ABEL RESUMMATION, REGULARIZATION, RENORMALIZATION AND INFINITE SERIES
|
|
- Piers Hall
- 5 years ago
- Views:
Transcription
1 ABEL RESUMMATION, REGULARIZATION, RENORMALIZATION AND INFINITE SERIES Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EHU (Uiversity of Bsque coutry) I Solid Stte Physics Addres: Prctictes Ad y Grilb 5 G P.O Portuglete Vizcy (Spi) Phoe: () E-mil: osegrc@yhoo.es ABSTRACT: We Study the use of Abel summtio pplied to the evlutio of ifiite series d ifiite (diverget) itegrls, we give severl emples of how we c obti regulriztio for the cse of diverget sums d itegrls. Keywords: = Abel sum formul,abel-pl formul, poles, ifiities, reormliztio, regulriztio, multiple itegrls, Csimir effect. Abel summtio for diverget series: Give power series of the form < we defie the Abel resummtio of the series which is coverget o the regio s the limit = = lim A( S), if such limit eist we will sy tht the series summble to the vlue A(s). is Abel As emple let be the series [6] + d ( ) = = = B + d + + () Ufortutely the series is NOT Abel summble, this is due to the pole t = of the fuctio ( ), however Guo [5] studied this series d gve the followig idetity usig epoetil regultor. ε d Γ ( + ) Z( ) ( ) e = = + ε ε + dε e ε =! ()
2 Where we hve used iside () the Tylor epsio ivolvig Beroulli s umber = B d the epressio for egtive vlues of the Riem e =! B zet fuctio ζ ( ) =. To evlute the Riem zet iside () for egtive vlues we will eed the s π s Riem s fuctiol equtio defied by ζ ( s) = ( π ) Γ( s)cos ζ ( s), π with Γ( s) Γ( s) = si( π s) They itroduce smll prmeter epsilo d fter clcultios te ε, ufortutely for = - Guo s method gives oly ifiite swer e ε = log ε, this is becuse the followig epressios for the -th Hrmoic umber d for the Lplce trsform of the logrithm H = γ + log 3 εt γ + log ε dte log t = (3) ε Where γ = is the Euler-Mscheroi costt. ε If we te () d igore the pole prt we hve tht f. p e = ζ ( ) for every ecept =-, this is precisely the vlue of the series obtied vi Zet regulriztio, so Abel resummtio d Zet regulriztio re lied d give the sme swer for the diverget series provide we igore the poles To study emple of how the regulriztio d reormliztio of the poles is mde we will study the Csimir Effect ε o Csimir effect: The Csimir effect is physicl force due to the qutiztio of Electromgetic fields, see [7], i the simplest versio of the Csimir effect the vcuum Eergy of the system per uit of Are A is give by / π h π 3 π rdr r 3 E hc c = + = A 4π 6 (4) h Here, π 8 c = 3 m / s is the speed of light i the vcuum. 34 h = =.54 J. s is the reduced Plc s costt d
3 If we use Zet regulriztio [3] we fid the vlue 3 =, if we isert this vlue iside (4) we get the correct eperimetl vlue of Csimir effect E cπ = h Fc d E cπ so = = h. 3 4 A 7 A d A 4 The physicists pproch to Csimir effect is bit more complicte, for emple they use reormliztio d compute the qutity δ hcπ 3 ε 3 εt E = Ediscrete E = e dtt e 3 6 (5) This differece c be computed with the id of the Euler-Mcluri sum formul B (6) ( )! ( ) 3 ε f ( ) f ( ) d = f () f ( ) = e = Or usig the Abel-Pl sum formul with ε f () f ( it) f ( it) 3 ε f ( ) f ( ) d = + i dt f ( ) = e πt e (7) t Γ ( + ) If we retur to Guo s formul (), d we use the idetity dte ε t = + we ε fid the followig. Z( ) Γ ( + ) ε tε e == ( ε ) + dte t = + =! (8) ε So lthough the Abel regulriztio is ot vlid for the series differece, the (9) ε tε = e dte t = ζ ( ) ε Mes perfect sese d is lwys FINITE, lso for the cse =- we fid tht the Hrmoic series is summble d its sum is equl to Euler-Mscheroi costt = γ fter removig the regultor e ε. 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 3
4 i, h = i ( g g ) ( ) E = ctrce g () The opertor iside () is the Lplce-Beltrmi opertor d i, g = det g g g,, g,, is determit of mtri, equtio () is the epressio for the vcuum eergy of 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= r m r ( m r + ) d r= B Γ ( m + ) ( r)! Γ( m r + ) () Where is positive iteger, d the ifiite sum iside (9) must be uderstood i the sese of Abel regulriztio, so i e ε i= Also this recurrece () is fiite for 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, i this cse we d c lso use the idetity = m m vlid for Re( m ) > m The cse m=- is ot icluded d must be te seprtely, if we te the fiite prt f. p e ε ε e = γ or if we use the epressio f ( ) = iside the + Euler-Mcluri summtio formul f ( ) + f ( ) B ( ) ( ) f ( ) = f ( ) d + ( f ( ) f ( ) ) ( )! () + = Ad tig ito ccout the followig series epsio for the Digmm fuctio Γ '( ) B Ψ ( ) = = log + r () γ Γ( ) r= Ψ = (3) 4
5 We get the reormlized result for the itegrl with logrithmic divergece i d the form = log, this mes tht i regulrized/reormlized + reorm sese the 3 itegrls d + d d d re equl to For the cse =, which is the first term iside the recurrece formul () we f ( ) + f ( ) d = + e ε = fid tht this is becuse the vlue ζ ( ) = for the Riem zet fuctio, so if we te the fiite prt of the diverget sum we get the fiite vlue f. p e ε = ζ () 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 y perso could uderstd sice it uses very simple mthemtics, this method of reormliztio regulriztio is bsed o the resummtio of diverget series of power of the positive itegers d lso o reltioship i the form of recurrece equtio betwee the diverget itegrl diverget series couterprt d d its discrete, the method is the followig. Split the itegrl bove ito fiite prt f ( ) d plus diverget prt f ( ) d, this c lwys be mde Epd the itegrd iside f ( ) d ito Luret series of the form with coefficiets give by itegrl over the comple ple f ( z) usig Cuchy s theorem [] = dz + πi z Apply itegrtio o ech term of which is vlid d well defied for m C d = m m+ d the formul m 5
6 Use the regulriztio for the Hrmoic series logrithmic itegrl the the diverget logrithmic itegrl = γ d of the d = log to regulrize d give fiite meig Use formul () to regulrize the diverget itegrls m=,,,..., with Abel resummtio for every m of this series is ust m e ε m d for every, the reormlized vlue m ε e = ζ ( m) so Abel d Zet regulriztio give both the sme results, ecept for the hrmoic series 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 reorm ε ε e e d whe the regultor epsilo is te to, this results is logue to zet regulriztio. As emple, let be the diverget itegrl d, with c >, the + c reormlized vlue of this itegrl usig formul () would be d = d c d + c d d = c log c c + 6 reg + c + c + c (4) A more complicte -loop itegrl d dy y c be computed with our + y + reormliztio method bsed o the regulriztio d study of diverget series, i this cse, the itegrl hs sub divergece i the vrible which should be reormlized first, the reormlized vlue of this itegrl is d dy = + y + d + + ( + y + )( + ) dyy ( y ) ydy (5) The itegrl iside (5) d = f ( ) is fiite for every positive, to ( + y + )( + ) 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 [] d e y ( y + ) y ( y + ) ω (6) ( + y + )( + ) + + y + ( ) ( ) = 6
7 Now, ech term iside (6) deped o y so we hve to reormlize the diverget itegrls ( is the umber of poits of the qudrture formul used) e ( y + ) y ω dy 4, this hs qurtic divergece Λ, this c be + + y + ( ) ( ) = see if we itroduce cut-off term i the itegrl, we hve coverted -loop itegrl ito ordiry itegrl by usig Numericl method d pplyig the y Abel resummtio d formul () to our origil itegrl d dy + y + o Uderstdig the Csimir effect reormliztio d why the diverget series = ζ ( ) hs fiite physicl vlue: Let be the boudry vlue problem D f d f = f () = f ( π ) = D f = E f d E = (7) The, if we defie the opertor T = D, the sums re the trces of the powers of the opertor T i terms of the spectrl zet fuctio of the Eergies of the eigvlue problem iside (7) = Trce ( T ) = ζ T, L = π ( ) T s ζ s, L = π = E = ζ ( s) (8) The spectrum of problem (7) is discrete, sice we hve imposed the boudry coditios for the eigefuctios f () = f ( L = π ) =, if we te the limit L the spectrum is o loger discrete d the trces re give by itegrl isted of discrete sum, Trce ( T ) = t dt = ζ T, L L, This itegrl is still diverget but if we te the differece betwee the ( epoetil regultor is ssumed), d we c defie reormlized vlue of the diverget series ε tε ζ T, L = π ζt, L = = e dtt e = ζ ( ) (9) Ad for the cse of the Hrmoic series, the differece is ζ T, L = π ζ T, L = = γ which is gi reormliztio of the diverget Hrmoic series, so i the ed we hve oly fiite vlue. 7
8 This method is use i the evlutio of the fuctiol determit of opertor with discrete set of eigevlues det( A ) = λ, i geerl the epressio log λ is diverget but we c defie the logrithm of the fuctiol determit s the fiite differece (substrctio of the divergece). log A LogC Z(, ) Z(,) = + Z( s, ) = ( + λ ) s s s () Ad C is fiite costt, this method is used for emple to epd the Gmm fuctio d the sie fuctio ito ifiite product over their zeros. π = + Γ ( + ) si( π ) = π () 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 , 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 [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 [6] Hrdy, G. H. (949), Diverget Series, Oford: Clredo Press. [7] Prellberg T Mthemtics of Csimir effect vlible olie t [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 8
9 [9] Shirov D. ; Fifty Yers of the Reormliztio Group, I.O.P Mgzies. 9
Abel Resummation, Regularization, Renormalization & Infinite Series
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
More informationZETA REGULARIZATION METHOD APPLIED TO THE CALCULATION OF DIVERGENT INTEGRALS
ZETA REGULARIZATION METOD APPLIED TO TE CALCULATION OF DIVERGENT INTEGRALS s Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EU (Uiversity of Bsque coutry) I Solid Stte Physics Addres: Prctictes
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationOptions: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.
O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8
More informationChapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Dr. Hamid R. Rabiee Fall 2013
Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Dr. Hmid R. Rbiee Fll 03 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Outlie Itroductio to the -Trsform Properties
More information2. Infinite Series 3. Power Series 4. Taylor Series 5. Review Questions and Exercises 6. Sequences and Series with Maple
Chpter II CALCULUS II.4 Sequeces d Series II.4 SEQUENCES AND SERIES Objectives: After the completio of this sectio the studet - should recll the defiitios of the covergece of sequece, d some limits; -
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationZETA REGULARIZATION APPLIED TO THE PROBLEM OF RIEMANN HYPOTHESIS AND THE CALCULATION OF DIVERGENT INTEGRALS
ZETA REGULARIZATION APPLIED TO THE PROBLEM OF RIEMANN HYPOTHESIS AND THE CALCULATION OF DIVERGENT INTEGRALS Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EHU (Uiversity of Bsque coutry) I Solid
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationFrequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
More information0 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
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationEigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):
Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationChapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Hamid R. Rabiee Arman Sepehr Fall 2010
Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Hmid R. Riee Arm Sepehr Fll 00 Lecture 5 Chpter 0 Outlie Itroductio to the -Trsform Properties of the ROC of
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationRemarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
More informationPower Series Solutions to Generalized Abel Integral Equations
Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp. 5-54 http://www.isciece.org/jourl/ijmcs Power Series Solutios to Geerlized Abel Itegrl Equtios Rufi Abdulli * Deprtmet of Physics d Mthemtics,
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probbility d Stochstic Processes: A Friedly Itroductio for Electricl d Computer Egieers Roy D. Ytes d Dvid J. Goodm Problem Solutios : Ytes d Goodm,4..4 4..4 4..7 4.4. 4.4. 4..6 4.6.8 4.6.9 4.7.4 4.7.
More information1 Section 8.1: Sequences. 2 Section 8.2: Innite Series. 1.1 Limit Rules. 1.2 Common Sequence Limits. 2.1 Denition. 2.
Clculus II d Alytic Geometry Sectio 8.: Sequeces. Limit Rules Give coverget sequeces f g; fb g with lim = A; lim b = B. Sum Rule: lim( + b ) = A + B, Dierece Rule: lim( b ) = A B, roduct Rule: lim b =
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationDIGITAL SIGNAL PROCESSING LECTURE 5
DIGITAL SIGNAL PROCESSING LECTURE 5 Fll K8-5 th Semester Thir Muhmmd tmuhmmd_7@yhoo.com Cotet d Figures re from Discrete-Time Sigl Processig, e by Oppeheim, Shfer, d Buck, 999- Pretice Hll Ic. The -Trsform
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More informationCrushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric
More informationSurds, Indices, and Logarithms Radical
MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More information9.5. Alternating series. Absolute convergence and conditional convergence
Chpter 9: Ifiite Series I this Chpter we will be studyig ifiite series, which is just other me for ifiite sums. You hve studied some of these i the pst whe you looked t ifiite geometric sums of the form:
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl
More informationTHE THEORY OF DISTRIBUTIONS APPLIED TO DIVERGENT INTEGRALS OF THE FORM
THE THEOY OF DISTIBUTIONS APPLIED TO DIVEGENT INTEGALS OF THE FOM ( ) u ( b) Jose Jvier Gri Moret Grdute studet of Physis t the UPV/EHU (Uiversity of Bsque outry) I Solid Stte Physis Address: Address:
More informationLecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More informationCourse 121, , Test III (JF Hilary Term)
Course 2, 989 9, Test III (JF Hilry Term) Fridy 2d Februry 99, 3. 4.3pm Aswer y THREE questios. Let f: R R d g: R R be differetible fuctios o R. Stte the Product Rule d the Quotiet Rule for differetitig
More informationReview of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
More informationDiscrete Mathematics I Tutorial 12
Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More information( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
More informationSUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
SUTCLIFFE S NOTES: CALCULUS SWOKOWSKI S CHAPTER Ifiite Series.5 Altertig Series d Absolute Covergece Next, let us cosider series with both positive d egtive terms. The simplest d most useful is ltertig
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More informationMathematical Notations and Symbols xi. Contents: 1. Symbols. 2. Functions. 3. Set Notations. 4. Vectors and Matrices. 5. Constants and Numbers
Mthemticl Nottios d Symbols i MATHEMATICAL NOTATIONS AND SYMBOLS Cotets:. Symbols. Fuctios 3. Set Nottios 4. Vectors d Mtrices 5. Costts d Numbers ii Mthemticl Nottios d Symbols SYMBOLS = {,,3,... } set
More informationdenominator, think trig! Memorize the following two formulas; you will use them often!
7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges
More informationCertain sufficient conditions on N, p n, q n k summability of orthogonal series
Avilble olie t www.tjs.com J. Nolier Sci. Appl. 7 014, 7 77 Reserch Article Certi sufficiet coditios o N, p, k summbility of orthogol series Xhevt Z. Krsiqi Deprtmet of Mthemtics d Iformtics, Fculty of
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationCALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS
CALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS Abstrct Mirce I Cîru We preset elemetry proofs to some formuls ive without proofs by K A West d J McClell i 99 B
More informationFast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation
Algorithm Desig d Alsis Victor Admchi CS 5-45 Sprig 4 Lecture 3 J 7, 4 Cregie Mello Uiversit Outlie Fst Fourier Trsform ) Legedre s Iterpoltio ) Vdermode Mtri 3) Roots of Uit 4) Polomil Evlutio Guss (777
More informationUNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)
UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationEVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
More informationMAS221 Analysis, Semester 2 Exercises
MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationPRODUCT OF DISTRIBUTIONS AND ZETA REGULARIZATION OF DIVERGENT INTEGRALS
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
More informationDouble Sums of Binomial Coefficients
Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationLinear Programming. Preliminaries
Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio
More informationChapter 11 Design of State Variable Feedback Systems
Chpter Desig of Stte Vrible Feedbck Systems This chpter dels with the desig of cotrollers utilizig stte feedbck We will cosider three mjor subjects: Cotrollbility d observbility d the the procedure for
More informationSUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationConvergence rates of approximate sums of Riemann integrals
Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More informationUsing Quantum Mechanics in Simple Systems Chapter 15
/16/17 Qutiztio rises whe the loctio of prticle (here electro) is cofied to dimesiolly smll regio of spce qutum cofiemet. Usig Qutum Mechics i Simple Systems Chpter 15 The simplest system tht c be cosidered
More informationELG4156 Design of State Variable Feedback Systems
ELG456 Desig of Stte Vrible Feedbck Systems This chpter dels with the desig of cotrollers utilizig stte feedbck We will cosider three mjor subjects: Cotrollbility d observbility d the the procedure for
More informationLimits and an Introduction to Calculus
Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit
More informationInternational Journal of Mathematical Archive-5(1), 2014, Available online through ISSN
Itertiol Jourl of Mthemticl Archive-5( 4 93-99 Avilble olie through www.ijm.ifo ISSN 9 546 GENERALIZED FOURIER TRANSFORM FOR THE GENERATION OF COMPLE FRACTIONAL MOMENTS M. Gji F. Ghrri* Deprtmet of Sttistics
More informationy udv uv y v du 7.1 INTEGRATION BY PARTS
7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product
More informationChapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
More informationIn an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
More informationSequence and Series of Functions
6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite
More informationLinford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)
Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationis continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a
. Cosider two fuctios f () d g () defied o itervl I cotiig. f () is cotiuous t d g() is discotiuous t. Which of the followig is true bout fuctios f g d f g, the sum d the product of f d g, respectively?
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION
School Of Distce Eductio Questio Bk UNIVERSITY OF ALIUT SHOOL OF DISTANE EDUATION B.Sc MATHEMATIS (ORE OURSE SIXTH SEMESTER ( Admissio OMPLEX ANALYSIS Module- I ( A lytic fuctio with costt modulus is :
More informationB. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationSOME SHARP OSTROWSKI-GRÜSS TYPE INEQUALITIES
Uiv. Beogrd. Publ. Elektroteh. Fk. Ser. Mt. 8 006 4. Avilble electroiclly t http: //pefmth.etf.bg.c.yu SOME SHARP OSTROWSKI-GRÜSS TYPE INEQUALITIES Zheg Liu Usig vrit of Grüss iequlity to give ew proof
More informationF x = 2x λy 2 z 3 = 0 (1) F y = 2y λ2xyz 3 = 0 (2) F z = 2z λ3xy 2 z 2 = 0 (3) F λ = (xy 2 z 3 2) = 0. (4) 2z 3xy 2 z 2. 2x y 2 z 3 = 2y 2xyz 3 = ) 2
0 微甲 07- 班期中考解答和評分標準 5%) Fid the poits o the surfce xy z = tht re closest to the origi d lso the shortest distce betwee the surfce d the origi Solutio Cosider the Lgrge fuctio F x, y, z, λ) = x + y + z
More information2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r
Z-Trsforms. INTRODUCTION TO Z-TRANSFORM The Z-trsform is coveiet d vluble tool for represetig, lyig d desigig discrete-time sigls d systems. It plys similr role i discrete-time systems to tht which Lplce
More informationStudents must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...
Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More informationAPPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES
Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics,
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationIndices and Logarithms
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic
More informationNational Quali cations SPECIMEN ONLY
AH Ntiol Quli ctios SPECIMEN ONLY SQ/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig.
More informationMath 2414 Activity 17 (Due with Final Exam) Determine convergence or divergence of the following alternating series: a 3 5 2n 1 2n 1
Mth 44 Activity 7 (Due with Fil Exm) Determie covergece or divergece of the followig ltertig series: l 4 5 6 4 7 8 4 {Hit: Loo t 4 } {Hit: } 5 {Hit: AST or just chec out the prtil sums} {Hit: AST or just
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More informationTheorem 5.3 (Continued) The Fundamental Theorem of Calculus, Part 2: ab,, then. f x dx F x F b F a. a a. f x dx= f x x
Chpter 6 Applictios Itegrtio Sectio 6. Regio Betwee Curves Recll: Theorem 5.3 (Cotiued) The Fudmetl Theorem of Clculus, Prt :,,, the If f is cotiuous t ever poit of [ ] d F is tiderivtive of f o [ ] (
More informationAlgebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents
Algebr II, Chpter 7 Hrdig Chrter Prep 06-07 Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit
More informationNumerical Integration
Numericl tegrtio Newto-Cotes Numericl tegrtio Scheme Replce complicted uctio or tulted dt with some pproimtig uctio tht is esy to itegrte d d 3-7 Roerto Muscedere The itegrtio o some uctios c e very esy
More information