Even and odd functions and half-range Fourier series 629
Hence the half-range Fourier sine series forf(x)in the
range 0 toπis given by:
f(x)=
8
3 π
sin2x+
16
15 π
sin4x
+
24
35 π
sin6x+···
or f(x)=
8
π
(
1
3
sin2x+
2
( 3 )( 5 )
sin4x
+
3
( 5 )( 7 )
sin6x+···
)
Now try the following exercise
Exercise 231 Further problems on
half-range Fourier series
- Determine the half-range sine series for the
function defined by:
f(x)=
⎧
⎨
⎩
x, 0 <x<
π
2
0 ,
π
2
<x<π
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
f(x)=
2
π
(
sinx+
π
4
sin2x
−
1
9
sin3x
−
π
8
sin4x+···
)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- Obtain (a) the half-range cosine series and
(b) the half-range sine series for the function
f(t)=
⎧
⎪⎨
⎪⎩
0 , 0 <t<
π
2
1 ,
π
2
<t<π
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(a) f(t)=
1
2
−
2
π
(
cost
−
1
3
cos3t
+
1
5
cos5t−···
)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡
⎢
⎢
⎢
⎢
⎢⎢
⎢
⎣
(b) f(t)=
2
π
(
sint−sin2t
+
1
3
sin3t+
1
5
sin5t
−
1
3
sin6t+···
)
⎤
⎥
⎥
⎥
⎥
⎥⎥
⎥
⎦
- Find (a) the half-range Fourier sine series and
(b) the half-range Fourier cosine series for the
functionf(x)=sin^2 xin the range 0≤x≤π.
Sketch the function within and outside of the
given range.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(a) f(x)=
8
π
(
sinx
( 1 )( 3 )
−
sin3x
( 1 )( 3 )( 5 )
−
sin5x
( 3 )( 5 )( 7 )
−
sin7x
( 5 )( 7 )( 9 )
−···
)
(b) f(x)=
1
2
( 1 −cos2x)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- Determine the half-range Fourier cosine series
in the rangex=0tox=πfor the function
defined by:
f(x)=
⎧
⎪⎪
⎨
⎪⎪
⎩
x, 0 <x<
π
2
(π−x),
π
2
<x<π
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
f(x)=
π
4
−
2
π
(
cos2x
+
cos6x
32
+
cos10x
52
+···
)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦