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nxא#א x^4+

=uU!3^)א 2I c 4^2 S&u3א8<)OTUא#אI)א )nx u x3 d4^אI

S7JאW

nxא#א x3 d6^א /%* g] [0;+∞ #W

] [ ( )1

10; n

n nx x

n x −′∀ ∈ +∞ =

=uU!3^)א 2I c 6^2 S&u3א8 /%* g 0g<I)א )OTUא#א )nx u x3 d6^א /%* gI

#W ( )( )( )

( )[ ] 1n

nn

u xx I u x

n u x −

′′∀ ∈ =

) W(/א#+ ) na x x a∈ ∈ =

nv(!U(#v(!U(#v(!U(#v(!U(# nDX(#DX(#DX(#DX(# 0a > nS a= ;n nS a a= −

0a = 0S = 0S =

0a < nS a= − S = ∅

9TB Q77)+(?@`אא7W

=pr

q=M$#K? ZW *p#)א ∈ *q ∈

] [0,

p

q qr px x x x∀ ∈ +∞ = = I).W

• ] [1

0; n nx x x∀ ∈ +∞ = xD2fD//$KV,-!1)א#) •,8U!W ( ) ( ) ( )[ ]* rr f x u x∈ =

D.W( ) 0u x > /f uD x x D= ∈ ∈

• ( )( ) ( )( ) ( ) ( )[ ]1 1

11'n

n nu x u x u x u xn

−′ ′ = = × ×

=!4+xy=*+!4+=rr ′=*

• ( ) ' 'rr r rx x ×= • ' 'r r r rx x x +× =

• r r

rx x

y y

= • ( )r r rx y x y× = ×

• '

'

1 rr

xx

−= • r

r r

r

xx

x

′−′

=

18

05)אK42 א#א)+

Wא#אZ)–א#א,8

$ $ $ $ @#. @#. @#. @#.

1!-1!-1!-1!- 1n nu u r+ = + rq .א@

1n nu q u+ = × qq .א@

א#א א#א א#א א#א ( )n pu u n p r= + −

( )p n≤

n pn pu u q −

×=

( )p n≤

(#$:,2(#$:,2(#$:,2(#$:,2

1 1...

1

n p

p n p

qu u u

q

− + − + + = × −

1 1...

1

n p

p n p

qu u u

q

− + − + + = × −

( )1q ≠ abc

(#$)*)(#$)*)(#$)*)(#$)*)

2b a c= + ²b a c= ×

* :א#Y*א#–א#א#

=( )n n Iu ∈(#

• ( ) nn n Iu n I u M∈ ⇔ ∀ ∈ ≤ (# <M

• ( ) nn n Iu n I u m∈ ⇔ ∀ ∈ ≥ (# V4m

• ( )n n Iu ∈V4<( )n n I

u ∈ ⇔(#7

+(81W

=( )n n Iu ∈(#

• ( ) 1n nn n Iu n I u u+∈ ⇔ ∀ ∈ ≤ 4g -

• ( ) 1n nn n Iu n I u u+∈ ⇔ ∀ ∈ -א# ≤

• ( ) 1n nn n Iu n I u u=+∈ ⇔ ∀ ∈ )

19

=W

)=א# )nαa)W *α ∈ W

0α > 0α <

limn

nα

→+∞= +∞ lim 0

nnα

→+∞=

(Zא#א=( )nqa)W q ∈ W

1q > 1q = 1 1q− < < 1q ≤−

lim n

nq

→+∞= +∞ lim 1n

nq

→+∞= lim 0n

nq

→+∞= א( )nq

" CG

+bא7^W

-א#> •+8 / D.

• / D.V44g - +8

lim lim

lim

n n n

n nn n

nn

v u w

v u

v

→+∞ →∞

→+∞

≤ ≤ = ⇒ ==

l l

l

limlim 0

n n

nnn

n

u vu

v →∞→+∞

− ≤ ⇒ ==

l

l

limlim

n n

nnn

n

u v

uv →+∞

→+∞

≤ ⇒ = −∞= −∞ lim

lim

n n

nnn

n

u v

uv →+∞

→+∞

≥ ⇒ = +∞= +∞

)א! )1nu f un+ =W

)Yא )nuD ,8U!אW

( )0

1n n

u a

u f u+

= =

M$f3 )IMf)א2^4 )f I I⊂a=4!אI c )&Sא8 )nuOTU / " "l( ,+$W( )f x x=

20

05)אK42 אNc)א4א

4;DWא)א4אNc)א

:1W

=f3 2^U!(#א(I

OP3/Fא#>Pא(D.f3 dא^I

O Oאh!pאe/yאS&W

• F/%* g3 d6^א I

• ( ) ( )'x I F x f x∀ ∈ =

NHW

3 dאאZ.^>P3-/>+)א 8+)א2^4

=f3 2^U!(#א(I

c f3)אP<#אS&Fא8dא^IOTUW

Hzf^U!IDא#א3א<#אWW ( ) ( )x F x k k+ ∈

=f+</-(#3)א)א 2^>PI

=0x=4!אI0y=4!א

$>Pא(#-#Fא#f3 dא^I

D#<א!pאe/yW ( )0 0F x y=

Ncא4א(אWE!+א0d-Q77)+(W@)אF+א

NHW

=fg(#L3)א 2^LU!LIk #)א$//

c g3f)א>PLL#אGLS&Fא8dא^IOTUDאא^W

• F G+א#>Pא(f g+^3 dאI

• kFא#>Pא(kf^3 dאI

21

+Jא)א4א<Nc4א)א4א( W

( )F x ( )f x

ax k+ a ∈

1²

2x k+ x

1k

x

−+

1

²x

2 x k+ 1

x

1

1

rxk

r

++

+ rx ( )* -1r ∈ −

cosx k− + sinx

sinx k+ cosx

tanx k+ 1

1 tan ²cos ²

xx

+ =

ln x k+ 1

x

( )k ∈ xke + xe

Wא4אNcאeN40אS7J%))8>א)

( )F x ( )f x

( )2 u x k+ ( )

( )

'u x

u x

( )

1k

v x+

( )

( )[ ]

'

²

v x

v x

−

( )[ ] 1

1

ru xk

r

++

+

( ) ( )[ ]' ru x u x× ( )* -1r ∈ −

( )ln u x k+ ( )

( )

'u x

u x

( )u xke + ( )

( )' u xu x e×

( )1

sin ax b ka

+ + ( )cos ax b+ ( )0a ≠

( )k ∈ ( )1

cos ax b ka

− + + ( )sin ax b+ ( )0a ≠

22

Q05)אK42א,^א

9TD1/+אW :1W

=f3 f3)אP<#אFI)א2^4dא^I

ab3 d4!==אI +א#א-f=ak&bD//א#)א.W

( ) ( )[ ] ( ) ( )ba

bf x dx F x F b F a

a= = −∫

NHW 9fאW

( ) 0a

f x dxa

=∫ ( ) ( )a b

f x dx f x dxb a

= −∫ ∫

( ) ( ) ( )b b

k kf x dx k f x dxa a

∈ =∫ ∫ ( ) ( )[ ] ( ) ( )b b b

f x g x dx f x dx g x dxa a a

+ = +∫ ∫ ∫

4T.W

( ) ( ) ( )b c b

f x dx f x dx f x dxa a c

= +∫ ∫ ∫

B1Cא/Wא

O ] S&Wא8 ] ( ), 0x a b f x∀ ∈ ≥

OTUW( ) 0b

f x dxa

≥∫

O ]S&Wא8 ] ( ) ( ),x a b f x g x∀ ∈ ≤

OTUW( ) ( )b b

f x dx g x dxa a

≤∫ ∫

Wא07א#9

=f3 ])א2^4 ],a b

3 d@#0א^אא/,אD.D//א#)אW ( )1 b

f x dxab a− ∫

Fא\ c8 Wא#=uv3 2^6 /%*L gLא(ILא#אMfu′v′ dא^L43I

ab3 d4!==אI

( ) ( ) ( ) ( )[ ] ( ) ( )ba

b bu x v x dx u x v x u x v x dx

a a′ ′= −∫ ∫

\))

# ok& )=א ), ,o i j

$ u. $#א A(#0+אא$ D.0/ oL"|אi

j

1. .u A i j= ×

23

=f3 ])א2^4 ],a b n $אאL 4אfC+> U7א

,. ( =ZאL,/אW x a=x b=

D.W ( ) . .b

f x dx u Aa

∫

=fg3 2^L4Lא([ ],a b Ln $אאL 4אfCgC7

> Uא ,. ( =ZאL,/א+Wx a=x b=D.W

D.W ( ) ( ) . .b

f x g x dx u Aa

− ∫

NHJ)W o@o@o@o@ DnA-DnA-DnA-DnA- $* $* $* $* $אא $אא $אא $אאD|<D|<D|<D|ao@!o@!o@!o@!D.D.D.D.WWWWאaאaאaא>

f< 3 dא^[ ],a b

( ) . .b

f x dx u Aa

∫

f< @ 3 dא^[ ],a b

( ) . .b

f x dx u Aa

− ∫

• f<

3 dא^[ ],a c

• f< @

3 dא^[ ],c b

( ) ( ) . .c b

f x dx f x dx u Aa c

+ − ∫ ∫

( )fC6U#( )gC 3 dא^[ ],a b

( ) ( )( ) . .b

f x g x dx u Aa

− ∫

• ( )fC6U( )gC

3 dא^[ ],a c • ( )gC6U( )fC

3 dא^[ ],c b

( ) ( )( ) ( ) ( )( ) . .c b

f x g x dx g x f x dx u Aa c

− + − ∫ ∫

O)^):

^n)$|oאodא# #אOא )fC73$3 2a 8(+> Uא[ ];a b

.W ( )( )² .b

V f x dx u vaπ

= ∫

uvW|א#$

24

0g2א)א4אK405)א

0g<א)אאא :1W

1א? oא>D.vKא#אא<#א)אx

x3 dא^] [0; +∞

a#-rא1 ! C!Wln NH אW

ln 1e = ln1 0=

] [ ] [0; 0;x y∀ ∈ +∞ ∀ ∈ +∞

ln lnx y x y= ⇔ =

ln lnx y x y> ⇔ >

] [0;

ln y

x y

x y x e

∀ ∈ +∞ ∀ ∈

= ⇔ =

] [ ] [

( )

( )

0; 0;

ln ln ln

ln ln

1ln ln

ln ln ln

r

x y

xy x y

x r x

xx

xx y

y

∀ ∈ +∞ ∀ ∈ +∞

= +

=

= − = −

( )r ∈

O S&nא8Xא(# OTUW( )* ln lnnx x n x∀ ∈ = א:; 0W

fDא#אא#אא#אא#א,8U!D ,8U!D ,8U!D ,8U!WWWW 2fD.D.D.D.WWWW,-!1א#א2,-!1א#א2,-!1א#א2,-!1א#א ( ) ( )[ ]lnf x u x= ( ) 0u x و < /f uD x x D= ∈ ∈

( ) ( )( )2lnf x u x=

( ) ( )lnf x u x= ( ) 0u x و ≠ /f uD x x D= ∈ ∈

A=W

( )lim lnx

x→+∞

= +∞ lnlim 0n

x

x

x→+∞=

( )0

lim lnx

x→

= −∞>

( )0

lim ln 0n

xx x

→=

>

1

lnlim 1

1x

x

x→=

−

( )0

ln 1lim 1x

x

x→

+=

( )n *∈

41JאW

lnxא#א x3 d4^א] [0;+∞

=uU!3^)א 2I c 4^2 S&u3א8 0g<Iא#אOTU( )[ ]lnx u x3 d4^אI

25

S7JאW

lnxא#א x6 /%* g^] [0;+∞ #:

] [ ( )1

0; lnx xx

′∀ ∈ +∞ =

=uU!3^)א 2I c 6^2 S&u3א8 /%* g 0g<I)א

OTUWא#א( )[ ]lnx u x3 d6^א /%* gI #W ( )[ ]( )

( )

( )

''ln

u xx I u x

u x∀ ∈ =

Q#א0/אW

'*lnW

+∞ 1 0 x

+

- lnx

W(aaאUhOgא)א * 1a+

∈ −

:1W

q @o !aא#אא? C!rא#אאD.Waogl

M$W] [ ( )ln

0;ln

a

xx og x

a∀ ∈ +∞ = l

NH אW 1 0

1

a

aa

og

og

=

=

l

l

] [ ] [0; 0;

og og

og

a a

ra

x y r

x y x y

x r x a

∀ ∈ +∞ ∀ ∈ +∞ ∀ ∈

= ⇔ =

= ⇔ =

l l

l

] [ ] [

( )

( )

0; 0;

1

a a a

ra a

a a

a a a

x y

og xy og x og y

og x r og x

og og xx

xog og x og y

y

∀ ∈ +∞ ∀ ∈ +∞

= +

=

= −

= −

l l l

l l

l l

l l l

( )r ∈

=1&:

1a > 0 1a< <

a aog x og y x y> ⇔ >l l a aog x og y x y< ⇔ <l l

0

lim

lim

ax

ax

og x

og x+

→+∞

→

= +∞

= −∞

l

l

0

lim

lim

ax

ax

og x

og x+

→+∞

→

= −∞

= +∞

l

l

7X#אW

] [ ( ) 10, '

lnax og x

x a∀ ∈ +∞ =l

26

c2א)א4אK405)א

0g<א)אאא :1W

א>Kא? ,#אא.Dא#אא>K@#אאא

C!! Wexp +HBx=( )exp xx e=

NH אW 0xx e∀ ∈ >

( )ln xx e x∀ ∈ =

] [ ln0, xx e x∀ ∈ +∞ =

] [0;

lnx

x y

e y x y

∀ ∈ ∀ ∈ +∞

= ⇔ =

( ); ² x y

x y

x y e e x y

e e x y

∀ ∈ = ⇔ =

⇔

> > > > > > > >

x y∀ ∈ ∀ ∈

x y x ye e e +× =

( )r ∈ ( )rx rxe e=

1 xx e

e

−=

xx y

y

ee

e

−=

א: 0;W

fDא#אא#אא#אא#א,8U!D ,8U!D ,8U!D ,8U!WWWW 2fD.D.D.D.WWWW,-!1א#א2,-!1א#א2,-!1א#א2,-!1א#א ( ) x

f x e= fD =

( )( )u x

f x e= /f uD x x D= ∈ ∈

A=W

lim x

xe

→+∞= +∞

lim 0x

xe

→−∞=

limx

nx

e

x→+∞

= +∞

( )lim 0n x

xx e

→−∞=

0

1lim 1

x

x

e

x→

−=

( )n *∈

41JאW

xא#אx e^4

=uא(U!^3 2I c 3אdS&u^4א8Iא#אOTU( )u x

x e3 d4^אI

27

S7JאW

xא#אx e g 6 /%*^ #W( )x xx e e′∀ ∈ =

=uU!3^)א 2I c S&u^6א8/%* g d3א IOTUWאא#( )u x

x e3 d6^א /%* gI

#W ( )( ) ( )( )'u x u x

x I ue x e′∀ ∈ = ×

lnWא0/א#Q)א

W1a(aaאUhcא)א ∗

+∈ −

:1W

aog lq#אאא#א@ ! a-,^א#אא@ C!Wexpa

+HBx=( )exp xa x a=

NHWא lnx x ax a e∀ ∈ =

( )xaog a x=l

] [ ( )0; og xax a a∀ ∈ +∞ =

l

( ) 2; x yx y a a x y∀ ∈ = ⇔ =

] [0;x y∀ ∈ ∀ ∈ +∞

( )aog y lxa y x= ⇔ =

( ) 2;x y∀ ∈ x y x ya a a +× =

( )r ∈ ( )rx rxa a=

1 xx a

a

−=

xx y

y

aa

a

−=

=1&W

1a > 0 1a< < x ya a x y⇔> > x ya a x y⇔< <

lim x

xa

→+∞= +∞

lim 0x

xa

→−∞=

lim 0x

xa

→+∞=

lim x

xa

→−∞= +∞

0

1lim ln

x

x

aa

x→

−=

7X#אW ( ) ( )lnx xa a a′ = ×

28

)א+א7)c2 אK4 05)א

D.#/2,א#א)אW ² 1i = −( ) / ; ²z a ib a b= = + ∈ i(7+(P@8אWא

=z a ib= +$ )MW#)א/# ); ²a b ∈ • a ib+v#/א(#Y;א z-,^א

!z,^א;א//a(#Dא#) • j!W( )Re z

!z,^א;אb(#D#)א • j!W ( )Im z

jNHj)W • O ) S&Wא8 )Im 0z =OTUzD//$(#.

• O ) S&Wא8 )Re 0z =( )Im 0z ≠OTUz U!> ,^#)א E(7+(i1W

=zz ′L#/=(# ( ) ( )Im Imz z ′=( ) ( )Re Rez z z z′ ′= ⇔ =

i(7א0/א#+(QW oJ# ok& v#/אא=( )1 2, ,o e e

i(7+(bkאW =z a ib= +M$ )W#)א/# ); ²a b ∈

Wz.א#)א/#zv!אeUא#) a ib= −

( )M z( )M z′D//אn,< O*) ,

• ' 'z z z z+ = + • ' 'z z z z× = × • n nz z=( )*n ∈

• 1 1

' 'z z

=

• ' '

z z

z z

= ( )' 0z ≠

• z z z⇔ =D//$(# • z z z⇔ = −9!>D(# • ( )2 Rez z z+ = • ( )2 Imz z i z− = • ( )[ ] ( )[ ]² ²Re Imzz z z= +

i(7+(W

=z a ib= +M$ )W#)א/# ); ²a b ∈ v#/א#)א !z0/ ( ),M a b

)Wzt-,^<א#)Mא/^,M0eא/z0א#) • )M z

Z8^,OMeא|"zא#) •

tW( )OM z

P( )z Aff OM=

=z a ib= +M$ )W#)א/# ); ²a b ∈ v#/א#)א ztW².א#)א//Dא ²z zz a b= = +

29

( )

( )

*

' 0

n nz z n

z z

z zz

z z

= ∈

− =

= ≠′ ′

1 1

z z z z

z z

z z

′ ′× = ×

=

=′ ′

/XאQ#א(>gi(7+(c8א Wא

=zj->#K? M#)א/# v#/א#)א#,z.θ"אאא @ g#$PW( ),1 OMe

! j!Wargz tW[ ]arg 2z θ π=

=z#K? #)א/# HBr z=[ ]arg 2z θ π=

• #DEEא+pאv#/א(z.W ( ) [ ]cos sin ,z r i rθ θ θ= + =

• v#/א@#)א zD.Wizא re θ=

NHJ)W D//$(#EEא #aאK?

0a > 0a <

[ ], 0a a=

,2

ai aπ

= +

[ ],a a π= −

,2

ai aπ

= − −

• ( ) ( )[ ]arg ' arg arg ' 2zz z z π≡ +

• [ ]arg arg 2z z π≡ −

• ( )[ ]arg arg 2z zπ π− ≡ +

• [ ]arg arg 2nz n z π≡

• [ ]1

arg arg 2zz

π≡ − − − −

• ( ) [ ]arg arg arg ' 2'

zz z

zπ≡ −

• [ ] [ ] [ ], ', ' '; 'r r rrθ θ θ θ× = +

• [ ] [ ], ,r rθ θ= −

• [ ] [ ], ,r rθ π θ− = +

• [ ], ;n nr r nθ θ =

• [ ]

1 1; '

'; ' 'r rθ

θ

= −

• [ ]

[ ];

; ''; ' '

r r

r r

θθ θ

θ

= −

• ( )''' ' ii ire r e rr e θ θθ θ +× =

• i ire reθ θ−=

• ( )iire re π θθ +− =

• ( )ni n inre r eθ θ=

• ''

1 1

''

ii

err e

θθ

−=

• ( )'' ''

ii

i

re re

rr e

θθ θ

θ−=

[ ] [ ], 2 ,k r k rθ π θ∀ ∈ + =

• argz z kπ⇔ د =

• arg2

z z kπ

π⇔ = + ) ف د )k ∈

kאYNW >AYNW

( ) ( ) ( )cos sin cos sinn

n

i n n i nθ θ θ θ

∀ ∈

+ = +

( )1cos

2i ie eθ θ

θ θ−∀ ∈ = +

( )1sin

2i ie e

i

θ θθ

−= −

²z(/א#+ z a∈ = a)( )a ∈W ( WWWWא )א )א )א ),3$א,3$א,3$א,3$א2222( ( ( WWWW

0a > ;S a a= −

0a = 0S = ²z z a∈ =

0a < ;S i a i a= − − −

30

W ²(/א#+ 0z az bz c∈ + + = a)Wab c77)+א(A( )0a ≠

( WWWWא )א )א )א ( 2WWWW,3$א )2,3$א )2,3$א )2,3$א

0∆ > ;2 2

b bS

a a

− − ∆ − + ∆ =

0∆ = 2

bS

a

−=

( )

2

2

0

4

z az bz c

b ac

∈ + + =

∆ = −

0∆ < ;2 2

b i b iS

a a

− − −∆ − + −∆ =

)א+א7)cא%9(6O6&W

D@#Cא"אD@#Cא"אD@#Cא"אD@#Cא"א א*gא/#א*gא/#א*gא/#א*gא/# U ABא B AAB z z= −

I0/14א[ ];A B 2

A BI

z zz

+=

qאאg( );AB AC

( ) [ ]; arg 2c A

B A

z zAB AC

z zπ

− ≡ −

ABC,// C A

B A

z z

z z

−∈

−

ABCDא#/ D A B C

B A D C

z z z z

z z z z

− −× ∈

− −PD A D C

B A B C

z z z z

z z z z

− −× ∈

− −

א*gא/#א*gא/#א*gא/#א*gא/# D@#Cא"אD@#Cא"אD@#Cא"אD@#Cא"א

Az z r− =

( )0r >

• AM r= • M .8!rא#א!אk&D,-A " %r

A Bz z z z− = − • AM BM= • M@אk&D,-[ ]AB

;2

C A

B A

z zr

z z

π− = ± −

ABCaאאo gMEA

[ ]1;C A

B A

z z

z zθ

−=

− ABCv MEaLg Aא

1;2

C A

B A

z z

z z

π− = ± −

ABCaאאo gLg vאMEA

1;3

C A

B A

z z

z z

π− = ± −

ABC:*Aאv ME

<(7.l+Jא.א%W

+nא+nא+nא+nא .v#/אjEx.v#/אjEx.v#/אjEx.v#/אjExWWWW Suאא|"tאXא$

z z b′ = +M$b"|אeu

D8 nאh]8!vZאΩj<k ( )z k zω ω′ − = −M$ω0/Ωeא Oאא#r]8!vZאΩjאXθ ( )iz e zθω ω′ − = −M$ω0/Ωeא

31

"א#+Jא&

2K

405)א

A WWWWא )א Aא )א Aא )א Aא )א )א Aאאאא, )א A+א, )א A+א, )א A+א, WWWW+א

'y ay b= +

( )0a ≠

( ) ax by x e

aα= −

( )α ∈

A WWWWא )א Aא )א Aא )א Aא )א א,"( א,"( א,"( א,"( WWWW WWWWא )א,-/>+א )א,-/>+א )א,-/>+א )א,-/>+ A )א, )א Aא+א, )א Aא+א, )א Aא+א, WWWWא+א

0∆ >

L//$L$ L 1r2r

( ) 1 2r x r xey x e βα +=

M$W( ), ²α β ∈

0∆ =

$#א//$*$r ( ) ( ) rx

ey x x βα += M$W( ), ²α β ∈

'' ' 0y ay by+ + = ( )

² 0

² 4

r ar b

a b

+ + =

∆ = −

0∆ <

L/UאQL#/L$W

1r p iq= −

2r p iq= +

( ) ( )cos sin pxy x qx qx eα β= + M$W( ), ²α β ∈

32

Mm&א(Z2 אK4 05)א

oJ# ok& Bאא=אZ.6 @a!% <( ), , ,o i j k

=Qא@)אFא#J=OIJא@)אFאWQ0אYא%4

=( ), ,u a b c( )', ', 'v a b c

=L"|3ϑ

• . ' ' 'u v aa bb cc= + +

• ² ² ²u a b c= + +

•

'' ' '

'' ' '

'

i a ab b a a a a

u v j b b i j kc c c c b b

k c c

∧ = = − +

kWא#

0/L ULאABD.W

( ) ( ) ( )² ² ²B A B A B AAB x x y y z z= − + − + −

U )=M/0א )Pj( 0ax by cz d+ + + =D.W

( )( ),² ² ²

M M Max by cz dd M

a b c

+ + +Ρ =

+ +

0 Uא/M=o/( ),A u∆D.W ( )( ),

AM ud A

u

∧∆ =

`+W

( ) ( ), , : 0n a b c ax by cz d⇔ Ρ + + + =,^א"|( )P

c K?/ABCOTU,/BS&Aא8 AC∧

,^א"|( )ABC )א##y=m( )ABCD U~א @5 W

( ) ( ). 0M ABC AM AB AC∈ ⇔ ∧ =

k+W

.8!U( ( ), ,a b cΩ " %RD.W

( ) ( ) ( )² ² ² ²x a y b z c R− + − + − =

33

U( ( )S . 0gP#$P[ ]AB @5 .##y=mD U~א W ( ) . 0M S AM BM∈ ⇔ =

$*Wא( )S .8!Ω14[ ]AB " %2

AB

knR71( ),S RΩ`( ) : 0ax by cz dΡ + + + =

=H,8א/א!,v(Ωא^( )Ρ HBW( )( );d H d= Ω = Ω Ρ

)א )Pא( )S 5O h /

)א )Pq J( )S 0/aHא

)א )PאH0/( )S )eU)א! )C

.8!WH

" %W2 2r R d= −

knR71( ),S RΩO7( )∆W

=H8!,v(,א/אΩo/א^( )∆ HBW( )( );d H d= Ω = Ω ∆

o/א( )א∆( )S

5O h / o/א( )∆q J( )S

0/aHא o/א( )/H0א∆( )S

LL0/a

34

05)אK42א)א+

0;QMW

:1W

",2DE,dא!> (#.E! j!WCardE NH)W 0Card∅ =

NHW

ABO "O ,2 ( ) ( )Card A B CardA CardB Card A B∪ = + − ∩

0;O0W

:1W

=A"א=2,E o,A ,|,<E! C!rא,dאD.WA M$ /A x E x A= ∈ ∉

I).W

• A A∩ = ∅

• A A E∪ =

• cardA cardE cardA= − Wא#)AאQc)א+

"| t0- !Ypא ) א\ )*p ∈ &SFo3א 81nא8 Oא5\

O 8 Dא5\EאoF2n8 .........................................

O 8 8oFpnpא5\ W 1א,.א;#אOTU#)א 2 3 ... pn n n n× × × ×

8א1C8)- א1Cאjא8W

81CאאW

=np==!4*( )p n≤ <-Qא(#pL =!4 !א3n.!4Wpn

35

8א1Cj8)א W

=np==!4*( )p n≤ <-Qא(## pL =!4O-!א3n.!4W

( ) ( ) ( )1 2 ... 1pnA n n n n p= × − × − × × − +

p+אא= NH)W

# <-!-+8nL =!4O-!א3n3#<-Z8^,-!4n!4 .(#W( ) ( )! 1 2 ... 2 1n n n n= × − × − × × ×

Wא]&

=E .!> (#",2n +8A=E]!> (#p( )p n≤

3l-^,pL =!4n4!

. lא]Z.(#W !

pp nn

AC

p=

)א+cאW!npnAp

nC

( ) ( )! 1 2 ... 2 10! 1

n n n n n∗∈ = × − × − × × ×=

( )!

! !pn

nC

p n p=

−

( )!

!pn

nA

n p=

−

1nnC = 1

nC n= 0 1nC = 1nnC n− =

p n pn nC C −= 1

1p p pn n nC C C−

++ =

B%א!אA<8W

ntpL =!4n!4( )p n≤ D aא;#3אWא

tnא:tnא:tnא:tnא:WWWW .א, <nא(#.א, <nא(#.א, <nא(#.א, <nא(#WWWW t-Qאt-Qאt-Qאt-Qא D p

nC o"K? 3*$T H pn "o

3*$&O# H pnA o"

36

J0)J2אK405)א

%9

i04אi04אi04אi04אD WWWWא$5, Dא$5, Dא$5, Dא$5, ] ] ] ] WWWW ! pא +</- !+8!E8P|=

ΩO8 א א,|! pא .2D,א #$A A א=O8אΩ

D#אא #$ 4!א$#א8+$#=,B e/yAא# B∩ O Oaא$#BS&Aאe/yא#( e/yAא# B∪ e/yאS&APB ,.P

#n( B#אאA )A.א# )A A A A∩ = ∅ ∪ = Ωو ABL,|K?O )#$ A B∩ = ∅

o()א7אJ)40)אo(W :1W

=Ωאp ! &O8 • D#א#$3א /!א$,# iωj,gaipOP3/#3א א$, iω.W ip

tW ( )i iP pω = 5א5 #אאZ.O-rאא# • א$, #$3.2,:א$5,

O S&vPא8 1 2 3; ; ;...; nA ω ω ω ω== )#$Ω#3א OTUA.Wא$, ( ) ( ) ( ) ( ) ( )1 2 3 ... np A p p p pω ω ω ω= + + + +

NHW

=Ωאp ! &O8 • ( ) 0p ∅ = ( ) 1p Ω =

• ( )0 1p A≤ ≤#$+A=Ω

• L)#$( y3א )$#(Lא$,y3א )$#(Lא$,y3א )$#(Lא$,y3א WWWWא$, L)#$+AB=Ω

( ) ( ) ( ) ( )p A B p A p B p A B∪ = + − ∩ ( ) ( ) ( )p A B p A p B∪ = +O |,BLS&Aא8K?

• ( B#א3א )א$,B#א3א )א$,B#א3א )א$,B#א3א WWWWא$, #$+A=ΩW ( ) ( )1p A p A= −

J0)Jאi1"kW :1W

&Sא8 Hzcא$#אא5 #א" &O8אp !a3 א$5,Ω

#$+83 ) OTUA=Ω.Wא$, ) cardAp A

card=

Ω

37

QRX40א)JאJE-().7אW :1W =ABMfאp|! אאG L0<-!L)#$W ( ) 0p A ≠

#$3 OPא#Bא$,,A(#א.e/7W( ) ( ) ( )( )

p A BBp B pA p AA

∩= =

W

+L)#$ABMfאp|! אאG L0<-!W ( ) ( ) 0p A p B× ≠ #W( ) ( ) ( ) ( ) ( )B Ap A B p A p p B p BA

∩ = × = ×

:1W

L)#$+ABאp|! אאG L0<-! ( ) ( ) ( )A p A B p A p B⇔ ∩ = ×BO*/O )#$

NHW

=Ωאp ! &O81Ω2Ω3 Ω )1 2Ω ∪ Ω = Ω ( 1 2Ω ∩ Ω = ∅

#$+A=ΩW( ) ( ) ( ) ( ) ( )1 21 2

A Ap A p p p p= Ω × + Ω ×Ω Ω

*WאHJאא# =A,$אאp !a )#$j p

.Z[א|! S&#Pאn U!O#אe/y3 kא$, A ,.<B !W ( ) ( ) ( )1k n kk

nk n C p p−≤ −

QMאX>Y40)אjTW ^ ! pא=ΩKVאpא &O8

KV3א Oא$,g##nDאpאXL WH<א!$Lא • ##y( ) 1 2 3; ; ;...; nX x x x xΩ =WKVא .Z\lrאo/2,אX

• 3 )twא$5, )ip X x=+i,dא= 1;2;...;n Q"א/cאJ*Y#אJQMאX>Y#ipא[א9אqJאW

nx ... 3x 2x 1x ix =Xj g KVאpא D ;#3א 9!W

np ... 3p 2p 1p ( )ip X x=

KV,DA XWWWWא+א! ,DAKVא+א! ,DAKVא+א! ,DAKVא+א! ( ) 1 1 2 2 3 3 ... n nE X x p x p x p x p= × + × + × + + × KV,! VאKV,! VאKV,! VאKV,! VאXWWWW ( ) ( ) ( )[ ]² ²V X E X E X= −

:1W

KV,vXא9א0!א!w5אKV,vXא9א0!א!w5אKV,vXא9א0!א!w5אKV,vXא9א0!א!w5אXWWWW ( ) ( )X V Xσ = Qא,)אj7אW

=p#$3 !Z.#n[א|! p !aאAא$, DאpאKVאXא א#אvZ! 8+| #)א!א"Ue/nrA] 0@ $#אX-^,np

# ( ) ( )0;1;2;...; 1 n kk knk n p X k C p p

−∀ ∈ = = × × − ( )E X n p= × ( ) ( )1V X np p= −

38

Q#א,^א)>?1(2K405)א

+א)אM*א#JאO74א( W

Wא.TE8אBא#

-1 cos 1

-1 sin 1

cos ² sin ² 1

x

x

x x

≤ ≤

≤ ≤

+ =

sintan

cos1

1 tan ²cos ²

xx

x

xx

=

+ =

( )

( )

( )

cos 2 cos

sin 2 sin

tan tan

x k x

x k x

x k x

π

π

π

+ =

+ =

+ =

AJ+W

- 2x a kπ= + Pcos cos 2x a x a kπ= ⇔ = +

( )- 2x a kπ π= + P sin sin 2x a x a kπ= ⇔ = +

( ) tan tanx a x a k kπ= ⇔ = + ∈

2

π

3

π

4

π

6

π 0 x

1 3

2

2

2

1

2 0 sinx

0 1

2

2

2

3

2 1 cosx

3 1 3

3 0 tanx

2x

π+

2x

π− +xπ xπ− x−

cosx cosx -sinx sinx -sinx sin

sinx− sinx - cosx - cosx cosx cos

39

!0;/_eNW ( )

( )

( )

cos cos cos - sin sin

sin sin cos cos sin

tan tantan

1 - tan tan

a b a b a b

a b a b a b

a ba b

a b

+ = × ×

+ = × + ×

++ =

×

( )

( )

( )

cos - cos cos sin sin

sin - sin cos - cos sin

tan - tantan -

1 tan tan

a b a b a b

a b a b a b

a ba b

a b

= × + ×

= × ×

=+ ×

LMW

cos 2 cos ² - sin ²

2 cos ² - 1

1 - 2 sin ²

sin 2 2 sin cos

2 tantan 2

1 - tan ²

a a a

a

a

a a a

aa

a

=

=

=

= ×

=

1 cos 2cos ²

21 - cos 2

sin ²2

aa

aa

+=

=

HA W tan2

at =

2sin

1 ²

1 - ²cos

1 ²2

tan1 - ²

ta

t

ta

t

ta

t

=+

=+

=

!0;r'Fא( /_WFא( r'!0;/_W

( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ]

1cos cos cos cos -

21

sin sin cos cos2

1sin cos sin sin

21

cos sin sin - sin2

a b a b a b

a b a b a b

a b a b a b

a b a b a b

× = + +

× = − + − −

× = + − −

× = + −

cos cos 2cos cos2 2

cos cos 2 sin sin2 2

sin sin 2 sin cos2 2

sin sin 2cos sin2 2

p q p qp q

p q p qp q

p q p qp q

p q p qp q

+ − + = + − − = −

+ − + = + − − =

/_Wcos sina x b x+( ) ( ), 0,0a b ≠

( )

cos sin ² ² cos sin² ² ² ²

² ² cos

a ba x b x a b x x

a b a b

a b x α

+ = + + + +

= + −

M$αe/D//$(#W

sin² ²

b

a bα =

+cos

² ²

a

a bα =

+