Fonctions complexes et théorème de d`Alembert.

∼
C
C
R2
C R2z∈Cx
y∈R2
C∋1≃1
0∈R2C∋i≃0 1∈R2,
C∋z=x+iy ≃x
y∈R2, x, y ∈R.
zk=xk+iykk= 1,2xk, yk∈R C = (C,+, .)
z1+z2= (x1+x2) + i(y1+y2)
z1z2= (x1x2−y1y2) + i(x1y2+x2y1).
i2=−1
z=x+iy ¯z=x−iy |z|=x2+y2=|| x
y||R2z
x
yz¯z=|z|2
z̸= 0 1
z=¯z |z|2=x x2+y2−iy x2+y2.
eiθ = cos θ+isin θ z
z=ρeiθ,
ρ=|z|θ(x, y)
x z =x+iy =ρeiθ
z1=ρ1eiθ1z2=ρ2eiθ2
z1z2= (ρ1ρ2)ei(θ1+θ2).
R2z=ρeiθ ρ
θ z−1=1
ρe−iθ z=ρeiθ ̸= 0 R2

z−11
ρ−θ
R2x1
y1x2 y2=x1x2−y1y2 x1y2+x2y1
x
yR2
x
y−1
=1
x2+y2x
−y.
C|.|:z∈C→ |z| ∈ R+
R2
B(z0, R) = {z∈C:|z−z0|< R}
≃B(x0
y0, R) = {x y∈R2: (x−x0)2+ (y−y0)2< R2}.
UR2CU
f:U→CUC
UR2P Q :U→R
⃗
f:x
y∈U→⃗
f(x, y) = P(x, y)
Q(x, y)∈R2.
∂⃗ f
∂x (x, y) = ∂P
∂x (x, y)
∂Q
∂x (x, y)(= lim
h→0
⃗
f(x+h, y)−⃗ f(x, y)
h),
∂⃗ f
∂y (x, y) = ∂P
∂y (x, y)
∂Q
∂y (x, y)(= lim
h→0
⃗
f(x, y +h)−⃗ f(x, y)
h).
⃗
f(x0, y0)
⃗
f(x, y) = ⃗ f(x0, y0) + d⃗ f(x0, y0).x−x0
y−y0+o(||(x−x0, y−y0)||R2
=⃗ f(x0, y0) + ∂P
∂x (x0, y0)∂P ∂y (x0, y0)
∂Q
∂x (x0, y0)∂Q ∂y (x0, y0).x−x0
y−y0+o(||(x−x0, y−y0)||R2).
⃗
f
UCf:U→C
f(z) = P(x, y) + iQ(x, y)z=x+iy
f:U→Cz0∈U
lim
z→z0
f(z)−f(z0)
z−z0
,
f′(z0)∈C
lim
z→z0
f(z)−f(z0)
z−z0
=f′(z0).

f f
f(z) = f(z0) + (z−z0)f′(z0) + o(z−z0).
z=x+iy z0=x0+iy0f′(z0) = α0+iβ0
P(x, y) + iQ(x, y) = P(x0, y0) + iQ(x0, y0) + ((x−x0) + i(y−y0))(α0+iβ0) + o(z−z0),
P(x, y)
Q(x, y)=P(x0, y0) Q(x0, y0)+α0−β0 β0α0x−x0 y−y0+o(z−z0).
f:U→Cz∈U
∂P
∂x (x, y) = ∂Q ∂y (x, y),
∂Q
∂x (x, y) = −∂P ∂y (x, y).
f′(z) = ∂P
∂x (x, y) + i∂Q ∂x (x, y) = ∂Q ∂y (x, y)−i∂P ∂y (x, y)
=∂P
∂x (x, y)−i∂P ∂y (x, y) = ∂Q ∂y (x, y) + i∂Q ∂x (x, y).
f′(z) = α+iβ
f
U f
f(z) = f(z0) + f′(z0)(z−z0) + o(z−z0)
f(z0)f(z)z0f′(z0) =
α0+iβ0=ρ0eiθ0(z−z0)R2
◦α0−β0
β0α0 x−x0 y−y0◦
ρ2
0cos θ0−sin θ0
sin θ0cos θ0
f(z)−f(z0) = ρ0eiθ0(z−z0) + o(z−z0)
⃗
f:U→R2
◦
C→C
R2→R2
f+g fg f ◦g
f′+g′f′g+fg′(f′◦g)g′
(f◦g)′= (f′◦g)g′
(f◦g)′(z) = f′(g(z))g′(z)f′g′
f′g′
⃗
f , ⃗g :R2→R2⃗ f=f1
f2⃗g =g1 g2f=f1+if2g=
g1+ig2fk, gk:R2→Rd(⃗ f◦⃗g) = d⃗ f◦⃗g.d⃗g
◦
◦R2→R2
f′◦g.g′:C→C

C
f(z) = znn≥0n≥1zn−zn
0= (z−z0)(zn−1+zn−2z0+... +zzn−2
0+zn−1 0
(zn)′=nzn−1
C
C
1
zn−1 zn
0=zn
0−zn
znzn
0(1
zn)′=−n1 zn+1 (z−n)′=−nz−n−1
f:z→¯z
⃗
f:x
y→x −y
x
1 0
0−1◦
fR2→C˜ f:
U→C˜ f(x, y) = f(z)z=x+iy
d˜ f(x, y) = ∂˜
f
∂x (x, y)dx +∂˜
f
∂y (x, y)dy ∈L(R2;C),
˜
f(x, y) = ˜ f(x0, y0) + df(x0, y0).x−x0
y−y0+o(||(x−x0, y−y0)||R2)
=˜ f(x0, y0) + ∂˜
f
∂x (x0, y0) (x−x0) + ∂˜
f
∂y (x0, y0) (y−y0) + o(||(x−x0, y−y0)||R2),
∂˜ f
∂x (x0, y0)∂˜
f
∂y (x0, y0)
∂˜ f
∂x (x0, y0) = α0+iβ0=f′(z0),
∂˜ f
∂y (x0, y0) = −β0+iα0=if ′(z0).
∂f
∂x (z) = ∂˜
f
∂x (x, y) (= ∂P ∂x (x, y) + i∂Q ∂x (x, y)),
∂f
∂y (z) = ∂˜
f
∂y (x, y) (= ∂P ∂y (x, y) + i∂Q ∂y (x, y)).
∂⃗ f
∂x
∂⃗ f
∂y C
x, y ∈Rz=x+iy ¯z=x−iy z ¯z
x=1
2(z+ ¯z)y=1 2i(z−¯z)z¯z x
y
∂
∂z
∂
∂¯z
∂f
∂x =∂f ∂z
∂z
∂x +∂f ∂¯z
∂¯z
∂x ,
∂f
∂y =∂f ∂z
∂z
∂y +∂f ∂¯z
∂¯z
∂y ,
∂f
∂x =∂f ∂z +∂f ∂¯z,
∂f
∂y =i∂f ∂z −i∂f ∂¯z.
∂
∂z =1 2(∂ ∂x −i∂ ∂y ),
∂
∂¯z=1 2(∂ ∂x +i∂ ∂y ).

∂f
∂z (z) = f′(z) (= 1 2(∂(P+iQ) ∂x (x, y)−i∂(P+iQ) ∂y (x, y))),
∂f
∂¯z(z) = 0 (= 1 2(∂(P+iQ) ∂x (x, y) + i∂(P+iQ) ∂y (x, y))).
df(z) = ∂f
∂z (z)dz =f′(z)dz.
∂f
∂¯z= [∂P ∂x −∂Q ∂y ] + i[∂Q ∂x +∂P ∂y ]
∂f
∂¯z= 0
R2⃗r :t∈[a, b]→⃗r(t) = x(t)
y(t)∈R2
Γ = Im(⃗r)C⃗r
z:t∈[a, b]→z(t) = x(t) + iy(t).
d⃗r =dx
dy =x′(t) y′(t)dt dz
dz =dx +idy z′(t)dt = (x′(t) + iy′(t)) dt.
f∈C0(Ω; C) Ω CΓ
Γ
f dz =b
t=a
f(z(t)) z′(t)dt.
f(z) = P(x, y) + iQ(x, y)∈Cz=x+iy x, y ∈RP, Q ∈C0(Ω; R) Ω
R2
Γ
f dz =b
t=a
[P(x(t), y(t)) x′(t)−Q(x(t), y(t)) y′(t)] dt
+ib
t=a
[Q(x(t), y(t)) x′(t) + P(x(t), y(t)) y′(t)] dt.
⃗
f: (x, y)∈Ω⊂R2→⃗ f(x, y) = P(x, y)
Q(x, y)∈R2f(z) =
P(x, y) + iQ(x, y)∈Cz=x+iy
C∋Γ f dz ≃⃗ F=b
t=ax′(t)−y′(t)
y′(t)x′(t).⃗
f(x(t), y(t))dt ∈R2,
◦z′∈C≃x′−y′
y′x′⃗
fΓ
⃗
F=F1
F2Γf dz =F1+iF2
f(z) = −y+ix z =x+iy ⃗r(t) = Rcos t
Rsin t
Γt∈[0,2π]
Γ
f dz =R22π
0
−sin t(−sin t)−(cos t)(cos t)dt +iR22π
0
−sin t(cos t) + cos t(−sin t)dt = 0.
Γf dz Γ