Methods of Mathemetical Physics I
Assignment II
Due on Friday, September 29, 2023.
Solve all the problems. Each problem carries 10 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. 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 4. The heat equation of a rod of length L reads
∂T(x, t)
∂t
= D
∂
2
T (x, t)
∂x
2
,
where D is the thermal diusivity. Write down a Fourier seires of trigonometric functions satisfying
the boundary condition T (x = 0) = T (x = L) = T
0
, all t. What is the time dependence of the
coecients?