# Handy Fourier tricks

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```					                           B3D Handout 18: Handy Fourier Tricks
Here are just a couple of things we can do with the Fourier series: the standard Fourier series for a
function with period 2L is
∞
1                 nπx          nπx
f (x) =             a0 +     an cos     + bn sin                                     .
2      n=1
L            L

1       Integration of a Fourier series
The Fourier series for f (x) can be integrated term by term provided that f (x) is piecewise continuous
in the period 2L (i.e. only a ﬁnite number of jumps):
β                          β                         ∞            β                                   β
1                                                  nπx                          nπx
f (x) dx =                    a0 dx +     an                       cos             dx + bn            sin       dx .
α                       α        2         n=1                     α                 L                α           L

2       Parseval’s identity
We can multiply f (x) by itself and integrate:
2L                                     2L                      ∞
1                 nπx          nπx
f (x)f (x) dx       =                        a0 +     an cos     + bn sin                                  f (x) dx
0                                      0            2      n=1
L            L
2L                         ∞               2L                                    2L
a0                                                                 nπx                                   nπx
=                          f (x) dx +                an               cos       f (x) dx + bn               sin       f (x) dx
2        0                          n=1             0               L                     0               L
∞                                                   ∞
a0                                  a2
=           La0 +     (an Lan + bn Lbn ) = L 0 +     (a2 + b2 ) .
n    n
2        n=1
2   n=1

To put it another way,
2L                                       ∞
1                                         1 2
f (x)f (x) dx =           a +   (a2 + b2 ).
L       0                                 2 0 n=1 n     n

This is Parseval’s identity.

Example

Remember the square wave, of height 1 and period 2π:

6

s                                             s                                       s                               s -
−2π               −π                          0                   π                  2π         3π                   4π

The Fourier series for this function was
 4

∞
n odd
f (x) =                    bn sin nx                 with             bn =   nπ
0         n even.

1

Parseval’s identity gives
2π                              ∞
1                                                                    16   1  1   1
f 2 (x) dx =                     b2 ;
n             2=         1+ +   +   + ···
π   0                                   n=1
π2   9 25 49
which tells us that
1   1   1         π2
1+          +   +   + ··· =    .
9 25 49           8

18

```
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