Delta Function and Expansion in Hermite

Gauge Institute Journal H. Vic Dannon

Delta Function, and Expansion in Hermite Functions H. Vic Dannon [email protected] June, 2012

Abstract Let f()x be defined on the real numbers, and let

Hxn() be the Hermite Polynomials on the real numbers,

2 3 Hx0()= 1, Hx1()= 2 x, Hx2()=− 4 x 2, Hx3()=− 8 x 12 x,… The Hermite Series associated with f()x is

aH00( x )+++ aH 11 ( x ) aH 22 ( x ) .... where ξ=∞ 1 2 ae= −ξ f()ξξH ()dξ nnn ∫ 2!n π ξ=−∞ are the Hermite coefficients. The Hermite Series Theorem supplies the conditions under which the Hermite Series associated with f()x equals f()x .

It is believed to hold in the Calculus of Limits for smooth enough function. In fact,

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The Theorem cannot be proved in the Calculus of Limits under any conditions, because the summation of the Hermite Series requires integration of the singular Hermite Kernel. Plots of partial sums of the Hermite Series speak volumes about the sensibility of the claims to have infinity bound by epsilon. In Infinitesimal Calculus, the Hermite Kernel

2 11eHHx−ξ (ξξ ) ( )++ ... HHx ( ) ( )+ ... π {}00 2!n n nn is the Delta Function, δξ()− x .

δξ(− x) equals its Hermite Series, and the Hermite Series associated with any hyper-real integrable f ()x , equals f ()x

Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real, infinite Hyper-real, Infinitesimal Calculus, Delta Function, Hermite Polynomials, Hermite Coefficients, Delta Function, Hermite Series, Hermite Kernel, Expansion in Hermite Functions,

2000 Mathematics Subject Classification 26E35; 26E30;

26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15; 46S20; 97I40; 97I30.

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Contents

0. The Origin of the Hermite Series Theorem 1. Divergence of the Hermit Kernel in the Calculus of Limits 2. Hyper-real line. 3. Integral of a Hyper-real Function 4. Delta Function 5. Convergent Series 6. Hermite Sequence and δξ()− x

7. Hermite Kernel and δξ()− x .

8. Hermite Series of δξ()− x

9. Hermite Series Theorem References

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The Origin of the Hermite Series Theorem

The Hermite Polynomials on (−∞, ∞)

2 3 Hx0()= 1, Hx1()= 2 x, Hx2()=− 4 x 2, Hx3()=− 8 x 12 x,…, are orthogonal so that

x =∞ −xn2 ∫ eHxHxdxnmn() ()= 2 ! πδmn. x =−∞ The Hermite Polynomials can be generated by expanding

2 exxx222xαα− =+1 [2αα − ] +11 [2 αα − ]2 + [2 αα −2 ]3 + ... 2! 3!

=+12xxxαα −22234 +1 [4 α − 4 α + α ] + 2!

+−+−1 [8xxx33αααα 12 24 6 5 6 ]+ ... 3! the coefficient of 1 α0 is 0!

Hx0()= 1, the coefficient of 1 α1 is 1!

Hx1()= 2 x, the coefficient of 1 α2 is 2! 2 Hx2()=− 4 x 2, the coefficient of 1 α3 is 3! 3 Hx3()=− 8 x 12 x, …………………………

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0.1 Schrodinger Equation for atomic size particle in

linear harmonic motion

An atomic size particle with mass m , oscillates along a segment of wire [−AA,], at frequency ν , under the force −kx .

The particle’s position is xt()= A cosω t, ωπ= 2 ν. Thus, xA =−ωωsin t

xAt =−ωωω22cos =− x The force equation is −=kx mx = m() −ω2 x .

Hence, the force constant is km= ω2 , and the potential energy of the particle is

Vkxm==1122ω x2. 22 De Broglie associated with the moving particle a wave of length

h λ = , mv where v is the velocity of the particle, and h is Planck’s constant. The wave’s frequency is

vvmv2 ν == = . λ h h mv The wave’s angular frequency is

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mv2 ωπνπ==22 h In terms of the De Broglie wave, the particle’s energy is a multiple of Planck’s radiation energy,

Eh==εν εω= , = = h , ε is the multiplier. 2π The kinetic energy of the particle is

1 mv2 =− E V . 2 Hence, mv=−2( m E V ),

h λ = , 2(mE− V )

=ω v ==λνN . 1 ω 2(mE− V ) 2π

12(mE− V) = v222= ω Schrodinger postulated a complex valued potential

Ψ=(,)xtψ () xeitω that satisfies the wave equation

221 ∂Ψxt(,)xt = ∂Ψ (,) xt . v2 Then,

221 0(,)(=∂xt Ψxt − ∂Ψ xt,) v2

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2(mE− V ) =−ψψ"(xe ) itωω(x )( −ω2 ) eit. =22ω The Schrodinger equation for the linear harmonic oscillator is

2m ψψ"(xEVx )+− ( ) ( ) = 0 . =2 Substituting E , and V ,

2m ψεωωψ"(+−= 1 mx22 )=0 =2 2 = Multiplying by , mω = mω ψε"(xxx )+− (22 ) ψ ( ) = 0 . mω = ξ2

The change of variable ξ = mωx , gives = dddψψξ ==ψξ'( ) mω , dx dξ dx =

dd2ψξ d =={}ψξ'( ) mmωω ψ''( ξ ) , dx 2 ddξ ==x and the equation becomes

ψξ"( )+− (2 ε ξψ2 ) (x ) = 0 .

0.2 Hermite Differential Equation

The Schrodinger equation

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ψξ"( )−=− ξψ2 (xx ) 2 εψ ( ) can be factored

()()()2DDξξ−+ξξψεψ x =−()x.

To solve the homogeneous equation

()()()DDξξ−+ξξψ x =0, we solve ψ ' 1 ξ2 ()()Dx−=ξψ 0 ⇒ = ξ ⇒ log ψξ=+1 2 c ⇒ ψ = Ce 2 . ξ ψ 2 1

As ξ →∞, ψ1 →∞, and is discarded.

ψ ' −1 ξ2 ()()Dx+=ξψ 0 ⇒ =−ξ ⇒ log ψξ=−1 2 +c ⇒ ψ = Ce 2 . ξ ψ 2 2 Now, substituting −1 ξ2 ψξ()= He () ξ 2 in ψξ"( )+− (2 ε ξψξ2 ) ( ) = 0 , we have

−−11ξξ22 0()(2)()=+DH22ξεξξ e22−H e ξ ()

−−11ξξ22 −1ξ2 =−+DH'(ξξξεξξ ) e22 H ( ) e (2−2 )H ( ) e2 ξ ()

−−−111ξξξ222−1ξ2 =−H''()ξξξξξξ e222 2 H '() e −+ He ()2 He () 2+

−1 ξ2 +−(2εξ2 )He ( ξ ) 2

−1 ξ2 ⎡⎤2 =−⎣⎦HH''(ξξξε ) 2 '( ) +− (2 1) He ( ξ ) . The Schrodinger equation becomes Hermit Differential Equation HH''(ξξξε )−+− 2 '( ) (2 1) H ( ξ )= 0,

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Substituting in it

21ll++ l2 Hccc()ξξξξξξ=+01 + 2 ++... ccll +++1 + c l 2 +..., we have

ll=∞ =∞ l=∞ 2 ll l Dcξξ∑∑llξξ−+−2( Dc ξ21 ε) ∑c l ξ=0, ll ==00 l = 0 ll=∞ =∞ ll−−21 ∑∑(1)llc− llξξlc ll==21

l =∞ l ∑ {(llclcc++ 1)( 2)ll+2 −+− 2 (2εξ 1) l } = 0 , l =0

(1)(2)[212]llc++ll+2 −+−= lε c0

212l +−ε cc= ll+2 (1)(2)ll++

The solution is

Hccc()ξξξ=+ +12−−εε23 + c 32ξ + 01 012⋅⋅ 123

++cc(12)(52)−−εεξξ45 (32)(72) −− εε+... 011234⋅⋅⋅ 2345⋅⋅⋅

=+c {1 12− ε ξξ24 +(12)(52)−−εε +...} + 0 12⋅⋅1234⋅⋅

++c ξξ{1 32− ε 24 +(3−− 2εε )(7 2 ) ξ +...}. 1 23⋅⋅2345⋅⋅ To keep the solution from diverging at ξ →∞, for n= 2k, the c0 series terms vanish for 2ε =+ 1,5,9,13,...4k 1,..., and we obtain the H2k ()ξ Hermite Polynomials.

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for nk=+21, the c1 series terms vanish for 2ε =+ 3,7,11,..., 4k 3,... and we obtain the H21k+ ()ξ Hermite Polynomials. A solution for ψξ() is the infinite linear combination

−−11ξξ22 − 1 ξ2 22 2 αξ00He()++ αξ11 He () αξ22 He ()+ ....

0.3 The Hermite Series Associated with f ()x

Let f ()x be defined on (,−∞ ∞) , and let Hxn() be the Hermite Polynomials

2 3 Hx0()= 1, Hx1()= 2 x, Hx2()=− 4 x 2, Hx3()=− 8 x 12 x,… The Polynomials are orthogonal on (−∞, ∞). That is,

x =∞ −xn2 ∫ eHxHxdxnmn() ()= 2 ! πδmn x =−∞ We define the Orthonormalized Hermite Functions

1 −1x 2 2 ϕnn()xe= 1 H()x (2n n !π )2

If f ()x can be expanded in the ϕn()x ,

f ()xxxx=++αϕ00 () αϕ 11 () αϕ 22 ()+ ..., Then,

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xx=∞ =∞

∫∫f ()xϕαnn () x dx =+++ {00ϕα ()x 11ϕα () x 22ϕ () x ...}()ϕ x dx xx=−∞ =−∞

xxx=∞ =∞ =∞ =++αϕϕαϕϕαϕϕ()xxdx () () xxdx () () xxdx ()+ .. 00∫∫∫nn 11 22n xxx =−∞ =−∞ =−∞

δδδ012nn n

= αn .

Thus, the Hermite coefficients with respect to the ϕn()x are

ξ=∞

αξϕnn= ∫ f () ()ξξd . ξ=−∞

The Orthonormal Hermite Series associated with f ()x is

αϕ00()xxx++ αϕ 11 () αϕ 22 ()+ ....

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Divergence of the Hermit Kernel in the Calculus of Limits

Calculus of Limits Conditions for the Hermite Series to equal its function reflect the belief that a smooth enough function equals its Hermite Series. In fact, in the Calculus of Limits, no smoothness of the function guarantees even the convergence of the Hermite Series.

1.1 The Hermite Kernel is either singular or zero

In the Calculus of Limits, the Hermite Series is the limit of the sequence of Partial Sums

HSermite n {}f ()xx=++αϕ00 () ... αϕn n ()x

⎛⎞⎛ξξ=∞ =∞ ⎞ ⎜⎜⎟⎟ =+⎜⎜f ()ξϕ () ξdx ξ⎟⎟ ϕ () ..+ f ()ξϕ () ξ d ξ ϕ ()x ⎜⎜∫∫00⎟⎟nn ⎝⎠⎝⎜⎜ξξ=−∞ ⎟⎟=−∞ ⎠

ξ=∞

=+∫ f ()ξϕξϕ{}00 () ()xx ...+ ϕξϕnn () ()d ξ. ξ=−∞

As n →∞, the orthonormal Hermite Sequence

ϕξϕ00() ()xx++ ... ϕξϕnn () ()

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becomes the orthonormal Hermite Kernel,

ϕξϕ00( ) (xx )++ ... ϕξϕnn ( ) ( ) + ..., To see that it is singular at ξ = x , we apply the Christoffel Summation Formula, [Sansone, p.371],

n + 1 ϕξϕϕξϕ() ()xx− () () ϕξϕ() ()xx++ ... ϕξϕ () () = nn++11 nn. 00 nn 2 ξ − x

For ξ → x ,

22 ϕξϕ00( ) (xxx )++ ... ϕξϕnn ( ) ( ) → ϕ0 ( ) ++ ... ϕn (x ), and nn++11ϕξϕϕξϕ() ()xx− () () 0 nn++11 nn→ . 22ξ − x 0 Applying Bernoulli’s rule to the indeterminate limit,

ϕξϕϕξϕ() ()xx− () () DxDxϕξϕ() ()− ϕξϕ () () lim nn++11 nn= lim ξξnn++11 nn ξξ→→xx ξξ−−xDξ()x

=−lim[ϕξϕϕξϕnnnn++11 '( ) (xx ) '( ) ( )] ξ→x

=−ϕϕϕϕnnnn++11'(xx ) ( ) '( x ) ( x ) Therefore,

n + 1 ϕϕ22()xx++ ... () = [ ϕϕϕϕ '()() xxx− '() ()]x. 01nn2 ++nnn1

Since ϕn()x , and ϕn+1()x solve the differential equation, [Szego, p.105, #5.5.2],

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2 1N ⋅+⋅++−=zx ''( ) 0N zx '( ) (2 n 1 xzx ) ( ) 0, ax() bx() we have, − bx()dx ∫ ax() ϕϕϕϕnnnn++11'(xx ) ( )−= '( x ) ( x ) ( conste )

0⋅dx = ()const e∫

= const , for any −∞

n + 1 ϕϕ22()xx++ ... () = const 0 n 2 and the Hermite Kernel diverges to ∞ at any ξ = x . Therefore, while the partial sums of the Hermite Series exist, their limit does not. That is, due to the singularity at ξ = x , the Hermite Series does not converge in the Calculus of Limits. Avoiding the singularity at ξ = x , by using the Cauchy Principal Value of the integral does not recover the Theorem, because at any ξ ≠ x , the Hermite Kernel vanishes, and the integral will be identically zero, for any function f ()x .

To see that the kernel vanishes for any ξ ≠ x , we apply the Christoffel Summation Formula, with ξ ≠ x .

ϕξϕϕξϕ() ()xx− () () ϕξϕ() ()xx++ ... ϕξϕ () () =n+1 nn++11 nn. 00 nn 2 ξ − x

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We have 1 −1x 2 2 ϕnn()xe= 1 H()x (2n n !π )2

1 1x 2 2 2 −x = 1 ee Hxn() (2n n !π )2

By [Szego, p. 105, #5.5.3],

−−xn22nx eHxnx()=− ( 1) De. Thus, n (1)− 1x 2 2 2 nx− ϕnx()xe= 1 D{}e, (2n n !π )2

n+1 (1)− 1x 2 2 2 nx+−1 ϕnx+1()xe= 1 D{}e (2n+1 (n + 1)!π )2 and

ϕξϕϕξϕ() ()xx− () () n+1 nn+1 nn+1= 2 ξ − x

21n+ 1(1n+1 − )nnxnn+−11ξξ22 − − 2 +−x2 =−{Dξξ{}{} e Dexx De {} D {} e } ξ − x 2 22!(1)!n πnn+ →→∞0, n →→∞0, n

→→0, as n ∞. That is, the Hermite Kernel vanishes for any ξ ≠ x . Plots of the Hermite Sequence confirm that In the Calculus of Limits, the Hermite Kernel is either singular or zero

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2 1.2 Plots of 11eHHx−ξ { (ξξ ) ( )++ ... HHx ( ) ( )} π 00 2!n n nn

223 2 In Maple, plot( 11 e−x * HermiteH (,.5)* i HermiteH (, i x ), x =− 23..23) ∑ π 2!i i i=0

223 2 In Maple, plot(*(11 e−x HermiteH i,−=1)*( HermiteH i, x),2 x −3..23) ∑ π 2!i i i =0

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2 e−x that suppresses oscillations away from the origin, enhances them at the origin. Thus, a singularity away from the origin needs more terms

223 2 In Maple, plot( 11 e−x * HermiteH ( i , 2) * HermiteH ( i , x ), x =− 23..23) ∑ π 2!i i i =0

The plots confirm that the Hermite Series Theorem cannot be proved in the Calculus of Limits.

1.3 Infinitesimal Calculus Solution

By resolving the problem of the infinitesimals [Dan2], we obtained the Infinite Hyper-reals that are strictly smaller than ∞, and constitute the value of the Delta Function at the singularity. The controversy surrounding the Leibnitz Infinitesimals derailed the development of the Infinitesimal Calculus, and the Delta

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Function could not be defined and investigated properly. In Infinitesimal Calculus, [Dan3], we can differentiate over jump discontinuities, and integrate over singularities. The Delta Function, the idealization of an impulse in Radar circuits, is a Discontinuous Hyper-Real function which definition requires Infinite Hyper-reals, and which analysis requires Infinitesimal Calculus. In [Dan5], we show that in infinitesimal Calculus, the hyper-real

ω=∞ 1 δω()xe= ixω d 2π ∫ ω=−∞ is zero for any x ≠ 0 , it spikes at x = 0 , so that its Infinitesimal Calculus

x =∞ integral is ∫ δ()xdx= 1, x =−∞

1 and δ(0) =<∞. dx Here, we show that in Infinitesimal calculus, the Hermite Kernel is a hyper-real Delta Function.

And the Hermite Series LSegendre {f ()x } associated with a Hyper- real function f ()x , equals f ()x .

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Hyper-real Line

Each real number α can be represented by a Cauchy sequence of rational numbers, (,rrr123, ,...) so that rn → α . The constant sequence (ααα, , ,...) is a constant hyper-real.

In [Dan2] we established that, 1. Any totally ordered set of positive, monotonically decreasing

to zero sequences (ιιι123 , , ,...) constitutes a family of infinitesimal hyper-reals. 2. The infinitesimals are smaller than any real number, yet strictly greater than zero. 3. Their reciprocals (111,,,...) are the infinite hyper-reals. ιιι123 4. The infinite hyper-reals are greater than any real number, yet strictly smaller than infinity. 5. The infinite hyper-reals with negative signs are smaller than any real number, yet strictly greater than −∞. 6. The sum of a real number with an infinitesimal is a non-constant hyper-real. 7. The Hyper-reals are the totality of constant hyper-reals, a family of infinitesimals, a family of infinitesimals with

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negative sign, a family of infinite hyper-reals, a family of infinite hyper-reals with negative sign, and non-constant hyper-reals. 8. The hyper-reals are totally ordered, and aligned along a line: the Hyper-real Line. 9. That line includes the real numbers separated by the non- constant hyper-reals. Each real number is the center of an interval of hyper-reals, that includes no other real number. 10. In particular, zero is separated from any positive real by the infinitesimals, and from any negative real by the infinitesimals with negative signs, −dx . 11. Zero is not an infinitesimal, because zero is not strictly greater than zero. 12. We do not add infinity to the hyper-real line. 13. The infinitesimals, the infinitesimals with negative signs, the infinite hyper-reals, and the infinite hyper-reals with negative signs are semi-groups with respect to addition. Neither set includes zero.

14. The hyper-real line is embedded in \∞ , and is not homeomorphic to the real line. There is no bi-continuous one-one mapping from the hyper-real onto the real line.

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15. In particular, there are no points on the real line that can be assigned uniquely to the infinitesimal hyper-reals, or to the infinite hyper-reals, or to the non-constant hyper- reals. 16. No neighbourhood of a hyper-real is homeomorphic to

an \n ball. Therefore, the hyper-real line is not a manifold. 17. The hyper-real line is totally ordered like a line, but it is not spanned by one element, and it is not one-dimensional.

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Integral of a Hyper-real Function

In [Dan3], we defined the integral of a Hyper-real Function. Let f ()x be a hyper-real function on the interval [,ab].

The interval may not be bounded. f ()x may take infinite hyper-real values, and need not be bounded. At each ax≤≤b, there is a rectangle with base [,xx−+dx dx ], height f ()x , and area 22 f ()xdx.

We form the Integration Sum of all the areas for the x ’s that start at x= a, and end at xb= , ∑ f ()xdx. xab∈[,]

If for any infinitesimal dx , the Integration Sum has the same hyper-real value, then f ()x is integrable over the interval [,ab].

Then, we call the Integration Sum the integral of f ()x from xa= , to x= b, and denote it by

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xb= ∫ f ()xdx. xa= If the hyper-real is infinite, then it is the integral over [,ab],

If the hyper-real is finite,

xb= ∫ fxdx( )= real part of the hyper-real ., xa=

3.1 The countability of the Integration Sum

In [Dan1], we established the equality of all positive infinities: We proved that the number of the Natural Numbers,

Card` , equals the number of Real Numbers, Card\ = 2Card` , and we have

22Card` Card` Card``=====( Card ) .... 2 2 ... ≡∞. In particular, we demonstrated that the real numbers may be well-ordered. Consequently, there are countably many real numbers in the interval [,ab], and the Integration Sum has countably many terms.

While we do not sequence the real numbers in the interval, the summation takes place over countably many f ()xdx.

The Lower Integral is the Integration Sum where f ()x is replaced

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by its lowest value on each interval [,xx−+dx dx ] 22 ⎛⎞ ⎜ ⎟ 3.2 ∑ ⎜ inff (tdx )⎟ ⎜ xtx−≤≤+dx dx ⎟ xab∈[,]⎝⎠22

The Upper Integral is the Integration Sum where f ()x is replaced by its largest value on each interval [,xx−+dx dx ] 22 ⎛⎞ ⎜ ⎟ ⎜ supf (tdx )⎟ 3.3 ∑ ⎜ ⎟ ⎜ xtx−≤≤+dx dx ⎟ xab∈[,]⎝⎠22

If the integral is a finite hyper-real, we have

3.4 A hyper-real function has a finite integral if and only if its upper integral and its lower integral are finite, and differ by an infinitesimal.

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Delta Function

In [Dan5], we have defined the Delta Function, and established its properties 1. The Delta Function is a hyper-real function defined from the

⎪⎧⎫1 ⎪ hyper-real line into the set of two hyper-reals ⎨⎪0, ⎬⎪. The ⎩⎭⎪⎪⎪dx ⎪ hyper-real 0 is the sequence 0, 0, 0, ... . The infinite hyper-

1 real depends on our choice of dx . dx 2. We will usually choose the family of infinitesimals that is

1 1 1 spanned by the sequences , , ,… It is a n n2 n3 semigroup with respect to vector addition, and includes all the scalar multiples of the generating sequences that are non-zero. That is, the family includes infinitesimals with

1 negative sign. Therefore, will mean the sequence n . dx Alternatively, we may choose the family spanned by the

1 1 1 1 sequences , , ,… Then, will mean the 2n 3n 4n dx

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sequence 2n . Once we determined the basic infinitesimal

dx , we will use it in the Infinite Riemann Sum that defines an Integral in Infinitesimal Calculus. 3. The Delta Function is strictly smaller than ∞ 1 4. We define, δ()xx≡ χ⎡⎤−dx, dx (), dx ⎣⎢⎥22⎦

⎧⎪1,x ∈−⎡ dx , dx ⎤ ⎪ ⎣⎢ 22⎦⎥ where χ⎡⎤−dx, dx ()x = ⎨ . ⎣⎦⎢⎥22 ⎪ 0, otherwise ⎩⎪ 5. Hence, for x < 0 , δ()x = 0

dx 1 at x =− , δ()x jumps from 0 to , 2 dx 1 for x ∈−⎡⎤dx, dx , δ()x = . ⎣⎦⎢⎥22 dx 1 at x = 0 , δ(0) = dx dx 1 at x = , δ()x drops from to 0. 2 dx for x > 0 , δ()x = 0 .

xxδ()= 0

1 6. If dx = , δ()xxx= χχχ[,]−−−11 (),2 [,]11 (),3 [,]11 ()...x n 22 44 66

12 3 7. If dx = 2 , δ()x = , , ,... n 2cosh22xxx 2cosh 2 2cosh 2 3

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8. If dx = 1 , δχχχ()xe= −−xxx,2 e23 , 3 e − ,... n [0,∞∞∞ ) [0, ) [0, )

x =∞ 9. ∫ δ()xdx= 1. x =−∞

k =∞ 1 10. δξ()−=xe−−ik()ξ x dk 2π ∫ k =−∞

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Convergent Series

In [Dan8], we defined convergence of infinite series in Infinitesimal Calculus

5.1 Sequence Convergence to a finite hyper-real a

an → a iff aan −=infinitesimal .

5.2 Sequence Convergence to an infinite hyper-real A

an → A iff an represents the infinite hyper-real A.

5.3 Series Convergence to a finite hyper-real s

aa12++→... s iff aas1 ++... n − =infinitesimal .

5.4 Series Convergence to an Infinite Hyper-real S

aa12++→... S iff

a1 ++... an represents the infinite hyper-real S .

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Hermite Sequence and δξ()− x

6.1 Hermite Sequence Definition

If f ()x can be expanded in the Hxn(),

f ()xaHxaHxaHx=++00 () 11 () 22 ()+ ..., Then,

x =∞ 2 efxHxdx−x () () = ∫ n x =−∞

x =∞ 2 =+f (xe )−x { aH ( x ) aH ( x )+ aH ( x )+ ...} H ( xdx ) ∫ 00 11 22 n x =∞

xx=∞ =∞ −−xx22 =+aeHxHxdxaeHxHxdx00∫∫()nn () 11() ()+ .. xx =−∞ =−∞ 01 20! πδ01nn21! πδ

n = 2!naπ n . The Hermite Series partial sums

HSermite n {}f ()xaHxaH=++00 () ...n n ()x

ξ=∞ 2 =+f ()ξξ11eHHx−ξ () () ...+ HHxd () ξ () ξ. ∫ π {}00 2!n n nn ξ=−∞ give rise to the Hermite Sequence

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2 Hx(,ξξ )=+11 e−ξ HHx()() ...+ HHx ()ξ () . nnπ {}00 2!n n n

6.2 Hermite Sequence is a Delta Sequence

For each n = 0, 1, 2, 3, ...

2 Hx(,)ξξ=+11 e−ξ HHx( ) ( ) ...+ HHx (ξ ) ( ) , nnπ {}00 2!n n n 1. has the sifting property

ξ=∞ 2 11eHHx−ξ ()ξξ ()++ ... HHxd () ()ξ= 1 ∫ π {}00 2!n n nn ξ=−∞

2. is a continuous function

3. peaks for each ξ → x to const⋅+ n 1 Proof of (1)

ξ=∞ 2 11eHHx−ξ (ξξ ) ( )++ ... HHxd ( ) ( ) ξ= ∫ π {}00 2!n n nn ξ=−∞

ξξ=∞ =∞ 2 2 =+Hx()11 e−−ξξ H()ξξ d ...+ Hx () 1 e H()ξξ d N00ππ∫∫N nn2!n n 11 ξξ=−∞ =−∞ 1 By [Spanier, p.222, #24:10:5], for kn= 1,2,..., ,

ξ=∞ ⎪⎧ 0, k = 1,3,5,... −ξ2 ⎪ eHbd()ξξ= 1 ∫ k ⎨ 2 n ⎪ πnb!(−= 1)2 , k 2, 4,6,... ξ=−∞ ⎩⎪ Therefore, for kn= 1,2,..., ,

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ξ=∞ −ξ2 ∫ eHk ()ξξ d= 0. ξ=−∞

Hence,

ξ=∞ 2 11eHHx−ξ ()ξξ ()++ ... HHxd () ()ξ= 1., ∫ π {}00 2!n n nn ξ=−∞

Proof of (3) By the Christoffel Summation Formula, [Sansone, p.371],

n + 1 ϕξϕϕξϕ() ()xx− () () ϕξϕ() ()xx++ ... ϕξϕ () () = nn++11 nn, 00 nn 2 ξ − x where 1 −1x 2 2 ϕnn()xe= 1 H()x. (2n n !π )2 For ξ → x ,

nn++11ϕξϕϕξϕ() ()xx− () () 0 nn++11 nn→ . 22ξ − x 0 Applying Bernoulli’s rule to the indeterminate limit,

ϕξϕϕξϕ() ()xx− () () DxDxϕξϕ() ()− ϕξϕ () () lim nn++11 nn= lim ξξnn++11 nn ξξ→→xx ξξ−−xDξ()x

=−lim[ϕξϕϕξϕnnnn++11 '( ) (xx ) '( ) ( )] ξ→x

=−ϕϕϕϕnnnn++11'(xx ) ( ) '( x ) ( x ) Therefore,

31 Gauge Institute Journal H. Vic Dannon

n + 1 ϕϕ22()xx++ ... () = [ ϕϕϕϕ '()() xxx− '() ()]x. 01nn2 ++nnn1

Since ϕn ()x , and ϕn+1()x solve the differential equation, [Szego, p.105, #5.5.2],

2 1N ⋅+⋅++−=zx ''( ) 0N zx '( ) (2 n 1 xzx ) ( ) 0 , ax() bx() we have, − bx()dx ∫ ax() ϕϕϕϕnnnn++11'(xx ) ( )−= '( x ) ( x ) ( conste )

0⋅dx = ()const e∫

= const , for any −∞

22 ϕϕ0(xxn )++ ...n ( ) = + 1const Therefore, substituting

11x 2 n 22 Hxnn()= (2 n !πϕ ) e () x,

11ξ2 n 22 Hnenn()ξπ= (2 ! )ϕξ ()

2 11eHHx−ξ { (ξξ ) ( )++ ... HHx ( ) ( )} = π 00 2!n n nn

−−1()ξ22x 2 =+ex{}ϕξϕ00() () ...+ ϕξϕnn () ()x

22 →++{}ϕϕ0()xx ...n () =+ncons1 t., ξ→x

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Hermite Kernel and δξ()− x

7.1 Hermite Kernel in the Calculus of Limits

The Hermite Series partial sums

ξ=∞ 2 HS 11−ξ ermite n {}f ()xfeHHxHHx=+ ()ξξ{}00 () () ...+n nn () ξ ()dξ. ∫ π 2!n ξ=−∞ Hermite Sequence give rise to the Hermite Sequence. The limit of the Hermite Sequence is an infinite series called the Hermite Kernel

2 H ()ξξ−=xeHHxHHx11−ξ ( ) ( )++ ... ( ξ ) ( )+ ... ermite π {}00 2!n n n n

7.2 In the Calculus of Limits, the Hermite Kernel does not have

the sifting property Proof: for ξ → x ,

11−ξ2 eHHx00(ξξ ) ( )++ ...n HHxnn ( ) ( )+ .. = lim ncon + 1 st π {}2!n n→∞ →∞ n→∞ That is, for ξ → x ,

2 11eHHx−ξ (ξξ ) ( )++ ... HHx ( ) ( )+ ... is singular., π {}00 2!n n nn

33 Gauge Institute Journal H. Vic Dannon

7.3 Hyper-real Hermite Kernel in Infinitesimal Calculus

2 H ()ξξ−=xeHHxHHx11−ξ ( ) ( )++ ... ( ξ ) ( )+ ... ermite π {}00 2!n n n n ⎧ ⎪ nx , ξ = = ⎨ ⎪ 0 , ξ ≠ x ⎩⎪ =−δξ(x).

Proof:

2 H ()ξξ−=xeHHxHHx11−ξ ( ) ( )++ ... ( ξ ) ( )+ ... ermite π {}00 2!n n n n ⎧ ⎪ nx , ξ = = ⎨ . ⎪ 0 , ξ ≠ x ⎩⎪

Denoting by 1 the infinite hyper-real n , dx

⎪⎧ 0, ξ ≠ x = ⎨⎪ ⎪ 1 , ξ = x ⎩⎪ dx =−δξ()x .,

34 Gauge Institute Journal H. Vic Dannon

Hermite Series and δξ()− x

8.1 Hermite Series of a Hyper-real Function

Let f ()x be a hyper-real function integrable on (−∞, ∞).

Then, for each n = 0, 1, 2, 3, ... , the integrals

x =∞ 2 ae= 1 −x f()xH ()xdx. nn2!n n π ∫ x =−∞ exist, with finite, or infinite hyper-real values. The an are the Hermite Coefficients of f ()x .

The Hermite Series associated with f ()x is

HSermite {}f ()xaHxaHxaHx=++00 () 11 () 22 ()+ ... For each x , it may assume finite or infinite hyper-real values.

8.2 HSermite {}δξ()−=−xx δξ () Proof:

HSermite {}δξ(−=xaHxaHxaHx )00 () + 11 () + 22 () + ... where x =∞ 2 aex=−1 −x δξ()()Hxdx nn2!n n π ∫ x =−∞

35 Gauge Institute Journal H. Vic Dannon

2 = 1 eH−ξ ()ξ . 2!n n π n Therefore,

2 HS{}δξ()−=xeHHxHHx11−ξ (ξ ) ( )++ ... (ξ ) ( )+ ... ermite π {}00 2!n n n n

=Hermite(ξ −x), by 7.3, =−δξ(x), by 7.3.,

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Hermite Series Theorem

The Hermite Series Theorem for a hyper-real function, f ()x , is the

Fundamental Theorem of Hermite Series. It supplies the conditions under which the Hermite Series associated with f ()x equals f ()x .

It is believed to hold in the Calculus of Limits under the Picone Conditions, or under the Hobson Conditions [Sansone]. In fact, The Theorem cannot be proved in the Calculus of Limits under any conditions, because the summation of the Hermite Series requires integration of the singular Hermite Kernel.

9.1 Hermite Series Theorem cannot be proved in the

Calculus of Limits

Proof: Let f ()x be integrable on (,−∞ ∞).

In the Calculus of Limits, the Hermite Series is the limit of the sequence of Partial Sums

HSermite n {}f()xaHxaH=++00 () ...n n ()x

37 Gauge Institute Journal H. Vic Dannon

⎛⎞ξ=∞ ⎜ −ξ2 ⎟ =+⎜ 1 feH()ξξξ () dHx⎟ () ... ⎜ π ∫ 00⎟ ⎝⎠⎜ ξ=−∞ ⎟

⎛⎞ξ=∞ ⎜ −ξ2 ⎟ ... + ⎜ 11 f ()()()ξξξeH dHx⎟ ⎜ 2!n n π ∫ nn⎟ ⎝⎠⎜ ξ=−∞ ⎟

ξ=∞ 2 =+f ()ξξ11eHHx−ξ () () ...+ HHxd () ξ () ξ. ∫ π {}00 2!n n nn ξ=−∞

As n →∞, the Hermite Sequence

2 11eHHx−ξ ()ξξ ()++ ... HHx () () π {}00 2!n n nn becomes the Hermite Kernel,

2 11eHHx−ξ (ξξ ) ( )++ ... HHx ( ) ( )+ ... , π {}00 2!n n nn By 7.2, the Hermite Kernel diverges to infinity at any ξ = x . Therefore, while the partial sums of the Hermite Series exist, their limit does not. Conditions by Uspensky [Sansone] failed to comprehend the sifting through the values of f ()ξ by the Hermite

Kernel, and the picking of f ()ξ at ξ = x .

Avoiding the singularity at ξ = x , by using the Cauchy Principal Value of the integral does not recover the Theorem, because for any ξ ≠ x , the Hermite Kernel vanishes, and the integral is identically zero, for any function f ()x .

38 Gauge Institute Journal H. Vic Dannon

Thus, the Hermite Series Theorem cannot be proved in the Calculus of Limits.,

9.2 Calculus of Limits Conditions are irrelevant to Hermite

Series Theorem

Proof: The Uspensky Conditions [Sansone, p.371] are 1. f ()x integrable in any bounded interval

2. f ()x integrable in (−∞, ∞)

It is clear from 9.1 that these conditions on f ()x do not resolve the singularity of the Hermite kernel, and are not sufficient for the Hermite Series Theorem.,

In Infinitesimal Calculus, by 7.3, the Hermite Kernel is the Delta Function, and by 8.2, it equals its Hermite Series. Then, the Hermite Series Theorem holds for any Hyper-Real Function:

8.3 Hermite Series Theorem for Hyper-real f ()x

If f ()x is hyper-real function integrable on(−∞, ∞)

Then, f()xf= HSermite {}()x Proof:

39 Gauge Institute Journal H. Vic Dannon

ξ=∞ fx()=−∫ f ()(ξδξ xd ) ξ ξ=−∞

Substituting from 7.3,

2 δξ()−=xeHHxHHx11−ξ (ξ ) ( )++ ... (ξ ) ( )+ ... , π {}00 2!n n nn

ξ=∞ 2 f (xfeHHxHHx )=+ (ξξ )11−ξ ( ) ( ) ...+ ( ξ ) ( )+ ... dξ ∫ π {}00 2!n n nn ξ=−∞

This Hyper-real Integral is the summation,

ξ=∞ 2 f (ξξ )11eHHx−ξ ( ) ( )++ ... HHxd ( ξ ) ( )+ ... ξ ∑ π {}00 2!n n nn ξ=−∞ which amounts to the hyper-real function f ()x ,and is well-defined.

Hence, the summation of each term in the integrand exists, and we may write the integral as the sum

⎛⎞ξ=∞ ⎜ 1 −ξ2 ⎟ =+⎜ feH()ξξξ () dHx⎟ () ... ⎜ ∫ 00⎟ ⎝⎠⎜ π ξ=−∞ ⎟

a0 ⎛⎞ξ=∞ ⎜ 1 −ξ2 ⎟ ... ++⎜ feH(ξξξ ) ( ) dHx⎟ ( ) ... ⎜ n ∫ nn⎟ ⎝⎠⎜ 2!n π ξ=−∞ ⎟

=++aH00() x aH 11 () x aH 22 () x + ...

= HSermite {}f ()x .,

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In particular, the Delta Function violates Uspensky’s Conditions The Hyper-real δ()x , is not defined in the Calculus of Limits,

and is not integrable in any interval. But by 8.2, δξ(− x) satisfies the Hermite Series Theorem.

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References

[Abramowitz] Abramowitz, M., and Stegun, I., “Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables”, U.S. Department of Commerce, National Bureau of Standards, 1964. [Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities, and the Continuum Hypothesis” in Gauge Institute Journal Vol. 6 No. 2, May 2010; [Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6 No. 4, November 2010; [Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal Vol. 7 No. 4, November 2011; [Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann Hypothesis Origin, the Factorization Error, and the Count of the Primes”, in Gauge Institute Journal of Math and Physics, Vol. 5, No. 4, November 2009. [Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal Vol. 8, No. 1, February, 2012; [Dan6] Dannon, H. Vic, “Riemannian Trigonometric Series”, Gauge Institute Journal, Volume 7, No. 3, August 2011. [Dan7] Dannon, H. Vic, “Delta Function the Fourier Transform, and the Fourier Integral Theorem” in Gauge Institute Journal Vol. 8, No. 2, May, 2012; [Dan8] Dannon, H. Vic, “Infinite Series with Infinite Hyper-real Sum ” in Gauge Institute Journal Vol. 8, No. 3, August, 2012; [Ferrers] Ferrers, N., M., “An Elementary treatment on Spherical Harmonics”, Macmillan, 1877.

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[Gradshteyn] Gradshteyn, I., S., and Ryzhik, I., M., “Tables of Integrals Series and Products”, 7th Edition, edited by Allan Jeffery, and Daniel Zwillinger, Academic Press, 2007 [Hardy] Hardy, G. H., Divergent Series, Chelsea 1991. [Hobson] Hobson, E., W., “The Theory of Spherical and Ellipsoidal Harmonics”, Cambridge University Press, 1931. [Jackson] Jackson, Dunham, “Fourier Series and Orthogonal Polynomials”, Mathematical association of America, 1941. [Magnus] Magnus, W., Oberhettinger, F., Sony, R., P., “Formulas and Theorems for the Special Functions of Mathematical Physics” Third Edition, Springer-Verlag, 1966. [Sansone] Sansone, Giovanni, “Orthogonal Functions”, Revised Edition, Krieger, 1977. [Spiegel] Spiegel, Murray, “Mathematical Handbook of formulas and tables” Schaum’s Outline Series, McGraw Hill, 1968. [Spanier] Spanier, Jerome, and Oldham, Keith, “An Atlas of Functions”, Hemisphere, 1987. [Szego2] Szego, Gabor, “Orthogonal Polynomials” Revised Edition, American Mathematical Society,1959. [Szego4] Szego, Gabor, “Orthogonal Polynomials” Fourth Edition, American Mathematical Society,1975. [Todhunter] Todhunter, I., “An Elementary Treatment on Laplace’s Functions, Lame’s Functions, and Bessel’s Functions” Macmillan, 1875. [Weisstein], Weisstein, Eric, W., “CRC Encyclopedia of Mathematics”, Third Edition, CRC Press, 2009.