Matthias Beck Gerald Marchesi Dennis Pixton Lucas Sabalka

Transkript

1 Matthias Beck Gerald Marchesi Dennis Pixton Lucas Sabalka Version.53

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8 x 2 + = 0. i i 2 + = 0 i 2 = i x 3 + px + q q q p3 27

9 p q C := {(x, y) : x, y 2 R}, (x, y)+(a, b) := (x + a, y + b) (x, y) (a, b) := (xa yb, xb+ ya). C R (x,0) (x,0)+(y, 0) =(x + y, 0) (x,0) (y, 0) =(xy, 0).

10 C (C, +, ) (x, y), (a, b), (c, d ) 2 C (x, y)+(a, b) 2 C (x, y)+(a, b) +(c, d )=(x, y)+ (a, b)+(c, d ) (x, y)+(a, b)=(a, b)+(x, y) (x, y)+(0, 0)=(x, y) (x, y)+( x, y)=(0, 0) (x, y) (a, b)+(c, d ) =(x, y) (a, b)+(x, y) (c, d ) (x, y) (a, b) 2 C (x, y) (a, b) (c, d )=(x, y) (a, b) (c, d ) (x, y) (a, b)=(a, b) (x, y) (x, y) (, 0)=(x, y) (x, y) 2 C \{(0, 0)} : (x, y) Ä x x 2 +y, 2 y x 2 +y 2 ä =(, 0) (C, +) (0, 0) (C \{(0, 0)}, ) (, 0) R (x, y)+( x, y) =(x +( x), y +( y)) = (0, 0). (0, ) (0, ) =(, 0).

11 (a, 0) (x, y) =(ax, ay) (x, y) =(x,0)+(0, y) =(x,0) (, 0)+(y, 0) (0, ). R C (x,0) (y, 0) x y (x, y) (, 0) (0, ) x y (, 0) (0, ) i (x, y) x + y i x + iy. x y x + iy Re(x + iy)=x Im(x + iy)=y i 2 =. C d d C x + iy

12 k W D (x, y) R 2 x y R 2 z + z 2 z z 2 R 2 z = x + iy r = z := x 2 + y 2, z = x + iy φ 2 R x = r cosφ y = r sinφ. z = x +iy = + 0i 0 2π 4π 2π 2πk k 0 = 0 + 0i 0 φ

13 k D 0 z z 2π z = z, z 2 2 C R 2 d (z, z 2 ) R 2 d (z, z 2 )= z z 2 = z 2 z. z = x + iy z 2 = x 2 + iy 2 d (z, z 2 )= (x x 2 ) 2 +(y y 2 ) 2. z z 2 (x x 2 ) 2 =(x 2 x ) 2 (y y 2 ) 2 = (y 2 y ) 2 z 2 z z z 2 4z z 2 z z 2 = z 2 z z z 2 z 2 z x + iy r φ x 2 + iy 2 r 2

14 x f F... φ 2 x + iy =(r cosφ )+i(r sinφ ) x 2 + iy 2 = (r 2 cosφ 2 )+i(r 2 sinφ 2 ) (x + iy )(x 2 + iy 2 )=(r cosφ + ir sinφ )(r 2 cosφ 2 + ir 2 sinφ 2 ) =(r r 2 cosφ cosφ 2 r r 2 sinφ sinφ 2 )+i(r r 2 cosφ sinφ 2 + r r 2 sinφ cosφ 2 ) = r r 2 (cosφ cosφ 2 sinφ sinφ 2 )+i(cosφ sinφ 2 + sinφ cosφ 2 ) = r r 2 cos(φ + φ 2 )+i sin(φ + φ 2 ). r r 2 φ + φ 2 φ + φ 2 φ z z φ... z z 2 cosφ + i sinφ φ e iφ := cosφ + i sinφ. 2π

15 e 7πi 8 s s e πi 4 = p 2 + i p 2 e πi s 2 = i e iφ φ,φ,φ 2 2 R e iφ e iφ 2 = e i(φ +φ 2 ) e i0 = e i(φ+2π) = e iφ e iφ = e = e iφ iφ d dφ e iφ = ie iφ. e iφ d dφ e iφ = d dφ (cosφ + i sinφ) = sinφ+i cosφ = i (cosφ + i sinφ) = ieiφ. e 2πi = sin(2π)=0 cos(2π)= π i e

16 (e 2πi m n ) n = m n > 0 e 2πiq q 2 Q z n = e 2πi m n m n > 0 ζ ζ n = n ζ n th n ζ n = ζ n th 4 th ± ±i = e ± πi 2 4 th x + iy r φ x + iy = re iφ. p x 2 + y 2 apple p x 2 apple x apple p x 2 apple p x 2 + y 2 z applere(z) apple z z appleim(z) apple z. x + iy 2 = x 2 + y 2 =(x + iy)(x iy).

17 x + iy re i θ (x, y) z y x z r θ x + iy x iy x iy x + iy x + iy := x iy. z z z, z, z 2 2 C z ± z 2 = z ± z 2 z 2 = z z z z 2 = z z 2 Ä z z 2 ä = z z 2 z = z z = z Re(z)= 2 (z + z) Im(z)= 2i (z z) e iφ = e iφ. z = x + iy z 2 = x 2 + iy 2 z z 2 = (x x 2 y y 2 )+i(x y 2 + x 2 y )=(x x 2 y y 2 ) i(x y 2 + x 2 y ) =(x iy )(x 2 iy 2 )=z z 2.

18 z = z = z z 2. R n z, z 2 2 C z + z 2 apple z + z 2. z + z 2 2 = (z + z 2 ) (z + z 2 ) = (z + z 2 )(z + z 2 ) = z z + z z 2 + z 2 z + z 2 z 2 = z 2 + z z 2 + z z 2 + z 2 2 = z 2 + 2Re(z z 2 ) + z 2 2 apple z z z 2 + z 2 2 = z z z 2 + z 2 2 = z z z 2 + z 2 2 = ( z + z 2 ) 2, z, z 2,...,z n 2 C ±z ± z 2 apple z + z 2. ±z ± z 2 z z 2. nx z k k= apple nx z k. k= ±z = z

19 C R 2 y C [2 + i,2] D[ 2, 3 ] ~ x z w z w a r z 2 C z a = r r a a r C [a, r ] := {z 2 C : z a = r }. a r D[a, r ] := {z 2 C : z a < r }. D[a, r ] C [a, r ]

20 C G C a 2 G G a G b 2 C G b G G c 2 C G c G c d 2 G G d G d G G G G r > 0 a 2 C {z 2 C : z {z 2 C : z a > r } a < r } = D[a, r ] D[a, r ] := {z 2 C : z a appler } C? G G G G G G G [ G D[a, r ] D[a, r ] D[a, r ] C [a, r ] G G D[0, r ] r

21 R y x [0, ) (, 2] X, Y C A, B C X A Y B G C G A B X Y X =[0, ) Y =(, 2] A B A = D[0, ] B = D[2, ] X [Y =[0, 2]\{} C γ : [a, b]! C [a, b] R γ γ(t ), a apple t apple b

22 γ γ 0 γ : [a, b]! C t 0 2 [a, b] γ t 0 lim γ(t )=γ(t t!t 0 ), 0 γ 0 (t 0 )= lim t!t0 γ(t ) γ(t 0 ) t t 0. y 3 + i(t 2) 0 apple t apple 3 γ (t )= e it, π 2 apple t apple 2π γ 2 (t )= 6 t + i 2 (t ) 3 apple t apple 5 x γ γ 2 γ γ 3 (t )= e it, 0 apple t apple 3π 2, γ γ 3 γ γ 3 γ γ : [a, b]! C γ 0 (t ) a < t < b lim t!a + γ 0 (t ) lim t!b γ 0 (t )

23 [a, b] C R 2 x(t ) y(t ) γ(t )=x(t )+iy(t ) γ : [a, b]! C γ(t ) γ(a)=γ(b) γ : [a, b]! C γ(a)=γ(b) C [0, ] γ(t )=e it, 0 apple t apple 2π G C G G G C G G G G G z 0 z z n z k z k+ G k = 0,,...,n D[0, ] D[0, ] D[0, ] D[0, ] G = D[0, ] \ {0} D[0, ] G G

24 S C S z = + 2i w = 2 z + 3w i z 3 z 2 + z + i w z Re(w 2 + w) z a z+a 3+5i 7i+ a 2 R Ä +i p 3 2 ä 3 i n n i 3 i p 2+3i (2 + i)(4 + 3i) ( + i) 6 2i i p 5 i + i 3 + p 3 i (2 i) 2 3 4i Ä ä 4 p i 3 p 2 e i 3π 4 e i250π 34 e i π 2 2 e 4πi e ln(5) i d dφ e φ+iφ

25 a, b, c 2 R a 6= 0 az 2 + bz + c = 0 b ± p b 2 2a 4ac. p b 2 4ac = i p b 2 + 4ac b 2 4ac z = 0 2z 2 + 2z + 5 = 0 5z 2 + 4z + = 0 z 2 z = z 2 = 2z z 2 + 2z +( i)=0 a 2 C b 2 R z 2 + Re(az)+b = 0 a 2 4b z 6 = z 4 = 6 z 6 = 9 z 6 z 3 2 = 0 z = z = z z z = z z (z) 2 = z 2 z z 2 = 0 z = 0 z 2 = 0

26 n z n = z = e 2πi m n m 2 z n = z = e 2πi a n a 2 R a = m + b m 0 apple b < b z 5 = (z ) z 2 + 2z cos π 5 + z 2 2z cos 2π 5 + cos π 5 cos 2π 5 n w z n = w w cos(3φ)=cos 3 φ 3cosφ sin 2 φ sin(3φ)=3cos 2 φ sinφ sin 3 φ ñ x x, y 2 R M (x, y) := y ô y x M (x, y)+m (a, b) =M (x + a, y + b) M (x, y) M (a, b) =M (xa yb, xb+ ya). {M (x, y) : x, y 2 R} C = {(x, y) : x, y 2 R}

27 {z 2 C : z + i = 2} {z 2 C : z + i apple 2} {z 2 C : z = z + } {z 2 C : z = 2 z + } {z 2 C :Re(z + 2 2i)=3} z 2 C :Re(z 2 )= {z 2 C : z i + z + i = 3} z 2 C :Im(z 2 )= p p(z)= p (z) p(z)=0 p (z) = 0 z z 2 z z 2. z C [0, 2] z 2 apple 3 z + 3 < 2 Im(z) < z + z + = 2 z + z + < 3 0 < z < 2 z Re(z)+

28 G z 2 C z 2 < z < z < z = z = 2 G G G G G G A B B A B A B A A B C [ + i,] i 2i C [0, 34] ± ± 2i {z 2 C : z + z + = 4} G 2 < z < 3 G G

29 34 = i = π = 2 + 2i = 2 p 3 + i = 2 e i0 = 3 e πi 2 = 2 e i π = e 3πi 2 = 2 e 3πi 2 =

30 z + z + 2 2z z z 2 iz z z 2 Re(z) Im(z) i Im(z) z z i

31 f G C C f : G! C G f z 2 G z f (z) R m R n f (z) =z f (z)=2z + i f (z)=z 3 f (z)= z C z = 0 z f (x, y)=x 2iy f (x, y)=y 2 ix f (r,φ)=2r e i(φ+π)

32 f : G! C z 0 G w 0 ε > 0 δ > 0 z 2 G 0 < z z 0 < δ f (z) w 0 < ε w 0 f z z 0 lim z!z 0 f (z) =w 0. z 0 z z 0 0 < z z 0 z 0 f z 0 f f (z 0 ) lim z 2 = z!i ε > 0 δ > 0 0 < z i < δ z 2 + < ε z 2 + = z i z + i < δ z + i. δ z +i δ < min{ ε 3, } 0 < z i < δ z 2 + < 3δ < ε. ε > 0 0 < δ < min{ ε 3, } 0 < z z + i = z i + 2i apple z i + 2i < 3, i < δ z 2 ( ) = z 2 + = z i z + i < 3δ < ε. lim z!i z 2 =

33 w 0 z z 0 f : G! C lim z!z0 f (z)=w 0 G e G z 0 G e f e f G e lim e z!z0 f (z) w 0 z z z! 0 z! 0 z = x 2 R lim z!0 z z x = lim x!0 x = lim x x!0 x =. z = iy y 2 R lim z!0 z z iy = lim y!0 iy iy = lim y!0 iy =. z z z! 0 f g G z 0 G c 2 C lim z!z0 f (z) lim z!z0 g (z) lim z!z0 ( f (z)+cg(z)) = lim z!z0 f (z)+c lim z!z0 g (z) lim z!z0 ( f (z) g (z)) = lim z!z0 f (z) lim z!z0 g (z) lim z!z0 f (z) g (z) = lim z!z 0 f (z) lim z!z0 g (z) lim z!z0 g (z) 6= 0

34 c 6= 0 L = lim z!z0 f (z) M = lim z!z0 g (z) ε > 0 δ, δ 2 > 0 0 < z z 0 < δ f (z) L < ε 2 0 < z z 0 < δ 2 g (z) M < ε 2 c. δ = min{δ, δ 2 } 0 < z z 0 < δ ( f (z)+cg(z)) (L + cm) apple f (z) L + c g (z) M < ε. lim z!z0 ( f (z)+ cg(z)) = L + cm f : G! C z 0 2 G z 0 G lim z!z 0 f (z) = f (z 0 ) f z 0 f E G f z 2 E ε δ f : G! C z 0 2 G f z 0 ε δ f (z) f (z 0 ) < ε z 2 G z z 0 < δ. f : C! C f (z)=z 2 z = i C

35 g : C! C 8 < z z z 6= 0, g (z) := : z = 0. g z = 0 g : G! C { g (z) : z 2 G}. g f : H! C f g : G! C ( f g )(z) := f ( g (z)). g : G! C H f : H! C z 0 G lim z!z0 g (z) =w 0 2 H f w 0 lim z!z0 f ( g (z)) = f (w 0 ) lim f ( g (z)) = f z!z 0 Å ã lim g (z). z!z 0 ε > 0 η > 0 w w 0 < η f (w) f (w 0 ) < ε. η δ > 0 0 < z z 0 < δ g (z) w 0 < η. 0 < z z 0 < δ f ( g (z)) f (w 0 ) < ε. lim z!z0 f ( g (z)) = f (w 0 )

36 z z f (z) z! 0 z 0 0 f : G! C z 0 G f z 0 f 0 (z 0 ) := lim z!z0 f (z) f (z 0 ) z z 0, f z 0 f z 0 f z 0 f E G E C f : C! C f (z) =z 3 C z 0 2 C f (z) f (z lim 0 ) z 3 z0 3 (z 2 + zz = lim = lim 0 + z0 2)(z z 0 ) = 3z 2 z!z 0 z z z!z0 0 z z z!z0 0 0 z z. 0

37 f 0 f (z (z 0 )=lim 0 + h) f (z 0 ). h!0 h h f : C! C f (z)=(z) 2 0 f 0 z = z 0 + re iφ z 2 z z 0 2 = z 0 + reiφ = (z 0 + re iφ ) 2 z 0 z z 0 z 0 + re iφ z 0 re iφ = z z 0 re iφ + r 2 e 2iφ 2 z 0 re iφ = 2 z 0 e 2iφ + re 3iφ. 2 = 2 z 0 re iφ + r 2 e 2iφ re iφ z 0 6= 0 f (z) z! z 0 2 z 0 e 2iφ + re 3iφ r! 0 2 z 0 e 2iφ φ z z 0 z 0 = 0 re 3iφ = z e 3iφ lim z!0 z 2 z = lim z!0 z e 3iφ = lim z!0 z = 0, z 2 z lim 0 = lim z!z 0 z z 0 2 z z!0 z 2 = 0. f : C! C f (z)=z z z lim 0 z z = lim 0 z = lim z!z 0 z z z!z0 0 z z z!0 0 z,

38 f g z 2 C h g (z) f (z)+cg(z) 0 = f 0 (z)+cg 0 (z) c 2 C f (z) g (z) 0 = f 0 (z) g (z)+ f (z) g 0 (z) f (z) 0 = f 0 (z) g (z) f (z) g 0 (z) g (z) 2 6= 0 g (z) g (z) 2 z n 0 = nz n n g z h( g (z)) 0 = h 0 ( g (z)) g 0 (z). f (z)g (z) 0 f (z + h) g (z + h) f (z) g (z) = lim h!0 h f (z + h) ( g (z + h) g (z)) + ( f (z + h) f (z)) g (z) = lim h!0 h g (z + h) g (z) f (z + h) f (z) = lim f (z + h) + lim g (z) h!0 h h!0 h = f (z) g 0 (z)+ f 0 (z) g (z). f (z) γ(t ) f a 2 C f 0 (a) 6= 0 γ γ 2 a φ f γ γ 2 f (a) φ

39 γ (t ) γ 2 (t ) γ (0)= γ 2 (0)=a γ 0(0) γ a γ2 0(0) γ 2 a γ γ 2 f f (γ ) f (a) d dt f (γ (t )) = f 0 (γ (0)) γ 0 (0) = f 0 (a) γ 0 (0), t =0 f (γ 2 ) f (a) f 0 (a) γ2 0 (0) f γ 0(0) γ0 2 (0) f 0 (a) f 0 (a) f 0 (a) f : G! H w 2 H z 2 G f (z) =w w 2 H z 2 G z 2 G f (z) =w f : G! H g : H! G f f ( g (z)) = z z 2 H f g H G, H C f : G! H g : H! G f z 0 2 H f g (z 0 ) f 0 ( g (z 0 )) 6= 0 g z 0 g z 0 g 0 (z 0 )= f 0 ( g (z 0 )).

40 f ( g (z)) = z z 2 H g 0 (z 0 )= lim z!z0 g (z) g (z 0 ) z z 0 g (z) g (z = lim 0 ) z!z0 f ( g (z)) f ( g (z 0 )) = lim f ( g (z)) f ( g (z 0 )) g (z) g (z 0 ) z!z 0. w 0 = g (z 0 ) 8 f (w) f (w >< 0 ) w 6= w φ(w) := w w 0 0 >: f 0 (w 0 ) w = w 0. w 0 lim z!z0 g (z)=w 0 g g 0 (z 0 )= lim z!z0 φ ( g (z)) = Å φ lim g (z) z!z 0 ã = f 0 (w 0 ) = f 0 ( g (z 0 ). f : R 2! R f x (x 0, y f 0 ) y (x 0, y 0 ) (x 0, y 0 ) 2 R 2 f (z) f 0 (z 0 ) z 0 =(x 0, y 0 ) 2 C f 0 (z 0 ) f x (z 0 ) := lim x!x 0 f (x, y 0 ) f (x 0, y 0 ) x x 0

41 f y (z f (x 0 ) := lim 0, y) f (x 0, y 0 ) y!y 0 y y 0 f z 0 = x 0 + iy 0 f f x (z 0 )= i f y (z 0 ). f f x f y z 0 z 0 f z 0 f 0 f 0 (z 0 )= f x (z 0 ). f f (z)= f (x, y)=u(x, y)+iv(x, y) u f v f x = f x f y = f y f x = u x + iv x if y = i(u y + iv y )=v y iu y. u x (x 0, y 0 ) = v y (x 0, y 0 ) u y (x 0, y 0 ) = v x (x 0, y 0 ).

42 f z 0 = x 0 + iy 0 u v z 0 f = u + iv G u v G u v u xx (x 0, y 0 )=v yx (x 0, y 0 )=v xy (x 0, y 0 )= u yy (x 0, y 0 ), u xx (x 0, y 0 )+u yy (x 0, y 0 )=0 v G C G f G u v f : C! C f (z) =z 3 =(x + iy) 3 = x 3 3xy 2 + i 3x 2 y y 3. f x (z) =3x 2 3y 2 + 6ixy f y (z) = 6xy+ 3ix 2 3iy 2 C f x = if y f (z)=z 3 f : C! C f (z) =(z) 2 =(x iy) 2 = x 2 y 2 2ixy. f x (z) =2x 2iy f y (z) = 2y 2ix,

43 f x = if y z = 0 f (z)=(z) 2 C \{0} f z 0 =(x 0, y 0 ) f 0 (z 0 )= lim z!0 f (z 0 + z) f (z 0 ). z z z = x f 0 (z 0 )= lim x!0 f (z 0 + x) f (z 0 ) x = lim x!0 f (x 0 + x, y 0 ) f (x 0, y 0 ) x = f x (x 0, y 0 ). z = i y f 0 (z 0 )= lim i y!0 f (z 0 + i y) f (z 0 ) i y = lim y!0 i f (x 0, y 0 + y) f (x 0, y 0 ) y = i f y (x 0, y 0 ). f 0 (z 0 )= f x (z 0 )= if y (z 0 ) f x f y z 0 f 0 (z 0 )= f x (z 0 ) f x (z 0 )= x + i y z f x (z 0 )= x z f x (z 0 )+ y z if x (z x 0 )= z f x (z 0 )+ y z f y (z 0 ). f 0 (z 0 ) f (z 0 + z) f (z 0 ) z = f (z 0 + z) f (z 0 + x)+ f (z 0 + x) f (z 0 ) z = f (z 0 + x + i y) f (z 0 + x) + f (z 0 + x) f (z 0 ). z z

44 lim z!0 f (z 0 + z) f (z 0 ) f z x (z 0 ) y f (z0 + x + i y) = lim z!0 z y f (z 0 + x) f y (z 0 ) x f (z0 + x) f (z + lim 0 ) z!0 z x f x (z 0 ). 0 x y z z 0 0 f x (z 0 )= lim x!0 f (z 0 + x) f (z 0 ) x z! 0 x! 0 x y z! 0 u(z) v(z) f (z) 0 < a, b < u(x 0 + x, y 0 + y) u(x 0 + x, y 0 ) y v(x 0 + x, y 0 + y) v(x 0 + x, y 0 ) y = u y (x 0 + x, y 0 + a y) = v y (x 0 + x, y 0 + b y).

45 f (z 0 + x + i y) f (z 0 + x) f y y (z 0 ) u(x0 + x, y = 0 + y) u(x 0 + x, y 0 ) y v(x0 + x, y + i 0 + y) v(x 0 + x, y 0 ) y u y (z 0 ) v y (z 0 ) = u y (x 0 + x, y 0 + a y) u y (x 0, y 0 ) +i v y (x 0 + x, y 0 + b y) v y (x 0, y 0 ). u y v y (x 0, y 0 ) lim u z!0 y (x 0 + x, y 0 + a y) =u y (x 0, y 0 ) lim v z!0 y (x 0 + x, y 0 + b y) =v y (x 0, y 0 ), z! 0 0 I f : I! R f 0 (x) 0 x 2 I c 2 R f (x)=c x 2 I x, y 2 I f (y) f (x) = f 0 x + a(y x) (y x)

46 0 < a < f 0 (x + a(y x)) = 0 f (y)= f (x) x, y 2 I f I 0 f f f I f : {x + iy2 C : x 6= 0}!C 8 < Re z > 0, f (z) := : 2 Re z < 0, f 0 (z)=0 z f f C G C f : G! C f 0 (z) 0 z 2 G f f G H G y 0 2 R z 2 H Im(z)=y 0 u(z) f (z) z 2 H Im(z)=y 0 H u(z)=u(x, y 0 ) x z = x +iy 0 f 0 (z)=0 z 2 H u x (z)=re( f 0 (z)) = 0 u(z) H v(z) f (z) H v x (z)=im( f 0 (z)) = 0 H f (z) H f (z) H

47 f G x y G f (x)= f (y) f G z o 2 C lim z!z0 (az+b)=az 0 +b iz lim 3 z!i z+i lim (x + i(2x + y)) z! i f : G! C z 0 G f (z)=0 lim z!z0 f (z) = 0 lim z!z0 f (z)= x 2 y x 4 + y 2 z = x + iy6= 0. f 0 lim z!0 f (z) y = x 2

48 f : C! C f (z)=z 2 C g : C! C C \{0} 8 < z z z 6= 0, g (z)= : z = 0 f : C! C 8 < 0 z = 0 z f (z)= : q z = p q 2 Q \{0} 8 < 0 z = 0, f (z)= : sinφ z = re iφ 6= 0. z 0 G z 0 G f : C \{0}!C f (z) = z f 0 (z)= z 2 T (z) := az+b cz+d a, b, c, d 2 C ad bc 6= 0 T 0 (z)=0 f (z) z f f (z) x = Re z y = Im z

49 u v f (z) = u(x, y)+iv(x, y) u v f f (z)=e x e iy f (z)=2x + ixy 2 f (z)=x 2 + iy 2 f (z)=e x e iy f (z)=cos x cosh y f (z)=im z i sin x sinh y f (z)= z 2 = x 2 + y 2 f (z)=z Im z f (z)= ix+ y f (z)=4(re z)(im z) i(z) 2 f (z)=2xy i(x + y) 2 f (z)=z 2 z 2 f (z) =0 Re(z) Im(z) =0 f (z) = Re(z) Im(z) 6= 0 f z = 0 f z = 0 f G C f G f 0 = 0 f (z) f (z) G C f (z) G f f (z)=v(x)+iu(y) f x = Re(z) f y = Im(z) f (z)=az + b a 2 R b 2 C f u v u(z) v(z)= 3 z f

50 u = v u r r φ r = v φ r. u v u + iv u(x, y)=x 2 y 2 u(x, y)=cosh(y) sin(x) u(x, y)=2x 2 + x + 2y 2 u(x, y)= x x 2 +y 2 u(x, y)= x x 2 +y C u(x, y)= x 2 2 x 2 +y 2 u(x, y)=ax 2 + bxy+ cy 2, a b c u a = c u f (z)=az 2 A 2 C A a b c f 0 G C f : G! C f 00 (z) 0 z 2 G f (z)=az+ b a, b 2 C f 0 (z)=a f (z) az

51 f (z) = az + b cz+ d a, b, c, d 2 C ad bc 6= 0 f z f (z)= az+b cz+d C \{ d c } c = 0 f c 6= 0 az+b cz+d = a c ad bc = 0 f (z)= az+b cz+d a c

52 a, b, c, d 2 C c 6= 0 f : C \{ d c }!C\{a c } f (z)= az+b cz+d f : C \{ a c }!C\{ d c } f (z) = dz b cz+ a. f (z) c = 0 f C f (z)= z iz+i C \{ } z iz+ i = w () z = iw+ iw+, f (z)= iz+ iz+ C \{ i} f f (z )= f (z 2 ) az + b cz + d = az 2 + b cz 2 + d, (az + b)(cz 2 + d )=(az 2 + b)(cz + d ) (ad bc)(z z 2 )=0. ad bc 6= 0 z = z 2 f g (z)= dz b cz+a f g f : C \{ d c }!C\{a c } f (z)= az+b cz+d f 0 (z) = a(cz+ d ) c(az + b) (cz+ d ) 2 = ad bc (cz+ d ) 2

53 f (z) =z + b f (z) =az f (z) = z f (z) = az+b cz+d c = 0 c 6= 0 f (z) = a d z + b d, f (z) = bc ad c 2 z + d c + a c. f (z)= z iz+i φ 2 R f (e iφ )= e iφ ie iφ + i = (e iφ )(e iφ + ) i e iφ + 2 = e iφ e iφ i e iφ + 2 = 2Im(e iφ ) e iφ + 2 = 2 sinφ e iφ + 2, f f (z)= z x 0 +iy 0 r (x x 0 ) 2 +(y y 0 ) 2 = r 2 α(x 2 + y 2 )+βx + γ y + δ = 0

54 α β γ δ β 2 + γ 2 > 4 αδ α = 0 z = x + iy u + iv := z u + iv = x iy x 2 + y, 2 x 0 = α + β x 2 + y + y 2 γ x 2 + y + δ 2 x 2 + y 2 = α + βu γv + δ(u 2 + v 2 ). u + iv f f : G! C lim z!z0 f (z)= M > 0 δ > 0 z 2 G 0 < z z 0 < δ f (z) > M lim z! f (z)=l ε > 0 N > 0 z 2 G z > N f (z) L < ε lim z! f (z)= M > 0 N > 0 z 2 G z > N f (z) > M z 0 G G B > 0 z 2 G z > B

55 lim z!0 z 2 = M > 0 δ := p M 0 < z < δ f (z) = az+b cz+d lim z! f (z)= a c f (z) = z 2 > δ 2 = M. c 6= 0 L := ad bc ε > 0 N := L c + d 2 ε c z > N cz+ d c z d c z d > L c ε f (z) a c = c(az + b) a(cz+ d ) c(cz+ d ) = L c cz+ d < ε. C ± R C lim z!z0 f (z)= lim z!z0 g (z)=a lim z!z0 ( f (z)+ g (z)) = Ĉ := C [ {} a 2 C + a = a + = a 6= 0 a = a = a = 0 = a 6= 0 a 0 =. CP +

56 z +( z)=0 z! 0 z z f (z) = z z = 0 z = f (0) = f () =0 f (z) Ĉ f (z) =z + b z = f () = + b = f (z) =az a 6= 0 z = f () =a = Ĉ Ĉ ad bc 6= 0 c 6= 0 f : Ĉ! Ĉ 8 az+b d >< cz+d z 2 C \ c, f (z) := z = d c >:, a c z =. f c = 0 f ()= f (z)= z iz+i f ( )= f ()= f Ĉ! Ĉ i f (z)= z iz+i f C \{ } z = f Ĉ Ĉ z = f C Ĉ

57 Ĉ C z f : Ĉ! Ĉ f (z)= iz+i 7! 0, i 7!, 7!. i Ĉ f R Ĉ z z z 2 z 3 Ĉ z z 2 z 3 [z, z, z 2, z 3 ] := (z z )(z 2 z 3 ) (z z 3 )(z 2 z ). [z 3, z, z 2, z 3 ]= z z z 2 z 3 [z,, z 2, z 3 ]= z 2 z 3 z z 3 f (z) = z iz+i f (z) = [z,, i, ] f : Ĉ! Ĉ f (z) =[z, z, z 2, z 3 ] f (z )=0, f (z 2 )=, f (z 3 )=. g g (z )=0 g (z 2 )= g (z 3 )= f g f h := g f

58 h(0)= g ( f (0)) = g (z )=0 h()= h()= h(z)= az+b cz+d 0 = h(0) = b d = h() = a c = h() = a + b c + d = a d = a d =) b = 0 =) c = 0 =) a = d h(z) = az + b cz+ d = az d = a d z = z, h = g f f g Ĉ 0 z z 2 z 3 w w 2 w 3 z z 2 z 3 Ĉ w w 2 w 3 Ĉ h h(z )=w h(z 2 )=w 2 h(z 3 )=w 3 h = g f f (z) =[z, z, z 2, z 3 ] g (w) =[w, w, w 2, w 3 ] h z j w j h f (z j )=w j a b c d C R 3

59 C (x, y) R 3 C = {(x, y, 0) 2 R 3 } x +iy S 2 := (x, y, z) 2 R 3 : x 2 + y 2 + z 2 =. {(x, y, 0) : x 2 + y 2 = } N :=(0, 0, ) S 2 S :=(0, 0, ) S 2 Ĉ N φ: S2! Ĉ P 2 S 2 \{N } z P N P C Q φ(p ) := Q φ(n ) := φ 8 < x z φ(x, y, z)=, y z, 0 z 6=, : z =.

60 φ ( p, q, 0) = φ ()=(0, 0, ) 2 p p 2 + q 2 +, 2q p 2 + q 2 +, p2 + q 2 p 2 + q 2 + P =(x, y, z) 2 S 2 \{N } N P r (t )=N + t (P N )=(0, 0, )+t [(x, y, z) (0, 0, )] = (tx, ty, + t (z )) t 2 R r (t ) C 0 t = z t r φ φ Q =(p, q, 0) 2 C P =(x, y, z) 2 S 2 φ(p )=Q P x 2 + y 2 + z 2 x = φ(p )=Q z = p y z = q p 2 + q 2 = x 2 + y 2 ( z) 2 = z 2 ( z) 2 = + z z. p 2 + q 2 = +z z z x = p( z) y = q( z) φ φ φ φ φ S 2 Ĉ γ S2 2φ(γ) φ(γ) C N 2 γ S 2 S 2 H (x 0, y 0, z 0 ) H k H H = (x, y, z) 2 R 3 : (x, y, z) (x 0, y 0, z 0 )=k = (x, y, z) 2 R 3 : xx 0 + yy 0 + zz 0 = k.

61 k (x 0, y 0, z 0 ) 2 S 2 0 apple k apple k < 0 (x 0, y 0, z 0 ) ( x 0, y 0, z 0 ) k > H \ S 2 =? H \S 2 ( p, q, 0) φ φ ( p, q, 0) H φ ( p, q, 0) (z 0 k) p 2 +(2x 0 ) p +(z 0 k)q 2 +(2y 0 )q = z 0 + k. z 0 k = 0 ( p, q) ( p, q) z 0 = k N 2 H φ z 0 k 6= 0 p + x q + y 2 0 = k 2 z 0 k z 0 k (z 0 k). 2 ( p, q) k < k = k > S 2 ( p, q) R 3 f (z)=e i θ z φ S 2 f (z) =rz f (z) =z + b f (z)= z S 2 (x, y, z) 2 S 2

62 φ f S 2 φ φ(x, y, z)=( x z, y z, 0) p + iq = x z + i y z. p 2 + q 2 = +z z x 2 + y 2 = ( + z)( z) f Å ã x z + i y z = z x + iy = ( z)(x iy) x 2 + y 2 = x + z i y + z. φ φ(x, y, z)= ( x z, y z, 0) (x, y, z) f (z) = z S2 (x, y, z) (x, y, z) x f (z)= z C S 2 φ f (z) = z φ f (z)= z (0, 0, ) φ C S 2 x N φ C S 2 R 3 S 2 C z N Ĉ S 2 S 2 N

63 e it = cos t + i sin t exp : C! C z = x + iy exp(z) := e x (cos y + i sin y) = e x e iy. z, z, z 2 2 C exp (z ) exp (z 2 ) = exp (z + z 2 ) exp(z) = exp ( z) exp (z) = exp (Re z) exp(z) 6= 0 exp (z + 2πi) = exp (z) d dz exp (z) = exp (z). 2πi exp exp f (z)=exp(z) f x = e x (cos y + i sin y) f y = e x ( sin y + i cos y). e x = P k 0 k! x k

64 C f x (z) = i f y (z) z 2 C f (z)=exp(z) f 0 (z) = f (z) =exp(z). x z = x 2 R exp(x) =e x (cos 0 + i sin 0) = e x. sin z := 2i (exp(iz) exp( iz)) cos z := 2 (exp(iz)+exp( iz)), tan z := sin z cos z = i exp(2iz) exp(2iz)+ cot z := cos z sin z = i exp(2iz)+ exp(2iz), exp(z)exp( z)=exp(0)= exp sin cos

65 z = x 2 R sin z = 2i (exp(ix) exp( ix)) = 2i (cos x + i sin x cos( x) i sin( x)) = sin x. z, z, z 2 2 C sin( z)= sin z cos( z)=cos z sin(z + 2π)=sin z tan(z + π)=tan z cos(z + 2π)=cos z cot(z + π)=cot z sin(z + π 2 )=cos z cos(z + π 2 )= sin z sin (z + z 2 ) = sin z cos z 2 + cos z sin z 2 cos (z + z 2 ) = cos z cos z 2 sin z sin z 2 cos 2 z + sin 2 z = cos 2 z sin 2 z = cos(2z) d d sin z = cos z dz cos z = sin z. dz sin(iy) y!± sinh z = 2 (exp(z) exp( z)) cosh z = 2 (exp(z)+exp( z)) tanh z = sinh z cosh z = exp(2z) coth z = cosh z exp(2z)+ sinh z = exp(2z)+ exp(2z). d dz sinh z = cosh z d cosh z = sinh z. dz

66 sinh(iz)=i sin z cosh(iz)=cos z. exp((z)) = z = (exp z). exp f (z)=(z)+ 2πi exp exp G : G! C exp( z)=z G z = re iφ z = u(z)+iv(z) exp( z) =e u e iv = re iφ = z, e u = r e iv = e iφ v = φ + 2πk k 2 x ln(x) u = ln z z = ln z + i z z z Arg z z 6= 0 ( π, π] z Log :

67 C \{0}!C Log(z) := ln z + i Arg(z). Log Log(2) =ln(2)+i Arg(2) =ln(2) Log(i) =ln()+i Arg(i) = πi 2 Log( 3) =ln(3)+i Arg( 3) =ln(3)+πi Log( i) =ln( p 2)+i Arg( i) = 2 ln(2) πi 4. Log C\R apple0 G G \{0} z 0 Log(z 0 ) 2πi (exp z)=z z = x + iy Log(exp z) =ln e x e iy + i Arg(e x e iy )=ln e x + i Arg(e iy )=x + i Arg(e iy ). z = x + iy y 2 ( π, π] z (exp z) 6= z ln(xy)=ln(x)+ln(y) C Log(i)+Log(i ) =i π 2 + ln p 2 + 3πi 4 = 5πi 2 ln Log(i(i )) = Log( i) = 2 ln 2 3πi 4.

68 arg z := z log z := ln z + i arg z. arg log exp(log z)=z log G G d dz (z) = z. exp H := {(z) : z 2 G} f : H! G f (z)=exp(z) g : G! H g (z)=(z) g f g 0 (z) = exp 0 ( z) = exp( z) = z. a, b 2 C a 6= 0 a b a b := exp(b Log(a)). a b log a b

69 f (z)=e z e e = lim n! + n n f (x)=e x a z e Arg e = 0 e z = exp(z Log(e)) = exp (z (ln e + i Arg(e))) = exp (z ln(e)) = exp (z). a b a b e z 6= exp(z) f (z)= az+b cz+d f (z)= dz b cz+a c = 0 f : C! C f (z) = az+b d f (z) β 2 + γ 2 > 4 αδ x + iy u + iv f (z) = z iz+i f z f (z)=z e 2πi =

70 f (z)= +z z z = f G f (G) f a 2 C a < f a (z) := z a az. f a (z) f a (z)= f a (z) f a (z) D[0, ] ñ a A = c ô b d 2 2 ad bc Ĉ T az+b A (z) = cz+d T A T B = T A B Ĉ a 2 C lim z!z0 f (z)= lim z!z0 g (z)=a lim z!z0 ( f (z)+ g (z)) =. lim z!z0 f (z)= lim z!z0 g (z)=a 6= 0 lim z!z0 ( f (z) g (z)) =. lim z!z0 f (z)=lim z!z0 g (z)= lim z!z0 ( f (z) g (z)) =. lim z!z0 f (z)= lim z!z0 g (z)=a lim z!z0 g (z) f (z) = 0. lim z!z0 f (z)=0 lim z!z0 g (z)=a 6= 0 lim z!z0 g (z) f (z) =.

71 c 0, c,...,c d 2 C lim + c d + c d 2 z! z z + + c 0 =. 2 z d f (z)= 2z z+2 z w 0 ±2 i x y x = y 2 0 az+b cz+d! 0, 2!, 3!! 0, + i!, 2! 0! i,!,! i. z k C [ + i,] γ γ γ 0 2

72 z < w = iz i z+ x > 0, y > 0 w = z 0 < x < w = z z i z+i {x +iy2 C : x + y > 0} u = u(x, y) v = v(x, y) u x v x u y v y f = u + iv f 0 (z) 2 5. Ĉ f (z)= z 2 2z+ f f 0!!!0 f!! i i! f x y = x y y = x {x + iy 2 C : x + y = 0} {z 2 C : z < } {x + iy2 C : x + y > 0} {x + iy2 C : x + y < 0},

73 a 2 R \{0} y = a i 2a 2a z z 2 z 3 Ĉ z z z 2 z 3 [z, z, z 2, z 3 ] φ φ φ φ φ (0, 0, ), (0, 0, ), (, 0, 0), (0,, 0), (,, 0) H x + y z = 0 H H \ S 2 φ Ĉ S 2 φ S 2 NS N S N sin(z)=sin(z) cos(z)=cos(z) z = x + iy sin z = sin x cosh y + i cos x sinh y cos z = cos x cosh y i sin x sinh y sin z π

74 exp(z) z = iy, 0 apple y apple 2π z = + iy, 0 apple y apple 2π {z = x + iy2 C :0apple x apple, 0 apple y apple 2π}. z = x + iy sin z 2 = sin 2 x + sinh 2 y = cosh 2 y cos z 2 = cos 2 x + sinh 2 y = cosh 2 y cos 2 x sin 2 x cos x = 0 cot z 2 = cosh2 y cosh 2 y apple. y cot z 2 apple sinh2 y + sinh 2 y = + sinh 2 y apple + sinh 2 apple 2. tan(iz)=i tanh(z) π {z 2 C : 2 < Re z < π 2 } x = A sin t y = B cos t x = A cosh t y = B sinh t

75 Log(2i) ( ) i Log( + i). x + iy e i π e π i i e sin(i) exp(log(3 + 4i)) ( + i) 2 p 3 ( i) Ä i+ p 2 ä 4. arg(z) = arg(z) Arg(z)= Arg(z) log(z 2 ) 2 log z z z 2 sin z z 3 + (z 2i + ) (z)=ln z + i (z) 0 apple (z) < 2π exp(z) (z 3) i i z 3. Log(z)= πi 2 Log(z)= 3πi 2

76 exp(z)=πi sin(z)=cosh(4) cos(z)=0 sinh(z)=0 exp(iz)=exp(i z) z 2 = + i. < z < e Log C \ R apple0. G H := {(z) : z 2 G} : G! H f (z) : H! G f (z)=exp(z) f H a z = a Re z a c 2 C \{0} f (z)=z c exp(b log a) b z n b exp y = x x = t, y = t t! t! f (z)=z 2 f f

77 T i 2i 0 T f (T ) f (z)=z 2 Q i 2i f (Q ) i 2 + 2i 2i z(t )=2 + it u + iv = f (z(t )) (u, v)

78 / 0 2 5π 6 π 3 0 π 3 5π 6 exp

79 R b a f (x) dx a b a b a, b 2 R g : [a, b]! C b a g (t ) dt := b a Re g (t ) dt + i b a Im g (t ) dt. R 2 γ γ γ(t ), a apple t apple b f γ f γ b f = f (z) dz := f (γ(t ))γ 0 (t ) dt. γ γ a

80 γ γ(t ), a apple t apple b [a, c ] [c, c 2 ] [c n, c n ] [c n, b] f γ γ f := c a f (γ(t ))γ 0 (t ) dt + c2 b f (γ(t ))γ 0 (t ) dt + + f (γ(t ))γ 0 (t ) dt. c c n f : C! C f (z)=z 2 + i γ 0 + i γ(t )=t + it, 0 apple t apple γ 0 (t )= + i f (γ(t )) = (t it) 2 γ f = 0 (t it) 2 ( + i) dt =( + i) 0 = (t 2 2it 2 t 2 ) dt 2i( + i) 3 = 2 ( i). 3 γ y = x i γ(t )=t + it 2, 0 apple t apple γ 0 (t )= + 2it f (γ(t )) = (t it) 2 = t 2 t 4 2it 3, γ f = 0 (t 2 t 4 2it 3 ) ( + 2it) dt = 0 = (t 2 + 3t 4 2it 5 ) dt 2i 6 = 4 5 i 3. γ γ 0 γ 2 +i γ (t )=t, 0 apple t apple γ 2 (t )=+it, 0 apple t apple c i

81 γ f = γ γ 2 f + f = 0 t 2 dt + = 3 + i ( 2it t 2 ) dt = 3 + i Å 2i 2 0 ã 3 0 = i. ( it) 2 idt γ γ(t ), a apple t apple b σ(t ), c apple t apple d σ γ [c, d ] [a, b] γ σ σ = γ τ γ(t ), a apple t apple b σ(t ), c apple t apple d γ d c f (σ(t ))σ 0 (t ) dt = b a f (γ(t )) γ 0 (t ) dt. γ(t )=e it, 0 apple t apple 2π, σ(t )=e 2πi sin(t ), 0 apple t apple π 2, R γ f γ f = i 2π 0 f e it e it dt γ f = 2πi π 2 0 f Ä e 2πi sin(t )ä e 2πi sin(t ) cos(t ) dt.

82 γ length(γ) := b a γ 0 (t ) dt γ(t ) a apple t apple b γ 0 + i γ(t )=t + it 0 apple t apple γ 0 (t )= + i length(γ) = 0 + i dt = 0 p 2 dt = p 2. γ γ(t )=e it 0 apple t apple 2π γ 0 (t )=ie it length(γ) = 2π 0 i e it dt = 2π 0 dt = 2π. γ f g γ c 2 C ( f + cg)= f + c g. γ γ γ γ γ(t ), a apple t apple b γ γ(t ) := γ(a + b t ), a apple t apple b γ f = γ f.

83 γ γ 2 γ γ 2 γ γ 2 γ 2 γ γ γ 2 γ γ 2 f = f + f. γ f apple max f (z) length(γ). z2γ γ γ s = a + b t f = γ = = b a b a a b f (γ(a + b t ))(γ(a + b t )) 0 dt f (γ(a + b t )) γ 0 (a + b t ) dt f (γ(s)) γ 0 (s) ds = b a f (γ(s)) γ 0 (s) ds = γ f. γ(t ) γ γ 2 γ [a, b ] γ 2 [a 2, b 2 ] 8 < γ γ(t ) := (t ) a apple t apple b, : γ 2 (t b + a 2 ) b apple t apple b + b 2 a 2,

84 [a, b + b 2 a 2 ] γ γ 2 s = t b + a 2 γ γ 2 f = b +b 2 a 2 a b f (γ(t ))γ 0 (t ) dt = f (γ(t ))γ 0 (t ) dt + a = f + f. γ γ 2 b +b 2 a 2 b f (γ(t ))γ 0 (t ) dt γ [a, b ] γ γ [b, b + a 2 ] γ 2 b 2 φ = Ä Arg R γ f ä R γ f = R γ f e iφ R γ f 2 R γ f = e iφ b = apple a b a γ f b = Re e iφ a Re f (γ(t ))e iφ γ 0 (t ) dt f (γ(t ))e iφ γ 0 (t ) dt = apple max aapplet appleb f (γ(t )) b a γ 0 (t ) dt f (γ(t ))γ 0 (t ) dt b a f (γ(t )) γ 0 (t ) dt = max f (z) length(γ). z2γ γ dz z w = 2πi, γ w 2 C 2πi γ r 2π = γ dz z w apple max z2γ z w length(γ) = r 2πr.

85 C F G C F 0 (z) = f (z) z 2 G F f G f G F (z) =z 2 f (z)=2z F f G C F (z)=z 2 + c c 2 C Å d dz 2i (exp(iz) exp( iz)) ã = (exp(iz)+exp( iz)), 2 F (z)=sin z f (z)=cos z C F (z) =Log(z) f (z) = z C \ R apple0 f C \{0} f G C γ G γ(t ) a apple t apple b f G F f G γ f = F (γ(b)) F (γ(a)). d dt F (γ(t )) = f (γ(t )) γ0 (t ) γ f = b a f (γ(t )) γ 0 (t ) dt = F (γ(b)) F (γ(a)).

86 F (z)= 2 z 2 f (z)=z C γ f = 2 ( + i) = i γ γ(a)=γ(b) G C γ G f G G γ f = 0. f : C \{0}!C f (z)= z R γ f = 2πi γ C \{0} f C \{0} G C z 0 2 G f : G! C R f = 0 γ G γ F : G! C F (z) := γ z f, γ z G z 0 z f G F F z 0 z F 0 (z)= f (z) z 2 G G C z 0 2 G f : G! C R γ f = 0 γ G R σ f σ G z 0 z 2 G σ σ 2

87 σ σ 2 f f = f = 0. σ 2 σ σ 2 F (z) := f σ γ z F (z + h) F (z) = γ z+h f γ z f = γ f γ G z z + h z C R = h γ F (z + h) h F (z) f (z) = h γ f (w) dw f (z) h = h γ dw γ ( f (w) f (z)) dw. h λ z z + h G F (z + h) F (z) f (z) = f (w) f (z)) dw = h h γ( ( f (w) f (z)) dw. h λ h! 0 ε > 0 δ > 0 w z < δ =) f (w) f (z) < ε f z δ h < δ ( f (w) f (z)) dw apple h λ h max f (w) f (z) length(λ) w2λ = max f (w) f (z) < ε. w2λ

88 G C z 0 2 G f : G! C R f = 0 γ G γ F : G! C F (z) := γ z f, γ z G z 0 z f G R γ 0 γ G C γ 0 (t ), 0 apple t apple γ (t ), 0 apple t apple γ 0 G γ h : [0, ] 2! G s, t 2 [0, ] h(t, 0)=γ 0 (t ), h(t, )=γ (t ), h(0, s)=h(, s). γ G γ 2 γ G γ 2 h(t, s) s h(t, s) t s 0

89 γ 0 γ (C \{0}) ±3 ± 3i h(t, s) := ( 8 + 8it 0 apple t apple 8, 2 8t + i >< 8 apple t apple 3 8, s)e 2πit + 3s + 4i( 2t ) 3 8 apple t apple 5 8, 8t 6 i 5 8 >: apple t apple 7 8, + 8i(t ) 7 8 apple t apple h(t, s) 6= 0 0 apple t, s apple (C \ {0}) (C \{0}) G G G

90 G C f G γ 0 γ G γ 0 G γ γ 0 f = γ f. f γ dz z = 2πi γ f (z)= z G = C\{0} γ G f 2πi f 0 G h γ 0 γ

91 h 8 h (t, s) 0 apple t apple t, >< h 2 (t, s) t apple t apple t 2, h(t, s)= >: h n (t, s) t n apple t apple, h j (t, s) 0 apple s apple γ s h(t, s), 0 apple t apple I : [0, ]! C I (s) := γ s f, I (0)= R γ 0 f I ()= R γ f I I (0)=I () d ds I (s) = = = = d ds f (h(t, s)) h t dt = f 0 (h(t, s)) h s f 0 (h(t, s)) h t t f (h(t, s)) h s 0 s h t + f (h(t, s)) 2 h s t h s + f (h(t, s)) 2 h t s dt. f (h(t, s)) h t dt dt dt h f 0 h h t = t i

92 d ds I (s) = 0 t f (h(t, s)) h dt s = f (h(, s)) h (, s) s h f (h(0, s)) (0, s) =0, s h(0, s)=h(, s) s G C γ G γ G γ G 0 (C \ R) G C f G γ γ G 0 f = 0. γ

93 Log G = C\R apple0 γ G γ Log(z) dz = 0. C f γ γ f = 0. γ z 2 dz 2z γ γ dz z 2 2z = 2 γ dz z 2 dz 2 γ z. f (z)= z 2 f C\{2} γ (C\{2}) 2πi γ dz z 2 2z = πi.

94 C [a, r ]={z 2 C : z a = r } D[a, r ]={z 2 C : z a < r } D[a, r ]={z 2 C : z a apple r } a 2 C r > 0 C [a, r ] f D[w, R] f (w) = f (z) 2πi C [w,r] z w dz. f (z) z C [w, R] f (w) f = u + iv D[w, R] f (w) = 2π f w + Re it dt, 2π 0 u(w) = 2π u w + Re it dt v(w) = 2π v w + Re it dt. 2π 0 2π 0 f G D[w, R] f (z) z w H := G \{w} 0 < r < R C [w, r ] H C [w, R],

95 C [w,r] f (z) dz 2πi f(w) = z w = apple C [w,r ] C [w,r ] max z2c [w,r ] = max z2c [w,r ] f (z) dz dz f (w) z w C [w,r ] z w f (z) z f (w) dz w f (z) z f (z) f (w) r f (w) length (C [w, r ]) w 2πr = 2π max f (z) f (w). z2c [w,r ] ε > 0 f w δ > 0 z w < δ f (z) f (w) < ε 2π. z 2 C [w, δ 2 ] r = δ 2 C [w,r] f (z) dz 2πi f(w) < ε. z w ε f (w) = 2π 2πi 0 f (w + Re it ) w + Re it w ireit dt = 2π 2π 0 f w + Re it dt, u(w)+iv(w) = 2π 2π 0 u w + Re it dt + i 2π 2π 0 v w + Re it dt. C [i,] dz z 2 +.

96 f (z)= z+i C \{ i} D[i,] C [i,] dz z 2 + = C [i,] z+i z i dz = 2πi f(i) =2πi 2i = π. C [w, R] γ w f G γ R D[w, R] G f (z) z w H := G \{w} γ H C [w, R] f (w) = 2πi γ f (z) z w dz. γ γ γ γ f (w) f (w) γ γ γ D[w, R] γ γ C [w, R] γ D[w, R]

97 γ G γ G γ C [w, R] f G γ w γ γ G 0 f (w) = 2πi γ f (z) z w dz. γ G G γ γ G 0 G γ D[w, R] γ G γ C [w, R] C [w, R] G 0 γ G 0 G γ w γ G g (z) = z w z 6= w g G γ G 0 R g (z) dz = 0 γ R > 0 D[w, R] γ C\{ w } γ C [w, R] R g (z) dz = 2πi γ γ dz z 2 + = π

98 γ i (C \{ i}) C [0,3] exp(z) z 2 2z dz C [0,3] exp(z) z 2 2z dz = exp(z) 2 C [0,3] z 2 dz exp(z) dz. 2 C [0,3] z f (z)=exp(z) γ C [0,3] exp(z) z 2 2z dz = 2 2πi exp(2) 2 2πi exp(0) = πi e 2. γ(t )=3t + i apple t apple γ(t )=i sin(t ) π apple t apple π γ(t )=i + e i πt 0 apple t apple γ(t )=t ie it 0 apple t apple 2π C [ + i,] i 2i C [0, 34] ± ± 2i

99 f (z)=z dz z R γ γ w 2 C r > 0 C [w,r ] dz z w = 2πi. C [0, 2] f (z)=z + z f (z)=z 2 2z + 3 f (z)= z 4 f (z)=xy R xdz R ydz R zdz R zdz γ γ γ γ z z x ± iy γ 0 γ = C [0, ] γ = C [a, r ] a 2 C i R exp(3z) dz γ γ i γ = C [0, 3] γ y = x 2 x = 0 x = R f f γ γ f (z)=z 2 γ(t )=t + it 2 0 apple t apple f (z)=z γ i f (z)=exp(z) γ 0 z 0 f (z)= z 2 γ i

100 f (z)=z + z γ γ(t ) 0 apple t apple Im γ(t ) > 0 γ(0)= 4 + i γ()=6 + 2i f (z)=sin(z) γ i π γ γ σ τ R f σ = γ τ σ f g G γ G γ(a) γ(b) γ fg 0 = f (γ(b))g (γ(b)) f (γ(a))g (γ(a)) γ f 0 g. I (k) := R 2π e ikt dt 2π 0 I (0)= I (k)=0 k I ( 2 ) R C [0,2] z 2 dz. R z n dz = 0 γ γ n 6= n γ n = m γ R z dz = 2mπi γ

101 z 0 2 C γ z 0 γ n γ (z z 0 ) n dz = 8 < 2πi n =, : 0 R γ z exp(z 2 ) dz = 0 γ F (z)= i i 2 Log(z + i) Re(z) > 0 F (z) arctan z 2 Log(z i) +z 2 γ 4i z + dz z 2 dz γ z γ dz sin 2 (z) dz z 2 + z γ γ z i dz γ (t )=e it π, 2 apple t apple π 2 γ z i dz γ 2 (t )=e it, π 2 apple t apple 3π 2 γ 2 ±3 ± 3i a 2 C γ 0 γ a γ 0 C\{a} γ G

102 γ G γ(t ), t 2 [0, ] τ [0, ] [0, ] γ G γ τ τ s (t )= s τ(t )+( s)t 0 apple s apple C C G C f G f 0 γ γ G 0 γ f (z) dz = γ (u + iv)(dx + idy)= γ udx vdy + i γ vdx+ udy a 2 C I (r ) := C [0,r ] dz z a. r < a r > a γ r C \ {a} p(z) z γ C R γ p = 0. C [0,2] dz z 3 + = 0 C [0, 2] C [0, r ] r > R dz C [0,r ] z 3 + r!

103 2π 0 dφ 2 + sinφ z = e iφ 0 < r < 2π 2π 0 r 2 2r cos(φ)+r 2 dφ =. P r (φ) := r 2 2r cos(φ)+r 2 f g G γ G f (z)= g (z) z 2 γ f (z)= g (z) z γ r 6=, 3 I (r ) := C [ 2i,r ] dz z 2 + C [0,r ] dz z 2 2z 8 r = r = 3 r = 5 r = 3

104 z 2 C [,2] 4 z dz 2 sin z dz z C [0,] C [0,2] C [0,4] exp(z) z(z 3) dz exp(z) z(z 3) dz f (z) = z 2 γ = C [, ] σ = C [, ] R f = R f γ σ γ 6 G σ G = C \{±} f G z w G G f G f 0 γ w γ γ G 0 g : [0, ]! C g (t ) := γ f (w + t (z z w w)) dz. g 0 = 0 f t (z + t (w z)) g (0) g () G

105 f 0 f 00 f G γ G w γ f 00 (w) f 0 (w) = 2πi γ f (z) (z w) 2 dz. f 00 (w) = πi γ f (z) (z w) 3 dz.

106 f (w + w) w f (w) = w 2πi = 2πi γ γ f (z) z (w + w) dz 2πi γ f (z) (z w w)(z w) dz. f (z) z w dz w! 0 f (w + w) w f (w) = 2πi f (z) 2πi γ (z w) dz 2 f (z) γ (z w w)(z w) = w 2πi γ f (z) dz (z w) 2 f (z) dz. (z w w)(z w) 2 w! 0 w! 0 γ length(γ) M := max z2γ f (z) δ > 0 D[w, δ] \ γ =? z w δ z γ z 2 γ f (z) (z w w)(z w) 2 apple f (z) ( z w w ) z w 2 apple M (δ w )δ 2, w! 0 f 0 f 00 f f

107 C [0,] sin(z) z 2 dz = 2πi d dz sin(z) z=0 = 2πi cos(0) =2πi. i γ 2 / s ^ ] s γ j C [0,2] dz z 2 (z ), γ γ 2

108 C [0,2] dz z 2 (z ) = C [0,] cos(z) z 3 γ = 2πi = 0. dz z 2 (z ) + γ 2 d dz dz = πi z z=0 + 2πi d 2 dz 2 cos(z) z=0 dz z 2 (z ) = γ 2 = 2πi z z dz + 2 ( ) 2 = πi ( cos(0)) = πi. γ 2 + 2πi z 2 z dz f G f 00 f 0 G f 0 f 00 f 000 f (n) n f f (n) f G f G f x y f G γ f = 0 γ G f G

109 F f G F G f G γ G γ G G C γ G 0 γ G D[a, r ] C \ R apple0 C\R apple0 C \{0} (C \ {0}) f f G C G f : C! C f (z)=exp(z 2 ) F C F f (z)= z C \{0} C \{0} f (z) = z C \ R apple0 Log(z)

110 f G R γ f γ G γ(a) γ(b) z 2 {z 2 C : z < 2} z 2 z p(z) d z d p(z) d a d R 2 a d z d apple p(z) apple 2 a d z d z z R p(z) d a d a d z d p(z) = a d z d + a d z d + a d 2 z d a z + a 0 = a d z d + a d a d z + a d 2 a d z a a d z d + a 0 a d z d. z! 2 2 z C

111 p p(z) 6= 0 z 2 C p(z) p(0) = 2πi C [0,R] R > 0 d p(z) a d R p(z) z dz, p(0) = 2π C [0,R] dz zp(z) apple 2π max z2c [0,R] zp(z) 2πR apple 2 a d R. d R R p(0) = 0 p z a a p a p(z) z a C R p(x)=2x 4 + 5x R p C p(x) = x 2 + 2x = (x + i)(x i) Ä p 2 x + p 3 i ääp 2 x p 3 i ä

112 f (z) applem z 2 C w 2 C C [w, R] R > 0 f f 0 (w) = 2πi C [w,r] f (z) dz apple (z w) 2 2π max z2c [w,r] f (z) (z w) 2 2πR = max z2c [w,r] f (z) R apple M R. R f 0 = 0 f p p(z) 6= 0 z 2 C f (z)= p(z) f! 0 z! f f p dx x 2 + = π. σ R [ R, R] R R γ R R R R R >

113 γ R R s s R σ R σ R dz z 2 + = π. R > R! γ R dz z 2 + apple max z2γ R z R! π = lim R! σ R dz z 2 + = lim R! [ R,R] πr apple max z2γ R dz z lim R! γ R πr = z 2 dz z 2 + = πr R 2 dx x 2 +. É ±4 ± 4i exp(z 2 ) exp(3z) dz z 3 (z πi) dz 2 É É

114 É sin(2z) dz (z π) 2 É exp(z)cos(z) (z π) 3 dz f 00 f 0 (w) f 00 (w) f 0 (w) f (z) f 00 (w) C [0, 3] Log(z 4i) exp z z 3 z 2 z C [0,2] cos z z i z 3 2 sin z (z )2 (z + 4)(z 2 + ) exp(2z) (z ) 2 (z 2) exp z dz w w 6= (z w) 2 f : D[0, ]! C f (z) := [0,] dw wz f D[0, ]

115 f : R! R f (x) := 8 < x 2 sin( x ) x 6= 0, : 0 x = 0 R f 0 f (z)=z 2 exp(sin z) C f (z) =exp( z ) f (x)=e x f C lim z! f (z) f lim z! f (z) =L R > 0 f (z) L < z > R f (z) < L + z > R f (z) z apple R p n > 0 c, z, z 2,...,z k j,..., j k j + + j k = n p(z) =c (z z ) j (z z 2 ) j2 (z z k ) j k, f f (z) apple p z z 2 C f f M > 0 f (z) f M z 2 C

116 f f (z)=u(z)+iv(z) M > 0 u(z) applem z 2 C f exp( f (z)) f a b f (z) applea z +b z 2 C f f : D[0, ]! D[0, ] z < f 0 (z) apple z. dx x 4 +. f (z) = exp(iz) z 2 + R > R f = π σ R e σ R [ R, R] γ R R R R exp(iz) apple z f (z) apple 2 z z 2 R lim R! f = 0 lim γ R R! R[ R,R] f = π e cos(x) x 2 + dx = π e. cos(x) x 4 + dx.

117 G C f G γ G \{0} G γ(t ) a apple t apple b γ f (z) dz = σ Å ã f z z dz 2 σ(t ) := γ(t ) a apple t apple b lim z!0 f z z = L H := 2 z g : H [{0}!C g (z) := 8 < f z z 2 z 2 H, : L z = 0. g H [{0} γ f = σ g. : z 2 G \{0} R g R f σ γ R γ z n dz n 6= R γ z n dz = 0 n 6=

118 G C u : G! R G G u xx + u yy = 0. u(x, y)=xy C u xx +u yy = 0+0 = 0 u(x, y)=e x cos(y) C u xx + u yy = e x cos(y) e x cos(y) =0.

119 f = u + iv G u v G u v u v u x = v y u y = v x G u xx + u yy = (u x ) x + u y y = v y x + ( v x ) y = v yx v xy = 0. v v G G u(x, y)=xy C Im( f )=u f (z) = 2 z 2 = 2 x 2 y 2 + ixy u(x, y)=e x cos(y) C Re( f )=u f (z) =exp(z) =e x cos(y)+ie x sin(y) G G

120 u G v G f = u + iv G v u f v = Im f g := u x iu y. g g f g u Re g = u x Im g = u y u Re g Im g (Re g ) x = u xx = u yy = (Im g ) y (Re g ) y = u xy = u yx = (Im g ) x. g G h g G G h h = a + ib g = h 0 = a x + ib x = a x ia y. g u x u x = a x u(x, y)=a(x, y)+c(y) c y g h 0 u y = a y u(x, y)=a(x, y)+c(x) c x c u(x, y)=a(x, y)+c f (z) := h(z)+c G u g

121 f f = u + iv f 0 = u x + iv x = u x iu y. u G u x g = u x iu y G G u(x, y)=xy g := u x iu y = y ix = iz h(z) = i 2 z 2 = xy i 2 x 2 y 2 u u v(x, y) := 2 x 2 y 2 C u C y v(x, y) := 0 u (x, t ) dt x x 0 u (t, 0) dt y u

122 u + iv v y (x, y)= u(x, y), x u v y 2 u (x, y)= x 0 x u u = (x, y) y 2 (x, t ) dt u y (x,0)= y (x,0) y 0 u y (x,0)= 2 u u (x, t ) dt t2 u (x, y). y y (x,0) u G z 0 2 G u (n) (z 0 ) n r > 0 D[z 0, r ] G D[z 0, r ] f D[z 0, r ] u = Re f D[z 0, r ] f D[z 0, r ] u f G D[z 0, r ] f

123 u G D[w, r ] G u(w) = 2π u w + re it dt. 2π 0 R D[w, r ] D[w, R] G D[w, R] f D[w, R] u = Re f D[w, R] f f (w) = 2π f w + re it dt. 2π 0 G C u : G! R w 2 G D[w, r ] G u(z) apple u(w) z 2 D[w, r ] u(z 0 ) < u(w) z 0 2 D[w, r ] u G G w G w z 0 u(z 0 ) < u(w) r := z 0 w u(w) = 2π u w + re it dt. 2π 0 u(w) z 0 = w + re it 0 0 apple t0 < 2π u(z 0 ) < u(w) u [t 0, t ] [0, 2π] u(w + re it ) < u(w) t 0 apple t apple t

124 u(w) = 2π 2π = 2π 0 Ç t0 0 u w + re it dt t 2π å u w + re it dt + u w + re it dt + u w + re it dt t 0 t u(w) u(w) < Ç t0 t 2π å u(w) dt + u(w) dt + u(w) dt 2π 0 t 0 t = u(w), u u u f G f G ln f (z) G G f (z) ln u G w D[w, r ] G z 2 D[w, r ] u(z) apple u(w) f G G

125 u G sup u(z) =max u(z) inf u(z) =min u(z) z2g z2 G z2g z2 G G G u G u G u G u G u(z) apple sup u(z) =max u(z) =0 z2g z2 G u(z) inf u(z) =min u(z) =0, z2g z2 G u v G u(z) =v(z) z 2 G u(z) =v(z) z 2 G u v G G u v G u G G u G u G u u û e iφ := u e iφ û re iφ := 2π 2π 0 u e it P r (φ t ) dt r <,

126 P r (φ) û u u(x, y) v(x, y) G c 2 R u(x, y)+ cv(x, y) G u(x, y)=e x sin y u C f Re( f )=u u(x, y)=ln x 2 + y 2 u C \{0} u C \{0} f G ln f (x, y) G u(x, y) R 2! R x u

127 v(x, y) x 3 + y 3 + iv(x, y) f G C H := { f (z) : z 2 G} u H u( f (z)) G u(r,φ) R 2! R u(r,φ) r u r + u rr + r 2 u φφ = 0. u(r,φ)=r 2 cos(2φ) C u(r,φ) r u u(r,φ) φ u u C u u(x, y) x y u x y P r (φ) = r 2 2r cos(φ)+r 2, 0 < r < u D[0, ] r < u re iφ = 2π 2π 0 u e it P r (φ t ) dt. u D[0, ] R 0 > u D[0, R 0 ] f a (z) = z a az,

128 a 2 C a < u( f a (z)) D[0, R ] D[0, ] u( f a (z)) w = 0 u(a) = u( f a (z)) dz. 2πi C [0,] z f a (z) a = re iφ u(a) = 2π u e it a 2 2π 0 e it a 2 dt. G D[a, r ] G R > r D[a, r ] D[a, R] G G = C R = r + w 2 C \ G M = w a K = D[a, M ] \ G K z 0 2 K f (z)= z a K R = z 0 a

129 C [2,3] exp(z) sin(z) dz, s s π π

130 0 π a n a(n) (a n ) n= (a n ) n (a n ) n n (a n ) L 2 C ε > 0 N n N a n L < ε (a n ) L lim a n! n = L. L (a n ) lim n N i n 0 = i n n n n! i n n = 0 ε > 0 N > ε = i n n = n apple N < ε.

131 (a n ) L 2 C ε > 0 N n a n L ε (a n = i n ) L 2 C ε = 2 Re(L) 0 N n N a n = a 4k+2 = i 4k+2 = k 0 a n L = + L > 2. Re(L) < 0 N n N a n = a 4k = i 4k = k > 0 a n L = L > > 2. (a n = i n ) (a n ) (b n ) c 2 C lim a n! n + c lim b n! n = lim (a n! n + cb n ) lim a n! n lim b n! n = lim (a n! n b n ) lim n! a n lim n! b n an = lim n! b n lim a n! n = lim a n! n+ lim n! b n 6= 0 f : G! C L := lim n! a n a n 2 G lim f (a n! n )= f (L).

132 R (a n ) a n+ n a n+ apple a n n a n (a n ) a n := n!. a n apple 3 (a n ) e := + lim n! a n. 0 apple r < lim n! r n = 0 (a n = r n ) 0 L := lim n! r n L = lim n! r n = lim n! r n+ = r lim n! r n = rl. ( r )L = 0 r 6= 0 L = 0

133 x N x n p(n) c 2 C c > p(n) lim = 0. n! c n c 2 C c n lim n! n! = 0. (a n ) a n = P n k= b k a n = P n k=0 b k (b k ) a n = P n k= b k a n = P n k=0 b k R

134 L lim a n! n = lim n! nx b k = L. ε > 0 N n N k= nx b k L < ε. k= L = X b k k X b k L = k= z 2 C z < P k z k X z k z = z. k nx z k = z + z z n = z z n+, z k= n! z < X nx Å k k 2 + k = lim n! k= k Å = lim n! Å = lim n! n + ã k + ã = n ã n +

135 b k 2 R 0 X b k k b k = k! nx b k = k= nx k= k! X k k! = e R b k c k 0 k X b k X k c k. k P n k= b k nx c k k= apple nx b k. k= P k c k X k b k lim n! b n = 0. lim b n! n 6= 0 lim b n! n X b k k z P k z k lim n! z n

136 X k b k 0 = lim n! nx b k k= X n lim b n! k k= Ç X n = lim n! n b k k= k= å X b k = lim n! b n. P k k 0 L L = > = = L, f (x) f (x) f () f (2) f (3) f (4) f () f (2) f (3) f (4) f (5) x x

137 f : [, )! R 0 f (t ) dt apple X k f (k) apple f ()+ f (t ) dt. f [k, k + ] f (k) f (k + ) n n n + f : [, )! R 0 Pk f (k) R f (t ) dt R f (t ) dt = P n k= f (k) P k f (k) R f (t ) dt P n k= f (k) P k f (k) P k k p > p < p p = dx x p a p+ = lim a! p + + p p > X k b k X b k k

138 P k b k b k 8 8 < b + k := b k b k 0, < b b : k := k b k < 0, 0 : 0 0 apple b + k apple b k 0 apple b k apple b k k X X b + k k b k k X b k = X b + k + X b k. k k k b k 2 C b k = c k + id k 0 apple c k apple b k k P k c k P k d k P k c k P k d k X b k = X c k + i X d k. k k k ζ(z) := X k k z Re(z) > X k k z = X k k Re(z) z ζ(z) ζ(z) C\{}

139 P k ( ) k+ k X ( ) k+ k k = = Å ã + 2 Å ã Å 5 ã + 6 2k 2k = 2k(2k ) apple (2k ) apple 2 k, 2 P ( ) k+ k k P k ( ) k+ k G C f n : G! C n f : G! C z 2 G ( f n ) lim f n! n (z) = f (z). ( f n ) f : G! C ε > 0 N z 2 G n N f n (z) f (z) < ε. H G ( f n ) H 0 < Re(z) < Re(z)= 2

140 8 9 G 8 ε > 0 8 z 2 G 9 N 8 n N f n (z) f (z) < ε, G 8 ε > 0 9 N 8 z 2 G 8 n N f n (z) f (z) < ε. N z N z 2 G f n : D[0, ]! C f n (z)=z n f : D[0, ]! C f (z)=0 z = 0 ε > 0 0 < z < N > ln(ε) ln z n N f n (z) f (z) = z n 0 = z n apple z N < ε. f n : D[0, 2 ]! C f n (z)=zn f : D[0, 2 ]! C f (z)=0 ε > 0 z < 2 N > ln(2) ln(ε) n N f n (z) f (z) = z n apple z N < 2 N < ε.

141 ( f n ) G z 0 2 G lim lim f n! z!z n (z) = lim 0 z!z0 lim f n! n (z). G C f n : G! C n ( f n ) f : G! C f z 0 2 G f z 0 ε > 0 N z 2 G n N f n (z) f (z) < ε 3. f n ε > 0 δ > 0 z z 0 < δ f n (z) f n (z 0 ) < ε 3. f (z) f (z 0 ) = f (z) f n (z)+ f n (z) f n (z 0 )+ f n (z 0 ) f (z 0 ) apple f (z) f n (z) + f n (z) f n (z 0 ) + f n (z 0 ) f (z 0 ) < ε. f z 0 f n : [0, ]! R f n (x)=x n f : [0, ]! R 8 < 0 0 apple x <, f (x)= : x =.

142 f n : G! C n ( f n ) f : G! C γ G lim n! γ f n = γ ε > 0 N z 2 G n N γ f. f n (z) f (z) < ε length(γ). γ f n γ f = γ f n f apple max f z2γ n (z) f (z) length(γ) < ε. P k f k (z) G γ G X f k (z) dz = X k γ k γ f k (z) dz. M f k : G! C k f k (z) applem k z 2 G P k M k P k f k P k f k G P k f k

143 z P k f k (z) ε > 0 N n N X nx M k M k = X M k < ε. k k= k>n z 2 G n N X f k (z) k nx f k (z) = X k>n k= f k (z) apple X f k (z) apple X M k < ε, k>n k>n f k f k P k f k P k z k z z < X z k = k z z. f k (z) =z k 0 < r < M k = r k f k (z) = z k apple r k z appler, P k r k P k z k z appler P k z k z < z appler r < z 0 X c k (z k 0 z 0 ) k

144 c 0, c, c 2,...2 C X z k = k 0 z. z 0 = 0 c k = k 0 z < z appler r < P k 0 z k z P k 0 c k (z R 0 R = z 0 )k P k 0 c k (z z 0 )k z z 0 < R P k 0 c k (z z 0 )k z z 0 appler r < R P k 0 c k (z z 0 )k z z 0 > R P k 0 c k (z z 0 )k z z 0 = R R P k 0 c k (z z 0 )k D[z 0, R] R = C P k 0 c k (w z 0 )k P k 0 c k (z z 0 )k z z 0 < w z 0 r := w z 0 P k 0 c k (w z 0 )k lim k! c k (w z 0 ) k = 0 c k (w z 0 ) k = c k r k apple M.

145 z z 0 < w z 0 X k 0 c k (z z 0 ) k = X k 0 z c k r k z0 k r apple M X z z0 k. r k 0 z P k 0 c k (z z 0 )k z 0 < r S := x 2 R 0 : X c k x k. k S S P k 0 c k (z z 0 )k z z 0 appler r R = x 2 S x > r P k 0 c k r k c k (z z 0 ) k apple c k r k S R R = 0 P k 0 c k (z z 0 )k z = z 0 R > 0 z z 0 < R R S r 2 S z z 0 < r apple R. P k 0 c k (w z 0 )k w = z 0 + r P k 0 c k (z z 0 )k z z 0 appler r < R x 2 S r < x apple R P k 0 c k r k P k 0 c k (z z 0 )k z z 0 appler z z 0 > R r /2 S R apple r < z z 0. P k 0 c k r k P k 0 c k r k P k 0 c k (z z 0 )k

146 k lim k! ck P k 0 c k (z z 0 )k 8 < k lim k! ck = 0, R = : lim k! k c k R R = R P k 0 c k (z z 0 )k z z 0 < R z z 0 > R k r := z z 0 < R lim k! ck = R 2 R+r > R N k ck < 2 R+r k N k c k (z z 0 ) k = c k z z 0 k = Ä k ck r ä k Å ã 2r k < R + r P k=n c k (z z 0 ) k 2r R+r < P 2r k k 0 R+r Pk 0 c k (z z 0 )k k r = z z 0 > R lim k! ck = R 2 R+r < R N k ck > 2 R+r k N k c k (z z 0 ) k = Ä k ck r ä k Å ã 2r k > >, R + r c k (z z 0 ) k P k 0 c k (z z 0 )k X kz k k 0 lim k! k ck = lim k! kp k = lim k! e k ln(k) = e lim k! ln(k) k = e 0 =,

147 X k 0 k! z k. lim k! c k+ c k = lim k! k! (k + )! = lim k! k + = 0, P k 0 k! z k C P k 0 c k (z z 0 )k R > 0 D[z 0, R] w 2 D[z 0, R] r < R w 2 D[z 0, r ] R 6= r = w z 0 +R 2 P k 0 c k (z z 0 )k D[z 0, r ] D[z 0, r ] w P k 0 c k (z z 0 )k R > 0 γ D[z 0, R] γ γ X c k (z z 0 ) k dz = X c k (z z 0 ) k dz. k 0 k 0 γ γ X c k (z z 0 ) k dz = 0. k 0 r := max z2γ γ(z) z 0 γ D[z 0, r ] r < R P k 0 c k (z z 0 )k D[z 0, r ]

148 a n = e πin 4 a n = cos(n) a n = sin( n ) a n = ( )n n a n = 2 in 2 2n 2 + X + i n p X + 2i n p n 3 n 5 X Å ã n n n i X n n 3 + i n X n n 2 + 2n. lim a n! n = a =) lim a n! n = a lim a n! n = 0 () lim a n! n = 0 lim n! a n M a n applem n (a n ) a ε > 0 N 2 >0 n > N a n a < ε

149 k! apple 3 k(k+) k n n! apple 3. (c n ) (Re c n ) (Im c n ) Q a n apple b n apple c n n lim n! a n = lim n! c n = L lim n! b n = L Re e 2πit : t 2 Q \. (c n ) P n 0 c n P k 0 (c 2k +c 2k+ ) (c n ) 0 X b k lim k n! X k n b k = 0.

150 f n (x) := sin n (x) X k X k k k 2 + k k 3 + G C f n : G! C n (a n ) R lim n! a n = 0 n f n (z) applea n z 2 G. ( f n ) G G C f n : G! C n ( f n ) G (z n ) G lim f n! n (z n )=0. (z n = e n )

151 f n : [0, π]! R f n (x)=sin n (x) n ( f n ) f : [0, π]! R 8 < x = π 2 f (x)=, : 0 x 6= π 2, (nz n ) z n n +nz Re(z) 0 f n (x)=n 2 xe nx lim n! f n (x) =0 x 0 x = 0 x > 0 n x R lim n! 0 f n (x) dx 0 z 0 z 0 + 4z 3 z 2 z 2 (4 z) 2 cos z cos(z 2 ) z 2 sin z (sin z) 2

152 (c k ) P k 0 c k (z z 0 )k (c k ) 0 P k 0 c k (z z 0 )k f (z)= z f (z)=log(z) M X k X k 0 z k D[0, ] k 2 z k {z 2 C : z 2} z 2 C r > z X k 0 z k X k 0 z k + r < z w D[0, r ] 0 apple k w w r R = c lim k+ k! c P k k 0 c k (z z 0 )k 8 < c lim k+ k! c = 0, R = k : lim k! c k c k+ X a k 2 z k a 2 C X k n z k k 0 k 0 n 2

153 X k 0 z k! X k z k k k X k ( ) k k z k(k+) X cos(k) z k k 0 X 4 k (z 2) k k 0 X k 0 z 2k k! X k (z ) k X k(k k k 2 ) z k f : D[0, ]! C f dw f (z) := [0,] wz. f n : R 0! R f n (t )= n e t n n f n (t ) n f n (t ) R 0 R 0 f n (t ) dt 0 n!

154 C [2,3] exp(z) sin(z) dz. exp(z) sin(z) π exp(z) sin(z) π f (z)= P k 0 c k (z f D[z 0, R] z 0 )k R > 0 f D[z 0, R] γ D[z 0, R] R f = 0 γ

155 f (z)= P k 0 c k (z z 0 )k R > 0 f 0 (z) = X kc k (z z 0 ) k z 2 D[z 0, R], k R z 2 D[z 0, R] z z 0 < R R z z 0 < R < R γ := C [z 0, R ] D[z 0, R] z γ f D[z 0, R] f 0 f 0 (z) = 2πi γ = X k 0 c k 2πi f (w) (w z) dw = 2 2πi = X kc k (z z 0 ) k. k γ (w z 0 ) k (w z) dw = X c 2 k k 0 γ X c (w z) 2 k (w k 0 d dw (w z 0 )k w=z z 0 ) k dw (z z 0 ) k f 0 (z) R f 0 z z 0 < R R f (z) (z z 0 ) f 0 (z) f (z) f (z) = X k 0 z k k!.

156 f C f (z)=exp(z) f 0 (z) = d dz X k 0 z k k! = X k z k (k )! = X k 0 z k k! = f (z). d dz f (z) exp(z) = d dz ( f (z)exp( z)) = f 0 (z)exp( z) f (z)exp( z) =0, f (z) exp(z) z = 0 f (z)=exp(z) exp(z) sin z = 2i (exp(iz) exp( iz)) = Ç å X (iz) k X ( iz) k 2i k 0 k! k 0 k! = X Ä (iz) k ( ) k (iz) kä = X 2(iz) k 2i k 0 k! 2i k 0 k! = X (iz) 2 j + = X i 2 j z 2j+ = X ( ) j i j 0 (2j + )! j 0 (2 j + )! j 0 (2 j + )! z 2 j + = z z 3 3! + z 5 5! z 7 7! +. f 0 f 00 f (z)= P k 0 c k (z z 0 )k c k = f (k) (z 0 ). k!

157 f (z 0 )=c 0 f 0 (z 0 )=c f 0 f 00 (z) = X k(k ) c k (z z 0 ) k 2 k 2 f 00 (z 0 )=2 c 2 f 000 (z 0 )=6 c 3 f 0000 (z 0 )= 24 c 4 f D f D P k 0 c k (z z 0 )k P k 0 d k (z z 0 )k z 0 c k = d k k 0 f (z) =exp(z) z 0 = π f (k) (z 0 )=exp(z) z=π = e π, X k 0 e π k! (z π)k, z 2 C f D[z 0, R] f z 0 R f (z) = X k 0 c k (z z 0 ) k c k = 2πi γ f (w) dw, (w z 0 ) k+ γ D[z 0, R] z 0 γ

158 g (z) := f (z + z 0 ) g D[0, R] z 2 D[0, R] r := z +R 2 g (z) = g (w) 2πi C [0,r ] w z dw. w z w 2 C [0, r ] z w < w z = w z w = w X k 0 z w k w 2 C [0, r ] g (z) = 2πi C [0,r ] f (z)= g (z f (z) = X k 0 g (w) w z dw = g (w) X z 2πi C [0,r ] w k 0 w = X k 0 2πi C [0,r ] k dw g (w) dw w k+ z 0 ) Ç å f (w) dw (z z 2πi C [z 0,r ] (w z 0 ) k+ 0 ) k. z k. C [z 0,r ] f (w) dw = (w z 0 ) k+ γ f (w) dw. (w z 0 ) k+ f : G! C f z 0 2 G R

159 G G C z 0 2 G z 0 G G { z z 0 : z 2 G} z 0 G f : G! C z 0 2 G f z 0 z 0 G f : C \{±i}!c f (z) := z 2 + z 0 = 0 f ±i f f (z) = z 2 + = X k 0 z 2 k X = ( ) k z 2k, k 0 R f G γ w γ γ G 0 f (k) (w) = k! 2πi γ f (z) dz. (z w) k+ f D[w, R] f (z) applem z 2 D[w, R] f (k) (w) apple k! M. R k

160 r < R f (k) (w) = k! f (z) dz 2πi C [w,r ] (z w) k+ apple k! 2π apple k! 2π max z2c [w,r ] f (z) length(c [w, r ]) (z w) k+ M k! M 2πr =. r k+ r k r R G z 0 2 G G f : G! C z 0 2 G R > 0 c 0, c, c 2,...2 C X c k (z z 0 ) k k 0 D[z 0, R] f (z) D[z 0, R] f z 0 f G f G G G G p(z) d > 0 a p(a) =0 p(z) z a p(z)=(z a) q(z) q(z) d q(z) a (z a)

161 p(z) p(z) =(z a) m g (z) m apple d g (z) a m a p(z) f : G! C f a 2 G f D a f (z)=0 z 2 D m g : G! C g (a) 6= 0 f (z) =(z a) m g (z) z 2 G. a D[a, r ] f m f a a R > 0 f (z) = X c k (z a) k z 2 D[a, R], k 0 c 0 = f (a)=0 c k = 0 k 0 m c k = 0 k < m c m 6= 0 f (z)=0 z 2 D[a, R] z 2 D[a, R] f (z) =c m (z a) m + c m+ (z a) m+ + =(z a) m (c m + c m+ (z a)+ ) =(z a) m X c k+m (z a) k. k 0

162 g : G! C 8X c k+m (z a) >< k z 2 D[a, R], k 0 g (z) := f (z) >: z 2 G \ {a}. (z a) m z 2 D[a, R]\{a} g D[a, R] g G \ {a} g (a)=c m 6= 0 f (z) =(z a) m g (z) z 2 G. g (a) 6= 0 r > 0 g (z) 6= 0 z 2 D[a, r ] D[a, r ] f m f a G f : G! C f (a n )=0 (a n ) G f G f g G f (a k )= g (a k ) w 2 G a k 6= w k f (z)= g (z) z G G X := {a 2 G : r f (z)=0 z 2 D[a, r ]} Y := {a 2 G : r f (z) 6= 0 z 2 D[a, r ] \{a}}. f (a) 6= 0 f a f a 2 Y

163 f (a)=0 a 2 X a f a 2 Y G X Y X Y G X Y lim n! a n Y X G = X u : G! R w D[w, r ] G z 2 D[w, r ] u(z) apple u(w) f G f G f G sup f (z) = max f (z). z2g z2 G f f z 0 2 G max z2g f (z) = f (z 0 ) sup z2g f (z) applemax z2g f (z) z 0 G z 0 z 0 62 G z 0 G f G f a G f (a)=0 u G G

164 a 2 G R > 0 f (a) f (z) z 2 D[a, R] f f (a) =0 f (z) =0 z 2 D[a, R] f f (a) 6= 0 g : G! C g (z) := f (z) f (a) g (z) apple g (a) = z 2 D[a, R], g (a) = g r apple R Re( g (z)) > 0 z 2 D[a, r ] h : D[a, r ]! C h(z) := Log( g (z)) h(a) =Log( g (a)) = Log() =0 Re(h(z)) = Re(Log( g (z))) = ln( g (z) ) apple ln() =0. h D[a, r ] g (z)= exp(h(z)) exp(0)= z 2 D[a, r ] f (z)= f (a) g (z) f (a) z 2 D[a, r ] f G exp( z ) Å ã exp z = X k 0 Å ã k k! z = X k 0 k! z k,

165 X a k := X a k + X a k. k2 k 0 k a k 2 C z 0 X c k (z z 0 ) k. k2 exp( z ) 0 c k = 0 k < 0 z X c k (z z 0 ) k = X c k (z z 0 ) k + X c k (z z 0 ) k. k2 k 0 k R 2 {z 2 C : z z 0 < R 2 } {z 2 C : z z 0 appler 2 } r 2 < R 2 z z 0 < R R {z 2 C : z z 0 r } r > R A := {z 2 C : R < z z 0 < R 2 }

166 R < R 2 {z 2 C : r apple z z 0 appler 2 } R < r < r 2 < R 2. sin(z) z 0 = 0 g : D[0, π]! C 8 < sin(z) z z 6= 0, g (z) := : 0 z = 0. g lim z!0 sin(z) z z = 6. 8 < cos(z) g 0 sin (z)= 2 (z) + z z 6= 0, 2 : 6 z = 0, g D[0, π] g g (z) = 6 z z z 5 + z < π 0 < z < π sin(z) = z + 6 z z z 5 + g D[0, π] sin(z) z D[0, π] \{0} ζ(z)= P k k z C \{}

167 f A := {z 2 C : R < z z 0 < R 2 }. f A z 0 f (z) = X c k (z z 0 ) k c k = f (w) dw, k2 2πi C [z 0,r ] (w z 0 ) k+ R < r < R 2 C [z 0, r ] γ A C [z 0, r ] γ 2 γ γ g (z)= f (z + z 0 ) g {z 2 C : R < z < R 2 } R < r < z < r 2 < R 2 γ γ := C [0, r ]