:

:

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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