Delta Function - Gauge-Institute

Gauge Institute Journal, Volume 8, No. 1, February 2012 H. Vic Dannon 

The Delta Function 

H. Vic Dannon 

vic0@comcast.net 

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 =∞ 

∫ 

x =−∞ 

δ( 

xdx ) = 1, 

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 

1


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 xδ( x) 

defined at x = 0 ? 

How is the Delta Function the derivative of a Step Function? 

How do we integrate the Delta Function? 

2 

What is δ (x) 

? 

2 

What is δ ( x) 

? 

3 

What is δ( 

x ) ? 

3 


? 

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. 

2


Contents 

Introduction 

1. Hyper-real Line 

2. Delta Function Definition 

3. Delta Function Plots 

4. Delta Function Properties 

5. Delta Sequence 

6. Delta Sequence 

n 1 

δ n( 

x) 

= 

2 2 cosh nx 

−nx 

n( x) ne ( x) 

[0, ) 

δ χ = 

∞ 

7. Primitive of Delta Function 

8. 

9. 

10. 

δ(()) 

fx 

n 

δ( 

x ) 

n n 

δ( x − ( dx) 

) 

11. Integral of 

δ( 

x) 

12. The Principal Value Derivative of Delta: The Dipole Function 

13. 2 nd Principal Value Derivative of Delta: the 4-Pole Function 

14. Higher Principal Value Derivatives of Delta 

References 

3


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 

Denoting 

k =∞ ⎛ ξ=∞ 

⎞ 

1 ⎜ ⎟ − ξ 

f ( x) = f( ξ) e ξ 

⎟ 

⎜ ⎟ 

2π 

⎜ ⎟ 

k =−∞ ⎜ ⎟ 

⎝ξ=−∞ ⎠⎟ 

ik ikx 

∫ ∫ d e dk 

ξ=∞ 

⎛ k =∞ ⎞ 

⎜ 1 

⎟ 

−ik( ξ−x) 

= f () ξ ⎜ e dk 

⎟ 

∫ ⎜ ⎟ ξ 

⎜ 2π 

∫ ⎟d 

⎟ 

ξ=−∞ 

⎜⎝ k =−∞ ⎠⎟ 

1 

2π 

k =∞ 

∫ 

k =−∞ 

−ik( ξ−x) δξ 

e dk ≡ ( −x 

) , 

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( 

ξδξ ) ( −x) 

d ξ. 

ξ=−∞ 

In the derivation of his Zeta Function, Riemann (1859) uses this 

sifting property repeatedly, without using a function notation for 

4


k =∞ 

1 −ik( −x) 

the integral 

2π 

∫ 

k =−∞ 

e ξ 

dk 

that represents the Delta function 

δξ− ( 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, 

and the integral 

diverges. 


ξ = x ⇒ e = 1, 

1 

2π 

k =∞ 

∫ 

k =−∞ 


e dk 

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. 

5


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 = cτ. 

1 

2 

Thus, a transmission lasts very short time. Then, the Radar 

system converts into a receiver for 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. 

1 

Since the illuminating power dissipation is proportional to 2 r 

, 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, 

and 

x =∞ 

∫ 

x =−∞ 

δ( 

xdx ) = 1, 

6


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( 

xdx ) may exist. Then, 

x =−∞ 

1 ⎛ x =− 

n 

x =∞ ⎞ 

⎟ 

⎜ 

lim 

⎜ ⎟ 

δ( xdx ) δ( 

xdx ) ⎟ 0 

→∞ ⎜ + ⎟ = 

⎜ ⎟ 

⎜⎝x=−∞ x = ⎠⎟ 

n 

x =∞ 

in contradiction to δ( 

xdx ) = 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 −∞, 

7 

1 

n


x=∞ x=∞ 

∫ ∫ 

Y '( x) ϕ( x) dx =− Y( x) ϕ'( 

x) dx 

x=−∞ x=−∞ 

Thus, 

x =∞ 

∫ 

=− ϕ'( xdx ) =−ϕ( 

x) 

x = 0 

= ϕ(0) 

x =∞ 

= ∫ δ( x) ϕ( 

x) dx 

x =−∞ 

Y ' = δ . 

ϕ(0) = ∫ δ( x) ϕ( 

x) dx 

x =∞ 

x = 0 

Since Yx ( ) 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 =∞ 

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 

8


1 ⎛ x =− 

n 

x =∞ ⎞ 

⎟ 

⎜ 

lim 

⎜ ⎟ 

⎜ δ( x) ϕ( x) dx δ( x) ϕ( 

x) dx ⎟ 0 

→∞ ⎜ + ⎟ = 

⎜ ∫ ∫ 

⎟ 

⎜⎝x=−∞ x = 

⎠⎟ 

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, 

= 1 1 = [ , ] ⎨ 

2n 2n 

⎪⎪⎩ 

δn( x) nχ ( x) 

− 

1 

n 

⎧⎪ 

⎪nx 

, ∈− [ , 

0, otherwise 

Then, the delta Function is defined as the limit 

δ( x) = lim δ ( x) 

. 

n→∞ 

1 1 

2n 2n 

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 δ (0) = lim n = ∞. 

n 

n 

n→∞ n→∞ 

Then, the integration of δ( x) 

has to skip the point x = 0 . 

9 

] 

.


That is, only the Cauchy Principal Value of the integral 

x =∞ 

∫ 

x =−∞ 

δ( 

xdx ) 

may exist. The Principal Value is 

1 ⎛ x =− 

n 

x =∞ ⎞ 

⎟ 

⎜ 

lim 

⎜ ⎟ 

δ( xdx ) δ( 

xdx ) ⎟ 0 

→∞ ⎜ + ⎟ = 

⎜ ⎟ 

⎜⎝x=−∞ x = ⎠⎟ 

n 

x =∞ 

∫ 

∫ ∫ . 

That is, the sifting δ ( xdx ) = 1, 

is not preserved for the limit 

x =−∞ 

of the Delta sequence, δ( x) = lim δ ( x) 

.� 

n 

n→∞ 

0.5 The Hyper-real Delta Function 

The above attempts failed because the Delta Function is a hyper- 

real 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. 

10 

n 

1 

n


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 xδ( x) 

defined at x = 0 ? 

How is the Delta Function the derivative of a Step Function? 

How do we integrate the Delta Function? 

2 

What is δ (x) 

? 

2 


? 

3 

What is δ( 

x ) ? 

3 


? 

11


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 ( ααα , , ,...) is a constant hyper-real. 

In [Dan2] we established that, 

1. Any totally ordered set of positive, monotonically decreasing 

to zero sequences ( ι , ι , ι ,...) constitutes a family of 

infinitesimal hyper-reals. 

1 2 3 

2. The infinitesimals are smaller than any real number, yet 

strictly greater than zero. 

1 1 1 

3. Their reciprocals ( , , ,... 

ι1 ι2 ι ) are the infinite hyper-reals. 

3 

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 

12


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. 

13 

∞


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 

n 

an � 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. 

14


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 0, 0, 0,... . 

The infinite hyper-real 1 

dx 

depends on our choice of dx . We will 

usually choose the family of infinitesimals that is spanned by the 

1 

n 

sequences 1 

n , 2 

1 , 3 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, 1 

dx 

will mean the sequence n . 

Alternatively, we may choose the family spanned by the sequences 

1 

1 

1 

2 n 

2 n , 2 

2 n , 3 

,… 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 

15


Infinitesimal Calculus. 

2.2 The Delta Function is strictly smaller than 

Proof: Since dx > 0, 

1 

dx


⎧⎪ 1, x ∈ ⎡− , ⎤ 

= 

⎥⎦ ⎨ is the sequence 

dx ⎪⎪⎩ 

0, otherwise 

dx dx 

1 

2.4 δ( 

x) 

⎪ ⎢⎣ 

2 2 

where dx = in 

. 

⎧⎪∈ ⎡ 

− 

⎨ 

⎪ 

⎪⎩ 

0, otherwise 

1 

in in 

⎪ , x , 

n i ⎢ 

⎣ 2 2 ⎥ 

⎦ 

Namely, as a hyper-real function the value of Delta at the 

singularity is the infinite hyper-real 1 

dx 

infinite vector with countably many components. 

17 

which is a sequence, an 

⎤


3. 

Delta Function Plots 

3.1 Delta Plot for dx = 

If 

1 

n 

1 i n = , Delta is the infinite Hyper-Real number, 

n 

δ χ χ χ 

( x) = ( x),2 ( x),3 ( x),... 

[ −1,1] [ − , ] [ − , ] 

1 1 1 1 

2 2 3 3 

We plot in Maple the 10 th component with 

Similarly, we use 

18


to plot an the 100 th component of Delta 

3.2 Delta with 

We use 

1 

2 n dx = is 

χ[ − , ] [ − , ] [ − , ] 

δ χ χ 

( x) = 2 ( x), 4 ( x),8 

( x ),... 

to plot the 4 th component of Delta 

1 1 1 1 1 1 

4 4 8 8 16 16 

19


Similarly, we use 

to plot the 6 th component of Delta, 

20


4. 

Delta Function Properties 

4.1 

xδ ( x) 

= 0 

Proof: x ≠ 0 ⇒ δ ( x) 

= 0 ⇒ xδ ( x) 

= 0. 

4.2 

4.3 

x = 

0 

⇒ 

0 

xδ( x) 

= 0 δ(0) 

= = 0, 

since dx > 0. 

dx 

1 

( x x ) ( x) 

δ − 0 ≡ χ⎡x0− dx, x dx ⎤ 

dx ( x 

2 0+ 

− 

2 

0) 

⎢⎣ ⎥⎦ 

n 

= 

That is, δ (0) spikes to 

4.4 

1 

dx 

0 , dx 

dx 2 0 2 

χ⎡ − + ⎤ 

⎢⎣ ⎥⎦ 

( ) 

x x x 

n 1 

δ ( x) = ( x) 

, n = 2, 3,... 

( ) 

dx, dx 

n 

dx 2 2 χ⎡−⎤ ⎢⎣ ⎥⎦ 

1 

( ) n 

dx 

, which is greater than 1 

dx . 

1 1 

(, xy) ≡ ()() x y = [ dx, dx () x () 

] dy dy y 

− 

dx [ , ] 

2 2 dy − 

2 2 

δ δ δ χ χ 

21


5. 

Delta Sequence 

1 

δ n( x) = n 

2cosh 

Depending on the choice of the infinitesimal dx = in 

, there are 

2 

nx 

many Delta Sequences that lead to the Delta Function, δ( 

x) 

. 

5.1 Each 

1 

δ n( x) = n 

2cosh 

2 

nx 

� has the sifting property 

� is continuous 

� peaks at x = 0 to δ (0) = 

n 

x =∞ 

∫ 

x =−∞ 

n 

2 

x =∞ x =∞ 

∫ 

δ ( xdx ) = 1 

1 tanhnx 1 

= = 1 −( − 1) = 1.� 

2 2cosh nx 2n2 Proof: n dx n 

( ) 

x =−∞ 

n 

x =−∞ 

The sequence represents the hyper-real Delta Function 

5.2 If 

in 

2 

= , 

n 

1 2 3 

δ ( x) 

= , , ,... 

2 2 2 

2 cosh x 2 cosh 2x 2 cosh 3x 

22


plots in Maple, the 50 th component, 

plots in Maple the 200 th component, 

23


6. 

Delta Sequence 

6.1 Each 

−nx 

δ ( x) = ne χ 

n 

−nx 

n( x) ne ( ) 

[0, ) 

δ χ = x 

∞ 

[0, ∞) 

� has the sifting property 

x =∞ 

∫ 

x =−∞ 

� is continuous hyper-real function 

� peaks at x = 0 to δ (0) = 

x=∞ x=∞ 

Proof: −nxχ[0, ∞) 

−nx 

x=−∞ x= 

0 

n 

n 

δ ( xdx ) = 1 

e 

ne ( x) dx = ne dx = n = 1 

−n 

n 

−nx 

x =∞ 

∫ ∫ .� 

x = 0 

The sequence represents the hyper-real Delta Function 

6.2 If 

in 

1 

= , 

n 

−x −2x −3x 

[0, ∞) [0, ∞) [0, ∞) 

δ( x) = e χ ,2 e χ , 3 e χ ,... 

24




25


7. 

Primitive of Delta Function 

δ( 

x) 

7.1 is the derivative of 

⎧⎪− ⎪ 1, x < 0 

gx ( ) = ⎪ 

⎨ 0, x= 

0 

⎪ 

⎪⎩ 

1, x > 0 

Proof: 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 

1 

g '(0 − ) = . 

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 

1 

g '(0 + ) = . 

dx 

26


Since the right and left derivatives are equal, the derivative at 

x = 0 is 

1 

g '(0) = = δ(0) 

. � 

dx 

Since at x ≠ 0 , g'( x ) = 0 , we have, 

1 

g'( x) 

= 

⎣ 

− , 

⎦ . � 

dx dx 

dx 2 2 

χ ⎡ ⎤ 

⎢ ⎥ 

⎧0, x ≤ 0 

7.2 δ( 

x) 

is the Principal Value derivative of hx ( ) = ⎪ 

⎨ 

⎪ 

⎪⎩ 

1, x > 0 

Proof: For any dx , 

h(0) −h( −dx) 

0 

= = 0 ⇒ h '(0 − ) = 0 . 

dx dx 

hdx ( ) −h(0) 1−0 1 

= = 

dx dx dx 

1 

⇒ h '(0 + ) = . 

dx 

hx x = 0 � 

Therefore, ( ) has no derivative at . 

But since 

h( dx) −h( −dx) 2 2 1−0 1 

= = , 

dx dx dx 

hx ( ) x = 0 

The principal value derivative of at is 

p.v. h '(0) 

1 

= . � 

dx 

1 

Since at x ≠ 0 , p.v. h'( x ) = 0 , we have, p.v. h'( x) 

dx , dx 

dx 2 2 

χ = ⎡− ⎤ 

⎢⎣ ⎥⎦ 

.� 

27


8. 

δ (()) fx 

8.1 

1 

δ( ax) = δ( 

x) 

a 

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 ξ 1 is the only zero of f ( x ) , and f '( ξ1) ≠ 0, 

1 

(()) ( ) 

Then, δ fx = δ x−ξ 

1 

f '( ξ1) 

Proof: δ(()) f x = δ( 

f() x − f( 

ξ ) ) 

For x − ξ = infinitesimal , 

By 8.1, 

1 

1 

( f '( )( x ) ) 

= δ ξ − ξ 

1 

1 1 

1 

= δ( x − ξ1) 

.� 

f '( ξ ) 

28


8.3 If ξ1, ξ 2 are the only zeros of f ( x ) , and f '( ξ1), f '( ξ2) ≠ 0 

1 1 

(()) ( ) ( ) 

Then, δ fx = δ x− ξ1 + δ x−ξ2 

f '( ξ1) f '( ξ2) 

Proof: δ δ( ξ ) δ( 

( f ( x)) = f( x) − f( ) + f( x) − f( 

ξ ) 

1 2 

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 '( ξ )( x − ξ ) + δ f '( ξ )( x −ξ ) 

1 1 2 

1 1 

= δ( x − ξ ) + δ( x −ξ 

) .� 

f '( ξ ) f '( ξ ) 

1 2 

1 2 

2 2 1 1 

8.4 δ( x − a ) = δ( x − a) + δ( 

x + a) 

2 a 2 a 

1 1 

δ ( x −a)( x − b) = δ( 

x − a) + δ ( x + a) 

a −b b −a 

8.5 ( ) 

8.6 If 1 are the only zeros of 

,... ξ ξ n 

f ( x ) , and f '( ξ1),.., f '( ξn) ≠ 0 

1 1 

Then, δ(()) fx = δ( x− ξ1) + ... + δ( 

x− 

ξn) 

f '( ξ ) f '( ξ ) 

1 

29 

) 

n 

2 

)


8.7 If ξ1, ξ 2,.. 

. are zeros of f ( x ) , and f '( ξ1), f '( ξ2),... ≠ 0 

8.8 

1 1 

Then, δ( fx ( )) = δ( x− ξ1) + δ( 

x−ξ 

n ) + ... 

f '( ξ ) f '( ξ ) 

1 

δ(sin x) = ... + δ( x + 2 π) + δ( x + π) + δ( x) + δ( x − π) + δ( x − 2 π) 

+ ... 

Proof: The zeros of si n x are ... −2 π, −π, 

0, π,2 π,... 

and cos( nπ ) = 1.� 

30 

n


9. 

n 

δ ( x ) 

9.1 

Proof: 

1 

( x ) = ( x), 

2xdx 

2 δ χ⎡−xdx, xdx ⎤ 

⎣ ⎦ 

2 δ = χ ( 

2 

) ( 

2 

2 

⎡ d x d x ) ⎤ 

dx ( ) ⎢−, ⎥ 

⎢ 2 2 

⎣ ⎦⎥ 

2 

x > 

1 

( x ) ( x) 

1 

( ) 

2xdx 

χ = ⎡ ⎤ , 

⎢−⎥ ( 

2 

) ( 

2 x 

d x d x ) 

, 

2 2 

⎢⎣ ⎦⎥ 

where to ensure dx ( ) > 0, 

we must have 

x > 0 . 

The amplitude and domain of the δ( 

x ) spike depend on x . 

For instance, 

9.2 

1 

2xdx 

χ⎡ xdx, xdx ⎤ 

⎣ 

− 

⎦ x = = 

( x) δ( 

x) 

9.3 −xdx, 

xdx 

( ) 2 ( ) 2 

dx 

2 2 dx dx 

xdx x 

, 

2 ( dx) 

− 

2 2 

1 

2 

χ⎡ ⎤ χ⎡ 

⎤ 

1 1 

2 

( x) = ( x) 

≤ δ ( x) 

⎣ ⎦ = ⎢ ⎥ 

⎢⎣ ⎥⎦ 

31 

2 

0


1 

( x ) ( x) 

n 

9.4 = 

( 

n 

) ( 

n 

n d x d x ) 

dx ( ) − , 

2 2 

δ χ⎡ ⎤ 

⎢ ⎥ 

⎢⎣ ⎦⎥ 

1 

= ( x) 

, x > 0 

− 

⎡ x dx x dx⎤ 

⎣⎢ ⎦⎥ 

n n 1 1 

1 

, n n 

n nx dx 2 2 χ − − 

− 

The amplitude and domain of the ( ) spike depend on x . 

n 

δ x 

For instance, 

9.5 

1 

χ⎡ n n−1 n 1 , n − 

− x dx x dx⎤ 

1 

⎣ 2 2 ⎦ = 1 

n 

n−1 

⎢ ⎥ 

nx dx 

( x) = δ( 

x) 

x 1 ( ) n− 

32


10. 

n n 

δ( x − ( dx) 

) 

2 dx 2 While x − ( ) is infinitesimally close to , 

different from δ( 

x ) . 

2 2 

10.1 ( ) 

2 

2 

dx 1 dx 1 dx 

δ x − ( ) = δ( 

x − ) + δ ( x + ) 

2 dx 2 dx 2 

Proof: By 8.4, since dx > 0.� 

10.1 has two positive spikes. For instance, 

10.2 

2 

2 2 

x ( x ( dx) 

2 ) 

⎡ 1 1 ⎤ 1 1 

⎢ δ( x − ) + δ( x + ) ⎥ = δ( 

− ) + ( 

⎢ 

⎣dx dx ⎥ 

⎦ dx dx 

δ 

dx dx dx dx 

2 2 2 

2 

x = 0 

) 

δ − is 

1 1 

= χ 2 [ −dx,0] 

( x) + χ 2 [0, dx ] ( x) 

. 

( dx) ( dx) 

Similarly, ( has three positive spikes. 

3 3 

δ x − ( dx) 

) 

2 2 

3 3 1 

i π 2i 

π 

10.3 δ 3 3 

( x ( dx) ) 2 ( δ( x dx) δ( x e dx ) δ( 

x e dx ) ) 

− = − + − + − . 

3( dx) 

3 3 

Proof: x − ( dx) 

has the three zeros 

33


Since 

x1= dx 

, 

2 

f '( x ) = 3x 

, and since 

2 

i 

2 

3 

π 

x = e dx, 

and 

3 

2 2 2 

2i 

2 

3 

π 

x = e dx. 

x2 = x3 = ( dx 

) , by 6.6 we obtain 

2 2 

3 3 1 

i π 2i 

π 

3 3 

( x ( dx) ) 2 ( ( x dx) ( x e dx ) ( x e dx ) ) 

δ − = δ − + δ − + δ − . � 

3( dx) 

n n ( x ( dx) 

) 

δ − has n positive spikes. 

1 

2π 2 

n n 

( n−1) 

π 

10.4 δ( x − ( dx) ) = 

1 ( δ( x − dx) + δ n ( x − e dx ) + ... + δ 

n 

n 

( x − e dx 

− 

) ) 

n 

n 

ndx ( ) 

Proof: x − ( dx) 

has the n zeros 

Since 

x = dx 

, 

1 

n 1 

f '( x) nx − = , and since 

by 6.6 we obtain 

2 

i 

2 

n 

π 

( n−1) i 

n 

2π 

x = e dx,…, 

x = e dx. 

n 

n−1 n−1 n−1 

n 

x1 = ... = x = ( dx ) , 

1 

2π 2 

n n 

( n−1) 

π 

( x ( dx) ) 1 ( ( x dx) n ( x e dx ) ... 

n 

n 

( x e dx 

− 

) ) 

δ − = δ − + δ − + + δ − . � 

ndx ( ) 

34


11. 

Integral of 

δ( 

x) 

11.1 Integral of a Hyper-real Function 

Let f () x be a hyper-real function on the interval [ ab , ] . 

f () x may take infinite hyper-real values, and need not be 

bounded. 

At each 

a ≤ x ≤b, 

there is a rectangle with base dx dx 

[ x − , x + ], height f () x , and area 

2 

2 

f ( xdx. ) 

We form the Integration Sum of all the areas for the x ’s that 

start at x = a, 

and end at x = b, 

∑ 

x∈[ a, b] 

f ( xdx ) . 

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 x = a, 

to x = b, 

and denote it by 

35


x= b 

∫ 

x= a 

f ( xdx ) . 

If the hyper-real is infinite, then it is the integral over [ ab , ] , 

If the hyper-real is finite, 

x= b 

∫ fxdx ( ) = 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, 2 , and 

Card� 

Card � = 

we have 

2 Card� 

2 

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 

36


by its lowest value on each interval 

11.3 

∑ 

x∈[ a, b] 

dx dx 

2 2 

[ x − , x + ] 

dx dx 

2 2 

⎛ ⎞ 

⎜ inf f ( t) ⎟ 

⎜ ⎟dx 

⎜⎜⎝ ⎟ 

x− ≤t≤ x+ 

⎠⎟ 

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

by its largest value on each interval 

11.4 

∑ 

x∈[ a, b] 

dx dx 

2 2 

[ x − , x + ] 

dx dx 

2 2 

⎛ ⎞ 

⎜ ⎟ 

⎜ sup f ( t) ⎟ 

⎜ ⎟dx 

⎜ ⎟ 

⎝x− ≤t≤ x+ 

⎠⎟ 

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( 

xdx ) = 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 

37


12. 

The Principal Value Derivative of 

Delta: the Dipole Function 

We have seen in 2.3 that 

at 

for 

at 

Here, we show 

dx 

x =− , δ( x) 

jumps from 0 to 

2 

x ∈ ⎡− ⎤ 

dx dx 

⎢⎣ , 

2 2 ⎥⎦ 

, 

dx 

x = , δ( 

x) 

drops from 

2 

1 

dx , 

1 

δ ( x) 

= . In particular, 

dx 

1 

dx 

to . 0 

� in 12.1, that δ( 

x) 

has no derivative at 

Principal Value Derivative over the jump at 

Positive Impulse Function. 

� in 12.2, that δ( 

x) 


Principal Value Derivative over the jump at 

Negative Impulse Function. 

We sum up 12.1, and 12.2. in 12.3. 

38 

1 

δ (0) = 

dx 

1 x =− dx , but the 

2 

1 x =− dx , is a 

2 

x = dx, 

but the 

1 

2 

x = dx, 

is a 

1 

2


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 

δ( 

x) 


dx 

x =− . 

2 

The Principal Value Derivative of δ( 

x) 

at 

1 

Impulse function χ[ − dx,0]. 

2 

( dx) 

Proof: The left derivative of δ( 

x) 

at x =− dx 

is 

( −dx) − ( −dx) 1 −0 

2 

dx 2 

= = 

− dx + dx dx 

2 

( dx) 

2 2 

δ δ 

The right derivative of δ( 

x) 

at x =− dx 

is 

δ(0) −δ( − ) − 

= = 0 . 

0 −− ( ) 

1 

2 

dx 1 1 

2 dx dx 

dx dx 

2 2 

x =− dx 

1 

2 

1 x =− dx , is the Positive 

2 

Since the left and right derivatives are unequal, δ( 

x) 

has no 

derivative at 

x =− dx .� 

1 

2 

39 

1 

2


The Principal Value Derivative at 

x =− dx 

is 

1 

δ(0) −δ( −dx) 

− 0 

dx 1 

= = 

0 −− ( dx) dx ( dx) 

It is the Positive Impulse Function 

1 

1 

2 

2 

( dx) 2 

. 

χ[ − dx,0] 

.� 


δ( 

x) 


x = dx. 


x) 

at 

Impulse Function 

1 

[0, dx] 

2 

( dx) χ − . 

Proof: The Left Derivative of δ( 

x) 

at x = dx 

is 

1 

2 

( dx) 

− 

2 

(0) 1 − 1 

dx dx 0 

dx 

2 

dx 

2 

dx 

δ δ 

1 

2 

x = 

dx 

2 

x = dx, 

is the Negative 

1 

2 

= = = 0 . 

The Right Derivative of δ( 

x) 

at x = dx 

is 

( dx) 

− ( dx) 

0− 1 

2 

dx 2 

= = − 

dx − dx dx 

2 

( dx) 

2 2 

δ δ 

Since the Left and Right Derivatives are unequal, δ( 

x) 

has no 

derivative at 

x = dx.� 

1 

2 

40 

1 

2 

.


The Principal Value Derivative at 

x = dx 

is 

1 

δ( dx) 

− δ(0) 0 − 

dx 1 

= = − . 

dx) dx 2 ( dx) 

It is the Negative Impulse Function 

1 

2 

1 

[0, dx] 

2 ( dx) χ − .� 


δ( x) 

has no derivative at x = 0 . 

The Principal Value Derivative of δ ( x) 

, p.v.D δ( 

x) 

is the Dipole Function 

1 1 

Dipole( x) = χ[ −dx,0] − χ[0, 

dx] 

. 

2 2 

( dx) ( dx) 

Proof: The Left Derivative of δ( x) 

at x = 0 is 

1 

δ(0) −δ( −dx) 

− 0 

dx 1 

= = 

dx dx ( dx) 

The Right Derivative of δ( x) 

at x = 0 is 

1 

δ( dx) 

− δ(0) 0 − 

dx 1 

= = − . 

dx dx 2 ( dx) 

2 

x = 

Since the left and right derivatives are unequal, δ( 

x) 

has no 

derivative at x = 0 .� 


x) 

is 

41 

0


δ( x + dx) −δ( x − dx) 

2 2 1 dx 1 dx 

= δ( x + ) − δ( 

x − ). 

dx dx 2 dx 2 

It is the Dipole Function 

If 

1 dx = , this is the sequence 

n 

1 1 

χ[ −dx,0] − χ[0, 

dx] 

.� 

2 2 

( dx) ( dx) 

2 1 2 1 

χ χ 

n n 

Dipole( x) = n [ − ,0] − n [0, ] . 

Then, a Maple plot of the 10 th component of Dipole( x) 

is 

42


13. 

The 2 nd Principal Value Derivative 

of Delta: the 4-Pole Function 

The 2 nd Principal Value Derivative of δ( 

x) 

is the 4-pole Function 

1 

2 

13.1 (p.v.D) δ( x) = ( δ( x + dx) − 2 δ( x) + δ( 

x − dx) 

) 

Proof: 

2 

( dx) 

1 3dx dx dx dx dx 3dx 

= { χ[ − , − ] −2 χ[ − , ] + χ[ 

, ] } . 

3 2 2 2 2 2 2 

( dx) 

Dipole( x + ) −Dipole( x − 

(p.v.D) δ( 

x) 

= 

dx 

dx dx 

2 2 2 

1 

= + − + − ) 

2 

( dx) 

) ` 

( δ( x dx) 2 δ( x) δ( 

x dx ) 

1 3dx dx dx dx dx 3dx 

= { χ[ − , − ] −2 χ[ − , ] + χ[ 

, ] } .� 

3 2 2 2 2 2 2 

( dx) 

The 4-pole Function has four Impulse Functions 

1 

� a Positive Impulse δ ( x + dx) 

centered at x =−d 

x, 

2 

( dx) 

43


If 

1 

� two Negative Impulses − 2 δ( 

x) 

centered at x = 0 , 

2 

( dx) 

1 

� a Positive Impulse δ( x − dx) 

centered at x = dx 

. 

2 

( dx) 


n 

3 3 1 3 1 1 3 1 3 

χ χ 

2n 2n 2n 2n 2n 2n 

4 pole( x) = n [ − , − ] − 2 n [ − , ] + n χ[ 

, ] 

Then, a Maple plot of a component of 4 pole( x) 

is 

The x axis units are 1 

n 

3 

. The y axis units are n . 

44 

.


14. 

Higher Principal Value 

Derivatives of Delta 

14.1 The 3 rd Principal Value Derivative of δ( 

x) 

, 

If 


3 

(p.v.D) δ( 

x) 

1 

8 pole( x) = ( χ[ −2 dx, −dx] −3 χ[ − dx,0] + 3 χ[0, dx] − χ[ 

dx,2 dx] 

) . 

4 ( dx) 


n 

4 2 1 4 1 4 

χ χ χ 1 1 2 

n n n n n n 

8 pole( x) = n [ − , − ] − 3 n [ − ,0] + 3 n [0, ] − n [ , ] 

4 χ 


is 


n 

4 


45 

.


14.2 The 4 th Principal Value Derivative of δ( 

x) 

, (p.v.D) δ( 

x) 

If 


1 5dx3dx 3dx 

dx 

16 pole( x) 

= ( χ[ − , − ] − 4 χ[ 

− , − ] + 

5 2 2 2 2 

( dx) 


n 

( 5 5 

+6 χ[ − , ] − 4 χ[ , ] + χ[ 

, ] . 

5 3 3 1 

χ χ 

2n 2n 2n 2 

dx dx dx 3dx 3dx 5dx 

2 2 2 2 2 2 

16 pole( x) = n [ − , − ] −4 n [ − , − ] + 

5 1 1 5 1 3 5 3 5 

χ χ χ 

2n 2n 2n 2n 2n 2n 

+6 n [ − , ] − 4 n [ , ] + n [ , ] . 


is 


n 

5 


46 

4 

) 

)


Using the Binomial coefficients, 

k x 

14.3 The k th Principal Value Derivative of δ ( x) 

, (p .v.D) δ( 

) is 

k 

2 pole( x) 

= 

If 

k−1 dx k−1 

pole x + − pole x + dx 

2 2 

2 ( ) 2 ( 

dx 

1 ⎛ ⎛k⎞ = 

⎜ 

χ[ , ] 

⎟ 

⎜ − − − 

⎜ ⎟χ[ 

− , − 

k+ 

1 

( dx) 

⎜ 

⎜ 

⎜ ⎜1 ⎟ 

⎝ ⎜⎝ 

⎠⎟ 

( k+ 1) dx ( k−1) dx ( k−1) dx ( k−3) dx 

2 2 2 2 

⎛k⎞ ( k−1) dx ( k−3) dx k ( k− 1) dx ( k+ 1) dx 

+ 

⎜ ⎟ 

⎜ ⎟ 

⎜ 

χ[ , ] ... ( 1) χ[ 

, ] 

2 

⎟ − − + + − 

2 2 2 2 ) . 

⎜⎝ ⎠⎟ 


n 

1 ( k 1) ( k 1) n 

k+ + − k+ 

1 

( n−1) dx ( n−3) dx 

n χ[ , ] n χ[ 

, ] 

2n 2n 1 2 2 

⎛ ⎞ − − − 

⎜ ⎟ 

⎜ ⎟ 

⎜ ⎟ − − 

⎜⎝ ⎠⎟ 

⎛k⎞ k+ 1 k 1 k 3 k 1 k 1 k 1 

n 

⎜ ⎟ − − k+ 

− + 

+ ⎜ 

χ[ , ] ... ( 1) n χ[ 

, ] 

2 

⎟ − − + + − 

⎜⎝ ⎟ 2n 2n 2n 2n 

⎠ 

47 

) 

+ 

]+ 

)


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 eye- 

opener 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


[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

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