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z = x+iy x y
i i=√−1
i2 =−1
f(z) =u(x, y) +iv(x, y)
u(x, y) v(x, y)
x y
f(z) =z2
f(z) =z2
= (x+iy)2 =x2
−y2+i2xy
u(x, y) =x2−y2 v(x, y) = 2xy
f(z)
f(z)
f�(z) = lim ∆z→0
f(z+∆z)−f(z) ∆z
∆z=∆x+i∆y ∆z
f(z)
∂2u
∂x2 +
∂2u
∂y2 = 0
∂2v
∂x2 +
∂2v
∂y2 = 0
∇2u= 0
f(z) = u +iv u v
u v
f(z)
z
f(z)
u(x, y) =x2−y2
∇2u= ∂
2u
∂x2 +
∂2u
∂y2 = 2−2 = 0
v(x, y) u+iv z
∂v ∂y =
∂u ∂x = 2x
y
v(x, y) = 2xy+g(x)
g(x) x x
∂v
∂x = 2y+
dg dx =−
∂u ∂y = 2y
dg
dx = 0, g=
f(z)
�
C
f(z)dz= 0
�z2
z1 f(z)dz
�z2
z1 f(z)dz=F(z2)−F(z1) F�(z) =f(z) f(z)
f(z) = 2z
�
C
2z dz = 0
� 1+i
2i
2z dz=z2|1+2i i= (1 +i)2−(2i)2= 2i+ 4
f(z) a
�
C
f(z)
z−a
dz = 2πi f(a)
a a
�
C
f(z) (z−a)n+1
dz = 2πi
n! f
(n)
(a)
f(n)(z) n f(z)
�
C
sinz
2z−π
dz
R.
b
• b f(z) z=z0 z0
• b �= 0 b bn f(z)
z=z0 n= 1 f(z)
• b f(z)
z=z0
• b1 z−1z0 f(z) z=z0
z0 f(z) �Cf(z)dz
C z0 z0
�
C
f(z)dz= 2πi· f(z) C
C z0, z1,
z2, . . . C0, C1
C2 . . . f(z) C
�
C
f(z)dz = 2πi· f(z) C
C
f(z)
f(z) z=z0
R(z0) b1 (z−1z0)
f(z) = ez
z−1 z= 1
ez z−1 =
e·ez−1
z−1 = e z−1
�
1 + (z−1) + (z−1)
2
2! +. . .
�
= e
z−1 +e+ e
2(z−1) +. . .
1
f(z) z=z0
f(z) (z−z0) z=z0
f(z) = z
(2z+1)(5−z) R(−12) R(5)
R(−12) f(z) (z + 12) (2z+ 1) z=−12
(z+1
2)f(z) = (z+ 1 2)
z
(2z+ 1)(5−z) = z 2(5−z)
R(−1 2) =
−12
2(5 + 12) =− 1 22
R(5)
(z−5)f(z) = (z−5) z
(2z+ 1)(5−z) =− z 2z+ 1
R(5) =−115
f(z) hg((zz)) g(z) z=z0
h(z0) = 0 h�(z0)�= 0
R(z0) =
g(z0)
h�(z0)
h�(z
0) h(z) z=z0.
f(z) = z
(2z+1)(5−z) g(z) = z h(z) = (2z+ 1)(5−z)
h�(z) = 2(5−z) + (2z+ 1)(−1) =−4z+ 9
R(z0) =
z0 −4z0+ 9
R(−1 2) =
−12 2 + 9 =−
1 22
R(5) = 5
