Théorème de Weierstrass

(an)nlim
n∞an=∞
(bn)n
FC
anF
n anbn
(qn)nr0≤r < 1
+∞
X
n=0
|bn|rqn<+∞
n≥1
ln |bn|
qn
≤ln µ1 + 1
n¶
n≥1
lim sup
m∞
|bm|1/qm≤1 + 1
n
lim sup
m∞
|bm|1/qm≤1
P
n≥0
bnzqn≥1
+∞
P
n=0
|bn|rqn<+∞
(an)nan
+∞
X
n=0
bnzpn
apn
n(z−an)pn=qn−1
n z =an
bnapn
n
apn
n
=bn
R > 0n|an| ≤ R r
0< r < 1N n ≥N
|an| ≥ 1
r
R
|an|≤r(|an|> R)
z|z| ≤ R n ≥N
¯
¯
¯
¯
bnzpn
apn
n(z−an)¯
¯
¯
¯
≤|bn|
1−r
rpn
|an|≤|bn| 1−rrqn
+∞
P
n=N
bnzpn
apn
n(z−an)|z| ≤ R
SCZ P
z S m(z)
FC
Z F
P F
z∈S m(z)z

g S
z∈S m(z)z∈Z−m(z)z∈P
∆C\S z0∈∆z∈∆γ z0
z∆
Rγg(z)dz γ δ z0z∆
λ=γ−δ
Zγ
g(z)dz −Zδ g(z)dz ∈2iπZ
exp ³Rγg(z)dz´z γ
F∆
F
ω0∈∆r > 0D=D(ω0, r[ ∆ g
G D G(ω0) = 0 ω∈∆
F(ω) = F(ω0) exp (G(ω))
F D
F0(ω) = F(ω0) exp (G(ω)) G0(ω) = F(ω0) exp (G(ω)) g(ω) = F(ω)g(ω)
z∈∆
F0(z)
F(z)=g(z)
z∈Z P F0
Fm(z)−m(z)
z∈Z m(z)
z∈P m(z)
C
FC
F F =F/1
g F F
f=gF
F=f/g