## Cauchy integral formula on the distinguished boundary with values in complex universal Clifford algebra.(English)Zbl 1476.30164

Summary: As an integral representation for holomorphic functions, Cauchy integral formula plays a significant role in the function theory of one complex variable and several complex variables. In this paper, using the idea of several complex analysis we construct the Cauchy kernel in universal Clifford analysis, which has generalized complex differential forms with universal Clifford basic vectors. We establish Cauchy-Pompeiu formula and Cauchy integral formula on the distinguished boundary with values in universal Clifford algebra. This work is the basis for studying the Cauchy-type integral and its boundary value problem in complex universal Clifford analysis.

### 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
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### References:

 [1] Brackx, F.; Delanghe, R.; Sommen, FC, Clifford analysis, Research Notes in Mathematics (1982), Boston: Pitman (Advanced Publishing Program), Boston · Zbl 0529.30001 [2] Delanghe, R.; Sommen, FC; Soucek, V., Clifford Algebra and Spinor-Valued Functions (1992), Dordrecht: Kluwer, Dordrecht · Zbl 0747.53001 [3] Du, JY; Xu, N.; Zhang, ZX, Boundary behavior of Cauchy-type integrals in clifford analysis, Acta Math. Scientia., 29B, 1, 210-224 (2009) · Zbl 1199.30267 [4] Du, J.Y., Zhang, Z.X.: A Cauchy’s integral formula for functions with values in a universal Clifford algebra and its applications. Complex Var. Elliptic Equ. 47(10), 915-928 (2002) · Zbl 1061.30043 [5] Du, JY; Xu, N., On boundary behavior of the Cauchy type integrals with values in a universal Clifford algebra, Adv. Appl. Clifford Algebras., 21, 49-87 (2011) · Zbl 1217.30043 [6] Huang, S.; Qiao, YY; Wen, GC, Real and Complex Clifford Analysis (2006), Berlin: Springer, Berlin · Zbl 1096.30042 [7] Heinrich, B.; Zhang, ZX; Du, JY, On Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra, Acta Math. Scientia., 23B, 1, 95-103 (2003) · Zbl 1145.30314 [8] Iftimie, V., Functions hypercomplex, Bull. Math. Soc. Sc. R. S. R., 4, 279-332 (1965) [9] Ku, M.; Du, JY; Wang, DS, Some properties of holomorphic Cliffordian functions in complex Clifford analysis, Acta. Math. Sci., 30, 3, 747-768 (2010) · Zbl 1240.22009 [10] 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 [11] Krantz, SG, Function Theory of Several Complex Variables (1992), New York: Wadsworth Brooks and Cole Advanced and Software, New York · Zbl 0776.32001 [12] Li, ZF; Yang, HJ; Qiao, YY; Guo, BC, Some properties of T-operator with bihypermonogenic kernel in Clifford analysis, Complex Var. Elliptic., 62, 7, 938-956 (2017) · Zbl 1371.30045 [13] Li, ZF; Yang, HJ; Qiao, YY, A new Cauchy integral formula in the complex Clifford analysis, Adv. Appl. Clifford Algebras., 28, 75, 1-12 (2018) · Zbl 1401.30060 [14] Li, SS; Leng, JS; Fei, MG, Spectrums of functions associated to the fractional Clifford-Fourier transform, Adv. Appl. Clifford Algebras., 30, 1, 1-6 (2020) · Zbl 1433.42006 [15] Lu, QK; Zhou, XY, Introduction to Functions of Multiple Complex Variables (2018), Beijing: Science Press, Beijing [16] Lu, JK, Boundary value problem of analytic function (2004), Wuhan: Wuhan University Press, Wuhan [17] Ryan, J., Complexied Clifford analysis, Complex Variables., 1, 119-149 (1982) [18] Ryan, J., Singularities and Laurent expansions in complexied Clifford analysis, Appl. Anal., 15, 33-49 (1983) · Zbl 0536.32003 [19] Shi, HP; Yang, HJ; Li, ZF; Qiao, YY, Fractional Clifford Fourier transform and its application, Adv. Appl. Clifford Algebras., 30, 68, 1-17 (2020) · Zbl 1451.30098 [20] Shi, HP; Yang, HJ; Li, ZF; Qiao, YY, Two-dided Fourier transform in Clifford analysis and its application, Adv. Appl. Clifford Algebras., 30, 67, 1-23 (2020) [21] Shi, JH, Fundamentals of Variable Function Theory (1996), Beijing: Higher Education Press, Beijing [22] Xu, ZY, Riemann problem for regular functions with values on Clifford analysis, Sci. Bull., 32, 23, 476-477 (1987) [23] Yang, HJ; Qiao, YY; Huang, S., Some properties of Cauchy-Type singular integrals in Clifford analysis, J. Math. Res. Appl., 32, 2, 189-200 (2012) · Zbl 1265.30190 [24] Yang, HJ; Qiao, YY; Xie, YH; Wang, LP, Cauchy integral formula for k-monogenic function with $$\alpha$$-weight, Adv. Appl. Clifford Algebras., 28, 2, 1-11 (2018) [25] Youssef, E.H.: Titchmarsh’s Theorem in Clifford Analysis. Adv. Appl. Clifford Algebras. 31(10), 1-15 (2021) · Zbl 1457.42018 [26] Zhang, ZX, The Schwarz type lemma in upper half space in Clifford analysis, Adv. Appl. Clifford Algebras., 28, 98, 1-15 (2018)
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