EULER-MACLAURIN SUM FORMULA AND ITS GENERALIZATIONS AND APPLICATIONS

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1 EULER-MACLAURI SUM FORMULA AD ITS GEERALIZATIOS AD APPLICATIOS Joe Javier Garcia Moreta Graduate tudet of Phyic at the UPV/EHU (Uiverity of Baque coutry) I Solid State Phyic Addre: Practicate Ada y Grijalba 5 G P.O Portugalete Vizcaya (Spai) Phoe: () joegarc@yahoo.e MSC : 4C5, 4-, 4H5, 47G99 ABSTRACT: We tudy everal poible geeralizatio of the Euler-Maclauri formula, for everal variable ad ifiite-dimeioal pace ad it poible applicatio to umber theory ad other brache of mathematic. Alo we tudy how uig thi Euler-Maclauri ummatio we ca provide a approimate differetial equatio whoe olutio i the ee of ditributio theory i jut the Merte fuctio M() or the um of Liouville fuctio λ(). Keyword:= Euler-Maclauri um formula, ditributio EULER-MACLAURI SUM FORMULA AD ITS POSSIBLE GEERALIZATIOS AD APPLICATIOS The Euler-MacLauri um formula relatig the um of a ifiite erie to a defiite itegral over the iterval, i the form: B ( r)! f () f ( ) f ( ) D f ( ) D () df ( ) () r If the fuctio f() ad it derivative vaihe a, the formula () ca be ued to obtai approimate reult for the ifiite erie o the left, for other cae we ca k k k make the replacemet D h D ad multiply the whole left term by h (tep) o the ifiite erie i ow f ( h), the mot traight proof for the Euler-Maclauri ummatio formula i to ue the idetity:

2 D I B e e.... D () D D D D e D r ( r)! d D I i the idetity operator, D with e f ( ) f ( ), formula () provide d aother ueful iformatio ice we could replace D by ay differetial or itegral operator, for eample for R if we defie the calar k. k ( a, a,..., a ) ad i the Gradiet, if we iert iide () equatio with the followig property k. e f ( r ) f ( r k ) with r, k R, we have B r ( r)! () ( ) f f k k. f () k. f () r (3) the otatio i the vector whoe compoet are all, however if we defie a iitial vector r the formula (3) till hold eve for ifiite-dimeioal pace, the k. f ( r ) r,,..., R mea that we hould olve the liear otatio g PDE ai. f ( r ) i abece of further boudary or iitial value problem ad if the i ri multivariable Fourier traform for f eit the i k. f d e F( ) k. i. r F(w) = Fourier traform of f (4) certaily if we coider that a fuctio i a vector that belog to a fuctioal pace ad ue the et geeralizatio from uual derivative to Fuctioal oe: d. j d ( ) i j i i i i R F F[ ( y)] F[ ] lim (5) u the if we ue a a vector k, the oe whoe compoet are ki i (Kroecker delta) for a fied value u formula (3) for Fuctioal (Path) itegral read F[ ] D F h h F h u R (6) [ ] [ ] [ ( )] B F[ ] ( )! ( ) ( )... ( ) R h d d... d r r (7) agai h i a tep of the erie (6) ad the fuctioal derivative F[ ] hould be evaluated at the poit, ote that the ( ) ( )... ( ) epreio for the Remaider R i i geeral a ditributio ad will eed to be

3 regularized, the epreio (6) i the Euler-Maclauri ummatio formula for fuctioal pace ad path itegral, the itegral i (6) will be take over all the poible fuctio ad ca be coidered a ifiite dimeioal geeralizatio of the multiple itegral: D[ ] F[ ] d d d... d F(,,..., ) (8) 3 c c c3 c ad ( ) u c for j=,,3,4,..formula (6) ca be ued to defie ome kid of j j Riema um for the path itegral formalim (although due to the delta derivative i cae fuctioal are o-liear we would have ome problem with ditributio theory) aother kid of approach to path itegral i the Poio ummatio formula is[ ] im is[ m ( y)] D[ ] e e ( = tep) W[ ] epid m m (9) with the periodic fuctioal W[ ] W[ ( y)], here S[ ] i claical actio of the ytem ad D[ ] D[ ] M[ ] with M a fuctioal ad D [ ] a tralatioal ivariat meaure that remai ivariat uder the chage ( ) ( ) k ( y) if M i differet from we hould replace the actio iide the epoetial by the epreio ivolvig the logarithm of M a S[ ] i l M[ ], for other kid of Fuctioal itegral we may ue the idetity for Fuctioal itegral is[ ] is[ ] is[ ] D[ ]. e D[ ]. e D[ ]. e S[ ] d. L(, ) (). o Euler um formula ad a differetial idetity for the Merte ad Liouville fuctio: The mai applicatio of Euler-Maclauri formula i to obtai the approimate um of ifiite erie, if f() ad all it derivative vaih a the () give u the approimate um of the ifiite erie if we kow how to obtai the itegral dtf ( t ), i may other cae if we kew a approimate aymptotic epreio of a itegral a ( ) a( ) dtg( t) h( ) a( )... lim a ( ) i i () the from () ad () we could get a aymptotic epaio for erie f ( t). Aother importat property of Euler-Maclauri um formula i that i cae m i a iteger m >, it give a lik betwee the zeta regularized um of the diverget erie ( m) m R i ad the diverget itegral i d m with i the form 3

4 B I m m I m m a m r I m r () (, ) ( / ) (, ) ( ) mr ( r ( r)! ) (, ) ( m ) Ref. [3], amr r ( m ),hece the recurrece i () i fiite if m i a iteger ad ( u) for u=,,3,4,., thee coefficiet are o-zero ule m Euler-Maclauri um formula ca be eteded to iclude ditributio, if we chooe a uitable et of tet fuctio o ( ) C (ifiitely differetiable fuctio) with a coverget um ( ) for every, ad ay ditributio M() M ( ) B M M M ( r)! ( ) ( ) ( ) (3) r M=M() i a ditributio, f g i the calar product o the iterval,, o (3) i the geeralizatio of idetity () to iclude ditributio theory, with the aumptio ad that the idetity e M ( ) M ( a) till hold wheever M i a ditributio. If we coider Euler-Maclauri um formula epreed i the form f ( ) B r f f dtf r r t t t t ( )! (4) i a fied parameter ad lim f, epreio (3) yield to a differitegral t t equatio i variable, to eplore it utility let coider the et itegral idetitie ci ci d ( ) M ( ) d ( ) i ( ) i ( ) (5) ci M() i the Merte fuctio ad ( ) i Liouville fuctio (ee ref. [] ), Let uppoe we wat to tudy the olutio of the differetial-itegral equatio ci f ( ) B f dtf H (l ) f r ( r)! t t t t (6) 4

5 d H() i the Heaviide tep fuctio with Melli traform (l ) H, takig the d Melli traform to both ide o (6), ice. f ( ) F( ) we reach to the cocluio that the igular olutio to (5) i jut y( ) M ( ) Merte fuctio, if we put itead H(l) followig the ame tep we would fid that the igular olutio would be y( ) A( ) ( ) Liouville fuctio the fact of a ditributio olvig a liear ODE hould ot be urpriig,for eample the ditributio g( ) ''( ) olve the differetial equatio (igular olutio) ( ) g ''( ) (4 6 ) g '( ) 6 g( ) for every real value of If we differetiate repect to iide the formula (6) it become y y J ( r)! t t ( ) B r ( ) ( ) f. r t (7) Y() M() A( ) ( ) J() H(l) I all cae the derivative repect to t are evaluated at the poit t=, ad J() i jut the ource of the differetial equatio (6), depedig o the value of J() we fid H ( ) a differet olutio,for other cae of Dirichlet erie of the form, where ( ) H ( ) the Melli ivere eit ad ca be eaily calculated, the the ource J() iide ci (7) i i J ( ) dh ( ), ad the olutio to (6) i jut y( ) a B( ), ci aother eample i the Chebyhev fuctio that ca be obtaied by ettig y( ) ( ) ( ) d ( ) H ( ) d J ( ) d u l( u) (8) Ufortuately, equatio (5) ad () are oly equal i the limit, o the olutio to (6) will deped o the ued y( ) y ( ) i the limit tedig to ifiity we will recover the Merte ad Chebyhev fuctio,uig a uitable ource J() epreed i mathematical laguage we have the limit, ee Ref [7] 5

6 lim y ( ) M ( ) lim y ( ) ( ) J ( ) H (l ), d u l( u) (9) d [] i the floor fuctio ad [ ] ( ), depedig o the value of the d approimatio (7) ca be wore or better for eample if = we have the approimatio B B B 4 y y '( ) y ''( ) y '( ) r ( r)! t t t 4 () Referece: [] Apotol T., "A Elemetary View of Euler' Summatio Formula", America Mathematical Mothly, volume 6, umber 5, page (May 999). [] Apotol T. Itroductio to aalytic umber theory, Spriger ew York (976) [3] Elizalde E Romeo A. Ad Zerbii S.. Zeta regularizatio techique with applicatio World Scietific Pre (994) [4] Gapard Pierre, "r-adic oe-dimeioal map ad the Euler ummatio formula", Joural of Phyic A, 5 (99) [5] Hage K. Path Itegral i Quatum Mechaic, Statitic, Polymer Phyic, ad Fiacial Market, 4th editio, World Scietific (Sigapore, 4) [6] More, P. M. ad Fehbach, " Method of Theoretical Phyic, Part I. ew York: McGraw-Hill, p. 43, 953. [7] Strichartz S. (994). A Guide to Ditributio Theory ad Fourier Traform CRC Pre. ISB [8] Zi Juti, ; Path Itegral i Quatum Mechaic, Oford Uiv. Pre (4) 6