论文标题
在非轴对称域中的延伸定理和表示公式，用于切片常规函数
Extension theorem and representation formula in non-axially symmetric domains for slice regular functions
论文作者
论文摘要
切片分析是对四个复杂变量与四元变量的全体形态函数理论的概括。 在这种情况下出现的新现象中，有一个事实是，$ f（q）=σ_{n \ in \ mathbb {n}}}}（q-p）^{*n} a_n $的收敛域，$σ$ -BALL $σ（p，r）$ in $ oble in $ mathbb { $ p \ in \ mathbb {r} $。这激发了我们在本文中调查切片常规功能的自然拓扑。事实证明，自然拓扑是所谓的切片拓扑，它与欧几里得拓扑不同，并且很好地适应了四元组的切片结构。我们将片段规则函数的功能理论扩展到切片拓扑中的任何域。 在我们的一般环境中，轴向对称域的经典切片分析中的许多基本结果失败。 我们甚至可以构造反例，以表明域中的切片常规函数不能扩展到轴向对称域。 为了提供积极的结果，我们需要考虑所谓的路径套件而不是切片功能。 沿着这条线，我们可以在切片域中建立一个扩展定理和表示公式。
Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of $f(q)=Σ_{n\in\mathbb{N}}(q-p)^{*n} a_n$, given by a $σ$-ball $Σ(p,r)$, is not open in $\mathbb{H}$ unless $p\in\mathbb{R}$. This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along this line, we can establish an extension theorem and a representation formula in a slice-domain.
