Линейные формы и статистические критерии І
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| Datum: | 1953 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1953
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7732 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512725285732352 |
|---|---|
| author | Линник, Ю. В. Линник, Ю. В. |
| author_facet | Линник, Ю. В. Линник, Ю. В. |
| author_sort | Линник, Ю. В. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T09:55:16Z |
| description | - |
| first_indexed | 2026-03-24T03:33:21Z |
| format | Article |
| fulltext |
1953 YKPAHHCKHfi MATEMATH!LIECKHfi /KYPHAJI
HHCTHTYT MATEMATHKH
JluueHHble cpopMbl H CTaTHCTH'IeCKHe KpHrepHH. I.
!0. B. JhlHH.UK
1.
JhiHeHHhie ljlopMbJ
§ 1.
T. V, Ng 2
DycTb Xt, x2 , .•• , x, rrpe,I:LcTaBJIH!OT co6oH. He3aBHCHM'ble Ha6JI!O,I:LeHHH
Ha,I:L cJiyqaifr!Oi'I BeJIHqHHOH X, pacrrpe,I:LeJieHHOH rro 3aKoHy F (x).
DycTb
H ) ( 1, 1)
,I:LBe JIHHei'IHble <j) op~.lbl C ITOCTOHH!1'b!MH peaJibHblMH K03 <j)<j)HIJ,HeHT3M H OT H3-
6JIIO,I:LeHHH (mmeHHble CT3THCTHKH).
Mbr 6y,I:LeM ,I:LonycKaTb o6pamemre HeKOTOpbiX K03<j)<j)HIJ,HeHTOB ai 11 bi
,. r
B HYJib, HO H e BCeX cpa3y, T3K qTo ~ az =/= 0 H
i=l
Mbr 6y,I:LeM HHTepecoBaTbCH TeMrr 3 aKOHaMH F (x), rrpH KOTOpbiX Harnrr
cpopMbi L 1 (x) rr L 2 (x) OK33hiBaiOTCH o,I:LHHaKoBo p acnpe,I:LeJieHI-IhiMH . .3To Mbi
6y,I:LeM o6o3HaqaTb L1 (x) ~ L2 (x). TaKHe 33KOHhi o6pa3y!OT rrp11 3a,I:LaH
H biX L 1 H L 2 HeKOTOpbie MHOMeCTBa 3a KOHOB Q L L • Tpe6oBarme o,I:LrrHaKO
noi1 pacrrpe,I:LeJieHHOCTI1 L 1 (x) H L2 (x) M OMHO HHT~p~peTHpoBaTb KaK Tpe6o
BaHrre O,l:LHOpO,I:LHOCTH 'COOTBeTCTBYIOlll,HX Bbi60pOK H Ha '3TOM OCHOB3HHH
IT OCTpOHTb CT3THCTnqeCKHH KpHTe prrif ,l:LJIH HCITbiT3HHH flliTOTe3bl 0 rrpn
H3.LJ:Jie)!<HOCTH 33KOHa F( x ) K l\lHOMeCTBY QL, ,L,- B qacTHOCTH, T3KllM
o6pa30M y,I:LaeTC51 DOCTpOHTb HOBbiH KpHTepllH DpH!Ia.LJ:Ji e)!<HOCTH 33KOH3
K HOpM3JibHOMY THITy.
B rr e pBoif qacTH pa6oTbi 6y.LJ:yT H3JIOMeHbi Teop eTHKO-BepoHTHOCTHbie
T eopeMbl 11 llX ,l:LOK33aTeJibCTB3, a BO BTOpOH - HX ITpHJIO)!{e HHH K MaTeMa
THqecKOH CT3THCTHKe.
§ 2.
Ll.JIH ,I:LaHHbiX ,I:LBYX cpopM L1 (x) = a1x1 + a2x2 + ... + a,x,, H L2 (x) =
= b 1x 1 + b2x 2 + ... + b,X, BBe,I:Lel\1 IJ,eJiylO <j)yHKIJ,HlO KOMDJieKCHOfO rrepe
MeHHOfO
a <z> = I a 1 lz + I a2 :z + · · · + I a, lz - 1 b1 lz- · · · - 1 br iz,
KOTopy!O Mbi 6y,I:LeM Ha3biB3Tb on p e ,I:L e JI 5I 10 Ill e H. <P y H K II. H e H..
~07
T eo p e M a 1. flyerb )J,aHbi )J,Be JIHHeiiHbre eTaTHeTHKH
L1 (x) = a1x1 + ... + a,x,
Lz (x) = b1XJ + ... + b,x,,
y )J,OBJieTBopmorn.He yeJioBmo 1
sup ( lat l, la2 /,. · ·, [ar l) *sup ( lb1 l, l b~ /, ... , / br l). (2,1)
.Umi 3KBHBaJieHTHOeTH ,ll,Byx YTBep:tK,ll,eHHH -
(A) Ha6JIIO)J,eHHfl X; npriHa,;vie:tKaT K HOpMaJJbHOMY THTIY pacnpe,ll,e.'!eHHH,
(B) eTaTHeTHKH L1 (x) H L2 (x) o)J,HHaKoBo paenpe,ll,eJJeHbi - He o 6 x o
)1, 11 M o H )J, o eTa To q H o BbinOJJHeHHe cJJe,ll,yiOW.HX yCJioBHii:
1) a1 + az + ... + ar = b1 + bz + ... + b,;
2) a(2)=0;
3) Bee TIOJJO)I{l!TeJJbHbJe KOpHH .a (z), HBJJ5IIOW.Hee5I QeJJbiMH '-ll!Co1 aMH,
)J,E'.'lHlll.HMHeH Ha 4, )J,OJJ:tKHbl 6biTh npoeTbJMH;
4) Bee TIOJIO:tKHTeJJbHbie KOpHH a (z), 5IBJI5I!OW:Hee5I QeJJblMH l!eTHb!MH
'-IHCJJaMH = 2 (mod 4), J(O il:tKHbl HMeTb KpaTHOeTb He eBbTUie 2, npHTOM, eColH
eCTb TaKOH ,ll,ByKpaTHbiH KOpeHb, OH )J,OJJ:tKe H 6b!Tb e)J,HHeTBeHHbiM li 1\!ai<CH
MaJibHbiM H3 Beex TIOJIO:tKHTeJJbHbiX KOpHeH a (z);
5) eeJJH a (z) HMeeT TIOJIO:tKHTeJJbHbiH KOpeHb y, He HBJJHIOIUHHeH QeJJbll\1
l!eTHbiM l!HeJJOM, TO OH )J,OJJ:tKeH 6b!Tb e)J,HHeTBeHHblM, MaKeHMaJJbHb!M 11 3
noJJO:tKHTeJibHbiX KopHeli, npoeTbiM, H [ ~ J I u.eJraH qaeTb j I )J,OJJ)KHa 6biTb
Hel!eTHOli.
IJ:oKa3aTeJJbeTBO TeopeMbl npOBO)J,HTeH B § 3-54 li 3aKaHl!HBaeTCH
B §54.
B pa3,ll,eJJe II 6y,ll,yT )J,aHbi npHJJO:tKeHHH TeopeMbi 1 K eTaTaeTHIIeCKHM
KpHTepHHM.
Teo p eM a II. TlyeTb 3a)J,aHbi JJHHeHHbre eTaTHeTHKH
L 1 (x) = a 1x1 + ... + a,x,
Lz (x) = b1x1 + ... + b,x,,
no)J, yCJioBHeM
sup ( I a1 I, ... , I a, j) * sup (j b1 [, ... , i b, i). ( 2,1)
fiycTb ,ll,JJH HeKoToporo 3aKOHa F (x) pacnpe,ll,eJieHHH Ha6JIIO)J,eHHH xi
HMeeM
L 1 (x) ~ L2 (x).
ITycTb r - MaKCHM3JibHblH peaJibHb!H KopeHb onpe,ll,eJI .HIOW:eH cpyHKU.HH a (z).
Tor)J,a r > 0. EeJJH 3aKOH F(x) HMeeT 2m-li MOMeHT, r)J,e m = [~ + 1]' TO
F (x) - HOpMaJJbHbiH 3aKoH (6b!Tb MO:tKeT, Heeo6eTBeHHbiii).
)loKa3aTeJJbeTBO 3TOH TeopeMbl npOBO)J,HTeH B § 3-26 li 3aK3Hl!HBaeTCH
B § 26.
I B MOeii 3aMeTKe ,JlHHeHHbie CTaTHCTHKH H HOpMaJibHb!H 3aKOH pacnpe.neJieHHH"
(.UAH CCCP, T. LXXXIII, N2 3) 3TO ycJIOBHe no onm6Ke onyll!eHo. CJiylfaii ero Hapywe
HHH ,!\OJIMeH 6b!Tb H3ylfeH .L\OHOJIHHTeJll,HO.
208
B pa3)l.eJie II 6y)l.yT )l.aHbi ee npHJIO:tKemur K MaTeMaTwieCKOH cTaTH
CTHKe H K Bb!BO)l.y 3 aKOHa MaKCBeJIJia B KJiaCCHlieCKOH cpH3HKe )l.JIH pacnp e
)l.e.TJeHHH CKOpOCTH )l.BH:tKeHHH MOJieKyJI [1).
ECJIH OT6pOCHTb Tpe6oBai-IHe ( 2,1), TO IIOJiyliaeTCH CXO)l.HaH
c II TeopeM a.
Teo p eM a II'. OycTb )l.JIH HeKoToporo 3aKoHa F (x) pacnp e)l.eJieHHH
Ha6JIIO)l.eJ-IHH X; HMeeM L1 (x) ;;~ L2 (x) , Ho Tpe6osaHHe (2,1) He HaJiaraeTcH.
f1ycTb o eCTb TOqHaH BepXHHH rpaHb a6CU,HCC HyJieH onpe)l.eJIHI-Olll.eii
cpyHKU,HH
EcJIH 3aKoH F (x) HMeeT 2m-i1 MOMeHT, r)l.e
TO F (x) - HopMa.ribHbiH 33KOH (6b iTb Mo:tKeT, Heco6cTBeHHbi1'!}. 3aMeTHM,
'ITO ecJIH u(z) =!=- 0, TO sce r)l.a a < oo. TeopeMa II' )l.OKa3biBae TC51 B § 3-26_
C JI e )l. C T B I1 e. ECJIH a (z) =/= 0 H F (x) HMeeT BCe MOMeHTbi, TO !1 3
TeopeMbi ll' CJie)l.yeT, liTO F(x) HOpMa.TJbHb!H (6b!Tb MO:tKeT, Heco6CTBeHHb!H)
3aKOH pacnp e)l.eJieHHH.
3To c.rJ e)l.CTBHe B HeCKOJibKO HHOH cpopMy.rmpoBKe )l.OKa3aHo Mapu,HHKe
BHli e M [2). f!pH '3TOM Mapu,HHKeBHli CT3BHT B0!1p0C 0 TOM, MO:tKHO JIH
B cpopMyJIHpOBKe Tp e 6oBaTb Cylll.eCTBOB3Hl1H TOJibKO BTOpOrO MO MeHT3
'"
a~=~ x'dF(x). 0TBeT Ha 3TOT sonpoc oT!'mu,aTeJI.bHbiH. noJiee Toro, cyme-
- 00
CTBOB3HHe JII06oro cpHKCHpOBaHHOfO KOJIHlieCTBa MOM e HTOB 3aKOHa F(x) IIpH
ycJIOBHH a(z) =/= 0 11 L 1 (x) ~, L 2 (x) e m e H e o6ecneliHBaeT HOpMaJibHOCTH F (x).
3TO )l.OK33biBaeTCH B § 55.
f1pe)l.bl)l.Ylll.He TeopeMbl TpaKTYIOT CJIYli3H HOpM3JlbHOCTI1. f1epei1)l.eM
K 6oJiee o6meMy cJiyllaiO .
Teo p eM a III. f1ycTb 3a)l.aHbi liHCJia y1, y2, .. . , y1 no)l. ycJIOBHeM
(2,4)
Tor)l.a MO:tKHO IIOCTp011Tb 6eCKOHeliHO MHOrO lieTBepOK .TJHHe HHbiX CT3THCTI1K
L;=a~nx1 +···+a~ilx.; i=l,2,3,4, a~l :.;;> O,
T3Kl1X, liTO eCJIH
(2,5) ·
)l.JIH K3I<OrO-Jli160 Cl1MMeTp11llHOrO 33KOI-Ia ([) (x) C xapaKTepHCTHlieCKOH cpyHK
U,HeH f(u) , TO f(u) HMeeT BH)l.
f(u)=exp (- j: A;) u l1j), (2,6)
j = l
r)l.e A 1, - HeKOTOpbie KOHCTaHTbi, npHlieM A1, :.;;> 0 11 Are ): 0.
TollHO TaK :tKe, eCJII1 33)l.aH C11MMeTp!1lleCKI1i'r 33KOH w(x) c X. cp. I
l JlJIH KpaTKOCTH Mhl BCIO.!ly .!l3Jibllle l!HllleM X. cp. BMeCTO XapaKTepHCTH'I e CK3H
¢YHKUHH-
209
BH,ZJ.a (2,6), HanpHMep, KOMD03HU.H51 ycToiiqHBb!X 33KOHOB C X. cp.
!(u)=exp(-±A·1 \u\11 ), r.z:r.e A 11 >0,
f = i
TO ,ZJ.JIH llJ(x) 6y.z:r.eM :HMeTb
L 1 (x) ~ L~ (x); L 3 (x) ~ L~ (x).
3Ta TeopeMa ,ZJ.OKa3biBaeTC51 B § 56.
MHO:tf<eCTBO X. cp. THDa (2,6), K CO)K3JleHHIO, He HCqepnbiBaeTCH KOMDO·
3HU.HHMH yCToifqHBhiX 33KOHOB. Era cocTaB noHCHHeTCH TeopeMoli IV.
Teo p eM a IV. flycTh 3a,ZJ.aHbi noJIOlliHTeJihHbre qHcJia -yr, y2, ... , Ye
no.z:r. yCJioBHHMH
j'I < '/2 < · · · < Ye> 0 < '/ I < 2, (2,7)
nOJIO)KHTeJibHOe qHCJIO Ae > 0 H npoH3BOJihHbre peaJibHbie l!HCJia
E2, E 3, ••. , Er-I· Tor.z:r.a npH BCHKOM .z:r.ocTaTol!Ho 6aJihillOM A cpyHKU.HH
f (u) = exp (-A /u/1' + E 2 / u /1' +···+Ee-l [u/''e-l- A. / u i;'•) (2,8)
6y.z:r.eT xapaKTepHCTHqeCIWH cpyHKU.HeH ,ZJ.JIH HeKOTOporo CHMMeTpHqecKoro
3 3KOHa llJ(x).
,UoKa3aTeJihCTBO 8TOH TeopeMbl .z:r.aeTCH B § 44-49.
06ume TeopeMbi o6 O.D.HHaKoBo pacnpe.z:r.eJieHHhiX cpopMax BbirJIH.D.HT
,u:oBOJihHO rpoM03.D.KO. Mbr npHBe.z:r.eM .z:r.se H3 HHX.
Teo p e M a V. flycTh .z:r.aHbi .z:r.se JIHHeHHbie H OJJ:HHaKOBO pacnpe.z:r.e
JieHHbie CT3THCTHKH
Lr (x) H Lz(x) c ycJIOBHeM
sup ( /a 1 /, ••• , /ar /) =I= sup ( / b1 /, •• ·, / br / ). (2, 1)
Tor.z:r.a f (u) =1= 0 Ha Bceli ocii. 0603HaqaH 1/J (u)=ln /! (u)l'. noJiyqHM
lfJ(u)=l/J(-u) H ,ZJ.JIH u ~> 0
u k ~
) 1/J~U) du= J~ u;'i · p 11 (In u) + ]:;
0
Sn,(u), (2,2)
r.z:r.e y11 Y2, ••• , 'Yk - peaJibHbie HYJIII u(z), P 1 . (ln u) - noJIHHOMhl cTeneHII
I }
-<: k2, H
~ -
S,. (u) = ~ [u?i • P~1 (1n u) + u?J • P~1 (ln u)],
j = l
r.z:r.e kr, k2, k3 .. . - KOHCT3HTbl, 33BHCHll.LHe TOJ!bKO OT 0 (z), P?1(ln U) -
ITOJIHHOMbl CTeiTeHH < k 2 ; •Qi - KOMDJieKCHbie KOpHH o(z) B BepXHeH DO.JIY·
ITJIOCKOCTH z = x + iy, Melli.D.Y op.z:r.IIHaTaMH y = m + Oo H y = m + 1 + Oo,
npiiqeM o0 MO)KHO H HYlliHO Bhi6paTb TaK, qTo6bl see KOpHH o (z) 6hiJIH Ha
paccTOHHIIH > o1 aT ropH30HTaJieli y = m + o. PH.D. (2,2) 6y.z:r.ei cxo,ZJ.HTbCH
paBHOMepHO . BO BCHKOM cerMeHTe ITOJiyrrpHMOH [O,oo).
,UaJiee, eCJIH OTMeTHM Cpe,ZJ.H TOqeK yr, y2, . •• , '/k• TaKHe TOqKH, ,ZJ.JIH
KOTOpb!X COOTBeTCTBYIOll.LHe ITOJIHHOMbl P.1. {In u) He HCqe3310T TO)K,ZJ.eCTBeHHO,
I
H eCJIH 0! H 02 > o1 - KpaHHHe H3 3THX TOqeK, TO a6CU.IICCbl BCeX KOMO.JieKC-
Hb!X HyJieH (!i' ,ZJ..JIH KOTOpb!X P e/ln u) =/= 0, JielliaT Melli.D.Y 01 H 02. flpH
::>TOM or y.z:r.oBJieTBOpHeT ycJIOBHIO
0 < 01 < 2. (2,3)
210
TeopeMa V .li.OKa3bJB.1.eTc5I B § 3-27, npwieM TqM coo6rnaioTCH B BH,ll.e
JICMM eme HCKOTOpblC CBC,li.CHHH 0 pH,li.C (2,2).
B cJiyqae, KOr,ll.a ycJiosue (2, 1) HapyweHo, npe,ll.cTaBJieHne ( 2,2),
B006rne rOBOpH, ,li.CHCTBYCT TOJibKO B6JIH3li 0. T04Hee, HMCCT MCCTO CJie,ll.y!O
lU3H TeopeMa.
Teo p eM a V'. B ycJIOBHHX TeopeMbi V, 3a HCKJII04CHHCM (2, 1), HMeeM
rrpe,ll.cTaBJieHHe (2,2) rrpn 0 < u < u0 , r,ll.e u0 - HCKOTopoe 4HCJIO, npnqeM
npn I u I> uo H 0 < a1 < 1 f (u) MO.IKeT, soo6me rosopH, 6biTb JII06oii HeBo
rHyToii KHH3Y qeTHOH cpyHKILHeii, o6pama10rneiicH B 0 npn lu I > Ut > 0.
,llOK333TCJibCTBO :noli TeopeMbl 6y,ll.eT ,li.3HO B § 57.
0TMCTHM erne, 4TO BBH.li.Y li3BeCTHOH .li.Y3Jib!-IOCTH MC.IK.li.Y cpyHKUHeH li ee
Tp31-1CcpOpM3UHCH ¢ypbe, npH KOTOpOH YM!-10:/:KCHHIO cpyHKUHH OTBC43CT
cnepTKa Tp3HccpopM311.HH <Pypbe n o6paTHO, MO)KHO ycTaHOBHTh HeKoTopyiO
.li.Y3JibHOCTb MC.IK,ll.y YM!-10./KCHHCM X. cp., OTBe4310lUHM CJIO.IKCHHIO !-IC33BHCH
MbiX H36JIIO,IJ:eHHH, li YMHO.IKC!-IHCM HH TerpaJib!-lbiX 33KOHOB, OTBC43l0lli.HM
paccMoTpeHHIO MaKCHMyMa Ha6JIIO,ll.eHui1. BBH.li.Y 3Toro .TJHHCHHbiM cpopMaM
GtXt + ... + a ,,.Xn MO.IKHO COITOCT3BHTb HC3!-13JIHT!14eCKHe CT3TI1CTHKH
SUp (atXt, a 2X2, ... , anXn) li paCCMaTpHB3Tb TCOpeMbi 06 O,li.HH3KOBOM pac
npe,ll.eJieHHli T3KHX CT3THCTHK.
CooTBeTCTBYIOlUHe TeopeMbi H BbiTeKaiOmue H3 HHx CTaTHCTH'IeCKHe KpH
Tepnu 6y.ll.yT pa3o6paHw B .ll.pyroii pa6oTe.
§ 3.
Dpn 33)l,3HHbiX cpopMax Lt (x) H L2(x) ( 1,1) coBoKynHOCTb 33KOHOB
F (x), ,li.JIH KOTOpbiX Lt (x) ·;:;;;; L2 (x) o6pa3yeT, oqeBH.li.HO, accounaTHBHYIO
CHCTeMy C O,li.!-IHM ,li.CHCTBHeM, CCJIH 33 :no ,li.CHCTBHC B3HT3 onepaUHH KOMTI0 -
3 liiJ.lili 33KOHOB, OTBe4310lU35I C.TJO:/KeHHIO He33BHCHMbiX CJIY43HHb!X BCJIH4HH .
.LlaJiee, ecJIH Lt (x) ~ L2 (x) ,li..TJH Ha6JIIO;I.e HHH xi c 3aKOH0~1 pacnpe,ll.eJieHHH
F(x), TO O.li.HHaKoBaH p3cnpe,ll.eJieHHOCT b H3WHX cpopM coxp3HHTCH H rrpH
33MCHe Xi H3 (-Xi), (i = 1, 2, ... , r) 11 no-npe,ll.bl.li.YlUCMy - li ,li.JIH Ha6JIIO
,li.CHHH Xi- X{, r,ll.e Xi II X{ - HC33BliCliMbiC li O,li.HH3KOBO pacnpe,li.C.1C!Ulble
BCJIH4li!-lbl. BeJIH4Hl-lbl Xi- X{ o6JJa,li.310T CHMMCTpH4CCKHM 33KOHOM
W(x) =F(x) ,:, (1-F(-x)) (*3l-13K KOMTI03HUHH).
B ,li.3JibHeiiweM Mbi 6y.ll.eM 113yqaTb CHMMeTpHqecKne 33KOHbi p3crrpe,ll.e
JieHHH H36JIIO.li.CHHH, .li.JIH KOTOpbiX Lt (x) ~ L2 (x). .Umr TaKnx 33KOHOB
HMCCM
BBH.li.Y 3TOro MQ)!(HO CqHT3Tb, He HapyW3H 06lUHOCTH, qTo BCe K03cpcpH
UHei-!Tbl ai li bi I-ICOTpHI1.3TeJibHbl.
PaccMoTpHM x . cp. Ha6JIIO.li.CHHH xi
"'
f (u) = E(eiuxi) = ~ eiux d (]) (x) .
3T3 cpyHKIJ.HH peaJibHa, qeTH3, Henpepb!Bl-13 l-13 BCeH OCH, II f (0) = 1.
Mbi c,ll.eJI3CM Tenepb B3.1KHOe rrpe,ll.noJio.IKeHHe
sup (a1 , a2 , • • ·, a.) =1=- sup (b 1 , bz, . .. , b.), (3,1)
oTKa3 oT KOToporo 6y,ll.eM oroB3pHB3Tb oco6o.
211
l13 YCJIOBHH Lt (x) ·~ L2 (x) Haxo.n:HM, 'ITO
I (a1u) I (a2u) ... I (a2u) =I (b1u) ... I (b2u). (3,2)
By.n:eM CIJHTaTb, He HapylllaH o6mHoCTH, 'ITO a 1 = sup ( a1, a2, 0 0 0 , a,,
b1, b2, 0 0 0, b,). Tor.n:a B CHJIY ycJIOBHH (3,1) HMeeM
b;<ai (i=1,2,ooo,r)o (3,3)
BBO)J.H HOBYfO rrepeMeHHYfO a1U, KOTopyiO MO:tKHO CHOBa 0603Ha'll1Tb 6yK
BOH u, 110JIY'll1M !13 (3,2)
I (u)l (a1 u) ... I (a,u) =I (b1n) ... I (bru), (3,4)
r.n:e
a;<;;1 (i=1, 2, •.. , r); b;<1 (i=1, 2, ... , r)o (3,5)
§ 40
JI eMMa 1. <f>yHKUHH I (u) =!= 0 Ha sce!'f peaJibHOH oc110
ll o K a 3 aTe JI b c T Boo ITycTb B (3,4) HMeeM a1, a2, 0 0 0 , ak = 1;
an+l, 0 0 0 ' a,< 1) (k MQ)KeT COBrra.n:aTb c 1). Tor.n:a ITOJIY'IHM
(! (u) )k I (an+l u) ... I (a,.u) =I (b1u) o •• I (bru). (4,1)
ITycTb ~ MHO:tKeCTBO TOIJeK, )J.JIH KOTOpbiX f(u) = Oo Mbi .n:oJI:tKHbi
D:OKa3aTb, LJTO ~ - ITYCTOe MHO:tKeCTBOo
ITycTb 3TO He TaKo B CHJIY HerrpepbiBHOCTH f ( u) ~ - 3aMKHyToe MHO
)KeCTBO. TaK KaK f (0) = 1, HeKOTOpaH OKpeCTHOCTb Hy.'IH He BXO)J.HT B ~.
Cyw:ecTByeT 6JIH:tKai'!IllaH crrpasa K 0 TOLJKa ~; rrycTb 3TO 6y.n:eT ua;
f(uo) = 0, Uo > Oo BBHD:Y ycJIOBHH, HaJIO)KeHHoro Ha a1, .. 0, a2, b1, 0 0 0, bz,
HMeeM l(a;u) =!=O; l(bp)=!=O; i=k+ 1, .. 0, r; j= 1, 2,000' r.
BBHD:Y 3TOrO l13 ( 4,1) ITOJIYIJHM 11pl1 BCHKOM )J.OCTaTO'IHO MaJIOM s > 0
(! (uo-s))k = f (bdu"o-s) )._ .. I (b,.(u 0 -s)) .
I (ak+1 (uo-~)) ... I (ar (uo-- c)
I (b1 uo) ···I (b,ua)
IT pH s ·~ 0 .'leBaH IIaCTb -+ 0, a rrpaBaH IJaCTb - K I I =!= 0.
-=- (ak 1u0) • • • (a,u0)
oTO rrpOTHBOpeii:He l1 )J.OK33bJBaeT JieMMYo +
Terrepb B paBeHCTBe (3,4) MO:tKHo npoH3BOD:HTb coKpameHHH, ecJIH
HeKOTOpb!e !13 'IHCeJI a; COBITaD:a!OT C HeKOTOpbiMH l13 'IHCe.'l b jo Ll,onyCTHM,
lJTO T3KOe COKpameHHe npOl13BeD:eHO, 11 Mbl IIOJIY'IHM
(4,2)
r.n:e y)Ke ,IJ,aJiee coKpaw:aTb HeJib3Ho
ITpH 3TOM Mo:tKeT oKa3aTbCH, 'ITO rroJiyiiHTCH paseHCTBO
(f(u))k= 1.
JlerKO BH)J.e_Tb B CiiJIY Henpepb!BHOCTH f ( U) l1 paBeHCTBa f (Q) = 1, 'ITO
/(u) = 1 H OTBeiiafOlll.HH eli 3aKOH W(x) = s(x) Heco6cTBeHHbiHo TaKoil
c.riyiiali 6y.n:eM CIJHTaTb TPHBHaJibHbiM 11 oT6pacbiBaTbo ITocJie 3Toro MO:tKHO
CIIHTaTb, 'ITO IJHCJia a; H b h BXO.Ll.Hlll.He B ( 4,2), y.n:oBJieTBOpHIOT HepaBeHCTBaM
O<a;<;;l; O<bi<1 (i=l, 2,0 .. , r, j =1, 2,000' q) (4,3)
212
§ 5.
BBH.ll.Y .ll.OK333HHOfO Bb!llle f(u) II0"1 0)KliTCJ!bH3 Ha BCCH OCH H Herrpe
pbiBHa. 0Ha l.JCTHa, T3K l.JTO .lJ.OCT3TOl.JHO II3Yt.J3Tb ee npH U > 0.
OoJIO?KH ~t Tenepb a; = In a;; (l; = In bi; u = e 1; Tor.na a 1 = 0,
ai < 0, /) i < 0; t MCHHCTCH OT - 00 .ZJ.O 00.
Bse.neM HOBYIO ct>YHKU.HIO cp ( t) = In f ( u) = In f ( e1); Mbi 3aMel!aeM, t.JTO
cp (t) Henpepb!BHa Ha BCeH OCH t H l!TO
cp (t) < 0. ( 5,1)
YpasHe rme ( 4,2) rreperrHllleM TaK:
p(t)+p(t+o:1) + ... + tp(t+o:,.) = p(t+ (Jl) + ... + g;(t+{Jq). (5,2)
3.ZJ.CCb Bee ai OTJ!Hl!Hbl OT BCeX f3i·
Mbr .ZJ.OJDKHbi paccMaTpHBaTb HerrpepbiBHbie pellleHHH ypasHeHHH ( 5,2).
YpasHeHIIH BH.ZJ.a (5,2) HBJIHIOTCH l!aCTHbiM cJiyl!aeM JIHHeilHhrx .ZJ.Hct>ct>epeH
U.HaJibHhrx ypasHeHHil co C.ZJ.BHHYTbiM apryMeHTOM, H3yl!eHHhrx A . .ll. Mbilll
KHCOM [3].
AHaJIHTHl!eCKHe p e llleHHH TaKHX H :~ Hal!IITeJihHO 6oJiee o6w:Hx ypasHe
HHH, TpaKTyeMbiX KaK JIHHeHHbie .ZJ.Hct>ct>epeHu.HaJibHbie ypasHeHHH 6ecKoHel!HO
BbiCOKoro rropH.ZJ.Ka, H3yl!aJmcb A. 0. f eJibct>oH.ZJ.OM [4], .naBlllHM pH.ll. sa)!{HbiX
npHJIO)KCH HH IIX B TeopHH ct>YHKU.HH KO \<mJieKCHoro rrepeMeHHoro. B l!aCT
HOCTH, ypaBHCHHH THIIa ( 5,2) ( CH36MCI·!Hb!C IIOCTOHHHb!Mll K03ct>ct>HU.HeHT3Mll
Ak. Aj rrpl! <p(t+a") li tp (t+f3i)), KaK OKa3aJJOCb, HrpaiOT BaMHYIO
pOJib B TeOpHll U.CJ! b!X cpyHKU.l!H [5].
P e llleHHH ypasrreHHH THrra ( 5,2) ecTeCTBeHHO I-Ja3biB3Tb o6o6I.ll,eHHO
rrepHO.ZJ.Ht.IecKHMll ct>YHKl.J.HHMll, T3K K3K OHH OXB3Tb!B3!0T 06b!l!Hble n epHO.lJ.ll·
LICCKHe ct>YHKl.J.Hll, 33.ZJ.3B3CMblC ypaBHCHHeM
cp (t) = cp (t + (3).
3aMCTHM 3.ZJ.eCb )!(C, l!TO 3TH peWCHHH, B006W:e fOBOpH, H e HBJ! HIOTCH IIOl.JTll
rrepHO.ZJ.Hl.JCCKH M H ct>YHKU.HHMH l1 He OXB3Tbll3310T BCCX H e npe pb!BHb!X I104Tll
n e pHO.ll.Hl!CCKHX cpyHKU.HH.
Yl3 BCeX HenpepbiBHblX p e weHHH ypaBHCHHH (5,2) Mbl .ll.OJI:>!Hlbl Bbl.ZJ.C
JIHTb Te, KOTOpb!e MOfYT 6b!Tb JIOrapHct>Ma Ml! p eaJi bHblX X . cp. CJi y l!ailHblX
BeJIHl!HH. BBH.ll.Y TOfO, l.JTO f (0) = 1, H MeeM , IIOMl!MO (5, 1),
'J •(t) .....,-. 0 np11 t-+- oo. ( 5,3 )
EcJI H ycJIOBHe (3, 1) He BbiiiOJIHeHo, TO (5,2) MO)!(HO rapaHTHpoBaTb np~I
i < to.
§ 6.
Mbl 6y.neM p e waTb ypasHeHHe (5,2) c rroMOW:biO TpaHccpopMaU.HH Jla
nJiaca, ,ll,JIH t.J e ro !10H3.ZJ.06HTCH HCKOTOpbiC anpxopHbJC OU.eHKH p e weHHH.
J1 eMMa II. ¥hreiOT MeCTO ou.eHKH:
IIP (t) I< C,ec•t rrpH t > 0,
IIP (t) [ < C, npH t < 0.
{B )J,aJibHei1weM C1, C2, ... ; K,, K2, . . ·. , rroJIOMHTeJibHbie KOHCTaHTbi).
(6, 1)
(6,2)
213
.lJ. 0 K a 3 aTE' J1 b C T B 0 . BBHJI.Y CBOikTB 4HCeJI ai H (3; ypaBHeHHe (5,2)
MO:>KHO rrepermcaTb Tare
k
mq;(i) = ~ Eirp(t- B ;), r)l,e m - 4HCJIO cosrra)l,aiOill.HX ai = 0, e; = + 1:.
i=l
B = I a; I HJIH J (3; J; k < r + q.
0TCIO)l,a
k b
Jrp (t) I< ~ lq;(t- B,.) l < ~ M(t- Bn),
n=l n = l
r)l,e 4epe3 M(t) o6o3Ha4eHa ¢YHKU.HH M(t) = sup lrp(v).
BBHJI.Y MOHOTOHHOCTH M (f) IIOJiyl:JaeM v -<; t
k
M(t)<;;;: ~ M(t-Bn)
n= l
H, )l,aJiee, M (t) <; kM (t- B1), r)l,e 4epe3 B1 o6o3Ha4eHo MHHHMaJihHOe H3
4HCeJI Bi.
TaK KaK q;(t) ·----') 0 IIpH t--oo, TO M (t) < c3 IIpH t < BJ. DycTb
t = qB1, r)J.e q > 1 u.eJioe 4HCJIO. Tor.na corJiacHo HaweMy HepaseHCTBY )l,JIH
t
M (t) rroJiy4HM M (qB1) <; C3kq = C3k 8 '. EcJIH ;
1
He u.eJioe, TO
H BBH.!J.Y MOHOTOHHOCTH M (f) IIOJIY4HM
4TO li )l,OKa3biBaeT (6, 1) ; ( 6,2) - TpHBHaJibHOe CJie)l,CTBHe ( 5,3).
§ 7.
YpasHeHHe ( 5,2) .noiiycKaeT MHoro HerrpepbiBHhiX 4aCTHbiX peweHHiL
B l:JaCTHOCTH, eCJIH A. - KopeHb u.eJioli <PYHKU.HH
(7,1)
II eCJIH A HMeeT KpaTHOCTb y )' 1, TO ( 5,2) HMeeT qacTHOe peweHHe BH.D.a
e21 • P2 (t) + e-;:1 • P2 (t), r.ne P2 (t) JII06oli rroJIHHOM CTerreHH Y- 1.
06mee Heiipepb!BHOe pellieHHe (.5,2) MO:>KHO IIOJIY4HTb C IIOMO!l.Lb!O
TpaHc¢opMaU.HH JiarrJiaca, rrpH yqeTe JieMMbi II.
DpH 3TOM .nocTaT04HO BbiiiOJIHeHHe ypasHeHHH (5,2) Ha rroJiyiipHMoH:
f > fo HJIH t < to )l,JIH IIOJiy4eHHH rrpe,nCTaBJieHHH <p (t) qepe3 Ha3BaHHbie
Bhiille 4aCTHhie pellieHHH Ha COOTBeTCTBYIO!l.Leli IIOJiyrrpHMOH.
PaccMoTpHM rroJiyrrpHMYIO t ;>to H IIOJIO)fCHM rp(t +to) = qJI (t), TaK
4TO <p i (t) 6y,neT y,noBJieTBOpHTh Ha IIOJiyiipHMOH f > 0 ypaBHeHHIO
qJ1 (f-fo) + (/!1 (t+a2-fo) + · · · + rp1 (t+ar-fo) =
= rp1 (t+ f31 - t,) + • • • + 'P1 (t+(Jq-to). (7,2)
214
I3y,neM C'IHT3Tb to ~ 0, T3K K3K JJ:JI5I HallleH 33JJ:3'IH HaM ITOHa,no65ITC5I JIHlllb .
rroJiyrrpHMhie t ~ to ~ 0 H t < to < 0. JleMMa II ,naeT B03MOMHOCTb rrpHMe
HHTh K C{J l (f) OJJ:HOCTOpOHHlOlO TpaHCcpopMa~HlO JlarrJJaCa.
TioJiaraH z = x + iy Hai.f,nel\1 H 3 (6,1), 'ITO npH x ~ x0 , HHTerpaJI
"'
X (z) = ~ e-zt (/!1 (f) dt
l)
CXOJJ:HTC5I H 5IBJI5IeTC5I cpyHK~HeH peryJIHpHOH B X> Xo. ,lJ.aJiee, rrpH BC5IKOM
cpHKCHpOB3HHOM a HMeeM
oo a
~ e-zt(/!1 (t+o:)dt=eza. x(z)-eza ~ e-zt. f/11 (t)dt.
0 0
TaKHM o6pa30M, rroJiaraH, KaK paHee,
a(z) = ea,z + ... + ea,z_e(3,z- •.• - e(Jqz (o:, =0)
af.. (Jk
e-1o:zo(z)x(z)- k~ e"kz. ~ e-ztrp1 (t)dt + k~ e(JA,z. ~ e-z1rp1 (t)dt=O.
llaJiee, HMeeM
0 0
,. q
E(z) = ~ E(z, o:i,)-~ E(z, (3i,),
k=l k=l
E (z) e1<>"
x(z)=--- .
o(z)
§ 8.
(7 ,2)..
(7,3}
,lJ.JI5I JJ:3JibHeHlllero IlpHMeHHeM TeopeMy f. ,lJ.eTllla [6]: eCJIH OJJ:HOCTOpOH
H5IH TpaHCcpopMa~HH JlarrJiaCa HerrpepbiBHOH cpyHK~HH (/Jl (t) CXOJJ:HTCH IlpH
x = Rz>xo, TO
t .r+i«>
~ q\ (~) dr=V.p. 2~i ~ etz. X ;z) dz, (8,1}
0 x-ico
.r+ig
r.ZI.e V. p. (valeur principale) =lim \ .
u~• )
x-ig
21 Si
Dp!i BCHKOM peaJibHOM A, cpyHKIJ,liH e-At· g;1 (t) 6y)l,eT HMeTb TpaHccpop ·
MaiJ,HIO JlariJiaca X ( z +A), cxo)l,Hlll,YIOCH B HeKoTopoli rroJiyrrJioCKOCTH,
BBH)l,y qero (8, 1) )],aCT
t x+i oo
~g;l(r)e-'1"dr=V.p. 2~i ~ etz. x(z:A) ·dz. (8,2)
0 x-ioo
TaKHM o6pa3oM, Haxo)l,HM rrp!i t > 0, A :?:- 0
t x + ioo
(/Jl ~· e t: = , p. - . elz-t-10(z+A) --- dz = ~ ( ) - A" d V 1 ~ E(z+ A)
2rrz za (z +A)
0 x-ioo
x+ioo
= V. p. ~ \ e(t+to) z-At E(z) dz
27Tl J (z-A) a(z)
(8,3)
x-ioo
rrp!i X > Xo >A.
JI eM M a III. Bee HYJIH a (z) pacrroJio:tKeHbi B HeKoTopoli rroJioee
Xt < x < X2. KpaTHOCTb Ka:tK)l,Oro HYJIH He rrpesoexo)l,HT k0, II KOJIIIqecTBO
HyJieH B Ka:tK)l,OM Kpyre pa)l,HyCa 1 He rrpeBOCXO)l,liT k 1 •
.ZJ. 0 K a 3 a T e JI b C T B 0. I1MeeM
(8,4)
li rrepBoe YTBep:tK)l,eHHe cJie)l,yeT li3 Bli)l,a 3TOH cpyHKIJ,HH. BTopoe yTBep:t!<)l,e
HIIe CJie)l,yeT li3 TOro, qTo a (z) y)l,OBJieTBOpHeT Ha BCeH IIJIOCKOCT!i JII1HCH
HOMY O)l,IiOpO)l,HOMY )l,!icpcpepeHIJ,IIaJihHoMy ypasHeHIIIO rropH)l,Ka r + q c no
CTOHHHbiM!i K03cpcp!ill,HeHTaMH. ECJIH 6bi )l,JIH HeKOToporo z H ko = r + q + 1
HMeJIH 6bi cr(z0 ) =a' (z0 ) = ... = a<r+q> (z0 ) = 0, TO OTCIO)l,a cJie)l,oBaJio 6hz
o (z) :::= 0 Ha Beeli IIJIOCKOCTll.
TpeTbe yTsep:tK)l,eHHe cJie)l,yeT H3 H3BeCTHOH TeopeMbi HeHeeHa [7], rrp!i
.MeHeHHOH K KpyraM e IJ,eHTpaMll X2 + 1 + iy li pa)l,HyeaMH 2 (X2- XI + 1) .
§ 9.
JI eM M a IV. EcJIH ToqKa zo y)l,aJieHa OT scex HyJreli o (z) 6oJiee qeM
Ha &o > 0, T. e. i z- Qt I > Eo (!/k HYJIH o (z)), TO
I o(zo)l > Co( t.o ). (9,1)
.lJ. o K a 3 a T e JI b c T B o. I13 Bbipa:tKeHHH )l,JIH a ( z) 6y)l,.eM HMeTb
I o (z) I> Co rrp!i x ;> X3 li rrp!i x < - X3 )l,JIH rro)l,XO)l,Hll.Lero X 3.
flyeTb B noJioee I x I < x3 HMeeTeH ToqKa zo, y)l,aJieHHaH oT seex HyJieli
o (z) 6oJibWe, qeM Ha Eo. Orr!iWeM BOKpyr TOqK!i Kpyr I z- Zo I < X4, r)l,e X4
Bbi6paHo rro)l, yCJIOB!ieM: x 4 ;> 4x3 , H Bee TOLJK!i oKpy:tKHOCTH Kpyra y)l,aJieHbi
OT seex HyJieli o (z) 6oJiee qeM Ha co.
floCJie)l,Hee yeJIOBHe Beer )],a BbiiiOJ!Hl!MO IIp!i )l,OeTaTOqHO MaJIOM f :D
B CHJIY JieM Mbi II I.
PaeeMaTp!iBaeM )l,aJiee cpyHKIJ,liiO
a1 (z) = ( ) ( _ _ ) . z- u1 • • • z u,,.
a (z)
216
0Ha peryJIHpHa B HaweM Kpyre, H IJ:a ero oKpy:tKHOCTH
I (}1 <z) I < 06 .
c~t
(9,2)
Do npHHU.Hny MOJI.YJIH TO :tKe CBOikTBO 6y,LI.eT BbinOJIHHTbCH BO BCeM
Kpyre.
OoJiaraH L(z) = In(}' (z) BHJJ.HM, liTO L(z) peryJIHpHa B HaweM Kpyre
H liTO
R L(z)= In [at (z) i < C, +mIn_!_ < C8 (eo)
Eo
(m orpaHHlleHo no JieMMe III) .
0TCIOJJ.a 3aKJIIOllaeM no TeopeMe KapaTeo,LI.opH - JlaH,LI.ay [7], liTO B Kpyre
\ z - Zo I < ~1 ; HMeeM
I L (z)- L(zo) l < 4Gs(1'o).
Ho Kpyr i Z- Zol < X2 CO,LI.ep:tKHT TOliKH fiOJIOCbl X > X 3, H ,LI.JIH TaKHX TOlJeK
z Hawero Kpyra I L (z) I < Cg.
0TCIO,LI.a noJiylJHM i L(zo) l < Cg(so). BBHJJ.Y ycJIOBHH ,LI.JIH TOlJKH Zo HMeeM :
In o(z0 ) = L (zo) + ln(zo - x1 ) .. . (zo- x,n)
l in (zo- UJ} ... (zo- u,) I < G10 (co) ,
OTKyJJ.a l lno(zo) I< C"(so); lo (zo)I > Go(so), liTO H Tpe6oBaJIOCb BbiBeCTH.
§ 10.
J1 eM M a V. DpH x < - x 5 (oocTOHHHoe lJHCJIO) HMeeM
IE (z) I< c12 . eto I X I •
a(z)
.U o K a 3 a T e JI b c T B o. Ec.1n
Ao =sup(la2 1, la3 1, ... , !ar l, !/111, ... , IPsl),
r,LI.e ai> (1; - liHCJI3 H3 (8 ,2), TO npH X <- X5
(10,1)
la(z) !> C13 eAoX. (10,3)
,UaJiee, HMeeM
a
E(z, a)= eaz ~ e-zucp(u)du.
Dpn x = Rz < 0 noJiylJHM:
to
IE(z, a) ]< C13 ea I xI 8 to I x I,
E(z) < cl< eAo l x i+ lo l X I, {10,4)
(10,3) II (10,4) BMeCTe ,[(310T (10,1).
B ,[(aJibHeilweM 6yL\eM o6o3HalJaTb 6yKBOH B orpaHHllemiOe no MOL\YJIIO
KOJIHlJ eCTBO, H·e Bcer,[(a OL\HO H TO :tKe. Tor,LI.a ( 10,1) HannweM
IE (z) I = Beto l x I. (10,5)
o(z)
7. YKpaHHCKHil MaTeMaT. n<ypHaJI, T. V, N2 2. 217
§ 11.
CooTBeTCTBeHHO JieMMe III o6o3Ha'IHM
xi = inf Re, xz = sup R(!. (11,1)
Me)KJI.Y Ka)!{,ll.biMH Jl.BYMH rrpHMbiMH y = n n y = n + 1 rrpoBe,ll.eM sep
THKaJihHbie rrpHMhie y = Yn· ynaJieHHbie OT scex HyJieli a (z) 6oJiee 'leM Ha
Eo> 0. Tor,n:a corJiacHo (8,3)
t xa+iyn
\ 'Pl (~)e-Md<£"=lim -1-. \ e(t+to)z-At. E(z)dz '
) n~oo2nz ) (z-A)a(z)
(11 ,2)
0 xa-iYn
xa+iYn
rne X3 > Xo. PaccMoTpHM ~ • .llorroJIHHH ero ropH30HTaJibHbiMH KOHTypaMn
Xa-iYn
Z =X+ iyn, X< X 3, n > 1, Hali,n:eM, 'ITO Ha 3THX KOI-!Typax COrJiaCHO
JieMMe IV
I a(z) I >Co (.:0), IE (z) I = Beto lx I
a(z) '
-oo+~. ~
I ~ eHto)z-At (z ~)!(z)dxl,~! ~ exp (t0 lxl + (t+to)x-At)dx =
x3 +iYn -GO
B = nt exp (t(xs- A)+ 2io I~ I). ( 11,3)
EyneM C'IHTaTb, 'ITO A He eCTb HYJib o(z), H A> x 3 . O ycTh Sn(t) eCTb
CYMMa BbJ'IeTOB IIO,ll.HHTerpaJibHOH cpyHKI.I,HH Me)!{,ll.y Jl.BYMH rrapaMH IIpHMbJX:
y=yn, y=yn+l; y=Y-n-l• y=y-n; So(t) - cyMMa BbJ'IeTOB Me)!{,n:y
Y = YI H Y = Y-I· Torna rroJiy'IHM, rroJiaraH X3 = xz + t:,
t
'P2 (t, A)= ~ rpl (~>e-M dr= m~O Srn (t) e-At + Rn (f)
0
IRn(t)l = ~iE)exp(t(x2 +E-A)+2t0 l x2 +E I ),
peaJibH35I cpyHKI.I,HH t, KOTopaH (,:OCTOHT H3 'IJieHOB BH,ll.a
(11,4)
(11,5)
e1 ("J+ 1~Jl • Pi(t) + et(ort~;l. Pi(t), (11,6)
rne P;(t) - IIOJIHHOM CTerreHH He Bbillie k0 ; a.+ h . rrpo6eraeT HYJIH o(z)
J J
B BepxHeH llOJIYllJIOCKOCTH.
§ 12.
06o3Ha'IHM 11epe3 Ln rrpHMoyroJibHbiH KOHTYP Me)KJI.Y rrpHMbiMH Y = Ym
y=yn+lt X= XI- f, X= X2 + e, Lo - KOHTYP Me)!{Jl.Y Y = y~, Y = Y-I;
x = x 1 - t:, x = x2 + E H 11epe3 L_n (n < f) - 3epKaJibHble H3o6pa)!{eHHH
KOHTypoB Ln B peaJibHOH ocn. Torna, ecJIH A> x2 + E
1 f'h E(z)
S,. (t) = 2ni J e(t+to) z (z- A) a(z) dz.
L,.
218
B pe3ynbTaTe nonyqaeTCH npH
00
f > 0 H q;s{f, A) = ~ rp1 (r) e-M d,;
" 1 E( )
'! · (t A)=- e-At '\"' - rfl e(t+ful z z dz- R (t)
" ' L.J 2ni;J (z-A)u(z) n
m=-n
L,.
(12,1)
(12,2)
§ 13.
HaM Hy:»mo nonyqHTb aHanorwmyiO <!JopMyny ,ll.nH cnyqaH, KOr,ll.a q;(t),
3a,ll.atiHaH Ha Bceli ocH, Y.li.OBneTBOpHeT npH t <( to <( 0 ypaBHeHHIO (5,2).
0pOBO,li.H paccy:»<,ll.eHHH, COBepweHHO aHanOrH4HbJe npe,ll.bl.li.Yill.HM, HMeeM:
rrpH t<O H q;1 (t)=rp(t+t9), A<x1 -e
I
rpa {t, A) = ~ rp1 (r) e-M dr =
-oo
~ _1_ l"he(t+tolz E(z)dz + R (t)
L.J 2rci -;:t' (z-A)a(z) n
m=-n
( 13, 1)
(13,2)
B cnyqae, Kor,ll.a cp(t) Y.li.OBneTBOpHeT ypaBHeHHIO (5,2) Ha Bceli ocH, rrony
qaeM <!JopMyny (to = 0, A = 0)
00
\ oo 1 1 E(z)dz
) qJ(r)d7:=- n]:;oo 2ni g> ez za(z) '
Ln
BepHyiO npH t > 0 H t < 0. OcTaTOl!Hbie qneHbi nocne cyMMHpoBaHHH OT
- n ,ll.O n 6y.ll.yT BbipaMaTbCH <!JopMynoli ( 12,2) npH f > 0 H (13,2) npH
t<O.
Bee BbilieTbi no,li.HHTerpanbHOH <PYHKUHH Ha,ll.o BbJqHcnHTb B TOl!Kax z = A
H z = (!, r,ll.e Q npo6eraeT HynH a (z). Docne,li.HHe BbJqeTbi ,ll.aiOT BbipaMeHHe
BH,ll.a ( 11 ,6). DpH .9TOM ,ll.nH HeKOTOpbiX (! = aj + h1 MOMeT cnyqHTbCH
TaK, qTo Pi(t) == 0. B CBH3H c .9THM, TaKHe HynH o(z), ,ll.nH KOTOpbiX BbJqeTbi
He Hcqe3aiOT, 6y,ll.eM Ha3b!BaTb a K T H B H bl M H HynHMH o(z).
BepHeMCH CHOBa K a6CUHCCaM Xt H X2, orpaHHljHBa!OilJ.HM nonocy, r,ll.e
neMaT HynH a (z) (.9TY nonocy 6y.ll.eM Ha3biBaTh KpHTHqecKoli).
0T6Hpa.H cpe,li.H Hyneli KpHTHqeCKOH IlOnOCbl aKTHBHb!e HynH, o6pa3yeM
KpHTHqecKyiO rronocy aKTHBHbiX Hyneli [at, a2]; 3,ll.ecb x 1<u1 <u2<x2; u1 H u2
COOTBeTCTBYIOIIJ:He TOljHbie HHMHHH H BepXHHH rpaHh aKTHBHb!X Hyneli.
QqeBH,li.HO, B <!JopMynax ( 12,1)- ( 13,2) MOMHO 3aMeHHTb X 1 H X 2 Ha
OJ H 02.
Tenepb Mbl ,li.OnMHbl HCilOnb30BilTb TOT <!JaKT, 4TO <p(t) <( 0 Ha BCeH OCH.
Mbr ,li.OKaMeM neMMy.
219
JI eMMa VI. EcJIH <p(t) y,ll,oBJieTaopHeT ypaBHeHHIO (5,2) Ha aceH:
oc11, TO a6cJJ,HCCbi 01 11 az HBJIHIOTCH aKTHBH'biMH HYJIHMH a ( z). EcJIH ({J ( t)
y,li,OBJieTBopHeT (5,2) Ha npasoH: noJiynpHMOH t > t0 , TO az HBJIHeTCH aKTHB
HhTM HyJieM a ( z), a ecJIH <p ( t) y,li,OBJieTBopHeT (5,2) Ha JieaoH: noJiynpHMOH,
TO 0! HBJIHeTCH aKTHBHbiM HyJieM 0 (z).
n 0 K a 3 aTe JI b C T B 0. PaCCMOTpHM cnepBa CJiyl!aH, CBH33HHblH C npa
BOH noJiynpHMOH t > to > 0. B 3TOM cJiy<Iae HMeeM cooTHOllleHHe ( 12, I)
11 ( 12,2). noKa.IKeM, l!TO oz 3KTHBHb!H HYJib a (z). nycTb ·3TO He TaK. Tor,ll,a
cymeCTBYIOT ,li,Ba KOMnJieKCHhiX HYJIH fh + i'C 11 p-i" TaKHe, l!TO npH ,li,OCTa
TO'IHO MaJIOM E! > 0 B TIOJIYTIJIOCKOCTH X > f£ + C! HeT peaJibHb!X 3KTHBHb!X
HyJieil a (z). OpoBe,ll,eM qepe3 3TH ,li,Ba HYJIH OKPYR<HOCTh c JJ,eiiTpoM a Ha
peaJihHOH oc11, 11 6y,ll,eM saprmpoaaTb ee pa,li,Hyc. Op11 ,li,OcTaTOl!HO 6oJihlliOM
pa,li,Hyce BHYTPH 3Tofr OKpy)KHOCTH 11 Ha Hei1 He 6y,ll,eT peaJihHhTX HyJieif,
a JJ,eHTP ee 6y,ll,eT npaBee oz. Cpe,li,H Bcex Ko~mJieKCHhiX HyJJeH: a (z) BHYTPH
:noli oKpy.IKHOCTli 11 H3 Heif OTMeTHM 6JIH.IKaHlliHe K JJ,eHTPY OKPY.IKHOCTH a
H cpe,li,H HHX - napy conpH.IKeHHblX HyJieH C MHHHM3JlbHb!MH no MO,li,YJIIO
op,li,HHaTaMH. Opose.neM q e pe3 3TH HYJIH OKPY.IKHOCTh c JJ,eHTpoM a. BHyTpH
Hee He 6y,ll,eT HyJieif a (z). EcJIH HX He 6y,ll,eT 11 Ha OKPY.IKHOCTH, 6y,ll,eM
C'IHTaTb npoJJ,eCC 3aKOHl!eHHbiM. ECJIH OHH TaM npHCYTCTBYIOT, TO ,ll,a,li,HM
pa,li,Hycy oKpyR<HOCTH, npoxo,li,Hll\eH: qepe3 HalliH ,li,Ba HYJ1H MaJiyiO sapHa
JJ,HIO, npH KOTopoil Ha oKpyR<HOCTH 11 BHYTPH H ee y)Ke He oKa.N<eTCH HHhiX
HYJieif.
TaK Mbi noJiy'IHM oKpy.IKHOCTh c JJ,eHTpoM A> oz. Op11 3TOM 6y,ll,eT
•cymeCTBOB3Tb ,li,Ba, 11 TOJihKO ,li,Ba, aKTHBHhiX HYJIH q0 =ft 0 +ho H (?o=ft0 -h,
6JJH.IKaHlliHX K A.
Tenepb o6paTI1MCH K ( 12,1). OpaBbie CTeHKH KOHTypoB npoBe,ll,eM npH
o2 +A
a6cJJ,Hcce - 2- Op11 n-'? = pH,ll, (12, 1) 6y,ll,eT cxo,li,HTCH paBHOMepHo
B Ka)K,li,OM <i>HKCHpOB311HO.M cerMeHTe noJiynpHMOH t > 0.
§ 14.
Pa3o6beM aKTHBHbie HYJIH a (z) Ha Tpli KJiacca: 'llo - HYJIH eo li Q~,
6JIH)K3HlliHe K A, c pacCTOHHHeM OT A do= [ (!0 - A [ = I (I-;;- Af; 'll0 -
HYJIH, JJe.IKall(He BI-IYTPH TaKHX KOHTypoB L"' Bee TO'IKH KOTOpbiX Haxo,li,HTCH
Ha paCCTOHHHH OT A, npeBbJlllaiOmeM .2(do + 1); 'll1 - ocTaJibHbie HYJJH.
I1x 6y,ll,eT KOHe'IHOe l!HCJIO, H BCe OHH 6y,ll,yT JJe.IKaTb B <i>HKCHpOBaHHOM
Kpyre C I~eHTpOM A H Ha paCCTOHHHli OT A, npeBbilllaiOU(eM do.
0603 H3'IHM Tenepb
00
g(t) = ~ f/J1 ('C)e-Md~;
,(12,1) 11 (12,2) nepenHllleM
~ 1 gJ etoz E(z) g(t) =- ~. et<z-A) dz.
2m (z-A)a(z)
(nJ
( 14, 1)
Ln
220
(J2+A
OpaBbie CTeHKH KOHTypoB Ln HMe!OT a6Cli.HCCY - 2- , H BBHJl.Y CKa-
3a~IHOrO Bbiille 0 CXO)l.HMOCTH pH)l.a H TOrO, 'ITO
MOMHO rro'lJieHHO rrpoHHTerpupoBaTb pHJl. OT t > 0 Jl.O oo. OoJiy'lHM
00
(' ( )d - ~ 1 tfi t(z-A) et.,z. E(z)
) g t t - {:;r 2ni ;) e (z- A)2 u(z)"
t Ln
HoBbiH pHJl. 6y)l.eT yMe a6coJIIOTHO CXOJI.HIII.HMCH. .3Ty orrepaumo MO)KHO
rroBTOpHTb. Bse)l.eM orrepaTop
00 00
Lr(g) = ~ df. • • ~ g(f)df. (14,2)
t t
'-_-'
r paa
(14,3)
OpH r > 2 pHJl. 6y)l.eT a6coJIIOTHO H paBHOMepHo cxo)l.HTbCH npu t > 0.
§ 15.
PaccMurpHM HYJIH MHOMeCTsa ~2 • CooTBeTCTBYIOIII.He HM 'lJieHbi pH,ll.a
( 14,3) 6y)l.yT HMeTb ITO a6COJIIOTHOH BeJIH'lHHe Oll.eHKY
+ "z-A 1
Be 2 M (15,1} • n • L:Jr'
ll
r)l.e Mn=sup I ~(~ll If L111 =inf lz-AI Ha KOHType L,.
no ITOCTpoeHHIO KmnypoB Ln HMeeM Mn < CI4; )l.aJiee, BBH)l.Y TOro, 'ITO
paccMaTpHBaiOTCH HYJIH H3 ~2 , HMeeM
L1n > 2 (do+ 1) (n <;Cis)
BBHJl.Y :nora KOHTYPbi, cooTBeTCTBYIOIII.He HYJIHM H3 ~2, )l.a)l.yT B ( 14,3)
cyMMY C Oll.eHKOH
(
()2- .. /1) exp t-2-
B·(2(d0+1))r+1" (15,2)
Terrepb rrepei1)l.eM K H3Y'INIHIO 'laCTH cyMMbi ( 14,3), co3)l.asaeMoi1 HY
JIHMH E '2{1• l1x 6y)l.eT cpHKCHpOBaHHOe 'IHCJIO, If Mbl MOMeM rrepeHyMepo
BaTb HX (6e3 y'leTa KpaTHOCTH): (?I, (/2, .. . , (!k,· J\.aM)l.Oe (!j (j <; k2) 06Be)l.eM
oKpyMHOCTb!O CQi pa,UHyca, CTOJib MaJioro, 'ITO oHa 6y ,UeT 3aKJIIO'laTh
Jllflllb O)l.HH HYJib l! paCCTOHHHe BCeX TO'leK C~i OT TO'IKH Z = A 6y)l.eT
221
> do+~. r.n;e TJ=inf{(~i-AI-I£?0-A I }. Mhl IIOJiyqHM infla(z)l =
= E"' > 0 no JieMMe III. floJiaraH
1 rh et<z- A). etozE(z)
D~/f) = 2ni 'j' (z-A)'41 a(z) dz,
c~j
a,-A
1 - 2-
1 D (f) I= B e ~j "' _( ___ '1_)-,.+.,-:;1 '
do+2
(15,3)
r.n;e B'1 qJicJio, orpaHH<IeHHOe rrpH 'Y'J cpHKCHpoBaHHOM. BBHAY ·::nora ou.eHKa
qJieHOB, BHOCHMbiX B CYMMY ( 14,3) HYJIHMH H3 'il(l' 6y.n;eT
(15,4)
§ 16.
OcTaeTCH H3YqHTb rrpHJIO)KeHHe orrepaTopa L, K BbJqeTaM B ITOJIIOCax eo
e1o• E (z)
H .C!o cpyHKUHH - z _A 0 (z)
e 1 leo-A) • P0 (f)+ e t!eo- A) P0 (f) ( 16,1)
P0 (/) = a0t• + a1f"- 1 + · · ·+a.; a0 =/= 0, ,, > O.
"' 00
L, (e4 "1 • P0 (f))= ~ df ... ~ e4 "1 Po (t) dt.
t t
flo nocTpoeHHIO A, Rd < 0. floaToMy
L, (edt) = ( -1)' e;;t .
naJiee, )J.JIH n > 0 I.l,eJIOrO HaXO)J.HM
fn-m)
X r(r+l) ... (r+m-1) J+»• .
floJIO)KHM crrepBa, qTo Y = 0 (TaK 6y.n.eT, eCJIH Qo H eo
HYJIH a (z), HO, p a3yMeeTCH, He TOJibKO B aToM CJiyqae).
222
(16,2)
(16,3)
IIpOCTb!e
B yKa3aHHOM c.nyqae rrprwo:tKemie orrepaTopa L, K ( 16,1) .n.acT
2R( exp(d·t-ir(ex+n)))=
ao dr+!
0
_ 2 . (I 1exp[(d0 cosex+id0 sinex)t-ir(ex+n)]')
- R aa dr-l-1 •
•
(16,4)
Dpl:l .n.aHHbiX r l:l t ou.eHKH (15,2) n (15,4) .n.a.n.yT B cyMMe BeJinqnHy, He
rrpeBOCXO.li,HW:YIO
t~z-A
e 2 c --- (16,5)
15 ( dJ + ]f+l .
3aMeTnM, qTo sin a> 0, cos a< 0 B ( 16,4) no rrocTpoeHn!O. Bhr6epeM
Tenepb r CTOJib 60Jibilll:IM, qTo6b! 6b!JIO
( 2n u2-A) ex ·--
1 ( 4n ) p dG sin ex 2
2 I ao I d~ exp do cos ex • do sin ex > 2Cl5 (do+ ~r+l (16,6)
3acj:JIIKCHpyeM 3TO r. OocJie .3TOrO Bbi6epeM f = f1 B cerMeHTe
4n J
r 2n
d0 sin ex '
d . T3K, qTO
0 Sln o:
exp [ ( d 0 cos a + ido sin a) t - i (a + n) r J = exp (do cos ex· t).
Tor;~a (16,6) HaBepHoe 6y.n.yT BbiiTOJIHeHhi H noJiyqHM
1 ( 4n ) L, (g(t)) >I ao I d'+I exp d 0 cos ex· d--.- > 0
0 stn ex
(16,7)
npn t = t 1. Ho Mbi HMeJil:l <Cf'l (1:) < 0 npl:l t > 0. 0TCIO.ll,a g(t) < 0
n L, (g) < 0 npn Bcex t. DoJiyq¥mrueecH npoTnBopeqne .n.oKa3biBaeT, qTo
o2 - aKTHBHbiH HYJib a (z).
DycTh Tencpb JJ =/= 0, TaK qTo JJ > 1. ( 16,3) .n.aeT Tor.n.a
L,(e"Jot • P.(t)) = (-1}-r(r+l) ... (r+y-1) ;::v(l+B(.d)l~lv)a0 (-1)",
r.n.c B - orpamrqeHHoe qnCJIO npn cj:mKcnpoBaHHOM Ll, t > ~:0 > 0. CHoBa
Bbi6l:lpaH cnepBa no.n.xo.n.mr~ee, .n.ocTaToqHo 6oJibll!Oe r, qmKcnpyeM ero, l:l, Bbi·
I 2n 4n J 6npaH rro.n.xo.n.mu.ee t B cerMeHTe d . d . , rroJiyql:IM, qTo npn TaKOM
0 Sln ~ ' 0 Sln ex
t = t1, L, (g) > 0. CHoBa rroJiyqaeTCH rrpoTnBopeqne, .n.oKa3hiBaiOmee, qTo
a, :-- aKTHBHbiH HYJib a (z).
§ 17.
B cJiyqae, Kor.n.a ypaBHeHne ( 5,2) y.n.oBJieTBOpHeTcH cpyHKU.HeH cp (t) Ha
TIOJiynpHMOH t < to < 0, TO, .ll,eHCTBYH C COOTHOWeHHHMl:l ( 13,1) l:l ( 13,2)
T3K :>Ke, K3K Mbl .ll,eHCTBOB3Jil:l C (12,1) l:l (12,2), ITOJiyqnM, qTQ a6Cll,l:ICC3
o 1 5IBJI5ICTC5l 3KTl:IBHb!M HyJieM a ( z) (.li.JIH Bb!BO.ll,a y.n.o6HO ITOJIO:tKHTb t = - u;
223
u>O, ~:p(-u) =f(u)). ECJH! qJ(t) y,UOBJieTBopHeT (5,2) Ha BCeH OCI1, TO 01
11 a2 - aKTI1BHbie HYJII1 a (z), II JieMMa VI ,UoKa3aHa rroJIHOCTbiO.
flycTb Terrepb II MeeT MeCTO ( 12,1) II ITYCTb
res.,, ( e<t+to)z ... ~(~~) = e"'1 • P 2 (f), (17, I)
r,Ue P(t) = aot' + alr-t + ... + av - ITO.Jli1HOM, 3 aBI1CHll.LHH, Boo6me
rOBOpH, OT fo.
Mbi XOTIIM ,UOKa3aTb, IJTO a 0 < 0, II BbiCKa3aTb aHaJiorHIIHOe YTBepm,Ue
HI1e OTHOCIITe.JlbHO o 1 rrp11 YCJIOBHI1 ( 13,1), T. e. CcpOpMyJI11pOB3Tb JieMMY:
§ 18.
J1 e M M a VII, flpii Tex me ycJIOBHHX, rrpH KOTOpbiX ,UoKa3aHo cyme
CTBOBaHHe aKTliBHbiX HyJieH 02 li.Jlii or, BbiiJeTbl IIMeiOT Bli,ll,
1<1 ·
e 1 • P/t) (j= 1, 2),
r,Ue P2(t) rroJII1HOM c OTpHuaTeJibHbiM CTaprnHM K03cpcpHUHeHTOM,
a Pr (t) - ITO.JlliHOM CTeiTeHII m ;> 0, CTaplll11H K03cpcp11U11eHT KOTOpOrO IIMeeT
3HaK (- 1)m+ I .
.ll o K a 3 aTe JIb c T B o . PaccMOTpHM crrepBa cJiyqaif Ha.'JHqi1H ( 12,1)
11 cymecTBOBaHHe aKTIIBHoro HYJIH o2•
Bbi6epeM A B TOIIHOCTH TaKIIM me, KaK rrpH ,U0~3aTeJibCTBe JieMMbi VI
B § 13-16, 11 coxpaHIIM Te me 0603HaiJeHHH, (!O II (! 0 6y,UyT 6JIIImaHlllHMli
K IJIICJIY A cpe,UH KOMIT.JleKCHb!X aKTliBH'biX HyJieH o (z), HO 11X paCCTOHHHe
d0 = 1(10 -AI 6y,UeT 60Jibllle paccTOHHHH Oo=A-o2 :
do> Oo. ( 18, 1)
Terrepb o6paTHMCH K BbipameHHIO
"' 00
~ e-41 e" P2(t)dt > ~0 ~ e ·
0TCIO,Ua COr JiaCHO § 16
~ a,-A t
Ao e-"ot e 2
Lr(g) > 2oo ~ + B (--)r+l +
0 do+~
224
r.rr.e c~. II c~. MaJib!e KPY.>KKII BOKpyr (!o II (!o· Ilpii .[I,OeTaToqHoi:i MaJioeTH
3THX Kpy)!<KOB eOOTBeTeTBYIOlll.He HHTerpaJibl IJOJiyqaJOT OIJ:eHKY
a,-d1
(! 2
(18,3)
Bhi6epeM t = t 1, 11, 3a<fJIIKeHposas ero, sw6epeM r eTOJih 6oJihlliiiM,
qTo
1 A 0 e-"·1
B ¢opMy.'le (18,2) 2 26 •. Jr- 6oJihWe seex OeTaJibHb!X qJieHOB .
( 18,2). Torpa noJiyqHM
( Ao cr 1
L,. g) > 4 e- ul, • cl" > 0,
0
qTo HeB03MO.>KHO . .3TIIM .Ll.OKa3aHa nepsaH qaeTh JieMMhi VII.
,ll.JIH .rr.oKa3aTeJiheTBa BTopoH: qaen1 6epeM (13, I) B ¢opMe
I
g (f)= ( m (f)e-Md't= ~ rh e(t+tolz-d· t E(z)dz + R (f) (18,4}
1 ) -rl ~ ';) (z-A) <J(z) n
-oo m == -n Lm
IR (t) i =B(c)e(6,-A - •)t. A <:rh (18,5)
" n (t) '
(3peeh t0 <fJIIKeiiposaHo). ct>opMyJia ro.[I,Ha np« t < 0. Bse.[I,H onepaTop
l t
Lr (gl) = ~ df · • · ~ gl (t)df
-oo -<:» - r pa3
H Bbi6IIpa5I A < 0"1 , ,li.OeTaToqHo 6JIH3KHM K o 1, .[I,OKa.>KeM H BTopyiO qaeTb
JieMM'bl.
§ 19 .
.n e M M a VIII. 4IICJIO
O"j > 0. ( 19, I)
D. o K a 3 aTe JI he T B o. Mbi HenoJih3yeM TOT ¢aKT, qTo .cp (t) ---'? 0 npn
t---'?- co. TiycTb crrepsn. 01 < 0. <PHKCHpyeM A= 1, 1. 01, < 0. 06pamaHcb
K (17,5) H (17,6), B03bMeM e B (17,6) CTOJ!b MaJibiM, qTQ o 1 - A-t:> 0.
B ( 17,5) npose.[l,eM JieBhie CTeHKH KOHTypos Lm JI.JIH I m I > I npH a6euHcee
a 1 - SI, r.[l,e e1 eTOJ!b MaJIO, qTQ paeeTOHHHe IIOJiyqeHHbiX CTeHOK, a CT3JIO
6b!Tb, H seex ToqeK KOHTypos Ln OT qiieJia z = A 6ypeT > bo + TJ, r.[l,e
IAI ai+A
<50 = o 1 -A= lO II 7J > 0. IlpH 3TOM nyeTh Ot - e1 > - 2- .
,ll.aJiee, see HYJIH Qi BHYTPH KOHTypa Lo, 3a IIeKJIJOqeHHeM HYJIH <Jt, 3a
KJIJOqaeM B KPY)KKH eTOJib MaJihre, qTo paeeTOHHHe seex ToqeK HX oKpy.>K
HoeTeif OT Z =A 6ypeT > Oo + 7] H Bee OHH Jie)!<aT B IJOJIYITJIOeKOCTH
a1 +A
x <:: -2-.
22fi
Tenepb paccMoTpHM np11 t < 0 onepaTop
t t
Lr(g1) = ~ df • • • ~ df,
-oo -oo --.-.-
r pa3
Ero npHMeHeHI1e K npaBoii: qacTH ( 17,5) ·)].acT
t <11-A
Lr(g1)=Lr( ~ e-Ate"'1 P1 (t)dt) + B(o:~:+l' (19,2)
-oo
llaJiee, corJiaCHO JieMMe VII, P 1 (t) = A 0tm + ... + Am, r)l.e Ao (- 1) m+l > 0.
llaJiee, np11 t < - T1 < 0 P1 (t) coxpaHHeT cBoii: 3HaK, 11 I P1 (t) I> I ~0 I .
BB11.Il.Y 3Toro np11 t <- T1
t t
I Lr ~ e-·11 e "'1 p1 (t) dt I = Lr ( ~ e-Ate u,t I pl (t) I dt) >
-oo -oo
a1- A
1 I Ao I e"o·t e- 2- t
2 2~>B(oo+11Y+1 ' (19,3)
B 113 npaBoii: qacTI1 ( 19,2) - orpaHI1qeHHOe qi1CJIO; I B I< cl 6· BBI1.Il.Y
3Toro noJiyqaeM np11 t =- T1, r > ro(TI)
~t
IL( I IAolerYot _ IAol~~ r+l (19,5)
r gl) > 4 o~+l - 4 IA I"+l 10 .
C .LI.pyroii: cTopOHhi, Mhi 11MeeM, B BI1.Il.Y Toro, qTo I -g;1 (t) I < 1 np11 t ,;;;; - T1,
t t
ILr(gl)l< ~ df. · · ~ dte-M I<pJr) l d'L'<;:
-oo
(19,6)
Dp11 .LI.OCTaToqHo 6oJihlllOM r ( 19,5) 6y.LI.eT 6oJihllle ( 19,6), qTo np11BO
.Il.11T K npOTI1BOpeql1~ .
I1TaK, o 1 > 0. JloKa:>KeM, qTQ CJiyqaif o 1 = 0 T8K:>Ke llp11BO)l.I1T K llpOTI1-
BOpeql1~.
no 38)1.8HHOMY eo Bhi6epeM Tl > 100 TaKoe, qTo I Cf'l (1:) I <eo np11
?: < - rj. Bhr6epeM A< 0; I AI < 0,001 11 38cp11KC11pyeM. I A
llaJiee, np11BO.LI.I1M paccy:>K.LI.eHI1H, KaK 11 Bhillle, cqHTaH, qTo I P1(t) I> --0
2
np11 t <- Tl 11 qTo PI (t) coxpaHHeT 3HaK Ha noJiynpHMOH t <- Tl.
DoJiaraH Oo = 01 - A= I A I, noJiyqi1M, KaK B ( 19,5),
I Ao I erYot. I Lr (gi) I> 4 d~+l' f=- T1• (19,7}
226
C ,ll,pyroii: CTOpOHbi, rrpn t =- T1.
JLr(Yl)l< f dt. · · f e-A< I(]!l(r)ld'£" -<. coLr(e-M) =e0 et1ot npH t=-T1 • (19,8) J J cJ~+l
-IX- -00
EcJIH .:o 6biJIO Bhr6pauo MeHblile I ~0 I, TO ( 19,8) BCTyrraeT B rrpoTHBopetme
C ( 19,7), 4TO H ,ll,OKa3biBaeT JieMMy.
§ 20.
Mbr 6y ,ll,eM rrpo,ll,OJI:tKaTb 3aHHMaTbC5I KJiaccoM <PYHKU.HH g; (t), y,ll,OBJie
TBopmomnx ( 5,2) Ha ITOJiyrrp5IMOH t < to, H pa3 biCKHBaTb cpe,ll,H HHX TaKHe,
KOTOpbre 5IBJI5IIOTC5I JIOrapH<t>MaMH X. <P· CHMMeTpH4eCKHX CJiy4aHHbiX BeJIH-
4HH. Mbr BH,ll,eJIH, qyo ,ll,JIH HHX 01 > 0 H rroToMy B pa3Jio:tKeHHH ( 17,5)
MO:tKHO B35ITb A = 0 H HaiTH CaTb
I R (t) I= B(.:) eta,-•)t. t < 0
" n l tl ' ·
(20, 1)
(20,2)
Mbr HMeJIH
r,ll,e
u=e", Uo=e1•; O<u.-<1, o<u~ < l.
<l>yHKU.HH lii(uu0 ), a 3Ha4HT H P(u) , 5IBJI5IeTC5I Jiorapn<t>MoM x. <P- llpH TaKOH
3aMeHe rrepeMeHHbiX ( 20,1) H (20,2) rr e peiTHiliYTC5I TaK:
u
~ P(uu0)du= ~ _1_ tt;(uuo)zE(z)dz+R (u)
u ~ 2ni 'j" za(z) " '
0 lm l-> n
Lm
(20,3)
R u - B(s)
I n ( ) I - n Jln u I uo·-•; (20,4)
JieBbie CTe HKH KOHTypoB Lm C4HTaeM rrpoBe,ll,eHHbiMH rrpH a6cu.ncce c 1 > 0.
,UJIH ,ll,aJibHeiiwero uaM HY:tKHO nccJie,ll,oBaTb BJIHHHHe ua pacrroJio:tKe HHe
HYJIH a 1, COBITa,ll,aiO!ll,e ro C HH:tKHeH rpaHblO a6CU.HCC BCeX aKTHBHblX HyJieH
o(z), uaJIH4H5I rrpoH3BO,ll,HbiX P(u) rrpH u=O.
BBe,ll,eM eme o6o3Ha4eHHe o 3 H a4 ,li,JIH HH:tKHeii H Bepxueii rpaHH
a6CU.HCC BCeX aKTHBHb!X K 0 M IT JI e K C H bl X HyJieH a (z).
TaKHM o6pa3oM,
(20,5)
llepeuyMepyeM see aKTHBHbre peaJibHbie HYJIH o (z); rrycTb sTo 6y,ll,yT
s 1 = a 1, s2, ... , sP < a2. (20,6)
3HaK paBeHCTBa ,ll,Jl 5I Sp 6y,ll,eT, B qaCTHOCTH, TOr,ll,a, KOf,ll,a cp(t) y,ll,OBJieTBO
p5IeT (5,2) Ha KaKOH-JIH60 rrpaBOH ITOJiyrrpHMOH.
227
Bee KopHH si Jie)KaT BHYTPH KOHTypa La. 0603HaqHJvl cyMMY cooTBeT
CTBYIOlll,HX IIM BbJqeToB ,11}15! cpyHKIJ.HII (uu0 )zE((z) qepe3 S(u):
a Z)
S(u) = ~ res5 . (<uu0yE(z)). (20,7)
si 1 a(z)
ll
T (J! > 0, \ S(uu) du Or,ll,a ITOJiyqiiM: BBH,ll,Y J
0
ll
~ S (u) _ _ ~ , ( , E(z)) --du-S1 (u)- ~res5 . (uzzo) ~().
u 8 . ' za z
0 J
t20,8)
BHoCH 3TO B (20,3), rroJiyqiiM
ll
( du 1 ~' A;: (uu 0)z E(z)
.) P(uu0) u = 8 1 (u)+ 2ni ~ ~ za(z) dz+ Rn(u),
0 lm l <nLm
(20,9)
r,ll,e nnpiix npH 3HaKe cyMMbi rroKa3biBaeT, qTo B ~ He HY.IKHO yqiiTbiBaTI>
BbJqeTOB B peaJibHbiX KOpHHX Sj . L
3aMeTIIM T3K./Ke, qTO p33JIO./KeHIIe TIII13 (20,9) ~,ll,HHCTBeHHO, K3K B 3TOM
JierKO y6e,ll,HTbCH.
§ 21.
,ll.JIH ,ll,aJibHei:fwero Heo6xo,ll,HMO yKa3aTb HOBYIO ou.eHKY ,ll,JIH Rn(u),
Bbipa.tKeHHYIO qepe3 a6cu.IIccy a 3•
Herrocpe,ll,CTBeHHO HCHO, qTo B cyMMe Z' II3 (20,9) MO)!<HO rrpoBecnJ
JieBble CTeHKH BCeX KOHTypoB L.n I1pH 30CIJ.IICCe ·0 3 - c rrpii CKOJ!b-yro,ll,HO
M3JIOM cpiiKCHpOB3HHOM E.
B03BpamaHCb OT rrepeMeHHoi:f u K rrepeMeHHoi:f t = In u II rrpoBe,ll,H pac
cy.tK,ll,etmH, ITOJIHOCTbiO aHaJIOrHqHbie paccy.tK,ll,eHIIHM § 11, H3H,ll,eM, qTO
B ou.eHKe ,ll,JIH R11 (u) MO)KHO 3aMeHIITb 01 Ha 03, TaK qTo rroJiyqHTCH
(21 '1)
Mbi ,ll,OKa)KeM Terrepb cJie,ll,yiOmyiO JieMMy.
JI eMMa IX. Dycrb cpyHKU.IIH P(u) IIMeeT n-10 rrpoii3BO,ll,HYIO P<nl(zz),
0 < u < u0 , npiiqeM rrpii u --:. + 0, p<n) (u) ~ lJf~l (0), T. e. cymecrByer
npe,ll,eJI crrpaBa.
Tor,ll,a HMeer Mecro HepaBeHCTBO
03;;;;, n.
KpoMe roro, re peaJibHbie aKTIIBHbie HYJIH o (z), KOTOpbie MeHbrne n
si< n,
(21 ,2)
CYTb U.eJibie ITOJIO./KHTeJibHbie qiiCJia, II TOqKII Sj CYTb rrpOCTble ITOJIIOCbl
QlYHKU.IIH
228
(uu<•)= E(z)
a(z)
Bsn.uy HeB03MO:>KHOCTH ITO'IJICHHO .un¢¢epeHI..I,HpoBaTb pHA (20,9), rrpn
XO,UHTCH rrpH6eraTb K CJIO:>KHOMY OKOJlbHOMY ITYTH ,UJl5.1 ,UOK333TCJlbCTBa ::noi1
JICMMbl.
Crrepsa .uoKa:tKeM TaKyiO scrroMoraTeJibHYIO JieMMy.
DycTb g(u) 3a,UaHa B [0,1 ]; g(O) = 0 II g<nJ (u) cymeCTsyeT II Herrpe
pbiBHa B [0, 1 ]. CocTaBHM Bbrpa:rKeHHe
1 1
( ·) _ ~ dar ~ dar-1 X U,I- - --- ...
<Xr <Xr-1
0 0
1
\ dal g (ala2 ... arll).
~ al
0
Tor.ua, ecJIH n > 1, (21,3) HMeeT CMbiCJI, II
1 I
(n)( ) - c d \ d n-1 n-1 ,,_1 g(n) ( u) X u, r - J a1 ... J a.a, a, ... a, a1a 2 ... a, .
0 0
(21,3)
(2 1,4
Jl.oKaJJ<CM, 'ITO ( 21 ,3) HMCCT CMbiCJl. 3aMeTHM, 'ITO B CHJIY cymeCTBOB3H!iS!
II HerrpepbiBHOCTH g' ( U) H TOfO, 'ITO g ( 0) = 0, ¢YHKIJ,H5.1 g (zz) Herrpepb!BHa
u u
B [0, 1 ], H 11HTerpaJI
BCHKOM V 1 > 0
~ g~u) du cymecTsyeT. BBH.UY ::noro HMeeM: npu
0
=lim c'g(z)dz= f'g(z)dz=gl(vl).
<--++0 J z J z
i'Vt 0
DoJIO:>KHM vr = a2v2; HMeeM g: (vr) = g(v1) cymecTByeT 11 HerrpepbiBHa
vl
n [0,1], H g 1 (0) = 0. Bsn.uy 3Toro rro-rrpe,Uhr.UymeMy g2 (0)=0 II g;(v2 )
cymeCTsyeT H HerrpephrBHa. Dpo.uoJJJJ<aH .uaJiee 3TO paccy:>K,UeH11e, .uoKa:tKeM
Hallie YTBep:>K,UeH11e.
Terrepb ,UOK3)KCM, 'ITO
1
~ a7-1 g (n ) (a1 V1) da1 = g~') (vl).
0
Dpn n = 1 HY:>KHO .uoKa3aTb
1
~ '( )d _ '() g(vl) g a1v1 a1-g, V1 = --
vl
0
1
(vi> 0). (21,5)
3TO paBHOCl1•1bHO paBeHCTBY ~ g' ( (!.l v1 ) da 1V1 = g(V1 ), KOTopoe Herrocpe,UCTBCHHO
rrposepHeTCH. 0
.UaJiee, B ( 21,5) .uorrycTIIMo .uaJJhHei1!llee .un¢¢epeHunposamte no,n: 3Ha-
KOM 1'fHTerpaJJa, ecJJH da g' (au1) = v1g" (av1) cymecTsyeT H HenpepbiBHa
al
ITO J{lJ ll Vr, 'ITO COBITa,UaeT C cymeCTBOB3HHCM 11 Herrpepb!BHOCThiO g"(u)
229
npu ue [0,1]0 I1epexo,n: K (21,4) coseprnaeTcH rrpu n = 1 HH,n:yKurrerr rro r,
H 3aTeM 3aKOHHbiM ,n:mpcpepeHUHpOBaHIIeM (21 ,4) rrpu n = 1 no,n: 3HaKOM
HHTerpaJiao
06o3HaquM IJI(uu0 ) = iJI1 (u). Qqesu,n:Ho, BBu,n:y Uo =I= 0 yCJioBH5I ,n:ucj:>-
cj:>epeHUHPYeMocTu W(u), ,n:aHHbre B cj:>opMyJmpoBKe TeopeMbr, 6y,n:yT BbiiTOJI
H5ITbC5I u ,n:JI5I 'P1 ( u) B cerMeHTe .[0, 1]0 L{aJiee, W1 (0) = iJI(O) = Oo Ha ocHo-
1 u
BaHHH CKa3aHHOrO Bb!llie HMeCM \ lpl (au) da= \ ipl (u) du. 0603HaqHM 3TOT
) (t ) u
0 0
HHTerpaJI qepe3 IJI2 (u). Tor,n:a ,n:JI5I iJI1 (u) 6y,n:eT sepHo pa3JIO)KeHue (20,9) 0
L{aJiee, eCJili z~, Z2, 0 0 0' zl' - JII06biC cj:>HKCHpOBaHHbie KOMITJieKCHbiC qucJia;
{11, {12, o o o, {J qifCJia li3 (0,1), TO HMCCM ,«
fzjw2(flju) =£ziSdfJp)+ 2~i ~· cj' (f~fl;) X
]=1 ]=1 / m I <; n Lm }=!
E(z) M
X (mz0 )= - (- )dz + ~ Z.Rn(f3ou).
Z(J Z ~ 1 I
j=l
§ 220
M
(21,6)
0603Haqa5I ll(z) = ~ ~f3;, Bb!pa3HM ITO,QliHTerpaJ1bHYIO cpyHKUli!O (21,6)
B Bli,QC i=l
q_j(z) = IT(z) (uu.)z E((z)) 0 (22,1)
Z(J Z
Mbr XOTHM ,n:oKa3aTb TaKyiO JieMMy: rrycTb Q0 , (!J, • . ·o, 1?1< - KaKHe-Jiu6o
aKTHBHbie HYJIH (J (z), KOTOpbie 5IBJI5IIOTC5I ITOJIIOCaMH ~(~1 COOTBeTCTBeHHbiX
nop5I,n:KOB lo, l~, 0 o 0, lko Tor,n:a MO)KHO rro,n:o6paTb fh, /h 0 0 0, fJk, e (0,1)
H Z r, Z2, 0 0 0 , Zkt, rr, cTaJio 6biTb, II (z) Ta:c, trT::> (?, Q 2 , o .• , (Jk He 6y,n:yT
ITOJIIOCaMH f]j(z), a l.]o 6y,n:eT noJIIOCOM nop5I,n:Ka e0 , r,n:e eo < lo - 3a,n:aHHOe
ueJIOC qifCJIOo
L{JIH ,n:oKa3aTeJibCTBa Bbr6epeM qucJio ~ > 0, CTOJib MaJioe, qTo
e-~~j, =f= e-;~j, npH it =f= h, H COCTaBHM CHCTeMy ypaBHeHHH
k=l
~ Zm e-mli~j =or (22,2)
m=O
3,n:ecb j = 1, 2, 0 • 0 , k, oo = 1, oi = 0 (j > 0). CuCTeMa pa3pernuMa, TaK
KaK ee ,n:eTepMHHaHT eCTb ,n:eTepMHHaHT BaH,n:epMoH,n:a =1=0.
k-1
I1oJiara5I ll1 (z) = ~ Zme-mliz, Hali,n:eM, qTo ecJIH /=sup(/1 ,12 , ••• , !")
m=O
(Bee 3TH KpaTHOCTH orpaHuqeHbl KOHCTaHTOH), TO (II1 (z) )1 = II2 (z) 6yJI.eT
HMeTh IJi HYJIHMH KpaTHOCTH, no Kpa:iiHeli Mepe, lj npu f>-1, H Il2 (Q0 ) =I= 0.
ECJIH Terrepb e0 Ha3HaqeHo < l0 , TO BBe,n:eM eme cpyHKUHIO (e-z-e-~•) 1·-e•;
Tor,n:a II(z) = ll2 (z) (e-z-e-~o)'o-eo 6y,n:eT HMeTb (!>() HyJieM KpaTHOCTH
k,
p6BHO [0 - e0 • L{aJiee, II (z) MO)KHO H3o6pa3HTb B BH,n:e ll(z) = ~ Z/lj,
J=l
r,n:e 0 < flt < 1, H qepe3 Zi o603HaqeHbl HOBbie KOMITJICKCHbie qucJiao
Z30
§ 23.
Ha<meM c noKa3aTeJihCTBa Toro, 'ITO o3 > n. OycTb o3 < n. Torna Mhi
CMO)KeM H3HTH KOMTIJieKCHbiH aKTHBHbiH HYJib Qo = bo + h 0 TIO,ll, YCJIOBHHMH
bo = n- TJo, 'Y]o > 0; 03- bo < 0,01. (23, I)
06o3Hal!HM I ·Qo I = oo. OycTb KpaTHOCTh Q0 ecTh [0• Bhi6epeM Teneph K rro.u
_ 1 _ _ 1
yCJioBneM: K > 2 + oo + 2o~-bo . Bee HYJIH, KOTOpbie Haxo,ZJ,HTCH B rrpH
MoyroJibHHKe 01 < x < .a2 ; I y I < 2Koo, 3a HCKJnol!eHHeM HYJIH •Qo, rrepeHy
MepyeM, KaK Q1 , Q2 • • • (h,.
OoCTpOHM corJiaCHO rrpe,ZJ,bi,ZJ,YW:eMy TaKyiO II (z), 'ITO eo 6y.ueT rrpo
CThiM rroJIIOCOM (/j (z), a Q1, Q2, • •• , Qk, - He 6ynyT rroJIIOCaMn. To rna H3
(21,6) IIOJIY'IHM
E (z) k,
X (uu0 )z -( ) + ~ Z.Rn ({:1 u)
za z ~ 1 '
j = 1
(cyMMa conep)KamaH S1(u) HCl!e3aeT).
BBe,ZJ,eM Terrepb (r- I) -KpaTHhiH HHTerpaJI
HMeeM 'P1 (0) =0 B CHJIY (21,1). €orJiaCHO cKa3aHHOMY Bhiiiie
HMeeT cMhiCJI. BBH,ZJ,Y paBHOMepHoli cxo,ZJ,HMOCTH (23,2) rroJiy'IHM
1 z l?(z)
x(u)= 2ni ~ rh ll(z)(uuo) z'u(z)dz.
(m) ;j'
L,.
(23,2)·
(23,3)
(23,3)
(23,4)
DoJiyl!eHHbiH pHn 6yneT CXOJI.HThCH a6coJIIOTHO H paBHoMepHo rrpn r > 2.
EcJin r > n + 2, ero MO)KHO n pa3 rrol!JieHHO nn¢¢epeHIJ.HpoBaTb rro u ..
B 3THX yCJioBHHX HMeeM
.x<nl(u) = 2~i~ <j> ll(z)uz-nu~ z~~~)z(z-1) ... (z-n+l)dz.
(m) Lm
(23,5)·
OoJiaraH
w.(z) = ll(z)u~ ~~j z(z-1) ... (z-n+ 1),
(23,6)-
23E
CorJiaCHO (23,2) II § 21 6y)l.eM HMeTb
II
1 k, 1
.xCnl(u) = £; zj ~ dr:x.r-1· •.
0
\ d 8" n-1 n-1 n-1 ,Tr{n)(,q ) ) at j u a.' . • . a.r-1 "¥' I" jua., O:t • • • u,._l .
0
BBH)l.Y CKa3aHHOro BbiWe o zp~nl(u) II Toro, 4TO 4IICJia a, a 1, •• • , a,-1, /Ji
nee E (0,1), HMeeM
I X {ll) (u) I< c17 (23,7)
np11 BCeX Z > n + 2 II BCeX U E (0, 1).
Tenepb npe)l.IJO.TJOJKHM, 4TO o3 < n. Mbi )l.OJI)KHbi npHTTH K npoTHBO
pe4HIO.
Bee IJO.lJIOCbl ml (z)' KpOMe npOCTOro IJOJIIOCa (Jo, y)l.aJieHbl OT 0 Ha pac-
CTOHHHe > 2Koo, r)l.e
(23,8)
0Kpy)]{llM HX Kpy)]<K3Mll GQj CTOJib M3Jlb!Mll, 4TO T04KH HX KOHTypoB Jie:tKaT
B noJiynJIOCKOCTII x > o3 - 0,01 n paccToHHIIe IIX oT 0 6y)l.eT > Koo.
B bi4eT IJO)l.HHTerpaJibHOii <PYHKUHH ( 23,6) B (!o 6y)l.eT nMeTb BH)l.
(23,9)
r)l.e F0 - <PHKCHpOBaHHoe KOMIJJieKCHoe 4HCJIO. BBH)l.Y 3Toro (23,6) nepe
IJHIIIeM:
(23,10)
§ 24.
CorJiacuo npe)l.bi)l.YIUIIM cBe)l.eHIIHM o pacnoJio:tKeHHH f!i 11 JieMMe IV
(9, 1) IJOCJie)l.H5I5I CYMMa B (23, 1 0) M3:tKOpiipyeTC5I Bbipa:tKeHHeM
KOTOpoe BBII)l.y TOfO, 4TO 03- bo < 0,01, MeHbllle, 4eM
ubo-0,02-n
018 (KooY ·
noJ10:tKIIM u = (Koo)-'. Tor)l.a
232
I. zz~o-n I ubu-n r
Fo~·- = [F0 [• -~ = [Fo[· K ·
(24,1)
(:24,2)
JJ.anee,
ubo-0,02-n ubo-n u-0,02r
Cis (Koo)" = C1s -~ · ~ ·
BBH)l.y Toro, lJTO K > Oo,
u-o,o2
Kr
---'----------';:--- < o = ~ K- -3- < K- 3.
(Koo)o,o2r 0o,o2, (0 )o,o2r , ,.
Kr K ~ Ko,o2r K
no3TOMY (24,3) He npeBOeXO)l,HT
1'
CIS I Fo I f(" K 3 •
Op1-1 .uoeTaTOlJHO 6oJibWOM r 3TO BbipameHHe 6y.ueT
(24,3)
(24,4)
06paw.aHeb K (23,10) 1-1 ytJHTbJBaH (24,2) 1-1 (24,4), I-rali.ueM np1-1
U = (Ko0 )- r 1-1 Beex )l,OeTdTOlJHO 60JibWI-IX r
I x'"l (u) I > } I Fo I Kr.
Op1-1 .uoeTaTOlJHO 6oJibWOM r 3To BCTynaeT B npOTHBopetJHe e (23,7). 9To
II ,li,OK33b!B3eT nepByiO qaeTb JieMMbr:
(21 ,2)
LI,JIH .uoKa3aTeJibeTBa BTopoi\ qaeTl-1 JieMMbl - o peaJJbHb!X aKTI-IBHbiX HYJTHX
Mb! MO)!(eM 33MeTI-!Tb eJie,UyiOW.ee.
Een1-1 aKTI-IBHbrli peaJibHbiH HYJib sj y.uosJTeTsopHeT yeJTOBl-110 si < n
1-1 eeJJl-1 Sj He eeTb U.eJTOe 41-!eJTO, TO Mbl MO)!(eM nposeeTl-1 )l,JIH Hero )Ke pae
cym,UeHl-151, 4TO 1-1 )l.JTH HYJTH eo B § 22-24. Tor.ua noJTyYHTeH npoTHBopeYHe,
.UOK33b!BaiOIU.ee, 4TO si > n. EeJTH me sj - u,e;roe 4HeJTo, TO noeTpoeHHe
B006W,e HeB03MO)[<HO BeJTe,UeTBHe ·npHeyTeTBHSl B f/J1 (z) MH0)[{11TeJIH
z(z- 1) ... (z - n+ l), HMeiOw.ero npoeTbie HYJTH B u.eJT biX TOlJKax. 9To
OTBe 4aeT TOMy, 4TO ,li,JIH u,eJTOH
B [0, l ]. EeJT H )[<e epeAH TOlJeK
110p5I)l.Ka {i > 2 )l.JIH cpyHKLJ,Hl-1
eTeneHl-1 usi see npoH3BO,li,Hbie HenpepblBHbl
Sj < II BCTpeYaeTeH XOTH 6bl Oj~HH nOJTIOe
E(z)
uz --- TO MHO)!(IHeJib Z (z- I) ...
• a (z) '
(z - n + I) noHH3HT ero nopH.LlOi< JJHUib Ha I; .uaJJee no.u6opoM eooTBeT
eTBYIOUJ.ei\ cpyHKUI111 n (z) ero MO)!(HO C,UeJJ3Tb llOJJIOCOM nepBoro nopH,D,Ka
H npoBeeTH .UJJH Hero paeey)[{,UeHHH § 22- 24.
9nJM JJeMMa IX noJJHoeTbiO .uoKa3aHa.
§ 25.
OyeTb eyw.eCTsyeT HenpepblBHaH p<nJ (u) npH ue [O,l ], 1-1 nycTb n > y,
r.ue y- MaKeHMaJJbHbiH peaJJbHbiH HYJJb "(z) (6e3 Tpe6osaHHH aKTHBHOeTH).
,IJ.aJJee, nyeTb ypaBHeHHe (5,2) y.uosJJeTsopHeTeH Ha seei\ oeH. Tor.ua
HMeeM: o2 - aKTHBHb!H HYJJb o(z), 1-1 see aKTHBHbre HYJIH JJe)[{aT B [uh a2].
TaK KaK n > (52 11 o :~ ~ a2, TO KOMnJJeKeHbie aKTI-IBI!ble HYJIH 6y.ll.yT oTeyT
eTBOBaTb, a see peaJJbHbie aKTHBHbre HYJJH 6y.ll.yT ueJIOYHeJreHHbiMl-1 11 npo·
eTbiMH nomoeaMH !~;;.
t<. YKJWIII!CKltH M<neMa T. *Y!HtaJl, T . V, N~ :l 233
Ha Bceli rroJiyrrpHMo:li u > 0 6yJI.eM HMeTb
u
~ 'P~u) du = A 1 un+ A,un•+ ... + Apu"P,
0
rJI.e n; - u.eJibie LIHCJia A 1 < 0, Ap < 0. 0TcroJI.a
'P(u)=E1u"•+E2un,+···+Epu''P, E 1 <0, Ep < O. (25,1)
.li,aJiee, W(u) = lrr f(u), rJI.e f(u) - X. cp. CJiylla:liHOH CHMMeTpHlleCKO:li
BeJIHliHHbr. TaKHM o6pa3oM, Jl.JIH u < 0 W(u)= W( I u I). CJieJI.oBaTeJibHO, Jl.JIH
Bcex 3HalleHn:li
f (u) = e 'P' (u); 'P (u) = E 1 I u I"• + E2l U I"• + · · · + Ep I u I"P. (25,2)
EcJIH Mbl Jl.OITYCTHM, LITO f<n> (u) cymeCTByeT H HerrpepbiBHa Jl.JIH Bcex
ue(-oo, oo), To zp<n>(u) TaK>Ke cymecTByeT n HerrpepbiBHa BO Bcex TOli
Kax, BKJIIOlla5l U = 0. Ho BBHJl.Y TOrO, liTO n > np, BCe CTerreHH ni B (25,2)
Jl.OJILKHbl 6b!Tb LieTHbiMH, HHaLie IJ>(n) (0) He cymecTBOBaJia 6bi. J1TaK, 1J5(u)
HMeeT BHJl.
W(u)=E1 u2m•+ E 2u2m, + · · · + Epu""P, (2.5,3)
rJI.e mi - U.eJible LIHCJia, T. e. 'P(u) eCTb ITOJIHHOM OT U C lleTJ-IbiMH
cTerreHHMH.
l13BeCTHO [2], LITO eCJIH f(u) = e'F(u) X. cp. CJiyllaUHOH BeJIHliHHbl, TO
'P(u) - KBaJI.paTJ-IblH ITOJ111HOM, TaK liTO HMeeT MeCTO HOpMaJibHblH 3aKOH.
§ 26.
Mbr MOLKeM Tenepb Jl.OKa3aTb TeopeMy II § 2.
OyCTb
Lt = a1x1 + ... + a,x,
L2 = btXt + ... + b,x,
Jl.Be JlHHeHHbie cpopMbl, rrpHl!eM
sup { I a 1 l, ... , I a,. I} =F sup { I b1 l .. · ·, I b,.l }.
OycTb L1 ~ L2 H X; HMeeT 2m-li MOMeHT, rJI.e
H y MaKCHMaJlbHhlH peaJibHhlll KopeHb
EcJIH f(u) X. cp. X;, TO HMeeM
I a1 I Y1 + · · · + I a2l Y~ ~ I bl I Y1 + · · · + I b,. l y.,
rJI.e Y;- He3aBHCHMbie BeJIHLIHHbi c x. cp. lf (u)l 2 = ft(u). EcJIH X; HMeiOT
2m-i1 MOMeHT, TO H seJIHl!HHbi y; ero HMeiOT. B 3TOM cJiyqae, KaK H3Becnro,
f< 2 m> (u) cymecTsyeT 11 HenpepbiBHa . .UaJiee, 2m > y, n Bee paccy.>KJl.eHHH
§ 25 6yJI.yT HMeTb MeCTO. l1TaK, Y; 6yJI.yT pacnpeJI.eJieHbl ITO HOpM3JlbHOMY
THrry. Ho Y; = X;- x;', rJI.e X; H x;' He3aBHCHMbi H OJl.HHaKoBo pacnpeJI.e
JieHbi. no H3BeCTHOH TeopeMe r. KpaMepa [8], X; 6yJJ.yT TaK.>Ke HOpMaJibHbl.
3ro JJ,OKa3biBaeT TeopeMy II § 2. B ycJIOBHHX TeopeMbi II', ecJIH o2 oTcyr
CTByer B pa3JIOLKeHHH (20,9), 3aMeHHM era Ha BepXHIOIO rpaHb a6CU.HCC
1\0MITJieKCHbiX HyJieU 04 H rrpOBeJJ,eM TO LKe paccy.>KJJ,eHHe 1.
1 0 pH 31'0M BMeCTO JieMMbl 0 e •r (u) H3 [2] Ha,ll,o npHJIO)KHTb 6oJiee CHJ!bHYJO
JieMMY XII H3 § 42, He 3aBHCHMYIO OT ocTaJibHoro.
234
§ 21.
BepHeMCH CHOBa K pa3JIOR<eHHJO ( 20,3) n .rr,oKaJKeM JieMMy.
JI eMMa X. LincJio ·Ol y.rr,oBJieTaopHeT HepaBeHCTBaM:
0 <ol < 2 (27, l)
.li. o K a 3 a T e JI b c T B o. Mbi 3HaeM H3 npe.rr,bi.rr,ymero, qro o1 > 0.
OcTaeTCH .rr,oKa3aTb, qTo npe.rr,noJioJKeHHC
01 >2 (27,2)
Be.rr,er K npornaopeqnJO .
.lJ,JIH ~noro noJIOR<HM B (20,9) n = l, W(uu0 ) = P 1 (u). ¥13 (20,9)
Bb!BO,li,HM: npH U > 0, U < U
u
~ l[f1u(u) du =B(E) • u"•-E.
0
Bbi6npaH E .rr,ocraroqHo MaJibiM n yqHTbiBaH (27,2), Hau.rr,eM
u
~ l[f1 ( u) du = Bu2+'1· 0 0 < / , "'> , u , u1• u
0
(27,3)
Ban.rr,y 3HaKonocroHHCTBa P(u) 11, craJio 6biTb, '111 (u) Ha aceii ocn,
3aKJIIOqaeM:
u u
~~ P1~u)du!~l~ ~ P 1 (u)duj.
0 0
OTKy.rr,a COrJiaCHO (27,3)
u
u
~ P 1 (u)du=Bu3+'1.
0
(27,4)
OrcJO.rr,a ~ W1 (u) du = Bu3+'1, Ban.rr.y HenpepbiBHOCTI1 W1 (u) no reopeMe
u
JlarpaHJK! Haii.rr,eTCH TaKoe uj,E [~• U], liTO ~Pl(uk)=Bu~+'1, 11JII1
'P1 (uk) = Bu'fc+'1. (27,5)
Ban.rr,y qeTHOCTI1 P 1 (u) nMeeM
P1 (- uk) = W1 (uk) = Bu~+'~. (27,6)
I1TaK, cyw,ecrayeT nocJie.rr,oaareJibHOCTb qnceJI uk ~ 0, TaKHX, qro BbinOJI
HHJOTCH (27,5) 11 (27,6).
ct>yHKIJ,HH f(u) = e'~" (u) ecTh x. cp. .rr,JIH Hei<oToporo 3ai<oHa F(x),
a IIOTOMy f (uu0)=e'F,(ul 6y.rr,eT x. cp . .LI.JIH Hei<oToporo 3ai<oHa F 1 (x).
0603HaqnM f(uuo) = !1 (u) = er,(u>; Haii.rr,eM
235
/1 (uk) + 11 ( -ui) -2ft (0)
uZ (27,7)
YqHTblB3H (27,5) H (27,6), H3XO)J.HM, qTQ JieBaH qaCTb npe,nCTaBJIHeTCH
B BH)J.e
Bu2+•1
__ k_ = Bu·~~o npu k~ oo . uz k
J1TaK,
"' 4 sin' ukx
~ -uz~ dF1(x) = Ek~o npu k~ oo.
-oo
B cuJiy HeoTpuu.aTeJibHOCTH no.n.uHTerpaJJbHOif ¢YHKI.J,HH J-I axo.n.uM
1
Uk 1
~ 4 . 2 ukk uk
sm 2 8 ~
E > ---- -- dF (x) ~ - x 2 dF (x).
k ut 1 9' n' 1
1 1
Opu k ~ oo Haxo.n.uM
-oo
9To MO:>KeT 6biTb JJHWb ,nJIH J-Ieco6cTBemwro 3aKoHa F1 (x) nocTOHHHoif
BeJiuqHHbi, paBHoif 0. Tmme 33KOHbi Mbi OT6pocHJIH c caMoro HaqaJJa. 9THM
JieMMa X ,noKa3aHa .
§ 28.
Bee BbnueyKa3aHHOe .n.oKa3biBaeT nepsy10 tJaCTb TeopeMbi V § 2.
JhopaH qacTb TeopeMbi V 6y,neT ,noKa3aua B .n.aJibHeiiweM .
Ha nyTu K .n.oKa3aTeJibCTBY TeopeMbi I HY)KHO .n.oKa3aTb eme o.n.Hy
cymecTBeHHYJO .D.JIH .n.aJibHeifwero JieMMy.
JI e M M a XI. n ycTb 0! = 2 HBJIHeTCH npoCTbiM nOJIIOCOM ¢YHKli,HH
E(z)
a(z) ·
0yCTb Cylll.eCTByeT peaJibHb!M llOJIJOC Sz > a 1 = 2 ( aKTHBHb!H Hym.
a (z)), KOTOpblH He HBJIHeTCH ll,eJiblM q eTHbiM qHCJIOM H O)J.HOBpeMeHHO npo-
CTbiM quCJIOM.
Tor.n.a
0 3;;:;,: S2 - 2 1. (28,1)
ECJJH T3KHX llOJIJOCOB S 2 HeT, TO aKTHBHbiX KOMnJieKCHbi X HyJieif HeT, H
f/J (u) = Au2 , A< 0. (28,2)
)loKa3aTeJibCTBO 3TOM JieMMbl noTpe6yeT nepBOHaqaJibHOfO o60CHOBaHHH
HeKOTOpbiX BCllOMOraTeJibHbiX YTBep:>K.D.eHHH.
l a3, KaK H paHee, - TOqHaH HH)KHHH rpaHb a6CU:HCC aKTHBHb!X KOMnJieKCHb!X HyJieH.
236
,UoKa:tr<eM cne pBa c.rre,nyiOLQee yTBep:t!<,neHHe: B ycJIOBIUIX JieMMbi XI
cyLQeCTByeT W"(u), HenpepbiBHaH Ha scei'I ocH, 11 cyLQeCTsyeT ,nucnepcmr
COOTBeTCTBYIOLQero pacnpe,neJieHHH. I1MeeT MCCTO OQeH Ka
IJI(u) = Bu2 npH I u I<; 1. (28,3)
YTBe p:tK,LJ.eHI1e (28,3) B CBOIO 04epe,nb Mbl BbiBe,LJ.eM 113 CJie,nyiOLQero, 6oJiee
cJia6oro, YTBep:tK,n:eJIHH: cyrll,eCTByeT nocJie,noBaTeJibHOCTb noJIO:tKHTeJibHbiX
LJHCeJI u" ____,. 0, TaKaH, '-ITO
I1P(u,J I<: Ct9u,,2. (28,4)
,[LJIH ,n:oKa3aTeJibCTBa o6pamaeMCH K (20,9) 11 (21, I) . EcJJI1 npH 3TOM
o3 > o1 = 2, To (28,4) TpHBH3JJbHO. YlMeHHO, 6epH n =I, noJJy'-laeM npH
O<u<;I
ll ll
\ IJI(two) ~u = ~
0 0
BBHJJ:Y TOfO, '-ITO S 1 (u)= Cu2 +0(u2+ "~); '7 > 0, C - KOHCT3HT3. 0TCIO,[(a
BBHJI.Y 3H3KOOOCTOHHCTB3 ![Jl (zz)
ll
11 \ 1Jf1 (u)du=Bu 3 • 0Tcio,n:a, paccym;~aH, r<aK B § 27, npH BbiBO,ne 113
0
( 27,4) CJi e,[(CTBHH ( 27,6), 33KJI I043eM, 'ITO cyLQeCTByeT DOCJie,LJ.OB3TeJJbl-IOCTb
u1/----;> 0, ,[(JIH KOTOpbiX sepHO
lif(uk') = Buk'2 •
fioJJaraH uk = uk'u0- 1, npHxo,n:HM K (28,4) . CJJy'-lai'I, Kor,na 03 = 01 = 2,
Tpe6yeT 6o"1 ee cJJomHbiX paccym,neHHH.
§ 29.
fiycTb o3 = 01 = 2.
( 21 , I) , noJJy'-IHM
06paTHMC51 K (20,9), ll, yCTp e MJJ5151 n----;. <X> C y'-leTOM
r,ne
1
T(u)= 2ni
00
~·
(29, l)
f"h E(z)
'j" (uuo)• za (z) dz (29,2)
11 WTpHX 03H343eT, '-ITO 6epyTC51 Bbi'-IeTbl JIHllib B KOMOJJeKCHbiX 3KTHBHb!X
HYJJHX.
l13 ( 29, I) 3aKJJI04aeM, '-ITO
237
HerrpepbiBHa H HMeeT HerrpepbiBHYIO rrpoH3BO.ll.HYIO T' ( u) B [0, l]. Ll.aJiee,
T (0} = 0. BBH.ll.Y eToro, rroJiaraH
Tt(u) = uT'(u), (29,3)
u
T ( u) = ~ T1 ( u) ~u .
0
Mbi xoTHM Terrepb .noKa3aTb, 'ITO cymecTsyeT 6eCKOHe'IHaH rroCJie.nosa
TeJibHOCTb ITOJIO.lKHTeJibHbiX 'IHCeJI Uk ·~ 0, T3KHX, 'ITO
Tt(Uk) > 0. (29,4)
}lJIH )I.OK333TeJibCTBa BBO)I.HM 33MeHy rrepeMeHHbiX:
u=e1; u0 =e1• ; Tt<u)= T1 (e1)= W(t). (29,5)
Tor.na (29,2) rrpHMeT BH.ll.
t
~ W(f) df = 2~i ~~ p e<t+to)z !~:~ dz
-oo Lm
(29,6)
)I.JIH f < 0. IlyCTb {J < 0 - JII06oe 'IHCJIO. IloJiaraH
t
W1 (t) = ~ w(t) dt,
-oo
W (t+{J)= _!___, ~· tfi e(t+to)., • ef1z • E(z) dz.
1 2nz ~ ";j zu(z)
m
BBH.ll.Y ::noro, ecJIH /]1, /]2, ... , fJP- - JII06bie oTpm~aTeJibHbie 'IHCJia,
a A 1, A 2, ••• , A~" - JII06bie HeoTpHuaTeJibHbie 'IHCJia, TO
~ A.W (t+tf.) = __!_, ~· e(t+tolz • ll(z) E (z) dz (29,7)
~ 1 1 1 2m ~ za(z) '
1=1 m=-co
f)I.e qepe3 fl (z) o603H3'IeHa !flyHKUHH
M
~ Aie{Jfz.
j=l
(29,8)
IlpH 3TOM, eCJIH )I.J!H HeKOTOpblX f JieBaH 'I3CTb (29,7) OK33biBaeTCH ITOJIO
.lKHTeJibHOH, TO 3TO .lKe BepHO XOTH 6bl )I.JIH O)I.HOrO H3 'IJieHOB W 1 (t + /Ji).
PaccMoTpHM see KOMITJieKCHbie noJIIOCbi E((z)). .[lei:fcTByH, KaK B § 13,
za z
Hai:f.neM 'IHCJIO A< OJ, T3KOe, 'ITO )I.Ba KOMnJieKCHbiX nOJIIOCa eo H Qo
6y,nyT 6JIH.lK3HlliHMH K 'IHCJIY A, Bee OCT3JlbHbie 6y,nyT Ha 60JibllieM
paCCTOHHHH.
IloJiaraH L1 =eo - A = doeia, 6y.neM HMeTb
E(z). >O
npO'IHX ITOJIIOCOB za(z), 1J •
I'! ·- A I ;;>do+ 11 )I.JISI I .
238
,lloKa)KeM Terrepb, 'ITO MO)KHO Bbi6paTb TaKyJO <iJYHKUHIO fl (z) . THITa
(29,8), 'ITO eo H Qo 6y.nyT HYJIHMH n (z) rrpe.nrrncaHHOH KpaTHOCTH v.
EcJIH (lo = bo + h 0 , TO bo > 2; 6y.neM Cl:JHTaTb, 'ITO -r0 > 0.
n _nbo
floJIO)KHM: f1 =- -; A=e ~ •. Tor.na ¢YHKUHH A+ ef3z 6y.neT HMeTb
-z;o
Z = (lo npoCTbiM KOpHeM, a <iJYHKUHHfi(z)=(A +ef3z)~- KOpHeM KpaTHOCTH v.
n (z) ITOCJie pacKpbiTHH CK060K 6y.neT HMeTb ¢opMy (29,8). BBH,LI,Y :noro
Mbl M01KeM BCer,LI,a Bbi6paTb fl (z) THITa (29,8) TaK, 'ITO (!o 6y,neT npOCTblM
ITOJIIOCOM <!JYHKUHH
tP(z) = e1.zfl(z) E(z).
za(z)
E(z) _
EcJIH eo 6bJJI npoCTbiM ITOJIIOCOM - -( - ), TO ITOJiaraeM fl(z)= 1.
ZO' Z
06o3Ha'laH Tenepb
M
w~ (t) = ~ AjW1 (!+ fl;),
j=l
r.ne Ai> fJ; OTBe'laJOT Bbi6pamwii fl(z), noJJy'IHM
00
1 '\"'. n; w2 (t) = 2ni ..:::... ;r etz (p (z) dz.
m=-co '
Lm
(29,9)
(29,10)
AKTHBHble KOMITJieKCHble HYJJH Qj =I= Qo OKPY)KHM KPY)KO'!KaMH c~i
CTOJJb MaJibiMH, 'ITO BCe TO'mH HX KOHTypoB HaXO,li,HTCH B ITOJJYITJIOCKOCTH
~+A ~
x > --2- H Ha paccTOHHHH > d, + 2- OT A. Ha KOHTypax KPY)KKOB
,n:OJI)KHO 6biTb KpoMe Toro I a(z) I> eo (eM. JieMMY IV). B CHJIY CKa3aH
Horo Bblllle ITOJJY'IHM
BBe,neM onepaTop
§ 30.
t t
L2 (/ (t)) = ~ dt .. • ~ db · f (t),
-oo -oo ..__,_.,
r pa3
(29,11)
KOTOpbi}i 6y,LI,e M npHMeHHTb KOr,na ITOCJie,LI,OBaTeJibHble HHTerpa.llbl 6y.nyT CyiiJ.e
CTBOBaTb 11 ,naBaTb HenpepblBHble <!JyHKUHH.
IToJJaraH I (t) = w2 (t) e-At, ITOJIY'IHM H3 ( 2'9, I I) cor JiaCHO § I 9
et (~o-A)
L~ (W2 ( t) e-At) = Z0 (Qo _ A)2 +
- e1 !~u- · 1> I ' gJ (p (z) dz + Zo + -2 . '\"' et(z-A) .,.---~~
(!!o- A) 2 nz .0:::.. .· (z- AY ·
ce;
(30,1)
239
TipH r ;> 2 (s CHJIY <flopMbi @(z) ,n:a:tKe rrpH r ;> 1) pH,n: 6y,n:eT a6coJJIOTHO
CXOJJ:HTbCH, H CYMMa B ( 30,1) 6y,n:eT M3:tKOpHpOB3TbCH qHCJIOM
rrpH mo6brx t < 0 H r ;> 2 . .[LaJiee HMeeM, rroJiaraH
2R(z exp(f(Qo-:!))) =
0 (eo- A)2
= :r R ( I Zo I exp (fdo cos a+ fd0 i sin a- iar).
0
(30,2)
(30,3)
TipH BCHKHX QlHKCHpOB3HHb!X T < 0 H r B cerMeHTe
Haii,n:eTCH T3KOe f, qyo ( 30,3) IIOJIO:tKHTeJibHO H paBHO
[T- d 2n a' T]
0 cos
2 2
dor I Zo I exp (fdo cos a) > d.r I Zo I exp ( Td0 COS a- 2n) (30,4)
B CHJIY TOfO, qyo COS a> 0 IIO IIOCTpOeHHIO A.
TiycTb ga,n:aHo T < 0. Bbr6epeM r CTOJib 6oJiblllHM, qyo (30,4)
(
rJ- A) exp T _ll_2_
> 2C20 ( r
d 0 + ~-)
H 33QlHKCHpyeM ero. Tioc.Tie sToro Bbr6epeM te [ T- ~~ - T l TaKoe, qTo
cos a , 1
(30,3) paBHO Jiesoii qacTH (30,4). Tor,n:a rrpaBaH qacTb (30,1) 6y,n:eT 6oJibllle
1
dor I Z0 I exp (fdo COS a) > 0. (30,5)
Tor,n:a, oqeBHJJ:HO, ,11,0Jl:tKHO cyw,ecTBOB3Tb Tl < T, JJ:JIH KOTOporo w2 ( T) > 0,
HHaqe cooTHOlllemre Lr (W2 (f) e-At)> 0 6biJIO 6bi HeB03MO)KHbiM. B CHJIY
QlOpMbl W2 (f) 33KJI!OqaeM, qyo XOTb JJ:JIH OJJ:I-IOfO H3 H3lllHX qHCCJI {Ji < 0
l+lij
W1 (t+.B) > O. BsH,n:y Toro, qyo W1 (t+.Bi) = ~ W(t)dt > O, Haii,n:eTCH t1 < T1,
,n:JJH KOToporo W (t1) > 0. EepH Terrepb T = 2t1 < 0 H IIOBTOpHH Te
:tKe paccy:tK,n:eHHH, Haii,n:eM f2 < 2t~, ,n:JIH KOToporo W(t2) > 0. lleiicTBYH
T3KHM o6pa30M, IIOCTpOHM IIOCJie,n:oBaTeJibHOCTb fk -:) - CXl, JJ:JIH K3:tKJJ:OfO
H3 KOTOpbiX W (tk) > 0.
Jl:OK33biBaeM (29,4).
240
§ 31.
lb ( 29,1) noJiyqaeM npH u > 0:
'IJ\;u) = S/ (u)+T' (u) =S/ (u) + T1~u);
S/ (u) = 2Cu+O(u1+'1); OTCIO,II,a
W1 (uk) = 2Cuz + T1 (uk) + o (uz+'l) .
Ho G < 0 no JieMMe VII. .llaJiee, lfi1 (u) < 0 npH u > 0, BBH,II,Y q e ro, B
CHJIY TOro, 'ITO TI(uk) > O, H3XO,ll,HM j P1 (Uk)\<;: I 2C j uz+O(u~+'1), HJIH
I lpl (uk) ! < c21 ui. Eho ,II,OK33biBaeT (28,4). 0TCIO,II,a MO)!{HO Bb!BCCTH (28,3)
,ll,Jlll BCeH OCH, UOJib3YHCb MeTO,II,OM § 27. ECJIH f(u) = e'F(u) H F(x)- COOT
BCTCTBYIOmHH 3il.KOH pacnpe,II,eJieHHll, TO
I (uk)+ I (-uk)- 21 (0)
Uk (31' 1)
Mbr BH,II,HM, 'ITO JieBaH qacTb (31,1) HeoTpm.J.aTeJibHa. B CHJIY (28,4) OHa
He npeBOCXO,ll,HT 0 22 . 0TCIO,ll,3
1
u k 1
~ 4 sin2 u~x 8 u~k
------,2,---dF(x)~---, x 2 dF(x).
Uk n ,
1 1
fJpH k ---7 oo, u,, ._ 0, H Mbl H3XO,ll,HM
00
~ xi dF (x) < C22.
-oo
B 3TOM cJiyqae, KaK H3BeCTHO, f" (u) cymeCTByeT BCio,II,y, orpaHH'!eHa H He
npepbiBHa Ha BCCH OCH. To )!{C K3C3CTCll f' (u); DpH 3TOM f' (0) = 0, B CHJIY
CHMMeTpHH 33KOHa F (x) .
.llaJiee, HMeeM
W(u)=In 1 (a); P' (u) = ~c~j; P" (u)== ~~~}- ( ~~c~!r.
0Tcro,II,a Haxo,II,HM: W' (u) H lp'" (u) - HenpepbmHbi 11, cTaJio 6biTb, orpaHH
'leHbi B [0, 1]; lp' (0) = 0. 0TCIO,ll,3 HMCCM
W(u) = P(O)+ uP' (0) + ~ P" (0,_.) = ~2 W" (Ou).
TipH I u I < 1 B CHJIY H3JIOlliCHHoro Bbrwe 3To npHBO,II,HT K (28,3).
241
§ 32.
n.Jur .LJ:OKasaTeJibCTBa JieMMbi XI HY:>KHhr .n;aJibHeifrmre paccy:>K.n;eHIUI.
Mbi TIOJIY'IHM OII.eHKY (2'8,3) .D:JIH BCeX CJ!yqaeB pacnOJIO:>KeHHH o3, BKJIIO'laH
0 3 =OJ= 2.
B cJiy<Iae, Kor.n;a 0 3 > OJ = 2, o6pamaHCb K paBeHCTBY ( 29,1), paso6heM
SJ (u) - cyMMY Bbi'leTOB no peaJJbHblM aKTHBHblM HYJJHM - Ha .n;Be l!aCTH:
St (u) = Vo (u) + Vt(u), (32, 1)
k
r.n:e V0 (u) = ~ ex"'u2m - cyMMa no Bhi'leTaM B npoCTbiX nOJJIOCax, paBHbiX
m= J
ll.eJibiM tieTHbiM tiHCJiaM; V t (u) - cyMMa no ocTaJJbHhiM noJIIOCaM.
BBH.LJ:y (20,9) H (21, 1) TIOJJytiaeM npH 0 <; U <; 1: eCJJH o3 > 2, TO npH
BCHKOM e > 0
u
~ lp1~u)du=V0 (u)+V1 (u)+B(F)u"•-'; (32,2)
0
03 - e > 2 rrpH MaJJOM e.
,llaJJee, ).I.JlH BCeX CJiytiaeB paCIIOJIO:>KeHHH 0 3
lJI(u) = & 2 rrpH I u I <; 1. (28,3)
HcCJie).I.yeM crrepBa HaH6oJiee TPY.LJ:HbiR CJiy<Iaii 2 <; .a3 < 4. Mbr HMeeM :
f(u) = e'Ftl u l l eCTb x. cp. HeKoToporo s aKoHa F(x). flpH 9TOM lJ1 ( j u I) <; 0
Ha BCeii OCH. B TaKOM CJiytiae QJYHKUHH /(u)e-u' =e 'F(I u l) -u' HMeeT OUeHKY
co
1 c lft(u)j < e-u' Ha Bceii OCH. BBH.LJ:Y 9Toro 2n J e-iux f 1 (u)du=y(x)
-oo
a6COJIIOTHO CXO.LJ:HTCH H H306pa:>KaeT ll.eJJyiO QJYHKil.HIO X, orpaHH'leHHYIO Ha
00
BCeii OCH. fi09TOMY f 1 (u) = ~ eiux y (x) dx, r.n:e y(x)
-oo
TIJJOTHOCTb BepOHTHOCTH. fiOJJO:>KHM IIpH U > 0
u u
\ 1]12 (u) du= \ tp1 (u) du- u2.
J u J u 2
0 0 •
HMeeM Terre ph, rrpH 0 < u <; 1,
/ 1 (u)=e'F,(u)= 1 + IJI2(u)+~ (u),
r.n;e
;(u) = e 'F, (u)- 1 - IJI2 (u).
HerrpepbiBHaH
(32,3)
(32,4)
(32,5)
TiycTh .n;aJiee K (a) - cpyHKil.HH napaMeTpa a rro.n: yCJIOBHHMH K (a) > 0;
L
K (ex) - HerrpepbiBHaH cpyHKil.HH;
ex
Ha3hlBaTb H).I.pOM. L
~ K (ex) da = 1. TaKyiO QJYHKil.HIO 6y.n;eM
0
<l>yHKil.HH / 2 (u)= ~ K(a)f 1 (au)da OT H).I.pa H X. cp. ft 6y.n;eT TaK:>Ke
0
xapaKTepHCTH'leCKOR QJYHKil.HeR, OTBetiaiOlll,eR IIJJOTHOCTH BepOHTHOCTH
L
~ K~a)y (;)da (3aMeTHM, LITO y(x) orpmmqeHo Ha sceli ocu). Ha,ll. ¢YHKI.J.Heii
lJ
f2( u) MO)KHO nposecTn no,ll.o6HyiO )Ke onepauuiO, u noJiyqnTh HOBYIO x. ¢.
f 3 (u,) u T. )1. Ha 3TOM u 6y)].eT ocHoBaHo ,ll.OKa3aTeJibCTBO JieMMbi XI.
(0KOH'!aHHe B CJie,a:yiOIJ.J.eM HOMepe )!{ypHaJia).
JU1TEPATYPA
I. 10. B. J1 H H H H K, 3aMe'!aHHH no noso;'l.y KJiaCCH'!eCKoro Bbmo,a:a 3aKoHa MaKe·
aeJIJia, .LI.AH, T. &5, N~ 6, 1952.
2. J. M a r c, in k i e w i c z, Sur une propriete de Ia loi de Gauss. Math. Zeitschrift,
Bd. 44, H . 4-5, 1938.
3. A . .LI.. M hi III K H c, J1nHeHHbie ,a:n¢¢epeHI.LHaJihHbie ypasHeHHH c 3ana3,a:biBaiOW.HM
apryMeHTOM, ITTH, 1951.
4. A. 0. r e Jl b ¢ 0 H ,a:, J1HHCHHbiC ,a:H<jJ<jJepeHUHaJibHbie ypaBHCHHH c 110CT051HHblMH
K03<jJ<jJHUHeHTaMH 6eCKOHe'-!HOfO 110pH,a:Ka H aCHM11TOTH'!CCKHC nepHOtl:bl UCJ!b!X <jJyHKl.\HH,
Tp. MHAH, T. XXXVIII, 1951.
5. A. 0. r e Jl b ¢ 0 H ,a:, 0 l.\CJIO'-!HCJieHHb!X aHaJIHTH'!CCKHX <jJyHKl.\HHX, .LI.AH, T. 81,
1951.
6. G. Doetsch, Theorie und Anwendung der Laplace, Transformation, 1937.
7. E. T H T 'I Map m, TeopHH ¢YHKUHH, fTTH, 1951.
8. r. K paM e p, CJiy'-laiiHhie BeJIH'IHHbi H pacnpe,a:eJieHHH aepoHTHOCTei1, fl1I1J1, 1947.
Ilo.~ y'-leHa 27 ,a:eKa6pH 1952 r.
J1eHHHrpa,a:.
0207
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|
| id | umjimathkievua-article-7732 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:21Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/53/193612ce7e859ffbd0d53ead385e7553.pdf |
| spelling | umjimathkievua-article-77322023-08-15T09:55:16Z Линейные формы и статистические критерии І Линейные формы и статистические критерии І Линник, Ю. В. Линник, Ю. В. - - Institute of Mathematics, NAS of Ukraine 1953-05-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7732 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 2 (1953); 207-243 Український математичний журнал; Том 5 № 2 (1953); 207-243 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7732/9428 Copyright (c) 1953 Ю. В. Линник |
| spellingShingle | Линник, Ю. В. Линник, Ю. В. Линейные формы и статистические критерии І |
| title | Линейные формы и статистические критерии І |
| title_alt | Линейные формы и статистические критерии І |
| title_full | Линейные формы и статистические критерии І |
| title_fullStr | Линейные формы и статистические критерии І |
| title_full_unstemmed | Линейные формы и статистические критерии І |
| title_short | Линейные формы и статистические критерии І |
| title_sort | линейные формы и статистические критерии і |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7732 |
| work_keys_str_mv | AT linnikûv linejnyeformyistatističeskiekriteriií AT linnikûv linejnyeformyistatističeskiekriteriií |