infinitesimal complex calculus

Gauge Institute Journal, Volume 10, No. 4, November 2014 H. Vic Dannon

Infinitesimal Complex Calculus

H. Vic Dannon [email protected] November, 2010

Revised November, 2014

Abstract We develop here the Infinitesimal Complex

Calculus to obtain results that are beyond the reach of the

Complex Calculus of Limits.

1) In the Calculus of Limits, Cauchy’s Theorem that any loop

integral of a Complex ( )f z on a Simply-Connected domain,

vanishes, requires only Continuity of ( )f z .

Then, the derivation of the Cauchy Formula requires only

continuity.

And since Cauchy Formula guarantees differentiability, it

follows that Continuity implies Differentiability.

But the continuous ( )f z = z is not differentiable.

Thus, the derivation of Cauchy Formula in the Calculus of

Limits leads to a falsehood, and must be flawed.

In contrast, the derivation of Cauchy Formula in

1


Infinitesimal Complex Calculus requires Differentiability

of ( )f z , and avoids the contradiction.

2) Infinitesimal Complex Calculus supplies us with a

discontinuous complex function that has a derivative.

No such result exists in the Calculus of Limits.

3) The Cauchy Integral Formula holds for Hyper-Complex

Function analytic in an infinitesimal disk in the Hyper-

Complex Domain. No infinitesimal disk exists in the

Complex Plane, and no such result can exist in the Calculus

of Limits.

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.

2


Contents

Introduction

1. Hyper-Complex Plane

2. Hyper-Complex Function

3. Hyper-Complex Continuity

4. 1z

, 0z ≠

5. Log , z 0z ≠

6. Complex Derivative

7. The Step Function

8. Cauchy-Riemann Equations

9. Hyper-Complex Path-Integral

10. The Fundamental Theorem of Path Integration

11. Path Independence and Loop Integrals

12. Cauchy Integral Theorem

13. Cauchy Integral Formula in an infinitesimal disk

15. Cauchy Integral Formula

References

3


Introduction

0.1 The Cauchy Integral Formula

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

analytic

γ

( )f z as the convolution of with f1 1

2 iπ ζ.

1 ( )( )

2f

f z di z

γ

ζζ

π ζ=

−∫

1 1

( )2

f di z

γ

ζ ζπ ζ

=−∫

Thus, the Cauchy Integral Formula recovers the value of a

complex function ( )f ζ at the point in the interior of a loop

, by sifting through the values of

z

γ ( )f ζ on . γ

In the Calculus of Limits, the derivation of the Cauchy

Integral Formula raises two difficulties:

0.2 The Problem with taking 0ε ↓

The Calculus of Limits entertains the notion that the

singularity at can be bypassed by tracing a circular

path around , even when the radius of the circle, ,

vanishes.

zζ =

z zζ −

4


But in the Calculus of limits, 1( )h

zζ

ζ=

− , is defined only

out of a disk of radius ε about , and a vanishing

radius, requires , and

zζ =

0ε ↓1ε→ ∞ .

To see the flaw in the Calculus of Limits evaluation of

0

1lim

z

dzε

ζ ε

ζζ→

− =−∫ ,

put iz e φζ ε− =

id i e dφζ ε= φ . Then,

2

0

1 1 ii

z

d iz e

φ πφ

φζ ε φ

ζ εζ ε

=

− = =

=−∫ ∫ e dφ

2

0

1i dφ π

φ

ε φε

=

=

= ∫

12 iπ ε

ε= .

Whenever , we have 0ε >

11ε

ε= , and 1

2z

d iz

ζ ε

ζ πζ

− =

=−∫ .

But for , we have , and 0ε ↓0

lim 0ε

ε→

=

5


0

00

lim 0lim

lim 0ε

εε

εεε ε

→

→→

= = ,

which is undefined.

Therefore, 0

1lim dε

ζ ε

ζζ↓

=∫ is undefined.

In the Calculus of Limits,

0lim 0ε

ε→

= ,

and the limit process , drives to , without stopping

at some positive value, so that may be cancelled out.

0ε ↓ ε 0

ε

On the real line, there is no such ε that can decrease to zero,

and have a nonzero limit.

ε alludes to the hyper-real infinitesimals. But infinitesimals

do not exist on the real line, or in the complex plane, and

cannot be used in the Calculus of Limits.

Thus, to derive the Cauchy Integral Formula, we need the

Complex Infinitesimals.

0.3 Problem of Continuity implying Differentiability

The derivation of the Cauchy Formula, uses Cauchy’s

Theorem by which any loop integral of a complex ( )f z on a

6


simply connected domain, vanishes.

Cauchy’s Theorem has a proof that seems to require only

Continuity of ( )f z on the domain.

And the flawed proof of Cauchy Integral Formula in the

Calculus of Limits, requires only continuity.

Since by Cauchy Formula, ( )f z is analytic, it seems that

Continuity can imply Differentiability, which is impossible:

The continuous z is not differentiable.

In contrast, the proof of the Cauchy Integral Formula in

Infinitesimal Complex Calculus, requires differentiability.

We develop here the Infinitesimal Hyper-Complex Calculus.

In particular, we show that the Hyper-complex step function

has an infinite Hyper-Complex valued derivative at its

discontinuity.

We derive the Cauchy Integral Formula for Hyper-Complex

Function analytic in an infinitesimal disk in the Hyper-

Complex Domain. This result cannot be obtained in the

Complex Calculus of Limits.

7


Finally, we derive the Cauchy Integral Formula requiring

the differentiability of ( )f z in a simply connected hyper-

complex domain.

8


1.

Hyper-Complex Plane The Hyper-Complex Plane is the cross product of a Hyper-

real line, with a hyper-real line which elements are

multiplied by 1i = − .

Each complex number can be represented by a

Cauchy sequence of rational complex numbers,

iα + β

1 1 2 2 3 3, , ...r is r is r is+ + + so that . n nr is iα+ → + β

The constant sequence ( is a

Constant Hyper-Complex Number.

, , ,...)i i iα β α β α β+ + +

Following [Dan2] we claim that,

1. Any set of sequences , where

belongs to one family of infinitesimal hyper

reals, and belongs to another family of

infinitesimal hyper-reals, constitutes a family of

infinitesimal hyper-complex numbers.

1 1 2 2 3 3( , , ,...)i i iι ο ι ο ι ο+ + +

1 2 3( , , ,...)ι ι ι

1 2 3( , , ,...)ο ο ο

9


2. Each hyper-complex infinitesimal has a polar

representation , where

is an infinitesimal, and .

*( ) idz dr e eφ ο= = iφ*dr ο=

arg( )dzφ =

3. The infinitesimal hyper-complex numbers are smaller

in length, than any complex number, yet strictly

greater than zero.

4. Their reciprocals ( )1 1 2 2 3 3

1 1 1, , ,...i i iι ο ι ο ι ο+ + +

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

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.

10


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 line, and the imaginary as an i

coordinate. Radial symmetry leads to Polar

Coordinates.

r θ

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

∞

11


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

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

elements, and is not two-dimensional.

12


2.

Hyper-Complex Function

2.1 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

2.2 Every function from complex numbers into complex

numbers is a hyper-complex function.

2.3 sin( )dzdz

has the constant hyper-complex value 1

13


Proof: 3 5( ) ( )

sin( ) ...3! 5!dz dz

dz dz= − + −

2 4sin( ) ( ) ( )

1 ...3! 5!

dz dz dzdz

= − + −

2.4 cos( has the constant hyper-complex value 1 )dz

Proof: 2 4( ) ( )

cos( ) 1 ...2! 4 !dz dz

dz = − + −

2.5 has the constant hyper-complex value 1 dze

Proof: 2 3 4( ) ( ) ( )

1 ...2! 3! 4 !

dz dz dz dze dz= + + + + +

2.6 1dze is an infinite hyper-complex, and

1 1 cosdz dre e

φ= .

Proof: 1 1 1Re[ ] cosidz dr dr

ee e eφ φ−

= = .

2.7 log( is an infinite hyper-complex, and )dz 1log( )dr

dz >

Proof: 2 2 1log( ) [log( )] log( )dr

dz dr drφ= + > >

14


3.

Hyper-Complex Continuity 3.1 Hyper-Complex Continuity Definition ( )f z is continuous at 0z iff for any , ( ) idz dr e θ=

0 0( ( ) ) ( )= infinitesimi alf z dr e f zθ+ − .

3.2 2( )f z z= is Continuous at 1z =

Proof: 2 2(1 ( ) ) (1) (1 ( ) ) 1i if dr e f dr eθ θ+ − = + −

2 22( ) ( )i idr e dr eθ θ= +

. infinitesimal=

3.3 0, 1

( )1, 1

zh z

z

⎧ ≤⎪⎪= ⎨⎪ >⎪⎩ is discontinuous on . iz e φ=

Proof: h e . ( ( ) ) ( ) 1 0i i idr e h eφ θ φ+ − = −

3.4 ( )f z = z is continuous at any 0z

15


Proof: 0 0( ) ( )i iz dr e z dr e drθ θ+ − ≤ = .

3.5 ( )g z z= is discontinuous at any 0z

Proof: 0 0 0 0( ) ( )i iz dr e z z z dr eθ θ−+ − = − +

. infinitesimal≠

16


4.

1z

, 0z ≠

In the Calculus of Limits, the function

1( )f z

z= is defined for all . 0z ≠

We avoid , because the oscillation of 0z =1

( )f zz

= over a

disk that includes , is infinite. 0z =

However, 1

0zz

→ ⇒ → ∞ .

Therefore, 1( )f z

z= has to avoid a disk of radius ε , that

includes . Namely, 0z =

4.1 In the Calculus of limits, 1( )f z

z= , is defined only out

of a disk of radius ε about . 0z =

In Infinitesimal Calculus, if 1n

dz = , then 1n

dz= < ∞ ,

17


and we have,

4.2 In Infinitesimal Calculus, the Hyper-Complex function

1( )f z

z= , is defined, for any . 0z ≠

4.3 1( )f z

z= is discontinuous at ( ) . id e φρ

because

(( ) ( ) ) (( ) )i i if d e dr e f d eφ θ φρ ρ+ − =

1 1

( ) ( ) ( )i id e dr e d eφ θρ ρ= −

+ iφ

2 2 ( )

( )

( ) ( )( )

i

i i

dr e

d e d dr e

θ

φ θρ ρ +=

+ φ

2( ) ( )( )i i

dr

d e d dr eφ θρ ρ=

+

1dρ

∼ .

18


5.

Log( )z , 0z ≠

In the Calculus of Limits, the function

( ) Log( ) logf z z z θ= = + i is defined for all . 0z ≠

We avoid , because the oscillation of 0z = log z over a disk

that includes , is infinite. 0z =

However, for , 0ε >

3 51 1 1 1 1 1

log ...2 1 3 1 5 1

ε ε εε

ε ε ε

⎛ ⎞ ⎛ ⎞− − −⎟ ⎟⎜ ⎜− = + + +⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜+ + +⎝ ⎠ ⎝ ⎠

To first order 11

1ε

ε≈ −

+, and we have,

1 1 1

log 1 2 ...2 3 5

ε ε ε ε⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜− ≈ − + − + − +⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

Therefore,

0 logε ε→ ⇒ → −∞.

Consequently, the domain of ( ) Log( )f z = z has to avoid a

disk of radius about . Namely, ε 0z =

5.1 In the Calculus of limits, ( ) Log( )f z = z , is defined

only out of a disk of radius about . ε 0z =

19


In Infinitesimal Calculus, if 1n

dz = , then

1log( ) log logn

dz n n= = − > − > −∞

Consequently, we have,

5.2 In Infinitesimal Calculus, the Hyper-Complex function

( ) Log( )f z = z , is defined for any . 0z ≠

20


6.

Complex Derivative

6.1 Complex Derivative Definition

( )f z defined at , has a Complex Derivative at , 0z 0z 0'( )f z ,

iff for any complex infinitesimal dz ,

0 0( ) ( )f z dz f z

dz

+ −

equals a unique hyper-complex number.

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

is the complex derivative 0'( )f z .

If that number is a finite Non-Constant Hyper-complex, then

it is the sum of a constant hyper-complex and a complex

infinitesimal. Then, the constant Hyper-Complex part is the

Complex Derivative 0'( )f z .

6.2 Derivative of 3( )f z z= at 1z =

For any dz ,

3 32(1 ) (1)

3 3 ( )dz

dz dzdz

+ −= + + .

21


Therefore, 3( )f z z= has derivative '(1) 3f = .

6.3 ( )f z = z has no derivative at 0z =

For , 2( ) idz dr e drπ= =

0( ) (0)1

dzf dz f drdz dr dr

−−= = =

−

.

For , ( ) idz dr e drπ= =

0( ) (0)1

dzf dz f drdz dr dr

−−= = =

− −− .

Thus, the derivative of ( )f z = z at , does not exist. 0z =

6.4 ( )g z z= has no derivative with respect to z at any 0z

Proof: dz adz

= ⇒dx idy

adx idy

−=

+ ⇒

dx adx

idy iady

=− =

⇒1

1

a

a

== −

22


7.

Step Functions 7.1 the Step-Up Function Definition

we define 0, 0

( )1, 0

zh z

z

⎧ =⎪⎪= ⎨⎪ >⎪⎩.

gives its plot on the plane in Maple. 0Z =

7.2 the Step-Down function definition

We define the step-down function as1, 0

0, 0

z

z

⎧ =⎪⎪⎨⎪ >⎪⎩,

7.3 The Step Function is discontinuous at 0z =

The discontinuity jump of the step-up function, is seen with

23


7.4 { }

1 ,1( ) ( )

0,dz

z dz

z ddh z z

otherwisedz dzχ ≤

⎧⎪ ≤⎪= = ⎨⎪⎪⎩

z

Proof: For any dz , ( ) (0) 1 0 1h dz hdz dz dz− −

= = .

7.5 The step-up function is differentiable at its discontinuity

at . Its derivative is the infinite hyper-complex 0z = 1dz

.

24


8.

Cauchy-Riemann Equations

8.1 If ( ) ( , ) ( , )f z u x y iv x y= + has derivative at 0 0z x i= + 0y

Then, , ,

, ,x y

y x

u v

u v

=

= − at 0 0( , )x y

Proof: ( ) ( )x iyu iv u iv∂ + = ∂ + ⇒ x y

y x

u v

u v

⎧ ∂ = ∂⎪⎪⎨⎪∂ = −∂⎪⎩

8.2 ( )f z = z

= +

y x

u v

u v

∂ = = ∂∂ = = −∂

satisfies Cauchy-Riemann equations at any z

Proof: z x ⇒ u x , v y iy = =

⇒ Cauchy Riemann equations hold. 1

0x y

8.3 ( )f z = z has no derivative with respect to z at any 0z

Proof: By 6.3, ( )f z = z has no derivative at , 0z =

At , 0z ≠ 2 2yz x= + ⇒ 2 2u x= + y , 0v =

25


2 2

2 2

0

0

x y

y x

xu v

x yy

u vx y

∂ = ∂ =+

∂ = ∂ =+

Cauchy Riemann equations do not hold, at and by ⇒ 0z ≠

8.1 there is no derivative.

8.4 ( )g z z= has no derivative with respect to at any z 0z

Proof: z x iy= − ⇒

yv

u x , v y = = −

⇒ Cauchy Riemann equations 1 1xu∂ = ≠ − = ∂

do not hold, and by 8.1 there is no derivative

8.5 0, 0

( )1, 0

zh z

z

⎧ =⎪⎪= ⎨⎪ >⎪⎩ satisfies the Cauchy Riemann

equations at anyz .

Proof: , . 0, 0

1, 0

ru

r

⎧ =⎪⎪= ⎨⎪ >⎪⎩0v =

26


9.

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

the Hyper-Complex Integral of ( )f z over a path ,

, in its domain, is the sum of the areas

( )z t

[ , ]t α β∈ ( ) '( )f z z t dt of

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

9.1 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 , , be a path, , so that , and

is continuous.

( )z t [ , ]t α β∈ ( , )a bγ '( )dz z t dt=

'( )z t

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

2( )z t

2[ ( ) , ]dz dzz t − + , height ( )f z , and area ( ( )) '( )f z t z t dt .

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

, and end at , ( )z α = a ( )z bβ =

27


[ , ]

( ( )) '( )t

f z t z t dtα β∈∑ .

If for any infinitesimal , the Integration Sum

equals the same hyper-complex number, then

'( )dz z t dt=

( )f z is Hyper-

Complex Integrable over the path . ( , )a bγ

Then, we call the Integration Sum the Hyper-Complex

Integral of ( )f z over the , and denote it by( , )a bγ( , )

( )a b

f z dzγ∫ .

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

then it equals ( , )

( )a b

f z dzγ∫ .

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

equals( , )

( )a b

f z dzγ∫ .

The Integration Sum may take infinite hyper-complex

values, such as 1dz

, but may not equal to ∞ .

The Hyper-Complex Integral of the function 1( )f z

z= over a

path that goes through diverges. 0z =

28


9.2 The Countability of the Integration Sum

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

infinities:

We proved that the number of the Natural Numbers,

Card , equals the number of Real Numbers,

, and we have 2CardCard =

2 2( ) .... 2 2 ...CardCardCard Card= = = = = ≡ ∞ .

In particular, we demonstrated that the real numbers may

be well-ordered.

Consequently, there are countably many real numbers in the

interval [ , , and the Integration Sum has countably many

terms.

]α β

While we do not sequence the real numbers in the interval,

the summation takes place over countably many ( )f z dz .

9.3 Continuous ( )f z is Path-Integrable

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

Proof:

Let , , be a path, , so that , and

is continuous. Then,

( )z t [ , ]t α β∈ ( , )a bγ '( )dz z t dt=

'( )z t

29


( )(( ( )) '( ) ( ( ), ( )) ( ( ), ( )) '( ) '( ))f z t z t u x t y t iv x t y t x t iy t= + +

( )

( ( ), ( )) '( ) ( ( ), ( )) '( )

U t

u x t y t x t v x t y t y t⎡ ⎤= −⎣ ⎦ +

( )

( ( ), ( )) '( ) ( ( ), ( )) '( )

V t

i u x t y t y t v x t y t x t⎡ ⎤+ +⎣ ⎦

, ( ) ( )U t iV t= +

where , and are Hyper-Real Continuous on [ , .

Therefore, by [Dan3, 12.4], , and are integrable on

.

( )U t ( )V t ]α β

( )U t ( )V t

[ , ]α β

Hence, ( ( )) '( )f z t z t is integrable on [ , . ]α β

Since

( , )

( ( )) '( ) ( )t

t a b

f z t z t dt f z dzβ

α γ

=

=

=∫ ∫ ,

( )f z is Path-Integrable on . ( , )a bγ

30


10.

The Fundamental Theorem of

Path Integration

The Fundamental Theorem of Path Integration guarantees

that Integration and Differentiation are well defined inverse

operations, that when applied consecutively yield the

original function.

The Fundamental Theorem requires Hyper-Complex

Integrability of the Hyper-Complex Function.

10.1 The Fundamental Theorem

Let ( ( ))f z t be Hyper-Complex Integrable on [ , ]a bγ

Then, for any , [ , ]z aγ∈ b

0

( ) ( )

( ) ( )

( ( )) ( ) ( ( ))( )

u z t

u a

df u du f z t

dz t

τ

τ α

τ τ=

=

=∫

Proof:

0

( ) ( )

( ) ( )

( ( )) ( )( )

u z t

u a

df u du

dz t

τ

τ α

τ τ=

=

=∫

31


1 12 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ( )) ( ) ( ( )) ( )

( )

z t dz t z t dz t

a a

f d f d

dz t

ζ τ ζ τ

ζ τ α ζ τ α

ζ τ ζ τ ζ τ ζ τ= + = −

= =

−

=∫ ∫

2 2[ , ] [ , ]

( ( )) '( ) ( ( )) '( )

'( )

dt dtt t

f d f

z t dt

τ α τ α

ζ τ ζ τ τ ζ τ ζ τ τ∈ + ∈ −

−

=

∑ ∑ d

( ( )) '( )

'( )f z t z t dtz t dt

=

( ( ))f z t= .

32


11.

Path Independence, and Loop

Integrals

The Fundamental Theorem of Path Integration implies

Path Independence. we have,

11.1 If the Hyper-Complex ( )f z is Path-Integrable on a

Hyper-Complex Domain.

Then, ( , )

( )a b

f z dzγ∫ is Path-independent

Proof:

By 10.1, the Principal Value Derivative of ( , )

( )a z

f dγ

ζ ζ∫ with

respect to z is ( )f z , for any path . ( , )a zγ

Therefore,( , )

( )a b

f z dzγ∫ does not depend on the path .

Only on the endpoints, a , and b .

( , )a bγ

33


Path independence is equivalent to the vanishing of the

Circulation of ( )f z .

11.2 Let the Hyper-Complex ( )f z be defined on a Hyper-

Complex Domain. Then the following are equivalent

A. ( , )

( )a b

f z dzγ∫ is Path-independent

B. For any loop γ with interior in the domain,

. ( ) 0f z dzγ

=∫

34


12.

Cauchy Integral Theorem

By Cauchy Integral Theorem any loop integral of a

Differentiable ( )f z on a Simply-Connected domain, vanishes.

It seems that a Continuous ( )f z on its Domain may suffice.

The argument is as follows

By 9.3, The Continuity of ( )f z with respect to z , on the

Domain , guarantees that D ( )f z is Path-Integrable on D .

By 11.1, for any path in D , ( , )a bγ

( , )

( )a b

f z dzγ∫ is Path-independent.

By 11.2,

For any loop with interior in the domain, γ ( ) 0f z dzγ

=∫ .

However, Cauchy Integral Theorem leads to the Cauchy

Integral Formula for ( )f z , and to the conclusion that ( )f z is

differentiable. But the continuous function ( )f z = z is not

differentiable.

35


Consequently, the Cauchy Integral Theorem requires

differentiability of ( )f z , and we present a proof that

requires differentiability:

12.1 Cauchy Integral Theorem

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

Complex Simply Connected Domain D

Then, for any loop with interior in the domain, γ

( ) 0f z dzγ

=∫ .

Proof:

( )( ) ( )f z dz u iv dx idyγ γ

= + +∫ ∫

udx vdy i vdx udyγ γ

= − + +∫ ∫

Simple-Connectedness allows the use of Green’s Theorem,

int int

( )y x x y

x y x y

u v u v

dxdy i dxdyu v v u

γ γ

− + −

∂ ∂ ∂ ∂= +

−∫∫ ∫∫ ,

which vanishes by Cauchy Riemann equations.

36


13.

Cauchy Integral Formula in an

Infinitesimal Disk

13.1 0

0

12

z dr

d iz

ζ

ζ πζ

− =

=−∫ .

Proof: Put

0 ( ) iz dr e φζ − =

( ) id i dr e dφζ φ= . Then,

0

2 2

0 0 0

1 1( ) 2

( )i

iz dr

d i dr e d i dz dr e

φ π φ πφ

φζ φ φ

ζ φζ

= =

− = = =

= =−∫ ∫ ∫ iφ π=

because ( ) , for any infinitesimal dr , and any . 0idr e φ ≠ φ

The precision of 13.1, enables us to obtain the Cauchy

Integral Formula in an infinitesimal disk: A result that

cannot be obtained in the Complex calculus of Limits.

37


13.2 Cauchy Integral Formula in 0z dζ − ≤ r

If ( )f z is Hyper-Complex function Differentiable at 0z z=

Then, 0

00

1 (( )

2z dr

)ff z d

i zζ

ζζ

π ζ− =

=−∫ ,

Proof:

Since is differentiable at , then, on the circle

,

f 0z

0 ( ) iz dr e θζ − =

0 0 0( ( ) ) ( ) '( )( )i if z dr e f z f z dr eφ φ+ = + ,

Therefore,

0 0

0 0

0 0

( ) '( )( )( ) i

z dr z dr

f z f z dr efd d

z z

φ

ζ ζ

ζζ ζ

ζ ζ− = − =

+=

− −∫ ∫ ,

0 0

0 00 0

2

1( ) '( )( )

i

z dr z dr

i

ef z d f z dr

z z

φ

ζ ζ

π

ζ ζζ ζ

− = − =

= +− −∫ ∫ d

Substitute

0 ( ) iz dr e φζ − =

( ) id i dr e dφζ φ= . Then,

2

0 00

12 ( ) '( )( ) ( )

( )i i

iif z f z dr e i dr e d

dr e

φ πφ φ

φφ

π φ=

=

= + ∫

38


2

0 00

0

2 ( ) '( )( ) iif z f z dr e dφ π

φ

φ

π φ=

=

=

= + ∫

= . 02 ( )if zπ

Since the Formula can be differentiated at with

respect to z , to any order, we conclude

0z z=

13.2 a Hyper-Complex function, Differentiable at , is

differentiable to any order at .

0z z=

0z z=

39


14.

Cauchy Integral Formula

14.1 Cauchy Integral Formula

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

Complex Simply-Connected Domain D .

Then, 1 ( )( )

2f

f z di z

γ

ζζ

π ζ=

−∫ ,

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

Proof:

The Hyper-Complex function ( )fz

ζζ −

is Differentiable on the

Hyper-Complex Simply-Connected domain D , and on a path

that includes and an infinitesimal circle about z . γ

40


Then, the integral over the infinitesimal circle has a an

opposite sign because its direction is opposite to the direction

on . γ

By Cauchy Integral Theorem, we have

2 ( )

( ) ( )0

z dr

if z

f fd dz z

γ ζ

π

ζ ζζ ζ

ζ ζ− =

− =− −∫ ∫ .

Since the Formula can be differentiated with respect to z ,to

any order, we conclude

14.2 A Hyper-Complex ( )f z Differentiable on a Hyper-

Complex Simply-Connected Domain is differentiable to any

order.

41


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;

[Riemann] Riemann, Bernhard, “On the Representation of a Function

by a Trigonometric Series”.

(1) In “Collected Papers, Bernhard Riemann”, translated

from the 1892 edition by Roger Baker, Charles

Christenson, and Henry Orde, Paper XII, Part 5,

Conditions for the existence of a definite integral, pages

231-232, Part 6, Special Cases, pages 232-234. Kendrick

press, 2004

(2) In “God Created the Integers” Edited by Stephen

Hawking, Part 5, and Part 6, pages 836-840, Running

Press, 2005.

42

http://www.gauge-institute.org/continuum/OneInfinity.pdf

http://www.gauge-institute.org/continuum/OneInfinity.pdf

http://www.gauge-institute.org/Infinitesimal/infinitesimals.pdf

http://www.gauge-institute.org/Infinitesimal/infinitesimalCalculus.pdf

infinitesimal complex calculus

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