Workbook E-mail: kawahira@math.nagoya-u.ac.jp 2004 8 9, 10, 11 1 2 1 2 a n+1 = pa n + q x = px + q a n better 2 a n+1 = aan+b ca n+d 1 (a, b, c, d) =(p, q, 0, 1) 1 = 0 3 2 2 2 f(z) =z 2 + c a n+1 = a 2 n + c ver. 20040818 1
1 1.1 (numerical sequence) a 1,a 2,a 3,..., a 1000,a 1001,..., a 10000,a 10001,... a n = n 1, 2, 3,..., 1000, 1001,..., 10000, 10001,... a n =(n ) 2, 3, 5, 7, 11,... a n =(n ) n a n ( general term formula) a n = n 3 6n 2 +12n 6 a 1 =1, a 2 =2, a 3 =3 a 4 4 n 1000 a n = n n>1000 a n n a n =(n ) 1.2 2 (recursive formula) a n+1 a n a n+1 = a n +1; a 1 =0 a n+1 =5a n ; a 1 = π a n+1 =2a n +1; a 1 =1 a n+1 =(n +1)a n ; a 1 =1 a n+1 = a 2 n 2a n ; a 1 =1 2
1.3 2 a n+1 = pa n + q ; a 1 = a p =1 p 1 Step 1 x = px + q x = α := q 1 p Step 2 α = pα + q a n+1 α = p(a n α). Step 3 b n = a n α Step 2 b n+1 = pb n ; b 1 = a 1 α = a α. Step 4 {b n } b 1 = a α p b n =(a α)p n 1. Step 5 a n = b n + α =(a α)p n 1 + α. (1) a n+1 =2a n +1; a 1 =1 (2) a n+1 = a n 4 +3; a 1 =4 1.4 a n+1 =2a n +1; a 1 =1 3
b n+1 =2b n ; b 1 =2 b 0,b 1,b 2,... B a n b n = a n +1 a 0,a 1,a 2,... A A B B b n b n+1 =2b n ; b 1 =2 P B n P b n P 1 b n+1 =2b n 1 P b 1 =2 B B..., 2, 1, 0, 1, 2, 3, 4,... P X 1 g(x) =2X B g(x) =2X (dynamical system) P 1 X = 3 P 1 3 3 2 3 2 2 3 2 3... X = 3 g(x) (orbit) 1. a n b n 4
2. A B A x 3. A f(x) 4. A B A B A B P P A B a n+1 = pa n + q ; a 1 = a A f(x) x = px + q 5
1.5 R 1, 2, 3,... 0 y = f(x) x 1 f(x) R f (dynamics) x x f(x) f(f(x)) f(f(f(x)))... x f (orbit) f(f(f(x))) f 3 (x) f(x) 3 {f(x)} 3 x (initial value) f x = f(x) ( fixed point) f(x) =2x+1 x = 1 n f n (x) ={f(x)} n p y = f(x) p f(p) f(f(p)) f(f(f(p)))... y = f(x) xy (p, p) (f(p),f(p)) Step 1 (p, p) y = f(x) (p, f(p)) Step 2 y = x (f(p),f(p)) y = x p (graphical analysis) web diagram 6
y = f(x) y = x y f(p) y = x p y = f(x) O p f(p) x y =2x, y =2x +1 p 1.6 a n+1 = 3a n +2 a n +2 ; a 1 = 1 (1.0) 3x +2 x = x = 1, 2 x = 1 x +2 a n+1 ( 1) = 3a n +2 a n +2 ( 1) a n+1 +1= an +1 a n +2 x =2 (1.1) (1.2) a n+1 2= 3a n +2 a n +2 2 a n+1 2= an 2 a n +2 a n+1 +1 a n+1 2 = an +1 a n 2 7 (1.1) (1.2)
b n = {(a n +1)/(a n 2)} b 1 = (a 1 +1)/(a 1 2) = 2 b n = a n +1 a n 2 = a n =. 8
2 a n+1 = 3a n +2 a n +2 ; a 1 = 1 (1.0) 3x +2 A x f(x) = x +2 b n B X g(x) =4X A B a n+1 =2a n +1 A B 2 2.1 R + ˆR R xy x R (X, 0) X (0, 1 2 ) 1 2 C (0, 1) N N (X, 0) C N P X. C N C N R X X = 100000 00 1 P N X 100000 00 P N N 9
y C N P O X x R 1: ˆR N (point at infinity) C C N=R C = R ˆR 1 X C (x, y) X C (x, y) X ˆR C = ˆR 0 1 2 3 4 R C = ˆR ˆR x + x = x + = 1 (extended reals) Cf. C = R { } 10
x X X + x, x + X, X C N= X 0 x x = x = x =0 x 0 =, ±, /, 0. x = (1) x +1 x 2 x +1 x 2 = 1+ 1 x 1 2 x (3) 1 x (2) 3x +1 (4) x2 1 x 2 4 T (x) = x +1 xy x 2 T (x) ˆR ˆR 11
a, b, c, d ad bc 0 ˆR T (x) = ax + b cx + d 2 T,Möbius transformation with real coefficients T R T (x) = x = 1 x +0 0 x +1 ad bc =0 1 A= R f(x) =2x +1 B= R g(x) =2X ˆR X = T (x) =x +1 2 ˆR T 12
x ˆR Â X ˆR ˆB +2 Â f(x) =3x x +2 ˆB g(x) =4X X = T (x) = x +1 x 2 P ˆR P P f g P f(x) =x +1 g(x) =4x f(x) = ax + b cx + d (ad bc 0) x = f(x) x = ax + b cx + d (a, b, c, d) =(1, 0, 0, 1) ˆR (*) 13
c =0 ad bc = ad 0 d 0 f(x) = ax + b d (*) =: px + q x = f(x) x = px + q 2 ˆR 3 x = x = x = c 0 q 1 p x = ax + b cx + d cx2 +(d a)x b =0 2 (*) 3 Case 1 ˆR Case 2 ˆR Case 3 ˆR 2 3 p =1 Case 1 Case 2 Case 1 g(x) =rx (r 0, 1) 3x +2 r f(x) = x +2 Case 2 g(x) =X + d (d 0) 3 R 14
d f(x) = 3x +1 x +5 Case 3 (*) ˆR (*) x = x +1 x +1 f(x) = x +1 x +1 x 2 = 1 (1) R a n+1 = a n +1 a n +1 ; a 1 = a ˆR a 4 1 7 Case 3, f ˆR ˆR 15
2 x cos θ + sin θ 1 8 y = (θ =2π/5) x sin θ + cos θ sin, cos 1. 5 2. 5 ˆR 16
3 2 3.1 xyz xy C (X, Y, 0) Z = X + Yi (0, 0, 1) 2 1 S N :(0, 0, 1) Z 2 S P P Z S N C S N=C 2 Z P N N S = C { } (Riemann sphere) Ĉ S N P Z C 2: Ĉ a n =(2i) n n = 0, 1, 2,... {a n } = {1, 2i, 4, 8i, 16, 32i,...}. Z = a n S = Ĉ P N 3 a n =( i/2) n S (x, y, z) Z x, y, z Z = X+Yi S Z 17
i Ri z =1 i 1 0 1 1 0 1 R i i 3: Z =(x + yi)/(1 z),x= X/(1 + Z 2 ), y = Y/(1 + Z 2 ),z = Z 2 /(1 + Z 2 ) 3.2 a, b, c, d ad bc 0 Ĉ T (z) = az + b cz + d Ĉ Ĉ a, b, c, d ˆR ˆR f(z) = az + b cz + d 18 (ad bc 0)
z = f(z) z = az + b cz + d (a, b, c, d) (1, 0, 0, 1) (*) 3 Case 1 ˆR Case 2 ˆR Case 3 ˆR 2 f (*) 4: Case 1 Case 3 (*) (*) ˆR f(z) = z +1 z +1 z = z +1 z +1 z 2 = 1 z = ±i. (2) 2 ±i f f Z = T (z) = z i z + i i, i f 19
5: Case 2 6: Case 3 45 60 20
0 1 1 0 0 T 1 f 1 T i i f f 1 0 T i 1 i T i f T i 1 g(z) =iz g T (f(z)) = g(t (z)) g(z) =iz g 4 (Z) = i 4 Z = Z Z 4 f z 4 T ˆR T Ĉ Ĉ f Case 3 1 λ 1 g(z) =λz g Z Z λ Z λ 2 Z λ 3... 1 arg λ f ˆR 3.3 2 f(z) =z 2 + c A 0,B,C 2 f(z) =Az 2 + Bz + C f( ) = 21
3 1 A Af(z) =A 2 z 2 + ABz + Ac =(Az + B 2 )2 B2 4 + AC Af(z)+ B 2 =(Az + B 2 )2 B2 4 + B 2 + AC. T (z) =Az + B 2 c = B2 4 + B 2 T (f(z)) = {T (z)} 2 B2 4 + B 2 + AC + AC z f f(z) T T (z) T {T (z)} 2 + c T f T (z) {T (z)} 2 + c Z = T (z) f g(z) =Z 2 + c 2 2 1 c c 2 c =0 c z 2 + c R Java OTIS http://www.math.nagoya-u.ac.jp/~kawahira/programs/otis.html 22
c f c (z) =z 2 + c z z f c z 2 + c f c (z 2 + c) 2 + c f c ( ) z z z > 100 f c (z) = z 2 (1 + c z ) = z 2 1+ c 1+ c. > 10000 2 z 2 z 2 c < 10 c = c < 0.001 z 2 z 2 0.1 10000 f(z) z K c =( ( ) z ) z K c K c K c f c (filled Julia set) K c f c (Julia set) c =0 f c (z) =f 0 (z) =z 2 K 0 K c c K c K c M := (K c c ) (Mandelbrot set) c 7 c K c K c (fractal) 23
7: 90 24
4 1 1. y =2x +1 10 5-10 -5 5 10-5 -10 25
2. y =2x 10 5-10 -5 5 10-5 -10 26
3. y = x 3 2x 4 3 2 1-4 -2 2 4-1 -2-3 -4 27
4. y = 3x +2 x +2 10 5-10 -5 5 10-5 -10 28
5. y =4x 10 5-10 -5 5 10-5 -10 29
6. y = x +1 x 2 10 5-10 -5 5 10-5 -10 30
7. y = x +1 x +1 10 5-10 -5 5 10-5 -10 31
8. y = x cos θ + sin θ x sin θ + cos θ (θ =2π/5) 10 5-10 -5 5 10-5 -10 32
9. y = x 2 5 4 3 2 1-4 -2 2 4-1 -2-3 33
10. y = x 2 +0.25 5 4 3 2 1-4 -2 2 4-1 -2-3 34
11. y = x 2 +0.35 5 4 3 2 1-4 -2 2 4-1 -2-3 35
12. y = x 2 +0.10 5 4 3 2 1-4 -2 2 4-1 -2-3 36
13. y = x 2 1 5 4 3 2 1-4 -2 2 4-1 -2-3 37
5 2 (complex number)z 2 x, y i 2 = 1 i z = x + yi y =0 x C z = x + yi xy (x, y) (complex plane) x R (real axis) z z 0 x 2 + y 2 z (absolute value) (modulus) z arg z z (argument) y z = x + yi arg z 0 x z R i i 2 1 2 z = x + yi, w = u + vi z + w =(x + u)+(y + v)i z w =(x u)+(y v)i z + w w r z (r >1) z 0 z = x + yi x yi 38
2 z = x + yi, w = u + vi zw =(xu yv)+(xv + yu)i 1 z = x yi ( z 0) x 2 + y2 z w = z 1 w = (xu + yv)+( xv + yu)i u 2 + v 2 ( w 0) zw z = x + yi r r z = r(x + yi) =rx + ryi r r i i z = x + yi i z = i(x + yi) =xi + yi 2 = y + xi 90 iz = y + xi z = x + yi 0 R 1 180 i i = 1 1 i i 90 90 =180 39
2 z = x + yi, w = u + vi z w =(x + yi)w = xw + y(iw). z w z y iw y iw x w 0 x w 0 zw = z w zw w w z arg z arg zw = arg z + arg w. 360. 2 2 ax 2 + bx + c =0 D = b 2 4ac 0 x = b ± D 2a D <0 x = b D 2a ± 2a i 2 A z = x + yi (1) i 3 (2) (2 + i)(2 i) (3) ( i 2 ) 2 40
(4) 1 1+i (5) ( 1+ 3i) 3 B z = x+yi z 2 z w = z 2 41
C (1) x 2 +1=0 (2) z 2 + z +1=0 (3) z 3 =1 (4) z 4 =1 (5) z 5 =1 A: (1) i (2) 5 (3) 1 4 (4) 1 i (5) 8. 2 B: z 2 = z z z 2 z 2 z 1 w = z 2 2 C (1) ±i (2) 1 ± 3i (3) z 3 1=(z 1)(z 2 + z +1)=0, 2 1 (4), (5) 5 42
6 (1) 30 1989 (2) 1982 (3) I II (1998?) (4) (2002) (2) (3) II (4) (5) 1996 (6) 1995 (7) 1987 (8) R.L.Devaney 2 (2003) (9) (1997) (6) (7) (8) (9) (8) (8) (9) CG Web http://www.math.nagoya-u.ac.jp/~kawahira/ 43