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Hyperholomorphic functions and hyper-conjugate harmonic functions of octonion variables
Journal of Inequalities and Applications volume 2013, Article number: 77 (2013)
We represent hyperholomorphic functions on octonionic function theory and octonionic differential operators. We research the properties of hyperholomorphic functions, hyper-conjugate harmonic functions and the integral calculus of hyperholomorphic functions by octonion forms.
MSC:32A99, 30G35, 32W50, 11E88.
The octonions in Clifford algebra are a normed division algebra with eight dimensions over the real numbers larger than the quaternions. The octonions are non-commutative and non-associative but satisfy a weaker form of associativity. The octonions were discovered in 1843 by John T. Graves and constructed in 1845 by A. Cayley. They are referred to as Cayley numbers or the Cayley algebra. The octonions have been applied in fields such as string theory, special theory of relativity and quantum theory. Dentoni and Sce  gave a definition of octonionic regular functions and several properties of octonionic regular functions in 1973.
In 2004 and 2006, Kajiwara, Li and Shon [2, 3] obtained some results for the regeneration in complex, quaternion and Clifford analysis, and for the inhomogeneous Cauchy-Riemann system of quaternion and Clifford analysis in ellipsoid.
In 2011, Koriyama and Nôno  gave three regularities (HK-holomorphy, HF-holomorphy, -holomorphy) of octonionic functions based on holomorphic mappings in a domain in . Naser  and Nôno [6, 7] gave some properties of quaternionic hyperholomorphic functions. For any complex harmonic function in a domain of holomorphy D in , we can find a function such that will be a function hyperholomorphic in D and the Cauchy theorem of hyperholomorphic functions in quaternion analysis. The aim of this paper is to define hyperholomorphic functions with octonion variables in and investigate the properties of the hyperholomorphic functions of octonion variables. We give the condition of harmonicity in . Then for any complex-valued functions and satisfying the condition of harmonicity in a pseudoconvex domain Ω in , we can find hyper-conjugate harmonic functions and , respectively, on Ω such that is a hyperholomorphic function on Ω. Also, we investigate the Cauchy theorem of hyperholomorphic functions in octonion analysis.
The field of octonions
is an eight-dimensional non-commutative and non-associative R-field generated by eight base elements , , , , , , and with the following non-commutative multiplication rules:
The element is the identities of and identifies the imaginary unit in the C-field of complex numbers. An octonion z given by (1) is regarded as , where , , and are complex numbers in C. Thus, we identify with .
For the equation in the complex plane C, the three solutions are −2, , in C.
In the octonion , the equation has solutions whose forms are as follows:
Then the equation satisfies . That is, , , , , , , , . This means that the equation has infinitely many solutions
with in .
For two octonions and , the inner product is defined as follows:
Also, the octonionic conjugation , the absolute value of z and an inverse of z in are defined, respectively, by
Thus, the octonion and the octonion conjugation have the following forms:
where , , and .
We use the following differential operators:
where , () are usual differential operators used in complex analysis. And we use the following octonionic differential operators:
is the usual complex Laplacian Δ.
3 Some properties of hyperholomorphic functions on
Let Ω be an open set in . The function is defined by the following form in Ω with value in :
where and , , and are complex-valued functions.
Definition 3.1 Let Ω be an open set in . A function is said to be L(R)-hyperholomorphic in Ω if the following two conditions are satisfied:
() are continuously differential functions in Ω, and
When we deal with an L-hyperholomorphic function in , for simplicity, we often say that is a hyperholomorphic function in .
Equation (2) is applied to as follows:
If the following equations:
are satisfied, the function is a hyperholomorphic function in Ω. The equations in (3) are the corresponding o-Cauchy-Riemann equations in .
Remark 3.2 We redefine equations (3) as follows:
We call that equations (4) are the condition of harmonicity.
Remark 3.3 We redefine equations (4) in as follows:
where , , and for real-valued functions ().
If the function is hyperholomorphic in an open set Ω in , then the functions , , and are of class in Ω.
If the function satisfies the condition of harmonicity in an open set Ω in , then the functions , , and are harmonic in Ω.
and the functions , and are proved by a similar method as in the proof of the case of . And, by (i), () are of class functions in Ω. □
Definition 3.5 Let be an open set with a boundary. Let , where ρ is in in a neighborhood of and grad on b Ω. Then Ω is pseudoconvex if
for all and satisfying .
Consider an automorphism γ:
of . A domain Ω in is said to be pseudoconvex with respect to the complex variables , if is a pseudoconvex domain of the space of four complex variables in the sense of complex analysis.
Theorem 3.6 Let Ω be a domain in , which is a pseudoconvex domain with respect to the complex variables and let and be complex-valued functions of class on Ω satisfying the condition of harmonicity (4). Then there exist hyper-conjugate harmonic functions and , respectively, of class on Ω such that is a hyperholomorphic function on Ω.
Proof We consider the 1-forms and the differential operator on :
We operate the operator δ from the left-hand side of the 1-forms and on :
By the condition of harmonicity (4), all coefficients vanish. From Hörmander , the δ-closed forms and of are δ-exact forms on . Since Ω is a pseudoconvex domain, there exist hyper-conjugate harmonic functions and of class on Ω with -closed forms and on Ω of are -exact -forms on Ω such that is a hyperholomorphic function on Ω (see Krantz ). □
Theorem 3.7 Let Ω be a domain in , which is a pseudoconvex domain with respect to the complex variables and let be a complex-valued function of class on Ω satisfying the condition of harmonicity (4). Then there exists a hyper-conjugate harmonic function of class on Ω such that is a hyperholomorphic function on Ω.
Proof We consider the 1-form and the differential operator on :
We operate the operator δ from the left-hand side of the 1-form ψ on :
By the same method as the proof of Theorem 3.6, our result is proved. □
Theorem 3.8 Let be a hyperholomorphic function in a domain G of and let
Then for any domain with smooth boundary b Ω,
where τg is the octonion product of the form τ on the function .
By the rule of octonion multiplications,
where , and by the condition of harmonicity (4), . By Stoke’s theorem, we have
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The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009646), and by the Research Fund Program of Research Institute for Basic Sciences, Pusan National University, Korea, 2012, Project No. RIBS-PNU-2012-101.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Lim, S.J., Shon, K.H. Hyperholomorphic functions and hyper-conjugate harmonic functions of octonion variables. J Inequal Appl 2013, 77 (2013). https://doi.org/10.1186/1029-242X-2013-77
- hyperholomorphic function
- Clifford analysis
- pseudoconvex domain