Bicomplex Version of Laplace Transform

Transcription

1 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5- Bicomplx Vrsion of Lplc Trnsform * Mr. Annd Kumr, Mr. Prvindr Kumr *Dprmn of Applid Scinc, Roork Enginring Mngmn Tchnology Insiu, Shmli INDIA Dprmn of Elcronics nd Communicion Enginring, Roork Enginring Mngmn Tchnology Insiu, Shmli INDIA E-mil: * nnd67_ibs@rdiffmil.com, i.prvs@gmil.com Absrc In his ppr w hv sudid h bicomplx vrsion of Lplc Trnsformion LT, condiion of xisnc nd xmind h Rgion of Convrgnc ROC of bicomplx Lplc Trnsformion gomriclly wih h hlp of projcion on hyprbolic pln. Also w r promoing som usful propris of Lplc Trnsform in h bicomplx vribl from complx vribl. Ths rsuls cn b highly pplicbl in h fild of Signl Procssing. Indx Trms Bicomplx Numbr, Lplc Trnsform for complx vribl, ROC. INTRODUCTION In 89, in srch for spcil lgbrs, Corrdo Sgr [] publishd ppr in which h rd n infini fmily of lgbrs whos lmns r commuiv gnrlizion of complx numbrs clld bicomplx numbrs, ricomplx numbrs,.c. Inroducion pr conins hr following Scions. Min work will sr from Scion 4.. Crin Bsics Thory of Bicomplx Numbrs: Sgr dfind bicomplx numbr s- = x + i x + i x + ii x, whr x, x, x, x r rl numbrs, i = i = nd i i = ii. Th s of bicomplx numbrs is dnod s C. In h hory of bicomplx numbrs, h ss of rl numbrs nd complx numbrs r dnod s C nd C rspcivly. Thus C = { : = + i + i + ii,,,, C } Or C = { : = z + i z z, z C}.. Idmpon Rprsnion: Thr r wo non-rivil idmpon lmns in C, dnod by nd nd dfind s- + ii ii =, = ; + = nd = =. Evry lmn of C cn b uniquly xprssd s complx combinion of nd, viz. = z + i z = z i z + z + i z.this rprsnion of bicomplx numbr is known s h Idmpon Rprsnion of. Furhr, h complx cofficins z iz nd z + iz r clld h Idmpon Componns of h bicomplx numbr = z + i z.. Singulr Elmns: An lmn = z + iz is singulr if nd only if z + z =. Th s of singulr lmns is dnod s nd is chrcrizd s- O = { C : is h collcion of ll complx mulipls of nd }. Norm: Th norm +. : C C of bicomplx numbr is dfind s follows: whr + C dno h s of ll non-ngiv rl numbrs. If = z + i z hn - C z iz + z + iz = { z + z } = = x + x + x + x 4 / O. C is Bnch spc which is no Bnch lgbr bcus, in gnrl, η η holds insd of h sndrd condiion, viz. η η In his sns, C,,,,. + is rd s modifid Bnch lgbr. Th Auxiliry complx spcs A nd.4 Auxiliry Complx Spcs: A r dfind s follows: A= { z iz z, z C}, A= { z+ iz z, z C} ISSN : Jun - July 5

2 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5-.5 Crsin S: A crsin s drmind by X nd X in A nd A rspcivly is dnod s X X nd is dfind s: X X = z + i z C : z + i z = w + w, w X, w X. { } By h hlp of idmpon rprsnion w dfin som funcions such s- h : C A, h : C A, s follows:- p z + iz = h[ z iz + z + iz ] = z iz A z + iz C ; p z + iz = h[ z iz + z + iz ] = z + iz A z + i z C ; Hyprbolic projcion: H ρ = + i + i + ii = + ii. Crin Bsics of Bicomplx Anlysis: In 98 nd 9, Michiji Fugw origind h concp of holomorphic funcions of bicomplx vribl, in sris of pprs [], [4]. In 94, Drgoni [5] gv som bsic rsuls in h hory of bicomplx holomorphic funcions. A full ccoun of h updd hory cn b hld from Pric[6]. Som glimpss of h richnss of h hory cn b sn in Srivsv [7].. Smn: F is convrgn in domin D iff F nd F r convrgn in domin P : D D nd P : D D rspcivly.. Som Rsuls of Bicomplx numbrs-. = + = +. cos = cos + = cos + cos. sin = sin + = sin + sin n 4. n n n = + = + n 5. η = + η η = η + η = η + η n n n n 6. + ; η O = = + η η + η η η 7. η = + η + η = η + η 8. n n n n n n n n n n + η = + + η + η = + η + + η 9. = +. f d f d f d D D D d d d f = f + f d d d 4. Conjuncur:- ; Hr P : D D, P : D D L f b rl vlud funcion which hs xponnil ordr K such h S = = xis hr S C nd convrgn for R S > K nd k nohr Lplc L f f d F S Trnsform for S C L f = f S d = F S such s w hv linr combinion of F S nd F S wih nd such s:- xis nd convrgn for R S > K. Now S S S S + F S + F S = f d + f d f d f d F = = = F xis for R S > K nd R S > K or R P : > K nd R P : > K. Sinc F S nd F S r complx vlud funcions which r convrgn for R S > K nd R S > K rspcivly, so bicomplx vlud funcion F = F S + F S will b convrgn in h rgion D nd his is dfin s: D = { : = S + S; R } P: > K R P: > K. S = x + ix S = x + ix. Thus R S = x > K R S = x > K, hn L 4 + ii ii x+ x x+ x4 x4 x x x = x + ix + x + ix = x + ix + x + ix = + i + i + ii 4 4 ISSN : Jun - July 6

3 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5- Now hr r hr possibl css:. If hn x x nd x + x = x = x = x = x > K Hnc if = + i + i + ii, hn > K nd =. Thus R P : S = R P : S =.. If x x x > x hn > x > x nd x + x K + x K + x K x x x x x > > + = K + > K > K hus > K + > K > K nd >.. If x x x < x hn < x < x nd x + x K + x K + x K x x x x x > > + = K > K > K + hus > K > K > K + nd <. Ths hr condiions mking hr ss. { : } { : } { : } D = = + i + i + ii > K = D = = + i + i + ii > K + > D = = + i + i + i i > K < Thus R P > K nd : R P : > K implis D = D D D condiions in h s D, D nd D cn b wrin s > K + nd his shows h D nd D dfind s D = : H rprsn Righ hlf pln > K + { ρ } H ρ Hyprbolic projcion of. If K > hn hyprbolic projcion of li in h rgion s givn in blow figur., If K < hn hyprbolic projcion of li in h figur.. Fig.. Fig.. If = K + hn F hs pols, hs pols r shown on h rd lins s in following figurs. If K > :- If K = Fig.. Fig..4. ISSN : Jun - July 7

4 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5- If K < Convrs Rsul:- I will b h gurn of xisnc of Lplc Trnsform- If S D = D D D hn R P r grr hn K. Firs of ll w xprss in indpndn componn form such s- Fig.5. R P nd = = = ; R P = + R P = Cs. L S D hn > K, =, R P = + = > K.Similrly R P = > K. i i j i i i i i Cs. L S D hn > K + > hn + > > K. Hnc h rsul. Cs. L S D hn > K < hn > + > K. Hnc h rsul. LAPLACE TRANSFORM:- Now w r rdy o dfin h Lplc Trnsform for Bicomplx vribl. L f b rl vlud funcion of xponnil ordr K. Thn Lplc Trnsform of f for cn b dfin s- L{ f } = f d = F. Hr which hs H ρ in h Righ hlf pln H ρ hyprbolic projcion bcus r fr from rsricion. Thorm : EXISTANCE OF LAPLACE TRANSFORM:- If F = f d, Whr Crsin rgion of righ hlf plns R > K nd R > K Proof:- F = f d = f d + f d F is xis nd convrgn for ll D = D D D or > K +. In D hr r infini which hv sm f is of xponnil ordr K, hn is Lplc rnsform L{ f } F = +, h dfining ingrl for > nd >. So R K R K F xis for x x > K. Thorm : UNIQUENESS OF LAPLACE TRANSFORM:- L h funcion f nd g hv Lplc Trnsform F nd f = g F F F G G G = is givn by- F xis poins in h.boh h ingrls r xis whn = + or = x + ix + ix + iix, whr G, rspcivly. If F G Proof:- L- = + = +. If F G possibl iff F = G F = G f d = g d f d = g d f = g.. I s only possibl if =, hn = hn i is PROPERTIES OF LAPLACE TRANSFORM:- Thorm : LINEARITY OF LAPLACE TRANSFORM:- L h funcion f nd g hv Lplc Trnsform F nd G, rspcivly. If nd b r consns, hn L{ f + b g } = F + bf ISSN : Jun - July 8

5 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5- Proof:- L K b chosn so h boh F nd G r dfind for D = D D D, hn { + } = { + } = { } { } = f d + b f d + f d + b f d = F + bg + F + bg = F + F + + = F + + bg + = F + bg Thus L{ f + b g } = F + bf L f b g f b g d f b g d f b g d b G G NOTE:- = = + = + = + F F f d f d f d F F Thorm 4: LAPLACE TRANSFORM OF DERIVATIVES:- L f nd f ' b coninuous for F = L{ f } Proof:- L K b lrg nough h boh f nd ' of xponnil ordr K, hn L{ f ' } F f = whr f r of xponnil ordr K. L{ f ' } is givn by { '} ' ; = = f f d f F Thus L{ f ' } = F f f, f ' nd f '' r of xponnil ordr, hn { ''} = '' ; = ' ' = ' + ' = ' + { '} f ' F f F f f ' L{ f ''' } = F f f ' f '' c. L f f d D f f d = + = + Corollry:- If L f f d D f f d f f d f L f = + = Similrly- Thorm5: LAPLACE TRANSFORM OF INTEGRATIVES:- L f b coninuous for Thn F L{ f d } = nd b of xponnil ordr K nd l F b is Lplc rnsform. Proof:- L g = f d hn g ' = f nd g = f d =, Sinc f is of K xponnil ordr K, hr xis M > K so h- f M. K M K K g f d M d = M. Thus g is lso of xponnil ordr K. K Thn- F = L{ f } = L{ g ' } = L{ g } g o = L{ g } F Thus L{ g } L f d = = { } Thorm 6: MULTIPLICATION BY : Prov h L{ f } = F ' Proof:- W hv Libniz s rul in complx nlysis [8] Now- F ' = f ' + f ' = f d + f d By Libniz s rul w cn wri i s- = f d + f d = f d + f d = f d = L f { } = ' Thus L f F ISSN : Jun - July 9

6 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5- Thorm 7: DIVISION BY l F dno h Lplc rnsform of Proof:- W hv η η F f d =. So η η + η = + = + o, F F f d f d η = + F η dη F η dη F η dη η η = f d d + f d dη η f f. If Lim xis, hn - f L F = η dη W rvrs h ordr of ingrion in h doubl ingrl of h quion o obin-.now w k ingrion wih limi f f η η= η + η = + F d f dd f dd d d η η η η f f = d + d f f = d = L f Thus L = F η dη Thorm 8: SHIFTING THE VARIABLE - F is h Lplc rnsform of If f f = d+ d f, hn L { f } = F { } { } { } = Proof:- w hv L f = F S = f d L f = f d = f d = F Thus L f F Thorm 9: SHIFTING THE VARIABLE :- If F is h Lplc rnsform of f nd, U f r illusrd in figur- hn L{ U f } F, = whr f nd Proof:- Fig. 6. s W hv + s F = f S ds = f S ds L + s =, hn- = = + Bcus U f = for < nd U f = f for, hn = + = Thus L U f = F F f d f d f d F U f d U f d U f d { } ISSN : Jun - July

7 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5- NOTE:- = + = + U f d U f d U f d f d Som Lplc Trnsform Pirs wih ROC in Complx nd Bicomplx Vribl r givn in h following bl- In his bl w us- = + i C, S = S + is C, = + i+ i + ii = + C. U - Uni sp Funcion, δ - Uni Impuls funcion. S. No. f Exponnil ordr F S F ROC of F S Wih idmpon Componns ROC of F Wih Rl Componns δ All S All All U U m U U U m U m U S R S > Or S > S R S < Or S < m! R S > m S + m + R S > R = S+ + R S < S+ + R S > R = m S+ m+ + R S < m S+ m+ + S R S > S + ω + ω ω ω R S > S + ω + ω S+ + R S > R = S+ + ω + + ω Cosω. U Sinω. U Cosω U. Sinω U. ω ω + + ω S+ + ω R S R R > R > R < R < R > R > R > R > R < R < R > R > R < R < R > R > R > R > > = R > R > R > R > > < > > + < + > + < + > > > + > +. s U R S > R > > S R > 4. δ ' S All S All All ISSN : Jun - July

8 Annd Kumr l. / Inrnionl Journl of Enginring nd Tchnology Vol.,, 5- FUTURE WORK: W cn s h ffc on ROC if w k ω s complx nd purly imginry nd ROC lso ffcd if w hv hyprbolic funcions. This ppr is promod for Linr Tim Invrin LTI sysms. REFERENCES [] C. Sgr L rpprsnzioni rl dll form complss Gli Eni Iprlgbrici, Mh. Ann., 4 89, [] W. Rudin Rl nd Complx Anlysis, McGrw-Hill, 987. [] M. Fugw On h hory of funcions of qurnry vribl-i Tohoku Mh. J., 9 98, 75-. [4] M. Fugw On h hory of funcions of qurnry vribl-ii. Tohoku Mh. J., 5 9, 69-. [5] G.S.Drgoni Sull funzioni olomorf di un vribl bicomplss Rl Acd.d Ili Mm. Clss Sci. Fic. M. N., 594, [6] G. B. Pric An Inroducion o mulicomplx spcs nd funcions, Mrcl Dkkr Inc., Nw York, 99. [7] Rjiv K. Srivsv Bicomplx Anlysis: A Prospciv gnrlizion of h hory of spcil funcions, Proc. Soc. Of Spcil Funcions nd hir Appl. SSFA 5, [8] John H.Mhws, Russll W. Howll Complx Anlysis for Mhmics Enginring Nros publicion Hous-6. AUTHORS BIOGRAPHIES: Mr. Annd Kumr - H hs don M. Phil from Dr. B.R.Ambdkr Univ.,Agr U.P. H rcivd his M.Sc nd B.Sc dgr from M. J. P. Rohilkhnd Univ., Brilly U.P. H hs yrs of ching xprinc. His rsrch inrs r is bicomplx numbr. H hs publishd on ppr in inrnionl journl nd lso hs good G Scor-. H is working s Assisn Profssor in REMTch. Insiu, Shmli U.P. Mr. Prvindr Kumr H hs don M.Tch. in Digil Communicion D.C from Ambdkr Insiu of Tchnology, Dlhi I.P. Univ., Dlhi. H rcivd his B.Tch Dgr in Elcronics nd Communicion Enginring from Idl Insiu of Tchnology, Ghzibd U.P nd B.Sc. dgr from C. C. S. Univ., Mru. H hs yrs of ching xprinc. His ching nd rsrch inrss r h wirlss communicion nd digil signl procssing. H hs publishd four pprs in Inrnionl Journl nd on in IEEE. H is working s n Assisn Profssor in REMTch. Insiu, Shmli U.P. ISSN : Jun - July