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Gauge Institute Journal
H. Vic Dannon
The Fourier-Bessel Integral
of the Polar Delta of a Complex
Variable and of an Analytic
Function
H. Vic Dannon
[email protected]
January, 2011
Revised December, 2014
Abstract The Fourier-Bessel Transform applies to a function of
two variables that has a polar symmetry
g(x , y ) = f (r) .
The Transform
r =¥
Bess { f (r)} = 2p
ò
f (r)J 0 (2pnr)rd r
r =0
is the integration sum over 0 £ r < ¥ of the infinitesimal
projections 2p f (r)rd r on the Polarly symmetric Bessel functions
J 0 (2pnr) .
The Fourier-Bessel Transform, and Integral would be particularly
applicable to functions f (z ) of Complex variable z = rei q
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But No attempt was made to define it.
We define here the Fourier-Bessel Transform of the Complex
Polar Delta Function, and the Fourier-Bessel Transform of a
hyper-complex Analytic function f (z ) , along a closed path gz in
the complex plane z .
Then,
1. the Fourier-Bessel Integral of Hyper-complex Polar Delta is
d(z - z )
= (2p)2
z
ò
J 0 (2pnz )J 0 (2pnz )nd n .
n =dr
2. the Cauchy Integral Formula
f (z ) =
1
2pi
ò
gz
f (z )
dz ,
z -z
yields the Fourier-Bessel Integral Theorem
æ
ö÷
çç
÷
f (z ) = 2p ò
 ççç 2p ò f (z )J 0(2pnz )zdz ÷÷÷÷J 0(2pnz )ndn .
n =1 è
z -z =1
ø
3. The convergence of the Fourier Integral implies the existence of
the Fourier Transform of f (z ) , and its inverse transform:
If f (z ) is a hyper-complex analytic function on a hypercomplex domain that includes the circle z - z = 1 ,
Then,
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1) the hyper-complex integral
2p
ò
f (z )J 0 (2pnz )zd z
z -z =1
converges to
F (n ) º ( Bessel {f (z )} )(n )
1) the hyper-complex integral
2p
ò
F (n )J 0 (2pnz )nd n
n =1
converges to
-1
f (z ) º ( Bessel
{F (n )} )(z )
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.
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.
Hyper-Complex Plane
3.
Hyper-Complex Function
4.
Hyper-Complex Path Integral
5.
Hyper-real Delta Function
6.
Hyper-Complex Delta Function d(z )
7.
Cauchy Integral Formula
8.
Hyper-real Fourier Transform
9.
Hyper-real Polar Delta Function
d(r - s)
s
10. Hyper-real Fourier-Bessel Transform
11. Hyper-Complex Bessel Function J 0 (z )
12. Hyper-Complex Polar Delta Function
13. Fourier-Bessel Transform of
14. Fourier-Bessel Integral of
d(z - z )
z
d(z - z )
z
d(z - z )
z
15. Fourier-Bessel Integral of Analytic f (z )
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16. Fourier-Bessel Transform of Analytic f (z )
References
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Introduction
0.1 The Fourier-Bessel Transform
The Fourier-Bessel Transform applies to a function of two
variables that has a polar symmetry
g(x , y ) = f (r) .
The Transform
r =¥
Bess { f (r)} = 2p
ò
f (r)J 0 (2pnr)rd r
r =0
is the integration sum over 0 £ r < ¥ of the infinitesimal
projections 2p f (r)rd r on the Polarly symmetric Bessel functions
J 0 (2pnr) .
Calculus of Limits Conditions for the Fourier-Bessel Integral
Theorem stated in [Watson, p. 458], require the existence of
r =¥
lim
l ¥
ò
r =0
æ w =l
ö÷
çç
f (r)ç ò J 0 (ws )J 0 (wr)wd w ÷÷÷rd r
çç
÷
è w =0
ø÷
We have shown in [Dan10] that as l  ¥ , the Bessel functions
integral with respect to w = 2pn is singular.
Hence, in the
Calculus of Limits, the Transform is not defined.
The Fourier-Bessel Transform would be particularly applicable to
functions of Complex variable
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z = re iq
But No attempt was made to define it.
f (z ) , as defined by Titchmarsh, does
The Fourier Transform of
not suggest a path to follow:
0.2 The Fourier Transform of f (z )
The Fourier Transform of a function of Complex variable is
defined in [Titchmarsh, p. 44], in
THEOREM 26.
Let f (z ) be an analytic function, regular for
-a < y < b ,
where a > 0 , b > 0 .
In any strip interior to -a < y < b , and for any e > 0 , let
ìï O(e-(l-e)x ), (x  ¥)
,
f (z ) = ïí
ïïO(e(m-e)x ), (x  -¥)
î
where l > 0 , m > 0 .
Then,
F (w ) =
1
z =¥
ò
2p z =-¥
f (z )e i wzdz
satisfies conditions similar to those imposed on f (z ) ,
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with a, b, l, m replaced by l, m, b, a ;
and
f (z ) =
1
w =¥
ò
2p w =-¥
F (w)e -i wzd w
In [Dan11], we noted that the Conditions of Theorem 26 above,
limit the integration path to the real line.
Instead of integrating along a closed path such as
which may be homologous to a unit circle, the Conditions require
the function to approximately vanish out of a compact interval,
and limit the admissible functions that may have a Fourier
Transform.
For an Analytic Function these Conditions are unnecessary,
because Cauchy Theorem, Cauchy Integral Formula, and the
Residue Theorem hold along paths in the complex plane.
The limitation is particularly awkward considering that an
Analytic function is inseparable from its disk of convergence, and
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its derivative may be obtained in any radial direction.
Under the limitations, the Fourier Transform F (w) is an analytic
function in a strip parallel to the x axis.
At the borders of the strip, the Analytic F (w) which is inseparable
from its disk of convergence, will have one sided derivative.
Nevertheless, similar conditions that define the Fourier-Bessel
Transform of f (z ) , were not found.
0.3 Definition along a Closed Path
Here, we show that Cauchy Integral Formula yields the FourierBessel Integral of an Analytic Function.
That is, for an Analytic function, the Cauchy Integral Formula,
and the Fourier-Bessel Integral, coincide.
The Fourier-Bessel Integral defines the Fourier-Bessel Transform
of an Analytic function f (z ) along a closed path.
We start by recalling the Hyper-real line, and the Hyper-Complex
Plane.
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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-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.
Following [Dan2] we claim 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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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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3.
Hyper-Complex Function
3.1 Definition of a hyper-complex function
f (z ) is a hyper-complex function, iff it is from the hyper-complex
numbers into the hyper-complex numbers.
This means that any number in the domain, or in the range of a
hyper-complex f (x ) is either one of the following
 complex
 complex + infinitesimal
 infinitesimal
 infinite hyper-complex
3.2 Every function from complex numbers into complex numbers
is a hyper-complex function.
3.3
sin(dz )
has the constant hyper-complex value 1
dz
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(dz )3 (dz )5
Proof: sin(dz ) = dz +
- ...
3!
5!
sin(dz )
(dz )2 (dz )4
= 1+
- ...
dz
3!
5!
3.4 cos(dz ) has the constant hyper-complex value 1
(dz )2 (dz )4
+
- ...
Proof: cos(dz ) = 1 2!
4!
3.5 edz has the constant hyper-complex value 1
Proof: e
dz
(dz )2 (dz )3 (dz )4
= 1 + dz +
+
+
+ ...
2!
3!
4!
1
1
1
3.6 e dz is an infinite hyper-complex, and e dz = e dr
1
1
Proof: e dz = e dr
Re[e-if ]
1
= e dr
cos f
cos f
.
.
3.7 log(dz ) is an infinite hyper-complex, and log(dz ) > dr1
Proof:
log(dz ) =
[log(dr )]2 + f2 > log(dr ) >
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1
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H. Vic Dannon
4.
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 ) .
4.1 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
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.
4.2 The Countability of the Integration Sum
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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 .
4.3 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 ) . 
21
f (z )dz ,
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H. Vic Dannon
5.
Hyper-real Delta Function
In [Dan5], we defined the Hyper-real 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,
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 úû
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
1
dx
, d(x ) jumps from 0 to
,
dx
2
1
x Î éêë - dx2 , dx2 ùûú , d(x ) =
.
dx
 at x = 0 ,
 at x =
d(0) =
1
dx
1
dx
, d(x ) drops from
to 0 .
dx
2
 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 ) =
H. Vic Dannon
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 ]
d(x )dx = 1 .
x =-¥
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6.
Hyper-Complex Delta Function
d(z )
In [Dan9], we introduced the Complex Delta Function:
1) The Hyper-Complex Delta Function is defined from the HyperComplex plane into the set of two hyper-complex numbers,
ìïï
1 üïï
í 0,
ý.
ïîï 2pidz ïþï
The hyper-complex 0 is the sequence
The infinite hyper-complex
Arg z = f .
0, 0, 0,... .
1 1
1 1 -if
=
e
depends on
2pi dz
2pi dr
1
will mean the sequence n .
dr
2) d(z ) is an infinite hyper-complex on the infinitesimal
hyper-complex disk z £ dr . In particular, d(z ) < ¥
3) For any infinitesimal dz ,
 on the disk, z - z 0 £ dr , d(z - z 0 ) =
25
1 1
.
2pi dz
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H. Vic Dannon
 off the disk, for z - z 0 > dr , d(z - z 0 ) = 0 .
4)
d(z - z 0 ) =
c
1 1 -i(f-f0 )
e
{ z -z 0 £dr }(z ) ,
2pi dr
where f = arg z , f0 = arg z 0
ì
0, z - z 0 > dr
ï
ï
(
z
)
.
=
í
{ z -z 0 £dr }
ï
1,
z
z
dr
£
0
ï
î
c
1
1
e -inf
5)
(d(z ))n =
6)
d(z - z ) =
7)
d
1
1
d(z - z ) =
dz
2pi (z - z )2
n
n
(2pi ) (dr )
c{
d 1
( Log(z - z ) )
dz 2pi
 in the disk z - z £ dr ,
c{
8)
dz k
d(z - z ) =
c{
, n = 2, 3, ...
z -z £dr }(z )
z -z £dr }(z )
d
1 1 -2i q
d(z - z ) =
.
e
dz
2pi (dr )2
 off the disk, in z - z > dr ,
dk
z £dr }(z )
1
k!
2pi (z - z )k +1
d
d(z - z ) = 0 .
dz
c{
k
z £dr }(z )
 in the disk z - z £ dr , d k d(z - z ) =
dz
26
k!
1
e -i (k +1)q ,
+
1
k
2pi (dr )
Gauge Institute Journal
H. Vic Dannon
 off the disk, in z - z > dr ,
9) d(az ) =
10)
dz k
d(z - z ) = 0 .
1
d(z )
a
z1 = only zero of f (z ) , f '(z1 ) ¹ 0 
 d( f (z )) =
11)
dk
1
d(z - z1 )
f '(z1 )
z1, z 2 are the only zeros of f (z ) ; f '(z1 ), f '(z 2 ) ¹ 0 

d( f (z )) =
12) d(z 2 - a 2 ) =
1
1
d(z - z1 ) +
d(z - z 2 )
f '(z1 )
f '(z 2 )
1
1
d(z - a ) + d(z + a )
2a
2a
13) d ( (z - a )(z - b) ) =
1
1
d(z - a ) +
d(z - b)
a -b
b -a
14) 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 )
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15)
H. Vic Dannon
z1, z 2 ,... are zeros of f (z ) , f '(z1 ), f '(z 2 ),... ¹ 0 
d( f (z )) =
1
1
d(z - z1 ) +
d(z - zn ) + ...
f '(z1 )
f '(zn )
16)
d(sin z ) = .. + d(z + 2p) - d(z + p) + +d(z ) - d(z - p) + d(z - 2p) + ..
ò
17)
d(z - z )d z = 1
z -z =dr
18)
If f (z ) is Hyper-Complex Differentiable function at z
ò
Then,
f (z )d(z - z )d z = f (z )
z-z =dr
19)
20)
d
f (z ) =
dz
dk
dz k
f (z ) =
ò
f (z )
z -z =dr
ò
d
d(z - z )dz
dz
f (z )
z -z =dr
28
dk
dz k
d(z - z )dz
Gauge Institute Journal
H. Vic Dannon
7.
Cauchy Integral Formula
7.1 Cauchy Integral Formula
If
f (z ) is Hyper-Complex Differentiable function on a Hyper-
Complex Simply-Connected Domain D .
Then,
f (z ) =
1
2pi
f (z )
ò z - z d z ,
g
for any loop g , and any point z in its interior.
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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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 . 
2
i
z
p
z

d (z -z )


f (z )
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H. Vic Dannon
8.
Hyper-real 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
w =-¥
ei wxd w
=0
x ¹0
4. Fourier Integral Theorem
k =¥ æ x =¥
÷÷ö ikx
çç
1
-ik x
f (x ) =
ò ççç ò f (x )e d x ÷÷÷÷e dk
2p k =-¥
è x =-¥
ø
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
7. 2-Dimesional Inverse Fourier Transform

-1
{ F (wx , wy )} =
wy =¥ wx =¥
1
2
(2p)
ò
ò
wy =-¥ wx =-¥
32
F (wx , wy )e
i ( wx x + wy y )
d wxd wy
Gauge Institute Journal
H. Vic Dannon
ny =¥ nx =¥
ò
=
ò
F (2pnx , 2pny )e
2 pi (nx x +nyy )
d nxd ny ,
ny =-¥ nx =-¥
8.
wx = 2pnx
wy = 2pny
2-Dimesional Fourier Integral Theorem
æ h =¥ x =¥
÷÷ö i (wx x + wyy )
çç
-i wx x -i wy h
f (x , y ) =
d xd h ÷÷e
d wxd wy
ò ò ç ò ò f (x, h)e
÷÷
(2p)2 w =-¥ w =-¥ ççè h =-¥ x =-¥
ø
y
x
1
wy =¥ wx =¥
w =¥
w =¥
æ
ö
æ
÷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
è wx =-¥
ø è wy =-¥
h =-¥ x =-¥
ø
h =¥ x =¥
ö÷
æ nx =¥
ö÷ æç ny =¥
wx = 2pnx
çç
2 pi ny (y - h )
÷
2 pi nx (x -x )
÷
ç
d nx ÷÷d x çç ò e
d ny ÷÷d h ,
= ò
ò f (x, h)ççç ò e
÷
÷
wy = 2pny
è nx =-¥
ø÷ èçç ny =-¥
h =-¥ x =-¥
ø÷
h =¥ x =¥
9.
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
è wx =-¥
øè wy =-¥
ø÷
ö
æ nx =¥
öæç ny =¥
÷
wx = 2pnx
÷
çç
p
n
2
i
y
÷
= ç ò e 2pi nx xd nx ÷÷÷ççç ò e y d ny ÷÷ ,
÷÷ wy = 2pny
çç
÷÷çç
è nx =-¥
øè ny =-¥
ø
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H. Vic Dannon
9.
d(r - s )
Hyper-real Polar Delta
s
In [Dan10], we defined the Hyper-real Polar Delta Function, and
established its properties:
Denoting
d(r - s ) =
1
dr
c[s-
d(f - q) =
1
df
c[q-
(r) , r - s ³ 0 ,
dr
,s + d r ]
2
2
(f) , 0 £ f - f0 £ 2p ,
df
,q + d f ]
2
2
1) Transforming between Polar and Cartesian Coordinates
x = r cos f
,
y = r sin f
x = s cos q
,
h = s sin q
d(x - x )d(y - h ) =
2)
s > 0,
1
d(r - s )d(f - q) ,
s
2pd(x - x )d(y - h ) =
34
d(r - s)
s
Gauge Institute Journal
H. Vic Dannon
d(r - s)
= 2pd(x - x )d(y - h)
s
3)
is the Polarly Symmetric Delta Function,
4) Transforming between Polar and Cartesian Coordinates
x = r cos q
y = r sin q
wx = w cos g
,
wy = w sin g
x = s cos f
h = s sin f
n =¥
d(x - x )d(y - h ) = (2p)2
ò
J 0 (2pnr)J 0 (2pns)ndn
n =0
d (r - s )
5)
s
n =¥
3
= (2p)
ò
J 0 (2pnr)J 0 (2pns )nd n
n =0
w =¥
= 2p
ò
J 0 (wr)J 0 (ws)wd w
w =0
6) Sifting by dPolar (r - s)
s =¥
ò
s =0
d(r - s )
sd s = 1
s
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H. Vic Dannon
w =¥
d(r - s) = 2ps
ò
J 0 (wr)J 0 (ws )wd w
w= 0
7)
n =¥
3
= (2p) s
ò
n =0
36
J 0 (2pnr)J 0 (2pns)nd n
Gauge Institute Journal
H. Vic Dannon
10.
Hyper-real 2-D Fourier-Bessel
Transform
In [Dan10], we defined the Hyper-real 2-D Fourier-Bessel
Transform, established its properties, and showed that the
Fourier-Bessel Integral holds:
1) The 2-D Fourier-Bessel Transform
r =¥
Bess { f (r)} = 2p
ò
f (r)J 0 (rw)rd r
r =0
r =¥
= 2p
ò
f (r)J 0 (2pnr)rd r ,
w = 2pn
r =0
2)
The 2-D Inverse Fourier-Bessel Transform
-1
Bess
1
{ F (w ) } =
2p
w =¥
ò
F (w)J 0 (rw)wdw ,
w=
wx2 + wy2
w =0
n =¥
= 2p
ò
F (2pn )J 0 (2pnr)nd n , w = 2pn
n =0
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Gauge Institute Journal
3)
H. Vic Dannon
Fourier-Bessel Integral Theorem
If
f (r) is Hyper-real function,
Then, the Fourier-Bessel Integral Theorem holds.
n =¥ æ
s =¥
ö÷
çç
f (r) = 2p ò ç 2p ò f (s)J 0 (2pns )sd s ÷÷÷J 0 (2pnr)nd n
ç
÷
è s =0
ø÷
n =0 ç
s =¥
=
ò
s =0
4)
n =¥
æ
÷ö÷
çç
2
f (s )ç (2p) ò J 0 (2pns )J 0 (2pnr)nd n ÷÷ sds .
çç
÷÷
è
ø
n =0
Existence of the Transform, and its Inverse
If f (r) is Hyper-real function,
Then,
s =¥
 2p
ò
f (s)J 0 (2pns )sd s converges to F (2pn )
s =0
n =¥
 2p
ò
F (2pn )J 0 (2pnr)nd n converges to f (r)
n =0
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H. Vic Dannon
11.
Hyper-complex Bessel Function
J 0 (z )
for any complex number z in  ,
z = x + iy = re if ,
The
Hyper-Complex
Bessel
Function
J 0 (z )
is
defined
[Abramowitz] as the sum of its Taylor Series,
J 0 (z ) = 1 -
1
4
z2
(1!)2
+
( 14 z 2 )2
(2!)2
-
( 14 z 2 )3
(3!)2
+ ...
and in [Watson, p.16],
z2
z4
z6
= 1 - 2 + 2 2 - 2 2 2 + ...
2
24
246
= J 0 (-z )
which is even in z .
The Series converge, and J 0 (z ) is Analytic for any z in  .
39
in
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H. Vic Dannon
12.
Hyper-Complex Polar Delta
d(z - z )
Function
z
Denote
z = x + iy = reif ,
z = x + i h = sei q ,
n =¥
12.1
d(z - z )
1
(2p)2 ò J 0 (2pnr)J 0 (2pns )nd n ,
=
q
i
z
i se
n =0
=
Proof:
1 d(r - s ) -i Arg(z )
e
2pi
s
d(z - z )
1
d(x - x + i[y - h ])
=
if
z
se
=
1
se if
d(x - x )d(i[y - h ]) .
Since d(i[y - h ]) = 1i d(y - h ) ,
=
1
i se if
d(z - z )
is
z
d(x - x )d(y - h )
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Gauge Institute Journal
H. Vic Dannon
n =¥
Since by 9.4, d(x - x )d(y - h ) = (2p)2
ò
J 0 (2pnr)J 0 (2pns)ndn ,
n =0
n =¥
d(z - z )
1
(2p)2 ò J 0 (2pnr)J 0 (2pns )nd n
=
f
i
z
i se
n =0
=
1
2pi se if
n =¥
3
(2p)
ò
J 0 (2pnr)J 0 (2pns )nd n
n =0
n =¥
Since by 9.7, d(r - s ) = (2p)3
ò
J 0 (2pnr)J 0 (2pns )nd n ,
n =0
d(z - z )
1 d(r - s ) -i Arg(z )
.
e
=
2pi
z
s
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H. Vic Dannon
13.
Fourier-Bessel Transform of
d(z - z )
z
13.1 The Fourier-Bessel Transform of hyper-complex f (z )
The Fourier-Bessel Transform of the hyper-complex function
f (r) = u(r) + iv(r) of a hyper-real r is the Integration Sum
r =¥
2p å f (r)J 0 (2pnr)rd r .
r =0
As r varies, the infinitesimal projections of 2p f (r)d r on J 0 (2pnr) ,
namely,
2p f (r)J 0 (2pnr)d r ,
sum
up
to
the
Fourier-Bessel
Transform of f (r) .
We define the Fourier-Bessel Transform of a hyper-complex
function f (z )
( Bessel {f (z )} )(n )
along a closed path g(z ) by the Integration Sum
2p å f (z )J 0 (2pnz )zd z .
z Îg
13.2 Fourier-Bessel Transform of
42
d(z - z )
z
Gauge Institute Journal
Bessel {
H. Vic Dannon
d(z - z )
} = 2p å d(z - z )J 0 (2pnz )d z .
z
z Îg
Without loss of generality, g may be the unit circle z = eif .
Therefore,
Bessel {
13.3
Proof:
Bessel {
d(z - z )
} = 2pJ 0 (2pnz )
z
d(z - z )
} = 2p
z
ò
d(z - z )
J 0 (2pnz )zd z
z
ò
d(z - z )J 0 (2pnz )d z
z =1
= 2p
z =1
By Cauchy Theorem,
= 2p
1
2pi
ò
z =dr
1
J (2pnz )d z
z -z 0
By Cauchy Integral Theorem,
= 2pJ 0 (2pnz ) z =z
= 2pJ 0 (2pnz ) . 
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H. Vic Dannon
14.
Fourier-Bessel Integral of
d(z - z )
z
14.1 Inverse Complex Fourier-Bessel Transform of HyperComplex F (n )
We define the Inverse Fourier-Bessel Transform of a hypercomplex function F (n )
-1
{F (n )} )(z )
( Bessel
along a closed path g(n ) by the Integration Sum
2p å F (n )J 0 (2pnz )nd n .
n Îg
14.2
Fourier-Bessel Integral of
d(z - z )
= (2p)2
z
ò
d(z - z )
z
J 0 (2pnz )J 0 (2pnz )nd n
n =dr
Proof: Since
Bessel {
d(z - z )
} = 2pJ 0 (2pnz ) ,
z
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H. Vic Dannon
the inverse Transform of 2pJ 0 (2pnz ) is
d(z - z )
.
z
That is,
-1
{2pJ 0 (2pnz )} =
Bessel
d(z - z )
.
z
By 14.1, the Inverse Transform along a closed path g(n ) is the
Integration Sum
2p å {2pJ 0 (2pnz )}J 0 (2pnz )nd n .
n Îg
Therefore,
d(z - z )
= 2p å {2pJ 0 (2pnz )}J 0 (2pnz )nd n
z
n Îg
Taking g(n ) to be the infinitesimal circle n = dr
= 2p
ò
{2pJ 0 (2pnz )}J 0 (2pnz )nd n . 
n =dr
By Cauchy Integral Theorem, the integration path may be along
the unit circle.
Hence,
14.3
Fourier-Bessel representation of d(z - z )
d(z - z ) = (2p)2 z
ò
J 0 (2pnz )J 0 (2pnz )ndn
n =1
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15.
Fourier-Bessel Integral of an
Analytic f (z )
15.1 Fourier-Bessel Integral Theorem for hyper-complex
Analytic f (z ) , along infinitesimals paths
If
f (z ) is a hyper-complex analytic function,
Then, the Complex Fourier-Bessel Integral Theorem holds:
ö÷
æ
çç
÷
f (z ) = 2p ò
 çç 2p ò f (z )J 0(2pnz )zdz ÷÷÷÷J 0(2pnz )ndn ,
n =1 ç
è z -z =e
ø
where e is infinitesimal
Proof:
By the Cauchy Integral Formula, 5.1,
f (z ) =
1
2pi
f (z )
ò z - z d z ,
g
By Cauchy Integral Theorem,
=
1
2pi
ò
z -z =dr
46
f (z )
dz ,
z -z
Gauge Institute Journal
H. Vic Dannon
By the definition of d(z ) ,
=
ò
f (z )d(z - z )d z ,
z -z = e
Substituting from 14.3, d(z - z ) = (2p)2 z
ò
J 0 (2pnz )J 0 (2pnz )nd n ,
n =1
æ
ö÷
çç
÷
2
f (z ) = ò
 f (z )ççç (2p) z ò J 0(2pnz )J 0(2pnz )nd n ÷÷÷÷d z .
z -z = e
n =1
è
ø
By changing the Summation order,
æ
ö÷
çç
÷
f (z ) = 2p ò
 ççç 2p ò f (z )J 0(2pnz )zdz ÷÷÷÷J 0(2pnz )nd n . 
n =1 è
z -z = e
ø
Similarly, we obtain
15.2 Fourier Integral Theorem for f (z ) on Unit Circles
If f (z ) is a hyper-complex analytic function, in a hyper-complex
domain that contains the unit circle z - z = 1
Then, the Complex Fourier-Bessel Integral Theorem holds.
æ
ö÷
çç
÷
f (z ) = 2p ò
 ççç 2p ò f (z )J 0 (2pnz )zd z ÷÷÷÷J 0(2pnz )nd n
n =1 è
z -z =1
ø
Proof:
By the Cauchy Theorem, the Fourier-Bessel summation
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Gauge Institute Journal
H. Vic Dannon
2p
ò
f (z )J 0 (2pnz )zd z ,
z -z = e
along the infinitesimal circle z - z = e , can be done along the
unit circle. Hence it equals
2p
ò
f (z )J 0 (2pnz )zd z .
z -z =1
Therefore,
ö÷
æ
çç
÷
f (z ) = 2p ò
 ççç 2p ò f (z )J 0 (2pnz )zdz ÷÷÷÷J 0 (2pnz )nd n
n =1 è
z -z = e
ø
æ
÷÷ö
çç
= 2p ò
 çç 2p ò f (z )J 0(2pnz )zd z ÷÷÷÷J 0(2pnz )nd n . 
n =1 ç
è z -z =1
ø
It follows that due to 14.3, the Fourier-Bessel representation of
d(z - z ) ,
15.3 For an Analytic Hyper-Complex Function f (z ) , the
Fourier-Bessel Integral is the Cauchy Integral
Formula for f (z )
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H. Vic Dannon
16.
Fourier-Bessel Transform of an
Analytic f (z )
The convergence of the Fourier Integral of a hyper-complex
analytic function f (z ) , implies the existence of the Fourier
Transform of f (z ) , and its inverse transform
16.1
If f (z ) is a hyper-complex analytic function on a hypercomplex domain that includes the circle z - z = 1 ,
Then,
2) the hyper-complex integral
2p
ò
f (z )J 0 (2pnz )zd z
z-z =1
converges to
F (n ) º ( Bessel {f (z )} )(n )
3) the hyper-complex integral
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Gauge Institute Journal
H. Vic Dannon
2p
ò
F (n )J 0 (2pnz )nd n
n =1
converges to
-1
f (z ) º ( Bessel
{F (n )} )(z )
Proof:
The convergence
æ
ö÷
çç
÷
= 2p ò
 ççç 2p ò f (z )J 0(2pnz )zd z ÷÷÷÷J 0(2pnz )nd n = f (z )
n =1 è
z -z =1
ø
mandates that
1) The Complex Fourier-Bessel Transform of f (z ) ,
2p
ò
f (z )J 0 (2pnz )zd z ,
z-z =1
converges to a hyper-complex function F (n ) , some of its values
may be infinite hyper-complex, like the complex Delta Function.
2) The Inverse Complex Fourier-Bessel Transform of F (n )
2p
ò
F (n )J 0 (2pnz )nd n
n =1
converges to the hyper-complex function f (z ) . 
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References
[Abramowitz] Abramowitz, M., and Stegun, I., “Handbook of Mathematical
Functions
with
Formulas
Graphs
and
Mathematical
Tables”,
U.S.
Department of Commerce, National Bureau of Standards, 1964.
[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.
[Dan9] “Complex Delta Function” in Gauge Institute Journal
51
Gauge Institute Journal
H. Vic Dannon
[Dan10] “Polar Delta Function, the 2-D Fourier-Bessel Transform, and the 2D Fourier-Bessel Integral” in Gauge Institute Journal
[Dan11] “The Fourier Integral of the Complex Delta Function and of an
Analytic Function” in Gauge Institute Journal
[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.
nd
[Watson] Watson, G. N., A Treatise on the Theory of Bessel Functions, 2
Edition, Cambridge, 1958.
52