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 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 a_2k = 0 and a_2k+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 c_n = -i/[np] for n ≠ 0 and c₀ = 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'.