-
.. , .. , ..
- -
2015
Copyright & A K-C
-
517.58(075.8)
22.1675
39
.. , , ,
.. , ,
, . .
40 : - - / .. , .. , .. ; - . , . . . -. : , 2015. 91 .: .
517.58(075.8)
22.1675
.., .., .., 2015
. , 2015
Copyright & A K-C
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3
1.
..................................................................................
4 1.1
.................................................................................................................
4 1.2
...........................................................................................
7
2.
..................................................................................................................
15 2.1 -
..........................................................................................................
15 2.2 -
..............................................................................................................
23
3.
.......................................................................................................
29 3.1 . ................. 29 3.2
.................................................................................................
31 3.3
....................................................................
34 3.4
.............................................................................
36 3.5
................................................................................
38
4. .. 42
4.1 u = 1a2
2ut2
+but+Cu
...........................................................................................................................................
42 4.2 .
...........................................................................................................................
44 4.3
...................................................................................................
47
5. -
............................................................................................................
50 5.1 -
...............................................................................................
50 5.2 , -
...............................................................................................................................................
53
6. .................................................... 58 6.1
....................................................................
58 6.2
............................................................................................
60 6.3 .
...................................................................................................................................................
61
7.
....................................................................................................................................................
64 7.1 .................................... 64 7.2
.......................................................................................................
67 7.3
...................................................................................................................
69
............ 73
..........................................................
74
........................................................................
76
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91
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4
1. 1.1
.
,
,
= ,
k .
= !",
.
= ! = !,
= !!!!!
= !! !!!! ,
.
,
- ! !!
. ,
:
sh =! !!
2,
ch =! + !!
2,
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5
th =! !!
! + !!,
cth =! + !!
! !!,
sch =2
! + !!,
csch =2
! !!.
, ,
, ,
.
sh = sh ch ch sh,
ch = ch ch sh sh,
sh2 = 2sh ch,
ch2 = ch! + sh!,
:
ch! sh! = 1.
, = ch, = sh
! ! = ! . ,
= a cos , = sin ! + ! = ! .
.
R, ch
, sh .
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6
1.1 = ch = sh
sh ch , :
sh + ch = !,
ch! sh! = 1,
ch2 = 1 + 2sh!,
sh2 = 2sh ch.
th =shch
cth =chsh
.
th , cth
= 0 ;
, .
y = sh x
y =chx
0
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7
1.2 th
1.3 cth
1.2
, = sh , = th = ch, 0
, = ch, 0 .
.
-1 0 1
-1
1
y = th x
y = cth x
1
-1O
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8
:
1. , ,
sh arsh = : arsh = ln + ! + 1 ;
2. , ,
ch (arch) = , 1: Arch = ln( + ! 1);
3. , ,
: Arth = !!ln !!!
!!!;
4. ,
, : Arcth = !!ln !!!
!!!;
5. , ,
: Arsch = ln !! !!!!
!;
6. ,
, :
Arsh =ln1 1 + !
< 0;
ln1 + 1 + !
> 0;
.
,
, . ( ) ,
; arsh ; arch , arth .
:
arsh =1arc sin
arch = arc cos
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9
crth =1crc th
arsh = ln + 1 + ! ,
arth =12ln1 + 1
< 1
arch! = ln ! 1 , 1
arch! = ln + ! 1 , 1
:
ch ! = sh sh = ch +
sh ! = ch ch = sh +
th ! = sech! sech! = th +
cth ! = csch! csch! = cth +
(sech) = sech th sech th = sech +
csch ! = csch cth csch cth = csch +
1
ch2 + 3sh
.
.
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10
ch2 + 3sh
=
= 2 + 3sh = 3ch
=3
=3=13
=13ln 2 + 3sh + .
2
sh!
.
ch! sh! = 1, , , sh! = ch! 1,
= sh! = sh!sh = ! 1 sh
,
= ch! 1 sh = = ch = sh =
= ! 1 =!
3 + =
ch!3
ch + .
3
sh.
.
: = .
sh = = = sh = = ch = ch ch = ch sh + .
4
!sh.
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11
.
sh =! !!
2,
!sh = !! !!
2 =
12
(!! 1) =12
12!! + =
=!!
42+ .
5
1 + ch
.
.
,
ch =! + !!
2.
,
1 +
=
1 + !!!!!!
!
=2
2 + ! + !!= 2
!2! + !! + 1
=
= 2!
2! + !! + 1= 2
! + 1! + 1 !
= 2
! + 1+ .
6
sh + 2ch
.
.
,
sh =! !!
2, ch =
! + !!
2 .
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12
,
=
sh + 2ch=
!!!!!!
!+ 2 !
!!!!!
!
=2
! !! + 2! + 2!!=
= 2
3! + !!= 2
!3!! + 1
.
.
= 2!
3!! + 1= =
!
= ! = 2
3! + 1=23
! + !!
=
=23
! + !!
! =23
31
arctg!!
+ =23arctg 3 + =
=23arctg 3! + .
7
sh2 ch3.
.
sh =! !!
2, ch =
! + !!
2
sh2ch3 =!! !!!
2!! + !!!
2 =
=14
!! !!! !! + !!! =
=14
!!!!! !!!!!! + !!!!! !!!!!! =
=14
!! ! + !! !!! =14
!!
5 ! !! +
!!!
5+ =
=110
!! + !!!
212
! + !!
2+ =
ch510
ch2+ .
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8
sh cos.
.
,
sh cos = = cos = sh = sin = ch =
= ch cos ch sin = chcos + ch sin =
= = sin = ch = cos = sh =
= ch cos + sh sin sh cos
sh cos = ch cos + sh sin sh cos.
sh cos,
sh cos =ch cos + sh sin
2+ .
1)
a) sh + = sh ch + ch sh,
b) ch + = ch ch + sh sh.
2)
1) ch
2 + 3sh; 2) sh!;
3) sh; 4) !sh;
5)
1 + ch; 6)
sh + 2ch
;
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7) sh 2 ch3; 8) sh cos;
9) cth!; 10) sh
11) sh!; 12) ch!;
13)
sh!ch!; 14) !ch3;
15) !sh; 16) sh2 arcsh ;
17) ch2 arch ; 18) th2 arth;
19) cth arcth. 20) ch
.
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15
2.
2.1 -
. -
= !!!!!!
!
(2.1)
c = 0 = .
-
1. () 0,+ .
- (2.1)
:
= !!!!!!
!
+ !!!!
!
!! = ! + !. (2.2)
! > 0 0,
!!!!! !!!! !!!
0 1.
!!!!
!
> 0 0 .
! .
0 < !!!!! !!/!
1, = ,
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!!!
!
!
= 2!!!.
, (2.1) > 0 0.
2.
.
, (2.1)
[,] 0,+ , 0 < < < +. , :
0 < !!!!! !!!!! ,
0 1;
0 < !!!!! !!!!! ,
1
!!!!!!
!
, !!!!!!
!
. (2.2)
(2.1) [, ]. [, ]
. [,] 0,+ ,
.
> 0. ,
!! = !!! ln !!
> 0 > 0, , :
!! , =!
!
!!! ln !! =!
!
= !!! ln !! = !!! ln !! = ! + !
!
!
!
!
.
(2.3)
[,] 0,+ .
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, 0 < < . ! ln 0 0 + 0.
> 0 , ! ln 0,1 .
0,1 !!! ln !! !!!!!
.
! [, ].
> 0: !! ln 0 .
! > 0 , !! ln ! 1
,
!!! ln !! !! ln !!!!!!! ! !!!!!!! .
!
! , . , (2.3)
, , ,
> 0 :
! = !!! ln !!!!
!
. (2.4)
(2.4), , :
() = !!!ln! !!!!
!
.
,
0,+ -
:
(!) = !!!(ln )!!!.!
!
3.
:
+ 1 = (2.5)
(2.5) + 1
:
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+ 1 = !!!!
!
= = !
= !! = !!! = !!
=
= !!! !! + !!!!! = .!
!
, (2.5)
-. -
. , (2.5),
+ = + 1 + 2 + (2.6)
, (0,1] ,
, > 0 . (2.6) , = 1
,
1 = !! = 1,!
!
+ 1 = + 1 3 2 1 = !
, (2.5) ,
0 + 0:
=( + 1)
~(1)
=1
, + 0 + 0.
- , !! > 0
> 0., !! , (0,+).
1 = 2 = 1, 1,2
!! ! 1,2 . ! < 0 < !
! > 0 > ! , .. () , !
(!,+) ; = ! .
,
! = 1,4616 ,min = ! = 0,8856. . ..
> 2 (2.5) , + 1 = 2 = ,
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+ +.
(2.5)
, 1,2, ,, . .. .
1 < < 0 :
= + 1
(2.7)
1,0 . ,
(1,0)
1 + 0, 0 0 .
, , (1,0), (2.7)
(2,1) . ,
+
1 0 2 + 0. ,
, .
2.1.
2.1
4.
12= .
-5 -4 -3 -2 -1 0 1 2 3 4
-2
-1
1
2
3
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!!! =!!
!
2
,
!!! = = , ! = 0
=2
, ! = =
!!
2 =
12
!
!
!!!!! =
12
!
!
!
!
12=
2
.
!!= .
5.
+12=1 3 5 2 1
2!,
.
32=
2
52=3 2!
,
72=3 52!
,
. .,
+12=3 5 2 1
2!.
6.
12= 2 .
12= !
!+ 1
!!
= 2 ,
32=( !
!+ 1)
!!
=23
12=2!
3.
7. (0) = (1) = = () = .
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0 = 10
=10= ,
1 =(0)1
= ,
2 =(1)2
= ,
..
8.
1 =
sin.
9.
+12= 2!!!! 2 .
10.
+1
. . . + 1
= (2)!!!!
!!!!" .
11.
1 =!!!
1 +
!
!
, 0 < < 1 .
12.
1 =
sin, 0 < < 1 .
, - :
1
!!!!!!
!
=! = ! =
=
!
! =
1!
1!!!!
= 0 = 0; =
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=
!
!
!!!
!
1!
1!!!! =
1 !!!!
!!
!
!
!!!!!! =
=1
!!!!!!!! !!!! =
1 !!!!
+ 1
.!
!
2
!! !!! =! = = =
12!
!!
= 0 = 0; =
!
!
!!! 12!
!! =
!
!
=12
!!!!
!
!
!! =12
!!!!!! !! =
12 +
12=
!
!
=3 5 (2 1)
2!!!.
3
! !!! =! = =
!! =
1!!!!
= 0 = 0; =
!
!
= !! !!
1!!!! =
1
!!!! !!!! =
1 + 1
.!
!
!
!
4
! !!" ln() = = =
=
= 0 = 0;
!
!
=
=!
!
!
!
!! ln =
1!!!
!!! ln =1
!!! + 1 .
!
!
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5
!!!! !!! =! = =
!! =
12!
!!
= 0 = 0;
!
!
=
= !!!!!!
!
!
12!
!! =
12
!!!!!! =
12
!!!!!!!! =
12 +
12
.!
!
!
!
2.2 -
. -
:
, = !!! (1 )!!!!
!
(2.8)
. < 1, < 1,
(2.8) , .
= 0 = 1,
.
-
1. (, )
(, ), > 0 > 0,
(2.8) .
2. (, ) > 0 > 0
.
3. (, )
, . . , = , .
= 1
(2.8)
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, = !!!!
!
(1 )!!! = = 1 = = 0 = 1 ; = 1, = 0 =
= (1 )!!!!
!
!!! = !!!!
!
(1 )!!! = , .
4. -
:
, =!!!
(1 + )!!! =
!
!
!!! + !!!
(1 + )!!!
!
!
.
(2.8) = (1 )!! ,
0,1 1,+
!!= .
5. ,
:
+ 1, =
+ , , , + 1 =
+
, .
+ 1, .
.
6. 0 < < 1 :
, 1 =!!!
1 + =
sin
.!
!
, , = 1 = !!
, !!
, !!= .
7. , -
, =()()( + )
.
-
:
Copyright & A K-C
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sin
= , 1 = 1
1= 1 ,
, = 0,1,2
1 =
sin.
= !!
12= !
!!!! =
!
!
.
= ! ,
:
!!! =2
!
!
.
8. , = 2!!!! !!
, .
- -, :
+12= 2!!!! 2 .
9. :
( + 1) = ! = 2
1 +1
12+
1288!
139
51840!+ 0(!) .
10. , = 1
+ 1 , 1 =
1 + 1
1, .
11. , =( 1)! ( 1)!
( + 1)!
12. , 1 =!!!
1 + , 0 < < 1 .
!
!
13. , =( 1)!
+ 1 ( + 1).
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14. , =( 1)!
+ 1 + 1.
, -.
1
! = 1 = !!! 1 !!! =!
!
!
!
!
!
2,2 = 2 (2)
(4)=
=13!=16
.
2
! =!
!
!!(1 )
!!
!
!
= !!!!(1 )
!!!! =
32
,32
!
!
=
= !
! !
! 3
= 1 + !
! 1 + !
!2!
=8
.
3
! ! ! =!
!
= !!
!1
!
! =
!
!= ! = ! =
2
= 0 = 0; = = 1=
!
!
=!
2!!!!(1 )
!!!! =
!
2
!
!
32
,32=!
2 !
! !
! 3
=!
16.
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4
1 !!
=! = =
!! =
12!!!!
= 0 = 0; = = 1=12
!!!! (1 )!
!!
!
!
!
!
=
=12
!!!!(1)
!!!! =
12
12
,12=12
12
,12= !
! !
!2 1
= 2
.!
!
5
!!(1 + )!! =
!!!
(1 + )!!! = ,
!
!
=!
!
=!!!!
(1 + )!!!
!!
= 54
;34= !
! !
!(2)
= 14
14+12=
!
!
= +12= 2!!!! 2 = 2!!!
!!
24= 2.
1)
1) !
!
1 !! ; 2) 1 !!
;!
!
3) 1 !!
;!
!
4) 1 !
!
!
;
5) 1 !!!
!
; 6) cos !!! sin !!!;
!!
!
7) !!!;!
!
8) tg !;
!!
!
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9) sin !!!! cos !!!!;
!!
!
10) 1 !!
;!
!
11) 1 !!
.!
!
12) , , = + 1, + (, + 1)
13) , + 1, = !!(, + 1)
14) ,
(1 + )!!!(1 )!!! = 2!!!!! , .!
!!
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3.
3.1 .
,
:
!!
+!!
+!!
= 0. (3.1)
: = ,
= , = , (3.1) :
!!
+1
+
1!
!!
+!!
= 0. (3.2)
: ,
,
, :
= () () (),
, , .
. (3.2) :
!! + !!
! + !!!
!! + !! = 0,
( )
+1
!
+
1!
!!
+!!
= 0.
:
1
1!
=
,
, , , ;
, .
:
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= ; !! = 0;
1
1!
= ;
+1
+ =
1!
;
! !! + ! + !
=
,
; ,
. :
= , !! + = 0;
! !! + ! + !
= , ! !! + ! + ! = 0.
, , ,
:
! !! + ! + ! = 0,
!! + = 0, !! = 0, (3.3)
,
.
, , , (3.3), =
(3.2). ,
(3.2) :
+1
+
1!
+=+1
!+ =
=! !! + ! + ( ! )
! = 0
, (3.2),
,
, = , , , (3.3)
, .
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(3.3) = 1, 0
. = !,
( ), ( ), ,
:
! !! + ! + ! ! = 0. (3.4)
. , , ,
.
3.2
(3.4) :
= ! !!!!!
!!!
.
! = + ! !!!!!
!!!
,
! !! = + + 1 ! !!! ,!!
!!!
! ! = ! !!!!!!!
!!!
! ! !!!!!
!!!
=
= !!! !!! ! ! !!!!!
!!!
,!!
!!!
! !! + ! + ! ! =
= ( + )! ! ! !!! + !!! !!! =!!
!!!
!!
!!!
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= 2 + ! !!! + !!! !!!!!
!!!
.!!
!!!
,
2 + 1 ! !!! + 2 + ! + !!! !!! = 0!!
!!!
2 + 1 ! = 0, 2 + ! + !!! = 0 ,
= 2,3,4 :
2 + 1 ! = 0,
3 2 + 3 ! + ! = 0,
5 2 + 5 ! + ! = 0,
2 2 + 2 ! + ! = 0,
4 2 + 4 ! + ! = 0,
6 2 + 6 ! + ! = 0,
. . .
, ! = 0, ! = 0, ! = 0,
! ; !, !, !,
( ).
! =1
2! ( + 1),
:
! = !
4 ( + 1)=
12!!! ( + 1) ( + 1)
= 1
2!!! 1! ( + 2),
! = !
4 2 ( + 2)=
12!!! 2! ( + 2) ( + 2)
= 1
2!!! 2! ( + 3),
! = !
4 3 ( + 3)=
12!!! 3! ( + 3) ( + 3)
= 1
2!!! 3! ( + 4),
(3.4) :
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=1
2! + 1 !
12!!! 1! + 2
!!! +1
2!!! 2! + 3 !!!
=!!
!
+ 1
!!
!!!
1! + 2+
!!
!!!
2! + 3 =
= 1 ! 1
! + + 1
!!
!!!
2
!!!!.
, (3.4),
, ,
(3.4) 0 < < +, < < +.
! = (1)! ( 2)
!!!!
! ( + + 1)
!!
!!!
, (3.5)
.
(3.4).
:
! = (1)! 2
!!!!
! + !,
!!
!!!
(3.5')
, ,
! = (1)! ( 2)
!!
(!)!, ! = (1)!
( 2)!!!!
! + 1 !
!
!!!
!!
!!!
.
! = (1)! ( 2)
!!!
! + !
!
!!!
. ! = (1)! ( 2)
!!!!
! ( + + 1)
!
!!!
1- .
!() !!()
(3.4). ,
, , , ,
. ,
:
Copyright & A K-C
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= ! ! + ! !! . (3.6)
= ( ), ,
(3.5), (, 1/() = 0,1,2, ,
:
!! = (1)! ( 2)
!!!!!
! ( + + 1)= (1)!
( 2)!!!!!
! + !,
!!
!!!
!!
!!!
(3.5'')
, + ,
!! = (1)! (1)! ( 2)
!!!!
+ ! != (1)! ! ,
!!
!!!
(3.7)
, !!() !()
! !! + ! + ! ! = 0.
(3.6)
(3.4).
3.3
! =(1)! (/2)!!!!
! ( + + 1)
!!
!!!
; !()!
=12!
(1)! (/2)!!
! ( + + 1)
!!
!!!
;
!()!
=12!
(1)! (/2)!!!!
1 ! ( + + 1),
!!
!!!
= + 1;
!()!
=12!
(1)!!! (/2)!!!!
! ( + + 2)=
!!
!!!
=
2!!!
(1)! (/2)!!
! ( + + 2)=
!!!()!!!
.!!
!!!
,
!()!
= !!!()!!!
. (3.8)
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, !!!"
(
1/ ), !!(!)!!
,
. ,
, :
! ! !
= 1 ! !!! !!!
. (3.8')
! + ! = 2! (1)! (/2)!!!!!
! ( + + 1)= 2!
(1)! (/2)!!!!!
! ( + ) ( + )
!!
!!!
!!
!!!
;
! !() = 2! (1)! (/2)!!!!!!!
! ( + )
!!
!!!
=
= 2!!! (1)! (/2)! !!! !!!
! ( + )= !!! !!!()
!!
!!!
,
! ! = !!! !!! (3.9)
, !!!"
, ! !(),
v . m , :
!
! !() = !!! !!!() (3.9')
. (3.8),
:
!!
!
= !!!!
; !! !!!!
= !!!!
; !
! = !!!.
, , , ! = !. (3.9), :
! ! ! = ! !!!; ! ! + !!! ! = ! !!!; ! +
! = !!!.
:
2 ! = !!! !!!, (3.10)
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2
! = !!! + !!! (3.11)
(3.11)
!, !. , (3.11) ( = 1):
! =2 2
!!! !!!, (3.11')
:
! =2
! !,
! =4
! ! =8! 1 !
4
!,.
3.4
, ,
, .
+ 1/2, -.
. :
!! =
1 ! !!
!!!!!
! + !!
!!
!!!
, +32=1 3 5 (2 + 1)
2!!! ,
+32=1 3 5 (2 + 1)
2!!! ,
,
!! =
(1)! (/2)!!!!! 2!!!
! 1 3 5 (2 + 1) =
(1)! !!!!!
2!!!! 1 3 5 (2 + 1)
!!
!!!
!!
!!!
.
2! ! 1 3 5 2 + 1 = 2 + 1 !, :
!! =
2
1 ! !!!!
2 + 1 !=
2
sin!!
!!!
. (3.12)
Copyright & A K-C
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37
!!! =
1 ! !!
!!!!!!
! + !!
,!!
!!!
+12=1 3 5 (2 1)
2! ,
,
!!! =
1 ! !!
!!!!!! 2!
! 1 3 5 2 1 =
!!
!!!
=1 ! !
!!!!!
2!!!! ! 1 3 5 2 1
!!
!!!
.
2! ! 1 3 5 2 1 = 2 !,
!!! =
2
1 ! !!
2 !=
2
cos!!
!!!
(3.13)
(3.8') :
! !!
!!
= 1 ! !!!!
!!!!
,
(3.12)
!!()
!!
=2
sin
,
,
!!!! = 1 !
2
!!!!
! sin
(3.14)
(3.9') :
!!
!! !!!
= !!!!! !!!!!
,
(3.13)
!!! !!!
=2
cos
,
Copyright & A K-C
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, ,
! !!!! =
2
!!!!
! cos
(3.15)
3.5
,
,
2- !(), ,
(, 0)
! =! cos !!()
sin (3.16)
:
! = lim!! ! . (3.17)
! !() -,
= !! + !! (3.18)
2-
, 1- , :
!!! + !!! =2!
!!! !!! = 2!!
!! = !!!!
!!! = !!!!!
(3.19)
..
Copyright & A K-C
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!! = 1 !! , = 0,1,2, ., (3.20)
, .
!! () !
! () ,
(, 0),
!! = ! + ! , !
! = ! ! , (3.21)
!() !() - 1- 2- .
,
!() !()
x
.
3- ( )
,
(, 0) . ,
!() , (3.1)'
:
!! + !!! = !! + !!
! = !!! +!!
! , (3.22)
!,! ,!,!,!,! - .
!() !() ,
3- ,
, ..
Copyright & A K-C
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!!!! + !!!
! =2!
! ,
!!!! !!!
! = 2!
!
,
!!! = !!!!
! ,
!!!! = !!!!!
!
(3.22)
..
1) , ! = ! .
2) , ! ! = ! .
3) !!! + ! + ! !! = 0. .
:
= !+ !
=
= !!!! 1
!
3!+!
5!!
7!+ + !
!!! 1 !
2!+!
4!
4) !! + ! + = 0 0 = 1, ! 0 = 0.
:
= 1 !
2!+
!
1 2 !2!+
!
1 2 3 !2!++ 1 !
!!
! !2!!+
5) !! + !! + 4 = 0.
1) ! 2 , 2) ! 2 .
Copyright & A K-C
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6) I, II, III :
) !!! + ! + ! 2 = 0
) !!! + ! + ! 12 = 0
7)
) !!! + ! + ! 4 = 0,
) !!! + ! + ! !!"
= 0,
) !!! + ! + ! !! = 0,
) !! + !!! + 4 = 0.
Copyright & A K-C
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4.
.
.
4.1 u = 1a2
2ut2
+but+Cu
=1!
!!
+ + , (4.1)
- , , , , .
,
, ..
(,, )
(, , ) :
= cos
= sin
=
(0 , < , +)
(4.2)
Copyright & A K-C
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1
!
+1!
!!
+!!
=1!
!!
+ + (4.3)
,
= . (4.4)
(4.4) (4.3)
1
! +1!"+" =
1
1!
" + .
,
,
(!)
1!
" + + ! = 0 (4.5)
1
! + ! +1!"=
"
.
, ,
, (!)
" (! + ) = 0
!1
! + ! + ! = "
(4.6)
!,
" + ! = 0 (4.7)
1
! + ! !
! = 0 (4.8)
,
(4.4), (, , ) ,
.
Copyright & A K-C
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44
(4.4)
(4.5), (4.7),
(4.8).
=!!
+!!
+!!
= 0
. .
.
4.2 .
, ,
,
:
I ! =
II !
=
III+
!=
(4.9)
- ,
.
, ,
(4.9).
, ,
,
,
.
Copyright & A K-C
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45
,
1 !!! = 0, !!! = !, !!! = ! ,
2 !!! = , !!! = !!! = 0. (4.10)
(,, ) ,
,
( 4.1), , ,
.
4.1
,
, .. ! = !(), ! = !() = ().
,
= ,
:
l
l
aO
z
Copyright & A K-C
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1
! + ! = 0, " ! = 0. (4.11)
:
= ! + ! , = ch + sh , (4.12)
!() !() .
.. !() 1 , !() , 0 ,
= 0 ,
!() ,
:
= ! =!
,
! !().
,
:
= ! = !ch!+ !sh
!
! !
, = 1,2 (4.13)
, ,
.
(4.2) ,
= 0
, !"#!
( = 1,2). (4.12)
:
= ! + !
, = sin
(4.14)
!() !() .
!!"! 0, = 0, ,
= ! = !! sin
, = 1,2 (4.15)
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= !!
!"#!
!!"#
!
sin
,
!
!!!
(4.16)
! () sin!"#
!:
! =2
sin
.
!
!
(4.17)
, ,
-, .
4.3
,
! = () ( )
, .
.
= (4.18)
,, ,
,
|!!! = () (4.19)
!!!
= !!! , (4.20)
- .
Copyright & A K-C
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, , (4.18)
= ,
! + ! = 0,1
! + ! = 0 =
,
= ! + ! , = !!!
! ! .
0, ! 1, ! ,
= 0, (4.20)
, ,
! ! = 0, =!!
= ,
! ! = 0, (4.21)
, ,
.
(4.21) ! 0 < ! < ! <
,
! =!
, , (4.18),
,
= ! = !! !!
!
!!!! ! !
, = 1,2 (4.22)
,
= !! !!
!
!!!!! !
,!
!!!
(4.23)
, (4.19), !
,
Copyright & A K-C
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= !! !
0 < .!
!!!
(4.24)
()
, .
,
! =2
! !! ! + !! ! ! !
,!
!
(4.25)
(4.23).
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50
5. -
5.1 -
= !() = 0,
= 0, ,
! = 1.!!
!!
! ;
, ,
:
1) ! =1
1
1 + ! ,
2) ! =1
!!! .
.
5.1.
5.1
-2 -1 0 1 2
1
1
1)
2)
Copyright & A K-C
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,
= ! .
, 1)
! =
1
1 + ()! .
, , ,
,
, . . .
, . .
! = ! = = = ! = ! .!!
!!
!!
!!
!!
!!
!!
!!
0 = !
,
! m ,
. , !
, !!
( , ).
.
, , ! =!!
!!!(!!)!
0 ()! 1, . .
.
! =!!!(!")! 0.
= 0 . ! = ! 0 ,
! 0 = ! 0 .
,
:
1) 0 < 0 > 0.
2) = 0.
Copyright & A K-C
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3) ! = 1.!!
!!
, , -
.
.
, - ,
,
, 3). , ,
= ![!]
.
:
= = 0 .!!
!!
(5.1)
, = 0, 0,
= = ,!!
!!
!!
!!
.
( 2 ),
0 ,
= = 0 = 0 = 0 1 = 0 .!
!!
!
!!
!
!!
, (5.1) .
: (5.1) .
, , (
= ).
, :
= .!!
!!
(5.2)
Copyright & A K-C
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.
=1 , = 0 . (5.3)
=1
!(!) ! , (5.4)
() = !.
-
.
= . (5.5)
1
! 3 = 3! = 9.!!
!!
2
:
) ! + 3 + 5 = [ 5)! + 3 + 5 = 28 + 5 .
) 2 8 = 2 4 = !!
4 .
) ! + 2 = 1 + 2 = !! 1 + !
! + 2 ;
5.2 , -
, - .
- .
Copyright & A K-C
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1) =
0, < 0, > 0, = 0
= 1 !!
!!
;
) ! ! = + ( + )
2 ;
) = + ( )
;
4) () 1, ! , !! !, ! !; 5) !!", ! !!", , ! !!!".
6) = ()!
!!
,
.
7) =
!! = 2!
,
..,
(!)() = 1 !!!
.
8) = 0, !! = 0, , !!! ! = 0;
9) = (0)();
10) = ;
11) = ( )!!
!!
;
12) ! ! = ! ! , = !
!!
!!
.
13) ! !!
!!
= ! ;
14) ()!( )
!
!!
!!
= (1)! ! ,
Copyright & A K-C
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! = .
15) ! = (1)! ! (0)!!
!!
.
16) ,! !!
!!
!!
! .
17) - - ( !) :
=12
! !"# !!
!!
=12
!"!!
! =12
!!
!!
!!"# =12
!!"(!!!!).!!
!!
18) : ,! = ! ! ( !).
19) ,! =!! ! ( !) .
20) ,! =!!! !
!!"#!
( !)( !)
.
21) ! !()
! =12
!!"#!
!
,
! =12
!!"#!
!!
=12
!"#!
!
, ! + ! = ! = ! .
1
:
) ! + 4 + 5 = 5 ! + 4 + 5 = 29 + 5 .
Copyright & A K-C
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) 2 10 = 2 5 =12 5 .
) (! + 6 = [ + 3 2 =15 + 3 +
15 2 .
2
:
) ! 4 = 4! = 16;!!
!!
) (!!!
!!
+ 3 + 2) 2 = 2! + 6 + 2 = 12;
) ! + 3 + 2!!
!!
! 2 = 7.
3
:
) !! = , 0 = 0, ! 0 = 0
. :
!! =
! = = 1
=1!
= , ! = 1
, -
, ,
= 1 = 0 = 1.
) !! + ! = ! + ! , 0 = !, ! 0 = !!.
Copyright & A K-C
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.
Z !! + ! = ! + !
!() ! + ! = !!!" + !!" ,
=1
! + != ! + ! + !!" ! + .
, :
= ! cos +!
sin +!
sin + cos .
1)
a) 5 ;
b) ! + 5 + 6 ;
c) ! + 5 + 6 ;
d) ! + 5 + 6 2 ;
e) ! 4 ;
f) 2 3 ;
2) :
a) !! = ! , 0 = ! 0 = 0.
b) !! = , 0 = ! 0 = 0.
c) !! + 2! + = !! , 0 = ! 0 = 0.
d) !! + + sin = ! , 0 = ! 0 = 0.
e) ! + = + ! + !
! + = + + ! , 0 = 0 = 0.
f) !! + 2! + =
! ! 2 + 2 = 1 2.
g) !! + 2 = 0
!! 2 = , 0 = 0, ! 0 = 1, 0 = ! 0 = 0.
h) !! + 2! + = , 0 = ! 0 = 0.
Copyright & A K-C
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6.
6.1
() ,
:
=2
!!!,!
! (6.1)
,
= . ,
, = 0
= . () ,
,
.
, (6.1)
!!! . :
=2
1 !!!
!
!
!!!
=2
1 !!!!!
! 2 + 1 , <
!
!!!
!
!
, (6.2)
(6.2) , () .
()
,
6.1.
Copyright & A K-C
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6.1
(0) = 0 ,
= 1,
!!! =2
!
!
. (6.3)
()
1 =2
!!! 2
!
!
!!! =!
!
2
!!!!!
!
. (6.4)
, , ,
.
, () ,
,
!" = !!!
!
!
=2 ,
0 0,5 1 1,5 2 2,5 3
1
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!"# = !!!
!!
!
=2
1 () .
.
6.2
,
, .. ,
( , + )
1 2
(!!!)!
!!! , (6.5)
- .
,
, , :
< < =1
2!
!!!!!! =
!!!
!!!
1
!!!!"
!! !
! !! !
=
=12
2
2
(6.6)
.
, = , = +, = 1, .
= , = , ,
.
= 1
2 (6.7)
,
, :
> = 1
2 (6.8)
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= !, :
= > ,
. (6.7) (6.8) ,
:
! 2
=12
,
, ! = 0,67449 .
1
, ,
10 ,
= 1 , 1 .
.
10 0,1 = 12= 0,683,
68% .
0,7 .
6.3 .
,
> 0 , !
= 0 ,
.
= ,
=
, =
,
Copyright & A K-C
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;
;
;
,
=!!
(6.9)
!!! = 0 (6.10)
!!!= 0, !! ! . (6.11)
= !!
,
!!"!
!
=!!
!!"!
!
,
!!
= !
!!!= 0,
!!!
.
(6.12)
=!
1
+ !! ! , (6.13)
> 0 > 0.
, .
!!! ! !!!
!! !!!
!
!
= !!!! 1
2
!!!
! !
!
,
Copyright & A K-C
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!!! !!!! 1 ,
.
1) , ,
= +1!!! + .
2) ,
= !!"!
!
=!!
1
2
.
3)
) = !!! sin 2 !
!
; ) ! = 1 4
!!! !!!!
1 + !
!
!
;
) =2
!!!sin 2
!
!
; ) 1 ! =4
!!! !!!!
1 + !
!
!
, arg 4
.
4)
) 1 =2!!!! !!!!!!"
!
!
;
) 1 ! =4!!!! !!!!!!" !
!
!
.
5)
) =12
!!!
!!!
!
!
,
) =12
!!
!!!
!
!
.
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7.
7.1
. Ei
Ei =!
!
!!
, arg < , (7.1)
,
( 7.1).
7.1
,
Ei .
, (7.1)
Ei =!!
!!
!
, arg < , (7.2)
z
O
L
Copyright & A K-C
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Ei , 0 .
= > 0 7.2.
7.2.
Ei 0 = +
Ei = 0.
,
, (7.1)
Ei =!
!!
!!
+! 1
!
!!
+! 1
!
!
+!
!
!!
,
, .
= !!, = ,
C =1 !! !
!!
!
!
, (7.3)
, = = 0,5772157.
,
Ei = + ln +! 1
!
!
, arg < . (7.4)
0 1 2 3
1
Copyright & A K-C
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,
,
, ,
. ,
.
! 1
!
!
=!!!
!
!
!!!
!
!
=!
!
!
!!!
, < ,
,
Ei = + ln +!
!
!
!!!
, arg < . (7.5)
,
.
(7.5) , Ei
Ei 0 = Ei! , > 0,
Ei! ,
Ei! =12
Ei + 0 + Ei 0 = + ln +!
!
!
!!!
, > 0. (7.6)
Ei!
Ei = Ei! , > 0.
.
.
!
! 1
!!
=!
2Ei + Ei 1 .
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, (7.1),
Ei =!
!!
!
!!!
+ ! , (7.7)
! = + 1 ! !!e!
!!!
!
!!
, arg < . (7.8)
! + 1 ! !!!
. (7.9)
, Ei
,
.
7.2
Ei
= > 0 ,
Si Ci ,
.
, ,
:
Si() =sin
!
!
, Ci =cos
!
!!
, arg < , (7.10)
, - , ,
, ,
( 7.3).
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Si() - , Ci() - ,
, 0 .
= > 0
.
Si()
=sin
,Ci()
=cos
,
, Si() = = 1,2, . . ,
Ci = + !! = 0,1,2 .
Si =2
, Ci = 0,
Si 0 = 0, Ci +0 = . (7.11)
Si() i() 7.3
7.3 Si() i()
Ei , Si Ci .
Ei = Ci 2 Si , > 0 (7.12)
0 10
-2
-1
1
2
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Ei = Ci + 2 Si , > 0
Ci =12
Ei !"! + Ei !
!"! ,
Si =2
12
Ei !!"! Ei !
!"! ,
arg 0, ) !!"()
!() (0)
! () ! !!!!! ! (0)!!!
!!!
()
1()
1 !!() !()!
()
()!
!
! !()
! !( )!
!
! !()
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1
1.
=12ln + 1 3.
2.
ch 3 5 sh 3 + 1
.
3. -
tg! !
!
.
4. I, II III
!!! + ! + ! 12 = 0.
5.
!!! + ! + ! 81 = 0.
6.
!! = , 0 = 0, ! 0 = 0.
Copyright & A K-C
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2
1.
= 2sin 3+ 1.
2.
ch! .
3. -
1 !"!"!
!
.
4. -
cos!
!!
!
sin! .
5. I, II III
!!! + ! + ! 3 = 0.
6.
!!! + ! + ! 1 = 0.
7.
!! = , 0 = 0, ! 0 = 0.
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3
1.
=12cos +
6 1.
2.
sh 3 ch + 2
.
3. -
cos!"
!!
!
sin! .
4. -
!!!!!
!
.
5. I, II III
!!! + ! + ! 15 = 0.
6.
!!! + ! + ! 144 = 0.
7.
!! = , 0 = ! 0 = 0.
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4
1.
=12sin +
6 1.
2.
ch 3 sh + 2
.
3. -
cos!
!!
!
sin!" .
4. -
!!!!!
!
.
5. I, II III
!!! + ! + ! 13 = 0.
6.
!!! + ! + ! 100 = 0.
7.
!! + + sin = , 0 = ! 0 = 0.
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5
1.
= 2ln + 2 1.
2.
ch .
3. -
! 27 !!
!
.
4. -
cos!"
!!
!
sin!! .
5. I, II III
!!! + ! + ! 2 = 0.
6.
!!! + ! + ! 14
= 0.
7.
!! + 2! + = , 0 = ! 0 = 0.
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6
1.
= 2ln 2 + 1.
2.
! ch .
3. -
! 8 !!
!
.
4. -
cos!
!!
!
sin! .
5. I, II III
!!! + ! + ! 4 = 0.
6.
!!! + ! + ! 121 = 0.
7.
! + = + ! + !
! + = + + ! , 0 = 0 = 0.
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7
1.
= 3 sin 6+ 2.
2.
1 + sh
.
3. -
! 1 !!!
!
.
4. -
tg!
!!
!
.
5. I, II III
!!! + ! + ! 7 = 0.
6.
!!! + ! + ! 25 = 0.
7.
!! 2 = !! + 2 = 0, 0 =
! 0 = 0, 0 = 0, 0 = 1 .
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8
1.
=12cos +
3 2.
2.
ch + 2sh
.
3. -
! 1 !!
!
.
4. -
!!!!!
!
.
5. I, II III
!!! + ! + ! 10 = 0.
6.
!!! + ! + ! 49 = 0.
7.
!! + 2! + = , 0 = ! 0 = 0.
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9
1.
= 2 ! !! + 3.
2.
ch 2 sh 3 .
3. -
1 !!
!
!
.
4. -
cos!
!!
!
sin! .
5. I, II III
!!! + ! + ! 6 = 0.
6.
!!! + ! + ! 16 = 0.
7.
! + = ! 2 = 2, 0 = 0 = 1.
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10
1.
= 2 ! !! 3.
2.
ch sh .
3. -
! ! !
!
.
4. -
cos!
!!
!
sin! .
5. I, II III
!!! + ! + ! 8 = 0.
6.
!!! + ! + ! 36 = 0.
7.
! 8 + 7 .
Copyright & A K-C
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86
11
1.
= 2 ln 1 + 3.
2.
ch 2 3sh 2 1
.
3. -
! 128 !!
!
.
4. -
tg!
!!
!
.
5. I, II III
!!! + ! + ! 5 = 0.
6.
!!! + ! + ! 9 = 0.
7.
! 8 + 7 3 .
Copyright & A K-C
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12
1.
= 2 cos 3+ 1.
2.
sh 2ch + 3
.
3. -
! 32 !!
!
.
4. -
tg!"
!!
!
.
5. I, II III
!!! + ! + ! 14
= 0.
6.
!!! + ! + ! 64 = 0.
7.
! + 3 + 2 3 !!
!!
.
Copyright & A K-C
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13
1.
= 2 ! !! 1.
2. sh 7 2ch 7 + 2
.
3. -
1 !!
!
.
4. -
!!!!!
!
.
5. I, II III
!!! + ! + ! 32
= 0.
6.
!!! + ! + ! 949
= 0.
7.
! + 3 + 2 3 !!
!!
.
Copyright & A K-C
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89
14
1.
= 3 ln 2 + 4.
2.
ch 7 2sh 7 + 2
.
3. -
1 !!" !
!
.
4. -
!!!!!
!
.
5. I, II III
!!! + ! + ! 23
= 0.
6.
!!! + ! + ! 49
= 0.
7.
! + 3 + 5 .
Copyright & A K-C
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15
1.
= 2 ! !! + 3.
2.
!! sh .
3. -
! 1 !! !
!
.
4. -
cos!
!!
!
sin! .
5. I, II III
!!! + ! + ! 169
= 0.
6.
!!! + ! + ! 94
= 0.
7.
! 3 !!
!!
.
Copyright & A K-C
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1. , . . / ..
. -. . . : , 2015. -
358 .
2. , . .
/ . . . : , 1984. 382.
3. , . . . .
. / . . . :
, 1973. 336 .
4. , . . / . . .
: , 2003. 254.
5. , .. :
/ .. .
: , 1960. 196 .
Copyright & A K-C
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Copyright & A K-C
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