Hyperholomorphic functions and hyper-conjugate harmonic functions of octonion variables
© Lim and Shon; licensee Springer 2013
Received: 6 November 2012
Accepted: 26 January 2013
Published: 1 March 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 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.
with in .
where , , and .
is the usual complex Laplacian Δ.
3 Some properties of hyperholomorphic functions on
where and , , and are complex-valued functions.
() 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 .
are satisfied, the function is a hyperholomorphic function in Ω. The equations in (3) are the corresponding o-Cauchy-Riemann equations in .
We call that equations (4) are the condition of harmonicity.
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. □
for all and satisfying .
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.
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.
By the same method as the proof of Theorem 3.6, our result is proved. □
where τg is the octonion product of the form τ on the function .
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.
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