[pdf]

Z+∞
0
sin t
tdt
π/2
f: [0 ; +∞[→R
Z+∞
0
f(t) sin(t) dt
x7→ Rx
0sin(et) dt+∞
Z+∞
−∞
eit2dt
+∞
f(λ) = Z1
0
eiλx2dx
x > 0
f(x) = Zx
0
eit2dt=Zx
0
cos(t2) dt+iZx
0
sin(t2) dt
f(x) = eix2−1
2ix +1 2iZx
0
eit2−1
t2dt
f λ +∞
g(x) = λ−f(x)x > 0
g(x) = 1
2iZ+∞
x
eit2
t2dt−eix2 2ix
+∞
g(x) = −eix2
2ix + O 1 x3
f: [0 ; +∞[→R
Z+∞
0
f(t) dt
lim
x→+∞
1
xZx
0
tf(t) dt
f: [1 ; +∞[→R
Z+∞
1
f(t) dt=⇒Z+∞
1
f(t)
tdt
f: [0 ; +∞[→Rα > 0
Z+∞
0
f(t) dt=⇒Z+∞
0
f(t)
1 + tαdt

f: [a; +∞[→RfC1+∞
g: [a; +∞[→RM∈R+
∀x∈[a; +∞[,Zx
a
g(t) dt≤M
Z+∞
a
f(t)g(t) dt

]0 ; +∞[
]0 ; 1]
ZA
1
sin t
tdt=−cos t tA
1−ZA
1
cos t
t2dt
A→+∞
Z+∞
1
sin t
tdt
(Sn)
Sn=Znπ
0
f(t) sin(t) dt
Sn=
n−1
X
k=0 Z(k+1)π
kπ
f(t) sin(t) dt
Z(k+1)π
kπ
f(t) sin(t) dt=Zπ
0
f(t+kπ) sin(t+kπ) dt= (−1)kvk
vk=Zπ
0
f(t+kπ) sin(t) dt
f(vk)
f(vk)
f+∞
0≤vk≤f(kπ)π
(vn)
(−1)kvk(Sn)S
X≥0nXX/π
ZX
0
f(t) sin(t) dt=SnX+ZX
nXπ
f(t) dt
0≤ZX
nXπ
f(t) dt≤ZX
nXπ
f(nXπ) dt=f(nXπ)(X−nXπ)≤f(nXπ)π
X→+∞nX→+∞SnX→S
ZX
nXπ
f(t) dt→0
ZX
0
f(t) sin(t) dt→S
Zx
0
sin(et) dt=Zx
0
etsin(et)e−tdt=−cos(et)e−tx
0−Zx
0
cos(et)e−tdt
cos(ex)e−x−−−−−→
x→+∞0
t7→ cos(et)e−t[0 ; +∞]
t2cos(et)e−t−−−−→
t→+∞0
R+∞
0sin(et) dt
Z+∞
0
eit2dt=Z+∞
0
2t
2teit2dt

Z+∞
0
eit2dt=Z+∞
−∞
2t
2teit2dt="eit2−1 2it#+∞
0
+1
2i Z+∞
0
eit2−1
t2dt
2teit2
Z+∞
0
eit2dt
C1t=√λx
f(λ) = 1
√λZ√λ
0
eit2dt
u=t2
ZA
0
eit2dt=1
2ZA2
0
eiu
√udu
ZA
0
eit2dt=1
2 eiu−1
i√uA
0
+1
2ZA2
0
eiu−1
iu3/2du!
ZA
0
eit2dt=i
4ZA2
0
1−eiu
u3/2du
ZA
0
eit2dt−−−−−→
A→+∞
C
C
f(λ)∼
λ→+∞
C
√λ
C
x > a > 0
Zx
a
eit2dt=Zx
a
2it
2iteit2dt="eit2−1 2it #x
a
+Zx
a
eit2−1
2it2dt
a→0
Zx
a
eit2dt→Zx
0
eit2dt, eia2−1
2ia ∼a 2→0
Zx
a
eit2−1
2it2dt→Zx
0
eit2−1
2it2dt
f(x) = eix2−1
2ix +1 2iZx
0
eit2−1
t2dt
eix2−1
2ix ≤1 x→0Zx
0
eit2−1
2it2dt→Z+∞
0
eit2−1
2it2dt
f(x)→λ=Z+∞
0
eit2−1
2it2dt
g(x) = λ−f(x) = 1
2iZ+∞
x
eit2−1
t2dt−eix2−1 2ix
g(x) = 1
2iZ+∞
x
eit2
t2dt−1 2iZ+∞
x
1
t2dt−eix2−1 2ix
g(x) = 1
2iZ+∞
x
eit2
t2dt−eix2 2ix

Z+∞
x
eit2dt
t2=Z+∞
x
teit2dt
t3="eit2 2it3#+∞
x
+3
2iZ+∞
x
eit2dt
t4
Z+∞
x
eit2dt
t2
=−eix2
2ix3+3 2iZ+∞
x
eit2dt
t4≤1 2x3+3 2Z+∞
x
dt
t4=1 x3
Z+∞
x
eit2dt
t2= O 1 x3
F f
F(x)−−−−−→
x→+∞`=Z+∞
0
f(t) dt
1
xZx
0
tf(t) dt=F(x)−1
xZx
0
F(t) dt
1
xZx
0
F(t) dt−`≤1
xZx
0|F(t)−`|dt
ε > 0A∈R+
∀t≥A, |F(t)−`| ≤ ε
[0 ; A]|F(t)−`|M > 0
x≥max(A, AM/ε)
1
xZx
0|F(t)−`|dt=1
xZA
0|F(t)−`|dt+1
xZx
A|F(t)−`|dt≤2ε
1
xZx
0
F(t) dt−−−−−→
x→+∞`
lim
x→+∞
1
xZx
0
tf(t) dt= 0
f
f[1 ; +∞[
f F f
+∞
ZA
1
f(t)
tdt=F(t) tA
1
+ZA
1
F(t)
t2dt
F(A)/A −−−−−→
A→+∞
0t7→ F(t)/t2[1 ; +∞[F
+∞t7→ f(t)/t
[1 ; +∞[
F f [0 ; +∞[
Z+∞
0
f(t)
tα+ 1 dt=F(t) tα+ 1+∞
0
+αZ+∞
0
F(t)tα−1
(tα+ 1)2dt
R+∞
0f(t) dt F
+∞
+∞F[0 ; +∞[ +∞
t→+∞
F(t)tα−1
(tα+ 1)2= O 1 tα+1 
α > 0
R+∞
0
f(t)
1+tαdt
G(x) = Zx
a
g(t) dt
Zx
a
f(t)g(t) dt= [f(t)G(t)]x
a−Zx
a
f0(t)G(t) dt