Higher Engineering Mathematics

FOURIER SERIES OVER ANY RANGE 681

L

Problem 5. Find the half-range Fourier sine

series for the functionf(x)=xin the range

0 ≤x≤2. Sketch the function within and outside

of the given range.

A half-range Fourier sine series indicates an odd
function. Thus the graph off(x)=xin the range 0
to 2 is shown in Fig. 72.5 and is extended outside of
this range so as to be symmetrical about the origin,
as shown by the broken lines.

24 6

2

− 2

f(x)

f(x) = x

− 4 − 2 0 x

Figure 72.5

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

f(x)=

∑∞

n= 1

bnsin

(nπx

L

)

bn=

2

L

∫L

0

f(x) sin

(nπx

L

)

dx

=

2

2

∫ 2

0

xsin

(nπx

L

)

dx

=

⎡

⎢

⎣

−xcos

(nπx

2

)

(nπ

2

) +

sin

(nπx

2

)

(nπ

2

) 2

⎤

⎥

⎦

2

0

=

⎡

⎢

⎣

⎛

⎜

⎝

−2 cosnπ

(nπ

2

) +

sinnπ

(nπ

2

) 2

⎞

⎟

⎠

−

⎛

⎜

⎝^0 +

sin 0

(nπ

2

) 2

⎞

⎟

⎠

⎤

⎥

⎦

=

−2 cosnπ

nπ

2

=

− 4

nπ

cosnπ

Henceb 1 =

− 4

π

(−1)=

4

π

b 2 =

− 4

2 π

(1)=

− 4

2 π

b 3 =

− 4

3 π

(−1)=

4

3 π

and so on.

Thus the half-range Fourier sine series in the range

0to2isgivenby:

f(x)=

4

π

[

sin

(πx

2

)

−

1

2

sin

(

2 πx

2

)

+

1

3

sin

(

3 πx

2

)

−

1

4

sin

(

4 πx

2

)

+···

]

Now try the following exercise.

Exercise 245 Further problems on half-

range Fourier series over rangeL

1. Determine the half-range Fourier cosine
   series for the functionf(x)=xin the range
   0 ≤x≤3. Sketch the function within and
   outside of the given range.
   ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
   f(x)=

3

2

−

12

π^2

{

cos

(

πx

3

)

+

1

32

cos

(

3 πx

3

)

+

1

52

cos

(

5 πx

3

)

+···

}

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2. Find the half-range Fourier sine series
   for the function f(x)=x in the range
   0 ≤x≤3. Sketch the function within and
   outside of the given range.
   ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
   f(x)=

6

π

(

sin

(πx

3

)

−

1

2

sin

(

2 πx

3

)

+

1

3

sin

(

3 πx

3

)

−

1

4

sin

(

4 πx

3

)

+···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

3. Determine the half-range Fourier sine series
   for the function defined by:

f(t)=

{ t,0<t< 1

(2−t), 1<t< 2