The Connection between the Basel Problem and a Special Integral

Transcription

1 Applied Mahemaics Published Olie Sepember 4 i SciRes hp://wwwscirporg/joural/am hp://ddoiorg/436/am45646 The Coecio bewee he Basel Problem ad a Special Iegral Haifeg Xu Jiuru Zhou School of Mahemaical Scieces Yaghou Uiversiy Jiagsu Chia hfu@yueduc houjr98@homailcom Received 9 Jue 4; revised July 4; acceped Augus 4 Copyrigh 4 by auhors ad Scieific Research Publishig Ic This wor is licesed uder he Creaive Commos Aribuio Ieraioal Licese (CC BY) hp://creaivecommosorg/liceses/by/4/ Absrac By usig Fubii heorem or Toelli heorem we fid ha he ea fucio value a is equal o a special iegral Furhermore we fid ha his special iegral is wo imes of aoher special iegral By usig his fac we give a easy way o calculae he value of he aleraig sum of wihou usig he Fourier epasio Also we discuss he relaioship bewee Geocchi umbers ad Beroulli umbers ad ge some resuls abou Beroulli polyomials Keywords Basel Problem Zea Fucio Beroulli Numbers Beroulli Polyomials Iroducio Basel problem ass for he precise value of he progressio I was firs posed by Piero Megoli i π 644 ad solved by Leohard Euler i 735 [] The value is ow as 6 There are more geeral resuls [] abou he progressio Le ζ Moreover usig Fourier epasio of a i becomes he + b a b a ( + a) lim d b a How o cie his paper: Xu HF ad Zhou JR (4) The Coecio bewee he Basel Problem ad a Special Iegral Applied Mahemaics hp://ddoiorg/436/am45646

2 H F Xu J R Zhou we will ge + π cos + 4 ( ) 3 + π () I he ed of Secio 5 we give aoher proof of () by usig he relaioship of wo special iegrals which are iroduced i Secios 3 ad 4 Also ispired by his i Secio 6 we discuss abou Beroulli umbers ad Geocchi umbers We obai some properies of Beroulli umbers ad Beroulli polyomials Basic Properies The covergece of he ifiie series is obvious We ca use various mehods o prove i Especially whe we cosider Riema-Zea fucio ζ ( s) s ( s ) he progressio diverges whe s ad coverges whe s > Also we ca use he esimae of he parial sum of he series < N N N N ( ) whe N + Or we ca use he Cauchy priciple I fac for > hus whe + The he progressio coverges 3 Calculaio of ζ < ( ) + p + p < < < p There are various proofs of he Basel problem ad Robi Chapma wroe a survey [3] abou hese Some are elemeary ad some will use advaced mahemaics such as Fourier aalysis comple aalysis or mulivariable calculus Here we review he mehod of Jiaqiag Mei [4] which is raher elemeary ad easy o udersad There is also a elemeary proof o he Wii [] Repeaed use of he equaio we ge Noe ha cos + si si + 4 π 4si cos + si si + si 4 π π π 3π + si si si si si si si π + () 57

3 H F Xu J R Zhou we may rewrie he Equaio () as where Usig he iequaliy we ge he esimaio Le ( ) π + π + π π + si si π si + E + si + π + π si E + π + π si cos π π < + < si si we obai he followig equaio π < E < π si ( + π) The above progressio is uiformly coverge i ay closed ierval o coaiig { π} wrie as Especially we have Therefore 4 As a Special Case of Power Series + + π si ( + π) ( π) lim 3 si π π ζ ( ) 6 For he power series we calculae he domai of covergece Sice he radius of covergece equals is coverge If R If lim a lim ad ca be he power series becomes he progressio which which is also coverge Therefore he he power series becomes 57

4 H F Xu J R Zhou he domai of covergece is S Suppose Muliply boh sides by We ca do he derivaio iem by iem i he ieval Derivae boh sides agai we ge Thus S S + S S ( S S ) + We obai a secod order ordiary differeial equaio If we se y S Muliplie boh side by d Le h y The Tha is S + S (3) he Equaio (3) is covered o a firs order equaio we have y + y y d + y d d dh d ( ) + Usig he iiial codiios y S h y h ( ) The if Tha is Noe ha S The Paricularly h l C we have l ( ) l y ( ) l S ( ) l S S S S d d l ( ) l ( ) u l u S d du d u u u 573

5 H F Xu J R Zhou Therefore for he improper iegral l u d u we ow is value is equal o u π 6 5 From he Special Iegral o he Basel Problem I his secio we will calculae he special iegral arised i he las secio ie For Thus l u d u u l d l d l d l lim l d + + l d ( ) Le f l ( ] Obviously f L ([ ] ) For simpliciy we deoe L ([ ] ) by L By Miowsi iequaliy [5] we have The L L where is Lebesgue measure f f (4) l d l d ζ ( ) We will prove he equaliy holds i our case Firs we have he followig lemma Lemma f + g f + g if ad oly if here is a real valued fucio h ha is oegaive ae L L L such ha whe boh f ad g are o he g hf ae Proof Please refer o [6] Lemma ca be geeralied o ifiie summaio case if ad oly if here is a real valued fucio f ad a series of real valued L L Lemma f f fucios g which have he same sigs such ha f g f ae Proof Firs by iducio he lemma holds for fiie sum Tha is Le E: { f have he same sig} The Sice he measure of N N f f L L N N N + f f f L ( E) f L ( E) L ( E) L ( E) c E is ero we have Combie Equaio (4) we complee he proof O he oher had we observe ha for f f L L l l l 574

6 H F Xu J R Zhou The we ge l d l d l d ζ ( ) We give a remar abou he secod equaliy of he above equaio I ca be ifered by Fubii heorem or Toelli s heorem [7] Ifac µ ν [ ] l d l d d where µ is he couig measure o ad ν is he Lebesgue measure o Obviously hey are boh σ -fiie measures Ad sice l is o-egaive by Toelli s heorem [ ] µ ν µ ν [ ] l d d l d d l d There are oher ways o ge his relaioship from his special iegral o ζ esablished by James P Leso ad Wedy D Smih [] r a > ad b we have Lemma 3 For [ ) Especially whe a ad b (5) yields By his lemma r Firs recall he lemma b a r ru d u a + b a (5) ru r du l ( r) ru ( r) r ( ) ( ) ( ) l l d dr dudr dd ur r r ru ru The by moooe covergece heorem i equals o See [3] Or we ca do i i his way l l ( ) ( ) ( ) ( ) r r d dr dr r r r dr r d r 6 Relaioship bewee Two Special Iegrals We will use a resul from ([8] Eer ) Lemma 4 Assume ha he fucio f ( ) is moooe o he ierval ; we assume however ha he improper iegral f d pois he The for our case he iegral lim f f f f d l I eed o be bouded a he eiss Uder hese codiios d eiss ad saisfies he codiios I fac le f l 575

7 H F Xu J R Zhou Le g + l he ( ) l ( ) + l ( ) ( ) f g + l + l < Sice lim g we have g( ) > for So f ( ) is moooe o he ierval by Lemma 4 we have If we cosider he improper iegral Le g l( ) l l l l d lim lim l ( ) l ( ) d Le f he g l f < ( ) + + Sice lim g lim l ( ) g( ) < Thus f < So f ( ) is also moooe o he ierval l l l ( ) d lim lim lim l l lim Ne we deduce he followig equaio ad give aoher discripio of () l l d d + l ( ) Lemma 5 Le h d [ ] Proof Le h h h( ) ϕ By chagig variables The The By Lemma 4 we have The (6) + he h h h( ) ϕ + i is easy o oe ha is derivae l ( ) u l ( + u) h d d u u ϕ equals o ero ad 576

8 H F Xu J R Zhou Here oe ha ( u) + u ( + u) l h( ) du l ( u) d l u u + l u l ( + u) l u l ud l ( u) d u + + u lim l + l u By Lemma 5 l l ( ) ( ) l d d h h d + l For he iegral d we have a releva resul + l Lemma 6 Le h d + > he h + h ( l ) Proof l h d l d l ( + ) + l ( + ) l l ( + ) d he l ( + ) h l l + d Hece ad The secod equaliy holds because Observe ha Thus h h + l l ( + ) l ( l ( + ) l ) l ( + ) l ( + ) d + ( l ) ( l ) ( l ) ( + u ) ( + ) d l l d d+ ( l ) l ( ) + ( ) ( ) d d d l ( + ) l l d d d ζ ( ) + Applyig he same argume i begiig of his secio we ge d lim l l l l d l lim + 577

9 H F Xu J R Zhou Remar I mus be very ieresig if we could calculae he iegral Remar Similarly we ca prove ha ( l ) d ζ 3 l d o usig he progressio 7 Beroulli Numbers ad Beroulli Polyomials Recall some facs of Beroulli umbers ad for more iformaio please refer o [4] [9]-[] The Beroulli umbers B are defied by he power series epasio The e B! m B B ( e ) B! m m!!! Thus we ge a recursio formula for he Beroulli umbers amely We ge B From he ideiy he fucio + e Someimes people prefer o use B B if + + +!! ( )!!! ( )! if > B e + e + e + e e e e ah is a eve fucio Hece i has oly eve erms i is power series epasio B + e! o deoe B ( ) + e We have various ways o ge he impora equaio Lemma 7 Proof By replacig by πi i he ideiy we ge he! π π ζ ( ) B!! e + e e e B! π πco π B! 578

10 H F Xu J R Zhou The by aig he logarihmic derivaive of he produc epasio for he sie we ge he epasio of πco π si π π + co ( ) π π + + ζ + Comparig he coeffices of we ge (7) Aoher way is o ae he logarihmic derivaive of he ideiy [4] which yields The we ge (7) by comparig wih Proposiio Proof where I is easy o prove ha I Le us cosider he epasio where sih + π m m ( ) ζ ( m + ) m+ coh + m+ π B coh + ah e! ( ) π 3 6 ( ) l se s s s s e! s l s s d d e + d s G are Geocchi umbers The ( ) e s ds sde s e s s ds +! ( ) se s ed s s + s ed s s! 3 + I! s ed I s s! by iducio Therefore l 3 d + ( ) e + G! 579

11 H F Xu J R Zhou G G ( e + ) + m! m m!! + G G!! Thus G ad G+ G which ifers ha G For > Tha is Noe ha Taig epasio we have he + G!! G G G ( ) > (7) e e + e B G B ( )!!! G This give a quic way o compue Beroulli umbers sice i (7) we have If le deoe ( ) Proposiio G Proof By chagig variables Noe ha Se I ed he B (8) π G B l e d d + + e e + + G! + + l G G d ed ed! +! he I ( ) I By iducio we prove ha I l d ( ) G ! Therefore 58

12 H F Xu J R Zhou Togeher wih (6) we have π G Remar Sice B ad G he proposiio ca also be wrie as Beroulli polyomials B The fucios B Similarly we defie G The fucios G Comparig he coefficies of ad G π π ( ) + are defied by he formula e e are polyomials i ad by he formula B! B B are polyomials i I fac e G (9) e +! G ( ) G e! e! +! we have G G G () G Usig (7) for > we have O he oher had by defiiio Do he addiio Comparig he coefficies of G G G + G G + G G ( + ) G( ) e G( + ) e e +! e +! G( + ) + G( ) + e!! we have G + + G Le 3 ad summaio hese equaios we ge ( ) G + G + + G + G + i From he equaio G + G we ifer ha i G G G 58

13 H F Xu J R Zhou Therefore If m If m+ G G + + G 3 + G G G G G G ( ) + ( ) ( ) + ( ) G G G i G i m ( ) ( ) ( ) i G i m + m 3 + m G m+ i G i m + m + m Wheher is odd or eve we always have he followig rivial ideiy + + ( + ) G G i G i i G G i By differeiaig o a boh sides of (9) we also have Bu beig differe from B d we have G G G G G G d d G Proposiio 3 i) ( ) + G G G B B + G B B ii) iii) iv) B B + + B Proof i) hus ii) ( ) ( ) e ( ) ( ) + e G G! e + e +! + ( ) ( ) G G e e e + e e e 58

14 H F Xu J R Zhou By comparig he coeffies of iii) By comparig he coeffies of we ge we ge B G B + e e e + + e e e + B + G B iv) by ii) ad iii) Remar ) Especially we have B 4 B 4 sice G ) Le or i iv) we have ( B ) B Thus B ( ) B 4 3) Le i iii) we will ge G ( ) B Equaio (8) ca also be deduced i he followig way Usig we obai Similarly This ifers ha e + e e B ( ) B G e e + e B B!!! G B B By subsiuig () i he above formula we obaide G B > () Acowledgemes We epress our graiude o David Harvey who poi ou ha he umbers D i our mauscrip (here is G ) are esseially he Geocchi umbers see [3] The wor is parially suppored by Naioal Naural Sciece Foudaio of Chia (NSFC) Tiayua fud for Mahemaics No 646 ad he Uiversiy Sciece Research Projec of Jiagsu Provice (3KJB9) Refereces [] hp://ewiipediaorg/wii/basel_problem [] Leso JP ad Smih WD (3) A Laplace Trasform Techique for Evaluaig Ifiie Series Mahemaics Magaie [3] Chapma R (3) Evaluaig ζ hp://secamlocaleacu/people/saff/rjchapma/ec/eapdf 583

15 H F Xu J R Zhou [4] Mei JQ () Calculus Higher Educaio Press Beijig [5] Grafaos L (4) Classical Fourier Aalysis Graduae Tes i Mahemaics (Boo 49) d Ediio Spriger New Yor [6] Burard E () Mah 9C Homewor hp://mahucredu/~edwardb/graduae%classes/mah%9c/9c%hwpdf [7] hp://ewiipediaorg/wii/fubii's_heorem [8] Pólya G ad Segö G (4) Problems ad Theorems i Aalysis I Spriger Berli [9] Lag S (3) Ellipic Fucios Spriger-Verlag Berli [] Pólya G ad Segö G (4) Problems ad Theorems i Aalysis II Spriger Berli [] Sei EM ad Shaarchi R (3) Fourier Aalysis Priceo Uiversiy Press Priceo [] Sei EM ad Shaarchi R (3) Comple Aalysis Priceo Uiversiy Press Priceo [3] hp://ewiipediaorg/wii/geocchi_umber 584

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