Homework 10   Math427K-H   Due Thursday, April 23, 2009


1. Find the solutions of the eigenvalue problem  y" + \lambda y = 0

on the interval [0,L] which satisfy the boundary conditions y'(0) = 
y'(L) = 0.


2.  Consider Fourier series on the interval  [-L,L]  and assume the functions
below are defined on this interval.
Compute the Fourier series for  |x|  on this interval.


3. Compute the Fourier series for f(x) = sgn(x) on this  same interval.
How are they related?  Note:  sgn(x) means
the sign of x, +1 for x> 0 and -1 for x < 0.

4.  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.

5. Using your answer from #2 and #3 , compute the cosine series of
L - |x| on the   interval [0,L] and the sine series for f(x) = 2 on the
same interval.








Extra Credit: Due Friday April 24.

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