**MATH3271**

**FW03**

**Answers to
Assignment 1 **

1 See MAPLE output

2.(a) If f(x) and g(x) are both T-periodic
functions. Show that the following
functions are also T-periodic: (i) f(x) - g(x),
(ii) f(x)g(x), (iii) f(x)/g(x) if
g(x) ¹ 0, h(f(x)) where h(x) is any
function (not necessarily periodic).

We know that f(x+T) = f(x) and
g(x+T) = g(x).

Let d(x) = f(x) - g(x). Then d(x+T) = f(x+T) - g(x+T) = f(x) - g(x) =
d(x) so s is T-periodic.

Similarly, let p(x) = f(x)g(x) so
that p(x+T) = f(x+T)g(x+T) = f(x)g(x) = p(x) so p is T-periodic.

Also, let r(x) = f(x)/g(x) assuming
g(x) ¹ 0.
Then r(x+T) = f(x+T)/g(x+T) = f(x)/g(x) = r(x) so r is T-periodic.

Finally, let H(x) = h(f(x)). Then H(x+T) = h(f(x+T)) = h(f(x)) = H(x) so H
is T-periodic.

(b) If f(x) is T-periodic, show that f(px) is
periodic and determine its period.

Let F(x) = f(px) and assume that it
is U-periodic. Then F(x+U) = f(p(x+U)) =
f(px + pU) = f(px) = F(x) if pU = T since f(x+T) = f(x). Thus F is periodic with period U = T/p.

3.(a) Find the Fourier series for each of the
following functions:

(i) f(x) = cos(x/2) if -p £ x < p,

f is an even
function with period 2p. Hence b_{n} = 0 and _{}_{}Thus the Fourier
series is _{}

(ii) f(x) = 1 if -2 £ x < -1, f(x) = 0 if -1 £ x < 1, f(x) =1 if 1 £ x < 2.

In this case f has period 4 (i.e. p = 2) and is an even function. Thus we can write _{}

_{}

Thus a_{2k} = 0 and a_{2k+1} = 2(-1)^{k+1}/(2k+1)π. Also b_{n} = 0 and the Fourier series
is

_{}

(b) For both of the functions given in part (a)
plot the function and the partial sums s_{4} and s_{6} on the
same graph.

See MAPLE output

4.(a) For the function defined as f(x) = 1 - x if 0
£ x < 1 find both the sine and cosine half-range
expansions.

The period for
this expansion is 2 so that the
coefficients for the sine half-range expansion are

_{}

so that the odd
extension can be written as

_{}

For the
coefficients of the cosine half-range expansion we have

_{}

_{}

Thus a_{2k}
= 0 and a_{2k+1} = 4/(2k+1)^{2}π^{2} so that the even
extension can be written as

_{}

(b) In separate figures, plot both the even and
odd extensions of the function from part (a) and their partial sums s_{3}
and s_{5}.

See MAPLE output.

5. Write the Fourier series from question 4 in
complex form.

The complex form for
the odd extension is

_{}

where the complex coefficients are c_{n} = -i/[np] for n ≠ 0 and c_{0 }= 0.

The complex form
for the even extension is

_{}

where the complex coefficients are c_{k} = 2/[(2k+1)p]^{2}.

6. Suppose that f(x) and g(x) are both
2p-periodic functions and that f has Fourier coefficients a_{n} and b_{n}
while g has Fourier coefficients a_{n}' and b_{n}'. Show that the function 5f(x) + 2g(x) has
Fourier coefficients 5a_{n} + 2a_{n}' and 5b_{n} + 2b_{n}'.

Let h(x) = 5f(x) + 2g(x). Then the Fourier cosine coefficients for h
are

_{}

Similarly the
sine coefficients for h can be shown to be 5b_{n} + 3b_{n}'.