УрМатФиз / УрМатФиз с теорией

w~ =

z

z1 z1 + 1

z

+ z1 1 z1

12.

D =

f

z : z

2

CI

z

1

0

:

62

g

13.

D =

z : z

CI

z

1

z

0 +

:

f

2

62 1

62

1

g

14.

D =

f

z : z

2

CI

z

0 p2 1 + i =2

g

:

62

15.

D = fz : z

2CI

z

62 jfzj

= 1

arg z

0 gg:

16.

D = fz : z

2CI

Re z 0

z

2

i 1 + i g:

17.

D =

f

z : z

2

CI

z

i 0

:

62

g

18.

D =

f

z : z

2

CI

z

i

1

i

z

0

+i

1

:

62

62

g

19.

D = fz : z

2CI

z

62 jfzj

= 1

Im z

Re z

0 gg:

20.

D = fz : z

2CI

z

62 jfzj

= 1

Re z

0 gg:

21.

D =

f

z : z

2

CI

Im z

Re z 0

z

0 p2

1 + i =2

g

:

62

22.

D =

f

z : z

2

CI

Im z

0

z

i

1

2i

z

i 0

g

:

62

62

23.

D =

f

z : z

2

CI

z

i

i

z

1

i +

1

i

g

:

62 1

62

24.

D =

f

z : z

2

CI

Re

0

z

0

1

z

2

+

1

g

:

62

62

25.

D =

z : z

CI

Im z 0

z

1 1 + i z

1 + 2i 1 + i

:

26.

D =

f

z : z

2CI

62

z

0

p

62

:

1

g

Im z + Re z

0

2 1 + i =2

f

2CI

2 0

p

62

g

27.

D = fz : z

2

z

2 1 + i =2 g:

28.

D = fz : z

2CI

z

62 jfzj

= 1

Re z + Im z

0 gg:

29.

D =

f

z : z

2

CI

Re z

0

z

i

2

i

z

1

i

i

g

:

62 1

62

30.

D =

f

z : z

2

CI

z

1

i

1

1

z

1 + i

1 + i

1

g

:

62

62

2.7.4. - ! w = w z , #$

# % & ' ( D

' ( G :

1 D = fz : z 2CI

2 Im z

Re z 1g

G = fw : w

2CI

jwj 1 Im w 0g)

2 D = fz : z 2CI

2 Im z

1

Re z 0g

G = fw : w

2CI

jwj 1 Re w

0

Im w 0g)

p

p

3 D = fz : z 2

CI

jz + 1

ij

2

jz

+ 2 2ij 2

2g

G = fw : w

2CI

Re w

0

Im w 0g)

4 D = fz : z 2CI

jz + 1j

1

jz + 2j

2 Im z 0g

G = fw : w

2CI jwj

1 Im w 0g)

p

Re z + Im z 0g

5 D = fz : z 2CI jz + 1 + ij

2

G = fw : w 2CI Re w 0

Im w 0g:

.

w = ez = exeiy = excos y + i sin y

:

! " " CI z = 1

" #

!$ Dk =

fz : z

2 CI

2 k Im z 2 k + 1g k

2 ZZ: % &

Dk '-

& ! ' ' ! ! &

$ "

j

w

= ea

Re w

6

=ea !' Im z = y = b

j

x ib

x

+1.

Dk ' & " w = e e 1

1 0 ' ! ' &

' . 2.7.4, . + "

i =4 p

!' ' & w1 = z + 2i e

2

| - 2i,

2

! ' '- " ' ! "

' - =4

' & 4 p2=2.

i =4 p2

w1

= z + 2i e

2

!

!

2

w2 = ew1

w = w

1

w2

+ 1

!

!

w2 = e , -

ABC A0 B0 C0, EF H |

E0 F 0 H0.

! ! - " w = #w2

1%=#w2 + 1%

H0 = A0 = 0 B0 = 1 C0

= E0 = 1

A

= 1 B = 0 C = 1:

. w =

exp( #1 i%#z + 2i%=2* 1

:

exp( #1 i%#z + 2i%=2* + 1

2% + ",

.

2.7.4, . . " w1 = #z + 3i%

|

! 3i, 0

=2:

2

w = ew . 1

AB , 2 =2 ! "-

v1 = =2

t

i =2

, AB : u1 = t

1 t 0 $ A B : w = e e

1 t 0:

4 BC ! " | BC :

u1 = 0

v1 = t

=2 t $

B C : w = eit =2 t : 4

u1 = t

CE : v1 =

0 t 1 , 2

,

! " #

$: w1 =

z w2 = ew1

w = iw2:

2

w1 = e i =4z

w2 =

1

w1

!

!

. w = exp 0

2p

ei =4

1

= exp 0

2 1 + i 1

2

:

@

z

A

@

z

A

.

D fw~ :

w~ 2CI

Im w~ =2g

- , -

! " # $! $. &-

1=2, " $ $

-

" =2 1 , 4

w2 =

= )w1 + 1=2,ei =24 = )w1 + 1=2,4 i:

2 w = ew2 :

u2 = t

3

A : v2 = 0 1 t 0

$ $

0

0

u2 = 0

A

B

: w = et 1 t 0: 3

BC : v2 = t

0 t

$ $ B0 C0 : w = eit 0 t ,

u2 = t

" CE : v2 =

0 t 1 $

1

1

w = exp 4 z

+

2

:

w = e

4

+

= e

4=z

e

2

= e

4=z

:

#. 0 , # z1 = 1 + i z2 = 2 z3 = 4

# w~1 = 1 w~2 = 0 w~3 = i:

p

5 2 D = fz : z 2 CI

jz + 1 + ij

2

Re z + Im z 0g . % 2 fw2 : wi=2

42CI

0 Im w2

= 2g

" #$ % w1 = e

z

w2 = p2 w1:

3 - w3 = w2

i

#-

* #$ w = e

w3

: w3

2 CI

, # + fw3

Im w3 2g G = fw : w 2CI

Re w 0

Im w

0g

. 2.7.4, .

. w = exp

0

1

i

i

1

:

@

z

A

.

:

w = e

i =z i = e

i =z e i = e i =z :

1.

D = fz : z 2CI 1 Re z 2g

G = fw : w 2CI jwj 1 Re w 0g:

2.

D = fz : z 2CI 1 Im z 2 Re z 0g

G = fw : w 2CI jwj 1 Re w 0 Imw 0g:

3.

D = fz : z 2CI jz ij 1 jz 2ij 2g

G = fw : w 2CI Re w 0 Im w 0g:

4.

D = fz : z 2CI jz 1j 1 jz 2j 2 Im z 0g

G = fw : w 2CI jwj 1 Imw 0g:

5.

D = fz : z 2CI jz 1j 1 Re z 0g

G = fw : w 2CI Re w 0 Im w 0g:

6.

D = fz : z 2CI 1 Im z 2g

G = fw : w 2CI jwj 1 Re w 0g:

7.

D = fz : z 2CI 1 Re z 2 Im z 0g

G = fw : w 2CI jwj 1 Re w 0 Imw 0g:

8.

D = fz : z 2CI jz 1j 1 jz 2j 2g

G = fw : w 2CI Re w 0 Im w 0g:

G = fw : w 2CI jwj 1 Re w 0g:

10.

D = fz : z 2CI

jz ij 1

Im z 0g

G = fw : w 2CI

Re w 0

Imw 0g:

11.

D = fz : z 2CI

2 Re z 1g

G = fw : w 2CI

jwj 1 Imw 0g:

12.

D = fz : z 2CI

1 Im z 2 Re z 0g

G = fw : w 2CI jwj 1 Re w 0 Im w 0g:

13.

D = fz : z 2CI

jz + ij 1 jz + 2ij 2g

G = fw : w 2CI

Re w 0

Imw 0g:

14.

D = fz : z 2CI

jz + 1j 1

jz + 2j 2

Im z 0g

G = fw : w 2CI

jwj 1 Imw 0g:

15.

D = fz : z 2CI

jz + 1j 1

Re z 0g

G = fw : w 2CI

Re w 0

Imw 0g:

16.

D = fz : z 2CI

2 Im z 1g

G = fw : w 2CI

jwj 1 Imw 0g:

17.

D = fz : z 2CI

1 Re z 2 Im z 0g

G = fw : w 2CI

jwj 1 Re w 0 Imw 0g:

18.

D = fz : z 2CI

jz + 1j 1

jz + 2j 2g

G = fw : w 2CI

Re w 0

Imw 0g:

19.

D = fz : z 2CI

jz + ij 1

jz + 2ij 2

Re z 0g

G = fw : w 2CI jwj 1 Re w 0g:

20.

D = fz : z 2CI

jz + ij 1

Im z 0g

G = fw : w 2CI

Re w 0

Imw 0g:

21.

D = fz : z 2CI

1 Im z Re z 2g

G = fw : w 2CI jwj 1 Re w 0g:

22.

D = fz : z 2CI

2 Im z 1 Re z 0g

G = fw : w 2CI

jwj 1 Re w 0 Imw 0g:

23.

D = fz : z 2CI

2 Re z 1

Im z 0g

G = fw : w 2CI

jwj 1 Re w 0 Imw 0g:

24.

D = fz : z 2CI

jz 1j 1

jz 2j 2

Im z 0g

G = fw : w 2CI jwj 1 Re w 0g:

D = fz : z 2CI

jz 1 ij p

25.

2

Im z + Re z 0g

G = fw : w 2CI

Re w 0

Imw 0g:

26.

D = fz : z 2CI

1 Re z + Im z 2g

G = fw : w 2CI jwj 1 Re w 0g:

jz 1 ij p

2ij 2p

g

27.

D = fz : z 2CI

2

jz 2

2

G = fw : w 2CI

Re w 0

Imw

0g:

28.

D = fz : z 2CI

jz 1 + ij p

jz 2 + 2ij 2p

g

2

2

G = fw : w 2CI

Re w 0

Imw

0g:

29.

D = fz : z 2CI

jz ij 1

jz 2ij 2

Re z 0g

G = fw : w 2CI

jwj 1 Imw 0g:

jz + 1 ij p

30.

D = fz : z 2CI

2

Im z Re z 0g

G = fw : w 2CI

Re w 0

Imw 0g:

1+ D =

f

z : z

2

CI

z

j

1 z

,

1

1=2- z

,i=2 i-

.

j

62

,

62

i-

g

2+ D =

f

z : z

2

CI

z

j

1 z

,1 2- z

62

i

1

:

j

62

g

% %$"- $/! #$ %& " w = w*z+, % # "-

$" ! D ( 1 "" $ % ! G = fw : w

2CI

Im w 0g

:

3+ D =

: z

1

Im z

Re z 0

,p

i+=6 p

z

CI

z

z

2*1

2*1

i+=2-

.

4+ D =

f

z

: z

2 j j

1

Re z

0

62

p

i+=2

p

g

: