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By Cinzia Bisi, Caterina Stoppato (auth.), Graziano Gentili, Irene Sabadini, Michael Shapiro, Franciscus Sommen, Daniele C. Struppa (eds.)

This quantity is meant to assemble vital examine effects to the lectures and discussions which came about in Rome, on the INdAM Workshop on diverse Notions of Regularity for features of Quaternionic Variables in September 2010. This quantity will acquire fresh and new effects, that are attached to the subject coated in the course of the workshop. The paintings goals at bringing jointly overseas major experts within the box of Quaternionic and Clifford research, in addition to younger researchers drawn to the topic, with the assumption of featuring and discussing contemporary effects, interpreting new traits and strategies within the region and, usually, of marketing medical collaboration. specific cognizance is paid to the presentation of other notions of regularity for capabilities of hypercomplex variables, and to the research of the most good points of the theories that they originate.

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It C. com G. it G. Gentili et al. 1007/978-88-470-2445-8_4, © Springer-Verlag Italia 2013 55 56 C. Della Rocchetta et al. setting. For instance we can cite, among the most basic, results that concern the Cauchy Representation Formula and the Cauchy kernel, the Identity Principle, the Maximum Modulus Principle, the Open Mapping Theorem and a new notion of analyticity [3, 9–11, 13]. Moreover, the theory of slice regular functions has been extended and generalized to the Clifford Algebras setting, and to the more general setting of alternative algebras, originating a class of functions that is also under deep investigation at the moment (see [4] and references therein, and [15, 16]).

Clifford Algebra and Spinor-Valued Functions— A Function Theory for the Dirac Operator. Kluwer Academic, Dordrecht (1992) 9. : Clifford Algebra and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991) 10. : Table of Integrals, Series, and Products, 7th edn. Translated from Russian, translation edited and with a preface by A. Jeffrey and D. Zwillinger. Elsevier, Amsterdam (2007) 11. : Groups and Geometric Analysis. Pure and Applied Mathematics. Academic Press, Orlando (1984) 12.

Brackx et al. Proceeding in a similar way, it is verified that the functions A2 (x0 , x ) and B2 (x0 , x ) defined by A2 (x0 , x ) = a0 (·) ∗ A1 (x0 , ·)( x ) = b0 (·) ∗ B1 (x0 , ·)( x ) B2 (x0 , x ) = a0 (·) ∗ B1 (x0 , ·)( x ) = b0 (·) ∗ A1 (x0 , ·)( x ) are conjugate harmonic potentials in Rm+1 of the function C1 (x0 , x ). The distributional limits for x0 → 0+ are is a monogenic potential in Rm+1 + given by ⎧ ⎨ a2 ( x ) = lim A2 (x0 , x ) = a0 ∗ a1 ( x ) = b0 ∗ b1 ( x ) x0 →0+ ⎩ b2 ( x ) = lim B2 (x0 , x ) = a0 ∗ b1 ( x ) = b0 ∗ a1 ( x ) x0 →0+ which may be calculated explicitly to be a2 ( x ) = − 1 1 1 T∗ π (m − 1)(m − 3) σm+1 −m+3 and b2 ( x ) = 1 1 1 U∗ .

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