Sturm-Liouville Expansions. of the Delta Function

Transcription

1 Gauge Istitute Joural H. Vic Dao Sturm-Liouville Expasios of the Delta Fuctio H. Vic Dao May, 04 Abstract We expad the Delta Fuctio i Series, ad Itegrals of Sturm-Liouville Eige-fuctios. Keywords: Sturm-Liouville Expasios, Ifiitesimal, Ifiite- Hyper-Real, Hyper-Real, ifiite Hyper-real, Ifiitesimal Calculus, Delta Fuctio, Fourier Series, Laguerre Polyomials, Legedre Fuctios, Bessel Fuctios, Delta Fuctio, 000 Mathematics Subject Classificatio 6E5; 6E0; 6E5; 6E0; 6A06; 6A; 0E0; 0E55; 0E7; 0H5; 46S0; 97I40; 97I0.

2 Gauge Istitute Joural H. Vic Dao Cotets 0. Eige-fuctios Expasio of the Delta Fuctio. Hyper-real lie.. Hyper-real Fuctio. Itegral of a Hyper-real Fuctio 4. Delta Fuctio 5. Coverget Series 6. Hyper-real Sturm-Liouville Problem 7. Delta Expasio i No-Normalized Eige-Fuctios 8. Fourier-Sie Expasio of Delta Associated with y"( x) + ly( x) = 0 & a = b = 0 9. Fourier-Cosie Expasio of Delta Associated with y"( x) + ly( x) = 0 & a = b = p 0. Fourier-Sie Expasio of Delta Associated with y"( x) + ly( x) = 0 & a = 0. Fourier-Bessel Expasio of Delta Associated with - 4 u"( x) + ( l - ) u( x) = 0 & a = b = 0 x. Delta Expasio i Orthoormal Eige-fuctios. Fourier-Hermit Expasio of Delta Associated with u x x u x "( ) + ( l - ) ( ) = 0 for ay real x 4. Fourier-Legedre Expasio of Delta Associated with

3 Gauge Istitute Joural H. Vic Dao u"( q) + [ l + ] u( q) = 0, - p < q < p 4 cos q 5. Fourier-Legedre Expasio of Delta Associated with y"( q) + [ l + ( - m ) ] y( q) = 0, - p < q < p 4 cos q 6. Fourier-Bessel Expasio of Delta Associated with - 4 u"( x) + ( l - ) u( x) = 0 & 0 x < x < b 7. Fourier-Bessel Expasio of Delta Associated with y"( x) + ( l - x) y( x) = 0, 0 < x < & a = 0 8. Fourier-Bessel Expasio of Delta Associated with y"( x) + ( m - x) y( x) = 0, 0 < x < & a = p 9. Delta Expasio i a Itegral over a Cotiuum of Eige- Fuctios 0. Fourier-Cosie Itegral of Delta Associated with y"( x) + ly( x) = 0, 0 < x <. Weber Formula for Fourier-Bessel Itegral of Delta Associated with - 4 y"( x) + ( l - ) y( x) = 0, a < x <, a > 0 x. Hakel Formula for Fourier-Bessel Itegral of Delta Associated with - 4 y"( x) + ( l - ) y( x) = 0, 0 < x < x. Fourier-Bessel Itegral of Delta Associated with y"( x) + ( l + x) y( x) = 0, 0 x < 4. Fourier-Bessel Itegral of Delta Associated with

4 Gauge Istitute Joural H. Vic Dao x y"( x) + ( l + e ) y( x) = 0, - < x < 5. Fourier-Bessel Itegral of Delta Associated with x y"( x) + ( l - e ) y( x) = 0, - < x < Refereces 4

5 Gauge Istitute Joural H. Vic Dao 0. Eige-Fuctios Expasio of the Delta Fuctio Ulike Taylor s expasio of a fuctio that requires its derivatives of ay order, Fourier Itegral or Fourier Series represetatio require o derivatives. The, the fuctio s projectios o the orthogoal sequece of eige-fuctios sum up to the fuctio. It is little kow that the Fourier Series represetatio, ad the Fourier Itegral represetatio result from a expasio of the Delta Fuctio: For istace, the Fourier Itegral represetatio, x = ò f ( x) = f( xd ) ( x -x) d x x =- x= w= iw( x-x) f () x e dwd p x=- w=- = ò ò x w= x= = ò ò p w=- x=- -iwx iwx f () x e dxe d F( w) w Here, the Fourier Trasform i x F( w ) is the projectio of f ( x ) o e w. 5

6 Gauge Istitute Joural H. Vic Dao The eige-fuctio expasios of the Delta Fuctio are at the root of the eige-fuctio expasio of ay fuctio. 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, r,...) 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, i,...) costitutes a family of. The ifiitesimals are smaller tha ay real umber, yet strictly greater tha zero.. Their reciprocals (,,,...) i i i 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. 0. I particular, zero is separated from ay positive real by the ifiitesimals, ad from ay egative real by the ifiitesimals with egative sigs, - dx.. Zero is ot a ifiitesimal, because zero is ot strictly greater tha zero.. We do ot add ifiity to the hyper-real lie.. 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 () x 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 ( x ) 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. 0

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

12 Gauge Istitute Joural H. Vic Dao x= b ò f ( xdx ). x= a If the hyper-real is ifiite, the it is the itegral over [, ab], If the hyper-real is fiite, x= b ò fxdx ( ) = real part of the hyper-real. x= a. 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 ( xdx. ) The Lower Itegral is the Itegratio Sum where f ( x ) is replaced

13 Gauge Istitute Joural H. Vic Dao by its lowest value o each iterval [ x -, x + ] dx dx. å xî[ a, b] æ ö if f ( t) dx ç çèx- t x+ ø dx dx The Upper Itegral is the Itegratio Sum where f ( x ) is replaced by its largest value o each iterval [ x -, x + ] dx dx. å xî[ a, b] æ ö sup f ( t) dx ç èx- t x+ ø dx dx If the itegral is a fiite hyper-real, we have.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.

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 í0, ï ý. The ïî dx ïþ hyper-real 0 is the sequece 0, 0, 0,.... The ifiite hyperreal dx depeds o our choice of dx.. We will usually choose the family of ifiitesimals that is spaed by the sequeces,,, 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, dx will mea the sequece. Alteratively, we may choose the family spaed by the sequeces,,, The, 4 dx will mea the 4

15 Gauge Istitute Joural H. Vic Dao sequece. Oce we determied the basic ifiitesimal dx, we will use it i the Ifiite Riema Sum that defies a Itegral i Ifiitesimal Calculus.. The Delta Fuctio is strictly smaller tha 4. We defie, c d ( x) º dx ( ), dx x dx é ù, ê- ë úû where c é ê- ë dx, dx ì, x Î é-dx, dx ù ( x) = ï ê ú í ë û. ïï 0, otherwise î ù úû 5. Hece, for x < 0, d ( x) = 0 at for dx x =-, d ( x) jumps from 0 to dx, dx, dx ê ë ú û, ( x) x Î é- ù d =. dx at x = 0, d (0) = dx at dx x =, d ( x) drops from dx for x > 0, d ( x) = 0. to 0. xd ( x) = 0 6. If dx =, ( x) = ( x), ( x), ( x )... [-, ] [-, ] [-, ] d c c c If dx =, d ( x) =,,,... cosh x cosh x cosh x 5

16 Gauge Istitute Joural H. Vic Dao 8. If -x -x -x dx =, d( x) = e c [0, ), e c[0, ), e c[0, ),... x = ò 9. d( xdx ) =. x =- k = -ik( x -x ) 0. dx ( - x) = 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. 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. 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 yx ( ), - y"( x) + q( x) = ly( x), o a iterval that may be bouded, or may be the whole Hyper-real lie. The hyper-real fuctio qx ( ) 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 complex), allows the equatio with boudary coditios, at the iterval edpoits to become a eige-value problem: l is a eige-value, ad y ( x ) is the correspodig Hyper-real eige-fuctio iff - y "( x)( x) + q( x) y ( x) = l ( x) y. The eige-fuctios are orthogoal, over the iterval. x= b ò y ( x) y ( x) dx = 0, for ¹ m. m x= a 8

19 Gauge Istitute Joural H. Vic Dao 7. Delta Expasio 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 (, x l ), with f (, a l ) = sia, ad f '( a, l ) =-cosa, a a c (, x l ), with c (, b l ) = sib, ad c '( b, l ) =-cosb,, b which are etire fuctios of l. The, b a b d dx f ( x, l) c ( x, l) f '( x, l) c '( x, l) f ( x, l) c ( x, l) a b a b a b = + f '( x, l) c '( x, l) f '( x, l) c '( x, l) f "( x, l) c "( x, l) a b a b a b W [ f, c ] 0 a b = fc "-cf " a b b a = f ( q -l) c -c ( q -l) f a b b a = 0. Hece, the Wroskia W[ f a, c b ] is a fuctio of l aloe: W a b [ f, c ] = w( l). 9

20 Gauge Istitute Joural H. Vic Dao Now, if the oly zeros of wl ( ) are the simple zeros The, for each = 0,,,,... l,,, l,, 0 l l f (, x l ) c (, x l ) a b 0 = wl ( ) =. f '( x, l ) c '( x, l ) That is, for each = 0,,,,... there is a umber k so that a c (, x l ) = k f (, x l ). b Titchmarsh applied the Residue Theorem to obtai the coefficiets i the Sturm-Liouville expasio of f ( x ). Followig Titchmarsh, we coclude that the Hyper-real Sturm- a b Liouville expasio of a Hyper-real fuctio f ( x ) i the Hyperreal eige-fuctios fa (, x l ) is = æ x= b k ö fx ( ) = å f( xf ) ( xl, ) dx f( x, l) a a = 0 ç w'( l ) ò çè x = a ø = æ x = b k ö = å f() xf(, xl) dx f(, l) ç å a a w'( l ) x çè ø = 0 x = a Exchagig summatio order x = b= k = åå f (, x l ) f (, x l )() f x dx a a w'( l ) x = a = 0 x = b = k ò x a a 0 w'( l ) x = a = d( x -x) = å f (, l ) f (, x l )() f x dx 0

21 Gauge Istitute Joural H. Vic Dao Therefore, 7. The Hyper-real Delta Fuctio Expasio i o-ormalized eige-fuctios is = k d( x - x) = å f ( x, l ) f ( x, l ) a a w'( l ) = 0 That is, 7. The Hyper-real Delta Fuctio is the ifiite sequece k 0 a 0 a 0 a a '( ) w'( l ) 0 w l k f (, x l ) f (, x l ) f (, x l ) f (, x l ).

22 Gauge Istitute Joural H. Vic Dao 8. Fourier-Sie Expasio of Delta Associated with y"( x) + ly( x) = 0 & a = b = 0 Two idepedet solutios are cos l x, ad si l x. For a = 0, f l l (, x ) =- si ( x -a) 0 l satisfies the Boudary coditios ad 0 f (, a l ) = si0 = 0, 0 f '( a, l ) =- cos 0 =-. For b = 0, c l l satisfies the Boudary coditios (, x ) = si ( b -x) 0 l ad 0 c (, b l ) = si0 = 0, 0 c '( b, l ) =- cos 0 =-, Therefore,

23 Gauge Istitute Joural H. Vic Dao si l( x -a) si l( b -x) - wl ( ) = l l -cos l( x -a) -cos l( b -x) = si l( b - a). l Hece, the zeros of wl ( ) are l æ p ö = çèb - a ø, =,,,... 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 (, x l ) 0 f (, x l ) 0 si l ( b - x) =- si l ( x - a) si =- si b-x b-a x-a b-a p p x-a si[ p - p] b-a =- x-a si p b-a

24 Gauge Istitute Joural H. Vic Dao x-a x-a = {- si( p)cos p + cos( p)si p } x-a b-a b-a si p b-a = (-). 0 (-) Therefore, = k d( x - x) = å f ( x, l ) f ( x, l ) a a w'( l ) = = = å (-) b - a ( - ) l si l ( x -a) si l ( x -a) = l l = æ x -aö x a si p æ - ö = si p å b -a çè b -a ø çè b -a ø = 8. The Fourier Sie expasio of Delta Fuctio i [ ab], Namely, = æ x -aö x a d( x x) si p æ - si p ö - = å b a b a - çè - ø çè b -a ø = 8. The Hyper-real Delta i [ ab], is the ifiite sequece æ x -a si si x a x a x a p ö æ - p ö æ -... si p ö æ - ö + + si p b -a çè b -a ø çè b -a ø çè b -a ø çè b -a ø = = 4

25 Gauge Istitute Joural H. Vic Dao 9. Fourier-Cosie Expasio of Delta Associated with y"( x) + ly( x) = 0 & a = b = p Similarly to the former expasio, for a = b = p, we obtai 9. Fourier Cosie expasio of the Delta Fuctio i [ ab], ì = x a x a ü d( x x) ï æ - ö cos p æ - ö - = í + cos p ï ý b a å = b a - çè - ø çè b a ïî - ø ïþ Namely, 9. The Hyper-real Delta i [ ab], is the ifiite sequece æ x -aö æ a x a a cos cos x - ö æ - ö æ x - ö + p p cos p cos p b -a çè b -a ø çè b -a ø çè b -a ø çè b -a ø = = 5

26 Gauge Istitute Joural H. Vic Dao 0. Fourier-Sie Expasio of Delta Associated with y"( x) y( x) 0 + l = & a = 0 For a = 0, si f (, x l) =- satisfies the Boudary coditios 0 lx l ad 0 f (, a l ) = si0 = 0, 0 f '( a, l ) =- cos 0 =-. The solutio c l = l - b + l -x b (, x ) si ( b x)cos cos[ ( b )]si b l satisfies the Boudary coditios ad c (, b l ) = sib, b c '( b, l ) =- cos b, b Therefore, at x = b, wl ( ) - si lb si b l = - cos lb - cos b 6

27 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, = 0 w'( l ) = bcos bcos( b l ){ + l ta b + ta b} k l b = cos bcos( b l ){ + l ta b } = k d( x - x) = å f ( x, l ) f ( x, l ) a a w'( l ) = b + l ta b si( x l )si( x l ) l b b = å = 0 + ta + ta b 0. The Fourier Sie expasio of Delta i [ ab], Namely, + l ta b d( x - x) = å si( x l )si( x l ) b l b b = = 0 + ta + ta b 0. The Hyper-real Delta i [ ab], is the ifiite sequece b = j= + l ta b j å si( x l )si( x l ) j j j = 0 + l ta b + tab j b = 0 7

28 Gauge Istitute Joural H. Vic Dao. Fourier-Bessel Expasio of Delta Associated with - 4 u"( x) ( l ) u( x) = & a = b = 0 x Put ux ( ) = xyx ( ), ad obtai Bessel s equatio y"( x) + y'( x) + ( l - ) y( x) = 0. x x Two idepedet solutios to Bessel s equatio are J ( x l), ad Y ( x l ). Two idepedet solutios to - 4 u"( x) + ( l - ) u( x) = 0 are x xj ( x l), ad xy ( x l ). For a = 0, f ( x, l) = p ax{ J ( x l) Y ( a l) - Y ( x l) J ( a l) } 0 satisfies the Boudary coditios 0 f (, a l ) = si0 = 0, 8

29 Gauge Istitute Joural H. Vic Dao ad f '( a, l ) =- cos 0 =-. 0 For b = 0, c ( x, l) = p bx{ J ( x l) Y ( b l) - Y ( x l) J ( b l) } 0 satisfies the Boudary coditios ad 0 c (, b l ) = si0 = 0, 0 c '( b, l ) =- cos 0 =-, [Titchmarsh, p.8] obtais wl ( ) f (, x l) c (, x l) 0 0 = f (, x l) c (, x l) x 0 x 0 = 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 ) where l are the zeros of wl ( ). Therefore, = k d( x - x) = å f ( x, l ) f ( x, l ) a a w'( l ) = 0 9

30 Gauge Istitute Joural H. Vic Dao p l J ( b l ) = = xx å = 0 J ( a l )- J ( b l ) { J ( x l ) Y ( a l ) Y ( x l ) J ( a l ) } - { J ( x l ) Y ( a l ) Y ( x l ) J ( a l ) } -. The Fourier Bessel expasio of Delta i [ ab], p l J ( b l ) d( x - x) = x = x å = 0 J ( a l )- J ( b l ) { J ( x l ) Y ( a l ) Y ( x l ) J ( a l ) } - { J ( x l ) Y ( a l ) Y ( x l ) J ( a l ) } - Namely,. The Hyper-real Delta i [ ab], is the ifiite sequece lj ( b l ) j= j j xx å j = 0 J ( a l )- J ( b l ) j j p { J ( x l ) Y ( a l ) Y ( x l ) J ( a l ) j j j j } - { J ( x l ) Y ( a l ) Y ( x l ) J ( a l ) j j j j } = 0 - = 0

31 Gauge Istitute Joural H. Vic Dao. Delta Expasio 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 ( x), y ( x), y ( x), 0 x= b ò y ( x) y ( x) dx = d. m m x= a The, a hyper-real fuctio f ( x ) may be expaded i them by = æ x = b ö f () x = f() xy() xdx y() = 0 ç çè x = a ø å ò x. = x = b = åå = 0 x = a Exchagig summatio order, f xy() xdxy() x x = b ì= ü = ï y () x y () x ï åí f() x ý x = a å dx ïî = 0 ïþ x = b = = ò å y () x y ()() x f x dx. x = a = 0 d( x -x)

32 Gauge Istitute Joural H. Vic Dao Therefore,. The Hyper-real Delta Fuctio expasio i orthoormal Sturm-Liuoville eige-fuctios is = d( x - x) = å y ( x) y ( x ). = 0 That is,. The Hyper-real Delta Fuctio is the ifiite sequece y ( x) y ( x) + y( x) y( x) y ( x) y ( x). 0 0

33 Gauge Istitute Joural H. Vic Dao. Fourier-Hermit Expasio of Delta Associated with u"( x) + ( l - x ) u( x) = 0, for ay x real Put - x ux ( ) = e yx ( ), ad obtai Hermit s equatio y"( x) + xy'( x) + ( l - ) y( x) = 0, The eige values are l = +, = 0,,,,.. ad the correspodig eige fuctios are Hermit Polyomials of degree, Therefore, - x e H ( x) H ( x ). solve u"( x) + ( l - x ) u( x) = 0. [Titchmarsh, p. 75] shows that the Normalized eige fuctios are Therefore, - x y ( x) = e H ( x ).! p

34 Gauge Istitute Joural H. Vic Dao = d( x - x) = å y ( x) y ( x ) = 0 - x - x = å = e e H ( x) H ( x) p = 0!. The Fourier-Hermit expasio of Delta i realx, ad x = - x - x d( x - x) = e e å H ( x) H ( x) p = 0! Namely,. The Hyper-real Delta i real x, ad x is the ifiite sequece - x - x e e H () x H () x + H () x H ()... x + H () x H () x 0 0 p! = = 0 4

35 Gauge Istitute Joural H. Vic Dao 4. Fourier-Legedre Expasio of Delta Associated with u "( q) + [ l + ] u( q) = 0, p q 4 cos q - < < p Legedre s equatio with m = 0 is (- x ) y"( x) + xy'( x) + ( l - ) y( x) = 0. 4 The eige values are l = ( + ), = 0,,,,... ad the correspodig eige fuctios are Legedre Polyomials of degree, Put ad obtai Put ad obtai P ( x ). x = si q, y"( q) - y'( q)ta q + ( l - ) y( q) = 0. u 4 y() q = u() q, cosq "( q) [ l ( ta q)] u( q) =. Substitutig cos q ta q = +, 5

36 Gauge Istitute Joural H. Vic Dao u"( q) + [ l + ] u( q) = 0, 4 cos q Therefore, cos qp (si q ) solve u"( q) + ( l + ) u( q) = 0. [Titchmarsh, p. 79] shows that 4. The Normalized eige-fuctios are Therefore, = dq ( - j) = å y( qy ) ( j) = 0 4cos y () q = + cos qp (si q). = å = cos q cos j ( + ) P (si q) P (si j ) = 0 q 4. The Fourier-Legedre expasio of Delta i - p < q < p = å P P ) = 0 dq ( - j) = cosq cos j ( + ) (si q) (sij Namely, 4. The Hyper-real Delta 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 ) = 0 0 = 0 6

37 Gauge Istitute Joural H. Vic Dao 5. Fourier-Legedre Expasio of Delta Associated with u "( q ) + [ l + ( - m ) ] u ( q ) = 0, p q 4 cos q - < < p Legedre s equatio with for m = 0,,,..., is (- x ) y "( x ) + xy '( x ) + [ l - - m ] y ( x ) = 0. The eige values are 4 -x m l = ( - m + ), = m,m +,m +,... ad the correspodig eige fuctios are Legedre Fuctios Put ad obtai Put ad obtai P ( x ). m x = si q, y "( q ) - y '( q )ta q + [ l - - m ] y ( q ) = 0. 4 cos q y() q = u() q, cos q u"( q) + [ l + ( + ta q) - m ] u ( q) = cos q Substitutig cos q = + ta q, 7

38 Gauge Istitute Joural H. Vic Dao u "( q ) + [ l + ( - m ) ] u ( q ) = 0, 4 cos q Therefore, m cos q (si q ) solve u "( q ) + [ l + ( - m ) ] u ( q ) = 0. P 4 cos [Titchmarsh, p. 80] shows that 5. The Normalized eige-fuctios are ( - m)! m y () q = cos (si ) - m + q P q, = m,m +,...! Therefore, = dq ( - j) = å y( qy ) ( j) = m = ( - m)! m m = cos q cos jå ( - m + ) (si ) (si ) P q P j! = m q 5. The Fourier-Legedre expasio of Delta i - p < q < p, for m = 0,,,..., is ( - m)! dq ( - j) = cosq cos j ( - m + ) P (si q) P (sij = m m å ) = m! Namely, 5. The Hyper-real Delta i - p < q, j < p is the sequece k= ( k - m)! m m cosq cos j å ( k - m + ) (si ) (si ) P q k P j k k! k= m = = m 8

39 Gauge Istitute Joural H. Vic Dao 6. Fourier-Bessel Expasio of Delta Associated with y"( x) + [ l - ( - )] y( x) = 0, 0 x 4 < x < b Two idepedet solutios are For ³ xj ( x l), ad xy ( x l ) By [Titchmarsh, p. 8], the eige-values l are the zeros of J ( b l ), =,,,... ad the ormalized eige-fuctios are Therefore, bj J '( b l ) ( x l ). 6. For ³ the Fourier-Bessel expasio of Delta i 0 < x <b is = d( x - x) = xxå J ( x l ) J ( x l ), ³ '( l ) b = J b 9

40 Gauge Istitute Joural H. Vic Dao Namely, 6. For ³ the Hyper-real Delta i 0 < x, x <b is the ifiite sequece k= xx å J ( x l ) J ( x l ) k k '( l ) b k = J b k = = For 0 < <, ¹ By [Titchmarsh, p. 8], the eige-values where c = c ost. l are the zeros of cj ( b l) - l J ( b l), - 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 p b - r xj ( b l ){ cl J ( x l )- J ( x l ) }. - sip Therefore, 6. If 0 < <, ¹, the the Fourier-Bessel expasio of Delta i 0 < x <b is 40

41 Gauge Istitute Joural H. Vic Dao p b d - x = x l l l - l = ( x ) x ( ) ( ) ( ) å r J b c J x J x 4si p = - { - } - { cl J ( x l ) J ( x l ) - } - Namely, 6.4 If 0 < <, ¹, the the Hyper-real Delta i 0 < x, x <b is the ifiite sequece p b 4si p k= k = - { - } xx å r J ( b l ) cl J ( x l )- J ( x l ) k k k k k - { cl J ( x l ) J ( x l ) k k - k } - = = 6.5 If 0 < <, ¹, the For c = ifiite hyper-real, the expasio is i J : p b d - x = xå l l l x l ) = - ( x ) c x r J ( b ) J ( x ) J ( 4si p = For c = ifiitesimal, the expasio is i J : - p b d - x = xå l l x l ) = ( x ) 4si p x r J ( b = ) J ( x - ) J ( - 4

42 Gauge Istitute Joural H. Vic Dao 7. Fourier-Bessel Expasio of Delta Associated with y"( x) + [ l - x] y( x) = 0,, with 0 < x < a = 0 By [Titchmarsh, p. 9], the eige-values l are the zeros of l J l - ad the ormalized eige-fuctios are - l é J ( l ) J ( l ) ù ê - - ë ú û Therefore, J ( ) + ( ), =,,,.. { ( [ ]) ( [ ]) - } l -x J l - x + J l -x 7. the Fourier-Bessel expasio of Delta i 0 < x < is = d( x - x) = å l -x l - x = l é J ( l ) J ( l ) ù ê - - ë ú û { J ( [ l x] ) J ( [ l x] ) - } { J ( [ l x] ) J ( [ l x] ) - }

43 Gauge Istitute Joural H. Vic Dao Namely, 7. the Hyper-real Delta i 0 < x, x < is the ifiite sequece k= å l -x l -x k k k = l é J ( l ) ( l ) ù k k êë - J - úû { J ( [ l x]) J ( [ l x]) k - k } { J ([ l x]) J ([ l x]) k - k } = = 4

44 Gauge Istitute Joural H. Vic Dao 8. Fourier-Bessel Expasio of Delta Associated with y"( x) + [ m - x] y( x) = 0, 0 < x <, with a = p By [Titchmarsh, p. 9], the eige-values m are the zeros of m J m - ad the ormalized eige-fuctios are - m é J ( m ) J ( m ) ù ê + - ë ú û Therefore, J ( )- ( ), =,,,... { ([ ]) ([ ]) - } m -x J m - x + J m -x 8. the Fourier-Bessel expasio of Delta i 0 < x < is = d( x - x) = å m -x m - x = m é J ( m ) J ( m ) ù ê + - ë ú û { J ( [ m x] ) J ( [ m x] ) - } { J ( [ m x] ) J ( [ m x] ) - }

45 Gauge Istitute Joural H. Vic Dao Namely, 8. the Hyper-real Delta i 0 < x, x < is the ifiite sequece k= å m -x m -x k k k = m é J ( m ) ( m ) ù k k êë + J - úû { J ([ m x]) J ([ m x]) k - k } { J ( [ m x]) J ( [ m x]) k - k } = = 45

46 Gauge Istitute Joural H. Vic Dao 9. Delta Expasio i a Itegral over a Cotiuum of 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 expasio is represeted by a itegral. Titchmarsh applied the Residue Theorem to obtai the projectios of a fuctio f ( x ) o the eige-fuctios of Sturm-Liouville problems, with cotiuous spectrum of eige-values. Followig Titchmarsh, we expad the Delta Fuctio i Itegrals of Sturm-Liuoville eige-fuctios. 46

47 Gauge Istitute Joural H. Vic Dao 0. Fourier-Cosie Itegral of Delta Associated with y"( x) + ly( x) = 0, 0 < x < The eige-values l º w, are the iterval of hyper-real positive umbers (0, ). By [Titchmarsh, p. 7], the Hyper-real fuctio f ( x ) is give for ay hyper-real x by x= l= fx ( ) = cos( x l)cos( x l) dlf( x) dx + l p ò ò x=- l= 0 ì ï = + p ïî x= l= + si( x l)si( x l) dl ( x) l p ò ò f dx x=- l= 0 x= l= ï {cos( x l)cos( x l) si( x l)si( x l)} dlï ò í ý ( x) ò f dx l x=- ï l= 0 cos[( x-x) l] ï x= w= = ò cos w( x x) dwf( x) d p ò - x x=- w= 0 Therefore, d( x -x) üï ïþ 47

48 Gauge Istitute Joural H. Vic Dao 0. The Fourier-Cosie Hyper-real Itegral of Delta i - < x, x < is w= d( x - x) = cos w( x -x) p ò dw w= 0 48

49 Gauge Istitute Joural H. Vic Dao. Weber Formula for Fourier-Bessel Itegral of Delta Associated with y"( x) + [ l - ( - )] y( x) = 0, a < x < a > 0 x 4 The eige-values - l º s, are the iterval of hyper-real egative umbers (-,0). By [Titchmarsh, p. 87], the Hyper-real fuctio f ( x ) is give for hyper-real a < x < by x = s = fx ( ) = x J( xsy ) ( as) -J( asy ) ( xs) ( ) ( ) J as + Y as x = a s= 0 { } ò ò { x x } J ( sy ) ( as) -J ( asy ) ( s) sds xf( x) dx Therefore,. The Fourier-Bessel Hyper-real Itegral of Delta i a < x, x < is 49

50 Gauge Istitute Joural H. Vic Dao s = d( x- x) = xx ò { J ( xsy ) ( as) - J ( asy ) ( xs) } ( ) ( ) J as + Y as s = 0 { ( x ) ( ) ( ) ( x )} J sy as -J asy s sds 50

51 Gauge Istitute Joural H. Vic Dao. Hakel Formula for Fourier- Bessel Itegral of Delta Associated with y"( x) + [ l - ( - )] y( x) = 0, 0 x 4 < x < > By [Titchmarsh, p. 88], the Hyper-real fuctio f ( x ) is give for ay positive hyper-real x by Therefore, x = s = f ( x) x J ( xs) J ( xs) sds xf( x) = ò ò dx x = 0 s = 0. For >, the Fourier-Bessel Hyper-real Itegral of Delta i hyper-real positive x, ad x is s = d( x - x) = xx ò J ( xs) J ( xs) sds s = 0 5

52 Gauge Istitute Joural H. Vic Dao 0 < <, ad c < 0 By [Titchmarsh, p. 89], the Hyper-real fuctio f ( x ) is give for ay positive hyper-real x, ad a costat c < 0 by x = s = ò ò { } 4 - c - cs cosp + s x = 0 s = 0 fx ( ) = x cj( xs) -s J ( xs) Therefore,. For 0 < <, ad c < 0, { } cj ( xs) -s J ( xs) sds xf ( x) dx the Fourier-Bessel Hyper-real Itegral of Delta i - hyper-real positive x, ad x is s = d( x - x) = xx ò { cj ( xs) - s J ( xs) } 4 - c - cs cosp + s s = 0 { ( x ) ( x )} cj s -s J s sds - 0 < <, ad c > 0 By [Titchmarsh, p. 90], the Hyper-real fuctio f ( x ) is give for ay positive hyper-real x, ad a costat c > 0 by x = s = ò ò { } p [ c + log s] + x = 0 s = 0 p fx ( ) = x cj( xs) -Y( xs) J( xs)logs 5

53 Gauge Istitute Joural H. Vic Dao { } cj ( xs) - Y ( xs) + J ( xs)log s sds xf ( x) dx 0 0 p 0 x = - - pc p c -pc + e xk ( xe ) ò xk ( xe ) f( x) dx 0 0 x = 0 Therefore,. For 0 < <, ad c > 0, the Fourier-Bessel Hyper-real Itegral of Delta i hyper-real positive x, ad x is s = d( x - x) = xx cj ( xs) - Y ( xs) + J ( xs)logs ò [ c + log s] + s = 0 p { } 0 0 p 0 { ( x ) ( x ) ( x )log } cj s - Y s + J s s sds 0 0 p 0 pc - - pc -pc + e xxk ( xe ) K ( xe ) 0 0 5

54 Gauge Istitute Joural H. Vic Dao. Fourier-Bessel Itegral of Delta Associated with y"( x) + [ l + x)] y( x) = 0, 0 x < The eige-values l º- m, are the iterval of hyper-real egative umbers (-,0). By [Titchmarsh, p. 9], ìï J ( x -m ) -I ( m ), m J ( x - m ) I ( m ) p - - f(, x m) = m x -m ï í ïïï I ( x - m ) I ( m ), m ï ³ I ( x - m ) I ( m ) - - ïî x x ad the Hyper-real fuctio f ( x ) is give for 0 x < by x= ì m= ü f ( x) = ï f( x, m) f( x, m) dm ï ò í ý ( x) x p ò f d x= 0 ïî m= 0 ïþ Therefore, m= x m= d( x - x) = f(, x m)(, f x m) dm + f(, x m)(, f x m) p p ò ò dm m= 0 m= x 54

55 Gauge Istitute Joural H. Vic Dao m= 0 ( -m ) - ( m - m m m= x 4p J x I d( x - x) = x 7 ò m -m J ( x ) I ( - - ) ) ìï J ( x -m ) -I ( m ), m x J ( x - m ) I ( m ) - - x -m ï í dm ïïï I ( x - m ) I ( m ), m x ï ³ I ( x - m ) I ( m ) - - ïî ( - m ) ( m - m m m= 4p I x I + x 7 ò m -m I ( x ) I ( m= x - - ) ) ìï J ( x -m ) -I ( m ), m x J ( x - m ) I ( m ) - - x -m ï í dm ïïï I ( x - m ) I ( m ), m ³ x ï I ( x - m ) I ( m ) - - ïî 55

56 Gauge Istitute Joural H. Vic Dao 4. Fourier-Bessel Itegral of Delta Associated with x y"( x) + [ l + e )] y( x) = 0, x - < < By [Titchmarsh, p. 95], the Hyper-real fuctio f ( x ) is give for hyper-real - < x < x= l= ìï = x x fx ( ) = ï í4 J ( e) J ( e) + ï å x=- = ïî l= 0 4 sih( p l) by ò ò x x x x { J ( e ) + J ( e )}{ J ( e ) + J ( e )} d l f ( x) d x. l - l l - l i i i i } Therefore, 4. The Fourier-Bessel Hyper-real Itegral of Delta i x ad x is = l= x x å ò = l= 0 d( x - x) = 4 J ( e ) J ( e ) + 4sih( p l) x x x x { J ( e ) + J ( e )}{ J ( e ) + J ( e )} dl i l -i l i l -i l } Alteratively, f ( x ) is give for hyper-real - < x < by 56

57 Gauge Istitute Joural H. Vic Dao x= l= ìï = x x fx ( ) = ï ò í (4+ ) J ( e) J ( e) ï å ò x=- = ïî l= 0 4 sih( p l) x x x x { J ( e )- J ( e )}{ J ( e )- J ( e )} d l f ( x) d x. l - l l - l i i i i } Therefore, 4. The Fourier-Bessel Hyper-real Itegral of Delta i x ad x is = l= x x å + + ò = l= 0 d( x - x) = (4 + ) J ( e ) J ( e )- 4sih( p l) x x x x { J ( e ) -J ( e )}{ J ( e ) -J ( e )} dl i l -i l i l -i l } 57

58 Gauge Istitute Joural H. Vic Dao 5. Fourier-Bessel Itegral of Delta Associated with x y"( x) + [ l - e )] y( x) = 0, x - < < By [Titchmarsh, p. 96], the Hyper-real fuctio f ( x ) is give for hyper-real Therefore, - < x < x= l= x=- l= 0 by f ( x ) K ( e x ) K ( e x = ò )sih( p l) d l ( x) x i l i l p ò f d 5. The Fourier-Bessel Hyper-real Itegral of Delta i x ad x is l= ( x ) K ( e x ) K ( e x d - x = )sih( p l) i l i l p ò dl l= 0 58

59 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 00; 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 00; Ifiitesimals [Da] Dao, H. Vic, Ifiitesimal Calculus i Gauge Istitute Joural Vol. 7 No. 4, November 0; 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 009. 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, 0; The Delta Fuctio [Da6] Dao, H. Vic, Riemaia Trigoometric Series, Gauge Istitute Joural, Volume 7, No., August 0. Riemaia Trigoometric Series 59

60 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, 0; 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., August, 0; Ifiite Series with Ifiite Hyper-real Sum [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, 007 [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, 977. [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,

61 Gauge Istitute Joural H. Vic Dao [Szego4] Szego, Gabor, Orthogoal Polyomials Fourth Editio, America Mathematical Society,975. [Titchmarsh], E. C. Titchmarsh, Eigefuctio Expasios Associated with Secod-order Differetial Equatios, Part I, Secod Editio, Oxford, 96. [Weisstei], Weisstei, Eric, W., CRC Ecyclopedia of Mathematics, Third Editio, CRC Press,