15—Fourier Analysis 380Problems15.1 Invert the Fourier transform,g, in Eq. (15.7).
15.2 What is the Fourier transform ofeik^0 x−x
(^2) /σ 2
? Ans: A translation of thek 0 = 0case
15.3 What is the Fourier transform ofxe−x
(^2) /σ 2
?
15.4 What is the square of the Fourier transform operator? That is, what is the Fourier transform of
the Fourier transform?
15.5 A function is defined to be
f(x) =
{
1 (−a < x < a)
0 (elsewhere)What is the convolution off with itself? (f∗f)(x)And graph it of course. Start by graphing both
f(x′)and the other factor that goes into the convolution integral.
Ans:(2a−|x|)for (− 2 a < x <+2a), and zero elsewhere.
15.6 Two functions are
f 1 (x) =
{
1 (a < x < b)
0 (elsewhere) and f^2 (x) =
{
1 (A < x < B)
0 (elsewhere)What is the convolution off 1 withf 2? And graph it.
15.7 Derive these properties of the convolution:
(a)f∗g=g∗f (b)f∗(g∗h) = (f∗g)∗h (c)δ(f∗g) =f∗δg+g∗δfwhereδf(t) =tf(t),
δg(t) =tg(t),etc. (d) What areδ^2 (f∗g)andδ^3 (f∗g)?
15.8 Show that you can rewrite Eq. (15.9) as
F(f∗g) =F(f).F(g)
where the shorthand notationF(f)is the Fourier transform off.
15.9 Derive Eq. (15.10) from Eq. (15.9).
15.10 What is the analog of Eq. (15.10) for two different functions? That is, relate the scalar product
of two functions,
〈
f 1 ,f 2
〉
=
∫∞
−∞f 1 *(x)f 2 (x)
to their Fourier transforms. Ans:
∫
g 1 *(k)g 2 (k)dk/ 2 π
15.11 In the derivation of the harmonic oscillator Green’s function, and starting with Eq. (15.15), I
assumed that the oscillator is underdamped: thatb^2 < 4 km. Now assume the reverse, the overdamped
case, and repeat the calculation.
15.12 Repeat the preceding problem, but now do the critically damped case, for whichb^2 = 4km.
Compare your result to the result that you get by taking the limit of critical damping in the preceding
problem and in Eq. (15.17).