Leonhard Euler s. De fractionibus continuis Wallisii

Transcription

1 Trsltio o Leohrd Euler s De rtioius otiuis Wllisii E7 O the otiued rtios o Wllis Origilly pulished i Meoires de l'deie des siees de St-Petersourg 8 pp - lso see Oper Oi: Series Volue pp Kristi Msters Christopher Tippie d Thos J Osler Trsltor s Itrodutio Kristi Msters tehes Lti d Christopher Tippie tehes thetis t Millville Seior High Shool i New Jersey Christopher is lso grdute studet o thetis t Row Uiversity where To Osler is proessor o thetis I this ollortio we hve ttepted to provide ot oly urte trsltio ut to lso oprehed d ouite Euler s disoveries To this ed we kept reul otes s we podered Euler s ides d illed i the issig detils These ides re otied i our opio pper Reletios d Notes o E7 I dditio we hve writte shorter Syopsis o E7 to ssist the reder ll o these e oud o the Euler rhive We thk Doii Klyve d his Euler rhive or providig hoe o the iteret or this d other trsltios ter Willi Brouker oud his eorle otiul rtio or the qudrture o irle d ter he shred it with Joh Wlle without epltio Wllis devoted ore tie to it so tht he ould uover the soure o Brouker s oted orul Wllis rekoed tht there ws use or these etrordiry oruls whih he rought to light i his work ritheti o Iiities The he produed through little too struse lultios ot oly Brouker s otiul rtio ut lso eve ore

2 outless other oes like it oth o whih ertily iludig Brouker s epressio re deeed worthy to e resued ro olivio However the thigs whih perti to this ro Wllis ritheti o Iiities whih were rought to light log eore the lysis o Iiities ws disovered re ow le to e represeted i suh wy tht whe the liits o itegrl oruls re eteded ro 0 to the ollowig qudrtures re produed: et I rrged these oruls i the third olu i suh wy tht the deoitors lerly llow iterpoltio d so it oly reis tht the uertors re trsored i suh wy tht they llow iterpoltio eqully I suh series progresses ordig to the lw o uiority o ourse i it were ivestigted s B C D E F et it would eoe: B ; BC ; CD; DE; et This very thig Wllis reveled through the highest wisdo o his geius I will ler up this ivestigtio uh ore geerlly through uh esier lultio

3 However whe this series o letters B C D et is oud the whole usiess will hve ee opletely settled For whe s the ollowig tle revels: BC BC BCDE BCDE BCDEFG BCDEFG et Iterpoltio provides or us the ollowig qudrtures: B BCD BCDEF et Sie ow π where π deotes the irueree o irle whose dieter equls whose vlue I write or the ske o revity π q ll vlues o the letters B C D et or this qutity q will e epressed i the ollowig er: Dieree

4 00 q B q 7079 C 79 q 9 q D 9 E 707 9q 9 F q Here I tthed third olu whih ehiits the ueri vlues o these letters to show ore lerly the etet tht these uers irese ordig to the lw o uiority This does ot hppe i I tke lse vlue i the ple o q Now tht these thigs hve ee eplied I will id i uh esier er otiued rtios or eh idividul letter Wht is ore I will rrge this ivestigtio i uh ore geerlity while I resolve the ollowig prole: Prole: To id series o letters B C D et progressig y the lw o uiority i suh wy tht B ; BC ; CD ; et 7 Solutio: It is ieditely ler ro this poit tht ws utio o tht very s it ought to e tht B is utio o tht very the oreover C is utio o ; D is utio o ; d so o

5 Oe this rule hs ee oserved i we ssue s it ought to lso e ssued tht B s ; where the letters d B ought to hold the se rtio B etwee theselves I this wy B should rise ro i is writte i the ple o Thereore whe these rtios re reoved sie s d s B the produt o these oruls is equl to Now this equtio rises B ro the rtios: B s B s ss 0 Thereore let us ssue tht s d the equtio divided y should e B B whih is esily tored i the ollowig wy: B 8 Now i oth letters d B re equl the let prt should e B ollowig the ove ptter we should id tht d B Whe you sustitute these the il equtio tkes the or: B B Whe epded d the rtios reoved the ollowig equtio rises:

6 9 B B 0 Thereore it is ssued here tht 9 d it ollows tht : B B 9 whih gi is le to e represeted y tors s: B 9 Now oe gi the *iddle vlue* etwee d B is urtherore let us ssue tht d B d oe this sustitutio is B de we will get: B Whe epded it eoes: B B 0 ssue tht s d this equtio will tke o the or: B B whih is tored s: B 0 gi s eore let it e ssued tht d IV B Oe this sustitutio is de we get IV B 7 7 IV IV B Whe this equtio is ipulted s eore we will get IV B IV IV IV 7 B 7 9

7 7 where o ourse we put s 9 The i tored or it will e IV IV 7 B 7 Fro here it is ler how these opertios re to e otiued urther Whe these thigs re resoed tht s 9 9 et we oti or the ollowig otiued rtio: 9 9 et I we write i the ple o the series et we will rete siilr otiued rtios or B C D et whih re: B 9 9 et C 9 9 et D et

8 8 I ow we put d we will produe the very se hdled y Wllis Fro this Wllis oud his otiued rtios whe their vlues re epressed through the qudrture o irle they will e the ollowig: WLLIS CONTINUL FRCTIONS 9 q π 9 et B q π 9 9 et 8 C 9 q π et 9q 9π D et

9 9 8 E 9 9 9q 9π et The irst o these is the very se otiued rtio disovered y Brouker However this is y o es siilr to the roudout wy Brouker e to his orul Isted I elieve tht he derived it ro the osidertio o his ous series: π et 7 9 This series is ooly ustoed to e ttriuted to Leiiz; however it ws disovered uh erlier y Jo Gregory Fro Gregory s work Brouker ws le to derive it O ourse it ws possile to oe out through suiietly esy d essile opertios i the ollowig wy: Whe you isert π α α β β γ It eoes α π α α α 9β 9 α β β β γ β γ γ γ

10 0 γ δ 7 7 9δ γ 7δ 7δ 7 δ But i these reetly oud vlues re sustituted i the ple o et α β γ Brouker s otiued rtio is otied π 9 9 et Now this rigs e to y geerl solutio o the prole Oe eve epress the vlues o idividul otiued rtios through erti qudrtures whih I will show i the ollowig prole: Prole To ivestigte the vlues o the idividul letters epressed irst through otiued produts the epressed through itegrl oruls or the sequee B C D et tht otiues ordig to the lw o uiority i suh wy tht B ; BC ; CD ; et Solutio Thereore let ; B ; C ; et B C D

11 Whe these vlues re sustituted otiuously oe ids tht et et iiitely Beuse tor reis either i the deoitor or i the uertor wheever the series is roke o o liitig vlue rises this wy This ioveiee however will e reoved i I rrge the siple tors i the ollowig wy: et I this wy tors will otiuously pproh oe d iiitely they will equl oe; d thus this epressio will ertily hve liitig vlue However whe I will show how oe ought to redue the vlue to itegrl epressios I will ll upo this *le* or help: Whe itegrls re evluted ro 0 to we will get: k k k k k k k Whe I pply this le to y prole sie our eers idividul tors tke irese o thus it ought to e ssued tht The whe it is ssued tht d k we will hve: There is typogrphil error o the let hd side The rdil i the deoitor should e idetil to the orrespodig rdil o the right hd side

12 This epressio oe djusted will yield previous tors o idividul eers For the et equtios let us ssue tht d k reis whe this is doe we get: 7 It is ow evidet tht the seod orul oe it hs ee divided y the previous orul will show y otiul produt Whe iiite itegrls re elled out o oth sides we get: : Siilrly strightwy we id B : C : et But this ivestigtio still e geerlized s the ollowig prole will show MORE GENERL PROBLEM To id series B C D et tht proeeds uiorly i suh wy tht ; ; ; ; DE CD BC B Where the letter is iresed y qutity There is error i eh o the ove three equtios The right hd sides should e ultiplied y d respetively See setio 7

13 PREVIOUS SOLUTION FOR CONTINUED FRCTIONS 8 gi here it is evidet tht is utio o d B ought to e utio o ; C is utio o ; D is utio o d so o Thereore whe B i d B re equl whe is oitted B I is osidered less th B ought to e osidered greter th Fro here whe B will equl Beuse oes ro B i is writte i the ple o it ought to e tht B Fro here we olude tht ½ ; d so the priipl prts or d B will e - ½ d B ½ or d B Thereore or the ollowig letters we id C ; D ; E 7 et 9 Whe these priipl vlues hve ee oud we will osider tht s s ;B ; B But suitle vlue will eerge or s Thereore ro here we get: s s ss B B B This equtio oe the rtios re reoved will tke o the or: B s B s ss 0 Now let us ssue tht s d oe divisio is de we get: B B

14 This equtio e epressed i tored or s: B 0 Siilrly s it e reogized through dedutio i d B re equl the eer o the let would e 0 d thereore B Beuse B ought to rise ro i is writte i the ple o it is evidet tht the priipl prts re d B Thereore let us ssue tht d B B Fro here i these vlues re sustituted the preedig equtio will tke o this or i tors: B This equtio oe the epsio hs ee de will rig us to this equtio: 9 B B Oe the rtios hve ee reoved we get: B 9 B 0 Thereore oe it is ssued tht s 9 d the divisio hs ee de this equtio rises: B B 9 This e represeted y tors i the ollowig wy:

15 B Beuse this equtio is siilr to the previous oe gi i this se i B we will get Furtherore it is ssued tht d B B Fro here the il equtio will e i tor or: B But oe the epsio hs ee de d the rtios hve ee reoved it will produe: B B 0 Thereore y ssuig tht s d y dividig y s it will eoe: B B or i produt or: B Furtherore it will e estlished tht B d B d whe these vlues re sustituted the preedig equtio will e i produt or: 7 7 B Whe these se opertios hve ee repeted d whe it hs ee ssued tht s 9 this equtio is redued to the ollowig: B 7 B 7 9

16 Or i tored or it will e: 7 B 7 Fro this it is ow udtly ler how the il lultio ought to proeed Thereore oe these vlues re suessively sustituted euse s ; 9 ; ; 9 ; et we oti the ollowig otiued rtio or : 9 9 et Siilrly B 9 9 et C 9 9 et D et

17 7 NOTHER SOLUTION FOR CONTINUED PRODUCTS Whe we get: ; ; ; ; et DE CD BC B et et But i this epressio wheever you stop there will e tor let either i the uertor or i the deoitor This will eoe lerer i we sustitute i the letter F s ollows F However whe we sustitute i the ollowig letter G it will eoe: G But i thereore these two epressios re otiued iiitely d re lulted ito theselves the il letter tor whih here is F G will equl oe Sie i this se the uer leves tor i the uertor o oe I will write its irst tor prt i the rot prt d the produt will e epressed i the ollowig wy: et

18 8 Now the tors er iiity pproh oe d thus this epressio proeeds ordig to the lw o uiority Here guish etwee two ses: i Cse # : Where - I the irst se y tor llows itsel to e resolved i two wys Thereore i we irst estlish tht - the otiued rtio e epressed i the ollowig wy: however it is ittig to disti is positive or egtive uer 7 7 et d i the ple o this epressio we ow hve sequee i otiued tors; or the siple letter it is: et ywhere i the i tors o this epressio the su o the tors i the uertor Beuse o this epressed i itegrl orul is equl to the su tor i the deoitor these tors re le to e There is isprit i the lst tor It should e

19 9 7 It is greed tht i this itegrl orul: k is evluted ro the vlue is redued to the ollowig iiite produt: 0 to k k k Thereore whe we pply this k orul to our epressio sie sigle tors re iresed i the ollowig tor y the qutity oe ssue tht However whe it is ssued tht d k it is oud to e: vious tors o its eers For the ltter oes however i it reis tht it is ssued tht d k This epressio oe iverted otis the pre Oe this is doe it will rete this equtio: I thereore this equtio is divided y the preedig oe lter itegrl tors el eh other out d it will produe iiite produt givig the vlue tht is epressed y two itegrls s: : Note tht this tie he hs the orret tor whih he egleted i setio 7 whe 0

20 0 8 Let e show this y the ollowig eple I I ssue tht we get these vlues: B BC 8 CD DE et I this se our otiued rtio is: 77 et But it will e this i otiued produts: et The it will e this wy i itegrl oruls: : It is greed tht or y liits o itegrtio ro 0 to it is d π π Fro here is see to e This its spledidly with Wllis π produt where et 7 CSE # : Where 9 Let us solve ow other se whose otiul rtio tkes o this or: 9 9 et

21 However whe is writte i the ple o i the previous otiued produt the ew or is epressed igirily: et However it is evidet tht whe prgrph is pplied to the se epressio tht e writte i the ple o ro whih we get: et Thereore the produt o these two epressios eoes rel it will e: et This epressio is ogruous with the previous oe oud i prgrph 0 But this epressio eoes igiry through itegrl oruls For i is writte i the ple o i the orul i prgrph 7 the ollowig epressio rises: : However oe the sig o the igiry uers hve ee hged it eoes: : Here there is o dout tht i oth epressios the igiry uers el eh other out lthough there is o ppret ethod to show the utul elltio o igiry uers

22 However i oth epressios re rought together the this elltio e show esily The the produt is It e deostrted tht igiry uers el out i oth the uertor d the deoitor Beuse it is suiiet to hve show this o r the deoitor sie the uertor rises ro it y writig i the ple o So tht this deostrtio e delt with ore suitly let us ssue or the ske o revity tht V Whe this hs ee doe d oe the igiry uers hve ee delt with the deoitor o y epressio will e V V Now let us ssue the tors: The su: p V q V Dieree: qq pp V V d it is oted tht the proposed produt will e: Now I will show how oth pp d qq e redued to rel qutities lutio i the ple o i possile igiry uers let us write so it eoes I this ev i e

23 p e i e i V q e i e i V Here Euler writes l or our log Sie we kow tht φ φ φ φ e e osφ d e e siφ ssue or the ske o lrity tht i φ d get p V osφ d q V siφ Fro here the deoitor eoes: pp qq V osφ V siφ whih lerly is rel epressio Here the vlue o the uertor esily is rought together to eoe: V osφ V siφ I suh wy tht our epressio where the igiry uers hve ee reoved is represeted i rel uers i the ollowig wy: V osφ V osφ V siφ V siφ W e get : V dφ l

24 I lysis however ethod is still issig or delig with itegrtig oruls o this kid: osl d si l Mewhile however i the deoitor is set oth oruls re le to e itegrted whih jewel o y work shows i the ollowig wy For this e ehiited with the help o ous redutio Q PQ Q I P osl d Q is ssued or the previous orul we get: P P osl osl si l However or the other orul i P sii d Q is ssued it will e si l si l osl Fro here we get y sustitutig osl osl si l; si l si l osl However whe iludig the deoitor othig else is kow eept to epress the itegrl s kid o trsedetl qutity tht is still ukow Rell tht Euler uses l log