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4 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z)

5 xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q)

6 1 1 (b, d) (a, c) a = d, c = b ( a c ) c a

7 w = { z = x + iy u + iv v = { u = ax cy + p cx + ay + q w = ax cy + p + i(cx + ay + q) = a(x + iy) + ic(x + iy) + (p + iq) = Az + B (A = a + ic, B = p + iq) A arg A B

8 f(x) f(x lim 0 ) = f (x x x 0 ) 0 x x 0 f(x) x 0 ( ) f(x) = f(x 0 ) + f (x 0 )(x x 0 ) + ɛ lim ɛ/(x x x x 0 ) = 0 0 ɛ x x 0 f(x) = f(x 0 ) + f (x 0 )(x x 0 )

9 f(z) = f(z 0 ) + f (z 0 )(z z 0 ) + ɛ ɛ z 0 f(z) = f(z 0 ) + f (z 0 )(z z 0 ) = f (z 0 )z + ( f(z 0 ) + f (z 0 )z 0 ) f (z 0 ) 0

10 f u, v u x = v y u y = x v a, b, c, d u 2 u x u y 2 = 0 x z

11 w = z 2 = r 2 (cos 2θ + i sin 2θ)

12 w = e z = e x (cos y + i sin y) π 2 i w = z 0 dz (z 5 1) 2/5

13

14 Riemann, Georg Friedrich Bernhard ( )

15 { w < 1} (

16 f u. u(x, y) [0, 1] u(0) = a, u(1) = b u (x) = 0 u(x) = (b a)x + a

17 a b 1

18 (x, y) u(x, y)

19

20 4 4 z A z D G z B z C

21 G G G { w = 1} 1 1 G 3 ζ 1, ζ 2, ζ 3 { w = 1} 3 ξ 1, ξ 2, ξ 3 1

22 2 4 (G; z A, z B, z C, z D ), (G ; z A, z B, z C, z D ) G G z A, z B, z C, z D z A, z B, z C, z D z D z A z D f z A z C z C z B z B G G

23 4 (G; z A, z B, z C, z D ) (G ; z A, z B, z C, z D ) 4 (G; z A, z B, z C, z D ) 4 (G ; ξ A, ξ B, ξ C, ξ D ) z dz w = 0 (z ξ A )(z ξ B )(z ξ C )(z ξ D ) (G ; z A, z B, z C, z D )

24 z A z D z B z C z A z D z B z C

25 4 A D A D B C B C z A z D z D z A z C z B z C z B

26 2

27

28

29 A D A D B C BC AB = B C A B B C z A z D z D z A z C z B z C z B 4

30 4 BC AB 4 z A z B z C z D 4 z A z B z C z D

31 Γ D D ρ(z) Γ Γ γ ρ(z) 1 + γ ρ(z) dz 1 D D (ρ(z))2 dxdy Γ

32 a γ ρ(x, y) b = γ ρ(z) dz 1 D ρ2 dxdy ρ o (x, y) = 1 b D ρ2 0 dxdy = ab b 2

33 [ ] 2 4 (G; z A, z B, z C, z D ), (G; z A, z B, z C, z D ) z A z B z C z D z A z B z C z D

34 2 2 2

35 2

36 2

37 ?

38 2πi r R e z log r2 R log r log R e z = e x (cos y + i sin y)

39 f r R r R f f

40 R r = R r r R r R 2

41 r R 2 {r < z < R} 2 2 log R r 2π 2

42 2 2

43

44 2π 2π

45

46

47 2 3 2

48 e z 2π log r log R

49

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

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36 3 D f(z) D z f(z) z Taylor z D C f(z) z C C f (z) C f(z) f (z) f(z) D C D D z C C 3.: f(z) 3. f (z) f 2 (z) D D D D D f (z) f 2 (z) D D f (z) f 2 ( 3 3. D f(z) D D D D D D D D f(z) D f (z) f (z) f(z) D (i) (ii) (iii) f(z) = ( ) n z n = z + z 2 z 3 + n= z < z < z > f (z) = e t(+z) dt Re z> Re z> [ ] f (z) = e t(+z) = (Rez> ) +z +z t= z < f(z) Taylor

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http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ

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z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a z z x = y = /x lim y = + x + lim y

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7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7

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z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =

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140 120 100 80 60 40 20 0 115 107 102 99 95 97 95 97 98 100 64 72 37 60 50 53 50 36 32 18 H18 H19 H20 H21 H22 H23 H24 H25 H26 H27 1 100 () 80 60 40 20 0 1 19 16 10 11 6 8 9 5 10 35 76 83 73 68 46 44 H11

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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx

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2 p1 i 2 = 1 i 2 x, y x + iy 2 (x + iy) + (γ + iδ) = (x + γ) + i(y + δ) (x + iy)(γ + iδ) = (xγ yδ) + i(xδ + yγ) i 2 = 1 γ + iδ 0 x + iy γ + iδ xγ + yδ xδ = γ 2 + iyγ + δ2 γ 2 + δ 2 p7 = x 2 +y 2 z z p13

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1 : f(z = re iθ ) = u(r, θ) + iv(r, θ). (re iθ ) 2 = r 2 e 2iθ = r 2 cos 2θ + ir 2 sin 2θ r f(z = x + iy) = u(x, y) + iv(x, y). (x + iy) 2 = x 2 y 2 + 1.3 1.4. (pp.14-27) 1 1 : f(z = re iθ ) = u(r, θ) + iv(r, θ). (re iθ ) 2 = r 2 e 2iθ = r 2 cos 2θ + ir 2 sin 2θ r f(z = x + iy) = u(x, y) + iv(x, y). (x + iy) 2 = x 2 y 2 + i2xy x = 1 y (1 + iy) 2 = 1

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(ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z B 4 24 7 9 ( ) :,..,,.,. 4 4. f(z): D C: D a C, 2πi C f(z) dz = f(a). z a a C, ( ). (ii), a D, a U a,r D f. f(z) = A n (z a) n, z U a,r, n= A n := 2πi C f(ζ) dζ, n =,,..., (ζ a) n+, C a D. (iii) U a,r

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arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = = arctan arctan arctan arctan 2 2000 π = 3 + 8 = 3.25 ( ) 2 8 650 π = 4 = 3.6049 9 550 π = 3 3 30 π = 3.622 264 π = 3.459 3 + 0 7 = 3.4085 < π < 3 + 7 = 3.4286 380 π = 3 + 77 250 = 3.46 5 3.45926 < π < 3.45927

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1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 : 9 ( ) 9 5 I II III A B (0 ) 5 I II III A B (0 ), 6 8 I II A B (0 ), 6, 7 I II A B (00 ) OAB A B OA = OA OB = OB A B : P OP AB Q OA = a OB = b () OP a b () OP OQ () a = 5 b = OP AB OAB PAB a f(x) = (log

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.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

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