The question relates to the complex Fourier Series. This is just a rewriting of the ordinary Fourier
Series
∞
a0 X
an cos nx + bn sin nx
(1)
f (x) =
+
2
n=1
in terms of exponentials as follows:
∞
c0 X
f (x) =
+
cn einx + c∗n e−inx
2
n=1
(2)
with complex coefficients cn . Actually, from your lectures you know that the cn are related to the an
and bn by
1
(3)
cn = (an − ibn )
2
which is going to be important at one stage of this question. To find the complex coefficients we use
(assuming the function is periodic and defined between −π and π)
Z π
1
cn =
f (x)e−inx ,
(4)
2π −π
remembering that it is often necessary to check that the result from this calculation works for n = 0;
otherwise we calculate c0 by hand as the integral of just f (x) between −π and π, divided of course
by the same factor of 2π.
To get practice with the complex Fourier Series we will calculate the coefficients for a simple function:
f (x) = x (−π ≤ x ≤ π)
(5)
which is given in Q6 (ii) on your question sheet and matches Q2 (ii) (which will come in handy
below). The first part of the question is
1. Show that the coefficients cn for the above function are given by
(−1)n+1
cn =
in
2. Use this result to determine the corresponding an and bn and compare to the result you found
when carrying out the calculation last term for Q2. Hint: Straightforward inspection of real
and imaginary parts, or you can express exp (inx) = cos nx + i sin nx etc.
3. By evaluating f (x) at x =
π
2
and making use of the answer to part 2, deduce that
X (−1) n−1
2
π
1 1
=
= 1 − + − ···
4 n=1,3,···
n
3 5
n−1
Hint: sin nπ
can be written for odd integers as (−1) 2 (and is trivial for even integers) and
2
some gymnastics with powers of −1 are needed for the closed form - if your result agrees with
the final expression above then don’t worry too much!
Parseval’s theorem is a very interesting and important theorem which gives a relationship between
the integral of the function f (x) and the set of coefficients an and bn . Specifically
Z π
∞
1
a0 1 X 2
2
|f (x)| dx =
+
an + b2n .
(6)
2π −π
4
2 n=1
1
The theorem can be proved by substituting in the Fourier series of f (x) on the left hand side and then
carrying out the integral, using the trigonometric identities which are used to derive the expressions
for the coefficients in the first place. In practice questions involving the theorem simply require you
to evaluate both sides of this equation for a particular function and to play games with some algebra.
As an example, use the above function and Parseval’s theorem to derive
∞
π2 X 1
=
.
6
n2
n=1
(7)
For a Mars Bar (or similar chocolate bar of your choice) have a go at proving Parseval’s theorem in
the manner indicated above.
2