The Quaternionic Exponential (and beyond)

TheQuaternionicExponential(andbeyond)[e-mail]~Hubert.Holin08/12/1999Motivation......................................................................................................................................2Chapter1Quaternionsredux.....................................................................................................21-Whattofindhere.....................................................................................................................22-ThenatureoftheBeast...........................................................................................................23-Quaternions’kin......................................................................................................................44-Quaternionsandrotations......................................................................................................55-Miscellany.................................................................................................................................8Chapter2BuildingtheQuaternions.......................................................................................111-Whattofindhere...................................................................................................................112-Cayleyalgebra,alternativealgebra...................................................................................113-TheCayleydoublingprocedure..........................................................................................124-R,C,H,O,X.......................................................................................................................135-ThefullCayleyladderallatonce........................................................................................14Chapter3TheExponential.......................................................................................................181-Whattofindhere...................................................................................................................182-Definition................................................................................................................................183-Linkswithdifferentiation...................................................................................................184-TheclosedformulafortheexponentialinCR().............................................................195-Somepropertiesoftheexponentialandfurtherconsequences..................................206-Conclusion...............................................................................................................................20Bibliography...............................................................................................................................21Softwareindex.............................................................................................................................211InterestingURLs.......................................................................................................................21MotivationIfelttheneedtotakeacloserlookatquaternionswhen,sometimeback,IwaslookingfornewapplicationstoHarthong-Reebcircles(onwhichIwasworkingatthetime),andcameacross[D.Pletincks(1989)].Thatpaper,ononehand,didindicateonepotentialapplicationforthatmethod,but,ontheotherhand,alludedtosomeoddconstructionsinvolvingquaternions,thevalidityofwhichwaspropitiouslyleftintheshadows.Thepresenttextisthereforeacompilationofmanywell-knownbutapparentlyscatteredresultsaboutquaternions(andrelatedentities),aswellassomenewdevelopments,notablytheexplicitformulaforthequaternionicexponential(andfriends).Incidentally,theseresultsenablesonetosolvetheproblemfoundin[D.Pletincks(1989)],butwithouttheunsalvageableconstructions.Chapter1Quaternionsredux1-WhattofindhereThischapteronlycontainsaquick-and-dirty(butsufficientformostuses)presentationofthequaternions,alongwiththeirmostclassicalproperties,inspiredverylargelyby[D.Leborgne(1982)],[J.Lelong-Ferrand,J.M.Arnaudiès(1978)]and[M.Berger(1990)].Thisapproach,however,obscuresthedeeprelationshipwhichlinksthequaternions,thecomplexandrealnumbersandmoreexoticthingsknownasoctonions;thisrelationshipwillbethethrustofthenextchapter.Itshouldbesaidthatotherimportantusesofquaternionsexist([K.Gürlebeck,W.Spössig(1989)],...),butthattheywillnotbetoucheduponhere.Aswell,quaternionicanalysis([A.Sudbery(1979)])andgeometry([S.Salamon(1982)]),thoughperhapsnotasvibrantastheircomplexcounterparts,dokeepevolving;thoughtheseusuallyinvolvefairlysophisticatedmathematicalmachinery,veryniceresultscanalsobehadwithveryelementaryones([P.deCasteljau(1987)],...).Allarebeyondthescopeofthisarticle,however.2-ThenatureoftheBeastLetH=R4withtheusualfour-dimensionalvectorspacestructureoverR.Wedefinee=()1000,,,,i=()0100,,,,j=()0010,,,andk=()0001,,,.Thefirstimportantthingweneedisamultiplication,denoted*,whichwedefinetobea(non-commutative)R-bilinearoperationonHsuchthatiijjkke*=*=*=-,ijjik*=-*()=,jkkji*=-*()=andkiik