(Issued Wed, Nov. 8)

Due Wed, Nov 15

Reading: Boas Ch. 2, Ch 7.

1) Starting from the general form of the Fourier expansion of a function f(x) with period 2[[pi]], , write out the details of the derivation of the formula . (Hint: Use Boas 7.5.2!)

2a) Boas Problem 7.5.3. Be sure to sketch your function over several pds. Boas gives the answer, so please show your work carefully and thoroughly.

2b) Given g(x) = -x for (assume period=2[[pi]]). Sketch g(x) over an interval of at least from -3[[pi]] to 3[[pi]]. Find the Fourier sine-cosine series for g(x).

3. Use Dirichlet's theorem to find the value to which the Fourier series in problem 2a converges at x=0, , , .

4. Repeat #2b, but do a complex exponential Fourier series. Verify, (use Euler's formula) that your answer is equivalent to the one you found in 2b above.

b) Show that if any real function f(x) is expanded in a complex exponential Fourier series , then ( is the complex conjugate of )

5. Write the following functions as the sum of an even function and an odd function: , and . Are your answers unique? 6a) Construct a function f(n,x) in Mma to give the sum of the first n terms of the Fourier series expansion of problem 2a. Also construct a Mma function f(x) which directly gives the function itself, from problem 2a. (This may require a Which[], or If[], or Mod[]) (Note: To make life easier, you may assume we will only want f over the range -2[[pi]] < x < 2[[pi]]!) Show f[5,x], to check with Boas' answer. Then make plots of f(x) and f(n,x) together, over -2[[pi]] < x < 2[[pi]], for n=1, 2, 5, and 25. ( 4 plots total) Describe what you see happening at the endpoints, and the discontinuity at [[pi]]/2. (This last might take a couple of minutes - you can always type comments (or start the next part) while Mma is "thinking")

6b) Do the exact same for g(x) from #2b. I.e, write a Mma function g(x), which is the given function (again, assume we will only want it over the range -2[[pi]] < x < 2[[pi]]) and g(n,x) which is the sum of the first n terms of the series expansion of that function. Show g[5,x] (So, you can check your analytic answer!) . Make the same 4 plots as above. (This shows nicely how the Fourier series converges!)