Algèbre - Mathematiques pour les PLP.

nk
n
t7−→ tkRt7−→ e−t
R
RR
t+∞et
t
et∈O
t→+∞tk+2
tke−t∈O
t→+∞
1
t2
t7−→ 1
t2+∞
t7−→ tke−tR
t7−→ 1
t2+∞
Z+∞
0
tke−tdt
fPnfRt7−→ e−t
R
RR
t+∞fk≤n
t7−→ tk
f(t)∈O
t→+∞tk
f(t)e−t∈O
t→+∞
1
t2
t7−→ f(t)e−t+∞
t7−→ f(t)e−t+∞
x+∞
x g(x) = exZ+∞
x
f(t)e−tdt
g(x)RR
Φ(eo)
t7−→ −e−tt7−→ e−t
lim
t→+∞e−t= 0,
[Φ(eo)](x) = exZ+∞
x
e−tdt
=exh−e−ti+∞
x
= 1

Φ(e1)
t7−→ e1(t)C1Rt7−→ e−t
R[x, A]
ZA
x
t e−tdt=h−t e−tiA
x+ZA
x
e−tdt
=h−t e−tiA
x+h−e−tiA x
=−A e−A+x e−x−e−A+e−x
A+∞eA
A
lim
A→+∞
Ak
eA= 0 ∀k∈N
A e−Ae−AA
+∞
Z+∞
x
t e−tdt=x e−x+e−x
[Φ(e1)](x) = x+ 1
Φ(e2)
t7−→ e2(t)C1Rt7−→ e−t
R[x, A]
ZA
x
t2e−tdt=h−t2e−tiA
x+ZA
x2t e−tdt
[Φ(e2)](x) = x2+ 2 x+ 2
t7−→ ek+1(t) = tk+1 C1Rt7−→ e−t
R[x, A]
ZA
x
tk+1 e−tdt=h−tk+1 e−tiA
x+ (k+ 1) ZA
x
tke−tdt
=−Ak+1 e−A+xk+1 e−x+ (k+ 1) ZA
x
tke−tdt
A
+∞Ak+1 e−A
Z+∞
x
tk+1 e−tdt=xk+1 e−x+ (k+ 1) Z+∞
x
tke−tdt
ex
Φ(ek+1) = ek+1 + (k+ 1)Φ(ek)
0 ! = 1
Φ(e0) = e0
Φ(e1) = 1 ! 0
0 ! e0+1 1 ! e1
Φ(e2) = 2 !
2
X
p=0
1
p!ep

Φ(ek) = k!
k
X
p=0
1
p!epk≤n−1
Φ(ek+1)
Φ(ek+1)=(ek+1)+(k+ 1) ×k!
k
X
p=0
1
p!ep
= (k+ 1) !
k+1
X
p=0
1
p!ep
kk+ 1
k= 0 k= 1 k= 2
k0≤k≤n
Φ(ek) = k!
k
X
p=0
1
p!ep∀k0≤k≤n
Φekk
B={e0, e1, . . . , en}R
PnfPn
ek
f=
n
X
k=0
αkek
[x, A]
ZA
x
f(t)e−tdt=ZA
x
n
X
k=0
αkek(t)e−tdt
=
n
X
k=0
αkZA
x
ek(t)e−tdt
Z+∞
x
f(t)e−tdt=
n
X
k=0
αkZ+∞
x
ek(t)e−tdt
ex
[Φ(f)](x) =
n
X
k=0
αk[Φ(ek)](x)
Φ(f)
nnΦ(f)Pn
ΦPn
Z+∞
x
f(t)e−tdtx f Pn
Z+∞
x(α f1+β f2)(t)e−tdt=αZ+∞
x
f1(t) dt+βZ+∞
x
f2(t)e−tdt∀(α, β, x)∈R3
exZ+∞
x(α f1+β f2)(t)e−tdt=α exZ+∞
x
f1(t) dt+β exZ+∞
x
f2(t)e−tdt
Φ(f)Pn
R−Pn

fKer(Φ) ΦfB
f=
n
X
k=0
αk.ekΦ(f) = 0
Φ Φ(ek)
n
X
k=0
αk.k !
k
X
p=0
ep= 0
n
X
k=0
αk.k !
k
X
p=0
epαn.n !.enαn= 0
αkf= 0
Ker(Φ) {0}Φ
mij i j M mij
iΦ(ej)B
mij = 0 j < i
mij =j!
i!i≤j
j M
Φj
MΦB={e0, e1, . . . , en}
mii = 1,∀i
M
N=M−1
Pn
ΦΦ
uPnU
ΦU=M×u u =N×U
0 !
0 ! u0+1 ! 0 ! u1+2 ! 0 ! u2+... +(n−1) ! 0 ! un−1+n! 0 ! un=U0
1 !
1 ! u1+2 ! 1 ! u2+... +(n−1) ! 1 ! un−1+n! 1 ! un=U1
2 !
2 ! u2+... +(n−1) ! 2 ! un−1+n! 2 ! un=U2
(n−1) !
(n−1) ! un−1+n! (n−1) ! un=Un−1
n!
n!un=Un
N=M−1ui
un=Un
un−1=Un−1−n un
=Un−1−n Un

un=Unuj=Uj−(j+ 1) Uj+1 k≤k < n
uk−1
uk−1=Uk−1−
n
X
j=k
j!
(k−1) ! uj
=Uk−1−
n−1
X
j=k
j!
(k−1) ! Uj+
n−1
X
j=k
(j+ 1) !
(k−1) ! Uj+1 −n! (k−1) ! Un
=Uk−1−k Uk
j=n−1j k ≤j < n
j=k−1
uj=Uj−(j+ 1) Uj+1 0≤j < n
nij i j N=M−1
nij
nij = 0 j < i
nii = 1 i=j
nij =−j j =i+ 1
nij = 0 i+ 1 < j
M=
1 1 2 6 24 120
0 1 2 6 24 120
0 0 1 3 12 60
0 0 0 1 4 20
0 0 0 0 1 5
0 0 0 0 0 1
M−1=
1−1 0 0 0 0
0 1 −2 0 0 0
001−3 0 0
0 0 0 1 −4 0
0 0 0 0 1 −5
0 0 0 0 0 1
λΦ
f
Φ(f) = λ f,
f λ
g= Φ(f)fR
g(x) = λ f(x)∀x∈R
g= Φ(f)fR
g0(x) = λ f0(x)∀x∈R,
g
d
dxexZ+∞
x
f(t)e−tdt=λ f0(x)
t7−→ f(t)e−t
d
dxZ+∞
x
f(t)e−tdt=−f(x)e−x