# MA22S3 Outline Solutions Tutorial Sheet 4. 28 October 2009 Questions

```MA22S3 Outline Solutions Tutorial Sheet 4.12
28 October 2009
Questions
1. (4) Express the following periodic function (L = 2π) as complex Fourier series

 0 −π &lt; t &lt; −a
1 −a &lt; t &lt; a
f (t) =

0 a&lt;t&lt;π
(1)
where a ∈ (0, π) is a constant.
P
Solution: f (x) = n∈Z cn eint with
Z π
Z a
1
1
−int
dt e
f (t) =
dt e−int
cn =
2π −π
2π −a
so that c0 = a/π and
a
1 e−int 1 eian − e−ian
1
cn =
=
=
sin an.
2π −in −a πn
2i
πn
2. (4) Compute the Fourier transform of f (t) = e−a|t| where a is a positive constant.
Solution: The idea here is to plug f (t) directly in to the equation for f˜(k) and then
use |t| = t for t &gt; 0 and |t| = −t for t &lt; 0.
Z ∞
1
˜
f (k) =
dt e−ikt f (t)
2π −∞
Z ∞
Z ∞
Z 0
1
1
−ikt −a|t|
−ikt−at
−ikt+at
=
dt e
e
=
dt e
+
dt e
2π &quot;−∞
2π 0 #
−∞
∞
0
e−t(a+ik) et(a−ik) 1
−
−
=
2π
a + ik 0
a − ik −∞
1
1
1
a
1
=
+
.
=
2π a + ik a − ik
π a2 + k 2
f can be represented as a Fourier integral
Z ∞
Z
a ∞
eikt
ikt ˜
f (t) =
dk e f(k) =
.
dk 2
π −∞
a + k2
−∞
1
2