УрМатФиз / УрМатФиз с теорией

28.

D = : 2CI 0

Re z

2 Im z 0

z 62 =2 =2 + i g

z 623 =2 3 =2 + i g, w = sin z:

29.

D = fz : z 2CI Re z

0 0

Im z

2

z 62 1 + i =2 i =2 g

z 62 1 + i3 =2 i3 =2 g, w = ch z:

30.

D = fz : z 2CI 0

Re z

2 Im z

0

z 62 =2 =2 i g

z 623 =2 3 =2 i g, w = cos z:

" # $ % & # ' ( ' ) ( *

4u = 0 +0 r

1 0 ' 2 , 9

+2:7:2,

u 1 = g+', +0

' 2 , :

=

+2:7:3,

1, g+', = cos 4'/

2, g+', =

sin 2'

:

5 + 3 cos '

1

2

1 r2

u+r ', =

Z

u+1 , d :

2

1 2r cos+' , + r

2

0

u+z, = Re

1

I u+&,

& + z

d& jzj 1

2 i

+& z,&

j&j=1

# * 6 j&j = 1 ( ( $ (

* .

1, 7 (3& 3& $ ( +2.7.3, ( #

u

= cos 4 =

ei4

+ e i4

= u+&, =

&4 +

& 4

&

8 + 1

:

2

2

=

2&4

r=1

73$

I = I

+&8 + 1,+& + z,

d& =

I f+&, d&:

+2:7:4,

2&

4

+& z,&

j&j=1

j&j=1

f & & = 0 & = z

j&j = 1 & = 1.

! " # I = 2 ires f

& + res f & I

= 2 i res f & :

0

z

1

& ' f & ' & = 1:

( ) f & & = 1 :

&91 + 1=&81 + z=&

1

3

2

1

z

1

f & =

= 2

&

+ &

z +

+

!

=

2&61 z=&

&5

&6

1 z=&

1

z

2

3

4

= 21 &3 + &2z +

+

! 01 + z& + z&2 + z&3 + z&

4 + : : :1 =

&5

&6

@

A

= : : : + 1& 21z4 + 21z4! + : : : :

&2

1

sin 2

1

3

&

1 =

u r=1= 5 + 3 cos = u & =

2i

2

5 + 2 & + & !

&4 1&

=

&4 1

:

=

i&23&2 + 10& + 3

i3& + 3& + 1=3&

7

I = I

&4 1& + z

d& =

I

f & d&:

2:7:5

3i&2 & + 3& + 1=3& z

j&j=1

j&j=1

f & & = 0 & = 1=3

& = z j&j

= 1

. . .

& = 3

j&j = 1 & = 1:

I = 2 ires f & + res f

& + res f &

0

1=3

z

I = 2 i res f & res f & :

3

1

! " f & # #$ # % & = 3:

res f & =

lim

&4

1& + 3& + z

=

10z 3

:

3i&2 &

+ 3& + 1=3& z

9i z

+ 3

3

&!3

! " f & & = 1:

) *

! " f & & = 1 :

&5 1

1

! 1 + z& !

&4

f & = 3i&5 1 + 3& ! 1 +

1

! 1 z& ! =

3&

1

1

z

2

1

1

3

2

=

1+ z

!01

3 + 32

+: : :1 1

+

+: : :!01+

+ z2 +: : :1=

4

5

2

2

3i

&

&

&

@

&

&

A

3&

3

&

@

&

&

A

1

1

1

1

1

10

=

: : :

+ &

z 3 3

+ z! + : : :! = : : : +

2z

! + : : : :

3i

&

3i

3

+ c 1 = 6z 10=9i | .!! " # 1=&, -

1 , res f & = 1 = 6z 10=9i :

2$ 2.7.5

1

2

3

3

1

10z

3

+

6z 10

6z2

2z

I = 2 ires f & + res f &=2 i

5

=2

:

4

9i z + 3

9i

9z

+ 3

6 * ,

u rei' = u r ' = Re 0

1

2

2 3z2 z

1 = Re

0

2

3z2

z7z + 3

1 =

2

@

2 i

9z + 3 A

@9i

jz + 3j

A

2

1

3zjzj2jzj2 +9z2 3z

2

1

3r3ei' r2 +9r2ei2' 3rei'

=

Re 0

2

1=

Re

0

2

1=

9

@i

jz + 3j

A

9

@i

r

+ 6r cos ' + 9

A

2

3r3 sin '+9r2 sin 2' 3r sin

'

= 9

r2 + 6r cos ' + 9

:

. u r ' =

2

r3 sin ' + 3r2 sin 2' r sin

'

:

3

r2 + 6r cos ' + 9

2.7.7.

-

4u = 0 !0 r 1 0 '

2 $ 9

u

1 = g!'$ !0 ' 2 $:

=

23.

g ' =

cos '

:

24.

g ' =

sin 2'

:

3 sin '

5

3 sin '

5

25.

g ' = 2 + sin 4':

26.

g ' =

1

:

5 + 3 cos '

27.

g ' =

cos 2'

:

28.

g ' =

1

:

5 + 3 cos

'

4 sin '

5

29.

g ' =

cos 2'

:

30.

g ' =

sin 2'

:

4 sin '

5

4 sin '

5

4u = 0 x y 2 D:

2

1

1 D = f x y 2 IR

:

1 x +1 y 0g u y=0=

:

x2 + 2x + 3

u y=1= x x 1 :

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

u r=1= 1

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

5 D =

f

x y

2

IR2

:

1

x +

1

y 0 y

.0 1/

62

g

u

y=0

= x + 1

x

u x=+0 = 1

u x= 0 = 0:

0 y 1

0 y 1

u x y =

1 +

y

u t 0

dt:

2:7:6

1 t

x 2 + y2

& !

' ( ' ( z = x + iy & = t + is ) '

2.7.6 ' + !,

u z = Re

1

+1u t 0

dt:

i 1

t

z

, -

u

=

R x

j

R x

j

A

x

! 1

y=0

jxj

!"

, # ! -

! !" " &k, # !" # -

:

n

0 R &k 1

u z =

2 Re

X

res

:

2:7:7

@&k z A

k=1 Im

&k 0

1

2.7.7 , ! x2 + 2x + 3 =

= x + 1 + ip2x + 1 ip2 :

u z = 2 Re res

& + 1 + ip

1

ip

=

2& + 1

2&

z

&

ip

= 1

2

1

1 + x

i p

+ y

=

2 Re

=

Re

2

=

i2p2

1

ip2

z

ip2 1 + x 2 + p2 + y 2

1

p2 + y

= p

1 + x 2 + p

+ y 2

:

2

2

p

+ y

1

2

. u x y = p

1 + x 2 + p

+ y 2

:

2

2

D = fz : z 2CI

Re z 0

Im z 0g -

! w =z2 "

8 4u~ = 0

w 2 G

0

u~

= #1 :

:

/ 0 1 * 2 " " !-

! 3 #2.7.6

1

+1

u~#t 0

1

dt

1

t

1

u~# =

dt =

=

arctg

=

2

2

2

2

1Z #t +

1Z

#t +

!

1

1

1

1

1

1

1

=

arctg

!

2

! = 2 +

arctg

! :

5 w = z2 + i = x2 y2 + 2iy

= x2 y2 = 2xy:

! * ) ) #x y, *

u~#x2 y2 2xy = u#x y = 1 + 1 arctg 0

1 x2 + y2 1 :

2

@

2xy

A

. u x y =

1

+

1

arctg

01

x2 + y2 1

:

2

@

2xy

A

8 4u~ = 0 w 2 G

=

0

u~

= # + e #

+ 1 + #

1:

:

0 1 2 , 3$ $ "-

" ! 4 #2.7.6

1 +1

u~#t 0

1

dt

+1

dt

u~# = 1Z

dt =

2 Z

+ Z

3 =

#t 2 +

2

#t 2 + 2

#t 2 +

2

6 e

1