On Some Double Integrals Involving H -Function of Two Variables and Spheroidal Functions

Transcription

1 IOSR Joural of Mathematis (IOSR-JM) e-iss: , -ISS: 9-765X. Volume 0, Issue Ver. IV (Jul-Aug. 0), PP O Some Double Itegrals Ivolvig H -Futio of Two Variables ad Sheroidal Futios Yashwat Sigh, Harmedra Kumar Madia. Deartmet of Mathematis, Govermet College, Kaladera, Jaiur, Raastha, Idia. Deartmet of Mathematis. M. L. (P.G.) College, Jhuhuu, Raastha, Idia Abstrat: The reset aer evaluates ertai double itegrals ivolvig H -futio of two variables [] ad Sherodial futios []. These double itegrals are of most geeral harater ow so far ad a be suitably seialized to yield a umber of ow or ew itegral formulae of muh iterest to mathematial aalysis whih are liely to rove uite useful to solve some tyial boudary value roblems. Key words: H -futio, H -futio of two variables, Sheroidal futio. I. Itrodutio The H -futio ourrig i the aer is defied ad rereseted by Bushma ad Srivastava [] as follows : where,, P M, M, ( a ; ; A ),( a ; ) P P,, ( b, ), M,( b, ; B ) M, H z H z ( ) M ( b ) ( a ) P B M ( b ) ( a ) i ( ) zd i (.) A Whih otais fratioal owers of the gamma futios. Here, ad throughout the aer a (,..., ) ad b (,..., ) are omlex arameters, 0(,..., P), 0(,..., ) (ot all zero simultaeously) ad exoets A (,..., ) ad B (,..., ) a tae o o iteger values. The followig suffiiet oditio for the absolute overgee of the defiig itegral for the H -futio give by euatio (.) have bee give by (Bushma ad Srivastava). M P A B 0 ad M i (.) (.) arg( z) (.) The behavior of the H -futio for small values of z follows easily from a result reetly give by (Rathie [8],.06,e.(6.9)). We have M, b H P, z 0 z, mi Re, z 0 If we tae A (,,..., ), B ( M,..., ) i (.), the futio H Fox s H -futio [7]. (.5) M, P, [.] redues to the The followig series reresetatio for the H -futio will be reuired i the seuel [see Rathie,[8].05-06,e.(6.8)]: M, a, ; A, a,,, P H P, z b,, b, ;, M B M, 55 Page

2 M O Some Double Itegrals Ivolvig H -Futio of Two Variables ad Sheroidal Futios M h, r h, r r b a ( ) z h r0 h Where hr, B P,, b a r! h r h r h M b r h. h The H -futio of two variables The H -futio of two variables itrodued by Sigh ad Madia [] will be defied ad rereseted i the followig maer:, :, :, a, ; A,,, ; K,,,, e,, E ; R, e,, E x o m m x, H x, y H, :, ;, yh y b, ; B, d,,, ;,,,, ;, d, m L f m, F f, m F S m,, ( ) ( ) x y dd (.7) = Where L L, a A A hr, a A b B K m d d m m R e E f F e E f F m L S (.8) (.6) (.9) (.0) Where x ad y are ot eual to zero (real or omlex), ad a emty rodut is iterreted as uity i, i, i, mare o-egative itegers suh that 0 i i, o m ( i,,;,). All the a (,,..., ), b (,,..., ), (,,..., ), d (,,..., ), e (,,..., ), f (,,..., ) are omlex arameters. 0(,,..., ), 0(,,..., ) (ot all zero simultaeously), similarly E 0(,,..., ), F 0(,,..., ) (ot all zero simultaeously). The exoets K (,,..., ), L ( m,..., ), R (,,..., ), S ( m,..., ) a tae o oegative values. The otour L is i -lae ad rus from the right ad the oles of to i i d (,,..., m ) lie to. The oles of K (,,..., ), a A (,,..., ) to the left of the otour. For K (,,..., ) ot a iteger, the oles of gamma futios of the umerator i (.9) are overted to the brah oits. 56 Page

3 The otour L is i -lae ad rus from to i f F m R. i The oles of (,,..., ) lie to the right ad the oles of e E (,,..., ), a A (,,..., ) to the left of the otour. For R (,,..., ) ot a iteger, the oles of gamma futios of the umerator i (.0) are overted to the brah oits. The futios defied i (.7) is a aalyti futio of x ad y, if (.) U 0 (.) V A E B F 0 The itegral i (.7) overges uder the followig set of oditios: m (.) L K 0 m m (.) A A F F S E R E B 0 m arg x, arg y (.5) The behavior of the H -futio of two variables for small values of z follows as: H[ x, y] 0( x y ),max x, y 0 (.6) Where d mi Re m For large value of z, f mi Re m F ' ' (.7) H[ x, y] 0 x, y,mi x, y 0 (.8) Where e ' max Re K, ' max Re R E (.9) Provided that U 0 ad V 0. If we tae K (,,..., ), L ( m,..., ), R (,,..., ), S ( m,..., ) i (.7), the H -futio of two variables redues to H -futio of two variables due to [5]. Sheroidal futio: Sheroidal futio ( z, ) of geeral order later by Chu ad Stratto [] are those solutios of the differetial euatio: '' ' z (, z) ( ) z (, z) b z (, z) 0 (.0) is defied ad ivestigated by Stratto [] ad That remais fiite at the sigular oits z. The sheroidal futio a be exaded i Bessel futio o (, ) ([9],. 90) 57 Page

4 a J ( z) i ( z, ) ' (.) V () 0, ( z) With eight eige values V (), valid for, whereas the rime over the summatio sigh idiates that the summatio is tae over oly eve or odd values of as is eve or odd. a si (.) ad the eige-values b () A reursio relatioshi ([], (9)) for determiig the, from the differetial euatio (.0). If ( z) is real ad fiite J ( z) ( z) absolutely ad uiformly overget. Moreover it reresets a otiuous futio for all z. If 0 ad z suh that z remais fiite ad the ormalizatio is hose to be suh that 0, are obtaied is bouded, hee it follows by M-test that the series i (.) is ' ia, (.) V The futio ( z, ) redues to J ( z) a. sie 0, ( z) The followig results ([5],. 7; [9],. 5; [],. 6) i the seuel will be used durig the roof of our mai results i a little simlifiatio: A B mi AB,,,,Re( ),Re( ) 0. x y f Ax By dxdy z f ( z) dz (.) = m,0 a,, x H, x dx b,, b, ;, m B m, m B b a m Provided m b b Re B 0,,,..., m; arg, where. B 0 m,0 z J v( z) z H 0, v v,,,; M, a, ; A, a,, M, a, / ;,, /, A a, P, P H P, P, x H x ; 0 b,, b, ;, /,, / ;, M B b M, b, M B M, (.) (.5) (.6) 58 Page

5 II. We establish the followig double itegrals: Mai Results x y H ux y Ax By v Ax By f Ax By dxdy 00 A B = 0 z, 0,0: m, : m, f () z H, :, ;, s a, ; A,,, ; K,, ;,, ;,,,, ;,,, e, E R e, E Mz, dz vz b, ; B, d,,, ;,, ;,,,, ;, d, m L f m, F f, m F S m, (.) s Where, for oveiee, M ua B, ad the futio f is so resribed that the itegral (.) overges. 00 m,0, x y Ax By H Ax By H ux y Ax By, v Ax By dxdy A B W 0, m: m, : m, = H MW, :, :, vw, e,,,, E ; R, e,, E, a, ; A, b ;,,, ; K,, ;,, ;,, b, ; B, a ;, d,, d, ; L,, ;, f, F, f, F ; S,, m m,, m m, s Where for oveiee,,, M ua B rovided that arg W where m 0, m b d i ' Re L S' 0 Di F ' f,..., m; i,..., m ; ',..., m ;,,, are all ositive uatities ad the oditios (.) to (.5), with x relaed by M ad y relaed by v, are satisfied. x y H ux y Ax By v Ax By Ax By 00, Ax By A B i W,W dxdy V (.) 59 Page

6 r0, ' a 0,: m, : m, MW H, :, :, vw s a, ; A, r ;,,, ; K,, ;,, ;,,, e, E ; R, e, E,,, b, ; B, d,, d, ; L,, ;, f, F, f, F ; S, m m,, m m,, (.) Where for oveiee,,, M ua B, d f i ' rovided that Re L S ' r 0, Di F' i,..., m ; ',..., m ; r 0,,... ;,,,, are all ositive uatities; W is ot eual to zero; ad the oditios (.) to (.5) with x relaed by M ad y relaed by v are satisfied. Proof of (.): To rove the itegral relatio (.), we first relae H by its Melli-Bares double otour itegral from (.7) with m 0. O ivertig the order of itegratio, whih is ustified due to absolute overgee of the itegrals ivolved i the roess, we obtai s, U U u v x y i L L 00 Ax By f Ax By dd. ow, by virtue of the familiar result (.) ad subseuetly to the oditio (.7), the right had side of (.) is readily verified. The imortae of the result (.) lies i the fat that may more iterestig double itegrals a be evaluated easily by hoosig f() z i oveiet form as show below. Proof of (.): I order to rove the result (.), we first set m,0 a, ; A, a,,, f () z z H, Wz b,, b, ;, m B m, I the euatio (.) ad evaluate the resultig itegral as follows: First exress H i the double otour itegral form (.7), iterhage the order itegratio whih is ermissible due to absolute overgee of the itegrals thus ivolved i the roess, ad evaluate the z - itegral with the hel of (.) after usig the roerty (.6). O iterretig the result thus obtaied by virtue of (.7) we arrive at the right had side of (.). Proof of (.): I order to rove the result (.), we first set f ( z) z,w z I the result (.) ad evaluate the resultig itegral as follows: First exress the sheroidal futio i the exaded form (.), hage the order of itegratio ad summatio whih is ustified due to the uiform overgee of the series reresetig sheroidal futios. By exressig the Bessel futio thus ivolved i the form of H -futio usig the result (.5), we iterret the H -futio i the otour itegral form (.7). Agai, hage the order of itegratio by virtue of De La Valle Poussi s well ow theorem ([],.50) due to absolute overgee of the itegrals thus ivolved. The, evaluatig the ier itegral by virtue of result (.) ad ivertig the double otour itegrals by defiitio (.7), we get the reuired result (.) Page

7 Regardig the overgee of the series o the right had side of (.) it would be worth metioig that the a r ratio is, ad the ratio of gammas ivolvig r (eve or odd) ([6],.7()), hee the series is ar r uiformly ad absolutely overget by M-test. Seial Cases Sie the H -futio of two variables ad sheroidal futios emerge may higher trasedetal futios ad olyomials, a large umber of ew ad iterestig results follow as seial ases but we reord here oly a few of them, for the la of sae. (i) I (.) if we ut A B,, s 0, L S ad relae, W, u, v,, resetively by,,,, h,, we arrive at the result obtaied by Guta ad Mittal ([0],.). (ii) I additio to (i), if we set 0, m, b,, A ( i,,..., ), B (,,..., ), we arrive at the i i result obtaied by Patha ([7],.) by virtue of relatio:,0 0, (0,) ax H ax e (iii) Agai, if we tae 0, A B, L (,..., ), S ( m,..., ) our itegral formula (.) would orresod to a ow result due to Pada ([6],., e. (.6)). (iv) If 0, z suh that ( z) remais fiite, (.) redues to the followig ew result by virtue of (.): 00, x y H ux y Ax By v Ax By Ax By A B W J W ( Ax By ) dxdy b, ; B, d,, d, ; L,, ;, f, F, f, F ; S H 0,: m, : m, MW, :, :, vw s a, ; A, r ;,,, ; K,, ;,, ;,,, e, E ; R, e, E,,,, m m,, m m,, (.) Valid uder the oditios as give for (.). (v) I (.), if we set, A, B,,,, A (,..., ), B (,..., ), L (,..., ), S ( m,..., ), F, m, eah eual to uity; s,, v,,, f eah eual to zero; ad relae by, we get the result obtaied by Guta ad Jai ([8],.605). Referees []. Agarwal, R.P.; A extesio of Meier s G -futio, Pro. at. Ist. Si. Idia, A,(965), []. Bromwih, T.J.I.A.; A Itrodutio to the Theory of Ifiite Series, MaMilla, Lodo, 965. []. Bushma, R.G. ad Srivastava, H.M.; The H futio assoiated with a ertai lass of Feyma itegrals, J.Phys.A:Math. Ge., (990), []. Chu, L.J. ad Stratto, J.A.; Elliti ad Sheroidal wave futios, J. Math. ad Phys., 0(9), [5]. Edwards. J.; A Treatise o the Itegral Calulus, vol. II, Chelsea Publiatio, ew Yor, 95. [6]. Erdelyi, A. et. al.; Higher Trasedetal Futios, vol.i, MGraw-Hill Boo Co., ew Yor, 95. [7]. Fox, C.; The G -futio ad H futio as symmetri Fourier erels, Tras. Amer. Math. So. 98, (96), [8]. Guta, K.C. ad Jai, U.C.;The H-futio II, Pro. at. Aad. Si., Idia 6A(966), [9]. Guta,K.C. ad Mittal, P.K.; The H -futio trasform, J. Australia Math. So., 98 (970), Page

8 [0]. Guta,K.C. ad Mittal, P.K.; A itegral ivolvig geeralized futio of two variables, Pro. Idia Aad. Si. LXXV(97), 7-. []. Guta,K.C., Jai, R. ad Sharma, A.; A study of uified fiite itegrals with aliatios. J. Raastha Aad. Phys. Si. Vol.():69-8(00). []. Guta, S.C.; Itegrals ivolvig roduts of G -futio, Pro. at. Aad. Si. Idia, 9A(969), 9-0. []. Lue, Y.L.; The Seial Futios ad Their Aroximatios, Vol.I, Aademi Press, ew Yor, 95. []. Mathur, A.B.; A study of geeralized trasforms i oe ad two variables with aliatios, A ThesisAroved Ph. D., Viram Ui., Uai (969). [5]. Mittal, P.K. & Guta, K.C.; A itegral ivolvig geeralized futio of two variables. Pro. Idia Aad. Si. Set. A( 75):67-7(96). [6]. Pada, R. ad Srivastava, H.M.; Some oeratioal tehiues i the theory of seial futios, ederl. Aad. Wetesh. Pro. Ser. A76=Idag. Math.5(97), [7]. Patha, R.S.; Some results ivolvig G ad H -futios, Bull. Calutta Math. So., Idia,6(970), [8]. Rathie,A.K. ;A ew geeralizatio of geeralized hyergeometri futios Le Mathematihe Fas.II,5,(997),97-0. [9]. Rhodes, D.R.; O the sheroidal futios, Jour. Res. at. Bureau of Stadrads-7, Math. Si. 7B, [0]. (970), []. Sharma, B.L.; O the geeralized futio of two variables-i, Aals de la So. Si. de Bruxelles, 79(965), 6-0. []. Sigh,Y. ad Madia, H.; A study of H -futio of two variables, Iteratioal Joural of Iovative researh i siee,egieerig ad tehology,vol.,(9)0,9-9. []. Srivastava, H.M., Guta, K.C. ad Goyal,S.P.; The H -futios of Oe ad Two Variables With Aliatios. South Asia Publishers, ew Delhi (98). []. Stratto, J.A.; Sheroidal futios, Pro. at. Aad. Si., (95), Page