Download Delta Function of a Complex Variable Abstract

Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
Delta Function of a Complex
Variable
H. Vic Dannon
[email protected]
January, 2011
Revised December, 2014
Abstract
Ignorance
of
Infinitesimal
Complex
Calculus
prevented the definition of the Complex Delta Function.
d(z - z ) is a Plane Delta Function that spikes to
1
on the
2pi(z - z )
Infinitesimal Disk z - z £ dr , and vanishes outside it.
For dr =
1
, r = z , and f = Arg(z )
n
d(z ) =
1 -if 1 -drr
e
e .
2pi
dr
Delta has the Bessel Integral Representation
W=¥
1 -if
d(z ) =
e r ò J 0 (Wr )Wd W
2pi
W= 0
The Circulation of Delta along the Infinitesimal circle z - z = dr
is
ò
d(z - z )d z = 1
z -z =dr
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H. Vic Dannon
And If f (z ) is Hyper-Complex Differentiable function at z
ò
Then,
f (z )d(z - z )d z = f (z ) .
z -z =dr
For an analytic function on a simply connected domain, this
Sifting through the function values by the Complex Delta
Function is the Cauchy Integral Formula
Also the Residue of a Laurent Expansion of a singular Function
follows from the sifting property of the Delta Function.
Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real,
Cardinal, Infinity. Non-Archimedean, Non-Standard Analysis,
Calculus,
Limit,
Continuity,
Derivative,
Integral,
Complex
Variable, Complex Analysis, Analytic Functions, Holomorphic,
Cauchy Integral Theorem, Cauchy Integral Formula, Contour
Integral.
Delta
Polar
Representation,
Delta
Exponential
Representation, Delta Fourier-Bessel Representation,
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;
46S20; 97I40; 97I30.
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Gauge Institute Journal Volume 11, No. 1, February 2015
Contents
Introduction
1.
Hyper-real Line
2.
Hyper-real Integral
3.
Delta Function
4.
The Fourier Transform
5.
Hyper-Complex Plane
6.
Hyper-Complex Path Integral
7.
Hyper-Complex d(z )
8.
d(z ) Plots
9.
d(z ) Properties
10. d( f (z ))
11. Polar Representation of d(z )
12. d(r)
13. d(r) and d(x )d(y )
14. Cartesian Representation of d(z )
15. Bessel Integral Representation of d(z )
16. Primitive and Derivatives of d(z )
17. Circulation of d(z )
18. Sifting by d(z - z ) , and
d
dz
d(z - z )
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H. Vic Dannon
Gauge Institute Journal Volume 11, No. 1, February 2015
19. d(z - z ) and the Cauchy Integral Formula
20. d(z - z ) and the Residue of a Laurent Expansion
References
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H. Vic Dannon
Introduction
0.1 Dirac’s Confusion about the Complex Delta Function
The confusion about the Complex Delta Function can be traced to
Dirac. In [Dirac] at the bottom of page 778, he writes
“…,we must use the formula
d
1
log z = - i pd(z ) ,
dz
z
(27)
in which the term
-i pd(z )
is required in order to make the integral of the right-hand
side of (27) between the limits a , and -a equal
log(-1) ,
the integral of
1
z
between these limits being assumed to be zero.”
Dirac made here four errors:
1st Error, log(-1) ¹ -ip
Indeed, integrating the left-hand side of (27), for a ¹ 0 ,
z =a
d
ò dz log z = log z
z =-a
z =a
z =-a
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Gauge Institute Journal Volume 11, No. 1, February 2015
= log a - log(-a )
= log
a
-a
= log(-1)
= log -1 + i Arg(-1)


p
=0
= ip .
z =a
2
nd
Error
1
dz ¹ 0
z
z =-a
ò
Indeed, by the Residue Theorem
z =a
1
dz =i [Arg(a ) - Arg(-a )]

z
z =-a
ò
p
= ip .
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Gauge Institute Journal Volume 11, No. 1, February 2015
3rd Error
H. Vic Dannon
d
1
log z ¹ - i pd(z )
dz
z
Indeed, it is well known that without any exception
d
1
log z = .
dz
z
z =a
4
th
Error
ò
d(z )dz ¹ 1
z =-a
Indeed, unlike the hyper-real Delta Function d(x ) , which sifts
through its singularity at x = 0 , the Hyper-Complex Delta
Function sifts along a line that encircles the singularity, and
avoids it. In the proceeding, it will become clear that
z =a
ò
d(z )dz =
z =-a
1
i [Arg(a ) - Arg(-a )]
2pi 
p
=
1
.
2
Dirac’s theory about the infinite distribution of electrons in the
theory of the positron is flawed, because of Dirac’s confusion about
the Complex Delta Function.
0.2 The Hyper-Complex Delta Function
For z in the interior of g , Cauchy Integral Formula gives an
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analytic f (z ) as the convolution of f with
f (z ) =
H. Vic Dannon
1 1
.
2pi z
1
1
ò f (z ) 2pi z - z dz
g
Thus,
1 1
2 pi z-z
recovers the value of a complex function f (z ) at the
point z in the interior of a loop g , by sifting through the values of
f (z ) on g .
However,
1 1
2 pi z-z
is the Hyper-Complex Delta Function only if the
integration path is in the infinitesimal disk
z - z £ dr ,
then, the sifting is performed by
1
1
2pi z - z
c{ z -z £dr }(z ) º d(z - z ) .
We call d(z - z ) the Hyper-Complex Delta Function.
In [Dan7], we showed that
ò
z -z =dr
Due to
1
1
dz = 1 .
2pi z - z
1
, the Complex Delta spikes to
z -z
1 1
1
=
e -i Arg(z -z )
2pi d z
2pidr
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on the infinitesimal disk z - z £ dr .
The primitive of the Complex Delta on the disk is
1
Log(z - z ) ,
2pi
and the derivative of the Complex Delta on the disk is
1
1
.
2pi (z - z )2
Since
z - z = (dr )e -i Arg(z -z ) ,
is a hyper-complex infinitesimal, we need to recall the HyperComplex Plane that we introduced in [Dan7].
Since the Complex Delta Function is an extension of the Real
Delta Function, we need to recall the Delta Function that we
introduced in [Dan4].
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1.
Hyper-real Line
Each real number a can be represented by a Cauchy sequence of
rational numbers, (r1, r2 , r3 ,...) so that rn  a .
The constant sequence (a, a, a,...) is a constant Hyper-real.
In [Dan2] we established that,
1. Any totally ordered set of positive, monotonically decreasing
to zero sequences (i1, i2 , i3 ,...) 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
, ,
i1 i2 i3
)
,... 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.
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7. The Hyper-reals are the totality of constant Hyper-reals, a
family of infinitesimals, a family of infinitesimals with
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.
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14.
H. Vic Dannon
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.
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 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.
Hyper-real Integral
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 [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 ,
å
f (x )dx .
x Î[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
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x =b
ò
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
ò
f (x )dx = real part of the hyper-real . 
x =a
2.1 The countability of the Integration Sum
In [Dan1], we established the equality of all positive infinities:
We proved that the number of the Natural Numbers,
Card , equals the number of Real Numbers, Card  = 2Card  , and
we have
Card 
Card  = (Card )2 = .... = 2Card  = 22
= ... º ¥ .
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 .
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The Lower Integral is the Integration Sum where f (x ) is replaced
by its lowest value on each interval [x - dx2 , x + dx2 ]
2.2
å
x Î[a ,b ]
æ
ö
çç
inf
f (t ) ÷÷÷dx
çè x -dx £t £x + dx
÷ø
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 ]
2.3
æ
ö÷
çç
f (t ) ÷÷dx
å çç x -dxsup
÷÷
dx
£t £x +
ø
x Î[a ,b ] è
2
2
If the integral is a finite Hyper-real, we have
2.4 A Hyper-real function has a finite integral if and only if its
upper integral and its lower integral are finite, and differ by an
infinitesimal.
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3.
Delta Function
In [Dan5], we defined the Delta Function, and established its
properties
1. The Delta Function is a Hyper-real function defined from the
ïì 1 ïü
Hyper-real line into the set of two Hyper-reals ïí 0, ïý . The
îïï dx þïï
Hyper-real 0 is the sequence
real
0, 0, 0,... . The infinite Hyper-
1
depends on our choice of dx .
dx
2. We will usually choose the family of infinitesimals that is
spanned by the sequences
1
1
1
,
,
,… It is a
2
3
n
n
n
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
will mean the sequence n .
dx
Alternatively, we may choose the family spanned by the
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1
sequences
2n
,
1
3n
1
,
4n
,… Then,
H. Vic Dannon
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 Infinitesimal Calculus.
3. The Delta Function is strictly smaller than ¥
d(x ) º
4. We define,
1
dx
where
c
é -dx , dx ù (x ) ,
êë 2 2 ûú
c
ïïì1, x Î éê - dx , dx ùú
ë 2 2 û.
é -dx , dx ù (x ) = í
ïï 0, otherwise
ëê 2 2 ûú
î
5. Hence,
 for x < 0 , d(x ) = 0
 at x =  for
dx
1
, d(x ) jumps from 0 to
,
2
dx
1
x Î éêë - dx2 , dx2 ùûú , d(x ) =
.
dx
 at x = 0 ,
 at x =
d(0) =
1
dx
dx
1
, d(x ) drops from
to 0 .
2
dx
 for x > 0 , d(x ) = 0 .
 x d(x ) = 0
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6. If dx =
7. If dx =
8. If dx =
1
n
2
n
1
n
, d(x ) =
c
, d(x ) =
2 2
1
,
ò
c
(x ), 3
[- 1 , 1 ]
4 4
2
,
(x )...
[- 1 , 1 ]
6 6
3
2 cosh2 x 2 cosh2 2x 2 cosh2 3x
,...
, d(x ) = e -x c[0,¥), 2e-2x c[0,¥), 3e-3x c[0,¥),...
x =¥
9.
c
(x ), 2
[- 1 , 1 ]
H. Vic Dannon
d(x )dx = 1 .
x =-¥
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4.
The Fourier Transform
In [Dan6], we defined the Fourier Transform and established its
properties
1.  { d(x )} = 1
2. d(x ) = the inverse Fourier Transform of the unit function 1
w =¥
1
=
ei wxd w
ò
2p w =-¥
n =¥
ò
=
e 2pixd n , w = 2pn
n =-¥
1
3.
2p
w =¥
ò
e i wxd w
w =-¥
=
x =0
1
= an infinite Hyper-real
dx
w =¥
ò
ei wxd w
w =-¥
=0
x ¹0
4. Fourier Integral Theorem
1
f (x ) =
2p
æ x =¥
ö÷
çç
÷
-ik x
ò ççç ò f (x )e d x ÷÷÷÷eikxdk
ø
k =-¥ è x =-¥
k =¥
does not hold in the Calculus of Limits, under any
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H. Vic Dannon
conditions.
5. Fourier Integral Theorem in Infinitesimal Calculus
If f (x ) is a Hyper-real function,
Then,
 the Fourier Integral Theorem holds.
x =¥

ò
f (x )e-i axdx converges to F (a)
x =-¥
1

2p
a =¥
ò
F (a)e -i axd a converges to f (x )
a =-¥
6. 2-Dimesional Fourier Transform
y =¥ x =¥
 { f (x , y )} =
ò
ò
f (x , y )e
-i wx x -i wyy
dxdy
y =-¥ x =-¥
y =¥ x =¥
=
ò
ò
f (x , y )e
-2 pi (nx x + nyy )
dxdy ,
y =-¥ x =-¥
wx = 2pnx
wy = 2pny
8. 2-Dimesional Inverse Fourier Transform

-1
{ F (wx , wy )} =
wy =¥ wx =¥
1
2
(2p)
ò
ò
wy =-¥ wx =-¥
20
F (wx , wy )e
i ( wx x + wy y )
d wxd wy
Gauge Institute Journal Volume 11, No. 1, February 2015
ny =¥ nx =¥
ò
=
ò
F (2pnx , 2pny )e
H. Vic Dannon
2 pi (nx x +nyy )
d nxd ny ,
ny =-¥ nx =-¥
9.
wx = 2pnx
wy = 2pny
2-Dimesional Fourier Integral Theorem
æ h =¥ x =¥
ö÷
çç
-i wx x -i wy h
÷ i (wx x + wyy )d w d w
(
,
)
f (x , y ) =
f
x
h
e
d
x
d
h
çç ò
÷÷÷e
x
y
ò
ò
ò
2
(2p) w =-¥ w =-¥ çè h =-¥ x =-¥
÷
ø
y
x
1
wy =¥ wx =¥
w =¥
w =¥
æ
æ
÷÷ö
÷÷ö çç 1 y
çç 1 x
i wy (y - h )
i wx (x -x )
= ò
d wx ÷÷d x çç
e
d wy ÷÷d h
ò f (x, h)ççç 2p ò e
÷÷ çç 2p ò
÷÷ø
è wx =-¥
ø è wy =-¥
h =-¥ x =-¥
h =¥ x =¥
ö÷
æ nx =¥
ö÷ æç ny =¥
wx = 2pnx
çç
2 pi ny (y - h )
÷÷
2 pi nx (x -x )
÷
ç
÷
,
= ò
f
(
x
,
h
)
e
d
n
d
x
e
d
n
d
h
ç
ç
÷ ç ò
x÷
y÷
ò
ç ò
÷
wy = 2pny
èç nx =-¥
ø÷ çèç ny =-¥
h =-¥ x =-¥
ø÷
h =¥ x =¥
10.
2-Dimesional Delta Function
wy =¥
w =¥
æ
ö÷çæ
÷ö÷
çç 1 x
1
i wyy
i wx x
÷
ç
d ( x, y ) = ç
e d wx ÷÷çç
e d wy ÷÷
çç 2p ò
÷÷çç 2p ò
÷ø÷
è wx =-¥
øè wy =-¥
ö
æ nx =¥
öæç ny =¥
÷
wx = 2pnx
÷
çç
p
n
2
i
y
÷
= ç ò e 2pinx xd nx ÷÷÷ççç ò e y d ny ÷÷ ,
÷÷ wy = 2pny
çç
÷÷çç
è nx =-¥
øè ny =-¥
ø
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5.
Hyper-Complex Plane
Each complex number a + i b can be represented by a Cauchy
sequence of rational complex numbers, r1 + is1, r2 + is2 , r3 + is 3 ...
so that rn + isn  a + ib .
The constant sequence (a + i b, a + i b, a + i b,...) is a Constant
Hyper-Complex Number.
In [Dan2] we established that,
1. Any set of sequences (i1 + i o1, i2 + i o2 , i3 + i o3 ,...) , where
(i1, i2 , i3 ,...) belongs to one family of infinitesimal hyper reals,
and (o1, o2 , o3 ,...) belongs to another family of infinitesimal
hyper-reals, constitutes a family of infinitesimal hypercomplex numbers.
2. Each hyper-complex infinitesimal has a polar representation
dz = (dr )eif = o*e if , where dr = o* is an infinitesimal, and
f = arg(dz ) .
3. The
infinitesimal hyper-complex numbers are smaller in
length, than any complex number, yet strictly greater than
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H. Vic Dannon
zero.
4. Their reciprocals
(
1
, 1 , 1
i1 +i o1 i2 +i o2 i3 +io3
)
,... are the infinite hyper-
complex numbers.
5. The infinite hyper-complex numbers are greater in length
than any complex number, yet strictly smaller than infinity.
6. The sum of a complex number with an infinitesimal hypercomplex is a non-constant hyper-complex.
7. The Hyper-Complex Numbers are the totality of constant
hyper-complex
numbers,
a
family
of
hyper-complex
infinitesimals, a family of infinite hyper-complex, and nonconstant hyper-complex.
8. The Hyper-Complex Plane is the direct product of a HyperReal Line by an imaginary Hyper-Real Line.
9. In Cartesian Coordinates, the Hyper-Real Line serves as an
x coordinate line, and the imaginary as an iy coordinate
line.
10.
In Polar Coordinates, the Hyper-Real Line serves as a
Range r line, and the imaginary as an iq coordinate. Radial
symmetry leads to Polar Coordinates.
11.
The Hyper-Complex Plane includes the complex
numbers separated by the non-constant hyper-complex
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numbers. Each complex number is the center of a disk of
hyper-complex numbers, that includes no other complex
number.
12.
In particular, zero is separated from any complex
number by a disk of complex infinitesimals.
13.
Zero is not a complex infinitesimal, because the length
of zero is not strictly greater than zero.
14.
We do not add infinity to the hyper-complex plane.
15.
The hyper-complex plane is embedded in ¥ , and is
not homeomorphic to the Complex Plane  . There is no bicontinuous one-one mapping from the hyper-complex Plane
onto the Complex Plane.
16.
In particular, there are no points in the Complex Plane
that can be assigned uniquely to the hyper-complex
infinitesimals, or to the infinite hyper-complex numbers, or
to the non-constant hyper-complex numbers.
17.
No neighbourhood of a hyper-complex number is
homeomorphic to a n ball.
Therefore, the Hyper-Complex
Plane is not a manifold.
18.
The Hyper-Complex Plane is not spanned by two
elements, and is not two-dimensional.
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6.
Hyper-Complex Path Integral
Following the definition of the Hyper-real Integral in [Dan3],
the Hyper-Complex Integral of f (z ) over a path z (t ) , t Î [a, b ] , in
its domain, is the sum of the areas f (z )z '(t )dt = f (z )dz (t ) of the
rectangles with base z '(t )dt = dz , and height f (z ) .
6.1 Hyper-Complex Path Integral Definition
Let f (z ) be hyper-complex function, defined on a domain in the
Hyper-Complex Plane. The domain may not be bounded.
f (z ) may take infinite hyper-complex values, and need not be
bounded.
Let z (t ) , t Î [a, b ] , be a path, g(a, b ) , so that dz = z '(t )dt , and z '(t )
is continuous.
For each
t , there is a hyper-complex rectangle with base
[z (t ) - dz2 , z (t ) + dz2 ] , height f (z ) , and area f (z (t ))dz (t ) .
We form the Integration Sum of all the areas that start at
z (a) = a , and end at z (b ) = b ,
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å
H. Vic Dannon
f (z (t ))dz (t ) .
t Î[ a, b ]
If for any infinitesimal dz = z '(t )dt , the Integration Sum equals
the same hyper-complex number, then f (z ) is Hyper-Complex
Integrable over the path g(a, b ) .
Then, we call the Integration Sum the Hyper-Complex Integral of
f (z ) over the g(a, b ) , and denote it by
ò
f (z )dz .
g (a ,b )
If the hyper-complex number is an infinite hyper-complex, then it
ò
equals
f (z )dz .
g (a ,b )
If the hyper-complex number is finite, then its constant part
equals
ò
f (z )dz .
g (a ,b )
The Integration Sum may take infinite hyper-complex values,
such as
1
dz
, but may not equal to ¥ .
The Hyper-Complex Integral of the function f (z ) =
1
over a path
z
that goes through z = 0 diverges.
6.2 The Countability of the Integration Sum
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H. Vic Dannon
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
= ... º ¥ .
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 (z )dz .
6.3 Continuous f (z ) is Path-Integrable
Hyper-Complex f (z ) Continuous on D is Path-Integrable on D
Proof:
Let z(t ) , t Î [a, b ] , be a path, g(a, b ) , so that dz = z '(t )dt , and z '(t )
is continuous. Then,
f (z (t ))z '(t ) = ( u(x (t ), y(t )) + iv(x (t ), y(t )) )( x '(t ) + iy '(t ) )
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= éë u(x (t ), y(t ))x '(t ) - v(x (t ), y(t ))y '(t ) ùû +

U (t )
+i éë u(x (t ), y(t ))y '(t ) + v(x (t ), y(t ))x '(t ) ùû

V (t )
= U (t ) + iV (t ) ,
where U (t ) , and V (t ) are Hyper-Real Continuous on [a, b ] .
Therefore, by [Dan3, 12.4], U (t ) , and V (t ) are integrable on [a, b ] .
Hence, f (z (t ))z '(t ) is integrable on [a, b ].
Since
t =b
ò
f (z (t ))z '(t )dt =
t =a
ò
g (a ,b )
f (z ) is Path-Integrable on g(a, b ) . 
28
f (z )dz ,
Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
7.
Hyper-Complex Delta d(z )
7.1 Domain and Range of d(z )
The Hyper-Complex Delta Function is defined from the HyperComplex plane into the set of two hyper-complex numbers,
ì
1 üïï
ï
ï
0,
í
ý.
ï
ï
p
idz
2
ï
ï
î
þ
The hyper-complex 0 is the sequence
0, 0, 0,... .
The infinite hyper-complex
1 1
1 1 -if
=
e
2pi dz
2pi dr
depends on
Arg z = f ,
and on our choice of the infinitesimal dr .
We will usually choose the family of infinitesimals that is spanned
by the sequences
1
1
1
,
,
,… It is a semigroup with
n
n2
n3
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.
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Gauge Institute Journal Volume 11, No. 1, February 2015
Therefore,
H. Vic Dannon
1
will mean the sequence n .
dr
Alternatively, we may choose the family spanned by the sequences
1
2n
,
1
22n
,
1
23n
1
will mean the sequence 2n .
dr
,… Then,
7.2 d(z ) is an infinite hyper-complex function on the infinitesimal
hyper-complex disk z £ dr , In particular,
d(z ) < ¥
Proof: Since dr > 0 , we have
1
1
e -if < ¥ .
< ¥ , and
dr
2pidr
7.3 Definition of d(z - z 0 )
For any dz ,
d(z - z 0 ) º
1
2pidz
c{
z -z 0 £ dz
}(z ) ,
ì
ï
1, z - z 0 £ (dx )2 + (dy )2
1
ï
=
í
2pi(dx + idy ) ï
0,
otherwise
ï
ï
î
Thus,
 on the disk, z - z 0 £ dr , d(z - z 0 ) =
1 1
.
2pi dz
 off the disk, for z - z 0 > dr , d(z - z 0 ) = 0 .
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H. Vic Dannon
8.
d(z ) Plots
8.1 Delta Plot for
d(z ) =
1 -if
e
2pi
c{ z £1}(z ), 2c{ z £ }(z ), 3c{ z £ }(z ),...
1
2
1
3
is a sequence of shrinking disks with increasing heights
31
,
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H. Vic Dannon
The disks are in the body generated by revolving
1
1
= , over the
z
r
Complex plane
8.2 Delta Plot for
d(z ) =
1 -if
e
2pi
c{ z £ }(z ), 2c{ z £ }(z ), 4c{ z £ }(z ), 8c{ z £ }(z )...
1
20
1
21
32
1
22
1
23
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H. Vic Dannon
is a sequence of shrinking disks with increasing heights that lie in
the body of revolution of
1
z
2
=
1
2x
.
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H. Vic Dannon
9.
d(z ) Properties
9.1
d(z - z 0 ) =
c
1 1 -i(f-f0 )
e
{ z -z 0 £dr }(z ) ,
2pi dr
f = arg z , f0 = arg z 0
1
2pidr
9.2
d(0) =
9.3
(d(z ))n =
1
1
(2pi )n (dr )n
e -inf
c{
34
z £dr }(z )
, n = 2, 3,...
Gauge Institute Journal Volume 11, No. 1, February 2015
10.
d( f (z ))
10.1
d(az ) =
1
d(z )
a
Proof: d(az ) =
1 1
c
(z )
2pi d (az ) { az £dr }
=
1 1 1
c
(z )
a 2pi dz { z £ a1 dr }
=
1 1 1
c
(z )
a 2pi dz { z £ dz }
=
1 1
d(z ) . 
a 2pi
10.2
z1 = only zero of f (z ) , f '(z1 ) ¹ 0 
 d( f (z )) =
Proof:
1
d(z - z1 )
f '(z1 )
d( f (z )) = d ( f (z ) - f (z1 ) )
For z - z1 = infinitesimal ,
35
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= d ( f '(z1 )(z - z1 ) )
=
1
d(z - z1 ) , by 10.1. 
f '(z1 )
10.3 z1, z 2 are the only zeros of f (z ) ; f '(z1 ), f '(z 2 ) ¹ 0

d( f (z )) =

1
1
d(z - z1 ) +
d(z - z 2 )
f '(z1 )
f '(z 2 )
Proof: d( f (z )) = d ( f (z ) - f (z1 ) ) + d ( f (z ) - f (z 2 ) )
For z - z1 = infinitesimal , z - z 2 = infinitesimal ,
d( f (z )) = d ( f '(z1 )(z - z1 ) ) + d ( f '(z 2 )(z - z 2 ) )
=
10.4
10.5
1
1
d(z - z1 ) +
d(z - z 2 ) . 
f '(z1 )
f '(z 2 )
d(z 2 - a 2 ) =
1
1
d(z - a ) + d(z + a )
2a
2a
d ( (z - a )(z - b) ) =
1
1
d(z - a ) +
d(z - b)
a -b
b -a
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
10.6 z1,...zn are the only zeros of f (z ) ; f '(z1 ),.., f '(zn ) ¹ 0 
d( f (z )) =
1
1
d(z - z1 ) + ... +
d(z - zn )
f '(z1 )
f '(zn )
10.7
d(sin z ) = .. + d(z + 2p) - d(z + p) + +d(z ) - d(z - p) + d(z - 2p) + ..
Proof: The zeros of sin z are ... - 2p, -p, 0, p, 2p,...
and
…, cos(-p) = -1 , cos(0) = 1 , cos(p) = -1 ,… 
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
11.
Polar Representation of d(z )
1 -i Arg(z )
e
d( z )
2pi
d(z ) =
11.1
Proof:
1
d(z ) =
2pidz
=
c{
z £ dz
dx - idy
2pi[(dx )2 + (dy )2 ]
dx
=
}(z ) ,
(dx )2 + (dy )2
where
c{
c{
z £ (dx )2 +(dy )2
}
z £ dz
ìï 0, z > dz
ï
}(z ) = í
ïï 1, z £ dz
î
(z )
dy
-i
(dx )2 + (dy )2
2
2
2pi (dx ) + (dy )
c
cos f - i sin f 1
=
z £d r }(z ) ,
2pi
d
r {

c{
z £ (dx )2 +(dy )2
}
(z )
d r = (dx )2 + (dy )2
where
f = Arg(z )
d(r )
=
1 -i Arg(z )
e
d(r) ,
2pi
r = z = x + iy =
38
x 2 + y2 .
Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
12.
d(r)
12.1
Each component of
1
,
2
,
3
e r e 2r e 3r
,...
r =¥
 has the sifting property:
ò
ne
e-n r
dr = n
-n
r =0
 is continuous Hyper-real function
 peaks at r = 0 to n
Therefore,
12.2
e -r , 2e -2r , 3e -3r ,... represents
the Hyper-Real Delta Function d(r)
12.3
1
For d r =
,
n
1 -drr
d(r) =
e
dr
39
r =¥
-n r
= 1.
r =0
Gauge Institute Journal Volume 11, No. 1, February 2015
12.4
H. Vic Dannon
plots in Maple
the 100th component of d(r ) ,
12.5
plots in Maple
the 200th component of d(r )
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Gauge Institute Journal Volume 11, No. 1, February 2015
41
H. Vic Dannon
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H. Vic Dannon
13.
d(r) and d(x )d(y )
13.1
d(r) =
1
dr
c[-
13.2
d(f) =
1
df
c[-
(r) , r ³ 0
dr dr
, ]
2 2
(f) , 0 £ f £ 2p
df df
, ]
2 2
Transforming from Polar to Cartesian Coordinates,
13.3
d(r)d(f) = rd(x )d(y )
Proof:
¶(x , iy )
d(r) d(if) = d(x ) d(iy )

 ¶(r, if)
1
d (f )
i
1
d (y )
i
d(r)d(f) = d(x )d(y )
cos f
i sin f
.
i r sin f r cos f

r
Integrating over f ,
13.4
x = r cos f
iy = i r sin f
d(r) = 2prd(x )d(y )
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Gauge Institute Journal Volume 11, No. 1, February 2015
f =2 p
Proof:
d(r)
ò
f =2p
d(f)d f = rd(x )d(y )
ò
df . 
f =0


f=0

1
2p
43
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
14.
Cartesian Representation of d(z )
æ
1 -i Arg(z ) çç 1
d(z ) = e
rç
çç 2p
i
è
14.1
Proof:
wx =¥
ò
e
wx =-¥
i wx x
wy =¥
ö÷
ö÷æç
1
i wyy
÷
÷
ç
d wx ÷÷çç
e d wy ÷÷
ò
÷÷
÷÷çç 2p
øè wy =-¥
ø
By 12.1,
d(z ) =
1 -i Arg(z )
e
d(r)
2pi
By 13.4,
=
1 -i Arg(z )
e
2prd(x )d(y )
2pi
1
= e-i Arg(z )rd(x )d(y )
i
By 4.10,
æ
1 -i Arg(z ) çç
rç
= e
i
çèç
Denoting
wx = 2pnx
wy = 2pny
ö÷
öæç ny =¥
÷
2 pi nyy
÷
2 pi nx x
÷
ç
ò e d nx ÷÷÷÷ççç ò e dny ÷÷÷ ,
÷ø
øçè ny =-¥
nx =-¥
nx =¥
,
æ
1 -i Arg(z ) çç 1
rç
= e
çç 2p
i
è
wy =¥
ö÷
öçæ
÷
1
i wyy
÷
i wx x
÷
ç
ò e d wx ÷÷÷÷ççç 2p ò e dwy ÷÷÷÷ . 
øçè wy =-¥
wx =-¥
ø
wx =¥
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
15.
Bessel Integral Representation of
d(z )
u =¥
15.1
d(x )d(y ) = 2p
ò
J 0 (2pur)ud u ,
u = n x + i ny
u =0
1
=
2p
W=¥
ò
J 0 (Wr)Wd W ,
W = wx + i wy
W= 0
= Inverse 2D-Bessel-Fourier Transform of 1 .
w = 2pn , w = wx + i wy , n = nx + i ny
Proof: By 4.2,
ö
æ nx =¥
öæç ny =¥
÷
÷
çç
2
i
y
p
n
÷
d(x )d(y ) = ç ò e 2pinx xd nx ÷÷÷ççç ò e y d ny ÷÷ .
÷÷
çç
÷÷çç
è nx =-¥
øè ny =-¥
ø
Substitute
x = r cos f
y = r sin f
r = x + iy
nx = u cos b
ny = u sin b
u = n x + i ny
Then,
nx x + nyy = ur(cos b cos q + sin b sin q) = ur cos(b - q) .
Integrating with respect to u , and b ,
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Gauge Institute Journal Volume 11, No. 1, February 2015
u =¥ æ b =2 p
ç
= ò çç
ç
è
u =0 ç
ò
e
H. Vic Dannon
ö÷
d b ÷÷÷ ud u
÷
ø÷
2 pi ur cos( b -q )
b =0
Denoting a = b - f ,
b =2p
ò
a = 2 p -f
e
2 pi ur cos( b -f )
db =
b =0
ò
e 2pi ur cos ad a .
a =-f
Since e 2pi ur cos a is periodic with period 2p ,
a =2 p
=
ò
e 2pi ur cos ad a ,
a=0
a =2p
=
ò
a =0
æ
ö
2pi ur cos a (2pi ur cos a)2 (2pi ur cos a)3
+
+
+ ... ÷÷÷d a .
ççç 1 +
÷ø
çè
1
2!
3!
The integrals of the odd powers vanish, and we have
= 2p -
(2pur)2
2
2p +
2
(2pur)4
2
2 ⋅4
2
2p -
(2pur)6
2
2
2
2 ⋅4 ⋅6
2p + ...
= 2pJ 0 (2pur) .
Therefore,
u =¥
d(x )d(y ) = 2p
ò
J 0 (2pur)ud u ,
u = n x + i ny
u =0
Put w = 2pn , w = wx + i wy , n = nx + i ny
1
=
2p
W=¥
ò
J 0 (Wr)Wd W ,
W= 0
46
W = wx + i wy . 
Gauge Institute Journal Volume 11, No. 1, February 2015
Thus,
W=¥
d(r) = r
15.2
ò
J 0 (Wr)Wd W
W= 0
Proof: By 13.4,
d(r) = 2prd(x )d(y )
By 15.1,
1
= 2pr
2p
W=¥
ò
J 0 (Wr)Wd W . 
W= 0
Hence,
W=¥
15.3
Proof:
1 -i Arg(z )
d(z ) =
e
r ò J 0 (Wr)Wd W
2pi
W= 0
By 11.1,
d(z ) =
1 -i Arg(z )
e
d(r)
2pi
By 15.2,
1 -i Arg(z )
=
e
r
2pi
W=¥
ò
J 0 (Wr)Wd W . 
W= 0
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Gauge Institute Journal Volume 11, No. 1, February 2015
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16.
Primitive and Derivatives of d(z )
The Hyper-Complex Log Function is the primitive of d(z ) , on the
Hyper-Complex Plane
16.1
16.2
d(z - z ) =
d 1
( Log(z - z ) )
dz 2pi
d
1
1
d(z - z ) =
dz
2pi (z - z )2
c{
c{
z -z £dr }(z )
z -z £dr }(z )
That is,
 on the disk z - z £ dr ,
d
1 1 -2i q
d(z - z ) =
.
e
dz
2pi (dr )2
 off the disk, in z - z > dr ,
16.3
dk
dz k
d(z - z ) =
d
d(z - z ) = 0 .
dz
1
k!
2pi (z - z )k +1
That is,
48
c{
z £dr }(z )
Gauge Institute Journal Volume 11, No. 1, February 2015
k
 on the disk, z - z £ dr , d k d(z - z ) =
dz
 off the disk, on z - z > dr ,
dk
dz k
H. Vic Dannon
k!
1
e -i (k +1)q ,
2pi (dr )k +1
d(z - z ) = 0 .
16.4 d(z ) is differentiable, and integrable to any order,
But it has no Taylor Series, it is Not analytic,
Hence, its integral along a closed path does not vanish:
ò d(z )dz ¹ 0
g
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H. Vic Dannon
17.
Circulation of d(z )
The Hyper-complex Delta Function is Not Analytic.
It is singular on the infinitesimal disk z - z £ dr .
Integrating along a path that encircles its singularity at z = z ,
the Circulation of Delta along the infinitesimal circle is 1 .
ò
17.1
d(z - z )d z = 1 .
z -z =dr
Proof: Put
z - z = (dr )e i a
d z = i(dr )ei ad a .
Then,
1
2pi
1
1
d
z
=
ò z - z
2pi
z -z =dr
a =2 p
ò
a =0
1
(dr )e
ia
i(dr )e i ad a ,
Since (dr )e i a ¹ 0 , for any infinitesimal dr , and any a ,
1
i
=
2pi
a =2 p
ò
a=0
50
da = 1 .
Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
18.
Sifting by d(z - z ) and
d
dz
d(z - z )
18.1 Sifting by d(z - z )
If f (z ) is Hyper-Complex Differentiable function at z
ò
Then,
f (z )d(z - z )d z = f (z )
z -z =dr
Proof:
Since f (z ) is differentiable at z , on the circle z - z = (dr )ei a ,
z = z + (dr )e i a ,
d z = i(dr )e i ad a
f (z + (dr )e i a ) = f (z ) + f '(z )(dr )e i a ,
ò
f (z )d(z - z )d z =
z -z =dr
= f (z )
ò
d(z - z )d z + f '(z )dr
z -z =dr
ò
eia
z -z =dr

1
1
2pi z
-z
1 e -i a
dr
1
a =2 p
1
1
= f (z ) +
f '(z )(dr ) ò e i a
i(dr )e i ad a
i
a
2pi
(dr )e
a =0
51
dz
i (dr )e i ad a
Gauge Institute Journal Volume 11, No. 1, February 2015
1
= f (z ) +
f '(z )(dr )
2pi
a =2 p
ò
e i ad a
a = 0

0
= f (z ) . 
18.2
Proof:
d
f (z ) =
dz
d
1
f (z ) =
2pi
dz
=
ò
f (z )
z -z =dr
f (z )
ò
dz
2
(
z
)
z
z -z =dr
ò
f (z )
z -z =dr
18.3
Proof:
dk
dz k
ò
f (z ) =
=
d
d(z - z )d z . 
dz
f (z )
z -z =dr
d
k!
f (z ) =
2pi
dz
d
d(z - z )dz
dz
ò
z -z =dr
ò
z -z =dr
dk
f (z )
dz
k +1
(z - z )
f (z )
d(z - z )dz
dz k
dk
dz
k
52
d(z - z )d z . 
H. Vic Dannon
Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
19.
d(z - z ) and the Cauchy Integral
Formula
19.1 The Cauchy Integral Formula is Sifting by Delta
If
f (z ) is Hyper-Complex Differentiable function on a Hyper-
Complex Simply-Connected Domain D .
Then, for any loop g , and any point z in its interior
f (z ) =
ò
f (z )d(z - z )d z =
z -z =dr
1
2pi
ò
g
f (z )
dz
z -z
f (z )
is Differentiable
z -z
on the Hyper-Complex Simply-Connected domain D , and on
Proof: The Hyper-Complex function
a path that includes g and an infinitesimal circle about z .
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
Then, the integrals on the lines between g and the circle have
opposite signs and cancel each other.
The integral over the circle has a negative sign because its
direction is clockwise, and by Cauchy Integral Theorem,
f (z )
ò z - z d z - ò
g
z -z =dr
f (z )
dz = 0 .
z -z
Therefore,
ò
g
f (z )
dz =
z -z
ò
z -z =dr
= 2pi
f (z )
dz
z -z
ò
f (z )
z -z =dr
1
1
dz . 
p
z
i
z
2

d (z -z )


f (z )
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
20.
d(z - z )
and
the
Residue
of
a
Laurent Expansion
20.1 Laurent Expansion of a Singular f (z )
If
f (z ) is Hyper-Complex Differentiable function on a Hyper-
Complex disk 0 < z - z 0 < r
Then, f (z ) = ... + a-3 (z - z 0 )-3 + a-2 (z - z 0 )-2 + a-1(z - z 0 )-1 +
+a0 + a1(z - z 0 ) + a2 (z - z 0 )2 + ...
where for any loop g , and for any point z ¹ z 0 in its interior
a-k =
1
2pi
ò f (z )(z - z 0 )k -1d z ,
k = 1, 2,... ,
g
a-3 =
a-2 =
1
2pi
ò f (z )(z - z 0 )2d z
1
2pi
ò f (z )(z - z 0 )d z ,
a-1 =
g
g
1
2pi
ò f (z )d z ,
55
g
Gauge Institute Journal Volume 11, No. 1, February 2015
a0 =
a1 =
a2 =
ak =
Proof:
1
2pi
1
2pi
1
2pi
1
2pi
ò
g
f (z )
dz ,
z - z0
f (z )
ò (z - z
g
g
dz ,
2
0)
f (z )
ò (z - z
g
f (z )
ò (z - z
k +1
0)
H. Vic Dannon
dz ,
3
0)
dz
k = 0,1, 2,...
The Hyper-Complex Differentiable f (z ) satisfies Cauchy
Integral Formula in the Hyper-Complex domain D , bounded by a
path that includes g and an infinitesimal circle about z 0
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
Then, the integrals on the lines between g and the circle have
opposite signs and cancel each other.
The integral over the circle has a negative sign because its
direction is clockwise, and by Cauchy Integral Formula,
æ
1 çç
f (z )
f (z ) =
dz çç ò

2pi ç z - z
èg
For z along g ,
ö÷
f (z )
÷
ò z - z dz ÷÷÷÷ .
z -z =dr
ø
1
1
1
1 æç
=
=
ç1 +
z -z
z - z 0 1 - z -z 0
z - z 0 çè
z -z
z -z 0
z -z 0
+
ö
+ ... ÷÷÷ .
ø÷
( )
z -z 0
2
z -z 0
0
Then,
f (z )
f (z )
f (z )
f (z )
z
(
)
d
z
z
d z (z - z 0 )2 + ...
+

0
ò
2
3
(z - z 0 )
0)
g
g





ò z - z d z = ò z - z 0 d z + ò (z - z
g
g

2 pia0 (z 0 )
2 pia1(z 0 )
2 pia2 (z 0 )
For z along the circle z - z = dr ,
-
1
1
1
1 æç
=
=
ç1 +
z -z
z - z 0 1 - z -z 0
z - z 0 çè
z -z
z -z 0
z -z 0
+
( )
z -z 0
2
z -z 0
ö
+ ... ÷÷÷ .
ø÷
0
Then,
-f (z )
ò z - z d z
=
1
ò f (z )d z z - z

+
0
2 pia-1 (z 0 )
f (z )
ò z - z
dz
0

2pia-2 (z 0 )
1
2
(z - z 0 )
+
f (z )
ò (z - z
2
dz
0 )  (z

1
- z 0 )3
+ ...
2 pia-3 (z 0 )
By the Cauchy Integral Theorem the integrals of a-1 , a-2 , a-3 ,…
can be taken along g . 
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
20.2 d(z - z ) and the Residue of a Laurent Expansion
If
f (z ) is Hyper-Complex function on a Hyper-Complex disk
0 < z - z 0 < r so that
f (z ) = ... + a-3 (z - z 0 )-3 + a-2 (z - z 0 )-2 + a-1(z - z 0 )-1 +
+a0 + a1(z - z 0 ) + a2 (z - z 0 )2 + ...
Then, for any loop g , around z 0
ìï 0,
k ¹ -1
ïï
k
ò (z - z 0 ) dz = íï 2pi ò d(z - z 0 )dz = 2pi, k = -1
ïï
g
z -z 0 =dr
ïî
Hence,
1
2pi
ò f (z )d z = a-1 ,
g
Proof:
For any integer k ¹ -1 ,
1
(
)
(z - z 0 )k +1
z
z
dz
=
0
ò
k +1
z =a
k
g
=0
z =a
For k = -1 ,
ò (z - z 0 )-1dz
g
= 2pi
1
1
dz = 2pi . 
p
2
i
z
z
z -z 0 =dr 0
ò
d (z -z 0 )
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Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
References
[Bremermann] Hans Bremermann, “Distributions, Complex Variables, and
Fourier Transforms” Addison-Wesley, 1965
[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities,
and the Continuum Hypothesis” in Gauge Institute Journal Vol.6 No 2, May
2010;
[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6 No
4, November 2010;
[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal
Vol.7 No 4, November 2011;
[Dan4] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal
Vol.8 No 1, February 2012;
[Dan5] Dannon, H. Vic, “Infinitesimal Vector Calculus” in Gauge Institute
Journal
[Dan6] Dannon, H. Vic, “Circular and Spherical Delta Functions” in Gauge
Institute Journal
[Dan7] Dannon, H. Vic, “Infinitesimal Complex Calculus” in Gauge Institute
Journal Vol.10 No 4, November 2014;
[Dan8] H. Vic Dannon, “Delta Function, the Fourier Transform, and Fourier
Integral Theorem”, in Gauge Institute Journal Vol.8 No 2, May 2012.
[Dirac] P. A. M. Dirac, “Discussion of the infinite distribution of electrons in
the theory of the positron” in Proceedings of the Cambridge Philosophical
Society, Volume 30, part II, 30 April 1934, pp. 150-63.
Reprinted in “The
Collected Works of P. A. M. Dirac, 1924-1948, Edited by R. H. Dalitz. p. 771.
59
Gauge Institute Journal Volume 11, No. 1, February 2015
H. Vic Dannon
[Needham] Tristan Needham, “Visual Complex Analysis” Oxford U. Press,
1998 (with corrections)
[Paley, Wiener]
Raymond Paley, and Norbert Wiener, “Fourier Transforms
in the Complex Plane” American Mathematical Society, 1934
[Sneddon] Ian Sneddon, “Fourier Transforms”, McGraw-Hill, 1959.
[Titchmarsh] E. C. Titchmarsh “Introduction to the theory of Fourier
Integrals”, Third Edition, Chelsea, 1986.
60