az+ az +b cz+ cz+d
ϕ (z ) = G
• • Hol (G)
A
G Mer (G)
∈ GL2, hA (z) A, B ∈ GL2 , hA ◦ hB = hAB ∀λ = 0,0 , hλA = hA
Har (G)
G C (G) • C (
hλdd¯ = hI
G ϕ
2
z1 , z2 , z3 , z4
hA
∈C
G
⇐⇒
S c
∈C
z2 , z3 , z4
G
X a, b
∈
X = G G.
γ
X =
∅
⇒
f : G R [s, t] (f ) f )
→
⊆
→C
1 2πi
X G
\
zj , 1
≤j≤4
dz γ z−z0
0
γ
f =
´
γ1
= 2πi 2 πi
·
f ◦γ
f γ
γ
∈
f
γ
∈ Hol (G) f ∈ Hol (G) , γ ⊂ G ∀z0 ∈/ G, log f ∈ Hol (G) G1 , G2 ⊂ C f : G1 → G2 ⇐⇒ G2
(z0 ) = 0
f : D
D
sup
f
z →z0
√ log f, f
⇐⇒
´
f = 0
γ
⇐⇒ C\G f
⇐⇒
f (0) f (0) = 0
→D
⇐⇒
G
limn→∞
| |
θ
[0, 2π] |a| < 1, θ ∈ [0,
D
G C G= f : G D f : D
f g (w0 ) f ( f (z0 )
→
C
f ( f (a) = 0, R
G1
x0
2
n
n
z−a 1−az
d dt
∞ n=0 cn (z
t0 f
≤ z −w 1−zw
g (f ( f (z ))
z =z0
=
c c
γ (t0 )
γ
→C
f : G
f : G
|f |
G
f
f (t) = 0, z
∀ ∈G
→C
f
a=
∂f ∂z
f : C (0); b = f : G
→∂f C
(0) ∂ ¯ ∂ z¯
→C
z0 = 0 ∞ z n+1 n=0 cn n+1
∈ Hol (G) (f ) f ) ,
f
→C u:G→R
f n : G
∈ Hol (G)
G=C G
±c
f (z ) =
c z
n→∞
→
−
´ b
→C
a
→C
f ( f (t) dt =
C 1
´
γ
|
n
1
n
limRn→1 f ( f (x) = L
−1| sup |1z−|z < −|z | n
n
→
∞
limn→∞ f ( f (zn) =
γ : [a, [ a, b]
→G
´ b
f ( f (t) dt + i
a
´ b a
f ( f (t) dt
f f n
´
γ 1
γ1
f n f
→ ´ γ f
f = G
|
γ : [a, [ a, b] C 1 π = a = t0 < .. . < tN = b
δ>0
{
´ f −
N −1 f (zi ) (zi+1 i=0 f (
γ
G
⊂C
G v
− z0)n−k
(γ ) supt∈[a,b] f (γ ( γ (t)) a,b] f (
f ∂u ∂z
∞ n! n=k (n−k)! cn (z
f
γ 1
→C
(f ) f )
u:G
− z0)n−1
f : G C f (z ) dz = f ( f (γ ( γ (b)) f ( f (γ ( γ (a))
γ π
f
= ck
zn
· G ⊂ C f : G → C
f
R
(z0 ) k!
f
f (k) (z ) =
D (z0 , R)
∞ n=1 cn n (z
γ
´ f ≤ γ
f (z ) = az + bz¯ ∀z ∈ C f (
R
(k)
γ 2 γ : [a, [ a, b]
f (γ 1 ) , f ( f (γ 2 )) = (γ 1 , γ 2 ) (f (
[0, ∞] ∈ [0,
0
f
→C
C ∞
f
⇐⇒ ∀z ∈ G : f (z) = 0
f
− z0 )n
c
´
◦
R−1 =
f (z ) =
n
(f g ) =
cn cn+1
f (z )
| | n→∞ → 0
n
R
n cn
=L
n
R = lim limn→∞
G (z ) =
f : [a, [ a, b]
→C
{M n}∞n=0
− z0)n
z = z0 :
γ
f : G
D (z0 , δ )
G
D (z0 , R)
´
f : G
∃δ > 0
D (z0 , R)
|z − z0| > R
[0, 2π ] ∈ [0,
f ( f (z)−f ( f (w) 1−f (z )f ( f (w)
w0 = f ( f (z0 )
sup |a | f ( f (z) =
f (a) > 0
2
γ
u
G
⇐⇒
f (γ ( γ (t0 )) γ ( γ˙ (t0 )
1 z
∞
f (z )
f (z ) =
−|f (z )| |f (z)| ≤ 1−|f 1−|z −|z |
→D z0 g
f
l (z ) = log(f log(f ))
∈ G, f ( 0 f (z0 ) =
z0
∞ n n=0 cn z
f ( f (z ) =
⊂
3, 4
∈R
u
∞ n=0 cn (z
f (z)| ≤ |z| |f (
f (0) f (0) = 0
→D
→D
2
− u2 ≡ l ∈ Hol (G)
∞ n=0 un (z )
f −1
||
1
z∈G un < M n ∞ n=0 M n < +
(z0 ) = 0
Hol (D) f : D f ( f (z) = λz, λ = 1 Hol (D) f : f ( f (z) = eiθ z
´
G γ
4
a
f = 0
F (z ) =
z0 / G
3
{un (z)}∞n=0 ´
∈ Hol (G)
∈ Hol (G)
u1
f
G f
2
(0)
C
γ
→
[z , z , z z ] , z , z , z = [z
(z1 , z2 , z3 , z4 )
log (f ) f )
f dz γ f
∗ z, [z ]C 2
´
⊂G C 1 f ∈ Hol (γ ) ´ f ∈ Hol (G) ; γ 0 , γ 1 : [α, [ α, β ] → G γ 0 ∼ γ 1 ⇒ γ G
∗ 1 C
[z ]
C
⊆ C u1 , u2 G⊆C l:G→C f (: f (: G → C) ∈ Hol (G) ∀z ∈ G.u (z) = f f ((zz)) f (: G → C) ∈ Hol (G)
eψ(t) = γ ( γ (t)
[ a, b] → C ∃ψ : [a,
γ
∀
∗ z, [z ]C 1
G
α,β.f (α) = ∃α,β.f (
´
hI
b
f
→C ±2πi
(z0 ) =
⊆G a
f : G
s, f ( f (β ) = t γ : [a, [ a, b]
∈C C 1 −→ C 2
ϕ
3
[z1 , z2 , z3, z4 ] = [ϕ (z1 ) , ϕ (z2 ) , ϕ (z3 ) , ϕ (z4 )] ϕ 1≤j ≤4 T zj = wj
w1 , w2 , w3, w4
S
C
⊂C G ⊂ C
f : G
γ
→ ´ C
γ
f : G
´
− zi) < ε
→C f : G → C
f = 0
ε> 0 λ (π ) < δ
}
zi = γ ( γ (ti ) f π
{}
´
γπ
T
G
f = 0 f
G γ
− ´ γ < ε
´
γ2
f
z0
∀z ∈ D
f ( f (z ) =
´ 2π
1 2πi
0
f f z + eit dt
D
⊂C Aut (D)
f
C ∞ ∞ n=0 an (z
D (z0 , R) , f ( f (z ) = n! 2πi
an =
f
− z0 )n
D (z0 , R)
C 1
[ a, b] → D f : D → D, γ : [a,
∀z ∈
LH (γ ) = LH (f γ )
◦
LH [f γ ]
◦ ≤ LH [γ ]
∂ D
f (z )dz ∂D( ∂D (z0 ,R) ,R) (z −z0 )n+1
´
f ( f (z )dz ∂D( ∂D (z0 ,R) ,R) (z −z1 )n+1
n! 2πi
f (n) (z1 ) =
´
1 z = 2 ; zw = zw z w = zw z z
G
| || | | |
||
f P
∈ C [z]
P
• an z n • g, h
p
s p f = a−1 z0
g z0 h
h
g(z0 ) h (z0 )
=
f : C
→D →C →C
C
• •
f : CP1 f : C
G
f f
´
∂T
f ( f (z ) dz = 0 I
(z
− z0 )
G
f
⊂G
f
C (G) ∈ Hol (G\I ) ∩ C (
z0
g (z )
T
Hol (G)
g (z) =
(z f − z ) 0
−m
am
f
∈ Hol (G)
f = g
A
⊂G f
f
∈ Hol (G)
(k )
k f n
→ f (k)
∈ (R1 , R2) f
f =
an =
∈ Hol (A)
´
f f
A = R1 < z < R 2
{
A = R1 < z < R 2
{
||
D
≡g
→C
∞ n n=−∞ an z f ( f (z) dz |z|=r z n+1
1 2πi
∞ n n=−∞ an z
f n : G
f
\ {z0 }
∈ Hol (D (z0, r) \ {z0})
z0
z0
}
z0 =
∞
f (z )| = ∞ ⇐⇒ limz→z |f ( ⇐⇒ ∀ε > 0 f {D (z0 , ε) \ {z0 }} 0
f
z0
C
f
||
{an}∞n=−∞
}
#
{C\f {D (z0 , ε) \ {z0}}} ≤ 1 f ∈ Hol (D (z0 , r) \ {z0 }) f (z ) dz = 2πi 2 πi · z f ∀0 < ε < r, |z−z |=ε f ( ´ f f ∈ Mer (G) ∩ C G ⇒ ∂G f = 2πi 2 πi (Z G − P G ) ∂G f, g ∈ Hol (G) ∩ C G ∀z ∈ ∂G |f ( f (z ) − g (z)| < |f ( f (z )| Z f f = Z g ´
0
0
G
f −1
1
f (z) =
f f −1
1 (z ))
f (f
∈ Hol (G) f : G → C
−
´
g(γ )
f
∞
f ( f (z ) dz =
´
γ
f ( f (g (w )) g (w ) dw f (z ) dz = 2πi 2 πi · ∀r > R, − ´ |z|=r f (
∈ Hol ({z | |z| > R })
f ) ∞ (f )
C
f f
∈ Hol (C) ∈ M er (C)
∞ ⇒ f ∞ ⇒ f f
m m
∈ Hol (G)
z0 G, w0 = f ( f (z0 ) ε> 0 w
∈
δ>0
G´
G
f : G
∂G
f ( f (z ) dz = 0
G
\ {a1, . . . , an}
´
∂G
f = 2πi 2 πi
Γ
2πi f ( f (z0 ) =
Γ (ak )
∈ ∈
ak f
´
Γ
γ 1 2πi
→C
ak f
⊂G
´ Df (z⊆) C
z0
∈ G\γ
γ
(z0 ) f ( f (z0 ) =
1 2πi
f =
f (z ) γ z −z0 dz
´
¯ D
f
∀z0 ∈ D
∂D z −z0
G
0
| − w0| < ε
|z − z0 | < δ
z /G f ( f (z ) z G
Hol (G)
∩ C G ⊃
f : G
→
C
1 2πi
f (ζ ) dζ ∂G ζ −z
´
n
• • •
limz→∞ f ( f (z ) = 0 ∞ f = z0 f g =
◦
0
1 w2
1 (n
=
∞ f =
−
dn−1 ((z ((z 1)! z→c dz n−1 lim
z0
limz→∞ z f ( f (z)
1 w
− f
(f ( f (g (z0 )) g (z0 ))
• r = limn→∞ |a−n|1/n • R1 = limn→∞ |an|1/n
f ( f (z ) = r
f (z ) =
A
m
∞
G
R
c
G
∈ Hol (G) m
z0
c f =
f ( f (z) z = z0 z = z0
f f, g
⊂
G
f m
∈ Hol (G) ∩ C G
f
·
f (z)) − c)n f (
f z0 f
=m
