Experiments with Fourier Series

Part 3: Fourier Approximations to a Saw-Tooth Wave

Now we consider a function f that is not continuous everywhere:

where f is extended to be 2 pi-periodic. The extension makes the function discontinuous at integer multiples of pi.

1. Give a graphical explanation of why a₀ = 0.
   
   
2. Calculate the other coefficients a_k. How could we have known in advance what these coefficients would be?
   
   
3. Calculate the coefficients b_k. Write down the simplified form of these coefficients you obtain by using the fact that k is an integer.
   
   
4. Describe the values of cos(pi k) for k = 1, 2, ... .
   
   
5. Compare the Fourier approximations with the function f. What is happening at the points of discontinuity? What is the value of each of the approximations at these points?

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