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2 H D Ω C [1] p

3 Ω C 2 g(z) = 1/z g(z) = 1/z g(z) = 1/z z 0 g(z) = 1/z g(z) = 1/z 3

4 7 E M /R [1] Arsenity V.Akopyan Geometry of the Cardioid [2] 2 1. Ω 1, Ω 2 R 2 f : Ω 1 Ω 2 Ω 1 Γ 1, Γ 2 p Ω 1 Γ 1, Γ 2 p f(ω 1 ), f(ω 2 ) f(p). [3] p w = f(z) z 0 C df dz (z 0) 0 z 0 C 1, C 2 w = f(z) C 1, C 2 Γ 1, Γ 2 z 0 C 1, C 2 f(z 0 ) Γ 1, Γ 2 Proof. C 1, C 2 z 1, z 2 C. f(z 0 ) = w 0, f(z 1 ) = w 1 f(z 2 ) = w 2 f(z 1 ) f(z 0 ) z 1 z 0 = w 1 w 0 z 1 z 0 f(z 2 ) f(z 0 ) z 2 z 0 = w 2 w 0 z 1 z 0 4 = df dz (z 0) + ε 1 = df dz (z 0) + ε 2

5 z i z 0 ε i 0 { z1 z 0 = r 1 e iθ 1, (1) z 2 z 0 = r 2 e iθ 2, (2) { w1 w 0 = p 1 e iϕ 1, (3) w 2 w 0 = p 2 e iϕ 2. (4) p 1 e i(ϕ 1 θ 1 ) = df r 1 dz (z 0) + ε 1, p 2 e i(ϕ 2 θ 2 ) = df r 2 dz (z 0) + ε 2. z 1, z 2 z 0 ε 1, ε 2 0 p 1 r 1 e i(ϕ 1 θ 1 ) = p 2 r 2 e i(ϕ 2 θ 2 ) (z 1, z 2 z 0 ) lim z z 0 (ϕ 1 θ 1 ) = lim z z0 (ϕ 2 θ 2 ). (5) (5) α C 1, C 2 α Γ 1, Γ 2 f(z 0 ) z 0 C 1, C 2 f(z 0 ) Γ 1, Γ 2 df dz (z 0) Ω D = {z C : z < 1} D C Ω C Ω C Ω C F : Ω D 5

6 2. Ω C z 0 Ω F : Ω D F (z 0 ) = 0, F (z 0 ) > 0 [1] p H D H D H D H = {z C : Im(z) > 0}, D = {z C : z < 1}. F : H D F (i) = 0 F (i) 0 F (z) = i z i + z (6) F (i) = 0, F (i) = i 2 0 F G(w) = i 1 w 1 + w 1. (6) F H D Proof. F (z) z H i i i z < i + z F (z) < 1 F (z) H z D G(w) H 6

7 Im(G(w)) > 0 w = u + iv ( Im(G(w)) = Im i 1 u iv ) 1 + u + iv ( ) 1 u iv = Re 1 + u + iv ( ) (1 u iv)(1 + u iv) = Re (1 + u) 2 + v ( ) 2 1 u 2 v 2 2iv = Re (1 + u) 2 + v 2 = 1 u2 v 2 (1 + u) 2 + v 2 w < 1 u 2 + v 2 < 1 1 u 2 + v 2 > 0 1 u 2 v 2 (1 + u) 2 + v 2 > 0. G(w) H F (G(w)) = i i 1 w 1+w i + i 1 w 1+w G(F (z)) = i 1 i z i+z 1 + i z i+z = w, = z. F G (6) F H D H D 3 kardia 1 r = 1 cos,. D 7

8 1 r = 1 cos θ w = x + iy x + iy = (1 cos θ) cos θ + i(1 cos θ) sin θ 1: z = e iθ x + iy = (1 cos θ) cos θ + i(1 cos θ) sin θ = cos θ cos 2 θ + i(sin θ sin θ cos θ) ( ) cos 2θ + 1 = cos θ + i sin θ i( 1 sin 2θ) 2 2 = 1 2 cos 2θ i sin 2θ + cos θ + i sin θ 2 = 1 2 (ei2θ ) + e iθ 1 2 = z 1 2 z2 1 2 = 1 (z 1)2 2 8

9 f(z) f(z) = 1 2 (z 1)2 (7) 1. (7) f(z) Proof. z 1 u z 1 = 1 u = z u reiα 2: u v u v be iα r > b > (z 1)2 1 2 (reiα ) (beiα ) 2 2 r > b 1 2 (reiα ) 2 > 1 2 (beiα ) 2 u v 9

10 1 2 (z 1)2 z = 2iw 1/2 + 1 ( 1) 1/2 = i (7) f(z) 1 2 (z 1)2 f(z) 3 f(z) 3: f(z) 4 [2] W A 1 2 W P P W m 4 10

11 m A A m B B r = 1 cos θ 4: 5: θ π < θ < 3 π OB W Q 2 2 BQ OQ O B A A m BQOO BQ = AA = 1 OQ θ cos θ OB θ OB = OQ + QB = 1 + ( cos θ) = 1 cos θ θ 0 < θ < π 5 OB 2 W Q OB m G O AA H AH θ 1 cos θ 2 OG 1 1 cos θ 2 2 OB OG ( 1 OB = ) 2 cos θ = 1 cos θ 11

12 θ 3 π < θ < 2π B 2 OB = 1 cos θ (1 cos θ)e iθ B r = 1 cos θ B W W 6 6: 4.2 P BP 7 P P a C z a = BP = a P 8 12

13 7: BP 8: 2. {C a : z a = BP = a (a W )} Proof. BP 7 θ sin 1 θ P 2 a 7 W a = 1 2 (eiθ 1) 1 2 (eiθ 1) = 1 2 cos θ i1 2 sin θ z = x + iy (x 1 2 cos θ 1 2 )2 + (y sin θ) 2 sin θ = x 2 + y 2 + x x cos θ y sin θ 1 2 cos θ sin2 1 2 θ = x 2 + y 2 + (1 cos θ)x (1 cos θ) y sin θ sin2 1 2 θ = x 2 + y 2 + (1 cos θ)x y sin θ = 0 x cos θ + y sin θ x 2 y 2 x = 0 (8) 0 θ 2π (8) [4] 13

14 2 ([4], 2.2). f(x, y; θ) = 0 f(x, y; θ) = 0, θ f(x, y; θ) = 0 θ (8) (8) f = x sin θ y cos θ = 0 θ (9) (8)(9) θ ( ) ( ) ( ) ( ) x y cos θ x 2 + y 2 + x = = (x 2 + y x) y x sin θ 0 0. ( ) cos θ sin θ ( ) ( ) ( ) ( ) cos θ = x2 + y 2 + x x y 1 = x2 + y 2 + x x sin θ x 2 + y 2 y x 0 x 2 + y 2 y. x 2 +y 2 +x x 2 +y 2 H { cos θ = Hx, (10) cos 2 θ + sin 2 θ = 1 sin θ = Hy. (11) H 2 (x 2 + y 2 ) = 1, (x 2 + y 2 + x) 2 = x 2 + y 2, (x 2 + y 2 )(x 2 + y 2 + 2x) y 2 = 0. (12) (12) r = 1 cos θ x, y 14

15 9: 10: [2] p.146 [2] 5.1 g(z) : C C g(z) = 1/z w = 1/ z g(z) = 1/z 1 2 (z 1)2 v = 2 (z 1) 2 (13) 3. z C v v = 2 (z 1) 2 15

16 Proof. z z = e iθ (13) v = 2 (z 1) 2 = = = = 2 (e iθ 1) = 2 2 (cos θ + i sin θ 1) 2 2 cos 2 θ sin 2 θ + 2i sin θ cos θ 2 cos θ 2i sin θ e i2θ 2e iθ + 1 = 2e iθ e iθ 2 + e = 2e iθ iθ 2 cos θ 2 e iθ 1 cos θ. e iθ (14) 1 cos θ (14) 11 Re(z) = 1 11 I, D V 11: V

17 w = 1/ z g(z) = 1/z 12: 5.2 [3] 4 ( ). f(z) = az + b cz + d (z, w C) C C C 1. b/c C C 2. b/c C C 4.2 C a 17

18 5. {C a : z a = a } g(z) = 1/z Proof. g(z) = 1/z g(z) = 1/z θ P z a = a P 7 g(z) : 14: 6 g(z) = 1/z 6. M, N C a M, N C 1, C 2 C 1, C 2 W W L 1,, 2 15 P, P C 1, C 2 P C 1, C 2 W 5.2 C a 18

19 15: M, N M, N [5] 6.2 Proof. Re(z) = 1 M, N 16 M, N l 1, l 2 l 1, l 2 g(z) l 1, l 2 g(z) z 0 z 0 l 1, l 2 L 1, L 2 L 1, L 2 Re(z) = 1 g(z) = 1/z W W 19

20 16: 17: W P, P W 19 L 1 L 1, L 2 18: 19: L 1 20

21 7 Ω C x, y g(z) = 1/z g(z) = 1/z 19 L 1, L 2 L 1, L 2 21

22 [1] E M /R [2] Arsenity V.Akopyan,Geometry of the Cardioid,Amer.Math Monthly,vol122,pp ,MAA,2015. [3] [4] [5]