Relation of Finite Mellin Integral Transform. with Laplace and Fourier Transforms

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1 Cotmpo Egiig Si Vol. 4 o Rltio o Fiit Mlli Itgl Tom with Lpl d Foui Tom S. M. Khi R. M. Pi* d J. N. Sluk** Dptmt o Mthmti Mhht Adm o Egiig Aldi-45Pu Idi *Dptmt o Mthmti (A.S.&H. R.G.I.T. Vov Adhi (WMumbi-53 Idi **Dptmt o Mthmti Noth Mht UivitJlgo-Idi Abtt Th im o thi pp i to div th ltio btw th Fiit Mlli itgl tom with th Lpl tom b uig th doubl Lpl d Foui Fiit Mlli itgl. Popti lik liit popt lig opt pow popt d (xb lo divd. Th hitig d ivio thom o Lpl-Fiit Mlli itgl om d Foui-Fiit Mlli itgl tom lo diud.

2 7 S. M. Khi R. M. Pi d J. N. Sluk Mthmti Subjt Cliitio: 44A 47D3 46FO5 47G3 33C45; 44A35 6D99 Kwod: Lpl Tom Foui Tom Mlli Itgl Tom Fiit Mlli Itgl Tom : Itodutio Th Lpl tom d Foui Tom widl ud o olvig ditil d itgl qutio. I phi d giig it i ud o li o li tim -ivit tm uh ltil iuit hmoi oillto optil dvi d mhil tm. I thi li th Lpl tom i ot itptd tomtio om th tim-domi i whih iput d output utio o tim to th qu-domi wh th m iput d output utio o omplx ul qu i di p uit tim. Giv impl mthmtil o utiol diptio o iput o output to tm. Th Lpl tom povid lttiv utiol diptio tht ot implii th po o lzig th bhvio o th tm o i thizig viw tm bd o t o piitio. Foui Tom i ot u i igl poig. Th tho o itgl h ptd dit d tmti thiqu o th olutio o ti tp o lil boud d iitil vlu poblm.to b uul th tom mut b doptd to th om o th ditil optoto b limitd wll to th g o itt d th oitd boud oditio. Th umou o whih o uitbl tom xit. H w oid Lpl Fiit Mlli itgl tom to th movl o th pol opto tht ou wh Lpl opto i xpd i ith phil o pl pol oodit Th Doubl Lpl Tom b ud to id th Lpl Fiit Mlli Itgl Tom i th g.d. Foui Lpl tom i ud to id th Foui- Fiit Mlli itgl tom i th g.d.

3 Fiit Mlli itgl tom 7 : Plimi Rult.: Rltio O Th Fiit Mlli Itgl Tom With Lpl Tom Th Lpl tom o th utio (x o x i dotd b L(x d did L(x x ( x dx whv thi itgl i xit o > i th pmt Th iv o th Lpl tom i dotd b L ( x ( x did x L ( x ( x L ( x d πi i Th Mlli itgl tom o th utio o ( o i dotd b ( d i did ( ( d whv thi itgl i xit o > pmt. Th ivio o th Mlli itgl tom i dotd b M ( ( d did M ( ( ( d πi i Th Fiit Mlli itgl tom o ( o i dotd b M ( ( i d did M ( ( ( d whv thi itgl i xit o > pmt b uig x lo i th Lpl tom. It iv i dotd b ( M ( d did ` ( M ( M ( d πi i o > whv thi itgl i xit. Th doubl Lpl tom i dotd b L ( x z d did i ( x z L ( x z ( x z dxdz whv thi doubl itgl i xit o > d > pmt. d

4 7 S. M. Khi R. M. Pi d J. N. Sluk Subtitut z -lo th z d dz - i z th d i z th ( x z L ( x z ( x z dxdz x z ( x z dxdz x z ( ( x z dxdz x d ( ( x lo dx( x L ( x z ( x dxd Thi i th ltio btw Fiit Mlli itgl d Lpl tom o (x with pmt > d > i th g ; d i dotd ( x L wh x L ( x ( x dxd ( p x d.: Rltio O Th Fiit Mlli Itgl Tom With Foui Tom Th Foui tom i dotd b F{(xL d b did ix F ( x L ( x dx whv thi itgl i xit. o pmt > Th iv o th Foui Tom i dotd b F ( x ( x i did ix F ( x ( x F ( x d πi whv pmt > i d

5 Fiit Mlli itgl tom 73 Th Mlli Itgl Tom i th ollowig w M ( x x ( x dx α < R( < β wh α d β l umb dtig th mximum g o vlu o R(z uh tht th itgl ovg. Th ivio o th Mlli itgl tom i dotd b M ( ( i d did M ( ( ( d πi Th Fiit Mlli itgl tom o ( o i dotd b M ( ( d did M ( ( ( d whv thi itgl i xit i o > pmt b uig x lo i th Lpl tom. It iv i dotd b ( M ( d did ` ( M ( M ( d πi i o > whv thi itgl i xit. i Th Foui Lpl tom FL ( x z b did z FL ( x z x z (ix ( dxdz whv thi itgl i xit. o pmt > > z d ubtitut z -lo/ dz I z th d i z th lo ix d FL ( x z ( x lo dx(

6 74 S. M. Khi R. M. Pi d J. N. Sluk lo ix d ( x lo dx( ix d ( x lo ( dx( ix FL ( x z ( x dxd Thi i th Foui Fiit Mlli Itgl Tom i i wh i gt th zo (>. Thi itgl i dotd b F ( x.i ; wh ix F ( x ( x dxd ( p x d 3: LEMMAS ( Lpl-Fiit Mlli itgl Tom-LFMIT 3.:Lmm. 3..: Th LFMIT i x L M (x ( x dxd th 3..: L M α (x β x α L M (x β L M x 3..: L M (x L ( M x z Poo x 3..: I L M (x ( x dxd th x L M α (x β x α ( x β x dxd

7 Fiit Mlli itgl tom 75 L M t t α ( x z dtdz β x z dtdz x x α ( x dxd β x dxd α L M (x β L M x α (x β x α L M (x β L m x (3 3..: Th Lpl-Fiit Mlli itgl tom i ubtitut x L M (x ( x dxd th L M (x x ( x dxd z z d z dz I th z d i th z L M (x t ( x ( x dx z dz x ( x z z dxdz L ( M x z L M (x L ( M x z (4 3.:Lmm- Th LFMIT i t L M (t z ( t z z dtdz th 3.. L M (x d L M (tz/ d d 3..:L M (xd p L M *x/d d

8 76 S. M. Khi R. M. Pi d J. N. Sluk Poo 3..:Th Lpl-Fiit Mlli itgl tom i x L M (x ( x dxd th x L M (x d ( x d dxd Subtitut xt d dz th zt/ d z/d dxdt/ d ddz/d I x th tz d i x th t d i th zd L M (x d d d t d ( t z z ( d t ( t z z dtdz d L M (x d L M (tz/ d (5 d 3..: Th Lpl-Fiit Mlli itgl tom i dtdz x L M (x ( x dxd th x L M (xd ( x d( d dxd t ( x dx d d ubtitut xt d dz th xt/ d z/d dxdt/ d ddz/d I x th tz d i x th t d i th zd d t ( t t dt d ( t dt z z( d d z z L(t/zd d L M (xd L M (t/zd (6 d dz d dz

9 Fiit Mlli itgl tom 77 4: LEMMAS ( Foui-Fiit Mlli Itgl Tom-FFMIT 4.: Lmm Lt (x x R ix F ( x ( x dxd th 4..: α ( x β x α F ( x β x F 4..: F ( x F ( x Poo 4.. :Lt (x d x x R d α d β ott th F ix F ( x ( x dxd bom F ix α ( x β x α ( x β x dxd ix ix α x dxd β x dxd ix ix α α ( x dxd β x dxd α F ( x β x F α ( x β x α F ( x β F x (7 4..: Lt (x d x x R d α d β ott th ix F ( x ( x dxd bom ix F ( x ( x dxd Subtitut z z z dz I th z d i th z d F

10 78 S. M. Khi R. M. Pi d J. N. Sluk ix F ( x ( x z ( z dx z dz ( x z ix z dxdz F ( x F ( x p q F ( x (8 4.: Lmm Lt d g b two l vlud utio d ix F ( x ( x dxd th 4..: F ( x d F ( M t z d b 4..: F ( x b F ( v M u b Poo (4.. Lt d g b two l vlud utio th ix F ( x ( x dxd bom ix F ( x d ( x dxd ubtitut I xt xt/ dxdt/ d dz z/d ddz/d i x th tz d i x th t d i th zd F ( x d d d ( t z z ( d it / d ( t z z it / dtdz d F ( M t z d F ( x d p q F ( M t z (9 d 4..: Lt d g b two l vlud utio th dtdz

11 Fiit Mlli itgl tom 79 ix F ( x ( x dxd bom ix F ( x b ( x b dxd ix ( x dx b d I xu xu/ dxdu/ d bv v/b ddv/b I x th uv d x th u d i th vb F ( x b b b F ( x b b ( u b iu / ( u du v ( b du v dv b b iu / dv b F ( u v b F ( u v ( 5. SHIFTING THEOREMS 5. Shitig Thom Fo Lpl-Fiit-Fiit Mlli Itgl Tom Th hitig thom o Lpl- Fiit Mlli itgl tom i Poo I x L M (x ( x dxd th L M bx (x L M (x x L M (x ( x dxd th L M bx bx x (x ( x dxd

12 8 S. M. Khi R. M. Pi d J. N. Sluk L M bx ( x ( x dxd (x L M (x ( 5. Shitig Thom Fo Th Foui-Fiit Mlli Itgl Tom Lt (x b th two vibl utio i x d th th LFMIT i Poo ix F ( x ( x dxd th F ( x F ( x ix I F ( x ( x dxd th ix ix ix F ( x ( x dxd i( x ( x dxd ix F ( x F ( x ( 6: Ivio Thom 6.: Ivio Thom Fo Th Lpl-Fiit Mlli Itgl Tom I th Lpl tom o th utio (x i did L(x >i th pmt th it iv tom i i x ( x dx whv thi itgl i xit x (x L ( x d i. π i Th Mlli itgl tom o ( i to i did

13 Fiit Mlli itgl tom 8 ( p d p> i th pmt.th it ivio omul i i ( ( d πi i o th Lpl- Fiit Mlli tom i d it ivio omul i Poo Aum tht ( d whv thi itgl i xit x L M (x ( x dxd (x i i x L ( x ; dd (πi p ii L M (x ; i gul utio i th tip R( < ( to b l umb o th -pl d tht <<ν -i i wh i ott d R(p < q ( q to b l umb o th p-pl d tht <<ν -i p i wh i ott I th Lpl- Fiit Mlli itgl tom i th x L M (x ( x dxd L M (x x x Lm ( x ; dd (πi x dxd x Lm ( x ; dd (πi x Lm ( x ; dd (πi x dx p x dxd d

14 8 S. M. Khi R. M. Pi d J. N. Sluk x Lm ( x ; dd (πi x L ( x ; dd (πi p A N d N Thi i th ivio o th Lpl-Fiit Mlli itgl tom d it i dotd (x th i i x (x L ( x ; dd ( i (3 π p i 6.: Ivio Thom Fo Th Foui-Fiit Mlli Itgl Tom Lt (x b th two vibl utio i x d th th LFMIT p ix F ( x ( x dxd Th th ivio ult i i i (x F M ix ( x F ( x dd (πi i i i Poo Aum tht ( x i gul utio i th tip F R( < ( b l umb o th -pl d tht <<ν -i i wh i ott d R(p < q ( q b l umb o th p-pl d tht <<ν -i p i wh i ott Lt (x b th two vibl utio i x d th th FFMIT i ix F ( x ( x dxd th ix ix L ( x F ( x dd dxd (πi ix ix F ( x dd dxd (πi

15 Fiit Mlli itgl tom 83 ( ( d dx dd x M F i it ix π it ix i dd x M F i ( ( π ( ( ( dd x M F i ix π ( ( ( i dd x M F i x π dd x M F i i ix ( ( π Thi i th ivio o th Lpl Ftiol Mlli itgl tom i to N d N i dotd b (x ( x M L (x ( x M F dd x M F i i ix i i i i ( ( π (4

16 84 S. M. Khi R. M. Pi d J. N. Sluk 7: TABLES 7.:Lpl Tom Ad Fiit Mlli Itgl Tom Lpl Tom Fiit Mlli Itgl Tom Th Lpl Tom o (x i dotd b L(x d did Th Fiit Mlli itgl tom o th utio o ( o i dotd b L(x ( x dx ( d i did whv thi itgl i xit o > i th pmt ( ( d whv thi itgl i xit o > i th pmt Iv Lpl Tom Iv Fiit Mlli Itgl Tom Th iv o th Lpl tom Th ivio o th Mlli itgl i dotd b L ( x ( x tom i dotd b d did M ( ( L ( x ( x d did M ( ( x L ( x d πi i i ( d πi i

17 Fiit Mlli itgl tom 85 7.: Foui Tom Ad Fiit Mlli itgl Tom Foui Tom Fiit Mlli Itgl Tom Th Foui tom i dotd Th Fiit Mlli itgl tom o b F{(xL d did th utio o ( o i dotd b ( d i did ix F ( x L ( x dx whv thi itgl i xit. o pmt > ( ( d whv thi itgl i xit o > i th pmt Iv Foui Tom Iv Fiit Mlli Itgl Tom Th iv o th Foui Tom Th ivio o th Mlli itgl i dotd b F ( x ( x tom i dotd b d did M ( ( F ( x ( x d did i M ( ( ix F ( x d πi i i ( d whv pmt > πi i 7.3: Rltio btw Fiit Mlli Itgl Tom With Lpl Ad Foui Tom Th Lpl- Fiit Mlli Itgl Tom x L ( x ( x dxd Th Foui- Fiit Mlli Itgl Tom ix F ( x ( x dxd

18 86 S. M. Khi R. M. Pi d J. N. Sluk 7.4: Rltio btw Lpl Tom d Fiit Mlli Itgl Tom Lpl Fiit Mlli Foui-Fiit Mlli Tom Tom Lmm::3.. Lmm:4.. L M α (x β x F α ( x β x α F ( x β F x α L M (x β L m x Lmm:3.. L M (x L ( M x z 3 Lmmq:3.. L M (x d d L M (tz/ d 4 Lmm:3.. L M (xd L M d (t/zd Lmm:4.. F ( x p q F Lmm:4.. F ( x d p q d Lmm:4.. F ( x b b b F ( u v ( x F ( t z

19 Fiit Mlli itgl tom : Shitig Thom Fo Lpl d Foui Tom With Fiit Mlli Itgl Tom Th Shitig Thom o Lpl- Fiit Mlli Itgl Tom L M bx (x L M (x Th Shitig Thom o Foui- Fiit Mlli Itgl Tom ix F ( x F ( x 7.6: Ivio Thom Fo Lpl Ad Foui Tom With Fiit Mlli Itgl Tom Th Iv o Lpl- Fiit Mlli Itgl Tom i i x F(x L ( x ; dd (πi p i Th Iv o Foui- Fiit Mlli Itgl Tom i i (x ix F M ( x F ( x dd (πi i i i R.Dk Nlo O Mlli Tp Itgl Tom Joul o Mthmti d Mhi (963 vol. No.. I N. Sddo Th u o Itgl Tom TMH ditio S. Dod Robut d Eiit Foui-Mlli Tom Appoximtio o G Lvl img Rotutio d Complt ivit Diptio Comput Viio d Img U ( 4.I.S. Rd Th Mlli Tp Doubl Itgl Cmbidg Lodo. 5. Lokth Dbth Itgl Tom Ad Thi Applitio995CRC P I.

20 88 S. M. Khi R. M. Pi d J. N. Sluk 6.D.J. Bdighm Dimiol Rguliztio d Mlli ummtio i High-Tmptu Clultio Thotil Phi Th Blktt Lboto Impil Collg Pi Coot RodSW7 BWUK. A Xiv:ph/x NOV. 7.A. H. Zmi Th Ditibutiol Lpl d Mlli Tomtio J. SIAM Vol.4. No.. J. 98 Pitd i U.S.A. 8..J.M. Mdz d J.R. NgiO th iit Hkl Swhwtz Tomtio o Ditibutio Gt. Vol.39 No C. Fox Applitio o Mlli Tomtio to Itgl Equtio 3 d Mh 934 pp C.Fox Applitio o Mlli Tomtio to th itgl qutio (933. S. M. Khi R.M. Pi & J. N. Sluk Stud o Th Sumudu Mlli Itgl Tom d it Applitio It. J. o Mt. Si. & Egg. Appl. (IJMSEA Vol. 4 No. IV ( Otob. pp Rivd: Apil

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