Linear Differential Equations. Driven by Delta Functions

Transcription

1 Gauge Istitute Joural H. Vic Dao Liear Differetial Equatios Drive by Delta Fuctios H. Vic Dao May, 4 Abstract We solve the liear differetial equatio LG ( ) (, ) = d( -), for the Hyper-real Gree fuctio G (, ), where L ( ) is a liear differetial operator, ad d( - ) is the hyper-real Delta fuctio. Keywords: Sturm-Liouville Epasios, Ifiitesimal, Ifiite- Hyper-Real, Hyper-Real, ifiite Hyper-real, Ifiitesimal Calculus, Delta Fuctio, Fourier Series, Laguerre Polyomials, Legedre Fuctios, Bessel Fuctios, Delta Fuctio, Mathematics Subject Classificatio 6E35; 6E3; 6E5; 6E; 6A6; 6A; 3E; 3E55; 3E7; 3H5; 46S; 97I4; 97I3.

2 Gauge Istitute Joural H. Vic Dao Cotets. Liear differetial Equatio Drive by Delta Fuctios. Hyper-real lie.. Hyper-real Fuctio 3. Itegral of a Hyper-real Fuctio 4. Delta Fuctio 5. Coverget Series 6. Gree s Fuctio for y'( ) + P( ) y( ) = f( ), with y () = 7. Gree s Fuctio for y"( ) + P( ) y'( ) + Q( ) y( ) = f( ), with y () =, & y '() = 8. Gree s Fuctio for y"( ) + P( ) y'( ) + Q( ) y( ) = f( ), with ya () =, & yb () = 9. Hyper-real Sturm-Liouville Problem. Delta Epasio i No-Normalized Eige-Fuctios. Gree s Fuctio Epasio i No-Normalized Eige- Fuctios. Gree Fuctio for Sie Sturm-Liouville y"( ) + ly( ) = & a = b = 3. Gree Fuctio for Cosie Sturm-Liouville y"( ) + ly( ) = & a = b = p 4. Gree s Fuctio for Bessel Sturm-Liuoville

3 Gauge Istitute Joural H. Vic Dao y"( ) + ly( ) = & a = 5. Gree s Fuctio for Bessel Sturm-Liuoville - 4 u"( ) + ( l - ) u( ) = & a = b = 6. Delta Epasio i Orthoormal Eige-fuctios 7. Gree s Fuctio Epasio i Orthoormal Eige-fuctios 8. Gl (,, ) Aalytic Cotiuatio i l ito the Comple Plae 9. Gree Fuctio for Hermit Sturm-Liuoville u u "( ) + ( l - ) ( ) = for ay real. Gree Fuctio for Legedre Sturm-Liuoville u"( q) + [ l + ] u( q) =, - p < q < p 4 cos q. Gree Fuctio for Legedre Sturm-Liuoville y"( q) + [ l + ( - m ) ] y( q) =, - p < q < p 4 cos q. Gree Fuctio for Bessel Sturm-Liuoville - 4 u"( ) + ( l - ) u( ) = & < < b 3. Gree Fuctio for Bessel Sturm-Liuoville y"( ) + ( l - ) y( ) =, < < & a = 4. Gree Fuctio for Bessel Sturm-Liuoville y"( ) + ( m - ) y( ) =, < < & a = p 5. Gree s Fuctio Itegral Epasio i Eige-fuctios 6. Gree s Fuctio Itegral Epasio i Cosie Fuctios of y"( ) + ly( ) =, < < 3

4 Gauge Istitute Joural H. Vic Dao 7. Gree s Fuctio Itegral Epasio for Weber Equatio - 4 y"( ) + ( l - ) y( ) =, a < <, a > 8. Gree s Fuctio for Hakel s - 4 y"( ) + ( l - ) y( ) =, < < 9. Gree s Fuctio for Bessel s y"( ) + ( l + ) y( ) =, - < < 3. Gree s Fuctio for Bessel s y"( ) + ( l + e ) y( ) =, - < < 3. Gree s Fuctio for Bessel s y"( ) + ( l - e ) y( ) =, - < < Refereces 4

5 Gauge Istitute Joural H. Vic Dao. Liear Differetial Equatio drive by Delta Fuctio A liear Differetial Equatio for the Hyper-real fuctio y ( ), forced by the Hyper-real fuctio f ( ) is LDy (, )() = f(), where the Differetial Operator LD (, ) is liear. If there is a Hyper-real Gree fuctio, that satisfies the equatio G (, ) LDG (, ) (, ) = d(, ), with the same Boudary coditio that y ( ) satisfies the, G (, ) is the Kerel of a Itegral Operator that trasforms a fuctio f ( ) ito the solutio y ( ) of the liear differetial equatio L (,Dy )() = f(). Ideed, if the = b y ( ) = ò G (, ) f( ) d, = a 5

6 Gauge Istitute Joural H. Vic Dao = b LDy (, )() = LD (, ) ò G (, )() f d, = a = b = LD (, ) åg (, )() f d, = b = a = å LDG (, ) (, )() f d = a = f ( ). d( -) We shall see that the epasio of d( - ) i series, or itegrals guaratees the eistece of a Gree Fuctio that ca be epaded similarly. The Delta Fuctio ca be defied oly as a hyper-real fuctio i ifiitesimal Calculus. We proceed with the defiitio of the Hyper-real lie. 6

7 Gauge Istitute Joural H. Vic Dao. Hyper-real Lie Each real umber a ca be represeted by a Cauchy sequece of ratioal umbers, ( r, r, r3,...) so that r a. The costat sequece ( aaa,,,...) is a costat hyper-real. I [Da] we established that,. Ay totally ordered set of positive, mootoically decreasig to zero sequeces ifiitesimal hyper-reals. ( i, i, i3,...) costitutes a family of. The ifiitesimals are smaller tha ay real umber, yet strictly greater tha zero. 3. Their reciprocals (,,,...) i i i 3 are the ifiite hyper-reals. 4. The ifiite hyper-reals are greater tha ay real umber, yet strictly smaller tha ifiity. 5. The ifiite hyper-reals with egative sigs are smaller tha ay real umber, yet strictly greater tha The sum of a real umber with a ifiitesimal is a o-costat hyper-real. 7. The Hyper-reals are the totality of costat hyper-reals, a family of ifiitesimals, a family of ifiitesimals with 7

8 Gauge Istitute Joural H. Vic Dao egative sig, a family of ifiite hyper-reals, a family of ifiite hyper-reals with egative sig, ad o-costat hyper-reals. 8. The hyper-reals are totally ordered, ad aliged alog a lie: the Hyper-real Lie. 9. That lie icludes the real umbers separated by the ocostat hyper-reals. Each real umber is the ceter of a iterval of hyper-reals, that icludes o other real umber.. I particular, zero is separated from ay positive real by the ifiitesimals, ad from ay egative real by the ifiitesimals with egative sigs, - d.. Zero is ot a ifiitesimal, because zero is ot strictly greater tha zero.. We do ot add ifiity to the hyper-real lie. 3. The ifiitesimals, the ifiitesimals with egative sigs, the ifiite hyper-reals, ad the ifiite hyper-reals with egative sigs are semi-groups with respect to additio. Neither set icludes zero. 4. The hyper-real lie is embedded i, ad is ot homeomorphic to the real lie. There is o bi-cotiuous oe-oe mappig from the hyper-real oto the real lie. 8

9 Gauge Istitute Joural H. Vic Dao 5. I particular, there are o poits o the real lie that ca be assiged uiquely to the ifiitesimal hyper-reals, or to the ifiite hyper-reals, or to the o-costat hyperreals. 6. No eighbourhood of a hyper-real is homeomorphic to a ball. Therefore, the hyper-real lie is ot a maifold. 7. The hyper-real lie is totally ordered like a lie, but it is ot spaed by oe elemet, ad it is ot oe-dimesioal. 9

10 Gauge Istitute Joural H. Vic Dao. Hyper-real Fuctio. Defiitio of a hyper-real fuctio f () is a hyper-real fuctio, iff it is from the hyper-reals ito the hyper-reals. This meas that ay umber i the domai, or i the rage of a hyper-real f ( ) is either oe of the followig real real + ifiitesimal real ifiitesimal ifiitesimal ifiitesimal with egative sig ifiite hyper-real ifiite hyper-real with egative sig Clearly,. Every fuctio from the reals ito the reals is a hyper-real fuctio.

11 Gauge Istitute Joural H. Vic Dao 3. Itegral of a Hyper-real Fuctio I [Da3], we defied the itegral of a Hyper-real Fuctio. Let f () be a hyper-real fuctio o the iterval [ ab],. The iterval may ot be bouded. f ( ) may take ifiite hyper-real values, ad eed ot be bouded. At each a b, there is a rectagle with base [ -, + d ], height f ( ), ad area d f ( d. ) We form the Itegratio Sum of all the areas for the s that start at = a, ad ed at = b, å f ( d ). Î[ a, b] If for ay ifiitesimal d, the Itegratio Sum has the same hyper-real value, the f ( ) is itegrable over the iterval [ ab],. The, we call the Itegratio Sum the itegral of f () from = a, to = b, ad deote it by

12 Gauge Istitute Joural H. Vic Dao = b ò f ( d ). = a If the hyper-real is ifiite, the it is the itegral over [, ab], If the hyper-real is fiite, = b ò fd ( ) = real part of the hyper-real. = a 3. The coutability of the Itegratio Sum I [Da], we established the equality of all positive ifiities: We proved that the umber of the Natural Numbers, Card, equals the umber of Real Numbers, Card = Card, ad we have Card Card Card = ( Card ) =... = = =... º. I particular, we demostrated that the real umbers may be well-ordered. Cosequetly, there are coutably may real umbers i the iterval [ ab],, ad the Itegratio Sum has coutably may terms. While we do ot sequece the real umbers i the iterval, the summatio takes place over coutably may f ( d. ) The Lower Itegral is the Itegratio Sum where f ( ) is replaced

13 Gauge Istitute Joural H. Vic Dao by its lowest value o each iterval [ -, + ] d d 3. å Î[ a, b] æ ö if f ( t) d ç çè- t + ø d d The Upper Itegral is the Itegratio Sum where f ( ) is replaced by its largest value o each iterval [ -, + ] d d 3.3 å Î[ a, b] æ ö sup f ( t) d ç è- t + ø d d If the itegral is a fiite hyper-real, we have 3.4 A hyper-real fuctio has a fiite itegral if ad oly if its upper itegral ad its lower itegral are fiite, ad differ by a ifiitesimal. 3

14 Gauge Istitute Joural H. Vic Dao 4. Delta Fuctio I [Da5], we have defied the Delta Fuctio, ad established its properties. The Delta Fuctio is a hyper-real fuctio defied from the hyper-real lie ito the set of two hyper-reals ì ï ü í, ï ý. The ïî d ïþ hyper-real is the sequece,,,.... The ifiite hyperreal d depeds o our choice of d.. We will usually choose the family of ifiitesimals that is spaed by the sequeces,, 3, It is a semigroup with respect to vector additio, ad icludes all the scalar multiples of the geeratig sequeces that are o-zero. That is, the family icludes ifiitesimals with egative sig. Therefore, d will mea the sequece. Alteratively, we may choose the family spaed by the sequeces, 3,, The, 4 d will mea the 4

15 Gauge Istitute Joural H. Vic Dao sequece. Oce we determied the basic ifiitesimal d, we will use it i the Ifiite Riema Sum that defies a Itegral i Ifiitesimal Calculus. 3. The Delta Fuctio is strictly smaller tha 4. We defie, c d ( ) º d ( ), d d é ù, ê- ë úû where c é ê- ë d, d ì, Î é-d, d ù ( ) = ï ê ú í ë û. ïï, otherwise î ù úû 5. Hece, for <, d ( ) = at for d =-, d ( ) jumps from to d, d, d ê ë ú û, ( ) Î é- ù d =. d at =, d () = d at d =, d ( ) drops from d for >, d ( ) =. to. d ( ) = 6. If d =, ( ) = ( ), ( ),3 ( )... [-, ] [-, ] [-, ] d c c c If d =, 3 d ( ) =,,,... cosh cosh cosh 3 5

16 Gauge Istitute Joural H. Vic Dao 8. If d =, d( ) = e c [, ), e c[, ), 3 e c[, ),... = ò 9. d( d ) =. =- k = -ik( - ). d ( - ) = e dk p ò k =- 6

17 Gauge Istitute Joural H. Vic Dao 5. Coverget Series I [Da8], we defied covergece of ifiite series i Ifiitesimal Calculus 5. Sequece Covergece to a fiite hyper-real a a a iff a - a = ifiitesimal. 5. Sequece Covergece to a ifiite hyper-real A a A iff a represets the ifiite hyper-real A. 5.3 Series Covergece to a fiite hyper-real s a + a +... s iff a a - s = ifiitesimal. 5.4 Series Covergece to a Ifiite Hyper-real S a + a +... S iff a a represets the ifiite hyper-real S. 7

18 Gauge Istitute Joural H. Vic Dao 6. Gree s Fuctio for y'( ) + P( ) y( ) = f( ), with y () = q= q= q= ò - ò ò q= q= q= 6. G (, ) = e e = e Proof: We solve the equatio P( q) dq P( q) dq P( q) dq G (, ) + PG () (, ) = d ( - ), with G(, ) = Multiplyig both sides by the itegratig factor q= ò e q= P( q) dq q= q= q= ò P( q) dq ò P( q) dq ò P( q) dq q= q= q= e G(, ) + e P() G(, ) = e d( - ) q= q= ò = P( q) dq P( q) dq { e q= G (, )} e q= d( ò -) Itegratig with respect to, q= q= P( q) dq = ò ò P( q) dq q= q= e G(, ) = e d( -) d ò =- 8

19 Gauge Istitute Joural H. Vic Dao q= = P( q) dq å ( ) d q= = d - =- ò e Therefore, q= ò = e q= P( q) dq q= q= ò G (, ) = e e ò P( q) dq - P( q) dq q= q= q= ò P( q) dq q= = e. 6. = ò P( q) dq q= y ( ) = ò e f( d ) = q= 9

20 Gauge Istitute Joural H. Vic Dao 7. Gree s Fuctio for y"( ) + P( ) y'( ) + Q( ) y( ) = f( ) ad y '() = y () =, with, 7. y"( ) + P( ) y'( ) + Q( ) y( ) = f( ), with y () =, y '() = u ( )solves u"( ) + P( ) u'( ) + Q( ) u( ) =, with u () = v ( )solves v"( ) + P( ) v'( ) + Q( ) v( ) =, with v () = The, G (, ) = u() v() u ( ) v ( ) u() v() u'( ) v'( ) Proof: We solve the equatio G (, ) + P () G (, ) + QG () (, ) = d( - ), with G(, ) =, ad G (, ) = = Let u ( ) solve the equatio The, u"( ) + P( ) u'( ) + Q( ) u( ) = ug " + PuG ' + QuG = ud( - ),, with u () =

21 Gauge Istitute Joural H. Vic Dao Subtractig, Gu " + PGu ' + QGu =, ug "- Gu " + P( ug '- Gu ') = ud( -) ( ug '-Gu ')' ò P( q) dq q= Multiplyig by the itegratig factor e, P( ) d P( ) d { e ò q q [ u ( ) G '(, ) G (, ) u '( )]} e ò q q q= q= - = u ( ) d( -). Itegratig, P( q) dq t= ò ò P( q) dq q= q= e [ u( ) G'(, ) - G(, ) u'( )] = ò e u( t) d( t -) dt. (I) q= q= q= q= q= t t = Similarly, for v ( ) that solves the equatio v"( ) + P( ) v'( ) + Q( ) v( ) =, with v () = q= q= t P( q) dq t = ò ò P( q) dq q= q= e [() v G'(, ) - G(, )'()] v = ò e v()( t d t -) dt. (II) t = To elimiate G ', subtract (I) v ( )-(II) u( ), q= q= t P( q) dq t = ò P( ) d u ( ) v ( ) ò q q [ u( t) v( t) q= q= e G(, ) = e d( ) u'( ) v'( ) ò t -dt u ( ) v ( ) Substitutig, by Abel equality, t= q= ò e q= P( q) dq u ( ) v ( ) u() v() =, u'( ) v'( ) u'() v'() ad

22 Gauge Istitute Joural H. Vic Dao q= t ò e q= P( q) dq = u() v() u'() v'() u( t) v( t) u'( t) v'( t) t = u() v() t= u() v() u'() v'() [ u( t) v( t) G (, ) = ( ) d u'() v'() ò d t - t u( t) v( t) u( ) v( ) u'( t) v'( t) t= [ u( t) v( t) G (, ) = ò d( t -) dt u( t) v( t) u ( ) v ( ) t = u'( t) v'( t) [ u( ) v( ) u ( ) v ( ) =. u() v() u'( ) v'( )

23 Gauge Istitute Joural H. Vic Dao 8. Gree s Fuctio for y"( ) + P( ) y'( ) + Q( ) y( ) = f( ) ad yb () = ya () =, with, 8. y"( ) + P( ) y'( ) + Q( ) y( ) = f( ), with ya () =, yb () =, u"( ) + P( ) u'( ) + Q( ) u( ) =, with ua () =, ub () ¹, v"( ) + P( ) v'( ) + Q( ) v( ) =, with va () ¹, vb () = Proof: The, G (, ) ìï ()(), u v a = ï í u() v() ïï ïuu ( ) ( ), b³ ³ u'( ) v'( ) ïî We solve the equatio Let u ( ) solve the equatio y"( ) + P( ) y'( ) + Q( ) y( ) = f( ), with ya () =, ad yb () =. u"( ) + P( ) u'( ) + Q( ) u( ) = ad let v ( ) solve the equatio, with ua () =, ad ub () ¹, v"( ) + P( ) v'( ) + Q( ) v( ) =, with va () ¹, ad vb () =. 3

24 Gauge Istitute Joural H. Vic Dao By 7 a particular solutio to the equatio for y ( ) is = [ u( ) v( ) y ( ) = ( ) p ò f d, u() v() u ( ) v ( ) = a ad the geeral solutio is u'( ) v'( ) y ( ) = cu ( ) + cv ( ) + y( ). = ya ( ) = c ua ( ) + c va ( ) + y( p a) c = ¹ p = yb ( ) = c ub ( ) + c vb ( ) + y( p b) ¹ c =- y ub () p() b = b =- ub () ò u() v() = a u'( ) v'( ) [ u( ) v( ) f () d ub () vb () u( ) v( b) -u( b) v( ) = b = ò = a v() u() v() u'( ) v'( ) f () d = b = v() u()() v - u()() v y( ) = u( ) ò f( ) d + f( ) d u() v() ò u() v() = a = a u'( ) v'( ) u'( ) v'( ) 4

25 Gauge Istitute Joural H. Vic Dao Therefore, = b = uv ( ) ( ) u( ) v ( ) = ò f () d + f() u() v() ò d u() v() = = a u'( ) v'( ) u'( ) v'( ) G (, ) ìï u()(), v a = ï í u() v() ïï ïuu ( ) ( ), b³ ³ u'( ) v'( ) ïî. 5

26 Gauge Istitute Joural H. Vic Dao 9. Hyper-real Sturm-Liouville Problem The Hyper-real Sturm-Liouville equatio is the secod order Hyper-real liear differetial equatio for the Hyper-real fuctio y ( ), - y"( ) + q( ) = ly( ), o a iterval that may be bouded, or may be the whole Hyper-real lie. The hyper-real fuctio q ( ) is (assumed i the literature to be) cotiuous o the iterval, ad bouded at its edpoits. The choice of the umber l (which may be real or comple), allows the equatio with boudary coditios, at the iterval edpoits to become a eige-value problem: l is a eige-value, ad y ( ) is the correspodig Hyper-real eige-fuctio iff - y "( )( ) + q( ) y ( ) = l ( ) y. The eige-fuctios are orthogoal, over the iterval. = b ò y ( ) y ( ) d =, for ¹ m. m = a 6

27 Gauge Istitute Joural H. Vic Dao. Delta Epasio i No-Normalized Eige-fuctios As described i [Titchmarsh, Chapter I], give umbers a, ad b, the Sturm-Liouville Problem o the iterval with edpoits a, ad b has solutios f (, l ), with f (, a l ) = sia, ad f '( a, l ) =-cosa, a a c (, l ), with c (, b l ) = sib, ad c '( b, l ) =-cosb,, b which are etire fuctios of l. The, b a b d d f (, l) c (, l) f '(, l) c '(, l) f (, l) c (, l) a b a b a b = + f '(, l) c '(, l) f '(, l) c '(, l) f "(, l) c "(, l) a b a b a b W [ f, c ] a b = fc "-cf " a b b a = f ( q -l) c -c ( q -l) f a b b a =. Hece, the Wroskia W[ f a, c b ] is a fuctio of l aloe: [ f, c ] = w( l). W a b 7

28 Gauge Istitute Joural H. Vic Dao Now, if the oly zeros of wl ( ) are the simple zeros The, for each =,,, 3,... l,,, l,, l l 3 f (, l ) c (, l ) a b = wl ( ) =. f '(, l ) c '(, l ) That is, for each =,,, 3,... there is a umber k so that a c (, l ) = k f (, l ). b Titchmarsh applied the Residue Theorem to obtai the coefficiets i the Sturm-Liouville epasio of f ( ). Followig Titchmarsh, we coclude that the Hyper-real Sturm- a b Liouville epasio of a Hyper-real fuctio f ( ) i the Hyperreal eige-fuctios fa (, l ) is = æ = b k ö f ( ) = å f( f ) ( l, ) d f(, l) a a = ç w'( l ) ò çè = a ø = æ = b k ö = å f() f(, l) d f(, l) ç å a a w'( l ) çè ø = = a Echagig summatio order = b= k = åå f (, l ) f (, l )() f d a a w'( l ) = a = = b = k ò a a w'( l ) = a = d( -) = å f (, l ) f (, l )() f d 8

29 Gauge Istitute Joural H. Vic Dao Therefore,. The Hyper-real Delta Fuctio Epasio i o-ormalized eige-fuctios is = k d( - ) = å f (, l ) f (, l ) a a w'( l ) = That is,. The Hyper-real Delta Fuctio is the ifiite sequece k a a a a '( ) w'( l ) w l k f (, l ) f (, l ) f (, l ) f (, l ). 9

30 Gauge Istitute Joural H. Vic Dao. Gree s Fuctio Epasio i No-Normalized Eige-fuctios Gree s Fuctio is defied by DG(,, l) + [ l- q ()] G(,, l) = d( - ) with the Boudary coditios at the itervals edpoits. Substitutig Delta from., k DG(,, ) [ q ()] G(,, ) (, ) (, ) = l + l- l = å f l f l a a = w'( l ) Thus, Gree s Fuctio must have a epasio i f (, l ) f (, l ), = G (,, l) = å a() lf(, l) f(, l). = a a a a To determie the a ( l ), we substitute the epasio ito the Sturm-Liouville equatio for Gl (,, ), k (){ l f "(, l ) [ l ()] f (, l )} f (, l ) f (, l ) f (, l ) = = åa + - q = a a a å a = = w'( l ) -[ l-q ( )] f (, ) a l [ l-l] fa (, l) k ()[ l l l ] f (, l ) f (, l ) f (, l ) f (, l ) = = a - = a a a a = = w'( l ) å å, Equatig the coefficiets, for each =,,, 3,... a 3

31 Gauge Istitute Joural H. Vic Dao k a ( l)[ l - l ] =, w'( l ) a ( l) = k l - l w'( l ).. The Hyper-real Gree Fuctio is = k G (,, l) = å f(, l) f(, l). a a l - l w'( l ) = That is,. The Hyper-real Gree Fuctio is the ifiite sequece k k f (, l ) f (, l ) f (, l ) f (, l ) l -l w l l -l w l = a a a a '( ) '( ) =..3 The Hyper-real yl (, ) for No-Normalized Eige-fuctios = b y (, l) = ò G (,, l)() f d = a Proof: We substitute it ito the Sturm-Liouville equatio = b "(, l) (,, l) ( ) = a y = ò D G f d, = b y"(, l) + [ l - q()](, y l) = ò { D G(,, l) + [ l -q()] G(,, l)}() f d = a d( -) = f ( ). 3

32 Gauge Istitute Joural H. Vic Dao. Gree Fuctio for Sie Sturm- Liuoville y"( ) + ly( ) = & a = b = Two idepedet solutios are cos l, ad si l. For a =, f l l (, ) =- si ( -a) l satisfies the Boudary coditios ad f (, a l ) = si =, f '( a, l ) =- cos =-. For b =, c l l satisfies the Boudary coditios (, ) = si ( b -) l ad c (, b l ) = si =, c '( b, l ) =- cos =-, Therefore, 3

33 Gauge Istitute Joural H. Vic Dao si l( -a) si l( b -) - wl ( ) = l l -cos l( -a) -cos l( b -) = si l( b - a). l Hece, the zeros of wl ( ) are l æ p ö = çèb - a ø, =,,3,... d w'( l) = si ( - ) l dl { l b a } { écos ( ) b a l b a ù - l si l( b a) } = l ê ë úû l l w'( l ) = ( b -a)cos l ( b -a) l = - - ). ( b a)( l p. k = c (, l ) f (, l ) si l ( b - ) =- si l ( - a) si =- si b- b-a -a b-a p p -a si[ p - p] b-a =- -a si p b-a 33

34 Gauge Istitute Joural H. Vic Dao -a -a = {- si( p)cos p + cos( p)si p } -a b-a b-a si p b-a = (-). (-) Therefore,. The Hyper-real Gree Fuctio i [ ab], = æ -a a G (,, l) sip ö æ - sip ö = å b -a ç = l è b -a ø èç b a - - ø p ( ) b-a = k Proof: G (,, l) = å f(, l) f(, l) a a l - l w'( l ) = = = å (-) (-) p ( ) b-a = l - b-a l si l ( -a) si l ( -a) l l Namely, = æ -aö æ -a = å si p si p ö = l b -a çè b -a ø èç b a - - ø. p ( ) b-a. The Hyper-real Gree Fuctio i [ ab], is the ifiite sequece æ aö æ aö si - p - si p +... b -a çè b -a ø çè b a - - ø p ( ) l b-a æ -aö æ -aö + si p si p ç ç l - = p ( ) è b -a ø è b -a ø b- a = 34

35 Gauge Istitute Joural H. Vic Dao 3. Gree Fuctio for Cosie Sturm- Liuoville y"( ) + ly( ) = & a = b = p Similarly to the former sectio, for a = b = p, we obtai 3. The Hyper-real Gree Fuctio i [ ab], ì = ü a a G (,, l) ï æ - cosp ö æ - cosp ö = í + ý ï b a å - l = p l ( ) ç b a ç b a - è - ø è - ø ïî b-a ïþ Namely, 3. The Hyper-real Gree Fuctio i [ ab], is the ifiite sequece æ -aö æ -aö cos p cos p b -a l ç l b -a ç b a - - = p ( ) è ø è ø b- a = 35

36 Gauge Istitute Joural H. Vic Dao 4. Gree s Fuctio for Bessel Sturm-Liuoville y"( ) ly( ) & + = a = For a =, si f (, l) =- satisfies the Boudary coditios l l ad f (, a l ) = si =, f '( a, l ) =- cos =-. The solutio c l = l - b + l - b (, ) si ( b )cos cos[ ( b )]si b l satisfies the Boudary coditios ad c (, b l ) = sib, b c '( b, l ) =- cos b, b Therefore, at = b, wl ( ) - si lb si b l = - cos lb - cos b 36

37 Gauge Istitute Joural H. Vic Dao Hece, the zeros of wl ( ) are the roots of That is, si lbcos b =-cos lbsi b. l ta l b =- l ta b. [Titchmarsh, p. 7] obtais Therefore, w'( l ) = bcos bcos( b l ){ + l ta b + ta b} k l b = cos bcos( b l ){ + l ta b } 4. The Gree Fuctio epasio i [ ab], is + l ta b G (,, l) = si( l)si( l) b l l l b b = å = - + ta + ta b Proof: = k G (,, l) = å f(, l) f(, l). a a l - l w'( l ) = Namely, 4. The Hyper-real Gree Fuctio i [ ab], is the ifiite sequece = j= + l ta b j å si( l )si( l ) j j j = l - l + l ta b + tab j b = b 37

38 Gauge Istitute Joural H. Vic Dao 5. Gree s Fuctio for Bessel Sturm Liuoville - 4 u"( ) + ( l - ) u( ) = & a = b = Put u ( ) = y ( ), to obtai Bessel s y"( ) + y'( ) + ( l - ) y( ) =. Two idepedet solutios to Bessel s equatio are J ( l), ad Y ( l ). Two idepedet solutios to - 4 u"( ) + ( l - ) u( ) = are For a =, J ( l), ad Y ( l ). f (, l) = p a{ J ( l) Y ( a l) - Y ( l) J ( a l) } satisfies the Boudary coditios ad For b =, f (, a l ) = si =, f '( a, l ) =- cos =-. c (, l) = p b{ J ( l) Y ( b l) - Y ( l) J ( b l) } satisfies the Boudary coditios 38

39 Gauge Istitute Joural H. Vic Dao ad c (, b l ) = si =, c '( b, l ) =- cos =-, [Titchmarsh, p.8] obtais wl ( ) f (, l) c (, l) = f (, l) c (, l) = p ab J a ly b l -Y a l J b l. { ( ) ( ) ( ) ( )} ì a J '( a l) b J '( b l) ü ab ìj ( b l) J ( a ) w'( l) ï ï w( l) ï l = ü ï í ý í ïýïï ï l J ( b l) l J ( b l) l î J ( a l) J ( b l) ïþ ïî þ ab ì J ( b l ) J ( a l ü w'( l ) ï - ) =- í ï ý l ï J ( a l ) J ( b l ) î ïþ k b J ( b l ) =, a J ( a l ) l are the zeros of wl ( ). 5. The Hyper-real Gree Fuctio i [ ab], is G (,, l) p l J ( b l ) = = å = l - l J ( a l )- J ( b l ) { J ( l ) Y ( a l ) Y ( l ) J ( a l ) } - { J ( l ) Y ( a l ) Y ( l ) J ( a l ) } - 39

40 Gauge Istitute Joural H. Vic Dao Proof: = k G (,, l) = å f(, l) f(, l). a a l - l w'( l ) = Namely, 5. The Hyper-real Gree Fuctio i [ ab], is the ifiite sequece p lj ( b l ) ( ) ( ) j= j j å j = l - l J a l - J b l j j { J ( l ) Y ( a l ) Y ( l ) J ( a l ) j j j j } - { J ( l ) Y ( a l ) Y ( l ) J ( a l ) j j j j } = - = 4

41 Gauge Istitute Joural H. Vic Dao 6. Delta Epasio i Orthoormal Eige-fuctios The Hyper-real eige-fuctios of a Hyper-real Sturm-Liouville problem over the iterval with edpoits a, ad b, ca be ormalized so that y ( ), y ( ), y ( ), = b ò y ( ) y ( ) d = d. m m = a The, a hyper-real fuctio f ( ) may be epaded i them by = æ = b ö f () = f() y() d y() = ç çè = a ø å ò. = = b = åå = = a Echagig summatio order, f y() dy() = b ì= ü = ï y () y () ï åí f() ý = a å d ïî = ïþ = b = = ò å y () y ()() f d. = a = d( -) 4

42 Gauge Istitute Joural H. Vic Dao Therefore, 6. The Hyper-real Delta Fuctio epasio i orthoormal Sturm-Liuoville eige-fuctios is = d( - ) = å y ( ) y ( ). = That is, 6. The Hyper-real Delta Fuctio is the ifiite sequece y ( ) y ( ) + y( ) y( ) y ( ) y ( ). 4

43 Gauge Istitute Joural H. Vic Dao 7. Gree s Fuctio Epasio i Orthoormal Eige-fuctios Gree s Fuctio is defied as the solutio to Sturm-Liuoville problem. Namely, the Sturm-Liouville equatio DG(,, l) + [ l- q ()] G(,, l) = d( - ) with the Boudary coditios at the itervals edpoits. That is, = DG(,, l) + [ l- q ()] G (,, l) = y() y() å = Hece, Gree s Fuctio must have a epasio i y ( ) y ( ), = G (,, l) = å a() ly() y(). = To determie the a ( l ), we substitute ito the Sturm-Liouville equatio for G(,, l), = = å a ( l){ y "( ) + [ l - q( )] y ( )} y ( ) = y ( ) y ( ) = = -[ l -q ( )] y ( ) [ l-l ] y ( ) å = = åa ()[ l l - l ] y () y () = () () å y y, = = 43

44 Gauge Istitute Joural H. Vic Dao Equatig the coefficiets, for each =,,, 3,... a ( l)[ l - l ] =, a ( l ) = l - l. 7. The Hyper-real Gree Fuctio is = G (,, l) = å y() y(). l - l = That is, 7. The Hyper-real Gree Fuctio is the ifiite sequece y ( ) y ( ) + y( ) y( ) y ( ) y ( ). l l l l l l = = 7.3 The Hyper-real yl (, ) for Orthoormal Eige-fuctios Proof:.3. = b y (, l) = ò G (,, l)() f d = a 44

45 Gauge Istitute Joural H. Vic Dao 8. Gl (,, ) Aalytic Cotiuatio i l ito the Comple Plae 8. Fi, ad, cosider l i the comple plae, ad let = G (,, z) = å y() y() z - l = be the Hyper-Comple Aalytic Cotiuatio of Gl (,, ) ito the z Hyper-comple plae. Let g be a cotour that ecloses all the l s. ò The G (,, z) dz= pd i( -) g First Proof: (that uses the Residue Theorem) By Sturm-Liuoville Oscillatio Theorem [Ic..6], l are the simple poles of Gz (,, ) o the positive real lie. Applyig the Residue Theorem = ò G (,, z) dz= piå Res { G (,, z) }. z= l g = Sice the Residue of Gz (,, ) at each l is y ( ) y ( ), = = piå y ( ) y ( ) = 45

46 Gauge Istitute Joural H. Vic Dao = pd i ( - ). Sice we do ot supply here a proof of the Residue Theorem, we shall give a direct proof: Secod Proof (that uses the Circular Comple Delta) g = G (,, z) dz= ò å y() y() dz z - l ò g = = = å y ( ) y ( ) d ò z. z - l = g Now, by [Da], pi z - l is the Hyper-comple Circular Delta Fuctio, ad Therefore, ò dz =, for each pi z - l g =,, 3,... g = G (,, z) dz= å y() y() d ò z = z - l g ò = = piå y ( ) y ( ) = = pd i ( - ). pi 46

47 Gauge Istitute Joural H. Vic Dao 9. Gree Fuctio for Hermit Sturm- Liouville Put u"( ) + ( l - ) u( ) =, real - u ( ) = e y ( ), ad obtai Hermit s equatio y"( ) + y'( ) + ( l - ) y( ) =, The, the eige values are l = +, =,,, 3,.. ad the correspodig eige fuctios are Hermit Polyomials of degree, H ( ). Therefore, - e H ( ) solve u"( ) + ( l - ) u( ) =. [Titchmarsh, p. 75] shows that 9. The Normalized eige fuctios are Therefore, - y ( ) = e H ( ).! p 47

48 Gauge Istitute Joural H. Vic Dao 9. = - - G (,, l) = e e å H() H() p = ( l - -)! = Proof: G (,, l) = å y() y(). l - l = Namely, 9.3 The Hyper-real Gree Fuctio i real, ad is the ifiite sequece - - e e H ( ) H ( ) H ( ) H ( ) p ( l - ) ( l - -)! = = 48

49 Gauge Istitute Joural H. Vic Dao. Gree Fuctio for Legedre Sturm-Liouville u "( q) + [ l + ] u( q) =, 4 cos q - p < q < p Legedre s equatio with m = is The eige values are (- ) y"( ) + y'( ) + ( l - ) y( ) =. 4 l = ( + ), =,,, 3,... ad the correspodig eige fuctios are Legedre Polyomials of degree, Put to obtai Put to obtai P ( ). = si q, y"( q) - y'( q)ta q + ( l - ) y( q) =. u 4 y() q = u() q, cos q "( q) [ l ( ta q)] u( q) =. Substitutig cos q = + ta q, 49

50 Gauge Istitute Joural H. Vic Dao Therefore, u"( q) + [ l + ] u( q) =. 4 cos q cos qp (si q ) solve u"( q) + ( l + ) u( q) =. [Titchmarsh, p. 79] shows that. The Normalized eige fuctios are 4cos y () q = + cos qp (si q). q Therefore,. for - p < q ad j < p G(, qjl, ) = cosq cos j ( + ) (si q) (si j) = å P P = l -( + ) Namely,.3 The Hyper-real Gree Fuctio i - p < q, j < p is the ifiite sequece cos q cos j P(si q) P(si j) ( + ) P (si q) P (si j) l = l ( ) 4 = 5

51 Gauge Istitute Joural H. Vic Dao. Gree Fuctio for Legedre Sturm-Liouville u "( q ) + [ l + ( - m ) ] u ( q ) =, - p < q < p 4 cos q Legedre s equatio with for m =,,, 3..., is (- ) y "( ) + y '( ) + [ l - - m ] y ( ) =. The eige values are 4 - m l = ( - m + ), = m,m +,... ad the correspodig eige fuctios are Legedre Fuctios Put to obtai Put to obtai P ( ). m = si q, y"( q) - y'( q)ta q + [ l - - m ] y( q) =. 4 cos q y() q = u() q, cos q u"( q) + [ l + ( + ta q) - m ] u ( q) =. 4 4 cos q Substitutig cos q = + ta q, 5

52 Gauge Istitute Joural H. Vic Dao Therefore, u"( q) + [ l + ( - m ) ] u( q) =. 4 cos q m cos q (si q ) solve u"( q) + [ l + ( - m ) ] u( q) =. P [Titchmarsh, p. 8] shows that. The Normalized eige fuctios are 4 cos q Therefore, ( - m)! m y () q = cos (si ) - m + q P q.!. For - p < q j <, = m p, ad for m =,,, 3..., G(, qjl, ) = cosq cosj = ( - m)! ( m ) P m (si ) m å - + q P (si j ) l -( - m + )! Namely,.3 The Hyper-real Gree Fuctio i - p < q, j < p is the ifiite sequece cos q cosj k= ( k - m)! ( ) m (si ) m å k - m + P q k P k (si j ) l -( - m + ) k! k= m = = m 5

53 Gauge Istitute Joural H. Vic Dao. Gree Fuctio for Bessel Sturm- Liouville y"( ) + [ l - ( - )] y( ) =, 4 < < b Two idepedet solutios are J ( l), ad Y ( l ) By [Titchmarsh, p. 85], J ( b l) J ( l) y (, l) = p J( )() f d ò l J ( b l) Y ( b l) Y ( l) = = Therefore,. + = b J ( b l) J ( l) p J ( l) f ( ) d ò J ( b l) Y ( b l) Y ( l) = G (,, l) p = í J ( b l) ïïïï ìï J ( b l) J ( l) J ( l), ï Y ( b l) Y ( l) ï J ( b l) J ( l) J ( l), ³ ï Y ( b l) Y ( l) ïî 53

54 Gauge Istitute Joural H. Vic Dao Alteratively, we may epad Gl (,, ) i eige fuctios: For ³ By [Titchmarsh, p. 8],. The eige-values l are the zeros of J ( b l ), =,, 3,... ad the ormalized eige-fuctios are Therefore, bj.3 For ³, i < < b b = - J b J ( l ). '( b l ) = G (,, l) = å J( l) J( l), ³ ( l l ) '( l ) Namely,.4 For ³ the Hyper-real Gree Fuctio i <, < b is the ifiite sequece k= å J ( l ) J ( l ) = k k b k = ( l - l ) J '( b l ) k = For < <, ¹ 54

55 Gauge Istitute Joural H. Vic Dao By [Titchmarsh, p. 83],.5 The eige-values l are the zeros of cj ( b l) - l J ( b l), where c = c ost. - r ì - c l J '( b l) - l J '( b l) ü - =- Re sï í l + ï ý - ï c l J ( b l) - l J ( b l) b î ïþ - l= l ad the ormalized eige-fuctios are Therefore, p b - r J ( b l ){ cl J ( l )- J ( l ) }. - sip < < ¹, the i < < b.6 If, p b G = r J b = (,, l) ( ) å l 4si p = l - l Namely,.7 If - - { cl J ( l ) J ( l )}{ cl J ( l ) J ( l ) - - } - - < <, ¹, the the Hyper-real Gree Fuctio i <, <b is the ifiite sequece p b 4si p k= - å r J ( b l ){ cl J ( l )- J ( l ) k k k k - k } l - l k = 55

56 Gauge Istitute Joural H. Vic Dao - { cl J ( l ) J ( l ) k k - k } - = =.8 If < <, ¹, the For c = ifiite hyper-real, the epasio is i J : p b G c r J b J J = - (,, l) = ( ) ( ) ( ) å l l l l 4si p = l - l For c = ifiitesimal, the epasio is i J : - p b G r J b J J = (,, l) = ( ) ( ) ( ) å l l l - - 4si p = l - l 56

57 Gauge Istitute Joural H. Vic Dao 3. Gree Fuctio for Bessel Sturm Liouville y"( ) + [ l - ] y( ) =, < <, with a = By [Titchmarsh, p. 9], 3. The eige-values l are the zeros of l J 3 l - 3 J ( ) + ( ), =,, 3,... ad the ormalized eige-fuctios are - l é J ( l ) J 3 ( l ) ù ê ë ú û Therefore, { ( [ ]) ( [ ]) 3-3 } l - J l - + J l - 3. The Hyper-real Gree Fuctio i < < is G (,, l) = l - l = å l l - = - l é J ( l ) ( l ) ù êë - J 3-3 { J ( [ l ] ) J 3 ( [ l ] ) - 3 } { J ( [ l ] ) J 3 ( [ l ] ) - 3 } úû 57

58 Gauge Istitute Joural H. Vic Dao Namely, 3.3 The Hyper-real Gree Fuctio i <, < is the ifiite sequece k= å l - l - k k k = l - l l é J ( ) ( ) l k l ù k êë - J 3-3 úû { J ( [ l ]) J 3 ( [ l ]) k - 3 k } { J ([ l ]) J 3 ([ l ]) k - 3 k } = = 58

59 Gauge Istitute Joural H. Vic Dao 4. Gree Fuctio for Bessel Sturm- Liouville y"( ) [ m ] y( ), + - = < < with a = p By [Titchmarsh, p. 9], 4. The eige-values m are the zeros of J ( )- ( ), =,, 3,... m J 3 m - 3 ad the ormalized eige-fuctios are - m é J ( m ) J 3 ( m ) ù ê ë ú û Therefore, { ([ ]) 3 ([ ]) - 3 } m - J m - + J m - 4. the Hyper-real Gree Fuctio i < < is G (,, m) = m - m = å - = m - m m é J ( ) ( ) m m ù êë + J 3-3 { J ( [ m ] ) J 3 ( [ m ] ) - 3 } { J ( [ m ] ) J 3 ( [ m ] ) - 3 } úû 59

60 Gauge Istitute Joural H. Vic Dao Namely, 4.3 the Hyper-real Gree Fuctio i <, < is the ifiite sequece k= å m - m - k k k = m - m m é J ( ) ( ) m k m ù k êë + J 3-3 úû { J ([ m ]) J 3 ([ m ]) k - 3 k } { J ( [ m ]) J ( [ m ]) 3 k - 3 k } = = 6

61 Gauge Istitute Joural H. Vic Dao 5. Gree s Fuctio Itegral Epasio i Eige-fuctios If the eige-values are oly ifiitesimally separated from each other, the series summatio of eige-fuctios over discrete eige values is replaced by Hyper-real itegratio over the cotiuum of eige-values. The, the epasio is represeted by a itegral. Titchmarsh applied the Residue Theorem to obtai the projectios of a fuctio f ( ) o the eige-fuctios of Sturm-Liouville problems, with cotiuous spectrum of eige-values. This yields the Itegral epasio of the Delta Fuctio i Sturm- Liuoville eige-fuctios. The Delta Fuctio Epasio determies the Gree s Fuctio itegral epasio i Sturm-Liuoville eige-fuctios. 6

62 Gauge Istitute Joural H. Vic Dao 6. Gree s Fuctio Itegral Epasio i Cosie Fuctios of y"( ) + ly( ) =, < < We may solve Gree s Equatio by Fourier Trasformig it DG(,, l) + lg(,, l) = d( -) DG(,, l) + lg (,, l) = d( -) = ( w) ˆ( w,, l) ˆ( w,, l) d( ) i G G - e d = e ò =- -iw -iw ( ) ˆ(,, ) ˆ -i iw G wl + lg( wl,, ) = e w, ˆ(,, ) -i G wl = e w, l - w The, Iverse Trasformig it, w= ì ü -iw iw G (,, l) = ï e ïe dw p ò í ý ïîl - w ïþ w=- 6

63 Gauge Istitute Joural H. Vic Dao = w= iw( -) ò e dw. - w=- p l w That is, 6. The Hyper-real Gree Fuctio is w= iw( -) G (,, l) = e d p ò l - w w=- w Sice Fourier Trasformig applies oly to Liear Differetial Equatios with costat coefficiets, we demostrate a method that applies to equatios with variable coefficiets: The eige-values l º w, are the iterval of hyper-real positive umbers (, ). By [Titchmarsh, p. 7], the Hyper-real fuctio f ( ) is give for ay hyper-real by = l= f ( ) = cos( l)cos( l) dlf( ) d + l p ò ò =- l= ì ï = + p ïî = l= + si( l)si( l) dl ( ) l p ò ò f d =- l= = l= ï {cos( l)cos( l) si( l)si( l)} dlï ò í ý ( ) ò f d l =- ï l= cos[( -) l] ï üï ïþ 63

64 Gauge Istitute Joural H. Vic Dao = w= = ò cos w( ) dwf( ) d p ò - =- w= d( -) Hece, the Hyper-real Delta i - <, < is w= d( - ) = ò cos w( -) dw p w= ( iw( -) -iw( -) e + e ) w= = p ò p Sice Gree s Fuctio satisfies w=- e iw( -) dw w= iw( -) = å e dw. w=- DG(,, l) + lg (,, l) = d( - ), i ( ) the Hyper-real Gree Fuctio must have a epasio i - : l = å w w. p w= iw( -) G (,, ) a( ) e d w=- Substitutig i the Gree s Fuctio Differetial Equatio å a( w){ - w + l} e dw = å e p p w= w= iw( -) iw( - w=- w=- ) dw. e w Equatig each term, a( w){ - w + l} =, a( w) = l - w 64

65 Gauge Istitute Joural H. Vic Dao å w, w= iw( -) G (,, l) = e d p w=- l - w = w= iw( -) ò e dw. - w=- p l w 65

66 Gauge Istitute Joural H. Vic Dao 7. Gree s Fuctio Itegral Epasio for Weber Equatio y"( ) + [ l - ( - )] y( ) =, a < < a > 4 The eige-values - l º s, are the iterval of hyper-real egative umbers (-,). By [Titchmarsh, p. 87], the Hyper-real fuctio f ( ) is give for hyper-real a < < by = s = f ( ) = J( sy ) ( as) -J( asy ) ( s) ( ) ( ) J as + Y as = a s= { } ò ò { } J ( sy ) ( as) -J ( asy ) ( s) sds f( ) d Therefore, the Hyper-real Delta i a <, < is s= J ( sy ) ( as) - J ( asy ) ( s) d( - ) = J ( sy ) ( as) -J ( asy ) ( s) sds ò s= J ( as) + Y ( as) Sice Gree s Fuctio satisfies { } 66

67 Gauge Istitute Joural H. Vic Dao - 4 G (,, l) + [ l- ] G (,, l) = d( - ), the Hyper-real Gree Fuctio must have a epasio i J ( sy ) ( as) - J ( as) Y ( s) y (,, s) = J ( s) Y ( as) -J ( as) Y ( s) { } J ( as) + Y ( as). Substitutig s = G (,, l) = å as (){ y(,,)} s sds s = i the Gree s Fuctio Differetial Equatio, - as (){ y [ l ] y } sds y (,, ssds ) s= s= = s= s= å å. Now, s is the eige-value associated with J ( s), with Y ( s) with J ( s) Y ( as) - J ( as) Y ( s), ad thus with y (,,) s. Hece, - 4 y = -[ s - ] y. Substitutig i the Gree s Fuctio differetial Equatio s= s= å å as (){ l - s} y sds= y (,, ssds ) s= s=, Equatig each term, as (){ l - s} =, as () = l - s 67

68 Gauge Istitute Joural H. Vic Dao s = G (,, l) = å y(,,) ssds. l - s s = 7. s = J ( sy ) ( as) - J ( asy ) ( s) G(, l, ) = ò { J ( s) Y ( as) -J ( as) Y ( s) } sds l - + s = s J ( as) Y ( as) 68

69 Gauge Istitute Joural H. Vic Dao 8. Gree s Fuctio for Hakel s y"( ) + [ l - ( - )] y( ) =, 4 < < By [Titchmarsh, p. 88], = () ( ) =- p ( l) ( ) ( ) ò l = y i H J f d Therefore, - = () ip J ( l) H ( ) f( ) d ò l = 8. G (,, l) ìï () H ( l) J ( l), ip =- ï í ïïï () H ( l) J ( l), ³ ïî 69

70 Gauge Istitute Joural H. Vic Dao 9. Gree s Fuctio for Bessel s y"( ) + [ l + )] y( ) =, - < < By [Titchmarsh, p. 93], () () y (, l) =- ip + lh ([ + l]) + lh ([ + l])() f d = 6 =- ò Therefore, 9. = () () - ip + lh ( [ + l] ) + lh ( [ + l] ) f( ) d 6 = ò ìï () () H ([ + l]) H ([ + l]), ip G (,, l) =- + l + lï í 6 ïïï () () ïh ( [ + l]) H 3 ( [ + l]), 3 ïî ³ 7

71 Gauge Istitute Joural H. Vic Dao 3. Gree s Fuctio for Bessel s y"( ) + [ l + e )] y( ) =, - < < By [Titchmarsh, p. 95], ip y (, l) =- { J ( e) + J ( e)} J ( e) f( ) d i l -i l ò -i l sih( p l) = =- Therefore, = ip - J ( e ) { J ( e ) J ( e )} f( ) d -i l ò + i l -i l sih( p l) =- 3. G (,, l) ìï { J ( e ) + J ( e )} J ( e ), i l -i l -i l ip =- ï í sih( p l) ïï { ( ) ( )} ( ï J e + J e J e i i i ), ³ ïî l - l - l 7

72 Gauge Istitute Joural H. Vic Dao 3. Gree s Fuctio for Bessel s y"( ) + [ l - e )] y( ) =, - < < By [Titchmarsh, p. 96], = = ò ò y(, l) =-K ( e ) I ( e )() f d -I ( e ) K ( e )() f d Therefore, i l -i l -i l i l =- =- 3. G (,, l) ìï K ( e ) I ( e ), i l -i l =-í ï ïï ïk ( e ) I ( e ), ³ ïî i l -i l 7

73 Gauge Istitute Joural H. Vic Dao Refereces [Abramowitz] Abramowitz, M., ad Stegu, I., Hadbook of Mathematical Fuctios with Formulas Graphs ad Mathematical Tables, U.S. Departmet of Commerce, Natioal Bureau of Stadards, 964. [Da] Dao, H. Vic, Well-Orderig of the Reals, Equality of all Ifiities, ad the Cotiuum Hypothesis i Gauge Istitute Joural Vol. 6 No., May ; Well-Orderig of the Reals, Equality of all Ifiities, ad the Cotiuum Hypothesis [Da] Dao, H. Vic, Ifiitesimals i Gauge Istitute Joural Vol.6 No. 4, November ; Ifiitesimals [Da3] Dao, H. Vic, Ifiitesimal Calculus i Gauge Istitute Joural Vol. 7 No. 4, November ; Ifiitesimal Calculus [Da4] Dao, H. Vic, Riema s Zeta Fuctio: the Riema Hypothesis Origi, the Factorizatio Error, ad the Cout of the Primes, i Gauge Istitute Joural of Math ad Physics, Vol. 5, No. 4, November 9. Riema Zeta Fuctio: the Riema Hypothesis Origi, the Factorizatio Error, ad the Cout of the Primes [Da5] Dao, H. Vic, The Delta Fuctio i Gauge Istitute Joural Vol. 8, No., February, ; The Delta Fuctio [Da6] Dao, H. Vic, Riemaia Trigoometric Series, Gauge Istitute Joural, Volume 7, No. 3, August. Riemaia Trigoometric Series 73

74 Gauge Istitute Joural H. Vic Dao [Da7] Dao, H. Vic, Delta Fuctio the Fourier Trasform, ad the Fourier Itegral Theorem i Gauge Istitute Joural Vol. 8, No., May, ; Delta Fuctio, the Fourier Trasform, ad Fourier Itegral Theorem [Da8] Dao, H. Vic, Ifiite Series with Ifiite Hyper-real Sum i Gauge Istitute Joural Vol. 8, No. 3, August, ; Ifiite Series with Ifiite Hyper-real Sum [Da9] Dao, H. Vic, Sturm-Liuoville Epasios of the Delta Fuctio Sturm-Liouville Epasios of the Delta Fuctio [Da] H. Vic Dao, Comple Delta Fuctio, Comple Fourier Trasform, ad Fourier Itegral Theorem. Comple Delta Fuctio, Comple Fourier Trasform, ad Fourier Itegral Theorem [Ferrers] Ferrers, N., M., A Elemetary treatmet o Spherical Harmoics, Macmilla, 877. [Gradshtey] Gradshtey, I., S., ad Ryzhik, I., M., Tables of Itegrals Series ad Products, 7 th Editio, edited by Alla Jeffery, ad Daiel Zwilliger, Academic Press, 7 [Hardy] Hardy, G. H., Diverget Series, Chelsea 99. [Jackso] Jackso, Duham, Fourier Series ad Orthogoal Polyomials, Mathematical associatio of America, 94. [Magus] Magus, W., Oberhettiger, F., Soy, R., P., Formulas ad Theorems for the Special Fuctios of Mathematical Physics Third Editio, Spriger-Verlag, 966. [Sasoe] Sasoe, Giovai, Orthogoal Fuctios, Revised Editio, Krieger,

75 Gauge Istitute Joural H. Vic Dao [Spiegel] Spiegel, Murray, Mathematical Hadbook of formulas ad tables Schaum s Outlie Series, McGraw Hill, 968. [Spaier] Spaier, Jerome, ad Oldham, Keith, A Atlas of Fuctios, Hemisphere, 987. [Szego] Szego, Gabor, Orthogoal Polyomials Revised Editio, America Mathematical Society,959. [Szego4] Szego, Gabor, Orthogoal Polyomials Fourth Editio, America Mathematical Society,975. [Titchmarsh], E. C. Titchmarsh, Eigefuctio Epasios Associated with Secod-order Differetial Equatios, Part I, Secod Editio, Oford, 96. [Weisstei], Weisstei, Eric, W., CRC Ecyclopedia of Mathematics, Third Editio, CRC Press, 9. 75