Methods of Mathematical Physics I
Assignment II
Due on Wednesday, September 25, 2024.
Solve all the problems. Each problem carries 12 marks.
Problem 1. Find the discrete Fourier transform (DFT)
˜
A(p) of the following function, with
2N + 1 points
A(n) =
1 0 ≤ n ≤ W −1
1 2N + 1 − W ≤ n ≤ 2N
0 otherwise
Figure 1. Sketch of the function A(n)
Apply the appropriate theorem if required to simplify the evaluation of the DFT. Take the limit
of this DFT when N → ∞ and W = x
√
2
N
+ 1
and
p
=
k
√
2
N
+ 1
.
Problem 2. Derive the convolution theorem for discrete Fourier transforms, dened as
A(n) =
1
√
N
N−1
X
p=0
˜
A(p)e
2πi
N
pn
a) That is, if
˜
C(p) =
˜
A(p)
˜
B(p), express C(n) in terms of A(n) and B(n).
b) Use this to derive the discrete Fourier transform (DFT) of the following array:
C(n) =
(
1 −
|n|
5
for|n| ≤ 5
0 otherwise
Figure 2. Sketch of C(n)
2
Problem 3. Determine the Fourier transform
Z
∞
−∞
(ℓ
Γ
(x))
2
e
−ikx
dx,
where ℓ
Γ
(x) is the Lorentzian function, dened as
ℓ
Γ
(x) =
Γ
π
1
x
2
+ Γ
2
,
i. by the method of residues,
ii. using the Fourier transform of ℓ
Γ
(x), denoted by
˜
ℓ
Γ
(k), after evaulating it and invoking the
appropriate theorem,
iii. by dierentiating ℓ
Γ
and
˜
ℓ
Γ
w.r.t Γ.
0.1in
Problem 4. Evaluate the Fresnel diraction integral
e
ikz
C
ZZ
F (x
′
, y
′
)e
ik
2z
[(x−x
′
)
2
+(y−y
′
)
2
]
dx
′
dy
′
for a gaussian
F = e
−(x
2
+y
2
)/2σ
2
.
a) What does C have to be so that the integral reduces to F (x, y) when z → 0?
b) At nite z, interpret F(x, y, z) as a beam with a gaussian amplitude distribution with σ(z) and
a phase distribution (x
2
+ y
2
)/2R(z). What does R(z) mean, in terms of wavefronts?
c) As one travels along the z-axis, x = 0, y = 0, is the phase of the beam just kz? If not, why
not?
Problem 5. The temperature on the ends of a thin regular rod is maintained constant and equals
zero. For simplicity, let the x-axis be oriented along the rod and its ends have the coordinates
x = 0 and x = l. The thermal diusivity coecient D of the material of the rod equals a
2
. Find
spatial and time distribution T (x, t)(0 ≤ x ≤ l, t > 0) of the temperature along the rod for the
two cases:
a) at time moment t = 0 the temperature of the rod is constant, T (x, 0) ≡ T
0
at 0 < x < l;
b) in the center of the rod, a point source of intensity Q is switched on at time instant t = 0, and
T (x, 0) ≡ 0 at 0 ≤ x ≤ l.
You could write the solution in the separable form as:
T (x, t) = X(x)U(t)
and obtain the eigenvalue solutions for the given initial and boundary conditions.