Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 68 Article Jose J. G. Moret Abstrct I this pper, Abel summtio method is pplied to evlute ifiite series d diverget itegrls. Severl emples of how oe c obti regulriztios re give. Key Words: Abel sum formul, Abel-Pl formul, poles, ifiities, reormliztio,, multiple itegrls, regulriztio, Csimir effect. Abel summtio for diverget series Give power series of the form which is coverget o the regio, we defie the Abel resummtio of the series s the limit lim A( S) If the previous limit eists, the the series As emple, let be the series [6] is sid to be Abel-summble to the vlue A(s). d ( ) 3... B d () Ufortutely, the series. is NOT Abel-summble due to the pole t = of the fuctio However, Guo [5], usig epoetil regultor, studied this series d gve the followig idetity d ( ) Z( ) ( ) e d e! () Correspodece: Jose J. G. Moret, Idepedet Resercher, Spi. E-Mil: osegrc@yhoo.es ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 68 where both, the Tylor epsio ivolvig Beroulli s umber B d the e! B epressio for egtive vlues of the Riem zet fuctio ( ) were used. To evlute the Riem zet iside () for egtive vlues, oe eeds the Riem s fuctiol s equtio defied by s ( s) ( s)cos ( s), with ( s) ( s). si( s) Guo itroduces smll prmeter epsilo d fter clcultios te the limit, Ufortutely for = - Guo s method gives oly ifiite swer e log, this ll is becuse the followig epressios for the -th Hrmoic umber d the Lplce trsform for the logrithm H... log 3 t log dte log t (3) where.577.. is the Euler-Mscheroi costt. If oe igores the pole prt i (), oe hs f. p e ( ) for every ecept =-. This is precisely the vlue of the series obtied vi Zet regulriztio. So, Abel resummtio d Zet regulriztio re relted d give the sme swer for the diverget series provided tht oe igores the pole prt proportiol to. As emple, we will study the Csimir Effect to see how the regulriztio d reormliztio of the diverget sum is mde. o Csimir effect: The Csimir effect is physicl force due to the qutiztio of Electromgetic fields (see, e.g., [7]). I the simplest versio of the Csimir effect, the vcuum Eergy of the system per uit of Are A is give by / 3 rdr r 3 E c c A 4 (4) 6 h where of light i the vcuum. 34.54 Js. is the reduced Plc s costt d c 8 3 m/ s is the speed ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 683 3 If we use Zet regulriztio [3], we fid the vlue. After we isert this vlue iside E c (4), we get the correct eperimetl vlue for Csimir effect. 3 A 7 So Fc d E c. 4 A d A 4 The physicists s pproch to Csimir effect is bit more complicted. For emple, they use reormliztio d compute the qutity (differece) c 3 3 t E Ediscrete E e dtt e 3 6 (5) This differece c be computed with the id of the Euler-Mcluri sum formul B () 3 f ( ) f ( ) d f () f ( ) e (6) ( )! Or with the Abel-Pl sum formul,with f () f ( it) f ( it) 3 f ( ) f ( ) d i dt f ( ) e t e (7) t ( ) If we retur to Guo s formul () d use the idetity dte t, we fid the followig result Z( ) t ( ) e ( ) dte t! (8) So, lthough the Abel regulriztio is ot vlid for the series, the differece (9) t e dte t ( ) ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 684 mes perfect sese d is lwys FINITE. Also, for the cse =- we fid tht the Hrmoic series is summble d its sum is equl to Euler-Mscheroi costt removig the regultor e. fter So, both methods reormliztio d zet regulriztio gives the sme fiite swer. However, Zet regulriztio is esier d fster method d c be geerlized to the cse of more geerl opertors. For emple, E i, ctrce i gg where the opertor is the Lplce-Beltrmi opertor d g () i,,, g det g g g g is determit of,, mtri, the qutity () is the Vcuum eergy for the Lplci opertor i two dimesios. Abel summtio d diverget itegrls Abel summtio formul c be eteded to obti fiite results for diverget itegrls too, first we eed the formul m m m m m m d d i i i i B r( m) mr ( m r ) d r ( r)! ( m r ) () where is positive iteger d the ifiite sum iside () must be uderstood i the Abel regulriztio sese i e i Also, the recurrece () is fiite if is positive iteger due to the poles of the Gmm fuctio t the egtive itegers. I cse is positive d rel umber, the recurrece () is ifiite d it must be tructed, d we c lso use iside () the idetity,which is vlid for Re( m). m m m ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 685 The cse m=- must be cosidered seprtely. If we te the fiite prt the e f. p e, or if we use the epressio f( ) iside the Euler-Mcluri summtio formul f ( ) f ( ) B () () ( ) ( ) ( ) ( ) f f d f f ( )! () d te ito ccout the followig series epsio for the Digmm fuctio '( ) B ( ) log r () ( ) r (3) We get the reormlized result for the itegrl with logrithmic divergece d log,which mes tht i regulrized/reormlized sese the 3 itegrls d reorm d d d re equl to. f ( ) f ( ) For the cse =, we fid tht d e becuse the vlue for the Riem zet fuctio, this mes tht this series hs the fiite prt f. p e () o Reormliztio/regulriztio theory from diverget series Usig Abel-summtio d formul (), we c give esy method to regulrize diverget itegrls of the form f ( ) d, which is esy to uderstd. This method of reormliztio d regulriztio is bsed o the resummtio of diverget series of power of positive itegers d reltioship i the form of recurrece equtio betwee the diverget itegrl d d its discrete diverget series couterprt method to regulrize diverget itegrls would be the the followig, the ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 686 Split the itegrl bove ito fiite prt c lwys be mde; f ( ) d plus diverget prt f ( ) d, this Epd the itegrd iside f ( ) d ito Luret series of the form the coefficiets of this epsio re give by itegrl over the comple ple (Cuchy s f( z) theorem [] ) dz i ; z Apply itegrtio o ech term of the form which is vlid d well defied for m ; Use the regulriztio for the Hrmoic series logrithmic itegrl logrithmic divergece C m d d use the formul m m d the regulriztio of the d log to regulrize d give fiite meig to the Use formul () to regulrize the diverget itegrls the series m e d m for every m=,,,...,, for, the reormlized vlue for every m of these is ust m e ( m) so Abel d Zet regulriztio give both the sme results, ecept reorm for the hrmoic series which is ot zet regulrizble Aother defiitio of the reormlized ifiite series is mde with the Abel-pl sum formul, use Abel-Pl formul to compute the reormlized vlue of the series e e d, whe the regultor epsilo is te to, this results is logue to zet regulriztio. As emple, let the diverget itegrl be itegrl usig formul () would be d, with c >, the reormlized vlue of this c reg c c c d d c d c d d c log c c 6 (4) ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 687 A more complicted -loop itegrl d dy y c be computed withi our y reormliztio method bsed o the regulriztio d study of diverget series. I this cse, the itegrl hs subdivergece i the vrible which should be reormlized first, the reormlized vlue of this itegrl is d dy = y d dyy ( y ) ydy (5) ( y )( ) d The itegrl iside (5) f( ) is fiite for every positive. ( y )( ) To simplify the clcultios, we c replce (pproimte) this itegrl by qudrture formul with -poits so the sum (qudrture) is esier to wor with. For emple, if we use the Lguerre qudrture formul, vlid for [, ) (see, e.g., []): d y y y y e ( ) ( ) (6) ( y )( ) y Sice ech term iside (6) deped o y, we hve to reormlize the itegrls e ( y ) y dy y, which hve ll them qurtic divergece see if we itroduce cut-off term i the itegrl. 4. This c be We hve coverted -loop itegrl ito ordiry itegrl by usig umericl method d hve lso pplied the Abel resummtio d formul () to the origil itegrl y d dy to get fiite reormlized vlue for it. y o Uderstdig the Csimir effect reormliztio d why the diverget series ( ) hvig fiite physicl vlue Let be the boudry vlue problem Df d f f() f( ) D f E f d E (7) ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 688 Now, if we defie the opertor T D, the sums re the the trces of the powers of the opertor T i terms of the spectrl zet fuctio of the Eergies of the eigevlue problem iside (7) TrceT T, L T s s, L E ( s) (8) The spectrum of problem (7) is discrete, sice we hve imposed the boudry coditios for the eigefuctios f () f ( L ). If we te the limit L, the spectrum is o loger discrete d the trces re give by itegrl isted of discrete sum, TrceT t dt T, L L. This itegrl is still diverget but, if we te the differece betwee the two ( epoetil regultor is ssumed), we c defie reormlized vlue of the diverget series t T, L T, L e dtt e ( ) (9) For the cse of the Hrmoic series, the differece is T, L T, L,which is gi reormliztio of the diverget Hrmoic series. So, i the ed we hve oly fiite vlue for every diverget sum d itegrl. Whe this method is used i the evlutio of the fuctiol determit of opertor with discrete set of eigevlues det( A), the epressio log is diverget i geerl. But we c defie the logrithm of the fuctiol determit s the fiite differece (substrctio of the divergece) Z( s, ) log A LogC Z(, ) Z(,) s s s () where C is fiite costt. For emple, this method c be used to epd the Gmm fuctio d the sie fuctio ito ifiite product over their zeros ( ) si( ) () ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 689 Refereces [] Abrmowitz, M. d Stegu, I. A. (Eds.). "Riem Zet Fuctio d Other Sums of Reciprocl Powers." 3. i Hdboo of Mthemticl Fuctios. New Yor: Dover, pp. 87-88, 97. [] Berdt. B Rmu's Theory of Diverget Series, Chpter 6, Spriger-Verlg (ed.), (939) [3] Elizlde E. Te Physicl Applictios of Spectrl Zet Fuctios, Lecture Notes i Physics. New Series M35 (Spriger-Verlg, 995) [4] Grci J.J ; A commet o mthemticl methods to del with diverget series d itegrls e-prit vlible t http://www.wbbi.et/sciece/moret.pdf [5] Guo L d Zhg B. Differetil Algebric Birhoff Decompositio Ad th reormliztio of multiple zet vlues Jourl of Number Theory Volume 8, Issue 8, August 8, Pges 38 339 http://d.doi.org/.6/.t.7..5 [6] Hrdy, G. H. (949), Diverget Series, Oford: Clredo Press. [7] Prellberg T Mthemtics of Csimir effect vlible olie t http://www.mths.qmul.c.u/~tp/tls/csimir.pdf [8] Shri A. The geerlized Abel-Pl formul pplictio to Bessel fuctios d the Csimir effect e-prit t CERNhttp://cds.cer.ch/record/48795/files/39.pdf [9] Shirov D., Fifty Yers of the Reormliztio Group, I.O.P Mgzies. ISSN: 53-83 Prespcetime Jourl Published by QutumDrem, Ic. www.prespcetime.com