Higher Engineering Mathematics, Sixth Edition

628 Higher Engineering Mathematics

Problem 7. Find the half-range Fourier sine

series to represent the functionf(x)= 3 xin the

range 0≤x≤π.

From para. (c), for a half-range sine series:

f(x)=

∑∞

n= 1

bnsinnx

Whenf(x)= 3 x,

bn=

2

π

∫π

0

f(x)sinnxdx=

2

π

∫π

0

3 xsinnxdx

=

6

π

[

−xcosnx

n

+

sinnx

n^2

]π

0

by parts

=

6

π

[(

−πcosnπ

n

+

sinnπ

n^2

)

−( 0 + 0 )

]

=−

6

n

cosnπ

Whennis odd,bn=

6

n

Henceb 1 =

6

1

,b 3 =

6

3

,b 5 =

6

5

andsoon.

Whennis even,bn=−

6

n

Henceb 2 =−

6

2

,b 4 =−

6

4

,b 6 =−

6

6

and so on.

Hence the half-range Fourier sine series is given by:

f(x)= 3 x= 6

(

sinx−

1

2

sin2x+

1

3

sin3x

−

1

4

sin4x+

1

5

sin5x−···

)

Problem 8. Expandf(x)=cosxas a half-range

Fourier sine series in the range 0≤x≤π, and sketch

the function within and outside of the given range.

When a half-range sine series is required then an

odd function is implied, i.e. a function symmetrical

about the origin. A graph ofy=cosx is shown in

Fig. 68.6 in the range 0 toπ.Forcosxto be symmetrical

about the origin the function is as shown by the broken

lines in Fig. 68.6 outside of the given range.

From para. (c), for a half-range Fourier sine series:

f(x)=

∑∞

n= 1

bnsinnxdx

2 

21

0  2 x

1 y^5 cos x

f(x)

Figure 68.6

bn=

2

π

∫π

0

f(x)sinnxdx

=

2

π

∫π

0

cosxsinnxdx

=

2

π

∫π

0

1

2

[sin(x+nx)−sin(x−nx)]dx

=

1

π

[

−cos[x( 1 +n)]

( 1 +n)

+

cos[x( 1 −n)]

( 1 −n)

]π

0

=

1

π

[(

−cos[π( 1 +n)]

( 1 +n)

+

cos[π( 1 −n)]

( 1 −n)

)

−

(

−cos0

( 1 +n)

+

cos0

( 1 −n)

)]

Whennis odd,

bn=

1

π

[(

− 1

( 1 +n)

+

1

( 1 −n)

)

−

(

− 1

( 1 +n)

+

1

( 1 −n)

)]

= 0

Whennis even,

bn=

1

π

[(

1

( 1 +n)

−

1

( 1 −n)

)

−

(

− 1

( 1 +n)

+

1

( 1 −n)

)]

=

1

π

(

2

( 1 +n)

−

2

( 1 −n)

)

=

1

π

(

2 ( 1 −n)− 2 ( 1 +n)

1 −n^2

)

=

1

π

(

− 4 n

1 −n^2

)

=

4 n

π(n^2 − 1 )

Henceb 2 =

8

3 π

,b 4 =

16

15 π

,b 6 =

24

35 π

andsoon.