УрМатФиз / УрМатФиз с теорией

8.

f r = 18r3 2 cos 2

D = fr ag:

9.

f r = 5r

3 cos 2 cos 3

De = fr ag:

10.

f r = 5r2 cos + 2 cos 3

D = fr ag:

11.

f r = r

5 2 + cos

De = fr ag:

12.

f r = 2r 2 cos

D = fr ag:

13.

f r = 9r

6 cos 2

De = fr ag:

14.

f r = 3r2 sin2 + 3

D = fr ag:

15.

f r = 3r

4 cos 2 1

De = fr ag:

16.

f r = 9r cos 2 + 2

D = fr ag:

17.

f r = 5r

5 cos 3 cos

De = fr ag:

18.

f r = 5r3 cos 3 + cos

D = fr ag:

19.

f r = 2r

5 3 cos

De = fr ag:

20.

f r = 9r4 2 cos 2

D = fr ag:

21.

f r = 9r

6 sin2

De = fr ag:

22.

f r = 2r2 3 + cos

D = fr ag:

23.

f r = 5r

4 cos 3 + 2 cos

De = fr ag:

24.

f r = 9r3 cos 2 3

D = fr ag:

25.

f r = 3r

4 2 + sin2

De = fr ag:

26.

f r = 2r 2 + cos

D = fr ag:

27.

f r = 3r

4 3 + cos 2

De = fr ag:

28.

f r = 5r3 cos 3 + cos

D = fr ag:

29.

f r = 2r

5 5 + cos

De = fr ag:

30.

f r = 3r2 cos 2 + 3

D = fr ag:

.

D

G

2.7.5 !

-

.

v z, |

" a

@u

@v

z

=

= g z

v z

= Z g z ds:

@n# z2@D

@s# z2@D

z2@D

z0

"

:

4v = 0 zz 2 D

8 v

=

g z ds

z2@D

zZ0

:

v x y -

w = 1=z-

3 D = fz : z 2CI

jz 3ij 3 Im z 0g * w = 1=z-

4 D = fz : z 2CI

0 Im z Re z 2g * w = 1=z:

I#*

C =

f1g

I#

CI

C: 2 .

2.7.2

*

* 5

AB * D

x

= t

. 2.7.1, . 6

5 AB : y = 2

t +1 1: 7* 5 A0 B0 :

w =

= 1 + i t + i2 + 1 = t 1 + i t + 2 = u t + iv t . .

-

0

0

u = t 1

A

B

:

v = t + 2 - t

+1

1: 8 (

v u = 3:

2 w = 1=z :

"

w=1=z !CI f1g=CI

"

"

# CI

CI $

! " #

BC

x = t

t 2 0.

y = 0

B0 C0 :

u = 1=t

2 t 0

v = 0

B = 2 $ B0

= 1=2 C = 0 $ C0

= 1:

x = 0

CE :

y = t -

t 0 1:

0

0

u = 0

C

E

:

v = 1=t 0

t 1,

= 0

$ C0 = 1 E = 1

$ E0 = 0:

A x = 2: , x = Re z,

1

z + z

= 2 z = 1=w, !

w = u +1iv :

2

1

1 1

1

2 z + z = 2

+

! = 2

2w + w = 2w w

2

w

w

u = 2u2 + v2

u

41!2 + v2 = 41!2 :

$%& % ,

x = 2 ' % ( % )

x = t

3 %( % ! A & y

= 0

!

1 t

0, ' (

% )

%

u = 1=t

0

0

: v = 0

! 1 t 0, e % % %

A

B

( A = 1

$ A0 = 0 = 0 $ 0 = 1:

4 % ) ! % jz 3ij = 3 '-

% ( &, &

1

3i

= 3 , j1 3iwj = j3wj ,

w +

i

= jwj:

w

3

CIw

i=3 0, . .

Im w = 1=6:

x = t

0

0

! " # EF : y = 0 0 t +1

& ' E

F

:

u = 1=t

0 t +1, =0 $ 0 =+1 F =+1

$ F0 =0:

v = 0

# # ( . -

x = t

! " # ABC : y = t 1 t +1 & -

0

0

0

u = 1=2t

A

B

C

: v = 1=2t

1 t +1, .

, A=1

$ A0 =0 =0 $ 0 =1 =1 $ 0 =0:

0 , y = Im z x = Re z 1 2

1

(z z3+

1

(z

+ z3+ = 2: - & 2 & '

2

2i

w = 1=z:

1

1

1

1

1

1

1

1

!

+

! = 2 ,

w

w 2 w + w = 2ww ,

2i

w

w

2

w

w

2i

, v u = 2 u

2

2

,

u +

1

!

2

1

!

2

1

+ v

4

+ v + 4

= 8:

1+i =4 p

=4: -

2

! -

1.

1 D = fz : z 2CI

Re z 1g w = 1 + i z + 1+

2 D = fz : z 2CI

0 Re z 1=2 Im z 0g

w = 1=z:

2.

1 D = fz : z 2CI

Im z 1g w = 2

i z + 1+

2 D = fz : z 2CI

0 Im z 1

Re z 0g

w = 1=z:

3.

1 D = fz : z 2CI

Re z 1g w = 1 i z + 1+

2 D = fz : z 2CI

1 Re z 0 Im z 0g w = 1=z:

4.

1 D = fz : z 2CI

Re z 2g w = 1 + i z + 1+

2 D = fz : z 2CI

jz ij 1 jz 2ij 2g

w = 1=z:

5.

1 D = fz : z 2CI

Im z 1g

w = 2 i z + 1+

2 D = fz : z 2CI

jz 1j 1 jz 2j

2g

w = 1=z:

6.

1 D = fz : z 2CI

Re z 1g w = 1

i z + 1+

2 D = fz : z 2CI

1 Re z 0 Im z 0g w = 1=z:

7.

1 D = fz : z 2CI

Im z 1g w = 2 + i z + 1+

2 D = fz : z 2CI

0 Im z 1=2 Re z 0g

w = 1=z:

8.

1 D = fz : z 2CI

Re z 1g w = 1 + i z + 1+

2 D = fz : z 2CI

jz + ij 1 jz + 2ij 2g

w = 1=z:

9.

1 D = fz : z 2CI

Re z 2g w = 1

i z + 1+

2 D = fz : z 2CI

0 Re z 1

Im z 0g

w = 1=z:

10.

1 D = fz : z 2CI

Im z 1g

w = 1 + 2i z + 1+

2 D = fz : z 2CI

0 Re z + Im z 1g w = 1=z:

11.

1 D = fz : z 2CI

Re z 1g

w =

1

+ 2i z + 1

2 D = fz : z 2CI

jz + 1j 1

jz + 2j 2g

w = 1=z:

12.

1 D = fz : z 2CI

Im z 1g

w =

1 2i z + 1

2 D = fz : z 2CI

1 Im z 0

Re z 0g w = 1=z:

13.

1 D = fz : z 2CI

Re z 1g

w =

1 2i z + 1

2 D = fz : z 2CI

1 Re z + Im z 0g

w = 1=z:

14.

1 D = fz : z 2CI

Re z 2g

w =

2 + i z + 1

2 D = fz : z 2CI

0 Im z Re z 1g w = 1=z:

15.

1 D = fz : z 2CI

Im z 1g

w =

1 + i z + 1

2 D = fz : z 2CI

1 Im z Re z 0g

w = 1=z:

16.

1 D = fz : z 2CI

Re z 1g

w =

1 2i z + 1

2 D = fz : z 2CI

jz ij 1

Im z 0g

w = 1=z:

17.

1 D = fz : z 2CI

Im z 1g

w =

1 + 2i z + 1

2 D = fz : z 2CI

1 Im z 0

Re z 0g w = 1=z:

18.

1 D = fz : z 2CI

Re z 1g

w =

1 + 2i z + 1

2 D = fz : z 2CI

1 Re z 2 Im z 0g

w = 1=z:

19.

1 D = fz : z 2CI

Re z 2g

w =

2 i z + 1

2 D = fz : z 2CI

jz 1j 1

Re z 0g

w = 1=z:

20.

1 D = fz : z 2CI

Im z 1g

w =

1 i z + 1

2 D = fz : z 2CI

jz 2ij 2

Im z 0g

w = 1=z:

21.

1 D = fz : z 2CI

Re z 1g

w =

2 + i z + 1

2 D = fz : z 2CI

jz 2j 2

Re z 0g

w = 1=z:

22.

1 D = fz : z 2CI

Im z 1g

w =

1 + i z + 1

2 D = fz : z 2CI

jz + ij 1

Im z 0g

w = 1=z:

23.

1 D = fz : z 2CI

Re z 1g

w =

2 i z + 1

2 D = fz : z 2CI

jz + 2ij 2

Im z 0g

w = 1=z:

24.

1 D = fz : z 2CI

Re z 2g

w = 1

2i z + 1

2 D = fz : z 2CI

2 Im z 1

Re z 0g w = 1=z:

25.

1 D = fz : z 2CI

Im z 1g

w = 2 + i z + 1

2 D = fz : z 2CI

2 Re z 4 Im z 0g

w = 1=z:

26.

1 D = fz : z 2CI

Im z 1g

w = 1

i z + 1

2 D = fz : z 2CI

1 Re z + Im z 1g

w = 1=z:

27.

1 D = fz : z 2CI

Re z 1g

w = 1 + i z + 1

2 D = fz : z 2CI

jz + 1j 1

Re z 0g

w = 1=z:

28.

1 D = fz : z 2CI

Im z 1g

w = 1 2i z + 1

2 D = fz : z 2CI

0 Re z 1 Im z 0g

w = 1=z:

29.

1 D = fz : z 2CI

Re z 2g

w = 2 + i z + 1

2 D = fz : z 2CI

jz + 2j 2

Re z 0g

w = 1=z:

30.

1 D = fz : z 2CI

Im z 2g

w = 2

i z + 1

2 D = fz : z 2CI

0 Im z Re z 1g w = 1=z:

1 D = fz : z

2CI

jzj 1g G = fw : w 2CI Re w 1g

w i = 1 w 0 = 2

2 D = fz : z

2CI

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

jwj 2

Im w 0g w 0 =

2 w i = 0

jwj 1

3 D = fz : z

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

Im w Re wg w 0 = p2 1 + i =2 w 1 = 0:

.

. - "# *

az + b

w = cz + d

. c 6= 0 ad bc 6= 0

/,) '* - .