УрМатФиз / УрМатФиз с теорией

2

= 0

i'

1

2

= 0

i'

i'

1

2

w =

1 + z

!

1 + rei'

1 + rei'1

re

i'

=

1 z

@1

re

A

@1 re

1

re

A

=

1

r2 + 2ir sin ' 2

1 + r2

2r cos ' 2

:

=

1

r2 2

4r2 sin2 '

=

4r1

r2 sin '

:

2:7:8

1 + r2

2r cos ' 2

1 + r2

2r cos ' 2

1

1

1

!

u r ' =

2 +

arctg

!

2.7.8 .

w = z

+ 1=z =2: * + +

u z = u zw

u~ w

8 4u~ = 0 w 2 G

=

0

u~

=

+ 1:

:

. / 0 1 + + 2

1

+1

u~t 0

+1

dt

1 1

1 +

u~ =

Z

dt

=

Z

= 2 + arctg

!

t

2 + 2

t

2 + 2

1

1

w = + i =

1

rei' +

1e i'

!

=

1 + r2 cos ' + i1 r2 sin ':

2

r

2r

2r

D =

f

z : z

2

CI Im z 0 z

60 i7

1 '%&

62

g 2

2

+ 1 && %

! $+ w = pz

G = fw : w

2

2 CI Im w 0g, w = p | + %

+ . 2.7.8, .

u x y

.

" u x y

# $ #-

$ u z = u zw

u~ w

#$

8

4u~ = 0

w 2 G

= +

p

2

+ 1 +

1 :

u~

0

:

,

2.7.6

1

+1

u~ t 0

1

1

t

1

1

t

u~ =

Z

dt =

Z

+

Z

=

t +

t

2

+

2

t

2

+

2

1

p2

0

=

1 2arctg

t

1

+ arctg

t

1

3 =

4

! p

! 0

5

2

= 1 2arctg 0p

+ 1

1 +

1

3 :

2

arctg

!

+ arctg

!

arctg

4

@

A

!5

2

2

4

y2 2 + 4x2y2 exp i

!

w = pz2

+ 1 = qx2 + 1

y2 + 2ixy = q x2

+ 1

2

1

p

+

1

1 +

1

2

arctg

!+arctg

u x y =

2arctg 0

! arctg !3

4

@

A

5

2.7.9 .

D =

f

z : z

2

CI

Im z 0

j

z

i=2

j

1=2

g

$ $) $ -

1=z+i

=

e

=z

%%

+(% $

' , w = e

$( G = fw : w 2CI

Im w 0g . 2.7.8,

.

w = e =z

!

. 2.7.8,

/ $ $

' ), $

,

$0 u x y ,.

u x y -

u z = u zw u~ w

8 4u~ = 0

w 2 G

=

1:

u~

:

0

! " # " -

$ % 2.7.6

1

+1

u~t 0

1

+

dt

1

1

+1

u~ =

Z

dt=

=

arctg

!

=

t

2 + 2

t

2 + 2

1

1

1

1

1

1

! :

= 2

arctg

w =

ez

=

e x

iy = x2+y2

=

e= x2+y2 e iy=

x2+y2 -

:

=

=

x2+y2

cos y= x

2

+ y

2

e

=

x2+y2

sin y= x

2

2

:

2:7:10

= e

+ y

1

1

1

! !

u x y

2

arctg

2.7.10

7 #

$ $ z = x + iy: % & & '

D = fz : z 2CI

jz i=2j 1=2

jz ij

1g

( -

)'$

! & * w = e2z

$$ -

' G = fw : w 2CI

Im w 0g . 2.7.8,

4.

D = f r cos ' r sin ' 2IR2 : 0 r 1

0 ' g

u 1 = 1

u '=0= u '= = 0:

5.

D =

f

x y

2

IR2

:

1

x +

1

y 0 y

0 1

62

g

u

y=0

= x

x

1

u x= 0 = 0

u x=+0 = 1:

0 y 1

0 y 1

11.

D = f r cos ' r sin ' 2IR2 : 0 r 1

0 ' g

u r=1= 0

u '=0= u '= = 1:

12.

D =

f

x y

2

IR2

:

1

x +

1

y 0 y

0 1

62

g

u

y=0

= 0 u x= 0 = 1:

0 y 1

13.

D = f x y

2 IR2

:

y 0 y 1=2 2 + x2 1=4g

u y=0= x

u y1=2 2+x2=1=4 = 0:

14.

D = f x y

2 IR2

:

y

1 2 + x2 1

y 1=2 2 + x2 1=4g

u x=p

= 1 u x=

p

= 0 u y 1=2 2+x2=1=4 = 0:

1 y 1 2

1 y 1 2

15.

D = f x y

2 IR

2

:

x +1

y 0g

u y=0=

x

:

4 + x2

16.

D = f x y 2IR2 :

x 0

y 0g

u y=0= x

1

u x=0= y

1 :

17.

D = f x y 2 IR2 :

x +1

0 y 1g

u y=0= x

u y=1= x :

18.

D = f r cos ' r sin ' 2IR2 :

0 r 1 0 ' g

u r=1= 0

u '=0= 1 u '= = 0:

19.

D =

f

x y

2

IR2

:

x +

1

y 0

y

0 1

62

g

u

y=0

= x

x

1

u x=+0 = 1

u x= 0 = 0:

0 y 1

0 y 1

20.

D = f x y 2IR2 :

y 0

y

1=2 2+x2 1=4g

u y=0=

x

u y1=2 2+x2=1=4 = 0:

21.

D = f x y 2 IR2 :

y

1 2 + x2 1

y 1=2 2 + x2 1=4g

u x=

p

= 1 u x=p

= u x2+ y 1=2 2=1=4 = 0:

1 y 1 2

1 y 1 2

22.

D = f x y 2 IR2 :

x +1

y 0g

u y=0=

1

:

x2

x + 1

23.

D = f x y 2 IR2 :

x 0

y 0g

u y=0= x

x

1

u x=0= 0:

24.

D = f x y 2 IR2 :

x +1

0 y 2g

u y=0=

x

u y=2= 0:

25.

D = f r cos ' r sin ' 2IR2 :

0 r 1 0 ' g

u r=1= ' =2 u '=0= u '= = 0:

26.

D =

f

x y

2

IR2

:

x +

1

y 0

y

0 1

62

g

u

y=0

= 1 u x= 0 = 0:

0 y 1

27.

D = f x y 2 IR2

:

y 0 y

1=22 + x2 1=4g

u

y=0

= 1

u y1=2 2+x2=1=4 = 0:

28.

D = f x y 2 IR2

:

y

12 + x2 1

y

1=22 + x2 1=4g

u y1 2+x2=1=4 = 1 u y 1 2+x2=1 = 0:

29.

D = f x y 2 IR2

:

x +1

y 0g

u

=

x

:

y=0

1 + 4x2

30.

D =f x y 2 IR2 :

x 0

y 0g

u

=0

u

= y

y 1:

y=0

x=0

L+v, + x- v = 0

x- 2 D

2:8:1

# .)## ! ! / ! ! .

@v

+ v!

@n-

@D= 0

j j + j j 6= 0

2:8:2

L v = div kgradx

v q x v

x

1

x 2 C D k x

2 C

D

x x 0 x 0 x

x 2 C @D n |

@D.

-

.

2 "

# $ n:

1

2

3 : : :

n : : : :

. 1 2

x D

"

D, * " # f x, 1-

* D

$" " "

2

x dx =

2

2

f x

fk jjvkjj