On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

Transcription

1 Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn M Kazminia M Mhrjoo Dparmn of Tlcommunicaions Unirsiy of Sisan Balouchsan Zahdan Iran Corrsponding Auhor: M Kazminia ABSTRACT: W prsn a closd-form prssion for h firs ordr driais of Bssl modifid Bssl funcions wih rspc o boh h ordr h argumn i whn h ordr h argumn ar dpndn Th highr ordr driais ar drid asily from h obaind firs ordr on Kywords: Driai; Bssl funcions; Modifid Bssl funcions INTRODUCTION Bssl funcions ar inold in signal procssing dirs problm samns apparing in physics nginring mahmaical physics A wid rang of phnomna in lcriciy magnism microwa opical ransmission ha conducion acousical ibraions ar rprsnd by Bssl funcions Du o h as applicaions Bssl funcions hir driais propris ha bn insigaing in h liraur (Gradshyn 007; Abramowiz Sgun 970; Wason 944) Th driais of Bssl modifid Bssl funcions wih rspc o h ordr or h argumn is in h liraur Howr h driais of Bssl modifid Bssl funcions wih rspc o a paramr ha appars in boh h ordr h argumn ha no bn compud Rcnly in som paricular applicaions modifid Bssl funcions of h scond kind appar whr h ordr h argumn boh ar funcions of a common indpndn paramr For insanc in fr spac opical ransmission o find h maimum liklihood simaion of h opical flucuaion disribuion w dal wih h driai of h modifid Bssl funcion wih rspc o boh h ordr h argumn (Kazminia Mhrjoo 03) Moror h closd-form driais of Bssl modifid Bssl funcions wih rspc o boh h ordr h argumn do no is in rfrnc abls of h spcial funcions (Gradshyn 007; Abramowiz Sgun 970) Th rs of h papr is organizd as follows In scion a proposiion som ssnial dfiniions ar proidd In scion 3 closd-form formulas for h driais of Bssl modifid Bssl funcions wih rspc o boh h ordr h argumn ar drid Prliminaris In his scion w prsn ssnial dfiniions known rsuls rquird o compu h driais Bssl funcions of h firs kind J (z) h scond kind Y (z) (also calld Numann s funcions) h hird kind H () (z) H () (z) (also calld Hankl s funcions) ar h soluions of h following scond ordr diffrnial quaion: d u du z z ( z ) u 0 dz dz () Similarly Modifid Bssl funcions of h firs kind I (z) h scond kind K (z) ar h soluions of h following diffrnial quaion: d u du z z ( z ) u 0 dz dz () Prior o sa h main rsuls of his papr w rcall a proposiion which will b usd succssily in h proof procding (Almkis Zilbrgr 990)

2 Inl Rs J Appl Basic Sci Vol 4 () Proposiion L X b an opn subs of T b a masur spac Suppos h : X T saisfis h following condiions: () h( ) is a masurabl funcion of for ach X () For almos all T h driai h( )/ iss for all X (3) Thr is an ingrabl funcion :T such ha h( ) ( ) for X Thn d h( ) h( ) d X X Proposiion is known as h mhod of diffrniaion undr h ingral sign (3) MATERIAL METHODS MAIN RESULTS In his scion closd-form formulas for h firs ordr driais of Bssl modifid Bssl funcions wih rspc o boh h ordr h argumn ar prsnd in horms 3 o 36 Thorm 3 L f() b a diffrniabl funcion h driai of J (f()) wih rspc o is J ( f ( )) f () J( f ( )) J( f ( )) f () n f() f ( ) f ( ) Jn( f ( )) J ( f ( )) ln ( ) ( ) (3) n0 n!( n ) Proof W sar wih formula (Gradshyn 007 p93 84) (0 ) f ( ) f ( ) J ( f ( )) p (3) i 4 Th driai of (3) wih rspc o rsuls in ( ( )) ( ) (0 ) (0 ) ( ) ( ) ( ) J f f f f f p p 4 4 i (33) i Th driais of ach summ in (33) ar f ( ) f ( ) f ( ) f ( ) ln (34) i i f ( ) (0 ) (0 ) (0 ) f ( ) ( ) ( ) ( ) ( ) p ln( ) f f f p f p (35) rspcily Th firs ingral in h righ h sid of (35) is calculad using proposiion Assum f() h( ) ( )p (36) 4 Hnc ln( ) p ( )p 4 4 (0 ) (0 ) f ( ) f ( ) d d ( )p 4 (0 ) f () d i J ( f ( )) d f()

3 Inl Rs J Appl Basic Sci Vol 4 () i f ( ) ln( ) J ( f ( )) J ( f ( )) 0 f() (37) Th scond ingral in h righ h sid of (35) is (0 ) f ( ) i p J ( f ( )) 4 f() (38) By subsiuing (37) (38) in (35) rplacing (34) (35) in (33) w achi J ( f ( )) f () J ( f ( )) J( f ( )) J( f ( )) (39) f () whr (Gradshyn 007 p96 847) J ( f ( )) J( f ( )) J( f ( )) J( f ( )) f ( ) f ( ) (30) (Gradshyn 007 p ) n f() f ( ) f ( ) Jn( f ( )) J( f ( )) J( f ( )) ln ( ) ( ) n0 n!( n ) (3) Thorm 3 If f() is a diffrniabl funcion hn Y ( f ( )) f () Y ( f ( )) Y ( f ( )) f () co( ) J ( f ( )) csc( ) ( ( )) csc( ) ( ( )) or ingr J f Y f n n n Proof Rcalling formula (Gradshyn 007 p94 845) f f ( ) / cos ( ) Y ( f ( )) R( ) ( ) 0 f / ( ) ( ) ( ) (33) w ak h driai of Y (f()) wih rspc o : Y ( f ( )) f ( ) / cos f ( ) f ( ) / cos f ( ) / / ( ) ( ) ( ) ( ) ( ) ( ) (34) whr f ( ) / f ( ) / f ( ) f ( ) ln ( ) ( ) ( ) ( ) ( ) f ( ) (35) cos f ( ) f() sin f ( ) cos f ( ) ln( ) / / / ( ) ( ) ( ) (36) Th firs ingral in h righ h sid of (36) is (Gradshyn 007 p ) sin f ( ) ( ) Y / ( f ( )) ( ) f( ) (37) Using proposiion h scond ingral in h righ h sid of (36) is compud as follows (3) 49

4 Inl Rs J Appl Basic Sci Vol 4 () cos f ( ) ( ) f( ) ln( ) Y ( ( ) ln ( ) ( ( )) / f Y f 0 ( ) f() (38) By insring (37) (38) in (36) plugging (35) (36) in (34) w ha Y ( f ( )) f () Y ( f ( )) Y ( f ( )) Y ( f ( )) f () (39) whr (Gradshyn 007 p96 847) Y ( f ( )) Y ( f ( )) Y ( f ( )) Y ( f ( )) f ( ) f ( ) (30) (Gradshyn 007 p ) Y ( f ( )) co( ) J( f ( )) csc( ) J ( f ( )) csc( ) Y ( f ( )) n or n n ingr (3) Thorm 33 For a diffrniabl funcion f() h driai of H () (f()) wih rspc o is () H ( f ( )) f () () () H ( f ( )) H ( f ( )) + J( f ( )) i Y( f ( )) f () Proof Rcalling formula (Gradshyn 007 p95 845) ( ) () i f ( ) / if H ( f ( )) R( ) ( ) 0 f / ( ) ( ) ( ) (33) W compu h driai of H () (f()) wih rspc o as follows: () ( ) / if ( ) ( ( )) ( ) / if ( ) H f f f i i / / ( ) ( ) ( ) ( ) ( ) ( ) whr f ( ) / i f ( ) / f ( ) f ( ) i ln ( ) ( ) ( ) ( ) ( ) f ( ) (35) if ( ) if ( ) if ( ) f () i ln( ) / / / (3) (34) ( ) ( ) ( ) (36) Th firs ingral in h righ h sid of (36) quals if () cos( f ( ) ) sin( f ( ) ) i / / / ( ) ( ) ( ) ( ) J ( f ( )) iy ( f ( )) f( ) ( ) () H ( f ( )) (37) f( ) Th scond ingral in h righ h sid of (36) is calculad by proposiion : 430

5 Inl Rs J Appl Basic Sci Vol 4 () if ( ) ( ) () f ( ) () ln( ) H ( ( ) ln ( ) / f H ( f ( )) ( ) i f() 0 (38) Thn w us quaions (34) o (38) o obain h following formula: () H ( f ( )) f () () () () H ( f ( )) H ( f ( )) H ( f ( )) f () (39) whr (Gradshyn 007 p96 847) () () () () H ( f ( )) H ( f ( )) H ( f ( )) H ( f ( )) f ( ) f ( ) (330) () H ( f ( )) J ( f ( )) i Y ( f ( )) (33) Thorm 34 L f() b dfind as in horm 3 Thn h driai of H () (f()) is () H ( f ( )) f () () () H ( f ( )) H ( f ( )) J( f ( )) i Y( f ( )) (33) f () Proof W us a similar formula o (33) o achi h driai of h H () (f()) wih rspc o (Gradshyn 007 p95 846): ( ) () i f ( ) / if H ( f ( )) R( ) ( ) 0 f / ( ) ( ) ( ) (333) Hnc h driai of H () (f()) wih rspc o is calculad wih h sam approach as h on s of H () (f()) Thorm 35 If f() is a diffrniabl funcion hn I ( f ( )) f () I( f ( )) I( f ( )) f () k f ( ) f ( ) ( k ) I ( f ( ))ln n or n n ingr (334) k! ( k ) k0 Proof W sar wih (Gradshyn 007 p96 843) f( ) / / f ( ) I ( f ( )) ( ) R( ) 0 ( ) ( ) (335) Th driai of I (f()) wih rspc o is I ( ( )) ( ) / / ( ) ( ) / f f f f / f ( ) ( ) ( ) ( ) ( ) ( ) ( ) (336) whr 43

6 Inl Rs J Appl Basic Sci Vol 4 () f ( ) / f ( ) / f ( ) f ( ) ln ( ) ( ) ( ) ( ) ( ) f ( ) (337) / f ( ) f() / f ( ) / f ( ) ( ) ( ) ln( )( ) (338) To compu h firs ingral rm in (338) w rplac h ponnial funcion wih is powr sris: n / f ( ) f() / n ( ) ( ) n! n0 (339) If n is n hn h ingral in h righ h sid of (339) quals zro Thrfor for n=k+ w find k / f ( ) f() / k ( ) ( ) (k )! 0 k0 k f( ) 3 B( k ) (k )! k0 k ( ) ( k ) f() (k )! ( k ) k0 k ( ) f ( ) (k )! (k )! ( k ) k k0 ( k )! k ( f( ) / ) ( )( / f ( )) k! ( k ) k0 ( )( / f ( )) I ( f ( )) (340) whr B is h ba funcion Using proposiion h scond ingral of (338) is obaind: / f ( ) f( ) / f( ) ln( )( ) I ( f ( )) ln ( ) I ( f ( )) 0 ( ) ( ) (34) By subsiuing h drid quaions (337) (338) in (336) h driai of I (f()) wih rspc o is achid: I ( f ( )) f () I ( f ( )) I( f ( )) I( f ( )) (34) f () whr (Gradshyn 007 p ) I ( f ( )) I( f ( )) I ( f ( )) I( f ( )) f ( ) f ( ) (343) (Gradshyn 007 p ) k f ( ) f ( ) ( k ) I( f ( )) I( f ( ))ln n or n n ingr k! ( k ) k0 (344) Thorm 36 L f() b a diffrniabl funcion hn h driai of K (f()) is 43

7 Inl Rs J Appl Basic Sci Vol 4 () K ( f ( )) f () K( f ( )) K( f ( )) f () I( f ( )) I ( f ( )) co( ) K ( f ( )) csc( ) n or n n ingr (345) Proof Rcall formula (Gradshyn 007 p ) f ( ) 4 f 0 ( ) K ( f ( )) arg f ( ) R f ( ) 0 Th driai of K (f()) wih rspc o is f ( ) 4 f ( ) 4 K ( f ( )) f ( ) f ( ) 0 0 whr f ( ) f ( ) f ( ) f ( ) f ( ) ln (346) (347) (348) f ( ) 4 f ( ) 4 f ( ) 4 f ( ) f ( ) ln( ) (349) Th firs ingral in h righ h sid of (349) quals ( ) 4 f K ( f ( ) 0 f() (349) Using proposiion w compu h scond ingral in h righ h sid of : (350) f ( ) 4 ln( ) ln K ( ( )) ( ( )) 0 f K f f ( ) f ( ) 0 (35) By subsiuing (350) (35) in (349) hn plugging (348) (349) in (347) w ha K ( f ( )) f () K ( f ( )) K( f ( )) K( f ( )) (35) f () Whr (Gradshyn 007 p ) K ( f ( )) K( f ( )) K( f ( )) K( f ( )) f ( ) f ( ) (353) (Gradshyn 007 p ) I( f ( )) I ( f ( )) K( f ( )) co( ) K( f ( )) csc( ) n or n n ingr (354) Th rquird driais wih rspc o h ordr h argumn in horms 3 o 36 ar prsnd Thrfor h proofs of h main rsuls compl REFERENCES Abramowiz M Sgun IA (Eds)970 Hbook of Mahmaical Funcions Appl Mahs Sris ol 55 Naional Burau of Sards Nw York Almkis G Zilbrgr D 990 Th mhod of diffrniaing undr h ingral sign Jour Symb Comp 0: Gradshyn IS Ryzhik IM 007 Tabls of Ingrals Sris Producs snh d AcadmicPrss Nw York Kazminia M Mhrjoo M 03 A Nw Mhod for Maimum Liklihood Paramr Esimaion of Γ-Γ Disribuion Journal of Lighwa Tchnology 39: Wason GN944 A Trais on h Thory of Bssl Funcions scond d Cambridg Unirsiy Prss Cambridg London NwYork 433