solution b)/
spliì1e ìnctho(Ì
80 S, SALLAM
,l.tthle .tr.J
(n :5, 1t : 0.9f109tì90, /cr: .0250867, h: ¡140) solutiolr Ìtr- ilre
gcn. Avor'. tìtc[ìto(l
l\,IÅ.ttItiI{A].tc.\
_
fìlì\_ulf t),El.
DE r. Fric) n rri nn ì'.ìI#;
ìoT,^ì.TrlltnÏ,
u,,"
tr,,JTNÁT,YSE NUN' RNTQUN
BT LA TI'ÉORID I}IÌ
L'APPIIO XT]II,,T.J]ION,l.ouc
.I5, Nol,
19{16,pp. g[_g3
,TIIE
ÄSY},J
rNoRÐArsrNc Wl PifOlII€ ltu
Á 1.,l.ET],ELÄîIO}[S zllRos oF Rr\r\ 1lE Cop¡riicUõñäis FOR I-,ûilNOìní¿
,.1.I.IEÏI{ DEIìINIiT'ELY
¡-,sBosKo S. TOI,IIÓ (Bctgladc)
.
-'\s an asyrtrlltofic
exJ;ressionfo'
a..polyrromialwi,r
altc,rrrate coeffi_cr c^l,s
.l' tlcg'ec
u ( tr,) iì)'ri..
tol."-ur"
rò'o"."ì"î^îr,i""îri.l (1)
ur,,e)t'_
{tr!)n 1|
r1..,¡.,,_:t_,,Ë
t _7)u a,,,6n_rrn _-+ _l æ.rni¿r,rs
t|itlìi""*rï'tl
ro l;hist.i.or.ial r'"lon*i¿o"
afinite
scquence of porvnor(2)
11r(r, ctu).- í.rj .!ln_1(r:, a,ò
l- (_r)r,
(Lo,,lt: 8,4,5,.
..,
,ru,arn
f
a,r.""l'ol1l 2' 1'r ',P) of
the.Po^lyno-,
r,
" 1ï
å'iü,:;îå r""rn
the
trr e or¡' "'f f''þí;i;: ïï ::Ï;
t
L
:, i;;
I"{-l)'
Q,,, iv?v:
0,((ru}
0)
rx
nl4 -r)
3;t 14 n 5rl4 3r12 7rl4 ,)-
0.3ó25831lì-j 00 .0.31it7.s07E_0ô
-0
352ã83711 1 00.-0 4990201Jì+00
-
(,.352582611 ¡00 0 tlri(1226311-0(i
r) . :J5258il1tj -l- 00 0.49902t)llì l o0
soìrr Lion ltr [he gr:n A\.0t'. lnotho<l
0 . 3 547(i 96tt -l_ (x)
0.21391õtiE 02
- 0 .:1521ì97illj*00
- 0 4992.13:lE_i 00
-0 .i15239{ì2It l 00 -0.3,1:ì1366Iì_(,2 0.:ì51 ó110I-t j 00 0 .,t9998641.t 1 00
0.3534291E+00 --0.2067024Ij_04
-
0.35349398 +00- 0. 500383611 + 00
0 . :15352378 _l_ 00
r) . 261 ô1 55E _ 03 0.3it39488E+00 0.c00803bE F00 'l'ul¡le 4.4.
0.3511344E1_00
-
0 .4748825n_02-0
.3575117r]+00-
0 .5013575E+00-
0 .3560276E+00 0.27404378_-02 0.35471078-l-00 0.5007516rì+00Diflolcncc
0.8460283E-03 0.2035156ìi _0.1 0.9101629E_03 0.1363575E_02 0,94709778_t):l 0.260ô594E _0:ì 0.1364648E _ 02
0. 178:J490u _ 02
l) iffct'c r rc t:
o.30i9414r, 02
0.6887983I.t_02 0.51143768 02
0 .2709r70F. _O2
0.36iJ1711Èt 02 0.5571804ri_02 0.3199696E__02 0 .7652040Iì_ 03 (nr - 5, l.t -. (t.250tt24, /1, : 6.6164,] B, It _ nl4¡)
3tr 12
2¡
r,'nr
l'2
?T
l;¡
7n12
4¡
IÌt.ltrtìlìltNctis
1. ll rr r'li o rv s l{ i, Ìr, .J. :rntl C o rv a lt. O l).
. ol _st:r:otttl ordcr dillcrenliul tlilfcrertcc cqurtt
2 ìr I l ¿ì g g â r,.{., ^Sol¿¿/ions lrctrntonirltris ct
.- 0,'l'hc 14tli. l\lrn. OonI. o¡t St¿rt. (ìorlr.
lJ l) ol z ó 1<1, L Lt, , Att r:flicit¡t[ ntunet,icttl n
4. rI orcler no¡t li¡tear
- I lrtcn. Ins. vol., lialcquu-
r. te¡y spline
funeÍiin
(i. ,,,. ti.,rg,rriii:a, 6íi. sorurìons.
c¡., ¡iut0 s.¡:lirtt eJ)J:t,ox.in ttl.r¡(tt) _ 0j.St;r[isiic¡ ìi,,,ä.riL¡,,
i ltc cocfljr,i(Ìtì1, 40 o1, i,he hrgllc,sl jro\\\)i, lcrrlrls
iq
zrr¡¡"llicir
¡:¡r,ol
i.ll¿.r 16o1,s-o1'llrt,r.r¡uiilio.r
113) irrcr,r,lrs<,i jli,l,,iiiri{;r,l.,..irr lrirsrrlirtr,r.;tlrrr,.L1,r 1hjs
1ì;c,ot,r,r lc
It,lìtrs
tli,{i(ìì,1¿ìilr tic
1
tht' qtt<.s1ig¡
ol, ,ilr Ibsr;lutc r_¿l r
1l (3)lìccoivccl l7 \l 10f14. ) !, t
I lr nnl ít s I )e l.¡rtI t;tt t¡ I
l ¡tit¡tt:'itll ol littttitiI l).(). Jlo.¡. ,i!)t;I .,
/i ¡r ì.ìj-4 / 7'
6-íi.2t7S soluliou b¡-
spline rnctìrorl
82 B, S. TOMIC
Ilor
ecluation (3) rvith alternate coefficients wc shall ltrove thefollotr- ing
theot'erns :1
.
Th,e gea,test real root of equatiott, (3), wltose coeJf,icients areall
altar-na,te
,
tenil,sto læ
as ao tends to zero(4) ro,,
(øo)* i-cor
ao ---+!
0.2. In
cøse q,ll tlte rootsof
equa,tion (3)toith
alternate coeffic'ients are contpLen, lwo conjugcr,ta compl,er roots witlt, the greatest reu,l ar¿el, þositiue purt llacome req|L und, tlte ¡¡reater one of tltese real rools tenils tl¿en to!
oo as uotentls
to
aet'o.We
consideÌ'one rnore finite
sequetlceof
ecluations(5)
þ'No{-f)'tr,,a)P-v :0, p :2,3,4,...,n(a">O).
Fol
equations(5) we shall prove the following
theoroms :3. 'Ihe
t'oots t'r,,r(p:213r4,...,tt), whiclt
tendto )-æ,
are d,istri-butecl ts.f'ollotos
(6) 0{?'zz{r++1...<
t.
Betueert,tlte
roots t"rr,rr, ü1'td, r"zr)+t, ,n*, tlr,ere enists tl¡e asqmptotic t'¿tlut'ion,(7)
t'zp, zt, @o)-
't'zt,*t,zr*r(Øo), øo-+ f
0.i¡.
I,tt"o¡n (6) and,(7)'ít
JrilIotos th,at th,e roots, tend,ingto !æ, of
equ,u-I,ions
(5)
ctre si.tuatecl,,in the i.nteraa,L ('t,rr,r,"")which tenrÌ,slo
zero (rs aoten,ils to zero,1. lYe
consiclera larnily of
parabolasof
degree(n-12) with
alter,-nate
coefficientsin its
equation(8)
U,,+z(t)t"r):'å.(-
L)u u,., x;,,+z-u, (øu>
0)where øo
is
the parametelof the
liamily.For polynomial (8)
we introducethe asymptotic
relation(9)
an+z (fi¡ ao)- n"'
(øon'- ãtfl *
az), fr-* +
æas
well
asfor the first
anclthe
secontl derivatel( i')
&o nz- (" I ')
ü,n .r ('i )'',]
t ASYMPTOTIC RELA'IONS .OR ?''E INDEFIN,ITELY IùI.R,EASING ZEROS 83
lVith
regarclto fonnula
(2) lve consicler tìre parabolas(2r)
yr(n,),) :
tt'lå¡
a2-
rcrt:+
q.z, (r.o:
#,
À)
1cutting
the ,Y-axis. x'orthis reaso'e
introclucc the parameter Àin
placeof rn..Ily r,
_*-e crcnote thex-axis
of pg,raboia(2r).-\4i;;ir
r,riis systdin"ái coor<linatestìre
sysr-em_!_X):,rf., by izi,
ioã,Àoliiii., íìJ*, ¡¡ by ,r
audlrom this subtract ør,
$;e äbtai'n(2r)
yr(n, )t): ++
4ar),,tf -
a,rnz| ørr -
u,r.Parabola
(2r) is referled to t
putting, in the
system (Xr),at the
clistance øo. We obtainthe parallel to the
axisX'
belo.rvth
corresponcling manner also
for the f formula
(2).For
the.cal
rootsb.t
trifferentfrom
zero orthe equation u+zllz(fr, üo): O,
wllet,ecto: -4-, 4arlt
' ÀÞ1,
we obtain(18)
r2r(À): -^ (, - f+), n, tt
rrrlx¡ : ?%\
'-+) ,ì.2t
rzr( À)
-, o,
,
À-+
J,_ooQ,7
rrr(ì,)
--->* oo,
À _+foo
&0, Tzz (øo) ---
a'
ao-> |
0..!'or the equation
u+zUL(fr, úoj:0 we
obtainrír(\,rr, - ?',:r\ +# ? -ll
'2
2r¡n_2,
f
__n*æ
(14)
(15)
(16) (17)
(18)
(,19)
rLz(t,,n¡: ?:rL
(n f 1)¿
À0'r
1+
'!lia2(r,
u')-
(10)
Yli-,
, (r,
ø)-
(11)
frn-
t,
fr-t
-¡' oo('*V
nt
J- 2ntr,2
!
2nu,t n n!1 +2
I
[(" ;')
uo*' - (" *rt)
" "*(';) "']
I À>I
1
l
With
regarclto relatioti
(9) we considor t,he sequence of functions(L2) ,,*z!/z(.ürüo): fi"'(*on' -
Q,7s:,* az)rtt,:lr2r3, ...
(ø*1)'
À 1)t)-]-(20)
rirQ,,n)-,!' . '
n"Q'r n+1 , À+ fco
i
J I\SYMI'TOI-JC ÌìELT\TION^S FOIì T'llE INDE¡'INITELY INCRE.A.SING ZEROS 85
rT in thc
j'olrnul¿¡e(r4)
arrd(1iì)
u.e cxparLclthe
sqrrarg r,oobsin
¿l'binomi¿r,l ser,ics,
l'c¡
obtaùr(21) r"rr()',n)n
-l-o¡,
À ---+ -l-æ' Foi' the cquation
u+zlllz' (üt øn): 0
rve obtainv
(22)
r'!r(À,
l) :
,(:13 )
B. S. 'I'OMIÓ J
t'rr(),) --
r!rr(),, n)-1+t: -);2{ -(''i)(-i l.('l')(I)'* }-
t''"),,
tt): 4;:'rtt
;în (, -
nrr,(rr,I
tt,f l) --2
L)')' t (t";"," )
rifl
1
rt,!2
1- ),>1
,)''ÞI
2tl2
1
a{
rl2
¡,
2u)'
,ILIn--2 1\
1-^^ )
1+
(L^ tt -
1,r;l
()., n)+ "z , )'+ f
oo0t 'll' -f L
r:,iQ',
ri') ---+ -þoo,
À --++
oo'lhc
Tirral conclu.iiotr ott '¡he dis1,r'il>rrtionof the
coLrsitlclcd ¿bscissasfrt¡m tvhich
I'elatiol
(27). Thc algurnent^bionfol lclation
(ZB) issimilt¡,to i,hat foi' r'eiation
(27).2 ilt tlte
poi.nts .q.= 4; e,, tt)
wtr,rln : rl,
(),,tt) tlte
r.¡rtlhua,tes"f
ou,rur:
(r2)
i,ncreuse 'i,n'tleÍituc'tely 'ir¡ u,bsohtle ,uulue-iull¿en, ),->
l-co; fgr
crrri-oenier¿ce rt.f' th,e calou,løt,io+t ,¿De orr,rù talce
n:2p -
1 ,¿oitltou,t, d,ln¿i.qíshl¡n¡1 1tp, ¡¡ ¡i,¡¡rzt' t,l, ¿¡¡g ¡1,
zpt.l/ztr'jr(À, 21t
-
1t), tJ, .={lr ll -f á rr, ('rr,
f
1)(24)
(2r-r )
¡r -s
fe]]6rvs-¡'^t'
. .+"?-(pa,
I
V D¡"'(par-l-u,r-VÐ)
(21t
t
7)2Irt,z (2.p
!
1) a,i4p"-r
([t X
t)
< rji
(À, n){
r,ir()',l) i lzr()'){
zn'in()')'1
I'j!(À, rr'){r'1"(À'
ir'){ r"i}')
i!) X
) I
).
'llre
inecltrtllitvra;,.,,,(À)<r'i!(À, tt')e-risl'sÏot tt'73
a'ticlÏor'
À) P-rr- All otheÌ incqualities ale
v¿Lidfol'
rt, =.!, 2,
3,. .. atitl tol. each
lo¿ll riurnbor ).>
1-. rnequalities (26) al'g ptol'cclby
mczlnsof
elcment"i'r-v c¿t'1-;iìilîo"|-; iumr"Uy t'', starts Îror^^ thc ïoirnulac t¡ernselvt,s io..
thtrr¡.entionecl abscis¡ias.
Ämotrg the
u,bscissàs 2erùritr ()'), r'jj(À, tt'),r',r(tr'rr') atrtl t'"()')
thcrcorisi,
1,hcÍollowing asymptotic
rolations(27)
?'rr(À)-
t"=r(ì',n')- ,.:?::,
þt,!
2)a' )''->-l
oo,(28)
r'=r(),,n)-rü(t',ù - fø,-¡,,', À- +-
ljl¡.,I
t¡,-
2.(
'l1r)
,r!,1-(),,rt) -
"tr'',.,,'(i') - ,,", '
r¿
-i Z, l'- ' 'l- oo' tt ?
'>(J6)
,(iì0) 'l'zz
r'r"('t,, rt)
--
r!t!t(),, n.)ltertce tlicrie tlxisl;s tlto
lnr;:o.nnll (1). 'Ilt'r"
paínt r'":r(t,,rt)
Li.ts upproni"m'otcl't1itt
tlt'r: nt'xl'tlltt ,hefween(tt
't)oitl'ts li'r( i', rr,) and rrr(),)wlutt
)t+
-f co'rllll') zt,+lt¿[rJr(,¡.,2¡t I), r j* -oo, À+ f
co.illho olclinate
of the culye (8) at the poini, n:
j.'zz(À,r)
increascsincìefinitely in
absolute r.a,luervllen
À-
-Foo.llhis
orclíñate is,(:12,\
lln+z {r'rz(À, tt,)À].:
n+¿!/z {,tóz(À,z),
À}.n,. ("r,,,n(.l))r-r -'
¡¡:p2u! ,
flonr whichct,,,
--
4p't(35)
:l
il
))"- t
froln
-whicìr r'rr(ì', n) uor'rzr(ì.,, n-
errLz()',,n) | arl
'(33) !lu*2{ri2(\, n),
tr}* -
oo,
), --+f
co'ln I
rltril":r si¡trilai. lniìnnet n'r: obtùiir,(3'j)
L,+z'tr'¿'()r, n), Ài- -
co, ),+ *-
oo ;'lllOt'('o\'{rì', l'e lt¿f't' lt,lsó (À)
-
rir(À, 'n)-|t À* *
ooruu + l{ 2t,,, r,, ( À) , À} ._ 2fi,r),
0r')
u,r(.--^ I
1))'*
+e2arlt
(lt
'nr('-À l- l) --
__cor
),-,
_l_ oc.86 B. S. TOMIC
Finally, lhe
ordina,teof the
curl¡e(8) at tho point
u:
rr"(),)inclefinitely in
altsolutc valuelvhen
)i- +oo. 'Iho
ordinate ot(8) at
1,hepoint ø:'rsz(À)is
'1
As.!¿lvrpror-rc REr-AT[oNs ¡'oR' TIIE rNDE¡'rNrîEL]. rNcÈEAsrNGzERos
g7.and frorn this
(r,,,t.rr(À),r f 1Ì :
(r{r.rr(À),1} * y,
.Àr22(À): r(aa)
ar{2(rL+
2XÀ+ltñ _-)ì _ 1) +
1} _-_+1- co, ).+f
oo.fl'lre tangent to the curvo
,U:.!/r.,2(fr, À) at the point, r:
?.rs(À)folnrs
¿¡rL acute a,ngle rvhichtcntls
tol
or thi,;
irL,rcrion";';;;;;;':ï",r,ï:.-
-r
oo' 'r'rrcrirst
crc'ivate(r15 )
rr;;t()).rì,rrr(À)
li*trLt'r,
(À), ÀÌ:
6
Increases.
the
curve'!1,+ztrrzz (À), ),1
:
r'iz(-t) {uor!, (À)- ttrrrr()') {
ør\¡-
&" Ì.,11, (7,)+ (116) |
u,nr!);z(À)+...+ (-1)'+z ûu+z: ri;'(),) -e¡ -|
+ -o:--. T
. .. I (- .[)'*'
1.":1,,l-, *,
).--,
-r- oo.'
i 'rr(7,) ritt
( À)l
3. ilre
sh,ct'll represent cet'ttti+t, th,eorents concterninq th,e arc of th,e curue.U: !/r*r(*,ì,)
i.n th'e interaal (r'i*r,r(À),*oo),
wh'ere the poi,ntu:
r'rl*r,r(ìt)' i,sthe
l,astltoi,ttt
oJ inflection.With
regard 1,o tho asymptotic relation (10) rve cotrsicler 1,he sequenceof
parabolas(37) ll :'Qz(n,n):
(tr'|
2)uur2-
(n*
L)ørtt! ttcr,r, tu:I,2,3,...
All
parabolas (37) pass l,hlougJrthe fixed points P(rr'
rYrr)ancl 80rr,.
Nr;)
rvhosc oldinaters a¡rc(38)
Nrr.(üo,n)=
nz {rrr(À),tt'1¡:2ør(i' -l/it --'- f) < 0, ),.¡
1,(39) Nrr(ao,n)='4r{rrr(ì'.),tt) :2ør(l'l /l', - ¡- 1) }
0, À> 1;
it is
easyto provc
1,ha,1, thesequantitics are not
clcpenclenton
ru and therefore wewrite them without the
argument ra,for
instance(40)
J/rr(øo)= tYrr(ao,n):
N"r(ao,1) :
Saor!r()')-
2ctrrrr(À)J-
nr.[t
we use the parameter À, thenwewlite ]rr(À).X'or À:1
rve har.eNzz(l):
:
0.In virtue of
(15)antl
(17) we h¿ì,\'e(41)
^]\[rYrr(r)
: +oo'
With
regartlto
the asyrnptotic relation (11) we consicler l,he secluence,of
parabolas E:
(r{(w, nJ=
(42)ütfrI
ü2, Lr2,À ___+
f
oo,ftortr u'hich it
follorvs'(46)
Ui*r't:'rz(Ào), Àoi:
r1 (Ào)>
o,'where
"l(À") is a.llositi'e
¿ir.r ar,'r-¡it'a.iry. grtrir,t rrurnber clepcncli'gou
Àorrvhitc Àn
is a suffiòien'y
sreãl;;i;;'of
thepara'reter
À.since
vi*,(.r,
Ào) is a poivnomial, rvcü"", ;,;;l;;*o
rheorern,UL*r@, Ào)
* *
oo, ø __+ _þoo3;åt",1.,rî::tlli:f,,îï î#:j
in the neighbourhood of rhe ponr n: {
æ'$r) r. ?li*, (n',
)ro))
u{ (Àn)(48) 2. n' : rrr(ì,,) > 0,
7.,)
À0.In virtue of
(18)we
have(49) o {
I'zz (Ào)<
r.rr(},,)for
À,}
À0,frorn which, in virtue of (4g), it
follows,(50) o<rrr(t)<n,.
ï'rom (42)
ancl (50)it
followsfllrmonru (
tt,'{;;;::":#;,l,7iu?,,ì, }; ¡: i:ii,liíi"r,,^i.,,n",,::,:;:{{:";:::!!
Fi',,;; "riai" ¡\L' r : r"(l'')' (51)
Ur*z{r.rr(Ào), Ào}_ _
.E(Ào)<
O.i'1ril|:iil',(;îïî
be chosen as bo be the samein
rerario,rs (a6) and (þr).'rhr"isr';.rñå:ï,1ËÄ?irl3'.=I';i'li;làht""lr*;""ìlriliî3ä",î;r
I
1- (p-7) ørr!;r(X) -¡
-(-
1) r+r &t, +,r3;'(À)
. Àtr, (À)*
-l- oo,n+2
2
'n-l
L)"'* - (
211,
2 't'ù ü.
rn
virbue of uroiormura(;:
)-r (,;', ) : (j]
î 1)
.çr.e hå,r.e,(2(n, rù
l-
1):
(2(r,n) |
'r¡r(u,n')'and florn thc
last,fornttla it
follorvs immetliately(43)
(2(n,r't' +- 1).- -(r(,r,1)'f É'¡r(r,
u)S8 E}, S. TOM]Ó
thus tlre
point
of ini,crseotiolrt'.p¡2 p+z( )') oΡthi,s pa:a,bolarvit'h l,hc ¿ùxis rY?+2is si1,na1,cil to thc rigìrt of tÌre
poirlt
rrr(),), nàìnel¡'ï,t2)
0<r'rr(À)
{?ì,+2,r,+z().).Orving
to (10), frorn
(l-r2) i1,follot's
(53) r, ,.r*r(
)') ---+{ oo,
),+ * æ'
,I'he l'ir,st r'[erir'¿1,e
tf
lJro palabola,!:]l
t,+z(c'¡ À) zr,ttho poiltl, ø:"
:
ríz(),r ?r) isy', * r{t""r(\, p), ?' )
:
{I'1,( ),, 1t)'rt:' - z-1-
(-1¡'*t' from l'hich it
follows(t""r(),,
p))'-"
Y'rr*"1,r''.r()', P),
)'I -t -cor
7'+ *
@r thus(54)
u'o¡or,r''"(),o,?),
)'u)t- -
lJ( )'n)<
0''Ihc
va,lue ?16 cilìr so bo ghoser ¿Ìsto
lte the s¿uno il.i rela'tiotrs (46), (52) anc'lilf7j. f.to,u
i+O) anci (ir4)it
follorvs tha,tin thc intclva'l
{r'ir(À0,-?)'lrr(}o)}
|íãiii.*C,tcrì,'ítc
Uioir('r,, Ào) charrgesthe sig.
ancl t'akes tire. r':¡luc', 0, 1,hus(55)
'!tL*r'rr|*r,r*'()',,), )'o]:
(),rvhereb¡r t'L*".o*r(Ào)
is the
gre,àtest-r'oot of the
equiìtiorì.!lL¡z(r', )'ò.:
--0. O\"iíg ió (¡¿)
anct (46),fol thc
¡tbsciss¿'I'i,*r.r*,.(\ò l'hero ¡oltls
the inequa,lit¡r(56)
0<
rir(À0,?) {
t''1,+z,n+t( )'o)<
r'rr()'o)'ilhc
sccond rlerivai,e of' 1,ltc pa,l'a,llolaI :11r,*"(t',)') it't' thc
pointn :
rí'"(À, p) isll',i*
" It'J'"Q', /t),
)') : (";').,
9 ASYMPTOTIC RDLAI'IONS FOR THE INDEFINITELY INCIìtrASING ZEROS 89
illhc
secont'l rlcrir.¿r,teoll thc
para,bola, !/:
ll ,*r(.;tr,À) a,t lJtc point
¡;
-
t'Lz(À, p) is8i
\(5E)
'y',1,"{t""r(),,?), \} -: 2
lt""r(),, p)}1,-2 . (r[?'j.r(]., ?),,11l It
2 2
p-2
2
est'Lor-"(I,
?)T
. .. -(-l¡, )
\ (Lp1
(tl't'|r()', p),
pl :
(Lzl"r!-z()t,
p) '
.(r{'r:rr()',p), 1t)(z
f
1)'^*v rr pØ+2) I r.-- itrl.r.)-t, \
{
-,,, -
I ),r,+ (,',:=e;
1ì
I
I
I
'p+2
ìI
1ir?)
lt'r,' *"It"l'"(,7',,rr), 7'i- -@t i' -+
i'-oo' from rvhiclho,nsuc*rcrrlr¡, ,,,. ,;ï:;ì11;lt;t ^'t;*Lii:],i,j]*"' ar
i,rrepoi.r, ø:.
.-
rJl(),u.7i) f'o:'lr
stti'l'icionl,lr- glczttylrlue 7.n. : .
:t/'í*ztt'!z(
),,p),
).1r
-F oo,).->J co,
,.,"rr,,.rtÏff'
)'+
-'Loo;t(59)
!'ri *rttt'lr(),0, 'p), i.ul: -l-
1i-(À0)>
{).(lorrseclucltl¡-
thc
parabolzlll -,!/t,+z(r" iu) is
conc¿ù\rc ¿rttlrtr point r J
== t'3zQ,o, ?).
._ _Fror'
(õ7) anrl (59)it follo*'s
lh¿r,tin
t,lrt, i.l,e'r.¿rIrrt,ílr(),,,,p), i,j,().0.,p)) the
scconcl derir'¿r,te changcsthc
sigrrnnd llht¡s
tLrtr 'í'ätüó"0,'ii,..i
,(
G{l)
utri*"{r,rl ,,r(),u), Àr.}:
0,whcrcb¡'r'i*_r,n(Àu_) is
tirc
groatcstroot, oI tho
eqnzrtiottt/i,*r(t:, Io) :0.
rJn'irrg to (57) and (59), for thc abscissa tii*r,r(Ào) l,ìicrle
hold,i"ilre
ircclualit¡,(61)
0<
,ril(À,,?) {
,r,.} *r.r(),0){
r.,rz(),0, p).Sirrco
l,i:.r,r\À)is
1,hela,sl,rea,l clo ol flre poì¡,nomal
,4,i,*r(n;, À), t,hisnreâns
that
y'ri*:(a, À),
1;r.r,ìiirg orrl¡.posil;ivt r.aliùs. tendí'tf -¡t".r,s
,rincreas<¡s
ïïoltt
lrl¡2.-,1, (À) 1ofco. ,;,*, (r,
Àjlenrains
pelnrancntl¡'
posil,iveas +;;
áncil,his rneans Ural, the parabola i,s
c
L¡,1 ,"i.
Conse-c¡uentl¡' in-
the irrtctr.al
{r'ri*r.r(,
.,son
thc¿tl'c cr1'
the
para,bol,\ ll.:.1¡,,-r(i,
),) -a,nt1thetrfole, r.lìcn
). incrcases,¡o
nûw zcìr'o can atisr¡jlt'hirrd the
zelt¡ )'1t+z.u+z( ).). ltìhisis thc
concepl,ol
thc lir.sLzclo. tlhus ir'e
lra,r'r' tlro.illnnonrilr
(13)._ll'ol a, sttJJíc.icntl .t¡ lrn'{tc ),,à
).o l,lte unt.:e !/:
!/r,+z'( 'r;, ).') c¡tts th,e
urís
x1,+, ttt Ìh,e poitt,t ;t'.= t't,+2,t,+,(i,') uirrutÌ¡ttt.),'i,nñ)i"t,,ii
[r.t itrct'eu,se, no nelt) zero cfl,lt. ctr,ise be]ti¡¿tl tiis-öert¡"
At thc point
D{t"r(},),7¡r*rl.t,, tarigcirL
to the cuÌl¡e ? :
.ttr,+z(t,-),)- 1,his curvcis
conca¡.eilt
1,hc-inten,ai cur'\rolics
¿lltor-ethc
tangerrtai; tir
the al¡sciss¿r, oli the poin1, o1i iitt,ersecrti wellat
e(62)
0{
r'zz(À){
11, 12,1,*, (À){ dr*,(},), ). }
).0.nl'he
poirtt of
interseciic¡rLii thc lrcnfionetl
1'auÍìerit r,r'ith 1,hrr u,ris -lu.,,, .hasthc
altsciss¿i,i(63)
¡: =-ilt,-,(t) : r-:z(À) .þU]U(ÀI
r:Ìtt l, *o', r,,"()'), ),t, (ú, +'t
(" ;').'' f'rí'(t"PD'-'{t -
.)
2
0,,þ
1
t)
('; a2
riL()', P)+ (--t)n p-l I
)
90 B. S' ToMrc
[n virl,ue of (16) alcl (45) tve
hâ\'ev ..2{r2r(À), À
10r 11 ASYMPTOTIC R,DLAIIONS IIOR, TIIE INDEFùNITELY INCREASING zEn¡os 91
¿¡ncl l,hcrefore
flom (63) it
follorvsthat
(64) {d,*r(À) -
rrr(À)} ->0 as
À-' * æ'
Frorn
(62),in r.irtuc of
(64),we
obtain(65)
?'e2(),)-
l'tt+z,tt+z (À)as
À-' +oo'
'Ihe eqnations
UL*r(n, À): 0 antl
(7t!
2)aon¿-.
\'tti
,\)n.tn^¡
lrri
0'n"toog'io the sa-é finít,'
*equcrrccof
equations (5) sincetheil first
t'hretr;.',ñîâ"e
equal. Thcreforoio their
gleâtestloots
l,h.ere canbc
zr,ppliectr' r,he asyrnpl,oticrelalion
({i.5) rvhich now has the followingfoml
(66) t {
r''r*r,o*t (À)- t"rr(\,
P),\- -f
oo'In tlte
sametnânner
1,hc equal,iotls,rri
+z?,,:;,'),) 0
anrl(':')
&onz- (oIt)".*('r) ", :
oEirrc
tho
rclation(67) o<ri*r,r(À)-fj;(À,1), À-'f
co.By
(33), (34), (35)antl
(36)the
absolute Yzr,lues o1 i,ht¡ oÏrlinates of'the pai'aÈoÉ
u:
ùor."(,^,,À) at the
poinl,s -rr:zffu.Ì', (À).L
r;:',(^,p)S
<',"'";l\,p) {r;;:^)" itiéi"is" inclcfinitcl¡' rvith ]'
a1c1 zrccolclingto
theaforeïaíciihe
arìcof l,his
parabola,is
conc¿lt't:in the itt¡r'val
þ"'o*r,n(\),i', * ,,, + 2( À)) .
''
'"'"if'tnä
isquàr'e rool,sin folmulae
(14) anti (23) are exptrtrcletlin
series'or
torrnulae 121¡ a,ntt (2,3)alc
atlclccltìpr
-\\re obi,¿r,inr',,()')
t'!,1,(\, n')- Tffi;,
^"-
fooalrcl
in
viervof
1,ht', as¡rrppNoi,ic rolations (65) ancl (67) tve have(68)
t't'r¡2,e+z ().)-
1'!,i*".,
()'))-+ f oo'
7,- *æ'
llhe arc oI the
palaboln,'!J
!lr,+z(n, À)iu the
inte,rval{r^í-tr,r(À), t'o*",r..r(),)\is
callecltlie ,,last a,ic" of lhis
parabola'thus,
1'he Îollorving theorcm has 'becn Plovecl :llnnonnlr (4)t:
!l'h'e ,,Icr,st tr,r'a"oJ
theparabolú y-- ttr+z(3t.!)
I'ies'he¡otu Lltø anis
X'r*,
ctnd,tiíe
qbsoh,u,te, acLlucol
crclt, -ord,inateoJ tlt'is
cw!cts wel,l øs
the
tri"'¡¡ín{lr*r'*r(i') -- t",l*r,r(T)) of theinterual {ri*r,r(\),
r o *2,e*z(^)1t increase'i ndeJirui,telg
willt'
)''" ''-fn'íiôw
of ret¿i,tiooÑ 1OS¡,"100) anct (6?),flom the
asymptoticrt{ation
(30)we
obtain{69l !l-".r-rtÙ - ,'r.r,r, - 1,
}. ---+ ì-oo.t"tt+2,t¡+r(1.)
- t'jl*r,,
(À)4. ||he inequality rrr.zr(\)
{,t,zrt+t,2t¡t-'(À)
antlthe
asyrnptotic lela-Liotr't'2r, ,2o (ìt)
-
t'zn *t, ,,*,
( À).In virtue of
teorem (4)it
follows: l'he ,,last ârc" of the
parabola.Jl
:
Uzt,(r,
À) touchesthe
axis,Y* from the
sicleof
the positive ordinatesfor the value
),.:
o'zpof
the parameter, À. Ir'romthis
rve d.ecluce the_ 'lÌrr¡onnrr (!)
o.f euen d,egreedecrect,ses untl
if all
th¿n ¿¡r¡t Ttolyno-m,ial has ot least
ott,
d, ,ualuei :
qz,of the parameter
À
can at'i,se zoh,eñ,L
cotttimtesto
increo,se incl,eJinitelu.ft
is knormr from thc theory of algebraic equatioirs [5 | :thc
presence.of a positive
minimum or of
a negative maximumof a
polynornial mea,nstho
existenceof trvo
conjugate compìexroots.
Consequentlytwo'
coniu- ,grì,te complexroots
are bounclto the ,,last
a,rc"iÎ tÌris arc
cloesnot
cutbhe axi-q
Xrr. Ileirce we
har.elnnonnlt (6). Tlte
truo cortluçlute atntple:t' t'oots wlticlt ut'e bounil, to lh,eí{
the posil,i,ac put'ameter rni,n,í,rnum oJ th,et'
ta,lres llt,e ,,Iast a,rc)' t:q,ltte},,:
oJ tlte ltcwa,bol,u dz, u,ncl tltesetuo ll -
cott¡ttgate compler!l¿t, (u, ],.) aanislt, t'oots changeto
one ilou,hlc rectl t'oot.Äs long as the
palameter
)' has a vA,lue rvhicrh isinfinitely little
cliffe-rent
(9-*
O¡frorrr and
isarbitralill'closc srnaller than to
anclarrrthe
above cur\rcthc
axis! : Xroírt
llzt, (n, theneighSou'hoodì,)rl, :
o.r,-
e,of the last
clouble '-eal zerut. 'Llhus,for'
À:
oroI e the
complex zerosbelonging
to the
positivcminirnum
are situatecl-inttre
neighbourhood of l,helast
double real zero, n'hile the other complex zeros,if they
exist, have ,smallcl realpalts.
IIor ), )
o.r, th.e,,last Alctt of the
curve(70) ll:!lrr,*r(n,
À): *.yzn(fr,\) -ürr,*r,
À)øru
påi,sses
through the
poinl;s 1'ro,rr,_r(\) at;id.rrr,rr(À) of the
axisXr,
arrc{ lies belorvthis axis in
1;heinterval
between thesetwo poinl,s. The arc
ofthe
culve (70) isin
theinterval
lrr"r,rr(l'),f
oorì cor cave ancl nronotonously ìncreasing. Owingto this, the culve
(70) cuts theaxis Xro*r,lying
abovethe
axis X22,7 a.'t, the poinl, rzp+r,zt,+t(7,) situateclto the rig[t
of the straight\inen:!'zr,zr(À),
À)
arr,, i.e.{71)
o{ rrr,rr(},) 1r'rr,*r,rr*r(l),
), )> d-zn.'Ilre
point
of intelscctiorr of the tangent to thecutve
U:
!yp1,a1(n, Ì'),at tho point
1'{rrn,ro()'),-
(Lzt,+t} of the s}.stern(Irr*r),
wiLh thc axis Xgr,+r,tas the
abscissa'(72) úrr*r(À)
_-'t'2t,.2t,(^)
- ¡:1"^¡^l^ t t' Þ
a2r''We wish to pror.e th:lt
.rJi,,*r{rro,2r,(),),À]
incleasesinclefinitel¡' with
?t.Ay
(53) +vc have(73) I'ro.rr(\)**oo, À*tæ
+0r )'---++oo
Y i, *rt
rrr(ì'),
)'\92 B, S. TO1VIIE 12:
nnd L)y (52)
$+)
0{
?'r, (À)< rrr,rr,(}'),,
'i,}
a-¡t,.llro
va,lue oTthc lilsl,
r'lclir.aley'"r*r(r,
À)at
thepoiltt ,u:
r'rr,.',,( 7) is(?5)
3l'rn*r',rzr,zo( ?'), ),1 =-r\i,ri,$) '
'qr{r'rr,2r,(}'),27t-
!1,'( ì.
.{r t' - t
'Jlhe palrl:olir,
(?6) y :'q"(rr'12p -
1)=
(2pf [)c'0,r2
27ta,t+ \2p -
])4.,crtt¡
thc axis,lo
at 1ùc ¡ioirtl's.lrr-
?"r( ),,27t- 1)alrtl
';r2--
rt2(À,27r-
-l-)-In viltue
oT 1,he inequa,Iit¡' (26) rve ha,r'c0 <
r'1,r()',2p - l) { t'rr().),
).> I,
¿urd
n'ith
lee'ltt'd 1,o 1hc¡ inrrqttalil,¡. (74)it
lcrllol's\77) 0 q ljr(r',
2p- t) 1lrr(
).){ i'r,,,or(),), it
-). 'r.zr¡.P¿¡r'a,bol¿1, (76),
for'
,t,,-
t'zz(À), hrts 1,hc ol'dinatcr(?s) r,-ir'rr(À).:Ì7r ll -- 2ør(I
F |/7,'z-
)ì- l)+
-i-oo,
).+ |
co.lll-tt', olclinatcs ol'pzr,rtbolil (76) alcr 1;osii'ive
fol
rrri
t'',r(),r2'p--
1) a,rld tìre¡' i¡cleasg rnolototrr)ttsly as ,u- l-
co.lhus,
olt'jttg Lo (74)r l'hc¡ro <¡rists tlÌe' irrequalil,y'Qzttl'rr.rr\ì,)r 27t
- lI >
'rt"',1'zr(7,),2p-
lì > 0. )'
)>' tr",r,rtrul, ou'itrg
to
(?8), l't'orn 1.hisit
follorvs(79)
r,,[t'2;,.2r,()'),2]t- 1]
-1-oo,
)' ---+f
co.l-i:orr¡ (7õ)
u,ntl(7fì)
i1,l'ollotr'¡: .
. ,i(E0) ,
;U|p+: it'2r,2r,(À), 7')-' -l
co, À-' + co.
.':lhc fonirulile (7:ì)
:tncl(80) git'c :.
i(81)
',trrnt(},)-
t'2r,21,(À)l+ 0,
).-+
-þ coa,rrtl, sinccr thc, ctn'r't'
ll -.
i!/z.t)+,( ,', À) is croltciìt't' i1 tþe ilt1,e¡r.al 1r"lrtr,"r-t(\)r.-l-"oÌ, ilt
vir'l,rLt'oT t,his concai-it¡-rn(ì in virl,uc
ol'(71)we
obtir,in(S2) (l
.1 t'zn.rn(7.)-r ,':,,.,.rr,,(À) {
/z,,nr(À),
),}
o.2r,.thus, florn
(82),in
virl.nutif
(81), rvcob1'ain
'(8ll) t'21,.21,\)r) - t'21t+t rrn,(),), ), -> ..l- oo,
l'or'
1,hc r-rcga,1,irt'.
r'ootsof tlìo
g'rcfttostlt'bsolute
\/a,ltto /'r, *r,r( À) rìn{l 1'r,,,,(}.)
of' r'o¿rl ocÌrt¿¡tionsu'itlt
posii,ivcooolificictrts
:I'
\
tt,,,;'tt'-'-: 0, p :
21tt2n!
1,)-=0
13 ÂSYI\{PTOTIC R,IILAîJONS T'OR TIIE INDEFINITEËY INCN,EASING ZEROS [ì(i
ib car bc plovctl by
ureansof
i,ìro s¿r,rne rnt¡thocl[a]
th¿r,tfol
thern l,helc erisl,stho
as-vrnpt,oi;ic lola,l,ion1'2,+t,t(À)
-
?'2,,,r(7r),
).+ f
co.Sincc tlre proofs for the inctlurr,Iities
)'2e,21,(1,).i
r2r,¡,.2p+z( )') :l,ìr(f i"2?)+3,2,*r(À) {
I'2.p¡1,21t+1(À)ale
r-er¡'lcngthyr
\\'(', shir,llgil'c thetrt
itt sell,ìr'alrc par)er.ft
shoulcl l¡t-rlnentiont'tl thnt
1,he proofsfor'thcse
iut:r¡ut-iifil¡s
l1¿¡'1r¡r i,heir' ìrasisin
'Jìhoolenr (4).lì lil l.'Ii r,. ti N o ti s
1. lì a n (ì i ó, I , I¡iia alael¡ra lt iztlarr.lc. Narrðna l<rrjiga, Booglart 1954, pp irlr ó(i.
2. l, a rr t'e rr 1, FI., 'frttilt| d'AIgi:br¿. lonrcr ì ll, Gzrr.rthicr \¡iÌlnt's ot Fils, lmplinrenls Li lrr':rilos, I)n liulcau clcs Ìolg-iLuclos, dc l'(rcole poì¡-[c.chÌriquc, I)alis 189.1, pp. 112-11.1
ll. l\I:r l cl c r. \i., 'l'|rc Cìeonrclry ol llrc Zcros of a PoL¡¡trontittl in tt Oom¡tlc;t Yarinl¡lc. -\Iatht'ltr¿r
ticll SrLlvors. Nurnbel III, r\nrelicurr II¿'Lthcrrr¡Lical Socic[r', Nol'Yolk 1949, pp ](19- 110
4. 'l- o rrt i ó ll. li., Les rclnlions usgntplolír¡trcs ¡xtur lcs :ét os ct oÌssunl irtdéfinirtuttL des polt¡ti)
¡ncs tt cocll'iri<'nls posilils. Acad. jìo¡'. l:lclÍ1., llull. Cl. Sci.. 5(! S(r'ic, 'l'onrc Xt,lX, [JL'uxtl]cs
1 9(i3 8, lr p. 799 81 8
:i. Zr\!,rtsltitt,\1.L,.,Spraooòttilipoüislcttitn¡nt'lotltttttrt:sr:n.ji.jualgthruíct:sl:ilti l¡¡tttscendcn!¡tilti ut rtntt.jcu.ii.i. lìizrììrìtg..iz, Nloskvn 1960, p. 5¡J.
lìct:cir,ccì J \ t I983 Li tt i uc r s i I ¡¡ Bel¡¡ rarLc I:rttttllt¡ t'¡ )[rtlnutirrrl Iittgirt,, r itt,¡
ri,
\', '.