On the real differential of a slice regular function

Abstract

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 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.

Slice Regularity and Harmonicity on Clifford Algebras

“…Remark 6.6. As it was proved in [1], the harmonic functions f ′ s (y) and ∂f ′ s ∂x (y) are, respectively, the first and the second coefficients of the spherical expansion at y of a slice-regular function f (see [26,14]).…”

Twistor interpretation of slice regular functions

“…In the next part of this section we will recall some theorems regarding the nature of the real differential of a slice regular function. These results can be found in [29,14] and their generalization in [3]. Firstly we will expose a representation of the differential.…”

“…The approach that we will use is the one introduced by R. Ghiloni and A. Perotti in [21], which exploit the use of the so-called stem functions to define the class of continuous functions to which we will apply the definition of regularity. The use of stem functions might seems unnecessarily technical, however, by using precisely these techniques many results in the theory of slice regular functions were extended to a more general setting and many others were proven for the first time [1,2,3,21,22,23]. We will, then, describe our family of functions, directly by using this approach.…”

“…In [3,Proposition 10] it is proved that Definition 12 is well posed, i.e. if D is symmetric with respect to the real axis, then…”

“…More precisely we will use the concepts of slice function and of stem function introduced in [21] in a more general context. Using these instruments (that will be defined in Section 3), it is possible to extend some rigidity and differential results, that hold for regular functions defined on domains which intersects the real axis (see [2,3]).…”