关于四元数函数的一种对数定义
On a definition of logarithm of quaternionic functions
AI总结：
本文针对切片正则四元数函数的经典指数不保持正则性的问题，研究其$*$-对数的存在性，证明存在性依赖函数向量部分零点集与定义域拓扑，局部非零函数必有$*$-对数，并给出球域反例。
摘要：
对于切片正则四元数函数$f$，经典指数函数$\text{exp}\,f$通常不具备切片正则性。学界已提出一种替代的指数函数定义——$*$-指数$\text{exp}_*$：若$f$是切片正则函数，则$\text{exp}_*(f)$也为切片正则函数。出于基础研究层面的动因，对切片正则函数$f$的$*$-对数$\text{log}_*(f)$的研究具有重要价值，本文即开展了相关研究。主要结果表明，除定义域的拓扑结构外，$\text{log}_*(f)$的存在性仅取决于切片正则函数$f=f_0+f_v$的向量部分$f_v$的零点集结构。我们还证明，局部而言，每个非零切片正则函数都存在$*$-对数；最后，我们给出一个反例：球面上的一个非零切片正则函数在该球域上不存在$*$-对数。
英文摘要：
For a slice--regular quaternionic function $f,$ the classical exponential function $\exp f$ is not slice--regular in general. An alternative definition of exponential function, the $*$-exponential $\exp_*$, was given: if $f$ is a slice--regular function, then $\exp_*(f)$ is a slice--regular function as well. The study of a $*$-logarithm $\log_*(f)$ of a slice--regular function $f$ becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a $\log_*(f)$ depends only on the structure of the zero set of the vectorial part $f_v$ of the slice--regular function $f=f_0+f_v$, besides the topology of its domain of definition. We also show that, locally, every slice--regular nonvanishing function has a $*$-logarithm and, at the end, we present an example of a nonvanishing slice--regular function on a ball which does not admit a $*$-logarithm on that ball.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
