The Cauchy Integral Formula for Biregular Function in Octonionic Analysis

In this paper, we mainly study the Cauchy integral formula and mean value theorem for biregular function in octonionic analysis. Octonion is the extension of complex number to non-commutative and non-associative space. Because of the non-associative properties of multiplication, octonion plays an important role in wave equation, Yang-Mills equations, operator theory and so on. In recent years, octonion has become a hot topic for scholars at home and abroad and got many rich results, such as Fourier transform, Bergman kernel, Taylor series and its applications in quantum mechanics. On the basis of two Stokes theorems, we get Cauchy integral formula for biregular function in octonionic analysis by using the methods in dealing with the Cauchy integral formula for biregular function in Clifford analysis and regular function in octonionic analysis. As a direct result we also get the mean value theorem for biregular function in octonionic analysis. This will generalize the corresponding conclusion in complex analysis and Clifford analysis, and lays a solid foundation for the application of octonionic analysis in physics.


Introduction
Octonion is a generalization of quaternion to nonassociative algebra which has played a very important role in physical phenomena of black hole [1], supersymmetry and duality, extended supersymmetry [2], supergravity models etc [3][4][5][6][7]. Wang wei professor and his collaborators discussed the octonion Heisenber group [8]. Calin, O. and Chang, D. C. and Markina, I studied the geometric analysis on H-type groups related to division algebras [9]. In 1998, X. M. Li studied octonionic analysis [10]. Octonion has been widely studied by Baez [11]. Recently, Many experts and scholars are dedicated to octonionic analysis and obtain some results such as Cauchy integral formula for regular function, Hardy space, Bergman space [12][13][14][15]. H. Y. Wang and his collaborators studied the right inverse of Dirac in octonion space and generalized octonionic analysis to octonionic analysis of several variables [16][17]. J. X. Wang, X. M. Li described the octonion Bergman kernel for the unit ball [18]. In this paper, we will study regular function of two variables, called biregular function. But the octonions are neither commutative nor associative, which bring barriers to the study of the problems in biregular function. Therefore, we have biregular Cauchy integral formula and mean value theorem of octonions by use the associator to overcome the difficulties.

Octonion
The octonions O are the nonassociative, noncommutative, normed division algebra over the real generated by 1 7 e ,...,e [1], where the multiplication rules between the basis are given as follows [2,10]: e e + e e =-2 , , 1,...,7, For each x O ∈ , it can be written a  In order to illustrate the relationship between octonion multiplication, shown in Figure 1. Fano mnemonic where for simplicity we have used which consists of 7-points and 7-directed lines, then 7-points represent the standard basis for octonions.
Octonion multiplication are the nonassociative, noncommutative. But the subalgebra generated by any two elements is associative, namely, the octonions are alternative. So, for any , , x y z O ∈ , the associator [x, y, z] is defined by [x, y, z]=(xy)z−x (yz). Octonions obey some weakened associative laws, including the so-called Moufang identities, for any , , x y z O ∈ , it satisfies [1,19].

Biregular Function
The corresponding Dirac operator is defined as More precisely, then the function is said to be left regular function (right regular function) in Ω . In short, left regular function is also called regular function.
then the function f is called as a biregular function in Ω .

Cauchy Integral Formula
We introduce the Cauchy kernel, which satisfies the relation π is the area of the unit sphere in 8 Proof: Applying the divergence theorem, we know that Lemma 1 [16] For any , , Proof: Since , .

{ }
The first term of the above proof is due to the nature of kernel function 1 ( In the following proof, we will get Using the Stokes theorem 1, we thus obtain The proof is similar to (20).
Therefore, for sufficiently small δ , we have ( , ) The proof is similar to the above manner. Hence

Conclusion
In this paper, using the methods in dealing with the Cauchy integral formula of biregular function in Clifford analysis and regular function in octonionic analysis, we obtain the Cauchy integral formula and mean value theorem for biregular function in octonionic analysis.

Funding
The work was supported by the National Natural Science Foundation of China (No. 11601390) and the foundation of TUTE (KYQD14041, KJ15-18).