az+b
� (z) = C
cz+d
" G
A " GL2, hA (z)
" Hol (G) G Mer (G) Har (G)
A, B " GL2, hA ć% hB = hAB
G
" = 0, hA = hA

hdd = hI
Ż
" C (G) G
� 2 hA 3 hI
z1, z2, z3, z4 " C [z1, z2, z3,z4] = [� (z1) , � (z2) , � (z3) , � (z4)]
w1, w2, w3,w4 " C � "1d"jd"4T zj = wj
S �!�! Sc � C1 - C2 z, [z]" z, [z]"
C1 C2
G " C
z2, z3, z4 C [z1]" , z2, z3, z4 = [z1, z2, z3,z4]
C
G X ą" G X \ G
zj, 1 d" j d" 4 (z1, z2, z3, z4) " R
X = G X = "
a, b " G. a b
f : G C f
G ą" C u1, u2 u1 - u2 a"
f : G R "ą, �.f (ą) =
1
G ą" C l : G C l " Hol (G) l (z) =
s, f (�) = t �! [s, t] ą" (f)
z
f (: G C) " Hol (G) u log (f) u
ł : [a, b] C "� : [a, b] C e�(t) = ł (t)
f (z)
"z " G.u (z) =
ą2Ąi
f(z)
�
1 dz
(z0) =
f (: G C) " Hol (G) z0 " G, f (z0) = 0 "� > 0 D (z0, �)

ł
2Ąi ł z-z0
log (f)
�
f
ł �" G C1 f " Hol (ł) dz = 2Ąi � (0)
fć%ł
ł f
� �
f " Hol (G) ; ł0, ł1 : [ą, �] G ł0 <" ł1 �! f = f
ł0 ł1
G
a
ł C
{un (z)}" G {Mn}"
n=0 n=0
G
�
supz"G |un| < Mn
ł f " Hol (G) f = 0
ł
"
Mn < +"
�
n=0
f " Hol (G) F (z) = f
zz0
"
un (z) G
n=0
ł z0 " G (z0) = 0
/
ł
"
"
f " Hol (G) log f, f cn
cnzn R = limn" cn+1 R-1 =
n=0
n
limn" sup |an|
f " Hol (G) , ł �" G G �!�! C\G �!�!
�
"z0 " G, (z0) = 0 �!�! f = 0 �!�! f G
/
ł "
ł
f (z) = cn (z - z0)n R " [0, "]
n=0
log f " Hol (G)
D (z0, R)
G1, G2 �" C f : G1 G2 f-1 G1
|z - z0| > R
�!�! G2
"
f (z) D (z0, R) f (z) = cnn (z - z0)n-1
Hol (D) f : D D f (0) = 0 |f (z)| d" |z| n=1
f (z)
f (z) = z, || = 1
Hol (D) f : D D f (0) = 0 � " [0, 2Ą]
f (z) = ei�z
" "
n!
f (z) = cn (z - z0)n C" D (z0, R) f(k) (z) = cn (z - z0)n-k
n=0 n=k (n-k)!
z-a
f(k)(z0)
f : D D D |a| < 1, � " [0, 2Ą] f (z) =
1-az z = z0 : = ck
k!
z0 = 0 f
G �" C G = C

"
f : G D f (a) = 0, R f (a) > 0
G (z) = cn zn+1
n=0 n+1
1-|f(z)|2 f(z)-f(w) n"
z-w
f : D D |f (z)| d" d"
n |cn| 0 f (z) = cnzn limR n1 f (x) = L
1-zw
1-|z|2 1-f(z)f(w)
cn = L
|zn-1|
cn zn n" 1 sup < " limn" f (zn) =

1-|zn|
d
cn
f z0 g w0 = f (z0) g (f (z)) z=z0 =
dt
g (w0) f (z0)
f x0 �!�!
� f : G C ł : [a, b] G
f (z) dz = f (ł (b)) - f (ł (a))
ł
�
ł t0 f ł (t0) (f ć% g) = f ł
ł
f (ł (t0)) ł (t0)
Ł
0
� � �
b b b
f : [a, b] C f (t) dt = f (t) dt + i f (t) dt
a a a
� �
f : G C f �!�! "z " G : f (z) = 0
ł2 ł1 f ł1 f = f
ł1 ł2
f : G C f
fn : G C fn �! f G
� �
f : G C |f| G f ł : [a, b] C C1 fn f
ł ł
�
f : G C f (t) = 0, "z " G f

f d" (ł) � supt"[a,b] |f (ł (t))|
(f (ł1) , f (ł2)) = (ł1, ł2) ł
f : C C R "z " C f (z) = az + bz G �" C f : G C ł : [a, b] C1 � > 0
Ż
"f "f
� > 0 Ą = {a = t0 < . . . < tN = b}  (Ą) < �
a = (0) ; b = (0)
"z "z
Ż
�
N-1
f : G C R f
f - f (zi) (zi+1 - zi) < � zi = ł (ti)
i=0
ł
� �
łĄ f {Ą} - < �
łĄ ł
f " Hol (G) f f
G �" C f : G C T G
(f) , (f) �
f = 0
ł
"u
u : G C " Hol (G)
"z
G �" C f : G C f G
u : G R
G �" C f : G C ł
�
f = 0
G = C v ł
G ąc
� 2Ą f D �" C
1
z0 "z " D f (z) = f z + eit dt
2Ąi 0
Aut (D) f : D D, ł : [a, b] D C1 LH (ł) = LH (f ć% ł)
f LH [f ć% ł] d" LH [ł]
C"
f D (z0, R) "z "
" "D
D (z0, R) , f (z) = an (z - z0)n
n=0
�
f(z)dz
n!
an =
2Ąi "D(z0,R)
(z-z0)n+1
�
f(z)dz
n!
f(n) (z1) =
2Ąi "D(z0,R)
(z-z1)n+1
1 z
= ; zw = zw |z| |w| = |zw|
G
z
|z|2
f
P " C [z] P
" anzn p spf = a-1
g
" g, h z0 h =
z0 h
g(z0)
h (z0)
f : C D C
f
" f z0 m = m
z0 f
f : CP1 C
f : C C
" c n
G f " Hol (G) )" C G
1 dn-1
f = lim ((z - c)n f (z))
c
zc
(n - 1)! dzn-1
f G T �" G
�
f (z) dz = 0 f G
"
"T
I �" G f " Hol (G\I) )" C (G) f " Hol (G)
" limz" f (z) = 0 f = limz" z � f (z)
"
1 1
" f = - f
" 0
w2 w
" f ć% g = (f (g (z0)) g (z0))
z0 z0
f z0 f m f (z) =
(z - z0)-m f (z) z = z0
(z - z0)m g (z) Hol (G) g (z) =
am z = z0

" r = limn" |a-n|1/n
f G 1
" = limn" |an|1/n
R
f, g " Hol (G) f = g A �" G A f a" g
R
fn : G C
f
f " Hol (G)
(k)
k fn f(k)
"
f (z) = anzn A = {R1 < |z| < R2}
n=-"
�
f(z)
1
r " (R1, R2) an = dz
2Ąi |z|=r zn+1
f " Hol (A) A = {R1 < |z| < R2} {an}"
n=-"
"
f = anzn
n=-"
f D\ {z0} z0 z0 = "
f " Hol (D (z0, r) \ {z0}) z0 �!�! limzz0 |f (z)| = "
f z0 �!�! "� > 0 f {D (z0, �) \ {z0}}
C
f # {C\f {D (z0, �) \ {z0}}} d" 1
�
f " Hol (D (z0, r) \ {z0}) "0 < � < r, f (z) dz = 2Ąi � f
z0
|z-z0|=�
�
f
f " Mer (G) )" C G �! = 2Ąi (ZG - PG) "G
"G f
f, g " Hol (G) )" C G "z " "G |f (z) - g (z)| < |f (z)| Zf = Zg
G
f " Hol (G) f
f : G C f-1
1
f-1 (z) =
f (f-1(z))
� �
f (z) dz = f (g (w)) g (w) dw
g(ł) ł
�
" f " Hol ({z | |z| > R}) "r > R, - f (z) dz = 2Ąi � (f)
"
|z|=r
C
f " Hol (C) " �! f
f " Mer (C) " �! f
f " Hol (G) z0 " G, w0 = f (z0)
m � > 0 � > 0 |w - w0| < �
m |z - z0| < �
G� f : G C
G f (z) dz = 0
"G
�
G\ {a1, . . . , an} f = 2Ąi f
ak
"G
�
� f =
�
2Ąi (ak) f
� ak
�
f(z)
1
ł �" G z0 " G\ł (z0) f (z0) =
ł
2Ąi ł z-z0dz
Ż
D ą" C f D "z0 " D
�
f(z)
1
f (z0) =
2Ąi "D z-z0
�
f(ś)
1
G Hol (G) )" C G �" f : G C dś =
2Ąi "G ś-z
0 z " G
/
f (z) z " G


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