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Gauge Institute Journal H. Vic Dannon
Delta Function, and Expansion in Hermite
Functions H. Vic Dannon
vic0@comcast.net June, 2012
Abstract Let ( )f x be defined on the real numbers, and let
be the Hermite Polynomials on the real numbers, ( )nH x
0( ) 1H x = , , , ,… 1( ) 2H x x= 22( ) 4 2H x x= − 3
3( ) 8 12H x x x= −
The Hermite Series associated with ( )f x is
0 0 1 1 2 2( ) ( ) ( ) ....a H x a H x a H x+ + + where
21( ) ( )
2 !n nna e f
n
ξξ
ξ
ξ ξπ
=∞−
=−∞
= ∫ H dξ
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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Gauge Institute Journal H. Vic Dannon
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
{ }21 10 0 2 !( ) ( ) ... ( ) ( ) ...
n n nne H H x H H xξ
πξ ξ− + + +
is the Delta Function, . ( )xδ ξ −
( xδ ξ − ) 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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Gauge Institute Journal H. Vic Dannon
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 ( )xδ ξ −
7. Hermite Kernel and . ( )xδ ξ −
8. Hermite Series of ( )xδ ξ −
9. Hermite Series Theorem
References
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Gauge Institute Journal H. Vic Dannon
The Origin of the Hermite Series
Theorem The Hermite Polynomials on ( ,−∞ ∞)
0( ) 1H x = , , , ,…, 1( ) 2H x x= 22( ) 4 2H x x= − 3
3( ) 8 12H x x x= −
are orthogonal so that
2( ) ( ) 2 !
xx n
m n mx
e H x H x dx n πδ=∞
−
=−∞
=∫ n .
The Hermite Polynomials can be generated by expanding 22 2 2 21 1
2! 3!1 [2 ] [2 ] [2 ] ...xe x x xα α α α α α α α− = + − + − + − +2 3
2 2 2 3 412!
1 2 [4 4 ]x x xα α α α α= + − + − + +
3 3 2 4 5 613!
[8 12 6 ] ...x x xα α α α+ − + − +
the coefficient of 010!α is
0( ) 1H x = ,
the coefficient of 111!α is
1( ) 2H x x= ,
the coefficient of 212!α is
22( ) 4 2H x x= − ,
the coefficient of 313!α is
33( ) 8 12H x x x= − ,
…………………………
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Gauge Institute Journal H. Vic Dannon
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 [ , at frequency ν , under the force . ,A A− ]
ν
t
x
kx−
The particle’s position is
( ) cosx t A tω= , . 2ω π=Thus,
sinx Aω ω= −
2 2cosx A tω ω ω= − = −
The force equation is 2( )kx mx m xω− = = − .
Hence, the force constant is 2k mω= ,
and the potential energy of the particle is
2 21 12 2
V kx mω= = 2x .
De Broglie associated with the moving particle a wave of length
hmv
λ = ,
where v is the velocity of the particle, and h is Planck’s constant.
The wave’s frequency is
2
hmv
v v mvh
νλ
= = = .
The wave’s angular frequency is
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Gauge Institute Journal H. Vic Dannon
2
2 2mvh
ω πν π= =
In terms of the De Broglie wave, the particle’s energy is a multiple
of Planck’s radiation energy,
E hε ν ε ω= = , 2hπ
= , is the multiplier. ε
The kinetic energy of the particle is
212mv E V= − .
Hence, 2 ( )mv m E V= − ,
2 ( )
h
m E Vλ =
−,
12
2 ( )v
m E Vπω
ωλ ν= =
−.
2 2 2
1 2 (m E V
v ω
−=
)
Schrodinger postulated a complex valued potential
( , ) ( ) i tx t x e ωψΨ =
that satisfies the wave equation
2 22
1( , ) ( , )x tx t x t
v∂ Ψ = ∂ Ψ .
Then,
2 22
10 ( , ) (x tx t x t
v= ∂ Ψ − ∂ Ψ , )
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Gauge Institute Journal H. Vic Dannon
22 2
2 ( )"( ) ( )( )i t i tm E Vx e x eω ωψ ψ
ω
−= − −ω .
The Schrodinger equation for the linear harmonic oscillator is
2
2"( ) ( ) ( ) 0
mx E V xψ ψ+ − = .
Substituting E , and V ,
2 2122
2" ( )m
m xψ ε ω ω ψ+ − 0=
Multiplying by mω
,
2
2"( ) (2 ) ( ) 0m
x x xm
ξ
ωψ ε ψ
ω+ − = .
The change of variable m xωξ = , gives
'( ) md d ddx d dx
ωψ ψ ξψ ξ
ξ= = ,
{ }2
2'( ) ''( )m md d d
d dxdxω ωψ ξ
ψ ξ ψ ξξ
= = ,
and the equation becomes
2"( ) (2 ) ( ) 0xψ ξ ε ξ ψ+ − = .
0.2 Hermite Differential Equation
The Schrodinger equation
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Gauge Institute Journal H. Vic Dannon
2"( ) ( ) 2 ( )x xψ ξ ξ ψ εψ− = − can be factored
( )( ) ( ) 2D D xξ ξξ ξ ψ εψ− + = − ( )x
0
D xξ ξ ψ− =
.
To solve the homogeneous equation
( )( ) ( )D D xξ ξξ ξ ψ− + = ,
we solve
( ) ( ) 0 ⇒ 'ψξ
ψ= ⇒ 21
2log cψ ξ= + ⇒
212
1 Ce ξψ = .
As , , and is discarded. ξ → ∞ 1ψ → ∞
( ) ( ) 0D xξ ξ ψ+ = ⇒ 'ψξ
ψ= − ⇒ 21
2log cψ ξ= − + ⇒
212
2 Ceξψ −= .
Now, substituting 21
2( ) ( )H eξψ ξ ξ −=
in , we have 2"( ) (2 ) ( ) 0ψ ξ ε ξ ψ ξ+ − =
( )2 21 12 22 20 ( ) (2 ) ( )D H e H eξ ξ
ξ ξ ε ξ ξ− −= + −
( )2 21 12 2 2'( ) ( ) (2 ) ( )D H e H e H eξ ξ
ξ ξ ξ ξ ε ξ ξ− −= − + −21
2ξ−
2 2 21 1 1
2 2 2 2''( ) 2 '( ) ( ) ( )H e H e H e H eξ ξ ξξ ξ ξ ξ ξ ξ− − −= − − +21
2ξ− +
2122(2 ) ( )H e ξε ξ ξ −+ −
21
2''( ) 2 '( ) (2 1) ( )H H H e ξξ ξ ξ ε ξ −⎡ ⎤= − + −⎣ ⎦ .
The Schrodinger equation becomes Hermit Differential Equation
''( ) 2 '( ) (2 1) ( ) 0H H Hξ ξ ξ ε ξ− + − = ,
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Gauge Institute Journal H. Vic Dannon
Substituting in it
2 10 1 2 1 2( ) ... ...l l l
l l lH c c c c c cξ ξ ξ ξ ξ ξ+ ++ += + + + + + + +2
0l =
0
,
we have
2 1
2 1
2
0 0 0
( 1)
2 (2 1)
l ll l
l ll l
l l ll l
l l ll l l
l lc lc
D c D c cξ ξ
ξ ξ
ξ ξ ξ ε ξ
=∞ =∞− −
= =
=∞ =∞ =∞
= = =
−
− + −
∑ ∑
∑ ∑ ∑ ,
20
{( 1)( 2) 2 (2 1) } 0l
ll l l
l
l l c lc cε ξ=∞
+=
+ + − + − =∑ ,
2( 1)( 2) [2 1 2 ]l ll l c l cε++ + − + − =
22 1 2
( 1)( 2)l ll
c cl l
ε+
+ −=
+ +
The solution is
2 31 2 3 20 1 0 11 2 2 3
( )H c c c cε εξ ξ ξ− −⋅ ⋅
= + + + +ξ
(1 2 )(5 2 ) (3 2 )(7 2 )4 50 11 2 3 4 2 3 4 5
...c cε ε ε εξ ξ− − − −⋅ ⋅ ⋅ ⋅ ⋅ ⋅
+ + +
(1 2 )(5 2 )2 41 20 1 2 1 2 3 4{1 ...}c ε εε ξ ξ− −−
⋅ ⋅ ⋅ ⋅= + + + +
(3 2 )(7 2 )2 43 21 2 3 2 3 4 5
{1 ...}c ε εεξ ξ ξ− −−⋅ ⋅ ⋅ ⋅
+ + + + .
To keep the solution from diverging at ξ , → ∞
for , the series terms vanish for 2n = k 0c
2 1,5,9,13,...4 1,...kε = + ,
and we obtain the Hermite Polynomials. 2 ( )kH ξ
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Gauge Institute Journal H. Vic Dannon
for , the series terms vanish for 2n k= + 1 1c
2 3,7,11,..., 4 3,...kε = +
and we obtain the Hermite Polynomials. 2 1( )kH ξ+
A solution for is the infinite linear combination ( )ψ ξ
2 21 1 12 2 2
0 0 1 1 2 2( ) ( ) ( ) ....H e H e H eξ ξ ξα ξ α ξ α ξ− − −+ +2
+
0.3 The Hermite Series Associated with ( )f x
Let ( )f x be defined on ( , , and let be the Hermite
Polynomials )
)
−∞ ∞ ( )nH x
0( ) 1H x = , , , ,… 1( ) 2H x x= 22( ) 4 2H x x= − 3
3( ) 8 12H x x x= −
The Polynomials are orthogonal on ( . That is, ,−∞ ∞
2( ) ( ) 2 !
xx n
m n mx
e H x H x dx n πδ=∞
−
=−∞
=∫ n
We define the Orthonormalized Hermite Functions
212
12
1( ) ( )
(2 ! )
xn n
nx e
nϕ
π
−= H x
If ( )f x can be expanded in the , ( )n xϕ
0 0 1 1 2 2( ) ( ) ( ) ( ) ...f x x x xα ϕ α ϕ α ϕ= + + + ,
Then,
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Gauge Institute Journal H. Vic Dannon
0 0 1 1 2 2( ) ( ) { ( ) ( ) ( ) ...} ( )x x
n nx x
f x x dx x x x x dxϕ α ϕ α ϕ α ϕ=∞ =∞
=−∞ =−∞
= + + +∫ ∫ ϕ
n +
0 1 2
0 0 1 1 2 2( ) ( ) ( ) ( ) ( ) ( ) ..
n n n
x x x
n nx x x
x x dx x x dx x x dx
δ δ δ
α ϕ ϕ α ϕ ϕ α ϕ ϕ=∞ =∞ =∞
=−∞ =−∞ =−∞
= + +∫ ∫ ∫
. nα=
Thus, the Hermite coefficients with respect to the are ( )n xϕ
( ) ( )n nf dξ
ξ
α ξ ϕ=∞
=−∞
= ∫ ξ ξ .
The Orthonormal Hermite Series associated with ( )f x is
0 0 1 1 2 2( ) ( ) ( ) ...x x xα ϕ α ϕ α ϕ+ + + .
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Gauge Institute Journal H. Vic Dannon
1.
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
{ } 0 0( ) ( ) ... ( )ermite n n nf x xα ϕ α ϕ= + +H S x
0 0( ) ( ) ( ) .. ( ) ( ) ( )n nf d x f dξ ξ
ξ ξ
ξ ϕ ξ ξ ϕ ξ ϕ ξ ξ ϕ=∞ =∞
=−∞ =−∞
⎛ ⎞ ⎛⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= + +⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎝ ⎠ ⎝∫ ∫ x
⎞
⎠
{ }0 0( ) ( ) ( ) ... ( ) ( )n nf x xξ
ξ
ξ ϕ ξ ϕ ϕ ξ ϕ ξ=∞
=−∞
= + +∫ d .
As n , the orthonormal Hermite Sequence → ∞
0 0( ) ( ) ... ( ) ( )n nx xϕ ξ ϕ ϕ ξ ϕ+ +
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Gauge Institute Journal H. Vic Dannon
becomes the orthonormal Hermite Kernel,
0 0( ) ( ) ... ( ) ( ) ...n nx xϕ ξ ϕ ϕ ξ ϕ+ + + ,
To see that it is singular at , we apply the Christoffel
Summation Formula, [Sansone, p.371],
xξ =
1 10 0
( ) ( ) ( ) ( )1( ) ( ) ... ( ) ( )
2n n n n
n n
x xnx x
x
ϕ ξ ϕ ϕ ξ ϕϕ ξ ϕ ϕ ξ ϕ
ξ+ +−+
+ + =−
.
For , xξ →
2 20 0 0( ) ( ) ... ( ) ( ) ( ) ... ( )n n nx x xϕ ξ ϕ ϕ ξ ϕ ϕ ϕ+ + → + + x ,
and
1 1( ) ( ) ( ) ( )1 12 2
n n n nx xn nx
ϕ ξ ϕ ϕ ξ ϕξ
+ +−+ +→
−00
.
Applying Bernoulli’s rule to the indeterminate limit,
1 11 1 ( ) ( ) ( ) ( )( ) ( ) ( ) ( )lim lim
( )n n n nn n n n
x x
D x D xx x
x Dξ ξ
ξ ξ ξ
ϕ ξ ϕ ϕ ξ ϕϕ ξ ϕ ϕ ξ ϕ
ξ ξ+ ++ +
→ →
−−=
− − x
1 1lim[ '( ) ( ) '( ) ( )]n n n nxx x
ξϕ ξ ϕ ϕ ξ ϕ+ +→
= −
1 1'( ) ( ) '( ) ( )n n n nx x x xϕ ϕ ϕ ϕ+ += −
Therefore,
2 20 1
1( ) ... ( ) [ '( ) ( ) '( ) ( )]
2n n n nn
x x x x xϕ ϕ ϕ ϕ ϕ ϕ+ ++
+ + = − 1n x .
Since , and solve the differential equation, [Szego,
p.105, #5.5.2],
( )n xϕ 1( )n xϕ +
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Gauge Institute Journal H. Vic Dannon
2
( ) ( )
1 ''( ) 0 '( ) (2 1 ) ( ) 0a x b x
z x z x n x z x⋅ + ⋅ + + − = ,
we have, ( )
( )1 1'( ) ( ) '( ) ( ) ( )
b xa xdx
n n n nx x x x const eϕ ϕ ϕ ϕ−
+ +∫− =
0( )
dxconst e
⋅∫=
, const=
for any . x−∞ < < ∞
Hence,
2 20
1( ) ... ( )
2nn
x xϕ ϕ+
+ + = const
and the Hermite Kernel diverges to ∞ at any . xξ =
Therefore, while the partial sums of the Hermite Series exist,
their limit does not. That is, due to the singularity at , the
Hermite Series does not converge in the Calculus of Limits.
xξ =
Avoiding the singularity at , by using the Cauchy Principal
Value of the integral does not recover the Theorem, because at any
, the Hermite Kernel vanishes, and the integral will be
identically zero, for any function
xξ =
xξ ≠
( )f x .
To see that the kernel vanishes for any , we apply the
Christoffel Summation Formula, with .
xξ ≠
xξ ≠
1 110 0 2
( ) ( ) ( ) ( )( ) ( ) ... ( ) ( ) n n n nn
n n
x xx x
x
ϕ ξ ϕ ϕ ξ ϕϕ ξ ϕ ϕ ξ ϕ
ξ+ ++ −
+ + =−
.
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Gauge Institute Journal H. Vic Dannon
We have
21
212
1( ) ( )
(2 ! )
xn n
nx e
nϕ
π
−= H x
21 22
12
1( )
(2 ! )
x xn
ne e H x
n π
−=
By [Szego, p. 105, #5.5.3], 2 2
( ) ( 1)x nn xe H x D e− −= − n x .
Thus, 21 2
212
( 1)( ) { }
(2 ! )
nx n x
n xn
x e Dn
ϕπ
−−= e ,
21 2
212
11
11
( 1)( ) { }
(2 ( 1)! )
nx n x
n xn
x en
ϕπ
++ −
++
−=
+D e
and
112
( ) ( ) ( ) ( )n n n nn x x
x
ϕ ξ ϕ ϕ ξ ϕξ
++ −=
−1+
{ }2 2 22 1
1 112
0, 0,
1 ( 1){ } { } { } { }
2 2 !( 1)!
nn n x n nn
x xn
nn
D e D e D e D ex n n
ξ ξξ ξξ π
++ − − − + −+
→ →∞→ →∞
−= −
− +
2x
. 0, as n→ → ∞
That is, the Hermite Kernel vanishes for any . xξ ≠
Plots of the Hermite Sequence confirm that
In the Calculus of Limits,
the Hermite Kernel is either singular or zero
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Gauge Institute Journal H. Vic Dannon
1.2 Plots of 21 1
0 0 2 !{ ( ) ( ) ... ( ) ( )}
n n nne H H x H H xξ
πξ ξ− + +
In Maple,
2
2231 1
2 !0
( * ( ,.5) * ( , ), 23..23)ix
ii
plot e HermiteH i HermiteH i x xπ
−
=
= −∑
In Maple, 2223
1 12 !
0
( * ( , 1) * ( , ), 2i
xi
i
plot e HermiteH i HermiteH i x xπ
−
=
− =∑ 3..23)−
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Gauge Institute Journal H. Vic Dannon
2xe− that suppresses oscillations away from the origin, enhances
them at the origin. Thus, a singularity away from the origin
needs more terms
In Maple, 2223
1 12 !
0
( * ( ,2) * ( , ), 23..23)i
xi
i
plot e HermiteH i HermiteH i x xπ
−
=
= −∑
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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Gauge Institute Journal H. Vic Dannon
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
1( )
2i xx e
ωω
ω
δ ωπ
=∞
=−∞
= ∫ d
is zero for any , 0x ≠
it spikes at , so that its Infinitesimal Calculus
integral is ,
0x =
( ) 1x
x
x dxδ=∞
=−∞
=∫
and 1(0)
dxδ = < ∞
}
.
Here, we show that in Infinitesimal calculus, the Hermite Kernel
is a hyper-real Delta Function.
And the Hermite Series { ( )egendre f xL S associated with a Hyper-
real function ( )f x , equals ( )f x .
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Gauge Institute Journal H. Vic Dannon
2.
Hyper-real Line Each real number α can be represented by a Cauchy sequence of
rational numbers, so that . 1 2 3( , , ,...)r r r nr α→
The constant sequence ( is a constant hyper-real. , , ,...)α α α
In [Dan2] we established that,
1. Any totally ordered set of positive, monotonically decreasing
to zero sequences constitutes a family of
infinitesimal hyper-reals.
1 2 3( , , ,...)ι ι ι
2. The infinitesimals are smaller than any real number, yet
strictly greater than zero.
3. Their reciprocals (1 2 3
1 1 1, , ,...ι ι ι ) are the infinite hyper-reals.
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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Gauge Institute Journal H. Vic Dannon
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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Gauge Institute Journal H. Vic Dannon
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 ball. Therefore, the hyper-real line is not a manifold. n
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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Gauge Institute Journal H. Vic Dannon
3.
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 [ , . ]a b
The interval may not be bounded.
( )f x may take infinite hyper-real values, and need not be
bounded.
At each
a x≤ ≤ b ,
there is a rectangle with base 2
[ ,dx dxx x− +2], height ( )f x , and area
( )f x dx .
We form the Integration Sum of all the areas for the x ’s that
start at x , and end at x b , a= =
[ , ]
( )x a b
f x dx∈∑ .
If for any infinitesimal dx , the Integration Sum has the same
hyper-real value, then ( )f x is integrable over the interval [ , . ]a b
Then, we call the Integration Sum the integral of ( )f x from ,
to x , and denote it by
x a=
b=
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Gauge Institute Journal H. Vic Dannon
( )x b
x a
f x dx=
=∫ .
If the hyper-real is infinite, then it is the integral over [ , , ]a b
If the hyper-real is finite,
( ) real part of the hyper-realx b
x a
f x dx=
=
=∫ .
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, , and
we have
2CardCard =
2 2( ) .... 2 2 ...CardCardCard Card= = = = = ≡ ∞ .
In particular, we demonstrated that the real numbers may be
well-ordered.
Consequently, there are countably many real numbers in the
interval [ , , and the Integration Sum has countably many terms. ]a b
While we do not sequence the real numbers in the interval, the
summation takes place over countably many ( )f x dx .
The Lower Integral is the Integration Sum where ( )f x is replaced
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Gauge Institute Journal H. Vic Dannon
by its lowest value on each interval 2 2
[ ,dx dxx x− + ]
3.2 2 2[ , ]
inf ( )dx dxx t xx a b
f t dx− ≤ ≤ +∈
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∑
The Upper Integral is the Integration Sum where ( )f x is replaced
by its largest value on each interval 2 2
[ ,dx dxx x− + ]
3.3 2 2[ , ]
sup ( )dx dxx t xx a b
f t dx− ≤ ≤ +∈
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∑
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.
24
Gauge Institute Journal H. Vic Dannon
4.
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
hyper-real line into the set of two hyper-reals 1
0,dx
⎧ ⎫⎪⎪⎨⎪⎪ ⎪⎩ ⎭
⎪⎪⎬⎪. The
hyper-real is the sequence 0 0,0, 0,... . The infinite hyper-
real 1dx
depends on our choice of dx .
2. We will usually choose the family of infinitesimals that is
spanned by the sequences 1n
,2
1
n,
3
1
n,… It is a
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
negative sign. Therefore, 1dx
will mean the sequence n .
Alternatively, we may choose the family spanned by the
sequences 1
2n,
1
3n,
1
4n,… Then, 1
dx will mean the
25
Gauge Institute Journal H. Vic Dannon
sequence 2n . Once we determined the basic infinitesimal
, we will use it in the Infinite Riemann Sum that defines
an Integral in Infinitesimal Calculus.
dx
3. The Delta Function is strictly smaller than ∞
4. We define, 2 2,
1( ) ( )dx dxx x
dxδ χ⎡ ⎤−⎢ ⎥⎣ ⎦
≡ ,
where 2 2
2 2,
1, ,( )
0, otherwisedx dx
dx dxxxχ⎡ ⎤−⎢ ⎥⎣ ⎦
⎧ ⎡ ⎤⎪ ∈ −⎢ ⎥⎪ ⎣ ⎦= ⎨⎪⎪⎩.
5. Hence,
for , 0x < ( ) 0xδ =
at 2dx
x = − , jumps from to ( )xδ 01dx
,
for 2 2
,dx dxx ⎡ ⎤∈ −⎢ ⎥⎣ ⎦ , 1
( )xdx
δ = .
at , 0x =1
(0)dx
δ =
at 2dx
x = , drops from ( )xδ1dx
to . 0
for , . 0x > ( ) 0xδ =
( ) 0x xδ =
6. If 1n
dx = , 1 1 1 1 1 12 2 4 4 6 6
[ , ] [ , ] [ , ]( ) ( ),2 ( ), 3 ( )...x x xδ χ χ χ− − −= x
7. If 2n
dx = , 2 2 2
1 2 3( ) , , ,...
2 cosh 2cosh 2 2cosh 3x
x x xδ =
26
Gauge Institute Journal H. Vic Dannon
8. If 1n
dx = , 2 3[0, ) [0, ) [0, )( ) ,2 , 3 ,...x x xx e e eδ χ χ χ− − −
∞ ∞ ∞=
9. . ( ) 1x
x
x dxδ=∞
=−∞
=∫
10. ( )1( )
2
kik x
k
x e ξδ ξπ
=∞− −
=−∞
− = ∫ dk
27
Gauge Institute Journal H. Vic Dannon
5.
Convergent Series In [Dan8], we defined convergence of infinite series in
Infinitesimal Calculus
5.1 Sequence Convergence to a finite hyper-real a
na → a iff infinitesimalna a− = .
5.2 Sequence Convergence to an infinite hyper-real A
iff na → A na represents the infinite hyper-real A .
5.3 Series Convergence to a finite hyper-real s
1 2 ...a a+ + → s iff 1 ... infinitesimalna a s+ + − = .
5.4 Series Convergence to an Infinite Hyper-real S
iff 1 2 ...a a+ + → S
1 ... na + + a represents the infinite hyper-real S .
28
Gauge Institute Journal H. Vic Dannon
6.
Hermite Sequence and ( )xδ ξ −
6.1 Hermite Sequence Definition
If ( )f x can be expanded in the , ( )nH x
0 0 1 1 2 2( ) ( ) ( ) ( ) ...f x a H x a H x a H x= + + + ,
Then,
2( ) ( )
xx
nx
e f x H x dx=∞
−
=−∞
=∫
2
0 0 1 1 2 2( ) { ( ) ( ) ( ) ...} ( )x
xn
x
f x e a H x a H x a H x H x dx=∞
−
=∞
= + + +∫
2 2
0 10 1
0 0 1 1
2 0! 2 1!
( ) ( ) ( ) ( ) ..
n n
x xx x
n nx x
a e H x H x dx a e H x H x dx
πδ πδ
=∞ =∞− −
=−∞ =−∞
= +∫ ∫ +
2 !nnn aπ= .
The Hermite Series partial sums
{ } 0 0( ) ( ) ... ( )ermite n n nf x a H x a H= + +H S x
{ }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( )n n nn
f e H H x H H x dξ
ξπ
ξ
ξ ξ ξ=∞
−
=−∞
= + +∫ ξ .
give rise to the Hermite Sequence
29
Gauge Institute Journal H. Vic Dannon
{ }21 10 0 2 !
( , ) ( ) ( ) ... ( ) ( )nn nn
H x e H H x H H xξπ
ξ ξ−= + + nξ .
6.2 Hermite Sequence is a Delta Sequence
For each 0,1,2,3,...n =
{ }21 10 0 2 !
( , ) ( ) ( ) ... ( ) ( )nn nn
H x e H H x H H xξπ
ξ ξ−= + + nξ ,
1. has the sifting property
{ }21 10 0 2 !( ) ( ) ... ( ) ( ) 1
n n nne H H x H H x d
ξξ
πξ
ξ ξ=∞
−
=−∞
+ + =∫ ξ
2. is a continuous function
3. peaks for each to xξ → 1const n⋅ +
Proof of (1)
{ }21 10 0 2 !( ) ( ) ... ( ) ( )
n n nne H H x H H x d
ξξ
πξ
ξ ξ=∞
−
=−∞
+ + =∫ ξ
2 21 1 10 0 2 !1 1
1
( ) ( ) ... ( ) ( )nn nn
H x e H d H x e H dξ ξ
ξ ξπ πξ ξ
ξ ξ ξ ξ=∞ =∞
− −
=−∞ =−∞
= + +∫ ∫
By [Spanier, p.222, #24:10:5], for , 1,2,...,k n=
2
122
0, 1, 3,5,...( )
!( 1) , 2, 4,6,... nk
ke H b d
n b k
ξξ
ξ
ξ ξπ
=∞−
=−∞
⎧ =⎪⎪⎪= ⎨⎪ − =⎪⎪⎩∫
Therefore, for , 1,2,...,k n=
30
Gauge Institute Journal H. Vic Dannon
2( ) 0ke H d
ξξ
ξ
ξ ξ=∞
−
=−∞
=∫ .
Hence,
{ }21 10 0 2 !( ) ( ) ... ( ) ( ) 1
n n nne H H x H H x d
ξξ
πξ
ξ ξ=∞
−
=−∞
+ + =∫ ξ .
Proof of (3)
By the Christoffel Summation Formula, [Sansone, p.371],
1 10 0
( ) ( ) ( ) ( )1( ) ( ) ... ( ) ( )
2n n n n
n n
x xnx x
x
ϕ ξ ϕ ϕ ξ ϕϕ ξ ϕ ϕ ξ ϕ
ξ+ +−+
+ + =−
,
where 21
212
1( ) ( )
(2 ! )
xn n
nx e
nϕ
π
−= H x .
For , xξ →
1 1( ) ( ) ( ) ( )1 12 2
n n n nx xn nx
ϕ ξ ϕ ϕ ξ ϕξ
+ +−+ +→
−00
.
Applying Bernoulli’s rule to the indeterminate limit,
1 11 1 ( ) ( ) ( ) ( )( ) ( ) ( ) ( )lim lim
( )n n n nn n n n
x x
D x D xx x
x Dξ ξ
ξ ξ ξ
ϕ ξ ϕ ϕ ξ ϕϕ ξ ϕ ϕ ξ ϕξ ξ
+ ++ +
→ →
−−=
− − x
1 1lim[ '( ) ( ) '( ) ( )]n n n nxx x
ξϕ ξ ϕ ϕ ξ ϕ+ +→
= −
1 1'( ) ( ) '( ) ( )n n n nx x x xϕ ϕ ϕ ϕ+ += −
Therefore,
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Gauge Institute Journal H. Vic Dannon
2 20 1
1( ) ... ( ) [ '( ) ( ) '( ) ( )]
2n n n nn
x x x x xϕ ϕ ϕ ϕ ϕ ϕ+ ++
+ + = − 1n x .
Since , and solve the differential equation, [Szego,
p.105, #5.5.2],
( )n xϕ 1( )n xϕ +
2
( ) ( )
1 ''( ) 0 '( ) (2 1 ) ( ) 0a x b x
z x z x n x z x⋅ + ⋅ + + − = ,
we have, ( )
( )1 1'( ) ( ) '( ) ( ) ( )
b xa xdx
n n n nx x x x const eϕ ϕ ϕ ϕ−
+ +∫− =
0( )
dxconst e
⋅∫=
, const=
for any . x−∞ < < ∞
Hence,
2 20( ) ... ( ) 1nx x nϕ ϕ+ + = + const
Therefore, substituting 21 1
2 2( ) (2 ! ) ( )xn
n nH x n e xπ ϕ= ,
21 12 2( ) (2 ! ) ( )n
n nH n e ξξ π= ϕ ξ
21 10 0 2 !
{ ( ) ( ) ... ( ) ( )}n n nn
e H H x H H xξπ
ξ ξ− + + =
{ }2 21
2( )
0 0( ) ( ) ... ( ) ( )xn ne xξ ϕ ξ ϕ ϕ ξ ϕ− −= + + x
{ }2 20( ) ... ( )nxx x
ξϕ ϕ
→→ + + 1n const= + .
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Gauge Institute Journal H. Vic Dannon
7.
Hermite Kernel and ( )xδ ξ −
7.1 Hermite Kernel in the Calculus of Limits The Hermite Series partial sums
{ } { }21 10 0 2 !
Hermite Sequence
( ) ( ) ( ) ( ) ... ( ) ( )nermite n n nn
f x f e H H x H H xξ
ξπ
ξ
ξ ξ ξ=∞
−
=−∞
= + +∫H S dξ .
give rise to the Hermite Sequence.
The limit of the Hermite Sequence is an infinite series called the
Hermite Kernel
{ }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( ) ...nermite n nn
x e H H x H H xξπ
ξ ξ ξ−− = + + +H
7.2 In the Calculus of Limits, the Hermite Kernel does not have
the sifting property
Proof: for , xξ →
{ }21 10 0 2 !( ) ( ) ... ( ) ( ) .. lim 1
n n nn ne H H x H H x n conξ
πξ ξ−
→∞+ + + = + st
n→∞→ ∞
That is, for , xξ →
{ }21 10 0 2 !( ) ( ) ... ( ) ( ) ...
n n nne H H x H H xξ
πξ ξ− + + + is singular.
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Gauge Institute Journal H. Vic Dannon
7.3 Hyper-real Hermite Kernel in Infinitesimal Calculus
{ }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( ) ...nermite n nn
x e H H x H H xξπ
ξ ξ ξ−− = + + +H
,
0 ,
n x
x
ξ
ξ
⎧⎪ =⎪= ⎨⎪ ≠⎪⎩
. ( xδ ξ= − )
Proof:
{ }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( ) ...nermite n nn
x e H H x H H xξπ
ξ ξ ξ−− = + + +H
,
0 ,
n x
x
ξ
ξ
⎧⎪ =⎪= ⎨⎪ ≠⎪⎩.
Denoting by 1dx
the infinite hyper-real n ,
1
0,
,dx
x
x
ξ
ξ
⎧ ≠⎪⎪= ⎨⎪ =⎪⎩
. ( )xδ ξ= −
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Gauge Institute Journal H. Vic Dannon
8.
Hermite Series and ( )xδ ξ −
8.1 Hermite Series of a Hyper-real Function
Let ( )f x be a hyper-real function integrable on ( . ,−∞ ∞)
Then, for each , the integrals 0,1,2, 3,...n =
212 !
( ) ( )n
xx
n nnx
a e f xπ
=∞−
=−∞
= ∫ H x dx .
exist, with finite, or infinite hyper-real values. The are the
Hermite Coefficients of
na
( )f x .
The Hermite Series associated with ( )f x is
{ } 0 0 1 1 2 2( ) ( ) ( ) ( ) ...ermite f x a H x a H x a H x= + +H S +
For each x , it may assume finite or infinite hyper-real values.
8.2 { }( ) ( )ermite x xδ ξ δ ξ− = −H S
Proof:
{ } 0 0 1 1 2 2( ) ( ) ( ) ( ) ...ermite x a H x a H x a H xδ ξ − = + + +H S
where
212 !
( ) ( )n
xx
n nnx
a e xπ
δ ξ=∞
−
=−∞
= −∫ H x dx
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Gauge Institute Journal H. Vic Dannon
21
2 !( )
n nne Hξ
πξ−= .
Therefore,
{ } { }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( ) ...nermite n nn
x e H H x H H xξπ
δ ξ ξ ξ−− = + + +H S
, by 7.3, (ermite xξ= H )−
) , by 7.3. ( xδ ξ= −
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Gauge Institute Journal H. Vic Dannon
9.
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
{ } 0 0( ) ( ) ... ( )ermite n n nf x a H x a H= + +H S x
37
Gauge Institute Journal H. Vic Dannon
21
0 0( ) ( ) ( ) ...f e H d H xξ
ξπξ
ξ ξ ξ=∞
−
=−∞
⎛ ⎞⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫
21 1
2 !... ( ) ( ) ( )
n n nnf e H d H x
ξξ
πξ
ξ ξ ξ=∞
−
=−∞
⎛ ⎞⎟⎜ ⎟⎜ ⎟+ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫
{ }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( )n n nn
f e H H x H H x dξ
ξπ
ξ
ξ ξ ξ=∞
−
=−∞
= + +∫ ξ .
As n , the Hermite Sequence → ∞
{ }21 10 0 2 !( ) ( ) ... ( ) ( )
n n nne H H x H H xξ
πξ ξ− + +
becomes the Hermite Kernel,
{ }21 10 0 2 !( ) ( ) ... ( ) ( ) ...
n n nne H H x H H xξ
πξ ξ− + + + ,
By 7.2, the Hermite Kernel diverges to infinity at any . xξ =
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 . xξ =
Avoiding the singularity at , by using the Cauchy Principal
Value of the integral does not recover the Theorem, because for
any , the Hermite Kernel vanishes, and the integral is
identically zero, for any function
xξ =
xξ ≠
( )f x .
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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, { }( ) ( )ermitef x f= H S x
Proof:
39
Gauge Institute Journal H. Vic Dannon
( ) ( ) ( )f x f x dξ
ξ
ξ δ ξ ξ=∞
=−∞
= −∫
Substituting from 7.3,
{ }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( ) ...n n nn
x e H H x H H xξπ
δ ξ ξ ξ−− = + + + ,
{ }21 10 0 2 !
( ) ( ) ( ) ( ) ... ( ) ( ) ...n n nn
f x f e H H x H H xξ
ξπ
ξ
ξ ξ ξ=∞
−
=−∞
= + +∫ dξ+
This Hyper-real Integral is the summation,
{ }21 10 0 2 !
( ) ( ) ( ) ... ( ) ( ) ...n n nn
f e H H x H H x dξ
ξπ
ξ
ξ ξ ξ=∞
−
=−∞
+ + +∑ 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
2
0
0 01
( ) ( ) ( ) ...
a
f e H d H xξ
ξ
ξ
ξ ξ ξπ
=∞−
=−∞
⎛ ⎞⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫
21
... ( ) ( ) ( ) ...2 !
n
n nn
a
f e H d H xn
ξξ
ξ
ξ ξ ξπ
=∞−
=−∞
⎛ ⎞⎟⎜ ⎟⎜ ⎟+ +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫
0 0 1 1 2 2( ) ( ) ( ) ...a H x a H x a H x= + + +
{ }( )ermite f x= H S .
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Gauge Institute Journal H. Vic Dannon
In particular, the Delta Function violates Uspensky’s Conditions
The Hyper-real , is not defined in the Calculus of Limits,
and is not integrable in any interval.
( )xδ
But by 8.2, satisfies the Hermite Series Theorem. ( xδ ξ − )
41
Gauge Institute Journal H. Vic Dannon
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[Abramowitz] Abramowitz, M., and Stegun, I., “Handbook of Mathematical
Functions with Formulas Graphs and Mathematical Tables”, U.S.
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[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
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[Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann Hypothesis
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[Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal Vol.
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Gauge Institute Journal H. Vic Dannon
[Gradshteyn] Gradshteyn, I., S., and Ryzhik, I., M., “Tables of Integrals Series
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43
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