Mathematical Tools for Physics

5—Fourier Series 122

and Eq. ( 4 ) then becomes

〈

un,um

〉

=

{

0 n 6 =m

L/ 2 n=m where un(x) = sin

(nπx

L

)

(9)

The Fourier series manipulations become

1 =

∑∞

1

anun, then

〈

um, 1

〉

=

〈

um,

∑∞

1

anun

〉

=

∑∞

n=1

an

〈

um,un

〉

=an

〈

un,un

〉

(10)

This is a far more compact notation than you see in the steps between Eq. ( 5 ) and Eq. ( 7 ). Youstillhave to
evaluate the integral, but when you master this notation you’ll likely make fewer mistakes in figuring out what
integral you have to do.
The analogy between the vectors such asxˆand functions such as sine is really far deeper, and it is central
to the subject of the next chapter. In order not to to get confused by the notation, you have to distinguish
between a whole functionf, and the value of that function at a point,f(x). The former is the whole graph of
the function, and the latter is one point of the graph.
The scalar product notation of Eq. ( 8 ) is not necessarily restricted to the interval 0 < x < L. Depending
on context it can be over any interval that you happen to be considering at the time. In Eq. ( 8 ) there is a complex
conjugation symbol. The functions here have been real, so this made no difference, but you will often deal with
complex functions and then the fact that the notation

〈

f,g

〉

includes a conjugation is important. This notation
is really a special case of a far more general development, but that can wait until section6.6.

5.3 Choice of Basis
When you work with components of vectors in two or three dimensions, you will choose the basis that is most
convenient for the problem you’re working with. If you do a simple mechanics problem with a mass moving on
an incline, you can choose a basisˆxandˆythat are arranged horizontally and vertically. OR, you can place them
at an angle so that they point down the incline and perpendicular to it. The latter is often a simpler choice in
that type of problem.
The same applies to Fourier series. The interval on which you’re working is not necessarily from zero toL,
and even on the interval 0 < x < Lyou can choose many bases:

sinnπx/L (n= 1, 2 ,...) as above, or you can choose a basis