Periodic Delta Function. in Bessel Functions

Transcription

1 Gauge Istitute Joural H. Vic Dao Periodic Delta Fuctio ad Fourier Epasio i Bessel Fuctios H. Vic Dao vic@comcast.et Jue, Abstract Let f ( be itegrable o [,]. The zeros of the Bessel Fuctio J (. < l < l < l <... 3 defie the orthogoal sequece of fuctios J ( l, J (, J ( l, l 3 The Bessel Series associated with f ( is where aj ( l + aj ( l + aj ( l = = a = ò f( J ( l d = ò f( J ( l d [ J '( ] [ ( ] l J l = = are the Bessel coefficiets. The Bessel Series Theorem supplies the coditios uder which the Bessel Series associated with f ( equals f (.

2 Gauge Istitute Joural H. Vic Dao It is believed to hold i the Calculus of Limits uder Hobso s Coditios. I fact, The Theorem caot be proved i the Calculus of Limits uder ay coditios, because the summatio of the Bessel Series requires itegratio over the sigular Bessel Kerel. Plots of partial sums of the Bessel Series speak volumes about the sesibility of the claims to have ifiity boud by epsilo. I Ifiitesimal Calculus, the Bessel Kerel J ( l J( l + J ( l J( l +... ý J( l J( l is the Periodic Delta, { } d ( - =... + d ( d ( - + d ( Periodic { cos p ( cos p (...} = {... i ( - - i ( - i ( - e e e e i ( -...} p p p p = The Periodic Delta equals its Bessel Series, ad the Bessel Series associated with ay hyper-real itegrable f (, equals f ( For Bessel s Fuctio J (, where may be a fractio, the zeros < l < l < l <..., 3 defie the orthogoal sequece of fuctios J ( l, J (, ( l, l J 3

3 Gauge Istitute Joural H. Vic Dao The -Bessel Series associated with f ( is where aj ( l + aj ( l + aj ( l = = ak = ò f( J ( l d = ò f( J ( l k d [ J '( ] [ ( ] k lk J l = + k = The, the -Bessel Kerel is the Periodic Delta, J( l J( l J( l J( l ý J+ ( l J+ ( l Periodic { } d ( - = + cos p ( - + cos p ( ad the -Bessel Series associated with ay hyper-real itegrable f (, equals f (. Keywords: Ifiitesimal, Ifiite-Hyper-Real, Hyper-Real, ifiite Hyper-real, Ifiitesimal Calculus, Delta Fuctio, Periodic Delta, Bessel Fuctios, Bessel Coefficiets, Fourier- Bessel Series, Bessel Kerel, Fourier-Bessel Epasio. Mathematics Subject Classificatio 6E35; 6E3; 6E5; 6E; 6A6; 6A; 3E; 3E55; 3E7; 3H5; 46S; 97I4; 97I3. 3

4 Gauge Istitute Joural H. Vic Dao Cotets. The Origi of the Bessel Series Theorem. Divergece of the Bessel Kerel i the Calculus of Limits. Hyper-real lie. 3. Itegral of a Hyper-real Fuctio 4. Delta Fuctio d- ( 5. Periodic Delta d ( - 6. Coverget Series Periodic 7. Bessel Sequece ad d ( - Periodic 8. Bessel Kerel ad d ( -. Periodic 9. Bessel Series of d ( - Periodic. Bessel Series Theorem Refereces 4

5 Gauge Istitute Joural H. Vic Dao The Origi of the Bessel Series Theorem For Bessel s Fuctio J (, where may be a fractio, the zeros < l < l < l <..., 3 defie the orthogoal sequece of fuctios o ³, so that J ( l, J (, ( l, l J 3 = ò J( lmj ( lkd = J + ( lk dmk. = The Fuctios may be geerated by epadig = å =- ( a- J ( a = e a 3 ( a ( a ( a ( a 4 = é ù é ù é ù a êë - a úû + 3 êë a úû 34 êë a úû..., the coefficiet of the coefficiet of the coefficiet of a a a - is 4 6 J ( = , is J (, is J - (, 5

6 Gauge Istitute Joural H. Vic Dao. Euler (766 The problem of the vibratios of a stretched membrae, led Euler to the derivatio of the Bessel Equatio, ad to a ifiite series solutio which is a Bessel Fuctio.. Fourier (8 Fourier assumed the Bessel Series Theorem i the Theory of Heat coductio, ad from derived f ( = a J ( l + a J ( l + a J ( l +... a m 3 3 = = ò f( J( l m d [ J ( ] l m =.3 Bessel Equatio for a particle trapped i a sphere A subatomic particle with mass m, is trapped i a sphere of radius a, uder the potetial, r < a Vr ( =, N, r ³ a where N is a ifiite Hyper-real umber. De Broglie associated with the particle a wave of legth h l =, mv where v is the velocity of the particle, ad h is Plack s costat. 6

7 Gauge Istitute Joural H. Vic Dao The wave s frequecy is v v mv = = = l h h mv The wave s agular frequecy is w = p = p mv h I terms of the De Broglie wave, the wave s eergy is a multiple of Plack s radiatio eergy. That is, E = e h = ew, =, e is the multiplier. h p The kietic eergy of the particle is Hece, mv = E. mv = me, h l =, me v w = l =. me w p me me = = v w w Schrodiger postulated a comple valued potetial Y (, r qf,, t = y(, r qf, e iwt that satisfies the wave equatio 7

8 Gauge Istitute Joural H. Vic Dao The, Y (, r qf,, t = Y(,,, t r qft. v = Y( r, qf,, t - Y(,, t r qf, t v iwt me = y( r, q, f e - y( r, q, f( -w e w The Schrodiger equatio for the trapped particle is mw y(, r q, f + ey(, r q, f =. iwt. I spherical coordiates mw y + q y + y - ey =. r( r (si r r r si q q q r si q f Assumig that y(, r q, f = R( r Q( q F ( f, the Schrodiger equatio becomes The, Substitutig + Q + F - mw =. ( rr'' (si q '' '' er R Q si q F si q R mw ( rr'' - e r = costº C. l Rr ( = r, Hece, mw ll ( + - e r = C. 8

9 Gauge Istitute Joural H. Vic Dao ll ( + = C. Thus, the Schrodiger Radial Equatio is mw R e ( rr'' + r = ll ( + mw rr'' + rr' + e r R - ll ( + R =, r Deotig r = r mw e, dr dr dr dr mw '( = = R'( r R r e, r r e r rr '( r = R '( m w = R '( mw e mw d { } d r R"( r = R'( r e = R''( r dr dr r, mw e, mw r mw = r e = r mw e rr "( r R ''( R "( r The, the Radial Schrodiger equatio becomes r R''( r + rr'( r + r R - l( l + R =. To obtai Bessel s Equatio put - R( r = r J( r e R'( r =- r J + r J ' 9

10 Gauge Istitute Joural H. Vic Dao rr'( r =- r J + r J ', R''( r = r J - r J ' + r J '' 3 4 Substitutig i the equatio, r R''( r = r J - r J ' + r J '' 3 - r R - l( l + R = r J - l( l + r J r J - r J ' + r J ''- r J + r J ' + r J - l( l + r J =, 3 4 J - rj ' + r J ''- J + rj ' + r J - l( l + J =, 4 r J '' + rj ' + [ r - l( l + - ] J = r J '' + rj ' + [ r -( l + ] J = The solutios are the Bessel Fuctios J J r w m ( r = ( e l+ + l Requirig the wave fuctio to vaish o the sphere, r = a, J a w e m ( = l+. That is, there is a sequece e so that Thus, J a w e m l ( = + l = a mw e. m are the zeros of J ( r w e. l +

11 Gauge Istitute Joural H. Vic Dao Therefore, the solutios of the Bessel equatio are the orthogoal Bessel Fuctios J r J mw r ( e = ( l l+ l+ a l a. The solutio for J( r is the ifiite series a J ( l + a J ( l + a J ( l +... r r r l+ a l+ a 3 l+ 3 a.4 The Bessel Series Associated with f ( Let f ( be defied o [,]. The zeros of Bessel s Fuctio J (, defie the sequece of fuctios J < l < l < l <... The fuctios are orthogoal, = ò = 3 ( l, J (, J ( l, l 3 ( lm ( l = dm [ '( lm] = dm [ ( lm] J J d J J If f ( ca be epaded i the Bessel Fuctios,. The, f ( = a J ( l + a J ( l + a J ( l +..., 3 3 = = ò ò f( J ( l d = aj { ( l + aj ( l +...} J ( d = = l

12 Gauge Istitute Joural H. Vic Dao = = ò = a f( J ( l J ( l d + a f( J ( l J ( l d +... = = = a [ J ( l ]. d d [ J ( ] [ ( ] Thus, the Bessel coefficiets are l J l ò = = a = f( J ( l d = f( J ( l [ J '( ] [ ( ] ò ò d. l J l = = The Bessel Series associated with f ( is aj ( l + aj ( l + aj ( l The Bessel Series Theorem supplies the coditios uder which the Bessel Series associated with f ( equals f (..5 The -Bessel Series Associated with f ( For the Bessel Fuctio J (, where is a fractio, the zeros defie the sequece of fuctios J < l < l < l <..., 3 ( l, J (, ( l, The fuctios are orthogoal, l J 3

13 Gauge Istitute Joural H. Vic Dao = ò = ( lk ( l l = dkl [ '( ] [ lk = dkl + ( lk] J J d J J The Bessel Series associated with f ( is where aj ( l + aj ( l + aj ( l = = ak = ò f( J ( l d = ò f( J ( l k d [ J '( ] [ ( ] k lk J l = + k = The Bessel Series Theorem supplies the coditios uder which the Bessel Series associated with f ( equals f (.. 3

14 Gauge Istitute Joural H. Vic Dao. Divergece of the Bessel Kerel i the Calculus of Limits Calculus of Limits Coditios for the Bessel Series to equal its fuctio reflect the belief that a smooth eough fuctio equals its Bessel Series. I fact, i the Calculus of Limits, o smoothess of the fuctio guaratees eve the covergece of the Bessel Series. I the Calculus of Limits, the Bessel Series is the limit of the sequece of Partial Sums { f } = aj l + aj l + + a J ( ( (... ( l essel æ = = ö æ ö = f( J ( l d J( l.. f( J ( l d + + J( l ò J( l ò ç J ( l è = ø çè = ø = J( l J( l J( l J( l J( l J( l = f (.. ò ýd. J( l J( l J( l As =, the Bessel Sequece J( l J( l J( l J( l J( l J( l ý J( l J( l J( l 4

15 Gauge Istitute Joural H. Vic Dao becomes the Bessel Kerel, J( l J( l J( l J( l J( l 3 J( l ý, J( l J( l J( l3 To see that it diverges for =, we apply the Schlafli Asymptotic Summatio Formula [Watso, p. 585]: For large, J( l J( l J( l J( l J( l J( l ý = J( l J( l J( l Sice si( + q si é si( + p ( - si( + p ( + ù - si p ( si p. êë - ( + úû q = + cos q + cos q +...cosq, J( l J( l J( l J( l J( l J( l ý J( l J( l J( l { + cos p ( - + cos p ( cos p ( - } { cos p ( cos p (... cos p ( } By [Hardy, p., #(..3], + cos q + cos q + cos 3 q +... =, for ay q ¹ Therefore, for large eough, ad ¹ ¹, J( l J( l J( l J( l J( l J( l ý J( l J( l J( l 5

16 Gauge Istitute Joural H. Vic Dao For, Hece, { + cos p ( - + cos p ( cos p ( - } J( l J( l J( l ý [ + cos +...cos] J( l J( l J( l. J( l J( l J( l ý = + lim, J( l J( l J( l3 ad the Bessel Kerel diverges to at ay = ¹. Therefore, while the partial sums of the Bessel Series eist, their limit does ot. That is, due to the sigularity at =, the Bessel Series does ot coverge i the Calculus of Limits. Avoidig the sigularity at =, by usig the Cauchy Pricipal Value of the itegral does ot recover the Theorem, because at ay ¹ ¹, the Bessel Kerel vaishes, ad the itegral will be idetically zero, for ay fuctio f ( : For ay ¹ ¹, J( l J( l J( l J( l J( l 3 J( l ý = J( l J( l J( l3 { cos p ( cos p ( cos3 p (...} = ad the Bessel Kerel vaishes. Plots of the Bessel Sequece cofirm that I the Calculus of Limits, 6

17 Gauge Istitute Joural H. Vic Dao the Bessel Kerel is either sigular or zero. Plots of J ( l J ( l J( lk J ( lk ý J( l J( lk I Maple, æ 37 BesselJZeros(, i BesselJ(, ö plot BesselJ(, BesselJZeros(, i *,..3 å = ç çè i = ébesselj(, BesselJZeros(, i ù ë û ø The pulses arrow, ad the vaishig betwee them icreases, with more terms: I Maple, æ BesselJZeros(, i BesselJ(, ö plot BesselJ(, BesselJZeros(, i *,..3 å = ç çè i = ébesselj(, BesselJZeros(, i ù ë û ø 7

18 Gauge Istitute Joural H. Vic Dao Plottig with the asymptotic sequece æ æ ö I Maple, cos ( ( ö plot i p,..3 + å - = ç è ç è i= ø ø 8

19 Gauge Istitute Joural H. Vic Dao The plots cofirm that the Bessel Series Theorem caot be proved i the Calculus of Limits..3 -Bessel Kerel divergece i the Calculus of Limits For the Bessel Fuctio J (, where is a fractio, the -Bessel Series is the limit of the sequece of Partial Sums { } f ( = a J ( l + a J ( l a J ( l As essel k k k = J( l J( l J( lk J( l k = f (... ò + + ýd. J ( l J ( lk = + +, the -Bessel Sequece becomes the -Bessel Kerel, J( l J( l J( l J( l J( l3 J( l ý, J+ ( l J+ ( l J+ ( l3 To see divergece at =, apply the Schlafli Formula [Watso, p. 585]: For large k, we choose l,k < Ak < l, k + so that J( l J( l J( lk J( l k ý J+ ( l J+ ( lk é si A k( si A k( ù si p ( si p ( êë - + úû By [Watso, p.584], we may set A = kp + ( + p. Ad k followig Dii, [Watso, p.577], Watso chose also + ³, 9

20 Gauge Istitute Joural H. Vic Dao We will set A = ( K + p, where K ³ k is a atural umber Sice k si( K + q si q = + cosq + cos q +...coskq J( l J( l J( l J( l J( lk J( l k ý J+ ( l J+ ( l J+ ( lk { + cos p ( - + cos p ( cos Kp ( - } By [Hardy, p., #(..3], { cos p ( cos p (...cos Kp ( } cos q + cos q + cos 3 q +... =, for ay q ¹ Therefore, for k large eough, ad ¹ ¹, J( l J( l J( l J( l J( lk J( l k ý J+ ( l J+ ( l J+ ( lk For, { + cos p ( - + cos p ( cos Kp ( - } J( l, J( l, J( l, k.. [ ý + cos +...cos ] J + ( l, J + ( l, J + ( l, k. K Hece, J( l J( l J( l ý = + lim K, J( l J( l J( l3 k

21 Gauge Istitute Joural H. Vic Dao ad the -Bessel Kerel diverges to at ay = ¹. Therefore, while the partial sums of the -Bessel Series eist, their limit does ot. That is, due to the sigularity at -Bessel Series does ot coverge i the Calculus of Limits. Avoidig the sigularity at =, the =, by usig the Cauchy Pricipal Value of the itegral does ot recover the Theorem, because at ay ¹ ¹, the -Bessel Kerel vaishes, ad the itegral will be idetically zero, for ay fuctio f ( : For ay ¹ ¹, J( l J( l J( l J( l J( l3 J( l ý = J+ ( l J+ ( l J+ ( l3 { cos p ( cos p ( cos3 p (...}, = ad the -Bessel Kerel vaishes. Thus, the -Bessel Series Theorem caot be proved i the Calculus of Limits..4 Ifiitesimal Calculus Solutio By resolvig the problem of the ifiitesimals [Da], we obtaied the Ifiite Hyper-reals that are strictly smaller tha, ad costitute the value of the Delta Fuctio at the sigularity.

22 Gauge Istitute Joural H. Vic Dao The cotroversy surroudig the Leibitz Ifiitesimals derailed the developmet of the Ifiitesimal Calculus, ad the Delta Fuctio could ot be defied ad ivestigated properly. I Ifiitesimal Calculus, [Da3], we ca differetiate over jump discotiuities, ad itegrate over sigularities. The Delta Fuctio, the idealizatio of a impulse i Radar circuits, is a Discotiuous Hyper-Real fuctio which defiitio requires Ifiite Hyper-reals, ad which aalysis requires Ifiitesimal Calculus. I [Da5], we show that i ifiitesimal Calculus, the hyper-real w= ( e i w d = w p ò d w=- is zero for ay ¹, it spikes at =, so that its Ifiitesimal Calculus = ò itegral is d( d =, ad =- d ( = <. d Here, we show that i Ifiitesimal calculus, the Bessel Kerel is a hyper-real Periodic Delta: A trai of Delta Fuctios. Ad the Bessel Series { f ( associated with a Hyper-real fuctio f (, equals f (. essel }

23 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 3

24 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. 4

25 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. 5

26 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 6

27 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 7

28 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. 8

29 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, ý d. The 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 9

30 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, Î é-d, d ù ( = ê ú ë û., otherwise 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 ( = d c c c 6. If d =, ( = (, (,3 (... [-, ] [-, ] [-, ] If d =, 3 d ( =,,,... cosh cosh cosh 3 3

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

32 Gauge Istitute Joural H. Vic Dao 5. Periodic Delta d ( - 5. Periodic Delta Defiitio Periodic { } d ( - =... + d ( d ( - + d ( Periodic { cos p ( cos p (...} = {... i ( - - i ( - i ( - e e e e i ( -...} p p p p = is a trai of hyper-real Delta fuctios. 5. Periodic Delta peaks at - = m, for ay m =,,,... ad has the siftig property over[,] Proof of Siftig: = ò { + cos p ( - + cos p ( } d = = = = { cos ( cos (...} = å + p- + p- + d By [Hardy, p., #(..3], + cos q + cos q + cos 3 q +... =, for ay q ¹ Therefore, for ay ¹, the summatio terms, 3

33 Gauge Istitute Joural H. Vic Dao { cos ( cos (...} + p- + p- + d, vaish. For =, the summatio yields { p p } + cos ( + cos ( +... d. The Hyper real umber is the sequece + cos p( + cos p( =. d Cosequetly, the itegral equals d =, d ad the Periodic Delta has the siftig property. 33

34 Gauge Istitute Joural H. Vic Dao 6. Coverget Series I [Da8], we defied covergece of ifiite series i Ifiitesimal Calculus 6. Sequece Covergece to a fiite hyper-real a a a iff a - a = ifiitesimal. 6. Sequece Covergece to a ifiite hyper-real A a A iff a represets the ifiite hyper-real A. 6.3 Series Covergece to a fiite hyper-real s a + a +... s iff a a - s = ifiitesimal. 6.4 Series Covergece to a Ifiite Hyper-real S a + a +... S iff a a represets the ifiite hyper-real S. 34

35 Gauge Istitute Joural H. Vic Dao 7. Bessel Sequece ad d ( - Periodic 7. Bessel Sequece Defiitio The Bessel Series partial sums = J( l J( l J( l k J( lk essel k { f ( } == f(.. ò + + ýd. J( l J( lk = give rise to the Bessel Sequece J( l J( l J( l k J( lk ý. J( l J( lk J( l J( l J( l k J( lk 7. The Bessel Sequece ý J( l J( lk is asymptotically the Periodic Delta Sequece { + cos p ( - + cos p ( cos kp ( -} Proof: I., it follows that for large k, J( l J( l J( l J( l J( l k J( lk ý J( l J( l J( lk { + cos p ( - + cos p ( cos kp ( - }. 35

36 Gauge Istitute Joural H. Vic Dao J( l J( l J( lk J( l k 7.3 The Bessel Sequece ý J+ ( l J+ ( lk is asymptotically the Periodic Delta Sequece { + cos p ( - + cos p ( cos kp ( -} Proof: same as

37 Gauge Istitute Joural H. Vic Dao 8. Bessel Kerel ad d ( - Periodic 8. Bessel Kerel i the Calculus of Limits The Bessel Series partial sums essel k { f } = J( l J( l J( l k J( lk ( = f(.. ò + + ýd. J ( l J( lk = Bessel Sequece give rise to the Bessel Sequece. Its limit is the Bessel Kerel J( l J( l J( l J( l J( l 3 J( l ý J( l J( l J( l3 8. I the Calculus of Limits, the Bessel Kerel does ot have the siftig property Proof: for, J( l J( lk ý + cos ( cos ( J( l J( lk { p kp } That is, ³ ( +k k J( l J( l J( l J( l ý is ot itegrable. J( l J( l 37

38 Gauge Istitute Joural H. Vic Dao 8.3 Hyper-real Bessel Kerel i Ifiitesimal Calculus J( l J( l J( l J( l essel erel (, = ý J( l J( l { cos p ( cos p (...} = {... i ( - - i ( - i ( - e e e e i ( -...} p p p p = = +, m =, - ¹ m {... d ( d ( d (.. } = = d ( Periodic -. Proof: By., J( l J( l J( l J( l essel erel (, = ý J( l J( l { cos p ( cos p (...} = = +, m =, - ¹ m Deotig the hyper-real + by, d,,,... - ¹ - - ¹ - ¹ = , +, ý - = - - =, - = d d d {... d ( d ( d (...} =

39 Gauge Istitute Joural H. Vic Dao 8.4 Hyper-real -Bessel Kerel i Ifiitesimal Calculus Let + be a ifiite Hyper-real. The Hyper-real -Bessel Kerel is J( l J( l J( l J( l ý = J+ ( l J+ ( l { cos p ( cos p (...} = {... i ( - - i ( - i ( - e e e e i ( -...} p p p p = = +, m =, - ¹ m {... d ( d ( d (.. } = = d ( Periodic -. 39

40 Gauge Istitute Joural H. Vic Dao 9. Bessel Series ad d ( - Periodic 9. Bessel Series of a Hyper-real Fuctio Let f ( be a hyper-real fuctio itegrable o [,]. The zeros of the Bessel Fuctio J (. < l < l < l <... 3 defie the orthogoal sequece of fuctios For each J ( l, J (, J ( l, l 3 =,,, 3,..., the hyper-real itegrals a = = ò f( J( l d [ J ( ] l = eist, with fiite, or ifiite hyper-real values. The a are the Bessel Coefficiets of f (. The Bessel Series associated with f ( is { } = l + l + 3 l3 essel f ( a J ( a J ( a J ( +... For each, it may assume fiite or ifiite hyper-real values. 9. { } Proof: d ( - = d ( - essel Periodic Periodic 4

41 Gauge Istitute Joural H. Vic Dao { d } l l 3 l3 essel Periodic( - = a J ( + a J ( + a J ( +... where = ò dperiodic lk ( lk = ak = ( -J ( d J Substitutig from 8.3, J( l J( l J( l J( l dperiodic( - = ý, J( l J( l = ò J( l J( l J( l J( l +. ý lk ( lk ( l ( l.. ( = ak = + J J J J = = J ( l ò J ( l J ( lk d +... J( lk J( l = = m m J( lk J( lm = d... + J ( l ò J ( l J ( lk d +... By [Spiegel, p.43, #4.95] for ay m ¹ k, = ò J ( l J ( l d = { l J ( l J( l -l J ( l J( l } = m k m k m k m k l k - l = m By [Spiegel, p.43, #4.96] for m = k, Hece, = ò J kj kd= J k + J m = J = ( l ( l ( l ( l ( l k. 4

42 Gauge Istitute Joural H. Vic Dao = ak = J( l k ò J( lkj ( lkd J( lk J( lk = = J ( l Therefore, k J ( l k J ( l J( l J( l essel { dperiodic( - } = J ( l + J ( l +... J k ( l J( l = d ( -. Periodic Proof: same as 8.. { } d ( - = d ( - essel Periodic Periodic 4

43 Gauge Istitute Joural H. Vic Dao. Bessel Series Theorem The Bessel Series Theorem for a hyper-real fuctio, f (, is the Fudametal Theorem of Bessel Series. It supplies the coditios uder which the Bessel Series associated with f ( equals f (. It is believed to hold i uder Hobso s coditios i the Calculus of Limits. I fact, The Theorem caot be proved i the Calculus of Limits uder ay coditios, because the summatio of the Bessel Series requires itegratio of the sigular Bessel Kerel.. Bessel Series Theorem caot be proved i the Calculus of Limits Proof: Let f ( be itegrable o [,]. I the Calculus of Limits, the Bessel Series is the limit of { f } = aj l + aj l + + a J ( ( (... ( l essel æ = ö = æ ö = f( J ( l d J( l.. f( J ( l d + + J( l ò J( l ò ç J ( l è = ø èç = ø 43

44 Gauge Istitute Joural H. Vic Dao = J( l J( l J( l J( l J( l J( l = f (.. ò ýd. J ( l J( l J( l = As Bessel Sequece, the Bessel Sequece becomes the Bessel Kerel, J( l J( l J( l J( l J( l 3 J( l ý, J( l J( l J( l3 By., the Bessel Kerel diverges to ifiity at ay =. Therefore, while the partial sums of the Bessel Series eist, their limit does ot. Calculus of Limits fails to comprehed the siftig through the values of f ( by the Bessel Kerel, ad the pickig of f ( at =. Avoidig the sigularity at =, by usig the Cauchy Pricipal Value of the itegral does ot recover the Theorem, because for ay ¹, the Bessel Kerel vaishes, ad the itegral is idetically zero, for ay fuctio f (. Thus, the Bessel Series Theorem caot be proved i the Calculus of Limits.. Calculus of Limits Coditios are isufficiet for the Bessel Series Theorem Proof: The Hobso Coditios [Watso, p.59] are. f ( t defied i (, 44

45 Gauge Istitute Joural H. Vic Dao. tf( t, ad tf( t are itegrable o [,] 3. at < a < < b <, (i sup f ( t - if f( t is bouded for ay e > -<<+ e t e -<<+ e t e (ii the limits f ( -, ad f ( + eist. It is clear from. that these coditios o f ( do ot resolve the sigularity of the Bessel kerel, ad are isufficiet for the Bessel Series Theorem. I Ifiitesimal Calculus, by 8.3, the Bessel Kerel is the Delta Fuctio, ad by 9., it equals its Bessel Series. The, the Bessel Series Theorem holds for ay Hyper-Real Fuctio:.3 Bessel Series Theorem for Hyper-real f ( If f ( is hyper-real fuctio itegrable o[,], The, f ( = { f( } Proof: = ò = essel { } f ( = f(... + d( d( - + d( d Substitutig from 8.3, d Periodic ( -, where the period of Delta is 45

46 Gauge Istitute Joural H. Vic Dao we have J( l J( l J( l J( l dperiodic( - = ý, J( l J( l = J( l J( l J( l J( l f ( = f(... ò + + ýd J( l J( l = This Hyper-real Itegral is the summatio, J ( J ( J ( J ( ( ýd = f l l l l å = J( l J( l which amouts to the hyper-real fuctio f (,ad is well-defied. Hece, the summatio of each term i the itegrad eists, ad we may write the itegral as the sum æ = ö æ = ö = f( J ( l d J( l f( J ( l d + J( l... ò J( l ò + ç J ( l è = ø èç = ø a = aj ( l + aj ( l +... essel { f ( } =. I particular, the Delta Fuctio violates Hobso s Coditios The Hyper-real d ( t,is ot defied i the Calculus of Limits, ad is ot itegrable i [,]. a But by 9., d ( - Periodic satisfies the Bessel Series Theorem. 46

47 Gauge Istitute Joural H. Vic Dao.4 -Bessel Series Theorem for Hyper-real f ( If f ( is hyper-real fuctio itegrable o[,], The, f ( = { f( } Proof: same as.3. essel 47

48 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. [Bowma] Bowma, Frak, Itroductio to Bessel Fuctios, Dover, 958. [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 ; [Da] Dao, H. Vic, Ifiitesimals i Gauge Istitute Joural Vol.6 No. 4, November ; [Da3] Dao, H. Vic, Ifiitesimal Calculus i Gauge Istitute Joural Vol. 7 No. 4, November ; [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. [Da5] Dao, H. Vic, The Delta Fuctio i Gauge Istitute Joural Vol. 8, No., February, ; [Da6] Dao, H. Vic, Riemaia Trigoometric Series, Gauge Istitute Joural, Volume 7, No. 3, August. [Da7] Dao, H. Vic, Delta Fuctio the Fourier Trasform, ad the Fourier Itegral Theorem i Gauge Istitute Joural Vol. 8, No., May, ; [Da8] Dao, H. Vic, Ifiite Series with Ifiite Hyper-real Sum i Gauge Istitute Joural Vol. 8, No. 3, August, ; [Gradshtey] Gradshtey, I., S., ad Ryzhik, I., M., Tables of Itegrals Series 48

49 Gauge Istitute Joural H. Vic Dao ad Products, 7 th Editio, edited by Alla Jeffery, ad Daiel Zwilliger, Academic Press, 7 [Gray] Gray, Adrew, ad Mathews, G. B., A treatise o Bessel Fuctios ad their applicatios to Physics, d Editio, Dover 966 [Hardy] Hardy, G. H., Diverget Series, Chelsea 99. [Relto] Relto, F.E., Applied Bessel Fuctios, Dover, 965 [Spiegel] Spiegel, Murray, Mathematical Hadbook of formulas ad tables Schaum s Outlie Series, McGraw Hill, 968. [Todhuter] Todhuter, I., A Elemetary Treatmet o Laplace s Fuctios, Lame s Fuctios, ad Bessel s Fuctios Macmilla, 875. [Trater] Trater, C. J., Bessel Fuctios with some physical Applicatios, Hart, 969 [Watso] Watso, G. N., A treatise o the theory of Bessel Fuctios, Secod Editio, Cambridge, 944. [Weisstei], Weisstei, Eric, W., CRC Ecyclopedia of Mathematics, Third Editio, CRC Press, 9. 49