A new Cauchy integral formula in the complex Clifford analysis. (English) Zbl 1401.30060

Li, Zunfeng; Yang, Heju; Qiao, Yuying

Adv. Appl. Clifford Algebr. 28, No. 4, Paper No. 75, 12 p. (2018).

Summary: In this paper, we construct an analogue of Bochner-Martinelli kernel based on theory of functions of several complex variables in complex Clifford analysis, which has generalized complex differential forms with Clifford basis vectors. Using these complex differential forms, we obtain the Stoke’s formula of complex Clifford functions which are defined on a domain \(\Omega \subset C^{n+1}\) and take values in a complex Clifford algebra \(Cl_{0,n}(C)\). Then, we give a Stoke’s formula which has a classical form and an analogue of Cauchy-Pompeiu formula which is represented by Bochner-Martinelli kernel, and establish an analogue of Cauchy integral formula in complex Clifford analysis. It is possible to promote these results to complex manifold’s corresponding results in the Clifford analysis using the representation by generalized complex differential forms.

MSC:

30G35	Functions of hypercomplex variables and generalized variables
30E20	Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30E25	Boundary value problems in the complex plane

Keywords:

Clifford analysis; Cauchy formula; Pompeiu formula

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