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kt.cabaynatssuoruod.wwwcab-aynat-ssuoruod/moc.bf.www
imazmazze assi yb daelpu
2 2
4 -
5 -
6
8
01
21 - -
31
41
61n
81
02
22
42
62
82
13
23
43
63
83
3
4
( )
0 a) ( +b xa:
- +ba
x -
+b xa a a
0 a) ( + +c xb xa: + + = Rc xb xa x) (:
:
R = R 0 x x) (
Rx) ( Rx) (
f = S 0 <
x - +
Rx) ( a
Sb 0 = a2-
=
+b
a2-
- x
a
Rx) ( a
) (b
x a xa2
+ = R
ca4 - b =
0 >
=x , x S2 1} { :
1b
xa2
D - - =
2b
xa2
D + - =
x - x1 x2 +
a
a
Rx) ( a
- - = Rx x x x a x2 1) () ( ) (
R= + + 0 c xb xa x 0 a) (: x2 x1
2 1: x xb
a-
2 1 = +x xca
=
5
( )
. ba
+ + = +b ba2 a b a2 2 2) ( + - = -b ba2 a b a2 2 2) (
+ - = -b a b a b a2 2) () (
+ + + = +b ba3 b a3 a b a3 2 2 3 3) ( - + - = -b ba3 b a3 a b a3 2 2 3 3) (
+ + - = -b ba a b a b a2 2 3 3) () ( + - + = +b ba a b a b a2 2 3 3) () (
: x f
: f
= Df R =x x f) ( ) (
) () ( ) (x
x fx QR
=0 x Q / x Df} {) ( =
R =0 x / x Df} {) ( R =x x f) ( ) (
) ( ) () (x
x fx Q
R > =0 x Q / x Df} {) ( =
) ( ) () (x
x fx Q
R R =0 x / x Df{) ( >0 x Q} ) ( =
) () ( ) (x
x fx QR
f) () ( 0 x Q} ) ( =x
0 / x Dx Q
=R
6
( )
: ax x a x x * nn ) ( n
0 x0 x mil
=0 x0 x mil
=
>
xn
0 mil1-x
=
xn
0 mil1+x
=
xx mil+
+ =
x
0 mil1+x
=
: n : n n
xx mil+
+ =
nxx mil-
+ =
0 xn
mil1x
+ =>
0 xn
mil1x
+ =
0 xn
mil1x
- =l 0l 0
8
( )
:
: 0) ( ) (
0
x f x f mil fx x
= >
x0
0) ( ) (0
x f x f mil fx x
= a0
nan mil+
+ =na0 n mil+
=
:R q:qn) ( - 1 q < < -1 q 1 =1 q >1 q
nnq mil+
n + =n1 q mil+
n =n0 q mil+
qn) ( =
:
n n n
n nn n
nn
w u vu mil v mil
w mil +
+
= =
=
l l
l
n nnnnn
v u 0 v milu mil
+
== -
l l
n nnnnn
v u+ v milu mil
+
- =- =
n nnnnn
v u+ v milu mil
+
+ =+ =
:= +u f un 1 n) ( : un) (
) (0
n 1 n
a u+u f u
==
I a I I f) ( I f =x x f) ( l un) (
02
( )
:
: I f
I f F :
I F = "x f x 'F I x) ( ) (
:
I f : I f F
: I f a +k k x F x) ( ) (
I f y0 I x0
: I f F =y x F0 0) (
: - :
: k I g f
I g f G F :
I +g f +G F I fk Fk
21
: ( )F x ( )f x
ax k+ a 1 x + k2
x
1 kx-
+ 1x
2 x k+ 1x
r 1x kr 1
++
+
rx { }( )r * 1 - -
cos x k- + sin x sin x k+ cos x tan x k+ 11 tan x
cos x+ =
ln x k+ 1x
xe k+ xe :
( )F x ( )f x ( ) ( )u x v x k+ + ( ) ( )u ' x v ' x+
( )a u x k+ ( )a u ' x ( )a ( ) ( )u x v x k + ( ) ( ) ( ) ( )u ' x v x u x v ' x +
( )1 k
v x+ ( )
( )v ' x
v x -
( )( )
u xk
v x+ ( ) ( ) ( ) ( )
( )u ' x v x u x v ' x
v x -
( )2 u x k+ ( )( )
u ' x
u x
( ) r 1u xk
r 1
+ ++
( ) ( )
ru ' x u x { }( )r * 1 - -
( )ln u x k+ ( )( )
u ' xu x
( )u xe k+ ( ) ( )u xu ' x e
( )1 sin ax b ka
+ + ( )cos ax b+ ( )a 0
( )k R
( )1 cos ax b ka
- + + ( )sin ax b+ ( )a 0
22
( )
: _ !b,a] [ f F _ !b,a] [ f
: b a f - = = aaa F b F x F xd x fb b) ( ) ( ) ( ) (
: :
) ( ) ( = a0 xd x f a) (b a
xd x f xd x fa b
- =
) ( ) ( ) ( ) ( = a axd x f k xd x fk kb b) ( ) ( ) (b b b
xd x g xd x f xd x g x fa a a
+ = + :
+ =c a axd x f xd x f xd x fb c b) ( ) ( ) (
: "0 x f b,a x) ( ] [ :
a0 xd x fb) (: "x g x f b,a x) ( ) ( ] [:
a axd x g xd x fb b) ( ) (: :
b,a] [ f -a baxd x f 1b) ( : b,a] [
: 'g 'f #b,a] [ g f
#b,a] [
- = a aaxd x ' g x f x g x f xd x g x ' fbb b) ( ) ( ) ( ) ( ) ( ) ( :
r rj ,i,o) ( A . u :
i o r
j r
j i = A . u.1r r
32
b;a] [ f Cf
: :=b y=a x
aA.u. xd x f b) (
b;a] [ g f Cf
Cg :=b y=a x:
aA.u. xd x g x f b) ( ) ( -
:
f) ( b,a] [
bA.u. xd x f
a
f) ( b,a] [
bA.u. xd x f
a -
f c,a] [
f b,c] [
c aA.u. xd x f xd x fb c) ( ) ( - +
Cg) ( Cf) ( b,a] [
aA.u. xd x g x f b) () ( ) ( -
Cg) (Cf) ( c,a] [
Cf) (Cg) ( b,c] [
c aA.u. xd x f x g xd x g x fb c) () ( ) ( ) () ( ) ( - + -
:
Cf) ( b,a] [
av.u xd x f V b) () ( : =
p
24
) (
: : 1x
xa ] [0;+ 1
:ln :
ln e 1= ln1 0=
] [ ] [x 0;+ y 0;+" " ln x ln y x y= = ln x ln y x y> >
] [y
x 0;+ y
ln x y x e
" "
=
=
( )r
] [ ] [
r
x 0;+ y 0;+
ln xy ln x ln y
ln x r ln x
1ln ln x
xx
ln ln x ln yy
" "
= +
=
= -
= -
n : nx * ln x n ln x" = :
f ( )f x ln x= ] [fD 0;+=
( ) ( )f x ln u x= ( ) }u x > 0 {f uD x / x D= : : :
lim ln xx
= ++
( ) ( )0 0
lim u x lim ln u xx x x x
= + = +
lim ln xx 0+
= -
( ) ( )0 0
lim u x 0 lim ln u xx x x x
+= = -
nln x
lim 0x x
=+
( )n * ( )( )
( ) n0 0ln u x
lim u x lim 0x x x x u x
= + =
nlim x ln x 0x 0+
=
( ) ( ) ( )n0 0
lim u x 0 lim u x ln u x 0x x x x
+= =
ln xlim 1
x 1 x 1=
- ( ) ( )( )0 0
ln u xlim u x 1 lim 1
x x x x u x 1= =
-
( )ln x + 1lim 1
x 0 x=
( ) ( )( )0 0ln u x 1
lim u x 0 lim 1x x x x u x
+= =
0x 0x + -
52
: +;0[ ] ax nl x
I a x u nl x) ( I u ax nl x
+,0[ ]
x nl ;0 x1 ) ( [ ] :x
= + "
u I a x u nl x) ( I
) () ( ) () ( :'x ' u
x u nl I xx u
= "
:nl :
- +1 a*} {: a
: gola: a
a [ ]:x nl
x gol ;0 xa nl
= + "
lgo : lgo01 : :
a
a
0 1 go
1 a go
=
=
l
l
[ ]r
a
r ;0 x
a x r x go
" + "
= =
l
[ ] [ ]a a a
a a
a a
a a a
;0 y ;0 x
y g n x go yx gox go r x gor
1x go go
x
xy go x go go
y
+ " + "
+ =
=
- =
- =
l l l
l l
l l
l l l
r) (
: 1 < a < 0 1 > a
l ly < x y go > x goa a l ly > x y go > x goa aa
a
x go milx
x go mil+0 x
+ =+
- =
l
l
a
a
x go milx
x go mil+0 x
- =+
+ =
l
l
: a) ( [ ]
1' x go ,0 x
a nl x l = + "
x 0 1 + x nl - +
62
( )
:
nl ae xx :
x
x
0 e x
x e nl
"
=
>
= + "x e ;0 xx nl [ ][ ]
x
;0 y x
y nl x y e
+ " "
= =
y x ) (y x
y x e e y;x
y x e e
= = "
> >
) (
) (y x y x
xr xr
xx
xy x
y
r y;x
e e e
e e
e 1ee ee
+
-
-
" "
=
=
=
=
: f
= Df =e x fx ) ( =D x / x Du f} { =e x fx u) ( ) (
: : :
xxe mil+
x u) ( ) ( + =0 0x x x xe mil x u mil
+ = + =
xx0 e mil-
x u) ( ) ( =0 0x x x x0 e mil x u mil
= - =
x
xn
mile+x
) ( * n) ( + =) (
) (
x u
0 0x x x xn
mil x u mile x u
+ = + =
x nx0 e x mil-
x u n) ( ) ( ) ( =0 0x x x x0 e x u mil x u mil
= - =
x
0 x
1 mil1 ex-
=
) () (
) (x u
0 0x x x x
mil 0 x u mil1 e x u
- + = =
x0 x0 - +
: ae xx
I ae xx u) ( I u
72
ae xx = "e e xx x) (:
I u I ae xx u) (
= " e x u e I xx u x u) ( ) ( ) () (: :
- +1 a*} {: a :
a gola aa xx :
) (a nlx x
xa
e a x
x a gol
= "
=
[ ] = + "x a ;0 xa golx
= =y x a ay x[ ]
a x) (
;0 y x
y gol x y a
+ " "
= =
) (
) (y x y x
xr xr
xx
xy x
y
y;x
a a a
a a
a 1aa aa
+
-
-
"
=
=
=
=
r) (
: < 1 a
< >y x a ay x > >y x a ay xx
xa mil+
+ =x
x0 a mil-
=
xx0 a mil+
=x
xa mil-
+ =x
0 x
a nl mil1 ax-
=
: = a a nl ax x) ( ) (
82
( )
+ = = b;a / bi a z{) ( - =1 i} : :
b;a) ( : + =bi a z z +bi a z eR) (: z a z mI) (: z b
z =0 z mI) (: : z =0 z eR) (:
: zz
= = z eR z eR z z) ( ) ( =z mI z mI) ( ) ( :
e, e,o2 1) ( ruu ru
:
z ' M) ( z M) (
+ = +' z z ' z z = ' z z ' z z
* n) ( =z znn
1 1' z ' z
=
z z' z ' z
=
0 ' z) (
= z z z - = z z z = +z eR2 z z) ( = -z mI i2 z z) (
+ =z mI z eR zz ) ( ) (
:
b;a) (: + =bi a z b ,a M) ( z
z M) (: z M M z MO z
ruuuz MO) (:
ruuu ruuu =MO ffA z) (
b;a) (: + =bi a z - =bi a z: z
b;a) ( : + =bi a z + = =b a zz z: z
29
( )
( )
nn *z z n
z z
zz z ' 0z z
=
- =
=
z z z z
z z
1 1z z
=
=
=
:
:
a
( ) ( ) [ ]arg zz ' arg z arg z ' 2 + p [ ]arg z arg z 2 - p ( )[ ]arg z arg z 2- p + p [ ]narg z n arg z 2 p
[ ]1arg arg z 2z
p-
( ) [ ]zarg arg z arg z ' 2z '
- p
[ ] [ ] [ ]r, r ', ' rr '; 'q q = q + q [ ] [ ]r, r,q = -q [ ] [ ]r, r,- q = p + q [ ]n nr, r ;n q = q
[ ]
1 1 ; 'r '; ' r '
= -q q
[ ]
[ ]r; r ; '
r '; ' r 'q = q - q q
( )i 'i i 're r 'e rr 'e q+qq q =
i ire req - q=
( )iire re p+qq- =
( )ni n inre r eq q= i '
i '1 1 e
r 'r 'e- q
q=
( )i
i 'i '
re r er 'r 'e
qq-q
q=
[ ] [ ]k r, 2k r," q + p = q Z z arg z k = p z arg z k
2p
= + p ( )k Z
: :
( ) ( ) ( )nn
cos i sin n cos n isin n
"
q + q = q + q
( )i i1cos e e2 q - q"q q = +R ( )i i1sin e e2i q - qq = -
z z a = ( )a :
a > 0 { }S i a;i a= - a = 0 { }S 0= z z = a a < 0 { }S i a;i a= - - -
z M z q :( ), OM1e uuuuruur
:arg z :[ ]arg z 2= q p z r z= [ ]arg z 2= q p
z: ( ) [ ]z r cos i sin r,= q + q = q
z :iz re q=
a 0> a 0< [ ]a a,0=
ai a,2p
= +
[ ]a a,= - p
ai a,2p
= - -
03
0 a) ( cba: = + + 0 c zb za z: : :
; Sb b 0 > a2 a2
= D + - D - -
Sb 0 = a2 = -
) (
2 2
0 = c + zb + za z
ca4 - b =
; Si b i b 0 < a2 a2
= D- + - D- - -
:
- =z z BAA B BA I
B;A] [ B A
Iz z
z2+
=
A c] [ ) ( ruuu ruuuCA;BA) ( A B
z z2 gra CA;BA
z z-
- p
ruuu ruuu
CBAA C
A B
z zz z
-
-
DCBAA DC B
C D A B
z zz zz z z z
--
- -A DC D R
C B A B
z zz zz z z z
--
- - R
) (r z zA
0 r
= -
>
=r MA M
r A =MB MA - = -z z z zB A
BA] [ M A C
A B
z z;r
2 z z - = p -
A CBA
A C] [A B
z z;1
z z-
q =-
A CBA
A C
A B
z z;1
2 z z - = p -
A C
A B
z z;1
z z - = p -
3
CBA A
CBA
:
: :u b:+ = b z z ut:
r
W w:w - = w - z k z) ( k; h) (: W w:w - = w - qz e zi) ( ; r) (:
) (b ya 'y
0 a+ =
) (
) (
e x yb xaa
a
- a =
R
:
0 >
r2 r1 b + a =e e x yx r x r2 1 ) (
R b a ,) (:
r 0 = b + a =e x x yxr ) ( ) (
R b a ,) (:
)
(
0 = yb + 'ya + ''y) (0 = b + ra + r
b4 - a =
0 <
: - =qi p r1
+ =qi p r2
b + a =e xq nis xqsoc x yxp ) ( ) (
R b a ,) (:
23
( )
r r rk,j , i ,o) ( : - - :
J3 r' c,' b,' a v) ( rc,b,a u) ( r r+ + =' cc ' bb ' aa v.u
r+ + =c b a u
' a a i' a a ' a a ' b b
k j i ' b b j v u' b b ' c c 'c c
' c c k+ - = =
rr rr r r r r
: : B A
- + - + - = = z z y y x x BA BAA B A B A B) ( ) ( ) ( ruuu
: = + + +0 d zc yb xa: P) ( M
,M dd zc yb xaM M M ) () (c b a
+ + += R
+ +
: r Du,A) ( M
) () (u MA
,A du
= D
r ruuuu r
: R) ( r= + + + R 0 d zc yb xa : c,b,a n) ( ) (
CA BA CBA ruuu ruuu
CBA) ( : CBA) (
ruuu ruuu ruuuu= 0 CA BA .MA CBA M) ( ) ( :
: R Wc,b,a) ( = - + - + -R c z b y a x) ( ) ( ) (
33
= + + + R0 d zc yb xa : ) ( WR, S) ( R) ( W H
R W = W =; d H d) () (:
R) ( S) (
R) ( S) (
H
R) ( C) ( S) ( H:
- =d R r: :D) ( WR, S) ( D) ( W H
D W = W =; d H d) () (:
R) (
S) ( D) (
H S) ( S) ( D) (
BA] [ S) ( = 0 MB . MA S M) (:
ruuu ruuuu
BA BA] [ W S) ( :2
43
( )
: :
EdraC: E E = 0 draC:
: BA
- + = B A draC BdraC AdraC B A draC) ( ) (
: :
E A A: E A =A x / E x A} {
: = A A = E A A - =Adrac Edrac Adrac
: * p) (p
n1 n2
......................................... np p
n ... n n np 3 2 1:
: - :
n p) (* pn np: n p
53
: :
n p) (* pn : n p
+ - - - =1 p n ... 2 n 1 n n Anp) ( ) ( ) ( p
: n n n
- - =1 2 ... 2 n 1 n n !n) ( ) (: :
n E n p) ( p E A
n p
: p
n pn
CA!p
=
:Cnp Anp!n: 1 !0 * n1 2 ... 2 n 1 n n !n) ( ) (
- - = =
) (pn
C!n! p n !p
=-
) (
pn
A!n! p n
=-
n =1 Cn
1 =n Cn
0 =1 Cn
1 nn Cn
= -p n pC Cn n
C C C1 n n np 1 pp =--
= ++
: n n
= + +n n n n3 2 1) ( A n1 B n2 C n3
: 3 2 1
!n ! n ! n ! n
: : n p) ( n p
: p
Cn np
p An
63
( )
W
W A A
B A B A B A B A
W = = A A A A) ( A A = B A BA
: - :
W pi: wi} { pi wi} {
= wp Pi i) (} {:
: A W w w w w =;...; ; ; An 3 2 1} { w + + w + w + w =p ... p p p A pn 3 2 1) ( ) ( ) ( ) ( ) (
: W
= W1 p) ( = 0 p) ( W A 1 A p 0) ( :
W B A - + = B A p B p A p B A p) ( ) ( ) ( ) (
B A + = B p A p B A p) ( ) ( ) ( :
- =A p 1 A p) ( ) (: W A :
: W
A pAdrac ) ( : W A drac
=W
: - :
0 A p) (: B A
) () ( ) ( ) (: A B AA p A p B pBB A p
= =
73
: 0 B p A p) ( ) (: B A
= = B Ap B p p A p B A pA B) ( ) ( ) ( ) ( ) ( : :
B A B = B p A p B A p A) ( ) ( ) (
: W W2 W1 W
W = W W = W W2 1 2 1) ( :W A
W W W + W =p p p p A pA A2 12 1) ( ) ( ) ( ) ( ) ( :
W X : X
X : = Wx;...; x; x; x Xn 3 2 1} { ) ( n;...;2;1} { i =x X pi) ( : - -
X :
: + + + + =p x ... p x p x p x X En n 3 3 2 2 1 1 ) ( X
- = X E X E X V) ( ) ( ) ( X = sX V X) ( ) ( X
: A p
n A X
p n - = = "- p 1 p C k X p n;...;2;1;0 knk kk n) ( ) ( } {:
=p n X E) ( - =p 1 pn X V) ( ) (
xi x1 x2 x3 ... xn
=x X pi) ( p1 p2 p3 ... pn
38
)( ) (
:
:
-1 cosx 1-1 sinx 1cosx + sinx = 1
sinxtanx =cosx
11 + tanx = cosx
( )( )( )
cos x + 2k = cosxsin x + 2k = sinxtan x + k = tanx
:
x = -a + 2k cosx = cosa x = a + 2k ( )x = - a + 2k sinx = sina x = a + 2k
( )tanx = tana x = a + k k Z
2
3
4
6 0 x
1 32
22
12
0 sinx
0 12
22
32
1 cosx
3 1 33
0 tanx
+ x2
- x2
+ x - x -x
cosx cosx -sinx sinx -sinx sin
-sinx sinx -cosx -cosx cosx cos
39
:
( )( )
( )
cos a + b = cos a cos b -sin a sin bsin a + b = sin a cos b + cos a sin b
tan a + tan btan a + b =1- tan a tan b
( )( )
( )
cos a - b = cosa cosb + sina sinbsin a - b = sina cosb - cosa sinb
tana - tanbtan a - b =1+ tana tanb
:
cos 2a = cos a -sin a = 2cos a -1 =1- 2sin asin 2a = 2sin acos a
2tan atan 2a = 1- tan a
1 + cos 2acos a =2
1- cos 2asin a = 2
:at = tan2
2tsin a = 1 + t1- tcos a = 1 + t2ttan a =
1- t
: :
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1cos acos b = cos a + b + cos a - b21sin asin b = cos a + b cos a b2
1sin acos b = sin a + b sin a b21cos a sin b sin a + b -sin a b2
- - -
- -
-
=
p + q p qcos p + cos q = 2cos cos
2 2p + q p qcos p cos q = 2sin sin
2 2p + q p qsin p + sin q = 2sin cos
2 2p + q p qsin p sin q = 2cos sin
2 2
-
-- -
-
--
: acosx bsin x+ ( ) ( )a,b 0,0
( )
a ba cos x bsin x a b cos x sin xa b a b
a b cos x
+ = + ++ +
+ - a =
a : bsin =
a + b acos =
a + b
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