239. Intégrales dépendant d`un paramètre. Exemples et

(E, A, µ) (X, d)f:E×X→C
x t 7→ f(t, x)F(x) = REf(t, x)dt
E
t f(t, .)x0
x f(., x)
|f(t, x)|6g(t)L1
x0f(t, .)
Γ(x) = R∞
0e−ttz−1dt ΓRe(z)>0
R∞
0xe−xtdt 0
L1
F(t) = R∞
1
e−xt
√t
R+∗[a, +∞[
Rn
f(t, .)x0i x0fi(., x)
x
x f(., x)
|fi(t, x)|6g(t)L1
x0REfi(t, x0)dt
F
REfi(t, x0)dt
dF (x)(h) = REdf(t, x)(h)dt

F(x) = x2R1
0e−xtdt C1C1
F(x) = R1
0e−xtdt
F(t, x)
g f(t, x)=1x6tg(x)
F(t) = Rt
0g(x)dx. g f(t, x) (t, x)
ΓC1C∞R+∗
F(t) = RR
dt
t+x2C∞R∗+C∞
In=R∞
0
dt
(1 + x2)n
I(x) = R∞
0
sin(xt)
te−tdt =arctan(x)J(x) = R∞
0e−teixtTα−1dt = Γ(α)(x2+1)−α/2eiαarctan(x)
F(t) = R∞
0
e−tx −e−x
xC1R+∗R∞
0e−xt =−1
t
−log(t)
f(t, .) Ω
f(., x)
|f(t, x)|6g(t)L1Ωfz(z) = REfz(t, x)dt
ΓRe(Z)>0
F G
F∗G(p) = RγF(p−q)g(q)dq γ
F∗G
F anG bnF∗G
Pck
nakbn−k
K E RX
af(t, x)dx F (t)X
G(t) = R∞
0e−tx sin(x)
xdx
[0,∞[C1]0,∞[
R0∞sin(x)
x=π 2
F(t) = R∞
1eitx2f(x)dx f C0
]0,∞[
(t, x)Rb
af0(x, t)dx
C1Rb
af0(x, t)dx
F(a) = RRe−ax2Re(z)>0
|a|>ε Re(a)>0

L2
fΩ
γ
I(γ, z0)f(k)(z0) = 1
2iπ Zγ
f(z)dz
(z−z0)k+1
|f(n)(a)|6n!||f||∞,C(a,R)
RnfΩD(a, R)
Ω
|f(z)|6a+bzkz k
(E, A, µ) (F, B, ν)σ
f:E×FCF(t) = RFf(t, x)dν G(x) = REf(t, x)dµ
f¯ R+
¯
R+
ZE
F(t)dµ =intFG(x)dν =ZZE×F f(t, x)dµ ×dν
f
L1
ZE
F(t)dµ =intFG(x)dν =ZZE×F f(t, x)dµ ×dν
σ−finies
E=RF=Rf(x, t) = 1x=t
F(x) = xRF=∞G(t)=0 RG= 0
L1
f(x, t) = x2−y2
(x2+y2)
2
[0,1]2f(0,0) = 0 R1
0F=π
4R1
0G=−π
4
f L1F G
f(0,0) = 0 f(x, y) = xy
(x2+y2)2f[−1,1]2
L1