Nombres complexes

C
C
a+ib, a ∈R, b ∈R
i i2=−1
z=a+ib (a, b)∈R2
•aza= (z)
•bzb= (z)
a+ib
ib
i2=−1
(a+ib)+(a0+ib0)=(a+a0) + i(b+b0)
(a+ib)×(a0+ib0)=(aa0−bb0) + i(ab0+ba0)

z=a+ib z
z z =a−ib
z=a+ib z0=a−ib
z×z=a2+b2
(z) = z+z
2(z) = z−z 2i
z=z
z⇐⇒ z=z z ⇐⇒ z=−z
z+z0=z+z0z×z0=z×z0
1
z=1 a+ib =1 a+ib ×a−ib a−ib =a−ib a2+b2=z a2+b2
z=a+ib z|z|
|z|=√a2+b2
z z0
z×z=|z|2
|z|=|z|
(z)6|z|(z)6|z|
|z×z0|=|z|×|z0|
z z0C|z+z0|6|z|+|z0|
|z+z0|2= (z+z0)(z+z0) = zz +z0z0+zz0+zz0=|z|2+|z0|2+ 2 (zz0)6|z|2+|z0|2+ 2|zz0|= (|z|+|z0|)2

C1 0
U={z∈C/|z|= 1}
Cθ= (−→ i , −−→ OM)
(cos(θ),sin(θ)) z∈U
∃θ∈R/ z = cos(θ) + isin(θ)
θ z
z
z
|z|z ρ =|z|
θ
θ∈Reiθ eiθ = cos(θ) + isin(θ)z=a+ib
z
ez=ea+ib =eaeib =ea(cos(b) + isin(b))
θ∈Rcos(θ) = (eiθ) sin(θ) = (eiθ)
θ∈R(eiθ) = e−iθ
|eiθ|= 1
eiθ ×eiθ0=ei(θ+θ0)
θ∈Rn∈Z(eiθ)n=einθ
(cos(θ) + isin(θ))n= cos(nθ) + isin(nθ)
θ∈R
cos(θ) = eiθ +e−iθ
2,sin(θ) = eiθ −e−iθ 2i

z∈C∗z θ
z=|z|(cos θ+isin θ) = |z|eiθ
(z)z=|z|ei(z)
z
z z0
z=z0⇐⇒ |z|=|z0|
∃k∈Z/arg(z) = arg(z0)+2kπ
z
z
2π
z[0,2π[ ] −π, π]
[a, b[ ]a, b] 2π
C
C
z1z2n∈N
(z1+z2)n=
n
k=0
n
kzk
1zn−k 2
an−bn
z1z2n∈N
zn
1−zn 2= (z1−z2)
n−1
k=0
zk
1zn−1−k 2

nC
Z∈C∗n∈N∗n Z z zn=Z
n= 2 z z2=Z
i−1
√z z
z1/2
n
n n 1
ωk= exp ik 2π
n, k ∈J0, n −1K
nUnn1
z z =ρeiθ
zn= 1 ⇐⇒ ρneinθ = 1ei0
⇐⇒ ρn= 1
∃k∈Z/ nθ = 2kπ
⇐⇒ ρ= 1
∃k∈Z/ z = exp ik 2π
n=ωk
n
ωk=ω`⇐⇒ ∃p∈Z/2kπ
n=2`π n+ 2pπ
⇐⇒ ∃p∈Z/ k =`+pn
ωkω`` k n
n ωii∈J0, n −1K
n1
Z n n
z0=n|Z|ei(Z)/n
z0n
n∈N∗n n
n−1
k=0
exp 2ikπ
n= 0