# The $*$-exponential as a covering map

Amedeo Altavilla  Altavilla Amedeo: Dipartimento di Matematica, Università
degli Studi di Bari “Aldo Moro”, via Edoardo Orabona, 4, 70125, Bari, Italy.
<EMAIL_ADDRESS>and  Samuele Mongodi  Mongodi Samuele: Dipartimento
di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via
Roberto Cozzi, 55, 20125, Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

We employ tools from complex analysis to construct the $*$-logarithm of a
quaternionic slice regular function. Our approach enables us to achieve three
main objectives: we compute the monodromy associated with the $*$-exponential;
we establish sufficient conditions for the $*$-product of two $*$-exponentials
to also be a $*$-exponential; we calculate the slice derivative of the
$*$-exponential of a regular function.

###### Key words and phrases:

Slice-regular functions, quaternionic exponential, quaternionic logarithm,
Baker-Campbell-Hausdorff, covering maps, monodromy

###### 2020 Mathematics Subject Classification:

Primary 30G35, 30C25; secondary 30B50, 33B10, 58K10, 32A10

Partially supported by PRIN 2022MWPMAB - “Interactions between Geometric
Structures and Function Theories” GNSAGA of INdAM and by the INdAM project
“Teoria delle funzioni ipercomplesse e applicazioni”.

##  1\. Introduction

One of the fundamental ideas in the early stages of real analytic geometry and
in the study of flat real analytic CR manifolds is the complexification: real
analyticity ensures that most of the features we are interested in will be
reflected in the complexification, at least on a formal and algebraic level.

In this respect, the theory of slice regular functions benefits no less, if
not more, from this idea. As it is known, slice regular functions on a
symmetric open domain $U\subseteq{\mathbb{H}}$ can be viewed as a special
family of holomorphic functions (namely stem functions) from an open domain in
${\mathbb{C}}$ to ${\mathbb{C}}\otimes{\mathbb{H}}$ (i.e. the
_complexification_ of ${\mathbb{H}}$ as an associative algebra with unity).

In [Mongodi:HolSR], this idea was employed to account for the many
similarities between the theory of slice regular functions and the theory of
holomorphic functions of one complex variable; this argument was pushed
further in [Mongodi:AssAlg] to describe the link between the complex geometry
of the set of square roots of $-1$ of an associative algebra and the space of
slice regular functions.

On the other hand, the purely algebraic properties resulting from the
structure of ${\mathbb{C}}\otimes{\mathbb{H}}$ were used to a great extent in
[AltavillaPAMS, AltavillaAMPA, AltavillaLAA, altavillaLOG], where many
properties of the $*$-exponential were studied, the structure of the
$*$-product was better understood, and the problem of the existence of a
$*$-logarithm was initiated.

Building on the same intuition and exploiting more thoroughly the equality of
the analytic expressions of algebraic operations of ${\mathbb{H}}$ and
${\mathbb{C}}\otimes{\mathbb{H}}$, in [AM:powercover] we showed how the
problem of finding $*$-roots of a slice regular function can be translated
into a problem of lifting functions through a holomorphic covering map. The
number and the structure of such $*$-roots were then linked to the group of
deck transformations of the covering map.

Again, we would like to emphasize that this is possible because the analytic
expressions of the multiplication in ${\mathbb{H}}$ and in
${\mathbb{C}}\otimes{\mathbb{H}}$ in terms of the coordinates with respect to
some basis of ${\mathbb{H}}$ and the corresponding complexified basis of
${\mathbb{C}}\otimes{\mathbb{H}}$ are the same, reflecting the fact that a
real analytic function on, say, the real line has a unique extension to the
complexification of the real line given by a power series with the same
coefficients.

We present here another instantiation of this consideration: we treat the case
of $*$-logarithms by considering the map $\exp:{\mathbb{H}}\to{\mathbb{H}}$
and lifting it to the complexification, to a map with the same analytic
expression. The study of the local inverses of $\exp$ again becomes a problem
in complex analytic covering maps, from whose solution we also recover what we
already proved in the case of $*$-roots. Using a geometric approach we will
see that under natural topological hypotheses the exponential map in
${\mathbb{C}}\otimes{\mathbb{H}}$ is a covering map (see Theorem 4.2) and we
will be able to write down its monodromy. Then, thanks to the standard
relation between holomorphic stem functions and slice regular functions, given
a never-vanishing slice regular function $f:U\to{\mathbb{H}}$ such that its
“vector part” is never-vanishing, we will be able to construct a $2$-parameter
family of $*$-logarithms ($1$-parameter family if
$U\cap{\mathbb{R}}\neq\emptyset$), see Corollaries 5.2 and 5.3 for the results
and Remark 5.1 for the explicit description of the monodromy.

Such a study extends what is already contained in [altavillaLOG, GPV1, GPV2]
by showing the geometric nature of the many problems encountered in the search
for a good notion of logarithm in the non-commutative setting.

As already mentioned, the proof of many results contained in the present paper
follows topological strategies, which then produce natural hypotheses and
conditions, simplifying many proofs contained in the aforementioned papers. On
the other hand, since it is not the specific aim of this work, we will only
give a glimpse of how the remaining residual cases should be treated, i.e. how
some of the hypotheses could be relaxed.

In an effort to highlight the impact that this simple idea can produce, we
analyze the problem of when a product of exponential is an exponential itself;
the question for quaternions is easily settled by using a simplified version
of the Baker-Campbell-Hausdorff formula (or, if one interprets quaternions as
rotations, by a standard application of Rodrigues’ formula). Once the problem
is _analytically_ solved for quaternions, we formally consider the same
solution for the same problem in ${\mathbb{C}}\otimes{\mathbb{H}}$, where stem
functions take their values. This gives us a solution to the same problem at
the level of stem functions, hence for slice regular functions. The same idea
is used to compute the slice derivative of the $*$-exponential of a slice
regular function. Even though this computation is quite natural, it has not
been implemented yet, possibly due to the lack of a strategy like the one we
use here. In these last two tasks, we will use a simple formula inspired by
standard linear algebra, which allows, given a generic slice regular function
$f$, to write any other function $g$ as a sum of a component in the
“direction” of $f$ and another “orthogonal” part. This will allow us to write
much simpler formulas and to identify possible future generalizations.

##  2\. Preliminaries

###  2.1. Algebraic structures of ${\mathbb{H}}$ and of
${\mathbb{H}}\otimes{\mathbb{C}}$

In this paper we will deal with many different imaginary units, not only those
contained in the space of quaternions, but also with others coming from
different algebras. Starting from complex numbers, the symbol ‘$\imath$’ will
denote the standard imaginary unit in ${\mathbb{C}}$ (and hence will be used
when working in ${\mathbb{C}}^{N}$, $N\geq 1$). The symbol ‘$i$’ will denote
the first imaginary unit in the definition of the space ${\mathbb{H}}$ of
quaternions:

${\mathbb{H}}:=\\{q=q_{0}+q_{1}i+q_{2}j+q_{3}k\,|\,q_{\ell}\in{\mathbb{R}},\,\ell=0,1,2,3,\,i^{2}=j^{2}=k^{2}=-1,\,ij=k=-ji\\}.$

We will make use of the standard conjugation in ${\mathbb{H}}$ denoted by the
superscript $c$:

$q=q_{0}+q_{1}i+q_{2}j+q_{3}k\mapsto q^{c}=q_{0}-(q_{1}i+q_{2}j+q_{3}k).$

Using this conjugation, given any quaternion $q$, it is possible to define its
scalar and vector parts as follows

$q_{0}=\frac{q+q^{c}}{2},\quad q_{v}=\frac{q-q^{c}}{2},$

so that $q=q_{0}+q_{v}$. Obviously, if $q$ is represented in the form
$q=q_{0}+q_{1}i+q_{2}j+q_{3}k$, then $q_{0}$ is the scalar part of $q$ and
$q_{v}=q_{1}i+q_{2}j+q_{3}k$. Using this representation, we can express the
product of two quaternions $q=q_{0}+q_{v}$ and $p=p_{0}+p_{v}$ in a more
understandable way:

(1) $qp=q_{0}p_{0}-\langle
q_{v},p_{v}\rangle+q_{0}p_{v}+p_{0}q_{v}+q_{v}\wedge p_{v},$

where $\langle\cdot,\cdot\cdot\rangle$ and $\wedge$ denote the standard
Euclidean and cross products. In particular, the square norm of $q$ can be
computed as $|q|^{2}=qq^{c}$ and we have that $q_{v}^{2}=-|q_{v}|^{2}$.

Whenever $q_{v}\neq 0$, we are able to represent $q$ in another convenient
form:

$q=\alpha+I\beta,$

where $\alpha=q_{0}$, $I=\frac{q_{v}}{|q_{v}|}$ and $\beta=|q_{v}|$. In
particular, if we denote the set of imaginary units as follows

${\mathbb{S}}:=\\{I\in{\mathbb{H}}\,|\,I^{2}=-1\\}=\\{\alpha_{1}i+\alpha_{2}j+\alpha_{3}k\,|\,\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}=1\\},$

and we denote by
${\mathbb{C}}_{I}=span(1,I)=\\{\alpha+I\beta\,|\,\alpha,\beta\in{\mathbb{R}}\\}$
the complex plane generated by $1$ and $I$, we have that

${\mathbb{H}}=\bigcup_{I\in{\mathbb{S}}}{\mathbb{C}}_{I}.$

This last representation comes in handy when working with slice functions, and
in order to do that we need to talk about we need to discuss the
complexification of ${\mathbb{H}}$ (in particular, we will follow the approach
of [GP:AltAlg]).

The symbol ‘$\sqrt{-1}$’ will denote the complex imaginary unit defining the
complexification of ${\mathbb{H}}$, i.e.

${\mathbb{C}}\otimes{\mathbb{H}}:=\\{q+\sqrt{-1}p\,|\,q,p\in{\mathbb{H}}\\}.$

The algebraic structure of ${\mathbb{C}}\otimes{\mathbb{H}}$ is defined in the
usual way: if
$q_{1}+\sqrt{-1}p_{1},q_{2}+\sqrt{-1}p_{2}\in{\mathbb{C}}\otimes{\mathbb{H}}$,
then:

$(q_{1}+\sqrt{-1}p_{1})(q_{2}+\sqrt{-1}p_{2})=q_{1}q_{2}-p_{1}p_{2}+\sqrt{-1}(q_{1}p_{2}+p_{1}q_{2}).$

By fixing a (orthogonal) basis of ${\mathbb{H}}$ containing $1$ and by writing
any quaternion in its $4$ real coordinates we get a biholomorphism between
${\mathbb{C}}\otimes{\mathbb{H}}$ and ${\mathbb{C}}^{4}$: if
$q=q_{0}+q_{1}i+q_{2}j+q_{3}k$ and $p=p_{0}+p_{1}i+p_{2}j+p_{3}k$, then we
define $\phi:{\mathbb{C}}\otimes{\mathbb{H}}\to{\mathbb{C}}^{4}$ as

$\phi(q+\sqrt{-1}p)=(q_{0}+\imath p_{0},q_{1}+\imath p_{1},q_{2}+\imath
p_{2},q_{3}+\imath p_{3}).$

In particular, this biholomorphism induces an algebraic structure on
${\mathbb{C}}^{4}$ that is defined exactly as that of ${\mathbb{H}}$:

${\mathbb{C}}\otimes{\mathbb{H}}=\\{z=z_{0}+z_{1}i+z_{2}j+z_{3}k\,|\,z_{\ell}\in{\mathbb{C}}\,,\ell=0,1,2,3,\,i^{2}=j^{2}=k^{2}=-1,\,ij=k=-ji\\}.$

In ${\mathbb{C}}\otimes{\mathbb{H}}$ it is possible to define two commuting
conjugations:

$\displaystyle q+\sqrt{-1}p\mapsto$
$\displaystyle(q+\sqrt{-1}p)^{c}=q^{c}+\sqrt{-1}p^{c},$ $\displaystyle
q+\sqrt{-1}p\mapsto$ $\displaystyle\overline{q+\sqrt{-1}p}=q-\sqrt{-1}p.$

If we work in ${\mathbb{C}}^{4}$, these two conjugations translates as follows

$\displaystyle z=z_{0}+z_{1}i+z_{2}j+z_{3}k\mapsto$ $\displaystyle
z^{c}=z_{0}-(z_{1}i+z_{2}j+z_{3}k),$ $\displaystyle
z=z_{0}+z_{1}i+z_{2}j+z_{3}k\mapsto$
$\displaystyle\overline{z}=\bar{z}_{0}+\bar{z}_{1}i+\bar{z}_{2}j+\bar{z}_{3}k.$

Exactly as before, we can define the “scalar” and “vector” part of
$z\in{\mathbb{C}}\otimes{\mathbb{H}}$ as

$z_{0}=\frac{z+z^{c}}{2},\quad\underline{z}=\frac{z-z^{c}}{2},$

Within this language, the product of two elements
$z,w\in{\mathbb{C}}\otimes{\mathbb{H}}$ can be written formally as in Formula
1:

$zw=z_{0}w_{0}-\langle\underline{z},\underline{w}\rangle+z_{0}\underline{w}+w_{0}\underline{z}+\underline{z}\wedge\underline{w},$

where $\langle\cdot,\cdot\cdot\rangle$ and $\wedge$ are the formal
generalization of the Euclidean and cross product. In particular if
$z=z_{0}+\underline{z}=z_{0}+z_{1}i+z_{2}j+z_{3}k$, setting

$\underline{z}^{2}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2},$

we have that $zz^{c}=\langle
z,z\rangle=z_{0}^{2}+\underline{z}^{2}\in{\mathbb{C}}$ and it is a real number
only if the four components of $z$ are real numbers, i.e. only if
$z\in{\mathbb{H}}$. However, since the product in ${\mathbb{C}}$ is
commutative, for any $z,w\in{\mathbb{C}}\otimes{\mathbb{H}}$ we have

(2) $(zw)(zw)^{c}=zww^{c}z^{c}=(zz^{c})(ww^{c}).$

If $z$ is such that $zz^{c}\neq 0$, then $z\neq 0$, but unfortunately
${\mathbb{C}}\otimes{\mathbb{H}}$ contains zero divisors. So, in particular,
there are $z\neq 0$ such that $zz^{c}=0$.

###  2.2. Stem functions, slice functions and regularity

We are now ready to introduce and discuss slice functions. As already said, we
will rely on the approach of stem functions developed in [GP:AltAlg] and in
subsequent works by the same authors. We also refer to [AM:powercover] to
deepen our specific point of view. We start with the following definition.

######  Definition 2.1.

Let $\mathcal{U}\subset{\mathbb{C}}$ be such that
$\overline{\mathcal{U}}=\mathcal{U}$. A function
$F:\mathcal{U}\to{\mathbb{C}}\otimes{\mathbb{H}}$ is said to be a stem
function if, for any $z\in\mathcal{U}$, we have $F(\bar{z})=\overline{F(z)}$.

If we write $F:\mathcal{U}\to{\mathbb{C}}\otimes{\mathbb{H}}$ as
$F(z)=F_{ev}(z)+\sqrt{-1}F_{od}(z)$, then the condition
$F(\bar{z})=\overline{F(z)}$ is reflected in the following two equalities
$F_{ev}(\bar{z})=F_{ev}(z)$ and $F_{od}(\bar{z})=-F_{od}(z)$. If, instead, we
read $F$ as a function taking values in ${\mathbb{C}}^{4}$,
$F(z)=(F_{0}(z),F_{1}(z),F_{2}(z),F_{3}(z))$, then the stem condition must be
satisfied by all four components, i.e.
$F_{\ell}(\bar{z})=\overline{F_{\ell}(z)}$, for $\ell=0,1,2,3$.

######  Definition 2.2.

Let $U\subset{\mathbb{H}}$ be such that if $q=\alpha+I\beta\in U$ then
$\alpha+J\beta\in U$, for any $J\in{\mathbb{S}}$ and let
$\mathcal{U}=\\{\alpha+\imath\beta\,|\,\alpha+I\beta\in U\\}$. A function
$f:U\to{\mathbb{H}}$ is said to be a slice function if there exists a stem
function
$F=F_{ev}+\sqrt{-1}F_{od}:\mathcal{U}\to{\mathbb{C}}\otimes{\mathbb{H}}$ such
that
$f(\alpha+I\beta)=F_{ev}(\alpha+\imath\beta)+IF_{od}(\alpha+\sqrt{-1}\beta)$;
in this case we will write $f=\mathcal{I}(F)$ and we will say that $f$ is
induced by $F$.

If $U$ is a domain and $F$ is a holomorphic function, then $f$ is said to be a
slice regular function.

The definition of stem functions guarantees the well-definition of slice
functions: in fact, since $F(\bar{z})=\overline{F(z)}$, then the value of $f$
at $\alpha+(-I)(-\beta)$ is not different from that of $f$ at $\alpha+I\beta$.
Examples of slice regular functions are polynomials and converging power
series in the quaternionic variable $q$ with right quaternionic coefficients.

The main property of slice functions is the so-called Representation Formula
contained in the following statement (see [GSS:RegFunc, Theorem 1.16]). It
essentially says that a slice function can be recovered from its values on two
different semislices ${\mathbb{C}}_{I}^{+}$ and ${\mathbb{C}}_{K}^{+}$, where
the apex ‘$+$’ indicates the upper half plane.

######  Theorem 2.3 (Representation Formula).

Let $f:U\to{\mathbb{H}}$ be a slice function and let $J,K\in{\mathbb{S}}$ be
such that $J\neq K$. Then, for every $\alpha+I\beta\in U$ the following
formula holds

$f(\alpha+I\beta)=(I-K)((J-K)^{-1}f(\alpha+J\beta))-(I-J)((J-K)^{-1}f(\alpha+K\beta)).$

It is well known that the pointwise product of two slice functions does not
preserve regularity, however, the pointwise product of two stem functions is a
stem functions, therefore it is natural to introduce a new notion of product
as follows.

######  Definition 2.4.

Let $f=\mathcal{I}(F)$ and $g=\mathcal{I}(G)$ be two stem functions defined on
the same domain $U$. The $*$-product of $f$ and $g$ is defined as the slice
function

$f*g=\mathcal{I}(FG):U\to{\mathbb{H}}.$

Since the $*$-product is defined from the pointwise product in a non-
commutative algebra (namely in ${\mathbb{C}}\otimes{\mathbb{H}}$), it is non-
commutative itself. However, if we consider a slice function
$f=\mathcal{I}(F_{ev}+\sqrt{-1}F_{od}):U\to{\mathbb{H}}$, such that $F_{ev}$
and $F_{od}$ take only real values, then $F$ is a legit complex function of
one complex variable and, for any other slice function $g$ defined on $U$, we
have that

$f*g=fg=g*f.$

A function $f=\mathcal{I}(F)$ with the above property is said to be slice
preserving. In fact, as $F_{ev}$ and $F_{od}$ are real valued, then, for any
$q=\alpha+I\beta\in U$, the element $f(q)$ belongs to the same slice
${\mathbb{C}}_{I}$ of $q$. Written as a complex curve in ${\mathbb{C}}^{4}$
the stem function $F$ of a slice preserving function takes the following form

$F(z)=(F_{ev}+\imath F_{od},0,0,0)=(F_{0},0,0,0).$

At this stage we can apply all the formalism and properties described in the
previous part of this Section and obtain that, if $f=\mathcal{I}(F)$ and
$g=\mathcal{I}(G)$, $F=F_{0}+F_{1}i+F_{2}j+F_{3}k$ and
$G=G_{0}+G_{1}i+G_{2}j+G_{3}k$, $f_{\ell}=\mathcal{I}(F_{\ell})$ and
$g_{\ell}=\mathcal{I}(G_{\ell})$ for $\ell=0,1,2,3$,
$f_{v}=\mathcal{I}((F+F^{c})/2)$ and $g_{v}=\mathcal{I}((G+G^{c})/2)$, then

$f*g=f_{0}g_{0}-\langle
f_{v},g_{v}\rangle_{*}+f_{0}g_{v}+g_{0}f_{v}+f_{v}\operatorname{\wedge\mspace{-9.0mu}_{*}\
}g_{v},$

where $\langle f_{v},g_{v}\rangle_{*}=f_{1}g_{1}+f_{2}g_{2}+f_{3}g_{3}$ and
$f_{v}\operatorname{\wedge\mspace{-9.0mu}_{*}\
}g_{v}=(f_{2}g_{3}-f_{3}g_{2})i+(f_{3}g_{1}-f_{1}g_{3})j+(f_{1}g_{2}-f_{2}g_{1})k$
and, of course, all $f_{\ell}$ and $g_{\ell^{\prime}}$ are slice preserving
functions. These last two operators can be defined in an intrinsic way by
means of the so-called regular conjugation: given a slice function
$f=\mathcal{I}(F):U\to{\mathbb{H}}$, we define its regular conjugate as the
function $f^{c}:U\to{\mathbb{H}}$ defined as $f^{c}=\mathcal{I}(F^{c})$. Then,
if $g:U\to{\mathbb{H}}$ is another slice function, we have that

$\langle f,g\rangle_{*}=\frac{f*g^{c}+g*f^{c}}{2},\qquad
f\operatorname{\wedge\mspace{-9.0mu}_{*}\
}g=f_{v}\operatorname{\wedge\mspace{-9.0mu}_{*}\
}g_{v}=\frac{f*g-g*f}{2}=\frac{[f,g]}{2}.$

This representation of the product highlights how many algebraic features of
slice functions directly come from those of quaternions (or of quaternionic
curves). For instance, two non slice preserving functions $f$ and $g$ commute
if and only if $f_{v}\operatorname{\wedge\mspace{-9.0mu}_{*}\ }g_{v}\equiv 0$
if and only if there exist two slice preserving functions $\alpha$ and $\beta$
not both identically zero, such that $\alpha f_{v}+\beta g_{v}\equiv 0$ (see
e.g. [AltavillaPAMS, Proposition 2.10]). A particular instance of this
phenomenon is when $f$ and $g$ are both ${\mathbb{C}}_{I}$-preserving for some
$I\in{\mathbb{S}}$, i.e., for any $q=\alpha+I\beta$ in the domain of $f$ and
$g$, we have that $f(q),g(q)\in{\mathbb{C}}_{I}$. Slice preserving functions
are ${\mathbb{C}}_{I}$-preserving for any $I\in{\mathbb{S}}$. Keeping this
parallelism between the algebraic features of ${\mathbb{H}}$ and those of
${\mathbb{C}}\otimes{\mathbb{H}}$, we notice that the role of the Euclidean
norm of ${\mathbb{R}}^{4}\simeq{\mathbb{H}}$ is taken here by the so-called
symmetrization of $f$: given a slice function $f=f_{0}+f_{1}i+f_{2}j+f_{3}k$,
its symmetrization is the function
$f^{s}:=\mathcal{I}(FF^{c})=f_{0}^{2}+f_{1}^{2}+f_{2}^{2}+f_{3}^{2}$. The
symmetrization of $f$ has an important role in the study of zeroes of $f$ (see
[GSS:RegFunc, Chapter 3]).

We now recall an important example of slice preserving regular function.

######  Example 2.1.

Let $\mathcal{J}:{\mathbb{H}}\setminus{\mathbb{R}}\to{\mathbb{H}}$ be the
function such that

$\mathcal{J}(q)=\frac{q_{v}}{|q_{v}|}.$

This is clearly a slice preserving function and it is constant on each
semislice ${\mathbb{C}}_{I}^{+}$. In fact, if
$q=\alpha+I\beta\in{\mathbb{H}}\setminus{\mathbb{R}}$, and $\beta>0$, then
$\mathcal{J}(q)=I$. This particular function plays an important role in the
theory of slice regular functions with domain that does not intersect the real
axis. In fact, thanks to the fact that
$\mathcal{J}*\mathcal{J}=\mathcal{J}^{2}\equiv-1$, we are able to define slice
regular idempotent functions (and hence zero divisors) as

$\ell_{+},\ell_{-}:{\mathbb{H}}\setminus{\mathbb{R}}\to{\mathbb{H}},\qquad\ell_{\pm}=\frac{1\mp\mathcal{J}i}{2}.$

For a complete study of these functions see [AltavillaLAA].

Starting from the previous example it is worth noticing that, given any slice
regular function $f=f_{0}+f_{v}$, such that $f_{v}\not\equiv 0$ and
$\sqrt{f_{v}^{s}}$ is well defined (e.g. when $f_{v}^{s}$ is never vanishing
see [AltavillaPAMS, Corollary 3.2]), then the function

$\frac{f_{v}}{\sqrt{f_{v}^{s}}},$

is such that

$\frac{f_{v}}{\sqrt{f_{v}^{s}}}*\frac{f_{v}}{\sqrt{f_{v}^{s}}}\equiv-1\,.$

So, at least two (intrinsically) different functions, take the role of the
imaginary unit in the setting of slice functions.

As said before, a slice regular function is a slice function such that its
stem function is holomorphic. In fact, if $f=\mathcal{I}(F):U\to{\mathbb{H}}$
is a slice function of class $\mathcal{C}^{1}$ defined on a domain $U$, then
the function $\partial F/\partial\bar{z}$ and $\partial F/\partial z$ are stem
functions as well. In particular, the function
$\partial_{c}f:=\mathcal{I}(\partial F/\partial z)$ is called slice derivative
of $f$. As it is clear from the definition, the slice derivative of a slice
regular function controls the behavior “along slices”. Thus, to have complete
information at first order of a slice regular function $f$ we need to consider
another operator, namely the spherical derivative: given a slice function
$f=\mathcal{I}(F):U\to{\mathbb{R}}$, we define
$\partial_{s}f:U\setminus{\mathbb{R}}\to{\mathbb{R}}$ as the slice function
$\partial_{s}f(\alpha+I\beta)=\mathcal{I}\left(\frac{F_{od}(\alpha+\imath\beta)}{\beta}\right)$.
Even if it does not look like a derivative, the spherical derivative can also
be obtained as the result of a differential operator applied to $f$ (see
[perotti]).

We close this preliminary section by recalling the definition of the
$*$-exponential of a slice regular function.

######  Definition 2.5.

Let $f:U\to{\mathbb{H}}$ be any slice function. We denote by $f^{*2}=f*f$ and,
for any $N>2$ we define

$f^{*N}=f*f^{*(N-1)}.$

If $f$ is slice regular, then we define the function
$\exp_{*}(f):U\to{\mathbb{H}}$ as

$\exp_{*}(f)=\sum_{n\in{\mathbb{N}}}\frac{f^{*n}}{n!}.$

Many properties and representations of the $*$-exponential of a slice regular
function are discussed in [AltavillaPAMS, altavillaLOG, GPV1, GPV2].

##  3\. The quaternionic exponential as a covering map

It is well known that the quaternionic exponential map
$\exp:{\mathbb{H}}\to{\mathbb{H}}$ is a covering map. However, in order to be
self-contained we propose here a proof of this fact using its slice regular
nature. In fact, the function $\exp$ is induced by the stem function
$E:{\mathbb{C}}\to{\mathbb{C}}\otimes{\mathbb{H}}$ defined as
$E(\alpha+\imath\beta)=e^{\alpha+\sqrt{-1}\beta}=e^{\alpha}(\cos\beta+\sqrt{-1}\sin\beta)$
or, with our usual abuse of notation, viewed as a curve in ${\mathbb{C}}^{4}$,
as $E(z)=(e^{z},0,0,0)$. Notice that, from the definition, the function $\exp$
is slice preserving.

We now introduce the following family of sets where $\exp$ will result to be
non-singular. For any $k\in{\mathbb{N}}$ set

$\mathcal{U}_{k}:=\\{q\in{\mathbb{H}}\,|\,k\pi<|q_{v}|<(k+1)\pi\\}.$

We are now ready to state and prove the first result.

######  Theorem 3.1.

The real differential of $\exp$ is non-singular at $q$ if and only if
$q\in\mathcal{U}_{k}$ for some $k\in{\mathbb{N}}$. Moreover, for each
$k\in{\mathbb{N}}$, the restriction $\exp_{|\mathcal{U}_{k}}$ is a
diffeomorphism onto its image, which is ${\mathbb{H}}\setminus{\mathbb{R}}$.

###### Proof.

Following [AM:powercover, Lemmas 3.1 and 3.3] or [GSS:RegFunc, Proposition
8.19], if $f:\Omega\to{\mathbb{H}}$ is slice regular and
$q=\alpha+I\beta\in\Omega$, then the real differential of $f$ is singular at
$q$ if and only if $\partial_{c}f(q)=0$ or $\partial_{s}f(q)=0$ or
$\partial_{c}f(q)(\partial_{s}f(q))^{c}\in({\mathbb{C}}_{I})^{\perp}$. We have
that $\partial_{c}\exp=\exp$, so it is never vanishing. The spherical
derivative of $\exp$ at $q=\alpha+I\beta$ is equal to
$e^{\alpha}\frac{\sin\beta}{\beta}$ and therefore it vanishes if and only if
$\beta=h\pi$, where $h\in{\mathbb{Z}}^{*}$. As
$((\partial_{c}\exp)(q))\cdot((\partial_{s}\exp)(q))^{c}=(\exp(q))\cdot
e^{\alpha}\frac{\sin\beta}{\beta}\in{\mathbb{C}}_{I}$, for all
$q\in{\mathbb{H}}$, we have that the set of critical points, i.e. the set of
points where the real differential has not maximum rank, is given by

$C_{0}(\exp)=\\{q\in{\mathbb{H}}\,|\,|q_{v}|=h\pi,h\in{\mathbb{Z}}^{*}\\}=\\{\pi(z,I)\,|\,\mathsf{Im}(z)=h\pi,\,h\in{\mathbb{Z}}^{*},\,I\in{\mathbb{S}}\\}.$

It is easy to see that $\exp(C_{0}(\exp))={\mathbb{R}}$ and therefore, the set
of singular points $S(\exp)=\exp^{-1}(\exp(C_{0}(\exp)))$ is given by

(3)
$S(\exp)=\\{\pi(z,I)\,|\,\mathsf{Im}(z)=h\pi,\,h\in{\mathbb{Z}},\,I\in{\mathbb{S}}\\}.$

Collecting everything, we get the following equality

${\mathbb{H}}\setminus S(\exp)=\bigcup_{h\in{\mathbb{N}}}\mathcal{U}_{h}.$

Now, as $\exp$ is slice-preserving and $\exp({\mathbb{H}}\setminus
S(\exp))\subseteq{\mathbb{H}}\setminus{\mathbb{R}}$, given
$q=\alpha+I\beta,q^{\prime}=\alpha^{\prime}+I^{\prime}\beta^{\prime}\in{\mathbb{H}}\setminus
S(\exp)$, we have that $\exp(q)=\exp(q^{\prime})$ if $I=I^{\prime}$. In fact,
if we denote by
$\mathcal{T}:{\mathbb{H}}\setminus{\mathbb{R}}\to{\mathbb{H}}\setminus{\mathbb{R}}$
the map defined by

$\mathcal{T}(\alpha+I\beta)=\alpha+I(\beta+\pi)$

for $\beta>0$ and $I\in{\mathbb{S}}$, by the standard properties of the
complex exponential, we obtain that $q=\mathcal{T}^{(h)}(q^{\prime})$ for some
$h\in{\mathbb{N}}$, where the superscript denotes iterates.

More generally, $\mathcal{T}$ is a diffeomorphism from $\mathcal{U}_{h}$ and
$\mathcal{U}_{h+1}$ for all $h$; therefore, $\exp$ is injective from
$\mathcal{U}_{h}$ to ${\mathbb{H}}\setminus{\mathbb{R}}$ and, again by the
properties of the complex exponential, also surjective. Being a bijective
local diffeomorphism, $\exp$ is a diffeomorphism between $\mathcal{U}_{h}$ and
${\mathbb{H}}\setminus{\mathbb{R}}$. ∎

######  Remark 3.1.

The open domains $\mathcal{U}_{h}$ are all disjoint, therefore the map

$\exp_{|{\mathbb{H}}\setminus S(\exp)}:{\mathbb{H}}\setminus
S(\exp)\to{\mathbb{H}}\setminus{\mathbb{R}}$

is a covering map in the trivial way (i.e. there is no ramification).

##  4\. The exponential in ${\mathbb{C}}\otimes{\mathbb{H}}$

In this section we are going to study the exponential function of the algebra
${\mathbb{C}}\otimes{\mathbb{H}}$. As ${\mathbb{C}}\otimes{\mathbb{H}}$ is
biholomorphic to ${\mathbb{C}}^{4}$, this study is made in order to apply the
ideas of complex analysis to our quaternionic context. Given
$z=z_{0}+z_{1}i+z_{2}j+z_{3}k\in{\mathbb{C}}^{4}\cong{\mathbb{C}}\otimes{\mathbb{H}}$
we recall that, $\underline{z}=z_{1}i+z_{2}j+z_{3}k$ and
$\underline{z}^{2}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}$. Moreover, given a slice
regular function $f=\mathcal{I}(F)$, we also recall the following set of
relations already introduced in the preliminary section:

$f_{0}=\mathcal{I}(F_{0}),\quad f_{v}=\mathcal{I}(\underline{F}),\quad
f^{c}=\mathcal{I}(F_{0}-\underline{F}),\quad
f^{s}=\mathcal{I}(F_{0}^{2}+\underline{F}^{2}),\quad
f_{v}^{s}=\mathcal{I}(\underline{F}^{2}).$

In [AM:powercover] we introduced the analog of the quaternionic $n$th
$*$-power as
$\sigma_{n}:{\mathbb{C}}\otimes{\mathbb{H}}\to{\mathbb{C}}\otimes{\mathbb{H}}$,
where

$\sigma_{n}(z)=(p_{0}^{n}(z_{0},\underline{z}^{2}),z_{1}p_{1}^{n-1}(z_{0},\underline{z}^{2}),z_{2}p_{1}^{n-1}(z_{0},\underline{z}^{2}),z_{3}p_{1}^{n-1}(z_{0},\underline{z}^{2})),$

where $p_{0}^{n}$ and $p_{1}^{n-1}$ are the usual Chebischev polynomials, such
that

(4) $(x+iy)^{n}=p_{0}^{n}(x,y^{2})+ip_{1}^{n-1}(x,y^{2}).$

We are now able to introduce the exponential function of the algebra
${\mathbb{C}}\otimes{\mathbb{H}}$ as the function
$\varepsilon:{\mathbb{C}}\otimes{\mathbb{H}}\to{\mathbb{C}}\otimes{\mathbb{H}}$
defined by

$\varepsilon=\sum_{n=0}^{\infty}\frac{\sigma_{n}}{n!}\,.$

From the definition of $\sigma_{n}$, we have that

$\varepsilon(z)=\sum_{n=0}^{\infty}\frac{p_{0}^{n}(z_{0},\underline{z}^{2})}{n!}+\underline{z}\sum_{n=0}^{\infty}\frac{p_{1}^{n-1}(z_{0},\underline{z}^{2})}{n!}\,.$

Moreover, thanks to Formula 4, we have that

$\displaystyle\sum_{n=0}^{\infty}\frac{p_{0}^{n}(x,y^{2})}{n!}=\mathrm{Re}\sum_{n=0}^{\infty}\frac{(x+Iy)^{n}}{n!}$
$\displaystyle=e^{x}\cos(y)\,,$ $\displaystyle
y\sum_{n=0}^{\infty}\frac{p_{1}^{n-1}(x,y^{2})}{n!}=\mathrm{Im}\sum_{n=0}^{\infty}\frac{(x+Iy)^{n}}{n!}$
$\displaystyle=e^{x}\sin(y)\,.$

The aim of the following pages is to prove that, under suitable hypotheses,
$\varepsilon$ is a covering map. In order to obtain such a result, we ‘lift’
our construction to another space where we are able to use standard techniques
from complex analysis. This was the fruitful strategy already used in
[AM:powercover] in order to better understand $*$-roots of slice functions.

###  4.1. Lift to ${\mathbb{C}}^{2}\times\mathcal{S}$

As in [AM:powercover] we consider the set $\mathcal{S}$ of imaginary units
contained in ${\mathbb{C}}\otimes{\mathbb{H}}$

$\mathcal{S}:=\\{z=\underline{z}\in{\mathbb{C}}\otimes{\mathbb{H}}\,|\,z^{2}=\underline{z}^{2}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=-1\\},$

and the map
$\rho:{\mathbb{C}}^{2}\times\mathcal{S}\to{\mathbb{C}}\otimes{\mathbb{H}}$
defined by

$\rho((u_{0},u_{1}),s)=u_{0}+u_{1}s.$

######  Remark 4.1.

With the language and symbols of tensor product the set $\mathcal{S}$ contains
those elements $z=p+\sqrt{-1}q$, with $p,q\in{\mathbb{H}}$, such that
$-1=(p+\sqrt{-1}q)^{2}=p^{2}-q^{2}+\sqrt{-1}(pq+qp)$. Therefore, $p$ and $q$
satisfy the following system (see also [AM:powercover, Remark 4.2]),

$\begin{cases}p^{2}-q^{2}=-1,\\\ pq+qp=0.\end{cases}$

The map $\rho$ is a local diffeomorphism and, if we set
$\mathcal{W}^{\prime}:=\\{(u_{0},u_{1})\in{\mathbb{C}}^{2}\,|\,u_{1}\neq 0\\}$
and

$\Omega^{\prime}=\rho(\mathcal{W}^{\prime}\times\mathcal{S})=\\{(z_{0},\underline{z})\in{\mathbb{C}}^{4}\,|\,\underline{z}^{2}\neq
0\\},$

then, the restriction of $\rho$ to $\mathcal{W}^{\prime}\times\mathcal{S}$ is,
in fact, a double cover to its image $\Omega^{\prime}$. In fact, for any
$(z_{0},\underline{z})\in\Omega^{\prime}$, we have that

$\rho^{-1}(z_{0},\underline{z})=\left(\left(z_{0},\pm\sqrt{\underline{z}^{2}}\right),\pm\frac{\underline{z}}{\sqrt{\underline{z}^{2}}}\right).$

In [AM:powercover] we made a large use of this double cover because the total
space is ‘large enough’ to allow many classical properties to hold. Now we
consider the map
$\mathfrak{e}:{\mathbb{C}}^{2}\times\mathcal{S}\to{\mathbb{C}}^{2}\times\mathcal{S}$
defined as

$\mathfrak{e}((u_{0},u_{1}),s)=((e^{u_{0}}\cos(u_{1}),e^{u_{0}}\sin(u_{1})),s).$

This map is defined in order to have that
$\varepsilon\circ\rho=\rho\circ\mathfrak{e}$,

$\begindc{\commdiag}[3]\obj(0,180)[A]{{\mathbb{C}}^{2}\times\mathcal{S}}\obj(280,180)[B]{{\mathbb{C}}^{2}\times\mathcal{S}}\obj(0,0)[C]{{\mathbb{C}}\otimes{\mathbb{H}}}\obj(280,0)[D]{{\mathbb{C}}\otimes{\mathbb{H}}}\mor{A}{B}{\mathfrak{e}}[\atleft,\solidarrow]\mor{A}{C}{\rho}[\atright,\solidarrow]\mor{C}{D}{\varepsilon}[\atright,\solidarrow]\mor{B}{D}{\rho}[\atleft,\solidarrow]\enddc$

i.e., $\mathfrak{e}$ can be viewed as the lift of $\varepsilon$ in
${\mathbb{C}}^{2}\times\mathcal{S}$. We want to prove that $\mathfrak{e}$ is a
covering map onto its image. This result will allow us to state that
$\varepsilon$ is a covering map as well. Before stating the result, we
introduce the following set

$\mathcal{W}:=\\{(u_{0},u_{1})\in{\mathbb{C}}^{2}\,|\,u_{0}^{2}+u_{1}^{2}=0\\}.$

######  Theorem 4.1.

The complex differential of $\mathfrak{e}$ is everywhere non-singular.
Moreover, the function $\mathfrak{e}$ is a covering map onto its image, which
is
$\mathfrak{e}({\mathbb{C}}^{2}\times\mathcal{S})=({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}$.

###### Proof.

Setting $(u^{\prime},s)=\mathfrak{e}(u,s)$, the differential of $\mathfrak{e}$
at $(u,s)$ is a map

$D\mathfrak{e}_{(u,s)}:T_{u}{\mathbb{C}}^{2}\times T_{s}\mathcal{S}\to
T_{u^{\prime}}{\mathbb{C}}^{2}\times T_{s}\mathcal{S}$

that can be represented by the following matrix

$\begin{pmatrix}e^{u_{0}}\cos(u_{1})&-e^{u_{0}}\sin(u_{1})&\mathbf{0}\\\
e^{u_{0}}\sin(u_{1})&e^{u_{0}}\cos(u_{1})&\mathbf{0}\\\
\mathbf{0}^{\top}&\mathbf{0}^{\top}&\mathbf{I_{\mathcal{S}}}\end{pmatrix},$

where $\mathbf{0}=(0,0)$ and, as $\dim_{{\mathbb{C}}}\mathcal{S}=2$,
$\mathbf{I_{\mathcal{S}}}$ the $2\times 2$ identity matrix. We have
$\det(D\mathfrak{e}_{(u,s)})=e^{2u_{0}}$, so $D\mathfrak{e}$ is always
invertible, i.e. $\mathfrak{e}$ is a local diffeomorphism between
${\mathbb{C}}^{2}\times\mathcal{S}$ and itself.

We now pass to look at the image of $\mathfrak{e}$. Given
$(w_{0},w_{1})\in{\mathbb{C}}^{2}$ we consider the following system

(5) $\begin{cases}e^{u_{0}}\cos(u_{1})=w_{0}\\\
e^{u_{0}}\sin(u_{1})=w_{1}.\end{cases}$

We have that

$w_{0}+\imath w_{1}=e^{u_{0}}e^{\imath u_{1}},\qquad w_{0}-\imath
w_{1}=e^{u_{0}}e^{-\imath u_{1}},$

and hence

$\displaystyle u_{0}$ $\displaystyle=\frac{\log(w_{0}+\imath
w_{1})+\log(w_{0}-\imath w_{1})}{2}+(h_{1}+h_{2})\imath\pi,$ $\displaystyle
u_{1}$ $\displaystyle=\frac{\log(w_{0}+\imath w_{1})-\log(w_{0}-\imath
w_{1})}{2\imath}+(h_{1}-h_{2})\pi,$

with $h_{1},h_{2}\in{\mathbb{Z}}$. Therefore, the system in Formula (5) has a
solution if and only if $w_{0}\pm\imath w_{1}\neq 0$, i.e. if and only if
$w_{0}^{2}+w_{1}^{2}\neq 0$. Hence,
$\mathfrak{e}({\mathbb{C}}^{2}\times\mathcal{S})=({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}$.

We now pass to prove that $\mathfrak{e}$ is a covering map onto its image.
Having proved that it is a local diffeomorphism, we are left to prove that the
lifting property is satisfied, i.e., given a continuous curve
$\gamma:[0,1]\to({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}$, we
will show that it is possible to construct a continuous
$\tilde{\gamma}:[0,1]\to{\mathbb{C}}^{2}\times\mathcal{S}$ such that
$\mathfrak{e}\circ\tilde{\gamma}=\gamma$, i.e., such that the following
diagram commutes.

$\begindc{\commdiag}[3]\obj(280,180)[B]{{\mathbb{C}}^{2}\times\mathcal{S}}\obj(0,0)[C]{[0,1]}\obj(280,0)[D]{({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}}\mor{C}{B}{\tilde{\gamma}}[\atleft,\solidarrow]\mor{C}{D}{\gamma}[\atright,\solidarrow]\mor{B}{D}{\mathfrak{e}}[\atleft,\solidarrow]\enddc$

Given
$((\tilde{u}_{0},\tilde{u}_{1}),\tilde{s})\in{\mathbb{C}}^{2}\times\mathcal{S}$
let us consider a curve
$\gamma=((\gamma_{0},\gamma_{1}),\gamma_{s}):[0,1]\to({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}$,
such that $\gamma(0)=\mathfrak{e}((\tilde{u}_{0},\tilde{u}_{1}),\tilde{s})$,
then $\gamma_{0}(t)^{2}+\gamma_{1}(t)^{2}\neq 0$ for all $t\in[0,1]$. Thus, if
we define

$\alpha(t)=\gamma_{0}(t)+\imath\gamma_{1}(t),\qquad\beta(t)=\gamma_{0}(t)-\imath\gamma_{1}(t),$

we have that $\alpha$ and $\beta$ are continuous paths in ${\mathbb{C}}^{*}$.
But then, as $z\mapsto e^{z}$ is a covering map from ${\mathbb{C}}$ to
${\mathbb{C}}^{*}$, we can construct $\tilde{\alpha}$ and $\tilde{\beta}$ such
that

$e^{\tilde{\alpha}}=\alpha,\qquad e^{\tilde{\beta}}=\beta,$

with $\tilde{\alpha}(0)=\tilde{u}_{0}+\imath\tilde{u}_{1}$ and
$\tilde{\beta}(0)=\tilde{u}_{0}-\imath\tilde{u}_{1}$.

In conclusion, the path
$\tilde{\gamma}:[0,1]\to{\mathbb{C}}^{2}\times\mathcal{S}$ defined by

$\tilde{\gamma}(t)=\left(\left(\frac{\tilde{\alpha}(t)+\tilde{\beta}(t)}{2},\frac{\tilde{\alpha}(t)-\tilde{\beta}(t)}{2}\right),\gamma_{s}(t)\right),$

is continuous, is such that
$\tilde{\gamma}(0)=((\tilde{u}_{0},\tilde{u}_{1}),\tilde{s})\in\mathfrak{e}^{-1}(\gamma(0))$
and $\mathfrak{e}\circ\tilde{\gamma}=\gamma$, i.e. the map
$\mathfrak{e}:{\mathbb{C}}^{2}\times\mathcal{S}\to({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}$
is a covering map. ∎

######  Remark 4.2.

As a byproduct of the proof of Theorem 4.1, we get that the fundamental group
of ${\mathbb{C}}^{2}\setminus\mathcal{W}$ is isomorphic to ${\mathbb{Z}}^{2}$;
indeed, for $(h_{1},h_{2})\in{\mathbb{Z}}^{2}$, the monodromy is given by the
following action

$(h_{1},h_{2})\cdot(u_{0},u_{1})=(u_{0}+(h_{1}+h_{2})\imath\pi,u_{1}+(h_{1}-h_{2})\pi).$

Moreover, notice that as $h_{1}+h_{2}$ and $h_{1}-h_{2}$ have the same parity,
this result is coherent to [altavillaLOG, Theorem 1.2] (see the relation
between the indices $m$ and $n$ at point (2) in the second bullet of the
referred result).

###  4.2. Back to ${\mathbb{C}}\otimes{\mathbb{H}}$

We now move back to the study of $\varepsilon$. Let us recall from
[AM:powercover, Section 4] the definition of the following two sets:

$\displaystyle V_{-1}$
$\displaystyle:=\\{(z_{0},\underline{z})\in{\mathbb{C}}\otimes{\mathbb{H}}\,|\,z_{0}^{2}+\underline{z}^{2}=0\\}=\rho(\mathcal{W}\times\mathcal{S}),$
$\displaystyle V_{\infty}$
$\displaystyle:=\\{(z_{0},\underline{z})\in{\mathbb{C}}\otimes{\mathbb{H}}\,|\,\underline{z}^{2}=0\\}=({\mathbb{C}}\otimes{\mathbb{H}})\setminus\Omega^{\prime}.$

From these two sets, we define

$\Omega:=\varepsilon^{-1}({\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup
V_{\infty}))=\\{(z_{0},\underline{z})\in{\mathbb{C}}\otimes{\mathbb{H}}\,|\,\underline{z}^{2}\neq
h^{2}\pi^{2},\mbox{ for }h\in{\mathbb{Z}}\\}.$

Notice that $\Omega$ is the exact transpose in the context of
${\mathbb{C}}\otimes{\mathbb{H}}$ of the set ${\mathbb{H}}\setminus S(\exp)$,
where $S(\exp)$ is defined in Formula (3). Eventually, recalling that
$\varepsilon\circ\rho=\rho\circ\mathfrak{e}$ and collecting the fact that
$\mathfrak{e}:{\mathbb{C}}^{2}\times\mathcal{S}\to({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}$
and $\rho:\mathcal{W}^{\prime}\times\mathcal{S}\to\Omega^{\prime}$ are
covering maps, we have proved the following theorem.

######  Theorem 4.2.

The function
$\varepsilon:\Omega\to{\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup
V_{\infty})$ is a covering map and its monodromy group is isomorphic to
${\mathbb{Z}}^{2}$.

The case when $h=0$ is somehow special as described in the following remark.

######  Remark 4.3.

The further restriction of $\varepsilon$ to $V_{\infty}$ is a covering map
onto its image
$\varepsilon(V_{\infty})=V_{\infty}\setminus\\{(0,\underline{z})\,|\,\underline{z}^{2}=0\\}$.
In fact, if $(z_{0},\underline{z})\in V_{\infty}$, i.e. when $h=0$, we have
that $\varepsilon(z_{0},\underline{z})=e^{z_{0}}(1,\underline{z})$ (compare
with [AltavillaPAMS, Corollary 4.6]), which, again, belongs to $V_{\infty}$;
moreover it is easy to see that $(z_{0},\underline{z})\mapsto e^{z_{0}}(1,z)$
is a covering map. However, while in this case each element in $V_{\infty}$
has a one-parameter family of preimages, thanks to Remark 4.2, in the general
case of Theorem 4.2 any element in
${\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup V_{\infty})$ has a two-
parameters family of preimages in $\Omega$. So, in a sense, in the case
described in this remark, we lose a bunch preimages. In particular, we have
the following isomorphisms of the fundamental groups:

$\pi_{1}({\mathbb{C}}\otimes{\mathbb{H}}\setminus
V_{\infty})\simeq{\mathbb{Z}},\qquad\pi_{1}({\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup
V_{\infty}))\simeq{\mathbb{Z}}^{2}.$

Thanks to the previous theorem, we can construct global ‘logarithms’ with
respect to $\varepsilon$.

######  Corollary 4.3.

Let ${\mathcal{U}}$ be a simply connected domain and let
$F:{\mathcal{U}}\to{\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup
V_{\infty})$ be a continuous function. Then there exist a two-parameter family
of continuous functions
$F_{(h_{1},h_{2})}:{\mathcal{U}}\to{\mathbb{C}}\otimes{\mathbb{H}}$, for
$(h_{1},h_{2})\in{\mathbb{Z}}^{2}$, such that $\varepsilon\circ
F_{h_{1},h_{2}}=F$.

Of course the previous corollary applies, in particular, to stem functions.

##  5\. Global $*$-logarithms

In this short section we collect a series of consequences of the previous
section, allowing us to define global $*$-logarithms of a slice regular
function $f=\mathcal{I}(F)$ such that $f^{s}\neq 0\neq f_{v}^{s}$ or,
equivalently, such that
$F\in{\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup V_{\infty})$. As in
[AM:powercover, Section 5] we declare the following assumption.

######  Assumption 5.1.

From now on, the set of definition $\mathcal{U}=\overline{\mathcal{U}}$ of our
stem functions will be open and simply connected or the union of two simply
connected domains (if $\mathcal{U}\cap{\mathbb{R}}=\emptyset$).

In [GPV1, GPV2] the quaternionic domains $U$ coming from the sets
$\mathcal{U}$ just described in the previous Assumption are called basic
domains.

Thanks to Corollary 4.3, if $f$ is a slice function, such that its stem
function $F$ does not intersects $V_{-1}\cup V_{\infty}$, we virtually have a
two-parameters countable family of $*$-logarithms. We only need to check that
the resulting logarithms at the level of ${\mathbb{C}}\otimes{\mathbb{H}}$ are
stem functions as well. Later we will see that if the domain of $f$ contains
real points, then we lose a parameter, obtaining a closer analogy with the
complex case.

As done in [AM:powercover] for the case of $n$-th $*$-powers, given a stem
function
$F:{\mathcal{U}}\to{\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup
V_{\infty})$, we define the following set

$\mathcal{G}:=\\{G:{\mathcal{U}}\to{\mathbb{C}}\otimes{\mathbb{H}}\,|\,\varepsilon\circ
G=F\\}.$

Since $\varepsilon(\bar{z})=\overline{\varepsilon(z)}$, if $G\in\mathcal{G}$
then the function $\hat{G}:{\mathcal{U}}\to{\mathbb{C}}\otimes{\mathbb{H}}$
defined by $\hat{G}(z)=\overline{G(\bar{z})}$ belongs to $\mathcal{G}$ as
well, in fact

$(\varepsilon\circ\hat{G})(z)=\varepsilon(\hat{G}(z))=\varepsilon(\overline{G(\bar{z})})=\overline{\varepsilon(G(\bar{z}))}=\overline{F(\bar{z})}=F(z).$

Note that $G$ is a stem function if and only if $\hat{G}=G$. Therefore, we can
prove the following result, the proof of which goes exactly as that of
[AM:powercover, Theorem 5.3].

######  Corollary 5.2.

Let $U$ be a basic domain such that $U\cap{\mathbb{R}}=\emptyset$ and let
$f:U\to{\mathbb{H}}$ be a slice regular function such that $f^{s}(q)\neq 0\neq
f_{v}^{s}(q)$, for all $q\in U$. Then, there exists a two-parameters family of
slice functions $f_{(h_{1},h_{2})}:U\to{\mathbb{H}}$, for
$(h_{1},h_{2})\in{\mathbb{Z}}^{2}$, such that

$\exp_{*}(f_{h_{1},h_{2}})=f.$

######  Remark 5.1.

Following Remark 4.2, if $g=g_{0}+g_{v}$ is a $*$-logarithm of $f$, then, for
any couple of integers $h_{1}$ and $h_{2}$, the function

$g_{0}+(h_{1}+h_{2})\mathcal{J}\pi+(\sqrt{g_{v}^{s}}+(h_{1}-h_{2})\pi)\frac{g_{v}}{\sqrt{g_{v}^{s}}}=g+\left[(h_{1}+h_{2})\mathcal{J}+(h_{1}-h_{2})\frac{g_{v}}{\sqrt{g_{v}^{s}}}\right]\pi,$

is a $*$-logarithm of $f$ as well (see [altavillaLOG, Theorem 1.2]).

We now pass to analyze the case in which the function $f$ is defined on a
domain which intersects the real axis. Under this hypothesis, it is clear that
the function $\mathcal{J}$ cannot appear in the set of solutions. In fact, as
explained later, we will obtain that the two parameters $h_{1}$ and $h_{2}$
shall be related by the equality $h_{1}=-h_{2}$.

The proof of the following corollary goes as that of [AM:powercover, Theorem
5.4].

######  Corollary 5.3.

Let $U$ be a basic domain such that $U\cap{\mathbb{R}}\neq\emptyset$ and let
$f:U\to{\mathbb{H}}$ be a slice regular function such that $f^{s}(q)\neq 0\neq
f_{v}^{s}(q)$, for all $q\in U$. Then, there exist a one-parameter family of
slice functions $f_{h}:U\to{\mathbb{H}}$, for $h\in{\mathbb{Z}}$, such that

$\exp_{*}(f_{h})=f.$

Exactly as in [AM:powercover], thanks to our construction, the previous two
corollaries can be stated without the hypothesis of regularity.

######  Remark 5.2.

As anticipated before, if the domain of $f$ contains real points a one-
parameter of solution is missing. In fact, if $x^{0}\in U\cap{\mathbb{R}}$,
then
$F(x^{0})\in{\mathbb{R}}^{4}\subset{\mathbb{C}}^{4}\simeq{\mathbb{C}}\otimes{\mathbb{H}}$
and there exists $y^{0}\in{\mathbb{R}}^{4}$ such that
$\varepsilon(y^{0})=F(x^{0})$. We have $y^{0}=\rho((u_{0},u_{1}),s)$ with
$(u_{0},u_{1})\in{\mathbb{R}}^{2}$. Therefore, we obtain

$\displaystyle\varepsilon^{-1}(F(x^{0}))$
$\displaystyle=\\{\rho((h_{1},h_{2})\cdot((u_{0},u_{1}),s))\,|\,(h_{1},h_{2})\in{\mathbb{Z}}^{2}\\}$
$\displaystyle=\\{\rho((u_{0}+(h_{1}+h_{2})\imath\pi,u_{1}+(h_{1}-h_{2})\pi),s)\,|\,(h_{1},h_{2})\in{\mathbb{Z}}^{2}\\},$

whose only real points are those obtained when $h_{1}+h_{2}=0$ (see the first
component), i.e. when $h_{2}=-h_{1}$. To each point $y$ in
$\varepsilon^{-1}(F(x_{0}))$ we associate $G\in\mathcal{G}$ such that
$G(x^{0})=y$; the $G$’s described in the previous result are those
corresponding to real points in $\varepsilon^{-1}(F(x^{0}))$.

All the results contained in this paper so far are coherent with those
contained in [altavillaLOG, GPV1]. As already pointed out in the introduction,
the main difference here is the idea of using the complex geometry of
${\mathbb{C}}^{4}$ and of ${\mathbb{C}}^{2}\times\mathcal{S}$ to reveal the
nature of $\varepsilon$ and of $\mathfrak{e}$ as covering maps. In a broad
sense, the strategy of [altavillaLOG] is that of “solving” the $*$-logarithm
mostly in algebraic terms, while in [GPV1] the same problem is addressed by
considering a sort of “$*$-logarithm variety” and by analyzing the geometry of
curves contained in it. As already noted in [AM:powercover], in our opinion
our approach seems to be more suitable to generalizations to other contexts,
while giving a global view of the geometric structure lying beneath the
specific issue.

We conclude this section by highlighting how it is possible to recover results
about the $*$-roots, starting from the $*$-logarithm. The starting point is a
quite standard argument from one complex variable but, as we will see, the
computation of the monodromy needs some deeper investigation.

######  Remark 5.3.

As already said, in [AM:powercover] we widely studied the existence of a
$n$-th $*$-rooth of a slice regular function. Exactly as in the complex case,
almost all the work done in [AM:powercover] can be recovered from the study of
the $*$-logarithm. In fact, if $U$ is a basic domain and $f:U\to{\mathbb{H}}$
is a slice function such that $f^{s}(q)\neq 0\neq f_{v}^{s}(q)$, for all $q\in
U$, then, for any $n\in\mathbb{R}$ we are able to define

$(f)^{*\frac{1}{n}}:=\exp_{*}\left(\frac{1}{n}\log_{*}(f)\right),$

where the apex $*$, means that we are considering the power with respect to
the $*$-product. We will show in the next section how to recover the monodromy
of $n$-th $*$-root from that of the $*$-logarithm.

##  6\. Automorphisms of $\mathfrak{e}$ and of $\varepsilon$

In this section we will give a description of the deck transformations of
$\mathfrak{e}$ and of $\varepsilon$, i.e. the automorphisms of
${\mathbb{C}}^{2}\times\mathcal{S}$ and of $\Omega$ fixing the fibers of
$\mathfrak{e}$ and of $\varepsilon$, respectively. To be precise, given a
covering map $\pi:X\to Y$, we are interested in the set

$\operatorname{Aut}_{\pi}:=\\{f:X\to X\,|\,\pi\circ f=\pi\\}.$

In particular, we will study $\operatorname{Aut}_{\mathfrak{e}}$ and
$\operatorname{Aut}_{\varepsilon}$. We recall from [AM:powercover] that
$\operatorname{Aut}_{\rho}=\\{\mathrm{id},\Gamma\\}$, where
$\Gamma:{\mathbb{C}}^{2}\times\mathcal{S}\to{\mathbb{C}}^{2}\times\mathcal{S}$
is given by

$\Gamma((u_{0},u_{1}),s)=((u_{0},-u_{1}),-s).$

Thanks to the content of the previous section we are able to represent
$\operatorname{Aut}_{\mathfrak{e}}$ in a convenient way. In fact, let us
define
$T_{\ell}:{\mathbb{C}}^{2}\times\mathcal{S}\to{\mathbb{C}}^{2}\times\mathcal{S}$
as the function

$T_{\ell}((u_{0},u_{1}),s)=((u_{0}+\imath\pi,u_{1}+\ell\pi),s),\quad\ell=-1,1.$

Then, following the proof of Theorem 4.1, we have that

$\operatorname{Aut}_{\mathfrak{e}}=\\{T_{(a,b)}:=aT_{1}+bT_{-1}\,|\,a,b\in{\mathbb{Z}},\,a\equiv_{2}b\\}.$

In particular, given $(h_{1},h_{2})\in{\mathbb{Z}}^{2}$ from Remark 4.2, we
get $a=h_{1}+h_{2}$ and $b=h_{1}-h_{2}$, while, given $T_{(a,b)}$, then
$h_{1}=\frac{a+b}{2}$ and $h_{2}=\frac{a-b}{2}$.

We now pass to study $\operatorname{Aut}_{\varepsilon}$. Recall that
$\varepsilon\circ\rho=\rho\circ\mathfrak{e}$ and notice that $\Gamma\circ
T_{1}=T_{-1}\circ\Gamma$. Therefore, if $(a,b)\in{\mathbb{Z}}^{2}$, we get

$\Gamma\circ(aT_{1}+bT_{-1})=a\Gamma\circ T_{1}+b\Gamma\circ
T_{-1}=aT_{-1}\circ\Gamma+bT_{1}\circ\Gamma=(bT_{1}+aT_{-1})\circ\Gamma.$

Hence, $T_{(a,b)}$ descend to a map $S\in\operatorname{Aut}_{\varepsilon}$, if
and only if $(a,b)=(b,a)$, i.e. if and only if
$T_{(a,b)}=T_{(a,a)}=a(T_{1}+T_{-1})$, $a\in{\mathbb{Z}}$. But then it follows
that, if we define
$S_{0}:{\mathbb{C}}\otimes{\mathbb{H}}\to{\mathbb{C}}\otimes{\mathbb{H}}$ as
the map

$S_{0}(z_{0},\underline{z})=(z_{0}+2\imath\pi,\underline{z}),$

then

$\operatorname{Aut}_{\varepsilon}=\langle
S_{0}\rangle_{{\mathbb{Z}}}\simeq{\mathbb{Z}}.$

We have just proven the following result which is analogous to [AM:powercover,
Corollary 6.4 and Proposition 6.5]

######  Proposition 6.1.

The covering map $\mathfrak{e}$ is regular, while $\varepsilon$ is not.

###  6.1. Monodromy of $*$-roots

We now want to recover the monodromy of $*$-roots by means of that of
$*$-logarithm. This computation was already performed in [AM:powercover] with
different techniques. Here we will use what we just learned from the study of
$*$-logarithm.

Since, for any $z\in{\mathbb{C}}\otimes{\mathbb{H}}$, we have that
$\varepsilon(2z)=\varepsilon(z)^{2}$, where sum and product are the algebra
operations of ${\mathbb{C}}\otimes{\mathbb{H}}$, for any $n\in{\mathbb{N}}$,
then we also have that

$\varepsilon(nz)=\sigma_{n}(\varepsilon(z)).$

Given $G\in\mathcal{G}$, let
$H_{G}:\mathcal{U}\to{\mathbb{C}}^{2}\times\mathcal{S}$ be such that
$\rho\circ H_{G}=G$. Then we have

$\varepsilon\circ\rho\circ T_{\ell}\circ H_{G}=\rho\circ\mathfrak{e}\circ
T_{\ell}\circ H_{G}=\rho\circ\mathfrak{e}\circ H_{G}=\varepsilon\circ\rho\circ
H_{G}=\varepsilon\circ G=F\;.$

Under the hypotheses of Corollary 4.3, the set $\mathcal{G}$ is a two-
parameters family of logarithms with respect to $\varepsilon$. Therefore, we
can represent each element of $\mathcal{G}$ as $G_{(a,b)}=\rho\circ
T_{(a,b)}\circ H_{\tilde{G}}$, with $(a,b)\in{\mathbb{Z}}^{2}$, where
$G_{(0,0)}=:\tilde{G}$ is any particular solution of $\varepsilon\circ X=F$.

Now,
$\varepsilon\circ(\frac{1}{n}{G}_{(a,b)})=\varepsilon\circ(\frac{1}{n}{G}_{(c,d)})$
with $(a,b),\ (c,d)\in{\mathbb{Z}}^{2}$, if and only if

$\varepsilon\circ\rho\circ\frac{1}{n}T_{(a,b)}\circ
H_{\tilde{G}}=\varepsilon\circ\rho\circ\frac{1}{n}T_{(c,d)}\circ
H_{\tilde{G}}\;.$

Set $H_{\tilde{G}}(z)=((u_{0}(z),u_{1}(z)),s(z))$, then

$\frac{1}{n}T_{(a,b)}\circ
H_{\tilde{G}}=\left(\left(\frac{1}{n}u_{0}(z)+\frac{a+b}{n}\imath\pi,\frac{1}{n}u_{1}(z)+\frac{a-b}{n}\pi\right),s(z)\right).$

Assume, also, that $\\{u_{1}\neq 0\\}$, then the equality

$\rho\circ\mathfrak{e}\circ\frac{1}{n}T_{(a,b)}\circ
H_{\tilde{G}}=\rho\circ\mathfrak{e}\circ\frac{1}{n}T_{(c,d)}\circ
H_{\tilde{G}}$

holds true if and only if

$\mathfrak{e}\circ\frac{1}{n}T_{(a,b)}\circ
H_{\tilde{G}}=\mathfrak{e}\circ\frac{1}{n}T_{(c,d)}\circ H_{\tilde{G}}$

that is equivalent to say that there exists $(e,f)\in{\mathbb{Z}}^{2}$ such
that

$T_{(e,f)}\circ\frac{1}{n}T_{(a,b)}\circ
H_{\tilde{G}}=\frac{1}{n}T_{(c,d)}\circ H_{\tilde{G}},$

i.e.

$\left(\left(\frac{1}{n}u_{0}(z)+\left((e+f)+\frac{a+b}{n}\right)\imath\pi,\frac{1}{n}u_{1}(z)+\left((e-f)+\frac{a-b}{n}\right)\pi\right),s(z)\right)\\\
=\left(\left(\frac{1}{n}u_{0}(z)+\frac{c+d}{n}\imath\pi,\frac{1}{n}u_{1}(z)+\frac{c-d}{n}\pi\right),s(z)\right),$

i.e.

$\frac{(c+d)-(a+b)}{n}=e+f\qquad\frac{(c-d)-(a-b)}{n}=e-f$
$\frac{c-a}{n}+\frac{d-b}{n}=e+f\qquad\frac{c-a}{n}-\frac{d-b}{n}=e-f$

i.e.

$\frac{c-a}{n}\in{\mathbb{Z}}\qquad\frac{d-b}{n}\in{\mathbb{Z}}\;.$

i.e. $n|(c-a)$ and $n|(d-b)$, i.e. $(a,b)\equiv(c,d)\bmod n$.

Consider the subgroup of automorphisms

$\mathsf{I}_{n}=\left\langle T_{(a,b)}\,|\,a\equiv b\equiv 0\bmod
n\right\rangle$

and let $W_{n}={\mathbb{C}}^{2}\times\mathcal{S}/\mathsf{I}_{n}$; the
projection $\mathfrak{e}_{n}:{\mathbb{C}}^{2}\times\mathcal{S}\to W_{n}$ is a
covering map. As $\mathsf{I}_{n}$ is a subgroup of
$\mathrm{Aut}_{\mathfrak{e}}$ and as $\mathfrak{e}$ is a Galois covering, we
can factor $\mathfrak{e}$ via $\mathfrak{e}_{n}$: we consider the map
$\mathfrak{s}_{n}:W_{n}\to({\mathbb{C}}^{2}\setminus\mathcal{W})\times\mathcal{S}$
such that $\mathfrak{e}=\mathfrak{s}_{n}\circ\mathfrak{e}_{n}$;
$\mathfrak{s}_{n}$ is in fact a covering map of degree $n^{2}$ (the index of
$\mathsf{I}_{n}$ in $\mathrm{Aut}_{\mathfrak{e}}$).

As $\mathrm{Aut}_{\rho}$ does not intersect $\mathrm{Aut}_{\mathfrak{e}}$, we
can induce a map $\tilde{\rho}:W_{n}^{\prime}\to\Omega_{n}$ on a suitable open
set $W^{\prime}_{n}\subseteq W_{n}$ such that

  * •

$\tilde{\rho}$ is a double cover

  * •

there is a (unique) covering map
$\sigma_{n}:\Omega_{n}\to({\mathbb{C}}\otimes{\mathbb{H}})\setminus(V_{-1}\cup
V_{\infty})$ with $\sigma_{n}\circ\tilde{\rho}=\rho\circ\mathfrak{s}_{n}$

  * •

there is a (unique) covering map $\epsilon_{n}:\Omega\to\Omega_{n}$ such that
$\epsilon=\sigma_{n}\circ\epsilon_{n}$.

It is easy to notice that
$\mathfrak{e_{n}}((u_{0},u_{1}),s)=((u_{0}/n,u_{1}/n),s)$,
$\epsilon_{n}(z)=\epsilon(z/n)$, $\tilde{\rho}=\rho$, $\sigma_{n}(z)=z^{n}$
(and the induced $\mathfrak{s}_{n}$) satisfy the previous requirements.
Therefore, $W^{\prime}_{n}$ and $\Omega_{n}$ can be realizes as subdomains of
${\mathbb{C}}^{2}\times\mathcal{S}$ and ${\mathbb{C}}\otimes{\mathbb{H}}$
respectively.

Carryig out the computations, one could find the definitions given in
[AM:powercover].

Finally, we compute the monodromy of $\mathfrak{s}_{n}$. By simple arithmetic,
the group ${\mathbb{Z}}_{n}\times{\mathbb{Z}}_{n}={\mathbb{Z}}^{2}/\mathsf{I}$
is generated by the classes $[(1,1),(1,-1)]$ when $n$ is odd and by the
classes $[(1,1),(1,-1),(1,0)]$ when $n$ is even.

Given $a,b\in\\{0,\ldots,n-1\\}$

$\mathfrak{e}_{n}(T_{(a,b)}((u_{0},u_{1}),s))=\mathfrak{e}\left(\left(\frac{1}{n}u_{0}+\frac{a+b}{n}\imath\pi,\frac{1}{n}u_{1}+\frac{a-b}{n}\pi\right),s\right)=$
$=\left(\left(e^{\frac{u_{0}}{n}}e^{\frac{a+b}{n}\imath\pi}\cos\left(\frac{u_{1}}{n}+\frac{a-b}{n}\pi\right),e^{\frac{u_{0}}{n}}e^{\frac{a+b}{n}\imath\pi}\sin\left(\frac{u_{1}}{n}+\frac{a-b}{n}\pi\right)\right),s\right)=\xi\cdot\left(A_{\eta}\frac{(u_{0},u_{1})}{n},s\right)$

where $\xi=e^{\frac{a+b}{n}\imath\pi}$ is a $n$-th root of unity and

$A_{\eta}=\begin{pmatrix}\cos(\frac{a-b}{n}\pi)&-\sin(\frac{a-b}{n}\pi)\\\
\sin(\frac{a-b}{n}\pi)&\cos(\frac{a-b}{n}\pi)\end{pmatrix}$

is the $2\times 2$ matrix representation of the complex number
$\eta=e^{\frac{a-b}{n}\imath\pi}$.

So, for $n$ odd, the generators of the deck transformations of
$\mathfrak{s}_{n}$ are $\xi$ (corresponding to $[(1,1)]$) and $A_{\eta}$
(corresponding to $[(1,-1)]$) with $\xi$, $\eta$ primitive $n$-th roots of
$1$; for $n$ even we have these two and $\xi\cdot A_{\eta}$ (corresponding to
$[(1,0)]$) with $\xi$, $\eta$ primitive $2n$-th roots of $1$.

##  7\. Product of two $*$-exponentials

In this section we will give sufficient conditions for the product of two
exponentials to be an exponential. This topic clearly deals with the so-called
Baker-Campbell-Hausdorff (BCH) formula for the $*$-exponential.

In its more general formulation the BCH formula states that, whenever it
exists, the product $e^{X}e^{Y}$ equals $e^{Z}$, where

(6)
$Z=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]]+\cdots$

Clearly, depending on the context, it is possible to give sufficient
conditions for the sum on the right hand side of Formula (6) to be convergent
(see for instance [BiagiBCH, Proposition 2.2] for Banach algebras or
[bonfiglioli] for a general overview). In the context of quaternions, the
situation is much more clear: if $p=p_{0}+p_{v}$ and $q=q_{0}+q_{v}$, with
$p_{v}\neq 0\neq q_{v}$, then

$\displaystyle\exp(p)\exp(q)=$
$\displaystyle\left[e^{p_{0}}\left(\cos|p_{v}|+\sin|p_{v}|\frac{p_{v}}{|p_{v}|}\right)\right]\left[e^{q_{0}}\left(\cos|q_{v}|+\sin|q_{v}|\frac{q_{v}}{|q_{v}|}\right)\right]$
$\displaystyle=$ $\displaystyle
e^{p_{0}+q_{0}}\left[\cos|p_{v}|\cos|q_{v}|-\sin|p_{v}|\sin|q_{v}|\langle\frac{p_{v}}{|p_{v}|},\frac{q_{v}}{|q_{v}|}\rangle+\right.$
$\displaystyle+\left.\cos|p_{v}|\sin|q_{v}|\frac{q_{v}}{|q_{v}|}+\cos|q_{v}|\sin|p_{v}|\frac{p_{v}}{|p_{v}|}+\sin|p_{v}|\sin|q_{v}|\frac{p_{v}}{|p_{v}|}\wedge\frac{q_{v}}{|q_{v}|}\right]$
$\displaystyle=$ $\displaystyle\exp(w_{0}+w_{v}),$

where $w_{0}=p_{0}+q_{0}$ and $w_{v}$ solves

$\begin{cases}\cos|w_{v}|=\cos|p_{v}|\cos|q_{v}|-\sin|p_{v}|\sin|q_{v}|\langle\frac{p_{v}}{|p_{v}|},\frac{q_{v}}{|q_{v}|}\rangle,\\\
\sin|w_{v}|\frac{w_{v}}{|w_{v}|}=\cos|p_{v}|\sin|q_{v}|\frac{q_{v}}{|q_{v}|}+\cos|q_{v}|\sin|p_{v}|\frac{p_{v}}{|p_{v}|}+\sin|p_{v}|\sin|q_{v}|\frac{p_{v}}{|p_{v}|}\wedge\frac{q_{v}}{|q_{v}|}.\end{cases}$

Notice that, in this case, the existence of the solution is granted because
$\exp(p)\exp(q)\neq 0$ and hence it is possible to define $w$.

Now, as ${\mathbb{H}}$ and ${\mathbb{C}}\otimes{\mathbb{H}}$ have the same
algebraic structure, the same equalities hold true for the complexification.
However, recall that the euclidean norm must be translated into its purely
algebraic form, i.e., if
$u=u_{0}+\underline{u},u^{\prime}=u^{\prime}_{0}+\underline{u^{\prime}}\in{\mathbb{C}}\otimes{\mathbb{H}}$,
then

$\displaystyle\exp(u)\exp(u^{\prime})=$ $\displaystyle
e^{u_{0}+u^{\prime}_{0}}\left[\cos\sqrt{\underline{u}^{2}}\cos\sqrt{\underline{u^{\prime}}^{2}}-\sin\sqrt{\underline{u}^{2}}\sin\sqrt{\underline{u^{\prime}}^{2}}\langle\frac{\underline{u}}{\sqrt{\underline{u}^{2}}},\frac{\underline{u^{\prime}}}{\sqrt{\underline{u^{\prime}}^{2}}}\rangle+\right.$
$\displaystyle+\left.\cos\sqrt{\underline{u}^{2}}\sin\sqrt{\underline{u^{\prime}}^{2}}\frac{\underline{u^{\prime}}}{\sqrt{\underline{u^{\prime}}^{2}}}+\cos\sqrt{\underline{u^{\prime}}^{2}}\sin\sqrt{\underline{u}^{2}}\frac{\underline{u}}{\sqrt{\underline{u}^{2}}}+\right.$
$\displaystyle+\left.\sin\sqrt{\underline{u}^{2}}\sin\sqrt{\underline{u^{\prime}}^{2}}\frac{\underline{u}}{\sqrt{\underline{u}^{2}}}\wedge\frac{\underline{u^{\prime}}}{\sqrt{\underline{u^{\prime}}^{2}}}\right].$

In this case, the solution
$p=p_{0}+\underline{p}\in{\mathbb{C}}\otimes{\mathbb{H}}$ of the equation
$\exp(u)\exp(u^{\prime})=\exp(p)$ exists provided
$\exp(u)\exp(u^{\prime})\in{\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup
V_{\infty})$. From the previous computations and also from [AltavillaPAMS,
Theorem 4.14], we already know that if $u$ commutes with $u^{\prime}$ or, if
$\underline{u}^{2}=\pi^{2}n^{2}$ and $\underline{u^{\prime}}^{2}=\pi^{2}m^{2}$
with $n$ and $m$ satisfying a certain parity condition, then
$\exp(u)\exp(u^{\prime})=\exp(u+u^{\prime})$. Therefore, we are interested in
understanding when these conditions are satisfied excluding the cases listed
in the already mentioned result [AltavillaPAMS, Theorem 4.14]. In order to
proceed, we need a couple of preliminary lemmas.

######  Lemma 7.1.

Let $\underline{z}\in{\mathbb{C}}\otimes\mathsf{Im}({\mathbb{H}})$ be such
that $\underline{z}^{2}\neq 0$, then, for any
$\underline{w}\in{\mathbb{C}}\otimes\mathsf{Im}({\mathbb{H}})$, there exist
$w_{1}\in{\mathbb{C}}$ and
$w_{\perp}\in{\mathbb{C}}\otimes\mathsf{Im}({\mathbb{H}})$, such that

$\underline{w}=w_{1}\underline{v}+w_{\perp},\qquad\mbox{and}\qquad\langle\underline{z},w_{\perp}\rangle=0.$

Moreover, it holds

$\underline{(\underline{z}\wedge
w_{\perp})}^{2}=\underline{z}^{2}\underline{w_{\perp}}^{2}.$

###### Proof.

By standard linear algebra, it is sufficient to define

$w_{1}=\frac{\langle\underline{w},\underline{z}\rangle}{\langle\underline{z},\underline{z}\rangle}=\frac{\langle\underline{w},\underline{z}\rangle}{\underline{z}^{2}}.$

∎

Now, if $z,w\in{\mathbb{C}}\otimes{\mathbb{H}}$, then
$(zw)_{0}^{2}+\underline{(zw)}^{2}=(z_{0}^{2}+\underline{z}^{2})(w_{0}^{2}+\underline{w}^{2})$
(see Formula (2)), therefore, if $z_{0}^{2}+\underline{z}^{2}\neq 0\neq
w_{0}^{2}+\underline{w}^{2}$, then $(zw)_{0}^{2}+\underline{(zw)}^{2}\neq 0$.
Hence, the following result complete the characterization we are looking for.

######  Theorem 7.2.

Let $z,w\in{\mathbb{C}}\otimes{\mathbb{H}}\setminus(V_{-1}\cup V_{\infty})$,
then, $\underline{(zw)}^{2}=0$ if and only if

$w_{0}=-z_{0}w_{1}\pm\sqrt{-1}\sqrt{\frac{z_{0}^{2}+\underline{z}^{2}}{\underline{z}^{2}}}\sqrt{\underline{w_{\perp}}^{2}},$

where $w_{1}$ and $w_{\perp}$ are the elements defined in Lemma 7.1.

###### Proof.

First of all, thanks to Lemma 7.1, we can write
$\underline{w}=w_{1}\underline{z}+w_{\perp}$, with $w_{1}\in{\mathbb{C}}$ and
$\langle w_{\perp},\underline{z}\rangle=0$. Therefore,

$\displaystyle zw$
$\displaystyle=(z_{0}+\underline{z})(w_{0}+w_{1}\underline{z}+w_{\perp})$
$\displaystyle=(z_{0}w_{0}-\underline{z}^{2}w_{1})+(z_{0}w_{1}\underline{z}+z_{0}w_{\perp}+w_{0}\underline{z}+\underline{z}\wedge
w_{\perp}),$

and so

$\displaystyle\underline{(zw)}^{2}$
$\displaystyle=\underline{(z_{0}w_{1}\underline{z}+z_{0}w_{\perp}+w_{0}\underline{z}+\underline{z}\wedge
w_{\perp})}^{2}$
$\displaystyle=\underline{(z_{0}w_{1}\underline{z}+z_{0}w_{\perp}+w_{0}\underline{z})}^{2}+\underline{(\underline{z}\wedge
w_{\perp})}^{2}$
$\displaystyle=(z_{0}w_{1}+w_{0})^{2}\underline{z}^{2}+(z_{0}^{2}+\underline{z}^{2})\underline{w_{\perp}}^{2}.$

Therefore, $\underline{(zw)}^{2}=0$ if and only if
$(z_{0}w_{1}+w_{0})^{2}\underline{z}^{2}+(z_{0}^{2}+\underline{z}^{2})\underline{w_{\perp}}^{2}=0$,
which is equivalent to

$(z_{0}w_{1}+w_{0})^{2}=-\frac{z_{0}^{2}+\underline{z}^{2}}{\underline{z}^{2}}\underline{w_{\perp}}^{2},$

and hence, we get the thesis. ∎

The previous result can be applied to slice regular functions recalling their
decomposition in “scalar-vector” parts. We start by rewriting Lemma 7.1 in
terms of slice regular functions.

######  Corollary 7.3.

Let $f=f_{0}+f_{v}:U\to{\mathbb{H}}$ be a slice regular function such that,
for any $q\in U$, $f_{v}^{s}(q)\not\equiv 0$. Then, for any slice regular
function $g=g_{0}+g_{v}$, there exist two slice regular function
$g_{1},g_{\perp}$, $g_{1}$ being slice preserving, such that

$g=g_{1}f_{v}+g_{\perp},\qquad\mbox{and}\qquad\langle
f_{v},g_{\perp}\rangle=0.$

Moreover, it holds

$(f_{v}\wedge g_{\perp})^{s}=f_{v}^{s}{g_{\perp}^{s}}.$

###### Proof.

Exactly as in the previous result, for any $q\in U$ such that
$f_{v}^{s}(q)\neq 0$ it is sufficient to define $g_{1}:=\frac{\langle
g_{v},f_{v}\rangle}{f_{v}^{s}}$. Assume now that
$q_{0}=\alpha_{0}+i\beta_{0}\not\in{\mathbb{R}}$ and $f_{v}^{s}(q_{0})=0$.
Define $D_{q_{0}}(\epsilon)$ as the disk in $U\cap{\mathbb{C}}_{i}$ centered
at $q_{0}$ with radius $\epsilon$, such that
$\overline{D_{q_{0}}(\epsilon)}\subset U\cap{\mathbb{C}}_{i}$ such that
$f_{v}^{s}(q)\neq 0$ for any
$q\in\overline{D_{q_{0}}(\epsilon)}\setminus\\{q_{0}\\}$. Then we can define
$g_{1}(q_{0})$ by means of the Cauchy formula

$g_{1}(q_{0}):=\frac{1}{2\pi i}\int_{\partial
D_{q_{0}}(\epsilon)}\frac{g_{1}(\alpha+i\beta)}{\alpha+i\beta-
q_{0}}d(\alpha+i\beta).$

By repeating the same construction at $q_{0}^{c}$ and using the Representation
Formula, we obtain the thesis. The previous argument can be performed at
$q_{0}\in{\mathbb{R}}$. ∎

######  Corollary 7.4.

Let $f,g$ be two slice regular functions defined on $U$ such that $f^{s}\neq
0\neq g^{s}$ and $f_{v}^{s}\neq 0\neq g_{v}^{s}$, then, $(f*g)_{v}^{s}(q)=0$
if and only if

(7)
$(f_{0}(q)g_{1}(q)+g_{0}(q))^{2}+\frac{f^{s}(q)}{f_{v}^{s}(q)}{g_{\perp}^{s}(q)}=0$

where $g_{1}$ and $g_{\perp}$ are the functions defined in Corollary 7.3. In
particular, if $U\cap{\mathbb{R}}\neq\emptyset$, then there exists an open
neighborhood $U$ of $U\cap{\mathbb{R}}$ such that ${(f*g)_{v}^{s}}_{|U}\neq
0$.

###### Proof.

The first part of the statement is a direct consequence of Theorem 7.2. For
the second part, assume that $x\in U\cap{\mathbb{R}}$, then Formula (7)
evaluated at $x$, gives no solutions since the left hand side is strictly
positive. Therefore, there exists an open neighborhood $U$ of
$U\cap{\mathbb{R}}$ where the function
$(f_{0}g_{1}+g_{0})^{2}+\frac{f^{s}}{f_{v}^{s}}{g_{\perp}^{s}}$ is never
vanishing, and hence $(f*g)_{v}^{s}\neq 0$ on $U$. ∎

Thanks to the previous two corollaries we can reverse engineer several
examples of slice regular functions $f,g$ with $f^{s}\neq 0\neq g^{s}$ and
$f_{v}^{s}\neq 0\neq g_{v}^{s}$ but $(f*g)_{v}^{s}(q)=0$.

######  Example 7.1.

Assume for simplicity that $U\cap{\mathbb{R}}=\emptyset$. Then, given $f$
satisfying the hypotheses of previous corollary, we define
$g=g_{0}+g_{v}=g_{0}+g_{1}f_{v}+g_{\perp}$ as follows:

$g_{0}=-f_{0}g_{1}\pm\mathcal{J}\sqrt{\frac{f^{s}}{f_{v}^{s}}}\sqrt{g_{\perp}^{s}},$

with $g_{1}$ and $g_{\perp}$ be such that
$g_{v}^{s}=g_{1}^{2}f_{v}^{s}+g_{\perp}^{s}\neq 0$ and
$g_{0}^{2}+g_{1}^{2}f_{v}^{s}+g_{\perp}^{s}\neq 0$. Clearly if
$g_{\perp}^{s}\equiv 0$, then it is sufficient to take
$g=g_{1}(-f_{0}+f_{v})+g_{\perp}$. For instance, if $f$ is
${\mathbb{C}}_{i}$-preserving, i.e. $f=f_{0}+f_{1}i$, with
$f_{0}^{2}+f_{1}^{2}\neq 0\neq f_{1}$, then, if $g=-f^{c}+\ell_{+,i}*j$, we
have that $f*g=-f^{s}+f*\ell_{+,i}*j$, $(f*g)_{v}=f*\ell_{+,i}*j$ and,
therefore $(f*g)_{v}^{s}\equiv 0$.

Another readable case is when $f_{0}\equiv 0$. In this case, given $f=f_{v}$,
it is sufficient to consider
$g=\pm\mathcal{J}\sqrt{g_{\perp}^{s}}+g_{1}f_{v}+g_{\perp}$, with
$g^{s}=g_{1}^{2}f_{v}^{s}\neq 0$, i.e. $g_{1}\neq 0$, and
$g_{1}^{2}f_{v}^{s}+g_{\perp}^{s}\neq 0$.

Clearly, the previous example allows to construct functions $f$ and $g$, such
that $(f*g)_{v}^{s}\equiv 0$, while the condition in Corollary 7.4 si given
point wise.

At this point we are able to give sufficient conditions for $f$ and $g$ in
order to have that $\exp_{*}(f)*\exp_{*}(g)$ is an exponential function. As we
said, this happens if $(\exp_{*}(f)*\exp_{*}(g))_{v}^{s}(q)\neq 0$ for all
$q$. Clearly, we can separate the “scalar” part of $\exp_{*}(f)$ and of
$\exp_{*}(g)$ and only consider $(\exp_{*}(f)_{v}*\exp_{*}(g)_{v})_{v}^{s}$.

We get the following result.

######  Corollary 7.5.

Let $f,g:U\to{\mathbb{H}}$ be slice regular function such that $f_{v}$ do not
commute with $g_{v}$ and for all $q\in U$
$f_{v}^{s}(q),g_{v}^{s}(q)\not\in\\{\pi^{2}n^{2}\,|\,n\in{\mathbb{Z}}\\}$.
Write $g_{v}=g_{1}\frac{f_{v}}{\sqrt{f_{v}^{s}}}+g_{\perp}$. If for any $q\in
U$

$(g_{1}\cos\sqrt{f_{v}^{s}}\sin\sqrt{g_{v}^{s}}+\cos\sqrt{g_{v}^{s}})^{2}(\sin\sqrt{f_{v}^{s}})^{2}+(\sin\sqrt{g_{v}^{s}})^{2}g_{\perp}^{s}\neq
0,$

then there exists a slice regular function $h:U\to{\mathbb{H}}$ such that

$\exp_{*}(f)*\exp_{*}(g)=\exp_{*}(h).$

Notice that in the last result the function $h=h_{0}+h_{v}$ is determined by
$h_{0}=f_{0}+g_{0}$ and $h_{v}$ solves

$\begin{cases}\cos\sqrt{h_{v}^{s}}=\cos\sqrt{f_{v}^{s}}\cos\sqrt{g_{v}^{s}}-\sin\sqrt{f_{v}^{s}}\sin\sqrt{g_{v}^{s}}\langle\frac{f_{v}}{\sqrt{f_{v}^{s}}},\frac{g_{v}}{\sqrt{g_{v}^{s}}}\rangle,\\\
\sin\sqrt{h_{v}^{s}}\frac{h_{v}}{\sqrt{h_{v}^{s}}}=\cos\sqrt{f_{v}^{s}}\sin\sqrt{g_{v}^{s}}\frac{g_{v}}{\sqrt{g_{v}^{s}}}+\cos\sqrt{g_{v}^{s}}\sin\sqrt{f_{v}^{s}}\frac{f_{v}}{\sqrt{f_{v}^{s}}}+\sin\sqrt{f_{v}^{s}}\sin\sqrt{g_{v}^{s}}\frac{f_{v}}{\sqrt{f_{v}^{s}}}\wedge\frac{g_{v}}{\sqrt{g_{v}^{s}}}.\end{cases}$

##  8\. Slice derivative of the $*$-exponential

In this section we will provide a formula for the slice derivative of
$\exp_{*}(f)$, $f$ being a slice regular function. As in the previous section,
let us begin with a short description of the general algebraic case. If $X$ is
a matrix, the differential of $e^{X}$, is given by the following formula

(8)
$e^{-X}de^{X}=dX-\frac{1}{2!}\left[X,dX\right]+\frac{1}{3!}[X,[X,dX]]-\frac{1}{4!}[X,[X,[X,dX]]]+\cdots$

Assume now that $q:[0,1]\to{\mathbb{H}}$ is a differentiable curve and denote
by $\dot{q}=\frac{dq}{dt}$. Therefore, Formula 8 can be written in the
quaternionic setting as

$\displaystyle e^{-q(t)}\frac{de^{q(t)}}{dt}$
$\displaystyle=\dot{q}(t)-\frac{1}{2!}\left[q(t),\dot{q}(t)\right]+\frac{1}{3!}[q(t),[q(t),\dot{q}(t)]]-\frac{1}{4!}[q(t),[q(t),[q(t),\dot{q}(t)]]]+\cdots$
$\displaystyle=\dot{q}(t)+\sum_{m=2}^{\infty}\frac{(-1)^{m-1}}{m!}[q^{(m-1)},\dot{q}](t),$

where $[q^{(n)},\dot{q}](t)$ stands for

$[\smash{\underbrace{q(t)[q(t)[\dots[q(t)}_{n\text{ times}}},\dot{q}(t)]]]].$

Therefore,

$e^{-q(t)}\frac{de^{q(t)}}{dt}=\dot{q}(t)-\sum_{h=1}^{\infty}\frac{1}{(2h)!}[q^{(2h-1)},\dot{q}](t)+\sum_{h=1}^{\infty}\frac{1}{(2h+1)!}[q^{(2h)},\dot{q}](t).$

Now, as for any $p,q\in{\mathbb{H}}$, we have that $[p,q]=2p\wedge
q=2p_{v}\wedge q_{v}$, then we have

$\displaystyle[q,\dot{q}]$ $\displaystyle=2q_{v}\wedge\dot{q}_{v}$
$\displaystyle[q^{(2)},\dot{q}]$ $\displaystyle=2^{2}[\langle
q_{v},\dot{q}_{v}\rangle q_{v}-|q_{v}|^{2}\dot{q}_{v}]$
$\displaystyle[q^{(3)},\dot{q}]$
$\displaystyle=2^{3}(-1)|q_{v}|^{2}q_{v}\wedge\dot{q}_{v}$
$\displaystyle[q^{(4)},\dot{q}]$ $\displaystyle=2^{4}(-1)|q_{v}|^{2}[\langle
q_{v},\dot{q}_{v}\rangle q_{v}-|q_{v}|^{2}\dot{q}_{v}]$
$\displaystyle[q^{(5)},\dot{q}]$
$\displaystyle=2^{5}(-1)^{2}(|q_{v}|^{2})^{2}q_{v}\wedge\dot{q}_{v}$
$\displaystyle[q^{(6)},\dot{q}]$
$\displaystyle=2^{6}(-1)^{2}(|q_{v}|^{2})^{2}[\langle q_{v},\dot{q}_{v}\rangle
q_{v}-|q_{v}|^{2}\dot{q}_{v}]$ $\displaystyle\dots$

where, in order to simplify the notation, we removed the dependence from the
parameter $t$. Hence,

$\displaystyle e^{-q}\frac{de^{q}}{dt}=$
$\displaystyle\dot{q}-\left[\sum_{h=1}^{\infty}\frac{(-1)^{h-1}2^{2h-1}}{(2h)!}|q_{v}|^{2(h-1)}\right]q_{v}\wedge\dot{q}_{v}$
$\displaystyle+\left[\sum_{h=1}^{\infty}\frac{(-1)^{h-1}2^{2h}}{(2h+1)!}|q_{v}|^{2(h-1)}\right][\langle
q_{v},\dot{q}_{v}\rangle q_{v}-|q_{v}|^{2}\dot{q}_{v}]$ $\displaystyle=$
$\displaystyle\dot{q}-\frac{\sin^{2}(|q_{v}|)}{|q_{v}|^{2}}q_{v}\wedge\dot{q}_{v}+\frac{|q_{v}|-\cos(|q_{v}|)\sin(|q_{v}|)}{|q_{v}|^{3}}\left[\langle
q_{v},\dot{q}_{v}\rangle q_{v}-|q_{v}|^{2}\dot{q}_{v}\right]$ $\displaystyle=$
$\displaystyle\dot{q}-\frac{1-\cos(2|q_{v}|)}{2|q_{v}|}\frac{q_{v}}{|q_{v}|}\wedge\dot{q}_{v}+\left[1-\frac{\sin(2|q_{v}|)}{2|q_{v}|}\right]\left[\left\langle\frac{q_{v}}{|q_{v}|},\dot{q}_{v}\right\rangle\frac{q_{v}}{|q_{v}|}-\dot{q}_{v}\right]$
$\displaystyle=$
$\displaystyle\dot{q}+\left[1-\frac{\sin(2|q_{v}|)}{2|q_{v}|}\right]\left[\left\langle\frac{q_{v}}{|q_{v}|},\dot{q}_{v}\right\rangle\frac{q_{v}}{|q_{v}|}-\dot{q}_{v}\right]-\frac{1-\cos(2|q_{v}|)}{2|q_{v}|}\frac{q_{v}}{|q_{v}|}\wedge\dot{q}_{v}.$

Clearly, if $q_{v}=0$, we obtain the usual formula. Moreover, if $q_{v}\neq 0$
and $\dot{q}_{v}$ commutes with $q_{v}$, i.e. there exists a real valued
function $\alpha$ such that $\dot{q_{v}}=\alpha q_{v}$, then

$e^{-q}\frac{de^{q}}{dt}=\dot{q},$

as expected.

If we write $\dot{q}=q_{1}\frac{q_{v}}{|q_{v}|}+q_{\perp}$, then we obtain

(9) $\displaystyle e^{-q}\frac{de^{q}}{dt}$
$\displaystyle=\dot{q}-\frac{1-\cos(2|q_{v}|)}{2|q_{v}|}\frac{q_{v}}{|q_{v}|}\wedge{q}_{\perp}-\left[1-\frac{\sin(2|q_{v}|)}{2|q_{v}|}\right]q_{\perp}$
(10)
$\displaystyle=\dot{q}_{0}+q_{1}\frac{q_{v}}{|q_{v}|}+\frac{\sin(2|q_{v}|)}{2|q_{v}|}q_{\perp}-\frac{1-\cos(2|q_{v}|)}{2|q_{v}|}\frac{q_{v}}{|q_{v}|}\wedge{q}_{\perp}.$

Now, exactly as we have done before, this last relation extends to the
complexification of ${\mathbb{H}}$, where in place of a curve
$q:[0,1]\to{\mathbb{H}}$ we consider a complex curve
$F:D\subset{\mathbb{C}}\to{\mathbb{C}}\otimes{\mathbb{H}}$, the derivative
with respect to $t$ is changed into the derivative with respect to $z\in D$
and the usual exponential function is changed into $\varepsilon$. After these
modifications we have the following formula.

######  Proposition 8.1.

Let $f:U\to{\mathbb{H}}$ be a slice regular function. Then we have the
following formula

$\displaystyle\exp_{*}(f)^{-*}*\partial_{c}(\exp_{*}(f))=$
$\displaystyle\partial_{c}f+\left[\sum_{h=1}^{\infty}\frac{(-1)^{h-1}2^{2h}}{(2h+1)!}(f_{v}^{s})^{(h-1)}\right][\langle
f_{v},(\partial_{c}f)_{v}\rangle_{*}f_{v}-f_{v}^{s}(\partial_{c}f)_{v}]+$
$\displaystyle-\left[\sum_{h=1}^{\infty}\frac{(-1)^{h-1}2^{2h-1}}{(2h)!}(f_{v}^{s})^{(h-1)}\right]f_{v}\operatorname{\wedge\mspace{-9.0mu}_{*}\
}(\partial_{c}f)_{v}.$

From this proposition it is possible to derive some convenient corollaries.

######  Corollary 8.2.

Let $f:U\to{\mathbb{H}}$ be a slice regular function and let $q_{0}\in U$ be
any point. If $f_{v}^{s}(q_{0})=0$, then

$\Big{(}\exp_{*}(f)^{-*}*\partial_{c}(\exp_{*}(f))\Big{)}(q_{0})=(\partial_{c}f)(q_{0})-(f_{v}\operatorname{\wedge\mspace{-9.0mu}_{*}\
}(\partial_{c}f)_{v})(q_{0})+\frac{2}{3}\Big{(}\langle
f_{v},(\partial_{c}f)_{v}\rangle_{*}\Big{)}(q_{0})f_{v}(q_{0})$

In the case when $f_{v}^{s}$ is never-vanishing, many equivalent formulas can
be derived.

######  Corollary 8.3.

Let $f:U\to{\mathbb{H}}$ be a slice regular function such that $f_{v}^{s}$ is
never-vanishing, then we have

$\displaystyle\partial_{c}(\exp_{*}(f))=\exp_{*}(f)*$
$\displaystyle\left\\{\partial_{c}f+\left[1-\frac{\sin(2\sqrt{f_{v}^{s}})}{2\sqrt{f_{v}^{s}}}\right]\left[\left\langle\frac{f_{v}}{\sqrt{f_{v}^{s}}},(\partial_{c}f)_{v}\right\rangle\frac{f_{v}}{\sqrt{f_{v}^{s}}}-(\partial_{c}f)_{v}\right]+\right.$
$\displaystyle\left.-\frac{1-\cos(2\sqrt{f_{v}^{s}})}{2\sqrt{f_{v}^{s}}}\frac{f_{v}}{\sqrt{f_{v}^{s}}}\operatorname{\wedge\mspace{-9.0mu}_{*}\
}(\partial_{c}f)_{v}\right\\}$

As said before, the formula contained in the last corollary is just one of the
possible generalizations we have seen in this section. With the same spirit,
it is clearly possible to generalize Formula (9) or (10).

######  Remark 8.1.

Many of the previous formulas can also be related to the function
$\nu:{\mathbb{H}}\to{\mathbb{H}}$ introduced in [altavillaLOG, Definition
2.16] as

$\nu(q)=\sum_{m\in{\mathbb{N}}}\frac{(-1)^{m}q^{2m+1}}{(2m+1)!},$

and noticing that $\nu(q^{2})q=\sin(q)$.

######  Remark 8.2.

Exactly as in the case of a quaternionic curve, even in this case the formula
for the slice derivative of the $*$-exponential of a slice regular function
simplifies to the usual one when $(\partial_{c}f)_{v}$ and $f_{v}$ commute,
i.e., getting rid of the trivial cases, when there exists a slice preserving
function $\gamma$ such that

$(\partial_{c}f)_{v}=\gamma f_{v}.$

Examples of functions satisfying this relations are slice constant functions
(i.e. functions with everywhere vanishing slice derivative),
${\mathbb{C}}_{I}$-preserving function (for any $I\in{\mathbb{S}}$) or
functions of the form $f_{v}=\exp(\gamma(q)q)c$, where $c$ is any purely
imaginary quaternion.

## References