634 Higher Engineering Mathematics

4. Determine the Fourier series for the half
   wave rectified sinusoidal voltage Vsinωt
   defined by:

f(t)=

⎧

⎪⎨

⎪⎩

Vsinωt, 0 <t<

π

ω

0 ,

π

ω

<t<

2 π

ω

which is periodic of period

2 π

ω ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

f(t) =

V

π

+

V

2

sinωt

−

2 V

π

(

cos2ωt

( 1 )( 3 )

+

cos4ωt

( 3 )( 5 )

+

cos6ωt

( 5 )( 7 )

+···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

69.2 Half-range Fourier series for

functions defined over rangeL

(a) By making the substitution u=

πx

L

(see

Section 69.1), the rangex=0tox=L corre-

sponds to the rangeu=0tou=π. Hence a

function may be expanded in a series of either

cosine terms or sine terms only, i.e. ahalf-range

Fourier series.

(b) Ahalf-range cosine seriesin the range 0 toLcan

be expanded as:

where

f(x)=a 0 +

∑∞

n= 1

ancos

(nπx

L

)

a 0 =

1

L

∫L

0

f(x)dx and

an=

2

L

∫L

0

f(x)cos

(nπx

L

)

dx

(c) Ahalf-range sine seriesin the range 0 toLcan

be expanded as:

f(x)=

∑∞

n= 1

bnsin

(nπx

L

)

where bn=

2

L

∫L

0

f(x)sin

(nπx

L

)

dx

Problem 4. Determine the half-range Fourier

cosine 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 cosine series indicates an even

function. Thus the graph off(x)=xin the range 0 to

2 is shown in Fig. 69.4 and is extended outside of this

range so as to be symmetrical about the f(x)axis as

shown by the broken lines.

f (x )

f (x ) 5 x

(^2422) x
2
0642
Figure 69.4
From para. (b), for a half-range cosine series:
f(x)=a 0 +
∑∞
n= 1
ancos
(nπx
L
)
a 0 =
1
L
∫L
0
f(x)dx=
1
2
∫ 2
0
xdx

1
2
[
x^2
2
] 2
0
= 1
an=
2
L
∫L
0
f(x)cos
(nπx
L
)
dx

2
2
∫ 2
0
xcos
(nπx
2
)
dx

⎡
⎢
⎣
xsin
(nπx
2
)
(nπ
2
) +
cos
(nπx
2
)
(nπ
2
) 2
⎤
⎥
⎦
2
0

⎡
⎢
⎣
⎛
⎜
⎝
2sinnπ
(nπ
2
) +
cosnπ
(nπ
2
) 2
⎞
⎟
⎠−
⎛
⎜
⎝^0 +
cos0
(nπ
2
) 2
⎞
⎟
⎠
⎤
⎥
⎦

⎡
⎢
⎣
cosnπ
(nπ
2
) 2 −
1
(nπ
2
) 2
⎤
⎥
⎦

(
2
πn
) 2
(cosnπ− 1 )