Last Home work 427K-H   Due Wednesday May 10

1.  Consider Fourier series on the interval  [-L,L]  and assume the functions
below are defined on this interval.
Compute the Fourier series for  |x|  and the Fourier series for 
f(x) = sgn(x) on this interval.  How are they related?  Note:  sgn(x) means
the sign of x, +1 for x> 0 and -1 for x < 0.

2.  Apropos of the question of what the value of sgn(0) is, choose f
to be any nice function on the interval [-L,L]. Select a new function
g(x)=f(x) if    x =/0 and g(0) = a.  show that the Fourier series of
g is the same of the Fourier series of f no matter what a is.

3. Using your answer from #1, compute the cosign series of L - |x| on the
interval [0,L].

4. Separate variables in the wave equation 

            (d/dt)^2 u  -  (d/dx)^2 u =0

on the interval  [0,l].   Assume u(0,t) = 0 and u(L,t)= 0.  We will use
you answer in class Wednesday to talk about pitch. (Yes, it is in the
book, but I want you to carry uot the computations).

5. Page 593 #17


Extra credit


6. Compute the Fourier series of the function  (sin(x))^3  on the interval 
[-\pi, \pi].  (Hint: try your hand at trig identities).