MA22S3, sample for my four questions on the
schol exams
Eight questions do six in a three hour exam.
1. Prove that for two periodic functions f (t) and g(t) with the same period
L then
Z ∞
Z
1 L/2
cn d∗n
f (t)g ∗ (t)dt =
L −L/2
n=−∞
where cn and dn are the coefficients in the complex Fourier series for
f (t) and g(t) respectively. Deduce Parseval’s theorem from this.
2. Write the triangular pulse



At/T + A
−T < t < 0
f (t) =  −At/T + A 0 < t < T

0
|t| > T
as a Fourier integral.
3. Solve the Euler-Cauchy equation
t2
dy
d2 y
+ 3t + y = 0.
2
dt
dt
4. Using the method of Fröbenius, or otherwise, find the general solution
to
d2 y
dy
t 2 + 2 − αty = 0
dt
dt
where α > 0.
Some useful formula
• A function with period L has the Fourier series expansion
f (t) =
∞
∞
X
2πnt
2πnt
a0 X
bn sin
an cos
+
.
+
2
L
L
n=1
n=1
where
a0 =
2 Z L/2
f (t)dt
L −L/2
1
bn
Z
2 L/2
2πnt
f (t) cos
dt
L −L/2
L
Z
2πnt
2 L/2
dt
f (t) sin
=
L −L/2
L
an =
• A function with period L has the Fourier series expansion
f (t) =
∞
X
cn exp
n=−∞
where
2iπnt
.
L
1 Z L/2
−2iπnt
cn =
dt
f (t) exp
L −L/2
L
• The Fourier integral or Fourier transform:
f (t) =
fg
(k) =
Z
∞
−∞Z
1
2π
2
dk fg
(k)eikt
∞
−∞
dt f (t)e−ikt