# Stable and Holomorphic Implementation of Complex Functions Using a Unit Circle-Based Transform

Authors: Binesh Thankappan
Journal: Journal of Mathematics
 (2021)

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## Abstract

A stable and holomorphic implementation of complex functions inℂ plane making use of a unit circle-based transform is presented in this paper. In this method, any complex number or function can be represented as an infinite series sum of progressive products of a base complex unit and its conjugate only, where both are defined inside the unit circle. With each term in the infinite progression lying inside the unit circle, the sum ultimately converges to the complex function under consideration. Since infinitely large number of terms are present in the progression, the first element of which may be deemed as the base unit of the given complex number, it is addressed as complex baselet so that the complex number or function is termed as the complex baselet transform. Using this approach, various fundamental operations applied on the original complex number in ℂ are mapped to equivalent operations on the complex baselet inside the unit circle, and results are presented. This implementation has unique properties due to the fact that the constituent elements are all lying inside the unit circle. Out of numerous applications, two cases are presented: one of a stable implementation of an otherwise unstable system and the second case of functions not satisfying Cauchy–Riemann equations thereby not holomorphic in ℂ plane, which are made complex differentiable using the proposed transform-based implementation. Various lemmas and theorems related to this approach are also included with proofs.


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## Body

## 1. Introduction

Complex analysis has long been regarded as a powerful tool for solving problems in pure mathematics as well as in applied streams and engineering. Detailed studies in complex variables and analytic functions along with transforms and holomorphic analysis have been reported in [1–13]. Generalizations of Cauchy–Riemann equations, stability of polynomials, transfer functions, and analysis of complex spatial variables have been also been investigated extensively [14–18]. Significant results making use of complex analysis in applied mathematics and physics have also been reported with special relevance to electric circuits [19], quantum mechanics [20, 21], and quantum field theory [22, 23].Theorem1 introduces the basic concept of the proposed transform-based implementation of a complex number or an element of any complex function defined in ℂ plane. According to this theorem, any arbitrary function defined outside the unit circle can be mapped to a basic complex element defined inside the unit circle. Alternatively, this random complex function can be generated back, using its base element and the conjugate only, using an infinite progressive series that always converges.Theorem 1.
IfZBis a complex number defined outside the unit circle inℂthat can also be an element of an arbitrary complex function and can be represented in the formZB=xb2+xb+yb2/1−xb2−yb2+iyb/1−xb2−yb2=x+iy, thenZBcan be generated by the infinite progressive seriesZB=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ, where the base unitzb=xb+iyb=reiθand all other components of the series being defined inside the unit circle, withzb=r<1. Consequently, the entireℂplane outside the unit circle can be mapped to inside the unit circle.Proof.
The proof consists of three parts:Part 1, Part 2, and Lemma 1Part 1.
Defining an infinite series with sumS and elements as progressive products of a base unit zb=xb+iyb=reiθ and its complex conjugate zb∗=xb−iyb=re−iθ as follows:(1)S=xb+iyb+xb+iybxb−iyb+xb+iybxb−iybxb+iyb+⋯infterms,S=xb+iyb+g+gxb+iyb+g2+⋯infterms,where g=r2=xb2+yb2=zb2, with g<1.
The sum converges as follows:(2)S=xb+iyb11−g+g1−g,S=xb+iyb+g1−g=11−xb2−yb2xb2+xb+yb2+iyb,S=xb2+xb+yb21−xb2−yb2+iyb1−xb2−yb2=ZB.
Therefore,ZB can be represented as above, using a base unit zb inside the unit circle, provided zb exists.Part 2.
Proof for existence of base unit inside the unit circle for an arbitrary complex number or function inℂ.
FromPart 1, any arbitrary complex number outside the unit circle can be expressed in the form as follows:(3)x+iy=ZB=xb2+xb+yb21−xb2−yb2+iyb1−xb2−yb2.
The real part is expressed as follows:(4)x=xb+xb2+yb21−xb2−yb2,where xb2+yb2<1.
The imaginary part is expressed as follows:(5)y=yb1−xb2−yb2.
The solutions forxb and yb can be extracted from the following set of equations and conditions of inequalities:(6)xb=1±1−4x+1−1±1+4yy−xb2/2y2x+1−x2x+1,yb=−1±1+4yy−1±1−4x−1x+1yb2/2x−122y.
The additional conditions of inequalities are as follows:(7)yb2<4x−xx−1x+12,xb2<−4x+3x+1x+12.
Therefore, the base unitxb+iyb of the complex number or function ZB in ℂ exists inside the unit circle.Lemma 1.
The integer∞defined in the complex planeℂfor the vanishing progression is even, and therefore the integer∞−1is odd.Proof.
FromPart 1 of proof, the sum to infinite terms is(8)S=ZB=xb+iyb+g+gxb+iyb+g2+g2xb+iyb+…infterms.
Analyzing the lemma, withxb<1, yb<1, and also g<1, the series is converging, and assuming at ∞ (infinity), an odd term vanishes to zero. If that happens, the next even term and all subsequent terms will be zero, and therefore, for the series, ∞ has to be defined at that odd integer (n) term under consideration. Alternatively, for the series, if this odd integer n is defined as ∞, then the term gnxb+iyb has to vanish. This happens only when gn is vanished to zero because xb and yb are finite within the unit circle. This implies that if an odd integer is defined as ∞ for the series, gn in that term would have already vanished. However, gn constitutes the even term just before the nth term, i.e., the first term that vanishes—which would be the last term of the progression—cannot be an odd integer term in the series, and therefore, the integer ∞ defined in the complex plane ℂ for the vanishing progressive series is even. Consequently, the integer ∞−1 will be odd.UsingPart 1, Part 2, and Lemma 1, the transform can be represented as follows: ZB=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ, where the base unit zb=xb+iyb=reiθ and all other components of the series being defined inside the unit circle, with the condition zb<1. So, the entire ℂ plane outside the unit circle can be mapped to inside the unit circle.Thus, the theorem is proved.


## 2. Terminologies: Complex Baselet and Complex Baselet Transform

Because an infinite progressive series representation of a complex number or function is used in the approach, the first element of which may be deemed as the base unit of the given complex number, the terminology complex baselet [3] denoted by zb is used in the paper and the complex number or function is termed as the complex baselet transform [3] of zb, denoted by ZB.The complex baselet transform ofzb is formally defined from Theorem 1 as follows: ZB=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ, where zb=xb+iyb=reiθ and all other elements of the series being defined inside the unit circle, with zb<1. The given complex number ZB outside the unit circle becomes the complex baselet transform of zb. Additionally, the inverse complex baselet transform can be defined recursively as follows: zb=1/∑n=1,n⟶odd∞−11/ZBrn−1+∑n=2,n⟶even∞rn−1e−iθ/ZB even though estimation of zb can be carried out using Part 2 of proof in the aforementioned theorem. However, this recursive relation is useful for further analysis.Lemma 2.
Ifzb=xb+iyb=reiθ, defined inside the unit circle is the complex baselet ofZB=x+iy, inℂoutside the unit circle, thenlimn+1⟶∞zb∗zbne−in−1θ=limn⟶∞zbne−inθ=0, wherezb∗represents the complex conjugate ofzb.Proof.
Proof of Lemma2 follows from Theorem 1 and Lemma 1


## 3. Theorems on Complex Baselet Transform

Theorem 2.
Conjugate operation of a complex number inℂplane is transformed as is to the complex baselet inside the unit circle, i.e., whenZB=x+iybecomesZB∗=x−iyinℂplane, the complex baselet gets transformed tozb∗=xb−iyb=re−iθfromzb=xb+iyb=reiθinside the unit circle.Proof.
Assume as proposed in the theorem,zb∗=xb−iyb=re−iθ is the complex baselet of ZB∗=x−iy. Next, the basic generation series becomes(9)S=xb−iyb+xb−iybxb+iyb+xb−iybxb+iybxb−iyb+⋯infterms,S=xb−iyb1/1−g+g/1−g, where g=r2=xb2+yb2=zb2 with g<1,(10)S=xb−iyb+g1−g=11−xb2−yb2xb2+xb+yb2−iyb,and S=xb2+xb+yb2/1−xb2−yb2−iyb/1−xb2−yb2=ZB∗=x−iy in ℂ.
Therefore, conjugate operation onZB in ℂ is transformed as is to zb inside the unit circle.Theorem 3.
LetZBbe a complex number defined outside the unit circle inℂ, which is the complex baselet transform ofzbdefined inside the unit circle, with the scalarr=zbandr<1, then the operation onZB,FZB=1/∑n=1,n⟶odd∞∑k=1Ne−1kiθ1/ZBrn−1+∑n=2,n⟶even∞−1∑k=1Ne−1kiθ1/ZBrn−1e−iθcarried out inℂproduces a transformationztron the complex baseletzbofZBinside the unit circle, as the solution ofZB=Ncosθ∑n=1,n⟶odd∞−1ztrsecθ/Nne−in−1θ+∑n=2,n⟶even∞ztrsecθ/Nne−inθwithztrsecθ/N<1, for N even and(11)ZB=1N−1cosθ+e−iθ∑n=1,n⟶odd∞−1ztrsecθNne−in−1θ+∑n=2,n⟶even∞ztrsecθNne−inθ,withztr/N−1cosθ+e−iθ<1, for N odd.Proof.
Let the given operation onZB produce a transformation Tzb on zb as follows:(12)Tzb=1∑k=1Ne−1kiθ1∑n=1,n⟶odd∞1/ZBrn−1+∑n=2,n⟶even∞−11/ZBrn−1e−iθ.
This can be expressed as follows:(13)Tzb=1e−iθ+eiθ+e−iθ+⋯Nterms∑n=1,n⟶odd∞1/ZBrn−1+∑n=2,n⟶even∞−11/ZBrn−1e−iθ.
ForN even,(14)Tzb=1Ncosθ1∑n=1,n⟶odd∞1/ZBrn−1+∑n=2,n⟶even∞−11/ZBrn−1e−iθ,i.e., Tzb=1/Nzbsecθ, for N even, with the condition zbsecθ/N<1 for the complex baselet to be inside the unit circle.
ForN odd,(15)Tzb=1N−1cosθ+e−iθ1∑n=1,n⟶odd∞1/ZBrn−1+∑n=2,n⟶even∞−11/ZBrn−1e−iθ,and Tzb=zb/N−1cosθ+e−iθ, with the condition zb/N−1cosθ+e−iθ<1 for the complex baselet to be inside the unit circle.
Therefore, the operation produces a new element inℂ:(16)ZBtrans=ZBNcosθ=∑n=1,n⟶odd∞−1ztrsecθNne−in−1θ+∑n=2,n⟶even∞ztrsecθNne−inθ,with ztrsecθ/N<1 for N even.
Alternatively,ztr is the solution of(17)ZB=Ncosθ∑n=1,n⟶odd∞−1ztrsecθNne−in−1θ+∑n=2,n⟶even∞ztrsecθNne−inθ,with ztrsecθ/N<1, for N even.
Similarly,ztr is the solution of(18)ZB=1N−1cosθ+e−iθ∑n=1,n⟶odd∞−1ztrN−1cosθ+e−iθne−in−1θ+∑n=2,n⟶even∞ztrN−1cosθ+e−iθne−inθ,with the condition ztr/N−1cosθ+e−iθ<1, for N odd.
Thus, the theorem is proved.Corollary 1.
Let a multiplication operationR=Z2ZB=r2eiϕZB be defined in ℂ, for ZB, a complex number defined outside the unit circle, which is the complex baselet transform of zb defined inside the unit circle, then the operation transforms the original complex baselet zb of ZB to ztrr2eiϕ, while ztr is the solution of(19)R=∑n=1,n⟶odd∞−1ztrr2ne−in−1θ+ϕ+∑n=2,n⟶even∞ztrne−inθ+ϕ,with the condition ztrr2<1 and argztr=θ+ϕ.Proof.
From Theorem3, the operation becomes a transformation Tzb on zb as(20)Tzb=r2eiϕ∑n=1,n⟶odd∞1/ZBrn−1+∑n=2,n⟶even∞−1rn−1e−iθ/ZB.
Next,zb is modified to a complex quantity inside the unit circle ztrr2eiϕ, with ztr being the solution of(21)R=∑n=1,n⟶odd∞−1ztrr2ne−in−1θ+ϕ+∑n=2,n⟶even∞ztrne−inθ+ϕ,along with the condition ztrr2<1 and argztr=θ+ϕ.Corollary 2.
Let a division operationR=ZB/Z2=ZB/r2eiϕ be defined in ℂ, on ZB, a complex number defined outside the unit circle, which is the complex baselet transform of zb defined inside the unit circle, then the operation transforms the original complex baselet zb of ZB to ztr/r2eiϕ, while ztr is the solution of(22)R=∑n=1,n⟶odd∞−1ztrr2ne−in−1θ−ϕ+∑n=2,n⟶even∞ztrr2ne−inθ−ϕ,with the condition ztr/r2<1 and argztr=θ−ϕ.Proof.
The proof follows from Corollary1, with zb modified to ztr/r2eiϕ, where ztr is the solution of(23)R=∑n=1,n⟶odd∞−1ztrr2ne−in−1θ−ϕ+∑n=2,n⟶even∞ztrr2ne−iθ−ϕ,with the condition ztr/r2<1 and argztr=θ−ϕ.Theorem 4.
IfZB1andZB2are two complex numbers or functions defined outside the unit circle inℂ, with their respective complex baseletszb1andzb2defined inside the unit circle, the summation operation ofzb1andzb2produces a functionzs=FZB1,ZB2such that(24)zs=ZB11−zb12−zb12+ZB21−zb22−zb22.Proof.
Let two complex numbers or functions inℂ be ZB1 and ZB2, and their respective complex baselets be zb1 and zb2, and the sum of complex baselets be zs.
From Theorem1, let for any complex number or function ZB in ℂ,(25)S=xb+iyb11−g+g1−g.
Substituting and simplifying, the following relation can be obtained:(26)zb=1−zb2ZB−zb21−zb2,with the condition zb<1.
ForZB1,(27)zb1=1−zb12ZB1−zb121−zb12.
ForZB2,(28)zb2=1−zb22ZB2−zb221−zb22.
The sum of two complex baselets which are individually defined inside the unit circle is related toZB1 and ZB2 as follows:(29)zs=1−zb12ZB1−zb121−zb12+1−zb22ZB2−zb221−zb22,zs=ZB11−zb12−zb12+ZB21−zb22−zb22.Corollary 3.
IfZB1 and ZB2 are two complex numbers or functions defined outside the unit circle in ℂ, with their respective complex baselets zb1 and zb2 defined inside the unit circle, then the operation in ℂ: Zsum=ZB1+ZB2 corresponds to a complex baselet zs inside the unit circle as follows:(30)zs=1−zs2ZB1+ZB2−zs21−zs2,with zs<1.Proof.
The inverse complex baselet transform equation is derived as follows:(31)zb=1−zb2ZB−zb21−zb2.
Let the complex baselet transform bezs, for Zsum=ZB1+ZB2. Therefore, the relation modifies to the following:(32)zs=1−zs2ZB1+ZB2−zs21−zs2,with zs<1.
Thus, the theorem is proved.Corollary 4.
IfZB1 and ZB2 are two complex numbers or functions defined outside the unit circle in ℂ, with their respective complex baselets zb1 and zb2 defined inside the unit circle, then the operation outside unit circle in ℂ: Zdiff=ZB1−ZB2 corresponds to a complex baselet zd inside the unit circle as follows:(33)zd=1−zd2ZB1−ZB2−zd21−zd2,with zd<1.Proof.
The inverse complex baselet transform equation is derived as follows:(34)zb=1−zb2ZB−zb21−zb2.
ForZdiff=ZB1−ZB2, let the complex baselet transform be zd. Then, the relation modifies to the following:(35)zd=1−zd2ZB1−ZB2−zd21−zd2,with zd<1.
Thus, the theorem is proved.Theorem 5.
Unstable systems whose poles of the transfer functionHZlie outside the unit circle inZdomain can be implemented in a stable configuration, with no poles outside the unit circle, making use of complex baselet transform.Proof.
A system whose transfer function defined in theZ domain is stable if all its poles lie within the unit circle in the Z plane. Let the transfer function of the unstable system be expressed in the form: HZ=AZ+BCZ+D…/Z−p1Z−p2…, where the poles p1,p2,… are complex quantities lying outside the unit circle. Transform HZ=rnreiα/rdr1eiβ1rdr2eiβ2…=rnr/rdreiα−β1−β2−…=rnr/rdreiα−β, where rdr=rdr1,rdr2,… and β=β1+β2+…. Let zb=reiθ be the complex baselet of the numerator of the transfer function, defined inside the unit circle. Due to the presence of denominator whereby poles are present outside the unit circle for this function, the new transformed complex baselet denoted by z tr will be the solution of the equation with converging RHS due to a necessary condition, expressed as follows. Therefore,(36)HZ=∑n=1,n⟶odd∞−1ztrrdrne−in−1θ−β+∑n=2,n⟶even∞ztrrdrne−iθ−β,with the condition ztr/rdr<1.
When implemented inside the unit circle as in equation (36), the transfer function HZ does not have any poles outside the unit circle and is stable. Therefore, those unstable systems whose poles lie outside the unit circle in Z domain can be implemented in a stable configuration, using complex baselet transform.Theorem 6.
A counter clockwise rotation defined inside the unit circle on the complex baseletf=zbeiϕsatisfies the Cauchy–Riemann equations with the partial derivatives defined with respect toxbandyband hence differentiable with respect tozband holomorphic inside the unit circle. The derivative offbecomes the same counter clockwise rotation of the derivative ofzb, which becomeseiϕ.Proof.
Let a counter clockwise rotation be defined on the complex baseletzb inside the unit circle as follows:(37)f=zbeiϕ.
Next,f=xb+iybcosϕ+isinϕ, lying inside the unit circle can be expressed as follows:(38)f=xbcosϕ−ybsinϕ+ixbsinϕ+ybcosϕ.
Applying Cauchy–Riemann equations inside the unit circle,(39)∂U∂xb=cosϕ,∂V∂yb=cosϕ,∂U∂yb=−sinϕ,−∂V∂xb=−sinϕ.
Therefore,(40)dfdzb=∂U∂xb+i∂V∂xb=cosϕ+isinϕ=eiϕ.
From equations (37) and (40), df/dzb=eiϕdzb/dzb=eiϕ.Theorem 7.
The first derivative of a counter clockwise rotation defined inside the unit circle on thenthpower of the complex baselet,f=zbneimϕ, where n and m are integers, becomes the same counter clockwise rotation of the derivative ofzbn, which becomeseimϕnzbn−1.Proof.
Forf=zbeimϕ, using de Moivre’s theorem and Theorem 6 as follows:(41)dfdzb=eimϕdzbdzb=eimϕ.
Forf=zbneimϕ,(42)dfdzb=dzbeimϕzbn−1dzb.
Differentiating using product rule and also using equation (41),(43)dfdzb=zbeimϕdzbn−1dzb+zbn−1eimϕ,dfdzb=zbeimϕn−1zbn−2+zbn−1eimϕ.
Therefore,df/dzb=eimϕnzbn−1=eimϕdzbn/dzb.Theorem 8.
A clockwise rotation defined inside the unit circle on the complex baselet asf=zbe−iϕsatisfies the Cauchy–Riemann equations, with the partial derivatives defined with respect toxbandyband thus differentiable with respect tozband holomorphic inside the unit circle. The derivative offbecomes a clockwise rotation of the derivative ofzbby the same angle, which will bee−iϕ.Proof.
Let a clockwise rotation be defined on the complex baseletzb as follows:(44)f=zbe−iϕ.
Then,f=xb+iybcosϕ−isinϕ, inside the unit circle.(45)f=xbcosϕ+ybsinϕ+iybcosϕ−xbsinϕ.
For verifying the function to be holomorphic, applying Cauchy–Riemann equations inside the unit circle,(46)∂U∂xb=cosϕ,∂V∂yb=cosϕ,∂U∂yb=sinϕ,−∂V∂xb=sinϕ.
Therefore,(47)dfdzb=∂U∂xb+i∂V∂xb=cosϕ−isinϕ=e−iϕ.
From equations (44) and (47), df/dzb=e−iϕdzb/dzb=e−iϕ.Theorem 9.
The first derivative of a clockwise rotation defined inside the unit circle on the nthpower of the complex baselet,f=zbne−imϕ, where n and m are integers, will be equivalent to a clockwise rotation of the derivative ofzbn, by the same angle, which becomese−imϕnzbn−1.Proof.
Forf=zbe−imϕ, using de Moivre’s theorem and Theorem 8,(48)dfdzb=e−imϕdzbdzb=e−imϕ.
Forf=zbne−imϕ,(49)dfdzb=dzbe−imϕzbn−1dzb.
Differentiating using product rule and also using Theorem8,(50)dfdzb=zbe−imϕdzbn−1dzb+zbn−1e−imϕ,dfdzb=zbe−imϕn−1zbn−2+zbn−1e−imϕ.
Therefore,df/dzb=e−imϕnzbn−1=e−imϕdzbn/dzb.Theorem 10.
The nonanalytic functionfZ=ZB∗defined inℂoutside the unit circle and not satisfying the Cauchy–Riemann equations which constitute a necessary and sufficient condition for it to be complex differentiable inℂ, when implemented using the complex baselet, can be made complex differentiable inside the unit circle. Consequently, any complex differential operation onZB∗inℂ, which does not exist, is made equivalent to a differential operation on the complex baselet-based implementation inside the unit circle, which always converges and therefore exists.Proof.
First, complex derivative ofZB outside the unit circle is computed as follows:(51)ZB=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ.
Furthermore,(52)dZBdZ=dZBdzbdzbdZ=∑n=1,n⟶odd∞−1nzbn−1e−in−1θ+∑n=2,n⟶even∞nzbn−1e−inθdzbdZ,dZBdZ=1zb∑n=1,n⟶odd∞−1nzbne−in−1θ+1zb∑n=2,n⟶even∞nzbne−inθdzbdZ,dZBdZ=1zbzb11−zb2+2zb21−zb22+1zb2zb21−zb2+2zb41−zb22dzbdZ.
This is simplified as follows:(53)dZBdZ=11−zb2+2zb21−zb22+1zb2zb21−zb2+2zb41−zb22dzbdZ,dZBdZ=1zbzb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ,where dzb/dZ and zb will be quantities lying inside the unit circle; therefore, the RHS is the equivalent first derivative for ZB in terms of the complex baselet zb of ZB. Anything above first derivative is redundant in implementation because the second or higher derivatives can be implemented inside the unit circle with their own complex baselets. However, the complex conjugate ZB∗ is not complex differentiable in the complex plane Z except at Z=0 and also not holomorhic. ZB∗ can also be implemented using the complex baselet zb of ZB as follows:(54)ZB∗=∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ.
This implementation is complex differentiable from Theorem9, and also applying the chain rule,(55)dZB∗dZ=dZB∗dzbdzbdZ=∑n=1,n⟶odd∞−1nzbn−1e−in+1θ+∑n=2,n⟶even∞nzbn−1e−inθdzbdZ,dZB∗dZ=1zb∑n=1,n⟶odd∞−1nzbne−in+1θ+1zb∑n=2,n⟶even∞nzbne−inθdzbdZ,dZB∗dZ=1zbzb∗11−zb2+2zb21−zb22+1zb2zb21−zb2+2zb41−zb22dzbdZ,which can be simplified to the following expression:(56)dZB∗dZ=zb2zb211−zb2+2zb21−zb22+1zb2zb21−zb2+2zb41−zb22dzbdZ,dZB∗dZ=zb2zb3zb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ.
Therefore, using the complex baselet-based implementation,dZB∗/dZ exists for all zb except for zb=0. ZB∗ which exists in ℂ outside the unit circle does not satisfy Cauchy–Riemann equation and not analytic because it is not complex differentiable except at Z=0 is made complex differentiable using the complex baselet transform approach. It is infinitely differentiable and hence holomorphic in this implementation.Theorem 11.
The nonanalytic functionZB2defined inℂ, not satisfying the Cauchy–Riemann equations which constitute a necessary and sufficient condition for it to be complex differentiable inℂ, can be made complex differentiable inside the unit circle using the complex baselet implementation and so any differential operation onZB2inℂ, that does not exist, is made equivalent to a differential operation on the complex baselet-based implementation inside the unit circle, which always converges and hence exists.Proof.
The complex baselet implementation ofZB inside unit circle is as follows:(57)ZB=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ.
Furthermore,ZB2=ZBZB∗ in ℂ, outside the unit circle,(58)dZB2dZ=dZB2dzbdzbdZ.
However,dZB2/dzb=dZBZB∗/dzb.(59)dZB2dzb=ZBdZB∗dzb+ZB∗dZBdzb.
From Theorem10,(60)ZB∗=∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ,dZBdzb=1zbzb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ,dZB∗dzb=zb2zb3zb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ.
Therefore,(61)dZB2dzb=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθzb2zb3zb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ+∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1zbzb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ.
Then, the following relation is obtained:(62)dZB2dZ=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθzb2zb3zb+2zb21−zb2+2zbzb2+zb41−zb22+∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1zbzb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ,dZB2dZ=zb+2zb21−zb2+2zbzb2+zb41−zb22zb2zb3∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ+1zb∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθdzbdZ,which exists and always converges. Thus, using the complex baselet implementation, dZB2/dZ exists for all zb except for zb=0.
ZB2, which exists in ℂ outside the unit circle does not satisfy Cauchy–Riemann equation and not holomorphic because it is not complex differentiable, is made complex differentiable using the complex baselet transform approach. It is infinitely differentiable and hence holomorphic in this implementation.Theorem 12.
The nonanalytic functionZBdefined inℂ, not satisfying the Cauchy–Riemann equations which constitute a necessary and sufficient condition for it to be complex differentiable inℂ, can be made complex differentiable inside the unit circle using the complex baselet implementation. Thus, any differential operation onZBinℂ, that does not exist, is made equivalent to a differential operation on the complex baselet-based implementation inside the unit circle, which always converges and therefore exists.Proof.
The complex baselet implementation ofZB inside unit circle is as follows:(63)ZB=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ.
Furthermore,ZB=ZBZB∗1/2=ZB1/2ZB∗1/2 in ℂ, outside the unit circle,(64)dZBdZ=dZBdzbdzbdZ.
However,dZB/dzb=dZB1/2ZB∗1/2/dzb.(65)dZBdzb=ZB1/2dZB∗1/2dzb+ZB∗1/2dZB1/2dzb,(66)ZB1/2=∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ1/2.
Equation (66) is converging.(67)dZB1/2dzb=12∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ1/2dZBdzb=12∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ1/21zbzb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ,(68)ZB∗1/2=∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1/2.
Equation (68) is converging.(69)dZB∗1/2dzb=12∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1/2dZB∗dzb,dZB∗1/2dzb=12∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1/2zb2zb3zb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ.
Substituting equations (66), (67), (68), and (69) in equation (65),(70)dZBdzb=zb2zb3∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ1/22∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1/2zb+2zb21−zb22+2zbzb2+zb41−zb22dzbdZ+∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1/22zb∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ1/2zb+2zb21−zb2+2zbzb2+zb41−zb22dzbdZ.
Therefore,(71)dZBdzb=12zb1+2zb21−zb2+2zb2+zb41−zb22zb2zb2∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ1/2+∑n=1,n⟶odd∞−1zbne−in+1θ+∑n=2,n⟶even∞zbne−inθ∑n=1,n⟶odd∞−1zbne−in−1θ+∑n=2,n⟶even∞zbne−inθ1/2dzbdZ.
This exists and always converges. Consequently,dZB/dZ exists for all zb except for zb=0.
Therefore,ZB that exists in ℂ outside the unit circle and not satisfying the Cauchy–Riemann equations because it is not complex differentiable is made complex differentiable using the complex baselet transform approach. The transformed expression is infinitely differentiable and hence holomorphic in this implementation.


## 4. Implementation Unit of Complex Baselet Transform

The implementation unit ofZB using its complex baselet, zb, is displayed in Figure 1. This structure follows directly from Part 1 of the proof of Theorem 1. A toggle switch with initial position at 1 and toggle frequency f=1/tmult, where tmult is the time of computation of the multiplier, forms the first input of the multiplier. The second input constitutes a feedback loop of the multiplier output as shown, with the initial condition for this input being unity. The accumulator unit consecutively sums the components of the progressive series. The series being infinite but converging in nature demands only a finite and limited memory storage of the accumulator, that is, designed to have a settling time Ts, after which only progressive series is made available at the output. This settling time allows sufficient number of terms being summed up in the infinite series so that the output will be the complex baselet transform implementation of ZB, from its complex baselet input zb, with the designed accuracy. The symbolic block representation of complex baselet transform is displayed in Figure 2.Figure 1
Implementation unit of ZB using complex baselet.Figure 2
Symbolic block representation of complex baselet transform.


## 5. Conclusions

Stability along with holomorphicity constitutes a significant and desirable property set for many complex functions inℂ plane. A transform-based implementation using complex baselet transform is proposed that comprises an infinite converging series of progressive products of a base unit called complex baselet and its conjugate only. Various operations applied on the original complex function are mapped to equivalent operations on the complex baselet inside the unit circle. This method of implementation has got a distinct feature that the component elements are all lying inside the unit circle. Theorems and lemmas related to the proposed approach are also presented with proofs. A stable implementation of a normally unstable complex function is developed using the approach. Additionally, functions not satisfying Cauchy–Riemann equations and consequently not holomorphic in ℂ are made complex differentiable with the proposed transform–based implementation. Analysis can be extended further for the proposed approach, which can reveal the in-depth geometrical interpretation and patterns.



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*Source: 7817026-2021-12-23.xml*