Odoardo Volonterio 1, Michele Nardelli 2

Transcription

1 ON SOME APPLICATIONS OF THE VOLONTERIO S TRANSFORM: SERIES DEVELOPMENT OF TYPE N+M AND MATHEMATICAL CONNECTIONS WITH SOME SECTORS OF THE STRING THEORY Odoardo Voloero Mchele Nardell Polecco d Mlao Paa Leoardo da Vc 0 Mlao Ialy Darmeo d Scee della Terra Uverà degl Sud d Naol Federco II Largo S. Marcello Naol Ialy Abrac I h wor we have decrbed a ew mahemacal alcao cocerg he dcree ad he aalyc fuco: he Voloero Traform (V Traform) ad he Voloero Polyomal. We have decrve varou mahemacal alcao ad roere of hem recely he ere develome of he ye N+M. Furhermore we have howed alo varou eamle ad he oble mahemacal coeco wh ome ecor of Number Theory ad Srg Theory.

2 Ide ON SOME APPLICATIONS OF THE VOLONTERIO S TRANSFORM: SERIES DEVELOPMENT OF TYPE N+M AND MATHEMATICAL CONNECTIONS WITH SOME SECTORS OF THE STRING THEORY... VOLONTERIO S TRANFORM GENERALIZED AND SERIES DEVELOPMENT OF TYPE N+M... Geeraled defo of raform V... 6 Geeraled defo of vere raform V. 6 RELATION BETWEEN THE VOLONTERIO S TRANFORM OF LAPLACE AND ZETA.8 Defo ad roere of he Voloero olyomal 0

3 Traformed V of a dcree erodc fuco SERIES DEVELOPMENT OF THE TYPE N+M... 6 Aumo... 6 Proof... 7 CASE (N < M):... CASE (N = M)... CASE (N > M)... PARTICULAR CASES... 5 Cae wh... 5 Cae wh... 7 EXAMPLES... 7 Eamle... 7 Eamle... 8 Eamle... 9 Eamle... 0 ON SOME MATHEMATICAL CONNECTION WITH SOME SECTORS OF STRING THEORY.. Referece..56

4 VOLONTERIO S TRANFORM GENERALIZED AND SERIES DEVELOPMENT OF TYPE N+M Defo (raformed geeraled V) The raform V of a dcree fuco a aalyc fuco of a real varable (or comle) hrough whch oble o a from he world of dcree or fe mahemac he world of dffereal mahemac. The raformed V rovde a overvew hgher ha ca rovde a geerag fuco. The raformed caocal dguhed from geeraled becaue eece baed o couou fuco ad fely dffereable for = 0 whle he geeraled baed o a couou fuco ad fely dffereable a (where for we oba obvouly he raform caocal). The roere of raformao ad a-raformao of he raform V are deede from he fac ha we coder he raformed caocal or geeraled. Defo (vere geeraled raform V) The vere raform V of a aalyc fuco of a real varable coue he ero ad fe me dffereable a = 0 (caocal) or a = (geeraled) ( oher word a fuco develoable MacLaur or Taylor ere ) a dcree fuco defed hrough whch oble he rao from he dffereal world o he world of dcree or fe mahemac.

5 DEFINITION OF TRANSFORM V Le a dcree fuco he we ca defe he raformao a follow: (.) CONDITION OF EXISTENCE AND UNIQUENESS OF THE TRANSFORM V To eure he codo of eece of he raform mu be eured he followg relao: (a) where he radu of he covergece whle e he Eulero-Neero coa. The relao (a) a eceary codo ha ha bee demoraed elog he codo of he roo of Cauchy- Hadamard whle he codo of uquee ca be arbued o he roere of ere of ower where e he Euler-Neero coa. Defo of vere raform V We defe wh vere raform of he dcree fuco obaed by he followg defo: (b) Defo of vere raform V Or by he followg formula alerave o he (b): (c) where eceary ad uffce codo becaue (c) vald ha afed he codo where he radu of covergece (a). 5

6 Defo of vere raform V (.) or for N 0 (ee c): T d 0 V : R V e co (d) Geeraled defo of raform V A follow we defe he geeraled raformed :. Geeraled defo of vere raform V We defe a geeraled vere raform of he dcree fuco obaed by he followg defo:. Th defo arcularly ueful all hoe cae where he fuco ca o e 0. Aoher alerave defo he followg:.5 6

7 Fudameal roere of he raform V Amog he geeraled raform ad he caocal raform ueful o ee md he followg dey:

8 RELATION BETWEEN THE VOLONTERIO S TRANFORM OF LAPLACE AND ZETA We coder he followg defo of Gamma fuco Lalace Traform ad Zea Traform:.9 we have:.0 hece:.. bu for he lef-had de we oe ha:. hece:. whle for he rgh-had de ug:.5 ug we have:.6.7 8

9 vce vera.8 more geerally he cae of geeraled raform V we have:.9.0 9

10 ADDITIONAL DEFINITIONS I order o read ad erre able comlee ee clarfcao eeded o he fuco ad abbrevao ha have bee roduced ad alo wll be eeal of he eamle ha follow afer he able. I ay cae before roceedg o he l of raformao ueful o coder he followg relao defo ad fuco. Defo of oeraor Wh he ymbol we defe he followg oeraor: (.) where alcao eraed me o a deermed fuco he followg way ereed by he oeraor of he alcao for eamle we coder V hece: Defo ad roere of he Voloero olyomal (.) oher formula o deerme he olyomal (ee TF N ad 0) are he followg: (.) or by he followg recurve formula: (.) (.5) or: (.6) From he above defo by he Voloero raform: 0

11 (.7) Defo of Beroull Polyomal: The Beroull olyomal (ee TF N ) : (.8) The geerag fuco of he Beroull olyomal : (.9) Defo of Eulero Polyomal: Eulero Polyomal (ee TF N ): (.0) The geerag fuco of he Eulero olyomal : (.) Defo of Laguerre Polyomal: Laguerre Polyomal (ee TF N 5 ad N ) (.) Defo of he Beel Polyomal of he fr d: (.) Defo of he Herme Polyomal (.)

12 PROPERTIES OF THE TRANSFORM V PF N. Fuco Defo

13 TRANSFORMATION V OF SOME KNOWN FUNCTIONS TF N. Fuco Defo (dem for )

14 TRANSFORMATION V OF SOME KNOWN FUNCTIONS TF N. Fuco Defo wh 9

15 FUNDAMENTAL PROPERTIES OF THE INVERSE TRANSFORM V PA N. 5 Fuco Defo

16 AF N. INVERSE TRANSFORM OF KNOWN FUNCTIONS Fuco Defo GTF N. Fuco GENERALIZED TRANSFORMATION OF NOTE FUNCTIONS Defo

17 GTF N. Fuco GENERALIZED TRANSFORMATION OF NOTE FUNCTIONS Defo wh 7

18 GTF N. Fuco GENERALIZED TRANSFORMATION OF NOTE FUNCTIONS Defo 9 0 8

19 GENERALIZED INVERSE TRANSFORMATION OF NOTE FUNCTIONS GAF N. Fuco Defo

20 From he GAF.6 we oberve:.5 Where eecally f we wa o ue h roery we mu remember o ru he re of he calculao wh a eed equvale baed o he roere (.6) (.7) ad (.8)

21 Eamle PROBLEM Solve he followg equao o he fe dfferece of he d order. (.8) Now o olve uch a mle equao o he fe dfferece of he ecod order homogeeou wh coa coeffce may be ued varou mehod cludg he mehod of he geerag fuco ad he mehod ug he raform realed here. * * * * * SOLUTION a) METHOD OF THE GENERATING FUNCTION We coder he followg geerag fuco: (.9) (.0) (.) (.) (.) ag o accou he followg obervao: (.) From whch we ge he ew geerag fuco (have already bee codered he al codo): (.5)

22 b)resolving METHOD BY TRANSFORMED V Callg wh T h raformao from he varable o he varable ad lacg wh we oba: (.6) (.7) The characerc equao aocaed : (.8).e. he oluo of he dffereal equao wh he al eg : (.9) where ow a-raform obag he oluo earched: (.50)

23 Traformed V of a dcree erodc fuco: Le be a arcular dcree fuco whch vald he followg relao:.5 Tha o he Fourer ere he dcree doma:.5 we ca aly he raformao V o he (.5) a (.5):.5 for he raformao able we oba: Now relacg of he (.5) he eq. (.56) we oba he eq. (.57):.57.58

24 hece eadg he erm we oba:.59 Now for he ame defo of raformed we have:.60.e. for hece e amely:.6.6 from whch we deduce he followg relao:.6.6 I he arcular cae where he erod N of he erodc dcree fuco very large or eve edg o fy we roceed a follow:

25 where he e amely hece rewre he (.6) a follow:.65 ad for we have hece rewre he eq. (.65) a follow: (.66).e.: (.67) Gve ha he raformed of we oba he equaly: (.68) from whch we have ha: (.69) (.70) 5

26 SERIES DEVELOPMENT OF THE TYPE N+M Aumo The e o fd a form equvale o he fe um below wh ad : (.) We wll olve h roblem by ug he Voloero Traform. I hee roof ad eamle we wll coder he covergece roblem here he oluo ad crera adoed. 6

27 Proof Coder he followg eamle wh ad eeded o uderad he roof ha wll follow where a dcree erodc fuco of value of erod hfed by M. Table Table Thece for he (.0) ad o he Table ad where he relao bewee ad we oba he followg Table: Table / - -/ -/ 0 / / / 5/ Wh he Table eay o uderad he followg equvalece: (.) Now for he defo () of he Voloero Traform we ca wre he followg relao (ee able ad defo aached): (.) Of coure he relao (.) he oe ha wll lead u o oba he geeraled oluo of he ereo (.). 7

28 We coder he followg ow relaoh [A.V. Oehem R.W. Schafer Elaboraoe umerca de egal (Dgal Sgal Proceg) Fraco Agel Edo]: (.) ad more geeral wh M N we defe he followg ereo: (.5) of he (.) we mu arrow he feld o oly o-egave eger amely he (.) here mu reur he ero for each eger value egave o we have o rewre (.) a follow: (.6) where a e dcree fuco of Heavde defed a: (.7) furhermore we oberve ha: (.8) Now lacg we deduce he dcree fuco (.9) ad he hrough he obervao (.8): (.0) Or he equvale ereo: (.) 8

29 So o ule he raformed V ueful o rewre (.0) he followg way: (.) he we ca rewre he relao (.) a follow: ad hece: (.) From he PF N. 9 whch gve here for coveece: (.) we have: (.5) Whle for he PF N.5 whch gve here wh : (.6) ad hece: (.7) We oba fally he followg equvalece: (.8) To faclae he uderadg of he eamle ha follow wll call wh 9

30 ad wh : maulag he he followg way: (.9) we oberve ha he de brace of he (.9) recely he fuco (.0) 0

31 CASE (N < M): Eamle.: where goe from 0 o / / / hece: (.) Eamle.: wh from 0 o 0: / / / hece: (.)

32 Eamle.: where goe from 0 o / / / / / hece: (.)

33 CASE (N = M) Eamle.: where goe from 0 o / / / / / / / hece: (.) Eamle.: where goe from 0 o / / / / / / / hece: (.5)

34 CASE (N > M): Eamle.: where goe from 0 o / / / / / / hece: (.6) *** I geeral from hee obervao by duco we coclude for deduco he followg equvalece: (.7) Where wh mler form: reree he fuco ha reur he eger ar of ad he we ca rewre he (.8) a (.8) where f ad arcular for M 0 we have hece: (.9)

35 PARTICULAR CASES Cae wh Cae wh : (.0) I ca be how ealy from (.) ad (.) ha he fuco afe he followg dffereal equao: (.) hece: (.) we have for he TF N. furhermore we have: (.) hece: (.) afer everal e we oba: (.5) where here we earae he real ar from he magary ar: (.6) follow: 5

36 (.7) ad hece: (.8) I arcular for we have: (.9) From. how ha: ad Pug we oba bu ad hece 6

37 Cae wh Rewrg he eq. (.8) a follow: (.0) h eq. we relacg he followg ereo ad hece (.) *** EXAMPLES Eamle Calculae Soluo We coder wh N= ad M= ad we relace he (.) obag: (.) he hrd ummao of he (.) ull becaue M<M hece: bu he we roceed he e: 7

38 (.) (.) (.5) (.6) for he TF.5 we have: (.7) (.8) Eamle Calculae. We hall ee how h eamle he Voloero olyomal led hemelve o rovdg a geeraled oluo of h roblem. Soluo I h cae M=0 hu he la erm of he (.0) ull: (.9) For he TF. N. we oba: 8

39 (.50) where he Voloero olyomal of order. Pug ad we have : (.5) (.5) (.5) Here we ca coec wh he Ramauja equao cocerg he umber 8 ha a Fboacc umber ad led o he hycal vbrao of he uerrg.e. W 0 / cow w e d 0 a log coh w w e w w coh log h. (.5b) Eamle Calculae. Soluo (.5) (.55) 9

40 (.56) (.57) Now o calculae he raform we ue he TF No. 7: (.58) Thece: (.59) Afer ome calculao we have: (.60) Eamle Calculae Soluo Well ee how h eamle relaed o he Laguerre olyomal. (.6) (.6) (.6) from he TF N. 5 we have: (.6) he relacg: 0

41 (.65) Pug N= ad M= we have : (.66) (.67) (.68) (.69) (.70) Th la equao ca be coeced wh he Euler Gamma Fuco ad wh he umber led o he hycal vbrao of he booc rg.e.. Ideed wh regard he umber h relaed o he mode ha correod o he hycal vbrao of he booc rg by he followg Ramauja fuco: cow w e d 0 a log coh w e w w log w. Thece we have he followg ereo:

42 w w e d e w a W w w w h 9 6 coh log coh co log : 7 0. (.70b) ON SOME MATHEMATICAL CONNECTION WITH SOME SECTORS OF STRING THEORY I 968 Veeao rooed he followg heurc awer A (.) wh 0. Euler Gamma fuco ha ole he egave real a a eger value wh redue...! (.) Hece a fed he amlude ha fely may ole a 0 for 0 or 0 M (.) wh redue

43 A!...!. (.) I he booc rg he mle vere oeraor he oe for he achyo ae 0 N hece / M. We have: V d g e d g X ; 0; V. (.5) Wh regard he -o achyo amlude we have he followg equao: l j l j m m m l j d C SL Vol g. (.6) Seg m we ed u wh l j l j l j d C SL Vol g. (.7) Afer fg he SL C varace by ug he ero o a 0 ad we ed u wh d g (.8) ug Gamma fuco dee h ereo ca be gve a ce form. Oe mu ue he egral rereeao c b a c b a d b a (.9) where c b a. Wh h (.8) ca be how o be equal o / / / / / / u u g (.0) erm of he Madelam varable

44 ; ; u (.) whch afy o hell (.e. ue he achyo ma / M ) 6 M u (.) We ca wre alo he followg mahemacal coeco: d g / / / / / / u u g 6 M (.) Th ereo ca be relaed wh he followg Ramauja modular equao led wh he mode (.e. 8 ha alo a Fboacc umber) ha correod o he hycal vbrao of he uerrg: log coh co log 8 0 w w e d e w a w w w. (.) Thece we have he followg relaoh: d g / / / / / / u u g 6 M

45 log coh co log 0 w w e d e w a w w w. (.5) We oe ha h relaoh ca be relaed alo wh he eq. (d).e. he vere raform of V hece we oba h furher mahemacal coeco: 0 co : d e V V T R d g / / / / / / u u g 6 M log coh co log 0 w w e d e w a w w w. (.5b) b)he oe rg caerg Wh regard he oe rg caerg he amlude comued wh oeraor ero alog he boudary of he d whch ma oo he real a of he comle lae. The equao of he amlude : ˆ ˆ... X X e e d R SL Vol g 6 l j j j d R SL Vol g. (.6) For a gve orderg he redual ymmery ca be ued o f o o 0 0 ad. The reulg ereo coa a gle egrao for 0

46 6 0 d g. (.7) Th egral relaed o he Euler Bea fuco (hece wh he Euler Gamma fuco) 0 b a b a d b a B b a. (.8) Whece ug ow he achyo ma / M oe recover he Veeao amlude g. (.9) Thece we have he followg oble mahemacal relaoh bewee.6) (.7) ad (.9): ˆ ˆ... X X e e d R SL Vol g 6 l j j j d R SL Vol g 0 d g g. (.0) Alo h relaoh ca be relaed wh eq. (d) hece we oba h furher mahemacal coeco: 0 co : d e V V T R ˆ ˆ... X X e e d R SL Vol g 6 l j j j d R SL Vol g 0 d g g. (.0b) c) Four o amlude for he achyo from CFT The groud ae achyo he wed ecor correod o:

47 M N (.) For he ear margal achyo he large N lm whch are he N h rereeao ecor he vere oeraor he ~ H ~ H N N V e e e e e.. (.) The four o amlude for hee lowe lyg achyo ca ow be comued by ag wo verce he 00 rereeao ad wo he rereeao. ~ ~ C d V 0 C ~ ~ e TFe TFV V e TFe TFV. (.) C g The coa c C where C relaed o gc by C gc. (.) Th amlude ca ow be comued ad gve by I C. d C F (.5) where F he hyergeomerc fuco F N N 0 N F ;; dyy y N y N (.6) ad 7.

48 8 I he large N aromao /... N O N F. (.7) Noe ha he erm rooroal o N / (.7) hf he -chael ole. There a addoal facor of. due o whch he coac erm from ay of he erm of (.7) aar from would a lea be of / N O. Wh h obervao he egral ca ow be erformed for F.. C I m u g c. (.8) Now ug m u 8 m m m g I c. (.9) where we have o ead he gamma fuco. Alo here we ca wre he followg relaoh bewee (.5) ad (.9): C F d C I.

49 9 8 m m m g c (.0) Alo h ereo ca be relaed wh he eq. (d) ad wh he Ramauja modular equao cocerg he umber 8 ad hece we oba h furher mahemacal coeco: 0 co : d e V V T R C F d C. 8 m m m g c log coh co log 0 w w e d e w a w w w (.0b) d) ereo cocerg he four achyo amlude CSFT Wh regard a cloed aalycal ereo for he off-hell four achyo amlude CSFT Gddg gave a elc coformal ma ha ae he Rema urface defed by he We dagram o he adard dc wh four achyo vere oeraor o he boudary. Th coformal ma defed erm of four arameer. The four arameer are o deede varable. They afy he relao (.) ad 0 0 (.) where 0 defed by

50 50 0 F K E K F E. (.) I (.) K ad E are comlee ellc fuco of he fr ad ecod d F he comlee ellc egral of he fr d. The arameer ad afy (.). (.5) By ug he egral rereeao of he ellc fuco oble o wre he equao (.) a ueful form / / d K d E. (.6) To ead (.6) for mall ad we have o dvde he egrao rego o hree erval uch a way ha he quare roo he deomaor of (.6) ca be coely eaded ad he egral erformed. For eamle coder he egral he fr erm of (.6) ca be rewre a / / d d d d (.7) I each egral of he rh he egrao doma coaed he covergece radu of he Taylor eao of he quare roo coag o ha hey ca be afely eaded ad he egral erformed. Wh h rocedure oe ge he followg equao equvale o (.6): 0!! E

51 5 0!! l K l (.8) Thece from (.6) ad (.8) we ca wre he followg mahemacal relaoh: / / d K d E 0!! E 0!! l K l. (.9) Alo h ereo ca be relaed wh he eq. (d) ad hece we oba h furher mahemacal coeco: 0 co : d e V V T R / / d K d E

52 5 0!! E 0!! l K l (.0) e) hycal erreao of he orval ea ero erm of achyoc rg ole The four-o dual rg amlude obaed by Veeao wa R B d A u A A A (.) where he Regge rajecore he reecve u chael are: u u. (.) The coervao of he eergy-momeum yeld: 0. (.) We have alo ha he um 8 u (.) ma u of m Plac = whe all he four arcle are achyo ad oe ha he o-hell codo:

53 m m Plac = (.5) he aural u LPlac = uch ha he rg loe arameer hoe u gve by LPlac = / ad he rg ma ecrum quaed mulle of he Plac ma mplac =. From he coervao of eergy-momeum (.) ad he achyo o-hell codo eq. (.5) oe ca deduce ha:. (.6) Therefore from eq. (.) (.6) raghforward o how: u 8 (.7) Th relaoh amog u m 8 wll be crucal wha follow e. From eq. (.) (.) ad (.7) we lear ha:. (.8) There e a well-ow relao amog he fuco (Euler Gamma fuco) erm of fuco A u (Rema ea fuco) aearg he ereo for A u cae he egrao rego he real le ha defe lead o he very mora dey whe fall de he crcal r. I h eq. (.) ca be dvded o hree ar ad A u B (.9) where ad are cofed o he eror of he crcal r. The dervao behd eq. (.9) rele o he codo eq. (.8) ad he dee 5

54 co co co (.50) (.5) lu he remag cyclc ermuao from whch oe ca fer (.5) (.5). (.5) Therefore eq. (.50) (.5) allow u o reca he lef had de of (.9) a A u B co co co. (.55) Ad fally he ow fucoal relao co (.56) cojuco wh he codo A u uch ha (.9) ereg elcly he rg amlude wha eablhe he mora dey eher erm of ea fuco or erm of fuco. I cocluo we have he followg ereg relaoh bewee he eq. (.) (.9) ad (.55): A A A Au d B R 5

55 55 B u A co co co (.57) from whch we ca o oba he followg equvale ereo: R B d A 8 co 8 co 8 co (.58) I h ereo here are boh π ad 8.e. he umber ha coeced wh he mode ha correod o he hycal vbrao of a uerrg by he followg Ramauja fuco: log coh co log 8 0 w w e d e w a w w w. (.59) Thece he fal mahemacal coeco: R B d A 8 co 8 co 8 co log coh co log 0 w w e d e w a w w w (.60)

56 Referece Odoardo Voloero Mchele Nardell Fraceco D Noo O a ew mahemacal alcao cocerg he dcree ad he aalyc fuco. Mahemacal coeco wh ome ecor of Number Theory ad Srg Theory. Feb- 0 h://emlocal.e.ac.u/eole/aff/mrwa//ea/ardell0.df 56