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).