Circular Complex Delta Function - Gauge-institute.org

Gauge Institute Journal H. Vic Dannon 

Complex Delta Function, 

Complex Fourier Transform, 

and Fourier Integral Theorem 

H. Vic Dannon 

vic0@comcast.net 

January, 2011 

Abstract The Cauchy Integral Formula for the function 

f () z = 1on 

the Infinitesimal Circle ζ − z = dr 

follows from the 

sifting property for a Circular Hyper-Complex Delta Function in 

Hyper-Complex Calculus. 

The Hyper-Complex Circular Delta Function is a Plane Delta 

Function that spikes to 

ζ − z = dr. 

1 

2 πζ− ( z) 

on the Infinitesimal Circle 

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, 

1


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. 

Contents 

Introduction 

1. Hyper-Complex Plane 

2. Hyper-Complex Path Integral 

3. Circular Hyper-Complex Delta 

4. Circular Delta Function Plots 

5. Circular Delta Properties 

6. Primitive and Derivatives of the Circular Delta 

7. δ (()) f z 

8. Circulation of the Circular Delta 

9. Sifting by ( z , and 

δ ζ − ) d δCircle ( ζ − z) 

Circle 

10. Representing f () z by Cauchy Integral Formula 

11. Fourier Transform of 

δ() 

z 

12. Fourier Integral Theorem for f () z 

References 

dz 

2


Introduction 

For z in the interior of γ , Cauchy Integral Formula gives an 

analytic 

Thus, 

f () z as the convolution of f with 

1 1 

2πi ζ− z 

1 1 

2πiζ . 

1 1 

f () z = ∫ f() 

ζ d ζ 

2πi 

ζ − z 

γ 

recovers the value of a complex function f () ζ at the 

point z in the interior of a loop γ , by sifting through the values of 

f () ζ on γ . 

If the integration path is the infinitesimal circle 

the sifting is performed by 

1 1 

2 i − z 

ζ − z = dr, 

{ ζ z dr} () ζ δCircle( ζ z) 

. 

π ζ χ − = ≡ − 

We call δ ( ζ − z) 

the Hyper-Complex Circular Delta Function. 

Circle 

Unlike the Hyper-Real Linear Delta function [Dan4], the 

integration path does not cross the Infinitesimal Hyper-Complex 

disk. Instead, the path encircles the singularity. 

In [Dan7], we showed that 

3


∫ 

ζ− 

z = dr 

1 1 

dζ 

= 1. 

2πi 

ζ − z 

Due to 1 

, the Complex Delta spikes to 

ζ − z 

1 1 1 

ζ 

= e 

2πidζ 2πidr 

on the infinitesimal Circle ζ − z = dr. 

−iArg( −z) 

The primitive of the Complex Delta on the Circle is 

1 

Log( ζ − z) 

, 

2πi 

and the derivative of the Complex Delta on the disk is 

1 1 

2 πi ( ζ − z) 

Since on the infinitesimal circle, ζ − z is a hyper-complex 

−iArg( ζ−z) 

infinitesimal ( dr) e , we recall the Hyper-Complex Plane 

that we introduced in [Dan7]. 

4 

2 

.


1. 

Hyper-Complex Plane 

Each complex number α + iβ 

can be represented by a Cauchy 

sequence of rational complex numbers, 

so that r + is → α + iβ. 

n n 

r + is , r + is , r + is ... 

1 1 2 2 3 3 

The constant sequence ( α + iβ, α + iβ, α + iβ,...) 

is a Constant 

Hyper-Complex Number. 

Following [Dan2] we claim that, 

1. Any set of sequences ( ι + iο , ι + iο , ι + iο 

,...) , where 

( ι1, ι2, ι3,...) 

belongs to one family of infinitesimal hyper reals, 

and ( ο , ο , ο ,...) belongs to another family of infinitesimal 

1 2 3 

hyper-reals, constitutes a family of infinitesimal hyper- 

complex numbers. 

1 1 2 2 3 3 

2. Each hyper-complex infinitesimal has a polar representation 

iφ 

( ) * , where is an infinitesimal, and 

iφ 

dz = dr e = ο e 

dr = ο* 

φ = 

arg( dz) 

. 

5


3. The infinitesimal hyper-complex numbers are smaller in 

length, than any complex number, yet strictly greater than 

zero. 

1 1 1 

4. Their reciprocals ( , , ,... 

ι1+ iο1 ι2+ iο2 ι3+ iο 

) are the infinite hyper- 

3 

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 hyper- 

complex 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 non- 

constant hyper-complex. 

8. The Hyper-Complex Plane is the direct product of a Hyper- 

Real Line by an imaginary Hyper-Real Line. 

9. In Cartesian Coordinates, the Hyper-Real Line serves as an 

x 

line. 

coordinate line, and the imaginary as an iy coordinate 

10. In Polar Coordinates, the Hyper-Real Line serves as a 

Range r line, and the imaginary as an i θ coordinate. Radial 

symmetry leads to Polar Coordinates. 

6


11. The Hyper-Complex Plane includes the complex 

numbers separated by the non-constant hyper-complex 

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 bi- 

continuous 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 ball. Therefore, the Hyper-Complex 

Plane is not a manifold. 

n 

7 

∞


18. The Hyper-Complex Plane is not spanned by two 

elements, and is not two-dimensional. 

8


2. 

Hyper-Complex Path Integral 

Following the definition of the Hyper-real Integral in [Dan3], 

the Hyper-Complex Integral of f () z over a path zt () , t ∈ [ αβ , ] , in 

its domain, is the sum of the areas 

rectangles with base z '( t) dt = dz , and height f () z . 

f ()'() zz tdt= fzdzt () () of the 

2.1 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 zt () , t ∈ [ αβ , ] , be a path, γ(,) ab , so that dz = z '( t) dt , and z'( t) 

is continuous. 

For each t , there is a hyper-complex rectangle with base 

[() zt − ,( zt) 

+ , height f () z , and area f (()) zt dzt () . 

2 

2 ] 

dz dz 

We form the Integration Sum of all the areas that start at 

z( α ) = a , and end at z( β ) = b, 

9


∑ 

t∈[ 

αβ , ] 

f (()) zt dzt () . 

If for any infinitesimal dz = z '( t) dt , the Integration Sum equals 

the same hyper-complex number, then 

Integrable over the path γ(,) 

ab . 

f () z is Hyper-Complex 

Then, we call the Integration Sum the Hyper-Complex Integral of 

f () z over the γ(,) 

ab , and denote it by 

∫ 

γ(,) 

ab 

f () zdz. 

If the hyper-complex number is an infinite hyper-complex, then it 

equals 

∫ 

γ(,) 

ab 

f () zdz. 

If the hyper-complex number is finite, then its constant part 

equals 

∫ 

γ(,) 

ab 

f () zdz. 

The Integration Sum may take infinite hyper-complex values, 

such as 1 , but may not equal to ∞ . 

dz 

The Hyper-Complex Integral of the function 

that goes through z = 0 diverges. 

fz () 

2.2 The Countability of the Integration Sum 

1 

= over a path 

z 

In [Dan1], we established the equality of all positive infinities: 

10


We proved that the number of the Natural Numbers, 

Card , equals the number of Real Numbers, 2 , and 

Card 

Card = 

we have 

2 Card 

2 

Card 

Card = ( Card) 

= .... = 2 = 2 = ... ≡ ∞. 

In particular, we demonstrated that the real numbers may be 

well-ordered. 

Consequently, there are countably many real numbers in the 

interval [ αβ , ] , 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 () zdz. 

2.3 Continuous f () z is Path-Integrable 

Hyper-Complex f () z Continuous on D is Path-Integrable on D 

Proof: 

Let zt () , t ∈ [ αβ , ] , be a path, γ(,) ab , 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 

= ⎡uxt ((), yt ()) x'() t −vxt 

((), yt ()) y'() t ⎤ 

⎣ ⎦ 

+ 

Ut () 

11


+ iuxt ⎡ ((), yt ()) y'() t + vxt ((), yt ()) x'() t⎤ 

⎣ ⎦ 

= Ut () + iVt () 

, 

V() t 

where Ut () , and Vt () are Hyper-Real Continuous on [ αβ , ] . 

Therefore, by [Dan3, 12.4], Ut () , and Vt () are integrable on [ αβ , ] . 

Hence, f (()) zt z'() t is integrable on [ αβ , ] . 

Since 

t= 

β 

∫ f (()) zt z'() tdt= ∫ f() zdz, 

t= α γ(,) 

a b 

f () z is Path-Integrable on γ(,) 

ab . 

12


3. 

Circular Hyper-Complex Delta 

3.1 Domain and Range of Circular Delta 

The Hyper-Complex Circular Delta Function is defined from the 

Hyper-Complex plane into the set of two hyper-complex numbers, 

⎧ ⎪ 1 ⎫⎪ 

⎨0, ⎪ 

⎬. 

⎪ ⎪⎩ 2πidz ⎪ ⎪⎭ 

The hyper-complex 0 is the sequence 0, 0, 0,... . 

The infinite hyper-complex 

depends on 

1 1 1 1 

= e 

2πidz 2πidr 

Arg z = φ , 

and on our choice of the infinitesimal dr . 

We will usually choose the family of infinitesimals that is spanned 

by the sequences 1 

n , 

1 

n 

2 

, 

1 

n 

3 

−iφ 

,… It is a semigroup with 

respect to vector addition, and includes all the scalar multiples of 

the generating sequences that are non-zero. That is, the family 

includes infinitesimals with negative sign. 

13


Therefore, 1 

dr 

will mean the sequence n . 

Alternatively, we may choose the family spanned by the sequences 

1 

2 n 

, 

1 

2 2 n 

, 

1 

3 2 n 

,… Then, 1 

dr will mean the sequence 2n . 

3.2 is an infinite hyper-complex on the infinitesimal 

z δ 

Circle () 

hyper-complex circle z = dr, 

In particular, 

Proof: Since dr > 0 , we have 

3.3 Definition of 

For any infinitesimal dz , 

δ 

This means that 

δ


off the circle, for z −z0≠ dr , δ ( z − z ) = 0. 

3.4 Delta in Polar Coordinates 

iArg( z−z ) 

Circle 0 

If z − z 0 

0 = ( dr) e , then, for any infinitesimal dr , 

δ 

where 

1 −iArg( z−z0) ( z − z ) = e { z− z0= dr} 

( z), 

2πidr 

Circle 0 

χ − = 

χ 

⎧⎪ 0, z −z0≠ dr 

{ z z0 dr} 

() z = ⎪ 

⎨ . 

⎪ 

⎪⎩ 

1, z − z0= dr 

15


4. 

Circular Delta Function Plots 

4.1 Delta Plot for dr = 

δ 

1 

n 

1 −iφ 

() z = e χ{ z = } (),2 z χ 1 1 

{ } 

(),3 z χ{ 

} 

(),... z , 

z = z = 

2πi 

2 3 

Circle 1 

is a sequence of shrinking circles with increasing heights 

16


The circles lie on the surface of revolution of 

4.2 Delta Plot for dr = 

δ 

Circle 

1 

2 n 

1 1 

= . 

z r 

χ χ χ χ 

{ z = 1 1 1 1 

0} { z = 

1} { z = 

2} { z = 

3} 

1 −iφ 

() z = e (),2 z (),4 z (),8 z ()... z 

2πi 

2 2 2 2 

17


is a sequence of shrinking circles with increasing heights that lie 

on the surface of revolution of 

1 1 

= . 

z x 2 2 

18


5. 

Circular Delta Properties 

5.1 

δ (0) = 0 

Circle 

Proof: z = 0 is off the infinitesimal circle z = dr. 

1 1 −i( φ−φ 5.2 0 ) 

δCircle( 

z − z0) = e { z− z0= dr} 

( z), 

2πidr 

χ 

φ = arg z , φ 0 = arg z0 

n 1 1 −inφ 

5.3 ( δCircle( 

z)) = e { z dr} 

( ) 

n n 

= z , n = 2, 3,... 

(2 πi) 

( dr) 

19 

χ


6. 

Primitive and Derivatives of the 

Circular Delta 

The Hyper-Complex Log Function is the primitive of the Circular 

Hyper-Complex Delta, on the Hyper-Complex Plane 

d 1 

δ ( ζ − z) = Log( ζ − z) 

( ζ ) 

dz 2 i 

6.1 Circle 

( ) { ζ z dr} 

π χ − = 

χ − = 

d 

1 1 

6.2 δCircle( ζ − z) = 

2 { ζ z dr} 

( z) 

dz 2 πi ( ζ − z) 


d 

1 1 −2iθ 

on the circle ζ − z = dr, 

δCircle( ζ − z) = e . 

dz 2 πi 

2 

( dr) 

off the circle, in z d 

k 

ζ − ≠ r, Circle 

d 

δ ( ζ − z) 

= 0. 

dz 

χ = 

d 1 k! 

6.3 δCircle( ζ − z) = 

1 { z dr} 

( z) 

k 2 i k 

dz π + ( ζ − z) 

20



d k! 

1 

θ 

Circle − = , 

k 2 i k 1 

dz π + ( dr) 

− ik ( + 1) 

on the circle, ζ − z = dr, 

δ ( ζ z) e 

off the circle, on z d 

k 

ζ − ≠ r, 

k Circle 

21 

d 

dz 

k 

δ ( ζ − z) 

= 0.


7. 

Circle (()) fz δ 

1 

( az) = ( z) 

a 

7.1 δCircle δCircle 

1 1 

Proof: δCircle( 

az) = χ { az = dr } ( z) 

2 πidaz 

( ) 

1 1 1 

= χ 

a 2πidz = 

a 

1 1 1 

= 

a 2πidz 

{ z 1 dr} 

χ = 

() z 

{ z dz } () z 

1 1 

Circle() 

z 

a 2 i δ = . 

π 

7.2 z = only zero of () , 

1 

fz 1 

f '( z ) ≠ 0 ⇒ 

1 

δ (()) f z = δ( 

z − z ) 

Circle 1 

f '( z1) 

Proof: δ (()) f z = δ ( f() z − f( z ) ) 

For z − z = infinitesimal , 

1 

Circle Circle 1 

( f '( z )( z z ) ) 

= δ 

− 

Circle 1 1 

22


1 

= δCircle( 

z − z 1) 

, by 6.1. 

f '( z ) 

7.3 z1, z 2 are the only zeros of f () z ; f '( z1), f '( z2) ≠ 0 ⇒ 

1 

1 1 

δ (()) f z = δ( 

z − z ) + δ( z −z 

) 

Circle 1 2 

f '( z1) f '( z2) 

(()) f z = f() z − f( z ) + f() z − f( 

z ) 

Proof: δ δ ( ) δ ( 

Circle Circle 1 Circle 2 

For z − z = infinitesimal , z − z2= 

infinitesimal , 

1 

( ) ( 

δ ( f ( z)) = δ f '( z )( z − z ) + δ f '( z )( z − z ) 

Circle Circle 1 1 Circle 2 2 

1 1 

= δ ( z − z ) + δ ( z − z ) . 

f '( z ) f '( z ) 

Circle 1 Circle 2 

1 2 

1 1 

( z − a ) = ( z − a) + ( z + a) 

2a 2a 

2 2 

7.4 δCircle δCircle δCircle 

7.5 

1 1 

δCircle ( ( z −a)( z − b) ) = δCircle( 

z − a) + δ Circle( 

z −b) 

a −b b −a 

7.6 1 are the only zeros of 

,... z z n 

f () z ; f '( z1),.., f '( zn) ≠ 0 ⇒ 

23 

) 

)


1 1 

δ ( f ( z)) = δ ( z − z ) + ... + δ ( z −z 

) 

Circle Circle 1 Circle 

f '( z1) f '( zn) 

7.7 z1, z 2,.. 

. are zeros of f () z , f '( z1), f '( z2),... ≠ 0 ⇒ 

7.8 

1 1 

δ ( f( z)) = δ ( z − z ) + δ ( z − z ) + ... 

Circle Circle 1 Circle 

f '( z1) f '( zn) 

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

+ 

Circle Circle Circle 

+ δ ( z) −δ ( z − π) + δ ( z −2 

π) 

+ ... 

Circle Circle Circle 

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

0, π,2 π,... 

and …,co s( − π) 

= −1, co s(0) = 1, 

co s( π ) =−1,… 

 

24 

n 

n


8. 

Circulation of Circular Delta 

The Circular Hyper-complex Delta Function is singular on 

the infinitesimal circle ζ − z = dr. 

Its integration path encircles the singularity of 

ζ = z . 

1 

ζ − z 

at 

The Circulation of Delta along the infinitesimal circle is 1. 

8.1 Circle 

Proof: Put 

Then, 

∫ 

ζ− 

z = dr 

δ ( ζ − zd ) ζ = 1. 

( ) i z dr e α 

ζ − = 

( ) i 

d i dr e d 

α 

ζ = α. 

α= 2π 

1 1 1 1 iα 

dζ i( 

) α 

2πi ∫= ζ − z 2 πi 

∫ dr e d , 

iα 

( dr) e 

i 

ζ− z = dr 

α= 

0 

Since ( dr) e 0, 

for any infinitesimal dr , and any α , 

α ≠ 

α= 2π 

1 

= i dα= 

1 

2πi 

∫ . 

25 

α= 

0


9. 

Sifting by δ ( ζ − z) 

and 

d 

dz 

δ ( ζ − z) 

Circle 

9.1 Sifting by 

Proof: 

( ) z δ ζ − 

Circle 

Circle 

If f () z is Hyper-Complex Differentiable function at z 

∫ 

Then, f ()( ζδζ− zd ) ζ= 

f() z 

ζ− 

z = dr 

Since f () ζ is differentiable at z , on the circle ( ) , 

i 

z dr e α 

ζ − = 

∫ 

ζ− 

z = dr 

f()( ζδζ− z) dζ= 


ζ = + , 

( ) i 

d i dr e d α 

ζ = α 

iα iα 

f ( z + ( dr) e ) = f( z) + f '( z)( dr) e , 

 

iα 

1 1 

= fz () ∫ δCircle( ζ − zd ) ζ + f'() zdr∫ e dζ 

2πi ζ − z 

− = − = 

iα 

( ) 

 

ζ z dr ζ z dr idredα 1 

26 

1 −iα 

e 

dr


α= 2π 

1 iα 1 iα 

= f () z + f '()( z dr) e i( dr) e α 

2 πi 

∫ d 

iα 

( dr) e 

α= 

0 

α= 2π 

1 

iα 

= f () z + f '()( z dr) e α 

2πi 

∫ d 

= f () z . 

 

α= 

0 

0 

d d 

9.2 f () z = f() 

ζ δ( ζ ) 

dz ∫ 

−z 

dζ 

dz 

Proof: 

ζ− 

z = dr 

d 1 f ( ζ) 

f () z = 

dζ 

dz 2 πi ∫ 

2 

( ζ − z) 

ζ− 

z = dr 

d 

= ∫ f () ζ δCircle( ζ − zd ) ζ. 

dz 

ζ− 

z = dr 

k k 

d d 

9.3 f () z = f() 

Circle( 

) 

k ∫ ζ δ ζ −z 

dζ 

k 

dz dz 

Proof: 

ζ− 

z = dr 

! ( ) 

() 

2 1 

( ) k 

d k f ζ 

f z = 

dζ 

dz πi ∫ 

+ 

ζ − z 

ζ− 

z = dr 

k 

d 

= ∫ f () ζ δCircle( ζ − zd ) ζ. 

k dz 

ζ− 

z = dr 

27


10. 

Representing fz () by Cauchy 

Integral Formula 

10.1 Cauchy Integral Formula 

If f () z is Hyper-Complex Differentiable function on a Hyper- 

Complex Simply-Connected Domain D . 

Then, 

1 f ( ζ) 

f () z = 

dζ 

2πi∫ 

, 

ζ − z 

γ 

for any loop γ , and any point z in its interior. 

f () ζ 

Proof: The Hyper-Complex function is Differentiable 

ζ − z 

on the Hyper-Complex Simply-Connected domain D , and on 

a path that includes γ and an infinitesimal circle about z . 

28


Then, the integrals on the lines between γ 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, 

Therefore, 

f () ζ f () ζ 

dζ − dζ 

= 0 

ζ −z ζ −z 

∫ ∫ 

. 

γ ζ− 

z = dr 

f () ζ f () ζ 

dζ = 

dζ 

ζ −z ζ −z 

∫ ∫ 

γ ζ− 

z = dr 

1 1 

= 2 πi ∫ 

f( ζ) dζ. 

2πi 

ζ − z 

ζ− 

z = dr 

δ ( ζ−z) 

Circle 

29 

f ( z)


11. 

Fourier Transform of 

δ() 

z 

The Fourier Transform of the hyper-complex function f () θ of a 

hyper-real variable θ is the Integration Sum 

θ=∞ 

∑ 

θ=−∞ 

− ωθ 

i f () θ e dθ. 

i 

As θ varies, the exponential e spirals along its multi-plane 

ωθ − 

Riemann Surface. Such spiral, can be graphed by say 

ParametricPlot3D[{3 Cos[t],3 Sin[t],1/11 t},{t,-10 π,10 π}] 

2 

0 

-2 

-2 

-2 

ωθ 

0 

2 

0 

i 

The products f () θ e d − θ sum up to the Fourier Transform of f () θ . 

30 

2


f () θ may be complex valued, but so far, θ has been taken to be 

real. [Titchmarsh, p.42]. 

For δ() 

z , the main branch vanishes anywhere except on the 

infinitesimal circle 

( ) i z dr e φ = , 

and we define the Fourier Transform of the main branch of a 

hyper-complex Delta function by 

i z 

11.1 {()} z () z e ω − 

F δ = δ dz 

∫ 

z = dr 

This suggests the choice of the infinitesimal circle path to define 

the Fourier Transform of the main branch of a hyper-complex f () z 

i z 

11.2 {()} f z f() z e ω − 

F 

∫ 

= dz 

z = dr 

We define the Inverse Fourier Transform of the main branch of 

fˆ ( ω ) by 

11.3 

F 

{( ˆ )} ˆ iωz f ω = f( ω) 

e dω 

∫ 

−1 

1 

2π 

ω = dr 

31


11.4 F {( δζ− z)} 

= 1 

Proof: Put 

Then, 

∫ 


ζ − = 

( ) i 

d i dr e d 

α 

ζ = α. 

( ) ( ) i 

−iωζ−z −iωdr 

e 

e = e 

i ( z) 

{( z)} ( z) e ωζ − − 

F δζ− = δζ− d ζ 

By 8.1, 

11.5 

ζ− 

z = dr 

i 1 i ( dr) e ... α 

= − ω + 

= 1 + infinitesimal 

1 1 

= dζ 

infinitesimal 

2πi 

∫ 

+ 

, 

ζ − z 

ζ− 

z = dr 

= 1 + infinitesimal . 

∫ 

i ( z) 

( z) e ωζ− 

δζ− = dω 

1 

2π 

ω = dr 

Proof: the complex Delta is the inverse Transform of ˆ f ( ω ) = 1. 

32 

α


12. 

Fourier Integral Theorem for f () z 

12.1 Fourier Integral Theorem for f () z 

If f () z is the main branch of a hyper-complex function, 

Then, the Complex Fourier Integral Theorem holds. 

⎛ ⎞ 

1 ⎜ ⎟ 

− 

f () z = ⎜ f() ζ e ζ 

⎟ 

2π 

∫ ⎜ ⎟ 

⎜ ∫ 

⎟ 

⎜ ⎟ 

ω = η⎜⎝ ζ− z = ε ⎠⎟ 

iωζ izω 

d e dω 

Proof: By 9.1, f () z is the integration sum, 

∫ 

f () z = f()( ζδz −ζ) dζ. 

ζ− z = ε 

Substituting from 11.4, ( z) 1 

2 

i ( z) 

e 

dr 

ωζ 

π 

− 

ω = 

∫ 

δζ− = dω, 

we have 

⎛ ⎞ 

1 ⎜ ⎟ 

f () z = f() ζ ⎜ e dω ⎟ 

e 

2π 

∫ ⎟ 

⎜ ∫ ⎟ 

⎜ ⎟ 

ζ− z = ε ⎜⎝ ω = η ⎠⎟ 

iωz −iωζ 

dζ. 

By changing the Summation order, 

⎛ ⎞ 

1 ⎜ ⎟ 

−iωζ 

izω 

f () z = ⎜ f() ζ e dζ ⎟ 

e dω 

2π 

∫ ⎜ ⎟ 

⎜ ∫ 

⎟ 

ω = η⎜ ⎟ 

⎜⎝ ζ− z = ε ⎠⎟ 

Therefore, 

. 

33


12.2 If f () z is the main branch of a hyper-complex function, 

Then, 

iωζ 

1) the hyper-complex integral f () ζ e d − 

∫ ζ converges to ˆ f ( ω ) 

ζ− z = ε 

2) the hyper-complex integral 1 fˆ ( ω) 

e dω 

converges to f () z 

2 

Proof: 

∫ 

π 

ω = η 

1) The Complex Fourier transform of the main branch of f () z , 

∫ 

ζ− z = ε 

−iωζ 

f () ζ e dζ, 

converges to a hyper-complex function ˆ f ( ω ) , some of its values 

may be infinite hyper-complex, like the complex Delta Function. 

2) The Inverse Fourier Transform of the main branch of ˆ f ( ω ) 

1 ˆ iz 

f ( ) e d 

2 

ω 

∫ ω ω 

π 

ω = η 

izω 

converges to the hyper-complex function f () z . 

34


References 

[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 

[Dan8] H. Vic Dannon, “Delta Function, the Fourier Transform, and Fourier 

Integral Theorem”, in Gauge Institute Journal Vol.8 No 2, May 2012. 

[Needham] Tristan Needham, “Visual Complex Analysis” Oxford U. Press, 

1998 (with corrections) 

[Titchmarsh] E. C. Titchmarsh “Introduction to the theory of Fourier 

Integrals”, Third Edition, Chelsea, 1986. 

35

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