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Advances in Hypercomplex Analysis by Cinzia Bisi, Caterina Stoppato (auth.), Graziano Gentili,

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

This quantity is meant to gather vital study effects to the lectures and discussions which came about in Rome, on the INdAM Workshop on various Notions of Regularity for features of Quaternionic Variables in September 2010. This quantity will acquire contemporary and new effects, that are hooked up to the subject coated throughout the workshop. The paintings goals at bringing jointly foreign prime experts within the box of Quaternionic and Clifford research, in addition to younger researchers attracted to the topic, with the assumption of proposing and discussing contemporary effects, interpreting new developments and methods within the region and, commonly, of selling medical collaboration. specific consciousness is paid to the presentation of alternative notions of regularity for features of hypercomplex variables, and to the research of the most beneficial properties of the theories that they originate.

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Now putting, for m even and n = 0, 1, 2, . . , ∂ −m−n δ = Em+n , and hence ∂ −m−n [ . ] = ∂ −m−n δ ∗ [ . ] = Em+n ∗ [ . ] we indeed have ∂ −m−n E−m−n = ∂ −m−n δ ∗ ∂ m+n δ = Em+n ∗ ∂ m+n δ = δ. So the operator ∂ μ [ . ] eventually is defined for all μ ∈ C, and there holds in distributional sense ∂ μ [Eμ ] = ∂ μ ∂ −μ δ = δ, or, at the level of the operators: ∂ μ ∂ −μ μ∈C = 1. 5 A New Operator Recalling the following distributional boundary values of the conjugate harmonic potentials studied in [7] ⎧ 1 Γ ( m−2k ⎪ 2 ) ∗ ⎪ T−m+2k , k ∈ Z, 2k < m ⎪ m+2k ⎨ a2k−1 = 22k π 2 ⎪ 1 Γ ( m−2k ⎪ 2 ) ∗ ⎪ ⎩ b2k = U−m+2k+1 , k ∈ Z, 2k < m 22k+1 π m+2k+2 2 it becomes clear, in view of the results in Sect.

Summarizing, ∂ μ is defined for all μ ∈ C, except for μ = −m, −m − 1, . . when m is even. We will define ∂ μ for those exceptional values further on. First we prove the following fundamental property. Proposition 2 For μ, ν ∈ C when m is odd or for μ, ν ∈ C such that μ, ν and μ + ν are different from −m, −m − 1, −m − 2, . . when m is even, one has ∂ μ δ ∗ ∂ ν δ = ∂ μ+ν δ Proof Using definition (4) for ∂ μ δ and ∂ ν δ, the convolution at the left-hand side decomposes into four terms. They are respectively given by m+μ+ν 1 + eiπμ 1 + eiπν μ+ν m Γ ( 2 ) ∗ π 2 m−μ m−ν T−m−μ−ν 2 2 2 π 2 π 2 for the first one, − m+μ+ν+1 ) ∗ 1 + eiπμ 1 − eiπν μ+ν m Γ ( π 2 m−μ 2m−ν+1 U−m−μ−ν 2 2 2 π 2 π 2 for the second, − m+μ+ν+1 ) ∗ 1 − eiπμ 1 + eiπν μ+ν m Γ ( π 2 m−μ+12 m−ν U−m−μ−ν 2 2 2 π 2 π 2 for the third, and m+μ+ν 1 − eiπμ 1 − eiπν μ+ν m Γ ( 2 ) ∗ 2 π 2 m−μ m−ν T−m−μ−ν 2 2 π 2 π 2 for the fourth.

The Kernel Function and Conformal Mapping. Am. Math. , Providence (1970) 2. : Kernel Functions and Elliptic Differential Equations in Mathematical Physics. Academic Press, New York (1953) 3. : Hypercomplex function theory and Hilbert modules with reproducing kernel. Proc. Am. Math. Soc. 37, 545–576 (1978) 4. : Clifford Analysis. Pitman Research Notes in Mathematics, vol. 76 (1982) 5. : A structure formula for slice monogenic functions and some of its consequences. In: Hypercomplex Analysis. Trends in Mathematics, pp.

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