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