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1

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

() 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 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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x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + 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

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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2

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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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 =, [ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b

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5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................ 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)

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II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (

II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x ( II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )

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1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

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l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r 2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2

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I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

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ψ(, v = u + v = (5.1 u = ψ, v = ψ (5.2 ψ 2 P P F ig.23 ds d d n P flow v : d/ds = (d/ds, d/ds 9 n=(d/ds, d/ds ds 2 = d 2 d v n P ψ( ψ

ψ(, v = u + v = (5.1 u = ψ, v = ψ (5.2 ψ 2 P P F ig.23 ds d d n P flow v : d/ds = (d/ds, d/ds 9 n=(d/ds, d/ds ds 2 = d 2 d v n P ψ( ψ KENZOU 28 8 9 9/6 4 1 2 3 4 2 2 5 2 2 5.1............................................. 2 5.2......................................... 3 5.2.1........................................ 3 5.2.2...............................

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x (0, 6, N x 2 (4 + 2(4 + 3 < 6 2 3, a 2 + a 2+ > 0. x (0, 6 si x > 0. (2 cos [0, 6] (0, 6 (cos si < 0. ( 5.4.6 (2 (3 cos 0, cos 3 < 0. cos 0 cos cos

x (0, 6, N x 2 (4 + 2(4 + 3 < 6 2 3, a 2 + a 2+ > 0. x (0, 6 si x > 0. (2 cos [0, 6] (0, 6 (cos si < 0. ( 5.4.6 (2 (3 cos 0, cos 3 < 0. cos 0 cos cos 6 II 3 6. π 3.459... ( /( π 33 π 00 π 34 6.. ( (a cos π 2 0 π (0, 2 3 π (b z C, m, Z ( ( cos z + π 2 (, si z + π 2 (cos z, si z, 4m, ( si z, cos z, 4m +, (cos z, si z, 4m + 2, (si z, cos z, 4m + 3. (6.

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0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (, [ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =

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20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................

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2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( ( (. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2

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211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

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x ( ) x dx = ax

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