[Nachlass] of the Theory of the Arithmetic-Geometric Mean and the Modulus Function.

Transcription

1 [Nchlss] of he Theory of he Arihmeic-Geomeric Me d he Modulus Fucio Defiiio d Covergece of he Algorihm [III 6] Le d e wo posiive rel mgiudes d le From hem we form he wo sequeces: i such wy h y wo correspodig erms re he mes of he wo precedig d i fc he erms of he upper series re o e he rihmeic me he lower he geomeric me ie v v We presuppose h squre roos will ll e e s posiive [wih posiive sigs]; he he series coiues idefiiely ll is erms re eirely deermied d hve posiive rel vlues Furher from his direcly follows: If ll erms of he wo series re However if he i follows he iequliy: < d similrly < < K h is ech erms of he lower series is smller he he correspodig erm of he upper series < > < > K Th is he upper series coiully decreses he upper icreses; hece i ecomes cler h oh series hve limiig vlue d respecively which lie ewee d

2 From he relioship/depedecy [Beziehug] follows Similrly < < < < v v < v Th is he mgiudes form cosly descedig series wih he limiig vlue Thus Th is he upper d lower series hve he sme limi d his is smller h ll erms of he firs series u greer h ll he erms of he secod We deoe his limi he Arihmeic-Geomeric Me gm ewee d d wrie M foooe Bcwrds eesio [Rücwärsverlägerug] of he Algorihm we will ow eed he lgorihm cwrds For h purpose we oe h he roos of he qudric equio re rel d posiive s resul of he precodiio d h is he rihmeic he geomeric me ewee oh hese roos Thus we se c c c so h - is o idice/sigify h erm of he upper series which precedes he erm d - s h erm precedig of he lower series; hrough - - he lgorihm hs hus ee eeded cwrds We ow similrly se 6 c ; c c c ; c c

3 c ; c c c e regrded s he coiuio of he upper series o he lef d similrly s he correspodig eesio of he lower series so h ow wo series re hus oied which re coiued [forsez] o oh sides d ifiium: 7 All erms re rel d posiive [d] ech erm of he upper series is greer h he correspodig erms of he lower The firs series decreses from lef o righ he secod ehves ecly opposie [gerde ugeehr-direcly flipped roud] O he righ oh series hve he sme limi u o he lef he upper grows ps ll limis d he lower srys owrds zero ecep whe Th is c c > > c c > c c c > 6c from which i is cler h he series c c c cd ccordigly lso mus overshoo/eceed ech limi Furher < < lim v> < ie he limi vlue of he series is zero From which is see h he series grdully pproches geomeric progressio wih he quoie From he defiiio of he gm is he followig proposiio/heorem so evide h i eeds o furher elucidio: The gm ewee y wo correspodig erms of he upper d lower series 7 is he sme s h ewee d ie 8 lim lim M M M > > Moreover he ruh of he followig relios is immediely relized M M ; 9 M M M M

4 A umericl emple is see s o how fs he lgorihm coverges owrd he limi vlue For emple wih he clculio of M he erms d firs differ i he h deciml plce d d i he erd plce; is lredy smller h More progressio of he lgorihm Ideiies I he sme wy we ow form he lgorihm which leds o Mc d hus se c c c c c c c c c whose correspodig cwrds eesio we oi from he formuls : c c c c c c d i pplies for ll relioships lim lim c M c M c M c i epsio of he relioships 6 d we se geerlly c c c vlid for ll posiive d egive v d ideify c c c The simple relios eis ewee ll hese mgiudes Firs of ll c ; c c d his formul ideed followig 6 lso pplies for egive v Moreover his ecomes c c c c c c c c < c c c

5 c c Furhermore from follows 6 c c c c c c The ls formul is ideicl wih 6 for egive v Correspodig relios pply for he lgorihm ledig o Mc mely: c c c c ; c ; 7 c c < c i follow hus from 7 d c 9 c c furher from d 6 rises c c c c c c c c c c c ; > from 8 is oied i he sme mer hus h ; ; > c c

6 c c he ggrege of d gives: ; ; c c Lsly we oe y sill oe more relio which is ewee he squre roo composed of he elemes of he sme lgorihm: c ; hus rises he formuls c d logously c c Boh relios 8 d c hus e epded i he followig wy: lim lim lim lim M M M M M ; c 6 lim lim c lim lim M c M c M c M c M c Trigoomeric Cosrucio/Delieio/Demosrio/Digrm of he Algorihm [X ] A rigoomeric epressio of he lgorihm of he gm c e give which is usully mos suile for clculio We se: 7 cos M cos M cos M he:

7 cos M cos M cos M cos M Bewee M M M eis esily ssigle relios mely 8 M si M M si M M si M Thus re successively clculed wih he followig Schem: cos M cos M 9 si M M cos M cos M si M M cos M cos M The he gm is direcly give hrough M cos M cos M cos M cos M The Power Series for M [III 6] We posi ow he prolem o fid for he me M series uil o he [forschreiede] powers of Sice[d] M we pply: M Where re cos coefficies; heir deermiio crried ou hrough he fuciol equio sprug from 8 M M We se mely: he from follows M M hus

8 8 8 From which hrough gherig lie coefficies he followig equios for rise: d from h is cquired: Thus he sough series 9 6 M Sice however heir coefficies dhere o o cler lw we igore his series d dop oher wy h wrrs forue oucome 6 The Power Series for M d M [III66-69] We solve he prolem M hrough he power series uil o M M Ad hece he sough series is oied from i which is susiued for ; hrough which rises: M ± d from here: M The coefficies of his series c lso e eslished/cquired/deermied idepedely of i he followig wy I is sid:

9 The M M M M We dd o o his: 6 8 M β β β β --i is immediely oied which due o he ivrice of M compred o/cross from/opposed o/ of permuio of wih- he odd powers of c o occur-- Thus will: 6 β β β K 8 6 β β β β From which hrough he equios of he coefficies follows: β β 6β β β 8β 6β 96β β 6β β β d from here oi: β β β β 6 6 hus we gi come c o he series Ii his series s well he coefficies re sujec o o sigle lw; o he oher hd divide i io oe d ge: K M K where he coefficies hus progress followig very simple lw However his mehod y mes of which we cquired his quie ice resul depeds solely o he uheiciy/coclusiveess[beweisrf] of coclusio y iducio; he deducive proof is he ojecive of he followig cosiderio We me he proposiio: M The fuciol equio he ecomes:

10 M M hus 6 8 i is however i geerl: λ λ! λ λ λ λ λ λ λ ; Thus ecomes: From his follows wo differe clsses of equios for he deermiio of h is 6 Ou of his is oied he relios: } {

11 The preheicl epressio simplifies hrough he ideiy: hus is foud: ±K 6 6 K i pplies however/hus furher wihi: d hus y: 6 ±

12 6 ± ± ± 9 6 ± ± This equio pplies for eve d odd if he wih eve ide re se equl o zero[?] Ou of i follows for successively he relios: wherefrom is oied: 6 9 ' K K ; hrough his is he suspeced lw of coefficies of he series is i fc deomosred 7 The Differeil Equio [III7] The jus ied cocise series sufices s simple s well s eleg differeil equio which is for he eire heory of ereme imporce [eischeideder Bedeuug] We se for he mome: K 6 9 y M K 6 9 so:

13 dy d d y d d y d dy d 9 9K K K 6K K 9K 6K 9 9K 6K K K K d y dy 9 9 y K d d 6 9K K 6K he he ideiy is: Thus is recogized he eisece of he Relio: d y dy d y dy y d d d d Th is y sisfies he Differeil equio d y dy 7 y d d Oher h M hese differeil equios hve furher priculr/idividul [priculeres] iegrl They evidely/pprely remi uchged whe is replced wih For we se ξ So dy ξ dy d y ξ d y dy d ξ dξ d ξ dξ ξ dξ hus d y d - y d dy d y dy ξ ξ ξ ξy ξ dξ dξ Th is 7 urs io

14 y d dy d y d ξ ξ ξ ξ ξ ξ Now M M y is soluio from 7 we oi hus wice/douly priculr [priculeres] iegrl M M M ξ ξ ξ η Ds llgemeie 8 M c M c 8 The complee Ellipicl Iegrl of he firs clss As is geerlly ow hese relios hold φ φ cos d cos φ φ d cos φ φ d K K Thece 9 φ φ cos d φ φ φ φ d 6 6 cos 6 cos cos K K 6 6 M M We hve hus he reciprocl vlue of he gm used i comiio/coecio wih he complee ellipicl iegrl if he firs id 6 Nurlly his mehod/operio c lso del wih iegrls of geerl form For emple if > γ he iegrl ϕ γ β ϕ β ϕ ϕ γ β cos cos d d γ β β β γ β M M

15 Thus he sme reciprocl gm from he reciprocl mimum d miimum of he iegrls [Iegrde] I his form he formul c lso e ssiged/crried ou i he cse h γ < Th is we se ψ ϕ so d ϕ β γ ϕ cos dψ β γ γ cos ψ Also filly/eveully/lsly β β γ M γ d ϕ dψ ψ ϕ β γ cosϕ β γ γ cos ψ β γ β γ d ψ M β γ γ cos ψ Oe recogizes from his h he vlue of complee ellipicl iegrl hs o e evlued y he followig rule: mimum d miimum vlue υ d υ ' of he iegrls i he iervl of iegrio re eslished he he vlue of he iegrl is equl o M υ υ' From his i ecomes cler h if more epressios of he form i he β γ cosϕ iegrl re [Iegriosgeie] hve equl erem heir complee iegrls come o he sme vlue The solved[geloese] prolem c e geerlized i which o s he vlue of he udermied ellipicl iegrl ϕ cos ϕ I would e dvgeous o proceed such h h he iegrd wih he cosies of muliple ϕ is developed: cos P ϕ P cos ϕ P Where P deped oly o The dϕ cos ϕ P cos6ϕ ϕ dϕ cos ϕ

16 P P si ϕ P si ϕ P si 6ϕ Ad sill hus ied oly from he deermiio of P The firs coefficie we lredy ow: P ϕ dϕ M cos ϕ For he deermiio of oher coefficies we will ler give simple mehod d similrly geerlized proof of his phrse [Seze] which is cquired from my url[ursprueglichere] priciples 7 There re lwys people h do o ow of he sulimiy of he eerl ruhs d heir divie euy d herefore hve oly lered o pprecie he vlue of mhemicl ivesigios from heir ppliciliy i he field of pplied sciece; he ove devoeme will hve he use o me hese people more plesed our ivesigio I fc i is geerlly ow of how gre uiliy such rpidly coverge epsio/progressio s h developed from he ove seeces is i physicl sroomy or he heory of plery perurios The coherece/comiio/coecio wih he Ellipse qudr A ellipse is preseed wih semi es > ; he legh of ellipse qur is o e q We he deoe c 67 ; ' ' The s is geerlly ow q is educile hrough complee ellipse iegrl of wo clsses: q d 68 d K d i geerl K d * K we fid for q he series epsio 8 * q * [X 78] 6 K 6 K ;

17 Similrly s he iegrl of he firs clss c lso epress his Iegrl of wo clsses i simple mer hrough he gm We geere mely followig M ' M K K K K s is d M ' M ' d K K d { M ' M ' } d K K wih which follows from 69 he relio d 7 q { M ' M ' } d ie he sough equio We c give i coveie form for clculio From 67 is ' 7 d d ' hus oi d dm ' M ' M ' d ' d ' dm ' q M ' ' M ' d' d he from 66 is dm ' M ' K d' ' Filly we cquire q M ' 7 [X78] K The formul 7 is he so clled Klsseeziehug ewee he complee ellipicl iegrl of firs d secod id 9

18 Cosrucio of he coiuous frcio [X ] The eire ivesigio of he precedig prgrphs ws sed upo he lysis of cse I which we hd ccomplished hrough lysis[zerleug] of he susiuio S i he produc of S m There is however sill oe oher cio/procedure/operio o complee his cse i which ' is developed i coiuous frcio For h we wrie: βi ' γi δ γi δ βi If is sill o e deermied whole umer he we se: γ γ ; δ δ β he ' γ δi i i β I is see immediely h he [followig] cogrueces pply: γ γ mod δ δ mod d h he deermie of he ls frcio is: βγ δ δ βγ We deermie he umer such h δ < β We ow se logously: γ ; β β δ he will e γ δi i β i β i β i γ δi γ i δi where mod β β mod δ βγ δ βγ We choose he whole umer so h 7 β < δ < β I is he hus comied/cosolided ' i βi i γ i δ

19 where he ls frcio iself sisfies covergece d deermi codiios s h from which we hs proceeded Oly wih his frcio c we liewise proceed d oi wih proprie choice he whole umers βi γ i δ i β i i γ i δ mod β γ mod δ mod δ β γ < δ β β < Through coiuio of he procedure eveully frcio for he β hus δ is oied which hus hs he form: γ γ i i Therefore ech susiuio ' c e represeed i he form of coiuous frcio βi 7 ' γi δ i i O i Where he forms odd [ugerde] umer/quiy of rel whole umers Also iversely ech coiuous frcio of he form 7 represes rsformio of he s clss The e is i i i i i i furher susiuio wih he sme properies d hece from he ierio of his coclusio rises he fc h ech coiuous frcio of he form 7 represes susiuio of he s clss wih he deermi By mes of he represeio 7 i is ow esy o emie he ehvior of p q r wih he susiuio i i ; Ki ;

20 he followig 6 8: ' p ' p p K K p p ' p p K K K i q r ' q r K K i K r which compleely correspods wih our formul 6 Oe c ow lwys furher discuss how o cover he ove i coecio o he formul 76 however we will o go io his more sice i is he siclly he sme ri of hough The Qudric Form The Fudmel [ereich] re/divisio/field/rge/scope/domi There remis impor relio ewee he heory of he here reed lier frciol susiuio d h he qudric forms wih egive deermi If F d F re wo qudric forms i he vriles u v d u v wih whole umer coefficies: 77 F u uυ cυ F ' ' u' ' u' υ' c' υ' The oh forms ecome he oher hrough whole umer susiuio of he vrile wih deermi hus oh he forms re clled irisiclly equivele u' u βυ υ ' γu δυ δ βγ A eccesry codiio for he equivelece of wo forms is evidely he equliy of is deermis: d c d' ' c' Oe qudric form F is clled defiie if i lwys hs he sme sig oher h for uv for ll possile rel vlues of u v h is if is roos re o rel he is demi is egive If ow F d F re wo posiive defiie equivele forms h is > '> d d'< u u' From which lso follows c> c >; he mus ewee he vlues d for υ υ' vishig of F d F o follow of he equivelece defiiio relio of he form: u β u' υ υ' u γ δ υ Wih whole rel β γ δ he deermi remis We will hece deoe wo defiie qudric forms of oe vrile:

21 c d ' ' c' As equivele if he roos he oe i d i he oher reled hrough rel lier reciol rel umer susiuio wih he deermi h is if i β ' i γ i δ ' or βi ' γ i δ Thus d rele hrough susiuio of he form Now is y riol vlue λ µ i σi τ Where γ µ σ τ re whole rel umers d umeror d deomior wihou commo fcors The quric form c e foud whose roo is i; h is se: σ τ A λσ µτ B λ µ C AC B λτ µσ D The: σi τ λi µ B Di 8 i σ τ A A roo of he posiive defiie form 8 A B C A B C Which he egive deermie B AC D eiis A equivele form c o his oe 8 c ow lwys e foud for 8 c This form is deoed s he reduced of he origil or lso s he simples equivele form o 8 To rrive his reduced form we proceed wih he followig Algorihm We se 8 D B AA B B ha h whole umer The A C d B we chose s he solue les remider of B mod A hus he h B A Evidely ow D B D B mod A ; We hus se D B A A Ad if o lredy A A will deermie B from he equio B B h A h whole umer Furher will s solue smlles remider of B mod A so h B A

22 The furher is Ad i ecomes If o A A D B mod A D B he furher deermie from B B h A A A h whole umer B s solue smlles remider of B mod A hus B A Ad coiue so forh from D B A A Arises A A wh mus A B A K B Di i A D ib iha A hi hi A D Bi hi h i A B A B A A I ' I ' h g h g g g ' σ : ' σ : ' i σ : ' σ : ' σ : ' σ 6 : ' i i i I I

23 ' K K τ τ τ τ τ τ e e e e q p p q ; ' ' p q p p r p q ' K m K τ τ τ τ e e e e ' cos A p q

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26 Ellipicl iegrl ifiie umer of vlues of he gm Lemisce fucio 9 rsformio of he ellipicl iegrl du/r of ellipse J iegrl of ellipse