
motor variable

ganithashaasthrathil, oru sankeernna veriyabilinte pravarthanangalil saadhaarana sankeernna samkhyakal ulppedunnathupole, splitt-complexes nambar playinil aargyumentukalum moolyangalum ulla oru functionaanu motor veriyabilinte function. vilyam kingdon clifford thante "baikvaattarniyansinte praadhamika rekhaachithrathil" (1873) oru kainamattik oopparettare soochippikkunna motor enna padam upayogichu. addeham thante splitt-bikkaadarniyanukalil skailerukalkkaayi splitt-complexes samkhyakal upayogichu. "swarabhedathinum paarambaryatthinum "splitt-complexes variablinu" pakaram "motor variable" ivide upayogikkunnu.
udaaharanathinu,
oru motor veriyabilinte pravartthanangal yadhaartha vishakalanam vipuleekarikkunnathinum playininte maappingukalude othukkamulla praathinidhyam nalkunnathinum oru sandarbham nalkunnu. ennirunnaalum, eee sidhaantham saadhaarana sankeernna thalatthile pravartthana sidhaanthathil ninnu valare kuravaanu. ennirunnaalum, parambaraagatha sankeernamaaya vishakalanathinte chila vashangalkk motor veriyabilukalumaayum kooduthal saadhaaranayaayi hyper complexes vishakalanathilum nalkiyirikkunna oru vyaakhyaanamundu.
praadhamika pravartthanangal.
dii = formula _ 2, splitt-complexes thalam. thaazhepparayunna udaaharana functionukalaaya "f" nuu domain shreniyum d yill undu
hyperbolic versor formula _ 3-nte pravartthanam vivarthanavumayi samyojippichu affine parivartthanam srishtikkunnu.
saadhaarana sankeernamaaya ganithathil squaring functionu saadrishyamilla. anuvadikkuka.
naalu kwaadrantukal aidantitti gadakamaayi mapp cheythirikkunnu ennathaanu falam
formula _ 8 yoonit hyperbola formula _ 9 roopappeduthunnu. athinaal, 
parasparaprathikaranam
sii yile vrithathinu vipareethamaayi hyperbola reference vakramaayi ulppedunnu.
leanier fraakshanal parivartthanangal.
oru valayathinu mukalilulla projective line enna ashayam upayogichu projective line pii (dii) roopappedunnu. nirmmaanam splitt-complexes nambar gadakangalulla ekeekritha kordinettukal upayogikkunnu. projective line p (d) leanier fraakshanal parivartthanangaliloode parivartthanam cheyyappedunnu
praadhamika rekheeya bhinnaroopaanthareekaranangal ulppedunnu
iva oronninum oru vipareethamundu, koodaathe rachanakal rekheeya bhinnaroopaanthareekaranangalu oru koottam niraykkunnu. motor veriyabilinte savisheshatha athinte dhruva kordinettukalile hyperbolic angle aanu, saadhaarana sankeernna thalatthinte mobius parivarthanangalaal vrithaakrithiyilulla angle samrakshikkappedunnathu eee angle motor variable leanier fraakshanal parivartthanangaliloode samrakshikkappedunnu. konukal samrakshikkunna parivartthanangale konformal ennu vilikkunnu, athinaal leanier fraakshanal parivartthanangal konformal maappukalaan.
parivartthanangalude athirthi pradeshangal thaarathamyam cheyyaam udaaharanathinu, saadhaarana sankeernamaaya thalatthil, kayley parivartthanam mukalile pakuthi thalam yoonit diskilekku kondupookunnu, angane athine bandhippikkunnu. dii yude aidantitti gadakamaaya yuu 1 oru deerghachathurathilekku mapp cheiyunnathu thaarathamyappeduthaavunna oru bounding aakshan nalkunnu
ivide "t" = {"z" = "x" + j "y": |"vaii"| <"x" <1 allengil |"vaii"| <2-"x" 1 ≤ "x" <2} aayirikkumbol.
projective lainile baijakshanukal enna nilayil leanier fraakshanal parivartthanangal thirichariyaan dii yude compactification upayogikkunnu. thaazhe koduthirikkunna bhagam kaanuka.
expi, logue, square root.
exponential function muzhuvan thalavum "d" "u" 1 lake vahikkunnu
athinaal "x" = "b" j aayirikkumbol, ex oru hyperbolic versoraanu. pothuvaaya motor variable "z" = "a" + "b" j-nuu, oraalkk undu
oru motor veriyabilinte functionukalude sidhaanthathil square root, logaritham functionukal ennivaykku prathyeka shradha nalkanam. prathyekichum, splitt-complexes nambarukalude thalam naalu connectucheytha gadakangalude formula vipareethamillaatha ekavachana pooyintukalude koottavum ulkkollunnu "z" = "x" ± "x" j ", x" ±r. aidantitti ghadakam, athaayath {"z": "x"> |"vaii"| } = u1, squaring functionteyum exponentialinteyum sreniyaanu. athinaal ithu square roottinteyum logaritham functionukaludeyum domainanu. square rootum logarithavum squaring functionteyum exponential functionteyum vann-tuu-vann invezhsukalaayi nirvachikkappedunnathinaal mattu moonnu kwaadrantukal domainil ulppedunnilla.
dii yude logarithathinte graphik vivaranam motor & rosa avarude lekhanamaaya "hyperbolic calculusil" (1998) nalkiyittundu.
dii-holomorphic functionukal.
sankeernamaaya thalatthile oru domainile holomorphic functionukalude swabhaava savisheshathakal nalkunna kaachi-reeman samavaakyangalkk oru motor veriyabilinte functionukalkku oru analog undu. oru wirtinger derivative upayogichu dii-holomorphic functionukalilekkulla oru sameepanam motor & rossa nalkki 
f" = "u" + j "v" enna functione d-holomorphic ennu vilikkunnu.
yadhaarthavum saangalppikavumaaya gadakangal pariganikkunnathiloode, oru dii-holomorphic function thripthippeduthunnu.
eee samavaakyangal 1893-ll jorg skifezhsu prasidheekarichu, athinaal avaye skifezhsu vyavasthakal ennu vilikkunnu.
harmonic function sidhaanthathile thaarathamyappeduthaavunna sameepanam peettar durren oru vaachakathil kaananakum.
gadakangal "yuu" aanennu vyakthamaan." 
oru dii-holomorphic function "eph" yude "vii", taranga samavaakyathe thripthippeduthunnu, ithu bandhappettirikkunnu 
dii 'alambert, athesamayam sii-holomorphic functionukalude gadakangal laaplesinte samavaakyam thripthippeduthunnu.
plaita paadangal.
1935ll laa plaita deseeya sarvakalashaalayil jeey.sii. ananthamaaya shrenikalude samyojanathil vidagdhanaaya wignox sarvakalaashaalayude vaarshika aanukaalikathil motor veriyabilinekkurichulla naalu lekhanangal sambhavana cheythu. aamukhathinte eaka rachayithavaaya addeham thante vakupp medaaviyumaayi koodiyaalochichu. mattullavarekkurichu duranona vaii media. sobre laas seerees dii numeros complejos hyperbolicos" ennathil addeham parayunnu (p. 123):
udaaharanathinu, kaachi, aabel, mertons, hardy enniva kaaranam motor veriyabilinte domainilekku sidhaanthangal saamaanyavalkkarikkaan addeham munnott pookunnu.
chuvade udharichirikkunna praadhamika lekhanathil, dii-holomorphic functionukalum dii 'alambarttinte samavaakyathinte samthrupthi avayude gadakangalum addeham pariganikkunnu. y" = "x", "y" = − "x" ennee konukalkku samaantharamaayi vashangalulla oru deerghachathurathe, athinte vashangal isotropic rekhakalilaayathinaal, oru "isotropic deerghachathuram" ennu addeham vilikkunnu.
eee vaakkukalode addeham thante sangraham avasaanippikkunnu
bernstain polynomialukal oru yoonit isotropic deerghachathurathile dii-holomorphic functionukalude ekadeshathekkurichulla aarr pejulla kurippode wignox thante seerees poortthiyaakki. eee shreniyil chila typographical pishakukalum rund saankethika edarchakalum undengilum, yadhaarthavum saadhaaranavumaaya sankeernamaaya vishakalanathin edayilulla sidhaanthathinte pradhana varikal roopappeduthunnathil wignox vijayichu. gadakangalil ninnulla maathrukaaparamaaya vikasanam kaaranam vidyaarthikalkkum adhyapakarkkum oru prabodhana rekha enna nilayil eee vaachakam prathyekichum aakarshakamaan. koodaathe, muzhuvan vinodasanchaaravum athinte prachodanatthinu adivarayidunnathinaayi "emilinte boralinte jyamithiyumaayulla bandhatthil" veroonniyathaanu.
bairal variable.
1892ll corado segre tesserine beejaganithathe "bicomplexes nambarukal" ennu thirichuvilichu. swaabhaavikamaayum yadhaartha tessarinukalude upa beejaganitam uyarnnuvarikayum athine bairal nambarukal ennu vilikkukayum cheythu.
1946ll yuu. erattasamkhyakaleyum splitt-complexes nambarukaleyum kurichu benziwenga oru lekhanam prasidheekarichu, avide addeham bairal nambar enna padam upayogichu. bairal variablinte chila function sidhaanthangalum addeham vivarichu. 1949ll britiish columbia sarvakalashaalayil jiophree phoox thante maastezhs theesis "hyper complexes veriyabilinte praadhamika pravartthana sidhaanthavum hyperbolic playinile konformal mapping sidhaanthavum" ezhuthiyappol eee lekhanam padichu. 46-aamm pejil kurukkan report cheiyunnu "oru bairal veriyabilinte oru function hyperbolic playinine svayam mapp cheiyunnuvennu benziwenga kaanichu, oru functionte derivative nilanilkkukayum aprathyakshamaakathirikkum cheiyunna pooyintukalil hyperbolic konukal mappingil samrakshikkappedunnu".
jii. kurukkan oru bairal veriyabilinte dhruva vighadanam nalkunnathinaayi munnott pookukayum hyperbolic orthogonaalitti charcha cheyyukayum cheiyunnu. 57-aamm pejil addeham theliyikkunna mattoru nirvachanathil ninnu aarambikkunnu.
phoox "bilinear transformations" formula _ 22 ll shradha kendreekarikkunnu, ivide formula _ 23 bairal sthiraankangalaanu. ekathvathe neridaan addeham ananthathayil orotta pooyint upayogichu thalam vardhippikkunnu (peju 73).
function sidhaanthathile addehathinte puthiya sambhaavanakalil onnu "interlocked sisttam" enna aashayamaanu. oru bairalinu "keey" thripthikaramaanennu kurukkan kaanikkunnu 
hyperbolus 
parasparam murichukadakkaruthu (oru interlock cheytha sisttam undaakkuka). oru bairal veriyabilinte dviroopeekarana parivartthanangaliloode eee property samrakshikkappedunnuvennu addeham kaanikkunnu.
compactification.
gunana vipareeta pravartthanam valare pradhaanamaan, athu deferential jyaamithiyude mappingil ulppeduthaan theevra nadapadikal kaikkollunnu. udaaharanathinu, sankeernamaaya thalam saadhaarana sankeernamaaya ganithathinaayi riman golatthilekku uruttunnu. splitt-complexes ganithashaasthrathinu oru golatthinu pakaram oru hyperboloid upayogikkunnu formula _ 24 riman golatthilennapole, reethi "p" = (0,0,1) muthal "t" = ("x", "y", 0) vare hyperboloidilake stereographic projakshanaanu. s" poojyamaakumbol "p" umm "s" onnaayirikkumbol "t" umm kadannupokunna vidhathil "l" = "pt" enna variye soothravaakyathil "s" kondu paaraameettar cheiyunnu.
h" θ "l" ll ninnu athu pinthudarunnu
t" shoonya konilaanengil, "s" = 2 umm (2 "x", ± 2 "x",-1) "h" lum, vipareeta pooyintukal (2 "x", ± 2 "x", 1) ananthathayil prakasha koonn undaakkunnu, athu vipareethathinu keezhilulla shoonya koninte chithramaanu.
formula _ 27 "s" ulla "t" negatteevu aanennu shradhikkuka. p" muthal "t" vareyulla back-reey "h" ennathil pooyint nalkunnu ennathaanu ardham. eee pooyintukal "tii" yoonit hyperbola hyperbola konjugettinu mukalilum thaazheyumaan.
w" = 1 ithuvare upayogicha affine space ("x, y, z") vyakthamaakkunna ekeekritha kordinettukal ("w, x, y, z") upayogichu compactification p3r ll poortthiyaakkanam. hyperboloid "echu" oru compact space aaya projective conic formula _ 28-ll agiranam cheyyappedunnu.
hans beck kaaranam oru mapping upayogichaanu walter benz compactification nadathiyathu. isak yaglom mukalil paranjathupole rund gattangalulla oru compactification chithreekarichu, pakshe hyperboloidilekkulla splitt-complexes playin taanjant upayogichu. 2015 ll emuello & nolder aadyam motor playin oru torressil ulppeduthi, thudarnnu antipodal pooyintukal thiricharinju kondu projective aakki compactification nadatthi.

