Mathematical Tools for Physics

5—Fourier Series 127

am=

4

(2m+ 1)π

Then the series is
4
π

[

sin

πx

2 L

+

1

3

sin

3 πx

2 L

+

1

5

sin

5 πx

2 L

+···

]

(20)

Parseval’s Identity
Letunbe the set of orthogonal functions that follow from your choice of boundary conditions.

f(x) =

∑

n

anun(x)

Evaluate the integral of the absolute square offover the domain.

∫b

a

dx|f(x)|^2 =

∫b

a

dx

[

∑

m

amum(x)

]*[

∑

n

anun(x)

]

=

∑

m

a*m

∑

n

an

∫b

a

dxum(x)*un(x) =

∑

n

|an|^2

∫b

a

dx|un(x)|^2

In the more compact notation this is
〈
f,f

〉

=

〈∑

m

amum,

∑

n

anun

〉

=

∑

m,n

a*man

〈

um,un

〉

=

∑

n

|an|^2

〈

un,un

〉

(21)

The first equation is nothing more than substituting the series forf. The second moved the integral under the
summation. The third equation uses the fact that all these integrals are zero except for the ones withm=n.
That reduces the double sum to a single sum. If you have chosen to normalize all of the functionsunso that the
integrals of|un(x)|^2 are one, then this relation takes on a simpler appearance.
What does this say if you apply it to a series I’ve just computed? Take Eq. ( 7 ) and see what it implies.
〈
f,f

〉

=

∫L

0

dx1 =L=

∑∞

k=0

|ak|^2

〈

un,un

〉

=

∑∞

k=0

(

4

π(2k+ 1)

) 2 ∫L

0

dxsin^2

(

(2k+ 1)πx

L

)

=

∑∞

k=0

(

4

π(2k+ 1)

) 2

L

2