MATH3271

FW03

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 bn = 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 a2k = 0 and a2k+1 = 2(-1)k+1/(2k+1)π.  Also bn = 0 and the Fourier series is

(b)  For both of the functions given in part (a) plot the function and the partial sums s4 and s6 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 a2k = 0 and a2k+1 = 4/(2k+1)2π2 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 s3 and s5.

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 cn = -i/[np] for n ≠ 0 and c0 = 0.

The complex form for the even extension is

where the complex coefficients are ck = 2/[(2k+1)p]2.

6.  Suppose that f(x) and g(x) are both 2p-periodic functions and that f has Fourier coefficients an and bn while g has Fourier coefficients an' and bn'.  Show that the function 5f(x) + 2g(x) has Fourier coefficients 5an + 2an' and 5bn + 2bn'.

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 5bn + 3bn'.