Epreuve 1 : MP

Concours National Commun
Epreuve 1 : MP
Session 2003
CRR C
C0(R)Cp(R)C∞(R)CpC∞
f∈ CRb f(x) = Z+∞
−∞
e−ixtf(t)dt.
b
f f
x, α
t7→ 1−e−t
t]0, α]
t7→ e−t
t[x, +∞[
ϕR∗
+ϕ(x) = Z+∞
x
e−t
tdt
x0< ϕ(x)<e−x
x
ϕR∗
+ϕ0
x0+ϕ(x) + ln x C =ϕ(1) −Z1
0
1−e−t
tdt.
ln x
x > 0ϕ(x) + ln x=C+Zx
0
1−e−t
tdt,
ϕ(x) + ln x=C+
+∞
X
k=1
(−1)k−1
k
xk
k!.
ψR∗
ψ(x) = 1
2ϕ(|x|).
ψ]− ∞,0[ ]0,+∞[
x∈Rψ(x)
b
ψ(x) = Z+∞
0
ϕ(t) cos(xt)dt.

b
ψC∞R
x
b
ψ(x) = 1
xZ+∞
0
e−t
tsin(xt)dt,
b
ψ(0)
Φ : x7→ R+∞
0
e−t
tsin(xt)dt ]0,+∞[
Φ0(x)x > 0
b
ψ(x) = arctan x
x.
fR
xb f(x)b f
fb f
F:ϕ7→ bϕ,
R CR
R
a fa(t) = f(t−a)af(t) = f(at)
xb fa(x) = e−iax b f(x)c
af(x) =
1
|a|b
fx
a(a6= 0).
t7→ f(t)eiat
f
f
[0,+∞[
fC1(R)f f0R
f±∞
∀x∈R,b f0(x) = ix b f(x),b f±∞
g:t7→ tf(t)Rb f
C1R∀x∈R,b f0(x) = −ibg(x).

h t 7→ e−t2R+∞
−∞ h(t)dt =√π.
b
hR
y0+x
2y= 0.(1)
b
h
t7→ e−εt2, ε > 0.
fRb f
R(εn)n
v∈ C0(R)R
lim
n−→+∞R+∞
−∞ v(y)e−εny2dy =R+∞ −∞ v(y)dy.
w∈ C0(R)x∈R
lim
n−→+∞R+∞
−∞ w(x+εny)e−y2dy =w(x)√π.
n∈Nx∈RR+∞
−∞ f(t)R+∞ −∞ e−iy(t−x)−εny2dydt = 2√πR+∞ −∞ f(x+
2√εns)e−s2ds.
x
(p, q)ε > 0Rp
−peixy−εy2Rq −qf(t)e−iytdtdy =
Rq
−qf(t)Rp −pe−iy(t−x)−εy2dydt.
ε > 0 lim
q−→+∞R+∞
−∞ eixy−εy2Rq −qf(t)e−iytdtdy =R+∞ −∞ eixy−εy2R+∞ −∞ f(t)e−iytdtdy.
q ε > 0 lim
p−→+∞Rq
−qf(t)Rp −pe−iy(t−x)−εy2dydt =
Rq
−qf(t)R+∞ −∞ e−iy(t−x)−εy2dydt.
ε > 0R+∞
−∞ f(t)R+∞ −∞ e−iy(t−x)−εy2dydt =R+∞ −∞ eixy−εy2b
f(y)dy.
x∈Rf(x) = 1
2πR+∞
−∞ eixy b
f(y)dy.
FIN
c
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Mamouni My Ismail PCSI 2 Casablanca Maroc