New Results Involving a Class of Generalized Hurwitz- Lerch Zeta Functions and Their Applications

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1 Turkih Joural of Aalyi ad Nur Thory 3 Vol No 6-35 Availal oli a hp://pucipuco/a///7 Scic ad Educaio Pulihig DOI:69/a---7 Nw Rul Ivolvig a Cla of Gralid Hurwi- Lrch Za Fucio ad Thir Applicaio H M Srivaava Mi-Ji Luo R K Raia 3 Dpar of Mahaic ad Saiic Uivriy of Vicoria Vicoria Briih Coluia Caada Dpar of Applid Mahaic Doghua Uivriy Shaghai Popl Rpulic of Chia 3 Dpar of Mahaic M P Uivriy of Agriculur ad Tchology Raaha Idia Corrpodig auhor: hariri@ahuvicca Rcivd Spr 5 3; Rvid Ocor 3; Accpd Novr 3 Arac I hi papr w udy a crai cla of gralid Hurwi-Lrch a fucio W driv vral w ad uful propri of h gralid Hurwi-Lrch a fucio uch a (for xapl hir parial diffrial quaio w ri ad Mlli-Bar yp coour igral rpraio ivolvig Fox H- fucio ad a pair of uaio forula Mor iporaly y coidrig hir applicaio i Nur Thory w coruc a w coiuou aalogu of Lippr Hurwi aur So aiical applicaio ar alo giv Kyword: Hurwi-Lrch a fucio arihic diy of ur hory parial diffrial quaio ri ad Mlli-Bar yp coour igral rpraio Fox H-fucio uaio forula gralid Hurwi auur proailiy diy fucio o graig fucio Ci Thi Aricl: H M Srivaava Mi-Ji Luo ad R K Raia Nw Rul Ivolvig a Cla of Gralid Hurwi-Lrch Za Fucio ad Thir Applicaio Turkih Joural of Aalyi ad Nur Thory o (3: 6-35 doi: 69/a---7 Iroducio Dfiiio ad Prliiari W gi y rcallig h failiar gral Φ a which i dfid Hurwi-Lrch Za fucio y ( for xapl []; alo [89] Φ : ( ( a ( + a ( a \ ; wh < ; ( > wh Spcial ca of h Hurwi-Lrch Za fucio Φ a iclud (for xapl h Ria Za fucio ad h Hurwi (or gralid Za fucio ( a dfid y ( for dail [ Chapr I] ad [ Chapr ] ad ( ( : Φ ( ( ( ( > : Φ ( a ( + a ( > a ; \ ( a (3 rpcivly Ju a i aforiod pcial ca a h Hurwi-Lrch Za fucio ad ( a Φ dfid y (4 ca coiud roorphically o h whol coplx -pla xcp for a ipl pol a wih i ridu I i alo kow ha [[] Equaio (3] Φ ( a Γ a d ( a Γ d (4 ( ( a > ; > wh ( ; ( > wh Rcly h followig odifid (ad lighly gralid vrio of h igral i (4 wa iroducd ad udid y Raia ad Chhad [[6] Equaio (6]: ( a ; : xp a ( d Γ ( i { ( a ( } > ; ( ; ; or whr w hav aud furhr ha > wh ad ( ( > wh ad (5

2 Turkih Joural of Aalyi ad Nur Thory 7 providd of cour ha h igral i (5 xi A a ar of fac h aforiod ivigaio y Raia ad Chhad [6] wa oivad y h followig pcial ca of h fucio ( a ; dfid y (5: ( a ; ( a Φ a d Γ ( ( ( a > ; > wh ( ( > wh whr h fucio Φ ( a dfid y ( ( a+ (6 ; ( a : (7 Φ! wa udid y Goyal ad Laddha [[4] Equaio (5] Hr ad i wha follow ( ν ν do h Pochhar yol (or h hifd facorial which i dfid i r of h failiar Gaa fucio y ( v Γ ( + ν : Γ ( ( ν { } ( + ( + + ( ; ; \ ν whr i i udrood covioally ha ( : ad aud acily ha h G-quoi xi ig h of poiiv igr I ay of ir o orv i paig ha i r of h Ria-Liouvill fracioal drivaiv opraor D dfid y ( for xapl [377] D { f ( } ( f ( d ( ( < Γ( d { D { f ( }} ( ( < ( d h ri dfiiio i ( ad (7 radily yild ( a Φ D { Φ ( a } ( ( > Γ ( (8 which (a alrady rarkd y Li ad Srivaava [8] xhii h irig (ad uful fac ha h fucio Φ ( a i ially a Ria-Liouvill fracioal drivaiv of h claical Hurwi-Lrch fucio Φ a O ohr pcial ca of h fucio ( a ; dfid y (5 occur wh w ad i h dfiiio (5 W hu oai ( a ( a ; : whr ( a Γ xpa d (9 i h xdd Hurwi a fucio dfid i [] I fac u a i i alrady poid ou i [] h ri rpraio ( [[6] Equaio (] giv for h fucio ( a ; i (5 i icorrc Oviou furhr pcialiaio i (6 ad (9 would idialy rla h fucio wih h Ria a ad h Hurwi (or gralid a fucio fucio ( a dfid y ( ad (3 rpcivly By uig h ri xpaio of h ioial occurrig i h igrad of (5 ad valuaig h rulig igral y a of h corrcd vrio of a kow igral forula [[3] Equaio (53] i r of Fox H-fucio dfid y ( low h followig ri ad Mlli-Bar yp coour igral rpraio of h fucio ( a ; dfid y (5 wr oaid i []: ad ( a ; Γ ( a + H a+ >! ( ( ( i Γ Γ( ( a ; π i Γ Γ( i ( a ( ( ( H a+ d > i ig aud ha ach r of h ario ( ad ( xi Rark Th H-fucio ivolvd o h righ-had id of ( ad ( ar paricular ca of h clrad Fox H-fucio which i dfid a follow Dfiiio Th wll-kow Fox H-fucio i dfid hr y ( for dail [[3] Dfiio ]; alo [[634] H H ( ( ap A p Hpq ( p Bq ( a A ( ap Ap ( B ( p Bq pq pq Ξ π i L d (

3 8 Turkih Joural of Aalyi ad Nur Thory whr Hr Γ + Γ Ξ ( q p Γ Γ ( B ( a A ( B ( a A { } \ wih arg < π (3 a py produc i irprd a p ad q ar igr uch ha q ad p ( ( ( p β ( q A > p ad B > q α ad ad L i a uial Mlli-Bar yp coour paraig h pol of h gaa fucio { Γ ( + B } fro h pol of h gaa fucio { Γ( a A } I our pr ivigaio w coidr crai aiical applicaio of h gralid Hurwi-Lrch a fucio ( a ; dfid y (5 W fir driv a parial diffrial quaio aifid y h fucio i ( W h oai aohr ri rpraio ad rlad rul for hi gralid Hurwi-Lrch a fucio Th rul drivd hr ar alo applid i our ivigaio cocrig h gralid Hurwi-Lrch a aur ad i rlad aiical cocp Diffrial Equaio of h Gralid Hurwi-Lrch Za Fucio ( a ; I hi cio w will how ha h gralid Hurwi-Lrch a fucio ( α a aifi a parial diffrial quaio wh h parar i giv y ( W fir prov h followig la which will ud i h proof of our ai hor La (Drivaiv Propry Th followig drivaiv forula hold ru: d + ( a + ; { ( a ; } ( > ( d ad ; d + a+ a ; ( ( d Proof Th proof of h drivaiv forula ( ad ( ar dirc For xapl y applyig h ri rpraio ( o aily fid ha d d { ( a ; } ( Γ ( a+ H ( a+ ( Γ + + ( + ( a! ( H ( a+ + (! + ( a + ; ( > which i prcily h fir rul ( ard y h La Th cod ario ( follow idialy fro ( upo ig ( ad ( Our fir ai rul i coaid i h followig hor Thor Th gralid Hurwi-Lrch a fucio a ( ; ( aifi h followig parial diffrial quaio: + ( D ( a ; ( a+ θ (3 whr h diffrial opraor D ad θ ar giv y ad D : θ( θ θ θ (4 θ : rpcivly Proof W fir rwri h H-fucio occurrig i h Mlli-Bar yp coour igral rpraio ( a follow: H ( a ( w w ( w ( a dw πi Γ + Γ L (5 whr L i a uial Mlli-Bar yp coour igral i h coplx w-pla By ig

4 Turkih Joural of Aalyi ad Nur Thory 9 ( ad ( i h aov quaio (5 ad h applyig h wllkow (Gau-Lgdr uliplicaio forula: w fid ha π Γ Γ + ; H ( a ( ( w Γ ( + w Γ( w ( a dw π i L ( π Γ ( + Γ + ( L π i (6 w w w a dw ( π + G + ( a (7 whr ( + G + a i a vry pcialid ca of Mir G-fucio G ( pq which i ur i a pcial ca of Fox H-fucio dfid y ( ha i w hav h followig rlaiohip ( for dail [4]; alo [5]: Gpq ( Gpq H pq ( a ( p q ( a ( ap ( ( p W kow ha h fucio W dfid y W : G pq ( a ( p q (8 aifi h followig diffrial quaio of ordr ax( pq ( for xapl [[] Equaio 54(]: whr W p ( ( ϑ a + ( ϑ ap + ( ϑ ( ϑ q d ϑ d Hc clarly h fucio giv y (7 aifi h followig diffrial quaio: + ( ( a θ( θ θ θ (9 + G + ( a whr a alrady ad i Thor θ Now if w wri [ alo Equaio (4] : ( D θ θ θ θ ( θ : h h quaio (9 co + D G a a ( + ( ( + ( + ( + G a Applyig h diffrial opraor D o h fucio a ; + giv y ( wih ( ad ( w fid y uig ( ha ; D a π i ( π i Γ Γ( Γ Γ i ( a + ( + D G a d + ( ( π ξ +i Γ Γ π iγ Γ( ξ i ( a ( a d + G + ( a + :( ( ai I ( whr h fir igral I i acually h gralid Hurwi-Lrch a fucio giv y a ; I ( Th valuaio of h cod igral I giv y

5 3 Turkih Joural of Aalyi ad Nur Thory ( π i Γ + Γ I : π iγ Γ( i ( ( a G + (3 + d a i or coplicad Sic h ridu of Γ ( + a h pol k ( k ar copud y { Γ ( + } R k ( k li ( + k Γ ( + k ( k! h Ridu Thor ipli ha ( π ( ( k I Γ Γ ( a+ k Γ + k k { Γ ( + } R k G + ( a k + + ( π Γ ( + k ( k k Γ Γ( k ( k! ( a+ k G + ( a k + + ( π Γ ( + k+ k ( Γ Γ ( k + + k! ( a k (4 G + ( a k ( k + ( Γ k ( a+ + k k! (5 H ( a+ + k ( ( a ; + + Applyig ( i (5 w g d a ; I d (6 Now upo uiuig fro ( ad (6 io ( w oai D a ; ( ; + a a + ( a ; + which afr a lil iplificaio co + ( a θ a; D θ : Fially y ig ad (7 (8 i h la quaio (8 w arriv a h dird rul (3 ard y Thor I i irig o coidr a pcial ca of Thor wh Thu if w wri ( a ( a ; : ; (9 h w hav h followig corollary Corollary Th gralid Hurwi-Lrch a fucio ( a ; aifi h followig parial diffrial quaio: ( a ; ( ( a+ Furhror; h fucio ( a ; coidrd a a aalyic fucio of h varial aifi h followig rlaio: a ; ( a+ + ( a + ; 3 Furhr Sri Rpraio ad Rlad Rul ( I hi cio w fir giv a w ri rpraio of h gralid Hurwi-Lrch a fucio ( a ; ivolvig h failiar Lagurr polyoial of ordr (idx α ad dgr i x which ar grad y α x ( α xp L ( x ( < ; α (3

6 Turkih Joural of Aalyi ad Nur Thory 3 Idd upo ig ad x i (3 w g ( + xp L α α (3 W ow ak u of (3 ad h ri xpaio of h ioial ( ad ( occurrig i h igrad of (5 By valuaig h rulig Eulria igral w hu arriv a h ri rpraio giv y Thor low Thor Each of h followig ri rpraio hold ru for h gralid Hurwi Lrch a fucio ( a ; : ad ( a ; Γ l l α ( α L ( + α ( + + Γ ( ( a > ; ( + α > + l l a + l Γ (33 ( a ; Γ ( α ( ( α L ( Φ ( + ( α + + a ( ( a > ; ( + α > (34 providd ha ach r of h ario (33 ad (34 xi Φ ( a ig giv y (7 Proof A alrady oulid aov our doraio of h fir ario (33 of Thor i ad ially upo h rpraio (3 ad h followig wllkow Eulria igral: ρ σ Γ d ( ρ ( { ( ρ ( σ } i > (35 ρ σ Th cod ario (34 follow fro h fir ario (33 wh w irpr h l -ri i (34 y a of h dfiiio (7 I our drivaio of ach of h uaio forula (33 ad (34 i i aud ha h rquird ivrio of h ordr of uaio ad igraio ar uifid y aolu ad uifor covrgc of h ri ad igral ivolvd Th fial rul (33 ad (34 would hu hold ru whvr ach r of h ario (33 ad (34 of Thor xi Rark For h xdd Hurwi a fucio ( a dfid y (9 i i aily dducd fro h ario (34 of Thor wh ad ha a Γ Γ α ( α L ( ( + α + + a ( ( a > ; ( + α > providd ha ach r of (36 xi ( a (36 ig h Hurwi (or gralid a fucio giv y (3 Th oviou furhr pcial ca of (36 wh a ad α would yild h corrcd vrio of a kow rul ( [[] Equaio (778] W ow giv a pair of uaio forula ivolvig h gralid Hurwi-Lrch a fucio ( a ; Thor 3 Each of h followig uaio forula hold ru for h gralid Hurwi-Lrch a fucio ( a ; : ( a + ( a ( a ; (! ; ; + ad ( a ( a ; ; ( + a ; ( +! (37 (38 providd ha ach r of h ario (37 ad (38 xi Proof Makig u of h igral rpraio i (5 for h fucio ( a ; w g ( a ; ( a ; + ( ( ( + + xp a Γ Sic ( (! (39 d + + (3 y uiuig fro (3 io (39 ad irchagig h ordr of uaio ad igraio w fid ha ( a ; + ( a ; ( xp ( a + d! Γ ( a τ xp ( d τ τ +! Γ τ ( a + ; (! (3 which oviouly prov h ario (37 of Thor 3 Th ario (38 of Thor 3 ca prov i a ar aalogou o ha daild aov

7 3 Turkih Joural of Aalyi ad Nur Thory Rark 3 If w i (37 ad (38 h ri occurrig o hir righ-had id would ria Upo ig ad a a w hu oai ad ( a ( a ; + ; ( a ; ( a ( a ; ; a + ; (3 (33 I paricular if w i h la wo uaio forula (3 ad (33 w g ad ( a ; ( a ( a ; ; ( a ; ( a ; a + ; (34 (35 rpcivly I i furhr pcial ca wh h uaio forula (34ca how o corrpod o kow rul ( for xapl [[] Thor 79]; alo [] 4 A Graliaio of h Hurwi Maur Suppo ha ( χ do h characriic fucio A of h u A of h of poiiv igr (or i h laguag of proailiy hory h idicaor fucio of h v A Th i i wll kow ha h followig arihic diy of ur hory: d A k A (4 k k li χ ( do o dfi a aur o h of poiiv igr I ordr o rdy hi dficicy Golo [5] dfid a proailiy o h apl pac a follow: Q ( A ( χ A (4 whr do h Ria a fucio dfid y ( ad h characriic (or idicaor fucio χ ω i giv y A χ A ( ( ω A ( ω A (43 Furhror Golo [5] howd ha if h u A of ha a arihic diy h li Q A d A (44 hry allowig ur-horic fac rgardig dii of of poiiv igr o prov y proailiic a ad h howig ha uch propri ar prrvd i h lii I a irig qul o Golo ivigaio [5] Lippr [9] gav a aalogou dfiiio of h proailii P wh h i rplacd y h of all ral ur grar ha Thu for a Borl A ( Lippr Hurwi aur of h A i dfid y ( for dail [[9] Dfiiio ] P( A χ ( A a + x dx (45 or quivally y P A χ x ( A xd x (46 whr i r of h Hurwi (or gralid a a dfid y (3 fucio ( x ( x : ad d x x d x dx ( + (47 I hi cio w propo o iroduc a w coiuou aalogu of Lippr Hurwi aur i (45 y uig a pcial ca of h gralid Hurwi-Lrch a fucio ( a ; dfid y (5 Dfiiio For a Borl A ( h gralid Hurwi aur of h A i dfid y P( A χ ( ; A a + a da ( ; or quivally y whr ad (48 P A χ ; a ( A ad a (49 ; : ( a ( a ; ( ; (4 ( a ; d ( a ; ( ; d ( + a; da ( ; ic i i aily fro h dfiiio (5 ha (4 d ( a ; ( + a ; (4 da I viw of h followig rlaiohip:

8 Turkih Joural of Aalyi ad Nur Thory 33 (( ( ; ( a ( P d a li ; ; a h gralid Hurwi aur P ( A i (48 or (49 alo dfi a proailiy aur o ( Rark 4 For ad y lig w hav which ipli ha Γ li H a ( a li ( a ; li ( ( x : ( x Thu clarly ( x y ( a ; Propoiio Th aur ( a ; (43 (44 ca coiuouly approxiad followig diffrc quaio: ( a+ ; ( a ; H a ( a Γ ( ; ( > ; a > ; > ; > aifi h (45 Proof Fro h ri rpraio ( of ( a + ; (wih ad w hav ( a ; + ( a ( a H a+ + Γ + + H a+ Γ + ( a; H a ( a Γ (46 Th diffrc quaio (45 ow follow o coiig (4 ad (46 Rark 5 For ad y lig h diffrc quaio (45 rduc o h followig for: whr ( x ( a+ ( a (47 a i giv y (47 For op v h gralid Hurwi aur P A i (48 or (49 ca valuad y uig (49 ad h aov Propoiio Th rul ar ig ad a Thor 4 low Thor 4 If A ( aa + h (( + P A P aa H a ( a Γ ( ; (48 Mor grally; h gralid Hurwi aur of a A i giv y op P( A P( ( ai i i I i I whr A a a ; i I i I ( a i; ( i; ( ; ( [ i i i i (49 Th followig hor how ha h gralid Hurwi aur P ( A i (48 or (49 aically ihri all propri of Lippr Hurwi aur giv y (45 or (46 Thor 5 Corrpodig o h gralid Hurwi aur giv y (49 l Th A ( ε ( ii+ ε ( ε [ ] (4 i ( li P A ε ε (4 Proof Fro (49 w hav ( i ; ( i + ε; P ( A (4 i ( ; By xpadig h fucio ( i + ε y a of Taylor ri ad uig h drivaiv forula (4 w g P ( A ε ( + i ; i (43 ( ; ε ( + ( + i ; + i W ow coidr ach u i (43 paraly W hu fid ha + i ( i ; H ( i + ( + Γ ( + (44 + i ( i+ H ( + + ( + Γ ( + + ( + +

9 34 Turkih Joural of Aalyi ad Nur Thory Sic h ur of o-gaiv igr oluio of h Diophai quaio + N i N + N + h doul uaio i (44 ca rplacd y a igl uaio ha i + i ( i ; + + H ( N ( Γ ( + + N ( N + ( + ; (45 W hu oai li P ( A ( ; ( ; ( + ; ( ( ; ( + ; ( ( ε li ε + + ε ε + li ; + (46 W o ha wh ; i h ri for divrg ad h ri for ( ; + i covrg Thrfor all ohr r vaih i (46 xcp h ladig r Coquly w g li P A ε (47 which copl h proof of Thor 5 A i h hory of proailiy w iroduc h followig dfiiio Dfiiio 3 A rado varial ξ i aid o gralid Hurwi diriud if i proailiy diy fucio (pdf i giv y ( + a ; ( a f ( a : ξ ( ; (48 ( ohrwi Thor 6 L ξ a coiuou rado varial ξ wih i pdf dfid y (48 Th h o graig fucio M ( of h rado varial ξ i giv y M wih h o : ξ ξ! ξ of ordr giv y (49 ( ;! Γ k k ξ (43 k ( k! Γ ; Proof Th ario i (49 follow aily y uig h xpoial ri for ξ If w u igraio y par w fid fro h dfiiio ha ( ; ξ a f a da ξ a ( + a ; da a d ( ( a ; ( ; ( ; ; ( ; a ( a ; ( ; a a + a ( ; da li + a + a ( a ; d a ( ; + a ( a; da ( ( ; (43 whr w hav alo ud h drivaiv propry (4 ad h followig lii forula: ( a li a ; a xpa a li d a Γ xp a li a d a Γ ( (43 Coquly w hav h followig rducio forula for ξ : ( ; ( ; ξ ξ + ( (433 By iraig h rcurrc (433 w arriv a h dird rul (43 ard y Thor 6 Rark 6 Th ario (43 of Thor 6 provid a graliaio of a kow rul [[9] Propoiio 3] Rfrc [] M A Chaudhry ad S M Zuair O a Cla of Icopl Gaa Fucio wih Applicaio Chapa ad Hall (CRC Pr Copay Boca Rao Lodo Nw York ad Wahigo DC [] A Erd lyi W Magu F Orhigr ad F G Tricoi Highr Tracdal Fucio Vol I McGraw-Hill Book Copay Nw York Toroo ad Lodo 953 [3] A Erd lyiw Magu F Orhigr ad F G Tricoi Tal of Igral Trafor Vol II McGraw-Hill Book Copay Nw York Toroo ad Lodo 954 [4] S P Goyal ad R K Laddha O h gralid Za fucio ad h gralid Lar fucio Ga ia Sadh ( [5] S W Golo A cla of proailiy diriuio o h igr J Nur Thory (

10 Turkih Joural of Aalyi ad Nur Thory 35 [6] A A Kila ad M Saigo H-Trafor: Thory ad Applicaio Chapa ad Hall (CRC Pr Copay Boca Rao Lodo Nw York ad Wahigo DC 4 [7] A A Kila H M Srivaava ad J J Truillo Thory ad Applicaio of Fracioal Diffrial Equaio Norh-Hollad Mahaical Sudi Vol 4 Elvir (Norh-Hollad Scic Pulihr Arda Lodo ad Nw York 6 [8] S-D Li ad H M Srivaava So faili of h Hurwi- Lrch Za fucio ad aociad fracioal drivaiv ad ohr igral rpraio Appl Mah Copu 54 ( [9] R A Lippr A proailiic irpraio of h Hurwi a fucio Adv Mah 97 ( [] M-J Luo ad R K Raia So w rul rlad o a cla of gralid Hurwi a fucio A Polo Mah (uid for pulicaio [] A M Mahai ad R K Saxa Th H-Fucio wih Applicaio i Saiic ad Ohr Dicipli Wily Ear Liid Nw Dlhi 978 [] A M Mahai ad RK Saxa Gralid Hyprgoric Fucio wih Applicaio i Saiic ad Phyical Scic Lcur No i Mahaic Vol 348 Sprigr-Vrlag Brli Hidlrg ad Nw York 973 [3] A M Mahai R K Saxa ad H J Hauold Th H-Fucio: Thory ad Applicaio Sprigr Nw York Dordrch Hidlrg ad Lodo [4] F W J Olvr D W Loir R F Boivr ad C W Clark (Edior NIST Hadook of Mahaical Fucio [Wih CD-ROM (Widow Macioh ad UNIX] U S Dpar of Corc Naioal Iiu of Sadard ad Tchology Wahigo D C ; Caridg Uivriy Pr Caridg Lodo ad Nw York [5] A P Prudikov Yu A Brychkov ad O I Marichv Igral ad Sri Vol 3: Mor Spcial Fucio Gordo ad Brach Scic Pulihr Lodo ad Nw York 99 [6] R K Raia ad P K Chhad Crai rul ivolvig a cla of fucio aociad wih h Hurwi a fucio Aca Mah Uiv Coiaa 73 (4 89- [7] S G Sako A A Kila ad O I Marichv Fracioal Igral ad Drivaiv: Thory ad Applicaio Tralad fro h Ruia: Igral ad Drivaiv of Fracioal Ordr ad So of Thir Applicaio ( Naukai Tkhika Mik 987 Gordo ad Brach Scic Pulihr Radig Tokyo Pari Brli ad Laghor (Pylvaia 993 [8] H M Srivaava So forula for h Broulli ad Eulr polyoial a raioal argu Mah Proc Caridg Philo Soc 9 ( [9] H M Srivaava So graliaio ad aic (or q- xio of h Broulli Eulr ad Gocchi polyoial Appl Mah Ifor Sci 5 ( [] H M Srivaava Graig rlaio ad ohr rul aociad wih o faili of h xdd Hurwi-Lrch Za fucio SprigrPlu (3 Aricl ID :67-4 [] H M Srivaava ad J Choi Sri Aociad wih h Za ad Rlad Fucio Kluwr Acadic Pulihr Dordrch Boo ad Lodo [] H M Srivaava ad J Choi Za ad q-za Fucio ad Aociad Sri ad Igral Elvir Scic Pulihr Arda Lodo ad Nw York [3] H M Srivaava K C Gupa ad S P Goyal Th H-Fucio of O ad Two Varial wih Applicaio Souh Aia Pulihr Nw Dlhi ad Madra 98 [4] H M Srivaava ad H L Maocha A Trai o Graig Fucio Hald Pr (Elli Horwood Liid Chichr Joh Wily ad So Nw York Chichr Bria ad Toroo 984

Note 6 Frequency Response

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