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《Springer.Borwein.J.M.Ramanujan's.Arithmetic.Geometric.Mean.Continued.Fractions.and.Dynamics.Dalhousie.Colloquium.Springer.2003.(78s)》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、Ramanujan'sArithmetic-GeometricMeanContinuedFractionsandDynamicsJonathanM.Borwein,FRSCPreparedforDalhousieColloquiumSpring2003ResearchChair,FacultyofComputerScienceDalhousieUniversity,Halifax,NSCanadawww.cecm.sfu.ca/~jborwein/talks.htmlwww.cs.dal.ca/~jborweinRevised:February4,20031SrinivasaRaman
2、ujan(1887{1920)²G.N.Watson(1886{1965),onreadingRa-manujan'swork,describes:athrillwhichisindistinguishablefromthethrillIfeelwhenIentertheSagres-tiaNuovooftheCapellaMediciandseebeforemetheausterebeautyofthefourstatuesrepresenting`Day,'`Night,'`Evening,'and`Dawn'whichMichelan-gelohassetoverthetombo
3、fGuilianode'MediciandLorenzode'Medici.21.AbstractTheRamanujanAGMcontinuedfractionaR´(a;b)=b2´+4a2´+9b2´+´+...enjoysattractivealgebraicpropertiessuchasastrikingarithmetic-geometricmeanrelation&elegantlinkswithelliptic-functiontheory.²Thefractionpresentsacomputationalchallenge,whichwecouldnotresis
4、t.²JointworkwithRichardCrandallandvari-ouslyD.Borwein,G.Fee,R.LukeandR.Mayer.3
5、MuchofthisworkistoappearinExperi-mentalMathematics[CoLabPreprints27,29and253.]1.InPartI:weshowhowtorapidlyeval-uateRforanypositiverealsa;b;´.Theproblematiccasebeinga¼b
6、thensubtletransformationsallowrapidevaluation.²On
7、routewe¯nd,e.g.,thatforrationala=b,R´isanL-serieswitha'closed-form.'²Weultimatelyexhibitanalgorithmyield-ingDdigitsofRinO(D)iterations.¤2.Finally,inPartIIofthistalk,weaddressthehardertheoreticalandcomputationaldilemmasarisingwhenparametersareal-lowedtobecomplex.¤Thebig-Oconstantisindependentofth
8、epositive-realtriplea;b;´.42.PreliminariesPARTI.Entry12ofChapter18ofRamanu-jan'sSecondNotebook[BeIII]givesthebeau-tiful:aR´(a;b)=(1:1)b2´+4a2´+9b2´+´+...whichweinterpret
9、inmostofthepresenttreatment
10、forreal,positivea;b;´>0.Remarkably,forsuchparameters,Rsatis¯esanAGMrelationµp¶a+bR´(a;b)+R´(b;a)R´
11、;ab=(1:2)2251.(1.2)isoneofmanyrelationswedevelopforcomputationofR´.2.Thehardcasesoccurwhenbisneartoa,"includingthecasea=b.3.Weeventuallyexhibitanalgorithmuniformlyofgeometric/linearconvergenceacrossthepositivequadranta;b>0.
