I = [a, b] R γ : I C γ(a) = γ(b) z C \ γ(i) 1(4) γ z winding number index Ind γ (z) = φ(b, z) φ(a, z) φ 1(1) (i)(ii) 1 1 c C \ {0} B(c; c ) L c z B(c;

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21 1 http://www.ozawa.phys.waseda.ac.jp/index2.html ( ) 1. I = [a, b] R γ : I C γ γ(i) z 0 C \ γ(i) (1) ε > 0 φ : I B(z 0 ; ε) C (i) B(z 0 ; ε) γ(i) = (ii) (t, z) I B(z 0 ; ε) exp(φ(t, z)) = γ(t) z (2) ε > 0 φ j : I B(z 0 ; ε) C, j = 1, 2, (i) B(z 0 ; ε) γ(i) = (ii) (t, z) I B(z 0 ; ε) exp(φ j (t, z)) = γ(t) z, j = 1, 2 n Z (t, z) I B(z 0 ; ε) φ 1 (t, z) φ 2 (t, z) = n (3) γ C 1 (1) (i)(ii) φ : I B(z 0 ; ε) C z B(z 0 ; ε) φ(, z) : I t φ(t, z) C C 1 t φ(t, z) = γ (t)/(γ(t) z) (4) γ(a) = γ(b) (1) (i)(ii) φ : I B(z 0 ; ε) C φ(b, z) φ(a, z) z B(z 0 ; ε) 1

I = [a, b] R γ : I C γ(a) = γ(b) z C \ γ(i) 1(4) γ z winding number index Ind γ (z) = φ(b, z) φ(a, z) φ 1(1) (i)(ii) 1 1 c C \ {0} B(c; c ) L c z B(c; c ) exp(l c (z)) = z ( ) l(z) = n=1 z n n B(0; 1) l : B(0; 1) z l(z) C B(0; 1) B(0; 1) l (z) = z n 1 = n=1 n=0 z n = 1 1 z d dz (exp(l(z))(1 z)) = exp(l(z))[l (z)(1 z) 1] = 0 z B(0; 1) exp(l(z))(1 z) = exp(l(0))(1 0) = exp(l(0)) = 1 1 z = exp( l(z)) c C \ {0} z B(c; c ) z c < c (z c)/c < 1 (c z)/c B(0, 1) z B(c; c ) l((c z)/c) exp( l((c z)/c)) = 1 (c z)/c = z/c z B(c; c ) z = c exp( l((c z)/c)) 2

c C \ {0} = exp(c) b C c = e b z = exp(b l((c z)/c)) ( ) c z L c (z) = b l = b c n=1 B(c; c ) L c B(c; c ) exp L c = id 1 n ( c z c ) n 1 (1) {z 0 }, γ(i) ε > 0 B(z 0 ; 2ε) γ(i) = γ a = t 0 < t 1 < < t n = b {t j } n j=0 j 1 t, s I j [t j 1, t j ] γ(t) γ(s) ε B 0 = B(z 0 ; ε) Γ : I B 0 C \ {0} Γ(t, z) = γ(t) z, (t, z) I B 0 ε t I Γ(t, z 0 ) = γ(t) z 0 2ε 1 j n j B j = B(Γ(t j 1, z 0 ); 2ε) z B j t I z Γ(t j, z 0 ) < 2ε Γ(t, z 0 ) z B(Γ(t j, z 0 ); Γ(t j, z 0 ) ) B j B(Γ(t j, z 0 ); Γ(t j, z 0 ) ) 0 t I B(Γ(t, z 0 ); Γ(t, z 0 ) ) 1 j L j : B j C z B j exp(l j (z)) = z 3 n j=1 B j

j 1 (t, z) I j B 0 Γ(t, z) Γ(t j 1, z 0 ) γ(t) γ(t j 1 ) + z z 0 < 2ε Γ(I j B 0 ) B j z B 0 Γ(t j, z) Γ([t j 1, t j ] B 0 ) Γ([t j, t j+1 ] B 0 ) = Γ(I j B 0 ) Γ(I j+1 B 0 ) B j B j+1 L j : B j C, L j+1 : B j+1 C exp(l j (Γ(t j, z))) = Γ(t j, z) = exp(l j+1 (Γ(t j, z))) z B 0 L j (Γ(t j, z)) L j+1 (Γ(t j, z)) Z B 0 z L j (Γ(t j, z)) C, B 0 z L j+1 (Γ(t j, z)) C B 0 N j Z z B 0 φ : I B 0 C L j (Γ(t j, z)) L j+1 (Γ(t j, z)) = N j φ I 1 B 0 = L 1 Γ, j 1 φ I j B 0 = L j Γ + N k, k=1 2 j n n n ([t 0, t 1 ) B 0 ) ((t j 1, t j ) B j ) ({t j 1 } B) j=2 {N j } n 1 j=1 φ : I B 0 C I j B 0 exp(φ(t, z)) = exp(l j (Γ(t, z))) = γ(t) z (2) φ 1 φ 2 : I B(z 0 ; ε) C (t, z) I B(z 0 ; ε) (2) exp(φ 1 (t, z) φ 2 (t, z)) = 1 (3) γ C 1 J z B(z 0 ; ε) t, t + h J h 0 j=2 φ(t + h, z) φ(t, z) h = γ(t + h) γ(t) h φ(t + h, z) φ(t, z) exp(φ(t + h, z)) exp(φ(t, z)) 4

h 0 φ φ(t + h, z) φ(t, z) w e w w = φ(t, z) 1/ exp(φ(t, z)) = 1/(γ(t) z) γ (t) t J t φ(t, z) t φ(t, z) = γ (t) γ(t) z J t φ(t, z) C C 1 (4) γ(a) = γ(b) exp(φ(a, z)) = γ(a) z = γ(b) z = exp(φ(b, z)) φ(b, z) φ(a, z) Z φ : I B(z 0 ; ε) C B 0 φ(b, z) φ(a, z) = N N Z (1) (i)(ii) ψ : I B(z 0 ; ε) C (2) (t, z) I B(z 0 ; ε) n ψ(t, z) φ(t, z) = n ψ(b, z) φ(b, z) = n = ψ(a, z) φ(a, z) ψ(b, z) ψ(a, z) = φ(b, z) φ(a, z) I = [a, b] γ : I C curve γ(a) γ(b) γ γ(a) = γ(b) γ closed curve, loop I L (I) L (I) = {γ C(I; C); γ(a) = γ(b)} L (I) γ L (I) γ = sup{ γ(t) ; t I} L (I) L (I) d(γ 1, γ 2 ) = γ 1 γ 2 2 I = [a, b] 5

(1) γ L (I) Ind γ : C \ γ(i) z Ind γ (z) Z C \ γ(i) C \ γ(i) Ind γ 0 (2) γ 0 L (I) z 0 C \ γ 0 (I) Ind (z 0 ) : γ Ind γ (z 0 ) γ 0 γ 0, z 0 δ > 0 Ind γ1 (z 0 ) = Ind γ0 (z 0 ) γ 1 γ 0 < δγ 1 L (I) z 0 C \ γ 1 (I) ( ) (1) 1 Ind γ γ(t) γ C \ B(0; γ ) = {z C; z > γ } C \ γ(i) C \ γ(i) C \ γ(i) B(0; γ ) C \ B(0; γ ) C \ γ(i) 1 C 1 B( 1; 1) L : B( 1; 1) C z C \ B(0; γ ) t I e w = z w C φ(t) = w + L(z 1 γ(t) 1) φ L (I) exp(φ(t)) = exp(w) exp(l(z 1 γ(t) 1)) = z(z 1 γ(t) 1) = γ(t) z Ind γ (z) = φ(b) φ(a) = L(z 1 γ(b) 1) L(z 1 γ(a) 1) = 0 Ind γ C \ B(0; γ ) 0 (2) {z 0 } γ 0 (I) = δ inf{ γ 0 (t) z 0 ; t I} > 0 γ 1 γ 0 < δ γ 1 L (I) t I (γ 0 (t) z 0 ) (γ 1 (t) z 0 ) γ 1 γ 0 < δ γ 0 (t) z 0 γ 1 (t) z 0 > 0 6

z 0 C \ γ 1 (I) γ : I C γ(t) = γ 1(t) z 0 γ 0 (t) z 0 γ 0, γ 1 L (I) γ L (I) γ(t) 1 = (γ 1 (t) z 0 ) (γ 0 (t) z 0 ) γ 0 (t) z 0 < 1 γ(i) B(1; 1) C \ B(1, 1) C \ γ(i) 0 (1) Ind γ (0) = 0 t I γ(t) = exp(φ(t)) γ 0 (t) z 0 = exp(φ 0 (t)) φ, φ 0 : I C t I Ind γ1 (z 0 ) (γ 0 (t) z 0 )γ(t) = exp(φ(t) + φ 0 (t)) (γ 0 (t) z 0 )γ(t) = γ 1 (t) z 0 Ind γ1 (z 0 ) = (φ(b) + φ 0(b)) (φ(a) + φ 0 (a)) = φ(b) φ(a) + φ 0(b) φ 0 (a) = Ind γ (0) + Ind γ0 (z 0 ) = Ind γ0 (z 0 ) 3 I = [a, b] γ : I C C 1 z C \ γ(i) Ind γ (z) = 1 1 γ ζ z dζ ( ) 1 (3) I a = t 0 < t 1 < < t n = b j 1 γ : [t j 1, t j ] C φ(, z) : [t j 1, t j ] C C 1 t [t j 1, t j ] t φ(t, z) = γ (t)/(γ(t) z) 7

1 γ = 1 1 ζ z dζ = 1 n j=1 tj n j=1 tj t j 1 t j 1 t φ(t, z)dt = 1 γ (t) γ(t) z dt n (φ(t j, z) φ(t j 1, z)) j=1 = 1 (φ(t n, z) φ(t 0, z)) = Ind γ (z) I = [a, b] U C γ 0, γ 1 L (I) (1) γ 0 γ 1 U (homologous in U) (a) γ 0 (I) γ 1 (I) U (b) C \ U Ind γ0 = Ind γ1 z C \ U Ind γ0 (z) = Ind γ1 (z) (2) U γ 0, γ 1 γ 1 γ 0 U 0 (a) γ 0 (I) U (b) z C \ U Ind γ0 (z) = 0 (3) γ 0 γ 1 U (homotopic in U) (a) γ 0 (I) γ 1 (I) U (b) H : [0, 1] I (θ, t) H(θ, t) C H([0, 1] I) U H(0, t) = γ 0 (t), t I H(1, t) = γ 1 (t), t I H(θ, a) = H(θ, b), θ [0, 1] (4) U γ 0, γ 1 γ 1 γ 0 U 0 (a) γ 0 (I) U (b) H : [0, 1] I (θ, t) H(θ, t) C z 0 U H([0, 1] I) U 8

H(0, t) = γ 0 (t), t I H(1, t) = z 0, t I H(θ, a) = H(θ, b), θ [0, 1] 4 I = [a, b], U C γ 0, γ 1 L (I) γ 0 γ 1 U γ 0 γ 1 U ( ) H : [0, 1] I (θ, t) H(θ, t) C H([0, 1] I) U H(0, t) = γ 0 (t), t I H(1, t) = γ 1 (t), t I H(θ, a) = H(θ, b), θ [0, 1] θ [0, 1] γ θ (t) = H(θ, t), t I γ θ : I t γ θ (t) C γ θ (a) = H(θ, a) = H(θ, b) = γ θ (b) γ θ L (I) H : [0, 1] I C sup γ θ γ θ = θ θ δ sup θ θ δ = sup θ θ δ sup γ θ (t) γ θ (t) t I sup H(θ, t) H(θ, t) t I sup H(θ, t) H(θ, t ) (θ,t) (θ,t ) δ δ 0 0 [0, 1] θ γ θ L (I) 2 (2) z C \ U [0, 1] θ Ind γθ (z) Z [0,1] Ind γθ (z) θ [0, 1] Ind γ0 (z) = Ind γ1 (z) z C \ U 5 γ 0, γ 1 L (I), I = [a, b] (1) γ 0 γ 1 C \ {0} (2) γ 0 γ 1 C \ {0} ( ) (1) (2) : 4 9

(2) (1) : I γ j (I) γ j (I) 0 1 φ j : I C t I exp(φ j (t)) = γ j (t) H(θ, t) = exp(θφ 1 (t) + (1 θ)φ 0 (t)) H : [0, 1] I C H([0, 1] I) γ 0 (I) γ 1 (I) C \ {0} H(0, t) = exp(φ 0 (t)) = γ 0 (t), t I H(1, t) = exp(φ 1 (t)) = γ 1 (t), t I θ [0, 1] H(θ, a) = H(θ, b) [θφ 1 (b) + (1 θ)φ 0 (b)] [θφ 1 (a) + (1 θ)φ 0 (a)] Z (2) [θφ 1 (b) + (1 θ)φ 0 (b)] [θφ 1 (a) + (1 θ)φ 0 (a)] = θ[φ 1 (b) φ 1 (a)] + (1 θ)[φ 0 (b) φ 0 (a)] = θ Ind γ1 (0) + (1 θ) Ind γ0 (0) = θind γ0 (0) + (1 θ)ind γ0 (0) = Ind γ0 (0) Z 6 U C U f : U C \ {0} (1) f U L : U C exp L = f (2) U γ Ind f γ (0) = 0 U L C exp f C \ {0} ( ) (1) (2) : I = [a, b] t γ(t) U γ(a) = γ(b) f γ : I t f(γ(t)) C 0 C \ (f γ)(i) t I exp((l γ)(t)) = (f γ)(t) 10

Ind f γ (0) = (f γ)(b) (f γ)(a) = 0 (2) (1) : U (U λ ) λ Λ U = λ Λ U λ (2) λ Λ U γ Ind f γ (0) = 0 (1) λ Λ f U λ exp L λ = f U λ L λ : U λ C ( λ Λ L U λ = L λ ) (2) (1) U U ( ) z 0 U f(z 0 ) C \ {0} = exp(c) w 0 C f(z 0 ) = e w 0 U z U γ z C([0, 1]; U) γ z (0) = z 0, γ z (1) = z f γ z : [0, 1] C 1 φ z : [0, 1] C t [0, 1] t = 0, 1 f(γ z (t)) = exp(φ z (t)) f(z 0 ) = f(γ z (0)) = exp(φ z (0)), f(z) = f(γ z (1)) = exp(φ z (1)) f(z 0 ) = exp(w 0 ) m(z) Z z U w 0 φ z (0) = m(z) L(z) = φ z (1) + m(z) exp(l(z)) = exp(φ z (1)) = f(γ z (1)) = f(z) L : U z L(z) C exp L = f L : U C (2) z 1 U z 1 L f z 1 B(f(z 1 ); f(z 1 ) ) = {w C; w f(z 1 ) < f(z 1 ) } f(z 1 ) r > 0 B(z 1 ; r) U, B(z 1 ; r/2) U, f(b(z 1 ; r)) B(f(z 1 ); f(z 1 ) ) f(b(z 1 ; r/2)) B(f(z 1 ); f(z 1 ) ) 11

1 l z1 : B(f(z 1 ); f(z 1 ) ) C n(z 1 ) Z exp(l z1 (f(z 1 ))) = f(z 1 ) = f(γ z1 (1)) = exp(φ z1 (1)) l z1 (f(z 1 )) φ z1 (1) = n(z 1 ) z B(z 1 ; r/2) γ z (1 3t), t [0, 1/3] γ(t) = γ z1 (3t 1), t [1/3, 2/3] (3 3t)z 1 + (3t 2)z, t [2/3, 1] φ z (1 3t), t [0, 1/3] φ(t) = φ z1 (3t 1) + (m(z 1 ) m(z)), t [1/3, 2/3] l z1 (f((3 3t)z 1 + (3t 2)z)) + (m(z 1 ) m(z) n(z 1 )), t [2/3, 1] γ L (I) γ z (0) = z 0 = γ z1 (0), γ z1 (1) = z 1, z = γ z (1) φ z (0) = w 0 m(z) = (φ z1 (0) + m(z 1 )) m(z) = φ z1 (0) + (m(z 1 ) m(z)), φ z1 (1) + (m(z 1 ) m(z)) = l z1 (f(z 1 )) n(z 1 ) + (m(z 1 ) m(z)) = l z1 (f(z 1 )) + (m(z 1 ) m(z) n(z 1 )) φ : [0, 1] C 0 C \ (f γ)([0, 1]) Ind f γ (0) = φ(1) φ(0) (1) z B(z 1 ; r/2) exp(φ(t)) = (f γ)(t), t [0, 1] = l z1 (f(z)) + (m(z 1 ) m(z) n(z 1 )) φ z (1) L(z) = φ z (1) + m(z) = l z1 (f(z)) + (m(z 1 ) n(z 1 )) 12

f : B(z 1 ; r/2) B(f(z 1 ), f(z 1 ) ) l z1 : B(f(z 1 ); f(z 1 ) ) C L : B(z 1 ; r/2) C 6 1. U C 0 U γ(i) U, I = [a, b] γ L (I) H C([0, 1] I; C) z 0 U H([0, 1] I) U H(0, t) = γ(t), t I H(1, t) = z 0, t I H(θ, a) = H(θ, b), θ [0, 1] f C(U; C \ {0}) L C(U; C) exp L = f ( ) h = f H h C([0, 1] I; C \ {0}) h([0, 1] I) f(u) C \ {0} h(0, t) = (f γ)(t), t I h(1, t) = f(z 0 ) C \ {0, }, t I h(θ, a) = f(h(θ, a)) = f(h(θ, b)) = h(θ, b), θ [0, 1] h C \ {0} f γ γ 0 : I t z 0 C \ {0} 4 Ind f γ (0) = Ind γ0 (0) Ind γ0 (0) = 0 Ind f γ (0) = 0 6 1 6 2. (1) K = {z C; Re z 1, Im z 1} f C(K; C \ {0}) L C(K; C) exp L = f (2) K = B(0; 1) = {z C; z 1} f C(K; C \ {0}) L C(K; C) exp L = f ( ) 13

(1) U = {z C; Re z < 1, Im z < 1} f C(K; C \ {0}) F K = f F C(C; C \ {0}) inf z C F (z) = inf z K f(z) = min z K f(z) > 0 F 1 (C \ {0}) C z K F (z) = f(z) C \ {0} z F 1 (F (z)) F 1 (C \ {0}) K F 1 (C \ {0}) K (C \ F 1 (C \ {0}) = δ d(k, C \ F 1 (C \ {0})) > 0 K (1 + δ/2)u F 1 (C \ {0}) (1+δ/2)U 0 γ L (I) H(θ, t) = (1 θ)γ(t), t I 6 1 L C((1 + δ/2)u; C) K (2) U = B(0; 1) = {z C; z < 1} (1) 7 K C f : K C \ {0} (1) f C \ {0} H : [0, 1] K (θ, z) H(θ, z) C w 0 C \ {0} H([0, 1] K) C \ {0} H(0, z) = f(z), z K H(1, z) = w 0, z K (2) f C f : C C \ {0} f K = f (3) f L : K C exp L = f ( ) 2 (X, d) K X F X K F K F K C \ {0} f C(K; C \ {0}), g C(F ; C \ {0}), H C([0, 1] K; C \ {0}) 14

H(0, x) = f(x), x K H(1, x) = g(x), x K f : K C \ {0} F f : F C \ {0} f K = f ( 2 ) [0, 1] X C = ([0, 1] K) ({1} F ) H : [0, 1] K C \ {0} H ({1} F ) = g C H : C C \ {0} Re H, Im H H : C C \ {0} H : [0, 1] X C \ {0} H(C) = H(C) C \ {0} H 1 (C \ {0}) C [0, 1] X [0, 1] K C H 1 (C \ {0}) [0, 1] X [0, 1] K [0, 1] X ([0, 1] X) \ H 1 (C \ {0}) dist([0, 1] K, ([0, 1] X) \ H 1 (C \ {0})) δ > 0 j 1 K j {x X; dist(x, K) 1/j} [0, 1] K = [0, 1] j 1 K j C H 1 (C \ {0}) j 2/δ [0, 1] K j H 1 (C \ {0}) x F dist(x, K) 1/j x K j {(min(1, jdist(x, K)), x) [0, 1] X; x F } ([0, 1] K j ) ({1} F ) H 1 (C \ {0}) x F f(x) = H(min(1, j dist(x, K)), x) F dist(,k) j R 1 R [0, 1] id F [0, 1] F H C \ {0} f f f f K = H ({0} K) = f 15

( 7 ) (1) (2) : 2 X = F = C, g : C z w 0 C \ {0} (2) (3) : C 0 γ L (I) H(θ, t) = (1 θ)γ(t), t I H γ 0 C 6 1 U = C, f C(C; C \ {0}) (3) (3) (1) : H(θ, z) = exp((1 θ)l(z)), θ [0, 1], z K H : [0, 1] K C f w 0 = 1 Robert B. Burckel, An Introduction to Classical Complex Analysis Vol.1, Pure and Applied Mathematics 82, Academic Press. [ ] H. 16

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