# Regular Functions with Values in Ternary Number System on the Complex Clifford Analysis

Authors: Ji Eun Kim; Su Jin Lim; Kwang Ho Shon
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We define a new modified basisi^ which is an association of two bases, e1 and e2. We give an expression of the form z=x0+i^z0-, where x0 is a real number and z0- is a complex number on three-dimensional real skew field. And we research the properties of regular functions with values in ternary field and reduced quaternions by Clifford analysis.


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## Body

## 1. Introduction

The noncommutative three-dimensional real fieldℝ3 of the hypercomplex numbers is called a ternary number system 𝕋. The quaternions are represented by the form z=∑j=03‍ejxj, where xj (j=0,…,3) are real numbers on four dimensional real field ℝ4. Fueter [1] has given a definition of quaternionic functions in ℝ4 and Deavours [2] and Sudbery [3] have developed theories of quaternionic analysis. Naser [4] investigated some properties of hyperholomorphic functions and Koriyama et al. [5] researched properties of hyperholomorphic functions and holomorphic functions in quaternionic analysis. Nôno [6] obtained several results for regular functions which have a complex number form in quaternion analysis. Cho [7] researched some properties of Euler’s formula and De moivre’s formula for quaternions. Sangwine and Bihan [8] obtained some results for the quaternionic polar representation with a complex modulus and complex argument inspired by the cayley-dickson form. Fueter [9] obtained some properties of the three variables which are called the Fueter variables and researched the fact that structures lead to the set of all Fueter-regular functions in the general cases of Clifford analysis. By Brackx et al. [10], the theory of Fueter-regularity has been developed and generalized as quaternionic variables for theories of Clifford-valued regular functions.Lim and Shon [11–13] researched the existence of hyperconjugate harmonic functions of octonion variables, properties of dual quaternion functions, and regularity of functions with values in a noncommutative subalgebra of complex matrix algebras.We consider that ternary numbers are generated by a new basisi^ and give some properties of regular functions with values in 𝕋. Also, we represent the corresponding Euler’s formula for the form z=x0+i^z0¯ and investigate calculating rules for regular functions in Clifford analysis. We research new representations of Fueter variables in reduced quaternions with i^ and some characteristics of regularity of functions on the Fueter variable system.


## 2. Preliminaries

The ternary number system𝕋 is a three dimensional non-commutative and associative real field by three bases e0, e1, and e2 with the following rules:
(1)e12=e22=-1,e1e2=-e2e1,e0¯=e0,ej¯=-ej(j=1,2).
The element e0 is the identity of 𝕋 and e1 identifies the imaginary unit -1 in the complex field. We consider an association of two bases e1 and e2 as follows:
(2)i^≔ae1+be2a2+b2=αe1+βe2withi^2=-1,
where α≔a/a2+b2, β≔b/a2+b2, and a, b are real numbers except both zeros.The number of the skew field𝕋 is
(3)z=x0+e1x1+e2x2=x0+i^z0¯,
where xj (j=0,1,2) are real variables, z0¯=γ(x1-x2e1e2), and γ:=α+βe1e2.We define the ternary number system(4)𝕋≔{z∣z=x0+i^z0¯}.
The conjugate number z* of z in 𝕋 is given by the form:
(5)z*=x0-i^z0¯.
And the norm |z| of z and the inverse z-1 of z are given by the following forms:
(6)|z|=zz*=x02+z0¯z0=∑j=02‍xj2,z-1=z*|z|2(z≠0),
where z0=γ¯(x1+x2e1e2) and γ¯=α-βe1e2.We define the addition and multiplication of two ternary numbersz=x0+i^z0¯ and w=y0+i^w0¯ as follows:
(7)z+w=(x0+y0)+i^(z0¯+w0¯),zw=(x0y0-z0w0¯)+i^(x0w0¯+z0¯y0).Theorem 1.
Letz be an arbitrary number in 𝕋. Then the corresponding Euler formula for z is
(8)ez=ex0(cos|z0|+z0|z0|i^sin|z0|).
Moreover, taking logarithms of both sides, one obtains the equation as follows:
(9)lnz=ln|z|+z0|z0|i^cos-1(x0|z|).Proof.
For the numberz=x0+i^z0¯ in 𝕋, we get |i^z0¯|=|z0¯|=|z0| and ((z0/|z0|)i^)2=-1. Then,
(10)ez=ex0+i^z0¯=ex0e(i^z0¯/|i^z0¯|)|i^z0¯|=ex0(cos|z0|+z0|z0|i^sin|z0|).
From
(11)z=|z|(x0|z|+z0|z0|i^|z0||z|)=|z|{(cos-1(x0|z|))z0|z0|cos(cos-1(x0|z|))000000+z0|z0|i^sin(cos-1(x0|z|))},
we have
(12)lnz=ln|z|+z0|z0|i^cos-1(x0|z|).
We consider the following differential operators:(13)D≔12∑j=02‍ej¯∂∂xj=12(∂∂x0-i^∂∂z0),D*=12∑j=02‍ej∂∂xj=12(∂∂x0+i^∂∂z0),
where ∂/∂z0=γ(∂/∂x1-e1e2(∂/∂x2)) and ∂/∂z0¯=γ¯(∂/∂x1+e1e2(∂/∂x2)). Then the Laplacian operator is
(14)4Δ≔DD*=D*D=∂2∂x02+∂2∂z0∂z0¯=∑j=02‍∂2∂xj2.
LetΩ be an open set in ℝ3. The function f(z) that is defined by the following form in Ω with values in 𝕋:
(15)f:Ω⟶𝕋
satisfies
(16)z=(x0,z0¯)∈Ω⟼f(z)=u0(x0,z0¯)+i^f0¯(x0,z0¯)∈𝕋,
where uj (j=0,1,2) are real-valued functions and
(17)f0=γ¯(u1+u2e1e2),f0¯=γ(u1-u2e1e2)
are complex-valued functions with values in 𝕋.Remark 2.
The operatorsD and D* act for the function f(z) on 𝕋 as follows:
(18)Df=12{(∂u0∂x0+∂f0¯∂z0¯)+i^(∂f0¯∂x0-∂u0∂z0)},D*f=12{(∂u0∂x0-∂f0¯∂z0¯)+i^(∂f0¯∂x0+∂u0∂z0)},fD=12{(∂u0∂x0+∂f0∂z0)+i^(∂f0¯∂x0-∂u0∂z0)},fD*=12{(∂u0∂x0-∂f0∂z0)+i^(∂f0¯∂x0+∂u0∂z0)}.


## 3. Properties of Regular Functions with Values in𝕋

Definition 3.
LetΩ be an open set in ℝ3. A function f(z)=u0(x0,z0¯)+i^f0¯(x0,z0¯) is said to be L(R)-regular in Ω, if the following two conditions are satisfied:(i)
u0 and f0 are continuously differential functions on Ω;(ii)
D*f(z)=0 (f(z)D*=0) on Ω.Remark 4.
The left equation (ii) of Definition3 is equivalent to the following:
(19)∂u0∂x0=∂f0¯∂z0¯,∂f0¯∂x0=-∂u0∂z0.
The equations in (19) are called the corresponding Cauchy-Riemann system for f(z) in 𝕋. The right equation (ii) of Definition 3 is equivalent to (19). When the function f(z)=u0(x0,z0¯)+i^f0¯(x0,z0¯) is a L-regular function on Ω⊂ℝ3, simply we say that f(z) is a regular function on Ω⊂ℝ3. In this case, we often say that f(z) is a biregular function on Ω⊂ℝ3.Remark 5.
LetΩ be an open set in ℝ3 and let f(z) be a regular function on Ω. Then we can replace the corresponding Cauchy-Riemann system in ℝ3 as follows:
(20)∂u0∂x0=∂u1∂x1+∂u2∂x2,∂u1∂x2=∂u2∂x1,∂u0∂x1=-∂u1∂x0,∂u0∂x2=-∂u2∂x0,
where uj (j=0,1,2) are real-valued functions.Theorem 6.
LetΩ be an open set in ℝ3 and let f be a regular function on Ω. Then the derivative f′ of f defined by Df is
(21)f′=∂f∂x0=-i^∂f∂z0
on Ω.Proof.
By the definition of regular function with values in𝕋, we have
(22)Df=12{(∂u0∂x0+∂f0¯∂z0¯)+i^(∂f0¯∂x0-∂u0∂z0)}=∂u0∂x0+i^∂f0¯∂x0=∂f∂x0
on Ω. And
(23)Df=∂f0¯∂z0¯-i^∂u0∂z0=-i^(∂∂z0i^f0¯+∂u0∂z0)=-i^∂f∂z0
on Ω.Theorem 7.
LetΩ be an open set in ℝ3 and let f=u0+i^f0¯ be a function with values in 𝕋. Suppose that ∂f/∂x0 and ∂f/∂z0 exist and are continuous on Ω. If
(24)∂f∂x0=-i^∂f∂z0
on Ω, then f is regular on Ω.Proof.
Since∂f/∂x0=-i^(∂f/∂z0), we have
(25)∂f∂x0=∂u0∂x0+i^∂f0¯∂x0.
Hence, we have D*f=0 and then f is regular on Ω.Definition 8.
LetΩ be an open set in ℝ3. A function f=u0+i^f0¯ is said to be harmonic on Ω if all its components u0 and f0¯ of f are harmonic on Ω.Proposition 9.
LetΩ be an open set in ℝ3. If the function f is regular on Ω, then f is harmonic on Ω.Proof.
Sincef is regular function on Ω, we have
(26)DD*f0¯=14{(∂∂x0∂f0¯∂x0+∂∂z0¯∂f0¯∂z0)0000+i^(∂∂x0∂f0¯∂z0-∂∂z0∂f0¯∂x0)}=0.
Similarly, we can prove that DD*u0=0. So, we obtain the result.Proposition 10.
LetΩ be an open set in ℝ3 and let f=u0+i^f0¯ and g=v0+i^g0¯ be regular functions on Ω. Then the following properties hold:(i)
fαis regular onΩ, ifαis any ternary constant;(ii)
αfis not regular onΩ, ifαis any ternary constant;(iii)
f±gis regular onΩ;(iv)
fgis not regular onΩ. Moreover, ifgis a real-valued function, thenfgis regular onΩ.Proof.
It is sufficient to show the second condition of Definition3.(i)
Letα be a ternary constant with α=a0+i^α0¯, where
(27)α0=c1a1+c2a2c12+c22+c2a1-c1a2c12+c22e1e2
and a0, a1, a2, c1, and c2 are real numbers. Then the equation
(28)D*(fα)=12(∂∂x0+i^∂∂z0)×{(u0a0-f0α0¯)+i^(u0α0¯+f0¯a0)}=12((∂u0∂x0a0-∂f0∂x0α0¯-∂u0∂z0¯α0¯-∂f0¯∂z0¯a0)00000+i^(∂u0∂x0α0¯+∂f0¯∂x0a0+∂u0∂z0a0-∂f0∂z0α0¯))=0.
Hence, fα is regular on Ω.(ii)
Since(29)D*(αf)=12(∂∂x0+i^∂∂z0)×{(a0u0-α0f0¯)+i^(a0f0¯+α0¯u0)}=12((a0∂u0∂x0-α0∂f0¯∂x0-α0¯∂f0∂x0-∂u0∂z0α0)00000+i^(a0∂f0¯∂x0+α0¯∂u0∂x0+a0∂u0∂z0-α0∂f0¯∂z0)α0∂f0¯∂z0))
is not zero, αf is not always regular on Ω.(iii)
Since(30)D*(f±g)=12(∂∂x0+i^∂∂z0){(u0±v0)+i^(f0¯±g0¯)}=12((∂u0∂x0±∂v0∂x0-∂f0¯∂z0¯∓∂g0¯∂z0¯)00000+i^(∂u0∂z0±∂v0∂z0+∂f0¯∂x0a0±∂g0¯∂x0))=0,f±g is regular on Ω.(iv)
Since(31)D*(fg)=12(∂∂x0+i^∂∂z0)×{(u0v0-f0g0¯)+i^(u0g0¯+f0¯v0)}=12((∂u0∂x0-∂f0¯∂z0¯)v0+u0(∂v0∂x0-∂g0¯∂z0¯)00000-(∂f0∂x0+∂u0∂z0¯)g0¯-(f0∂g0¯∂x0+f0¯∂v0∂z0¯)00000+i^{(∂u0∂x0-∂f0∂z0)g0¯+u0(∂g0¯∂x0+∂v0∂z0)000000000+(∂f0¯∂x0+∂u0∂z0)v0000000000+(f0¯∂v0∂x0-f0∂g0¯∂z0)}(∂u0∂x0-∂f0¯∂z0¯))=12(-(f0∂g0¯∂x0+f0¯∂v0∂z0¯)+i^(f0¯∂v0∂x0-f0∂g0¯∂z0))
is not zero, fg is not always regular on Ω.Theorem 11.
LetΩ be an open set in ℝ3 and let f and g be regular functions on Ω. Then we have the following equations:
(32)2D*(fg)=(D*f)g+f∂g∂x0+i^(u0∂g∂z0+i^f0¯∂g∂z0¯).(33)2D(fg)=(Df)g+f∂g∂x0-i^(u0∂g∂z0+i^f0¯∂g∂z0¯).Proof.
From the proof of Proposition10, we have the following equations:
(34)2D*(fg)={(∂u0∂x0-∂f0¯∂z0¯)+i^(∂f0¯∂x0+∂u0∂z0)}(v0+i^g0¯)-(f0∂g0¯∂x0+f0¯∂v0∂z0¯)+i^(f0¯∂v0∂x0-f0∂g0¯∂z0)={(∂u0∂x0-∂f0¯∂z0¯)+i^(∂f0¯∂x0+∂u0∂z0)}(v0+i^g0¯)+u0(∂v0∂x0-∂g0¯∂z0¯)-(f0∂g0¯∂x0+f0¯∂v0∂z0¯)+i^u0(∂g0¯∂x0+∂v0∂z0)+i^(f0¯∂v0∂x0-f0∂g0¯∂z0)=(D*f)g+f∂g∂x0+i^(u0∂g∂z0+i^f0¯∂g∂z0¯).
Similarly, we can prove (33).
We let(35)k=e1e212dz0∧dz0¯+e2αdx0∧dz0¯-e1βdx0∧dz0¯.Theorem 12.
LetΩ be an open set in ℝ3 and U be any domain in Ω with smooth boundary bU such that U⊂Ω. If f=u0+i^f0¯ is a regular function on Ω, then
(36)∫bUkf=0,
where kf is the ternary product of the form k on the function f(z).Proof.
Since the functionf=u0+e1αf0¯+e2βf0¯ exists, we have
(37)kf=(e1e212u0-e212αf0¯+e112βf0¯)dz0∧dz0¯+(e2αu0-e1βu0)dx0∧dz0¯+(-e1e2α2f0¯-e1e2β2f0¯)dx0∧dz0.
Then
(38)d(kf)=e1e2(∂u0∂x0-α2∂f0¯∂z0¯-β2∂f0¯∂z0¯)dV+e2(-α∂f0¯∂x0-α∂u0∂z0)dV+e1(β∂f0¯∂x0+β∂u0∂z0)dV+(-αβ∂f0¯∂z0¯+αβ∂f0¯∂z0¯)dV={e1e2(∂u0∂x0-∂f0¯∂z0¯)-e2α(∂f0¯∂x0+∂u0∂z0)000+e1β(∂f0¯∂x0+∂u0∂z0)}dV,
where dV=dx0∧dz0∧dz0¯ in U, and by the corresponding Cauchy-Riemann system for f(z) in 𝕋, d(kf)=0. By Stokes theorem, we obtain the result.Remark 13.
Since(39)(i^z0¯)k={(-1)k/2(|z0|)k,k:even(-1)[k/2]i^(|z0|)k-1z0¯,k:odd,
we have
(40)zn=∑k=0n‍α(k)x0n-k|z0|[k/2]z0¯δk,
where
(41)α(k)={(nk)(-1)k/2,k:even(nk)(-1)[k/2]i^,k:odd,δk={0,k:even1,k:odd.
And [k/2] is the greatest integer that is less than or equal to k/2.Theorem 14.
Letf be a homogeneous polynomial of degree n with respect to the variables x0 and z0¯. If f is regular on Ω, then
(42)f(z)=1n!∂nf(z)∂x0nzn,(43)f(z)=(-i^)n1n!∂nf(z)∂z0n-r∂z0¯rzn,
where r is a nonnegative integer.Proof.
Sincef(z) is a homogeneous polynomial, then
(44)f(z)=1n∂f(z)∂x0z.
Also, since ∂f(z)/∂x0 is a homogeneous polynomial of degree n-1, we have
(45)∂f(z)∂x0=1n-1∂2f(z)∂x02z.
Then we have
(46)f(z)=1n(n-1)∂2f(z)∂x02z2.
Continuing this process, we can get the result (42). Similarly, we obtain the result (43).


## 4. Properties of Regular Functions with Values in𝕋(ℂ)

We define the number system(47)𝕋(ℂ)={z∣z=i^γ(z1-e1e2z2)},
where z1=x1-(1/2)e1x0 and z2=x2-(1/2)e2x0.The non-commutative multiplication of two numbersz=i^γ(z1-e1e2z2) and w=i^γ(w1-e1e2w2) is defined by
(48)zw=-{(z1w1+z2w2)+e1e2(z2¯w1-z1¯w2)},wz=-{(w1z1+w2z2)+e1e2(w2¯z1-w1¯z2)}.The conjugate numberz* of z in 𝕋(ℂ) is given by the following:
(49)z*=-i^γ(z1¯-e1e2z2¯).
And the norm |z| of z and the inverse z-1 of z are given by the following forms:
(50)|z|=zz*=z*z|z|=(z1z1¯+z2z2¯)+e1e2(z2¯z1¯-z1¯z2¯)|z|=∑j=02‍xj2,z-1=z*|z|2(z≠0).We consider the following differential operators:(51)D=-12i^γ(Dz1-e1e2Dz2),D*=12i^γ(Dz1¯-e1e2Dz2¯),
where
(52)Dz1=12e1∂∂x0+∂∂x1,Dz2=12e2∂∂x0+∂∂x2.
Then the Laplacian operator is
(53)4Δ≔DD*=D*D=∑j=02‍∂2∂xj2.LetG be an open set in ℂ2. The function f(z) that is defined by the following form in G with values in 𝕋(ℂ):
(54)f:G→𝕋(ℂ)
satisfies
(55)z=(z1,z2)∈G⟼f(z)=f(z1,z2)=i^γ(f1(z1,z2)-e1e2f2(z1,z2)),
where f1=u1-(1/2)e1u0 and f2=u2-(1/2)e2u0 are complex-valued functions with values in 𝕋(ℂ) and uj (j=0,1,2) are real-valued functions.Remark 15.
The operatorsD and D* act for a function f(z) on 𝕋(ℂ) as follows:
(56)Df=-i^2{(Dz1f1+Dz2f2)+e1e2(Dz2¯f1-Dz1¯f2)},D*f=i^2{(Dz1¯f1+Dz2¯f2)+e1e2(Dz2f1-Dz1f2)}.We define a commutative multiplication of two numbersz=i^γ(z1-e1e2z2) and w=i^γ(w1-e1e2w2) by
(57)z⊙w=w⊙z=12(zw+wz)=12i^2{(z1w1+z2w2+w1z1+w2z2)0000000+e1e2(z2¯w1-z1¯w20000000000000+w2¯z1-w1¯z2)}.Remark 16.
The operatorsD and D* act for a function f(z) on 𝕋(ℂ) as follows:
(58)D⊙f=12(Df+fD)={12(Dz1f1+Dz2f2)000+12e1e2(f2¯Dz2¯f1-Dz1¯f200000000000+f2¯Dz1-f1¯Dz2)12}={12(Dz1f1+Dz2f2+12∂u0∂x0)000+12e1e2(f1¯Dz2¯f1-Dz1¯f200000000000+Dz1f2¯-Dz2f1¯)12∂u0∂x0)},D*⊙f=12(D*f+fD*)=-{12(Dz1¯f1+Dz2¯f2)0000+12e1e2(Dz2f1-Dz1f20000000000000+f2¯Dz1¯-f1¯Dz2¯)12}=-{(Dz1¯f1+Dz2¯f2-12∂u0∂x0)0000+12e1e2(Dz2f1-Dz1f2000000000000+Dz1¯f2¯-Dz2¯f1¯)∂u0∂x0)}.Definition 17.
LetG be a domain in ℂ2. A function f=i^γ(f1-e1e2f2) is said to be dot-regular in G if the following two conditions are satisfied:(i)
f1 and f2 are differential functions in G,(ii)
D*⊙f=0 in G.Remark 18.
The above equation (ii) of Definition17 is equivalent as follows:
(59)Dz1¯f1+Dz2¯f2=12∂u0∂x0,Dz2f1-Dz1f2=Dz2¯f1¯-Dz1¯f2¯.Theorem 19.
LetG be an open set in ℂ2 and let f be a dot-regular function on G. Then the derivative f′ of f defined by D⊙f is
(60)f′=2i^γ(Dz1¯-Dz1)f=2e1(Dz1¯-Dz1)f,f′=-2i^γ(Dz2-Dz2¯)f=2e2(Dz2¯-Dz2)f.Proof.
By the definition of a dot-regular function with values in𝕋(ℂ), we have
(61)D⊙f=(Dz1¯f1+Dz2¯f2+e1∂u1∂x0+e2∂u2∂x0+32∂u0∂x0)+12e1e2(∂u2∂x0Dz2f1-Dz1f2+Dz1¯f2¯00000000000-Dz2¯f1¯-2e2∂u1∂x0+2e1∂u2∂x0)=2i^γ(Dz1¯-Dz1)f
on G. And, similarly, we have
(62)D⊙f=-2i^γ(Dz2-Dz2¯)f
on G.



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