2009/10 annual paper, all questions, draft version.
1. (a) The Fourier series relies on the orthogonality of the sine and cosine
functions: as an example, show
Z
π
sin nt cos mtdt = 0
−π
for integers n and m.
(b) In a similar way it can be shown that
Z
π
Z
sin nt sin mtdt = πδnm ,
−π
π
cos nt cos mtdt = πδnm
−π
How are these identities used to derive formulas for the a0 , an and
bn in the Fourier series.
(c) Find the Fourier series of the function f (t) with period 2π defined
by
f (t) = t
2. Write the on-off pulse



as a Fourier integral.
3. What are
Z
∞
(t4 +t2 −1)δ(t)dt,
−∞
1
−π < t < 0
f (t) = −1 0 < t < π


0
otherwise
Z
∞
(t4 +t2 −1)δ(t+1)dt,
−∞
Z
∞
(t4 +t2 −1)δ(t+1)dt
0
where δ(t) is the Dirac delta function. Show that the Fourier intergral
representation of δ(t) is
δ(t) =
Starting with
Z
∞
−∞
2
|f (t)| dt =
Z
∞
−∞
1
2π
Z
Z
∞
eikt dk
−∞
∞
f (t)f ∗ (t′ )δ(t − t′ )dtdt′
−∞
substitute the Fourier integral representation of δ(t − t′ ) and derive the
Plancherel formula.
1
4. Show that y(t) = 1 is a particular solution of
ÿ + 4ẏ + 4y = 4
and find the general solution. What is the solution if y(0) = ẏ(0) = 0.
5. Using the substitution t = ez or otherwise, find the general solution of
Euler equation
t2 ÿ + 3tẏ + y = t2
6. Calculate the first four non-zero terms in the power series solution to
(1 − t2 )ẏ − 2ty = 0
with y(0) = 2.
Some useful formula
• A function with period L has the Fourier series expansion
∞
∞
X
a0 X
2πnt
2πnt
f (t) =
bn sin
an cos
+
.
+
2
L
L
n=1
n=1
where
Z
2 L/2
f (t)dt
L −L/2
Z
2πnt
2 L/2
dt
f (t) cos
=
L −L/2
L
Z
2 L/2
2πnt
=
dt
f (t) sin
L −L/2
L
a0 =
an
bn
• A function with period L has the Fourier series expansion
f (t) =
∞
X
cn exp
n=−∞
where
cn =
1
L
Z
L/2
f (t) exp
−L/2
2iπnt
.
L
−2iπnt
dt
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