PC* l`intervalle d`intégration. Continuité (x, t) est continue sur A. f(x, t

K=R C A I R
A I
f:A×I−→ K(x, t)∈A×I f(x, t)
f
t I x 7−→ f(x, t)A
x A t 7−→ f(x, t)I I
φ I
∀(x, t)∈A×I|f(x, t)| ≤ φ(t)
F x A F (x) = I f(x, t)dt A
•c)I
t7→ f(x, t)b)
•x∈A
A
I
A
•f A×I
f:A×I−→ K(x, t)∈A×I f(x, t)
f
x A t 7−→ f(x, t)I
∂f
∂x A×I
t∈I x 7→ f(x, t)x7→ ∂f
∂x (x, t)A
x∈A t 7→ ∂f
∂x (x, t)I I
φ1I
∀(x, t)∈A×I
∂f
∂x (x, t)
≤φ1(t)
F x A F (x) = I f(x, t)dt c1A
∀x∈A F ′(x) = I
∂f
∂x (x, t)dt
•e)c)
•a)f

•f, ∂f
∂x φ, φ1
I
I[a, b]
A×[a, b]
[α, β]×[a, b] [α, β]A
[a, b]
f c1A×[a, b]F c1A
Γ
Γ(x) = +∞
0
e−ttx−1dt
Γc1R+∗
[a, b] 0 < a < b
∀(x, t)∈[a, b]×R+∗, f(x, t) = tx−1e−t=e(x−1) ln te−t
∂f
∂x (x, t) = ln t e(x−1) ln te−t= ln t tx−1e−t
•t∈]0,1] ln t≤0x∈[a, b]⇒a−1≤x−1≤b−1
x∈[a, b]⇒(a−1) ln t≥(x−1) ln t≥(b−1) ln t
x∈[a, b]⇒0≤tx−1e−t≤ta−1e−t
•t≥1lntgeq0x∈[a, b]⇒0≤tx−1e−t≤tb−1e−t
•∀x∈[a, b],∀t∈R+∗,|f(x, t)| ≤ e−t(ta−1+tb−1) = φ(t)
•
∀x∈[a, b],∀t∈R+∗,
∂f
∂x (x, t)
≤ |ln t|e−tta−1+tb−1=φ1(t)
I=]0,+∞[
c1Γ [a, b] Γ c1
]0,+∞[
∀x > 0,Γ′(x) = +∞
0
ln t tx−1e−tdt
Γc∞]0,+∞[
∀p∈N,∀x > 0,Γ(p)(x) = +∞
0
(ln t)ptx−1e−tdt
x > 0 Γ(x+ 1) = xΓ(x)
∀n∈N∗Γ(n) = (n−1)!
Γ1
2=+∞
0
e−tt−1/2dt =+∞
0
2e−u2du =√π