MA22S3, sample for my questions on the summer exams
Six questions do four in a two hour exam.
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
−π
Z π
sin nt sin mtdt = πδnm ,
−π
cos nt cos mtdt = πδnm
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. Determine the Fourier transform of the one-off pulse:



0
|t| > T
f (t) =  −A −T < t < 0

A
0<t<T
3. What are
1 + t2
dt,
δ(t)
1 − t2
Z−∞
Z ∞
Z ∞
5
2
δ(t − 3)t dt,
−5
Z
δ(t − 1)et−1 dt,
−∞
1
δ(t − 3)t2 dt
−1
Calculate the Fourier transform of δ(t).
4. Find the general solution of
d2 y dy
−
− 6y = 0,
dt2
dt
1
d2 y dy
−
− 6y = e5t .
2
dt
dt
5. What is the solution of
d2 y
dy
− 2 + y = et
2
dt
dt
with y(0) = ẏ(0) = 1?
6. Find a series solution for
(1 + t)
dy
+ y = 0.
dt
Some useful formula
• A function with period L has the Fourier series expansion
∞
∞
X
a0 X
2πnt
2πnt
f (t) =
an cos
bn sin
+
+
.
2
L
L
n=1
n=1
where
a0
an
bn
2 Z L/2
f (t)dt
=
L −L/2
Z L/2
2
2πnt
=
f (t) cos
dt
L −L/2
L Z L/2
2
2πnt
=
f (t) sin
dt
L −L/2
L
• 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 =
f (t) exp
dt
L −L/2
L
• The Fourier integral or Fourier transform:
f (t) =
fg
(k) =
Z ∞
dk fg
(k)eikt
−∞Z
1
2π
2
∞
−∞
dt f (t)e−ikt