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ec ecoo ayoo epa, . 2 (23), 2004 MATEMATA 517.5 METO XAPATEPCTECX H B OEHBAH MATEMATECOO OAH CA HX BEH C ECOHEHO CEPCE A.B. aa (1), E.B. axao (1), A.C. Aa (1, 2) email: lappa@csu.ru (1) ec ocyapce yepce, . ec, Pocc (2) ec ocyapce cy aepo xpyp, . ec, Pocc Ca ocya 4 a 2004 .

Beee Hacoa paoa ocea oea aeaecoo oa cyax e c ecoeo cepce, y pacpeee oopx ee ceey acoy.

Toee, paccapae ooepy eopaey cyay ey c ye pacpeee F(x) = P < x, yoeope o yco { } 1- F(x) = O x-, x ; > 1, (1) ( ) o eo acoy cya 1- F(x) = cx- + o x-, x ; 1< < 2, ; c > 0. (2) ( ) coe (1) ec ca: oca cpeaxc poex eopaex cyax e c oe aeaec oae M cyecye > 1, oeceaee (1). cepc D o cyae oe cyecoa, ec O(x- ) cpec y ocaoo cpo.

coe (2) ec oee opae, p o cea D =. Cyae e oo a cpeac o ox oex peax e, acoc, e, ooe, acpoo [14]. oy y pae p oeo eoa Moeapo pee ypae epeoca ye. B acoc, eca oaa oea aoca [5] pacea ooc ooa ye aao oe ec eopaeo cyao eo a (2) c = 3 / 2 [68]. Bcee ooc ooa coc oea aeaecoo oa o cyao e.

Oeae aeaecoo oa paccapaex cyax e ec aae eapaepeco cac, ocoy pacpeee (1) (2) e opeec ooao ae oeoo ca apaepo. Caapo oeo aeaecoo oa cyao e o cyae ec opooe cpeee n = k, (3) n k =2 A.B. aa, E.B. axao, A.C. Auua e (1,..., n ) eaca opa peaa. a oea cocoea eceea p o pacpeee c oe M. p D < acoa pacpeee opaa; pe caap coco oe caceco opeoc - M (epe oeae D ); acoa ya o opeoc opaa: n-1/ 2, n ; xapaep pe M yco: oe poc aoepo.

p D = xopoe coca epc: acoa pacpeee, ooe oop, e opaa ( cyae (2) oa pae accy ycox aoo); poco caap coco oe caceco opeoc epe; opeoc yae eeee, e n-1/ 2 (a n- (-1) / cyae (2)); xapaep pe M eyco:

epo oe poc, pe cya ycyyec c yeee. Bce o cyeceo apye paecoe cooae opooo cpeeo oea aeaecoo oa cyae ecoeo cepc.

epecee eae coca opooo cpeeo c pooee eo yaoo ococa: peoc cya ea c oe aeaec oae. ocpoe oee aecex oeo ececeo oaac o ao yepcaoc, oac yec, o oooc oee oo, eyc apopy opa o pacpeee ocpeeo cyao e, o ec yco (1) (2) ae cyae.

Oa aa oea eopaeo cyao e a (2) ocpoea paoax [7, 8]. cooa ec peya: pacpeeee cpeeo apeecoo eacx peaa cyao e a (2), cxoc ycooy aoy c oaaee, y pacpeee oopoo ee eoopoo caapoo pacpeee c y apaepa: ca acaa. B oo pacpeee opeeec oo apaepo, apaep ca coaae c M. Aop [7,8] peo pa cxoy opy peaa a eepeceaec py oaooo paepa, coca oy opy cpex apeecx py pe aece oe M ayy xopoy oey apaepa ca ycooo pacpeee. co, o a oeoc py aa oea ec ceeo, o ceee oo cea co yoo a, yea paep py. B aece xopoe oe paoax [7, 8] a cooaa eyppoaa oea oa (ea oa opox cac c ao cepce) [9].

B acoe paoe coyec pyo eo, aac caceco oea xapaepceco y cyao e cooa acoecx ce ey xapaepceco ye aeaec oae, oax yco (1) (2). B epo apapae ca (pecae caocoe epec) peeo acoecoe paoee xapaepceco y poooo pacpeee, eeo acoecoe paoee ceeoo a, a ace cya (1), (2). o peya aee coyec ocpoe yx oeo aeaecoo oa, pex cyax e o (1) (2). Oe oaa po ococ o cpae c opo cpe : oeo cepce, opao cxooc ( n-1/ 2 p n ), ycooc o ooe poca p., o eae oyee oe epce pa.

1. Aco xapaepceco y Copypye caaa oo ooe coooee xapaepceco y paccapaex cyax e.

ea co eopaeo cyao e c aeaec oae ye pacpeee F a (1) cpaeo ceyee pecaee xapaepceco y:

Memo xapamepucmuecux yu oeuauu ameamuecoo ouau cyax euu g(t) = 1+ it + it F(x) eitx -1 dx. (4) ( ) (1- ) co, o ea cpaea aco cyae (2). Cey peya, oyac c cooae o e, ycaaae c ey cee acoec paoe xapaepceco y y pacpeee.

Teopea yc y pacpeee F eopaeo cyao e ee :

N j 1- F(x) = x + (x), x > 0; N = 0, 1,...; 1< 1 <... < N < 2; cj 0, (5) j c j =e (x) = o(x-N ), x ; 0 (1, 2). ec aee cy a oaac pa y.

j = Toa xapaepceca y o cyao e pecaa e:

N j g(t) = 1+ it + b j (t) t + (t), (6) c j j =e n = M, b(t) = Re A + i Im Asign(t), A = i z- eiz -1 dz, (t) = o t, t 0. (7) ( ) ( ) B yx acx cyax, oa 1) (x) = O(x-), x, >N 2) (x) = o(x-), x, N ee:

O t < 2,, ( ) ( ) o t, < 2, O o 1) (t) = t2 ln t, = 2, t 0; 2) (t) = t2 ln t, = 2, t 0, (8) ( ) ( ) Kt2 + o t2 > 2, Kt2 + o t2 > 2,,, ( ) ( ) e K - (x)x dx, K <.

opya (6) cae xapaepcecy y g aeaecoe oae cyao e a (5). p N = 0,1 ee ceye a acx peyaa, coyee aee.

Cecmue co eopaeo cyao e c aeaec oae ye pacpeee F a (1) cpaeo ceyee acoecoe pecaee xapaepceco y:

g(t) = 1+ it + 1(t), (9) 4 A.B. aa, E.B. axao, A.C. Auua e O t, < 2, ( ) O 1(t) = t2 ln t, = 2, t 0; K - F(x) x dx, K <. (10) ( ) (1- ) Kt2 + o t2 > 2,, ( ) Cecmue co eopaeo cyao e c aeaec oae ye pacpeee F a (2) cpaeo ceyee acoecoe pecaee xapaepceco y:

g(t) = 1+ it + cb(t) t + 2(t), (11) e b(t) opeeec opya (7) o t < 2,, ( ) 2(t) = t2 ln t, = 2, t 0, K - 1-F(x)-cx- x dx, K <. (12) o ( ) () Kt2 +o t2, > 2, ( ) 2. Oe aeaecoo oa Paccop eep opoc o caceco oea aeaecoo oa M = x dF(x) eopaeo cyao e c ye pacpeee a (1) (2). e eoa aaec caceco oea ee xapaepceco y g p eoopo cpoao t cooa ce ey g , ycaoex Cecx 1 2. e c a e pae oe, coye pay apopy opa o y pacpeee, aey paoex (1) (2). Moo o oe coeco, aca pae g Cecx 1, 2 oe e:

Gt) g(t)-1= it +c b(t) t + (t), j =1,2, (13) ( jj 0, j = e ooae c b e pe cc (eopea 1 Cece 2), =.

j 1, j = p j = 1 o paee ec peya Cec 1, cpae p oe (1);

p j = 2 peya Cec 2, cpae p oe (2). y 1(t) ee (10), a 2(t) (12).

epee (13) oeo eceo o ac c yeo a y ( G (t) (Gt) = ReG(| t |) + i ImG(| t |) sign(t), (t) = Re (| t |) + i Im (| t |)sign(t) ):

j jj j ReG(| t |) = c Re b(| t |) t + Re (| t |), jj (14) ImG(| t |) = | t | +c j Imb(| t |) t + Im j (| t |).

Oca p cpoao t 0 :

ImG(| t |) - B ReG(| t |) Im (| t |) - B Re j (| t |) jj j =-, (15) | t | | t | Memo xapamepucmuecux yu oeuauu ameamuecoo ouau cyax euu e Im A B =. (16) Re A ae, o paec ao aco cyae = 3 / 2 ecea a ac epaa A = i z- eiz -1 dz coaa: Re A3/ 2 = Im A3/ 2 = - 2 , ceoaeo, ( ) B3/ 2 = 1.

pe aece eoopoo pe epoe caaeoe (15):

ImG(| t |) - BR eG(| t |) j (t) =. (17) j | t | opeoc o apoca:

Imj (| t |) - B Re (| t |) jj b(t) (t) - =. (18) j j | t | ocpoe caceco oe aeaecoo oa ee cyae e Im ei|t| -1 - B Re ei|t| -( ) j ( ) (t) = (19) j sin(| t | ) - jB cos(| t | ) + jB.

| t | | t | (17) cacecoo cca xapaepceco y ( g(t) = Meit ) ceye, o M (t) = (t). (20) j j pe aece oeo aeaecoo oa opooe cpeee cyax e (t) :

j nn Im exp i | t | k -1 B Re exp i | t | k -( ) ( ) ()- () j (t) = (t) =, (21) j j,k nn | t | k =1 k =e (,..., ) eaca opa peaa.

j,1 j,n j Bpaee (21) oee e oe:

n Im exp(i | t | k ) -1 () 1(t) =, (22) n | t | k =n Im exp i | t | k -1 - B Re exp i | t | k -( ) ( ) () () 2(t) =, (23) n | t | k = O c co oea M. Toee, pae (22), (23) p pax cpoax aex apaepa t opee 2 ceeca oeo. Oe , pe, j pe p x oex t 0. Ho, ocoy o e o t ( (-t) = (t) ), pe jj aece oac oox ae apaepa t oeco ooex ce (,).

Oea 1 pea oea M cyae (1) (, payeec, eo aco cyae (2)), o ec oca cpeaxc poex eopaex cyax e c oe aeaec oae. p o cepc D oe oeo 6 A.B. aa, E.B. axao, A.C. Auua ecoeo. Ba ococo oe 1 ec yepcaoc ee a: o e ca c apaepa pacpeee ocpeeo cyao e, a apaep (1).

Oea 2 pea oea M oo cyae (2), oop ae oo cyae e c ecoeo cepce. Oea coye oy apopy opa o pacpeee, e 1, ooy o ee ceye oa oe ooc. B o oe, ececeo, eee yepcae, o paec ao, o o e ac o oca c, yppye yco (2). O ca oo c apaepo (2), acoec opo ya 1- F, oop oo opeeec cax ox pecae o ocpy cyao e.

aee oyee oe aeaecoo oa cyax e a (1), (2) oaa e po ococ o cpae c opo cpe. B oe o oceeo o e oey cepc p D = , a cece, o oaa oe copoc ya caceco opeoc, pe caap coco ee oea, x cxooc ee oee peyp xapaep. Bao, o oe poc peaa. ao a peyeca ec ceeoc ae ecocoeoc oeo p o ae apaepa t > 0. Ho eoca cea acoe p t 0, o eae oe epec c paeco o pe. ye aex acoecx coc oeo ocea ceya ca aopo.

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