Higher Engineering Mathematics

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 697

L

From equation (17) above,cn=−j

2

nπ

( 1 −cosnπ)

Whenn=1,

c 1 =−j

2

(1)π

( 1 −cosπ)

=−j

2

π

(

1 −(−1)

)

=−

j 4

π

Whenn=2,

c 2 =−j

2

2 π

( 1 −cos 2π)=0;

in fact, all even values ofcnwill be zero.

Whenn=3,

c 3 =−j

2

3 π

( 1 −cos 3π)

=−j

2

3 π

( 1 −(−1))=−

j 4

3 π

By similar reasoning,

c 5 =−

j 4

5 π

,c 7 =−

j 4

7 π

, and so on.

Whenn=−1,

c− 1 =−j

2

(−1)π

( 1 −cos(−π))

=+j

2

π

( 1 −(−1))=+

j 4

π

Whenn=−3,

c− 3 =−j

2

(−3)π

( 1 −cos(− 3 π))

=+j

2

3 π

( 1 −(−1))=+

j 4

3 π

By similar reasoning,

c− 5 =+

j 4

5 π

,c− 7 =+

j 4

7 π

, and so on.

Since the waveform is odd,c 0 =a 0 = 0.
From equation (18) above,

f(x)=

∑∞

n=−∞

−j

2

nπ

( 1 −cosnπ)ejnx

Hence,

f(x)=−

j 4

π

ejx−

j 4

3 π

ej^3 x−

j 4

5 π

ej^5 x

−

j 4

7 π

ej^7 x−···+

j 4

π

e−jx+

j 4

3 π

e−j^3 x

+

j 4

5 π

e−j^5 x+

j 4

7 π

e−j^7 x+···

=

(

−

j 4

π

ejx+

j 4

π

e−jx

)

+

(

−

j 4

3 π

e^3 x+

j 4

3 π

e−^3 x

)

+

(

−

j 4

5 π

e^5 x+

j 4

5 π

e−^5 x

)

+···

=−

j 4

π

(

ejx−e−jx

)

−

j 4

3 π

(

e^3 x−e−^3 x

)

−

j 4

5 π

(

e^5 x−e−^5 x

)

+···

=

4

jπ

(

ejx−e−jx

)

+

4

j 3 π

(

e^3 x−e−^3 x

)

+

4

j 5 π

(

e^5 x−e−^5 x

)

+···

by multiplying top and bottom byj

=

8

π

(

ejx−e−jx

2 j

)

+

8

3 π

(

ej^3 x−e−j^3

2 j

)

+

8

5 π

(

ej^5 x−e−j^5 x

2 j

)

+···

by rearranging

=

8

π

sinx+

8

3 π

sin 3x+

8

3 x

sin 5x+···

from equation (4), page 690

i.e.

f(x)=

8

π

(

sinx+

1

3

sin 3x+

1

5

sin 5x

+

1

7

sin 7x+···

)

Hence,

f(x)=

∑∞

n=−∞

−j

2

nπ

( 1 −cosnπ)ejnx

≡

8

π

(

sinx+

1

3

sin 3x+

1

5

sin 5x

+

1

7

sin 7 x+···

)