Solve 7.26, by Laplace transforming the problem as given in
time. This is a good way to practice back transform methods. Note
that one factor in is a simpler function at a shifted
value of coordinate .
Solve 7.35 by Laplace transform in time. Clean up completely;
only the given function may be in your answer, no Heaviside
functions or other weird stuff. There is a minor error in the
book’s answer.
Derive the (real) Fourier series for the -periodic
function that satisfies
If possible, check your answer versus 24.17 in the math handbook.
Plot the Fourier series for increasing number of terms,
using 360 equally spaced plot points in
the range from 0 to .
Based on these plots, discuss whether the Fourier series really
converges to the given function for high enough number of terms:
When does an actual jump first show up?
Are there points where the Fourier series does not converge?
If so, give the values for which it does not converge.
If it converges everywhere, the maximum difference between the
Fourier series and the given function should obviously become
zero. You might want to plot the function for
using 1800 equally spaced plot points before
you answer that. Explain.