5—Fourier Series 1375.8 In the two problems 5 and 6 you improved the convergence by choosing boundary conditions that better
matched the function that you want. Can you do better? The functionxvanishes at the origin, but its derivative
isn’t zero atL, so try boundary conditionsu(0) = 0andu(L) =Lu′(L). These conditions match those ofx
so this ought to give even better convergence, but first you have to verify that these conditions guarantee the
orthogonality of the basis functions. You have to verify that the left side of Eq. ( 12 ) is in fact zero. When you
set up the basis, you will examine functions of the formsinkx, but you will not be able to solve explicitly for the
values ofk. Don’t worry about it. When you use Eq. ( 19 ) to get the coefficients all that you need to do is to
use the equation thatksatisfies to do the integrals. You do not have to havesolvedit. If you do all the algebra
correctly you will probably have a surprise.
5.9 Use the periodic boundary conditions on−L < x <+Land basiseπinx/Lto writex^2 as a Fourier series.
Sketch the sums up to a few terms. Evaluate your result atx=Lwhere you know the answer to beL^2 and
deduce from this the value ofζ(2).
5.10 On the interval−π < x < π, the functionf(x) = cosx. Expand this in a Fourier series defined byu′′=λu
andu(−π) = 0 =u(π). If you use your result for the series outside of this interval you define an extension of
the original function. Graph this extension and compare it to what you normally think of as the graph ofcosx.
5.11 Represent a functionfon the interval−L < x < Lby a Fourier series using periodic boundary conditions
f(x) =∑∞
−∞anenπix/L(a) If the functionfis odd, prove that for alln,a−n=−an
(b) If the functionfis even, prove that alla−n=an.
(c) If the functionfis real, prove that alla−n=a*n.
(d) If the function is both real and even, characterizean.
(e) If the function is imaginary and odd, characterizean.
5.12 Derive the series Eq. ( 29 ).