PRODUCT OF DISTRIBUTIONS AND ZETA REGULARIZATION OF DIVERGENT INTEGRALS

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Emerald Thornton
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1 PRODUCT OF DISTRIBUTIONS AND ZETA REGULARIZATION OF DIVERGENT INTEGRALS s AND FOURIER TRANSFORMS Jose Jvier Grci Moret Grdte stdet of Physics t the UPV/EHU (Uiversity of Bsqe cotry) I Solid Stte Physics Address: Address: Prctictes Ad y Grijlb 5 G P.O Portglete Vizcy (Spi) Phoe: () E-il: josegrc@yhoo.es ABSTRACT: Usig the theory of distribtios d Zet lriztio we ge to give defiitio of prodct for Dirc delt distribtios, we show how the fct of oe c be defie coheret d fiite prodct of ddirc delt distribtios is relted to the lriztio of diverget itegrls s d Forier series, for Forier series ig Tylor sbstrctio we c defie lr prt F ( ) defied s fctio for every pls dirc delt series we show the how ( i ) () forl d lytic cotitio for the series N ( i) ci ( ), which is diverget for =, i c be lrized sig cobitio of Eler-Mclri i i ( ) ( PRODUCT OF DIRAC DELTA DISTRIBUTIONS ) ( ) ( ) ( ) Oe of the probles with distribtios, s proved by Schrtz (see ref [] ) is tht we c ot (i geerl) defie coheret prodct of distribtios, for eple P P ( ) P li ()

2 For the cse of the prodct of Heviside step fctio H() with the derivtives of the Delt fctio (d its derivtives ) we hve to del with the proble of diverget ( ) qtities, for eple ccordig to [] we c defie the prodct H, with the id of test fctio ( ) C ( R) s the recrrece ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H () () ( ) '( ) () The cse = is jst H, d coes fro cosiderig the Heviside fctio H() to be the derivtive of ( ), so H ( ) ( ) H ( ) H ( ) If we se the Covoltio theore [5 ] i forl sese, so it c be rded s vlid eve for the cse tht the Forier trsfor re defied ONLY s distribtios (3) ( ) i D ( ) D ( ) F AF dtt ( t) Here A is orliztio (fiite) costt tht depeds o the defiitio yo te for d the Forier trsfor, bt it c ot be depedet o or d D. Uofrttely (3) es o sese sice the itegrl over t is DIVERGENT d eeds to be lrized, if we se the Bioil theore o t ( t) for d itegres i D ( ) D ( ) = i AD ( )( ) i D () (4) i The proble here is tht D () is ifiite d wold eed to be lrizd i order to e sese iside (3) or (4), for + beig Odd iteger, sig Cchy s pricipl vle defiitio P. v (this iposes the coditio tht oly + or - c pper iside (4) ), the proble is tht is still diverget, the se proble hppeed iside () where oe eeds to to lrize ( epressios ) () i order to defie coherete prodct of distribtios ivolvig Heviside step-fctio d Dirc delt d its derivtives. I geerl (4) will be ocottive so we c i geerl epect ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) eple bt () () ( ) ( ) () ( ) ( ) ( ) ( ) () (5) () ()

3 The lst eqlity i (5) coes fro the fct tht () is by sig Cchy s pricipl vle, the cse == is jst the sqre of delt fctio A, this c be obtied fro the zet lriztio s we will see i the et sectio o Zet lriztio for diverget itegrls: I or previos pper [4] we sed the Eler-Mclri stio forl with s f ( ) i order to stblish s s s s s ( s ) B r( s ) rs ( r s) r ( r)! ( r s) > (6) The ide is, give fied we defie s sfficietly lrge so the itegrl s d the series ( s ) i coverge, d the se the lytic cotitio to i eted the defiitio of the s s the egtive vle of the Rie Zet ( ) i, i order to lrize, sig (5) the diverget itegrls, if is i iteger we c set = d (5) becoes esier epressio s B r ( ) r ( ) ( r ) (7) ( r)! ( r ) r The cse -s=- iside (6) c ot be lrized ieditly de to the pole () i, hece to lrize we itegrte with respect to to fid i C log( ), sig Eler-Mclri stio forl pls the lriztio of Hrwitz Zet fctio gi The first three ters of the recrrece (7) re log( ) (, ) d differetitio with respect to s 3

4 / () ( ) B ( ) (9) With r ( ) d ( r ) B beig the Berolli bers e! B, fro the defiitio of or prodct of Dirc delt distribtios give i (4) d sice we wt the idetity H to be tre for every test fctio, we c idetify () (), () () () ( ) B d fro the poit of view of Zet lriztio. Althogh we hve sed oly defiitio for distribtios o R, it c be geerlized to R by sig the defiitio of Dirc delt fctio d Heviside fctio i severl vribles cse we hve chose the lriztio j ( ) i () ( ) j j H ( j ), i y for iteger odd or eve, other defiitio for the Forier trsfor c e fctor differet to pper i (4) for eple Let be e i ( ) REGULARIZATION OF FOURIER INTEGRAL USING DISTRIBUTIONS R, the we c lrize the Forier trsfor tylor series sbstrctio with the defiitio i. d e f F R i. i. e f d e f ( ) ( ) vi ( ) ( ) (see [6] ) R ( ) () () i.. e f f e f N! N! i ()...!!.!...!.... is the lti-ide ottio to write dow the defiitio of Tylor series (9) The Tylor series is fiite d is trcted fter give N so R d f ( ) (ltrviolet divergece ct-off ), this llows s to write dow lr prt of the Forier trsfor pls distribtiol prt for the Forier trsfor i. F ( ) d e f ( ) f () N! R C f () i ( ) N! () (lrized prt = fctio ) (siglr prt = distribtio ) N 4

5 The proble with () coes wheever the itegrl is diverget d we set =, i this cse we shold hve to evlte d other diverget qtities, lso sice two distribtios c ot i geerl be ltiplied the F G oly the lr prts of both F d G F G or G defiitio to c NOT be defied, F, c be defied, here we fid the proble of givig lrized for itegers (,), this ws discssed i (4) (5) (6) d (7) d (8) forle icldig o how to del with with the ifiite ters Zet-lriztio, sll proble we fid here is tht depedig o the defiitio of the Dirc delt fctio vi Forier trsfor etr ter proportiol to or siilr cold pper, this hppes becse slly the defiitio of the Forier trsfor is ot iversl (p to fctor proprtiol to or sqre root of ). So i geerl depedig o the defiitio for the Forier trsfor we shold e the replceet to get the correct reslts. vi PRODUCT OF DISTRIBUTIONS P, P P, P Applyig the covoltio pls the zet lriztio lgorith d the Forier i trsfor for the Heviside fctio H ( ) e ( ) ip we c eted or defiitio of (lrized) prodct of distribtio to iclde the Pricipl vle distribtio P relted to Cchy s pricipl vle of the itegrl ( P pv ), sig gi the Forier trsfor covoltio theore P : i this cse sig the covoltio defiitio ( ) ( ) ip AF dth ( t) ia ( ) () A ( ) () Prodct of P ( ) : sig gi the Forier trsfor for H() () A ( ) ip ( ) AF dth ( t) ( ) (3) 5

6 P P : this cse is fr bit ore coplicted to obti this prodct we eed the idetity dth ( t) H ( t) H () H ( ) H()= / ( ) ip ( ) ip AH () i ( ) AH () P (4) i ( ) ( ) ip d ( ) ip i ( ) ppropite for of the covoltio theore, gi sig the ( ) i ( ) ( ) ip AI ( ) A (5) ( ) ip i ( ) ia () ( ) I ( ) A (6) P d P ( ) i : sig e H ( ) i ( ) P( ) d the covoltio theore we c write dow ( ) ( ) ip i ( ) i ( ) () A I ( ) (7) i ( ) i ( ) P ( ) A AI ( ) (8) P P : sig (4) (7) (8) d the prodct i ( ) P i ( ) P AF dth ( t) H ( t)( t) t (9) i A ( ) The lst epressio i (9) is jst AP 4,gi we hve sed the 6 idetity dth ( t) H ( t) H () H ( ) together with (4) d (5) i order to give fiite eig for the prodct P P, ote tht i epressios (-8) we ( ) ( ) eed to evlte prodcts of the for ( ) ( ) which eed to be lrized by (4) 6

7 Depedig o the order i which covoltio is te we y fid H(-t) or H(t) (-t) or siply t iside (-8), here s lwys A is ber itrodced by the defiitio () te for the covoltio d ( ) I, () re fiite correctios (lriztios ) for the diverget itegrls tht pper whe we try to defie correct prodct of distribtios, fro these forle bove together with the Lebiiz forl (cosidered to be vlid t lest i forl sese) d A B da db B A, we c defie lso P or siilr prodcts d d d ( ) ( ) P d ( ) P ( ) for rbitrry, H ( ) P ( ) ( ) ( ) i i ( ) P AF dth ( t) t ( )( ) I () Here ( ) ( ) P i ( ) AF dt( t) H ( t) A ( ) Ii ( ) () I t dt these itegrls c be lrized vi forl (6) or (7) However if we pt =- i order to evlte H P iside () d () we will fid severl oddities tht prevet s fro defiig coheret epressio, however the derivtive of this prodct of distribtio stisfy d H P P H P H P P P H () Aother possibility is to defie H P T ( ) so its derivtive dt ( P ), sig the Tylor distribtiol series give i [] ( ) ( ) ( ) ( ), sig forle () d () d itegrtio with respect! to we c get H P T ( ) p to soe costt C. Also if we ew how to ltiply H P T ( ) for soe >, (to void the siglr poit ( ) ( ) = ), sig the Tylor distribtiol series ( ) H ( ) H ( ) d! 7

8 the sig () (). Althogh we hve oly cosidered the -D cse, the Covoltio theore, Bioil theore d siilr c be defied lso i R, lso we st te ito ccot tht i geerl for diverget itegrls chge of vrible cold ot wor y dy dr r d si cos, the best ethod wold be to se Fey pretriztio to defie the prodct of itegrls ( )! d d d (3) A A A A A With A ( ) beig diverget itegrl tht c be lrized ( Zet-lriztio ) vi CONCLUSIONS AND FINAL REMARKS Usig the zet lriztio lgorith (6) (7) we hve ged to give fiite (Nocottive) prodct of dirc delt distribtios ( ) ( ), ( ) ( ) d ( ) ( ) H ( ), with H beig the Heviside step-fctio, sice the prodct is ocottive we shold lso te cre whe tig the prodcts ( ) ( ) ( ) ( ( ) ) ( ) ( ) so ssocitivity will ot lwys hold, sig the Covoltio theore pls the se of Forier trsfor, with the -th d -th powers of F * AF dt( t) t A = orliztio costt, will llow s to ( ) ( ) ( copte the prodct p to severl diverget qtities ) (), which re proportiol to the diverget itegrl, this itegrl c be lrized [4] sig the zet lriztio lgorith i order to sbstrct fiite qtities proportiols to ( ) =,,,3,.... Althogh we hve oly eited the cse of dirc delt d its derivtives, i severl cses it cold pper the distribtio i e d si g( ) H ( ) H ( ) (4) d Althogh we hve ot etioed the cse the clcltio of Forier itegrl by settig f ( ) e i, this itegrl c be redced to f ( ) H ( ) f ( ) H ( ) f ( ) f ( ) g( ) i i g( ) e f ( ) e (5) 8

9 ( I this cse we will ecoter diverget ters ) (), whe sig the Leibiz s d d f d H forl to perfor the Tylor sbstrctio er = f. H. sice the derivtive of step fcito ivolves dirc delt, gi we will eed forl (5) (6) d (7) to get soe fiite reslts. If the itegrl of f() hs soe logrithic divergece so f ( ) log y hve to lrize the distribtio H ( ) s, the we H ( ) () ( ) P. f () log (6) Ad the igorig ll the diverget ters proportiol to log (vi coterters) iside () so oly fiite cotribtios will pper iside f ( ) Refereces: [] Colobe, J. F., Eleetry itrodctio to ew geerlized fctios. North- Holld, Asterd, 985. [] Estrd R. Kwl R. A distribtiol pproch to syptotics Bosto Birhäser Birhäser () ISBN: [3] Elizlde E. ; Zet-fctio lriztio is well-defied, Jorl of Physics A 7 (994), L [4] Grci J.J ; Zet Reglriztio pplied to the proble of Rie Hypothesis d the Clcltio of diverget itegrls e-prit vlible t [5] Kler, D. W. A First Corse i Forier Alysis. Upper Sddle River, NJ: Pretice Hll,. [6] Schrf G, Fiite Qt Electrodyics: The Csl Approch, d editio, Spriger, New Yor (995) [7] Zeidler E. Qt Field theory Vol.;A Bridge betwee Mtheticis d Physicists Spriger (9) ISBN:

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0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k. . Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric