f(w) f(z) = C f(z) = z z + h z h = h h h 0,h C f(z + h) f(z)

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5 Ω f: Ω C l C z Ω f f(w) f(z) z a w z = h 0,h C f(z + h) f(z) h = l. z f l = f (z) Ω f Ω f Ω H(Ω) n N C f(z) = z n h h 0 h z + h z h = h h C f(z) = z f (z) = f( z) f f: Ω C Ω = { z; z Ω} z, a Ω

6 f (z) f (a) z a z a f( z) f(ā) = ( ) = f z a z ā (ā). Ω fg f + g Ω g f {z Ω; g(z) 0} f g g(ω ) g: Ω C f: Ω 1 C (f g) (z) = f (g(z))g (z) Ω f g Ω 1 f(z) = U(x, y)+ V (x, y) z 0 = x 0 + y 0 Ω f: Ω C V = f U = f (x 0, y 0 ) V U z 0 f U x (x 0, y 0 ) = V y (x 0, y 0 ) U y (x 0, y 0 ) = V x (x 0, y 0 ). f y (z 0) = f x (z 0). h 0 f(z 0 +h) f(z 0 ) h = f (z 0 ) z 0 = x 0 + y 0 f h

7 f U(x 0 + h, y 0 ) U(x 0, y 0 ) (z 0 ) = + V (x 0 + h, y 0 ) V (x 0, y 0 ) h 0 h h = U x (x 0, y 0 ) + V x (x 0, y 0 ) = f x (x 0 + y 0 ). t R h = t f U(x 0, y 0 + t) U(x 0, y 0 ) (z 0 ) = + V (x 0, y 0 + t) V (x 0, y 0 ) t 0 h t = U y (x 0, y 0 ) + V y (x 0, y 0 ) = f y (z 0). f U(x 0 + s, y 0 + t) U(x 0, y 0 ) = s U x (x 0, y 0 ) + t U y (x 0, y 0 ) + h ε 1 (h) V (x 0 + s, y 0 + t) V (x 0, y 0 ) = s V x (x 0, y 0 ) + t V y (x 0, y 0 ) + h ε (h) h 0 ε 1 (h) = h 0 ε (h) = 0 h = s + t f(z 0 +h) f(z 0 ) = s U x (x 0, y 0 )+t U y (x 0, y 0 )+ (s V x (x 0, y 0 )+t V y (x 0, y 0 ))+ h ε(h), ε = ε 1 + ε f(z 0 + h) f(z 0 ) = Ah + h η(h), f 0 h 0 η(h) A = U x (x 0, y 0 ) + V x (x 0, y 0 ) f (z 0 ) = A z 0,

8 f = U + V Ω C C V U U Ω U = U x + U y f (z) = f (x, y) = f (x, y) x y U = V = 0 V Ω f = U V (x, y) + V (x, y) = (x, y) U (x, y) x x y y = U V (x, y) U (x, y) = (x, y) + V (x, y), x y y x z = x + y f(z) = U(x, y) = x y f: C C xy V y = x y f V V x = y + g (x) = ( y x) = x + y V = xy y + g(x) f(z) = (1 + )z + C V = x y + xy + C f(z) = x 3 3xy xy 1 f: C C V y = 3x 3y f V V x = 6xy + g (x) = 6xy + x V = 3x y y 3 y + g(x) y V = 3x y y 3 y + x + C f (z) = f x = 3x 3y y+ (6xy+x) = 3(x y + xy)+ (x+ y) = c R f(z) = z 3 + z c 3z + z

9 Ω f Ω f 0 f f 0 f f 0 D(z 0, r) Ω r > 0 z 0 Ω f(z 0 ) = z = z 1 + z 0 D(z 0, r) z 1 D(z 0, r) f(z 0 ) = f(z 1 ) f y = 0 f(z ) = f(z 1 ) f x = 0 f(z ) Ω C f: Ω C Ω f Ω f Ω f Ω f Ω f 1) ) ) 3) 3) 4) ) 3) Ω f f = 0 Ω f = c f f f = c f = c 0 f f f Ω f

10 R > 0 n 0 a n z n f R n 1 na n z n 1 g f (z) = g(z) D(0, R) f n N 0 < h r h C z C (z + h) n z n nhz n 1 h ( z + r)n r n z n 1 1 r (( z + r)n + z n ) (z + h) n z n nhz n 1 = n Cnh k k z n k z n nhz n 1 n = C k nh k z n k k=0 h n k= C k n z n k h k h r n k= k= C k n z n k r k h r ( z + r)n. (z + h) n z n nhz n 1 nr z n 1 z n ( z + r) n nr z n 1 z n + ( z + r) n + (z + r) n z n nrz n 1 z n + ( z + r) n.

11 n 1 na n z n 1 R R R na n z n 1 z + r < R r > 0 D(0, R) n 1 na n z n 1 1 r ( a n ( z + r) n + a n z n ) f(z + h) f(z) g(z) h h r R R R = R n=1 a n ( z + r) n, z D(0, R) f (z) = g(z) h 0 f(z) = n=0 f (n) (0) z n a n = f (n) (0) f C (D(0, R)) f(z) = a n z n n! n! n=0 0 f e z z n = n! z n n! ez n=0 n 0 ( z ) = z z+w = z w z C z z = 1 x R x > 0 x R 0 < x < 1 z = z (x, y) R x+ y = x y R y = 1 z 0, z C z z (C, +) (C, )

12 z = z + z z = z z i z = z + z z = z + z z y x+ y = x ( y + y) z = z + z θ = θ + θ z + z = 1 z z = e z z + z = e z z z = 1 z z = e z z + z = e z ( z) = z ( z) = z z = r( θ+ θ) = re θ θ [0, π[ r > 0 z = x+ y C\{0} r = x + y θ + θ = 1 r = 1 f(θ) = θ y 0 0 x < 1 θ ]0, π ] x [0, 1[ [0, 1] [0, π ] x = θ x + y = y = 1 x = θ y 0 x + y = 1 θ + θ x 1 y 0 x 0 x y = θ + θ θ ]0, π ] π θ [ 3π, π[ θ θ = x + y = (π θ) + (π θ) y 0 x 1 x 0 x + y = x y = θ + θ θ ] 3π, π] θ π [ π, π[ ( π + θ) + ( π + θ) y 0 x 1 x 0 x y = θ + θ θ ]0, π ]

13 θ + π ]π, 3π ] x + y = (π + θ) + (π + θ) x + y = y = 0 x = 1 x + y = π + π y = 0 x = 1 z π [0, π[ θ r > 0 z C \ {0} α R z = r( θ + θ) θ [α, α + π[ A: C \ R + ]0, π[ A(z) = A(r( θ + θ)) = θ θ ]0, π[ z = r( θ + θ) = x + y x = r θ = r (π θ) = r ( π θ ) + r y = r θ = r (π θ) = r ( π θ ) ( π θ ) r x = r ( π θ ) ) ( π θ ) y = r ( π θ y x + x + y = (π θ ) π θ = 1 y ( x + y x ) θ = π 1 y ( x + y x ), A C \ {t α, t 0} ]α, α + π[ A α (z) = A(r( θ + θ)) = θ f: Ω C C Ω

14 f (z) 0 z Ω C Ω f(ω) = Ω Ω f Ω f 1 Ω f 1 (f 1 ) (w) = 1 f (f 1 (w)), w Ω. z Ω w Ω z 0 = f 1 (w 0 ) Ω w 0 Ω w = f(z) f 1 (w) f 1 (w 0 ) w w 0 = z 0 z w 0 w f 1 z z 0 f(z) f(z 0 ) 1 w w 0 f (z 0 ) = 1 f (f 1 (w 0 )). f (z) 0 f 1 C Ω z Ω A t = {x + y C; t π < y < π + t, x R} z z A t C \ J t t R J t = {r t, r 0} C \ J t t A(z) C \ J 0 A 0 z = x + y + 1 y x + x + y, z = x + y C \ J 0 z C \ J t t R ( t z) = 1 z (C \ J t ) (C \ J t ) f(z) = t (z) t (z)

15 (z 1.z ) = z 1 + z z 1 + z = 3 π π z 1z = π z 1z = 3 π = π z 1 = 3 π 4 = z (1 z) (1 z) = f (z) = n=0 n=1 z n n z n = 1 1 z z < 1 f(z) = + z < 1 n=1 z n n f(0) = (1 0) = 0 f(z) + (1 z) = 0 α C C Ω z α = e α (z) Ω z 0 V z 0 Ω Ω f: Ω C z V f(z) = a n (z z 0 ) n a n (z z 0 ) n n=0 n 0 f(z) = R > 0 n 0 a n z n D(0, R) n=0 a n z n

16 m=0 n=0 m=0 n=0 a n,m = a n,m < + (a n,m ) n,m N n=0 m=0 m=0 n=0 a n,m = N,M N a n,m = n=0 m=0 [ a n,m. M m=0 n=0 N a n,m ], f(z) = n=0 f (n) (z 0 ) (z z 0 ) n z 0 = r 0 < R z 0 D(0, R) n! z D(z 0, R r 0 ) z z 0 < r r 0 < R r 0 z C S(z) = k=0 n=k C k na n z n k 0 (z z 0 ) k k=0 n=k C k na n z n k 0 (z z 0 ) k R(z) = k=0 n=k C k n a n z 0 n k z z 0 k R(z) k=0 n=k C k n a n r n k 0 (r r 0 ) k = n=0 a n ( n k=0 C k nr n k 0 (r r 0 ) k ) = n=0 a n r n < S +

17 S(z) = k 0 f (k) (z 0 ) (z S(z) = k! S(z) = n=0 a n ( n k=0 k=0 C k nz n k 0 (z z 0 ) k ) = z D(z 0, R r 0 ) f(z) = 1 k! (z z 0) k n! ( (n k)! a nz0 n k ), n=k z 0 ) k k=0 n=0 a n z n = f(z). f (k) (z 0 ) (z z 0 ) k k! z 0 Ω C Ω f Ω f 0 V f 0 z 0 V f (n) (z 0 ) = 0 n 0 ) 1) z A = {z Ω; f 0 Ω A a Ω A (z n ) n f (k) (a) = 0 k N f (k) (z n ) = 0 z n A a f 0 f Ω C g f Ω Ω g f

18 Ω f Ω C f Ω f f A = f 1 {0} f A f (k) (z 0 ) 0 k z 0 A f (k) (z 0 ) 0 k f(z) = a k (z z 0 ) k + (z z 0 ) k g(z) z 0 f g(z) = n=1 a n+k (z z 0 ) n g(z 0 ) = 0 a k 0 z V g(z) < a k z 0 V g(z 0 ) = 0 z V \ {z 0 } f(z) z z 0 k ( a k g(z) ) > 0 V f z 0 Ω Ω f f Ω g f Ω Ω

19 f(z) = f(re θ ) = F : C C U(r, θ) + V (r, θ) z z = 1 z + z = 1 (z+w) = z w+ (z+w) = z w+ z w z w (z+w) = z w+ z w (z+w) = z w z w z = y + x = y x = 1 ( y + x) z = x + y z = y + x = y x = 1 ( y x) z y z 0 = 4 z 0 = π/ + (4 + 15) a Ω Ω g f n + z n = a n N f(z n ) = g(z n ) z n a Ω (z n ) n Ω f g f( 1 n ) = 1. 0 f n f( 1 n ) = n. 0 f n + 1 z 0 C g

20 g n 0 g (n) (z 0 ) ]x 0 R, x 0 + f(x) = n=0 C a n (x x 0 ) n f D(x 0, R) f R[ n N f (n) (0) M R f C f z D(0, R) f(z π n ) = f(z) D(0, R) f n N f(z) = g(z n ) D(0, R n ) g

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