Geometric function theory over quaternionic slice domains

Theorem 2.4. (Identity Principle). Let f, g be slice regular functions on a slice domain $\Omega$ . If, for some $I \in \mathbb{S}$ , f and g coincide on a subset of $\Omega_I$ having an accumulation point in $\Omega_I$ , then f = g in $\Omega$ .

Theorem 2.6. (Representation Formula). Let f be a slice regular function on a symmetric slice domain $\Omega$ and let $x + y\mathbb{S} \subset \Omega$ . For all $I, J, K \in \mathbb{S}$ with $J \neq K$ $$f(x+yI) = (J-K)^{-1} [Jf(x+yJ) - Kf(x+yK)] + (2)$$ + $I(J-K)^{-1} [f(x+yJ) - f(x+yK)].$ Moreover, the quaternion $b:=(J-K)^{-1}\left[Jf(x+yJ)-Kf(x+yK)\right]$ and the quaternion $c:=(J-K)^{-1}\left[f(x+yJ)-f(x+yK)\right]$ do not depend on J,K but only on x,y. Another version of the Representation Formula was proven in [20, Theorem 2.4], yielding that every slice regular function on a symmetric slice domain is real analytic (see [18, Proposition 7]). The next two results are also very useful. We quote the second one in the slightly corrected version of [13, Theorem 2.1].

Lemma 2.7. (Extension Lemma). Let $\Omega$ be a symmetric slice domain and let $I \in \mathbb{S}$ . If $f_I : \Omega_I \to \mathbb{H}$ is holomorphic then there exists a unique slice regular function $g : \Omega \to \mathbb{H}$ such that $g_I = f_I$ in $\Omega_I$ . The function g is denoted by $\text{ext}(f_I)$ and called the regular extension of $f_I$ .

Theorem 2.8. (Extension Formula). Let J, K be distinct imaginary units; let T be a domain in $L_J$ , such that $T_J^+$ is connected and $T \cap \mathbb{R} \neq \emptyset$ ; let $U := \{x + yK : x + yJ \in T\}$ . Choose holomorphic functions $r : T \to \mathbb{H}, s : U \to \mathbb{H}$ such that $r_{|T \cap \mathbb{R}} = s_{|U \cap \mathbb{R}}$ . Let $\Omega$ be the symmetric slice domain such that $\Omega_J^+ = T_J^+, \Omega \cap \mathbb{R} = T \cap \mathbb{R}$ and set, for all $x + yI \in \Omega$ with $x, y \in \mathbb{R}, y \geq 0$ and $I \in \mathbb{S}$ , $$f(x+yI) := (J-K)^{-1} [Jr(x+yJ) - Ks(x+yK)] + I(J-K)^{-1} [r(x+yJ) - s(x+yK)]$$ (3) The function $f: \Omega \to \mathbb{H}$ is the (unique) slice regular function on $\Omega$ that coincides with r in $\Omega_J^+$ , with s in $\Omega_K^+$ and with both r and s in $\Omega \cap \mathbb{R}$ . The study over slice domains that are not symmetric has not been further developed for several years, possibly because it was commonly believed that every slice regular function on a slice domain $\Omega$ could be extended in a unique fashion to the symmetric completion $\widetilde{\Omega}$ of $\Omega$ , in accordance with the next definition.

Lemma 2.11. Let Y be an open subset of $\mathbb{H}$ and let $J_0 \in \mathbb{S}$ . Let C be a compact and path-connected subset of $Y_{J_0}$ such that $C \cap \mathbb{R}$ is a closed interval and $\emptyset \neq C \setminus \mathbb{R} \subset Y_{J_0}^+$ . Let $q_0 \in C$ be such that $\max_{p \in C} |\operatorname{Im}(p)| = |\operatorname{Im}(q_0)|$ . Then there exists $\varepsilon > 0$ such that $$\Gamma(C,\varepsilon) := \bigcup_{p \in C \setminus \mathbb{R}} B\left(p, \frac{|\operatorname{Im}(p)|}{|\operatorname{Im}(q_0)|}\varepsilon\right) \cup \bigcup_{p \in C \cap \mathbb{R}} B(p,\varepsilon)$$ is a slice domain and $C \subset \Gamma(C, \varepsilon) \subseteq Y$ .

Theorem 2.12. (Local Extension). Let f be a slice regular function on a slice domain $\Omega$ . For every $p_0 \in \Omega$ , there exist a symmetric slice domain N with $N \cap \mathbb{R} \subset \Omega$ , a slice domain $\Lambda$ with $p_0 \in \Lambda \subseteq \Omega \cap N$ , and a slice regular function $\tilde{f}: N \to \mathbb{H}$ such that $\tilde{f}$ coincides with f in $N \cap \mathbb{R}$ , whence in $\Lambda$ . We will state and prove a stronger version of this result in Theorem 11.4. Theorem 2.12 has the following consequence.

Corollary 2.13. Every slice regular function on a slice domain is real analytic. Motivated by Theorem 2.12, we give a new definition.

Theorem 2.15. (Local Representation Formula). Let $\Omega$ be a slice domain and let $f: \Omega \to \mathbb{H}$ be a slice regular function. For all $J, K \in \mathbb{S}$ with $J \neq K$ and all $x, y \in \mathbb{R}$ with $y \geq 0$ such that $x + yJ, x + yK \in \Omega$ , let us set $$b(x+yJ, x+yK) := (J-K)^{-1} [Jf(x+yJ) - Kf(x+yK)],$$ $$c(x+yJ, x+yK) := (J-K)^{-1} [f(x+yJ) - f(x+yK)].$$ For every $p_0 \in \Omega$ , there exists a slice domain $\Lambda$ with $p_0 \in \Lambda \subseteq \Omega$ such that the following properties hold for all $x, y \in \mathbb{R}$ with $y \geq 0$ : - If $U = (x+y\mathbb{S}) \cap \Lambda$ is not empty, then b, c are constant in $(U \times U) \setminus \{(u, u) : u \in U \setminus \mathbb{R}\}.$ - If $I, J, K \in \mathbb{S}$ with $J \neq K$ are such that $x + yI, x + yJ, x + yK \in \Lambda$ , then $$f(x+yI) = b(x+yJ, x+yK) + Ic(x+yJ, x+yK).$$ (5) In the previous statement, for y=0 we get well-defined b(x,x), c(x,x) because, regardless of the choice of $J,K\in\mathbb{S}$ with $J\neq K$ , it holds $(J-K)^{-1}[Jf(x)-Kf(x)]=f(x)$ and $(J-K)^{-1}[f(x)-f(x)]=0$ . In [13], we also proved an Extension Theorem valid on a special class of slice domains.

Theorem 2.17. (Extension). Let f be a slice regular function on a simple slice domain $\Omega$ . There exists a unique slice regular function $\widetilde{f}:\widetilde{\Omega}\to\mathbb{H}$ that extends f to the symmetric completion of its domain. In [13], we observed that the slice domain $\Omega$ of Example 2.10 is not simple. We also described, without giving a formal proof, a large class of examples of simple domains. We include a proof here.

Proposition 2.18. If an open subset $\Omega$ of $\mathbb{H}$ is starlike with respect to a point $x_0 \in \Omega \cap \mathbb{R}$ , then it is a simple slice domain.

Proposition 3.5. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. Pick $p \in \Omega$ and let $df_p$ denote the real differential of f at p. - 1. If $p \in \Omega \cap \mathbb{R}$ , then $df_p(v) = vf'_c(p)$ for all $v \in T_p\Omega \simeq \mathbb{H}$ . As a consequence, $df_p$ is singular if, and only if, $f'_c(p) = 0$ . - 2. If $p \in \Omega \setminus \mathbb{R}$ , if $I \in \mathbb{S}$ is such that $p \in \Omega_I$ and if we split $T_p\Omega \simeq \mathbb{H}$ as $L_I \oplus L_I^{\perp}$ , then $$df_p(v+w) = vf'_c(p) + wf'_s(p)$$ for all $v \in L_I$ and $w \in L_I^{\perp}$ . As a consequence, $df_p$ is singular if, and only if, $f'_c(p)\overline{f'_s(p)} \in L_I^{\perp}$ .

Theorem 4.3. Let $\Omega$ be a slice domain in $\mathbb{H}$ . Then $+,,^c$ are operations on the set of slice regular functions on $\Omega$ and turn it into a real \-algebra. Moreover, if $f,g:\Omega\to\mathbb{H}$ are slice regular functions, then the following properties hold: - 1. if f is slice preserving, then f \ g(q) = f(q)g(q) = g \ f(q) in $\Omega$ ; - 2. if g is constantly equal to c, then f \* g(q) = f(q)c for all $q \in \Omega$ ; - 3. $f^s$ is a slice preserving regular function and $$f^s = f f^c = f^c f.$$

Proposition 4.4. Let $\Omega$ be a slice domain in $\mathbb{H}$ , let $f: \Omega \to \mathbb{H}$ be a slice regular function and pick $p = x + yI \in \Omega$ . If f(p) = 0, then f \* g(p) = 0. Otherwise, the point $f(p)^{-1}pf(p)$ belongs to $x + y\mathbb{S}$ and $$f * g(p) = f(p) \widetilde{g}(f(p)^{-1}pf(p))$$ where $\widetilde{g}(q) = g_s^{\circ}(p) + \operatorname{Im}(q)g_s'(p)$ for all $q \in x + y\mathbb{S}$ (whence $\widetilde{g}$ coincides with g in the connected component of $(x + y\mathbb{S}) \cap \Omega$ that includes p). Proof. In the previous proof, we had found an extension triplet $(\widetilde{f}, N, \Lambda)$ for f with $p \in \Lambda$ and an extension triplet $(\widetilde{g}, N', \Lambda')$ for $g_{|_{\Lambda}} : \Lambda \to \mathbb{H}$ with $p \in \Lambda'$ . We had also found a symmetric slice domain M with $p \in \Lambda' \subseteq M \subseteq N \cap N'$ , such that the functions f \ g and $\widetilde{f}_{|_{M}} \widetilde{g}_{|_{M}}$ coincided in $\Lambda'$ . In particular, $f g(p) = \widetilde{f}_{|_{M}} \widetilde{g}_{|_{M}}(p)$ . Let us apply [14, Theorem 3.4] to $\widetilde{f}_{|_{M}}$ , $\widetilde{g}_{|_{M}}$ : if $\widetilde{f}(p) = 0$ , then $\widetilde{f}_{|_{M}} * \widetilde{g}_{|_{M}}(p) = 0$ ; otherwise, $$\widetilde{f}_{|_{M}}*\widetilde{g}_{|_{M}}(p)=\widetilde{f}(p)\,\widetilde{g}\left(\widetilde{f}(p)^{-1}p\widetilde{f}(p)\right)$$ Remembering that $\widetilde{f}(p) = f(p)$ , we derive the first two equalities appearing in our statement. Moreover, by construction, for all $q \in x + y\mathbb{S}$ it holds $\widetilde{g}(q) = g_s^{\circ}(p) + \operatorname{Im}(q)g_s'(p)$ . The latter expression coincides with g(q) for all q in the connected component of $(x + y\mathbb{S}) \cap \Omega$ that includes p. We conclude this section with the following remarks, though not essential for the present work. If we modify Definition 4.1 assuming f,g to be locally slice functions on $Y \subseteq \mathbb{H}$ , we get well-defined locally slice functions $f g, f^c, f^s : Y \to \mathbb{H}$ , which we may call slice product of f and g, slice conjugate of f and normal function* of f. This generalizes [18, Definitions 9 and 11]. Analogs of Theorem 4.3 and Proposition 4.4 hold.

Proposition 5.1. Let $\Omega$ be a slice domain in $\mathbb{H}$ , let $f: \Omega \to \mathbb{H}$ be a slice regular function and consider its zero set Z(f). Fix any 2-sphere $S = x + y\mathbb{S}$ (with $x, y \in \mathbb{R}, y \neq 0$ ). For $p \in S$ , the equality f(p) = 0 is equivalent to $$f_s^{\circ}(p) = -\operatorname{Im}(p)f_s'(p)$$ . If the previous equality holds and if C denotes the connected component of $\Omega \cap S$ that includes p, then either - 1. $f'_s(p) \neq 0$ and $Z(f) \cap C = \{p\}$ ; or - 2. $f'_s(p) = 0$ and $C \subseteq Z(f)$ . The former possibility is excluded if f is slice preserving.

Theorem 5.2. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. Either $f \equiv 0$ or, for any $S = x + y\mathbb{S}$ intersecting $\Omega$ and for any connected component C of $S \cap \Omega$ , the set $Z(f) \setminus C$ has no accumulation points in C.

Corollary 5.3. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function with $f \not\equiv 0$ . Suppose $p \in Z(f)$ , let $S = x + y\mathbb{S}$ include p and let C be connected component of $S \cap \Omega$ that includes p. If either $p \in \mathbb{R}$ or $f'_s(p) \neq 0$ , then p is an isolated point in Z(f). If, instead, $p \not\in \mathbb{R}$ and $f'_s(p) = 0$ , then Z(f) includes C and $Z(f) \setminus C$ has no accumulation points in C.

Proposition 5.4. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f,g:\Omega\to\mathbb{H}$ be slice regular functions. Then $$Z(f) \subseteq Z(f * g)$$ . Moreover. $$Z(f * g) \setminus Z(f) = \{ p \in \Omega : g_s^{\circ}(p) = -f(p)^{-1} \operatorname{Im}(p) f(p) g_s'(p) \}.$$

Proposition 5.6. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. The zero sets Z(f) and $Z(f^s)$ are related, as follows: - 1. Fix any 2-sphere $S = x + y\mathbb{S}$ (with $x, y \in \mathbb{R}, y \neq 0$ ). If $p \in Z(f)$ and if C denotes the connected component of $\Omega \cap S$ that includes p, then $C \subseteq Z(f^s)$ . - 2. The equality $Z(f) \cap \mathbb{R} = Z(f^s) \cap \mathbb{R}$ holds. Thus, if $f^s \equiv 0$ , then $f \equiv 0$ . - Proof. 1. Since $f^s = f * f^c$ , Proposition 5.4 implies that $Z(f) \subseteq Z(f^s)$ . Since $f^s$ is slice preserving, by Proposition 5.1, if $p \in Z(f^s)$ and if C denotes the connected component of $\Omega \cap S$ that includes p, then $C \subseteq Z(f^s)$ . - 2. If $q \in \Omega \cap \mathbb{R}$ , then $f^s(q) = |f(q)|^2$ vanishes if, and only if, f(q) = 0. Now, if $f^s \equiv 0$ , then $\Omega \cap \mathbb{R} \subseteq Z(f)$ . By the Identity Principle 2.4, it follows that $f \equiv 0$ . For future reference, we make the following remark. Remark 5.7. Let $\Omega$ be a slice domain in $\mathbb{H}$ . If $f:\Omega\to\mathbb{H}$ is a slice regular function with $f\not\equiv 0$ , then we can combine several results in this section to conclude that $Z(f^s)$ consists of: isolated real points; or whole connected components of $(x+y\mathbb{S})\cap\Omega$ , each isolated from the rest of $Z(f^s)$ . As a consequence, $\Omega\setminus Z(f^s)$ is a slice domain. Although not strictly needed for the present work, we can make the following remarks. Proposition 5.1, Proposition 5.4 and Remark 5.5 still hold true if we relax the assumptions on f, g, assuming them to be locally slice functions on an arbitrary subset Y of $\mathbb{H}$ . Under the same assumptions, property 1. in Proposition 5.6 holds true, while property 2. is reduced to the equality $Z(f) \cap \mathbb{R} = Z(f^s) \cap \mathbb{R}$ . Indeed, it may well happen that $f^s \equiv 0$ while $f \not\equiv 0$ : Example 5.8. Consider the slice regular functions $f_0$ and $f_1$ defined in Example 2.10 and their difference $D := f_1 - f_0$ on the interior of the solid torus $\widetilde{C}_1$ . For $x, y \in \mathbb{R}$ with $y \geq 0$ and $J \in \mathbb{S}$ such that x + Jy belongs to the interior of $\widetilde{C}_1$ , it holds $D(x + Jy) = \pi(I + J) = \pi I + yJ\frac{\pi}{y}$ . Thus, $D^c(x + Jy) = \pi I - yJ\frac{\pi}{y}$ $$D^{s}(x + Jy) = \pi^{2} - y^{2} \frac{\pi^{2}}{y^{2}} + 2yJ \operatorname{Re}\left(-\frac{\pi^{2}}{y}I\right) \equiv 0.$$

Proposition 6.2. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function such that $f \not\equiv 0$ . Then $f^{-*}$ is a slice regular function on $\Omega' = \Omega \setminus Z(f^s)$ and the following equalities hold in $\Omega'$ : $$f^{-}(q) = (f^s)^{-} * f^c(q) = f^s(q)^{-1} f^c(q),$$ $$f f^{-} = f^{-} f \equiv 1.$$ Moreover, if f is slice preserving, then $f^{-}$ is slice preserving and $f^{-}(q) = f(q)^{-1}$ for all $q \in \Omega'$ .

Proposition 6.6. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f, g : \Omega \to \mathbb{H}$ be slice regular functions. For each $p = x + yI \in \Omega \setminus Z(f^s)$ , it holds $$f^{-} g(p) = \widetilde{f}(T_f(p))^{-1} \widetilde{g}(T_f(p))$$ where $T_f(p) := f^c(p)^{-1} p f^c(p)$ , while $\widetilde{f}(q) = f_s^{\circ}(p) + \operatorname{Im}(q) f_s'(p)$ and $\widetilde{g}(q) = g_s^{\circ}(p) + \operatorname{Im}(q) g_s'(p)$ for all $q \in x + y \mathbb{S}$ .

Theorem 7.1. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. Fix a 2-sphere $S = x_0 + y_0 \mathbb{S}$ (with $x_0, y_0 \in \mathbb{R}$ and $y_0 > 0$ ) that intersects $\Omega$ and consider the slice domain $\Omega' := \Omega \setminus \mathcal{O}(S)$ . Then, there exists a slice regular function $h: \Omega' \to \mathbb{H}$ such that $$f(q) = [(q - x_0)^2 + y_0^2] * h(q) = [(q - x_0)^2 + y_0^2]h(q)$$ throughout $\Omega'$ . Moreover, h cannot be continuously extended to any open set intersecting $\mathcal{O}(S)$ .

Theorem 7.2. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. Fix $p \in \Omega$ and consider the slice domain $\Omega' := \Omega \setminus \mathcal{O}(p)$ . Then there exists a slice regular function $g: \Omega' \to \mathbb{H}$ such that $$f(q) = (q - p) * g(q)$$ throughout $\Omega'$ . Moreover, for any slice domain $\Lambda$ that intersects $\mathcal{O}(p)$ , there cannot exist a slice regular $g: \Lambda \to \mathbb{H}$ such that the previous equality holds in $\Lambda$ .

Proposition 7.3. Let $\Omega$ be a slice domain in $\mathbb{H}$ , let $f: \Omega \to \mathbb{H}$ be a slice regular function and fix $x \in \Omega \cap \mathbb{R}$ . It holds f(x) = 0 if, and only if, there exists a slice regular function $g: \Omega \to \mathbb{H}$ such that $$f(q) = (q - x) * g(q) = (q - x)g(q)$$ throughout $\Omega$ .

Proposition 8.4. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f, g : \Omega \to \mathbb{H}$ be slice regular functions. Fix an $S := x + y\mathbb{S}$ intersecting $\Omega$ and a connected component C of the intersection $S \cap \Omega$ . - 1. Z(f\*g) is the union between Z(f) and the set of points $p \in \Omega \setminus Z(f)$ such that $q f(p)^{-1}pf(p)$ divides g(q) near p. - 2. $Z(f^s)$ intersects C if, and only if, it includes it; this happens if, and only if, there exists $\widetilde{p} \in S$ such that $q \widetilde{p}$ divides f(q) near C. - 3. There exists $\widetilde{p} \in S$ such that $q \widetilde{p}$ divides f(q) near C if, and only if, there exists $\widehat{p} \in S$ such that $q \widehat{p}$ divides $f^c(q)$ near C.

Theorem 8.6. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function with $f \not\equiv 0$ . Fix an $S:=x+y\mathbb{S}$ (with $x,y\in\mathbb{R},y>0$ ) intersecting $\Omega$ and a connected component C of the intersection $S\cap\Omega$ . There exist $m\in\mathbb{N}$ such that $[(q-x)^2+y^2]^m$ divides f(q) near C but $[(q-x)^2+y^2]^{m+1}$ does not. Moreover, for the function h such that $$f(q) = [(q-x)^2 + y^2]^m h(q)$$ near C, there exist a number $n \in \mathbb{N}$ , points $p_1, \ldots, p_n \in S$ (with $p_i \neq \bar{p}_{i+1}$ for all $i \in \{1, \ldots, n-1\}$ ), and a slice regular function g (with $g_{l,c}^s \not\equiv 0$ ) such that $$h(q) = (q - p_1) (q - p_2) \cdots (q - p_n) g(q)$$ $near\ C$ .

Theorem 8.8. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. Fix $p \in \Omega$ and set h := f - f(p). - 1. If $p \in \mathbb{R}$ , then $df_p$ is singular if, and only if, the Cullen derivative $f'_c(p)$ vanishes. This happens if, and only if, $m_h(p) \geq 2$ . - 2. Suppose, instead, p = x + yI with $x, y \in \mathbb{R}$ and y > 0 and let C denote the connected component of $(x + y\mathbb{S}) \cap \Omega$ including p. Let us denote by 2m the spherical multiplicity of h at C and by n the isolated multiplicity of h at p, relative to p. Then p is singular if, and only if, p if, p is p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in p in

Theorem 9.2. Choose any sequence $\{a_n\}_{n\in\mathbb{Z}}$ in $\mathbb{H}$ . Let $R_1,R_2\in[0,+\infty]$ be such that $R_1=\limsup_{m\to+\infty}|a_{-m}|^{1/m},1/R_2=\limsup_{n\to+\infty}|a_n|^{1/n}$ . For all $p\in\mathbb{H}$ the regular Laurent series centered at p associated to $\{a_n\}_{n\in\mathbb{Z}}$ , namely, $$f(q) = \sum_{n \in \mathbb{Z}} (q - p)^{*n} a_n,$$ converges absolutely and uniformly on the compact subsets of $\Sigma(p, R_1, R_2)$ and it does not converge at any point of $T(p, R_1)$ nor at any point of $\mathbb{H} \setminus \overline{\Sigma(p, R_2)}$ . Furthermore: if $\Omega(p, R_1, R_2) \neq \emptyset$ , then the sum of the series defines a slice regular function $f: \Omega(p, R_1, R_2) \to \mathbb{H}$ .

Theorem 9.3. (Regular Laurent Expansion). Let $\Omega$ be a domain in $\mathbb{H}$ , let $f: \Omega \to \mathbb{H}$ be a slice regular function and let $p \in \mathbb{H}$ . There exists a sequence $\{a_n\}_{n\in\mathbb{Z}}$ in $\mathbb{H}$ such that, for all $0 \le R_1 < R_2 \le +\infty$ with $\Sigma(p, R_1, R_2) \subseteq \Omega$ , $$f(q) = \sum_{n \in \mathbb{Z}} (q - p)^{*n} a_n \tag{6}$$ in $\Sigma(p, R_1, R_2)$ . If, moreover, $\Sigma(p, R_2) \subseteq \Omega$ , then $a_n = 0$ for all n < 0 and equality (6) holds throughout $\Sigma(p, R_2)$ . We now give some new definitions and results. For the case of symmetric slice domains, see [14, §5.3] (derived from [27, 30]).

Theorem 9.7. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let f be semiregular in $\Omega$ . Fix $p = x + yI \in \Omega$ and let C denote the connected component of $(x + y\mathbb{S}) \cap \Omega$ that includes p. Let $m := \operatorname{ord}_f(p)$ and $n := \max_{\widetilde{p} \in C} \operatorname{ord}_f(\widetilde{p})$ . There exist a slice domain $\Lambda$ with $C \subseteq \Lambda \subseteq \Omega$ and a slice regular function $g : \Lambda \to \mathbb{H}$ such that $$f(q) = [(q-x)^2 + y^2]^{-n} (q-p)^{(n-m)} g(q)$$ in $\Lambda \setminus (x + y\mathbb{S})$ . Moreover, if n > 0 then $g(p) \neq 0$ .

Proposition 9.8. The set of semiregular functions on a slice domain $\Omega$ is a real \-algebra with respect to +, \, °. Moreover, it is a division ring.

Theorem 9.9. Let $\Omega$ be a slice domain in $\mathbb{H}$ , let f be semiregular in $\Omega$ and suppose $f \not\equiv 0$ . Fix an $S := x + y\mathbb{S}$ intersecting $\Omega$ and a connected component C of $S \cap \Omega$ . There exist $m \in \mathbb{Z}, n \in \mathbb{N}, p_1, ..., p_n \in S$ (with $p_i \neq \bar{p}_{i+1}$ for all $i \in \{1, ..., n-1\}$ ) such that $$f(q) = [(q-x)^2 + y^2]^m (q-p_1) (q-p_2) \dots (q-p_n) g(q)$$ (7) in $\Omega$ , for some semiregular function g on $\Omega$ that is slice regular near C and has $g_{|C}^s \not\equiv 0$ .

Theorem 10.1. (Maximum Modulus Principle). Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. If |f| has a relative maximum at $p \in \Omega$ , then f is constant. We now prove the Minimum Modulus Principle over slice domains, exploiting the analogous result for symmetric slice domains, namely [14, Theorem 7.3].

Theorem 10.2. (Minimum Modulus Principle). Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. If |f| has a relative minimum at $p \in \Omega$ , then either f(p) = 0 or f is constant. Proof. We can apply Theorem 2.12 to find an extension triplet $(\tilde{f}, N, \Lambda)$ for f, with $p \in \Lambda$ . Clearly, $|\tilde{f}|$ has a relative minimum at p. By [14, Theorem 7.3], either $\tilde{f}(p) = 0$ or $\tilde{f}$ is constant. In the former case, f(p) = 0. In the latter case, f is constant in $\Lambda$ , whence in $\Omega$ by the Identity Principle 2.4. We now come to the Open Mapping Theorem. For the case of symmetric slice domains, see [14, §7.2]. We begin with some preliminary material.

Proposition 10.4. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. Either f is constant or $D_f$ is a proper analytic subset of $\Omega \setminus \mathbb{R}$ . In the latter case, the dimension of $D_f$ cannot exceed 3. Proof. Since $f'_s$ is a real analytic function, $D_f$ is an analytic subset of $\Omega \setminus \mathbb{R}$ . If this subset is proper, then its dimension is less than, or equal to, 3. Assume, instead, $D_f = \Omega \setminus \mathbb{R}$ . Consider an extension triplet $(\widetilde{f}, N, \Lambda)$ for f. Clearly, $D_{\widetilde{f}}$ includes the nonempty open set $\Lambda \setminus \mathbb{R}$ . If we apply [21, Theorem 6.4] (or [1, Proposition 4.12]) to the function $\widetilde{f}$ on the symmetric slice domain N, we can conclude that $\widetilde{f}$ is constant. Thus, f is constant in $\Lambda$ , whence in $\Omega$ by the Identity Principle 2.4. We point out that, when $\Omega$ is not symmetric, the degenerate set $D_f$ needs not be symmetric. Indeed, the spherical derivative $f_s'$ may vanish identically on a connected component of $(x+y\mathbb{S})\cap\Omega$ while not vanishing elsewhere in $(x+y\mathbb{S})\cap\Omega$ . This happens, for instance, in Example 8.2. We are now ready for the announced theorem. In the statement, $\overline{D_f}$ denotes the closure in $\Omega$ of $D_f \subseteq \Omega \setminus \mathbb{R}$ . We point out that the added set $\overline{D_f} \setminus D_f$ is included in the intersection between the singular set of f and $\mathbb{R}$ , whence in the discrete set $Z(f'_c) \cap \mathbb{R}$ .

Theorem 10.5. (Open Mapping). Let $\Omega$ be a slice domain in $\mathbb{H}$ , let $f: \Omega \to \mathbb{H}$ be a nonconstant slice regular function and let $D_f$ be its degenerate set. Then $f: \Omega \setminus \overline{D_f} \to \mathbb{H}$ is open. Proof. Let U be an open subset of $\Omega \setminus \overline{D_f}$ , pick $q_0 \in U$ and let us prove that $p_0 = f(q_0)$ is an interior point of f(U). To do so, we will find $\varepsilon > 0$ such that f(U) includes the open Euclidean ball $B(p_0, \varepsilon)$ of radius $\varepsilon$ centered at $p_0$ . By construction, either $q_0 \in \mathbb{R}$ or $f'_s(q_0) \neq 0$ . In either case, by Corollary 5.3, the point $q_0$ is an isolated zero of the function $f(q) - p_0$ . Thus, there exists r > 0 such that $\overline{B(q_0, r)} \subseteq U$ and $f(q) - p_0 \neq 0$ for all $q \in \overline{B(q_0, r)} \setminus \{q_0\}$ . This implies that $$\min_{|q-q_0|=r} |f(q) - p_0| \ge 3\varepsilon$$ for some $\varepsilon > 0$ . If $p \in B(p_0, \varepsilon)$ and if $|q - q_0| = r$ , it holds $$|f(q) - p| \ge |f(q) - p_0| - |p - p_0| \ge 3\varepsilon - \varepsilon = 2\varepsilon > \varepsilon \ge |p_0 - p| = |f(q_0) - p|.$$ It follows that $$\min_{|q-q_0|=r} |f(q)-p| > |f(q_0)-p|,$$ whence |f(q)-p| has a local minimum point $q_1 \in B(q_0,r)$ . Taking into account that f(q)-p is not constant, Theorem 10.2 implies that $f(q_1)=p$ . We conclude that $f(U) \supseteq B(p_0,\varepsilon)$ , as desired.

Lemma 11.1. Let $f: \Omega \to \mathbb{H}$ be a slice regular function, let $I \in \mathbb{S}$ and let $U_I$ be a bounded Jordan domain in $L_I$ , with $\overline{U_I} \subset \Omega_I$ . If $\partial U_I$ is rectifiable then $$f(z) = \frac{1}{2\pi I} \int_{\partial U_L} \frac{ds}{s - z} f(s)$$ for all $z \in U_I$ . For the previous statement, the proof of [14, Lemma 6.3] holds verbatim without assuming $\Omega$ to be a symmetric slice domain (an assumption made in the original statement). We now derive a local version of the Cauchy Formula, valid on all slice domains. Another version will be provided in Theorem 11.5. For each $s \in \mathbb{H}$ , let $(s-q)^{-*}$ denote the regular reciprocal of $q \mapsto s - q$ , i.e. $$(s-q)^{-*} = (|s|^2 - q2Re(s) + q^2)^{-1}(\bar{s}-q).$$

Proposition 11.2. (Local Cauchy Formula I). Let $\Omega$ be a slice domain in $\mathbb{H}$ , let $f: \Omega \to \mathbb{H}$ be a slice regular function and let $(\widetilde{f}, N, \Lambda)$ be an extension triplet for f. If U is a bounded symmetric open subset of $\mathbb{H}$ with $\overline{U} \subseteq N$ , if $I \in \mathbb{S}$ and if the boundary $\partial U_I$ is a finite union of disjoint rectifiable Jordan curves, then $$f(q) = \int_{\partial U_I} (s - q)^{-*} (2\pi I)^{-1} ds \, \widetilde{f}(s)$$ for all $q \in U \cap \Lambda$ .

Proposition 11.3. (Local Volume Cauchy Formula). Let $\Omega$ be a slice domain in $\mathbb{H}$ , let $f: \Omega \to \mathbb{H}$ be a slice regular function and let $(\widetilde{f}, N, \Lambda)$ be an extension triplet for f. Let U be a bounded symmetric open subset of $\mathbb{H}$ with $\overline{U} \subseteq N$ and assume the boundary $\partial U$ to be $C^1$ . For $w \in \partial U$ , let $\mathbf{n}(w)$ denote the outer normal versor of $\partial U$ at w and let $d\sigma_w$ denote the standard 3-volume form on $\partial U$ . Then $$f(q) = \int_{\partial U} C(q, w) \mathbf{n}(w) \widetilde{f}(w) d\sigma_w$$ for all $q \in U \cap \Lambda$ .

Theorem 11.4. (Local Extension). Let f be a slice regular function on a slice domain $\Omega$ and let $J_0 \in \mathbb{S}$ . Let $\mathcal{C}$ be a symmetric, compact and path-connected subset of $\mathbb{H}$ such that $$\emptyset \neq \mathcal{C}_{J_0}^+ \subset \Omega_{J_0}^+$$ . If $C \cap \mathbb{R}$ is not empty, suppose it is a closed interval included in $\Omega$ . Then there exist an extension triplet $(\widetilde{f}, N, \Lambda)$ for f and a real number $\delta > 0$ such that $$\bigcup_{|J-J_0|\leq \delta} \overline{\mathcal{C}_J^+} \subset \Lambda.$$ In particular, $\widetilde{f}$ coincides with f in a neighborhood of $\overline{\mathcal{C}_{J_0}^+}$

Theorem 11.5. (Local Cauchy Formula II). Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f:\Omega\to\mathbb{H}$ be a slice regular function. Let U be a symmetric open subset of $\mathbb{H}$ such that $\overline{U}$ is compact and path-connected. If $\overline{U}\cap\mathbb{R}\neq\emptyset$ , we assume the same intersection to be a closed interval of $\mathbb{R}$ included in $\Omega$ . Suppose, for some $J_0\in\mathbb{S}$ , that the boundary $\partial U_{J_0}$ is a finite union of disjoint rectifiable Jordan curves and that $$\overline{U}_{J_0}^+ \subset \Omega_{J_0}^+$$ . Then, there exists a real number $\varepsilon > 0$ such that, for all $I \in \mathbb{S}$ and for all $$q \in \bigcup_{|J-J_0|<\varepsilon} U_J^+ \cup (U \cap \mathbb{R}),$$ it holds $$f(q) = \int_{\partial U_I} (s - q)^{-*} (2\pi I)^{-1} ds \, \widetilde{f}(s) \,,$$ where, for all $x + yJ \in \partial U$ (with $x, y \in \mathbb{R}, y > 0$ and $J \in \mathbb{S}$ ), it holds $$\widetilde{f}(x+yJ) = f_s^{\circ}(x+yJ_0) + yJf_s'(x+yJ_0)$$ and for all $x \in \partial U \cap \mathbb{R}$ it holds $\widetilde{f}(x) = f_s^{\circ}(x) = f(x)$ .

Theorem 12.1. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f:\Omega \to \mathbb{H}$ be a slice regular function. If $x_0 + y_0 \mathbb{S}$ intersects $\Omega$ and if C is a connected component of the intersection, then there exist an open neighborhood U of C and quaternions $\{a_k\}_{k\in\mathbb{N}}$ such that $$f(q) = \sum_{n \in \mathbb{N}} [(q - x_0)^2 + y_0^2]^n [a_{2n} + qa_{2n+1}]$$ for all $q \in U$ .

Theorem 12.2. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f:\Omega\to\mathbb{H}$ be a slice regular function. Suppose $x_0,y_0,r_1,r_2\in\mathbb{R}$ to fulfill the inequalities $0< r_1< r_2\leq y_0$ and set $U:=U(x_0+y_0\mathbb{S},r_1,r_2)$ . If there exists $J_0\in\mathbb{S}$ such that $$\overline{U_{I_0}^+} \subset \Omega$$ , then there exist a real number $\varepsilon > 0$ and quaternions $\{a_k\}_{k \in \mathbb{Z}}$ such that $$f(q) = \sum_{n \in \mathbb{Z}} [(q - x_0)^2 + y_0^2]^n [a_{2n} + qa_{2n+1}]$$ for all q belonging to the open neighborhood $$\bigcup_{|J-J_0|<\varepsilon} U_J^+$$ of $U_{J_0}^+$ .

Theorem 12.4. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. Suppose $x_0, y_0, r_2 \in \mathbb{R}$ to fulfill the inequalities $0 < r_2 \le y_0$ and set $$V := U(x_0 + y_0 \mathbb{S}, 0, r_2) = U(x_0 + y_0 \mathbb{S}, r_2) \setminus (x_0 + y_0 \mathbb{S}).$$ If there exist $I_0 \in \mathbb{S}$ and $\delta > 0$ such that $$\bigcup_{|J-I_0|<\delta} \overline{V_J^+} \setminus \{x_0+y_0J\} \subset \Omega\,,$$ then there exist quaternions $\{a_k\}_{k\in\mathbb{Z}}$ such that $$f(q) = \sum_{n \in \mathbb{Z}} [(q - x_0)^2 + y_0^2]^n [a_{2n} + qa_{2n+1}]$$ for all q belonging to $$\bigcup_{|J-I_0|<\delta} V_J^+ \, .$$

Definition 2.1. Let f be a quaternion-valued function defined on a domain $\Omega$ . For each $I \in \mathbb{S}$ , let $f_I := f_{|_{\Omega_I}}$ be the restriction of f to $\Omega_I$ . The restriction $f_I$ is called holomorphic if it has continuous partial derivatives and $$\bar{\partial}_I f(x+yI) := \frac{1}{2} \left( \frac{\partial}{\partial x} + I \frac{\partial}{\partial y} \right) f_I(x+yI) \equiv 0.$$ (1) The function f is called slice regular if, for all $I \in \mathbb{S}$ , $f_I$ is holomorphic. For future reference, we recall the following definition, valid for any slice regular function $f: \Omega \to \mathbb{H}$ . After setting $$\partial_I f(x+yI) := \frac{1}{2} \left( \frac{\partial}{\partial x} - I \frac{\partial}{\partial y} \right) f_I(x+yI)$$ for $x + yI \in \Omega_I$ , the Cullen derivative (or complex derivative) $f'_c : \Omega \to \mathbb{H}$ of f is defined as the slice regular function whose restriction $(f'_c)_I$ equals $\partial_I f$ for each $I \in \mathbb{S}$ . Example 2.2. For any $n \in \mathbb{N}$ and any choice of $a_0, a_1, \ldots, a_n$ in $\mathbb{H}$ , the quaternionic polynomial $f(q) = a_0 + qa_1 + \ldots + q^n a_n$ is a slice regular function $\mathbb{H} \to \mathbb{H}$ . It holds $f'_c(q) = a_1 + q2a_2 + \ldots + q^{n-1}na_n$ . Power series centered at 0 provide further examples of slice regular functions, see [14, Theorem 1.6]. The analogs at points p other than 0 have been studied and slice regular functions turned out to admit power series expansions at all points p of their domains, in an appropriate sense, see [14, Chapter 2] (derived from [12]). The latter property is subsumed in Theorem 9.3 of the present work. The theory of slice regular functions turned out to be particularly interesting over the class of domains specified in the next definition, because of the subsequent results.

Definition 2.3. Let $\Omega$ be a domain in $\mathbb{H}$ that intersects the real axis. $\Omega$ is called a slice domain if, for all $I \in \mathbb{S}$ , the intersection $\Omega_I$ with the complex plane $L_I$ is a domain of $L_I$ .

Definition 2.5. A set $T \subseteq \mathbb{H}$ is called (axially) symmetric if, for all points $x+yI \in T$ with $x,y \in \mathbb{R}$ and $I \in \mathbb{S}$ , the set T contains the whole sphere $x+y\mathbb{S}$ .

Definition 2.9. The (axially) symmetric completion of a set $T \subseteq \mathbb{H}$ is the smallest symmetric set $\widetilde{T}$ that contains T. In other words, $$\widetilde{T} := \bigcup_{x+yI \in T} (x+yS). \tag{4}$$ However, the recent work [5] proved that not every slice regular function on a slice domain $\Omega$ extends to $\widetilde{\Omega}$ , providing the following counterexample. Example 2.10. Fix an imaginary unit $I \in \mathbb{S}$ . For $J \in \mathbb{S}$ , consider the half line $$h_J := (-\infty, -2] + 2J = \{s + 2J : s \in (-\infty, -2]\}$$ and the closed disk $$D_J := \{ z \in L_J : |z + 1 - 2J| \le 1 \}.$$ For any $t \in [0,1]$ and all $J \in \mathbb{S}$ , consider the following arc, with endpoints -2 + 2J and 2J, within $D_J$ : $$\alpha_{t,J} := -1 + 2J + \{(1-t)e^{2\pi Js} + te^{-2\pi Js} : s \in [0, 1/2]\}.$$ In particular: $\alpha_{0,J}$ is the upper half of the circle $\partial D_J$ ; $\alpha_{\frac{1}{2},J}$ is the line segment [-2,0]+2J; and $\alpha_{1,J}$ is the lower half of the circle $\partial D_J$ within $L_J^+$ . For $t \in [0,1]$ , consider the unique holomorphic function $\phi_t : L_I \setminus (h_I \cup \alpha_{t,I}) \to L_I$ such that $\phi_t(x+2I) = \ln(x)$ for all $x \in (0,+\infty)$ . For all $x, y \in \mathbb{R}$ such that $x + yI \notin h_I \cup \alpha_{t,I}$ , it holds $$\phi_t(x+yI) := \ln \sqrt{x^2 + (y-2)^2} + I \arg_t(x+I(y-2)),$$ where $\arg_t$ denotes the argument on the domain obtained from $L_I$ by erasing both the half line $(-\infty, -2]$ and the arc $\alpha_{t,I} - 2I$ . Note that $\phi_t$ cannot be holomorphically extended through any point of $\alpha_{t,I}$ . For each $t \in (0,1]$ , let $C_t$ denote the closed subset of $D_I$ bounded by $\alpha_{0,I}$ and $\alpha_{t,I}$ . Then the intersection of the domains of $\phi_t$ and $\phi_0$ , namely $L_I \setminus (h_I \cup \alpha_{t,I} \cup \alpha_{0,I})$ , has two connected components: the interior of $C_t$ and the set $L_I \setminus (h_I \cup C_t)$ . By direct inspection, $\phi_t$ and $\phi_0$ coincide in $L_I \setminus (h_I \cup C_t)$ , while $\phi_t - \phi_0 \equiv 2\pi I$ in the interior of $C_t$ . If we denote the restriction of $\phi_t$ to $\Phi_t := L_I \setminus (h_I \cup \overline{h_I} \cup \alpha_{t,I} \cup \overline{\alpha_{t,I}})$ again by $\phi_t$ , we may set $$f_t := \operatorname{ext}(\phi_t) : \widetilde{\Phi}_t \to \mathbb{H}$$ . For each $t \in (0,1]$ , the intersection $\widetilde{\Phi}_t \cap \widetilde{\Phi}_0$ has two connected components: the interior of $\widetilde{C}_t$ and $\mathbb{H} \setminus (\widetilde{h_I} \cup \widetilde{C}_t)$ . Clearly, $f_t$ and $f_0$ coincide in the latter connected component. Moreover, $$f_t(x+Jy) - f_0(x+Jy) = (2I)^{-1} [I2\pi I + I0] + J(2I)^{-1} [2\pi I - 0]$$ = $\pi(I+J)$ for all $x, y \in \mathbb{R}$ with $y \geq 0$ and for all $J \in \mathbb{S}$ such that x + Jy belongs to the interior of $\widetilde{C}_t$ . In particular, the difference is zero when J = -I. Let $\Omega$ be the slice domain with the following properties: $\Omega \supset \mathbb{R}$ ; moreover, for every $J \in \mathbb{S}$ , $$\Omega_J^+ := L_J^+ \setminus (h_J \cup a_J)$$ where, after setting $T(J) := \min\{|J - I|, 1\}$ , the arc $a_J$ is defined to equal $\alpha_{T(J),J}$ . In particular, $a_I = \alpha_{0,I}$ is the upper half of the circle $\partial D_I$ within $L_I^+$ and, for every imaginary unit J with $|J - I| \ge 1$ (including J = -I), $a_J = \alpha_{1,J}$ is the lower half of the circle $\partial D_J$ within $L_I^+$ . Define the slice regular function $f: \Omega \to \mathbb{H}$ as follows: for each $J \in \mathbb{S}$ , f is defined to equal $f_{T(J)}$ in $\Omega_J^+$ ; in $\mathbb{R}$ , f equals $f_0$ (whence all $f_t$ with $t \in [0,1]$ ). Then f admits no regular extension to $\widetilde{\Omega} = \mathbb{H} \setminus (\widetilde{h_I} \cup \{2I\})$ : if it did, then $f_I = (f_0)_I = \phi_0$ would extend holomorphically through $a_I = \alpha_{0,I}$ minus its endpoints; we already noted that this is impossible. In [13], we proved the following results, valid on all quaternionic slice domains.

Definition 2.14. Let f be a slice regular function on a slice domain $\Omega$ . If $\widetilde{f}$ is a slice regular function on a symmetric slice domain N with $N \cap \mathbb{R} \subset \Omega$ and if there exists a slice domain $\Lambda \subseteq \Omega \cap N$ where f and $\widetilde{f}$ coincide, then $(\widetilde{f}, N, \Lambda)$ is called an extension triplet for f. In [13], we proved the next result.

Definition 2.16. A slice domain $\Omega$ is simple if, for any choice of $J, K \in \mathbb{S}$ , the set $$\Omega_{J,K}^+:=\{x+yJ\in\Omega_J^+:x+yK\in\Omega_K^+\}$$ is connected.

Definition 3.1. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f: \Omega \to \mathbb{H}$ be a slice regular function. The spherical value of f is the function $f_s^{\circ}: \Omega \to \mathbb{H}$ mapping each $p_0 \in \Omega$ to $b(p_0)$ . The spherical derivative of f is the function $f_s': \Omega \setminus \mathbb{R} \to \mathbb{H}$ mapping each $p_0 = x_0 + y_0 I_0 \in \Omega \setminus \mathbb{R}$ (with $y_0 > 0$ ) to $y_0^{-1}c(p_0)$ . If $\Omega$ is a symmetric slice domain, then the previous definition is consistent with [14, Definition 1.18] (which, in turn, derived from [18]). The following properties hold by construction. We recall that a slice regular function $f: \Omega \to \mathbb{H}$ is called slice preserving if $f(\Omega_I) \subseteq L_I$ for all $I \in \mathbb{S}$ . Remark 3.2. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f : \Omega \to \mathbb{H}$ be a slice regular function. By construction, the following properties hold: - 1. $f_s^{\circ}, f_s'$ are real analytic functions; - 2. for each sphere $x_0 + y_0 \mathbb{S}$ intersecting $\Omega \setminus \mathbb{R}$ , the functions $f_s^{\circ}$ , $f_s'$ are locally constant in $(x_0 + y_0 \mathbb{S}) \cap \Omega$ ; - 3. for all $q \in \Omega \setminus \mathbb{R}$ , it holds $$f(q) = f_s^{\circ}(q) + \operatorname{Im}(q) f_s'(q) ,$$ while, for all $q \in \Omega \cap \mathbb{R}$ , it holds $f(q) = f_s^{\circ}(q)$ ; 4. f is slice preserving if, and only if, $f_s^{\circ}$ , $f_s'$ are real-valued. Example 3.3. For $p \in \mathbb{H} \setminus \mathbb{R}$ , consider the binomial B(q) := q - p: then $B(\bar{q}) = \bar{q} - p$ . Thus, in each $x + y\mathbb{S}$ , it holds $B_s^{\circ} \equiv x - p$ and $B_s' \equiv 1$ . The next example displays a spherical value and derivative which are locally constant but not constant in $(x_0 + y_0 \mathbb{S}) \cap \Omega$ . Example 3.4. Let $f: \Omega \to \mathbb{H}$ be the slice regular function of Example 2.10. The 2-sphere $-1+2\mathbb{S}$ intersects $\Omega$ in two disjoint caps $C^+, C^-$ : the former including the point p:=-1+2I of the arc $\alpha_{\frac{1}{2},I}$ and the latter including $\bar{p}$ . For all $J \in \mathbb{S}$ , it holds: $$\begin{split} f(-1+2J) &= f_{T(J)}(-1+2J) \\ &= (2I)^{-1} \left[ I\phi_{T(J)}(p) + I\phi_{T(J)}(\bar{p}) \right] + J(2I)^{-1} \left[ \phi_{T(J)}(p) - \phi_{T(J)}(\bar{p}) \right] \\ &= \frac{1}{2} \left[ \phi_{T(J)}(p) + \phi_{T(J)}(\bar{p}) \right] + \frac{JI}{2} \left[ \phi_{T(J)}(\bar{p}) - \phi_{T(J)}(p) \right] \,, \end{split}$$ where $\phi_{T(J)}(p) = I \arg_{T(J)}(-1)$ equals $-\pi I$ for $T(J) < \frac{1}{2}$ and $\pi I$ for $T(J) > \frac{1}{2}$ , while $\phi_{T(J)}(\bar{p}) = \phi_0(\bar{p}) = \ln \sqrt{17} + I \arg_0(-1 - 4I)$ irrespective of J. Thus, $$(f_s^{\circ})_{|_{C^+}} \equiv \frac{1}{2} \left[ \phi_0(\bar{p}) - \pi I \right] ,$$ $(f_s')_{|_{C^+}} \equiv \frac{1}{4} \left[ I \phi_0(\bar{p}) - \pi \right] ,$ while $$(f_s^{\circ})_{|_{C^-}} \equiv \frac{1}{2} \left[ \phi_0(\bar{p}) + \pi I \right] ,$$ $(f_s')_{|_{C^-}} \equiv \frac{1}{4} \left[ I \phi_0(\bar{p}) + \pi \right] .$ We are now ready to study the real differentials of slice regular functions on slice domains. For the case of symmetric slice domains, see [14, Remark 8.15], [21, Corollary 6.1], and [25, Proposition 2.3]. In the next statement, we use the notion of Cullen derivative $f'_c$ , which we recalled in Section 2.

Definition 3.6. Let $Y \subseteq \mathbb{H}$ . Then $f: Y \to \mathbb{H}$ is a locally slice function if there exist functions $b, c: Y \to \mathbb{H}$ such that, for all $S:=x_0+y_0\mathbb{S}$ (with $x_0, y_0 \in \mathbb{R}$ and $y_0 \geq 0$ ), the following properties hold: - 1. b, c are constant in each connected component of $S \cap Y$ ; - 2. for all $I \in \mathbb{S}$ such that $x_0 + y_0 I \in Y$ , it holds $$f(x_0 + y_0 I) = b(x_0 + y_0 I) + Ic(x_0 + y_0 I).$$ If this is the case, the spherical value $f_s^{\circ}: Y \to \mathbb{H}$ and spherical derivative $f_s': Y \setminus \mathbb{R} \to \mathbb{H}$ are defined by setting $f_s^{\circ}(x_0 + y_0 I) := b(x_0 + y_0 I)$ if $x_0 + y_0 I \in Y$ and $f_s'(x_0 + y_0 I) := y_0^{-1}c(x_0 + y_0 I)$ if $x_0 + y_0 I \in Y \setminus \mathbb{R}$ . If f is locally slice then, by construction, $f(q) = f_s^{\circ}(q) + \operatorname{Im}(q)f_s'(q)$ for all $q \in Y \setminus \mathbb{R}$ and $f(q) = f_s^{\circ}(q)$ for all $q \in Y \cap \mathbb{R}$ . The function f is slice preserving, i.e., $f(Y_I) \subseteq L_I$ for all $I \in \mathbb{S}$ , if, and only if $f_s^{\circ}, f_s'$ are real-valued. Of course, all slice regular functions on slice domains are locally slice functions. It is possible to produce other examples of locally slice functions by considering linear combinations of slice regular functions that are defined on slice domains whose intersection Y is not a slice domain. This is the case with the difference $D := f_1 - f_0$ between the two slice regular functions $f_0$ and $f_1$ defined in Example 2.10, which is a slice function on the union between the interior of the solid torus $\widetilde{C}_1$ and the open set $\mathbb{H} \setminus (\widetilde{h_I} \cup \widetilde{C}_1)$ .

Definition 4.1. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f, g : \Omega \to \mathbb{H}$ be slice regular functions. We define the regular product $f * g : \Omega \to \mathbb{H}$ by setting $$f * g(q) := f_s^{\circ}(q) g_s^{\circ}(q) + \operatorname{Im}(q)^2 f_s'(q) g_s'(q) + \operatorname{Im}(q) (f_s^{\circ}(q) g_s'(q) + f_s'(q) g_s^{\circ}(q))$$ for all $q \in \Omega \setminus \mathbb{R}$ and $f * g(q) := f_s^{\circ}(q) g_s^{\circ}(q) = f(q)g(q)$ for all $q \in \Omega \cap \mathbb{R}$ . We define the regular conjugate $f^c: \Omega \to \mathbb{H}$ by setting $$f^{c}(q) := \overline{f_{s}^{\circ}(q)} + \operatorname{Im}(q)\overline{f_{s}'(q)}$$ for all $q \in \Omega \setminus \mathbb{R}$ and $f^c(q) := \overline{f_s^{\circ}(q)} = \overline{f(q)}$ for all $q \in \Omega \cap \mathbb{R}$ . The symmetrization $f^s : \Omega \to \mathbb{H}$ is defined as $$f^{s}(q) = |f_{s}^{\circ}(q)|^{2} - |\operatorname{Im}(q)f_{s}'(q)|^{2} + 2\operatorname{Im}(q)\operatorname{Re}(f_{s}^{\circ}(q)\overline{f_{s}'(q)}),$$ for all $q \in \Omega \setminus \mathbb{R}$ and as $f^s(q) = |f_s^{\circ}(q)|^2 = |f(q)|^2$ for all $q \in \Omega \cap \mathbb{R}$ . If $\Omega$ happens to be a symmetric slice domain, then the previous definition is consistent with [14, Definitions 1.27, 1.34, 1.35] by [22, Remark 3.2]. This is true, in particular, for quaternionic polynomials: Example 4.2. For $p = x_0 + y_0 I \in \mathbb{H}$ , consider the binomial B(q) = q - p. It holds $$B^{c}(q) = q - \bar{p}$$ , $B^{s}(q) = q^{2} - q^{2}\operatorname{Re}(p) + |p|^{2} = (q - x_{0})^{2} + y_{0}^{2}$ . In each $x + y\mathbb{S}$ with $x, y \in \mathbb{R}$ and y > 0, it holds $$(B^c)_s^{\circ} \equiv x - \bar{p} = x - x_0 + y_0 I, \quad (B^c)_s' \equiv 1,$$ $(B^s)_s^{\circ} \equiv |x - x_0|^2 + y_0^2 - y^2, \quad (B^s)_s' \equiv 2(x - x_0).$ The set of slice regular functions on a symmetric slice domain is a real \-algebra with the operations $+, , ^c$ (see, for instance, [22, page 4741]). We extend this property to all slice domains.

Definition 6.1. For all $a, b \in \mathbb{H}$ with $|a| \neq |b|$ or $\operatorname{Re}(a\bar{b}) \neq 0$ , set $$\Phi(a,b) := \frac{|a|^2 \bar{a} + \bar{b}a\bar{b}}{(|a|^2 - |b|^2)^2 + (2\operatorname{Re}(a\bar{b}))^2}.$$ Let $\Omega$ be a slice domain in $\mathbb{H}$ . If $f: \Omega \to \mathbb{H}$ is a slice regular function with $f \not\equiv 0$ , consider the slice domain $\Omega' := \Omega \setminus Z(f^s)$ . The regular reciprocal $f^{-*}: \Omega' \to \mathbb{H}$ of f is defined as follows. $$f^{-*}(q) := \Phi \left( f_s^{\circ}(q), |\operatorname{Im}(q)| \, f_s'(q) \right) \, - \, \, \frac{\operatorname{Im}(q)}{|\operatorname{Im}(q)|} \, \, \Phi \left( |\operatorname{Im}(q)| \, f_s'(q), f_s^{\circ}(q) \right)$$ for all $q \in \Omega' \setminus \mathbb{R}$ and $f^{-*}(q) := f_s^{\circ}(q)^{-1} = f(q)^{-1}$ for all $q \in \Omega' \cap \mathbb{R}$ . The last displayed formula is well defined if, and only if, $|f_s^{\circ}(q)| \neq |\operatorname{Im}(q)f_s'(q)|$ or $\operatorname{Re}(f_s^{\circ}(q)f_s'(q)^c) \neq 0$ , which is equivalent to $f^s(q) \neq 0$ by Definition 4.1. This is the reason why the domain of $f^{-*}$ is $\Omega' := \Omega \setminus Z(f^s)$ . We can now prove the following result.

Definition 6.5. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f,g:\Omega\to\mathbb{H}$ be slice regular functions. We define the regular quotient of f and g as $f^{-}g:\Omega\setminus Z(f^s)\to\mathbb{H}$ .

Definition 7.4. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f:\Omega\to\mathbb{H}$ be a slice regular function. If there exist slice regular functions $h,g:\Omega\to\mathbb{H}$ such that $$f(q) = h(q) * g(q)$$ in $\Omega$ , we say that h(q) (left) divides f(q). Now suppose C to be a connected component of $(x+y\mathbb{S})\cap\Omega$ : we say that h(q) (left) divides f(q) near C if there exist a slice domain $\Lambda$ with $C\subseteq \Lambda\subseteq \Omega$ and slice regular functions $h,g:\Lambda\to\mathbb{H}$ such that $$f(q) = h(q) * g(q)$$ in $\Lambda$ . Finally, for any point $p_0 \in x + y\mathbb{S}$ , we say that h(q) (left) divides f(q) near $p_0$ if h(q) left divides f(q) near the connected component of $(x + y\mathbb{S}) \cap \Omega$ that includes $p_0$ . We now present examples of slice regular functions $g:\Omega\to\mathbb{H}$ and points $\widetilde{p}=x+yJ\in\Omega$ such that $q-\widetilde{p}$ divides g(q) - 1. near the connected component of $(x + y\mathbb{S}) \cap \Omega$ that includes $\widetilde{p}$ (whence $g(\widetilde{p}) = 0$ ); - 2. near a connected component of $(x+y\mathbb{S})\cap\Omega$ that does not include $\widetilde{p}$ (while g has no zeros in $x+y\mathbb{S}$ ); - 3. globally in $\Omega$ , despite the fact the $g_s^{\circ}, g_s'$ are not constant in $(x + y\mathbb{S}) \cap \Omega$ (a case when $g(\widetilde{p}) = 0$ , too). Example 7.5. Let $f: \Omega \to \mathbb{H}$ be the slice regular function of Examples 2.10 and 3.4. Recall that we have set p:=-1+2I. Let us pick $J \in \mathbb{S}$ and set $\widetilde{p}:=-1+2J$ . We define $$v := f_s^{\circ}(p) + \operatorname{Im}(\widetilde{p}) f_s'(p) = \frac{1}{2} [\phi_0(\overline{p}) - \pi I] + \frac{J}{2} [I\phi_0(\overline{p}) - \pi]$$ and notice that - 1. If $\widetilde{p} \in C^+$ , then $v = f_s^{\circ}(\widetilde{p}) + \operatorname{Im}(\widetilde{p}) f_s'(\widetilde{p}) = f(\widetilde{p})$ . - 2. If $\widetilde{p} \in C^- \setminus \{\overline{p}\}$ , then $v \neq f(\widetilde{p})$ because $f(\widetilde{p}) v = \frac{1}{2} [\phi_0(\overline{p}) + \pi I] + \frac{J}{2} [I\phi_0(\overline{p}) + \pi] v = \pi(I + J) \neq 0$ . - 3. If $\widetilde{p} = \overline{p}$ , then $v = f(\overline{p})$ because $f(\widetilde{p}) v = \pi(I I) = 0$ . We define $$g: \Omega \to \mathbb{H}, \quad q \mapsto f(q) - v.$$ By direct computation. $$(g_s^{\circ})_{|_{C^+}} \equiv -\frac{J}{2} \left[ I \phi_0(\bar{p}) - \pi \right] ,$$ $(g_s')_{|_{C^+}} \equiv \frac{1}{4} \left[ I \phi_0(\bar{p}) - \pi \right] ,$ whence the constant values of $(g_s^\circ)_{|_{C^+}}$ and $-2J(g_s')_{|_{C^+}} = -\operatorname{Im}(\widetilde{p})(g_s')_{|_{C^+}}$ coincide. On the other hand, $$(g_s^{\circ})_{|_{C^-}} \equiv -\frac{JI}{2}\phi_0(\bar{p}) + \pi \left(I + \frac{J}{2}\right),$$ $(g_s')_{|_{C^-}} \equiv \frac{1}{4}\left[I\phi_0(\bar{p}) + \pi\right].$ If J=-I, then the constant values of $(g_s^\circ)_{|_{C^-}}$ and $-\operatorname{Im}(\widetilde{p})(g_s')_{|_{C^-}}$ coincide. If, instead, $J\neq -I$ , we can prove that, for all $\widehat{p}\in (-1+2\mathbb{S})\cap \Omega$ , the constant values of $(g_s^\circ)_{|_{C^-}}$ and $-\operatorname{Im}(\widehat{p})(g_s')_{|_{C^-}}$ are distinct. Indeed, the constant value of $$\left| \left( g_s^{\circ} \right)_{|_{C^-}} \right|^2 - 4 \left| \left( g_s' \right)_{|_{C^-}} \right|^2 = \left| J(g_s^{\circ})_{|_{C^-}} \right|^2 - 4 \left| \left( g_s' \right)_{|_{C^-}} \right|^2$$ is different from 0, according to the following computation: $$\begin{split} & \left| \frac{I}{2} \phi_0(\bar{p}) + \pi \left( JI - \frac{1}{2} \right) \right|^2 - \left| \frac{I}{2} \phi_0(\bar{p}) + \frac{\pi}{2} \right|^2 \\ &= \pi^2 \left| JI - \frac{1}{2} \right|^2 + \pi \operatorname{Re} \left( \left( IJ - \frac{1}{2} \right) I \phi_0(\bar{p}) \right) - \frac{\pi^2}{4} - \pi \operatorname{Re} \left( \frac{I}{2} \phi_0(\bar{p}) \right) \\ &= \pi^2 (1 - \operatorname{Re}(JI)) + \pi \operatorname{Re} \left( (IJ - 1) I \phi_0(\bar{p}) \right) \\ &= \pi (1 - \operatorname{Re}(JI)) (\pi + \arg_0(-1 - 4I)) \neq 0 \,. \end{split}$$ The last equality holds because $I\phi_0(\bar{p}) = -\arg_0(-1-4I) + I \ln \sqrt{17} \in L_I$ . There are three possibilities: 1. If $\widetilde{p} \in C^+$ , then $q - \widetilde{p}$ divides g(q) near $C^+$ but not near $C^-$ . Moreover, $$Z(g) \cap C^+ = \{\widetilde{p}\}, \quad Z(g) \cap C^- = \emptyset$$ 2. If $\widetilde{p} \in C^- \setminus \{\overline{p}\}$ , then $q - \widetilde{p}$ divides g(q) near $C^+$ but not near $C^-$ . Moreover, $$Z(g) \cap C^+ = \emptyset, \quad Z(g) \cap C^- = \emptyset$$ 3. If $\widetilde{p} = \overline{p}$ , then $q - \overline{p}$ divides g(q) globally in $\Omega$ . Moreover, $$Z(g) \cap C^+ = \emptyset, \quad Z(g) \cap C^- = \{\bar{p}\}.$$ We conclude this section with the next definition, consistent with [14, Definition 3.26] (which, in turn, derived from [9]).

Definition 7.6. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f:\Omega\to\mathbb{H}$ be a slice regular function with $f\not\equiv 0$ . Fix an $S:=x+y\mathbb{S}$ (with $x,y\in\mathbb{R}$ ) intersecting $\Omega$ , a connected component C of the intersection $S\cap\Omega$ and a point $p\in S$ . We define the classical multiplicity of f at p relative to C, and denote by $m_f^C(p)$ , the number $n\in\mathbb{N}$ such that $(q-p)^{n}$ divides f(q) near C but $(q-p)^{(n+1)}$ does not. If, moreover, $S\cap\Omega$ has a unique connected component C, then we also call $m_f^C(p)$ the classical multiplicity of f at p and denote it as $m_f(p)$ . If $x \in \Omega \cap \mathbb{R}$ , then of course $x + 0\mathbb{S} = \{x\}$ intersects $\Omega$ in a connected set: namely, the singleton $\{x\}$ . Thus, it is automatically possible to write $m_f(x)$ without specifying a connected component C. The previous definition is well posed for the following reason. Supposing p = x + yI, any equality $f(q) = (q - p)^{n} g(q)$ valid in a slice domain $\Lambda$ implies an equality $f_I(z) = (z - p)^n g_I(z)$ valid for all $z \in \Lambda_I$ . Moreover, $f_I$ is a holomorphic function: if for all $n \in \mathbb{N}$ there exist holomorphic functions $\phi^{[n]}$ such that $f_I(z) = (z - p)^n \phi^{[n]}(z)$ , then $f_I \equiv 0$ . But in such a case, it holds $f \equiv 0$ by the Identity Principle 2.4. Example 7.7. Let us consider again the functions g and the points $\tilde{p}$ appearing in Example 7.5, in the three separate cases listed there. - 1. It holds $g(\widetilde{p}) = 0$ and $m_g^{C^+}(\widetilde{p}) \ge 1$ ; on the other hand, $m_g^{C^-}(\widetilde{p}) = 0$ . - 2. It holds $g(\widetilde{p}) \neq 0$ and $m_q^{C^+}(\widetilde{p}) \geq 1$ ; on the other hand, $m_q^{C^-}(\widetilde{p}) = 0$ . - 3. It holds $g(\widetilde{p}) = 0$ and $m_g^{C^+}(\widetilde{p}) \ge 1$ , $m_g^{C^-}(\widetilde{p}) \ge 1$ .

Definition 8.7. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f:\Omega\to\mathbb{H}$ be a slice regular function. In the situation described in Theorem 8.6, the number 2m is called the spherical multiplicity of f at C and the number n is called the isolated multiplicity of f at $p_1$ , relative to C. If, moreover, $S \cap \Omega$ has a unique connected component, then we also call 2m the spherical multiplicity of f at $S \cap \Omega$ and call n the isolated multiplicity of f at $p_1$ . For $x \in \Omega \cap \mathbb{R}$ , we define the isolated multiplicity of f at x to coincide with the classical multiplicity $m_f(x)$ . The previous definition is consistent with [14, Definition 3.37], which in turn derived from [11, 17]. We conclude this section using factorization to characterize the set of points p such that the real differential $df_p$ of a slice regular function f at p is singular. The corresponding result for symmetric slice domains is [14, Theorem 8.20] (from [8]). We follow here a different approach, analogous to that of [26].

Definition 9.1. The functions $\sigma, \tau : \mathbb{H} \times \mathbb{H} \to \mathbb{R}$ are defined by setting, for all $p, q \in \mathbb{H}$ , $$\sigma(q, p) := \begin{cases} |q - p| & \text{if } p, q \text{ lie in the same plane } L_I \\ \omega(q, p) & \text{otherwise} \end{cases}$$ $$\tau(q, p) := \begin{cases} |q - p| & \text{if } p, q \text{ lie in the same plane } L_I \\ \sqrt{\left[\text{Re}(q) - \text{Re}(p)\right]^2 + \left[|\operatorname{Im}(q)| - |\operatorname{Im}(p)|\right]^2} & \text{otherwise} \end{cases}$$ where $$\omega(q, p) := \sqrt{\left[\text{Re}(q) - \text{Re}(p)\right]^2 + \left[|\text{Im}(q)| + |\text{Im}(p)|\right]^2}.$$ For $0 \le R_1 \le R_2 \le +\infty$ , the following sets are defined: $$\begin{split} &\Sigma(p,R_2) := \{q \in \mathbb{H} : \sigma(q,p) < R_2\} \\ &T(p,R_2) := \{q \in \mathbb{H} : \tau(q,p) < R_2\} \\ &\Omega(p,R_2) := \{q \in \mathbb{H} : \omega(q,p) < R_2\} \\ &\Sigma(p,R_1,R_2) := \{q \in \mathbb{H} : \tau(q,p) > R_1,\sigma(q,p) < R_2\} \\ &\Omega(p,R_1,R_2) := \Omega(p,R_2) \setminus \overline{T(p,R_1)} \,, \end{split}$$ where $\overline{T(p,R_1)}$ denotes the closure of $T(p,R_1)$ .

Definition 9.4. Let $f: \Omega \to \mathbb{H}$ be a slice regular function and fix $p \in \mathbb{H}$ . We say that p is a singularity for f if there exists R > 0 such that $\Sigma(p, 0, R) \subseteq \Omega$ , whence the Laurent expansion of f at p, $f(q) = \sum_{n \in \mathbb{Z}} (q-p)^{*n} a_n$ , has 0 as its inner radius of convergence and has a positive outer radius of convergence. Suppose this to be the case: we say that p is a pole for f if there exists an $m \geq 0$ such that $a_{-k} = 0$ for all k > m; the minimum such f is called the order of the pole and denoted by $ord_f(p)$ . If f is not a pole, then we call it an essential singularity for f and set $ord_f(p) = +\infty$ . Additionally, we call f a removable singularity if f extends to a neighborhood of f as a slice regular function. Every removable singularity has order 0, but the converse implication is false. In the next example, a connected component of $(x + y\mathbb{S}) \cap \Omega$ consists of nonremovable singularities, including one having order 0. Example 9.5. Let us consider again the function g of Example 8.1, which was divisible by q-p in the connected component $C^+$ of $(-1+2\mathbb{S})\cap\Omega$ but not in the complementary connected component $C^-$ . Let us define a function $h:\Omega\setminus(-1+2\mathbb{S})\to\mathbb{H}$ by setting $$h(q) := (q-p)^{-} g(q) = (q^2 + 2q + 5)^{-1} (q - \bar{p}) * g(q).$$ Then every $\widetilde{p} = -1 + 2J \in C^+$ is a removable singularity for h and $ord_h(\widetilde{p}) = 0$ . On the other hand, every $\widetilde{p} = -1 + 2J \in C^-$ is a nonremovable singularity for h. Moreover, $ord_h(\widetilde{p}) \geq 1$ for all $\widetilde{p} \in C^-$ except $\overline{p}$ : indeed, the equality $h_I(z) = (z - p)^{-1}g_I(z)$ , valid for all $z \in \Omega_I \setminus \{\overline{p}\}$ , yields that $ord_h(\overline{p}) = 0$ .

Definition 9.6. Let $\Omega$ be a slice domain in $\mathbb{H}$ . A function f is semiregular in $\Omega$ if it is slice regular in a slice domain $\Omega' \subseteq \Omega$ such that every point of $\Omega \setminus \Omega'$ is a pole (or a removable singularity) for f and such that, for any $x, y \in \mathbb{R}$ and for each connected component C of $(x+y\mathbb{S}) \cap \Omega$ , the supremum $\sup_{p \in C} \operatorname{ord}_f(p)$ is finite Notice that, if f is semiregular in $\Omega$ and if $\mathcal{S}$ is the set of its nonremovable poles, then, for all $I \in \mathbb{S}$ : the slice $\mathcal{S}_I$ is discrete; the restriction $f_I$ is holomorphic in $\Omega_I \setminus \mathcal{S}_I$ ; and $f_I$ is meromorphic in $\Omega_I$ . We point out that $\Omega \setminus \mathcal{S}$ is a slice domain. We will see in Section 12 (specifically, in Remark 12.5) that the hypothesis $\sup_{p \in C} \operatorname{ord}_f(p) < +\infty$ is not restrictive if $\mathcal{S}$ is a cap within a 2-sphere of the form $x + y \mathbb{S}$ .

Definition 9.10. In the situation presented in Theorem 9.9: if $m \leq 0$ then we call -2m the spherical order of f at C, and write $ord_f^C(x + y\mathbb{S}) = -2m$ ; whenever n > 0, we say that n is the isolated multiplicity of f at $p_1$ , relative to C. If, moreover, $S \cap \Omega$ has a unique connected component, then we also call -2m the spherical order of f at $S \cap \Omega$ and denote it as $ord_f(x + y\mathbb{S})$ ; we call n the isolated multiplicity of f at $p_1$ .

Definition 10.3. Let $\Omega$ be a slice domain in $\mathbb{H}$ and let $f:\Omega\to\mathbb{H}$ be a slice regular function. The zero set of the spherical derivative $f'_s:\Omega\setminus\mathbb{R}\to\mathbb{H}$ is called the degenerate set of f and denoted by the symbol $D_f$ .