Dual Quaternion Functions and Its Applications

Abstract

A dual quaternion is associated with two quaternions that have basis elements 𝑒₀, 𝑒₁, 𝑒₂, 𝑒₃, and 𝜀. Dual numbers are often written in the form 𝑧=𝜁+𝜀𝜁^∗, where 𝜀 is the dual identity and has the properties 𝜀²=0(𝜀≠0). We research the properties of some regular functions with values in dual quaternion and give applications of the extension problem for dual quaternion functions.

1. Introduction

Let 𝒯 be the quaternion algebra constructed over a real anti-Euclidean quadratic four-dimensional vector space. Brackx [1], Deavours [2], and Sudbery [3] researched properties of theories of a quaternion function. Naser [4] gave properties of hyperholomorphic functions, and Nôno [5, 6] gave properties of various hyperholomorphic functions. They obtained basic theorems such as Cauchy Theorem, Morera’s Theorem, and Cauchy Integral Formula with respect to Clifford analysis. Also, we [7–10] have investigated certain properties of hyperholomorphic functions and some regular functions in Clifford analysis.

A dual quaternion algebra 𝒟ℋ is an ordered pair of quaternions and is constructed from real eight-dimensional vector spaces. A dual quaternion can be represented in the form 𝑧=𝜁+𝜀𝜁^∗, where 𝜁 and 𝜁^∗ are ordinary quaternions and 𝜀 is the dual symbol. The quaternion can represent only rotation, while the dual quaternion can do both rotation and translation. So, the dual quaternion is used in applications to 3D computer graphics, robotics, and computer vision. Kenwright [11] gave characteristics of dual quaternions; Pennestrì and Stefanelli [12] researched some properties using dual, and Kula and Yayli [13] investigated dual split quaternions and screw motion in Minkowski 3-space.

Son [14–16] gave the extension problem for the solutions of partial differential equations in 𝐑^𝑛 and it is generalized for the solutions of the Riesz system. In this paper, we give some regular functions with values in dual quaternions and research the extension problem for regular functions with values in dual quaternions. Also, we give some applications for these problems.

2. Preliminaries

We consider associated Pauli matrices 𝑒₀=1001,𝑒₁=,𝑒𝑖00−𝑖₂=01−10,𝑒₃=.0𝑖𝑖0 (1) Then the associated Pauli matrices satisfy the triple rule as follows: 𝑒²_𝑗=−1,𝑒_𝑗𝑒_𝑘+𝑒_𝑘𝑒_𝑗=−𝛿_𝑗𝑘𝑗,𝑘=1,2,3, (2) where 𝛿_𝑗𝑘 is Kronecker delta. And we let the dual symbol 𝜀=0100 (3) be a nonzero and satisfy 0𝜀=𝜀0=0, 1𝜀=𝜀1=𝜀, 𝜀²=0. The element 𝑒₀ is the identity and the element 𝜀 is the dual identity of 𝒯.

The dual quaternion algebra 𝒟ℋ is a noncommutative and associative one of the quaternion algebra. Then 𝒟ℋ∶=𝑧=³_𝑗=0𝑒_𝑗𝑥_𝑗+𝑒_𝑗𝑥^∗_𝑗𝜀∣𝑥_𝑗,𝑥^∗_𝑗=∈𝐑𝑗=0,1,2,3𝑧=𝜁+𝜁^∗𝜀∣𝜁,𝜁^∗∈𝒯≅𝒯×𝒯, (4) where 𝜁=∑³_𝑗=0𝑒_𝑗𝑥_𝑗, 𝜁^∗=∑³_𝑗=0𝑒_𝑗𝑥^∗_𝑗, and 𝑥^∗_𝑗 is a dual quaternion component of 𝑥_𝑗. We can identify 𝒟ℋ with 𝐂⁴. The numbers of the skew field 𝒟ℋ of dual quaternions are 𝑧=³_𝑗=0𝑒_𝑗𝑥_𝑗+𝑒_𝑗𝑥^∗_𝑗𝜀=³_𝑗=0𝑒_𝑗𝜉_𝑗=𝜉₀+𝑖𝜉₁𝜉₂+𝑖𝜉₃−𝜉₂+𝑖𝜉₃𝜉₀−𝑖𝜉₁,𝑤=³_𝑗=0𝑒_𝑗𝑦_𝑗+𝑒_𝑗𝑦^∗_𝑗𝜀=³_𝑗=0𝑒_𝑗𝜂_𝑗=𝜂₀+𝑖𝜂₁𝜂₂+𝑖𝜂₃−𝜂₂+𝑖𝜂₃𝜂₀−𝑖𝜂₁, (5) where 𝜉_𝑗=𝑥_𝑗+𝜀𝑥^∗_𝑗 and 𝜂_𝑗=𝑦_𝑗+𝜀𝑦^∗_𝑗(𝑗=0,1,2,3). The dual quaternion conjugate 𝑧^∗ of 𝑧 is 𝑧^∗=³_𝑗=0𝑒_𝑗𝑥_𝑗+𝑒_𝑗𝑥^∗_𝑗𝜀=³_𝑗=0𝑒_𝑗𝜉_𝑗=𝜉₀−𝑖𝜉₁−𝜉₂−𝑖𝜉₃𝜉₂−𝑖𝜉₃𝜉₀+𝑖𝜉₁, (6) where 𝑒_𝑗=−𝑒_𝑗. The absolute value |𝑧| of 𝑧 and the inverse 𝑧⁻¹ of 𝑧 are, respectively, |𝑧|=√𝑧𝑧^∗=³_𝑗=0𝜉²_𝑗,𝑧⁻¹=𝑧^∗|𝑧|²(𝑧≠0). (7)

Let Ω be an open subset of 𝐂²×𝐂² and let the dual quaternion function 𝑓∶Ω⟶𝒟ℋ (8) satisfy (𝑧)=𝑧∈Ω⟼𝑓_3𝑗=0𝑒_𝑗𝑓_𝑗𝜁,𝜁^∗=𝑓₀𝜁,𝜁^∗+𝑖𝑓₁𝜁,𝜁^∗𝑓₂𝜁,𝜁^∗+𝑖𝑓₃𝜁,𝜁^∗−𝑓₂𝜁,𝜁^∗+𝑖𝑓₃𝜁,𝜁^∗𝑓₀𝜁,𝜁^∗−𝑖𝑓₁𝜁,𝜁^∗∈𝒟ℋ, (9) where 𝑓_𝑗(𝜁,𝜁^∗)=𝑢_𝑗(𝜁,𝜁^∗)+𝜀𝑢^∗_𝑗(𝜁,𝜁^∗) and 𝑢_𝑗, 𝑢^∗_𝑗  (𝑗=0,1,2,3) are real-valued functions.

We use the following dual quaternion differential operators in 𝒟ℋ: 𝐷=³_𝑗=0𝑒_𝑗𝜕𝜕𝑞_𝑗=𝜕𝜕𝑞₀𝜕−𝑖𝜕𝑞₁−𝜕𝜕𝑞₂𝜕−𝑖𝜕𝑞₃𝜕𝜕𝑞₂𝜕−𝑖𝜕𝑞₃𝜕𝜕𝑞₀𝜕+𝑖𝜕𝑞₁, (10) and the dual quaternion conjugates differential operators 𝐷^∗=³_𝑗=0𝑒_𝑗𝜕𝜕𝑞_𝑗=𝜕𝜕𝑞₀𝜕+𝑖𝜕𝑞₁𝜕𝜕𝑞₂𝜕+𝑖𝜕𝑞₃−𝜕𝜕𝑞₂𝜕+𝑖𝜕𝑞₃𝜕𝜕𝑞₀𝜕−𝑖𝜕𝑞₁, (11) where 𝜕/𝜕𝑞_𝑗=𝜕/𝜕𝑥_𝑗+𝜀(𝜕/𝜕𝑥^∗_𝑗)(𝑗=0,1,2,3) and 𝑞_𝑗=𝑥_𝑗+(1/𝜀)𝑥^∗_𝑗. Then we have 𝐷𝐷^∗=³_𝑗=0𝜕²𝜕𝑞²_𝑗=Δ_𝑞. (12)

Definition 1. Let Ω be an open set in 𝐂²×𝐂². A function 𝑓(𝑧) is said to be 𝜀-regular in Ω if the following two conditions are satisfied:(a)𝑓_𝑗(𝑗=0,1,2,3) are continuously differential functions in Ω,(b)𝐷^∗𝑓(𝑧)=0 in Ω.

Definition 2. Let Ω be an open set in 𝐂²×𝐂². A function 𝑓(𝑧) is said to be 𝜀-biregular in Ω if the following two conditions are satisfied:(a)𝑓_𝑗(𝑗=0,1,2,3) are continuously differential functions in Ω, (b)𝐷^∗𝑓(𝑧)=0 and 𝑓(𝑧)𝐷^∗=0 in Ω.

The operators act for a function 𝑓(𝑧) on 𝒟ℋ as follows: 𝐷^∗𝑓(𝑧)=³_𝑗=0𝑒_𝑗𝜕𝜕𝑞_𝑗³_𝑗=0𝑒_𝑗𝑓_𝑗=𝐷^∗₁𝐷^∗₂𝐷^∗₃𝐷^∗₄, (13) where 𝐷^∗₁=𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−𝜕𝑓₂𝜕𝑞₂−𝜕𝑓₃𝜕𝑞₃+𝑖𝜕𝑓₁𝜕𝑞₀+𝜕𝑓₀𝜕𝑞₁+𝜕𝑓₃𝜕𝑞₂−𝜕𝑓₂𝜕𝑞₃,𝐷^∗₂=𝜕𝑓₂𝜕𝑞₀−𝜕𝑓₃𝜕𝑞₁+𝜕𝑓₀𝜕𝑞₂+𝜕𝑓₁𝜕𝑞₃+𝑖𝜕𝑓₃𝜕𝑞₀+𝜕𝑓₂𝜕𝑞₁−𝜕𝑓₁𝜕𝑞₂+𝜕𝑓₀𝜕𝑞₃,𝐷^∗₃=−𝜕𝑓₂𝜕𝑞₀+𝜕𝑓₃𝜕𝑞₁−𝜕𝑓₀𝜕𝑞₂−𝜕𝑓₁𝜕𝑞₃+𝑖𝜕𝑓₃𝜕𝑞₀+𝜕𝑓₂𝜕𝑞₁−𝜕𝑓₁𝜕𝑞₂+𝜕𝑓₀𝜕𝑞₃,𝐷^∗₄=𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−𝜕𝑓₂𝜕𝑞₂−𝜕𝑓₃𝜕𝑞₃+𝑖−𝜕𝑓₁𝜕𝑞₀−𝜕𝑓₀𝜕𝑞₁−𝜕𝑓₃𝜕𝑞₂+𝜕𝑓₂𝜕𝑞₃,𝑓(𝑧)𝐷^∗=³_𝑗=0𝑒_𝑗𝑓_𝑗³_𝑗=0𝑒_𝑗𝜕𝜕𝑞_𝑗=𝐷^∗₅𝐷^∗₆𝐷^∗₇𝐷^∗₈, (14) where 𝐷^∗₅=𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−𝜕𝑓₂𝜕𝑞₂−𝜕𝑓₃𝜕𝑞₃+𝑖𝜕𝑓₁𝜕𝑞₀+𝜕𝑓₀𝜕𝑞₁−𝜕𝑓₃𝜕𝑞₂+𝜕𝑓₂𝜕𝑞₃,𝐷^∗₆=𝜕𝑓₂𝜕𝑞₀+𝜕𝑓₃𝜕𝑞₁+𝜕𝑓₀𝜕𝑞₂−𝜕𝑓₁𝜕𝑞₃+𝑖𝜕𝑓₃𝜕𝑞₀−𝜕𝑓₂𝜕𝑞₁+𝜕𝑓₁𝜕𝑞₂+𝜕𝑓₀𝜕𝑞₃,𝐷^∗₇=−𝜕𝑓₂𝜕𝑞₀−𝜕𝑓₃𝜕𝑞₁−𝜕𝑓₀𝜕𝑞₂+𝜕𝑓₁𝜕𝑞₃+𝑖𝜕𝑓₃𝜕𝑞₀−𝜕𝑓₂𝜕𝑞₁+𝜕𝑓₁𝜕𝑞₂+𝜕𝑓₀𝜕𝑞₃,𝐷^∗₈=𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−𝜕𝑓₂𝜕𝑞₂−𝜕𝑓₃𝜕𝑞₃+𝑖−𝜕𝑓₁𝜕𝑞₀−𝜕𝑓₀𝜕𝑞₁+𝜕𝑓₃𝜕𝑞₂−𝜕𝑓₂𝜕𝑞₃. (15)

Remark 3. Equations (b) of Definition 2 are equivalent to the following system: 𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−𝜕𝑓₂𝜕𝑞₂−𝜕𝑓₃𝜕𝑞₃=0,𝜕𝑓₁𝜕𝑞₀=−𝜕𝑓₀𝜕𝑞₁,𝜕𝑓₃𝜕𝑞₂=𝜕𝑓₂𝜕𝑞₃,𝜕𝑓₂𝜕𝑞₀=−𝜕𝑓₀𝜕𝑞₂,𝜕𝑓₃𝜕𝑞₁=𝜕𝑓₁𝜕𝑞₃,𝜕𝑓₃𝜕𝑞₀=−𝜕𝑓₀𝜕𝑞₃,𝜕𝑓₂𝜕𝑞₁=𝜕𝑓₁𝜕𝑞₂. (16)

3. Extension Problem for the Dual Quaternion Functions

Definition 4. Let Ω be a domain in 𝐂^𝑛×𝐂^𝑛  (𝑛≥1). A function 𝑓(𝑧)=∑^𝑛−1_𝑗=0𝑒_𝑗𝑓_𝑗(𝑧) is said to be regular in Ω if 𝐷𝑓(𝑧)=0, (17) where 𝐷=∑^𝑛−1_𝑗=0𝑒_𝑗(𝜕/𝜕𝑞_𝑗) on Ω.

Theorem 5 (uniqueness theorem for regular functions). If two regular functions 𝑓 and 𝑔 in a domain Ω⊂𝐂^𝑛×𝐂^𝑛  (𝑛≥1) and coincide on a nonempty open set 𝐺⊂Ω, then 𝑓≡𝑔 in Ω.

Rocha-Chávez et al. [17] obtained the following remark.

Remark 6. For a regular function 𝑓 in the domain Ω⊂𝐂^𝑛×𝐂^𝑛(𝑛≥1) and a bounded domain 𝐺 with smooth boundary 𝑏𝐺, such that 𝐺⊂Ω, one has 𝑓(𝑧)=1𝑎_𝑛𝑏𝐺𝜁−𝑧𝜁−𝑧^𝑛𝑑𝜎_𝜁𝑓(𝜁),𝑧∈𝐺, (18) with 𝑎_𝑛 the area of the unit sphere in 𝐂^𝑛 and 𝑑𝜎_𝜁 a Clifford algebra valued differential form of order 𝑛−1.

Let Ω=Ω₁×Ω₂ be a domain in 𝐂⁴×𝐂^𝑛−4(𝑛≥5) where Ω₁ is a domain in 𝐂⁴(𝜉₀,𝜉₁,𝜉₂,𝜉₃) and Ω₂ is a domain in 𝐂^𝑛−4(𝜉₄,𝜉₅,…,𝜉_𝑛−1). Let 𝑈 be an open connected neighborhood of 𝑏Ω.

Proposition 7. If 𝑓(𝑧) is a regular function in 𝑈⊂𝐂⁴×𝐂^𝑛−4(𝑛≥5) which satisfies the condition 𝐷^∗𝑓(𝑧)=0, (19) then 𝑓(𝑧) can be extended continuously to a regular function in the whole domain of Ω. That is, there exists a regular function 𝑓(𝑧) in Ω such that 𝑓(𝑧)=𝑓(𝑧) in 𝑈.

Proof. By Remark 6 and the proof of the main extension theorem of Son [15], it is proved.

We consider the system of an extension of the system (16) 𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−⋯−𝜕𝑓_𝑛−1𝜕𝑞_𝑛−1=0,𝜕𝑓_𝑗𝜕𝑞₀=−𝜕𝑓₀𝜕𝑞_𝑗,𝜕𝑓_𝑙𝜕𝑞_𝑘=𝜕𝑓_𝑘𝜕𝑞_𝑙𝑗,𝑘,𝑙=1,…,𝑛−1, (20) where 𝑓(𝑧)={𝑓₀(𝑧),𝑓₁(𝑧),…,𝑓_𝑛−1(𝑧)} are the unknown functions.

By using the same technique as in Son [15], we have the following theorem.

Theorem 8. Let 𝑓(𝑧)={𝑓₀(𝑧),𝑓₁(𝑧),…,𝑓_𝑛−1(𝑧)} be a given 𝒞²-solution of the system (20) in 𝑈⊂𝐂⁴×𝐂^𝑛−4(𝑛≥5), which satisfies the system (16) in Remark 3. If the functions 𝑓₄(𝑧),𝑓₅(𝑧),…,𝑓_𝑛−1(𝑧) depend only on 𝜉₀,𝜉₁,𝜉₂,𝜉₃, and 𝑈 is an open neighborhood of the boundary of the domain Ω⊂𝐂⁴×𝐂^𝑛−4(𝑛≥5), then 𝑓(𝑧) can be extended to a solution of the system (20) in the whole domain of Ω.

Proof. Let the function 𝜙(𝑧) with values in Clifford algebra be defined by 𝜙(𝑧)=^𝑛−1_𝑗=0𝑓_𝑗𝑒_𝑗. (21) Then we have 𝐷𝜙(𝑧)=0 and 𝐷^∗𝜙(𝑧)=0. By Proposition 7, the result follows.

We consider the following system: ^𝑛−1_𝑗=0^𝑚−1_𝑘=0𝐻^(𝑃)_𝑗𝑘(𝑧)𝜕𝑓_𝑗𝜕𝑞_𝑘=0𝑃=1,…,𝑝, (22) where 𝐻^(𝑃)_𝑗𝑘(𝑧) are holomorphic functions and 𝑓={𝑓₀,𝑓₁,…,𝑓_𝑛−1} are the unknown functions of the system (22).

Let Ω be an open set in 𝐂^𝑛  (𝑛≥2) and let 𝐾 be a compact subset of Ω such that Ω⧵𝐾 is simply connected. We consider the system ^𝑛−1_𝑗=0^𝑚−1_𝑘=0𝐻^(𝑃)_𝑗𝑘(𝑧)𝜕𝜔_𝑗𝜕𝑞_𝑘=𝜑^(𝑃)_(𝑧)𝑃=1,…,𝑝, (23) where 𝜑^(𝑃)_(𝑧)∈𝒞^∞(Ω).

By using the same technique as in Son [16], we have the following theorem and example.

Theorem 9. If every 𝜑={𝜑⁽¹⁾,…,𝜑^(𝑝)}∈𝒞^∞(Ω) in the inhomogeneous system (23) has a solution 𝜔=𝜔₀,…,𝜔_𝑛−1∈𝒞^∞(Ω), (24) then every solution 𝑓 of (22) given in Ω⧵𝐾 can be extended to a solution of this system (23) in the whole domain of Ω.

Proof. This result follows from the theorem in [18, page 30].

Example 10. We give an application of Theorem 9 to the system (20) and recall the system (16) as follows: 𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−⋯−𝜕𝑓_𝑛−1𝜕𝑞_𝑛−1=0,𝜕𝑓_𝑗𝜕𝑞₀+𝜕𝑓₀𝜕𝑞_𝑗=0,𝜕𝑓_𝑙𝜕𝑞_𝑘−𝜕𝑓_𝑘𝜕𝑞_𝑙.=0𝑗,𝑘,𝑙=1,…,𝑛−1 (25)

Assume that 𝜕𝑓_𝑛−1𝜕𝑞_𝑛−1=0. (26) Then we have the following form: 𝜕𝑓₀𝜕𝑞₀−𝜕𝑓₁𝜕𝑞₁−⋯−𝜕𝑓_𝑛−2𝜕𝑞_𝑛−2=0,𝜕𝑓_𝑗𝜕𝑞₀+𝜕𝑓₀𝜕𝑞_𝑗=0,𝜕𝑓_𝑙𝜕𝑞_𝑘−𝜕𝑓_𝑘𝜕𝑞_𝑙.=0𝑗,𝑘,𝑙=1,…,𝑛−2 (27) The corresponding inhomogeneous system of (27) has the following form: 𝜕𝜔₀𝜕𝑞₀−𝜕𝜔₁𝜕𝑞₁−⋯−𝜕𝜔_𝑛−2𝜕𝑞_𝑛−2=𝜑,𝜕𝜔_𝑗𝜕𝑞₀+𝜕𝜔₀𝜕𝑞_𝑗=𝜑_𝑗,0,𝜕𝜔_𝑗𝜕𝑞_𝑘−𝜕𝜔_𝑘𝜕𝑞_𝑗=𝜑_𝑗,𝑘,𝜕𝜔_𝑘𝜕𝑞₀+𝜕𝜔₀𝜕𝑞_𝑘=𝜑_𝑘,0,𝜕𝜔_𝑗𝜕𝑞_𝑙−𝜕𝜔_𝑙𝜕𝑞_𝑗=𝜑_𝑗,𝑙,𝜕𝜔_𝑘𝜕𝑞_𝑙−𝜕𝜔_𝑙𝜕𝑞_𝑘=𝜑_𝑘,𝑙𝑗,𝑘,𝑙=1,…,𝑛−2, (28) where 𝜑,𝜑_𝑗,0,𝜑_𝑗,𝑘,𝜑_𝑘,0,𝜑_𝑗,𝑙,𝜑_𝑘,𝑙∈𝒞^∞(Ω). Then we can get the system from (28) as 𝜕𝜔_𝑘𝜕𝑞_𝑗=𝜕𝜔_𝑗𝜕𝑞_𝑘−𝜑_𝑗,𝑘,𝜕𝜔_𝑘𝜕𝑞₀=−𝜕𝜔₀𝜕𝑞_𝑘+𝜑_𝑘,0,𝜕𝜔_𝑙𝜕𝑞_𝑗=𝜕𝜔_𝑗𝜕𝑞_𝑙−𝜑_𝑗,𝑙,𝜕𝜔_𝑙𝜕𝑞_𝑘=𝜕𝜔_𝑘𝜕𝑞_𝑙−𝜑_𝑘,𝑙,𝜕²𝜔_𝑘𝜕𝑞_𝑗𝜕𝑞₀+𝜕𝜑_𝑗,𝑘𝜕𝑞₀𝜕=−²𝜔₀𝜕𝑞_𝑗𝜕𝑞_𝑘+𝜕𝜑_𝑗,0𝜕𝑞_𝑘,𝜕²𝜔_𝑗𝜕𝑞_𝑙𝜕𝑞_𝑘−𝜕𝜑_𝑗,𝑙𝜕𝑞_𝑘=𝜕²𝜔_𝑘𝜕𝑞_𝑗𝜕𝑞_𝑙−𝜕𝜑_𝑘,𝑙𝜕𝑞_𝑗. (29)(30) From (29), we have 𝜕𝜕𝑞_𝑗−𝜕𝜔₀𝜕𝑞_𝑘+𝜑_𝑘,0+𝜕𝜑_𝑗,𝑘𝜕𝑞₀𝜕=−²𝜔₀𝜕𝑞_𝑗𝜕𝑞_𝑘+𝜕𝜑_𝑗,0𝜕𝑞_𝑘,𝜕𝜕𝑞_𝑙𝜕𝜔_𝑘𝜕𝑞_𝑗+𝜑_𝑗,𝑘−𝜕𝜑_𝑗,𝑙𝜕𝑞_𝑘=𝜕²𝜔_𝑘𝜕𝑞_𝑗𝜕𝑞_𝑙−𝜕𝜑_𝑘,𝑙𝜕𝑞_𝑗. (31) Thus, we can have the system 𝜕𝜑_𝑘,0𝜕𝑞_𝑗+𝜕𝜑_𝑗,𝑘𝜕𝑞₀=𝜕𝜑_𝑗,0𝜕𝑞_𝑘,𝜕𝜑_𝑗,𝑘𝜕𝑞_𝑙−𝜕𝜑_𝑗,𝑙𝜕𝑞_𝑘=−𝜕𝜑_𝑘,𝑙𝜕𝑞_𝑗. (32) We let 𝜔=^𝑛−2_𝑗=0𝑒_𝑗𝜔_𝑗,𝑧=^𝑛−2_𝑗=0𝑒_𝑗𝜉_𝑗. (33) The system (29) has the form 𝐷𝜔=𝐹𝑧,𝜉_𝑛−1,𝜕𝜔₀𝜕𝑞_𝑘+𝜕𝜔_𝑘𝜕𝑞₀=𝜑_𝑘,0,𝜕𝜔_𝑗𝜕𝑞_𝑙−𝜕𝜔_𝑙𝜕𝑞_𝑗=𝜑_𝑗,𝑙,𝜕𝜔_𝑘𝜕𝑞_𝑙−𝜕𝜔_𝑙𝜕𝑞_𝑘=𝜑_𝑘,𝑙, (34)(35) where 𝐹𝑧,𝜉_𝑛−1=𝜑+^𝑛−2_𝑗=1𝑒_𝑗𝜑_𝑗,0+𝑛2−1^𝑛−2_𝛼=1^𝑛−2_{𝑗,𝑘=1𝑗,𝑘≠𝛼}𝑒_𝛼𝜑_𝑗,𝑘. (36) We put 𝐹₁∶=𝜑+^𝑛−2_𝑗=1𝑒_𝑗𝜑_𝑗,0,𝐹₂𝑛∶=2−1^𝑛−2_𝛼=1^𝑛−2_{𝑗,𝑘=1𝑗,𝑘≠𝛼}𝑒_𝛼𝜑_𝑗,𝑘. (37) From the system (28), we have 𝜔₀=𝜑+𝜕𝜔₁𝜕𝑞₁+𝜕𝜔₂𝜕𝑞₂+⋯+𝜕𝜔_𝑛−2𝜕𝑞_𝑛−2𝑑𝑞₀,𝜔_𝑘=𝜑_𝑘,0−𝜕𝜔₀𝜕𝑞_𝑘𝑑𝑞₀(𝑘=1,…,𝑛−2). (38) By the systems (29) and (38), we get 𝜔=𝐹𝑑𝑞₀+^𝑛−2_𝑗=1𝜕𝜔_𝑗𝜕𝑞_𝑗𝑑𝑞₀−^𝑛−2_𝑗=1𝑒_𝑗𝜕𝜔₀𝜕𝑞_𝑗𝑑𝑞₀−𝐹₂𝑑𝑞₀=𝐹𝑑𝑞₀−^𝑛−2_𝑗=1𝑒_𝑗𝜕𝜔𝜕𝑞_𝑗𝑑𝑞₀. (39) By Cauchy Integral Formula, 𝐹𝑍,𝜉_𝑛−1=1(2𝜋𝑖)^𝑛𝐹𝑍,𝜉_𝑛−1𝑍−𝜁𝑑𝑍, (40) where 𝑍=∑^𝑛−2_𝑘=0𝑒_𝑘𝑍_𝑘. Thus we have 1𝜔=(2𝜋𝑖)^𝑛𝐹𝑍,𝜉_𝑛−1𝑍−𝜁𝑑𝑍₀𝑑𝑍, (41) and 𝜔=0 when |𝜉_𝑛−1| is large enough. Also, {𝜔,0}={𝜔₀,𝜔₁,…,𝜔_𝑛−2,0} is a solution of the system (22) outside a compact set 𝐾 of Ω. From Theorem 5, it follows that 𝜔=0 is outside the compact set 𝐾 of Ω or 𝜔∈𝒞^∞(Ω). That is, 𝜔₀,𝜔₁,…,𝜔_𝑛−2∈𝒞^∞(Ω). It follows from the system (35) that 𝜔_𝑛−1=𝜕𝜔_𝑚−1𝜕𝑞_𝑛−1−𝜑_{𝑚−1,𝑛−1}𝑑𝑞_𝑚−1. (42) Since 𝜔_𝑚−1∈𝒞^∞(Ω), we get 𝜕𝜔_𝑚−1/𝜕𝑞_𝑛−1∈𝒞^∞(Ω). We can choose 𝜔_𝑛−1∈𝒞^∞(Ω) which satisfy the system (35). From (42), we find that 𝜕𝜔_𝑛−1𝜕𝑞_𝑃=𝜕²𝜔_𝑚−1𝜕𝑞_𝑃𝜕𝑞_𝑛−1−𝜕𝜑_{𝑚−1,𝑛−1}𝜕𝑞_𝑃𝑑𝑞_𝑚−1=𝜕𝜕𝑞_𝑛−1𝜕𝜔_𝑃𝜕𝑞_𝑚−1+𝜕𝜑_{𝑚−1,𝑃}𝜕𝑞_𝑛−1−𝜕𝜑_{𝑚−1,𝑛−1}𝜕𝑞_𝑃𝑑𝑞_𝑚−1=𝜕𝜕𝑞_𝑛−1𝜕𝜔_𝑃𝜕𝑞_𝑚−1−𝜕𝜑_{𝑃,𝑛−1}𝜕𝑞_𝑚−1𝑑𝑞_𝑚−1=𝜕𝜔_𝑃𝜕𝑞_𝑛−1−𝜑_{𝑃,𝑛−1}. (43) Hence, 𝜔_𝑛−1 satisfies the system (29). This means that the function 𝜔=(𝜔,𝜔_𝑛−1)=(𝜔₀,…,𝜔_𝑛−1) is a solution of the system (29) and 𝜔∈𝒞^∞(Ω).

Acknowledgment

The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013R1A1A2008978).