Gauge Institute Journal, Volume 8, No. 1, February The Delta Function. H. Vic Dannon. September, 2010

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1 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao The Delta Fuctio H. Vic Dao vic0@comcast.et September, 00 Abstract The Dirac Delta Fuctio, the idealizatio of a impulse i Radar circuits, is a Hyper-Real fuctio which defiitio ad aalysis require Ifiitesimal Calculus, ad Ifiite Hyper-reals. The cotroversy surroudig the Leibitz Ifiitesimals derailed the developmet of the Ifiitesimal Calculus, ad the Delta Fuctio could ot be defied ad ivestigated properly. For istace, it is labeled a Geeralized Fuctio although it geeralizes o fuctio. Dirac s ituitive defiitio by Delta s samplig property x = δ( x ) =, x = that avoids specifyig δ(0), remais the mai defiitio of the delta fuctio, although the Delta Fuctio is ot Riema itegrable i the Calculus of Limits, ad is ot Lebesgue itegrable i Measure Theory. I fact, i the Calculus of Limits, oly the Cauchy Pricipal Value

2 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Itegral of the Delta Fuctio exists, ad it equals zero. Oly i Ifiitesimal Calculus, ca the Delta Fuctio be defied, differetiated, ad itegrated. Ifiitesimal Calculus allows us to resolve ope problems such as What is δ(0)? How is xδ( x) defied at x = 0? How is the Delta Fuctio the derivative of a Step Fuctio? How do we itegrate the Delta Fuctio? What is δ (x )? What is δ ( x)? 3 What is δ( x )? What is δ 3 ( x)? The Delta Fuctio eables us to defie the Fourier Trasform with miimal requiremets o the trasformed fuctio. Keywords: Ifiitesimal, Ifiite-Hyper-Real, Hyper-Real, Cardial, Ifiity. No-Archimedea, No-Stadard Aalysis, Calculus, Limit, Cotiuity, Derivative, Itegral, 000 Mathematics Subject Classificatio 6E35; 6E30; 6E5; 6E0; 6A06; 6A; 03E0; 03E55; 03E7; 03H5; 46S0; 97I40; 97I30.

3 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Cotets Itroductio. Hyper-real Lie. Delta Fuctio Defiitio 3. Delta Fuctio Plots 4. Delta Fuctio Properties 5. Delta Sequece δ ( x) = cosh x 6. Delta Sequece x χ [0, ) δ ( x ) = e ( x ) 7. Primitive of Delta Fuctio 8. δ(()) fx 9. δ( x ) 0. δ( x ( ) ). Itegral of δ( x). The Pricipal Value Derivative of Delta: The Dipole Fuctio 3. d Pricipal Value Derivative of Delta: the 4-Pole Fuctio 4. Higher Pricipal Value Derivatives of Delta Refereces 3

4 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Itroductio 0. Cauchy, Poisso, ad Riema Cauchy (86), ad Poisso (85) derived the Fourier Itegral Theorem by usig the siftig property of the Delta Fuctio. By Fourier Itegral Theorem Deotig π k = ξ= ikξ f ( x) = f( ξ) e ξ π k = ξ= ikx d e dk = ξ= k = ik( ξ x) f () ξ e dk ξ π d = k = k = ik( ξ x ) e dk δξ ( x ), k = the Delta Fuctio is the Fourier Trasform of the costat fuctio, Ad Fourier Itegral theorem states the siftig property for the Delta Fuctio ξ= f ( x) = f( ξδξ ) ( x) d ξ. ξ= I the derivatio of his Zeta Fuctio, Riema (859) uses this siftig property repeatedly, without usig a fuctio otatio for ξ 4

5 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao k = ik( ξ x ) the itegral e π dk k = that represets the Delta fuctio δξ ( x). The derivatios are i [Da4, p.84, p.90, p.97]. The derivatios were ot supplied by Riema. Riema s 859 paper, as well as much of Riema s published writigs, outlies ideas, ad states results without proof. I particular, the represetatio of Delta that follows from the Fourier Itegral Theorem does ot hold i the Calculus of Limits. Ideed, ξ ik( ξ x ) = x e =, ad the itegral π k = k = e ik( ξ x ) dk diverges. Avoidig the sigularity at ξ = x does ot recover the Theorem, because without the sigularity the itegral equals zero. Thus, the Fourier Itegral Theorem caot be writte i the Calculus of Limits. I other words, the idetermiate ature of sigularities i the Calculus of Limits does ot allow the Fourier Itegral Theorem to hold. 5

6 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 0. Dirac The Delta Fuctio ca be realized as a Radar trasmissio Pulse. A Radar Trasmissio has to be pulsed because a cotiuous wave trai will ot allow us to measure the time iterval τ betwee trasmissio ad receptio, ad determie the rage of the target by r = cτ. Thus, a trasmissio lasts very short time. The, the Radar system coverts ito a receiver for the reflected sigal. This process of trasmittig ad receivig repeats thousads of time per secod, i order to follow a movig target. The Radar pulse evelops a carrier wave of very short wavelegth. Radar carrier waves wet dow from cetimeters to micrometers of light wavelegth. Sice the illumiatig power dissipatio is proportioal to r, the short electromagetic wave-trai has a electric field that seems early ifiite, although the pulse power is fiite. Dirac (930) was familiar with Radar Pulses whe he defied the Delta Fuctio i [Dirac, p.7] through the siftig property, ad x = δ( x ) =, x = 6

7 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao δ ( x) = 0, for all x 0. Dirac defiitio left ope the questio of the early ifiite amplitude at x = 0. That is, δ(0) was left udefied. The, the siftig property does ot hold. Ideed, sice δ (0) is ifiite, the itegratio of δ( x) has to skip the poit itegral x = 0, ad oly the Cauchy Pricipal Value of the x = x = δ( x ) may exist. The, x = x = lim δ( x ) δ( x ) 0 + = x = x =, x = i cotradictio to δ( x ) =. x = 0.3 Lauret Schwartz Lauret Schwartz presets his Delta Distributio as follows [Schwartz, p. 8] 0, x < 0 Let Y =., x > 0 The, for ay ϕ( x) ifiitely differetiable, that vaishes at, ad at, 7

8 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao x= x= x= Thus, Y '( x) ϕ( x) = Y( x) ϕ'( x) x= x = = ϕ'( x ) = ϕ( x) x = 0 = ϕ(0) x = = δ( x) ϕ( x) x = x = x = 0 Y ' = δ. Sice Yx ( ) is ot defied at x = 0, Y '(0) is ot defied, ad the coclusio Y ' = δ, avoids δ(0). That is, Schwartz Defiitio is as icomplete as Dirac s. Furthermore, the equality x = ϕ(0) = δ( x) ϕ( x) x = is the defiitio of the Delta Fuctio by its siftig property that does ot hold. Sice δ(0) is ot defied, the itegratio has to skip the poit x = 0, ad the itegral is the Cauchy Pricipal Value Itegral 8

9 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao x = x = lim δ( x) ϕ( x) δ( x) ϕ( x) 0 + = x = x = Like the Dirac Delta, the Schwartz Delta avoids δ(0), ad postulates the siftig property. 0.4 Delta Sequece Attempts to get back to the sigular Delta Fuctio, replaced the Delta Fuctio by a Delta Sequece of fuctios that coverge to the Delta Fuctio. For istace, x, [, δ ( x) χ ( x) 0, otherwise = = [, ] ]. The, the delta Fuctio is defied as the limit δ( x) = lim δ ( x). The sequetial approach is reviewed i [Mikusiski], ad is used i Mathematical Physics texts. However, the Delta Sequece cotradicts Dirac s defiitio. Ideed, as, δ(0) = lim δ (0) = lim =. The, the itegratio of δ( x) has to skip the poit x = 0. 9

10 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao That is, oly the Cauchy Pricipal Value of the itegral x = x = δ( x ) may exist. The Pricipal Value is x = x = lim δ( x ) δ( x ) 0 + = x = x =. That is, the siftig x = x = δ ( x ) =, is ot preserved for the limit of the Delta sequece, δ( x) = lim δ ( x). 0.5 The Hyper-real Delta Fuctio The above attempts failed because the Delta Fuctio is a hyperreal fuctio. A fuctio from the hyper-reals ito the hyperreals. By resolvig the problem of the ifiitesimals [Da], we obtaied the Ifiite Hyper-reals that are strictly smaller tha, ad ca serve to supply the value of the Delta Fuctio at the sigularity. The attempts to get by with Calculus restricted to the real lie, deprived Calculus of its full power. I Ifiitesimal Calculus, [Da3], we differetiate over a jump discotiuity of a step fuctio, ad obtai the Delta Fuctio. We ca itegrate over a sigularity, ad obtai a fiite value. 0

11 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Here, we preset the Delta Fuctio, ad the properties of the Delta Fuctio i Ifiitesimal Calculus. I particular, we resolve ope problems such as What is δ(0)? How is xδ( x) defied at x = 0? How is the Delta Fuctio the derivative of a Step Fuctio? How do we itegrate the Delta Fuctio? What is δ (x )? What is δ ( x)? 3 What is δ( x )? What is δ 3 ( x)?

12 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao. Hyper-real Lie Each real umber α ca be represeted by a Cauchy sequece of ratioal umbers, ( r, r, r,...) so that r α. 3 The costat sequece ( ααα,,,...) is a costat hyper-real. I [Da] we established that,. Ay totally ordered set of positive, mootoically decreasig to zero sequeces ifiitesimal hyper-reals. ( ι, ι, ι3,...) costitutes a family of. The ifiitesimals are smaller tha ay real umber, yet strictly greater tha zero. 3. Their reciprocals (,,,... ι ι ι 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. 6. 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

13 Gauge Istitute Joural, Volume 8, No., February 0 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,.. 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. 3

14 Gauge Istitute Joural, Volume 8, No., February 0 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. 4

15 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao. Delta Fuctio Defiitio. Domai ad Rage The Delta Fuctio is a hyper-real fuctio defied from the hyper-real lie ito the set of two hyper-reals The hyper-real 0,. 0 is the sequece 0, 0, 0,.... The ifiite hyper-real depeds o our choice of. 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, will mea the sequece. Alteratively, we may choose the family spaed by the sequeces,, 3, The, will mea the sequece. Oce we determied the basic ifiitesimal, we will use it i the Ifiite Riema Sum that defies a Itegral i 5

16 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Ifiitesimal Calculus.. The Delta Fuctio is strictly smaller tha Proof: Sice > 0, <..3 Defiitio of the Delta Fuctio We defie, where δ ( x) χ ( ), x, χ, ( x), x, =. 0, otherwise This meas that for x <, δ ( x ) = 0 at x =, δ( x) jumps from 0 to, for,, ( x) x δ =. at x = 0, δ (0) = at x =, δ( x) drops from for x >, δ ( x) = 0. to 0. 6

17 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao, x,.4 δ( x) = 0, otherwise is the sequece i i, x, i 0, otherwise where = i. Namely, as a hyper-real fuctio the value of Delta at the sigularity is the ifiite hyper-real which is a sequece, a ifiite vector with coutably may compoets. 7

18 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 3. Delta Fuctio Plots 3. Delta Plot for = If i =, Delta is the ifiite Hyper-Real umber, δ χ χ χ ( x) = ( x), ( x),3 ( x),... [,] [, ] [, ] 3 3 We plot i Maple the 0 th compoet with Similarly, we use 8

19 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao to plot a the 00 th compoet of Delta 3. Delta with = is We use χ[, ] [, ] [, ] δ χ χ ( x) = ( x), 4 ( x),8 ( x ), to plot the 4 th compoet of Delta 9

20 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Similarly, we use to plot the 6 th compoet of Delta, 0

21 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 4. Delta Fuctio Properties 4. xδ ( x) = 0 Proof: x 0 δ ( x) = 0 xδ ( x) = 0. x = 0 0 xδ( x) = 0 δ(0) = = 0, sice > δ ( x x ) χ ( x ) 0 x ( x, 0) 0 x 0+ = χ ( ) 0, x x x χ δ ( x) = ( ), x, =, 3,... ( ) That is, δ (0) spikes to ( ), which is greater tha. 4.4 (, xy) ()() x y = [, () x () ] dy dy y [, ] dy δ δ δ χ χ

22 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 5. Delta Sequece δ ( ) = x cosh x Depedig o the choice of the ifiitesimal = i, there are may Delta Sequeces that lead to the Delta Fuctio, δ( x). 5. Each δ ( ) = x cosh x x = has the siftig property δ ( x ) = is cotiuous x = peaks at x = 0 to δ (0) = x = x = tahx = = ( ) =. cosh x Proof: ( ) x = x = The sequece represets the hyper-real Delta Fuctio 5. If i =, 3 δ ( x) =,,,... cosh x cosh x cosh 3x

23 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao plots i Maple, the 50 th compoet, plots i Maple the 00 th compoet, 3

24 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 6. Delta Sequece δ x χ [0, ) ( ) = e ( x) x 6. Each δ x ( x ) e = χ [0, ) x = has the siftig property δ ( x ) = x = is cotiuous hyper-real fuctio peaks at x = 0 to δ (0) = x= x= x x e Proof: e χ[0, ) ( x) = e = = x= x= 0 x x =. x = 0 The sequece represets the hyper-real Delta Fuctio 6. If i =, x x 3x [0, ) [0, ) [0, ) δ( x) = e χ, e χ, 3 e χ,... 4

25 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao plots i Maple the 00 th compoet, plots i Maple the 00 th compoet, 5

26 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 7. Primitive of Delta Fuctio δ( x) 7. is the derivative of, x < 0 gx ( ) = 0, x= 0, x > 0 Proof: At the jump over [, 0], from to 0, for ay, Therefore, the left derivative at g(0) g(0 ) 0 ( ) = = x = 0 is g '(0 ) =. At the jump over [0, ], from 0 to, for ay, Therefore, the right derivative at g(0 + ) g(0) 0 = = x = 0 g '(0 + ) =. is 6

27 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Sice the right ad left derivatives are equal, the derivative at x = 0 is g '(0) = = δ(0). Sice at x 0, g'( x ) = 0, we have, g'( x) =, χ. 7. δ( x) is the Pricipal Value derivative of Proof: For ay, hx ( ) 0, x 0 =, x > 0 h(0) h( ) 0 = = 0 h '(0 ) = 0. h ( ) h(0) 0 = = h '(0 + ) =. Therefore, hx ( ) has o derivative at x = 0. But sice h( ) h( ) 0 = =, hx ( ) x = 0 The pricipal value derivative of at is p.v. h '(0) =. Sice at x 0, p.v. h'( x ) = 0, we have, p.v. h'( x) =, χ. 7

28 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 8. δ (()) fx 8. δ( ax) = δ( x) a Proof: δ( ax) a = δ( ax) d( ax) = = δ( x). We divide both sides by a, ad put a, because the Delta s o both sides are positive. 8. If ξ is the oly zero of ( ) f x, ad f '( ξ) 0, The, δ(()) fx = δ( x ξ ) f '( ξ ) Proof: δ(()) f x = δ( f() x f( ξ )) For x ξ = ifiitesimal, By 8., ( f '( )( x )) = δ ξ ξ = δ( x ξ). f '( ξ ) 8

29 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 8.3 If ξ, ξ are the oly zeros of f ( x ), ad f '( ξ), f '( ξ) 0 The, δ (()) fx = ( ) ( ) f '( ξ ) δ x ξ + f '( ξ ) δ x ξ Proof: δ δ( ξ ) δ( ( f ( x)) = f( x) f( ) + f( x) f( ξ ) If x ξ = ifiitesimal, f ( x) f( ξ) = f '( ξ)( x ξ) If x ξ = ifiitesimal, f ( x) f( ξ) = f '( ξ)( x ξ) Either way, ( ) ( δ( f ( x)) = δ f '( ξ )( x ξ ) + δ f '( ξ )( x ξ ) = ( x ) ( x ) f '( ξ ) δ ξ + '( ) δ ξ. f ξ ) ) 8.4 δ ( x a ) = ( ) ( a δ x a + a δ x + a ) ( x a)( x b) = δ( x a) + δ ( x + a) a b b a 8.5 δ( ) 8.6 If ξ,... ξ are the oly zeros of ( ) f x, ad f '( ξ),.., f '( ξ ) 0 The, δ (()) fx = ( x )... ( x ) f '( ξ ) δ ξ + + f '( ξ ) δ ξ 9

30 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 8.7 If ξ, ξ,... are zeros of ( ) f x, ad f '( ξ), f '( ξ),... 0 The, δ ( fx ( )) = ( x ) ( x )... f '( ξ ) δ ξ + f '( ξ ) δ ξ δ(si x) =... + δ( x + π) + δ( x + π) + δ( x) + δ( x π) + δ( x π) +... Proof: The zeros of si x are... π, π, 0, π, π,... ad cos( π ) =. 30

31 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 9. δ ( x ) 9. δ ( x ) χ x, x ( x x = ), x > 0 Proof: δ( x ) = χ ( x) ( ) d( x ) d( x ), = χ ( ), x d( x ) d( x ) x, where to esure ( ) > 0, we must have x > 0. The amplitude ad domai of the δ( x ) spike deped o x. For istace, 9. x χ x, x x = x = ( ) δ( x) χ χ 9.3 x, x ( x) = ( ) ( ) ( x) δ ( x) x x = ( ), 3

32 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao δ( ) χ ( x) 9.4 x = ( ) ( d x d x ) ( ), χ x, x x = ( x), x > 0 The amplitude ad domai of the δ x ) spike deped o x. For istace, ( 9.5 x χ x, x = ( x) = δ( x) x ( ) 3

33 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 0. δ( x ( ) ) While x ( ) is ifiitesimally close to, differet from δ( x ). x ( x ( ) ) δ is 0. δ( x ) Proof: By 8.4, sice > 0. ( ) = δ( x ) + δ ( x + ) 0. has two positive spikes. For istace, 0. δ( x ) + δ( x + ) = δ( ) + ( δ ) x = 0 = χ [,0] ( x) + χ [0, ]( x). ( ) ( ) Similarly, ( 3 3 δ x ( ) ) has three positive spikes. 3 3 i 0.3 ( ) ( ( π i ) ( π δ x ( ) δ( x ) δ x e 3 δ x e 3 )) = ( ) Proof: 3 x ( ) 3 has the three zeros 33

34 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao x =, i π 3 x = e, ad 3 i π 3 x = e. Sice f '( x ) = 3x, ad sice x x3 ( d ) = = x, by 6.6 we obtai 3 3 i ( ) ( ( π i ) ( π x ( ) ( x ) x e 3 x e 3 )) δ = δ + δ + δ. 3( ) ( x ( ) ) δ has positive spikes. π 0.4 ( ) ( ( ) ( ( ( ) ( )... ) π δ x = δ x + δ x e + + δ x e )) ( ) Proof: x ( ) has the zeros x =, i π ( ) i π x = e,, x = e. Sice f '( x) = x, ad sice x =... = x = ( ), by 6.6 we obtai π ( ) ( ( ) ( ( ( ) ( )... ) π x x x e x e )) δ = δ + δ + + δ. ( ) 34

35 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao. Itegral of δ( x). Itegral of a Hyper-real Fuctio Let f () x be a hyper-real fuctio o the iterval [, ab]. 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 + ], height f () x, ad area f ( x. ) We form the Itegratio Sum of all the areas for the x s that start at x = a, ad ed at x = b, f ( x ). x [ a, b] If for ay ifiitesimal, 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 35

36 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao x= b f ( x ). x= a If the hyper-real is ifiite, the it is the itegral over [, ab], If the hyper-real is fiite, x= b fx ( ) = 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 ) =... = = =.... 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 ( x. ) The Lower Itegral is the Itegratio Sum where f ( x ) is replaced 36

37 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao by its lowest value o each iterval.3 x [ a, b] [ x, x + ] if f ( t) x t x+ The Upper Itegral is the Itegratio Sum where f ( x ) is replaced by its largest value o each iterval.4 x [ a, b] [ x, x + ] sup f ( t) x t x+ If the itegral is a fiite hyper-real, we have.5 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. x =.6 δ( x ) =. x = Proof: The oly term i the itegratio Sum is =. Both the upper itegral, ad the lower itegral are equal to =. 37

38 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao. The Pricipal Value Derivative of Delta: the Dipole Fuctio We have see i.3 that at x =, δ( x) jumps from 0 to, for,, ( x) x δ =. I particular, δ (0) = at Here, we show x =, δ( x) drops from to 0. i., that δ( x) has o derivative at x =, but the Pricipal Value Derivative over the jump at x =, is a Positive Impulse Fuctio. i., that δ( x) has o derivative at x =, but the Pricipal Value Derivative over the jump at x =, is a Negative Impulse Fuctio. We sum up., ad.. i.3. 38

39 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Namely, δ( x) has o derivative at x = 0, but the Pricipal Value Derivative over the two jumps is a Dipole Fuctio. That Dipole fuctio is a positive Impulse Fuctio followed by a egative Impulse fuctio. Both the positive, ad the egative impulses have jumps far greater tha the jump of the geeratig delta fuctio.. The Pricipal Value Derivative of Delta at x = δ( x) has o derivative at x =. The Pricipal Value Derivative of δ( x) at x =, is the Positive Impulse fuctio χ[,0]. ( ) Proof: The left derivative of ( x) at δ x = is ( ) δ( ) 0 = = + ( ) δ The right derivative of δ( x) at x = is δ(0) δ( ) = = 0 ( ) 0. Sice the left ad right derivatives are uequal, δ( x) has o derivative at x =. 39

40 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao The Pricipal Value Derivative at x = is δ(0) δ( ) 0 = = 0 ( ) ( ). It is the Positive Impulse Fuctio ( ) χ[,0].. The Pricipal Value Derivative of Delta at x = δ( x) has o derivative at x =. The Pricipal Value Derivative of δ( x) at x =, is the Negative Impulse Fuctio χ[0, ]. ( ) Proof: The Left Derivative of ( x) at δ δ x ( ) δ(0) 0 = is = = = 0. The Right Derivative of δ( x) at x = is ( ) δ( ) 0 = = ( ) δ Sice the Left ad Right Derivatives are uequal,. δ( x) has o derivative at x =. 40

41 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao The Pricipal Value Derivative at x = is δ( ) δ(0) 0 = =. ) ( ) It is the Negative Impulse Fuctio ( ) χ[0, ]..3 The Pricipal Value Derivative of Delta at x = 0 δ( x) has o derivative at x = 0. The Pricipal Value Derivative of δ ( x), p.v.d δ( x) is the Dipole Fuctio Dipole( x) = χ[,0] χ[0, ]. ( ) ( ) Proof: The Left Derivative of δ( x) at x = 0 is δ(0) δ( ) 0 = = ( ) The Right Derivative of δ( x) at x = 0 is δ( ) δ(0) 0 = =. ( ) Sice the left ad right derivatives are uequal, δ( x) has o derivative at x = 0. The Pricipal Value Derivative of δ( x) is 4

42 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao δ( x + ) δ( x ) = δ( x + ) δ( x ). It is the Dipole Fuctio χ[,0] χ[0, ]. ( ) ( ) If =, this is the sequece χ χ Dipole( x) = [,0] [0, ]. The, a Maple plot of the 0 th compoet of Dipole( x) is 4

43 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 3. The d Pricipal Value Derivative of Delta: the 4-Pole Fuctio The d Pricipal Value Derivative of δ( x) is the 4-pole Fuctio 3. (p.v.d) δ( x) = ( δ( x + ) δ( x) + δ( x ) ) ( ) { 3 3 = χ[, ] χ[, ] + χ[, ]}. 3 ( ) Proof: Dipole( x + ) Dipole( x (p.v.d) δ( x) = )` = + + ) ( ) ( δ( x ) δ( x) δ( x ) { 3 3 = χ[, ] χ[, ] + χ[, ]}. 3 ( ) The 4-pole Fuctio has four Impulse Fuctios a Positive Impulse δ ( x + ) cetered at x = d x, ( ) 43

44 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao two Negative Impulses δ( x) cetered at x = 0, ( ) a Positive Impulse δ( x ) cetered at x =. ( ) If =, this is the sequece χ χ 4 pole( x) = [, ] [, ] + χ[, ]. The, a Maple plot of a compoet of 4 pole( x) is The x axis uits are. The y axis uits are 3. 44

45 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 4. Higher Pricipal Value Derivatives of Delta 4. The 3 rd Pricipal Value Derivative of ( x), δ 3 (p.v.d) δ( x) is the 8-pole Fuctio 8 pole( x) = ( χ[, ] 3 χ[,0] + 3 χ[0, ] χ[, ] ). 4 ( ) If =, this is the sequece χ χ χ 8 pole( x) = [, ] 3 [,0] + 3 [0, ] 4 χ[, ]. The, a Maple plot of a compoet of 8 pole( x) is The x axis uits are. The y axis uits are 4. 45

46 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao 4. The 4 th Pricipal Value Derivative of ( x), (p.v.d) δ( x) δ 4 is the 6-pole Fuctio pole( x) = ( χ[, ] 4 χ[, ] + 5 ( ) +6 χ[, ] 4 χ[, ] + χ[, ] ) If =, this is the sequece ( 5 χ χ 3 6 pole( x) = [, ] 4 [, ] χ χ χ +6 [, ] 4 [, ] + [, ]. The, a Maple plot of a compoet of 6 pole( x) is ) The x axis uits are. The y axis uits are 5. 46

47 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Usig the Biomial coefficiets, 4.3 The k th Pricipal Value Derivative of δ ( x), (p.v.d) δ( x) is k k pole( x) = k k pole x + pole x + ( ) ( k = χ[, ] [, k+ ( ) ( k+ ) ( k ) ( k ) ( k 3) χ ) ]+ k ( k ) ( k 3) k ( k ) ( k+ ) + χ [, ]... ( ) [, ] + + χ ). If =, this is the sequece ( k ) ( k ) k+ + k+ ( ) ( 3) χ[, ] χ[, ] + k k+ k k 3 k k k k+ + + χ[, ]... ( ) χ[, ] + + ) 47

48 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao Refereces [Da] Dao, H. Vic, Well-Orderig of the Reals, Equality of all Ifiities, ad the Cotiuum Hypothesis i Gauge Istitute Joural of math ad Physics, Vol.6 No, May 00; [Da] Dao, H. Vic, Ifiitesimals i Gauge Istitute Joural of math ad Physics, Vol.6 No 4, November 00; [Da3] Dao, H. Vic, Ifiitesimal Calculus i Gauge Istitute Joural of Math ad Physics, Vol.7 No, February 0; [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; [Dirac] Dirac, P. A. M. The Priciples of Quatum Mechaics, Secod Editio, Oxford Uiv press, 935. [He] Hele, James M., ad Kleiberg Eugee M., Ifiitesimal Calculus, MIT Press 979. [Hosk] Hoskis, R. F., Stadard ad Nostadard Aalysis, Ellis Horwood, 990. [Keis] Keisler, H. Jerome, Elemetary calculus, A Ifiitesimal Approach, Secod Editio, Pridle, Weber, ad Schmidt, 986, pp [Laug] Laugwitz, Detlef, Curt Schmiede s approach to ifiitesimals-a eyeopeer to the historiography of aalysis Techische Uiversitat Darmstadt, Preprit Nr. 053, August 999 [Mikusiski] Mikusiski, J. ad Sikorski, R., The elemetary theory of distributios, Rosprawy Matematycze XII, Warszawa

49 Gauge Istitute Joural, Volume 8, No., February 0 H. Vic Dao [Rad] Radolph, Joh, Basic Real ad Abstract Aalysis, Academic Press, 968. [Riema] Riema, Berhard, O the Represetatio of a Fuctio by a Trigoometric Series. () I Collected Papers, Berhard Riema, traslated from the 89 editio by Roger Baker, Charles Christeso, ad Hery Orde, Paper XII, Part 5, Coditios for the existece of a defiite itegral, pages 3-3, Part 6, Special Cases, pages Kedrick press, 004 () I God Created the Itegers Edited by Stephe Hawkig, Part 5, ad Part 6, pages , Ruig Press, 005. [Schwartz] Schwartz, Lauret, Mathematics for the Physical Scieces, Addiso-Wesley, 966. [Temp] Temple, George, 00 Years of Mathematics, Spriger-Verlag, 98. pp