Weak slice regular functions on the $n$-dimensional quadratic cone of octonions
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| Main Authors: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866911934019796992 |
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| author | Dou, Xinyuan Ren, Guangbin Sabadini, Irene Yang, Ting |
| author_facet | Dou, Xinyuan Ren, Guangbin Sabadini, Irene Yang, Ting |
| contents | In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called $n$-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_15089 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Weak slice regular functions on the $n$-dimensional quadratic cone of octonions Dou, Xinyuan Ren, Guangbin Sabadini, Irene Yang, Ting Complex Variables 30G35 (Primary), 32A10 (Secondary) In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called $n$-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion. |
| title | Weak slice regular functions on the $n$-dimensional quadratic cone of octonions |
| topic | Complex Variables 30G35 (Primary), 32A10 (Secondary) |
| url | https://arxiv.org/abs/2010.15089 |