CR Analye Complexe

7/24/2019 CR Analye Complexe

1/12

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.


2/12

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


3/12

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)|


4/12

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


5/12

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


6/12

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


7/12

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


8/12

U S UaS ma N U S aS ma

(fn)n X C

(1 fn) X

fn

fn n,fn 1c


9/12

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 .


10/12

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


11/12

: [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


12/12

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

CR Analye Complexe

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