x - Math Plp

α n
f f :x7−→ xn(Acos(α x) + Bsin(α x))
fRf
F(x) = Zx
0
tn(Acos(α t) + Bsin(α t)) dt
=AZx
0
tncos(α t) dt+BZx
0
tnsin(α t) dt
x7−→ xnR
x7−→ cos(α x)x7−→ sin(α x)R
F(x) = ·A
αtnsin(α t)¸x
0
−A n
αZx
0
tn−1sin(α t) dt
−·B
αtncos(α t)¸x
0
+B n
αZx
0
tn−1cos(α t) dt
n= 1
F(x) = A
αxsin(α x)−A αZx
0
sin(α t) dt−B
αxcos(α x) + B αZx
0
cos(α t) dt
=µA
αx+B α2¶sin(α x)−µB αx−A α2¶cos(α x)−A α2
n= 1 n−1
n−1
x7−→ A n
αxn−1sin(α x)−B n αxn−1cos(α x)
x7−→ P(x) cos(α x) + Q(x) sin(α x) + K P Q
n−1K
2≤n
F(x) = A
αxnsin(α x)−A n αZx
0
tn−1sin(α t) dt−B
αxncos(α t) + B n αZx
0
tn−1cos(α t) dt
=A
αxnsin(α x)−B αxncos(α x)−Zx
0µA n
αtn−1sin(α t)−B n αtn−1cos(α t)¶dt

F(x) = µA
αxn−Q(x)¶sin(α x)−µB αxn−P(x)¶cos(α x)−K
n
n= 1
n
F f
F(x) = P(x) cos(α x) + Q(x) sin(α x) + K
P Q n K
n= 2
Gn(x) = ·1
αtnsin(α t)¸x
0
−n
αZx
0
tn−1sin(α t) dt1≤n
=1
αxnsin(α t)−n αZx
0
tn−1sin(α t) dt
Zx
0
tn−1sin(α t) dt=Pn−1(x) cos(α x) + Qn−1(x) sin(α x) + Kn−1
Pn−1Qn−1n−1
Gng:x7−→ xncos(α x)
Gn(x) = Pn(x) cos(α x) + Qn(x) sin(α x) + Kn
Pnn−1Qnn
Qn
1
αxn
PnQn
Pn(x) cos(α x) + Qn(x) sin(α x) + Kn=Zx
0
tncos(α t) dt
Pnn−1Qnn
P0
n(x) cos(α x)−α Pn(x) sin(α x) + Q0 n(x) sin(α x) + α Qn(x) cos(α x) = xncos(α x)
½P0
n(x) + α Qn(x) = xn
Q0
n(x)−α Pn(x)=0
½P0
n(x) + α Qn(x) = xn
Q00
n(x) = α P 0 n(x)
PnQn
Qn
Q00
n(x) + α2Qn(x) = α xn(Eq)

ϕ
ϕ(x) = Acos(α x) + Bsin(α x)
n
ai
Φ(x) =
n
X
i=0
aixi
(Eq)
α2an=α
α2an−1= 0
i(i−1) ai+α2ai−2= 0 2 ≤i < n −1
an−2i=(−1)i
α2i+1 ×n! (n−2i) ! 0≤i≤n 2
an−2i−1= 0 0 ≤i < n
2
(Eq)
Acos(α x) + Bsin(α x) + X
0≤i≤n
2
(−1)i
α2i+1 ×n! (n−2i) ! ×xn−2i
(Eq)
Φ(x) = X
0≤i≤n
2
(−1)i
α2i+1 ×n! (n−2i) ! ×xn−2i
Qn(x) = X
0≤i≤n
2
(−1)i
α2i+1 ×n! (n−2i) ! ×xn−2i
Pn(x) = X
0≤i≤n−1
2
(−1)i
α2 (i+1) ×n! (n−2i−1) ! ×xn−2i−1

x7−→ Gn(x) = Pn(x) cos(α x) + Qn(x) sin(α x) + K
n
x7−→ xncos(α x)n
n
n= 0 Zx
−x
cos(α t) dt=·1
αsin(α t)¸x
−x
=2
αsin(α x)
nZx
−x
tncos(α t) dt= 0
nZx
−x
tncos(α t) dt=Zx
0
tncos(α t) dt−Z−x
0
tncos(α t) dt= 2 Zx
0
tncos(α t) dt
Zx
−x
tncos(α t) dt= 2 (Pn(x) cos(α x) + Qn(x) sin(α x))
x
x Pn
Qn
n= 2 Q2(x) = 1
αx2−2 α3P2(x) = 1 αQ0
2(x) = 2
α2x
Zπ
−π
t2cos(α t) dt=4π
α2cos(α π) + µ2 απ2−4 α3¶sin(α π)

f f(x) = 1 −x2
π2]−π, π]
−2 2 4 6 8
0
−0.5
0.5
1.5
y = f(x)
y
x
3 ππ−π−4
1−bπb
<−π∗π < π < 3∗π−2∗π−4∗π
f]−π, π]
f0(x) = −2x
π2x∈]−π, π]
]π, 2,π]f
f(x) = 1 −(x−2π)2
π2x∈]π, 3π]
f0(x) = −2 (x−2π)
π2x∈]π, 3π]
x π
lim
x→π−
f(x) = lim
x→π+f(x) = 0
f π ]−π, 3π]