Review of Fourier Series

Part 3: Half-Range Expansions

In this part we consider the approximation of functions by either sine functions alone or cosine functions alone. (The constant function 1 = cos(0x) is included in the cosine functions here.)

1. Now let f be the function defined on the interval from 0 to 4 with the graph given below.
   Graph of f on [0, 4]
   

   First we will extend f to an odd function g on the interval [-4, 4]. Then consider g to be extended to become 8-periodic.

   Graph of g on [-4, 4]
   
2. Compare the graphs of the first 10 approximations determined by the Fourier coefficients for g with the graphs of f and g.

3. Now consider f to be extended to be an even function h on [-4, 4].
   Graph of h on [-4, 4]
   

4. Compare the graphs of the first 10 approximations determined by the Fourier coefficients for h with the graphs of f and h.