082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

23.6 Half-range series 747
Half-range cosine series:
a 0
= 4 T
a n
= 4 T
b n
= 0
∫ T/2
0
∫ T/2
and then f (t)isgiven by
0
f(t)dt (23.8)
f(t)cos 2nπt
T
fornapositive integer
f(t)= a ∞
0
2 + ∑
a n
cos 2nπt
T
n=1
dt fornapositive integer (23.9)
Example23.19 BydefininganappropriateperiodicextensionofthefunctionillustratedinFigure23.24,
find the half-range cosineseries representation.
Solution The function illustrated in Figure 23.24 is given by the formula f (t) =t for 0 <t < π
and is undefined outside this interval. Since the cosine series is required an even periodic
extension must be formed. This is illustrated in Figure 23.25. Taking T = 2π in
Equations(23.8) and (23.9),wefinda 0
anda n
.
∫ π
a 0
= 2 tdt
π 0
= 2 [ ] t
2 π
π 2
0
= π
∫ π
a n
= 2 tcosntdt
π 0
= 2 {[ ] tsinnt π ∫ π
}
sinnt
− dt
π n
0 0 n
[ cosnt
= 2 π
n 2 ] π
0
y
π
y
π
π
t
–2π
–π 0
π
2π
t
Figure23.24
Graph forExample 23.19.
Figure23.25
GraphforExample23.19.