Mathematical Tools for Physics

5—Fourier Series 139

5.19 In the calculation leading to Eq. ( 31 ) I assumed thatf(t)is real and then used the properties ofanthat
followed from that fact. Instead, make no assumption about the reality off(t)and compute

〈

|f(t)|^2

〉

=

〈

f(t)*f(t)

〉

Show that it leads to the same result as before,

∑

|an|^2.

5.20 The series
∑∞

n=0

ancosnθ (|a|<1)

represents a function. Sum this series and determine what the function is. While you’re about it, sum the similar
series that has a sine instead of a cosine. Don’t try to do these separately; combine them and do them as one
problem. And check some limiting cases of course. Ans:asinθ/

(

1 +a^2 − 2 acosθ

)

5.21 Apply Parseval’s theorem to problem 9 and see what you can deduce.

5.22 If you take all the elementsunof a basis and multiply each of them by 2, what happens to the result for
the Fourier series for a given function?

5.23 In the section5.3several bases are mentioned. Sketch a few terms of each basis.

5.24 A function is specified on the interval 0 < t < T to be

f(t) =

{

1 ( 0 < t < t 0 )

0 (t 0 < t < T)

0 < t 0 < T

On this interval, choose boundary conditions such that the left side of the basic identity ( 12 ) is zero. Use the
corresponding choice of basis functions to writefas a Fourier series on this interval.

5.25 Show that the boundary conditionsu(0) = 0andαu(L) +βu′(L) = 0make the bilinear concomitant in
Eq. ( 12 ) vanish. Are there any restrictions onαandβ?