Download - CR Analye Complexe
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7/24/2019 CR Analye Complexe
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f C
R2
f
Jf(df)
z0U limzz0
f(z)f(z0)zz0
=f(z0)
f z = x+ iy z0 = x0, y0)
ux
(x0, y0) = vy
(x0, y0)vx
(x0, y0) =uy (x0, y0)
z =
1
2
x i
y
z =
1
2
x+ i
y
.
f fz
= 0
C
:[a, b]C (a) = (b) 1 2 C
1 C1
1 = 2 C1 f C1
f=N1i=1
ti+1ti
f((t))(t)dt
ti
f C1
f
supU
|f| . ().
1 2 C1
C1
1
f=
2
f.
C
f
f
U{z0}
T
f= 0.
C1 f U {z0}
f= 0.
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1 2 [a, b] 1 2 H: [0, 1]
[a, b]
U H(0, t) = 0(t)
H(1, t) = 1(t)
H(s, a) = H(s, b) s .
H(s, a) = 0(a) = 1(a)H(s, b) = 0(b) = 1(b)
f U {z0} 0 1 C1
0
f= 1
f.
0 C1
0
f= 0
C1 z0 /Im() z0
Ind(z0) = 1
2i
dz
z z0 .
z0
Z z Ind(z)0
f z0U C1 z0 /I m()
Ind(z0)f(z0) = 1
2i f(z)z z0 dz. f z0 U z U, |f| |f(z0)| f
f z0U f z0
f(z) =
an(z0)(z z0)n zB(z0, r) r= d(z0, U).
f
f U z0U
Ind(z0)f(k)(z0) =
k!
2i
f(z)
(z z0)k+1 dz .
f U {z0} z0 f
f
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f C f f
C
C
C1
z /
U Indz() = 0 f
z0U Im()
Ind(z0)f(z0) = 1
2i
f(z)
z z0 dz.
f f
C1 {z,Indz() = 1} {z,Indz() = 0} K ={z,Indz() = 1} I m() f K Im() f
1
2i
f(z)
f(z)dz.
C
C f
f U T U T
f= 0.
C
U B(z0, R)f g U zC(z0, R) |f(z) g(z)|
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f : B B f(0) = 0 z B |f(z)| =|z| |f(0)| 1
f
a: z za1az
a B R = a : zeiz bB,
R = b
f B B z1 = z2 1= f(z1) 2 = f(z2) 1 21 12
z1 z21 z1z2 f(z1)1 |1|2
1 |z1|2
f
F
K Mk f F, zK, |f(z)| MK F
U=, = C
U= C
U B 0
={: UB, (0) = 0}
(U) = B |(0)|= max
|(0)|
C
U
f U C log f U
f U C
f U
f C1
f= 0
U
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f C(a, r1, r2) f a nZ
Cn = 1
2i
B+(a,r)
f(z)
(z a)n+1 dz, r]r1, r2[
f a
nZ
Cn(z a)n
r
f C(0, r1, r2)
Cnzn
n0
Cnzn
B(0, r2)
n
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f U S U
f U S f S
U
U C1 U f U S f Im()
f= 2i
aSRes(f, a)Ind(a),
Res(f, a) f a 1/(z a) f a
C C
U C U C(U,R+) C2 U U
(z) =
log (z)
2
(z)
,
= 4zz= 2
x2+
2
y2
B
0(z) = 2
1 |z|2 0(z) =zz log 0
20(z) =1,zB
U1, U2 C U2 f U1 U2 f f U1
zU1, f(z) =|f(z)| (f(z))
f : U1 U2 U2
f(z) = (f(z)),zU1
U C 1 f
f(z)0(z),zB
U2 U2
(z) A < 0,zU2 C U2
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f f
f
f(C)C {0, 1} f
F
U F (fn)n (gn)n
(gn)n KU
(gn)n
KU
g
1
n
F
U F {f0, f F} KU
U aU z1/(z a) U {a} U
K S CK A={ S} A K
U K U (i)i=1..n f UzK
f(z) = 1
2i
ni=1
i
f(z)
z z dz .
K C K K
K CK
K K
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U S UaS ma N U S aS ma
(fn)n X C
(1 fn) X
fn
fn n,fn 1c
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U C U U
u C2 U
u= 0
u U f V U u f V f
C
U u U U u = Re(f) f
U u U U C
R U C f R U z0= x0 +iy0
f(x + iy) =p,q
Cp,q(x x0)p(y y0)q .
U C u U u R U
u U z0U r B(z0, r)U
u(z0) = 1
2
20
u(z0+ rei)d .
u U z0 U, z U, |u(z)| |u(z0)| u u U U u U
D z0 r D
PD : D D R+(, z) PD(, z) PD(, z) =
| z0|2 |z z0|2( z)2 .
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D= B (0, 1)
Pa() = P(, a) =|a()| Pr(e
it) =
nZ r|n|eint
D
u
D
zD, u(z) =D
PD(, z)u() d
|D | .
K U u U
(k1, k2)
N
2
>0
c > 0 supK
ku c supK
u ,
u K K
U (un) U (un) u
u B(z0, r) r < r t[0, 2]
r rr+ r
u(z0)u(z0+ reit) r+ r
r r u(z0).
U (un)
(un) U u
(un) U
u= 0u U , u= U
B = B(0, 1)
U C
U
zUP(z) =
D
P(z, )(z)d
2 .
U
P(z) =
D
P(z, )d().
B
0B
limz0
P(z) = (0)
B
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: [a, b]C |[a,b[
C
C
C
S U C1 U f U
f= 0
ln(f)
f f :U C
U B
D f DB f DB
D D
U C u U
z0
U
n0> 0
r < r0
u(z0) = 1
2
20
u(z0+ rei)d .
u U
U C u: U[, +[ U
u c R, {z| u(z) < c}
z0U,r0> 0,r < r0, u(z0) 12 20
u(z0+ rei)d .
u, v U
max u, v lambda0 u + v [; +[ u
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U C u U z0U, zU, u(z)u(z0) u
U u U U
maxU
u= maxU
u .
U C u U
u
V U h V V u
h V u
h V
DU zD
u(z)D
PD(z, )u() d
|D|
u
B(z0, R)
r < R Ir(u) = 12
u(z+ reid