636 Higher Engineering Mathematics
- Determine the half-range Fourier sine series
for the function defined by:
f(t)=
{
t, 0 <t< 1
( 2 −t), 1 <t< 2
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
f(t)=
8
π^2
(
sin
(
πt
2
)
−
1
32
sin
(
3 πt
2
)
+
1
52
sin
(
5 πt
2
)
−···
)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- Show that the half-range Fourier cosine series
for the functionf(θ )=θ^2 in the range 0 to 4
is given by:
f(θ )=
16
3
−
64
π^2
(
cos
(
πθ
4
)
−
1
22
cos
(
2 πθ
4
)
+
1
32
cos
(
3 πθ
4
)
−···
)
Sketch the function within and outside of the
given range.