EXTRA CREDIT

• Due September 21:
  • Prove you cannot totally order the complex numbers. This is problem #4 on page 25.
  • Find a function f of two real variables whose partial derivatives exist but f is not differentiable (originally typed continuous instead of differentiable)..
• Due September 24:
  • Page 29, #17.
  • Chapter 2: For Goursat's theorem, could we modify the proof to work for a general hexagon? If yes, can we deduce the theorem for triangles from this?
• Due October 1:
  • Integrate the function f(z) = zⁿ over the triangle with vertices 0, 1 and 1+ia for any n; you must do the integrals over the three sides directly, and not appeal to any theorems. Then integrate over the same region the function g(z) = x - i y (complex conjugation).
• Due November 5:
  • Find a sequence a_n of real numbers that is conditionally convergent but not absolutely convergent such that the product Prod_n (1 + a_n) diverges, or show that no such series exists.
  • Let f: (-1,1) to (-1,1) be a real analytic automorphism. What is the largest f'(0) can be? We showed in class that it could be as large as pi/2, and then as large as 2. Can you do better?
• Due December 3:
  • Consider a look-up table for cosine, where we record the value of cos(theta_n) for theta_n = 2 pi n / N, n = 0, 1, ..., N-1. For now, consider the following look-up procedure. For a given theta, find the largest n such that theta_n <= theta. What is the average error? In other words, what is (1/2pi) Int_{0 to 2pi} (cos theta - cos theta_n)^2 d theta as a function of N? I have a very nice expression for this. More generally, what if we take the closer of the two entries in our look-up table, or, even better, if we interpolate?