National Conference on Recent Trends in Synthesis and Characterization of Futuristic Material in Science for the Development of Society

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1 ABSTRACT Naoa Coferece o Rece Tred Syhe ad Characerzao of Fuurc Maera Scece for he Deveome of Socey (NCRDAMDS-208) I aocao wh Ieraoa Joura of Scefc Reearch Scece ad Techoogy Some New Iegra Reao of I- Fuco Jyo Mhra Dearme of Mahemac, Gya Gaga Iue of Tech ad Scece, Jabaur, Madhya Pradeh, Ida Th aer dea wh ome ew egra reao of I- fuco of oe varabe Keyword: I- fuco, Muvarabe oyoma I INTRODUCTION The I- fuco of oe varabe defed by Saxea [6] ad we ha reree here he foowg maer: [( a, j, j ), ],[( aj, ), ] m j I[ z] I, q: r z [( bj, j ), m],[( bj, j) q ] = ( ) z d, 2 w L where ( ), z( 0) a comex varabe ad z ex[ {og z w arg z}] (2) I whch og z reree he aura ogarhm of z ad arg z o eceary he rce vaue A emy roduc erreed a uy, ao, m ( bj j) ( a j j) j j ( ), r q (3) ( bj j) ( a j j) jm j, ad (,r) are o egave eger afyg 0 < <, 0 < m < q, (,r), α j, (j=, ; =, r) ad β j (j =, q ; =, r) are aumed o be ove quae for adardzao uroe Ao a j ( j =,, ; =,,r) ad b j ( j =,, q ; =,,r) are comex umber uch ha oe of he o () S {( b v) }, h,, m; v 0,,2,, (4) h whch are he oe of (b β S), h =, m ad he o S = {( a ),, ; 0,,2,, (5) whch are he oe of ( a ) cocde wh oe aoher, e wh ( b v) b ( a ) (6) for, h = 0,,2, ; h =,, m; =,, IJSRST430 NCRDAMDS Jauary-February-208 [(4): 26-3 ] 26

2 Furher, he coour L ru from - o + Such ha he oe of ( b ), h =, m; e o he rgh of L ad he oe ( a ), =,, e o he ef of L The egra () coverge, f arg z < ½ B π (B>0), A < 0, where ad A a q j j j j (7) (8) B, j j j j j j j jm m (,,r) Gradhey IS, Ryzhk, IM [4] gve abe of Iegra, ere, Sharma [7] evauaed he egra vovg geera ca of oyoma wh H-fuco, Srvaava, HM ad Garg, M [9] eabhed ome egra vovg a geera ca of oyoma ad he muvarabe H- fuco Recey,Sayaarayaa, B ad Pragah Kumar, Y [5] ha evauaed Some fe egra vovg muvarabe oyoma, Agarwa, P [] cabhed egra vovg he roduc of Srvaava oyoma ad geerazed Me-Bare ye of coour egra, Bhaar ad Bhargava [3] eabhed ome egra formua vovg wo H - fuco ad muvarabe geera ca of oyomya Sayaarayaa, B ad Pragah Kumar, Y [5] ha evauaed Some fe egra vovg muvarabe oyoma Foowg he w evauaed ome ew egra vovg muvarabe oyoma, ad I-fuco of oe varabe II FORMULA REQUIRED The foowg formua w be requred our vegao () The ecod ca of muvarabe oyoma gve by Srvaava (9) defed a foow: [ / U] [ / U] k U,, U x x S,, [ x,, x ] ( ) ( ) A[, ;;, ] (524) U k Uk k K 0 K 0 k k (2) () The fr ca of muvarabe oyoma roduced by Srvaava ad Garg (9) defed a foow : x x [,, ] ( ) ( ) A[, ;;, ] (22) k k U k U K k U U,, S x x U k Uk k K k o q III SOME NEW FINITE INTEGRALS FORMULAE I h eco we rove wo egra formuae, whch vovg muvarabe oyoma, ad I-fuco of oe varabe ( ) ( ) S U,, U [y ( ) m ( ),, y ( ) m ( ) x x ],, x x x x g h [( aj, j ), ],[( aj, j ), ] I, : ( ) ( ) [(, ), ],[(, ), ] d q r z x x bj j b x j j [ /U ] [ /U ] 2 ( ) ( ) A[, k ; K 0 K 0 Uk UK oume 4 Iue Ieraoa Joura of Scefc Reearch Scece ad Techoogy (wwwjrcom) 27

3 I m,, q : r z2 hg ( m k h;),( k, g;)[( a, ) ],[( a, ) ], j j, j j, [( b, ) ],[( b, ) ]( ( m ) k, hg;) j j, j j,q (3 ) where m >0 (=,,), >0 ( =,,) h 0,g 0 (o boh are zero muaeouy) U U m m ( x) ( x) S [ y ( x) ( x),, y ( x) ( x) ] g h [( aj, j ), ],[( aj, j ), ] I, : ( ) ( ) [(, ), ],[(, ), ] d q r z x x bj j b x j j y y 2 ( ) (, ;; ) 2 I U k U k k k ( m ) k U k Uk A k k, k 0 k 2 2,q : r ( mk, h;),( k, g;),[( a j, j ), ],[( a j, j ), ] hg [( b, ) ],[( b, ) ],( ( m ) k, hg;) j j,m j j q (32) Provded he codo aed reu (3) are afed Proof : To eabh egra (3), we exre I-fuco occurrg ef had de erm of Me- Bare coour egra gve by (3 ), he ecod ca of oyoma gve by (2) The erchage he order of egrao of ummao ad egrao, we arrve he foowg : [ / U] [ / U] k y y ( ) U ( ) ( k U, ;;, ) k A k k k k 0 0 gmk hxk x ( ) z ( x) ( x) d 2 L y y [ / U] [ / U] k ( ) U ( ) ( k U, ;;, ) k A k k k 0 k 0 k 2 ( h k ) ( g m k ) h k g mk ( ) z d 2 L ( h k g m k 2) [ / U] [ / U] k ( m ) k y y U k Uk A k ;, ) 2 k 0 k 0 k k = ( ) ( ) (, ; 2 L ( z ) ( h k ) ( g m k ) ( h k g m k 2) hg ( z2 ) d Now we evauae he above egra wh he of egra (22) Ierreg he reug coour egra of H- fuco we ca eay arrve a dered reu (3) oume 4 Iue Ieraoa Joura of Scefc Reearch Scece ad Techoogy (wwwjrcom) 28

4 To eabh egra (32) ca be eay eabhed o he ame e mar o he roof of (3) I SPECIAL CASES OF (3) AND (32) Take A (, k ; ;,k ) = A (,k ) A (,k ) (3) he muvarabe oyoma U U S ( x,, x) reduced U o he roduc of we kow geera ca of oyoma S () x ad he reu (3) reduced o foowg form v U m ( x) ( x) S [ y ( x) ( x) ] m g h [( a, ) ],[( a, ) ] I q r z x x b b, j j, j j,, : ( ) ( ) [(, ), ],[(, ),q ] j j j j [ / U] [ / U ] k k ( m ) k ( ) U ( ) ( k U,, ) 2 k A k k 0 k 0 k I 2 2,q : r hg (a) Subug r= (3), we oba : ( m k, h,),( k, g;)[( a, ) ],[( a, ) ] j j, j j, [( b, ) ],[( b, ) ],( ( m ) k, hg;) j j,m j j q y y (4) (b) ( ) ( ) S U,, U [y ( ) m ( ),, y ( ) m ( ) x x ],, x x x x g h ( a j, j ), H, q z( x) ( x) ( b, ) j j,q [ /U ] [ /U ] k k ( ) ( ) Uk UK y y 2 A[, k;;, k ] ( m ) k k k k k m, H q, z2 K 0 K 0 hg ( m k h;),( k, g;)( a, ), j j, ( b, ) ( ( m ) k, hg;) j j,q (42) Subug α j =β j = (42) we oba S U U m m ( x) ( x) [ y( x) ( x) y ( ) ( ) ] x x G ( a ) g h j, z( x) ( x) bj,q,q ( ) [ / U] [ / U] k ( m ) + + k 2 y y ( ) U ( ) ( k U,,, ) 2 k A k k 0 k 0 k k oume 4 Iue Ieraoa Joura of Scefc Reearch Scece ad Techoogy (wwwjrcom) 29

5 G 2 2,q (a) Subug r= (32), we oba : ( mk, h,),( k, g;)( a j ), hg ( bj ),q,( ( m ) k, hg;) (43) (b) q U U m m ( x) ( x) S [ y ( x) ( x),, y ( x) ( x) ] g h ( a j, j), H, q z( x) ( x) ( b, ) j j,q y y 2 ( ) (, ;, ) 2 H U k U k k k ( m ) k U k U A k k, k 0 k 2 2,q Subug α j =β j = (44), we oba ( mk, h,),( k, g;),( a j, j ), hg ( b, ),( ( m ) k, hg;) j j,q (44) U U m m ( x) ( x) S [ y ( x) ( x),, y ( x) ( x) ] g h ( a j ), G, q z( x) ( x) ( b ) j,q y y 2 ( ) (, ;, ) 2 G U k U k k k ( m ) k U k Uk A k k, k 0 k 2 2,q REFERENCES z [] Agarwa, P O a ufed egra vovg he roduc of Srvaava' oyoma ad geerazed Me-Bare ye of coour egra Advace Mechaca Egeerg ad Acao (AMEA), (202) 58 o 2, No 2 [2] Agarwa, Pravee ad Chad, Mehar New heorem vovg he geerazed Me-Bare ye of coour egra ad geera ca of oyoma GJSFRM, (202), o 2 ( m k, h,),( k, g;),( a ) j, hg 2 ( b ),( ( m ) k, hg;) j,q [3] Bhaar ad Bhargava Iegra formuae' vovg wo - fuco ad muvarabe' geera ca of oyomya ISCA Buhma, (204) [4] Gradhey, IS ad Ryzhk, IM Tabe of Iegra, ere ad roduc, 6/e Academc re, (200), New Deh [5] Sayaarayaa, B ad Pragah Kumar, Y Some fe egra vovg muvarabe oyoma, H-fuco of oe varabe ad H- fuco of r' varabe Afrca Joura of Mahemac ad Comuer Scece Reearch o 4(8), (20), oume 4 Iue Ieraoa Joura of Scefc Reearch Scece ad Techoogy (wwwjrcom) 30

6 [6] Saxea, S Forma ouo of cera ew ar of dua egra equao vovg H-fuco Proc Na Acad Sc Ida 52, (982), A III, [7] Sharma, RP O fe egra vovg Jacob oyoma ad he H-fuco Kyugook Mah J, (2006), 46: [8] Srvaava, A The egrao of cera roduc erag o he H-fuco wh geera oyoma Gaa Sadeh, (200), o3/32, 5-58 [9] Srvaava, HM ad Garg, M Some egra vovg a geera ca of oyoma ad he muvarabe H-fuco Rev Roamae Phy, (983),32: [0] Srvaava, HM ad Sgh, NP The egrao of cera roduc of he muvarabe H-fuco wh a geera ca of oyoma Red Crc Ma Paermo, (983), 32 oume 4 Iue Ieraoa Joura of Scefc Reearch Scece ad Techoogy (wwwjrcom) 3