15, 2013 at the beginning of class
Finish reading Chapter 8 on ordinary differential equations. So you should start looking
at it.
1. Boas 14.7.34
2. Show that the Fourier Transform of f (x) = e
done (essentially) this problem before):
−b x
with b > 0 is given by (you have
1$ b '
&
) . Then€find the inverse transform of g(α) and show that you obtain
π % b2 + α 2 (
f (x) back again. I want you to do the integral for the inverse transform as a contour
integration in the complex plane. First you need to find the singularities, then you
need to choose a contour and argue that the integral over parts of the contour will go
to zero as certain things go to infinity. The contour changes depending upon whether
x is positive or negative, but the obvious contours are the right ones to choose.
g(α ) =
€
3. This problem is an introduction to finding the Green’s function for an ODE.
d 2Φ(x) 2
− k Φ = δ (x), Φ(±∞) = Φ'(±∞) = 0
2
Consider the differential equation dx
. We
want to transform the equation to Fourier space by taking the Fourier Transform of
the differential equation itself.
a) Begin by showing that if g(α) is the Fourier transform of ϕ(x), then (iα)g(α) is the
Fourier transform of ϕ’(x).
b) Use part a) and the properties of delta functions to obtain the Fourier transform of
−1
g(α ) =
2π (k 2 + α 2 ) .
the differential equation. Show that this gives,
c) Now use the result in part b) to obtain an integral expression for ϕ(x) and use
contour integration to evaluate the integral. Referring to the last problem can really
simplify this a lot.
4. Boas 8.2.9
5. Boas 8.2.16
6. Boas 8.3.3
7. Boas 8.5.5 (you don’t need to compare to computer solutions)
8. Boas 8.5.9
9. Boas 8.5.29