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2

3

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

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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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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.

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0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

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