New Results Involving a Class of Generalized Hurwitz- Lerch Zeta Functions and Their Applications
|
|
- Alicia Pierce
- 5 years ago
- Views:
Transcription
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
No 6 Frqucy Rpo Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada. alyical Exprio
More informationInfinite Continued Fraction (CF) representations. of the exponential integral function, Bessel functions and Lommel polynomials
Ifii Coiu Fraio CF rraio of h oial igral fuio l fuio a Lol olyoial Coiu Fraio CF rraio a orhogoal olyoial I hi io w rall h rlaio bw ifi rurry rlaio of olyoial orroig orhogoaliy a aroria ifii oiu fraio
More information1973 AP Calculus BC: Section I
97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f
More informationNumerical Simulation for the 2-D Heat Equation with Derivative Boundary Conditions
IOSR Joural of Applid Chmisr IOSR-JAC -ISSN: 78-576.Volum 9 Issu 8 Vr. I Aug. 6 PP 4-8 www.iosrjourals.org Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios Ima. I. Gorial parm of Mahmaics
More informationx, x, e are not periodic. Properties of periodic function: 1. For any integer n,
Chpr Fourir Sri, Igrl, d Tror. Fourir Sri A uio i lld priodi i hr i o poiiv ur p uh h p, p i lld priod o R i,, r priodi uio.,, r o priodi. Propri o priodi uio:. For y igr, p. I d g hv priod p, h h g lo
More informationContinous system: differential equations
/6/008 Coious sysm: diffrial quaios Drmiisic modls drivaivs isad of (+)-( r( compar ( + ) R( + r ( (0) ( R ( 0 ) ( Dcid wha hav a ffc o h sysm Drmi whhr h paramrs ar posiiv or gaiv, i.. giv growh or rducio
More informationPart B: Transform Methods. Professor E. Ambikairajah UNSW, Australia
Par B: rasform Mhods Profssor E. Ambikairaah UNSW, Ausralia Chapr : Fourir Rprsaio of Sigal. Fourir Sris. Fourir rasform.3 Ivrs Fourir rasform.4 Propris.4. Frqucy Shif.4. im Shif.4.3 Scalig.4.4 Diffriaio
More informationAdvanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.
Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%
More informationECEN620: Network Theory Broadband Circuit Design Fall 2014
ECE60: work Thory Broadbad Circui Dig Fall 04 Lcur 6: PLL Trai Bhavior Sam Palrmo Aalog & Mixd-Sigal Cr Txa A&M Uivriy Aoucm, Agda, & Rfrc HW i du oday by 5PM PLL Trackig Rpo Pha Dcor Modl PLL Hold Rag
More informationWhat Is the Difference between Gamma and Gaussian Distributions?
Applid Mahmaics,,, 85-89 hp://ddoiorg/6/am Publishd Oli Fbruary (hp://wwwscirporg/joural/am) Wha Is h Diffrc bw Gamma ad Gaussia Disribuios? iao-li Hu chool of Elcrical Egirig ad Compur cic, Uivrsiy of
More information3.2. Derivation of Laplace Transforms of Simple Functions
3. aplac Tarform 3. PE TRNSFORM wid rag of girig ym ar modld mahmaically by uig diffrial quaio. I gral, h diffrial quaio of h ordr ym i wri: d y( a d d d y( dy( a a y( f( (3. d Which i alo ow a a liar
More informationApproximately Inner Two-parameter C0
urli Jourl of ic d pplid Scic, 5(9: 0-6, 0 ISSN 99-878 pproximly Ir Two-prmr C0 -group of Tor Produc of C -lgr R. zri,. Nikm, M. Hi Dprm of Mmic, Md rc, Ilmic zd Uivriy, P.O.ox 4-975, Md, Ir. rc: I i ppr,
More information15. Numerical Methods
S K Modal' 5. Numrical Mhod. Th quaio + 4 4 i o b olvd uig h Nwo-Rapho mhod. If i ak a h iiial approimaio of h oluio, h h approimaio uig hi mhod will b [EC: GATE-7].(a (a (b 4 Nwo-Rapho iraio chm i f(
More informationBoyce/DiPrima 9 th ed, Ch 7.9: Nonhomogeneous Linear Systems
BoDiPrima 9 h d Ch 7.9: Nohomogou Liar Sm Elmar Diffrial Equaio ad Boudar Valu Prolm 9 h diio William E. Bo ad Rihard C. DiPrima 9 Joh Wil & So I. Th gral hor of a ohomogou m of quaio g g aralll ha of
More information, R we have. x x. ) 1 x. R and is a positive bounded. det. International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:10 No:06 11
raioal Joral of asic & ppli Scics JS-JENS Vol: No:6 So Dirichl ors a Pso Diffrial Opraors wih Coiioall Epoial Cov cio aa. M. Kail Dpar of Mahaics; acl of Scic; Ki laziz Uivrsi Jah Sai raia Eail: fkail@ka..sa
More information( A) ( B) ( C) ( D) ( E)
d Smsr Fial Exam Worksh x 5x.( NC)If f ( ) d + 7, h 4 f ( ) d is 9x + x 5 6 ( B) ( C) 0 7 ( E) divrg +. (NC) Th ifii sris ak has h parial sum S ( ) for. k Wha is h sum of h sris a? ( B) 0 ( C) ( E) divrgs
More informationEE Control Systems LECTURE 11
Up: Moy, Ocor 5, 7 EE 434 - Corol Sy LECTUE Copyrigh FL Lwi 999 All righ rrv POLE PLACEMET A STEA-STATE EO Uig fc, o c ov h clo-loop pol o h h y prforc iprov O c lo lc uil copor o oi goo y- rcig y uyig
More informationImproved estimation of population variance using information on auxiliary attribute in simple random sampling. Rajesh Singh and Sachin Malik
Imrovd imaio of oulaio variac uig iformaio o auxiliar ariu i iml radom amlig Rajh igh ad achi alik Darm of aiic, Baara Hidu Uivri Varaai-5, Idia (righa@gmail.com, achikurava999@gmail.com) Arac igh ad Kumar
More informationThe Development of Suitable and Well-founded Numerical Methods to Solve Systems of Integro- Differential Equations,
Shiraz Uivrsiy of Tchology From h SlcdWorks of Habibolla Laifizadh Th Dvlopm of Suiabl ad Wll-foudd Numrical Mhods o Solv Sysms of Igro- Diffrial Equaios, Habibolla Laifizadh, Shiraz Uivrsiy of Tchology
More informationSome Applications of the Poisson Process
Applid Maaics, 24, 5, 3-37 Publishd Oli Novbr 24 i SciRs. hp://www.scirp.org/oural/a hp://dx.doi.org/.4236/a.24.59288 So Applicaios of Poisso Procss Kug-Ku s Dpar of Maaics, Ka Uivrsiy, Uio, USA Eail:
More informationAnalyticity and Operation Transform on Generalized Fractional Hartley Transform
I Jourl of Mh Alyi, Vol, 008, o 0, 977-986 Alyiciy d Oprio Trform o Grlizd Frciol rly Trform *P K So d A S Guddh * VPM Collg of Egirig d Tchology, Amrvi-44460 (MS), Idi Gov Vidrbh Iiu of cic d umii, Amrvi-444604
More informationAnalysis of Non-Sinusoidal Waveforms Part 2 Laplace Transform
Aalyi o No-Siuoidal Wavorm Par Laplac raorm I h arlir cio, w lar ha h Fourir Sri may b wri i complx orm a ( ) C jω whr h Fourir coici C i giv by o o jωo C ( ) d o I h ymmrical orm, h Fourir ri i wri wih
More informationChapter 3 Linear Equations of Higher Order (Page # 144)
Ma Modr Dirial Equaios Lcur wk 4 Jul 4-8 Dr Firozzama Darm o Mahmaics ad Saisics Arizoa Sa Uivrsi This wk s lcur will covr har ad har 4 Scios 4 har Liar Equaios o Highr Ordr Pag # 44 Scio Iroducio: Scod
More informationSOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C
Joural of Mathatical Aalysis ISSN: 2217-3412, URL: www.ilirias.co/ja Volu 8 Issu 1 2017, Pags 156-163 SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C BURAK
More informationChapter4 Time Domain Analysis of Control System
Chpr4 im Domi Alyi of Corol Sym Rouh biliy cririo Sdy rror ri rpo of h fir-ordr ym ri rpo of h cod-ordr ym im domi prformc pcificio h rliohip bw h prformc pcificio d ym prmr ri rpo of highr-ordr ym Dfiiio
More informationPoisson Arrival Process
1 Poisso Arrival Procss Arrivals occur i) i a mmorylss mar ii) [ o arrival durig Δ ] = λδ + ( Δ ) P o [ o arrival durig Δ ] = 1 λδ + ( Δ ) P o P j arrivals durig Δ = o Δ for j = 2,3, ( ) o Δ whr lim =
More informationTrigonometric Formula
MhScop g of 9 FORMULAE SHEET If h lik blow r o-fucioig ihr Sv hi fil o your hrd driv (o h rm lf of h br bov hi pg for viwig off li or ju coll dow h pg. [] Trigoomry formul. [] Tbl of uful rigoomric vlu.
More informationMAT3700. Tutorial Letter 201/2/2016. Mathematics III (Engineering) Semester 2. Department of Mathematical sciences MAT3700/201/2/2016
MAT3700/0//06 Tuorial Lr 0//06 Mahmaics III (Egirig) MAT3700 Smsr Dparm of Mahmaical scics This uorial lr coais soluios ad aswrs o assigms. BARCODE CONTENTS Pag SOLUTIONS ASSIGNMENT... 3 SOLUTIONS ASSIGNMENT...
More informationOn the approximation of the constant of Napier
Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of
More informationWeb-appendix 1: macro to calculate the range of ( ρ, for which R is positive definite
Wb-basd Supplmary Marials for Sampl siz cosidraios for GEE aalyss of hr-lvl clusr radomizd rials by Sv Trsra, Big Lu, oh S. Prissr, Tho va Achrbrg, ad Gorg F. Borm Wb-appdix : macro o calcula h rag of
More information) and furthermore all X. The definition of the term stationary requires that the distribution fulfills the condition:
Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 oblm For R b X a raom variabl havig ormal isribuio wih ma µ a variac σ (his is wri as ~ (,) X. by: R a. Is X ) a urhrmor all
More information2 Dirac delta function, modeling of impulse processes. 3 Sine integral function. Exponential integral function
Chapr VII Spcial Fucios Ocobr 7, 7 479 CHAPTER VII SPECIAL FUNCTIONS Cos: Havisid sp fucio, filr fucio Dirac dla fucio, modlig of impuls procsss 3 Si igral fucio 4 Error fucio 5 Gamma fucio E Epoial igral
More informationBMM3553 Mechanical Vibrations
BMM3553 Mhaial Vibraio Chapr 3: Damp Vibraio of Sigl Dgr of From Sym (Par ) by Ch Ku Ey Nizwa Bi Ch Ku Hui Fauly of Mhaial Egirig mail: y@ump.u.my Chapr Dripio Ep Ouom Su will b abl o: Drmi h aural frquy
More informationTypes Ideals on IS-Algebras
Ieraioal Joural of Maheaical Aalyi Vol. 07 o. 3 635-646 IARI Ld www.-hikari.co hp://doi.org/0.988/ija.07.7466 Type Ideal o IS-Algebra Sudu Najah Jabir Faculy of Educaio ufa Uiveriy Iraq Copyrigh 07 Sudu
More informationUNIT I FOURIER SERIES T
UNIT I FOURIER SERIES PROBLEM : Th urig mom T o h crkh o m gi i giv or ri o vu o h crk g dgr 6 9 5 8 T 5 897 785 599 66 Epd T i ri o i. Souio: L T = i + i + i +, Sic h ir d vu o T r rpd gc o T T i T i
More informationPoisson Arrival Process
Poisso Arrival Procss Arrivals occur i) i a mmylss mar ii) [ o arrival durig Δ ] = λδ + ( Δ ) P o [ o arrival durig Δ ] = λδ + ( Δ ) P o P j arrivals durig Δ = o Δ f j = 2,3, o Δ whr lim =. Δ Δ C C 2 C
More informationON H-TRICHOTOMY IN BANACH SPACES
CODRUTA STOICA IHAIL EGA O H-TRICHOTOY I BAACH SPACES Absrac: I his papr w mphasiz h oio of skw-oluio smiflows cosidrd a gralizaio of smigroups oluio opraors ad skw-produc smiflows which aris i h sabiliy
More informationLaguerre wavelet and its programming
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 agurr l ad is prograig B Sayaaraya Y Pragahi Kuar Asa Abdullah 3 3 Dpar o Mahaics Acharya Nagarjua Uivrsiy Adhra pradsh Idia Dpar o Mahaics
More informationMathematical Preliminaries for Transforms, Subbands, and Wavelets
Mahmaical Prlimiaris for rasforms, Subbads, ad Wavls C.M. Liu Prcpual Sigal Procssig Lab Collg of Compur Scic Naioal Chiao-ug Uivrsiy hp://www.csi.cu.du.w/~cmliu/courss/comprssio/ Offic: EC538 (03)5731877
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More informationResponse of LTI Systems to Complex Exponentials
3 Fourir sris coiuous-im Rspos of LI Sysms o Complx Expoials Ouli Cosidr a LI sysm wih h ui impuls rspos Suppos h ipu sigal is a complx xpoial s x s is a complx umbr, xz zis a complx umbr h or h h w will
More informationModeling of the CML FD noise-to-jitter conversion as an LPTV process
Modlig of h CML FD ois-o-ir covrsio as a LPV procss Marko Alksic. Rvisio hisory Vrsio Da Comms. //4 Firs vrsio mrgd wo docums. Cyclosaioary Nois ad Applicaio o CML Frqucy Dividr Jir/Phas Nois Aalysis fil
More informationELECTROMAGNETIC COMPATIBILITY HANDBOOK 1. Chapter 12: Spectra of Periodic and Aperiodic Signals
ELECTOMAGNETIC COMPATIBILITY HANDBOOK Chapr : Spcra of Priodic ad Apriodic Sigals. Drmi whhr ach of h followig fucios ar priodic. If hy ar priodic, provid hir fudamal frqucy ad priod. a) x 4cos( 5 ) si(
More informationMixing time with Coupling
Mixig im wih Couplig Jihui Li Mig Zhg Saisics Dparm May 7 Goal Iroducio o boudig h mixig im for MCMC wih couplig ad pah couplig Prsig a simpl xampl o illusra h basic ida Noaio M is a Markov chai o fii
More informationSeries of New Information Divergences, Properties and Corresponding Series of Metric Spaces
Srs of Nw Iforao Dvrgcs, Proprs ad Corrspodg Srs of Mrc Spacs K.C.Ja, Praphull Chhabra Profssor, Dpar of Mahacs, Malavya Naoal Isu of Tchology, Japur (Rajasha), Ida Ph.d Scholar, Dpar of Mahacs, Malavya
More informationOn a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G.
O a problm of J. d Graaf coctd with algbras of uboudd oprators d Bruij, N.G. Publishd: 01/01/1984 Documt Vrsio Publishr s PDF, also kow as Vrsio of Rcord (icluds fial pag, issu ad volum umbrs) Plas chck
More informationPractice papers A and B, produced by Edexcel in 2009, with mark schemes. Practice Paper A. 5 cosh x 2 sinh x = 11,
Prai paprs A ad B, produd by Edl i 9, wih mark shms Prai Papr A. Fid h valus of for whih 5 osh sih =, givig your aswrs as aural logarihms. (Toal 6 marks) k. A = k, whr k is a ral osa. 9 (a) Fid valus of
More informationEXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar
Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded
More informationA FAMILY OF GOODNESS-OF-FIT TESTS FOR THE CAUCHY DISTRIBUTION RODZINA TESTÓW ZGODNOŚCI Z ROZKŁADEM CAUCHY EGO
JAN PUDEŁKO A FAMILY OF GOODNESS-OF-FIT TESTS FO THE CAUCHY DISTIBUTION ODZINA TESTÓW ZGODNOŚCI Z OZKŁADEM CAUCHY EGO Abrac A w family of good-of-fi for h Cauchy diribuio i propod i h papr. Evry mmbr of
More informationAE57/AC51/AT57 SIGNALS AND SYSTEMS DECEMBER 2012
AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER Q. Drmi powr d rgy of h followig igl j i ii =A co iii = Solio: i E P I I l jw l I d jw d d Powr i fii, i i powr igl ii =A cow E P I co w d / co l I I l d wd d Powr
More informationPhys463.nb Conductivity. Another equivalent definition of the Fermi velocity is
39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h
More informationAn Asymptotic Expansion for the Non-Central Chi-square Distribution. By Jinan Hamzah Farhood Department of Mathematics College of Education
A Asypoic Expasio fo h o-cal Chi-squa Disibuio By Jia Hazah ahood Dpa of Mahaics Collg of Educaio 6 Absac W div a asypoic xpasio fo h o-cal chi-squa disibuio as wh X i is h o-cal chi-squa vaiabl wih dg
More informationarxiv: v1 [math.nt] 13 Dec 2010
WZ-PROOFS OF DIVERGENT RAMANUJAN-TYPE SERIES arxiv:0.68v [mah.nt] Dec 00 JESÚS GUILLERA Abrac. We prove ome diverge Ramauja-ype erie for /π /π applyig a Bare-iegral raegy of he WZ-mehod.. Wilf-Zeilberger
More informationz 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
More informationMa/CS 6a Class 15: Flows and Bipartite Graphs
//206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d
More informationFourier Series: main points
BIOEN 3 Lcur 6 Fourir rasforms Novmbr 9, Fourir Sris: mai pois Ifii sum of sis, cosis, or boh + a a cos( + b si( All frqucis ar igr mulipls of a fudamal frqucy, o F.S. ca rprs ay priodic fucio ha w ca
More informationLaw of large numbers
Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs
More informationIt is quickly verified that the dynamic response of this system is entirely governed by τ or equivalently the pole s = 1.
Tim Domai Prforma I orr o aalyz h im omai rforma of ym, w will xami h hararii of h ouu of h ym wh a ariular iu i ali Th iu w will hoo i a ui iu, ha i u ( < Th Lala raform of hi iu i U ( Thi iu i l bau
More informationTHE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE 1
TH ROAL TATITICAL OCIT 6 AINATION OLTION GRADAT DILOA ODL T oci i providig olio o ai cadida prparig or aiaio i 7. T olio ar idd a larig aid ad old o b a "odl awr". r o olio old alwa b awar a i a ca r ar
More information(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is
[STRAIGHT OBJECTIVE TYPE] l Q. Th vlu of h dfii igrl, cos d is + (si ) (si ) (si ) Q. Th vlu of h dfii igrl si d whr [, ] cos cos Q. Vlu of h dfii igrl ( si Q. L f () = d ( ) cos 7 ( ) )d d g b h ivrs
More informationMarket Conditions under Frictions and without Dynamic Spanning
Mar Codiio udr Friio ad wihou Dyai Spaig Jui Kppo Hlii Uivriy of hology Sy Aalyi aboraory O Bo FIN-5 HU Filad hp://wwwhufi/ui/syaalyi ISBN 95--3948-5 Hlii Uivriy of hology ISSN 78-3 Sy Aalyi aboraory iblla
More information1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
More informationRing of Large Number Mutually Coupled Oscillators Periodic Solutions
Iraioal Joural of horical ad Mahmaical Physics 4, 4(6: 5-9 DOI: 59/jijmp446 Rig of arg Numbr Muually Coupld Oscillaors Priodic Soluios Vasil G Aglov,*, Dafika z Aglova Dparm Nam of Mahmaics, Uivrsiy of
More informationChapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationPWM-Scheme and Current ripple of Switching Power Amplifiers
axon oor PWM-Sch and Currn rippl of Swiching Powr Aplifir Abrac In hi work currn rippl caud by wiching powr aplifir i analyd for h convnional PWM (pulwidh odulaion) ch and hr-lvl PWM-ch. Siplifid odl for
More informationChapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
More informationThe Exile Began. Family Journal Page. God Called Jeremiah Jeremiah 1. Preschool. below. Tell. them too. Kids. Ke Passage: Ezekiel 37:27
Faily Jo Pag Th Exil Bg io hy u c prof b jo ou Shar ab ou job ab ar h o ay u Yo ra u ar u r a i A h ) ar par ( grp hav h y y b jo i crib blo Tll ri ir r a r gro up Allo big u r a i Rvi h b of ha u ha a
More informationChapter 7 INTEGRAL EQUATIONS
hapr 7 INTERAL EQUATIONS hapr 7 INTERAL EUATIONS hapr 7 Igral Eqaios 7. Normd Vcor Spacs. Eclidia vcor spac. Vcor spac o coios cios ( ). Vcor Spac L ( ) 4. ach-baowsi iqali 5. iowsi iqali 7. Liar Opraors
More information1a.- Solution: 1a.- (5 points) Plot ONLY three full periods of the square wave MUST include the principal region.
INEL495 SIGNALS AND SYSEMS FINAL EXAM: Ma 9, 8 Pro. Doigo Rodrígz SOLUIONS Probl O: Copl Epoial Forir Sri A priodi ri ar wav l ad a daal priod al o o od. i providd wi a a 5% d a.- 5 poi: Plo r ll priod
More informationELG3150 Assignment 3
ELG350 Aigmt 3 Aigmt 3: E5.7; P5.6; P5.6; P5.9; AP5.; DP5.4 E5.7 A cotrol ytm for poitioig th had of a floppy dik driv ha th clodloop trafr fuctio 0.33( + 0.8) T ( ) ( + 0.6)( + 4 + 5) Plot th pol ad zro
More informationChapter 11 INTEGRAL EQUATIONS
hapr INTERAL EQUATIONS hapr INTERAL EUATIONS Dcmbr 4, 8 hapr Igral Eqaios. Normd Vcor Spacs. Eclidia vcor spac. Vcor spac o coios cios ( ). Vcor Spac L ( ) 4. achy-byaowsi iqaliy 5. iowsi iqaliy. Liar
More informationThe Asymptotic Form of Eigenvalues for a Class of Sturm-Liouville Problem with One Simple Turning Point. A. Jodayree Akbarfam * and H.
Joral of Scic Ilaic Rpblic of Ira 5(: -9 ( Uirity of Thra ISSN 6- Th Ayptotic For of Eigal for a Cla of Str-Lioill Probl with O Sipl Trig Poit A. Jodayr Abarfa * ad H. Khiri Faclty of Mathatical Scic Tabriz
More informationThomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple
5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2,
More informationTechnical Support Document Bias of the Minimum Statistic
Tchical Support Documt Bias o th Miimum Stattic Itroductio Th papr pla how to driv th bias o th miimum stattic i a radom sampl o siz rom dtributios with a shit paramtr (also kow as thrshold paramtr. Ths
More informationECE351: Signals and Systems I. Thinh Nguyen
ECE35: Sigals ad Sysms I Thih Nguy FudamalsofSigalsadSysms x Fudamals of Sigals ad Sysms co. Fudamals of Sigals ad Sysms co. x x] Classificaio of sigals Classificaio of sigals co. x] x x] =xt s =x
More informationThe universal vector. Open Access Journal of Mathematical and Theoretical Physics [ ] Introduction [ ] ( 1)
Ope Access Joural of Mahemaical ad Theoreical Physics Mii Review The uiversal vecor Ope Access Absrac This paper akes Asroheology mahemaics ad pus some of i i erms of liear algebra. All of physics ca be
More informationFourier Techniques Chapters 2 & 3, Part I
Fourir chiqus Chaprs & 3, Par I Dr. Yu Q. Shi Dp o Elcrical & Compur Egirig Nw Jrsy Isiu o chology Email: shi@i.du usd or h cours: , 4 h Ediio, Lahi ad Dog, Oord
More informationSignal & Linear System Analysis
Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Sigal & Liar Sym Aalyi Sigal Modl ad Claificaio Drmiiic v. Radom Drmiiic igal: complly pcifid fucio of im. Prdicabl, o ucraiy.g., < < ; whr A ad ω ar
More informationInstructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems
Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi
More informationLinear Systems Analysis in the Time Domain
Liar Sysms Aalysis i h Tim Domai Firs Ordr Sysms di vl = L, vr = Ri, d di L + Ri = () d R x= i, x& = x+ ( ) L L X() s I() s = = = U() s E() s Ls+ R R L s + R u () = () =, i() = L i () = R R Firs Ordr Sysms
More informationChap.3 Laplace Transform
Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl
More informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
More informationAsymptotic Behaviors for Critical Branching Processes with Immigration
Acta Mathmatica Siica, Eglih Sri Apr., 9, Vol. 35, No. 4, pp. 537 549 Publihd oli: March 5, 9 http://doi.org/.7/4-9-744-6 http://www.actamath.com Acta Mathmatica Siica, Eglih Sri Sprigr-Vrlag GmbH Grmay
More informationISSN: [Bellale* et al., 6(1): January, 2017] Impact Factor: 4.116
IESRT INTERNTIONL OURNL OF ENGINEERING SCIENCES & RESERCH TECHNOLOGY HYBRID FIED POINT THEOREM FOR NONLINER DIFFERENTIL EQUTIONS Sidhshwar Sagram Bllal*, Gash Babrwa Dapk * Dparm o Mahmaics, Daaad Scic
More informationRuled surfaces are one of the most important topics of differential geometry. The
CONSTANT ANGLE RULED SURFACES IN EUCLIDEAN SPACES Yuuf YAYLI Ere ZIPLAR Deparme of Mahemaic Faculy of Sciece Uieriy of Aara Tadoğa Aara Turey yayli@cieceaaraedur Deparme of Mahemaic Faculy of Sciece Uieriy
More informationUNIT III STANDARD DISTRIBUTIONS
UNIT III STANDARD DISTRIBUTIONS Biomial, Poisso, Normal, Gomric, Uiform, Eoial, Gamma disribuios ad hir roris. Prard by Dr. V. Valliammal Ngaiv biomial disribuios Prard by Dr.A.R.VIJAYALAKSHMI Sadard Disribuios
More informationRevised Variational Iteration Method for Solving Systems of Ordinary Differential Equations
Availabl at http://pvau.du/aa Appl. Appl. Math. ISSN: 9-9 Spcial Iu No. Augut 00 pp. 0 Applicatio ad Applid Mathatic: A Itratioal Joural AAM Rvid Variatioal Itratio Mthod for Solvig St of Ordiar Diffrtial
More informationSome Newton s Type Inequalities for Geometrically Relative Convex Functions ABSTRACT. 1. Introduction
Malaysia Joural of Mahemaical Scieces 9(): 49-5 (5) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/joural Some Newo s Type Ieualiies for Geomerically Relaive Covex Fucios
More informationOverview. Introduction Building Classifiers (2) Introduction Building Classifiers. Introduction. Introduction to Pattern Recognition and Data Mining
Ovrv Iroduco o ar Rcogo ad Daa Mg Lcur 4: Lar Dcra Fuco Irucor: Dr. Gova Dpar of Copur Egrg aa Clara Uvry Iroduco Approach o uldg clafr Lar dcra fuco: dfo ad urfac Lar paral ca rcpro crra Ohr hod Lar Dcra
More informationThe Solution of Advection Diffusion Equation by the Finite Elements Method
Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 88 T Soluio of Advcio Diffusio Equaio by Fii Els Mod Hasa BULUT, Tolga AKTURK ad Yusuf UCAR Dpar of Maaics, Fira Uivrsiy, 9, Elazig-TURKEY Dpar
More informationLecture 4: Laplace Transforms
Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions
More informationMeromorphic Functions Sharing Three Values *
Alied Maheaic 11 718-74 doi:1436/a11695 Pulihed Olie Jue 11 (h://wwwscirporg/joural/a) Meroorhic Fucio Sharig Three Value * Arac Chagju Li Liei Wag School o Maheaical Sciece Ocea Uiveriy o Chia Qigdao
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
More informationwww.vidrhipu.com TRANSFORMS & PDE MA65 Quio Bk wih Awr UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART-A. Oi pri diffri quio imiig rirr co d from z A.U M/Ju Souio: Giv z ----- Diff Pri w.r. d p > - p/ q > q/
More information1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4)
7 Differeial equaios Review Solve by he mehod of udeermied coefficies ad by he mehod of variaio of parameers (4) y y = si Soluio; we firs solve he homogeeous equaio (4) y y = 4 The correspodig characerisic
More informationFurther Results on Pair Sum Graphs
Applid Mathmatis, 0,, 67-75 http://dx.doi.org/0.46/am.0.04 Publishd Oli Marh 0 (http://www.sirp.org/joural/am) Furthr Rsults o Pair Sum Graphs Raja Poraj, Jyaraj Vijaya Xavir Parthipa, Rukhmoi Kala Dpartmt
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationControl Systems. Transient and Steady State Response.
Corol Sym Trai a Say Sa Ro chibum@oulch.ac.kr Ouli Tim Domai Aalyi orr ym Ui ro Ui ram ro Ui imul ro Chibum L -Soulch Corol Sym Tim Domai Aalyi Afr h mahmaical mol of h ym i obai, aalyi of ym rformac i.
More information