THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 691
L
Since e^0 =1, thec 0 term can be absorbed into the
summation since it is just another term to be added
to the summation of thecnterm whenn=0. Thus,
f(x)=
∑∞
n= 0
cnej
2 πnx
L +
∑∞
n= 1
c−ne−j
2 πnx
L (10)
Thec−nterm may be rewritten by changing the limits
n=1ton=∞ton=−1ton=−∞. Sincenhas
been made negative, the exponential term becomes
ej
2 πnx
L andc−nbecomescn. Thus,
f(x)=
∑∞
n= 0
cnej
2 πnx
L +
−∞∑
n=− 1
cnej
2 πnx
L
Since the summations now extend from−∞to− 1
and from 0 to+∞, equation (10) may be written as:
f(x)=
∑∞
n=−∞
cnej
2 πnx
L (11)
Equation (11) is thecomplexorexponential form
of the Fourier series.
74.3 The complex coefficients
From equation (7), the complex coefficientcnwas
defined as:cn=
an−jbn
2
However,anandbnare defined (from page 630) by:
an=
2
L
∫L 2
−L 2
f(x) cos
(
2 πnx
L
)
dx and
bn=
2
L
∫L
2
−L 2
f(x) sin
(
2 πnx
L
)
dx
Thus, cn=
⎛
⎜
⎝
2
L
∫L 2
−L 2
f(x) cos
( 2 πnx
L
)
dx
−j^2 L
∫L 2
−L 2
f(x) sin
( 2 πnx
L
)
dx
⎞
⎟
⎠
2
=
1
L
∫L
2
−L 2
f(x) cos
(
2 πnx
L
)
dx
−j
1
L
∫L
2
−L 2
f(x) sin
(
2 πnx
L
)
dx
From equations (3) and (4),
cn=
1
L
∫L
2
−L 2
f(x)
(
ej
2 πnx
L +e−j
2 πnx
L
2
)
dx
−j
1
L
∫L
2
−L 2
f(x)
(
ej
2 πnx
L −e−j
2 πnx
L
2 j
)
dx
from which,
cn=
1
L
∫L
2
−L 2
f(x)
(
ej
2 πnx
L + e−j
2 πnx
L
2
)
dx
−
1
L
∫ L 2
−L 2
f(x)
(
ej
2 πnx
L −e−j
2 πnx
L
2
)
dx
i.e. cn=
1
L
∫L
2
−L 2
f(x)e−j
2 πnx
L dx (12)
Care needs to be taken when determiningc 0 .Ifn
appears in the denominator of an expression the
expansion can be invalid whenn=0. In such cir-
cumstances it is usually simpler to evaluatec 0 by
using the relationship:
c 0 =a 0 =
1
L
∫ L
2
−L 2
f(x)dx (from page 676). (13)
Problem 1. Determine the complex Fourier
series for the function defined by:
f(x)=
{
0, when− 2 ≤x≤− 1
5, when− 1 ≤x≤ 1
0, when 1 ≤x≤ 2
The function is periodic outside this range of
period 4.
This is the same Problem as Problem 2 on page 677
and we can use this to demonstrate that the two forms
of Fourier series are equivalent.
The functionf(x) is shown in Figure 74.1, where
the period,L=4.
From equation (11), the complex Fourier series is
given by:
f(x)=
∑∞
n=−∞
cnej
2 πnx
L