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t
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*SES btfn ,4S : ltguydo CAph Te&n
SAS !&ltg &fd*r I€p r = ,$$S Set ?#
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Trg o&:
*r
?8 -Ir&u ilung Sqrl :- tr{* Ngi
$i,y a6i: S88Ei
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114 1l'3 t ? 4
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ddng tr6n l* vr,f,t c5p si cQng, iid,ng tlu*i i* mOt
cdp s6 nhf,n.
- Trongle6c e0ng trirrh iriq.ahilurnh$ to&a trge
* rne tg x,YII, duu xea *jnren (lrj{?}. phesnla
(teer); Meocsto (1868) {te n&ursqr tifn h} giEa gi{
trl lig r y01 digu tieh gsa nngt Utoh &ri,Ie si*i
hga phla du&i b&i nrgt ph&c cila lrqc hoAah-trt
1 $4u r, hei b€n b&i rhi?ng pb&r eria durlng
th{ug aong aong '..rii trqre tun#, edn n}ii a tron
- lr&i rndt phEa e&e Trh*trh i:3p**Lr6u gr;ilrr (ph&n
' e6 ki gleh & hieh 1). Sr; li8n h( nty duqe hibu
thi b&i csng ltir,rc;
ri:,t I
##H WW#m* Tif ." l{Affi 5* {"*,&&$ir efi'#" H&ffi 5s ftti
Ys SrO'tqS. ?ffuY
{--ffr: d6n nny, ehftng hi6u bi€t baa d}u ri: 16-
\" karit dug* minh id3.' iu mgl b&ng so a*nlr
6i&* rngt'cdp a6 cQng vi rrgt e6p sd nbin.
Vl$c so s*nh nS5' duqc n6u ra dE iAu, trt trrrde
C n&m t61a lS n{n xust li&nrbSng l'6serir d&u
ti&a -- rngt cOng trinh lao dong bEn bi audt 34
*6rn irrli cira J. N6pe,'r Ching hgu, *fr thh tial.
th{y tlong t*e pl$.nr ( Nguiln g6c ed he.ru } (t{?3}
e&a rh& tcf,n hqe irlng An Krrsi" hrile, chi ti6t
h,rn:,.-,tro:rg eu6n { Sd hge tlru}.u tri.y } eia &{i-
khais St{p}rcr, *" d6 c6 b&ag :
Trong Lrli giing nbt tiOog ( Toin srr c{lp vdi
garaa {itnr ,eao etlf l nEm l$S7 :- 1908. l(1,3y dfr
Ct ngiri rlrlng cOng thrt c ui.r llm . o si iIF
x$y dung l-f thul'0,t lOgerlt !ron6 gilo iriuh loarr
phb thtlrrg, Cdn diah ughil hlnr a6 !0grril nhrr
lir h&m s6 xglrqe eria him s,'i mti niru 6 trtr&rs
p[b- thOng higrr ncy {lnrpe Gle ndu r.r nHnr i7',8
7t *:-.-:*,
,l?
vl
iit!!il.i
sil ]Ogsrig
te kh&o sit dL, thi hi.m s6 I = l:la
hEa & gi6 tri fr duong. ?r€a hinh
h&;:r sd Lj a 1l* v*i r )-0.
lroqg +*S* {$h{f} ;:r*s Si&i !1+h uE'hqn*. }ihu
vA- la ph&i tr'i.i r;u* iil,! nIin kb ii"r kbi 16garit
tluoc dE nghi dua ra *rI dUng l161{)' ehirtg ta
rnCrr rde dinh tliair ngiiia l6garit llrn rG s& eiro
BAi b*o nAy triah-b&y Jri tttug1t ilnh *oc cr?a
l6gr1t;1 dgr tbco quaa tlilm eia Kl0y. Nd khOng
itd, h6t nh&ng hibu,bi6t vE ticb pbdn ciiug nhu
gi6l h"dn, rrsi eb& ydu ehi aitn ddn nh&'ng ki$n
thti:e & l&p d&u bgo FTTI{. Kb&i' ai$m lOgarit
duqe diah ughic b&ug hinh hp* ch..r ph6! $riy ra
aga3'duge'm$t ed {lsh ahdi eira n6 ilbag th*i
eSn cho phdp tn'd6, su] ra khAi nipm vd linl:
ehdt efia liy thrta.
ED don giia, ch{rng ta e hl gt6i h.gu E trf thdng
l&garit vdi eo sd l6n hsn 1. l{h& eSng *.}rirc
loggr = lnrllna
mil llinh nghia logarit .v0i ecr c6 a ) t {xem
m$e 1) fiuq* 6ir dgag eho ci trudng hqp 0{a{1"
!. Enffi sd ilibu di6n'cli$n tieh m$t hi*hft
fr*6c h€t ta di trt erQt h*'rrl sd fion gifrn aISt
{trAm hing) cung ttb th! e*a nd d.b dcn rnft hArn
sd khd: bi$u di6n diQn tieh m$t hinh.
Tr€a trirrh 2 bitn di&e drdng tb&ng y = 3 so$B
eong vdi trqe O;r. Cft vdi rn6i gi6 tri du*ng e
d8u ir*g v&i tung tl0 bing 3 v* v&i di$n iieh
hinb eh& nh{t ?r. Nhu vgy, diQa tich e&e. irlnh
ehfi nh$t !r&a hlnh.Q, dugo bilu di€u b&i hnm
a6 b$e ah5t y = 2,1.
c v& il€o tuc th** chit vi s&* hinh' n6u tgrroe
.l-iu- i'iri 6brg i:o ti:i di{:* iie }r gerqc xerY{ i* dr+rrg
cduadq.ttag ehitsu kirn dbng hb thi rerrr l{ im'
.qbu v&!, o u?t tti dg tr€n, hlm es ?r v{ ct biBrr
ci6a ei+a ituu'**g v6i c& trir&ng hqp r { 0'
d,
,fr"
+ ll
i1
li
li
I t'i
t
*
S. *&m
tsly gid
v$ ehi gi6i
4 1n db thi
['
I
i
I
i
i.
t
Hlnh 2
. Te ehuy&E Ealrg hlnhS biEu di6n rlu*i:g ih&ug
i/ = 2*. Y&! m6t giS tri dron6 r': trly f dBu trng
v0i lihdag nb&tg tung d0 ]a-lrril ci rii{,'n tich
tarrr gi6e -uuOrg 1.t2. p - 1r '- :r''o. Dc dl6 di$n tieh
e,ic tim' ,giAe -trto hinh i duqrs bi?": di6n b'i I
llflms$9=-''
?ro*6 hai thf tIiS tren, EB 8ie thi$t .r') 0' NSr;
;; { t thl & trwong !:qp thrl riir{t'-di*e t$*h hill.tt
e !:& nh$l t$oeg &ag li Om {Es < S}" c&n trong
tru&Bg hqp thri hai. diBn {ieh larn gi6o l&dueng
{r8 } 0). sE x{e diah d{u eho nrgi tr*&ng hop.
ic qar xdc: di t*r 5#e O itdn *itm s6 he&ah dQ
ilinh l' ,
Chung ta xery diQr tlch hish .18CD Id n:Qt n
h&rn sd uriu o. 6 dsy, ta bflt SBu tish tren truc -
0r ttr diBm ,1 <:6 hoAnh dS ,. , i
tr)i{r tieh hinh n&y xf,e d{iah rn$i irtci s5 m&i
ggi IA /r&r4 ,sd idgnrif tfi.a c, ki hi$u : lugt-
Tu diuh nghia rrily s&y r{. ngey- &uge r&ng
logr iluong ktri e )> I v& &rn khi r ( l. Srong
nSi€a khi ri = I thl eli$r tieh binh.b&ng 0: iogtiuc0'
{lhu j r*.*g ktui c't[:rg thl di$e iit:h htrt:h r:&eg t&ng ;
logr t6eg tlreo s.
Ta e6 ih} tlnk "gid tri.S&n $*eg afia tggc vrli
il$ ehiah x6e cho trr:6o. Ch&ng b1u. tc. ph&i tinh
Icg*. Hu**r vBy* ie ehia €oqu [t; Al tr*n o:c
r
ilirrlt;i
{Atnlr l} iir&e* n:gt ar-1 1:}rilu bi:lg rrhaii, ,:},i*g
ba* lAr* 10 ph&n, tu*ng &ng !.Si l* rl*r$ng vscfu
sons so.Bg v*i Oy. I}b tinir ghn dLiirg. ta ti*l:
$i$r: ti*i:r *5ie hisrh tiri! l.ihi:"1 dr*qe 5i*i ti6a b&i
eii:* drrdrng virtl!t. SSr c:iiq$ ;:hi3u r:lir.1 iii tuug Sit
:3og riirdng v6*& b&a ir&i, tn *o sS rl* di$p tlirh
3&n dur'rg th*a. +r)n r,du riqr 1A Srrfrnil vqch
b6* ph&i, ta Suryc gi* tr! gin ,5ir*g thi6u. tr{$1
qu& ta cr$ hei hi:t* Irdc f lrt;rg nr& gi6 ,ri Iogt
gbm gifre rSc rr5 do di*cr {tch }:ai hiah l:ily, Bi€!
ahi&u.agang mii hinh *hrt ah{t ln 6,1, ta *$ 5!d
tri g&u dir*g lhi6u i:Sr:g :
I k. i: i; !; l;
0,1 I --- * - _ +-*+*-,i*- -1-
\ l,t r,B r,g 1,1 l,s
io66,1 * *"tr,$0?SS*; lo50,t a* *!,60g{1li;
iogS,o * ""0,698I54. rr.r.".
"" H6n ta ehqa bB sd /. khl.c nhsu, ei:ff.ug bqu
11",.: 1i 5-;..' thl s€ duqe nh*ng him s&'i*ga-
rit khf,q xh*u v&, d&u dgn6 6rry* $& ihi r:*a ri:*-l:g
ir & rn s6 dy.
- tr]b lti ragt b&mr s6 i0garir lA nt$r du&ng eun51
ii8n t$* rrhtr d&' trlnh trty t:.o*g s*+Lr ci&+r kh;os.
teiu plr& th6ng ir.uug huc.
o, {.fr s{,
, V6i k = l. ia duqc idgrrrl ! lry t;lti, n, kl ii!grr
tn ia (i v6 n l& hai ctrF dEu ti&s erie lsg*rllh-
i*us yii aatrrlsi!). ?heo k6t qu& {r tr6o ie g *
{;1^69315; in 3 e l.S986l i... Irigl:ia I& inlr { I ..,#i
;,h = ? vA Ia * F 1 vlri r -, i. ra& hAni s€r r:&_y
i+., l6ng khi;e tflng t& I d{n B. Cho r&n,6iFa "J
vi $ t*.t ph&i *S m11i gi*" tri r d& In r'd{ng
ir$ng t. C6 ih& *hiing rii*ir r&ug tDn tai giA rri
d6 N6.b$xg 1,?18?8..., din e ki higu b'&i oh* ,
i.A dgl l* ed Ndpe, ri.a irh* rr,* n lrr:6 =L 6 *1
iLS .ki Slli. liiru vg.r,
. {: ry 3,71818 y& i*e .2" tr
"
l,'rri h+ s6 .t-dueag b$i kg, la ,i5 d6eg th*y
rang.: !(}s." = k Iu c.
,Ie{.gid ia c6 thh c}iqn /i sc,i 0h,., }og.; bfisg 1
vii g{& rr! cba tru&e e&o €Sy e&a " 5t. Ch,#g
li1*, 1du nra.'!n iinq* ihm a6 ligarit rhso gi&
lrl I {rng v&i r .--" 16 thl ehl eBm clh,9rl A aas ehs
logl0 s 1r ta t0 = i
aglria i*.A ar I : in 10 ry 1r'!,i$l$S a*,.g"{S4::$.
Ii.li d6, u6u&i ta B?i la t0galiJ thgp y:tzdtn, kg
tri{u Ig.
Pibr-c r4-t ig u e {i.{B{2g ie m. Ch*rg iigu
tg3 e o,.tsr:t$ .n ? + 0.3tit0S ;
, i' t 1r, /,. /r1
1-'---l*.* -[..-"r-*;^
1,6 t,'; i,8 ,"s g I
* s ijii{},tI -1- *.SS -l- s,?? *i* *,?tr -t ti,s? + 8,fi3 +
+ i,,** + {J,$e + 0.-13 + j,;0) * 0,$?ri.
Tirsr*g tE. gi/,r ir; gia driag thtle b$
j\iir* r'&s ti,6?Ilr { }og:i { O,;Z;f
C;e. tr, irurg bi;rb cgug e 0 * tiul kSl
* : c.:':t Lt lg9 e S,{Sil$ la 3 * f,{??;f y.v
*.,." -r.-,-. N6l chung, d6i r,6i a0
+uq t:tV ia eht.,rr {H8's g6 /. src ch.,
b5t h.l- e )y t, 16 rhb
gt& tri tr*lrg rlrrg **a
i&garit bSag I
t i*g* ::: l: icr{z = .l
i(hl l = 1..'!tii:, te c0 L&rn e$ Icrgt, ahSn 6;a
t:i I khi.?od,'fs ki hriu i* trg.'i. sd " i,ri.rc
6Ai J& trs :;fi r.niu ltiy*tt!., Sliu v&.v
iegori' - hr/l*ri vi log,rr,, , I.
togS ry &,{i*S-L
Nr,:r t qnir .Jr,.r qrin !rluh tr*a tr.r hh{,ng phJ:i d:^tgc
eilia lil:n 18 ru* chia 1*ti 4:hSrr t&i eS dr:qr* k€t
i1*&: ehinh xiir li,;ya .l(} lria C:S tr"i g.&r.r rlrirxS eira
IciiS iiri $ r,nt ,d lhfr;: u:hAu lC :
iog".r a ij {jSJll/
LLlh.g s€ru iL,r, rle e,l th& iinli gld tLi glrr tliilp
l':g*rlt e &* *A* :srX irLS;r.'i hrtug hrn legS,* t.U$S*1S;
iogi e I'Su!{41; Iogl8 * X,8$159/: v.v... R{i rdss !tr chish !,lsaril tts ahjes ro co c6 l.
Ta *6e ed ih} tinb sr<r* isgarit e6a rl:ft*g *** l*g*ril th{p ph&r: i$ es sd lir 10.
s6 flird hox L l,r.ir !:ifr€ €i6 lrilo -t*1, ebu *
, L,.tn trlrr)
erJi mdg, rm,,{c *t,$.c
I3,{ qun t*a soqn b{o q-Ia{n }tqe v& tual
ir,6 3 nh$n duEe thu e&u t{p tht ehi€a ci
horil lhu 68'- 4S{ (XgtS finu} h$i ring:
6il*,.bao nlrfu dudng rnlt n6t dE u€ dxge irlnft
.tcar"j
F* f" i'
Y* Ai,*s mdsa
,{inlr j
CIuqc hlnh 1 b*og 8 durlag mgt aEi. B&ag ltp
lu{,a tuong tF te th5y hiBn n}ri€n:.
Dinh li I : ilfu mQt tzlnh c6 In dinh ti thi
e&n &hdng .tt hon n dudng mrlt n{l dB uE ro.
C&u h6i dugo iHt ra l& lifn cO ih} ,*€ ia6t hinh
c6 tn Elsh l& b,[n6 {Ifrng ntlrr*ng m6t net kirSng'.)
BiBu il6 dE d&ng cbuyfn vb e&u hdl tttoag duong
rlng ligu raQt htnh kh$ug er iliah 16 n{o vA
liin thdng (trfc tA t* rngt dinh ed rhe rrti r{t re
eia ilinh kh&oi e6 thB v€ b&ng mgt ilu&ag mgt
. a€t k6p kin kh6ng (bing ea';i: t}Orn vio gi*a
rn$t e{p diqh 16 mgt canir n:r}i ntli ehung)? eau
trl ldi trong trudng irqp nA.y I& cd :
Dinh l! 2t Aat.hinh Jiin lt1ng ttti l;hdng cu
di.nh 16 ndo d?u cd tht uE rltrJe birtg drdng ngt
nil khtp kin
, Ta ohfag rainh dinh I-f nty b$ag e6eh sri dgag
nguy6n lf eryc iI+|. 'ft} gl& thi{r li&a th6ng ra ruF
re tlln tqt uh&ug du&ag m$t ndt khep ktn (kh0sg
Sinh eB llie ta i6i rngt dlrh dE tii -qua (n{u
V& nguy&n t&e l&i Si&l bAi to&n nay hhOng uh{t thi6t p&&i ehfra t{t e& }*a .goi efia triabi.
kh&o ldi giai bi,i to&",n'? ebite e{q & lhinh B&i vi n6u xu{t ph6.t tr} r*gr dinh ato d6 vi
Ksaigsbrrg, m$t bf,i l*&* iluqc tlte gi&l quy6t di tbee e6o equL t6i c1[e dlulr kh6e thl nh{t
& rhs k, r8"
'lrirne bdt tn pl'r&a logl e{e d.lah e&a hioh ra kh6u-g tht dinh eu6i e*ng 16.i!u;;g i egnh xudt
l&ra B loqi, tuqi din]l /d v& loei di.nh chdn. ph6t). Troag tdt c& rto dtuir;rg mr;t ndt hh€p kie
Dleh t6 l& *Ae dieh *$ m6t o6 l€ rloan xuf,i ta r€t 6u&rg c6 ohiEu egah nh{r,
phit tr! a6o vil cliqh ch&n ld'eae illah cdn lqi Bgn ilpc ed tliB eh&ng mlnLr d6 rliag riag khi
Ndu v€ mgt dud*g mgt adt khdng kh€p kln d6 duSng rn$t ndt aiy ehria t{.t ei cAe eqnh e&a
{tte ta dl+h iIXu va tllnh eudi ph&x b gt nhru) hinh. DB e0 tbb chrtag rninb li6p, e&e bge tham
v:* rseb nt1 rs i(h$i iriuh, thl rioh cr,in !& erla Z khf,s th0m ldi gili bAi tong B/t3S d[ng tr$n tO
dinh el&u mrlt e&a durrng thay dbi edn eria e$e . Ed' lll$65 bg,s ('IoAn hgc vA. tubi tr6 n,
ilinh khie drqo gi& ugiy&u. T& do euy ra rodu T{t abi&n e6 baa s€ hiri ( 1rf,y v&i nh*rg hiah
e6 th& r'E hiah 1 $"'&cg cAe du0ng m6 s6t thi ta e6 ?n * i dirih 16 rhi ssa ? i). Xia nhudng e ho
&u li rbst td S dtr&rg rnQt n{t ahu v{y ;r[go&i *d,c bgrr eh*og nr:ir:h r&ng r*pi hi:rh aes rtl eo
':3*:i:.i:3-Y:-Yr *:: 3.:1" "Y' ::
\
+*\, { S
J r- -::rt :i_
a'1= 3 F \l
;'r i::.!,ti,,rj:i .J;;"ll
?) iru41 = i,*r, + u;), r,stl = ju"*n==,
2 u1*us
n = l, 2,..
Ch*ng ntinh riing
,u
rrs** * Bn$* { ts /:o y(tt mpt ;r,
l-Si giri;.' Yl u1 ). il, ur F S ruy ra uo p S,
uu)0 Vn,
!s t+t eO
eln-Fi * ve** = {lu, * u*i472is1,, -i' us} > S
- l*S5, ut = 1985.
Vr = 1, Ir..., su)- ra ug ) rrg Yk = 2,3,,.,
, V+y 0{(rit c us}13{u& trtl) { I, Vlr*?,3,,..
"Ttl dd ) : :
l
ru&+r'' ur&t sr {ar l'1iu}af&{cg + r,'r} ( rls * rrg
r/198CI < ?.
M{t kh*.c d6 thsy 2n > 1 * n rio {16
tl tr
17 p 2"+t *J 22 1lo ;F 3.
. E&i vny .: ,
u
lrs*t * un+t ( 2 < 3zl:lo $ll = I, ?...
. R I{,T.
E*i 2/t.5U. (ha cdt s?j thr.tc .rr..,., ,rr..
0(.r,1 (cz('..{*o{1.'
K$ hi}u 3:o = 0r ;rn+l = l. Giu sii rdc,sd nrig
thda mdn i '
n+?
T 1/h:1 * r.,) = o
j*0, j+i
Ch*ng n**h r&ng ca4.1-i = I - s d6i udi
i = 7, 2,.,., &;
Loi gidl: Gel P(.r,) s (a - c'6) (r * wt)..,
D+1
Pr(.d = x, '(.r ,'t)
i*0, t+k
u4l
DE tLdy. F'(r) = E rr(c) vt
" k=0
. U+l
j k:()
k+j
tu (lo
: n*I
p"(.ri} = : pl(.r,;)l(ei -. rj) _
-i, k =o
r=ts j
*.X -Pi(clji{;u - rr} * o
k=0
k+i
do dliEu kign eria dEu bii vri,i i = 1, !,..., r.
Do d6 *(o * 1).p"{*) = (n + CXn+I}p(*) (l}
- SE deryg th6y r&rg clrl tba *gi duy :nh{t mgt
ila tb*e b{e rl * ! v$i ,hS sd eao nh{.r blug I
th&a aiEn (l).
Mf,i kh*c tla thrir fre.i = (*tlo. p(l - ru)
th6s rnilri phuong tinh (t). gi.r.) 16 6a thrlc b{.c
n * 2-vCri hg *'{ c*o rhdt b&ng t. YSy ,
(*tlup(l ;*)*p{,rp).
Trt.66 vI r& gi& thtdt
' ?e Bil!: rt iiiBu Ph&i" cliirns,uiinh
' Air6r.r rdJ: EAi l*&u o&y chtr c6 h*i b4n grii
lel gi6i ddn u?iung ldi gi&i 6Eu khdng driag.
D.H 1'
6ii 3/l5g: Cio ddg s0 an, Un darrc sric dlrth
theo qui luQl:
Jn,= .ta*! * Igr- 1o!n2o
Pn = Sa*1 f 216-1 cos'o
uhi 'ru * 0". 9o ='co.9d: Chtirtg atinil r#ng;
:1n
. - n;sys = -:-, : c!!-t 'rr:i 1:*)
- * i=[
(C[ =' ri !l(n * r) ! r' !r "
Ldi gtdi; (e*a -\gd Brio Chdu * Ael0 DHT[I).
. { *o=f,n-t*2ge-1sinza
'I'a rc t ,* = uo*r * 1:tt**1 cosza
Hhia 2 vd: cfis d&ng thrle tr€a v*i ess,ft, efie
df ag thftc dudi voi sin{r ta duqe
f to.ota : la-!(o$i[ + gs-1eind' sin2a'.
I go ,ina = gr"-fsinfl * tit*'1*oto. cip2fl
j roeose[ * gasiu* '-
* I - 1ro-rro.o * Yn'raias)(l * sin2a)
I coooto - Unrina -
t -- (*o-ruosc - yr*t sinaXl * rin2a)
' Trty h8i dhs''t165 r,.o ta.ed
[ *r eosa * Yi rlnt :
I o (*o.o*a * Yocina)(l + sin2a'o
| *o.oto - !/n cina =
[ = (rocoaa - ys oina)(1 *'sin2a)o
f tru eorer * Yrrsinatz *
[-j_*io':o(r*rin2e]?o
I4
+(
| * lainttz(t - cfn2sj2' '
lr
Trrl 3 {!*eg \hto trGn theo t€ tht cd
{rjynoi$ft. sasP k ..
r.
-- i aial2a[(t * ain:e]so -- (t * Bin:d)sal
.,4
r ( rt t
= + sinz2*i i .1i-t (rin 2oi2'*1i
:l (i=t .J