
Methods of Mathemetical Physics I
Final Examination
06 October 2023
Total marks: 40
Time: 2 hours
Solve any four problems.
Problem 1. Demonstrate explicitly for the odd (about x = 0) square-wave function that
Parseval’s theorem is valid. You will need to use the relationship
∞
n=0
1
(2n + 1)
2
=
π
2
8
.
Show that a lter that transmits frequencies only up to 8π/T will still transmit more than 90%
of the power in a square-wave voltage signal of period T .
(Parseval’s theorem states the equivalence of the energy of a signal to the sum of the squares of
its Fourier coecients.)
Problem 2. Dene the Heaviside step function
Θ(x) =
0 x < 0
1 x > 0
1
2
x = 0
a) Justify the statement
˜
Θ(k) =
1
i(k − iϵ)
(ϵ → 0 from above).
b) Then nd the Fourier transform of
Θ(
x
−
a
)
e
−bx
, where
a, b >
0
, using the above result and
taking ϵ → 0 and applying the shift/translation property of Fourier transform.
The convention is
˜
A(k) =
∞
−∞
dx A(x) e
−ikx
.
Problem 3. Find the normalization factor A in the function
f(x) = A e
−
x
2
2σ
2
for it to be a valid probability density function.
Then, use the value of A to deduce the value of the gamma function Γ(
1
2
) with an appropriate
change of variable. The gamma function is dened as the integral
Γ(n) =
∞
0
t
n−1
e
−t
dt.