Information for different problem types, exam 1.

Problem 1.
  The equation can change, for example, set c=1.
  The zero conditions u(0,t)=u(1,t)=0 won't change.
  The length of the string can change, e.g., L=2.
  The function in u(x,0) can change, e.g., u(x,0)=0.
  The function in u_t(x,0) can change, e.g., u_t(x,0)=sin(Pi x).

Problem 2.

  (a) Products and sums are possible, as well as compositions. 
      Examples: sin(x)cos(2x), sin(x)^2, exp(sin(x)), sin(x)+cos(x)
  (b) f(x) can be the odd or even extension of period 2T, with base 
      function given on [0,T]. All such examples require computation 
      of the extension f(x) at a point outside the base interval.
  (c) The answer can be yes or no. 
      Examples: exp(cos(x)) odd or even? cos(x)sin(x)+tan(x) odd or even?
                x sin(x) odd or even? (1-x^2)/2 odd or even? Which of these
                are periodic?

Problem 3. The function f(x) will change. Maybe the interval will change
           from -Pi to Pi to -1 to 1, or 0 to 4.
  
  (a) Compute the coefficient of the third term in the series (given).
      Compute the coefficient of sin(Pi x).
      Compute the coefficient of cos(2 Pi x).
  (b) Find the places where Gibb's overshoot occurs in |x| < 2.
      Give an example of a function f(x) with no Gibb's overshoot.
      Draw a figure, showing what Gibb's overshoot looks like at x=1.

Problem 4. Let f(x) be an even function on -Pi to Pi. Let f(x) be its 
           2Pi-periodic extension to the whole real line. Write the
           formula for the coefficient of cos(7x), but don't evaluate
           any integrals.

           Example. Same problem, change -Pi to Pi to -1 to 1, and find
                    a formula for the 4th Fourier coefficient.

           Example. Same problem, change -Pi to Pi to -1 to 1, change
                    f(x) to an odd function of period 2, and find
                    a formula for the 10th Fourier coefficient. 
 
Problem 5. 
  (a) Display Dirichlet's kernel formula and sketch how it is proved.
      Estimate the magnitude of |1/2+cos(x)+...+cos(nx)| for n=1000.
  (b) When does the Fourier convergence theorem return the value of the 
      Fourier series sum as exactly f(x)?
      At which points is it possible for Gibb's overshoot to occur?
      If a Fourier series sums to f(x), but f(x) is discontinuous, then
      what equation must hold at a point x of discontinuity?
  (c) Give an example of a periodic function of period 2 with a Gibb's
      overshoot at x=1.
      Is there a smooth function f(x) with a Gibb's overshoot?