Chapter 69
Fourier series over any range
69.1 Expansion of a periodicfunction
of periodL
(a) A periodic function f(x) of period L
repeats itself when x increases by L,i.e.
f(x+L)=f(x). The change from functions
dealt with previously having period 2πto func-
tions having periodLis not difficult since it may
be achieved by a change of variable.
(b) To find a Fourier series for a function f(x)in
the range−
L
2
≤x≤
L
2
anewvariableuis intro-
duced such that f(x), as a function ofu,has
period 2π.Ifu=
2 πx
L
then, when x=−
L
2
,
u=−π and when x=
L
2
,u=+π. Also, let
f(x)=f
(
Lu
2 π
)
=F(u). The Fourier series for
F(u)is given by:
F(u)=a 0 +
∑∞
n= 1
(ancosnu+bnsinnu),
wherea 0 =
1
2 π
∫π
−π
F(u)du,
an=
1
π
∫π
−π
F(u)cosnudu
and bn=
1
π
∫π
−π
F(u)sinnudu
(c) It is however more usual to change the formula of
para. (b) to terms ofx.Sinceu=
2 πx
L
,then
du=
2 π
L
dx,
and the limits of integration are−
L
2
to+
L
2
instead of from−π to+π. Hence the Fourier
series expressed in terms ofxis given by:
f(x)=a 0 +
∑∞
n= 1
[
ancos
(
2 πnx
L
)
+bnsin
(
2 πnx
L
)]
where, in the range−
L
2
to+
L
2
:
and
a 0 =
1
L
∫ L
2
− 2 L
f(x)dx,
an=
2
L
∫ L 2
−L
2
f(x)cos
(
2 πnx
L
)
dx
bn=
2
L
∫ L 2
−L
2
f(x)sin
(
2 πnx
L
)
dx
The limits of integration may be replaced by any interval
of lengthL, such as from 0 toL.
Problem 1. The voltage from a square wave
generator is of the form:
v(t)=
{
0 , − 4 <t< 0
10 , 0 <t< 4
and has a period of 8ms.
Find the Fourier series for this periodic function.