A representation formula for slice regular functions over slice-cones in several variables

Abstract

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form ${\mathbb {R}}^{2n}$. This is a new setting which contains and encompasses in a nontrivial way other cases already studied in the literature and which requires new tools. To this end, we define a cone ${\mathcal {W}}_{\mathcal {C}}^d$ in $[{\text {End}}({\mathbb {R}}^{2n})]^d$ and we extend the slice topology $\tau _s$ to this cone. Slice regular functions can be defined on open sets in $\left( \tau _s,{\mathcal {W}}_{\mathcal {C}}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

1 Introduction

Quaternionic and more in general Clifford analysis are a classical subject which started over a hundred years ago, when complex analysts were looking for function theories generalizing holomorphic functions in one variable to higher dimensional cases. Functions “hyperholomorphic” in a suitable sense, with values in an algebra extending that one of complex numbers, emerged as possible theories alternative to the one of holomorphic functions in several complex variables. The list of contributors is quite long and we refer the interested reader to the recent [10] and the references therein.

In more recent times, inspired by a work of Cullen [11], Gentili and Struppa introduced a definition of hyperholomorphicity [19] over the quaternions ${\mathbb {H}}$. The corresponding class of functions includes convergent power series in the quaternionic variable q

$$\begin{aligned} \sum _{m\in {\mathbb {N}}} q^m a_m,\qquad a_m\in {\mathbb {H}}. \end{aligned}$$

The original definition of slice regularity in [19] requires that for any imaginary unit I in ${\mathbb {H}}$, the restriction of f to the complex plane ${\mathbb {C}}_I$ is holomorphic. In our terminology, this analysis is called weak slice analysis. Weak slice analysis has been extended to Clifford algebras, see [6], and octonions [20]. This function theory has been widely developed in the past decade especially because of various applications to operator theory (see [2, 7] and the references therein), geometry (see e.g. [3, 17]), geometric function theory (see e.g. [26,27,28]) and since we have no pretense of completeness, we refer the reader to the references in the aformentioned books and in [9, 16, 18].

It is crucial to note that a fundamental tool in the study of slice regular functions is the so-called representation formula, whose validity requires extra assumptions on the open sets on which the functions are defined. Moreover, if one is willing to consider slice regularity in the case of real alternative algebras, the proof of the representation formula involves questions about the invertibility of some elements. Another definition of slice regularity, introduced by Ghiloni and Perotti in [21], is given for functions with values in a real alternative $*$-algebra A and defined on a domain $\Omega$ in the quadratic cone of A, and it is based on the fact that these functions have a holomorphic stem function. This class of functions automatically satisfy the representation formula but, according to the open set where the functions are defined, may be different from the one previously discussed. In this paper, the functions as in [21] are called strong slice regular and strong slice analysis is the corresponding theory. This analysis can be extended to the case of several variables in the case of quaternions, Clifford algebras, octonions and real alternative $*$-algebras, see [8, 22, 23, 29], respectively.

In the recent paper [14] we introduced a new topology $\tau _s$, called slice topology, in ${\mathbb {H}}$. This topology is finer than the Euclidean topology, see [13, 14], and it is used to further extend the class of weak slice regular functions which can be defined on open sets in $\tau _s$. We also proved that over non-axially symmetric domains weak slice regular functions may not satisfy the representation formula and in [14] we prove a new formula suitable in this case. Thus, it is clear that slice regular functions in non-axially symmetric domains cannot be considered using the methods in [21]. We note that the slice topology can be extended to the several variables case by using an idea firstly introduced in [29] in the context of octonions. In that paper, the authors consider a cone

$$\begin{aligned} {\mathbb {O}}_s^d:=\bigcup _{I\in {\mathcal {S}}_{\mathbb {O}}}{\mathbb {C}}_I^d\ \left( \subset {\mathbb {O}}^d\right) ,\qquad d\in {\mathbb {N}}_+, \end{aligned}$$

where

$$\begin{aligned} {\mathbb {S}}_{\mathbb {O}}=\{I\in {\mathbb {O}}:I^2=-1\} \end{aligned}$$

(1.1)

is the set of imaginary units in octonions and ${\mathbb {C}}_I^d=({\mathbb {R}}+{\mathbb {R}}I)^d$ is a d-dimensional complex space.

Then, in [15], the slice topology is extended to a cone ${\mathbb {O}}_s^d$ in ${\mathbb {O}}^d$, so one can define weak slice regular functions in several octonionic variables. Following this approach, in this paper we extend the notion of weak slice regularity to a completely new and rather general setting. The main novelties are basically two: the first one is that we consider functions with values in a finite dimensional real vector space of even dimension, namely - up to isomorphisms - ${\mathbb {R}}^{2n}$. According to this choice of the codomain, the domain of the functions will be a set ${\mathcal {W}}_{\mathcal {C}}^d$ in $\left[ {\text {End}}({\mathbb {R}}^{2n})\right] ^d$ that we call weak slice-cone, and this is the second novelty. The cone ${\mathcal {W}}_{\mathcal {C}}^d$ will be equipped with the slice topology.

Thus the weak slice regular functions studied in this work are ${\mathbb {R}}^{2n}$-valued and defined on open sets in the topological space $({\mathcal {W}}_{\mathcal {C}}^d,\tau _s)$. For these functions, we prove various results among which a representation formula which generalizes the one in [4, 5, 14, 15]. A key tool for its proof is the Moore-Penrose inverse.

This new and general approach can be applied to a wide set of even-dimensional real algebras which are called left slice complex structure algebras, LSCS algebras for short, which include quaternions, octonions, Clifford algebras, real alternative $*$-algebras, some left alternative algebras, sedenions. Thus, not only we obtain the weak slice analysis covering all the previous known cases, but we also provide completely new settings.

Now, we describe the structure of the paper. In Sect. 2, we define a weak slice-cone ${\mathcal {W}}_{\mathcal {C}}^d$ in $\left[ {\text {End}}({\mathbb {R}}^{2n})\right] ^d$ and we extend the slice topology $\tau _s$ on it, also proving some of its properties. In Sect. 3, weak slice regular functions are defined on open sets in the slice topology $\tau _s$. We prove a splitting lemma, an identity principle and the maximum modulus principle for these functions. In the case of the latter result, we discuss an issue coming from the definition of a norm in ${\mathbb {R}}^{2n}$. Section 4 is devoted to an extension lemma as well as some other useful results. In Sect. 5, we prove a representation formula for weak slice regular functions, we define path-slice functions on subsets of ${\mathcal {W}}_{{\mathcal {C}}}^d$ and we show that weak slice regular functions are path-slice. In Sect. 6, we define a kind of real algebras, called LSCS algebras which include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also one-sided alternative algebras and sedenions. We show that the results in the previous sections apply to this very general framework.

2 Weak slice-cone

In this section, we define a cone, which we call the n-dimensional weak slice-cone ${\mathcal {W}}_{\mathcal {C}}^d$ of ${\mathcal {C}}\subseteq {\mathfrak {C}}_n$, where ${\mathfrak {C}}_n$ is the set of complex structures on ${\mathbb {R}}^{2n}$. ${\mathcal {W}}_{\mathcal {C}}^d$ is a subset of $\left[ {\text {End}}\left( {\mathbb {R}}^{2n}\right) \right] ^d$ with a slice structure and it plays the same role as the quadratic cone in a real alternative $*$-algebra when $d=1$, where ${\text {End}}\left( {\mathbb {R}}^{2n}\right)$ denotes the set of real linear maps from ${\mathbb {R}}^{2n}$ to ${\mathbb {R}}^{2n}$. Then, we generalize the slice topology in [14] to the weak slice-cone ${\mathcal {W}}_{\mathcal {C}}^d$ and we prove some properties similar to those of the slice topology in [14].

Definition 2.1

Let ${\mathcal {C}}\subset {\mathfrak {C}}_n$. We call

$$\begin{aligned} {\mathcal {W}}_{\mathcal {C}}^d:=\bigcup _{I\in {\mathcal {C}}}{\mathbb {C}}_I^d \end{aligned}$$

the d-dimensional weak slice-cone of ${\mathcal {C}}$, where

$$\begin{aligned} {\mathbb {C}}_I^d:=\left( {\mathbb {R}}+{\mathbb {R}}I\right) ^d. \end{aligned}$$

By construction, we have ${\mathbb {C}}_I^d \subset \left[ {\text {End}}\left( {\mathbb {R}}^{2n}\right) \right] ^d$ for any $I\in {\mathcal {C}}\subset {\mathfrak {C}}_n$ so that ${\mathcal {W}}_{\mathcal {C}}^d\subset \left[ {\text {End}}\left( {\mathbb {R}}^{2n}\right) \right] ^d$.

Definition 2.2

The set ${\mathcal {C}}\subset {\mathfrak {C}}_n$ is called symmetric, if

$$\begin{aligned} {\mathcal {C}}=-{\mathcal {C}}:=\{-c\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) :c\in {\mathcal {C}}\}. \end{aligned}$$

We note that for each ${\mathcal {C}}\subset {\mathfrak {C}}_n$,

$$\begin{aligned} {\mathcal {W}}_{\mathcal {C}}^d={\mathcal {W}}_{{\mathcal {C}}^*}^d, \end{aligned}$$

where ${\mathcal {C}}^*:={\mathcal {C}}\cup (-{\mathcal {C}})$. Thus, in the sequel we will assume the following:

Assumption 2.3

Without loss of generality, we can assume that ${\mathcal {C}}$ is a non-empty symmetric subset of ${\mathfrak {C}}_n$. Thus, we can fix a subset ${\mathcal {C}}^+$ of ${\mathcal {C}}$ such that

$$\begin{aligned} {\mathcal {C}}=\left( -{\mathcal {C}}^+\right) \bigsqcup {\mathcal {C}}^+. \end{aligned}$$

Let A be a (real) algebra. We denote by $L_a:A\rightarrow A$ the left-multiplication corresponding to $a\in A$, that is

$$\begin{aligned} L_a (b)=ab,\qquad \forall \ b\in A. \end{aligned}$$

Example 2.4

We consider the case of the non-commutative algebra of real quaternions ${\mathbb {H}}$. As a real vector space it is isomorphic to ${\mathbb {R}}^4$. It is easy to check that the set

$$\begin{aligned} {\mathcal {C}}_{{\mathbb {H}}}:=\left\{ L_q\in {\text {End}}\left( {\mathbb {R}}^4\right) \,\ q\in {\mathbb {H}},\ q^2=-1\right\} \end{aligned}$$

is a non-empty symmetric subset of ${\mathfrak {C}}_2$ (the set of complex structure on ${\mathbb {R}}^{4})$. Writing J instead of $L_J$, with a slight abuse of notation, we have that the set

$$\begin{aligned} ({\mathbb {H}}\cong )\ L ({\mathbb {H}}):=\left\{ L_a\,\ a\in {\mathbb {H}}\right\} =\bigcup _{J\in {\mathcal {C}}_{{\mathbb {H}}}}{\mathbb {C}}_J\ \big (={\mathcal {W}}_{{\mathcal {C}}_{{\mathbb {H}}}}^1\big ) \end{aligned}$$

is a 1-dimensional weak slice-cone of ${\mathcal {C}}_{{\mathbb {H}}}$.

Denote by $\tau \left( \left[ {\text {End}}\left( {\mathbb {R}}^{2n}\right) \right] ^d\right)$ the Euclidean topology on the real vector space $\left[ {\text {End}}\left( {\mathbb {R}}^{2n}\right) \right] ^d$. For each subset $U\subset \left[ {\text {End}}\left( {\mathbb {R}}^{2n}\right) \right] ^d$, denote by $\tau (U)$ the subspace topology induced by $\tau \left( \left[ {\text {End}}\left( {\mathbb {R}}^{2n}\right) \right] ^d\right)$. It is immediate that the two topologies $\tau$ and $\tau _s$ coincide on ${\mathbb {C}}_I^d$ for any $I\in {\mathcal {C}}\subseteq {\mathfrak {C}}_{n}$. We will refer to the subspace topology $\tau$ on U as to the Euclidean topology on U.

We now generalize the notion of slice topology that was first introduced in [14] in the quaternionic case:

Definition 2.5

We call

$$\begin{aligned} \tau _s\left( {\mathcal {W}}_{{\mathcal {C}}}^d\right) :=\left\{ \Omega \subset {\mathcal {W}}_{{\mathcal {C}}}^d\,\ \Omega _I\in \tau ({\mathbb {C}}_I^d),\ \forall \ I\in {\mathcal {C}}\right\} , \end{aligned}$$

the slice topology on ${\mathcal {W}}_{{\mathcal {C}}}^d$, where

$$\begin{aligned} \Omega _I:=\Omega \cap {\mathbb {C}}_I^d. \end{aligned}$$

It is immediate to check that the slice topology $\tau _s\left( {\mathcal {W}}_{{\mathcal {C}}}^d\right)$ is the final topology with respect to the inclusion maps $\left\{ \imath _I:{\mathbb {C}}_I^d\rightarrow {\mathcal {W}}_{\mathcal {C}}^d\right\} _{I\in {\mathcal {C}}}$.

Remark 2.6

It is important to note that slice topology is strictly finer than the topology induced by $\sigma$-distance defined in [17] (see [14, Proposition 3.3] for the proof and [13] for further details).

Convention: Let $\Omega \subset {\mathcal {W}}_{{\mathcal {C}}}^d$. Denote by $\tau _s(\Omega )$ the subspace topology induced by $\tau _s\left( {\mathcal {W}}_{{\mathcal {C}}}^d\right)$. Open sets, connected sets and paths in $\tau _s(\Omega )$ are called slice-open sets, slice-connected sets and slice-paths in $\Omega$, respectively. A domain in the slice topology is called slice topology-domain or, in short, st-domain.

Proposition 2.7

Let $\ell \in {\mathbb {N}}_+$, $I,J\in {\mathcal {C}}$ with $I\ne \pm J$. Then

$$\begin{aligned} {\mathbb {C}}_I^\ell \cap {\mathbb {C}}_J^\ell ={\mathbb {R}}^\ell . \end{aligned}$$

Proof

Note that ${\mathbb {C}}_I$ can be identified with a complex plane. The equation $z^2=-1$, $z\in {\mathbb {C}}_I$, only has two solutions $z=\pm I$.

Suppose that $\lambda +\mu J\in {\mathbb {C}}_I$, for some $\lambda \in {\mathbb {R}}$ and $\mu \in {\mathbb {R}}\backslash \{0\}$. It is clear that

$$\begin{aligned} J=\mu ^{-1}[(\lambda +\mu J)-\lambda ]\in {\mathbb {C}}_I. \end{aligned}$$

Since $J^2=-1$, we have $J=\pm I$, which is a contradiction. Hence ${\mathbb {C}}_I\cap {\mathbb {C}}_J={\mathbb {R}}$ and then ${\mathbb {C}}_I^\ell \cap {\mathbb {C}}_J^\ell ={\mathbb {R}}^\ell$. $\square$

Remark 2.8

It is easy to check that, for each $q\in {\mathcal {W}}_{{\mathcal {C}}}^d\backslash {\mathbb {R}}^d$, there is a unique triple $(J,x,y)\in {\mathcal {C}}^+\times {\mathbb {R}}^d\times {\mathbb {R}}^d$ such that $q=x+yJ$. Let

$$\begin{aligned} \textrm{Re}(q):=x,\qquad \textrm{Im}(q):=yJ,\qquad \text{ and }\qquad {\mathbb {C}}_q:={\mathbb {C}}_{J}^d. \end{aligned}$$

The $\sigma$-distance defined in [17] can be extended to our case as follows:

$$\begin{aligned} \sigma (q,p):=\left\{ \begin{aligned}&|q-p|,{} & {} \quad \exists \ I\in {\mathbb {S}},\ \mathrm{such\ that}\ p,q\in {\mathbb {C}}_I^d, \\ {}&\sqrt{|\textrm{Re}(q-p)|_{{\mathbb {R}}^d}^2+|\textrm{Im}(q)|_{{\mathbb {C}}_{q}}^2+|\textrm{Im}(p)|_{{\mathbb {C}}_{p}}^2},{} & {} \quad \textrm{otherwise}, \end{aligned}\right. \end{aligned}$$

where ${\mathbb {C}}_p:={\mathbb {R}}^d$ for $p\in {\mathbb {R}}^d$, and $|\cdot |_{{\mathbb {C}}_q}$ is the Euclidean distance on the complex (or real) plane ${\mathbb {C}}_d$. Denote by $({\mathcal {W}}_{{\mathcal {C}}}^d,\tau _\sigma )$ the topology induced by the $\sigma$-distance.

Following the reasoning in [14, Proposition 3.3], one can check that

$$\begin{aligned} \tau \left( {\mathcal {W}}_{{\mathcal {C}}}^d\right) \subseteq \tau _\sigma \left( {\mathcal {W}}_{{\mathcal {C}}}^d\right) \subseteq \tau _s\left( {\mathcal {W}}_{{\mathcal {C}}}^d\right) . \end{aligned}$$

The first inclusion is strict when $|{\mathcal {C}}|=2$ (i.e. when ${\mathcal {W}}_{{\mathcal {C}}}^d={\mathbb {C}}_I^d$, $\forall \ I\in {\mathcal {C}}$) while the second inclusion is strict when $|{\mathcal {C}}|<+\infty$. Since $\tau$ is Hausdorff, so is $\tau _s$.

It is easy to check that

$$\begin{aligned} \tau \left( {\mathbb {R}}^d\right) =\tau _s\left( {\mathbb {R}}^d\right) \end{aligned}$$

and

$$\begin{aligned} \tau \left( {\mathbb {C}}_I^d\right) =\tau _s\left( {\mathbb {C}}^d_I\right) ,\qquad \forall \ I\in {\mathcal {C}}. \end{aligned}$$

(2.1)

Definition 2.9

A subset $\Omega$ of ${\mathcal {W}}_{{\mathcal {C}}}^d$ is called real-connected, if

$$\begin{aligned} \Omega _{{\mathbb {R}}}:=\Omega \cap {\mathbb {R}}^d \end{aligned}$$

is connected in ${\mathbb {R}}^d$. In particular, when $\Omega \cap {\mathbb {R}}^d=\varnothing$, $\Omega$ is real-connected, since the empty set is connected, by assumption.

Proposition 2.10

Let $\Omega$ be a slice-open set in ${\mathcal {W}}_{{\mathcal {C}}}^d$ and $q\in \Omega$. Then there is a real-connected st-domain $U\subset \Omega$ containing q.

Proof

The proof is similar to the proof of analogous statements in [14, 15]. We repeat here the main arguments: if $q\in {\mathbb {R}}^d$, then denote by D the connected component of $\Omega _{{\mathbb {R}}}$ containing q in ${\mathbb {R}}^d$; otherwise, set $D:=\varnothing$. Let U be the slice-connected component of $(\Omega \backslash \Omega _{{\mathbb {R}}})\cup D$ containing q. It is easy to check that $q\in U\subset \Omega$ and U is a real-connected st-domain. $\square$

Now we describe slice-connectedness by means of slice-paths.

Definition 2.11

A path $\gamma$ in $\left( {\mathcal {W}}_{{\mathcal {C}}}^d,\tau \right)$ is called on a slice, if $\gamma \subset {\mathbb {C}}_I^d$ for some $I\in {\mathcal {C}}$.

Proposition 2.12

Each path in $\left( {\mathcal {W}}_{{\mathcal {C}}}^d,\tau \right)$ on a slice is a slice-path.

Proof

This proposition holds directly by (2.1). $\square$

Proposition 2.13

Let $\Omega$ be a real-connected st-domain in ${\mathcal {W}}_{{\mathcal {C}}}^d$. Then the following statements hold:

• (i) If $\Omega _{\mathbb {R}}=\varnothing$, then $\Omega \subset {\mathbb {C}}_I^d$ for some $I\in {\mathcal {C}}$.

• (ii) If $\Omega _{\mathbb {R}}\ne \varnothing$, then for any $q\in \Omega$ and $x\in \Omega _{\mathbb {R}}$, there is a path on a slice from q to x.

Proof

(i). If $\Omega _{\mathbb {R}}=\varnothing$, then (see Assumption 2.3 for the definition of ${\mathcal {C}}^+$)

$$\begin{aligned} \Omega \subset \bigsqcup _{J\in {\mathcal {C}}^+}\left( {\mathbb {C}}_J^d\backslash {\mathbb {R}}^d\right) . \end{aligned}$$

Note that ${\mathbb {C}}_J^d\backslash {\mathbb {R}}^d$ is slice-open in ${\mathcal {W}}_{{\mathcal {C}}}^d$ for each $J\in {\mathcal {C}}$. Since $\Omega$ is slice-connected, $\Omega \subset {\mathbb {C}}_I^d$ for some $I\in {\mathcal {C}}$.

(ii). Let $q\in \Omega$ and $x\in \Omega _{\mathbb {R}}$. Then $q\in {\mathbb {C}}_I^d$ for some $I\in {\mathcal {C}}$. Since $\Omega$ is a st-domain in ${\mathcal {W}}_{{\mathcal {C}}}^d$, it follows by definition that $\Omega _I$ is an open set in ${\mathbb {C}}_I^d$. Denote by V the connected component of $\Omega _I$ containing q. Then V is also open in ${\mathbb {C}}_I^d$. By definition, ${\mathbb {C}}_I^d\backslash {\mathbb {R}}^d$ and $\bigcup _{J\in {\mathcal {C}}\backslash \{\pm I\}}\left( {\mathbb {C}}_J^d\backslash {\mathbb {R}}^d\right)$ are slice-open.

If $V_{\mathbb {R}}=\varnothing$, then

$$\begin{aligned} V=\Omega \cap \left( {\mathbb {C}}_I^d\backslash {\mathbb {R}}^d\right) \qquad \text{ and }\qquad \Omega \backslash V=\Omega \cap \left[ \bigcup _{J\in {\mathcal {C}}\backslash \{\pm I\}}\left( {\mathbb {C}}_J^d\backslash {\mathbb {R}}^d\right) \right] \end{aligned}$$

are slice-open. Since $\Omega$ is slice-connected and nonempty, it follows from

$$\begin{aligned} \Omega =V\ \bigsqcup \ (\Omega \backslash V) \end{aligned}$$

that $V=\Omega$. Therefore $\Omega _{\mathbb {R}}=V_{\mathbb {R}}=\varnothing$, which is a contradiction.

Otherwise, $V_{\mathbb {R}}\ne \varnothing$. Fix $x_0\in V_{\mathbb {R}}$. Since V is the connected component of $\Omega _I$ containing q, there is a path $\alpha$ in V from q to $x_0$. Because $\Omega$ is real-connected, we have a path $\beta$ in $\Omega _{\mathbb {R}}$ from $x_0$ to x. It is clear that $\alpha \beta$ is a path on a slice from q to x. $\square$

Corollary 2.14

Let $\Omega$ be a real-connected st-domain in ${\mathcal {W}}_{{\mathcal {C}}}^d$. Then the following statements hold:

• (i) $\Omega _I$ is a domain in ${\mathbb {C}}_I^d$ for each $I\in {\mathcal {C}}$.

• (ii) Let $p,q\in \Omega$. Then there are two paths $\gamma _1,\gamma _2$ in $\Omega$ such that each of them is on a slice, $\gamma _1(1)=\gamma _2(0)$, and $\gamma _1\gamma _2$ is a slice-path from p to q.

Proof

This follows directly from Proposition 2.13. $\square$

Remark 2.15

(i) A set $\Omega \in \tau _s\left( {\mathcal {W}}_{\mathcal {C}}^d\right)$ is a real-connected st-domain if it either intersects ${\mathbb {R}}^d$ along a connected set and each $\Omega _I$, $I\in {\mathcal {C}}$, is connected, or if it does not intersect ${\mathbb {R}}^d$ then the set lies entirely in one slice ${\mathbb {C}}_I^d$ and $\Omega _I$ is a domain in ${\mathbb {C}}_I$.

(ii) Note that the slice-connected component of a real-connected slice-open set is still real-connected. According to Corollary 2.14 (ii), two points in different slices of the real-connected slice-open set $\Omega$ are in the same slice-connected component if and only if they can be connected by a slice-path through a point in ${\mathbb {R}}^d\cap \Omega$.

(iii) By (ii), the set

$$\begin{aligned} \left( \bigcup _{I\in {\mathcal {C}}}B_I\big ((I,\ldots ,I),2\big )\right) \bigcup \left( \bigcup _{I\in {\mathcal {C}}}B_I\big ((5I,\ldots ,5I),1\big )\right) \end{aligned}$$

is real-connected but not slice-connected and hence it is not a st-domain.

Other examples are discussed in [13].

Proposition 2.16

The topological space $\left( {\mathcal {W}}_{{\mathcal {C}}}^d,\tau _s\right)$ is connected, locally path-connected and path-connected.

Proof

It follows from Proposition 2.10 and Corollary 2.14 (ii) that $\left( {\mathcal {W}}_{{\mathcal {C}}}^d,\tau _s\right)$ is locally path-connected. Since ${\mathcal {W}}_{{\mathcal {C}}}^d\cap {\mathbb {C}}_I^d={\mathbb {C}}_I^d\supset {\mathbb {R}}^d$ for each $I\in {\mathcal {C}}$, $({\mathcal {W}}_{{\mathcal {C}}}^d,\tau _s)$ is path-connected. It implies that $({\mathcal {W}}_{{\mathcal {C}}}^d,\tau _s)$ is also connected. $\square$

3 Weak slice regular functions

As we wrote in the Introduction, the slice regular functions defined in [19] are called weak slice regular functions in this paper. In [14], we generalized these functions to open sets in the slice topology $\tau _s$ on ${\mathbb {H}}$. In this section, we define ${\mathbb {R}}^{2n}$-valued weak slice regular functions on open sets in the slice topology $\tau _s({\mathcal {W}}_{\mathcal {C}}^d)$, and we prove a splitting lemma and an identity principle. To consider functions with values in a real vector space may seem to be reductive, and in fact in Sect. 6 we will show that the study can be generalized to the case of functions with values in a suitable algebra; but on the contrary, it shows that part of the theory, including some powerful tools like the representation formula, in fact depends only on the vector space structure of the set of values not on its algebra structure.

Definition 3.1

Let $\Omega \in \tau _s({\mathcal {W}}_{\mathcal {C}}^d)$. A function $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is called weak slice regular if and only if for each $I\in {\mathcal {C}}$, $f_I:=f|_{\Omega _I}$ is (left I-)holomorphic, i.e. $f_I$ is real differentiable and for each $\ell =1,2,\ldots ,d$,

$$\begin{aligned} \frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+I\frac{\partial }{\partial y_\ell }\right) f_I(x+yI)=0,\qquad \text{ on }\qquad \Omega _I. \end{aligned}$$

Example 3.2

The monomial

$$\begin{aligned} (x_1+y_1 I,\ldots ,x_d+y_d I)\shortmid \!\longrightarrow (x_1+y_1 I)^{m_1}\cdots (x_d+y_d I)^{m_d} a, \end{aligned}$$

is a weak slice regular function, where $x_1,\ldots ,x_d,y_1,\ldots ,y_d\in {\mathbb {R}}$, $a\in {\mathbb {R}}^{2n}$, $m_1,\ldots ,m_d\in {\mathbb {N}}$ and $I\in {\mathcal {C}}$.

Lemma 3.3

(Splitting Lemma) Let $\Omega \in \tau _s\left( {\mathcal {W}}_{\mathcal {C}}^d\right)$. A function $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is weak slice regular if and only if for any $I\in {\mathcal {C}}$ and I-basis $\{\xi _1,....,\xi _n\}$, there are n holomorphic functions $F_1,\ldots ,F_n:\Omega _I\rightarrow {\mathbb {C}}_I$, such that

$$\begin{aligned} f_I=\sum _{\ell =1}^n(F_\ell \xi _\ell ). \end{aligned}$$

Proof

(i) Suppose that f is weak slice regular. Let $I\in {\mathcal {C}}$ and $\{\xi _1,....,\xi _n\}$ be an I-basis. Then

$$\begin{aligned} {\mathbb {R}}^{2n}={\mathbb {C}}_I\xi _1\oplus \cdots \oplus {\mathbb {C}}_I\xi _n. \end{aligned}$$

(3.1)

Then there are n functions $F_1,\ldots ,F_n:\Omega _I\rightarrow {\mathbb {C}}_I$, such that

$$\begin{aligned} f_I=\sum _{\ell =1}^n(F_\ell \xi _\ell ). \end{aligned}$$

Since the composition of maps is associative in ${\text {End}}\left( {\mathbb {R}}^{2n}\right)$ and f is weak slice regular, we have for any $\imath \in \{1,\ldots ,d\}$,

$$\begin{aligned} \begin{aligned} 0=&\frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) f_I =\sum _{\ell =1}^n \frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) (F_\ell \xi _\ell ) \\=&\sum _{\ell =1}^n\left[ \frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) F_\ell \right] \xi _\ell . \end{aligned} \end{aligned}$$

Note that

$$\begin{aligned} \left[ \frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) F_\ell \right] \xi _\ell \in {\mathbb {C}}_I\xi _\ell ,\qquad \ell =1,\ldots ,n. \end{aligned}$$

It follows from (3.1) that for any $\imath \in \{1,\ldots ,d\}$ and $\ell \in \{1,\ldots ,n\}$,

$$\begin{aligned} \left[ \frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) F_\ell \right] \xi _\ell =0. \end{aligned}$$

(3.2)

Since ${\mathbb {R}}^{2n}$ is an n-dimensional ${\mathbb {C}}_I$ (complex) vector space, for any $a\in {\mathbb {R}}^{2n}\backslash \{0\}\cong \mathbb C_I^n\backslash \{0\}$ and $z\in {\mathbb {C}}_I$, we have $z(a)=0$ if and only if $z=0$. Thus for any $\imath \in \{1,\ldots ,d\}$ and $\ell \in \{1,\ldots ,n\}$, (3.2) yields

$$\begin{aligned} \frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) F_\ell =0. \end{aligned}$$

Hence $F_1,\ldots ,F_n$ are holomorphic.

(ii) Suppose that for any choice of $I\in {\mathcal {C}}$ and of an I-basis $\{\xi _1^I,....,\xi _n^I\}$, there are n holomorphic functions $F_1,\ldots ,F_n:\Omega _I\rightarrow {\mathbb {C}}_I$, such that

$$\begin{aligned} f_I=\sum _{\ell =1}^n(F_\ell \xi _\ell ^I). \end{aligned}$$

Then for any $\imath \in \{1,\ldots ,d\}$,

$$\begin{aligned} \begin{aligned} \frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) f_I =\sum _{\ell =1}^n\left[ \frac{1}{2}\left( \frac{\partial }{\partial x_\imath }+I\frac{\partial }{\partial y_\imath }\right) F_\ell \right] \xi _\ell ^I=0, \end{aligned} \end{aligned}$$

and so f is weak slice regular by definition. $\square$

Lemma 3.4

Let $\Omega$ be a real-connected st-domain in ${\mathcal {W}}_{\mathcal {C}}^d$, and $f,g:\Omega \rightarrow {\mathbb {R}}^{2n}$ be weak slice regular. Then the following statements holds.

• (i) If $\Omega _{\mathbb {R}}\ne \varnothing$ and f, g coincide on a non-empty open subset of $\Omega _{\mathbb {R}}$, then $f=g$ on $\Omega$.

• (ii) If f, g coincide on a non-empty open subset of $\Omega _I$ for some $I\in {\mathcal {C}}$, then $f=g$ on $\Omega$.

Proof

• (i) Suppose that f, g coincide on a non-empty open subset U of $\Omega _{\mathbb {R}}$. Let $p\in U$ and $I\in {\mathcal {C}}$. By Splitting Lemma 3.3, $f_I,g_I$ have same Taylor series at p. Hence there is an open set V in ${\mathbb {C}}_I^d$ such that $f=g$ on V. By Corollary 2.14 (i), $\Omega _I$ is a non-empty domain in ${\mathbb {C}}_I^d$. Therefore $f=g$ on $\Omega _I$. Since the choice of I is arbitrary, $f=g$ on $\Omega =\cup _{I\in {\mathcal {C}}}\Omega _I$.

• (ii) Suppose that f, g coincide on a non-empty open subset of $\Omega _I$ for some $I\in {\mathcal {C}}$. By the classical Identity Principle (in several complex variables), $f=g$ on $\Omega _I$. If $\Omega _{{\mathbb {R}}}=\varnothing$, then by Proposition 2.13 (i), $\Omega =\Omega _I$ so that $f=g$ on $\Omega$. Otherwise, $\Omega _I\cap {\mathbb {R}}^d\ne \varnothing$, hence $f=g$ on a non-empty open set $\Omega _{\mathbb {R}}=\Omega _I\cap {\mathbb {R}}^d$ in $\Omega _{\mathbb {R}}$. Therefore $f=g$ on $\Omega$ by (i).

$\square$

Theorem 3.5

(Identity Principle) Let $\Omega$ be a st-domain in ${\mathcal {W}}_{\mathcal {C}}^d$ and $f,g:\Omega \rightarrow {\mathbb {R}}^{2n}$ be weak slice regular. Then the following statements holds.

• (i) If $f=g$ on a non-empty open subset D of $\Omega _{{\mathbb {R}}}$, then $f=g$ on $\Omega$.

• (ii) If $f=g$ on a non-empty open subset D of $\Omega _I$ for some $I\in {\mathcal {C}}$, then $f=g$ on $\Omega$.

Proof

Let us consider the set

$$\begin{aligned} E:=\{x\in \Omega :\exists \ V\in \tau _s(\Omega ),\ \text{ s.t. }\ x\in V\ \text{ and }\ f=g\ \text{ on }\ V\}, \end{aligned}$$

which is a slice-open set in $\Omega$, by its definition.

According to Proposition 2.10, there is a real-connected st-domain U in ${\mathcal {W}}_{\mathcal {C}}^d$ such that $U\cap D\ne \varnothing$ and $U\subset \Omega$. Since U is slice-open, $U_I$ is open in ${\mathbb {C}}_I^d$ and $U_{\mathbb {R}}$ is open in ${\mathbb {R}}^d$. This fact implies that $U\cap D$ is non-empty and open in ${\mathbb {C}}_I^d$ (by (ii)) or in ${\mathbb {R}}^d$ (by (i)). It follows from Lemma 3.4 that $f=g$ on U. Hence $U\subset E$ and E is nonempty.

Let now $q\in \Omega \backslash E$. By Proposition 2.10, there is a real-connected st-domain V with $q\in V\subset \Omega$. Since E and V are slice-open, so is $E\cap V$. We have two cases: if $E\cap V \ne \varnothing$, then $f=g$ on the non-empty slice-open $E\cap V$. By Lemma 3.4, $f=g$ on V. It implies that $q\in E$, which is a contradiction.

Otherwise, $E\cap V =\varnothing$ and it follows from $E,V\subset \Omega$ that $V\subset \Omega \backslash E$. Hence $q\in V$ is a slice-interior point in $\Omega \backslash E$. Therefore $\Omega \backslash E$ is slice-open so that E is slice-closed in $\Omega$. Since $\Omega$ is slice-connected, we deduce that $E=\Omega$ and the thesis follows. $\square$

Definition 3.6

For any $I\in {\mathfrak {C}}_n$, the set $\{\xi _1,\ldots ,\xi _n\}\subset {\mathbb {R}}^{2n}$ is called an I-basis of ${\mathbb {R}}^{2n}$ if

$$\begin{aligned} \{\xi _1,\ldots ,\xi _n,I(\xi _1),\ldots ,I(\xi _n)\} \end{aligned}$$

is a basis of ${\mathbb {R}}^{2n}$ as a real vector space.

Proposition 3.7

(Maximum Modulus Principle) Suppose $\Omega$ be a st-domain in ${\mathcal {W}}_{\mathcal {C}}^d$ and $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is a weak slice regular function. Let $|\cdot |$ be a norm on ${\mathbb {R}}^{2n}$. If for some $p\in \Omega$,

$$\begin{aligned} \sup _{q\in \Omega }|f(q)|=|f(p)| \end{aligned}$$

(3.3)

then f is constant, that is $f\equiv f(p)$.

Proof

Suppose that $p=(p_1,\ldots ,p_d)\subset {\mathbb {C}}_I^d$ and $\overline{P_{{\mathbb {C}}_I^d}(p,r)}\subset \Omega _I$ for some $I\in {\mathcal {C}}$ and $r\in {\mathbb {R}}_+$, where $P_{{\mathbb {C}}_I^d}(p,r):=\{(z_1,\ldots ,z_d)\in {\mathbb {C}}_I^d:|z_\ell -p_\ell |_{{\mathbb {C}}_I^d}<r\}$ is a polydisc in ${\mathbb {C}}_I^d$, and $\overline{P_{{\mathbb {C}}_I^d}(p,r)}$ is the closure of $P_{{\mathbb {C}}_I^d}(p,r)$. The function

$$\begin{aligned} z_1\rightarrow f(z_1,q_2,\ldots ,q_d) \end{aligned}$$

is holomorphic on the closed ball $\overline{B_{{\mathbb {C}}_I}(p_1,r)}$. By the Splitting Lemma 3.3 and the mean value theorem for holomorphic functions in one variable,

$$\begin{aligned} f(p_1,p_2,\ldots ,p_d)=\frac{1}{2\pi }\int _0^{2\pi }f(p_1+re^{I\theta },p_2,\ldots ,p_d)d\theta . \end{aligned}$$

(3.4)

According to Jensen’s inequality,

$$\begin{aligned} |f(p)|\le \max _{\theta \in [0,2\pi )}|f(p_1+re^{I\theta },p_2,\ldots ,p_d)|. \end{aligned}$$

By Eq. (3.3),

$$\begin{aligned} |f(p)|= \max _{\theta \in [0,2\pi )}|f(p_1+re^{I\theta },p_2,\ldots ,p_d)|. \end{aligned}$$

(3.5)

Since the restriction of f to $\Omega _I$ is continuous, it follows from Eqs. (3.4) and (3.5), that there is $C\in {\mathbb {R}}^{2n}$ such that

$$\begin{aligned} C=f(p_1+re^{I\theta },p_2,\ldots ,p_d),\qquad \forall \ \theta \in [0,2\pi ). \end{aligned}$$

By the identity principle of holomorphic functions in one variable

$$\begin{aligned} C=f(z_1,p_2,\ldots ,p_d),\qquad \forall \ z_1\in \overline{B_{{\mathbb {C}}_I}(p_1,r)}, \end{aligned}$$

and hence $f(p)=C$.

For any fixed $z_1\in B_{{\mathbb {C}}_I}(p_1,r)$, the function

$$\begin{aligned} z_2\rightarrow f(z_1,z_2,q_3,\ldots ,q_d) \end{aligned}$$

holomorphic on $\overline{B_{{\mathbb {C}}_I}(p_2,r)}$, again attains its maximum modulus at $(z_1,z_2,p_3,\ldots ,p_d)$, the center of $\overline{B_{{\mathbb {C}}_I}(p_2,r)}$, and hence is constant on $\overline{B_{{\mathbb {C}}_I}(p_2,r)}$. Iterating this procedure we obtain that $f(z)=f(p)$ for all $z\in \overline{P_{{\mathbb {C}}_I^d}(p,r)}$. By the Identity Principle 3.5, $f(z)=f(p)$ for all $z\in \Omega$. $\square$

Remark 3.8

The above proof basically follows from that one of several complex variables. A major difference is that we use a real norm $|\cdot |$ which is not necessarily a complex norm for some slice ${\mathbb {C}}_I^d$ as it happens in complex analysis. In fact, in general, we cannot find a real norm whose restriction to each slice ${\mathbb {C}}_I^d$ is a complex norm. To convince the reader, we discuss an example. Let I be a complex structure on ${\mathbb {R}}^2$, and ${\mathcal {N}}_I$ be the set of norms on the complex Banach space ${\mathbb {R}}^2$ with respect to I. Fix $a\in {\mathbb {R}}^2\backslash \{0\}$ and define a complex structure

$$\begin{aligned} J:\lambda a+\mu [a+I(a)]\mapsto -\mu a+\lambda [a+I(a)],\qquad \forall \ \lambda ,\mu \in {\mathbb {R}}. \end{aligned}$$

It is easy to check that ${\mathcal {N}}_I\cap {\mathcal {N}}_J=\varnothing$, since

$$\begin{aligned} {\left\{ \begin{array}{ll} |a|_I=\frac{\sqrt{2}}{2}\big |a+I(a)\big |_I, \\ |a|_J=\big |a+I(a)\big |_J, \end{array}\right. }\qquad \forall \ |\cdot |_I\in {\mathcal {N}}_I\ \text{ and }\ |\cdot |_J\in {\mathcal {N}}_J. \end{aligned}$$

4 Extension lemma

In this section we prove an extension lemma for weak slice regular functions with values in ${\mathbb {R}}^{2n}$. As a byproduct, we shall obtain other results of independent interest.

For each $I\in {\mathcal {C}}$, we define an isomorphism $\Psi _i^I$ by

(We note that the same isomorphism will be used in Sect. 6 where we will work in an algebra A and ${\mathcal {C}}$ will be denoted by ${\mathcal {S}}_A$). Let $\gamma :[0,1]\rightarrow {\mathbb {C}}^d$ and $I\in {\mathcal {C}}$. Define a corresponding path in ${\mathbb {C}}_I^d$ by

$$\begin{aligned} \gamma ^{I}:=\Psi _i^{I}\circ \gamma . \end{aligned}$$

We now introduce the following set:

$$\begin{aligned} {\mathscr {P}}({\mathbb {C}}^d):=\{\gamma :[0,1]\rightarrow {\mathbb {C}}^d,\ \gamma \ \text{ is } \text{ a } \text{ path } \text{ s.t. } \gamma (0)\in {\mathbb {R}}^d\}; \end{aligned}$$

for any $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$ we define

$$\begin{aligned} {\mathscr {P}}\left( {\mathbb {C}}^d,\Omega \right) :=\left\{ \delta \in {\mathscr {P}}\left( {\mathbb {C}}^d\right) :\exists \ I\in {\mathcal {C}}, \text{ s.t. } Ran(\delta ^{I})\subset \Omega \right\} , \end{aligned}$$

and, finally, for an arbitrary, but fixed $\gamma \in {\mathscr {P}}\left( {\mathbb {C}}^d\right)$ we define

$$\begin{aligned} {{\mathcal {C}}(\Omega ,\gamma )}:=\left\{ I\in {\mathcal {C}}: Ran(\gamma ^{I})\subset \Omega \right\} , \end{aligned}$$

where $Ran(\cdot )$ is the image of a map.

Lemma 4.1

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$, $\gamma \in {\mathscr {P}}\left( {\mathbb {C}}^d,\Omega \right)$ and $J=(J_1,\ldots ,J_k)\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$. Then there is a domain U in ${\mathbb {C}}^d$ containing $\gamma ([0,1])$ such that

$$\begin{aligned} \Psi _i^{J_\ell }(U)\subset \Omega ,\qquad \ell =1,\ldots ,k. \end{aligned}$$

(4.1)

Proof

Let $\ell \in \{1,\ldots ,k\}$. Since $J\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$, we have $\gamma ^{J_\ell }\subset \Omega _{J_\ell }$. Hence for each $t\in [0,1]$, there is $r_{t,\ell }\in {\mathbb {R}}_+$ such that

$$\begin{aligned} B_{J_\ell }(\gamma ^{J_\ell }(t),r_{t,\ell })\subset \Omega , \end{aligned}$$

where $B_{J_\ell }(\gamma ^{J_\ell }(t),r_{t,\ell })$ is the ball with center $\gamma ^{J_\ell }(t)$ and radius $r_{t,\ell }$ in ${\mathbb {C}}_{J_\ell }^d$. Let us set

$$\begin{aligned} r_t:=\min _{\ell =1,\ldots ,k}\{r_{t,\ell }\}. \end{aligned}$$

Therefore

$$\begin{aligned} U:=\bigcup _{t\in [0,1]}B(\gamma (t),r_t) \end{aligned}$$

is a domain in ${\mathbb {C}}^d$ containing $\gamma ([0,1])$ and satisfying Eq. (4.1), since

$$\begin{aligned} \begin{aligned} \Psi _i^{J_\ell }\left( U\right) =&\Psi _i^{J_\ell }\left( \bigcup _{t\in [0,1]}B(\gamma (t),r_t)\right) =\bigcup _{t\in [0,1]}B_{J_\ell }\left( \gamma ^{J_\ell }(t),r_{t}\right) \\\subset&\bigcup _{t\in [0,1]}B_{J_\ell }\left( \gamma ^{J_\ell }(t),r_{t,\ell }\right) \subset \Omega \end{aligned} \end{aligned}$$

for all $\ell \in \{1,..,k\}$. $\square$

A map $A\in {\text {End}}\left( {\mathbb {R}}^{2n}\right)$ is represented by a $2n\times 2n$ real matrix, after fixing a real basis $\vartheta$ of ${\mathbb {R}}^{2n}$.

Let $M=(M_{\imath \jmath })\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) ^{k \times \ell }$ be an ${\text {End}}\left( {\mathbb {R}}^{2n}\right)$-valued matrix, where $M_{\imath \jmath }\in {\text {End}}\left( {\mathbb {R}}^{2n}\right)$, $1\le \imath \le k$, $1\le \jmath \le \ell$. Denote by $M^T:=(M_{\jmath \imath })\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) ^{\ell \times k}$ the transpose of M. We treat M as a $2nk\times 2n\ell$ real matrix with respected to the basis $\vartheta ^k(=\vartheta \times \cdots \times \vartheta$ k times) of ${\mathbb {R}}^{2nk}$ and the basis $\vartheta ^\ell$ of ${\mathbb {R}}^{2n\ell }$. We denote by $M^*$ the adjoint matrix of the $2n\ell \times 2nk$ matrix M. Obviously, in the case of matrix with real real entries $M^*=M^T$.

As it is well known in linear algebra, given M there is a unique $2n\ell \times 2nk$ real matrix $M^+\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) ^{\ell \times k}$, called the Moore–Penrose inverse of M, satisfying the Moore-Penrose conditions:

• (i) $MM^+M=M$.

• (ii) $M^+MM^+=M^+$.

• (iii) $(MM^+)^*=MM^+$.

• (vi) $(M^+M)^*=M^+M$.

We recall some useful properties of the Moore-Penrose inverse:

• (i) If $\ell =k$ and M is invertible, then

  $$\begin{aligned} M^{-1}=M^+. \end{aligned}$$

  (4.2)

• (ii) If $P\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) ^{k \times k}$, $Q\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) ^{\ell \times \ell }$ are real unitary matrices, then

  $$\begin{aligned} (PMQ)^+=Q^{-1}M^+P^{-1}. \end{aligned}$$

  (4.3)

For any $I\in {\mathfrak {C}}_n$, let us choose a fixed I-basis of ${\mathbb {R}}^{2n}$ which is denoted by

$$\begin{aligned} \theta ^I:=\{\theta ^I_1,\ldots ,\theta ^I_n\}, \end{aligned}$$

(4.4)

and let us consider the ${2n\times 2n}$ real matrix $D_I\in {\text {End}}\left( {\mathbb {R}}^{2n}\right)$ given by

$$\begin{aligned} D_I:=\begin{pmatrix}\theta _1^I&\cdots&\theta _n^I&I\theta _1^I&\cdots&I\theta _n^I\end{pmatrix}\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) , \end{aligned}$$

(4.5)

where the real $2n\times 2n$ real matrix $D_I$ is treated as a linear map from ${\mathbb {R}}^{2n}$ to ${\mathbb {R}}^{2n}$ with respected to the fixed real basis $\vartheta$ of ${\mathbb {R}}^{2n}$.

It is easy to check that

$$\begin{aligned} I=D_I{\mathbb {J}}_{2n} D_I^{-1}, \end{aligned}$$

(4.6)

where

$$\begin{aligned} {\mathbb {J}}_{2n}=\begin{pmatrix} &{} -{\mathbb {I}}_{n\times n}\\ {\mathbb {I}}_{n\times n}&{} \end{pmatrix}\in {\text {End}}\left( {\mathbb {R}}^{2n}\right) . \end{aligned}$$

(4.7)

Let $J=(J_1,\ldots ,J_k)\in {\mathcal {C}}^k$, and consider

$$\begin{aligned} D_J:=\begin{pmatrix} D_{J_1}\\ {} &{}\ddots \\ {} &{}&{}D_{J_k} \end{pmatrix}, \qquad {\text {diag}}(J):=\begin{pmatrix} J_1\\ {} &{}\ddots \\ {} &{}&{}J_k \end{pmatrix}, \\ \zeta (J):=\begin{pmatrix} 1&{}J_1\\ \vdots &{}\vdots \\ 1&{}J_k \end{pmatrix}\qquad \text{ and }\qquad \sigma :=\begin{pmatrix} &{}-1\\ 1 \end{pmatrix}. \end{aligned}$$

We call

$$\begin{aligned} \zeta ^+(J):=[D_J\cdot \zeta (J)]^+ D_J \end{aligned}$$

(4.8)

the J-slice inverse of $\zeta (J)$, where $[D_J\cdot \zeta (J)]^+$ is the Moore-Penrose inverse of $D_J\cdot \zeta (J)$.

Proposition 4.2

Let $I\in {\mathcal {C}}$ and $J=(J_1,\ldots ,J_k)\in {\mathcal {C}}^k$. Then the following statements hold.

• (iv) $I(1,I)=-(1,I)\sigma$,

• (i) ${\text {diag}}(J)\cdot \zeta (J)=-\zeta (J)\sigma$,

• (iii) $D_J{\text {diag}}(J)(D_J)^{-1}$ is unitary,

• (iv) $I\left[ (1,I)\zeta ^+(J)\right] =\left[ (1,I)\zeta ^+(J)\right] {\text {diag}}(J)$,

where I(1, I) is short for $\begin{pmatrix} I\\ {} &{}I \end{pmatrix}(1,I)$.

Proof

(i) It is immediate to verify that

$$\begin{aligned} I(1,I)=(I,-1)=(1,I)\begin{pmatrix} &{}-1\\ 1 \end{pmatrix}=(1,I)\sigma . \end{aligned}$$

(ii) According to (i),

$$\begin{aligned} {\text {diag}}(J)\zeta (J)=\begin{pmatrix} J_1(1,J_1)\\ \vdots \\ J_k(1,J_k) \end{pmatrix}=\begin{pmatrix} -(1,J_1)\sigma \\ \vdots 1,J_k)\sigma \end{pmatrix}=-\zeta (J)\sigma . \end{aligned}$$

(iii) By (4.6),

$$\begin{aligned} \begin{aligned} D_J{\text {diag}}(J)(D_J)^{-1}=&\begin{pmatrix} D_{J_1}J_1(D_{J_1})^{-1}\\ {} &{}\ddots \\ {} &{}&{}D_{J_k}J_k(D_{J_k})^{-1} \end{pmatrix} \\=&\begin{pmatrix} {\mathbb {J}}_{2n}\\ {} &{}\ddots \\ {} &{}&{}{\mathbb {J}}_{2n} \end{pmatrix}. \end{aligned} \end{aligned}$$

It is clear that $D_J{\text {diag}}(J)(D_J)^{-1}$ is unitary.

(iv) Since $\sigma$ is unitary, it follows from Eq. (4.3) that

$$\begin{aligned} -\sigma [D_J\cdot \zeta (J)]^+=\sigma ^{-1}[D_J\cdot \zeta (J)]^+=[D_J\cdot \zeta (J)\sigma ]^+. \end{aligned}$$

(4.9)

By (iii), $D_J{\text {diag}}(J)(D_J)^{-1}$ is unitary. Again according to Eq. (4.3),

$$\begin{aligned} \begin{aligned} \left[ D_J{\text {diag}}(J)\zeta (J)\right] ^+=&\left[ D_J{\text {diag}}(J)(D_J)^{-1} D_J\zeta (J)\right] ^+ \\=&\left[ D_J\cdot \zeta (J)\right] ^+\left[ D_J{\text {diag}}(J)(D_J)^{-1}\right] ^{-1} \\=&\left[ D_J\cdot \zeta (J)\right] ^+D_J[-{\text {diag}}(J)](D_J)^{-1}. \end{aligned} \end{aligned}$$

(4.10)

We then deduce the following chain of equalities

$$\begin{aligned} \begin{aligned}&I\left[ (1,I)[D_J\cdot \zeta (J)]^+ D_J\right] =-(1,I)\sigma [D_J\cdot \zeta (J)]^+ D_J \\&\quad =(1,I)[D_J\cdot \zeta (J)\sigma ]^+D_J =-(1,I)[D_J{\text {diag}}(J)\zeta (J)]^+D_J \\&\quad =-(1,I)\left[ D_J\cdot \zeta (J)\right] ^+D_J[-{\text {diag}}(J)](D_J)^{-1}D_J \\&\quad =\left[ (1,I)\left[ D_J\cdot \zeta (J)\right] ^+D_J\right] {\text {diag}}(J), \end{aligned} \end{aligned}$$

where the first equality holds by (i), the second, third and fourth equalities follow from Eqs. (4.9), (ii), and (4.10), respectively. We conclude that (iv) holds. $\square$

Let $J=(J_1,\ldots ,J_k)\in {\mathcal {C}}^k$, $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$ and $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$. We define

$$\begin{aligned} {\mathcal {C}}_{ker}(J):=\left\{ I\in {\mathcal {C}}:\ker (1,I)\supset \bigcap _{\ell =1}^k\ker (1,J_\ell )\right\} , \end{aligned}$$

and

$$\begin{aligned} {\mathcal {C}}(\Omega ,\gamma ,J):={\mathcal {C}}(\Omega ,\gamma )\cap {\mathcal {C}}_{ker}(J), \end{aligned}$$

where $\ker (\cdot )$ is the kernel of a map, and $\ker (1,J_\ell )$ stands for $\ker ((1,J_\ell ))$.

Remark 4.3

Intuitively, a complex structure I in ${\mathcal {C}}_{ker}(J)$ is a complex structure satisfying some algebraic conditions (such as Eq. (4.12) below) to allow the validity of Path-representation Formula. Similarly, a complex structure I in ${\mathcal {C}}(\Omega ,\gamma )$ satisfies a geometric condition, namely that $\gamma ^I$ should be in $\Omega$, for the same purpose. Finally, a complex structure I in ${\mathcal {C}}(\Omega ,\gamma ,J)$ is required to satisfy both algebraic conditions and the geometric condition.

Proposition 4.4

Let $J=(J_1,\ldots ,J_k)\in {\mathcal {C}}^k$. Then

$$\begin{aligned} Ran\left[ \zeta ^+(J)\zeta (J)-id_{({\mathbb {R}}^{2n})^{2\times 1}}\right] = \ker [\zeta (J)]=\bigcap _{\ell =1}^k \ker (1,J_\ell ), \end{aligned}$$

(4.11)

where $id_{({\mathbb {R}}^{2n})^{2\times 1}}$ is the identity map on $({\mathbb {R}}^{2n})^{2\times 1}$.

Moreover, for each $I\in {\mathcal {C}}_{ker}(J)$,

$$\begin{aligned} (1,I)\zeta ^+(J)\zeta (J)=(1,I). \end{aligned}$$

(4.12)

Proof

This proposition can be proof by the similar method of [24, Theorem 2].

(i) By the Moore-Penrose condition (i),

$$\begin{aligned}{}[D_J\cdot \zeta (J)]=[D_J\cdot \zeta (J)][D_J\cdot \zeta (J)]^+[D_J\cdot \zeta (J)]. \end{aligned}$$

Since $D_J$ is invertible, it follows from Eq. (4.8) that

$$\begin{aligned} \zeta (J)=\zeta (J)\zeta ^+(J)\zeta (J). \end{aligned}$$

Hence

$$\begin{aligned} \zeta (J)\left[ \zeta ^+(J)\zeta (J)-id_{({\mathbb {R}}^{2n})^{2\times 1}}\right] =0. \end{aligned}$$

It implies that

$$\begin{aligned} Ran\left[ \zeta ^+(J)\zeta (J)-id_{({\mathbb {R}}^{2n})^{2\times 1}}\right] \subset \ker [\zeta (J)] \end{aligned}$$

(4.13)

On the other hand, for each $w\in \ker [\zeta (J)]$,

$$\begin{aligned} \left[ \zeta ^+(J)\zeta (J)-id_{({\mathbb {R}}^{2n})^{2\times 1}}\right] (-w)=\zeta ^+(J)0+w=w, \end{aligned}$$

It is clear that Eq. (4.11) holds by the above equality and Eq. (4.13).

(ii) Let $I\in {\mathcal {C}}(\Omega ,\gamma ,J)$. By definition and Eq. (4.11),

$$\begin{aligned} \ker (1,I)\supset \bigcap _{\ell =1}^k \ker (1,J_\ell )= Ran\left[ \zeta ^+(J)\zeta (J)-id_{({\mathbb {R}}^{2n})^{2\times 1}}\right] . \end{aligned}$$

It implies that

$$\begin{aligned} (1,I)\zeta ^+(J)\zeta (J)-(1,I)=(1,I)\left[ \zeta ^+(J)\zeta (J)-id_{({\mathbb {R}}^{2n})^{2\times 1}}\right] =0. \end{aligned}$$

Hence equality Eq. (4.12) holds. $\square$

Remark 4.5

We note that $\zeta ^+(J)\zeta (J)$ is a projection, since by the Moore-Penrose condition (i) it follows that

$$\begin{aligned} \begin{aligned}&(\zeta ^+(J)\zeta (J))^2=\{[D_J\zeta (J)]^+D_J\zeta (J)\}^2\\&\qquad =[D_J\zeta (J)]^+ D_J\zeta (J)[D_J\zeta (J)]^+D_J\zeta (J)=[D_J\zeta (J)]^+D_J\zeta (J)=\zeta ^+(J)\zeta (J). \end{aligned} \end{aligned}$$

Thus (4.12) yields that the restriction $\left[ \zeta ^+(J)\zeta (J)\right] |_{Im[(1,I)]}$ of the projection is the identity map on Im[(1, I)].

Lemma 4.6

(Extension Lemma) Let $U\in \tau ({\mathbb {C}}^d)$, $I\in {\mathcal {C}}$ and $J=(J_1,\ldots ,J_k)\in {\mathcal {C}}^k$. If $g_\ell :\Psi _i^{J_\ell }(U)\rightarrow {\mathbb {R}}^{2n}$, $\ell =1,\ldots ,k$ are holomorphic, then the function $g[I]:\Psi _i^I(U)\rightarrow {\mathbb {R}}^{2n}$ defined by

$$\begin{aligned} g[I](x+yI)=(1,I) \zeta ^+(J) g(x+yJ),\qquad \forall \ x+yi\in U, \end{aligned}$$

(4.14)

where

$$\begin{aligned} g(x+yJ)=\begin{pmatrix} g_1(x+yJ_1)\\ \vdots \\ g_k(x+yJ_k) \end{pmatrix} \end{aligned}$$

(4.15)

is holomorphic.

Moreover, if $U_{\mathbb {R}}:=U\cap {\mathbb {R}}^d\ne \varnothing$, $g_1=\cdots =g_k$ on $U_{\mathbb {R}}$ and $I\in {\mathcal {C}}_{ker}(J)$, then

$$\begin{aligned} g[I]=g_1=\cdots =g_k\qquad \text{ on }\qquad U_{\mathbb {R}}. \end{aligned}$$

(4.16)

Proof

(i) By Proposition 4.2 (iv), for each $\ell \in \{1,\ldots ,d\}$ and $x+yi\in U$,

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+I\frac{\partial }{\partial y_\ell }\right) g[I](x+yI) \\&\quad =\frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+I\frac{\partial }{\partial y_\ell }\right) (1,I)\zeta ^+(J)g(x+yJ) \\&\quad =(1,I)\zeta ^+(J)\begin{pmatrix} \frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+J_1\frac{\partial }{\partial y_\ell }\right) \\ &{}\quad \ddots \\ &{}&{}\frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+J_k\frac{\partial }{\partial y_\ell }\right) \end{pmatrix}\begin{pmatrix} g(x+yJ_1)\\ \vdots \\ g(x+yJ_k) \end{pmatrix} \\&\quad =(1,I)\zeta ^+(J)\begin{pmatrix} \frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+J_1\frac{\partial }{\partial y_\ell }\right) g(x+yJ_1)\\ \vdots \\ \frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+J_k\frac{\partial }{\partial y_\ell }\right) g(x+yJ_k) \end{pmatrix}=0. \end{aligned} \end{aligned}$$

Hence g[I] is holomorphic.

(ii) Suppose that $U_{\mathbb {R}}\ne \varnothing$, $g_1=\cdots =g_k$ on $U_{\mathbb {R}}$ and $I\in {\mathcal {C}}_{ker}(J)$. Then

$$\begin{aligned} \begin{pmatrix} g_1(x)\\ \vdots \\ g_k(x) \end{pmatrix}=\begin{pmatrix} 1&{}J_1\\ \vdots &{}\vdots \\ 1&{}J_k \end{pmatrix}\begin{pmatrix}g_1(x)\\ 0\end{pmatrix} =\zeta (J)\begin{pmatrix}g_1(x)\\ 0\end{pmatrix}. \end{aligned}$$

On the other hand, by Eq. (4.12), we have

$$\begin{aligned} (1,I)\zeta ^+(J)\zeta (J)=(1,I). \end{aligned}$$

Hence for each $x\in U_{\mathbb {R}}$,

$$\begin{aligned} \begin{aligned} g[I](x)&=(1,I)\zeta ^+(J) \begin{pmatrix} g_1(x)\\ \vdots \\ g_k(x) \end{pmatrix} =(1,I)\zeta ^+(J)\zeta (J)\begin{pmatrix} g_1(x)\\ 0 \end{pmatrix} \\&=(1,I)\begin{pmatrix} g_1(x)\\ 0 \end{pmatrix}=g_1(x). \end{aligned} \end{aligned}$$

It follows that Eq. (4.16) holds. $\square$

5 Path-representation formula

In this section, we prove a weak path-representation formula for weak slice regular functions. We also define path-slice functions and show that weak slice regular functions are path-slice.

Theorem 5.1

(Path-representation Formula) Let $\Omega$ be a slice-open set in ${\mathcal {W}}_{\mathcal {C}}^d$, $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$, $J=(J_1,J_2,\ldots ,J_k)\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ and $I\in {\mathcal {C}}(\Omega ,\gamma ,J)$. If $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is weak slice regular, then

$$\begin{aligned} f\circ \gamma ^I=(1,I)\zeta ^+(J)(f\circ \gamma ^J), \end{aligned}$$

(5.1)

where

$$\begin{aligned} f\circ \gamma ^J:= \begin{pmatrix} f\circ \gamma ^{J_1}\\ \vdots \\ f\circ \gamma ^{J_k} \end{pmatrix}. \end{aligned}$$

(5.2)

Proof

By Lemma 4.1, there is a domain U in ${\mathbb {C}}^d$ containing $\gamma ([0,1])$ such that

$$\begin{aligned} \Psi _i^{K}(U)\subset \Omega ,\qquad K=I,J_1,...J_k. \end{aligned}$$

Since f is weak slice regular, $f|_{\Psi _i^{J_\ell }(U)}$, $\ell =1,\ldots ,k$ are holomorphic.

Note that $I\in {\mathcal {C}}_{ker}(J)$. By Lemma 4.6, there is a holomorphic function $g:\Psi _i^{I}(U)\rightarrow {\mathbb {R}}^{2n}$, such that for each $x+yi\in U$,

$$\begin{aligned} g(x+yI)=(1,I)\zeta ^+(J)f(x+yJ), \end{aligned}$$

and

$$\begin{aligned} g=f|_{\Psi _i^{J_1}(U)}=f=f_I,\qquad \text{ on }\qquad U_{\mathbb {R}}, \end{aligned}$$

where

$$\begin{aligned} f(x+yJ)=\begin{pmatrix} f(x+yJ_1)\\ \vdots \\ f(x+yJ_k) \end{pmatrix}. \end{aligned}$$

Since $g,f_I$ are holomorphic on a domain $\Psi _i^{I}(U)$ in ${\mathbb {C}}_I^d$ and $g=f_I$ on $U_{\mathbb {R}}\subset \Psi _i^{I}(U)$, it follows from the Taylor series expansion and the identity principle in complex analysis that

$$\begin{aligned} g=f_I,\qquad \text{ on }\qquad \Psi _i^{I}(U). \end{aligned}$$

Let $t\in [0,1]$ and consider

$$\begin{aligned} \gamma (t)=x_t+y_t i,\qquad \text{ for } \text{ some } x_t,y_t\in {\mathbb {R}}^d. \end{aligned}$$

Then

$$\begin{aligned} \gamma ^K(t)=x_t+y_t K,\qquad K=I,J_1,\ldots ,J_k. \end{aligned}$$

It is clear that for each $t\in [0,1]$,

$$\begin{aligned} \begin{aligned} f\circ \gamma ^I(t)&=f(x_t+y_t I)=g(x_t+y_t I) \\&=(1,I)\zeta ^+(J) \begin{pmatrix} f(x_t+y_tJ_1)\\ \vdots \\ f(x_t+y_tJ_k) \end{pmatrix} =(1,I)\zeta ^+(J)\begin{pmatrix} f\circ \gamma ^{J_1}(t)\\ \vdots \\ f\circ \gamma ^{J_k}(t) \end{pmatrix}. \end{aligned} \end{aligned}$$

It implies that (5.1) holds. $\square$

Definition 5.2

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$, $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$ and $J=(J_1,\ldots ,J_k)\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$. We say that J is a slice-solution of ${\mathcal {C}}(\Omega ,\gamma )$ if

$$\begin{aligned} {\mathcal {C}}(\Omega ,\gamma )={\mathcal {C}}(\Omega ,\gamma ,J). \end{aligned}$$

Example 5.3

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$ and $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$. If $\pm I\in {\mathcal {C}}(\Omega ,\gamma )$, then $(I,-I)$ is a slice-solution of ${\mathcal {C}}(\Omega ,\gamma )$. This is because

$$\begin{aligned} \begin{pmatrix} 1&{}I\\ 1&{}-I \end{pmatrix}^{-1}= \frac{1}{2}\begin{pmatrix} 1&{}1\\ -I&{}I \end{pmatrix}, \end{aligned}$$

and

$$\begin{aligned} \ker (1,I)\cap \ker (1,-I)=\ker \left[ \begin{pmatrix} 1&{}I\\ 1&{}-I \end{pmatrix}\right] =0. \end{aligned}$$

Similarly, if $J_1,J_2\in {\mathcal {C}}(\Omega ,\gamma )$ with $J_1-J_2$ being invertible, then $(J_1,J_2)$ is a slice-solution of ${\mathcal {C}}(\Omega ,\gamma )$ (cf. (5.6) below).

Proposition 5.4

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$ and $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$. Then there is at least one slice-solution $J\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ of ${\mathcal {C}}(\Omega ,\gamma )$ for some $k\in {\mathbb {N}}_+$.

Proof

Let $\ell \in {\mathbb {N}}_+$. Define

$$\begin{aligned} m_{\ell }:=\min \left\{ \dim _{{\mathbb {R}}}\left( \bigcap _{\jmath =1}^\ell \ker (1,K_\jmath )\right) :K_\jmath \in {\mathcal {C}}(\Omega ,\gamma )\right\} , \end{aligned}$$

and

$$\begin{aligned} m_{0}:=\dim _{{\mathbb {R}}}\left( \bigcap _{K_\jmath \in {\mathcal {C}}(\Omega ,\gamma )} \ker (1,K_\jmath )\right) . \end{aligned}$$

It is clear that $m_0\le m_{\ell }$ for each $\ell \in {\mathbb {N}}_+$.

Suppose that $I=(I_1,\ldots ,I_\ell )\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^\ell$ such that

$$\begin{aligned} m_{\ell }=\dim _{{\mathbb {R}}}\left( \bigcap _{\jmath =1}^\ell \ker (1,I_\jmath )\right) . \end{aligned}$$

If $m_\ell >m_0$, then there is $I_{\ell +1}\in {\mathcal {C}}(\Omega ,\gamma )$ such that $\ker (1,I_{\ell +1})\nsupseteq \bigcap _{\jmath =1}^\ell \ker (1,I_\jmath )$. Therefore,

$$\begin{aligned} m_{\ell +1}\le \dim _{{\mathbb {R}}}\left( \bigcap _{\jmath =1}^{\ell +1} \ker (1,I_\jmath )\right) \le \dim _{{\mathbb {R}}}\left( \bigcap _{\jmath =1}^\ell \ker (1,I_\jmath )\right) -1 = m_\ell -1. \end{aligned}$$

It implies that there is $k\in {\mathbb {N}}_+$ such that $m_k=m_0$. By definition there is $J=(J_1,\ldots ,J_k)\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$, such that $m_0=m_k=\dim _{{\mathbb {R}}}\left( \bigcap _{\jmath =1}^k \ker (1,J_\jmath )\right)$. It follows that for each $I\in {\mathcal {C}}(\Omega ,\gamma )$,

$$\begin{aligned} \bigcap _{\jmath =1}^K \ker (1,J_\jmath )=\bigcap _{K_\jmath \in {\mathcal {C}}(\Omega ,\gamma )}\ker (1,K_\jmath )\subset \ker (1,I). \end{aligned}$$

By definition, for each $I\in {\mathcal {C}}(\Omega ,\gamma )$, we have $I\in {\mathcal {C}}(\Omega ,\gamma ,J)$. Hence ${\mathcal {C}}(\Omega ,\gamma ,J)={\mathcal {C}}(\Omega ,\gamma )$ and J is a slice-solution of ${\mathcal {C}}(\Omega ,\gamma )$. $\square$

Now we generalize the notion of path-slice functions to ${\mathcal {W}}_{{\mathcal {C}}}^d$. We also show that weak slice regular functions are path-slice.

Definition 5.5

Let $\Omega \subset {\mathcal {W}}_{{\mathcal {C}}}^d$. A function $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is called path-slice if there is a function $F:{\mathscr {P}}({\mathbb {C}}^d,\Omega )\rightarrow ({\mathbb {R}}^{2n})^{2\times 1}$ such that

$$\begin{aligned} f\circ \gamma ^{I}(1)=(1,I)F(\gamma ), \end{aligned}$$

(5.3)

for any $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$ and $I\in {\mathcal {C}}(\Omega ,\gamma )$, where $f\circ \gamma ^J$ is defined by (5.2).

We call F a path-slice stem function of f.

Definition 5.6

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$. A function $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is called path-pseudoslice if for any $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$, $J\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ and $I\in {\mathcal {C}}(\Omega ,\gamma ,J)$,

$$\begin{aligned} f\circ \gamma ^I=(1,I)\zeta ^+(J)(f\circ \gamma ^J), \end{aligned}$$

where $f\circ \gamma ^J$ is defined by Eq. (5.2).

Definition 5.7

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$. A function $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is called path-solution slice, if for any $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$ and slice-solution $J\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ of ${\mathcal {C}}(\Omega ,\gamma )$,

$$\begin{aligned} f\circ \gamma ^I=(1,I)\zeta ^+(J)(f\circ \gamma ^J),\qquad \forall \ I\in {\mathcal {C}}(\Omega ,\gamma ), \end{aligned}$$

(5.4)

where $f\circ \gamma ^J$ is defined by Eq. (5.2).

Let $\gamma \in {\mathscr {P}}({\mathbb {C}}^d)$ and $t\in [0,1]$. Define the path $\gamma [t]:[0,1]\rightarrow {\mathbb {C}}^d$ by setting

$$\begin{aligned} \gamma [t](s)=\gamma (ts),\qquad \forall \ s\in [0,1]. \end{aligned}$$

It is clear that $\gamma [t]\in {\mathscr {P}}({\mathbb {C}}^d)$.

Proposition 5.8

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$ and $f:\Omega \rightarrow {\mathbb {R}}^{2n}$. Then the following statements are equivalent.

• (i) f is path-slice,

• (ii) f is path-pseudoslice,

• (iii) f is path-solution slice.

Proof

(i)$\Rightarrow$(ii) Suppose that f is path-slice. Then there is a path-slice stem function $F:{\mathscr {P}}({\mathbb {C}}^d,\Omega )\rightarrow ({\mathbb {R}}^{2n})^{2\times 1}$ of f. Let $t\in [0,1]$, $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$, $J=(J_1,\ldots ,J_k)\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ and $I\in {\mathcal {C}}(\Omega ,\gamma ,J)$. It is easy to check that

$$\begin{aligned} J\in \left[ {\mathcal {C}}(\Omega ,\gamma [t])\right] ^k\qquad \text{ and }\qquad I\in {\mathcal {C}}\left( \Omega ,\gamma [t],J\right) \subset {\mathcal {C}}_{ker}(J). \end{aligned}$$

Since f is path-slice, we have

$$\begin{aligned} f\circ \gamma ^{K}(t)=f\circ \gamma [t]^{K}(1)=(1,K)F(\gamma [t]),\qquad K=I,J_1,\ldots ,J_k. \end{aligned}$$

Hence

$$\begin{aligned} {\left\{ \begin{array}{ll} f\circ \gamma ^I(t)=(1,I)F(\gamma [t]), \\ f\circ \gamma ^J(t)=\zeta (J)F(\gamma [t]). \end{array}\right. } \end{aligned}$$

According to $I\in {\mathcal {C}}_{ker}(J)$ and Eq. (4.12),

$$\begin{aligned} f\circ \gamma ^I(t)=(1,I)F(\gamma [t])=(1,I)\zeta ^+(J)\zeta (J)F(\gamma [t])=(1,I)\zeta ^+(J)f\circ \gamma ^J(t). \end{aligned}$$

It is clear that f is path-pseudoslice, since the choice of t is arbitrary.

(ii)$\Rightarrow$(iii) Suppose that f is path-pseudoslice. Let $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$ and $J\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ be a slice-solution of ${\mathcal {C}}(\Omega ,\gamma )$. Then ${\mathcal {C}}(\Omega ,\gamma )={\mathcal {C}}(\Omega ,\gamma ,J)$. Since f is path-pseudoslice, by definition it follows that for each $I\in {\mathcal {C}}(\Omega ,\gamma )={\mathcal {C}}(\Omega ,\gamma ,J)$,

$$\begin{aligned} f\circ \gamma ^I=(1,I)\zeta ^+(J)(f\circ \gamma ^J). \end{aligned}$$

Therefore f is path-solution slice.

(iii)$\Rightarrow$(i) Suppose that f is path-solution slice. By Proposition 5.4, for each $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$, we can choose a slice-solution $J^{\gamma }\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ of ${\mathcal {C}}(\Omega ,\gamma )$. Define a function $F:{\mathscr {P}}({\mathbb {C}}^d,\Omega )\rightarrow ({\mathbb {R}}^{2n})^{2\times 1}$ by

$$\begin{aligned} F(\gamma ):=\zeta ^+(J^\gamma )(f\circ \gamma ^{(J^\gamma )}). \end{aligned}$$

It is clear by Eq. (5.4) that F is a path-slice stem function of f and f is path-slice. $\square$

Corollary 5.9

Each weak slice regular function defined on a slice-open set in ${\mathcal {W}}_{\mathcal {C}}^d$ is path-slice and path-solution slice.

Proof

This Corollary follows directly from Theorem 5.1 and Proposition 5.8. $\square$

Below we present a standard path-representation formula, see (5.7) in Corollary 5.11), of the form given in [14]. However, we note that this formula only works when $J_1-J_2$ is invertible. In contrast, our new formula (5.1) works in more general cases.

Corollary 5.10

(Another form of path-representation Formula) Let $\Omega \in \tau _s({\mathcal {W}}_{\mathcal {C}}^d)$, $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$, and $J\in \left[ {\mathcal {C}}(\Omega ,\gamma )\right] ^k$ be a slice-solution of ${\mathcal {C}}(\Omega ,\gamma )$. If $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ is weak slice regular, then for each $I\in {\mathcal {C}}(\Omega ,\gamma )$,

$$\begin{aligned} f\circ \gamma ^I=(1,I)\zeta ^+(J)(f\circ \gamma ^J), \end{aligned}$$

(5.5)

where $f\circ \gamma ^J$ is defined by Eq. (5.2).

Proof

This corollary follows directly from Corollary 5.9. $\square$

Let $J_1,J_2\in {\mathcal {C}}$. Note that

$$\begin{aligned} J_1(J_1-J_2)=-(J_1-J_2)J_2\ \left( =-1-J_1J_2\right) . \end{aligned}$$

If $J_1-J_2$ is invertible, then

$$\begin{aligned} (J_1-J_2)^{-1}J_1=-J_2(J_1-J_2)^{-1}. \end{aligned}$$

One can easily verify that

$$\begin{aligned} \begin{pmatrix} 1&{}J_1\\ 1&{}J_2 \end{pmatrix}^{-1} =\begin{pmatrix} (J_1-J_2)^{-1}J_1&{}-(J_1-J_2)^{-1}J_2\\ (J_1-J_2)^{-1}&{}-(J_1-J_2)^{-1} \end{pmatrix}. \end{aligned}$$

(5.6)

Corollary 5.11

(Standard path-representation Formula) Let $\Omega \in \tau _s({\mathcal {W}}_{\mathcal {C}}^d)$, $f:\Omega \rightarrow {\mathbb {R}}^{2n}$ be a weak slice regular function, $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$ and $I,J_1,J_2\in {\mathcal {C}}(\Omega ,\gamma )$ such that $J_1-J_2$ is invertible. Then

$$\begin{aligned} f\circ \gamma ^I=(1,I) \begin{pmatrix} 1&{}J_1\\ 1&{}J_2 \end{pmatrix}^{-1} \begin{pmatrix} f\circ \gamma ^{J_1}\\ f\circ \gamma ^{J_2} \end{pmatrix}. \end{aligned}$$

(5.7)

Proof

Set $J=(J_1,J_2)$. Since $J_1-J_2$ is invertible, it follows from (5.6) that

$$\begin{aligned} \zeta (J)=\begin{pmatrix} 1&{}J_1\\ 1&{}J_2 \end{pmatrix} \end{aligned}$$

is invertible, so is $D_J\cdot \zeta (J)$. By (4.2),

$$\begin{aligned} \left[ D_J\cdot \zeta (J)\right] ^+=\left[ D_J\cdot \zeta (J)\right] ^{-1}=\zeta (J)^{-1}D_J^{-1}. \end{aligned}$$

Hence

$$\begin{aligned} \zeta ^+(J)=\left[ D_J\cdot \zeta (J)\right] ^+D_J=\zeta (J)^{-1}D_J^{-1}D_J=\zeta (J)^{-1}. \end{aligned}$$

(5.8)

On the other hand, by Example 5.3, J is a slice-solution of ${\mathcal {C}}(\Omega ,\gamma )$. Therefore equality Eq. (5.7) follows directly from Eqs. (5.5) and (5.8). $\square$

6 Weak slice regular functions over LSCS algebras

Despite what happens for the various types of slice regular functions treated in the literature, the weak slice regular functions we considered in the previous sections have values in a real vector space, not in an algebra, and their domain ${\mathcal {W}}_{{\mathcal {C}}}^d$ is not a subset (proper or improper) of the codomain. In order to consider functions with values in a real algebra it is necessary to require additional hypothesis on the algebra at hand. We call such algebras left slice complex structure algebras, or LSCS algebras for short.

The weak slice analysis for LSCS algebras includes the slice analysis already known and studied in the literature in the case of quaternions, Clifford algebras, octonions, but it also provides new frameworks, for example the case of left alternative algebras and of sedenions, which form a Cayley-Dickson algebra.

6.1 LSCS algebras

In this subsection, we define the so-called LSCS algebras. We choose a cone ${\mathcal {W}}_A^d$ for the LSCS algebra A and define the slice topology on it.

Let A be a real algebra, whose binary multiplication operation is not assumed to be associative and recall that $L_a:\ A\rightarrow A$ denotes the multiplication on the left by $a\in A$.

Definition 6.1

A finite-dimensional real, unital algebra $A\ne \{0\}$ is called a left slice complex structure algebra, LSCS algebra for short, if there is $a\in A$ such that $L_a$ is a complex structure on A.

Example 6.2

Complex numbers, quaternions, octonions, Clifford algebras $\mathbb R_{n,m}$ ($(n,m)\ne (1,0)$) and real alternative $*$-algebras are examples of LSCS algebras. Moreover, also sedenions and other Cayley–Dickson algebras are LSCS algebras.

Assume that A is an LSCS algebra. Since there is a complex structure on A, the real dimension of A is even. Hence we can set

$$\begin{aligned} n:=\frac{1}{2}\dim (A), \end{aligned}$$

so that $A\cong {\mathbb {R}}^{2n}$ as a real vector space.

Definition 6.3

We call

$$\begin{aligned} {\mathcal {W}}_A^d:=\bigcup _{I\in {\mathcal {S}}_A}{\mathbb {C}}_I^d \end{aligned}$$

the (left) weak-cone of A. Let

$$\begin{aligned} {\mathcal {S}}_A:=\left\{ a\in A:L_a^2=-id_A\right\} \end{aligned}$$

(6.1)

the so-called the set of (left) slice-units of A.

It is immediate that

$$\begin{aligned} L({\mathcal {S}}_A):=\{L_I\,\ I\in {\mathcal {S}}_A\} \end{aligned}$$

is a symmetric subset of ${\mathfrak {C}}(A)$ $\left( \cong {\mathfrak {C}}\left( {\mathbb {R}}^{2n}\right) \right)$, and

$$\begin{aligned} L[d]\left( {\mathcal {W}}_A^d\right) =\{(L_{q_1},\ldots ,L_{q_d}):(q_1,\ldots ,q_d)\in {\mathcal {W}}_A^d\} \end{aligned}$$

is a weak slice-cone of A identified with ${\mathbb {R}}^{2n}$.

As we did above, also in this case we introduce

$$\begin{aligned} \tau _s\left( {\mathcal {W}}_A^d\right) :=\{\Omega \subset {\mathcal {W}}_A^d:\Omega _I\in \tau ({\mathbb {C}}^d_I),\ \forall \ I\in {\mathcal {S}}_A\}. \end{aligned}$$

(6.2)

namely, the slice topology on ${\mathcal {W}}_{A}^d$.

6.2 Weak slice regular functions over LSCS algebras

In this subsection, we define weak slice regular functions on slice-open sets in ${\mathcal {W}}_A^d$, i.e. sets belonging to $\tau _s\left( {\mathcal {W}}_{A}^d\right)$, and with values in an LSCS algebra. Properties in Sects. 3, 4 and 5 also hold for the weak slice regular functions over LSCS algebras, including a splitting lemma, an identity principle and a representation formula. We state some lemmas and theorems without proofs, since they are similar to the corresponding ones in Sects. 3, 4 and 5.

Definition 6.4

Let $\Omega \in \tau _s({\mathcal {W}}_{A}^d)$. A function $f:\Omega \rightarrow A$ is called weak slice regular if and only if for each $I\in {\mathcal {S}}_A$, $f_I=f|_{\Omega _I}$ is real differentiable and for any $\ell =1,2,\ldots ,d$,

$$\begin{aligned} \frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+I\frac{\partial }{\partial y_\ell }\right) f_I(x+yI)=0,\qquad \text{ on }\qquad \Omega _I. \end{aligned}$$

(6.3)

It is evident that the monomial

$$\begin{aligned} (x+y I)\shortmid \!\longrightarrow (x+yI)^{\alpha } a=(x_1+y_1I)^{\alpha _1}\cdots (x_d+y_dI)^{\alpha _d} a, \end{aligned}$$

is a weak slice regular function, where $x,y\in {\mathbb {R}}^d$, $a\in A$, $\alpha =(\alpha _1,\ldots , \alpha _d)\in {\mathbb {N}}^d$ and $I\in {\mathcal {S}}_A$.

Let $\Omega \in \tau _s({\mathcal {W}}_A^d)$. Note that for each weak slice regular function $f:\Omega \rightarrow A$, equality Eq. (6.3) holds for each $I\in {\mathcal {S}}_A$. It implies that

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+L_I\frac{\partial }{\partial y_\ell }\right) f\circ L[d]^{-1}|_{L[d](\Omega )} (x+yL_I) \\&\quad =\frac{1}{2}\left( \frac{\partial }{\partial x_\ell }+I\frac{\partial }{\partial y_\ell }\right) f_I(x+yI)=0, \end{aligned} \end{aligned}$$

for any $L_I\in L({\mathcal {S}}_A)$, $x+yL_I\in L(\Omega )$ and $\ell =1,2,\ldots ,d$. It is clear that

$$\begin{aligned} f\circ L[d]^{-1}|_{L[d](\Omega )}:L[d](\Omega ) \longrightarrow A\cong {\mathbb {R}}^{2n} \end{aligned}$$

is a ${\mathbb {R}}^{2n}$-valued weak slice regular function over the weak slice-cone ${\mathcal {W}}_{L({\mathcal {S}}_A)}^d$ of ${\mathbb {R}}^{2n}$, where $L[d](\Omega )\in \tau _s\left( {\mathcal {W}}_{L({\mathcal {S}}_A)}^d\right)$.

Similarly, if $f\circ L[d]^{-1}|_{L[d](\Omega )}$ is a weak slice regular function, f is also a weak slice regular function. In summary, the following proposition holds.

Proposition 6.5

Let $\Omega \in \tau _s\left( {\mathcal {W}}_A^d\right)$. A function $f:\Omega \rightarrow A$ is weak slice regular if and only if $f\circ L[d]^{-1}|_{L[d](\Omega )}$ is weak slice regular.

Proposition 6.5 implies that various properties of weak slice regular functions over weak slice-cones of ${\mathbb {R}}^{2n}$ also hold for weak slice regular functions over A, among which the Splitting Lemma and the Identity Principle that we do not repeat here. We only adapt the notations and the statement of the path-representation formula.

Let $\Omega \subset {\mathcal {W}}_{A}^d$ and $\gamma \in {\mathscr {P}}\left( {\mathbb {C}}^d\right)$. Define

$$\begin{aligned} {\mathscr {P}}\left( {\mathbb {C}}^d,\Omega \right) :=\left\{ \delta \in {\mathscr {P}}\left( {\mathbb {C}}^d\right) :\exists \ I\in {\mathcal {S}}_A, \text{ s.t. } Ran(\delta ^{I})\subset \Omega \right\} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {S}}_A(\Omega ,\gamma ):=\left\{ I\in {\mathcal {S}}_A:Ran(\gamma ^{I})\subset \Omega \right\} . \end{aligned}$$

Let $J=(J_1,\ldots ,J_k)\in {\mathcal {S}}_A^k$. Define

$$\begin{aligned} {\mathcal {S}}_A^{ker}(J):=\left\{ I\in {\mathcal {S}}_A:\ker (1,L_I)\supset \bigcap _{\ell =1}^k\ker (1,L_J)\right\} . \end{aligned}$$

Let $\Omega \subset {\mathcal {W}}_{\mathcal {C}}^d$ and $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$. Set

$$\begin{aligned} {\mathcal {S}}_A(\Omega ,\gamma ,J):={\mathcal {S}}_A(\Omega ,\gamma )\cap {\mathcal {S}}_A^{ker}(J). \end{aligned}$$

Theorem 6.6

(Path-representation Formula) Let $\Omega$ be a slice-open set in ${\mathcal {W}}_{A}^d$, $\gamma \in {\mathscr {P}}({\mathbb {C}}^d,\Omega )$, $J=(J_1,J_2,\ldots ,J_k)\in \left[ {\mathcal {S}}_A(\Omega ,\gamma )\right] ^k$ and $I\in {\mathcal {S}}_A(\Omega ,\gamma ,J)$. If $f:\Omega \rightarrow A$ is weak slice regular then

$$\begin{aligned} f\circ \gamma ^I=(1,L_I)\zeta ^+(L_J)(f\circ \gamma ^J), \end{aligned}$$

where $f\circ \gamma ^J$ is defined by Eq. (5.2).

6.3 The case of left alternative algebras

In [21], Ghiloni and Perotti define a notion of slice regular functions in one variable for functions with values in a real alternative $*$-algebra and using stem functions. This class does not coincide with the one considered in this work and in fact in this section we show that the class of weak slice regular functions on a finite-dimensional real left alternative algebra A is a generalization of the class of slice regular functions defined in [21].

We recall that a real algebra A is called left alternative, if for any $a,b\in A$,

$$\begin{aligned} a(ab)=(aa)b. \end{aligned}$$

The interested reader may consult [1] in which the author introduces the notion of right alternative algebra that can be easily adapted to our case.

Assumption 6.7

Assume that $A\ne \{0\}$ is a finite-dimensional real unital left alternative algebra with ${\mathcal {S}}_A\ne \varnothing$.

We extend the quadratic cone in [21] to a real unital left alternative algebra A by

$$\begin{aligned} Q_A:=\bigcup _{I\in {\mathbb {S}}_A}{\mathbb {C}}_I, \end{aligned}$$

where ${\mathbb {S}}_A$ is the set of imaginary units, defined by

$$\begin{aligned} {\mathbb {S}}_A:=\{a\in A:a^2=-1\}. \end{aligned}$$

(6.4)

Since for every $I\in {\mathbb {S}}_A$,

$$\begin{aligned} (L_I)^2(a)=I(Ia)=(II)a=-a, \end{aligned}$$

it follows that $L_I$ is a complex structure on A. Hence

$$\begin{aligned} {\mathbb {S}}_A={\mathcal {S}}_A,\qquad \text{ and }\qquad Q_A={\mathcal {W}}_A^1. \end{aligned}$$

(6.5)

Obviously, a left alternative algebra A is an LSCS algebra, hence the definitions and propositions in Sects. 6.1 and 6.2 are valid in this case.

Remark 6.8

All the real alternative $*$-algebras considered in [21] are left alternative algebras satisfying Assumption 6.7.

Let us set $d=1$. Then, using [21, Proposition 8] and (6.5), one may check that the slice regular functions as in [21, Definition 8] (called strong slice regular functions in this paper) are weak slice regular functions according to our Definition 6.4. Moreover, strong slice regular functions are slice functions. However, a weak slice regular function may not be a slice function, see [14, Section 8]. Therefore, the class of weak slice regular functions strictly includes the class of strong slice regular functions.

When $d>1$, we have ${\mathcal {W}}_A^d\subsetneq Q_A^d$, so the functions considered in [23] and ours are different. More specifically, for any strong slice regular function $f:\Omega \in \tau (Q_A^d)\rightarrow A$, its restriction $f|_{\Omega \cap {\mathcal {W}}_A^d}$ is a weak slice regular function.

Remark 6.9

In [25], Prezelj and Vlacci study a class of functions denoted by ${\mathcal {H}}_{rhs}[z_1,\ldots ,z_d]$. This class of function restrictions is included in the class of weak slice regular functions but it is smaller. In fact every function $f:U\in \tau ({\mathbb {H}}^n)\rightarrow {\mathbb {H}}$ in ${\mathcal {H}}_{rhs}[z_1,\ldots ,z_d]$ has a series expansion of the form

$$\begin{aligned} f(z_1,\ldots ,z_d)=\sum _{k\in {\mathbb {N}}}z_{\imath _{1}^k}\cdots z_{\imath _{m_k}^k} a_k,\qquad \forall \ (z_{1},\ldots ,z_{d})\in U \end{aligned}$$

(6.6)

where $\imath _\ell ^k\in \{1,\ldots ,d\}$ and $a_k\in {\mathbb {H}}$. If $z_1,\ldots ,z_d\in {\mathbb {C}}_I$ for some $I\in {\mathbb {S}}_{{\mathbb {H}}}$, then $z_\imath z_\jmath =z_\jmath z_\imath$. It is immediate that $f|_{U\cap {\mathbb {H}}_s^n}$ is weak slice regular, where ${\mathbb {H}}_s^n=\cup _{I\in {\mathbb {S}}_{\mathbb {H}}}{\mathbb {C}}_I^n={\mathcal {W}}_{\mathbb {H}}^n$.

However, there are weak slice regular functions $g:\Omega \in \tau ({\mathbb {H}}_s^n)\rightarrow {\mathbb {H}}$ such that there is no $h\in {\mathcal {H}}_{rhs}[z_1,\ldots ,z_d]$ with $h|_\Omega =g$. For example, consider the domain in ${\mathbb {H}}_s^n$ given by $\Omega :=\{q=(q_1,\ldots ,q_n)\in {\mathbb {H}}_s^n:q_1\notin (-\infty ,0]\}$. Consider a weak slice regular function $g:\Omega \rightarrow {\mathbb {H}}$ with

$$\begin{aligned} g(x_1,q_2,\ldots ,q_n)=ln(x_1),\qquad \forall \ (x_1,q_2,\ldots ,q_n)\in \Omega \text{ with } x_1\in (0,+\infty ). \end{aligned}$$

Assume that there is $h\in {\mathcal {H}}_{rhs}[z_1,\ldots ,z_d]$ such that $h|_\Omega =g$. Then, it follows from Eq. (6.6) that for each fixed $I\in {\mathbb {S}}$, the complex logarithm function $g_I:\Omega _I\rightarrow {\mathbb {C}}_I$ with respect to the first complex variable has a Taylor series on $\Omega _I=({\mathbb {C}}_I\backslash {\mathbb {R}})\times {\mathbb {C}}_I^{d-1}$, which is a contradiction.

Finally, we show that the Moore-Penrose inverse is necessary in the case of real alternative $*$-algebras or more general cases for weak slice regular functions. The standard-type of representation formula (such as [4, Theorem 3.2], [14, Theorem 2.11] and [21, Proposition 6]) can be expressed as follows:

Proposition 6.10

(Standard path-representation Formula) Let $\Omega$ be a slice-open set in ${\mathcal {W}}_A^1$, $f:\Omega \rightarrow A$ be a weak slice regular function, $\gamma \in {\mathscr {P}}({\mathbb {C}}^1,\Omega )$ and $I,J_1,J_2\in {\mathcal {S}}_A(\Omega ,\gamma )$ with $L_{J_1}-L_{J_2}$ being invertible. Then

$$\begin{aligned} f\circ \gamma ^I=(1,L_I) \begin{pmatrix} 1&{}L_{J_1}\\ 1&{}L_{J_2} \end{pmatrix}^{-1} \begin{pmatrix} f\circ \gamma ^{J_1}\\ f\circ \gamma ^{J_2} \end{pmatrix}, \end{aligned}$$

(6.7)

where

$$\begin{aligned} \begin{pmatrix} 1&{}L_{J_1}\\ 1&{}L_{J_2} \end{pmatrix}^{-1} =\begin{pmatrix} (L_{J_1}-L_{J_2})^{-1}L_{J_1}&{}-(L_{J_1}-L_{J_2})^{-1}L_{J_2}\\ (L_{J_1}-L_{J_2})^{-1}&{}-(L_{J_1}-L_{J_2})^{-1} \end{pmatrix}. \end{aligned}$$

The above formula is enough for studying the slice regular function $f:\Omega \in \tau _s({\mathcal {W}}_A^d)\rightarrow A$ with $\Omega$ being axially symmetric. However, if $\Omega$ is not axially symmetric, then for some $\gamma \in {\mathscr {P}}({\mathbb {C}}^1,\Omega )$ with $|{\mathcal {S}}_A(\Omega ,\gamma )|\ge 2$, there may be no $J_1,J_2\in {\mathcal {S}}_A(\Omega ,\gamma )$ such that $J_1-J_2$ is invertible. Standard path-representation formula does not work in above situation. Correspondingly, Proposition 5.4 guarantees the new one, Path-representation Formula 6.6 holds for any $\gamma \in {\mathscr {P}}({\mathbb {C}}^1,\Omega )$.

Remark 6.11

When A is a Clifford algebra, since A is associative, the left-multiplication $L_I$ by $I\in {\mathcal {S}}_A$ can be replaced by I. For example, (6.7) can be written as

$$\begin{aligned} f\circ \gamma ^I=(1,I) \begin{pmatrix} 1&{}J_1\\ 1&{}J_2 \end{pmatrix}^{-1} \begin{pmatrix} f\circ \gamma ^{J_1}\\ f\circ \gamma ^{J_2} \end{pmatrix}. \end{aligned}$$

6.4 The case of Cayley-Dickson algebras

The algebras produced by Cayley-Dickson construction [12] are known as Cayley–Dickson algebras, and examples of such algebras are complex numbers $A_1={\mathbb {C}}$, quaternions $A_2={\mathbb {H}}$, octonions $A_3={\mathbb {O}}$ and sedenions $A_4$. Slice analysis has been established in quaternions and octonions. Since Cayley-Dickson algebras are LSCS algebras, the theory in Sects. 6.1 and 6.2 apply to Cayley-Dickson algebras. However, a new concrete example can only be given in the case of sedenions $A_4$ since

$$\begin{aligned} {\mathcal {S}}_{A_4}=\left\{ a+be_8\in {\mathbb {S}}_{A_4}:ab=ba,\ \text{ for }\ a,b\in {\mathbb {O}}\right\} \end{aligned}$$

while a concrete expression for ${\mathcal {S}}_{A_k}$ is still missing for $k>4$.