L oa £B^ , , P J 1 Cl
da thi^c x8p xi tong qu5t Q(x) cua ham trUM tuvng Ax Cac da thí^c nay co dgng chung l a :
thí^c nay co dgng chung l a :
QCx) == ^o y^o(x) > Ol f^U) + . . . + Cn"/^n ( ^ ) . trong do cac ham f±(x) co dging : trong do cac ham f±(x) co dging :
1/^^(x) = y ^ ( x ) ^^^(x) . . . ^ i ( x ) , ( i - o , n )
con ^'(^Cx), ^ ^ ( x ) , . . . , ^\^x) l a h | hâi x a c ^inh t r o n g khSng
g i a n dang khao o a t , co n i a n g i a t r i cung t h u ^ c voo khong g i a n
dọ Cac he GO C^, Q^ , . . . , C^ se du^'c x a c dinh t u y t h e o t > n g phittíns phojj c\i the*
•Csc phi^ông phap dU(^c khao s a t t r o n g chiitmg nay l a dgng
t o n g q u a t cíio n g t v a i phuang phap du^c khao s a t t r o n g cnỉóng I va n g t s5 k 6 t qu^ cua t a c g i a k h a c du^c coi nhú nhúng trú»6?ng
hg^ d|ic b i ^ t .
§ ! • ^^IGt v a i k h a i nịe-n ve t y s a i phan suy ronf; toni^ c u a t cho t o a n tu" va cou^ thirc ngi suy Iliu^tôn ST^T rpnR*
Tiong phSn nay t a se xay d^'ng k h a i n i ^ n ve t y s a l phan
euy rgng t o n g q u a t cho t o a n vụ
Xet ham t n r u t'J'g'ng Ax, ham nay chuyen khong g i a n dinh chuan X vao khong g i a n djnh chudn X.
Ky h i ^ u c6c khong g i a n cua nhỉng t o a n tir t\xy6n t i n h X vao Y l a /fX -> I J .
I^ýng h0 ham ^ Q ^ ^ ^ ' /1 ^^^ »• • • > 7 k - 1 ^^^ ' t r o n g do
^ j ( x ) l a c a c Snh x§ b i e n X vao chinh n o .
TJC dem g i a n , t a ky h i p u ' i =" / C'[ o» * • •» 1 k - 1 ^ ' Í-^'óng t\^ nhir chutmg I , t a thanh l ^ p khong g i a n t i c h :
S2 = X.X ( t i c h B§ C5c)
3z = X . X . X . , . . . , Ev = . ^ • ^ • • • ^ I t r o n g do k l a , . . ^ k i S n
m£)t so tú n h i e n . ' ^ ' ^ " '• '-Xii ':VX\ • • " • • '
PhSn tu cua I^^ ^ ' ^ ^ ^ ^ " ^ ^^^^ ^^^ ^ ^ » -^o-^' ^^^
X^ 6 X» ^^ " o*"!)* T'.íÔng t l / , ph9n t^i cua Ệ '^g^c v i e t d b ^
Gia sir t o n t a i lagt coan to* t u y e n t i n h Ăx. , x- ; 'r ) •^1 -^o
chuyen c a c phSn tú cua kh8ng g i a n Eo vao cac p h a n t'*'' cua khong
g i a n ^ X - > I J ' va t h o a Juan d i e u ki^m t
(1.1) ĂXi^^, x^^; f ) U^(^^J = ^^^ " ^^:^^i^-
l o o n v't Ăx^ , Xj » 4 ) dirg'c g g i l a t y s a l phan b^-c nhfit cua
^o ^1
ham trúu tirg^ng Ax 'rng vói h^ han -i dug'c i S y t y l c5c phan ti*?
X. , X. f- X.
^o ^1 ^' (adsbygoogle = window.adsbygoogle || []).push({});
Ihay X. bang mgt ohSn ti^ b S t ky x t ^ J t*? ( 1 . 1 ) t a suy r a t
( 1 . 2 ) Ax = Ax. + Ăx, X. \ Cf ) 4 \ ( x ) .
^•o ^ 0 ' ^^-0
C6ng thiJc ( 1 . 2 ) l a cong thu^c n g i suy F i u - t c ^ co p h a n du cua ham t t i ) u t i r g ^ Ax 6ns vói ^^ ] » duVc i S y t ^ l c a c p h a n tiJ
X , X. *^ X .
Gia sir tBn t j i mgt t o a n tú song tuySn t i n h
Ăx. , X- , X. ; -^ )i t r o n g do x. ^ X, ( j = o T S ) , ch\Tycn
2 - ^ 1 - ^ 0 0
c a c phSn tu* cua khong g i a n .^- vao cac phSn tu cua khong g i a n / " X --^ £'X -^IJ J va thoa man d i e u k i ^ n :
( 1 . 3 ) A ( x ^ ^ , x^ ; . ) - Ăx^ , 3^ ; 1^ ) =
d o o M
To5n tu Ăxj^ , Xj^ , x^ ; ^ ) so ^--g^c ggi l a ty s a i p h e n b ^ c
h a i ciie ham t r i ' u tirg'ng Ax ifeg vol h^ / duVc i S y t^:! c5c p>^.ln
tó X, tr Xj (d = 0 , 2 ) .
i
Toe dyng C8 h a i ve ( I . J ) l e n W^. (x^. ) t a rVug'c :
' ^0 - 2
( 1 . 4 ) AC^^. , X. ; f ) f . (x. ) - Ăx. . X. ; T ) Y i ( x - : . ) =
•^2 ^o "-0 ^2 ^1 ^o -"o 'cl
= Ăx,^,, X. , X. ; f ) Yi-iC^^^) ^ i (2^ J .
"2 "1 • o ^ o fX
Thay x^^ bDjiig mgt phan tú' b S t ky x ': X va d^i^a vao ( 1 . 1 ) , tủ
( 1 . 4 ) suy !•& ;
( 1 . 5 ) Ax ^ Ax. I- /i(x,. , X. ; V- ) ^ ^ (-0 + ^ o ^ 1 ^o ^ ^0
+ Ăx, X. , X. 5 ^ ) -^ (x) ^ - ( x ) . ^ 1 -^o ^ -^1 ' -^o
G6ng th^o ( 1 . 5 ) l a coiog thil'c n g i crj^'- ITiu-tc;n sviy r j n g CO phSn du cua ham tru*u tu^g^'^ Ax uns vói hy han j duVc I 3 y
t ^ i cSc phl^ii tiV X, ^ ^ . c ^> ^^ " o ' / f ) .
Hoon t o a n tutmg t ^ , t a co t h e d i n h n^^la cho t y s a i
p h a n bf c k : ( 1 . 6 ) Ăx^ , x^ » • • • , X- , XJ ? V ) - A ( x ^ . , . . . . x . , x ; O ^ k ^ k - 5 ^1 ^ o ' -^k-1 ^1 % ^ = Ăx, , x^ , . . . , X, , x^ ; / ) t i ^^ ^' ^k ^ k - 1 -1 - o ^ ^ hc-1 ^ k Lan l i ^ t t a c f?§ng ca h a i ve ( 1 . 6 ) l e n ({A C^^i ) 4^' ( x . ) . . . w-'. (r^. ) t a nh^n :
^k k - 2 ^1 -^o ' k - 2 -^k o k (adsbygoogle = window.adsbygoogle || []).push({});
- A ( x , ^ . . . X, , X. ; (f ) ( f , (X,, ) . . . ' f i ( ^ i > =
-^k - 1 "' I o k - 2 i-: o k
- Ăx. , . . . , X. , X ; t ^ ) ^ , / - i . ^ • • - ^ 1 <^i^^- Thay x^, beng mgt phan t*? b ? t ky x ^ X va d^a vao cac cong th'j'c ( 1 . 1 ) - ( 1 . 4 ) , tir ( 1 . 7 ) t a sioy r a t ( 1 . 8 ) Ax = Ax. + Ăx- , X. ; ^ ) Cf. ( x ) -f- ^o 1 o o + Ăx. , X, , X, ; f ) f i (x) ^f, (x) -f "2 - 1 *o H -^0 (x, zc. , . . . , - i » ^ i f ^ ^ 1 , / ^ ^ - " ^ ^ ^^^ ^ k - 1 ^1 ^0 k - 1 -^o + . . . + A
C5ng t^^^c ( 1 . 8 ) l 3 cong tv^u^c n g i suy ITlu-tón co ph^n dỉ cua
ham Ax img v 6 l hf ham ^f t ^ i cac p h a n tú x , x i ^ ^ X
( j - ^TF-T).
I r u - ^ g hgp d$c blC^t, xoSu chgn hg ^±(^0 = x - x^ ( c a c anh xg tpnh ti^n) t h i f^ cac dj-nh n g h i a ( 1 . 1 ) - ( 1 . S ) se suy r a
cac dinh n g h i a ( 1 . 1 ) - ( 1 , 4 ) cua S 1 Chu^c;ng I .
Tỉ(tog t!/ nhu^ § 1 chifc^ng I , co th£ chv?ng minh di^g'c c5c
t y s a l phan t o n g q u a t t^^uy rgng l e dSi xi^ng thoo cac n 5 c x ^ . § 2 . VS mgt vai ohi^nix jh&o n y i suy t;6m: o u a t suy r^nE p i a l .gan dunr: phL^^n^^ t r i n h d l ) vs B\J' hOl t u cue chun/:*
Gia :ji5 phtrcteg t r i n h (1 ) co nghlgm d\xns l a : ^ , x * ^ ^ ^ ^
uOi = i X : 11 X - :^lj 4 <r ; ^ l a h?.ng s6 duvng-^. Cho ti-rc'c x^ I s xSp : d ban :*Su -^i gSn x * . x^u dyng cac d^nli ngh'ia ve t y
sẹi phfxn suy rvng tdng q u a t cWo t o a n t'> da t r i n h bay o- § 1 , t a xSy dv*ng mgt s8 phiiwng phap l^v-p sau đy s
TTgi dun^^ cac PhiTcyn.^ Phap va c6c ^tJẢ^ l y y§ t S c ^g h g i tyị a/ írúông hg^) k = 1 .
Xu3\; p h S t tu p h e n t»> x ^ , t a yS.y d'^Jng ph!ln vJ :
( 2 . 1 ) x ° = x^ f MAx^ , o </M_:^ 1 .
Xet t y s a l phen b g c hex A ( x , x ° , x ^ ; <^ )
(2.2) ACx, X?, x^; f ) f ^ ( x ) f . (x) -
- Ăx, X? ? y ) f . , ( x ) ^ Ăx^, x°5 f ) ^ , ^ ( x ) . D'/a vao ( 1 . 1 ) tú ( 2 . 2 ) suy r e :
( 2 . 5 ) Ax = . ^ + Ăx^, x° ; f ) ^ ^ ( x ) +
+ A ( x , x^^ x ^ ; f ) ^f^(x) ^ ^ ( x ) .
^ a y li*> :
( 2 . 4 ) Ax - Ax^ ^ A ( x ? , x ^ ^ ) ^ , - ( x ) - A ( : : ^ , x ° ; ^. ) ^; , j ( x ° ) + A ( x ? , x ^ ; f ) f ^ ( x ^ ) + Ăx, x ^ . x ^ ( H f ô-^> f 1 ^^^)- J);'a vao ''»-inh n g h i a ( 1 . 1 ) § 1 chu^ông I va ( 1 . 1 ) , tú ( 2 . 4 ) suy r a :
(2.:^) AX = Ax° f A ( x ° , r r ^ i f ) f . , ( x , x ° ) ( x - x « ) v
+ Ăx, >f, x ° ; f ) c^^(x) . i ^ ( r ) . 2h5 nhr-tig s
( 2 . 6 ) i r ^ ^ ( x , x°) - f , , ( x ° , x ° ) 7 (x - x ^ ) = = q ' , ( x , >:^, x°) (x - x ° ) (ỵ - x°) = q ' , ( x , >:^, x°) (x - x ° ) (ỵ - x°) Do do : (2.7) (fA::, x°) (x - x ° ) =^ f . ( 4 , x") 0 . - .:°) > //) / O 0 \ ^ 0> / ON + ^^(^> x^, X ; u^ - X ; vx - : J ^ ; ,
t??ong do 4^1 ^'^o» ^ ' • • • » ^-.^ 1^^ ty oai phan b^c j cua ham ^ i ( x ) . ^ i ( x ) . Thay ( 2 . 7 ) vao (2.5) t a '^(^c : (2.S) Ax = Ax^ f K x ^ , x ^ f ) q^iz:^, x^) (x - x^) + ^- Ăx^, x ^ ^ ) ^f^(x, .:^, x ° ) ( : : - x ^ ) ( x - x ^ ) + + Ăx, x^, x^; f ) ^ ^ ( x ) f , ( x ) . Xap x i Ax bang da th'^c QQ(X) : ( 2 . 9 ) Ax ^ QQ(X) = Ax^ + Ăx^, x^; / ) ; ^ ( x ° , x°) (x - x^) Gia su* ''.^^(x ) - G \*a ton t ? i cac tear, tỉ nghich dao
A" (x^, x^; j ) va •J^Ux?, x ^ ) , tir ( 2 . 9 ) t a suy ra : x'' = x^ -- ^ A ( x ^ ^ x^; 7 ) ^ ; . ( x ? , x^) 7 "^ Ax^ x'' = x^ -- ^ A ( x ^ ^ x^; 7 ) ^ ; . ( x ? , x^) 7 "^ Ax^ C?5ng c u 5 t , nS^-' tr- r^^.t :
r " - * i n ' i ^ - ^ - ---"^ + •T-/ ^•^-^ Tl = o -^ '^
t h i l y luCn hoar, toan túcrig t^' n^-ir t r e n t a as nh^n : (adsbygoogle = window.adsbygoogle || []).push({});
( 2 . 1 1 ) Ax = k'jX' + Ăx^, x ^ f ) ^\(yXl, x^-) (>: - 2;^-) +
va J
( 2 . 1 2 ) ~^^'^'^ ^ -CAi^, x ^ ^ ) ^ . , ( 4 , :-:'')7"^ '^x^\
n = 0 , 1 , 2 , . . .
Tx^ccng hg^ -^^^c b i ^ t , neu ^--f^C^) = (x - :cj) t h i t;> ( 2 . 1 2 ) ta se nhg.n qua t r i n h li^p t'/a Aitlccnctephenxen trong £"15_7; ^^eu se nhg.n qua t r i n h li^p t'/a Aitlccnctephenxen trong £"15_7; ^^eu ^ ^ ( x ) = (x - x^) va xj = 2 x^' - x ° , trong do x ^ v i x^ l a h a i
phan tú ban dau lay du gan nhau va r^n gan x' , t h i t ^ ( 2 . 1 2 ) t a sc nh'Jn qua t r i n h l$p ci\a C'^^^^J- sc nh'Jn qua t r i n h l$p ci\a C'^^^^J-
Sy hgi tv Clia qua t r i n h 1-Jp (2.12) du*g'c the hi^^n b -^inh l y sau dixy : l y sau dixy :
^ n h l y 2 ^ .