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Gauge Institute Journal
H. Vic Dannon
The Fourier Series of a
Singular Function of a
Complex Variable
H. Vic Dannon
[email protected]
November, 2014
Revised January, 2015
Abstract To date, the Fourier Series of a function of a complex
variable was never defined.
Following Real Analysis methods, the Fourier Transform of a
function of a complex variable was integrated along the real line.
But the coefficients of the Fourier Series of a function of complex
variable are Fourier Transforms integrated along closed paths in
the Complex plane.
Recently, we defined the Fourier Transform along a closed path in
the Complex Plane, and we proceed here to define, and obtain the
properties of the Fourier Series of a function of a complex variable:
Let the Hyper-Complex, Path Integrable function f (z ) be defined
on the circle
z = z 0 + rei a ,
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H. Vic Dannon
where
r = z - z0 ,
a = Arg(z - z 0 ) .
Fixing z , fixes r , and
f (z ) = f (re i a ) º j(a) .
Denote
1
2p
a=p
ò
j(a)e -in ad a º cn
a =-p
The Complex Fourier Series associated with f (z ) is
 { f (z )} = .. + c-ne i (-n )q + .. + c-1e i (-1)q + c0 + c1e i (1)q + .. + cne i (n )q + .. ,
where
q = Arg(z - z 0 ) .
Then, we show that on the Hyper-Complex Pierced disk
0 < z - z 0 < r , the Fourier Series of an Analytic Function is its
Laurent Series.
Consequently,
For an Analytic Function on the Hyper-Complex
Pierced disk 0 < z - z 0 < r ,
the Fourier Series Theorem f (z ) =  { f (z )} , holds
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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.
Laurent Series
9.
Hyper-real Fourier Transform
10. Fourier Transform of d(z ) , and of an Analytic f (z )
11. Fourier Series of a Hyper-Complex Path-Integrable function
12. Fourier Series of a singular Function is its Laurent Series
References
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Introduction
The Fourier Series of an Analytic function f (z ) was never defined.
In [Titchmarsh, p.42], the Fourier Transform is defined by
integration along the real line. Integration along closed paths in
the Complex Domain is never considered.
Since the coefficients of the Fourier Series of an analytic function
are Fourier Transforms integrated along a closed path, the
Fourier Series of an analytic function could not be defined.
In [Dan9] we defined the Fourier Transform of an Analytic
function f (z ) , and the Fourier Integral of an Analytic Function,
along closed paths in the Hyper-Complex plane.
That enables us to define here the Fourier Series associated with
a Path-Integrable function f (z ) , and we show that it is the
Laurent Series of f (z ) on the pierced disk 0 < z - z 0 < r .
That is, for an Analytic Function on the Hyper-Complex Pierced
disk 0 < z - z 0 < r , the Laurent Series, and the Fourier Series
coincide:
In [Dan7], we applied Infinitesimal Complex Calculus to derive
the Cauchy Integral Formula in an infinitesimal disk, and in
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[Dan9] we introduced the Fourier Transform of an analytic
function f (z ) .
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 + i b .
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 i q 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 ) >
14
1
dr
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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 ) . 
18
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
ì 1 ï
ü
ï
Hyper-real line into the set of two Hyper-reals ï
í 0, ï
ý . The
ïï dx þ
ïï
î
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
n
n2
n3
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 Hyper-Complex Delta Function of a
Complex Variable d(z ) :
1) The Hyper-Complex Delta Function d(z ) is defined from the
Hyper-Complex plane into the set of two hyper-complex
ì
1 ïüï
ï
numbers, ïí 0,
ý.
ï
ïþï
2
idz
p
ï
î
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 ) < ¥
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3)
H. Vic Dannon
d(z - z 0 ) =
where
c
1 1 -i Arg(z -z 0 )
e
{ z -z 0 £dr }(z ) ,
2pi dr
ìï 0, z - z 0 > dr
ï
.
{ z -z 0 £dr }(z ) = í
ïï 1, z - z 0 £ dr
î
c
 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 .
1
1
e -inf
4)
(d(z ))n =
5)
d(z - z ) =
6)
d
1
1
d(z - z ) =
dz
2pi (z - z )2
(2pi )n (dr )n
c{
z £dr }(z )
d 1
( Log(z - z ) )
dz 2pi
 in the disk z - z £ dr ,
c{
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 ,
23
d
d(z - z ) = 0 .
dz
Gauge Institute Journal
dk
7)
dz
k
H. Vic Dannon
1
k!
2pi (z - z )k +1
d(z - z ) =
c{
z £dr }(z )
k
 in the disk z - z £ dr , d k d(z - z ) =
dz
 off the disk, in z - z > dr ,
8)
d(az ) =
dk
dz k
k!
1
e -i (k +1)q ,
2pi (dr )k +1
d(z - z ) = 0 .
1
d(z )
a
9) z1 = only zero of f (z ) , f '(z1 ) ¹ 0 
 d( f (z )) =
10)
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 )) =
11) 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
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H. Vic Dannon
12) d ( (z - a )(z - b) ) =
1
1
d(z - a ) +
d(z - b)
a -b
b -a
13) z1,...zn are the only zeros of f (z ) ; f '(z1 ),.., f '(zn ) ¹ 0 
d( f (z )) =
14)
1
1
d(z - z1 ) + ... +
d(z - zn )
f '(z1 )
f '(zn )
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 )
15)
d(sin z ) = .. + d(z + 2p) - d(z + p) + +d(z ) - d(z - p) + d(z - 2p) + ..
ò
16)
d(z - z )d z = 1
z -z =dr
17)
If f (z ) is Hyper-Complex Differentiable function at z
Then,
ò
f (z )d(z - z )d z = f (z )
z -z =dr
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Gauge Institute Journal
18)
19)
H. Vic Dannon
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
26
dk
dz
k
d(z - z )dz
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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 . 
pi z - z
2
d (z -z )


f (z )
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H. Vic Dannon
8.
Laurent Series
8.1 Laurent Series of a Singular f (z )
If
f (z ) is Hyper-Complex Differentiable function on a Hyper-
Complex pierced 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
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 =
a0 =
g
g
1
2pi
1
2pi
ò f (z )dz ,
ò
g
29
g
f (z )
dz ,
z - z0
Gauge Institute Journal
H. Vic Dannon
a1 =
a2 =
ak =
1
2pi
1
2pi
1
2pi
f (z )
ò (z - z
g
f (z )
ò (z - z
g
f (z )
ò (z - z
g
dz ,
2
0)
k +1
0)
dz ,
3
0)
dz
k = 0,1, 2,...
for any loop g , and for any point z ¹ z 0 in its interior.
Proof:
The Hyper-Complex Differentiable function 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
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 èç
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 )
d
z
(
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 èç
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 )dz z - z

2 pia-1 (z 0 )
+
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 )
Note that by the Cauchy Integral Theorem the integrals of a-1 ,
a-2 , a-3 ,… can be taken along g . 
31
Gauge Institute Journal
H. Vic Dannon
9.
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÷
çç
1
÷ ikx
-ik x
f (x ) =
çç ò f (x )e d x ÷÷e dk
ò
÷
2p k =-¥ çè x =-¥
ø÷
does not hold in the Calculus of Limits, under any
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Gauge Institute Journal
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 =-¥
33
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÷
çç
-i wx x -i wy h
i(w x + w y )
f (x , y ) =
f (x, h )e
d xd h ÷÷÷e x y d wxd wy
çç ò
ò
ò
ò
2
÷
(2p) w =-¥ w =-¥ çè 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
= ò
d nx ÷÷d x çç ò e
d ny ÷÷d h ,
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
÷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
34
Gauge Institute Journal
H. Vic Dannon
10.
Fourier Transform of d(z ), and of
an Analytic f (z )
In [Dan11], we defined the Fourier Transform of the HyperComplex Delta Function, and of a Hyper-complex Analytic
Function
1) The Fourier Transform of the function f (q) = u(q) + iv(q) of a
hyper-real q is the Integration Sum over the infinitesimal
projections of f (q)d q on e -i wq
q =¥
å
f (q)e -i wqd q .
q =-¥
2) In the complex plane, an integration path from -¥ , to ¥ is a
closed path through ¥ .
Therefore, we define the Fourier Transform of a hyper-complex
function f (z ) along a closed path g by the Integration Sum
 {f (z )} =
å f (z )e-i wzdz ,
z Îg
where g may be the unit circle z = eif .
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Gauge Institute Journal
H. Vic Dannon
3) Fourier Transform of d(z )
ò
 {d(z )} =
d(z )e -i wzdz ,
z =1
= 1.
4) Fourier Integral of d(z - z )
d(z - z ) =
5)
1
2p
ò
ei w(z -z )d w
w =1
Fourier Integral Theorem for f (z ) on Infinitesimal
Circles
If
f (z ) is a hyper-complex analytic function,
Then, the Complex Fourier Integral Theorem holds:
1
f (z ) =
2p
æ
ö÷
çç
÷
ò ççç ò f (z )e-i wzd z ÷÷÷÷eiz wdw ,
w = h è z -z = e
ø
where e , and h are infinitesimals
6)
Fourier Integral Theorem for f (z ) on Unit Circles
If f (z ) is a hyper-complex analytic function, in a hypercomplex domain that includes the unit circle z - z = 1
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Gauge Institute Journal
H. Vic Dannon
Then, the Complex Fourier Integral Theorem holds.
1
f (z ) =
2p
7)
The Fourier Integral of an Analytic Hyper-Complex
f (z )
8)
æ
ö÷
çç
÷
-i wz
ò ççç ò f (z )e d z ÷÷÷÷eiz wdw
w =1 è z -z =1
ø
is the Cauchy Integral Formula for f (z )
Existence of the Fourier Transform of f (z )
If f (z ) is a hyper-complex analytic function on a hypercomplex domain that includes the circle z - z = 1 ,
Then,
1) the hyper-complex integral
ò
f (z )e-i wzd z
z -z =1
converges to fˆ(w)
2) the hyper-complex integral
1
2p
ò
fˆ(w)eiz wd w
w =1
converges to f (z )
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Gauge Institute Journal
H. Vic Dannon
11.
Fourier Series of Hyper-Complex
Path-Integrable Function
In [Dan10], we presented the Fourier Series for a hyper-real f (x )
integrable on [-p, p ] , so that f (-p) = f (p) : The Fourier Series
associated with f (x ) is
 { f (x )} = ... + c-ne i (-n )x + ... + c-1e i (-1)x + c0 + c1e i (1)x + ... + cne i (n )x + ...
where for each n = ..., -3, -2, -1, 0,1, 2, 3,... , the integrals
u =p
1
f (u )e -inudu º cn
ò
2p u =-p
exist, with finite, or infinite hyper-real values. The cn are the
Fourier Coefficients of f (x ) .
For each x ,  { f (x )} may assume finite or infinite hyper-real
values.
We show in [Dan10] that the Calculus of limits does not supply
relevant conditions to the equality of the Fourier Series the
function, while in Infinitesimal Calculus the function does equal
its Fourier Series
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Gauge Institute Journal
H. Vic Dannon
Let the Hyper-Complex function, f (z ) be Path-Integrable, hence,
analytic on a Hyper-Complex domain that (without loss of
generality) includes the unit circle
z = ei a .
Then,
f (z ) = f (ei a ) = j(a) .
The Path Integral
1
2p
a=p
ò
j(a)e -in ad a
a =-p
exists as a Fourier Transform of an analytic function:
In fact, it equals
1
2pi
ò
-i a
j(a)e e

z =1 f (z ) z -1

ia
-in
ae -i a e
w
z
ia
ie
d a

dz
g (z )
which is the Fourier Transform of the function
f (z )
,
z
g(z ) =
at
w=
na
.
z
Thus, denote
39
Gauge Institute Journal
H. Vic Dannon
1
cn º
2p
a=p
ò
j(a)e-in ad a ,
a =-p
That is,
………………………………..
c-3
c-2
1
=
2p
1
=
2p
a=p
ò
j(a)e 3i ad a
a =-p
a=p
ò
j(a)e 2i ad a
a =-p
a=p
c-1
1
=
j(a)e i ad a
ò
2p a =-p
1
c0 =
2p
1
c1 =
2p
a=p
ò
j(a)d a
a =-p
a=p
ò
j(a)e -i ad a
a =-p
a=p
1
c2 =
j(a)e-2i ad a
ò
2p a =-p
a=p
1
c3 =
j(a)e-3i ad a
ò
2p a =-p
…………………………………
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Gauge Institute Journal
H. Vic Dannon
The Complex Fourier Series associated with f (z ) is
 { f (z )} = .. + c-ne i (-n )q + .. + c-1ei (-1)q + c0 + c1e i (1)q + .. + cnei (n )q + .. ,
where
q = Arg(z - z 0 ) .
We proceed to show that this Series satisfies the Fourier Series
Theorem for an analytic Function on the Hyper-Complex pierced
disk 0 < z - z 0 < r .
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Gauge Institute Journal
H. Vic Dannon
12.
Fourier
Series
of
a
singular
Function
12.1 The Fourier Series of an analytic Function on the
Hyper-Complex Pierced disk 0 < z - z 0 < r is its
Laurent Series
Proof:
By 8.1, If
f (z ) is Hyper-Complex Differentiable function on a
Hyper-Complex pierced 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
a-k =
1
2pi
ò f (z )(z - z 0 )k -1d z ,
k = 1, 2,... ,
g
a-3 =
1
2pi
ò f (z )(z - z 0 )2d z
g
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Gauge Institute Journal
H. Vic Dannon
a-2 =
1
2pi
a-1 =
a0 =
a2 =
1
2pi
g
1
2pi
1
2pi
1
2pi
a1 =
ak =
ò f (z )(z - z 0 )d z ,
1
2pi
ò f (z )d z ,
ò
g
g
f (z )
dz ,
z - z0
f (z )
ò (z - z
g
g
0)
f (z )
ò (z - z
g
f (z )
ò (z - z
dz ,
2
k +1
0)
3
0)
dz ,
dz
k = 0,1, 2,...
for any loop g , and for any point z ¹ z 0 in its interior.
Let g be the circle
z = z 0 + rei a ,
where
r = z - z0 ,
a = Arg(z - z 0 ) .
Then, fixing z , fixes r . Thus,
f (z ) = f (re i a ) = j(a) ,
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Gauge Institute Journal
H. Vic Dannon
d z = riei ad a ,
and
q = Arg(z - z 0 ) .
Therefore,
a-3 (z - z 0 )-3
æ
ç 1
= çç
çç 2pi
è
ö
÷
1
÷
ò f (z )(z - z 0 )2d z ÷÷÷÷ (z - z )3 ,
0
z -z 0 = r
ø
æ
ç 1
= çç
ç 2pi
çè
ö÷
÷÷ 1
j
a
r
e
i
r
e
d
a
(
)(
)
÷÷ iq 3
ò
z -z 0 = r
ø÷ (re )
æ
ç 1
= çç
çç 2p
è
ia 2
ia
ö
÷
÷
ò j(a)e 3iad a ÷÷÷÷e-3iq
z -z 0 = r
ø
a=p
æ
÷÷ö -3iq
çç 1
3i a
=ç
j(a)e d a ÷÷e
çç 2p ò
÷÷
è a =-p
ø

c-3
-2
a-2 (z - z 0 )
æ
ç 1
= çç
çç 2pi
è
÷÷ö
1
ò f (z )(z - z 0 )d z ÷÷÷÷ (z - z )2 ,
0
z -z 0 = r
ø
æ
ç 1
= çç
çç 2pi
è
÷÷ö 1
ò j(a)(re )i re d a ÷÷÷÷ (reiq )2
z -z 0 = r
ø
ia
44
ia
Gauge Institute Journal
H. Vic Dannon
a=p
æ
ö÷
çç 1
2i a
=ç
j(a)e d a ÷÷÷e -2i q
çç 2p ò
÷
è a =-p
ø÷

c-2
-1
a-1(z - z 0 )
æ
ç 1
= çç
ç 2pi
çè
÷÷ö 1
ò f (z )dz ÷÷÷÷ z - z ,
0
z -z 0 = r
ø
æ
ç 1
= çç
çç 2pi
è
÷÷ö 1
ò j(a)i re d a ÷÷÷÷ (reiq )2
z -z 0 = r
ø
ia
a=p
æ
ö÷
çç 1
ia
=ç
j(a)e d a ÷÷÷e-i q
çç 2p ò
÷
è a =-p
ø÷

c-1
a0 =
=
1
2pi
1
2pi
1
dz ,
z - z0
ò
f (z )
ò
j(a)
z -z 0 = r
z -z 0 = r
1
re
ia
i rei ad a
a=p
æ
÷÷ö
çç 1
=ç
j(a)d a ÷÷
çç 2p ò
÷÷
è a =-p
ø

c0
æ
ç 1
a1(z - z 0 ) = çç
çèç 2pi
÷÷ö
ò f (z ) (z - z )2 d z ÷÷÷÷(z - z 0 )
0
z -z 0 = r
ø
1
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Gauge Institute Journal
æ
ç 1
= çç
ç 2pi
çè
H. Vic Dannon
ö÷
÷
ò j(a) (reia )2 i re d a ÷÷÷÷ reiq
z -z 0 = r
ø
1
ia
a=p
æ
÷÷ö i q
çç 1
-i a
=ç
j(a)e d a ÷÷e
çç 2p ò
÷÷
è a =-p
ø

c1
æ
ç 1
a2 (z - z 0 ) = çç
çç 2pi
è
÷÷ö
ò f (z ) (z - z )3 dz ÷÷÷÷(z - z 0 )2
0
z -z 0 = r
ø
æ
ç 1
= çç
ç 2pi
çè
÷÷ö 2iq
ò j(a) (reia )3 i re d a ÷÷÷÷ re
z -z 0 = r
ø
2
1
1
ia
a=p
æ
÷÷ö 2iq
çç 1
-2i a
=ç
j(a)e
d a ÷÷e . 
çç 2p ò
÷÷
è a =-p
ø

c2
Consequently,
12.2 For an Analytic Function on the Hyper-Complex
Pierced disk 0 < z - z 0 < r ,
the Fourier Series Theorem f (z ) =  { f (z )} , holds
Proof: The singular f (z ) equals its Laurent Series, which, by 12.1
equals its Fourier Series. 
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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.
[Dan9] “Complex Delta Function” in Gauge Institute Journal
[Dan10] “Periodic Delta Function and Dirichlet Summation of Fourier Series”
in Gauge Institute Journal
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Gauge Institute Journal
H. Vic Dannon
[Dan11] ”The Fourier Integral of Delta Function of a Complex Variable, 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.
48