The Trigonometric Representation of Complex Type Number System

Transcription

1 Ieriol Jourl of Scieific d Reserch Pulicios Volume 7 Issue Ocoer ISSN The Trigoomeric Represeio of Complex Type Numer Sysem ThymhyPio Jude Nvih Deprme of Mhemics Eser Uiversiy Sri Lk Asrc- A exesio o he clssicl complex umer sysem from he spce of rel sequeces s proposed We sho h he e complex ype umer sysem forms field over he rel or complex field ih ifiie dimesio We explore some possile expoeil d rigoomeric formulios his e complex ype umers d ivesige heir properies The complex ype umers re used o represe hree dimesiol physicl spces ih Cresi coordies Idex Terms- Cuchy-Riem codiios Euler s formul Hrmoic polyomils Hyper-complex umers T I INTRODUCTION he rel umer sysem urlly ideified d ko from he erly period of philosophicl er d o deoed y R is field ih o sic iry operios mely ddiio d muliplicio hvig he ell-ko properies shorly lised s closedess commuive ssociive d disriuive properies exisece of ideiies d iverses for oh operios log ih he ssumpio h he ddiive d muliplicive ideiies re disic The developme of mhemics is so hevily depede o his fudmel umer sysem h y oher umer sysems soughre evlued for heir usiliy sed o hvig hese sic field properies some or ll of hem I his coecio he oly umer sysem hvig he ll of heove meioed field properies is he complex umer sysem Ispired y he vs usiliy of complex umer sysem i my res serch for higher dimesiol exesios of his kid hd ee cive reserch ieres i he ps The erly exesios ere defied o fiie dimesiol spces As i is o ko d proved h he oly fiie dimesiol spces ih ll he field properies re he rel he complex umer sysems hese emps for higher dimesiol umer sysems hve oviously filed Hmilo4 dediced his eire life for his filed emp Filly i 847 he cme up ih 4-dimesiol querios hich lck oe of he properies of field he commuive propery Sice he higher dimesiol exesios of complex umers ere focused o relxig some of he field properies The commuive d ssociive properies re relxed i querio5 d ocoio5 respecively Clifford Alger lso relxes he commuive propery Recely Fleury e l5 defied muli-complex umer sysem relxig he exisece of iverse of ll ozero mui-complex umers I his pper e propose clss of umer sysems of ifiie dimesio hich reis ll he properies of field Alhough his umer sysem is ifiie dimesiol i c e pplied o represe pois d fucios i hree d higher dimesiol Cresi spces We cll our e umer sysem he Complex Type Numer sysem The moivio of he defiiio of complex ype sysem comes from emp for geerlizio of he fc h he complex vlued lyic fucios of he o dimesiol Cresi coordie vriles x d y cosiue pir of rel vlued hrmoic fucios Our complex ype represeio gives coicl exesio of his propery o fucios of higher dimesiol coordie vriles Beyod hese y he complex ype field ih some geerlizios oher sic properies of complex ype umers such s rigoomeric fucios represeio De Moivre s formul d Euler s formul for expoeil represeios re sisfied I y fiie dimesiol spce our cosrucio of complex ype umers c e redily exeded o sisfy he hrmoic fucio propery for fucios I his pper e resric our discussio o fucio i hree dimesiol vriles oly hich c e direcly exeded o higher dimesios I his coex e see h our complex ype represeio is url exesio of he clssicl complex spce C i he sese h our complex ype spce iclude C d h he fudmel lgeric rigoomeric expoeil represeios d holomorphic properies of fucio re crried ou o he exeded complex ype spce ih some modificios This pper is orgized s follos: I Secio he defiiio of complex ype umer sysem d is sic properies re give I Secio 3 he exeded rigoomeric d expoeil represeio d heir ideiies re eslished I Secio 4 he lyic properies of complex ype vlued fucios sed o derivives of hese fucios re descried d filly coclusio is dr i Secio 5 isrporg

2 Ieriol Jourl of Scieific d Reserch Pulicios Volume 7 Issue Ocoer ISSN II DEFINITION AND BASIC PROPERTIES Assume e symolic vrile d R e he spce of ifiiely differeile fucios ih rel coefficies We defie Cs he se of pirs of fucios R Defiiio The complex ype spce C is defied s he se C { R } ih operios defied y here I i R i Here e hve omied he rgume symol i he fucios i Rd deoe s for reviy of oios Alhough e defie he spce cosisig of fucios of he symol e re mily ieresed i he coefficies of he poers of i he ifiie series represeio of he fucios I his coex he fucios re regrded s geerig fucios of heir coefficies We deoe he sequece of coefficies of fucio y iself so h he { } e hve We refer he pirs of fucios or he pirs of sequeces s he complex ype umers d deoe ; for he fucio form d ; for he sequece form We use oh oios ierchgely s hey poi o he sme cocep Alerively he elemes of he spce C c lso e expressed i he complex ype form here he imgiry symol sisfies For complex ype umer e cll d he rel ype d imgiry ype prs of f respecively The ddiio d muliplicio for complex ype umers re i such y h he rules of ddiio d muliplicio re cosise ih he rules of rel umers Thus if ff d ff e hve ff ff d ff ff Our prolem is o see heher he complex ype spce c e cdide for higher dimesiol umer sysem h llos lysis of fucio defied o he higher dimesiol spces Proposiio The spce of complex ype umers C forms field uder he defied operios The elemes of C c e expressed i he complex ype form here Proof: By lgeric mipulios closed ssociive commuive d disriuive properies of he operios re immedie The zero eleme is ; here d he ui eleme is here The iverse of o-zero complex ; f ype umer ff ; is The secod elemery sis eleme ; sisfies coupled relio I should e poied ou h he fudmel imgiry ui is fucio of he symolic vrile There is mi differece eee he prese complex ype formulio d he previous oher formulio h he fudmeluiesisfies he codiio of he forme for some poer Defiiio 3Giveff e defie is complex ype couge ff y ff f d is modulus solue sizeff y here d he rel ype d imgiry ype prs of f respecively From he ove defiiio e see h he sysem of complex umers hs coicl exesios of my properies of heir complex couerprs Proposiio 4For he complex ype umers vv Ce hve i ii v v iii iv v v if d oly if or v vi if d oly if Proof: By lgeric mipulio e c proof i o v isrporg

3 Ieriol Jourl of Scieific d Reserch Pulicios Volume 7 Issue Ocoer ISSN isrporg To proof vi i is coveie o defie geerig fucio for sequece } { s follos: For ech so h e hve d No for ih e prove iducively h for ll Equig he cos erm o zero e ge Hece No cosider Suppose iducively h i i for i We he ge The gives Thus e hve d hece This complees he proof To prove v e see from ii h vv oooo vv oooo vv We see h our e complex ype spce C is exesio of he complex spce C i he sece h C is icluded i C Specificlly he suspce C of C is he complex spce cosisig he cos coefficies of he rel d imgiry ype prs of complex ype umers III EXPONENTIAL AND TRIGONOMETRIC REPRESENTATIONS I our complex ype umers formulio e c formule polr form hrough he fuciol pproch The folloig resuls sho h he rigoomeric d expoeil fucios represeios d heir properies i he clssicl complex spce hve coicl exesios i he complex ype spce For his purpose e cosider defiig complex ype vlued fucio of complex ype vrile s follos:

4 Ieriol Jourl of Scieific d Reserch Pulicios Volume 7 Issue Ocoer 7 59 ISSN Defiiio3 Le F e ifiiely differeile rel vlued fucio of oe vrile d e complex ype vrile i C The fucio F is defied s F F α α! here αα is rel umer i he domi of FF Sice C is field oe c esily see h F is complex ype vlued fucio No e ill prove he folloig resuls cocerig he expoeil d logrihmic fucios of complex ype vriles Lemm 3The expoeil fucioeeeeee sisfy he relioee uuvv ee uu ee vv Proof By Defiiio 3 he expoeil fucio is give y he series form No for he complex ype vrile uu vv e cosider uu ee uuvv vv!! ee!! uu uu!! uu! vv ee uu ee vv! ρ e Theorem 33The complex ype umers C c e expressed i polr ype expoeilformulios here ρρ ρρre fucios of he symolic vrileih rel vlued coefficies Proof We defie l usig Defiiio 3 ih αα Hece l c e expressed i complex ype form l λλ I he meime i is esy o see h expl y he sme Defiiio ove y he expsio of he rel vlued fucio expl xx xx ou αα gi This gives expl Hece e hve y Lemm 3 ee λλ ee λλ ee ρρee Where ρρ ρρ Theorem 34:A complex ype umer c e expressed i rigoomeric ype Euler formulio give y ρρcos si d he rigoomeric ype fucios re defied s cos : cos ρ si si : d ρ Furher he rigoomeric ype fucios sisfy he De Moivre s formul: For ieger cos si cos si Proof I vie of Theorem 33 e hve ρρee ρρee ii ρρ cos ii si ρρcos si here he rigoomeric ype fucios re defied By proposiio 4 ee ee ii d heceρρ Here he relios eee he expoeil d rigoomeric ype fucios re eslish regrdig hem s ifiie series The De Moivre s ype formul is he immedie cosequece y iducio I is o recogized heher he rel coefficies of hve y geomeric or rigoomeric meig ecuse e hve ideified from he coceps of ifiie series represeios of he fucios ivolved I is hoever oserved h he firs coefficie is he clssicl gulr rgume of he complex umer f We derive some ideiies for he rigoomeric fucios h re similr o he correspodig ideiies o heir rigoomeric couerprs isrporg

5 Ieriol Jourl of Scieific d Reserch Pulicios Volume 7 Issue Ocoer 7 59 ISSN Theorem 35The rigoomeric ype fucios sisfy he folloig ideiies: cos si cos ± cos cos si si 3 4 si ± si cos ± cos si cos cos si si cos si si cos 5 Proof: I c e proved y direc mipulio of heir clssicl versio IV CONCLUSION A complex ype spce for rel sequece is proposed The complex ype spce is he coicl exesio of he clssicl complex spce i he sese h he fudmel rigoomeric lgeric d expoeil properies re crried o he proposed spce ih some exesios REFERENCES Cr E Ecyclop Des scieces mhem Pures e ppliqu es Guhier-Vills Pris 99 Fleury N Rusch De Trueerg M Ymleev R M Commuive exeded complex umers d coeced rigoomery J Mh Al Appl Fleury N Rusch De Trueerg M Ymleev R M Exeded complex umer lysis d coforml-like rsformios J Mh Al Appl Hmilo W R Elemes of querios Logms Gree: Hrvrd uiversiy press Horv h Bsic ses of polyomil soluios for pril differeil equios Proc Amer Mh Soc Joh C Bez The Ocoios Bullei ofamericl Mhemicl Sociey Kechum P W A complee soluio of Lplce s equio y ifiie hypervrile Amer Jour Mh Miles E P Willims Jr d E A sic se of homogeeous hrmoic polyomils i k vriles Proc Amer Mh Soc Nsir H M Sphericl Hrmoics i No-polr Co-ordie sysem d Applicio o Fourier Series i -Sphere Mh Meh Appl Sci Wiley Iersciece Nsir H M A e clss of mulicomplex lger ih pplicios J Mh Sci Price G B A iroducio o muli-complex spces d fucios Mrcel Dekker Ic 99 AUTHORS Firs Auhor ThymhyPio Jude Nvih Deprme of Mhemics Eser Uiversiy Sri Lk isrporg