VERIFYING AND DISCOVERING BBP-TYPE FORMULAS. Submitted by. Melissa Larson. Applied and Computational Mathematics

Transcription

1 VERIFYING AND DISCOVERING BBP-TYPE FORMULAS Sbmitted by Meliss Lrson Applied nd Compttionl Mthemtics In prtil flfillment of the reqirements For the Degree of Mster of Science University of Minnesot Dlth Dlth MN Spring

2 Chpter : Introdction There re nmeros well-nown formls for inclding C r (where C=circmference nd r=rdis) lim n n n n n n n n from [ p ] 9 from [ p forml ] nd from [ p forml ] Another notble forml is () credited to Dvid Biley Peter Borwein nd Simon Ploffe in 99 It is often clled the BBP-forml They wrote On the Rpid Compttion of Vrios Polylogrithmic Constnts in which the forml ws first mentioned [ p ] The forml is significnt s it permits the compttion of the n th hedeciml digit of withot clcltion the preceding n digits The lgorithm is eplined in [

3 section p-] [ p9] nd [ p] Since its discovery formls of similr form hve been discovered nd hve become nown s BBP-type formls The generl BBP-type forml hs the form p( ) b q( ) where is constnt p nd q re polynomils with integer coefficients deg( p) deg( q) p( ) / q( ) is nonsinglr for nonnegtive nd b is n integer [ p ] Mny nice BBP-type formls cn be written n n n n n n where is well-nown constnt n re constnts (slly integers) nd n We cll n the bse of the BBP form This is the only form I considered in this b project A few emples of BBP-type formls re the following: which is nown s the Leibniz forml ln nd

4 rctn from [ p ] The following re not formls of BBP-type: ( ) sin () ( )! from [9 p forml ] nd n n! from [ p 9 (9)] (n )! The objective of this project ws to verify nd discover BBP-type formls for scled vles of In chpter I begin by verifying previosly discovered BBP-type formls sing their reltionship to definite integrls The chpter begins by eplining the method sed followed by two emples nd then tble contining the BBP-type formls verified The chpter ends with verifiction of two formls conjectred in [ p formls & ] Net chpter shows how to serch for BBP-type formls An lgorithm is given nd detiled emple of its se is provided The net section chpter provides serch of the simplest cses where the bse is or The serches for bse nd bse formls re presented in chpters nd respectively The net chpter is the conclsion with description of strengths limittions nd ftre wor relted to this project The pper ends with two ppendies The first contins tble smmrizing chpters nd while the second incldes copy from Mthemtic s otpt for integrting the non-lternting bse cse

5 Chpter : Unified Method to Verify A techniqe ws sed to verify vrios BBP-type formls This chpter will provide detil eplntion of the techniqe s steps Also provided is tble of formls from the litertre I hve verified by this method Recll in this project I m interested in formls of the type n n n n n n where is well-nown constnt n re constnts (slly integers) nd is constnt The ey formls re presented in Theorem nd Theorem Theorem : If or if nd n i i then n n n n d n n n n n n Proof: The series is power series with rdis of convergence If we cn integrte nd differentite s stted in [ p Theorem ] When nd n i i the reslting sm will converge by the limit comprison test with We only verify the cse when below Let n n n n n n

6 We bre this into n smmtions n n n n n n n n Ting ech term individlly let f ( ) n n Mltiplying both sides by gives f ( ) n n By ting the derivtive we get n ( ) d f n n n d This is geometric series which cn be smmed d f ( ) d By the Fndmentl n Theorem of Clcls f( ) dt which implies f n ( ) dt t n when t Similrly for the i th smmtion let f ( ) i n i n i As before i ( ) n i i d i n i i d f d d n i geometric series tht sms to i i n Integrting gives i i i t fi( ) dt Ths n t i t i fi ( ) dt n t Adding the n terms

7 t t dt n n n n n n n n n n t n Finlly sing the sbstittion t we obtin n n n n d Therefore verifying n n n n n n is eqivlent to verifying n n n n d We were lso interested in the lternting cse of the form n n n n n n We hve: Theorem : If or then n n n n d n n n n n n Proof: Agin we only give proof in the cse where

8 The proof is entirely nlogos to tht of Theorem ecept tht the geometric series is i i n i i ( ) n so dding gives (fter the sme chnge of vribles in the integrl) n n n n d From here we will focs mostly on c where c is n lgebric constnt It shold be mentioned tht this lgorithm wors for other constnts tht involve rctngents nd logrithms As n emple we verify Leibniz s forml In this emple n nd the series is lternting Using Theorem we get ( ) d which cn be esily integrted d rctn( ) [rctn() rctn()]

9 9 This verifies The originl BBP forml is Here n nd it is non- lternting To verify this forml is eqivlent to verifying tht d by Theorem Fctoring the nmertor nd denomintor we cncel ot common terms ( )( ) d d ( )( ) d Using prtil frctions we get ( ) d d two qdrtic terms which cn be integrted by hnd We hve

10 ( ) v d dw dv w v ln ln ( ) rctn( ) ln ln As in the previos emple when verifying BBP-type formls it is common tht the rtionl polynomil s nmertor nd denomintor will hve common fctors tht cncel ot Also the remining rtionl polynomil slly hs nice prtil frctions decomposition For convenience we se the nottion BPP( n( )) n n n n n n n n n n n d nd n n ABPP( n( n)) n n n n n n n n d verified The following tbles contin BBP-type formls from the litertre which I hve

11 Tble - BBP-type formls verified for the bses nd BBP -Type Forml Prtil Frctions Bredown ABBP ( ) ( ) 9 BBP ( ) ( ) BBP ( ) 9 ( ) ( ) ( ) ( ) c BBP ( ) ( ) b BBP ( ) ( ) b BBP ( ) b -These formls cme from [ p forml nd p forml ] b-these formls cme from [ formls nd ] c-this forml comes from combintion of formls from [ p forml mins p forml 9] For emple for the forth row

12 BBP ( ) ( ) d ln ln ln rctn ( ln ) (ln ln ) (ln ln ) The second tble incldes the formls where the bse is Tble - BBP-type formls verified for bse ( ) BBP n n Prtil Frctions Bredown 9 9 BBP ( 9 ) ( ) ( ) BBP ( 9 ) ( ) ( ) b BBP (9 ) ( ) ( ) b BBP ( ) -These formls cme from [ p forml nd p forml respectively] b-these formls cme from [ formls nd respectively]

13 We lso verified two more complicted cses conjectred in the pper A Compendim of BBP-type Formls for Mthemticl Constnts by Dvid H Biley He conjectred tht in [ p formls & respectively] In or nottion we wish to show BBP ( 9 ) nd BBP ( 9 ) re both zero Using Theorem

14 BBP ( 9 ) 9 9 ( ) ( ) d 9 d ln( ) rctn ln( ) (ln ln ) ln ln ln rctn ln( ) nd BBP ( 9 ) d ( ) d ln( ) rctn ln( ) (ln ln ) ln ln ln rctn ln( ) In fct most of Biley s conjectres for degree one BBP-type formls from [ p nd ] cn be proven sing Theorem nd formls for combintions of rctngents

15 Chpter : Serching nd Discovering BBP-Type Formls In the previos chpter we described how to verify previosly fond BBP-type formls In this chpter we describe n lgorithm for finding sch formls This lgorithm is lso bsed on Theorem nd Theorem Using Theorem for emple we strt with () n n n n d n n n n n n We integrte the right hnd side of () symboliclly s fnction of n nd loo for vles of the prmeters tht give nice nswer For smll cses this cn be done by-hnd bt for most cses softwre progrm ie Mthemtic ws sed Once integrted rctngents nd logrithms of fnctions of pper We se one of the rctngents to give (slly by setting its rgment to be ) This typiclly fies or vle of Plgging in nice vle we get fnction in terms of n We constrct system of liner eqtions involving n to eliminte non- vles Rowredcing these eqtions in mtri we get set of soltions tht reslt in BBP-type formls for scled vles of We hve For emple sppose we wish to find nice forml of type ABPP( ( )) ABPP( ( )) d

16 from Theorem Using Mthemtic to perform the integrl ( ) ABPP( ( )) ( )rctn ( )ln( ) ( )ln( ) ( ) We cn get n epression involving sing either or rctn When sing ( ) we need to eliminte other terms The rctngent term cn only be eliminted by setting (since we cnnot se ( ) or or vle wold be eliminted lso) We hve ABPP ( ) ln ln 9 Ths we need nd Row redcing we get implying nd reslting in the BBP-type forml () ABBP () 9 Alterntively when

17 rctn rctn This reslts in ( ) ( ) ( )ln ABPP 9 Ths we need There re two degrees of freedom which implies two BBPtype formls Letting we get () ABBP( ) 9 If then we get () ABBP( ) 9 Hence p to liner combintions of () () nd () we hve fond three BBP-type formls for the cse ABPP( ( )) There re certinly more formls if yo llow more eotic vles In mny cses I hd to simplify Mthemtic s otpt by hnd The otpt in this pper is not the ect otpt from Mthemtic See Appendi for n emple of the ctl Mthemtic printot for the non-lternting bse cse Simplifiction ws needed for generl vle nd when serching for system of liner eqtions to eliminte the non- vles The simplified version is shown throghot this pper

18 Chpter : Emining Cses Where the Bse is or Using the lgorithm described in the lst section we serched for BBP-type formls for non-lternting nd lternting bses nd In this chpter we present the reslts for the simplest cses bses nd The simplest cse of ll is the lternting bse cse From Theorem we hve ( ) ABBP d After integrting we get ABBP ( ) rctn( ) ln( ) tht From this eqtion we loo for nice -vles The criteri for to be nice re n needs to be rtionl nd its se llows for the evlte or eliminte n rctngent In order to eliminte n rctngent we need to be reslt of the integrl which does not lwys occr In the bove emple the two vles or led to interesting formls since rctn nd rctn When ABBP ( ) ln

19 9 Ths which reslts in the BBP-type forml () ABBP ( ) When ABBP ( ) ln() Ths which gives () ABBP ( ) the Leibniz forml gin For non-lternting formls with bse n from Theorem we hve BPP( ( )) d Using Mthemtic to integrte BPP( ( )) ( ) i ( ) ( )rctn ( )ln( ) ( )ln( )

20 The only nice vle ppers to be since we then hve rctn With we hve BPP(( )) ( ) i ( ) ( )ln This implies tht nd Row-redcing yields Ths nd reslting in the BBP-type forml () BPP(( )) 9 The lternting bse formls were investigted in the previos chpter Net we will emine non-lternting bse formls BBP ( ) d Using Mthemtic

21 BBP ( ) ( ) i ( )rctn( ) ( )ln( ) ( )ln( ) ( )ln( ) The rctn( ) term cn be evlted if or When BBP ( ) ( ) ( ) ln ( ) ln ( 9 )ln ln We row-redce 9 which gives Ths nd Therefore the BBP-type forml is BBP ( ) 9 which is the sme s times the reslt in () this With ( ) BBP only converges if Using

22 BBP ( ) ( ) ( )ln Hence we row-redce nd get This prodces the BBP-type formls BBP ( ) which is () gin nd BBP ( ) The lternting bse cse gives ( ) ( )rctn ( ) ln( ) ABBP ( ) rctn ( ) ln( )

23 The only obvios nice vle is which elimintes rctn This implies tht or we will eliminte or only term When ABBP ( ) ( )rctn ( )ln ln By row-redcing we hve Ths we hve the BBP-type forml ABBP ( ) It trns ot tht lso prodces BBP-type formls This is becse rctn( ) nd rctn( ) When we hve ( ) ( ) ( )ln ABBP ( )ln

24 Row-redcing yields Therefore we hve ABBP() ( ) which is the Leibniz forml () in disgise nd () ABBP () ( )

25 Chpter : Emining the Cses with Bse We fond no nice BBP-type formls with bse In this chpter we present reslts for bse For the non-lternting version ( ) ( ) ( ) BBP i ( ) rctn ( ) rctn ( )ln( ) ( ) ln( ) ( )ln( ) ( )ln( ) The rctn term sggests since rctn or since rctn rctn The rctn term lso evltes when () since rctn rctn When BBP ( ) ( ) rctn 9 ln ln ln

26 Row-redcing 9 we get Ths BBP ( ) 9 nd BBP ( ) When we need for convergence Using this BBP ( ) ( ) ( )ln ( )ln Row-redcing

27 yields Ths we hve BBP ( ) BBP ( ) nd BBP ( ) The second of these is eqivlent to the forml () for bse In the lternting cse we hve ABBP ( ( ) ( ) ( )rctn ( ) rctn ( ) rctn ( ) ln ( )ln ( )ln

28 We cn se nd in the rgment rctn( ) We cn lso eliminte rctn when When the terms rctn nd rctn evlte since rctn nd rctn When ABBP ( ) rctn 9 ln ln 9 ln Row-redcing gives We hve

29 9 9 ABBP (9 ) nd ABBP ( ) When ABBP ( ) 9 rctn 9 rctn ln 9 ln ln Row-redcing gives 9

30 yielding ABBP ( 9 ) 9 9 Finlly when ABBP ( ) ( ) ( )ln ( ) ln Row-Redcing gives Hence this yields for BBP-type formls ABBP() which is Leibniz s forml in disgise gin ABBP ()

31 ABBP () 9 nd ABBP () The third nd lst formls re eqivlent to forml () nd forml () respectively

32 Chpter : Emining Cses with Bse This chpter dels with BBP-type formls with bse There do not pper to be ny nice formls with bse In the non-lternting cse ln ) ( ln ) ( )rctn ( )rctn ( ) )ln( ( ) ) rctn( ( ) )ln( ( ) )ln( ( ) ( ) ( ) ( i BBP Possible -vles re nd from ) rctn( We cn lso eliminte ) rctn( if The term ) rctn( lso evltes when = since rctn( ) nd rctn( ) When ( ) 9 9 rctn 9 rctn BBP ln ( 9 )ln ( 9 ) ln ( 9 ) ln ( 9 ) l n

33 Row-redcing gives 9 Ths we hve the BBP-type forml 9 9 BBP ( 9 ) This forml is eqivlent to the forml in () When

34 )ln ( )ln ( ln )ln ( )ln ( rctn ) ( ) rctn() ( ) ( ) ( BBP Row-redcing we get This gives two BBP-type formls ) ( BBP which is the originl BBP forml () nd

35 ) ( BBP When ) ( BBP only converges if Using this ) ln( ln ln ) ( ) ( BBP Row-redcing yields Ths we hve the BBP-type formls () ) ( ) ( BBP

36 ( ) BBP () ( ) ( ) BBP ( ) BBP () ( ) nd () BBP( ) ( ) For more elegnt reslts we cn te three times () nd sbtrct () to get BBP( ) We cn lso te () nd dd for times () to get BBP( ) Also () mins () is BBP( ) Lstly we hve two times () mins () which is

37 BBP( ) For the lternting cse we hve eliminted the severl pges of the integrtion ABBP ( ) involves the following rctngents rctn rctn rctn nd rctn These rctngents evlte if s follows rctn rctn

38 rctn nd rctn Using this informtion we were ble to simplify ABBP ( ) eliminte the logrithms we row-redced To ( ) which yields This leds to the following BBP-type formls

39 9 ABBP() ( ) ABBP() ( ) ABBP() ( ) nd ABBP() ( ) The first forml is eqivlent to Leibniz s forml () nd the third forml is eqivlent to ()

40 Chpter : Conclsions In this project I focsed on BBP-type formls for scled vles of It might be interesting to crry ot this lgorithm for other constnts sch s rctngents logrithms nd zero The constnt zero reltes the nll spce of the system of liner eqtions Ecept for n emple in the lternting bse cse nd bse cses or method only tilized single rctngent term Since there re mny formls for combintions of rctngents lie rctn rctn A more sophisticted pproch wold te this into ccont BBP-type formls cn be etended to the formls of the form n ( n ) s ( n ) s n ( n n) s where s is n integer This form leds to BBP-type formls for constnts sch s the following emple s in ( ) ( ) ( ) ( ) ( ) ( ) ( ) from [ p forml 9] Or lgorithm hndles only cses where s Ftre wor cold eplore this re From [ p ] we hve the eqtion

41 The bove forml llows for the individl he or binry digits of to be clclted It is over % fster thn forml () It wold be of interest to serch for fster eqtion or of BBP-type formls tht re combintions of different bses with different -vles Another limittion to or lgorithm is tht s we increse the bse we complicte the integrtion Bse is the highest cse tht or lgorithm cn hndle withot too mch compleity Bse ppers to hve BBP-type formls; it wold hve been nice to te closer loo t it Or lgorithm is sefl s it is combintion of elementry concepts We simply sed Theorem nd long with row-redcing mtrices of systems liner eqtions to eliminte non- constnts Also sing integrtion provided sefl tool in discovering nice -vles Althogh or lgorithm for serching for BBP-type formls does not wor well for higher bses it does wor well for verifying most cses previosly discovered of degree one

42 References [] S Abbott Understnding Anlysis Springer New Yor [] J Arndt nd C Henel Pi Unleshed Springer Berlin [] D Biley A Compendim of BBP-Type Formls for Mthemticl Constnts vilble t [] D Biley nd J Borwein Mthemtics by Eperiment: Plsible Resoning in the st Centry A K Peters MA [] D Biley P Borwein nd S Ploffe On the Rpid Compttion of Vrios Polylogrithmic Constnts Mthemtics of Compttion vol no 99 p 9-9 [] P Becmnn A History of Pi The Golem Press Bolder CO 9 [] L Berggren J Borwein nd P Borwein Pi: A Sorce Boo rd ed Springer New Yor [] I Lehmnn nd A S Posmentier A Biogrphy of the World s Most Mysterios Nmber Promethes Boos Amherst NY [9] J Stewrt Clcls: Concepts nd Contets nd ed Broos/Cole Pcific Grove CA [] []

43 Appendi : Tble Smmrizing Bses nd The following tble smmrizes the reslts of BBP-type formls discovered in chpters nd Tble - BBP-type formls discovered # ( ) BBP n n Prtil Frctions Bredown ABBP ( ) ABBP () BBP ( ) 9 * ABBP ( ) 9 * 9 9 ABBP ( ) ABBP ( ) * * BBP ( ) 9 BBP ( ) 9 BBP ( ) *

44 # ( ) BBP n n Prtil Frctions Bredown ABBP ( ) ABBP () ( ) ( ) ABBP () ( ) ( ) * ( ) 9 BBP ( ) ( ) BBP ( ) BBP ( ) ( ) ( ) * BBP ( ) ( ) ( ) BBP ( ) ( ) ( ) * ABBP (9 ) 9 ABBP ( ) * ABBP ( 9 ) * ABBP() ( ) ( ) ( )

45 # ( ) BBP n n Prtil Frctions Bredown ABBP () ( ) ( ) ( ) * 9 ABBP () ( ) ( ) ABBP ( ) ( ) ( ) 9 BBP ( 9 ) ( ) BBP ( ) BBP ( ) BBP ( ) * ( ) ( ) ( ) ( ) 9 BBP ( ) * ( ) ( ) ( ) ( ) BBP ( ) ( ) ( ) BBP ( ) * ( ) ( ) ( ) ( ) BBP ( ) * ( ) ( ) ( ) ( )

46 # ( ) BBP n n Prtil Frctions Bredown ( ) ABBP ( ) ABBP * ) ( ( ) ABBP ( ) ABBP * We will refer to ech forml by its nmber corresponding to the first colmn Formls nd re eqivlent to the power series of tn Formls nd re eqivlent to Leibniz s Forml () Formls nd re eqivlent Formls nd re eqivlent Formls nd re eqivlent Formls nd cn be fond in [ p forml nd p formls & 9 respectively] Formls nd re eqivlent Formls nd cn be fond in [ formls & respectively] Formls nd re possibly new BBPtype formls s we did not see them in the litertre They re lso mred with n * in the bove tble If two formls re eqivlent only the forml from with smller bse is considered new

47 Appendi : Mthemtic s Otpt for the Non-Alternting Bse This is the ctl Mthemtic otpt for integrting the non-lternting bse cse The restrictions on cn be disregrded since we re concerned with nd when we dd the necessry constrints Also b c d e nd f re eqivlent to nd respectively For simplifiction epressions were epnded nd constnts were fctored Logrithms were lso epnded ie f Log is eqivlent to which ws epnded to ln( ) ln ln( ) ln( ) ln( ) This often ffected the system of liner eqtions nd dded degree of freedom Similr simplifiction ws sed once n vle ws plgged in In the bse cse prtil frctions were sed to convert the integrl into for integrls The for integrls were integrted seprtely nd their reslts were smmed together

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