Foundational aspects of a new matrix holomorphic structure

Abstract

Purpose

In this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis.

Design/methodology/approach

We introduce and investigate the complex space HC consisting of all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1, (z1,w1,z2,w2)∈C4.

Findings

We develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties.

Originality/value

We give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for holomorphicity of a function of one complex matrix variable ξ∈HC. In particular, we show that we have a lot of holomorphic functions of one matrix quaternion variable.

Citation

Khedhiri, H. and Mkademi, T. (2024), "Foundational aspects of a new matrix holomorphic structure", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-08-2023-0002

1. Introduction

The theory of quaternionic analysis was founded in 1928 and is devoted especially to the study of the so-called regular functions introduced by R. Fueter in 1935 [1] which satisfy the (left) Cauchy–Fueter equation

∂∂x1(.)+i∂∂x2(.)+j∂∂x3(.)+k∂∂x4(.)=0,

where {1, i, j, k} is the standard basis of the four-dimensional real algebra H of the quaternions numbers constructed in 1843 by W. R. Hamilton [2]. All such quaternion numbers have the representation

x0+x1i+x2j+x3k,(x0,x1,x2,x3)∈R4

which provide in fact a foundation for simple mathematical representation of rotations. Therefore, they are powerfully used in the fields of mechanics, magnetism, aerospace, software development, etc. Thus, many mathematicians show a great interest in studying quaternionic analysis and particularly quaternionic analysis over a complex structure. Among many research papers about quaternionic analysis, for instance, we may observe the various versions of their works presented in Refs. [3–5].

In this paper, we are deeply interested in the field MH of quaternion numbers represented by all 2 × 2 complex matrices having the form

z=z1z2−z‾2z‾1,(z1,z2)∈C2.

such MH will be regarded here as an R-algebra isomorphic to the R-algebra H of W. R. Hamilton. We propose to introduce the complex space HC of the so-called complex quaternion numbers of the form

ξ=z+iw,(z,w)∈MH×MH,

so that

HC={ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1:(z1,w1,z2,w2)∈C4}.

We present HC as the left MH-vector space of complex quaternion numbers with basis

{1,i1}=1001,i00i.

so that HC has a splitting into the direct sum MH⊕iMH. Moreover, we develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties.

The main parts of this work are organized as follows. In Section 2, we present the construction of the space HC of the complex quaternion numbers and we prove the following.

Theorem 2.1. Let HC,+,. denote the set MH×MH equipped with the operations (+) and (.) defined such that for all (z,w),(z′,w′)∈HC, for all λ=a+ib∈C, we have:

1. (z, w) + (z′, w′) = (z + z′, w + w′).

2. λ.(z, w) = (a + ib)(z, w) = (az − bw, aw + bz).

Then, it holds that

1. HC,+,. is a C-vector space.

2. HC has a splitting into a direct sum H1⊕iH1 where H1 is an R-sub-vector space of HC isomorphic to MH.

3. If (×) denotes the usual multiplication of square matrices, then the space HC,+,.,× is a C-algebra and the space MH,+,.,× is an R-sub-algebra.

The above structure on HC has its own particular features, it induces a C-algebra structure. So, we have included in this section the basic correspondent algebraic properties. Moreover, we define a conjugation in HC for the one HC can be viewed as an inner product space. In Section 3, we give the fundamental operational calculus on functions of one complex quaternion (or complex matrix) variable that take values in a vector space E∈{R,C,MH,HC}. In particular, the concepts of real and complex quaternionic derivatives are introduced. In Section 4, the meaning of a quaternionic holomorphic function is given due to the following operators

∂ξ≕∂=∂∂z−i∂∂w and∂‾ξ≕∂‾=∂∂z‾−i∂∂w‾

which act on differentiable quaternionic functions of one variable ξ∈HC. Therefore, we provide a characterization of quaternionic holomorphic functions by means of sufficient and necessary conditions in terms of Cauchy–Riemann type equations.

Theorem 4.1. Let E∈{R,C,MH,HC} and Φ:D→E be a complex quaternionic function of one complex quaternion variable ξ = z + iw, with (z,w)∈MH×MH, defined on an open subset D in HC. Suppose that

Φ(ξ)=Φ(z+iw)=f(z,z‾,w,w‾)+ig(z,z‾,w,w‾).

Then, Φ is holomorphic on D, if and only if the following Cauchy–Riemann type equations ∂f∂z‾=∂g∂w‾and∂f∂w‾=−∂g∂z‾ are satisfied.

Such new version of complex structure gives an other way of studying quaternionic analysis. In addition, it is quite different from which provided in Ref. [4] and can be viewed from a complex matrix analysis viewpoint. Furthermore, several different concrete computational methods provided throughout this work show that the presented matrix (quaternionic) complex structure is flexible and is close to the standard complex structure. In fact, this can be shown with the help of Theorem 4.1, providing a non trivial example of holomorphic quaternionic function.

Theorem 4.3. Let Φ: ξ↦ξ⁻¹ be the complex quaternionic inversion function defined for all ξ∈HC\S, where S={ξ∈HC:detξ=0}. Then, it holds that,

1. Φ has a decomposition into f + ig where f and g are functions of two variables (z,w)∈MH×MH satisfying the Cauchy–Riemann type equations ∂f∂z‾=∂g∂w‾ and ∂f∂w‾=−∂g∂z‾.

2. Φ is a biholomorphism from HC\S to HC\S.

Theorem 4.3 shows that we have a lot of holomorphic functions of one matrix variable. In fact, the complex matrix analysis is the theory of such functions. The other results of this paper can offer potential methods and stimulate activity in the theory of complex quaternionic analysis. On the other hand, we illustrate our abstract study by several examples to insist that the presented quaternionic holomorphic structure can induce a new aspect of pluripotential theory in quaternionic plurisubharmonic functions as provided in Ref. [6]. Moreover, it should be denoted that our paper can be useful for authors working on subjects studied in Refs. [7–9].

Finally, let us recall that according to Ref. [2], the algebra of quaternion numbers is the non-commutative field

H=R.1⊕R.i⊕R.j⊕R.k,

which has a structure of a four-dimensional R-vector space, with basis {1, i, j, k}, for which a binary composition law is equipped and defined as a bilinear form, such that 1 is the unity and

i2=j2=k2=−1,ij=−ji=k,jk=−kj=i,ki=−ik=j.

on the other hand, the field H can be described as the sub-algebra MH of the R-algebra M2(C) of dimension 8 consisting of all complex square matrices (see Ref. [10]). In addition, the isomorphism

H→MHa+bi+cj+dk↦a1+bJ+cK+dL

between H and MH, shows that MH is a 4-dimensional R-vector space with basis {1,J,K,L}=1001,i00−i,01−10,0ii0.

2. The complex quaternionic space HC

With a complexification method applied on the product R-vector space MH×MH, we shall introduce the complex quaternionic space HCto be the C−vector space consisting of all 2 × 2 complex matrices written uniquely as ξ=z+iw,(z,w)∈MH×MH.

Theorem 2.1.

Let HC,+,. denote the set MH×MH equipped with the operations (+) and (.) defined such that for all (z,w),(z′,w′)∈HC, for all λ=a+ib∈C, we have:

• •

  (z, w) + (z′, w′) = (z + z′, w + w′).

• •

  λ.(z, w) = (a + ib)(z, w) = (az − bw, aw + bz).

Then, it holds that

• (i)

  HC,+,. is a C-vector space.

• (ii)

  HC has a splitting into a direct sum H1⊕iH1 where H1 is an R-sub-vector space of HC isomorphic to MH.

• (iii)

  If (×) denotes the usual multiplication of square matrices, then the space HC,+,.,× is a C-algebra and the space MH,+,.,× is an R-sub-algebra of HC,+,.,×.

Proof. Statement (i) holds since the followings are immediately satisfied.

1. HC,+ is an Abelian group with zero element 0=0000.

2. The map C×HC→HC(λ,ξ)↦λ.ξ satisfies the following rules

   • (a)

     ∀(α,β)∈C2, ∀ξ∈HC:α.(β.ξ)=(αβ).ξ.

   • (b)

     ∀ξ∈HC:1.ξ=ξ.

3. ∀ λ∈C, ∀(ξ1,ξ2)∈HC×HC:λ.(ξ1+ξ2)=λ.ξ1+λ.ξ2.

4. ∀(λ,μ)∈C2, ∀ξ∈HC:(λ+μ).ξ=λ.ξ+μ.ξ.

Statement (ii) holds since each of the following maps

φ:MH→HCz↦z+i0 andiφ:MH→HCw↦0+iw

is R-linear and injective. We let H1=φ(MH) and iH1=iφ(MH). Then, we verify at once that H1∩iH1={0} and H1+iH1=HC. The first part of the statement (iii) holds since by statement (i), (HC,+,.) is a C-vector space and the multiplication law (×) in HC is associative with unity 1. Furthermore, the multiplication law (×) in HC is distributive over the addition law (+). Moreover, for all complex quaternion numbers (ξ,η)∈HC×HC and for any complex number λ∈C, we have (λ.ξ) × η = ξ × (λ.η) = λ.(ξ × η). The second part of the statement (iii) holds since for all (z,w)∈MH×MH and for any real number λ∈R, we have z+w∈MH,z×w∈MH andλ.z∈MH. In addition 1∈MH, where 1=1001 is the the square unity matrix.□

2.1 Basic algebraic properties of the space HC

Proposition 2.2.

Each of the following statements holds in HC.

• (i)

  For all ξ∈HC, there exists a unique (z,w)∈MH×MH such that ξ = z + iw. As such HC can be identified to the direct sum MH⊕iMH.

• (ii)

  The space HC is formed by all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾+iw‾2 where (z1,w1,z2,w2)∈C4.

• (iii)

  The space HC is 4-dimensional and admits the family

1001,i00−i,01−10,0ii0as a basis overC.

Proof. Statement (i) holds since the map Φ:MH×MH→HC(z,w)↦z+iw is R-linear and injective between finite dimensional R-vector spaces. Indeed, let z=z1z2−z‾2z‾1 and w=w1w2−w‾2w‾1 be in MH, (z1,w1,z2,w2)∈C4. By a direct computation, we have

z+iw=z1z2−z‾2z‾1+iw1w2−w‾2w‾1=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1.

In addition, the equation z+iw=0HC is equivalent to the followings

z1+iw1=0z‾1+iw‾1=0z2+iw2=0z‾2+iw‾2=0⇔z1=−iw1z1=iw1z2=−iw2z2=iw2⇔z1=0z2=0w1=0w2=0

⇔z=w=0MH.

Hence, kerΦ={(0MH,0MH)} so Φ is an isomorphism. Statement (ii) is a consequence of statement (i). Statement (iii) is also a consequence of statement (i) since any vector ξ∈HC can be written uniquely as the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1,(z1,w1,z2,w2)∈C4. Setting z_k = a_k + ib_k, w_k = c_k + id_k, (ak,bk)∈R2, (ck,dk)∈R2 for k ∈ {1, 2}, we have

ξ=a1−d1+i(b1+c1)a2−d2+i(b2+c2)−a2−d2+i(b2−c2)a1+d1−i(b1−c1)=(a1+ic1).1001+(b1+id1).i00−i+(a2+ic2).01−10+(b2+id2).0ii0

Hence the family 1001,i00−i,01−10,0ii0 generates the space HC and obviously is linearly independent.□

The following gives another specificity of the space HC∼MH×MH.

Proposition 2.3.

Each of the following statements holds:

• (i)

  HC,+,× is a left MH-vector space with basis {1,i1}.

• (ii)

  MH×MH,+,× is a left MH-vector space with basis {(1,0),(0,1)}.

Proof. Statement (i) holds since the following properties are satisfied:

1. ∀(z,w)∈MH×MH, ∀ξ∈HC: (z + w) × ξ = z × ξ + w × ξ.

2. ∀z∈MH, ∀(ξ,η)∈HC×HC:z×(ξ+η)=z×ξ+z×η.

3. ∀(z,w)∈MH×MH, ∀ξ∈HC:(z×w)×ξ=z×(w×ξ).

4. ∀ξ∈HC,1×ξ=ξ.

Statement (ii) holds since the following properties are satisfied:

1. ∀(z,w)∈MH×MH, ∀(a,b)∈MH×MH:

   (z+w)×(a,b)=z×(a,b)+w×(a,b).

2. ∀z∈MH, ∀(a,b),(c,d)∈MH×MH2:

   z×(a,b)+(c,d)=z×(a,b)+z×(c,d).

3. ∀(z,w)∈MH×MH, ∀(a,b)∈MH×MH:

   (z×w)×(a,b)=z×w×(a,b).

4. ∀(a,b)∈MH×MH,1×(a,b)=(a,b).□

Proposition 2.4.

Let θ:MH→HC be the injection map and let l be an R-linear map from MH to another C-vector space F, then there is a unique C-linear map l̃ from HC to F such that the following diagram

θMH→HC↘l↓l̃F

commutes (l̃°θ=l).

Proof. Let l̃ be the map such that

l̃(ξ)=l̃(z+iw)=l(z)+il(w)for allξ=z+iw∈HC.

Then, for all λ=a+ib∈C, (a,b)∈R2, for all ξ=z+iw∈HC, we have

l̃(λ.ξ)=l̃[(az−bw)+i(bz+aw)]=l(az−bw)+il(bz+aw)=al(z)+il(w)+ibl(z)+il(w)=(a+ib).[l(z)+il(w)]=(a+ib).l̃(z+iw)=λ.l̃(ξ).

Hence the map l̃ is C-linear. Further, l̃ is unique since the existence of two maps l̃1 and l̃2 such that l̃1°θ=l̃2°θ=l, provides that

Im(θ)⊂ker(l̃1−l̃2)andIm(iθ)⊂ker(l̃1−l̃2),

where iθ is the map defined by iθ(z) = iz for all z∈MH. Which makes that ker(l̃1−l̃2) is an R-subspace of HC, containing both MH and iMH so that ker(l̃1−l̃2)=MH+iMH=HC. Thus l̃1=l̃2.□

2.2 The conjugation in HC

Let ξ=z+iw∈HC be a complex quaternion (z,w)∈MH×MH. We define the conjugate of ξ to be the complex quaternion

ξ‾=z‾−iw‾,(z‾,w‾)∈MH×MH.

Therefore, for all (z1,w1,z2,w2)∈C4, we have

ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾+iw‾2⇔ξ‾=z‾−iw‾=z‾1−iw‾1z‾2−iw‾2−z2+iw2z−iw2.

From now on, the notation Tr(ξ) stands for the trace of ξ, while ^tξ stands for the transpose of ξ.

Proposition 2.5.

For all ξ = z + iw and all ξ′ = z′ + iw′ in HC with (z, w) and (z′, w′) in MH×MH, and for all λ∈C, the conjugation map ξ↦ξ‾ satisfies on HC the following rules:

• (i)

  ξ‾‾=ξ,ξ±ξ′‾=ξ‾±ξ‾′;ξ×ξ′‾=ξ‾×ξ′‾ and λ.ξ‾=λ‾.ξ‾.

• (ii)

  The map ξ↦^tξ is a C-isomorphism of HC that satisfies ξt‾=ξ‾t.

• (iii)

  There exists a unique (ξ1,ξ2)∈HC×HC such that ξ = ξ₁ + ξ₂ with ξ‾1=iξ1 and ξ‾2=−iξ2.

• (iv)

  Trw×tz‾=Trz×tw‾.

• (v)

  ξ×tξ‾=z×tz‾+w×tw‾+iw×tz‾−z×tw‾.

Proof. Statements (i) and (ii) are obvious since they can be proved by a direct computation. Let us prove statement (iii). The map ξ↦ξ‾, on HC, induces on MH×MH, an R-linear mapping J defined by J(z, w) = (−w, z) and satisfies J² = −Id. Thus, J has two eigenvalues {−i, + i}. Let H⁻ be the eigenspace corresponding to the eigenvalue −i and let H⁺ be the eigenspace corresponding to the eigenvalue + i, then HC=H−⊕H+ and hence any ξ=z+iw∈HC can be written as ξ = ξ₁ + ξ₂ where ξ₁ ∈ H⁻ and ξ₂ ∈ H⁺. Indeed, we may take ξ1=z−w‾2+i−z‾+w2 and ξ2=z+w‾2+iz‾+w2. Finally, statements (iv) and (v) can be obtained by simple computations.□

2.3 HC as an inner product space

Proposition 2.6.

Let ⟨,⟩ denote the map

HC×HC→C(ξ,η)↦〈ξ,η〉=Tr(ξ×tη‾).

• (i)

  For all η∈HC, the map ξ↦⟨ξ, η⟩ is C-linear.

• (ii)

  For all (ξ,η)∈HC×HC, we have 〈ξ,η〉=〈η,ξ〉‾.

Therefore, HC;〈,〉 is an inner product space.

Proof. Statement (i) holds since for all λ∈C and all ξ1,ξ2,η∈HC, we have

〈λ.ξ1+ξ2,η〉=Tr(λ.ξ1+ξ2)×tη‾=Trλ.ξ1×tη‾+Trξ2×tη‾=λ〈ξ1,η〉+〈ξ2,η〉.

Statement (ii) holds since for all ξ,η∈HC, we have

〈ξ,η〉=Trξ×tη‾=Tr(ξ×tη‾)t=Trη‾×tξ=Trη×tξ‾‾=〈η,ξ〉‾.

□

Proposition 2.7.

In the inner product space HC;〈,〉, each of the following statements holds.

• (i)

  For all (ξ,η)∈HC×HC, if ξ = x + iv and η = y + iw with (x,v),(y,w)∈MH×MH, then we have

(2.1) 〈ξ,η〉=〈x,y〉+〈v,w〉+i(〈v,y〉−〈x,w〉).

• (ii)

  For all (z,w)∈MH×MH, we have 〈iz,iw〉=〈z,w〉=〈w,z〉∈R.

• (iii)

  For all ξ=z+iw=z1+iw1z2+iw2−z‾2−w‾2z‾1+iw‾1∈HC, we have

(2.2) 〈ξ,ξ〉=|z1+iw1|2+|z1−iw1|2+|z2+iw2|2+|z2−iw2|2.

• (iv)

  For all ξ=z+iw∈HC, with (z,w)∈MH×MH, we have

(2.3) 〈ξ,ξ〉=〈z,z〉+〈w,w〉.

• (v)

  For all ξ=z+iw∈HC, with (z,w)∈MH×MH, we have

(2.4) detξ=det(z+iw)=detz−detw+iTr(ztw‾).

Furthermore, if for all ξ∈HC, we put ‖ξ‖=〈ξ,ξ〉, then HC;‖.‖ is a C-normed vector space.

Proof. Formula (2.1) in statement (i) is a consequence of the equality

〈ξ,η〉=Tr(ξ×tη‾)=Trx×ty‾+v×tw‾+iTrv×ty‾−x×tw‾=〈x,y〉+〈v,w〉+i(〈v,y〉−〈x,w〉)..

Statement (ii) holds since for z=z1z2−z‾2z‾1 and w=w1w2−w‾2w‾1 in MH, (z1,z2,w1,w2)∈C4, we have

〈iz,iw〉=Tr(iz)×(iw)‾t=−i2Tr(z×tw‾)=〈z,w〉=z1w‾1+w1z‾1+z2w‾2w2z‾2=〈w,z〉=Tr(w×tz‾).

Formula (2.2) in statement (iii) can be obtained by a direct computation of Tr(ξ×tξ‾). Formula (2.3) in statement (iv) holds since we have

〈ξ,ξ〉=〈z+iw,z+iw〉=〈z,z〉+〈z,iw〉+〈iw,z〉+〈iw,iw〉=〈z,z〉+〈w,w〉+i〈z,w〉−〈w,z〉=〈z,z〉+〈w,w〉.

Formula (2.4) in Statement (v) holds since we have

detξ=det(z+iw)=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1=(z1+iw1)(z‾1+iw‾1)+(z2+iw2)(z‾2+iw‾2)=|z1|2+|z2|2−|w1|2−|w2|2+i(z1w‾1+w1z‾1+z2w‾2+z‾2w2)=detz−detw+iTr(ztw‾).

□

Remark 2.8.

(1) Since ∀(ξ,η)∈HC×HC,t(ξ×η)‾=η‾t×tξ‾ and ∀ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1∈HC, 〈ξ,ξ〉=Tr(ξ×tξ‾), formula (2.2) can be transformed to

(2.5)Trξ×tξ‾=|z1+iw1|2+|z2+iw2|2+|z1−iw1|2+|z2−iw2|2

and gives

(2.6)∀(ξ,η)∈HC×HC,‖ξ×η‖≤‖ξ‖‖η‖.

Indeed, let η=(z1′+iw1′z2′+iw2′−z‾2′−iw‾2′z‾1′+iw‾1′)∈HC. In vertu of the associativity of multiplication in HC, the trace proprieties and formula (2.5), we have

Tr(ξ×η)t(ξ×η‾)=Trξ×(η×tη‾)×tξ‾=Tr(ξ‾t×ξ)(η×tη‾)=|z1′+iw1′|2+|z2′+iw2′|2|z1−iw1|2+|z2−iw2|2+|z1+iw1|2+|z2+iw2|2|z1′−iw1′|2+|z2′−iw2′|2.

On the other hand, again by (2.5), we have

‖ξ‖2‖η‖2=Tr(ξ×tξ‾)Tr(η×tη‾)=|z1+iw1|2+|z2+iw2|2+|z1−iw1|2+|z2−iw2|2×|z1′+iw1′|2+|z2′+iw2′|2+|z1′−iw1′|2+|z2′−iw2′|2≥|z1′+iw1′|2+|z2′+iw2′|2|z1−iw1|2+|z2−iw2|2+|z1+iw1|2+|z2+iw2|2|z1′−iw1′|2+|z2′−iw2′|2=‖ξ×η‖2.

• (2)

  For all (z,w)∈MH×MH, an easy computation provides

  (2.7)det(z+w)=detz+detw+Trz×tw‾.

• (3)

  For z=z1z2−z‾2z‾1 and w=w1w2−w‾2w‾1 in MH, we have Trz×tw‾=z1w‾1+z2w‾2+w1z‾1+w2z‾2. Then, by the Cauchy–Schwarz inequality we deduce that

  (2.8)|Trz×tw‾|2≤(detz)(detw).

Moreover, if ξ=z+iw∈HC, then (2.4) provides that.

(2.9)|det(z+iw)|≤detz+detw.

Indeed, following (2.4) and (2.8), we have

|det(z+iw)|2=|detz−detw+iTrz×tw‾|2=(detz−detw)2+Trz×tw‾2≤(detz−detw)2+(detz)(detw)≤(detz+detw)2.

3. Functions of one complex quaternion variable and complex quaternionic differentiability

3.1 Functions of one complex quaternion variable

Let E∈{R,C,MH,HC} and D be an open subset in HC. We say that

f:D→Eξ↦f(ξ)

is a function of one complex quaternion variable ξ∈HC (or a complex quaternionic function or a complex matrix function), if f is an association which associates to each element ξ∈D an element f(ξ) ∈ E. In case E=HC, which means that f(ξ)∈HC for all ξ∈HC, we deduce that for all (z,w)∈MH×MH, there exist g1(z,w)=g1(z,z‾,w,w‾)∈MH and g2(z,w)=g2(z,z‾,w,w‾)∈MH such that

f(ξ)=g1(ξ)+ig2(ξ)=g1(z+iw)+ig2(z+iw)=g1(z,w)+ig2(z,w)=g1(z,z‾,w,w‾)+ig2(z,z‾,w,w‾).

3.2 Quaternionic R-differentiability

Definition 3.1.

For E∈{R,C,MH,HC}, let f:D→E be a complex quaternionic function defined on an open subset D of HC.

• (i)

  We say that f is quaternionic R-differentiable (or simply R-differentiable) at point ξ0∈D, if there exists an R-linear map f′(ξ0):HC→E such that

limh→01‖h‖f(ξ0+h)−f(ξ0)−f′(ξ0)(h)=0.

Which is equivalent to

limh→0‖f(ξ0+h)−f(ξ0)−f′(ξ0)(h)‖‖h‖=0.

• (ii)

  We say that f is R-differentiable on D if it is R-differentiable at any point ξ∈D.

Example 3.2.

Let us illustrate Definition 3.1 by the following examples:

• (1)

  The function f:HC→Rξ↦|ξ|2 is R-differentiable and for all h∈HC, we have

f′(ξ)(h)=Trh×tξ‾+ξ×th‾.

In particular, at point h=1, we have f′(ξ)(1)=Trξ+tξ‾. Indeed, for all h∈HC, we have

‖ξ+h‖2=‖ξ‖2+2R〈h,ξ〉+‖h‖2.

Further, the map h↦2R〈h,ξ〉=Trh×tξ‾+ξ×th‾ is R-linear on HC and the function h↦ɛ(h) = ‖h‖² satisfies limh→0ε(h)‖h‖=0, so that, f is R-differentiable.

• (2)

  The function f:HC→Cξ↦det(ξ) is R-differentiable and its differential is defined by

∀h=h1+ih2∈HC,f′(ξ)(h)=〈ξ,h̃〉,

where h̃=h1−ih2,(h1,h2)∈MH2. In particular, for all ξ∈HC, at point h=1, f′(ξ)(1)=〈ξ,1〉=Trξ. Indeed, by formula (2.4) in Proposition 2.7, for all h=h1+ih2∈HC and for all ξ=z+iw∈HC, we have

det(ξ+h)=detz+h1+i(w+h2)=detξ+deth+〈ξ,h̃〉

where 〈ξ,h̃〉=Trz×th‾1−w×th‾2+i(h1×tw‾+z×th‾2).

It is clear that

h↦〈ξ,h̃〉=Tr(ξ×th̃‾)

is an R-linear map on HC. In addition, due to (2.1), the function h↦ɛ(h) = det h satisfies ɛ(h) = o(‖h‖) since limh→0ε(h)‖h‖=0. In fact, by (2.2) in Proposition 2.7, we know that ‖h‖² = |h₁|² + |h₂|². Moreover, by (2.9) given in Remark 2.8, for ‖h‖ ≠ 0, we have

|det(h)|2‖h‖2=|det(h1+ih2)|2‖h‖2≤(deth1+deth2)2‖h‖2=(|h1|2+|h2|2)2‖h‖2=‖h‖2.

Which affirms that limh→0ε(h)‖h‖=0.

3.3 Quaternionic directional derivative

Definition 3.3.

For E∈{R,C,MH,HC}, let

f:D→Ez+iw↦f(z+iw)

be a complex quaternionic function of one variable ξ=z+iw∈HC defined on an open subset D of HC. If η∈HC\{0} is a nonzero complex quaternion, then we say that f has a quaternionic directional derivative at point ξ0∈D, in the direction of the vector η, if the function of one real variable t∈R↦f(ξ0+tη) is differentiable at point 0. We denote fη′(ξ0) the derivative of f at point ξ₀ in the direction of η. Hence,

fη′(ξ0)=limt→01tf(ξ0+tη)−f(ξ0).

3.4 Quaternionic partial derivative

As we have HC∼MH×MH, we may suppose for the present, that D is an open subset of MH×MH. Let E∈{R,C,MH,HC} and

f:D→E(z,w)↦f(z,w)

be a complex quaternionic function defined on D such that

f(z,w)=f(z,z‾,w,w‾),∀(z,w)∈D.

We say that f has a quaternionic (or matrix) partial derivative ∂f∂z(z,w) with respect to the variable z, if it is differentiable in the direction of the vector (1,0) and we say that f has a quaternionic partial derivative ∂f∂w(z,w) with respect to the variable w, if it is differentiable in the direction of the vector (0,1). Therefore, we have

∂f∂z(z,w)=limh→0,h∈R\{0}1hf(z+h1,w)−f(z,w),∂f∂w(z,w)=limh→0,h∈R\{0}1hf(z,w+h1)−f(z,w).

Similarly, if (z‾,w‾)∈D for all (z,w)∈D, then we say that f has a quaternionic partial derivative ∂f∂z‾(z,w) with respect to the variable z‾, if the function

f̃:D→E(z‾,w‾)↦f̃(z‾,w‾)=f(z,z‾,w,w‾)

is differentiable in the direction of the vector (1,0) and we say that f has a partial derivative ∂f∂w‾(z,w) with respect to the variable w‾, if the function

f̃:D→E(z‾,w‾)↦f̃(z‾,w‾)=f(z,z‾,w,w‾)

is differentiable in the direction of the vector (0,1). Therefore, we have

∂f∂z‾(z,w)=limh→0,h∈R\{0}1hf(z,z‾+h1,w,w‾)−f(z,z‾,w,w‾)∂f∂w‾(z,w)=limh→0,h∈R\{0}1hf(z,z‾,w,w‾+h1)−f(z,z‾,w,w‾).

Example 3.4.

(1) The function f:MH×MH→MH(z,w)↦(z+w)2 has partial derivatives such that ∂f∂z(z,w)=∂f∂w(z,w)=2(z+w).

• (2)

  The function f:MH×MH→MH(z,w)↦(z+w‾)t has partial derivatives such that ∂f∂z(z,w)=∂f∂w‾(z,w)=1.

• (3)

  The function f:MH×MH→C(z,w)↦Tr(ztw‾) satisfies the fact that f(z‾,w‾)=f(z,w)=f(w,z) and an easy computation gives

∂f∂z‾(z,w)=∂f∂z(z,w)=Tr(w‾)=Tr(w)∂f∂w(z,w)=∂f∂w‾(z,w)=Tr(z‾)=Tr(z).

• (4)

  The function f:MH×MH→C(z,w)↦Tr(z+w) satisfies the fact that f(z‾,w‾)=f(z,w)=f(w,z) and an easy computation gives

∂f∂z‾(z,w)=∂f∂z(z,w)=Tr(1)∂f∂w‾(z,w)=∂f∂w(z,w)=Tr(1).

• (5)

  Using formula (2.7) in Remark 2.8, expressing that

∀(z,w)∈MH×MH,det(z+w)=detz+detw+Tr(ztw‾).

Hence, the partial derivatives of f:MH×MH→R+(z,w)↦det(z+w) are such that ∂f∂z(z,w)=∂f∂w(z,w)=Tr(z+w).

3.5 Complex quaternionic derivative

For E∈{R,C,MH,HC}, let f:D→E be a complex quaternionic function defined on an open subset D of HC. We shall define what means by a quaternionic C-differentiable function of one complex quaternion variable.

Definition 3.5.

(i). We say that f has a complex quaternionic derivative (or complex matrix derivative) at point ξ0∈D, if there exists a C-linear map f′(ξ0):HC→E such that

limh→01‖h‖f(ξ0+h)−f(ξ0)−f′(ξ0)(h)=0.

which means that

limh→0‖f(ξ0+h)−f(ξ0)−f′(ξ0)(h)‖‖h‖=0.

• (ii).

  We say that f is complex quaternionic differentiable on D if it has a complex quaternionic derivative at any point ξ∈D. In that case f is said to be quaternionic holomorphic (or matrix holomorphic) on D.

Remark 3.6.

Statement (i) in Definition 3.5 can be written as follows. The function f has a complex quaternionic derivative at point ξ0∈D, if there exists a C-linear map f′(ξ0):HC→E and a function h↦ɛ(h) defined on a neighborhood V of 0 satisfying lim_h→0ɛ(h) = 0 and such that for all h∈V, we have f(ξ₀ + h) = f(ξ₀) + f′(ξ₀)(h) + ‖h‖ɛ(h). Moreover, a complex quaternionic differentiable function is obviously continuous.

Example 3.7.

(1) The function f:HC→Rξ↦‖ξ‖2 is not holomorphic on HC since it is complex differentiable only at ξ = 0. Indeed,

∀ξ∈HC,∀h∈HC,‖ξ+h‖2=‖ξ‖2+2R〈h,ξ〉+‖h‖2.

Moreover, h↦ɛ(h) = ‖h‖² satisfies limh→0ε(h)‖h‖=0, but, for all ξ∈HC\{0}, the map h↦2R〈h,ξ〉=Tr(htξ‾+ξth‾) is not C-linear.

• (2)

  If S={ξ∈HC:detξ=0}, then HC\S is an open subset of HC since the function ξ↦ det ξ is continuous on HC. Let us prove that the function f:HC\S→HC\Sξ↦ξ−1. is C-differentiable on HC\S and f′(ξ) is such that

∀ξ∈HC\S,∀h∈HC,f′(ξ)(h)=−ξ−1hξ−1.

In particular, at point h=1, we have f′(ξ)(1)=−ξ−2. So that f is holomorphic on HC\S. Indeed, following Example 3.2, ∀ξ∈HC and ∀h=h1+ih2∈HC, we have

det(ξ+h)=detξ+deth+〈ξ,h̃〉,whereh̃=h1−ih2.

By the Cauchy–Schwarz inequality and the Example 3.2 (2), we have

|〈ξ,h̃〉|≤‖ξ‖‖h‖and|deth|≤‖h‖2.

So that, if ‖h‖ is small enough, then | det(h)| and |〈ξ,h̃〉| are also small enough. Hence the sum (ξ + h) is invertible in HC whenever ξ is invertible in HC and h is small enough. Furthermore, for all ξ∈HC\S and all h∈HC we have

(ξ+h)(ξ−1−ξ−1hξ−1)=1−hξ−1hξ−1.

Therefore, ∀ξ∈HC\S and ∀h∈HC small enough,

(ξ+h)−1=ξ−1−ξ−1hξ−1+(ξ+h)−1hξ−1hξ−1.

Let ɛ(h) = (ξ + h)⁻¹hξ⁻¹hξ⁻¹, since h↦ξ⁻¹hξ⁻¹ is a C-linear map on HC, to prove that f is C-differentiable at point ξ, it is sufficient to prove that ɛ(h) = o(‖h‖). On the other hand, using inequality (2.6) in Remark 2.8 we get:

‖ε(h)‖=‖(ξ+h)−1hξ−1hξ−1‖≤‖(ξ+h)−1‖‖h‖‖ξ−1‖‖h‖‖ξ−1‖≤‖(ξ+h)−1‖‖ξ−1‖2‖h‖2.

hence, limh→0|ε(h)||h|=0, and so f is complex differentiable on HC\S.

3.6 Operations on complex quaternionic differentiable functions

Proposition 3.8.

Let E∈{R,C,MH,HC}, f:D→E and g:D→E be two complex quaternionic functions defined on the open subset D of HC and let λ∈C.

• (i)

  If f and g are both complex differentiable at ξ∈D, then so is the function f + λg and for all h∈HC, we have

(f+λg)′(ξ)(h)=f′(ξ)(h)+λg′(ξ)(h).

• (ii)

  If f and g are both complex differentiable at ξ∈D, then so are the functions f.g and g.f. Moreover, for all h∈HC, we have

(f.g)′(ξ)(h)=f′(ξ)(h).g(ξ)+f(ξ).g′(ξ)(h).

Proof. Statement (i) is obvious. Let us justify statement (ii). Since f and g are both C-differentiable at point ξ, we have

f(ξ+h)=f(ξ)+f′(ξ)(h)+|h|ε1(h),limh→0ε1(h)=0.g(ξ+h)=g(ξ)+g′(ξ)(h)+|h|ε2(h),limh→0ε2(h)=0.

Computing the product of the above equalities, gives that

f.g(ξ+h)=f.g(ξ)+f′(ξ)(h).g(ξ)+f(ξ).g′(ξ)(h)+|h|ε3(h),limh→0ε3(h)=0.

□

Corollary 3.9.

Any quaternionic polynomial function f(ξ)=∑n=0danξn of degree d ≥ 1, with coefficients in HC, is C-differentiable and its complex quaternionic derivative is given at any ξ∈HC by

(3.1) ∀h∈HC,f′(ξ)(h)=∑n=1d−1an∑k=0n−1ξkhξn−k−1.

Proof. By Proposition 3.8, it suffices to prove the result for a quaternionic monomial f(ξ) = ξⁿ, which can be proved by induction on n∈N. First, let show the existence of a function h↦ɛ(h) such that for all ξ∈HC and for all h∈HC, we have

(ξ+h)n=ξn+∑k=0n−1ξkhξn−k−1+ε(h)andε(h)=o(|h|).

This is obvious for n = 1, (we take ɛ(h) = 0). Suppose the statement holds for all k∈N,1≤k≤n where n > 1 is a natural number. Then we have

(ξ+h)n+1=(ξ+h)ξn+∑k=0n−1ξkhξn−k−1+ε(h)=ξn+1+(ξ+h)∑k=0n−1ξkhξn−k−1+hξn+(ξ+h)ε(h)=ξn+1+∑k=0n−1ξk+1hξn−k−1+hξn+h∑k=1n−1ξkhξn−k−1+(ξ+h)ε(h)=ξn+1+∑k=0(n+1)−1ξkhξ(n+1)−k−1+ε1(h)

where

ε1(h)=h∑k=1n−1ξkhξn−k−1+(ξ+h)ε(h).

Since the map defined on HC by

h↦∑k=0n−1ξkhξn−k−1

is C-linear and since limh→0ε1(h)‖h‖=0. Then, f is holomorphic on HC.□

Remark 3.10.

(1) If a∈HC\S(deta≠0) and b∈HC, then the quaternionic polynomial function f(ξ) = aξ + b of degree 1, is holomorphic on HC and ξ₀ = −a⁻¹b is its unique zero.

• (2)

  At point h=1, formula (3.1) provides f′(ξ)(1)=∑n=1d−1nanξn−1.

Proposition 3.11.

Let D and D′ be two open subsets of HC and let f and g be complex quaternionic functions defined on D and D′ respectively. Suppose that f(D)⊂D′, f is complex quaternionic differentiable at ξ∈D and g is complex quaternionic differentiable at f(ξ), then the composite function g°f is complex quaternionic differentiable at ξ and its complex quaternion derivative is given by

∀h∈HC,(g°f)′(ξ)(h)=g′(f(ξ))°f′(ξ)(h).

Proof. Since f is C-differentiable at ξ, then for all h∈HC we have

f(ξ+h)=f(ξ)+f′(ξ)(h)+‖h‖ε1(h),limh→0ε1(h)=0,

and since g is C-differentiable at f(ξ), for all k∈HC, we have

g(f(ξ)+k)=g(f(ξ)))+g′(f(ξ))(k)+‖k‖ε2(k),limk→0ε2(k)=0.

Hence by composition we find that

g°f(ξ+h)=g(f(ξ+h))=gf(ξ)+f′(ξ)(h)+‖h‖ε1(h)=g(f(ξ)))+g′(f(ξ))×(f′(ξ)(h))+‖h‖ε3(h)=g(f(ξ)))+g′(f(ξ))°f′(ξ)(h)+‖h‖ε3(h)

where, lim_h→0ɛ₃(h) = 0. Thus, g°f is C-differentiable at ξ.□

4. Quaternionic holomorphic structure on HC

Let ∂_ξ and ∂‾ξ‾ be the operators defined on the space of differentiable quaternionic functions of one variable ξ∈HC by

∂ξ≕∂=∂∂z−i∂∂wand∂‾ξ≕∂‾=∂∂z‾−i∂∂w‾.

Let Φ be a complex quaternionic differentiable (complex matrix differentiable) function of one variable ξ=z+iw∈HC where (z,w)∈MH×MH. Then Φ can be written Φ(z + iw) = f(z, w) + ig(z, w) and we get

∂(Φ)=∂Φ∂ξ=∂∂z−i∂∂wf+ig=∂f∂z−i∂f∂w+i∂g∂z−i∂g∂w=∂f∂z+∂g∂w+i∂g∂w−∂f∂w∂‾(Φ)=∂Φ∂ξ‾=∂∂z‾−i∂∂w‾f+ig=∂f∂z‾−i∂f∂w‾−i∂g∂z‾−i∂g∂w‾=∂f∂z‾−∂g∂w‾−i∂f∂w‾+∂g∂z‾.

4.1 Cauchy–Riemann quaternionic differential equations

The following criterion provides a necessary and sufficient condition for the holomorphicity of complex quaternionic functions.

Theorem 4.1.

Let E∈{R,C,MH,HC} and Φ:D→E be a complex quaternionic function of one complex quaternion variable ξ = z + iw with (z,w)∈MH×MH, defined on an open subset D in HC. Suppose that

Φ(ξ)=Φ(z+iw)=f(z,w)+ig(z,w),

then, Φ is holomorphic on D, if and only if the following Cauchy–Riemann type equations ∂f∂z‾=∂g∂w‾and∂f∂w‾=−∂g∂z‾ are satisfied.

Proof. Since the spaces HC and its dual THC are isomorphic as two dimensional MH-vector spaces, then the family ∂∂ξ,∂∂ξ‾ constitutes a basis of the tangent space THC over MH. Therefore, the function Φ is holomorphic on D, if and only if for all ξ∈D, the map Φ′(ξ) is C-linear. Since Φ′(ξ) can be written in the basis ∂∂ξ,∂∂ξ‾, then the C-linearity condition of the map Φ′(ξ) is equivalent to ∂‾ξΦ=0 which is equivalent to the following

∂f∂z‾−∂g∂w‾−i∂f∂w‾+∂g∂z‾=0⇔∂f∂z‾−∂g∂w‾=∂f∂w‾+∂g∂z‾=0⇔∂f∂z‾=∂g∂w‾and∂f∂w‾=−∂g∂z‾.

□

Example 4.2.

Let us illustrate Theorem 4.1 with the following examples:

1. The function Φ(ξ)=ξ‾t satisfies for all ξ=z+iw∈HC,

Φ(z+iw)=z‾t−iw‾t=f(z,w)+ig(z,w).

We have ∂f∂z‾=Tr(1)≠∂g∂w‾=−Tr(1)and∂f∂w‾=−∂g∂z‾=0. Thus, Φ is not holomorphic on HC.

• (2)

  The function Φ:HC→Cz+iw↦det(z+iw) satisfies

Φ(z+iw)=det(z)−det(w)+iTr(ztw‾)=f(z,w)+ig(z,w).

Since f(z,w)=f(z‾,w‾) and g(z,w)=g(z‾,w‾), then, an easy computation gives ∂f∂z‾=∂g∂w‾=Tr(z) and ∂f∂w‾=−∂g∂z‾=−Tr(w). Hence Φ is holomorphic on HC.

• (3)

  The function Φ:HC→HCξ↦ξ2 is complex differentiable on HC. Indeed, Φ can be decomposed into Φ = f + ig where f(z, w) = z² − w² and g(z, w) = zw + wz, (z,w)∈MH×MH. In addition, an easy computation shows that ∂f∂z‾=∂g∂w‾=0 and ∂f∂w‾=−∂g∂z‾=0. Therefore, Φ is holomorphic on HC.

• (4)

  The function Φ:HC→HCξ↦ξ‾2 is complex differentiable only at ξ = 0. Moreover, Φ can be decomposed into Φ = f + ig such that f(z,w)=z‾2−w‾2 and g(z,w)=−z‾w‾−w‾z‾, (z,w)∈MH×MH. An easy computation yields

∂f∂z‾=2z‾,∂g∂w‾=−2z‾,∂f∂w‾=−2w‾and−∂g∂z‾=2w‾.

The equations ∂f∂z‾=∂g∂w‾ and ∂f∂w‾=−∂g∂z‾ are simultaneously satisfied only at ξ=(z,w)=(0MH,0MH). Hence Φ is not holomorphic.

The following result is a consequence of Theorem 4.1. It provides a principal example of complex quaternionic holomorphic function of one variable. Moreover, it shows that we have a lot of complex quaternionic holomorphic functions of one complex quaternion variable.

Theorem 4.3.

If S={ξ∈HC:detξ=0} and Φ(ξ) = ξ⁻¹ is the complex quaternionic inversion function defined for all ξ∈HC\S, then it holds that,

• (i)

  Φ has a decomposition into f + ig where f and g are functions of two variables (z,w)∈MH×MH satisfying the Cauchy–Riemann type equations ∂f∂z‾=∂g∂w‾ and ∂f∂w‾=−∂g∂z‾.

• (ii)

  Φ is a biholomorphism from HC\S to HC\S.

Proof. First of all, it is clear that Φ is one-to-one since Φ°Φ=IdHC\S. Furthermore, if ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1∈HC\S, then ξ⁻¹ = Φ(ξ) can be written as follows

ξ−1=1detz−detw+iTr(ztw‾)z‾1+iw‾1−z2−iw2z‾2+iw‾2z1+iw1=1detz−detw+iTr(ztw‾)z‾t+i.w‾t=f(z,w)+ig(z,w),

where f and g are functions of tow variable (z,w)∈MH×MH such that

(4.1)f(z,w)=detz−detw(detz−detw)2+(Tr(ztw‾))2.z‾t+Tr(ztw‾)(detz−detw)2+(Tr(ztw‾))2.w‾t=u(z,z‾,w,w‾).z‾t+v(z,z‾,w,w‾).w‾t

(4.2)g(z,w)=−Tr(ztw‾)(detz−detw)2+(Tr(ztw‾))2.z‾t+detz−detw(detz−detw)2+(Tr(ztw‾))2.w‾t=−v(z,z‾,w,w‾).z‾t+u(z,z‾,w,w‾).w‾t.

Furthermore, the partial derivatives of f and g are such that

(4.3)∂f∂z‾(z,w)=∂u∂z‾(z,w).z‾t+u(z,w).1+∂v∂z‾(z,w).w‾t∂g∂w‾(z,w)=−∂v‾∂w(z,w).z‾t+u(z,w).1+∂u∂w‾(z,w).w‾t.

(4.4)∂f∂w‾(z,w)=∂u∂w(z,w)z‾t+v(z,w).1+∂v∂w(z,w).w‾t−∂g∂z‾(z,w)=∂v∂z(z,w).z‾t+v(z,w).1−∂u∂z(z,w).w‾t.

Since the equations ∂f∂z‾=∂g∂w‾ and ∂f∂w‾=∂g∂z‾ induces equalities of matrices of the form ab−b‾a‾ in MH, then following (4.1), (4.2), (4.3) and (4.4), f and g satisfy simultaneously the above Cauchy–Riemann equations, if and only if each of the followings holds

(4.5)z‾1∂u∂z+w‾1∂v∂z=−z‾1∂v∂w+w‾1∂u∂wz2∂u∂z+w2∂v∂z=−z2∂v∂w+w2∂u∂w.

(4.6)z‾1∂u∂w+w‾1∂v∂w=z‾1∂v∂z−w‾1∂u∂zz2∂u∂w+w2∂v∂w=z2∂v∂z−w2∂u∂z.

An easy computation of the partial derivatives ∂u∂z and ∂v∂w‾ yields

∂u∂z‾(z,w)=A(z,w)−B(z,w)C(z,w)and∂v∂w(z,w)=A(z,w)−B′(z,w)C(z,w)

where

A=(detz−detw)2+(Tr(ztw‾))2.Tr(z)B=2(detz−detw)(detz−detw).Tr(z)+Tr(ztw‾).Tr(w)C=[(detz−detw)2+(Tr(ztw‾))2]2B′=2(detw−detz).Tr(w)+Tr(ztw‾).Tr(z)[Tr(ztw‾)].

Moreover, the partial derivatives

∂u∂w‾(z,w)and∂v∂z‾(z,w)

can be obtained directly using the facts that

u(z‾,w‾)=−u(w‾,z‾)andv(z‾,w‾)=v(w‾,z‾).

We get ∂u∂w‾(z,w)=−∂u∂z‾(w,z) and∂v∂z‾(z,w)=∂v∂w‾(w,z). Since we have ∀(z,w)∈MH×MH,C(z,w)=C(w,z)>0, then formula (4.5) and (4.6) are equivalent to

(4.7)z‾1(2A−B−B′)(z;w)=w‾1(B+B′−2A)(w;z)z2(2A−B−B′)(z;w)=w2(B+B′−2A)(w;z).z‾1(B+B′−2A)(w;z)=w‾1(B+B′−2A)(z;w)z2(B+B′−2A)(w;z)=w2(B+B′−2A)(z;w).

On the other hand, an easy computation provides immediately the equalities

(B+B′)(z,w)=2A(z,w) and(B+B′)(w,z)=2A(w,z).

Which permits to conclude that (4.7) are automatically satisfied and so that

∂f∂z‾(z,w)=∂g∂w‾(z,w) and∂f∂w‾(z,w)=−∂g∂z‾(z,w).

Hence, Φ and Φ⁻¹ = Φ both are holomorphic on HC\S by Theorem 4.1.□

Financial interests: The authors declare they have no financial interests.

Conflict of interests: The authors declare they have no conflicts of interests.