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T m t t n i d u n g

S p h c l m t s m r n g h o n h o c a t p s t h c , l m t t r - n g s m v n

g i i p h - n g t r n h i s - c g i i q u y t t r i t v n h n g y u c u k h c c a t o n h c .

C h n h v v y s p h c c v a i t r q u a n t r n g t r o n g t o n h c , c b i t l l n h v c t o n h c

n g d n g .

G i i t c h p h c l m t m n h c n g h i n c u v c c s p h c , n - c h n h t h n h d a

t r n n h n g n h l v n g u y n l c b n m n h n g n g u y n l t r n h b y t r o n g t i u l u n

n y l m t p h n t r o n g s .

N i d u n g c h n h c a t i u l u n n y n h - s a u :

P h n 1 : t r n h b y n g u y n l m u n c c i v n g u y n l b o t o n m i n .

P h n 2 : t r n h b y c c n g d n g c a c c n g u y n l t r n .

D o s h n c h v t i l i u v k h n n g n n t i u l u n c n n h i u k h i m k h u y t . T c g i

r t m o n g n h n - c s g p c a T h y C v c c b n .

Q u i N h n , n g y 0 8 t h n g 0 6 n m 2 0 0 9

B i V n C n h

1

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M c l c

1 N h n g n g u y n l c b n 1

1 . 1 N g u y n l b o t o n m i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 . 2 N g u y n l m u n c c i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

n g d n g 3

2 . 1 N g u y n l a r g u m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 . 2 B S c h w a r z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 . 3 n h l L i o u v i l l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 . 4 n h l c b n c a i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 . 5 M t s n g d n g k h c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2

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T r l i n h l , n u f Ta c o n s t A j f j Ta c o n s t ( d o n h n x t t r n ) .

G i s M a m x

z P G

j f @ z A j v i G a G @ G .

G i s z

l m t i m t h u c G m j f @ z

A j a M

t E a f z P G g j j f @ z A a M j A z

P E . S u y r a E l t p n g ( d o j f @ z A j l i n t c ) .

T a c n c h n g m i n h E m , t c l z

l i m t r o n g c a G .

G i s z

k h n g l i m t r o n g c a E . K h i t r o n g m i l n c n c a z

p h i c i m z k h n g

t h u c E . S u y r a j f @ z A j < M ( d o n h n g h a E ) . L y v n g t r n t m z

, b n k n h , t r n c

z

0

X j f @ z

0

A j < M . V v y W > H X j z z

0

j < t h j f @ z A j < M .

G i C

l c u n g t r n t r n v n g t r n t m z

, b n k n h n m t r o n g l n c n c a z

0

, t a c

f @ z

A a

I

P

2

0

f

z

C e

i

d

j f @ z

A j a

I

P

2

0

f

z

C e

i

d

a

I

P

C

f

z

C e

i

d C

C n C

f

z

C e

i

d

I

P

H

f

d

C

f

z

C e

i

d C

C n C

f

z

C e

i

d

I

g

e

8/14/2019 Giai tich phuc

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2 n g d n g

2 . 1 N g u y n l a r g u m e n t

n h l 2 . 1 . G i s f @ z A c h n h h n h t r o n g m i n G , g i i h n b i - n g c o n g n n g C , t r

r a t i h u h n c c i m t r o n g G , c n t r n C , h m f @ z A c h n h h n h v k h c 0 . K h i t a c

I

P i

C

f

H

@ z A

f @ z A

d z a N P

t r o n g N l s k h n g i m c a f @ z A t r o n g G , P l s c c i m t r o n g G , n g t h i m i k h n g

i m h o c c c i m - c t n h m t s l n b n g b i c a n .

C h n g m i n h . G i s f c c c k h n g i m : b

1

; b

2

; : : : ; b

s

v c c c c c i m : a

1

; : : : ; a

r

.

X t h m

@ z A a

f

H

@ z A

f @ z A

G i s z

0

l k h n g i m c p m c a f @ z A . K h i

f @ z A a a

m

@ z z

0

A

m

C : : : ; a

m

Ta H a @ z z

0

A

m

g @ z A ; g @ z

0

A Ta H

.

f

H

@ z A

f @ z A

a @ l o g f @ x A A

H

a m @ z z

0

A

1

C

g

H

@ z A

g @ z A

V g @ z A Ta H n n

g

0

( z )

g ( z )

c h n h h n h . S u y r a

r e s

z

0

f

H

@ z A

f @ z A

a m :

N u z

0

l c c i m c p k c a f @ z A , t n g t t r n , t a c f @ z A @ z z

0

A

k

a g @ z A c h n h h n h v

k h c 0 t i z

0

. D o

f

H

@ z A

f @ z A

a @ l o g f @ x A A

H

a

k

z z

0

C

g

H

@ z A

g @ z A

S u y r a

r e s

z

0

f

H

@ z A

f @ z A

a k

V v y

I

P i

C

f

H

@ z A

f @ z A

a

z P G

r e s

z

f a N P

3

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2 . 2 B S c h w a r z

n h l 2 . 2 . G i s h m f @ z A c h n h h n h t r o n g h n h t r n n v f j z j

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T r l i p h p c h n g m i n h n h l :

G i s f @ z A l c h n h h n h t r n t o n m t p h n g p h c . L y z

0

PC

.

V f @ z A g i i n i n n j f @ z A j M ; V z PC

. K h i x t n g t r n b n k n h r , t m z

0

, p d n g

c n g t h c ( 1 ) , t a c

j f

H

@ z

0

A j M :

I

P r

2

: P r a

M

r

C h o r 3 I , t a c f

H

@ z

0

A a H t r n C

. i u n y c n g h a f l h n g s t r n C

.

2 . 4 n h l c b n c a i s

n h l 2 . 4 . M i a t h c b c n @ n ! I A h s p h c u c n n g h i m . ( m i n g h i m - c t n h

m t s l n b n g b i c a n )

C h n g m i n h . L y f @ z A a z

n

C z

1

z

n 1

C : : : C a

n

; n ! I . C n c h n g m i n h f @ z A c t n h t m t

n g h i m .

G i s f @ z A k h n g c n g h i m . K h i f @ z A Ta H V z PC

; g @ z A a

I

f @ z A

c h n h h n h t r n C

. M

l i m

z 3 I

j f @ z A j a I A l i m

z 3 I

g @ z A a H

S u y r a t n t i m t s R > H l n s a o c h o k h i j z j ! R , t a c j g @ z A j I .

M t k h c , g c h n h h n h n n V j z j R , t h e o n g u y n l m u n c c i , t a c

j g @ z A j m x

j z j = R

j g @ z A j I

D o , t h e o n h L i o u v i l l e , t a c g l h n g s . T s u y r a f p h i l h m h n g . i u n y v

l v f c b c n ! I . V y f p h i c t n h t m t n g h i m z

1

.

B n g c c h v i t l i f t h e o d n g f @ z A a @ z z

1

A q

n 1

@ z A , t a t i p t c l p l u n c h o q

n 1

@ z A ,

r i l i t c h r a n h t r n . T a c t h v i t l i f d i d n g f @ z A a a @ z z

1

A : : : @ z z

n

A . V y f c

n n g h i m .

2 . 5 M t s n g d n g k h c

n h l 2 . 5 . C h o

f @ z A a a

0

C a

1

z C : : : C a

d

z

d

l m t a t h c k h c h n g s , v a

d

Ta H . K h i c m t s p h c z

0

s a o c h o f @ z

0

A a H .

C h n g m i n h . G i s t r i l i . K h i

I

f @ z A

c x c n h v i m i z , v n l h m g i i t c h .

B n g c c h v i t

f @ z A a a

d

z

d

a

0

a

d

z

d

C : : : C I

t a t h y r n g

l i m

j z j 3 I

I

f @ z A

a H

5

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L y l m t s p h c s a o c h o f @ A Ta H . C h n m t s R d n g l n s a o c h o j z j < R , v

n u j z j ! R t h

I

j f @ z A j

@ I ; S A

2

. V y k h n g t n t i f n h t h .

n h l 2 . 7 . G i s f c h n h h n h t r o n g f j z j R g . t M @ r ; f A a m x

j z j r

j f @ z A j . K h i t a c

T @ r ; f A l o g

+

M @ r ; f A

R C r

R r

T @ R ; f A ; H r < R

C h n g m i n h . D o f c h n h h n h n n k h n g c c c i m , s u y r a N @ r ; f A a H . M T @ r ; f A a

m @ r ; f A . T h e o n h n g h a m , t a c

T @ r ; f A a

I

P

2

0

l o g

+

f

r e

i

d l o g

+

M @ r ; f A

S u y r a b t n g t h c u t i n .

N u M @ r ; f A I t h l o g

+

M @ r ; f A a H t h b t n g t h c t h h a i h i n n h i n n g .

G i s M @ r ; f A > I . T h e o n g u y n l m u n c c i , t n t i ( t h u c b i n ) v z

0

a r e

i

s a o

c h o

j f @ z

0

A j a M @ r ; f A

T a c

l o g

+

M @ r ; f A a l o g

+

j f @ z

0

A j a l o g j f @ z

0

A j

p d n g c n g t h c P o i s s o n - J e n s e n

l o g j f @ z

0

A j a

I

P

2

0

l o g

f

R e

i

R

2

r

2

R

2

P R r o s @ A C r

2

d C

M

= 1

l o g

R @ z a

A

R

2

a z

6

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I

P

2

0

l o g

f

R e

i

R

2

r

2

R

2

P R r o s @ A C r

2

d

R C r

R r

I

P

2

0

l o g

f

R e

i

d a

R C r

R r

T @ R ; f A

n h l 2 . 8 . C h o f @ z A l c h n h q u y , v j f @ z A j M t r o n g h n h t r n j z a j R ; v g i s

r n g f @ a A Ta H . K h i n u c c s z

m

; m a I ; : : : ; n t r o n g h n h t r n j z a j

1

3

R m l m c h o

f @ z

m

A a H t h n s k h n g v - t q u A l o g @ M = j f @ z A j A .

C h n g m i n h . T a c t h g i s r n g a a H . L y z

1

; : : : ; z

n

l c c s p h c s a o c h o f @ z

i

A a H

t r o n g j z j

1

3

R v l y

g @ z A a

f @ z A

n

m = 1

I

z

z

m

K h i g @ z A l c h n h q u y v i j z j R , v t r n j z j a R t a c j z = z

m

j ! Q v i m a I ; : : : ; n . V

v y

j g @ z A j

M

n

m = 1

@ Q I A

a P

n

M

v i j z j a R , v v v y c n g v i j z j < R ( t h e o n g u y n l m u n c c i ) . T r o n g t r n g h p

r i n g i u n y c n g n g c h o z a H . B i v g @ H A a f @ H A n n s u y r a r n g

j f @ H A j P

n

M

v v v y

n

I

l o g P

l o g

M

j f @ H A j

y c h n h l i u c n c h n g m i n h .

N h n x t : M t k t q u y h n c t h t h u c t n h l J e n s e n . N u c c s r

1

; : : : ; r

n

t h u c j z j R m f @ r

m

A a H ; m a I ; : : : ; n . K h i

l o g

R

n

r

1

: : : r

n

a

I

P

2

0

l o g

f

e

i

d

l o g j f @ H A j l o g M l o g j f @ H A j

L y c c s r

1

; : : : ; r

n

t h u c j z j R ; H <

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n h l 2 . 9 . ( n h l b a - n g t r n c a H a d a m a r d )

L y f @ z A l m t h m g i i t c h v c h n h q u y t r o n g h n h v n h k h n R

1

j z j r

3

. L y

r

1

< r

2

< r

3

v t M

1

; M

2

; M

3

l c c i c a j f @ z A j t r n v n g t r n j z j a r

1

; r

2

; r

3

t - n g

n g . K h i

M

l o g ( r

3

= r

1

)

2

M

l o g ( r

3

= r

2

)

1

M

l o g ( r

2

= r

1

)

3

( 2 )

C h n g m i n h . L y ' @ z A a z

f @ z A , t r o n g l m t h n g s s c x c n h s a u . K h i

' @ z A l c h n h q u y t r o n g v n h k h n g i a j z j a a r

1

v j z j a r

3

. V v y c c i c a j ' @ z A j x y

r a t r n b i n c a n g t r n ( t h e o n g u y n l m u n c c i ) , t c l

' @ z A m x

r

1

M

1

; r

3

M

3

:

V v y t r n j z j a r

2

,

j f @ z A j m x

r

1

r

2

M

1

; r

3

r

2

M

3

( 3 )

B y g i t a c h n

a

l o g

M

3

M

1

l o g

r

3

r

1

t h a y v o ( 3 ) ,

M

2

r

2

r

1

M

1

v v v y

M

l o g ( r

3

= r

1

)

2

@ r

2

= r

1

A

l o g ( M

3

= M

1

)

M

l o g ( r

3

= r

1

)

1

a M

l o g ( r

3

= r

2

)

1

M

l o g ( r

2

= r

1

)

3

:

8

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T i l i u t h a m k h o

[ 1 ] S . L a n g C o m p l e x a n a l y s i s ( 4 e d ) . S p r i n g e r , 1 9 9 9 .

[ 2 ] T i t c h m a r s h E . C . T h e t h e o r y o f f u n c t i o n s ( 2 e d ) . O x f o r d , 1 9 3 9 .

9