243 - Convergence des séries entières, propriétés de la - IMJ-PRG

1
+∞= 0 1 0= +∞R+
Panznan∈CNz∈C
P∞
n=0 anzn
z0∈C(anzn
0)n∀r∈[0,|z0|[,Panzn
B(0, r)
PanZnR= sup{r≥
0|(anrn)n} ∈ R+
Pzn
1
1−z
∀z∈C,|z|< R, Panzn
∀z∈C,|z|> R, Panzn
∀r < R, PanznB(0, r)
PnznPzn
n2
PanznPbnznRa
RbDaDbfa
fb
Panzn
0
R=|z0|

Panzn
lim 
an+1
an
=l∈R+Ra=1
l
P∈C[X]PP(n)zn
Pzn/n! +∞
Ra=lim|an|1/n−1
λ∈Cα∈RPanznPλanznPnαanzn
PnanznPanzn
Pn≥1ϕ(n)
nnzn+∞
Pn≥1n
ϕ(n)nzn
an=O(bn)Ra≤Rb
an∼bnRa=Rb
Pln(1 + 1/n)zn
P(an+bn)zn≥min(Ra, Rb)
z0∈Da∩DbP∞
n=0(an+bn)zn
0=P∞
n=0 anzn
0+P∞
n=0 bnzn
0
Ra6=Rbmin(Ra, Rb)
cn=Pn
k=0 akbn−kPcnzn
≥min(Ra, Rb)z0∈Da∩DbP∞
n=0 cnzn
0=
P∞
n=0 anzn
0 P∞
n=0 bnzn
0Pcnzn
fa
f(p)
a(z) = P(n+p)!
n!an+pzn
exp0= exp
PanznPn≥1
an−1
nzn
an=f(n)
a(0)/n!Ra>0
(bn)nfa(x) = PbnxnR
(an)n

ΩR C x0∈Ωg: Ω →C
g
x0r > 0B(x0, r)⊂ΩPanzn
R > r g(z+x0) = P∞
n=0 anznB(0, r) (an)n
ang x0g
Ω
z0∈C, z 7→ 1
z−z0
C\ {z0}
ΩC
f∈H(Ω) g x0∈Ω
D x0Ω
a∈D f(a) = P∞
n=0(a−x0)nR∂D
f(z)
(z−z0)n+1 dz
2πi
f∈H(Ω) =⇒f0∈H(Ω)
f: Ω →Cf∈
H(Ω) TΩR∂T f= 0
f∈
H(Ω) Ω f
f∈H(Ω) |f|
z0∈Ωf z0
Ω
f∈H(Ω) z0∈Ω
f(z0)=0 p∈N∗f(p)(z0)6= 0
v z0Ωh∈H(V)h(z0)6= 0 g(z)=(z−z0)ph(z)

g V =]x0−α, x0+α[x0
g C∞V∀x∈V, Rn(x) = g(x)−Pn
k=0
(x−x0)k
k!g(k)(x0)→n0
kg(k)k∞
p V
g C∞
Pg(n)(x0)zn
gΩC > 0
kg(k)k∞≤Cn+1n!
a > 0f:] −a, a[→R
C∞∀k∈N,∀x∈]−a, a[, f(2k)(x)≥0f
]−a, a[
xα
(arcsin x)2
R1
0xxdx
PanznPan
θ0∈[0, π/2[ ∆θ0={z∈Da| ∃ρ > 0, θ ∈
[−θ0, θ0] : z= 1 −ρeiθ}limz→1,z∈∆θ0fa(z) = P∞
n=0 an
∆θ0
P∞
n=1
(−1)n
n= ln(2)
P∞
n=1
(−1)n
2n+1 =π 4
P∞
n=1
sin(nt)
n=π−t 20<t<2π
an=O(1/n)
fa(z)→z→1,z∈[0,1[ lPanl

PanznDa
∂Dag
P∞
n=0 |an|2R2n=1
2πR2π
0|g(θ)|2dθ
Da
Da
z∈∂Dafa
z z
Panzn
aPf(n)(a/2)
n!zn
an
Γ