MATH3271

FW03

Assignment 1 Due date: Monday, Sept. 29

1.(a) Use MAPLE (or some other facility to plot functions such as Mathematica or MATLAB) to plot the following periodic functions:

(i) cos(x) + sin(3x), (ii) cos(x)sin(3x), (iii) cos(2px/3), (iv) [x] - 2[x/2].

Choose a suitable range for x so that the periodic nature of each function is displayed. (Note that the MAPLE function for [x] is floor(x)).

(b) What are the periods of each of the functions in part (a)?

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).

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

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

(i) f(x) = cos(x/2) if -p £ x ≤ p,

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

(b) For both of the functions given in part (a) plot the function and the partial sums s₄ and s₆ on the same graph. Note the MAPLE function Heaviside(x) which is 0 if x < 0 and 1 if x > 1. You can also use the absolute value function abs(x) to create a function like Heaviside.

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

(b) In separate figures, plot both the even and odd extensions of the function from part (a) and their partial sums s₃ and s₅.

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

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'.

Hand in a printout of your MAPLE worksheet containing the plots with the rest of your assignment.