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

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.


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
Full Text: DOI


[1] Brack, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pinman Book Limited, Boston (1982) · Zbl 0529.30001
[2] Eriksson, SL, Integral formulas for hypermonogenic functions, Bull. Belg. Math. Soc., 11, 705-718, (2004) · Zbl 1071.30047
[3] Eriksson, S.L., Leutwiler, H.: Hypermonogenic function in Clifford algebras and their applications in mathematical physics, vol. 2, pp. 287-302. Birkhauser, Boston (2000)
[4] Eriksson, Sirkka-Liisa; Leutwiler, Heinz, Hypermonogenic functions and their Cauchy-type theorems, 97-112, (2004), Basel · Zbl 1057.30040
[5] Eriksson, SL; Leutwiler, H., An improved Cauchy formula for hypermonogenic functions, Adv. Appl. Clifford Algebras, 19, 269-282, (2009) · Zbl 1172.30027
[6] Huang, S., Qiao, Y.Y., Wen, G.C.: Real and Complex Clifford Analysis. Springer, New York (2006) · Zbl 1096.30042
[7] Ku, M.; Du, JY; Wang, DS, Some properties of holomorphic Cliffordian functions in complex Clifford analysis, Acta. Math. Sci., 30, 747-768, (2010) · Zbl 1240.22009
[8] Ku, M.; Du, JY; Wang, DS, On generalization of Martinelli-Bochner integral formula using Clifford analysis, Adv. Appl. Clifford Algebras, 20, 351-366, (2010) · Zbl 1206.30069
[9] Li, ZF; Yang, HJ; Qiao, YY; Guo, BC, Some properties of T-operator with bihypermonogenic kernel in Clifford analysis, Complex Var. Elliptic Equ., 62, 938-956, (2017) · Zbl 1371.30045
[10] Qiao, YY; Bernstein, S.; Eriksson, SL, Function theory for Laplace and Dirac-Hodge operators on hyperbolic space, J. D’Anal. Math., 98, 43-63, (2006) · Zbl 1135.58013
[11] Qiao, YY; Ryan, J., Orthogonal projections on hyperbolic space, Harm. Anal. Signal Process. Complex., 238, 111-120, (2005) · Zbl 1086.30042
[12] Ryan, J., Complexied Clifford analysis, Complex Var. Theory Appl., 1, 119-149, (1982)
[13] Ryan, J., Singularities and Laurent expansions in complexied Clifford analysis, Appl. Anal., 16, 33-49, (1983) · Zbl 0536.32003
[14] Ryan, J., Iterated Dirac operators in \(C^{n}\), Z. Anal. Anwendungen, 9, 385-401, (1990) · Zbl 0758.47039
[15] Ryan, J., Intrinsic Dirac operators in \(C^{n}\), Adv. Math., 118, 93-133, (1996) · Zbl 0852.32005
[16] Steven, G.: Function Theory of Several Complex Variables. Wadsworth Brooks and Cole Advanced and Software, New York (1992) · Zbl 0776.32001
[17] Wen, G.C.: Clifford Analysis and Elliptic System. World Scientific, Singapore (1991)
[18] Xie, YH, Boundary properties of hypergenic-Cauchy type integrals in Clifford analysis, Complex Var. Elliptic Equ., 59, 599-615, (2014) · Zbl 1291.30017
[19] Xie, YH; Yang, HJ; Qiao, YY, Complex \(k\)-hypermonogenic functions in complex Clifford analysis, Complex Var. Elliptic Equ., 58, 1467-1479, (2013) · Zbl 1284.30053
[20] Xie, YH; Zhang, XF; Tang, XM, Some properties of \(k\)-hypergenic functions in Clifford analysis, Complex Var. Elliptic Equ., 61, 1614-1626, (2016) · Zbl 1345.30077
[21] Yang, HJ; Qiao, YY; Huang, Sha, Some properties of Cauchy-type singular integrals in Clifford analysis, J. Math. Res. Appl., 32, 189-200, (2012) · Zbl 1265.30190
[22] Yang, HJ; Qiao, YY; Wang, LP, Some properties of a kind of higher order singular teodorescu operator in \(R^{n}\), Appl. Anal., 91, 1-14, (2012) · Zbl 1262.30055
[23] Yang, HJ; Zhao, XH, The fixed point and Mann iterative of a kind of higher order singular teodorescu operator, Complex Var. Elliptic Equ., 60, 1658-1667, (2015) · Zbl 1327.30060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.