The Delta Function Abstract

Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
The Delta Function
H. Vic Dannon
[email protected]
September, 2010
Abstract
The Dirac Delta Function, the idealization of an
impulse in Radar circuits, is a Hyper-Real function which
definition and analysis require Infinitesimal Calculus, and
Infinite Hyper-reals.
The controversy surrounding the Leibnitz Infinitesimals derailed
the development of the Infinitesimal Calculus, and the Delta
Function could not be defined and investigated properly.
For instance, it is labeled a “Generalized Function” although it
generalizes no function.
Dirac’s intuitive definition by Delta’s sampling property
x =∞
∫
δ(x )dx = 1 ,
x =−∞
that avoids specifying δ(0) , remains the main definition of the
delta function, although the Delta Function is not Riemann
integrable in the Calculus of Limits,
and is not Lebesgue
integrable in Measure Theory.
In fact, in the Calculus of Limits, only the Cauchy Principal Value
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H. Vic Dannon
Integral of the Delta Function exists, and it equals zero.
Only in Infinitesimal Calculus, can the Delta Function be defined,
differentiated, and integrated.
Infinitesimal Calculus allows us to resolve open problems such as
What is δ(0) ?
How is x δ(x ) defined at x = 0 ?
How is the Delta Function the derivative of a Step Function?
How do we integrate the Delta Function?
What is δ(x 2 ) ?
What is δ 2 (x ) ?
What is δ(x 3 ) ?
What is δ 3 (x ) ?
The Delta Function enables us to define the Fourier Transform
with minimal requirements on the transformed function.
Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real,
Cardinal, Infinity. Non-Archimedean, Non-Standard Analysis,
Calculus, Limit, Continuity, Derivative, Integral,
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;
46S20; 97I40; 97I30.
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Contents
Introduction
1. Hyper-real Line
2. Delta Function Definition
3. Delta Function Plots
4. Delta Function Properties
5. Delta Sequence δn (x ) =
n
1
2 cosh2 nx
6. Delta Sequence δn (x ) = ne −nx
χ
[0,∞ )
(x )
7. Primitive of Delta Function
8. δ( f (x ))
9. δ(x n )
10. δ(x n − (dx )n )
11. Integral of δ(x )
12. The Principal Value Derivative of Delta: The Dipole Function
13. 2nd Principal Value Derivative of Delta: the 4-Pole Function
14. Higher Principal Value Derivatives of Delta
References
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Introduction
0.1 Cauchy, Poisson, and Riemann
Cauchy (1816), and Poisson (1815) derived the Fourier Integral
Theorem by using the sifting property of the Delta Function.
By Fourier Integral Theorem
k =∞ ⎛ ξ =∞
⎞⎟
⎜⎜
1
⎟ ikx
−ik ξ
f (x ) =
⎜⎜ ∫ f (ξ )e d ξ ⎟⎟e dk
∫
⎟
2π k =−∞ ⎜⎝ ξ =−∞
⎠⎟
k =∞
⎛
⎞⎟
⎜⎜ 1
−ik (ξ −x )
= ∫ f (ξ )⎜
e
dk ⎟⎟⎟d ξ
∫
⎜⎜ 2π
⎟
⎝ k =−∞
⎠⎟
ξ =−∞
ξ =∞
Denoting
k =∞
1
e −ik (ξ −x )dk ≡ δ(ξ − x ) ,
∫
2π k =−∞
the Delta Function is the Fourier Transform of the constant
function 1 ,
And Fourier Integral theorem states the sifting property for the
Delta Function
ξ =∞
f (x ) =
∫
f (ξ )δ(ξ − x )d ξ .
ξ =−∞
In the derivation of his Zeta Function, Riemann (1859) uses this
sifting property repeatedly, without using a function notation for
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1
the integral
2π
H. Vic Dannon
k =∞
∫
e −ik (ξ −x )dk that represents the Delta function
k =−∞
δ(ξ − x ) . The derivations are in [Dan4, p.84, p.90, p.97].
The derivations were not supplied by Riemann. Riemann’s 1859
paper, as well as much of Riemann’s published writings, outlines
ideas, and states results without proof.
In particular, the representation of Delta that follows from the
Fourier Integral Theorem does not hold in the Calculus of Limits.
Indeed,
ξ = x ⇒ e −ik (ξ −x ) = 1 ,
and the integral
1
2π
k =∞
∫
e−ik (ξ −x )dk
k =−∞
diverges.
Avoiding the singularity at ξ = x does not recover the Theorem,
because without the singularity the integral equals zero.
Thus, the Fourier Integral Theorem cannot be written in the
Calculus of Limits.
In other words, the indeterminate nature of singularities in the
Calculus of Limits does not allow the Fourier Integral Theorem to
hold.
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0.2 Dirac
The Delta Function can be realized as a Radar transmission
Pulse. A Radar Transmission has to be pulsed because a
continuous wave train will not allow us to measure the time
interval τ between transmission and reception, and determine the
range of the target by r = 12 cτ .
Thus, a transmission lasts very short time.
system converts into a receiver for
Then, the Radar
the reflected signal.
This
process of transmitting and receiving repeats thousands of time
per second, in order to follow a moving target.
The Radar pulse envelops a carrier wave of very short wavelength.
Radar carrier waves went down from centimeters to micrometers
of light wavelength.
Since the illuminating power dissipation is proportional to
1
r2
, the
short electromagnetic wave-train has an electric field that seems
nearly infinite, although the pulse power is finite.
Dirac (1930) was familiar with Radar Pulses when he defined the
Delta Function in [Dirac, p.71] through the sifting property,
x =∞
∫
δ(x )dx = 1 ,
x =−∞
and
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H. Vic Dannon
δ(x ) = 0 , for all x ≠ 0 .
Dirac definition left open the question of the nearly infinite
amplitude at x = 0 . That is, δ(0) was left undefined.
Then, the sifting property does not hold.
Indeed, since δ(0) is infinite, the integration of δ(x ) has to skip
the point x = 0 , and only the Cauchy Principal Value of the
x =∞
integral
∫
δ(x )dx may exist. Then,
x =−∞
⎛ x =−n1
⎞⎟
x =∞
⎜⎜
⎟
lim ⎜⎜ ∫ δ(x )dx + ∫ δ(x )dx ⎟⎟⎟ = 0 ,
n →∞ ⎜
⎟
⎜⎝ x =−∞
x=1
⎠⎟
n
x =∞
in contradiction to
∫
δ(x )dx = 1 .
x =−∞
0.3 Laurent Schwartz
Laurent Schwartz presents his Delta Distribution as follows
[Schwartz, p. 82]
⎪⎧ 0, x < 0
Let Y = ⎪
.
⎨
⎪⎪ 1, x > 0
⎩
Then, for any ϕ(x )
infinitely differentiable, that
vanishes at ∞ , and at −∞ ,
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x =∞
∫
H. Vic Dannon
x =∞
∫
Y '(x )ϕ(x )dx = −
x =−∞
Y (x )ϕ '(x )dx
x =−∞
x =∞
=−
∫
x =0
x =∞
ϕ '(x )dx = − ϕ(x ) x =0
= ϕ(0)
x =∞
=
∫
δ(x )ϕ(x )dx
x =−∞
Thus,
Y ' = δ.
Since Y (x ) is not defined at x = 0 , Y '(0) is not defined, and the
conclusion Y ' = δ , avoids δ(0) .
That is, Schwartz’ Definition is as incomplete as Dirac’s.
Furthermore, the equality
x =∞
ϕ(0) =
∫
δ(x )ϕ(x )dx
x =−∞
is the definition of the Delta Function by its sifting property that
does not hold.
Since δ(0) is not defined, the integration has to skip the point
x = 0 , and the integral is the Cauchy Principal Value Integral
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⎛ x =−n1
⎞⎟
x =∞
⎜⎜
⎟
lim ⎜⎜ ∫ δ(x )ϕ(x )dx + ∫ δ(x )ϕ(x )dx ⎟⎟⎟ = 0
n →∞ ⎜
⎟
⎜⎝ x =−∞
x=1
⎠⎟
n
Like the Dirac Delta, the Schwartz Delta avoids
δ(0) , and
postulates the sifting property.
0.4 Delta Sequence
Attempts to get back to the singular Delta Function, replaced the
Delta Function by a Delta Sequence of functions that converge to
the Delta Function. For instance,
δn (x ) = n
χ
⎧
⎪
n, x ∈ [− 21n , 21n ]
⎪
(x ) = ⎨
.
[− 1 , 1 ]
⎪
2n 2n
0,
otherwise
⎪
⎩
Then, the delta Function is defined as the limit
δ(x ) = lim δn (x ) .
n →∞
The sequential approach is reviewed in [Mikusinski], and is used
in Mathematical Physics texts.
However, the Delta Sequence contradicts Dirac’s definition.
Indeed, as n → ∞ ,
δ(0) = lim δn (0) = lim n = ∞ .
n →∞
n →∞
Then, the integration of δ(x ) has to skip the point x = 0 .
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That is, only the Cauchy Principal Value of the integral
x =∞
∫
δ(x )dx may exist. The Principal Value is
x =−∞
⎛ x =−n1
⎞⎟
x =∞
⎜⎜
⎟
lim ⎜⎜ ∫ δ(x )dx + ∫ δ(x )dx ⎟⎟⎟ = 0 .
n →∞ ⎜
⎟
⎜⎝ x =−∞
x=1
⎠⎟
n
x =∞
That is, the sifting
∫
δn (x )dx = 1 , is not preserved for the limit
x =−∞
of the Delta sequence, δ(x ) = lim δn (x ) . ,
n →∞
0.5 The Hyper-real Delta Function
The above attempts failed because the Delta Function is a hyperreal function.
A function from the hyper-reals into the hyper-
reals.
By resolving the problem of the infinitesimals [Dan2], we obtained
the Infinite Hyper-reals that are strictly smaller than ∞ , and can
serve to supply the value of the Delta Function at the singularity.
The attempts to get by with Calculus restricted to the real line,
deprived Calculus of its full power. In Infinitesimal Calculus,
[Dan3], we differentiate over a jump discontinuity of a step
function, and obtain the Delta Function. We can integrate over a
singularity, and obtain a finite value.
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Here, we present the Delta Function, and the properties of the
Delta Function in Infinitesimal Calculus.
In particular, we resolve open problems such as
What is δ(0) ?
How is x δ(x ) defined at x = 0 ?
How is the Delta Function the derivative of a Step Function?
How do we integrate the Delta Function?
What is δ(x 2 ) ?
What is δ 2 (x ) ?
What is δ(x 3 ) ?
What is δ 3 (x ) ?
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1.
Hyper-real Line
Each real number α can be represented by a Cauchy sequence of
rational numbers, (r1, r2 , r3 ,...) so that rn → α .
The constant sequence (α, α, α,...) is a constant hyper-real.
In [Dan2] we established that,
1. Any totally ordered set of positive, monotonically decreasing
to zero sequences (ι1, ι2 , ι3 ,...) constitutes a family of
infinitesimal hyper-reals.
2. The infinitesimals are smaller than any real number, yet
strictly greater than zero.
3. Their reciprocals
(
1 1 1
, ,
ι1 ι2 ι3
)
,... 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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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 nonconstant 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.
H. Vic Dannon
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 hyperreals.
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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2.
Delta Function Definition
2.1 Domain and Range
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 0 is the sequence
The infinite hyper-real
1
dx
0, 0, 0,... .
depends on our choice of dx . We will
usually choose the family of infinitesimals that is spanned by the
1
n
sequences
,
1
n2
,
1
n3
,… 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,
1
dx
will mean the sequence n .
Alternatively, we may choose the family spanned by the sequences
1
2n
,
1
22n
,
1
23n
,… Then,
1
dx
will mean the sequence 2n .
Once we determined the basic infinitesimal dx , we will use it in
the Infinite Riemann Sum that defines an Integral in
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Infinitesimal Calculus.
2.2 The Delta Function is strictly smaller than ∞
1
< ∞.
dx
Proof: Since dx > 0 ,
2.3 Definition of the Delta Function
We define,
δ(x ) ≡
1
dx
χ⎡⎣⎢ −
dx , dx
2 2
⎤ (x ) ,
⎥⎦
where
⎪⎪⎧1, x ∈ ⎡⎢ − dx , dx ⎤⎥
⎣ 2 2 ⎦.
⎡ −dx , dx ⎤ (x ) = ⎨
⎪⎪ 0, otherwise
⎣⎢ 2 2 ⎦⎥
⎩
χ
This means that
for x < − 12 dx , δ(x ) = 0
at x = − 12 dx , δ(x ) jumps from 0 to
for
1
,
dx
1
.
x ∈ ⎡⎢⎣ − dx2 , dx2 ⎤⎥⎦ , δ(x ) =
dx
at x = 0 ,
δ(0) =
1
dx
at x = 12 dx , δ(x ) drops from
for x > 12 dx , δ(x ) = 0 .
16
1
to 0 .
dx
Gauge Institute Journal, Volume 8, No. 1, February 2012
⎪1, x ∈ ⎡⎢ − dx , dx ⎤⎥
1 ⎧
⎣ 2 2 ⎦ is the sequence
2.4 δ(x ) = ⎨⎪
0,
otherwise
dx ⎪
⎪
⎩
H. Vic Dannon
⎧⎪ 1 , x ∈ ⎡ − in , in ⎤
⎪⎪ i n
⎢⎣ 2 2 ⎥⎦
⎨
⎪⎪ 0, otherwise
⎪⎩
where dx = in .
Namely, as a hyper-real function the value of Delta at the
singularity is the infinite hyper-real
1
dx
which is a sequence, an
infinite vector with countably many components.
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3.
Delta Function Plots
3.1 Delta Plot for dx =
1
n
If in = n1 , Delta is the infinite Hyper-Real number,
δ(x ) =
χ[−1,1](x ), 2χ[− , ](x ), 3χ[− , ](x ),...
1 1
2 2
We plot in Maple the 10th component with
Similarly, we use
18
1 1
3 3
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to plot an the 100th component of Delta
3.2 Delta with dx =
1
2n
is
χ[− , ](x ), 4χ[− , ](x ), 8χ[−
δ(x ) = 2
1 1
4 4
1 1
8 8
We use
to plot the 4th component of Delta
19
(x ),...
1,1]
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Similarly, we use
to plot the 6th component of Delta,
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4.
Delta Function Properties
4.1
x δ(x ) = 0
Proof: x ≠ 0 ⇒ δ(x ) = 0 ⇒ x δ(x ) = 0 .
x = 0 ⇒ x δ(x ) = 0δ(0) =
4.2
δ(x − x 0 ) ≡
=
4.3
1
d (x − x 0 )
1
dx
χ⎡⎣⎢ x −
δ n (x ) =
0
4.4
0
dx ,x + dx
2 0
2
1
(dx )n
That is, δ n (0) spikes to
χ⎡⎣⎢ x −
0
= 0 , since dx > 0 .
dx
dx ,x + dx
2 0
2
⎤ (x )
⎦⎥
χ
1
(dx )n
⎤ (x )
⎦⎥
⎡ −dx , dx ⎤ (x ) ,
⎢⎣ 2 2 ⎥⎦
n = 2, 3,...
, which is greater than
δ(x , y ) ≡ δ(x )δ(y ) =
1
.
dx
1
χ[−dx ,dx ](x ) 1 χ[−dy ,dy ](y )
2 2
dx
dy
2 2
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5.
Delta Sequence δn (x ) = n
1
2 cosh 2 nx
Depending on the choice of the infinitesimal dx = in , there are
many Delta Sequences that lead to the Delta Function, δ(x ) .
5.1
Each
δn (x ) = n
1
2 cosh2 nx
x =∞
∫
ƒ has the sifting property
δn (x )dx = 1
x =−∞
ƒ is continuous
ƒ peaks at x = 0 to δn (0) =
x =∞
Proof:
n
2
tanh nx
∫ n 2 cosh2 nx dx = n 2n
x =−∞
1
x =∞
=
x =−∞
1
( 1 − (−1) ) = 1 . ,
2
The sequence represents the hyper-real Delta Function
5.2 If in =
2
,
n
δ(x ) =
1
,
2
,
3
2 cosh2 x 2 cosh2 2x 2 cosh2 3x
22
,...
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plots in Maple, the 50th component,
plots in Maple the 200th component,
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6.
Delta Sequence δn (x ) = ne−nx χ[0,∞) (x )
6.1
Each δn (x ) = ne −nx χ[0,∞)
x =∞
ƒ has the sifting property
∫
δn (x )dx = 1
x =−∞
ƒ is continuous hyper-real function
ƒ peaks at x = 0 to δn (0) = n
x =∞
Proof:
∫
x =∞
ne
x =−∞
−nx
χ[0,∞)(x )dx =
∫
x =0
ne
−nx
e −nx
dx = n
−n
x =∞
= 1 .,
x =0
The sequence represents the hyper-real Delta Function
6.2 If in =
1
, δ(x ) = e −x χ[0,∞), 2e −2x χ[0,∞), 3e−3x χ[0,∞),...
n
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plots in Maple the 100th component,
plots in Maple the 200th component,
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7.
Primitive of Delta Function
7.1
Proof:
⎧
⎪
−1, x < 0
⎪
⎪
δ(x ) is the derivative of g(x ) = ⎪⎨ 0, x = 0
⎪
⎪
1, x > 0
⎪
⎪
⎩
At the jump over [−dx , 0] , from −1 to 0 , for any dx ,
g(0) − g(0 − dx )
0 − (−1)
1
=
=
dx
dx
dx
Therefore, the left derivative at x = 0 is
g '(0−) =
1
.
dx
At the jump over [0, dx ] , from 0 to 1 , for any dx ,
g(0 + dx ) − g(0) 1 − 0
1
=
=
dx
dx
dx
Therefore, the right derivative at x = 0 is
g '(0+) =
26
1
.
dx
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Since the right and left derivatives are equal, the derivative at
x = 0 is
g '(0) =
1
= δ(0) . ,
dx
Since at x ≠ 0 , g '(x ) = 0 , we have,
g '(x ) =
1 ⎡ dx dx ⎤
χ − , .,
dx ⎢⎣ 2 2 ⎥⎦
⎧ 0, x ≤ 0
⎪
7.2 δ(x ) is the Principal Value derivative of h(x ) = ⎪
⎨
⎪
1, x > 0
⎪
⎩
Proof:
For any dx ,
h(0) − h(−dx )
0
=
=0
dx
dx
⇒ h '(0−) = 0 .
h(dx ) − h(0) 1 − 0
1
=
=
dx
dx
dx
⇒ h '(0+) =
1
.
dx
Therefore, h(x ) has no derivative at x = 0 . ,
But since
h(dx2 ) − h(− dx2 )
dx
=
1−0
1
,
=
dx
dx
The principal value derivative of h(x ) at x = 0 is
p.v.h '(0) =
1
.,
dx
Since at x ≠ 0 , p.v.h '(x ) = 0 , we have, p.v.h '(x ) =
27
1 ⎡ dx dx ⎤
χ − , .,
dx ⎢⎣ 2 2 ⎥⎦
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8.
δ( f (x ))
1
δ(x )
a
8.1
δ(ax ) =
Proof:
δ(ax )adx = δ(ax )d (ax ) = 1 = δ(x )dx .
We divide both sides by adx , and put a , because the Delta’s on
both sides are positive. ,
8.2
If ξ1 is the only zero of f (x ) , and f '(ξ1 ) ≠ 0 ,
Then,
Proof:
δ( f (x )) =
1
δ(x − ξ1 )
f '(ξ1 )
δ( f (x )) = δ ( f (x ) − f (ξ1 ) )
For x − ξ1 = infinitesimal ,
= δ ( f '(ξ1 )(x − ξ1 ) )
By 8.1,
=
1
δ(x − ξ1 ) . ,
f '(ξ1 )
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If ξ1, ξ2 are the only zeros of f (x ) , and f '(ξ1 ), f '(ξ2 ) ≠ 0
8.3
Then, δ( f (x )) =
Proof:
1
1
δ(x − ξ1 ) +
δ(x − ξ2 )
f '(ξ1 )
f '(ξ2 )
δ( f (x )) = δ ( f (x ) − f (ξ1 ) ) + δ ( f (x ) − f (ξ2 ) )
If x − ξ1 = infinitesimal , f (x ) − f (ξ1 ) = f '(ξ1 )(x − ξ1 )
If x − ξ2 = infinitesimal , f (x ) − f (ξ2 ) = f '(ξ2 )(x − ξ2 )
Either way,
δ( f (x )) = δ ( f '(ξ1 )(x − ξ1 ) ) + δ ( f '(ξ2 )(x − ξ2 ) )
=
1
1
δ(x − ξ1 ) +
δ(x − ξ2 ) . ,
f '(ξ1 )
f '(ξ2 )
1
1
δ(x − a ) +
δ(x + a )
2a
2a
8.4
δ(x 2 − a 2 ) =
8.5
δ ( (x − a )(x − b) ) =
1
1
δ(x − a ) +
δ(x + a )
a −b
b −a
8.6 If ξ1,...ξn are the only zeros of f (x ) , and f '(ξ1 ),.., f '(ξn ) ≠ 0
Then, δ( f (x )) =
1
1
δ(x − ξ1 ) + ... +
δ(x − ξn )
f '(ξ1 )
f '(ξn )
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Gauge Institute Journal, Volume 8, No. 1, February 2012
8.7
If
ξ1, ξ2 ,... are zeros of f (x ) , and f '(ξ1 ), f '(ξ2 ),... ≠ 0
Then, δ( f (x )) =
8.8
H. Vic Dannon
1
1
δ(x − ξ1 ) +
δ(x − ξn ) + ...
f '(ξ1 )
f '(ξn )
δ(sin x ) = ... + δ(x + 2π) + δ(x + π) + δ(x ) + δ(x − π) + δ(x − 2π) + ...
Proof: The zeros of sin x are ... − 2π, −π, 0, π, 2π,...
and cos(n π) = 1 . ,
30
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
9.
δ(x n )
δ(x 2 ) =
9.1
1
χ⎡
⎤ (x ) ,
2xdx ⎣ −xdx ,xdx ⎦
x >0
Proof:
δ(x 2 ) =
=
1
d (x 2 )
1
2xdx
χ⎡⎢−
⎢⎣
χ⎡⎢−
⎣⎢
d (x 2 ) d (x 2 ) ⎤
⎥
,
2
2 ⎥
(x )
⎦
d (x 2 ) d (x 2 ) ⎤
⎥
,
2
2 ⎥
(x ) ,
⎦
where to ensure d(x 2 ) > 0 , we must have
x > 0.
The amplitude and domain of the δ(x 2 ) spike depend on x .
For instance,
9.2
1
2xdx
χ⎡⎣ −xdx ,xdx ⎤⎦ (x ) x =
9.3
1
2xdx
χ⎡⎣ −xdx,xdx ⎤⎦ (x ) x =
1
2
dx
2
= δ(x )
=
1
2
(dx )
31
χ⎡⎢ −
⎢⎣
(dx )2 (dx )2
,
2
2
⎤ (x )
⎥
⎥⎦
≤ δ 2 (x )
Gauge Institute Journal, Volume 8, No. 1, February 2012
δ(x n ) =
9.4
=
1
d (x
n
H. Vic Dannon
χ
)
1
nx n −1dx
⎡ d (x n ) d (x n ) ⎤ (x )
⎢−
⎥
,
2
2 ⎥
⎢⎣
⎦
χ
⎡ −n x n −1dx , n x n −1dx ⎤ (x ) ,
2
⎣⎢ 2
⎦⎥
x >0
The amplitude and domain of the δ(x n ) spike depend on x .
For instance,
9.5
1
nx n −1dx
χ⎡⎣⎢ − x
n n −1dx , n x n −1dx
2
2
1
⎤ (x )
⎦⎥
x =( 1 )n −1
n
32
= δ(x )
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
10.
δ(x n − (dx )n )
δ ( x 2 − (dx2 )2 ) is
While x 2 − (dx2 )2 is infinitesimally close to x 2 ,
different from δ(x 2 ) .
10.1
Proof:
δ ( x 2 − (dx2 )2 ) =
1
1
δ(x − dx2 ) +
δ(x + dx2 )
dx
dx
By 8.4, since dx > 0 . ,
10.1 has two positive spikes. For instance,
10.2
⎡ 1
⎤
1
1
1 dx
dx ⎥
dx
⎢ δ(x − dx ) +
(
)
(
)
δ
x
δ
δ( 2 )
+
=
−
+
2
2 ⎥
2
⎢⎣ dx
dx
dx
dx
⎦x =0
=
1
χ[−dx ,0](x ) +
2
(dx )
1
(dx )2
χ[0,dx ](x ) .
Similarly, δ ( x 3 − (dx )3 ) has three positive spikes.
10.3 δ ( x 3 − (dx )3 ) =
Proof:
( δ(x − dx ) + δ ( x − e
3(dx )
1
2
x 3 − (dx )3 has the three zeros
33
i 23π
) (
dx + δ x − e
2i 23π
dx
)) .
Gauge Institute Journal, Volume 8, No. 1, February 2012
i 2π
x1 = dx , x 2 = e 3 dx ,
and
2
H. Vic Dannon
x3 = e
2i 2 π
3
dx .
2
Since f '(x ) = 3x 2 , and since x 2 = x 3 = (dx )2 , by 6.6 we obtain
δ ( x 3 − (dx )3 ) =
1
3(dx )2
(
(
i 2π
) (
δ(x − dx ) + δ x − e 3 dx + δ x − e
2i 23π
dx
)) . ,
δ ( x n − (dx )n ) has n positive spikes.
10.4 δ ( x n − (dx )n ) =
1
n(dx )n −1
( δ(x − dx ) + δ ( x − e dx ) + ... + δ ( x − e
2π
n
(n −1)2nπ
dx
))
Proof: x n − (dx )n has the n zeros
i 2π
x1 = dx , x 2 = e n dx ,…, xn = e
Since f '(x ) = nx n −1 , and since x1
n −1
(n −1)i 2 π
= ... = x n
n
n −1
dx .
= (dx )n −1 ,
by 6.6 we obtain
δ ( x n − (dx )n ) =
1
n(dx )n −1
( δ(x − dx ) + δ ( x − e dx ) + ... + δ ( x − e
2π
n
34
(n −1)2nπ
dx
)) . ,
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
11.
Integral of δ(x )
11.1 Integral of a Hyper-real Function
Let f (x ) be a hyper-real function on the interval [a, b ] .
f (x ) may take infinite hyper-real values, and need not be
bounded.
At each
a ≤ x ≤b,
there is a rectangle with base [x − dx2 , x + dx2 ] , 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 = a , and end at x = b ,
∑
f (x )dx .
x ∈[a ,b ]
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 x = a ,
to x = b , and denote it by
35
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
x =b
∫
f (x )dx .
x =a
If the hyper-real is infinite, then it is the integral over [a, b ] ,
If the hyper-real is finite,
x =b
∫
f (x )dx = real part of the hyper-real . ,
x =a
11.2 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
Card `
Card ` = (Card `)2 = .... = 2Card ` = 22
= ... ≡ ∞ .
In particular, we demonstrated that the real numbers may be
well-ordered.
Consequently, there are countably many real numbers in the
interval [a, b ] , 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 (x )dx .
The Lower Integral is the Integration Sum where f (x ) is replaced
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Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
by its lowest value on each interval [x − dx2 , x + dx2 ]
11.3
∑
x ∈[a ,b ]
⎛
⎞
⎜⎜
inf
f (t ) ⎟⎟⎟dx
⎜⎝ x −dx ≤t ≤x + dx
⎠⎟
2
2
The Upper Integral is the Integration Sum where f (x ) is replaced
by its largest value on each interval [x − dx2 , x + dx2 ]
11.4
⎛
⎞⎟
⎜⎜
f (t ) ⎟⎟dx
∑ ⎜⎜ x −dxsup
⎟
dx
≤t ≤x +
⎠⎟
x ∈[a ,b ] ⎝
2
2
If the integral is a finite hyper-real, we have
11.5 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.
x =∞
11.6
∫
δ(x )dx = 1 .
x =−∞
Proof: The only term in the integration Sum is
1
dx = 1 .
dx
Both the upper integral, and the lower integral are equal to
1
dx = 1 . ,
dx
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Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
12.
The Principal Value Derivative of
Delta: the Dipole Function
We have seen in 2.3 that
at x = −
for
1
dx
, δ(x ) jumps from 0 to
,
dx
2
1
1
. In particular, δ(0) =
x ∈ ⎡⎢⎣ − dx2 , dx2 ⎤⎦⎥ , δ(x ) =
dx
dx
at x =
dx
1
, δ(x ) drops from
to 0 .
2
dx
Here, we show
™ in 12.1, that δ(x ) has no derivative at x = − 12 dx , but the
Principal Value Derivative over the jump at x = − 12 dx , is a
Positive Impulse Function.
™ in 12.2, that δ(x ) has no derivative at
x = 12 dx , but the
Principal Value Derivative over the jump at x = 12 dx , is a
Negative Impulse Function.
We sum up 12.1, and 12.2. in 12.3.
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Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
Namely, δ(x ) has no derivative at x = 0 , but the Principal Value
Derivative over the two jumps is a Dipole Function.
That Dipole function is a positive Impulse Function followed by a
negative Impulse function.
Both the positive, and the negative impulses have jumps far
greater than the jump of the generating delta function.
12.1 The Principal Value Derivative of Delta at x = − 12 dx
δ(x ) has no derivative at x = −
dx
.
2
The Principal Value Derivative of δ(x ) at x = − 12 dx , is the Positive
Impulse function
Proof:
1
(dx )2
χ[−dx , 0] .
The left derivative of δ(x ) at x = − 12 dx is
δ(− dx2 ) − δ(−dx )
− dx2 + dx
=
1
dx
−0
dx
2
=
2
(dx )2
The right derivative of δ(x ) at x = − 12 dx is
δ(0) − δ(− dx2 )
0 − (− dx2 )
=
1
dx
− dx1
dx
2
= 0.
Since the left and right derivatives are unequal,
derivative at x = − 12 dx . ,
39
δ(x ) has no
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
The Principal Value Derivative at x = − 12 dx is
1
dx
δ(0) − δ(−dx )
=
0 − (−dx )
−0
dx
=
1
It is the Positive Impulse Function
2
(dx )
1
2
(dx )
.
χ[−dx , 0] . ,
12.2 The Principal Value Derivative of Delta at x =
dx
2
δ(x ) has no derivative at x = 12 dx .
The Principal Value Derivative of δ(x ) at x = 12 dx , is the Negative
Impulse Function −
Proof:
1
(dx )2
χ[0, dx ].
The Left Derivative of δ(x ) at x = 12 dx is
δ(dx2 ) − δ(0)
dx
2
=
1
dx
− dx1
dx
2
=
0
= 0.
dx
The Right Derivative of δ(x ) at x = 12 dx is
δ(dx ) − δ(dx2 )
dx − dx2
=
0 − dx1
dx
2
=−
2
(dx )2
.
Since the Left and Right Derivatives are unequal,
derivative at x = 12 dx . ,
40
δ(x ) has no
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
The Principal Value Derivative at x = 12 dx is
0 − dx1
1
δ(dx ) − δ(0)
=
=−
.
2
dx )
dx
(dx )
It is the Negative Impulse Function −
1
2
(dx )
χ[0, dx ]. ,
12.3 The Principal Value Derivative of Delta at x = 0
δ(x ) has no derivative at x = 0 .
The Principal Value Derivative of δ(x ) , p.v.Dδ(x )
is the Dipole Function
Dipole(x ) =
Proof:
1
2
(dx )
χ[−dx , 0] −
1
2
(dx )
χ[0, dx ] .
The Left Derivative of δ(x ) at x = 0 is
δ(0) − δ(−dx )
=
dx
1
dx
−0
dx
=
1
(dx )2
The Right Derivative of δ(x ) at x = 0 is
0 − dx1
1
δ(dx ) − δ(0)
=
=−
.
dx
dx
(dx )2
Since the left and right derivatives are unequal,
derivative at x = 0 . ,
The Principal Value Derivative of δ(x ) is
41
δ(x ) has no
Gauge Institute Journal, Volume 8, No. 1, February 2012
δ(x + dx2 ) − δ(x − dx2 )
dx
It is the Dipole Function
If dx =
1
n
=
1
(dx )2
H. Vic Dannon
1
1
δ(x + dx2 ) − δ(x − dx2 ) .
dx
dx
χ[−dx , 0] −
1
(dx )2
χ[0, dx ] . ,
, this is the sequence
Dipole(x ) = n 2χ[− n1 , 0] − n 2χ[0, n1 ] .
Then, a Maple plot of the 10th component of Dipole(x ) is
42
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
13.
The 2nd Principal Value Derivative
of Delta: the 4-Pole Function
The 2nd Principal Value Derivative of δ(x ) is the 4-pole Function
13.1 (p.v.D)2δ(x ) =
1
(dx )2
=
2
Proof: (p.v.D) δ(x ) =
=
=
1
3
(dx )
( δ(x + dx ) − 2δ(x ) + δ(x − dx ) )
{ χ[− 3dx2 , − dx2 ] − 2χ[− dx2 , dx2 ] + χ[dx2 , 3dx2 ]} .
Dipole(x + dx2 ) − Dipole(x − dx2 )
dx
1
(dx )2
1
3
(dx )
`
( δ(x + dx ) − 2δ(x ) + δ(x − dx ) )
{ χ[− 3dx2 , − dx2 ] − 2χ[− dx2 , dx2 ] + χ[dx2 , 3dx2 ]} . ,
The 4-pole Function has four Impulse Functions
ƒ a Positive Impulse
1
(dx )2
δ(x + dx ) centered at x = −dx ,
43
Gauge Institute Journal, Volume 8, No. 1, February 2012
ƒ two Negative Impulses −2
ƒ a Positive Impulse
If dx =
1
n
1
(dx )2
1
(dx )2
H. Vic Dannon
δ(x ) centered at x = 0 ,
δ(x − dx ) centered at x = dx .
, this is the sequence
4 pole(x ) = n 3χ[− 23n , − 21n ] − 2n 3χ[− 21n , 21n ] + n 3χ[ 21n , 23n ] .
Then, a Maple plot of a component of 4 pole(x ) is
The x axis units are
1
n
. The y axis units are n 3 .
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Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
14.
Higher Principal Value
Derivatives of Delta
The 3rd Principal Value Derivative of δ(x ) , (p.v.D)3δ(x )
14.1
is the 8-pole Function
8pole(x ) =
If dx =
1
n
1
(dx )4
( χ[−2dx, −dx ] − 3χ[−dx, 0] + 3χ[0, dx ] − χ[dx, 2dx ]) .
, this is the sequence
8pole(x ) = n 4 χ[− n2 , − n1 ] − 3n 4 χ[− n1 , 0] + 3n 4 χ[0, n1 ] − n 4 χ[ n1 , n2 ] .
Then, a Maple plot of a component of 8pole(x ) is
The x axis units are
1
n
. The y axis units are n 4 .
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Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
14.2 The 4th Principal Value Derivative of δ(x ) , (p.v.D)4δ(x )
is the 16-pole Function
16pole(x ) =
1
5
(dx )
( χ[− 5dx2 , − 3dx2 ] − 4χ[− 3dx2 , − dx2 ] +
+ 6χ[− dx2 , dx2 ] − 4χ[dx2 , 3dx
] + χ[ 3dx
, 5dx
]) .
2
2
2
If dx =
1
n
, this is the sequence
16pole(x ) = ( n 5χ[− 25n , − 23n ] − 4n 5χ[− 23n , − 12 ] +
+ 6n 5χ[− 21n , 21n ] − 4n 5χ[ 21n , 23n ] + n 5χ[ 23n , 25n ] ) .
Then, a Maple plot of a component of 16pole(x ) is
The x axis units are
1
n
. The y axis units are n 5 .
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Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
Using the Binomial coefficients,
14.3 The kth Principal Value Derivative of δ(x ) , (p.v.D)k δ(x ) is
k
2 pole(x ) =
=
2k −1 pole(x + dx2 ) − 2k −1 pole(x + dx2 )
dx
⎛
⎛ ⎞
⎜⎜ χ[− (k +1)dx , − (k −1)dx ] − ⎜⎜ k ⎟⎟ χ[− (k −1)dx , − (k −3)dx ] +
⎜⎜ 1 ⎟⎟
2
2
2
2
(dx )k +1 ⎜⎜⎝
⎝ ⎠
1
⎛ k ⎞⎟
(k −1)dx
(k −3)dx
(k −1)dx (k +1)dx
+ ⎜⎜⎜ ⎟⎟ χ[− 2 , − 2 ] + ... + (−1)k χ[ 2 , 2 ] .
⎜⎝ 2 ⎠⎟
)
If dx =
n
k +1
1
n
, this is the sequence
(k +1)
(k −1)
χ[− 2n , − 2n ]
− n
⎛ n ⎞⎟
(n −1)dx
(n −3)dx
⎜⎜ ⎟⎟ χ[− 2 , − 2 ] +
⎜⎝ 1 ⎠⎟
k +1 ⎜
⎛k ⎞
+ n k +1 ⎜⎜⎜ ⎟⎟⎟ χ[− k2−n1 , − k2−n3 ] + ... + (−1)k n k +1χ[ k2−n1 , k2+n1 ] )
⎜⎝ 2 ⎠⎟
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Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
References
[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities,
and the Continuum
Hypothesis” in Gauge Institute Journal of math and
Physics, Vol.6 No 2, May 2010;
[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal of math
and Physics, Vol.6 No 4, November 2010;
[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal
of Math and Physics, Vol.7 No 1, February 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;
[Dirac] Dirac, P. A. M. The Principles of Quantum Mechanics, Second Edition,
Oxford Univ press, 1935.
[Hen] Henle, James M., and Kleinberg Eugene M., Infinitesimal Calculus,
MIT Press 1979.
[Hosk] Hoskins, R. F., Standard and Nonstandard Analysis, Ellis Horwood,
1990.
[Keis] Keisler, H. Jerome, Elementary calculus, An Infinitesimal Approach,
Second Edition, Prindle, Weber, and Schmidt, 1986, pp. 905-912
[Laug] Laugwitz, Detlef, “Curt Schmieden’s approach to infinitesimals-an eyeopener to the historiography of analysis” Technische Universitat Darmstadt,
Preprint Nr. 2053, August 1999
[Mikusinski] Mikusinski, J. and Sikorski, R., “The elementary theory of
distributions”, Rosprawy Matematyczne XII, Warszawa 1957.
48
Gauge Institute Journal, Volume 8, No. 1, February 2012
H. Vic Dannon
[Rand] Randolph, John, “Basic Real and Abstract Analysis”, Academic Press,
1968.
[Riemann] Riemann, Bernhard, “On the Representation of a Function by a
Trigonometric Series”.
(1)
In “Collected Papers, Bernhard Riemann”, translated from
the 1892 edition by Roger Baker, Charles Christenson, and
Henry Orde, Paper XII, Part 5, Conditions for the existence of a
definite integral, pages 231-232, Part 6, Special Cases, pages
232-234. Kendrick press, 2004
(2)
In “God Created the Integers” Edited by Stephen Hawking,
Part 5, and Part 6, pages 836-840, Running Press, 2005.
[Schwartz] Schwartz, Laurent, Mathematics for the Physical Sciences,
Addison-Wesley, 1966.
[Temp] Temple, George, 100 Years of Mathematics, Springer-Verlag, 1981.
pp. 19-24.
49