1 1 1 1 1 1
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)
xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q)
1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b ( a c ) c a
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
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 )
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
f u, v u x = v y u y = x v a, b, c, d u 2 u x 2 + 2 u y 2 = 0 x z
w = z 2 = r 2 (cos 2θ + i sin 2θ)
w = e z = e x (cos y + i sin y) π 2 i w = z 0 dz (z 5 1) 2/5
Riemann, Georg Friedrich Bernhard (1826.9.17-1866.7.20)
{ w < 1} (
f u. u(x, y) [0, 1] u(0) = a, u(1) = b u (x) = 0 u(x) = (b a)x + a
a b 1
(x, y) u(x, y)
4 4 z A z D G z B z C
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
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
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 )
z A z D z B z C z A z D z B z C
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
2
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
4 BC AB 4 z A z B z C z D 4 z A z B z C z D
Γ D D ρ(z) Γ Γ γ ρ(z) 1 + γ ρ(z) dz 1 D D (ρ(z))2 dxdy Γ
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
[ ] 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
2 2 2
2
2
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2πi r R e z log r2 R log r log R e z = e x (cos y + i sin y)
f r R r R f f
R r = R r r R r R 2
r R 2 {r < z < R} 2 2 log R r 2π 2
2 2
2 1 4 4 2
2π 2π 2 2 4 2
3 3 3 2
3 3 2 2 1 2 2
2 3 2
e z 2π log r log R