The complex or exponential form of a Fourier series 645
Rearranging gives:
f(x)=a 0 +
∑∞
n= 1
[(
an−jbn
2
)
ej
2 πLnx
+
(
an+jbn
2
)
e−j
2 πLnx
]
(5)
The Fourier coefficientsa 0 ,anandbnmay be replaced
by complex coefficientsc 0 ,cnandc−nsuch that
c 0 =a 0 (6)
cn=
an−jbn
2
(7)
and c−n=
an+jbn
2
(8)
wherec−nrepresents the complex conjugate ofcn(see
page 216).
Thus, equation (5) may be rewritten as:
f(x)=c 0 +
∑∞
n= 1
cnej
2 πnx
L +
∑∞
n= 1
c−ne−j
2 πnx
L (9)
Since e^0 =1, thec 0 term can be absorbed into the sum-
mation 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 πLnx
+
−∞∑
n=− 1
cnej
2 πLnx
Since the summations now extend from−∞to−1and
from 0 to+∞, equation (10) may be written as:
f(x)=
∑∞
n=−∞
cnej
2 πnx
L (11)
Equation (11) is thecomplexorexponential formof
the Fourier series.
71.3 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 πLnx
+e−j
2 πLnx
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 πLnx
dx (12)
Careneeds to betaken when determiningc 0 .Ifnappears
in the denominator of an expression the expansion can
beinvalidwhenn=0.In such circumstances 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 630). (13)