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

Abstract

We define a new modified basis 𝑖̂ which is an association of two bases, 𝑒₁ and 𝑒₂. We give an expression of the form 𝑧=𝑥₀+𝑖𝑧₀, 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.

1. Introduction

The noncommutative three-dimensional real field ℝ³ of the hypercomplex numbers is called a ternary number system 𝕋. The quaternions are represented by the form 𝑧=∑³_𝑗=0𝑒_𝑗𝑥_𝑗, where 𝑥_𝑗 (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 basis 𝑖̂ and give some properties of regular functions with values in 𝕋. Also, we represent the corresponding Euler’s formula for the form 𝑧=𝑥₀+𝑖𝑧̂₀ and investigate calculating rules for regular functions in Clifford analysis. We research new representations of Fueter variables in reduced quaternions with 𝑖̂ and some characteristics of regularity of functions on the Fueter variable system.

2. Preliminaries

The ternary number system 𝕋 is a three dimensional non-commutative and associative real field by three bases 𝑒₀, 𝑒₁, and 𝑒₂ with the following rules: 𝑒²₁=𝑒²₂=−1,𝑒₁𝑒₂=−𝑒₂𝑒₁,𝑒₀=𝑒₀,𝑒_𝑗=−𝑒_𝑗𝑗=1,2. (1) 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: 𝑖̂∶=𝑎𝑒₁+𝑏𝑒₂𝑎²+𝑏²=𝛼𝑒₁+𝛽𝑒₂with𝑖²̂=−1, (2) where 𝛼∶=𝑎/𝑎²+𝑏², 𝛽∶=𝑏/𝑎²+𝑏², and 𝑎, 𝑏 are real numbers except both zeros.

The number of the skew field 𝕋 is 𝑧=𝑥₀+𝑒₁𝑥₁+𝑒₂𝑥₂=𝑥₀+𝑖𝑧̂₀, (3) where 𝑥_𝑗 (𝑗=0,1,2) are real variables, 𝑧₀=𝛾(𝑥₁−𝑥₂𝑒₁𝑒₂), and 𝛾∶=𝛼+𝛽𝑒₁𝑒₂.

We define the ternary number system 𝕋∶=𝑧∣𝑧=𝑥₀+𝑖𝑧̂₀. (4) The conjugate number 𝑧^∗ of 𝑧 in 𝕋 is given by the form: 𝑧^∗=𝑥₀−𝑖𝑧̂₀. (5) And the norm |𝑧| of 𝑧 and the inverse 𝑧⁻¹ of 𝑧 are given by the following forms: |𝑧|=√𝑧𝑧^∗=𝑥²₀+𝑧₀𝑧₀=²_𝑗=0𝑥²_𝑗,𝑧⁻¹=𝑧^∗|𝑧|²(𝑧≠0), (6) where 𝑧₀=𝛾(𝑥₁+𝑥₂𝑒₁𝑒₂) and 𝛾=𝛼−𝛽𝑒₁𝑒₂.

We define the addition and multiplication of two ternary numbers 𝑧=𝑥₀+𝑖𝑧̂₀ and 𝑤=𝑦₀+𝑖𝑤̂₀ as follows: 𝑧+𝑤=𝑥₀+𝑦₀+𝑖̂𝑧₀+𝑤₀,𝑧𝑤=𝑥₀𝑦₀−𝑧₀𝑤₀+𝑖̂𝑥₀𝑤₀+𝑧₀𝑦₀. (7)

Theorem 1. Let 𝑧 be an arbitrary number in 𝕋. Then the corresponding Euler formula for 𝑧 is 𝑒^𝑧=𝑒^𝑥0cos𝑧₀+𝑧₀𝑧₀𝑖̂sin𝑧₀. (8) Moreover, taking logarithms of both sides, one obtains the equation as follows: |𝑧|+𝑧ln𝑧=ln⁰𝑧₀𝑖ĉos⁻¹𝑥₀|𝑧|. (9)

Proof. For the number 𝑧=𝑥₀+𝑖𝑧̂₀ in 𝕋, we get |𝑖𝑧̂₀|=|𝑧₀|=|𝑧₀| and ((𝑧₀/|𝑧₀|)𝑖)̂²=−1. Then, 𝑒^𝑧=𝑒^𝑥0+𝑖̂_𝑧0=𝑒^𝑥0𝑒⁽𝑖̂_𝑧0/|𝑖̂_𝑧0|)|𝑖̂_𝑧0|=𝑒^𝑥0cos𝑧₀+𝑧₀𝑧₀𝑖̂sin𝑧₀. (10) From |𝑧|𝑥𝑧=⁰|𝑧|+𝑧₀𝑧₀𝑖̂𝑧₀|𝑧|=|𝑧|coscos⁻¹𝑥₀|𝑧|+𝑧₀𝑧₀𝑖̂sincos⁻¹𝑥₀|𝑧|, (11) we have |𝑧|+𝑧ln𝑧=ln⁰𝑧₀𝑖ĉos⁻¹𝑥₀|𝑧|. (12)
We consider the following differential operators: 1𝐷∶=2²_𝑗=0𝑒_𝑗𝜕𝜕𝑥_𝑗=12𝜕𝜕𝑥₀−𝑖̂𝜕𝜕𝑧₀,𝐷^∗=12²_𝑗=0𝑒_𝑗𝜕𝜕𝑥_𝑗=12𝜕𝜕𝑥₀+𝑖̂𝜕𝜕𝑧₀, (13) where 𝜕/𝜕𝑧₀=𝛾(𝜕/𝜕𝑥₁−𝑒₁𝑒₂(𝜕/𝜕𝑥₂)) and 𝜕/𝜕𝑧₀=𝛾(𝜕/𝜕𝑥₁+𝑒₁𝑒₂(𝜕/𝜕𝑥₂)). Then the Laplacian operator is 4Δ∶=𝐷𝐷^∗=𝐷^∗𝜕𝐷=²𝜕𝑥²₀+𝜕²𝜕𝑧₀𝜕𝑧₀=²_𝑗=0𝜕²𝜕𝑥²_𝑗. (14)
Let Ω be an open set in ℝ³. The function 𝑓(𝑧) that is defined by the following form in Ω with values in 𝕋: 𝑓∶Ω⟶𝕋 (15) satisfies 𝑧=𝑥₀,𝑧₀∈Ω⟼𝑓(𝑧)=𝑢₀𝑥₀,𝑧₀+𝑖𝑓̂₀𝑥₀,𝑧₀∈𝕋, (16) where 𝑢_𝑗 (𝑗=0,1,2) are real-valued functions and 𝑓₀=𝛾𝑢₁+𝑢₂𝑒₁𝑒₂,𝑓₀=𝛾𝑢₁−𝑢₂𝑒₁𝑒₂ (17) are complex-valued functions with values in 𝕋.

Remark 2. The operators 𝐷 and 𝐷^∗ act for the function 𝑓(𝑧) on 𝕋 as follows: 1𝐷𝑓=2𝜕𝑢₀𝜕𝑥₀+𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝑓₀𝜕𝑥₀−𝜕𝑢₀𝜕𝑧₀,𝐷^∗1𝑓=2𝜕𝑢₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀,1𝑓𝐷=2𝜕𝑢₀𝜕𝑥₀+𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝑓₀𝜕𝑥₀−𝜕𝑢₀𝜕𝑧₀,𝑓𝐷^∗=12𝜕𝑢₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀. (18)

3. Properties of Regular Functions with Values in 𝕋

Definition 3. Let Ω be an open set in ℝ³. A function 𝑓(𝑧)=𝑢₀(𝑥₀,𝑧₀)+𝑖𝑓̂₀(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 Definition 3 is equivalent to the following: 𝜕𝑢₀𝜕𝑥₀=𝜕𝑓₀𝜕𝑧₀,𝜕𝑓₀𝜕𝑥₀=−𝜕𝑢₀𝜕𝑧₀. (19) 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)+𝑖𝑓̂₀(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. Let Ω 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: 𝜕𝑢₀𝜕𝑥₀=𝜕𝑢₁𝜕𝑥₁+𝜕𝑢₂𝜕𝑥₂,𝜕𝑢₁𝜕𝑥₂=𝜕𝑢₂𝜕𝑥₁,𝜕𝑢₀𝜕𝑥₁=−𝜕𝑢₁𝜕𝑥₀,𝜕𝑢₀𝜕𝑥₂=−𝜕𝑢₂𝜕𝑥₀, (20) where 𝑢_𝑗 (𝑗=0,1,2) are real-valued functions.

Theorem 6. Let Ω 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 𝑓^′=𝜕𝑓𝜕𝑥₀=−𝑖̂𝜕𝑓𝜕𝑧₀ (21) on Ω.

Proof. By the definition of regular function with values in 𝕋, we have 1𝐷𝑓=2𝜕𝑢₀𝜕𝑥₀+𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝑓₀𝜕𝑥₀−𝜕𝑢₀𝜕𝑧₀=𝜕𝑢₀𝜕𝑥₀+𝑖̂𝜕𝑓₀𝜕𝑥₀=𝜕𝑓𝜕𝑥₀ (22) on Ω. And 𝜕𝐷𝑓=𝑓₀𝜕𝑧₀−𝑖̂𝜕𝑢₀𝜕𝑧₀=−𝑖̂𝜕𝜕𝑧₀𝑖𝑓̂₀+𝜕𝑢₀𝜕𝑧₀=−𝑖̂𝜕𝑓𝜕𝑧₀ (23) on Ω.

Theorem 7. Let Ω be an open set in ℝ³ and let 𝑓=𝑢₀+𝑖𝑓̂₀ be a function with values in 𝕋. Suppose that 𝜕𝑓/𝜕𝑥₀ and 𝜕𝑓/𝜕𝑧₀ exist and are continuous on Ω. If 𝜕𝑓𝜕𝑥₀=−𝑖̂𝜕𝑓𝜕𝑧₀ (24) on Ω, then 𝑓 is regular on Ω.

Proof. Since 𝜕𝑓/𝜕𝑥₀=−𝑖(̂𝜕𝑓/𝜕𝑧₀), we have 𝜕𝑓𝜕𝑥₀=𝜕𝑢₀𝜕𝑥₀+𝑖̂𝜕𝑓₀𝜕𝑥₀. (25) Hence, we have 𝐷^∗𝑓=0 and then 𝑓 is regular on Ω.

Definition 8. Let Ω be an open set in ℝ³. A function 𝑓=𝑢₀+𝑖𝑓̂₀ is said to be harmonic on Ω if all its components 𝑢₀ and 𝑓₀ of 𝑓 are harmonic on Ω.

Proposition 9. Let Ω be an open set in ℝ³. If the function 𝑓 is regular on Ω, then 𝑓 is harmonic on Ω.

Proof. Since 𝑓 is regular function on Ω, we have 𝐷𝐷^∗𝑓₀=14𝜕𝜕𝑥₀𝜕𝑓₀𝜕𝑥₀+𝜕𝜕𝑧₀𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝜕𝑥₀𝜕𝑓₀𝜕𝑧₀−𝜕𝜕𝑧₀𝜕𝑓₀𝜕𝑥₀=0. (26) Similarly, we can prove that 𝐷𝐷^∗𝑢₀=0. So, we obtain the result.

Proposition 10. Let Ω be an open set in ℝ³ and let 𝑓=𝑢₀+𝑖𝑓̂₀ and 𝑔=𝑣₀+𝑖𝑔̂₀ be regular functions on Ω. Then the following properties hold:(i)𝑓𝛼 is regular on Ω, if 𝛼 is any ternary constant;(ii)𝛼𝑓 is not regular on Ω, if 𝛼 is any ternary constant;(iii)𝑓±𝑔 is regular on Ω;(iv)𝑓𝑔 is not regular on Ω. Moreover, if 𝑔 is a real-valued function, then 𝑓𝑔 is regular on Ω.

Proof. It is sufficient to show the second condition of Definition 3.(i)Let 𝛼 be a ternary constant with 𝛼=𝑎₀+𝑖𝛼̂₀, where 𝛼₀=𝑐₁𝑎₁+𝑐₂𝑎₂𝑐²₁+𝑐²₂+𝑐₂𝑎₁−𝑐₁𝑎₂𝑐²₁+𝑐²₂𝑒₁𝑒₂ (27) and 𝑎₀, 𝑎₁, 𝑎₂, 𝑐₁, and 𝑐₂ are real numbers. Then the equation 𝐷^∗=1𝑓𝛼2𝜕𝜕𝑥₀+𝑖̂𝜕𝜕𝑧₀×𝑢₀𝑎₀−𝑓₀𝛼₀+𝑖̂𝑢₀𝛼₀+𝑓₀𝑎₀=12𝜕𝑢₀𝜕𝑥₀𝑎₀−𝜕𝑓₀𝜕𝑥₀𝛼₀−𝜕𝑢₀𝜕𝑧₀𝛼₀−𝜕𝑓₀𝜕𝑧₀𝑎₀+𝑖̂𝜕𝑢₀𝜕𝑥₀𝛼₀+𝜕𝑓₀𝜕𝑥₀𝑎₀+𝜕𝑢₀𝜕𝑧₀𝑎₀−𝜕𝑓₀𝜕𝑧₀𝛼₀=0. (28) Hence, 𝑓𝛼 is regular on Ω.(ii)Since 𝐷^∗=1𝛼𝑓2𝜕𝜕𝑥₀+𝑖̂𝜕𝜕𝑧₀×𝑎₀𝑢₀−𝛼₀𝑓₀+𝑖̂𝑎₀𝑓₀+𝛼₀𝑢₀=12𝑎₀𝜕𝑢₀𝜕𝑥₀−𝛼₀𝜕𝑓₀𝜕𝑥₀−𝛼₀𝜕𝑓₀𝜕𝑥₀−𝜕𝑢₀𝜕𝑧₀𝛼₀+𝑖̂𝑎₀𝜕𝑓₀𝜕𝑥₀+𝛼₀𝜕𝑢₀𝜕𝑥₀+𝑎₀𝜕𝑢₀𝜕𝑧₀−𝛼₀𝜕𝑓₀𝜕𝑧₀ (29) is not zero, 𝛼𝑓 is not always regular on Ω.(iii)Since 𝐷^∗=1𝑓±𝑔2𝜕𝜕𝑥₀+𝑖̂𝜕𝜕𝑧₀𝑢₀±𝑣₀+𝑖̂𝑓₀±𝑔₀=12𝜕𝑢₀𝜕𝑥₀±𝜕𝑣₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀∓𝜕𝑔₀𝜕𝑧₀+𝑖̂𝜕𝑢₀𝜕𝑧₀±𝜕𝑣₀𝜕𝑧₀+𝜕𝑓₀𝜕𝑥₀𝑎₀±𝜕𝑔₀𝜕𝑥₀=0, (30)𝑓±𝑔 is regular on Ω.(iv)Since 𝐷^∗=1𝑓𝑔2𝜕𝜕𝑥₀+𝑖̂𝜕𝜕𝑧₀×𝑢₀𝑣₀−𝑓₀𝑔₀+𝑖̂𝑢₀𝑔₀+𝑓₀𝑣₀=12𝜕𝑢₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀𝑣₀+𝑢₀𝜕𝑣₀𝜕𝑥₀−𝜕𝑔₀𝜕𝑧₀−𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀𝑔₀−𝑓₀𝜕𝑔₀𝜕𝑥₀+𝑓₀𝜕𝑣₀𝜕𝑧₀+𝑖̂𝜕𝑢₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀𝑔₀+𝑢₀𝜕𝑔₀𝜕𝑥₀+𝜕𝑣₀𝜕𝑧₀+𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀𝑣₀+𝑓₀𝜕𝑣₀𝜕𝑥₀−𝑓₀𝜕𝑔₀𝜕𝑧₀=12−𝑓₀𝜕𝑔₀𝜕𝑥₀+𝑓₀𝜕𝑣₀𝜕𝑧₀+𝑖̂𝑓₀𝜕𝑣₀𝜕𝑥₀−𝑓₀𝜕𝑔₀𝜕𝑧₀ (31) is not zero, 𝑓𝑔 is not always regular on Ω.

Theorem 11. Let Ω be an open set in ℝ³ and let 𝑓 and 𝑔 be regular functions on Ω. Then we have the following equations: 2𝐷^∗𝑓𝑔=𝐷^∗𝑓𝑔+𝑓𝜕𝑔𝜕𝑥₀+𝑖̂𝑢₀𝜕𝑔𝜕𝑧₀+𝑖𝑓̂₀𝜕𝑔𝜕𝑧₀.2𝐷𝑓𝑔=𝐷𝑓𝑔+𝑓𝜕𝑔𝜕𝑥₀−𝑖̂𝑢₀𝜕𝑔𝜕𝑧₀+𝑖𝑓̂₀𝜕𝑔𝜕𝑧₀. (32)(33)

Proof. From the proof of Proposition 10, we have the following equations: 2𝐷^∗𝑓𝑔=𝜕𝑢₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀𝑣₀+𝑖𝑔̂₀−𝑓₀𝜕𝑔₀𝜕𝑥₀+𝑓₀𝜕𝑣₀𝜕𝑧₀+𝑖̂𝑓₀𝜕𝑣₀𝜕𝑥₀−𝑓₀𝜕𝑔₀𝜕𝑧₀=𝜕𝑢₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀+𝑖̂𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀𝑣₀+𝑖𝑔̂₀+𝑢₀𝜕𝑣₀𝜕𝑥₀−𝜕𝑔₀𝜕𝑧₀−𝑓₀𝜕𝑔₀𝜕𝑥₀+𝑓₀𝜕𝑣₀𝜕𝑧₀+𝑖𝑢̂₀𝜕𝑔₀𝜕𝑥₀+𝜕𝑣₀𝜕𝑧₀+𝑖̂𝑓₀𝜕𝑣₀𝜕𝑥₀−𝑓₀𝜕𝑔₀𝜕𝑧₀=𝐷^∗𝑓𝑔+𝑓𝜕𝑔𝜕𝑥₀+𝑖̂𝑢₀𝜕𝑔𝜕𝑧₀+𝑖𝑓̂₀𝜕𝑔𝜕𝑧₀. (34) Similarly, we can prove (33).
We let 𝑘=𝑒₁𝑒₂12𝑑𝑧₀∧𝑑𝑧₀+𝑒₂𝛼𝑑𝑥₀∧𝑑𝑧₀−𝑒₁𝛽𝑑𝑥₀∧𝑑𝑧₀. (35)

Theorem 12. Let Ω be an open set in ℝ³ and 𝑈 be any domain in Ω with smooth boundary 𝑏𝑈 such that 𝑈⊂Ω. If 𝑓=𝑢₀+𝑖𝑓̂₀ is a regular function on Ω, then _𝑏𝑈𝑘𝑓=0, (36) where 𝑘𝑓 is the ternary product of the form 𝑘 on the function 𝑓(𝑧).

Proof. Since the function 𝑓=𝑢₀+𝑒₁𝛼𝑓₀+𝑒₂𝛽𝑓₀ exists, we have 𝑘𝑓=𝑒₁𝑒₂12𝑢₀−𝑒₂12𝛼𝑓₀+𝑒₁12𝛽𝑓₀𝑑𝑧₀∧𝑑𝑧₀+𝑒₂𝛼𝑢₀−𝑒₁𝛽𝑢₀𝑑𝑥₀∧𝑑𝑧₀+−𝑒₁𝑒₂𝛼²𝑓₀−𝑒₁𝑒₂𝛽²𝑓₀𝑑𝑥₀∧𝑑𝑧₀. (37) Then 𝑑𝑘𝑓=𝑒₁𝑒₂𝜕𝑢₀𝜕𝑥₀−𝛼²𝜕𝑓₀𝜕𝑧₀−𝛽²𝜕𝑓₀𝜕𝑧₀𝑑𝑉+𝑒₂𝜕−𝛼𝑓₀𝜕𝑥₀−𝛼𝜕𝑢₀𝜕𝑧₀𝑑𝑉+𝑒₁𝛽𝜕𝑓₀𝜕𝑥₀+𝛽𝜕𝑢₀𝜕𝑧₀+𝜕𝑑𝑉−𝛼𝛽𝑓₀𝜕𝑧₀𝜕+𝛼𝛽𝑓₀𝜕𝑧₀=𝑒𝑑𝑉₁𝑒₂𝜕𝑢₀𝜕𝑥₀−𝜕𝑓₀𝜕𝑧₀−𝑒₂𝛼𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀+𝑒₁𝛽𝜕𝑓₀𝜕𝑥₀+𝜕𝑢₀𝜕𝑧₀𝑑𝑉, (38) 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 𝑖𝑧̂₀^𝑘=(−1)^𝑘/2𝑧₀^𝑘(),𝑘:even−1^[𝑘/2]𝑖̂𝑧₀^𝑘−1𝑧₀,𝑘:odd, (39) we have 𝑧^𝑛=^𝑛_𝑘=0𝛼(𝑘)𝑥₀^𝑛−𝑘𝑧₀^[𝑘/2]𝑧₀^𝛿𝑘, (40) where 𝛼(𝑘)=𝑛𝑘(−1)^𝑘/2𝑛𝑘(),𝑘:even−1^[𝑘/2]̂𝛿𝑖,𝑘:odd,_𝑘=0,𝑘:even1,𝑘:odd. (41) And [𝑘/2] is the greatest integer that is less than or equal to 𝑘/2.

Theorem 14. Let 𝑓 be a homogeneous polynomial of degree 𝑛 with respect to the variables 𝑥₀ and 𝑧₀. If 𝑓 is regular on Ω, then 𝑓(𝑧)=1𝜕𝑛!^𝑛𝑓(𝑧)𝜕𝑥^𝑛₀𝑧^𝑛,𝑓(𝑧)=−𝑖̂^𝑛1𝜕𝑛!^𝑛𝑓(𝑧)𝜕𝑧₀^𝑛−𝑟𝜕𝑧₀^𝑟𝑧^𝑛, (42)(43) where 𝑟 is a nonnegative integer.

Proof. Since 𝑓(𝑧) is a homogeneous polynomial, then 𝑓(𝑧)=1𝑛𝜕𝑓(𝑧)𝜕𝑥₀𝑧. (44) Also, since 𝜕𝑓(𝑧)/𝜕𝑥₀ is a homogeneous polynomial of degree 𝑛−1, we have 𝜕𝑓(𝑧)𝜕𝑥₀=1𝜕𝑛−1²𝑓(𝑧)𝜕𝑥²₀𝑧. (45) Then we have 𝑓(𝑧)=1𝑛()𝜕𝑛−1²𝑓(𝑧)𝜕𝑥²₀𝑧². (46) Continuing this process, we can get the result (42). Similarly, we obtain the result (43).

4. Properties of Regular Functions with Values in 𝕋(ℂ)

We define the number system 𝕋(ℂ)=𝑧∣𝑧=𝑖𝛾̂𝑧₁−𝑒₁𝑒₂𝑧₂, (47) where 𝑧₁=𝑥₁−(1/2)𝑒₁𝑥₀ and 𝑧₂=𝑥₂−(1/2)𝑒₂𝑥₀.

The non-commutative multiplication of two numbers 𝑧=𝑖𝛾̂(𝑧₁−𝑒₁𝑒₂𝑧₂) and 𝑤=𝑖𝛾̂(𝑤₁−𝑒₁𝑒₂𝑤₂) is defined by 𝑧𝑤=−𝑧₁𝑤₁+𝑧₂𝑤₂+𝑒₁𝑒₂𝑧₂𝑤₁−𝑧₁𝑤₂,𝑤𝑧=−𝑤₁𝑧₁+𝑤₂𝑧₂+𝑒₁𝑒₂𝑤₂𝑧₁−𝑤₁𝑧₂. (48)

The conjugate number 𝑧^∗ of 𝑧 in 𝕋(ℂ) is given by the following: 𝑧^∗=−𝑖𝛾̂𝑧₁−𝑒₁𝑒₂𝑧₂. (49) And the norm |𝑧| of 𝑧 and the inverse 𝑧⁻¹ of 𝑧 are given by the following forms: |𝑧|=√𝑧𝑧^∗=√𝑧^∗𝑧=𝑧₁𝑧₁+𝑧₂𝑧₂+𝑒₁𝑒₂𝑧₂𝑧₁−𝑧₁𝑧₂=²_𝑗=0𝑥²_𝑗,𝑧⁻¹=𝑧^∗|𝑧|²(𝑧≠0). (50)

We consider the following differential operators: 1𝐷=−2𝑖𝛾̂𝐷_𝑧1−𝑒₁𝑒₂𝐷_𝑧2,𝐷^∗=12𝑖𝛾̂𝐷_𝑧1−𝑒₁𝑒₂𝐷_𝑧2, (51) where 𝐷_𝑧1=12𝑒₁𝜕𝜕𝑥₀+𝜕𝜕𝑥₁,𝐷_𝑧2=12𝑒₂𝜕𝜕𝑥₀+𝜕𝜕𝑥₂. (52) Then the Laplacian operator is 4Δ∶=𝐷𝐷^∗=𝐷^∗𝐷=²_𝑗=0𝜕²𝜕𝑥²_𝑗. (53)

Let 𝐺 be an open set in ℂ². The function 𝑓(𝑧) that is defined by the following form in 𝐺 with values in 𝕋(ℂ): 𝑓∶𝐺→𝕋(ℂ) (54) satisfies 𝑧=𝑧₁,𝑧₂∈𝐺⟼𝑓(𝑧)=𝑓𝑧₁,𝑧₂=𝑖𝛾̂𝑓₁𝑧₁,𝑧₂−𝑒₁𝑒₂𝑓₂𝑧₁,𝑧₂, (55) where 𝑓₁=𝑢₁−(1/2)𝑒₁𝑢₀ and 𝑓₂=𝑢₂−(1/2)𝑒₂𝑢₀ are complex-valued functions with values in 𝕋(ℂ) and 𝑢_𝑗 (𝑗=0,1,2) are real-valued functions.

Remark 15. The operators 𝐷 and 𝐷^∗ act for a function 𝑓(𝑧) on 𝕋(ℂ) as follows: 𝐷𝑓=−𝑖²̂𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂+𝑒₁𝑒₂𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂,𝐷^∗𝑓=𝑖²̂𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂+𝑒₁𝑒₂𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂. (56)

We define a commutative multiplication of two numbers 𝑧=𝑖𝛾̂(𝑧₁−𝑒₁𝑒₂𝑧₂) and 𝑤=𝑖𝛾̂(𝑤₁−𝑒₁𝑒₂𝑤₂) by 1𝑧⊙𝑤=𝑤⊙𝑧=2()=1𝑧𝑤+𝑤𝑧2𝑖²̂𝑧₁𝑤₁+𝑧₂𝑤₂+𝑤₁𝑧₁+𝑤₂𝑧₂+𝑒₁𝑒₂𝑧₂𝑤₁−𝑧₁𝑤₂+𝑤₂𝑧₁−𝑤₁𝑧₂. (57)

Remark 16. The operators 𝐷 and 𝐷^∗ act for a function 𝑓(𝑧) on 𝕋(ℂ) as follows: 1𝐷⊙𝑓=2=𝐷𝐷𝑓+𝑓𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂+12𝑒₁𝑒₂𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂+𝑓₂𝐷_𝑧1−𝑓₁𝐷_𝑧2=𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂+12𝜕𝑢₀𝜕𝑥₀+12𝑒₁𝑒₂𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂+𝐷_𝑧1𝑓₂−𝐷_𝑧2𝑓₁,𝐷^∗1⊙𝑓=2𝐷^∗𝑓+𝑓𝐷^∗=−𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂+12𝑒₁𝑒₂𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂+𝑓₂𝐷_𝑧1−𝑓₁𝐷_𝑧2=−𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂−12𝜕𝑢₀𝜕𝑥₀+12𝑒₁𝑒₂𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂+𝐷_𝑧1𝑓₂−𝐷_𝑧2𝑓₁. (58)

Definition 17. Let 𝐺 be a domain in ℂ². A function 𝑓=𝑖𝛾̂(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 Definition 17 is equivalent as follows: 𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂=12𝜕𝑢₀𝜕𝑥₀,𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂=𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂. (59)

Theorem 19. Let 𝐺 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 𝑓^′=2𝑖𝛾̂𝐷_𝑧1−𝐷_𝑧1𝑓=2𝑒₁𝐷_𝑧1−𝐷_𝑧1𝑓𝑓,_′=−2𝑖𝛾̂𝐷_𝑧2−𝐷_𝑧2𝑓=2𝑒₂𝐷_𝑧2−𝐷_𝑧2𝑓. (60)

Proof. By the definition of a dot-regular function with values in 𝕋(ℂ), we have 𝐷⊙𝑓=𝐷_𝑧1𝑓₁+𝐷_𝑧2𝑓₂+𝑒₁𝜕𝑢₁𝜕𝑥₀+𝑒₂𝜕𝑢₂𝜕𝑥₀+32𝜕𝑢₀𝜕𝑥₀+12𝑒₁𝑒₂𝐷_𝑧2𝑓₁−𝐷_𝑧1𝑓₂+𝐷_𝑧1𝑓₂−𝐷_𝑧2𝑓₁−2𝑒₂𝜕𝑢₁𝜕𝑥₀+2𝑒₁𝜕𝑢₂𝜕𝑥₀=2𝑖𝛾̂𝐷_𝑧1−𝐷_𝑧1𝑓 (61) on 𝐺. And, similarly, we have 𝐷⊙𝑓=−2𝑖𝛾̂𝐷_𝑧2−𝐷_𝑧2𝑓 (62) on 𝐺.

Acknowledgment

Kwang Ho Shon was supported by the Research Fund Program of Research Institute for Basic Sciences, Pusan National University, Korea, 2013, Project no. RIBS-PNU-2013-101.