On the real differential of a slice regular function
Abstract
In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato, and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called \textit{spherical expansion}), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.3993
 Bibcode:
 2014arXiv1402.3993A
 Keywords:

 Mathematics  Complex Variables;
 30G35;
 30B10;
 58A10
 EPrint:
 23 pages, some adjustment in the structure of the sections, some typos removed, last example reviewed