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методы решения интегральных уравнений

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  • .. , .. , ..

    - -

    2015

    Copyright & A K-C

  • 517.968 (075.8)

    22.161.675

    39

    .. , , ,

    .. , ,

    , . .

    39 : - - / .. , .. , .. ; - . , . . . -. : , 2015. 104 .: .

    517.968 (075.8)

    22.161.675

    .., .., .., 2015

    . , 2015

    Copyright & A K-C

  • 3

    ................................................................................................................................................ 3

    ..................................................................................................................................................... 4

    1 ......................................................................................... 7 1.1 ...................................................................................................................................... 7 1.2 ................................................................................................................................. 9 1.3 ........................ 12 1.4 ........................................................................................................................................................... 16 1.5. ................................... 19 1.6 - ................................................................................................................................................................ 25 1.7 (x,+) ................................................... 28 1.8 1- ......................................................................... 32

    2 2- ............................................................................................. 35 2.1 ................................................................................................................................... 35 2.2 .................................................................................................. 38 2.3 ................................................ 42 2.4 .................................................................... 52 2.5 .............................................................. 57 2.6 , ......................................................................................................................................................... 64 2.7 ..................... 67 2.8 ................................................................................... 68

    ...... 77

    .......................................................... 87

    ........................................................................ 89

    ........................................................................................ 104

    Copyright & A K-C

  • 4

    1.

    , , ,

    ,

    ()!

    !

    .

    , , () , ..

    ()!

    !

    .

    = (, )

    ( ! = (0, )) = , ( ! = 0

    , .

    (,)

    , , ,

    !()!

    !

    ().

    , !(, ) !.

    !

    1.

    .

    2. ! !.

    Copyright & A K-C

  • 5

    3. () ! ,

    () !

    4. () ! () ! ,

    ()()!

    !

    !

    !()!

    !

    ! !

    !

    (1)

    () ! () ! ,

    ,

    , = .!

    !

    (2)

    () !

    = (, ) = !()!

    !

    (3)

    5. !

    + + (4)

    6. . () ! , ! , , ! ,

    (, ).

    ! () ! = 0,!

    !

    , ! , ! ,

    , , ().

    !() ! (),

    () ! !() () .

    , !() !

    , > 0 > 0,

    ! () ! ,!

    !

    Copyright & A K-C

  • 6

    > > .

    . !()

    , ,

    . ! , ..

    ! ,

    !.

    () !(, )

    (, ), () . ,

    = (, ).

    (, )

    = , ,

    ! , < +!

    !

    .

    (, )

    = !(, )!

    !

    .

    Copyright & A K-C

  • 7

    1

    1.1

    = + (, )()!

    !

    , (1.1)

    (), (, ) , () ,

    , 2-

    . (, ) . () 0,

    (1.1)

    = (, )()!

    !

    , (1.2)

    2- .

    , = ()!

    !

    , (1.3)

    , 1-

    . , ,

    .

    (1.1), (1.2) (1.3)

    , , ,

    ( ).

    . ,

    =1

    1 + !!!

    Copyright & A K-C

  • 8

    =1

    1 + !+23

    1 + !

    ()!

    !

    (1.4)

    . () (1.4) !(!!!!)

    ! !,

    11 + !

    +

    1 + !1

    1 + ! ! !

    !

    !

    =1

    1 + !+23

    1

    1 + !32

    1 + ! ! !!!!

    !!!

    =

    =1

    1 + !+

    1

    1 + ! ! !

    11 + !

    =1

    1 + ! ! != .

    , = !(!!!!)

    ! ! (1.4)

    : 1

    (1 + !)!!=

    1

    (1 + !)!!

    , , = !(!!!!)

    ! !

    (1.4).

    ,

    .

    1 =1

    (1 + !)!!

    , =3 + 2!

    3 1 + ! !

    3 + 2! 1 + ! !

    !

    !

    .

    2 = ! ! !sin! ,

    = 1 !! cos 1 !! sin 1 + + 1 !! .!

    !

    3 = ! , = sin + 2 cos .!

    !

    Copyright & A K-C

  • 9

    4 = !

    6, = sh .

    !

    !

    5 = 1 , !!! = .!

    !

    6 = 3, ! = ! .!

    !

    7 =12

    , ()

    = .!

    !

    8 =1

    ,

    ()

    = 1!

    !

    1.2

    !!

    + ! !!!!!!

    ++ ! = , (1.5)

    ! = 1,2, ,

    0 = !, ! 0 = !,, !!! 0 = !!!, (1.6)

    2- .

    2-o

    !!

    + ! + ! = , (1.5')

    0 = !, ! 0 = !. (1.6')

    !!

    = . (1.7)

    (1.6'), :

    Copyright & A K-C

  • 10

    = + !, = + ! + !

    !

    !

    !

    !

    (1.8)

    -

    !

    !!

    !

    !!

    !

    !!

    =1

    1 ! !!!

    !

    !!

    (1.7) (1.8), (1.5') :

    + ! !

    !

    + !! +

    + ! + !! + !! = ()!

    !

    + ! + ! =!

    !

    = !! !! !!().

    (1.9)

    , = ! + ! , (1.10)

    = !! !! !! . (1.11)

    (1.9)

    = , + ()!

    !

    , (1.12)

    . . 2-ro .

    (1.12)

    (1.5') (1.6')

    = 0.

    , (1.12) ,

    (1.10) (1.11), , ()

    Copyright & A K-C

  • 11

    (1.8),

    (1.5'), (1.6').

    . ,

    !! + ! + = 0 (1.13)

    0 = 1, ! 0 = 0. (1.14)

    .

    !!

    = . (1.15)

    = + ! 0 = ,

    !

    !

    !

    !

    = + 1!

    !

    . (1.16)

    (1.15) (1.16) ,

    + + + 1 = 0,!

    !

    !

    !

    = 1 2 .!

    !

    ,

    :

    9. !! + = 0; 0 = 0, ! 0 = 1.

    10. ! = 0; 0 = 1.

    11. !! + = cos ; 0 = ! 0 = 0.

    12. !! 5 + 6 = 0; 0 = 0, ! 0 = 1.

    Copyright & A K-C

  • 12

    13. !! + = cos ; 0 = 0, ! 0 = 1.

    14. !! ! !"# ! + ! = ; 0 = 1, ! 0 = 1.

    15. !! + 1 + ! = cos ; 0 = 0, ! 0 = 2.

    16.!!! + !! + ! = ! + 1; 0 = ! 0 = 1, !! 0 = 0.

    17. !!! 2 = 0; 0 = !!, ! 0 = ! 0 = 1.

    1.3

    2-

    = + , ,!

    !

    (1.17)

    (, ) 0 , 0 ,

    0 .

    (l.17)

    :

    = ! + ! + !! ++ !! +. (1.18)

    (1.17),

    ! + ! + !! ++ !! + =

    = + , ! + ! ++ !! + .!

    !

    (1.19)

    , :

    ! = ,

    (1.20) ! = , ! = (, )()

    !

    !

    !

    !

    ,

    Copyright & A K-C

  • 13

    ! = , ! = (, ) (, !)(!)!!

    !

    !

    !

    !

    !

    ,

    .

    (1.20)

    !(). ,

    (, ) (1.18)

    0,

    (1.17).

    , (1.20) :

    ! = , ,!

    !

    (1.21)

    ! = , , ! ! !

    !

    !

    =!

    !

    = (!)! , , ! = ! , ! ! !,!

    !

    !

    !!

    !

    !

    (1.22)

    ! , ! = , , ! .!

    !!

    (1.23)

    ,

    ! = ! , , !

    !

    = 1,2, .

    (1.24)

    ! ,

    .

    ! , = , ,

    !!! , = , ! , ,!

    !

    = 1,2, . (1.25)

    Copyright & A K-C

  • 14

    (1.24) (1.25), (1.18) :

    = + ! ! , .!

    !

    !

    !!!

    (1.26)

    , ; ,

    , ; = !!

    !!!

    !!! , (1.27)

    ( )

    (1.17). (1.27) ,

    .

    ,

    .

    , ;

    :

    , ; = , + (, )(, ; )!

    !

    . (1.28)

    (1.17)

    = + (, ; )()!

    !

    (1.29)

    .

    (, ) 1.

    . ! , = , = 1. , (1.25)

    ! , = , ! , = = ,!

    !

    !

    !

    ! , = 1 =( )!

    2

    !

    !

    ,

    Copyright & A K-C

  • 15

    ! , = 1 !

    2 =

    !

    3!,

    !

    !

    .

    ! , = 1 !!! , = 1 !!!

    2 ! =

    ( )!!!

    1 !

    !

    !

    !

    !

    .

    ,

    , ; = !!!! , =!( )!

    != !(!!!).

    !

    !!!

    !

    !!!

    :

    :

    36 = ! + !!!()!

    !

    37 = sin + 2 !!!()!

    !

    19 , =

    20 , = !!!

    21 , = !!!!!

    22 , =1 + !

    1 + !

    23 , =2 + cos2 + cos

    24 , =chch

    25 , = !!!, > 0

    Copyright & A K-C

  • 16

    38 = 3! 3!!!()!

    !

    39 = ! +2 + cos2 + cos

    ()!

    !

    40 = 1 2 !!!!!()!

    !

    41 = !!!!! + 2 !!!!!()!

    !

    42 = 1 + ! +1 + !

    1 + !()

    !

    !

    43 =1

    1 + !+ sin ( )()

    !

    !

    44 = !!

    ! + ! !!! ()!

    !

    45 = !! + ! !!! sin !

    !

    ()

    1.4

    2-

    = + , .!

    !

    (1.30)

    , 0, , ,

    0 , 0 .

    Copyright & A K-C

  • 17

    - 0, ! .

    (1.30) ! ,

    ! = + (, )!()!

    !

    .

    !

    0, . ,

    ! , ! , , ! ,,

    ! = + , !!! .!

    !

    ,

    {! ()} ()

    (1.29).

    , , !() , ! ()

    (1.19) . 1.3,

    (1.30).

    !() {! ()}

    .

    .

    = 1 + ,!

    !

    ! 0.

    . ! 0, ! = 1. ,

    ! = 1 + 1 !

    !

    = 1 + ,

    ! = 1 + 1 + !

    !

    = 1 + +!

    2,

    Copyright & A K-C

  • 18

    ! = 1 + 1 + +!

    2 =

    !

    !

    1 + +!

    2!+!

    3!.

    ,

    ! = 1 +1!+!

    2!++

    !!!

    1 !.

    , ! n-

    !

    != !

    !

    !!!

    .

    , !() !.

    , = !

    ,

    ! = 1 + !,!

    !

    ! = 1 + ! !!!!!! = 1 + ! !,

    ! = ! .

    :

    46 = , !() 0.!

    !

    47 = 1 + , ! = 1.!

    !

    48 = 1 ,! 0.!

    !

    49 = + 1 ,!

    !

    a) ! = 1, ) ! = + 1.

    Copyright & A K-C

  • 19

    50 =!

    2+ ,

    !

    !

    a) ! = 1, ) ! = ,

    ) ! =!!

    !+ .

    51 = 1 + + , ! = 1.!

    !

    52 = 2 + 2 ,!

    !

    a) ! = 1, ) ! = 2.

    53 = 2! + 2 ,!

    !

    ) ! = 2, ) ! = 2.

    54 =!

    3 2 , ! = !.

    !

    !

    55. ,

    ! , < +.!

    !

    !

    !

    ,

    , = 0!

    !

    () 0 ! 0, .

    1.5.

    ! !() ,

    0. !(),

    ! = ! ! .!

    !

    (1.31)

    Copyright & A K-C

  • 20

    , 0 ,

    . ! !() -

    ,

    ! = ! !, (1.32)

    ..

    ( ).

    2-

    = + ,!

    !

    (1.33)

    .

    (1.33) .

    ,

    ,

    () !!!! , !!!! (1.34)

    , ,

    () (1.34):

    () !!!! .

    , ,

    () ,

    Re = > max(!, !, !).

    , , .

    (1.33)

    ,

    = + . (1.35)

    =()

    1 () 1 .

    Copyright & A K-C

  • 21

    ()

    (1.33).

    .

    = sin + 2 cos .!

    !

    . ,

    sin 1

    ! + 1, cos

    ! + 1

    .

    .

    ( ),

    =1

    ! + 1+

    2! + 1

    .

    1 2

    ! + 1=

    1! + 1

    =1

    ( 1)! ! .

    ,

    = ! .

    :

    59 = ! !!! .!

    !

    60 = !!! .!

    !

    Copyright & A K-C

  • 22

    61 = !! + !!! .!

    !

    62 = .!

    !

    63 = cos ( ) cos .!

    !

    64 = 1 + + !! !!! .!

    !

    65 = + sin .!

    !

    66 = sin + .!

    !

    67 = sh .!

    !

    68 = 1 2 4! + 3 6 4 ! .!

    !

    69 = sh ch .!

    !

    70 = 1 + 2 cos .!

    !

    71 = ! + 2 cos .!

    !

    72 = cos + .!

    !

    Copyright & A K-C

  • 23

    ! = ! + !" ! = 1, 2, , , !

    !

    !

    !!!

    (1.36)

    !" , !() ,

    .

    (1.36) ,

    ! = ! + !" ! ,!

    !!!

    ( = 1, 2, , ) (1.37)

    !().

    , !(),

    (1.36).

    .

    ! = 1 2 ! !!! ! + ! ,!

    !

    !

    !

    ! = 4 ! + 4 ! .!

    !

    !

    !

    (1.38)

    .

    ,

    ! =1

    2 2

    ! +1! ,

    ! =4!1! +

    4!! .

    !() !(),

    ! =

    ( + 1)!=

    1 + 1

    1

    ( + 1)!

    ! =3 + 2

    ( 2)( + 1)!=89

    1

    2+13

    1

    + 1 !89

    1

    + 1.

    !() !()

    Copyright & A K-C

  • 24

    ! = !! !!

    ! =89!! +

    13!!

    89!!

    ! ,!()

    (1.38).

    :

    73

    ! = sin + ! ,!

    !

    ! = 1 cos !()!

    !

    74

    ! = !! + ! ,!

    !

    ! = 1 ! !!! ! .!

    !

    75

    ! = ! + ! !!!! ,!

    !

    !

    !

    ! = ! + !()!

    !

    !

    !

    76

    ! = ! ! !

    !

    + 4 !!!!()!

    !

    ! = 1 !!!! + ! .!

    !

    !

    !

    Copyright & A K-C

  • 25

    1.6 -

    -

    ! ! + ! !!! ++ ! +

    + ! , ! = ,!

    !

    !

    !!!

    (1.39)

    77

    ! = + !()!

    !

    ! = 1 ! !

    !

    ! = sin +12

    ( )!()!

    !

    78

    ! = 1 ! !

    !

    ! = cos 1 + !()!

    !

    ! = cos + !()!

    !

    79

    ! = + 1 + !()!

    !

    ! = + ( )!()!

    !

    ! = cos 1 ! .!

    !

    Copyright & A K-C

  • 26

    ! , ,!(), , !(, ) = 0,1, , , ()

    .

    - (1.39),

    , ()

    0 = !, ! 0 = !! , , !!! 0 = !!!! . (1.40)

    (1.39) ! = ( = 0, 1, ,)

    ! , = ! ( = 0, 1, , ), .. !

    . , ! = 1. (1.39)

    ! + ! !!! ++ ! +

    + ! ! = !

    !

    !

    !!!

    , !, , ! . (1.41)

    () !() -

    ! ! ( = 0, 1, , )

    ()

    () ()

    (1.41) .

    ! ! !!!! !!!!! !!!!

    ( = 0, 1, ,). (1.42)

    ! ! !

    !

    !

    () ! !!!! !(!!!)

    = 0,1, , .

    (1.43)

    (1.41) :

    ! + !!!! ++ ! + !!

    !!!

    = , (1.44)

    Copyright & A K-C

  • 27

    .

    (1.44) (1.41)

    (1.40). , () -

    (1.41), (1.40).

    . -

    !! + + ! !!! ! = !!!

    !

    !

    !

    (1.45)

    0 = ! 0 = 0. (1.46)

    . () (). (1.46)

    !() (),

    !! ! .

    (1.45)

    ! +

    2 =

    1 2

    (1.47)

    1 !

    2=

    1 2

    . (1.48)

    (1.48)

    =1

    ( 1)! ! ! + 1.

    , - (1.45),

    (1.46),

    = ! ! + 1

    - :

    80 !! + ! !!! ! = !! ,!

    !

    0 = 0, ! 0 = 1

    Copyright & A K-C

  • 28

    81 ! + ! = !

    !

    !

    !

    0 = 1.

    82

    !! 2! + +

    + 2 cos !! 2 sin ! = cos ,!

    !

    !

    !

    ! 0 = 0 = 0.

    83 !! + 2! + !! 2 sin ! = cos

    !

    !

    !

    !

    ,

    ! 0 = 0 = 0.

    84 !! + + sh + ch ! = ch

    !

    !

    !

    !

    ,

    ! 0 = 0 = 0.

    1.7 (,+)

    = + ,!

    !

    (1.49)

    ,

    .

    .!

    !

    (1.50)

    ,

    ( )()!!

    !!

    = 2 , (1.51)

    G(), () .

    Copyright & A K-C

  • 29

    = _ ..

    = 0, > 0 , < 0

    = ! = , > 00, < 0 (1.52)

    (1.51) :

    ( )()!!

    !

    = 2_()!() , (1.53)

    ( ,

    ).

    ,

    ,

    = 2 !() . (1.54)

    , (1.53) (1.54)

    ( )()!

    !

    = 2 _()!() . (1.55)

    2_()

    :

    2_()= !!" = ()!"

    !

    !

    !

    !!

    .

    = (),

    2_()= = !".

    !

    !

    ,

    ( )()!

    !

    = . (1.56)

    (1.49).

    (1.49),

    = +()() (1.57)

    Copyright & A K-C

  • 30

    ( )

    =()

    1 ()

    1 , (1.58)

    = !".!

    !

    (1.59)

    =12

    ()1 ()

    !"

    !!!!

    !!!!

    (1.60)

    (1.49). ,

    (1.57) (1.60) , ,

    () .

    .

    = + ! !!! ()!

    !

    (1.61)

    . = , = !!.

    =1!

    , = !!!!" =1

    2 , < 2.

    !

    !

    , :

    =1!+

    12

    ,

    ,

    = 2

    !( 1). (1.62)

    =12

    2!( 1)

    !"; (0 < < 2)

    !!!!

    !!!!

    . (1.63)

    Copyright & A K-C

  • 31

    (1.63) .

    = 0

    = 1 , > 1 ,

    (1.61)

    = ! !!! .!

    !

    :

    !!!

    2!( 1)

    !" = 2 + 1, res!!!

    2!( 1)

    !" = ! .

    , (1.61)

    = 2 + 1 + !

    ( - ).

    :

    86 = !! + .!

    !

    87 = !! + !!! .!

    !

    88 = cos + !!! .!

    !

    89 = 1 + ! !!! ; > 0 .!

    !

    Copyright & A K-C

  • 32

    1.8 1-

    1-

    , = , 0 = 0,!

    !

    (1.64)

    () - .

    , , , !"(!,!)!"

    , !()

    0 , 0 . (1.64) x,

    , + ,

    = ! .

    !

    !

    (1.65)

    0 () (1.64)

    , , (1.65). ,

    0 (1.65)

    (1.64).

    (, )

    0, , (1.65) :

    =!()(, )

    ! , ,

    ,!

    !

    (1.66)

    . . 2- ,

    .

    , 0,

    (1.65) . .

    . (, ) 0, ,

    = 0, (1.66) ,

    2- . (

    3- .) , ,

    .

    Copyright & A K-C

  • 33

    .

    cos = .!

    !

    (1.67)

    . = , , = cos( )

    .

    (1.6) ,

    cos 0 sin = 1!

    !

    = 1 + sin .!

    !

    (1.68)

    (1.68) 2- .

    , .

    ,

    =1+

    1! + 1

    ,

    =! + 1!

    =1+

    1! 1 +

    !

    2.

    = 1 + !!

    ! (1.68), ,

    (1.67),

    .

    1-

    2- :

    90 !!! = sin .!

    !

    Copyright & A K-C

  • 34

    91 3!!! = .!

    !

    92 !!!()!

    !

    = , 0 = 0.

    93 1 ! + ! =!

    2.

    !

    !

    94 2 + ! ! = !.!

    !

    95 sin = !!! 1.

    !

    !

    Copyright & A K-C

  • 35

    2 2-

    2.1

    2-

    , = !

    !

    , (2.1)

    - , , ,

    , , ,

    .

    , (2.1);

    , , ,

    ,

    (, ) !!

    !

    !

    !

    .

    0, (2.1) ;

    0,

    , = 0!

    !

    (2.2)

    .

    (2.1) (2.2)

    , .

    (2.1) (2.2)

    ,

    , .

    Copyright & A K-C

  • 36

    . , = sin !"!

    !

    4 , =

    2

    !

    !

    ,

    , =

    2 2

    , 0 ;

    2 2

    , 1.

    .

    !

    4 , =

    !

    !

    = !

    4 , +

    !

    !

    , !

    !

    =

    = !

    4 2

    2 +

    2 2

    !

    !

    !

    !

    =

    = !

    42 2

    !

    !

    +2

    2 !

    !

    .

    sin !"!

    ,

    sin2!

    42

    sin !"!

    2

    !

    !

    + 2 !

    !

    sin !"!

    2= sin

    2

    !

    42

    cos

    2+

    2!

    sin2 !!!

    !!!

    +

    + 2

    cos2

    2!

    sin2 !!!

    !!!

    =2

    .

    Copyright & A K-C

  • 37

    , !! !

    ! , , , = sin !"

    !

    .

    ,

    .

    137 = 1, + !" 1 = ! .!

    !

    138 = sin, + In(1 2cos + !) = 0.!

    !

    139 = 1 2 sin 1 !

    !

    , cos ( + ) !

    !

    = 1.

    140 = , , = 3

    !

    !

    +!

    6.

    , =

    (2 )2

    , 0 ,

    (2 )2

    , 1.

    141 = ! , + sin = 1.!

    !

    142 = cos , ! + cos = sin .!

    !

    143 = !! , 4 ! !!!!

    !

    = 1 !! .

    144 = cos 2, 3 , = 1.!

    !

    , = sin cos , 0 sin cos , .

    Copyright & A K-C

  • 38

    145 =4C

    sin + !! , = , 4

    sin sin!

    !

    !

    = 0.

    2.2

    2-

    , = ()!

    !

    (2.3)

    = + , ; ,!

    !

    (2.4)

    , (, ; ), (2.3),

    , ; =(, ; )()

    (2.5)

    , () 0. (, ; ) () :

    , ; = , +1 !

    !! , ! ,

    !

    !!!

    (2.6)

    = 1 +1 !

    !!!

    !

    !!!

    , (2.7)

    ! , =

    ,

    !,

    !,

    ! ,

    , ! !, ! !, !

    ! , !

    , ! !, ! !, !

    ! , !

    ! !!!

    !! . (2.8)

    Copyright & A K-C

  • 39

    ! , = , ;

    ! =

    !, ! !, ! !, !

    ! , !

    !, ! !, ! !, !

    ! , !

    !, ! !, ! !, !

    ! , !

    ! ! .!

    !

    !

    !

    (2.9)

    (, ; ) , ()

    . , (, )

    ! , !

    !

    !

    !

    , (2.6) (2.7) ,

    , .

    , ; =(, ; )()

    , ,

    (). (, ; ).

    .

    , = !; = 0, = 1.

    . ! , = !. ,

    ! , =! !!!! !!!

    ! = 0!

    !

    ,

    ! , =! !! !!!! !!! !!!!! !!! !!!

    !

    !

    !! = 0,!

    !

    . ,

    ! , = 0. !:

    Copyright & A K-C

  • 40

    ! = !, ! ! = !!!!

    !

    ! = 1!

    !

    ,

    ! =!!! !!!!!! !!!

    !

    !

    !! = 0!

    !

    .

    , ! = 0.

    (2.6) (2.7)

    , ; = , = !; = 1 .

    ,

    , ; =(, ; )()

    =!

    1 .

    !!

    !

    = 1 .

    (2.4)

    = + !

    1 ()

    !

    !

    .

    , = !!

    = !! +

    1 .

    .

    ,

    :

    146. , = 2 ; 0 1, 0 1.

    147. , = ! !; 0 1, 0 1.

    148. , = sin cos ; 0 2, 0 2.

    149. , = sin sin ; 0 2, 0 2.

    Copyright & A K-C

  • 41

    (2.8) (2.9) ! , !

    (2.6) (2.7) ,

    :

    ! , = ! , , !!! , ,!

    !

    (2.10)

    ! = !!! , .!

    !

    (2.11)

    , ! = 1 ! , = , (2.11)

    (2.10) !,! , ,!,! , ,! ..

    . (2.10) (2.11),

    , = 2, 0 1, 0 1.

    . ! = 1,! , = 2. (2.11),

    ! = !

    !

    = !

    2 !

    !

    12

    .

    (2.10)

    ! , = 2

    2 2 2 = + 2 +

    23

    .!

    !

    ! = 2 + 2! +23

    =13

    ,!

    !

    ! , = 23

    2 2 + 2 +23

    = 0,!

    !

    ! = ! = = 0,! , = ! , = = 0.

    ,

    = 1 +2+!

    6; , ; = 2 + + 2

    23

    .

    Copyright & A K-C

  • 42

    , ; = 2 + + 2 !

    !

    1 + !!+ !

    !

    !

    .

    (2.10) (2.11),

    :

    150. , = + + 1; 1 1, 1 1.

    151. , = 1 + 3; 0 1, 0 1.

    152. , = 4 !; 0 1, 0 1.

    153. , = !!!; 0 1, 0 1.

    154. , = sin( + ) ; 0 2, 0 2.

    155. , = sh(); 1 1, 1 1.

    :

    156. sin + !!

    !

    = 1.

    157. 2 !

    !

    =6

    .

    158. sin cos !!

    !

    = cos 2.

    159. !!!!

    !

    = ! .

    160. 4 !!

    !

    = .

    2.3

    Copyright & A K-C

  • 43

    , = ()!

    !

    (2.12)

    , (2.12)

    .

    = + ! ! ,!

    !!!

    (2.13)

    !() :

    ! = , ,!

    !

    ! = , ! = !(, )()!

    !

    !

    !

    ! 3 = , ! = !(, )()!

    !

    !

    !

    ..

    ! , = , ! , ,!

    !

    ! , = , ! , ,!

    !

    ! , = , !!! , ,!

    !

    (2.14)

    = 2, 3,, !(, ) (, ) . !(, ) ,

    (2.14), .

    Copyright & A K-C

  • 44

    ! , = ! , !!! , ,!

    !

    (2.15)

    , .

    (2.12)

    , ; = ! , !!!,!

    !!!

    (2.16)

    , , (, ).

    !!

    .

    :

    = 1.!

    !

    (2.19)

    (, ) 1, , ,

    ! = !(, )!

    !

    = = 1.!

    !

    !

    !

    !

    !

    Copyright & A K-C

  • 45

    , (2.17) , (2.16) < 1.

    (2.19), ,

    1 = 1,

    = ()!

    !

    .

    = 1 , , = 1

    (2.19) . , ,

    , (2.19)

    . > 1 (2.19) . ,

    1, = !!!!

    ,

    .

    (2.16)

    .

    : , (, ) .

    , :

    , , = 0; , , = 0!

    !

    !

    !

    (2.20)

    x t.

    . , = , = !! 1, 1 ,

    !! = ! ! = 0,!

    !!

    !

    !!

    !! = ! ! = 0.!

    !!

    !

    !!

    , .

    ! , 0,

    Copyright & A K-C

  • 46

    ! , . , ,

    (, ).

    . , = sin 2 ; 0 2, 0 2.

    sin!!

    !

    2 sin 2 =

    =12

    cos + 2 3 cos( 2 ) =!!

    !

    =1213sin + 2 3 + sin( 2 )

    !!!

    !!!!

    = 0

    , :

    , ; sin 2 ,

    (2.16) , ,

    .

    !(, )

    (, )

    ! , = , ! !, ! !!!, !!!!!

    !

    !

    !

    !

    !

    !

    . (2.21)

    !(, ) , !(, ) ,

    , ,

    (, ) .

    (, ) ,

    !(, ) .

    . , = ,

    = 0, = 1.

    . (2.13), :

    ! , = ,

    Copyright & A K-C

  • 47

    ! , = = + 2

    13

    !

    !

    ,

    ! , = !

    !

    + 2

    13

    = 12

    ,

    ! , = 112

    = 112! , =

    112

    + 2

    13

    !

    !

    ,

    ! , = 112

    + 2

    13

    = 112! , =

    12!

    !

    !

    ,

    ! , =112!

    =!(, )12!

    =112!

    + 2

    13

    !

    !

    .

    , :

    = 2 1:

    !!!! , =1 !!!

    12!!! ,

    = 2:

    !! , =(1)!!!

    12!!! + 2

    13

    ,

    = 1, 2, 3,.

    . ! , !(, ) , , =

    = !"# (!,!), = 0, = 1.

    .

    min , = , 0 ,, 1;

    , = ! , 0 ,

    ! , 1.

    , ..

    , = , .

    ! , = (, ). :

    Copyright & A K-C

  • 48

    ! , = , ! , = , , .!

    !

    !

    !

    , = ! , 0

    ! , 1

    , = ! , 0

    ! , 1

    (, ) , !(, )

    > .

    ! , = , , + , , + , , .!

    !

    !

    !

    !

    !

    (0, ) < < ,

    (, )(, )!

    !

    = !! = !! =!!

    2 !

    !!

    !

    !

    !

    =!! 1

    2.

    , < < ,

    , , = !! = !!! !! .!

    !

    !

    !

    , 1 > > ,

    , , = !! = 1 !!! .!

    !

    !

    !

    ,

    ! , = 2 !!! 1 + !!

    2 > .

    ! , < ,

    x t ! , > :

    ! , = 2 !!! 1 + !!

    2 < .

    ,

    Copyright & A K-C

  • 49

    ! , =2 !!!

    1 + !!

    2, 0

    2 !!! 1 + !!

    2, 1.

    . (, ) , ,

    , ,

    < .

    < :

    ! , = , , = + + .!

    !

    !

    !

    !

    !

    !

    !

    . ! , ! , , = 0, = 1

    , = + , 0 < , , < 1.

    . ! , = (, ),

    ! , = , , ,!

    !

    , = + , 0 < , , < 1, , = + , 0 < , < 1.

    (, ) , !(, )

    : 1) < 2) > .

    1) < .

    ! , = ! + ! + !,

    ! = + =!

    6+!2

    ,!

    !

    ! = + + =5!

    65!

    6+32!

    32!,

    !

    !

    Copyright & A K-C

  • 50

    ! = + =!

    6+!

    2 +

    2

    2+13,

    !

    !

    ,

    ! , = ! 23! ! + 2! +

    2

    +13

    ( < , )

    2) > .

    ! , = ! + ! + !,

    ! = + =32!

    5!

    6,

    !

    !

    ! = =!

    6

    !

    6!2+!

    2,

    !

    !

    ! = + = 56! +

    32! +

    2

    +13

    .!

    !

    ,

    ! , = 23! ! + ! + 2! +

    2

    +13

    > .

    ,

    ! , =23! + ! ! + 2! +

    2

    +13

    , 0 < ,

    23! ! + ! + 2! +

    2

    +13

    , < 1.

    ! ,

    ( = 3, 4, )

    .

    161 , = ; = 1, = 1,

    162 , = sin ; = 0, = !!

    ( = 2,3)

    163 , = ( )!; = 1, = 1 ( = 2,3)

    Copyright & A K-C

  • 51

    164 , = + sin ; = , =

    165 , = !; = 0, = 1.

    166 , = ! cos ; = 0, = .

    ! , :

    167. , = !!! ; = 0, = 1.

    168. , = ! !!; = 1, = 1.

    .

    = .!

    !

    (2.22)

    , = ; = 0, = 1. :

    ! , =

    ! , = =3

    ,!

    !

    ! , =13

    =3!

    ,!

    !

    ! , =3!!!

    .

    (2.16)

    , ; = ! , !!! = 3

    !!!

    =33

    ,!

    !!!

    !

    !!!

    < 3.

    (2.18) (2.22)

    = + 33

    .!

    !

    Copyright & A K-C

  • 52

    , =

    =3

    3 ,

    3.

    :

    169 , = !!!; = 0, = 1.

    170 , = sin cos ; = 0, = !!

    .

    171 , = !; = 1, = 1.

    172 , = 1 + 1 ; = 1, = 0.

    173 , = !!; = 1, = 1.

    174 , = ; = 1, = 1.

    2.4

    (, ) 2-

    ,

    , . .

    , = ! ! .!

    !!!

    (2.23)

    !() ! ( = 1,2, . . . ,)

    , .

    (2.23)

    ! ! !

    !!!

    = ()!

    !

    (2.24)

    .

    (2.24)

    Copyright & A K-C

  • 53

    = + !()!

    !!!

    !()()!

    !

    (2.25)

    ! = ! , = 1, 2, . . . , .!

    !

    (2.26)

    (2.25)

    = + !!

    !

    !!!

    , (2.27)

    ! ( ).

    ,

    ! = 1, 2, . . . , .

    (2.27) (2.24),

    ! ! + !! !

    !!!

    !

    !

    ! = 0.!

    !!!

    ! = 1,2, . . . ,

    ,

    ! ! + !! !

    !!!

    = 0!

    !

    ! ! ! ! = ! = 1,2, . . . , .!

    !

    !

    !

    !

    !!!

    !" = ! ! , !

    !

    !

    = ! ,!

    !

    ,

    ! !" ! = ! ( = 1,2, . . . ,)!

    !!!

    Copyright & A K-C

  • 54

    :

    1 !! ! !" ! !! ! = !!"! + 1 !! ! !! ! = !

    !!! !!! + (1 !!) ! = !

    (2.28)

    !

    .

    =

    1 !! !" !!!" 1 !! !!

    !! !! 1 !!

    . (2.29)

    0 , (2.28)

    !,!, . . . ,!,

    ! =1

    ()

    1 !! !!!!! !!!! !!!" !!!!! !!!! !!

    !! !"!!! !"!! 1 !!

    (2.30)

    = 1,2, . . . , .

    (2.24) () ,

    = + !!

    !

    !!!

    ,

    ! = 1,2, . . . , (2.30).

    . (2.28) ,

    (2.27) ! , ! , . . . , !()

    (2.27) ()

    (2.26), .

    .

    Copyright & A K-C

  • 55

    cos + ! sin + cos sin = .!

    !!

    (2.31)

    . :

    = () cos + sin !()!

    !!

    + cos () sin + .!

    !!

    !

    !!

    :

    ! = () cos ; ! = ! ;!

    !!

    !

    !!

    ! = () sin ,!

    !!

    (2.32)

    !,!,! . (2.31)

    = ! + ! sin + ! cos + . (2.33)

    (2.33) (2.32),

    ! = (! + ! sin + ! cos + )!

    !!

    cos ,

    ! = ! + ! sin + ! cos + !,!

    !!

    ! = (! + ! sin + ! cos + )!

    !!

    sin

    ! 1 cos !

    !!

    ! sin cos ! cos! = cos !

    !!

    !

    !!

    !

    !!

    ,

    ! !!

    !!

    + ! 1 ! sin !

    !!

    ! ! cos = !,!

    !!

    !

    !!

    ! sin !

    !!

    ! sin! + ! 1 cos sin !

    !!

    !

    !!

    = sin .!

    !!

    Copyright & A K-C

  • 56

    ,

    !,!,!: ! ! = 0,

    ! + 4! = 0,

    2! ! + ! = 2

    (2.34)

    =

    1 0

    0 1 4

    2 1

    = 1 + 2!! 0.

    (2.34)

    ! =2!

    1 + 2!!; ! =

    8!

    1 + 2!!; ! =

    21 + 2!!

    .

    !,!,! (2.33),

    :

    =2

    1 + 2!! 4 sin + cos + .

    :

    180 4 sin! = 2 .

    !!

    !

    181 e!"#$%& ! = tg .!

    !

    182 tg = ctg .

    !!

    !

    183 cos( ln )() = 1.!

    !

    Copyright & A K-C

  • 57

    184 arccos () =1

    1 !.

    !

    !

    185 ln1

    !

    = 1 ( > 1).!

    !

    186 ( ln ln ) =65(1 4).

    !

    !

    187 sin cos = sin .

    !!

    !

    188 sin = .!!

    !

    189 sin( ) = cos .!

    !

    190 (sin cos sin 2 cos 2 + sin 3 cos 3) = cos .!!

    !

    191 12

    12

    3! 1 +12(3! 1) = 1.

    !

    !!

    2.5

    2-

    , = 0!

    !

    (2.35)

    0 ,

    () .

    Copyright & A K-C

  • 58

    ,

    x 0 ,

    (2.35) , ,

    ,

    .

    = 0 , = 0

    (2.35) , 0.

    , = ,

    , ,

    ;

    .

    .

    ! !()!

    !!!

    = 0!

    !

    (2.36)

    =

    1 !! !" !!!" 1 !! !!

    !! !! 1 !!

    = 0 (2.37)

    . ( ) -

    1 !! ! !"! !!!= 0,

    !"! + 1 !! ! !!!= 0 .

    !! ! !!! + 1 !! != 0

    , (2.38)

    !" ! , = 1,2, , ,

    .

    Copyright & A K-C

  • 59

    (2.37) (1 ),

    (2.36) ;

    ! = 1,2, ,

    !! , !

    ! , , !! !,

    !! , !

    ! , , !! !

    ...............

    !(!), !

    (!), , !(!) !,

    (2.38).

    (2.36), .. ,

    ! = !!

    !

    !!!

    ! ,

    ! = !!

    !

    !!!

    ! ,

    ! = !!

    !

    !!!

    ! .

    , ,

    .

    ()

    , ..

    , , . , , ,

    , = 0,!

    !

    , !(!) , ( () =

    = !!!!, . 177).

    .

    , .. () ,

    Copyright & A K-C

  • 60

    , ,

    , ,

    .

    .

    cos! cos 2 + cos 3cos! = 0!

    !

    .

    .

    = cos! cos2 + cos3 !

    !

    !

    !

    cos! = 0

    != cos2 !

    !

    != cos!!

    !

    (2.39)

    = ! cos! + ! cos3. (2.40)

    (2.40) (2.39),

    ! 1 os!!

    !

    cos2 ! cos3 cos2 = 0,!

    !

    ! cos!!

    !

    + ! 1 cos! cos3 !

    !

    = 0.

    (2.41)

    cos!!

    !

    cos2 =4

    , cos3 cos2 = 0!

    !

    ,

    cos!!

    !

    = 0, cos!!

    !

    cos3 =8

    , (2.41)

    Copyright & A K-C

  • 61

    1 4

    != 0

    1 8

    != 0 (2.42)

    1 4

    0

    0 1 8

    = 0.

    ! =4

    , ! =8

    .

    = !! (2.39)

    0 != 0,12

    != 0,

    != 0, a ! . ! = !cos!

    , ! = 1, ! = cos!.

    ! =!! (2.38)

    (1) != 0,

    0 != 0,

    != 0, ! , ! = !cos3,

    , ! = 1, ! = cos3.

    , :

    ! =4

    , ! =8

    ;

    :

    ! = cos!, ! = cos3.

    .

    1.

    Copyright & A K-C

  • 62

    3 2 !

    !

    = 0.

    . ,

    = (3 2) .!

    !

    = !

    !

    , (2.43)

    = 3 2 . (2.44)

    (2.44) (2.43)

    1 3! 2 !

    !

    = 0. (2.45)

    3! 2 !

    !

    = 0,

    (2.45) = 0, , , 0.

    ,

    = 0, ,

    .

    2.

    = 0!

    !

    .

    = ! !, (2.46)

    Copyright & A K-C

  • 63

    != ,!

    !

    != .!

    !

    (2.47)

    (2.46) (2.47),

    1 25 ! +

    3!= 0

    2! + 1 +

    25 != 0

    (2.48)

    =1

    25

    3

    2

    1 +25= 1 +

    !

    150.

    , (2.48)

    != 0 != 0, ,

    , : 0. , (2.46)

    .

    :

    198 sin! = 0

    !!

    !

    .

    199 sin = 0!!

    !

    .

    200 sin sin = 0!!

    !

    .

    Copyright & A K-C

  • 64

    201 cos + = 0!

    !

    .

    202 45!In 9!In = 0!

    !

    .

    203 2 4! = 0.!

    !

    204 5! + 4! = 0!

    !!

    .

    205 5! + 4! + 3 = 0!

    !!

    .

    206 ch sh = 0!

    !!

    .

    207 ch !sh = 0!

    !!

    .

    208 ch ch = 0!

    !!

    .

    2.6 ,

    = ,!

    !!

    (2.49)

    ( ) ,

    OX,

    Copyright & A K-C

  • 65

    = . (2.50)

    , (2.49)

    !! = cos ,

    !! = sin ,

    = 1, 2, , (2.51)

    ! =1!

    = 1, 2, , (2.52)

    ! ():

    ! =1

    cos!

    !!

    , = 1, 2, . (2.53)

    , !

    cos , sin , !

    . ! 1

    (2.49),

    ! =1!

    , ! =1

    ,!

    !!

    , , cos

    =1!

    ,!

    !!

    (2.54)

    ! =1

    cos .!

    !!

    = ,

    cos!

    !!

    = cos =!!!

    !!!

    = cos cos!!!

    !!!

    + sin sin !!!

    !!!

    = ! cos,

    Copyright & A K-C

  • 66

    ,

    !

    ().

    ,

    cos = 1!

    cos ,!

    !!

    , cos (2.54).

    , sin

    (2.54),

    !!! !

    .

    225

    = cos!!

    !!

    .

    226 ,

    , =12

    1 !

    1 2 cos + ! ( , )

    < 1 1, sin, cos,

    1, 1 ! , 1 !.

    227

    = !

    !!

    ,

    = ! ( ) 2.

    Copyright & A K-C

  • 67

    2.7

    ! ! !

    !!!

    = 0,!

    !

    (2.55)

    (.. ()0),

    : () 0.

    (.. ()=0),

    (2.55) , ,

    .

    (2.55)

    .

    , !, !, , !

    (2.55), p ,

    , ,

    ! ( = 1, 2, , ).

    .

    cos! cos 2 + cos! cos 3 = 0.!

    !

    .

    = !!

    , ! =!!

    ,

    ! = cos!, ! = cos 3.

    = cos!; =4

    ;

    = cos 3; =8

    ;

    = 0; 4

    , 8

    ,

    Copyright & A K-C

  • 68

    .

    = 0.

    231 cos + = 0.!

    !

    232 arccos = 0.!

    !

    233 2()

    1 + cos 2 = 0.

    !!

    !

    234 14

    = 0.!

    !!

    235 + 6 ! 2 = 0.!

    !

    2.8

    2-

    , = ()!

    !

    (2.56)

    , (, ) :

    (, ) (, ) .

    ()

    ! ( = 1, 2, )

    Copyright & A K-C

  • 69

    , !

    !

    = 0, (2.57)

    (2.56)

    = !

    !

    !

    !!!

    ! . (2.58)

    ! (2.56),

    ! = ! ,!

    !

    (2.59)

    , (2.58),

    , .

    ,

    = !, ( !, ), (2.56),

    , . ,

    ,! = 0

    !

    !

    ! = 0, = 1, 2, , ,

    (2.60)

    .. ,

    !, .

    (2.45) ,

    = !

    !

    !

    !!!!!

    ! + !! ++ !! , (2.61)

    !, !,, ! .

    Copyright & A K-C

  • 70

    , = !

    !

    !!!

    ! (2.62)

    (2.58) (2.61)

    .

    (2.56), .. ,

    ! (2.57),

    (2.56) : = .

    .

    , !

    !

    = , (2.63)

    , = 1 , 0 ,

    1 , 1.

    .

    ! = !!; ! = sin, = 1, 2,

    !, (2.63)

    = !

    + !!

    !

    !!!

    sin. (2.64)

    ! :

    ! = sin !

    !

    = =

    = sin

    =

    = 1

    cos =

    = cos

    !

    !+

    1

    cos !

    !

    = cos

    sin! !

    !

    =1 !!!

    .

    (2.64),

    =

    (1)!!!

    ( + !!)

    !

    !!!

    sin.

    = !! (2.63) ,

    Copyright & A K-C

  • 71

    ! =1 !!!

    0.

    .

    , = cos,!

    !

    , = + 1 , 0 ,

    + 1 , 1.

    . :

    ! = 1, ! = !! = 1, 2,

    :

    ! = !, ! = sin + cos, ( = 1, 2,).

    1 !!,

    = !!

    1+

    ! + !!

    (sin + cos)!

    !!!

    ,

    ! = !!

    !

    cos = 1 + 1 + !

    ,

    ! = cos!

    !

    sin + cos = 0, 12

    , = 1,

    = cos + 1 + 1 + !

    !

    1

    2( + !)

    (sin + cos) .

    =1 = ! ( = 1) ,

    , .. cos ,

    ! = ! , ! = sin + cos.

    Copyright & A K-C

  • 72

    = !! , = 2,3, . .,

    , (2.61):

    = cos + 1 + 1 + !

    !

    1

    2( + !)

    (sin + cos) +

    + (sin + cos),

    .

    .

    , ,

    .

    .

    .

    , !

    !

    = ! , (2.65)

    , =

    sh sh( 1)sh1

    , 0 ,

    sh sh ( 1)sh 1

    , 1.

    .

    = ! +sh( 1)

    sh 1sh +

    sh sh 1

    !

    !

    sh 1 .!

    !

    (2.66)

    ,

    ! = ! +ch 1

    sh 1sh +

    sh 1sh 1

    !

    !

    sh +

    + ch sh 1

    sh 1 !

    !

    sh sh 1

    sh 1 ,

    Copyright & A K-C

  • 73

    !! = ! +sh 1

    sh 1sh +

    shsh 1

    !

    !

    sh 1 !

    !

    +

    + ch 1

    sh 1sh

    ch sh 1

    sh 1

    !! = ! + + .

    (2.66) = 0 = 1 , , 0 = 1 , 1 = .

    !! ( + 1) = ! , (2.67)

    0 = 1, 1 = . (2.68)

    :

    1) + 1 = 0, .. = 1. (2.67) !! = ! .

    = ! + ! + ! .

    (2.68),

    ! ! ! + 1 = 1,

    ! + ! + = ,

    != 0, != 0, , , = ! .

    2) + 1 > 0, .. > 1, 0. (2.67)

    = !ch 1 + + !sh 1 + !!

    !.

    (2.68) ! !

    ! 1= 1

    !ch 1 + + !sh 1 + = ,

    != 1 +1

    , != ch 1 + sh 1 +

    1 +1

    .

    Copyright & A K-C

  • 74

    = 1 +1

    sh 1 + (1 )

    sh 1 + !

    .

    3) + 1 < 0, . . < 1. + 1 = ! .

    (2.67)

    = ! cos + ! sin !

    1 + !.

    (2.68)

    ! +1

    1 + != 1,

    ! cos + ! sin = !

    1 + !.

    (2.69)

    :

    ) sin =0.

    !=!

    1 + !, !=

    ( cos)!

    1 + ! sin ,

    ,

    =!

    1 + !cos +

    cos sin

    sin +e!

    1 + !,

    = 1.

    ) sin = 0, .. = , = 1,2, .

    (2.69) , , (2.65)

    .

    + 1 + !! , !

    !

    = 0 (2.70)

    , ..

    ! = 1 + !!

    Copyright & A K-C

  • 75

    ,

    ! = sin

    (2.70).

    :

    236 !

    4 , =

    2

    ,!

    !

    , =

    (2 )2

    , 0 ,

    (2 )2

    , 1.

    237 + , = ! ,!

    !

    , =

    sh sh( 1)sh 1

    , 0 ,

    sh sh( 1)sh 1

    , 1.

    238 , = ! !,!

    !

    , = , 0 ,

    , 1.

    239 2 , = cos 2 ,

    !!

    !

    , =sin cos , 0 ,

    sin cos , 1.

    240 , = 1,!

    !

    , =sin cos , 0 ,

    sin cos , .

    241 , = ,!

    !

    , = + 1 ( 3), 0 ,

    + 1 ( 3), 1.

    242 , = ( )!,!

    !

    , =sin +

    4

    sin 4

    , 0 ,

    sin +4

    sin 4

    , .

    Copyright & A K-C

  • 76

    243 , = + 1,!

    !

    , =!!sh , 0 ,

    !!sh , 1.

    244 , !

    !

    = 2 1, , =

    ch ch( + 1)sh 1

    , 0 ,

    ch ch( + 1)sh 1

    , 1.

    Copyright & A K-C

  • 77

    1. 2- :

    = + , .!

    !

    (1)

    . (1)

    = + , ; .!

    !

    (2)

    (, ; ) - (1)

    , ; = !!!! , ,!

    !!!

    (3)

    !!!(, )

    !!! , = , ! , ,!

    !

    ! , = , , = 1, 2, .

    (4)

    . (1)

    !() , = 0, 1, 2,,

    ! = + , !!! .!

    !

    (5)

    !() ().

    2-

    Copyright & A K-C

  • 78

    = + .!

    !

    (6)

    .

    , () ,

    , ,() (. )

    (6)

    ,

    =()

    1 (),() 1. (7)

    () (6).

    1-

    , ()!

    !

    = , (8)

    (, ) 0,

    2-

    =!()(, )

    !! , ,

    .!

    !

    (9)

    1-

    = ()!

    !

    (10)

    . , ()

    -

    , , () ()

    , (10)

    ,

    Copyright & A K-C

  • 79

    =()()

    . (11)

    () (10).

    2. . 2- :

    , = .!

    !

    (12)

    . (12)

    = + , ; ,!

    !

    (13)

    , ; =(, ; )()

    ,() 0, (14)

    (12).

    , ; = , +1 !

    !! , ! ,

    !

    !!!

    (15)

    = 1 +(1)!

    !

    !

    !!!

    !! . (16)

    ! , , !

    ! , = !

    !

    , , ! (, !)

    !, !, ! (!, !)

    !, !, ! (!, !)

    .

    ! , ! , ! (! , !)

    !! ,!

    !

    (17)

    ! , = , .

    Copyright & A K-C

  • 80

    ! = !

    !

    !, ! !, ! (!, !)

    !, ! !, ! (!, !)

    !, ! !, ! (!, !)

    .

    ! , ! ! , ! (! , !)

    !! .!

    !

    (18)

    :

    ! , = ! , , !!! , ,!

    !

    (19)

    ! = !!! , , ( = 1, 2, )!

    !

    (20)

    ! = 1,! , = , .

    . (12)

    .

    = + ! ! ,!

    !!!

    (21)

    !() :

    !() = , ,!

    !

    ! = , ! = ! , ,!

    !

    !

    !

    ! = , ! = ! , ()!

    !

    !

    !

    ..

    ! , = , , ,!

    !

    Copyright & A K-C

  • 81

    ! , = , !(, )!

    !

    ! , = , !!!(, )!

    !

    , = 2, 3, . (22)

    ! , = (, ).

    ! , , (12),

    .

    :

    ! ! !

    !!!

    = .!

    !

    (23)

    (23)

    = + !! ,!

    !!!

    (24)

    !

    1 !! ! !"! !!! = !,

    !"! + 1 !! ! !!! = !,

    !!! !!! + 1 !! ! = ! .

    (25)

    !" = ! ! ; ! = ! !

    !

    !

    !

    (26)

    , = 1, 2, , .

    (25) , (23)

    (24).

    .

    0,

    2-

    Copyright & A K-C

  • 82

    , = 0!

    !

    (27)

    () 0,

    (21) , ,

    ,

    .

    ,

    = , ,

    .

    ! ! !

    !!!

    !

    !

    = 0. (28)

    =

    1 !! !" !!!" 1 !! !!

    !! !! 1 !!

    = 0, (29)

    () - (25); .

    (29) p (1 ) ,

    (27) p !, !, , !,

    ! = !! ! ( = 1, 2, , )

    !

    !!!

    (30)

    !(!) ( = 1, 2, ,) - (25),

    ! ( = 1, 2, , ) .

    ()

    (), .. (, ; ).

    Copyright & A K-C

  • 83

    , (, )

    ,

    .

    2- :

    , = ,!

    !

    (31)

    , = , .

    ! ( = 1, 2, ) - , ! ( = 1, 2, ) -

    , .

    a) ! ( = 1, 2, ),

    (31) ,

    = !

    !! ,

    !

    !!!

    (32)

    ! = ! .!

    !

    (33)

    (32) , .

    b) ,

    = ! , q ( ! ),

    (31) ,

    ()

    !, .. q

    ! = 0 = 1, 2, , .!

    !

    (34)

    Copyright & A K-C

  • 84

    = !

    !! + !! + !! ++ !! ,

    !

    !!!!!

    (35)

    !, ,! - , ! , ,!() -

    , ! .

    q (34) , ! ,

    (, ) (31) :

    = .

    2- ,

    :

    a) , ()!

    !

    = 0, (33)

    , =

    ( )

    , ,

    ( )

    , .

    :

    ! =

    !

    :

    ! = sin

    = 1, 2, .

    ) , ()!

    !!

    +12

    = 0,!

    !!

    , =

    12ln1 + 1

    , ,

    12ln1 + 1

    , .

    Copyright & A K-C

  • 85

    : ! = + 1 . :

    ! = !(), !() ,

    ! =1

    2!!!

    !(! 1)! .

    ! = 1 !(), ,

    + 1 !!! = 2 + 1 ! + !!! ,

    = 2, 3,.

    ) , = 0,!

    !

    , =

    12

    !, ,

    12

    !, .

    : ! = !! , ! -

    ! = 0. : ! = ! ! , , !()

    1- . 1-

    ! =2

    ! 1 !

    ! + + 12

    !!.

    !

    !!!

    ) , !!

    !

    = 0,

    , =

    !!!!

    !!

    , ,

    !!

    !

    !!!!

    !!

    , .

    !!

    !

    :

    Copyright & A K-C

  • 86

    ! = + 1.

    :

    ! = !!! ! ,

    !() - ,

    ! = !!

    !!!! .

    !!! = 2 + 1 ! !!!! 1 ,

    ! = 1

    ! = + 1.

    ) , = 0,!!

    !!

    , =

    1!!!!!

    ! !!! !!!, ,!!

    !

    !!

    !!

    1!!!!!

    ! !!! !!!, .!!

    !

    !

    !!

    : ! = 2 + 1 ( = 0, 1, 2, ).

    :

    !(0 = !!

    !

    ! ! ,

    !() - , :

    ! = 1 !!! !

    !!!! =

    = 2 ! 1

    1!(2)!!! +

    1 2 ( 3)2!

    (2)!!! +

    ! = 1 ! = 2,

    , :

    !!! = 2! 2!!! .

    Copyright & A K-C

  • 87

    () ()

    () 1

    !" 1

    ! !!

    !!" !

    !!!

    sin

    ! + !

    cos

    ! + !

    sh

    ! !

    ch

    ! !

    !" sin

    ! + !

    !" cos

    ! + !

    !"sh

    ! !

    !"ch

    ! !

    sin 2

    ! + ! !

    cos ! !

    ! + ! !

    sh 2

    ! ! !

    Copyright & A K-C

  • 88

    ch ! + !

    ! ! !

    1sin

    !

    !(! + !)

    2

    !! !

    ( > 0) 1

    !"() ( )

    ( > 0, ) !!"()

    !() (0)

    ! () ! !!!!! ! (0)!!!

    !!!

    ()

    1()

    1 !!() !()!

    ()

    ()!

    !

    ! !()

    ! !( )!

    !

    ! !()

    Copyright & A K-C

  • 89

    1

    1.

    = cos + .!

    !

    2.

    ! = 2 3 ! !!! ! + ! ,!

    !

    !

    !

    ! = 3 ! + 2 ! .!

    !

    !

    !

    3. -

    !! + + sh + ch ! = ch !

    !

    !

    !

    ,

    ! 0 = 1, 0 = 1.

    4.

    = !!! + .!

    !

    5. 1-

    ! !!! = sin 2.!

    !

    6. , = cos

    3 , = 1!

    !

    , , = sin cos, 0 ,sin + cos, .

    7. , = !cos,

    = 0, = .

    Copyright & A K-C

  • 90

    2

    1.

    = ! + 2 cos .!

    !

    2.

    ! = sin + ! ,!

    !

    ! = 1 cos ! .!

    !

    3. -

    !! + + sh + ch ! = ch !

    !

    !

    !

    ,

    ! 0 = 1, 0 = 1.

    4.

    = 3 + ! !!! .!

    !

    5. 1-

    cos = 2!.!

    !

    6. , = !!

    4 ! !!! = 1 !!!

    !

    .

    7. , = ! , = 0 ,

    = 1.

    Copyright & A K-C

  • 91

    3

    1.

    = 1 + 2 cos .!

    !

    2.

    ! = !! + ! ,!

    !

    ! = 1 ! !!! ! .!

    !

    3. -

    !! + 2! + !! sin ! = cos !

    !

    !

    !

    ,

    ! 0 = 0 = 0.

    4.

    = cos 3 + ! !!! .!

    !

    5. 1-

    sin = 2.!

    !

    6. , = cos

    ! + cos =!

    !

    sin .

    7. , = + sin ,

    = , = .

    Copyright & A K-C

  • 92

    4

    1.

    = sh ch .!

    !

    2.

    ! = ! + ! !

    !

    !!!! ,!

    !

    ! = ! !

    !

    + ! .!

    !

    3. -

    !! 2! + + 2 cos !! + sin ! = cos !

    !

    !

    !

    ,

    ! 0 = 0 = 0.

    4.

    = !!! + .!

    !

    5. 1-

    3 + ! ! !

    !

    = 3.

    6. , = !

    sin !

    !

    = 1.

    7. , = !,

    = 1, = 1.

    Copyright & A K-C

  • 93

    5

    1.

    = 1 2 4! + 3 6 4 ! .!

    !

    2.

    ! = ! + ! !

    !

    + 4 !!!! ,!

    !

    ! = 1 !!!! !

    !

    + ! .!

    !

    3. -

    ! + ! = !

    !

    !

    !

    ,

    0 = 1.

    4.

    = 2 + ! !!! .!

    !

    5. 1-

    1 ! + ! !

    !

    = !.

    6. , =

    , !

    !

    = 3+!

    6, , =

    (2 )2

    , 0 ,

    (2 )2

    , 1.

    7.

    sin !

    !

    = cos .

    Copyright & A K-C

  • 94

    6

    1.

    = sh .!

    !

    2.

    ! = + ! ,!

    !

    ! = 1 ! ,!

    !

    ! = sin +12

    ! !

    !

    .

    3. -

    ! + ! !! ! !

    !

    = !! ,

    0 = 0, ! 0 = 1.

    4. a

    = cos 2 + ! !!! .!

    !

    5. 1-

    ! !!! !

    !

    = 3.

    6.

    , = 2 , = 0, = 1.

    7.

    sin !!

    !

    = .

    Copyright & A K-C

  • 95

    7

    1.

    = sin + .!

    !

    2.

    ! = 1 ! ,!

    !

    ! = cos 1 + ! ,!

    !

    ! = cos + ! !

    !

    .

    3. -

    ! + + ! !! ! !

    !

    !

    !

    = !! ,

    0 = ! 0 = 0.

    8. a

    = !!! + ! !!! .!

    !

    9. 1-

    3! !!! !

    !

    = 2.

    10.

    , = sin cos , = 0, = 2.

    11.

    sin cos

    !!

    !

    = sin .

    Copyright & A K-C

  • 96

    8

    1.

    = + sin .!

    !

    2.

    ! = + 1 + ! ,!

    !

    ! = + ! ,!

    !

    ! = cos 1 ! !

    !

    .

    3. -

    ! + + sh !

    !

    + ch !

    !

    = ch ,

    0 = ! 0 = 0.

    4. a

    = !!! + .!

    !

    5. 1-

    ! !!! !

    !

    = sin 2.

    6.

    , = + + 1, = 1, = 1.

    7.

    ln ln !

    !

    =65

    1 4 .

    Copyright & A K-C

  • 97

    9

    1.

    = ! + .!

    !

    2.

    ! = ! + ! ,!

    !

    !

    !

    ! = 1 + !!!! !

    !

    + ! .!

    !

    3. -

    + ! !!! !

    !

    = !! ,

    0 = 0, ! 0 = 1.

    4. a

    = + ! !!! .!

    !

    5. 1-

    !!! !

    !

    = cos .

    6.

    , = !!!, = 0, = 1.

    7.

    cos ! sin + cos sin !

    !!

    = .

    Copyright & A K-C

  • 98

    10

    1.

    = + .!

    !

    2.

    ! = + ! !!!! ,!

    !

    !

    !

    ! = 1 + ! !

    !

    + ! .!

    !

    3. -

    ! + ! !

    !

    !

    !

    = ,

    0 = 1.

    4. a

    = !! + .!

    !

    5. 1-

    sin !

    !

    = cos .

    6.

    , = 1 + 3, = 0, = 1.

    7.

    tg

    !!

    !!!

    = ctg .

    Copyright & A K-C

  • 99

    11

    1.

    = 3 + ! + .!

    !

    2.

    ! = 3 ! + ! ,!

    !

    !

    !

    ! = 1 4 ! !

    !

    + 3 ! .!

    !

    3. -

    !! 2! + + 2 cos !! !

    !

    + 2 sin !

    !

    = cos ,

    0 = 0 = 0.

    4. a

    = !! + !!! .!

    !

    5. 1-

    ch !

    !

    = sh .

    6. C a

    sin !!

    !

    = 1.

    7.

    sin cos !!

    !

    = 0.

    Copyright & A K-C

  • 100

    12

    1.

    = 1 + ! + .!

    !

    2.

    ! = !! ! + 2 !!!! ,!

    !

    !

    !

    ! = + ! !!! ! !

    !

    + ! .!

    !

    3. -

    !! + 2! + !! !

    !

    2 sin ! !

    !

    = cos ,

    0 = ! 0 = 0.

    4. a

    = cos + !!! .!

    !

    5. 1-

    sin !

    !

    = 1 + !.

    6. C a

    2 !

    !

    =6

    .

    7.

    sin sin !!

    !

    = 0.

    Copyright & A K-C

  • 101

    13

    1.

    = 6 + ! + .!

    !

    2.

    ! = 2 ! 4 ! ,!

    !

    !

    !

    ! = 2 ! !

    !

    ! .!

    !

    3. -

    !! + sh !

    !

    + ch ! !

    !

    = ch ,

    0 = ! 0 = 0.

    4. a

    = 1 + ! !!! .!

    !

    5. 1-

    ch !

    !

    = .

    6. C a

    sin cos !!

    !

    = cos 2.

    7.

    cos + !

    !

    = 0.

    Copyright & A K-C

  • 102

    14

    1.

    = 5 + + .!

    !

    2.

    ! = 1 + ! + ! ,!

    !

    !

    !

    ! = 1 + ! !

    !

    ! .!

    !

    3. -

    !! + + sh !

    !

    + ch ! !

    !

    = ch ,

    0 = 1, ! 0 = 1.

    4.

    = 7 + ! !!! .!

    !

    5. 1-

    sh !

    !

    = ch .

    6. C a

    !!! !

    !

    = ! .

    7.

    45! ln 9! ln !

    !

    = 0.

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  • 103

    15

    1.

    = 1 + 2 + sin .!

    !

    2.

    ! = + ! + ! ,!

    !

    !

    !

    ! = 2 + ! !

    !

    + 2 ! .!

    !

    3. -

    !! + ! !!! !

    !

    = !! ,

    0 = 0, ! 0 = 1.

    4.

    = 2 + ! !!! .!

    !

    5. 1-

    ! !!! !

    !

    = ! .

    6. C a

    4 ! !

    !

    = .

    7.

    5! + 4! !

    !!

    = 0.

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  • 104

    1. , .. / .. , ..

    , .. . .: , 1968. 192 .

    2. , . . /. . , . .

    . - -. . .: , 2009. 160 c.

    3. , . . /

    . . . -. . .: , 2005. 176 c.

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