E F H0 A.F. Beardon Iterations of rational functions. Springer. Q Q Q Q H3/4 URL: kawahira/courses/5s-k

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1 E F H0 : Version :. Kawahira, Tomoki pdf 4 E F E F Q E Beltrami Beltrami 5 0 Beltrami 5 7 Beltrami (5 3 ) (6 7 ) Q F Fatou-Shishikura 7 5 Straightening 7 Sullivan 7 9 David 7 4 (7 6 ) (8 ) L.V. Ahlfors Lectures on quasiconformal mappings. AMS. L.V. URL: kawahira/courses/5s-kaiseki.html

html pdf 4 E F E F Q E 4 5 4 Beltrami 4 9 4 6 Beltrami 5 0 Beltrami 5 7 Beltrami 3 5 4 (5 3 ) (6 7 ) Q F

2 E F H0 A.F. Beardon Iterations of rational functions. Springer. Q Q Q Q H3/4 URL: kawahira/courses/5s-kaiseki.html

3 E F H0 Beltrami (4/5 4/6) : 06/05/7 Version :. 06/4/5 D f : D z = x + yi, w = f(z) = u + vi x, y, u, v f u = u(x, y), v = v(x, y) z 0 = x 0 + y 0 i D f x, y f x (z 0 ) := u x (x 0, y 0 ) + iv x (x 0, y 0 ), f y (z 0 ) := u y (x 0, y 0 ) + iv y (x 0, y 0 ) f(z) = f(z 0 ) + f x (z 0 )(x x 0 ) + f y (z 0 )(y y 0 ) + o(z z 0 ) (.) f z (z 0 ) := {f x(z 0 ) if y (z 0 )}, f z (z 0 ) := {f x(z 0 ) + if y (z 0 )}, z z (.) f(z) = f(z 0 ) + f z (z 0 )(z z 0 ) + f z (z 0 )(z z 0 ) + o(z z 0 ) (.) 3 z z z z f D z f z D 0 R 3 S S R 3 A = {z q : U q } q S f(z 0 + h) f(z 0) f(z 0 + hi) f(z 0) f x(z 0) := lim f y(z 0) := lim h 0,h R h h 0,h R h f z z x := x, y := y, z = z := ( x i y), z = z := ( x + i y) f z = zf = z f 3 dz = dx + dy i, dz = dx dy i df = f zdz + f zdz w = f(z) dw = w zdz + w zdz URL: kawahira/courses/6s-tokuron.html

0, y 0 ) f(z) = f(z 0 ) + f x (z 0 )(x x 0 ) + f y (z 0 )(y y 0 ) + o(z z 0 ) (.) f z (z 0 ) := {f x(z 0 ) if y (z 0 )}, f z (z 0 ) := {f x(z 0 ) + if y (z 0 )}, z z (.

4 E F H0 q S U q S q S R 3 A z q : U q z q (U q ) (local coordinate) U p U q (p, q S) z p (z q ) A S (altas) S A = {z q : U q } q S (S, A) U p U q (p, q S) z p (z q ). S q = (q, q, q 3 ) U = U q S = z = z q : U, p = (p, p, p 3 ) U, z = z(p) = x(p) + y(p)i 4 R 3 XY Z ds = dx + dy + dz 5 S ds S U x = x(p), y = y(p) ds = Edx + F dxdy + Gdy (.3) E = E(z) = E(x, y), F =, etc. E, G > 0 EG F > z = x+yi S p = (p (x, y), p (x, y), p 3(x, y)) Mathematica ParametricPlot3D 5 R 3 6 p S S URL: kawahira/courses/6s-tokuron.html

S q = (q, q, q 3 ) U = U q S = z = z q : U, p = (p, p, p 3 ) U, z = z(p) = x(p) + y(p)i 4 R 3 XY Z ds = dx + dy + dz 5 S ds S U x = x(p), y

5 E F H0 ( ) z q : U q (isothermal coordinate) E = E(z) = E(x, y) ds = E(dx + dy ) (= E dz ). A = {z q : U q } q S (S, A). q S A z : U, z = x + yi w : V, w = u + vi q U V E = E(z), Ẽ = Ẽ(w) ds = E dz = Ẽ dw w z dw = w z dz + w z dz w z w z > 0 w z > 0 E dz = Ẽ dw = Ẽ w zdz + w z dz = Ẽ w z dz + w z dz w z. z(u V ) E(z) = Ẽ(w) w z(z) > 0 w z (z) = 0 w z 7. ( ) S A z : U z(u) w : z(u) w z : U A = {z : U } Ã := {w z : U }. 8.3 () S R 3 S Ã S u, v R 3 z(u) R u = [Dz(p)](u), v = [Dz(p)](v) [Dz(p)] z : U R R 3, R 3 «u, v R 3 = t u E F v F G E, F, G p 7 8 S R 3 URL: kawahira/courses/6s-tokuron.html

dz = Ẽ w z dz + w z dz w z. z(u V ) E(z) = Ẽ(w) w z(z) > 0 w z (z) = 0 w z 7. ( ) S A z : U z(u) w : z(u) w z : U A = {z : U } Ã := {w z : U }. 8.

6 E F H0 06/4/ Beltrami. (.3). dz = dx + dy i, dz = dx dy i. ( ) z : U, z = x + yi ds (.3) Edx + F dxdy + Gdy = λ dz + µ dz λ = (E + G) + EG F 4 (E G) + F i µ = (E + G) + EG F. λ = λ(z) > 0, µ = µ(z) < z(u).. (Beltrami ) S A z : U z(u) w : z(u) (BE) w z (z) = µ(z) w z (z) µ = µ(z). w z : U (BE): w z (z) = µ(z)w z (z) Beltrami (Beltrami equation) 9. dw = w z dz + w z dz = w z dz + µ dz = w z Ẽ(w) := λ(z)/ w z(z) Ẽ ds = Ẽ( du + dv ) = Ẽ dw λ ds. Beltrami Beltrami Ω (measurable) µ : Ω D = {z z < } Beltrami f z = µf z f : Ω 9 w x w y URL: kawahira/courses/6s-tokuron.html

. (Beltrami ) S A z : U z(u) w : z(u) (BE) w z (z) = µ(z) w z (z) µ = µ(z). w z : U (BE): w z (z) = µ(z)w z (z) Beltrami (Beltrami equation) 9.

7 E F H0 Beltrami Beltrami. Ω f : Ω f(ω), w = f(z) α Ω z := z α f(α + z) = f(α) + f z (α) z + f z (α) z + o( z ) ( z 0) w := f(α + z) f(α) w f z (α) z + f z (α) z.: z = α f f α f(α) w = f(z) u + vi = f(x + yi) (x, y, u, v R) f (x, y) (u, v) w = f(z) α = a + bi J f (α) := u x (a, b)v y (a, b) u y (a, b)v x (a, b).3 ( Beltrami ) f : Ω f(ω) α Ω J f (α) = f z (α) f z (α) > 0 Ω f z (α) 0 f z (α) f z (α) < (Beltrami ) D µ f (α) := f z(α) f z (α) f α Beltrami (Beltrami coefficient) URL: kawahira/courses/6s-tokuron.html

8 解析学特論 E F 06 年前期プリント 0-6 担当教員川平友規研究室本館 H0 東京工業大学理学部数学科 Beltrami 係数の幾何学的意味点 α Ω を固定してそこでの Beltrami 係数 µ := µf (α) の幾何学的な意味を与えよういま全微分の式より w fz (α) z + µf (α) z が成り立つそこで z の長さは十分に小さいものとして z z + µ z = z + µ z が最大最小となる z の方向をもとめてみよう θ = arg z とおくと µ < および z/ z = より z z ( + µ ) z ( µ ) z + µ z = z + µ z が成り立つ右の等号は µ( z/ z) = µe θ > 0 のとき実現されるから最大値は arg µ + ( θ) = 0, π, arg z = arg µ arg µ + π, のときでありそのとき w fz (α) z ( + µ ) となる同様に左の等号は µ( z/ z) = µe θ < 0 のとき実現されるから最小値は arg µ + ( θ) = ±π, arg z = arg µ ± π のときでありそのとき w fz (α) z ( µ ) となる注意. 一般に向きを保つ写像 f は像の側にサイズが特定されない楕円による場いわば無限小楕円場 (a ﬁeld of inﬁnitesimal ellipses) を定める個々の無限小楕円は長径短径の方向と比によって一意的に定まりとくに長径と短径の比は Beltrami 係数 µf (z) によって表現されるのである URL: kawahira/courses/6s-tokuron.html

arg µ + π, のときでありそのとき w fz (α) z ( + µ ) となる同様に左の等号は µ( z/ z) = µe θ < 0 のとき実現されるから最小値は arg µ + ( θ) = ±π, arg z = arg µ ± π のときでありそのとき w fz (α) z ( µ ) となる注意.

9 E F H0 06/4/9 () f : Ω f(ω) α Ω µ f (α) := f z (α)/f z (α) D f α Beltrami f ( ) µ f (α) D K f (α) := + µ f (α) µ f (α) f α (dilatation) + µ f (z) K f := sup [, ] z Ω µ f (z) f (maximal dilatation) f K f K f f K f = K f (α) = ( α Ω) µ f (α) = 0 ( α Ω) f z (α) = 0 ( α Ω) f Ω K f (α) Beltrami z z a.e.= almost everywhere ( ) [a, b] R u : [a, b] R (absolutely continuous) ɛ > 0 δ > 0 a, a,..., a n b, b,..., b n a a < b a < b a n < b n b n b k a k < δ k= n u(b k ) u(a k ) < ɛ. k= 0 0 p39 URL: kawahira/courses/6s-tokuron.html

( α Ω) f Ω K f (α) Beltrami z z a.e.= almost everywhere ( ) [a, b] R u : [a, b] R (absolutely continuous) ɛ > 0 δ > 0 a, a,..., a n b, b,.

10 E F H0 3. ( ) u : [a, b] R a.e. x 0 [a, b] u(x 0 + x) u(x 0 ) u x (x 0 ) := lim x 0 x u(x 0 ) u(a) = x0 a u x (t) dt x 0 [a, b]. u x [a, b] u x u x : [a, b] R u n : [0, ] [0, ] : u 0, u, u, u 3, u 4 3 Λ Λ 0 Λ 0 δ Λ u 0 AL. u : [a, b] Re u(x), Im u(x) 3. a.e. x 0 [a, b] x x u x (x 0 ) = (Re u) x (x 0 ) + i(im u) x (x) a.e. x 0 [a, b] (AL) U f : U AL(absolutely continuous on lines) U x [a, b], y [c, d] Q = {x + yi x [a, b], y [c, d]} U x f(x, y 0 ) a.e. y 0 [c, d] y f(x 0, y) a.e. x 0 [a, b] URL: kawahira/courses/6s-tokuron.html

u : [a, b] Re u(x), Im u(x) 3. a.e. x 0 [a, b] x x u x (x 0 ) = (Re u) x (x 0 ) + i(im u) x (x) a.e. x 0 [a, b] (AL) U f : U AL(absolutely continuous on lines) U x [a, b], y [c, d] Q = {x + yi x [a, b], y [c, d]} U x f(x, y 0 ) a.

11 E F H0 3. f : Ω AL a.e. z 0 Ω f z (z 0 ) = {f x(z 0 ) if y (z 0 )}, f z (z 0 ) = {f x(z 0 ) + if y (z 0 )} (Gehring) (Lehto) 3. ( - ) f : Ω f(ω) a.e. z 0 Ω f z (z 0 ), f z (z 0 ) a.e. z 0 U f(z + z) = f(z) + f z (z) z + f z (z) z + o( z) ( z 0) 3.3 (AL Beltrami ) f : Ω AL a.e. α Ω Beltrami µ f (α) = f z (α)/f z (α) D a.e. α Ω f z (α) 0.3 L. Ω µ : Ω D 0 l < A(µ, l) := Area{z l µ(z) < } Area µ L µ = ess. sup µ(z) := z Ω inf {l A(µ, l) = 0} 0 l< ( ) f : Ω AL + µ f K f := sup [, ] z Ω µ f f (maximal dilatation) URL: kawahira/courses/6s-tokuron.html

e. α Ω Beltrami µ f (α) = f z (α)/f z (α) D a.e. α Ω f z (α) 0.3 L. Ω µ : Ω D 0 l < A(µ, l) := Area{z l µ(z) < } Area µ L µ = ess.

12 E F H0 06/4/6 () ( ) K < f : U V K- (K-quasiconformal mapping, K-qc) (A) (A) f AL K f K. K-qc AL a.e. a.e. K f K < K K f f.. f : Ω -qc K f : f(x + yi) = x + y i K-qc K. µ 0 D f : f(z) = z + µ 0z + µ 0 K 0 = ( + µ 0 )/( µ 0 )-qc Beltrami µ f (z) µ 0. f Beltrami 0. Ω = {z 0 < Im z < } f : Ω Ω K f (A) URL: kawahira/courses/6s-tokuron.html

µ 0 D f : f(z) = z + µ 0z + µ 0 K 0 = ( + µ 0 )/( µ 0 )-qc Beltrami µ f (z) µ 0. f Beltrami 0.

13 E F H0 / /3 /4. f : D, z z/( z ) K f =. Ω = {x + yi x, y [0, ]} f(x + yi) := x + (y + u(x))i AL µ f (z) = 0 K f = AL a.e ( ) f : Ω f(ω) = Ω K-qc (Q) f µ f (Q) Weyl -qc (Q3) f K-qc (Q4) g : Ω g(ω ) K -qc g f : Ω g(ω ) K K-qc. (Q5) f (Q6) f (/K)-Hölder (Q7) g : Ω g(ω ) µ g f = µ f. URL: kawahira/courses/6s-tokuron.html

14 E F H0 Beltrami (5/0-6/7) : 06/06/4 Version :. 06/5/0:Beltrami Ω u : Ω Lebesgue) f : Ω f(ω) p K u := ess. sup u(z) L - z Ω ( /p u p := u(z) dxdy) p : L p - L (Ω) := {u : Ω u u < }. L p (Ω) := { u : Ω u u p < }. 0 (Ω) := {u (Ω) supp(u) Ω} Ω = L p () L p f : Ω f(ω) () f h 0 (Ω) fh z dxdy = ϕh dxdy fh z dxdy = ψh dxdy. Ω Ω Ω ϕ, ψ L (Ω) f z = ϕ, f z = ψ (Sobolev ) f Sobolev W,p loc (Ω) Ω E f z p dxdy < f z p dxdy <. (5.) E p = f 5. ( ) f : Ω f(ω) K [, ) f K- (A) f W, loc (Ω) Ω E (A) K f K URL: kawahira/courses/6s-tokuron.html

0 (Ω) := {u (Ω) supp(u) Ω} Ω = L p () L p f : Ω f(ω) () f h 0 (Ω) fh z dxdy = ϕh dxdy fh z dxdy = ψh dxdy.

15 E F H0 Beltrami µ : D(D { z }) Beltrami f z = µf z 5. (Ahlfors-Bers Morrey) µ < µ : Ω f = f µ : f z = µf z, a.e.. f K-qc K = + µ µ. f(0) = 0, f() =. µ Ω µ µ(z) := { µ(z) (z Ω) 0 (z / Ω) f µ µ z t t µ t (z) ( ) t f µt (z) (P - ) p >, h L p P P h(ζ) := ( h(z) π z ζ ) dxdy. z 5.3 h L p (p > ) p K p ζ, ζ P h(ζ ) P h(ζ ) K p h p ζ ζ /p P h(ζ) ( /p)-hölder (T - ) h 0 () T auchy T h(ζ) := lim ϵ 0 ( π z ζ ϵ ) h(z) (z ζ) dxdy URL: kawahira/courses/6s-tokuron.html

16 E F H0 R > 0 supp(h) {z z ζ < R} (z ζ) dxdy = {ϵ z ζ R} = π R 0 π 0 ϵ e θi dθ r rdrdθ eθi R ϵ r dr = 0 ( ϵ ) T h(ζ) = π lim h(z) h(ζ) ϵ 0 {ϵ< z ζ <R} (z ζ) dxdy. ζ h 0 h R 0 D(ζ, R 0 ) = {z z ζ < R 0 } > 0 : z ζ < R = h(z) h(ζ) z ζ. D(ζ, R 0 ) h(z) h(ζ) (z ζ) ϵ 0 z ζ 5.4 (P - T -) h 0 T h( ) - () (P h) z = h () (P h) z = T h (3) T h = h 06/5/7 Beltrami 5.4 T 0 0 L 0 L T L P L T L p (p ) 6. (alderón-zygmund) p [, ) } () p := sup { T h p h 0 (), h p = <. () p p p p. URL: kawahira/courses/6s-tokuron.html

D(ζ, R 0 ) h(z) h(ζ) (z ζ) ϵ 0 z ζ 5.4 (P - T -) h 0 T h( ) - () (P h) z = h () (P h) z = T h (3) T h = h 06/5/7 Beltrami 5.

17 E F H0 6. T L p (p ) T 0 L p 6.3 h L p (p > ) = (P h) z = h, (P h) z = T h 0 L h L, h n 0 : h n h in L 6. T h T h n p p h h n p P (h h n )(ζ) K p h h n p ζ /p 5.4 ϕ 0 (P h n )ϕ z dxdy = h n ϕ dxdy n (P h)ϕ z dxdy = hϕ dxdy, (P h n )ϕ z dxdy = (T h n )ϕ dxdy n (P h)ϕ z dxdy = (T h)ϕ dxdy. µ (BE) k (BE) k (BE) k f z = µf z µ : D µ = k < 6.4 ( ) µ (BE) k f f(0) = 0, f z L p p p >, k p < f µ f z L p f z = µf z L p P f z := P (f z ) F := f P f z F z = f z (P (f z )) z = f z f z = 0 URL: kawahira/courses/6s-tokuron.html

4 ϕ 0 (P h n )ϕ z dxdy = h n ϕ dxdy n (P h)ϕ z dxdy = hϕ dxdy, (P h n )ϕ z dxdy = (T h n )ϕ dxdy n (P h)ϕ z dxdy = (T h)ϕ dxdy.

18 E F H0 F F = f z (P f z ) z = f z T f z F = (f z ) T f z L p L p = L p () 0 F F (0) = f(0) P f z (0) = 0 0 = 0 F (z) = z f(z) = P f z (z) z z- f z = T f z = T (µf z ). g g z = T (µg z ) = f z g z = T (µ(f z g z )) = f z g z p = T µ(f z g z ) p p µ(f z g z ) p k p f z g z p = f z = g z, a.e. = f z = g z, a.e. (BE) k = (f g) z = 0 (f g) z = 0, a.e. = f g f g = f g f(0) g(0) = 0 0 = 0 f = g 06/5/4:Beltrami (3) 6.4 h = T (µh) + T µ (7.) h 0 := T µ L p h L p T (h) := T (µh) T : L p L p (7.) h = T h + h 0 (I T )h = h 0 I T (7.) 6. () T h p = T (µh) p p µh p p µ h p k p h p k p < T kp <. (a) (I T ) : L p L p F F z 0 F URL: kawahira/courses/6s-tokuron.html

e. = f g f g = f g f(0) g(0) = 0 0 = 0 f = g 06/5/4:Beltrami (3) 6.4 h = T (µh) + T µ (7.) h 0 := T µ L p h L p T (h) := T (µh) T : L p L p (7.

19 E F H0 (b) (I T ) : L p L p (c) (I T ) T (I T ) = I + T + T + k p. h := (I T ) h 0 = (I + T + T + )T µ h (7.) = T µ + T µ(t µ) + T µ(t µ(t µ)) +. f(z) := P [µ(h + )](z) + z (BE) k µ(h + ) L p P [µ(h + )] f(z) := P [µ(h + )](z) + z 6.3 h (7.) f z = µ(h + ), f z = T (µ(h + )) + = (T µh + T µ) + = h + f z = µf z f (BE) k f f(0) = P [µ(h + )](0) + 0 = 0 f z = h + f z = h L p 6.4 ( 6.4) 7. () 6.4 µ f () f z p k p µ p () ζ, ζ f(ζ ) f(ζ ) K p k p µ p ζ ζ /p + ζ ζ. (): 6.4 h = ( I T ) h0 = ( I T ) T µ (c) h p f z = µ(h + ) k p p µ p f z p µh p + µ p µ h p + µ p k p k p µ p + µ p = k p µ p. URL: kawahira/courses/6s-tokuron.html

) f z = µ(h + ), f z = T (µ(h + )) + = (T µh + T µ) + = h + f z = µf z f (BE) k f f(0) = P [µ(h + )](0) + 0 = 0 f z = h + f z = h L p 6.4 ( 6.4) 7. () 6.

20 E F H0 () f(z) = P [µ(h + )](z) + z = P [f z ](z) + z f(ζ ) f(ζ ) P f z (ζ ) P f z (ζ ) + ζ ζ K p f z p ζ ζ /p + ζ ζ () K p k p µ p ζ ζ /p + ζ ζ. µ p >, k p < k (0, ) 7. ( µ ) µ, µ n : D (n =,, 3, ) µ n k, µ k. M µ n µ D(M) = { z < M} µ n µ 0 n. f n, f : µ n, µ f n f f z (f n ) z p 0 (n ) K- f n : f : µ fn µ f 0 (n ) ( K- f D(M) f z L p f(z) = z+o() ( z ) f n (z) = z + o() ( z ) f n f f(z) = P [µ(h + )](z) + z = P [µf z ](z) + z h L p f z = T [µf z ] + (f n ) z = T [µ n (f n ) z ] + f z (f n ) z p T [µf z µ n (f n ) z ] p f z (f n ) z p T [µf z µ n f z ] p + T [µ n f z µ n (f n ) z ] p p (µ µ n )f z p + k p f z (f n ) z p p k p (µ µ n )f z p. µ µ n D(M) µ n µ 0 f z (f n ) z p 0 f n (z) = P [µ n (f n ) z ](z) + z R > 0 ζ R f(ζ) f n (ζ) = P [µf z µ n (f n ) z ](ζ) K p µf z µ n (f n ) z p ζ /p K p R /p( µf z µ n f z p + µ n (f z (f n ) z ) p ) K p R /p( (µ µ n )f z p + k f z (f n ) z p ) 0 (n.) f n f URL: kawahira/courses/6s-tokuron.html

f n, f : µ n, µ f n f f z (f n ) z p 0 (n ) K- f n : f : µ fn µ f 0 (n ) ( K- f D(M) f z L p f(z) = z+o() ( z ) f n (z) = z + o() ( z ) f n f f(z) = P [µ(h + )](z) + z = P [µf z ](z) + z h L p f z =

21 E F H0 06/6/7:Beltrami (4) Beltrami (BE) k Beltrami 8. Beltrami (BE) k µ K := + k f K- k µ f 3. µ {µ n } n= 4 µ µ n 0 (n ). R > 0 µ n µ D(R) n µ n k.. µ n f n P - T - σ n (z) (f n ) z = e σn(z) 0 (f n ) z = µ n (f n ) z (f n ) z 3. f n µ n k (f n ) z (f n ) z = (f n ) z ( µ n ) f n f n K- 4. f n 7. K- K p µfn p µ n p /p ( k p ) ζ ζ ( k p ) +/p µ n p f n (ζ ) f n (ζ ) /p + f n (ζ ) f n (ζ ) kk p ( k p ) +/p f n(ζ ) f n (ζ ) /p + f n (ζ ) f n (ζ ). 5. f n (ζ ) = f n (ζ ) = ζ = ζ 7. µ n µ f n f n f(ζ ) = f(ζ ) = ζ = ζ 6. f f f 7. f z L p f z = (f z ) + L p loc. f z = µf z L p loc f W,p loc W, f K- loc 3 4 µ n(z) := (µ η n)(z) = µ( z)η n(z z)d xdỹ η n D(/n) D(/n) URL: kawahira/courses/6s-tokuron.html

22 E F H0 8. ((BE) k ) Beltrami (BE) k f µ : Ĉ Ĉ f µ + k k - f µ 0,, f µ f µ : 0 f µ Beltrami (BE) k (canonical solution) Beltrami µ (canonical qc mapping) ase. supp µ f f µ (ζ) := f(ζ) f() f µ (0) = 0, f µ () =, f µ (z) ( z ) ase. ϵ > D(ϵ) µ(z) = 0 ζ = τ(z) := /z τ(ζ) = /ζ µ(ζ) := (µ τ)(ζ) τ ( ) (ζ) ζ τ (ζ) = µ ζ supp µ ase f µ z = µf µ z 0,, f µ (ζ) f µ (z) := (τ f µ τ)(ζ) = { µ(z) (z D) ase 3. µ (z) := 0 (z D) µ (z) := ( µ µ µµ f µ z (f µ z ) ζ f µ (/z) ) (f µ ) (z) supp µ µ ase µ ase f µ f µ f µ (z) := f µ f µ (z) a.e. z f µ z = µ f z µ. (BE) k 0,, f, g µ f = µ g (= µ) a.e. ( ) µf µ g µ f g = gz g = 0 a.e. µ g µ f g z f g 0,, - URL: kawahira/courses/6s-tokuron.html

23 E F H0 ν L = L () 5 ϵ > 0 µ 0 L = L () µ 0 < (t, ζ) D(ϵ) L - {µ t (ζ) = µ(t, ζ)} t 0 ζ µ(t, ζ) = µ(0, ζ) + t ν(ζ) + o(t). o(t) µ t µ 0 tν t 0 (t 0) ϵ t D(ϵ) µ t µ 0 + t ( ν + o()) < Beltrami µ t f µt µ t µ 0 0 (t 0) 7. f µt (ζ) = f µ 0 (ζ) + o() (t 0) Beltrami t Beltrami t ζ V (ζ) f µt (ζ) = f µ 0 (ζ) + tv (ζ) + o(t) (t 0). yes 8.3 ( ) () µ 0 (ζ) = µ(0, ζ) 0 f µ 0 (ζ) = ζ ζ f µt (ζ) ζ V (ζ) = lim = ζ(ζ ) ν(z) t 0 t π z(z )(z ζ) dxdy. ζ(ζ ) () R(z, ζ) := z(z )(z ζ) µ 0 f µt (ζ) f µ0 (ζ) V (ζ) = lim = ν(z)r(f µ 0 (z), f µ 0 (ζ))(f µ 0 z (z)) dxdy t 0 t π 5 ν z ν(z) URL: kawahira/courses/6s-tokuron.html

I.2 z x, y i z = x + iy. x, y z (real part), (imaginary part), x = Re(z), y = Im(z). () i. (2) 2 z = x + iy, z 2 = x 2 + iy 2,, z ± z 2 = (x ± x 2 ) +

I.2 z x, y i z = x + iy. x, y z (real part), (imaginary part), x = Re(z), y = Im(z). () i. (2) 2 z = x + iy, z 2 = x 2 + iy 2,, z ± z 2 = (x ± x 2 ) + I..... z 2 x, y z = x + iy (i ). 2 (x, y). 2.,,.,,. (), ( 2 ),,. II ( ).. z, w = f(z). z f(z), w. z = x + iy, f(z) 2 x, y. f(z) u(x, y), v(x, y), w = f(x + iy) = u(x, y) + iv(x, y).,. 2. z z, w w. D, D.