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 coecients.)
Problem 2. Dene 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 )
(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 dened as the integral
Γ(n) =
0
t
n1
e
t
dt.
2
Problem 4. Find the Green’s function solution to the Poisson equation
2
φ = f(r) for a
spherically symmetric potential φ(r), when the source is f(r) = e
r
.
Problem 5. The moment generating function (MGF) of a random variable X is dened as
M
X
(s) = E[e
sX
]
We say that the MGF of X exists, if there exists a positive constant a such that M
X
(s) is nite
for all s [a, a].
a) Write M
X
(s) as a binomial series of moments, and use it to nd the k
th
moments of the
exponential density function
f
X
(x) = λe
λx
Θ(x)
b) Also use the series expansion to show that
E[X
k
] = M
(k)
X
(0).