616 Higher Engineering Mathematics
- For the waveform shown in Fig. 66.6 deter-
mine (a) the Fourier series for the functionand
(b) the sum of the Fourier series at the points
of discontinuity.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(a)f(x)=
1
2
+
2
π
(
cosx−
1
3
cos3x
+
1
5
cos5x−···
)
(b)
1
2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ x
f(x)
0
1
2 3
2
23
2
2
2
2
Figure 66.6
- For Problem 3, draw graphs of the first three
partial sums of the Fourier series and show that
as the series is added togetherterm by term the
result approximates more and more closely to
the function it represents. - Find the term representing the third harmonic
fortheperiodicfunctionofperiod2πgivenby:
f(x)=
{
0 , when −π<x< 0
1 , when 0<x<π
[
2
3 π
sin3x
]
- Determine the Fourier series for the periodic
function of period 2πdefined by:
f(t)=
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
0 , when −π<t< 0
1 , when 0<t<
π
2
− 1 , when
π
2
<t<π
The function has a period of 2π.
⎡
⎢⎢
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
f(t)=
2
π
⎛
⎜⎜
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
cost−
1
3
cos3t
+
1
5
cos5t−···
+sin2t+
1
3
sin6t
+
1
5
sin10t+···
⎞
⎟⎟
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎤
⎥⎥
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- Show that the Fourier series for the periodic
function of period 2πdefined by
f(θ )=
{
0 , when −π<θ< 0
sinθ, when 0<θ<π
is given by:
f(θ )=
2
π
(
1
2
−
cos2θ
( 3 )
−
cos4θ
( 3 )( 5 )
−
cos6θ
( 5 )( 7 )
−···
)