Higher Engineering Mathematics

(Greg DeLong) #1
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
Free download pdf