Higher Engineering Mathematics, Sixth Edition

618 Higher Engineering Mathematics

x

f(x) f(x) 52 x

22  2  0

22 



2 

2  3 

Figure 67.2

From Section 66.3(i),

a 0 =

1

2 π

∫π

−π

f(x)dx

=

1

2 π

∫π

−π

2 xdx=

2

2 π

[

x^2

2

]π

−π

= 0

an=

1

π

∫π

−π

f(x)cosnxdx=

1

π

∫π

−π

2 xcosnxdx

=

2

π

[

xsinnx

n

−

∫

sinnx

n

dx

]π

−π

by parts(see Chapter 43)

=

2

π

[

xsinnx

n

+

cosnx

n^2

]π

−π

=

2

π

[(

0 +

cosnπ

n^2

)

−

(

0 +

cosn(−π)

n^2

)]

= 0

bn=

1

π

∫π

−π

f(x)sinnxdx=

1

π

∫π

−π

2 xsinnxdx

=

2

π

[

−xcosnx

n

−

∫ (

−cosnx

n

)

dx

]π

−π

by parts

=

2

π

[

−xcosnx

n

+

sinnx

n^2

]π

−π

=

2

π

[(

−πcosnπ

n

+

sinnπ

n^2

)

−

(

−(−π)cosn(−π)

n

+

sinn(−π)

n^2

)]

=

2

π

[

−πcosnπ

n

−

πcos(−nπ)

n

]

=

− 4

n

cosnπ

When n is odd, bn=

4

n

. Thus b 1 =4, b 3 =

4

3

,

b 5 =

4

5

, and so on.

When n is even, bn=

− 4

n

. Thus b 2 =−

4

2

,

b 4 =−

4

4

,b 6 =−

4

6

, and so on.

Thus f(x)= 2 x=4sinx−

4

2

sin2x+

4

3

sin3x

−

4

4

sin4x+

4

5

sin5x−

4

6

sin6x+···

i.e. 2 x= 4

(

sinx−

1

2

sin2x+

1

3

sin3x−

1

4

sin4x

+

1

5

sin5x−

1

6

sin6x+···

)

(1)

for values of f(x)between−π andπ.Forvalues

of f(x)outside the range−πto+πthe sum of the

series is not equal tof(x).

Problem 2. In the Fourier series of Problem 1, by

lettingx=π/2, deduce a series forπ/4.

Whenx=π/2, f(x)=πfrom Fig. 67.2.

Thus, from the Fourier series of equation (1):

2

(π

2

)

= 4

(

sin

π

2

−

1

2

sin

2 π

2

+

1

3

sin

3 π

2

−

1

4

sin

4 π

2

+

1

5

sin

5 π

2

−

1

6

sin

6 π

2

+···

)

π= 4

(

1 − 0 −

1

3

− 0 +

1

5

− 0 −

1

7

−···

)

i.e.

π

4

= 1 −

1

3

+

1

5

−

1

7

+···

Problem 3. Obtain a Fourier series for the

function defined by:

f(x)=

{

x, when 0<x<π

0 , whenπ<x< 2 π.

ThedefinedfunctionisshowninFig.67.3between0and

2 π. The function is constructed outside of this range so

that it is periodic of period 2π, as shown by the broken

line in Fig. 67.3.

For a Fourier series:

f(x)=a 0 +

∑∞

n= 1

(ancosnx+bnsinnx)