FRACTIONAL MELLIN INTEGRAL TRANSFORM IN (0, 1/a)

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1 Ieol Jol o Se Reeh Pblo Volme Ie 5 y ISSN 5-5 FRACTIONAL ELLIN INTEGRAL TRANSFOR IN / S.. Kh R..Pe* J.N.Slke** Deme o hem hh Aemy o Egeeg Al-45 Pe I oble No.: 98576F No.: Deme o hem A.S.&H.* R.G.I.T. Veov Ahe W mb-5 I oble No F.No Deme o hem** Noh h UveyJlgo-I o.ble No Ab- The heoy o egl om eee e yem ehqe o he eeo o ll bol heoy. I h e he Lle oeo e e o ee The Fol ell egl om whh be ehqe o olvg boy l vle oblem Th Tom e he ol evl o /. Th wok whh ow o e how Lle oeo wol le o oee ooo heoem elo wh Fol ell egl om The m vew o o wok o gve oee om Lle Tom h o o be vl o Fol ell egl om. The el hve bee moe by lyg ble o whh le o he el Fol ell egl om he evl o / whee ove. To lle he vge e o h omo he el o Weyl ol om mmo o he ee ome mo eel eqo hve bee olve he e. The ghl oe eeee by gg ee vle o he mee by g ool o lb whh gve bghe vew o lo o Fol ell egl om. Ie Tem- Iegl om ell egl om Fe ell egl om Lle Tom AS hem Clo- 44A 47D T I. INTRODUCTION he Lle om e o ob he Fol ell egl om he ge o /. oee lke ley oey lg oey owe oey ooo lke o he o log g e / g e e by he Fol ell egl om o...theoem lke veo heoem ovolo heoem evl heoem hg heoem eo hg heoem e vl o he Fol ell egl om o / Alo o he FIT o Weyl Fol Tom mmo o he ee evve o he Fol ell egl om o / e lle. Solo o he oy eel eqo e. eeee ghlly by g lb. II. PRELIINARY RESULTS Le be gve o o whh ee o ll mee L e Sbg - log. - >. I he / he he / L Th egl eoe by /

2 Ieol Jol o Se Reeh Pblo Volme Ie 5 y ISSN 5-5 / / > > mee le bewee o eql o oe e. he / ol. Th Fol ell Iegl Tom wh keel ove mee. I he / ke ol vle. III. LEA.: Ley oey: Iegl Tom FIT le oeo Theoem: The FIT Le oeo h o y o g whoe ell Tye ol Iegl Tom e α β e o I / / he α + βg/ α /+ β g/.: Slg Poey: I / / he b 9/ b - b/ bq b> 4.: Powe Poey I / / b / b q /b / b b> 5.4: Pooo.4.: FIT o I / / by he bo z/ we hve / he he b q / 6 / z-+/ whee z--/ he ew ell Tye Iegl Tom he ge / o.e wok.4.: FIT o I.4.: F o log log log/ / / og/ e I / e / he / e / / e / e log / 7 e - he / e e / whee he New ell Tye Iegl Tom he ge /...4.4: FIT o egl I / / 8 he

3 Ieol Jol o Se Reeh Pblo Volme Ie 5 y ISSN whee / / y y / / 9 ele by +we hve / '/ / - I I I / / he 9./ I `/ / / - /. /.4.5: : FIT o egl I / g / g g lo g / g / z. / g / g / z. /.4.6: : FIT o egl Smlly lo / g / g / g z. / I he 4 / g / g z. / 6 5 g / g / z. / I - he om / g / g / z. / g / By he bo z we hve IV. AIN RESULTS 4.: Iveo heoem Theoem: The FIT o / 6

4 Ieol Jol o Se Reeh Pblo Volme Ie 5 y 4 ISSN 5-5 / / The veo oml N / > N Poo: Ame h / egl eqo he Re < o be el mbe o he -le h << - + whee o / N N N N / / / / / N / / N N / N / N N / 7 Le N me h / em boe lm whe Re he he egl o R.H.S o he eqo 7 e.hee / 8 / 4.: Covolo Theoem The FIT o / / he vee e / / g/ g he g- / 9 / g-/.4.: Ohogoly Pevl Theoem The FIT o / / g/ he vee e / / / g g / g / / I eoe by / N

5 Ieol Jol o Se Reeh Pblo Volme Ie 5 y 5 ISSN 5-5 g/ g he g / g / / g / 4.5: Deo Uh Se Fo I UHwhe > whe < he U o H kow he U Se Fo b Heve U Se Fo I U-H- whe > whe <he U- o H- kow he Heve U Se Fo. 4.6: F Shg Theoem The FIT o / / / he / + / 4.7: Seo Shg Theoem The FIT o / / / he -H- / - b/-b W The Weyl V. FRIT OF WEYL FRACTIONAL TRANSFOR F W om o he o eoe by F <Re < > he ee F F F F F W VI. APPLICATION OF THE FIT TO SUATION OF SERIES The FIT / vee The Hvz Ze o eoe by ee << Re> I ollow om he veo o he Fe ell egl om +

6 Ieol Jol o Se Reeh Pblo Volme Ie 5 y 6 ISSN The FIT. / vee / The Rem Ze o eoe by ee Re> I ollow om he veo o he Fe ell egl om ^ / / / / The FIT : / vee / I he / /

7 Ieol Jol o Se Reeh Pblo Volme Ie 5 y 7 ISSN 5-5 / / 6 By lyg o he eo-oe evve / / he o egg by ollow h. / / + - ''' / we ob /. / * 8 / ''/ VII. DERIVATIVES 7.: FIT o F Oe Devve: Theoem: Soe h oo o ll yg. o ome vle m h evve whh eewe oo o evey e evl he ge o. The he Fol ell Iegl Tom o he evve e whe > m ome o Poo: e o ll o Coeg he e whe oo o ll. The o egg by h ollow / / /. / / - - / + / +. / /. / / e e m h ell Tye ol Iegl Tom o evve obe. 7.: e : FIT o h oe Devve: +- '/ / he /... / / / / 9 Th he geelze el o he o The ell Tye ol Iegl Tom. h evve o. 8.. VIII. APPLICATIONS OF FRIT L F / /- / / / / + / - + / +/ he / Ug 4 obe / - / + / / he / / +

8 Ieol Jol o Se Reeh Pblo Volme Ie 5 y 8 ISSN 5-5 I R.H.S o 4 gve vle zeo. Fom 4 + / - / + he / / + / / whh e he hghe oe eel eqo wh o oee. The omlee olo gve by / / / whee o e evle by olvg l vle oblem oblem wh boy oo. IX. FUNCTIONS ABS RESULTS S. No. Fo / / e / / /!! 6 / / /! 5! / / / 5 8 / / /! 5! 5 6 / / / / / / / 7 8 log + 9 log - / / / / / / / log/ log/ /!

9 y- y- Ieol Jol o Se Reeh Pblo Volme Ie 5 y 9 ISSN 5-5 log o log log e 4 4 log/ / / / / / / / log / X. GRAPHICAL REPRESENTATION BY USING ATLAB ell Tye ol Iegl Tom gh loe bewee y o vo vle o mee. Hee he ogm h bee how wh oe vle o he mee. Coe he eqo 5 P- / /.: % log+ %log+ /+ % he ' / % / ; % he.69'.5 %.*.^.*.^ / / / Coe he eqo / / '/. % ^ %^ * % he 'y o> / '/ % he % he ' % / '/.^-.*.*- Ghl Reeeo lo o.5 lo o :: Ghl Reeeo lo o log+ lo o :: XI. REARKS. Deo o he Fol ell egl om. Lemm e ee ove. Pooo e ee ove 4. Theoem e e ove 5. Weyl om ee obe el by g FIT 6. Rel o mmo o he ee e obe by FIT 7. Devve e obe by g FI 8. Chy Le eel eqo e obe olo by g FIT 9. Fo Rel. Ghl eeeo 6

10 Ieol Jol o Se Reeh Pblo Volme Ie 5 y ISSN 5-5 XII. CONCLUSION Th Fol ell egl om el o olve he oy eel eqo by g l boy oo olo o he oy eel eqo how by ome emle how ghlly by g lb. REFERENCES Deek Nylo O ell Tye Iegl Tom Jol o hem eh 96 vol. No. C.Fo Alo o ell Tomo o he egl eqo 9 J..eez J.R.Neg O he e Hkel-Shwz Tomo o Dbo G 988 vol.9 No. 4 I N. Seo The e o Iegl TomTH eo C.Fo Alo o ell Tomo o Iegl Eqo h J..eez J.R.NegO he e Hkel Swhwz Tomo o DboG.Vol.9No A.H.ZemGeelze Iegl TomoIeee Pblo New Yok 968 Sohl AhmeLe Yg Ljo HzoShool o EC 8 Uvey o ohmoso7 BJUK 9 S.. Kh R.. Pe J. N. SlkeAlo O The ell Tye Iegl Tom I The Rge {/ IJSAVol. No. J D.J.BeghmDmeol Reglzo ell mmo Hgh- Temee ClloTheol Phy The Blke lbooy Imel College Pe Coo RoSW7 BWUK. Xv:h/ NOV. Ao De Se' ve Roheo'A F ell Sle TomHv Pblhg Cooo EURASIP Jol o Ave Sgl PoegVol.7Ale ID ge Do:55/7/897 J.J.Beo.T.Floe''A Pevl eqo geelze e Hkel Tomo''Comme.h.Uv.ole C.Fo''Iegl om be o ol Iego''Po.Comb.Phl.So Kh S.. Pe R.. Slkhe J. N. Alo o bol ell egl om he ge ;.

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