Mathematical Tools for Physics

5—Fourier Series 120

And then use the orthonormality of the basis vectors,xˆ.ˆy= 0etc. Take the scalar product of the preceding
equation withxˆ.
ˆx.A~=xˆ.

(

Axˆx+Ayˆy+Azzˆ

)

=Ax. (3)

This lets you get all the components ofA~.
There are orthogonality relations similar to the ones forˆx,ˆy, andzˆ, but for sines and cosines. Letnand
mrepresent integers, then
∫L

0

dxsin

(nπx

L

)

sin

(mπx

L

)

=

{

0 n 6 =m

L/ 2 n=m (4)

This is sort of likexˆ.ˆz= 0andˆy.ˆy= 1.

More Examples
For a simple example, take the functionf(x) = 1, the constant on the interval 0 < x < Land assume that there
is a series representation forfon this interval.

1 =

∑∞

1

ansin

(nπx

L

)

(0< x < L) (5)

Multiply both sides by the sine ofmπx/Land integrate from 0 toL.

∫L

0

dxsin

(mπx

L

)

1 =

∫L

0

dxsin

(mπx

L

)∑∞

n=1

ansin

(nπx

L

)

Interchange the order of the sum and the integral, and the integral that shows up is the orthogonality integral
just above. When you use the orthogonality of the sines, only one term in the infinite series survives.

∫L

0

dxsin

(mπx

L

)

1 =

∑∞

n=1

an

∫L

0

dxsin

(mπx

L

)

sin

(nπx

L

)

=

∑∞

n=1

an.

{

0 n 6 =m

L/ 2 n=m

(6)

=amL/ 2.