2009/10 Schol paper 2, my questions, draft
version.
This was the second schol paper, for students only doing one maths module.
1. Find the Fourier series for f (t) where f (t) = |t| when −π < t < π and
f (t + 2π) = f (t).
2. The convolution of two functions f (t) and g(t) is
f ∗ g(t) =
Z ∞
f (τ )g(t − τ )dτ
−∞
Show that
F(f ∗ g) = 2πF(f )F(g)
3. Find the general solution of
ÿ + 2ẏ + 2 = et
and write it in an explicitly real form.
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
∞
∞
X
a0 X
2πnt
2πnt
+
an cos
+
bn sin
.
2
L
L
n=1
n=1
f (t) =
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
1
• A function with period L has the Fourier series expansion
f (t) =
∞
X
cn exp
n=−∞
where
2iπnt
.
L
1 Z L/2
−2iπnt
f (t) exp
dt
L −L/2
L
cn =
• The Fourier integral or Fourier transform:
f (t) =
fg
(k)
Z ∞
dk fg
(k)eikt
−∞
1 Z∞
dt f (t)e−ikt
=
2π −∞
2