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z = x + iy x yi 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 = x2y2 + i2xyu(x, y) = x2 y2 v(x, y) = 2xy
f(z)f(z)
f (z) = limz0
f(z + z) f(z)z
z = x + iy z
f(z)
f(z) z = a f(z) z =a f(z)
f(z)
f(z) = u(x, y) + iv(x, y)
u
x=
v
yu
y= v
x
u(x, y) v(x, y) x yf(x)
f(z) = z3
f(z) = z3 = (x + iy)3 = x3 + 3x2iy + 3x(iy)2 + (iy)3 = x3 3xy2 + i(3x2y y3)
u(x, y) = x3 3xy2 v(x, y) = 3x2y y3
u
x= 3x2 3y2 ; u
y= 6xy ; v
x= 6xy ;
v
y= 3x2 3y2
u
x=
v
y
u
y= v
x
f(z) = z3
u v x y
2u
x2+
2u
y2= 0
2v
x2+
2v
y2= 0
2u = 0
f(z) = u + iv u vu
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) = u + iv = x2 y2 + i(2xy + )
f(z)
Cf(z)dz = 0
z2z1
f(z)dz z2z1
f(z)dz = F (z2) F (z1) F (z) = f(z) f(z)
f(z) = 2z
C2z dz = 0
1+i
2i2z dz = z2|1+i2i = (1 + i)
2 (2i)2 = 2i + 4
f(z) a
C
f(z)z a dz = 2i f(a)
a
a
C
f(z)(z a)n+1 dz =
2in!
f (n)(a)
f (n)(z) n f(z)
C
sin z2z dz
|z| = 1 |z| = 2
C
sin z2z dz =
C
sin z2(z 2 )
dz =12
C
sin zz 2
dz
a = 2 = 1, 57 a
a
12
C
sin zz 2
dz =122i sin
2= i
f(z) z = a
a
f(z) = f(a) + f (a)(z a) + f(a)2!
(z a)2 + f(a)3!
(z a)3 +
f(z) f(z)f(z) f(z)
f(z) z = a
C1 C2 z0 f(z)R f(z)
f(z) = a0 + a1(z z0) + a2(z z0)2 + +b1
z z0+
b2(z z0)2
+
an =1
2i
C
f(z) dz(z z0)n+1
; bn =1
2i
C
f(z) dz(z z0)n+1
z0
R
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 1zz0 f(z) z = z0
z0 f(z)C f(z)dz
C z0 z0
Cf(z)dz = 2i f(z) C
C z0, z1,z2, . . . C0, C1C2 . . . f(z) C
Cf(z)dz = 2i f(z) C
C
f(z)
f(z) z = z0R(z0) b1 1(zz0)
f(z) = ez
z1 z = 1
ez
z 1 =e ez1
z 1 =e
z 1
1 + (z 1) + (z 1)2
2!+ . . .
=e
z 1 + e +e
2(z 1) + . . .
1z1 R(1) = e
f(z) z = z0f(z) (z z0) z = z0
f(z) = z(2z+1)(5z) R(12) R(5)
R(12) f(z) (z +12) (2z + 1)
z = 12
(z +12)f(z) = (z +
12)
z
(2z + 1)(5 z) =z
2(5 z)
R(12) =
122(5 + 12)
= 122
R(5)
(z 5)f(z) = (z 5) z(2z + 1)(5 z) =
z
2z + 1
R(5) = 511
f(z) g(z)h(z) g(z) z = z0h(z0) = 0 h(z0) = 0
R(z0) =g(z0)h(z0)
h(z0) h(z) z = z0.
f(z) = z(2z+1)(5z) 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(12) =
122 + 9
= 122
R(5) =5
20 + 9 = 511
f(z) nf(z) (z z0)m m
n (m 1)(m 1)! z = z0
R(z0) =1
(m 1)!dm1
dzm1(z z0)mf(z)
z=z0
f(z) = z sin z(z)2 z =
m = 3 (m 1) = 2
R() =12!
d2
dz2z sin z
z=
=12(z sin z + 2 cos z)
z=
= 1
2
0
d
5 + 4 cos
z = ei 0 2z |z| = 1
z = ei dz = ieid = izd d = 1izdzcos = 12(e
i + ei) = 12(z +1z )
I = 2
0
d
5 + 4 cos =
C
1izdz
5 + 2(z + 1z )=
1i
C
dz
5z + 2z2 + 2=
1i
C
dz
(2z + 1)(z + 2)
C f(z) z = 12z = 2 z = 12 C f(z)
z = 12
R(12) =
1(2z + 1) + 2(z + 2)
z= 12
=1
4z + 5
z= 12
=13
I = 1i 2iR(12) =
23
0 13