Sc. Rs. hm. ommn.: (3, 0, 77-8 ISSN 77-669 ANALYTIITY THEOREM FOR FRATIONAL LAPLAE TRANSFORM P. R. DESHMUH * and A. S. GUDADHE a Prof. Ram Mgh Insttt of Tchnology & Rsarch, Badnra, AMRAVATI (M.S. INDIA a Dpartmnt of Mathmatcs, Gornmnt Vdarbha Insttt of Scnc & Hmants, AMRAVATI (M.S. INDIA (Rcd : 0.0.0; Rsd : 7.03.0; Accptd :.03.0 ABSTRAT anoncal transformaton whn appld to a qantm mchancs, n a grop nd to b xtndd from ral doman to complx. Torr A. had ntrodcd fractonal Laplac transform a spcal cas of lnar canoncal transform wth complx ntrs n th rprsntat matrx. Th fractonal Laplac transform (FrLT has bn sd n sral aras ncldng sgnal procssng, optcal commncaton tc. In ths papr w ha prod th analytcty thorm for th gnralzd fractonal Laplac transform. y words: Lnar canoncal transform, Fractonal Laplac transform INTRODUTION Now a day, th fractonal ntgral transform plays an mportant rol n sgnal procssng, mag constrcton, optcal commncaton, fltr dsgn tc. 5,6 Forr transform s on of th most frqntly sd tool n sgnal procssng and n many othr dscplns. Nams ntrodcd th fractonal Forr transform. Bhosal had xtndd fractonal Forr transform to th dstrbton of compact spport. Torr 4 had also ntrodcd fractonal Laplac transform as a spcal cas of complx lnar canoncal transform whn charactrstc matrx s cos sn. Th opratonal transform formla ha bn drd n sn cos 3. In ths papr w ha dfnd gnralzd fractonal Laplac transform. Th Analytcty thorm on fractonal Laplac transform s prod n scton. Lastly conclson s gn n scton 3. Fractonal Laplac transform Th fractonal Laplac transform s a gnralzaton of Laplac transform. It s a powrfl tool for th analyss of tm aryng fnctons. Th proprts and applcatons of th conntonal Laplac transform ar spcal cas of fractonal Laplac transform. Th fractonal Laplac transform of th sqar ntgrabl fncton f (t s dfnd n trms of a rnl, as Aalabl onln at www.sadgrpblcatons.com * Athor for corrspondnc; E-mal: pdshmh9@gmal.com, ala.gdadh@gmal.com
78 P. R. Dshmh and A. S. Gdadh: Analytcty Thorm for. L ( f ( t dt whr th rnl ( of th fractonal Laplac transform s gn by ( co+ co cot csc, s not mltpl of δ (t-, s a mltpl of Moror, th fractonal Laplac transform wth, corrsponds to classcal Laplac transform and wth 0, corrsponds to th dntty oprator. Analytcty on fractonal Laplac transform Th Gnralzd Fractonal Laplac Transform on E Lt Sa {t : t R n, t a, a > 0} and t Sa, 0 a <, f ( co + co csc cot, whr thn ( E(R n, f γ { ( } Sp < < D ( < Hnc E (R n s th tstng fncton spac. Th gnralzd fractonal Laplac transform of f (t E' (R n s dfnd as L {f (t} ( {f ( ( }, whr E' (R n s th dal spac of th tstng fncton spac. Analytcty thorm Lt f E' (R n and L ( {f (t}, ( }, thn L [f (t] ( s analytc on n f th spp f Sa {t : t R n, t a, a > 0} and for ach ε > 0, thr xst a constant and a post ntgr, sch that 0 <, L ( L ( ( + (acos + } { ((a + ε + I m cos-(a + ε I m } Moror L ( s dffrntabl and D L ( { f D }. Proof: Lt (,, 3,, n n Frst w pro t for, that s w show, L ( f ( Th gnral rslt follows by ndcton For fxd 0 choos two concntrc crcls and wth cntr and rads r and r rspctly, sch that 0 < r < r < }.
Sc. Rs. hm. ommn.: (3, 0 79 Lt Δ b a complx ncrmnt sch that 0 < Δ < r. Now consdr L ( + L ( f ( { f Δ whr ( t {,,...,... n + Δ and L ( f ( t d cot co + co csc f ( t dt ( t} (.. Whr ( cot co + co cosc [( ] sn co co sn sn + sn sn [( ] co + cos ( sn [( ] cos + cos [( ] sn cos + cos (.. For any t R n and ry fxd ntgr (,,. n, N 0 n (, { [( ] cos D t D + cos } ( (cos ( t cos ( s sμ ( t cos 0 s (..3 Whr ( (cos and ar th constant dpndng pon,,. Snc for any t R n, fxd ntgr and from, o to, D s analytc nsd and on ',
80 P. R. Dshmh and A. S. Gdadh: Analytcty Thorm for. By sng achy ntgral formla D ( t D z z ( z dz D ( z ( z dz Δ M ( z ( z dz z > (r r > 0 (..3 and z r Bt for all z and t rstrctd to a compact sbst of R n, o M ( s bondd by a constant. D Δ ( t ( r r r As 0, D Δ ( t 0 nformly on compact sbst of R n. (t onrgs n E (R n to zro. Snc f E', w concld that (.. tnds to zro. L ( s dffrntabl wth rspct to and t s tr for all I,,3..,n. Hnc L ( s analytc on n and D L ( {f( D (}. To pro th scond part spp f S a and lt ε > 0 choos ρ D (R n, sch that ρ(t on th nghborhood of S a and spp ρ S E' thn by bonddnss proprty of th gnralzd fncton thr xst a constant and a nonngat ntgr sch that [(L] ( f( - ( max sp β Rn Dβ t ] max max sp β R n sp β R n β D β β Dβ t D t D ] [( ] cos + cos ]
Sc. Rs. hm. ommn.: (3, 0 8 ( sn and sn a Im a Im (( + ε + cos( + ε β β max ν {( a + ε cos -} + a + ( { cos } (( ε + Im cos( ε Im Frthr procdng as n abo, t s clar that [( m L ] ( f ( t (, D [ ] D m m + a + ( { cos } (( ε + Im cos( ε Im Hnc L ( θ m for all R n θ m s th spac of all mltpls of S' (R n. ONLUSION In ths papr brf ntrodcton of gnralzd fractonal Laplac transform s gn and th analytcty thorm s prod. Th fractonal Laplac transform s sfl n solng dffrntal qatons occrrng n th branch of ngnrng. REFERENES. B. N. Bhosal and M. S. hodhary, Fractonal Forr Transform of Dstrbton of ompact Sppor 349-358 (00.. V. Namas, J. Inst. Math. Appl., 5, 4-65 (980. 3. P. R. Dshmh and A. S. Gdadh, Analytcal Stdy of a Spcal as of omplx anoncal transform, Global J. Math. Sc., (, 6-70 (00. 4. A. Torr, Lnar and Radcal anoncal Transform of Fractonal Ordr, J. omptatonal and Appl. Math., 53, 477-486 (003. 5. Tatana Ala and J. Bastaans Martn, On Fractonal Forr Transform Momnts, IEEE Sgnal Procssng Lttr, 7( (000. 6. Tatana Ala and J. Bastaans Martn, Wngr Dstrbton and Fractonal Forr Transform for Two Dmnsonal Symmtrc Bams, JOSA A, 7( (000 pp. 39-33.