The_Life_of_Pi(π)

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1、TheLifeofPiHistoryandComputationJonathanM.Borwein,FRSCPreparedforAUSTRALIANCOLLOQUIAJune21-July17,2003CanadaResearchChair&FoundingDirectorCECMCentreforExperimental&ConstructiveMathematicsSimonFraserUniversity,Burnaby,BCCanadawww.cecm.sfu.ca/~jborwein/talks.htmlpersonal/jborwein/picover.htmlRevi

2、sed:June1,20031TheLifeofPiMynameisPiscineMolitorPatelknowntoallasPiPatelForgoodmeasureIadded¼=3:14¤andIthendrewalargecirclewhichIslicedintwowithadiam-eter,toevokethatbasiclessonofgeometry."¤TheNotationof¼wasintroducedbyEulerin1737.2Abstract.Thedesire,andoriginallytheneed,tocalculateevermoreacc

3、uratevaluesof¼,theratioofthecircumferenceofacircletoitsdiameter,haschallengedmathematiciansformanycenturiesand,especiallyrecently,¼hasprovidedfascinatingexamplesofcompu-tationalmathematics.Itisalsopartofthepopularimagination.¤¤TheMacTutor"website,attheUniversityofSt.Andrews

4、myhometowninScotlan

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6、http://www-gap.dcs.st-and.ac.uk/~historyisratheragoodhistorysource.3TheSimpsons4Why¼isnot227EvenMapleorMathematica`knows'thissinceZ1(1¡x)4x422(1)0

7、,andtheanswerin(1)isanareaandsostrictlypositive,despitemillenniaofclaimsthat¼is22=7.Ofcourse22=7isoneoftheearlycontinuedfractionapproximationsto¼.The¯rst4are223333553;;;:71061135Inthiscase,theinde¯niteintegralprovidesim-mediatereassurance.WeobtainZtx4(1¡x)4(2)dx=01+x21726543t¡t+t¡t+4t¡4arctan(t

8、);733asdi®erentiationeasilycon¯rms,andthefun-damentaltheoremofcalculusproves(1).Onecantakethisideaabitfurther.NotethatZ11(3)x4(1¡x)4dx=;0630andweobservethat1Z1Z1(1¡x)4x4x4(1¡x)4dx

9、imedesfamouscompu-tation1010(5)3<¼<3:7170TheFigureshowstheestimategraphically.²Thederivationaboveseem¯rsttohavebeenwrittendowninEureka,theCam-bridgestudentjournalin1971.Theinte-gralin(1)wasshownbyKurtMahlertohisstudentsinthemidsix