f (x) dx

)x(f=y

O

0)

0

A

O

o o

o o

o o

: e-mail: strantoneas@gmail.com : http://users.sch.gr/stranton 6974593717 27310 28791

1

1. 3 1 5 2 28 51

1. 53 1 55 2 80 3 x0\ 101 4 106 5 o x0\ 128 6 136 7 148 8 159 182

2. 185 1 187 2 206 3 233 4 242 5 Rolle 247 6 261 7 272 8 281 9 291 10 305 11 de L Hospital 315 12 323 326

3. 331 1 333 2 344 3 357 4 F(x) = ( )

x

f t dt 368

5 378 6 407 423

428 449

1

__________________________________________________________

-

. - . -. - . . - . .

. . - , . - . , , . , 16 - , . , . , -

, . , . , . - , , , , . -

, . - , , . , -, . , . , - . - . Bolzano . - , [, ].

T

2

, - . ,

. - , , - . - Rolle, , , - . - - . . .

. . - , - . , , . - - , .

, , - . - . ,

, .

, 2012

, -

. - . . , , .

WILLIAM THRUSTON ,

3

1.

- ` ] _ \ ^

- . ( ). - - 3 4 - 16 .

- i. , - . - , .

- . - , . , - , , 16 .

-, . - - . . - .

1: & 5

1

, , , -, .

GOTTFRIED LEIBNIZ(16461716)

16 .

x = 2 4

2

.

- - . H - , + , , . , - , . , . , . -, , .

^ \ , :

6 1:

- -

, \ , (0) (1) .

i , i2 = 1. z ^ -

z = + i, , \ . + i, , \ -

i . O z Re(z). O z Im(z). ^ + 0i,

i 0 + i.

^

(z1 + z2) + z3 = z1 + (z2 + z3 ) (z1.z2).z3 = z1.(z2.z3 ) z1 + z2 = z2 + z1 z1.z2 = z2.z1

z + 0 = 0 + z = z z.1 = 1.z = z

z + (z) = (z) + z = 0 z . 1z = 1z

.z = 1, z 0 z1.(z2 + z3) = z1.z2 + z1.z3 z + z1 = z + z2 z1 = z2 z.z1 = z.z2 z1 = z2 , z 0

z + z1 = z2 z = z2 z1 z.z1 = z2 z = 21

zz , z1 0

+ i : + i = + i = = . 0 = 0 + 0i, : + i = 0 = 0 = 0. z = + i z = i - z. \ ^

\ ^ . - . . + i < + i

< = = 0. z = + i - - (, ) . - z. -

1: & 7

- -

. z = + i -

, OM , (, ). z = i z ,

(, ). MO (, ). z1 = + i z2 = + i

: z1 + z2 = ( + i) + ( + i) = ( + ) + ( + )i.

z2 = + i z1 = + i : z1 z2 = ( + i) ( + i) = ( ) + ( )i.

.

1 2

1(, ) 2(, ) + i + i - , ( + i) + ( + i) = ( + ) + ( + )i (+, +). ( 1) , OM

JJJJG = 1OMJJJJJG

+ 2OMJJJJJG

, : + i + i .

, ( + i) ( + i) = ( ) + ( )i (, ). ( 2) , ON

JJJG = 1OMJJJJJG

2OMJJJJJG

, : + i + i .

z1 = + i

z2 = + i : z1.z2 = ( + i).( + i) = + i + i + (i)(i) = + i + i + i2 = = + i + i = ( ) + ( + )i.

z1 = + i z2 = + i z2 0 :

8 1:

- -

1

2

zz =

i i++ =

( )( )( )( ) i i i i+ + = 2 2

( ) ( ) i

+ + + = 2 2

++ + 2 2

+ i.

> 1 z1 = z , z2 = z.z , - z = z1.z. z 0 , z0 = 1 , z = 1z . 4

:

i = i4+ = i4.i = (i4).i = 1.i = i =

1 , 0, 1

1 , 2, 3

i

i

= = = =.

^ \ .. (z w)2 = z2 2zw + w2 , (z + w)(z w) = z2 w2 , (z w)3 = z3 3z2w + 3zw2 + w3 , . - . = 1 + (1) , S = 2

(1 + ) = 2 [21 + (1)]

= 1 . 1 , S = 1( 1)

1

, 1.

! \ ^ . : x, y\ x2 + y2 = 0 x = y = 0.

(x2 + y2 = 0 x2 = y2 0. x2 0, x = 0 y2 = 0 y = 0) z, w^ z2 + w2 = 0 z, w -. , z = 1 w = i z2 + w2 = 12 + i2 = 1 1 = 0.

1.1. : z = + i z = i -

: (i) z+ z = 2a (ii) z z = 2i (iii) z z = 2 + 2

: (i) z + z = ( + i) + ( i) = + i + i = 2. (ii) z z = ( + i) ( i) = + i + i = 2i. (iii) z. z = ( + i)( i) = 2 (i)2 = 2 2i2 = 2 + 2. (i) : Re(z) = = 2

z z+ . (ii) : Im(z) = = 2z z

i .

1.2. : z :

(i) z\ z = z. (ii) z I z = z.

1: & 9

: z = + i, , \ : (i) z = z i = + i 2i = 0 = 0 z\ . (ii) z = z i = (+i) i = i 2 = 0 = 0 z I. 1.3. : z1 = + i z2 = + i ,

: (i) 1 2z z+ = 1z + 2z (ii) 1 2z z = 1z 2z (iii) 1 2z z = 1z . 2z (iv) 1

2

zz

= 1

2

zz , z2 0

: (i) 1 2z z+ = )()( ii +++ = i )()( +++ = (+) (+)i = + i i = (i) + (i)

= 1z + 2z . (ii) (1 )

1 2z z = ( ) ( ) i i+ + = ( ) ( ) i + = () ()i = i + i = (i) (i) = 1z 2z . (2 )

1 2z z = 1 2( )z z+ = 1z + ( )2z = 1z 2z . ( )2z = )( i+ = i = +i = (i) = 2z .

(iii) 1 2z z = ))(( ii ++ = i )()( ++ = () (+)i = = i i = (i) i(i) = (i)(i) = 1 2z z .

(iv) (1 )

1

2

zz

= i i+ + =

( )( )( )( ) i i i i+ + = 2 2

( ) ( ) i

+ + + = 2 2

++ 2 2

+ i.

1

2

zz =

i i =

( )( )( )( ) i i i i + + = 2 2

( ) ( ) i

+ + + = 2 2

++ 2 2

+ i.

12

zz

= 1

2

zz .

(2 )

1z = 1 22

z zz 1z =

1

2

zz

2z 1

2

zz

= 1

2

zz .

1.3.1. : :

1 2 ... z z z+ + + = 1z + 2z + + z . 1, 2, , \ 1 1 2 2 ... z z z + + + = 1. 1z + 2. 2z + + . z .

1 2 ... z z z = 1z . 2z . . z . z1 = z2 = = z = z z = ...

z z z

= ...

z z z

= ( )z .

10 1:

z = ( )1z = 1z = ( )1 z = ( ) z , z 0. z2 + z + = 0 , , \ 0 Vieta

z2 + z + =0 2z

+ za +

= 0 z2 + 2z 2

=

z2 + 2z 2

+ 2

2

= 2

24

2

2z

+ = 2

24

4

2

2z

+ = 2

4 = 2 4 .

:

> 0 : z1,2 = 2

.

= 0 : z = 2

.

< 0 24 = 2( 1)( )

4 = ( )22 2(2 )i =

2

2i

-

: 2

2z

+ = 2

2i

.

: z1,2 =

2 i

, -

.

: z1 + z2 =

z1 . z2 =

.

______________ ___________________________________________

, .

, .

z + i Re(z) , Im(z) , .

1.4. x, y :

(2x + 5y 6) + (5x 6y 2)i = (7+60i). :

(2x + 5y 6) + (5x 6y 2)i = 7 60i 2 5 6 75 6 2 60

x yx y+ = =

2 5 15 6 58

x yx y+ = = .

D = 2 55 6 = 12 25 = 39 Dx =

1 558 6 = 6 + 290 = 296

Dy = 2 15 58

= 116 + 5 = 111.

1: & 11

x = xDD = 296

37 = 8 y = yD

D = 11137

= 3.

1.5. z = (+i)(i+) (+i)(1+2i) + 2i. ) z +i. ) , z I. ) , : z\ .

: ) z = (+i)(i+) (+i)(1+2i) + 2i = 2i + + i2 + 2i (+2i+i+2i2) + 2i

= 2i + + 2i 2i i + 2 + 2i = (2) + (2+22+1)i. ) z 2 = 0 = 2 \ . ) z\ 2 + 2 2 + 1 = 0 ( 1)2 + 2 = 0 1 = 0 = 0

= 1 = 0.

1.6. w = (3i)(i+2) ii11+ . Re(w) Im(w).

:

w = (3 i)(i + 2) 11ii

+ = 3i + 6 i

2 2i 2(1 )

(1 )(1 )i

i i

+ = 3i + 6 + 1 2i 21 2

1 1i i ++

= 7 + i 1 2 12i = 7 + i 22

i = 7 + i + i = 7 + 2i. Re(w) = 7 Im(w) = 2.

.

-: i :4 -

0, 1, 2, 3. i 1, i, 1 i .

z z^ , z2, z3, -. , -, i. : (1 i)2004 . (1 i)2 = 1 2i + i2 = 1 2i 1 = 2i. (1 i)2004 = [(1 i)2]1002 = (2i)1002 = 21002.i1002 = 21002.i2 = 21002.

, -, .

1.7. ) i1962.

) w = i + i2 i3 + i1961 + i1962 -

12 1:

+i. ) w4 = 4.

: ) 1962:4 2, i1962 = i2 = 1. ) 1962 i i.

w = i1962( ) 1

1i

i = i

1962 11

ii + = i

1 11 i + =

21

ii

+ =

2 (1 )(1 )(1 )

i ii i

+

= 2 (1 )1 1i i + =

2 (1 )2

i i = i(1i) = i + i2 = 1i. ) 2w = (1+i)2 = (1)2 + 2(1)i + i2 = 12i 1 = 2i.

4w = ( )22w = (2i)2 = 4i2 = 4.

1.8. 4, (1+i)(1+i2) = 0.

: 1 : 4, : = 4+1, ` , (1+i)(1+i2) = (1+i4+1)(1+i8+2) = (1+i1)(1+i2) = (1+i)(11) = 0. = 4+2, ` , (1+i)(1+i2) = (1+i4+2)(1+i8+4) = (1+i2)(1+i0) = (11)(1+1) = 0. = 4+3, ` , (1+i)(1+i2) = (1+i4+3)(1+i8+6) = (1+i3)(1+i2) = (1i)(11) = 0. 2 : (1+i)(1+i2) = 1 + i + i2 + i3. - 1 = 1 = i 1, 4. ,

(1+i)(1+i2) = 14( ) 11

ii

= 4 1

1

ii =

1 11i

= 0

1.9. z = 32 + 12 i.

) z3. ) z2002 , z2019 . ) , : (+i)z2002 + z2 z2019 iz = 3 + i.

:

) z3 = 3

3 12 2 i

+ = ( )31 32 i

+ = 18 ( 3 +i)

3 = ( ) ( )3 2 2 31 3 3 3 3 38 i i i + + + = 18 (3 3 +9i3 3 i) =

18

.8i = i.

) z2002 = z3 . 667+1 = ( )6673z z = i667 z = i3 z = i z = i 3 12 2 i + = 12 32 i. z2019 = z3 . 673 = ( )6733z = i673 = i1 = i.

) 2z = 2

3 12 2 i

= 2

32

23 1

2 2 i + ( )212 i = 34 32 i 14 =

12

32 i .

- 1 - :

S = 111

.

2002:3 677:4 3. 2019:3

1: & 13

(+i).z2002 + 2z z2019 iz = 3 + i (+i) 1 32 2 i

+ 1 32 2 i

.i i 3 12 2 i

+ = 3 + i 1 32 2 i

(+i+i) i3 1

2 2 i + = 3 + i

1 32 2 i

(+2i) i 3 1

2 2 i + = 3 + i

2 + i 32

i + 3 32 i + 2

= 3 + i

2 + ( ) 32

i+ = 0 + = 0.

, .

1.10. : ) = (+i)4 (i)4. ) = (+i)4+2 + (i)4+2.

: ) (1 )

(+i)2 = 2 + 2i + (i)2 = 2 2 + 2i = p (i)2 = 2 2i + (i)2 = 2 2 2i = p. , = (+i)4 (i)4 = [(+i)2]2 [(i)2]2 = p2 (p)2 = p2 p2 = 0. (2 ) = (+i)4 (i)4 = (+i)4 (i2i)4 = (+i)4 [i(+i)]4 = (+i)4 i4(+i)4 = (+i)4 1.(+i)4 = (+i)4 (+i)4 = 0.

) (1 ) (+i)2 = p (i)2 = p. , = (+i)4+2 + (i)4+2 = (+i)2 (2+1) + (i)2 (2+1) = [(+i)2]2+1 + [(i)2]2+1 = p2+1 + (p)2+1 = p2+1 + (p2+1) = p2+1 p2+1 = 0. (2 ) = (+i)4+2 + (i)4+2 = (+i)4+2 + (i2i)4+2 = (+i)4+2 + [i(+i)]4+2 = (+i)4+2 + i4+2 (+i)4+2 = (+i)4+2 1.(+i)4+2 = (+i)4+2 (+i)4+2 = 0.

- .

z -, : z = z z

z = z z . z = w + w z = w . w z .

z = w w z . , -

14 1:

z = +i = 0 z\ = 0 z .

1.11. ) z = zz

1

2 zz

1

2 z .

) z

i w = ( )i ww + i1 z . :

) z = 1 12 2

z zz z

= 1

2

zz

1

2

zz

= 1

2

zz

1

2

zz =

1 1

2 2

z zz z

= z. z I.

) zi w = ( )i ww + 1i zi w = i ww + 1i z = i2w + w z = w + w . z\ .

- .

^ : 1

. , .

2 (z2 + z + = 0 , , \ 0) - = 2 4 - - . , 2 , Horner.

z = x + yi , x, y\ x, y.

1.12. , , :

) z

i2 + 1 = z i ) z

i1 z i

i2+ = z + i 112

. :

) 2z

i + 1 = z i z + 2 i = (2i)z i(2i) z + 2 i = 2z iz 2i + i2

z 2z + iz = 2 + i 2i 1 z + iz = 3 i (1 i)z = 3 + i z = 31ii

+

z = (3 )(1 )(1 )(1 )i ii i

+ + + =

3 3 11 1i i+ + + =

2 42

i+ = 2(1 2 )2i+ = 1 + 2i.

) 1z

i 2z i

i+ = z +

112

i (1 )(1 )(1 )z i

i i+

+ ( )(2 )(2 )(2 )z i i

i i + = z +

112

i

2z iz+ 2 2 15

z iz i = z + 112i 5(z + iz) 2(2z iz 2i 1) = 10z + 5(i 11)

1: & 15

5z + 5iz 4z + 2iz 4i + 2 = 10z + 5i 55 9z + 7iz = 57 + i (9 7i)z = 57 i z = 579 7

ii

=

(57 )(9 7 )(9 7 )(9 7 )

i ii i

+ + =

513 399 9 781 49

i i+ ++ =

520 390130

i+ = 520130 + 390130

i = 4 + 3i.

1.13. ) , , : z3 3z2 + 4z 2 = 0 ) : z3 + 4z = 3z2 + 2 z10 32z + 32 = 0 .

: ) z3 3z2 + 4z 2 = 0 : 1 , 2.

z3 3z2 + 4z 2 = 0 (z 1)(z2 2z + 2) = 0 z 1 = 0 z2 2z + 2 = 0

z = 1 z = 1 + i z = 1 i

[ = (2)2 4.1.2 = 4 8 = 4 z = 2 22i = 2(1 )2

i = 1 i. ] ) : z3 + 4z = 3z2 2 z3 3z2 + 4z 2 = 0 z = 1 z = 1 + i z = 1 i . .

z1 = 1 : 101z 32z1 + 32 = 110 32.1 + 32 = 1 32 + 32 = 1 0 z2 = 1 + i : 22z = (1 + i)2 = 1 + 2i 1 = 2i 102z = ( )522z = (2i)5 = 32 i5 = 32i 102z 32z2 + 32 = 32i 32(1 + i) + 32 = 32i 32 32i + 32 = 0

z3 = 1 i : 103z 32z3 + 32 = 102z 32 2z + 32 = 102 232 32z z + = 0 = 0. : 1 + i , 1 i .

1.14. ) z : (1+i)z + 4 z = 8+10i. ) w = z +z + z z+

z z +

2 2

3 35 2

36 +i. ) w14 .

: ) 1 : z = x + yi x, y\ . : (1 + i)z + 4 z = 8+10i (1 + i)(x + yi) + 4(x yi) = 8 + 10i x + yi + xi y + 4x 4yi = 8 + 10i (5x y) + (x 3y)i = 8 + 10i

==10385

yxyx

)5(

=+=

5015585

yxyx

14y = 42 y = 4214 y = 3

x 3(3) = 10 x + 9 = 10 x = 1. z = 1 3i.

1 3 4 2 1 1 2 2

1 2 2 0

16 1:

2 : zzi 4)1( ++ = i108+ (1 i) z + 4z = 8 10i. :

(1 ) 4 8 104 (1 ) 8 10

i z z iz i z i+ + = + + =

D = 1 44 1

ii

+ = (1 + i)(1 i) 16 = 1 + (1)

2 16 = 1 + 1 16 = 14.

Dz = 8 10 48 10 1

ii i

+ = (8 + 10i)(1 i) 4(8 10i) = 8 8i + 10i + 10 32 + 40i = 14 + 42i.

z = zDD = 14 42

14i +

= 1 3i. ) z + z = 2 , z z = 6i z. z = 1 + 9 = 10 ,

w = 2 2

3 35 2

36z z z z

z z+ + +

+ = 2

3 2 2( ) 2 5 2

( ) 3 3 36z z z z z z

z z z z z z+ + + + + =

2

3( ) 3 2

( ) 3 ( ) 36z z z z

z z z z z z+ + +

+ + = 2

3( ) 3 2

( ) 3 ( ) 36z z z z

z z z z z z+ + +

+ + = 2

32 3 10 2

( 6 ) 3 10( 6 ) 36i i+ +

+ + = 34 30 2

216 180 36i i+ +

+ = 36

216 180 36i i + = 36

36 36i+ = 36

36(1 )i+ = 1

1 i+ = 1

(1 )(1 )i

i i

+ = 11 1

i+ =

12

12 i .

) w = 21 (1 )2 i

= 14 (1 i)

2 = 14 (1 2i + i2) = 14 (1 2i 1) =

14 (2i) =

12 i

w14 = (w2)7 = ( )712 i = 712 i7 = 712 i3 = 712 (i) = 712 i. , w 14 = 14

1w

= 7

112

i =

72i = 2

128 ii = 1281

i = 128i I.

- .

(x,y) z = x+yi w ..., -, , , . z. x, y .

(x,y) z = x + yi , - - .

1.15. z=x+yi x, y\ w = (zi)( z +1).

1: & 17

) w +i. ) w\ , z . ) w I , z C. ) C - , . , .

: ) w = (z i)( z + 1) = z z + z i z i = x2 + y2 + x + yi i(x yi) i

= x2 + y2 + x + yi xi y i = (x2 + y2 + x y) + ( x + y 1)i . ) w\ x + y 1 = 0 y = x + 1 z : y = x + 1.

) w I x2 + y2 + x y = 0 x2 + 2.x. 12 + 14 + y

2 2.y. 12 + 14 =

14 +

14

( ) 212x+ + ( ) 212y = 12 . z C: ( ) 212x+ + ( ) 212y = 12 .

) C ( 12 ,12 ).

x = 12 y = 12 + 1 =

12 .

C . C . y = x + 1 : x2 + y2 + x y = 0 x2 + (x+1)2 + x (x+1) = 0 x2 + x2 + 2x +1 + x x 1 = 0 2x2 + 2x = 0 2x(x+1) = 0 x = 0 x+1=0 x = 0 x = 1. x = 0 y = x + 1 = 0 + 1 = 1 x = 1 y = x + 1 = 1 + 1 = 0. C (1,0) (0,1). () = ( ) ( )

2OA OB = 1 12

= 12 ..

1.16. f (z) = z ii z

4+2

z^ . ) f.

) z *^ { 12 i} : ( )f z1 = ( )f z1 . ) z

f (z)\ . :

) iz+2 0 iz 2 z 2i z 22( )ii

z 2i.

f ^ {2i}.

18 1:

) z 0 , z1 2i z 12i z 22

ii

z 21 i z

1 2 i z1 2 i

z1 2 i z 12i z 21 i.

z *^ { 12 i} : f ( z1 ) =

1 4

1 2

izi z

+

= 1 4

2

izz

i zz

+ =

1 42iz

i z+

f ( z1 ) = ( )

1 4

1 2

izi z

+

= 1 4

2

i zz

i zz

+ =

1 42iz

i z + =

4 12

izi z

+ .

1( )f z = 1 4

2iz

i z + =

1 42iz

i z+

+ = 4 1

2i zi z

+ = f (

1z ).

) z = x + yi x, y\ (x,y) (0,2). : f (z)\ )(zf = f (z) ( )42z iiz+ = 42z iiz+ 42z iiz+ + = 42z iiz+ ( z + 4i)(iz + 2) = (i z + 2)(z 4i) iz z + 2 z + 4i2z + 8i = iz z + 4i2 z + 2z 8i 2iz z 6z + 6 z + 16i = 0 iz z 3(z z ) + 8i = 0 i(x2 + y2) 3.2yi + 8i = 0 x2 + y2 6y + 8 = 0 x2 + (y3)2 = 1. x = 0 y = 2 : 02 + (23)2 = 0 + 1 = 1. z C: x2+(y3)2 = 1 (0,2).

___________________________________________________________________ 1.17. \ : (2 1) + (10 9 10)i = 0.

[.=-1] 1.18. x, y, :

3 2x

i 1y

i+ = 9 15i

i+

[.x=5 , y=2]

1.19. ) : i i+ +

i i+ = 2

2 2

2 2

+ , \ (, ) (0,0).

) z = 33

ii

+ +

33

ii

+ , Re(z).

) w = 121 153121 153

ii

+ +

121 153121 153

ii

+ , Im(w).

[.)1, )0] 1.20. z = 5 13i. z = x(1 + i) + y(1 i).

[.x=-4, y=9]

1: & 19

1.21. z1 = 1 + 2i z2 = 12 i. w

w = z1 + z2 , \ , : ) w = 5+2i ) w = 112 +

76 i.

[.)=3 , =4 )=1/4 , =- 23 ]

1.22. , , *\ 6 =

4 = 5

z = w , z = 2( ) + (2 )i

w = 23 + 2 i.

1.23. z1, z2^ Re(z1.z2) = Re(z1).Re(z2) z1\ z2\ . 1.24. x\ , z = 4 9

xix i++ .

[.x=6] 1.25. w = 2x i

y i+ x, y\ .

) x, y w\ . ) w , 2.

[.)x+2y=0] 1.26. z = 4

2x i

yi+ x] y *` . z . :

) x = 8y .

) z = 4y

) z 2 , x, y. [.(x,y)=(-8,1) (-4,2)]

1.27. 1, 2, 3 4 z1, z2, z3 z4 . 1234 z1 + z3 = z2 + z4 .

1.28. w = 16592009

1990

2i ii

.

[.i] 1.29. + i , , \ :

i) z = ( )963 77 3ii+ + ( )351 33 ii+ ii) w = ( )1336 55 6ii+ + ( )1679 22 9ii+ [.i)z=1-i ii)0]

1.30. :

) i

i+ + ( ) i i+ = i + i ) i i + + ( ) i i+ = (i) + (i).

1.31. :

20 1:

) S1 = i + i3 + i5 + + i1777. ) S2 = (1 + i) + (2 + 3i) + (22 + 5i) + + (299 + 199i). ) S3 = (20 + i) + (17 2i) + (14 + 4i) + + (31 217i). ) S4 = i + (2 + 3i) + (4 + 5i) + (6 + 7i) + + [(2 2) + (2 1)i] , *` .

[.)i )2100-1+104i )99-87381i )(-1)+2i] 1.32. z = 2 2i + 2 2 . ) z2 , z4 , z8. ) : z1792 = 21792.

1.33. z = 2 3 + i 2 3 + . ) z2 , z4 , z12. ) : z1980 = 21980.

1.34. z = 12 i 3

2 .

) z3, z1006, z1531 . ) w = i.z1006 + z1531 + 1254 3i z + i + i. ) w2010 = 22010.

[.)w =2i] 1.35. z z + 1z = 1, :

) : z3 = 1. ) w = z2015 + 2015

1z

. [.)1]

1.36. : ) (3 + 4i)2024 (4 3i)2024 = 0 ) (5 + 2i)2010 + (2 5i)2010 = 0 ) (2 + i)1992 (1 2i)1992 = (1 i)2138 + (1 + i)2138

1.37. i3+1 = 1.

[.=4+1] 1.38. = (2 + i2)(3 i) , ` . 1.39. z = 12i, :

i) 1z +1z ii) z

2 + 2z iii) zz + zz

iv) z3 + 3z v) z4 + 4z vi) 3z

z 3

zz

[.i)2/5 ii)-6 iii)-6/5 iv)-22 v)-14 vi)48i/125]

1.40. z = (5 + 9i)2010 + (5 9i)2010 . 1.41. , \ *` , z = ( + i) ( i) -.

1.42. x\ , w = ( )2020x ix i+ + ( )2020x ix i+ .

1: & 21

1.43. z = + i , \ . z , z

3 4 + 10 = 0 42 + 62 = 5.

1.44. z1, z2^ , z2 0 Re 12

zz

= 1 2 1 2

2 22z z z z

z z+ , Im 1

2

zz

= 1 2 1 2

2 22z z z z

iz z .

1.45. z1, z2^ , : ) z2 + 2z ) 1

2

zz

+ 12

zz

) 1z z+ + 1z z

+ 1.46. z1, z2^ , w = z1 2z + 1z z2 ,

u = z1 2z 1z z2 .

1.47. 3 52z i

z = ( )3 52z iz+ , z *^ z\ .

1.48. 2 93i zi z =

2 93

i zz

, z*^ z\ .

1.49. 45i z

iz = ( )45z iz+ , z *^ z .

1.50. z1, z2, z ^ z2 0 1 22

z z zz+ =

1

2

zz

(z1 + z2). z \ .

1.51. i z ww + = 21 i ( )wi w , z^ w *^ z -

.

1.52. 1 2

1z z =

1 2

1 2

z zz w z w

+ , z1 , z2 , w *^ w .

1.53. z, w w 0 z .w =1. : ) z + w 0 ) 1 zwz w

++ ,

z wz w+

1 zw iz w + .

1.54. z^ , w = iz 2 u = 3(z 2 + 6i) - .

[.z=3-5i] 1.55. z ,

1z

i+ 1

3 2z

i =

13 234 6

ii+ .

[.z=5-2i]

22 1:

1.56. : i) x2 8x + 25 = 0 ii) 36x2 + 48x + 25 = 0 iii) x2 2 3 x + 7 = 0 iv) x4 3x2 4 = 0 v) x4 + 14x2 + 45 = 0 vi) x3 x2 + 8x + 10 = 0

[.i)4 3i , ii)- 23 12 i , iii) 3 2i , iv) 2, i v) 3i, 5 i vi)-1, 1 3i] 1.57. : ) x2 2

4x

= 3 ) x2 + 24x

= 5

[.) 2, i ) i, 2i] 1.58. z (1 3i)iz (1 + i) z = 6 4i.

[.z=2-i] 1.59. z i.z + z + i = 3. 1.60. z^ 2 z + (1i 3 )z = i. 1.61. ^ z2 + 2 z + 1 = 0.

[.1, 1+2i, 1-2i]

1.62. (1 ) 4

2 (1 2 ) 6 3i z iw i

z i w i+ = + + = .

[.z=2-i , w=i]

1.63. 2 1 4

(1 2 ) 2 1 3z iw ii z i w i+ = + + = .

[.z=1+i , w=2+i] 1.64. x2 3x + 3 + i = 0 1 + i. 1 i. ;

1.65. 2x2 + x + = 0 , , \ , 2 i .

[.=-8, =10] 1.66. x2 + x + 5 = 0 , , \ , 1 + 1

2 i -

. [.=4, =-8]

1.67. 4x2 x + = 0 , , \ , 12

3i . -

. [.=4, =13]

1.68. z = 1i z5 + z3 + = 0 , , \ , = 2 = 8.

1.69. 2 , , - 1

3 i.

[.9x2-6x+10=0]

1: & 23

1.70. 2z3 z + 1 = 0 128z14 + 2z 1 = 0. [. 12 (1 i)]

1.71. (z) = z2 2z + 2 Q(z) = z3 + z2 + z 2 , \ .

i) z1 , z2 (z) 121z + 122z = 2

7. ii) P(z) Q(z), , .

( 2001) [.ii)=-3,=4] 1.72. ) z1, z2 z2 + 2z + 2 = 0, 201z

202z = 0.

) z1 () , , 1

z . [.)=4]

1.73. z : z = z2.

[.)0, 1, - 12 +32 i, -

12 -

32 i]

1.74. P(z) = z3 (3+i)z2 + (3+2i)z 1 3i. i , 1i

2 + i . 1.75. f (z) = z4 + (2 3i)z3 + (1 6i)z2 + (2 3i)z 6i ) : f (i) = f (i) = 0. ) f (z) = 0.

[.)-2, -i, i, 3i] 1.76. z : ) z2 = 3+4i ) z2 = 34i ) z2 = 1630i

[.)2+i,-2-i )1-2i, -1+2i )5-3i, -5+3i] 1.77. z. : ) z2 0, z\ . ) z2 < 0, z = i , *\ . ) z3 > 0, z 3 i , < 0.

1.78. z = 1 + 3i z1 , z2 y = 3 x y = 2x + 5.

[.z1=1+2i, z2=-2+i] 1.79. z = + i , , \ , - x y = 2 f () = i.z , ` . ) f (1600), f (1994), f (1729) . ) z () 2..

[.)z=1-i] 1.80. z2 + 2z + = 0 > > 0. ) . ) .

24 1:

1.81. z = x + yi, x, y\ \ : ( )22z z+ + ( )22z zi i = + (1)i. : ) Im(z) = 0 , = 1. ) = 0 , z2 + 1 = 0. ) 0 1. ) z , .

[.) , =1] 1.82. z , z2 1 , . OA OBJJJG JJJG , .

[.x=0 , x2+y2=1] 1.83. z .

, 1, z, 1 + z2 . [.y=0 (x-1)2+y2=1]

1.84. z . , 1, z, i.z .

[.(x- 12 )2+(y+ 12 )

2= 12 ] 1.85. z

Re(z2 z) = 1 Re(z). [.x2-y2=1]

1.86. z

z2 + 2z + 4 ( )2Im( )z = 2Re(z) 4.Im(z) [.(x- 12 )

2+(y+1)2= 54 ] 1.87. z :

z3 + 3z = 2.Re(z). [.O yy x2-3y2=1]

1.88. z w, w = 42

i zi z + .

) z w. ) z w I.

[.)z 2i )x2+(y+1)2=9 (0,2)] 1.89. z -: ) Re(z 1z ) =

12Re(z) ) Im(z 1z ) = 2

.Im(z) [.) yy (0,0) x2+y2=2 ) xx (0,0) x2+y2=1]

1.90. z w, w = z + iz , z 0. - z , Re(w) = Im(w).

[.y=x (0,0) x2+y2=1]

1: & 25

1.91. z 2 1

1zz++

. [.y=0 (-1,0) (x+1)2+y2=2]

1.92. z ,

2

1i zz+ .

[.y=0 (-1,0) (x+1)2+y2=1] 1.93. f (z) = 3

1z ii z+ z^ .

) f.

) z *^ {i} 1( )f z = f (1z ).

) z f (z)\ . [.)C-{-i} )x2+(y+2)2=1 (0,-1)]

1.94. f (z) = iz + 2 z 3i z^ . ) f 0. ) - f.

) f xx.

[.)-1-2i )- 32 - 32 i )x-2y-3=0]

1.95. f (z) = iz + 1 z^ . ) - f.

) () , - f , ..

[.) 12 +12 i]

1.96. z = (2 i) , *\ . ) 1z + i.

) (x,y) z , 1z ,

x + 2y = 25

, x 2y = 2.

) . [.)5x2-20y2=4]

1.97. ) z, w z+w\ zw , .

26 1:

) P(x) = .x + 1.x1 + + 1.x + 0 , i\ 0 i . z^ P(z) = + i P( z ) = i , , \ .

1.98. , ^ : i) z^ :

z , z = 0 , z ii) z, w, : (z + w) (z.w).

1.99. ) (1 + i)z2 + (1 5i)z 4 + 2i = 0 1 + i 1 + 2i. ) (z2 + z 4)2 + (z2 5z + 2)2 = 0.

1.100. , :

(1 + 2i) + (2 i) = 0. [.2]

1.101. (1 + i)(1 + i+1)(1 + i+2)(1 + i+3) = 0. 1.102. : x4 16x3 + 80x2 168x + 135 = 0

(2 + i2)(2 i) , ` . 1.103. z z = z3.

[.)0, -1, 1, i, -i]

1.104.

13 19

5 7

2 2

11

2

z wz w

z w

= = + = .

[.(i,i), (-i,-i)] 1.105. S = i + 2i2 + 3i3 + + i. : ) S i.S ) S

______________ _______________________________________ 1.106. . -

. I. z = + i w = + i,

. II. , ^ : = = 0 2 + 2 = 0. III. 2 3i < 2 + 3i.

IV. 23

2i = 3 22i = i3 = i.

V. , : i = i = . VI. z1, z2^ Re(z1 + z2) = 0 Re(z1) + Re(z2) = 0. VII. z1, z2^ Re(z1 z2) = 0 Im(z1 + z2) = 0 z1 = 2z . VIII. z1, z2^ Im(z1.z2) = Im(z1).Im(z2).

1: & 27

IX. z1, z2^ , z2 0 Re 12

zz

= 1

2

Re( )Re( )

zz .

X. z^ (z z )2 0. XI. z1 = + i , z2^ z1 + z2 = 2 , z2 = 1z . XII. z1 = + i , z2^ z1.z2 = 2 + 2 , z2 = 1z . XIII. Re(z) = 1 z

x = 1. XIV. 1, 2 z1 z2 xx

12 z1 = 2z .

XV. Im(z) = 4 z y = 4.

XVI. z w , - 2

z w+ . XVII. x2 4x + = 0 , \ , 1 + 2i 1 2i. XVIII. x2 + x + = 0 , , \ , 32i, 13

3 2i . 1.107. . -

.

I. ( )23 i = 1, : . 1 . 3 . 2 . 6 . 5

II. 2 + 3i 3 + 2i . x = 2 . y = 3 . y = x . y = 3 . x = 0

III. z2 + 8z = 0 , \ : . 3 + 4i . 4 + 3i . + i . 4 3i . 8 + 2i

IV. x2 + x + 13 = 0 , \ : . 3 + i . 1 3i . 3 + 4i . 2 2i . 3 + 2i

28 1:

2

, , - , .

ARTHUR CAYLEY (1821 1904)

-

. - - , , d(, ) = . \ - z ^ . z, z , . - .

2.1. : z = x + yi (x, y) . -

z - , z = OM = 2 2x y+ .

2.2. : z :

(i) z = z = z = z (ii) 2z = z z

: z = x + yi , x, y\ (i) z = x yi , z = x yi z = x + yi.

z = z = z = z = 2 2x y+ . (ii) zz = (x + yi)(x yi) = x2 (yi)2 = x2 y2i2 = x2 + y2 = 2z .

2: 29

2.3. : z1 , z2 : (i) 1 2z z = 1 2z z (ii) 1

2

zz =

1

2

zz

, z2 0. :

(i) : 1 2z z = 1 2z z 21 2z z = ( )21 2z z (z1.z2) ( )1 2z z = 2 21 2z z z1.z2. 1z . 2z = z1. 1z .z2. 2z , .

(ii) :

12

zz =

1

2

zz

2

1

2

zz =

21

2

zz

12

zz

1

2

zz

= 2

12

2

zz

1 12 2

z zz z =

1 1

2 2

z zz z

, . 2.4. : : 1 2 ... z z z = 1 2 ... z z z .

z1 = z2 = = z = z z = ...

z z z

= ...

z z z

=

z .

2.5. : z1 , z2

1 2z z 1 2z z+ 1z + 2z . : 1(z1) , 2(z2) (z1 + z2) z1 , z2 z1 + z2 . :

1 2OM OMJJJJG JJJJG 1 2OM OM+JJJJG JJJJG 1OMJJJJG + 2OMJJJJG . 1 2z z 1 2z z+ 1z + 2z . 2.6. :

. : 1(z1) , 2(z2) N(z1z2) z1 , z2 z1 z2 - . - 12 : (12) = 2 1M M

JJJJJJJG = ONJJJJG

= 1 2z z .

30 1:

0z z = z - z0 . - (z0) . 1z z = 2z z z z1 z2. - (z1) (z2). 1z z + 2z z = , > 0 1 2z z < - z 1 2 z1 z2 . 1 2z z z z = , > 0 1 2z z > - z - 1 2 z1 z2 .

______________ ___________________________________________

.

: .

, z z = + i .

z , z = z .

z w , -. w w.

2.7. z , w :

) z = 3 4

5(2+ ) (1 )

2 (1+2 )i i

i i ) w = i 7(1+ ) 2 (3 )1

i ii

5 ) 9+ 512i = 0.

:

) z = 3 4

5(2 ) (1 )

2 (1 2 )i i

i i+ + =

3 4

5

(2 ) (1 )

2 (1 2 )

i i

i i

+ + =

3 4

5

(2 ) (1 )

2 (1 2 )

i i

i i

+ + =

3 4

5

2 1

2 1 2

i ii i+ +

2: 31

= ( ) ( )

( )3 4

5

5 2

2 5

= ( )( )

4

2

2

2 5 = 42 5 =

25 .

) (1+i)2 = 12 + 2i + i2 = 1 + 2i 1 = 2i. (1+i)6 = [(1+i)2]3 = (2i)3 = 8i3 = 8i. (1+i)7 = (1+i)6(1+i) = 8i(1+i) = 8i8i2 = 88i.

w = (1+i)7 2 (3 )1i i

i 5 = 8 8i

2 (3 )(1 )(1 )(1 )i i i

i i + + 5

= 8 8i 22 (3 3 )

1 1i i i i +

+ 5 = 8 8i 2 (4 2 )

2i i + 5

= 8 8i 4i 2i2 5 = 5 12i. w = 5 12i = 2 25 ( 12)+ = 25 144+ = 169 = 13.

) 9 + 512i = 0 9 = 512

i .

9 = 512i 9 = 1512 = 9

1512 =

12 .

2.8. w w i = 3 , : ) z1 = 3w+4+4wi3i ) z2 = w22wi1

: ) 1 : z1 = 3w+4+4wi3i = 3(wi) + 4i(wi) = (wi)(3+4i). 1z = ( ) (3 4 )w i i + = 3 4w i i + = 2 23 3 4 + = 3.5 = 15. 2 :

z1 = 3w+4+4wi3i z14+3i = 3w+4wi z14+3i = w(3+4i) w = 1 4 33 4z i

i ++ .

w i = 3 1 4 33 4z i ii + + = 3

21 4 3 3 4

3 4z i i i

i +

+ = 3 13 4z

i+ = 3 1

3 4z

i+ =

3 12 23 4

z

+ = 3 1

5z = 3 1z = 15.

) z2 = w2 2wi 1 = w2 2wi + i2 = (wi)2. 2z =

2( )w i = 2w i = 32 = 9.

.

2z = zz . -

. ,

> 0 ,

32 1:

2.9. z^ , : z+4 = z2 1 z2 = 3. :

4z+ = 2 1z 24z+ = 22 1z (z+4) ( 4)z+ = (2z1) (2 1)z (z+4)( z +4) = (2z1)(2 z 1) z z + 4z + 4 z + 16 = 4z z 2z 2 z + 1 3z z + 6z + 6 z + 15 = 0 z z 2z 2 z 5 = 0 z z 2z 2 z = 5 z z 2z 2 z + 4 = 9 z( z 2) 2( z 2) = 9 ( z 2)(z2) = 9 (z2) ( 2)z = 9 22z = 9 2z = 3.

2.10. z^ , : z 1 + z 2 z + z 3 . : 1z + 2z z + 3z ( 1z + 2z )2 ( z + 3z )2 21z + 2 1z 2z + 22z 2z + 2 z 3z + 23z (z1)( z 1) + 2 2 3 2z z + +(z2)( z 2) z z + 2 2 3z z + (z3)( z 3) z z z z +1+2 2 3 2z z + + z z 2z2 z +4 z z +2 2 3z z + z z 3z3 z +9 2 2 3 2z z + 2 2 3z z +4 2 3 2z z + 2 3z z +2 , , .

2.11. z1 , z2 , , z . w

w = 2

1 2

1 2

+z zz z +

2

2 3

2 3

+z zz z + +

2

1

1

+

z zz z

, .

: : 1z = 2z = = z = .

1 1 : z = 2z = 2 z . z = 2 z = 2

z 1z + =

2

+1

z .

21

1

z zz z

++

+ = 2

1

1

z zz z

++

+ = 2

1

1

z zz z

++

+ =

22 2

12 2

1

z z z z

+

+

+ =

221

12

1

1

( )

( )

z zz z

z zz z

++

++

+ =

21

1

z zz z

++

+

= 2

1

1

z zz z

++

+ . w = w , w .

.

z -

2: 33

, :

2z = z2 z . 2z = z2 z

2.12. ) : z 2 = z2 z\ .

) z : z3 = z 2z z3 = z2 z .

z . :

) 2z = z2 z z = z2 z z z2 =0 z( z z) = 0 z=0 z = z z\ . ) 3z = z 2z z

3 = z 2z z ( z 2z) = 3 2z 2 z z = 3 2z = 3+2 z z.

z3 = z 2 z z(z 2 z ) = 3 z2 2 z z = 3 z2 = 3+2 z z. 2z = z2 . z .

.

, . . - . - 0z z = 1z z = 2z z z z0 - z1 z2 .

2.13. z w :

z w = z w+ z = w . z = w.

: 1 () z w . OA

JJJG OB

JJJG -

. : z w = z w+ OA OBJJJG JJJG = OA OB+JJJG JJJG OAJJJG OBJJJG z = w OAJJJG = OBJJJG . OAJJJG OBJJJG . , z = w. 2 () z = +i w = +i. z = w 2z = 2w 2 + 2 = 2 + 2. z w = z w+ 2z w = 2z w+ ( z w )2 = (z+w) ( )z w+

34 1:

2z 2 z w + 2w = (z+w)( z + w ) z z 2 2z + w w = z z + z w + w z + w w 2 2z = z w + zw 2 2z = 2Re(z w ) 2z = Re(z w ) (1) z w = (+i)(i) = i + i + = (+) + ()i. (1) 2 + 2 = ( + ) ( + )2 + ( + )2 = 2 + 2 + 2 + 2 + 2 + 2 = 2(2 + 2) + 2( + ) = 2( + ) + 2( + ) = 0. + = 0 + =0 = = z = w.

2.14. w *^ w , 2w + 32

iw , 2w 32

iw -

, . :

z1 = w , z2 = 2w + 32

iw z3 = 2w 32

iw ,

.

() = 1 2z z = 32 2w iww + =

32 2w iww+ = 3 32 2

w iw = 3 32 2iw

= w 3 32 2

i = w 9 34 4+ = w 3 . () = 1 3z z = 32 2

w iww = 3

2 2w iww+ + = 3 32 2

w iw+ = 3 32 2iw +

= w 3 32 2i+ = w 9 34 4+ = w 3 .

() = 2 3z z = 3 32 2 2 2w iw w iw + =

3 32 2 2 2w iw w iw + + + = 2 32

iw = 3iw

= w 3 . , () = () = () .

2.15. f (z) = 2z + 4z+ i z^ . : ) . ) z f . ) z , .

: ) f (z) = 2z + 4z i+ = 2 z + 4z i+ (2 ) ( 4 )z z i + + = 2 4z z i + + = 2 4i+

= 2 22 4+ = 4 16+ = 20 = 2 5 = f (2). f 2 5 . ) f (z) = 2z + 4z i+ = 2z + ( 4 )z i = () + () () = 2 5 . f - z , () + () = (), - z . -

=

y yx x =

0 42 0+ = 2. y y = (x x)

2: 35

y 0 = 2(x2) y = 2x 4. z = + (2 4)i , [0,2].

) 1 : z = 2 2(2 4) + = 2 24 16 16 + + = 25 16 16 + . z () = 52 16 + 16 - . = 162 5

=

85 [0,2]. z

z = 85 + (285 4)i =

85 + (

165 4)i =

85

45 i.

z = ( ) ( )2 28 45 5+ = 64 1625 25+ = 8025 = 4 55 . 2 : , . . = 1 .2 = 1 = 12 . : y y = (x x) y 0 = 12 (x 0) y =

12 x.

12

2 4

y x

y x

= =

12

1 2 42

y x

x x

= =

12

4 8

y x

x x

= =

12

5 8

y x

x

= =

1 8 42 5 5

85

y

x

= = =.

( 85 , 45 ). z =

85

45 i z =

4 55 .

2.16. z 4+3z i 2. ) z . ) 3 z 7.

:

) 4 3z i + 2 (4 3 )z i 2. - z (4,3) = 2. ) 1 () (, ) , - . () = 2 2(4 0) ( 3 0) + = 16 9+ = 25 = 5. () () () () () () () + ()

() () () + 5 2 z 5 + 2 3 z 7. z 7, 3. 2 () 4 3z i + 2 ( 4 3 )z i+ + 2.

4 3z i + ( 4 3 )z i+ + = 4 3z i + 2. , 4 3z i + 2 5z 2 2 z 5 2 2 + 5 z 2 + 5 3 z 7.

36 1:

2.17. z 10+5 +2i z 5 . ) z . ) .

:

) 10 5 2i z+ + 5 52

2( 5 )z i+ + 5 2 5

25z i+ + 5 5

2( 5 )z i

52 . z

(5, 52 ) =5

2 .

) 1 () (,) , . () () ()

. = K K

y yx x

=

5 025 0

= 12 . y y = (x x) y 0 = 12 (x 0) y =

12 x.

2 2

12

5 5( 5) ( )2 4

y x

x y

= + + + = ( )22

12

1 5 5( 5) 2 2 4

y x

x x

= + + + = ( )22

12

5 5( 5) 2 4

y x

xx

= + + + =

22

12

( 5) 5( 5) 4 4

y x

xx

= + + + =

2 2

12

4( 5) ( 5) 5

y x

x x

= + + + =

2

12

5( 5) 5

y x

x

= + =

2

12

( 5) 1

y x

x

= + =

2

12

( 5) 1

y x

x

= + =

1 12 2

5 1 5 1

y x y x

x x

= = + = + =

1 1( 4) ( 6)2 24 6

y y

x x

= = = = 2 3

4 6y yx x= = = = .

(4, 2) (6, 3). z = 42i z = 6 3i.

2 ()

z = ( ) ( )5 52 25 5z i i+ + + 525z i+ + + 525 i 52 + 2525 4+ = 52 + 5 254 = 52 +

5 52 =

6 52 = 3 5 . z 3 5 -

. - : z + 5 + 52 i = (5

52 i) z = 5

52 i 5 52 i z = (55) + (

52 52 )i , > 0

2: 37

52

5z i+ + = 52 ( )5 5 52 2 2( 5 5) 5 i i + + + = 52 5 5 52 2 2

5 5 5 i i i + + = 52 52

5 i = 52 2

2 2525 4 + = 52

25 254 = 52

5 52 = 52 =

15 =

15 .

z = (5 15 5) + (5 12 5

52 )i = 63i.

z = ( ) ( )5 52 25 5i z i + + + 5 52 25 5i z i + + 5 5 52 2 = 4 52 = 2 5 . z 2 5 - . -:

z+5+ 52 i = (552 i) z = 5

52 i 5 52 i z = 5(+1)

5( 1)2+ i , < 0

52

5z i+ + = 52 5 5 52 2 2

5 5 5 i i i + + = 52 52

5 i+ = 52 2

2 2525 4 + = 52

25 254 = 52

5 52 = 52 =

15 =

15 .

z = 5( 15 +1) 52 (

15 +1)i = 42i.

2.18. z w : z4 2 w i3 1. ) z, w ; ) 2 z w 8. ) z, w z w .

:

) 4z 2 3w i 1. z - (4,0) 1 = 2, - w - (0,3) 2 = 1. ) 1 () , (,1) (,2) . - , , .

() = 2 2(4 0) (0 3) + = 16 9+ = 25 = 5. () () () () () () () () + () + () () 1 2 () () + 1 + 2 5 2 1 z w 5 + 2 + 1 2 z w 8. z w 8, 2. 2 () z w = ( 4) (4 3 ) (3 )z i i w + + 4z + 4 3i + 3i w 2 + 5 + 1 = 8.

38 1:

z w = ( 4) (4 3 ) (3 )z i i w + + 4 3i ( 4z + 3i w ) = 4 3i 4z 3i w 5 2 1 = 2. 2 z w 8.

) = K K

y yx x

=

0 34 0 =

34 .

y y = (xx) y 0 = 34 (x4) y = 34 x + 3.

2 2

3 34( 4) 4

y x

x y

= + + = ( )2

3 343( 4) 3 44

y x

x x

= + + + = ( )22

3 3412 3( 4) 44

y x

xx

= + + =

22

3 343( 4)( 4) 44

y x

xx

= + + = 2

2

3 349( 4)( 4) 416

y x

xx

= + + =

2 2

3 3416( 4) 9( 4) 4 16

y x

x x

= + + =

2

3 3425( 4) 4 16

y x

x

= + =

2

3 344 16( 4) 25

y x

x

= + =

3 33 34 42 4 2 44 45 5

y x y x

x x

= + = + = =

3 28 3 123 34 5 4 528 125 5

y y

x x

= + = + = =

6 65 5

28 125 5

y y

x x

= = = = . ( 285 ,

65 ) (

125 ,

65 ).

2 2

3 34( 3) 1

y x

x y

= + + =

2 2

3 343( 3 3) 14

y x

x x

= + + + =

2 2

3 349 116

y x

x x

= + + =

2 2

3 3416 9 16

y x

x x

= + + =

2

3 3425 16

y x

x

= + =

2

3 341625

y x

x

= + = ( )3 43 4 33 4 54 5

4 45 5

yy

x x

= += + = =

12 185 54 45 5

y y

x x

= = = = .

( 45 ,125 ) (

45 ,

185 ).

z, w z w z = 125 +65 i w =

45 +

125 i

z, w z w z = 285 65 i w =

45 +

185 i.

2.19. z z1 1 z2 = 1,

z. :

1z 1 2z = 1, z (1,0) = 1 (2,0) = 1. , z q .

2: 39

, - . l = 900 (, ) . = 2, = 1 = 2 2 = 2 22 1 = 4 1 = 3 . z 3 , 1.

___________________________________________________________________ 2.20. :

) z1 = 5

6( 3 )(1 )

ii+ ) z2 =

(1 ) (1 2 )(1 3 )(1 )(2 )(3 )

i i ii i i

+ + +

[.)4 )1] 2.21. :

) z1 = 7 + i(2i) (12i)(3+i) 6i ) z2 = 3

2(1 )(1 )

ii

+ 1

) z3 = 21ii

+ + ( )21 21 ii++ ) z4 = 2(1 3 )1 ii+ (3 )(3 )2i ii +

[.) 10 ) 5 ) 61 /2 ) 122 ] 2.22. z, : ) z2 + 169i = 0 ) z3 125i = 0 ) z4 + 81i = 0

) z5 132 i = 0 ) z2004 + i = 0

[.)13 )5 )3 ) 12 )1]

2.23. z w 32z = 2 w = 9 29 6

zz

+ .

13w+ = 1. 2.24. z = 3 + 2(1 2)i , \ . ) z 4x + 3y = 6.

) z z 1,2. 2.25. z = ( 3) + 2(2 + )i , \ . ) z 2x y + 10 = 0.

) z z 2 5 . ) z .

2.26. z i , . z = 1, w =

2 ( )z z i iz+ 1.

40 1:

2.27. z0 0z 1 - 1 + z + z2 + + z1 = 0.

2.28. z^ z 1 : z = 1 11

zz+ .

2.29. z^ z , *+\ z z

+ z = .

2.30. z1, z2^ , : ) 21 22z z+ + 22 12z z = 5( 21z + 22z ). )

21 2z z+ + 22 1z z = (2 + 1)( 21z + 22z ) , \ .

2.31. z^ , : ) 2z iz+ = 2 2z + 2.Im(z2) ) 2z iz = 2 2z 2.Im(z2).

2.32. z^ : 2 1z = 2z z = 1. 2.33. z^ : 16z+ = 4 1z+ z = 4. 2.34. z : (5z 1)5 = (z 5)5. ) 5 1z = 5z ) z = 1 ) w = 5z + 1 , (w) .

2.35. z^ : 8z = 3 z 1z+ = 3. 2.36. z^ : 10z = 3 2z 1z = 3. 2.37. z^ : 3z+ = 2 z i+ 3 3 4z i + = 2 10 . 2.38. z^ , : ) 5z i = 5iz+ z = 1 ) z i = iz + z = 1 \ {1,1}. 2.39. z^ , : ) 9z = 9 1z z = 1 ) 4 1z = 4z z = 1. ) z = 1z z = 1 \ {1,1}. 2.40. z^ , : ) 49z+ = 7 1z + z = 7 ) 2z + = 1z + z = 0 1. 2.41. z (4 z)10 = z10 x = 2.

2.42. z, w : 100z w+ = 100z w . : ) z w+ = z w ) z w + z w = 0 ) Re( z w) = 0

2: 41

2.43. z, w *^ . : z w+ = z w zw . 2.44. z 1

2i w = 2

2 1z ii z+ .

w 1 z 1. 2.45. z^ , z2 + z + 1 = 0. z = 1z+ =1. 2.46. z^ , z = 1z+ = 1, : z2 + z + 1 = 0. 2.47. z1, z2.

21 2z z (1+ 21z )(1+ 22z )

2.48. z^ z 1

2 , : 3(1 )i z iz + < 3

4.

2.49. z^ : 1z + 3z + 6 + z + 2z .

2.50. z^ , : Re( ) Im( )2

z z+ z Re( )z + Im( )z . ; ;

2.51. z1 , z2 z3 1 , 2 4 . :

) z1 + z2 + z3 0 ) 1 2 3z z z+ + = 1 2 3

1 4 16z z z+ + .

2.52. z, w^ z2 + w2 = 0 , z w+ = z w . 2.53. z1, z2 0 -:

i) 1 2z z+ = 1z + 2z 12

zz > 0 ii) 1 2z z = 1z + 2z 12

zz < 0.

iii) 1 2z z = 1 2z z+ 12

zz < 0 iv) 1 2z z = 1 2z z 12

zz > 0.

2.54. ) z2 (3 + 2i)z + 6i = 0. ) f (z) = 1z z + 2z z z1, z2, . f.

[. )3,2i ) 13 ] 2.55. z^ , : 1z+ + 2z+ z + 3z+ 2.56. z^ , : 2z+ + 3z+ 1z+ + 4z+

42 1:

2.57. z1, z2, z3 *^ 1z = 2z = 3z = 1 z1 + z2 + z3 = 1, :

1

1z + 2

1z + 3

1z = 1.

2.58. z1, z2, z3 *^ 1z = 2z = 3z = z1 + z2 + z3 = , :

1

1z + 2

1z + 3

1z =

1

.

2.59. z, w, u 1. z + w + u

zw + wu + uz .

2.60. z1, z2 , w = 1 21 2

( )

z zz z++ , ` ,

. 2.61. z1, z2, z3^ 1z = 2z = 3z z1z2 + z2z3 + z3z1 = 0, z1 + z2 + z3 = 0. 2.62. . z1, z2, z3^ 1z = 2z = 3z z1+ z2+ z3 = 0, : ) z1z2 + z2z3 + z3z1 = 0 ) 21z +

22z +

23z = 0 )

31z +

32z +

33z = 3z1.z2.z3.

. w1, w2, w3 , , 1 2w w = 2 3w w = 3 1w w .

21w + 22w +

23w = w1w2 + w2w3 + w3w1.

2.63. z1, z2, z3^ z1 + z2 + z3 = 0 z1.z2 + z2.z3 + z3.z1 = 0, 1z = 2z = 3z . 2.64. ) z1, z2, z3^ z1 + z2 + z3 = 0 21z + 22z + 23z = 0 :

1z = 2z = 3z . ) , , , - w1, w2 w3 , : 21w +

22w +

23w = w1.w2 + w2.w3 + w3.w1 .

. 2.65. z w :

z w+ = z + w z = w . z = w.

2.66. , ,

z1 = 1 + 2i , z2 = 4 2i , z3 = 1 6i . .

2.67. ) , ^ , z3 = 8. ) .

2: 43

2.68. z1, z2^ , z2 0 , , z1+z2 , z1 z2 , z1 + z2i 3 , .

2.69. z^ 0z z = , z0^ , \

2z = 2Re( 0z z ) + 2 20z . 2.70. z ,

: ) z = 3 ) 2z i = 1 ) 3 4z i+ = 4 ) 2 4z i+ + = 6 ) 12 3 2z i = 3 ) 2 6z i = 5 ) 2 2 1z iz = 9

2.71. z, :

) z < 2 ) 3i z 1 ) 1 z 3 ) 2 < 8 2 4i z < 6 ) 10 10 3 5i z+ < 30 ) 1z iz+ 2 .

2.72. z : () Re( )z 3 () Im( )z 3 () z 3. z .

[. =9(4-)] 2.73. z , : ) 4z i = iz 2+ ) 2z+ = 6z i ) 6 2z i + = 3 4i z ) 4 4 12z i + = 16 2 4i z

2.74. ) z , : 4z + 4z+ = 10. ) z .

[.) 2x25 +2y9 =1 ) 3, 5]

2.75. ) z ,

: 15z i + 1

5z i+ = 226

25z + . ) z1 , z2 ) , .

[.) 2x144 +2y

169 =1 )26] 2.76. z , - 1z z + = 1z z+ .

[. yy (1,0) , (-1,0)] 2.77. z : 8z+ + 6z i = 10. ) z , .

44 1:

) z . [.)(-8,0) (0,6) )4,8]

2.78. w^ 2w + w i = 5 . ) w . ) w. ) w .

[.) (2,0) (0,-1) ) 2 55 )25 -

45 i]

2.79. ) () z : z = 2 Im(z) 0. ) , z (), - w = 1

2(z + 4z )

xx. [.) =2 ) (-2,0) (2,0)]

2.80. f (z) = 1

zi z + .

) z . ) ( )f z = ( )f z z\ . ) z , f (z) 1.

[.)z i )y= 12 ]

2.81. f f (z) = z iz+ , z z 0.

) ( )f z = ( )f z , z . ) ( )f z = 1, z . ) Re(f (z)) = 2 , z, .

[.)y=- 12 )x2+(y- 12 )

2= 14 (0,0)] 2.82. ) z, w - 2z i = 1 , 3 2w i + = 3. ) 1 z w 9.

2.83. ) z, w^ : 21zw 2z w = ( 2z 1)( 2w 1). ) z, w , zw , 1.

2.84. z1 z2 1z i+ = 1 2 4z = 2. 1 2z z .

[. 17 -3, 17 +3] 2.85. :

2: 45

i) A = {z / z^ 1z i+ 5}. ii) B = {z / z^ 6z i+ 4 5z i+ }. iii) = .

2.86. , \ z^ . : ) z i+ = z i+ Im(z) = 2

+ .

) z + = z + Re(z) = 2 + .

) z = z i .Re(z) .Im(z) = 2 22 .

2.87. z1, z2^ z2 = z1.z2 1z + 2z = 1 22z z z+ + 1 22

z z z+ + .

2.88. z, z1, z2^ z = 1 22 35z z+ , 1z z + 2z z = 1 2z z .

z, z1, z2 ;

2.89. z, z1, z2^ z = 1 2 z z +

+ , \ > 0, : 1z z + 2z z = 1 2z z . z, z1, z2 ;

2.90. ) z1, z2 :

21 2z z+ + 21 2z z = 2 21z + 2 22z .

. ) z1, z2^ 1 2z z = 1z = 2z , 1 2z z+ = 3 1z . ) z^ 1z+ = z = 1, : 1z = 3 . ) z1, z2^ 1z = 2z = 2 , :

21

12

zz z+ + 2

1 2 12

2

z z z

z

= 6.

2.91. z1, z2^ : 1 2z z+ = 2 1z 1 2z z = 2 2z . 2.92. z1, z2

21z +

22z =

21 2z z ,

1 2z z+ = 1 2z z . .

2.93. z^ 2z = 2 1z , Re(z2) = 12 .

2.94. z *^ z = 1z z+ , Re(z2) = 12 .

2.95. z, z , 1z z1 -

.

46 1:

[. 1 3 i22 ] 2.96. z , 1z = 3z = iz .

[.2+2i] 2.97. w : 2(w4+ 4w ) = (w2+ 2w )2 , w\ . 2.98. ) w^ , : w2+ 2w = 0 Re(w) = Im(w) Re(w) = Im(w). ) z, w : (zw)2 + (z w )2 = zw2 z + ( w z )2. z w .

2.99. z^ , :

i) z z+ + z z = 2 z z\ ii) z z+ + z z > 2 z z ( ) ^ \

2.100. ^ : ) 2z 2z + 2i = 0 ) 2z 4z = 1 + 8i.

[. )1+i )3-2i,1-2i] 2.101. ^ : ) 2z z2 8i = 2 ) 2z + z2 + 6i = 2.

[. )4-i,-4+i )1-3i,-1+3i] 2.102. ^ : ) z 2z + 3 + 6i = 0 ) z + z2 + 1 = 0.

[. )4+3i ) 1+ 52 i,-1+ 52 i]

2.103. z : 1z + i 1z + = z i + i.

[. z=0 z=-1-i] 2.104. z1, z2 1 , 1 2z z < 1 21 z z .

2.105. z1, z2^ , 1 21 21

z zz z+

+ z1, z2 1.

2.106. z 3 4z i+ = 2. z.

[.3, 7] 2.107. ) z^ z2 + z + 1 = 0 z = 1. ) w *^ w 1w = i , w = 1. ) z1, z2 *^ : 1

2

zz

2

1

zz = i. 1z = 2z .

2: 47

2.108. z : 4z 5

2 3z i 5

2.

[.z=2+ 32 i] 2.109. z :

z = 1 12iz = 2.

[. z=- 12 +12 i]

2.110. ) z^ ( )( )2 22 2z i z i+ 0 21 z+ 2 z . ) - z 2 z 21 z+ .

2.111. ) z^ ( )( )2 21 2 1 2z z+ 0 2 1z 2 z . ) - z 2 z 21 z+ .

2.112. z^ z 1. : 2 3z 4. z .

2.113. z^ z 2. : 2 4z 6. z - .

2.114. z^ z 3. : 2 5z i 8. .

2.115. z^ 1z 1. : 1 1+z 3. .

2.116. z^ 4z i+ 1, : 2 5 1 2z 2 5 +1. .

2.117. z^ 3z = 2 2z 2, 1 z 10 . .

2.118. z^ 1z+ =1 2z+ 1, 3 z 2. .

2.119. z z i = 1, 1z+ .

48 1:

2.120. ) z1 , z2 , z3 , z4 :

(z1 z4)(z2 z3) + (z2 z4)(z3 z1) + (z3 z4)(z1 z2) = 0. ) , , , z1 , z2 , z3 , z4 , - : ().() ().() + ().(). : - .

2.121. z, w 1. u, v u = z w zw + 1 v = z w zw + , \ . u, v .

2.122. z^ z = 1 + i = 22 11z izz iz+ + + , \

( 1)2 + 2 = 2 24

3z z+ + . 2.123. z1, z2, z3 1

3z1 + 4z2 + 5z3 = 0. : ) Re(z1. 2z ) = 0. )

21z +

22z = 0.

2.124. z, w^ , z.w = 1 z + w = 2, z + w .

2.125. 0, ^ 2z 2iz + 2(1+i) = 0.

[.z1,2=+(-1 2 1- -2 )i , 0 2 -1]

2.126. 3 5

2 4

01

z wz w

+ = =.

[.(z,w)=(-1,1), (z,w)=(1,-1)]

2.127. 3 7

5 11

01

z wz w

+ = =.

[.(z,w)=(i,i), (z,w)=(-i,-i)] 2.128. z1, z2^ , :

21 2z z+ (1+) 21z + (1+ 1 )2

2z .

2.129. z1, z2, z3^ : 1 2 3(1 )(1 )(1 )z z z 1 1z 2z 3z .

2: 49

2.130. z 1 z = 1, 11

i i

+ , \ .

2.131. z1, z2, , z z1 + z2 + + z

i 11

z iz i+ +

2

2

z iz i+ + +

z iz i+ < 1 , :

) z1, z2, , z .

) 1 21 2

. . .

. . .

z z z iz z z i+ + + + + + + < 1.

2.132. z.() + z1.(1) + + z. 1 = 0, *.` z0 0z >

12 .

2.133. z = + i, , _ z =1. 2 1z _ .

______________ _______________________________________ 2.134. . -

. I. .

II. : z = 3 z = 3z . III. z, w^ 2z + 2w = 0 z = w = 0. IV. z1.z2 =

22z z1 , z2 .

V. z i = 1iz+ . VI. 1z z = 2z z z^ z1, z2^ , . VII. z, , :

2 9z = 2 6z , . 2.135. . -

.

I. z = 3

2(1 2 )

(2 )(1 )i

i i+

:

. 52 . 52 .

54 .

52

. 5 52

II. z = 3 3

2 2(2 ) (2 ) 2(1 2 ) (1 ) 2

i i ii i i

+ + + :

. 4 . 202

. 5 . 5 22

. 3

50 1:

III. z = 12 + yi, y\ , z = 13, y : . 13 . 13 . 12 . 5 . 2 3

IV. 1z = 3 z2 = 3 + 4i z1 + z2 : . 5 . 8 . 9 . 12 . 14

V. 102z i+ = 10 : . . . . .

51

_________________________________________ _________________ 3.1. z , z i w = 2 1

zz + .

) w , z z = 1.

) , , 2 1z

z + = 3

3.

) z1, z2 (), -

= 2 21 1 2 2

2 21 1 2 2

( 1) ( 1)3

z z z zz z z z

+ + + .

[.ii) 32 12 i iii) 32 i - 32 i]

3.2. z1 = + i z2 = 11

22

zz

+ , , \ , 0. -

z2 z1\ . ) z2 z1 = 1. ) z1 . ) 21z . > 0, z1

(z1 + 1 + i)20 ( 1z + 1 i)20 = 0.

3.3. ) z1 , z2 ,

21z +2

2z = 2

1 2z z 1 2Re( )z z = 0. ) f : [, ]\ [, ]

z = 2 + i.f () , w = f () + i.2 . 0. 2w + 2z = 2w z - f (x) = 0 [, ].

( 1995) 3.4. f, g [, ] , (, ) g(x) ( )g x 0 ,

x (, ). z1 = f () + i.g() , z2 = g() + i.f () 1 2z z+ = 1 2z z , (, ) , ( )

( )f g +

( )( )

f g = 0.

3.5. ) z1, z2 :

z1 + z2 = 4 + 4i 2z1 2z = 5 + 5i.

) z, w 1z z 2 2w z 2 i) z, w , z = w ii) z w .

[.)3+i, 1+3i )i)z=w=2+2i ii)4 2 ] 3.6. z = 1 + i.x w = 1 + x + i , , > 0 x\ ,

2w z+ 2w z . = e.

52 1:

3.7. f : [, ]\ [, ] f (x) 0 , x [, ] z Re(z) 0 , Im(z) 0 Re( )z > Im( )z . z + 1z = f () z

2 + 21z

= f 2(), :

) z = 1 ) f 2() < f 2()

) x3.f () + f () = 0 (1,1).

3.8. z 14

z = 14

Re( )z + . ) z . ) o z w = z 1 .

[.)y2=x )z= z= 21 12 i]

3.9. f (x) = z ixw+ + i z ixw , z, w *^ x^ .

: ) zw I. ) f (x) .

3.10. f \ f (x) = 2 2

22

x z x zx z

++ , z -

z = + i, , \ , 0. ) lim

x+ f (x), limx f (x).

) f, 1z+ > 1z . ) f.

[.)0,0 )..f(- 2+2 )=2

2+

+2

22

.. f( 2+2 )=-2

2+

+2

22

)[-2

2+

+2

22

,2

2+

+2

22

]

53

1.

x0\

x0\

-

. - - . - , -, , , , , , . - .

, - . - - - . . - , , .

- , . - , - .

- [,] - Bolzano. , - - - , - .

1: 55

1

16 , -

,

Euler Introductio in analysin infinito-rum

. Euler1 Introductio in analysin infinitorum 1748. (function) f (x). ,

, . , , .

. , , , .

1.1. : \ . -

() f, x y. y f x f (x).

, : f : \ x f (x).

1 LEONARD EULER (17071783).

18 . - - . , . . Introductio in analysin infinitorum, Institutiones calculi

differentialis, Institutiones calculi integralis. , - , . , -. .

56 1: - -

x - , y, f x, .

f Df. f x, -

f f (). : f () = {y / y = f (x) x}.

1.2. : f xy

. (x, y) y = f (x), (x, f (x)) , xA, f Cf.

f xy - . (x,y) y = f (x), (x, f (x)) , xA, f Cf. xA y\ , - f . - f . - : o f -

Cf. o f f () -

Cf.

H f , xx, - f, (x, f (x)) (x, f (x)), xx.

1: 57

H f Cf xx , xx Cf .

, : 1. . f (x) = .x + 1.x1 + + 1.x + 0 , 0, 1, , \ ` . f \ 0 - . f (x) = 2x3 5x2 + 0,2x + 2 , g(x) = 23 x

4 + 9x3 3 5 x2 + 6 .

-.

f (x) = x +

f (x) = x2 , 0

f (x) = x3 , 0

2. .

f (x) = ( )( )

P xQ x , (x) Q(x) .

58 1: - -

f \ Q(x). f (x) =

5 3 2

43 6 8

2 3x x x

x x + + , g(x) =

3

23 1,2

5x x

x x + .

f (x) = x , 0

3. . - x. - - . F(x,y) = 0. f (x) = 2 3x , g(x) = 3 5x 2 . y2 2x + 3 = 0 (y + 2)2 x + 5 = 0 . f (x) = x

4. T .

f (x) = x , f (x) = x , f (x) = x f (x) = x.

5. f (x) = x , 0 < 1.

1: 59

6. f (x) = logx , 0 < 1.

. 1.3. : Kronecker

- , - . : I. f :\ f

f (x) = ( )f x xA. II. f, g :\

max{f, g}(x) =max{f (x),g(x)} = ( ) ( ) ( ) ( )2f x g x f x g x+ + , xA

min{f, g}(x) = min{f (x),g(x)} = ( ) ( ) ( ) ( )2f x g x f x g x+ , xA

III. f (x) = [x] , [x] , x.

IV. x. sign(x) = 1 0

0 01 0

xxx

x.sign(x) = ( 1) 0

0 01 0

x xx xx x

=

00 0

0

x xx

x x

= x .

, x.sign(x) = x sign(x) = xx =

xx , x

*\ . V. Dirichlet

f (x) = 1 0 ( )

xx

_\ _ .

.

60 1: - -

1.4. : f g :

x f (x) = g(x). f = g.

1.5. : (i) f, g ,

. x f (x) = g(x), f, g .

(ii) : , . f (x) = x g(x) = x3 xA = {1,0,1} , .

1.6. : f , -

xA : xA f ( x) = f (x).

- -

, (x, f (x)) - , (x, f (x)), - yy. -, yy.

1.7. : f , -

xA : xA f ( x) = f (x).

-

, (x, f (x)) - , (x, f (x)), - . , .

1.8. : f , -

*\ xA : x + A f (x + ) = f (x).

1: 61

- -

. - -. . , , . 1: f (x) = c . 2: f (x) = x g(x) = x = 2, h(x) = x (x) = x - = . 3: Dirichlet , . , _ x + _ x_ f (x + ) = 1 = f (x). x + ( ) \ _ x ( ) \ _ f (x + ) = 0 = f (x).

, -,

f, g . f, g f + g

o = o (f + g)(x) = f (x) + g(x) , x.

f, g f g o = o (f g)(x) = f (x) g(x) , x.

f, g f . g o = o (f . g)(x) = f (x) . g(x) , x.

f, g fg o = {x / x x , g(x) 0} o fg

(x) = ( )( )

f xg x , x.

f (x) = 11xx+ g(x) = 3

xx .

x f (x) g(x) -. f g f, g . x - 1 , 0 , 2 , 3 , 4 f (1) = 0 , f (0) = 1 , f (2) = 3 , f (3) = 2 , f (4) = 53 . x g

g(f (1)) = g(0) = 0 , g( f (0)) = g(1) = 14 , g(f (2)) = g(3) = ; , g(f (3)) = g(2)

62 1: - -

= 2 , g(f (4)) = g( 53 ) = 54 .

x f, Df = \ {1} f(x) g, Dg = \ {3}. - f g,

( )f

g

x Df x D

11 31

xxx

+ 1

3 3 1x

x x +

12 4xx

12

xx .

x\ {1,2}

g(f (x)) = g( 11xx+ ) =

11

11

3

xx

xx

+

+

=11

1 3( 1)1

xx

x xx

+

+

= 11 3 3x

x x+

+ + =1

4 2x

x+ .

1.9. : f, g , ,

f g, g D f, 1 = {xA / f (x) B} (g D f )(x) = g(f (x)).

f : f (A) g : B g(B).

g D f xA f (x), 1 = {xA / f (x)}. g D f 1 , f ()B .

1.10. : , f, g g D f f D g ,

. , g D f f D g. ( ).

f, g, h h D (g D f) , - (h D g) D f : h D (g D f) = (h D g) D f. ( ).

(x) = x f : I D f = f D I = f. ( ).

___________ ______________________________________________

1: 63

1.11. f x, y \ : f ( x y) = (x+4).f (y1) + (x+y)2 .

: f . x = 1 y = 1 : f ( 1 1) = (1+4).f(11) + (1+1)2 f (0) = 5.f (0) + 4 4.f (0) = 4 f (0) = 1 (1). x = 1 y = 1 : f ( 1 1) = (1+4).f (11) + (1+1)2 f (0) = 3.f(0) + 0 2.f (0) = 0 f (0) = 0 (2). (1) (2) f . , f .

1.12. f, g :

f (x) = 2

+1 1

x

x xx x

x x x

g(x) =

x x xx x

2

+

=

1 0 1( 1)2 0 1

( 2)( 1) 1 22

x

x x xx xx

x x x xx

+ < + >

= ( )

1 0 1

1 0 121 1 2

x

x xx

x

x x x

< >

.

fg

( )f Ag = ( ,1) (1,0) (0,+ ).

64 1: - -

1.13. f (x) = x 1 g(x) = x2 9 . :

i) g fD ii) f gD ii) f fD iv) g gD : f Df = [0,+ ). g , x2 9 0 x 3 x 3. g Dg = ( ,3] [3,+ ). i)

( )f

g

x Df x D

01 3 1 3

xx x

( )0

2 4x

x x

016

xx x 16. g D f g fD D = [16,+ ).

x g fD D (g D f )(x) = g( f (x)) = g( x 1) = 2( 1) 9x = 2 1 9x x + = 2 8x x .

ii) ( )

g

f

x Dg x D

23 3

9 0

x x

x

x 3 x 3. f Dg f gD D = ( ,3] [3,+ ). x f gD D (f Dg)(x) = f (g(x)) = f ( 2 9x ) = 2 9x 1 = 4 2 9x 1.

iii) ( )

f

f

x Df x D

01 0

xx

01

xx

01

xx x 1.

f D f f fD D = [1,+ ). x f fD D (f D f )(x) = f ( f (x)) = f ( x 1) = 1x 1.

iv)

( )g

g

x Dg x D

2 2( )3 3

9 3 9 3

x x

x x

23 39 9x x

x

23 318x x

x

2

3 3

18

x x

x

3 3

3 2x x

x

3 3

3 2 3 2x x

x x

x 3 2 x 3 2 .

g Dg g gD D = ( ,3 2 ] [3 2 ,+ ). x g gD D (g Dg)(x) = g(g(x)) = g( 2 9x ) = 2 2( 9 ) 9x = 2 9 9x

= 2 18x .

1.14. f (x) = x x

x x2 , 1

5 6 , 1 < g(x) = x 1+ .

: i) g fD ii) f gD : g Dg = [1,+ ).

x 3 3 + x29 + 0 0 +

1: 65

i) () f (x) = x 2 , x ( ,1) g(x) = 1x+ , x [1,+ ).

( )f

g

x Df x D

12 1x

x

66 1: - -

ii) g , (g D f )(x) = 22 12 2 1x

x x

+

f (x) = y 1x

x = y x = y yx x + yx = y (1+y)x = y y = 1 0.x = 1 .

y 1 x = 1y

y+ (1)

x 1 1y

y+ 1 y y+1 0 1 , (1) y 1.

(g D f )(x) = 22 12 2 1x

x x

+ g( f (x)) = 22 1

2 2 1x

x x

+ g(y) = 22 11

2 2 11 1

yy

y yy y

+ + + +

=

2

2

2 11

2 2 11(1 )

y yy

y yyy

+ +++

= 2 22

11

2 2 (1 ) (1 )(1 )

yy

y y y yy

+

+ + ++

= 2 2 2( 1)(1 )

2 2 2 1 2y y

y y y y y +

+ + + = 2

211

yy+ .

g(x) = 2

211

xx+ , x\ . : (g D f )(x) = g( f (x)) = g( 1

xx ) =

( )( )

2

2

11

11

xx

xx

+

=

2 2

2

2 2

2

(1 )(1 )

(1 )(1 )

x xx

x xx

+

= 2 2

2 21 21 2

x x xx x x + + + = 2

2 12 2 1

xx x

+ .

, g(x) = 2

211

xx+ , x\ .

1.16. f :[0,1] [1,+ ) : f 2(x) + x4 = 1 + 2x2.f(x) , x [0,1].

f. :

x [0,1] : f 2(x) + x4 = 1 + 2x2.f (x) f 2(x) 2x2.f (x) + x4 = 1 [f (x) x2]2 = 1 f (x) x2 = 1 f (x) x2 = 1 f (x) = x2+1 f (x) = x2 1. x [0,1] f (x) = x2 + 1 0 x 1 1 x2+1 2 1 f (x) 2 x [0,1] f (x) = x2 1 0 x 1 1 x21 0 1 f (x) 0 f (x) [1,+ ) x [0,1] f (x) = x2 + 1. , f (x) = x2 + 1, x [0,1].

1.17. f : \ \ : f (x+y) f (xy) + f (x) + f (y) = x2 + 4xy + y2 , x, y \ .

i) f (0) = 0. ii) f.

:

1: 67

i) x = y = 0 : f (0+0) f (00) + f (0) + f (0) = 02 + 4.0.0 + 02 f (0) f (0) + f (0) + f (0) = 0 2f (0) = 0 f (0) = 0.

ii) f . y = 0 : f (x+0) f (x0) + f (x) + f (0) = x2 + 4x.0 + 02 f (x) f (x) + f (x) + 0 = x2 + 0 + 0 f (x) = x2. x, y\ : f (x + y) f (x y) + f (x) + f (y) = (x + y)2 (x y)2 + x2 + y2 = x2 + 2xy + y2 (x2 2xy + y2) + x2 + y2 = x2 + 2xy + y2 x2 + 2xy y2 + x2 + y2 = x2 + 4xy + y2. , f (x) = x2, x\ .

1.18. f : \ \ : f (x+y) f (xy) + f (x) + f (y) = 2x + 3y + 2 , x, y \ .

: f . x = y = 0 : f (0+0) f (00) + f (0) + f (0) = 4.0 + 3.0 + 2 f (0) f (0) + f (0) + f (0) = 2 2f (0) = 2 f (0) = 1. y = 0 : f (x+0) f (x0) + f (x) + f (0) = 2x + 3.0 + 2 f (x) f (x) + f (x) + 1 = 2x + 2 f (x) = 2x + 1. x, y\ : f (x+y) f (xy) + f (x) + f (y) = 2(x+y) + 1 2(xy) 1 + 2x + 1 + 2y + 1 = 2x + 2y + 1 2x + 2y 1 + 2x + 1 + 2y + 1 = 2x + 6y + 2. , f .

1.19. , \ , : f (x).f (y) = f (x) + f (y) + 3 , x, y \ .

: f . y = x : f (x).f (x) = f (x) + f (x) + 3 f 2(x) 2f (x) 3 = 0 f (x) = 1 f (x) = 3. x1 , x2\ f (x1) = 1 f (x2) = 3, x = x1 y = x2 : f (x1).f (x2) = f (x1) + f (x2) + 3 1.3 = 1 + 3 + 3 3 = 5 . f (x) = 1 , x\ f (x) = 3 , x\ . x, y\ : f (x).f (y) = f (x) + f (y) + 3 1.(1) = 1 1+ 3 1 = 1 . x, y\ : f (x).f (y) = f (x) + f (y) + 3 3.3 = 3 + 3 + 3 9 = 9 . , : f (x) = 1, x\ f (x) = 3, x\ .

1.20. f : (0,+ ) \ f (1) = 0 x, y >0 : f (xy) lny + f (x).

: f . x = 1 :

68 1: - -

f (1.y) lny + f (1) f (y) lny + 0 f (y) lny. f (x) lnx, x > 0 (1) y = 1x :

f (x 1x ) ln1x + f (x) f (1) ln1 lnx + f (x) 0 lnx + f (x)

lnx f (x) x >0 (2) (1) (2) f (x) = lnx , x > 0. x, y (0,+ ) f (xy) = ln(xy) = lnx + lny = lny + f (x) lny + f (x). , f (x) = lnx, x > 0.

1.21. f : \ \ : .f (1+x) + .f (1x) = x2 + 5x 12 , x\ , (1,10) (3,6).

: f . Cf (1,10) (3,6) f (1) = 10 f (3) = 6. x = 2 x = 2 : .f (12) + .f (1+2) = (2)2 + 5(2) 12 .f (1) + .f (3) = 41012 106 = 18 5 + 3 = 9 .f (1+2) + .f (12) = 22 + 5.2 12 .f (3) + .f (1) = 4+1012 610 = 2 3+5 = 1.

5 3 93 5 1 + = + = . D =

5 33 5

= 259 = 16 , D = 9 31 5 = 45+3 = 48 ,

D = 5 93 1 = 527 = 32. =

DD =

4816 = 3 =

DD =

3216 = 2.

: 3.f (1+x) 2.f (1x) = x2+5x12 (1). x x : 3.f (1x) 2.f (1+x) = (x)2+5(x)12 3.f (1x) 2.f (1+x) = x25x12 (2) (1) (2) :

2

2

33 (1 ) 2 (1 ) 5 1222 (1 ) 3 (1 ) 5 12

f x f x x xf x f x x x

+ = + + + =

2

2

9 (1 ) 6 (1 ) 3 15 364 (1 ) 6 (1 ) 2 10 24f x f x x xf x f x x x

+ = + + + =

5.f (1+x) = 5x2+5x60 f (1+x) = x2+x12. x x1 : f (1+x1) = (x1)2+x112 f (x) = x22x+1+x112 f (x) = x2x12. x\ : 3.f (1+x) 2.f (1x) = 3[(1+x)2 (1+x) 12] 2[(1x)2 (1x) 12] = 3(1+2x+x2) 3(1+x) 36 2(12x+x2) + 2(1x) + 24 = 3+6x+3x233x362+4x2x2+22x +24 = x2+5x12. , f (x) = x2x12 , x\ .

1.22. f : \ \ ex+y f (x).f (y) f (x+y) , x, y\ .

) f (0) = 1.

) f (x) = 1( )f x , x\ . ) f.

1: 69

:

) x = y = 0 e0 f (0).f (0) f (0+0) 1 f 2(0) f (0). , f (0) 1 f 2(0) f (0) (0) 0f > f (0) 1. f (0) = 1.

) y = x e0 f (x).f (x) f (0) 1 f (x).f (x) 1. , f (x).f (x) = 1 f (x) = 1( )f x .

) f . y = 0 : ex f (x).f (0) f (x) ex f (x) f (x). f (x) ex x\ (1) x x : f (x) ex 1( )f x

1xe

ex f (x). f (x) ex x\ (2) (1) (2) f (x) = ex, x\ . , f (x) = ex, x\ . ___________________________________________________________________

1.23. f (x) =3 2

2

( 1) 5 1 1

x x xx x

+ + + .

. [. =5]

1.24. f : \ \ , :

f (x) + x.f (y) = x + y , x, y\ . 1.25. f \ , :

f (x + 1) + f (1 x) = 2x + 3 , x\ . 1.26.

) f (x) = 2x+ + 2 43xx

) f (x) = ln(1 5 x )

) f (x) = 21

1

xex

) f (x) = 3

2

xx x+

ln( 1)xx+

[.)={-2} [2,3) (3,+ ) )=(4,5] )=[0,1) (1,+ ) )=(-1,0) (0,1) (1,+ )] 1.27.

) f (x) = 2 1x ) g(x) = 1 1 xx

+

) h(x) = 4ln(4 )x

x x+ ) (x) =

13x

x

[.)=[1,5] )= \ -{x\ :x=2,] } )=(0, 2 ) )=(-3,-1] [1,3)]

70 1: - -

1.28.

) f (x) = 21 x + (x) ) g(x) = 3 21 1

x

x

[.)=[-1,- 12 ) (- 12 , 12 ) ( 12 ,1] )[-1,0) (0,1]] 1.29.

) f (x) = 2 8 9 24x x+ ) g(x) = ( )5 3log log