Les nombres complexes

θ∈Rn∈N
Cn=
n
X
k=0 n
kcos(kθ)Sn=
n
X
k=0 n
ksin(kθ)
BC
z∈B1−z+z2∈B1 + z+z2∈B
B
a, b, z
b
az−a z−b2
∈R∗
+

CnSn
n
X
k=0 n
keikθ = (1 + eiθ)n= 2neinθ
2cosnθ
2
Cn= 2ncos nθ
2cosnθ 2Sn= 2nsin nθ 2cosnθ 2
B={i,−i}
f:C→Cg:C→C
f(z) = 1 −z+z2g(z) = 1 + z+z2
|f(z)−i|=|z+ i||z−(1 + i)|,|f(z)+i|=|z−i||z−(1 −i)|
|g(z)−i|=|z−i||z+ 1 + i| |g(z)+i|=|z+ i||z+ 1 −i|
a∈B(zn)n≥0B z0=a
n∈N
zn+1 =(f(zn) Re(zn)≤0
g(zn) Re(zn)>0
un=z2
n+ 1=|zn−i||zn+ i|
Re(zn)≤0
un+1 =|f(zn)−i||f(zn)+i|=un|zn−(1 + i)||zn−(1 −i)|
zn
|zn−(1 + i)| |zn−(1 −i)|√2
un+1 ≥√2un
Re(zn)>0
u06= 0 (un)B
u0= 0 a=±iB⊂ {i,−i}
B6=∅i−i
B={i,−i}
u1/u = ¯u
(z−a)2= (z−a)1
¯z−1 ¯a=(z−a)(¯a−¯z) ¯a¯z=−a|z−a|2 ¯z
b
az−a z−b2
=|z−a|2
|z−b|2∈R∗
+