MATH0016 University College London Mathematics Methods 3 Fourier Series HW

Please complete Question 1a), Question 3 and Question 5.

Question 1. For each of the following functions, find its half-range sine series on [0, π]:

(a) * f(x) = x3 − π2x.

Question 3. *Find the half-range cosine series of f(x) = x4 − 2π2×2 on [0, π]. By considering f(0), show

that

􏰔∞ (−1)n+1 7π4n=1 n4 = 720 Question 5. *Find the Fourier series of f(x) = exp(x) on (−π,π). Hence find the Fourier series of g(x) =sinh(x) + 2cosh(x) on the same interval. Math0016 – Example Sheet 4
HAND IN QUESTIONS marked with *. Due on Friday November 1 , 2019 by
11am (end of lecture). Although only this subset of problems will count towards
your coursework grade, I encourage you to work through all the problems.
Question 1. For each of the following functions, find its half-range sine series on [0, π]:
(a) * f (x) = x3 − π 2 x.
2x
− 1.
π
(eπ − 1)x
(c) f (x) = ex −
− 1.
π
(


x
if x ∈ 0, π2


(d) f (x) =
π − x if x ∈ π2 , π .
(b) f (x) = cos x +
(e) f (x) = sin x.
Question 2. (a) Suppose that F (x) is an odd function on [−π, π]. Prove that
Rπ
2 0 F (x)2 dx.
Rπ
−π
F (x)2 dx =
Consider the odd function F (x) on [−π, π] equal to x(π − x) on [0, π]. The Fourier series of
F was found in lectures:
8 X sin(nx)
F (x) =
.
π
n3
n odd
(b) Use this to show that
T :=
1
1
1
π6
+
+
+
·
·
·
=
.
16 36 56
960
(c) Deduce that if
1
1
1
+ 6 + 6 + ···
6
1
2
3
and hence calculate S.
S :=
then S = T +
S
64
1
Question 3. *
Find the half-range cosine series of f (x) = x4 − 2π 2 x2 on [0, π]. By considering f (0), show
that
∞
X
(−1)n+1
7π 4
=
n4
720
n=1
Question 4.
Imagine there were a function δ(x) with the property that
Z 1
δ(x)g(x)dx = g(0)
−1
for any function g. What would the Fourier series of δ be? What might the graph of the
function δ look like? Let δN denote the approximation to δ obtained by summing the first
+1/2)πx)
N terms of its Fourier series. Show that δN (x) = sin((N
. Use a computer and (say) a
2 sin(πx/2)
python code to plot δN for some small values of N . As N increases, does the plot start to
look like the graph you imagined?
δ is called the Dirac delta function; it is not really a function, but fits into the theory of
“distributions”.
The sequence of truncations is called the Dirichlet kernel; analysis of the
R
integral δN (x)F (x + y)dx is involved in proving that the Fourier series of F converges in
the L2 -sense to F .
Question 5. *
Find the Fourier series of f (x) = exp(x) on (−π, π). Hence find the Fourier series of g(x) =
sinh(x) + 2cosh(x) on the same interval.
2